Electronic Devices and Circuits MILLMAN & HALKIAS INTERNATIONAL STUDENT EDI" McGRAW-HILL ELECTRICAL AND ELECTRONIC ENGINEERING SERIES Frederick Emmons Terman, Consulting Editor W. W. Harmon and J. G. Truxal, Associate Consulting Edno. Ahrendt and Savant ■ Servomechanism Practice Angelo • Electronic Circuits Aselline • Transform Method In Linear System Analyst* Atwater - Introduction to Microwave Theory Bailer and Gauli • Alternating -current Machinery Beranek ■ Acoustics Bracewefi • The Fourier Traniform and Iti Applications Brenner and iavid ■ Analysis of Electric Circuits Brown • Analysts of Llneor Time -invariant Systems Brum and Saunders • Analysis of Feedback Control Systems Caga • Theory and Application of Industrial Electronics Cauer • Synthesis of Linear Communication Networks Cften ■ The Analysis of Linear Systems Chen ■ Linear Network Design and Synthesis Chirlian ■ Analysis and Design of Electronic Circuits Chirtian and Zemanian ■ Electronics Clement and John ton ■ Electrical Engineering Science Cote and Oafces • Linear Vacuum-tube and Transistor Circuits Cuccio ■ Harmonics, Sidebands, and Transients In Communication Engineering Cunningham • Introduction to Nonlinear Analysis D'Azzo and Haupis ■ Feedback Control System Analysis and Synthesis Eastman ■ Fundamentals of Vacuum Tubes Elgerd • Control Systems Theory Eveleigh - Adaptive Control and Optimization Techniques Feinttein • Foundations of Information Theory Fitzgerald, Higginbotham, and Grabel ■ Basic Electrical Engineering Fitzgerald and Kingtley • Electric Mochinery Frank ■ Electrical Measurement Analysis Friedland, Wing, and Ash - Principles of Linear Networks Gebmtkh and Hammond ■ Electromechanical Systems Ghausi • Principles and Design of Linear Active Circuits Ghote • Microwave Circuit Theory and Analysis Greiner ■ Semiconductor Devices and Applications Hammond • Electrical Engineering Hancock ■ An Introduction to the Principles of Communication Theory Happell and Hettetberth -Engineering Electronics Mormon ■ Fundamentals of Electronic Motion Harmon ■ Principles of the Statistical Theory of Communication Harmon and lytic - Electrical and Mechanical Networks Harrington • Introduction to Electromagnetic Engineering Harrington - Time-harmonic Electromagnetic Fields Hayashi • Nonlinear Oscillations In Physical Systems Hayf • Engineering Electromagnetics Hoyt and Kemmerly • Engineering Circuit Analysis Hill ■ Electronics In Engineering JoWd and Brenner ■ Analysis, Transmission, and Filtering of Signals Jovid ond Brown • Field Analysis and Electromagnetics Johnson • Transmission Lines and Networks Koenig and Blackwell ■ Electromechanical System Theory Koenig, Tokad, and Kesavan • Analysis of Discrete Physical Systems Kraus • Antennas Kraut - Electromagnetics Kuh and Pederson > Principles of Circuit Synthesis Kvo - Linear Networks and Systems Ledley • Digital Computer and Control Engineering LePage • Analysis of Alternating-current Circuits LePoge • Complex Variables ond the Loplace Transform for Engineering LePage ond Seely • General Network Analysis Levi and Panzer ■ Electromechanical Power Conversion Ley, Lutz, and Rehberg - Linear Circuit Analysis Linvitl and Gibbons ■ Transistors and Active Circuits lit tatter ■ Pulse Electronics Lynch and Truxal • Introductory System Analysis Lynch and Truxal ■ Principles of Electronic Instrumentation Lynch and Truxal - Signals and Systems In Electrical Engineering McCfuskey • Introduction to the Theory of Switching Circuits Manning • Electrical Circuits Meftef • Principles of Electromechanical-energy Conversion Millman • Vacuum-tube and Semiconductor Electronics Millman and Hatktat • Electronic Devices ond Circuits Millman and Seely ■ Electronics Millman and Taub ■ Pulse and Digital Circuits MMmm and Taub ■ Pulse, Digital, and Switching Waveforms Mishkm and Bravn ■ Adaptive Control Systems Moore • Traveling-wave Engineering Nonovofi - An Introduction to Semiconductor Electronics Peltit ■ Electronic Switching, Timing, and Pulse Circuits Petti* ond MeWhorfer • Electronic Amplifier Circuits Pfeiffer ■ Concepts of Probability Theory Pfetffer ■ Linear Systems Analysis ftezo ■ An Introduction to Information Theory Rezo ond Seely • Modern Network Analysis Rogers ■ Introduction to Electric Fields fiuifon and Bordogna • Electric Networks: Functions, Filters, Analysis Ryder ■ Engineering Electronics Schwartz • Information Transmission, Modulation, and Noise Schwarz and Friedland ■ Linear Systems Seely • Electromechanical Energy Conversion Seely ■ Electron-tube Circuits Seely - Electronic Engineering Seely ■ Introduction to Electromagnetic Fields Seely ■ Radio Electronics Seifert and Sfeeg - Control Systems Engineering Sitkirid • Direct-current Machinery Sfcilh'ng ■ Electric Transmission Lines Sfcilfing • Transient Electric Currents Spangenberg ■ Fundamentals of Electron Devices Spang enberg ■ Vacuum Tubes Stevenson ■ Elements of Power System Analysis Stewart - Fundamentals of Signal Theory Sforer ■ Passive Network Synthesis Strauss - Wave Generation and Shaping So ■ Aetfve Network Synthesis Terman • Electronic and Radio Engineering TVrman and Pettit • Electronic Measurements Thaler • Elements of Servomeehanlsm Theory Thaler and Brown • Analysis and Design of Feedback Control Systems Thaler and Pastel ■ Analysis and Design of Nonlinear Feedback Control Systems Thompson • Alternating-current and Transient Circuit Analysis Tou - Digltol and Sampled-data Control Systems Tou - Modem Control Theory Trvxal ■ Automatic Feedback Control System Synthesis Turtle ■ Electric Networks: Analysis and Synthesis Vatdet • The Physical Theory of Transistors Van Model • Electromagnetic Fields Weinberg • Network Analysis ond Synthesis Williams and Young ■ Electrical Engineering Problems ELECTRONIC DEVICES AND CIRCUITS Jacob Millman, Ph.D. Professor of Electrical Engineering Columbia University Christos C. Halkias, Ph.D. Associate Professor of Electrical Engineering Columbia University INTERNATIONAL STUDENT EDITION McGRAW-HILL BOOK COMPANY* New York St. Louis San Francisco Diisseldorf London Mexico Panama Sydney Toronto KOGAKUSHA COMPANY, LTD. Tokyo ELECTRONIC DEVICES AND CIRCUITS INTERNATIONAL STUDENT EDITION Exclusive rights by Kogokusha Co., Ltd., for manufacture and export from Japan. This book cannot be re-exported from the country to which it it coniigned by Kogakusha Co., Ltd., or by McGraw-Hill Book Company or any of iti subsidiaries. XI Copyright © 1967 by McGraw-Hill, Inc. All Rights Re- served. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any meant, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permis- sion of the publisher. Library of Congress Catalog Card Number o7-16934 TOSHO J'HINTINQ CO., LTD., TOKYO, JAPAN PREFACE This book, intended as a text for a first course in electronics for elec- trical engineering or physics students, has two primary objectives: to present a clear, consistent picture of the internal physical behavior of many electronic devices, and to teach the reader how to analyze and design electronic circuits using these devices. Only through a study of physical electronics, particularly solid- state science, can the usefulness of a device be appreciated and its limitations be understood. From such a physical study, it is possible to deduce the external characteristics of each device. This charac- terization allows us to exploit the device as a circuit element and to determine its large-signal (nonlinear) behavior. A small-signal (linear) model is also obtained for each device, and analyses of many circuits using these models are given. The approach is to consider a circuit first on a physical basis, in order to provide a clear under- standing and intuitive feeling for its behavior. Only after obtaining such a qualitative insight into the circuit is mathematics (through simple differential equations) used to express quantitative relationships. Methods of analysis and features which are common to many different devices and circuits are emphasized. For example, Kirch- hoff's, Thevenin's, Norton's, and Miller's theorems are utilized through- out the text. The concepts of the load line and the bias curve are used to establish the quiescent operating conditions in many different circuits. Calculations of input and output impedances, as well as current and voltage gains, using small-signal models, are made for a wide variety of amplifiers. A great deal of attention is paid to the effects of feedback on input and output resistance, nonlinear distortion, frequency response, and the stabilization of voltage or current gains of the various devices and circuits studied. la order that the student appreciate the different applications of these circuits, the basic building blocks (such as untuned amplifiers, power amplifiers, feedback amplifiers, oscillators, and power suppliers) are discussed in detail. For the most part, real (commercially available) device charac- teristics are employed. In this way the reader may become familiar with the order of magnitude of device parameters, the variability of these parameters within a given type and with a change of temperature, the effect of the inevitable shunt capacitances in circuits, and the effect of input and output resistances and loading on circuit operation. These vii viii / PREFACE considerations are of utmost importance to the student or the practicing engi- neer since the circuits to be designed must function properly and reliably in the physical world, rather than under hypothetical or ideal circumstances. There are over 600 homework problems, which will test the student's grasp of the fundamental concepts enunciated in the book and will give him experience in the analysis and design of electronic circuits. In almost all numerical problems realistic parameter values and specifications have been chosen. An answer book is available for students, and a solutions manual may be obtained from the publisher by an instructor who has adopted the text. This book was planned originally as a second edition of Millman's "Vacuum-tube and Semiconductor Electronics" (McGraw-Hill Book Com- pany, New York, 1958). However, so much new material has been added and the revisions have been so extensive and thorough that a new title for the present text seems proper. The changes are major and have been made necessary by the rapid developments in electronics, and particularly by the continued shift in emphasis from vacuum tubes to transistors and other semi- conductor devices. Less than 25 percent of the coverage relates to vacuum tubes; the remainder is on solid-state devices, particularly the bipolar tran- sistor. In recognition of the growing importance of integrated circuits and the field-effect transistor, an entire chapter is devoted to each of these topics. But to avoid too unwieldy a book, it was decided not to consider gas tubes, silicon-controlled rectifiers, polyphase rectifiers, tuned amplifiers, modulation, or detection circuits. The companion volume to this book, Millman and Taub's "Pulse, Digital, and Switching Waveforms" (McGraw-Hill Book Company, New York, 1965), gives an extensive treatment of the generation and processing of nonsinusoidal waveforms. Considerable thought was given to the pedagogy of presentation, to the explanation of circuit behavior, to the use of a consistent system of notation, to the care with which diagrams are drawn, and to the many illustrative exam- ples worked out in detail in the text. It is hoped that these will facilitate the use of the book in self-study and that the practicing engineer will find the text useful in updating himself in this fast-moving field. The authors are very grateful to P. T. Mauzey, Professor H. Taub, and N. Voulgaris, who read portions of the manuscript and offered con- structive criticism. We thank Dr. Taub also because some of our material on the steady-state characteristics of semiconductor devices and on tran- sistor amplifiers parallels that in Millman and Taub's "Pulse, Digital, and Switching Waveforms." We acknowledge with gratitude the influence of Dr. V. Johannes and of the book "Integrated Circuits" by Motorola, Inc. (McGraw-Hill Book Company, New York, 1965) in connection with Chapter 15. We express our particular appreciation to Miss S. Silverstein, adminis- trative assistant of the Electrical Engineering Department of The City College, for her most skillful service in the preparation of the manuscript. We also thank J. T. Millman and S. Thanos for their assistance. Jacob Millman Christos C. Halkias CONTENTS Preface Electron Ballistics and Applications 1 1-1 Charged Particles 1 1-2 The Force on Charged Particles in an Electric Field 1-3 Constant Electric Field S 1-4 Potential 6 1-5 The eV Unit of Energy 7 1-6 Relationship between Field Intensity and Potential 1-7 Two-dimensional Motion 8 1-8 Electrostatic Deflection in a Cathode-ray Tube 10 1-9 The Cathode-ray Oscilloscope 12 1-10 Relativistic Variation of Mass with Velocity IS 1-11 Force in a Magnetic Field 15 1-12 Current Density 16 1-13 Motion in a Magnetic Field 17 1-14 Magnetic Deflection in a Cathode-ray Tube 20 1-15 Magnetic Focusing 21 1-16 Parallel Electric and Magnetic Fields 24 1-17 Perpendicular Electric and Magnetic Fields 26 1-18 The Cyclotron SI Energy Levels and Energy Bands 36 2-1 The Nature of the Atom 36 2-2 Atomic Energy Levels S8 2-3 The Photon Nature of Light 40 2-4 Ionization <{0 2-5 Collisions of Electrons with Atoms 41 2-6 Collisions of Photons with Atoms 41 2-7 Metastable States 42 2-8 The Wave Properties of Matter 48 2-9 Electronic Structure of the Elements 45 2-10 The Energy-band Theory of Crystals 47 2-11 Insulators, Semiconductors, and Metals 49 ix x / CONTENTS 3 5 6 Conduction in Metals 52 3-1 Mobility and Conductivity 62 3-2 The Energy Method of Analyzing the Motion of a Particle 54 3-3 The Potential-energy Field in a Metal 57 3-4 Bound and Free Electrons 69 3-5 Energy Distribution of Electrons 60 3-6 The Density of States 86 3-7 Work Function 68 3-8 Thermionic Emission 69 3-9 Contact Potential 70 3-10 Energies of Emitted Electrons 71 3-1 1 Accelerating Fields 74 3-12 High-field Emission 76 3-13 Secondary Emission 75 Vacuum-diode Characteristics 77 4-1 Cathode Materials 77 4-2 Commercial Cathodes 80 4-3 The Potential Variation between the Electrodes 4-4 Space-charge Current 82 4-5 Factors Influencing Space-charge Current 86 4-6 Diode Characteristics 87 4-7 An Ideal Diode versus a Thermionic Diode 4-8 Rating of Vacuum Diodes 89 4-9 The Diode as a Circuit Element 90 *» 7 80 88 8 Conduction in Semiconductors 95 5-1 Electrons and Holes in an Intrinsic Semiconductor 96 5-2 Conductivity of a Semiconductor 97 5-3 Carrier Concentrations in an Intrinsic Semiconductor 99 5-4 Donor and Acceptor Impurities 108 5-5 Charge Densities in a Semiconductor 105 5-6 Fermi Level in a Semiconductor Having Impurities 105 5-7 Diffusion 107 5-8 Carrier Lifetime 108 5-9 The Continuity Equation 109 5-10 The Hall Effect 113 Semiconductor-diode Characteristics 115 6-1 6-2 6-3 6-4 6-5 6-6 6-7 6-8 115 Qualitative Theory of the p-n Junction The p-n Junction as a Diode 117 Band Structure of an Open-circuited p-n Junction The Current Components in a p-n Diode 12$ Quantitative Theory of the p-n Diode Currents The Volt-Ampere Characteristic 127 The Temperature Dependence of p-n Characteristics Diode Resistance 1S2 9 120 124 ISO CONTENTS / xi 6-9 Space-charge, or Transition, Capacitance CV 6-10 Diffusion Capacitance 138 6-11 p-n Diode Switching Times 140 6-12 Breakdown Diodes 148 6-13 The Tunnel Diode 147 6-14 Characteristics of a Tunnel Diode 153 134 156 166 175 Vacuum-tube Characteristics 156 7-1 The Electrostatic Field of a Triode 7-2 The Electrode Currents 159 7-3 Commercial Triodes 161 7-4 Triode Characteristics 162 7-5 Triode Parameters 16$ 7-6 Screen-grid Tubes or Tetrodes 7-7 Pentodes 169 7-8 Beam Power Tubes 1 71 7-9 The Triode as a Circuit Element 173 7-10 Graphical Analysis of the Grounded-cathode Circuit 7-11 The Dynamic Transfer Characteristic 178 7-12 Load Curve. Dynamic Load Line 179 7-13 Graphical Analysis of a Circuit with a Cathode Resistor 181 7-14 Practical Cathode-follower Circuits 184 Vacuum-tube Small-signal Models and Applications 187 8-1 Variations from Quiescent Values 187 8-2 Voltage-source Model of a Tube 188 8-3 Linear Analysis of a Tube Circuit 190 8-4 Taylor's Series Derivation of the Equivalent Circuit 8-5 Current-source Model of a Tube 196 8-6 A Generalized Tube Amplifier 197 8-7 The Thevenin's Equivalent of Any Amplifier 199 8-8 Looking into the Plate or Cathode of a Tube 200 8-9 Circuits with a Cathode Resistor 204 8-10 A Cascode Amplifier 207 8-11 Interelectrode Capacitances in a Triode 209 8-1 2 Input Admittance of a Triode 211 8-13 Interelectrode Capacitances in a Multielectrode Tube 215 8-14 The Cathode Follower at High Frequencies 216 194 Transistor Characteristics 220 9-1 The Junction Transistor 220 9-2 Transistor Current Components 222 9-3 The Transistor as an Amplifier 225 9-4 Transistor Construction 226 9-5 Detailed Study of the Currents in a Transistor 9-6 The Transistor Alpha 230 9-7 The Common-base Configuration 23 1 227 xlf / CONTENTS CONTENTS / xitt M. 10 11 12 9-8 9-9 9-10 9-11 9-12 9-13 9-14 9-15 9-16 9-17 9-18 The Common-emitter Configuration 234 The CE Cutoff Region 237 The CE Saturation Region 239 Large-signal, DC, and Small-signal CE Values of Current Gain 242 The Common-collector Configuration 243 Graphical Analysis of the CE Configuration 244 Analytical Expressions for Transistor Characteristics £47 Analysis of Cutoff and Saturation Regions 251 Typical Transistor-junction Voltage Values 256 Transistor Switching Times 267 Maximum Voltage Rating 260 Transistor Biasing and Thermal Stabilization 263 10-1 The Operating Point 263 10-2 Bias Stability 285 10-3 Collector-to-Base Bias 268 10-4 Self-bias, or Emitter Bias 271 10-5 Stabilization against Variations in Vbe and § for the Self-bias Circuit 276 10-6 General Remarks on Collector-current Stability 280 10-7 Bias Compensation 28S 10-8 Biasing Circuits for Linear Integrated Circuits 285 10-9 Thermistor and Sensistor Compensation 287 10-10 Thermal Runaway 288 10-11 Thermal Stability 290 Small-signal Low-frequency Transistor Models 294 11-1 Two-port Devices and the Hybrid Model 294 11-2 Transistor Hybrid Model 296 1 1-3 Determination of the h Parameters from the Characteristics 298 11-4 Measurement of h Parameters 302 11-5 Conversion Formulas for the Parameters of the Three Transistor Configurations 305 11-6 Analysis of a Transistor Amplifier Circuit Using h Parameters S07 11-7 Comparison of Transistor Amplifier Configurations 312 11-8 Linear Analysis of a Transistor Circuit 316 11-9 The Physical Model of a CB Transistor S16 11-10 A Vacuum-tube-Transistor Analogy 319 Low- frequency Transistor Amplifier Circuits 323 12-1 Cascading Transistor Amplifiers 323 12-2 n-stage Cascaded Amplifier 327 12-3 The Decibel 332 12-4 Simplified Common-emitter Hybrid Model 333 12-5 Simplified Calculations for the Common-collector Configuration 335 13 12-6 Simplified Calculations for the Common-base Configuration SS9 12-7 The Common-emitter Amplifier with an Emitter Resistance 340 12-8 The Emitter Follower 346 12-9 Miller's Theorem 348 12-10 High- input-resistance Transistor Circuits 350 12-11 The Cascode Transistor Configuration 366 12-12 Difference Amplifiers 357 The High-frequency Transistor 363 13-1 The High-frequency T Model 363 13-2 The Common-base Short-circu it-current Frequency Response 366 13-3 The Alpha Cutoff Frequency 366 13-4 The Common-emitter Short-circuit-current Frequency Response S68 13-5 The Hybrid-pi (n) Common -emitter Transistor Model 369 13-6 Hybrid- pi Conductances in Terms of Low-frequency h Parameters 371 13-7 The CE Short-circuit Current Gain Obtained with the Hybrid-pi Model 376 13-8 Current Gain with Resistive Load S78 13-9 Transistor Amplifier Response, Taking Source Resistance into Account 380 14 Field-effect Tronsistors 384 390 15 14-1 The Junction Field-effect Transistor 14-2 The Pinch-off Voltage V P 388 14-3 The JFET Volt-Ampere Characteristics 14-4 The FET Small-signal Model 392 14-5 The Insulated-gate FET (MOSFET) 396 14-6 The Common-source Amplifier 400 14-7 The Common-drain Amplifier, or Source Follower 14-8 A Generalized FET Amplifier 403 14-9 Biasing the FET 406 14-10 Unipolar-Bipolar Circuit Applications 4** 14-11 The FET as a Voltage-variable Resistor (WE) 4*$ 14-12 The Unijunction Transistor 415 Integrated Circuits 418 15-1 Basic Monolithic Integrated Circuits 418 15-2 Epitaxial Growth 428 15-3 Masking and Etching 4%4 15-4 Diffusion of Impurities 4&5 15-5 Transistors for Monolithic Circuits 430 15-6 Monolithic Diodes 4$4 15-7 Integrated Resistors 436 402 xlv / CONTENTS *» 16 17 18 15-8 Integrated Capacitors and Inductors 488 15-9 Monolithic Circuit Layout 440 15-10 Integrated FieJd-effect Transistors 444 15-11 Additional Isolation Methods 449 Untuned Amplifiers 450 16-1 16-2 16-3 16-4 16-5 16-6 16-7 16-8 16-9 16-10 16-11 16-12 Classification of Amplifiers 460 Distortion in Amplifiers 46$ Frequency Response of an Amplifier 462 The AC-coupled Amplifier 455 Low-frequency Response of an /eC-ooupled Stage 467 High-frequency Response of a Vacuum-tube Stage 468 Cascaded CE Transistor Stages 460 Step Response of an Amplifier 466 Bandpass of Cascaded Stages 467 Effect of an Emitter (or a Cathode) Bypass Capacitor on U)w-frequency Response 468 Spurious Input Voltages 472 Noise 47S Feedback Amplifiers and Oscillators 480 17-1 17-2 17-3 17-4 17-5 17-6 17-7 17-8 17-9 17-10 17-11 17-12 17-13 17-14 17-15 17-16 17-17 17-18 17-19 17-20 17-21 Classification of Amplifiers 48O The Feedback Concept 48S General Characteristics of Negative-feedback Amplifiers 488 Effect of Negative Feedback upon Output and Input Resistances 491 Voltage-series Feedback 498 A Voltage-series Feedback Pair 602 Current-series Feedback 604 Current-shunt Feedback 508 Voltage-shunt Feedback 612 The Operational Amplifier 614 Basic Uses of Operational Amplifiers 517 Electronic Analog Computation 620 Feedback and Stability 522 Gain and Phase Margins 624 Sinusoidal Oscillators 625 The Phase-shift Oscillator 628 Resonant-circuit Oscillators 680 A General Form of Oscillator Circuit 582 Crystal Oscillators 686 Frequency Stability 687 Negative Resistance in Oscillators 588 Large-signal Amplifiers 542 18-1 Class A Large-aignal Amplifiers 542 18-2 Second-harmonic Distortion 644 18-3 Higher-order Harmonic Generation 546 CONTENTS / xv 18-4 The Transformer-coupled Audio Power Amplifier 18-5 Power Amplifiers Using Tubes 558 18-6 Shift of Dynamic Load Line 556 18-7 Efficiency 556 18-8 Push-Pull Amplifiers 668 18-9 Class B Amplifiers 660 18-10 Class AB Operation 564 549 **. 19 Photoelectric Devices 566 19-1 Photocmissivity 666 19-2 Photoelectric Theory 568 19-3 Definitions of Some Radiation Terms 571 19-4 Phototubes 578 19-5 Applications of Photodevices 575 19-6 Multiplier Phototubes 678 19-7 Photoconductivity 580 19-8 The Semiconductor Photodiode 588 19-9 Multiple-junction Photodiodes 586 19-10 The Photovoltaic Effect 687 20 Rectifiers and Power Supplies 592 20-1 A Half-wave Rectifier 692 20-2 Ripple Factor 597 20-3 A Full-wave Rectifier 698 20-4 Other Full-wave Circuits 600 20-5 The Harmonic Components in Rectifier Circuits 20-6 Inductor Filters 603 20-7 Capacitor Filters 606 20-8 Approximate Analysis of Capacitor Filters 609 20-9 L-section Filter 611 20-10 Multiple L-section Filter 616 20-1 1 11-section Filter 617 20-12 fl-section Filter with a Resistor Replacing the Inductor 620 20-13 Summary of Filters 621 20-14 Regulated Power Supplies 621 20-15 Series Voltage Regulator 623 20-16 Vacuum-tube-regulated Power Supply 629 602 Appendix A Probable Values of General Physical Constants 633 Appendix B Conversion Factors and Prefixes 634 Appendix C Periodic Table of the Elements 635 Appendix D Tube Characteristics 636 Problems 641 Index 745 1 ELECTRON BALLISTICS AND APPLICATIONS In this chapter we present the fundamental physical and mathemati- cal theory of the motion of charged particles in electric and magnetic fields of force. In addition, we discuss a number of the more impor- tant electronic devices that depend on this theory for their operation. The motion of a charged particle in electric and magnetic fields is presented, starting with simple paths and proceeding to more complex motions. First a uniform electric field is considered, and then the analysis is given for motions in a uniform magnetic field. This dis- cussion is followed, in turn, by the motion in parallel electric and mag- netic fields and in perpendicular electric and magnetic fields. 1-1 CHARGED PARTICLES The charge, or quantity, of negative electricity of the electron has been found by numerous experiments to be 1.602 X 10 - " C (coulomb). The values of many important physical constants are given in Appen- dix A. Some idea of the number of electrons per second that repre- sents current of the usual order of magnitude is readily possible. For example, since the charge per electron is 1.602 X 10~ 19 C, the number of electrons per coulomb is the reciprocal of this number, or approxi- mately, 6 X 10 18 . Further, since a current of 1 A (ampere) is the flow of 1 C/sec, then a current of only 1 pA (1 picoampere, or 10 -12 A) represents the motion of approximately 6 million electrons per second. Yet a current of 1 pA is so small that considerable difficulty is experi- enced in attempting to measure it. In addition to its charge, the electron possesses a definite mass. A direct measurement of the mass of an electron cannot be made, but the ratio e/m of the charge to the mass has been determined by a 1 2 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-2 number of experimenters using independent methods. The most probable value for this ratio is 1.759 X 10 11 C/kg. From this value of e/m and the value of e, the charge on the electron, the mass of the electron is calculated to be 9.109 X lO" 31 kg. The charge of a positive ion is an integral multiple of the charge of the electron, although it is of opposite sign. For the case of singly ionized parti- cles, the charge is equal to that of the electron. For the case of doubly ionized particles, the ionic charge is twice that of the electron. The mass of an atom is expressed as a number that is based on the choice of the atomic weight of oxygen equal to 16. The mass of a hypothetical atom of atomic weight unity is, by this definition, one-sixteenth that of the mass of monatomic oxygen. This has been calculated to be 1.660 X 10 -27 kg. Hen^e, in order to calculate the mass in kilograms of any atom., it is necessary only to multiply the atomic weight of the atom by 1.660 X 10~" kg. A table of atomic weights is given in Appendix C. The radius of the electron has been estimated as 10 -16 m, and that of an atom as 10~ 10 m. These are so small that all charges are considered as mass points in the following sections. Classical and Wave-mechanical Models of the Electron The foregoing description of the electron (or atom) as a tiny particle possessing a definite charge and mass is referred to as the classical model. If this particle is sub- jected to electric, magnetic, or gravitational fields, it experiences a force, and hence is accelerated. The trajectory can be determined precisely using New- ton's laws, provided that the forces acting on the particle are known. In this chapter we make exclusive use of the classical model to study electron ballistics. The term electron ballistics is used because of the existing analogy between the motion of charged particles in a field of force and the motion of a falling body in the earth's gravitational field. For large-scale phenomena, such as electronic trajectories in a vacuum tube, the classical model yields accurate results. For small-scale systems, however, such as an electron in an atom or in a crystal, the classical model treated by Newtonian mechanics gives results which do not agree with experi- ment. To describe such subatomic systems properly it is found necessary to attribute to the electron a wavelike property which imposes restrictions on the exactness with which the electronic motion can be predicted. This wave- mechanical model of the electron is considered in Chap. 2. 1-2 THE FORCE ON CHARGED PARTICLES IN AN ELECTRIC FIELD The force on a unit positive charge at any point in an electric field is, by definition, the electric field intensity £ at that point. Consequently, the force on a positive charge q in an electric field of intensity £ is given by q£, the resulting force Sec. 7-3 ELECTRON BALLISTICS AND APPLICATIONS / 3 being in the direction of the electric field. Thus, (1-1) where f« is in newtons, q is in coulombs, and £ is in volts per meter. Boldface type is employed wherever vector quantities (those having both magnitude and direction) are encountered. The mks (meter-kilogram-second) rationalized system of units is found most convenient for the subsequent studies. Therefore, unless otherwise stated, this system of units is employed. In order to calculate the path of a charged particle in an electric field, the force, given by Eq. (1-1), must be related to the mass and the acceleration of the particle by Newton's second law of motion. Hence dt (1-2) where m = mass, kg a = acceleration, m/sec* v = velocity, m/sec The solution of this equation, subject to appropriate initial conditions, gives the path of the particle resulting from the action of the electric forces. If the magnitude of the charge on the electron is e, the force on an electron in the field is f - -«S (1-3) The minus sign denotes that the force is in the direction opposite to the field. In investigating the motion of charged particles moving in externally applied force fields of electric and magnetic origin, it is implicitly assumed that the number of particles is so small that their presence does not alter the field distribution. 1-3 CONSTANT ELECTRIC FIELD Suppose that an electron is situated between the two plates of a parallel-plate capacitor which are contained in an evacuated envelope, as illustrated in Fig. 1-1- A difference of potential is applied between the two plates, the direction of the electric field in the region between the two plates being as shown. If the distance between the plates is small compared with the dimensions of the plates, the electric field may be considered to be uniform, the lines of force pointing along the negative X direction. That is, the only field that is present is £ along the -X axis. It is desired to investigate the characteristics of the motion, subject to the initial conditions »* = v a X = x when ( = (1-4) 4 / ELECTRONIC DEVICES AND CIRCUITS Sac. 7-3 ■i d- e-*- Fig. 1-1 The one-dimenstona) electric field between the plates of a parallel- plate capacitor. This means that the initial velocity v ex is chosen along e, the lines of force, and that the initial position x of the electron is along the X axis. Since there is no force along the Y or Z directions, Newton's law states that the acceleration along these axes must be zero. However, zero acceler- ation means constant velocity; and since the velocity is initially zero along these axes, the particle will not move along these directions. That is, the only possible motion is one-dimensional, and the electron moves along the X axis. Newton's law applied to the X direction yields or e£ = 7tta x 68 a, = — = const m d-5) where £ represents the magnitude of the electric field. This analysis indicates that the electron will move with a constant acceleration in a uniform electric field. Consequently, the problem is analogous to that of a freely falling body in the uniform gravitational field of the earth. The solution of this problem is given by the well-known expressions for the velocity and displacement, viz., v. = tv, + aj. x = x„ + v OI t + lad* d-6) provided that a- = const, independent of the time. It is to be emphasized that, if the acceleration of the particle is not a con- stant but depends upon the time, Eqs. (1-6) are no longer valid. Under these circumstances the motion is determined by integrating the equations dV; dl and dx dl = v x (1-7) These are simply the definitions of the acceleration and the velocity, respec- tively. Equations (1-6) follow directly from Eqs. (1-7) by integrating the latter equations subject to the condition of a constant acceleration. Sec 1-4 ELECTRON BALLISTICS AND APPLICATIONS / 5 EXAMPLE An electron starts at rest on one plate of a plane-parallel capacitor whose plates are 5 cm apart. The applied voltage is zero at the, instant the elec- tron is released, and it increases linearly from zero to 10 V in 0,1 Msec.f a. If the opposite plate is positive, what speed will the electron attain in 50 nsec? b. Where will it be at the end of this time? c. With what speed will the electron strike the positive plate? Solution Assume that the plates are oriented with respect to a cartesian system of axes as illustrated in Fig. 1-1. The magnitude of the electric field intensity is a. 6 = whence — — X — = 2 X 10 9 * 5 X 10-* 10" 7 V/m a* = ^ - - = — = (1.76 X 10»)(2 X 10»() at m M = 3.52 X 10 M ( m/sec J Upon integration, we obtain for the speed v, = T a x dt = 1.76 X 10*V At t = 5 X 10~ a sec, v x = 4.40 X 10* m/sec. 6. Integration of v x with respect to (, subject to the condition that x = when t = 0, yields x m j* Vz dt = P 1.76 X \0*H*dt = 5.87 X 10 ,9 f 3 At t m 5 X 10"" sec, x = 7.32 X 10~ 3 m = 0.732 cm, c. To find the speed with which the electron strikes the positive plate, we first find the time t it takes to reach that plate, or / x Y / 0.05 Y [ 1 = f J - 9.46 X 10' \5.87 X 10'7 \5.87 X 10'V Hence 1.76 X 10 M / S = 1.76 X 10»°(9.46 X 10" 8 )* - 1.58 X 10« m/sec 1-4 POTENTIAL The discussion to follow need not be restricted to uniform fields, but £ x may be a function of distance. However, it is assumed that E x is not a function t 1 /^ec = 1 microsecond = 10~»sec. 1 nsec = 1 nanosecond = 10 - *sec. Conversion factors and prefixes are given in Appendix B. 6 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1-4 of time. Then, from Newton's second law, e&a _ dv x m ~~ dt Multiply this equation by dx = v x dt, and integrate. This leads to / & x dx = v, dv* (1-8) 571 JXa JVoi The definite integral / * & x dx is an expression for the work done by the field in carrying a unit positive charge from the point x to the point x. By definition, the potential V (in volts) of point x with respect to point x„ is the work done against the field in taking a unit positive charge from x a to x. Thusf V m - £& x dx (1-9) By virtue of Eq. (1-9), Eq. (1-8) integrates to eV = §m(v x * - *,*) (1-10) where the energy eV is expressed in joules. Equation (1-10) shows that an electron that has "fallen" through a certain difference of potential V in going from point x a to point x has acquired a specific value of kinetic energy and velocity, independent of the form of the variation of the field distribution between these points and dependent only upon the magnitude of the potential difference V. Although this derivation supposes that the field has only one component, namely, 8* along the X axis, the final result given by Eq. (1-10) is simply a statement of the law of conservation of energy. This law is known to be valid even if the field is multidimensional. This result is extremely impor- tant in electronic devices. Consider any two points A and B in space, with point B at a higher potential than point A by V BA . Stated in its most genera] form, Eq. (1-10) becomes qVzA = fymA* — £wu>s* (1-11) where q is the charge in coulombs, qV B A is in joules, and v* and v B are the corresponding initial and final speeds in meters per second at the points A and By respectively. By definition, the potential energy between two points equals the potential multiplied by the charge in question. Thus the left-hand side of Eq. (1-11) is the rise in potential energy from A to B. The right-hand side repre- sents the drop in kinetic energy from A to B. Thus Eq. (1-11) states that the rise in potential energy equals the drop in kinetic energy, which is equivalent to the statement that the total energy remains unchanged. t The symbol ■ w used to designate "equal to by definition." Sec. 1-5 ELECTRON BALLISTICS AND APPLICATIONS / 7 It must be emphasized that Eq. (1-11) is not valid if the field varies with time. If the particle is an electron, then — e must be substituted for q. If the electron starts at rest, its final speed v, as given by Eq. (1-11) with v A — 0, v B = v, and V B a = V, is or -M v = 5.93 X 10 6 F* d-12) (1-13) Thus, if an electron "falls" through a difference of only 1 V, its final speed is 5-93 X 10 6 m/sec, or approximately 370 miles/sec. Despite this tremen- dous speed, the electron possesses very little kinetic energy, because of its minute mass. It must be emphasized that Eq. (1-13) is valid only for an electron starting at rest. If the electron does not have zero initial velocity or if the particle involved is not an electron, the more general formula [Eq. (1-11)] must be used. 1-5 THE eV UNIT OF ENERGY The joule (J) is the unit of energy in the mks system. In some engineering power problems this unit is very small, and a factor of 10 3 or 10 8 is introduced to convert from watts (1 W = 1 J/sec) to kilowatts or megawatts, respectively. However, in other problems, the joule is too large a unit, and a factor of 10~ 7 is introduced to convert from joules to ergs. For a discussion of the energies involved in electronic devices, even the erg is much too large a unit. This statement is not to be construed to mean that only minute amounts of energy can be obtained from electron devices. It is true that each electron possesses a tiny amount of energy, but as previously pointed out (Sec. 1-1), an enor- mous number of electrons is involved even in a small current, so that con- siderable power may be represented. A unit of work or energy, called the electron volt (eV), is defined as follows: 1 eV = 1.60 X 10- 19 J Of course, any type of energy, whether it be electric, mechanical, thermal, etc., may be expressed in electron volts. The name electron volt arises from the fact that, if an electron falls through a potential of one volt, its kinetic energy will increase by the decrease in potential energy, or by eV - (1.60 X 10- 19 C)(l V) = 1.60 X 10" 19 J = 1 eV However, as mentioned above, the electron-volt unit may be used for any type of energy, and is not restricted to problems involving electrons. The abbreviations MeV and BeV are used to designate 1 million and 1 billion electron volts, respectively. 8 / ELECTRONIC DEVICES AND CIRCUITS Sec, 1-6 1-6 RELATIONSHIP BETWEEN FIELD INTENSITY AND POTENTIAL The definition of potential is expressed mathematically by Eq. (1-9). If the electric field is uniform, the integral may be evaluated to the form - J* £« dx = -& x (x - Xo) = V which shows that the electric field intensity resulting from an applied potential difference V between the two plates of the capacitor illustrated in Fig. 1-1 is given by — V V £ * = x^J a = ~d (1-14) where 6, is in volts per meter, and d is the distance between plates, in meters. In the general case, where the field may vary with the distance, this equation is no longer true, and the correct result is obtained by differentiating Eq. (1-9). We obtain dV ax (1-15) The minus sign shows that the electric field is directed from the region of higher potential to the region of lower potential. 1-7 TWO-DIMENSIONAL MOTION Suppose that an electron enters the region between the two parallel plates of a parallel-plate capacitor which are oriented as shown in Fig. 1-2 with an initial velocity in the -f X direction. It will again be assumed that the electric field between the plates is uniform. Then, as chosen, the electric field £ is in the direction of the — Y axis, no other fields existing in this region. The motion of the particle is to be investigated, subject to the initial conditions fz = % x = (1-16) v v = y = ) when t = v, = z = Since there is no force in the Z direction, the acceleration in that direction is !«4- - r -=5 Fig, 1-2 Two-dimensional electronic motion in a uniform electric field. Sec. 1-7 ELECTRON BALLISTICS AND APPLICATION > / 9 zero. Hence the component of velocity in the Z direction remains constant. Since the initial velocity in this direction is assumed to be zero, the motion must take place entirely in one plane, the plane of the paper. For a similar reason, the velocity along the X axis remains constant and equal to v ox . That is, H = Mm from which it follows that x = v ex t (1-17) On the other hand, a constant acceleration exists along the Y direction, and the motion is given by Eqs. (1-6), with the variable x replaced by y; where v y = a v t e£„ Oy = = m V = W md (1-18) (1-19) and where the potential across the plates is V = V d . These equations indi- cate that in the region between the plates the electron is accelerated upward, the velocity component v v varying from point to point, whereas the velocity component v x remains unchanged in the passage of the electron between the plates. The path of the particle with respect to the point is readily determined by combining Eqs. (1-17) and (1-18), the variable ( being eliminated. This leads to the expression ^2 »«y (1-20) which shows that the particle moves in a parabolic path in the region between the plates. EXAMPLE Hundred-volt electrons are introduced at A into a uniform electric field of 10* V/m, as shown in Fig. 1-3. The electrons are to emerge at the point B in time 4.77 nsec. a. What is the distance AB? b. What angle does the electron beam make with the horizontal? Fig. 1-3 Parabolic path of an electron in a uniform electric field. J^ 10 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-8 Solution The path of the electrons will be a parabola, as shown by the dashed curve in Fig. 1-3, This problem is analogous to the firing of a gun in the earth's gravitational field. The bullet will travel in a parabolic path, first rising because of the muzzle velocity of the gun and then falling because of the downward attrac- tive force of the earth. The source of the charged particles is called an electron gun, or an ion gun. The initial electron velocity is found using Eq. (1-13). R, = 5.93 X 10 s -s/lOO = 5.93 X 10 s m/sec Since the speed along the X direction is constant, the distance AB = xte given by x = (v„ cos 6)t = (5.93 X 10 fi cos 0)(4.77 X 10~») = 2.83 X 10~ 2 cos 8 Hence we first must find 8 before we can solve for x. Since the acceleration a, in the Y direction is constant, then y = (v sin 8)t — ^Oyt* and y = at point B, or v„ sin 9 *i?"i© - 1(1.76 X 10") (10*) (4.77 X 10"*) - 4.20 X 10* m/sec and , . 