1 Op Amps For Everyone Ron Mancini, Editor in Chief Design Reference Advanced Analog Products August 2002 SLOD006B

2 IMPORTANT NOTICE Texas Instruments Incorporated and its subsidiaries (TI) reserve the right to make corrections, and services at modifications, enhancements, improvements, and other changes to its products any time and to discontinue any product or service without notice. Customers should obtain the latest relevant information before placing orders and should verify that such information is current and complete. All products are sold subject to TI’s terms and conditions of sale supplied at the time of order acknowledgment. TI warrants performance of its hardware products to the specifications applicable at the time of sale in accordance with TI’s standard warranty. Testing and other quality control techniques are used to the extent TI deems necessary to support this warranty. Except where mandated by requirements, testing of all parameters of each product is not necessarily performed. government TI assumes no liability for applications assistance or customer product design. Customers are responsible for their products and applications using TI components. To minimize the risks associated with customer products and applications, customers should provide adequate design and operating safeguards. TI does not warrant or represent that any license, either express or implied, is granted under any TI patent right, copyright, mask work right, or other TI intellectual property right relating to any combination, machine, or process in which TI products or services are used. Information published by TI regarding third party products or services does not constitute a license from TI to use such products or services or a warranty or endorsement thereof. Use of such information may require a license from a third party under the patents or other intellectual property of that third party, or a license from TI under the patents or other intellectual property of TI. Reproduction of information in TI data books or data sheets is permissible only if reproduction is without alteration and is accompanied by all associated warranties, conditions, limitations, and notices. Reproduction of this information with alteration is an unfair and deceptive business practice. TI is not responsible or liable for such altered documentation. Resale of TI products or services with statements different from or beyond the parameters stated by TI for that product or service voids all express and any implied warranties for the associated TI product or service and is an unfair and deceptive business practice. TI is not responsible or liable for any such statements. Mailing Address: Texas Instruments Post Office Box 655303 Dallas, Texas 75265 Copyright 2002, Texas Instruments Incorporated

3 Forward Everyone interested in analog electronics should find some value in this book, and an ef- fort has been made to make the material understandable to the relative novice while not too boring for the practicing engineer. Special effort has been taken to ensure that each can stand alone for the reader with the proper background. Of course, this causes chapter redundancy that some people might find boring, but it ’ s worth the price to enable the satis- faction of a diversified audience. Start at Chapter 1 if you are a novice, and read through until completion of Chapter 9. After Chapter 9 is completed, the reader can jump to any chapter and be confident that they are prepared for the material. More experienced people such as electronic technicians, digital engineers, and non-electronic engineers can start at Chapter 3 and read through engineers, and fledgling analog engi- Chapter 9. Senior electronic technicians, electronic neers can start anywhere they feel comfortable and read through Chapter 9. Experienced analog engineers should jump to the subject that interests them. Analog gurus should send their additions, corrections, and complaints to me, and if they see something that looks familiar, they should feel complimented that others appreciate their contributions. Chapter 1 is a history and story chapter. It is not required reading for anyone, but it defines the op amp ’ s place in the world of analog electronics. Chapter 2 reviews some basic phys- ics and develops the fundamental circuit equations that are used throughout the book. equations have been developed in other books, but the presentation here empha- Similar sizes material required for speedy op amp design. The ideal op amp equations are devel- oped in Chapter 3, and this chapter enables the reader to rapidly compute op amp transfer equations including ac response. The emphasis on single power supply systems forces the designer to bias circuits when the inputs are referenced to ground, and Chapter 4 gives a detailed procedure that quickly yields a working solution every time. Op amps can ’ t exist without feedback, and feedback has inherent stability problems, so feedback and stability are covered in Chapter 5. Chapters 6 and 7 develop the voltage feedback op amp equations, and they teach the concept of relative stability and com- pensation of potentially unstable op amps. Chapter 8 develops the current feedback op amp equations and discusses current feedback stability. Chapter 9 compares current feedback and voltage feedback op amps. The meat of this book is Chapters 12, 13, and 14 where the reader is shown how design the converter to transducer/actuator interface with the aid of op amps. The remaining chapters give support material for Chapters 12, 13, and 14. Chapter 18 was a late addition. Portable applications are expanding rapidly and they emphasize the need for low-voltage/low-power design techniques. Chapter 18 defines some parameters in a new way so they lend themselves to low voltage design, and it takes the reader through several low voltage designs. i

4 time to do Thanks to editor James Karki for his contribution. We never gave him enough detailed editing, so if you find errors or typos, direct them to my attention. Thanks to Ted Thomas, a marketing manager with courage enough to support a book, and big thanks for Alun Roberts who paid for this effort. Thomas Kugelstadt, applications manager, thanks for your support and help. Also many thanks to the contributing authors, James Karki, Richard Palmer, Thomas Ku- gelstadt, Perry Miller, Bruce Carter, and Richard Cesari who gave generously of their time. Regards, Ron Mancini Chief Editor ii

5 Contents Contents The Op Amp’s Place In The World 1-1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1 Review of Circuit Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction Laws of Physics 2-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 2.3 2-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Voltage Divider Rule 2.4 2-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Current Divider Rule Thevenin ’ 2.5 2-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s Theorem 2.6 Superposition 2-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Calculation of a Saturated Transistor Circuit 2-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Transistor Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10 Development of the Ideal Op Amp Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3-1 Ideal Op Amp Assumptions 3-1 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Noninverting Op Amp 3-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Inverting Op Amp 3-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 3-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Adder 3.5 The Differential Amplifier 3-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Complex Feedback Networks 3-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Amplifiers 3-9 3.8 Capacitors . 3-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary 3.9 . 4 Single Supply Op Amp Design Techniques 4-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Supply versus Dual Supply 4-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Circuit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 4.2 Simultaneous Equations 4-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Case 1: VOUT = +mVIN+b 4-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 b Case 2: VOUT = +mVIN – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13 4.3.2 4.3.3 Case 3: VOUT = mVIN + b 4-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – Case 4: VOUT = 4.3.4 mVIN – b 4-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 4.4 Summary 4-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feedback and Stability Theory 5.1 Why Study Feedback Theory? 5-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Block Diagram Math and Manipulations 5-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Feedback Equation and Stability 5-6 iii

6 Contents 5.4 5-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bode Analysis of Feedback Circuits Loop Gain Plots are the Key to Understanding Stability 5-12 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . 5-15 . . . . . . . . . . . . . . . . . The Second Order Equation and Ringing/Overshoot Predictions 5.6 References 5-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Development of the Non Ideal Op Amp Equations 6-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1 Review of the Canonical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 6-2 Noninverting Op Amps 6-5 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Inverting Op Amps 6-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Differential Op Amps 6-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Voltage-Feedback Op Amp Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 7-1 Internal Compensation 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2 7.3 External Compensation, Stability, and Performance 7-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Dominant-Pole Compensation 7-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 7-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gain Compensation 7.6 Lead Compensation 7-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Compensated Attenuator Applied to Op Amp 7-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Lead-Lag Compensation 7-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Comparison of Compensation Schemes 7-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-21 8 Current-Feedback Op Amp Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 8-1 CFA Model 8-1 8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Development of the Stability Equation 8-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The Noninverting CFA 8-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 8-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Inverting CFA 8.6 Stability Analysis 8-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Selection of the Feedback Resistor 8-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Stability and Input Capacitance 8-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 8-12 Stability and Feedback Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Compensation of CF and CG 8-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 8-14 Voltage- and Current-Feedback Op Amp Comparison 9-1 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction 9-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Precision 9-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 9-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bandwidth 9.4 Stability 9-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Impedance 9-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-8 9.6 Equation Comparison iv

7 Contents 10 Op Amp Noise Theory and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1 10-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Characterization 10-1 10.2.1 rms versus P-P Noise 10-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Noise Floor 10-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3 10.2.3 Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Multiple Noise Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3 10-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.5 Noise Units 10-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Types of Noise 10.3.1 Shot Noise 10-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Thermal Noise 10-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Flicker Noise 10-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-9 10.3.4 Burst Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.5 Avalanche Noise 10.4 Noise Colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-10 10.4.1 White Noise 10-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Pink Noise 10-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Red/Brown Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-12 10.5 Op Amp Noise 10-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 The Noise Corner Frequency and Total Noise 10-12 . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 The Corner Frequency 10-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-14 10.5.3 Op Amp Circuit Noise Model 10.5.4 Inverting Op Amp Circuit Noise 10-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-17 10.5.5 Noninverting Op Amp Circuit Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.6 Differential Op Amp Circuit Noise 10.5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-18 10.6 Putting It All Together 10-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 References 10-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Understanding Op Amp Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1 11.1 Introduction 11-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Operational Amplifier Parameter Glossary 11-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional Parameter Information 11.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8 11.3.1 Input Offset Voltage 11-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-10 11.3.2 Input Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Input Common Mode Voltage Range 11.3.4 Differential Input Voltage Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-11 11.3.5 Maximum Output Voltage Swing 11-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.6 Large Signal Differential Voltage Amplification 11-13 . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.7 Input Parasitic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-13 11.3.8 Output Impedance 11-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.9 Common-Mode Rejection Ratio 11-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.10 Supply Voltage Rejection Ratio 11-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-16 11.3.11 Supply Current v Contents

8 Contents 11.3.12 Slew Rate at Unity Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-16 11-17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.13 Equivalent Input Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.14 Total Harmonic Distortion Plus Noise 11-18 11.3.15 Unity Gain Bandwidth and Phase Margin 11-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.16 Settling Time 11-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-1 12 Instrumentation: Sensors to A/D Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-1 12-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Transducer Types 12-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Design Procedure 12.4 Review of the System Specifications 12-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Reference Voltage Characterization 12-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Transducer Characterization 12-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 ADC Characterization 12.8 Op Amp Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-15 12.9 Amplifier Circuit Design 12-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10 Test 12-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-23 12.12 References 12-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Wireless Communication: Signal Conditioning for IF Sampling 13-1 . . . . . . . . . . . . . . . . . . . . 13.1 Introduction 13-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Wireless Systems 13.3 Selection of ADCs/DACs 13-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-10 13.4 Factors Influencing the Choice of Op Amps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Anti-Aliasing Filters 13.6 Communication D/A Converter Reconstruction Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 13-13 13.7 External Vref Circuits for ADCs/DACs 13-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8 High-Speed Analog Input Drive Circuits 13-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-22 14 Interfacing D/A Converters to Loads 14-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction 14-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Load Characteristics 14-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 DC Loads 14-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 AC Loads 14-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-2 14.3 Understanding the D/A Converter and its Specifications . . . . . . . . . . . . . . . . . . . . . . . . — Understanding the Tradeoffs 14.3.1 Types of D/A Converters . . . . . . . . . . . . . . . . 14-2 14.3.2 The Resistor Ladder D/A Converter 14-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 The Weighted Resistor D/A Converter 14-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.4 The R/2R D/A Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-4 14.3.5 The Sigma Delta D/A Converter 14-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 D/A Converter Error Budget 14-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Accuracy versus Resolution 14-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-7 14.4.2 DC Application Error Budget vi

9 Contents 14.4.3 AC Application Error Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-8 14-10 14.4.4 RF Application Error Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 D/A Converter Errors and Parameters 14-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.1 DC Errors and Parameters 14-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.2 AC Application Errors and Parameters 14.6 Compensating For DAC Capacitance 14-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-19 . . . . . . . . . . . . . . . . . . . . . . . 14.7 Increasing Op Amp Buffer Amplifier Current and Voltage 14.7.1 Current Boosters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-20 14.7.2 Voltage Boosters 14-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.3 Power Boosters 14-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.4 Single-Supply Operation and DC Offsets 14-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-1 15 Sine Wave Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 What is a Sine Wave Oscillator? 15.2 Requirements for Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-1 15.3 Phase Shift in the Oscillator 15-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Gain in the Oscillator 15-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Active Element (Op Amp) Impact on the Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-5 15.6 Analysis of the Oscillator Operation (Circuit) 15-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Sine Wave Oscillator Circuits 15-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.1 Wien Bridge Oscillator 15-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.2 Phase Shift Oscillator, Single Amplifier 15.7.3 Phase Shift Oscillator, Buffered 15-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-17 15.7.4 Bubba Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.5 Quadrature Oscillator 15.7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-20 15.8 References 15-21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Active Filter Design Techniques 16-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-1 16.2 Fundamentals of Low-Pass Filters 16-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Butterworth Low-Pass FIlters 16-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.2 Tschebyscheff Low-Pass Filters 16-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.3 Bessel Low-Pass Filters 16-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4 Quality Factor Q 16-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.5 Summary 16.3 Low-Pass Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-11 16.3.1 First-Order Low-Pass Filter 16-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2 Second-Order Low-Pass Filter 16-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.3 Higher-Order Low-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-19 16.4 High-Pass Filter Design 16-21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 First-Order High-Pass Filter 16-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.2 Second-Order High-Pass Filter 16-24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.3 Higher-Order High-Pass Filter vii Contents

10 Contents 16.5 Band-Pass Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-27 16-29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.1 Second-Order Band-Pass Filter . . . . . . . . . . . . . . . . . . . . . 16.5.2 Fourth-Order Band-Pass Filter (Staggered Tuning) 16-32 16.6 Band-Rejection Filter Design 16-36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6.1 Active Twin-T Filter 16-37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-39 16.6.2 Active Wien-Robinson Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 All-Pass Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-41 16-44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7.1 First-Order All-Pass Filter 16-44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7.2 Second-Order All-Pass Filter 16.7.3 Higher-Order All-Pass Filter 16-45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8 Practical Design Hints 16-47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8.1 Filter Circuit Biasing 16-47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-50 16.8.2 Capacitor Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8.3 Component Values 16.8.4 Op Amp Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-53 16.9 Filter Coefficient Tables 16-55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.10 References 16-63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Circuit Board Layout Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-1 17.1 General Considerations 17-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1 The PCB is a Component of the Op Amp Design 17-1 . . . . . . . . . . . . . . . . . . . . . . . 17.1.2 Prototype, Prototype, PROTOTYPE! 17-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.3 Noise Sources 17-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 PCB Mechanical Construction 17.2.1 Materials — 17-3 . . . . . . . . . . . . . . . . . Choosing the Right One for the Application 17-4 17.2.2 How Many Layers are Best? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.3 Board Stack-Up — The Order of Layers 17-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Grounding 17-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 The Most Important Rule: Keep Grounds Separate . . . . . . . . . . . . . . . . . . . . . 17-7 17.3.2 Other Ground Rules 17-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.3 A Good Example 17-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.4 A Notable Exception 17-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 The Frequency Characteristics of Passive Components 17-11 . . . . . . . . . . . . . . . . . . . . . . . 17-11 17.4.1 Resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2 Capacitors 17-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.3 Inductors 17.4.4 Unexpected PCB Passive Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-14 17.5 Decoupling 17-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.1 Digital Circuitry — A Major Problem for Analog Circuitry 17-20 . . . . . . . . . . . . . . . . 17.5.2 Choosing the Right Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-21 17.5.3 Decoupling at the IC Level 17-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.4 Decoupling at the Board Level 17-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6 Input and Output Isolation 17-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.7 Packages 17-24 viii

11 Contents 17.7.1 Through-Hole Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-26 17-27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.7.2 Surface Mount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.7.3 Unused Sections 17-27 17.8 Summary 17-28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.8.1 General 17-28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-28 17.8.2 Board Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.8.3 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-28 17-29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.8.4 Routing 17-29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.8.5 Bypass 17.9 References 17-29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Designing Low-Voltage Op Amp Circuits 18-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Introduction 18-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-3 18.2 Dynamic Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Signal-to-Noise Ratio 18.4 Input Common-Mode Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-6 18.5 Output Voltage Swing 18-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Shutdown and Low Current Drain 18-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7 Single-Supply Circuit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-13 18.8 Transducer to ADC Analog Interface 18-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.9 DAC to Actuator Analog Interface 18-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.10 Comparison of Op Amps 18-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.11 Summary 18-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Single-Supply Circuit Collection A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1 Boundary Conditions A-4 A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Amplifiers A-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Inverting Op Amp with Noninverting Positive Reference A-5 . . . . . . . . . . . . . . . . . . A.3.2 A-6 . . . . . . . . . . . . . . . . . . . . . Inverting Op Amp with Inverting Negative Reference A.3.3 Inverting Op Amp with Noninverting Negative Reference A-7 . . . . . . . . . . . . . . . . . A.3.4 Inverting Op Amp with Inverting Positive Reference A-8 . . . . . . . . . . . . . . . . . . . . . . A.3.5 Noninverting Op Amp with Inverting Positive Reference A-9 . . . . . . . . . . . . . . . . . . A.3.6 Noninverting Op Amp with Noninverting Negative Reference A-10 . . . . . . . . . . . . . A.3.7 Noninverting Op Amp with Inverting Negative Reference . . . . . . . . . . . . . . . . A-11 Noninverting Op Amp with Noninverting Positive Reference . . . . . . . . . . . . . . A.3.8 A-12 Differential Amplifier A.3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-13 A.3.10 Differential Amplifier With Bias Correction A-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.11 High Input Impedance Differential Amplifier A-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.12 High Common-Mode Range Differential Amplifier . . . . . . . . . . . . . . . . . . . . . . . A-16 A.3.13 High-Precision Differential Amplifier A-17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.14 Simplified High-Precision Differential Amplifier A-18 . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.15 Variable Gain Differential Amplifier A-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.16 T Network in the Feedback Loop ix Contents

12 Contents A.3.17 Buffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-21 A-22 A.3.18 Inverting AC Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.19 Noninverting AC Amplifier A-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computing Circuits A-24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Inverting Summer A-24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.1 A.4.2 Noninverting Summer A-25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.3 Noninverting Summer with Buffers A-26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.4 Inverting Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-27 Inverting Integrator with Input Current Compensation A-28 A.4.5 . . . . . . . . . . . . . . . . . . . . A.4.6 Inverting Integrator with Drift Compensation A-29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.7 Inverting Integrator with Mechanical Reset A-30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.8 Inverting Integrator with Electronic Reset A-31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.9 Inverting Integrator with Resistive Reset A-32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.10 Noninverting Integrator with Inverting Buffer A-33 A.4.11 Noninverting Integrator Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-34 A.4.12 Inverting Differentiator A-35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.13 Inverting Differentiator with Noise Filter A-36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Oscillators A-37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.1 Basic Wien Bridge Oscillator A-37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.2 Wien Bridge Oscillator with Nonlinear Feedback A-38 . . . . . . . . . . . . . . . . . . . . . . . . A.5.3 Wien Bridge Oscillator with AGC A-39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.4 A-40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrature Oscillator A.5.5 Classical Phase Shift Oscillator A-41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.6 Buffered Phase Shift Oscillator A-42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.7 Bubba Oscillator A-43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.8 Triangle Oscillator A-44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1 B Single-Supply Op Amp Selection Guide x

13 Figures Figures – 1 ’ s Law Applied to the Total Circuit 2-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Ohm 2 2-2 ’ s Law Applied to a Component – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Ohm – Kirchoff ’ s Voltage Law 2-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 – 2 Kirchoff ’ s Current Law 2-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 5 Voltage Divider Rule 2-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – 2-4 Current Divider Rule 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 7 Original Circuit 2-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 2 – 8 Thevenin ’ s Equivalent Circuit for Figure 2 – 7 2-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 – Example of Thevenin ’ s Equivalent Circuit 2-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 – Analysis Done the Hard Way 2 2-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 11 Superposition Example 2-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 2 – 12 When V1 is Grounded 2-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 – When V2 is Grounded 2-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 14 Saturated Transistor Circuit 2-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 2 – 15 Transistor Amplifier 2-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 – 16 Thevenin Equivalent of the Base Circuit 2-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2 – 1 The Ideal Op Amp 3 – The Noninverting Op Amp 3-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 – The Inverting Op Amp 3-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 – 4 The Adder Circuit 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5 3 – 5 The Differential Amplifier 3-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 – 6 Differential Amplifier With Common-Mode Input Signal 3-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7 T Network in Feedback Loop 3-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 3 – 8 Thevenin ’ s Theorem Applied to T Network 3-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 – 9 Video Amplifier 3-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 3 3-10 10 Low-Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 – High-Pass Filter 3-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 – Split-Supply Op Amp Circuit 4-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 – 2 Split-Supply Op Amp Circuit With Reference Voltage Input 4 . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 4 – 3 Split-Supply Op Amp Circuit With Common-Mode Voltage 4-2 . . . . . . . . . . . . . . . . . . . . . . . . . . 4 – 4 Single-Supply Op Amp Circuit 4-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 Inverting Op Amp 4-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 4 – 6 Inverting Op Amp With VCC Bias 4-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 – 7 Transfer Curve for Inverting Op Amp With VCC Bias 4-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 – 8 Noninverting Op Amp xi Contents

14 Figures 4 9 Transfer Curve for Noninverting Op Amp 4-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – 10 4-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Schematic for Case1: VOUT = +mVIN + b 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12 – 4 Case 1 Example Circuit – Case 1 Example Circuit Measured Transfer Curve 4-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 12 – 13 Schematic for Case 2: VOUT = +mVIN – b 4-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 – Case 2 Example Circuit 4-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 – Case 2 Example Circuit Measured Transfer Curve 4-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 15 – 16 Schematic for Case 3: VOUT = – 4 4-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mVIN + b 4 – 17 Case 3 Example Circuit 4-17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 – 18 Case 3 Example Circuit Measured Transfer Curve 4-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 – Schematic for Case 4: VOUT = – mVIN – b 4-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case 4 Example Circuit 4-20 4 20 – 4-21 Case 4 Example Circuit Measured Transfer Curve 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5 1 Definition of Blocks 5-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 5 – 2 Summary Points 5-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 – Definition of Control System Terms 5-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 4 Definition of an Electronic Feedback Circuit 5-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 5 – 5 Multiloop Feedback System 5-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 Block Diagram Transforms 5-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – . . . . . . . . . . . . . . . . – 7 Comparison of Control and Electronic Canonical Feedback Systems 5-6 5 Low-Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 – 8 5-8 5 – Bode Plot of Low-Pass Filter Transfer Function 5-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 – Band Reject Filter 5-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 10 – 11 Individual Pole Zero Plot of Band Reject Filter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10 5 – 12 Combined Pole Zero Plot of Band Reject Filter 5-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 – 13 When No Pole Exists in Equation (5 – 12) 5-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 14 When Equation 5 – 12 has a Single Pole 5-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 5 – 15 Magnitude and Phase Plot of Equation 5 – 14 5-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 16 Magnitude and Phase Plot of the Loop Gain Increased to (K+C) 5-14 . . . . . . . . . . . . . . . . . . . . – 5 – 17 Magnitude and Phase Plot of the Loop Gain With Pole Spacing Reduced 5-14 . . . . . . . . . . . 18 5 – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Margin and Overshoot vs Damping Ratio 5-16 6 – Feedback System Block Diagram 6-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 – Feedback Loop Broken to Calculate Loop Gain 6-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 – 3 Noninverting Op Amp 6-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6 – Open Loop Noninverting Op Amp 6-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6 5 Inverting Op Amp 6-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 6 – 6 Inverting Op Amp: Feedback Loop Broken for Loop Gain Calculation 6-7 . . . . . . . . . . . . . . . . 6 – Differential Amplifier Circuit 6-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 1 Miller Effect Compensation 7-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 7 – 2 TL03X Frequency and Time Response Plots 7-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 – 3 Phase Margin and Percent Overshoot Versus Damping Ratio 7-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5 7 – 4 TL07X Frequency and Time Response Plots xii

15 Figures 7 5 TL08X Frequency and Time Response Plots 7-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – 6 7-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 TLV277X Frequency Response Plots 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7 – 7 TLV227X Time Response Plots – Capacitively-Loaded Op Amp 7-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 8 – 9 Capacitively-Loaded Op Amp With Loop Broken for Loop Gain (A β ) Calculation 7-9 . . . . . . . 7 7 – Possible Bode Plot of the Op Amp Described in Equation 7 – 7 7-11 . . . . . . . . . . . . . . . . . . . . . . 10 – Dominant-Pole Compensation Plot 7-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 11 – 12 Gain Compensation 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-12 7 – 13 Lead-Compensation Circuit 7-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 – 14 Lead-Compensation Bode Plot 7-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 – Inverting Op Amp With Lead Compensation 7-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 – Noninverting Op Amp With Lead Compensation 7-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 16 – 17 Op Amp With Stray Capacitance on the Inverting Input 7-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 – Compensated Attenuator Circuit 7-17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7 19 Compensated Attenuator Bode Plot 7-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 7 – 20 Lead-Lag Compensated Op Amp 7-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 – Bode Plot of Lead-Lag Compensated Op Amp 7-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7 22 Closed-Loop Plot of Lead-Lag Compensated Op Amp 7-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 8 – 1 Current-Feedback Amplifier Model 8-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 – 2 Stability Analysis Circuit 8-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 – 8-3 Stability Analysis Circuit 8 – 8-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Noninverting CFA 5 8-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – Inverting CFA 8 6 Bode Plot of Stability Equation 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 8-7 – 7 Plot of CFA RF, G, and BW 8-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 – 8 Effects of Stray Capacitance on CFAs 8-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 – 8 Bode Plot with CF 8-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 – Long-Tailed Pair 9-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1 – 2 Ideal CFA 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3 9 – 3 VFA Gain versus Frequency 9-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 – 4 CFA Gain vs Frequency 9-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 – 1 Gaussian Distribution of Noise Energy 10-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 – Shot Noise Generation 10-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 – Thermal Noise 10-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 – 4 Avalanche Noise 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-10 10 – 5 Noise Colors 10-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 – 6 TLV2772 Op Amp Noise Characteristics 10-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 7 Op Amp Circuit Noise Model 10-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 10 – 8 Equivalent Op Amp Circuit Noise Model 10-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 9 Inverting Op Amp Circuit Noise Model 10-16 – 10 – 10 Inverting Equivalent Op Amp Circuit Noise Model 10-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-17 10 – 11 Noninverting Equivalent Op Amp Circuit Noise Model xiii Contents

16 Figures 10 12 Differential Equivalent Op Amp Circuit Noise Model 10-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – 13 Split Supply Op Amp Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10-19 14 TLC2201 Op Amp Noise Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 10-20 10 15 TLC2201 Op Amp Circuit 10-21 10 – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 16 Improved TLC2201 Op Amp Circuit 10-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 – 1 Test Circuits for Input Offset Voltage 11-9 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Offset Voltage Adjust 11-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – IIB Test Circuit – 11 11-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 11 4V – 11-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OM ± 11 – 5 Input Parasitic Elements 11-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 – Effect of Output Impedance 11-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 – Figure 6. Slew Rate 11-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7 – 8 Figure 7. Simplified Op Amp Schematic 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-17 11 – 9 Typical Op amp Input Noise Spectrum 11-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 – 10 Output Spectrum with THD + N = 1% 11-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 Voltage Amplification and Phase Shift vs. Frequency 11-21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 11 – 12 Settling Time 11-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 – 1 Block Diagram of a Transducer Measurement System 12-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Significant Bits versus Binary Bits 12-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 3 Example of Spans That Require Correction 12-4 12 12-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 – 4 Voltage Divider Circuit for a Resistive Transducer 12 – Current Source Excitation for a Resistive Transducer 12-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 – Precision Current Source 12-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6 – 7 Wheatstone Bridge Circuit 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-8 12 – 8 Photodiode Amplifier 12-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 – 9 Phototransistor Amplifier 12-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 10 Photovoltaic Cell Amplifier 12-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 12 – 11 Active Full-Wave Rectifier and Filter 12-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 – 12 Reference and Transducer Bias Circuit 12-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 13 AIA Circuit 12-17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 12 – 14 Final Analog Interface Circuit 12-21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 13 13-2 1 A Typical GSM Cellular Base Station Receiver Block Diagram . . . . . . . . . . . . . . . . . . . . . 13 – An Implementation of a Software-Configurable Dual-IF Receiver 13-4 . . . . . . . . . . . . . . . . . . . 2 – Basic W-CDMA Cellular Base Station Transmitter Block 13-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 – 4 Communication DAC with Interpolation and Reconstruction Filters 13-5 13 . . . . . . . . . . . . . . . . . . 13 – 5 QPSK Power Spectral Density Without Raised Cosine Filter — W-CDMA 13-14 . . . . . . . . . . 13 – 6 Reconstruction Filter Characteristics 13-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7 A Single-Pole Reconstruction Filter 13-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 13 – 8 Voltage Reference Filter Circuit 13-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 9 Voltage Follower Frequency Response Plot 13-17 – 13 – 10 External Voltage Reference Circuit for ADC/DAC. 13-17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-18 13 – 11 Single-Ended to Differential Output Drive Circuit xiv

17 Figures 13 12 Differential Amplifier Closed-Loop Response 13-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – 13 ADC Single-Ended Input Drive Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13-20 14 Gain vs Frequency Plot for THS3201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 13-21 13 15 Phase vs Frequency for THS3201 13-21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 – 1 Resistor Ladder D/A Converter 14-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 14 – 2 Binary Weighted D/A Converter 14-3 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 R/2R Resistor Array 14-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – R/2R D/A Converter 14-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 – 5 Sigma Delta D/A Converter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-6 14 – 6 Total Harmonic Distortion 14-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 – 7 D/A Offset Error 14-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 – D/A Gain Error 14-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 – Differential Nonlinearity Error 14-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 9 – 10 Integral Nonlinearity Error 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-13 14 – 11 Spurious Free Dynamic Range 14-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 – 12 Intermodulation Distortion 14-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 13 D/A Settling Time 14-17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 14 – 14 D/A Deglitch Circuit 14-17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 – 15 Compensating for CMOS DAC Output Capacitance 14-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 – 16 D/A Output Current Booster 14-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 – 17 Incorrect Method of Increasing Voltage Swing of D/A Converters 14-21 . . . . . . . . . . . . . . . . . . 14-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 – 18 Correct Method of Increasing Voltage Range 14 – 14-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Single-Supply DAC Operation – Canonical Form of a Feedback System with Positive or Negative Feedback 15-2 . . . . . . . . . 15 1 – 2 Phase Plot of RC Sections 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-3 15 – 3 Op Amp Frequency Response 15-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 – 4 Op Amp Bandwidth and Oscillator Output 15-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 – Block Diagram of an Oscillator with Positive Feedback 15-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 15 6 Amplifier with Positive and Negative Feedback 15-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 15 – 7 Wien Bridge Circuit Schematic 15-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 – 8 Final Wien Bridge Oscillator Circuit 15-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 15 – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wien Bridge Output Waveforms 15-11 15 – 15-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Wien Bridge Oscillator with Nonlinear Feedback – – 10 15-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 11 Output of the Circuit in Figure 15 – 12 Wien Bridge Oscillator with AGC 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-13 15 – 13 Output of the Circuit in Figure 15 – 12 15-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 – 14 Phase Shift Oscillator (Single Op Amp) 15-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 – – 14 15-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Output of the Circuit in Figure 15 – 16 Phase Shift Oscillator, Buffered 15-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 15 – 17 Output of the Circuit Figure 15 – 16 15-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 – 18 Bubba Oscillator 15-17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 – 19 Output of the Circuit in Figure 15 – 18. xv Contents

18 Figures 15 20 Quadrature Oscillator 15-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – 21 Output of the Circuit in Figure 15 20 15-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 – 1 16-1 – . . . . . . . . . . . . . . 16 Second-Order Passive Low-Pass and Second-Order Active Low-Pass – First-Order Passive RC Low-Pass 16-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 – 3 16 16-3 . . . . . . . . . . . . . . . . . . . . Fourth-Order Passive RC Low-Pass with Decoupling Amplifiers 16 4 Frequency and Phase Responses of a Fourth-Order Passive RC Low-Pass Filter 16-4 . . . . – – Amplitude Responses of Butterworth Low-Pass Filters 16 16-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 16 6 Gain Responses of Tschebyscheff Low-Pass Filters 16-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 16 – 7 Comparison of Phase Responses of Fourth-Order Low-Pass Filters 16-8 . . . . . . . . . . . . . . . . 16 – Comparison of Normalized Group Delay (Tgr) of Fourth-Order Low-Pass Filters 16-8 . . . . . . 8 – Comparison of Gain Responses of Fourth-Order Low-Pass Filters 16-9 . . . . . . . . . . . . . . . . . . 16 9 – 10 Graphical Presentation of Quality Factor Q on a Tenth-Order 16 16-10 . . . . . . . . . . . . . . . . . . . . . . . Tschebyscheff Low-Pass Filter with 3-dB Passband Ripple 16 – 11 Cascading Filter Stages for Higher-Order Filters 16-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 12 First-Order Noninverting Low-Pass Filter 16-12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 13 First-Order Inverting Low-Pass Filter 16-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 16 – 14 First-Order Noninverting Low-Pass Filter with Unity Gain 16-14 . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 15 General Sallen-Key Low-Pass Filter 16-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 16 Unity-Gain Sallen-Key Low-Pass Filter 16-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 17 Second-Order Unity-Gain Tschebyscheff Low-Pass with 3-dB Ripple 16-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Adjustable Second-Order Low-Pass Filter 16 – 16-17 16 – 16-18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Second-Order MFB Low-Pass Filter – 16-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 20 First-Order Unity-Gain Low-Pass – 21 Second-Order Unity-Gain Sallen-Key Low-Pass Filter 16-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16 22 Fifth-Order Unity-Gain Butterworth Low-Pass Filter 16-21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 16 – 23 Low-Pass to High-Pass Transition Through Components Exchange 16-21 . . . . . . . . . . . . . . . 16 – 24 Developing The Gain Response of a High-Pass Filter 16-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 25 First-Order Noninverting High-Pass Filter 16-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 16 – 26 First-Order Inverting High-Pass Filter 16-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 27 General Sallen-Key High-Pass Filter 16-24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 28 Unity-Gain Sallen-Key High-Pass Filter 16-24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Second-Order MFB High-Pass Filter 16-25 16 – 16-27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Third-Order Unity-Gain Bessel High-Pass – 16-28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 31 Low-Pass to Band-Pass Transition – 32 Gain Response of a Second-Order Band-Pass Filter 16-29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16 33 Sallen-Key Band-Pass 16-30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 16 – 34 MFB Band-Pass 16-31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 35 Gain Responses of a Fourth-Order Butterworth Band-Pass and its Partial Filters 16-36 . . . . 16 36 Low-Pass to Band-Rejection Transition 16-37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 16 – 37 Passive Twin-T Filter 16-37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 38 Active Twin-T Filter 16-38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 39 Passive Wien-Robinson Bridge 16-39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-39 16 – 40 Active Wien-Robinson Filter xvi

19 Figures 16 41 Comparison of Q Between Passive and Active Band-Rejection Filters 16-41 . . . . . . . . . . . . . . – – 42 Frequency Response of the Group Delay for the First 10 Filter Orders . . . . . . . . . . . . . 16 16-43 43 First-Order All-Pass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 16-44 16 44 Second-Order All-Pass Filter 16-44 16 – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 45 Seventh-Order All-Pass Filter 16-46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 46 Dual-Supply Filter Circuit 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-47 16 47 Single-Supply Filter Circuit 16-47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – 16-48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 48 Biasing a Sallen-Key Low-Pass – 49 Biasing a Second-Order MFB Low-Pass Filter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-49 16 – 50 Biasing a Sallen-Key and an MFB High-Pass Filter 16-50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 51 Differences in Q and fC in the Partial Filters of an Eighth-Order Butterworth Low-Pass Filter 16-51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 16 52 Modification of the Intended Butterworth Response to aTschebyscheff-Type Characteristic 16-51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 53 Open-Loop Gain (AOL) and Filter Response (A) 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-53 17 – 1 Op Amp Terminal Model 17-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 – 2 Digital and Analog Plane Placement 17-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 – Separate Grounds 17-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 17 4 Broadcasting From PCB Traces 17-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 17 – 5 A Careful Board Layout 17-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 – 6 Resistor High-Frequency Performance 17-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-12 17 – 7 Capacitor High-Frequency Performance 17 – 17-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Inductor High-Frequency Performance 9 17-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – Loop and Slot Antenna Board Trace Layouts 17 10 PCB Trace Corners 17-16 17 – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 11 PCB Trace-to-Plane Capacitance Formula 17-17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 – . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-18 17 12 Effect of 1-pF Capacitance on Op Amp Inverting Input 17 13 Coupling Between Parallel Signal Traces 17-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – 17-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 14 Via Inductance Measurements – 15 Logic Gate Output Structure 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-21 17 – 16 Capacitor Self Resonance 17-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 – 17 Common Op Amp Pinouts 17-24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 17 18 Trace Length for an Inverting Op Amp Stage 17-25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 – 17-25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Mirror-Image Layout for Quad Op Amp Package – 17-26 . . . . . . . . . . . . . . . . . . . . . . 17 20 Quad Op Amp Package Layout with Half-Supply Generator – 21 Proper Termination of Unused Op Amp Sections 17-28 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 – 1 Op Amp Error Sources 18-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 – 2 Noninverting Op Amp 18-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Input Circuit of a NonRRI Op Amp 18-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 18 – 4 Input Circuit of an RRI Op Amp 18-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 – 5 Input Bias Current Changes with Input Common-Mode Voltage 18-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-10 18 – 6 Input Offset Voltage Changes with Input Common-Mode Voltage xvii Contents

20 Figures 18 7 RRO Output Stage 18-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – 8 18-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Data Acquisition System 9 18-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . – Schematic for the Transducer to ADC Interface Circuit 18 10 Final Schematic for the Transducer to ADC Interface Circuit 18-16 18 – . . . . . . . . . . . . . . . . . . . . . . – 11 Digital Control System 18-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 – 12 DAC Current Source to Actuator Interface Circuit 18-17 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 13 DAC Current Sink to Actuator Interface Circuit 18-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – Split-Supply Op Amp Circuit A-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 1 – 2 Split-Supply Op Amp Circuit With Reference Voltage Input A . . . . . . . . . . . . . . . . . . . . . . . . . A-2 A – 3 Split-Supply Op Amp Circuit With Common-Mode Voltage A-3 . . . . . . . . . . . . . . . . . . . . . . . . . . A – 4 Single-Supply Op Amp Circuit A-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – Inverting Op Amp with Noninverting Positive Reference A-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 – Inverting Op Amp with Inverting Negative Reference A-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 6 – 7 Inverting Op Amp with Noninverting Negative Reference A-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . A A – Inverting Op Amp with Inverting Positive Reference A-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 A 9 Noninverting Op Amp with Inverting Positive Reference A-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . – A – 10 Noninverting Op Amp with Noninverting Negative Reference A-10 . . . . . . . . . . . . . . . . . . . . . . A – Noninverting Op Amp with Inverting Positive Reference A-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 A 12 Noninverting Op Amp with Noninverting Positive Reference A-12 . . . . . . . . . . . . . . . . . . . . . . . – A – 13 Differential Amplifier A-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – 14 Differential Amplifier with Bias Correction A-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – 15 High Input Impedance Differential Amplifier A – A-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 High Common-Mode Range Differential Amplifier 17 A-17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – High-Precision Differential Amplifier A 18 Simplified High-Precision Differential Amplifier A-18 A – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 19 Variable Gain Differential Amplifier A-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – 20 T Network in the Feedback Loop A-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – A Buffer A-21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 – Inverting AC Amplifier A-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 22 – 23 Noninverting AC Amplifier A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-23 A – 24 Inverting Summer A-24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – 25 Noninverting Summer A-25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – A 26 Noninverting Summer with Buffers A-26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – Inverting Integrator A-27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 – Inverting Integrator with Input Current Compensation A-28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 28 – 29 Inverting Integrator with Drift Compensation A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-29 A – 30 Inverting Integrator with Mechanical Reset A-30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – 31 Inverting Integrator with Electronic Reset A-31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – Inverting Integrator with Resistive Reset A-32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A 33 Noninverting Integrator with Inverting Buffer A-33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – A – 34 Noninverting Integrator Approximation A-34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – 35 Inverting Differentiator A-35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-36 A – 36 Inverting Differentiator with Noise Filter xviii

21 Figures – 37 Basic Wien Bridge Oscillator A-37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A A – 38 Wien Bridge Oscillator with Nonlinear Feedback A-38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – 39 Wien Bridge Oscillator with AGC A-39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A – 40 Quadrature Oscillator – 41 Classical Phase Shift Oscillator A-41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 42 A – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Buffered Phase Shift Oscillator A-42 A – 43 Bubba Oscillator A-43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-44 A – 44 Triangle Oscillator xix Contents

22 Tables Tables – 1 3-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Basic Ideal Op Amp Assumptions 1 8-11 – 7 – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Data Set for Curves in Figure 8 1 Tabulation of Pertinent VFA and CFA Equations 9-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 – – 1 10 10-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noise Colors 11 1 Op Amp Parameter GLossary 11-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – Cross-Reference of Op Amp Parameters 11 11-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 12 1 Transducer Output Voltage 12-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 12 – 2 ADC Input Voltage 12-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 – Op Amp Selection 12-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 – 12-16 AIA Input and Output Voltages 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 12 5 Offset and Gain Error Budget 12-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 13 – 1 GSM Receiver Block System Budget 13-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 High-Speed Op Amp Requirements 13-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 14 – 1 DC Step Size for D/A converters 14-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 – 2 Converter Bits, THD, and Dynamic Range 14-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16-17 – 1 Second-Order FIlter Coefficients 16 – α For Different Filter Types and Different Qs 16-33 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Values of = 25 ° C, V = 5 V) 16-54 . . . . . . . . . . . . . . . . . . . 3 – Single-Supply Op Amp Selection Guide (T 16 A CC – 4 Bessel Coefficients 16-56 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16 5 Butterworth Coefficients 16-57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – – 6 Tschebyscheff Coefficients for 0.5-dB Passband Ripple 16-58 . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16 – 7 Tschebyscheff Coefficients for 1-dB Passband Ripple 16-59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 8 Tschebyscheff Coefficients for 2-dB Passband Ripple 16-60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – Tschebyscheff Coefficients for 3-dB Passband Ripple 16-61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 – 16-62 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 All-Pass Coefficients 17 1 PCB Materials 17-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 17 – 2 Recommended Maximum Frequencies for Capacitors 17-21 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1 Comparison of Op Amp Error Terms 18-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . – 2 Op Amp Parameters 18-21 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-4 A – 1 Ideal Op Amp Assumptions xx

23 Examples Examples 16 – 1 First-Order Unity-Gain Low-Pass Filter 16-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 16 – . . . . . . . . . . . . . . . . . . . . . . . . . Second-Order Unity-Gain Tschebyscheff Low-Pass Filter 16-16 16 – 3 Fifth-Order Filter 16-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-26 = 1 kHz – 4 Third-Order High-Pass Filter with f 16 C = 1 kHz 16-32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – 5 Second-Order MFB Band-Pass Filter with f m 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-34 – 6 Fourth-Order Butterworth Band-Pass Filter 16 – 7 2-ms Delay All-Pass Filter 16-45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi Contents

24 xxii

25 Chapter 1 The Op Amp’s Place In The World Ron Mancini In 1934 Harry Black[1] commuted from his home in New York City to work at Bell Labs in New Jersey by way of a railroad/ferry. The ferry ride relaxed Harry enabling him to do some conceptual thinking. Harry had a tough problem to solve; when phone lines were extended long distances, they needed amplifiers, and undependable amplifiers limited phone service. First, initial tolerances on the gain were poor, but that problem was quickly with an adjustment. Second, even when an amplifier was adjusted correctly at the solved factory, the gain drifted so much during field operation that the volume was too low or the incoming speech was distorted. attempts had been made to make a stable amplifier, but temperature changes and Many power supply voltage extremes experienced on phone lines caused uncontrollable gain drift. Passive components had much better drift characteristics than active components had, thus if an amplifier ’ s gain could be made dependent on passive components, the problem would be solved. During one of his ferry trips, Harry ’ s fertile brain conceived a novel solution for the amplifier problem, and he documented the solution while riding on the ferry. The solution was to first build an amplifier that had more gain than the application re- quired. Then some of the amplifier output signal was fed back to the input in a manner that makes the circuit gain (circuit is the amplifier and feedback components) dependent on the feedback circuit rather than the amplifier gain. Now the circuit gain is dependent on the passive feedback components rather than the active amplifier. This is called negative feedback, and it is the underlying operating principle for all modern day op amps. Harry had documented the first intentional feedback circuit during a ferry ride. I am sure unintentional feedback circuits had been built prior to that time, but the design- ers ignored the effect! I can hear the squeals of anguish coming from the managers and amplifier designers. I imagine that they said something like this, “ it is hard enough to achieve 30-kHz gain – bandwidth (GBW), and now this fool wants me to design an amplifier with 3-MHz GBW. But, he is still going to get a circuit gain GBW of 30 kHz ” . Well, time has proven Harry right, but there is a minor problem that Harry didn ’ t discuss in detail, and that is the oscillation 1-1

26 problem. It seems that circuits designed with large open loop gains sometimes oscillate when the loop is closed. A lot of people investigated the instability effect, and it was pretty well understood in the 1940s, but solving stability problems involved long, tedious, and intricate calculations. Y ears passed without anybody making the problem solution simpler or more understandable. In 1945 H. W. Bode presented a system for analyzing the stability of feedback systems by using graphical methods. Until this time, feedback analysis was done by multiplication and division, so calculation of transfer functions was a time consuming and laborious task. engineers did not have calculators or computers until the ’ 70s. Bode present- Remember, ed a log technique that transformed the intensely mathematical process of calculating a feedback system ’ s stability into graphical analysis that was simple and perceptive. Feed- design was still complicated, but it no longer was an art dominated by a few back system electrical engineers kept in a small dark room. Any electrical engineer could use Bode ’ s methods to find the stability of a feedback circuit, so the application of feedback to ma- chines began to grow. There really wasn ’ t much call for electronic feedback design until computers and transducers become of age. The first real-time computer was the analog computer! This computer used prepro- grammed equations and input data to calculate control actions. The programming was hard wired with a series of circuits that performed math operations on the data, and the hard wiring limitation eventually caused the declining popularity of the analog computer. The heart of the analog computer was a device called an operational amplifier because it could be configured to perform many mathematical operations such as multiplication, addition, subtraction, division, integration, and differentiation on the input signals. The name was shortened to the familiar , as we have come to know and love them. op amp The op amp used an amplifier with a large open loop gain, and when the loop was closed, the amplifier performed the mathematical operations dictated by the external passive components. This amplifier was very large because it was built with vacuum tubes and it required a high-voltage power supply, but it was the heart of the analog computer, thus its large size and huge power requirements were accepted as the price of doing business. early op amps were designed for analog computers, and it was soon found out that Many op amps had other uses and were very handy to have around the physics lab. At this time general-purpose analog computers were found in universities and large com- pany laboratories because they were critical to the research work done there. There was a parallel requirement for transducer signal conditioning in lab experiments, and op amps found their way into signal conditioning applications. As the signal conditioning applica- tions expanded, the demand for op amps grew beyond the analog computer require- ments, and even when the analog computers lost favor to digital computers, the op amp survived because of its importance in universal analog applications. Eventually digital computers replaced the analog computers (a sad day for real-time measurements), but the demand for op amps increased as measurement applications increased. 1-2

27 The first signal conditioning op amps were constructed with vacuum tubes prior to the introduction of ’ 50s, miniature vacu- transistors, so they were large and bulky. During the um tubes that worked from lower voltage power supplies enabled the manufacture of op amps that shrunk to the size of a brick used in house construction, so the op amp modules . Vacuum tube size and component size decreased until an op were nicknamed bricks ransistors were commercially amp was shrunk to the size of a single octal vacuum tube. T 60s, and they further reduced op amp size to several cubic inches, but developed in the ’ the nickname brick still held on. Now the nickname brick is attached to any electronic mod- ule that uses potting compound or non-integrated circuit (IC) packaging methods. Most of these early op amps were made for specific applications, so they were not necessarily general purpose. The early op amps served a specific purpose, but each manufacturer had different specifications and packages; hence, there was little second sourcing among the early op amps. ICs were developed during the late 1950s and early 1960s, but it wasn ’ t till the middle μ 1960s that Fairchild released the A709. This was the first commercially successful IC op amp, and Robert J. Widler designed it. The μ A709 had its share of problems, but any competent analog engineer could use it, and it served in many different analog applica- tions. The major drawback of the A709 was stability; it required external compensation μ and a competent analog engineer to apply it. Also, the μ A709 was quite sensitive because it had a habit of self destructing under any adverse condition. The self-destruction habit was so prevalent that one major military equipment manufacturer published a paper titled something like, The 12 Pearl Harbor Conditions of the μ A709 . The μ A741 followed the μ A709, and it is an internally compensated op amp that does not require external com- pensation if operated under data sheet conditions. Also, it is much more forgiving than the A709. There has been a never-ending series of new op amps released each year μ since then, and their performance and reliability has improved to the point where present day op amps can be used for analog applications by anybody. The IC op amp is here to stay; the latest generation op amps cover the frequency spec- trum from 5-kHz GBW to beyond 1-GHz GBW. The supply voltage ranges from guaran- teed operation at 0.9 V to absolute maximum voltage ratings of 1000 V. The input current and input offset voltage has fallen so low that customers have problems verifying the specifications during incoming inspection. The op amp has truly become the universal analog IC because it performs all analog tasks. It can function as a line driver, comparator (one bit A/D), amplifier, level shifter, oscillator, filter, signal conditioner, actuator driver, cur- rent ’ s problem is how source, voltage source, and many other applications. The designer to rapidly select the correct circuit/op amp combination and then, how to calculate the pas- sive component values that yield the desired transfer function in the circuit. This book deals with op amp circuits — not with the innards of op amps. It treats the cal- culations from the circuit level, and it doesn ’ t get bogged down in a myriad of detailed cal- culations. Rather, the reader can start at the level appropriate for them, and quickly move on to the advanced topics. If you are looking for material about the innards of op amps 1-3

28 you are looking in the wrong place. The op amp is treated as a completed component in this book. The op amp will continue to be a vital component of analog design because it is such a fundamental component. Each generation of electronics equipment integrates more t fear, as digi- functions on silicon and takes more of the analog circuitry inside the IC. Don ’ tal applications increase, analog applications also increase because the predominant supply of data and interface applications are in the real world, and the real world is an ana- log world. Thus, each new generation of electronics equipment creates requirements for new analog circuits; hence, new generations of op amps are required to fulfill these re- quirements. Analog design, and op amp design, is a fundamental skill that will be required far into the future. References Stabilized Feedback Amplifiers , BSTJ, Vol. 13, January 1934 1 Black, H. S., 1-4

29 Chapter 2 Review of Circuit Theory Ron Mancini 2.1 Introduction Although this book minimizes math, some algebra is germane to the understanding of analog electronics. Math and physics are presented here in the manner in which they are used later, so no practice exercises are given. For example, after the voltage divider rule is explained, it is used several times in the development of other concepts, and this usage constitutes practice. Circuits are a mix of passive and active components. The components are arranged in a manner that enables them to perform some desired function. The resulting arrangement of components is called a circuit or sometimes a circuit configuration. The art portion of analog design is developing the circuit configuration. There are many published circuit configurations for almost any circuit task, thus all circuit designers need not be artists. When the design has progressed to the point that a circuit exists, equations must be writ- ten to predict and analyze circuit performance. Textbooks are filled with rigorous methods for equation writing, and this review of circuit theory does not supplant those textbooks. But, a few equations are used so often that they should be memorized, and these equa- tions are considered here. There are almost as many ways to analyze a circuit as there are electronic engineers, and if the equations are written correctly, all methods yield the same answer. There are some simple ways to analyze the circuit without completing unnecessary calculations, and these methods are illustrated here. 2.2 Laws of Physics Ohm ’ s law is stated as V=IR, and it is fundamental to all electronics. Ohm ’ s law can be applied to a single component, to any group of components, or to a complete circuit. When the current flowing through any portion of a circuit is known, the voltage dropped across that portion of the circuit is obtained by multiplying the current times the resistance (Equa- tion 2 – 1). 2-1

30 Laws of Physics (2 1) – + V IR – 1, Ohm ’ In Figure 2 s law is applied to the total circuit. The current, (I) flows through the total resistance (R), and the voltage (V) is dropped across R. I R V Figure 2–1. Ohm’s Law Applied to the Total Circuit In Figure 2 – 2, Ohm ’ s law is applied to a single component. The current (I ) flows through R ) is dropped across R. Notice, the same formula is used the resistor (R) and the voltage (V R to calculate the voltage drop across R even though it is only a part of the circuit. I R R V V R Figure 2–2. Ohm’s Law Applied to a Component Kirchoff ’ s voltage law states that the sum of the voltage drops in a series circuit equals the sum of the voltage sources. Otherwise, the source (or sources) voltage must be dropped across the passive components. When taking sums keep in mind that the sum 3 and Equations ’ is an algebraic quantity. Kirchoff – s voltage law is illustrated in Figure 2 2 – 3. – 2 and 2 R 1 V R1 R V V 2 R2 Figure 2–3. Kirchoff’s Voltage Law (2 – 2) ȍ ȍ V V + SOURCES DROPS V + V V ) 3) (2 – R1 R2 ’ Kirchoff s current law states: the sum of the currents entering a junction equals the sum of the currents leaving a junction. It makes no difference if a current flows from a current 2-2

31 Voltage Divider Rule source, through a component, or through a wire, because all currents are treated identi- 4 and 2 ’ – 4 and Equations 2 cally. Kirchoff s current law is illustrated in Figure 2 – 5. – I I 3 4 I I 2 1 Figure 2 – 4. Kirchoff ’ s Current Law (2 – 4) ȍ ȍ I I + IN OUT I ) I I + ) I 5) – (2 4 2 3 1 2.3 Voltage Divider Rule When the output of a circuit is not loaded, the voltage divider rule can be used to calculate the circuit ’ s output voltage. Assume that the same current flows through all circuit ele- ). Equation + R ments (Figure 2 – 5). Equation 2 – 6 is written using Ohm ’ s law as V = I (R 2 1 2 – ’ s law across the output resistor. 7 is written as Ohm R 1 I I R V V 2 O Figure 2 – 5. Voltage Divider Rule V I + (2 6) – R ) R 1 2 (2 – IR 7) V + 2 OUT Substituting Equation 2 – 6 into Equation 2 – 7, and using algebraic manipulation yields Equation 2 – 8. R 2 – 8) (2 + V V OUT R ) R 1 2 A simple way to remember the voltage divider rule is that the output resistor is divided by the total circuit resistance. This fraction is multiplied by the input voltage to obtain the out- 2-3 Review of Circuit Theory

32 Current Divider Rule put voltage. Remember that the voltage divider rule always assumes that the output resis- tor is not loaded; the equation is not valid when the output resistor is loaded by a parallel component. Fortunately, most circuits following a voltage divider are input circuits, and input circuits are usually high resistance circuits. When a fixed load is in parallel with the output resistor, the equivalent parallel value comprised of the output resistor and loading resistor can be used in the voltage divider calculations with no error. Many people ignore the load resistor if it is ten times greater than the output resistor value, but this calculation can lead to a 10% error. 2.4 Current Divider Rule When the output of a circuit is not loaded, the current divider rule can be used to calculate – ). The currents I in Figure 2 and I 6 are the current flow in the output branch circuit (R 1 2 2 flowing in the branch circuits. Equation 2 9 is written with the aid of Kirch- assumed to be – ’ off – 10 with the aid of Ohm ’ s law. s current law. The circuit voltage is written in Equation 2 Combining Equations 2 – 9 and 2 – 10 yields Equation 2 – 11. I I 2 1 V I R R 2 1 Figure 2 – 6. Current Divider Rule (2 – 9) I + I I ) 1 2 (2 – 10) V I I R R + + 1 2 1 2 R ) R R 2 1 2 (2 – 11) ǒ Ǔ I ) + I + I I I ) I + 2 1 2 2 2 R R 1 1 Rearranging the terms in Equation 2 11 yields Equation 2 – 12. – R 1 12) – (2 ǒ Ǔ + I I 2 ) R R 2 1 The total circuit current divides into two parts, and the resistance (R ) divided by the total 1 . An easy method of remember- resistance determines how much current flows through R 2 ing the current divider rule is to remember the voltage divider rule. Then modify the voltage divider rule such that the opposite resistor is divided by the total resistance, and the frac- tion is multiplied by the input current to get the branch current. 2-4

33 Thevenin s Theorem ’ ’ 2.5 Thevenin s Theorem There are times when it is advantageous to isolate a part of the circuit to simplify the analy- sis of the isolated part of the circuit. Rather than write loop or node equations for the com- ’ plete circuit, and solving them simultaneously, Thevenin s theorem enables us to isolate the part of the circuit we are interested in. We then replace the remaining circuit with a simple series equivalent circuit, thus Thevenin ’ s theorem simplifies the analysis. There are two theorems that do similar functions. The Thevenin theorem just described s theorem is used when is the first, and the second is called Norton ’ s theorem. Thevenin ’ ’ s theorem is used when the input source the input source is a voltage source, and Norton is a current source. Norton ’ s theorem is rarely used, so its explanation is left for the reader to dig out of a textbook if it is ever required. The rules for Thevenin ’ s theorem start with the component or part of the circuit being re- toward placed. Referring to Figure 2 – 7, look back into the terminals (left from C and R 3 ) point XX in the figure) of the circuit being replaced. Calculate the no load voltage (V TH as seen from these terminals (use the voltage divider rule). X R R 3 1 C V R 2 X Figure 2 – 7. Original Circuit Look into the terminals of the circuit being replaced, short independent voltage sources, and calculate the impedance between these terminals. The final step is to substitute the – 8. Thevenin equivalent circuit for the part you wanted to replace as shown in Figure 2 X R R 3 TH C V TH X Figure 2 – 8. Thevenin ’ s Equivalent Circuit for Figure 2 – 7 The Thevenin equivalent circuit is a simple series circuit, thus further calculations are sim- s ’ plified. The simplification of circuit calculations is often suf ficient reason to use Thevenin 2-5 Review of Circuit Theory

34 Thevenin ’ s Theorem theorem because it eliminates the need for solving several simultaneous equations. The detailed information about what happens in the circuit that was replaced is not available using Thevenin ’ s theorem, but that is no consequence because you had no interest when in it. ) shown in As an example of Thevenin s theorem, let ’ s calculate the output voltage (V ’ OUT Figure 2 – 9A. The first step is to stand on the terminals X – Y with your back to the output ). This is a perfect opportunity to circuit, and calculate the open circuit voltage seen (V TH use the voltage divider rule to obtain Equation 2 – 13. R R R R 3 1 3 TH X X V V OUT V OUT R V 2 TH R R 4 Y Y 4 (a) The Original Circuit (b) The Thevenin Equivalent Circuit Figure 2 – 9. Example of Thevenin ’ s Equivalent Circuit R 2 V V + (2 – 13) TH R ) R 1 2 Still standing on the terminals X-Y, step two is to calculate the impedance seen looking into these terminals (short the voltage sources). The Thevenin impedance is the parallel 14. Now get off the terminals X-Y – as calculated in Equation 2 and R impedance of R 2 1 before you damage them with your big feet. Step three replaces the circuit to the left of . and R X-Y with the Thevenin equivalent circuit V TH TH R R 1 2 Ŧ R + R R + – 14) (2 2 1 TH ) R R 2 1 Note: Two parallel vertical bars ( || ) are used to indicate parallel components as – shown in Equation 2 14. The final step is to calculate the output voltage. Notice the voltage divider rule is used again. Equation 2 – 15 describes the output voltage, and it comes out naturally in the form of a series of voltage dividers, which makes sense. That ’ s another advantage of the volt- age divider rule; the answers normally come out in a recognizable form rather than a jumble of coefficients and parameters. 2-6

35 Thevenin s Theorem ’ R R R 4 2 4 Ǔ ǒ + V V V + – (2 15) TH OUT R ) ) R R R ) R R R 1 2 4 2 3 1 TH R ) ) R 3 4 R ) R 2 1 – 10, so you can see the advantage The circuit analysis is done the hard way in Figure 2 and I , are assigned to the circuit. of using Thevenin ’ s Theorem. Two loop currents, I 1 2 – 16 and 2 – 17 are written. Then the loop Equations 2 R R 1 3 R R 2 4 V OUT I I V 2 1 10. Analysis Done the Hard Way Figure 2 – ǒ Ǔ (2 16) – R V ) I + * I R R 2 2 1 2 1 Ǔ ǒ 17) (2 – I ) R ) R I + R R 3 2 2 1 4 2 Equation 2 – 17 is rewritten as Equation 2 – 18 and substituted into Equation 2 – 16 to obtain – 19. Equation 2 ) R R ) R 4 2 3 18) – (2 I I + 2 1 R 2 R ) R ) R 3 2 4 ǒ Ǔ 19) – (2 Ǔ ǒ V I R ) R + * I R 1 2 2 2 2 R 2 – 20. Ohm ’ s law is used to write Equation 2 – 21, The terms are rearranged in Equation 2 – 22. and the final substitutions are made in Equation 2 V I + – 20) (2 2 ) R ) R R 4 2 3 ǒ Ǔ R ) R * R 2 1 2 R 2 – 21) (2 R + I V 2 4 OUT R 4 V + V OUT (2 – 22) ǒ Ǔ Ǔǒ ) R R ) R ) R R 1 4 3 2 2 * R 2 R 2 This is a lot of extra work for no gain. Also, the answer is not in a usable form because the voltage dividers are not recognizable, thus more algebra is required to get the answer into usable form. 2-7 Review of Circuit Theory

36 Superposition 2.6 Superposition Superposition is a theorem that can be applied to any linear circuit. Essentially, when there are independent sources, the voltages and currents resulting from each source can be calculated separately, and the results are added algebraically. This simplifies the cal- culations because it eliminates the need to write a series of loop or node equations. An example is shown in Figure 2 – 11. R 1 R R 3 2 V OUT V 1 V 2 Figure 2 – 11.Superposition Example When V and the parallel combination of is grounded, V forms a voltage divider with R 2 1 3 R and R . The output voltage for this circuit (V ) is calculated with the aid of the volt- 2 1 OUT2 age divider equation (2 – 23). The circuit is shown in Figure 2 – 12. The voltage divider rule yields the answer quickly. R 3 R V R 1 2 2 V OUT2 12. When V is Grounded Figure 2 – 1 R ø R 1 2 23) – (2 + V V 2 OUT2 R R ) ø R 2 1 3 when V is grounded (Figure 2 – 13), V Likewise, forms a voltage divider with R and the 1 1 2 and R , and the voltage divider theorem is applied again to cal- parallel combination of R 3 2 culate V 24). (Equation 2 – OUT R 1 R V R 3 1 2 V OUT1 is Grounded 13. When V Figure 2 – 2 2-8

37 Calculation of a Saturated Transistor Circuit R ø R 3 2 – (2 24) + V V 1 OUT1 R ) R R ø 1 3 2 After the calculations for each source are made the components are added to obtain the final solution (Equation 2 – 25). R ø R R R ø 2 3 2 1 25) – (2 V + V V ) 2 1 OUT R R ø R ) R R ø R ) 3 1 2 3 2 1 The reader should analyze this circuit with loop or node equations to gain an appreciation for superposition. Again, the superposition results come out as a simple arrangement that is easy to understand. One looks at the final equation and it is obvious that if the sources = R , then the output voltage is zero. Conclu- are equal and opposite polarity, and when R 1 3 sions such as this are hard to make after the results of a loop or node analysis unless con- siderable effort is made to manipulate the final equation into symmetrical form. 2.7 Calculation of a Saturated Transistor Circuit The circuit specifications are: when V <0.05 <10 mA, and V = 12 V, V <0.4 V at I SINK IN IN OUT V, V 14. >10 V at I – = 1 mA. The circuit diagram is shown in Figure 2 OUT OUT 12 V I R C C I B V IN V R OUT B Figure 2 14. Saturated Transistor Circuit – The collector resistor must be sized (Equation 2 – 26) when the transistor is off, because it has to be small enough to allow the output current to flow through it without dropping more than two volts to meet the specification for a 10-V output. V * V ) 12 OUT 10 * 12 26) (2 – R + 2k + v C 1 I OUT When the transistor is off, 1 mA can be drawn out of the collector resistor without pulling the collector or output voltage to less than ten volts (Equation 2 – 27). When the transistor 28) to enable the input signal to drive is on, the base resistor must be sized (Equation 2 – enough base current into the transistor to saturate it. The transistor beta is 50. 2-9 Review of Circuit Theory

38 Transistor Amplifier V V * V ) 12 CE 12 ) – 27) (2 I I + + ) I I [ ) b L L B C R R C C V * V IN BE v R 28) – (2 B I B Substituting Equation 2 – 27 into Equation 2 – 28 yields Equation 2 – 29. Ǔ ǒ * V V b 0.6)50 V * (12 IN BE 35.6 k R v + + 29) (2 – B I 12 C ƪ ƫ ) (10) mA 2 the transistor goes on it sinks the load current, and it still goes into saturation. These When calculations neglect some minor details, but they are in the 98% accuracy range. 2.8 Transistor Amplifier The amplifier is an analog circuit (Figure 2 – 15), and the calculations, plus the points that must be considered during the design, are more complicated than for a saturated circuit. This extra complication leads people to say that analog design is harder than digital de- sign (the saturated transistor is digital i.e.; on or off). Analog design is harder than digital design because the designer must account for all states in analog, whereas in digital only two states must be accounted for. The specifications for the amplifier are an ac voltage gain of four and a peak-to-peak signal swing of 4 volts. 12 V 12 V R C R 1 V OUT V IN C IN R 2 R E1 C E R E2 Figure 2 – 15. Transistor Amplifier β I is selected as 10 mA because the transistor has a current gain ( ) of 100 at that point. C The collector voltage is arbitrarily set at 8 V; when the collector voltage swings positive 2-10

39 Transistor Amplifier 2 V (from 8 V to 10 V) there is still enough voltage dropped across R to keep the transistor C on. Set the collector-emitter voltage at 4 V; when the collector voltage swings negative 2 V (from 8 V to 6 V) the transistor still has 2 V across it, so it stays linear. This sets the ) at 4 V. emitter voltage (V E * V V ) 12 12 V * 8V C W v 400 + R + 30) – (2 C 10 mA I C V V V E E E 4V + ) ^ 400 + R + W + + R R 31) – (2 E2 E1 E ) I I I I 10 mA B E C C Use Thevenin ’ s equivalent circuit to calculate R 16. and R – as shown in Figure 2 2 1 R || R 1 2 I B R 2 V = 4.6 V B 12 + R R 2 1 Figure 2 – 16. Thevenin Equivalent of the Base Circuit I 10 mA C 32) (2 – + I + + 0.1 mA B 100 b 12R 2 33) (2 – V + TH ) R R 2 1 R R 2 1 (2 – 34) + R TH R ) R 2 1 We want the base voltage to be 4.6 V because the emitter voltage is then 4 V. Assume , so Equation 2 – 35 can be written. The drop across R a voltage drop of 0.4 V across R TH TH may not be exactly 0.4 V because of beta variations, but a few hundred mV does not mat- using the voltage divider rule (the and R ter is this design. Now, calculate the ratio of R 2 1 load current has been accounted for). 0.4 (2 – 35) 4k + k + R TH 0.1 R 2 – 36) (2 ) I R + ) V 12 V 0.4 + 4.6 + 5 + B B Th TH R R ) 1 2 7 – 37) (2 R + R 1 2 5 R is almost equal to R , thus selecting R as twice the Thevenin resistance yields approx- 1 2 1 = 11.2 k and R = 8 k. The ac gain is imately 4 K as shown in Equation 2 – 35. Hence, R 1 2 2-11 Review of Circuit Theory

40 Transistor Amplifier at high frequencies, so we can write approximately R /R shorts out R because C E E2 C E1 Equation 2 – 38. R 400 C (2 – 38) + + 100 W R + E1 4 G (2 – 39) W 300 R + + R 100 * R * + 400 E1 E2 E The capacitor selection depends on the frequency response required for the amplifier, but μ suffice for a starting point. and 1000 F for C F for C 10 μ IN E 2-12

41 Chapter 3 Development of the Ideal Op Amp Equations Ron Mancini 3.1 Ideal Op Amp Assumptions Ideal Op Amp is applied to this and similar analysis because the salient param- The name eters of the op amp are assumed to be perfect. There is no such thing as an ideal op amp, Ideal Op Amp analysis approaches but present day op amps come so close to ideal that analysis. Op amps depart from the ideal in two ways. First, dc parameters such as actual input offset voltage are large enough to cause departure from the ideal. The ideal as- sumes that input offset voltage is zero. Second, ac parameters such as gain are a function of frequency, so they go from large values at dc to small values at high frequencies. This assumption simplifies the analysis, thus it clears the path for insight. It is so much easier to see the forest when the brush and huge trees are cleared away. Although the ideal op amp analysis makes use of perfect parameters, the analysis is often valid be- cause some op amps approach perfection. In addition, when working at low frequencies, several kHz, the ideal op amp analysis produces accurate answers. Voltage feedback op amps are covered in this chapter, and current feedback op amps are covered in Chap- ter 8. Several assumptions have to be made before the ideal op amp analysis can proceed. First, assume that the current flow into the input leads of the op amp is zero. This assump- tion is almost true in FET op amps where input currents can be less than a pA, but this is not always true in bipolar high-speed op amps where tens of μ A input currents are found. Second, the op amp gain is assumed to be infinite, hence it drives the output voltage to any value to satisfy the input conditions. This assumes that the op amp output voltage can achieve any value. In reality, saturation occurs when the output voltage comes close to a power supply rail, but reality does not negate the assumption, it only bounds it. Also, implicit in the infinite gain assumption is the need for zero input signal. The gain drives the output voltage until the voltage between the input leads (the error voltage) is zero. This leads to the third assumption that the voltage between the input leads is zero. The implication of zero voltage between the input leads means that if one input is tied to 3-1

42 Ideal Op Amp Assumptions a hard voltage source such as ground, then the other input is at the same potential. The current flow into the input leads is zero, so the input impedance of the op amp is infinite. Fourth, the output impedance of the ideal op amp is zero. The ideal op amp can drive any load without an output impedance dropping voltage across it. The output impedance of most op amps is a fraction of an ohm for low current flows, so this assumption is valid in most cases. Fifth, the frequency response of the ideal op amp is flat; this means that the gain does not vary as frequency increases. By constraining the use of the op amp to the low frequencies, we make the frequency response assumption true. Table 3 – 1 lists the basic ideal op amp assumptions and FIgure 3 – 1shows the ideal op amp. Table 3 1. Basic Ideal Op Amp Assumptions – PARAMETER NAME PARAMETERS SYMBOL VALUE Input current I 0 IN Input offset voltage V 0 OS Z Input impedance ∞ IN 0 Z Output impedance OUT Gain a ∞ _ I = 0 B = 0 Z O V = 0 E V ∝ = Z OUT i ∝ a = + I = 0 B – 1. The Ideal Op Amp Figure 3 3-2

43 The Noninverting Op Amp 3.2 The Noninverting Op Amp The noninverting op amp has the input signal connected to its noninverting input (Figure – 3 2), thus its input source sees an infinite impedance. There is no input offset voltage be- = V = 0, hence the negative input must be at the same voltage as the positive cause V OS E until the negative input is at the voltage, input. The op amp output drives current into R F . This action causes V . to appear across R V G IN IN I = 0 B + V IN V a V OUT E _ R F V IN R G Figure 3 – 2. The Noninverting Op Amp The voltage divider rule is used to calculate V ; V is the input to the voltage divider, IN OUT is the output of the voltage divider. Since no current can flow into either op amp and V IN – 1 is written with the aid of the lead, use of the voltage divider rule is allowed. Equation 3 voltage divider rule, and algebraic manipulation yields Equation 3 – 2 in the form of a gain parameter. R G 1) (3 – V + V IN OUT ) R R F G R V R ) R F G OUT F (3 2) – 1 ) + + V R R IN G G When R – becomes very large with respect to R 2 reduces , (R 0 and Equation 3 /R ⇒ ) G F G F to Equation 3 3. – V + 1 (3 3) – OUT these conditions V Under = 1 and the circuit becomes a unity gain buffer. R is usually G OUT is deleted, R can also be deleted (RF achieve the same results, and when R deleted to G F must be shorted when it is deleted). When R and R are deleted, the op amp output is G F connected to its inverting input with a wire. Some op amps are self-destructive when R F is left out of the circuit, so R is used in many buffer designs. When R is included in a F F buffer circuit, its function is to protect the inverting input from an over voltage to limit the current through the input ESD (electro-static discharge) structure (typically < 1 mA), and can never be left out of the circuit it can have almost any value (20 k is often used). R F 3-3 Development of the Ideal Op Amp Equations

44 The Inverting Op Amp determines stability in current feed- in a current feedback amplifier design because R F back amplifiers. Notice that the gain is only a function of the feedback and gain resistors; therefore the feedback has accomplished its function of making the gain independent of the op amp parameters. The gain is adjusted by varying the ratio of the resistors. The actual resistor values are determined by the impedance levels that the designer wants to establish. = 10 k and R = 100 k and = 10 k the gain is two as shown in Equation 2, and if R If R F F G = 100 k the gain is still two. The impedance levels of 10 k or 100 k determine the cur- R G rent drain, the effect of stray capacitance, and a few other points. The impedance level /R does. does not set the gain; the ratio of R G F 3.3 The Inverting Op Amp The noninverting input of the inverting op amp circuit is grounded. One assumption made is that the input error voltage is zero, so the feedback keeps inverting the input of the op amp at a virtual ground (not actual ground but acting like ground). The current flow in the equals the cur- input leads is assumed to be zero, hence the current flowing through R G rent flowing through R 4; and the minus sign . Using Kirchoff ’ s law, we write Equation 3 – F is inserted because this is the inverting input. Algebraic manipulation gives Equation 3 – 5. I I 1 2 V IN R R G F = 0 I B _ V V a OUT E + I = 0 B Figure 3 – 3. The Inverting Op Amp V V OUT IN 4) – (3 I + I +* +* 1 2 R R F G V R OUT F – (3 5) +* V R IN G that the gain is only a function of the feedback and gain resistors, so the feedback Notice has accomplished its function of making the gain independent of the op amp parameters. The actual resistor values are determined by the impedance levels that the designer = 10 k the gain is minus one as shown in Equation = 10 k and R wants to est ablish. If R F G 3-4

45 The Adder – 3 = 100 k the gain is still minus one. The impedance levels = 100 k and R 5, and if R F G of 10 k or 100 k determine the current drain, the effect of stray capacitance, and a few other does. /R points. The impedance level does not set the gain; the ratio of R G F One final note; the output signal is the input signal amplified and inverted. The circuit input impedance is set by R because the inverting input is held at a virtual ground. G 3.4 The Adder An adder circuit can be made by connecting more inputs to the inverting op amp (Figure – 4). The opposite end of the resistor connected to the inverting input is held at virtual 3 the feedback; therefore, adding new inputs does not affect the response of the ground by existing inputs. V 1 R R F 1 _ V 2 R V 2 OUT + V N R N Figure 3 – 4. The Adder Circuit Superposition is used to calculate the output voltages resulting from each input, and the – 6 output voltages are added algebraically to obtain the total output voltage. Equation 3 – 7 and 3 8 are the other – are grounded. Equations 3 and V is the output equation when V 2 1 9. – superposition equations, and the final result is given in Equation 3 R F – (3 6) V V +* N OUTN R N R F (3 – 7) +* V V 1 OUT1 R 1 R F – 8) (3 V +* V 2 OUT2 R 2 R R R F F F 9) (3 – Ǔ ǒ V V ) ) +* V V 2 1 N OUT R R R 1 2 N 3-5 Development of the Ideal Op Amp Equations

46 The Differential Amplifier 3.5 The Differential Amplifier The differential amplifier circuit amplifies the difference between signals applied to the in- – 5). Superposition is used to calculate the output voltage resulting from puts (Figure 3 each input voltage, and then the two output voltages are added to arrive at the final output voltage. R V 1 + + V OUT V – R V _ 2 1 R R 3 4 V 2 – 5. The Differential Amplifier Figure 3 The op amp input voltage resulting from the input source, V , is calculated in Equa- 1 , and the – 10 and 3 – 11. The voltage divider rule is used to calculate the voltage, V tions 3 + 2) is used to calculate the noninverting output noninverting gain equation (Equation 3 – . voltage, V OUT1 R 2 – (3 10) + V V ) 1 ) R R 2 1 R ) R R 4 3 2 – 11) (3 ( ) ǒ Ǔ V + V G V + ) ) 1 OUT1 R ) R R 1 2 3 The inverting gain equation (Equation 3 5) is used to calculate the stage gain for V – OUT2 13. – 12. These inverting and noninverting gains are added in Equation 3 – in Equation 3 R 4 – 12) (3 ǒ Ǔ + * V V 2 OUT2 R 3 R ) R R R 4 3 2 4 13) – (3 ǒ Ǔ V V + * V 2 1 OUT R ) R R R 3 1 3 2 When R 14. = R – and R 13 reduces to Equation 3 = R – , Equation 3 3 1 4 2 R 4 Ǔ ǒ (3 – 14) + * V V V 1 2 OUT R 3 ), is multiplied by the stage gain, so It is now obvious that the differential signal, (V V – 1 2 the name differential amplifier suits the circuit. Because it only amplifies the differential 3-6

47 Complex Feedback Networks the input signal, it rejects the common-mode portion of the input signal. A com- portion of mon-mode signal is illustrated in Figure 3 – 6. Because the differential amplifier strips off or rejects the common-mode signal, this circuit configuration is often employed to strip dc or injected common-mode noise off a signal. R G + V OUT R F _ V 1 R R G F V 2 V CM Figure 3 – 6. Differential Amplifier With Common-Mode Input Signal The disadvantage of this circuit is that the two input impedances cannot be matched when it functions as a differential amplifier, thus there are two and three op amp versions of this circuit specially designed for high performance applications requiring matched input im- pedances. 3.6 Complex Feedback Networks When complex networks are put into the feedback loop, the circuits get harder to analyze because the simple gain equations cannot be used. The usual technique is to write and solve node or loop equations. There is only one input voltage, so superposition is not of ’ s theorem can be used as is shown in the example problem given any use, but Thevenin below. Sometimes it is desirable to have a low resistance path to ground in the feedback loop. inverting op amps can not do this when the driving circuit sets the input resistor Standard T value, and the gain specification sets the feedback resistor value. Inserting a network in the feedback loop (FIgure 3 – 7) yields a degree of freedom that enables both specifica- tions to be met with a low dc resistance path in the feedback loop. X V IN R R R 3 1 2 R 4 Y _ a V OUT + – 7. T Network in Feedback Loop Figure 3 3-7 Development of the Ideal Op Amp Equations

48 Complex Feedback Networks , and calculate the Y, stand on the terminals looking into R Break the circuit at point X – 4 – 15. The Thevenin equivalent imped- Thevenin equivalent voltage as shown in Equation 3 16. – ance is calculated in Equation 3 R 4 (3 – 15) V + V TH OUT R ) R 3 4 (3 16) – R + R ø R 4 3 TH Replace the output circuit with the Thevenin equivalent circuit as shown in Figure 5 – 8, and calculate the gain with the aid of the inverting gain equation as shown in Equation 3 – 17. V IN R R R 2 1 TH _ V a TH + Figure 3 8. Thevenin ’ s Theorem Applied to T Network – – 17 yields Equation 3 – 18. Substituting the Thevenin equivalents into Equation 3 ) R V R 2 TH TH (3 – 17) + * R V 1 IN ǒ Ǔ ø R R ) R V R R ) R R ) R ) R 4 3 2 OUT 3 4 2 3 4 TH – 18) (3 Ǔ Ǔ ǒ ǒ + + * R R R V R 1 4 4 1 IN Algebraic manipulation yields Equation 3 – 19. R R 3 2 R R ) ) 3 2 R V 19) – (3 4 OUT + * V R 1 IN Specifications for the circuit you are required to build are an inverting amplifier with an = 10 k), a gain of 100, and a feedback resistance of 20 K or input resistance of 10 k (R G must less. The inverting op amp circuit can not meet these specifications because R F = 485 k approximately = 10 k and R = R equal 1000 k. Inserting a T network with R 4 2 3 meets the specifications. 3-8

49 Video Amplifiers 3.7 Video Amplifiers signals contain high frequencies, and they use coaxial cable to transmit and receive Video signals. The cable connecting these circuits has a characteristic impedance of 75 Ω . To prevent reflections, which cause distortion and ghosting, the input and output circuit im- Ω cable. pedances must match the 75 Matching the input impedance is simple for a noninverting amplifier because its input im- can be selected as high values, = 75 Ω . R and R pedance is very high; just make R F G IN in the hundreds of Ohms range, so that they have minimal affect on the impedance of the , is placed in series with the op amp output input or output circuit. A matching resistor, R M , is placed at the input of Ω ; a terminating resistor, R to raise its output impedance to 75 T the next stage to match the cable (Figure 3 – 9). R + V M IN V a OUT _ R IN R R T F R G Figure 3 – 9. Video Amplifier The matching and terminating resistors are equal in value, and they form a voltage divider is selected equal to R so that the op amp is not loaded. Very often R of 1/2 because R T G F gain equals two. Then the system gain, which is the op amp gain multiplied by the divider gain, is equal to one (2 1/2 = 1). × 3.8 Capacitors s tool kit, thus a short discussion on Capacitors are a key component in a circuit designer ’ evaluating their affect on circuit performance is in order. Capacitors have an impedance = 1/2 π fC. Note that when the frequency is zero the capacitive impedance (also of X C known as reactance) is infinite, and that when the frequency is infinite the capacitive im- pedance is zero. These end-points are derived from the final value theorem, and they are get a rough idea of the effect of a capacitor. When a capacitor is used with a resis- used to tor, they form what is called a break-point. Without going into complicated math, just ac- cept that the break frequency occurs at f = 1/(2 π RC) and the gain is – 3 dB at the break frequency. The low pass filter circuit shown in Figure 3 – 10 has a capacitor in parallel with the feed- – 20. back resistor. The gain for the low pass filter is given in Equation 3 3-9 Development of the Ideal Op Amp Equations

50 Capacitors C F V IN R R F G _ V a OUT + Figure 3 – 10. Low-Pass Filter X R ø V F OUT C – 20) (3 +* R V IN G At very low frequencies X ⇒∞ , so R dominates the parallel combination in Equation C F . At very high 20, and the capacitor has no effect. The gain at low frequencies is – R /R F G 0, so the feedback resistor is shorted out, thus reducing the circuit gain ⇒ frequencies X C because complex im- = R the gain is reduced by √ 2 to zero. At the frequency where X F C pedances in parallel equal half the vector sum of both impedances. where it has the opposite effect makes a high Connecting the capacitor in parallel with R G pass filter (Figure 3 – – 21 gives the equation for the high pass filter. 11). Equation 3 V + IN V a OUT _ R C G G R F – 11.High-Pass Filter Figure 3 V R OUT F – 21) (3 1 + ) V ø R X IN C G At very low frequencies X ⇒∞ , so R dominates the parallel combination in Equation G C . At very /R 21, and the capacitor has no effect. The gain at low frequencies is 1+R 3 – G F ⇒ 0, so the gain setting resistor is shorted out thus increasing the high frequencies X C circuit gain to maximum. simple technique is used to predict the form of a circuit transfer function rapidly. Better This analysis techniques are presented in later chapters for those applications requiring more precision. 3-10

51 Summary 3.9 Summary When the proper assumptions are made, the analysis of op amp circuits is straightfor- ward. These assumptions, which include zero input current, zero input offset voltage, and the new op amps make them true in most infinite gain, are realistic assumptions because applications. When the signal is comprised of low frequencies, the gain assumption is valid because op amps have very high gain at low frequencies. When CMOS op amps are used, the in- put current is in the femto amp range; close enough to zero for most applications. Laser trimmed input circuits reduce the input offset voltage to a few micro volts; close enough to zero for most applications. The ideal op amp is becoming real; especially for unde- manding applications. 3-11 Development of the Ideal Op Amp Equations

52 3-12

53 Chapter 4 Single-Supply Op Amp Design Techniques Ron Mancini 4.1 Single Supply versus Dual Supply The previous chapter assumed that all op amps were powered from dual or split supplies, and this is not the case in today ’ s world of portable, battery-powered equipment. When – 1), the supplies are normally op amps are powered from dual supplies (see Figure 4 equal in magnitude, opposing in polarity, and the center tap of the supplies is connected to ground. Any input sources connected to ground are automatically referenced to the center of the supply voltage, so the output voltage is automatically referenced to ground. R R G F +V V IN _ R F – = V V IN OUT + R G V – Figure 4 – 1. Split-Supply Op Amp Circuit Single-supply systems do not have the convenient ground reference that dual-supply sys- tems have, thus biasing must be employed to ensure that the output voltage swings be- tween the correct voltages. Input sources connected to ground are actually connected to a supply rail in single-supply systems. This is analogous to connecting a dual-supply input to the minus power rail. This requirement for biasing the op amp inputs to achieve the de- sired output voltage swing complicates single-supply designs. When the signal source is not referenced to ground (see Figure 4 – 2), the voltage differ- ence between ground and the reference voltage is amplified along with the signal. Unless the reference voltage was inserted as a bias voltage, and such is not the case when the input signal is connected to ground, the reference voltage must be stripped from the signal so that the op amp can provide maximum dynamic range. 4-1

54 Single Supply versus Dual Supply R R G F V +V IN _ R F – (V = + V V ) REF OUT IN V REF R + G – V Figure 4 – 2. Split-Supply Op Amp Circuit With Reference Voltage Input An input bias voltage is used to eliminate the reference voltage when it must not appear , is in both input circuits, hence in the output voltage (see Figure 4 – 3). The voltage, V REF it is named a common-mode voltage. Voltage feedback op amps reject common-mode voltages because their input circuit is constructed with a differential amplifier (chosen be- cause it has natural common-mode voltage rejection capabilities). R R G F V +V IN _ R F = V V – IN OUT V V REF REF R + G V – R G V REF R F Figure 4 – 3. Split-Supply Op Amp Circuit With Common-Mode Voltage When signal sources are referenced to ground, single-supply op amp circuits exhibit a 4 shows a single-supply op amp circuit that large input common-mode voltage. Figure 4 – has its input voltage referenced to ground. The input voltage is not referenced to the mid- point of the supplies like it would be in a split-supply application, rather it is referenced to the lower power supply rail. This circuit does not operate when the input voltage is positive because the output voltage would have to go to a negative voltage, hard to do with a posi- tive supply. It operates marginally with small negative input voltages because most op amps do not function well when the inputs are connected to the supply rails. 4-2

55 Circuit Analysis R R F G +V V IN _ V OUT + – Figure 4 4. Single-Supply Op Amp Circuit ferent reference The constant requirement to account for inputs connected to ground or dif voltages makes it difficult to design single-supply op amp circuits. Unless otherwise speci- fied, all op amp circuits discussed in this chapter are single-supply circuits. The single- supply may be wired with the negative or positive lead connected to ground, but as long as the supply polarity is correct, the wiring does not affect circuit operation. Use of a single-supply limits the polarity of the output voltage. When the supply voltage 10. This limitation precludes ≤ = 10 V, the output voltage is limited to the range 0 ≤ V V CC out negative output voltages when the circuit has a positive supply voltage, but it does not preclude negative input voltages when the circuit has a positive supply voltage. As long as the voltage on the op amp input leads does not become negative, the circuit can handle negative input voltages. Beware of working with negative (positive) input voltages when the op amp is powered from a positive (negative) supply because op amp inputs are highly susceptible to reverse voltage breakdown. Also, insure that all possible start-up conditions do not reverse bias the op amp inputs when the input and supply voltage are opposite polarity. 4.2 Circuit Analysis The complexities of single-supply op amp design are illustrated with the following exam- ple. Notice that the biasing requirement complicates the analysis by presenting several conditions that are not realizable. It is best to wade through this material to gain an under- the problem, especially since a cookbook solution is given later in this chapter. standing of The previous chapter assumed that the op amps were ideal, and this chapter starts to deal amp deficiencies. The input and output voltage swing of many op amps are limited with op as shown in Figure 4 – 7, but if one designs with the selected rail-to-rail op amps, the input/ 5 is ana- output swing problems are minimized. The inverting circuit shown in Figure 4 – lyzed first. 4-3 Single-Supply Op Amp Design Techniques

56 Circuit Analysis V REF R R F G +V + _ V IN R L R G V OUT R F – 5. Inverting Op Amp Figure 4 – Equation 4 1 is written with the aid of superposition, and simplified algebraically, to ac- quire Equation 4 – 2. ) R R R R F G F F (4 – 1) Ǔǒ ǒ Ǔ V V V + * REF IN OUT ) R R R R F G G G R F ǒ Ǔ V + V * V 2) (4 – IN REF OUT R G , is a large value, it does not enter into the circuit calcula- As long as the load resistor, R L tions, but it can introduce some second order effects such as limiting the output voltage 4 – 3 is obtained by setting V equal to V , and there is no output volt- swings. Equation REF IN age from the circuit regardless of the input voltage. The author unintentionally designed a few of these circuits before he created an orderly method of op amp circuit design. Actu- ally, a real circuit has a small output voltage equal to the lower transistor saturation volt- age, which is about 150 mV for a TLC07X. R R F F Ǔ ǒ Ǔ ǒ V 0 * V V + + V * V + (4 3) – REF IN IN IN OUT R R G G When V = 0, V = -V (R /R ), there are two possible solutions to Equation 4 – 2. F G IN OUT REF should be negative voltage. The circuit can is any positive voltage, V First, when V IN OUT not achieve a negative voltage with a positive supply, so the output saturates at the lower is any negative voltage, the output spans the normal power supply rail. Second, when V IN – range according to Equation 4 5. (4 – 4) 0 V w 0, V + IN OUT R F Ť Ť + V v 0, V V 5) (4 – IN IN OUT R G When V 6, – equals the supply voltage, V 6. In Equation 4 , we obtain Equation 4 – CC REF ; that is impossible, so the output satu- is negative, V should exceed V when V IN OUT CC is positive, the circuit acts as an inverting amplifier. rates. When V IN 4-4

