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- 1. ANALYSIS AND DESIGNOF ANALOG INTEGRATEDCIRCUITSFourth EditionPAUL R. GRAYUniversity of California, BerkeleyPAUL J. HURSTUniversity of California, DavisSTEPHEN H. LEWISUniversity of California, DavisROBERT G. MEYERUniversity of California, BerkeleyJOHN WlLEY & SONS, INC.New York / Chichester / Weinheim / Brisbane /Singapore / Toronto -7 CL-,
- 2. ACQUISITIONS EDITOR William ZobristEDITORIAL ASSISTANT Susannah BarrSENIOR MARKETING MANAGER Katherine HepburnPRODUCTION SERVICES MANAGER Jeanine FurinoPRODUCTION EDITOR Sandra RussellDESIGN DIRECTOR Madelyn LesurePRODUCTION MANAGEMENT SERVICES Publication Services, Inc.Cover courtesy of Dr. Kenneth C. Dyer and Melgar Photography.This book was set in 10i12 Times Roman by Publication Services, Inc. and printed and bound byHamilton Printing Company. The cover was printed by Lehigh Press, Inc.This book was printed on acid-free paper. @Copyright 2001 O John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-mail: PERMREQ@WILEY.COM. To order books or for customer service please call 1-800-CALL-WILEY (255-5945).Library of Congress Cataloging-in-Publication DataAnalysis and design of analog integrated circuits 1Paul R. Gray. . .[et al.]. - 4th ed. p. cm. lncludes bibliographical references and index. ISBN 0-471-32 168-0 (cloth: alk. paper) 1. Linear integrated circuits-Computer-aided design. 2. Metal oxide semiconductors-Computer-aided design. 3. Bipolar transistors-Computer-aided design. I. Gray, Paul R., 1942- TK7874.A588 2000 621.38154~21 00-043583 Printed in the United States of America 10987654321
- 3. PrefaceIn the 23 years since the publication of the first edition of this book, the field df analogintegrated circuits has developed and matured. The initial groundwork was laid in bipolartechnology, followed by a rapid evolution of MOS analog integrated circuits. Further-more, BiCMOS technology (incorporating both bipolar and CMOS devices on one chip)has emerged as a serious contender to the original technologies. A key issue is that CMOStechnologies have become dominant in building digital circuits because CMOS digitalcircuits are smaller and dissipate less power than their bipolar counterparts. To reducesystem cost and power dissipation, analog and digital circuits are now often integratedtogether, providing a strong economic incentive to use CMOS-compatible analog circuits.As a result, an important question in many applications is whether to use pure CMOS or aBiCMOS technology. Although somewhat more expensive to fabricate, BiCMOS allowsthe designer to use both bipolar and MOS devices to their best advantage, and also al-lows innovative combinations of the characteristics of both devices. In addition, BiCMOScan reduce the design time by allowing direct use of many existing cells in realizing agiven analog circuit function. On the other hand, the main advantage of pure CMOS isthat it offers the lowest overall cost. Twenty years ago, CMOS technologies were only fastenough to support applications at audio frequencies. However, the continuing reduction ofthe minimum feature size in integrated-circuit (IC) technologies has greatly increased themaximum operating frequencies, and CMOS technologies have become fast enough formany new applications as a result. For example, the required bandwidth in video appli-cations is about 4 MHz, requiring bipolar technologies as recently as 15 years ago. Now,however, CMOS can easily accommodate the required bandwidth for video and is evenbeing used for radio-frequency applications. In this fourth edition, we have combined the consideration of MOS and bipolar cir-cuits into a unified treatment that also includes MOS-bipolar connections made possibleby BiCMOS technology. We have written this edition so that instructors can easily se-lect topics related to only CMOS circuits, only bipolar circuits, or a combination of both.We believe that it has become increasingly important for the analog circuit designer tohave a thorough appreciation of the similarities and differences between MOS and bipolardevices, and to be able to design with either one where this is appropriate. Since the SPICE computer analysis program is now readily available to virtuallyall electrical engineering students and professionals, we have included extensive use of ,SPICE in this edition, particularly as an integral part of many problems. We have usedcomputer analysis as it is most commonly employed in the engineering design process-both as a more accurate check on hand calculations, and also as a tool to examine complexcircuit behavior beyond the scope of hand analysis. In the problem sets, we have also in-cluded a number of open-ended design problems to expose the reader to real-world situa-tions where a whole range of circuit solutions may be found to satisfy a given performancespecification. This book is intended to be useful both as a text for students and as a reference bookfor practicing engineers. For class use, each chapter includes many worked problems; theproblem sets at the end of each chapter illustrate the practical applications of the materialin the text. All the authors have had extensive industrial experience in IC design as well
- 4. viii Preface as in the teaching of courses on this subject, and this experience is reflected in the choice of text material and in the problem sets. Although this book is concerned largely with the analysis and design of ICs, a consid- erable amount of material is also included on applications. In practice, these two subjects are closely linked, and a knowledge of both is essential for designers and users of ICs. The latter compose the larger group by far, and we believe that a working knowledge of IC design is a great advantage to an IC user. This is particularly apparent when the user mdst choose from among a number of competing designs to satisfy a particular need. An understanding of the IC structure is then useful in evaluating the relative desirability of the different designs under extremes of environment or in the presence of variations in supply voltage. In addition, the IC user is in a much better position to interpret a manufacturers data if he or she has a working knowledge of the internal operation of the integrated circuit. The contents of this book stem largely from courses on analog integrated circuits given at the University of California at the Berkeley and Davis campuses. The courses are un- dergraduate electives and first-year graduate courses. The book is structured so that it can be used as the basic text for a sequence of such courses. The more advanced mate- rial is found at the end of each chapter or in an appendix so that a first course in analog integrated circuits can omit this material without loss of continuity. An outline of each chapter is given below together with suggestions for material to be covered in such a first course. It is assumed that the course consists of three hours of lecture per week over a 15-week semester and that the students have a working knowledge of Laplace transforms and frequency-domain circuit analysis. It is also assumed that the students have had an introductory course in electronics so that they are familiar with the principles of transistor operation and with the functioning of simple analog circuits. Unless otherwise stated, each chapter requires three to four lecture hours to cover. Chapter 1 contains a summary of bipolar transistor and MOS transistor device physics. We suggest spending one week on selected topics from this chapter, the choice of topics depending on the background of the students. The material of Chapters 1 and 2 is quite important in IC design because there is significant interaction between circuit and device design, as will be seen in later chapters. A thorough understanding of the influence of device fabrication on device characteristics is essential. Chapter 2 is concerned with the technology of IC fabrication and is largely descriptive. One lecture on this material should suffice if the students are assigned to read the chapter. Chapter 3 deals with the characteristics of elementary transistor connections. The ma- terial on one-transistor amplifiers should be a review for students at the senior and gradu- ate levels and can be assigned as reading. The section on two-transistor amplifiers can be covered in about three hours, with greatest emphasis on differential pairs. The material on device mismatch effects in differential amplifiers can be covered to the extent that time allows. In Chapter 4, the important topics of current mirrors and active loads are considered. These configurations are basic building blocks in modern analog IC design, and this ma- terial should be covered in full, with the exception of the material on band-gap references and the material in the appendices. Chapter 5 is concerned with output stages and methods of delivering output power to a load. Integrated-circuit realizations of Class A, Class B, and Class AB output stages are described, as well as methods of output-stage protection. A selection of topics from this chapter should be covered. Chapter 6 deals with the design of operational amplifiers (op amps). Illustrative exam- ples of dc and ac analysis in both MOS and bipolar op amps are performed in detail, and the limitations of the basic op amps are described. The design of op amps with improved