4.20 X 10 s AmM a. x = 2.83 X 10-* X 0.707 = 2.00 X 10"* m = 2.00 cm 1-8 ELECTROSTATIC DEFLECTION IN A CATHODE-RAY TUBE The essentials of a cathode-ray tube for electrostatic deflection are illustrated in Fig. 1-4. The hot cathode A' emits electrons whieh are accelerated toward the anode by the potential V a . Those electrons which are not collected by the anode pass through the tiny anode hole and strike the end of the glass envelope. This has been coated with a material that fluoresces when bom- Anode Cathode ■*kS r Vertical-deflecting plates + V d + u s Fluorescent screen Fig. 1-4 Electrostatic deflection in a cathode-ray tube. See. 7-8 ELECTRON BALLISTICS AND APPLICATIONS / II barded by electrons. Thus the positions where the electrons strike the screen are made visible to the eye. The displacement D of the electrons is deter- mined by the potential V d (assumed constant) applied between the delecting plates, as shown. The velocity v ox with which the electrons emerge from the anode hole is given by Eq. (1-12), viz., \ m (1-21) on the assumption that the initial velocities of emission of the electrons from the cathode are negligible. Since no field is supposed to exist in the region from the anode to the point 0, the electrons will move with a constant velocity %* in a straight-line path. In the region between the plates the electrons will move in the para- bolic path given by y = ^{ajv^x 2 according to Eq. (1-20). The path is a straight line from the point of emergence M at the edge of the plates to the point P' on the screen, since this region is field-free. The straight-line path in the region from the deflecting plates to the screen is, of course, tangent to the parabola at the point M. The slope of the line at this point, and so at every point between M and P', is [from Eq. (1-20) J tan* = ^l m *J dxJz~i v ax 2 From the geometry of the figure, the equation of the straight line MP' is found to be (1-22) since x = I and y = ^aJ a /»„* at the point M . When y = 0, z = 1/2, which indicates that when the straight line MP' is extended backward, it will intersect the tube axis at the point O', the center point of the plates. This result means that O' is, in effect, a virtual cathode, and regardless of the applied potentials V a and V d , the electrons appear to emerge from this "cathode" and move in a straight line to the point P*. At the point P' t y = D, and x - L + $L Equation (1-22) reduces to ttjju T D = By inserting the known values of ay ( = eV d /dm) and v ox , this becomes lLV d D - 2dV a (1-23) This result shows that the deflection on the screen of a cathode-ray tube is directly proportional to the deflecting voltage V d applied between the plates. Consequently, a cathode-ray tube may be used as a linear-voltage indicating device. The electrostatic-deflection sensitivity of a cathode-ray tube is defined as 12 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1-9 the deflection (in meters) on the screen per volt of deflecting voltage. Thus B D IL S = V d = 2dV a (1-24) An inspection of Eq. (1-24) shows that the sensitivity is independent of both the deflecting voltage V d and the ratio e/m. Furthermore, the sensitivity varies inversely with the accelerating potential V a . The idealization made in connection with the foregoing development, viz., that the electric field between the deflecting plates is uniform and does not extend beyond the edges of the plates, is never met in practice. Consequently, the effect of fringing of the electric field may be enough to necessitate correc- tions amounting to as much as 40 percent in the results obtained from an application of Eq. (1-24). Typical measured values of sensitivity are 1.0 to 0.1 mm/V, corresponding to a voltage requirement of 10 to 100 V to give a deflection of 1 cm. U9 THE CATHODE-RAY OSCILLOSCOPE An electrostatic tube has two sets of deflecting plates which are at right angles to each other in space (as indicated in Fig. 1-6). These plates are referred to as the vertical-deflection and horizontal-deflection plates because the tube is ori- ented in space so that the potentials applied to these plates result in vertical and horizontal deflections, respectively. The reason for having two sets of plates is now discussed. Suppose that the sawtooth waveform of Fig. 1-6 is impressed across the horizontal-deflection plates. Since this voltage is used to sweep the electron beam across the screen, it is called a sweep voltage. The electrons are deflected Vertical-deflection plates Horizontal- deflection plates Vertical signal voltage v. Horizontal sawtooth voltage Electron beam Ftg. 1-5 A waveform to be displayed on the screen of a cathode-ray tube is applied to the vertical-deflection plates, and simultaneously a sawtooth voltage is applied to the hori- zontal-deflection plates. Sec. 1-70 ELECTRON BALLISTICS AND APPLICATIONS / 13 Voltage Fig. 1 -6 Sweep or sawtooth voltage for a cathode-ray tube. Time linearly with time in the horizontal direction for a time T. Then the beam returns to its starting point on the screen very quickly as the sawtooth voltage rapidly falls to its initial value at the end of each period. If a sinusoidal voltage is impressed across the vertical-deflection plates when, simultaneously, the sweep voltage is impressed across the horizontal- deflection plates, the sinusoidal voltage, which of itself would give rise to a vertical line, will now be spread out and will appear as a sinusoidal trace on the screen. The pattern will appear stationary only if the time T is equal to, or is some multiple of, the time for one cycle of the wave on the vertical plates. It is then necessary that the frequency of the sweep circuit be adjusted to synchronize with the frequency of the applied signal. Actually, of course, the voltage impressed on the vertical plates may have any waveform. Consequently, a system of this type provides an almost inertialess oscilloscope for viewing arbitrary waveshapes. This is one of the most common uses for cathode-ray tubes. If a nonrepeating sweep voltage is applied to the horizontal plates, it is possible to study transients on the screen. This requires a system for synchronizing the sweep with the start of the transient. 'f A commercial oscilloscope has many refinements not indicated in the schematic diagram of Fig. 1-5. The sensitivity is greatly increased by means of a high-gain amplifier interposed between the input signal and the deflection plates. The electron gun is a complicated structure which allows for acceler- ating the electrons through a large potential, for varying the intensity of the beam, and for focusing the electrons into a tiny spot. Controls are also pro- vided for positioning the beam as desired on the screen. 1-10 RELATIVISTIC VARIATION OF MASS WITH VELOCITY The theory of relativity postulates an equivalence of mass and energy accord- ing to the relationship W = mc* (1-25) where W = total energy, J m = mass, kg c = velocity of light in vacuum, m/sec t Superscript numerals are keyed to the References at the end of the chapter. T4 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-70 According to this theory, the mass of a particle will increase with its energy, and hence with its speed. If an electron starts at the point A with zero velocity and reaches the point B with a velocity v, then the increase in energy of the particle must be given by the expression eV, where V is the difference of potential between the points A and B, Hence eV = mc 2 — rrioC* (1-26) where m„c 3 is the energy possessed at the point A. The quantity m is known as the rest mass, or the electrostatic mass, of the particle, and is a constant, independent of the velocity. The total mass m of the particle is given by m = VI - »7c' (1-27) This result, which was originally derived by Lorentz and then by Einstein as a consequence of the theory of special relativity, predicts an increasing mass with an increasing velocity, the mass approaching an infinite value as the velocity of the particle approaches the velocity of light. From Eqs. (1-26) and (1-27), the decrease in potential energy, or equivalently, the increase in kinetic energy, is eV = m** ( X - i\ (1-28) This expression enables one to find the velocity of an electron after it has fallen through any potential difference F. By defining the quantity v x as the velocity that would result if the relativistic variation in mass were neglected, i.e., J2eV (1-28) can be solved for v, the true velocity of the particle. The Vn = (1-29) then Eq. result is v = c 1 - 1 "li (1-30) (1 + »ArV2c«)*_ This expression looks imposing at first glance. It should, of course, reduce to v = v N for small velocities. That it does so is seen by applying the binomial expansion to Eq. (1-30). The result becomes '*Mfr-W*-'") (1-31) From this expression it is seen that, if the speed of the particle is much less than the speed of light, the second and all subsequent terms in the expansion can be neglected, and then v = v N , as it should. This equation also serves as a criterion to determine whether the simple classical expression or the more formidable relativistic one must be used in any particular case. For example, Swc. I-W ELECTRON BALLISTICS AND APPLICATIONS / 15 S« if the speed of the electron is one-tenth of the speed of light, Eq. (1-31) shows that an error of only three-eighths of 1 percent will result if the speed is taken as Vft instead of v. For an electron, the potential difference through which the particle must fall in order to attain a velocity of 0.1c is readily found to be 2,560 V. Thus, if an electron falls through a potential in excess of about 3 kV, the relativistic corrections should be applied. If the particle under question is not an elec- tron, the value of the nonrelativistic velocity is first calculated. If this is greater than 0.1c, the calculated value of 0jy must be substituted in Eq. (1-30) and the true value of v then calculated. In cases where the speed is not too great, the simplified expression (1-31) may be used. The accelerating potential in high-voltage cathode-ray tubes is sufficiently high to require that relativistic corrections be made in order to calculate the velocity and mass of the particle. Other devices employing potentials that are high enough to require these corrections are x-ray tubes, the cyclotron, and other particle-accelerating machines. Unless specifically stated otherwise, nonrelativistic conditions are assumed in what follows. 1-11 FORCE IN A MAGNETIC FIELD To investigate the force on a moving charge in a magnetic field, the well- known motor law is recalled. It has been verified by experiment that, if a conductor of length L, carrying a current of /, is situated in a magnetic field of intensity B, the force /„ acting on this conductor is /. - BIL (1-32) where f m is in newtons, B is in webers per square meter (Wb/m 2 ),t / is in am- peres, and L is in meters. Equation (1-32) assumes that the directions of / and B are perpendicular to each other. The direction of this force is perpen- dicular to the plane of I and B and has the direction of advance of a right- handed screw which is placed at O and is rotated from I to B through 90°, as illustrated in Fig. 1-7. If I and B are not perpendicular to each other, only the component of I perpendicular to B contributes to the force. Some caution must be exercised with regard to the meaning of Fig. 1-7. If the particle under consideration is a positive ion, then I is to be taken along the direction of its motion. This is so because the conventional direction of the current is taken in the direction of flow of positive charge. If the current 's due to the flow of electrons, the direction of I is to be taken as opposite to the direction of the motion of the electrons. If, therefore, a negative charge t One weber per square meter (also called a testa) equals 10* G. A unit of more prac- tical size in most applications is the milliweber per square meter (mWb/m 1 ), which equals 10 G. Other conversion factors are given in Appendix B. 16 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1-12 Sec. M3 ELECTRON BALLISTICS AND APPLICATIONS / 17 L" o.^ -90' T Fig. 1-7 Pertaining to the determination of the direc- tion of the force f m on a charged particle in a magnetic field. lorv* moving with a velocity v~ is under consideration, one must first draw I anti- parallel to v~ as shown and then apply the "direction rule." If N electrons are contained in a length L of conductor (Fig. 1-8) and if it takes an electron a time T sec to travel a distance of L m in the conductor, the total number of electrons passing through any cross section of wire in unit time is N/T. Thus the total charge per second passing any point, which, by definition, is the current in amperes, is T = *!£ T (1-33) The force in newtons on a length L m (or the force on the N conduction charges contained therein) is BIL = BNeL Furthermore, since L/T is the average, or drift, speed v m/sec of the electrons, the force per electron is f» = eBv (1-34) The subscript m indicates that the force is of magnetic origin. To sum- marize: The force on a negative charge e (coulombs) moving with a component of velocity r (meters per second) normal to a field B (webers per square meter) is given by eBv~ (newtons) and is in a direction perpendicular to the plane of B and y~, as noted in Fig. 1-7. f 1-12 CURRENT DENSITY Before proceeding with the discussion of possible motions of charged particles in a magnetic field, it is convenient to introduce the concept of eurrent density. t In the crosa-product notation of vector analysis, f m m eB x v~. For a positive ion moving with a velocity v + , the force Is f m = ev+ X B. m N electrons D -i Fig. T-8 Pertaining to the determination of the magnitude of the force f m on a charged particle in a magnetic field. This concept is very useful in many later applications. By definition, the current density, denoted by the symbol J, is the current per unit area of the conducting medium. That is, assuming a uniform current distribution, "i (1-35) where J is in amperes per square meter, and A is the cross-sectional area (in meters) of the conductor. This becomes, by Eq. (1-33), r _ N * J TA But it has already been pointed out that T — L/v. Then _ _ Nev J ~LA (1-36) From Fig. 1-8 it is evident that LA is simply the volume containing the N electrons, and so N/LA is the electron concentration n (in electrons per cubic meter). Thus (1-37) N n = LA and Eq. (1-36) reduces to J = nev = pv (1-38) where p = ne is the charge density, in coulombs per cubic meter, and v is in meters per second. This derivation is independent of the form of the conducting medium. Consequently, Fig. 1-8 does not necessarily represent a wire conductor. It may represent equally well a portion of a gaseous-discharge tube or a volume element in the space-charge cloud of a vacuum tube or a semiconductor. Furthermore, neither p nor v need be constant, but may vary from point to point in space or may vary with time. Numerous occasions arise later in the text when reference ia made to Eq. (1-38). 1-13 MOTION IN A MAGNETIC FIELD The path of a charge particle that is moving in a magnetic field is now investi- gated. Consider an electron to be placed in the region of the magnetic field. If the particle is at rest, /„ = and the particle remains at rest. If the initial velocity of the particle is along the lines of the magnetic flux, there is no force acting on the particle, in accordance with the rule associated with Eq. (1-34). Hence a particle whose initial velocity has no component normal to a uniform magnetic field will continue to move with constant speed along the lines of flux. 18 / ELECTRONIC DEVICES AND CIRCUITS Sec. L13 Field-free region x X K Magnetic field Into * paper Fig. 1-9 Circular motion of an electron in a transverse magnetic field. Now consider an electron moving with a speed v to enter a constant uniform magnetic field normally, aa shown in Fig. 1-9. Since the force f m is perpendicular to v and so to the motion at every instant, no work is done on the electron. This means that its kinetic energy is not increased, and so its speed remains unchanged. Further, since v and B are each constant in magnitude, then f m is constant in magnitude and perpendicular to the direction of motion of the particle. This type of force results in motion in a circular path with constant speed. It is analogous to the problem of a mass tied to a rope and twirled around with constant speed. The force (which is the tension in the rope) remains constant in magnitude and is always directed toward the center of the circle, and so is normal to the motion. To find the radius of the circle, it is recalled that a particle moving in a circular path with a constant speed v has an acceleration toward the center of the circle of magnitude v 3 /R, where R is the radius of the path in meters. Then from which The corresponding angular velocity in radians per second is given by _ v_ _ eB R m The time in seconds for one complete revolution, called the period, is m _ 2t __ 2irni cd eB For an electron, this reduces to 3.57 X 10- 11 T = B (1-39) (1-40) (1-41) (1-42) In these equations, e/m is in coulombs per kilogram and B in webers per square meter. S*c. 1-13 ELECTRON BALLISTICS AND APPLICATIONS / T9 It is noticed that the radius of the path is directly proportional to the speed of the particle. Further, the period and the angular velocity are inde- pendent of speed or radius. This means, of course, that faster-moving particles will traverse larger circles in the same time that a slower particle moves in its smaller circle. This very important result is the basis of operation of numer- ous devices, for example, the cyclotron and magnetic-focusing apparatus. EXAMPLE Calculate the deflection of a cathode-ray beam caused by the earth's magnetic field. Assume that the tube axis is so oriented that it is normal to the field, the strength of which is 0.6 G. The anode potential is 400 V; the anode- screen distance is 20 cm (Fig. 1-10). Solution According to Eq. (1-13), the velocity of the electrons will be p m = 5.93 X 10* Vibo = 1.19 X 10 7 m/sec Since 1 Wb/m* = 10* G, then B = 6 X 10" B Wb/m a . From Eq. (1-39) the radius of the circular path is R = 1.19 X 10 7 = 1.12 m = 112 cm (e/m)B 2.76 X 10" X 6 X 10" 5 Furthermore, it is evident from the geometry of Fig. 1-10 that (in centimeters) 112 s = (112 - D)* + 20 2 from which it follows that D* - 2242) + 400 = The evaluation of D from this expression yields the value D = 1.8 cm. This example indicates that the earth's magnetic field can have a large effect on the position of the cathode-beam spot in a low-voltage cathode-ray tube. If Fig. 1-10 The circular path of an elec- tron in a cathode-ray tube, resulting from the earth's transverse magnetic field (normal to the plane of the paper). This figure is not drawn to scale. (112-0) 20 / ELECTRONIC DEVICES AND CIRCUITS Sec. I -U the anode voltage is higher than the value used in this example, or if the tube is not oriented normal to the field, the deflection will be less than that calculated. In any event, this calculation indicates the advisability of carefully shielding a cathode-ray tube from stray magnetic fields. 1-14 MAGNETIC DEFLECTION IN A CATHODE-RAY TUBE The illustrative example in Sec. 1-13 immediately suggests that a cathode- ray tube may employ a magnetic as well as an electric field in order to accom- plish the deflection of the electron beam. However, since it is not feasible to use a field extending over the entire length of the tube, a short coil furnishing a transverse field in a limited region is employed, as shown in Fig. 1-1 1. The magnetic field is taken as pointing out of the paper, and the beam is deflected upward. It is assumed that the magnetic field intensity B is uniform in the restricted region shown and is zero outside of this area. Hence the electron moves in a straight line from the cathode to the boundary of the magnetic field. In the region of the uniform magnetic field the electron experiences a force of magnitude eBv, where v is the speed. The path OM will be the arc of a circle whose center is at Q. The speed of the particles will remain constant and equal to J2eV a (1-43) The angle <p is, by definition of radian measure, equal to the length of the arc OM divided by R, the radius of the circle. If we assume a small angle of deflection, then ¥> « "5 where, by Eq. (1-39), mv R => eB (1-44) (1-45) In most practical cases, L is very much larger than I, so that little error will i^ Magnetic field *?i«*»l out of paper Fig. 1-1 T Magnetic deflection in a cathode-ray tube. S«. M5 ELECTRON BALLISTICS AND APPLICATIONS / 21 be made in assuming that the straight line MP', if projected backward, will pass through the center 0' of the region of the magnetic field. Then D « L tan <p « L<p By Eqs. (1-43) to (1-45), Eq. (1-46) now becomes (1-46) n r IL ILeB ILB D ~ L * = R=^ = ^r a 2m The deflection per unit magnetic field intensity, D/B, given by d = ih rr (1-47) W-tM is called the magnetic-deflection sensitivity of the tube. It is observed that this quantity is independent of B. This condition is analogous to the electric case for which the electrostatic sensitivity is independent of the deflecting potential. However, in the electric case, the sensitivity varies inversely with the anode voltage, whereas it here varies inversely with the square root of the anode voltage. Another important difference is in the appearance of e/m in the expression for the magnetic sensitivity, whereas this ratio did not enter into the final expression for the electric case. Because the sensitivity increases with L, the deflecting coils are placed as far down the neck of the tube as possible, usually directly after the accelerating anode. Deflection in a Television Tube A modern TV tube has a screen diameter comparable with the length of the tube neck. Hence the angle <p is too large for the approximation tan p *= p to be valid. Under these cir- cumstances it is found that the deflection is no longer proportional to B (Prob. 1-24). If the magnetic-deflection coil is driven by a sawtooth current waveform (Fig. 1-6), the deflection of the beam on the face of the tube will not be linear with time. For such wide-angle deflection tubes, special linearity- correcting networks must be added. A TV tube has two sets of magnetic-deflection coils mounted around the neck at right angles to each other, corresponding to the two sets of plates in the oscilloscope tube of Fig. 1-5. Sweep currents are applied to both coils, with the horizontal signal much higher in frequency than that of the vertical sweep. The result is a rectangular raster of closely spaced lines which cover the entire face of the tube and impart a uniform intensity to the screen. When the video signal is applied to the electron gun, it modulates the intensity of the beam and thus forms the TV picture. '-15 MAGNETIC FOCUSING As another application of the theory developed in Sec. 1-13, one method of measuring e/m is discussed. Imagine that a cathode-ray tube is placed in 22 / ELECTRONIC DEVICES AND CIRCUITS Sec. J -15 a constant longitudinal magnetic field, the axis of the tube coinciding with the direction of the magnetic field. A magnetic field of the type here con- sidered is obtained through the use of a long solenoid, the tube being placed within the coil. Inspection of Fig. 1-12 reveals the motion. The Y axis represents the axis of the cathode-ray tube. The origin is the point at which the electrons emerge from the anode. The velocity of the origin is v , the initial transverse velocity due to the mutual repulsion of the electrons being Vo X . It is now shown that the resulting motion is a helix, as illustrated. The electronic motion can most easily be analyzed by resolving the velocity into two components, v v and v 9t along and transverse to the magnetic field, respectively. Since the force is perpendicular to B, there is no accelera- tion in the Y direction. Hence v v is constant and equal to v» v . A force eBv t normal to the path will exist, resulting from the transverse velocity. This force gives rise to circular motion, the radius of the circle being mv 9 /eB t with v 9 a constant, and equal to y„- The resultant path is a helix whose axis is parallel to the Y axis and displaced from it by a distance R along the Z axis, as illustrated. The pitch of the helix, defined as the distance traveled along the direction of the magnetic field in one revolution, is given by V = v^T where T is the period, or the time for one revolution. (1-41) that 27T771 V = ~eB^ It follows from Eq. (1-48) If the electron beam is defocused, a smudge is seen on the screen when the applied magnetic field is zero. This means that the various electrons in the beam pass through the anode hole with different transverse velocities v„, and so strike the screen at different points. This accounts for the appear- ance of a broad, faintly illuminated area instead of a bright point on the screen. As the magnetic field is increased from zero the electrons will move in helices of different radii, since the velocity t>« that controls the radius of the path will be different for different electrons. However, the period, or the time to trace out the path, is independent of v ex , and so the period will be the same for all electrons. If, then, the distance from the anode to the screen is made equal to one pitch, all the electrons will be brought back to the Y axis (the point 0' in Fig. 1-12), since they all will have made just one revolution. Under these conditions an image of the anode hole will be observed on the screen. As the field is increased from zero, the smudge on the screen resulting from the defocused beam will contract and will become a tiny sharp spot (the image of the anode hole) when a critical value of the field is reached. This critical field is that which makes the pitch of the helical path just equal to the anode-screen distance, as discussed above. By continuing to increase Sec. T-T5 ELECTRON BALLISTICS AND APPLICATIONS / 23 Y Fig. 1-12 The helical path of an electron introduced at an angle (not 90°) with a constant magnetic field. Electronic path the strength of the field beyond this critical value, the pitch of the helix decreases, and the electrons travel through more than one complete revolution. The electrons then strike the screen at various points, so that a defocused spot is again visible. A magnetic field strength will ultimately be reached at which the electrons make two complete revolutions in their path from the anode to the screen, and once again the spot will be focused on the screen. This process may be continued, numerous foci being obtainable. In fact, the current rating of the solenoid is the factor that generally furnishes a practical limitation to the order of the focus. The foregoing considerations may be generalized in the following way: If the screen is perpendicular to the Y axis at a distance L from the point of emergence of the electron beam from the anode, then, for an anode-cathode potential equal to V a , the electron beam will come to a focus at the center of the screen provided that L is an integral multiple of p. Under these conditions, Eq, (1-48) may be rearranged to read e m 8ir 2 K~ n 2 L*B 2 (1-49) where n is an integer representing the order of the focus. It is assumed, in this development, that eV a ~ fynvey 2 , or that the only effect of the anode potential is to accelerate the electron along the tube axis. This implies that the transverse velocity x oz> which is variable and unknown, is negligible in comparison with v oy . This is a justifiable assumption. This arrangement was suggested by Busch, and has been used 2 to measure the ratio e/m for electrons very accurately. A Short Focusing Coil The method described above of employing a longitudinal magnetic field over the entire length of a commercial tube is not too practical. Hence, in a commercial tube, a short coil is wound around 24 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1-16 Rg. 1-13 Parallel electric and magnetic fields. the neck of the tube. Because of the fringing of the magnetic lines of flux, a radial component of B exists in addition to the component along the tube axis. Hence there are now two components of force on the electron, one due to the axial component of velocity and the radial component of the field, and the second due to the radial component of the velocity and the axial component of the field. The analysis is complicated, 8 but it can be seen qualitatively that the motion will be a rotation about the axis of the tube and, if conditions are correct, the electron on leaving the region of the coil may be turned sufficiently so as to move in a line toward the center of the screen. A rough adjustment of the focus is obtained by positioning the coil properly along the neck of the tube. The fine adjustment of focus is made by con- trolling the coil current. 1-16 PARALLEL ELECTRIC AND MAGNETIC FIELDS Consider the case where both electric and magnetic fields exist simultaneously, the fields being in the same or in opposite directions. If the initial velocity of the electron either is zero or is directed along the fields, the magnetic field exerts no force on the electron, and the resultant motion depends solely upon the electric field intensity £. In other words, the electron will move in a direction parallel to the fields with a constant acceleration. If the fields arc chosen as in Fig. 1-13, the complete motion is specified by v v = Vey — at y = v<n) t — frl* (1-50) where a = eZ/m is the magnitude of the acceleration. The negative sign results from the fact that the direction of the acceleration of an electron is opposite to the direction of the electric field intensity £. If, initially, a component of velocity v„ perpendicular to the magnetic field exists, this component, together with the magnetic field, will give rise to circular motion, the radius of the circular path being independent of £. However, because of the electric field £, the velocity along the field changes with time. Consequently, the resulting path is helical with a pitch that changes with the time. That is, the distance traveled along the Y axis per revolution increases with each revolution. Sec. 1-16 ELECTRON &ALLISTICS AND APPLICATIONS / 25 EXAMPLE Given a uniform electric field of 1.10 X 10* V/m parallel to and opposite in direction to a magnetic field of 7.50 X 10 -4 Wb/ra*. An electron gun in the XY plane directed at an angle <p = arctan f with the direction of the electric field introduces electrons into the region of the fields with a velocity v = 5.00 X 10 8 m/sec. Find: a. The time for an electron to reach its maximum height above the XZ plane 6. The position of the electron at this time c. The velocity components of the electron at this time Solution a. As discussed above, the path is a helix of variable pitch. The acceleration is downward, and for the coordinate system of Fig. 1-14, y = v ov t — £ai* v v = w 01f — at The electron starts moving in the +Y direction, but since the acceleration is along the — Y direction, its velocity is reduced to zero at a time t = t'. The particle will then reverse its K-directed motion. At maximum height i'„ = and f = v ov /a. Since v 0l) = v cos <p - (5 X 10*) (0.8) = 4 X 10 8 m/sec and e£ ay m — = (1.76 X 10") (1.10 X 10*) = 1.94 X 10" m/sec» m we find f tmZSm 4 X 10 8 1.94 X 10" = 2.06 X 10"* sec = 20.6 nsec 6. The distance traveled in the +Y direction to the position at which the reversal occurs is y - v ov t - iaf* = (4 X 10 8 )(2.06 X lO" 8 ) - £(1.94 X 1Q»)(4.24 X 10~ 18 ) * 4.13 X 10-* m = 4.13 cm It should be kept in mind that the term reversal refers only to the F-directed motion, not to -the direction in which the electron traverses the circular compo- nent of its path. The helical rotation is determined entirely by the quantities B and f„. The angular velocity remains constant and equal to a = — = (1.76 X 10") (7.50 X 10"*) - 1.32 X 10« rad/sec TO Pig. 1-14 A problem illustrating helical electronic •notion of variable pitch. 2d / ELECTRONIC DEVICES AND CIRCUITS -Z Sec. 1-17 Fig. 1-15 The projection of the path in the XZ plane is a circle. 180-e + Z\ u sin \p = u. By^use of either the relationship T = 2r/w or Eq. (1-42), there is obtained T = 4,75 X 10~ 8 sec, and hence less than one revolution is made before the reversal. The point P' in space at which the reversal takes place is obtained by con- sidering the projection of the path in the XZ plane (since the Y coordinate U already known). The angle 8 in Fig. 1-15 through which the electron has rotated is 9 - «rf - 1.32 X 10» X 2.06 X 10" 8 = 2.71 rad = 155° The radius of the circle is fi ^, = (5 X 10»)(0.6) to 1.32 X 10 8 From the figure it is clear that X = R sin (180 - $) = 2.27 sin 25° = 0.957 cm Z = R + R cos (180 - $) = 2.27 + 2.05 = 4.32 cm c. The velocity is tangent to the circle, and its magnitude equals v a sin *> = 5 X 10' X 0.6 = 3 X 10" m/sec. At 9 = 155°, the velocity components are 9, - -#«, cos (180 - 6) - -8 X 10 s cos 25° = -2.71 X 10 8 m/sec v ¥ = f. = v„ sin (180 - 6) = 3 X 10« sin 25° = 1.26 X 10 8 m/sec 1-17 PERPENDICULAR ELECTRIC AND MAGNETIC FIELDS The directions of the fields are shown in Fig. 1-16. The magnetic field is directed along the - 1' axis, and tho electric field is directed along the -X axis. The force on an electron due to the electric field is directed along the + X axis. Any force due to the magnetic field is always normal to B, and Sec. M7 ELECTRON BALLISTICS AND APPLICATIONS / 27 Fig. 1-16 Perpendicular electric and magnetic fields. hence lies in a plane parallel to the XZ plane. Thus there is no component of force along the Y direction, and the Y component of acceleration is zero. Hence the motion along Y is given by L = «U = 1*0 y = v ov t (1-51) assuming that the electron starts at the origin. // the initial velocity component parallel to B is zero, the path lies entirely in a plane perpendicular to B. It is desired to investigate the path of an electron starting at rest at the origin. The initial magnetic force is zero, since the velocity is zero. The electric force is directed along the +X axis, and the electron will be acceler- ated in this direction. As soon as the electron is in motion, the magnetic force will no longer be zero. There will then be a component of this force which will be proportional to the X component of velocity and will be directed along the +Z axis. The path will thus bend away from the +X direction toward the +Z direction. Clearly, the electric and magnetic forces interact with one another. In fact, the analysis cannot be carried along further, profitably, in this qualitative fashion. The arguments given above do, how- ever, indicate the manner in which the electron starts on its path. This path will now be shown to be a cycloid. To determine the path of the electron quantitatively, the force equations must be set up. The force due to the electric field £ is e& along the +X direc- tion. The force due to the magnetic field is found as follows: At any instant, the velocity is determined by the three components v x , v v> and v, along the three coordinate axes. Since B is in the Y direction, no force will be exerted on the electron due to v y . Because of v x , the force is eBv x in the -\-Z direc- tion, as can be verified by the direction rule of Sec. 1-11. Similarly, the force due to v, is eBv t in the —X direction. Hence Newton's law, when expressed in terms of the three components, yields j- dv x „ i, f x = m -j- = e8 — eBv z at dv. _ f ' = m dl = eBv ' By writing for convenience eB m = — and U = B (1-52) (1-53) 28 / ELECTRONIC DEVICES AND CIRCUITS the foregoing equations may be written in the form ~dl N oju — biVz dv t Tt = + m * Sec, 7-17 (1-54) A straightforward procedure is involved in the solution of these equations. If the first equation of (1-54) is differentiated and combined with the second, we obtain d 2 v x dv t „ (1-55) This linear differential equation with constant coefficients is readily solved for v x . Substituting this expression for v x in Eq. (1-54), this equation can be solved for v t . Subject to the initial conditions «, = »,= 0, we obtain v x = u sin (d p» = u — u cos tat (1-56) In order to find the coordinates x and z from these expressions, each equa- tion must be integrated. Thus, subject to the initial conditions x = z = 0, 4i tk» x = - (1 — cos at) z = ut — - sin o)t If, for convenience, 8 s at and Q = - then x = 0(1 - cos 8) z = Q(8 - sin 8) where u and a? are as defined in Eqs. (1-53). (1-57) (1-58) (1-59) Cycloid a! Path Equations (1-59) are the parametric equations of a com- mon cycloid, defined as the path generated by a point on the circumference of a circle of radius Q which rolls along a straight line, the Z axia. This is illustrated in Fig. 1-17. The point P, whose coordinates are x and z (y = 0), represents the position of the electron at any time. The dark curve is the locus of the point P. The reference line CC is drawn through the center of the generating circle parallel to the X axis. Since the circle rolls on the Z axis, then OC represents the length of the circumference that has already come in contact with the Z axis. This length is evidently equal to the arc PC (and equals Qd). The angle 8 gives the number of radians through which the circle has rotated. From the diagram, it readily follows that x = Q - Qcos8 z = Q& - Q sin 8 (1-60) which are identical with Eqs. (1-59), thus proving that the path is cycloidal as predicted. S«. M7 ELECTRON BALLISTICS AND APPLICATIONS / 29 Fig. 1-17 The cydoidol path of an electron in perpen- dicular electric and magnetic fields when the initial velocity is zero. The physical interpretation of the symbols introduced above merely as abbreviations is as follows: u represents the angular velocity of rotation of the Tolling circle. 8 represents the number of radians through which the circle has rotated. Q represents the radius of the rolling circle. Since u = wQ, then u represents the velocity of translation of the center of the rolling circle. From these interpretations and from Fig. 1-17 it is clear that the maximum displacement of the electron along the X axis is equal to the diameter of the rolling circle, or 2Q. Also, the distance along the Z axis between cusps is equal to the circumference of the rolling circle, or 2vQ. At each cusp the speed of the electron is zero, since at this point the velocity is reversing its direction (Fig. 1-17). This is also seen from the fact that each cusp is along the Z axis, and hence at the same potential. Therefore the electron has gained no energy from the electric field, and its speed must again be zero. If an initial velocity exists that is directed parallel to the magnetic field, the projection of the path on the XZ plane will still be a cycloid but the particle will now have a constant velocity normal to the plane. This path 30 / ELECTRONIC DEVICES AND CIRCUITS might be called a "cycloidal helical motion." (1-59), with the addition of Eqs. (1-51). Sec. 1-17 The path is described by Eqs. Straight Line Path As a special case of importance, consider that the elec- tron is released perpendicular to both the electric and magnetic fields so that v ox = v ay = and v ot ^ 0. The electric force is eS along the -\-X direction (Fig. 1-16), and the magnetic force is eBv<„ along the — X direction. If the net force on the electron is zero, it will continue to move along the Z axis with the constant speed u„. This conditions is realized when or e£ = eBv a Km = -g = u (1-61) from Eqs. (1-53). This discussion gives another interpretation to u. It represents that velocity with which an electron may be injected into perpendicular electric and magnetic fields and suffer no deflection, the net force being zero. Note that this velocity u is independent of the charge or mass of the ions. Such a system of perpendicular fields will act as a velocity filter and allow only those particles whose velocity is given by the ratio S/B to be selected. EXAMPLE A magnetic field of 0.01 Wb/m 2 is applied along the axis of a cathode- ray tube. A field of 10" V/m is applied to the deflecting plates. If an electron leaves the anode with a velocity of 10 B m/sec along the axis, how far from the axis will it be when it emerges from the region between the plates? The length I of the deflecting plates along the tube axis is 2.0 cm. Solution Choose the system of coordinate axes illustrated in Fig. 1-16. Then f . = v„ = v ov = 10 a m/sec As shown above, the projection of the path is a cycloid in the XZ plane, and the electron travels with constant velocity along the Y axis. The electron is in the region between the plates for the time l_ = 2 X IP"' » av 10 s = 2 X 10-" sec Then, from Eqs. (1-53) and (1-58), it is found that eB w = — = 1,76 X 10" X 10"* = 1.76 X 10° rad/sec m 8 10* tn , . u = — = • — - = 10 6 m/sec B 10-* Q = - = 10 6 = 5.68 X 10'* m = 0.0568 cm 1.76 X 10 s 6 = tDf = (1.76 X 10»)<2 X 10~ 8 ) = 35.2 rad sec. t-rs ELECTRON BALLISTICS AND APPLICATIONS / 31 Since there are 2ir rad/revolution, the electron goes through five complete cycles and enters upon the sixth before it emerges from the plate. Thus 35.2 rad = lOr + 3.8 rad Since 3.8 rad equals 218°, then Eqs. (1-59) yield x = Q(l - cos 0) = 0.0568(1 - cos 218°) = 0.103 cm z = Q(6 - sin 8) - 0.0568(35.2 - sin 218°) = 2.03 cm so that the distance from the tube axis is = Vx s + z* = 2.03 Trochoidal Paths If the initial-velocity component in the direction per- pendicular to the magnetic field is not zero, it can be shown* that the path is a trochoid. 6 This curve is the locus of a point on a "spoke" of a wheel rolling on a straight line, as illustrated in Fig. 1-18. If the length Q f of the spoke is greater than the radius Q of the rolling circle, the trochoid is called a prolate cycloid* and has subsidiary loops (Fig. l-19a). If Q' - Q, the path is called a common cycloid, illustrated in Fig. 1-17 or 1-196. If Q' is less than Q, the path is called a aviate cycloid, 6 and has blunted cusps, as indicated in Fig. l-19c. 1-18 THE CYCLOTRON The principles of Sec. 1-13 were first employed by Lawrence and Livingston to develop an apparatus called a magnetic resonator, or cyclptron.* This device imparts very high energies (tens of millions of electron volts) to positive ions. These high-energy positive ions are then allowed to bombard some substances, which become radioactive and generally disintegrate. Because of this, the cyclotron has popularly become known as an atom smasher. The basic principles upon which the cyclotron operates are best under- stood with the aid of Fig. 1-20. The essential elements are the "dees," the Rolling circle Angular velocity to Fig. 1 -1 8 The locus of the point P at the end of a "spoke" of a wheel rolling on a straight line is a trochoid. Track of rolling circle' Linear velocity of C la Qui = U 32 / ELECTRONIC DEVICES AND CIRCUITS See. 1. 