57 Circuit Analysis R F ǒ Ǔ (4 – 6) + V V V – IN CC OUT R G = 100 k The transfer curve for the circuit shown in Figure 4 – 6 (V , = 5 V, R Ω = R F CC G 7. – = 10 k Ω ) is shown in Figure 4 R L R R G F V CC V IN _ V OUT + V CC R G R L R F 6. Inverting Op Amp With V Bias – Figure 4 CC 5.5 5.0 4.5 TL072 4.0 V – 3.5 3.0 2.5 Input Voltage 2.0 – IN 1.5 V TLV2472 LM358 1.0 TLC272 0.5 TL072 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 3.5 4.5 1.0 0.0 5.0 0.5 4.0 1.5 3.0 2.5 2.0 Output Voltage V – V – OUT 7. Transfer Curve for Inverting Op Amp With V Bias Figure 4 – CC 4-5 Single-Supply Op Amp Design Techniques

58 Circuit Analysis Four op amps were tested in the circuit configuration shown in Figure 4 6. Three of the – old generation op amps, LM358, TL07X, and TLC272 had output voltage spans of 2.3 V to 3.75 V. This performance does not justify the ideal op amp assumption that was made in the previous chapter unless the output voltage swing is severely limited. Limited output or input voltage swing is one of the worst deficiencies a single-supply op amp can have because the limited voltage swing limits the circuit ’ s dynamic range. Also, limited voltage swing frequently results in distortion of large signals. The fourth op amp tested was the newer TLV247X, which was designed for rail-to-rail operation in single-supply circuits. The TLV247X plotted a perfect curve (results limited by the instrumentation), and it amazed the author with a textbook performance that justifies the use of ideal assump- tions. Some of the older op amps must limit their transfer equation as shown in Equation 4 – 7. R F ǒ Ǔ V w + for V V V * V w V 7) (4 – IN OUT OUT OH CC OL R G The noninverting op amp circuit is shown in Figure 4 – 8. Equation 4 – 8 is written with the aid of superposition, and simplified algebraically, to acquire Equation 4 9. – R ) R R R F G F F Ǔǒ Ǔ ǒ V V V * + 8) (4 – REF IN OUT ) R R R R F G G G R F ǒ Ǔ – (4 9) + V V V – REF IN OUT R G R F , there are two possible circuit solutions. First, when V + V When V = 0, V REF IN IN OUT R G is a negative voltage, V must be a negative voltage. The circuit can not achieve a neg- OUT ative output voltage with a positive supply, so the output saturates at the lower power sup- is a positive voltage, the output spans the normal range as ply rail. Second, when V IN 11. – shown by Equation 4 (4 10) – V v 0, V + 0 IN OUT – 11) (4 V + w 0, V V IN IN OUT = 100 k , 8 with V Ω = 5 V, R – = R The noninverting op amp circuit is shown in Figure 4 F CC G 9; a TLV247X . The transfer curve for this circuit is shown in Figure 4 Ω = 10 k – and R L serves as the op amp. 4-6

59 Circuit Analysis R R F G V CC V REF _ V OUT + V IN R G R F Figure 4 – 8. Noninverting Op Amp 5 4 V – TLV2472 3 2 Input Voltage – IN V 1 0 012345 V Output Voltage – V – OUT Figure 4 – 9. Transfer Curve for Noninverting Op Amp There are many possible variations of inverting and noninverting circuits. At this point many designers analyze these variations hoping to stumble upon the one that solves the circuit problem. Rather than analyze each circuit, it is better to learn how to employ simul- taneous equations to render specified data into equation form. When the form of the de- equation is known, a circuit that fits the equation is chosen to solve the problem. The sired resulting equation must be a straight line, thus there are only four possible solutions. 4-7 Single-Supply Op Amp Design Techniques

60 Simultaneous Equations 4.3 Simultaneous Equations Taking an orderly path to developing a circuit that works the first time starts here; follow these steps until the equation of the op amp is determined. Use the specifications given for the circuit coupled with simultaneous equations to determine what form the op amp equation must have. Go to the section that illustrates that equation form (called a case), solve the equation to determine the resistor values, and you have a working solution. A linear op amp transfer function is limited to the equation of a straight line (Equation 4 12). – (4 – 12) y +" mx " b The equation of a straight line has four possible solutions depending upon the sign of m, the slope, and b, the intercept; thus simultaneous equations yield solutions in four forms. Four circuits must be developed; one for each form of the equation of a straight line. The four equations, cases, or forms of a straight line are given in Equations 4 – 13 through 4 – 16, where electronic terminology has been substituted for math terminology. mV b ) +) V 13) – (4 IN OUT +) mV * b V – 14) (4 IN OUT V ) mV +* b – (4 15) IN OUT (4 – 16) V b * +* mV IN OUT , simultaneous equations are solved to Given a set of two data points for V and V IN OUT determine m and b for the equation that satisfies the given data. The sign of m and b deter- mines the type of circuit required to implement the solution. The given data is derived from the specifications; i. e., a sensor output signal ranging from 0.1 V to 0.2 V must be inter- faced into an analog-to-digital converter that has an input voltage range of 1 V to 4 V. = 4 V @ V = 0.1 V, V = 1 V @ V = 0.2 V) are inserted into data points (V These OUT OUT IN IN Equation 4 – 13, as shown in Equations 4 – 17 and 4 – 18, to obtain m and b for the specifica- tions. (4 – 17) 1 + m(0.1) ) b 18) (4 – 4 + ) b m(0.2) – Multiply Equation 4 – 18. 17 by 2 and subtract it from Equation 4 (4 – 19) 2 + m(0.2) ) 2b (4 20) – b +* 2 After algebraic manipulation of Equation 4 – 17, substitute Equation 4 – 20 into Equation – 21. 4 – 17 to obtain Equation 4 4-8

61 Simultaneous Equations 1 2 ) (4 21) – + + 30 m 0.1 – 13 yielding Equation 4 – 22. Now m and b are substituted back into Equation 4 (4 22) – V + 30V * 2 IN OUT Notice, although Equation 4 – 13 was the starting point, the form of Equation 4 – 22 is identi- cal to the format of Equation 4 14. The specifications or given data determine the sign – of m and b, and starting with Equation 4 – 13, the final equation form is discovered after m and b are calculated. The next step required to complete the problem solution is to de- velop a circuit that has an m = 30 and b = – 2. Circuits were developed for Equations 4 – 13 through 4 – 16, and they are given under the headings Case 1 through Case 4 respectively. There are different circuits that will yield the same equations, but these circuits were se- lected because they do not require negative references. 4.3.1 Case 1: V = +mV +b OUT IN 10. The – The circuit configuration that yields a solution for Case 1 is shown in Figure 4 figure includes two 0.01- F capacitors. These capacitors are called decoupling capaci- μ tors, and they are included to reduce noise and provide increased noise immunity. Some- F capacitors serve this purpose, sometimes more extensive filtering is μ times two 0.01- needed, and sometimes one capacitor serves this purpose. Special attention must be is used as a reference because when V paid to the regulation and noise content of V CC CC content of V will be multiplied by the circuit gain some portion of the noise CC . V CC 0.01 F μ + V R OUT 1 _ R 2 V IN R F μ 0.01 F V REF R R L G + b Figure 4 – 10. Schematic for Case1: V = +mV OUT IN The circuit equation is written using the voltage divider rule and superposition. 4-9 Single-Supply Op Amp Design Techniques

62 Simultaneous Equations R R ) R ) R R R F F G G 2 1 23) (4 – ǒ ǒ Ǔ Ǔ Ǔǒ Ǔǒ V ) + V V REF IN OUT ) R R ) R R R R 2 2 1 1 G G The equation of a straight line (case 1) is repeated in Equation 4 – 24 below so compari- sons can be made between it and Equation 4 – 23. (4 24) – V mV ) b + IN OUT – 25 and 4 – 26. Equating coefficients yields Equations 4 R ) R R F G 2 ǒ Ǔǒ Ǔ + m 25) – (4 R R R ) 1 2 G ) R R R F G 1 – 26) (4 ǒ Ǔǒ Ǔ b V + REF R ) R R 2 1 G Example; the circuit specifications are V = 0.01 V, V = 4.5 V at V = = 1 V at V IN OUT OUT IN = 10 k, five percent resistor tolerances, and V = 5 V. No reference voltage is 1 V, R L CC is used for the reference input, and V = 5 V. A reference voltage available, thus V REF CC source is left out of the design as a space and cost savings measure, and it sacrifices noise performance, accuracy, and stability performance. Cost is an important specifica- supply must be specified well enough to do the job. Each step in the sub- tion, but the V CC sequent design procedure is included in this analysis to ease learning and increase bore- dom. Many steps are skipped when subsequent cases are analyzed. The data is substituted into simultaneous equations. b 1 + m(0.01) ) 27) (4 – – 28) (4 4.5 m(1.0) ) b + Equation 4 – 27 is multiplied by 100 (Equation 4 – 29) and Equation 4 – 28 is subtracted – 30. from Equation 4 – 29 to obtain Equation 4 (4 – 29) m(1.0) ) 100b 100 + 95.5 – 30) (4 b + + 0.9646 99 The slope of the transfer function, m, is obtained by substituting b into Equation 4 – 27. b – 1 1 – 0.9646 31) – (4 + m + 3.535 + 0.01 0.01 Now that b and m are calculated, the resistor values can be calculated. Equations 4 – 25 )/R + R , and then they are set equal in Equation – and 4 26 are solved for the quantity (R F G G 33. – 4 – 32 thus yielding Equation 4 4-10

63 Simultaneous Equations R ) R R R R ) R ) F b G 1 1 2 2 – (4 32) Ǔ ǒ Ǔ ǒ m + + R R R V 2 1 CC G 3.535 – 33) (4 + R 18.316R + R 1 2 1 0.9646 5 Five percent tolerance resistors are specified for this design, so we choose R = 10 k Ω , 1 Ω is = 183.16 k Ω . The closest 5% resistor value to 183.16 k and that sets the value of R 2 . Being forced to yield to reality by Ω = 10 k Ω and R = 180 k ; therefore, select R 180 k Ω 2 1 choosing standard resistor values means that there is an error in the circuit transfer func- tion because m and b are not exactly the same as calculated. The real world constantly forces compromises into circuit design, but the good circuit designer accepts the chal- lenge and throws money or brains at the challenge. Resistor values closer to the calcu- lated values could be selected by using 1% or 0.5% resistors, but that selection increases cost and violates the design specification. The cost increase is hard to justify except in precision circuits. Using ten-cent resistors with a ten-cent op amp usually is false econo- my. and R . 32 is used to calculate R The left half of Equation 4 – F G R R ) R R ) F 180 10 ) G 1 2 Ǔ ǒ ǒ Ǔ + 3.73 3.535 + + m 34) – (4 180 R R 2 G (4 – 35) R 2.73R + F G The resulting circuit equation is given below. – 36) (4 0.97 V 3.5V ) + IN OUT The gain setting resistor, R , the closest 5% standard , is selected as 10 k Ω , and 27 k Ω G . Again, there is a slight error involved with value is selected for the feedback resistor, R F standard resistor values. This circuit must have an output voltage swing from 1 V to 4.5 V. The older op amps can not be used in this circuit because they lack dynamic range, so the TLV247X family of op amps is selected. The data shown in Figure 4 – 7 confirms the op amp selection because there is little error. The circuit with the selected component val- ues is shown in Figure 4 – 11. The circuit was built with the specified components, and the 12. – transfer curve is shown in Figure 4 4-11 Single-Supply Op Amp Design Techniques

64 Simultaneous Equations +5V F μ 0.01 + V = 1.0 to 4.5 V OUT R 1 _ R Ω 10 k 2 Ω 180 k = 0.01 V to 1 V V IN R F 0.01 F μ R 27 k Ω G R L +5V 10 k Ω Ω 10 k – 11.Case 1 Example Circuit Figure 4 1.0 0.9 0.8 0.7 V – TLV247x 0.6 0.5 0.4 Input Voltage – IN 0.3 V 0.2 0.1 0.0 012345 Output Voltage V – – V OUT Figure 4 – 12. Case 1 Example Circuit Measured Transfer Curve The transfer curve shown is a straight line, and that means that the circuit is linear. The intercept is about 0.98 V rather than 1 V as specified, and this is excellent perfor- V OUT mance considering that the components were selected randomly from bins of resistors. Different sets of components would have slightly different slopes because of the resistor The TLV247X has input bias currents and input offset voltages, but the ef tolerances. fect of these errors is hard to measure on the scale of the output voltage. The output voltage 4-12

65 Simultaneous Equations measured 4.53 V when the input voltage was 1 V. Considering the low and high input volt- age errors, it is safe to conclude that the resistor tolerances have skewed the gain slightly, but this is still excellent performance for 5% components. Often lab data similar to that shown here is more accurate than the 5% resistor tolerance, but do not fall into the trap of expecting this performance, because you will be disappointed if you do. Ω range arbitrarily. The resistors were selected in the k- fset specifications The gain and of determine the resistor ratios, but supply current, frequency response, and op amp drive capability determine their absolute values. The resistor value selection in this design is high because modern op amps do not have input current offset problems, and they yield reasonable frequency response. If higher frequency response is demanded, the resistor values must decrease, and resistor value decreases reduce input current errors, while supply current increases. When the resistor values get low enough, it becomes hard for another circuit, or possibly the op amp, to drive the resistors. 4.3.2 Case 2: V b = +mV – IN OUT The circuit shown in Figure 4 – 13 yields a solution for Case 2. The circuit equation is ob- tained by taking the Thevenin equivalent circuit looking into the junction of R and R . Af- 2 1 circuit is replaced with the Thevenin equivalent circuit, the gain is calculated , R ter the R 2 1 with the ideal gain equation (Equation 4 – 37). R R F 1 V R CC G μ 0.01 F _ R 2 V REF V OUT + R V L IN Figure 4 – 13. Schematic for Case 2: V = +mV OUT – b IN R R ø ) R ) R R R 1 2 F G 2 F 37) – (4 Ǔ ǒ Ǔ ǒ Ǔ ǒ * V V V + REF IN OUT R R ) R ø R R ) R R ø ) R 2 1 2 1 1 2 G G 14 enables the extraction of m and b. – Comparing terms in Equations 4 – 37 and 4 4-13 Single-Supply Op Amp Design Techniques

66 Simultaneous Equations R R ø R R ) ) 1 2 F G + m – 38) (4 R R ) R ø 1 2 G R R F 2 – (4 39) | | Ǔ ǒ Ǔ ǒ + V b REF R ) R R ø R R ) 1 2 1 2 G The specifications for an example design are: V = 4.5 = 1.5 V @ V = 0.2 V, V IN OUT OUT Ω = 10 k = 0.5 V, V = 5 V, R , and 5% resistor tolerances. The simulta- = V V @ V CC REF L IN 41), are written below. neous equations, (Equations 4 – 40 and 4 – (4 40) – + 0.2m ) b 1.5 – 41) (4 + 0.5m 4.5 ) b From these equations we find that b = -0.5 and m = 10. Making the assumption that ||R <

67 Simultaneous Equations +5V R 1 R F 75 k Ω Ω 180 k +5V R G F 0.01 μ R 20 k Ω 2 820 Ω _ 0.01 μ F V OUT + R L 10 k Ω V IN 14. Case 2 Example Circuit Figure 4 – 0.5 0.4 V – TLV247x 0.3 Input Voltage – IN V 0.2 0.1 012345 – V Output Voltage – V OUT Figure 4 – 15. Case 2 Example Circuit Measured Transfer Curve The TLV247X was used to build the test circuit because of its wide dynamic range. The transfer curve plots very close to the theoretical curve; the direct result of using a high per- formance op amp. 4-15 Single-Supply Op Amp Design Techniques

68 Simultaneous Equations 4.3.3 Case 3: V = – mV + b IN OUT – 16 yields the transfer function desired for Case 3. The circuit shown in Figure 4 R R F G V CC V IN _ V OUT + R 2 R L R V 1 REF Figure 4 – 16. Schematic for Case 3: V + b = – mV IN OUT The circuit equation is obtained with superposition. R R ) R R F G 1 F 46) (4 – ǒ Ǔ Ǔ ǒ Ǔǒ V V + – V ) IN REF OUT R R R ) R 1 2 G G – 45 and 4 – 15 enables the extraction of m and b. Comparing terms between Equations 4 R F – 47) (4 | | m + R G R ) R R F G 1 Ǔ ǒ Ǔǒ b V + 48) (4 – REF R R ) R 1 2 G = 1 V @ V = -0.1 V, The design specifications for an example circuit are: V OUT IN Ω = 6 V @ V = -1 V, V = V = 10 V, R = 100 , and 5% resistor tolerances. The V CC IN OUT REF L supply voltage available for this circuit is 10 V, and this exceeds the maximum allowable supply voltage for the TLV247X. Also, this circuit must drive a back-terminated cable that Ω looks like two 50- resistors connected in series, thus the op amp must be able to drive 6/100 = 60 mA. The stringent op amp selection criteria limits the choice to relatively new op amps if ideal op amp equations are going to be used. The TLC07X has excellent single- supply input performance coupled with high output current drive capability, so it is se- lected – 49 and 4 – 50), are written for this circuit. The simultaneous equations (Equations 4 below. (4 – 49) 1 ( – 0.1)m ) b + (4 – 50) 6 + ( – 1)m ) b – 5.6. From these equations we find that b = 0.444 and m = 4-16

69 Simultaneous Equations R F | | m + 5.56 + – (4 51) R G R + 5.56R (4 – 52) F G Ω = 10 k Let R , and then R is = 56.6 k Ω , which is not a standard 5% value, hence R F G F selected as 56 k Ω . R ) R R R F 56 10 G ) 1 1 (4 – 53) Ǔ ǒ Ǔǒ Ǔ ǒ ǒ Ǔ + 10 V + b CC 10 R ) R R R ) R 1 1 2 2 G 66 0.4444 – – 54) (4 R + 147.64R R + 1 2 1 0.4444 The final equation for the example is given below (4 – 55) V 0.444 +* 5.56V ) IN OUT Select R is not a standard 5% resistor value, = 2 k Ω and R Ω = 295.28 k Ω . Since 295.28 k 2 1 R is selected as 300 k Ω . The difference between the selected and calculated value of 1 – 17, and the has a nearly insignificant effect on b. The final circuit is shown in Figure 4 R 1 measured transfer curve for this circuit is shown in Figure 4 18. – R F Ω 56 k R G = 10 V V CC Ω 10 k μ F 0.01 V IN D1 _ V OUT + R L 0.01 μ F Ω 10 k = 10 V V CC R 2 R 1 2 k Ω 300 k Ω Figure 4 – 17. Case 3 Example Circuit 4-17 Single-Supply Op Amp Design Techniques

70 Simultaneous Equations 0.0 – 0.1 – – 0.2 0.3 – V – 0.4 – – 0.5 – 0.6 Input Voltage – IN 0.7 – V 0.8 – 0.9 – 1.0 – 01234567 Output Voltage V – – V OUT Figure 4 18. Case 3 Example Circuit Measured Transfer Curve – As long as the circuit works normally, there are no problems handling the negative voltage input to the circuit, because the inverting lead of the TLC07X is at a positive voltage. The positive op amp input lead is at a voltage of approximately 65 mV, and normal op amp operation keeps the inverting op amp input lead at the same voltage because of the as- is powered down while there is a nega- sumption that the error voltage is zero. When V CC tive voltage on the input circuit, most of the negative voltage appears on the inverting op amp input lead. , with its cathode on the inverting The most prudent solution is to connect the diode, D 1 op amp input lead and its anode at ground. If a negative voltage gets on the inverting op amp input lead, it is clamped to ground by the diode. Select the diode type as germanium or Schottky so the voltage drop across the diode is about 200 mV; this small voltage does can be split into two resistors not harm most op amp inputs. As a further precaution, R G with the diode inserted at the junction of the two resistors. This places a current limiting resistor between the diode and the inverting op amp input lead. 4-18

71 Simultaneous Equations b – mV 4.3.4 Case 4: V – = IN OUT The circuit shown in Figure 4 – 19 yields a solution for Case 4. The circuit equation is ob- tained by using superposition to calculate the response to each input. The individual re- 56. and V – are added to obtain Equation 4 sponses to V REF IN R F V R CC G1 F μ 0.01 _ V V R IN OUT G2 + R L V REF – Figure 4 – 19. Schematic for Case 4: V b = – mV IN OUT R R F F + V V V * – (4 56) – IN REF OUT R R G2 G1 56 and 4 – 16 enables the extraction of m and b. Comparing terms in Equations 4 – R F | | m + 57) – (4 R G1 R F | | – 58) (4 b + V REF R G2 The design specifications for an example circuit are: V = 1 V @ V = – 0.1 V, V OUT OUT IN = 5 V @ V , and 5% resistor tolerances. The = – 0.3 V, V Ω = V = 10 k = 5 V, R CC L IN REF – – 60, are written below. simultaneous Equations 4 59 and 4 1 b ( – 0.1)m ) + – 59) (4 (4 – 60) b ( 5 + ) – 0.3)m From 1 and m = – 20. Setting the magnitude of m equal – these equations we find that b = 57 yields Equation 4 to Equation 4 – 61. – R F | | m 20 + + 61) (4 – R G1 (4 – 62) R + 20R F G1 . Ω Let R = 20 k = 1 k Ω , and then R F G1 4-19 Single-Supply Op Amp Design Techniques

72 Simultaneous Equations R R F F (4 63) – | | Ǔ ǒ Ǔ ǒ 1 5 + + V + b CC R R G1 G2 R 20 F 100 k + R + + W 64) (4 – G2 0.2 0.2 The final equation for this example is given in Equation 4 – 63. – (4 65) 1 V * + – 20V IN OUT The final circuit is shown in Figure 4 – 20 and the measured transfer curve for this circuit is shown in Figure 4 – 21. R F 20 k Ω +5V R G1 F 0.01 μ Ω 1 k _ +5V V V R R IN OUT G2A G2B + Ω 51 k 51 k Ω R L 10 k Ω 0.01 μ F D1 – 20. Case 4 Example Circuit Figure 4 4-20

73 Simultaneous Equations – 0.10 0.15 – V – 0.20 – – 0.25 Input Voltage – IN V – 0.30 – 0.35 0123456 – Output Voltage – V V OUT 21. Case 4 Example Circuit Measured Transfer Curve Figure 4 – The TLV247X was used to build the test circuit because of its wide dynamic range. The transfer curve plots very close to the theoretical curve, and this results from using a high performance op amp. As long as the circuit works normally there are no problems handling the negative voltage input to the circuit because the inverting lead of the TLV247X is at a positive voltage. The positive op amp input lead is grounded, and normal op amp operation keeps the inverting op amp input lead at ground because of the assumption that the error voltage is zero. is powered down while there is a negative voltage on the inverting op amp input When V CC lead. , with its cathode on the inverting The most prudent solution is to connect the diode, D 1 op amp input lead and its anode at ground. If a negative voltage gets on the inverting op amp input lead it is clamped to ground by the diode. Select the diode type as germanium or Schottky so the voltage drop across the diode is about 200 mV; this small voltage does is split into two resistors (R Ω = 51 k ) with = R not harm most op amp inputs. R G2A G2B G2 a capacitor inserted at the junction of the two resistors. This places a power supply filter . in series with V CC 4-21 Single-Supply Op Amp Design Techniques

74 Summary 4.4 Summary Single-supply op amp design is more complicated than split-supply op amp design, but with a logical design approach excellent results are achieved. Single-supply design used to be considered technically limiting because older op amps had limited capability. The new op amps, such as the TLC247X, TLC07X, and TLC08X have excellent single-supply parameters; thus when used in the correct applications these op amps yield rail-to-rail performance equal to their split-supply counterparts. Single-supply op amp design usually involves some form of biasing, and this requires more thought, so single-supply op amp design needs discipline and a procedure. The rec- ommended design procedure for single-supply op amp design is: D Substitute the specification data into simultaneous equations to obtain m and b (the slope and intercept of a straight line). D Let m and b determine the form of the circuit. Choose the circuit configuration that fits the form. D D Using the circuit equations for the circuit configuration selected, calculate the resistor values. D Build the circuit, take data, and verify performance. D Test the circuit for nonstandard operating conditions (circuit power off while interface power is on, over/under range inputs, etc.). D Add protection components as required. D Retest. When this procedure is followed, good results follow. As single-supply circuit designers expand their horizon, new challenges require new solutions. Remember, the only equa- tion a linear op amp can produce is the equation of a straight line. That equation only has four forms. The new challenges may consist of multiple inputs, common-mode voltage rejection, or something different, but this method can be expanded to meet these chal- lenges. 4-22

75 Chapter 5 Feedback and Stability Theory Ron Mancini 5.1 Why Study Feedback Theory? The gain of all op amps decreases as frequency increases, and the decreasing gain re- sults in decreasing accuracy as the ideal op amp assumption (a ⇒∝ ) breaks down. In most real op amps the open loop gain starts to decrease before 10 Hz, so an understand- ing of feedback is required to predict the closed loop performance of the op amp. The real world application of op amps is feedback controlled, and depends on op amp open loop gain at a given frequency. A designer must know theory to be able to predict the circuit response regardless of frequency or open loop gain. Analysis tools have something in common with medicine because they both can be dis- tasteful but necessary. Medicine often tastes bad or has undesirable side effects, and analysis tools involve lots of hard learning work before they can be applied to yield results. Medicine assists the body in fighting an illness; analysis tools assist the brain in learning/ designing feedback circuits. The analysis tools given here are a synopsis of salient points; thus they are detailed enough to get you where you are going without any extras. The references, along with their counterparts, must be consulted when making an in-depth study of the thousands of field. Aspirin, home remedies, and good health practice handle the majority of health prob- lems, and these analysis tools solve the majority of circuit problems. Ideal op amp circuits can be designed without knowledge of feedback analysis tools, but these circuits are limited to low frequencies. Also, an understanding of feedback analysis tools is required to understand AC effects like ringing and oscillations. 5.2 Block Diagram Math and Manipulations Electronic systems and circuits are often represented by block diagrams, and block dia- grams have a unique algebra and set of transformations[1]. Block diagrams are used be- cause they are a shorthand pictorial representation of the cause-and-effect relationship between the input and output in a real system. They are a convenient method for charac- terizing the functional relationships between components. It is not necessary to under- stand the functional details of a block to manipulate a block diagram. 5-1

76 Block Diagram Math and Manipulations The input impedance of each block is assumed to be infinite to preclude loading. Also, the output impedance of each block is assumed to be zero to enable high fan-out. The systems designer sets the actual impedance levels, but the fan-out assumption is valid because the block designers adhere to the system designer ’ s specifications. All blocks multiply the input times the block quantity (see Figure 5 – 1) unless otherwise specified within the block. The quantity within the block can be a constant as shown in Figure function involving Laplace transforms. The blocks can 5 – 1(c), or it can be a complex math perform time-based operations such as differentiation and integration. OUTPUT V INPUT O (a) Input/Output Impedance Block A B Description (b) Signal Flow Arrows AK B B = AK (c) Block Multiplication dVI d V V = I O dt dt (d) Blocks Perform Functions as Indicated Figure 5 – 1. Definition of Blocks Adding and subtracting are done in special blocks called summing points. Figure 5 – 2 gives several examples of summing points. Summing points can have unlimited inputs, can add or subtract, and can have mixed signs yielding addition and subtraction within 3 defines the terms in a typical control system, and Fig- – a single summing point. Figure 5 – 4 defines the terms in a typical electronic feedback system. Multiloop feedback sys- ure 5 5) are intimidating, but they can be reduced to a single loop feedback sys- tems (Figure 5 – . An easier meth- /V tem, as shown in the figure, by writing equations and solving for V IN OUT od for reducing multiloop feedback systems to single loop feedback systems is to follow 6. – the rules and use the transforms given in Figure 5 5-2

77 Block Diagram Math and Manipulations C – + + + A A+B A A+B – C – B AA + + – B B B (c) Multiple Input Summary Points (b) Subtractive Summary Point (a) Additive Summary Point Figure 5 – 2. Summary Points Disturbance U Actuating Manipulated Control Signal Variable Plant Controlled Reference Elements Σ Output G Input 1 M C ± B RE =R + G 1 ± B Forward Path Primary Feedback Signal Feedback Elements H Feedback Path Figure 5 – 3. Definition of Control System Terms ERROR A V V Σ IN OUT E β Figure 5 – 4. Definition of an Electronic Feedback Circuit 5-3 Feedback and Stability Theory

78 Block Diagram Math and Manipulations G 3 + + + + R C G G G 2 1 4 + – H 1 H 2 + R + G ) (G G G C 1 2 3 4 1 – G H G 1 1 4 – H 2 Figure 5 – 5. Multiloop Feedback System Block diagram reduction rules: D Combine cascade blocks. Combine parallel blocks. D D Eliminate interior feedback loops. D Shift summing points to the left. D Shift takeoff points to the right. Repeat until canonical form is obtained. D Figure 5 6 gives the block diagram transforms. The idea is to reduce the diagram to its – canonical form because the canonical feedback loop is the simplest form of a feedback loop, and its analysis is well documented. All feedback systems can be reduced to the canonical form, so all feedback systems can be analyzed with the same math. A canonical loop exists for each input to a feedback system; although the stability dynamics are inde- pendent of the input, the output results are input dependent. The response of each input of a multiple input feedback system can be analyzed separately and added through super- position. 5-4

79 Block Diagram Math and Manipulations Before Transformation After Transformation Transformation Combine Cascade K K K K AB AB 1 2 1 2 Blocks + A K B Σ K K AB 1 ± 1 2 Combine Parallel ± Blocks K 2 + K 1 K A B A Σ B 1 1 ± K K Eliminate a 1 2 ± Feedback Loop K 2 + + K K C C Σ Σ A A Move Summer ± ± In Front of a Block B 1/K B + + K K C C Σ Σ A A Move Summer ± ± Behind a Block K B B K K B B A A Move Pickoff In Front of a Block B B K K K B B A A Move Pickoff Behind a Block I/K A A – 6. Block Diagram Transforms Figure 5 5-5 Feedback and Stability Theory

80 Feedback Equation and Stability 5.3 Feedback Equation and Stability 7 shows the canonical form of a feedback loop with control system and electron- – Figure 5 ic system terms. The terms make no difference except that they have meaning to the sys- tem engineers, but the math does have meaning, and it is identical for both types of terms. The electronic terms and negative feedback sign are used in this analysis, because sub- sequent chapters deal with electronic applications. The output equation is written in Equa- 1. tion 5 – + + + E E E G A A V R V C Σ Σ Σ OUT IN – – – β β H X R G C V A V IN OUT E = = = E = 1 1 R GH + GH + 1 + A 1 + A β β V IN (c) Feedback Loop is Broken to Calculate the Loop Gain (b) Electronics Terminology (a) Control System Terminology Figure 5 – 7. Comparison of Control and Electronic Canonical Feedback Systems V + EA (5 – 1) OUT 2. – The error equation is written in Equation 5 + V * b V E (5 – 2) IN OUT – – 2 yields Equation 5 Combining Equations 5 3. 1 and 5 – V OUT (5 – 3) V + V * b IN OUT A Collecting terms yields Equation 5 – 4. 1 ǒ Ǔ – (5 4) V + ) b V IN OUT A Rearranging terms yields the classic form of the feedback Equation 5 – 5. V OUT A – (5 5) + V 1 ) A b IN When the quantity A β in Equation 5 – 5 becomes very large with respect to one, the one can be neglected, and Equation 5 5 reduces to Equation 5 – 6, which is the ideal feedback – equation. Under the conditions that A β >>1, the system gain is determined by the feed- back factor β . Stable passive circuit components are used to implement the feedback fac- β tor, thus in the ideal situation, the closed loop gain is predictable and stable because is predictable and stable. 5-6

81 Bode Analysis of Feedback Circuits V OUT 1 6) (5 – + b V IN β is so important that it has been given a special name: loop gain. In Figure The quantity A 5 – 7, when the voltage inputs are grounded (current inputs are opened) and the loop is broken, the calculated gain is the loop gain, A β . Now, keep in mind that we are using com- plex numbers, which have magnitude and direction. When the loop gain approaches mi- ∝ . – 180 ° , Equation 5 – 5 approaches 1/0 nus one, or to express it mathematically 1 ⇒ ∠ The circuit output heads for infinity as fast as it can using the equation of a straight line. If the output were not energy limited, the circuit would explode the world, but happily, it is energy limited, so somewhere it comes up against the limit. Active devices in electronic circuits exhibit nonlinear phenomena when their output ap- proaches a power supply rail, and the nonlinearity reduces the gain to the point where the . Now the circuit can do two things: first it can become – loop gain no longer equals 1 ∠ ° 180 stable at the power supply limit, or second, it can reverse direction (because stored charge keeps the output voltage changing) and head for the negative power supply rail. The first state where the circuit becomes stable at a power supply limit is named lockup; the circuit will remain in the locked up state until power is removed and reapplied. The state where the circuit bounces between power supply limits is named oscillatory. second Remember, the loop gain, A β , is the sole factor determining stability of the circuit or sys- tem. Inputs are grounded or disconnected, so they have no bearing on stability. Equations 5 1 and 5 – 2 are combined and rearranged to yield Equation 5 – 7, which is the – system or circuit error equation. V IN – (5 7) + E 1 ) A b First, notice that the error is proportional to the input signal. This is the expected result because a bigger input signal results in a bigger output signal, and bigger output signals require more drive voltage. As the loop gain increases, the error decreases, thus large loop gains are attractive for minimizing errors. 5.4 Bode Analysis of Feedback Circuits H. W. Bode developed a quick, accurate, and easy method of analyzing feedback amplifi- ers, and he published a book about his techniques in 1945.[2] Operational amplifiers had not been developed when Bode published his book, but they fall under the general classi- fication of feedback amplifiers, so they are easily analyzed with Bode techniques. The mathematical manipulations required to analyze a feedback circuit are complicated be- cause they involve multiplication and division. Bode developed the Bode plot, which sim- plifies the analysis through the use of graphical techniques. 5-7 Feedback and Stability Theory

82 Bode Analysis of Feedback Circuits The Bode equations are log equations that take the form 20LOG(F(t)) = 20LOG(|F(t)|) + angle. Terms that are normally multiplied and divided can now be added and sub- phase tracted because they are log equations. The addition and subtraction is done graphically, thus easing the calculations and giving the designer a pictorial representation of circuit – – 8. performance. Equation 5 8 is written for the low pass filter shown in Figure 5 R V V O I C – Figure 5 8. Low-Pass Filter 1 V Cs OUT 1 1 + + + 1 t 1 ) s V RCs 1 ) IN R ) 8) – (5 Cs ω , j = √ ( – 1) , and RC = τ Where: s = j 2 2 Ǹ | ) | ( ń + 1 ń 1 V ) V tw The magnitude of this transfer function is . This magnitude, IN OUT | , and it is approximately = 0.1 τ = 1/ /V ω , it equals 0.707 when ≅ 1 when ω = 0.1/ τ |V OUT IN These points are plotted in Figure 5 – 9 using straight line approximations. = 10/ τ . when ω – 20 dB/decade or – 6 dB/octave. The magnitude curve is plotted as The negative slope is ω = 1/ τ . The negative slope begins a horizontal line until it intersects the breakpoint where at the breakpoint because the magnitude starts decreasing at that point. The gain is equal to 1 or 0 dB at very low frequencies, equal to 0.707 or 3 dB at the break frequency, and – it keeps falling with a – 20 dB/decade slope for higher frequencies. The phase shift for the low pass filter or any other transfer function is calculated with the aid of Equation 5 – 9. Real wt 1 * * 1 9) (5 – Ǔ ǒ Ǔ ǒ tangent f +* tangent + 1 Imaginary The phase shift is much harder to approximate because the tangent function is nonlinear. the phase information is only required around the 0 dB intercept point for an ac- Normally tive circuit, so the calculations are minimized. The phase is shown in Figure 5 – 9, and it , and 3 ° is 1, the tangent of 60 ° is √ is approximated by remembering that the tangent of 90 /3. the tangent of 30 ° is √ 3 5-8

83 Bode Analysis of Feedback Circuits = 10 = 0.1 / τω = 1 / τω ω / τ 0 dB ) I – 3 dB 20 dB/Decade – /V O 20 Log (V – 20 dB ° 0 – 45 ° Phase Shift ° 90 – Figure 5 – 9. Bode Plot of Low-Pass Filter Transfer Function A breakpoint occurring in the denominator is called a pole, and it slopes down. Converse- ly, a breakpoint occurring in the numerator is called a zero, and it slopes up. When the function has multiple poles and zeros, each pole or zero is plotted independently, transfer and the individual poles/zeros are added graphically. If multiple poles, zeros, or a pole/ zero combination have the same breakpoint, they are plotted on top of each other. Multiple poles or zeros cause the slope to change by multiples of 20 dB/decade. An example of a transfer function with multiple poles and zeros is a band reject filter (see Figure 5 – 10). The transfer function of the band reject filter is given in Equation 5 – 10. R V V OUT IN CC RR RC = τ Figure 5 10. Band Reject Filter – V ) )( ( ) 1 s s t ) 1 t OUT 10) – (5 + + G V s t t s IN ǒ Ǔ Ǔǒ 1 ) 1 ) 2 0.44 4.56 The pole zero plot for each individual pole and zero is shown in Figure 5 – 11, and the com- – 12. bined pole zero plot is shown in Figure 5 5-9 Feedback and Stability Theory

84 Bode Analysis of Feedback Circuits 40 dB/Decade dB τ = 1/ ω 0 LOG ( ω ) – 6 Amplitude 20 dB/Decade – τ ω = 4.56/ τ ω = 0.44/ 20 dB/Decade – Figure 5 – 11.Individual Pole Zero Plot of Band Reject Filter τ ω = 4.56/ τω = 0.44/ τ = 1/ ω LOG ( ω ) 0 dB – 6 dB Amplitude 25 ° 12 ° 0 Phase Shift ° 5 – Figure 5 – 12. Combined Pole Zero Plot of Band Reject Filter The individual pole zero plots show the dc gain of 1/2 plotting as a straight line from the – 6 dB intercept. The two zeros occur at the same break frequency, thus they add to a 40-dB/decade slope. The two poles are plotted at their breakpoints of τ and ω = 0.44/ = 4.56/ τ . The combined amplitude plot intercepts the amplitude axis at – 6 dB because ω of the dc gain, and then breaks down at the first pole. When the amplitude function gets to the double zero, the first zero cancels out the first pole, and the second zero breaks up. The upward slope continues until the second pole cancels out the second zero, and the amplitude is flat from that point out in frequency. When the separation between all the poles and zeros is great, a decade or more in fre- quency, it is easy to draw the Bode plot. As the poles and zeros get closer together, the plot gets harder to make. The phase is especially hard to plot because of the tangent func- tion, but picking a few salient points and sketching them in first gets a pretty good approxi- mation.[3] The Bode plot enables the designer to get a good idea of pole zero placement, and it is valuable for fast evaluation of possible compensation techniques. When the situa- tion gets critical, accurate calculations must be made and plotted to get an accurate result. 5-10

85 Bode Analysis of Feedback Circuits Consider Equation 5 11. – V OUT A – 11) (5 + V b 1 ) A IN – 11 yields Equation 5 – 12. Taking the log of Equation 5 V OUT 12) (5 – ) ( ǒ Ǔ ) 20Log(A) – 20Log 1 + A b 20Log V IN If A and β do not contain any poles or zeros there will be no break points. Then the Bode – plot of Equation 5 13, and because there are no 12 looks like that shown in Figure 5 – poles to contribute negative phase shift, the circuit cannot oscillate. dB 20 LOG(A) 20 LOG(1 + A β ) V OUT Amplitude Ǔ 20 LOG ǒ V IN ) LOG( ω 0 dB 12) – 13. When No Pole Exists in Equation (5 – Figure 5 All real amplifiers have many poles, but they are normally internally compensated so that they appear to have a single pole. Such an amplifier would have an equation similar to that given in Equation 5 – 13. a + A w (5 – 13) j 1 ) w a – 14. The plot for the single pole amplifier is shown in Figure 5 dB 20 LOG(A) 20 LOG(1 + A β ) x V Amplitude OUT ǒ 20 LOG Ǔ V IN ) LOG( ω 0 dB ω = ω ω a 12 has a Single Pole – Figure 5 – 14. When Equation 5 5-11 Feedback and Stability Theory

86 Loop Gain Plots are the Key to Understanding Stability The amplifier gain, A, intercepts the amplitude axis at 20Log(A), and it breaks down at a . The negative slope continues for all frequencies greater slope of – 20 dB/decade at ω = ω a . The closed loop circuit gain intercepts the amplitude axis at than the breakpoint, ω = ω a does not have any poles or zeros, it is constant until its β ), and because /V 20Log(V IN OUT intersects the amplifier gain at point X. After intersection with the amplifier gain projection curve, the closed loop gain follows the amplifier gain because the amplifier is the control- ling factor. Actually, the closed loop gain starts to roll off earlier, and it is down 3 dB at point X. At point 3 dB, thus accord- X the difference between the closed loop gain and the amplifier gain is – , hence 2 – 12 the term – 20Log(1+A ing to Equation 5 ) = – 3 dB. The magnitude of 3 dB is √ β 2 Ǹ Ǹ ( ) = 1. There is a method β , and elimination of the radicals shows that A 2 + A 1 b ) [4] of relating phase shift and stability to the slope of the closed loop gain curves, but only the Bode method is covered here. An excellent discussion of poles, zeros, and their inter- action is given by M. E Van Valkenberg,[5] and he also includes some excellent prose to liven the discussion. 5.5 Loop Gain Plots are the Key to Understanding Stability Stability is determined by the loop gain, and when A β = – 1 = |1| ∠ – 180 ° instability or os- cillation occurs. If the magnitude of the gain exceeds one, it is usually reduced to one by circuit nonlinearities, so oscillation generally results for situations where the gain magni- tude exceeds one. Consider oscillator design, which depends on nonlinearities to decrease the gain magni- tude; if the engineer designed for a gain magnitude of one at nominal circuit conditions, the gain magnitude would fall below one under worst case circuit conditions causing os- magnitude of one under cillation to cease. Thus, the prudent engineer designs for a gain case conditions knowing that the gain magnitude is much more than one under opti- worst mistic conditions. The prudent engineer depends on circuit nonlinearities to reduce the gain magnitude to the appropriate value, but this same engineer pays a price of poorer distortion performance. Sometimes a design compromise is reached by putting a nonlin- ear component, such as a lamp, in the feedback loop to control the gain without introduc- ing distortion. Some high gain control systems always have a gain magnitude greater than one, but they avoid oscillation by manipulating the phase shift. The amplifier designer who pushes the amplifier for superior frequency performance has to be careful not to let the loop gain phase shift accumulate to 180 ° . Problems with overshoot and ringing pop up before the loop gain reaches 180 ° phase shift, thus the amplifier designer must keep a close eye on loop dynamics. Ringing and overshoot are handled in the next section, so preventing os- 14 has the form of many loop gain cillation is emphasized in this section. Equation 5 – transfer functions or circuits, so it is analyzed in detail. 5-12

87 Loop Gain Plots are the Key to Understanding Stability ) ( K ( ) A b + 14) – (5 Ǔ Ǔǒ ǒ ( ) ) ( 1 1 ) t ) t s s 1 2 20 LOG(A ) β dB ) β 20 LOG(K) τ 1/ 1 Amplitude (A LOG(f) 0 dB ) GM β τ 1/ 45 – 2 – 135 Phase (A – 180 φ M Figure 5 – 15. Magnitude and Phase Plot of Equation 5 – 14 The quantity, K, is the dc gain, and it plots as a straight line with an intercept of 20Log(K). = ω The Bode plot of Equation 5 – 14 is shown in Figure 5 – 15. The two break points, ω 1 20 dB/decade = 1/ and ω = ω τ = 1/ τ – , are plotted in the Bode plot. Each breakpoint adds 2 1 2 the plot, and 45 ° phase shift accumulates at each breakpoint. This transfer func- slope to tion is referred to as a two slope because of the two breakpoints. The slope of the curve crosses the 0 dB intercept indicates phase shift and the ability to oscillate. Notice when it that a one slope can only accumulate 90 ° phase shift, so when a transfer function passes through 0 dB with a one slope, it cannot oscillate. Furthermore, a two-slope system can accumulate 180 ° phase shift, therefore a transfer function with a two or greater slope is capable of oscillation. A one slope crossing the 0 dB intercept is stable, whereas a two or greater slope crossing the 0 dB intercept may be stable or unstable depending upon the accumulated phase , and the gain margin, Figure 5 – 15 defines two stability terms; the phase margin, φ shift. M . Of these two terms the phase margin is much more important because phase shift G M is critical for stability. Phase margin is a measure of the difference in the actual phase shift and the theoretical 180 ° required for oscillation, and the phase margin measurement or calculation is made at the 0 dB crossover point. The gain margin is measured or calcu- lated at the 180 ° phase crossover point. Phase margin is expressed mathematically in Equation 5 – 15. 1 – (5 – 15) ( ) + 180 * tangent f A b M 5-13 Feedback and Stability Theory

88 Loop Gain Plots are the Key to Understanding Stability The phase margin in Figure 5 15 is very small, 20 ° , so it is hard to measure or predict from – ’ the Bode plot. A designer probably doesn ° phase margin because the system t want a 20 overshoots and rings badly, but this case points out the need to calculate small phase mar- gins carefully. The circuit is stable, and it does not oscillate because the phase margin is positive. Also, the circuit with the smallest phase margin has the highest frequency re- sponse and bandwidth. 20 LOG(K + C) ) β 20 LOG(K) β ) 20 LOG(A τ 1/ 1 Amplitude (A 0 dB LOG(f) ) τ 1/ β 2 45 – – 135 Phase (A – 180 φ = 0 M Figure 5 – 16. Magnitude and Phase Plot of the Loop Gain Increased to (K+C) 16 shifts the magnitude plot up. Increasing the loop gain to (K+C) as shown in Figure 5 – If the pole locations are kept constant, the phase margin reduces to zero as shown, and the circuit will oscillate. The circuit is not good for much in this condition because produc- tion tolerances and worst case conditions ensure that the circuit will oscillate when you want it to amplify, and vice versa. dB ) β β ) 20 LOG(A 20 LOG(K) 1/ τ 1 Amplitude (A LOG(f) 0 dB ) τ 1/ 2 β – 45 – 135 Phase (A = 0 φ M 180 – Figure 5 – 17. Magnitude and Phase Plot of the Loop Gain With Pole Spacing Reduced 5-14

89 The Second Order Equation and Ringing/Overshoot Predictions The circuit poles are spaced closer in Figure 5 17, and this results in a faster accumula- – tion of phase shift. The phase margin is zero because the loop gain phase shift reaches 180 before the magnitude passes through 0 dB. This circuit oscillates, but it is not a very ° the transition to 180 stable oscillator because phase shift is very slow. Stable oscillators ° have a very sharp transition through 180 ° . When the closed loop gain is increased the feedback factor, β , is decreased because , thus the stabil- /V β = 1/ β for the ideal case. This in turn decreases the loop gain, A V IN OUT ity increases. In other words, increasing the closed loop gain makes the circuit more stable. Stability is not important except to oscillator designers because overshoot and ringing become intolerable to linear amplifiers long before oscillation occurs. The over- shoot and ringing situation is investigated next. 5.6 The Second Order Equation and Ringing/Overshoot Predictions The second order equation is a common approximation used for feedback system analy- sis because it describes a two-pole circuit, which is the most common approximation used. All real circuits are more complex than two poles, but except for a small fraction, they can be represented by a two-pole equivalent. The second order equation is exten- [ ] 6 . sively described in electronic and control literature K ) 1 (1 ) A b ) + 16) (5 – ǒ Ǔǒ Ǔ t ) 1 1 ) t s s 1 2 17. After algebraic manipulation Equation 5 – 16 is presented in the form of Equation 5 – t ) t 2 1 K ) 1 2 (5 – 17) 0 ) s S ) + t t t t 1 2 2 1 Equation 5 – 17 is compared to the second order control Equation 5 – 18, and the damping ζ , and natural frequency, w ratio, are obtained through like term comparisons. N 2 2 (5 – 18) ) s zw 2 s ) w N N these equations yields formulas for the phase margin and per cent overshoot Comparing as a function of damping ratio. 1 K ) – (5 19) w + Ǹ N t t 2 1 t t ) 1 2 20) – (5 + c 2 w t t 1 2 N – 21 is valid. When the two poles are well separated, Equation 5 5-15 Feedback and Stability Theory

90 References 1 * (5 21) – ) ( c 2 tangent + f M The salient equations are plotted in Figure 5 – 18, which enables a designer to determine the phase margin and overshoot when the gain and pole locations are known. 1 Percent Maximum Overshoot 0.8 0.6 Phase Margin, φ M 0.4 Damping Ratio, 0.2 0 0102030405060 70 80 Figure 5 – 18. Phase Margin and Overshoot vs Damping Ratio Enter 18 at the calculated damping ratio, say 0.4, and read the overshoot at 25% – Figure 5 and the phase margin at 42 . If a designer had a circuit specification of 5% maximum over- ° shoot, then the damping ratio must be 0.78 with a phase margin of 62 ° . 5.7 References 1. DiStefano, Stubberud, and Williams, Theory and Problems of Feedback and Control Systems, Schaum ’ s Outline Series , Mc Graw Hill Book Company, 1967 2. Bode, H. W., Network , Analysis And Feedback Amplifier Design D. Van Nostrand, Inc., 1945 Intuitive Operational Amplifiers , 3. Frederickson, Thomas, McGraw Hill Book Company, 1988 4. Bower, J. L. and Schultheis, P. M., Introduction To The Design Of Servomechanisms , Wiley, 1961 5. Van Valkenberg, M. E., Network Analysis , Prentice-Hall, 1964 6. Del Toro, V., and Parker, S., Principles of Control Systems Hill, 1960. – Engineering, McGraw 5-16

91 Chapter 6 Development of the Non Ideal Op Amp Equations Ron Mancini 6.1 Introduction There are two types of error sources in op amps, and they fall under the general classifica- tion of dc and ac errors. Examples of dc errors are input offset voltage and input bias cur- The dc errors stay constant over the usable op amp frequency range; therefore, the rent. input bias current is 10 pA at 1 kHz and it is 10 pA at 10 kHz. Because of their constant and controlled behavior, dc errors are not considered until later chapters. AC errors are flighty, so we address them here by developing a set of nonideal equations that account for ac errors. The ac errors may show up under dc conditions, but they get worse as the operating frequency increases. A good example of an ac error is common – mode rejection ration (CMRR). Most op amps have a guaranteed CMRR specification, but this specification is only valid at dc or very low frequencies. Further inspection of the data sheet reveals that CMRR decreases as operating frequency increases. Several oth- er specifications that fall into the category of ac specifications are output impedance, power-supply rejection-ratio, peak-to-peak output voltage, differential gain, differential phase, and phase margin. Differential gain is the most important ac specification because the other ac specifications are derived from the differential gain. Until now, differential gain has been called op amp gain or op amp open loop gain, and we shall continue with that terminology. Let the data sheet call it differential gain. As shown in prior chapters, when frequency increases, the op amp gain decreases and errors increase. This chapter develops the equations that illustrate the effects of the gain changes. We start with a review of the basic canonical feedback system stability because the op amp equations are developed using the same techniques. Amplifiers are built with active components such as transistors. Pertinent transistor pa- rameters like transistor gain are subject to drift and initial inaccuracies from many sources, so amplifiers being built from these components are subject to drift and inaccura- 6-1

92 Review of the Canonical Equations cies. The drift and inaccuracy is minimized or eliminated by using negative feedback. The op amp circuit configuration employs feedback to make the transfer equation of the circuit independent of the amplifier parameters (well almost), and while doing this, the circuit transfer function is made dependent on external passive components. The external pas- sive components can be purchased to meet almost any drift or accuracy specification; only the cost and size of the passive components limit their use. Once feedback is applied to the op amp it is possible for the op amp circuit to become unstable. Certain amplifiers belong to a family called internally compensated op amps; they contain internal capacitors that are sometimes advertised as precluding instabilities. Although internally compensated op amps should not oscillate when operated under spe- cified conditions, many have relative stability problems that manifest themselves as poor phase response, ringing, and overshoot. The only absolutely stable internally compen- amp is the one lying on the workbench without power applied! All other internally sated op compensated op amps oscillate under some external circuit conditions. Noninternally compensated or externally compensated op amps are unstable without the addition of external stabilizing components. This situation is a disadvantage in many cases because they require additional components, but the lack of internal compensation enables the top-drawer circuit designer to squeeze the last drop of performance from the op amp. You have two options: op amps internally compensated by the IC manufacturer, or op amps externally compensated by you. Compensation, except that done by the op amp manufacturer, must be done external to the IC. Surprisingly enough, internally com- pensated op amps require external compensation for demanding applications. Compensation is achieved by adding external components that modify the circuit transfer function so that it becomes unconditionally stable. There are several different methods of compensating an op amp, and as you might suspect, there are pros and cons associat- ed with each method of compensation. After the op amp circuit is compensated, it must be analyzed to determine the effects of compensation. The modifications that compensa- tion have on the closed loop transfer function often determine which compensation scheme is most profitably employed. 6.2 Review of the Canonical Equations A block diagram for a generalized feedback system is repeated in Figure 6 – 1. This simple block diagram is sufficient to determine the stability of any system. + E A V V Σ OUT IN – β – 1. Feedback System Block Diagram Figure 6 6-2

93 Review of the Canonical Equations The output and error equation development is repeated below. (6 – 1) V + EA OUT (6 – 2) E + V V * b IN OUT Combining Equations 6 – 1 and 6 – 2 yields Equation 6 – 3: V OUT – 3) (6 * b V V + IN OUT A – 4: Collecting terms yields Equation 6 1 Ǔ ǒ (6 – 4) ) b V + V IN OUT A Rearranging terms yields the classic form of the feedback equation. V OUT A – 5) (6 + V 1 ) A b IN Notice that Equation 6 – 5 reduces to Equation 6 – 6 when the quantity A β in Equation 6 – 5 becomes very large with respect to one. Equation 6 6 is called the ideal feedback equa- – tion because it depends on the assumption that A β >> 1, and it finds extensive use when amplifiers are assumed to have ideal qualities. Under the conditions that A β >>1, the sys- tem gain is determined by the feedback factor β. Stable passive circuit components are used to implement the feedback factor, thus the ideal closed loop gain is predictable and β is predictable and stable. stable because V OUT 1 (6 6) – + b V IN The quantity A β is so important that it has been given a special name, loop gain. Consider Figure 6 – 2; when the voltage inputs are grounded (current inputs are opened) and the β. loop is broken, the calculated gain is the loop gain, A Now, keep in mind that this is a mathematics of complex numbers, which have magnitude and direction. When the loop 5 ap- approaches minus one, or to express it mathematically 1 gain – 180 ° , Equation 6 – ∠ The circuit output heads for infinity as fast as it can . proaches infinity because 1/0 ⇒∞ 6-3 Development of the Non Ideal Op Amp Equations

94 Review of the Canonical Equations using the equation of a straight line. If the output were not energy limited the circuit would the world, but it is energy limited by the power supplies so the world stays intact. explode + A AV Σ TEST – β V TEST V RETURN V RETURN β = A V TEST Figure 6 – 2. Feedback Loop Broken to Calculate Loop Gain Active devices in electronic circuits exhibit nonlinear behavior when their output ap- proaches a power supply rail, and the nonlinearity reduces the amplifier gain until the loop ° . Now the circuit can do two things: first, it could become 180 1 ∠ – gain no longer equals stable at the power supply limit, or second, it can reverse direction (because stored charge keeps the output voltage changing) and head for the negative power supply rail. The first state where the circuit becomes stable at a power supply limit is named lockup; the circuit will remain in the locked up state until power is removed. The second state where the circuit bounces between power supply limits is named oscillatory. Remember, the loop gain, A β , is the sole factor that determines stability for a circuit or system. Inputs are grounded or disconnected when the loop gain is calculated, so they have no effect on stability. The loop gain criteria is analyzed in depth later. 7, which gives – Equations 6 – 1 and 6 – 2 are combined and rearranged to yield Equation 6 an indication of system or circuit error. V IN – 7) (6 E + 1 ) A b First, notice that the error is proportional to the input signal. This is the expected result because a bigger input signal results in a bigger output signal, and bigger output signals require more drive voltage. Second, the loop gain is inversely proportional to the error. As the loop gain increases the error decreases, thus large loop gains are attractive for minimizing errors. Large loop gains also decrease stability, thus there is always a tradeoff between error and stability. 6-4

95 Noninverting Op Amps 6.3 Noninverting Op Amps 3. The dummy variable, V , is inserted to A noninverting op amp is shown in Figure 6 – B make the calculations easier and a is the op amp gain. + V a OUT _ V IN Z F Z G V B – 3. Noninverting Op Amp Figure 6 – 8 is the amplifier transfer equation. Equation 6 Ǔ ǒ – (6 8) V + V V * a IN B OUT The output equation is developed with the aid of the voltage divider rule. Using the voltage divider rule assumes that the op amp impedance is low. Z V OUT G – (6 9) V + + 0 for I B B ) Z Z F G 10. 9 yields Equation 6 Combining Equations 6 – 8 and 6 – – aZ V OUT G (6 10) – * aV V + IN OUT Z Z ) F G – 10 yields Equation 6 – 11, which describes the transfer Rearranging terms in Equation 6 function of the circuit. V a OUT + 11) – (6 aZ V IN G ) 1 ) Z Z F G Equation 6 – 5 is repeated as Equation 6 – 12 to make a term by term comparison of the equations easy. V OUT A (6 12) – + V β 1 ) A IN – By virtue of the comparison we get Equation 6 13, which is the loop-gain equation for the noninverting op amp. The loop-gain equation determines the stability of the circuit. The 6-5 Development of the Non Ideal Op Amp Equations

96 Inverting Op Amps also shows that the open loop gain, A, is equal to the op amp open loop gain, comparison a, for the noninverting circuit. aZ G (6 – 13) A b + Z ) Z F G Equation 6 – 13 is also derived with the aid of Figure 6 – 4, which shows the open loop non- inverting op amp. + aV a TEST _ Z F Z G V V RETURN TEST Figure 6 – 4. Open Loop Noninverting Op Amp The test voltage, V , is multiplied by the op amp open loop gain to obtain the op amp TEST . The voltage divider rule is used to calculate Equation 6 – 15, which output voltage, aV TEST is identical to Equation 6 – 14 after some algebraic manipulation. aV Z TEST G 14) (6 – + V RETURN Z ) Z F G V aZ RETURN G 15) (6 – + A b + V Z Z ) F TEST G 6.4 Inverting Op Amps The inverting op amp circuit is shown in Figure 6 – 5. The dummy variable (V ) is inserted A to make the calculations easier, and a is the op amp open loop gain. + V a OUT _ I B Z Z F G V IN V A Figure 6 – 5. Inverting Op Amp 6-6

97 Inverting Op Amps The transfer equation is given in Equation 6 – 16: 16) (6 – +* aV V A OUT The node voltage (Equation 6 – 17) is obtained with the aid of superposition and the volt- – 18 is obtained by combining Equations 6 – 16 and 6 – 17. age divider rule. Equation 6 Z V Z V G OUT F IN 17) (6 – ) + V for I + 0 B A ) Z ) Z Z Z F F G G aZ – F Z Z ) V F G OUT 18) – (6 + aZ V IN G 1 ) Z ) Z F G Equation 6 – 16 is the transfer function of the inverting op amp. By virtue of the comparison between Equations 6 – 18 and 6 – 14, we get Equation 6 – 15 again, which is also the loop gain equation for the inverting op amp circuit. The comparison also shows that the open loop gain (A) is different from the op amp open loop gain (a) for the noninverting circuit. – 6, and this circuit The inverting op amp with the feedback loop broken is shown in Figure 6 is used to calculate the loop-gain given in Equation 6 19. – + V a OUT _ Z F Z V G RETURN = a V + Z Z F TEST G Z G V V RETURN TEST Figure 6 – 6. Inverting Op Amp: Feedback Loop Broken for Loop Gain Calculation aZ V RETURN G (6 19) – + + A b Z ) V Z F G TEST things must be mentioned at this point in the analysis. First, the transfer functions Several – 13 and 6 – 18, are different. For a common for the noninverting and inverting Equations, 6 and Z values, the magnitude and polarity of the gains are different. Second, set of Z F G the loop gain of both circuits, as given by Equations 6 – 15 and 6 – 19, is identical. Thus, the stability performance of both circuits is identical although their transfer equations are different. This makes the important point that stability is not dependent on the circuit in- puts . Third, the A gain block shown in Figure 6 – 1 is different for each op amp circuit. By = aZ = a and A 5, 6 – 11, and 6 – 18 we see that A comparison of Equations 6 – INV – INV NON F ). + Z ÷ (Z G F 6-7 Development of the Non Ideal Op Amp Equations

98 Differential Op Amps 6.5 Differential Op Amps – 7. The dummy variable, V , is in- The differential amplifier circuit is shown in Figure 6 E serted to make the calculations easier, and a is the open loop gain. V 1 V – Z Z G F _ V E V a OUT + V 2 V Z + G Z F Figure 6 – 7. Differential Amplifier Circuit Equation 6 – 20 is the circuit transfer equation. (6 – 20) V + aV + V * V * ) E OUT The positive input voltage, V – 21 with the aid of superposition and , is written in Equation 6 + the voltage divider rule. Z F 21) – (6 + V V ) 2 Z Z ) F G The negative input voltage, V , is written in Equation 6 – 22 with the aid of superposition – and the voltage divider rule. Z Z G F 22) – (6 + V * V V * 1 OUT Z ) Z Z Z ) F F G G – Combining Equations 6 – 20, 6 – 21, and 6 23. 22 yields Equation 6 – Z V Z Z V V G OUT 2 1 F F (6 23) – ƪ ƫ V * * + a OUT Z ) Z Z ) ) Z Z Z F F F G G G 24. After algebraic manipulation, Equation 6 – 23 reduces to Equation 6 – aZ F Z ) Z V F G OUT (6 24) – + V V * aZ 1 2 G ) 1 ) Z Z F G The comparison method reveals that the loop gain as shown in Equation 6 – 25 is identical – 19. to that shown in Equations 6 – 13 and 6 6-8

99 Differential Op Amps aZ G (6 25) – b + A Z ) Z F G only a function of the closed loop, and Again, the loop gain, which determines stability, is independent of the inputs. 6-9 Development of the Non Ideal Op Amp Equations

100 6-10

101 Chapter 7 Voltage-Feedback Op Amp Compensation Ron Mancini 7.1 Introduction Voltage-feedback amplifiers (VFA) have been with us for about 60 years, and they have been problems for circuit designers since the first day. You see, the feedback that makes them versatile and accurate also has a tendency to make them unstable. The operational amplifier (op amp) circuit configuration uses a high-gain amplifier whose parameters are determined by external feedback components. The amplifier gain is so high that without these external feedback components, the slightest input signal would saturate the amplifi- er output. The op amp is in common usage, so this configuration is examined in detail, but the results are applicable to many other voltage-feedback circuits. Current-feedback amplifiers (CFA) are similar to VFAs, but the differences are important enough to warrant CFAs being handled separately. Stability as used in electronic circuit terminology is often defined as achieving a nonoscil- latory state. This is a poor, inaccurate definition of the word. Stability is a relative term, and this situation makes people uneasy because relative judgments are exhaustive. It is easy to draw the line between a circuit that oscillates and one that does not oscillate, so we can understand why some people believe that oscillation is a natural boundary be- tween stability and instability. Feedback circuits exhibit poor phase response, overshoot, and ringing long before os- cillation occurs, and these effects are considered undesirable by circuit designers. This chapter is not concerned with oscillators; thus, relative stability is defined in terms of per- formance. By definition, when designers decide what tradeoffs are acceptable, they de- termine what the relative stability of the circuit is. A relative stability measurement is the damping ratio ( ζ ) and the damping ratio is discussed in detail in Reference 1. The damping ratio is related to phase margin, hence phase margin is another measure of relative stabil- ity. The most stable circuits have the longest response times, lowest bandwidth, highest accuracy, and least overshoot. The least stable circuits have the fastest response times, highest bandwidth, lowest accuracy, and some overshoot. Op Amps left in their native state oscillate without some form of compensation. The first IC op amps were very hard to stabilize, but there were a lot of good analog designers 7-1

102 Internal Compensation around in the 60s, so we used them. Internally compensated op amps were introduced ’ ’ in the late 60s in an attempt to make op amps easy for everyone to use. Unfortunately, internally compensated op amps sacrifice a lot of bandwidth and still oscillate under some conditions, so an understanding of compensation is required to apply op amps. Internal compensation provides a worst-case trade-off between stability and perfor- mance. Uncompensated op amps require more attention, but they can do more work. Both are covered here. Compensation is a process of applying a judicious patch in the form of an RC network to make up for a less than perfect op amp or circuit. There are many different problems that can introduce instability, thus there are many different compensation schemes. 7.2 Internal Compensation Op amps are internally compensated to save external components and to enable their use by less knowledgeable people. It takes some measure of analog knowledge to compen- sate an analog circuit. Internally compensated op amps normally are stable when they are used in accordance with the applications instructions. Internally compensated op are not unconditionally stable. They are multiple pole systems, but they are internal- amps ly compensated such that they appear as a single pole system over much of the frequency range. Internal compensation severely decreases the possible closed-loop bandwidth of the op amp. Internal compensation is accomplished in several ways, but the most common method is to connect a capacitor across the collector-base junction of a voltage gain transistor (see Figure 7 – 1). The Miller effect multiplies the capacitor value by an amount approxi- equal to the stage gain, thus the Miller effect uses small value capacitors for com- mately pensation. V CC R C C V IN R B – 1. Miller Effect Compensation Figure 7 7-2

103 Internal Compensation 2 shows the gain/phase diagram for an older op amp (TL03X). When the gain – Figure 7 crosses the 0-dB axis (gain equal to one) the phase shift is approximately 108 ° , thus the op amp must be modeled as a second-order system because the phase shift is more than ° . 90 LARGE-SIGNAL DIFFERENTIAL VOLTAGE VOLTAGE-FOLLOWER AMPLIFICATION AND PHASE SHIFT SMALL-SIGNAL PULSE RESPONSE vs FREQUENCY 16 ° 0 100 k 12 10 k ° 30 mV V/mV A VD – 8 – 1 k ° 60 4 100 0 90 ° Phase Shift Phase Shift 4 – 120 ° 10 Output Voltage – Large-Signal Differential – 8 – ÁÁ O V VO 1 ° 150 – 12 VD Voltage Amplification ÁÁ Á AVD A 16 – 0.1 180 ° Á 1.4 1.0 0.2 0 0.6 10 1 k 100 10 k 100 k 1 M 10 M μ – – s t Time f Frequency – Hz – = 10 k Ω V = 15 V R ± Ω = 10 k = 15 V R ± V L CC L CC C ° = 100 pF T C = 25 C C ° = 25 = 25 pF T L A A L 2. TL03X Frequency and Time Response Plots – Figure 7 ° yields a phase margin of φ = 180 ° – 108 = 72 This ° , thus the circuit should be very stable. Referring to Figure 7 – 3, the damping ratio is one and the expected overshoot is zero. Fig- ure 7 – 2 shows approximately 10% overshoot, which is unexpected, but inspecting 2 further reveals that the loading capacitance for the two plots is different. The Figure 7 – pulse response is loaded with 100 pF rather than 25 pF shown for the gain/phase plot, and this extra loading capacitance accounts for the loss of phase margin. 7-3 Voltage-Feedback Op Amp Compensation

104 Internal Compensation 1 Percent Maximum Overshoot 0.8 0.6 φ Phase Margin, M 0.4 Damping Ratio, 0.2 0 0102030405060 80 70 Figure 7 – 3. Phase Margin and Percent Overshoot Versus Damping Ratio Why does the loading capacitance make the op amp unstable? Look closely at the gain/ phase response between 1 MHz and 9 MHz, and observe that the gain curve changes /decade. The radical slope drastically while the rate of phase change approaches 120 ° gain/phase slope change proves that several poles are located in this area. The loading works with the op amp output impedance to form another pole, and the new capacitance pole reacts with the internal op amp poles. As the loading capacitor value is increased, its pole migrates down in frequency, causing more phase shift at the 0-dB crossover fre- quency. The proof of this is given in the TL03X data sheet where plots of ringing and os- cillation versus loading capacitance are shown. Figure 7 – 4 shows similar plots for the TL07X, which is the newer family of op amps. Notice _ that the phase shift is approximately 100 when the gain crosses the 0-dB axis. This yields _ , which is close to unconditionally stable. The slope of the phase a phase margin of 80 changes to 180 curve _ /decade about one decade from the 0-dB crossover point. The rad- phase margin, furthermore the gain ical slope change causes suspicion about the 90 ° curve must be changing radically when the phase is changing radically. The gain/phase plot may not be totally false, but it sure is overly optimistic. 7-4

105 Internal Compensation LARGE-SIGNAL DIFFERENTIAL VOLTAGE OUTPUT VOLTAGE AMPLIFICATION AND PHASE SHIFT vs vs ELAPSED TIME FREQUENCY 6 10 28 Overshoot 24 5 10 Differential mV – 20 Voltage 90% 4 ° 0 10 Amplification 16 3 10 45 ° 12 8 Phase Shift 2 10 ° 90 Output Voltage Phase Shift Large-Signal Differential – 4 – Voltage Amplification 10% O 1 ° 135 10 VO V ÁÁ VD 0 AVD A t r ÁÁ 1 ° 180 – 4 10 M 1 10 0.7 100 1 k 10 k 100 k 1 M 0.6 0.1 0.2 0.3 0.4 0.5 0 f – Frequency – Hz t μ s – Elapsed Time – ± = 15 V = 5 V to V ± V ± V = 15 V R Ω = 10 k CC CC CC L R = 2 k Ω T = 25 ° C T ° C = 25 A L A Figure 7 – 4. TL07X Frequency and Time Response Plots The TL07X pulse response plot shows approximately 20% overshoot. There is no loading capacitance indicated on the plot to account for a seemingly unconditionally stable op amp exhibiting this large an overshoot. Something is wrong here: the analysis is wrong, the plots are wrong, or the parameters are wrong. Figure 7 – 5 shows the plots for the TL08X family of op amps, which are sisters to the TL07X family. The gain/phase curve and pulse response is virtually identical, but the pulse response lists a 100 pF loading ca- pacitor. This little exercise illustrates three valuable points: first, if the data seems wrong it probably is wrong, second, even the factory people make mistakes, and third, the load- ing capacitor makes op amps ring, overshoot, or oscillate. 7-5 Voltage-Feedback Op Amp Compensation

106 Internal Compensation LARGE-SIGNAL DIFFERENTIAL VOLTAGE OUTPUT VOLTAGE AMPLIFICATION AND PHASE SHIFT vs vs ELAPSED TIME FREQUENCY 6 28 10 24 5 10 mV Differential 20 – V/mV – Voltage 4 0 ° 10 Amplification 16 3 45 ° 12 10 8 Phase Shift 2 10 ° 90 Output Voltage 4 – Phase Shift Large-Signal Differential – O 1 135 ° 10 V 0 VD Voltage Amplification A – 4 180 ° 1 1.2 0.8 0.6 0.4 0.2 0 1.0 1 10 M 100 k 10 k 1 k 100 1 M 10 – Frequency – Hz f t Elapsed Time s – – μ V ± = 5 V to V = 2 k ± = 15 V Ω V = 15 V R ± CC L CC CC C R = 10 k Ω C T = 25 = 25 ° C ° = 100 pF T L A L A Figure 7 – 5. TL08X Frequency and Time Response Plots The frequency and time-response plots for the TLV277X family of op amps is shown in Figures 7 – 6 and 7 – 7. First, notice that the information is more sophisticated because the phase response is given in degrees of phase margin; second, both gain/phase plots are with substantial loading capacitors (600 pF), so they have some practical value; and done third, the phase margin is a function of power supply voltage. 7-6

107 Internal Compensation LARGE-SIGNAL DIFFERENTIAL VOLTAGE LARGE-SIGNAL DIFFERENTIAL VOLTAGE AMPLIFICATION AND PHASE MARGIN AMPLIFICATION AND PHASE MARGIN vs vs FREQUENCY FREQUENCY dB dB – – 300 100 300 100 240 80 240 80 A A VD VD 180 180 60 60 degrees degrees – – 120 40 120 40 Phase Phase 60 60 20 20 0 0 0 0 Phase Margin Phase Margin – – 60 – 20 – – 60 20 – m m φ φ 90 – – 40 90 40 – – Large-Signal Differential Amplification Large-Signal Differential Amplification 100 k 1 k 1 M 10 M 10 k 100 100 1 M 100 k 1 k 10 M 10 k – – – Frequency Hz – f – Frequency – Hz f VD VD A A ± = 5 V R = 600 k = 600 k Ω V = 2.7 V R V ± Ω DD DD L L C C = 25 ° C = 25 = 600 pF T C = 600 pF T ° L A A L – 6. TLV277X Frequency Response Plots Figure 7 INVERTING SMALL-SIGNAL VOLTAGE-FOLLOWER PULSE RESPONSE SMALL-SIGNAL PULSE RESPONSE 100 100 80 80 mV mV – – 60 60 40 40 20 20 0 0 Output Voltage Output Voltage – – 20 – 20 – O O V V 40 – 40 – – 60 – 60 012345 012345 Time – s μ – t s μ – t Time – = 2.7 V R V = 600 k Ω Ω = 600 k = 2.7 V R V L L DD DD C C C C ° ° = 25 = 100 pF T = 25 = 100 pF T L L A A = 1 = – 1 A A V V – 7. TLV227X Time Response Plots Figure 7 7-7 Voltage-Feedback Op Amp Compensation

108 External Compensation, Stability, and Performance At V = 5 V, the phase margin at the 0-dB crossover point is 60 ° , while it is 30 ° at V CC CC = 2.7 V. This translates into an expected overshoot of 18% at V = 5 V, and 28% at CC Unfortunately the time response plots are done with 100-pF loading capaci- = 2.7 V. V CC = 2.7 V overshoot is approx- tance, hence we can not check our figures very well. The V CC imately 2%, and it is almost impossible to figure out what the overshoot would have been with a loading capacitor. The small-signal pulse response is done with mV -signals, 600 pF and that is a more realistic measurement than using the full signal swing. Internally compensated op amps are very desirable because they are easy to use, and they do not require external compensation components. Their drawback is that the band- width is limited by the internal compensation scheme. The op amp open-loop gain eventu- ally (when it shows up in the loop gain) determines the error in an op amp circuit. In a non- = 2.7 V) inverting buffer configuration, the TL277X is limited to 1% error at 50 kHz (V CC because the op amp gain is 40 dB at that point. Circuit designers can play tricks such as bypassing the op amp with a capacitor to emphasize the high-frequency gain, but the error – 1 in mind because it defines the error. If the TLV277X were is still 1%. Keep Equation 7 not internally compensated, it could be externally compensated for a lower error at 50 kHz because the gain would be much higher. V IN – (7 1) E + ) 1 b A 7.3 External Compensation, Stability, and Performance Nobody compensates an op amp just because it is there; they have a reason to compen- sate the op amp, and that reason is usually stability. They want the op amp to perform a function in a circuit where it is potentially unstable. Internally and noninternally compen- sated op amps are compensated externally because certain circuit configurations do cause oscillations. Several potentially unstable circuit configurations are analyzed in this section, and the reader can extend the external compensation techniques as required. Other reasons for externally compensating op amps are noise reduction, flat amplitude response, and obtaining the highest bandwidth possible from an op amp. An op amp gen- erates noise, and noise is generated by the system. The noise contains many frequency components, and when a high-pass filter is incorporated in the signal path, it reduces high frequency noise. Compensation can be employed to roll off the op amp ’ s high frequency, closed-loop response, thus causing the op amp to act as a noise filter. Internally compen- sated op amps are modeled with a second order equation, and this means that the output voltage can overshoot in response to a step input. When this overshoot (or peaking) is undesirable, external compensation can increase the phase margin to 90 ° where there is no peaking. An uncompensated op amp has the highest bandwidth possible. External compensation is required to stabilize uncompensated op amps, but the compensation can be tailored to the specific circuit, thus yielding the highest possible bandwidth consis- tent with the pulse response requirements. 7-8

109 Dominant-Pole Compensation 7.4 Dominant-Pole Compensation We saw that capacitive loading caused potential instabilities, thus an op amp loaded with an output capacitor is a circuit configuration that must be analyzed. This circuit is called dominant pole compensation because if the pole formed by the op amp output impedance and the loading capacitor is located close to the zero frequency axis, it becomes domi- nant. The op amp circuit is shown in Figure 7 – 8, and the open loop circuit used to calcu- late the loop gain (A β ) is shown in Figure 7 – 9. + V IN Z O V ∆ V OUT V ∆ A C L – Z F POINT X Z G Figure 7 – 8. Capacitively-Loaded Op Amp V OUT V (Return) Z Z O F Z C G L ∆ V v = (Test) A ) Figure 7 – 9. Capacitively-Loaded Op Amp With Loop Broken for Loop Gain (A β Calculation The analysis starts by looking into the capacitor and taking the Thevenin equivalent cir- cuit. Va D (7 – 2) V + TH s 1 ) C Z L O Z O (7 – 3) Z + TH C 1 Z ) s L O Then the output equation is written. 7-9 Voltage-Feedback Op Amp Compensation

110 Dominant-Pole Compensation ȡ ȣ V Z Z TH Va D G G ȧ + ȧ + V – 4) (7 RETURN ȧ ȧ Z Z ) ) Z Z Z ) s C 1 O TH F G L O Z ) Z ) F G Ȣ s ) 1 C Z L O 5. Rearranging terms yields Equation 7 – aZ G Z Z Z ) ) V F O G RETURN – (7 5) + b A + V ǒ Ǔ Z C ) Z s Z TEST F L G O 1 ) Z ) ) Z Z F G O When the assumption is made that (Z 5 reduces to Equation + Z – ) >> Z Equation 7 O, G F 7 – 6. aZ G 1 – 6) (7 Ǔ ǒ b + A Z ) Z Z 1 ) C s F G L O Equation 7 – 7 models the op amp as a second-order system. Hence, substituting the se- cond-order a in Equation 7 – 6 yields Equation 7 – 8, which is the stability equation model for for the dominant-pole compensation circuit. K + a 7) – (7 ǒ Ǔǒ Ǔ t s ) t ) s 1 2 Z G 1 K – (7 8) b + A Z ) Z ) 1 s C Z Ǔǒ ǒ Ǔ s ) t t s ) F G L O 2 1 Several conclusions can be drawn from Equation 7 – 8 depending on the location of the poles. If the Bode plot of Equation 7 – 7, the op amp transfer function, looks like that shown ° phase margin, and there is approximately 48% overshoot. in Figure 7 – 10, it only has 25 and C moves towards the zero frequency axis it comes When the pole introduced by Z O L close to the τ pole, and it adds phase shift to the system. Increased phase shift increases 2 peaking and decreases stability. In the real world, many loads, especially cables, are ca- pacitive, and an op amp like the one pictured in Figure 7 – 10 would ring while driving a capacitive load. The load capacitance causes peaking and instability in internally com- pensated op amps when the op amps do not have enough phase margin to allow for the phase shift introduced by the load. 7-10

111 Dominant-Pole Compensation 100 dB ) β 20 Log (A 0-dB Frequency Log F 0dB τ W = 1 W = τ 2 – 45 ° – 135 ° Degrees Phase Shift – ° 155 Figure 7 – 10. Possible Bode Plot of the Op Amp Described in Equation 7 – 7 Prior to compensation, the Bode plot of an uncompensated op amp looks like that shown in Figure 7 – 11. Notice that the break points are located close together thus accumulating about 180 ° of phase shift before the 0 dB crossover point; the op amp is not usable and probably unstable. Dominant pole compensation is often used to stabilize these op amps. τ , is properly placed it rolls off the gain so that If a dominant pole, in this case ω D 1 introduces 45 phase at the 0-dB crossover point. After the dominant pole is introduced _ the op amp is stable with 45 ° phase margin, but the op amp gain is drastically reduced . This procedure works well for internally compensated op for frequencies higher than ω D amps, but is seldom used for externally compensated op amps because inexpensive dis- crete capacitors are readily available. dB ) β 20 Log (A 0dB Log(f) τ 1/ Dominant Pole τ 1/ 2 1 W D Figure 7 – 11.Dominant-Pole Compensation Plot << Z , the closed-loop transfer function is easy to calculate because Assuming that Z O F is enclosed in the feedback loop. The ideal closed-loop transfer equation is the same C L as Equation 6 – 11 for the noninverting op amp, and is repeated below as Equation 7 – 9. 7-11 Voltage-Feedback Op Amp Compensation

112 Gain Compensation V OUT a + 9) (7 – aZ V IN G 1 ) ) Z Z F G ⇒ ∞ Equation 7 – 9 reduces to Equation 7 – 10. When a V Z Z ) F OUT G + 10) (7 – Z V IN G As long as the op amp has enough compliance and current to drive the capacitive load, is small, the circuit functions as though the capacitor was not there. When the ca- and Z O pacitor becomes large enough, its pole interacts with the op amp pole causing instability. When the capacitor is huge, it completely kills the op amp ’ s bandwidth, thus lowering the noise while retaining a large low-frequency gain. 7.5 Gain Compensation When the closed-loop gain of an op amp circuit is related to the loop gain, as it is in voltage- feedback op amps, the closed-loop gain can be used to stabilize the circuit. This type of compensation can not be used in current-feedback op amps because the mathematical relationship between the loop gain and ideal closed-loop gain does not exist. The loop 11. Notice that the closed-loop gain parameters equation is repeated as Equation 7 gain – and Z are contained in Equation 7 – 11, hence the stability can be controlled by manip- Z F G ulating the closed-loop gain parameters. aZ G + A b 11) (7 – Z ) Z F G The original loop gain curve for a closed-loop gain of one is shown in Figure 7 – 12, and it is or comes very close to being unstable. If the closed-loop noninverting gain is changed to 9, then K changes from K/2 to K/10. The loop gain intercept on the Bode plot (Figure – 7 12) moves down 14 dB, and the circuit is stabilized. dB Loop Gain Curve K 20 Log 2 – 14 dB K 20 Log KZ 10 G 20 Log Z + Z G F Compensated Loop Gain Curve 0dB Log(f) τ 1/ τ 1/ 2 1 – 12. Gain Compensation Figure 7 7-12

113 Lead Compensation Gain compensation works for inverting or noninverting op amp circuits because the loop When the closed- gain equation contains the closed-loop gain parameters in both cases. gain is increased, the accuracy and the bandwidth decrease. As long as the applica- loop tion can stand the higher gain, gain compensation is the best type of compensation to use. Uncompensated versions of normally internally compensated op amps are offered for sale as stable op amps with minimum gain restrictions. As long as gain in the circuit you design exceeds the gain specified on the data sheet, this is economical and a safe mode of operation. 7.6 Lead Compensation Sometimes lead compensation is forced on the circuit designer because of the parasitic capacitance associated with packaging and wiring op amps. Figure 7 – 13 shows the cir- . That capacitor is often cuit for lead compensation; notice the capacitor in parallel with R F made by parasitic wiring and the ground plane, and high frequency circuit designers go to great lengths to minimize or eliminate it. What is good in one sense is bad in another, because adding the parallel capacitor is a good way to stabilize the op amp and reduce noise. Let us analyze the stability first, and then we will analyze the closed-loop perfor- mance. + V a OUT _ R F R G V IN C Figure 7 – 13. Lead-Compensation Circuit The loop equation for the lead-compensation circuit is given by Equation 7 – 12. R 1 Cs R ) ȡ ȣ G F K (7 – 12) ǒ Ǔ Ǔǒ b + A ȧ ȧ ) R R Cs ) 1 R ø R Ǔǒ Ǔ ǒ s ) t s ) t F G Ȣ F G 1 2 The compensation capacitor introduces a pole and zero into the loop equation. The zero always occurs before the pole because R . When the zero is properly placed it ||R >R F G F pole along with its associated phase shift. The original transfer function τ cancels out the 2 = 1/ , it τ ω C zero is placed at is shown in Figure 7 14 drawn in solid lines. When the R – F 2 20 dB/decade. – pole causing the bode plot to continue on a slope of cancels out the τ 2 40 dB/de- ||RG)C, this pole changes the slope to – When the frequency gets to ω = 1/(R F cade. Properly placed, the capacitor aids stability, but what does it do to the closed-loop 7-13 Voltage-Feedback Op Amp Compensation

114 Lead Compensation transfer function? The equation for the inverting op amp closed-loop gain is repeated be- low. aZ – F Z ) Z V F G OUT – 13) (7 + aZ V IN G ) 1 Z ) Z F G )) dB F Original Transfer Function + R G /(R ) 20 Log (A β G Modified Transfer Function 20 Log (KR 0dB Log(f) τ 1/ τ 1/ 1 2 1/R IIR C C 1/R G F F Figure 7 – 14. Lead-Compensation Bode Plot 14. a approaches infinity, Equation 7 – 13 reduces to Equation 7 – When V Z OUT F (7 14) – +* V Z IN IN 14 yields Equation 7 Substituting R ||C for Z 15, which is – for Z in Equation 7 – and R G F G F the ideal closed-loop gain equation for the lead compensation circuit. V R OUT 1 F (7 – 15) ǒ Ǔ +* R V Cs ) 1 R IN F G – 16. Compare Equation The forward gain for the inverting amplifier is given by Equation 7 – 13 with Equation 6 – 5 to determine A. 7 aR aZ 1 F F 16) – (7 Ǔ ǒ Ǔǒ + A + Z ) R A ) R ø R Cs ) 1 R F F G G F G The op amp gain (a), the forward gain (A), and the ideal closed-loop gain are plotted in Figure 7 15. The op amp gain is plotted for reference only. The forward gain for the invert- – ing op amp is not the op amp gain. Notice that the forward gain is reduced by the factor +R ), and it contains a high frequency pole. The ideal closed-loop gain follows the /(R R G F F breakpoint), and then it τ C breakpoint (same location as 1/ ideal curve until the 1/R F 2 7-14

115 Lead Compensation slopes 20 dB/decade. Lead compensation sacrifices the bandwidth between the down at – C breakpoint and the forward gain curve. The location of the 1/R C pole determines 1/R F F the bandwidth sacrifice, and it can be much greater than shown here. The pole caused , R , and C does not appear until the op amp ’ s gain has crossed the 0-dB axis, thus by R F G it does not affect the ideal closed-loop transfer function. 20 Log a Op Amp Gain aZ F 20 Log A + Z Z G F 1 (R )C || R G C Ideal Closed-Loop Gain Z F 20 Log Z G 0dB 1 1 1 τ and 1 τ R C 2 F Figure 7 – 15. Inverting Op Amp With Lead Compensation 11 to Equation – The forward gain for the noninverting op amp is a ; compare Equation 6 6 5. The ideal closed-loop gain is given by Equation 7 – 17. – V Z Cs R R ø 1 R ) ) ) Z R F F F OUT G G G (7 – 17) Ǔǒ ǒ Ǔ + + V Z R Cs ) 1 R IN G G F The plot of the noninverting op amp with lead compensation is shown in Figure 7 16. – There is only one plot for both the op amp gain (a) and the forward gain (A), because they are identical in the noninverting circuit configuration. The ideal starts out as a flat line, but it slopes down because its closed-loop gain contains a pole and a zero. The pole always > R ||R . The zero flattens the ideal occurs closer to the low frequency axis because R F G F closed-loop gain curve, but it never does any good because it cannot fall on the pole. The pole causes a loss in the closed-loop bandwidth by the amount separating the closed-loop and forward gain curves. 7-15 Voltage-Feedback Op Amp Compensation

116 Compensated Attenuator Applied to Op Amp 20 Log a + Z Z G F 20 Log Z G 0dB Log(f) 1 1 1 and R τ C τ F 2 1 1 (R )C || R G C Figure 7 – 16. Noninverting Op Amp With Lead Compensation the forward gain is different in the inverting and noninverting circuits, the closed- Although loop transfer functions take very similar shapes. This becomes truer as the closed-loop the op amp gain. This gain increases because the noninverting forward gain approaches relationship cannot be relied on in every situation, and each circuit must be checked to determine the closed-loop effects of the compensation scheme. 7.7 Compensated Attenuator Applied to Op Amp Stray capacitance on op amp inputs is a problem that circuit designers are always trying to get away from because it decreases stability and causes peaking. The circuit shown ,) connected from the inverting input to 17 has some stray capacitance (C – in Figure 7 G ground. Equation 7 – 18 is the loop gain equation for the circuit with input capacitance. + V a OUT _ R F R G C V G IN Figure 7 – 17. Op Amp With Stray Capacitance on the Inverting Input R ȣ ȡ G 1 K – (7 18) ǒ Ǔ Ǔǒ + b A ȧ ȧ R ) R R Cs ) 1 ø R ǒ Ǔ Ǔǒ s ) t s t ) 1 1 F G Ȣ F G 2 1 Op amps having high input and feedback resistors are subject to instability caused by stray capacitance on the inverting input. Referring to Equation 7 – 18, when the 7-16