- 5. Preface ixcharacteristics in both MOS and bipolar technologies is considered. This key chapter onamplifier design requires at least six hours. In Chapter 7, the frequency response of amplifiers is considered. The zero-value time-constant technique is introduced for the calculations of the -3-dB frequency of complexcircuits. The material of this chapter should be considered in full. Chapter 8 describes the analysis of feedback circuits. Two different types of analysisare presented: two-port and return-ratio analyses. Either approach should be covered infull with the section on voltage regulators assigned as reading. Chapter 9 deals with the frequency response and stability of feedback circuits andshould be covered up to the section on root locus. Time may not pennit a detailed discussionof root locus, but some introduction to this topic can be given. In a 15-week semester, coverage of the above material leaves about two weeks forChapters 10, 11, and 12. A selection of topics from these chapters can be chosen as follows.Chapter 10 deals with nonlinear analog circuits, and portions of this chapter up to Section10.3 could be covered in a first course. Chapter 11 is a comprehensive treatment of noisein integrated circuits, and material up to and including Section 11.4 is suitable. Chapter 12describes fully differential operational amplifiers and common-mode feedback and maybe best suited for a second course. We are grateful to the following colleagues for their suggestions for andlor eval-uation of this edition: R. Jacob Baker, Bemhard E. Boser, A. Paul Brokaw, John N.Churchill, David W. Cline, Ozan E. Erdogan, John W. Fattaruso, Weinan Gao, Edwin W.Greeneich, Alex Gros-Balthazard, Tiinde Gyurics, Ward J. Helms, Timothy H. Hu, ShafiqM. Jamal, John P. Keane, Haideh Khorramabadi, Pak-Kim Lau, Thomas W. Matthews,Krishnaswamy Nagaraj, Khalil Najafi, Borivoje NikoliC, Robert A. Pease, Lawrence T.Pileggi, Edgar Shnchez-Sinencio, Bang-Sup Song, Richard R. Spencer, Eric J. Swanson,Andrew Y. J. Szeto, Yannis P. Tsividis, Srikanth Vaidianathan, T. R. Viswanathan, Chomg-Kuang Wang, and Dong Wang. We are also grateful to Kenneth C. Dyer for allowing us touse on the cover of this book a die photograph of an integrated circuit he designed and toZoe Marlowe for her assistance with word processing. Finally, we would like to thank thepeople at Wiley and Publication Services for their efforts in producing this fourth edition. The material in this book has been greatly influenced by our association with Donald0. Pederson, and we acknowledge his contributions.Berkeley and Davis, CA, 2001 Paul R. Gray Paul J. Hurst Stephen H. Lewis Robert G. Meyer
- 6. ContentsCHAPTER 1 1.5.2 Comparison of Operating RegionsModels for Integrated-Circuit Active of Bipolar and MOS TransistorsDevices 1 45 1.5.3 Decomposition of Gate-Source1.1 Introduction 1 Voltage 471.2 Depletion Region of a pn Junction 1 1.5.4 Threshold Temperature Dependence 47 1.2.1 Depletion-Region Capacitance 5 1.5.5 MOS Device Voltage Limitations 1.2.2 JunctionBreakdown 6 481.3 Large-Signal Behavior of Bipolar 1.6 Small-Signal Models of the MOS Transistors 8 /Transistors 49 1.3.1 Large-Signal Models in the 1.6.1 Transconductance 50 Forward-Active Region 9 1.6.2 Intrinsic Gate-Source and 1.3.2 Effects of Collector Voltage on Gate-Drain Capacitance 51 Large-Signal Characteristics in the Forward-Active Region 14 1.6.4 Output Resistance 52 1.3.3 Saturation and Inverse Active Regions 16 1.6.5 Basic Small-Signal Model of the MOS Transistor 52 1.3.4 Transistor Breakdown Voltages 20 1.6.6 Body Transconductance 53 1.3.5 Dependence of Transistor Current 1.6.7 Parasitic Elements in the Gain PF on Operating Conditions Small-Signal Model 54 23 1.6.8 MOS Transistor Frequency1.4 Small-Signal Models of Bipolar Response 55 Transistors 26 1.7 Short-Channel Effects in MOS 1.4.1 Transconductance 27 Transistors 58 1.4.2 Base-Charging Capacitance 28 1.7.1 Velocity Saturation from the Horizontal Field 59 1.4.3 Input Resistance 29 1.7.2 Transconductance and Transition 1.4.4 Output Resistance 29 Frequency 63 1.4.5 Basic Small-Signal Model of the 1.7.3 Mobility Degradation from the Bipolar Transistor 30 Vertical Field 65 1.4.6 Collector-Base Resistance 30 1.8 Weak Inversion in MOS Transistors 1.4.7 Parasitic Elements in the 65 Small-Signal Model 3 1 1.8.1 Drain Current in Weak Inversion 1.4.8 Specification of Transistor 66 Frequency Response 34 1.8.2 Transconductance and Transition1.5 Large Signal Behavior of Frequency in Weak Inversion 68 Metal-Oxide-Semiconductor 1.9 Substrate Current Flow in MOS Field-Effect Transistors 38 Transistors 7 1 1.5.1 Transfer Characteristics of MOS A. 1.1 Summary of Active-Device Devices 38 Parameters 73
- 7. Contents xi CHAPTER 2 2.9.1 n-Channel Transistors 131 Bipolar, MOS, and BiCMOS Integrated-Circuit Technology 78 2.9.2 p-Channel Transistors 141 2.1 Introduction 78 2.9.3 Depletion Devices 142 2.2 Basic Processes in Integrated-Circuit 2.9.4 Bipolar Transistors 142 Fabrication 79 2.10 Passive Components in MOS 2.2.1 Electrical Resistivity of Silicon Technology 144 79 2.10.1 Resistors 144 2.2.2 Solid-State Diffusion 80 2.10.2 Capacitors in MOS Technology 2.2.3 Electrical Properties of Diffused 145 Layers 82 2.10.3 Latchup in CMOS Technology 2.2.4 Photolithography 84 148 2.2.5 Epitaxial Growth 85 2.11 BiCMOS Technology 150 2.2.6 Ion Implantation 87 2.12 Heterojunction Bipolar Transistors 152 2.2.7 Local Oxidation 87 2.13 Interconnect Delay 153 2.2.8 Polysilicon Deposition 87 2.14 Economics of Integrated-Circuit 2.3 High-Voltage Bipolar Fabrication 154 Integrated-Circuit Fabrication 88 2.14.1 Yield Considerations in 2.4 Advanced Bipolar Integrated-Circuit Integrated-Circuit Fabrication Fabrication 92 154 2.5 Active Devices in Bipolar Analog 2.14.2 Cost Considerations in Integrated Circuits 95 Integrated-Circuit Fabrication 2.5.1 Integrated-Circuit npn Transistor 157 96 2.15 Packaging Considerations for 2.5.2 Integrated-Circuit pnp Transistors Integrated Circuits 159 107 2.15.1 Maximum Power Dissipation 159 2.6 Passive Components in Bipolar 2.15.2 Reliability Considerations inI Integrated Circuits 115 2.6.1 Diffused Resistors 115 Integrated-Circuit Packaging 162 A.2.1 SPICE Model-Parameter Files 163 2.6.2 Epitaxial and Epitaxial Pinch Resistors 119 CHAPTER 3 2.6.3 Integrated-Circuit Capacitors 120 Single-Transistor and Multiple-Transistor 2.6.4 Zener Diodes 121 Amplifiers 170 2.6.5 Junction Diodes 122 3.1 Device Model Selection forI 2.7 Modifications to the Basic Bipolar Approximate Analysis of AnalogI Process 123 Circuits 171 2.7.1 Dielectric Isolation 123 3.2 Two-Port Modeling of Amplifiers 172 2.7.2 Compatible Processing for 3.3 Basic Single-Transistor Amplifier High-Performance Active Devices 124 Stages 174 2.7.3 High-Performance Passive 3.3.1 Common-Emitter Configuration Components 127 175 2.8 MOS Integrated-Circuit Fabrication 3.3.2 Common-Source Configuration 127 179 3.3.3 Common-Baseconfiguration 183 2.9 Active Devices in MOS Integrated 4 Circuits 131 3.3.4 Common-Gate Configuration 186
- 8. xii Contents 3.3.5 Common-Base and Common-Gate 3.5.6.5 Input Offset Current of Configurations with Finite r, 188 the Emitter-Coupled Pair 3.3.5.1 Common-Base and 235 Common-Gate Input 3.5.6.6 Input Offset Voltage of the Resistance 188 Source-Coupled Pair 236 3.3.5.2 Common-Base and 3.5.6.7 Offset Voltage of the Common-Gate Output Source-Coupled Pair: Ap- Resistance 190 proximate Analysis 236 3.3.6 Common-Collector Configuration 3.5.6.8 Offset Voltage Drift in the (Emitter Follower) 191 Source-Coupled Pair 238 3.3.7 Common-Drain Configuration 3.5.6.9 Small-Signal (Source Follower) 195 Characteristics of 3.3.8 Common-Emitter Amplifier with Unbalanced Differential Emitter Degeneration 197 Amplifiers 238 3.3.9 Common-Source Amplifier with A.3.1 Elementary Statistics and the Source Degeneration 200 Gaussian Distribution 2463.4 Multiple-Transistor Amplifier Stages 202 CHAPTER 4 3.4.1 The CC-CE, CC-CC, and Current Mirrors, Active Loads, and Darlington Configurations 202 References 253 3.4.2 The Cascode Configuration 206 4.1 Introduction 253 3.4.2.1 The Bipolar Cascode 206 4.2 Current Mirrors 253 3.4.2.2 The MOS Cascode 208 4.2.1 General Properties 253 3.4.3 The Active Cascode 211 4.2.2 Simple Current Mirror 255 3.4.4 The Super Source Follower 213 4.2.2.1 Bipolar 2553.5 Differential Pairs 215 4.2.2.2 MOS 257 3.5.1 The dc Transfer Characteristic of 4.2.3 Simple Current Mirror with Beta an Emitter-Coupled Pair 2 15 Helper 260 3.5.2 The dc Transfer Characteristic with 4.2.3.1 Bipolar 260 Emitter Degeneration 2 17 4.2.3.2 MOS 262 3.5.3 The dc Transfer Characteristic of a 4.2.4 Simple Current Mirror with Source-CoupledPair 2 18 Degeneration 262 3.5.4 Introduction to the Small-Signal 4.2.4.1 Bipolar 262 Analysis of Differential Amplifiers 22 1 4.2.4.2 MOS 263 3.5.5 Small-Signal Characteristics of 4.2.5 Cascode Current Mirror 263 Balanced Differential Amplifiers 224 4.2.5.1 Bipolar 263 3.5.6 Device Mismatch Effects in 4.2.5.2 MOS 266 Differential Amplifiers 23 1 4.2.6 Wilson Current Mirror 274 3.5.6.1 Input Offset Voltage and Current 23 1 4.2.6.1 Bipolar 274 3.5.6.2 Input Offset Voltage of 4.2.6.2 MOS 277 the Emitter-Coupled Pair 232 4.3 Active Loads 278 3.5.6.3 Offset Voltage of the 4.3.1 Motivation 278 Emitter-Coupled Pair: 4.3.2 Common-Emitter/Common-Source ~~~roximate Analysis Amplifier with Complementary 232 Load 279 3.5.6.4 Offset Voltage Drift in 4.3.3 Common-Emitter/Common-Source the Emitter-Coupled Pair Amplifier with Depletion Load 234 282