18 0* — Magnetic fleld (Into paper) 6-* fig. 1-1? The trocholdal paths of electrons in perpendicular electric and magnetic fields. two halves of a shallow, hollow, metallic "pillbox" which has been split along a diameter as shown; a strong magnetic field which is parallel to the axis of the dees; and a high-frequency ac potential applied to the dees. A moving positive ion released near the center of the dees will be acceler- ated in a semicircle by the action of the magnetic field and will reappear at point 1 at the edge of dee I. Assume that dee II is negative at this instant with respect to dee I. Then the ion will be accelerated from point 1 to point 2 across the gap, and will gain an amount of energy corresponding to the poten- tial difference between these two points. Once the ion passes inside the metal dee II, the electric field is zero, and the magnetic field causes it to move in the semicircle from point 2 to point 3. If the frequency of the applied ac poten- tial is such that the potential has reversed in the time necessary for the ion to Dees Particle orbit (schematic) Fig. 1-20 The cyclotron principle. South pole Vacuum chamber s*. its ELECTRON BALLISTICS AND APPLICATIONS / 33 go from point 2 to point 3, then dee I is now negative with respect to dee II, and the ion will be accelerated across the gap from point 3 to point 4. With the frequency of the accelerating voltage properly adjusted to this "resonance" value, the ion continues to receive pulses of energy corresponding to this difference of potential again and again. Thus, after each half revolution, the ion gains energy from the electric field, resulting, of course, in an increased velocity. The radius of each semi- circle is then larger than the preceding one, in accordance with Eq. (1-39), so that the path described by the whirling ion will approximate a planar spiral. EXAMPLE Suppose that the oscillator that supplies the power to the dees of a given cyclotron imparts 50,000 eV to heavy hydrogen atoms (deuterons), each of atomic number 1 and atomic weight 2.0147, at each passage of the ions across the accelerating gap. Calculate the magnetic field intensity, the frequency of the oscillator, and the time it will take for an ion introduced at the center of the chamber to emerge at the rim of the dee with an energy of 5 million electron volts (5 MeV). Assume that the radius of the last semicircle is 15 in. Solution The mass of the deuteron is at = 2.01 X 1.66 X 10~" = 3.34 X 10"" kg The velocity of the 5-MeV ions is given by the energy equation hrw* - (5 X 10 s ) (1.60 X 10"") = 8.00 X lO"" J /2 X 8.00 X 10- l> \* nnfl „. , \ 3.34 X 10-" / 3.34 X 10- The magnetic field, given by Eq. (1-39), (3.34 X 10-") (2.20 X 10 T ) B=™ = eR (1.60 X 1Q-»)(15 X 2.54 X 0.0!) = 1.20 Wb/m* is needed in order to bring these ions to the edge of the dees. The frequency of the oscillator must be equal to the reciprocal of the time of revolution of the ion. This is, from Eq. (1-41), . _ 1 _ eB T 2-rm 1.60 X 10-" X 1.20 2tt X 3.34 X 10"" = 9.15 X 10 B Hit = 9.15 MHi Since the ions receive 5 MeV energy from the oscillator in 50-keV steps, they must pass across the accelerating gap 100 times. That, is, the ion must make 50 complete revolutions in order to gain the full energy. Thus, from Eq. (1-41), the time of flight is 1= SOT - 50 X 1 9.15 X 10 8 * Hi « hertz = cycles per second m 5.47 X 10-* sec = 5.47 usee MHz = megahertz (Appendix B). 34 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1-78 In order to produce a uniform magnetic field of 1.2 Wb/m 2 over a circular area whose radius is at least 15 in., with an air gap approximately 6 in. wide, an enormous magnet is required, the weight of such a magnet being of the order of 60 tons. Also, the design of a 50- kV oscillator for these high frequencies and the method of coupling it to the dees present some difficulties, since the dees are in a vacuum-tight chamber. Further, means must be provided for introducing the ions into the region at the center of the dees and also for removing the high- energy particles from the chamber, if desired, or for directing them against a target. S»c. M« ELECTRON BALLISTICS AND APPLICATIONS / 35 hollow cylinder, since there is need for a magnetic field only transverse to the path. This results in a great saving in weight and expense. The dees of the cyclotron are replaced by a single-cavity resonator. Electrons and protons have been accelerated to the order of a billion electron volts (Bev) in synchro- trons. 8 The larger the number of revolutions the particles make, the higher will be their energy. The defocusing of the beam limits the number of allow- able cycles. With the discovery of alternating-gradient magnetic field focusing,* higher-energy-particle accelerators (70 BeV) have been constructed. 10 The bombardment of the elements with the high-energy protons, deu- terons, or helium nuclei which are normally used in the cyclotrons renders the bombarded elements radioactive. These radioactive elements are of the utmost importance to physicists, since they permit a glimpse into the consti- tution of nuclei. They are likewise of extreme importance in medical research, since they offer a substitute for radium. Radioactive substances can be fol- lowed through any physical or chemical changes by observing their emitted radiations. This "tracer," or u tagged-atom," technique is used in industry, medicine, physiology, and biology. F-M Cyclotron and Synchrotron It is shown in Sec. 1-10 that if an elec- tron falls through a potential of more than 3 kV, a relativistic mass correction must be made, indicating that its mass increases with its energy. Thus, if electrons were used in a cyclotron, their angular velocity would decrease as their energy increased, and they would soon fall out of step with the high-fre- quency field. For this reason electrons are not introduced into the cyclotron. For positive ions whose mass is several thousand times that of the elec- tron, the relativistic correction becomes appreciable when energies of a few tens of millions of electron volts are reached. For greater energies than these, the ions will start to make their trip through the dees at a slower rate and Blip behind in phase with respect to the electric field. This difficulty is overcome in the synchrocyclotron, or f-m cyclotron, by decreasing the frequency of the oscillator (frequency modulation) in accordance with the decrease in the angu- lar velocity of the ion. With such an f-m cyclotron, deuterons, a particles, and protons have been accelerated to several hundred million electron volts. 7 It is possible to give particles energies in excess of those for which the relativistic correction is important even if the oscillator frequency is fixed, provided that the magnetic field is slowly increased in step with the increase in the mass of the ions so as to maintain a constant angular velocity. Such an instrument is called a synchrotron. The particles are injected from a gun, which gives them a velocity approaching that of light. Since the radius of the orbit is given by R = mv/Be and since the ratio m/B is kept constant and v changes very little, there is not much of an increase in the orbit as the energy of the electron increases. The vacuum chamber is built in the form of a doughnut instead of the cyclotron pillbox. The magnet has the form of a REFERENCES 1. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," chaps. 14 and 19, McGraw-Hill Book Company, New York, 1965. 2. Goedicke, E. : Eine Neubestimmung der spezifischen Ladung des Electrons nach der Methode von H. Busch, Physik. Z., vol. 36, no. 1, pp. 47-63, 1939. 3. Cosslett, V. E.: "Introduction to Electron Optics," Oxford University Press, Fair Lawn, N.J., 1946. 4. Millman, J., and S. Seely: "Electronics," 2d ed., p. 35, McGraw-Hill Book Com- pany, New York, 1951. 5. James, G., and R. C. James: "Mathematics Dictionary," D. Van Nostrand Com- pany, Inc., Princeton, N.J., 1949. 6. Livingston, M. S.: The Cyclotron, I, J. Appl. Phys., vol. 15, pp. 2-19, January, 1944; The Cyclotron, II, ibid., pp. 128-147, February, 1944. Livingston, M. S.: Particle Accelerators, Advan. Electron., Electrochem. Eng., vol. 1, pp. 269-316, 1948. 7. Brobeck, W. M., E. 0. Lawrence, K. R. MaeKenzie, E. M. McMillan, R. Serber, D. C. Sewell, K. M. Simpson, and R. L. Thornton: Initial Performance of the 184- inch Cyclotron of the University of California, Phys. Rev., vol. 71, pp. 449-450, April, 1947. 8. Livingston, M. S., J. P. Blewett, G. K. Green, and L. J. Haworth: Design Study for a Three-Bev Proton Accelerator, Rev. Set. Tnstr., vol. 21, pp. 7-22, January, 1950. 9. Courant, E. D., M. S. Livingston, and H. 8. Snyder: The Strong-focusing Syn- chrotron: A New High Energy Accelerator, Phys. Rev., vol. 88, pp. 1190-1196, December, 1952. 10. Livingston, M. S., and J. P. Blewett: "Particle Accelerators," chap. 15, McGraw- Hill Book Company, New York, 1962. 2 /ENERGY LEVELS AND ENERGY BANDS In this chapter we begin with a review of the basic atomic properties of matter leading to discrete electronic energy levels in atoms. We also examine some selected topics in quantum physics, such as the wave properties of matter, the Schrodinger wave equation, and the Pauli exclusion principle. We find that atomic energy levels are spread into energy bands in a crystal. This band structure allows us to distinguish between an insulator, a semiconductor, and a metal. 2-1 THE NATURE OF THE ATOM In order to explain many phenomena associated with conduction in gases, metals, and semiconductors and the emission of electrons from the surface of a metal, it is necessary to assume that the atom has loosely bound electrons which can be torn away from it. Rutherford, 1 in 1911, found that the atom consists of a nucleus of positive charge that contains nearly all the mass of the atom. Sur- rounding this central positive core are negatively charged electrons. As a specific illustration of this atomic model, consider the hydrogen atom. This atom consists of a positively charged nucleus (a proton) and a single electron. The charge on the proton is positive and is equal in magnitude to the charge on the electron. Therefore the atom as a whole is electrically neutral. Because the proton carries practi- cally all the mass of the atom, it will remain substantially immobile, whereas the electron will move about it in a closed orbit. The force of attraction between the electron and the proton follows Coulomb's law. It can be shown from classical mechanics that the resultant closed path will be a circle or an ellipse under the action of such a force. This motion is exactly analogous to that of the planets about 36 $•**•» ENERGY LEVELS AND ENERGY BANDS / 37 the sun, because in both eases the force varies inversely as the square of the distance between the particles. Assume, therefore, that the orbit of the electron in this planetary model f the atom is a circle, the nucleus being supposed fixed in space. It is a simple matter to calculate its radius in terms of the total energy W of the electron. The force of attraction between the nucleus and the electron is e i /^ir€ r 2 , where the electronic charge e is in coulombs, the separation r between the two particles is in meters, the force is in newtons, and e<, is the permittivity of free space. f By Newton's second law of motion, this must be set equal to the product of the electronic mass m in kilograms and the acceleration v ! /r toward the nucleus, where is the speed of the electron in its circular path, in meters per second. Then 4fir€„r 3 r (2-1) Furthermore, the potential energy of the electron at a distance r from the nucleus is — e 2 /4ire r, and its the conservation of energy, W = Am« a 4ir t<,T kinetic energy is fymv 1 . Then, according to (2-2) where the energy is in joules. Combining this expression with (2-1) produces »3 W = (2-3) which gives the desired relationship between the radius and the energy of the electron. This equation shows that the total energy of the electron is always negative. The negative sign arises because the potential energy has been chosen to be zero when r is infinite. This expression also shows that the energy of the electron becomes smaller (i.e., more negative) as it approaches closer to the nucleus. The foregoing discussion of the planetary atom has been considered only from the point of view of classical mechanics, using the classical model for the electron. However, an accelerated charge must radiate energy, in accordance with the classical laws of electromagnetism. If the charge is performing oscil- lations of a frequency /, the radiated energy will also be of this frequency. Hence, classically, it must be concluded that the frequency of the emitted radiation equals the frequency with which the electron is rotating in its circular orbit. There is one feature of this picture that cannot be reconciled with experi- ment. If the electron is radiating energy, its total energy must decrease by the amount of this emitted energy. As a result the radius r of the orbit must decrease, in accordance with Eq. (2-3). Consequently, as the atom radiates energy, the electron must move in smaller and smaller orbits, eventually fall- t The numerical value of e„ is in Appendix B. 38 / ELECTRONIC DEVICES AND CIRCUITS Sec. 2-2 S*c>2-2 ENERGY LEVELS AND ENERGY BANDS / 39 ing into the nucleus. Since the frequency of oscillation depends upon the size of the circular orbit, the energy radiated would be of a gradually changing fre- quency. Such a conclusion, however, is incompatible with the sharply denned frequencies of spectral lines. The Bohr Atom The difficulty mentioned above was resolved by Bohr in 1913. 2 He postulated the following three fundamental laws: 1. Not all energies as given by classical mechanics are possible, but the atom can possess only certain discrete energies. While in states correspond- ing to these discrete energies, the electron does not emit radiation, and the electron is said to be in a stationary, or nonradiating, state. 2. In a transition from one stationary state corresponding to a definite energy W% to another stationary state, with an associated energy W\, radi- ation will be emitted. The frequency of this radiant energy is given by Wi - Wi f = (2-4) where h is Planck's constant in joule-seconds, the W's are expressed in joules, and / is in cycles per second, or hertz. 3. A stationary state is determined by the condition that the angular momentum of the electron in this state is quantized and must be an integral multiple of k/2w. Thus nh 7TWT = 2tt (2-5) where n is an integer. Combining Eqs. (2-1) and (2-5), we obtain the radii of the stationary states (Prob. 2-1), and from Eq. (2-3) the energy level in joules of each state is found to be W n = - me* 1 8fc V n 2 (2-6) Then, upon making use of Eq. (2-4), the exact frequencies found in the hydro- gen spectrum are obtained, a remarkable achievement. The radius of the lowest state is found to be 0.5 A. 2-2 ATOMIC ENERGY LEVELS Though it is theoretically possible to calculate the various energy states of the atoms of the simpler elements, these levels must be determined indirectly from spectroscopic and other data for the more complicated atoms. The experi- mentally determined energy-level diagram for mercury is shown in Fig. 2-1. The numbers to the left of the horizontal lines give the energy of these levels in electron volts. The arrows represent some of the transitions that 10.39 8.86, 8.61- 8.53- 8.38" 7.33 7.73 H fig. 2-1 The lower energy levels of atomic mercury. 5.46 4.88 4.66 Ionization level of mercury r~= 10140 3650 M I J_J Normal state of neutral mercury have been found to exist in actual spectra, the attached numbers giving the wavelength of the emitted radiation, expressed in angstrom units (10~ 10 m). The light emitted in these transitions gives rise to the luminous character of the gaseous discharge. However, all the emitted radiation need not appear in the form of visible light, but may exist in the ultraviolet or infrared regions. The meaning of the broken lines is explained in Sec. 2-7. It is customary to express the energy value of the stationary states in electron volts E rather than in joules W. Also, it is more common to specify the emitted radiation by its wavelength X in angstroms rather than by its frequency / in hertz. In these units, Eq. (2-4) may be rewritten in the form X m 12,400 Et — Ei (2-7) Since only differences of energy enter into this expression, the zero state may be chosen at will. It is convenient and customary to choose the lowest energy state as the zero level. This was done in Fig. 2-1. The lowest energy state is called the normal level, and the other stationary states of the atom are called excited, radiating, critical, or resonance levels. The most intense line in the mercury spectrum is that resulting from the transition from the 4.88-eV level to the zero state. The emitted radiation, as calculated from Eq. (2-7), is 12,400/4.88 = 2,537 A, as indicated in the diagram. It is primarily this line that is responsible for the ultraviolet burns which arise from mercury discharges. 40 / ELECTRONIC DEVICES AND CIRCUITS Sec. 2-3 2-3 THE PHOTON NATURE OF LIGHT The mean life of an excited state ranges from 10 -T to 10 -10 sec, the excited electron returning to its previous state after the lapse of this time. 3 In this transition, the atom must lose an amount of energy equal to the difference in energy between the two states that it has successively occupied, this energy appearing in the form of radiation. According to the postulates of Bohr, this energy is emitted in the form of a photon of light, the frequency of this radi- ation being given by Eq. (2-4). The term photon denotes an amount of radiant energy equal to the constant h times the frequency. This quantised nature of an electromagnetic wave was first introduced by Planck, 8 in 1901, in order to verify theoretically the blackbody radiation formula obtained experimentally. The photon concept of radiation may be difficult to comprehend at first. Classically, it was believed that the atoms were systems that emitted radi- ation continuously in all directions. According to the foregoing theory, how- ever, this is not true, the emission of light by an atom being a discontinuous process. That is, the atom radiates only when it makes a transition from one energy level to a lower energy state. In this transition, it emits a definite amount of energy of one particular frequency, namely, one photon hf of light. Of course, when a luminous discharge is observed, this discontinuous nature of radiation is not suspected because of the enormous number of atoms that are radiating energy and, correspondingly, because of the immense number of photons that are emitted in unit time. EX A M P LE Given a 50-W mercury-vapor lamp. Assume that 0.1 percent of the electric energy supplied to the lamp appears in the ultraviolet line, 2,537 A. Cal- culate the number of photons per second of this wavelength emitted by the lamp. Solution The energy per photon is, according to Eq. (2-7), 12,400 K 2,337 = 4.88 eV/photon The total power being transformed to the 2, 537- A line is 0.05 W, or 0.05 J/sec. Since 1 eV = 1.60 X 10~ 19 J, the power radiated is 0.05 J/sec - 3.12 x 10 17 eV/sec 1.60 X 10-" J/eV ' Hence the number of photons per second is 3.12 X 10»'eV/sec ' i-M . J . . oc ... . — - 6.40 X 10 18 photons/sec 4.88 eV/photon This is an extremely large number. 2-4 IONIZATION As the most loosely bound electron of an atom is given more and more energy, it moves into stationary states which are farther and farther away from the Sec. 2-6 ENERGY LEVELS AND ENERGY BANDS / 41 nucleus. When its energy is large enough to move it completely out of the field of influence of the ion, it becomes "detached" from it. The energy required for this process to occur is called the ionization potential and is represented as the highest state in the energy-level diagram. From an inspection of Fig. 2-1, this is seen to be 10.39 eV for mercury. The alkali metals have the lowest ionization potentials, whereas the inert gases have the highest values, the ionizing potentials ranging from approximately 4 to 25 eV. 2-5 COLLISIONS OF ELECTRONS WITH ATOMS The foregoing discussion has shown that, in order to excite or ionize an atom, energy must be supplied to it. This energy may be supplied to the atom in various ways, one of the most important of which is electron impact. Other methods of ionization or excitation of atoms are considered below. Suppose that an electron is accelerated by the potential applied to a dis- charge tube. When this electron collides with an atom, one of several effects may occur. A slowly moving electron suffers an "elastic" collision, i.e., one that entails an energy loss only as required by the laws of conservation of energy and momentum. The direction of travel of the electron will be altered by the collision although its energy remains substantially unchanged. This follows from the fact that the mass of the gas molecule is large compared with that of the electron. If the electron possesses sufficient energy, the amount depending upon the particular gas present, it may transfer enough of its energy to the atom to elevate it to one of the higher quantum states. The amount of energy neces- sary for this process is the excitation, or radiation, potential of the atom. If the impinging electron possesses a higher energy, say, an amount at least equal to the ionization potential of the gas, it may deliver this energy to an electron of the atom and completely remove it from the parent atom. Three charged particles result from such an ionizing collision: two electrons and a positive ion. It must not be presumed that the incident electron must possess an energy corresponding exactly to the energy of a stationary state in an atom in order to raise the atom into this level. If the bombarding electron has gained more than the requisite energy from the electric field to raise an atom into a par- ticular energy state, the amount of energy in excess of that required for exci- tation will be retained by the incident electron as kinetic energy after the collision. Or if the process of ionization has taken place, the excess energy divides between the two electrons. 2 "6 COLLISIONS OF PHOTONS WITH ATOMS Another important method by which an atom may be elevated into an excited energy state is to have radiation fall on the gas. An atom may absorb a photon of frequency / and thereby move from the level of energy Wi to the higher energy level W h where tF a = TFi + kf. 42 / ElECTRONIC DEVICES AND CIRCUITS Sec. 2-7 An extremely important feature of excitation by photon capture is that the photon will not be absorbed unless its energy corresponds exactly to the energy difference between two stationary levels of the atom with which it collides. Con- sider, for example, the following experiment: The 2, 537- A mercury radiation falls on sodium vapor in the normal state. What is the result of this irradi- ation? The impinging photons have an energy of 12,400/2,537 = 4.88 eV, whereas the first excitation potential of sodium is only 2.09 eV. It is con- ceivable that the sodium atom might be excited and that the excess energy 4.88 — 2.09 = 2.79 eV would appear as another photon of wavelength 12,400/2.79 = 4,440 A. Actually, however, the 2,537-A line is transmitted without absorption through the sodium vapor, neither of the two lines appear- ing. We conclude, therefore, that the probability of excitation of a gas by photon absorption is negligible unless the energy of the photon corresponds exactly to the energy difference between two stationary states of the atoms of the gas. When a photon is absorbed by an atom, the excited atom may return to its normal state in one jump, or it may do so in several steps. If the atom falls into one or more excitation levels before finally reaching the normal state, it will emit several photons. These will correspond to energy differences between the successive excited levels into which the atom falls. None of the emitted photons will have the frequency of the absorbed radiation! This fluorescence cannot be explained by classical theory, but is readily understood once Bohr's postulates are accepted. If the frequency of the impinging photon is sufficiently high, it may have enough energy to ionize the gas. The photon vanishes with the appearance of an electron and a positive ion. Unlike the case of photoexcitation, the photon need not possess an energy corresponding exactly to the ionizing energy of the atom. It need merely possess at least this much energy. If it possesses more than ionizing energy, the excess will appear as the kinetic energy of the emitted electron and positive ion. It is found by experiment, however, that the maximum probability of photoionization occurs when the energy of the photon is equal to the ionization potential, the probability decreasing rapidly for higher photon energies. 2-7 METASTABLE STATES Stationary states may exist which can be excited by electron bombardment but not by photoexcitation. Such levels are called metastable states. A tran- sition from a metastable level to the normal state with the emission of radiation has a very low probability of taking place. The 4.66- and 5.46-eV levels in Fig. 2-1 are metastable states. The forbidden transitions are indicated by dashed arrows on the energy-level diagram. Transitions from a higher level to a metastable state are permitted, and several of these are shown in Fig. 2-1. The mean life of a metastable state is found to be very much longer than s*c. 2-8 ENERGY IEVE1S AND ENERGY BANDS / 43 the mean life of a radiating level. Representative times are 10~ 2 to 10~* sec for metastable states and 10 -7 to 10~ 10 sec for radiating levels. The long lifetime of the metastable states arises from the fact that a transition to the normal state with the emission of a photon is forbidden. How then can the energy of a metastable state be expended so that the atom may return to its normal state? One method is for the metastable atom to collide with another molecule and give up its energy to the other molecule as kinetic energy of translation, or potential energy of excitation. Another method is that by which the electron in the metastable state receives additional energy by any of the processes enumerated in the preceding sections. The metastable atom may thereby be elevated to a higher energy state from which a transition to the normal level can take place, or else it may be ionized. If the metastable atom diffuses to the walls of the discharge tube or to any of the electrodes therein, either it may expend its energy in the form of heat or the metastable atoms might induce secondary emission. 2-8 THE WAVE PROPERTIES OF MATTER In Sec. 2-6 we find that an atom may absorb a photon of frequency / and move from the energy level Wi to the higher energy level W%, where W 3 = Wi + hf Since a photon is absorbed by only one atom, the photon acts as if it were concentrated in one point in space, in contradiction to the concept of a wave associated with radiation. In Chap. 19, where we discuss the photoelectric effect, it is again necessary to assign to a photon the property of a particle in order to explain the results of experiments involving the interaction of radi- ation and matter. According to a hypothesis of De Broglie, 3 in 1924, the dual character of wave and particle is not limited to radiation alone, but is also exhibited by particles such as electrons, atoms, molecules, or macroscopic masses. He calculated that a particle of mass m traveling with a velocity v has a wave- length X associated with it given by X - — - - mv p (2-8) where p is the momentum of the particle. The existence of such matter waves was demonstrated experimentally by Davisson and Germcr in 1927 and Thom- son in 1928. We can make use of the wave properties of a moving electron to establish Bohr's postulate that a stationary state is determined by the con- dition that the angular momentum must be an integral multiple of h/2ir. It seems reasonable to assume that an orbit of radius r will correspond to a sta- tionary state if it contains a standing-wave pattern. In other words, a stable orbit is one whose circumference is exactly equal to the electronic wavelength X, 44 / ELECTRONIC DEVICES AND CIRCUITS Soc. 2-6 or to nX, where n is an integer (but not zero). Thus 2ttt = n\=— (2-9) Clearly, Eq. (2-9) is identical with the Bohr condition [Eq. (2-5)], Wave Mechanics Schrodinger carried the implication of the wave nature of the electron further and developed a branch of physics called wave mechanics, or quantum mechanics. He argued that, if De Brogue' s concept is correct, it should be possible to deduce the properties of an electron system from a mathematical relationship such as the wave equation of electromagnetic theory, optics, mechanical vibrations, etc. Such a wave equation is v 2 Bt 2 (2-10) where 3z 2 x By 2 B 2 + — ^ Bz 2 and v is the velocity of the wave, and t is time. The physical meaning of <j> depends upon the problem under consideration. It may be one component of electric field, the mechanical displacement, the pressure, etc., depending upon the physical problem. We can eliminate the time variable by assuming a solution of the form *(*, V, *i - Mx, V, z)e*< (2-11) where a = 2wf is the angular frequency. Then Eq. (2-10) becomes VV+^*-0 (2-12) where X = v/f = the wavelength. From De Broglie's relationship [Eq. (2-8)], X s h* h 2 K } (2-13) where use has been made of the fact that the kinetic energy p 2 /2m is the difference between the total energy W and the potential energy U. Substi- tuting Eq. (2-13) in (2-12) gives the time-independent Schrodinger equation 87r 2 m vv + ^-jF (W - W = (2-14) The $ in Eq. (2-14) is called the wave function, and it must describe the behavior of the particle. But what is the physical meaning of ^? It is found that the proper interpretation of \p is that it is a quantity whose square gives the probability of finding the electron. In other words, \\fr\ 2 dz dy dz is pro- portional to the probability that the electron is in the volume dz dy dz at the point P(z, y, z) in space. The wave function $ must be normalized, that is, ///l^l 2 dxdy dz over all space equals unity, indicating that the probability of S*. 2-9 ENERGY LEVELS AND ENERGY BANDS / 45 finding the electron somewhere must be unity. Quantum mechanics makes no attempt to locate a particle at a precise point P in space, but rather the Schrodinger equation determines only the probability that the electron is to be found in the neighborhood of P. The potential energy U(z, y, z) specifies the physical problem at hand. For the electron in the hydrogen atom, U = — e 2 /lw€„r, whereas for a crystal, it is a complicated periodic function of space. The solution of Schrodinger's equation, subject to the proper boundary conditions, yields the allowed total energies W n (called characteristic values, or eigenvalues) of the particle and the corresponding wave functions ^ n (called eigenf unctions). Except for the very simplest potRntial functions (as in Sec. 3-6), there is considerable mathemati- cal complexity in solving for \ff. Hence we shall not obtain the solution of the Schrodinger equation for the hydrogen atom, but shall state the important result that such a solution leads to precisely the energy levels given in Eq. (2-3) which were obtained from the simpler Bohr picture of the atom. 2-9 ELECTRONIC STRUCTURE OF THE ELEMENTS The solution of the Schrodinger equation for hydrogen or any multielectron atom need not have radial symmetry. The wave functions may be a function of the azimuthal and polar angles as well as of the radial distance. It turns out that, in the general case, four quantum numbers are required to define the wave function. The total energy, the orbital angular momentum, the component of this angular momentum along a fixed axis in space, and the electron spin are quantized. The four quantum numbers are identified as follows : 1. The principal quantum number n is an integer 1, 2, 3, . . . and deter- mines the total energy associated with a particular state. This number may be considered to define the size of the classical elliptical orbit, and it corre- sponds to the quantum number n of the Bohr atom. 2. The orbital angular momentum quantum number I takes on the values 0, 1, 2, , . . , (n — 1). This number indicates the shape of t he classical orbit. The magnitude of this angular momentum is s/{l)(l + 1) (h/2ir). 3. The orbital magnetic number mi may have the values 0, ± 1, ±2, ... , ± I. This number gives the orientation of the classical orbit with respect to an applied magnetic field. The magnitude of the component of angular momentum along the direction of the magnetic field is mi{h/2ir). 4. EUctron spin. In order to explain certain spectroscopic and magnetic phenomena, Uhlenbeck and Goudsmit, in 1925, found it necessary to assume that, in addition to traversing its orbit around the nucleus, the electron must also rotate about its own axis. This intrinsic electronic angular momentum is called electron spin. When an electron system is subjected to a magnetic field, the spin axis will orient itself either parallel or antiparallel to the direc- 46 / ELECTRONIC DEVICES AND CIRCUITS Sec. 2-? S«. 2-10 tion of the field. The spin is thus quantized to one of two possible values. The electronic angular momentum is given by m.(A/2ir), where the spin quan- tum number m, may assume only two values, +i or — £. The Exclusion Principle The periodic table of the chemical elements (given in Appendix C) may be explained by invoking a law enunciated by Pauli in 1925. He stated that no two electrons in an electronic system can have the same set of four quantum numbers, n, I, m h and m,. This statement that no two electrons may occupy the same quantum state is known as the Pauli exclusion principle. Electronic Shells All the electrons in an atom which have the same value of n are said to belong to the same electron shell. These shells are identified by the letters K, L, M, N, . . . , corresponding to n = 1, 2, 3, 4, ... , respectively. A shell is divided into subskells corresponding to different values of I and identified as s, p, d, f, . . . , corresponding to I = 0, 1, 2, 3, . . . , respectively. Taking account of the exclusion principle, the distribution of electrons in an atom among the shells and subshells is indicated in Table 2-1. Actually, seven shells are required to account for all the chemical elements, but only the first four are indicated in the table. There are two states for n = 1 corresponding to I - 0, m x = 0, and m* = ±$. These are called the Is states. There are two states correspond- ing to n = 2, I « 0, m t = 0, and m s = +|. These constitute the 2s sub- shell. There are, in addition, six energy levels corresponding to n = 2, I = 1, mi = -1, 0, or +1, and m, = ±|. These are designated as the 2p subshell. Hence, as indicated in Table 2-1, the total number of electrons in the L shell is 2 + 6 = 8. In a similar manner we may verify that a d subshell contains a maximum of 10 electrons, an / subshell a maximum of 14 electrons, etc. The atomic number Z gives the number of electrons orbiting about the nucleus. Let us use superscripts to designate the number of electrons in a particular subshell. Then sodium, Na, for which Z = 11, has an electronic configuration designated by ls*2s*2p«Zs\ Note that Na has a single electron m the outermost unfilled subshell, and hence is said to be monovalent. This TABLE 2-1 Electron shells and si bshell: Shell K 1 L 2 M 3 N 4 / s s 1 P 3 1 P 2 d s 1 P 2 d Subshell 3 / Number | of | 2 2 6 2 6 10 2 6 10 14 electrons) 2 8 18 32 ENERGY IEVEIS AND ENERGY BANDS / 47 TABLE 2-2 Electronic configuration in Group IVA Element Atomic number Configuration C Si Ge Sn 6 14 32 50 ls»2s J 2p* laW2p B 3s s 3p 8 3di°4s*4p s Is^s^^Ss^pW^sH^d'oS^Sp* same property is possessed by all the alkali metals (Li, Na, K, Rb, and Cs), which accounts for the fact that these elements in the same group in the periodic table (Appendix C) have similar chemical properties. The inner-shell electrons are very strongly bound to an atom, and cannot be easily removed. That is, the electrons closest to the nucleus are the most tightly bound, and so have the lowest energy. Also, atoms for which the electrons exist in closed shells form very stable configurations. For example, the inert gases He, Ne, A, Kr, and Xe all have either completely filled shells or, at least, completely filled subshells. Carbon, silicon, germanium, and tin have the electronic configurations indicated in Table 2-2. Note that each of these elements has completely filled subshells except for the outermost p shell, which contains only two of the six possible electrons. Despite this similarity, carbon in crystalline form (diamond) is an insulator, silicon and germanium solids are semiconductors, and tin is a metal. This apparent anomaly is explained in the next section. 2-10 THE ENERGY-BAND THEORY OF CRYSTALS X-ray and other studies reveal that most metals and semiconductors are crystalline in structure. A crystal consists of a space array of atoms or molecules (strictly speaking, ions) built up by regular repetition in three dimensions of some fundamental structural unit. The electronic energy levels discussed for a single free atom (as in a gas, where the atoms are sufficiently far apart not to exert any influence on one another) do not apply to the same atom in a crystal. This is so because the potential U in Eq. (2-14), charac- terizing the crystalline structure, is now a periodic function in space whose value at any point is the result of contributions from every atom. When atoms form crystals it is found that the energy levels of the inner-shell elec- trons are not affected appreciably by the presence of the neighboring atoms. However, the levels of the outer-shell electrons arc changed considerably, since these electrons are shared by more than one atom in the crystal. The new energy levels of the outer electrons can be determined by means of quantum Mechanics, and it is found that coupling between the outer-shell electrons of ^e atoms results in a band of closely spaced energy states instead of the 48 / ELECTRONIC DEVICES AND CIRCUITS See. 2-70 widely separated energy levels of the isolated atom (Fig. 2-2). A qualitative discussion of this energy-band structure follows. Consider a crystal consisting of N atoms of one of the elements in Table 2-2. Imagine that it is possible to vary the spacing between atoms without altering the type of fundamental crystal structure. If the atoms are so far apart that the interaction between them is negligible, the energy levels will coincide with those of the isolated atom. The outer two subshells for each element in Table 2-2 contain two s electrons and two p electrons. Hence, if we ignore the inner-shell levels, then, as indicated to the extreme right in Fig. 2-2a, there are 2N electrons completely filling the 2N possible s levels, all at the same energy. Since the p atomic subshell has six possible states! our imaginary crystal of widely spaced atoms has 2N electrons, which fill only one-third of the 67V possible p states, all at the same level. If we now decrease the interatomic spacing of our imaginary crystal (moving from right to left in Fig. 2-2a), an atom will exert an electric force on its neighbors. Because of this coupling between atoms, the atomic-wave functions overlap, and the crystal becomes an electronic system which must obey the Pauli exclusion principle. Hence the 2N degenerate s states must spread out in energy. The separation between levels is small, but since N is very large (~10 23 cm" 8 ), the total spread between the minimum and maximum energy may be several electron volts if the interatomic distance is decreased sufficiently. This large number of discrete but closely spaced energy levels is called an energy band, and is indicated schematically by the lower shaded Isolated atom f 6N states ^*4 ' I 2N electrons ri«i , , J 2N p electrons /'{QN s tates [Energy tfe % electrons 8 a P y\ 2 N s tates 1 2 A' states | 2N electrons I Inner-shell atomic energy levels unaffected by crystal formation (4N states electrons Conduction band Interatomic spacing, d (a) (4W states < 4N electrons [ Valence band j, Crystal lattice I / spacing (6) Fig. 2-2 Illustrating how the energy [evels of isolated atoms are split into energy bands when these atoms are brought Into close proximity to form a crystal. &K. 2-" ENERGY LEVELS AND ENERGY SANDS / 49 region in Fig. 2-2a. The 2N states in this band are completely filled with 2/V electrons. Similarly, the upper shaded region in Fig. 2-2a is a band of 6# states which has only 2/V of its levels occupied by electrons. Note that there is an energy gap (a forbidden band) between the two bands discussed above and that this gap decreases as the atomic spacing decreases. For small enough distances (not indicated in Fig, 2-2a but shown in Fig. 2-26) these bands will overlap. Under such circumstances the 6iV upper states merge with the 2/V lower states, giving a total of 8/V levels, half of which are occupied by the 2N + 2N = 4/V available electrons. At this spacing each atom has given up four electrons to the band; these electrons can no longer be said to orbit in s or p subshells of an isolated atom, but rather they belong to the crystal as a whole. In this sense the elements in Table 2-2 are tetravalent, since they contribute four electrons each to the crystal. The band these electrons occupy is called the valence band. If the spacing between atoms is decreased below the distance at which the bands overlap, the interaction between atoms is indeed large. The energy- band structure then depends upon the orientation of the atoms relative to one another in space (the crystal structure) and upon the atomic number, which determines the electrical constitution of each atom. Solutions of Schrodinger's equation are complicated, and have been obtained approximately for only rela- tively few crystals. These solutions lead us to expect an energy-band diagram somewhat as pictured 4 in Fig. 2-26. At the crystal-lattice spacing (the dashed vertical line), we find the valence b&nd filled with 4/V electrons separated by a forbidden band (no allowed energy states) of extent Eq from an empty band consisting of 4/V additional states. This upper vacant band is called the con- duction band, for reasons given in the next section. 