117 Compensated Attenuator Applied to Op Amp ) pole moves close to 1/(R C the stage is set for instability. Reasonable compo- ||R τ 2 G G F values for a CMOS op amp are R = 1 M = 1 M Ω , R = 10 pF. The resulting nent Ω , and C G F G for many op τ occurs at 318 kHz, and this frequency is lower than the breakpoint of pole 2 phase the 1/(R ° ||R C) pole adds 45 _ of phase shift resulting from τ amps. There is 90 G F 1, starts to add another 45 ° phase shift at about 600 kHz. This circuit 318 kHz, and τ shift at 2 is unstable because of the stray input capacitance. The circuit is compensated by adding a feedback capacitor as shown in Figure 7 – 18. + V a OUT _ R F R G C F V IN C G Figure 7 – 18. Compensated Attenuator Circuit The loop gain with C added is given by Equation 7 – 19. F R G ȳ ȱ C R 1 ) s ȣ ȡ G G K ȧ ȧ (7 – 19) ȧ ȧ A b + ȧ ȧ R R Ǔǒ Ǔ ǒ ȧ ȧ 1 ) s ) s t 1 t Ȣ G F 1 2 ) s ) 1 C R s R ) C 1 ȴ Ȳ F F G G 20. If R – C 19 reduces to Equation 7 = R – C Equation 7 F F G G R ȣ ȡ G K ǒ Ǔ A + b ȧ ȧ 20) – (7 ) R R Ǔ ǒ Ǔǒ 1 t s s ) 1 ) t F G Ȣ 2 1 The compensated attenuator Bode plot is shown in Figure 7 – 19. Adding the correct 1/R breakpoint; the loop gain is independent of breakpoint cancels out the 1/R C C F G F G can be the capacitors. Now is the time to take advantage of the stray capacitance. C F running a wide copper strip from the output of the op amp over the ground plane formed by ; do not connect the other end of this copper strip. The circuit is tuned by remov- under R F ing some copper (a razor works well) until all peaking is eliminated. Then measure the copper, and have an identical trace put on the printed-circuit board. 7-17 Voltage-Feedback Op Amp Compensation

118 Lead-Lag Compensation Without C F dB ) β With C F 20 Log (A 1/ τ 1 ) Log(f) 0dB ° β τ 1/ 2 1 – ° 45 R C IIR G F G ° 135 – With C F FHASE (A – 180 ° Without C F Figure 7 – 19. Compensated Attenuator Bode Plot The inverting and noninverting closed-loop gain equations are a function of frequency. = C 21 is the closed-loop gain equation for the inverting op amp. When R Equation 7 – F F R – 22, which is independent of the breakpoint. 21 reduces to Equation 7 C – , Equation 7 G G This also happens to the noninverting op amp circuit. This is one of the few occasions when the compensation does not affect the closed-loop gain frequency response. R F ) s R C 1 V F F OUT 21) – (7 +* V R IN G s R C ) 1 G G V R OUT F 22) (7 – = R C When R C ǒ Ǔ +* G F F G V R IN G 7.8 Lead-Lag Compensation Lead-lag compensation stabilizes the circuit without sacrificing the closed-loop gain per- often used with uncompensated op amps. This type of compensation pro- formance. It is vides excellent high-frequency performance. The circuit schematic is shown in Figure 7 – 20, and the loop gain is given by Equation 7 – 23. 7-18

119 Lead-Lag Compensation + V OUT a _ C R R R F G V IN 20. Lead-Lag Compensated Op Amp Figure 7 – R RCs 1 ) G K b + A ) R R 23) – (7 ǒ Ǔ Ǔǒ Ǔ ǒ R ) R RR RR ) 1 ) 1 t t ) s s F G F F G G 2 1 ) 1 Cs Ǔ ǒ ) R R F G Referring to = 1/RC, and this pole reduces the gain ω Figure 7 – 21, a pole is introduced at 3 dB at the breakpoint. When the zero occurs prior to the first op amp pole it cancels out = 1/RC pole. The phase shift is completely canceled be- ω the phase shift caused by the fore the second op amp pole occurs, and the circuit reacts as if the pole was never introduced. Nevertheless, A β is reduced by 3 dB or more, so the loop gain crosses the 0-dB axis at a lower frequency. The beauty of lead lag compensation is that the closed- loop ideal gain is not affected as is shown below. The Thevenin equivalent of the input – 24, the circuit gain in terms of Thevenin equivalents is circuit is calculated in Equation 7 – 25, and the ideal closed-loop gain is calculated in Equation calculated in Equation 7 7 – 26. ) β 20 Log (A Before Compensation 20 Log (aR + R )) /(R G G F 20 Log (A ) β After Compensation Amplitude Log(f) 0dB 1/ τ 1/ τ 1/(RC) 2 1 Compensation Network (RR + R ) R + R R F F G G C 1 / + R ) (R F G 21. Bode Plot of Lead-Lag Compensated Op Amp – Figure 7 7-19 Voltage-Feedback Op Amp Compensation

120 Comparison of Compensation Schemes 1 1 Ǔ ǒ R ) R R ) G Cs Cs – 24) (7 + R V + V TH TH IN 1 1 R ) R ) R R ) ) G G Cs Cs R F – 25) (7 V V +* TH OUT R TH 1 R ) V R R Cs OUT F F + * + 1 V R 1 R ) ) R IN G (7 26) – ǒ Ǔ R ) R G Cs G Cs 1 ) R ) R G Cs Equation 7 – 26 is intuitively obvious because the RC network is placed across a virtual ground. As long as the loop gain, A β , is large, the feedback will null out the closed-loop effect of RC, and the circuit will function as if it were not there. The closed-loop log plot of the lead-lag-compensated op 22. Notice that the pole and zero amp is given in Figure 7 – from the compensation occur and are gone before the first amplifier poles come resulting on the scene. This prevents interaction, but it is not required for stability. 1/RC 20 Log A Amplitude R F 20 Log R G Log(f) 1/(R + R)C G τ 1/ 1/ τ 2 1 22. Closed-Loop Plot of Lead-Lag Compensated Op Amp Figure 7 – 7.9 Comparison of Compensation Schemes Internally compensated op amps can, and often do, oscillate under some circuit condi- tions. Internally compensated op amps need an external pole to get the oscillation or ring- ing started, and circuit stray capacitances often supply the phase shift required for insta- bility. Loads, such as cables, often cause internally compensated op amps to ring severe- ly. 7-20

121 Conclusions Dominant pole compensation is often used in IC design because it is easy to implement. It rolls off the closed-loop gain early; thus, it is seldom used as an external form of com- pensation unless filtering is required. Load capacitance, depending on its pole location, usually causes the op amp to ring. Large load capacitance can stabilize the op amp be- cause it acts as dominant pole compensation. The simplest form of compensation is gain compensation. High closed-loop gains are re- flected in lower loop gains, and in turn, lower loop gains increase stability. If an op amp circuit can be stabilized by increasing the closed-loop gain, do it. Stray capacitance across the feedback resistor tends to stabilize the op amp because it is a form of lead compensation. This compensation scheme is useful for limiting the circuit bandwidth, but it decreases the closed-loop gain. Stray capacitance on the inverting input works with the parallel combination of the feed- back and gain setting resistors to form a pole in the Bode plot, and this pole decreases the circuit ’ s stability. This effect is normally observed in high-impedance circuits built with CMOS op amps. Adding a feedback capacitor forms a compensated attenuator scheme that cancels out the input pole. The cancellation occurs when the input and feedback RC time constants are equal. Under the conditions of equal time constants, the op amp func- tions as though the stray input capacitance was not there. An excellent method of imple- menting a compensated attenuator is to build a stray feedback capacitor using the ground plane and a trace off the output node. Lead-lag compensation stabilizes the op amp, and it yields the best closed-loop frequen- cy performance. Contrary to some published opinions, no compensation scheme will in- crease the bandwidth beyond that of the op amp. Lead-lag compensation just gives the best bandwidth for the compensation. 7.10 Conclusions The stability criteria often is not oscillation, rather it is circuit performance as exhibited by peaking and ringing. The circuit bandwidth can often be increased by connecting an external capacitor in paral- lel with the op amp. Some op amps have hooks that enable a parallel capacitor to be con- nected in parallel with a portion of the input stages. This increases bandwidth because stages, but this method of com- it shunts high frequencies past the low bandwidth g m pensation depends on the op amp type and manufacturer. The compensation techniques given here are adequate for the majority of applications. When the new and challenging application presents itself, use the procedure outlined here to invent your own compensation technique. 7-21 Voltage-Feedback Op Amp Compensation

122 7-22

123 Chapter 8 Current-Feedback Op Amp Analysis Ron Mancini 8.1 Introduction Current-feedback amplifiers (CFA) do not have the traditional differential amplifier input structure, thus they sacrifice the parameter matching inherent to that structure. The CFA circuit configuration prevents them from obtaining the precision of voltage-feedback am- plifiers (VFA), but the circuit configuration that sacrifices precision results in increased bandwidth and slew rate. The higher bandwidth is relatively independent of closed-loop the constant gain-bandwidth restriction applied to VFAs is removed for CFAs. The gain, so slew rate of CFAs is much improved from their counterpart VFAs because their structure enables the output stage to supply slewing current until the output reaches its final value. In general, VFAs are used for precision and general purpose applications, while CFAs are restricted to high frequency applications above 100 MHz. Although CFAs do not have the precision of their VFA counterparts, they are precise enough to be dc-coupled in video applications where dynamic range requirements are not severe. CFAs, unlike previous generation high-frequency amplifiers, have eliminated the ac coupling requirement; they are usually dc-coupled while they operate in the GHz range. CFAs have much faster slew rates than VFAs, so they have faster rise/fall times and less intermodulation distortion. 8.2 CFA Model The CFA model is shown in Figure 8 – 1. The noninverting input of a CFA connects to the input of the input buffer, so it has very high impedance similar to that of a bipolar transistor noninverting VFA input. The inverting input connects to the input buffer ’ s output, so the inverting input impedance is equivalent to a buffer ’ s output impedance, which is very low. . The input Ω models the input buffer ’ s output impedance, and it is usually less than 50 Z B , is as close to one as IC design methods can achieve, and it is small buffer gain, G B enough to neglect in the calculations. 8-1

124 Development of the Stability Equation + NONINVERTING INPUT Z OUT G V B OUT I G OUT Z Z(I) B – INVERTING INPUT 1. Current-Feedback Amplifier Model Figure 8 – The output buffer provides low output impedance for the amplifier. Again, the output buffer , is very close to one, so it is neglected in the analysis. The output impedance gain, G OUT of the output buffer is ignored during the calculations. This parameter may influence the circuit performance when driving very low impedance or capacitive loads, but this is usual- ’ s output impedance can ly not the case. The input buffer ’ t be ignored because affects sta- bility at high frequencies. The current-controlled current source, Z, is a transimpedance. The transimpedance in a CFA serves the same function as gain in a VFA; it is the parameter that makes the perfor- mance of the op amp dependent only on the passive parameter values. Usually the trans- range, so the CFA gains accuracy by closing a feed- impedance is very high, in the M Ω back loop in the same manner that the VFA does. 8.3 Development of the Stability Equation 2. Remember, stability is inde- – The stability equation is developed with the aid of Figure 8 pendent of the input, and stability depends solely on the loop gain, A β . Breaking the loop , and calculating the return signal V develops the at point X , inserting a test signal, V TO TI stability equation. + CFA ; The Test Signal Output V Becomes V TO OUT _ Break Loop Here Z F Z ) Here Apply Test Signal (V G TI Figure 8 – 2. Stability Analysis Circuit The circuit used for stability calculations is shown in Figure 8 – 3 where the model of Figure 1 is substituted for the CFA symbol. The input and output buffer gain, and output buffer – 8 8-2

125 The Noninverting CFA output impedance have been deleted from the circuit to simplify calculations. This approx- imation is valid for almost all applications. I 2 = V V OUT TO I 1 + Z F Z Z Z I V TI 1 B G – 3. Stability Analysis Circuit Figure 8 – ” The transfer equation is given in Equation 8 s law is used to write 1, and the Kirchoff – Equations 8 – 3. 2 and 8 (8 – 1) V Z + I 1 TO ǒ Ǔ – 2) (8 I + ) Z V ø Z Z 2 TI B F G ǒ Ǔ – 3) (8 ø Z I I Z + Z 1 2 B B G – 4. Equations 8 – 2 and 8 – 3 are combined to yield Equation 8 Z Z B B Ǔ ǒ – 4) (8 Ǔ ǒ ǒ Ǔ I Z Z V ) Z 1 ) I + 1 ) Z + ø 1 1 F F TI B G Z Z ø Z G F G Dividing Equation 8 5, and this is the open loop – 1 by Equation 8 – 4 yields Equation 8 – transfer equation. This equation is commonly known as the loop gain. V TO Z A b + + V TI – 5) (8 Z B ǒ Ǔ Ǔ ǒ Z 1 ) F Z ø Z F G 8.4 The Noninverting CFA The closed-loop gain equation for the noninverting CFA is developed with the aid of Figure 8 – 4, where external gain setting resistors have been added to the circuit. The buffers are 4, but because their gains equal one and they are included within the shown in Figure 8 – feedback loop, the buffer gain does not enter into the calculations. 8-3 Current-Feedback Op Amp Analysis

126 The Noninverting CFA + V IN G = 1 + V OUT G = 1 Z B IZ I – Z F Z G V A – 4. Noninverting CFA Figure 8 – Equation 8 – 7 is the current equation at the inverting 6 is the transfer equation, Equation 8 node, and Equation 8 – 8 is the input loop equation. These equations are combined to yield the closed-loop gain equation, Equation 8 – 9. (8 6) – V + IZ OUT V – V V A OUT A (8 – 7) ǒ Ǔ ǒ Ǔ + – I Z Z F G 8) – (8 V + V – IZ IN B A Z F Z 1 ) Ǔ ǒ Z G Z B ) 1 Z ǒ Ǔ F Z ø Z – 9) (8 F G V OUT + V Z IN 1 ) Z B Z 1 ) ǒ Ǔ F Z ø Z F G When the input buffer output impedance, Z , approaches zero, Equation 8 – 9 reduces to B – 10. Equation 8 8-4

127 The Inverting CFA Z F Z 1 ) Ǔ ǒ Z Z G F ) 1 (8 – 10) Z Z V F G OUT + + V Z Z IN F ) 1 ) 1 Z F Z When the transimpedance, Z, is very high, the term Z /Z in Equation 8 – 10 approaches F zero, and Equation 8 – 10 reduces to Equation 8 – 11; the ideal closed-loop gain equation for the CFA. The ideal closed-loop gain equations for the CFA and VFA are identical, and the degree to which they depart from ideal is dependent on the validity of the assumptions. The VFA has one assumption that the direct gain is very high, while the CFA has two as- sumptions, that the transimpedance is very high and that the input buffer output imped- ance is very low. As would be expected, two assumptions are much harder to meet than one, thus the CFA departs from the ideal more than the VFA does. V Z OUT F 11) – (8 ) + 1 Z V IN G 8.5 The Inverting CFA The inverting CFA configuration is seldom used because the inverting input impedance ||Z is very low (Z +Z is made dominant by selecting it as a high resistance ). When Z G G F B . Z must also be selected as a high value to achieve at value it overrides the effect of Z B F least unity gain, and high values for Z result in poor bandwidth performance, as we will F causes is selected as a low value the frequency sensitive Z see in the next section. If Z B G the gain to increase as frequency increases. These limitations restrict inverting applica- tions of the inverting CFA. + G = 1 + V OUT G = 1 Z B IZ I – V IN Z Z G F V A 5. Inverting CFA – Figure 8 8-5 Current-Feedback Op Amp Analysis

128 The Inverting CFA The current equation for the input node is written as Equation 8 12. Equation 8 – 13 de- – fines the dummy variable, V – 14 is the transfer equation for the CFA. , and Equation 8 A These equations are combined and simplified leading to Equation 8 – 15, which is the closed-loop gain equation for the inverting CFA. V – V V – V A OUT A IN – 12) (8 I ) + Z Z F G – 13) (8 + IZ V – A B (8 – 14) IZ + V OUT Z Z B Z ) 1 Ǔ ǒ G Z ø Z F G V OUT (8 – 15) +* V Z IN ) 1 Z B Z 1 ) Ǔ ǒ F Z Z ø G F When Z 15 reduces to Equation 8 approaches zero, Equation 8 – 16. – B 1 Z V G OUT 16) (8 – + – 1 V 1 IN ) Z Z F When Z is very large, Equation 8 16 becomes Equation 8 – 17, which is the ideal closed- – loop gain equation for the inverting CFA. V Z OUT F (8 17) – + – Z V IN G The ideal closed-loop gain equation for the inverting VFA and CFA op amps are identical. Both configurations have lower input impedance than the noninverting configuration has, but the VFA has one assumption while the CFA has two assumptions. Again, as was the case with the noninverting counterparts, the CFA is less ideal than the VFA because of assumption always breaks down in bipolar junction the two assumptions. The zero Z B transistors as is shown later. The CFA is almost never used in the dif ferential amplifier con- s gross input impedance mismatch. ’ figuration because of the CFA 8-6

129 Stability Analysis 8.6 Stability Analysis – 18. The stability equation is repeated as Equation 8 V TO Z A + b + V TI – 18) (8 Z B Ǔ ǒ ǒ Ǔ Z 1 ) F Z Z ø F G Comparing Equations 8 – 9 and 8 – 15 to Equation 8 – 18 reveals that the inverting and non- inverting CFA op amps have identical stability equations. This is the expected result be- cause stability of any feedback circuit is a function of the loop gain, and the input signals have no affect on stability. The two op amp parameters affecting stability are the trans- . The external components af- impedance, Z, and the input buffer ’ s output impedance, Z B and Z . The designer controls the external impedance, although fecting stability are Z G F stray capacitance that is a part of the external impedance sometimes seems to be uncon- trollable. Stray capacitance is the primary cause of ringing and overshoot in CFAs. Z and are CFA op amp parameters that can ’ t be controlled by the circuit designer, so he has Z B to live with them. Prior to determining stability with a Bode plot, we take the log of Equation 8 – 18, and plot – the logs (Equations 8 6. 19 and 8 – 20) in Figure 8 – Z B Ť Ť Ǔ ǒ Z * | b + 20 LOG |A 20 LOG |Z| 1 ) 20 LOG 19) (8 – F Z ø Z F B 1 * (8 20) – ) ( + f TANGENT A b This enables the designer to add and subtract components of the stability equation graph- ically. 120 20LOGIZI Ω (1 + Z )I IIZ /Z 20LOGIZ F B G F 61.1 58.9 Composite Curve AMPLITUDE (dB ) LOG(f) 0 1/ τ 2 τ 1/ 1 60 – – 120 = 60 ° φ M PHASE 180 – (DEGREES) 6. Bode Plot of Stability Equation – Figure 8 8-7 Current-Feedback Op Amp Analysis

130 Stability Analysis The plot in Figure 8 6 assumes typical values for the parameters: – 1M W Z + – (8 21) ǒ Ǔ Ǔǒ t S ) 1 ) t S 1 2 1 (8 – 22) Z + 70 W B (8 – 23) 1k Z W + Z + F G The transimpedance has two poles and the plot shows that the op amp will be unstable without the addition of external components because 20 LOG|Z| crosses the 0-dB axis reduce the loop gain 61.1 dB, so the circuit Z , and Z , . Z ° after the phase shift is 180 B G F is stable because it has 60 ° -phase margin. Z is the component that stabilizes the circuit. F and Z contribute little to the phase margin because Z The parallel combination of Z F G B and Z have little effect on stability. is very small, so Z G B The manufacturer determines the optimum value of R during the characterization of the F exceeds the optimum value recom- – 6, it is seen that when R IC. Referring to Figure 8 F mended by the IC manufacturer, stability increases. The increased stability has a price is less than the optimum value recom- called decreased bandwidth. Conversely, when R F mended by the IC manufacturer, stability decreases, and the circuit response to step in- puts is overshoot or possibly ringing. Sometimes the overshoot associated with less than is tolerated because the bandwidth increases as R decreases. The peaked optimum R F F associated with less than optimum values of R can be used to compensate for response F cable droop caused by cable capacitance. . Under these conditions = Z/R = 0 Ω and Z β = R the loop gain equation is; A When Z F B F F Z and R can always be found to stabilize the circuit. determine stability, and a value of R F F The transimpedance and feedback resistor have a major impact on stability, and the input increases with an in- buffer ’ s output impedance has a minor effect on stability. Since Z B crease in frequency, it tends to increase stability at higher frequencies. Equation 8 – 18 is rewritten as – Equation 8 24, but it has been manipulated so that the ideal closed-loop gain is readily apparent. Z A b + R 24) (8 – F Ǔ ǒ 1 ) ) Z Z B F R G The closed-loop ideal gain equation (inverting and noninverting) shows up in the denomi- – 24, so the closed-loop gain influences the stability of the op amp. nator of Equation 8 approaches zero, the closed-loop gain term also approaches zero, and the op When Z B de- amp becomes independent of the ideal closed-loop gain. Under these conditions R F termines stability, and the bandwidth is independent of the closed-loop gain. Many people claim that the CFA bandwidth is independent of the gain, and that claim ’ s validity is depen- /Z being very low. dent on the ratios Z B F 8-8

131 Selection of the Feedback Resistor is given be- is important enough to warrant further investigation, so the equation for Z Z B B low. b s ȡ ȣ 0 1 ) w R T B ȧ ȧ ^ Z ) h 25) – (8 ib B ȧ ȧ b 1 ) S b 0 0 ) 1 ǒ Ǔ b 1 w ) Ȣ 0 T At low frequencies h . Z = 50 Ω varies in accordance Ω /( β = 75 +1) = 25 Ω , so Z and R B 0 B ib B – 25 with Equation 8 – 25 at high frequencies. Also, the transistor parameters in Equation 8 is vary with transistor type; they are different for NPN and PNP transistors. Because Z B the output transistors being used, and this is a function of the quadrant the dependent on is a small factor in the equation, has an extremely wide variation. Z output signal is in, Z B B but it adds a lot of variability to the current-feedback op amp. 8.7 Selection of the Feedback Resistor and it af fects closed-loop bandwidth, so it must The feedback resistor determines stability, be selected very carefully. Most CFA IC manufacturers employ applications and product non- . They measure each who spend a great deal of time and ef fort selecting R engineers F inverting gain with several different feedback resistors to gather data. Then they pick a that yields stable operation with acceptable peaking, and that compromise value of R F is recommended on the data sheet for that specific gain. This procedure is value of R F repeated for several different gains in anticipation of the various gains their customer ap- or the gain is changed from plications require (often G = 1, 2, or 5). When the value of R F the values recommended on the data sheet, bandwidth and/or stability is affected. When the circuit designer must select a different R value from that recommended on the F decreases stabil- data sheet he gets into stability or low bandwidth problems. Lowering R F decreases bandwidth. What happens when the designer needs to ity, and increasing R F operate at a gain not specified on the data sheet? The designer must select a new value for the new gain, but there is no guarantee that new value of R is an optimum value. of R F F selection problem is to assume that the loop gain, A β , is a linear One solution to the R F ) for for a gain of one equals (A β ) function. Then the assumption can be made that (A β N 1 a gain of N, and that this is a linear relationship between stability and gain. Equations 8 – 26 27 are based on the linearity assumption. – and 8 8-9 Current-Feedback Op Amp Analysis

132 Selection of the Feedback Resistor Z Z + 26) (8 – Z Z F1 FN ǒ Ǔ Ǔ ǒ Z ) 1 ) Z 1 ) ) Z Z B F1 FN B Z Z GN G1 Z Z F1 FN ǒ Ǔ ǒ Ǔ Z ǒ ) 1 + Z Ǔ ) 1 Z * ) – 27) (8 B F1 FN Z Z G1 GN can easily be chosen for each Equation 8 – 27 leads one to believe that a new value for Z F new gain. This is not the case in the real world; the assumptions don ’ t hold up well enough to rely on them. When you change to a new gain not specified on the data sheet, Equation , but you must test to determine the final value – 8 27, at best, supplies a starting point for R F . of R F When the R ’ t be used, an alternate method value recommended on the data sheet can F is to use graphical techniques. The graph shown in Fig- for R of selecting a starting value F 7 is a plot of the typical 300-MHz CFA data given in Table 8 – 1. ure 8 – GAIN and BANDWIDTH vs FEEDBACK RESISTOR 10 130 Gain vs. 9 120 Feedback Resistance 8 110 7 100 Bandwidth MHz – 6 90 vs. Feedback Gain Resistance 5 80 Bandwidth 4 70 3 60 2 50 1 40 400 100 900 0 700 800 500 600 200 300 1k Ω Feedback Resistor – , G, and BW 7. Plot of CFA R Figure 8 – F 8-10

133 Stability and Input Capacitance Table 8 1. Data Set for Curves in Figure 8 – 7 – Ω R GAIN (A ( ) ) BANDWIDTH (MHz) CL F + 1 1000 125 + 2 681 95 + 10 65 383 Enter the graph at the new gain, say A = 6, and move horizontally until you reach the CL intersection of the gain versus feedback resistance curve. Then drop vertically to the re- in this example). Enter the graph at Ω (500 sistance axis and read the new value of R F the new value of R , and travel vertically until you intersect the bandwidth versus feedback F resistance curve. Now move to the bandwidth axis to read the new bandwidth (75 MHz in this example). As a starting point you should expect to get approximately 75 MHz BW . Although this technique yields more reliable solutions = 500 Ω with a gain of 6 and R F than 27 does, op amp peculiarities, circuit board stray capacitances, and wir- Equation 8 – ing make extensive testing mandatory. The circuit must be tested for performance and stability at each new operating point. 8.8 Stability and Input Capacitance When designer lets the circuit board introduce stray capacitance on the inverting input to become reactive. The new impedance, node to ground, it causes the impedance Z G Z , is given in Equation 8 – 28, and Equation 8 – 29 is the stability equation that describes G the situation. R G Z + (8 28) – G 1 ) R s C G G Z + A b Z (8 – 29) F ) Z B 2 Z Z ) Z B G G Z A b + 30) (8 – R B Ǔ ǒ ǒ Ǔ R C R ø R 1 ) s 1 ) ø R B F F G G R ø R F G – 29 is the stability equation when Z Equation 8 consists of a resistor in parallel with stray G , is capacitance between the inverting input node and ground. The stray capacitance, C G a fixed value because it is dependent on the circuit layout. The pole created by the stray because it dominates R and R . R fluctuates with capacitance is dependent on R G B F B C pole placement is subject to IC manufacturing tol- manufacturing tolerances, so the R B G combination becomes larger, the pole moves towards the zero fre- C erances. As the R B G 8-11 Current-Feedback Op Amp Analysis

134 Stability and Feedback Capacitance quency axis, lowering the circuit stability. Eventually it interacts with the pole contained in Z, 1/ τ , and instability results. 2 The effects of stray capacitance on CFA closed-loop performance are shown in Figure – 8 8. AMPLITUDE vs FREQUENCY = 2 pF C IN = 2 pF C F No Stray Amplitude (3 dB/div) Capacitance 100 10 1 MHz Frequency – – f 8. Effects of Stray Capacitance on CFAs – Figure 8 causes more than 3 dB peaking in the CFA frequency Notice that the introduction of C G response plot, and it increases the bandwidth about 18 MHz. Two picofarads are not a lot of capacitance because a sloppy layout can easily add 4 or more picofarads to the cir- cuit. 8.9 Stability and Feedback Capacitance When a stray capacitor is formed across the feedback resistor, the feedback impedance 32 gives the loop gain when a feedback capacitor is given by Equation 8 – 31. Equation 8 – has been added to the circuit. 8-12

135 Compensation of CF and CG R F – (8 31) + Z F ) R C s 1 F F Ǔ ǒ s 1 ) R C Z F F + A b (8 – 32) R B Ǔ ǒ ǒ Ǔ s 1 R ø ) R C 1 ) R R ø B F F F G R ø R F G This loop gain transfer function contains a pole and zero, thus, depending on the pole/zero – 9. The placement, oscillation can result. The Bode plot for this case is shown in Figure 8 original and composite curves cross the 0-dB axis with a slope of – 40 dB/decade, so either curve can indicate instability. The composite curve crosses the 0-dB axis at a higher fre- quency than the original curve, hence the stray capacitance has added more phase shift to the system. The composite curve is surely less stable than the original curve. Adding capacitance to the inverting input node or across the feedback resistor usually results in , thus here is largely influences the location of the pole introduced by C instability. R F B another case where stray capacitance leads to instability. )I /Z IIZ 20LOGIZI – 20LOGIZ (1 + Z G F B F Ω POLE/ZERO Curve Composite Curve AMPLITUDE (dB ) 0 LOG(f) f f Z P Figure 8 9. Bode Plot with C – F Figure 8 – 8 shows that C = 2 pF adds about 4 dB of peaking to the frequency response F and C are the plot. The bandwidth increases about 10 MHz because of the peaking. C F G major causes of overshoot, ringing, and oscillation in CFAs, and the circuit board layout must be carefully done to eliminate these stray capacitances. 8.10 Compensation of C and C G F When C and C both are present in the circuit they may be adjusted to cancel each other F G – 33. is Equation 8 and C out. The stability equation for a circuit with C F G 8-13 Current-Feedback Op Amp Analysis

136 Summary Ǔ ǒ s C R 1 ) Z F F + b A 33) – (8 R B Ǔ ǒ Ǔ ǒ Ǔ ǒ 1 ) s 1 C ) ø R ) ø R C R R F B F F G G R ø R F G If the zero and pole in Equation 8 – 33 are made to cancel each other, the only poles re- 33 equal yields Equation 8 maining are in Z. Setting the pole and zero in Equation 8 – 34 – after some algebraic manipulation. Ǔ ǒ R R R C C ø + 34) – (8 F B F G G R and R , so Equation 8 – 34 is reduced to dominates the parallel combination of R B G B – Equation 8 35. (8 – 35) C C R + R F B F G R is an IC parameter, so it is dependent on the IC process. R it is an important IC param- B B eter, but it is not important enough to be monitored as a control variable during the has widely spread, unspecified parameters, thus depending manufacturing process. R B for compensation is risky. Rather, the prudent design engineer assures that the cir- on R B , and that the resulting frequency re- cuit will be stable for any reasonable value of R B sponse peaking is acceptable. 8.11 Summary Constant gain-bandwidth is not a limiting criterion for the CFA, so the feedback resistor is adjusted for maximum performance. Stability is dependent on the feedback resistor; is decreased, stability is decreased, and when R goes to zero the circuit becomes as R F F unstable. As R is increased stability increases, but the bandwidth decreases. F The inverting input impedance is very high, but the noninverting input impedance is very low. This situation precludes CFAs from operation in the differential amplifier configura- tion. Stray capacitance on the inverting input node or across the feedback resistor always leads to peaking, usually to ringing, and sometimes to oscillations. A prudent circuit de- signer scans the PC board layout for stray capacitances, and he eliminates them. Bread- boarding and lab testing are a must with CFAs. The CFA performance can be improved with a good layout, good decoupling capacitors, and low inductance com- immeasurably ponents. 8-14

137 Chapter 9 Voltage- and Current-Feedback Op Amp Comparison Ron Mancini and James Karki 9.1 Introduction The name, operational amplifier, was given to voltage-feedback amplifiers (VFA) when they were the only op amps in existence. These new (they were new in the late ’ 40s) ampli- fiers could be programmed with external components to perform various math operations on a signal; thus, they were nicknamed op amps. Current-feedback amplifiers (CFA) have been around approximately twenty years, but their popularity has only increased in the last several years. Two factors limiting the popularity of CFAs is their application dif ficulty and lack of precision. The VFA is familiar component, and there are several variations of internally compen- sated VFAs that can be used with little applications work. Because of its long history, the VFA comes in many varieties and packages, so there are VFAs applicable to almost any job. VFA bandwidth is limited, so it can ’ t function as well at high signal frequencies as the CFA can. For now, the signal frequency and precision separates the applications of the two op amp configurations. The VFA has some other redeeming virtues, such as excellent precision, that makes it the desirable amplifier in low frequency applications. Many functions other than signal amplification are accomplished at low frequencies, and functions like level-shifting a sig- nal require precision. Fortunately, precision is not required in most high frequency ap- plications where amplification or filtering of a signal is predominant, so CFAs are suitable to high frequency applications. The lack of precision coupled with the application difficul- ties prevents the CFA from replacing the VFA. 9-1

138 Precision 9.2 Precision The long-tailed pair input structure gives the VFA its precision; the long-tailed pair is shown in Figure 9 – 1 V V CC CC R R C C V O2 V O1 V V 2 1 I – V EE Figure 9 – 1. Long-Tailed Pair The transistors, Q and Q , are very carefully matched for initial and drift tolerances. Care- 1 2 ful attention is paid to detail in the transistor design to insure that parameters like current and , are matched between the input transistors, Q gain, β , and base-emitter voltage, V BE 1 Q the current, I, splits equally between the transistors, and V = V . When V . = V B2, O1 2 O2 B1 As long as the transistor parameters are matched, the collector currents stay equal. The causes a mismatch in the collector currents with respect to V slightest change of V B1 B2 V |. – and a differential output voltage |V B1 B2 When temperature or other outside influences change transistor parameters like current gain or base-emitter voltage, as long as the change is equal, it causes no change in the differential output voltage. IC designers go to great lengths to ensure that transistor pa- rameter changes due to external influences do not cause a differential output voltage change. Now, the slightest change in either base voltage causes a differential output volt- age change, and gross changes in external conditions do not cause a differential output voltage change. This is the formula for a precision amplifier because it can amplify small input changes while ignoring changes in the parameters or ambient conditions. This is a simplified explanation, and there are many different techniques used to ensure transistor matching. Some of the techniques used to match input transistors are parame- ter trimming, special layout techniques, thermal balancing, and symmetrical layouts. The long-tailed pair is an excellent circuit configuration for obtaining precision in the input cir- cuit, but the output circuit has one fault. The output circuit collector impedance has to be high to achieve high gain in the first stage. High impedance coupled with the Miller capaci- tance discussed in Chapter 7 forms a quasidominant pole compensation circuit that has poor high frequency response. 9-2

139 Bandwidth 2) connects to a buffer input inside the – The noninverting input of the CFA (see Figure 9 op amp. The inverting input of the CFA connects to a buffer output inside the CFA. Buffer inputs and outputs have dramatically different impedance levels, so any matching be- moot point. The buf fer can ’ t reject common-mode voltages introduced by param- comes a drifts because it has no common-mode rejection capability. The input current causes eter , and there is no way that a voltage drop across the input buffer ’ s output impedance, R B this voltage drop can be distinguished from an input signal. NONINVERTING INPUT + G = 1 OUTPUT R B Z I I INVERTING INPUT – 2. Ideal CFA – Figure 9 The CFA circuit configuration was selected for high frequency amplification because it has current-controlled gain and a current-dominant input. Being a current device, the CFA does not have the Miller-effect problem that the VFA has. The input structure of the CFA sacrifices precision for bandwidth, but CFAs achieve usable bandwidths ten times the us- able VFA bandwidth. 9.3 Bandwidth The bandwidth of a circuit is defined by high frequency errors. When the gain falls off at high frequencies unequal frequency amplification causes the signal to become distorted. The signal loses its high frequency components; an example of high frequency signal degradation is a square wave with sharp corners that is amplified and turned into slump cornered semi sine wave. The error equation for any feedback circuit is repeated in Equa- tion 9 – 1. V IN 1) – (9 + E 1 ) A b This equation is valid for any feedback circuit, so it applies equally to a VFA or a CFA. The 2. – loop gain equation for any VFA is repeated as Equation 9 9-3 Voltage- and Current-Feedback Op Amp Comparison

140 Bandwidth aR G – 2) (9 A b + R R ) F G Equation 9 – 2 is rewritten below as Equations 9 – 3 and 9 – 4 for the noninverting and invert- represent the closed and G ing circuits respectively. In each case, the symbol G CLNI CLI loop gain for the noninverting and inverting circuits respectively. a a A b + + G ) R R – (9 3) F G CLNI R G a a + + b A 4) – (9 ) 1 G R ) R F G CLI R G In both cases the loop gain decreases as the closed loop gain increases, thus all VFA er- rors increase as the closed loop gain increases. The error increase is mathematically coupled to the closed loop gain equation, so there is no working around this fact. For the VFA, effective bandwidth decreases as the closed loop gain increases because the loop gain decreases as the closed loop gain increases. 3. Referring A plot of the VFA loop gain, closed loop gain, and error is given in Figure 9 – to Figure 9 – 3, the direct gain, A, is the op amp open loop gain, a, for a noninverting op ))). The Miller effect causes + Z /(Z The direct gain for an inverting op amp is (a(Z amp. F F G the direct gain to fall off at high frequencies, thus error increases as frequency increases because the ef fective loop gain decreases. At a given frequency, the error also increases when the closed loop gain is increased. 20 LOG A Direct Gain β ) 20 LOG (1 + A Gain in dB 20 LOG(G ) CL CLOSED LOOP GAIN f LOG Figure 9 – 3. VFA Gain versus Frequency 9-4

141 Bandwidth The CFA is a current operated device; hence, it not nearly as subject to the Miller effect resulting from stray capacitance as the VFA is. The absence of the Miller effect enables the CFA ’ s does. A plot of the CFA s frequency response to hold up far better than the VFA ’ loop gain, transimpedance, and error is given in Figure 9 – 4. Notice that the transimpe- dance stays at the large low frequency intercept value until much higher frequencies than the VFA does. TRANSIMPEDANCE 20 LOG Z 20 LOG (1 + A ) β Gain in dB 20 LOG(G ) CL CLOSED LOOP GAIN f LOG Figure 9 – 4. CFA Gain vs Frequency The loop gain equation for the CFA is repeated here as Equation 9 – 5. Z + A b R 5) – (9 B ǒ Ǔ ) 1 R F R ø R F G 5 reduces to Equa- fer output resistance approaches zero, Equation 9 When the input buf – tion 9 – 6. Z 6) (9 – A b + R F – 6 shows that the closed-loop gain has no effect on the loop gain when R = Equation 9 B 0, so under ideal conditions one would expect the transimpedance to fall off with a zero slope. Figure 9 – 4 shows that there is a finite slope, but much less than that of a VFA, and Ω is usually 50 not being equal to zero. For example, R when the slope is caused by R B B = 50/500 ||R /R = 1000 Ω at A , and R = 1. If we let R Ω = R = 500 then R ||R R F G G, F CL B F F G = 0.1. 9-5 Voltage- and Current-Feedback Op Amp Comparison

142 Stability 7 is almost this value into Equation 9 – 6 yields Equation 9 – 7, and Equation 9 – Substituting identical to Equation 9 – 6. R does cause some interaction between the loop gain and the B transimpedance, but because the interaction is secondary the CFA gain falls off with a faster slope. Z (9 – 7) A b + 1.1 R F The direct gain of a VFA starts falling off early, often at 10 Hz or 100 Hz, but the transimpe- dance of a CFA does not start falling off until much higher frequencies. The VFA is constrained by the gain-bandwidth limitation imposed by the closed loop gain being incor- , does not porated within the loop gain. The CFA, with the exception of the effects of R B have this constraint. This adds up to the CFA being the superior high frequency amplifier. 9.4 Stability Stability in a feedback system is defined by the loop gain, and no other factor, including the inputs or type of inputs, affects stability. The loop gain for a VFA is given in Equation 9 – 2. Examining Equation 9 – 2 we see that the stability of a VFA is depends on two items; . /Z the op amp transfer function, a, and the gain setting components, Z F G The op amp contains many poles, and if it is not internally compensated, it requires exter- nal compensation. The op amp always has at least one dominant pole, and the most phase margin that an op amp has is 45 ° . Phase margins beyond 60 ° are a waste of op they can compensate and Z amp bandwidth. When poles and zeros are contained in Z F G, for the op amp phase shift or add to its instability. In any case, the gain setting components always affect stability. When the closed-loop gain is high, the loop gain is low, and low loop gain circuits are more stable than high loop gain circuits. Wiring the op amp to a printed circuit board always introduces components formed from stray capacitance and inductance. Stray inductance becomes dominant at very high fre- quencies, hence, in VFAs, it does not interfere with stability as much as it does with signal handling properties. Stray capacitance causes stability to increase or decrease depend- ing on its location. Stray capacitance from the input or output lead to ground induces insta- bility, while the same stray capacitance in parallel with the feedback resistor increases stability. , is given in Equation The loop gain for a CFA with no input buffer output impedance, R B 9 – 6. Examining Equation 9 – 6 we see that the stability of a CFA depends on two items: . The op amp contains the op amp transfer function, Z, and the gain setting component, Z F many poles, thus they require external compensation. Fortunately, the external com- . The factory applications engineer does extensive pensation for a CFA is done with Z F for a given gain. This value should be used testing to determine the optimum value of R F in all applications at that gain, but increased stability and less peaking can be obtained . Essentially this is sacrificing bandwidth for lower frequency perfor- by increasing R F mance, but in applications not requiring the full bandwidth, it is a wise tradeoff. 9-6

143 Impedance The CFA stability is not constrained by the closed-loop gain, thus a stable operating point can be found for any gain, and the CFA is not limited by the gain-bandwidth constraint. If the optimum feedback resistor value is not given for a specific gain, one must test to find the optimum feedback resistor value. Stray capacitance from any node to ground adversely affects the CFA performance. Stray capacitance of just a couple of pico Farads from any node to ground causes 3 dB or more of peaking in the frequency response. Stray capacitance across the CFA feedback resis- tor, quite unlike that across the VFA feedback resistor, always causes some form of insta- bility. CFAs are applied at very high frequencies, so the printed circuit board inductance associated with the trace length and pins adds another variable to the stability equation. Inductance cancels out capacitance at some frequency, but this usually seems to happen in an adverse manner. The wiring of VFAs is critical, but the wiring of CFAs is a science. Stay with the layout recommended by the manufacturer whenever possible. 9.5 Impedance The input impedance of a VFA and CFA dif fer dramatically because their circuit configura- tions are very different. The VFA input circuit is a long-tailed pair, and this configuration gives the advantages that both input impedances match. Also, the input signal looks into an emitter-follower circuit that has high input impedance. The emitter-follower input im- is a discrete emitter resistor. At low input currents, R + R ) where R is pedance is β (r E e E E very high and the input impedance is very high. If a higher input impedance is required, 2 + R (r ). β the op amp uses a Darlington circuit that has an input impedance of e E So far, the implicit assumption is that the VFA is made with a bipolar semiconductor pro- cess. Applications requiring very high input impedances often use a FET process. Both and CMOS processes offer very high input impedance in any long-tailed pair con- BIFET figuration. It is easy to get matched and high input impedances at the amplifier inputs. Do not confuse the matched input impedance at the op amp leads with the overall circuit input , and the imped- impedance. The input impedance looking into the inverting input is R G looking into the noninverting input is the input impedance of the op amp. While these ance ferent impedances, they are mismatched because of the circuit not the op amp. are two dif The CFA has a radically different input structure that causes it to have mismatched input impedances. The noninverting input lead of the CFA is the input of a buffer that has very high input impedance. The inverting input lead is the output of a buffer that has very low impedance. There is no possibility that these two input impedances can be matched. . Once the circuit Again, because of the circuit, the inverting circuit input impedance is R G is to increase R . But, R is determined by a gain is fixed, the only way to increase R F F G tradeoff between stability and bandwidth. The circuit gain and bandwidth requirements , hence there is no room to further adjust R to raise the resistance of R . If the fix R F F G when the closed-loop gain is two, then Ω = 100 ’ s data sheet says that R manufacturer F 9-7 Voltage- and Current-Feedback Op Amp Comparison

144 Equation Comparison R = 100 Ω or 50 Ω depending on the circuit configuration. This sets the circuit input im- G . This analysis is not entirely accurate because R pedance at 100 Ω adds to the input im- B pedance, but this addition is very small and dependent on IC parameters. CFA op amp circuits are usually limited to noninverting voltage applications, but they serve very well in inverting applications that are current-driven. The CFA is limited to the bipolar process because that process offers the highest speed. The option of changing process to BIFET or CMOS to gain increased input impedance is not attractive today. Although this seems like a limiting factor, it is not because CFAs . Also, are often used in low impedance where the inputs are terminated in 50 Ω or 75 Ω most very high-speed applications require low impedances. 9.6 Equation Comparison The pertinent VFA and CFA equations are repeated in Table 9 1. Notice that the ideal – closed-loop gain equations for the inverting and noninverting circuits are identical. The ideal equations for the VFA depend on the op amp gain, a, being very large thus making β large compared to one. The CFA needs two assumptions to be valid to obtain the ideal A equations. First, the ideal equations for the CFA depend on the op amp transimpedance, must be very small Z, being very large thus making A β large compared to one. Second, R B ||Z . compared to Z F G The ideal gain equations are identical, but the applications are very different because the VFA is best applied to lower frequency precision jobs while the CFA applications are in the very high frequency realm. The transimpedance in a CFA acts much like the gain does in a VFA. In each case, transimpedance or gain, it is the parameter that enables the use of feedback. 9-8

145 Equation Comparison Table 9 – 1. Tabulation of Pertinent VFA and CFA Equations CURRENT VOLTAGE CIRCUIT FEEDBACK AMPLIFIER CONFIGURATION FEEDBACK AMPLIFIER NONINVERTING Ǔ ǒ Z 1 ) Z ń Z Forward or direct F G a gain ǒ Ǔ ) Z ń Z ø Z 1 Z B F F G Z B Z 1 ) ǒ Ǔ F Z G Z B 1 ) Z ǒ Ǔ F a Z ø Z F G Actual closed loop aZ G gain ) 1 Z Z ø Z ) 1 F G Z B ȡ ȣ Z ) 1 B Z ) 1 F ȧ ȧ ø Z Z F G Ȣ Z Closed loop gain 1 ) Z ń ń Z Z 1 ) F F G G INVERTING Z Forward or direct Ǔ ǒ Z ) Z aZ ń ǒ Ǔ F F G gain Z Z ø Z Z ) 1 ń F B G G Ǔ ǒ ǒ Ǔ Ideal loop gain ń Z ń Z ø Z Z ) aZ Z ń 1 Z ) Z F F F B G G G Z B Ǔ ǒ * Z 1 ) G aZ * F Z Z ø F G Z Z ) Actual closed loop F G Z gain aZ ) 1 G 1 ) Z B Z ø Z Z ) 1 F G Ǔ ǒ F ø Z Z F G Z Closed loop gain * Z Z ń Z * ń F F G G 9-9 Voltage- and Current-Feedback Op Amp Comparison

146 9-10

147 Chapter 10 Op Amp Noise Theory and Applications Bruce Carter 10.1 Introduction The purpose of op amp circuitry is the manipulation of the input signal in some fashion. Unfortunately in the real world, the input signal has unwanted noise superimposed on it. Noise is not something most designers get excited about. In fact, they probably wish the topic would go away. It can, however, be a fascinating study by itself. A good under- whole standing of the underlying principles can, in some cases, be used to reduce noise in the design. 10.2 Characterization Noise is a purely random signal, the instantaneous value and/or phase of the waveform cannot be predicted at any time. Noise can either be generated internally in the op amp, from its associated passive components, or superimposed on the circuit by external sour- ces. External noise is covered in Chapter 17, and is usually the dominant effect. 10.2.1 rms versus P-P Noise Instantaneous noise voltage amplitudes are as likely to be positive as negative. When plotted, they form a random pattern centered on zero. Since noise sources have ampli- tudes that vary randomly with time, they can only be specified by a probability density function. The most common probability density function is Gaussian. In a Gaussian prob- ability function, there is a mean value of amplitude, which is most likely to occur. The probability that a noise amplitude will be higher or lower than the mean falls off in a bell- shaped curve, which is symmetrical around the center (Figure 10 – 1). 10-1

148 Characterization Mean Value 3 +1 +2 +3 – – 1 – 2 Noise Signal rms Value 99.7% Probability Signal Will Be <= 6 X rms Value Figure 10 – 1. Gaussian Distribution of Noise Energy σ is the standard deviation of the Gaussian distribution and the rms value of the noise volt- age and current. The instantaneous noise amplitude is within ± 1 σ 68% of the time. Theoretically, the instantaneous noise amplitude can have values approaching infinity. the probability falls off rapidly as amplitude increases. The instantaneous noise However, ± 3 σ of the mean 99.7% of the time. If more or less assurance is de- amplitude is within σ sired, it is between ± 2 σ 95.4% of the time and ± 3.4 99.94% of the time. 2 is the average mean-square variation about the average value. This also means that σ 2 2 , is the same as the or e the average mean-square variation about the average value, i 2 variance σ . Thermal noise and shot noise (see below) have Gaussian probability density functions. The other forms of noise do not. 10-2

149 Characterization 10.2.2 Noise Floor all input sources are turned off and the output is properly terminated, there is a level When that determines the smallest signal for which the circuit is noise floor of noise called the useful. The objective for the designer is to place the signals that the circuit processes above the noise floor, but below the level where the signals will clip. 10.2.3 Signal-to-Noise Ratio The noisiness of a signal is defined as: S rms signal voltage (f) 1) – (10 + N rms noise voltage (f) In other words, it is a ratio of signal voltage to noise voltage (hence the name signal-to- noise ratio ). 10.2.4 Multiple Noise Sources When there are multiple noise sources in a circuit, the total root-mean-square (rms) noise signal that results is the square root of the sum of the average mean-square values of the individual sources: 2 2 2 Ǹ 2) – (10 e + ) e e E ) AAA n rms Totalrms 1rms 2rms Put another way, this is the only “ break ” that the designer gets when dealing with noise. If there are two noise sources of equal amplitude in the circuit, the total noise is not dB). It only increases by 3 dB. Consider a very simple case, two doubled (increased by 6 : noise sources with amplitudes of 2 V rms Ǹ 2 2 Ǹ 3) – (10 8 ) 2 + 2 + 2.83 V E + rms Totalrms Therefore, when there are two equal sources of noise in a circuit, the noise is 2.83 20 log + instead 3.01 dB higher than if there were only one source of noise — 2 of double (6 dB) as would be intuitively expected. This relationship means that the worst noise source in the system will tend to dominate and another is the total noise. Consider a system in which one noise source is 10 V rms : 1 V rms Ǹ 2 2 Ǹ 4) – (10 E 10.05 V ) 1 + + + 101 10 rms Totalrms There is hardly any effect from the 1-V noise source at all! 10-3 Op Amp Noise Theory and Applications

150 Types of Noise 10.2.5 Noise Units Noise is normally specified as a spectral density in rms volts or amps per root Hertz, Ǹ Ǹ V ń Hz Hz ń or A “ user-friendly ” units. A frequency range is needed . These are not very to relate these units to actual noise levels that will be observed. For example: Ǹ A TLE2027 op amp with a noise specification of 2.5 nV Hz D ń is used over an audio frequency range of 20 Hz to 20 kHz, with a gain of 40 dB. The output voltage is 0 dBV (1 V). Ǹ To begin with, calculate the root Hz part: 20000 * 20 D + 141.35. D Multiplying this by the noise spec: 2.5 141.35 + 353.38 nV, which is the equivalent input noise (EIN). The output noise equals the input noise multiplied by the gain, which is 100 (40 dB). The signal-to-noise ratio can be now be calculated: D V m 353.38 nV 100 + 35.3 D Signal-to-noise (dB) = (10 – 5) 20 log(1V B 35.3 m V) + 20 log(28329) + 89 dB The TLE2027 op amp is an excellent choice for this application. Remember, though, that passive components and external noise s ources can degrade performance. There is also a slight increase in noise at low frequencies, due to the 1/f effect (see below). 10.3 Types of Noise There are five types of noise in op amps and associated circuitry: 1) Shot noise 2) Thermal noise 3) Flicker noise 4) Burst noise 5) Avalanche noise Some or all of these noises may be present in a design, presenting a noise spectrum unique to the system. It is not possible in most cases to separate the effects, but knowing general causes may help the designer optimize the design, minimizing noise in a particu- be- lar bandwidth of interest. Proper design for low noise may involve a “ balancing act ” tween these sources of noise and external noise sources. 10-4

151 Types of Noise 10.3.1 Shot Noise The name shot noise is short for Schottky noise. Sometimes it is referred to as quantum noise . It is caused by random fluctuations in the motion of charge carriers in a conductor. Put another way, current flow is not a continuous effect. Current flow is electrons, charged particles that move in accordance with an applied potential. When the electrons encoun- ter a barrier, potential energy builds until they have enough energy to cross that barrier. When they have enough potential energy, it is abruptly transformed into kinetic energy as they cross the barrier. A good analogy is stress in an earthquake fault that is suddenly released as an earthquake. As each electron randomly crosses a potential barrier, such as a pn junction in a semicon- ductor, energy is stored and released as the electron encounters and then shoots across pop as its stored energy is released when it the barrier. Each electron contributes a little crosses the barrier (Figure 10 – 2). – Figure 10 2. Shot Noise Generation The aggregate effect of all of the electrons shooting across the barrier is the shot noise. Amplified shot noise has been described as sounding like lead shot hitting a concrete wall. Some characteristics of shot noise: D Shot noise is always associated with current flow. It stops when the current flow stops. D Shot noise is independent of temperature. 10-5 Op Amp Noise Theory and Applications

152 Types of Noise D Shot noise is spectrally flat or has a uniform power density, meaning that when plotted versus frequency it has a constant value. Shot noise is present in any conductor — not just a semiconductor. Barriers in D conductors can be as simple as imperfections or impurities in the metal. The level of shot noise, however, is very small due to the enormous numbers of electrons moving in the conductor, and the relative size of the potential barriers. Shot noise in semiconductors is much more pronounced. The rms shot noise current is equal to: Ǹ 6) – (10 I + (2qI ) 4qI )B 0 dc sh Where: – 19 q = Electron charge (1.6 x 10 coulombs) Average forward dc current in A = I dc = Reverse saturation current in A I o B = Bandwidth in Hz If the pn junction is forward biased, I is zero, and the second term disappears. Using o Ohm ’ s law and the dynamic resistance of a junction, kT (10 – 7) + r d qI dc the rms shot noise voltage is equal to: 2B – 8) (10 + kT E Ǹ sh qI dc Where: – 23 Joules/ K) _ ’ s constant (1.38 x 10 k = Boltzmann – 19 Electron charge (1.6 x 10 q = coulombs) _ K T = Temperature in I = Average dc current in A dc Bandwidth in Hz B = For example, a junction carries a current of 1 mA at room temperature. Its noise over the audio bandwidth is: 20) * 2(20000 23 * (10 – 9) Ǹ 144 dBV 10 298 1.38 + – + 65 nV + E sh * 3 19 * ) ) (1 10 10 (1.6 Obviously, it is not much of a problem in this example. Look closely at the formula for shot noise voltage. Notice that the shot noise voltage is inversely proportional to the current. Stated another way, shot noise voltage decreases 10-6

153 Types of Noise as average dc current increases, and increases as average dc current decreases. This can be an elegant way of determining if shot noise is a dominant effect in the op amp circuit being designed. If possible, decrease the average dc current by a factor of 100 and see if the overall noise increases by a factor of 10. In the example above: 2(20000 * 20) * 23 10) (10 – Ǹ 10 124 dBV + 298 – + + 650 nV 1.38 E sh * * 19 5 ) 10 ) (1 (1.6 10 The shot noise voltage does increase by a factor of 10, or 20 dB. 10.3.2 Thermal Noise Thermal noise is sometimes referred to as Johnson noise after its discoverer. It is gener- ated by thermal agitation of electrons in a conductor. Simply put, as a conductor is heated, it will become noisy. Electrons are never at rest; they are always in motion. Heat disrupts response to an applied potential. It adds a random component to their mo- ’ the electrons 3). Thermal noise only stops at absolute zero. – tion (Figure 10 Figure 10 – 3. Thermal Noise shot noise, thermal noise is spectrally flat or has a uniform power density (it is Like white ), but thermal noise is independent of current flow. At frequencies below 100 MHz, thermal noise can be calculated using Nyquist ’ s relation: Ǹ (10 – 11) E + 4kTRB th or 10-7 Op Amp Noise Theory and Applications

154 Types of Noise 4kTB – 12) (10 Ǹ I + th R Where: Thermal noise voltage in Volts rms E th = I = Thermal noise current in Amps rms th – 23 ) Boltzmann k = ’ s constant (1.38 x 10 T = Absolute temperature (Kelvin) R = Resistance in ohms B = Noise bandwidth in Hertz (f ) f – max min The noise from a resistor is proportional to its resistance and temperature. It is important not to operate resistors at elevated temperatures in high gain input stages. Lowering re- sistance values also reduces thermal noise. For example: K) over the audio frequency range of 20 Hz The noise in a 100 k Ω resistor at 25 _ C (298 _ to 20 kHz is: Ǹ E + 4kTRB th * 23 Ǹ + (1.38 10 20) * ) 4 298 100, 000 (20, 000 – 13) (10 V m + 5.73 + 104.8 dBV – Decreasing the temperature would reduce the noise slightly, but scaling the resistor down Ω (a factor of 100) would reduce the thermal noise by 20 dB. Similarly, increasing to 1 k Ω would increase the thermal noise to – the resistor to 10 M 84.8 dBV, a level that would affect a 16-bit audio circuit. The noise from multiple resistors adds according to the root- mean-square law in Paragraph 10.2.4. Beware of large resistors used as the input resistor of an op amp gain circuit, their thermal noise will be amplified by the gain in the circuit (Paragraph 10.4). Thermal noise in resistors is often a problem in portable equipment, where resistors have been scaled up to get power consumption down. 10.3.3 Flicker Noise Flicker noise is also called 1/f noise . Its origin is one of the oldest unsolved problems in pervasive in nature and in many human endeavors. It is present in all active physics. It is and many passive devices. It may be related to imperfections in crystalline structure of semiconductors, as better processing can reduce it. 10-8

155 Types of Noise Some characteristics of flicker noise: D It increases as the frequency decreases, hence the name 1/f D It is associated with a dc current in electronic devices D It has the same power content in each octave (or decade) f f max max – 14) (10 Ǔ ǒ Ǔ ǒ K + ln E ln I Ǹ + K Ǹ v n n i f f min min Where: are proportionality constants (volts or amps) representing E and I and K K e i n n at 1 Hz and f are the minimum and maximum frequencies in Hz f max min Flicker noise is found in carbon composition resistors, where it is often referred to as ex- cess noise because it appears in addition to the thermal noise that is there. Other types of resistors also exhibit flicker noise to varying degrees, with wire wound showing the least. Since flicker noise is proportional to the dc current in the device, if the current is kept low enough, thermal noise will predominate and the type of resistor used will not change the noise in the circuit. Reducing power consumption in an op amp circuit by scaling up resistors may reduce the 1/f noise, at the expense of increased thermal noise. 10.3.4 Burst Noise Burst noise, also called popcorn noise, is related to imperfections in semiconductor mate- rial and heavy ion implants. It is characterized by discrete high-frequency pulses. The pulse rates may vary, but the amplitudes remain constant at several times the thermal noise amplitude. Burst noise makes a popping sound at rates below 100 Hz when played through a speaker — it sounds like popcorn popping, hence the name. Low burst noise is achieved by using clean device processing, and therefore is beyond the control of the designer. Modern processing techniques at Texas Instruments has all but eliminated its occurrence. 10.3.5 Avalanche Noise Avalanche noise is created when a pn junction is operated in the reverse breakdown mode. Under the influence of a strong reverse electric field within the junction ’ s depletion region, electrons have enough kinetic energy that, when they collide with the atoms of the 4). These collisions crystal lattice, additional electron-hole pairs are formed (Figure 10 – are purely random and produce random current pulses similar to shot noise, but much more intense. 10-9 Op Amp Noise Theory and Applications

156 Noise Colors Figure 10 4. Avalanche Noise – When electrons and holes in the depletion region of a reversed-biased junction acquire enough energy to cause the avalanche effect, a random series of large noise spikes will be generated. The magnitude of the noise is difficult to predict due to its dependence on the materials. Because the zener breakdown in a pn junction causes avalanche noise, it is an issue with op amp designs that include zener diodes. The best way of eliminating avalanche noise is to redesign a circuit to use no zener diodes. 10.4 Noise Colors While the noise types are interesting, real op amp noise will appear as the summation of some or all of them. The various noise types themselves will be difficult to separate. Fortu- nately, there is an alternative way to describe noise, which is called color . The colors of noise come from rough analogies to light, and refer to the frequency content. Many colors are used to describe noise, some of them having a relationship to the real world, and some of them more attuned to the field of psycho-acoustics. White noise is in the middle of a that runs from purple to blue to white to pink spectrum and red/brown. These colors correspond to powers of the frequency to which their spec- – 1. trum is proportional, as shown in Table 10 – Table 10 1. Noise Colors COLOR FREQUENCY CONTENT 2 Purple f Blue f White 1 1 Pink f 1 Red/Brown 2 f 10-10

157 Noise Colors There are an infinite number of variations between the colors. All inverse powers of fre- quency are possible, as are noises that are narrowband or appear only at one discrete frequency. Those, however, are primarily external sources of noise, so their presence is an important clue that the noise is external, not internal. There are no pure colors; at high frequencies, all of them begin to roll off and become pinkish. The op amp noise sources described above appear in the region between white noise and red/brown noise (Figure – 5). 10 NOISE COLORS Pink White Brown Red 2 Frequency 1 1/f 1/f Spectrum 3 dB/octave 6 dB/octave – – Johnson Brownian Types of Thermal Avalanche* Flicker Noise Shot Popcorn* * Approximate Figure 10 – 5. Noise Colors 10.4.1 White Noise White noise is noise in which the frequency and power spectrum is constant and indepen- dent of frequency. The signal power for a constant bandwidth (centered at frequency f ), o is varied. Its name comes from a similarity to white light, which has does not change if f o equal quantities of all colors. When plotted versus frequency, white noise is a horizontal line of constant value. Shot and thermal (Johnson) noise sources are approximately white, although there is no such thing as pure white noise. By definition, white noise would have infinite energy at infinite frequencies. White noise always becomes pinkish at high frequencies. Steady rainfall or radio static on an unused channel approximate a white noise character- istic. 10.4.2 Pink Noise Pink noise is noise with a 1/f frequency and power spectrum excluding dc. It has equal energy per octave (or decade for that matter). This means that the amplitude decreases logarithmically with frequency. Pink noise is pervasive in nature — many supposedly ran- dom events show a 1/f characteristic. noise displays a 1/f characteristic, which also means that it rolls off at 3 dB/octave. Flicker 10-11 Op Amp Noise Theory and Applications

158 Op Amp Noise 10.4.3 Red/Brown Noise Red noise is not universally accepted as a noise type. Many sources omit it and go straight to brown, attributing red characteristics to brown. This has more to do with aesthetics than it does anything else (if brown noise is the low end of the spectrum, then pink noise should be named tan). So if pink noise is pink, then the low end of the spectrum should be red. Red noise is named for a connection with red light, which is on the low end of the visible light spectrum. But then this noise simulates Brownian motion, so perhaps it should be called Brown. Red/brown noise has a – 6 dB/octave frequency response and a frequency 2 excluding dc. spectrum of 1/f Red/brown noise is found in nature. The acoustic characteristics of large bodies of water approximate red/brown noise frequency response. Popcorn and avalanche noise approximate a red/brown characteristic, but they are more defined as pink noise where the frequency characteristic has been shifted down correctly as far as possible in frequency. 10.5 Op Amp Noise This section describes the noise in op amps and associated circuits. 10.5.1 The Noise Corner Frequency and Total Noise Op amp noise is never specified as shot, thermal, or flicker, or even white and pink. Noise for audio op amps is specified with a graph of equivalent input noise versus frequency. These graphs usually show two distinct regions: D Lower frequencies where pink noise is the dominant effect D Higher frequencies where white noise is the dominant effect Actual measu rements for the TLV2772 show that the noise has both white and pink char- acteristics (Figure 10 – 6). Therefore, the noise equations for each type of noise are not able to approximate the total noise out of the TLV2772 over the entire range shown on the graph. It is necessary to break the noise into two parts — the pink part and the white and then add those parts together to get the total op amp noise using the root- part — mean-square law of Paragraph 10.2.4. 10-12

159 Op Amp Noise 1000 Hz Vrms/ – 100 Noise Voltage Input Noise Voltage – 1/f Noise n f nc V White Noise 10 10 100 1 k 10 k Frequency – f Hz – Figure 10 – 6. TLV2772 Op Amp Noise Characteristics 10.5.2 The Corner Frequency The point in the frequency spectrum where 1/f noise and white noise are equal is referred – 6 that the actual noise to as the noise corner frequency, f . Note on the graph in Figure 10 nc due to the root-mean-square addition of noise sources as defined higher at f voltage is nc in Paragraph 10.2.4. 6. It appears a little above can be determined visually from the graph in Figure 10 – f nc 1 kHz. This was done by: D Taking the white noise portion of the curve, and extrapolating it down to 10 Hz as a horizontal line. Taking the portion of the pink noise from 10 Hz to 100 Hz, and extrapolating it as a D straight line. the point where the white noise and pink D The point where the two intercept is f nc, Ǹ noise are equal in amplitude. The total noise is then 2 x white noise specification Ǹ (from Paragraph 10.2.4). This would be about 17 nV Hz ń for the TLV2772. This is good enough for most applications. As can be seen from the actual noise plot in Figure 10 – 6, small fluctuations make precise calculation impossible. There is a precise method, however: Determine the 1/f noise at the lowest possible frequency. D 10-13 Op Amp Noise Theory and Applications

160 Op Amp Noise D Square it. D Subtract the white noise voltage squared (subtracting noise with root-mean- squares is just as valid as adding). Multiply by the frequency. This will give the noise contribution from the 1/f noise. D D Then divide by the white noise specification squared. The answer is f . nc For example: Ǹ ń Hz The TLV2772 has a typical noise voltage of 130 nV at 10 Hz (from 5-V plot on data sheet). Ǹ (from data sheet) The typical white noise specification for the TLV2772 is 12 nV Hz ń 2 2 ȳ ȱ 130 nV 12 nV 2 2 15) (10 – ǒ ǒ Ǔ Ǔ 167560(nV) + @10Hz * + 10 Hz f noise ń 1 ȧ ȧ Ǹ Ǹ Hz Hz ȴ Ȳ 2 2 noise f ń 1 @10Hz 167560(nV) + + 1164 Hz + f nc 2 2 white noise – (10 16) 12 nV Ǔ ǒ Ǹ Hz Once the corner frequency is known, the individual noise components can be added to- gether as shown in Paragraph 10.2.2. Continuing the example above for a frequency range of 10 Hz to 10 kHz: f max 17) (10 – ( ) + E f * f ) f 1n E Ǹ max n nc min whitenoise f min 4 10 12 nV 4 – 18) (10 ǒ Ǔ Ǹ Hz + – + ) 116 dBV 10 V 1164 Hz * 10 Hz 1n + 1.611 m E n Ǹ 10 Hz The example above presupposed that the bandwidth includes f . If it does not, all of the nc contribution will be from either the 1/f noise or the white noise. Similarly, if the bandwidth , the contribution of the 1/f noise is very large, and extends to three decades or so above f nc can be ignored. 10.5.3 Op Amp Circuit Noise Model Texas Instruments measures the noise characteristics of a large sampling of devices. This information is compiled and used to determine the typical noise performance of the de- vice. These noise specifications refer the input noise of the op amp. Some noise portions 10-14

161 Op Amp Noise can be represented better by a voltage source, and some by a current source. Input volt- age noise is always represented by a voltage source in series with the noninverting input. Input current noise is always represented by current sources from both inputs to ground (Figure 10 – 7). _ _ inn Noiseless Op Amp inp + + en 7. Op Amp Circuit Noise Model – Figure 10 In practice, op amp circuits are designed with low source impedance on the inverting and noninverting inputs. For low source impedances and CMOS JFET inputs, only the noise voltage is important; the current sources are insignificant in the calculations because they are swamped in the input impedances. – The equivalent circuit, therefore, reduces to that shown in Figure 10 8: _ _ Noiseless Op Amp + + en 8. Equivalent Op Amp Circuit Noise Model – Figure 10 10-15 Op Amp Noise Theory and Applications

162 Op Amp Noise 10.5.4 Inverting Op Amp Circuit Noise If the previous circuit is operated in an inverting gain stage, the equivalent circuit becomes that shown in Figure 10 – 9: R 2 e2 _ Ein _ R 1 e1 Noiseless E0 Op Amp + + R en 3 e3 Figure 10 – 9. Inverting Op Amp Circuit Noise Model The additional voltage sources e1 through e3 represent the thermal noise contribution from the resistors. As stated in Paragraph 10.3.2, the resistor noise can also be dis- counted if the values are low. Resistor noise will be omitted in the examples that follow. R3 is also not usually present, unless low common-mode performance is important. De- leting it and connecting the noninverting input directly to (virtual) ground makes the com- mon mode response of the circuit worse, but may improve the noise performance of some There will be one less noise source to worry about. Therefore, the equivalent cir- circuits. cuit becomes that shown in Figure 10 – 10: R 2 _ Ein _ R 1 Noiseless E0 Op Amp + + en 10. Inverting Equivalent Op Amp Circuit Noise Model – Figure 10 10-16

163 Op Amp Noise This simplifies the gain calculation: 2 2 R R 2 2 (10 – 19) Ǔ ǒ ǒ Ǔ ǒ Ǔ Ǹ E E e + ) 1 ) n 0 in R R 1 1 where e = the total noise over the bandwidth of interest. n 10.5.5 Noninverting Op Amp Circuit Noise Taking the simplified equivalent op amp circuit from Paragraph 10.5.2 as the base, the noise equivalent of a noninverting op amp circuit is shown in Figure 10 11: – R 2 _ _ R 1 Noiseless E0 Op Amp + + Ein en Figure 10 11. Noninverting Equivalent Op Amp Circuit Noise Model – The gain of this circuit is: 2 2 R R 2 2 – (10 20) ǒ ǒ Ǔ Ǔ ǒ ǒ Ǔ Ǔ Ǹ E + ) 1 e E ) 1 ) n 0 in R R 1 1 10-17 Op Amp Noise Theory and Applications

164 Op Amp Noise 10.5.6 Differential Op Amp Circuit Noise Taking the simplified equivalent op amp circuit from Paragraph 10.5.2 as the base, the noise equivalent of a differential op amp circuit is shown in Figure 10 – 12: R 2 _ Ein _ 1 R 1 Noiseless E0 Op Amp + + Ein 2 R 3 en R 4 Figure 10 – 12. Differential Equivalent Op Amp Circuit Noise Model Assuming that R1 = R3 and R2 = R4, the gain of this circuit is: 2 2 R R 2 2 21) – (10 ) ( ǒ Ǔ Ǔ ǒ ǒ Ǔ Ǹ E 1 ) E e + * ) E n 0 in1 in2 R R 1 1 10.5.7 Summary The previous examples, though trivial, illustrate that noise always adds to the overall out- put of the op amp circuit. Reference 1 provides a much more in-depth derivation of op amp noise in circuits, including resistive effects. 10-18

165 Putting It All Together 10.6 Putting It All Together example is provided for analysis only — actual results depend on a number of other This factors. Expanding on the techniques of Paragraph 10.2.5: A low-noise op amp is needed over an audio frequency range of 20 Hz to 20 kHz, with a gain of 40 dB. The output voltage is 0 dBV (1V). The schematic is shown in Figure 10 – 13: 10 M +5 V +5 V 100 k 100 k _ V IN V OUT + TLE2027 100 k Figure 10 – 13. Split Supply Op Amp Circuit Ǹ . The data sheet, It would be nice to use a TLE2027 with a noise figure of 2.5 nV ń Hz reveals that this is a ± 15-V part, and that noise figure is only specified at ± 15 V. however, (see Chapter 11) show that it can only Furthermore, the specification for V and V – OM OM+ swing to within approximately 2 V of its voltage rails. If they are +5 V and ground, the op amp is close to clipping with a 1-V output signal. This illustrates a common fallacy: the designer chooses an op amp based on one parameter only, without checking others that affect the circuit. An expert analog designer must develop an attention to details or be pre- pared to spend a lot of time in the lab with false starts and unexpected problems. So, the only choice is to select a different op amp. The TLC2201 is an excellent choice. 14 appears right It is a low-noise op amp optimized for single supply operation. Figure 10 – on the first page of the data sheet, which should be extremely significant to the designer. 10-19 Op Amp Noise Theory and Applications

166 Putting It All Together TYPICAL EQUIVALENT INPUT NOISE VOLTAGE vs FREQUENCY 60 = 5 V V Hz DD R Ω = 20 S = 25 ° C T 50 A nV/ Hz – 40 30 20 10 Equivalent Input Noise Voltage – n V Vn 0 10 k 1 k 10 100 1 Frequency – – Hz f Figure 10 14. TLC2201 Op Amp Noise Performance – The first circuit change in this example is to change the TLE2027 to a TLC2201. V isually, appears to be somewhere around 20 Hz (from Paragraph the corner frequency f nc 10.5.2), the lower frequency limit of the band we are interested in. This is good, it means Ǹ for all practical purposes the 1/f noise can be discounted. It has 8 nV ń Hz noise instead Ǹ , and from Paragraph 10.2.5: of 2.5 nV ń Hz Ǹ + 141.35. D root Hz part: 20000 * 20 To begin with, calculate the D Multiplying this by the noise spec: 8 141.35 + 1.131 m V, which is the equivalent ). The output noise equals the input noise multiplied by the gain, input noise (E IN which is 100 (40 dB). The signal-to-noise ratio can be now be calculated: D 1.131 m V 100 V + 113.1 m Signal-to-noise (dB) = D (10 – 22) 113.1 log(1V B V) m 20 + 20 log(8842) + 78.9 dB Pretty good, but 10 dB less than would have been possible with a TLE2027. If this is not s acceptable (lets say for 16-bit accuracy), one is forced to generate a ± 15-V supply. Let ’ suppose for now that 78.9 dB signal-to-noise is acceptable, and build the circuit. 10-20

167 Putting It All Together When it is assembled, it oscillates. What went wrong? To begin with, it is important to look for potential sources of external noise. The way the schematic in – 14 is drawn provides a visual clue to the culprit: a long connection Figure 10 from the half-supply voltage reference to the high-impedance noninverting input. Added to that is a 50-k Ω source impedance, which does not effectively swamp external noise sources from entering the noninverting input. There is a big difference between simply providing a correct dc operating point, and providing one that has low impedances where they are needed. Most designers know the “ fix ” , which is to decouple the noninverting in- put as shown in Figure 10 – 15: 10 M +5 V +5 V 100 k 100 k _ V IN V OUT + TLE2201 100 k 0.1 F Figure 10 – 15. TLC2201 Op Amp Circuit Better it stopped oscillating. Probably a nearby noise source radiating into the nonin- — verting input was providing enough noise to put the circuit into oscillation. The capacitor lowers the input impedance of the noninverting input and stops the oscillation. There is much more information on this topic in Chapter 17, including layout effects and compo- nent selection. For now, it is assumed that all of these have been taken into account. The circuit is still slightly noisier than the 78.9 dB signal-to-noise ratio given above, espe- cially at lower frequencies. This is where the real work of this example begins: that of elimi- nating component noise. 15 has 4 resistors. Assuming that the capacitor is noiseless (not The circuit in Figure 10 – always a good assumption), that means four noise sources. For now, only the two resis- the voltage divider that forms the voltage reference will be considered. The capaci- tors in tor, however, has transformed the white noise from the resistors into pink (1/f) noise. From Paragraph 10.3.2 and 10.2.5, the noise from the resistors and the amplifier itself is: 2 2 2 Ǹ 23) (10 – 5.73 m V V ) 5.73 m V V ) 113.1 m m 113.1 + + E rms Totalrms Signal-to-noise (dB) = (10 – 24) log(1V 20 78.9 dB B 113.1 m V) + 20 log(8842) + So far, so good. The amplifier noise is swamping the resistor noise, which will only add a very slight pinkish component at low frequencies. Remember, however, that this noise 10-21 Op Amp Noise Theory and Applications

168 Putting It All Together voltage is multiplied by 101 through the circuit, but that was previously taken into account for the 78.9 dB signal-to-noise calculation above. Reducing the value of the resistors to decrease their noise is an option. Changing the volt- Ω to 1 k Ω age divider resistors from 100 k μ V capacitor the same, while leaving the 0.1 changes the corner frequency from 32 Hz to 796 Hz, right in the middle of the audio band. Note: Resist fect the temptation to make the capacitor larger to move the pinkish ef below the lower limits of human hearing. The resulting circuit must charge the large capacitor up during power up, and down during power down. This may cause unexpected results. If the noise from the half-supply generator is critical, the best possible solution is to use a low-noise, low-impedance half-supply source. Remember, however, that its noise will be multiplied by 101 in this application. The effect of the 100-k Ω resistor on the inverting input is whitish, and will appear across the entire bandwidth of the circuit. Compared to the amplifier noise, it is still small, just like the noise from the noninverting resistors on the input. The noise contribution of resistors will be discounted. resistor used as the feedback resistor. The Of much more concern, however, is the 10-M Ω noise associated with it appears as a voltage source at the inverting input of the op amp, therefore, is multiplied by a factor of 100 through the circuit. From Paragraph 10.3.2, and, Ω resistor is – 84.8 dBV, or 57.3 V. Adding this and the 100-k the noise of a 10-M Ω resistor μ noise to the amplifier noise: 2 2 Ǹ – (10 25) + 5.73 m V 77.9 dBV ) 113.1 m V – + + 126.8 m V E rms Totalrms Signal-to-noise (dB) = (10 – 26) + log(7887) 20 log(1V B 126.8 m V) + 20 77.9 dB Ω resistor subtracts 1 dB from the signal-to-noise The noise contribution from the 10-M Ω ratio. Changing the 10-M Ω resistor to 100 k , and the input resistor from 100 k Ω to 1 k Ω 16: – preserves the overall gain of the circuit. The redesigned circuit is shown in Figure 10 10-22

169 References 100 k +5 V +5 V 1 k 100 k _ V IN V OUT + TLE2201 0.1 F 100 k Figure 10 16. Improved TLC2201 Op Amp Circuit – For frequencies above 100 Hz, where the 1/f noise from the op amp and the reference resistors is negligible, the total noise of the circuit is: 2 2 2 Ǹ 27) (10 – m V 78.9 dBV ) 5.73 m V – ) 113.1 m V + 0.57 + 113.2 m V + E rms Totalrms Signal-to-noise (dB) = 28) (10 – 78.9 dB 20 log(1V B 113.2 m V) + 20 log(8830) + Proper selection of resistors, therefore, has yielded a signal-to-noise ratio close to the theoretical limit for the op amp itself. The power consumption of the circuit, however, has increased slightly, which may be unacceptable in a portable application. Remember, too, that this signal-to-noise ratio is only at an output level of 0 dBV, an input level of – 40 dBV. If the input signal is reduced, the signal-to-noise ratio is reduced proportionally. Music, in particular almost never sustains peak levels. The average amplitude may be down 20 dB to 40 dB from the peak values. This erodes a 79 dB signal-to-noise ratio to 39 dB in quiet passages. If someone “ cranks up the volume ” during the quiet passages, noise will become audible. This is done automatically with automatic volume controls. The only way a designer can combat this is to increase the voltage levels through the individual stages. If the preceding audio stages connecting to this example, for instance, could be scaled to provide 10 dB more gain, the TLC2201 would be handling an output level of 3.16 V instead of 1 V, which is well within its rail-to-rail limit of 0 V to 4.7 V. This would in- crease the signal-to-noise gain of this circuit to 88.9 dB — almost the same as would have been possible with a TLE2027 operated off of ± 15V! But noise in the preceding stages would also increase. Combatting noise is a difficult problem, and there are always trade- offs involved. 10.7 References (1) Texas Instruments Application Report, Noise Analysis in Operational Amplifier Circuits , SLVA043A, 1999 10-23 Op Amp Noise Theory and Applications

170 10-24

171 Chapter 11 Understanding Op Amp Parameters Bruce Carter 11.1 Introduction This chapter is about op amp data sheet parameters. The designer must have a clear un- derstanding of what op amp parameters mean and their impact on circuit design. The arranged for speedy access to parameter information. Their definitions, typical chapter is abbreviations, and units appear in Section 11.2. Section 11.3 digs deeper into important parameters for the designer needing more in-depth information. While these parameters are the ones most commonly used at Texas Instruments, the same parameter may go by different names and abbreviations at other manufacturers. Not every parameter listed here may appear in the data sheet for a given op amp. An op amp that is intended only for ac applications may omit dc offset information. The omission “ of information is not an attempt to ” anything. It is merely an attempt to highlight the hide parameters of most interest to the designer who is using the part the way it was intended. There is no such thing as an ideal op amp — or one that is universally applicable. The selection of any op amp must be based on an understanding of what particular parame- ters are most important to the application. If a particular parameter cannot be found in the data sheet, a review of the application may well be in order and another part, whose data sheet contains the pertinent information, more suitable. T exas Instruments manufactures a broad line of op amps that can might be implement almost any application. The inexperienced designer could easily select an op amp that is totally wrong for the application. Trying to use an audio op amp with low total harmonic distortion in a high-speed video circuit, for example, will not work — no matter how superlative the audio performance might be. Some parameters have a statistically normal distribution. The typical value published in the data sheet is the mean or average value of the distribution. The typical value listed σ value. This means that in 68% of the devices tested, the parameter is found to is the 1 be ± the typical value or better. Texas Instruments currently uses 6 σ to define minimum and maximum values. Usually, typical values are set when the part is characterized and never changes. 11-1

172 Operational Amplifier Parameter Glossary 11.2 Operational Amplifier Parameter Glossary ab- There are usually three main sections of electrical tables in op amp data sheets. The table and the recommended operating conditions table list solute maximum ratings constraints placed upon the circuit in which the part will be installed. Electrical characteris- tables detail device performance. tics maximum ratings are those limits beyond which the life of individual devices may Absolute be impaired and are never to be exceeded in service or testing. Limits, by definition, are maximum ratings, so if double-ended limits are specified, the term will be defined as a range (e.g., operating temperature range). Recommended operating conditions have a similarity to maximum ratings in that opera- tion outside the stated limits could cause unsatisfactory performance. Recommended op- erating conditions, however, do not carry the implication of device damage if they are ex- ceeded. Electrical characteristics are measurable electrical properties of a device inherent in its They are used to predict the performance of the device as an element of an electri- design. cal circuit. The measurements that appear in the electrical characteristics tables are based on the device being operated within the recommended operating conditions. Table 11 – 1 is a list of parameters and operating conditions that are commonly used in TI op amp data sheets. The glossary is arranged alphabetically by parameter name. An ab- breviation cross-reference is provided after the glossary in Table 11 – 2 to help the designer find information when only an abbreviation is given. More detail is given about important parameters in Section 11.3. Table 11 – 1. Op Amp Parameter GLossary PARAMETER ABBV UNITS DEFINITION INFO Bandwidth for 0.1 dB flat- 0.1 dB of ± The range of frequencies within which the gain is MHz ness the nominal value. Case temperature for 60 Usually specified as an absolute maximum — It is meant to ° C be used as guide for automated soldering processes. seconds Common-mode input ca- Input capacitance a common-mode source would see to pF C 11.3.7.1 ic ground. pacitance Common-mode input im- The parallel sum of the small-signal impedance between each Ω Z ic pedance input terminal and ground. Common-mode input volt- V The average voltage at the input pins. 11.3.3 V IC age The ratio of differential voltage amplification to common-mode CMRR Common-mode rejection ra- voltage amplification. Note: This is measured by determining or dB 11.3.9 tio the ratio of a change in input common-mode voltage to the k CMR resulting change in input offset voltage. 11-2

173 Operational Amplifier Parameter Glossary UNITS DEFINITION INFO ABBV PARAMETER Usually specified as an absolute maximum. It is the power that can be dissipated by the op amp package, including the mW Continuous total dissipation load power. This parameter may be broken down by ambient temperature and package style in a table. The ratio of the change in output voltage of a driven channel to the resulting change in output voltage from another channel X Crosstalk dBc T that is not driven. The change in ac gain with change in dc level. The ac signal is 40 IRE (0.28 V ± ) and the dc level change is 100 IRE PK A Differential gain error % D ( ± 0.7 V). Typically tested at 3.58 MHz (NTSC) or 4.43 MHz (PAL) carrier frequencies. Differential input capaci- pF 11.3.7.1 C (see common mode input capacitance) ic tance The small-signal resistance between two ungrounded input Ω r Differential input resistance id terminals. The voltage at the noninverting input with respect to the in- V Differential input voltage V ID verting input. The change in ac phase with change in dc level. The ac sig- nal is 40 IRE (0.28 V 100 IRE ) and the dc level change is ± PK Differential phase error Φ ° D ± 0.7 V). Typically tested at 3.58 MHz (NTSC) or 4.43 MHz ( (PAL) carrier frequencies. Differential voltage amplifi- 11.3.6 A (see open loop voltage gain) dB VD cation The time required for an output voltage step to change from Fall time t ns f 90% to 10% of its final value. Duration of short-circuit cur- Amount of time that the output can be shorted to network — usually specified as an absolute maximum. rent ground The range of common-mode input voltage that, if exceeded, Input common-mode volt- may cause the operational amplifier to cease functioning V 11.3.3 V ICR age range properly. This is sometimes is taken as the voltage range over which the input offset voltage remains within a set limit. The amount of current that can be sourced or sinked by the usually specified as an absolute maximum op amp input — Input current I mA I rating. pA The internal noise current reflected back to an ideal current Input noise current I 11.3.13 n source in parallel with the input pins. Ǹ Hz The internal noise voltage reflected back to an ideal voltage nV 11.3.13 V Input noise voltage n Ǹ source in parallel with the input pins. Hz The product of the open-loop voltage gain and the frequency Gain bandwidth product MHz 11.3.13 GBW at which it is measured. The reciprocal of the open-loop voltage gain at the frequency Gain margin A dB m where the open-loop phase shift first reaches – 180 ° . The highest positive op amp output voltage for the bias condi- 11.3.5 V V High-level output voltage OH tions applied to the power pins. The average of the currents into the two input terminals with 11.3.2 Input bias current I A μ IB the output at a specified level. 11-3 Understanding Op Amp Parameters

174 Operational Amplifier Parameter Glossary UNITS DEFINITION INFO PARAMETER ABBV The capacitance between the input terminals with either input c 11.3.7.1 pF Input capacitance i grounded. The difference between the currents into the two input termi- Input offset current μ A 11.3.2 I IO nals with the output at the specified level. V The dc voltage that must be applied between the input termi- IO, Input offset voltage mV 11.3.1 nals to cancel dc offsets within the op amp. V OS The ratio of the change in input offset voltage to the change Input offset voltage long- m V 11.3.1 term drift time. It is the average value for the month. month The dc resistance between the input terminals with either Ω r M Input resistance 11.3.7.1 i input grounded. The range of input voltages that may be applied to either the Input voltage range V V 11.3.15 I IN+ or IN – inputs Large-signal voltage amplifi- A dB (see open loop voltage gain) V cation Lead temperature for 10 or Usually specified as an absolute maximum. It is meant to be ° C 60 seconds used as guide for automated and hand soldering processes. The current into an output with input conditions applied that according to the product parameter will establish a low level Low-level output current I mA OL at the output. The smallest positive op amp output voltage for the bias con- Low-level output voltage V 11.3.5 V OL ditions applied to the power pins. The maximum peak-to-peak output voltage that can be ob- Maximum peak output volt- tained without clipping when the op amp is operated from a V V 11.3.5 ± OM age swing bipolar supply. Maximum peak-to-peak out- The maximum peak-to-peak voltage that can be obtained V V O(PP) put voltage swing without waveform clipping when the dc output voltage is zero. The range of frequencies within which the maximum output voltage swing is above a specified value or the maximum Maximum-output-swing frequency of an amplifier in which the output amplitude is at MHz B 11.3.15 OM bandwidth the extents of it ’ s linear range. Also called full power band- width. The ratio of the total noise power at the output of an amplifier, Noise figure NF dB referred to the input, to the noise power of the signal source. In a transimpedance or current feedback amplifier, it is the frequency dependent ratio of change in output voltage to the Z Ω Open-loop transimpedance M t frequency dependent change in current at the inverting input. In a transimpedance or current feedback amplifier, it is the ratio of change in dc output voltage to the change in dc cur- R M Ω Open-loop transresistance t rent at the inverting input. The ratio of change in output voltage to the change in voltage across the input terminals. Usually the dc value and a graph Open -loop voltage gain A dB OL showing the frequency dependence are shown in the data sheet. Temperature over which the op amp may be operated. Some of the other parameters may change with temperature, lead- Operating temperature T ° C A ing to degraded operation at temperature extremes. 11-4