- 9. Contents xiii 4.3.4 Common-Emitter/Common-Source 5.2 The Emitter Follower As an Output Amplifier with Diode-Connected Stage 344 Load 284 5.2.1 Transfer Characteristics of the 4.3.5 Differential Pair with Emitter-Follower 344 Current-Mirror Load 287 4.3.5.1 Large-Signal Analysis 5.2.2 Power Output and Efficiency 347 287 5.2.3 Emitter-Follower Drive 4.3.5.2 Small-Signal Analysis Requirements 354 288 5.2.4 Small-Signal Properties of the 4.3.5.3 Common-Mode Rejection Emitter Follower 355 Ratio 293 5.3 The Source Follower As an Output4.4 Voltage and Current References 299 Stage 356 4.4.1 Low-Current Biasing 299 5.3.1 Transfer Characteristics of the 4.4.1.1 Bipolar Widlar Current Source Follower 356 Source 299 5.3.2 Distortion in the Source Follower 4.4.1.2 MOS Widlar Current 358 Source 302 4.4.1.3 Bipolar Peaking Current 5.4 Class B Push-Pull Output Stage 362 Source 303 5.4.1 Transfer Characteristic of the 4.4.1.4 MOS Peaking Current Class B Stage 363 Source 304 5.4.2 Power Output and Efficiency of the 4.4.2 Supply-Insensitive Biasing 306 Class B Stage 365 4.4.2.1 Widlar Current Sources 5.4.3 Practical Realizations of Class B 306 Complementary Output Stages 4.4.2.2 Current Sources Using 369 Other Voltage Standards 5.4.4 All-npn Class B Output Stage 307 376 4.4.2.3 Self Biasing 309 5.4.5 Quasi-Complementary Output 4.4.3 Temperature-Insensitive Biasing Stages 379 317 5.4.6 Overload Protection 3 80 4.4.3.1 Band-Gap-Referenced Bias Circuits in Bipolar 5.5 CMOS Class AB Output Stages 382 Technology 3 17 5.5.1 Common-Drain Configuration 4.4.3.2 Band-Gap-Referenced 3 83 Bias Circuits in CMOS 5.5.2 Common-Source Configuration Technology 323 with Error Amplifiers 384A.4.1 Matching Considerations in Current 5.5.3 Alternative Configurations 391 Mirrors 327 5.5.3.1 Combined Common-Drain Common-Source A.4.1.1 Bipolar 327 Configuration 39 1 A.4.1.2 MOS 329 5.5.3.2 Combined Common-Drain Common-SourceA.4.2 Input Offset Voltage of Configuration with High Differential Pair with Swing 393 Active Load 332 5 S.3.3 Parallel Common-Source A.4.2.1 Bipolar 332 Configuration 394 A.4.2.2 MOS 334 CHAPTER 6 Operational Amplifiers with CHAPTER 5 Single-Ended Outputs 404 Output Stages 344 6.1 Applications of Operational Amplifiers 5.1 Introduction 344 405
- 10. xiv Contents 6.6 MOS Folded-Cascode Operational 6.1.1 Basic Feedback Concepts 405 Amplifiers 446 6.1.2 Inverting Amplifier 406 6.7 MOS Active-Cascode Operational 6.1.3 Noninverting Amplifier 408 Amplifiers 450 6.1.4 Differential Amplifier 408 6.8 Bipolar Operational Amplifiers 453 6.1.5 Nonlinear Analog Operations 409 6.8.1 The dc Analysis of the 741 6.1.6 Integrator, Differentiator 4 10 Operational Amplifier 456 6.1.7 Internal Amplifiers 41 1 6.8.2 Small-Signal Analysis of the 741 6.1.7.1 Switched-Capacitor Operational Amplifier 461 Amplifier 411 6.8.3 Input Offset Voltage, Input 6.1.7.2 Switched-Capacitor Offset Current, and Integrator 416 Common-Mode Rejection Ratio of the 741 4706.2 Deviations from Ideality in Real Oper- 6.9 Design Considerations for Bipolar ational Amplifiers 419 Monolithic Operational Amplifiers 6.2.1 Input Bias Current 419 472 6.2.2 Input Offset Current 420 6.9.1 Design of Low-Drift Operational 6.2.3 Input Offset Voltage 421 Amplifiers 474 / -,/6.2.4 Common-Mode Input Range 421 6.9.2 Design of Low-Input-Current Operational Amplifiers 476 $.2.5 Common-Mode Rejection Ratio / (CMRR) 421 / $2.6 Power-Supply Rejection Ratio CHAPTER 7 I 1/ (PSRR) 422 Frequency Response of Integrated 6.2.7 Input Resistance 424 Circuits 488 6.2.8 Output Resistance 424 7.1 Introduction 488 6.2.9 Frequency Response 424 7.2 Single-Stage Amplifiers 488 6.2.10 Operational-Amplifier Equivalent 7.2.1 Single-Stage Voltage Amplifiers Circuit 424 and The Miller Effect 4886.3 Basic Two-Stage MOS Operational 7.2.1.1 The Bipolar Differential Amplifiers 425 Amplifier: Differential- Mode Gain 493 6.3.1 Input Resistance, Output 7.2.1.2 The MOS Differential Resistance, and Open-Circuit Amplifier: Differential- Voltage Gain 426 Mode Gain 496 6.3.2 Output Swing 428 7.2.2 Frequency Response of the Common-Mode Gain for a 6.3.3 Input Offset Voltage 428 Differential Amplifier 499 6.3.4 Common-Mode Rejection Ratio 7.2.3 Frequency Response of Voltage 431 Buffers 502 6.3.5 Common-Mode Input Range 432 7.2.3.1 Frequency Response of the 6.3.6 Power-Supply Rejection Ratio Emitter Follower 503 (PSRR) 434 7.2.3.2 Frequency Response of the 6.3.7 Effect of Overdrive Voltages 439 Source Follower 509 7.2.4 Frequency Response of Current 6.3.8 Layout Considerations 439 Buffers 511 6.4 Two-Stage MOS Operational 7.2.4.1 Common-Base-Amplifier Amplifiers with Cascodes 442 Frequency Response 5 14 7.2.4.2 Common-Gate-Amplifier 6.5 MOS Telescopic-Cascode Operational Frequency Response 5 15 Amplifiers 444
- 11. Contents xv7.3 Multistage Amplifier Frequency 8.6.2 Local Shunt Feedback 591 Response 516 8.7 The Voltage Regulator as a Feedback 7.3.1 Dominant-Pole Approximation Circuit 593 5 16 7.3.2 Zero-Value Time Constant 8.8 Feedback Circuit Analysis Using Analysis 517 Return Ratio 599 7.3.3 Cascode Voltage-Amplifier 8.8.1 Closed-Loop Gain Using Return Frequency Response 522 Ratio 601 7.3.4 Cascode Frequency Response 8.8.2 Closed-Loop Impedance Formula 525 Using Return Ratio 607 7.3.5 Frequency Response of a Current 8.8.3 Summary-Return-Ratio Analysis Mirror Loading a Differential Pair 612 532 8.9 Modeling Input and Output Ports in 7.3.6 Short-Circuit Time Constants 533 Feedback Circuits 6 137.4 Analysis of the Frequency Response of the 741 Op Amp 537 CHAPTER 9 7.4.1 High-Frequency Equivalent Circuit Frequency Response and Stability of of the741 537 Feedback Amplifiers 624 7.4.2 Calculation of the -3-dB Frequency of the 741 538 9.1 Introduction 624 7.4.3 Nondominant Poles of the 741 9.2 Relation Between Gain and 540 Bandwidth in Feedback Amplifiers7.5 Relation Between Frequency 624 Response and Time Response 542 9.3 Instability and the Nyquist Criterion 626CHAPTER 8 9.4 Compensation 633Feedback 553 9.4.1 Theory of Compensation 6338.1 Ideal Feedback Equation 553 9.4.2 Methods of Compensation 6378.2 Gain Sensitivity 555 9.4.3 Two-Stage MOS Amplifier8.3 Effect of Negative Feedback on Compensation 644 Distortion 555 9.4.4 Compensation of Single-Stage8.4 Feedback Configurations 557 CMOS OP Amps 652 8.4.1 Series-Shunt Feedback 557 9.4.5 Nested Miller Compensation 656 8.4.2 Shunt-Shunt Feedback 560 9.5 Root-Locus Techniques 664 8.4.3 Shunt-SeriesFeedback 561 9.5.1 Root Locus for a Three-Pole 8.4.4 Series-Series Feedback 562 Transfer Function 664 9.5.2 Rules for Root-Locus Construction8.5 Practical Configurations and the Effect 667 of Loading 563 9.5.3 Root Locus for Dominant-Pole 8.5.1 Shunt-Shunt Feedback 563 Compensation 675 8.5.2 Series-Series Feedback 569 9.5.4 Root Locus for Feedback-Zero Compensation 676 8.5.3 Series-Shunt Feedback 579 9.6 Slew Rate 680 8.5.4 Shunt-Series Feedback 583 9.6.1 Origin of Slew-Rate Limitations 8.5.5 Summary 587 6808.6 Single-Stage Feedback 587 9.6.2 Methods of Improving Slew-Rate 8.6.1 Local Series Feedback 587 684
- 12. xvi Contents 9.6.3 Improving Slew-Rate in Bipolar 11.2.1 Shot Noise 748 Op Amps 685 9.6.4 Improving Slew-Rate in MOS Op 11.2.2 Thermal Noise 752 Amps 686 11.2.3 Flicker Noise (11f Noise) 753 9.6.5 Effect of Slew-Rate Limitations on 11.2.4 Burst Noise (Popcorn Noise) 754 Large-Signal Sinusoidal Performance 690 11.2.5 Avalanche Noise 755A.9.1 Analysis in Terms of Return-Ratio 11.3 Noise Models of Integrated-Circuit Parameters 69 1 Components 756 11.3.1 Junction Diode 756A.9.2 Roots of a Quadratic Equation 692 11.3.2 Bipolar Transistor 757 11.3.3 MOS Transistor 758CHAPTER 10 11.3.4 Resistors 759Nonlinear Analog Circuits 702 11.3.5 Capacitors and Inductors 75910.1 Introduction 70210.2 Precision Rectification 702 11.4 Circuit Noise Calculations 760 11-4.1 Bipolar Transistor Noise10.3 Analog Multipliers Employing the Performance 762 Bipolar Transistor 708 11.4.2 Equivalent Input Noise and the 10.3.1 The Emitter-Coupled Pair as a Minimum Detectable Signal 766 Simple Multiplier 708 11.5 Equivalent Input Noise Generators 10.3.2 The dc Analysis of the Gilbert 768 Multiplier Cell 7 10 11.5.1 Bipolar Transistor Noise 10.3.3 The Gilbert Cell as an Analog Generators 768 Multiplier 7 12 11.5.2 MOS Transistor Noise Generators 10.3.4 A Complete Analog Multiplier 773 715 11.6 Effect of Feedback on Noise 10.3.5 The Gilbert Multiplier Cell as a Performance 776 Balanced Modulator and Phase Dectector 7 16 11.6.1 Effect of Ideal Feedback on Noise Performance 77610.4 Phase-Locked Loops (PLL) 720 11.6.2 Effect of Practical Feedback on 10.4.1 Phase-Locked Loop Concepts Noise Performance 776 720 11.7 Noise Performance of Other Transistor 10.4.2 The Phase-Locked Loop in the Configurations 7 83 Locked Condition 722 10.4.3 Integrated-Circuit Phase-Locked 11.7.1 Common-Base Stage Noise Loops 731 Performance 783 10.4.4 Analysis of the 560B Monolithic 11.7.2 Emitter-Follower Noise Phase-Locked Loop 735 Performance 784 11.7.3 Differential-Pair Noise10.5 Nonlinear Function Symbols 743 Performance 785 11.8 Noise in Operational Amplifiers 788CHAPTER 1 1 11.9 Noise Bandwidth 794Noise in Integrated Circuits 748 11.10 Noise Figure and Noise Temperature11.1 Introduction 748 79911.2 Sources of Noise 748 11.lo. 1 Noise Figure 799 11.10.2 Noise Temperature 802