2-H INSULATORS, SEMICONDUCTORS, AND METALS A very poor conductor of electricity is called an insulator; an excellent con- ductor is a metal; and a substance whose conductivity lies between these extremes is a semiconductor, A material may be placed in one of these three classes, depending upon its energy-band structure. Insulator The energy-band structure of Fig. 2-26 at the normal lattice spacing is indicated schematically in Fig. 2-3a. For a diamond (carbon) crystal the region containing no quantum states is several electron volts high (£c « 6 eV). This large forbidden band separates the filled valence region trom the vacant conduction band. The energy which can be supplied to an electron from an applied field is too small to carry the particle from the filled lr *to the vacant band. Since the electron cannot acquire externally applied energy, conduction is impossible, and hence diamond is an insulator. Semiconductor A substance for which the width of the forbidden energy region is relatively small (~1 eV ) is called a semiconductor. Graphite, a 50 / ELECTRONIC DEVICES AND CIRCUITS Sk. 2-1 T E c * 6 eV . Holes T Conduction band JL Valence band (c) Fig. 2-3 Energy-band structure of (a) an insulator, (b) a semiconductor, and (c) a metal. crystalline form of carbon but having a crystal symmetry which is different from diamond, has such a small value of E G , and it is a semiconductor. The most important practical semiconductor materials are germanium and silicon, which have values of E 6 of 0.785 and 1.21 eV, respectively, at 0°K. Energies of this magnitude normally cannot be acquired from an applied field. Hence the valence band remains full, the conduction band empty, and these materials are insulators at low temperatures. However, the conductivity increases with temperature, as we explain below, and for this reason these substances are known as intrinsic semiconductors. As the temperature is increased, some of these valence electrons acquire thermal energy greater than E G and hence move into the conduction band. These are now free electrons in the sense that they can move about under the influence of even a small applied field. These free, or conduction, elec- trons are indicated schematically by dots in Fig. 2-36. The insulator has now become slightly conducting; it is a semiconductor. The absence of an electron in the valence band is represented by a small circle in Fig. 2-36, and is called a hole. The phrase "holes in a semiconductor" therefore refers to the empty energy levels in an otherwise filled valence band. The importance of the hole is that it may serve as a carrier of electricity, comparable in effectiveness with the free electron. The mechanism by which a hole contributes to conductivity is explained in Sec. 5-1. We also show in Chap. 5 that if certain impurity atoms are introduced into the crystal, these result in allowable energy states which lie in the forbidden energy gap. We find that these impurity levels also contribute to the conduction. A semi- conductor material where this conduction mechanism predominates is called an extrinsic (impurity) semiconductor. Since the band-gap energy of a crystal is a function of interatomic spacing (Fig. 2-2), it is not surprising that Eg depends somewhat on temperature. It has been determined experimentally that E G for silicon decreases with ENERGY LEVELS AND ENERGY BANDS / 51 temperature at the rate of 3.60 X 10 -4 eV/°K. Hence, for silicon, 5 E (T) = 1.21 - 3.60 X 10- 4 T (2-15) and at room temperature (300°K), E = 1.1 eV. Similarly, for germanium, 8 E (T) = 0.785 - 2.23 X 10~*r (2-16) and at room temperature, E G = 0.72 eV. Metal The band structure of a crystal may contain no forbidden energy region, so that the valence band merges into an empty band, as indicated in Fig. 2-3c. Under the influence of an applied electric field the electrons may acquire additional energy and move into higher energy states. Since these mobile electrons constitute a current, this substance is a conductor, and the empty region is the conduction band. A metal is characterized by a band structure containing overlapping valence and conduction bands. REFERENCES 1. Rutherford, E.: The Scattering of a and Particles by Matter and the Structure of the Atom, Phil. Mag., vol. 21, pp. 669-688, May, 1911. 2. Bohr, N\: On the Constitution of Atoms and Molecules, Part 2: Systems Containing Only a Single Nucleus, Phil Mag., vol. 26, pp. 476-502. September, 1913. 3. Richtmyer, F. K., E. H. Kennard, and T. Lauritsen: "Introduction to Modern Physics," McGraw-Hill Book Company, New York, 1955. 4. Adler, R. B., A. C. Smith, and R. L. Longini: "Introduction to Semiconductor Physics," vol. 1, p. 78, Semiconductor Electronics Education Committee, John Wiley & Sons, Inc., New York, 1964. 5. Morin, F. J., and J. P. Maita: Electrical Properties of Silicon Containing Arsenic and Boron, Phys. Rev., vol. 96, pp. 28-35, October. 1954. *• Morin, F. J., and J. P. Maita: Conductivity and Hall Effect in the Intrinsic Range of Germanium, Phys. Rev., vol. 94, pp. 1525-1529, June, 1954. 3/ CONDUCTION IN METALS In this chapter we describe the interior of a metal and present the basic principles which characterize the movement of electrons within the metal. The laws governing the emission of electrons from the surface of a metal are also considered. 3-1 MOBILITY AND CONDUCTIVITY In the preceding chapter we presented an energy-band picture of metals, semiconductors, and insulators. In a metal the outer, or valence, electrons of an atom are as much associated with one ion as with another, so that the electron attachment to any individual atom is almost zero. In terms of our previous discussion this means that the band occupied by the valence electrons may not be com- pletely filled and that there are no forbidden levels at higher energies. Depending upon the metal, at least one, and sometimes two or three, electrons per atom are free to move throughout the interior of the metal under the action of applied fields. Figure 3-1 shows the charge distribution within a metal, specifi- cally, sodium. 1 The plus signs represent the heavy positive sodium nuclei of the individual atoms. The heavily shaded regions represent the electrons in the sodium atom that are tightly bound to the nucleus. These are inappreciably disturbed as the atoms come together to form the metal. The unshaded volume contains the outer, or valence, elec- trons in the atom. It is these electrons that cannot be said to belong to any particular atom; instead, they have completely lost their indi- viduality and can wander freely about from atom to atom in the metal. Thus a metal is visualized as a region containing a periodic three-dimensional array of heavy, tightly bound ions permeated with 52 S9C. 3-1 CONDUCT/ON IN METALS / 53 Fig. 3-1 Arrangement of the sodium atoms in one plane of the metal. © © © D © © © d © © © © ) © © © • • m © 0X2345 1 ' a' ' ' ' A units a swarm of electrons that may move about quite freely. This picture is known as the electron-gas description of a metal. According to the electron-gas theory of a metal, the electrons are in continuous motion, the direction of flight being changed at each collision with the heavy (almost stationary) ions. The average distance between col- lisions is called the mean free path. Since the motion is random, then, on an average, there will be as many electrons passing through unit area in the metal in any direction as in the opposite direction in a given time. Hence the average current is zero. Let us now see how the situation is changed if a constant electtic field S (volts per meter) is applied to the metal. As a result of this electrostatic force, the electrons would be accelerated and the velocity would increase indefinitely with time, were it not for the collisions with the ions. However, at each inelastic collision with an ion, an electron loses energy, and a steady- state condition is reached where a finite value of drift speed v is attained. This drift velocity is in the direction opposite to that of the electric field, and its magnitude is proportional to S. Thus v = n& (3-1) where ^ (square meters per volt-second) is called the mobility of the electrons. According to the foregoing theory, a steady-state drift speed has been superimposed upon the random thermal motion of the electrons. Such a directed flow of electrons constitutes a current. If the concentration of free electrons is n (electrons per cubic meter), the current density J (amperes per square meter) is (Sec. 1-12) J = nev = nefi& = erS (3-2) 54 / ELECTRONIC DEVICES AND CIRCUITS Sec. 3-2 Sec. 3-2 CONDUCTION IN METALS / 55 where = net (3 _ 3) is the conductivity of the metal in (ohm-meter)" 1 . Equation (3-2) is recog- nized as Ohm's law, namely, the conduction current is proportional to the applied voltage. As already mentioned, the energy which the electrons acquire from the applied field is, as a result of collisions, given to the lattice ions. Hence power is dissipated within the metal by the electrons, and the power density (Joule heat) is given by J& = a& (watts per cubic meter). 3-2 THE ENERGY METHOD OF ANALYZING THE MOTION OF A PARTICLE A method is considered in Chap. 1 by which the motion of charged particles may be analyzed. It consists in the solution of Newton's second law, in which the forces of electric and magnetic origin are equated to the product of the mass and the acceleration of the particle. Obviously, this method is not applicable when the forces are as complicated as they must be in a metal. Furthermore, it is neither possible nor desirable to consider what happens to each individual electron. It is necessary, therefore, to consider an alternative approach. This method employs the law of the conservation of energy, use being made of the potential-energy curve corresponding to the field of force. The principles involved may best be understood by considering specific examples of the method. EXAMPLE An Idealized diode consists of plane-parallel electrodes, 5 cm apart. The anode A is maintained 10 V negative with respect to the cathode K. \n electron leaves the eathode with an initial energy of 2 eV. What is the maximum distance it can travel from the cathode? Solution This problem is analyzed by the energy method. Figure 3-2a is a linear Potential, V '.ru-rgy Potential energy V Total energy Jl' Distance, x Fig- 3-2 (o) Potential vs. distance in a plane-parallel diode, (b) The potential-energy barrier encountered by an electron in the retarding field. plot of potential vs. distance, and in Fig. 3-26 is indicated the corresponding potential energy vs. distance. Since potential is the potential energy per unit charge (Sec. 1-4), curve b is obtained from curve a by multiplying each ordinate by the charge on the electron (a negative number). Since the total energy W of the electron remains constant, it is represented as a horizontal line. The kinetic energy at any distance x equals the difference between the total energy W and the potential energy V at this point. This difference is greatest at 0, indicating that the kinetic energy is a maximum when the electron leaves the cathode. At the point P this difference is zero, which means that no kinetic energy exists, so that the particle is at rest at this point. This distance x„ is the maximum that the electron can travel from the cathode. At point P it comes momentarily to rest, and then reverses its motion and returns to the cathode. From geometry it is seen that x g /b = -i%, or x = 1 cm. Consider a point such as S which is at a greater distance than 1 cm from the cathode. Here the total energy QS is less than the potential energy RS, so that the difference, which represents the kinetic energy, is negative. This is an impos- sible physical condition, however, since negative kinetic energy (-^mv 2 < 0) implies an imaginary velocity. We must conclude that the particle can never advance a distance greater than OP' from the cathode. The foregoing analysis leads to the very important conclusion that the shaded portion of Fig. 3-26 can never be penetrated by the electron. Thus, at point P, the particle acts as if it had follided with a solid wall, hill, or barrier and the direction of its flight had been altered. Potential-energy barriers of this sort play important roles in the analyses to follow. It must be emphasized that the words "collides with" or "rebounds from" a potential "hill" are convenient descriptive phrases and that an actual encounter between two material bodies is not implied. As a second illustration, consider a mathematical pendulum of length £, consisting of a "point" bob of mass m that is free to swing in the earth's gravitational field. If the lowest point of the swing (point 0, Fig. 3-3) is chosen as the origin, the potential energy of the mass at any point P corre- sponding to any angle B of the swing is given by U = mgy = mgl(\ — cos 6) where g is the acceleration of gravity. This potential-energy function is illus- trated graphically in Fig. 3-4. fig. 3-3 Point F represents the mass •8 of a mathematical pendulum swinging in the earth's gravitational field. 56 / ELECTRONIC DEVICES AND CIRCUITS Sec. 3-2 Fig. 3-4 The potential energy of the moss m in Fig. 3-3 plotted as a function of the angle of swing. Consider the resultant motion of the bob if it is given a potential energy Ui by raising it through an angle B B and releasing it with zero initial velocity. If dissipation is neglected, the particle will swing back and forth through the angle 28 , going from 6 on one side to 8„ on the other side of the vertical axis. How might we analyze the motion of the physical system if only the potential- energy field of Fig. 3-4 were given without specifying the physical character of the system? The procedure is the same as that followed in the simple diode problem considered above. A horizontal line aebc is drawn at a height equal to the total energy 1*^ of the particle. At any point, such as e, the total energy is represented by eg = W x , and the potential energy is represented by fg. The difference between these two, namely, ef, represents the kinetic energy of the particle when the angle of swing, given by the intercept of eg on the axis, corresponds to Og. In other words, the difference between the total-energy line and the potential-energy curve at any angle represents the kinetic energy of the particle under these conditions. This difference is greatest at 0, indi- cating that the kinetic energy is a maximum at the bottom of the swing, an almost evident result. At the points a and b this difference is zero. This condition means that no kinetic energy exists, or that the particle is at rest at these points. This result is evident, since corresponding to the points a {6 = e B ) and b (8 = -$„), the particle is about to reverse its motion. Consider a point in the shaded region outside the range -B B to +0 O , such as A. Here the total energy ch is less than the potential energy dh. This impossible condition is interpreted by our previous reasoning to mean that the particle whose total energy is W i can never swing to the angle Oh, so that the motion must be confined to the region ah. The shaded portions of Fig. 3-4 represent the potential-energy barrier which can never be penetrated by the bob, if its total energy is no greater than W t . This type of constrained motion about a point is closely analogous to that of the so-called "bound" elec- trons in a metal, as shown in Sec. 3-4. Now consider the case when the mass has a total energy equal to W 2 , which is greater than the maximum of the potential-energy curve. Clearly, from Fig. %-4, the horizontal line corresponding to this energy cannot inter- sect the curve at any point. Consequently, the particle docs not "collide" with the potential-energy barrier, and its course is never altered, so that it S«. 3-3 CONDUCT/ON IN METALS / 57 moves through an ever-increasing angle. Of course, its kinetic energy varies over wide limits, being maximum for Q — 0, 2t, 4tt, . . . and minimum for D = *-, 3ir, Sir, . . . . Physically, this type of motion results when the bob has enough energy to set it spinning completely around in a circular path. This type of motion is somewhat analogous to that experienced by the so-called "free" electrons in a metal. This simple but powerful energy method facilitates the discussion of the motion of a particle in a conservative field of force, such as that found in the body of a metal. It is also applied to many other types of problem. For example, the method of analysis just considered is extremely useful in deter- mining whether electrons will possess sufficient energy to pass through grids and reach the various electrodes in a vacuum tube, whether or not electrons will be able to penetrate electron clouds in a vacuum tube, and whether charge carriers can cross a semiconductor junction. This method is now applied to the analysis of the motion of electrons in metals. THE POTENTIAL-ENERGY FIELD IN A METAL It is desired to set up the potential-energy field for the three-dimensional array of atoms that exists in the interior of a metal and to discuss the motion of electrons in this field. The resultant potential energy at any point in the metal is simply the sum of the potential energies produced at this point by all the ions of the lattice. To determine the potential energy due to one ion, it is noted that an atom of atomic number Z has a net positive charge Ze on its nucleus. Surrounding this nucleus is an approximately spherical cloud, or shell, of Z electrons. By Gauss' law the potential at a point at a distance r from the nucleus varies inversely as r and directly as the total charge enclosed within a sphere of radius r. Since the potential V equals the potential energy V per unit charge (Sec. 1-4), then U «= — eV. The minus sign is introduced since e represents the magnitude of the (negative) electronic charge. The potential of any point may be chosen as the zero reference of potential because it is only differences of potential that have any physical significance. For the present discussion it is convenient to choose zero potential at infinity, and then the potential energy at any point is negative. Enough has been said to make plausible the potential-energy curve illustrated in Fig. 3-5. Here a represents a nucleus, the potential energy of which is given by the curve aiot s . The vertical scale represents V, and the horizontal scale gives the distance r from the nucleus. It must be emphasized that r represents a radial distance from the nucleus, and hence can be taken in any direction. If the direction is horizontal but to the left of the nucleus, the dashed curve represents the Potential energy. To represent the potential energy at every point in space requires a four- dimensional picture, three dimensions for the three space coordinates and a fourth for the potential-energy axis. This difficulty is avoided by plotting U 58 / ELECTRONIC DEVICES AND CIRCUITS U Sec. 3-3 Fig, 3-5 The potential energy of an electron as a function of radial distance from an isolated nucleus, along some chosen line through the crystal, say, through a row of ions. From this graph and the method by which it is constructed it is easy to visualize what the potential energy at any other point might be. In order to build up this picture, consider first two adjacent ions, and neglect all others. The con- struction is shown in Fig. 3-6. a ia , is the U curve for nucleus a, and fluSj is the corresponding U curve for the adjacent nucleus 0. If these were the only nuclei present in the metal, the resultant U curve in the region between a and P would be the sum of these two curves, as shown by the dashed curve aidfa (since ad = ab + ac). It is seen that the resultant curve is very nearly the same as the original curves in the immediate vicinity of a nucleus, but it is lower and flatter than either individual curve in the region between the nuclei. Let us now single out an entire row of nuclei a, ft %$,«,... from the metallic lattice (Figs. 3-1 and 3-7) and sketch the potential energy as we pro- ceed along this line from one nucleus to the other, until the surface of the metal is reached. Following the same type of construction as above, but con- sidering the small influence of other nearby nuclei, an energy distribution somewhat as illustrated in Fig. 3-7 is obtained. According to classical electrostatics, which does not take the atomic structure into account, the interior of a metal is an equipotential region. The present, more accurate, picture shows that the potential energy varies apprecmbly in the immediate neighborhoods of the nuclei and actually tends to - « in these regions. However, the potential is approximately constant t/=o Fig. 3-6 The potential energy resulting from two nuclei, a and 0. Sec. 3-4 CONDUCTION IN METAtS / 59 Fig. 3-7 The potential-energy distribution within and at the surface of a metal. E/-0 for a very large volume of the metal, as indicated by the slowly varying por- tions of the diagram in the regions between the ions. Consider the conditions that exist near the surface of the metal. It is evident, according to the present point of view, that the exact position of the "surface" cannot be defined. It is located at a small distance from the last nucleus e in the row. It is to be noted that, since no nuclei exist to the right of e, there can be no lowering and flattening of the potential-energy curve such as prevails in the region between the nuclei. This leads to a most impor- tant conclusion; A potential-mergy "hill," or "barrier," exists at the surface of the metal. 3-4 BOUND AND FREE ELECTRONS The motion of an electron in the potential-energy field of Fig. 3-7 is now dis- cussed by the method given in Sec. 3-2. Consider an electron in the metal that possesses a total energy corresponding to the level A in Fig. 3-7. This electron collides with, and rebounds from, the potential walls at a and b. It cannot drift very far from the nucleus, but can move about only in the neigh- borhood ab of the nucleus. Obviously, this electron is strongly bound to the nucleus, and so is a bound electron. This particle is one of the inner-shell elec- trons of an isolated atom, discussed in Sec. 2-9. It is evident that these bound electrons do not contribute to the conductivity of the metal since they cannot drift in the metal, even under the stimulus of an externally applied electric field. These electrons are responsible for the heavy shading in the neighbor- hood of the nuclei of Fig. 3-1. Our present interest is in the free electrons in the metal rather than in the bound ones. A free electron is one having an energy such as level B of Fig. 3-7, corresponding to an energy in the conduction band. At no point vritkin the metal is its total energy entirely converted into potential energy. Hence, at no point is its velocity zero, and the electron travels more or less 60 / ELECTRONIC DEVICES AND CIRCUITS Energy, eV Sec. 3-5 Outside of metal Fig. 3-8 For the free electrons, Hie interior of a metal may be considered an equi- potentlal volume, but there is a potential barrier at the surface. Distance, x freely throughout the body of the metal. However, when the electron reaches the surface of the metal, it collides with the potential-energy barrier there. At the point C, its kinetic energy is reduced to zero, and the electron is turned back into the body of the metal. An electron having an energy correspond- ing to the level D collides with no potential walls, not even the one at the surface, and so it is capable of leaving the metal. Simplified Potential-energy Picture of a Metal In our subsequent dis- cussions the bound electrons are neglected completely since they in no way contribute to the phenomena to be studied. Attention is focused on the free electrons. The region in which they find themselves is essentially a potential plateau, or equipotential region. It is only for distances close to an ion that there is any appreciable variation in potential. Since the regions of rapidly varying potential represent but a very small portion of the total volume of the metal, we henceforth assume that the field distribution within the metal is equipotential and the free electrons are subject to no forces whatsoever. The present viewpoint is therefore essentially that of classical electrostatics. Figure 3-7 is redrawn in Fig. 3-8, all potential! variations within the metal being omitted, with the exception of the potential barrier at the surface. For the present discussion, the zero of energy is chosen at the level of the plateau of this diagram. This choice of the zero-energy reference level is valid since, as has already been emphasized, only difference of potential has physical sig- nificance. The region outside the metal is now at a potential equal to E B , the height of the potential-energy barrier in electron volts. 3-5 ENERGY DISTRIBUTION OF ELECTRONS In order to be able to escape, an electron inside the metal must possess an amount of energy at least as great as that represented by the surface barrier f This figure really represents potential energy, and not potential. However, the phrase "potential barrier" is much more common in the literature than the phrase "poten- tial-energy barrier." When no confusion is likely to arise, these two expressions are used interchangeably. Those barriers are measured in electron volts, and hence the symbol E replaces the U of the preceding sections. It must be emphasized that one unit of B repre- sents 1.60 X lO -1 * J of energy. S«.3.$ CONDUCTION IN METALS / 61 v It is therefore important to know what energies are possessed by the electrons in a metal. This relationship is called the energy distribution func- tion. We here digress briefly in order to make clear what is meant by a distri- bution function. Age Density Suppose that we were interested in the distribution in age of the people in the United States. A sensible way to indicate this relation- ship is shown in Fig. 3-9, where the abscissa is age and the ordinate is p A , the density of the population in age. This density gives the number dn A of people whose ages lie in the range between A and A + dA, or dn A = pa dA (3-4) The data for such a plot are obtained from census information. We see, for example, that the number of persons of ages between 10 and 12 years is repre- sented by dn A , with p A = 2.25 million per year chosen as the mean ordinate between 10 and 12 years, and dA is taken as 12 - 10 = 2 years. Thus dn A - PAdA = 4.50 million. Geometrically, this is the shaded area of Fig. 3-9. Evidently, the total population n is given by (3-5) n = J dn A = $p A dA or simply the total area under the curve. Energy Density We are now concerned with the distribution in energy of the free electrons in a metal. By analogy with Eq. (3-4), we may write dn E = pb dE 0-6) where dn B represents the number of free electrons per cubic meter whose energies lie in the energy interval dE, and p E gives the density of electrons in this interval. Since our interests are confined only to the free electrons, it is assumed that there are no potential variations within the metal. Hence there must be, a priori, the same number of electrons in each cubic meter of the metal. That is, the density in space (electrons per cubic meter) is Fig. 3-9 The distribution function in age of people in the United States. 40 60 Age, years 100 62 / ELECTRONIC DEVICES AND CIRCUITS Sec. 3-5 a constant. However, within each unit volume of metal there will be elec- trons having all possible energies. It is this distribution in energv that is expressed by PB (number of electrons per electron volt per cubic meter of metal) Ihe function Ps may be expressed as the product pb = f(E)N(E) (3-7) where N(E) is the density of states (number of states per electron volt per cub.c meter) m the conduction band, and/(£) is the probability that a quantum state with energy E is occupied by an electron. The expression for N(E) is derived in the following section and is given by N(E) - y E* where 7 is a constant defined by (3-8) 4ir y ~ h* ( 2m )'(1.60 X 10-»)l = 6.82 X 10" (3-9) The dimensions of y are (m->)(eV)-l; m is the mass of the electron in kilo- grams; and h is Planck's constant in joule-seconds. The Fermi-Dirac Function The equation for f(E) is called the Fermi- Dirac probability function, and specifies the fraction of all states at energy E (electron volts) occupied under conditions of thermal equilibrium From quantum statistics it is found 23 that /(£) = 1 1 + e< s -^)/W (3-10) where k = Boltzmann constant, eV/*K T = temperature, °K E P = Fermi level, or characteristic energy, for the crystal, eV Sin fiT!! * eve ' r T?,f ntS thG 6nergy 8tate ^ 50 P e ™ nt Probability of tZfxF r ll d ^ baUd eXiStS - The rea80n f0r this last ***™t is TzIp = *'■ /C ^ * f ° r any VaIue of ^Perature. A plot of f{E) boTh fo! T -ilT'T, f ^ 10a "^ ° f E ~ Ef VerSUS f{E) in Fig " Z ~ m > Doth tor T -OK and for larger values of temperature. When T = 0°K wo poss.ble conditions exist: (1) If E > E F , the exponential term becomes' infinite and f(E = Consequently, there is no probability of finding an occu- pied quantum state of energy greater than B, at absolute zero. (2) If E < Em he exponential in Eq (3-10) becomes zero and f{E) = 1. All ouantum levels with energies less than E F will be occupied at T = 0°K From Eqs. (3-7), (3-8), and (3-10), we obtain at absolute zero temperature »-{f for E < E F for E > E F (3-11) $k. 3 ' 5 f{E) * ^r=o°K 1.0 lx i r — r=300°K 0.8 ' NJ ^T=2500°K 0.6 t N 0.4 \ 0.2 i\., CONDUCTION IN METALS / 63 T=0"K -1.0 -0,6 -0.2 0.2 («) 0.6 1.0 E-E r ,eV 0.2 0.4 0.6 0.8 1.0 f(E) (&) Fig. 3-10 The Fermi-Dirac distribution function f{E) gives the probability that a state of energy E is occupied. Clearly, there are no electrons at 0°K which have energies in excess of E F . That is, the Fermi energy is the maximum energy that any electron may possess at absolute zero. The relationship represented by Eq. (3-11) is called the completely degenerate energy distribution function. Classically, all particles should have zero energy at 0°K. The fact that the electrons actually have energies extending from to E F at absolute zero is a consequence of the Pauli exclusion principle, which states that no two electrons may have the same set of quantum numbers (Sec. 2-9). Hence not all electrons can have the same energy even at 0°K. The application of Fermi-Dirac statistics to the theory of metals is due primarily to Sommerfeld. 3 A plot of the distribution in energy given by Eqs. (3-7) and (3-11) for metallic tungsten at T = 0°K and T = 2500°K is shown in Fig. 3-11. The area under each curve is simply the total number of particles per cubic meter of the metal; hence the two areas must be equal. Also, the curves for all temperatures must pass through the same ordinate, namely, ps = yE F */2, at the point E = E F , since, from Eq. (3-10), f(E) = £ for E = E F . A most important characteristic is to be noted, viz., the distribution function changes only very slightly with temperature, even though the tern- Fig. 3-11 Energy distribution in metallic tungsten otOand 2500° K. T=0°K 64 / ELECTRONIC DEVICES AND CIRCUITS Sec. 3-5 perature change is as great as 2500°K. The effect of the high temperature is merely to give those electrons having the high energies at absolute zero (those in the neighborhood of E F ) still higher energies, whereas those having lower energies have been left practically undisturbed. Since the curve for T = 2500°K approaches the energy axis asymptotically, a few electrons will have large values of energy. The Fermi Level An expression for E F may be obtained on the basis of the completely degenerate function. The area under the curve of Fig. 3-11 represents the total number of free electrons (as always, per cubic meter of the metal). Thus or f Er M3 (3-12) Inserting the numerical value (6.82 X 10") of the constant y in this expression there results E r = 3.64 X 10~> 9 rc* (3-13) Since the density n varies from metal to metal, E F will also vary among metals. Knowing the specific gravity, the atomic weight, and the number of free elec- trons per atom, it is a simple matter to calculate n, and so E F . For most metals the numerical value of E F is less than 10 eV. EXAMPLE The specific gravity of tungsten is 18.8, and its atomic weight is 184.04 Assume that there are two free electrons per atom. Calculate the numerical value of n and E F . Solution A quantity of any substance equal to its molecular weight in grams is a mole of that substance. Further, one mole of any substance contains the same number of molecules as one mole of any other substance. This number is Avo- gadro's number and equals 6.02 X 10 23 molecules per mole. Thus n = 6.02 X ]Q*a molecules x lmole x 188 JL x 2 electrons 1 atom mole 184 g ' cm 3 atom molecule - 12.3 X io»»? J<iCtron ? - i.23 X JQ'» electrons cm 3 m 3 since for tungsten the atomic and the molecular weights arc the same. There- fore, for tungsten, E F = 3.64 X 10" 13 (123 X 10") ! = 8.95 eV t The atomic weights of the elements are given in the periodic table (Appendix C). CONDUCT/ON *N METALS / 65 THE DENSITY OF STATES As a preliminary step in the derivation of the density function N(E) we first show that the components of the momentum of an electron in a metal are quantized. Consider a metal in the form of a cube, each side of which has a length L. Assume that the interior of the metal is at a constant (zero) poten- tial but that the potential-energy barrier (Fig 3-8) at the surface is arbitrarily high, so that no electrons can escape. Hence the wave functions representing the electrons must be zero outside the metal and at the surface. A one- dimensional model of the potential-energy diagram is given in Fig. 3-12a, and two possible wave functions are indicated in Fig. 3-126 and c. Clearly, this situation is possible only if the dimension L is a half-integral multiple of the De Broglie wavelength X, or r x L = n^ (3-14) where n x is a positive integer (not zero). From the De Broglie relationship (2-8), X = h/p x and the x component of momentum is _ n x h px ~2L (3-15) Hence the momentum is quantized since p* can assume only values which are integral multiples of k/2L. The energy W (in joules) of the electron in this one-dimensional problem is _ p,» _ n,'A« 2m SmL 2 (3-16) The wave nature of the electron has led to the conclusion that its energy must also be quantized. Since n x = 1, 2, 3, . . . , the lowest possible energy is k l /8mL 2 , the next energy level is h 2 /2mL 2 , etc. The Schrodinger Equation The above results may be obtained directly by solving the one-dimensional Schrodinger equation with the potential Fig- 3-12 (o) A one- dimensional problem in which the potential U is zero for a distance L but rises abruptly toward infinity at the boundaries x = and x = L. {b, c) Two possible wave functions f °r an electron in the system described by (a). L^ ^d (6) (c) 66 / ELECTRONIC DEVICES AND CIRCUITS Sac. 3-6 energy U set equal to zero. Under these circumstances Eq. (2-14) may be written dx 2 h 2 (3-17) The general solution of this second-order linear differential equation has two arbitrary constants, d and C%, and in the interval < z < L is given by where ^ = Ci sin ax + Ca cos ax Sr*mW h 2 (3-18) (3-19) Since for x = 0, ^ = 0, then C 2 ™ 0. Since for x = L, ^ = 0, sin oL = 0, or aL = n x w (3-20) where n x is an integer. Substituting from Eq, (3-20) into Eq. (3-19) and solving for W, we again obtain the quantized energies given in Eq. (3-16). The wave function is ^ = Ci sin {n x irx/L,). Since the probability of find- ing the electron somewhere in the metal is unity, then from Sec. 2-8, Ci 2 L / V& = l-/ L Ci«8in*^«fc« or d = (2/L)» and /2\i . u x tx (3-21) Note that n x cannot be zero since, if it were, ^ would vanish everywhere. For n x = 1 the function ^ is plotted in Fig. 3-12b, and for n x = 2 the wave function $ is as shown in Fig. 3-12c. Note also that a negative value of n x gives a value of \p which is the negative of the value of $ for the corresponding positive value of n x . Since only \$\ 2 has a physical meaning (Sec. 2-8), the state described by —n x is the same as that for +n z . Hence only positive integers are to be used for n x . The Uncertainty Principle We digress for a moment to make the point that the measurement of a physical quantity is characterized in an essential way by a lack of precision. For example, in the one-dimensional electronic problem discussed above, there is an inherent uncertainty Ap x in momentum because n x can have only integral values. The smallest value of An* = 1, and hence Ap x = h/2L, Since the electron is somewhere between x = and z — L, the uncertainty in position is Ax = L, Therefore Ap x Ax = - (3-22) This equation is a statement of the uncertainty principle, first enunciated by Heisenberg. He postulated that, for all physical systems (not limited to Sac. 3-6 CONOUCnON IN METALS / 67 electrons in a metal), there is always an uncertainty in the position and in the momentum of a particle and that the product of these two uncertainties is of the order of magnitude of Planck's constant h. Quantum States in a Metal The above results may be generalized to three dimensions. For an electron in a cube of metal, each component of momentum is quantized. Thus p x = n x p p y = n v p Pz = n t p (3-23) where p = h/2L, and n X) n y , and n L are positive integers. A convenient pic- torial representation may be obtained by constructing three mutually perpen- dicular axes labeled p x , p y , and p t . This "volume" is called momentum spate. The only possible points which may be occupied by an electron in momentum space are those given by Eq. (3-23). These are indicated in Fig. 3-13, where for clarity we have indicated points only in a plane for a fixed value of p* (say, p, = 2p). By the Pauli exclusion principle (See. 2-9), no two electrons in a metal may have the same four quantum numbers, n x , n y , n t , and the spin number s. Hence each dot in Fig. 3-13 represents two electrons, one for s — -j and the other for s = — £. We now find the energy density function N(E). Since in Fig. 3-13 there is one dot per volume p 3 of momentum space, the density of electrons in this space is 2/p 3 . The magnitude of the momentum is p = (p x 2 + p y 2 + p* 2 )*. The number of electrons with momentum between p and p + dp is those ng. 3-13 Momentum space. Each dot represents three quantum numbers, n x , n y , Qr >d n,. There are two electrons per dot, corre- sponding to the two possible values of spin. 68 / ELECTRONIC DEVICES AND CIRCUITS lying in the shaded spherical shell of Fig. 3-13. This number is r l\ 7rp 3 dp 8ttL ? p 2 dp (?)<**■ *>(§)- 5m. 3-7 (3-24) ^8/ (h/2Ly h* The factor £ introduced in the above equation is due to the fact that only positive values of n s , n v , and n, are permissible, and hence only that part of the shell in the first octant may be used. If W is the energy (in joules), then W = p 2 /2m. Hence p = (2mW)* pdp = mdW p 2 dp = 2*m*TF* dW (3-25) If N(W) is the density of states (per cubic meter), then, since the volume of the metal is L s , it follows from Eq. (3-24) that N(W ) dW = *^ (3-26) gives the number of electrons with momenta between p and p + dp, corre- sponding to energies between W and W + dW. Substituting for p 2 dp from Eq. (3-25) in Eq. (3-26), we finally obtain 4?r N(W) dW = ^ (2m)»TT» dW (3-27) If we use electron volts E instead of joules W as the unit of energy, then since W = 1.60 X 10^ 18 E (Sec. 1-5), the energy density N{E) is given by Eq. (3-8), with j defined in Eq. (3-9). 3-7 WORK FUNCTION In Fig. 3-14, Fig. 3-11 has been rotated 90° counterclockwise and combined with Fig. 3-8, so that the vertical axis represents energy for both sets of curves. At 0°K it is impossible for an electron to escape from the metal because this requires an amount of energy equal to E B , and the maximum energy possessed by any electron is only Er. It is necessary to supply an additional amount of energy equal to the difference between E B and E F in order to make this escape possible. This difference, written Ew, is known as the work function of the metal. Ew = Eg — Ep Enet T-2500K I ft eV Outside ""^ fig Distance, x (3-28) Fig. 3-14 Energy diagram used to define the work function. Sec. 3-5 CONDUCT/ON IN METALS / 69 Thus the work function of a metal represents the minimum amount of energy that must be given to the fastest-moving electron at the absolute zero of tem- perature in order for this electron to be able to escape from the metal. The experiments of Davisson and Germer 4 on the diffraction of electrons in passing through matter have verified the existence of the potential-energy barrier at the surface of the metal. In fact, based on the results of these experiments, together with experimentally determined values of Ew, it is possible to calculate the values of E F for the metals used. These data show fair agreement between the experimental and theoretical values. A second physical meaning of the term work function may be obtained by considering what happens to an electron as it escapes from a metal, with- out particular regard to the conditions within the interior of the metal. A negative electron will induce a positive charge on a metal from which it escapes. There will then be a force of attraction between the induced charge and the electron. Unless the electron possesses sufficient energy to carry it out of the region of influence of this image force of attraction, it will be returned to the metal. The energy required for the electron to escape from the metal is the work function Ew (based upon this classical electrostatic model). 3-8 THERMIONIC EMISSION The curves of Fig. 3-14 show that the electrons in a metal at absolute zero are distributed among energies which range in value from zero to the maxi- mum energy E F . Since an electron must possess an amount of energy at least as great as E B in order to be able to escape, no electrons can leave the metal Suppose now that the metal, in the form of a filament, is heated by sending a current through it. Thermal energy is then supplied to the elec- trons from the lattice of the heated metal crystal. The energy distribution of the electrons changes, because of the increased temperature, as indicated in fig. 3-14. Some of the electrons represented by the tail of the curve will have energies greater than E B and so may be able to escape from the metal. Using the analytical expression from the distribution function, it is possible to calculate the number of electrons which strike the surface of the metal per second with sufficient energy to be able to surmount the surface barrier and hence escape. Based upon such a calculation, 3 ' 6 the thermionic current in amperes is given by la = SA Th-**< tT w here S = area of filament, m 2 A = a constant, whose dimensions are A/(m 8 )(°K 2 ) T = temperature, °K k — Boltzmann constant, eV/°K Ew = work function, eV (3-29) 70 / ELECTRONIC DEVICES AND CIRCUITS Sec. 3-9 Equation (3-29) is called the thermionic-emission, Dushman, or Richardson equation. The work function E w is known also as the "latent heat of evapo- ration of electrons" from the metal, from the analogy of electron emission with the evaporation of molecules from a liquid. The- thermionic-emission equation has received considerable experimental verification. 