175 Operational Amplifier Parameter Glossary UNITS DEFINITION INFO PARAMETER ABBV The amount of current that is drawn from the op amp output. — Usually specified as an absolute maximum rating not for mA I Output current O long term operation at the specified level. The frequency dependent small-signal impedance that is placed in series with an ideal amplifier and the output termi- Output impedance Z Ω 11.3.8 o nal. The dc resistance that is placed in series with an ideal amplifi- r Output resistance Ω o er and the output terminal. The ratio of the largest deviation of the output voltage from its final steady-state value to the absolute value of the step after Overshoot factor – – a step change at the output. The absolute value of the open-loop phase shift at the fre- Φ 11.3.15 ° Phase margin m quency where the open-loop amplification first equals one. The absolute value of the ratio of the change in supply volt- ages to the change in input offset voltage. Typically both sup- Power supply rejection ratio PSRR dB 11.3.10 ply voltages are varied symmetrically. Unless otherwise noted, both supply voltages are varied symmetrically. The time required for an output voltage step to change from t Rise time nS r 10% to 90% of its final value. With a step change at the input, the time required for the out- put voltage to settle within the specified error band of the final t nS Settling time s value. Also known as total response time, t tot. The maximum continuous output current available from the amplifier with the output shorted to ground, to either supply, or mA I Short-circuit output current OS to a specified point. Sometimes a low value series resistor is specified. The rate of change in the output voltage with respect to time SR V/ μ s Slew rate 11.3.12 for a step change at the input. Temperature over which the op amp may be stored for long Storage temperature T ° C S periods of time without damage. terminal of the or V /V /V The current into the V CC+ DD DD+ – CC – mA Supply current /I I DD CC op amp while it is operating. / I CC – terminal of the /V /V or V The current into the V – DD CC CC+ – DD+ I mA Supply current (shutdown) – DD amplifier while it is turned off. SHDN Supply rejection ratio k 11.3.10 dB (see power supply rejection ratio) SVR k , SVS The absolute value of the ratio of the change in input offset ∆ V , ± CC Supply voltage sensitivity 11.3.10 dB voltage to the change in supply voltages. , V ∆ ± DD or V ∆ IO V / Bias voltage applied to the op amp power supply pin(s). Usu- CC Supply voltage V ± ally specified as a V value, referenced to network ground. DD The ratio of the change in input offset current to the change in Temperature coefficient of free-air temperature. This is an average value for the speci- 11.3.2 α I C ° A/ μ IO input offset current fied temperature range. 11-5 Understanding Op Amp Parameters

176 Operational Amplifier Parameter Glossary UNITS DEFINITION INFO PARAMETER ABBV The ratio of the change in input offset voltage to the change in Temperature coefficient of free-air temperature. This is an average value for the speci- 11.3.1 μ V/ ° C α V IO input offset voltage fied temperature range. Maximum current that can be supplied to the positive power Total current into V +/ CC terminal of the op amp — usually specified as an absolute mA V DD+ maximum. Maximum current that can be drawn from the negative power — terminal of the op amp usually specified as an absolute mA Total current out of V – DD maximum. The ratio of the RMS voltage of the first nine harmonics of the Total harmonic distortion THD dB fundamental signal to the total RMS voltage at the output. The ratio of the RMS noise voltage and RMS harmonic volt- Total harmonic distortion age of the fundamental signal to the total RMS voltage at the THD+N dB 11.3.14 plus noise output. The total dc power supplied to the device less any power delivered from the device to a load. Note: At no load: P = P Total power dissipation mW D D I × × I or P = V V DD+ D CC+ V The voltage required on the shutdown pin to turn the device IH – V Turn-on voltage (shutdown) on. SHDN V The voltage required on the shutdown pin to turn the device IL – Turn-off voltage (shutdown) V off. SHDN The time from when the turn-on voltage is applied to the shut- down pin to when the supply current has reached half of its Turn-on time (shutdown) μ s t EN final value. The time from when the turn-off voltage is applied to the shut- down pin to when the supply current has reached half of its Turn-off time (shutdown) μ s t DIS final value. The range of frequencies within which the open-loop voltage B MHz Unity gain bandwidth 11.3.15 1 amplification is greater that unity. Table 11 – 2. Cross-Reference of Op Amp Parameters ABBV PARAMETER α I Temperature coefficient of input offset current IO ∆ V / ± CC Supply voltage sensitivity V ∆ ± DD α V Temperature coefficient of input offset voltage IO V Supply voltage sensitivity ∆ IO A Differential gain error D A Gain margin m A Open loop voltage gain OL A Large-signal voltage amplification V Differential voltage amplification A VD 11-6

177 Operational Amplifier Parameter Glossary ABBV PARAMETER Unity gain bandwidth B 1 Maximum-output-swing bandwidth B OM c Input capacitance i Common-mode input capacitance C ic CMRR Common-mode rejection ratio Gain bandwidth product GBW / I CC SHDN – Supply current (shutdown) I SHDN DD – I /I Supply current DD CC I Input current I I Input bias current IB Input offset current I IO I Input noise current n I Output current O I Low-level output current OL I Short-circuit output current OS k Common-mode rejection ratio CMR k Supply rejection ratio SVR k Supply voltage sensitivity SVS Noise figure NF Total power dissipation P D Power supply rejection ratio PSRR r Input resistance i Differential input resistance R id Output resistance R o R Open-loop transresistance t SR Slew rate Operating temperature T A t Turn-off time (shutdown) DIS t Turn-on time (shutdown) EN THD Total harmonic distortion t Fall time f THD+N Total harmonic distortion plus noise t Rise time r t Settling time s T Storage temperature S /V Supply voltage V CC DD V Input voltage range I V Common-mode input voltage IC V Input common-mode voltage range ICR 11-7 Understanding Op Amp Parameters

178 Additional Parameter Information PARAMETER ABBV V Differential input voltage ID V Turn-on voltage (shutdown) IHSHDN V Turn-off voltage (shutdown) IL – SHDN V V Input offset voltage IO, OS V Input noise voltage n V Maximum peak-to-peak output voltage swing O(PP) High-level output voltage V OH V Low-level output voltage OL V Maximum peak output voltage swing ± OM X Crosstalk T Common-mode input impedance Z ic Z Output impedance o Z Open-loop transimpedance t Φ Differential phase error D Φ Phase margin m 11.3 Additional Parameter Information Depending on the application, some op amp parameters are more important than others. This section contains additional information for parameters that impact a broad range of designs. 11.3.1 Input Offset Voltage All op amps require a small voltage between their inverting and noninverting inputs to bal- ance mismatches due to unavoidable process variations. The required voltage is known . V is normally modeled as a voltage as the input offset voltage and is abbreviated V IO IO source driving the noninverting input. Figure 11 – 1 shows two typical methods for measuring input offset voltage — DUT stands for device under test. Test circuit (a) is simple, but since Vout is not at zero volts, it does not really meet the definition of the parameter. T est circuit (b) is referred to as a servo loop . The action of the loop is to maintain the output of the DUT at zero volts. Bipolar input op amps typically offer better offset parameters than JFET or CMOS input op amps. 11-8

179 Additional Parameter Information 10 k 10 k 1 k Vout V + IO 1000 _ + 10 nF 10 k _ Vout DUT DUT 10 10 _ + Vout + (b) (a) Figure 11 – 1.Test Circuits for Input Offset Voltage TI data sheets show two other parameters related to V ; the average temperature coeffi- IO cient of input offset voltage, and the input offset voltage long-term drift. , specifies the expected The average temperature coefficient of input offset voltage, α V IO is measured at the temperature C. V input of fset drift over temperature. Its units are μ V/ _ IO _ C. ∆ is computed as ∆ V / extremes of the part, and α V IO IO Normal aging in semiconductors causes changes in the characteristics of devices. The is expected to change with time. Its input offset voltage long-term drift specifies how V IO units are μ V/month. V is normally attributed to the input differential pair in a voltage feedback amplifier. Dif- IO processes provide certain advantages. Bipolar input stages tend to have lower off- ferent set voltages than CMOS or JFET input stages. Input offset voltage is of concern anytime that DC accuracy is required of the circuit. One way to null the offset is to use external null inputs on a single op amp package (Figure – 2). A potentiometer is connected between the null inputs with the adjustable terminal 11 connected to the negative supply through a series resistor. The input offset voltage is nulled by shorting the inputs and adjusting the potentiometer until the output is zero. _ – IN OUT N2 + IN + N1 V – CC – 2.Offset Voltage Adjust Figure 11 11-9 Understanding Op Amp Parameters

180 Additional Parameter Information 11.3.2 Input Current The input circuitry of all op amps requires a certain amount of bias current for proper op- , is computed as the average of the two inputs: eration. The input bias current, I IB Ǔ ǒ ) I I P N – (11 1) I + IB 2 CMOS and JFET inputs offer much lower input current than standard bipolar inputs. Fig- ure 11 3 shows a typical test circuit for measuring input bias currents. – The difference between the bias currents at the inverting and noninverting inputs is called – = I I . Offset current is typically an order of magnitude less the input offset current, I IO N P than bias current. S1 1 nF S1 Closed * Vout Vref 10 M + I P 7 _ 10 I N S2 Vout DUT + S2 Closed 1 nF I P Vref * Vout + I N 7 10 Vref 10 M Figure 11 – 3.Test Circuit – I IB Input bias current is of concern when the source impedance is high. If the op amp has high input bias current, it will load the source and a lower than expected voltage is seen. The best solution is to use an op amp with either CMOS or JFET input. The source impedance can also be lowered by using a buffer stage to drive the op amp that has high input bias current. In the case of bipolar inputs, offset current can be nullified by matching the impedance seen at the inputs. In the case of CMOS or JFET inputs, the offset current is usually not an issue and matching the impedance is not necessary. , specifies the expected α I The average temperature coefficient of input offset current, IO is measured at the temperature input offset drift over temperature. Its units are A/ ° C. I μ IO extremes of the part, and α I C. _ ∆ is computed as ∆ I / IO IO 11-10

181 Additional Parameter Information 11.3.3 Input Common Mode Voltage Range The input common voltage is defined as the average voltage at the inverting and nonin- verting input pins. If the common mode voltage gets too high or too low, the inputs will shut , speci- down and proper operation ceases. The common mode input voltage range, V ICR fies the range over which normal operation is guaranteed. Different input structures allow for different input common-mode voltage ranges: The LM324 and LM358 use bipolar PNP inputs that have their collectors connected to the negative power rail. This allows the common-mode input voltage range to include the neg- ative power rail. The TL07X and TLE207X type BiFET op amps use P-channel JFET inputs with the sources tied to the positive power rail via a bipolar current source. This allows the com- mon-mode input voltage range to include the positive power rail. TI LinCMOS op amps use P-channel CMOS inputs with the substrate tied to the positive power rail. This allows the common-mode input voltage range to include the negative power rail. Rail-to-rail input op amps use complementary N- and P-type devices in the differential in- puts. When the common-mode input voltage nears either rail, at least one of the dif feren- tial inputs is still active, and the common-mode input voltage range includes both power rails. of increasing concern. The trends toward lower, and single supply voltages make V ICR Rail-to-rail input is required when a noninverting unity gain amplifier is used and the input signal ranges between both power rails. An example of this is the input of an analog-to digital-converter in a low-voltage, single-supply system. High-side sensing circuits require operation at the positive input rail. 11.3.4 Differential Input Voltage Range Differential input voltage range is normally specified as an absolute maximum. Exceeding the differential input voltage range can lead to breakdown and part failure. Some devices have protection built into them, and the current into the input needs to be limited. Normally, differential input mode voltage limit is not a design issue. 11-11 Understanding Op Amp Parameters

182 Additional Parameter Information 11.3.5 Maximum Output Voltage Swing The maximum output voltage, V , is defined as the maximum positive or negative peak ± OM output voltage that can be obtained without wave form clipping, when quiescent DC out- is limited by the output impedance of the amplifier, the saturation o. V put voltage is zer ± OM voltage of the output transistors, and the power supply voltages. This is shown pictorially 4. in Figure 11 – +V CC +V CC Voltage drop across R1 + V of Q1 sat Q1 V OM+ R1 D1 0 V V O D2 V – OM R2 depends Note that V ± OM on the output load. Q2 of Q2 Voltage drop across R2 + V sat – V CC – V CC Figure 11 – 4.V OM This emitter follower structure cannot drive the output voltage to either rail. Rail-to-rail out- put op amps use a common emitter (bipolar) or common source (CMOS) output stage. With these structures, the output voltage swing is only limited by the saturation voltage (bipolar) or the on resistance (CMOS) of the output transistors, and the load being driven. Because newer products are focused on single supply operation, more recent data sheets and V to specify the maximum and from Texas Instruments use the terminology V OL OH minimum output voltage. Maximum and minimum output voltage is usually a design issue when dynamic range is the op amp cannot drive to the rails. This is the case in single supply systems where lost if the op amp is used to drive the input of an A to D converter, which is configured for full scale input voltage between ground and the positive rail. 11-12

183 Additional Parameter Information 11.3.6 Large Signal Differential Voltage Amplification Large signal differential voltage amplification, A , is similar to the open loop gain of the VD amplifier except open loop is usually measured without any load. This parameter is usual- vs. frequency. ly measured with an output load. Figure 11 – 11 shows a typical graph of A VD is a design issue when precise gain is required. The gain equation of a noninverting A VD amplifier: 1 1 – (11 2) + Gain 1 b 1 ) b A VD 1 b in the equa- is a feedback factor, determined by the feedback resistors. The term A b VD 1 , it will not greatly af is large in comparison with fect error term. As long as A tion is an VD b the gain of the circuit. 11.3.7 Input Parasitic Elements Both inputs have parasitic impedance associated with them. Figure 11 – 5 shows a model of the resistance and capacitance between each input terminal and ground and between the two terminals. There is also parasitic inductance, but the effects are negligible at low frequency. Input impedance is a design issue when the source impedance is high. The input loads the source. _ Vn Rn Cd Cn Rd Vp + Rp Cp – 5.Input Parasitic Elements Figure 11 11-13 Understanding Op Amp Parameters

184 Additional Parameter Information 11.3.7.1 Input Capacitance Input capacitance, C , is measured between the input terminals with either input groun- i is usually a few pF. In Figure 11 – ded. C || C = C . grounded, then C 5, if V i d p n i is is 5, if V is specified. In Figure 11 – Sometimes common-mode input capacitance, C ic p is the input capacitance a common mode source . C , then C || C = C shorted to V p n n ic ic would see referenced to ground. 11.3.7.2 Input Resistance r is the resistance between the input terminals with either input grounde- Input resistance, i 12 7 to 10 r Ω, = R de- || R Ω . r is grounded, then ranges from 10 d. In Figure 11 5, if V – i n d i p pending on the type of input. r , is specified. In Figure 11 – 5, if V is Sometimes common-mode input resistance, ic p is the input resistance a common mode source would r . , then r || R = R shorted to V p ic n n ic see referenced to ground. 11.3.8 Output Impedance data sheets list the output impedance under two different conditions. Some data Different output impedance while others list open-loop closed-loop sheets list output impedance, . both designated by Z o Z is defined as the small signal impedance between the output terminal and ground. Data o to 200 Ω . sheet values run from 50 Ω Common emitter (bipolar) and common source (CMOS) output stages used in rail-to-rail output op amps have higher output impedance than emitter follower output stages. Output impedance is a design issue when using rail-to-rail output op amps to drive heavy loads. If the load is mainly resistive, the output impedance will limit how close to the rails the output can go. If the load is capacitive, the extra phase shift will erode phase margin. is mostly Figure 11 – 6 shows how output impedance af the output signal assuming Z fects o resistive. Z Z O O V V O O 1 V + O Ǔ ǒ ) 1 f jf ń 0 R L A A A + V C R VD VD L L VD O Ǔ ǒ R Z ) Where: 0 L 1 f + O C 2 p Z 0 L Resistive Load Capacitive Load 6.Effect of Output Impedance Figure 11 – Some new audio op amps are designed to drive the load of a speaker or headphone di- rectly. They can be an economical method of obtaining very low output impedance. 11-14

185 Additional Parameter Information 11.3.9 Common-Mode Rejection Ratio Common-mode rejection ratio, CMRR, is defined as the ratio of the differential voltage /A . Ideally this ratio amplification to the common-mode voltage amplification, A DIF COM would be infinite with common mode voltages being totally rejected. The common-mode input voltage affects the bias point of the input differential pair. Be- cause of the inherent mismatches in the input circuitry, changing the bias point changes the offset voltage, which, in turn, changes the output voltage. The real mechanism at work / ∆ V . is ∆ V OS COM / In a Texas Instruments data she et, CMRR = ∆ V , which gives a positive number V ∆ OS COM in dB. CMRR, as published in the data sheet, is a dc parameter. CMRR, when graphed vs. frequency, falls off as the frequency increases. A common source of common-mode interference voltage is 50-Hz or 60-Hz ac noise. Care must be used to ensure that the CMRR of the op amp is not degraded by other circuit High values of resistance make the circuit vulnerable to common mode (and components. other) noise pick up. It is usually possible to scale resistors down and capacitors up to preserve circuit response. 11.3.10 Supply Voltage Rejection Ratio Supply voltage rejection ratio, k (AKA power supply rejection ratio, PSRR), is the ratio SVR of power supply voltage change to output voltage change. The power voltage affects the bias point of the input dif ferential pair. Because of the inher- ent mismatches in the input circuitry, changing the bias point changes the offset voltage, which, in turn, changes the output voltage. D V D V CC " " DD V ∆ . The term or K + + means that For a dual supply op amp, K ± CC SVR SVR D V D V OS OS the plus and minus power supplies are changed symmetrically. For a single supply op D V V D CC DD + or K + . amp, K SVR SVR V D D V OS OS Also note that the mechanism that produces k is the same as for CMRR. Therefore SVR is graphed as published in the data sheet is a dc parameter like CMRR. When k k SVR SVR vs. frequency, it falls off as the frequency increases. Switching power supplies produce noise frequencies from 50 kHz to 500 kHz and higher. is almost zero at these frequencies so that noise on the power supply results in noise k SVR on the output of the op amp. Proper bypassing techniques must be used (see Chapter 17) to control high-frequency noise on the power lines. 11-15 Understanding Op Amp Parameters

186 Additional Parameter Information 11.3.11 Supply Current current, I , is the quiescent current draw of the op amp(s) with no Supply exas load. In a T DD Instruments data sheet, this parameter is usually the total quiescent current draw for the whole package. There are exceptions, however, such as data sheets that cover single and is the quiescent current packaged op amps of the same type. In these cases, I multiple DD draw for each amplifier. In op amps, power consumption is traded for noise and speed. 11.3.12 Slew Rate at Unity Gain Slew rate, SR, is the rate of change in the output voltage caused by a step input. Its units are V/ μ s or V/ms. Figure 11 – 7 shows slew rate graphically. The primary factor controlling , which is added to make slew rate in most amps is an internal compensation capacitor C C 8, voltage change in the second – the op amp unity gain stable. Referring to Figure 11 . The stage is limited by the charging and discharging of the compensation capacitor C C . Es- maximum rate of change is when either side of the differential pair is conducting 2I E . Remember, however, that not all op amps have compensation ca- /C sentially SR = 2I C E pacitors. In op amps without internal compensation capacitors, the slew rate is deter- mined by internal op amp parasitic capacitances. Noncompensated op amps have great- er bandwidth and slew rate, but the designer must ensure the stability of the circuit by oth- er means. In op amps, power consumption is traded for noise and speed. In order to increase slew rate, the bias currents within the op amp are increased. t = 0 t = 0 + V IN dV V O _ dt SR = dV/dt 7.Figure 6. Slew Rate – Figure 11 11-16

187 Additional Parameter Information +V CC Q6 D1 V O D2 C Q1 Q2 V C n Q7 V p Q5 Q4 Q3 – V CC OUTPUT SECOND INPUT STAGE STAGE STAGE – 8.Figure 7. Simplified Op Amp Schematic Figure 11 11.3.13 Equivalent Input Noise Noise is covered in more detail in Chapter 10. All op amps have parasitic internal noise sources. Noise is measured at the output of an op amp, and referenced back to the input. Therefore, it is called equivalent input noise. , (or current, I ) per Equivalent input noise parameters are usually specified as voltage, V n n root Hertz. For audio frequency op amps, a graph is usually included to show the noise over the audio band. 11.3.13.1 Spot Noise The spectral density of noise in op amps has a pink and a white noise component. Pink noise is inversely proportional to frequency and is usually only significant at low frequen- cies. White noise is spectrally flat. Figure 11 – 9 shows a typical graph of op amp equivalent input noise. Usually spot noise is specified at two frequencies. The first frequency is usually 10 Hz where the noise exhibits a 1/f spectral density. The second frequency is typically 1 kHz 11-17 Understanding Op Amp Parameters

188 Additional Parameter Information pA nV rms rms (or where the noise is spectrally flat. The units used are normally for current Ǹ Ǹ Hz Hz noise). In Figure 11 – 9 the transition between 1/f and white is denoted as the corner fre- . quency, f nc 11.3.13.2 Broadband Noise , is the a peak to peak voltage over a specific frequency A noise parameter like V N(PP) band, typically 0.1 Hz to 1 Hz, or 0.1 Hz to 10 Hz. The units of measurement are typically nV P – P. Given the same structure within an op amp, increasing bias currents lowers noise (and increases SR, GBW, and power dissipation). Also the resistance seen at the input to an op amp adds noise. Balancing the input resist- ance on the noninverting input to that seen at the inverting input, while helping with of fsets due to input bias current, adds noise to the circuit. 1/f Noise Log Noise Voltage White Noise f nc Log Frequency Figure 11 – 9.Typical Op amp Input Noise Spectrum 11.3.14 Total Harmonic Distortion Plus Noise harmonic distortion plus noise, THD + N, compares the frequency content of the out- Total put signal to the frequency content of the input. Ideally, if the input signal is a pure sine the output signal is a pure sine wave. Due to nonlinearity and noise sources within wave, the op amp, the output is never pure. THD + N is the ratio of all other frequency components to the fundamental and is usually specified as a percentage: ( ) S Harmonic voltages ) Noise Voltages – 3) (11 ƫ ƪ THD + ) 100% N Fundamental Figure 11 – 10 shows a hypothetical graph where THD + N = 1%. The fundamental is the frequency as the input signal. Nonlinear behavior of the op amp results in harmonics same 11-18

189 Additional Parameter Information of the fundamental being produced in the output. The noise in the output is mainly due to the input noise of the op amp. All the harmonics and noise added together make up 1% of the fundamental. Two major reasons for distortion in an op amp are the limit on output voltage swing and slew rate. Typically an op amp must be operated at or below its recommended operating conditions to realize low THD. 99 Fundamental % Harmonics — OUT V Noise 0 f2f3f4f5f6f Frequency Figure 11 – 10. Output Spectrum with THD + N = 1% 11.3.15 Unity Gain Bandwidth and Phase Margin There are five parameters relating to the frequency characteristics of the op amp that are likely to be encountered in Texas Instruments data sheets. These are unity-gain band- ), gain bandwidth product (GBW), phase margin at unity gain ( ), gain margin φ width (B 1 m (A ). ), and maximum output-swing bandwidth (B OM m Unity-gain bandwidth (B ) and gain bandwidth product (GBW) are very similar. B speci- 1 1 of the op amp is 1: fies the frequency at which A VD ) ( + 1 f B + A 4) – (11 1 VD 11-19 Understanding Op Amp Parameters

190 Additional Parameter Information GBW specifies the gain-bandwidth product of the op amp in an open loop configuration and the output loaded: GBW + A f (11 5) – VD GBW is constant for voltage-feedback amplifiers. It does not have much meaning for cur- rent-feedback amplifiers because there is not a linear relationship between gain and bandwidth. ) is the difference between the amount of phase shift a sig- φ Phase margin at unity gain ( m nal experiences through the op amp at unity gain and 180 : _ ° f 180 * f @B1 + m 6) (11 – Gain margin is the difference between unity gain and the gain at 180 phase shift: _ ° Gain @180 phase shift * Gain margin + 1 7) – (11 Maximum output-swing bandwidth (B ) specifies the bandwidth over which the output OM is above a specified value: V B u while V f + 8) – (11 MAX, MIN O OM The limiting factor for B is slew rate. As the frequency gets higher and higher the output OM becomes slew rate limited and can not respond quickly enough to maintain the specified output voltage swing. , is purposely fabricated on chip in the In order to make the op amp stable, capacitor, C C – second stage (Figure 11 8). This type of frequency compensation is termed dominant pole compensation. The idea is to cause the open-loop gain of the op amp to roll off to 8 is very simpli- unity before the output phase shifts by 180 _ . Remember that Figure 11 – fied, and there are other frequency shaping elements within a real op amp. – 11 shows a typical gain vs. frequency plot for an internally compensated op Figure 11 amp as normally presented in a Texas Instruments data sheet. or GBW) is a design issue falls off with frequency. A (and thus B As noted earlier, A VD VD 1 when precise gain is required of a specific frequency band. ) and gain margin (A ) are different ways of specifying the stability of Phase margin ( φ m m the circuit. Since rail-to-rail output op amps have higher output impedance, a significant phase shift is seen when driving capacitive loads. This extra phase shift erodes the phase and for this reason most CMOS op amps with rail-to-rail outputs have limited abili- margin, ty to drive capacitive loads. 11-20

191 Additional Parameter Information 120 Dominant Pole 100 80 60 dB — 40 VD A 20 0 Gain Margin B1 20 – 0 45 — 90 Phase 135 φ m 180 225 100 0 10 1 M 10 k 100 k 10 M 1 k Frequency Hz — – 11.Voltage Amplification and Phase Shift vs. Frequency Figure 11 11-21 Understanding Op Amp Parameters

192 Additional Parameter Information 11.3.16 Settling Time It takes a finite time for a signal to propagate through the internal circuitry of an op amp. Therefore, it takes a period of time for the output to react to a step change in the input. In addition, the output normally overshoots the target value, experiences damped oscilla- , is the time required for the output voltage tion, and settles to a final value. Settling time, t s to settle to within a specified percentage of the final value given a step input. 12 shows this graphically: Figure 11 – Overshoot Tolerance Final Value OUT Damped V Oscillation Slew Rate Settling Time Time Figure 11 – 12. Settling Time Settling time is a design issue in data acquisition circuits when signals are changing rapid- ly. An example is when using an op amp following a multiplexer to buffer the input to an A to D converter. Step changes can occur at the input to the op amp when the multiplexer changes channels. The output of the op amp must settle to within a certain tolerance be- fore the A to D converter samples the signal. 11-22

193 Chapter 12 Instrumentation: Sensors to A/D Converters Ron Mancini 12.1 Introduction 1. The – The typical transducer measurement system block diagram is shown in Figure 12 transducer is the electronic system ’ s interface with the real world, and it issues data about a variable. The transducer converts the data into an electrical signal adequate for proc- essing by the circuitry that follows the transducer. Bias and excitation circuitry does the care and feeding of the transducer, thus this circuitry provides offset voltages, bias cur- rents, excitation signals, external components, and protection that is required for the transducer to operate properly. The output of the transducer is an electrical signal repre- senting the measured variable. DIgital Input Data Transducer ADC Amplifier Variable Out Digital Bias or Control Power Excitation Supply Circuits Figure 12 – 1. Block Diagram of a Transducer Measurement System The variables that must be measured are determined by the customer ’ s application, and the normally dictates the transducer selection. If the measured vari- measured variable able is temperature, then some sort of temperature sensing transducer must be employed, and the range of temperatures to be measured or the accuracy of the measure- ment is the primary factor influencing temperature transducer selection. Notice that the electrical output of the transducer is not a major concern at this point in the transducer ’ s electrical output is always a consideration, although picking selection. The transducer the right transducer for the job is the primary goal. The correct transducer for the job can change, or mV/ ° C change. All transducers have offset volt- V/ ° C have an Ω / ° C change, μ ages or currents, and they can be referenced to ground, either power supply rail, or some other voltage. The selection of the transducer is out of circuit designer ’ s hands; thus, the circuit designer must accept what the application demands. 12-1

194 The ADC selection is based on several system criteria such as resolution, conversion speed, power requirements, physical size, processor compatibility, and interface struc- ture. The ADC must have enough bits to obtain the resolution required by the accuracy specification. The formula for calculating the resolution of an ADC is given in Equation 1 where n is the number of significant bits contained in the ADC. 12 – n 1) – (12 + 2 RESOLUTION confusion exists about the word bits because the same word is used for binary bits Some and significant bits. Binary bits are ones and zeros used to calculate binary numbers; for n = 256 binary example, a converter with 8 different digital states has 8 significant bits and 2 bits – 2). The voltage value of a single bit, called a least significant bit (LSB), (see Figure 12 is calculated in Equation 12 – 2. 8-bit Converter Digital Code 1 0 0 1 1 1 1 0 8 n = 256 Number of Binary Bits = 2 = 2 Intermediate Bits MSB LSB – 2. Significant Bits versus Binary Bits Figure 12 FSV – 2) (12 LSB + n 2 FSV is the full-scale voltage of the converter in volts; hence, a 12-bit converter FSV = 10 12 = 2.441406 mV/bit. In an ADC, an LSB is the maximum volts has an LSB equal to 10/2 voltage change required to for a one-bit output change, and in an ADC an LSB is defined n 1). – as LSB = FSV/(2 Conversion speed is not critical in temperature measurement applications because tem- perature changes occur at slow rates. Directional control in a rocket traveling at Mach 2 happens much faster than temperature changes, so conversion speed is an important factor in rocket applications. The speed of an ADC is generally thought of as the conver- sion time plus the time required between conversions, and conversion speed dictates the converter structure. When conversion speed is a primary specification a flash converter is used, and flash converters require low impedance driving circuits. This is an example of a converter imposing a specification requirement, low output impedance, on the driving amplifier. 12-2

195 The system definition specifies the voltages available for design and the maximum avail- able current drain. Some systems have multiple voltages available, and others are limited to a single voltage. The available voltage impacts the converter selection. Size, processor compatibility, and interface structure are three more factors that must be considered when selecting the ADC. The available package that the converter comes in determines what footprint or size that the converter takes on the printed circuit board. Some applications preclude large power-hungry ADCs, so these applications are limited to recursive or Σ∆ -type ADCs. The converter must be compatible with the processor to preclude the addi- glue logic; thus, the processor dictates the ADC ’ s structure. This defines the inter- tion of face structure, sometimes the ADC structure, and the ADC timing. Notice that the amplifier designer has not been consulted during this decision process, but in actuality, the systems engineers do talk to the amplifier designers if only to pacify them. The selection of the ADC is out of circuit designer ’ s hands; thus, the circuit designer must accept what the application demands. There are many different transducer/ADC combinations, and each combination has a dif- ferent requirement. Although they may be natural enemies, any transducer may be coupled with any ADC, thus the amplifier must make the coupling appear to be seamless. There is no reason to expect that the selected transducer s output voltage span matches ’ ’ s input voltage span, so an amplifier stage must match the transducer the selected ADC output voltage span to the ADC input voltage span. The amplifier stage amplifies the transducer output voltage span and shifts its dc level until the transducer output voltage span matches the ADC input voltage span. When the spans are matched, the transducer/ ADC combination achieve the ultimate accuracy; any other condition sacrifices accuracy and/or dynamic range. The transducer output voltage span seldom equals the ADC input voltage span. Trans- ducer data is lost and/or ADC dynamic range is not fully utilized when the spans are un- equal, start at different dc voltages, or both. In Figure 12 3 (A), the spans are equal (3 – V), but they are offset by 1 V. This situation requires level shifting to move the sensor out- put voltage up by one volt so the spans match. In Figure 12 – 3 (B), the spans are unequal (2 V and 4 V), but no of fset voltage exists. This situation requires amplification of the sen- sor output to match the spans. When the spans are unequal (2 V versus 3 V) and offset (1 V), as is the case in Figure 12 – 3 (C), level shifting and amplification are required to match the spans. 12-3

196 4 V 4 V 4 V 3 V ADC ADC Input ADC Input Span Input Span 2 V 2 V Sensor Span Output Span Sensor Sensor 1 V 1 V Output Span Output Span 0 V 0 V 0 V 0 V (B) (C) (A) Figure 12 3. Example of Spans That Require Correction – The output span of the transducer must be matched to the input span of the ADC to achieve optimum performance. When the spans are mismatched either the transducer output voltage does not fit into the ADC input span thus losing sensor data, or the trans- ducer output voltage does not fill the ADC input span thus losing ADC accuracy. The latter situation requires an increase in ADC dynamic range (increased cost) because a higher bit converter must be used to achieve the same resolution. The best analog circuit avail- able for matching the spans is the op amp because it level shifts and amplifies the input voltage to make the spans equal. The op amp is so versatile that it shifts the signal s dc ’ level and amplifies the input signal simultaneously. A similar but different problem exists in the digital to analog converter (DAC) to actuator interface. The DAC output voltage or current span must match the actuator input voltage span to achieve maximum performance. The procedure for matching the DAC output the actuator input span can be quite different from the procedure for matching the span to transducer output span to the ADC input span. Transducer outputs are usually low-level signals, thus care must be taken to preserve their signal to noise ratio. Actuator input sig- nals may require significant power, thus robust op amps are required to drive some actua- tors. The system specifications eventually determine the transducer, ADC, and analog circuit specifications. System specifications are seen as absolute specifications; they must be met for the design to function in a satisfactory manner. Component specifications are di- vided into several categories; absolute maximum ratings (AMR), guaranteed minimum/ maximum specifications (V ), typical specifications (V), and guaranteed but or V MIN MAX not tested specifications (GNT). If any of the device parameters are taken beyond the AMR, the device can be destroyed (expect destruction). The manufacturer guard bands the AMR to guarantee safety and quality, and you should guard band the AMRs, too. Typical specifications are the most appealing, but throw them out because they are meaningless in most cases. In the vast majority of cases the typical specifications are not related to meaningful data; rather, they are marketing dreams. Never design with typical specifications unless you are in the habit of designing with meaningless data or have a good reason for believing that typical is close to reality. A violation of this rule is a specification like output voltage swing that de- 12-4

197 pends heavily on test conditions such as the value of the load resistor. When the load re- much higher than that specified for the test condition, the output voltage swing sistance is is closer to the typical rather than the guaranteed specification. The laws of physics guar- antee this truth, but the amount of extra voltage swing that is achieved is hard to calculate. Guaranteed min/max specifications define the limits of a parameter. The parameter will always exceed the minimum value, and it will never exceed the maximum value. The guaranteed min/max specifications are your design specifications. Guaranteed but not tested (GNT) specifications are usually applied to parameters that are very expensive to test. The manufacturer either tests some other parameter related to the specification or they sample test each lot to insure compliance. GNT specifications are design specifica- tions. There is a fifth specification called guaranteed by design (GBD), and if they are not critical specifications, GBD is a useful design specification. All specifications have conditions associated with testing. They specify ambient tempera- ture, supply voltage, test signals, test loads, and other conditions, and they define how the measurements are made. Inspect the test conditions carefully; an op amp that speci- fies a 5-V output swing with a 50- Ω load is much more capable of driving a load than an op amp that specifies a 5-V output swing with a 10-k load. Beware, you assume the risk Ω of a parameter being out of specification when you use devices at conditions other than the test conditions. An error budget is a logical and orderly method of tabulating errors, and it helps the de- signer keep track of the errors by error sources. Meeting the system specifications trans- lates into minimizing errors, choosing components with acceptable errors, and canceling or eliminating errors when possible. The error budget is first applied to the ADC and trans- ducer because they are the components that the designer has the least control over. When these two error budgets are combined and subtracted from the system error allow- ance the result is the error allowance for the amplifier and peripheral circuits. The designer must choose components and design wisely to stay within the error allowance, or the sys- tem specifications are not met. Sometimes the systems specifications can ’ t be met, and this fact is greeted with moans, name calling, and finger pointing. The error budgets are the design engineer ’ s only de- fense against subjective accusations. The error budgets document the design trail, and they show where changes need to be made to do the best job possible. It is very hard to maintain an error budget because errors must be converted to equivalent units (volts, bits, or amps), and the impact of the error term may not be calculable at that point in the design. When this situation occurs, tabulate the error terms in a table and calculate their effect later in the design process. This chapter teaches the designer how to characterize the transducer and ADC, how to determine amplifier and design specifications through the use of error budgets, and how to complete circuit design. The equations developed in this chapter are not nearly as im- portant as the design philosophy. 12-5

198 12.2 Transducer Types This is not a treatise on transducers, but an appreciation for the many different types of transducers gives a feel for the extent and complexity of the transducer characterization problem (following section). The variety of electrical output that transducers offer loosely groups transducers. Various types of transducer outputs are resistive, optical, ac-excited, junction voltage, and magnetic, and each of these outputs must be converted to an electri- cal signal that can be amplified to fit the input span of an ADC. There are excellent refer- 1 that deal with transducer characterization, but the transducer manufacturer ences should be your first source of reference material. The transducer manufacturer publishes data similar to that contained in an IC data sheet, and they take the same liberties with typical specifications that the IC manufacturers do. Sort through the data to determine the meaning of the various specifications, and then pay special attention to the test conditions that prevailed when the data was gathered. Look for application notes that show transducer excitation, bias, or interface circuits. Search several manufacturers for similar information because nobody manages to cover every aspect of a design Some transducers such as strain gages, thermistors, RTDs, and potentiometers sense R). The ∆ R sensing devices are used in at least three circuit con- a change in resistance ( ∆ figurations; the voltage divider, current excited, and Wheatstone bridge circuits shown in , and the change in trans- – 4, 12 Figures 12 5, and 12 – 6. The transducer resistance is R – T ∆ R. ducer resistance caused by a change in the measured variable is R 1 V OUT ∆ R V REF R T Figure 12 – 4. Voltage Divider Circuit for a Resistive Transducer The voltage divider circuit uses a stable reference voltage to convert the transducer resist- 3. ance into voltage, and its output voltage is given in Equation 12 – ) R R D T 3) – (12 V + V REF OUT D R ) R ) R 1 T the circuit has very low sensitivity because the circuit If R is comparable in value with R 1 T, must measure a small change in resistance in the presence of a large resistance. When 12-6

199 the bias resistor, R , is selected as a large value, V and R act as a current source, 1 1 REF and the transducer resistance can be neglected in the calculations thus yielding Equation – R) Equation 12 >> (R – + ∆ 4. 3 reduces to Equation 12 4. When R – 12 T 1 V REF – (12 4) Ǔ ǒ + R V D R ) T OUT R 1 Equation 12 – 5 is the equivalent of Equation 12 – 4, and it is obtained by exciting the trans- 5. The bias current can be made very ducer with a bias current as shown in Figure 12 – accurate by employing op amps in a current source configuration as shown in Figure >> (R R) need not enter the calculations. + ∆ 6, thus the approximation R 12 – T 1 R 1 V OUT ∆ R I R T Figure 12 5. Current Source Excitation for a Resistive Transducer – V CC R + _ V IN R 2 V * V IN CC + I R + ∆ R R T Figure 12 – 6. Precision Current Source Ǔ (12 5) – ǒ + D R ) R V I T OUT The Wheatstone bridge shown in Figure 12 – 7 is a precision device used to measure small changes in resistance. One leg of the bridge is made up of a voltage divider consisting and R ) and the reference voltage. When R and ∆ R equal of equal stable resistors (R 1 2 X R assumes ∆ . As the transducer resistance changes is selected equal to R zero, R T TX 12-7

200 some value, and R is switched until the bridge output voltage nulls to zero. At this point X the value of ∆ R is read from the R dial. Bridge circuits are used to convert resistive trans- X values to dial readings, but there are methods of using transducers in bridge circuits ducer that yield a voltage change proportional to the resistance change. The bridge circuit has a high output impedance, thus op amps configured in an instrumentation configuration (both inputs are equal high resistances) must be used to amplify the output voltage from bridge circuits. R X R 1 R TX V REF ∆ R R 2 V OUT R T – Figure 12 7. Wheatstone Bridge Circuit The three most popular optical transducers are the photoconductive cell, the photodiode and the photovoltaic cell. The photoconductive cell acts like a light sensitive resistor, thus – 7 that convert resistance changes – 5, or 12 – one of the circuits shown in Figures 12 4, 12 to voltage is used in photoconductive cell applications. The photodiode is a very fast diode with a small output current, and the circuit shown in Figure 12 – 8 is used to convert current to voltage. The photodiode is reversed biased with a constant voltage, so the photodiode terminating voltage stays constant thus maintaining linearity. The photodiode amplifier output voltage equation is Equation 12 6. – I R D F _ λ V PHOTODIODE OUT + V REF – 8. Photodiode Amplifier Figure 12 12-8

201 (12 – 6) I R V + D F OUT The phototransistor has a junction that is light sensitive, and the junction has a transparent cover so that it can sense ambient light. The collector-base junction of the transistor is reverse biased, and normal transistor action takes place with the ambient light induced – 9). base current taking place of the normal base current (see Figure 12 V CC R V OUT PHOTOTRANSISTOR λ Figure 12 9. Phototransistor Amplifier – The photovoltaic or solar cell circuit is shown in Figure 12 – 10. The circuit zero-biases the cell for minimum leakage current, and the cell ’ s output current is a linear function of the area exposed to light. When the photovoltaic cell is properly masked and evenly flooded – 10 and Equation with light, it operates as a linear distance transducer (see Figure 12 12 – 6). I D R F _ λ V OUT + Figure 12 – 10. Photovoltaic Cell Amplifier AC-excited transducers are usually used to make motion and/or distance sensors. In one type of ac-excited transducer, a stationary winding is excited with an ac current, and another winding is moved past the stationary winding inducing a voltage in the second winding. In a well-designed transducer, the induced voltage is proportional to distance, hence the output voltage is proportional to distance. Another ac-excited transducer uses two plates; one plate is excited with an ac current, and the other plate is ground. An object coming near the excited plate changes the capacitance between the plates, and the result is an output voltage change. 12-9

202 Resolvers and synchros are position transducers that indicate position as a function of the phase angle between the exciting signal and the output signal. Resolvers and syn- chros normally are multiple-winding devices excited from two or more sources. They indi- cate position very accurately, but their special circuitry requirements, cost, and weight lim- it them to a few applications such as airfoil control surfaces and gyros. AC-excited transducers require a rectifier circuit to make the output voltage unipolar prior to integration. Coarse transducers use a diode or diode bridge to rectify the output volt- age, but diodes are not adequate for precision applications because their forward voltage drop is temperature sensitive and poorly regulated. The diode problems are overcome through the use of feedback in the active full wave rectifier circuit shown in Figure 12 – 11 . An integrating capacitor, C, is added to the circuit so the output voltage is a dc voltage proportional to the average voltage value of the input voltage. R R V IN R R C R D1 D2 _ _ 2 V OUT + + Figure 12 – 11. Active Full-Wave Rectifier and Filter Semiconductor or wire junctions (thermocouples) are often used as temperature trans- ducers because there is a linear relationship between temperature and output voltage over a restricted temperature range. Thermocouples have small voltages varying from C, and they normally are configured with thermistors and zeroing resistors μ V/ ° C to mV/ ° in the output circuit. Thermocouples have small output voltages and high output resist- thus a special op amp called an instrumentation amplifier is required for thermocou- ance, ple amplification. An instrumentation amplifier has very high and equal input impedances, thus they don ’ t load the input signal source. Semiconductor junctions have a nominal temperature coefficient (TC) of – 2 mV/ ° C. The TC is linear, but it varies from diode to diode because of manufacturing techniques, semi- conductor materials, and bias currents. In a well-controlled application where thermal mass is insignificant, semiconductor junctions make excellent temperature transducers. The junction effect is so stable and linear that commercial temperature transducers have become available in a single IC. fields can be sensed by the Hall effect, and special semiconductors called Hall- Magnetic effect sensors have been developed to sense magnetic fields. Current is passed through 12-10

203 the semiconductor in a direction perpendicular to the magnetic field. A pair of voltage pick off leads is placed perpendicular to the direction of current flow, and the output voltage is proportional to the magnetic field strength. The manufacturing process for Hall effect transducers is a transduc- standard semiconductor manufacturing process, so Hall ef fect ers are offered for sale as transistors or ICs. 12.3 Design Procedure A step-by-step design procedure that results in the proper op amp selection and circuit design is given below. This design procedure works best when the op amp has almost ideal performance, thus the ideal op amp equations are applicable. When nonideal op amps are used, parameters like input current affect the design, and they must be ac- counted for in the design process. The latest generation of rail-to-rail op amps makes the ideal op amp assumption more valid than it ever was. No design procedure can anticipate all possible situations, and depending on the op amp selected, procedure modifications may have to be made to account for op amp bias cur- rent, input offset voltage, or other parameters. This design procedure assumes that sys- tem requirements have determined the transducer and ADC selection and that changing these selections adversely impacts the project. 1) Review the system specifications to obtain specifications for noise, power, current drain, frequency response, accuracy, and other variables that might affect the de- sign. 2) Characterize the reference voltage including initial tolerances and drift. 3) Characterize the transducer to determine its salient parameters including output voltage swing, output impedance, dc offset voltage, output voltage drift, and power requirements. These parameters determine the op amp ’ s required input voltage ), and input impedance requirements. The offset voltage and volt- to V range (V IN1 IN2 ’ age drift are tabulated as errors. At this point it is assumed that the selected op amp s voltage span is greater than the transducer ’ s output voltage excursion. Design input peripheral circuits if required. 4) Scrutinize the ADC ’ s specification sheet to determine it ’ s required input voltage range because this range eventually sets the op amp ’ s output voltage swing require- s input resistance, input capacitance, to V ’ ). Determine the ADC ment (V OUT2 OUT1 resolution, accuracy, full-scale range, and allowable input circuit charge time. Calcu- late the LSB value. 5) Create an error budget (in bits) for the transducer and ADC. Use the transducer/ ADC error budget to determine the value and range of the critical op amp parame- ters. Select an op amp, and justify the selection by creating an error budget for the op amp circuit. 12-11

204 6) Scan the transducer and ADC specifications, and make a set of analog interface am- plifier (AIA) specifications 7) Complete the AIA circuit design. 8) Build the circuit, and test it. 12.4 Review of the System Specifications 5% = 5 V ± The power supply has only one voltage available, and that voltage is 5V ± 250 mV. The power supply is connected with the negative terminal at ground and the positive . This is not a portable application, thus the allowed current drain, 50 mA, terminal at V CC is adequate for the job. No noise specifications are given, but the proposed power, ground, and signal traces are being done on high-quality circuit board material with planes and good size copper. A system of this quality should experience no more than 50 mV of noise on the logic power lines and 10 mV of noise on the analog power lines. temperature measuring system that requires updates every 10 seconds. Clearly, This is a ADC conversion speed or input charging rate is not cause for consideration. The low con- version speed translates into lower logic speed, and slow logic means less noise gener- ated. The temperature transducer is located at the end of a three-foot long cable, so ex- pect some noise picked up by the cable to be introduced into the circuit. Fortunately, the long time between ADC conversions enable extensive filtering to reduce the cable noise. The system accuracy required is 11 bits. The application measures several parameters so it is multiplexed, and a TLV2544 12 bits resolution ADC has been selected. The temper- to 100 ° C. ° C – ature transducer is a diode, and the temperature span to be measured is 25 The ambient temperature of the electronics package is held between 15 C to 35 ° C. ° 12.5 Reference Voltage Characterization A reference voltage is required to bias the transducer and act as a reference voltage for the analog interface amplifier (AIA). Selecting a reference with a total accuracy better than the accuracy specification (11 bits) does not guarantee meeting the system accuracy specification because other error sources exist in the design. Resistor tolerances, amplifi- er tolerances, and transducer tolerances all contribute to the inaccuracy, and the refer- ence can ’ t diminish these errors. The quandary here is a choice between an expensive reference and expensive accurate components, or an adjustment to null out initial errors. This quandary boils down to which is the lesser of two evils; expensive components or the expense of an adjustment. System engineering has decided that they want the adjustment, so the reference does not have to have 11-bit accuracy. A TL431A voltage reference is chosen for the design. 25 mV. This ± The output voltage specification at 25 ° C and 10-mA bias current is 2495 mV 12-12

205 reference has a temperature drift of 25 mV over 70 ° C, and this translates to 7.14 mV drift C temperature range. There is another drift caused by the cathode voltage over a 20 ° and this drift is 2.7 mV/V. The supply voltage regulation is 0.5 V, but much of this change, tolerance is consumed by the initial tolerance and wiring scheme, so the less than 0.1 V is due to regulator drift. The total drift is 7.14 mV + 0.27 mV = 7.41 mV. This yields a total drift of 0.3% maximum. The amplifier usually uses a fraction of the reference voltage, so the final AIA will not drift the full 0.3%. 12.6 Transducer Characterization The temperature transducer is a special silicon diode that is characterized for tempera- ± ture measurement work. When this diode is forward biased at 2.0 mA 0.1 mA its forward drop is 0.55 V ± 50 mV, and its temperature coefficient is – 2 mV/ ° C. The wide ac- voltage ceptable variation in bias current makes this an easy device to work with. The circuit for – 12. the bias calculations is shown in Figure 12 +5 V R B1 R B2 V OUT I I D REF D 1 – 12. Reference and Transducer Bias Circuit Figure 12 is calculated in Equation 12 – 7. Remember, the reference must The current through R B1 be biased at 10 mA, and the transducer must be biased at 2 mA. (12 – 7) I + I 12 mA ) + + I 10 ) 2 D REF is calculated in Equation 12 – 8, and the value of R is calculated in The value of R B1 B2 – 9. Equation 12 V * V ) 5 ref 2.495 * 5 8) – (12 208 + W + R + B1 12 I V ref 2.495 (12 – 9) R W 1247 + + + B2 2 I D Both resistors are selected from the list of 1% decade values, thus R = 210 Ω , 1%, and B1 , 1%. The resistor values have been established, so it is time to calculate Ω = 1240 R B2 12-13

206 the worst case excursions of I 10 and 12 – 11). The resistors are assumed (Equations 12 – D to have a 2% tolerance in these calculations. The extra 1% allows for temperature changes, vibration, and life. Three percent tolerances would have been used if the elec- tronics ambient temperature range were larger. ’ V REF(MIN) 0.025 2.47 * * 0.007 10) – (12 I + + 1.93 mA + D(MIN) ( ) R 1.02 1.24 B2(MAX) V REF(MAX) * 0.007 0.025 * 2.52 I 2.10 mA + + + (12 11) – D(MAX) ( ) R 1.24 0.98 B2(MIN) The bias current extremes do not exceed the transducer bias current requirements, so the transducer will meet the specifications advertised. The converter is 12 bits and the full-scale voltage is assumed to be 5 V, so the value of an LSB is calculated in Equation 12 – 12. The nominal transducer output voltage is 550 mV at an ambient temperature of 25 ° C. At – 25 ° C, the transducer output voltage is 550 mV+ ( – 2 mV/ ° C)( – 50 ° C) = 650 mV. – At 125 ° C, the transducer output voltage is 550 mV + ( 2 mV C) = 400 mV. This ° C)(75 ° data is tabulated in Table 12 – 1. 5 FSV + + LSB 1.22 mV + – (12 12) 12 N 2 2 – 1. Transducer Output Voltage Table 12 ANALOG INTERFACE TRANSDUCER TRANSDUCER TEMPERATURE OUTPUT VOLTAGE AMPLIFIER INPUT VOLTAGE 25 = 650 mV C 650 mV V – ° IN1 25 ° C 550 mV 550 mV 100 ° 400 mV V C = 400 mV IN2 The steady state (V ) offset voltage is ± 50 mV, thus transducer output voltage (V ) TOV TOS ranges from 350 mV to 700 mV. The offset voltage is stripped out by the adjustments in spans 100 mV, thus it is a 100 mV/1.22 mV/ the AIA, so it is not of any concern here. V TOS bit = 82 bit error unless it is adjusted out. The output impedance of the transducer is equivalent to the resistance of a forward biased diode (Equation 12 – 13). 26 26 (12 – 13) 13 + W + + R D 2 I At this stage of the design there are two parameters that influence the accuracy of the measurement, and they are the temperature coefficient of the transducer and the output 12-14

207 impedance of the tra nsducer. The temperature transducer has been biased correctly, thus ° – C. The output imped- its temperature coefficient should be the advertised value of 2 mV/ ance of the transducer forms a voltage divider with the input resistance of the AIA, but this ’ error can t be calculated until the AIA is selected. The final transducer error contribution that can ’ t be adjusted out, and this error is determined during is that portion of the V TOS the AIA design. 12.7 ADC Characterization This particular ADC was selected because it has a multiplexer and it enables different modes of operation. The temperature measurement is done in the single-shot mode be- this mode allows the user to set the charge time at the input to the converter. During cause ’ charging, the ADC s input resistance is low, but after the ADC input is charged the input resistance rises to 20 k Ω . This high input resistance does not load the AIA output circuit, thus the AIA achieves full rail-to-rail output voltage swing. The internal reference is used in this application, and the reference sets the input voltage span required to obtain full accuracy for the ADC. Using the internal reference, the input ) is ± 150 mV, and the voltage drift voltage span is 0 V to 4 V. The offset voltage (V ADCOS is 40 PPM/ ° C. The voltage drift over the full temperature range is 40 PPM/ ° C(20 ° C) = 800 PPM. There are 244 PPM/LSB in a 12-bit converter, so the drift voltage error is 800/244 ≈ 4 bits error. The ADC output is full scale (all bits 1) when the input voltage is 4 V, and it is zero (all bits 0) when the input voltage is 0 V. This data is tabulated in Table 12 2. Because the full scale – 12 ) = 976.6 V/bit. μ output voltage has changed to 4 volts the LSB is calculated to be 4/(2 Table 12 – 2. ADC Input Voltage ADC INPUT ANALOG INTERFACE DIGITAL OUTPUT AMPLIFIER OUTPUT VOLTAGE VOLTAGE 0 V 000000000000 V = 0 V OUT1 4 V 111111111111 V = 4 V OUT2 12.8 Op Amp Selection It is time to select the op amp, and the easiest way to do this is to list the known specifica- tions or requirements, list a candidate op amp ’ s specifications, and calculated the project- ed error that the candidate op amp yields. 12-15

208 – Table 12 3. Op Amp Selection DESIGN CANDIDATE OP AMP: ESTIMATED VALUE SPECIFICATION TLV247X 12 6 10 Ω 10 Ω R (13) IN V 350 mV to 700 mV – 0.2 V to 5.2 V TOV R 1.8 Ω OUT 0.15 V to 4.85 V 0 V to 4 V V INADC V ––– 2.2 mV OS I 100 pA ––– B 28 nV ––– V N Ǹ Hz 0.39 pA I ––– N Ǹ Hz Analog noise ––– 10 mV k ––– 63 dB SVR There should be almost no error from R because the transducer output impedance is IN very low. The high side of the op amp ’ s output voltage swing (4.85 V) is much higher than the ADC input voltage (4 V). The low side of the op amp ’ s output voltage swing (0.185 V) Ω is less than the ADC input voltage swing (0 V). The ADC input circuit is 20 k and that doesn ’ t load the op amp output stage, so the op amp output voltage swing is very close should present no problems acting as a voltage di- to the ADC input voltage range. R OUT vider with the ADC input resistance. V and I create offset voltages that add to the ref- OS IB erence offset voltage, and they have to be adjusted out as a group. The system noise overshadows the op amp noise, thus the op amp noise is accepted unless later calculation prove otherwise. 12.9 Amplifier Circuit Design Enough information exists for the AIA to be designed. The TLV247X op amp is selected because it meets all the system requirements. The first step in the design is to determine the AIA input and output voltages, and this has already been done. These voltages are taken from Tables 12 – 1 and 12 – 2, and repeated here as Table 12 – 4. Table 12 4. AIA Input and Output Voltages – INPUT VOLTAGE OUTPUT VOLTAGES st pair of data points = 650 mV V 1 = 0 V V OUT1 IN1 nd pair of data points = 400 mV V 2 = 4 V V OUT2 IN2 – 14. The equation of an op amp is the equation of a straight line as given in Equation 12 12-16

209 – 14) (12 ) b Y mX + – – 14 making Two pairs of data points shown in Table 12 4 are substituted in Equation 12 16. Equations 12 – 15 and 12 – – 15) (12 ) b 4 + 0.4m – (12 16) ) b 0 0.65m + – 16 to obtain Equation 12 – 17. – Equation 12 15 is solved and substituted into Equation 12 b * ǒ Ǔ 17) – (12 0.4 4 ) b + 0.65 Equation 12 – 15 yields m = – 16. Sub- – 17 yields b = 10.4, and solving Equation 12 Solving – 18, and Equation 14 yields Equation 12 stituting these values back into Equation 12 – 18 (the final equation for the AIA) is put in electronic terminology. 12 – 18) – (12 +* 16V ) 10.4 V IN OUT 18 is shown in Fig- – The circuit that yields the transfer function developed in Equation 12 13. ure 12 – R R G F V CC V IN _ V OUT + R 2 R V 1 REF 13. AIA Circuit Figure 12 – The equations for the AIA circuit are given below. R ) R R R F G F 1 19) (12 – Ǔǒ Ǔ ǒ Ǔ ǒ +* V ) V V REF IN OUT R ) R R R 2 1 G G R F | | 20) (12 – + m R G R R ) R F G 1 – 21) (12 Ǔ Ǔǒ ǒ V + b REF R ) R R 1 2 G Equation 12 – 18 gives the value for m as 16, and using Equation 12 . 20 yields R = 16R – F G because they are standard 1% resistor values, and Ω = 23.7 k = 383 k Ω and R Select R G F 12-17

210 this yields m = 16.16. The resistors R1 and R2 are calculated with the aid of Equations 12 – 23. – 22 and 12 R R b 23.7 G 10.4 1 Ǔ ǒ – (12 22) ǒ Ǔ + + 0.2424 + 383 ) 23.7 2.495 R R R ) R ) V 1 2 REF F G 0.2424 – (12 23) + R 0.32 R R + 2 1 2 0.7576 The parallel combination of R should equal the parallel combination of R and and R 1 2 F R so that the input voltage offset caused by the op amp input current is cancelled. Select G = 105 k Ω and R = 33.2 k Ω because they are standard 1% values, and then b = 10.3. R 1 2 The value of the parallel combination of R (R ||R = 25.22 k Ω ) almost matches the , R 1 1 2 2 = 22.3k (R ||R , R Ω ), and this is an adequate value of the parallel combination of R F F G G match for input current cancellation. The downsides of selecting large resistor values for are current noise amplification, increased resistor noise, smaller bandwidth because R F of stray capacitance, and increased of fset voltage due to input current. Bandwidth clearly is not a factor in this design. The op amp input current is 100 pA, so it won ’ t cause much with a 383-k Ω feedback resistor (38.3 offset V). The noise current and voltage are calcu- μ lated later when the error budget is made. The gain, m, and the intercept, b, are not accurate because the exact resistor values were not available in the 1%-resistor selection chart. This is a normal situation, and in less de- manding designs the small error either does not matter or is corrected someplace else in the signal chain. That error is critical in this design, so it must eliminated. There are sev- eral nondrift type errors that have accumulated up to this point, and now is the time to cor- rect all the nondrift errors with the addition of adjustments. Two adjustments are used; one adjustment controls the gain, m, and the other controls the intercept, b. The value of the adjustable resistor must be large enough to deliver an adequate adjustment range, but any value larger than that decreases the adjustment resolution. The data that determines the adjustment range required is tabulated in Table 12 – 5. Drift and gain errors are calculated in volts, but drift errors are calculated in bits because they are not eliminated by adjustments. Remember, a LSB for this system is 4/4096 = V/bit. μ 976.6 12-18

211 Table 12 – 5. Offset and Gain Error Budget ERROR INTERCEPT DRIFT GAIN PARAMETER V ± 25 mV REF drift 7.41 mV V ≈ 8 LSB REF 50 mV Transducer offset ± 13 ≈ 0 LSB Transducer R Ω OUT ± 150 mV 1 LSB ADC reference Total unadjusted 2 LSB ADC error 1.6 LSB Gain error ADC drift 4 LSB op amp V 2.2 mV OS op amp 100 pA I B 28 nV op amp V ≈ 1 LSB N Ǹ Hz 139 pA I op amp 0 LSB ≈ N Ǹ Hz PS noise 10 mV ≈ 2 LSB V NPS 0 LSB op amp 1.8 Ω ≈ R OUT V op amp 70 mV ≈ 72 LSB OUT LOW 18 LSB Total error The adjustment for the intercept, b, depends on R , R , and V . This adjustment has 1 REF 2 to account for the reference offset, the op amp input voltage offset, the op amp input cur- rent, and the resistor tolerances. The offset voltage inherent in the reference is given as ± 25 mV. The op amp input of fset voltage is 2.2 mV; usually op amp of fset voltage calcula- tions include multiplying this offset by the closed loop gain, but this isn ’ t done because the offset voltage is adjusted out in the input circuit. The op amp input current is converted to a common-mode voltage by the parallel combination of the reference resistors, so it is neglected in this calculation. , is calculated in Equa- The worst case reference input voltage for the op amp, V REF(MIN) tion 12 – 24, where the resistor tolerances are assumed to be 3%, and the reference volt- age error is 50 mV. 12-19

212 0.97R 1 ǒ Ǔ V * 50 mV V + REF REF(MIN) ) 1.03R 0.97R 1 2 (12 – 24) 0.97(33.2) ) ( + 2.495 * 0.05 0.566 V + 1.03(105) ) 0.97(33.2) The nominal reference voltage at the op amp input is 0.6 V, so the reference voltage has to have about 40 mV adjustment around the nominal, or a total adjustment range of = (2.495/(105 + 80 mV. The nominal current through the voltage divider is I DIVIDER 33.2) k = 0.018 mA. A 4444-k Ω resistor drops 80 mV, thus the adjustable resistor (a po- Ω , equal to Ω . Select the adjustable resistor, R tentiometer) must be greater than 4444 k 1A Ω because this is an available potentiometer value, and the offset adjustment is 45 5 k ± , and this subtraction to yield R mV. Half of the potentiometer value is subtracted from R 1 1B = 33.2 k Ω centers the adjustment about the nominal value of 0.6V. R – 2.5 k Ω = 30.7 1B . Ω as 30.9 k . Select R Ω k 1B The adjustment for the gain employs R and R to insure that the gain can always be set G F at the value required to insure that the transducer output swing fills the ADC input range. 18) is algebraically manipulated, worst case values are – The gain equation (Equation 12 substituted for m and b, and it is presented as Equation 12 – 25. V * 10.4 3.85 10.4 * OUT 25) – (12 + G + + 18.71 * 0.35 V * IN Equation 12 – 26 is Equation 12 – 27 with 3% resistor tolerances. Ǔ ǒ 26) (12 – + 18.71 1.03R 0.97R F G Doing the arithmetic in Equation 12 – = 19.86 R . Thus, on the high side the 26 yields R F G gain must go from 16 to 19.86, or it must increase by 3.86. Assuming that the low side gain variation is equal, and rounding off to 4 sets the gain variation from 12 to 20. When Ω = 23.7 k Ω R is divided into a potentiometer R varies from 284.4 k . R to 474 k Ω R G F F FA , thus the nominal gain can be varied from 11.8 to 20.2. Ω = 200 k Ω and R = 280 k FB 12-20

213 R R R G FA FB Ω 200 k Ω 23.7 k 280 k Ω +5 V +5 V TEMP D 1 R B2 SENSOR + 0.01 1 Ω 1240 R B1 μ F F μ Ω _ 210 R 2 Ω 105 k V OUT + TLV247x R 1B Ω 30.7 k TL431A R 1A + 10 0.01 Ω 5 k F μ F μ Figure 12 – 14. Final Analog Interface Circuit There is no easy method of setting two interacting adjustments because when the gain is changed the offset voltage changes. They quickest method of adjustment is to connect the transducer to the circuit, adjust the offset, and then adjust the gain. It takes several series of adjustments to get to the point where the both parameters are set correctly. The impedance and noise errors are calculated prior to completing the error budget. The op amp input impedance works against the transducer output impedance to act like a volt- age divider. The value of the voltage divider is calculated in Equation 12 – 27, and as Equa- tion 12 – 27 indicates, the output resistance of the transducer is negligible compared to the input resistance of the op amp. This is not always the case! R 6 10 IN – 27) (12 [ V V + V V + T D T T 6 R ) r ) 13 10 c IN The ADC input impedance works against the op amp output impedance to act like a volt- age divider. The value of the voltage divider is calculated in Equation 12 – 28. The voltage divider action introduces about a 0.009% error into the system, and this is within 13-bit accuracy, so it can be neglected. 3 Ǔ ǒ 10 20 20 – 28) (12 V + 0.99991 V + V + IN OUT OUT 3 Ǔ ǒ 20.0018 10 ) 1.8 20 0.5 ), and this must be converted to volts. There The noise specification is given in nV/(Hz are involved formulas for the conversion, but the simplest thing to do is assume the noise is wide band. If the numbers add up to a significant error, detail calculations have to be ) (G = V made. The voltage noise is multiplied by the closed loop gain, thus V N MAX NWB = 28 nV(20) = 560 nV = 0.56 μ V. The current noise is multiplied by the parallel combination Ω ) = 139 pA(22.5 k and R ||R ) = 3.137 nV. The system noise is (R I , thus I of R N NWB F G G F 10 mV, and this noise comes in through the inputs and the power supply. The power sup- 12-21

214 ply contribution is reduced by the power supply rejection ratio, and it is 10 mV/63 dB = 7.08 nV. This calculation assumes that high-frequency noise is not a prob- 10 mV/1412 = lem, but if this is not true, CMRR must be reduced per the data sheet CMRR versus fre- quency curves. Some of the system noise propagates through the inputs and is rejected by the common- mode rejection of the op amp. The op amp is not configured as a differential amplifier, so a portion of the closed loop gain will multiply some of the system noise. 29. The ac gain of the AIA is given in Equation 12 – R R R 1 F F Ǔǒ Ǔ ǒ + V a a V V * SN SN OUT R R ) R R ) R 2 1 F G G 29) – (12 ( ) a + V 4.12 16 + 11.8 a V * SN SN All of the system noise does not get in on the inputs, rather most of the system noise is found on the power supply. The fraction of the system noise that gets into the ground sys- α , is normally about 0.01 be- tem and onto the op amp inputs is very small. This fraction, cause the power supplies are heavily decoupled to localize the noise. Considering this, the system noise is 1.18 mV, or less than 2 LSBs. The op amp output voltage range does not include 0 volts, and the ADC output voltage low value is 0 volts, so this introduces another error. The guaranteed op amp low voltage is 185 mV at a load current of 2.5 mA. The output current in this design is 185 mV/20 k Ω μ = 9.25 A. This output current approximates a no load condition, hence the nominal low voltage typical specification of 70 mV is used. This leads to a 72 LSB error, by far the big- gest error. Referring to T able 12 – 5, notice that the total error is 90 LSB. Losing 90 LSBs out of 4096 total LSBs is approximately 11.97 bits accurate, so the 11-bit specification is met. The final circuit is shown in Figure 12 – 14. Notice that large decoupling capacitors have been added to the power supply and refer- ence voltage. The decoupling capacitors localize IC noise, prevent interaction between circuits, and help keep noise from propagating. Two decoupling capacitors are used, a large electrolytic for medium and low frequencies, and a ceramic for high frequencies. Al- though this portion of the design is low frequency, the op amp has a good frequency re- sponse, and the decoupling capacitors prevent local oscillations through the power lines. to form If cable noise is a problem, an integrating capacitor can be put in parallel with R F a low-pass filter. 12-22

215 12.10 Test The final circuit is ready to build and test. The testing must include every possible com- bination of transducer input and ADC output to determine that the AIA functions in all manufacturing situations. The span of the adjustments, op amp output voltage range, and ADC input range must be checked for conformance to the design criteria. After the design has been tested for the specification limits it should be tested for user abuse. What hap- pens when the power supply is ramped up, turned on instantly, or something between these two limits? What happens when the inputs are subjected to over voltage, or when the polarity is reversed? These are a few ideas to guide your testing. 12.11 Summary The systems engineers select the transducer and ADC, and their selection criterion is foreordained by the application requirements. The AIA design engineer must accept the selected transducer and ADC, and it is the AIA designer ’ s job to make these parts play together with adequate accuracy. The AIA design often includes the design of peripheral circuits like transducer excitation circuits, and references. The design procedure starts with an analysis of the transducer and ADC. The analysis is followed by a characterization of the transducer and reference. At this point enough in- formation is available to make an error budget and select candidate op amps. The op amp is selected in the next step in the procedure, and the circuit design follows. The output voltage span of the transducer and corresponding input voltage span of the ADC are coupled as two pairs of data points that form the equation of a straight line. The data point pairs are substituted into simultaneous equations, and the equations are solved to determine the slope and intercept of a straight line (an op amp solution). The op amp circuit configuration is selected based on the sign of the slope and the intercept. Finally, the passive components used in the op amp circuit are calculated with the aid of the op amp circuit design equations. The final circuit must be tested for conformance to the system specifications, but the pru- dent engineer tests beyond these specifications to determine the AIA ’ s true limits. 12.12 References 1 Wobschall, Darold, Circuit Design for Electronic Instrumentation , McGraw-Hill Book Company, 1979 12-23

216 12-24

217 Chapter 13 Wireless Communication: Signal Conditioning for IF Sampling Perry Miller and Richard Cesari 13.1 Introduction High-speed operational amplifiers (op amps) are used extensively in wireless commu- nication systems. These amplifiers typically operate at intermediate frequencies (IF) ≤ 500 MHz and most frequently operate below 25 MHz. Applications for high-speed op amps include filtering circuits in radio receivers, IF amplifiers, mixer circuits, and band- pass amplifiers. 13.2 Wireless Systems This chapter focuses on the requirements for the op amp and a number of techniques used in wireless communication systems to interface high-speed op amps to analog-to- digital converters (ADCs) and digital-to-analog converters (DACs). This section provides several examples of different op amp usage. Figure 13 – 1 shows an example of a dual-IF receiver. In this application, several stages with different IF frequencies are used to get the desired performance. The receiver con- the received radio frequency (RF) input from the antenna to a baseband signal. This verts type of system requires the ability to receive and operate over a wide range of signal strength. The inherent system noise level determines the lower operating limit and is a critical factor in the overall performance of the receiver. The receiver performance is mea- sured in terms of receiver sensitivity, which is defined as the ratio between the power of the wanted baseband signal at the output of the ADC and the total power of all unwanted signals (include random noise, aliasing, distortion, and phase noise contributed by the lo- cal oscillator) introduced by the different circuit elements in the receiver. A low-sensitivity receiver can cause signal saturation in the ADC input. 13-1

218 Wireless Systems 900 Signal @ MHz 104 dBm – Image 1st IF Rejection ( Interferer @ LNA 1st IF Duplexer Amp Filter 1 – 13 dBm ) SAW Filter 35 MHz B/W (660 MHz) RFLO Image Image 2nd 2nd Rejection Rejection IF Amp Mixer Filter 3 Filter 2 AGC LPF 17.3 MHz ADC 200 kHz B/W fs = 52 MHz 2nd (222.7 MHz) RFLO Figure 13 – 1. A Typical GSM Cellular Base Station Receiver Block Diagram The receiver contains two mixer stages reminiscent of a classic superhetrodyne receiver ood selectivity. The process of hetrodyning involves the translation of one frequency with g to another by the use of a mixer and local oscillator (LO) offset at the proper frequency to convert the RF signal to the desired IF. The LO signal is at a much higher level than the RF signal. In Figure 13 – 1, the 900 MHz RF signal is picked up by the antenna and amplified by a low-noise amplifier (LNA). After being sufficiently amplified by the LNA to overcome the noise level, the RF signal passes through a bandpass filter (BPF) used to provide image rejection and sufficient selectivity prior to the first stage mixing. High selectivity prevents adjacent channel energy from getting into the input of the ADC and decreasing the receiver dynamic range. A strong signal in an adjacent channel causes intermodulation products in the receiver that can result in loss of the received sig- nal. The band-pass filter is implemented with a surface acoustic wave (SAW) filter. The SAW filter provides very sharp edges to the passband, with minimum ripple and phase distortion. The first stage mixer down-converts the band-limited RF signal with the LO signal, pro- ducing a number of new frequencies in the spectrum, including the sum frequency compo- nent, the difference frequency component, and spurious responses. The first stage IF fil- ter provides sufficient filtering after the mixing circuit. It selects the difference frequency component while rejecting the sum frequency component and undesirable spurious re- sponses. Passing the difference frequency component on to the next stage of the receiver makes it much easier to provide the gain and filtering needed for proper receiver function- 13-2

219 Wireless Systems ality. Image rejection also places a – 20 MHz). constraint on the choice of the IF (10 MHz The spurious responses are the result of power supply harmonics and intermodulation products created during the mixing of the RF signal and the LO signal. If not substantially suppressed, spurious responses often corrupt the IF signal and cause it to be accepted as a valid IF signal by the IF amplifier. The first stage IF amplifier minimizes the effects of the first stage filter loss on the noise figure and amplifies the signal to a suitable level for the second stage mixer. The output from the second stage mixer is applied to the second stage IF amplifier, automatic gain producing a 1-V full-scale in- control (AGC) amplifier, and the subsequent low-pass filter, put to the ADC. The ADC samples and digitizes this baseband analog input. The AGC ensures that if the received signal amplitude goes up rapidly, the ADC is not satu- amplifier rated. At the other extreme, if there is a fast power ramp down, the AGC prevents the sig- nal quality from passing below an acceptable level. High-speed current-feedback opera- tional amplifiers (CFA) are typically used to filter and amplify the IF signals because this type of operational amplifier has good slew rate, wide bandwidth, large dynamic range, and a low noise figure. In this type of receiver, the ADC is a key component requiring sampling rates ≥ 40 MSPS with 12 b its to 14 bits of resolution and is usually a pipeline architecture device. The output ’ of the ADC is highly dependent on the ADC s sampling frequency, nonlinearities in the ADC and the analog input signal, and the converter maximum frequency. – Table 13 – 1 to the system 1 tabulates the contribution of each stage depicted in Figure 13 level budget for a typical GSM receiver. GSM is the global system for mobile communica- tions. It is one of the most popular digital cellular formats in the world. Table 13 1. GSM Receiver Block System Budget – † NOISE FIGURE (dB) GAIN (dB) ANF ELEMENT Duplexer 1 – 1 1 +18 0.51616 LNA 1.6 Image rejection Filter #1 2 0.00517 – 1st stage mixer 9.87 – 7 0.21853 Noise Filter – 2 0.07363 1st stage amplifier +49 1.7957 7.92 Image rejection filter #2 – 2 4.31E – 06 5.68E 2nd stage mixer 10.8 – 7 – 05 2nd stage image filter 2 0.0009 – AGC 11 +50 2.16E – 09 Anti-alias filter – 2 3.83E – 10 ADC 7.63 – 2 3.83E – 10 Total (5.7 dB) 3.61 † ANF = Adjusted noise figure (linear). 13-3 Wireless Communication: Signal Conditioning for IF Sampling

220 Wireless Systems – 2 shows a more flexible implementation for the receiver using a digital signal Figure 13 processor (DSP). Using a DSP allows a single receiver to access several different wire- less systems through changes in software configuration. 900 MHz IF – Amp LNA 1st IF BPF – RF IF SAW Filter Filter RFLO RFLO Baseband Processor LPF 2nd IF ADC FIR I c6X DSP Demod 2nd IF Amp LPF FIR ADC Q RFLO Figure 13 2. An Implementation of a Software-Configurable Dual-IF Receiver – – 3 shows a basic W -CDMA transmit chain. A voiceband CODEC (coder decod- Figure 13 – er), op amps, and a DSP are used to digitize and band-limit the audio signal. The digitized signal is then compressed to the appropriate data rate either in hardware or by a software program implemented on the DSP. Redundancy (error correction), encryption, and the appropriate form of modulation (QPSK for W – CDMA or GMSK for GSM) are added to the compressed digitized signal. This signal goes via an interpolating filter to the communica- tion DAC as shown in Figure 13 4. Eight times interpolation[1],[2] is shown in Figure – 13 – 4, but other multiple-of-2 interpolations are possible and quite often used. Assuming that the modulated bit stream is a 3.84 MSPS W – CDMA signal, for 8x interpolation, the sampling clock frequency would need to be 30.72 MHz. The DAC converts the modulated bit stream to analog, and the conversion is usually performed by a pair of DACs: one for – 3. The reconstruction filter, at the output I channel and one for Q channel — see Figure 13 13-4

221 Wireless Systems of the DAC, is usually a high-order Bessel or elliptic filter used to low-pass filter the analog output from the DAC. Antenna DAC PA DSP/ Σ ASIC RF DAC 0 Quadrature 90 Modulator LO1 Figure 13 – 3. Basic W-CDMA Cellular Base Station Transmitter Block The modulator block converts the baseband I and Q signal to the appropriate carrier fre- quency, typically 864 MHz. The up-converted 864 MHz signal is amplified to a suitable level by the power amplifier (PA) and sent out via the antenna over the air or to a nearby wireless base station. The RF power amplifier is a large-signal device with power gain and ef the order ficiency on of 50% for GSM and about 30% for code division multiple access (CDMA), an access method in which multiple users are permitted simultaneously on the same frequency. Reconstruction Interpolation filter filter 4 MHz BW 0 – THS5672 W-CDMA Comms DAC 3.84 MSPS Baseband x 8 signal f = 30.72 MHz S 4. Communication DAC with Interpolation and Reconstruction Filters – Figure 13 13-5 Wireless Communication: Signal Conditioning for IF Sampling

222 Selection of ADCs/DACs 13.3 Selection of ADCs/DACs In communication applications, the dc nonlinearity specifications that describe the con- verter s static performance are less important than the dynamic performance of the ADC. ’ The receiver (overall system) specifications depend very much on the ADC dynamic per- formance parameters: effective number of bits (ENOB), SFDR (spurious free dynamic range), THD (total harmonic distortion), and SNR (signal-to-noise ratio). Good dynamic performance and fast sampling rate are required for accurate conversion of the baseband analog signal at RF or IF frequencies. The SFDR specification describes the converter s ’ in-band harmonic characterization and it represents the converter ’ s dynamic range. SFDR is slew rate and converter input frequency dependent. The output from an ADC is highly dependent on the converter sampling frequency and the maximum frequency of the analog input signal. A low-pass or band-pass anti-aliasing filter placed immediately before the ADC band-limits the analog input. Band-limiting en- sures that the original input signal can be reconstructed exactly from the ADC ’ s output ) of twice the information bandwidth of the analog samples ƒ when a sampling frequency ( s input signal is used (Nyquist sampling). Undesirable signals, above ƒ /2, of a sufficient s level, can create spectrum overlap and add distortion to the desired baseband signal. This must not be allowed to dominate the distortion caused by ADC nonlinearities. Sampling at the Nyquist rate places stringent requirements on the anti-aliasing filter — usually a th or higher order filter is needed. steep transition 10 Oversampling techniques (sampling rate greater than the Nyquist rate) can be employed to drastically reduced the steepness of the anti-aliasing filter rolloff and simplify the filter design. However, whenever oversampling is used, a faster ADC is required to digitize the input signal. Very fast ADCs can be costly and they consume a fair amount of power ≥ 1000 mW). In a system application, such as a wireless base station, where large num- ( bers of ADCs are used, the individual device power consumption must be kept to the bare minimum ( ≤ 400 mW). High-resolution ADCs, with slower sampling rates, offer potential cost savings, lower power consumption, and good performance, and are often used in some applications. In this case, undersampling or bandpass sampling techniques (the ) of the analog signal digitization by the ADC exceeds half the sampling frequency ( ƒ s /2) are employed. ADC, but the signal information bandwidth is ≤ ƒ s Operating the ADC in a bandpass sampling application requires knowledge of the con- /2. In general, as the input signal verter ’ s dynamic performance for frequencies above ƒ s frequency to the converter increases, ENOB, SNR, SFDR, and harmonic performance degrades. The fact that the analog input to the ADC cannot be represented exactly with a limited number of discrete amplitude levels introduces quantization error into the output digital 2 1 2 samples. This error is given by the rms quantization error voltage e . = qns q s 12 13-6

223 Selection of ADCs/DACs 2 q s = power is is P where q is the quantization The mean squared quantization noise qn s 12R step size and R is the ADC input resistance, typically 600 Ω to 1000 Ω . Communication ADCs similar to the THS1052 and THS1265 typically have a full scale – input/ Ω p. Generally, wireless systems are based on a 50- to 2 Vp range p – (FSR) of 1 Vp output termination, therefore, the ADC input is made to look like 50 Ω . Based on this as- sumption, the quantization noise power for a 12-bit, 65 MSPS ADC (THS1265) is – 73.04 dBm. For a noise-limited receiver, the receiver noise power can be computed as the thermal noise power in the given receiver bandwidth plus the receiver noise figure NF[3]. _ C, and 4 dB to 6 dB NF, the receiver For 200 kHz BW (GSM channel), temperature 25 power is – 115 dBm. Therefore, to boost the receiver noise to the quantization noise noise power level requires a gain of 42 dB. 1, the GSM – – – 900 signal is at In Figure 13 104 dBm (GSM – 900 spec for smallest possible signal at which the raw bit-error rate must meet or exceed 1%) and, therefore, the signal- to-noise ratio (SNR) at baseband or at the converter and due to the thermal noise compo- E b 104 dBm +115 dBm = 9 dB. = = – nent is given by SNR thermal N 0 In order for the raw BER to be 1% in a GSM system, testing and standard curves [4] indi- cate, that a baseband SNR (derived from the sum of both thermal noise and ADC noise) of 9 dB is needed for this performance node. is defined as: The process gain G p 6 f s 52 10 2 24.15 dB + G 10 + + 2.6 + 1) – (13 p 3 BW 200 10 where GSM channel BW = 200 kHz and ƒ = 52 MHz (the ADC sampling frequency). s The converter noise at baseband should be much better than the radio noise ( = thermal process gain). Furthermore, the thermal noise alone brings the system only to the noise + reference bit-error rate (BER). + process gain G . Therefore the converter noise (at baseband) = SNR adc p The ADC SNR of the thermal noise should be 20 dB to 40 dB above the SNR thermal adc is selected to be 37 dB better than component (+9 dBm). In this example, ADC SNR adc of the thermal noise component (9 dB). the SNR thermal In other words, if the converter SNR is desired to be 37 dB better than the thermal noise adc ), the baseband converter is chosen to be 9 + 37 dB = 46 dB. component (SNR thermal ). ) + converter noise (N The total noise (N ) = thermal noise (N sum t conv 13-7 Wireless Communication: Signal Conditioning for IF Sampling

224 Selection of ADCs/DACs Therefore, the noise-to-signal ratio is: N N N sum conv t + ) S S S GSM GSM GSM 2) (13 – 1 1 ) + 0.9 4.6 10 10 S GSM The signal-to-noise ratio = 7.942 (linear), which is 8.999 dB. N sum This shows that the converter noise degenerates the baseband SNR due to thermal noise only 0.0001 dB (9.000 dBm 8.999 dB) when the signal is at reference sensitiv- alone, by – ity level. = 52 MSPS, the converter SNR 900 signal is – required to fit the GSM At f adc s (46 – 24.15) dB = 22 dB. The effective number of bits (ENOB) required for fitting the GSM 900 signal is – 1.76 * SNR ENOB + 3) – (13 6.02 – 900 signal. and thus 4 bits are needed for the GSM – Assuming that filter #3 attenuates the interferer by 50 dB, the interferer drops from 13 – 53 dBm, or 40 dB above the GSM signal. The requirements for the number of dBm to 40 dB = 6.3 bits. bits needed to accommodate the interferer is 6dB ń bit Approximately 6 bits are needed to accommodate the interferer, plus 2 bits of head room for constructive interference, for a total of 8 bits. The ADC requirements are 4 bits for the GSM signal plus 8 bits for the interferer for a total of 12 bits: 4 10 8 7 6 5 9 3 2 1 0 11 12-Bit ADC Upper 8 bits accommodate Lower 4 bits accommodate quantitized signal noise + interference. It follows from the above analysis that the GSM signal is 8 bits down, or 8 bits x 6 dB/bit = 48 dB down from the 1 V FSR of the ADC. The full-scale input power to an ADC having termination can be calculated as: Ω a 50- 2 2 (1) V 2 * 10 13 dBm (full scale or FSR) + + 2 + 4) – (13 50 R Therefore, the 4 bits for the GSM signal gives 13 dBm – (8 bits x 6 dB/bit) = – 35 dBm, the possible ADC input signal power with the interferer that will meet the GSM speci- smallest 13-8

225 Selection of ADCs/DACs fication. Without the interferer present, the signal level is chosen to be about 20 dB below – 8 dBm, to accommodate constructive interference and to accom- the ADC full scale, or any large ADC input signal that may arise from short-term errant gain due to gain modate settling in the AGC. Thus, with the smallest possible signal specified by the GSM – 900 spec, the signal is amplified from – 104 dBm to – 8dBm without the interferer, and to – 35 dBm with the interferer. – 1, an AGC is necessary to assure that the For a practical receiver, as shown in Figure 13 LSB represents a uniform noise input while the peak power does not exceed the ADC ’ s FSR. The receiver block shown in Figure 13 – 1 uses a high-speed, fairly wide bandwidth (100 MHz to 550 MHz) communication ADC to convert the baseband signal to a high- speed parallel bit stream for processing in a DSP. For an ADC to accurately produce a digital version of the baseband analog input, the device must have very good resolution and dynamic performance. 1 needs 95 dB of gain in order to bring the 104 dBm The signal path shown in Figure 13 – – GSM signal up to 9 dBm (equivalent to about 0.112 Vp – p across 50 Ω ), with allowances – for losses in the filters, and mixers. Usually this gain is split evenly between the RF and baseband, but baseband gain is less expensive and consumes less power. The LNA pro- vides 18 dB, which, after filtering and cable losses, yields about 16 dB, while the mixer provides – 7 dB of conversion gain/loss. Modern communication DACs are, effectively, an array of matched current sources opti- mized for frequency domain performance. To handle both strong and weak signals, com- munication DACs require large dynamic range. The dynamic specifications of most inter- est are SFDR, SNR, THD, IMD (two-tone intermodulation distortion), ACPR (adjacent channel power rejection), and settling time. Besides these, there are a number of dc parameters, such as integral nonlinearity (INL) and differential nonlinearity (DNL), that are considered important because of their influ- ence on the SFDR parameter. DNL errors occur only at certain points in the converter ’ s transfer function. INL and DNL errors appear as spurious components in the output spec- trum and can degrade the signal-to-noise ratio of the DAC. Typical SFDR figures for 12-bit DACs and 14-bit DACs, with a 5-MHz single tone input at 50 MSPS, range from 75 dB to 80 dB. In order to prevent adjacent communication chan- nels from interfering with each other, the DAC must exhibit a good SFDR specification. DACs normally have differential outputs, and the current-mode architec- Communication ture is used to give the DAC a higher update rate. 13-9 Wireless Communication: Signal Conditioning for IF Sampling

226 Factors Influencing the Choice of Op Amps 13.4 Factors Influencing the Choice of Op Amps IF amplifiers and filters can be built from discrete components, though most modern ap- plications use integrated circuits. High-speed wideband op amps are employed as buffer amplifiers in the LO circuit, at the front end of ADCs, at the output of the DAC, in the exter- nal voltage reference circuits for ADCs and DACs, and in the AGC amplifier and anti-alias- ing stage. Op amps operating at IF frequencies, such as the AGC amplifier in Figure 13 – 1, must attain a large gain control range. How well the amplifier handles large and small sig- nals is a measure of its dynamic range. The current-feedback op amp can be used every- where except for the anti-aliasing filter and in the reconstruction filter stage. The op amp must have a level gain response from almost dc to at least 500 MHz, after which a gentle rolloff is acceptable. Also, the phase response is important to avoid dispersing the signal this requires a linear phase response. — Several factors influence the choice of the current-feedback op amp (CFA) and voltage- feedback amplifier (VFA) for use in wireless communication systems: D The ADC/DAC resolution D ADC/DAC dynamic specification Operating frequencies D D Type of signal D Supply voltages and D Cost 3, the SFDR and In both the receiver and transmit circuits, shown in Figures 13 – 1 and 13 – IMD are the key ADC/DAC parameters that have the most influence on op amp selection. A minimum requirement is that the op amp s SFDR or THD parameter, measured at the ’ frequency of operation, should be 5 dB to 10 dB better than the converter ’ s SFDR. For a perfect 12-bit ADC, the SFDR is 72 dB, thus the op amp in front of the ADC should exhibit a SFDR (or THD) of 77 dB to 82 dB. When an op amp is used as a buffer amplifier, it must faithfully reproduce the input to a very high degree of accuracy. This requires that the amplifier be designed and optimized for settling time. Fast settling time is mandatory when driving the analog input of an ADC because the op amp output must settle to within 1 LSB of its final value (within a time peri- od set by the sampling rate) before the ADC can accurately digitize the analog input. The amplifier settling time determines the maximum data transfer rate for a given accuracy. For example, to settle within 1 LSB of full scale range implies that the settling accuracy ± of the ADC is ½ LSB. Hence, a 12-bit ADC will require the op amp to settle to 1 4 * V for a μ of final value, or 0.0122% of final value. An LSB = 244 1.22 10 + 12 2 2 12-bit ADC with 1 V full-scale range. Values for the settling time and other important op – 2. amp parameters as they relate to the receiver and transmit blocks are listed in Table 13 13-10