- 13. Contents xvii CHAPTER 12 12.5.3 CMFB Using Transistors in the Fully DifferentialOperational Amplifiers Triode Region 830 808 12.5.4 Switched-CapacitorCMFB 832 12.1 Introduction 808 12.6 Fully Differential Op Amps 835 12.2 Properties of Fully Differential 12.6.1 A Fully Differential Two-Stage Op Amplifiers 808 Amp 835 12.3 Small-Signal Models for Balanced 12.6.2 Fully Differential Telescopic Differential Amplifiers 811 Cascode Op Amp 845 12.6.3 Fully Differential Folded-CascodeI 12.4 Common-Mode Feedback 8 16 Op Amp 846 12.4.1 Common-Mode Feedback at Low 12.6.4 A Differential Op Amp with Two Frequencies 8 17 Differential Input Stages 847 12.4.2 Stability and Compensation 12.6.5 Neutralization 849 Considerations in a CMFB Loop 822 12.7. Unbalanced Fully Differential Circuits 850 12.5 CMFB Circuits 823 12.5.1 CMFB Using Resistive Divider 12.8 Bandwidth of the CMFB Loop 856 and Amplifier 824 12.5.2 CMFB Using Two Differential Index 865 Pairs 828
- 14. xviii Symbol Convention Symbol Convention Unless otherwise stated, the following symbol convention is used in this book. Bias or dc quantities, such as transistor collector current Ic and collector-emitter voltage V C Eare , represented by uppercase symbols with uppercase subscripts. Small-signal quantities, such as the incremental change in transistor collector current i,, are represented by lowercase symbols with lowercase subscripts. Elements such as transconductance g, in small-signal equivalent circuits are represented in the same way. Finally, quantities such as total collector current I,, which represent the sum of the bias quantity and the signal quantity, are represented by an uppercase symbol with a lowercase subscript.
- 15. Modelsfor Integrated-CircuitActive Devices1.1 IntroductionThe analysis and design of integrated circuits depend heavily on the utilization of suitablemodels for integrated-circuit components. This is true in hand analysis, where fairly simplemodels are generally used, and in computer analysis, where more complex models areencountered. Since any analysis is only as accurate as the model used, it is essential thatthe circuit designer have a thorough understanding of the origin of the models commonlyutilized and the degree of approximation involved in each. This chapter deals with the derivation of large-signal and small-signal models forintegrated-circuit devices. The treatment begins with a consideration of the properties ofpn junctions, which are basic parts of most integrated-circuit elements. Since this book isprimarily concerned with circuit analysis and design, no attempt has been made to producea comprehensive treatment of semiconductor physics. The emphasis is on summarizing thebasic aspects of semiconductor-device behavior and indicating how these can be modeledby equivalent circuits.1.2 Depletion Region of a pn JunctionThe properties of reverse-biased pn junctions have an important influence on the charac-teristics of many integrated-circuit components. For example, reverse-biased p n junctionsexist between many integrated-circuit elements and the underlying substrate, and thesejunctions all contribute voltage-dependent parasitic capacitances. In addition, a numberof important characteristics of active devices, such as breakdown voltage and output re-sistance, depend directly on the properties of the depletion region of a reverse-biased pnjunction. Finally, the basic operation of the junction field-effect transistor is controlled bythe width of the depletion region of a p n junction. Because of its importance and applica-tion to many different problems, an analysis of the depletion region of a reverse-biased p njunction is considered below. The properties of forward-biased pn junctions are treated in Section 1.3 when bipolar-transistor operation is described. Consider a pn junction under reverse bias as shown in Fig. 1.1. Assume constantdoping densities of ND atoms/cm3 in the n-type material and NA atoms/cm3 in the p-type material. (The characteristics of junctions with nonconstant doping densities will be described later.) Due to the difference in carrier concentrations in the p-type and n-typeregions, there exists a region at the junction where the mobile holes and electrons havebeen removed, leaving the fixed acceptor and donor ions. Each acceptor atom carries anegative charge and each donor atom carries a positive charge, so that the region near thejunction is one of significant space charge and resulting high electric field. This is called
- 16. 2 Chapter 1 Models for Integrated-Circuit Active Devices VR ),( Applied external Charge density p t reverse bias I Potential 4 I I Figure 1.1 The abrupt junc- tion under reverse bias V R .( a ) Schematic. (b)Charge density. (c) Electric field. (d) Electro- static potential. the depletion region or space-charge region. It is assumed that the edges of the depletion region are sharply defined as shown in Fig. 1.1, and this is a good approximation in most cases. For zero applied bias, there exists a voltage $0 across the junction called the built-in potential. This potential opposes the diffusion of mobile holes and electrons across the junction in equilibrium and has a value1 where the quantity ni is the intrinsic carrier concentration in a pure sample of the semiconductor and ni = 1.5 X 1 0 ~ c m -at 300°K for silicon. ~ In Fig. 1.1 the built-in potential is augmented by the applied reverse bias, V R ,and the + total voltage across the junction is ($0 V R ) . the depletion region penetrates a distance If W1 into the p-type region and Wz into the n-type region, then we require because the total charge per unit area on either side of the junction must be equal in mag- nitude but opposite in sign.
- 17. 1.2 Depletion Region of a pn Junction 3 Poissons equation in one dimension requires that =2 v- - = - q N ~ for d - wl < x < o dx2 E Ewhere p is the charge density, q is the electron charge (1.6 X 10-l9 coulomb), and E is thepermittivity of the silicon (1.04 X 10-l2 faradcm). The permittivity is often expressed aswhere Ks is the dielectric constant of silicon and € 0 is the permittivity of free space (8.86 Xl o w 4 Flcm). Integration of (1.3) giveswhere C1 is a constant. However, the electric field % is given bySince there is zero electric field outside the depletion region, a boundary condition is . % =0 for x = -W1and use of this condition in (1.6) gives 5 dx for -Thus the dipole of charge existing at the junction gives rise to an electric field that varieslinearly with distance. Integration of (1.7) givesIf the zero for potential is arbitrarily taken to be the potential of the neutral p-type region,then a second boundary condition is V = O for x = -W1and use of this in (1.8) givesAt x = 0, we define V = V 1 ,and then (1.9) givesIf the potential difference from x = 0 to x = W 2 is V2,then it follows thatand thus the total voltage across the junction is
- 18. 4 Chapter 1 . Models for Integrated-CircuitActive Devices Substitution of (1.2) in (1.12) gives From (1.13), the penetration of the depletion layer into the p-type region is Similarly Equations 1.14 and 1.15 show that the depletion regions extend into the p-type and n-type regions in inverse relation to the impurity concentrations and in proportion to d m . If either No or N A is much larger than the other, the depletion region exists almost entirely in the lightly doped region. EXAMPLE An abrupt pn junction in silicon has doping densities N A = 1015 atoms/cm3 and ND = 1016 atoms/cm3. Calculate the junction built-in potential, the depletion-layer depths, and the maximum field with 10 V reverse bias. From (1.1) From (1.14) the depletion-layer depth in the p-type region is = 3.5 p m (where 1 p m = 1 micrometer = m) The depletion-layer depth in the more heavily doped n-type region is Finally, from (1.7) the maximum field that occurs for x = 0 is Note the large magnitude of this electric field.