8 The graphical representation between the thermionic-emission current and the temperature is generally obtained by taking the logarithm of Eq. (3-29), viz., log /* - 2 log T = log SA - 0.434 ^f (3-30) where the factor 0.434 represents log e. Hence, if we plot log 7^ — 2 log T versus l/T, the result should be a straight line having a slope equal to — QA34Ew/k, from which the work function may be determined. By taking the derivative of the natural logarithm of Eq. (3-29), we obtain hh \ ■ Ew\ dT (3-31) For tungsten, Ew = 4.52 eV, and we calculate that at a normal operating tem- perature of 2400°K, the fractional change in current dlth/Ia. is 2 + 22 times the fractional change in the temperature. It is to be noted that the term 22 arises from the exponential term in the Dushman equation, and the term 2 arises from the 7' 2 term. We observe that the thermionic current is a very sensitive function of the temperature, since a 1 percent change hi T results in a 24 percent change in I lh . It must be emphasized that Eq. (3-29) gives the electron emission from a metal at a given temperature provided that there are no external fields present. If there are either accelerating or retarding fields at the surface, the actual current collected will be greater or less than the emission current, respectively. The effect of such surface fields is discussed later in this chapter. 3-9 CONTACT POTENTIAL Consider two metals in contact with each other, as at the junction C in Fig. 3-15. The contact difference of potential between these two metals is defined as the potential difference V AB between a point A just outside metal 1 and a Fig. 3-15 Two metals in contact at the junction C. S«. 3-10 CONDUCTION IN METALS / 71 point B just outside metal 2. The reason for the existence of the difference of potential is easily understood. When the two metals are joined at the boundary C, electrons will flow from the lower-work-function metal, say 1, to the other metal, 2. This flow will continue until metal 2 has acquired so much negative charge that a retarding field has built up which repels any further electrons. A detailed analysis 6 of the requirement that the number of electrons traveling from metal 1 across junction C into metal 2 is the same as that in the reverse direction across C leads to the conclusion that this equi- librium condition is attained when the Fermi energies E F of the two metals are located at the same height on the energy-level diagram. To satisfy this condition, the potential-energy difference E A n between points A and B is given by (Prob. 3-16) Eab = Ewz ~ Ewi (3-32) which means that ike contact difference of potential energy between two metals equals the difference between their work functions. This result has been verified experimentally by numerous investigators. Corresponding to the potential energy Eab, there is a contact potential (volts) which we designate by Vab — V and which is numerically equal to E A b- If metals 1 and 2 are similar, the contact potential between them is evi- dently zero. If they are dissimilar metals, the metal having the lower work function becomes charged positively and the higher-work-function metal becomes charged negatively. In a vacuum tube the cathode is usually the lowest-work-function metal. If it is connected to any other electrode exter- nally by means of a wire, the effective voltage between the two electrodes is not zero, but equals the difference in the work functions. This potential difference is in such a direction as to repel the electrons being emitted from the cathode. If a battery is connected between the two electrodes, the effec- tive potential is the algebraic sum of the applied voltage and the contact potential. 3-10 ENERGIES OF EMITTED ELECTRONS Since the electrons inside a metal have a distribution of energies, those which escape from the metal will also have an energy distribution. It is easy to demonstrate this experimentally. Thus consider a plane emitter and a plane- parallel collector. The current is measured as a function of the retarding voltage V, (the emitter positive with respect to the collector). If all the elec- trons left the cathode with the same energy, the current would remain con- stant until a definite voltage was reached and then it would fall abruptly to zero. For example, if they all had 2 eV energy, then, when the retarding voltage was greater than 2 V, the electrons could not surmount the potential barrier between cathode and anode and no particles would be collected. Experimentally, no such sudden falling off of current is found, but instead 72 / ELECTRONIC DEVICES AND CIRCUITS Sec. 3- 70 Sec. 3-10 CONDUCTION IN METALS / 73 there is an exponential decrease of current / with voltage according to the equation where V T is the "volt equivalent of temperature," defined by V m iT. = T T e 11,600 (3-33) (3-34) where k is the Boltzmann constant in joules per degree Kelvin. Note the distinction between k and k; the latter is the Boltzmann constant in electron volts per degree Kelvin. (Numerical values of £ and k are given in Appen- dix A. From Sec. 1-5 it follows that k = 1.60 X 10" 19 &.) The Volt-Ampere Characteristic Equation (3-33) may be obtained theoretically as follows: Since *« is the current for zero retarding voltage, the current obtained when the barrier height is increased by E T is determined from the right-hand side of Eq. (3-29) by changing E w to E w + E T . Hence / = SA Th- lB ' +E '> liT = I lk e-WT (3-35) where use was made of Eq. (3-29). Since F r is numerically equal to E r , and V T is numerically equal to kT, then kT (3-36) Hence Eq. (3-33) follows from Eq. (3-35). If V is the applied (accelerating) anode potential and if V is the (retard- ing) contact potential, then V r = V - V, and Eq. (3-33) becomes where I m I ei +viy r (3-37) (3-38) represents the current which is collected at zero applied voltage. Since > Vr, this current I is a small fraction of /*, If V is increased from zero, the current / increases exponentially until the magnitude of the applied voltage V equals the contact potential V. At this voltage V r = and the thermionic current is collected. If V > V, the field acting on the emitted electrons is in the accelerating direction and the current remains at the value In, A plot of the term log / versus V should be of the form shown in Fig. 3-16. The nonzero slope of this broken-line curve is (11,600 log e)/T = 5,030/jT. From the foregoing considerations, the potential represented by the distance from O to O' is the contact potential V. Because most commercial diodes do not. even approximate a plane cathode with a plane-parallel anode the volt-ampere characteristic indicated in Fig. 3-16 is only approached in practice. Fig. 3-16 To verify tke retarding- potential equation, log / is plotted versus V. Accelerating Furthermore, since the effect of space charge (Chap, 4) has been completely neglected, Eq. (3-33) is valid only for low values (microamperes) of current. For larger values of /, the current varies as the three-halves power of the plate potential (Sec. 4-4). EXAMPLE What percentage of the electrons leaving a tungsten filament at, 2700°K can surmount a barrier whose height is 1 eV? Solution Using Eq. (3-33), with V r = 1, and remembering that Vr = f/11,600, yields = e -Ul, 600X0/2,700 — f — 4.2a m 0.014 Ith Hence only about 1.4 percent of the electrons have surface-directed energies in excess of 1 e V. If the emitter is an oxide-coated cathode operating at 1000°K, a calcu- lation similar to the above gives the result that only about 0.001 percent of the electrons have a surface-directed energy in excess of 1 eV. A statistical analysis 3 - 6 shows that the average energy of the escaping electrons is given by the expression E = 2kT (3-39) F °r operating temperatures of 2700 and 1000°K, the average energies of the emitted electrons are 0.47 and 0.17 eV, respectively. These calculations demonstrate the validity of the assumption made in a P- 1 in the discussion of the motion of electrons in electric and magnetic . s > Vlz -, that the electrons begin their motions with very small initial veloei- ■ In most applications the initial velocities are of no consequence. 74 / ElfCTRON/C DEVICES AND CIRCUITS Sec. 3-11 3-11 ACCELERATING FIELDS Under normal operating conditions, the field applied between the cathode and the collecting anode is accelerating rather than retarding, and so the field aids the electrons in overcoming the image force at the surface of the metal. This accelerating field tends, therefore, to lower the work function of the metal, and so results in an increased thermionic emission from the metal. It can be shown 6 that the current / under the condition of an accelerating field of £ (volts per meter) at the surface of the emitter is / = Itf" ■«"»»• (3-40) where I& is the zero-field thermionic current, and T is the cathode tempera- ture in degrees Kelvin. The fact that the measured thermionic currents con- tinue to increase as the applied potential between the cathode and the anode is increased is often referred to as the Schottky effect, after the man who first pre- dicted this effect. Some idea of the order of magnitude of this increase can be obtained from the following illustration. EXAMPLE Consider a cylindrical cathode of radius 0.01 cm and a coaxial cylindrical anode of radius 1.0 cm. The temperature of the cathode is 2500°K. If an accelerating potential of 500 V is applied between the cathode and the anode, calculate the percentage increase in the zero- external-field thermionic-emission current because of the Schottky effect. Solution The electric field intensity (volts per meter) at any point r (meters) in the region between the electrodes of a cylindrical capacitor, according to classical electrostatics, is given by the formula £ - I In (r„/r k ) r (3-41) where In = logarithm to the natural base t V = plate voltage r a ~ anode radius rt = cathode radius Thus the electric field intensity at the surface of the cathode is 6 = 500 — = 1 .085 X 10 s V/m 2.303 log 100 10-* It follows from Eq. (3-40) that log 1 - (0-434)(0.44)(l.Q85 X 10°)* _ Q ^ 2,500 Hence ///, A - 1.20, which shows that the Schottky theory predicts a 20 percent increase over the zero-field emission current. S#c. 3- J3 CONDUCnON IN METALS / 75 HIGH-FIELD EMISSION 3-12 Suppose that the accelerating field at the surface of a "cold" cathode (one for which the thermionic-emission current is negligible) is very intense. Then, not only is the potential-energy barrier at the surface of the cathode lowered, but also it is reduced in thickness. For fields of the order of 10 8 V/m, the barrier may become so thin (~100 A) that an electron, considered as a De Broglie wave, may penetrate, or "tunnel," through the barrier (Sec. 6-13). Under these circumstances the variation of the emission-current density with the strength of the electric field intensity at the surface of the metal has been calculated by several investigators. 7 This tunneling effect is called high-field, cold-cathode, or autoelectronic emission. The electric field intensity at an electrode whose geometry includes a sharp point or edge may be very high even if the applied voltage is moderate. Hence, if high-field emission is to be avoided, it is very important to shape the electrodes in a tube properly so that a concentration of electrostatic lines of flux does not take place on any metallic surface. On the other hand, the cold-cathode effect has been used to provide several thousand amperes in an x-ray tube used for high-speed radiography. 3-13 SECONDARY EMISSION 8 The number of secondary electrons that are emitted from a material, either a metal or a dielectric, when subjected to electron bombardment has been found experimentally to depend upon the following factors: the number of primary electrons, the energy of the primary electrons, the angle of incidence of the electrons on the material, the type of material, and the physical condition of the surface. The yield, or secondary-emission ratio 8, denned as the ratio of the number of secondary electrons per primary electron, is small for pure metals, the maximum value being between 1.5 and 2. It is increased markedly by the presence of a contaminating layer of gas or by the presence of an electro- positive or alkali metal on the surface. For such composite surfaces, second- ary-emission ratios as high as 10 or 15 have been detected. Most secondary electrons are emitted with small (less than 3 eV) energies. The ratio & is a function of the energy E of the impinging primary elec- trons, and a plot of 5 versus E exhibits a maximum, usually at a few hundred electron volts. This maximum can be explained qualitatively as follows: or low-energy primaries, the number of secondaries that are able to over- come the surface attraction is small. As the energy of the impinging electrons ncreases, more energetic secondaries are produced and the yield increases. , nce > however, the depth of penetration increases with the energy of the incident electron, the secondaries must travel a greater distance in the metal T^fore they reach the surface. This increases the probability of collision in e metal, with a consequent loss of energy of these secondaries. Thus, if the 76 / ELECTRONIC DEVICES AND C/RCU/TS Sec, 3-13 primary energy is increased too much, the secondary-emission ratio must pass through a maximum. REFERENCES 1. Shockley, W. : The Nature of the Metallic State, J. Appl. Phys., vol. 10 ud 543-555 1939. 2. Fermi, E.: Zur Quantelung des idealen cinatomigen Gases, Z. Physik, vol. 36 dd 902-912, May, 1926. Dirac, P. A. M.: On the Theory of Quantum Mechanics, Proc. Roy. Soc. (London) vol. 112, pp. 661-677, October, 1926. 3. Sommerfeld, A., and H. Bethe: Elektronentheorie der Metalle, in "Handbuch der Physik," 2d ed., vol. 24, pt. 2, pp. 333-622, Springer Verlag OHG t Berli n , 1933. Darrow, K. K.: Statistical Theories of Matter, Radiation and Electricity, Bell System Tech. J., vol. 8, pp. 672-748, October, 1929. 4. Davisson, C. J., and L. H. Germer: Reflection and Refraction of Electrons by a Crystal of Nickel, Proc. Natl. Acad. Sci. U.S., vol. 14, pp. 619-627, August, 1928. 5. MUlman, J., and S. Seely; "Electronics," 2d ed., McGraw-Hill Book Company New York, 1951. 6. Dushman, S.: Thermionic Emission, Rev. Mod. Phys., vol. 2, pp. 381-476 October 1930. 7. Dyke, W. P., and W. W. Dolan: Field Emission, "Advances in Electronics," vol. 8, Academic Press Inc., New York, 1956. Fowler, R. H., and L, Nordheim: Electron Emission in Intense Electric Fields, Proc. Roy. Soc. {London), vol. 119, pp. 173-181, May, 1928. Oppenhcimer, J. R.: On the Quantum Theory of Autoelectric Field Circuits, Proc. Natl. Acad. Sci. U.S., vol. 14, pp. 363-365, May, 1928. 8. Spangenberg, K. R.: "Vacuum Tubes," McGraw-Hill Book Company, New York, 1948. McKay, K. G.: Secondary Electron Emission, "Advances in Electronics," vol. 1, pp. 65-130, Academic Press Inc., New York, 1948. An extensive review. 4 /VACUUM-DIODE CHARACTERISTICS The properties of practical thermionic cathodes are discussed in this chapter. In order to collect the emitted electrons, a plate or anode is placed close to the cathode in an evacuated envelope. If an acceler- ating field is applied, it is found that the plate current increases as the anode voltage is increased. When a large enough plate potential is applied to collect the thermionic-emission current !&, the anode cur- rent will remain constant at the value I a, even though the plate volt- age is increased further. The limitation of the current which can be collected in a diode at a given voltage because of the space charge of the electrons is discussed in detail in this chapter. Finally, practical diode volt-ampere characteristics are considered, and an analysis of a circuit containing a diode is given. 4-1 CATHODE MATERIALS The three most important practical emitters are pure tungsten, thori- ated tungsten, and oxide-coated cathodes. The most important prop- erties of these emitters are now discussed, and are summarized in Table 4-1. Tungsten Unlike the other cathodes discussed below, tungsten does not have an active surface layer which can be damaged by positive-ion bombardment. Hence tungsten is used as the cathode in high-voltage high- vacuum tubes. These include x-ray tubes, diodes for use as rectifiers above about 5,000 V, and large power- amplifier tubes for use in communication transmitters. Tungsten has the disadvantage that the cathode-emission efficiency, defined as the ratio of the emission current, in amperes, to the heating 77 78 / ELECTRONIC DEVICES AND CIRCUITS TABLE 4-1 Comparison of thermionic emitters Sec. 4-1 Type of cathode A. X 10-«, A/(m»)(°K') eV Approximate operating temperature, °K Efficiency, t A/W Plate voltage, V Gas or vacuum tube Tungsten. . . , Thoriated tungsten . . Oxide-coated 60.2 3.0 0.01 4.52 2.63 1.0 2,500 1,900 1,000 20-100 50-1 , 000 100-10,000 Above 5,000 750-5,000 Below 750 Vacuum Vacuum Vacuum or gas t K. R. Spangenberg, "Vacuum Tubes," McGraw-Hill Book Company, New York, 1948. power, in watts, is small. However, a copious supply of electrons can be pro- vided by operating the cathode at a sufficiently high temperature. The higher the temperature, the greater will be the evaporation of the filament during its operation and the sooner it will burn out. Economic considerations dictate that the temperature of the filament be about 2500°R, which gives it a life of approximately 2,000 hr. The melting point of tungsten is 3650°K. Thoriated Tungsten 1 In order to obtain copious emission of electrons at moderately low temperatures, it is necessary for the material to have a low work function. Unfortunately, the low-work-function metals, such as cesium, rubidium, and barium, in some cases melt and in other cases boil at tempera- tures necessary for appreciable thermionic emission. However, it is possible to apply a very thin (monatomic) layer of low-work-function material, such as thorium, on a filament of tungsten. Thoriated-tungsten filaments are obtained by adding a small amount (1 or 2 percent by weight) of thorium oxide to the tungsten. The base metal holds the adsorbed layer at high temperatures, even above the point at which the pure thorium would normally evaporate. Such a filament possesses emission properties that are considerably better than those of the pure tungsten. The limitation to the use of thoriated-tungsten emitters is the deacti- vation due to positive-ion bombardment. The effect of even a few ions is severe at high potentials, so that these filaments are confined to use in tubes that operate with potentials of less than about 5,000 V. Thoriated-tungsten filaments are used in a number of moderate-voltage transmitting tubes as well as in high-power beam-type microwave tubes. EXAMPLE At what temperature will a thoriated-tungsten filament give 5,000 times as much emission as a pure tungsten filament at the same temperature? The filament dimensions of the two emitters are the same. Uc.4-1 VACUUM-DIODE CHARACTERISTICS / 79 Solution It is required that I T -w = 5,000/n-. From Eq. (3-15) and Table 4-1, It-w - (S) (3.0 X 10<)(7 ,I )€- ! - 63 '*'' and I w = (5) (60.2 X 10*) (T*)t-*^ lkT Upon dividing these two equations, there results 1 T ~ w = 5 000 = — — e (<B2_2 - 63) ' (a - ,fl2X10 " 5T ' J I w ' 60.2 20.1 e 21,900/r where the value of k in electron volts per degree Kelvin given in Appendix A was used. We can solve for T with the aid of logarithms. Thus (0.434) (21 ,900) T = 1900°K - log (5,000) (20.1) = 5.00 Oxide-coated Cathodes 2 The modern oxide-coated cathode is the most efficient type of emitter that has been developed commercially. It consists of a metallic base of platinum, nickel, nickel with a few percent of cobalt or silicon, or Konal metal. Konal metal is an alloy consisting of nickel, cobalt, iron, and titanium. Konal-mctal sleeves are used very extensively as the indirectly heated cathode of radio receiving tubes. The wire filaments or the metallic sleeves are coated with oxides of the alkaline-earth group, espe- cially barium and strontium oxides. Four characteristics of the coating account for its extensive use: (1) It has a long life, several thousand hours under normal operating conditions being common. At reduced filament power, several hundred thousand hours has been obtained. (2) It can easily be manufactured in the form of the indirectly healed cathode. (3) It gives tremendous outputs under pulsed conditions. I hus it has been found that for (microsecond) pulses current densities in excess of 10 8 A/m 2 may be obtained. 3 (4) It has very high cathode efficiency. Oxide-coated cathodes are subject to deactivation by positive-ion bom- bardment, and so are generally used in low-voltage tubes only. The emission properties of an oxide-coated cathode are influenced by many factors, for Sample, the proportion of the contributing oxides, the thickness of the oxide coating, possibly the core material, and the details of the processing. Hence 'ho emission characteristics change with the age of the cathode and vary Markedly from tube to tube. How then can tubes using oxide-coated cathodes se rve satisfactorily in any circuit? It is shown in Sec. 4-4 that tubes usually °perate under conditions of space-charge limitation and not under conditions 01 temperature limitation. This statement means that the current is determined 80 / ELECTRONIC DEVICES AND CIRCUITS Sec. 42 Sec. 43 VACUUM-DIODE CHARACTERISTICS / 81 by the plate voltage and not. by the cathode temperature. Thus, despite their rather unpredictable emission characteristics, oxide-coated cathodes make excellent tube elements, provided only that their thermionic-emission current never falls below that required by the circuit. Oxide-coated cathodes are used in the greatest percentage of commercial electron tubes. Almost all receiving tubes, many low-voltage transmitting tubes, and practically all gas tubes use such cathodes. 4-2 COMMERCIAL CATHODES The cathodes used in thermionic tubes are sometimes directly heated filaments in the form of a V, a W, or a straight wire, although most tubes use indirectly heated cathodes. The indirectly heated cathode was developed so as to minimize the hum (Sec. 16-11) arising from the various effects of ac heater operation. The heater wire (tungsten) is contained in a ceramic insulator (oxides of beryllium and aluminum) enclosed by a nickel or Konal-metal sleeve on which the oxide coating is placed. The cathode as a unit is so massive that its temperature does not vary appreciably with instantaneous variations in the magnitude of the heater currents. Further, since the sleeve is the emitting surface, the cathode is essentially equipotential. The ceramic insulator which acts to iso- late electrically the heater wire from the cathode must, of course, be a good heat conductor. Under normal conditions of operation, the heater is main- tained at about 1000° C, which results in the cathode temperature being at approximately 850°C. Heaterless Cathodes Vacuum diode and multielectrode tubes have been constructed which contain no heater. The Thermionic Integrated Micro- Module, known as TIMM (General Electric trade name), obtains the heat needed to develop thermionic emission by conserving the normal dissipations of both active and passive components and containing this energy within a suitable insulated enclosure. A TIMM is constructed of special ceramic materials, with electrodes of titanium, and is operated at approximately 600°C. The oxide cathode coating is deposited upon platinum base metal, leading to chemical stability and long emitter life. 4-3 THE POTENTIAL VARIATION BETWFEN THE ELECTRODES Consider a simple thermionic diode whose cathode can be heated to any desired temperature and whose anode or plate potential is maintained at V P . It will be assumed that the cathode is a plane equipotential surface and that the collecting plate is also a plane parallel to it. The potential variations between Potential, V T 3 > T a > 7\ Fig. 4-1 The potential variation between plane-parallel elec- trodes for several values of cathode temperature. "For nonzero initial velocities the electrodes for various temperatures of the cathode are given in Fig. 4-1. The general shape of these curves may be explained as follows: At the tem- perature Ti at which no electrons are emitted, the potential gradient is con- stant, so that the potential variation is a linear function of the distance from the cathode to the anode. At the higher temperature T 2 , an appreciable density of electrons exists in the interelectrode space. The potential variation will be somewhat as illustrated by the curve marked T 2 in Fig. 4-1. The increase in temperature can change neither the potential of the cathode nor the potential of the anode. Hence all the curves must pass through the fixed end points K and A. Since negative charge (electrons) now exists in the space between K and A, then, by Coulomb's law, the potential at any point will be lowered. The greater the space charge, the lower will be the potential. Thus, as the temperature is increased, the potential curves become more and more concave upward. At T%t the curve has drooped so far that it is tangent to the X axis at the origin. That is, the electric field intensity at the cathode for this condition is zero. One may sketch the broken curve of Fig. 4-1 to represent the potential vari- ation at a temperature higher than T 3 . This curve contains a potential mini- mum. Such a condition is physically impossible if the initial velocities of the emitted electrons are assumed negligible. That this is so follows from the discussion given below. The Potential-energy Curves Since the potential energy is equal to the product of the potential V and the charge — e, the curves of Fig. 4-2 are simply se °f Fig. 4-1 inverted, the unit of the ordinates being changed to electron ^s. It is immediately evident that the broken curve represents a potential- energy barrier at the surface of the cathode. Several such potential-energy arriers have already been considered in Chap. 3. On the basis of our previ- 18 discussions, it is clear that only those electrons which possess an initial er gy greater than B m , the maximum height of the barrier, can escape from 82 / ELECTRONIC DEVICES AND CIRCUITS Sec. 4-4 the cathode and reach the anode. Consequently, the assumed condition of zero initial velocities of the emitted electrons precludes the possibility of any electrons being emitted. As a result, the barrier will be broken down, since the applied field will cause those electrons which produce the barrier to leave the intereleetrode space and become part of the anode current. This auto- matic growth and collapse of the potential barrier outside the cathode may be considered as a self -regulating valve that allows a certain definite number of electrons per second to escape from the cathode and reach the anode, for a given value of plate voltage. The Field Intensity at the Cathode It can be inferred from the fore- going argument that the maximum current that can be drawn from a diode for a fixed plate voltage and any temperature whatsoever is obtained under the condition of zero electric field at the surface of the cathode. Thus, for optimum conditions, & = dx at a; =0 (4-1) This condition is based on the assumption that the emitted electrons have zero initial energies. Because the initial velocities are not truly zero, the potential variation within the tube may actually acquire the form dlustrated by the broken curve of Fig. 4-1, However, since the potential minimum in Fig. 4-1 is usually small in comparison with the applied potential, it is neg- lected, and condition (4-1) is assumed to represent the true status when space- charge current is being drawn. 4-4 SPACE-CHARGE CURRENT We shall now obtain the analytical relationship between the current and volt- age in a diode. The electrons flowing from the cathode to the anode consti- Potential energy, eV Fig. 4-2 The potential-energy varia- tions corresponding to the curves of Fig. 4-1. Sec. 4-4 VACUUM-DIODE CHARACTERISTICS / 83 tute the current. The magnitude of the current density / in amperes per square meter is given by Eq. (1-38), viz., J = pv (4-2) where v is the drift velocity of these electrons in meters per second, and p is the volume density of electric charge in coulombs per cubic meter. Both p and v are functions of the distance from the origin (the cathode). However, the product is constant, since the number of electrons passing through unit area per second must be the same for all points between a plane cathode and a parallel anode. This statement expresses the principle of conservation of elec- tric charge. Therefore, at the cathode, where the velocity of the electrons is very small (the velocities being the initial velocities), the charge density must be very large. In the neighborhood of the anode, the velocity is a maximum; hence the charge density is a minimum. If the initial velocities are neglected, the velocity of the electrons at any point in the intereleetrode space may be determined from the equation that relates the kinetic energy of the particle with the potential through which it has fallen, viz., (4-3) (4-4) imv* = eV Poisson's equation is dW dx 2 , P_ «0 where x = distance from cathode, m V = potential, V p = magnitude of electronic volume charge density, C/m 3 e = permittivity of free space, mks system There results, from Eqs. (4-2) to (4-4), where ^!Z = t = L - J dx 2 t V€„ ~ [2(e/m)]*e Q K = 7-1 = KV-i [2(e/m)h (4-5) (4-6) *s a constant, independent of z. The Solution of Eq. (4-5) Let y = dV/dx, and this nonlinear differential equation may be solved by the separation-of-variables method. Thus or Hence dy = KV-i dx m KV-i — V ydy = KV-* dV 84 / ElfCTRON/C DEVICES AND CIRCUITS Sec. 4-4 Sec. 4-5 VACUUM-DIODE CHARACTERISTICS / 85 which integrates to %■ = 2KV* + Ci ■6 (4-7) The constant of integration C\ is zero because, at the cathode, V — and 2/ = dV/dx = 0, from Eq. (4-1). By taking the square root of Eq. (4-7) there results and V-* dV = 2KS dx ,_£_«.* This equation integrates to $F* = 2K*x + C 2 The constant of integration Ca is zero because V = at x = 0. Finally, F = (1)*KV (4-8) It is seen that the potential depends upon the four-thirds power of the interelectrode spacing. For example, the curve marked T% in Fig. 4-1 is expressed by the relation V = ax* (4-9) where a is readily found in terms of constanta and the current density J from the foregoing equations. However, a may also be written as Vp/d*, where d is the separation of the electrode and Vp is the plate potential. This is so because Eq. (4-9) is valid for the entire interelectrode space, including the boundary x = d, where V = Vp. The Three-halves-power Law The complete expression for the current density is obtained by combining Eqs. (4-8) and (4-6). The result is 9\ m) VI (4-10) In terms of the boundary values, this becomes, upon inserting the value of e/m for electrons and eo = 10~ 9 /36rr, J m 2.33 X 10" fl -^- Qr (4-11) Therefore the plate current varies as the three-halves power of the plate potential. This result was established by Langmuir, 4 although it had been previously published in a different connection by Child. 6 It is known by several differ- ent names, for example, the Langmuir-Child law, the three-kalves-power law, or simply, the space-charge equation. It will be noticed that this equation relates the current density, and so the current, in terms only of the applied potential and the geometry of the tube. The space-charge current does not depend upon either the temperature or the work function of the cathode. Hence, no matter how many electrons a cathode may be able to supply, the geometry of the tube and the potential applied thereto will determine the maximum current that can be collected by the anode. Of course, it may be less than the value predicted by Eq. (4-11) if the electron supply from the cathode is restricted (because the temperature is too low). To summarize, the plate current in a given diode depends only upon the applied potential, provided that this current is less than the tempera- ture-limited current. The velocity of the electrons as a function of position between the cathode and anode can be found from Eq. (4-3) with the aid of Eq. (4-10). Then the charge density as a function of x can be obtained from Eq. (4-2). It is found (Prob. 4-6) that v varies as the two-thirds power of x and that p varies inversely as the two-thirds power of x. This physically impossible result that at the cathode the charge density is infinite is a consequence of the assumption that the electrons emerging from the cathode all do so with zero initial velocity. Actually, of course, the initial velocities are small, but nonzero, and the charge density is large, though finite. Systems that possess plane-parallel electrodes were considered above because the simplicity of this geometry made it easy to understand the physical principles involved. However, such tube geometry is almost never met in practice. More frequently, tubes are constructed with cylindrical symmetry, the anode being in the form of a cylinder that is coaxial with a cathode of either the directly or the indirectly heated type. It is possible to demonstrate 6 that an expression of the form Ip = GV P { (4-12) where I P is the plate current, applies for any geometrical arrangement of cathode and anode, provided that initial velocities are neglected. The specific value of the constant G, called the perveance, that exists in this expression depends upon the geometry of the system. 4-5 FACTORS INFLUENCING SPACE-CHARGE CURRENT Several factors modify the equations for space charge given above, particu- larly at low plate voltages. Among these factors are: I. Filament Voltage Drop The space-charge equations are derived on the assumption that the cathode is an equi potential surface. This is not a valid assumption for a directly heated emitter, and the voltage across the ends of the filament causes a deviation from the three-halves-power equation. In & ct, the results depend on whether the plate current is returned to the positive °r to the negative end of the filament. Usually, the filament is heated with a ransformer, and the plate is returned to the center tap of the secondary binding. 86 / ELECTRONIC DEVICES AND CIRCUITS Sec. 45 2. Contact Potential lit every space-charge equation, the symbol V P must be understood to mean the sum of the applied voltage from plate to cathode plus the contact potential between the two. For plate voltages of only a few volts, this effect may be quite appreciable. 3. Asymmetries in Tube Structure Commercial tubes seldom possess the ideal geometry assumed in deriving the space-charge equations. 4. Gas The presence of even minute traces of gas in a tube can have marked effects on the tube characteristics. If the voltage is sufficiently high to cause ionization of the residual gas molecules, the plate current will rise above that demanded by the space-charge equations because the positive ions that are formed neutralize the electronic-charge density. Modern vacuum tubes are exhausted to pressures of about 10" e mm Hg. 5. initial Velocities of Emitted Electrons If the initial velocities of the electrons are not neglected, the variations of potential with interelectrode spacing will be somewhat as depicted by the broken curve of Fig, 4-1, which is reproduced in Fig. 4-3 for convenience. This represents a potential-energy barrier at the cathode surface, and so it is only those electrons whose energies are greater than the height E m = eV m of this barrier that can escape from the cathode. The height of this barrier is, from the results of Sec. 3-10, a fraction of 1 eV. At a distance x m from the surface of the thermionic emitter, the point of the potential minimum, the electric field intensity passes through zero. Hence the point M may be considered as the position of a "virtual" cathode. Evi- dently, the distance that will enter into the resulting space-charge equation will be d — x m , and not d. Likewise, the effective plate potential will be Vp + V m , and not Vp alone. Both of these factors will tend to increase the current above that which exists when the initial velocities are neglected. The exact mathematical formulation of the volt-ampere equation, taking into account the energy distribution of the electrons, is somewhat involved. 7 To Potential, V / • , . / t / o Vp / c / ** M / 1 v^i^' *Tu \ ■« d — V Fig. 4-3 The potential variation in a plane- parallef space-charge diode, with the initial velocities of the electrons taken into account. See- 4-6 VACUUM-DtODE CHARACTERISTICS / 87 summarize, the space-charge current in a diode is not strictly a function of the plate potential only, but does depend, to a small extent, upon the tem- perature of the cathode. 4-6 DIODE CHARACTERISTICS The two most important factors that determine the characteristics of diodes are thermionic emission and space charge. The first gives the temperature- saturated value, i.e., the maximum current that can be collected at a given cathode temperature, regardless of the magnitude of the applied accelerating potential. The second gives the space-charge-limited value, or the voltage- saturated value, and specifies the maximum current that can be collected at a given voltage regardless of the temperature of the filament. Large-voltage Characteristics The volt-ampere curves obtained experimentally for an oxide-coated cathode are shown in Fig. 4-4. It should be noted that the space-charge currents corresponding to the different tem- peratures do not coincide, but that the currents decrease slightly as the tem- perature decreases. Further, there is no abrupt transition between the space- charge-limited and the temperature-limited portions of the curves, but rather a gradual transition occurs. Also, the current for the temperature-limited regions gradually rises with increased anode potentials (because of the Schottky effect, Sec. 3-11). The shapes of these curves are determined by the factors mentioned in the preceding section. Low-voltage Characteristic The diode curve does not follow Eq, (4-12) for small currents or voltages because the initial velocities of the electrons and the contact potential cannot be neglected in this region. An expanded view of the volt-ampere curve near the origin is given in Fig. 4-5. Space charge is negligible at these small currents, and the volt-ampere relationship is given by Eq. (3-37), namely, Ip = I e v r'Vr •"ig. 4-4 Volt-ampere diode characteristics *or various filament temperatures. r s > Ti > T z > T t > TV (4-13) 100 V P ,V 88 / ELECTRONIC DEVICES AND CIRCUITS Sec. 4-7 Fig. 4-5 The volt-ampere characteristic of a vacuum diode for small voltages. 1.0 V,,V where /„ is the plate current at zero applied voltage V, and Vr = T/ 11,600 [Eq. (3-34)] is the volt equivalent of temperature. Note that the curve doea not pass through the origin. 4-7 AN IDEAL DIODE VERSUS A THERMIONIC DIODE An ideal diode is defined as a two-terminal circuit element having the following characteristics: (1) It offers no resistance to current flow if the plate is posi- tive with respect to cathode (zero forward resistance). (2) There is no current if the plate is negative with respect to the cathode (infinite reverse resistance). (3) The capacitance shunting the diode is zero. (4) These characteristics are independent of temperature. The volt-ampere characteristic of an ideal diode is shown in Fig. 4-6. A physical thermionic diode differs in the following important respects from the ideal diode: 1. The forward resistance is not zero, but lies in the approximate range of 100 to 1,000 £2. 2. The value of the resistance is not constant, but depends upon the applied voltage. Hence a distinction must be made between static and dynamic resistance. The statie resistance R is denned as the ratio Vp/Ip. At any point P on the volt-ampere characteristic of the diode, R is the recipro- cal of the slope of the line joining P to the origin. The static resistance varies widely with voltage, and hence is seldom used. For small-signal operation, r. Fig. 4-6 An ideal-diode characteristic. V F Sec. 4-8 VACUUM-DIODE CHARACTERISTICS I 89 an important parameter is the dynamic, incremental, or plate resistance, defined by dV P r„ = dh (4-14) This dynamic forward resistance will also be designated by R f . Of course, if the volt-ampere characteristic were a straight line passing through the origin, R f would equal R. Although r p varies with current, it is reasonable to treat this parameter as a constant in a small-signal model. 3. The back, or reverse, resistance R T is not infinite, although values of hundreds or even thousands of megohms are attainable even for small negative applied voltages. 4. The "break" in the characteristic (the division between the high- and low-resistance regions) is not sharp, and may not occur at zero applied voltage. 5. As already mentioned in Sec. 4-5, the volt-ampere characteristic is not strictly space-charge-limited, but does depend somewhat upon the filament temperature. Experiment reveals that there is a shift in the voltage at con- stant current of about -0.1 V for a 10 percent increase in the healer voltage. The higher the filament voltage, the more the curves shift to the left, because the increase in the initial velocities of the electrons with increase in tempera- ture results in higher currents at a given voltage. The shift with tube replace- ment or tube aging is found in practice to be of the order of ± 0.25 V. 6. Since a diode consists of two metallic electrodes (a cathode and an anode) separated by a dielectric (a vacuum), this device constitutes a capaci- tor. The order of magnitude of this capacitance is 5 pF. To this value must be added the wiring capacitance introduced when the diode is inserted into a circuit. 4-8 RATING OF VACUUM DIODES The rating of a vacuum diode, i.e., the maximum current that it may normally carry and the maximum potential difference that may be applied between the cathode and the anode, is influenced by a number of factors. 1. The plate eurrent cannot exceed the thermionic-emission current, 2. In order that the gas adsorbed by the glass walls should not be liber- ated, the temperature of the envelope must not be allowed to exceed the tem- perature to which the tube was raised in the outgassing process. 3. The most important factor limiting the rating of a tube is the allowa- ble temperature rise of the anode. When a diode is in operation, the anode becomes heated to a rather high temperature because of the power (IpV P ) hat must be dissipated by the anode. The temperature of the anode will rise Un til the rate at which the energy supplied to the anode just equals the rate at which the heat is dissipated from the anode in the form of radiation. Conse- 90 / ELECTRONIC DEVICES AND CIRCUITS Sec. 4-9 quently, the temperature will depend upon the area of the anode and the mate- rial of wMch it Ib constructed. The most common metals used for anodes are nickel and iron for receiving tubes and tantalum, molybdenum, and graphite for transmitting tubes. The surfaces are often roughened or blackened in order to increase the thermal emissivity and permit higher-power operation. These anodes may be operated at a cherry-red heat without excessive gas emission or other deleterious effects. For the larger tubes, it is necessary that the anodes be cooled either by circulating water through special cooling coils or by forced-air-cooling radiator fins attached to the anode. 