227 Anti-Aliasing Filters The op amp dynamic parameters in Table 13 – 2 represent the range of values to achieve low noise, good SFDR, high slew rate, good bandwidth, etc. – 2. High-Speed Op Amp Requirements Table 13 VALUE PARAMETER Ǹ Ǹ Noise voltage 2.7 nV to 8 nV ń Hz ń Hz Ǹ Ǹ Noise current Hz 1pA to 30 pA ń Hz ń THD 70 dBc to 95 dBc Slew rate 260 V/ μ V to 3500 V/ μ V Small signal bandwidth 200 MHz to 600 MHz Large signal bandwidth ≥ 100 MHz Common-mode input voltage 3 V 5 V ± Supply voltage Settling time 8 ns to 20 ns 40 mA to 100 mA Output current Output impedance ≤ Ω 20 PSRR – 60 dB CMRR – 70 dB Input offset voltage 10 mV (typical) Op amps operating from ± 5-V supplies typically have 6 V to 8 V of common-mode range. Single-supply op amps often handle much smaller voltage ranges, and in some commu- nication applications, could exhibit limited linear operation over a wide signal swing. With the exception of rail-to-rail op amps, most op amps can swing to within 1 V to 1.5 V of the positive rail. T ypically signal-to-noise ratio, slew rate, and bandwidth suffer for devices op- erating from low supply voltages. When selecting current-feedback op amps, the gain-bandwidth plots are essential. They are needed because with current feedback ordinary loop-gain-proportional bandwidth relationships do not hold. 13.5 Anti-Aliasing Filters Spurious effects in the receiver channel (Figure 13 – 1) appear as high frequency noise in the baseband signal present at the ADC. The spurious signals (> ƒ / 2) must be blocked s ) where they will cause aliasing errors from getting to the ADC (sampling at Nyquist rate, ƒ s in the ADC output. A suitable anti-aliasing low-pass analog filter placed immediately before the ADC can block all frequency components capable of causing aliasing from reaching the ADC. The 13-11 Wireless Communication: Signal Conditioning for IF Sampling

228 Anti-Aliasing Filters anti-aliasing filter cutoff frequency (f is set to the highest baseband signal frequency of c) = f ) so that f interest (f . Sampling theorem requires that the ADC minimum Nyquist max max c . This ensures that the original base band or IF signal rate sampling frequency fs = 2f max can be reconstructed exactly from the ADC ’ s digital outputs. It is important to know that only an brickwall -type response could fully satisfy the exacting anti-aliasing filter having a requirements imposed by the sampling theorem. The rolloff of real filters increases more gradually from cutoff to the stop band, and therefore, in practice, the ADC sampling fre- . ƒ quency is usually slightly higher than 2 max The anti-aliasing filter must reduce the out-of-band aliasing producing signals to less than 1 LSB of the ADC resolution, without introducing additional distortion of the baseband or IF signal in-band components and without predominating distortion due to the ADC nonli- nearities. The spectrum overlap (aliasing) requirements are determined by: D Highest frequency of interest D Sampling rate ADC resolution D The highest signal frequency of interest sets the filter cutoff frequency. For example, sup- pose the input signal is to be sampled to 12-bit accuracy with a sampling frequency of 52 – 3dB cutoff frequency could be chosen. MHz. If the IF signal is 17 MHz an 18 MHz filter ≤ ½ LSB, but gener- All frequencies above the Nyquist frequency should be attenuated to = (52 ’ s limit of resolution will be a problem; i.e., ƒ ally only frequencies above the ADC alias 17) MHz = 35 MHz. The frequency rolloff is 18 MHz to 35 MHz (about 1 octave) and the – required attenuation is 72 dB (12-bit ADC). A very high-order filter is required to accom- plish this task. Practical ant-aliasing filters are limited to fifth-order or sixth-order type be- cause of amplifier bandwidth, phase margin, layout parasitics, supply voltage, and com- ponent tolerances. Keep in mind that as the rolloff sharpens, the passband ripple and phase distortion increase. For communication applications, linear phase characteristic and gain accuracy (low pass- band ripple) are important. And normally, Chebychev or elliptic (Cauer) filter types are used for the anti-aliasing filter. For good transient response or to preserve a high degree of phase coherence in complex signals, the filter must be of linear-phase type (Bessel-type filter). The THS4011 or THS4021 voltage-feedback op amp is a good choice for implementing anti-aliasing filter in this example. The quality of the capacitors and resistors used to implement the design is critical for per- formance anti-aliasing filter. 13-12

229 Communication D/A Converter Reconstruction Filter 13.6 Communication D/A Converter Reconstruction Filter Modern communication DACs are, effectively, an array of matched current sources opti- mized for frequency domain performance. The most important dynamic specifications are SFDR, SNR, THD, IMD, ACPR and settling time. The dc parameters INL and DNL are considered important because of their influence on the SFDR parameter. Typical SFDR figures for 12-bit to 14-bit DACs, with a 5-MHz single-tone input at 50 MSPS, ranges from 75 dB to 80 dB. In order to prevent adjacent communication channels from interfering with each other, the DAC must exhibit a good SFDR specification. Communication DACs normally have differential outputs and current-mode architecture is used to give the DAC a higher update rate. Figure 13 4 shows an interpolating filter block before the DAC and the reconstruction – filter at the output of the DAC. The interpolating filter is a digital filter whose system analog clock frequency is an integer multiple of the filter input data stream and is employed to s in-band aliased images. This eases the job of the reconstruction filter reduce the DAC ’ at the output of the DAC. The filter is used to smooth the input data — the output waveform 5 shows the inter- frequency is the same as the input to the interpolating filter. Figure 13 – polation filter output. The system clock frequency of the DAC and the interpolating filter are running at the same rate; therefore, the frequency spectrum of the DAC output signal, integer multiples of the sampling rate, becomes increasingly separated as the repeated at sampling rate is increased. The further apart the repeated DAC output spectrum, then the less steep the attenuation characteristic of the anti-aliasing needs to be. Consequently, a simpler anti-aliasing filter with a less steep rolloff from pass band to stop band can be – 4, the system clock used without any increase in distortion due to aliasing. In Figure 13 is 30.772 MHz (3.84 MSPS x 8). Figure 13 – 6 shows the attenuation characteristics need- ed for the reconstruction anti-aliasing filter. 2) = 26.7 MHz, and the required attenuation is – = (28.7 The aliasing frequency is ƒ alias 84 dB (14-bit DAC). A third-order anti-aliasing low-pass elliptic or Bessel filter could be used to meet the attenuation requirements. Either type of high-order filter gives a relative- ly flat response up to just below ½ the sampling frequency, followed by a sharp cutoff. But this arrangement provides no correction for the sinc function (Sinx)/x falloff in amplitude naturally produced by the sample and hold function in the DAC. The tradeoff in building a simpler reconstruction filter is that a faster DAC is required to convert the input digital data stream to analog signal. 13-13 Wireless Communication: Signal Conditioning for IF Sampling

230 Communication D/A Converter Reconstruction Filter 40 20 0 20 – dB – – 40 Power 60 – 80 – 100 – 6 10 8 14 4 2 0 16 12 MHz – f – Frequency – 5. QPSK Power Spectral Density Without Raised Cosine Filter — W-CDMA Figure 13 Amplitude f ( MHz) 28.7 2 MHz 0 CLK 30.772 Figure 13 – 6. Reconstruction Filter Characteristics Figure 13 – 7 shows a first-order reconstruction (low-pass) filter consisting of a high-speed differential amplifier configured for unity gain. The DAC outputs are terminated into 50 Ω . 7, the value for the filter capacitor is given by the expression: – In Figure 13 13-14

231 External Vref Circuits for ADCs/DACs 4 * 3.1827 10 C + – 5) (13 fb f clk 2 C fb 49.9 499 THS5672 499 R 2 R +) 1 IOUT1 V DAC OUT IOUT2 – ) R 1 R THS4131 2 499 THS4141 THS4151 499 49.9 C fb V COM,OUT Figure 13 7. A Single-Pole Reconstruction Filter – 13.7 External V Circuits for ADCs/DACs ref Figure 13 – 8 shows an op amp voltage follower circuit that is often used to interface the external precision voltage reference supplying the ADC/DAC external reference voltage (see for example, Miller and Moore, [5], [6],1999, 2000 for a more detailed discussion on is the output from a precision voltage reference circuits used in ADC an DAC systems). V in voltage reference, such as the Thaler Corp. VRE3050. The low-pass filter (formed by 3 dB corner frequency of – R ) filters noise from the reference and op amp buffer. The C 1 1 R and the transfer function for this circuit can be written as the filter is 1/2 C π 1 1 Ǔ ǒ ) 1 R sC V 2 2 o + 6) – (13 R sC V 2 2 1 2 in ǒ Ǔ ) R C ) R C s 1 2 1 2 C R R C R R C C 2 1 1 2 2 2 1 1 R which has a zero at s = C 2 2. R With the approximation C R complex the denominator polynomial is solved for = 2C , 1 1 2 2 of the response, which results in: and p poles p 1 2 13-15 Wireless Communication: Signal Conditioning for IF Sampling

232 External Vref Circuits for ADCs/DACs 1 1 P +* j ) – 7) (13 1 R R 2C 2C 1 1 1 1 1 1 P * +* j 8) – (13 2 R R 2C 2C 1 1 1 1 TLE2227 V R 1 IN V – REF O (REF +) 42.2 1.2 F C 1 C 2 0.047 F R 2 2.15 k Figure 13 8. Voltage Reference Filter Circuit – The zero in the numerator of the transfer function improves the relative stability of the cir- should be kept fairly low, since a small amount of bias current flows cuit. Resistor R 2 through it and causes dc error and noise. Resistor R value ranges from 10 Ω to 50 Ω . 1 is in the feedback loop, so any small leakage current that is due to capacitor Resistor R 1 flows through R1and the voltage dropped across R is divided by the loop gain. For C 1 1 all practical purposes, the voltage across C is 0 V and hence give rise to negligible leak- 2 age current. A design example for a 3-kHz bandwidth filter is illustrated: F and R1 = 42.2 Choose C1 = 1.2 μ Ω . Having determined the value for C1 and R1, the capacitor C2 value is estimated to be value (C2 = 0.047 μ F) and resistor R2 is calculated using approximately 4% to 5% of C1 the approximation C = 2C R Ω) (R2 = 2.15 k R . 1 1 2 2 The calculated – 3-dB bandwidth for the circuit is 3.1 kHz and this value agrees with the 9. This circuit topology is good for circuit ’ s frequency response plot shown in Figure 13 – driving large capacitive loads. 13-16

233 External Vref Circuits for ADCs/DACs 50 20 0 0 50 – 20 – Phase – 100 – 40 dB – Degrees – Amplitude OUT 150 – V 60 – Phase – 200 – 80 – 250 – – 100 – 300 – 120 1 100 100 10 1 10 10 100 MHz kHz kHz MHz MHz Hz kHz Hz Frequency – f Figure 13 – 9. Voltage Follower Frequency Response Plot 10 shows an external reference circuit that provides a wide adjustment range – Figure 13 of the ADC full-scale range. Resistors R281 and R282 play two roles in this circuit: D Form part of the low-pass filter used to rolloff noise Isolate the ADC s reference input load capacitance from the buffer op amp output D ’ ) for the differential Potentiometer R247 sets the external common-mode voltage (V ocm – amplifier in Figure 13 11. TLE2227 R 282 10 REF – C 4.7 F 196 10 V V REF R R 296 247 V OCM R 9 2 k R 2 OUTPUT 1.1 k 10 POT 2.21 k R REF + 246 2 k 4.7 F C R 197 281 R 100 POT 10 2 k TLE2227 R POT 1 1.05 k – 10. External Voltage Reference Circuit for ADC/DAC. Figure 13 13-17 Wireless Communication: Signal Conditioning for IF Sampling

234 High-Speed Analog Input Drive Circuits 13.8 High-Speed Analog Input Drive Circuits Communication ADCs, for the most part, have differential inputs and require differential input signals to properly drive the device. Drive circuits are implemented with either RF transformers or high-speed differential amplifiers with large bandwidth, fast settling time, low output impedance, good output drive capabilities, and a slew rate of the order of 1500 μ S. The dif ferential amplifier is V/ usually configured for a gain of 1 or 2 and is used primari- ly for buffering and converting the single-ended incoming analog signal to dif ferential out- puts. Unwanted common-mode signals, such as hum, noise, dc, and harmonic voltages are generally attenuated or cancelled out. Gain is restricted to wanted differential signals, which are often 1 V to 2 V. The analog input drive circuit, as shown in Figure 13 – 11, employs a complementary bipo- lar (BiCom) THS4141 device. BiCom offers fast speed, linear operation over a wide fre- range, and wide power-supply voltage range, but draws slightly more current than quency a BiCMOS device. The circuit closed-loop response is shown in Figure 13 – 12, where the – 3-dB bandwidth is 120 MHz measured at the output of the amplifier. The analog input is ac-coupled to the THS4141 and the dc voltage V is the applied input common- V in ocm C34 are selected to meet the de- mode voltage. The combination R47 – C57 and R26 – sired f requency rolloff. If the input signal frequency is above 5 MHz, higher-order low-pass techniques (third-order or greater) are employed to reduce the op amp ’ s inherent filtering second harmonic distortion component. 1.8 pF C 45 511 R THS1470 R 28 47 R 32 V IN C 57 49.9 511 14 IN + ADC R 27 47 pF – – 49.9 – IN THS4141 R 26 CLK + CLK REF REF + R 31 C 45 49.9 47 pF 511 V OCM Input 0.01 F 0.1 F 1.8 pF C 27 R 36 523 Figure 13 – 11. Single-Ended to Differential Output Drive Circuit 13-18

235 High-Speed Analog Input Drive Circuits 10 0 = 120.8 MHz – 3dB 10 – dBV – 20 – OUT V – 30 – 40 THS4141 120.8 MHz Gain = 1 200 100 Phase at Degrees – 120.8 MHz = 38.9 0 Phase 100 – 100 1 10 100 1 kHz MHz GHz MHz MHz – Frequency f Figure 13 – 12. Differential Amplifier Closed-Loop Response – Figure 13 13 shows a design example of an ac-coupled, single-ended analog input drive circuit. This circuit uses a THS3201 current-feedback op amp and operates up to 975 MHz. The amplifier is configured as a noninverting amplifier with a gain of 2, where gain is 1 + R4/R3. In a current-feedback amplifier, the feedback resistor sets the amplifier 13-19 Wireless Communication: Signal Conditioning for IF Sampling

236 High-Speed Analog Input Drive Circuits bandwidth and the frequency response shape as well as defining the gain of the circuit. R4 and R3 cannot be arbitrary values. V OCM THS1470 R THS3201 C 66 V IN IN + R ADC 5 49.9 – IN 47 pF 100 nF 49.9 V COM R 4 0.1 F 619 C 16 R 7 2 k 4.7 pF R 3 619 Figure 13 – 13. ADC Single-Ended Input Drive Circuit , are shown in Figures Ω The frequency response plots for a gain of 2, with R3 = R4 = 619 – 13 and 13 – 14. R4 affects the amplifier bandwidth and frequency response peaking. 13 3-dB R3 has no effect on bandwidth and frequency peaking; it only affects the gain. The – bandwidth is 520 MHz. Increasing the resistance of R4 decreases the bandwidth. Con- versely, lowering R4 resistance increases the bandwidth at the expense of increased peaking in the ac response. The phase plot in Figure 13 – 14 exhibits a fairly linear phase shift (a flat group delay re- sponse) and hence the amplifier output should show excellent signal reproduction. Unlike a voltage-feedback amplifier, the power supply voltage affects the bandwidth of a current-feedback amplifier. For example, lowering the supply voltage of the THS3201 2.5 V reduces the bandwidth from 925 MHz to 350 MHz. ± from ± 5 V to 13-20

237 High-Speed Analog Input Drive Circuits 10 5 0 dB – 5 – OUT V – 10 – 15 – 20 100 1 10 100 1 kHz MHz GHz MHz MHz f – Frequency – 14. Gain vs Frequency Plot for THS3201 Figure 13 0 50 – 100 – 150 – Degrees – 200 – Phase – 250 300 – 1 1 10 100 100 MHz MHz GHz MHz kHz f – Frequency 15. Phase vs Frequency for THS3201 – Figure 13 13-21 Wireless Communication: Signal Conditioning for IF Sampling

238 References 13.9 References Theory and Application of Digital signal Processing [1] Rabiner, L.R., and Gold. B., , Pren- tice Hall, Englewood Cliffs, New Jersey, 1975 Digital Filter Design [2] Parks, T.W., and Burrus, C.S., , Wiley, New York, NY, 1987 [3] Miller, P., Aspects of Data Acquisition System Design , Analog Application Journal, Tex- as Instruments, August 1999 ork, 1983 [4] Proakis, J.G., , McGraw – Hill Book Company, New Y Digital Communications [5] Miller, P., and Moore, D ., Precision Voltage References , Analog Application Journal, Texas Instruments, November 1999 [6] Miller, P., and Moore, D ., The Design and Performance of a Precision Voltage Refer- , Analog Application Jour- and 16-Bit A-to-D and D-to-A Converters ence Circuit for 14-Bit nal, Texas Instruments, May 2000 rd Edition, Prentice Hall, – , 3 Analog [7] Daniel H. Sheingold, Digital Conversion Handbook Englewood Cliffs, New Jersey 07632 [8] Graeme J.G., Tobey G.E., and Huelsman L.P., Burr – Brown, Operation Amplifiers De- sign and Application , McGraw – Hill Book Company, New York, 1971 [9] Schartz, M., Information Transmission, Modulation and Noise , McGraw – Hill Book Company, New York, 1970 [10] Papoulis A., On The Approximation Problem on Filter Design , IRE conv. Record, Pt.2, – 1957 pp. 175 13-22

239 Chapter 14 Interfacing D/A Converters to Loads Bruce Carter 14.1 Introduction A digital to analog converter, or D/A, is a component that takes a digital word and converts it to a corresponding analog voltage. It has the opposite function of an A/D converter. The D/A is only capable of producing a quantized representation of an analog voltage, not an infinite range of output voltages. The application will almost always dictate the selection of the D/A converter, leaving the designer the task of interfacing that converter with the output load. A D/A converter interfaces with a buffer op amp. Most D/A converters are manufactured with a process that is incompatible with op amps. Therefore, the op amp cannot be manufactured on the same IC. It must be external, and its characteristics are an integral part of the conversion process. In most cases, the data sheet will make a recommendation for the selection of a buffer op amp. Follow the recommendation, unless there is a compel- ling reason not to do so. Performance can be improved only if you know exactly what op amp specifications need to be optimized. Signal conditioning — low pass filtering, DC offsets, and power stages — should all be placed after the recommended op amp buffer. Do not attempt to combine these functions with the buffer unless you are an experienced designer with a good grasp of all of the im- plications. 14.2 Load Characteristics There are two main types of loads that a D/A may have to drive — ac and dc. Each has different characteristics, and will require different interface circuitry. 14.2.1 DC Loads These include linear actuators such as those used on positioning tables, motors, pro- grammable power supplies, outdoor displays, and lighting systems. Large load currents 14-1

240 Understanding the D/A Converter and its Specifications and/or high voltages characterize some of these loads. DC accuracy is important, be- related to a series of desired mechanical positions or intensities in the load de- cause it is vice. 14.2.2 AC Loads These include linear actuators such as those used on positioning tables, motors, pro- grammable power supplies, outdoor displays, and lighting systems. Large load currents and/or high voltages characterize some of these loads. DC accuracy is important, be- cause it is related to a series of desired mechanical positions or intensities in the load de- vice. 14.3 Understanding the D/A Converter and its Specifications It is important to understand the D/A converter and its specifications before discussing interfaces. 14.3.1 Types of D/A Converters — Understanding the Tradeoffs D/A converters are available in several types, the most common of which is the resistor ladder type. There are several variations on the resistor ladder technique, with the R/2R configuration being the most common. 14.3.2 The Resistor Ladder D/A Converter 1 N – In this type of converter, a precision voltage reference is divided into 2 parts in an inter- nal voltage divider, where N is the number of bits specified for the converter. One switch at a time turns on, corresponding to the correct dc level (Figure 14 – 1). V REF B3 R B2 _ VOUT + R B1 R B0 – 1. Resistor Ladder D/A Converter Figure 14 14-2

241 Understanding the D/A Converter and its Specifications Unfortunately, the number of resistors and switches doubles for each additional bit of re- solution. This means that an 8-bit D/A converter would have 255 resistors and 256 switches, and a 16-bit D/A converter would have 65535 resistors and 65536 switches. For this reason, this architecture is almost never used for higher resolution D/A converters. 14.3.3 The Weighted Resistor D/A Converter This type of converter is very similar to the Resistor Ladder D/A converter. In this case, however, each resistor in the string is given a value proportional to the binary value of the bit it represents. Currents are then summed from each active bit to achieve the output 2). (Figure 14 – V REF 8R 4R 2R R B0 B3 B2 B1 0 1 0 1 0 1 1 0 R I1 I2 I3 I0 _ IIN VOUT + Figure 14 2. Binary Weighted D/A Converter – The number of resistors and switches reduced to one per bit, but the range of the resistors is extremely wide for high-resolution converters, making it hard to fabricate all of them on to the IC. The resistor used for B3 is the limiting factor for power dissipation from V REF ground. This converter architecture is often used to make logarithmic converters. In this case, the resistors are replaced with logarithmically-weighted resistors. R, 2R, 4R, 8R ... This type of converter, and the R/2R converter described in the next paragraph, use a feedback resistor fabricated on the D/A IC itself. This feedback resistor is not an optional convenience for the designer — it is crucial to the accuracy of the D/A. It is fabricated on the same silicon as the resistor ladder. Therefore, it experiences the same thermal drift as the resistor ladder. The gain of the buffer amplifier is fixed, with a full-scale output volt- . . If a different full-scale D/A output voltage is needed, change V age limited to V REF REF 14-3 Interfacing D/A Converters to Loads

242 Understanding the D/A Converter and its Specifications If the full scale V must exceed the maximum rating of the D/A reference voltage, use OUT a gain stage after the buffer op amp (see Paragraph 14.7.2). The op amp must be selected carefully, because it will be operated in much less than unity gain mode for some combinations of bits. This is probably one of the main reasons why this architecture is not popular, as well as the requirement for a wide range of resistor val- ues for high precision converters. 14.3.4 The R/2R D/A Converter An R/R2 network can be used to make a D/A converter (Figure 14 – 3). R 2R 2R I I I/2 I I I/2 V V V REF REF REF I/2 I/2 R R I/4 2R I/4 2R 2R I/4 I/2 I V REF I/4 I/2 2R R 2R 2R I/8 2R I/2 I V REF R I/2 2R I/16 R I/2 2R R Figure 14 – 3. R/2R Resistor Array For a given reference voltage V , a current I flows through resistor R. If two resistors, REF to ground, a current I/2 flows through each the same value (2R) are connected from V REF each leg of the circuit. But the same current will flow if one leg is made up of two resistors, each with the value of R. If two resistors in parallel whose value is 2R replace the bottom resistor, the parallel combination is still R. I/4 flows through both legs, adding to I/2. Ex- tending the network for 4 bits as shown on the right, the total current on the bottom leg is I/4 plus I/8 plus I/16 plus I/16 in the resistor to ground. Kirchoff ’ s current law is satisfied, and convenient tap points have been established to construct a D/A converter (Figure – 4): 14 14-4

243 Understanding the D/A Converter and its Specifications R R R V REF 2R 2R 2R 2R 2R B2 B1 B0 B3 1 0 0 1 0 1 1 0 R I0 I2 I1 I3 _ IIN VOUT + Figure 14 – 4. R/2R D/A Converter This converter architecture has advantages over the types previously mentioned. The resistors has doubled from the number required for the current-summing type, number of but there are only two values. Usually, the 2R resistors are composed of two resistors in series, again fabri- each with a value of R. The feedback resistor for the buffer amplifier is cated on the converter itself for maximum accuracy. Although the op amp is still not oper- ated in unity gain mode for all combinations of bits, it is much closer to unity gain with this architecture. The important op amp parameters for all resistor ladder D/As are: D Input offset voltage — the lower the better. It adds to the converter offset error. D Input bias current — the lower the better. The product of the bias current and the feedback resistance creates an output offset error. D Output voltage swing — it must meet or preferably exceed the zero to full-scale swing from the D/A. Settling time and slew rate — must be fast enough to allow the op amp to settle D before the next digital bit combination is presented to the D/A input register. 14.3.5 The Sigma Delta D/A Converter The sigma delta D/A converter takes advantage of the speed of advanced IC processes to do a conversion as a series of approximations summed together. A phase-locked loop- derived (PLL) sample clock operates at many times the overall conversion frequency — × . The PLL is used to drive an interpolation filter, in the case shown in Figure 14 – 5, it is 128 The conversion is done by using the density a digital modulator, and a 1-bit D/A converter. 14-5 Interfacing D/A Converters to Loads

244 D/A Converter Error Budget ratio of the voltage out of the 1-bit D/A as the analog signal. As the pattern of 1s and 0s is presented to the 1-bit converter, their time average at the sample frequency recreates the analog waveform. SAMPLE PLL CLOCK (Fs) 128 Fs 16 16 1 INTERPOLATION 1-BIT DIGITAL DATA D/A MODULATOR FILTER Figure 14 – 5. Sigma Delta D/A Converter Sigma delta converters are popular for audio frequencies, particularly CD players. The primary limiting factor is the sample clock. CD players operate at a sample rate of 44.1 kHz, which means that according to Nyquist sampling theory, the maximum audio fre- quency that can be reproduced is 22.05 kHz. If an audio frequency of 23.05 kHz is present in the recorded material, it will alias back into the audio output at 1 kHz — producing an annoying whistle. This places a tremendous constraint on the low-pass filter following the D/A in a CD player. It must reject all audio frequencies above 22.05 kHz while passing those up to 20 kHz, the commonly accepted upper limit of human hearing. While this can be done in conventional filter topologies, they are extremely complex (9 or more poles). Inevitably, phase shift and amplitude rolloff or ripple will start far below 20 kHz. The original because of this. CD players often sounded a bit ” “ harsh ” or “ dull The solution was to overclock the sample clock. To keep things simple, designers made × or even higher oversam- it a binary multiple of the original sampling frequency. T oday, 8 pling is standard in CD players. Little do the audio enthusiasts know that the primary rea- son why this was done was to substantially reduce the cost of the CD player! A faster sam- × oversampling, the CD player ple clock is very cheap. Nine-pole audio filters are not. At 8 only needs to achieve maximum rolloff at 352.8 kHz a very easy requirement. Instead — of the filter having to roll off in a mere 2 kHz of bandwidth, now it has 332 kHz of bandwidth to accomplish the rolloff. The sound of an oversampled CD player really is better, but it comes at the cost of increased radiated RFI, coming from the sample clock. Sigma delta converters introduce a great deal of noise onto the power rails, because the internal digital circuitry is continually switching to the power supply rails at the sample . clock frequency F S 14.4 D/A Converter Error Budget The system designer must do an error budget to know how many bits are actually needed ” to meet the system requirements — how much “ graininess or what step size is accept- able in the output signal. 14-6

245 D/A Converter Error Budget 14.4.1 Accuracy versus Resolution It is important for the designer to understand the difference between converter accuracy and converter resolution. The number of bits determines resolution of a converter. Insuffi- — it is a design characteristic of the D/A. If a given converter ’ s cient resolution is not error resolution is insufficient, use a converter with better resolution (more bits). Accuracy is the error in the analog output from the theoretical value for a given digital in- put. Errors are described in the next paragraph. A very common method of compensating for D/A error is to use a converter that has one or two bits more resolution than the applica- tion requires. With the cost of converters coming down, and more advanced models being introduced every day, this may be cost effective. 14.4.2 DC Application Error Budget DC applications will depend on the value of dc voltage coming out of the converter. THD and signal-to-noise will not be important because the frequency coming out of the convert- er is almost dc. ± 1/2 LSB, where an LSB is defined as: The resolution of a converter is V FS – (14 1) + 1LSB N 2 * 1 Where: Full-scale output voltage V FS Number of converter bits N The number of bits in a dc system determines the dc step size that corresponds to a bit. Table 14 – 1 shows the number of bits, and the corresponding voltage step size for three popular voltages: Table 14 – 1. DC Step Size for D/A converters BITS 3 V 5 V 10 V STATES 4 16 0.1875 0.3125 0.625 8 256 0.011719 0.019531 0.039063 0.009766 1024 10 0.004883 0.00293 12 0.001221 0.002441 0.000732 4096 14 16384 0.000183 0.000305 0.00061 65536 4.58 E – 05 7.63 E – 05 0.000153 16 18 262144 1.14 E – 05 1.91 E – 05 3.81 E – 05 20 1048576 2.86 E – 06 4.77 E – 06 9.54 E – 06 1.19 E 22 7.15 E – 07 4194304 – 06 2.38 E – 06 – 07 24 16777216 1.79 E – 07 2.98 E – 07 5.96 E 14-7 Interfacing D/A Converters to Loads

246 D/A Converter Error Budget The bit step size can get critical, especially for portable equipment. There is a requirement to operate off of low voltage, to minimize the number of batteries. The buffer amplifier, if it includes gain, will use large resistor values, lowering its noise immunity. Fortunately, the vast majority of dc applications are not portable; they are in an industrial environment. For example: A converter is used to position a drill on a table used to drill PCB holes. The positions of the holes are specified as 0.001 inches, 0.0003 inches. The actuators are ± centered on the table at zero volts, with full negative position of – 12 inches occurring at – 5 V, and full positive position of +12 inches occurring at +5 V. There are two actuators, one for vertical, and one for horizontal. This example has several aspects. The first is that the positioning voltage has to swing both positive and negative. In the real world, it may be necessary to add (or subtract in this case) a fixed offset to the D/A output. The output voltage has to swing over a 10-V range, which may mean that the output of the D/A has to be amplified. The actuators them- selves probably operate off of higher current than the D/A is designed to provide. Section 14.7 covers some methods for meeting these requirements. Assume, for now, that the D/A has the necessary offset and gain. A ± 12 inch position is circuitry. The 24-inch range must 5 V from the D/A 24 inches total, which corresponds to ± be divided into equal 0.0003-inch steps to meet the resolution requirement, which is 80,000 steps. From Table 14 1, an 18-bit D/A converter is required. The actual system – able to position with a step size of 0.0000916 inches. Two independent conversion will be systems are needed, one for horizontal and one for vertical. 14.4.3 AC Application Error Budget The error budget for an ac application will most likely be specified as total harmonic distor- tion, dynamic range, or signal-to-noise ratio. Assuming no internal noise, and no noise in the buffer op amp circuitry, the inverse of the dynamic range is the signal-to-noise ratio of the converter D/A. Of course, noise is always present, and is measured with all input data set to zero. Noise will make the S/N ratio decrease. The number of converter bits, however, is the overwhelming factor determining these parameters. Technically, they are not errors ” , because the design of the converter sets “ them. If the designer cannot live with these design limits, the only choice is to specify a converter with better resolution (more bits). 14.4.3.1 Total Harmonic Distortion The total harmonic distortion of an ideal D/A converter is the quantization noise due to the converter resolution. The number of bits of the converter determines the lowest possible total harmonic distortion. The greater the number of bits, the lower the amplitude of the – 6. harmonics, as shown in Figure 14 14-8

247 D/A Converter Error Budget 20 Fundamental 0 F 20 – Harmonics dB – 40 — – 60 80 – Amplitude H3 H9 H8 H7 H6 H4 H5 H2 100 – 120 – 130 – 10 k 8 k 4 k 5 k 7 k 6 k 3 k 2 k 1 k 9 k 0 Hz – Frequency Figure 14 – 6. Total Harmonic Distortion Assuming ideal D/A conversion, there is a direct relationship between the number of bits and the THD caused by the resolution: 1 2) (14 – THD(%) 100 + N 2 Where N is the number of converter bits. Of course, this is the limit for ideal conversion. 14.4.3.2 Dynamic Range There is also a direct relationship between the number of bits (n) and the maximum dy- – namic range of the D/A (Equation 14 – 2 and Table 14 2): – 3) (14 6.02 Dynamic Range + n ) 1.76 Notice that there is approximately a 6-dB improvement in dynamic range per bit. This is an easy way to figure out what improvement can be realized by increasing the number of bits from one value to another. 14-9 Interfacing D/A Converters to Loads

248 D/A Converter Errors and Parameters Table 14 2. Converter Bits, THD, and Dynamic Range – STATES Dynamic Range BITS THD 16 6.25% 4 25.8 8 256 0.390625% 49.9 10 1024 0.097656% 62.0 12 0.024414% 74.0 4096 14 16384 0.006104% 86.0 16 65536 0.001526% 98.1 18 262144 0.000381% 110.1 122.2 0.000095% 20 1048576 22 0.000024% 134.2 4194304 24 16777216 0.000006% 146.2 For example, if the designers of a CD player want to have a 90-dB signal-to-noise ratio, they would pick a 16-bit converter from Table 14 – 2. The THD is 0.0015% minimum. 14.4.4 RF Application Error Budget RF applications are a high-frequency subset of ac applications. RF applications may be concerned with the position and relative amplitude of various harmonics. Minimizing one harmonic at the expense of another may be acceptable if the overall RF spectrum is within specified limits. 14.5 D/A Converter Errors and Parameters The D/A errors described in this section will add to the errors caused by the resolution of the converter. This section is divided into dc and ac sections, but many of the dc errors masquerade as ac errors. A given D/A may or may not include either dc or ac error specifications. This give the designer a clue that the device is optimized for dc or ac applications. Like should any component, D/A converters are designed with tradeoffs. It is possible to misapply a converter meant for high-frequency ac operation in a dc application, etc. 14.5.1 DC Errors and Parameters The following paragraphs describe D/A dc errors and parameters. 14.5.1.1 Offset Error The analog output voltage range for the complete range of input bits may be shifted linear- V from 0 V ly from the ideal 0 to full-scale value (Figure 14 – 7). The offset error is the ±∆ that results when a digital code is entered that is supposed to produce 0. 14-10

249 D/A Converter Errors and Parameters 3 Actual Ideal 2 1 Analog Output Voltage Actual Offset Offset Error (+ 1 1/4 LSB) 0 001 010 011 000 Nominal Digital Output Code Offset – 7. D/A Offset Error Figure 14 the offset error is the offset error temperature coefficient, which is the change Related to in offset over temperature. This is usually specified in ppm/ _ C. Offset error is critical in dc applications. For this reason, a buffer op amp must be selected does not contribute to the problem that — its own offset voltages should be much less than that of the converter. In ac applications, the offset error is not important and can be igno- red. The buffer op amp can be selected for low THD, high slew rate, or whatever other parameters are important for the application. 14.5.1.2 Gain Error The gain of the D/A converter may greater than or less than the gain needed to produce 8). The gain error is the difference in the desired full-scale analog voltage (Figure 14 – slope between the ideal D/A output gain and the actual gain. 14-11 Interfacing D/A Converters to Loads

250 D/A Converter Errors and Parameters 7 Gain Error ( 1 1/4 LSB) – 6 5 Ideal Analog Output Value Actual Gain 4 0 110 000 100 101 111 Digital Input Code – 8. D/A Gain Error Figure 14 Related to the gain error is the gain error temperature coefficient, which is the change in gain over temperature. Gain error can be critical in both ac and dc applications. For example: D An RF predriver must not cause the output stage to exceed FCC license requirements. D A mechanical positioner must not stop short of or go past its intended position. The op amp buffer should be operated with the internal feedback resistor. If possible, full- . This way, tolerances and thermal scale amplitude adjustments should be made to V REF drift in external resistors do not contribute to the gain error. 14.5.1.3 Differential Nonlinearity Error V) is not the same for every combination of bits, When the increase in output voltage ( ∆ the converter has a differential nonlinearity error (DNL). If the DNL exceeds 1 LSB, the converter is nonmonotonic. This can cause a problem for some servo control loops. A 9 as a momentary dip in the analog output nonmonotonic D/A would appear in Figure 14 – characteristic. 14-12

251 D/A Converter Errors and Parameters 6 1 LSB 5 – DNL Error ( 1/4 LSB) 4 3 1 LSB 2 Analog Output Value DNL Error (+1/4 LSB) 1 0 001 101 100 011 010 000 Digital Input Code Figure 14 – 9. Differential Nonlinearity Error 14.5.1.4 Integral Nonlinearity Error The integral nonlinearity error is similar to the differential nonlinearity error, except it is a first-order effect that stretches across the entire range from 0 to full scale output voltage 10). (Figure 14 – 7 6 5 4 At Step 011 (1/2 LSB) 3 2 Analog Output Value End-Point At Step 001 Linear Error 1 (1/4 LSB) 0 101 110 111 000 010 011 100 001 Digital Input Code – 10. Integral Nonlinearity Error Figure 14 14-13 Interfacing D/A Converters to Loads

252 D/A Converter Errors and Parameters Both the INL and DNL errors affect ac applications as distortion and spectral harmonics (spurs). In dc applications, they will result in an error in the dc output voltage. The mechan- ical steps of a positioning table, for instance, may not be exact increments. 14.5.1.5 Power Supply Rejection Ratio The power supply rejection ratio is sometimes called the power supply sensitivity . It is the ability of the converter to reject ripple and noise on its power inputs. DC applications may not be adversely affected. Poor power supply rejection can cause spurs and harmonic distortion in ac applications, as external frequency components leak into the output and modulate with it. The designer must decouple the D/A and buffer op amp carefully to com- bat these problems. 14.5.2 AC Application Errors and Parameters The following paragraphs describe D/A ac errors and parameters. 14.5.2.1 THD+N There will always be some noise that is generated internally in the converter and buffer amp. A useful specification for audio and communication system designers is the THD+N (total harmonic distortion plus noise). The distortion plus noise (THD+N) is the ratio of the sum of the harmonic distortion and noise to the rms power of the input signal. As was the case with op amp parameters (Chapter 11), the noise sources add according to the root- mean-square law. The distortion and noise are measured separately and then added to- gether to form the ratio. The noise voltage relates to the measured bandwidth. 14.5.2.2 SINAD The signal-to-noise and distortion (SINAD) is the ratio of the input signal to the sum of the harmonic distortion and noise. The distortion and noise are measured separately and added together to form the ratio. The SINAD is the reciprocal to the THD+N. The SI- than NAD and THD+N are a good indication of the overall dynamic performance of the ADC, because all components of noise and distortion are included. 14.5.2.3 ENOB The SINAD is used to determine the effective number of bits (ENOB) of accuracy the con- verter displays at that frequency. For example, a nominal 8-bit resolution D/A may be spe- cified as having 45-dB SNR at a particular input frequency. The number of effective bits is defined as * 1.76 SNR REAL (14 – 4) ENOB + 7.2 bits + 6.02 The actual performance of the device is therefore less than its nominal resolution at this frequency. 14-14

253 D/A Converter Errors and Parameters 14.5.2.4 Spurious Free Dynamic Range Spurious free dynamic range is the dif ference in dB between the maximum signal compo- 11): nent and the largest distortion component (Figure 114 – 20 F — Fundamental 0 20 – dB SFDR 40 – — 60 – 80 – Amplitude 100 – – 120 – 130 0 7 k 6 k 5 k 8 k 2 k 1 k 9 k 10 k 3 k 4 k Hz Frequency – Figure 14 – 11. Spurious Free Dynamic Range It is an important specification in RF applications, where FCC regulations specify the mag- nitude of spurs. Improper decoupling may cause spurs. A notch filter can be used to eliminate a spur, but many RF applications are RF agile — changing the frequency of the spur as well. The notch must catch all spur frequencies, or it is useless. 14.5.2.5 Intermodulation Distortion The differential and integral nonlinearity errors described previously appear in a high-fre- – 12). quency ac application as intermodulation distortion (Figure 14 14-15 Interfacing D/A Converters to Loads

254 D/A Converter Errors and Parameters 20 0 20 – f2 f 1 dB – 40 — – 60 – 80 2f – f 1 2 + f f – 2f f Amplitude f f – 1 2 1 2 1 2 100 – – 120 – 130 1 k 3 k 6 k 7 k 5 k 4 k 8 k 2 k 9 k 10 k 0 – Hz Frequency Figure 14 – 12. Intermodulation Distortion The best method of combating intermodulation distortion is to make the buffer amplifier system as linear as possible (beware of rail-to-rail op amps that may not be linear near the voltage rails). Try to limit current through the internal feedback resistor in the DAC. See Paragraph 14.7.1 on increasing the voltage rail for suggestions about reducing inter- nal feedback resistor power dissipation. 14.5.2.6 Settling Time The settling time of a D/A converter is the time between the switching of the digital inputs of the converter and the time when the output reaches its final value and remains within 13). Settling time is the reciprocal of the maximum D/A a specified error band (Figure 14 – conversion rate. 14-16

255 D/A Converter Errors and Parameters Settling Time — t S Final Value Error Band Glitch Slew Rate Analog Output Voltage Delay Time t Digital Change Figure 14 – 13. D/A Settling Time When an output buffer op amp is used with a D/A, it becomes a part of the settling time/ conversion rate calculations. Related to the settling time is a glitch response that occurs when the digital code changes state. Even though this effect is transitory in nature, it can result in noise or harmonics when used in fast AC applications. The best way of reducing the glitch is to properly de- the D/A and op amp buf fer (see Chapter 17). In extreme cases, a deglitching circuit couple may be needed (Figure 14 14). – R C HOLD _ R DEGLITCHED BUFFER OUT OUTPUT + CONTROL Figure 14 – 14. D/A Deglitch Circuit This technique relies on the software designer to balance the timing of the control signal so it activates the hold function right before the D/A input code changes, then releases it must hold is critical — the hold right after the code has changed. The selection of C HOLD the buffer output without droop and without compromising system bandwidth. 14-17 Interfacing D/A Converters to Loads

256 Compensating For DAC Capacitance 14.6 Compensating For DAC Capacitance D/A converters are constructed of either bipolar or CMOS technology, with CMOS being the more common. CMOS transistors, however, have a lot of capacitance. This capaci- tance will add in D/A converters, depending on the number of resistors switched on or off. inverting op amp input is a good way to cause it to oscillate, especially Capacitance at an some buffer amplifiers will be operated at less than unity gain. The converter capaci- since 15). must be compensated for externally (Figure 14 – tance C O D/A RF CF _ VREF VOUT CO RS RIN D VREF/R + 15. Compensating for CMOS DAC Output Capacitance Figure 14 – The normal technique for compensating the buffer amplifier for output capacitance is to add a feedback capacitor C . C is calculated by the following: F F C O 1 – 5) (14 Ǹ C 2 + F 2 p R G F BW Where: C the output capacitance from the D/A data sheet O the feedback resistance from the D/A data sheet R F G the small signal unity gain bandwidth product of the output amplifier BW will both and the internal D/A capacitance C Unfortunately, the feedback capacitors C O F limit the conversion speed of the D/A. If faster conversion is needed, a D/A with a lower output capacitance, and therefore a lower feedback compensation capacitor will be need- ed. The overall settling time with the external capacitance is: 14-18

257 Increasing Op Amp Buffer Amplifier Current and Voltage Ǔ ǒ C ) C R F F O (14 – 6) Ǹ [ T S 2G BW Where: the D/A internal capacitance C O R the feedback resistor F the compensation capacitance C F the small signal unity gain bandwidth product of the output amplifier G BW 14.7 Increasing Op Amp Buffer Amplifier Current and Voltage Process limitations of op amps limit the power that can be dissipated at the output. Unfor- tunately, there are applications that will require the DAC to interface to loads that dissipate considerable power. These include actuators, position solenoids, stepper motors, loud- the possibilities are endless. — speakers, vibration tables, positioning tables several “ power op amps ” are available that can drive heavy loads, they usually com- While promise several other specifications to achieve the high power operation. Input voltage offset, input current, and input capacitance can be decades higher than the designer is accustomed to, and make these power op amps unsuitable for direct interface with a DAC as a replacement for the buffer op amp. The power booster stage can be designed discretely, or a prepackaged amplifier of some sort, depending on what is needed for the application. Sometimes high current is required for driving loads such as actuators and stepper motors. Audio applications can require a lot of wattage to drive loudspeakers. This implies a higher voltage rail than op amps commonly operate at. This and other high voltage applications can operate off of, and generate lethal voltages. The designer needs to be extremely careful not to create an un- safe product, or be electrocuted while developing it. The power stage is most often included in the feedback loop of the op amp circuit, so that the closed loop can compensate for power stage errors. This is not always possible if the voltage swing of the output exceeds that of the op amp voltage rails. In these cases, a voltage divided version of the output should be used. There are three broad categories of booster, the current booster, the voltage booster, and boosters that do both. All of them work on the same principle: anything that is put inside — the feedback loop of the op amp will be compensated for the output voltage will swing to whatever voltage it needs to make the voltage at the buffer op amp inputs equal. 14-19 Interfacing D/A Converters to Loads

258 Increasing Op Amp Buffer Amplifier Current and Voltage 14.7.1 Current Boosters These usually use some variation of the class-B push-pull amplifier topology (Figure 14 – 16). + 15 V R (bias) D/A NPN DIODE RF CF R D VREF/R (small) _ VREF CO RIN RS + R VOUT (small) DIODE PNP R (bias) 15 V – Figure 14 – 16. D/A Output Current Booster – The circuit in Figure 14 many resources are avail- 16 has been employed for decades — able that can be used to design exact component values. It boosts current because the output impedance of the op amp has been bypassed, and used as the driver for the base of the NPN and PNP power transistors. The two diodes compensate for the VBE drop in the transistors, whose bases are biased by two resistors off of the supplies. The output of the booster stage is fed back to the feedback resistor in the D/A to complete the feed- back loop. The output impedance of the stage is only limited by the characteristics of the output transistors and small emitter resistors. Modern power transistors have such high frequency response that this circuit may oscillate. The RC snubber network and a small — inductor in series with the load can be used to damp the oscillation or be omitted if os- cillation is not a problem. Beware of varying transistor betas, however. 14.7.2 Voltage Boosters If even more current is needed, or the output voltage swing must be more than ± 15 V, the booster stage can be operated at voltages higher than the buffer amplifier potentials. A – 17. designer might be tempted to try the circuit of Figure 14 14-20

259 Increasing Op Amp Buffer Amplifier Current and Voltage RF-EXT D/A +100 V +15 V RF CF D VREF/R _ VREF VOUT CO RS RIN + – – 100 V 15 V 17. Incorrect Method of Increasing Voltage Swing of D/A Converters Figure 14 – Any time there are higher voltage rails on the output section, there are potential hazards. The circuit above illustrates a common misapplication. to swing to a ± 100-V D The whole reason for using the booster amp is to allow the V OUT rail. If this circuit was operated in the unity gain mode (external R = 0), the VOUT F will only swing ± 15 V, maximum. There would be no need for the ± 100-V rail. That voltage rail is there to allow voltage gain. D If the circuit is operated with a gain (external R larger than 0), the external R adds F to the internal R to create the gain: F R ) R EXT F * INT F * – (14 7) + Gain R S The problem with this is that the wattage of the resistors increases as the external voltage rail increases. The designer has control over the wattage of the external R , but has no F . Because these resistors are fabricated on the or R control whatsoever over internal R F S IC, their wattage is limited. Even if the wattage rating of the internal resistors is meticulous- ly observed, they may have undesirable thermal coefficients if allowed to dissipate that wattage. Resistor self-heating will change the resistance according to its rated tempera- ture coefficient (maximum). The external resistor is sure to have a different thermal coeffi- cient from the internal resistors, causing a gain error. The designer may never have en- countered the effects of resistor self-heating before, because through-hole and surface- mount devices have enough bulk to minimize the effect of self-heating. At the geometries present on IC D/As, resistor self-heating is a much more pronounced ef fect. It will produce a nonlinearity error in the D/A output. This effect is most pronounced in high-resolution converters, where the geometry is the smallest. The designer, therefore, must limit the current in the feedback resistor if at all 18 shows a method of achieving gain control while keeping the high possible. Figure 14 – current path out of the internal feedback resistor: 14-21 Interfacing D/A Converters to Loads

260 Increasing Op Amp Buffer Amplifier Current and Voltage D/A +15 V +100 V RF CF D VREF/R _ VREF VOUT CO RS RIN + R3 – – 100 V 15 V R2 Figure 14 – 18. Correct Method of Increasing Voltage Range In Figure 14 – 18: D R3 and R2 are selected to ensure that the feedback voltage to the D/A internal R F can never exceed the D/A rated limits. D R3 and R2, of course, have to be the correct power rating. R2, in particular, has to be carefully selected. If it burns out, the feedback loop will present hazardous voltages to the D/A. R3, which drops the bulk of the voltage, will have to dissipate considerable wattage. If the combination of voltage swing and power ratings cannot be balanced to achieve a design, the only choice left to the designer will be to break the feedback loop and working live with the loss of accuracy. For ac applications, this may be acceptable. 14.7.3 Power Boosters The two types of boosters above can, of course, be combined to produce more power. In audio applications, for example, a ± 15-V power supply limits the output power to 1 12.5 W, absolute maximum, into an 8- load. In order to increase the power, the voltage rails Ω must also be increased, with all of the cautions of the previous paragraph observed. 14.7.4 Single-Supply Operation and DC Offsets A D/A power circuit is not the right place to try to apply single-supply design techniques. In audio applications, a single-supply design would force a large coupling capacitor, which would distort and limit low frequency response. In dc applications, a dc offset will continu- ally drive a load — which will have to dissipate the excess voltage through its internal re- sistance as heat. Nevertheless, there may be applications that require a dc offset. The designer is fortunate in that there is already a precision reference available in the circuit. The reference drives 14-22

261 Increasing Op Amp Buffer Amplifier Current and Voltage the resistor network in the D/A, and may be external or internal to it. In most cases, an internal reference is brought out to a pin on the device. It is important for the designer not to excessively load the reference, as that would directly affect D/A accuracy (Figure 14 – 19). D/A R RF CF +VCC D VREF/R _ VREF R CO RIN RS + VOUT R R Figure 14 – 19. Single-Supply DAC Operation – In the circuit in Figure 14 19, the output of the buffer amplifier is shifted up in dc level by ½ VREF (not ½ VCC). VREF was selected because it is much more stable and accurate than VCC. The four resistors in the level shifter circuit must be highly accurate and matched, or this circuit will contribute to gain and offset errors. Thermal errors, however, cannot be compensated for, because the external resistors are probably going to have a different thermal drift than those on the IC. This technique is limited to applications that will see only a small change in ambient temperature. 14-23 Interfacing D/A Converters to Loads

262 14-24

263 Chapter 15 Sine Wave Oscillators Ron Mancini and Richard Palmer 15.1 What is a Sine Wave Oscillator? — not the type that are sometimes unin- Op amp oscillators are circuits that are unstable tentionally designed or created in the lab — but circuits intentionally designed to remain in an unstable state. Oscillators are useful for creating uniform signals that are used as a reference in applications such as audio, function generators, digital systems, and com- munication systems. Two general classes of oscillators exist: sinusoidal and relaxation. Sinusoidal oscillators consist of amplifiers with RC or LC circuits that have adjustable oscillation frequencies, or crystals that have a fixed oscillation frequency. Relaxation oscillators generate triangu- lar, sawtooth, square, pulse, or exponential waveforms, and they are not discussed here. Op amp sine wave oscillators operate without an externally applied input signal. Some combination of positive and negative feedback is used to drive the op amp into an unsta- ble state, causing the output to transition back and forth at a continuous rate. The ampli- and the oscillation frequency are set by the arrangement of passive and active com- tude ponents around a central op amp. Op amp oscillators are restricted to the lower end of the frequency spectrum because op amps do not have the required bandwidth to achieve low phase shift at high frequencies. Voltage-feedback op amps are limited to a the low kHz range since their dominant, open loop pole may be as low as 10 Hz. The new current-feedback op amps have a much wider bandwidth, but they are very hard to use in oscillator circuits because they are sensitive to feedback capacitance and are beyond the scope of this chapter. Crystal oscillators are used in high frequency applications up to the hundreds of MHz range. 15.2 Requirements for Oscillation The canonical, or simplest form, of a negative feedback system is used to demonstrate the requirements for oscillation to occur. The block diagram of this system is shown in Fig- 15-1

264 Requirements for Oscillation – ure 15 1, and the corresponding classic expression for a feedback system is shown in 1. The derivation and explanation of the block diagram and equation can be – Equation 15 found in Chapter 5. + V A V Σ OUT IN _ β – 1. Canonical Form of a Feedback System with Positive or Negative Feedback Figure 15 V OUT A (15 – 1) + V 1 ) A b IN Oscillators do not require an externally applied input signal, but instead use some fraction of the output signal created by the feedback network as the input signal. It is the noise voltage that provides the inital boost signal to the circuit when positive feedback is employed. Over a period of time, the output builds up, oscillating at the frequency set by the circuit components[1]. Oscillation results when the feedback system is not able to find a stable state because its transfer function can not be satisfied. The system becomes unstable when the denomi- 1. The key to designing an oscillator, – nator in Equation 15 – 1 is 0. When (1 +A β ) = 0, A β = then, is to ensure that A = – 1. This is called the Barkhausen criterion. This constraint re- β _ quires the magnitude of the loop gain be 1 with a corresponding phase shift of 180 as indicated by the minus sign. An equivalent expression using complex math is β = 1 A – 180 _ for a negative feedback system. For a positive feedback system, the ex- ∠ pression becomes A β = 1 ∠ 0 _ and the sign is negative in Equation 15 – 1. Once _ and A β = |1|, the output voltage of the unstable system heads the phase shift is 180 for infinite voltage in an attempt to destroy the world, and is only prevented from succeed- ing by an energy-limited power supply. When the output voltage approaches either power rail, the active devices in the amplifiers change gain, causing the value of A to change so the value of A β ≠ 1; thus the charge to infinite voltage slows down and eventually halts. At this point, one of three things can occur. First, nonlinearity in saturation or cutoff can cause the system to become stable and lock up at the power rail. Second, the initial charge can cause the system to saturate (or cutoff) and stay that way for a long time before it becomes linear and heads for the opposite power rail. Third, the system stays linear and reverses direction heading for the opposite power rail. Alternative two produces highly distorted oscillations (usually quasi square waves), and the resulting oscillators are called relaxation oscillators. Alternative three produces sine wave oscillators. 15-2

265 Phase Shift in the Oscillator 15.3 Phase Shift in the Oscillator is introduced by active and passive _ β = 1 ∠ – 180 The 180 phase shift in the equation A _ components. Like any well-designed feedback circuit, oscillators are made dependent on passive component phase shift because it is accurate and almost drift-free. The phase shift contributed by active components is minimized because it varies with temperature, has a wide initial tolerance, and is device dependent. Amplifiers are selected such that they contribute little or no phase shift at the oscillation frequency. These constraints limit the op amp oscillator to relatively low frequencies. A single pole RL or RC circuit contributes up to 90 _ phase shift per pole, and because 180 of phase shift is required for oscillation, at least two poles must be used in the oscilla- _ tor design. An LC circuit has two poles, thus it contributes up to 180 _ phase shift per pole pair. But LC and LR oscillators are not considered here because low frequency inductors are expensive, heavy, bulky, and very nonideal. LC oscillators are designed in high fre- quency applications, beyond the frequency range of voltage feedback op amps, where the inductor size, weight, and cost are less significant. Multiple RC sections are used in low frequency oscillator design in lieu of inductors. Phase shift determines the oscillation frequency because the circuit oscillates at the fre- quency that accumulates 180 _ phase shift. The rate of change of phase with frequency, , determines frequency stability. When buffered RC sections (an op amp buffer pro- ω d φ /d vides high input and low output impedance) are cascaded, the phase shift multiplies by the number of sections, n (see Figure 15 – 2). 0 – 45 1 RC Section ° 90 – φ/ – 2 RC Sections 135 – 180 – 3 RC Sections 225 – – 270 Normalized Frequency 4 RC Sections – 315 – 360 100 10 0.1 0.01 1 ω/ω – Normalized Frequency C 2. Phase Plot of RC Sections – Figure 15 15-3 Sine Wave Oscillators

266 Gain in the Oscillator The frequency of oscillation is very dependent upon the change in phase at the point _ where the phase shift is 180 . A tight frequency specification will require a large change φ , for a small change in frequency, d ω , at 180 _ . Figure 15 – in phase shift, d 2 demonstrates that, although two cascaded RC sections eventually provide 180 _ phase shift, d φ /d ω at the oscillator frequency is unacceptably low. Thus, oscillators made with two cascaded RC sections have poor frequency stability. Three equal cascaded RC filter sections have a much higher d /d ω (see Figure 15 – 2), and the resulting oscillator has improved frequen- φ cy stability. Adding a fourth RC section produces an oscillator with an excellent d φ /d ω (see Figure 15 – 2); thus this is the most stable RC oscillator configuration. Four sections are the maximum number used because op amps come in quad packages, and the four-sec- tion oscillator section yields four sine waves 45 _ phase shifted relative to each other. This oscillator can be used to obtain sine/cosine or quadrature sine waves. ceramic resonators make the most stable oscillators because resonators have Crystal or an extremely high d /d ω resulting from their nonlinear properties. Resonators are used φ for high frequency oscillators, but low frequency oscillators do not use resonators be- cause of size, weight, and cost restrictions. Op amps are not generally used with crystal or ceramic resonator oscillators because op amps have low bandwidth. Experience shows that it is more cost effective to build a high frequency crystal oscillator, count the output down, and filter the output to obtain a low frequency than it is to use a low frequency resonator. 15.4 Gain in the Oscillator The oscillator gain must equal 1 (A β = 1 ∠ – 180 _ ) at the oscillation frequency. Under nor- mal conditions, the circuit becomes stable when the gain exceeds 1 and oscillations cease. However, when the gain exceeds 1 with a phase shift of 180 _ , the active device – nonlinearity reduces the gain to 1 and the circuit oscillates. The nonlinearity happens when the amplifier swings close to either power rail because cutoff or saturation reduces the active device (transistor) gain. The paradox is that worst case design practice requires nominal gains exceeding 1 for manufacturability, but excess gain causes more distortion of the output sine wave. When the gain is too low, oscillations cease under worst case conditions, and when the gain is too high, the output wave form looks more like a square wave than a sine wave. Distortion is a direct result of excess gain overdriving the amplifier; thus gain must be care- fully controlled in low-distortion oscillators. Phase shift oscillators have distortion, but they achieve low-distortion output voltages because cascaded RC sections act as distortion filters. Also, buffered phase shift oscillators have low distortion because the gain is con- trolled and distributed among the buffers. circuit configurations require an auxiliary circuit for gain adjustment when low-distor- Most tion outputs are desired. Auxiliary circuits range from inserting a nonlinear component in 15-4

267 Active Element (Op Amp) Impact on the Oscillator the feedback loop, to automatic gain control (AGC) loops, to limiting by external compo- nents such as resistors and diodes. Consideration must also be given to the change in gain due to temperature variations and component tolerances, and the level of circuit complexity will be determined based on the required stability of the gain. The more stable the gain, the better the purity of the sine wave output. 15.5 Active Element (Op Amp) Impact on the Oscillator Up to now, it has been assumed that the op amp has an infinite bandwidth and the output is not frequency dependent. In reality, the op amp has many poles, but it has been com- pensated so that they are dominated by a single pole over the specified bandwidth. Thus A β , is now frequency dependant through the op amp gain term, A. Equation 15 – 2 shows is the dominant pole fre- this dependance, where a is the maximum open loop gain, ω a ω quency, and 3 graphically depicts the frequen- is the frequency of the signal. Figure 15 – = 1/ β does not cy dependence of the op amp gain and phase. The closed loop gain, A CL contain any poles or zeros and its therefore constant over frequency to the point where . Here, the signal amplitude is attenuated ω it impacts the op amp open-loop gain at 3dB by 3 dB and the phase shift introduced by the op amp is 45 ° . The amplitude and phase , and the phase will continue really begin to change one decade below this point at 0.1 ω 3dB dB point. The gain – , one decade beyond the 3 ° at 10 ω to shift until it has reached 90 3dB continues to roll off at – 20 dB/decade until other poles and zeros come into play. The high- intercepts the op amp gain. er the closed-loop gin, the earlier A CL 1 A b CLideal A (15 2) – + A + + CL 1 A b 1 A ) CLideal ) 1 1 ) b A A OL The phase shift contributed by the op amp will impact the performance of the oscillator lowering the oscillation frequency, and the reduction in A circuit by < 1 and can make A β CL the oscillator will not oscillate. 15-5 Sine Wave Oscillators

268 Active Element (Op Amp) Impact on the Oscillator A CL 20 dB/ – Decade dB — Gain A CL ÉÉÉÉÉÉÉÉÉ Minimum Desired ÉÉÉÉÉÉÉÉÉ Range of f osc ÉÉÉÉÉÉÉÉÉ 0 0 ° ° / 45 ° — Decade 45 ° – Phase 90 ° – 10 f f 0.1 f C C C Frequency — Hz Figure 15 – 3. Op Amp Frequency Response . Most op amps are compensated and may have more than the 45 _ of phase shift at ω 3dB The op amp should therefore be chosen with a gain bandwidth that is at least one decade above the oscillation frequency, as shown by the shaded area of Figure 15 – 3. The Wien to maintain the gain and frequency bridge requires a gain bandwidth greater than 43 ω OSC 10% of the ideal values [2]. Figure 15 within 4 compares the output distortion versus fre- – quency of an LM328, a TLV247x, and a TLC071 op amp, which have bandwidths of 0.4 MHz and 2.8 MHz, and 10 MHz, respectively, in a Wien bridge oscillator with nonlinear feedback (see Section 15.7.1 for the circuit and transfer function) The oscillation frequen- cy ranged from 16 Hz to 160 kHz. The graph illustrates the importance of choosing the correct op smp for the application. The LM328 achieved a maximum oscillation of 72 kHz and was attenuated more than 75%, while the TLV247x achieved 125 kHz with 18% atten- The wide bandwidth of the TLC071 provided a 138 kHz oscillation frequency with uation. a mere 2% attenuation. The op amp must be chosen with the proper bandwidth or the out- put may oscillate at a frequency well below the design specification. 15-6

269 Analysis of the Oscillator Operation (Circuit) 8 7 TLC4501 6 5 % — 4 TLV247x 3 Distortion LM328 2 1 0 100 k 10 k 1 k 100 10 Hz Frequency — – 4. Op Amp Bandwidth and Oscillator Output Figure 15 Care must be taken when using large feedback resistors since they interact with the input capacitance of the op amp to create poles with negative feedback, and both poles and zeros with positive feedback. Large resistor values can move these poles and zeros into the proximity of the oscillation frequency and impact the phase shift [3]. A final consideration is given to the slew-rate limitation of the op amp. The slew rate must is the oscillation fre- is the peak output voltage and f , where V f V be greater than 2 π 0 P 0 P quency, or distortion of the output signal will result. 15.6 Analysis of the Oscillator Operation (Circuit) Oscillators are created using various combinations of positive and negative feedback. – 5 shows the basic negative feedback amplifier block diagram with a positive Figure 15 feedback loop added. When positive and negative feedback are used, the gain of the neg- ative feedback path is combined into one gain term (representing the closed loop gain) – – and Figure 15 1. The positive feedback network is then repre- 5 reduces to Figure 15 and subsequent analysis is simplified. When negative feedback is used, β = β sented by 2 is 0. The case of positive and neg- then the positive feedback loop can be ignored since β 2 ative feedback combined is covered here since the negative feedback case was reviewed in Chapters 5 and 6. 15-7 Sine Wave Oscillators

270 Analysis of the Oscillator Operation (Circuit) β 1 _ V A Σ OUT + β 2 – Figure 15 5. Block Diagram of an Oscillator: a) Positive and Negative Feedback Loops, and b) Simplified Diagram w A general form of an op amp with positive and negative feedback is shown in Figure 15 – 6a. The first step is to break the loop at some point without altering the gain of the cir- cuit. The positive feedback loop is broken at the point marked with an X. A test signal ) is measured ) is applied to the broken loop and the resulting output voltage (V (V TEST OUT with the equivalent circuit shown in Figure 15 – 6b. Z Z Z 4 Z 2 1 2 + _ V + + + + Z Z V V V 1 V 3 OUT + TEST OUT – – – I – + – V – Z Z ) 3 4 + I Z 1 (b) Loop Gain Calculation Equivalent Circuit (a) Original Circuit Figure 15 – 6. Amplifier with Positive and Negative Feedback 3, and then is treated as an input signal to a noninvert- V – is calculated first in Equation 15 + ing amplifier, resulting in Equation 15 – 3 is substituted for V 4. Equation 15 – into Equation + 15 – 5. The actual circuit elements are then 4 to get the transfer function in Equation 15 – substituted for each impedance and the equation is simplified. These equations are valid . when the op amp open-loop gain is large and the oscillation frequency is < 0.1 ω 3dB Z 4 – 3) (15 ǒ Ǔ V + V ) TEST ) Z Z 4 3 Z ) Z 2 1 – 4) (15 Ǔ ǒ + V V ) OUT Z 1 V Z ) Z Z OUT 3 1 2 5) – (15 ǒ Ǔ Ǔǒ + ) Z Z V Z 4 3 1 TEST 15-8

271 Sine Wave Oscillator Circuits Phase shift oscillators generally use negative feedback, so the positive feedback factor ( β ) becomes zero. Oscillator circuits such as the Wien bridge use both negative ( β ) and 2 1 ) feedback to achieve a constant state of oscillation. This circuit is analyzed β positive ( 2 in detain in Section 15.7.1 using Equation 15 – 5. 15.7 Sine Wave Oscillator Circuits There are many types of sine wave oscillator circuits and variations of these circuits — the choice depends upon the frequency and the desired purity of the output waveform. The focus of this section is on the more prominent oscillator circuits: Wien bridge, phase and quadrature. The transfer function is derived for each case using the techniques shift, described in Section 15.6 of this chapter and in Chapters 3, 5, and 6. 15.7.1 Wien Bridge Oscillator The Wien bridge is one of the simplest and best known oscillators and is used extensively 7 shows the basic Wien bridge circuit configu- in circuits for audio applications. Figure 15 – ration. This circuit has only a few components and good frequency stability. The major drawback of the circuit is that the output amplitude is at the rails, saturating the op amp output transistors and causing high output distortion. Taming this distortion is more of a challenge than getting the circuit to oscillate. There are a couple of ways to minimize this effect, which will be covered later. It is now time to analyze this circuit and come up with the transfer function. R F V CC _ V OUT + R G R C R C V REF Figure 15 – 7. Wien Bridge Circuit Schematic The Wien bridge circuit is of the form that is detailed in Section 15.6. The transfer function for the circuit is created using the technique described in that section. It is readily apparent 1/sC ). The loop is broken between = (R ) and Z + 1/sC , Z = (R = R , Z = R that Z 2 1 G 1 2 3 1 2 F 4 15-9 Sine Wave Oscillators

272 Sine Wave Oscillator Circuits is applied to Z , and V is calculated. The positive feedback the output and Z , V OUT 1 1 TEST , is calculated first in Equations 15 – 6 through 15 – 8. Equation 15 – 6 shows the voltage, V + s C simple voltage divider at the noninverting input. Each term is then multiplied by (R 2 2 to get Equation 15 – 7. + 1) and divided by R 2 R 2 ȡ ȣ ǒ Ǔ R C s 1 ) Z 2 2 4 (15 – 6) Ǔ ǒ + + V V V ȧ ȧ ) TEST TEST R ) Z Z 1 2 3 4 ǒ Ǔ Ǔ ǒ ) R ) 1 C R s ) 1 s C Ȣ 1 2 2 V ) 1 7) – (15 + R C V 1 1 2 TEST ) ) C ) S R ) 1 1 2 R R C s C 2 2 1 1 , where Substitute s = j ω C = 1/R ω ω is the oscillation frequency, ω , and = 1/R C 1 2 2 1 2 0 1 0 – 8. to get Equation 15 V ) 1 (15 – 8) + w C w R V 0 2 1 2 Ǔ ǒ TEST ) * ) j 1 ) w w R C 1 0 2 1 Some interesting relationships now become apparent. The capacitor in the zero, repre- , and the capacitor in the pole, represented by ω , must each contribute 90 _ ω sented by 1 2 of phase shift toward the 180 _ required for oscillation at ω and . This requires that C = C 1 0 2 ω equal to = R ω . Setting cancels the frequency terms, ideally removing any and ω R 1 2 2 1 0 change in amplitude with frequency since the pole and zero negate one another. An over- = 1/3 is the result (Equation 15 – 9). all feedback factor of β V ) 1 1 1 9) – (15 + + + w w w C 0 0 0 w R 3 V ǒ Ǔ Ǔ ǒ TEST * ) * ) j ) ) 1 3 j w w w w R C 0 0 0 The gain of the negative feedback portion, A, of the circuit must then be set such that A β to satisfy the condition. The must be set to twice the value of R = 1, requiring A = 3. R F G op amp in Figure 15 – 7 is single supply, so a dc reference voltage, V , must be applied REF to the positive to bias the output for full-scale swing and minimal distortion. Applying V REF restricts dc current flow to the negative feedback leg of the circuit. V through R input 2 REF was set at 0.833V to bias the output at the midrail of the single supply, rail-to-rail input and output amplifier, or 2.5 V. See Cahpter 4 for details on dc biasing single-supply op amps. is shorted to ground for split supply applications. V REF The final circuit is shown in Figure 15 – 8, with component values selected to provide an RC) = 15.9 kHz. The circuit oscillated π = 1/(2 , where f = 2 π f ω oscillation frequency of 0 0 0 at 1.57 kHz due to slightly varying component values with 2% distortion. This high value is due to the extensive clipping of the output signal at both supply rails, producing several 15-10

273 Sine Wave Oscillator Circuits ± 1%. Figure large odd and even harmonics. The feedback resistor was then adjusted 15 – 9 shows the output voltage waveforms. The distortion grew as the saturation in- was decreased by more , and oscillations ceased when R creased with increasing R F F than 0.8%. = 2R R F G 20 k Ω +5 V _ TLV2471 V OUT R + G R Ω 10 k 10 k Ω 10 nF C R 10 nF C Ω 10 k + V REF – 0.833 V Figure 15 – 8. Final Wien Bridge Oscillator Circuit V +1% R Ω = 20.20 k F V = 5 V CC V I V = 0.833 V REF Ω R = 20 k = 2 V/div F = 10.0 k Ω R G OUT V V 0.8% – = 19.84 k Ω R F μ s/div Time = 500 Figure 15 – 9. Wien Bridge Output Waveforms nonlinear feedback can minimize the distortion inherent in the basic Wien bridge Applying circuit. A nonlinear component such as an incandescent lamp can be substituted into 15-11 Sine Wave Oscillators

274 Sine Wave Oscillator Circuits as shown in Figure 15 – 10. The lamp resistance, R , is nominally se- the circuit for R LAMP G lected as half the feedback resistance, R , at the lamp current established by R and F F . When the power is first applied the lamp is cool and its resistance is small, so the R LAMP gain is large (> 3). The current heats up the filament and the resistance increases, lower- ing the gain. The nonlinear relationship between the lamp current and resistance keeps output voltage changes small. Figure 15 11 shows the output of this amplifier with a dis- – = 1.57 kHz. The distortion for this variation is reduced over the basic for f tortion of 1% OSC circuit by avoiding hard saturation of the op amp transistors. R F 377 Ω +5 V _ TLV247x V OUT – TI 327 + R L R Lamp 10 k Ω 10 nF C R 10 nF C Ω 10 k + V REF – 0.833 V Figure 15 – 10. Wien Bridge Oscillator with Nonlinear Feedback = 1 V/div OUT V Time = 500 μ s/div 10 – Figure 15 – 11. Output of the Circuit in Figure 15 15-12

275 Sine Wave Oscillator Circuits The impedance of the lamp is mostly due to thermal effects. The output amplitude is then very temperature sensitive and will tend to drift. The gain must be set higher than 3 to com- pensate for any temperature variations, which increases the distortion in the circuit [4]. This type of circuit is useful when the temperature does not fluctuate over a wide range or when used in conjunction with an amplitude limiting circuit. ap- (4). As f The lamp has an effective low frequency thermal time constant, t thermal OSC proaches t , distortion is greatly increased. Several lamps can be placed in series thermal and reduce distortion. The drawbacks are that the time required for to increase t thermal oscillations to stabilize is increased and the output amplitude is reduced. An automatic gain control (AGC) circuit must be used when neither of the two previous circuits yield low distortion. A typical Wien bridge oscillator with an AGC circuit is shown – 12, with the output waveform of the circuit shown in Figure 15 – 13. The AGC in Figure 15 is used to stabilize the magnitude of the sinusoidal output to an optimum gain level. The JFET serves as the AGC element, providing excellent control because of the wide range ), which is controlled by the gate voltage. The JFET of the drain to source resistance (R DS . This gate voltage is 0 V when the power is applied, and the JFET turns on with low R DS , raising the gain to 3.05, and oscillations begin +R +R in parallel with R places R S G2 G1 DS and gradually build up. As the output voltage gets large, the negative swing turns the , which provides a dc potential to the gate of Q . diode on and the sample is stored on C 1 1 limits the current and establishes the time constant for charging C Resistor R , which 1 1 . When the output voltage drifts high, R increases, should be much greater than f OSC DS lowering the gain to a minimum of 2.87 (1+R /R ). The output stabilizes when the gain G1 F reaches 3. The di stortion of the AGC is 0.8%, which is due to slight clipping at the positive rail. for a single-supply amplifier. A zener diode The circuit of Figure 15 – 12 is biased with V REF can be placed in series with D1 to limit the positive swing of the output and reduce distor- . tion. A split supply can be easily implemented by grounding all points connected to V REF There are a wide variety of Wien bridge variations that exist to more precisely control the and allow selectable or even variable oscillation frequencies. Some circuits use amplitude diode limiting in place of a nonlinear feedback component. The diodes reduce the distor- tion by providing a soft limit for the output voltage. 15-13 Sine Wave Oscillators

276 Sine Wave Oscillator Circuits V D1 + − Ω 10 k R + 1 C D1 1N4933 1 V C1 F μ 0.1 − R 18.2 k Ω R Ω 10 k R 2 F G2 Ω 11.3 k _ V OUT + J1 R G1 R R C S C Ω 10 k Ω 10 k R + − V = 2.5 V REF Figure 15 – 12. Wien Bridge Oscillator with AGC = 1 V/div OUT V Time = 500 μ s/div 12 – Figure 15 – 13. Output of the Circuit in Figure 15 15.7.2 Phase Shift Oscillator, Single Amplifier Phase shift oscillators have less distortion than the Wien bridge oscillator, coupled with good frequency stability. A phase shift oscillator can be built with one op amp as shown 14. Three RC sections are cascaded to get the steep d in figure 15 – φ slope as de- /d ω 3 to get a stable oscillation frequency. Any less and the oscillation scribed in Section 15 – frequency is high and interferes with the op amp BW limitations. 15-14

277 Sine Wave Oscillator Circuits R F Ω 1.5 M +5 V R G _ RR R V 55.2 k Ω OUT + Ω Ω 10 k 10 k 10 k Ω TLV2471 CC C 10 nF 10 nF 10 nF 2.5 V Figure 15 – 14. Phase Shift Oscillator (Single Op Amp) = 1 V/div OUT V s/div Time = 500 μ Figure 15 – 15. Output of the Circuit in Figure 15 – 14 3 1 (15 – 10) ǒ Ǔ A b + A 1 RCs ) The normal assumption is that the phase shift sections are independent of each other. – Then Equation 15 – 180 _ when the phase shift of 10 is written. The loop phase shift is _ f = 1.732/RC because the tangent of each section is – 60 π , and this occurs when ω = 2 3 , so the gain, A, must be equal to _ = 1.732. The magnitude of β at this point is (1/2) 60 8 for the system gain to be equal to one. The oscillation frequency with the component values shown in Figure 15 – 14 is 3.76 kHz rather than the calculated oscillation frequency of 2.76 kHz as shown in Figure 15 – 14. the gain required to start oscillation is 27 rather than the calculated gain of 8. These Also, 15-15 Sine Wave Oscillators

278 Sine Wave Oscillator Circuits discrepancies are partially due to component variations, but the biggest contributing fac- tor is the incorrect assumption that the RC sections do not load each other. This circuit configuration was very popular when active components were large and expensive. But now op amps are inexpensive, small, and come four in a package, so the single op amp phase shift oscillator is losing popularity. The output distortion is a low 0.46%, consider- ably less than the Wein bridge circuit without amplitude stabilization. 15.7.3 Phase Shift Oscillator, Buffered The buffered phase shift oscillator is much improved over the unbuffered version, the cost being a higher component count. The buffered phase shift oscillator is shown in Figure 16 and the resulting output waveform in Figure 15 15 17. The buffers prevent the RC – – from loading each other, hence the buffered phase shift oscillator performs clos- sections , loads the third RC er to the calculated frequency and gain. The gain setting resistor, R G the fourth buffer in a quad op amp buffers this RC section, the performance be- section. If comes ideal. Low-distortion sine waves can be obtained from either phase shift oscillator design, but the purest sine wave is taken from the output of the last RC section. This is a high-impedance node, so a high impedance input is mandated to prevent loading and frequency shifting with load variations. R F Ω 1.5 M +5 V R G _ R + R 180 k Ω + + R 10 k Ω _ V 10 k Ω OUT _ 10 k Ω 10 nF C 2.5 V 10 nF C C 10 nF ¼ TLV2474 TLV2474 ¼ ¼ TLV2474 Figure 15 – 16. Phase Shift Oscillator, Buffered 15-16

279 Sine Wave Oscillator Circuits = 200 mV/div OUT V μ s/div Time = 500 Figure 15 – 17. Output of the Circuit Figure 15 – 16 15.7.4 Bubba Oscillator The bubba oscillator in Figure 15 – 18 is another phase shift oscillator, but it takes advan- tage of the quad op amp package to yield some unique advantages. Four RC sections require 45 _ phase shift per section, so this oscillator has an excellent d φ /dt resulting in _ minimized frequency drift. The RC sections each contribute 45 phase shift, so taking out- puts from alternate sections yields low impedance quadrature outputs. When an output _ phase shifted sine waves. The loop is taken from each op amp, the circuit delivers four 45 ω given in Equation 15 – 11. When 11 reduces to Equa- = 1/RCs, Equation 15 – equation is tions 15 – 12 and 15 – 13. 4 1 (15 – 11) Ǔ ǒ + A A b RCs 1 ) 4 ȧ ȧ ȧ ȧ 1 1 1 (15 – 12) | | ǒ Ǔ b + + + ȧ ȧ 4 4 4 ) j Ǹ ȧ ȧ 2 1 * ° – 13) (15 ( ) + tan f 1 + 45 15-17 Sine Wave Oscillators

280 Sine Wave Oscillator Circuits R F Ω 1.5 M +5 V R G _ R + 360 k Ω + Ω 10 k _ C 10 nF 2.5 V 4/4 TLV2474 R 10 k Ω R + R + V _ OUT Ω 10 k _ Sine 10 k Ω 10 nF C C 10 nF 10 nF C V OUT Cosine 18. Bubba Oscillator – Figure 15 = 1 V/div OUT V μ s/div Time = 500 19. Output of the Circuit in Figure 15 Figure 15 – 18. – The gain, A, must equal 4 for oscillation to occur. The test circuit oscillated at 1.76 kHz rather than the ideal frequency of 1.72 kHz when the gain was 4.17 rather than the ideal and gain 4. The output waveform is shown in Figure 15 – 19. Distortion is 1% for V OUTSINE . With low gain, A, and low bias current op amps, the gain setting 0.1% for V OUTCOSINE 15-18

281 Sine Wave Oscillator Circuits resistor, R , does not load the last RC section, thus ensuring oscillator frequency accura- G cy. Very low distortion sine waves can be obtained from the junction of R and R . When G low-distortion sine waves are required at all outputs, the gain should be distributed be- tween all of the op amps. The noninverting input of the gain op amp is biased at 0.5 V to set the quiescent output voltage at 2.5 V for single supply operation, and should be ground but it has for split supply op amps. Gain distribution requires biasing of the other op amps, no effect on the oscillator frequency. 15.7.5 Quadrature Oscillator – The quadrature oscillator shown in Figure 15 20 is another type of phase shift oscillator, but the three RC sections are configured so each section contributes 90 of phase shift. _ provides both sine and cosine waveform outputs (the outputs are quadrature, or 90 This _ apart), which is a distinct advantage over other phase shift oscillators. The idea of the quadrature oscillator is to use the fact that the double integral of a sine wave is a negative sine wave of the same frequency and phase. The phase of the second integrator is then inverted and applied as positive feedback to induce oscillation [6]. C = R C = R C Equation The loop gain is calculated in Equation 15 14. When R – 2 1 3 1 2 3, = 1/RC, Equation 15 15 14 reduces to Equation 15 – 15. When ω – – 14 reduces to 1 ∠ – 180, = 2 f = 1/RC. The test circuit oscillated at 1.65 kHz rather than so oscillation occurs at ω π 21. This discrepancy is attributed to com- the calculated 1.59 kHz, as shown in Figure 15 – ponent variations. Both outputs have relatively high distortion that can be reduced with a gain stabilizing circuit. The sine output had 0.846% distortion and the cosine output had 0.46% distortion. Adjusting the gain can increase the amplitudes. The cost is bandwidth. R C s ) 1 3 3 1 14) – (15 ǒ Ǔ A b A + Ǔ ǒ s C R ǒ Ǔ s C 1 ) s R C R 1 1 3 2 3 2 2 1 (15 15) – ǒ Ǔ A b A + RCs 15-19 Sine Wave Oscillators

282 Sine Wave Oscillator Circuits 10 nF C 1 +5 V R 1 _ V OUT 10 k Ω Sine + Ω 10 k R 2 ½ TLV2474 ½ TLV2474 + V OUT C 2 _ Cosine 10 nF C 3 R 3 2.5 V 10 k Ω 10 nF 20. Quadrature Oscillator Figure 15 – V SINE OUT = 2 V/div V COSINE OUT OUT V Time = 500 μ s/div Figure 15 – 21. Output of the Circuit in Figure 15 – 20 15-20

283 Sine Wave Oscillator Circuits 15.7.6 Conclusion Op amp oscillators are restricted to the lower end of the frequency spectrum because they do not have the required bandwidth to achieve low phase shift at high frequencies. The new current-feedback op amps have a much greater bandwidth than the voltage-feed- back op amps, but are very difficult to use in oscillator circuits because of their sensitivity to feedback capacitance. Voltage-feedback op amps are limited to tens of Hz (at the most!) because of their low frequency rolloff. The bandwidth is reduced when op amps are cascaded due to the multiple contribution of phase shift. The Wien bridge oscillator has few parts and good frequency stability, but the basic circuit has a high output distortion. AGC improves the distortion considerably, particularly at the frequency range. Nonlinear feedback of fers the best performance over the mid and lower upper frequency ranges. The phase shift oscillator has lower output distortion and, with- out buffering, requires a high gain, which limits the use to very low frequencies. Decreas- ing cost of op amps and components has reduced the popularity of the phase shift oscilla- tors. The quadrature oscillator only requires two op amps, has reasonable distortion, and offers both sine and cosine waveforms. The drawback is the low amplitude, which may require a higher gain and a reduction in bandwidth, or an additional gain stage. May your oscillators always oscillate, and your amplifiers always amplify. 15-21 Sine Wave Oscillators

284 References 15.8 References [1] Gottlieb, Irving M., Practical Oscillator Handbook , Newnes, 1997. [2] Kennedy, E. J., Operational Amplifier Circuits, Theory and Applications, Holt Rhien- hart and Winston, 1988. , McGraw Hill Book Company, Optimizing Op Amp Performance [3] Graeme, Jerald, 1997. [4] Philbrick Researches, Inc., Applications Manual for Computing Amplifiers , Nimrod Press, Inc., 1966. [5] Graf, Rudolf F., Oscillator Circuits , Newnes, 1997. echniques [6] Graeme, Jerald, Applications of Operational Amplifiers, Third Generation T , McGraw Hill Book Company, 1973. 15-22

285 Chapter 16 Active Filter Design Techniques Thomas Kugelstadt 16.1 Introduction What is a filter? A filter is a device that passes electric signals at certain frequencies or . — Webster. frequency ranges while preventing the passage of others Filter circuits are used in a wide variety of applications. In the field of telecommunication, band-pass filters are used in the audio frequency range (0 kHz to 20 kHz) for modems and speech processing. High-frequency band-pass filters (several hundred MHz) are used for channel selection in telephone central offices. Data acquisition systems usually require anti-aliasing low-pass filters as well as low-pass noise filters in their preceding sig- nal conditioning stages. System power supplies often use band-rejection filters to sup- press the 60-Hz line frequency and high frequency transients. In addition, there are filters that do not filter any frequencies of a complex input signal, but just add a linear phase shift to each frequency component, thus contributing to a constant time delay. These are called all-pass filters. At high frequencies (> 1 MHz), all of these filters usually consist of passive components such as inductors (L), resistors (R), and capacitors (C). They are then called LRC filters. In the lower frequency range (1 Hz to 1 MHz), however, the inductor value becomes very large and the inductor itself gets quite bulky, making economical production difficult. In these cases, active filters become important. Active filters are circuits that use an op- erational amplifier (op amp) as the active device in combination with some resistors and 1). capacitors to provide an LRC-like filter performance at low frequencies (Figure 16 – C 2 R R LR 2 1 V V V IN OUT IN V OUT C C 1 Figure 16 – 1. Second-Order Passive Low-Pass and Second-Order Active Low-Pass 16-1

286 Fundamentals of Low-Pass Filters This chapter covers active filters. It introduces the three main filter optimizations (Butter- and Bessel), followed by five sections describing the most common worth, Tschebyscheff, active filter applications: low-pass, high-pass, band-pass, band-rejection, and all-pass fil- ters. Rather than resembling just another filter book, the individual filter sections are writ- ten in a cookbook style, thus avoiding tedious mathematical derivations. Each section with the general transfer function of a filter, followed by the design equations to cal- starts culate the individual circuit components. The chapter closes with a section on practical design hints for single-supply filter designs. 16.2 Fundamentals of Low-Pass Filters The most simple low-pass filter is the passive RC low-pass network shown in Figure 16 – 2. R V V IN OUT C Figure 16 – 2. First-Order Passive RC Low-Pass Its transfer function is: 1 RC 1 + A(s) + 1 1 ) sRC ) s RC , allows for any time variable signals. For where the complex frequency variable, s = j ω + σ σ , becomes zero and s = j ω pure sine waves, the damping constant, . s corner For a normalized presentation of the transfer function, s is referred to the filter ’ and has these relationships: frequency, or ω – 3 dB frequency, C, j w s f j + W + + j + s w w f C C C RC With the corner frequency of the low-pass in Figure 16 – 2 being f becomes = 1/2 π s , C and the transfer function A(s) results in: s = sRC 1 A(s) + ) s 1 The magnitude of the gain response is: 1 + |A| Ǹ 2 1 ) W For frequencies Ω >> 1, the rolloff is 20 dB/decade. For a steeper rolloff, n filter stages 3. To avoid loading effects, op amps, can be connected in series as shown in Figure 16 – operating as impedance converters, separate the individual filter stages. 16-2

287 Fundamentals of Low-Pass Filters R R V IN R R C C C V OUT C Figure 16 – 3. Fourth-Order Passive RC Low-Pass with Decoupling Amplifiers The resulting transfer function is: 1 + A(s) ǒ Ǔ Ǔǒ ) ( a a ) 1 ) a s s 1 AAA s 1 ) n 2 1 In the case that all filters have the same cut-off frequency, f , the coefficients become C n Ǹ Ǹ , and f times higher than f α of each partial filter is 1/ a a + + a + 2 a +AAA * 1 n C 1 C 2 of the overall filter. Figure 16 – 4 shows the results of a fourth-order RC low-pass filter. The rolloff of each par- tial filter (Curve 1) is 20 dB/decade, increasing the roll-off of the overall filter (Curve 2) – to 80 dB/decade. Note: Filter response graphs plot gain versus the normalized frequency axis ). Ω = f/f ( Ω C 16-3 Active Filter Design Techniques

288 Fundamentals of Low-Pass Filters 0 – 10 20 – 1st Order Lowpass 30 – dB — 40 – 4th Order Lowpass Gain — 50 – |A| – 60 Ideal 4th Order Lowpass 70 – – 80 100 1 0.01 0.1 10 Ω Frequency — 0 1st Order Lowpass Ideal 4th Order Lowpass 90 – degrees — – 180 Phase — φ 270 – 4th Order Lowpass 360 – 10 1 0.1 0.01 100 Ω — Frequency th th st Curve 1: 1 -order partial low-pass filter, Curve 2: 4 -order overall low-pass filter, Curve 3: Ideal 4 Note: -order low-pass filter Frequency and Phase Responses of a Fourth-Order Passive RC Low-Pass Filter 4. Figure 16 – The corner frequency of the overall filter is reduced by a factor of α ≈ 2.3 times versus the – 3 dB frequency of partial filter stages. 16-4