- 19. 1.2 Depletion Region of a pn Junction 51.2.1 Depletion-Region CapacitanceSince there is a voltage-dependent charge Q associated with the depletion region, we cancalculate a small-signal capacitance C, as follows:Nowwhere A is the cross-sectional area of the junction. Differentiation of (1.14) givesUse of (1.17) and (1.18) in (1.16) gives The above equation was derived for the case of reverse bias VR applied to the diode.However, it is valid for positive bias voltages as long as the forward current flow is small.Thus, if VD represents the bias on the junction (positive for forward bias, negative forreverse bias), then (1.19) can be written aswhere Cjo is the value of Cj for VD = 0. Equations 1.20 and 1.21 were derived using the assumption of constant doping inthe p-type and n-type regions. However, many practical diffused junctions more closelyapproach a graded doping profile as shown in Fig. 1.2. In this case a similar calculationyieldsNote that both (1.21) and (1.22) predict values of Cj approaching infinity as VD ap-proaches $0. However, the current flow in the diode is then appreciable and the equationsno longer valid. A more exact analysis2v3of the behavior of Cj as a function of VD givesthe result shown in Fig. 1.3. For forward bias voltages up to about $012, the values of Cjpredicted by (1.21) are very close to the more accurate value. As an approximation, somecomputer programs approximate Cj for VD > $012 by a linear extrapolation of (1.21) or(1.22).
- 20. 6 Chapter 1 rn Models for Integrated-Circuit Active Devices Charge density p p=m . - . I FX Distance ,.-- ..-I .I Figure 1.2 Charge density versus dis- tance in a graded junction. Reverse bias 1 Foward bias Figure 1.3 Behavior of p n junction depletion-layer capacitance C j as a function of bias voltage V D .rn EXAMPLE If the zero-bias capacitance of a diffused junction is 3 pF and $o = 0.5 V, calculate the capacitance with 10 V reverse bias. Assume the doping profile can be approximated by an abrupt junction. From (1.21) 2 1.2.2 Junction Breakdown From Fig. 1.lc it can be seen that the maximum electric field in the depletion region occurs at the junction, and for an abrupt junction (1.7) yields a value
- 21. 1.2 Depletion Region of a pn Junction 7Substitution of (1.14) in (1.23) gives I- max - [ ~ ~ N A N D 1V" R E (NA + No)where I!,~ has been neglected. Equation 1.24 shows that the maximum field increases asthe doping density increases and the reverse bias increases. Although useful for indicat-ing the functional dependence of %,, on other variables, this equation is strictly validfor an ideal plane junction only. Practical junctions tend to have edge effects that causesomewhat higher values of %,, due to a concentration of the field at the curved edgesof the junction. Any reverse-biased pn junction has a small reverse current flow due to the presenceof minority-carrier holes and electrons in the vicinity of the depletion region. These areswept across the depletion region by the field and contribute to the leakage current of thejunction. As the reverse bias on the junction is increased, the maximum field increases andthe carriers acquire increasing amounts of energy between lattice collisions in the depletionregion. At a critical field %,., the carriers traversing the depletion region acquire sufficientenergy to create new hole-electron pairs in collisions with silicon atoms. This is called theavalanche process and leads to a sudden increase in the reverse-bias leakage current sincethe newly created carriers are also capable of producing avalanche. The value of ,% is ,, about 3 X lo5 V/cm for junction doping densities in the range of 1015 to 1016 atoms/cm3,but it increases slowly as the doping density increases and reaches about lo6 V/cm for doping densities of 1018 atoms/cm3. A typical I-V characteristic for a junction diode is shown in Fig. 1.4, and the effect of avalanche breakdown is seen by the large increase in reverse current, which occurs as the reverse bias approaches the breakdown voltage BV. This corresponds to the maximum field %,, approaching %". It has been found empirically4 that if the normal reverse bias ,, current of the diode is IR with no avalanche effect, then the actual reverse current near thebreakdown voltage is Figure 1.4 Typical I-V characteristic of a junction diode showing avalanche breakdown.
- 22. 8 Chapter 1 . Models for Integrated-CircuitActive Devices where M is the multiplication factor defined by In this equation, V R is the reverse bias on the diode and n has a value between 3 and 6. The operation of a pn junction in the breakdown region is not inherently destruc- tive. However, the avalanche current flow must be limited by external resistors in order to prevent excessive power dissipation from occurring at the junction and causing dam- age to the device. Diodes operated in the avalanche region are widely used as voltage references and are called Zener diodes. There is another, related process called Zener breakdown, which is different from the avalanche breakdown described above. Zener breakdown occurs only in very heavily doped junctions where the electric field becomes large enough (even with small reverse-bias voltages) to strip electrons away from the valence bonds. This process is called tunneling, and there is no multiplication effect as in avalanche breakdown. Although the Zener breakdown mechanism is important only for breakdown voltages below about 6 V, all breakdown diodes are commonly referred to as Zener diodes. The calculations so far have been concerned with the breakdown characteristic of plane abrupt junctions. Practical diffused junctions differ in some respects from these results and the characteristics of these junctions have been calculated and tabulated for use by designers. In particular, edge effects in practical diffused junctions can result in breakdown voltages as much as 50 percent below the value calculated for a plane junction. EXAMPLE An abrupt plane pn junction has doping densities NA = 5 X 10 atoms/cm3 and ND = 1016 atoms/cm3. Calculate the breakdown voltage if %,*, = 3 X 10 Vlcm. The breakdown voltage is calculated using, % , , = % in (1.24) to give * , 1.3 Large-Signal Behavior of Bipolar Transistors In this section, the large-signal or dc behavior of bipolar transistors is considered. Large- signal models are developed for the calculation of total currents and voltages in transistor circuits, and such effects as breakdown voltage limitations, which are usually not included in models, are also considered. Second-order effects, such as current-gain variation with collector current and Early voltage, can be important in many circuits and are treated in detail. The sign conventions used for bipolar transistor currents and voltages are shown in Fig. 1.5. All bias currents for both npn and pnp transistors are assumed positive going into the device.