4. The voltage limitation of a high-vacuum diode is not always deter- mined by the permissible heating of the anode. Conduction may take place between the filament leads and the anode lead through the glass itself, if the voltage between these leads is high. For this reason, high-voltage rectifiers are generally constructed with the filament leads and the anode lead at opposite ends of the glass envelope. Peek Inverse Voltage The separation of the leads of high-voltage recti- fiers must be large enough to preclude flashover through the air. In fact, it is the highest voltage that may be safely impressed across the electrodes with no flow of charge which determines the safe voltage rating of a tube. Since, with an alternating potential applied between the cathode and anode, no cur- rent must exist during the portion of the cycle when the anode is negative with respect to the cathode, the maximum safe rating of a rectifying diode is known as the peak-inverse-voltage rating. Commercial vacuum diodes are made to rectify currents at very high voltages, up to about 200,000 V. Such units are used with x-ray equipment, high-voltage cable-testing equipment, and high-voltage equipment for nuclear- physics research. Semiconductor Diodes Because of their small size and long life and because no filament power is required, semiconductor diodes (Chap, (i) are replacing vacuum rectifiers in many applications. The tube must be used, however, if very high voltage or power is involved, if extremely low reverse currents are necessary, or if the diode is located in an unusual environ ment (high nuclear radiation or high ambient temperature). 4-9 THE DIODE AS A CIRCUIT ELEMENT The basic diode circuit of Fig. 4-7 consists of the tube in series with a load resistance Rl and an input-signal source i»,-. Since the heater plays no part in the analysis of the circuit, it has been omitted from Fig. 4-7, and the diode is indicated as a two-terminal device. This circuit is now analyzed to find the instantaneous plate current i P and the instantaneous voltage across the diode v P when the instantaneous input voltage is v,. Sec. 4-9 VACUUM-DtODE CHARACTERISTICS / 91 Fig, 4-7 The basic diode circuit. -v. » ®- *> Ri. > v » T The Load Line From Kirehhoff's voltage law, vp = Vi — ipRi, (4-15) where Rl is the magnitude of the load resistance. This one equation is not sufficient to determine the two unknowns vp and ip in this expression. How- ever, a second relation between these two variables is given by the static plate characteristic of the diode (Fig. 4-4). In Fig. 4-8a is indicated the simultane- ous solution of Eq. (4-15) and the diode plate characteristic. The straight line, which is represented by Eq. (4-15), is called the load line. The load line passes through the points ip = 0, v P = «,-, and ip = Vi/Ri, v P = 0. That is, the intercept with the voltage axis is «,-, and with the current axis is v { /Rt,. The slope of this line is determined, therefore, by Rl. It may happen that ip = Vi/RL is too large to appear on the printed volt-ampere characteristic supplied by the manufacturer. If I' does appear on this characteristic, one point on the load line is ip = I', Vp = f* — I'Ri, and the second point is ip ~ 0, Vp = V{. The point of intersection A of the load line and the static curve gives the current i A that will flow under these conditions. This con- struction determines the current in the circuit when the instantaneous input potential is v t . The Dynamic Characteristic Consider now that the input voltage is allowed to vary. Then the above procedure must be repeated for each volt- age value. A plot of current vs. input voltage, called the dynamic charac- teristic, may be obtained as follows; The current i A is plotted vertically above v i at point B in Fig. 4-86. As y, changes, the slope of the load line does not vary since R L is fixed. Thus, when the applied potential has the value v' i} the corresponding current is iV- This current is plotted vertically above v t at B'. The resulting curve OB'B that is generated as v* varies is the dynamic characteristic. It is to be emphasized that, regardless of the shape of the static charac- teristic or the waveform of the input voltage, the resulting waveform of the current in the output circuit can always be found graphically from the dynamic characteristic. This construction is indicated in Fig. 4-9. The input-signal Waveform (not necessarily sinusoidal) is drawn with its time axis vertically 92 / ELECTRONIC DEVICES AND CIRCUITS Sec. 4-9 Static curve Dynamic («) <*> Fig. 4-8 (o) The intersection A of the load line with the diode static charac- teristic gives the current i d corresponding to an instantaneous input voltage i\. (b) The method of constructing the dynamic curve from the static curve and the load line. downward, so that the voltage axis is horizontal. Suppose that the input voltage has the value indicated by the point A at an instant t'. The corre- sponding current is obtained by drawing a vertical line through A and noting the current a where this line intersects the dynamic curve. This current is then plotted at an instant of time equal to t'. Similarly, points 6, c, d, . . . of the current waveform correspond to points B, C, D, ... of the input- voltage waveform. Diode Appticotions The construction of Fig. 4-9 indicates that, for negative input voltages, zero output current is obtained. If the dynamic Output current b / g Fig. 4-9 The method of obtaining the output-current waveform from the dynamic curve for a given input- voltage waveform. Sec 49 VACUUM-DIODE CHARACTERISTICS / 93 characteristic is linear, the output voltage v = IpRl is an exact replica of the input voltage Vi except that the negative portion of ty is missing. In this application the diode acts as a clipper. If the diode polarity is reversed, the positive portion of the input voltage is clipped. The clipping level need not be at zero {or ground) potential. For example, if a reference battery Vr is added in series with R L of Fig. 4-7 (with the negative battery terminal at ground), signal voltages smaller than V R will be clipped. Many other wave- shaping circuits 8 employ diodes. One of the most important applications of a diode is rectification. If the input voltage is sinusoidal, the output consists of only positive sections (resem- bling half sinusoids). The important fact to note is that, whereas the average value of the input is zero, the output contains a nonzero dc value. Hence rectification, or the conversion from alternating to direct voltage, has taken place. Practical rectifier circuits are discussed in Chap. 20. Diodes also find extensive application in digital computers 8 and in circuits used to detect radio- frequency signals. REFERENCES 1. Dushman, S., and J. W. Ewald: Electron Emission from Thoriated Tungsten, Pkys. Rev., vol. 29, pp. 857-870, June, 1927. 2. Blewett, J. P.: Oxide Coated Cathode Literature, 1940-1945, J. Appl. Phys., vol. 17, pp. 643-647, August, 1946. Eisenstein, A. S.: Oxide Coated Cathodes, "Advances in Electronics," vol. 1, pp. 1-64, Academic Press Inc., New York, 1948. Hermann, G., and S. Wagner: "The Oxide-coated Cathode," vols. 1 and 2, Chapman & Hall, Ltd., London, 1951. Gewartowski, J. W., and H. A. Watson: "Principles of Electron Tubes," D. Van Nostrand Company, Inc., Princeton, N.J., 1965. 3. Coomes, E. A.: The Pulsed Properties of Oxide Cathodes, /. Appl. Phys., vol. 17, pp. 647-654, August, 1946. Sproull, R. L.: An Investigation of Short-time Thermionic Emission from Oxide- coated Cathodes, Phys. Rev., vol. 67, pp. 166-178, March, 1945. 4. Langmuir, I.: The Effect of Space Charge and Residual Gases on Thermionic Cur- rents in High Vacuum, Phys. Rev., vol. 2, pp. 450-486, December, 1913. 5. Child, C. D.: Discharge from Hot CaO, Phys. Rev., vol. 27, pp. 492-511, May, 1911. *• Langmuir, I., and K. T. Compton: Electrical Discharges in Gases, Part II: Funda- mental Phenomena in Electrical Discharges, Rev. Mod. Phys., vol. 3, pp. 191-257, April, 1931. 7 - Fry, T. C: The Thermionic Current between Parallel Plane Electrodes; Velocities of Emission Distributed According to Maxwell's Law, Phys. Rev., vol. 17, pp. 441- 452, April, 1921. 94 / ELECTRONIC DEVICES AND CIRCUITS Fry, T. C. : Potential Distribution between Parallel Plane Electrodes, ibid., vol. 22, pp. 445-446, November, 1923. Langmuir, L: The Effect of Space Charge and Initial Velocities on the Potential Distribution and Thermionic Current between Parallel Plane Electrodes, ibid., vol. 21, pp. 419-435, April, 1923. 8. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," McGraw- Hill Book Company, New York, 1965. 5 /CONDUCTION IN SEMICONDUCTORS In Chap. 2 we consider the energy-band structure of crystals and the classification of materials as insulators, conductors, and semicon- ductors. Because of their importance we examine semiconductors in this chapter, with special emphasis on the determination of hole and electron concentrations. The effect of carrier concentrations on the Fermi level and the transport of holes and electrons by conduction or diffusion are also investigated. 5-1 ELECTRONS AND HOLES IN AN INTRINSIC SEMICONDUCTOR 1 From Eq. (3-3) we see that the conductivity is proportional to the concentration n of free electrons. For a good conductor, n is very large ("~10 2S electrons/m 8 ) ; for an insulator, n is very small (~10 7 ); and for a semiconductor, n lies between these two values. The valence electrons in a semiconductor are not free to wander about as they are in a metal, but rather are trapped in a bond between two adjacent ions, as explained below. Germanium and silicon are the two most important semicon- ductors used in electronic devices. The crystal structure of these materials consists of a regular repetition in three dimensions of a unit cell having the form of a tetrahedron with an atom at each vertex. This structure is illustrated symbolically in two dimensions in Fig. 5-1. Germanium has a total of 32 electrons in its atomic structure, arranged in shells as indicated in Table 2-2. As explained in Sec. 2-10, each atom in a germanium crystal contributes four valence electrons, so that the atom is tetravalent. The inert ionic core of the germanium atom carries a positive charge of +4 measured in units 95 96 / ELECTRONIC DEVICES AND CIRCUITS Covalent r Valence oec. o* J Ge • *— * v aience bond. Ge /[electrons Je*H ■ *G » \ Ge it i i Fig. 5-T Crystal structure of germanium, illustrated symbolically in two dimensions. >Ge Ge of the electronic charge. The binding forces between neighboring atoms result from the fact that each of the valence electrons of a germanium atom is shared by one of its four nearest neighbors. This electron-pair, or covalent, bond is represented in Fig. 5-1 by the two dashed lines which join each atom to each of its neighbors. The fact that the valence electrons serve to bind one atom to the next also results in the valence electron being tightly bound to the nucleus. Hence, in spite of the availability of four valence electrons, the crystal has a low conductivity. At a very low temperature (say 0°K) the ideal structure of Fig. 5-1 is approached, and the crystal behaves as an insulator, since no free carriers of electricity are available. However, at room temperature, some of the covalent bonds will be broken because of the thermal energy supplied to the crystal, and conduction is made possible. This situation is illustrated in Fig. 5-2. Here an electron, which for the far greater period of time forms part of a covalent bond, is pictured as being dislodged and therefore free to wander in a random fashion throughout the crystal. The energy E required to break such a covalent bond is about 0.72 eV for germanium and 1.1 eV for silicon at room temperature. The absence of the electron in the covalent bond is represented by the small circle in Fig. 5-2, and such an incomplete covalent S* 5-2 CONDUCTION IN SEMICONDUCTORS / 97 Kg. 5-3 Th e mechanism by w tiich a hole contributes to the conductivity. (a) (6) o 10 O bond is called a hole. The importance of the hole is that it may serve as a carrier of electricity comparable in effectiveness to the free electron. The mechanism by which a hole contributes to the conductivity is quali- tatively as follows: When a bond is incomplete so that a hole exists, it is relatively easy for a valence electron in a neighboring atom to leave its covalent bond to fill this hole. An electron moving from a bond to fill a hole leaves a hole in its initial position. Hence the hole effectively moves in the direction opposite to that of the electron. This hole, in its new position, may now be filled by an electron from another covalent bond, and the hole will correspond- ingly move one more step in the direction opposite to the motion of the elec- tron. Here we have a mechanism for the conduction of electricity which does not involve free electrons. This phenomenon is illustrated schematically in Fig. 5-3, where a circle with a dot in it represents a completed bond, and an empty circle designates a hole. Figure 5-3o shows a row of 10 ions, with a broken bond, or hole, at ion 6. Now imagine that an electron from ion 7 moves into the hole at ion 6, so that the configuration of Fig. 5-36 results. If we compare this figure with Fig. 5-3a, it looks as if the hole in (a) has moved toward the right in (6) (from ion 6 to ion 7). This discussion indicates that the motion of the hole in one direction actually means the transport of a negative charge an equal distance in the opposite direction. So far as the flow of electric current is concerned, the hole behaves like a positive charge equal in magnitude to the electronic charge. We can consider that the holes are physi- cal entities whose movement constitutes a flow of current. In a pure semiconductor the number of holes is equal to the number of free electrons. Thermal agitation continues to produce new hole-electron pairs, whereas other hole-electron pairs disappear as a result of recombination. Ge* Ge* Ge 5-2 CONDUCTIVITY OF A SEMICONDUCTOR Fig. 5-2 Germanium crystal with a broken covalent bond. With each hole-electron pair created, two charge-carrying "particles" are formed. One is negative (the free electron), of mobility p n , and the other is positive (the hole), of mobility n P . These particles move in opposite directions ln an electric field £, but since they are of opposite sign, the current of each is in the same direction. Hence the current density J is given by (Sec. 3-1) J ~ (nun + pn P )e£ = <rS w here n — magnitude of free-electron (negative) concentration P = magnitude of hole (positive) concentration f = conductivity (5-1) 98 / ELECTRONIC DEVICES AND CIRCUITS Sec. 5-2 Sac 5-3 CONDUCTION IN SEMICONDUCTORS / 99 Hence <r « (nfin + pju P )e (5-2) For the pure (called intrinsic) semiconductor considered here, n — p = n,-, where n* is the intrinsic concentration. In pure germanium at room temperature there is about one hole-electron pair for every 2 X 10 9 germanium atoms. With increasing temperature, the density of hole-electron pairs increases [Eq. (5-21)], and correspondingly, the conductivity increases. In the following section it is found that the intrinsic concentration to,- varies with temperature in accordance with the relationship m* = A Th~ s ^ kT (5-3) The constants E 0Q , n n , /*„, and many other important physical quantities for germanium and silicon are given in Table 5-1. The conductivity of germanium (silicon) is found from Eq. (5-3) to increase approximately 6 (8) percent per degree increase in temperature. Such a large change in conductivity with temperature places a limitation upon the use of semiconductor devices in some circuits. On the other hand, for some applications it is exactly this property of semiconductors that is used to advantage. A semiconductor used in this manner is called a thermistor* Such a device finds extensive application in thermometry, in the measurement of microwave-frequency power, as a thermal relay, and in control devices actuated by changes in temperature. Silicon and germanium are not used as thermistors because their properties are too sensitive to impurities. Com- mercial thermistors consist of sintered mixtures of such oxides as NiO, Mn 2 3 , and Co 2 3 . TABLE 5-7 Properties of germanium and silicon! Property Atomic number Atomic weight Density, g/cm 3 . Dielectric constant (relative) Atoms/cm 1 Ego, eV, at 0"K Eg, eV, at 300°K «,- at 300°K, cm"' Intrinsic resistivity at 30Q°K, Jl-cm . }i n , cm V V-sec M P , cmVV-sec D„, cm ! /sec = ft n V T Z) p , cmVsec = p p Vt fG. L. Pearson and W. H. Brattain, History of Semiconductor Research, Proc. IRE, vol. 43, pp. 1794-1806, December, 1955. E. M. Conwell, Properties of Silicon and Germanium, Part II, Proc. IRE, vol. 46, no. 6, pp. 1281-1299, June, 1958, The exponential decrease in resistivity (reciprocal of conductivity) of a semiconductor should be contrasted with the small and almost linear increase in resistivity of a metal. An increase in the temperature of a metal results in greater thermal motion of the ions, and hence decreases slightly the mean free path of the free electrons. The result is a decrease in the mobibty, and hence in conductivity. For most metals the resistance increases about 0.4 percent/ C increase in temperature. It should be noted that a thermistor has a negative coefficient of resistance, whereas that of a metal is positive and of much smaller magnitude. By including a thermistor in a circuit it is possible to compen- sate for temperature changes over a range as wide as 100° C. 5-3 CARRIER CONCENTRATIONS IN AN INTRINSIC SEMICONDUCTOR In order to calculate the conductivity of a semiconductor from Eq. (5-2) it is necessary to know the concentration of free electrons to and the concentration of holes p. From Eqs. (3-6) and (3-7), with E in electron volts, dn = N(E)f{E) dE (5-4) where dn represents the number of conduction electrons per cubic meter whose energies he between E and E 4- dE. The density of states N(E) is derived in Sec. 3-6 on the assumption that the bottom of the conduction band is at zero potential. In a semiconductor the lowest energy in the conduction band is E c> and hence Eq. (3-8) must be generalized as follows: N(E) = y(E - Ec)* (5-5) The Fermi function f(E) is given by Eq. (3-10), namely, f(E) = I _|_ ^E-ErWiT (5-6) At room temperature kT « 0.03 eV, so that f(E) = if E - E F » 0.03 and f(E) = 1 if E - E F « 0.03 (Fig. 3-10). We shall show that the Fermi level lies in the region of the energy gap midway between the valence and con- duction bands, as indicated in Fig. 5-4. This diagram shows the Fermi-Dirac distribution of Eq. (5-6) superimposed on the energy-band diagram of a semi- conductor. At absolute zero (T = 0°K) the function is as shown in Fig. 5-4o. L t room temperature some electrons are excited to higher energies and some states near the bottom of the conduction band E c will be filled. Similarly, near the top of the valence band Ey, the probability of occupancy is decreased rom unity since some electrons have escaped from their covalent bond and re now in the conduction band. For a further increase in temperature the function is as shown by the curve in Fig. 5-46 marked "T = 1000°K." The concentration of electrons in the conduction band is, from Eq. (5-4), n = j~ c N{E)f{E)dE (5-7) 100 / ELECTRONIC DEVICES AND CIRCUITS Sec. 5-3 and- For E> E C ,E — E F y>kT and Eq. (5-6) reduces to f(E) = z-( E - E rW T n = (" y(E - E c )h-^- B ^' kT dE J Ee This integral evaluates to where L.60 X 10" I9 )i = 2 ( % fc..{S«v (5-8) (■3-9) (5-10) In deriving this equation the value of y from Eq. (3-9) is used, k is given in electron volts per degree Kelvin, and k is expressed in joules per degree Kelvin. (The relationship between joules and electron volts is given in Sec. 1-5.) The mass m has been replaced by the symbol m n , which represents the effective mass of the electron. Effective Mass 8 We digress here briefly to discuss the coucept of the effective mass of the electron and hole. It is found that, when quantum mechanics is used to specify the motion within the crystal of an electron or hole on which an external field is applied, it is possible to treat the hole and electron as imaginary classical particles with effective positive masses m p and m n , respectively. This approximation is valid provided that the externally applied fields are much weaker than the internal periodic fields produced by ~r ~~ Conduction band 9 Ey Eg & -J- E« Fig, 5-4 Fermi-Dirac distribution and energy-band diagram for an intrinsic semiconductor, (a) T = 0°K and (b) T = 300°K and T m 1000°K. Sbc. 5-3 CONDUCT/ON IN SEMICONDUCTORS / 101 the lattice structure. In a perfect crystal these imaginary particles respond only to the external fields. In conclusion, then, the effective-mass approximation removes the quan- tum features of the problem and allows us to use Newton's laws to determine the effect of external forces on the electrons and holes within the crystal. The Number of Holes in the Valence Band Since the top of the valence band (the maximum energy) is E v , the density of states [analogous to Eq. (5-5) J is given by N(E) - y(E v - E) — FM (5-11) Since a "hole" signifies an empty energy level, the Fermi function for a hole is 1 — f(E), where f(E) is the probability that the level is occupied by an elec- tron. For example, if the probability that a particular energy level is occupied by an electron is 0.2, the probability that it is empty (occupied by a hole) is 0.8. Using Eq. (5-6) for f{E), we obtain ,{E-e r ) ikT (5-12) 1 - f(E) = I + € IE-E,)I e -{Er-EyikT where we have made use of the fact that E F — E y> kT for E < E r (Fig. 5-4). Hence the number of holes per cubic meter in the valence band is (5-13) p = J* v m y(E r - E)*<-i*r-*iw dE This integral evaluates to p = N ¥tL -(£r~E v MT (5-14) where N v is given by Eq. (5-10), with m„ replaced by m„, the effective mass of a hole. The Fermi Level in an Intrinsic Semiconductor It is important to note that Eqs. (5-9) and (5-14) apply to both intrinsic and extrinsic or impure semiconductors. In the case of intrinsic material the subscript i will be added to n and p. Since the crystal must be electrically neutral, nt m Pi and we have from Eqs. (5-9) and (5-14) N ci ~{E c -E r )ikT = gffa-»jrWfim ■taking the logarithm of both sides, we obtain flic E(j -f- Ey — lEp (5-15) In a N, kT ence E = Ec + Ey kT Nc F 2 2 N v (5-16) 102 / HECTRONfC DEVICES AND CIRCUITS Sec. 5-4 If the effective masses of a hole and a free electron are the same, N~c = Mr, and Eq. (5-16) yields Ep = (5-17) Hence the Fermi level lies in the center of the forbidden energy band, as shown in Fig. 5-4. The Intrinsic Concentration Using Eqs. (5-9) and (5-14), we have for the product of electron-hole concentrations np = N c N v e-< E ^ E r» kT = N c Nv*- B °< kT (5-18) Note that this product is independent of the Fermi level, but does depend upon the temperature and the energy gap E G m E r — E v . Equation (5-18) is valid for either an extrinsic or intrinsic material. Hence, writing n = n,- and p — pi = n,-, we have the important relationship (called the mass-action law) np = nc (5-19) Note that, regardless of the individual magnitudes of n and p, the product is always a constant at a fixed temperature. Substituting numerical values for the physical constants in Eq. (5-10), we obtain N c - 4.82 X 1Q 21 &)' Ti (5-20) where Nc has the dimensions of a concentration (number per cubic meter). Note that Nv is given by the right-hand side of Eq. (5-20) with m n replaced by m p . From Eqs. (5-18) to (5-20), np m n t * = (2.33 X 10") / m n m p \l 'pi^-BalkT (5-21) As indicated in Eqs. (2-15) and (2-16), the energy gap decreases linearly with temperature, so that Eq — Ego — &T (5-22) where E co is the magnitude of the energy gap at 0°K. Substituting this relationship into Eq. (5-21) gives an expression of the following form: n,- 2 = A,TU-**>' k * This result has been verified experimentally. 4 and Eqo are given in Table 5-1. (5-23) The measured values of m 5-4 DONOR AND ACCEPTOR IMPURITIES If, to pure germanium, a small amount of impurity is added in the form of a substance with five valence electrons, the situation pictured in Fig. 5-5 results. S»C' s-4 CONDUCTION IN SEMICONDUCTORS / 103 G e /Free electron 4 Fig- 5-5 Crystal lattice with a germanium atom displaced by a pentavalent impurity atom. The impurity atoms will displace some of the germanium atoms in the crystal lattice. Four of the five valence electrons will occupy covalent bonds, and the fifth will be nominally unbound and will be available as a carrier of current. The energy required to detach this fifth electron from the atom is of the order of only 0.01 eV for Ge or 0.05 eV for Si. Suitable pentavalent impurities are antimony, phosphorus, and arsenic. Such impurities donate excess (negative) electron carriers, and are therefore referred to as donor, or n-type, impurities. When donor impurities are added to a semiconductor, allowable energy levels arc introduced a very small distance below the conduction band, as is shown in Fig. 5-6. These new allowable levels are essentially a discrete level because the added impurity atoms are far apart in the crystal structure, and hence their interaction is small. In the case of germanium, the distance of the new discrete allowable energy level is only 0.01 eV (0.05 eV in silicon) below the conduction band, and therefore at room temperature almost all of the "fifth" electrons of the donor material are raised into the conduction band. If intrinsic semiconductor material is "doped" with rc-type impurities, not only does the number of electrons increase, but the number of holes decreases below that which would be available in the intrinsic semiconductor. The reason for the decrease in the number of holes is that the larger number of electrons present increases the rate of recombination of electrons with holes. If a trivalent impurity (boron, gallium, or indium) is added to an intrinsic '9- 5-6 Energy-band diagram of n "type semiconductor. Conduction band 0.01 eV T E< , 1 8 c N Eg Donor energy level 4 *i Valence band . . 1 104 / ELECTRONIC DEVICES AND CIRCUITS See. 5-4 Ge Ge Q« \«/ / • / •\In Hole >Ge Fig. 5-7 Crystal lattice with a germa- nium atom displaced by an atom of a trivalertt impurity. -' ! '" J | ■ ! • • • Ge • Ge semiconductor, only three of the covalent bonds can be filled, and the vacancy that exists in the fourth bond constitutes a hole. This situation is illustrated in Fig. 5-7. Such impurities make available positive carriers because they create holes which can accept electrons. These impurities are consequently known as acceptor, or p-type impurities. The amount of impurity which must be added to have an appreciable effect on the conductivity is very small. For example, if a donor-type impurity is added to the extent of 1 part in 10 8 , the conductivity of germanium at 30° C is multiplied by a factor of 12. When acceptor, or p-type, impurities are added to the intrinsic semi- conductor, they produce an allowable discrete energy level which is just above the valence band, as shown in Fig. 5-8. Since a very small amount of energy is required for an electron to leave the valence band and occupy the acceptor energy level, it follows that the holes generated in the valence band by these electrons constitute the largest number of carriers in the semiconductor material. We have the important result that the doping of an intrinsic semiconductor not only increases the conductivity, but also serves to produce a conductor in which the electric carriers are either predominantly holes or predominantly electrons. In an n-type semiconductor, the electrons are called the majority carriers, and the holes are called the minority carriers. In a p-type material, the holes are the majority carriers, and the electrons are the minority carriers. r— ■ — Conduction band / Acceptor energy level | «^^ o.„!.v E, fi i Fig. 5-8 Energy-band diagram of p-type semiconductor. S«c- S-6 CONDUCTION IN SEMICONDUCTORS / 105 5 _ 5 CHARGE DENSITIES IN A SEMICONDUCTOR Equation (5-19), namely, np = n,- 2 (5-19) crives one relationship between the electron n and the hole p concentrations. These densities are further interrelated by the law of electrical neutrality, which we shall now state in algebraic form: Let No equal the concentration of donor atoms. Since, as mentioned above, these are practically all ionized, N D positive charges per cubic meter are contributed by the donor ions. Hence the total positive-charge density is N D -f- p. Similarly, if N A is the concen- tration of acceptor ions, these contribute N A negative charges per cubic meter. The total negative-charge density is N A + n. Since the semiconductor is electrically neutral, the magnitude of the positive-charge density must equal that of the negative concentration, or N D +p = N A + n (5-24) Consider an n-type material having N A = 0. Since the number of elec- trons is much greater than the number of holes in an n-type semiconductor (n^>p), then Eq. (5-24) reduces to n~ N D (5-25) In an n-type material the free-electron concentration is approximately equal to the density of donor atoms. In later applications we study the characteristics of n- and p-type materials connected together. Since some confusion may arise as to which type is under consideration at a given moment, we add the subscript n or p for an n-type or a p-type substance, respectively. Thus Eq. (5-25) is more clearly written n n = Nd (5-26) The concentration p» of holes in the n-type semiconductor is obtained from Eq. (5-19), which is now written n n p n = n< 2 . Thus n,* Similarly, for a p-type semiconductor, tt„p p = n, a **Nj (6-27) (5-28) 5-6 FERMI LEVEL IN A SEMICONDUCTOR HAVING IMPURITIES r ° m Eqs. (5-1) and (5-2) it is seen that the electrical characteristics of a semi- conductor material depend on the concentration of free electrons and holes. 106 / ELECTRONIC DEVICES AND CIRCUITS Sec. 5-6 The expressions for n and p are given by Eqs. (5-9) and (5-14), respectively, and these are valid for both intrinsic semiconductors and semiconductors with impurities. The only parameter in Eqs. (5-9) and (5-14) which changes with impurities is the Fermi level E F . In order to see how E F depends on temper- ature and impurity concentration, we recall that, in the case of no impurities (an intrinsic semiconductor), E F lies in the middle of the energy gap, indi- cating equal concentrations of free electrons and holes. If a donor-type impurity is added to the crystal, then, at a given temperature and assuming all donor atoms are ionized, the first N B states in the conduction band will be filled. Hence it will be more difficult for the electrons from the valence band to bridge the energy gap by thermal agitation. Consequently, the number of electron-hole pairs thermally generated for that temperature will be reduced. Since the Fermi level is a measure of the probability of occupancy of the allowed energy states, it is clear that E F must move closer to the conduction band to indicate that many of the energy states in that band are filled by the donor electrons, and fewer holes exist in the valence band. This situation is pictured in Fig. 5-9a for an n-type material. The same kind of argument leads to the conclusion that E F must move from the center of the forbidden gap closer to the valence band for a p-type material, as indicated in Fig. 5-96. If for a given concentration of impurities the temperature of, say, the «-type material increases, more electron-hole pairs will be formed, and since all donor atoms are ionized, it is possible that the concentration of thermally generated electrons in the conduction band may become much larger than the concen- tration of donor electrons. Under these conditions the concentrations of holes and electrons become almost equal and the crystal becomes essentially intrinsic. We can conclude that as the temperature of either n-type or p-type material increases, the Fermi level moves toward the center of the energy gap. A calculation of the exact position of the Fermi level in an n-type material *t Conduction band Kb f * Ea " Valeni :« band 5 1 (a) f{E) Fig. 5'9 Positions of Fermi level in (a) n-type and (b) p-type semiconductors. S«. 5-7 CONDUCTION IN SEMICONDUCTORS / 107 can be made if we substitute n = N" D from Eq. (5-25) into Eq. (5-9). We obtain tf D m N c e- iE c~ £ * VkT (5-29) or solving for E F , E F = E c - kT In ^ (5-30) Similarly, for p-type material, from Eqs. (5-28) and (5-14) we obtain E F = E v + kT In Ni (5-31) Note that, if N A - Nd, Eqs. (5-30) and (5-31) added together (and divided by 2) yield Eq. (5-16). 5-7 DIFFUSION In addition to a conduction current, the transport of charges in a semiconductor may be accounted for by a mechanism called diffusion, not ordinarily encoun- tered in metals. The essential features of diffusion are now discussed. We see later that it is possible to have a nonuniform concentration of particles in a semiconductor. Under these circumstances the concentration p of holes varies with distance x in the semiconductor, and there exists a concen- tration gradient dp/dx in the density of carriers. The existence of a gradient implies that, if an imaginary surface is drawn in the semiconductor, the density of holes immediately on one side of the surface is larger than the density on the other side. The holes are in a random motion as a result of their thermal energy. Accordingly, holes will continue to move back and forth across this surface. We may then expect that, in a given time interval, more holes will cross the surface from the side of greater concentration to the side of smaller concentration than in the reverse direction. This net transport of charge across the surface constitutes a flow of current. It should be noted that this net transport of charge is not the result of mutual repulsion among charges °f like sign, but is simply the result of a statistical phenomenon. This dif- fusion is exactly analogous to that which occurs in a neutral gas if a concen- tration gradient exists in the gaseous container. The diffusion hole- current density J p (amperes per square meter) is proportional to the concentration gradient, and is given by '• - -»•% (5-32) nere D p (square meters per second) is called the diffusion constant for holes. similar equation exists for diffusion electron-current density [p is replaced y n > and the minus sign is replaced by a plus sign in Eq. (5-32)]. Since both 108 / ELECTRONIC DEVICES AND CIRCUITS Sec. 5*8 diffusion and mobility are statistical thermodynamic phenomena, D and /* are not independent. The relationship between them is given by the Einstein equation Mp Mr. (5-33) where V T = IcT/e = T/l 1,600 is defined as in Eq, (3-34). At room temper- ature (300°K), ii = 39Z>. Measured values of y. and computed values of D for silicon and germanium are given in Table 5-1, on page 98. 5-8 CARRIER LIFETIME In Sec. 5-1 we see that in a pure semiconductor the number of holes is equal to the number of free electrons. Thermal agitation, however, continues to produce new hole-electron pairs while other bole-electron pairs disappear as a result of recombination. On an average, a hole (an electron) will exist for r v (r„) sec before recombination. This time is called the mean lifetime of the hole and electron, respectively. Carrier lifetimes range from nanoseconds (10 -9 sec) to hundreds of microseconds. These parameters are very impor- tant in semiconductor devices because they indicate the time required for elec- tron and hole concentrations which have been caused to change to return to their equilibrium concentrations. Consider a bar of n-type silicon illuminated by light of the proper fre- quency. As a result of this radiation the hole and electron concentrations will increase by the same amount. If p na and n^ are the equilibrium concen- trations of holes and electrons in the n-type specimen, we have p w - p w = n M - n no (5-34) where p^ and n„ e represent the carrier concentrations during steady irradiation. If we now turn off the source of light, the carrier concentrations will return to their equilibrium values exponentially and with a time constant t - t„ = r p . This result has been verified experimentally, and we can write Pn — Pno = {pno ~ P M )rT*' f n„ — nno = (fl no — nn>)e- th (5-35) (5-36) We should emphasize here that majority and minority carriers in a specific region of a given specimen have the same lifetime t. Using Eqs. (5-35) and (5-36), we can obtain the expressions for the rate of concentration change. For holes, we find from Eq. (5-35) d]>n _ Pn — pno _ d_ dt T ~ dt {Pn Vno) (5-37) For electrons, a similar expression with p replaced by n is valid. The quantity p n — Pno represents the injected, or excess, carrier density. The rate of change S« 5-9 CONDUCTION IN SEMICONDUCTORS / 109 f excess density is proportional to the density — an intuitively correct result. The minus sign indicates that the change is a decrease in the case of recombi- nation and an increase when the concentration is recovering from a temporary depletion. The most important mechanism through which holes and electrons recom- bine is the mechanism involving recombination centers*-* which contribute electronic states in the energy gap of the semiconductor material. These new states are associated with imperfections in the crystal. Specifically, metallic impurities in the semiconductor are capable of introducing energy states in the forbidden gap. Recombination is affected not only by volume impurities, but also by surface imperfections in the crystal. Gold is extensively used as a recombination agent by semiconductor- device manufacturers. Thus the device designer can obtain desired carrier lifetimes by introducing gold into silicon under controlled conditions. 78 5-9 THE CONTINUITY EQUATION In the preceding section it is seen that if we disturb the equilibrium concen- trations of carriers in a semiconductor material, the concentration of holes or electrons will vary with time. In the general case, however, the carrier con- centration in the body of a semiconductor is a function of both time and dis- tance. We now derive the differential equation which governs this functional relationship. This equation is based upon the fact that charge can be neither created nor destroyed. Consider the infinitesimal element of volume of area A and length dx (Fig. 5-10) within which the average hole concentration is p. If t„ is the mean lifetime of the holes, then p/r p equals the holes per second lost by recombination per unit volume. If e is the electronic charge, then, because of recombination, the number of coulombs per second Decreases within the volume = eA dx — (5-38) If g is the thermal rate of generation of hole-electron pairs per unit volume, the number of coulombs per second Increases within the volume = eA dx g (5-39) Fig, 5-10 Relating to the conservation of charge. I+dl x + dx 110 / ELECTRONIC DEVICES AND CIRCUITS Sec. 5-9 In general, the current will vary with distance within the semiconductor. If, as indicated in Fig. 5-10, the current entering the volume at x is / and leaving at x + dx is I + dl, the number of coulombs per second Decreases within the volume = dl (5-40) Because of the three effects enumerated above, the hole density must change with time, and the total number of coulombs per second Increases within the volume = eA dx -~ dt Since charge must be conserved, eAdx~- = —eA dz — + eA dx a — dl dt 7-* w (5-41) (5-42) The hole current is the sum of the diffusion current (Eq. (5-32)] and the drift current [Eq. (5-1)], or I = -AeD. dp dx + Apefi p & (5-43) where 8 is the electric field intensity within the volume. If the semiconductor is in thermal equilibrium with its surroundings and is subjected to no applied fields, the hole density will attain a constant value p . Under these conditions, 2" = and dp/dt = 0, so that, from Eq. (5-42), (5-44) This equation indicates that the rate at which holes are generated thermally just equals the rate at which holes are lost because of recombination under equilibrium conditions. Combining Eqs. (5-42), (5-43), and (5-44) yields the equation of conservation of charge, or the continuity equation, dp Tt P - p. ^ U * dx* th d{pZ) dx (5-45) If we are considering holes in the n-type material, the subscript n is added to p and p . Also, since p is a function of both t and *, partial derivatives should be used. Making these changes, we have, finally, dp n dt _ _ Pn - p„ + A 3 2 p„ dx 2 My 3 (p»g) dx (5-46) We now consider three special cases of the continuity equation. Concentration Independent of x and with Zero Electric Field We now derive Eqs. (5-35) and (5-37) using the continuity equation. Consider a situ- ation in which 8 = and the concentration is independent of x. For example, assume that radiation falls uniformly over the surface of a semiconductor and raises the concentration to p no , which is above the thermal-equilibrium value Sec. 5-9 CONDUCTION IN SEMICONDUCTORS /111 At t = the illumination is removed. How does the concentration vary with time? The answer to this query is obtained from Eq. (5-46), which now reduces to dp* = _ Pn — Pno (5-47) dt r p in agreement with Eq. (5-37). The solution of this equation is p„ - P*. - (Pnc ~ pno)«r ( ''- (5-48) which is identical with Eq. (5-35). We now see that the mean lifetime of the holes t p can also be interpreted as the time constant with which the concen- tration returns to its normal value. In other words, t p is the time it takes the injected concentration to fall to 1/e of its initial value. Concentration Independent of t and with Zero Electric Field Let us solve the equation of continuity subject to the following conditions: There is no electric field, so that 8 = 0, and a steady state has been reached, so that dpjdt = 0. Then (5-49) (5-50) (5-51) d 2 p n = Pn — Pno dx z Dp T p The solution of this equation is p n -Vno = K#*h + K&i** where Ki and Kt are constants of integration and L p m y/DpT v This solution may be verified by a direct substitution of Eq. (5-50) into Eq. (5-49). Consider a very long piece of semiconductor extending in the posi- tive X direction from x = 0. Since the concentration cannot become infinite as x — » oo, then Kt must be zero. The quantity p n — p no as P n (x) by which the density exceeds the thermal-equilibrium value is called the injected concen- tration and is a function of the position x. We shall assume that at x = 0, **■ = P n (Q) ~ p„(0) — p no . In order to satisfy this boundary condition, * J - J\(0). Hence P n (x) =p n - pno = P rt (0)*-"% (5-52) We see that the quantity L p (called the diffusion length for holes) represents the distance into the semiconductor at which the injected concentration falls to l/« of its value at x = 0. The diffusion length L p may also be interpreted as the average distance which an injected hole travels before recombining with an electron. This statement may be verified as follows: From Fig. 5-11 and Eq. (5-52), vp.