289 Fundamentals of Low-Pass Filters – 4 shows the transfer function of an ideal fourth-order low-pass func- In addition, Figure 16 tion (Curve 3). In comparison to the ideal low-pass, the RC low-pass lacks in the following characteris- tics: , thus amplifying the D The passband gain varies long before the corner frequency, f C upper passband frequencies less than the lower passband. D The transition from the passband into the stopband is not sharp, but happens gradually, moving the actual 80-dB roll off by 1.5 octaves above f . C D The phase response is not linear, thus increasing the amount of signal distortion significantly. of the one The gain and phase response of a low-pass filter can be optimized to satisfy following three criteria: 1) A maximum passband flatness, 2) An immediate passband-to-stopband transition, 3) A linear phase response. For that purpose, the transfer function must allow for complex poles and needs to be of the following type: A A 0 0 A(s) + + 2 2 2 P Ǔ ǒ Ǔǒ ǒ Ǔ 2 s ) a ) s ) b b s s 1 AAA 1 1 ) a ) s ) b a s Ǔ ǒ n n 1 ) s s b ) a 1 2 2 1 i i i where A is the passband gain at dc, and a and b are the filter coefficients. i 0 i Since the denominator is a product of quadratic terms, the transfer function represents and b being positive real coef- a series of cascaded second-order low-pass stages, with a i i ficients. These coefficients define the complex pole locations for each second-order filter stage, thus determining the behavior of its transfer function. The following three types of predetermined filter coefficients are available listed in table format in Section 16.9: D The Butterworth coefficients, optimizing the passband for maximum flatness D The Tschebyscheff coefficients, sharpening the transition from passband into the stopband D The Bessel coefficients, linearizing the phase response up to f C The transfer function of a passive RC filter does not allow further optimization, due to the complex poles. The only possibility to produce conjugate complex poles using pas- lack of 16-5 Active Filter Design Techniques

290 Fundamentals of Low-Pass Filters sive components is the application of LRC filters. However, these filters are mainly used at high frequencies. In the lower frequency range (< 10 MHz) the inductor values become very large and the filter becomes uneconomical to manufacture. In these cases active fil- ters are used. Active filters are RC networks that include an active device, such as an operational ampli- fier (op amp). Section 16.3 shows that the products of the RC values and the corner frequency must and b , to generate the desired transfer func- yield the predetermined filter coefficients a i i tion. The following paragraphs introduce the most commonly used filter optimizations. 16.2.1 Butterworth Low-Pass FIlters The Butterworth low-pass filter provides maximum passband flatness. Therefore, a But- low-pass is often used as anti-aliasing filter in data converter applications where terworth precise signal levels are required across the entire passband. Figure 16 – 5 plots the gain response of different orders of Butterworth low-pass filters ver- ); the higher the filter order, the longer the Ω sus the normalized frequency axis, Ω = f / f ( C passband flatness. 10 0 – 10 dB 20 – — 1st Order Gain 2nd Order 30 – — 4th Order |A| 40 – 10th Order 50 – – 60 0.01 1 100 10 0.1 Ω Frequency — Figure 16 – 5. Amplitude Responses of Butterworth Low-Pass Filters 16-6

291 Fundamentals of Low-Pass Filters 16.2.2 Tschebyscheff Low-Pass Filters The Tschebyscheff low-pass filters provide an even higher gain rolloff above f . However, C as Figure 16 – 6 shows, the passband gain is not monotone, but contains ripples of constant magnitude instead. For a given filter order, the higher the passband ripples, the higher the filter ’ s rolloff. 10 0 10 – 2nd Order dB – 20 — 4th Order Gain – 30 — |A| 40 – 9th Order 50 – 60 – 10 100 0.01 0.1 1 Ω Frequency — 6. Gain Responses of Tschebyscheff Low-Pass Filters – Figure 16 With increasing filter order, the influence of the ripple magnitude on the filter rolloff dimin- ishes. Each ripple accounts for one second-order filter stage. Filters with even order numbers generate ripples above the 0-dB line, while filters with odd order numbers create ripples below 0 dB. Tschebyscheff filters are often used in filter banks, where the frequency content of a si gnal is of more importance than a constant amplification. 16.2.3 Bessel Low-Pass Filters The Bessel low-pass filters have a linear phase response (Figure 16 – 7) over a wide fre- quency range, which results in a constant group delay (Figure 16 – 8) in that frequency range. Bessel low-pass filters, therefore, provide an optimum square-wave transmission behavior. However, the passband gain of a Bessel low-pass filter is not as flat as that of the Butterworth low-pass, and the transition from passband to stopband is by far not as 9). – sharp as that of a Tschebyscheff low-pass filter (Figure 16 16-7 Active Filter Design Techniques

292 Fundamentals of Low-Pass Filters 0 90 – degrees 180 – — Bessel Phase — φ 270 – Butterworth Tschebyscheff 360 – 100 10 1 0.1 0.01 Ω Frequency — – 7. Comparison of Phase Responses of Fourth-Order Low-Pass Filters Figure 16 1.4 1.2 s/s — 1 0.8 Tschebyscheff 0.6 0.4 Butterworth Normalized Group Delay — Bessel gr 0.2 T 0 100 10 1 0.1 0.01 Frequency Ω — Comparison of Normalized Group Delay (Tgr) of Fourth-Order Low-Pass Filters 8. Figure 16 – 16-8

293 Fundamentals of Low-Pass Filters 10 0 – 10 Bessel dB 20 – — Gain – 30 — Butterworth |A| 40 – Tschebyscheff 50 – 60 – 10 1 0.1 — Frequency Ω Figure 16 – 9. Comparison of Gain Responses of Fourth-Order Low-Pass Filters 16.2.4 Quality Factor Q n. Instead of de- The quality factor Q is an equivalent design parameter to the filter order th order Tschebyscheff low-pass, the problem can be expressed as designing signing an n a Tschebyscheff low-pass filter with a certain Q. For band-pass filters, Q is defined as the ratio of the mid frequency, f to the bandwidth m, at the two – 3 dB points: f m Q + * (f ) f 1 2 For low-pass and high-pass filters, Q represents the pole quality and is defined as: Ǹ b i Q + a i High Qs can be graphically presented as the distance between the 0-dB line and the peak point of the filter ’ s gain response. An example is given in Figure 16 – 10, which shows a tenth-order Tschebyscheff low-pass filter and its five partial filters with their individual Qs. 16-9 Active Filter Design Techniques

294 Fundamentals of Low-Pass Filters 40 30 20 Q 5 Overall Filter dB 10 — Gain 0 — 1st Stage |A| 2nd Stage – 10 3rd Stage – 20 4th Stage 5th Stage 30 – 10 1 0.1 0.01 Ω — Frequency 10. Graphical Presentation of Quality Factor Q on a Tenth-Order Figure 16 – Tschebyscheff Low-Pass Filter with 3-dB Passband Ripple The gain response of the fifth filter stage peaks at 31 dB, which is the logarithmic value of Q : 5 [dB] logQ 20 + Q · 5 5 Solving for the numerical value of Q5 yields: 31 20 + + 10 Q 35.48 5 which is 9, within 1% of the theoretical value of Q = 35.85 given in Section 16.9, Table 16 – last row. The graphical approximation is good for Q > 3. For lower Qs, the graphical values differ from the theoretical value significantly. However, only higher Qs are of concern, since the higher the Q is, the more a filter inclines to instability. 16.2.5 Summary The general transfer function of a low-pass filter is : A 0 1) (16 – + A(s) P 2 ǒ Ǔ a ) s ) b 1 s i i i The filter coefficients a and b distinguish between Butterworth, Tschebyscheff, and Bes- i i sel filters. The coef ficients for all three types of filters are tabulated down to the tenth order 10. – in Section 16.9, Tables 16 – 4 through 16 16-10

295 Low-Pass Filter Design th The multiplication of the denominator terms with each other yields an order polynomial n of S, with n being the filter order. * n · 20 dB ń decade While n determines the gain rolloff above f determine , a and b with i C i the gain behavior in the passband. Ǹ b i In addition, the ratio is defined as the pole quality. The higher the Q value, the Q + a i more a filter inclines to instability. 16.3 Low-Pass Filter Design Equation 16 – 1 represents a cascade of second-order low-pass filters. The transfer func- tion of a single stage is: A 0 (16 – 2) A (s) + i 2 Ǔ ǒ s s ) b 1 a ) i i For a first-order filter, the coefficient b is always zero (b =0), thus yielding: 1 A 0 – 3) (16 A(s) + 1 a s ) 1 The first-order and second-order filter stages are the building blocks for higher-order fil- ters. Often the filters operate at unity gain (A =1) to lessen the stringent demands on the op 0 amp ’ s open-loop gain. Figure 16 – 11 shows the cascading of filter stages up to the sixth order. A filter with an even number consists of second-order stages only, while filters with an odd order number order include an additional first-order stage at the beginning. 16-11 Active Filter Design Techniques

296 Low-Pass Filter Design 1st order 1st order a=1 2nd order 2nd order , b a 1 1 2nd order 1st order 3rd order a , b a 2 2 1 2nd order 2nd order 4th order a a , b , b 1 2 2 1 2nd order 1st order 2nd order 5th order , b , b a a a 1 2 3 2 3 2nd order 2nd order 2nd order 6th order a a a , b , b , b 2 2 3 1 1 3 Figure 16 11. Cascading Filter Stages for Higher-Order Filters – Figure 16 – 10 demonstrated that the higher the corner frequency of a partial filter, the high- er its Q. Therefore, to avoid the saturation of the individual stages, the filters need to be placed in the order of rising Q values. The Q values for each filter order are listed (in rising order) in Section 16 – 9, Tables 16 – 4 through 16 – 10. 16.3.1 First-Order Low-Pass Filter – Figures 16 12 and 16 – 13 show a first-order low-pass filter in the inverting and in the non- inverting configuration. R 1 V IN V OUT C 1 R 2 R 3 12. First-Order Noninverting Low-Pass Filter – Figure 16 16-12

297 Low-Pass Filter Design C 1 R 1 R 2 V IN V OUT – 13. First-Order Inverting Low-Pass Filter Figure 16 The transfer functions of the circuits are: R R 2 2 * ) 1 R R 1 3 and A(s) + + A(s) C s 1 ) w R s C ) w R 1 c c 1 2 1 1 phase shift from ° The negative sign indicates that the inverting amplifier generates a 180 the filter input to the output. – 3 yields: The coefficient comparison between the two transfer functions and Equation 16 R R 2 and 2 +* A A ) 1 + 0 0 R R 1 3 and a C R w + C + R w a c c 1 1 2 1 1 1 To dimension the circuit, specify the corner frequency (f ), the dc gain (A ), and capacitor C 0 , and then solve for resistors R : and R C 1 1 2 a a 1 and 1 + R R + 2 1 2 p f C 2 p f C c c 1 1 R 2 and ǒ Ǔ +* R A * R R 1 + 1 2 3 0 A 0 10 is taken from one of the coefficient tables, Tables 16 – 4 through 16 – The coefficient a 1 in Section 16.9. = 1. For higher filter orders, Note, that all filter types are identical in their first order and a 1 however, a ≠ 1 because the corner frequency of the first-order stage is different from the 1 corner frequency of the overall filter. 16-13 Active Filter Design Techniques

298 Low-Pass Filter Design Example 16 1. First-Order Unity-Gain Low-Pass Filter – For a first-order unity-gain low-pass filter with f = 47 nF, R calculates = 1 kHz and C C 1 1 to: a 1 1 R + W 3.38 k + + 1 * 3 9 C 2 f p 10 Hz · 47 · F 10 2 p · c 1 However, to design the first stage of a third-order unity-gain Bessel low-pass filter, assum- and C , requires a different value for R . In this case, obtain ing the same values for f 1 1 C 4 in Section 16.9 (Bessel coefficients) to for a third-order Bessel filter from Table 16 – a 1 : calculate R 1 a 0.756 1 + R + + W 2.56 k 1 3 * 9 C p f 2 · 2 p 10 · 10 Hz · F 47 c 1 When operating at unity gain, the noninverting amplifier reduces to a voltage follower (Fig- ure 16 – 14), thus inherently providing a superior gain accuracy. In the case of the inverting amplifier, the accuracy of the unity gain depends on the tolerance of the two resistors, R 1 . and R 2 R 1 V IN V OUT C 1 – 14. First-Order Noninverting Low-Pass Filter with Unity Gain Figure 16 16.3.2 Second-Order Low-Pass Filter There are two topologies for a second-order low-pass filter, the Sallen-Key and the Multi- ple Feedback (MFB) topology. 16.3.2.1 Sallen-Key Topology The general Sallen-Key topology in Figure 16 – 15 allows for separate gain setting via A – . However, the unity-gain topology in Figure 16 = 1+R /R 16 is usually applied in filter 4 3 0 designs with high gain accuracy, unity gain, and low Qs (Q < 3). 16-14

299 Low-Pass Filter Design C 2 R R 2 1 V IN V OUT C 1 R 4 R 3 – 15. General Sallen-Key Low-Pass Filter Figure 16 C 2 R R 2 1 V IN V OUT C 1 Figure 16 – 16. Unity-Gain Sallen-Key Low-Pass Filter The transfer function of the circuit in Figure 16 – 15 is: A 0 A(s) + 2 2 Ǔ Ǔ ǒ ǒ ƪ ƫ C R 1 ) R s C ) R 1 * A R w R ) C w C s ) c c 1 1 2 2 0 1 1 2 2 1 For the unity-gain circuit in Figure 16 – 16 (A =1), the transfer function simplifies to: 0 1 A(s) + 2 2 ǒ Ǔ C C s C R R R R s ) w ) w ) 1 c c 1 1 2 2 1 1 2 The coefficient comparison between this transfer function and Equation 16 – 2 yields: A + 1 0 Ǔ ǒ w R C ) + R a c 1 1 2 1 2 R + C b R C w c 2 1 1 2 1 Given C are calculated through: , the resistor values for R and C and R 1 1 2 2 2 2 Ǹ C C C # a C 4b * a 1 1 1 1 2 2 2 R + 1,2 C 4 p f C c 2 1 16-15 Active Filter Design Techniques

300 Low-Pass Filter Design In order to obtain real values under the square root, C must satisfy the following condi- 2 tion: 4b 1 C w C 2 1 2 a 1 Example 16 – 2. Second-Order Unity-Gain Tschebyscheff Low-Pass Filter schebyscheff low-pass filter with a corner The task is to design a second-order unity-gain T = 3 kHz and a 3-dB passband ripple. frequency of f C – From Table 16 9 (the Tschebyscheff coefficients for 3-dB ripple), obtain the coefficients a and b for a second-order filter with a = 1.9305. = 1.0650 and b 1 1 1 1 Specifying C of: as 22 nF yields in a C 2 1 4b 1.9305 · 4 1 * 9 150 nF w C + 22 · 10 ^ nF C · 1 2 2 2 1.065 a 1 results in: and b into the resistor equation for R Inserting a 1,2 1 1 2 * 9 * 9 9 * 9 * Ǹ Ǔ ǒ * · 150 · 10 10 1.065 * 4 · 1.9305 · 22 · 10 · · 150 150 · 1.065 10 · R + W 1.26 k + 1 * 3 9 * 9 · 10 10 3 · · · 10 · 22 · p 4 150 and 2 * 9 * * 9 9 9 * Ǹ Ǔ ǒ 4 · 150 · 10 22 * ) · 1.9305 · · 10 1.065 10 · 150 · · · 150 1.065 10 + R 1.30 k W + 2 3 * 9 * 9 · · 22 4 10 p · · 150 · 10 3 · 10 17. – with the final circuit shown in Figure 16 150n 1.26k 1.30k V IN V OUT 22n 17. Second-Order Unity-Gain Tschebyscheff Low-Pass with 3-dB Ripple Figure 16 – A special case of the general Sallen-Key topology is the application of equal resistor val- = C. = C = R = R and C ues and equal capacitor values: R 2 1 1 2 16-16

301 Low-Pass Filter Design The general transfer function changes to: A R 0 4 A(s) + with ) 1 A + 0 2 2 ǒ Ǔ R ) ( RC) s s 1 A * 3 RC w w ) c c 3 0 The coefficient comparison with Equation 16 – 2 yields: ǒ Ǔ + w A RC a 3 * c 1 0 2 Ǔ ǒ + w RC b c 1 results in: Given C and solving for R and A 0 Ǹ b a 1 1 1 + * 3 + * R A 3 + 0 C 2 p f Q and Ǹ c b 1 depends solely on the pole quality Q and vice versa; Q, and with it the filter type, Thus, A 0 is determined by the gain setting of A : 0 1 + Q 3 * A 0 The circuit in Figure 16 – 18 allows the filter type to be changed through the various resistor . /R ratios R 4 3 C RR V IN V OUT C R 4 R 3 – 18. Adjustable Second-Order Low-Pass Filter Figure 16 – 1 lists the coefficients of a second-order filter for each filter type and gives the Table 16 resistor ratios that adjust the Q. Table 16 – 1. Second-Order FIlter Coefficients SECOND-ORDER BESSEL BUTTERWORTH 3-dB TSCHEBYSCHEFF a 1.3617 1.4142 1.065 1 b 1.9305 0.618 1 1 Q 0.58 0.71 1.3 0.234 0.568 R 0.268 /R 3 4 16-17 Active Filter Design Techniques

302 Low-Pass Filter Design 16.3.2.2 Multiple Feedback Topology The MFB topology is commonly used in filters that have high Qs and require a high gain. R 2 R C R 1 3 1 V IN V OUT C 2 – 19. Second-Order MFB Low-Pass Filter Figure 16 – The transfer function of the circuit in Figure 16 19 is: R 2 R 1 A(s) +* R R 3 2 2 2 ǒ Ǔ R ) R R ) w ) C s C C 1 s ) w R c c 2 2 3 1 2 3 1 R 1 Through coefficient comparison with Equation 16 – 2 one obtains the relation: R 2 A +* 0 R 1 R R 3 2 Ǔ ǒ + R C ) R a ) w c 1 1 2 3 R 1 2 b + w C C R R c 1 2 2 3 1 R , and solving for the resistors R – and C : Given C 1 3 1 2 2 2 Ǹ Ǔ ǒ * a A * C 1 * 4b C C C a 2 1 1 1 2 2 1 0 + R 2 4 p f C C c 1 2 R 2 R + 1 * A 0 b 1 R + 3 2 2 p R f C C 4 c 1 2 2 16-18

303 Low-Pass Filter Design In order to obtain real values for R , C must satisfy the following condition: 2 2 Ǔ ǒ 1 * 4b A 1 0 w C C 2 1 2 a 1 16.3.3 Higher-Order Low-Pass Filters Higher-order low-pass filters are required to sharpen a desired filter characteristic. For that purpose, first-order and second-order filter stages are connected in series, so that the product of the individual frequency responses results in the optimized frequency re- sponse of the overall filter. for each filter and b In order to simplify the design of the partial filters, the coefficients a i i are listed in the coefficient tables (Tables 16 10 in Section 16.9), with 4 through 16 – type – each table providing sets of coefficients for the first 10 filter orders. Example 16 – 3. Fifth-Order Filter The task is to design a fifth-order unity-gain Butterworth low-pass filter with the corner fre- quency f = 50 kHz. C the coefficients for a fifth-order Butterworth filter are obtained from Table 16 – 5, Sec- First tion 16.9: a b i i Filter 1 a = 1 b = 0 1 1 a Filter 2 = 1.6180 = 1 b 2 2 a Filter 3 b = 1 = 0.6180 3 3 Then dimension each partial filter by specifying the capacitor values and calculating the required resistor values. First Filter R 1 V IN V OUT C 1 – 20. First-Order Unity-Gain Low-Pass Figure 16 = 1nF, With C 1 a 1 1 R + 3.18 k + W + 1 9 * 3 p C 2 f 10 Hz F · · 1 10 · p 50 2 · c 1 Ω . The closest 1% value is 3.16 k 16-19 Active Filter Design Techniques

304 Low-Pass Filter Design Second Filter C 2 R R 2 1 V IN V OUT C 1 – 21. Second-Order Unity-Gain Sallen-Key Low-Pass Filter Figure 16 With C = 820 pF, 1 4b 2 1 · 4 * 12 · 1.26 nF + + 820 · 10 C w F C 2 1 2 2 1.618 a 2 The closest 5% value is 1.5 nF. = 820 pF and C = 1.5 nF, calculate the values for R1 and R2 through: With C 1 2 2 2 2 2 Ǹ Ǹ a * a C C C 4b C * 4b C C ) C C * a a 2 1 2 2 2 2 2 2 1 2 2 2 2 2 R + + R 1 1 and C 4 C f f C p C 4 p c c 2 1 1 2 and obtain 2 9 * 12 * * 9 9 * Ǹ Ǔ ǒ * · 1.618 · 1.5 · 10 · 10 · 1.5 * 4 · 1 · 820 10 10 1.5 1.618 · · R W 1.87 k + + 1 * 12 3 * 9 10 · · 10 · 1.5 · 10 · 50 · p 4 820 2 * 12 * * 9 * 9 9 Ǹ Ǔ ǒ · 1.5 · 10 · * 4 1.618 1 · 820 ) 10 · · 1.5 · 10 · 1.5 10 · 1.618 R 4.42 k + W + 2 12 * 9 3 * · 820 · 10 p · · 1.5 · 10 50 · 10 4 R are available 1% resistors. and R 1 2 Third Filter The calculation of the third filter is identical to the calculation of the second filter, except and b are replaced by a and b , thus resulting in different capacitor and resistor that a 2 2 3 3 values. as 330 pF, and obtain C with: Specify C 2 1 4b 3 1 · 4 12 * C 3.46 nF · F + 330 · 10 w C + 1 2 2 2 0.618 a 3 The closest 10% value is 4.7 nF. 16-20

305 High-Pass Filter Design With C = 330 pF and C = 4.7 nF, the values for R1 and R2 are: 2 1 D R = 1.45 k Ω , with the closest 1% value being 1.47 k Ω 1 Ω , with the closest 1% value being 4.53 k Ω = 4.51 k D R 2 Figure 16 – 22 shows the final filter circuit with its partial filter stages. 1.5n 4.7n 3.16k 4.42k 1.87k V IN 1.47k 4.53k 1n V 820p OUT 330p 22. Fifth-Order Unity-Gain Butterworth Low-Pass Filter – Figure 16 16.4 High-Pass Filter Design By replacing the resistors of a low-pass filter with capacitors, and its capacitors with resis- tors, a high-pass filter is created. C 2 R 2 R R 2 1 C C V 2 1 IN V V IN OUT V C OUT 1 R 1 Figure 16 – 23. Low-Pass to High-Pass Transition Through Components Exchange To plot the gain response of a high-pass filter, mirror the gain response of a low-pass filter – 1. at the corner frequency, Ω =1, thus replacing Ω with 1/ Ω and S with 1/S in Equation 16 16-21 Active Filter Design Techniques

306 High-Pass Filter Design 10 A A ∞ 0 0 Highpass Lowpass dB — – 10 Gain — |A| – 20 30 – 0.1 10 1 — Frequency Ω Figure 16 – 24. Developing The Gain Response of a High-Pass Filter The general transfer function of a high-pass filter is then: A R – (16 4) A(s) + b a P i i ǒ Ǔ ) 1 ) 2 s i s being the passband gain. with A ∞ 4 represents a cascade of second-order high-pass filters, the transfer Since Equation 16 – function of a single stage is: A R 5) – (16 + (s) A i a b i i Ǔ ǒ 1 ) ) 2 s s With b=0 for all first-order filters, the transfer function of a first-order filter simplifies to: A 0 – 6) (16 + A(s) a i 1 ) s 16-22

307 High-Pass Filter Design 16.4.1 First-Order High-Pass Filter Figure 16 – 25 and 16 – 26 show a first-order high-pass filter in the noninverting and the in- verting configuration. C 1 V IN V OUT R 1 R 2 R 3 Figure 16 – 25. First-Order Noninverting High-Pass Filter R 2 C 1 R 1 V IN V OUT Figure 16 – 26. First-Order Inverting High-Pass Filter The transfer functions of the circuits are: R R 2 2 1 ) R R 3 1 and A(s) + A(s) +* 1 1 1 1 1 · ) · ) 1 s s w C R w R C c c 1 1 1 1 phase shift from The negative sign indicates that the inverting amplifier generates a 180 ° the filter input to the output. – 6 pro- The coefficient comparison between the two transfer functions and Equation 16 vides two different passband gain factors: R R 2 2 and 1 ) A A + +* R R R R 3 1 is the same for both circuits: while the term for the coefficient a 1 1 a + 1 C R w c 1 1 16-23 Active Filter Design Techniques

308 High-Pass Filter Design ), and capacitor ), the dc gain (A To dimension the circuit, specify the corner frequency (f ∞ C (C ), and then solve for R and R : 1 1 2 1 R + 1 C a 2 f p c 1 1 and ) ( R A R R +* * 1 + R A R R 3 2 1 2 16.4.2 Second-Order High-Pass Filter filters use the same two topologies as the low-pass filters: Sallen-Key and Mul- High-pass tiple Feedback. The only difference is that the positions of the resistors and the capacitors have changed. 16.4.2.1 Sallen-Key Topology The general Sallen-Key topology in Figure 16 – 27 allows for separate gain setting via = 1+R /R . A 0 3 4 R 2 C C 1 2 V IN V OUT R 1 R 4 R 3 Figure 16 – 27. General Sallen-Key High-Pass Filter The transfer function of the circuit in Figure 16 27 is: – a A(s) + R 4 ǒ Ǔ ) ( ) C C R ) C R 1 * a ) 1 + a with 2 2 1 2 1 1 1 1 ) · · ) 1 R 2 2 s w R C R C 3 s w R C R C c c 1 1 2 2 2 2 1 1 The unity-gain topology in Figure 16 – 28 is usually applied in low-Q filters with high gain accuracy. R 2 CC V IN V OUT R 1 – 28. Unity-Gain Sallen-Key High-Pass Filter Figure 16 16-24

309 High-Pass Filter Design α To simplify the circuit design, it is common to choose unity-gain ( = C = 1), and C = C. 2 1 The transfer function of the circuit in Figure 16 – 28 then simplifies to: 1 + A(s) 1 2 1 1 · ) · 1 ) 2 2 s 2 w R C s w R R C c c 1 2 1 5 yields: The coefficient comparison between this transfer function and Equation 16 – A + 1 R 2 + a 1 R C w c 1 1 b + 1 2 2 R C R w c 1 2 Given C, the resistor values for R and R are calculated through: 2 1 1 + R 1 Ca p f c 1 a 1 R + 2 p Cb 4 f c 1 16.4.2.2 Multiple Feedback Topology The MFB topology is commonly used in filters that have high Qs and require a high gain. and C assume the same value To simplify the computation of the circuit, capacitors C 3 1 29. = C – = C) as shown in Figure 16 (C 3 1 C 2 C C =C =C 1 3 R 1 V IN V OUT R 2 29. Second-Order MFB High-Pass Filter Figure 16 – The transfer function of the circuit in Figure 16 – 29 is: C * C 2 + A(s) 2C ) C 1 2 1 1 · · ) ) 1 2 2 s w C C R R R C C w s c 1 2 2 1 2 c 16-25 Active Filter Design Techniques

310 High-Pass Filter Design – 5, obtain the following relations: Through coefficient comparison with Equation 16 C A + R C 2 C ) 2C 2 + a 1 w CC R c 1 2 2C ) C 2 b + 1 R w CC c 1 2 Given capacitors C and C , and solving for resistors R and R : 2 1 2 2A * 1 R + R 1 2 p f a · C · c 1 a 1 R + 2 ) ( 2 p f C · b 1 * 2A R c 2 1 The passband gain (A ) of a MFB high-pass filter can vary significantly due to the wide ∞ . To keep the gain variation at a minimum, it is tolerances of the two capacitors C and C 2 necessary to use capacitors with tight tolerance values. 16.4.3 Higher-Order High-Pass Filter Likewise, as with the low-pass filters, higher-order high-pass filters are designed by cas- cading first-order and second-order filter stages. The filter coefficients are the same ones 4 used for the low-pass filter design, and are listed in the coefficient tables (Tables 16 – through 16 – 10 in Section 16.9). = 1 kHz Example 16 – 4. Third-Order High-Pass Filter with f C The task is to design a third-order unity-gain Bessel high-pass filter with the corner fre- = 1 kHz. Obtain the coefficients for a third-order Bessel filter from Table 16 – 4, quency f C Section 16.9: b a i i Filter 1 a b = 0.756 = 0 1 1 Filter 2 a = 0.9996 b = 0.4772 2 2 and compute each partial filter by specifying the capacitor values and calculating the re- quired resistor values. First Filter = 100 nF, With C 1 16-26

311 Band-Pass Filter Design 1 1 R W 2.105 k + + + 1 3 9 * C a f p 2 · 100 · 10 · Hz F 0.756 2 p · 10 c 1 1 Closest 1% value is 2.1 k Ω . Second Filter With C = 100nF, 1 1 + + R 3.18 k + W 1 * 9 3 Ca f p 0.756 100 · 10 · · 10 p · c 1 Closest 1% value is 3.16 k Ω . a 0.9996 1 1.67 k + + W R + 2 9 * 3 f p 4 Cb 10 p 4 · · 10 0.4772 · · 100 c 1 Closest 1% value is 1.65 k Ω . 30 shows the final filter circuit. Figure 16 – 1.65k 100n 100n 100n V IN V 2.10k OUT 3.16k Figure 16 – 30. Third-Order Unity-Gain Bessel High-Pass 16.5 Band-Pass Filter Design S In Section 16.4, a high-pass response was generated by replacing the term in the low- pass transfer function with the transformation 1/S . Likewise, a band-pass characteristic is generated by replacing the S term with the transformation: 1 1 (16 – 7) Ǔ ǒ s ) s DW In this case, the passband characteristic of a low-pass filter is transformed into the upper passband half of a band-pass filter. The upper passband is then mirrored at the mid fre- =1), into the lower passband half. Ω ( quency, f m 16-27 Active Filter Design Techniques

312 Band-Pass Filter Design |A| [dB] |A| [dB] 0 0 – 3 – 3 ∆Ω Ω Ω 0 Ω Ω 1 0 1 2 1 Figure 16 – 31. Low-Pass to Band-Pass Transition – The corner frequency of the low-pass filter transforms to the lower and upper 3 dB fre- and Ω . The difference between both frequencies is de- Ω quencies of the band-pass, 1 2 : fined as the normalized bandwidth ∆Ω DW + W W * 2 1 The normalized mid frequency, where Q = 1, is: W W · + 1 + W m 2 1 In analogy to the resonant circuits, the quality factor Q is defined as the ratio of the mid ) to the bandwidth (B): frequency (f m f f m m 1 1 (16 – 8) + + + + Q W * DW W B f * f 2 1 2 1 The simplest design of a band-pass filter is the connection of a high-pass filter and a low- filter in series, which is commonly done in wide-band filter applications. Thus, a first- pass order high-pass and a first-order low-pass provide a second-order band-pass, while a second-order high-pass and a second-order low-pass result in a fourth-order band-pass response. In comparison to wide-band filters, narrow-band filters of higher order consist of cascaded second-order band-pass filters that use the Sallen-Key or the Multiple Feedback (MFB) topology. 16-28

313 Band-Pass Filter Design 16.5.1 Second-Order Band-Pass Filter To develop the frequency response of a second-order band-pass filter, apply the trans- – 7 to a first-order low-pass transfer function: formation in Equation 16 A 0 + A(s) 1 ) s 1 1 Ǔ ǒ ) s Replacing s with s DW yields the general transfer function for a second-order band-pass filter: DW · · s A 0 (16 – 9) + A(s) 2 ) DW · s ) s 1 When designing band-pass filters, the parameters of interest are the gain at the mid fre- quency (A ) and the quality factor (Q), which represents the selectivity of a band-pass m filter. 7) and obtain: – with 1/Q (Equation 16 ∆Ω with A and Therefore, replace A 0 m A m · s Q – 10) (16 + A(s) 1 2 ) s s · 1 ) Q Figure 16 – 32 shows the normalized gain response of a second-order band-pass filter for different Qs. 0 – 5 Q = 1 10 – 15 – dB — 20 – Gain Q = 10 — 25 – |A| 30 – 35 – 45 – 10 1 0.1 Frequency — Ω Figure 16 – 32. Gain Response of a Second-Order Band-Pass Filter 16-29 Active Filter Design Techniques

314 Band-Pass Filter Design The graph shows that the frequency response of second-order band-pass filters gets steeper with rising Q, thus making the filter more selective. 16.5.1.1 Sallen-Key Topology R C R V IN V OUT C 2R R 2 R 1 Figure 16 – 33. Sallen-Key Band-Pass The Sallen-Key band-pass circuit in Figure 16 – 33 has the following transfer function: w RC G · · s m + A(s) 2 2 2 2 ) ( w RC ) 1 C w s · * G R · s ) 3 m m Through coefficient comparison with Equation 16 10, obtain the following equations: – 1 f + mid-frequency: m 2 p RC R 2 ) + 1 G inner gain: R 1 G + A gain at f m m: G 3 * 1 Q + filter quality: G * 3 The Sallen-Key circuit has the advantage that the quality factor (Q) can be varied via the inner gain (G) without modifying the mid frequency (fm). A drawback is, however, that Q cannot be adjusted independently. and A m Care must be taken when G approaches the value of 3, because then A becomes infinite m and causes the circuit to oscillate. and C and then solve for R: To set the mid frequency of the band-pass, specify f m 1 R + C 2 p f m 16-30

315 Band-Pass Filter Design Because of , there are two options to solve for R the dependency between Q and A ei- 2: m ther to set the gain at mid frequency: 2A 1 * m + R 2 1 ) A m or to design for a specified Q: * 1 2Q + R 2 Q 16.5.1.2 Multiple Feedback Topology C C R 1 R 2 V IN V OUT R 3 Figure 16 – 34. MFB Band-Pass The MFB band-pass circuit in Figure 16 – 34 has the following transfer function: R R 3 2 s · w C * m ) R R 3 1 + A(s) R R R R 2R 1 1 2 3 3 2 2 2 ) w s C w · · · s C ) 1 m m R R R R ) ) 3 1 3 1 The coefficient comparison with Equation 16 – 9, yields the following equations: R ) R 1 3 1 Ǹ f + mid-frequency: m R R R p C 2 2 3 1 R 2 + A * gain at f : m m 2R 1 R C Q + p f filter quality: m 2 1 B + bandwidth: R p C 2 The MFB band-pass allows to adjust Q, A , and f independently. Bandwidth and gain m m can be used to modify the mid frequency with- . Therefore, R factor do not depend on R 3 3 16-31 Active Filter Design Techniques

316 Band-Pass Filter Design out affecting bandwidth, B, or gain, A . For low values of Q, the filter can work without R3, m however, Q then depends on A via: m 2 2Q A * + m – 5. Second-Order MFB Band-Pass Filter with f Example 16 = 1 kHz m To design a second-order MFB band-pass filter with a mid frequency of f = 1 kHz, a quali- m 2, assume a capacitor value of C = 100 nF, and – = ty factor of Q = 10, and a gain of A m through R in the following sequence: solve the previous equations for R 1 3 Q 10 R W 31.8 k + + + 2 p · 1kHz · 100 nF C f p m R W 31.8 k 2 + W 7.96 k + + R 1 4 * 2A m * A R m W 7.96 k · 2 1 W 80.4 + + + R 3 2 200 * 2 2Q A ) m 16.5.2 Fourth-Order Band-Pass Filter (Staggered Tuning) – 32 shows that the frequency response of second-order band-pass filters gets Figure 16 steeper with rising Q. However, there are band-pass applications that require a flat gain response close to the mid frequency as well as a sharp passband-to-stopband transition. These tasks can be accomplished by higher-order band-pass filters. Of particular interest is the application of the low-pass to band-pass transformation onto a second-order low-pass filter, since it leads to a fourth-order band-pass filter. 7 gives the general transfer func- Replacing the S term in Equation 16 – 2 with Equation 16 – tion of a fourth-order band-pass: 2 2 ( ) · A s DW 0 b 1 (16 – 11) A(s) + 2 a a ( ) DW 1 1 4 2 3 ƪ ƫ DW · s s ) ) · s 1 ) ) ) 2 s DW · b b b 1 1 1 Similar to the low-pass filters, the fourth-order transfer function is split into two second-or- der band-pass terms. Further mathematical modifications yield: A A s mi mi · · a s a Q Q i i – (16 12) + · A(s) 2 2 s a s s 1 ƫ ƪ ) ( Ǔ ǒ ǒ Ǔ 1 a ) s ) ƪ ƫ ) ) 1 a a Q Q 1 i – Equation 16 12 represents the connection of two second-order band-pass filters in se- ries, where 16-32

317 Band-Pass Filter Design D A is the gain at the mid frequency, f , of each partial filter mi mi is the pole quality of each filter Q D i D and 1/ α are the factors by which the mid frequencies of the individual filters, f α m1 and f , of the overall bandpass. , derive from the mid frequency, f m m2 In a fourth-order band-pass filter with high Q, the mid frequencies of the two partial filters differ only slightly from the overall mid frequency. This method is called staggered tuning. Factor α needs to be determined through successive approximation, using equation – 13: 16 2 2 · DW · a a ( ) DW 1 1 2 (16 – 13) a + ƪ 0 ƫ ) ) * 2 * 2 1 2 ǒ Ǔ a b ) b a 1 1 being the second-order low-pass coefficients of the desired filter type. with a and b 1 1 To simplify the filter design, Table 16 – values 2 lists those coefficients, and provides the α for three different quality factors, Q = 1, Q = 10, and Q = 100. Table 16 2. Values of α For Different Filter Types and Different Qs – Butterworth Tschebyscheff Bessel a a 1.4142 a 1.0650 1.3617 1 1 1 b b 0.6180 b 1.9305 1.0000 1 1 1 100 Q 10 1 Q 100 10 1 Q 100 10 1 0.1 ∆Ω 0.1 1 ∆Ω 0.01 0.01 1 ∆Ω 0.01 0.1 1 1.39 1.0338 α 1.0032 1.0324 1.438 α 1.0035 1.036 1.4426 α 1.0033 16-33 Active Filter Design Techniques

318 Band-Pass Filter Design has been determined, all quantities of the partial filters can be calculated using the α After following equations: The mid frequency of filter 1 is: f m 14) – (16 + f m1 a the mid frequency of filter 2 is: (16 – 15) a f · + f m m2 with f being the mid frequency of the overall forth-order band-pass filter. m , is the same for both filters: The individual pole quality, Q i 2 Ǔ ǒ b 1 ) a 1 16) (16 – Q · + Q i a · a 1 with Q being the quality factor of the overall filter. The individual gain (A and f , is the same for both ) at the partial mid frequencies, f m2 m1 mi filters: Q A i m 17) (16 – + · A Ǹ mi B Q 1 with A being the gain at mid frequency, f , of the overall filter. m m Example 16 6. Fourth-Order Butterworth Band-Pass Filter – The task is to design a fourth-order Butterworth band-pass with the following parameters: D mid frequency, f = 10 kHz m bandwidth, B = 1000 Hz D = 1 and gain, A D m From Table 16 – 2 the following values are obtained: D a1 = 1.4142 D b1 = 1 = 1.036 α D 16-34

319 Band-Pass Filter Design In accordance with Equations 16 14 and 16 – 15, the mid frequencies for the partial filters – are: 10 kHz and + + + f f 10.36 kHz + 10 kHz · 1.036 9.653 kHz m2 mi 1.036 , and for this example results in Q = 10. The overall Q is defined as B ń Q + f m Using Equation 16 – 16, the Q of both filters is: i 2 ǒ Ǔ ) 1 1.036 · 1 · + Q 10 + 14.15 i 1.4142 1.036 · With Equation 16 – 17, the passband gain of the partial filters at f and f calculates to: m2 m1 14.15 1 Ǹ A + · + 1.415 mi 10 1 – 16 and 16 17 show that Q of the partial filters need to be inde- The Equations 16 and A – mi i pendently adjusted. The only circuit that accomplishes this task is the MFB band-pass fil- ter in Paragraph 16.5.1.2. To design the individual second-order band-pass filters, specify C = 10 nF, and insert the previously determined quantities for the partial filters into the resistor equations of the MFB band-pass filter. The resistor values for both partial filters are calculated below. Filter 2: Filter 1: Q Q 14.15 14.15 i i + + 43.5 k W + + + 46.7 k W R + R 22 21 · · p 10 nF 9.653 kHz · 10.36 kHz · 10 nF p p C p C f f m2 m1 R R 46.7 k 43.5 k W W 21 22 + + W + 16.5 k + R R + + 15.4 k W 11 12 1.415 1.415 * * · 2 2 · * * 2A 2A * * mi mi R R A * A * W · 16.5 k · 1.415 1.415 W 15.4 k 12 11 mi mi R 54.2 + W + + + 58.1 W R + + 31 32 2 2 2 2 · 14.15 14.15 2 · 2 ) ) 1.415 1.415 ) A ) 2Q 2Q A i i mi mi Figure 16 – 35 compares the gain response of a fourth-order Butterworth band-pass filter with Q = 1 and its partial filters to the fourth-order gain of Example 16 – 4 with Q = 10. 16-35 Active Filter Design Techniques

320 Band-Rejection Filter Design 5 A2 A1 0 Q = 1 – 5 Q = 10 – 10 dB — 15 – Gain — 20 – |A| – 25 30 – – 35 1 M 100 100 k 10 k 1 k Hz f — Frequency — Figure 16 – 35. Gain Responses of a Fourth-Order Butterworth Band-Pass and its Partial Filters 16.6 Band-Rejection Filter Design A band-rejection filter is used to suppress a certain frequency rather than a range of fre- quencies. Two of the most popular band-rejection filters are the active twin-T and the active Wien- Robinson circuit, both of which are second-order filters. To generate the transfer function of a second-order band-rejection filter, replace the S – 18: term of a first-order low-pass response with the transformation in 16 DW 18) – (16 1 s ) s which gives: 2 ǒ Ǔ A ) s 1 0 19) – (16 + A(s) 2 s DW · s ) ) 1 Thus the passband characteristic of the low-pass filter is transformed into the lower pass- band of the band-rejection filter. The lower passband is then mirrored at the mid frequen- 36). – =1), into the upper passband half (Figure 16 Ω ( cy, f m 16-36

321 Band-Rejection Filter Design |A| [dB] |A| [dB] ∆Ω 0 0 – 3 – 3 0 Ω 1 0 Ω Ω Ω 1 2 1 Figure 16 – 36. Low-Pass to Band-Rejection Transition 3-dB frequen- The corner frequency of the low-pass transforms to the lower and upper – . The difference between both frequencies is the and Ω cies of the band-rejection filter Ω 2 1 normalized bandwidth ∆Ω : DW + W W * max min Identical to the selectivity of a band-pass filter, the quality of the filter rejection is defined as: f m 1 + Q + DW B Therefore, replacing – 19 with 1/Q yields: ∆Ω in Equation 16 2 Ǔ ǒ s ) 1 A 0 20) – (16 + A(s) 1 2 · s ) s 1 ) Q 16.6.1 Active Twin-T Filter The original twin-T filter, shown in Figure 16 – 37, is a passive RC-network with a quality factor of Q = 0.25. To increase Q, the passive filter is implemented into the feedback loop of an amplifier, thus turning into an active band-rejection filter, shown in Figure 16 – 38. C C R/2 V V IN OUT RR 2C 37. Passive Twin-T Filter – Figure 16 16-37 Active Filter Design Techniques

322 Band-Rejection Filter Design C C R/2 V IN R R V OUT 2C R 2 R 1 Figure 16 – 38. Active Twin-T Filter The transfer function of the active twin-T filter is: 2 ǒ Ǔ ) s k 1 (16 21) – + A(s) 2 ( ) 2 · 2 * k 1 ) s ) s Comparing the variables of Equation 16 21 with Equation 16 – 20 provides the equations – that determine the filter parameters: 1 f + mid-frequency: m 2 p RC R 2 G + 1 ) inner gain: R 1 A + G passband gain: 0 1 Q + rejection quality: ( ) * G 2 2 The twin-T circuit has the advantage that the quality factor (Q) can be varied via the inner ). However, Q and A cannot be adjusted gain (G) without modifying the mid frequency (f m m independently. and C, and then solve for R: To set the mid frequency of the band-pass, specify f m 1 R + C 2 p f m , there are two options to solve for R Because of the dependency between Q and A ei- 2: m set the gain at mid frequency: ther to Ǔ ǒ + R A * 1 R 2 1 0 16-38

323 Band-Rejection Filter Design or to design for a specific Q: 1 Ǔ ǒ R R * 1 + 1 2 2Q 16.6.2 Active Wien-Robinson Filter The Wien-Robinson bridge in Figure 16 – 39 is a passive band-rejection filter with dif feren- tial output. The output voltage is the difference between the potential of a constant voltage divider and the output of a band-pass filter. Its Q-factor is close to that of the twin-T circuit. To achieve higher values of Q, the filter is connected into the feedback loop of an amplifier. V IN R 2R 1 C V OUT C R R 1 39. Passive Wien-Robinson Bridge Figure 16 – R 3 R 2R R 1 2 1 R 4 V IN C R V OUT C R Figure 16 – 40. Active Wien-Robinson Filter The active Wien-Robinson filter in Figure 16 – 40 has the transfer function: b 2 ǒ Ǔ 1 s ) ) a 1 22) – (16 +* A(s) 3 2 · s ) s 1 ) 1 ) a R R 2 2 with + a b + and R R 3 4 20 provides the equations – Comparing the variables of Equation 16 – 22 with Equation 16 that determine the filter parameters: 16-39 Active Filter Design Techniques

324 Band-Rejection Filter Design 1 + f mid-frequency: m 2 RC p b +* A passband gain: 0 1 ) a a ) 1 + Q rejection quality: 3 To calculate the individual component values, establish the following design procedure: and C and calculate R with: 1) Define f m 1 + R C 2 p f m 2) Specify Q and determine α via: 1 * a + 3Q 3) Specify A via: and determine β 0 +* · 3Q b A 0 4) Define R and calculate R with: and R 4 3 2 R 2 R + 3 a and R 2 R + 4 b In comparison to the twin-T circuit, the Wien-Robinson filter allows modification of the passband gain, A , without affecting the quality factor, Q. 0 If f is not completely suppressed due to component tolerances of R and C, a fine-tuning m is required. of the resistor 2R 2 – Figure 16 41 shows a comparison between the filter response of a passive band-rejec- tion filter with Q = 0.25, and an active second-order filter with Q = 1, and Q = 10. 16-40

325 All-Pass Filter Design 0 – 5 Q = 10 dB Q = 1 — 10 – Q = 0.25 Gain — |A| – 15 20 – 10 k 1 k 100 10 1 Ω Frequency — Figure 16 41. Comparison of Q Between Passive and Active Band-Rejection Filters – 16.7 All-Pass Filter Design In comparison to the previously discussed filters, an all-pass filter has a constant gain across the entire frequency range, and a phase response that changes linearly with fre- quency. Because of these properties, all-pass filters are used in phase compensation and signal delay circuits. Similar to the low-pass filters, all-pass circuits of higher order consist of cascaded first-or- der and second-order all-pass stages. To develop the all-pass transfer function from a with the conjugate complex denominator. low-pass response, replace A 0 The general transfer function of an allpass is then: P 2 ǒ Ǔ s 1 * a s ) b i i i (16 – 23) + A(s) P 2 Ǔ ǒ s 1 ) a b s ) i i i being the coefficients of a partial filter. The all-pass coefficients are listed in and b with a i i Table 16 – 10 of Section 16.9. Expressing Equation 16 – 23 in magnitude and phase yields: 16-41 Active Filter Design Techniques

326 All-Pass Filter Design 2 P 2 ja 2 * 2 Ǹ ǒ Ǔ b W W · e 1 ) a * i i i (16 – 24) A(s) + 2 P 2 2 2 ) ja Ǹ ǒ Ǔ 1 ) a W W * · e b i i i , of: φ This gives a constant gain of 1, and a phase shift, W a i (16 – 25) ȍ +* 2 +* a arctan f 2 2 b 1 * W i i To transmit a signal with minimum phase distortion, the all-pass filter must have a constant group delay across the specified frequency band. The group delay is the time by which the all-pass filter delays each frequency within that band. Ǹ – times its initial value is the corner The frequency at which the group delay drops to 1 ń 2 frequency, f . C The group delay is defined through: f d – (16 26) +* t gr d w To present the group delay in normalized form, refer t to the period of the corner frequen- gr cy, T , of the all-pass circuit: C t w gr c (16 – 27) + T · + t · f + t gr c gr gr 2 p T c Substituting t 26 gives: through Equation 16 – gr f d 1 – 28) (16 +* T · gr 2 p d W 16-42

327 All-Pass Filter Design φ term in Equation 16 – 25 into Equation 16 – 28 and completing the derivation, Inserting the results in: 2 Ǔ ǒ a W b 1 ) i i 1 29) (16 – ȍ T + gr p 2 2 4 2 ǒ Ǔ 1 ) * 2b b a · W W ) i 1 1 1 < 1, Setting Ω = 0 in Equation 16 – 29 gives the group delay for the low frequencies, 0 < Ω which is: 1 (16 – 30) ȍ a T + gr0 i p i The values for T – 10, Section 16.9, from the first to the tenth order. are listed in Table 16 gr0 – 42 shows the group delay response versus the frequency for the In addition, Figure 16 first ten orders of all-pass filters. 3.5 th Order 10 th 3 9 Order s/s th — Order 8 2.5 th 7 Order th 2 6 Order th Order 5 1.5 th 4 Order rd 1 3 Order Normalized Group Delay — nd 2 Order gr T 0.5 st 1 Order 0 1 0.1 0.01 10 100 — Frequency Ω 42. Frequency Response of the Group Delay for the First 10 Filter Orders – Figure 16 16-43 Active Filter Design Techniques

328 All-Pass Filter Design 16.7.1 First-Order All-Pass Filter – Figure 16 43 shows a first-order all-pass filter with a gain of +1 at low frequencies and a gain of – 1 at high frequencies. Therefore, the magnitude of the gain is 1, while the phase ° to – 180 ° changes from 0 . R R 1 1 V V IN OUT R C Figure 16 – 43. First-Order All-Pass The transfer function of the circuit above is: s · 1 * RC w c + A(s) s · RC w 1 ) c The coefficient comparison with Equation 16 – 23 (b =1), results in: 1 (16 – 31) 2 f a p + RC · c i and C and then solve for R: To design a first-order all-pass, specify f C a i 32) (16 – R + f 2 · C p c – 31 into 16 – 30 and substituting ω with Equation 16 – Inserting Equation 16 27 provides C the maximum group delay of a first-order all-pass filter: (16 – 33) + 2RC t gr0 16.7.2 Second-Order All-Pass Filter Figure 16 – 44 shows that one possible design for a second-order all-pass filter is to sub- tract the output voltage of a second-order band-pass filter from its input voltage. C R C R R 1 2 V R 3 IN V OUT R 44. Second-Order All-Pass Filter – Figure 16 16-44

329 All-Pass Filter Design – 44 is: The transfer function of the circuit in Figure 16 2 2 2 ǒ Ǔ 2R w * a R · C C w s · s ) R R 1 ) c c 1 2 1 2 A(s) + 2 2 2 C s · · s ) R R 1 C ) w 2R w c c 1 2 1 The coefficient comparison with Equation 16 – 23 yields: (16 – 34) p + 4 a f C R c 1 1 – 35) (16 b C + a R p f c 1 2 1 2 a 1 R 36) – (16 a + + b R 1 3 , C, and R, and then solve for the resistor values: To design the circuit, specify f C a 1 37) – (16 + R 1 p f 4 C c b 1 38) (16 – + R 2 a C p f c 1 R (16 – 39) R + 3 a – 27 Inserting Equation 16 – 34 into Equation16 – 30 and substituting ω with Equation 16 C gives the maximum group delay of a second-order all-pass filter: – 40) (16 t + 4R C 1 gr0 16.7.3 Higher-Order All-Pass Filter Higher-order all-pass filters consist of cascaded first-order and second-order filter stages. Example 16 – 7. 2-ms Delay All-Pass Filter A signal with the frequency spectrum, 0 < f < 1 kHz, needs to be delayed by 2 ms. To keep the phase distortions at a minimum, the corner frequency of the all-pass filter must be ≥ 1 kHz. f C – 27 determines the normalized group delay for frequencies below 1 kHz: Equation 16 t gr0 · + + 2.0 1kHz T + 2ms gro T C Figure 16 – 42 confirms that a seventh-order all-pass is needed to accomplish the desired = 2.1737. To set the group delay to precisely 2 ms, delay. The exact value, however, is T gr0 and obtain the corner frequency: solve Equation 16 – 27 for f C 16-45 Active Filter Design Techniques

330 All-Pass Filter Design T gr0 + + 1.087 kHz f C t gr0 To complete the design, look up the filter coefficients for a seventh-order all-pass filter, specify C, and calculate the resistor values for each partial filter. Cascading the first-order all-pass with the three second-order stages results in the de- sired seventh-order all-pass filter. C 2 R R 11 11 R 2 C 2 R R 12 22 R V 32 IN R R C 2 1 1 C 3 R 3 C 3 R R 13 23 R 33 R 3 C 4 R 4 C 4 R R 14 24 R 34 V OUT R 4 Figure 16 – 45. Seventh-Order All-Pass Filter 16-46

331 Practical Design Hints 16.8 Practical Design Hints This section introduces dc-biasing techniques for filter designs in single-supply applica- tions, which are usually not required when operating with dual supplies. It also provides recommendations on selecting the type and value range of capacitors and resistors as well as the decision criteria for choosing the correct op amp. 16.8.1 Filter Circuit Biasing The filter diagrams in this chapter are drawn for dual supply applications. The op amp op- erates from a positive and a negative supply, while the input and the output voltage are referenced to ground (Figure 16 46). – +V CC R 2 C 1 R 1 V IN V OUT – V CC Figure 16 – 46. Dual-Supply Filter Circuit For the single supply circuit in Figure 16 – 47, the lowest supply voltage is ground. For a output signal, the potential of the noninverting input is level-shifted to midrail. symmetrical +V CC R R 2 B C IN R 1 V IN V MID V OUT R B Figure 16 – 47. Single-Supply Filter Circuit The coupling capacitor, C 47, ac-couples the filter, blocking any unknown in Figure 16 – IN dc level in the signal source. The voltage divider, consisting of the two equal-bias resistors , divides the supply voltage to V and applies it to the inverting op amp input. R B MID For simple filter input structures, passive RC networks often provide a low-cost biasing the case of more complex input structures, such as the input of a second-order solution. In 16-47 Active Filter Design Techniques

332 Practical Design Hints low-pass filter, the RC network can affect the filter characteristic. Then it is necessary to either include the biasing network into the filter calculations, or to insert an input buffer between biasing network and the actual filter circuit, as shown in Figure 16 – 48. +V CC +V CC C 2 R B C IN V MID V R R V MID 2 1 MID V MID R V B IN C 1 V OUT – Figure 16 48. Biasing a Sallen-Key Low-Pass is derived from ac-couples the filter, blocking any dc level in the signal source. V C IN MID V via the voltage divider. The op amp operates as a voltage follower and as an imped- CC and R , to the noninverting input of the is applied via the dc path, R ance converter. V MID 2 1 filter amplifier. Note that the parallel circuit of the resistors, R create a high-pass filter. , together with C B IN To avoid any effect on the low-pass characteristic, the corner frequency of the input high- pass must be low versus the corner frequency of the actual low-pass. The use of an input buffer causes no loading effects on the low-pass filter, thus keeping the filter calculation simple. In the case of a higher-order filter, all following filter stages receive their bias level from the preceding filter amplifier. 49 shows the biasing of an multiple feedback (MFB) low-pass filter. – Figure 16 16-48

333 Practical Design Hints +V +V CC CC R 2 C R B 1 C IN V MID V R R MID 3 1 V R IN C B 2 V OUT R B V +V MID CC V MID R C B B to further filter stages Figure 16 – 49. Biasing a Second-Order MFB Low-Pass Filter The input buffer decouples the filter from the signal source. The filter itself is biased via the noninverting amplifier input. For that purpose, the bias voltage is taken from the output with low output impedance. The op amp operates as a difference am- generator of a V MID plifier and subtracts the bias voltage of the input buffer from the bias voltage of the V MID generator, thus yielding a dc potential of V at zero input signal. MID A low-cost alternative is to remove the op amp and to use a passive biasing network must be signifi- However, to keep loading effects at a minimum, the values for R instead. B cantly higher than without the op amp. The biasing of a Sallen-Key and an MFB high-pass filter is shown in Figure 16 – 50. The input capacitors of high-pass filters already provide the ac-coupling between filter and signal source. Both circuits use the V 50 for biasing. While generator from Figure 16 – MID the MFB circuit is biased at the noninverting amplifier input, the Sallen-Key high-pass is . In the ac circuit, the input signals travel biased via the only dc path available, which is R 1 via the low output impedance of the op amp to ground. 16-49 Active Filter Design Techniques

334 Practical Design Hints +V CC +V CC C 2 R 2 R 1 CC =C C C =C 1 3 V IN V V IN OUT R 1 V R OUT 2 V MID V MID +V CC R B V MID R C B B Figure 16 – 50. Biasing a Sallen-Key and an MFB High-Pass Filter 16.8.2 Capacitor Selection The tolerance of the selected capacitors and resistors depends on the filter sensitivity and on the filter performance. Sensitivity is the measure of the vulnerability of a filter ’ s performance to changes in com- , ponent values. The important filter parameters to consider are the corner frequency, f C and Q. For example, when Q changes by ± 5% change in the capacitance value, ± 2% due to a Q % 2% . The then the sensitivity of Q to capacity changes is expressed as: + + 0.4 s % 5% C following sensitivity approximations apply to second-order Sallen-Key and MFB filters: f f Q Q % c c [ [ 0.5 [ s s s [" s % R R C C Although 0.5 %/% is a small dif ference from the ideal parameter, in the case of higher-or- differences in each partial filter can signifi- der filters, the combination of small Q and f C cantly modify the overall filter response from its intended characteristic. Figures 16.51 and 16.52 show how an intended eighth-order Butterworth low-pass can turn into a low-pass with T schebyscheff characteristic mainly due to capacitance changes from the partial filters. Figure 16 – 51 shows the differences between the ideal and the actual frequency re- – 52. sponses of the four partial filters. The overall filter responses are shown in Figure 16 16-50

335 Practical Design Hints The difference between ideal and real response peaks with 0.35 dB at approximately 30 kHz, which is equivalent to an enormous 4.1% gain error can be seen. 9 A4R A4 7.5 6 4.5 dB — 3 Gain A3R A3 — A2R 1.5 A2 |A| 0 1.5 – A1 A1R – 3 10 k 100 k Hz — Frequency f — 51. Differences in Q and f in the Partial Filters of an Eighth-Order Butterworth Figure 16 – C Low-Pass Filter 0.4 A – Real 0 A – Ideal 0.4 – 0.8 – dB — – 1.2 Gain — 1.6 – |A| 2 – – 2.4 2.8 – 1 k 10 k 100 k — Frequency — Hz f 52. Modification of the Intended Butterworth Response to a Figure 16 – Tschebyscheff-Type Characteristic 16-51 Active Filter Design Techniques

336 Practical Design Hints If this filter is intended for a data acquisition application, it could be used at best in a 4-bit given with system. In comparison, if the maximum full-scale error of a 12-bit system is ½ LSB, then maximum pass-band deviation would be – 0.001 dB, or 0.012%. and Q, NPO (COG) ceramic capacitors are recommended To minimize the variations of f C for high-performance filters. These capacitors hold their nominal value over a wide tem- perature and voltage range. The various temperature characteristics of ceramic capaci- tors are identified by a three-symbol code such as: COG, X7R, Z5U, and Y5V. COG-type ceramic capacitors are the most precise. Their nominal values range from 0.5 pF to ± 0.25 pF for smaller values and approximately 47 nF with initial tolerances from up to ± 1% for higher values. Their capacitance drift over temperature is typically 30ppm/ ° C. X7R-type ceramic capacitors range from 100 pF to 2.2 μ F with an initial tolerance of +1% 15%. ± and a capacitance drift over temperature of For higher values, tantalum electrolytic capacitors should be used. precision capacitors are silver mica, metallized polycarbonate, and for high temper- Other atures, polypropylene or polystyrene. Since capacitor values are not as finely subdivided as resistor values, the capacitor val- ues should be defined prior to selecting resistors. If precision capacitors are not available to provide an accurate filter response, then it is necessary to measure the individual ca- pacitor values, and to calculate the resistors accordingly. For high performance filters, 0.1% resistors are recommended. 16.8.3 Component Values Resistor values should stay within the range of 1 k Ω to 100 k Ω . The lower limit avoids ex- cessive current draw from the op amp output, which is particularly important for single- amps in power-sensitive applications. Those amplifiers have typical output cur- supply op rents of between 1 mA and 5 mA. At a supply voltage of 5 V, this current translates to a minimum of 1 k Ω . The upper limit of 100 k Ω is to avoid excessive resistor noise. Capacitor values can range from 1 nF to several μ F. The lower limit avoids coming too close to parasitic capacitances. If the common-mode input capacitance of the op amp, used in a Sallen-Key filter section, is close to 0.25% of C1, (C1 / 400), it must be consid- ered for accurate filter response. The MFB topology, in comparison, does not require in- put-capacitance compensation. 16-52

337 Practical Design Hints 16.8.4 Op Amp Selection The most important op amp parameter for proper filter functionality is the unity-gain band- width. In general, the open-loop gain (A ) should be 100 times (40 dB above) the peak OL gain (Q) of a filter section to allow a maximum gain error of 1%. |A| [dB] A OL 40 dB ‘ A PEAK A A 0 0 f / Hz f f T P Figure 16 – 53. Open-Loop Gain (A ) and Filter Response (A) OL The following equations are good rules of thumb to determine the necessary unity-gain bandwidth of an op amp for an individual filter section. 1) First-order filter: f 100 · Gain + f · c T 2) Second-order filter (Q < 1): f ci f 100 · Gain · f + · k with k + c i T i f c 3) Second-order filter (Q > 1): 2 0.5 * Q f c i f + 100 · Gain · Ǹ T 2 a i 0.25 * Q i For example, a fifth-order, 10-kHz, T schebyscheff low-pass filter with 3-dB passband rip- = 2 has its worst case Q in the third filter section. With Q = 8.82 ple and a dc gain of A 0 3 and a = 0.1172, the op amp needs to have a unity-gain bandwidth of: 3 2 10 kHz 0.5 8.82 * Ǹ f 100 · + 2 · 17 MHz + T 2 0.1172 8.82 * 0.25 In comparison, a fifth-order unity-gain, 10-kHz, Butterworth low-pass filter has a worst is also lower and calculates = 1.62; a = 0.618. Due to the lower Q value, f case Q of Q 3 3 T to only: 16-53 Active Filter Design Techniques

338 Practical Design Hints 2 10 kHz 0.5 1.62 * Ǹ · + + f 100 1.5 MHz T 2 0.618 1.62 0.25 * Besides good dc performance, low noise, and low signal distortion, another important pa- rameter that determines the speed of an op amp is the slew rate (SR). For adequate full- power response, the slew rate must be greater than: · SR + p · V f PP C For example, a single-supply, 100-kHz filter with 5 V output requires a slew rate of at PP least: V · p · 5V 1.57 100 kHz + SR + m s Texas Instruments offers a wide range of op amps for high-performance filters in single supply applications. Table 16 – 3 provides a selection of single-supply amplifiers sorted in order of rising slew rate. = 5 V) Table 16 – 3. Single-Supply Op Amp Selection Guide (T = 25 ° C, V CC A Noise FPR BW SR V IO OP AMP (mV) Hz) s) (V/ (kHz) (MHz) (nV/ √ μ TLV2721 0.51 11 0.18 0.6 20 TLC2201A 0.6 1.8 159 2.5 8 TLV2771A 572 9 1.9 21 4.8 TLC071 10 1000 16 1.5 7 TLE2141 5.9 2800 45 0.5 10.5 6 7.5 THS4001 270 127 MHz (1V 400 ) PP 16-54

339 Filter Coefficient Tables 16.9 Filter Coefficient Tables The following tables contain the coefficients for the three filter types, Bessel, Butterworth and Tschebyscheff. The T schebyscheff tables (Table 16 – 9) are split into categories for the following passband ripples: 0.5 dB, 1 dB, 2 dB, and 3 dB. The table headers consist of the following quantities: is the filter order n is the number of the partial filter. i , b are the filter coefficients. a i i , to the corner is the ratio of the corner frequency of a partial filter, f k i Ci . This ratio is used to determine the frequency of the overall filter, f C unity-gain bandwidth of the op amp, as well as to simplify the test of . and comparing it to f a filter design by measuring f Ci C Q is the quality factor of the partial filter. i is the this ratio is used for test purposes of the allpass filters, where f / f f C i i frequency, at which the phase is 180 ° for a second-order filter, for a first-order all-pass. respectively 90 ° is the normalized group delay of the overall all-pass filter. T gr0 16-55 Active Filter Design Techniques

340 Filter Coefficient Tables Table 16 4. Bessel Coefficients – i a b n k Q = i i i i f / f Ci C 1 1.0000 0.0000 1.000 1 2 1.3617 0.6180 1.000 0.58 1 1 0.0000 3 1.323 0.7560 0.9996 0.4772 1.414 0.69 2 4 1 1.3397 0.4889 0.978 0.52 2 0.7743 1.797 0.81 0.3890 1 1.502 0.0000 5 0.6656 1.1402 0.4128 1.184 0.56 2 3 0.6216 0.3245 2.138 0.92 6 1.2217 0.3887 1.063 0.51 1 2 0.9686 0.3505 1.431 0.61 3 0.5131 0.2756 2.447 1.02 0.0000 7 1 0.5937 1.648 2 1.0944 0.53 0.3395 1.207 0.3011 1.695 0.8304 3 0.66 4 0.4332 0.2381 2.731 1.13 1 1.1112 0.3162 1.164 0.51 8 0.9754 2 1.381 0.56 0.2979 0.7202 0.2621 3 0.71 1.963 4 0.3728 0.2087 2.992 1.23 9 1 0.5386 0.0000 1.857 1.0244 2 0.2834 1.277 0.52 3 0.8710 1.574 0.59 0.2636 0.6320 0.2311 4 0.76 2.226 5 0.3257 0.1854 3.237 1.32 10 1 1.0215 0.2650 1.264 0.50 2 0.2549 1.412 0.54 0.9393 3 0.7815 0.2351 1.780 0.62 4 0.5604 0.2059 2.479 0.81 3.466 1.42 5 0.2883 0.1665 16-56

341 Filter Coefficient Tables Table 16 5. Butterworth Coefficients – i a b n k Q = i i i i f / f Ci C 1 1.0000 0.0000 1.000 1 2 1.4142 1.0000 1.000 0.71 1 1 0.0000 3 1.000 1.0000 1.0000 1.0000 1.272 1.00 2 4 1 1.8478 1.0000 0.719 0.54 2 0.7654 1.390 1.31 1.0000 1 1.000 0.0000 5 1.0000 1.6180 1.0000 0.859 0.62 2 3 0.6180 1.0000 1.448 1.62 6 1.9319 1.0000 0.676 0.52 1 2 1.4142 1.0000 1.000 0.71 3 0.5176 1.0000 1.479 1.93 0.0000 7 1 1.0000 1.000 2 1.8019 0.55 1.0000 0.745 1.0000 1.117 1.2470 3 0.80 4 0.4450 1.0000 1.499 2.25 1 1.9616 1.0000 0.661 0.51 8 1.6629 2 0.829 0.60 1.0000 1.1111 1.0000 3 0.90 1.206 4 0.3902 1.0000 1.512 2.56 9 1 1.0000 0.0000 1.000 1.8794 2 1.0000 0.703 0.53 3 1.5321 0.917 0.65 1.0000 1.0000 1.0000 4 1.00 1.272 5 0.3473 1.0000 1.521 2.88 10 1 1.9754 1.0000 0.655 0.51 2 1.0000 0.756 0.56 1.7820 3 1.4142 1.0000 1.000 0.71 4 0.9080 1.0000 1.322 1.10 1.527 3.20 5 0.3129 1.0000 16-57 Active Filter Design Techniques

342 Filter Coefficient Tables Table 16 6. Tschebyscheff Coefficients for 0.5-dB Passband Ripple – i a b n k Q = i i i i f / f Ci C 1 1.0000 0.0000 1.000 1 2 1.3614 1.3827 1.000 0.86 1 1 0.0000 3 0.537 1.8636 0.0640 1.1931 1.335 1.71 2 4 1 2.6282 3.4341 0.538 0.71 2 0.3648 1.419 2.94 1.1509 1 0.342 0.0000 5 2.9235 1.3025 2.3534 0.881 1.18 2 3 0.2290 1.0833 1.480 4.54 6 3.8645 6.9797 0.366 0.68 1 2 0.7528 1.8573 1.078 1.81 3 0.1589 1.0711 1.495 6.51 0.249 7 1 4.0211 0.0000 2 1.8729 1.09 4.1795 0.645 1.5676 1.208 0.4861 3 2.58 4 0.1156 1.0443 1.517 8.84 1 5.1117 11.960 0.68 0.276 8 7 2 2.9365 0.844 1.61 1.0639 0.3439 1.4206 3 3.47 1.284 4 0.0885 1.0407 1.521 11.53 9 1 5.1318 0.0000 0.195 2.4283 2 6.6307 0.506 1.06 3 0.6839 0.989 2.21 2.2908 0.2559 1.3133 4 4.48 1.344 5 0.0695 1.0272 1.532 14.58 10 1 6.3648 18.369 0.67 0.222 5 1.3582 4.3453 0.689 1.53 2 3 0.4822 1.9440 1.091 2.89 4 0.1994 1.2520 1.381 5.61 17.99 1.533 5 0.0563 1.0263 16-58

343 Filter Coefficient Tables Table 16 7. Tschebyscheff Coefficients for 1-dB Passband Ripple – i a b n k Q = i i i i f / f Ci C 1 1.0000 0.0000 1.000 1 2 1.3022 1.5515 1.000 0.96 1 1 0.0000 3 0.451 2.2156 0.5442 1.2057 1.353 2.02 2 4 1 2.5904 4.1301 0.540 0.78 2 0.3039 1.417 3.56 1.1697 1 0.280 0.0000 5 3.5711 1.1280 2.4896 0.894 1.40 2 3 0.1872 1.0814 1.486 5.56 6 3.8437 8.5529 0.366 0.76 1 2 0.6292 1.9124 1.082 2.20 3 0.1296 1.0766 1.493 8.00 0.202 7 1 4.9520 0.0000 2 1.6338 1.30 4.4899 0.655 1.5834 1.213 0.3987 3 3.16 4 0.0937 1.0432 1.520 10.90 1 5.1019 14.760 0.75 0.276 8 8 2 3.0426 0.849 1.96 0.8916 0.2806 1.4334 3 4.27 1.285 4 0.0717 1.0432 1.520 14.24 9 1 6.3415 0.0000 0.158 2.1252 2 7.1711 0.514 1.26 3 0.5624 0.994 2.71 2.3278 0.2076 1.3166 4 5.53 1.346 5 0.0562 1.0258 1.533 18.03 10 1 6.3634 22.746 0.75 0.221 8 1.1399 4.5167 0.694 1.86 2 3 0.3939 1.9665 1.093 3.56 4 0.1616 1.2569 1.381 6.94 22.26 1.532 5 0.0455 1.0277 16-59 Active Filter Design Techniques

344 Filter Coefficient Tables Table 16 8. Tschebyscheff Coefficients for 2-dB Passband Ripple – i a b n k Q = i i i i f / f Ci C 1 1.0000 0.0000 1.000 1 2 1.1813 1.7775 1.000 1.13 1 1 0.0000 3 0.357 2.7994 0.4300 1.2036 1.378 2.55 2 4 1 2.4025 4.9862 0.550 0.93 2 0.2374 1.413 4.59 1.1896 1 0.216 0.0000 5 4.6345 0.9090 2.6036 0.908 1.78 2 3 0.1434 1.0750 1.493 7.23 6 3.5880 10.464 0.373 0.90 1 8 2 0.4925 1.9622 1.085 2.84 3 0.0995 1.0826 1.491 10.46 0.154 7 1 6.4760 0.0000 2 1.3258 1.65 4.7649 0.665 1.5927 1.218 0.3067 3 4.12 4 0.0714 1.0384 1.523 14.28 1 4.7743 18.151 0.89 0.282 8 0 2 3.1353 0.853 2.53 0.6991 0.2153 1.4449 3 5.58 1.285 4 0.0547 1.0461 1.518 18.39 9 1 8.3198 0.0000 0.120 1.7299 2 7.6580 0.522 1.60 3 0.4337 0.998 3.54 2.3549 0.1583 1.3174 4 7.25 1.349 5 0.0427 1.0232 1.536 23.68 10 1 5.9618 28.037 0.89 0.226 6 0.8947 4.6644 0.697 2.41 2 3 0.3023 1.9858 1.094 4.66 4 0.1233 1.2614 1.380 9.11 29.27 1.531 5 0.0347 1.0294 16-60

345 Filter Coefficient Tables Table 16 9. Tschebyscheff Coefficients for 3-dB Passband Ripple – i a b n k Q = i i i i f / f Ci C 1 1.0000 0.0000 1.000 1 2 1.0650 1.9305 1.000 1.30 1 1 0.0000 3 0.299 3.3496 0.3559 1.1923 1.396 3.07 2 4 1 2.1853 5.5339 0.557 1.08 2 0.1964 1.410 5.58 1.2009 1 0.178 0.0000 5 5.6334 0.7620 2.6530 0.917 2.14 2 3 0.1172 1.0686 1.500 8.82 6 3.2721 11.677 0.379 1.04 1 3 2 0.4077 1.9873 1.086 3.46 3 0.0815 1.0861 1.489 12.78 0.126 7 1 7.9064 0.0000 2 1.1159 1.98 4.8963 0.670 1.5944 5.02 0.2515 1.222 3 1.0348 1.527 17.46 4 0.0582 4.3583 20.294 1 0.286 1.03 8 8 0.5791 2 0.855 3.08 3.1808 0.1765 1.4507 3 6.83 1.285 4 0.0448 1.0478 1.517 22.87 9 1 10.175 0.098 0.0000 9 2 1.4585 7.8971 0.526 1.93 3 0.3561 1.001 4.32 2.3651 0.1294 1.3165 4 8.87 1.351 5 0.0348 1.0210 1.537 29.00 10 1 5.4449 31.378 1.03 0.230 8 0.7414 4.7363 0.699 2.94 2 3 0.2479 1.9952 1.094 5.70 4 0.1008 1.2638 1.380 11.15 1.530 35.85 5 0.0283 1.0304 16-61 Active Filter Design Techniques

346 Filter Coefficient Tables Table 16 10. All-Pass Coefficients – i a b n f T / f Q i i i gr0 i C 1 0.6436 0.0000 1.554 0.204 1 9 2 1.6278 0.8832 1.064 0.58 0.518 1 1 3 1.1415 0.0000 0.876 0.843 1 7 2 1.5092 1.0877 0.959 0.69 4 1 2.3370 1.4878 0.820 0.52 1.173 8 2 1.1837 0.919 0.81 1.3506 1 0.771 0.0000 5 1.506 1.2974 0 2 2.2224 1.5685 0.798 0.56 3 1.2116 1.2330 0.901 0.92 6 2.6117 1.7763 0.750 0.51 1.839 1 5 2 2.0706 1.6015 0.790 0.61 3 1.0967 1.2596 0.891 1.02 2.173 1.3735 7 1 0.0000 0.728 7 2 0.742 0.53 2.5320 1.8169 1.6116 0.788 1.9211 3 0.66 4 1.0023 1.2743 0.886 1.13 1 2.7541 1.9420 0.718 0.51 2.508 8 4 2 1.8300 0.739 0.56 2.4174 1.7850 1.6101 3 0.71 0.788 4 0.9239 1.2822 0.883 1.23 9 1 1.4186 0.0000 0.705 2.843 4 2 2.6979 0.713 0.52 1.9659 2.2940 0.740 0.59 3 1.8282 1.6644 1.6027 4 0.76 0.790 5 0.8579 1.2862 0.882 1.32 10 1 2.8406 2.0490 0.699 0.50 3.178 6 2.6120 1.9714 0.712 0.54 2 3 2.1733 1.8184 0.742 0.62 4 1.5583 1.5923 0.792 0.81 0.881 1.42 5 0.8018 1.2877 16-62

347 References 16.10 References D.Johnson and J.Hilburn, Rapid Practical Designs of Active Filters , John Wiley & Sons, 1975. U.Tietze and Ch.Schenk, Halbleiterschaltungstechnik , Springer – Verlag, 1980. H.Berlin, , Howard W.Sams & Co., 1979. Design of Active Filters with Experiments M.Van Falkenburg, Analog Filter Design , Oxford University Press, 1982. Design with Operational Amplifiers and Analog Integrated Circuits S.Franko, , McGraw – Hill, 1988 16-63 Active Filter Design Techniques

348 16-64

349 Chapter 17 Circuit Board Layout Techniques Bruce Carter 17.1 General Considerations Prior discussions have focused on how to design op amp circuitry, how to use ICs, and the usage of associated passive components. There is one additional circuit component that must be considered for the design to be a success — the printed circuit board on which the circuit is to be located. 17.1.1 The PCB is a Component of the Op Amp Design Op amp circuitry is analog circuitry, and is very different from digital circuitry. It must be partitioned in its own section of the board, using special layout techniques. Printed circuit board effects become most apparent in high-speed analog circuits, but common mistakes described in this chapter can even affect the performance of audio cir- cuits. The purpose of this chapter is to discuss some of the more common mistakes made by designers and how they degrade performance, and provide simple fixes to avoid the problems. In all but very rare cases, the PCB layout for analog circuitry must be designed such that the effect of the PCB is transparent to the circuit. Any effect caused by the PCB itself should be minimized, so that the operation of the analog circuitry in production will be the same as the performance of the design and prototype. 17.1.2 Prototype, Prototype, PROTOTYPE! Normal design cycles, particularly of large digital boards, dictate layout of the PCB as soon as possible. The digital circuitry has been simulated, but in most cases, the produc- tion PCB itself is the prototype, and may even be sold to a customer. Digital designers can correct small mistakes by implementing cuts and jumpers , reprogramming gate arrays or flash memories, and go on to the next project. This is not the case with analog circuitry. Some of the common design mistakes discussed in this chapter cannot be corrected by the cut and jumper method. They can and will render the entire PCB unusable. It is very 17-1

350 General Considerations important digital designer, who is used to cuts and jumpers, to read and understand for the this chapter prior to releasing a board to a layout service. A little care, taken up front, can save a board worth thousands of dollars from becoming scrap because of blunders in a tiny section of analog circuitry. This author has been the unfortunate recipient of a simple analog circuit designed by another engineer, who was accustomed to the cut and jumper method of correcting his mistakes. This resulted in a design that was full of mistakes. Not only was the op amp hooked up with inverting and noninverting inputs reversed, but an RC time constant had to be added to prevent a race condition. Repercussions from these mistakes, and associated rework problems, caused literally hundreds of hours to be lost from a tight production schedule. Prototyping this cir- cuit would have taken less than a day. PROTOTYPE ALL ANALOG CIRCUITRY! 17.1.3 Noise Sources Noise is the primary limitation on analog circuitry performance. Internal op amp noise is covered in Chapter 10. Other types of noise include: D Conducted Emissions — noise that the analog circuitry generates through its connections to other circuits. This is usually negligible in analog circuitry, unless it is high power (such as an audio amplifier that draws heavy currents from its power supply). Radiated Emissions noise that the analog circuitry generates, or transmits, D — through the air. This is also usually negligible in analog circuitry, unless it is high frequency such as video. D Conducted Susceptibility — noise from external circuitry that is conducted into the analog circuit through its connections to other circuits. Analog circuitry must be connected to the “ outside world ” by at least a ground connection, a power connection, an input, and an output. Noise can be conducted into the circuit through all of these paths, as well as any others that are present. Radiated Susceptibility — noise that is received through the air (or transmitted into D the analog circuitry) from external sources. Analog circuitry, in many cases, resides on a PCB that may have high-speed digital logic including DSP chips. High-speed clocks and switching digital signals create considerable radio frequency interference (RFI). Other sources of radiated noise are endless: the switching power supply in a digital system, cellular telephones, broadcast radio and TV, fluorescent lighting, nearby PCs, lightning in thunderstorms, and so on. Even if the analog circuitry is primarily audio in frequency, RFI may produce noticeable noise in the output. 17-2

351 PCB Mechanical Construction 17.2 PCB Mechanical Construction It is important to choose a PCB with the right mechanical characteristics for the applica- tion. — Choosing the Right One for the Application 17.2.1 Materials PC Board materials are available in various grades, as defined by the National Electrical Manufacturers Association (NEMA). It would be very convenient for designers if this orga- was closely allied with the electronics industry nization controlling parameters such as — resistivity and dielectric constant of the material. Unfortunately, that is not the case. NEMA is an electrical safety organization, and the different PCB grades primarily describe the flammability, high temperature stability, and moisture absorption of the board. Therefore, specifying a given NEMA grade does not guarantee electrical parameters of the material. If this becomes critical for an application, consult the manufacturer of the raw board stock. Laminated materials are designated with (flame resistant) and G grades. FR – 1 is the FR least flame resistant, and FR – 5 is the most. G10 and G11 have special characteristics as described in Table 17 – 1. Table 17 – 1. PCB Materials GRADE DESIGNATION MATERIAL/COMMENTS Paper/phenolic: room temperature punchable, poor moisture resistance. FR 1 – FR 2 Paper/phenolic: suitable for single-sided PCB consumer equipment, good moisture re- – sistance. FR – 3 Paper/epoxy: designed for balance of good mechanical and electrical characteristics. FR – 4 Glass cloth/epoxy: excellent mechanical and electrical properties. FR – 5 Glass cloth/epoxy: high strength at elevated temperatures, self-extinguishing. G10 Glass cloth/epoxy: high insulation resistance, highest bond strength of glass laminates, high humidity resistance. Glass cloth/epoxy: high flexural strength retention at high temperature, extreme resist- G11 ance to solvents. Do not use FR – 1. There are many examples of boards with burned spots, where high wattage components have heated a section of the board for a period of time. This grade of PCB material has more in common with cardboard than anything else. FR – 4 is commonly used in industrial-quality equipment, while FR – 2 is used in high-vol- ume consumer applications. These two board materials appear to be industry standards. Deviating from these standards can limit the number of raw board material suppliers and PCB houses that can fabricate the board because their tooling is already set up for these materials. Nevertheless, there are applications in which one of the other grades may even be necessary to consider make sense. For very high frequency applications, it may 17-3 Circuit Board Layout Techniques

352 PCB Mechanical Construction Teflon or even ceramic board substrate. One thing can be counted on, however: the more exotic the board substrate, the more expensive it will be. In selecting a board material, pay careful attention to the moisture absorption. Just about every desirable performance characteristic of the board will be negatively impacted by moisture. This includes surface resistance of the board, dielectric leakage, high-voltage breakdown and arcing, and mechanical stability. Also, pay attention to the operating tem- perature. High operating temperatures can occur in unexpected places, such as in prox- imity to large digital ICs that are switching at high speeds. Be aware that heat rises, so if one of those 500-pin monster ICs is located directly under a sensitive analog circuit, both the PCB and circuit characteristics may vary with the temperature. After the board substrate material has been selected, the next decision is how thick to make the copper foil laminated to it. For most applications, 1-ounce copper is sufficient. ½ If the circuit consumes a lot of power, 2-ounce may be better. Avoid -ounce copper, be- cause it tends to break between the trace and the pad. 17.2.2 How Many Layers are Best? the complexity of the overall circuitry being designed, a designer must de- Depending on cide how many layers the PCB should be. 17.2.2.1 Single-Sided simple consumer electronics are sometimes fabricated on single-sided PCBs, keep- Very ing the raw board material inexpensive (FR – 1 or FR – 2) with thin copper cladding. These designs frequently include many jumper wires, simulating the circuit routing on a double- board. This technique is only recommended for low-frequency circuitry. For reasons sided below, this type of design is extremely susceptible to radiated noise. It is harder described to design a board of this type, because of the many things can go wrong. Many complex designs have been successfully implemented with this technique, but they require a lot of forethought. An example is a television set that puts all of the analog circuitry on a single-sided board at the bottom of the case, and uses the metallized CRT itself to shield the board from a separate digital tuning board near the top of the set. Be prepared to get creative if the design demands high volume, low cost PCBs. If a single-sided PCB is a requirement, remember the trace resistance! The op amp is not a 3-terminal device (inverting input, noninverting input, and output). It is a 7-terminal de- – 1. vice as shown in Figure 17 17-4

353 PCB Mechanical Construction V+ _ IN – OUT + IN + V – LOAD GROUND RETURN Figure 17 – 1. Op Amp Terminal Model Resistance in the power leads of the device must be taken into account, as well as resist- ance between the return for the inputs and the load. Both the input and load must return somewhere. If these are at different potentials due to trace resistance, there can be prob- lems. 17.2.2.2 Double-Sided The next level of complexity is double-sided. Although there are some double-sided FR – 2 4 material. The increased strength boards, they are more commonly fabricated with FR – of FR 4 material supports vias better. Doubled-sided boards are easier to route because – there are two layers of foil, and it is possible to route signals by crossing traces on dif ferent layers. Crossing traces, however, is not recommended for analog circuitry. Wherever pos- sible, the bottom layer should be devoted to a ground plane, and all other signals routed on the top layer. A ground plane provides several benefits: D Ground is frequently the most common connection in the circuit. Having it continuous on the bottom layer usually makes the most sense for circuit routing. It increases mechanical strength of the board. D D It lowers the impedance of all ground connections in the circuit, which reduces undesirable conducted noise. D It adds a distributed capacitance to every net in the circuit — helping to suppress radiated noise. D It acts a shield to radiated noise coming from underneath the board. 17-5 Circuit Board Layout Techniques

354 PCB Mechanical Construction 17.2.2.3 Multi-Layer Double-sided boards, in spite of their benefits, are not the best method of construction, especially for sensitive or high-speed designs. The most common board thickness is 1.5 mm. This separation is too great for full realization of some of the benefits listed above. Distributed capacitance, for example, is very low due to the separation. Critical designs call for multi-layer boards. Some of the reasons are obvious: D Better routing for power as well as ground connections. If the power is also on a plane, it is available to all points in the circuit simply by adding vias. Other layers are available for signal routing, making routing easier. D There will be distributed capacitance between the power and ground planes, D reducing high frequency noise. There are other reasons for multi-layer boards, however, that may not be obvious or intu- itive. D Better EMI/RFI rejection. There is due to the image plane effect , which has been known since the time of Marconi. When a conductor is placed close to a parallel surface, most of the high frequency currents will return directly under the conductive conductor, flowing in the opposite direction. This mirror image of the conductor within the plane creates a transmission line. Since currents are equal and opposite in the transmission line, it is relatively immune to radiated noise. It also couples the signal very efficiently. The image plane effect works equally well with ground and power planes, but they must be continuous. Any gap or discontinuity causes the beneficial effects to quickly vanish. There is more on this in the following paragraphs. D Reduced overall project cost for small production runs. Although multi-layer boards are more expensive to manufacture, EMI/RFI requirements from the FCC or other agencies may require expensive testing of the design. If there are problems, it can force a complete redesign of the PCB, leading to additional rounds of testing. A multi-layer PCB can have as much as 20-dB better EMI/RFI performance over a 2-layer PCB. If production volumes are going to be small, it makes sense to make a better PCB to begin with, than try to cut costs and take the risk of failing $25,000 to $50,000 tests. 17.2.3 Board Stack-Up The Order of Layers — There has been a lot of confusion in the past over what is the optimum order for PCB layers. Take, for example, a 4-layer board consisting of two signal layers, a power plane, and a ground plane. Is it better to route the signal traces between the layers, thus provid- ing shielding for the signal traces – or is it better to make the ground and power planes the two inner planes? In considering this question, it is important to remember that no matter what is decided, signals exposed on one or both of the top and bottom planes. The leads there will still be of the op amp PCB package, and the traces on the board leading to nearby passive com- 17-6

355 Grounding ponents and feed-throughs will be exposed. Therefore, any shielding effects are compro- far better to take advantage of the distributed capacitance between the power mised. It is and ground plane by making them internal. Another advantage of placing the planes internally is that the signal traces are available for probing and modification on the top and bottom layers. Anyone who has had to change connections on buried traces will appreciate this feature. For more than four layers, it is a general rule to shield higher speed signals between the ground and power planes, and route slower signals on the outer layers. 17.3 Grounding Good grounding is a system-level design consideration. It should be planned into the product from the first conceptual design reviews. 17.3.1 The Most Important Rule: Keep Grounds Separate Separate grounding for analog and digital portions of circuitry is one of the simplest and most ef fective methods of noise suppression. One or more layers on multi-layer PCBs are usually devoted to ground planes. If the designer is not careful, the analog circuitry will be connected directly to these ground planes. The analog circuitry return, after all, is the same net in the netlist as digital return. Autorouters respond accordingly and connect all of the grounds together, creating a disaster. separation of grounds on a mixed digital and analog board is almost impossi- the fact After ble. Every ground connection in the analog circuitry must be lifted from the board and con- nected together. For surface mount boards, this results in a colossal mess of “ tomb- stoned ” passive components and floating IC leads. 17.3.2 Other Ground Rules D Ground and power planes are at the same ac potential, due to decoupling capacitors and distributed capacitance. Therefore, it is important to isolate the power planes as well. D Do not overlap digital and analog planes (Figure 17 – 2). Place analog power coincident with analog ground, and digital power coincident with digital ground. If any portion of analog and digital planes overlap, the distributed capacitance between the overlapping portions will couple high-speed digital noise into the analog circuitry. This defeats the purpose of isolated planes. 17-7 Circuit Board Layout Techniques