- 23. 1.3 Large-Signal Behavior of Bipolar Transistors 9 -A VBE E Figure 1.5 Bipolar transistor sign PnP convention.1.3.1 Large-SignalModels in the Forward-ActiveRegionA typical npn planar bipolar transistor structure is shown in Fig. 1.6a, where collector,base, and emitter are labeled C , B, and E, respectively. The method of fabricating suchtransistor structures is described in Chapter 2. It is shown there that the impurity dopingdensity in the base and the emitter of such a transistor is not constant but varies withdistance from the top surface. However, many of the characteristics of such a device canbe predicted by analyzing the idealized transistor structure shown in Fig. 1.6b. In thisstructure the base and emitter doping densities are assumed constant, and this is sometimescalled a uniform-base transistor. Where possible in the following analyses, the equationsfor the uniform-base analysis are expressed in a form that applies also to nonuniform-basetransistors. A cross section AA is taken through the device of Fig. 1.6b and carrier concentrationsalong this section are plotted in Fig. 1 . 6 ~Hole concentrations are denoted by p and elec- .tron concentrations by n with subscripts p or n representing p-type or n-type regions. Then-type emitter and collector regions are distinguished by subscripts E and C, respectively.The carrier concentrations shown in Fig. 1 . 6 ~ apply to a device in the forward-active re-gion. That is, the base-emitter junction is forward biased and the base-collector junction isreverse biased. The minority-carrier concentrations in the base at the edges of the depletionregions can be calculated from a Boltzmann approximation to the Fermi-Dirac distributionfunction to give6 VBE np(0) = npoexp - VT np(WB) = npoexp - VBC 0 VT -where WB is the width of the base from the base-emitter depletion layer edge to the base-collector depletion layer edge and n,, is the equilibrium concentration of electrons in thebase. Note that VBC is negative for an npn transistor in the forward-active region andthus n,(WB) is very small. Low-level injection conditions are assumed in the derivation of(1.27) and (1.28). This means that the minority-carrier concentrations are always assumedmuch smaller than the majority-carrier concentration. If recombination of holes and electrons in the base is small, it can be shown that7the minority-carrier concentration np(x) in the base varies linearly with distance. Thus astraight line can be drawn joining the concentrations at x = 0 and x = WB in Fig. 1 . 6 ~ . For charge neutrality in the base, it is necessary that
- 24. 10 Chapter 1 a Models for Integrated-Circuit Active Devices , Carrier concentration ! I Depletion region II I A f e x A Y=o Emitter Base Collector (4 Figure 1.6 (a) Cross section of a typical npn planar bipolar transistor structure. (b) Idealized tran- sistor structure. (c) Carrier concentrations along the cross section AA of the transistor in (b). Uni- form doping densities are assumed. (Not to scale.) and thus where p,(x) is the hole concentration in the base and NA is the base doping density that is assumed constant. Equation 1.30 indicates that the hole and electron concentrations are separated by a constant amount and thus p p ( x ) also varies linearly with distance. Collector current is produced by minority-carrier electrons in the base diffusing in the direction of the concentration gradient and being swept across the collector-base depletion region by the field existing there. The diffusion current density due to electrons in the base is
- 25. 1.3 Large-Signal Behavior of Bipolar Transistors 11where D, is the diffusion constant for electrons. From Fig. 1 . 6 ~If Ic is the collector current and is taken as positive flowing into the collector, it followsfrom (1.32) thatwhere A is the cross-sectional area of the emitter. Substitution of (1.27) into (1.33) gives qAD n VBE zc = exp - WB VT = VBE Is exp - VTwhereand Is is a constant used to describe the transfer characteristic of the transistor in theforward-active region. Equation 1.36 can be expressed in terms of the base doping densityby noting thats (see Chapter 2 )and substitution of (1.37) in (1.36) giveswhere QB = WBNA is the number of doping atoms in the base per unit area of theemitter and ni is the intrinsic carrier concentration in silicon. In this form (1.38) appliesto both uniform- and nonuniform-base transistors and D, has been replaced by D,,which is an average effective value for the electron diffusion constant in the base. Thisis necessary for nonuniform-base devices because the diffusion constant is a functionof impurity concentration. Typical values of Is as given by (1.38) are from 10-l4 to10-l6 A. Equation 1.35 gives the collector current as a function of base-emitter voltage. Thebase current IB is also an important parameter and, at moderate current levels, consists oftwo major components. One of these ( I B l )represents recombination of holes and electronsin the base and is proportional to the minority-carrier charge Qe in the base. From Fig.1.6c, the minority-carrier charge in the base isand we have
- 26. 12 Chapter 1 4 Models for Integrated-CircuitActive Devices where rb is the minority-carrier lifetime in the base. IB1 represents a flow of majority holes from the base lead into the base region. Substitution of (1.27) in (1.40) gives 1 n,oW~qA VBE IBl = - exp - 2 Tb VT The second major component of base current (usually the dominant one in integrated- circuit npn devices) is due to injection of holes from the base into the emitter. This current component depends on the gradient of minority-carrier holes in the emitter and is9 where D, is the diffusion constant for holes and L, is the diffusion length (assumed small) for holes in the emitter. pnE(0) is the concentration of holes in the emitter at the edge of the depletion region and is If ND is the donor atom concentration in the emitter (assumed constant), then The emitter is deliberately doped much more heavily than the base, making ND large and p n ~small, so that the base-current component, IB2,is minimized. o Substitution of (1.43) and (1.44) in (1.42) gives qAD n2 VBE IB2 -- exp - = The total base current, IB, is the sum of ZB1 and IB2: Although this equation was derived assuming uniform base and emitter doping, it gives the correct functional dependence of IB device parameters for practical double-diffused on nonuniform-base devices. Second-order components of ZB, which are important at low current levels, are considered later. Since Ic in (1.35) and IBin (1.46) are both proportional to exp(VBE/VT) this anal- in ysis, the base current can be expressed in terms of collector current as where PF is the forward current gain. An expression for PF can be calculated by substi- tuting (1.34) and (1.46) in (1.47) to give where (1.37) has been substituted for n,,. Equation 1.48 shows that PF is maximized by minimizing the base width WB and maximizing the ratio of emitter to base doping
- 27. 1.3 Large-Signal Behavior of Bipolar Transistors 13densities NDINA.Typical values of PF for npn transistors in integrated circuits are 50 to500, whereas lateral p n p transistors (to be described in Chapter 2) have values 10 to 100.Finally, the emitter current iswhere The value of ( Y F can be expressed in terms of device parameters by substituting (1.48)in (1.50) to obtainwhereThe validity of (1.51) depends on WiI27bDn < 1 and (Dp/Dn)(WB/L,)(NA/N~) < 1, < <and this is always true if PF is large [see (1.48)]. The term y in (1.51) is called the emitterinjection efJiciency and is equal to the ratio of the electron current (npn transistor) injectedinto the base from the emitter to the total hole and electron current crossing the base-emitterjunction. Ideally y + 1, and this is achieved by making NDINAlarge and WB small. Inthat case very little reverse injection occurs from base to emitter. The term a~ in (1.51) is called the base transport factor and represents the fraction ofcarriers injected into the base (from the emitter) that reach the collector. Ideally cur + 1and this is achieved by making WB small. It is evident from the above development thatfabrication changes that cause a r and y to approach unity also maximize the value of PFof the transistor. The results derived above allow formulation of a large-signal model of the transis- tor suitable for bias-circuit calculations with devices in the forward-active region. One such circuit is shown in Fig. 1.7 and consists of a base-emitter diode to model (1.46) and a controlled collector-current generator to model (1.47). Note that the collector volt- age ideally has no influence on the collector current and the collector node acts as ahigh-impedance current source. A simpler version of this equivalent circuit, which isoften useful, is shown in Fig. 1.7b, where the input diode has been replaced by a bat- tery with a value VBE(,,,,),which is usually 0.6 to 0.7 V. This represents the fact that inthe forward-active region the base-emitter voltage varies very little because of the steep slope of the exponential characteristic. In some circuits the temperature coefficient of VBE(on) important, and a typical value for this is -2 mVPC. The equivalent circuits of is Fig. 1.7 apply for npn transistors. For p n p devices the corresponding equivalent circuits are shown in Fig. 1.8.
- 28. 14 Chapter 1 . Models for Integrated-CircuitActive Devices Figure 1.7 Large-signal models of npn transistors for use in bias calcula- E E tions. (a) Circuit incor- 1s VBE porating an input diode. zB = - exp - (b) Simplified circuit PF VT with an input voltage (b) (a) source. Figure 1.8 Large-signal models of pnp transistors corresponding to the (b) circuits of Fig. 1.7. 1.3.2 Effects of Collector Voltage on Large-Signal Characteristics in the Forward-Active Region In the analysis of the previous section, the collector-base junction was assumed reverse biased and ideally had no effect on the collector currents. This is a useful approximation for first-order calculations, but is not strictly true in practice. There are occasions where the influence of collector voltage on collector current is important, and this will now be investigated. The collector voltage has a dramatic effect on the collector current in two regions of de- vice operation. These are the saturation (VCE approaches zero) and breakdown (VCE very large) regions that will be considered later. For values of collector-emittervoltage VCE be- tween these extremes, the collector current increases slowly as VCE increases. The reason for this can be seen from Fig. 1.9, which is a sketch of the minority-carrier concentration in the base of the transistor. Consider the effect of changes in VCE on the carrier concen- tration for constant VBE. Since VBE is constant, the change in VcB equals the change in VCE and this causes an increase in the collector-base depletion-layer width as shown. The change in the base width of the transistor, AWB, equals the change in the depletion-layer width and causes an increase AIc in the collector current. From (1.35) and (1.38) we have q ~ ~ , n : VBE Ic = - - exp - - - QB VT Differentiation of (1.52) yields arc - - - - - qADnn; ~VCE Qi ( ) VBE ~ Q B exp - - VT ~ V C E and substitution of (1.52) in (1.53) gives