\-m<- clL * dx (5-53) 112/ RECTRONIC DEVICES AND CIRCUITS Sec, 5-9 5-JO CONDUCTION IN SEMICONDUCTORS / 113 Tfetsk ,.,..^, J. ■as*** i |p^. ■ i * ^^■■w^lv'i ;■.'■'■' • x=0 x -)dP„\ holes re com bine in the distance dx Fig. 5-11 Relating to the injected hole concentration in n-type material. |dP»| gives the number of injected holes which recombine in the distance between z and x + dx. Since each hole has traveled a distance x, the total distance traveled by \dP n \ holes is x \dP n \. Hence the total distance covered by all the holes is J Q x \dP n \. The average distance x equals this total distance divided by the total number P„(0) of injected holes. Hence /;*iip n i ! PM (5-54) thus confirming that the mean distance of travel of a hole before recombi- nation is L p . Concentration Varies Sinusoidally with I and with Zero Electric Field Let us retain the restriction £ = but assume that the injected concentration varies sinusoidally with an angular frequency w. Then, in phasor notation, Pn(x, I) = P n (x)*» (5-55) where the space dependence of the injected concentration is given by P n (x). If Eq. (5-55) is substituted into the continuity equation (5-46), the result is or d 2 P» 1 + jarr, dx* LJ n (5-56) where use has been made of Eq. (5-51). At zero frequency the equation of continuity is given by Eq. (5^49), which may be written in the form d*I\ dx 3 W A comparison of this equation with Eq. (5-56) shows that the ac solution at frequency u^O can be obtained from the dc solution (w = 0) by replacing L p by L p (l -f jWp)"*. This result is used in Chap. 13. 5-10 THE HALL EFFECT 1 If a specimen (metal or semiconductor) carrying a current I is placed in a transverse magnetic field B, an electric field £ is induced in the direction per- pendicular to both I and B. This phenomenon, known as the Hall effect, is used to determine whether a semiconductor is n- or p-type and to find the carrier concentration. Also, by simultaneously measuring the conductivity a, the mobility p. can be calculated. The physical origin of the Hall effect is not difficult to find. If in Fig. 5-12 I is in the positive X direction and B is in the positive Z direction, a force will be exerted in the negative Y direction on the current carriers. If the semiconductor is n-type, so that the current is carried by electrons, these electrons will be forced downward toward side 1 in Fig. 5-12, and side 1 becomes negatively charged with respect to side 2. Hence a potential V H) called the Hall voltage, appears between the surfaces 1 and 2. In the equi- librium state the electric field intensity E due to the Hall effect must exert a force on the carrier which just balances the magnetic force, or e£ = Bev (5-57) where e is the magnitude of the charge on the carrier, and a is the drift speed. From Eq. (1-14), £ = V H /d, where d is the distance between surfaces 1 and 2. From Eq. (1-38), J = pv = I/wd, where J is the current density, p is the charge density, and w is the width of the specimen in the direction of the magnetic field. Combining these relationships, we find V H = £d = Bvd = BJd H pw (5-58) If V H , B, I, and w are measured, the charge density p can be determined from Eq. (5-58). If the polarity of Vu is positive at terminal 2, then, as explained above, the carriers must be electrons, and p = ne, where n is the electron concentration. If, on the other hand, terminal 1 becomes charged positively with respect to terminal 2, the semiconductor must be p-type, and p = pe, where p is the hole concentration. It is customary to introduce the Hall coefficient Ru defined by Rh = - (5-59) Fi 9. 5-12 Pertaining to the Hall effect. •he carriers {whether electrons or holes) ar e subjected to a force in the negative Y direction. 4=2=35 114 / ELECTRONIC DEVICES AND CIRCUITS Sec. 5-10 Hence Rn = BI (5-60) If conduction is due primarily to charges of one sign, the conductivity <t is related to the mobility ju by Eq. (3-3), or a = pft (5-61) If the conductivity is measured together with the Hall coefficient, the mobility can be determined from (i = aRu (5-62) We have assumed in the foregoing discussion that all particles travel with the mean drift speed v. Actually, the current carriers have a random thermal distribution in speed. If this distribution is taken into account, it is found that Eq. (5-60) remains valid provided that Rn is defined by 3tt/8p. Also, Eq. (5-62) must be modified to m = (8<t/3it)Rh. REFERENCES 1. Shockley, W.: Electrons and Holes in Semiconductors, D. Van Nostrand Company, Inc., Princeton, N.J., reprinted February, 1963. Gibbons, J. F.: "Semiconductor Electronics," McGraw-Hill Book Company, New York, 1966. Middlebrook, R. D.: "An Introduction to Junction Transistor Theory," John Wiley & Sons, Inc., New York, 1957. 2. Becker, J. A., C. B. Green, and G. L. Pearson: Properties and Uses of Thermistors — Thermally Sensitive Resistors, Bell System Tech. J., vol. 26, pp. 170-212, January, 1947. 3. Adler, R, B,, A. C. Smith, and R. L. Longini: "Introduction to Semiconductor Physics," vol, 1, Semiconductor Electronics Education Committee, John Wiley & Sons, Inc., New York, 1964. 4. Conwell, E. M.: Properties of Silicon and Germanium: II, Proc. IRE, vol. 46, pp. 1281-1300, June, 1958. 5. Shockley, W., and W. T, Read, Jr. : Statistics of the Recombination of Holes and Electrons, Phys. Rev., vol 87, pp. 835-842, September, 1952. 6. Hall, R. N.: Electron-Hole Recombination in Germanium, Phys. Rev., vol. 87, p. 387, July, 1952. 7. Collins, C. B., R. O, Carlson, and C. J. Gallagher: Properties of Gold-doped Silicon, Phys. Rev., vol. 105, pp. 1168-1173, February, 1957. 8. Bemski, G.: Recombination Properties of Gold in Silicon, Phys, Rev,, vol. Ill, pp- 1515-1518, September, 1958. SEMICONDUCTOR-DIODE CHARACTERISTICS In this chapter we demonstrate that if a junction is formed between a sample of p-type and one of n-type semiconductor, this combination possesses the properties of a rectifier. The volt-ampere character- istics of such a junction are derived. Electron and hole currents as a function of distance are studied in detail. The capacitance across the junction is calculated. Although the transistor is a triode semiconductor, it may be con- sidered as one diode biased by the current from a second diode. Hence most of the theory developed in this chapter is utilized later in con- nection with our study of the transistor. 6-1 QUALITATIVE THEORY OF THE p-n JUNCTION 1 If donor impurities are introduced into one side and acceptors into the other side of a single crystal of a semiconductor, say, germanium, a p-n junction is formed. Such a system is illustrated in Fig. 6-la. The donor ion is indicated schematically by a plus sign because, after this impurity atom "donates" an electron, it becomes a positive ion. The acceptor ion is indicated by a minus sign because, after this atom "accepts" an electron, 'it becomes a negative ion. Initially, there are nominally only p-type carriers to the left of the junction and only ft-type carriers to the right. Because there is a density gradient across the junction, holes will diffuse to the right across the junction, and electrons to the left. As a result of the displacement of these charges, an electric field will appear across the junction. Equilibrium will be established when the field becomes large enough to restrain the process of diffusion. The general shape of the charge distribution may be as illustrated in 115 116 / ELECTRONIC DEVICES AND CIRCUITS 5«c. 6-1 Acceptor Hole («> Junction -4- Donor ion e © e e o o o © e e e o o o e e e e © © © © • • • © © © © • • ■ © © © © Electron p type n type Distance from junction ib) •HE-/ (c) U> (e) Electrostatic potential V or potential -energy barrier for holes j Distance from junction Potential -energy barrier for electrons i Distance from junction Fig. 6-1 A schematic diagram of a p-n junction, including the charge density, electric field intensity, and potential-energy barriers at the junction. (Not drawn to scale.) Sec. 6-2 SEMICONDUCTOR-DIODE CHARACTERISTICS / 117 Fig- 6-1&- The electric charges are confined to the neighborhood of the junc- tion and. consist of immobile ions. We see that the positive holes which neutralized the acceptor ions near the junction in the p-type germanium have disappeared as a result of combination with electrons which have diffused across the junction. Similarly, the neutralizing electrons in the n-type ger- manium have combined with holes which have crossed the junction from the p material. The unneutralized ions in the neighborhood of the junction are referred to as uncovered charges. Since the region of the junction is depleted of mobile charges, it is called the depletion region, the space-charge region, or the transition region. The thickness of this region is of the order of 10~ 4 cm = 10~ 8 m = 1 micron The electric field intensity in the neighborhood of the junction is indi- cated in Fig. 6-1 c. Note that this curve is the integral of the density func- tion p in Fig. 6-1&. The electrostatic-potential variation in the depletion region is shown in Fig. 6-ld, and is the negative integral of the function 8 of Fig. 6-lc. This variation constitutes a potential-energy barrier against the further diffusion of holes across the barrier. The form of the potential-energy barrier against the flow of electrons from the n side across the junction is shown in Fig. 6-le. It is similar to that shown in Fig. 6-ld, except that it is inverted, since the charge on an electron is negative. The necessity for the existence of a potential barrier called the contact, or diffusion, potential is now considered further. Under open-circuited conditions the net hole current must be zero. If this statement were not true, the hole density at one end of the semiconductor w T ould continue to increase indefinitely with time, a situation which is obviously physically impossible. Since the concentration of holes in the p side is much greater than that in the n side, a very large diffusion current tends to flow across the junction from the p to the n material. Hence an electric field must build up across the junction in such a direction that a drift current will tend to flow across the junction from the n to the p side in order to counterbalance the diffusion current. This equilibrium condition of zero resultant hole current allows us to calculate the height of the potential barrier V„ [Eq. (6-8)] in terms of the donor and acceptor concentrations. The numerical value for V a is of the order of magnitude of a few tenths of a volt. 6 " 2 THE p-n JUNCTION AS A DIODE Th e essential electrical characteristic of a p-n junction is that it constitutes a lode which permits the easy flow of current in one direction but restrains the 01, y m the opposite direction. We consider now, qualitatively, how this diode action comes about. Reverse Bias In Fig. 6-2, a battery is shown connected across the ttiinals of a p-n junction. The negative terminal of the battery is con- 118/ ELECTRONIC DEVICES AND CIRCUITS Sec. 6-2 S«. 6-2 SEMICONDUCTOR-DIODE CHARACTERISTICS / 119 Metal ohmic contacts <-^jy- V (a) Hp Fig. 6-2 (o) A p-n junction biased in the reverse direction, (b) The rectifkr symbol is used for the p-n diode. V (6) nected to the p side of the junction, and the positive terminal to the n side. The polarity of connection is such as to cause both the holes in the p type and the electrons in the n type to move away from the junction. Consequently, the region of negative-charge density is spread to the left of the junction (Fig. 6-16), and the positive-charge-density region is spread to the right. However, this process cannot continue indefinitely, because in order to have a steady flow of holes to the left, these holes must be supplied across the junction from the n-type germanium. And there are very few holes in the n-type side. Hence, nominally, zero current results. Actually, a small current does flow because a small number of hole-electron pairs are generated throughout the crystal as a result of thermal energy. The holes so formed in the n-type ger- manium will wander over to the junction. A similar remark applies to the electrons thermally generated in the p-type germanium. This small current is the diode reverse saturation current, and its magnitude is designated by I . This reverse current will increase with increasing temperature [Eq. (6-28)], and hence the back resistance of a crystal diode decreases with increasing temperature. The mechanism of conduction in the reverse direction may be described alternatively in the following way: When no voltage is applied to the p-n diode, the potential barrier across the junction is as shown in Fig. 6- Id. When a voltage V is applied to the diode in the direction shown in Fig. 6-2, the height of the potential-energy barrier is increased by the amount eV. This increase in the barrier height serves to reduce the flow of majority carriers (i.e., holes in p type and electrons in n type). However, the minority carriers (i.e., electrons in p type and holes in n type), since they fall down the potential- energy hill, are uninfluenced by the increased height of the barrier. The applied voltage in the direction indicated in Fig. 6-2 is called the reverse, or blocking, bias. Forward Bias An external voltage applied with the polarity shown in Fig. 6-3 (opposite to that indicated in Fig. 6-2) is called a forward bias. An ideal p-n diode has zero ohmic voltage drop across the body of the crystal. For such a diode the height of the potential barrier at the junction will be lowered by the applied forward voltage V. The equilibrium initially estab- lished between the forces tending to produce diffusion of majority carriers and the restraining influence of the potential-energy barrier at the junction Fig. 6-3 (a) A p-n junction biased in the forward direction, (b) The rectifier sym- bol is used for the p-n diode. /Metal contacts. V (a) V (b) will be disturbed. Hence, for a forward bias, the holes cross the junction from the p type to the n type, and the electrons cross the junction in the opposite direction. These majority carriers can then travel around the closed circuit, and a relatively large current will flow. Ohmic Contacts 1 In Fig. 6-2 (6-3) we show an external reverse (forward) bias applied to a p-n diode. We have assumed that the external bias voltage appears directly across the junction and has the effect of raising (lowering) the electrostatic potential across the junction. In order to justify this assump- tion we must specify how electric contact is made to the semiconductor from the external bias circuit. In Figs. 6-2 and 6-3 we indicate metal contacts with which the homogeneous p-type and n-type materials are provided. We thus see that we have introduced two metal-semiconductor junctions, one at each end of the diode. We naturally expect a contact potential to develop across these additional junctions. However, we shall assume that the metal- semiconductor contacts shown in Figs. 6-2 and 6-3 have been manufactured in such a way that they are nonrectifying. In other words, the contact potential across these junctions is approximately independent of the direction and magnitude of the current. A contact of this type is referred to as an ohmic contact. We are now in a position to justify our assumption that the entire applied voltage appears as a change in the height of the potential barrier. Inasmuch as the metal-semiconductor contacts are low-resistance ohmic contacts and the voltage drop across the bulk of the crystal is neglected, approximately the entire applied voltage will indeed appear as a change in the height of the potential barrier at the p-n junction. The Short-circuited and Open-circuited p-n Junction If the voltage V ^ Fig. 6_2 or 6-3 were set equal to zero, the p-n junction would be short- circuited. Under these conditions, as we show below, no current can flow k "* ^ anc * * ne e ke&N*totf* potential V„ remains unchanged and equal to e value under open-circuit conditions. If there were a current (J ^ 0), the T**' w ouId become heated. Since there is no external source of energy avail- Ie > the energy required to heat the metal wire would have to be supplied y the p-n bar. The semiconductor bar, therefore, would have to cool off. ear ly, under thermal equilibrium the simultaneous heating of the metal and 120 / ELECTRONIC DEVICES AND CIRCUITS See. 6-3 cooling of the bar is impossible, and we conclude that I = 0. Since under short-circuit conditions the sum of the voltages around the closed loop must be zero, the junction potential V must be exactly compensated by the metal-to-semiconductor contact potentials at the ohmic contacts. Since the current is zero, the wire can be cut without changing the situation, and the voltage drop across the cut must remain zero. If in an attempt to measure V„ we connected a voltmeter across the cut, the voltmeter would read zero voltage. In other words, it is not possible to measure contact difference of potential directly with a voltmeter. Large Forward Voltages Suppose that the forward voltage V in Fig. 6-3 is increased until V approaches V a . If V were equal to V 0) the barrier would disappear and the current could be arbitrarily large, exceeding the rating of the diode. As a practical matter we can never reduce the barrier to zero because, as the current increases without limit, the bulk resistance of the crystal, as well as the resistance of the ohmic contacts, will limit the current. Therefore it is no longer possible to assume that aH the voltage V appears as a change across the p-n junction. We conclude that, as the for- ward voltage V becomes comparable with V , the current through a real p-n diode will be governed by the ohmic-contact resistances and the crystal bulk resistance. Thus the volt-ampere characteristic becomes approximately a straight line. 6-3 BAND STRUCTURE OF AN OPEN-CIRCUITED p-n JUNCTION As in the previous section, we here consider that a p-n junction is formed by placing p- and n-type materials in intimate contact on an atomic scale. Under these conditions the Fermi level must be constant throughout the specimen at equilibrium. If this were not so, electrons on one side of the junction would have an average energy higher than those on the other side, and there would be a transfer of electrons and energy until the Fermi levels in the two sides did line up. In Sec. 5-6 it is verified that the Fermi level E F is closer to the conduction band edge E Cn in the n-type material and closer to the valence band edge E Vp in the p side. Clearly, then, the conduction band edge E Cp in the p material cannot be at the same level as E Cn , nor can the valence band edge E Vn in the a side line up with E Vp , Hence the energy-band diagram for a p-n junction appears as shown in Fig. $-4, where a shift in energy levels E„ is indicated. Note that E e » E Cp - E Cn = E Vp - Em - Ei + E t (6-1) This energy E g represents the potential energy of the electrons at the junction, as is indicated in Fig. 6-1 e. Sac. 6-3 SEMICONDUCTOR-DIODE CHARACTERISTICS / 121 Fig. 6-4 Band diagram for a p-n junction under open-circuit condi- tions. This sketch corresponds to Fig. 6-1 e and represents potential energy for electrons. The width of the forbidden gap is Bo in electron volts. The Contact Difference of Potential We now obtain an expression for E B . From Fig. 6-4 we see that Ep — Ey. = t%Eq — Ei and Ecn — Ep = \Ea — E% Adding these two equations, we obtain E„ = E x + E 2 = Eo - (E c « - Ep) - (E F - E Vp ) From Eqs. (5-18) and (5-19), E s = kT In NcNv From Eq. (5-30), nr E Cn -E F = kT In ' p rom Eq. (5-31), Ep - E Vp = kT In £p (6-2) (6-3) (6-4) (6-5) (6-6) (6-7) 122 / ELECTRONIC DEVICES AND CIRCUITS Sec. 6-3 S9C 6*A SEMICONDUCTOR-DIODE CHARACTERISTICS / 123 Substituting from Eqs. (6-5), (6-6), and (6-7) in Eq. (6-4) yields £, = fer f m ^_ ln ^_ ln £A = kTlJ^^XA = kTln?*4± (6-8) We emphasize that, in the above equations, the E*s are expressed in electron volts and k has the dimensions of electron volts per degree Kelvin. The con- tact difference in potential V Q is expressed in volts and is numerically equal to E . Note that V depends only upon the equilibrium concentrations, and not at aU upon the charge density in the transition region. Other expressions for E are obtained by substituting Eqs. (5-26), (5-27), and (5-28) in Eq. (6-8). We find E = kT In ^ = kT In — (6-9) where the subscripts o are added to the concentrations to indicate that these are obtained under conditions of thermal equilibrium. Using the reasonable values p^ = 10 16 cm"', p™ - 10* cm" 3 , and k T = 0.026 eV at room tempera- ture, we obtain E =* 0.5 eV. An Alternative Derivation 2 for V In Sec. 6-1 we indicate that an application of the equilibrium condition of zero resultant hole current allows a calculation of V to be made. We now carry out such an analysis. Since the net hole current density is zero, the negative of the hole diffusion current [Eq. (5-32)] must equal the hole drift current [Eq. (3-2)], or eDp dx = epLpp& (6 ~ 10) The Einstein relation [Eq. (5-33)] is ^ = V T (6-11) where the volt equivalent of temperature V T is defined by Eq. (3-34). Substi- tuting Eq. (6-11) in Eq. (6-10) and remembering the relationship (1-15) between field intensity and potential, we obtain dp _ &dx _ dV p Vt Vt (6-12) If this equation is integrated between limits which extend across the junction (Fig. 6-ld) from the p material, where the equilibrium hole concentration is Ppo, to the n side, where the hole density is p M) the result is Ppo - Pnot VJV T Since V e /V T = E„/kT, Eq. (6-13) is equivalent to Eq. (6-9). (6-13) Fig. 6-5 The hole- and electron-current compo- nents vs. distance in a p-n junction diode. The space- charge region at the Junc- tion is assumed to be negligibly small. Ipp, hole current I npt electron current (6) jc=0 Total current J 7 BB , electron current I pn , hole current Distance 6-4 THE CURRENT COMPONENTS IN A p-n DIODE In Sec. 6-2 it is indicated that when a forward bias is applied to a diode, holes are injected into the n side and electrons into the p side. The number of these injected minority carriers falls off exponentially with distance from the junction [Eq. (5-50)]. Since the diffusion current of minority carriers is proportional to the concentration gradient [Eq. (5-32)], this current must also vary exponentially with distance. There are two minority currents, /„* and In P , and these are indicated in Fig. 6-5. The symbol f /„«(£) represents the hole current in the n material, and l np (x) indicates the electron current in the p side as a function of x. Electrons crossing the junction at x = from right to left constitute a current in the same direction as holes crossing the junction from left to right. Hence the total current / at x = is / = U (6-14) .„ n (0)+/ np (0) Since the current is the same throughout a series circuit, / is independent of x, and is indicated as a horizontal line in Fig. 6-5. Consequently, in the p side, there must be a second component of current /„, which, when added to I „„, gives the total current /. Hence this hole current in the p side I„ (a majority carrier current) is given by I„(x) = I - I„ p (x) (6-15) This current is plotted as a function of distance in Fig. 6-5, as is also the corresponding electron current /„„ in the n material. This figure is drawn for a n unsym metrically doped diode, so that /,,„ ^ l np . Note that deep into the p side the current is a drift (conduction) current ipp of holes sustained by the small electric field in the semiconductor. As the t If the letters p and n both appear in a symbol, the first letter refers to the type of carrier, and the second to the type of material. 124 / ELECTRONIC DEVICES AND CIRCUITS See. 6-5 holes approach the junction, some of them recombine with the electrons, which are injected into the p side from the n side. Hence part of the current /„„ becomes a negative current just equal in magnitude to the diffusion current In P . The current l pp thus decreases toward the junction (at just the proper rate to maintain the total current constant, independent of distance). What remains of I pp at the junction enters the n side and becomes the hole diffusion current /„„. Similar remarks can be made with respect to current /„„. Hence, in a forward-biased p-n diode, the current enters the p side as a hole current and leaves the n side as an electron current of the same magnitude. We emphasize that the current in a p-n diode is bipolar in character since it is made up of both positive and negative carriers of electricity. The total current is constant throughout the device, but the proportion due to holes and that due to electrons varies with distance, as indicated in Fig. 6-5. 6-5 QUANTITATIVE THEORY OF THE p-n DIODE CURRENTS We now derive the expression for the total current as a function of the applied voltage (the volt-ampere characteristic). In the discussion to follow we neg- lect the depletion-layer thickness, and hence assume that the barrier width is zero. If a forward bias is applied to the diode, holes are injected from the p side into the n material. The concentration p« of holes in the n side is increased above its thermal-equilibrium value p no and, as indicated in Eq. (5-52), is given by y«{x) = Pno + Pn(0)r* ,L > (6-16) where the parameter L p is called the diffusion length for holes in the n material, and the injected, or excess, concentration at x = is P.(0) - p„(0) - p. (6-17) These several hole-concentration components are indicated in Fig. 6-6, which shows the exponential decrease of the density p n (x) with distance x into the n material. From Eq. (5-32) the diffusion hole current in the n side is given by ■* «n — ■**.£ (6-18) Taking the derivative of Eq. (6-16) and substituting in Eq. (6-18), we obtain I P n(x) - AeD P P »W € -*IL t (649) Li p This equation verifies that the hole current decreases exponentially with dis- tance. The dependence of I pn upon applied voltage is contained implicitly in the factor P»(0) because the injected concentration is a function of voltage. We now find the dependence of P„(0) upon V. Sec 6-5 SEMICONDUCTOR-DIODE CHARACTERISTICS / 125 I Concentration, p„ Rg. 6-6 Defining the several components of hole concentration in the n side f a forward-biased diode. The diagram is not drawn to scale since p„(0) » p n »- M0> n material Injected or excess charge Distance The Law of the Junction If the hole concentrations at the edges of the space-charge region are p p and p n in the p and n materials, respectively, and if the barrier potential across this depletion layer is V B , then (6-20) p p = p n € V » IV T This is the Boltzmann relationship of kinetic gas theory. It is valid 2 even under nonequilibrium conditions as long as the net hole current is small com- pared with the diffusion or the drift hole current. Under this condition, called low-level injection, we may to a good approximation again equate the magnitudes of the diffusion and drift currents. Starting with Eqs. (6-10) and (6-12) and integrating over the depletion layer, Eq. (6-20) is obtained. If we apply Eq. (6-20) to the case of an open-circuited p-n junction, then Pp ■ Pr°, Pn = p»», and V B = V* Substituting these values in Eq. (6-20), it reduces to Eq. (6-13), from which we obtain the contact potential V . Consider now a junction biased in the forward direction by an applied voltage V. Then the barrier voltage V s is decreased from its equilibrium value V by the amount V, or V B = V a - V. The hole concentration through- out the p region is constant and equal to the thermal equilibrium value, or P P = ppo. The hole concentration varies with distance into the n side, as indi- cated in Fig. 6-6. At the edge of the depletion layer, x = 0, p» - Pn(0). The Boltzmann relation (6-20) is, for this case, (6-21) Pvo = p.(0)«<*.-™^ Combining this equation with Eq. (6-13), we obtain p„(0) = Vno< ViV * This boundary condition is called the law of the junction. It indicates that, f °r a forward bias (V > 0), the hole concentration p„(0) at the junction is (6-22) 126 / ELECTRONIC DEVICES AND CIRCUITS Sec. 6-5 greater than the thermal-equilibrium value p„„. A similar law, valid for elec- trons, is obtained by interchanging p and n in Eq. (6-22). The hole concentration f\,(0) injected into the n side at the junction is obtained by substituting Eq. (6-22) in Eq. (6-17), yielding Pn(0) = p«,(e vlv r - 1) (6-23) The Forward Currents The hole current 7 pn (0) crossing the junction into the n side is given by Eq. (6-19), with x «=• 0. Using Eq. (6-23) for .Pti(O), we obtain V(0) AeD p p, ( e viv T _ x) (6-24) The electron current I np (0) crossing the junction into the p side is obtained from Eq, (6-24) by interchanging n and p, or AeD n n. /»p(0) = L n **1 ( € VIV T _ J) (6-25) Finally, from Eq. (6-14), the total diode current I is the sum of I pn (Q) and ^»p(0), or where / = LU^T - 1) (6-26) (6-27) If W p and W n are the widths of the p and n materials, respectively, the above derivation has implicitly assumed that W p y> L p and W„ S> L n . If, as some- times happens in a practical diode, the widths are much smaller than the dif- fusion lengths, the expression for I remains valid provided that L p and L n are replaced by W p and W n> respectively (Prob. 6-9). The Reverse Saturation Current In the foregoing discussion a positive value of V indicates a forward bias, The derivation of Eq. (6-26) is equally valid if V is negative, signifying an applied reverse-bias voltage. For a reverse bias whose magnitude is large compared with Y r ('-^26 mV at room tempera- ture), I —*■ —I . Hence I„ is called the reverse saturation current. Combining Eqs. (5-27), (5-28), and (6-27), we obtain Io - Ae \LpW D +l where n? is given by Eq. (5-23), n,- 2 = AeTh-Zootkr = A T*e- r °° lr r (6-28) (6-29) where V GO is a voltage which is numerically equal to the forbidden-gap energy Eqq in electron volts, and Vr is the volt equivalent of temperature [Eq. (3-34)]. For germanium the diffusion constants D p and D n vary approxi- SEMICONDUCTOR-DIODE CHARACTERISTICS / 127 mately 3 inversely proportional to T. Hence the temperature dependence of L is I. = JttSfV-WS* (6-30) where K\ is a constant independent of temperature. Throughout this section we have neglected carrier generation and recombi- nation in the space-charge region. Such an assumption is valid for a ger- manium diode, but not for a silicon device. For the latter, the diffusion cur- rent is negligible compared with the transition-layer charge-generation 3 - 4 current, which is given approximately by I = J ( € viiv r - 1) (6-31) where 57 « 2 for small (rated) currents and j? « 1 for large currents. Also, /„ is now found to be proportional to n,- instead of n*. Hence, if Kt is a constant, I = K t T l -*c v «>i* v T (6-32) The practical implications of these diode equations are given in the following sections. 6-6 THE VOLT-AMPERE CHARACTERISTIC The discussion of the preceding section indicates that, for a p-n junction, the current / is related to the voltage V by the equation I = I (€ v ^ v r - 1) (6-33) A positive value of I means that current flows from the p to the n side. The diode is forward-biased if V is positive, indicating that the p side of the junc- tion is positive with respect to the n side. The symbol n is unity for ger- manium and is approximately 2 for silicon. The symbol V T stands for the volt equivalent of temperature, and is given by Eq. (3-34), repeated here for convenience: V T = (6-34) 11,600 At room temperature (T = 300°K), V T = 0.026 V = 26 mV. The form of the volt-ampere characteristic described by Eq. (6-33) is shown in Fig. 6-7a. When the voltage V is positive and several times V T , the unity in the parentheses of Eq. (6-33) may be neglected. Accordingly, except for a small range in the neighborhood of the origin, the current increases e *ponentially with voltage. When the diode is reverse-biased and |F| is several times V T , I m —U. The reverse current is therefore constant, inde- pendent of the applied reverse bias. Consequently, h is referred to as the reverse saturation current. 128 / ELECTRONIC DEVICES AND CIRCUITS Sec. 6-6 LmA e 5 4 3 2 1 ♦ 0. 0.5 1.0 A v,v (b) Fig. 6-7 (a) The volt-ampere characteristic of an ideal p-n diode, (fa) The volt-ampere characteristic for a germanium diode redrawn to show the order of magnitude of currents. Note the expanded scale for reverse currents. The dashed portion indicates breakdown at Vz. For the sake of clarity, the current I in Fig. 6-7a has been greatly exag- gerated in magnitude. Ordinarily, the range of forward currents over which a diode is operated is many orders of magnitude larger than the reverse satu- ration current. In order to display forward and reverse characteristics con- veniently, it is necessary, as in Fig. 6-76, to use two different current scales. The volt-ampere characteristic shown in that figure has a forward current scale in milliamperes and a reverse scale in microamperes. The dashed portion of the curve of Fig. 6-76 indicates that, at a reverse- biasing voltage V Zi the diode characteristic exhibits an abrupt and marked departure from Eq. (6-33). At this critical voltage a large reverse current flows, and the diode is said to be in the breakdown region, discussed in Sec. 6-12. The Cutin Voltage V y Both silicon and germanium diodes are com- mercially available. A number of differences between these two types are relevant in circuit design. The difference in volt-ampere characteristics is brought out in Fig. 6-8. Here are plotted the forward characteristics at room temperature of a general-purpose germanium switching diode and a general- purpose silicon diode, the 1N270 and 1N3605, respectively. The diodes have comparable current ratings. A noteworthy feature in Fig. 6-8 is that there exists a cutin, offset, break-point, or threshold voltage V y below which the cur- rent is very small (say, less than 1 percent of maximum rated value). Beyond V y the current rises very rapidly. From Fig. 6-8 we see that V y is approxi- mately 0.2 V for germanium and 0.6 V for silicon. • Note that the break in the silicon-diode characteristic is offset about 0.4 V with respect to the break in the germanium-diode characteristic. The Sec. 6-6 SEMICONDUCTOR-DIODE CHARACTERISTICS / 129 /,mA 100 SO 60 40 20 / J Ge/ Si/ 0.2 0.4 0.6 0.8 1.0 V,V Fig. 6-8 The forward volt-ampere characteristics of a germanium (1N270) and a silicon (1N3605) diode at 25°C. reason for this difference is to be found, in part, in the fact that the reverse saturation current in a germanium diode is normally larger by a factor of about 1,000 than the reverse saturation current in a silicon diode of com- parable ratings. Thus, if T is in the range of microamperes for a germanium diode, I will be in the range of nanoamperes for a silicon diode. Since t) = 2 for small currents in silicon, the current increases as t v!2V r for the first several tenths of a volt and increases as e vlv r only at higher voltages. This initial smaller dependence of the current on voltage accounts for the further delay in the rise of the silicon characteristic. Logarithmic Characteristic It is instructive to examine the family of curves for the silicon diodes shown in Fig. 6-9. A family for a germanium diode of comparable current rating is quite similar, with the exception that corresponding currents are attained at lower voltage. From Eq. (6-33), assuming that V is several times Vt, so that we may drop the unity, we have log / = log /„ + 0.434 V/^V T . We therefore expect n Fig. 6-9, where log J is plotted against V, that the plots will be straight mes. We do indeed find that at low currents the plots are linear and corre- P°ud to ij a= 2. At large currents an increment of voltage does not yield as ar ge an increase of current as at low currents. The reason for this behavior 'to be found in the ohmic resistance of the diode. At low currents the m,c drop is negligible and the externally impressed voltage simply decreases e potential barrier at the p-n junction. At high currents the externally T30 / ELECTRONIC DEVICES AND CIRCUITS I, inA 1,000 500 100 50 10 5 1 0.5 0.1 0.05 A 50° C A l*€ -55 °C Sec. 6-7 Fig. 6-9 Volt-ampere characteristics at three different temperatures for a silicon diode (planar epitaxial passivated types 1N36G5, 1N3606, 1N3608, and1N3609). The shaded area indicates 25°C limits of controlled conductance. Note that the vertical scale is logarithmic and encom- passes a current range of 50,000. (Courtesy of General Electric Company.) 0.2 0.4 0.6 0.8 1.0 v,v impressed voltage is called upon principally to establish an electric field to overcome the ohmic resistance of the semiconductor material. Therefore, at high currents, the diode behaves more like a resistor than a diode, and the current increases linearly rather than exponentially with applied voltage. 6-7 THE TEMPERATURE DEPENDENCE OF p-n CHARACTERISTICS Let us inquire into the diode voltage variation with temperature at fixed current. This variation may be calculated from Eq. (6-33), where the tem- perature is contained implicitly in Vr and also in the reverse saturation cur- rent. The dependence of h on temperature T is, from Eqs. (6-30) and (6-32), given approximately by L = KT m <r v °oWT (6-35) where if is a constant and eVgo (e is the magnitude of the electronic charge) is the forbidden-gap energy in joules: For Ge: q = 1 For Si: ij = 2 m = 2 m = 1.5 Voo = 0.785 V Voo - 1.21 V Taking the derivative of the logarithm of Eq. (6-35), we find LdT d(ln h) dT _ m Voo T "'" v TVt (6-36) At room temperature, we deduce from Eq. (6-36) that d(ln L)/dT = 0.08°C- 1 for Si and 0.11°C~ l for Ge. The performance of commercial diodes is only approximately consistent with these results. The reason for the discrepancy Sac- 6 ' 7 SEMICONDUCTOR-DIODE CHARACTERISTICS / 131 is that, in a physical diode, there is a component of the reverse saturation current due to leakage over the surface that is not taken into account in Eq. (6-35)- Since this leakage component is independent of temperature, we may expect to find a smaller rate of change of I„ with temperature than that pre- dicted above. From experimental data we find that the reverse saturation current increases approximately 7 percent/°C for both silicon and germanium. Since (1.07) 10 * 2.0, we conclude that the reverse saturation current approxi- mately doubles for every 10°C rise in temperature. From Eq. (6-33), dropping the unity in comparison with the exponential, we find, for constant /, d I = v_ dT T ' T {ldh\ \I. dT) V - (Voo + m n V T ) (6-37) where use has been made of Eq. (6-36). Consider a diode operating at room temperature (300°K) and just beyond the threshold voltage V r (say, at 0.2 V for Ge and 0.6 for Si). Then we find, from Eq. (6-37), dV dT -2.1 mV/°C -2.3mV/°C for Ge for Si (6-38) Since these data are based on 'average characteristics," it might be well for conservative design to assume a value of dV dT = -2.5 mV/°C (6-39) for either Ge or Si at room temperature. Note from Eq. (6-37) that \dV/dT\ decreases with increasing T. The temperature dependence of forward voltage is given in Eq. (6-37) as the difference between two terms. The positive term V/T on the right-hand side results from the temperature dependence of Vt. The negative term results from the temperature dependence of I , and does not depend on the voltage V across the diode. The equation predicts that for increasing V, dV/dT should become less negative, reach zero at V = Voo + m^V T , and thereafter reverse sign and go positive. This behavior is regularly exhibited py diodes. Normally, however, the reversal takes place at a current which is higher than the maximum rated current. The curves of Fig. 6-9 also suggest this behavior. At high voltages the horizontal separation between curves °' different temperatures is smaller than at low voltages. Typical reverse characteristics of germanium and silicon diodes are given ltl *ig. 6-10a and 6. Observe the very pronounced dependence of current on Verse voltage, a result which is not consistent with our expectation of a con- ant saturated reverse current. This increase in /„ results from leakage across le surface of the diode, and also from the additional fact that new current rr iers may be generated by collision in the transition region at the junction. 1 the other hand, there are man}"- commercially available diodes, both ger- an nun and silicon, that do exhibit a fairly constant reverse current with 132 / ELECTRONIC DEVICES AND CIRCUITS Sec. 6-8 1.0 0.1 F *"""^ W r p^*"'* 55° C 25 °C _ i 5" c 2 40 60 SO 100 o.r. a 0.05 150"C 100"C 25 a C Reverse voltage, V (a) 30 60 90 120 Reverse voltage, V 150 Fig. 6-10 Examples of diodes which do not exhibit a constant reverse saturation current, (a) Germanium diode 1N270; (b) silicon 1N461. (Courtesy of Raytheon Company.) increasing voltage. The much larger value of h for a germanium than for a silicon diode, to which we have previously referred, is apparent in comparing Fig. 6-10a and b. Since the temperature dependence is approximately the same in both types of diodes, at elevated temperatures the germanium diode will develop an excessively large reverse current, whereas for silicon, I a will be quite modest. Thus we can see that for Ge in Fig. 6-10 an increase in tem- perature from room temperature (25°C) to 90°C increases the reverse current to hundreds of microamperes, although in silicon at 100°C the reverse current has increased only to some tenths of a microampere. 