356 Grounding WRONG RIGHT ANALOG + DIGITAL + ANALOG + DIGITAL + ÇÇÇÇ ÇÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇÇ ÇÇÇÇÇÇ ÇÇ ÇÇÇÇ ÇÇÇÇ DIGITAL ANALOG – – – – ANALOG DIGITAL Figure 17 – 2. Digital and Analog Plane Placement D Separate grounds does not mean that the grounds are electrically separate in the system. They have to be common at some point, preferably a single, low-impedance point. System-wise, there is only one ground, and it is the electrical safety ground in an ac-powered system or battery ground in a dc-powered system. Everything else to that ground. It would be a good idea to develop the discipline to refer to returns “ ” everything ground as a return . All returns should be connected together that is not a at a single point, which is system ground. At some point, this will be the chassis. It is important to avoid ground loops by multiple connections to the chassis. Ensuring only one chassis ground point is one of the most difficult aspects of system design. ANALOG CIRCUITRY DIGITAL CIRCUITRY _ + – + + – POWER SUPPLY and/or BATTERY Figure 17 – 3. Separate Grounds D If at all possible, dedicate separate connector pins to separate returns, and combine the returns only at system ground. Aging and repeated mating causes connector pins to increase in contact resistance, so several pins are needed. Many digital boards consist of many layers and hundreds or thousands of nets. The addition of one more net is seldom an issue, but the addition of several connector pins almost always is. If this cannot be done, then it will be necessary to make the two returns a single net on the PCB — with very special routing precautions. It is important to keep digital signals away from analog portions of the circuit. It D makes little sense to isolate planes, keep analog traces short, and place passive 17-8

357 Grounding components carefully if there are high-speed digital traces running right next to the sensitive analog traces. Digital signals must be routed around analog circuitry, and not overlap analog ground and power planes. If not, the design will include a new schematic symbol shown in Figure 17 – 4 the broadcasting antenna! — Most digital clocks are high enough in frequency that even small capacitances between traces and planes can couple significant noise. Remember that it is not only the fundamental frequency of the clock that can cause a potential problem, but also the higher frequency harmonics. ANALOG _ HIGH-IMPEDANCE GROUND ANALOG INPUT PLANE + HIGH-SPEED DIGITAL SIGNAL DIGITAL GROUND PLANE Figure 17 4. Broadcasting From PCB Traces – It is a good idea to locate analog circuitry as close as possible to the I/O connections D of the board. Digital designers, used to high current ICs, will be tempted to make a 50-mil trace run several inches to the analog circuitry thinking that reducing the resistance in the trace will help get rid of noise. What they have actually done is create a long, skinny capacitor that couples noise from digital ground and power planes into the op amp, making the problem worse! 17.3.3 A Good Example Figure 17 – 5 shows one possible board layout. In this system, all electronics, including the power supply, reside on one PCB. Three separate and isolated ground/power planes are employed: one for power, one for digital, and one for analog. Power and ground connec- tions from digital and analog sections of the board are combined only in the supply sec- tion, and are combined in close proximity. High-frequency conducted noise on the power lines is limited by inductors (chokes). In this case, the designer has even located low fre- quency analog circuitry close to low speed digital, keeping high frequency digital and ana- log physically apart on the board. This is a good, careful design that has a high likelihood — providing that good layout and decoupling rules are also followed. of success 17-9 Circuit Board Layout Techniques

358 Grounding POWER CONNECTOR HIGH SUPPLY V CC DIGITAL FREQUENCY LOW CONNECTOR ANALOG HIGH LOW FREQUENCY Figure 17 – 5. A Careful Board Layout 17.3.4 A Notable Exception one case where it is necessary to combine analog and digital signals on the ana- There is log ground plane. Analog to digital (A/D) and digital to analog (D/A) converters are pack- aged as ICs with analog and digital ground pins coming out of the package. One might assume, based on the previous discussion, that the digital ground pin should be con- nected to digital ground and the analog ground pin to analog ground. That, however, is not correct. refer to internal connections in the IC, The pin names analog ground and digital ground not the plane to which they should be connected. Both should connect to the analog ground plane. The connection would have been made inside the IC, but it is impossible to get low enough impedance at the typical geometries inside ICs. The IC designer actual- ly counts on the end user to supply a low impedance connection outside the IC. Other- wise, the performance of the converter will be worse than specified. One might suspect that the digital portions of the converter would make circuit perfor- mance worse by coupling digital switching noise onto the analog ground and power plane. Converter designers realize this, and design digital portions without a lot of output power to minimize switching transients. If the converter does not drive large fanouts, this should not be a problem. Be sure to properly decouple the logic supply for the converter to analog ground (see the following paragraphs). 17-10

359 The Frequency Characteristics of Passive Components 17.4 The Frequency Characteristics of Passive Components the right passive components for an analog design is important. In most cases, Choosing right passive component will fit on the same pads as a wrong a passive component, but not always. Start the design process by carefully considering the high frequency charac- teristics of passive components, and putting the correct part outline on the board from the start. Most designers are totally ignorant of the frequency limitations of the passive components they use in analog circuitry. Passive components have limited frequency ranges, and op- eration of the part outside of that range can have some very unexpected results. One might think that this discussion only applies to high-speed analog circuits. But high fre- quencies that are radiated or conducted into a low-speed circuit will affect passive compo- nents as well. For example: a simple op amp low-pass filter may well turn into a high-pass filter at RF frequencies. 17.4.1 Resistors High-frequency performance of resistors is approximated by the schematic shown in Fig- ure 17 6. – C P L R L Figure 17 – 6. Resistor High-Frequency Performance Resistors are typically one of three types: wire-wound, carbon composition, and film. It does not take a lot of imagination to understand how wire-wound resistors can become inductive because they are coils of resistive wire. Most designers are not aware of the in- ternal construction of film resistors, which are also coils of thin metallic film. Therefore, resistors are also inductive at high frequencies. The inductance of film resistors is low- film er, however, and values under 2 k Ω are usually suitable for high frequency work. The end caps of resistors are parallel, and there will be an associated capacitance. Usual- that the capacitance does ly, the resistance will make the parasitic capacitor so “ leaky ” not matter. For very high resistances, the capacitance will appear in parallel with the re- sistance, lowering its impedance at high frequencies. 17-11 Circuit Board Layout Techniques

360 The Frequency Characteristics of Passive Components 17.4.2 Capacitors High-frequency performance of capacitors is approximated by the schematic shown in FIgure 17 – 7. R P ESR L C Figure 17 – 7. Capacitor High-Frequency Performance Capacitors are used in analog circuitry for power supply decoupling and as filter compo- nents. For an ideal capacitor, reactance decreases by the formula: 1 – 1) (17 X + C ( ) 2 p fC Where: capacitive reactance in Ohms X C F frequency in Hz C capacitance in microfarads μΩ μ Therefore, a 10- at 10 kHz, and 160- F electrolytic capacitor has a reactance of 1.6 Ω at 100 MHz. Right? will never see the 160 μΩ with the electrolytic capacitor. Film and electrolytic In reality, one capacitors have layers of material wound around each other, which creates a parasitic inductance. Self-inductance effects of ceramic capacitors are much smaller, giving them a higher operating frequency. There is also some leakage current from plate to plate, which appears as a resistance in parallel with the capacitor, as well as resistance within the plates themselves, which add a parasitic series resistance. The electrolyte itself in electrolytic capacitors is not perfectly conductive (to reduce leakage current). These re- sistances combine to create the equivalent series resistance (ESR). The capacitors used for decoupling should be low ESR types, as any series resistance limits the ef fectiveness of the capacitor for ripple and noise rejection. Elevated temperatures also severely in- ESR, and can be permanently destructive to capacitors. Therefore, if an aluminum crease electrolytic will be subjected to high temperatures, use the high-temperature grade C), not the low temperature grade (85 _ C). _ (105 For leaded parts, the leads themselves also add a parasitic inductance. For small values of capacitance, it is important to keep the lead lengths short. The combination of parasitic inductance and capacitance can produce resonant circuits! Assuming a lead self-induc- F capacitor with two 1-cm leads tance of 8 nH/cm (see the following paragraphs), a 0.01- μ will resonate at 12.5 MHz. This effect was well known to engineers many decades ago, 17-12

361 The Frequency Characteristics of Passive Components who designed vacuum tube-based products with leaded components. Woe be to any hob- byist restoring antique radios that is unaware of this effect! If electrolytic capacitors are used in a design, make sure that the polarity is correctly ob- served. The positive terminal of the capacitor must be connected to the more positive of two dc potentials. If there is any doubt whatsoever which polarity is correct, design cal- culations must continue until it is known, or a prototype must be built. Incorrect polarity of electrolytic capacitors will cause them to conduct dc current, in most cases destroying the part and probably the rest of the circuit as well. If there is a rare case in which there — will be both polarities present, use a nonpolarized electrolytic (which is constructed by connecting two polarized electrolytic capacitors in series). Of course, one can always con- nect two capacitors in series on the PCB, keeping in mind that the effective capacitance will be cut in half for equal values of capacitor. 17.4.3 Inductors High frequency performance of inductors is approximated by the schematic shown in Fig- ure 17 – 8. C P LR S Figure 17 – 8. Inductor High-Frequency Performance Inductive reactance is described by the formula: – 2) (17 X + 2 p fL L Where: inductive reactance in Ohms X L F frequency in Hz L inductance in Henrys Therefore, a 10-mH inductor has a reactance of 628 Ω at 10 kHz, which increases to at 100 MHz. Right? Ω 6.28 M Ω 6.28 M with this inductor. Parasitic resistances are easy In reality, one will never see the to understand — the inductor is constructed of wire, which has a given resistance per unit length. Parasitic capacitance is harder to visualize, unless one considers the fact that turn of wire in the inductor is located next to adjacent turns, forming a capacitor. This each parasitic capacitance limits the upper frequency of this inductor to under 1 MHz. Even small wire-wound inductors start to become ineffective in the 10 MHz to 100 MHz range. 17-13 Circuit Board Layout Techniques

362 The Frequency Characteristics of Passive Components 17.4.4 Unexpected PCB Passive Components In addition to the obvious passive components above, the PCB itself has characteristics that form components every bit as real as those discussed previously — just not as ob- vious. 17.4.4.1 PCB Trace Characteristics The layout pattern on a PCB can make it susceptible to radiated noise. A good layout is one that minimizes the susceptibility of analog circuitry to as many radiated noise sources as possible. Unfortunately, there is always a level of RF energy that will be able to upset the normal operation of the circuit. If good design techniques are followed, that level will be one that the circuit never encounters in normal operation. Trace Antennas 17.4.4.1.1 A board is susceptible because the pattern of traces and component leads form antennas. Antenna theory is a complex subject, well beyond the scope of this book. Nevertheless, a few basics are presented here. One basic type of antenna is the whip, or straight conductor. This antenna works because a straight conductor has parasitic inductance, and therefore can concentrate flux from ex- ternal sources. The impedance of any straight conductor has a resistive and an inductive component: – (17 3) L Z R ) j w + For dc and low frequencies, resistance is the major factor. As the frequency increases, however, the inductance becomes more important. Somewhere in the range of 1 kHz to 10 kHz, the inductive reactance exceeds the resistance, so the conductor is no longer a low resistance connection, but rather an inductor. The formula for the inductance of a PCB trace is: ) W H 2X 4) – (17 ǒ Ǔ Ǔ ǒ ƪ ƫ 0.5 m ) H) + 0.0002X ) 0.2235 · L( In H ) W X Where: X length of the trace W width of the trace H thickness of the trace The inductance is relatively unaffected by the diameter, since it varies as the logarithm of the circumference. Common wires and PCB traces vary between 6 nH and 12 nH per centimeter. and an inductance of 8 nH For example, a 10 cm PCB trace has a resistance of 57 m Ω per cm. At 100 kHz, the inductive reactance reaches 50 m Ω . At frequencies above — not resistive. 100 kHz, the trace is inductive 17-14

363 The Frequency Characteristics of Passive Components A rule of thumb for whip antennas is that they begin to couple significant energy at about 1/20 of the wavelength of the received signal, peaking at ¼ the wavelength. Therefore, the 10-cm conductor of the previous paragraph will begin to be a fairly good antenna at frequencies above 150 MHz. Remember that although the clock generator on a digital PCB may not be operating at a frequency as high as 150 MHz, it approximates a square wave. Square waves will have harmonics throughout the frequency range where PCB become efficient antennas. If through-hole components are mounted in a way conductors leaves significant lead length, those component leads also become antennas, partic- that ularly if they are bent. Another major type of antenna is the loop. The inductance of a straight conductor is dra- matically increased by bending it into partial or complete loops. Increased inductance low- ers the frequency at which the conductor couples radiated signals into the circuit. Without realizing it, most digital designers are well versed in loop antenna theory. They not to make loops in critical signal pathways. Some designers, however, that would know never think of making a loop with a high speed clock or reset signal will turn right around and create a loop by the technique they use for layout of the analog section of the board. Loop antennas constructed as loops of wire are easy to visualize. What is not as obvious is that slot antennas are just as efficient. Consider the three cases shown in Figure 17 – 9. Figure 17 – 9. Loop and Slot Antenna Board Trace Layouts Version is a poor design. It does not use an analog ground plane at all. A loop is formed A by the ground and signal traces. An electric field E and perpendicular magnetic field H are created, and form the basis of a loop antenna. A rule of thumb for loop antennas is that the length of each leg is equal to half the most ef ficiently received wavelength. Remember, that even at 1/20 of the wavelength, the loop will still be a fairly ef ficient antenna. however, 17-15 Circuit Board Layout Techniques

364 The Frequency Characteristics of Passive Components Version is a better design, but there is intrusion into the ground plane, presumably to B make room for a signal trace. A much smaller slot antenna is formed by the difference in pathways between signal and return. A second loop is created by the cutout for the IC. Version is the best design. Signal and return are coincident with each other, eliminating C loop antenna effects completely. Note that there is still a cutout for the IC, but it is located away from the return path for the signal. 17.4.4.1.2 Trace Reflections Reflections and matching are closely related to loop antenna theory, but dif ferent enough to warrant their own discussion. When a PCB trace turns a corner at a 90 _ angle, a reflection can occur. This is primarily due to the change of width of the trace. At the apex of the turn, the trace width is increased to 1.414 times its width. This upsets the transmission line characteristics, especially the resulting in the reflection. It — distributed capacitance and self – inductance of the trace is a given that not all PCB traces can be straight, and so they will have to turn corners. Most CAD systems give some rounding effect on the trace. Sharp 90 corners in traces _ tape up days of PCB layout. The rounding effects of CAD programs, how- are a relic of the still do not necessarily maintain constant width as the trace rounds the corner. Figure ever, 17 – 10 shows progressively better techniques of rounding corners. Only the last example maintains constant trace width and minimizes reflections. Most CAD programs now sup- port these methods, but they can entail a little more work to master. BETTER BEST WORST 2W 1W min. W Figure 17 – 10. PCB Trace Corners A suggestion for the advanced PCB layout engineer: leave rounding to the last step before tear-dropping and flood-filling. Otherwise, the CAD program will slow down doing numeri- cal calculations as the traces are moved around during routing. 17-16

365 The Frequency Characteristics of Passive Components 17.4.4.2 Trace-to-Plane Capacitors PCB traces, being composed of foil, form capacitance with other traces that they cross on other layers. For two traces crossing each other on adjacent planes, this is seldom a problem. Coincident traces (those that occupy the same routing on different layers), form a long, skinny capacitor. The formula for capacitance is shown in Figure 17 – 11. A å C + 0.0085 R d d where: AREA A C + capacitance (pF) dielectric constant + å R 2 A + area of plate (mm ) separation of plates (mm) d + Figure 17 11. PCB Trace-to-Plane Capacitance Formula – For example, if the capacitance formula is applied to the following trace: D 4 Layer board — signal routing next to ground plane Board layer thickness: 0.188 mm D D Trace Width: 0.75 mm D Trace Length: 7.5 mm 4 PCB material is 4.5. Due to the variations of material from of FR – A typical value for E R which an FR 4 board can be fabricated, this value is not guaranteed, but should be in the – range of 4 to 5. The capacitance between these traces would be 1.1 pF. Of course, the antenna effect on a 7.5-mm trace would be devastating, so this example is a bit extreme. Ignoring the anten- na effects for now, there are cases in which even a very small parasitic capacitance like 1 pF is unacceptable. Figure 17 – 12 dramatically illustrates the effect of 1 pF capacitance occurring at the inverting input of the op amp. It causes a doubling of the output amplitude the bandwidth limit of the op amp. This is an invitation to oscillation, especially since near the trace is an efficient antenna above 180 MHz. 17-17 Circuit Board Layout Techniques

366 The Frequency Characteristics of Passive Components 7 = 1 pF C I 6 5 dB – 4 3 C I _ 2 + 1 V 50 Ω OUT Output Amplitude 50 Ω – 0 OUT V 1 – = 0 pF C I – 2 (Stray C Only) 3 – 1G 10M 1M 100K 100M f – Frequency – Hz 12. Effect of 1-pF Capacitance on Op Amp Inverting Input Figure 17 – There are numerous fixes to the problem above. The most obvious would be to shorten the length of the traces. Another not-so-obvious fix would be to use a different trace width. — it carries There is no reason why an inverting op amp trace has to be 0.75 mm wide almost no current. If the trace length is reduced to 2.5 mm (one third as long) and the trace width is changed to 0.188 mm (1/4 as wide), the capacitance becomes 0.1 pF — much less significant in the example above. Another fix is to remove the ground plane under the inverting input and the traces leading to it. The inverting input of op amps, particularly high-speed op amps, is especially prone to oscillation in high gain circuits. This is due to unwanted capacitance on the input stage. It is important to minimize capacitance on this input by reducing trace width and placing close as possible to this input. If this input still oscillates, it may be neces- components as sary to scale the input and feedback resistors lower by a decade or two to change the reso- nance of the circuit. Scaling the resistors up will seldom help, as the problem is also re- lated to the impedance of the circuit. If filter components are involved, they will also have to be scaled to avoid changing the filter characteristics of the circuit. The power consump- tion of the circuit will also increase if resistors are lowered. 17.4.4.3 Trace-to-Trace Capacitors and Inductors PCB traces are not infinitely thin. They have some finite thickness, as defined by the ounce parameter of the copper clad foil. The higher the number of ounces, the thicker the copper. If two traces run side-by-side, then there will be capacitive and inductive coupling 17-18

367 The Frequency Characteristics of Passive Components – 13). The formulas for these parasitic effects can be found in between them (Figure 17 transmission line and/or microstrip references, but are too complex for inclusion here. Figure 17 – 13. Coupling Between Parallel Signal Traces Signal lines should not be routed parallel to each other, unless transmission line or micro- strip effects are desired. Otherwise, a gap of at least three times the signal trace width should be maintained. Capacitance between traces in an analog design can become a problem if fixed resistors Ω ). Capacitance between the inverting and noninverting in the design are large (several M inputs of an op amp could easily cause oscillation. 17.4.4.4 Inductive Vias Whenever routing constraints force a via (connection between layers of a PCB, FIgure parasitic inductor is also formed. At a given diameter (d) the approximate induc- 17 – 14), a tance (L) of an via at a height of (h) may be calculated as follows: 4h h – 5) (17 ǒ Ǔ ǒ Ǔ L 1 ) +[ ln nH 5 d TOP LAYER TRACE h (mm) BOTTOM LAYER TRACE d (mm) – 14. Via Inductance Measurements Figure 17 17-19 Circuit Board Layout Techniques

368 Decoupling For example, a 0.4-mm diameter via through a 1.5-mm thick PCB has an inductance of 1.1 nH. Keep in mind that inductive vias combined with parasitic capacitance can form resonant The self-inductance of a via is small enough that these resonances are in the GHz circuits. range, but inductors add in series, lowering the resonant frequency. Do not put several vias on a critical trace of a high-speed analog circuit! Another concern is that the vias put holes in ground planes, potentially creating ground loops. They should be avoided — the best analog layout is one that routes all signal traces on the top layer of the PCB. 17.4.4.5 Flux Residue Resistance Yes, even an unclean board can affect analog circuit performance. — — even in the low M Ω Be aware that if the circuit has very high resistances special attention may need to be paid to cleaning. A finished assembly may be adversely af fected by flux or cleansing residue. The electronics industry in the past few years has joined the rest of the world in becoming environmentally responsible. Hazardous chemicals are be- — including flux that has to be cleaned with ing removed from the manufacturing process organic solvents. Water-soluble fluxes are becoming more common, but water itself can become contaminated easily with impurities. These impurities will lower the insulation the PCB substrate. It is vitally important to clean with freshly distilled wa- characteristics of ter every time a high-impedance circuit is cleaned. There are applications that may call for the older organic fluxes and solvents, such as very low power battery powered equip- Ω range. Nothing can beat a good vapor defluxing ma- ment with resistors in the 10s of M chine for ensuring that the board is clean. 17.5 Decoupling Noise, of course, can propagate into analog circuitry through the power pins of the circuit as a whole and op amp itself. Bypass capacitors are used to reduce the coupled noise by providing low impedance power sources local to the analog circuitry. 17.5.1 Digital Circuitry — A Major Problem for Analog Circuitry If analog circuitry is located on the same board with digital circuitry, it is important to under- stand a little about the electrical characteristics of digital gates. A typical digital output consists of two transistors connected in series between power and ground (Figure 17 – 15). One transistor is turned on and the other turned off to produce high and vice versa for logic low. Because one transistor is turned off for either logic logic state, the power consumption for either logic state is low, while the gate is static at that level. 17-20

369 Decoupling + Figure 17 – 15. Logic Gate Output Structure The situation changes dramatically whenever the output switches from one logic state to the other. There may be a brief period of time when both transistors conduct simulta- neously. During this period of time, current drawn from the power supply increases dra- matically — since there is now a low impedance path through the two transistors from power to ground. Power consumption rises dramatically and then falls, creating a droop on the power supply voltage, and a corresponding current spike. The current spike will radiate radio frequency (RF) energy. There may be dozens, or even hundreds of such out- puts on a digital IC, so the aggregate effect may be quite dramatic. It is impossible to predict the frequencies of these spikes, because the frequencies are affected by the propagation delays of the transistors in the gate. Propagation delay is af- fected by random factors that occur during manufacture. Digital switching noise will be broadband, with harmonics throughout the spectrum. A general rejection technique is re- quired, rather than one that rejects a specific frequency. 17.5.2 Choosing the Right Capacitor Table 17 – 2 is a rough guideline describing the maximum useful frequencies of common capacitor types. – Table 17 2. Recommended Maximum Frequencies for Capacitors TYPE MAX FREQUENCY Aluminum Electrolytic 100 kHz Tantalum Electrolytic 1 MHz Mica 500 MHz 1 GHz Ceramic 17-21 Circuit Board Layout Techniques

370 Decoupling Obviously from the table, tantalum electrolytic capacitors are useless for frequencies MHz. Effective high frequency decoupling at higher frequencies demands a ce- above 1 known and avoided or the ca- ramic capacitor. Self-resonances of the capacitor must be may not help, or even make the problem worse. Figure 17 – 16 illustrates the typical pacitor commonly used for bypassing — 10 μ F tantalum electro- self-resonance of two capacitors lytic and 0.01 μ F ceramic. 100,000 EQUIVALENT SERIES 10,000 INDUCTANCE EQUIVALENT 1000 SERIES RESISTANCE Ω — 100 μ F 10 L R 10 C 1 Impedance 10 nF 0.1 Z 0.01 1 Ǹ * Z + w L ) R 0.001 C w 100 k 10 M 100 M 1 G 1 M Hz Frequency — Figure 17 – 16. Capacitor Self Resonance Consider these resonances to be typical values, the characteristics of actual capacitors can vary from manufacturer to manufacturer and grade of part to grade of part. The impor- tant thing is to make sure that the self-resonance of the capacitor occurs at a frequency above the range of the noise that must be rejected. Otherwise, the capacitor will enter a region where it is inductive. Do not assume that a single 0.1- μ F capacitor will decouple all frequencies. Smaller capac- itors may work better at higher frequencies than larger ones. When poor decoupling at higher frequencies is suspected, try a smaller capacitor rather than a larger one. 17.5.3 Decoupling at the IC Level The method most often used to decouple the high-frequency noise is to include a capaci- tor, or multiple capacitors connected from the op amp power pin to the op amp ground pin. It is important to keep the traces on this decoupling capacitor short. If not, the traces on the PCB will have significant self-inductance, defeating the purpose of the capacitor. 17-22

371 Input and Output Isolation — whether it contains A decoupling capacitor must be included on every op amp package 1, 2 or 4 devices per package. The value of capacitor must be picked carefully to reject the type of noise present in the circuit. In particularly troublesome cases, it may be necessary to add a series inductor into the power supply line connecting to the op amp. This inductor is in addition to the decoupling capacitors, which are the first line of defense. The inductor should be located before, not after the capacitors. technique that is lower in cost is to replace the series inductor with a small resistor Another in the 10- to 100- Ω range. The resistor forms a low-pass filter with the decoupling capaci- Ω tors. There is a penalty to pay for this technique — depending on the power consumption of the op amp, it will reduce the rail-to-rail voltage range. The resistor forms a voltage di- vider with the op amp as a resistive active component in the lower leg of the divider. De- pending on the application, this may or may not be acceptable. 17.5.4 Decoupling at the Board Level There is usually enough low frequency ripple on the power supply at the board input to warrant a bulk decoupling capacitor at the power input. This capacitor is used primarily to reject low frequency signals, so an aluminum or tantalum capacitor is acceptable. An ceramic cap at the power input will decouple any stray high frequency switching additional noise that may be coupled off of the other boards. 17.6 Input and Output Isolation Many noise problems are the result of noise being conducted into the circuit through its input and output pins. Due to the high-frequency limitations of passive components, the response of the circuit to high frequency noise may be quite unpredictable. In situations in which conducted noise is substantially different in frequency from the nor- mal operating range of the circuit, the solution may be as simple as a passive RC low-pass filter that rejects RF frequencies while having negligible effect at audio frequencies. A good example is RF noise being conducted into an audio op amp circuit. Be careful, . More stages though. A low pass filter loses its characteristics at 100 to 1000 times f 3db may be required to cover different frequency ranges. If this is the case, make the highest frequency filter the one nearest to the source of noise. Inductors or ferrite beads can also be used in a noise rejection filter network to eliminate conducted noise. Ferrite beads are inductive up to a specified frequency, and then become resistive. The effect of radiated energy coupling into an analog circuit can be so bad that the only solution to the problem may be to completely shield the circuit from radiated energy. This , and must be carefully designed so that frequencies that Faraday Cage shield is called a 17-23 Circuit Board Layout Techniques

372 Packages are causing the problem are not allowed to enter the circuit. This means that the shield must have no holes or slots larger that 1/20 the wavelength of the offending frequency. This can be a demanding requirement. It is a good idea to design a PCB from the begin- ning to have enough room to add a metal shield if it becomes necessary. If a shield is used, frequently the problem will be severe enough that ferrite beads will also be required on all connections to the circuit. 17.7 Packages Op amps are commonly supplied one, two, or four per package. Single op amps often con- tain additional inputs for features such as offset nulling. Op amps supplied two and four per package only offer inverting and noninverting inputs, and the output. If the additional features are important, the only package choice is single. Be aware, though, that the off- set-nulling pins on a single op amp package can act as secondary inputs, and must be treated carefully. Consult the data sheet on the particular device being used. QUAD SINGLE DUAL SHDN N1 V 1OUT + 8 1 1 8 CC 14 1 1OUT 4OUT IN – V + 2OUT – 1IN 7 2 7 2 CC 2 13 1IN – 4IN – IN+ OUT 1IN+ 2IN – 3 6 3 6 12 3 1IN+ 4IN+ 5 4 V N2 4 5 – 2IN+ V – 11 4 CC CC V + V – CC CC 5 10 2IN+ 3IN+ N1, N2 = OFFSET 6 9 – 2IN 3IN – 7 8 2OUT 3OUT Figure 17 – 17. Common Op Amp Pinouts The single op amp package places the output on the opposite side from the inputs. This can be a disadvantage at high speeds, because it forces longer PCB traces. One way to overcome this difficulty is to use a surface-mount package, and mount the feedback com- the rear of the board. This, however, adds at least two vias to the design, and ponents on places a hole in the ground plane for a 2-layer design. It may make more sense to use a dual op amp package, even if the second op amp is unused (remember to terminate it properly — see paragraph 17.7.3). Figure 17 – 18 is an illustration of trace length for an inverting op amp stage. 17-24

373 Packages DUAL SINGLE SHORT TRACES LONG TRACES Figure 17 – 18. Trace Length for an Inverting Op Amp Stage It is popular to use dual op amps for stereophonic circuits, and quad op amps for filter stages with many sections. There is a penalty for doing so, however. Although modern processing techniques provide high levels of isolation between amplifiers on the same piece of silicon, there will be some crosstalk. If isolation between amplifiers is important, then single packages should be considered. Crosstalk problems are not limited to the IC – the dual and quad packages place a high density of passive components in close prox- imity to each other. This proximity will lead to some crosstalk. Dual and quad op amp packages offer some additional benefits beyond density. The am- plifier stages tend to be mirror images of each other. If similar stages are to be laid out on the PCB, the layout only needs to be done once, then it can be mirror-imaged to form the other stage. Figure 17 – 19 illustrates this effect for four inverting op amp stages imple- mented in a quad package: QUAD – Figure 17 19. Mirror-Image Layout for Quad Op Amp Package These illustrations, however, do not show all connections required for operation, in partic- ular, the half-supply generator for single-supply operation. Modifying the diagram of Fig- 20. – ure 17 – 19 to use the fourth op amp as a half supply generator is shown in Figure 17 17-25 Circuit Board Layout Techniques

374 Packages OUT1 IN1 + IN3 IN2 OUT3 OUT2 Figure 17 – 20. Quad Op Amp Package Layout with Half-Supply Generator This example shows all of the connections actually required to produce three independent stages. Note that the half-supply connection can be made entirely under the IC, inverting keeping the trace length short. This example is not meant to be used as a suggested PCB layout, it merely illustrates what can be done. The half-supply op amp, for example, could be any one of the four. The passive components can be selected so that they span the width equal to or less than lead pitch. Surface-mount 0402 packages, for example, span a a standard SO package. This can keep traces lengths extremely short for high frequency applications. Package styles for op amps include the dual-in-line (DIP) and small-outline (SO). Lead pitches for op amps have been continually decreasing, as has been the case for all ICs in general. Decreasing lead pitches have been accompanied by a decrease in the size of passive components as well. Decreasing the overall circuit dimensions reduces para- sitic inductance, which should allow higher frequency circuits, but it also increases the po- tential for crosstalk by placing conductors closer to each other where capacitive effects increase. 17.7.1 Through-Hole Considerations The older technology for op amps and other components is through-hole. Components are constructed with leads that insert through holes in the board — hence the name. Through-hole components, due to their size, are more suited to applications where space is not an issue. The components themselves are frequently lower in cost, but the PCB is more expensive due to the fact that the PCB fabrication house has to drill holes for compo- nent le ads. PCBs are primarily a mechanical fabrication — the number of holes and num- ber of different drills have a big impact on the price. The leads on a through-hole op amp are arranged on a 0.1-inch grid. Many PCB layout people like to maintain the 0.1-inch grid for the rest of the components as well. Resistors 17-26

375 Packages and other passive components can even be purchased with leads pre-bent to land on a 0.1 inch grid. Some electrolytic capacitors have leads that are on a 0.025-inch grid. These component sizes may force a lot of wasted area on the PCB. Components that should be placed close to the op amp itself may be forced several tenths of an inch ideally away, due to intervening components. Therefore, through hole circuitry is not recom- mended for high speed analog circuitry, or for analog circuitry in proximity to high speed digital. Some designers attempt to overcome the long trace length caused by resistors by placing the resistors on the board vertically, one lead of the resistor bent close to the body of the part. This is common in older consumer electronics. This allows for denser placement of parts, and may help some with trace length — but each resistor exposes almost 1 cm of one component lead to radiated signals, and lead self-inductance. An advantage of the through-hole approach of PCB layout is that the through-holes them- selves can serve as feedthroughs, reducing the number of vias in complex circuits. 17.7.2 Surface Mount Surface-mount circuitry does not require a hole for each component lead. Automated test- ing, however, may require vias on every node. The holes were never an issue with through-hole circuitry, because every component lead made a hole in the board. The PCB layout designer, who is used to designing a board with a minimum number of vias, now has to put a via on EVERY node of the circuit. This can make a Swiss cheese out of a nice negating many of the advantages it provides. continuous ground plane — “ via on every node ” requirement. This require- there is a close variation of the Fortunately, ment can often be met by putting a test pad on every node. The automated test station can then access the analog circuitry from the top of the board. A clamshell test fixture is significantly more expensive than one that accesses only one side of the board. The extra cost can be justified if there is documentation that circuit performance will be unaccept- able with vias. Signal connections to ground or the power supply may have to be made through a small resistor instead, so the automated equipment can access that pin of the IC and test fixed its function. 17.7.3 Unused Sections In many op amp designs, one or more op amps may be unused. If this is the case, the unused section must be terminated properly. Improper termination can result in greater power consumption, more heat, and more noise in op amps on the same physical IC. If the unused section of the op amp is connected as shown in the better side of Figure – 21, it will be easier to use it for design changes. 17 17-27 Circuit Board Layout Techniques

376 Summary GOOD BETTER ZERO OPEN _ _ HALF SUPPLY HALF + + SUPPLY ZERO Figure 17 – 21. Proper Termination of Unused Op Amp Sections 17.8 Summary Keep the following points in mind when designing a PCB for analog circuitry: 17.8.1 General D Think of the PCB as a component of the design. D Know and understand the types of noise the circuit will be subjected to. Prototype the circuit. D 17.8.2 Board Structure Use a high-quality board material such as FR – 4. D D Multi-layer boards are as much as 20-dB better than double-sided boards. D Use separate, nonoverlapping ground and power planes. D Place power and ground planes to the interior of the board instead of exterior. 17.8.3 Components D Be aware of frequency limitations of traces and other passive components. D Avoid vertical mounting of through-hole passive components in high-speed circuits. D Use surface mount for high-speed analog circuitry. D Keep traces as short as possible. D Use narrow traces if long traces are required. Terminate unused op amp sections properly. D 17-28

377 References 17.8.4 Routing D Keep analog circuitry close to the power connector of the board. D Never route digital traces through analog sections of the board, or vice versa. D Make sure that traces to the inverting input of the op amp are short. D Make sure that traces to the inverting and noninverting inputs of the op amp do not parallel each other for any significant length. It is better to avoid vias, but the self-inductance of vias is small enough that a few D should cause few problems. D Do not use right angle traces, use curves if at all possible. 17.8.5 Bypass D Use the correct type of capacitor to reject the conducted frequency range. D Use tantalum capacitors at power input connectors for filtering power supply ripple. Use ceramic capacitors at power input connectors for high frequency conducted D noise. D Use ceramic capacitors at the power connections of every op amp IC package. More than one capacitor may be necessary to cover different frequency ranges. Change the capacitor to a smaller value — not larger — if oscillation occurs. D D Add series inductors or resistors for stubborn cases. D Bypass analog power only to analog return, never to digital return. 17.9 References th 1. Analog Dialog 30 Anniversary Issue , Analog Devices, Ask the Applications Engineer – 10, James Bryant High-Speed Operational Amplifier Layout Made Easy 2. , Texas Instruments, Andy Hahn, SLOA046 17-29 Circuit Board Layout Techniques

378 17-30

379 Chapter 18 Designing Low-Voltage Op Amp Circuits Ron Mancini 18.1 Introduction t appreciate its value until your supply runs In one respect, voltage is like water: you don ’ low. Low-voltage systems, defined here as a single power supply less than 5 V, teach us ’ t the first electronic types to learn how valuable voltage to appreciate voltage. We aren is; about 15 years ago the audio console design engineers appreciated the relationship between voltage and dynamic range. They needed more dynamic range to satisfy their customers; thus, they ran op amps at the full rated voltage, not the recommended operat- ing voltage, so they could squeeze a few more dB of dynamic range from the op amp. These engineers were willing to take a considerable risk running op amps at the full rated voltage; but their customers demanded more dynamic range. The moral of this story is that dynamic range is an important parameter, and supply voltage is tied directly to dy- namic range. Knowing how to obtain and use the maximum dynamic range and input/output voltage range is critical to achieving success in low voltage design. We will investigate these sub- jects in detail later, but for now it is useful to review the history of op amps. Knowing how op amps evolved into the today ’ s marvels is interesting, and it gives designers an insight into system problems that they encounter as they design in the low voltage world. When power supplies were ± 15 V, the output voltage swing of an op amp didn ’ t seem im- portant. When the power supply was 30 V the typical circuit designer could afford to sacri- fice 3 V from each end of the output voltage swing (this was because of transistor satura- tion or cutoff). The transistors in the op amp need enough voltage across them to operate correctly, so why worry about 6 V out of 30 V. Also, the input transistors required base bias, so an op amp with 30-V supplies often offered a common-mode input voltage range of 24 V or less. These numbers come from the μ A741 data sheet; the μ A741 (about 1969) is the first internally compensated op amp to achieve wide popularity. A later generation op amp, the LM324, had better dynamic range characteristics than the μ A741. The LM324 ’ s output voltage swing is 26 V when operated from a 30-V power sup- ply, and the common-mode input voltage range is 28.5 V. The LM324 was big news be- cause it was specified to operate with a 5-V power supply. The LM324 ’ s output voltage 18-1

380 Introduction swing at V = 5 V is 3.48 V, and this presented problems for the early low voltage circuit CC designers because the output voltage swing was smaller than most analog-to-digital con- verter (ADC) input voltage ranges. You must fill the ADC input voltage range to obtain its – 1.5 V) dynamic range. The LM324 had an input voltage common-mode range of (V full CC to 0 V; at least this op amp could work with transducers connected to the lower power sup- ply rail if the transducer did not have an ac output voltage swing. The next incremental improvement in op amps was the LM10 because it operated on 1.1-V power supplies. It was introduced almost as an a fterthought because there was no pressing demand for it. Its brilliant designer, Robert J. Widlar, wrote “ IC op amps have reached a certain maturity in that there no longer seems to be a pressing demand for bet- ter performance. ” There were no pressing demands for a low voltage op amp in 1978 be- cause portable (portable means battery applications that are almost always single supply) did not become popular until the late 1980s or early 1990s. Cell phones, calculators, and portable instruments — not new battery technology — opened the market for low-voltage op amps, and when the portable concept caught on, the demand for low-voltage op amps increased. The increasing demand did not breed new companies committed to low-voltage IC design; rather, the established IC manufac- turers threw a few low voltage op amps into their portfolio. These op amps, like the LM324, could operate on a low voltage, but they were severely lacking in input common-mode voltage range and output voltage swing. Circuit designers had to be satisfied with this gen- amps until something better came along. Well, something better is here now! eration of op The next generation of low-voltage op amps has much better specifications. The TLV278X operates off a power supply ranging from 1.8 V to 3.6 V, and it has an output 1.63 V (when the power supply is 1.8 V) coupled with an input common- voltage swing of – 0.2 V to 2 V. The TLV240X operates off a power supply ranging mode voltage range of from 2.5 V to 16 V, and it has an output voltage swing of 2.53 V when the power supply 2.7 V. Also, when it is operated off a 2.7-V power supply, it has an input common- voltage is voltage range of – 0.1 V to 7.7 V. These new op amps are far superior to their prede- mode cessors when evaluated on the their merits, which are extended output voltage swing and input common voltage range. The latest op amps make it possible to design more accurate and cost ef fective electronic equipment, but there is one problem that they don ’ t solve. Low-voltage applications are defined here as single-supply applications, and in single-supply design, the op amp input /2). Un- voltage and output voltage is referenced to the midpoint of the power supply (V CC fortunately, most transducers are not connected to the midpoint of the power supply be- t ’ ground. It doesn and the majority of cases, this requires a third wire beyond V cause, in CC /2 at the transducer location (to save a wire) because it is not identical help to create V CC to the midpoint of the power supply (unresolved errors enter because of the reference volt- age difference). When the transducer in a single supply design is referenced to any volt- age other than the midpoint of the power supply, the reference voltage is amplified with the transducer voltage. 18-2

381 Dynamic Range The trick to designing single supply op amp circuits is using external biasing to strip off or null out the reference voltage. Designing op amp circuits with biasing normally involves an iterative cut and try approach where the designer assumes a circuit configuration, solves equations, changes the configuration, and repeats the process until a solution is found. A technique that solves the problem the first time is presented later. 18.2 Dynamic Range It is extremely hard to define dynamic range (DR) for an op amp, so lets start with a digital- to-analog converter (DAC) where DR is defined as the ratio of the maximum output volt- age to the smallest output voltage the DAC can produce (least significant bit or LSB). Dy- 1. – namic range is usually expressed in dB using the formula given in Equation 18 V OUT(MAX) (18 – 1) ǒ Ǔ 20 Log + DR 10 V OUT(MIN) The same definition of DR can be used for an op amp, and the maximum output voltage . This output voltage swing is defined as the maximum output volt- swing equals V OUTMAX ) minus the minimum output voltage the op amp can age the op amp can achieve (V OH achieve (V ). V and V are easily obtainable from an op amp IC data sheet. Normal- OH OL OL and V are guaranteed minimum and maximum parameters respectively. This ly, V OH OL yields Equation 18 – 2. (18 2) – V V * V + OH(MIN) OL(MAX) OUT(MAX) – 2 can be used to illustrate the role that power supply voltage plays in limiting Equation 18 is the most positive power supply voltage minus the voltage drop across the DR. V OH(MIN) is directly proportional to the most positive pow- the upper output transistor, thus V OH(MIN) er supply voltage. For any op amp, the output voltage swing is directly proportional to the power supply voltage, thus, in the same op amp, the DR is directly proportional to the pow- er supply voltage. At first thought, one might think that the smallest output voltage that an op amp can have is zero, and the natural conclusion based on this assumption is that the DR is equal to infinity. This is never the case because op amp and external circuit imperfections ensure that the smallest op amp output voltage is greater than zero. It turns out that V OUT(MIN) is actually determined by a series of error terms. These error terms are the op amp ’ s inter- and I ), external resistor noise (V ), ), power supply rejection ratio (k nal noise (V nR n SVR n ), current offset (I ), common-mode rejection ratio (CMRR), and voltage offset (V IO IO closed loop gain (G). Each of these error terms is referred to the input of the op amp, so they must be multiplied by the closed loop gain to be referred to the output (see Figure 1). – 18 18-3 Designing Low-Voltage Op Amp Circuits

382 Dynamic Range ±∆ V V CC+ I I IB+ n + R S I V IO V V V O IB nR n _ R 2 I I – IB n V IN R 1 1. Op Amp Error Sources Figure 18 – – 3, and this equation refers them The error sources are taken into account in Equation 18 to the op amp output by multiplying them by the op amp ’ s closed loop gain. + V OUT(MIN) 3) (18 – R I ) V IN IB P D V ǒ Ǔ G ) ) + V R ) a I ) V D T R ) V I ) I ) R n n nR S IO EQ CL IO IO k CMRR SVR – The maximum DR that can be achieved by an op amp is given in Equation 18 4. V V * V OH(MIN) OUT(MAX) OL(MAX) 4) – (18 ǒ Ǔ DR + 20 Log + 20 Log 10 10 V V OUT(MIN) OUT(MIN) The DR is reduced by the sum of the error terms, so it is proper to conclude that the maxi- mum power supply voltage and the op amp choice (this defines the error magnitude) both – 3 are dc error terms, establish the DR of an op amp. The first two terms in Equation 18 thus, they can be adjusted to zero by one of several methods not mentioned here. The input offset current and input bias current error terms were big factors with older genera- tion ICs, but today ’ s technology render them much less significant (see Table 18 – 1). 18-4

383 Signal-to-Noise Ratio Table 18 1. Comparison of Op Amp Error Terms – LM324 TLV240X UNITS TLV278X PARAMETER 1.5 9 V 4.5 mV IO α V No spec 8 3 μ V IO I pA 500,000 100 350 IB 50,000 100 300 pA I IO CMMR 50 50 65 dB dB k 65 70 100 SVR nV Vn No spec 18 (1 kHz) 500 (100 Hz) ń 2 1 ) ( Hz fA In No spec 0.9 (1 kHz) 8 (100 Hz) 2 ń 1 ) ( Hz The data in T able 18 – 1 indicates that older low voltage op amps are not capable of yielding the DR that that later technology op amps are. 18.3 Signal-to-Noise Ratio Noise sets a limit on the information and signals that can be handled by a system. The ability of an amplifier, receiver, or other device to discern a signal is degraded by noise. Noise mixed with the incoming signal, noise generated by the op amp, resistor noise, and power supply noise ultimately determine the size of the signal that can be recovered and measured. Noise fluctuates randomly over a period of time, so instantaneous signal or noise levels t describe the situation adequately. Averages over a long period of time (root mean ’ don squared or RMS) are used to describe both the signal and the noise. Signal-to-noise ratio (SNR) was initially established as a measure of the quality of the signal that exists in the presence of noise. This SNR was a power ratio, and it was established at the output of a circuit. The SNR that we are interested in is a voltage ratio because the impedance is constant, and it is established at the input to the op amp. This means that all noise volt- ages, including resistor noise voltage, must be calculated in RMS volts at the op amp in- put. The SNR is given in Equation 18 5. – V SIGNAL – (18 5) Ǔ ǒ + 20 Log SNR 10 V NOISE The signal is established by a transducer; a device that senses a change in a variable and converts that change into a voltage change. Transducers also convert some of their physi- cal surroundings into a noise voltage that is combined with the signal. Noise from the physical surroundings of the transducer, unless its nature is well known, is almost impos- 18-5 Designing Low-Voltage Op Amp Circuits

384 Input Common-Mode Range sible to separate from the transducer signal. When transducers are connected to the elec- tronics, cabling picks up noise, and some transducers like thermocouples can pick up noise from the connecting junctions. Thus, the signal is never clean as it enters the elec- , tronics. The noise generated by the op amp was defined in the previous section as V n , and this noise is added to the signal. R V/k , I ∆ R, and I nR SVR n EQ The transducer often has a very small output voltage swing, so when the transducer out- put voltage swing is converted to least significant bits (LSB) the noise voltage should be very small compared to an LSB. Consider a temperature transducer that has a 10-mV swing over its range. When the transducer output voltage swing is considered to be the – 6 full-scale voltage (FSV) of an ADC, the LSB is very small as is shown in Equation 18 for a 12-bit (N) ADC. 10 mV 10 mV FSV (18 – 6) LSB V + m 2.44 + + + 12 N 4096 2 2 The op amp for this application must be a very low noise op amp because an op amp with 1/2 equivalent input noise voltage and a bandwidth of 4 MHz contributes a 20-nV/(Hz) 40 μ V of noise. This high noise contribution is why extensive filtering and “ optimally ” low bandwidth is found desirable in the input stages of some electronic systems. If there is power supply noise, some of that noise passes through the op amp to its input. The power supply noise is divided by the power supply rejection ratio, but there is always a residual – 7 noise component of the power supply on the op amp input as shown in Equation 18 is 60 dB. where k SVR V 10 mV PS 7) – (18 m V + + V + 10 PS(INPUT) 1000 k SVR 18.4 Input Common-Mode Range Years ago the op amp ’ s input common-mode voltage range (V ) did not include the ICR +|V 6 V), and when the that was available was (V – | The best V power supply rails. EE CC ICR , distortion occurred. If the input voltage exceeded the input voltage approached V ICR power supply rails, the output stage might invert phase (it sometimes latched in the in- verted position causing control problems) or the IC might self destruct. The vast majority of transducers were connected to ground (0 V) because it was easy to make a ground connection and because a split supply op amp has inputs referenced to ground. In a split supply application with the transducer connected to ground, latch-up or self destruction is unlikely. when In special cases, transducers are connected to a power supply rail (usually V CC power supply current sensing) or some other voltage, and in this special case, additional bias circuitry was added to split power supply designs to keep the input voltage swing . Bias circuitry in conjunction with external components removed the effects within V ICR of the power supply rail connection. 18-6

385 Input Common-Mode Range Many low voltage op amp input signals come from transducers connected to a power sup- 2. ply rail like the circuit shown in Figure 18 – R R G F +V CC V 1 _ V OUT + R || R F G V 2 Figure 18 – 2. Noninverting Op Amp is the transducer input, the op amp must be capable of handling input = 0 and V When V 1 2 voltages that go to 0 V. Furthermore, the transducer voltage may be ac, so it swings above and below ground, thus the transducer voltage drops below the low power supply rail. This exceed the power supply voltage. Rail-to-rail in- requires that the op amp ’ s V situation ICR put (RRI) voltage capability is a necessary requirement for a low-voltage op amp that han- dles transducers connected to a power supply rail. When the input voltage is connected to ground and the input voltage swing is very small, 3 it can be seen that – a standard op amp like the LM324 suffices. Referring to Figure 18 the PNP input transistors are biased by the emitter current source. If the positive input is connected to ground bias current still flows and the transistor stays active. If the input tran- sistors are selected very carefully for operation with low collector-base junction reverse bias, the input voltage can go slightly below ground ( – 200 mV for the TLV278X) and the op amp will still operate correctly. The circuit operation is one sided though because when the input voltage approaches the positive supply rail, the emitter current source and input transistors turn off. This type of circuit does not offer rail-to-rail operation, but it does offer – 1.5V) operation. from rail to (V CC 18-7 Designing Low-Voltage Op Amp Circuits

386 Input Common-Mode Range V CC V Bias V BE – IN IN + GND Figure 18 – 3. Input Circuit of a NonRRI Op Amp An op amp with a NPN input stage works in a similar way around the positive supply rail. It can sense voltages close to V and maybe slightly above V , but it won ’ t work when CC CC it is within 1.5 V of ground. The solution for this problem is to include parallel input circuits as shown in Figure 18 4. – Bipolar Input CMOS Input V V CC CC IN+ IN+ IN – IN – GND GND Figure 18 – 4. Input Circuit of an RRI Op Amp The RRI op amps have parallel input stages. There are both PNP and NPN differential amplifiers used in the input stages of the RRI op amp, thus the RRI op amp can operate – above and below the power supply voltage. As Figure 18 4 shows, the parallel input stages can be made in bipolar or MOS technology. 18-8

387 Input Common-Mode Range The input stages operate in three different ranges. When the input voltage ranges from – 0.2 V to 1 V, the PNP differential amplifier is active and the NPN differen tial amplifi- about – 1 V), both the NPN er is cutoff. When the input voltage ranges from about 1 V to (V CC and PNP dif ferential amplifiers are active. When the input voltage ranges from about (V CC 1 V) to (V + 0.2 V), the NPN dif ferential amplifier is active and the PNP differential am- – CC ex- cut off. Inclusion of complementary differential input amplifiers achieves V plifier is ICR ceeding the power supply limits, but there is a penalty to pay in input bias current, input 5 and 18 offset voltage, and distortion. Figures 18 – 6 show the input bias current and – input offset voltage as a function of the input common-mode voltage. 4 TLV245X V =5 V DD 3 = 25 ° C T A 2 nA – 1 0 – 1 Input Bias Current – IB 2 – I 3 – – 4 3.5 4 1 0 3 1.5 2.5 2 0.5 4.5 5 – Common-Mode Input Voltage V V – IC Figure 18 – 5. Input Bias Current Changes with Input Common-Mode Voltage 18-9 Designing Low-Voltage Op Amp Circuits

388 Input Common-Mode Range 100 TLV245X 80 V =5 V CC = 25 ° C T A V 60 μ – 40 20 0 20 – – 40 Input Offset Voltage – IO V – 60 80 – – 100 5 4.5 4 3.5 3 1 1.5 2 0.5 0 2.5 V – Common-Mode Input Voltage – V IC 6. Input Offset Voltage Changes with Input Common-Mode Voltage Figure 18 – When both transistors are conducting current the input bias currents have a tendency to cancel, so in ± 1 V, the bias current is extremely low even when bipolar transis- the range of tors are used to make the op amp. Above this range, the PNP differential amplifier cuts off so the full bias current requirement of the NPN transistor becomes apparent. The same action happens below this range when the NPN differential amplifier cuts off. Notice that the PNP bias current is significantly larger than the NPN bias current; this is expected be- cause NPN transistors have better gain characteristics than PNP transistors. The base- emitter voltage of the NPN and PNP transistors is well matched because the magnitude of the input offset voltage at the extremes is almost equal. The bias current and of fset voltage variation with input signal amplitude cause errors and distortion of the input signal. Inserting a resistance equal to the parallel combination of and R into the positive op amp lead minimizes the effect of input bias current. The R F G , has the same voltage drop across it that the parallel combination of R and resistor, R F P R has, hence the bias current is converted to a common-mode voltage. The common- G is in the fractional nA range and R mode voltage is normally in the μ V-range because I IB P is in the tens of K Ω . The CMMR is approximately 60 dB, so the input bias current effect is reduced to the nV range where it is insignificant compared to the offset voltage. The ’ t , and it shows up as an input error. If the design can input of fset current is multiplied by R P tolerate these errors it is wise to switch to a CMOS op amp because its input currents are in the pA range. Another type of error creeps in when complementary differential amplifiers are used to DR, and this error is results from the different gain of the PNP and NPN transistors. obtain 18-10

389 Output Voltage Swing Op amps always suffer to a limited extent from distortion introduced by different gains when operating in different quadrants The positive quadrant is above V /2 where NPN CC /2 where the PNP transistors operate, and the negative quadrant is below V transistors CC operate. Normally, this is a very minor effect because only the gain of the output stage changes with quadrant, but with complementary input stages the input and output gains change with quadrant. These errors are small, and they are accepted as the sacrifice re- quired for obtaining RRI operation. 18.5 Output Voltage Swing output voltage swing (RRO) is desirable for at least two reasons. First, the DR Rail-to-rail can achieve the maximum obtainable value if the op amp is RRO. Second RRO op amps can drive any converter connected to the same power supply if the impedance is compat- ible. The schematic of a RRO op amp output stage, part of the TLC227X, is shown in Fig- ure 18 – 7. V CC S G D – Input Output stage + D S G Ground Figure 18 – 7. RRO Output Stage The RRO characteristic is achieved in the construction of the op amp output stage. A to- tem pole design that has upper and lower output transistors is used, and the output tran- type of tran- “ sistors are a complimentary pair. Each transistor in the pair is a ” self-locking sistor operating in the common-source mode. Consider the p-channel output transistor; as long as this transistor has a drain-source resistance it forms a voltage divider with the load resistance. When the load is a very large resistor or if the output current flow is very small, the voltage drop across the output transistor can be neglected. Output current flows ) across the drain- the output transistor, and because current drops a voltage (V through DS source resistor, the output voltage swing is reduced. The voltage drop subtracts from the power supply voltage, reducing the output voltage to less than RRO. t drive heavy loads and maintain their RRO capability because of the RRO op amps can ’ voltage dropped across the output transistors. Load resistance or output current is a test 18-11 Designing Low-Voltage Op Amp Circuits

390 Shutdown and Low Current Drain condition when the measurement of an op amp s output voltage swing is made. The size ’ ’ s ability to retain its RRO of the load resistor or output current is a measure of the op amp while sourcing or sinking an output current. When selecting a RRO op amp, the capability designer must consider the load resistance or output current required because these con- ditions control the output voltage swing. When an op amp is made that has RRI and RRO capability, it is called a rail-to-rail input/ output op amp. This long name is shortened to RRIO. 18.6 Shutdown and Low Current Drain Low voltage design often is accompanied by a requirement that the power supply current drain be low. The power supply current drain is kept low to decrease battery size and pro- long battery charge so recharging can be put off as long as possible. Many methods are employed to keep the current drain low including using high-value resistors, low bias cur- rent regulators/references, slow speed logic, keeping logic transitions to a minimum, low voltage power supplies, selecting op amps for low current drain, and shutting off unused ICs. High-value resistors have less current flowing through them than low value resistors do, and they can be used effectively in ratio applications, but there are some downsides to Ω , depending on using high-value resistors. When resistor values exceed 2 M Ω to 10 M the type of resistor, the temperature drift, vibration, and time-induced drift increases rapid- , works ly compared to that of lower value resistors. The input resistor to an op amp, R G with the stray capacitance from the input node to ground to form a pole in the loop gain. As the resistance increases, the pole moves towards the zero frequency intercept and the , works with circuit overshoots, rings, or becomes unstable. The feedback resistor, R F to form a low-pass filter. Sometimes this filter action stray capacitance in parallel with R F is desirable, but the filter often distorts the signal. Very often, low bias current regulators and references are just standard ICs specified at a lower current. These devices generally do not have the same small tolerances at low bias currents that they had at high bias currents. Although they are more often costly, re- designed low bias current regulators and references are becoming available. Ensure that the reference or regulator bias current used in the application is the same as that used to specify the device, because sometimes the error curves for references are nonlinear. Also, investigate the reference noise voltage to ensure that low bias current has not moved the device to a noisy portion of its operating curve. Saturated logic is the choice for low current drain applications because nonsaturated logic stays in the active region and has a higher current drain. Always pick the slowest logic gates that you can get away with. Speed in saturated logic requires enough current to drive low impedance loads, and that means high power supply currents coupled with log- ic-generated noise. High-speed logic has a low impedance totem pole output stage, and 18-12

391 Single-Supply Circuit Design every time the output is switched, both totem pole transistors are on causing a current spike through the power supply. Large decoupling capacitors are required to localize the current spike at the logic IC, thus preventing noise propagation. CMOS logic draws the quiescent current, and if the logic transitions are kept at a minimum, the current drain least stays small. One method of minimizing logic transitions is to use asynchronous logic. The op amp should be selected with current drain in mind. Three rail-to-rail op amps have widely differing current drains because they are designed for different applications. The μ A. TLV240X is designed for micropower applications, and its current drain is 1.29 μ A. The TL V287X The TLV411X is designed high output drive, and its current drain is 800 is designed for high speed, and its current drain is 820 μ A. These three op amps are low-voltage op amps, but they each serve a different application. The best method of conserving current is to shut the op amp down if you are not using it. Most op amps designed for low voltage applications have shutdown pins. A typical op μ A when operating, draws 1.7 μ A when it is shut down. The problem amp that draws 820 with shutdown is the time that it takes to wake the op amp up and knowing when to wake the op amp up. A typical low-voltage op amp turns on in less that 1 μ s, but the system designer usually has to choose the variable that eventually wakes the op amp up. 18.7 Single-Supply Circuit Design The op amp is a linear device, so it follows the equation of a straight line. The equation of a straight line has four forms as shown in Equation 18 – 8. (18 – 8) Y b " +" mx These four forms can be implemented with four single supply circuits. When the designer discovers the form of Equation 18 – 8 that yields the transform function required, it is a small task to find the corresponding circuit. Once the circuit and transfer function are es- tablished, the task reduces to matching coefficients between the transfer function and the circuit equation, and then calculating the resistor values. The key required to unlock the puzzle is to determine the form of Equation 18 – 8 that yields the required transfer function. This key is found in simultaneous equations because they define the equation of a straight line. Several examples of using simultaneous equations to determine the required form of the op amp transfer function are given in the next two sections. 18.8 Transducer to ADC Analog Interface An example is a transducer that needs to be interfaced to an ADC. The transducer specifi- . The ADC specifications are = 0.2 V, V = 0.5 V, and R = 600 Ω cations are V MIN MAX OUT . The system specifies a 5-V power = 20 k = 4.5 V, and R = 1.5 V, V Ω V IN(HIGH) IN(LOW) IN supply and 5% tolerance resistors. The transducer is connected to input of the amplifier 18-13 Designing Low-Voltage Op Amp Circuits

392 Transducer to ADC Analog Interface (see – 8), so its output voltage swing is renamed V Figure 18 , and the ADC is connected IN to the output of the amplifier, so its input voltage range is renamed V . Now, two data OUT = 4.5 V. = 0.2 V @ V = 1.5 V and V = 0.5 V @ V are constructed as V points IN2 OUT2 IN1 OUT1 The data points are substituted into the equation Y = mX + b; m is named the slope and b is named the X axis intercept or just the intercept for short. Don ’ t worry about the sign of m or b because it is determined by the math, and it is substituted into the equation that determines the transfer equation. The simultaneous equations are given below. Digital Amplifier ADC Transducer Output Figure 18 – 8. Data Acquisition System (18 – 9) 1.5 + 0.2m ) b (18 – 10) 0.5m b 4.5 + ) From these equations we find that b = – 0.5 and m = 10. The slope and intercept values – 8 to get Equation 8 are substituted into Equation 18 11. – (18 – 11) Y + 10X * 0.5 The mathematical terminology in Equation 18 – 11 is replaced by electronics terminology in Equation 18 – 12, and this is the transfer function required for the amplifier. The next step is to select the op amp, and this isn ’ t a hard task because there are many candidates that could do the job with these undemanding specifications, so let us not dwell on the selec- process. Assume that the selected op amp operates on a 5-V power supply, can drive tion the ADC input resistance of 20 k Ω with no voltage divider action, and that the op amp input impedance is so big that it doesn ’ t load the transducer. (18 – 12) V + 10V * 0.5 IN OUT – 9. The circuit that produces the desired transfer function is given in Figure 18 18-14

393 Transducer to ADC Analog Interface R R F 1 V R CC G F μ 0.01 _ R 2 V REF V OUT + R V L IN – 9. Schematic for the Transducer to ADC Interface Circuit Figure 18 The circuit equation is obtained with the aid of superposition. ø R R ) R R ) R R 2 1 F G F 2 (18 – 13) Ǔ ǒ Ǔ ǒ ǒ Ǔ V V V + * REF IN OUT R ) R ) ø R R ø R R R ) R 2 1 1 1 2 2 G G Comparing terms between Equations 18 – 12 and 18 – 13 enables the extraction of m and b. ) R ø R R ) R 1 2 F G – 14) (18 ǒ Ǔ m + R R ) R ø 1 2 G R R F 2 (18 – 15) | | ǒ Ǔ Ǔ ǒ V + b REF R R ) R ) R R ø 1 2 1 2 G Making the assumption that R ||R <

394 DAC to Actuator Analog Interface introduces about 3% error in the b coefficient, and this error shows up in the transfer of R 1 function as an intercept rather than a slope error. The parallel resistance of R and R is 2 1 , thus the earlier as- which is 20 k Ω and this is much less than R approximately 0 .82 k Ω G could have been selected as a smaller value, is justified. R ||R <

395 DAC to Actuator Analog Interface is to terminate the DAC output in a resistor that converts current into voltage, and then level shift and amplify the terminated voltage. The circuit that performs this function is shown in Figure 18 – 12. +V CC +V CC I + V OUT _ R T R R G F V REF – 12. DAC Current Source to Actuator Interface Circuit Figure 18 The DAC output current sources from I = 2 mA at an out- = 1 mA to I OUT(ONES) OUT(ZEROS) = 1 V to put compliance of 4.33 V. The actuator requires an input voltage swing of V IN1 V . The system specifications include = 4 V to drive it, and its input resistance is 100 k Ω IN2 one 5-V power supply and 5% resistors. The DAC is connected to input of the amplifier , and the actuator is con- (see Figure 18 11), so its output current swing is renamed I – IN . Now, two nected to the output of the amplifier, so its input voltage range is renamed V OUT = 1 V and I = 2 mA @ V = 1 mA @ V = data points are constructed as I OUT1 IN1 OUT2 IN2 4 V. The data points are substituted into the Equation 18 – 20. Don ’ t worry about the sign of m or b because it is determined by the math, and it is substituted into the equation that determines the transfer equation. The simultaneous equations are given below. (18 – 20) b V ) + mI IN OUT (18 21) – + ) b 1 m – (18 22) 4 + 2m ) b From these equations we find that b = – 2 and m = 3. The slope and intercept values are substituted into Equation 18 20 to get Equation 18 – 23. – (18 – 23) V + 3I * 2 IN OUT 12 is derived with the aid of superposition, The equation for the circuit shown in Figure 18 – 24. – and it is given below in Equation 18 R R F F (18 – 24) Ǔ ǒ 1 ) + I V R * V T REF IN OUT R R G G 24 enables the extraction of m Comparing terms between Equations 18 – 20 and 18 – and b. 18-17 Designing Low-Voltage Op Amp Circuits

396 DAC to Actuator Analog Interface R F (18 – 25) | | V b + REF R G R F 26) (18 – ǒ Ǔ m 1 ) + R T R G R F 2 (18 27) – + 5 R G R F ) 1 5 R G 28) – (18 2.14 R + + T R F 1 ) R G Ω These equations are written in terms of mA and k Ω , so R . There is no 2.14-k = 2.14 k Ω T resistors. R is split into 1.8-k Ω and 0.33-k Ω resistor in the 5% standard values; thus, R G T is selected as 51 k Ω = 20 k Ω . When I , so R = 2 mA V = 4.28 V, so the compliance F IN RT of the DAC is not violated. You might find that standard DACs are not so generous with their compliance specifications. When the current is sinked from the power supply by the DAC its sign reverses and the previous circuit is not usable. Consider these specifications: the DAC output sinks current 2 mA at an output com- = – 1 mA to I – = from the power supply I OUT(ONES) OUT(ZEROS) pliance of 4.33 V. The actuator requires an input voltage swing of V = 1 V to V = 4 V IN1 IN2 to drive it, and its input resistance is 100 k Ω . The system specifications include one 5-V power supply and 5% resistors. The DAC is connected to input of the amplifier (see Figure , and the actuator is connected to the 18 11), so its output current swing is renamed I – IN . Now, two data points output of the amplifier, so its input voltage range is renamed V OUT are constructed as I – = 4 V. The = 1 V and I = – 2 mA @ V = 1 mA @ V OUT1 IN1 IN2 OUT2 data points are substituted into the Equation 18 20. Don ’ t worry about the sign of m or – b because it is determined by the math, and it is substituted into the equation that deter- mines the transfer equation. The transfer function for the current sink DAC is given in Equation 18 – 29. (18 – 29) b ) V m + I IN OUT 18-18

397 DAC to Actuator Analog Interface The simultaneous equations are given below. +V R F +V _ V R O F + V REF V – Figure 18 – 13. DAC Current Sink to Actuator Interface Circuit – 30) (18 +* 1 ) b m – (18 31) +* 2 ) b 4 these equations we find that b = – 2 and m = – 3. The slope and intercept values are From substituted into Equation 18 – 28 to get Equation 18 – 32. (18 32) – ( ) + – I * V b m – b + mI * IN IN OUT The current equation for the circuit shown in Figure 18 – 13 is given below as Equation 18 – 33, and after algebraic manipulation it becomes Equation 18 – 34. V V OUT REF 33) – (18 + I * IN R R F G R F 34) (18 – V I R * V + F REF IN OUT R G Comparing terms between Equations 18 29 and 18 – 34 enables the extraction of m and – b. (18 – 35) | | R + m + 3 F R F (18 – 36) | | + V b REF R G R F 2 (18 – 37) + 5 R G Ω These equations are written in terms of mA and k Ω . R , so R is selected as 51 k = G F = 2 mA, the compliance of the DAC is 0.0 V. 20 k Ω . When I IN 18-19 Designing Low-Voltage Op Amp Circuits

398 Comparison of Op Amps When DAC interface circuits are designed, two parameters that have not been considered in detail can control the design. The DAC has a compliance voltage requirement, and that requirement must be met regardless of the circuit demands. If the DAC compliance re- quirements are not met, the DAC saturates or is starved for current, and either of these situations introduces considerable error. The actuators driven in these examples are quite benign because most actuators require considerably more current or voltage than is avail- able from an op amp. This fact does not negate the analysis given here. Regardless of the actuator current or voltage requirements, the design procedure is similar. Very often the low voltage device is plugged into a booster that supplies power to the actuator. One last item to consider is the output capacitance of DACs. The DAC output can have large amounts of stray capacitance that shows up as a capacitor across the op amp invert- – ing input node when the DAC is interfaced into circuits as shown in Figure 18 13. The to form a pole in the op DAC capacitance from the inverting node to ground acts with R G amp loop gain. Adding a pole to the op amp loop gain leads to overshoot, then ringing, and finally oscillation. The effect that the DAC capacitance has on stability must be investi- gated. Also, the DAC output capacitance is a function of the digital number addressing the DAC. This capacitance can range from near zero to hundreds of pF, thus the op amp must be compensated for the worst case which is the largest capacitance. Compensation schemes include connecting a capacitor across the feedback resistor. This compensation scheme is called a compensated attenuator, and if the RC time constants are equal, there will excellent performance at that DAC output capacitance. Alas, the circuit can only be ideally compensated at one point, and this point is normally chosen as the highest DAC output capacitance. The remainder of the DAC range suf fers from poor bandwidth because of overcompensation. 18.10 Comparison of Op Amps Since the author is only familiar with Texas Instruments op amps, a comparison involving actual op amp parameters would be unfair to other op amp manufacturers. Also, any com- parison using today ’ s production op amps becomes invalid in a short period of time. I write about Texas Instruments op amps, so I get plenty of samples, and the new product introductions come so fast that I have a hard time keeping up with them. The other point to consider is that teaching how to make the op amp comparison is a much more powerful thus a table containing the op amp parameters is established, and each of the param- tool, eters is discussed only in terms of low-voltage design. 18-20

399 Comparison of Op Amps 2. Op Amp Parameters – Table 18 PARAMETER RANGE OF VALUES UNITS DISCUSSION V 25 to 8000 mV DC parameter that can be adjusted out. IO o V Drift parameter that ends up as an error. 3 to 1000 μ V / α C IO Input bias current that can be cancelled out. I 0.1 to 9999 pA IB I 0.1 to 9999 pA Input offset current that can be adjusted out. IO G R Acts as a voltage divider with driving circuit. 0.0002 to 1 Ω IN dB Common-mode voltage is a nonlinear error. 0 to 90 CMRR A 20 to 140 dB Determines high-frequency errors. VD Input voltage range over which op amp works correctly with spe- – 1.5 V to V V +0.5 V V V CC ICR CC cified error. Maximum high output voltage swing. Load resistance or current 1.5 V to V – V V +0.5 V V CC CC OH is important. Limits dynamic range. Minimum low output voltage swing. Load resistance or current is – V v 0 + 1.5 V to 0 1.5 V OL important. Limits dynamic range. mA Current available to drive loads. I 1 to 100 O Power supply current. 0.1 to 10000 μ I A CC k 20 to 120 dB Power supply noise rejection. SVR I Power supply current in shutdown mode. 0.01 to 100 μ A CC(SHDN) nV 1 to 1000 Noise voltage limits dynamic range. V n 1 ń 2 ( ) Hz fA I Noise current limits dynamic range. 0.01 to 100 n 1 ń 2 ( ) Hz 0.5 to 10 μ s Op amp wake up time. t (ON) t . 0.1 to 5 μ s Time until power supply current reaches I CC(SHDN) (OFF) As the table shows, there are a lot of parameters to be considered in the selection of a low voltage op amp. The application weeds out some of these parameters thus easing the selection process. If the application is measuring the output from a low-bandwidth transducer such as a thermocouple, bandwidth is not a parameter of interest, so speed can immediately be sacrificed for power supply current. When the maximum supply voltage available is 3 V, rafts of op amps requiring more than 3 V operating voltage are eliminated. Taking this concept further, if dynamic range is an important specification in the 3-V design, those op amps that operate on 3 V without RRIO specifications can be eliminated. Because dynamic range is important, the noise voltage and current that detract from dynamic range are important parameters. If this 3-V applica- tion requires very low power supply current, the choice is narrowed down to a few candi- The application selects the op amp, and if this application puts a few more require- dates. ments on the 3-V op amp, we may have to design a new IC. 18-21 Designing Low-Voltage Op Amp Circuits

400 Summary ’ t be wavered. Always start the selection process with the parameter that absolutely can The next parameter considered in the selection process should be the next most impor- tant parameter, and this process is continued parameter by parameter until all require- ments are exhausted. Sometimes the supply of op amp candidates runs out before the parameter requirements do. When you reach this point, it is time to renegotiate the design specifications, find a new op amp, negotiate with op amp manufacturers, or announce that you won ’ t meet specifications. These designs are hard to work because the low power supply voltage requirement and the specifications usually leave very little room to work. As time marches on, more low power supply voltage op amps will come on the market, and these designs will get easier to work. 18.11 Summary It is extremely hard to achieve large dynamic range when the application is limited to a low power supply voltage. In an attempt to approach the dynamic range obtained by ± 30-V power supply designs, the new op amp designs put increased emphasis on the output voltage swing. The ratio of output voltage swing to power supply voltage was 0.8 for ± 30-V powered op amps, 0.7 for the first 5-V powered op amps, and it has risen to 0.9 for the newest family of 1.8-V powered op amps. The ratio output voltage swing to power supply voltage has increased with each new generation of low power supply volt- age op amps, but this improvement has reached the point of diminishing returns. fsets The op amp ’ s dc offset diminishes the output voltage swing, but in most cases the of are adjusted out, so they have less importance in the design. New op amp technology is not being pushed hard to improve in this area because the passive components continue to require the adjustments. Drift and noise continue to decrease the dynamic range. Op amp noise has decreased in new-generation op amps, and another decrease should put noise in the category of “ don ’ t care ” parameters. The signal-to-noise ratio has several components that have to be analyzed. The signal comes to the op amp with a noise burden caused by the transducer, cabling, and connec- tions. Making the op amp a filter/amplifier combination eliminates some of this noise. The biggest drawback to making the op amp a filter is the time required to charge the ADC input capacitance. ADC charging has not been investigated here because of scope limita- tions, but suffice it to say that filters slow down op amps. The internally-generated op amp noise is multiplied by the closed loop gain, and the SNR should be established in the front end, so the closed loop gain hurts one way and helps the other way. There is always sys- tem noise, and a portion of this noise propagates through the op amp into the signal. The system noise is minimized by extensive use of decoupling capacitors and a high power supply rejection ratio. In higher voltage systems, a resistor is placed in series with the power supply, thus making the decoupling capacitors more effective; low power supply voltage designs usually can ’ t afford that trick. RRI op amps are able to work with transducers connected to the power supply rails. As long as the ac component of the transducer output voltage does not exceed the input com- 18-22

401 Summary mon-mode range of the op amp, the design is reliable. RRI op amps are troubled by distor- tion introduced by the change in bias current, input offset voltage, and gain, but their con- s signal handling capability is invaluable. RRO op amps yield the tribution to the system ’ highest output voltage swing of any series of op amps. Beware: RRO op amps are speci- fied at a load resistance or current, and the output voltage swing decreases dramatically when the load resistance or current is increased. RRIO op amps contain the input and output features of RRI and RRO op amps. They also contain the drawbacks of both fea- tures. Shutdown is a current-saving feature that is becoming standard on RRIO op amps. When you are working off a battery, there is no reason to waste power when the electronics is not busy, and the shutdown feature accomplishes the power savings by turning the op amp off when it is not needed. The shutdown feature has a disadvantage in that it has a ’ t just depend on shutdown to re- finite wakeup time that the designer must allow for. Don duce current drain because there are other ways to reduce current drain, and some of these ways are using high value resistors, low speed logic, fewer logic transitions, and using low bias current regulators/references. The final thing to be considered is that low power supply voltage invariably means single- supply design, and single-supply design is tougher than split-supply design. Remember to get the two sets of data points, put them in simultaneous equations, solve for the slope and intercept, select the circuit configuration, and calculate the component values. DACs are a little different because you have to account for the polarity of the current, but their design generally follows the same procedure. A good reference for single supply design echniques is the T exas Instruments application report Single Supply Op Amp Design T (TI literature number SLOA030). 18-23 Designing Low-Voltage Op Amp Circuits

402 18-24

403 Appendix A Appendix A Single-Supply Circuit Collection Ron Mancini and Richard Palmer A.1 Introduction Portable and single-supply electronic equipment is becoming more popular each day. The demand for single-supply op amp circuits increases with the demand for portable elec- tronic equipment because most portable systems have one battery. Split- or dual-supply op amp circuit design is straightforward because op amp inputs and outputs are refer- enced to the normally grounded center tap of the supplies. In the majority of split-supply applications, signal sources driving the op amp inputs are referenced to ground, thus with – 1, there is no need one input of the op amp referenced to ground, as shown in Figure A to consider input common-mode voltage problems. R F A – 1A V +* V IN OUT R G + 5 V R R F G +V CC V IN _ 1 V/Div V OUT — + T OU V – V CC 5 V – V 1 V/Div — – 5 V + 5 V IN R = R = 10 k V = 10 Vp-p IN G F 5 V = 5 V – +V = V – CC CC Op Amp = TLC071 Figure A – 1. Split-Supply Op Amp Circuit A-1

404 Introduction – 2 and Equation A – 1B), the signal source is not referenced to ground (see Figure A When the voltage difference between ground and the reference voltage shows up amplified in the output voltage. Sometimes this situation is OK, but other times the difference voltage must be stripped out of the output voltage. R F ǒ Ǔ 1B – A V V V +* ) REF IN OUT R G + 5 V R R F G +V CC V IN _ 1 V/Div — V OUT + OUT V REF V – V CC – 5 V 1 V/Div — V – + 5 V 5 V IN = 10 Vp-p V = 1 V V REF IN R = 10 k = R +V = 5 V CC G F V – = – 5 V Op Amp = TLC071 CC Figure A – 2. Split-Supply Op Amp Circuit With Reference Voltage Input An input bias voltage is used to eliminate the difference voltage when it must not appear , is in both 3 and Equation A – 1C). The voltage, V in the output voltage (see Figure A – REF input like circuits, hence it is named a common-mode voltage. Voltage-feedback op amps, those used in this document, reject common-mode voltages because their input circuit is constructed with a differential amplifier (chosen because it has natural common-mode voltage rejection capabilities). R R ) R R R F G F F F ǒ Ǔ 1C A – Ǔ ǒ Ǔǒ +* V +* ) V V V ) V REF IN REF IN OUT R R R ) R R F G G G G A-2

405 Introduction + 5 V R R F G +V CC V IN _ 1 V/Div — V OUT + V OUT REF V – V CC R G – 5 V R V F REF — V 1 V/Div – 5 V + 5 V IN = 10 Vp-p V = 1 V V REF IN R = 10 k = 5 V = R +V G CC F Op Amp = TLC071 5 V – – V = CC – 3. Split-Supply Op Amp Circuit With Common-Mode Voltage Figure A When signal sources are referenced to ground, single-supply op amp circuits always have a large input common-mode voltage. Figure A – 4 shows a single-supply op amp circuit that has its input voltage referenced to ground. The input voltage is not referenced to the midpoint of the supplies like it would be in a split-supply application, rather it is referenced to the lower power supply rail. This circuit malfunctions when the input voltage is positive the output voltage would have to go negative — hard to do with a positive supply. because It operates marginally with small negative input voltages because most op amps do not function well when the inputs are connected to the supply rails. + 5 V R R F G +V CC 1 V/Div V IN _ — V OUT + OUT V 5 V – — 1 V/Div V 5 V + 5 V – IN V = 10 Vp-p R = 10 k = R IN G F Op Amp = TLV247x = 5 V V CC Figure A – 4. Single-Supply Op Amp Circuit A-3 Single-Supply Circuit Collection

406 Boundary Conditions The constant requirement to account for inputs connected to ground or other reference makes it difficult to design single-supply op amp circuits. This appendix presents voltages a collection of single-supply op amp circuits, including their description and transfer equa- tion. Those without a good working knowledge of op amp equations should reference the Understanding Basic Analog series of application notes available from Texas Instru- ments. Application note Understanding Basic Analog — Ideal Op Amps devel- SLAA068, ops the ideal op amp equations. Circuit equations in this appendix are written with the ideal op amp assumptions as specified in Understanding Basic Analog — Ideal Op Amps. The assumptions appear in Table A 1 for easy reference. – Table A – 1. Ideal Op Amp Assumptions PARAMETER NAME PARAMETER SYMBOL VALUE Input current I 0 IN Input offset voltage V 0 OS Z ∞ Input impedance IN Z 0 Output impedance OUT Op amp gain a ∞ Detailed information about designing single-supply op amp circuits appears in application note SLOA030, Single-Supply Op Amp Design Techniques . Unless otherwise specified, all op amp circuits shown here are single-supply circuits. The single supply may be wired with the negative or positive lead connected to ground, but as long as the supply polarity is correct, the wiring does not affect circuit operation. A.2 Boundary Conditions All op amps are constrained to output voltage swings less than or equal to their power Use of a single supply limits the output voltage to the range of the supply voltage. supply. equals +10 V, the output voltage is limited to For example, when the supply voltage V CC 10. This limitation precludes negative output voltages when the cir- ≤ ≤ the range 0 V OUT cuit has a positive supply voltage, but it does not preclude negative input voltages. As long as the voltage on the op amp input leads does not become negative, the circuit can handle negative voltages applied to the input resistors. Beware of working with negative (positive) input voltages when the op amp is powered from a positive (negative) supply because op amp inputs are highly susceptible to re- verse-voltage breakdown. Also, ensure that no start-up condition reverse biases the op amp inputs when the input and supply voltage are opposite polarity. It may be advisable to protect the op amp inputs with a diode (Schottky or germanium) connected anode to ground and cathode to the op amp input. A-4

407 Amplifiers A.3 Amplifiers types of amplifiers can be created using op amps. This section consists of a selec- Many some basic, single-supply op amp circuits that are available to the designer during tion of the concept stage of a design. The circuit configuration and correct single-supply dc bias- ing techniques are presented for the following cases: inverting, noninverting, differential, T-network, buffer and ac-coupled amplifiers. A.3.1 Inverting Op Amp with Noninverting Positive Reference The ideal transfer equation is given in Equation A – 1. R R ) R F G F A 1 – Ǔ ǒ ) V V V +* REF IN OUT R R G G 5) takes the form of Y = – mX + b. The trans- – The transfer equation for this circuit (Figure A fer function slope is negative, and the dc intercept is positive. R and R are contained G F in both halves of the equation, thus it is hard to obtain the desired slope and dc intercept . This is the minimum component count configuration for this trans- without modifying V REF fer function. When the reference voltage is 0, the input voltage is constrained to negative voltages because positive input voltages would cause the output voltage to saturate at ground. + 5 V R R F G +V CC V IN _ 1 V/Div — V OUT + OUT V V REF – 5 V 1 V/Div — V – 5 V + 5 V IN = 10 Vp-p V = 1 V V IN REF = R = 10 k = 5 V V R F G CC Op Amp = TLV247x – 5. Inverting Op Amp with Noninverting Positive Reference Figure A A-5 Single-Supply Circuit Collection

408 Amplifiers A.3.2 Inverting Op Amp with Inverting Negative Reference The transfer equation takes the form of Y = – mX + b and is given in Equation A – 2. R R F F 2 A – ) +* V V V REF IN OUT R R G1 G2 The transfer function slope is negative, and the dc intercept is positive (Figure A – 6). R G1 and R are contained in the equation, thus it is easy to obtain the desired slope and dc G2 intercept by adjusting the value of both resistors. Because of the virtual ground at the in- is the terminating impedance for V . When the reference voltage is verting input, R G2 REF 0, the input voltage is constrained to negative voltages because positive input voltages would cause the output voltage to saturate at ground. + 5 V R R G1 F +V CC R G2 _ 1 V/Div V IN — V OUT + V REF OUT V 5 V – — V 1 V/Div 5 V – + 5 V IN = = 10 Vp-p V 1 V – V REF IN = R V R = 10 k = R = 5 V F G1 CC G2 Op Amp = TLV247x Figure A – 6. Inverting Op Amp with Inverting Negative Reference A-6

409 Amplifiers A.3.3 Inverting Op Amp with Noninverting Negative Reference – mX – b and is given in Equation A – 3. The transfer equation takes the form of Y = ) R R R F G F 3 – A Ǔ ǒ +* V V V * IN REF OUT R R G G The transfer function slope is negative, and the dc intercept is negative. R are and R G F contained in both halves of the equation, thus it is hard to obtain the desired slope and . This is the minimum component count configuration dc intercept without modifying V REF for this transfer function. The slope and dc intercept terms in Equation A – 3 are both nega- tive, hence, unless the correct input voltage range is selected, the output voltage will satu- 400 mV because – rate at ground. The negative input voltage must be limited to less than op amp inputs either break down or have protection circuits that forward bias when large negative voltages are applied to the inputs. + 5 V R R G F +V CC _ 1 V/Div V IN — V OUT + OUT V V REF – 5 V 1 V/Div — V 5 V + 5 V – IN = 10 Vp-p – V 0.25 V = V IN REF = R = 10 k R = 5 V V F CC G Op Amp = TLV247x – 7. Inverting Op Amp with Noninverting Negative Reference Figure A A-7 Single-Supply Circuit Collection

410 Amplifiers A.3.4 Inverting Op Amp with Inverting Positive Reference The transfer equation takes the form of Y = mX b and is given in Equation A – – – 4. R R F F 4 A – V +* V * V IN REF OUT R R G1 G2 The transfer function slope is negative, and the dc intercept is negative. R and R are G1 G2 contained in the equation, thus it is easy to obtain the desired slope and dc intercept by adjusting the value of both resistors. Because of the virtual ground at the inverting input, is the terminating impedance for V . The slope and dc intercept terms in Equation R REF G2 – A 4 are both negative, hence, unless the correct input voltage range is selected, the out- put voltage will saturate. + 5 V R R F G1 +V CC R G2 1 V/Div _ V IN — V OUT + OUT V REF V 5 V – V 1 V/Div — – 5 V + 5 V IN = 10 Vp-p V = 1 V V REF IN = R = R V = 10 k = 5 V R G1 CC G2 F Op Amp = TLV247x Figure A – 8. Inverting Op Amp with Inverting Positive Reference A-8

411 Amplifiers A.3.5 Noninverting Op Amp with Inverting Positive Reference The transfer equation takes the form of Y = mX – b and is given in Equation A – 5. R ) R R F G F 5 A – + * V V V IN REF OUT R R G G The transfer function slope is positive, and the dc intercept is negative. This is the mini- mum component count configuration for this transfer function. The reference termination is the load across V are and R . R resistor is connected to a virtual ground, so R REF G G F contained in both halves of the equation, thus it is hard to obtain the desired slope and or placing an attenuator in series with V . dc intercept without modifying V REF IN + 5 V R R F G +V CC _ 1 V/Div V REF — V OUT + OUT V V IN – 5 V 1 V/Div V — + 5 V 5 V – IN = 10 Vp-p V = 1 V V REF IN R = 5 V = R = 10 k V CC G F Op Amp = TLV247x Figure A – 9. Noninverting Op Amp with Inverting Positive Reference A-9 Single-Supply Circuit Collection

412 Amplifiers A.3.6 Noninverting Op Amp with Noninverting Negative Reference – b and is given in Equation A – 6. The transfer equation takes the form of Y = mX ) R R R ) R R R F F G G 2 1 A – 6 ǒ Ǔ ǒ Ǔ + * V V V REF IN OUT ) R ) R R R R R 2 2 1 1 G G The transfer function slope is positive, and the dc intercept is negative. The reference is and R to obtain can be selected independent of R and R and R . R terminated in R 1 2 F 1 G 2 the desired slope and dc intercept. The price for the extra degree of freedom is two resis- tors. + 5 V R R G F +V CC 1 V/Div _ — R 1 V OUT + OUT V R 2 V IN V REF – 5 V — V 1 V/Div + 5 V 5 V – IN V = 10 Vp-p 1 V – = V REF IN Op Amp = TLV247x V = 5 V CC = R = R = 10 k = R R 2 G 1 F Figure A – 10. Noninverting Op Amp with Noninverting Negative Reference A-10

413 Amplifiers A.3.7 Noninverting Op Amp with Inverting Negative Reference The transfer equation takes the form of Y = mX + b and is given in Equation A – 7. R ) R R F G F 7 – A V + V V ) REF IN OUT R R G G The transfer function slope is positive, and the dc intercept is positive. This is the minimum component count configuration for this transfer function. The reference termination resis- and R are con- . R is the load across V tor is connected to a virtual ground, so R REF F G G tained in both halves of the equation. Thus it is hard to obtain the desired slope and dc or placing an attenuator in series with V . intercept without modifying V REF IN + 5 V R R F G +V CC _ 1 V/Div V REF — V OUT + OUT V V IN – 5 V 1 V/Div — V – + 5 V 5 V IN = 10 Vp-p 1 V – = V V REF IN = 10 k = R R V = 5 V F G CC Op Amp = TLV247x Figure A – 11. Noninverting Op Amp with Inverting Positive Reference A-11 Single-Supply Circuit Collection