- 29. 1.3 Large-Signal Behavior of Bipolar Transistors 15 Carrier concentration A I I Collector depletion I I I Figure 1.9 Effect of in- DX Emitter I Base creases in VCEon the collector depletion re- gion and base width of a bipolar transistor.For a uniform-base transistor QB = W E N A and (1.54) becomes ,Note that since the base width decreases as VCE increases, d WBldVcEin (1.55) is negativeand thus dIcldVCE is positive. The magnitude of d WBldVcEcan be calculated from (1.18)for a uniform-base transistor. This equation predicts that dWB/dVcE is a function of thebias value of V C E , the variation is typically small for a reverse-biased junction and butdWsldVcE is often assumed constant. The resulting predictions agree adequately withexperimental results. Equation 1.55 shows that dIcldVcs is proportional to the collector-bias current andinversely proportional to the transistor base width. Thus narrow-base transistors showa greater dependence of Ic on VCEin the forward-active region. The dependence ofdIcldVcE on Ic results in typical transistor output characteristics as shown in Fig. 1.10.In accordance with the assumptions made in the foregoing analysis, these characteristicsare shown for constant values of VBE. However, in most integrated-circuit transistors thebase current is dependent only on VBEand not on V C E ,and thus constant-base-currentcharacteristics can often be used in the following calculation. The reason for this is thatthe base current is usually dominated by the IB2 component of (1.45), which has no de-pendence on VCE. Extrapolation of the characteristics of Fig. 1.10 back to the VCEaxisgives an intercept V Acalled the Early voltage, whereSubstitution of (1.55) in (1.56) giveswhich is a constant, independent of Ic. Thus all the characteristics extrapolate to the samepoint on the VCEaxis. The variation of Ic with VCEis called the Early effect, and V A isa common model parameter for circuit-analysis computer programs. Typical values of V A
- 30. 16 Chapter 1 rn Models for integrated-Circuit Active Devices Figure 1.10 Bipolar transistor output characteristics showing the Early voltage, V A . for integrated-circuit transistors are 15 to 100 V. The inclusion of Early effect in dc bias calculations is usually limited to computer analysis because of the complexity introduced into the calculation. However, the influence of the Early effect is often dominant in small- signal calculations for high-gain circuits and this point will be considered later. Finally, the influence of Early effect on the transistor large-signal characteristics in the forward-active region can be represented approximately by modifying (1.35) to Ic = ( + v:,.) 7: Is 1 - exp - This is a common means of representing the device output characteristics for computer simulation. 1.3.3 Saturation and Inverse-Active Regions Saturation is a region of device operation that is usually avoided in analog circuits because the transistor gain is very low in this region. Saturation is much more commonly encoun- tered in digital circuits, where it provides a well-specified output voltage that represents a logic state. In saturation, both emitter-base and collector-base junctions are forward biased. Con- sequently, the collector-emitter voltage V C Eis quite small and is usually in the range 0.05 to 0.3 V.The carrier concentrations in a saturated npn transistor with uniform base doping are shown in Fig. 1.11. The minority-carrier concentration in the base at the edge of the depletion region is again given by (1.28) as but since VBCis now positive, the value of n p ( W B ) no longer negligible. Consequently, is changes in VCEwith VBEheld constant (which cause equal changes in V B C directly affect ) np(We).Since the collector current is proportional to the slope of the minority-carrier con- centration in the base [see (1.3 I ) ] ,it is also proportional to [ n p ( 0 - nP(WB)] ) from Fig. 1.11. Thus changes in n p ( W B )directly affect the collector current, and the collector node of the transistor appears to have a low impedance. As V C Eis decreased in saturation with V B E held constant, VBCincreases, as does n p ( W B from (1.59).Thus from Fig. 1.1 1 the collector )
- 31. 1.3 Large-Signal Behavior o f Bipolar Transistors 17 nnE j Carrier concentration l I P,(x) , I ! ! I Pnc P~E ,X I Emitter Base Collector ! Figure 1.1 1 Carrier concentrations in a saturated npn transistor. (Not to scale.)current decreases because the slope of the carrier concentration decreases. This gives riseto the saturation region of the Ic - VCEcharacteristic shown in Fig. 1.12. The slope ofthe Ic - VCE characteristic in this region is largely determined by the resistance in serieswith the collector lead due to the finite resistivity of the n-type collector material. A usefulmodel for the transistor in this region is shown in Fig. 1.13 and consists of a fixed voltagesource to represent VB,qon), a fixed voltage source to represent the collector-emitter andvoltage VCqsat). more accurate but more complex model includes a resistor in series Awith the collector. This resistor can have a value ranging from 20 to 500 a, depending onthe device structure. An additional aspect of transistor behavior in the saturation region is apparent fromFig. 1.11. For a given collector current, there is now a much larger amount of stored chargein the base than there is in the forward-active region. Thus the base-current contributionrepresented by (1.41) will be larger in saturation. In addition, since the collector-base junc-tion is now forward biased, there is a new base-current component due to injection ofcarriers from the base to the collector. These two effects result in a base current IB in sat-uration, which is larger than in the forward-active region for a given collector current Ic.Ratio Ic/IB in saturation is often referred to as the forced P and is always less than PF.As the forced / is made lower with respect to PF, the device is said to be more heavily 3saturated. The minority-carrier concentration in saturation shown in Fig. 1.11 is a straight linejoining the two end points, assuming that recombination is small. This can be representedas a linear superposition of the two dotted distributions as shown. The justification for thisis that the terminal currents depend linearly on the concentrations np(0) and np(WB).Thispicture of device carrier concentrations can be used to derive some general equations de-scribing transistor behavior. Each of the distributions in Fig. 1.11 is considered separatelyand the two contributions are combined. The emitter current that would result from npl (x)above is given by the classical diode equation ( ? ) : IEF = -IES exp - - 1where IESis a constant that is often referred to as the saturation current of the junction (noconnection with the transistor saturation previously described). Equation 1.60 predicts thatthe junction current is given by IEF = IESwith a reverse-bias voltage applied. However,
- 32. 18 Chapter 1 rn Models for Integrated-Circuit Active Devices Figure 1.12 Typical Ic-VcE characteristics for an npn bipolar transistor. Note the different scales for positive and negative currents and voltages. V B(on) ~ Vet. (sat) 0 - 0 E E (a) nPn (b) Figure 1.13 Large-signal models for bipolar transistors in the saturation region. in practice (1.60) is applicable only in the forward-bias region, since second-order effects dominate under reverse-bias conditions and typically result in a junction current several orders of magnitude larger than IES. The junction current that flows under reverse-bias conditions is often called the leakage current of the junction. Returning to Fig. 1.11, we can describe the collector current resulting from nP2(x) alone as ICR = -ICS ( exp - - 1 7 ) : where Ics is a constant. The total collector current Ic is given by ICR plus the fraction of IEF that reaches the collector (allowing for recombination and reverse emitter injection). Thus (?," ) Ic = a F I E s exp- - 1 - (7: Ics exp- - ) 1
- 33. 1.3 Large-Signal Behavior of Bipolar Transistors 19where a~ has been defined previously by (1.51). Similarly, the total emitter current iscomposed of IEF plus the fraction of ICR that reaches the emitter with the transistor actingin an inverted mode. Thus IE = (7:) ( " V " , " ) -IES exp-+ - 1 CYRICSexp- - 1where ( Y R is the ratio of emitter to collector current with the transistor operating inverted(i.e., with the collector-base junction forward biased and emitting carriers into the baseand the emitter-base junction reverse biased and collecting carriers). Typical values of f f Rare 0.5 to 0.8. An inverse current gain PR is also definedand has typical values 1 to 5. This is the current gain of the transistor when operatedinverted and is much lower than PF because the device geometry and doping densitiesare designed to maximize PF. The inverse-active region of device operation occurs forV C E negative in an n p n transistor and is shown in Fig. 1.12. In order to display thesecharacteristics adequately in the same figure as the forward-active region, the negativevoltage and current scales have been expanded. The inverse-active mode of operation israrely encountered in analog circuits. Equations 1.62 and 1.63 describe n p n transistor operation in the saturation regionwhen V B E and VBC are both positive, and also in the forward-active and inverse-activeregions. These equations are the Ebers-Moll equations. In the forward-active region, theydegenerate into a form similar to that of (1.33, (1.47), and (1.49) derived earlier. This canbe shown by putting V B E positive and VBc negative in (1.62) and (1.63) to obtain IE = ( ? ) : -IES exp- - 1 - ffRIcsEquation 1.65 is similar in form to (1.35) except that leakage currents that were previ-ously neglected have now been included. This minor difference is significant only at hightemperatures or very low operating currents. Comparison of (1.65) with (1.35) allows usto identify Is = a F I E S ,and it can be shown1° in general thatwhere this expression represents a reciprocity condition. Use of (1.67) in (1.62) and (1.63)allows the Ebers-Moll equations to be expressed in the general form ( ; : Ic = Is exp - - 1 - ) - ieXp - I)This form is often used for computer representation of transistor large-signal behavior. The effect of leakage currents mentioned above can be further illustrated as follows.In the forward-active region, from (1.66) ( 2 ) IES exp - - 1 = -IE - QRICS
- 34. 20 Chapter 1 .Models for Integrated-Circuit Active Devices Substitution of (1.68) in (1.65) gives where Ico = Ics(1 - ~ R ~ F I (1.69a) and Ico is the collector-base leakage current with the emitter open. Although Ico is given theoretically by (1.69a), in practice, surface leakage effects dominate when the collector- base junction is reverse biased and Ico is typically several orders of magnitude larger than the value given by (1.69a). However, (1.69) is still valid if the appropriate measured value for Ico is used. Typical values of Ico are from 10-lo to 10-l2 A at 25OC, and the magnitude doubles about every 8°C. As a consequence, these leakage terms can become very significant at high temperatures. For example, consider the base current IB. From Fig. 1.5 this is r, = -(lc + I E ) (1.70) If IE is calculated from (1.69) and substituted in (1.70), the result is But from (1SO) and use of (1.72) in (1.7 1) gives Since the two terms in (1.73) have opposite signs, the effect of Ico is to decrease the magnitude of the external base current at a given value of collector current. EXAMPLE If Ico is 10-lo A at 24"C, estimate its value at 120°C. Assuming that Ico doubles every 8"C, we have IC0(12O0C) = 10-lo x 212 = 0.4 pA 1.3.4 Transistor Breakdown Voltages In Section 1.2.2 the mechanism of avalanche breakdown in a pn junction was described. Similar effects occur at the base-emitter and base-collector junctions of a transistor and these effects limit the maximum voltages that can be applied to the device. First consider a transistor in the common-base configuration shown in Fig. 1 . 1 4 ~and supplied with a constant emitter current. Typical Ic - V C Bcharacteristics for an npn tran- sistor in such a connection are shown in Fig. 1.14b. For IE = 0 the collector-base junction breaks down at a voltage BVcBo, which represents collector-base breakdown with the emitter open. For finite values of IE, the effects of avalanche multiplication are apparent for values of VcB below BVcBo. In the example shown, the effective common-base current
- 35. 1.3 Large-Signal Behavior of Bipolar Transistors 21 A I,= 0 I v~~ 0 20 40 60 80 1 100 "Olts Figure 1.14 Common-base I I transistor connection. (a) Bvc~o Test circuit. (b) Zc - V C B (b) characteristics.gain a F = IC/IEbecomes larger than unity for values of VCBabove about 60 V. Operationin this region (but below BVcso) can, however, be safely undertaken if the device powerdissipation is not excessive. The considerations of Section 1.2.2 apply to this situation, andneglecting leakage currents, we can calculate the collector current in Fig. 1 . 1 4 as ~where M is defined by (1.26) and thusOne further point to note about the common-base characteristics of Fig. 1.14b is that forlow values of VcB where avalanche effects are negligible, the curves show very little of theEarly effect seen in the common-emitter characteristics. Base widening still occurs in thisconfiguration as VcB is increased, but unlike the common-emitter connection, it produceslittle change in Ic. This is because IEis now fixed instead of VBEor IB, and in Fig. 1.9,this means the slope of the minority-carrier concentration at the emitter edge of the baseis fixed. Thus the collector current remains almost unchanged. Now consider the effect of avalanche breakdown on the common-emitter characteris-tics of the device. Typical characteristics are shown in Fig. 1.12, and breakdown occurs ata value BVcEo, which is sometimes called the sustaining voltage LVcEo. As in previouscases, operation near the breakdown voltage is destructive to the device only if the current(and thus the power dissipation) becomes excessive. The effects of avalanche breakdown on the common-emitter characteristics are morecomplex than in the common-base configuration. This is because hole-electron pairs areproduced by the avalanche process and the holes are swept into the base, where they ef-fectively contribute to the base current. In a sense the avalanche current is then amplijied
- 36. 22 Chapter 1 .Models for Integrated-CircuitActive Devices by the transistor. The base current is still given by IB = -(Ic + IE) Equation 1.74 still holds, and substitution of this in (1.76) gives where 1 M = Equation 1.77 shows that Ic approaches infinity as M a F approaches unity. That is, the effective p approaches infinity because of the additional base-current contribution from the avalanche process itself. The value of BVcEo can be determined by solving If we assume that VcB = VCE, gives this and this results in and thus Equation 1.81 shows that BVcEo is less than BVcBo by a substantial factor. However, the value of BVcBo, which must be used in (1.81), is the plane junction breakdown of the collector-base junction, neglecting any edge effects. This is because it is only collector- base avalanche current actually under the emitter that is amplified as described in the pre- vious calculation. However, as explained in Section 1.2.2, the measured value of BVcBo is usually determined by avalanche in the curved region of the collector, which is remote from the active base. Consequently, for typical values of PF = 100 and n = 4, the value of BVcEo is about one-half of the measured BVcBo and not 30 percent as (1.81) would indicate. Equation 1.8 1 explains the shape of the breakdown characteristics of Fig. 1.12 if the dependence of PF on collector current is included. As VCEis increased from zero with IB = 0, the initial collector current is approximately PFICofrom (1.73);since Ico is typ- ically several picoamperes, the collector current is very small. As explained in the next section, PF is small at low currents, and thus from (1.81) the breakdown voltage is high. However, as avalanche breakdown begins in the device, the value of Ic increases and thus PF increases. From (1.81) this causes a decrease in the breakdown voltage and the characteristic bends back as shown in Fig. 1.12 and exhibits a negative slope. At higher collector currents, PF approaches a constant value and the breakdown curve with IB = 0 becomes perpendicular to the VCEaxis. The value of V C Ein this region of the curve is
- 37. 1.3 Large-SignalBehavior of Bipolar Transistors 23 usually defined to be BVcEo, since this is the maximum voltage the device can sustain. The value of PF to be used to calculate BVcEo in (1.81) is thus the peak value of PF. Note from (1.81) that high-/3 transistors will thus have low values of BVcEo. The base-emitter junction of a transistor is also subject to avalanche breakdown. How- ever, the doping density in the emitter is made very large to ensure a high value of PF[ND is made large in (1.45) to reduce IB2].Thus the base is the more lightly doped side of the junction and determines the breakdown characteristic. This can be contrasted with the collector-base junction, where the collector is the more lightly doped side and results in typical values of BVcBo of 20 to 80 V or more. The base is typically an order of magni- tude more heavily doped than the collector, and thus the base-emitter breakdown voltage is much less than BVcBo and is typically about 6 to 8 V. This is designed BVEBO. The breakdown voltage for inverse-active operation shown in Fig. 1.12 is approximately equal to this value because the base-emitter junction is reverse biased in this mode of operation. The base-emitter breakdown voltage of 6 to 8 V provides a convenient reference volt- age in integrated-circuit design, and this is often utilized in the form of a Zener diode. However, care must be taken to ensure that all other transistors in a circuit are protected against reverse base-emitter voltages sufficient to cause breakdown. This is because, un- like collector-base breakdown, base-emitter breakdown is damaging to the device. It can cause a large degradation in PF, depending on the duration of the breakdown-current flow and its magnitude. If the device is used purely as a Zener diode, this is of no consequence, but if the device is an amplifying transistor, the PF degradation may be serious.w EXAMPLE If the collector doping density in a transistor is 2 X 1015 atoms/cm3 and is much less than the base doping, calculate BVcEo for P = 100 and n = 4. Assume = 3 X lo5 V/cm. The plane breakdown voltage in the collector can be calculated from (1.24) using Zmax = gait: Since ND < N A , we have < From (1.81) 1.3.5 Dependence of Transistor Current Gain P,con Operating Conditions Although most first-order analyses of integrated circuits make the assumption that PF is constant, this parameter does in fact depend on the operating conditions of the transistor. It was shown in Section 1.3.2, for example, that increasing the value of VCE increases Ic while producing little change in le,and thus the effective PF of the transistor increases. In Section 1.3.4 it was shown that as VCE approaches the breakdown voltage, BVcEo, the collector current increases due to avalanche multiplication in the collector. Equation 1.77 shows that the effective current gain approaches infinity as VCEapproaches BVcEo.
- 38. 24 Chapter 1 rn Models for Integrated-Circuit Active Devices In addition to the effects just described, PF also varies with both temperature and transistor collector current. This is illustrated in Fig. 1.15, which shows typical curves of PF versus Ic at three different temperatures for an npn integrated circuit transistor. It is evident that PF increases as temperature increases, and a typical temperature coefficient for PF is +7000 ppmI0C (where ppm signifies parts per million). This temperature de- pendence of PF is due to the effect of the extremely high doping density in the emitter,12 which causes the emitter injection efficiency y to increase with temperature. The variation of PF with collector current, which is apparent in Fig. 1.15, can be divided into three regions. Region I is the low-current region, where PF decreases as Ic decreases. Region I1 is the midcurrent region, where PF is approximately constant. Region 111 is the high-current region, where PF decreases as Ic increases. The reasons for this behavior of PF with IC can be better appreciated by plotting base current IB and collector current Ic on a log scale as a function of V B E . This is shown in Fig. 1.16, and because of the log scale on the vertical axis, the value of In PF can be obtained directly as the distance between the two curves. At moderate current levels represented by region I1 in Figs. 1.15 and 1.16, both Ic and IB follow the ideal behavior, and VBE Ic =Is exp - VT IB - Is exp -VBE PFM VT where PFM is the maximum value of PF and is given by (1.48). At low current levels, Ic still follows the ideal relationship of (1.82), and the decrease in PF is due to an additional component in IB, which is mainly due to recombination of carriers in the base-emitter depletion region and is present at any current level. However, at higher current levels the base current given by (1.83) dominates, and this additional component has little effect. The base current resulting from recombination in the depletion region is5 VBE IBX = Isx exp - ~ V T where PF t Region I I Region 11 I Region 1 1 1 Figure 1.15 Typical curves of PI. versus Zc for an npn integrated- 0 1 1 - circuit transistor with 6 pm2 0.1 pA 1 pA 10 pA 100 pA 1 mA 10 mA emitter area.
- 39. 1.3 Large-Signal Behavior of Bipolar Transistors 25 Figure 1.16 Base and collector currents of a bipolar transistor plotted on a log scale versus VBEon a linear scale. The distance between I * "BE the curves is a direct (linearscale) measure of In p F .At very low collector currents, where (1.84) dominates the base current, the current gaincan be calculated from (1.82) and (1.84) asSubstitution of (13 2 ) in (1.85) givesIf rn = 2, then (13 6 ) indicates that PF is proportional to &at very low collector currents. At high current levels, the base current IB tends to follow the relationship of (1.83),and the decrease in PF in region I11 is due mainly to a decrease in Ic below the valuegiven by (1 22). (In practice the measured curve of IB versus VBEin Fig. 1.16 may alsodeviate from a straight line at high currents due to the influence of voltage drop across thebase resistance.) The decrease in Ic is due partly to the effect of high-level injection, andat high current levels the collector current approaches7 VBE Ic = ISHexp - ~ V TThe current gain in this region can be calculated from (13 7 ) and (13 3 ) asSubstitution of (13 7 ) in (13 8 ) givesThus PF decreases rapidly at high collector currents.
- 40. 26 Chapter 1 .Models for Integrated-Circuit Active Devices In addition to the effect of high-level injection, the value of PF at high currents is also decreased by the onset of the Kirk effect,13 which occurs when the minority-carrier con- centration in the collector becomes comparable to the donor-atom doping density. The base region of the transistor then stretches out into the collector and becomes greatly enlarged. 1.4 Small-Signal Models of Bipolar Transistors Analog circuits often operate with signal levels that are small compared to the bias currents and voltages in the circuit. In these circumstance

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