414 Amplifiers A.3.8 Noninverting Op Amp with Noninverting Positive Reference The transfer equation takes the form of Y = mX + b and is given in Equation A – 8. R ) R R ) R R R F F G G 2 1 A 8 – Ǔ ǒ Ǔ ǒ V + V ) V REF IN OUT R ) R R R R ) R 2 2 1 1 G G The transfer function slope is positive, and the dc intercept is positive. The reference is and R to obtain can be selected independent of R and R and R . R terminated in R 2 1 F 1 G 2 the desired slope and dc intercept. The price for the extra degree of freedom is two resis- tors. + 5 V R R G F +V CC _ 1 V/Div R 1 V — OUT + OUT V R 2 V IN V REF – 5 V V — 1 V/Div 5 V + 5 V – IN V = 1 V = 10 Vp-p V IN REF V = 5 V Op Amp = TLV247x CC = R = R = R = 10 k R F G 2 1 Figure A – 12. Noninverting Op Amp with Noninverting Positive Reference A-12

415 Amplifiers A.3.9 Differential Amplifier is set equal to R and R When R is set equal to R Equation A – 9 reduces to Equation 2 F 1, G A – 10. ) R R R R F G F 2 – 9 A Ǔ ǒ V + V V * IN2 IN1 OUT R R R ) R 2 1 G G R F ǒ Ǔ V + V * V 10 – A IN1 IN2 OUT R G These resistors must be matched very closely to obtain good differential performance. The mismatch error in these resistors reduces the common-mode performance, and the mismatch shows up in the output as an amplified common-mode voltage. Consider Equation A – 10. Note that only the difference signal is amplified, thus this config- uration is The differential amplifier is a popular circuit in pre- called a dif ferential amplifier. cision applications where it is used to amplify sensor outputs while rejecting common- mode noise. The inverting input impedance is R because of the virtual ground at the inverting op amp G input. The noninverting input impedance is R because the noninverting op amp in- + R F G put impedance approaches infinity. The two input impedances are different, and this leads to two problems with this circuit. First, mismatched input impedances preclude any attempts to cancel input bias currents through resistor matching. Often R so that the bias currents devel- is set equal to R || R 2 G F op equal common-mode voltages which the op amp rejects. This is not possible when R and R = R = R unless the source impedances are matched. Second, high output F 1 2 G sensors are often used, and when high output sensors work into mismatched impedance input impedances, errors occur. + 5 V R R G F +V CC _ 1 V/Div V IN1 R 1 V — OUT + OUT V V IN2 R 2 5 V – — V 1 V/Div 5 V – + 5 V IN2 V V = 10 = 1 V IN1 IN2 Vp-p Op Amp = TLV247x = 5 V V CC = 10 k = R = R = R R F G 2 1 – 13. Differential Amplifier Figure A A-13 Single-Supply Circuit Collection

416 Amplifiers A.3.10 Differential Amplifier With Bias Correction Equation A is set equal to R is set equal to R and R – 11 reduces to Equation When R 2 F G 1, 12. A – R ) R R ) R R R R F F G G 1 F 2 – 11 A Ǔ ǒ ǒ Ǔ V ) * V V + V REF IN1 IN2 OUT R R R ) R ) R R R 2 1 2 1 G G G R F Ǔ ǒ V V ) + V V * – 12 A IN1 IN2 REF OUT R G When an offset voltage must be eliminated from or added to the input signal, this dif feren- tial amplifier circuit is employed. The reference voltage can be positive or negative de- pending upon the polarity offset required, but care must be taken to protect the op amp inputs and not exceed the output range. + 5 V R R G F +V CC _ 1 V/Div V IN1 R 1 V — OUT + OUT V V IN2 R 2 5 V – V REF — 1 V/Div V – 5 V + 5 V IN2 V = 1 V = 10 Vp-p V IN1 IN2 – = 1 V V V = 5 V REF CC = 10 k = R = R = R R G F 2 1 Op Amp = TLV247x Figure A – 14. Differential Amplifier with Bias Correction A-14

417 Amplifiers A.3.11 High Input Impedance Differential Amplifier is set equal to R is set equal to R and R When R 13 reduces to Equation – Equation A 1 F G 2, – 14. A R ) R R R ) R F G 1 F 2 A – 13 ǒ Ǔ * V + V ) V IN1 IN2 OUT R R R 1 G G R ) R F G ǒ Ǔ V V V * + 14 – A IN1 IN2 OUT R G Each input signal is connected to an op amp noninverting input that is very high imped- ance. The input impedance of the circuit is very high, and it is matched, so this circuit is often used to interface to high-impedance sensors. Each op amp has a signal propagation experiences two propagation delays versus V ’ s one propagation delay. time, and V IN1 IN2 At high frequencies, the propagation delay becomes a significant portion of the signal pe- riod, and this configuration is not usable at that frequency. should be matched to achieve good common-mode rejection , and R and R and R R 1 F G 2 || R should be connected in se- capability. Bias current cancellation resistors equal to R F G ries with the input sources for precision applications. + 5 V R R R R 2 G 1 F +V +V CC CC _ _ 1 V/Div V — OUT + + OUT V V V IN1 IN2 – 5 V 1 V/Div — V + 5 V – 5 V IN2 V = – = 10 Vp-p V 1 V IN2 IN1 = 5 V V Op Amp = TLV247x CC = R = R = 10 k = R R F G 2 1 15. High Input Impedance Differential Amplifier – Figure A A-15 Single-Supply Circuit Collection

418 Amplifiers A.3.12 High Common-Mode Range Differential Amplifier – 15 reduces to Equation A – 16. When all resistors are equal, Equation A R R R ) R R F 2 1 2 F ǒ Ǔ Ǔ Ǔǒ ǒ Ǔǒ ) + V * ) V V * * REF1 IN2 OUT R R R R 1 1 G1 G1 – A 15 R ø R ) R R F G2 G1 F ǒ Ǔ ǒ Ǔ ) * V V IN1 REF2 R R ø R G2 G2 G1 ǒ Ǔ V + * V ) V 2V 3V * 16 – A IN1 REF1 IN2 REF2 OUT R are equal-value resistors terminated into a virtual ground, hence, the input and R 1 G2 sources are equally terminated. This configuration has high common-mode capability be- and R limit the current that can flow into or out of the op amp. Thus, the input cause R 1 G2 ’ s drive capability. The vol- voltage can rise to any value that does not exceed the op amp , are added for bias purposes. Without bias, the output and V tage references, V REF1 REF2 voltage of the op amps would saturate at ground, and the bias voltages keep the output voltage of the op amp positive. + 5 V R R 2 1 +V CC _ R R G1 F 0.5 V/Div V + IN2 — +V CC R G2 _ OUT V V REF1 V OUT + V IN1 V 0 V REF2 V — 0.5 V/Div 2.5 V – + 2.5 V IN2 V V = 1 V = 5 Vp-p IN2 IN1 = 1.25 V V = 1.67 V V REF1 REF2 = 5 V V Op Amp = TLV247x CC = R = 10 k = R = R = R R F 1 2 G1 G2 Figure A – 16. High Common-Mode Range Differential Amplifier A-16

419 Amplifiers A.3.13 High-Precision Differential Amplifier =V = R 17 reduces to Equation , R – = R Equation A , R When R = R , and V 4 1 REF2, 2 5 6 REF1 7 – 18. A ) R R R ) 2R R 6 2 4 3 7 Ǔ ǒ Ǔ Ǔǒ ǒ Ǔǒ V * V V + ) REF2 IN2 OUT R R R R ) 3 2 5 7 17 – A R ) R R R 2R R ) 3 6 6 2 5 1 ǒ Ǔ Ǔǒ Ǔ ǒ Ǔ ǒ ) V ) V V REF1 IN1 REF3 R R ) R R R 5 2 3 2 7 R 2R 6 1 Ǔ ǒ ǒ Ǔǒ Ǔ V * V V ) + ) 1 V 18 – A IN2 REF3 IN1 OUT R R 3 2 In this circuit configuration, both sources work into the input impedance of a noninverting op amp. This impedance is very high, and if the op amps are identical, both impedances are very nearly equal. The propagation delay is still equal to two op amp propagation de- lays, but the propagation delay is very nearly equal, so any distortion resulting from un- equal propagation delays is minimized. The equal resistors should be matched with more precision than is expected from the cir- com- Resistor matching eliminates distortion due to unequal gains, and it reduces the cuit. mon-mode voltage feed through. Resistors equal to (R || R )/2 may be placed in series 1 3 with the sources to reduce errors resulting from bias currents. This differential amplifier has the unique feature that the gain can be changed with only one resistor, and if the gain setting resistor is R3, no resistor matching is required to change gain. + 5 V +V CC R R + 6 2 _ V IN1 R 1 V +V REF1 CC 0.5 V/Div _ — R V 3 OUT OUT + V +V CC R 4 _ 0 V — 0.5 V/Div V 2.5 V – + 2.5 V + IN2 R 5 R 7 V V V = 5 Vp-p = 2.5 V IN2 IN2 IN1 V = 2.5 V = 2.5 V V REF1 REF2 = 2.5 V V V = 5 V REF3 CC V V REF2 REF3 = R = R = R = R = R = 10 k = R R 4 2 1 3 6 5 7 Op Amp = TLV247x Figure A – 17. High-Precision Differential Amplifier A-17 Single-Supply Circuit Collection

420 Amplifiers A.3.14 Simplified High-Precision Differential Amplifier is set equal to R When R , R 19 is set equal to R – and V , Equation A = V F REF2 1, G 2 REF1 – reduces to Equation A 20. R ) R R R F G F 2 Ǔ ǒ Ǔ ǒ Ǔǒ ǒ ǒ Ǔ Ǔ V ) V V ) V ) * + V IN1 REF2 REF1 IN2 OUT R R R ) R 2 1 G G – A 19 R ) R R F G 1 Ǔǒ ǒ Ǔ V REF3 ) R R R 2 1 G R F ǒ Ǔ Ǔ ǒ ) + V V * V V 20 A – REF3 IN1 IN2 OUT R G Both input sources are loaded equally with very high impedances in the simplified high precision differential amplifier. This configuration eliminates three resistors, two of which are matched, but it sacrifices flexibility in gain setting capability because the gain must be set with a matched pair of resistors. +V + 5 V CC R R + F G _ +V CC V IN1 _ 0.5 V/Div V OUT + — V REF1 +V CC OUT _ V + R 1 R 0 V 2 V IN2 V — 0.5 V/Div + 2.5 V 2.5 V – IN2 V V = 5 Vp-p = 2.5 V IN2 IN1 V V REF3 REF2 V = 2.5 V = 2.5 V V REF2 REF1 V V = 5 V = 2.5 V CC REF3 = R = 10 k = R = R R 1 F G 2 Op Amp = TLV247x – 18. Simplified High-Precision Differential Amplifier Figure A A-18

421 Amplifiers A.3.15 Variable Gain Differential Amplifier 21 reduces to Equation When R is set equal to R – and R Equation A is set equal to R 4, 3 1 2 A – 22. R R R ) R R R G G 4 3 4 2 Ǔǒ ǒ Ǔǒ Ǔ Ǔǒ Ǔ ǒ + V * V V 21 A – IN2 IN1 OUT R R R ) R R R 3 1 3 2 F F R R 4 F Ǔ ǒ Ǔǒ Ǔ ǒ + * V V V – A 22 IN1 IN2 OUT R R 3 G When a function is enclosed in a feedback loop, the function acts inverted on the closed attenuator. The circuit ends up being an /R transfer function. Thus, the gain stage R loop F G 19 can be used with any of the differential amplifiers to change gain shown in Figure A – , R and R , R must be matched to reduce the without affecting matched resistors. R 4 1 3 2 common-mode voltage. R R 1 2 V +V IN1 CC _ R V 3 OUT + R G R F V IN2 +V CC _ + R 4 V CC V REF 2 – 19. Variable Gain Differential Amplifier Figure A A-19 Single-Supply Circuit Collection

422 Amplifiers T Network in the Feedback Loop A.3.16 Sometimes it is desirable to have a low-resistance path to ground in the feedback loop. Standard inverting op amps cannot do this when the driving circuit sets the input resistor value and the gain specification sets the feedback resistor value. Inserting a T network in the feedback loop yields a degree of freedom that enables both specifications to be met with a low dc resistance path to ground in the feedback loop. R R R R 2 3 3 2 ȣ ȡ ȡ ȣ R ) R ) R ) R ) 3 2 3 2 R R 4 4 ) V +* V V ) 1 23 A – ȧ ȧ ȧ ȧ IN REF OUT R R 1 1 Ȣ Ȣ + 5 V R R R 1 2 3 R 4 +V CC V IN _ 0.5 V/Div — V OUT + OUT V V REF 0 V V — 0.5 V/Div – 2.5 V + 2.5 V IN = 114 mV V V = 250 mVp-p IN REF V = 5 V = 100 k R CC 2 = 10 k = R = R R 3 4 1 Op Amp = TLV247x Figure A – 20. T Network in the Feedback Loop A-20

423 Amplifiers A.3.17 Buffer The buffer input signal polarity must be unipolar because the output voltage swing is uni- polar. When this limitation precludes the buffer, a differential amplifier with the negative input correctly biased is used, or a reference voltage is added to the buffer to offset the must be included when the op amp inputs are not rated for the full sup- output voltage. R F limits the current into the op amp inputs, thus preventing latch ply voltage. In that case, R F up. Most new op amp inputs can withstand the full supply voltage, so they often leave R F out as cost savings. The main attraction of the buffer is that it has very high input imped- ance and very low output impedance. The impedance transformation capability is why buffers are often added to the input of other circuits. V V ) + V – 24 A IN REF OUT + 5 V R F +V CC _ 0.5 V/Div V OUT + — OUT V V IN 0 V V REF — V 0.5 V/Div – 2.5 V + 2.5 V IN V = 5 Vp-p V = 2.5 V IN REF Op Amp = TLV247x V = 5 V CC 21. Buffer – Figure A A-21 Single-Supply Circuit Collection

424 Amplifiers A.3.18 Inverting AC Amplifier /2 at the inverting input. R is connected to and resistors R set a dc level of V V CC CC G ground through a capacitor, thus the circuit functions as a buffer for dc. This causes the /2, so the quiescent output voltage is the middle of the supply dc output voltage to be V CC voltage, and it is ready to swing to either rail as the input signal commands. – 25. R The ac gain is given in Equation A and C form a coupling network for the ac signal. G Good coupling networks should be constant low impedance at the signal frequencies, so Equation A – 26 should be satisfied to get good low-frequency performance. The lowest , is determined by completing a Fourier se- frequency component of the input signal, f MIN = 100f in Equation A – 26 ensures that the 3-dB ries on the input signal. Then, setting f MIN and C is two decades lower than f . breakpoint introduced by R G MAX V R CC F V +* V ) 25 A – IN OUT 2 R G 1 + f – 26 A C 200 p R G 0 270 2 – 260 C R R – 4 250 G F dB 6 – 240 — – 8 230 +V V CC IN Degrees 10 – 220 _ — – 12 210 V OUT V Phase + OUT 14 – 200 Phase 190 16 – +V Output Amplitude CC – 18 180 R 20 – 170 R 100 100 k 1 k 10 10 k – f Frequency – Hz = 1 Vp-p = 0.01 F V C IN = R = 10 k R = R V = 5 V F G CC Op Amp = TLV247x Figure A – 22. Inverting AC Amplifier A-22

425 Amplifiers A.3.19 Noninverting AC Amplifier and the resistors (R) set a dc level of V /2 at the inverting input. R is connected V CC G CC to ground through a capacitor, thus the circuit functions as a buffer for dc. This causes the /2, so the quiescent output voltage is the middle of the supply dc output voltage to be V CC voltage, and it is ready to swing to either rail as the input signal commands. The ac gain is given in Equation A – 27. R and C create a coupling network for the ac sig- G nal. Good coupling networks should be a constant low impedance at the signal frequen- – 28 should be satisfied to get good low frequency performance. The cies, so Equation A , is determined by completing a Four- frequency component of the input signal, f lowest MIN = 100f in Equation A – 28 ensures that the ier series on the input signal. Then, setting f MIN . The breakpoint and C is two decades lower than f 3-dB breakpoint introduced by R G MAX and C is set in a similar manner. for R G 1 ) R V R ) R R F F CC G G Ǔ ǒ ) + V V 27 – A IN OUT 2 R R G G 1 f + – A 28 C 200 R p G 10 90 6 80 2 70 R R F G dB — – 2 60 6 – 50 +V C CC Degrees +V – 10 40 CC _ — – 14 30 V OUT V Phase OUT + 18 – 20 R Phase 10 22 – Output Amplitude – 26 0 – 30 10 – C 10 k 1 k 100 k 100 10 V IN R f Hz – Frequency – = 1 Vp-p V = 0.01 F C IN = R = 10 k = R R = 5 V V G F CC Op Amp = TLV247x Figure A – 23. Noninverting AC Amplifier A-23 Single-Supply Circuit Collection

426 Computing Circuits A.4 Computing Circuits Four versions of the inverting op amp and four versions of the noninverting op amp were given in the previous section. During the concept stage of the design, one of these eight op amp circuits is selected. Specifications for the input and output voltage are the selec- tion criteria that determines which circuit configuration is used. There are four versions of most of the circuits given in this and following sections, but just the simplest version of any circuit is included in this appendix because of space limita- tions. Each circuit configuration can be modified as required to fit specific applications. Look back to the first section to determine what bias is required to fit the application, and adapt that bias to the new circuit. A.4.1 Inverting Summer The three input voltages are inverted and added as Equation A – 29 shows. R should be B made equal in value to the parallel combination of R to convert the , R , and R , R G2 G3 F G1 sets the input bias current to a common-mode voltage so the op amp can reject it. V REF output voltage somewhere between the supply limits, and this allows negative addition (subtraction) to take place. R R R R F F F F ǒ Ǔ ǒ Ǔ V ) ) V ) V V )AAA 1 ) V +* A – 29 IN3 IN2 IN1 REF OUT R R R R AAA ø R ø R G1 G3 G2 G1 G2 G3 + 5 V R R F G1 R G2 R G3 V IN1 +V CC 0.5 V/Div _ V — IN2 V OUT V OUT IN3 + V R B 0 V V — 0.5 V/Div 2.5 V – + 2.5 V IN3 V REF = 5 V V = 1 V V CC IN1 V 1.5 V = 0.625 V – V = IN2 REF = 5 Vp-p V Op Amp = TLV247x IN3 = 10 k = R = R = R R G2 G3 G1 F 24. Inverting Summer – Figure A A-24

427 Computing Circuits A.4.2 Noninverting Summer This circuit adds the input voltages and multiplies them by the stage gain. R , and , R G2 G1 to cancel the input bias current in parallel with R in parallel should be equal to R R G F G3 is added to the circuit using the common-mode input voltage rejection technique. V REF to enable the addition of negative values. Ǔ ǒ ǒ Ǔ ǒ Ǔ V V ø ø R R R R ø R R V IN3 IN2 IN1 G3 G3 G2 G2 G1 G1 V ) ) +) ǒ Ǔ OUT R ) R R ø R ø R R ) ) R ø R R G2 G3 G2 G3 G1 G2 G1 G3 G1 A 30 – ) R R R F G F ǒ Ǔ * V REF R R G G + 5 V R G1 R G2 R G3 V IN1 +V CC 0.5 V/Div V — IN2 + V OUT _ V OUT IN3 V R F 0 V R G V — 0.5 V/Div + 2.5 V 2.5 V – IN3 = 5 V V = 1.5 V V CC IN1 V REF V = 1 V = – 1.25 V V – IN2 REF = 20 k R = 5 Vp-p V F IN3 Op Amp = TLV247x R = R = R = R = 10 k G2 G3 G G1 – 25. Noninverting Summer Figure A A-25 Single-Supply Circuit Collection

428 Computing Circuits A.4.3 Noninverting Summer with Buffers V and V are added to enable the buffers to handle positive input voltages. Their REF1 REF2 . This configuration uses output contribution to the last stage is cancelled out by V REF3 in parallel should be , and R , R fewer resistors at the expense of two op amps. R G2 G1 F to cancel the input bias current. made equal to R B 2R 2R R R R F F F F F ǒ Ǔ V ) V ) + V 1 V * V * V ) A – 31 REF3 REF2 IN2 REF1 IN1 OUT R R R R R ø R G1 G2 G2 G1 G1 G2 R R + 5 V V IN1 +V CC _ R G1 R/2 R F + 0.5 V/Div — +V V CC REF1 OUT _ V V OUT + R R V IN2 R B 0 V +V CC 0.5 V/Div V — 2.5 V – + 2.5 V IN3 _ R G2 V R/2 REF3 V V = 1.25 V = 5 Vp-p REF1 IN1 + = 1.25 V V V = 0.5 V IN2 REF2 = 5 V = 2.5 V V V REF3 CC V REF2 Op Amp = TLV247x R = 10 k = R = R = R F G1 G2 – 26. Noninverting Summer with Buffers Figure A A-26

429 Computing Circuits A.4.4 Inverting Integrator The Laplace operator, s= j ω , is used in Equation A – 32, and the mathematical operation 1/s constitutes an integration. Differentiation circuits are shown later, and the mathemati- cal operation, s, constitutes a differentiation. The integration time constant is RC, thus the magnitude crosses 0 dB on a log plot when RC = 1. Also the phase is – 45 ° when RC = 1. 1 V V +* A – 32 IN OUT RCs This integrator is not very practical because there is no method of discharging the capaci- tor; hence, any leakage current will eventually charge the capacitor until the circuit be- / 2 to center the out- comes saturated. The positive input of the integrator is biased at V CC put voltage at V / 2; thus allowing for positive and negative voltage swings. The bias CC resistors are selected as 2R so that the parallel combination equals R. This offsets the input current drawn through R. V C IN R F +V CC 1 V/Div V OUT V _ — IN V OUT + Voltage +V CC 2R 2R 100 s / Div — Time R = 10 k V = 2 Vp-p IN C = 10000 pF = 5 V V CC Op Amp = TLV247x 27. Inverting Integrator – Figure A A-27 Single-Supply Circuit Collection

430 Computing Circuits A.4.5 Inverting Integrator with Input Current Compensation Functionally, this circuit is the same as that shown in Figure A – 27, but a current com- bias the , and R , R pensation network has been added to offset the input current. V 1 2 CC / 2 to center the output voltage at V / 2; thus allowing for positive positive input at V CC CC and negative voltage swings. R and R are selected as relatively small values because the current flowing through R 2 A 1 also flows through the parallel combination of R and R . R forward biases the diode with 1 A 2 a constant current, thus the diode acts like a small voltage regulator. The diode voltage drop is temperature sensitive, and this factor works in our favor because the input transis- tors are temperature sensitive. The two temperature sensitivities cancel out if the diode is a large-value resistor that acts like a current source, current is selected correctly. R B correctly ensures so it is selected such that it supplies the input bias current. Selecting R B that no input current flows through the integration resistor, R. This integrator is not very practical because there is no method of discharging the capaci- tor. Hence, any input current will eventually charge the capacitor until the circuit becomes saturated. The bias circuit drastically reduces the input current flowing through R, thus it extends the integration time. A reset circuit is needed to make the integrator more practi- cal. This bias compensation scheme is set up for an op amp that has NPN input transistors. The diode must be reversed and connected to ground for op amps with PNP input circuits. 1 +* V V – A 33 IN1 OUT RCs C R +V CC V IN _ V OUT + R B +V CC +V CC R D A R 1 R2 – 28. Inverting Integrator with Input Current Compensation Figure A A-28

431 Computing Circuits A.4.6 Inverting Integrator with Drift Compensation Functionally, this circuit is the same as that shown in Figure A – 27, but it uses an RC circuit in the positive lead to obtain drift compensation. The voltage divider is made from a series ), and V biases the input in the center of the power supply. string of resistors (R A CC input current flows through R and C in parallel, so the positive input current drops Positive the same voltage across the parallel RC combination as the negative input current drops across its series RC combination. The common-mode rejection capability of the op amp rejects the voltages caused by the input currents. Much longer integration times can be /2, the achieved with this circuit, but when the input signal does not center around V CC compensation is poor. 1 V +* V A – 34 IN1 OUT RCs C R +V CC V IN _ +V V OUT CC + R C R A R A 29. Inverting Integrator with Drift Compensation – Figure A A-29 Single-Supply Circuit Collection

432 Computing Circuits A.4.7 Inverting Integrator with Mechanical Reset Functionally, this circuit is the same as that shown in Figure A – 27, but a method has been is a mechanical switch or relay and when provided to discharge (reset) the capacitor. S 1 the contacts close, they short the integrating capacitor forcing it to discharge. Some ca- is put in the discharge path to limit pacitors are sensitive to fast discharge cycles, so R S is absent from the circuit, the impulse of current that the initial discharge current. When R S occurs at the first instant of discharge causes considerable noise, so the selection of R S is also based on noise considerations. For all practical purposes, the time constant and C determines the discharge rate. formed by R S One advantage of mechanical discharge methods is that they are isolated from the re- mainder of the circuit. Their size, weight, time delay, and uncertain actuating time offset this advantage. When the disadvantages of mechanical reset outweigh the advantages, circuit designers go to electronic reset circuits. 1 V V +* 35 A – IN1 OUT RCs S R 1 S C R +V CC V IN _ 2R V OUT + +V CC 2R Figure A – 30. Inverting Integrator with Mechanical Reset A-30

433 Computing Circuits A.4.8 Inverting Integrator with Electronic Reset Functionally, this circuit is the same as that shown in Figure A – 27, but an electronic meth- is controlled by a gate drive od has been provided to discharge (reset) the capacitor. Q 1 is on, the gate-source resistance is signal that changes its state from on to off. When Q 1 low, less than 100 Ω. Α nd when Q about sev- is off, the gate-source resistance is high — 1 Ω . eral hundred M gate-source bias The source of the FET is at the inverting lead that is at ground, so the Q 1 is not affected by the input signal. Sometimes, the output signal can get large enough to correctly. Consult , so the designer must take care to bias Q leakage currents in Q cause 1 1 a transistor book for more detailed information on transistor reset circuits. A major prob- lem with electronic reset is the charge injected through the transistor ’ s stray capacitance. This charge can be large enough to cause integration errors. 1 +* V V 36 – A IN1 OUT RCs Reset Q 1 C R +V CC V IN _ 2R V OUT + +V CC 2R 31. Inverting Integrator with Electronic Reset – Figure A A-31 Single-Supply Circuit Collection

434 Computing Circuits A.4.9 Inverting Integrator with Resistive Reset – This circuit differs from that shown in Figure A 27 because it yields a breakpoint rather than a pure integration. On a log plot, the integrator slope is – 6 dB per octave at the 0 fre- quency intercept, and the 0 dB intercept occurs when f = 1/2 π RC. A breakpoint plots flat – 6 dB per octave. It is – on a log plot until the breakpoint where it breaks down at 3 dB when f = 1/2 RC. π is in parallel with the integrating capacitor, C, so it is continually discharging C. The R F low frequency attenuation that is the best attribute of the pure integrator is sacrificed for the reset circuit complexity R 1 F Ǔ ǒ +* V V 37 – A IN1 OUT R 1 C ) R G F 3 190 1 180 1 – 170 dB V Phase OUT R R G F – 3 160 — – 5 150 C Degrees – 7 140 V IN — 9 – 130 +V CC 11 – 120 Phase _ 110 – 13 R Output Amplitude V OUT + +V CC – 15 100 – 17 90 R 100 1 k 100 k 10 k 10 – Frequency – Hz f V = 1 Vp-p RG = RF = R = 10 k IN = 5 V V C = 0.01 F CC Op Amp = TLV247x Figure A – 32. Inverting Integrator with Resistive Reset A-32

435 Computing Circuits A.4.10 Noninverting Integrator with Inverting Buffer This circuit is an inverting integrator preceded by an inverting buffer. Eliminating the signal inversion costs an op amp and four resistors, but this is the easiest way to get true nonin- verting integrator performance. R 1 A 1 ǒ Ǔ V + V + * V A 38 – IN IN OUT R RCs RCs A C R R R A A V IN +V +V CC CC _ _ R A V OUT 2R + + +V CC +V CC R A 2R Figure A – 33. Noninverting Integrator with Inverting Buffer A-33 Single-Supply Circuit Collection

436 Computing Circuits A.4.11 Noninverting Integrator Approximation This circuit has fewer parts than the Noninverting Integrator With Inverting Buffer (Figure A – 33), but it is not a true integrator because there is a zero in the transfer equation. The RC, log plot starts rolling off at a – 6 dB per octave rate at low frequencies, but when f = 1/2 π the zero cuts in. The zero causes the log plot to flatten out because the slope decreases to 0 db per decade. This circuit functions as an integrator at very low frequencies, but at frequencies higher than f = 1/2 π RC, it functions as a buffer. 1 RCs ) V + 39 – A OUT RCs C R R A +V CC +V CC R A _ R V OUT + V IN 34. Noninverting Integrator Approximation – Figure A A-34

437 Computing Circuits A.4.12 Inverting Differentiator ferentiator is a positive slope of 6-dB per octave passing through 0 dB The log plot of the dif π RC. At extremely high frequencies, the capacitive reactance goes to very low at f = 1/2 values, thus the circuit gain approaches the op amp open-loop gain. This performance any system noise or noise generated by the op amp. The poor noise perfor- emphasizes mance of this circuit limits its application to a very few specialized situations. This configuration has a pole in the feedback loop. If the op amp has more than one pole, and most op amps have several poles, this configuration can become oscillatory. The V CC and R /2 should be selected circuit bias the output in the center of the power supplies. R A A so that input currents are canceled out. ||R equal to R F G V RCs +* V – 40 A IN OUT 25 – 80 Phase 20 – 90 15 – 100 dB — 10 – 110 C R 5 120 – Degrees 0 – 130 — +V CC – 5 140 – V IN V OUT + – 10 150 – Phase 2R V OUT _ 160 – – 15 +V Output Amplitude CC – 20 170 – 2R 25 – 180 – 100 1 1 k 10 Frequency – f Hz – V = 50 mVp-p R = 10 k IN = 5 V C = 0.001 F V CC Op Amp = TLV247x Figure A – 35. Inverting Differentiator A-35 Single-Supply Circuit Collection

438 Computing Circuits A.4.13 Inverting Differentiator with Noise Filter This circuit has a pure differentiator that rises at a 6-dB per octave slope from zero fre- C , the pole kicks in and the slope is reduced to zero. The pole has quency. At f = 1/2 π R F F two effects. First, it stabilizes the circuit by canceling zero ’ s phase shift. Second, it limits the circuit gain to 1 at high frequencies, so it acts like a noise filter. for good input current cancellation, and V coupled with R centers R/2 should equal R F CC the output voltage. Cs R F +* V V 41 – A IN OUT 1 ) C s R F F 25 – 80 Phase 20 – 90 15 – 100 dB C C R F 10 110 – — 5 120 – C F Degrees 0 V 130 – IN — 5 – 140 – V +V OUT CC 10 – – 150 Phase _ R 160 – 15 – Output Amplitude V OUT + +V CC 20 – 170 – – 25 180 – R 1 100 1 k 10 f – Frequency – Hz RF = R = 10 k = 50 mVp-p V IN = 5 V V C = 0.01 F CC Op Amp = TLV247x Figure A – 36. Inverting Differentiator with Noise Filter A-36

439 Oscillators A.5 Oscillators This section describes some general op amp sinewave oscillator circuits that fall under three main categories: Wien bridge, phase shift, and quadrature. A brief description and of each type is provided, along with one or two variations. Op amp sinewave oscillators are used to create references in applications such as audio and function/waveform gener- ators. A.5.1 Basic Wien Bridge Oscillator When ω = 2 π f = 1/RC, the feedback is in phase (this is positive feedback), and the gain = 2R the amplifier is 1/3, so oscillation requires an amplifier with a gain of 3. When R G F π RC. Normally, the gain is larger than 3 to ensure and oscillation occurs at f = 1/2 gain is 3 oscillation under worst case conditions. sets the output dc voltage in the center of the span. V REF The output sine wave is highly distorted because limiting by saturation and cutoff is con- trolling the output voltage excursion. The distortion decreases when the gain is de- creased, but the circuit may not oscillate under worst-case low gain conditions. V CC 2 V + A – 42 REF R F 1 ) R G R F +V CC _ V OUT + R G R C R C V REF 37. Basic Wien Bridge Oscillator – Figure A A-37 Single-Supply Circuit Collection

440 Oscillators A.5.2 Wien Bridge Oscillator with Nonlinear Feedback When the circuit gain is 3, R = R /2. L F Substituting a lamp (R ) for the gain setting resistor reduces distortion because the non – L linear lamp resistance adjusts the gain to keep the output voltage smaller than the power supply voltage. The output voltage never approaches the power supply rail, so distortion determine the lamp current (see Equations A 44). – 43 and A – and R t occur. R ’ doesn L F V OUT(RMS) + I 43 – A LAMP R R ) F L Ǔ ǒ V 2 OUT(RMS) R + 44 A – F ǒ Ǔ 3 I OUT(RMS) The lamp is selected by examining lamp resistance curves until a lamp with a resistance approximately equal to R /2 at I is found. The output voltage swing should be OUT(RMS) F must be greater than less than 75% of the maximum guaranteed voltage swing, and 3 R L should be V /5. the load resistance specified for the voltage swing specification. V CC REF R F +V CC _ V OUT Lamp + R L R C R C V REF – 38. Wien Bridge Oscillator with Nonlinear Feedback Figure A A-38

441 Oscillators A.5.3 Wien Bridge Oscillator with AGC The op amp is configured as an ac amplifier to ease biasing problems. The gain equation or R , but not both resistors, is required depending for the op amp is given below. R G1 G2 . on the selection of the Q 1 The diode, D , half-wave rectifies the output voltage and applies it to the voltage divider 1 and R . The voltage divider biases Q in its linear region, and they eventually formed by R 2 1 1 filters the rectified sine wave with a long time constant so that set the output voltage. C 1 must be selected large enough to act as a short at the output voltage stays constant. C 2 the oscillation frequency. increases. As the output voltage increases, the negative voltage across the gate of Q 1 to increase its drain-to-source resist- The increased negative gate voltage causes Q 1 ance. This results in increased op amp gain and an output voltage decrease. When the voltage divider and FET are selected properly, the output voltage swing is less than the guaranteed maximum swing, so distortion doesn ’ t occur. R F + G 45 – A R ø R ) R FET G2 G1 C R 2 1 1N4148 D 1 R F +V CC R G1 _ Q 1 V OUT + R G2 R C R C 2 1 C R V REF Figure A – 39. Wien Bridge Oscillator with AGC A-39 Single-Supply Circuit Collection

442 Oscillators A.5.4 Quadrature Oscillator Quadrature oscillators produce sine waves 90 ° out of phase, so they output sine/cosine, or quadrature waves. When R C = R π f = 1/RC. Both op amps act as = R C , the circuit oscillates at ω = 2 C 2 1 1 3 2 3 integrators causing two poles at 1/RC, thus the circuit oscillates when the loop gain cross- es the 0-dB axis. The integrators ensure that gain is always sufficient for oscillation. There is a slight bit of distortion at the sine output, and it is very hard to eliminate this distortion. C 1 +V CC R 1 _ V OUT(SINE) + R 2 +V CC + V OUT(COSINE) _ C 2 R 3 V CC 2 C 3 40. Quadrature Oscillator – Figure A A-40

443 Oscillators A.5.5 Classical Phase Shift Oscillator Theoretically, the three RC sections do not load each other, thus the loop gain has three identical poles multiplied by the op amp gain. – 180 ° The loop phase shift is – 60 ° , and this occurs when the phase shift of each section is when ω = 2 π f = 1.732/RC because the tangent of 60 ° = 1.73. The magnitude of β at this 3 , so the gain, A = R /R , must be greater or equal to 8 for the system gain point is (1/2) F G to be equal to 1. The assumption that the RC sections do not load each other is not entirely valid, thus the circuit does not oscillate at the specified frequency, and the gain required for oscillation is more than 8. This circuit configuration was very popular when an active component was large and expensive, but now that op amps are inexpensive, small, and come quad pack- ages, the classical phase shift oscillator is losing popularity. The classical phase shift oscillator has an undistorted sine wave available at the output of the third RC section. This is not a low-impedance output, and the signal amplitude is smallest here, but these sacrifices have to be made to get away from distortion. An undis- torted output can be obtained from the op amp if an AGC circuit similar to the one shown 39 is employed. The reference voltage is set according to the equation V in Figure A – REF /2. / 2(1+R /R ) to center the output voltage at V = V F G CC CC 3 1 ǒ Ǔ b + A – 46 A RCs ) 1 R F +V CC R G _ RR R V OUT + V CC CC C 18 41. Classical Phase Shift Oscillator – Figure A A-41 Single-Supply Circuit Collection

444 Oscillators A.5.6 Buffered Phase Shift Oscillator amp buffers each RC section in this oscillator. 46, repeated – Equation A A noninverting op >> R. below, truly represents the transfer function of this circuit if R G 3 1 Ǔ ǒ + A b – A 47 RCs ) 1 The loop phase shift is – 180 ° when the phase shift of each section is – 60 ° , and this occurs β when π f = 1.732/RC because the tangent 60 ° = 1.73. The magnitude of = 2 at this point ω 3 , so the gain, A = R , must be greater or equal to 8 for the system gain to be /R is (1/2) G F equal to one. The buffered phase shift oscillator has an undistorted sine wave available at the output of the third RC section. This is not a low-impedance output, and the signal amplitude is here, but these sacrifices have to be made to get away from distortion. An undis- smallest obtained from the op amp if an AGC circuit similar to the one shown torted output can be 39 is employed. – in Figure A There are three op amps, so the gain can be distributed among the op amps at the ex- pense of a few resistors, and the distortion is reduced. Another method of reducing distor- tion is to limit the output voltage swing softly with external components. The limiting tech- nique does not yield as good results as the AGC technique does, but it is less expensive. = V / 2(1+R /R ) to center The reference voltage is set according to the equation V F CC G REF the output voltage at V /2. CC R F +V CC R +V G CC _ +V R CC + R + + R _ V OUT _ C V C CC C 18 42. Buffered Phase Shift Oscillator – Figure A A-42

445 Oscillators A.5.7 Bubba Oscillator The Bubba oscillator is another phase shift oscillator, but it takes advantage of the quad op amp package to yield some unique advantages. Each RC section is buffered by an op >> R there is no loading in the circuit, and the circuit amp to prevent loading. When R G yields theoretical performance. phase shift. – 45 ° phase shift per section to accumulate – 180 ° Four RC sections require = 1/RC. The gain required for os- Each RC section contributes – 45 ° phase shift when ω 4 = 4. Taking outputs from alternate sections yields low-impedance (1/0.707) ≥ cillation is G quadrature outputs. When an output is taken from each op amp, the circuit delivers four ° 45 phase-shifted sine waves. The gain, A, must equal 4 for oscillation to occur. Very low distortion sine waves can be . When low-distortion sine waves are required at obtained from the junction of R and R G all outputs, the gain should be distributed among the op amps. Gain distribution requires biasing of the other op amps, but it has no ef fect on the oscillator frequency. This oscillator has the best d φ /df of the phase shift oscillators, so it has minimum frequency drift. The ) to center the / 2(1+R = V /R reference voltage is set according to the equation V CC REF F G /2. output voltage at V CC 4 1 Ǔ ǒ b A + A 48 – RCs ) 1 R F +V CC R +V G CC _ R + + _ C V CC 10 R +V CC +V CC R + R + _ V OUT _ C C C – 43. Bubba Oscillator Figure A A-43 Single-Supply Circuit Collection

446 Oscillators A.5.8 Triangle Oscillator The triangle oscillator produces triangle waves and square waves. The op amp functions as an integrator. When the output voltage of the comparator is low, the output of the op and R amp charges C until the output voltage exceeds the hysteresis voltage set by R F 1 and the reference voltage (V /2). At this point, the comparator output switches to a high CC state and the op amp integrates the voltage in a negative direction. The triangle wave (op amp output voltage swing) is given in Equation A – 49. The frequency of oscillation is given in Equation A 50. – R V V 1 CC CC V + " A 49 – OUT 2 2R F R F f + A 50 – 4CRR 1 The op amp reference voltage can be adjusted to equalize the triangle rise and fall times. C +V CC +V V CC _ CC R _ 2 + V OUT + V R F CC 2 R 1 – 44. Triangle Oscillator Figure A A-44

447 Appendix B Appendix A Single-Supply Op Amp Selection Guide This appendix contains a selection guide for single-supply op amps and abstracts from the data sheets for the TLC080, TLV2401, TLV2470, and TLV2760 single-supply op amps. B-1

448 Table B – 1. Single-Supply Operational Amplifier Selection Guide I V V DD IO DD I V CMRR SR GBW AMPS IB 1kHz n, DEVICE μ (V) A/Ch) μ V) ( ( (MHz) in (V/ (dB) (nV/ Hz s) ) (pA) μ √ (pA) (V/ s) Hz) √ (nV/ μ in (MHz) (dB) † FAMILY ‡ typ typ typ PKG typ typ max typ max typ max min (min) ≥ 1.8 V V V (min) ≥ 1.8 V DD DD 1.8 3.6 650 770 250 3000 2.5 8 2.8 100 18 TLV278x(A) S/D/Q 1.8 95 20 28 550 3500 3 0.5 0.2 70 S*/D/Q* TLV276x 3.6 (min) (min) 2.5 V V V ≥ ≥ 2.5 V DD DD 2500 TLV277x(A) 1000 2000 360 S/D/Q 2 5.1 10.5 96 17 2.5 5.5 S*/D 2.5 6 400 600 175 3500 0.3 2 1.57 63 26 TLV411x S/D/Q TLV240x 0.88 0.95 390 1200 100 0.0055 0.0025 120 n/a 2.5 16 0.0055 2.5 1.2 600 3000 100 1 0.002 100 n/a S/D/Q 12 TLV224x 2.7 6 500 575 150 2000 TLV246x(A) 5.2 1.6 80 11 S/D/Q 1300 S/D/Q 6 550 750 250 2200 2.7 2.8 1.4 78 15 TLV247x(A) 2 S 2.7 10 750 1200 750 3000 1 1.9 1.25 70 16 TLV2231 S 2.7 750 1200 750 3000 1 1.9 1.25 70 16 TLV2731 10 1 D/Q 725 1100 300 2000 10 1.75 1.3 75 18 TLV244x(A) 2.7 # 11000 S/D/Q 130 250 1700 7000 5.5 1 1 65 39 2.7 LMV321 12 2.7 8 200 250 300 2500 1 0.67 0.55 75 TLV226x(A) D/Q TLV243x(A) D/Q 2.7 10 98 125 300 2000 1 0.5 0.25 83 22 TLV2221 S 10 100 150 610 3000 1 0.48 0.18 82 20 2.7 S 2.7 100 150 610 3000 1 0.48 0.18 82 20 TLV2721 10 2.7 900 23 35 20 1500 S/D/Q 0.2 0.11 86 49 TLV245x(A) 6 1 D/Q 34 62 200 1500 8 0.187 0.1 75 19 TLV225x(A) 2.7 S 2.7 10 11 25 470 3000 1 0.056 0.025 83 TLV2211 22 TLV2711 2.7 10 11 25 470 3000 1 0.056 0.025 83 22 S D 23 10 50 75 300 2000 0.5 0.046 0.02 83 TLV2422(A) 2.7 ≥ 3.8 V V V (min) ≥ 3.8 V (min) DD DD 3 S 16 1500 2400 0.6 TLC2652(A) 4 1.9 2.8 140 23 3.8 TLC2654(A) S 3.8 16 1500 2400 5 20 50 1.9 2 125 13 TLC450x(A) S/D 6 1000 1500 – 80 1 4.7 2.5 100 12 4 1 D/Q 1100 1500 300 2500 16 2.18 3.6 75 9 TLC227x(A) 4.4 D/Q 4.4 16 200 250 300 TLC226x(A) 1 0.82 0.55 83 12 2500 TLC225x(A) 4.4 16 40 62 200 D/Q 1 0.21 0.12 88 19 1500 TLC07x(A) S/D/Q 4.5 16 1900 2500 60 1000 1.5 10 16 140 7 1000 TLC08x(A) 16 1800 2500 60 4.5 3 10 16 140 8.5 S/D/Q TLC220x(A) S/D 4.6 16 1000 1500 80 500 1 1.9 2.5 110 8 Note: = 25 ° C, V = 5 V, 2.7 V, and/or 1.8 V. Single-supply op amp selection guide parameters specified at T A DD * Devices in Product Preview stage of development at press time. Please contact TI or visit www.ti.com for current status. † (A) refers to precision grades with lower V and are part of the device number, e.g. TLV2470A. IO ‡ S = single-channel amplifier, D = dual-channel amplifier, Q = quad-channel amplifier (in the device name, replace x with 1, 2, or 4 for S, D, or Q respectively). § Variants of family with shutdown (in the device name, replace x with 0, 3, or 5 for single, dual, or quad amplifiers with shutdown). ¶ = 0.5 V from negative rail. Measured at V = 0.5 V from rails, except TLC07x and TLC08x where V O O # Includes LMV358 and LMV324 devices. B-2

449 Table B – 1. Single-Supply Operational Amplifier Selection Guide (Continued) ¶ RAIL I SHUT O DOWN to (mA) FEATURES § typ RAIL (min) ≥ 1.8 V V DD 10 I/O WBW, LV Y 4.8 I/O μ P, LV Y V ≥ 2.5 V (min) DD 6 Y O VHSR, HD 220 Y O VHD, SD 0.2 I/O nP, LV I/O P, LV 0.2 μ Y I/O WBW, LN, VHD 30 Y I/O WBW, LN, HD, LIBC 22 2.5 O WBW, LN 2.5 O WBW 2.5 LV, 600 Ω O – O SS 1.1 O LP, LN 2.5 O LP 1.8 O LP, SOT LP, SOT 1.8 O 4 Y μ P, R-R I/O I/O O μ 0.25 P, R-R O μ P, WVSR O 0.38 O μ P, WVSR 0.38 O LP, 600 Ω 2.5 V ≥ 3.8 V (min) DD O 1.4 PC O PC, LN 1.4 22 WBW, P, LN, HD O 2 O WBW, LN 1.3 O LP, LN 0.3 O P μ Y 55 WBW, FSR, LN 55 Y WBW, FSR O 1 P, LN Single-Supply Operational Amplifier Selection Guide Features Key: FSR fast slew rate nP nanopower HD high drive P precision LIBC PC precision (chopper) low input bias current LN low noise SOT SOT – 23 package LP low power SS second sourced WBW wide bandwith μ P micropower B-3 Single-Supply Op Amp Selection Guide

450 B-4

451 A B Band-pass filter AC loads, DAC, 14-2 Butterworth gain response graph, 16-36 AC parameters, DAC, 14-14 to 14-18 design, 16-27 to 16-37 See ACG, Automatic gain control gain response graph, 16-29 low-pass filter transition, 16-28, 16-37 Active filter, design techniques, 16-1 to 16-61 Sallen-Key circuit, 16-30 Actuator, analog interface to DAC, 18-16 to 18-21 second order, 16-29 to 16-39 Actuator interface Band-rejection filter, comparison graph active/pas- from DAC current sink circuit, 18-19 sive, 16-41 from DAC current source circuit, 18-17 Band-rejection filter ADC active twin-T filter, 16-37 to 16-40 analog interface from transducer, 18-13 to 18-14 active Wien-Robinson filter, 16-39 to 16-42 characterization, 12-15 design, 16-36 to 16-42 external ref circuits, 13-15 to 13-18 Bandwidth external voltage reference circuit, 13-17 current feedback amp, 9-3 to 9-7 input voltage, 12-15 current feedback amp graph, 8-10 selection, 13-6 to 13-10 voltage feedback amp, 9-3 to 9-7 single-ended input drive circuit, 13-20 Bessel high-pass filter, circuit, 16-27 ADC interface, from transducer circuit, 18-15, 18-16 Bessel low-pass filter, 16-7 to 16-10 Adder, 3-5 Bias circuit, 12-13 circuit, 3-5 Binary bits, versus significant bits, 12-2 Adjustable low-pass filter, circuit, 16-17 Black, Harry, Bell Labs, 1-1 Block diagram AIA circuit, 12-17 CDMA cellular base station, 13-5 All-pass filter cellular base station receiver, 13-2 design, 16-41 to 16-47 dual IF receiver, 13-4 first order, 16-44 feedback system, 6-2 first-order circuit, 16-44 GSM cellular base station, 13-2 high order, 16-45 to 16-47 loop gain, 6-4 second order, 16-44 to 16-46 oscillator with positive feedback, 15-8 second-order circuit, 16-44 Block diagram math, 5-1 to 5-6 seventh-order circuit, 16-46 control system terms, 5-3 Amplifier definition, 5-2 photodiode, 12-8 feedback circuit, 5-3 phototransistor, 12-9 multiloop feedback system, 5-4 transistor, 2-10 to 2-13 summary points, 5-3 photovoltaic cell, 12-9 transforms, 5-5 Analog input drive circuits, high-speed, 13-18 to Blocks, block diagram math, 5-2 13-22 Bode analysis, feedback, 5-7 to 5-13 Analog interface circuit, 12-21 Bode plot compensated attenuator, 7-18 Analog plane placement, 17-8 current feedback amp, 8-7 Antenna, loop and slot PCB layout, 17-15 dominant-pole compensation circuit, 7-11 Anti-aliasing filter, 13-11 to 13-13 feedback capacitance, 8-13 Automatic gain control, Wien bridge oscillator circuit, lead compensation, 7-14 15-13 lead-lag compensation, 7-19 Bode, H. W, 1-2 Avalanche noise, 10-9 to 10-11 Index-1

452 Broadcasting, from PCB traces, 17-9 Butterworth low-pass filter circuit with unity gain, 16-21 Bridge, wheatstone, 12-8 canonical feedback, 5-6 Broadband noise, 11-18 capacitively-loaded op amp, 7-9 Brown noise, 10-12 compensated attenuator, 7-17 Bubba oscillator, 15-17 to 15-19 complex feedback networks, 3-7 circuit, 15-17 control canonical feedback, 5-6 Burst noise, 10-9 to 10-11 current divider rule, 2-4 Butterworth band-pass filter, gain response graph, current feedback amplifier, 8-2 16-36 DAC current booster, 14-20 DAC current sink to actuator interface, 18-19 Butterworth low-pass filter, 16-6 DAC current source to actuator interface, 18-17 amplitude response graph, 16-6 DAC deglitch, 14-17 circuit with unity gain, 16-21 graph, 16-51 DAC voltage booster, 14-21, 14-22 Tschebyscheff-type characteristic, 16-51 differential amplifier, 3-6, 3-7 differential op amp, 6-8 dual-supply filter, 16-47 C electronic canonical feedback, 5-6 feedback, 5-6 Cable amplifier, 3-9 feedback loop broken, 6-7 Canonical equations, 6-2 to 6-5 first-order all-pass filter, 16-44 Canonical feedback, 15-2 first-order high-pass filter, 16-23 first-order inverting high-pass filter, 16-23 Capacitance effect on inverting op amp, 17-18 first-order inverting low-pass filter, 16-13 trace to plane, 17-17 first-order low-pass filter, 16-12 first-order noninverting low-pass filter, 16-14 Capacitively-loaded first-order passive RC low pass filter, 16-2 loop-gain calculation circuit, 7-9 fourth-order passive RC low pass filter, 16-3 op amp circuit, 7-9 high-frequency capacitor, 17-12 Capacitor, 3-9 to 3-11 high-frequency inductor, 17-13 high-frequency circuit, 17-12 high-pass filter, 3-10 self resonance, 17-22 high-performance resistor, 17-11 Cascading filter stages, 16-12 high-speed analog input drive, 13-18 to 13-22 CDMA cellular base station, transmitter block dia- ideal current feedback amp, 9-3 gram, 13-5 ideal op amp, 3-2 Cellular base station receiver, block diagram, 13-2 input current, 11-10 See current feedback amplifier CFA. input for an RRI op amp, 18-8 input for nonRRI op amp, 18-8 Characterization, transducer, 12-13 to 12-16 input offset voltage, 11-9 Circuit input offset voltage adjustment, 11-9 active full-wave rectifier and filter, 12-10 input parasitic elements, 11-13 active Wien-Robinson filter circuit, 16-39 inverting, 6-6, 6-7 ADC singled-ended input drive, 13-20 inverting current feedback amp, 8-5 adder, 3-5 inverting op amp, 3-4 AIA, 12-17 Kirchoff’s current law, 2-3 analog interface, 12-21 Kirchoff’s voltage law, 2-2 analysis done the hard way, 2-7 lead-lag compensation, 7-19 automatic gain control, 15-13 logic gate output, 17-21 band reject filter, 5-9 long-tailed pair, 9-2 Bessel high-pass filter, 16-27 bubba oscillator, 15-17 loop gain calculation, 7-9 Index-2

453 low pass filter, 16-1 single-supply filter, 16-47 slew rate, 11-16 low-pass filter, 3-10, 5-8 split-supply op amp, 10-19 low-pass filter circuit with unity gain, 16-19 stability analysis, 8-2, 8-3 maximum output voltage swing, 11-12 stray capacitance on inverting input, 7-16 MFB band-pass filter, 16-31 superposition example, 2-8 MFB high-pass filter biasing, 16-50 T network, 3-8 MFB low-pass filter biasing, 16-49 Thevenim equivalent of the base circuit, 2-11 Miller effect compensation, 7-2 s equivalent, 2-5, 2-6 ’ Thevenim noninverting current feedback amp, 8-4 TLC2201, 10-21, 10-23 noninverting op amp, 3-3, 18-7 traducer bias, 12-13 nonlinear feedback, 15-12 transducer to ADC interface, 18-15, 18-16 s law, 2-2 Ohm ’ transistor, 2-10 op amp design, 12-16 to 12-23 video amplifier, 3-9 op amp terminal model, 17-5 voltage divider rule, 2-3 op amp with negative feedback, 15-8 voltage feedback amp, 9-2 op amp with positive feedback, 15-8 voltage reference filter, 13-16 original amplifier, 2-5 Wein bridge oscillator, 15-12 passive twin-T filter, 16-37 Wein bridge oscillator with AGC, 15-13 passive Wien-Robinson bridge circuit, 16-39 wheatstone bridge, 12-8 phase shift oscillator, 15-14 Wien bridge oscillator, 15-9, 15-11 phase shift oscillator, buffered, 15-16 Wien-Robinson bridge circuit, 16-39 photodiode amplifier, 12-8 Wien-Robinson circuit, 16-39 phototransistor amplifier, 12-9 Circuit board layout, 17-1 to 17-30 photovoltaic cell amplifier, 12-9 component of op amp design, 17-1 quadrature oscillator, 15-19 noise sources, 17-2 reference bias, 12-13 prototype, 17-1 to 17-3 RRO output stage, 18-11 Sallen-Key band-pass filter, 16-30 Circuit Theory, op amps, 2-1 to 2-13 Sallen-Key high-pass filter, 16-24 Closed loop response, diff amp graph, 13-19 Sallen-Key high-pass filter biasing, 16-50 Colors, noise, 10-10 to 10-13 Sallen-Key low-pass filter, 16-15 Common-mode range, op amp, 18-6 to 18-10 Sallen-Key low-pass filter biasing, 16-48 Common-mode rejection ratio, 11-15 Sallen-Key low-pass filter circuit, 16-20 saturated transistor, 2-9 Compensated attenuator, 7-16 to 7-19 second-order all-pass filter, 16-44 circuit, 7-17 second-order low-pass filter, 16-17 Compensation second-order MFB high-pass filter, 16-25 comparison of schemes, 7-20 to 7-22 second-order MFB low-pass filter, 16-18 DAC, 14-18 to 14-20 second-order Tschebyscheff low-pass filter, 16-16 dominant pole graph, 7-11 seventh-order all-pass filter, 16-46 dominant-pole, 7-9 to 7-13 simplified op amp, 11-17 external, 7-8 simultaneous equation , 4-12, 4-15, 4-17, 4-20 feedback capacitance, 8-13 to 8-15 single supply, 4-1, 4-3 gain, 7-12 to 7-14 common-mode voltage, 4-2 gain capacitance, 8-13 reference voltage input, 4-2 graph, 7-12, 7-13 single supply inverting op amp, 4-4, 4-5 internal op amp, 7-2 to 7-9 single supply noninverting, 4-7 inverting op amp, 7-15 single-ended to diff output, 13-18 lead, 7-13 to 7-17 single-pole reconstruction filter, 13-15 lead-lag, 7-18 to 7-21 single-supply DAC, 14-23 Miller effect, 7-2 to 7-4 Index-3

454 Miller effect circuit, 7-2 damping ratio vs overshoot, 5-16 noninverting op amp, 7-16 damping ratio vs phase margin, 5-16 loop gain plot, 5-13, 5-14 Complex feedback network, 3-7 to 3-9 low-pass filter, 5-9 s theorem, 3-7 to 3-9 ’ Thevenin overshoot vs damping ratio, 5-16 Control system terms, block diagram math, 5-3 phase margin vs damping ratio, 5-16 Corner frequency, 10-13 to 10-15 single supply inverting op amp, 4-5 Current booster single supply noninverting op amp, 4-7 circuit for DAC, 14-20 transfer, 4-12, 4-15, 4-18, 4-21 op amps, 14-20 Current divider rule, 2-4 D Current drain, op amp, 18-12 to 18-14 Current feedback amplifier DAC analysis, 8-1 to 8-15 AC application error budget, 14-8 bandwidth, 9-3 to 9-7 AC errors, 14-14 to 14-18 bandwidth graph, 8-10 AC loads, 14-2 Bode plot, 8-7 AC parameters, 14-14 to 14-18 circuit, 8-2 accuracy vs resolution, 14-7 compensation, 8-13 to 8-15 actuator analog interface, 18-16 to 18-21 equation comparison, 9-8 to 9-10 compensation, 14-18 to 14-20 feedback capacitance, 8-12 to 8-14 current booster circuit, 14-20 feedback capacitance Bode plot, 8-13 DC application error budget, 14-7 feedback capacitance compensation, 8-13 to DC errors, 14-10 to 14-14 8-15 DC loads, 14-1 to 14-3 feedback resistor graph, 8-10 DC offsets, 14-22 feedback resistor selection, 8-9 to 8-12 DC parameters, 14-10 to 14-14 frequency graph, 9-5 deglitch circuit, 14-17 gain capacitance compensation, 8-13 to 8-15 differential nonlinearity error, 14-12 gain graph, 8-10, 9-5 differential nonlinearity error graph, 14-13 ideal circuit, 9-3 dynamic range, 14-9 to 14-11 impedance, 9-7 to 9-9 ENOB, 14-14 input capacitance, 8-11 to 8-13 error budget, 14-6 to 14-11 inverting, 8-5 to 8-7 errors, 14-10 to 14-16 inverting circuit, 8-5 external ref circuits, 13-15 to 13-18 model, 8-1 to 8-3 external voltage reference circuit, 13-17 nonnverting, 8-3 to 8-6 gain error, 14-11 to 14-13 noninverting circuit, 8-4 gain error graph, 14-12 precision, 9-2 to 9-4 integral nonlinearity error, 14-13 to 14-15 stability, 8-11 to 8-14, 9-6 to 9-8 intermodulation distortion, 14-15 to 14-17 stability analysis, 8-7 to 8-10 interpolation filter, 13-5 stability analysis circuit, 8-2, 8-3 nonlinearity error, 14-12 to 14-15 stability equation, 8-2 to 8-4 offset error, 14-10 to 14-12 stray capacitance graph, 8-12 offset error graph, 14-11 voltage feedback amp comparison, 9-1 to 9-10 parameters, 14-10 to 14-16 Current source, 12-7 power booster, 14-22 power supply rejection ratio, 14-14 Curve R/2R, 14-4 to 14-6 band reject filter, 5-10 R/2R circuit, 14-5 bode plot of low-pass filter, 5-9 R/R2 resistor array, 14-4 Bode plot, no poles, 5-11 reconstruction filter, 13-5, 13-13 to 13-16 Bode plot, with single pole, 5-11 Index-4

455 resistor ladder, 14-2 to 14-4 low-voltage op amp designs, 18-3 to 18-6 resistor ladder circuit, 14-2 RF application budget, 14-10 E selection, 13-6 to 13-10 settling time, 14-16 to 14-18 Effective number of bits, DAC, 14-14 sigma-delta , 14-5 to 14-7 Elapsed time SINAD, 14-14 graph for TL07x, 7-5 single-supply operation, 14-22 graph for TL08x, 7-6 specifications, 14-2 to 14-6 ENOB, DAC, 14-14 spurious free dynamic range, 14-15 Equivalent input noise, parameter, 11-17 to 11-19 THD, 14-8 to 14-10, 14-14 Error sources, op amp, 18-4 types, 14-2 voltage booster circuit, 14-21, 14-22 External compensation, 7-8 weighted resistor, 14-3 to 14-5 weighted resistor circuit, 14-3 F DAC current sink, actuator interface circuit, 18-19 DAC current source, actuator interface circuit, 18-17 Fairchild Robert J. Wilder, 1-3 Damping ratio, graph, 7-4 uA709, 1-3 DC loads, DAC, 14-1 to 14-3 Feedback, 5-6 to 5-8 DC offsets, DAC, 14-22 to 14-24 Bode analysis, 5-7 to 5-13 DC parameters, DAC, 14-10 to 14-14 Wien bridge oscillator circuit, 15-12 Deglitch circuit, DAC, 14-17 Feedback capacitance Design, op amp circuit, 12-16 to 12-23 Bode plot, 8-13 Design hints, 16-47 to 16-55 compensation, 8-13 to 8-15 current feedback amp, 8-12 to 8-14 Design procedure, 12-11 to 12-13 Feedback circuit, block diagram math, 5-3 Differential amp, closed loop response, 13-19 Feedback loop, broken circuit, 6-7 Differential input voltage range, parameter, 11-11 Feedback network, complex, 3-7 to 3-9 Differential nonlinearity error, 14-12 to 14-14 Feedback resistor graph, 14-13 graph, 8-10 Differential op amp, 3-6 to 3-8, 6-8 to 6-10 selection, 8-9 to 8-12 circuit, 3-6, 6-8 Feedback system, block diagram, 6-2 common-mode input signal circuit, 3-7 Filter noise, 10-18 anti-aliasing, 13-11 to 13-13 Differential output drive, from single-ended circuit, band reject circuit, 5-9 13-18 band reject curve, 5-10 Differential voltage amplification, parameter, 11-13 capacitor selection, 16-50 to 16-53 Digital plane placement, 17-8 cascading filter stages, 16-12 Distortion, total harmonic, 11-18 to 11-20 characteristics of reconstruction, 13-14 coefficients, 16-55 to 16-63 Dominant-pole compensation, 7-9 to 7-13 component values, 16-52 graph, 7-11 DAC recontracting, 13-13 to 13-16 Dominant-pole compensation circuit, bode plot, 7-11 description, 16-1 to 16-3 Dual IF receiver, block diagram, 13-4 dual-supply circuit, 16-47 Dual supply op amp, vs single supply, 4-1 to 4-4 low pass fundamentals, 16-2 to 16-11 Dual-supply filter, circuit, 16-47 low-pass design, 16-11 to 16-21 op amp selection, 16-53 to 16-55 Dynamic range quality factor, 16-9 to 16-11 DAC, 14-9 to 14-11 Index-5

456 single-pole reconstruction circuit, 13-15 current feedback amp, 8-10 single-supply circuit, 16-47 current feedback amp bandwidth, 8-10 voltage reference circuit, 13-16 current feedback amp frequency, 9-5 current feedback amp gain, 8-10, 9-5 Filter circuit biasing, 16-47 to 16-51 DAC gain error, 14-12 First-order all-pass filter, 16-44 DAC offset error, 14-11 First-order low-pass filter, 16-12 to 16-15 damping ratio, 7-4 Flicker noise, 10-8 to 10-10 diff amp closed loop response, 13-19 Flux residue resistance, 17-20 differential nonlinearity error, 14-13 Fourth-order band-pass filter, 16-32 to 16-37 dominant-pole compensation, 7-11 elapsed time for TL07x, 7-5 Frequency elapsed time for TL08x, 7-6 corner, 10-13 to 10-15 feedback resistor, 8-10 current feedback amp graph, 9-5 filter response, 16-53 graph for TL07x, 7-5 frequency for TL07x, 7-5 graph for TL08x, 7-6 frequency for TL08x, 7-6 graph for TLV277x, 7-7 frequency for TLV277x, 7-7 voltage feedback amp graph, 9-4 frequency response for group delay, 16-43 Full-wave rectifier, 12-10 frequency response of low-pass filter, 16-4 gain compensation, 7-12 gain of Tschebyscheff low-pass filter, 16-7 G gain response band-pass filter, 16-29 Gain group delay frequency response, 16-43 current feedback amp, 9-5 high-pass filter gain response, 16-22 current feedback amp graph, 8-10 input bias current changes, 18-9 oscillator, 15-4 to 15-6 input common-mode voltage, 18-9, 18-10 voltage feedback amp, 9-4 input noise spectrum, 11-18 input output voltage changes, 18-10 Gain capacitance, compensation, 8-13 to 8-15 integral nonlinearity error, 14-13 Gain compensation, 7-12 to 7-14 intermodulation distortion, 14-16 graph, 7-12 lead compensation, 7-13 Gain error lead-lag compensation, 7-20 DAC, 14-11 to 14-13 low pass to band pass transition, 16-28 graph, 14-12 maximum output voltage swing, 11-12 Gain error budget, 12-19 noise energy, 10-2 Gate output, circuit, 17-21 op amp bandwidth, 15-7 Gaussian distribution, noise graph, 10-2 op amp frequency response, 15-6 open loop response, 16-53 Glossary, parameter, 11-2 to 11-10 oscillator output, 15-7 Graph output spectrum, 11-19 band-rejection filter, 16-41 output voltage for TL07x, 7-5 Bassel low-pass filter, 16-8, 16-9 output voltage for TL08x, 7-6 Bode plot for dominant-pole compensation, 7-11 percent overshoot, 7-4 Bode plot for lead compensation, 7-14 phase margin, 7-4 Butterworth band-pass filter gain response, 16-36 phase margin for TLV277x, 7-7 amplitude of Butterworth low-pass filter, 16-6 phase response of phase response of low-pass Butterworth low-pass filter, 16-51 filter, 16-4 comparison of active/passive band-rejection filter, phase shift, 11-21 16-41 phase shift for RC, 15-3 comparison of normalized group delay graph, 16-8 phase shift for TL03x, 7-3 Index-6

457 phase shift for TL07x, 7-5 High-speed analog input drive circuits, 13-18 to phase shift for TL08x, 7-6 13-22 power spectral density, 13-14 pulse response for TL03x, 7-3 I pulse response for TLV277x, 7-7 settling time, 11-22, 14-17 Ideal op amp spurious free dynamic range, 14-15 assumptions, 3-1 stray capacitance, 8-12 characteristics, 3-2 time response for TL07x, 7-5 circuit, 3-2 time response for TL08x, 7-6 IF receiver, block diagram, 13-4 TL03x, 7-3 Impedance TL07x, 7-5 current feedback amp, 9-7 to 9-9 TL08x, 7-6 voltage feedback amp, 9-7 to 9-9 TLC2201 noise, 10-20 Inductive vias, 17-19 to 17-21 TLV2772 noise, 10-13 Tschebyscheff low-pass filter quality factor, 16-10 Inductor, high-frequency circuit, 17-13 voltage amplification, 11-21 Input capacitance, 11-14 voltage amplification for TL03x, 7-3 current feedback amp, 8-11 to 8-13 voltage amplification for TL07x, 7-5 Input common-mode range, op amp, 18-6 to 18-8 voltage amplification for TL08x, 7-6 Input common-mode voltage range, parameter, voltage amplification for TLV277x, 7-7 11-11 voltage feedback amp frequency, 9-4 Input current, 11-10 voltage feedback amp gain, 9-4 circuit, 11-10 voltage follower frequency, 13-17 Input noise spectrum, graph, 11-18 Wein bridge oscillator, 15-11 Input offset voltage Grounds, 17-8 adjust circuit, 11-9 Group delay, frequency response graph, 16-43 circuit, 11-9 GSM cellular base station, block diagram, 13-2 parameter, 11-8 to 11-10 GSM receiver block system budget, 13-3 Input parasitic elements, 11-13 to 11-15 circuit, 11-13 input capacitance, 11-14 H input resistance, 11-14 output impedance, 11-14 High-frequency capacitor, circuit, 17-12 output impedance circuit, 11-14 High-frequency inductor, circuit, 17-13 Input resistance, 11-14 High-frequency resistor, circuit, 17-11 Instrumentation amaplifiers, 12-1 to 12-24 High-pass filter Integral nonlinearity error, DAC, 14-13 Bessel circuit, 16-27 circuit, 3-10 Intermodulation distortion, DAC, 14-15 to 14-17 design, 16-21 to 16-27 Internal compensation, op amp, 7-2 to 7-9 first order, 16-23 to 16-25 Inverting from a low-pass filter, 16-21 current feedback amp, 8-5 to 8-7 gain response graph, 16-22 current feedback amp circuit, 8-5 higher order, 16-26 to 16-28 Inverting high-pass filter, circuit, 16-23 inverting circuit, 16-23 Inverting input, stray capacitance circuit, 7-16 MFB circuit, 16-25 Inverting low-pass filter, circuit, 16-13 noninverting circuit, 16-23 Inverting op amp, 3-4 to 3-6, 6-6 to 6-8 Sallen-Key circuit, 16-24 capacitance effect, 17-18 Sallen-Key circuit with unity gain, 16-24 circuit, 3-4, 6-6 second order, 16-24 to 16-27 Index-7

458 feedback back loop broken circuit, 6-7 fourth-order RC circuit, 16-3 noise, 10-16 to 10-18 frequency graph, 16-4 single supply, 4-3 to 4-6 fundamentals, 16-2 to 16-11 single supply circuit, 4-4, 4-5 high order, 16-19 to 16-22 transfer curve, 4-5 inverting circuit, 16-13 with lead compensation, 7-15 MFB circuit, 16-18 noninverting circuit, 16-12 Noninverting with unity gain circuit, 16-14 K phase response graph, 16-4 RC first-order circuit, 16-2 Kirchoff ’ s voltage law, 2-2 to 2-4 Sallen-Key circuit, 16-15 Sallen-Key circuit with unity gain, 16-15 Sallen-Key circuit with unity gain circuit, 16-20 L second-order, 16-14 to 16-20 second-order active, 16-1 Large signal differential voltage amplification, 11-13 second-order passive, 16-1 to 16-3 ’ Law, Ohm s, 2-1 to 2-3 to a high-pass filter, 16-21 Tschebyscheff, 16-7 Lead compensation, 7-13 to 7-17 Tschebyscheff gain response graph, 16-7 Bode plot, 7-14 Tschebyscheff unity gain circuit, 16-16 graph, 7-13 Tschebyscheff quality factor graph, 16-10 inverting op amp, 7-15 unity gain circuit, 16-19 noninverting op amp, 7-16 Low-pass filter circuit, 3-10 Lead-lag compensation, 7-18 to 7-21 Bode plot, 7-19 circuit, 7-19 M closed-loop plot, 7-20 Loading capacitance, making an op amp unstable, Math, block diagram, 5-1 to 5-6 7-4 Maximum output voltage swing Logic gate output, circuit, 17-21 graph, 11-12 Long-tailed pair, circuit, 9-2 parameter, 11-12 Loop antenna, PCB layout, 17-15 MFB band-pass filter, circuit, 16-31 Loop gain, block diagram, 6-4 MFB high-pass filter biasing circuit, 16-50 Loop gain plots, 5-12 to 5-16 circuit, 16-25 curve, 5-13, 5-14 MFB low-pass filter Low-pass filter biasing circuit, 16-49 adjustable circuit, 16-17 circuit, 16-18 band-pass filter transition, 16-28 band-pass transition, 16-37 Miller effect compensation, 7-2 to 7-4 Bessel, 16-7 to 16-10 circuit, 7-2 Butterworth, 16-6 Multiloop feedback system, block diagram math, 5-4 Butterworth amplitude response graph, 16-6 Multiple feedback, low-pass filter circuit, 16-18 Butterworth circuit with unity gain, 16-21 Multiple feedback band-pass filter, 16-31 to 16-33 circuit, 5-8 circuit, 16-31 comparison of gain responses graph, 16-9 Multiple feedback high-pass filter comparison of normalized group delay graph, biasing circuit, 16-50 16-8 circuit, 16-25 comparison of phase responses graph, 16-8 Multiple feedback low-pass filter, 16-18 to 16-20 design, 16-11 to 16-21 biasing circuit, 16-49 first-order, 16-12 to 16-15 Index-8

459 Nonlinear feedback, Wien bridge oscillator circuit, N 15-12 Nonlinearity error, 14-12 to 14-15 Noise, 10-8 to 10-10 Nonrail to rail op amp, input circuit, 18-8 avalanche, 10-9 to 10-11 Norton s theorem, 2-5 ’ broadband, 11-18 burst, 10-9 O characterization, 10-1 colors, 10-10 to 10-13 Offset budget, 12-19 corner, 10-12 to 10-14 Offset error differential op amp, 10-18 DAC, 14-10 to 14-12 equivalent input, 11-17 to 11-19 graph, 14-11 floor, 10-3 s Law, 2-1 to 2-3 ’ Ohm Gaussian distribution graph, 10-2 applied to a component, 2-2 graph, 11-18 applied to the total circuit, 2-2 inverting op amp, 10-16 Open-loop noninverting op amp, schematic, 6-6 multiple sources, 10-3 Operational amplifier noninverting op amp, 10-17 cable, 3-9 op amp, 10-1 to 10-24 canonical equations, 6-2 to 6-5 op amp model, 10-14 to 10-16 capacitively-loaded circuit, 7-9 pink, 10-11 circuit design, 12-16 to 12-23 red/brown, 10-12 circuit theory, 2-1 to 2-13 rms vs P-P, 10-1 to 10-3 comparison, 18-20 to 18-23 shot, 10-5 to 10-8 compensated attenuator, 7-16 to 7-19 spot, 11-17 to 11-19 current booster, 14-20 thermal, 10-7 to 10-9 current drain, 18-12 to 18-14 total, 10-12 to 10-14 current feedback analysis, 8-1 to 8-15 types, 10-4 to 10-11 designing low voltage, 18-1 to 18-24 units, 10-4 differential, 3-6 to 3-8, 6-8 to 6-10 white, 10-11 dominant-pole compensation, 7-9 to 7-13 Noise corner, 10-12 to 10-14 dynamic range, 18-3 to 18-6 Noise floor, 10-3 error sources, 18-4 external compensation, 7-8 Noise model, op amp, 10-14 to 10-16 frequency graph, 15-6 Noise sources, PCBs, 17-2 gain error budget, 12-19 Noninverting, current feedback amplifier, 8-3 to 8-6 gain compensation, 7-12 to 7-14 H. W. Bode, 1-2 Noninverting high-pass filter, circuit, 16-23 Harry Black, Bell Labs, 1-1 Noninverting low-pass filter history of, 1-1 to 1-5 circuit, 16-12 impact on oscillator, 15-5 to 15-8 circuit with unity gain, 16-14 ideal, 3-1 Noninverting op amp, 3-3 to 3-5, 4-6 to 4-8, 6-5 to increasing current, 14-19 to 14-24 6-7 input common-mode range, 18-6 to 18-10 circuit, 3-3, 18-7 internal compensation, 7-2 to 7-9 noise, 10-17 inverting, 3-4 to 3-6, 6-6 to 6-8, 7-15 open-loop, schematic, 6-6 inverting circuit, 6-6 schematic, 6-5 lead compensation, 7-13 to 7-17 single supply circuit, 4-7 lead-lag compensation, 7-18 to 7-21 transfer curve, 4-7 loading capacitance, 7-4 with lead compensation, 7-16 loop gain calculation circuit, 7-9 Index-9

460 noise, 10-12 to 10-19 Wien bridge circuit, 15-11, 15-12 noise characterization, 10-1 Wien bridge graph, 15-11 noise model, 10-14 to 10-16 Output impedance, 11-14 noise of differential, 10-18 circuit, 11-14 noise of inverting, 10-16 to 10-18 Output spectrum, graph, 11-19 noise of noninverting, 10-17 Output voltage noise theory, 10-1 to 10-24 graph for TL07x, 7-5 non-ideal, 6-1 to 6-11 graph for TL08x, 7-6 noninverting, 3-3 to 3-5, 6-5 to 6-7, 7-16 Output voltage swing noninverting circuit, 18-7 graph, 11-12 noninverting, schematic, 6-5 op amp, 18-11 to 18-13 nonrail to rail input circuit, 18-8 parameter, 11-12 offset budget, 12-19 Overshoot predictions, 5-15 to 5-17 open-loop noninverting, schematic, 6-6 output voltage swing, 18-11 to 18-13 parameter glossary, 11-2 to 11-10 P parameters, 11-1 to 11-23 pinouts, 17-24 P-P noise, vs rms noise, 10-1 to 10-3 proper termination, 17-28 Parameter rail to rail input circuit, 18-8 broadband noise, 11-18 rail to rail output circuit, 18-11 common-mode rejection ratio, 11-15 rms vs P-P noise, 10-1 to 10-3 differential input voltage range, 11-11 selection, 12-15 to 12-17, 13-10 to 13-12 equivalent input noise, 11-17 to 11-19 signal-to-noise ratio, 18-5 to 18-7 glossary, 11-2 to 11-10 single-supply circuit design, 18-13 input common-mode voltage range, 11-11 split-supply circuit, 10-19 input current, 11-10 terminal model circuit, 17-5 input offset voltage, 11-8 to 11-10 TL03x, 7-3 large signal differential voltage amplification, video, 3-9 11-13 uA709, 1-3 maximum output voltage swing, 11-12 uA741, 1-3 phase margin, 11-19 to 11-22 voltage booster, 14-20 to 14-23 settling time, 11-22 slew rate, 11-16 to 11-18 Oscillation, requirements, 15-1 to 15-3 spot noise, 11-17 to 11-19 Oscillator supply current, 11-16 analysis of circuit, 15-7 to 15-9 supply voltage rejection ratio, 11-15 bubba, 15-17 to 15-19 total harmonic distortion, 11-18 to 11-20 bubba circuit, 15-17 understanding, 11-1 to 11-23 buffered phase shift circuit, 15-16 unity gain bandwidth, 11-19 to 11-22 circuits for sine wave type, 15-9 to 15-21 Parameters, DAC, 14-10 to 14-16 description of sine wave, 15-1 Parasitic elements, 11-13 to 11-15 gain, 15-4 to 15-6 circuit, 11-13 impact of active element op amp, 15-5 to 15-8 PCB design output graph, 15-7 capacitors, 17-12 to 17-14 phase shift, 15-3 to 15-5, 15-14 to 15-17 decoupling, 17-20 to 17-24 phase shift circuit, 15-14 grounding, 17-7 to 17-11 positive feedback diagram, 15-8 inductors, 17-13 quadrature, 15-18 to 15-21 input isolation, 17-23 to 17-25 quadrature circuit, 15-19 ordering of layers, 17-6 to 17-8 sine wave, 15-1 to 15-21 output isolation, 17-23 to 17-25 Wein bridge, 15-9 to 15-15 Index-10

461 packages, 17-24 to 17-27 Precision current source, 12-7 parallel signal traces, 17-19 Pulse response, graph for TLV277x, 7-7 passive component characteristics, 17-11 to 17-21 Q resistors, 17-11 surface mount, 17-27 Quadrature oscillator, 15-18 to 15-21 through-hole considerations, 17-26 to 17-29 circuit, 15-19 trace antennas, 17-14 to 17-17 Quality factor, filter, 16-9 to 16-11 trace characteristics, 17-14 trace reflections, 17-16 to 17-21 PCB layout, 17-10 R loop and slot antenna, 17-15 R/2R trace corners, 17-16 circuit for DAC, 14-5 PCB mechanical conduction, 17-3 to 17-8 DAC, 14-4 to 14-6 double sided, 17-5 resistor array, 14-4 layers, 17-4 to 17-7 Rail-to-rail op amp, input circuit, 18-8 materials, 17-3 to 17-5 multi-layer, 17-6 Rail-to-rail output op amp, output stage circuit, 18-11 single layer, 17-4 to 17-6 RC PCB traces first-order passive low pass circuit, 16-2 broadcasting from, 17-9 fourth-order pass low pass circuit, 16-3 trace to plane capacitance, 17-17 low pass filter frequency graph, 16-4 low pass phase response graph, 16-4 Percent overshoot, graph, 7-4 RC , phase shift graph, 15-3 Phase margin, 11-19 to 11-22 graph, 7-4 Reconstruction filter graph for TLV277x, 7-7 characteristics, 13-14 circuit for single pole, 13-15 Phase shift DAC, 13-13 to 13-16 buffered oscillator circuit, 15-16 graph, 11-21 Red noise, 10-12 graph for TL07x, 7-5 Reference bias circuit, 12-13 graph for TL08x, 7-6 Reference circuit oscillator, 15-3 to 15-5 ADC, 13-15 to 13-18 Phase shift oscillator DAC, 13-15 to 13-18 buffered, 15-15 to 15-17 Reference voltage characterization, 12-12 to 12-14 circuit, 15-14 Resistance, flux residue, 17-20 single amplifier, 15-14 to 15-16 Resistor Photodiode amplifier, 12-8 current feedback amp, 8-9 to 8-12 Phototransistor amplifier, 12-9 high-performance circuit, 17-11 Photovoltaic cell amplifier, 12-9 Resistor ladder Pin outs, op amp, 17-24 circuit for DAC, 14-2 Pink noise, 10-11 DAC, 14-2 to 14-4 Plane, to trace capacitance, 17-17 Resistive transducer, 12-6, 12-7 Plane placement, digital and analog, 17-8 Resonance, capacitor, 17-22 Power booster, DAC, 14-22 Ringing predictions, 5-15 to 5-17 Power supply rejection ratio, DAC, 14-14 RMS noise, vs P-P noise, 10-1 to 10-3 Precision Rule current feedback amp, 9-2 to 9-4 current divider, 2-4 voltage feedback amp, 9-2 to 9-4 voltage divider, 2-3 Index-11

462 circuit, 4-1 S inverting circuit, 4-4, 4-5 circuit analysis, 4-3 to 4-8 Sallen-Key band-pass filter, circuit, 16-30 inverting, 4-3 to 4-6 Sallen-Key high-pass filter noninverting, 4-6 to 4-8 biasing circuit, 16-50 noninverting circuit, 4-7 circuit, 16-24 transfer curve, 4-5, 4-7 circuit with unity gain, 16-24 vs dual supply, 4-1 to 4-4 Sallen-Key low-pass filter Single-ended, to diff output drive circuit, 13-18 biasing circuit, 16-48 Single-ended input drive, circuit for ADC, 13-20 circuit, 16-15 Single-supply circuit design, 18-13 circuit with unity gain, 16-15, 16-20 Single-supply filter, circuit, 16-47 Sallen-Key topology, 16-14 to 16-18, 16-24 to 16-26 Single-supply operation, DAC, 14-22 to 14-24 band-pass filter, 16-30 to 16-32 Slew rate Saturated transistor circuit, 2-9 to 2-11 at unity gain, 11-16 Schematic, 6-5 circuit, 11-16 simultaneous equation, 4-9, 4-13, 4-16, 4-19 Slot antenna, PCB layout, 17-15 Second order equation, 5-15 to 5-17 See signal-to-noise ratio SNR. Second-order all-pass filter, 16-44 to 16-46 Spans, requiring correction, 12-4 Second-order band-pass filter, 16-29 to 16-33 Spectral density, graph, 13-14 multiple feedback topology, 16-31 to 16-33 Split-supply op amp, circuit, 10-19 Sallen-Key topology, 16-30 to 16-32 Spot noise, 11-17 to 11-19 Second-order high-pass filter, 16-24 to 16-27 Spurious free dynamic range, DAC, 14-15 multiple feedback topology, 16-25 to 16-27 Stability Sallen-Key topology, 16-24 to 16-26 current feedback amp, 8-11 to 8-13, 9-6 to 9-8 Second-order low-pass filter, 16-14 to 16-20 feedback capacitance, 8-12 to 8-14 coefficients, 16-17 op amp loading capacitance, 7-4 multiple feedback topology, 16-18 to 16-20 voltage feedback amp, 9-6 to 9-8 Sallen-Key topology, 16-14 to 16-18 Stability analysis Self resonance, capacitor, 17-22 circuit, 8-2, 8-3 Settling time, 11-22 current feedback amp, 8-7 to 8-10 DAC, 14-16 to 14-18 Stability equation, 8-2 to 8-4 graph, 11-22 Stray capacitance Shot noise, 10-5 to 10-8 graph for current feedback amp, 8-12 Shutdown, op amp, 18-12 to 18-14 on inverting input circuit, 7-16 Sigma-delta, DAC, 14-5 to 14-7 Summary points, block diagram math, 5-3 Signal conditioning, for IF sampling, 13-1 to 13-23 Superposition, 2-8 to 2-10 Signal-to-noise and distortion, DAC, 14-14 Supply current, 11-16 Signal-to-noise ratio, 10-3 Supply voltage rejection ratio, 11-15 op amp, 18-5 to 18-7 System specifications, review, 12-12 Significant bits, versus binary bits, 12-2 Simultaneous equations, 4-8 to 4-19 T SINAD, DAC, 14-14 Sine wave oscillator, 15-1 to 15-21 T network circuit circuits, 15-9 to 15-21 feedback loop, 3-7 description, 15-1 s theorem, 3-8 Thevenim ’ Termination, unused op amps, 17-28 Single supply op amp, 4-1 to 4-23 Index-12

463 Theorem Transducer bias circuit, 12-13 ’ Norton s, 2-5 Transforms, block diagram math, 5-5 Superposition, 2-8 to 2-10 Transistor amplifier, 2-10 to 2-13 Thevenin s, 2-5 to 2-8 ’ Tschebyscheff low-pass filter, 16-7 Thermal noise, 10-7 to 10-9 Butterworth response graph, 16-51 Thevenin s theorem, 2-5 to 2-8 ’ gain response graph, 16-7 complex feedback networks, 3-7 to 3-9 quality factor graph, 16-10 second order, 16-16 THS3201 gain vs frequence graph, 13-21 Twin-T filter phase vs frequency graph, 13-21 active circuit, 16-38 passive circuit, 16-37 TL03x, 7-4 frequency response graph, 7-3 op amp, 7-3 U phase shift graph, 7-3 small-signal pulse response graph, 7-3 uA709, op amp, 1-3 voltage amplification graph, 7-3 uA741, op amp, 1-3 voltage follower graph, 7-3 Unity gain, 11-16 TL07x, 7-4 to 7-6 Unity gain bandwidth, 11-19 to 11-22 graph, 7-5 TL08x, 7-5 to 7-7 V graph, 7-6 TLC2201, 10-19 to 10-24 Voltage feedback amplifier; voltage feed- See VFA. circuit, 10-21, 10-23 back amplifier noise graph, 10-20 Vias, inductive, 17-19 to 17-21 TLE2027, 10-19 Video amplifier, 3-9 TLV227x, 7-6 to 7-9 Video amplifier circuit, 3-9 graphs, 7-7 Voltage amplification TLV2772, 10-12 to 10-14 graph, 11-21 noise graph, 10-13 graph for TL07x, 7-5 typical noise, 10-14 graph for TL08x, 7-6 Total harmonic distortion, 11-18 to 11-20 graph for TLV277x, 7-7 DAC, 14-8 to 14-10, 14-14 parameter, 11-13 Voltage booster, Total noise, 10-12 to 10-14 op amp, 14-20 to 14-23 Trace circuit for DAC, 14-21, 14-22 length for inverting op amp, 17-25 Voltage divider rule, 2-3 to 2-5 length for quad op amp, 17-25 parallel signal coupling, 17-19 Voltage feedback amplifier, 7-1 to 7-23 quad op amp with half-supply generator, 17-26 bandwidth, 9-3 to 9-7 to plane capacitance, 17-17 current feedback amp comparison, 9-1 to 9-10 equation comparison, 9-8 to 9-10 Trace corners, PCB layout, 17-16 frequency graph, 9-4 Transducer, 12-1 to 12-6 gain graph, 9-4 ADC analog interface, 18-13 to 18-17 impedance, 9-7 to 9-9 ADC interface circuit, 18-15, 18-16 long-tailed pair circuit, 9-2 characterization, 12-13 to 12-16 precision, 9-2 to 9-4 measurement system, 12-1 stability, 9-6 to 9-8 output voltage, 12-14 Voltage follower frequency, graph, 13-17 resistive, 12-6, 12-7 Voltage reference circuit types, 12-6 to 12-12 Index-13

464 ADC, 13-17 graph, 15-11 DAC, 13-17 Wheatstone bridge, circuit, 12-8 Voltage reference filter, circuit, 13-16 White noise, 10-11 Wien bridge circuit, 15-9 W oscillator circuit, 15-11 Wien-Robinson filter, 16-39 to 16-42 Weighted resistor active circuit, 16-39 circuit for DAC, 14-3 passive circuit, 16-39 DAC, 14-3 to 14-5 Wilder, Robert J., Fairchild, 1-3 Wein bridge oscillator, 15-9 to 15-15 Wireless communication, 13-1 to 13-23 circuit with AGC, 15-13 Wireless systems, 13-1 to 13-6 circuit with nonlinear feedback, 15-12 Index-14