6-8 DIODE RESISTANCE The static resistance R of a diode is denned as the ratio V/I of the voltage to the current. At any point on the volt-ampere characteristic of the diode (Fig. 6-7), the resistance R is equal to the reciprocal of the slope of a line joining the operating point to the origin. The statie resistance varies widely with V and / and is not a useful parameter. The rectification property of a diode is indicated on the manufacturer's specification sheet by giving the maximum forward voltage Vf required to attain a given forward current If and also the maximum reverse current /« at a given reverse voltage Vr. Typi- st ^- fi SEMICONDUCTOR-DIODE CHARACTERISTICS / 133 ca l values for a silicon planar epitaxial diode are Vr = 0.8 V at /^ = 10 mA (corresponding to Rf = 80 fl) and Ir = 0.1 mA at Vr = 50 V (corresponding to Rb= 500 M). For small-signal operation the dynamic, or incremental, resistance r is an important parameter, and is defined as the reciprocal of the slope of the volt- ampere characteristic, r = dV/dl. The dynamic resistance is not a constant, but depends upon the operating voltage. For example, for a semiconductor diode, we find from Eq. (6-33) that the dynamic conductance g = 1/r is _ dJ I*rWT 9 ~ dV V V T I + I, nVr (6-40) For a reverse bias greater than a few tenths of a volt (so that |F/^Tr| 2> l)j g is extremely small and r is very large. On the other hand, for a forward bias greater than a few tenths of a volt, 7 » I 0) and r is given approximately by rfVr I (6-41) The dynamic resistance varies inversely with current; at room temperature and for y = 1, r = 26/7, where / is in milliamperes and.r in ohms. For a forward current of 26 mA, the dynamic resistance is 1 fi. The ohmic body resistance of the semiconductor may be of the same order of magnitude or even much higher than this value. Although r varies with current, in a small- signal model, it is reasonable to use the parameter r as a constant. A Piece wise Linear Diode Characteristic A large-signal approximation which often leads to a sufficiently accurate engineering solution is the piecewise linear representation. For example, the piecewise linear approximation for a semiconductor diode characteristic is indicated in Fig. 6-11. The break point is not at the origin, and hence V y is also called the offset, or threshold, voltage. 1 ho diode behaves like an open circuit if V < V y , and has a constant incre- mental resistance r = dV/dl if V > V y . Note that the resistance r (also designated as R f and called the forward resistance) takes on added physical 9- 6-11 The piecewise linear character- nation of a semiconductor diode. Slope, 134 / ELECTRONIC DEVICES AND CIRCUITS Sec. 6-9 significance even for this large-signal model, whereas the static resistance Rf = V/I is not constant and is not useful The numerical values V y and R f to be used depend! upon the type of diode and the contemplated voltage and current swings. For example, from Fig. 6-8 we find that, for a current swing from cutoff to 10 mA with a ger- manium diode, reasonable values are V y = 0.6 V and R f = 15 0. On the other hand, a better approximation for current swings up to 50 mA leads to the following values: germanium, V 7 = 0.3 V, R f = 6 S2; silicon, V y = 0.65 V, R/ = 5.5 ii. For an avalanche diode, discussed in Sec. 6-12, V r = V z , and Rf is the dynamic resistance in the breakdown region. 6-9 SPACE-CHARGE, OR TRANSITION, CAPACITANCE 1 C 7 As mentioned in Sec. 6-1, a reverse bias causes majority carriers to move away from the junction, thereby uncovering more immobile charges. Hence the thickness of the space-charge layer at the junction increases with reverse volt- age. This increase in uncovered charge with applied voltage may be con- sidered a capacitive effect. We may define an incremental capacitance Ct by Ct — dQ dV (6-42) where dQ is the increase in charge caused by a change dV in voltage. It follows from this definition that a change in voltage dV in a time dt will result in a current i ■ dQ/dt, given by t-r dV (6-43) Therefore a knowledge of Ct is important in considering a diode (or a transis- tor) as a circuit element. The quantity Ct is referred to as the transition- region, space-charge, barrier, or depletion-region, capacitance. We now consider Ct quantitatively. As it turns out, this capacitance is not a constant, but depends upon the magnitude of the reverse voltage. It is for this reason that C r is defined by Eq. (6-42) rather than as the ratio Q/V. An Alloy Junction Consider a junction in which there is an abrupt change from acceptor ions on one side to donor ions on the other side. Such a junction is formed experimentally, for example, by placing indium, which is trivalent, against n-type germanium and heating the combination to a high temperature for a short time. Some of the indium dissolves into the ger- manium to change the germanium from n to p type at the junction. Such a junction is called an alloy, or fusion, junction. It is not necessary that the concentration N A of acceptor ions equal the concentration No of donor impuri- ties. As a matter of fact, it is often advantageous to have an unsymmetrical S*. 6-9 SEMICONDUCTOR-DIODE CHARACTERISTICS / 135 P type n type Charge density Pig, 6-12 The charge- density and potential variation at a fusion p-n junction (W « 10 -4 cm). x = G junction. Figure 6-12 shows the charge density as a function of distance from an alloy junction in which the acceptor impurity density is assumed to be much smaller than the donor concentration. Since the net charge must be zero, then eN A W p = eN D W n (6-44) " Na « Ar B; then W p » W „. For simplicity, we neglect W n and assume that the entire barrier potential Vb appears across the uncovered acceptor ions. *ne relationship between potential and charge density is given by Poisson's equation, d*V dx 2 eN A 6 (6-45) ncre e is the permittivity of the semiconductor. If e r is the (relative) dielec- c constant and t„ is the permittivity of free space (Appendix B), then c = t,e , e electric lines of flux start on the positive donor ions and terminate on the gative acceptor ions. Hence there are no flux lines to the left of the bound- ry x = o in Fig. 6-12, and £ = — dV/dx = at x = 0. Also, since the zero Potential is arbitrary, we choose V = at x = 0. Integrating Eq. (6-45) T36 / ELECTRONIC DEVICES AND CIRCUITS subject to these boundary conditions yields V = eN * xi At x = W p « W , V - V B , the barrier height. Thus V B = ^ W* Sec. 6-9 (6-46) (6-47) If we now reserve the symbol V for the applied bias, then V B = V B — V, where V is a negative number for an applied reverse bias and V is the contact potential (Fig. 6- Id). This equation confirms our qualitative conclusion that the thickness of the depletion layer increases with applied reverse voltage. We now see that W varies as VV. If A is the area of the junction, the charge in the distance W is Q = eN A WA The transition capacitance CV, given by Eq. (6-42), is Ct = dQ dV - eN A A dW dV From Eq. (6-47), \dW/dV\ = e/eN A W, and hence Ct ~W It is interesting to note that this formula is exactly the expression which is obtained for a parallel-plate capacitor of area A (square meters) and plate separation W (meters) containing a material of permittivity e. If the concen- tration N D is not neglected, the above results are modified only slightly. In Eq. (6-47) W represents the total space-charge width, and 1/N A is replaced by 1/N A + 1/JW Equation (6-49) remains valid. A Grown Junction A second form of junction, called a grown junction, is obtained by drawing a single crystal from a melt of germanium whose type is changed during the drawing process by adding first p-type and then n-type impurities. For such a grown junction the charge density varies gradually (almost linearly), as indicated in Fig. 6-13. If an analysis similar to that Charge density Fig. 6-T3 The charge-density variation at a grown p-n junction. SEMICONDUCTOR-DIODE CHARACTERISTICS / T37 4.U 3.2 2.4 1.6 25° C ■■■■ 1N914 1N916 0.8 Fig. 6-M Typical barrier-capaci- tance variation, with reverse volt- age, of silicon diodes 1N914 and 1H916. (Courtesy of Fairchild Semiconductor Corporation.) 10 is 20 Reverse voltage, V given above is carried out for such a junction, Eq. (6-49) is found to be valid where W equals the total width of the space-charge layer. However, it now turns out that W varies as Vs k instead of Vb*. Varactor Diodes We observe from the above equations that the barrier capacitance is not a constant but varies with applied voltage. The larger the reverse voltage, the larger is the space-charge width W, and hence the smaller the capacitance C T - The variation is illustrated for two typical diodes in Fig. 6-14. Similarly, for an increase in forward bias (V positive), W decreases and C T increases. The voltage- variable capacitance of a p-n junction biased in the reverse direction is useful in a number of circuits. One of these applications is voltage tuning of an LC resonant circuit. Other applications include self-balancing bridge circuits and special types of amplifiers, called parametric amplifiers. Diodes made especially for the above applications which are based on the voltage- variable capacitance are called varactor s, varicaps, or voltacaps. A circuit model for a varactor diode under reverse bias is shown in Fig. 6-15. ve- to '9- 6-15 A varactor diode under reverse ,as - (a) Circuit symbol; (b) circuit model. Rr -AAA c~4 tf—L- C T -AAAr R. 0>) 138 / ELECTRONIC DEVICES AND CIRCUITS Sec. 6-10 The resistance R. represents the body (ohmic) series resistance of the diode. Typical values of C r and R. are 20 pF and 8.5 Q, respectively, at a reverse bias of 4 V. The reverse diode resistance R r shunting C T is large (>1 M), and hence is usually neglected. In circuits intended for use with fast waveforms or at high frequencies, it is required that the transition capacitance be as small as possible, for the following reason: A diode is driven to the reverse-biased condition when it is desired to prevent the transmission of a signal. However, if the barrier capacitance C T is large enough, the current which is to be restrained by the low conductance of the reverse-biased diode will flow through the capacitor (Fig. 6-156). 6-10 DIFFUSION CAPACITANCE For a forward bias a capacitance which is much larger than that considered in the preceding section comes into play. The origin of this capacitance is now discussed. If the bias is in the forward direction, the potential barrier at the junction is lowered and holes from the p side enter the n side. Similarly, electrons from the n side move into the p side. This process of minority- carrier injection is discussed in Sec. 6-5, where we see that the excess hole density falls off exponentially with distance, as indicated in Fig. 6-6. The shaded area under this curve is proportional to the injected charge. As explained in Sec. 6-9, it is convenient to introduce an incremental capaci- tance, defined as the rate of change of injected charge with applied voltage. This capacitance C D is called the diffusion, or storage, capacitance. Derivation of Expressions for C D We now make a quantitative study of the diffusion capacitance C D . For simplicity of discussion we assume that one side of the diode, say, the p material, is so heavily doped in comparison with the n side that the current / is carried across the junction entirely by holes moving from the p to the n side, or J = /^(O). The excess minority charge Q will then exist only on the n side, and is given by the shaded area of Fig. 6-6 multiplied by the diode cross section A and the electronic charge e. Hence Q = f* AeP n (Q)t-~iL f dx = AeLJ>„(0) and f> - d ® at dP n (0) dV —* dV The hole current / is given by l pR (x) in Eq. (6-19), with a; = 0, or , = AeD v P n {u) (6-50) (6-51) (6-52) Sec 6-l° and dP n (0) _ _L dV dl AeD P dV AeD, SEMICONDUCTOR-DIODE CHARACTERISTICS / 139 (6-53) where g = dl/dV is the diode conductance given in Eq. (6-40). Combining Eqs. (6-51) and (6-53) yields Since from Eq. (5-51) the mean lifetime for holes t p = t is given by 7 Z>„ (6-54) (6-55) (6-56) (6-57) then C D = rg From Eq. (6-41), g = IfaVt, and hence c - rI We see that the diffusion capacitance is proportional to the current I. In the derivation above we have assumed that the diode current / is due to holes only. If this assumption is not satisfied, Eq. (6-56) gives the diffusion capaci- tance Ca p due to holes only, and a similar expression can be obtained for the diffusion capacitance Co, due to electrons. The total diffusion capacitance can then be obtained as the sum of C Dp and C e , (Prob. 6-30). For a reverse bias g is very small and Co may be neglected compared with C r . For a forward current, on the other hand, C D is usually much larger than Cr- For example, for germanium (13 = 1) at I = 26 mA, g — 1 mho, and C D = t. If, say, r = 20 *»sec, then C D = 20 ftF, a value which is about a million times larger than the transition capacitance. Despite the large value of C D , the time constant rC D (which is of impor- tance in circuit applications) may not be excessive because the dynamic for- ward resistance r = l/g is small. From Eq. (6-56), rC D = r (6-58) Hence the diode time constant equals the mean lifetime of minority carriers, w hich lies in range of nanoseconds to hundreds of microseconds. The impor- tance of t in circuit applications is considered in the following section. Charge-control Description of a Diode From Eqs. (6-50), (6-52), and (6-55), 1 - Qrh = (6-59) Thi ^his very impjrtant equation states that the diode current (which consists of «oles crossing the junction from the p to the n side) is proportional to the 140 / ELECTRONIC DEVICES AND CIRCUITS Sac. 6- J J stored charge Q of excess minority carriers. The factor of proportionality is the reciprocal of the decay time constant (the mean lifetime t) of the minority carriers. Thus, in the steady state, the current I supplies minority carriers at the rate at which these carriers are disappearing because of the process of recombination. The charge-control characterization of a diode describes the device in terms of the current I and the stored charge Q, whereas the equivalent-circuit characterization uses the current J and the junction voltage V. One immedi- ately apparent advantage of this charge-control description is that the expo- nential relationship between I and V is replaced by the linear dependence / on Q. The charge Q also makes a simple parameter, the sign of which determines whether the diode is forward- or reverse-biased. The diode is forward-biased if Q is positive and reverse-biased if Q is negative. 6-11 p-n DIODE SWITCHING TIMES When a diode is driven from the reversed condition to the forward state or in the opposite direction, the diode response is accompanied by a transient, and an interval of time elapses before the diode recovers to its steady state. The forward recovery time t fT is the time difference between the 10 percent point of the diode voltage and the time when this voltage reaches and remains within 10 percent of its final value. It turns out 6 that t /T does not usually constitute a serious practical problem, and hence we here consider only the more impor- tant situation of reverse recovery, Diode Reverse Recovery Time When an external voltage forward- biases a p-n junction, the steady-state density of minority carriers is as shown in Fig. 6-16o (compare with Fig. 6-6). The number of minority carriers is very Junction Junction P type n type Fig. 6-16 Minority-carrier density distribution as a function of the distance x from a junction, (a) A forward-biased junction; (b) a reverse-biased junction. The injected, or excess, hole (electron) density is p„ - p na {n„ - n po ). Sec. o-M SEMICONDUCTOR-DIODE CHARACTERISTICS / T41 large. These minority carriers have, in each case, been supplied from the other side of the junction, where, being majority carriers, they are in plentiful supply. When an external voltage reverse-biases the junction, the steady-state density of minority carriers is as shown in Fig. 6-166. Far from the junction the minority carriers are equal to their thermal-equilibrium values p n <> and n po , as is also the situation in Fig. 6-16o. As the minority carriers approach the junction they are rapidly swept across, and the density of minority carriers diminishes to zero at this junction. The current which flows, the reverse saturation current I 0l is small because the density of thermally generated minority carriers is very small. If the external voltage is suddenly reversed in a diode circuit which has been carrying current in the forward direction, the diode current will not immediately fall to its steady-state reverse-voltage value. For the current cannot attain its steady-state value until the minority-carrier distribution, which at the moment of voltage reversal had the form in Fig. 6- 16a, reduces to the distribution in Fig. 6-166. Until such time as the injected, or excess, minority-carrier density p n — p™ (or n p - n^) has dropped nominally to zero, the diode will continue to conduct easily, and the current will be determined by the external resistance in the diode circuit. Storage and Transition Times The sequence of events which accom- panies the reverse biasing of a conducting diode is indicated in Fig. 6-17. We consider that the voltage in Fig. 6-176 is applied to the diode-resistor circuit in Fig. 6-I7a. For a long time, and up to the time hi the voltage Vi = Vr has been in the direction to forward-bias the diode. The resistance Ri is assumed large enough so that the drop across Rl is large in comparison with the drop across the diode. Then the current is i w V f /Rl = If- At the time t = ti the input voltage reverses abruptly to the value v = — Vr. For the reasons described above, the current does not drop to zero, but instead reverses and remains at the value i « — Vr/Rl = —Ir until the time t = t%. At t = t i} as is seen in Fig. 6-17c, the injected minority-carrier density at 2 = has reached its equilibrium state. If the diode ohmic resistance is Rd, then at the time h the diode voltage falls slightly [by (i> 4- Ir) Rd] but does not reverse. At t = h, when the excess minority carriers in the immediate neighborhood of the junction have been swept back across the junction, the diode voltage begins to reverse and the magnitude of the diode current begins to decrease. The interval h to k, for the stored-minority charge to become 2er o, is called the storage time t„ The time which elapses between & and the time when the diode has nominally recovered is called the transition time t t . This recovery interval W 'U be completed when the minority carriers which are at some distance from tne junction have diffused to the junction and crossed it and when, in addition, the junction transition capacitance across the reverse-biased junction has barged through R L to the voltage - V R . Manufacturers normally specify the reverse recovery time of a diode t„ 142 / ELECTRONIC DEVICES AND CIRCUITS •H— Rl- (a) -/„ =s -V. Forwardi . , storage, t, bias Sec. 6-1 1 h t \ (b) (c) (d) (e) Fig. 6-17 The waveform in (b) is applied to the diode circuit in (a); (c) the excess carrier density at the junction; (d) the diode current; (e) the diode voltage. in a typical operating condition in terms of the current waveform of Fig. 6-17<f. The time t„ is the interval from the current reversal at ( = (i until the diode has recovered to a specified extent in terms either of the diode cur- rent or of the diode resistance. If the specified value of R L is larger than several hundred ohms, ordinarily the manufacturers will specify the capaci- tance Cl shunting Ri in the measuring circuit which is used to determine t„* Thus we find, for the Fairchild IN 3071, that with I F = 30 mA and I K = 30 mA, the time required for the reverse current to fall to 1.0 mA is 50 nsec. Again we find, for the same diode, that with I? = 30 mA, — V R = — 35 V, R L » 2 K, and C L = 10 pF (-/« = -35/2 = -17.5 mA), the time required for the diode to recover to the extent that its resistance becomes 400 K is t„ = 400 nsec. Sac. 6-12 SEMICONDUCTOR-DIODE CHARACTERISTICS / 143 Commercial switching-type diodes are available with times t„ in the range from less than a nanosecond up to as high as 1 ^sec in diodes intended for switching large currents. 6-12 BREAKDOWN DIODES 8 The reverse-voltage characteristic of a semiconductor diode, including the breakdown region, is redrawn in Fig. 6- 18a. Diodes which are designed with adequate power dissipation capabilities to operate in the breakdown region may be employed as voltage-reference or constant- voltage devices. Such diodes are known as avalanche, breakdown, or Zener diodes. They are used characteristically in the manner indicated in Fig. 6-186. The source V and resistor R are selected so that, initially, the diode is operating in the break- down region. Here the diode voltage, which is also the voltage across the load Rl, is Vz, as in Fig. 6- 18a, and the diode current is I z - The diode will now regulate the load voltage against variations in load current and against variations in supply voltage V because, in the breakdown region, large changes in diode current produce only small changes in diode voltage. Moreover, as load current or supply voltage changes, the diode current will accommodate itself to these changes to maintain a nearly constant load voltage. The diode will continue to regulate until the circuit operation requires the diode current to fall to Izk, in the neighborhood of the knee of the diode volt-ampere curve. The upper limit on diode current is determined by the power-dissipation rating of the diode. Two mechanisms of diode breakdown for increasing reverse voltage are recognized. In one mechanism, the thermally generated electrons and holes acquire sufficient energy from the applied potential to produce new carriers by removing valence electrons from their bonds. These new carriers, in turn, produce additional carriers again through the process of disrupting bonds. I Vz ; 1 Izk V 1 («) h I —fc- R (6) l 9- 6-1 8 ( a ) The volt-ampere characteristic of an avalanche, or Zener, diode. lfa ) A circuit in which such a diode is used to regulate the voltage across Rl q 9atnst changes dve to variations in load current and supply voltage. 144 / ELECTRONIC DEVICES AND CIRCUITS Sac. 6-12 This cumulative process is referred to as avalanche multiplication. It results in the flow of large reverse currents, and the diode finds itself in the region of avalanche breakdown. Even if the initially available carriers do not acquire sufficient energy to disrupt bonds, it is possible to initiate breakdown through a direct rupture of the bonds because of the existence of the strong electric field. Under these circumstances the breakdown is referred to as Zener break- down. This Zener effect is now known to play an important role only in diodes with breakdown voltages below about 6 V. Nevertheless, the term Zener is commonly used for the avalanche, or breakdown, diode even at higher voltages. Silicon diodes operated in avalanche breakdown are available with maintaining voltages from several volts to several hundred volts and with power ratings up to 50 W. Temperature Characteristics A matter of interest in connection with Zener diodes, as with semiconductor devices generally, is their temperature sensitivity. The temperature dependence of the reference voltage, which is indicated in Fig, 6-19a and 6, is typical of what may be expected generally. In Fig. 0-1 9c the temperature coefficient oT the reference voltage is plotted as a function of the operating current through the diode for various different diodes whose reference voltage at 5 mA is specified. The temperature coef- ficient is given as percentage change in reference voltage per centigrade degree 10 15 20 25 30 35 40 45 50 35 Iz.mA (a) (6) Fig. 6-19 Temperature coefficients for a number of Zener diodes having different operating voltages (a) as a function of operating current, (b) as a function of operating voltage. The voltage V z is measured at I z = 5 mA {from 25 to 100°C). (Courtesy of Pacific Semiconductors, Inc.} 0.10 0.08 07 06 05 0.04 -0.01 -0.02 -0.03 -0.04 -0.05 -0 06 07 i / I I J -0.08 5 t r l V z @5 J i niA.V 5 SfMrCONDUCTOR-D/ODE CHARACTERISTICS / 145 change in diode temperature. In Fig. 6-196 has been plotted the tempera- ture coefficient at a fixed diode current of 5 mA as a function of Zener voltage. The data which are used to plot this curve are taken from a series of different diodes of different Zener voltages but of fixed dissipation rating. From the curves in Fig. 6-19a and b we note that the temperature coefficients may be positive or negative and will normally be in the range ±0.1 percent/°C. Note that, if the reference voltage is above 6 V, where the physical mechanism involved is avalanche multiplication, the temperature coefficient is positive. However, below 6 V, where true Zener breakdown is involved, the tempera- ture coefficient is negative. A qualitative explanation of the sign (positive or negative) of the temper- ature coefficient of V% is now given. A junction having a narrow depletion- layer width and hence high field intensity (<~~'10 6 V/cm even at low voltages) will break down by the Zener mechanism. An increase in temperature increases the energies of the valence electrons, and hence makes it easier for these electrons to escape from the covalent bonds. Less applied voltage is therefore required to pull these electrons from their positions in the crystal lattice and convert them into conduction electrons. Thus the Zener break- down voltage decreases with temperature. A junction with a broad depletion layer and therefore a low field intensity will break down by the avalanche mechanism. In this case we rely on intrinsic carriers to collide with valence electrons and create avalanche multiplication. As the temperature increases, the vibrational displacement of atoms in the crystal grows. This vibration increases the probability of collisions with the lattice atoms of the intrinsic particles as they cross the depletion width. The intrinsic holes and electrons thus have less of an opportunity to gain sufficient energy between collisions to start the avalanche process. Therefore the value of the avalanche voltage must increase with increased temperature. Dynamic Resistance and Capacitance A matter of importance in con- nection with Zener diodes is the slope of the diode volt-ampere curve in the operating range. If the reciprocal slope AVz/AJz, called the dynamic resist- ance, is r, then a change A/ z in the operating current of the diode produces a change A V z = r Al z in the operating voltage. Ideally, r = 0, corresponding to a volt-ampere curve which, in the breakdown region, is precisely vertical. ne variation of r at various currents for a series of avalanche diodes of fixed P° w er-dissipation rating and various voltages is shown in Fig. 6-20. Note l& rather broad minimum which occurs in the range 6 to 10 V, and note that large V z and small Iz, the dynamic resistance r may become quite large. lh «s we find that a TI 3051 (Texas Instruments Company) 200-V Zener diode yj* atin g at 1.2 mA has an r of 1,500 0. Finally, we observe that, to the left . e minimum, at low Zener voltages, the dynamic resistance rapidly becomes R 18° ^ e ' Some manufacturers specify the minimum current Izk (Fig. th i k^ ow which the diode should not be used. Since this current is on K&ee of the curve, where the dynamic resistance is large, then for currents 146 / ELECTRONIC DEVICES AND CIRCUITS S«. 6-12 no 100 yn 60 70 80 50 40 3 Cl 20 10 f) r ~ » I z = 1mA 6J 1 > f\ J^ 20 XjS^j^j^Ssdg^ 10 12 14 IG 18 20 22 24 26 28 30 32 v z ,v Fig, 6-20 The dynamic resistance at a number of cur- rents for Zener diodes of different operating voltages at 25" C. The measurements are made with a 60-Hz current at 10 percent of the dc current. (Courtesy of Pacific Semiconductors, Inc.) lower than Uk the regulation will be poor. Some diodes exhibit a very sharp knee even down into the microampere region. The capacitance across a breakdown diode is the transition capacitance, and hence varies inversely as some power of the voltage. Since Ct is propor- tional to the cross-sectional area of the diode, high-power avalanche diodea have very large capacitances. Values of C T from 10 to 10,000 pF are common. Additional Reference Diodes Zener diodes are available with voltages as low as about 2 V. Below this voltage it is customary, for reference and regulating purposes, to use diodes in the forward direction. As appears in Fig. 6-8, the volt-ampere characteristic of a forward-biased diode (sometimes called a stabistor) is not unlike the reverse characteristic, with the exception that, in the forward direction, the knee of the characteristic occurs at lower voltage. A number of forward-biased diodes may be operated in series to reach higher voltages. Such series combinations, packaged as single units, are available with voltages up to about 5 V, and may be preferred to reverse- biased Zener diodes, which at low voltages, as seen in Fig. 6-20, have very large values of dynamic resistance. When it is important that a Zener diode operate with a low temperature coefficient, it may be feasible to operate an appropriate diode at a current where the temperature coefficient is at or near zero. Quite frequently, such operation is not convenient, particularly at higher voltages and when the s*. 6-n SEMICONDUCTOR-DIODE CHARACTERISTICS / 147 diode must operate over a range of currents. Under these circumstances temperature-compensated avalanche diodes find application. Such diodes consist of a reverse-biased Zener diode with a positive temperature coefficient, combined in a single package with a forward-biased diode whose temperature coefficient is negative. As an example, the Transitron SV3176 silicon 8-V reference diode has a temperature coefficient of ±0.001 percent/ °C at 10 mA over the range — 55 to +100°C. The dynamic resistance is only 15 fl. The temperature coefficient remains below 0.002 percent/ C for currents in the range 8 to 12 mA. The voltage stability with time of some of these reference diodes is comparable with that of conventional standard cells. When a high-voltage reference is required, it is usually advantageous (except of course with respect to economy) to use two or more diodes in series rather than a single diode. This combination will allow higher volt- age, higher dissipation, lower temperature coefficient, and lower dynamic resistance. 6-13 THE TUNNEL DIODE A p-n junction diode of the type discussed in Sec. 6-1 has an impurity concen- tration of about 1 part in 10 s . With this amount of doping, the width of the depletion layer, which constitutes a potential barrier at the junction, is of the order of 5 microns (5 X lCr~ 4 cm). This potential barrier restrains the flow of carriers from the side of the junction where they constitute majority carriers to the side where they constitute minority carriers. If the concentration of impurity atoms is greatly increased, say, to 1 part in 10 3 (corresponding to a Fig. 6-21 Volt-ampere characteristic of a tunnel diode. 148 / ELECTRONIC DEVICES AND CIRCUITS Sec. 6-13 density in excess of 10 18 cm -3 ), the device characteristics are completely changed. This new diode was announced in 1958 by Esaki, 7 who also gave the correct theoretical explanation for its volt-ampere characteristic, depicted in Fig. 6-21. The Tunneling Phenomenon The width of the junction barrier varies inversely as the square root of impurity concentration [Eq. (6-47)] and there- fore is reduced from 5 microns to less than 100 A (10 -9 cm). This thickness is only about one-fiftieth the wavelength of visible light. Classically, a particle must have an energy at least equal to the height of a potential-energy barrier if it is to move from one side of the barrier to the other. However, for barriers as thin as those estimated above in the Esaki diode, the Schrodinger equation indicates that there is a large probability that an electron will penetrate through the barrier. This quantum-mechanical behavior is referred to as tunneling, and hence these high-impurity-density p-n junction devices are called tunnel diodes. This same tunneling effect is responsible for high-field emission of electrons from a cold metal and for radioactive emissions. We explain the tunneling effect by considering the following one-dimen- sional problem: An electron of total energy W (joules) moves in region 1, where the potential energy may be taken as zero, U = 0, At x = 0, there is a potential-energy barrier of height U„ > W, and as indicated in Fig. 6-22a, the potential energy remains constant in region 2 for x > 0. Region 1 The Schrodinger equation (2-14), dx* T h 2 w (6-60) has a solution of the form \p = Ce ±jiir * 1 * w "'* i * x , where C is a constant. The electronic wave function <f> = e-""'^ represents a traveling wave. In Sec. 2-8 the product of ^ and its complex conjugate ^* is interpreted as giving the probability of finding an electron between x and x + dx (in a one-dimensional space). Since \pip* = \4>\ 2 = C 2 = const, the electron has an equal proba- bility of being found anywhere in region 1. In other words, the electron is free to move in a region of zero potential energy. Fig. 6-22 (a) A potential- energy step of height U . The electronic energy is W < U . (b) A potential- energy hill of height U and depth d may be pene- trated by the electron provided that d is small enough. w x a x = d m $*c. 6-1 * SEMICONDUCTOR-DIODE CHARACTERISTICS / 149 Region 2 The Schrodinger equation for x > is dhp 8jt%i dx* h* (U e -WW = Since U > W, this equation has a solution of the form if, = A£-U***mlk*HU.-W)]lz _ J^g-mtm, where A is a constant and -si 8ir*m(U - W) * _ h r i f 4vl2m(U - IF) J (6-61) (6-62) (6-63) The solution of Eq. (6-61) is actually of the form ^ = Ae~* lzd * 4- Bt xl2d '. However, B = 0, since it is required that # be finite everywhere in region 2. The probability of finding the electron between x and x -f- dx in region 2 is ^* = Ah-* 1 *' (6-64) From Eq. (6-64) we see that an electron can penetrate a potential-energy barrier and that this probability decreases exponentially with distance into the barrier region. If, as in Fig. 6~22&, the potential-energy hill has a finite thick- ness d, then there is a nonzero probability A 2 e~ d/d * that the electron will pene- trate (tunnel) through the barrier. If the depth of the hill d is very much larger than d , then the probability that the electron will tunnel through the barrier is virtually zero, in agreement with classical concepts (Sec. 3-2). A calculation of d e for U - W = 1.60 X lO" 20 J (corresponding to 0.1 eV) yields d e « 3 A. For impurity densities in excess of those indicated above (10" cm -3 ), the barrier depth d approaches d B) and AH~ dld - becomes large enough to represent an appreciable number of electrons which have tunneled through the hill. Energy- band Structure of a Highly Doped p-n Diode The condition that d be of the same order of magnitude as d„ is a necessary but not a suf- ficient condition for tunneling. It is also required that occupied energy states exist on the side from which the electron tunnels and that allowed empty states exist on the other side (into which the electron penetrates) at the same energy level. Hence we must now consider the energy-band picture when the ln npurity concentration is very high. In Fig. 6-4, drawn for the lightly doped P~ft diode, the Fermi level E F lies inside the forbidden energy gap. We shall ow demonstrate that, for a diode which is doped heavily enough to make tunneling possible, E F lies outside the forbidden band. From Eq. (6-6), Ef = E c - kT In %£ 150 / ELECTRONIC DEVICES AND CIRCUITS Sac. 6-1 3 Space-charge region n side-* («) Valence band (6) Fig, 6-23 Energy bands in a heavily doped p-n diode (a} under open-circuited conditions and (fa) with an applied reverse bias. {These diagrams are strictly valid only at 0°K, but are closely approximated at room temperature, as can be seen from Fig. 3-10.) For a lightly doped semiconductor, N D < Nc, so that In (N c /Nd) is a positive number. Hence E F < E c , and the Fermi level lies inside the forbidden band, as indicated in Fig. 6-4. Since N c » 10 19 cm -3 , then, for donor concentrations in excess of this amount (N D > 10 19 cm" 3 , corresponding to a doping in excess of 1 part in 10 s ), In (N C /N D ) is negative. Hence E F > E c , and the Fermi level in the n-type material lies in the conduction band. By similar reasoning we conclude that, for a heavily doped p region, N A > Nv, and the Fermi level lies in the valence band [Eq. (6-7)]. A comparison of Eqs. (6-5) and (6-8) indicates that E > Eq, so that the contact difference of potential energy E, now exceeds the forbidden-energy-gap voltage E G . Hence, under open-circuit conditions, the band structure of a heavily doped p-n junction must be as pictured in Fig, 6-23o. The Fermi level E F in the p side is at the same energy as the Fermi level E F in the n side. Note that there are no filled states on one side of the junction which are at the same energy as empty allowed states on the other side. Hence there can be no flow of charge in either direction across the junction, and the current is zero, an obviously correct conclusion for an open-circuited diode. S*c.6-I3 SEMICONDUCTOR-DIODE CHARACTERISTICS / 151 The Volt-Ampere Characteristic With the aid of the energy-band picture of Fig. 6-23 and the concept of quantum-mechanical tunneling, the tunnel- diode characteristic of Fig. 6-21 may be explained. Let us consider that the P material is grounded and that a voltage applied across the diode shifts the n side with respect to the p side. For example, if a reverse-bias voltage is applied, we know from Sec. 6-2 that the height of the barrier is increased above the open-circuit value E . Hence the Ti-side levels must shift down- ward with respect to the p-side levels, as indicated in Fig. 6-23b. We now observe that there are some energy states (the heavily shaded region) in the valence band of the p side which lie at the same level as allowed empty states in the conduction band of the n side. Hence these electrons will tunnel from the p to the n side, giving rise to a reverse diode current. As the magnitude (a) (b) E v - K, fee (c) id) 24 The energy-band pictures in a heavily doped p-n diode for a forward * As the bias is increased, the band structure changes progressively from la) to (d). 152 / ELECTRONIC DEVICES AND CIRCUITS Sec. 6-13 S«r- 6-M SEMICONDUCTOR-DIODE CHARACTERISTICS / 153 of the reverse bias increases, the heavily shaded area grows in size, causing the reverse current to increase, as shown by section 1 of Fig. 6-25. Consider now that a forward bias is applied to the diode so that the potential barrier is decreased below E . Hence the n-side levels must shift upward with respect to those on the p side, and the energy-band picture for this situation is indicated in Fig. 6-24a. It is now evident that there are occupied states in the conduction band of the n material (the heavily shaded levels) which are at the same energy as allowed empty states (holes) in the valence band of the p side. Hence electrons will tunnel from the n to the p material, giving rise to the forward current of section 2 in Fig. 6-25. As the forward bias is increased further, the condition shown in Fig. 6-246 is reached. Now the maximum number of electrons can leave occupied states on the right side of the junction, and tunnel through the barrier to empty states on the left side, giving rise to the peak current I F in Fig. 6-25. If still more forward bias is applied, the situation in Fig. 6-24c is obtained, and the tunneling current decreases, giving rise to section 3 in Fig. 6-25. Finally, at an even larger forward bias, the band structure of Fig. 6-24d is valid. Since now there are no empty allowed states on one side of the junc- tion at the same energy as occupied states on the other side, the tunneling current must drop to zero. In addition to the quantum-mechanical current described above, the regular p-n junction injection current is also being collected. This current is given by Eq. (6-31) and is indicated by the dashed section 4 of Fig. 6-25. The curve in Fig. 6-256 is the sum of the solid and dashed curves of Fig. 6-25a, and this resultant is the tunnel- diode characteristic of Fig. 6-21. Fig. 6-25 (a) The tunneling current is shown solid. The injection current is the dashed curve. The sum of these two gives the tunnel-diode volt-ampere charac- teristic of Fig. 6-21, which is reproduced in (b) for convenience. 6 _!4 CHARACTERISTICS OF A TUNNEL DIODE 8 From Fig. 6-21 we see that the tunnel diode is an excellent conductor in the reverse direction (the p side of the junction negative with respect to the n side). Also- for small forward voltages (up to 50 mV for Ge), the resistance remains small (of the order of 5 fi). At the peak current Ip corresponding to the volt- age Vp, the slope dl/dV of the characteristic is zero. If V is increased beyond Vp, then the current decreases. As a consequence, the dynamic conductance g = dl/dV is negative. The tunnel diode exhibits a negative-resistance charac- teristic between the peak current Ip and the minimum value Iv, called the valley current. At the valley voltage V v at which I ~ Iv, the conductance is again zero, and beyond this point the resistance becomes and remains positive. At the so-called peak forward voltage Vp the current again reaches the value I p. For larger voltages the current increases beyond this value. For currents whose values are between Iv and Ip, the curve is triple- valued, because each current can be obtained at three different applied volt- ages. It is this multivalued feature which makes the tunnel diode useful in pulse and digital circuitry. 9 The standard circuit symbol for a tunnel diode is given in Fig. 6-26a. The small-signal model for operation in the negative-resistance region is indi- cated in Fig. 6-266. The negative resistance — R n has a minimum at the point of inflection between Ip and Iv- The series resistance R, is ohmic resistance. The series inductance L t depends upon the lead length and the geometry of the diode package. The junction capacitance C depends upon the bias, and is usually measured at the valley point. Typical values for these parameters for a tunnel diode of peak current value Ip = 10 mA are ~R n = -30 SI, R, = 1 fl, L, m 5 nH, and C = 20 pF. One interest in the tunnel diode is its application as a very high speed switch. Since tunneling takes place at the speed of light, the transient response is limited only by total shunt capacitance (junction plus stray wiring capacitance) and peak driving current. Switching times of the order of a nanosecond are reasonable, and times as low as 50 psec have been obtained. A second application 8 of the tunnel diode is as a high-frequency (microwave) oscillator. The most common commercially available tunnel diodes are made from germanium or gallium arsenide. It is difficult to manufacture a silicon tunnel diode with a high ratio of peak-to-valley current Ip/Iv. Table 6-1 summarizes the important static characteristics of these devices. The voltage values in th is table are determined principally by the particular semiconductor used and a re almost independent of the current rating. Note that gallium arsenide 9- 6-26 (a) Symbol for a tunnel 'ode ; (fc>) small-signal model In * negative-resistance region. o — VW-TTW^ is. L, c -JL (a) (b) 154 / ELECTRONIC DEVICES AND CIRCUITS Sec. 6-1 4 TABLE 6-1 parameters Typical tunnel-diode Ip/Iy . V F ,V V V ,V Ge 8 0.055 0.35 0.50 GaAfl 15 0.15 0.50 1.10 Si 3.5 0.065 0.42 0.70 has the highest ratio Ip/Iy and the largest voltage swing V p — V P «s 1.0 V as against 0.45 V for germanium. The peak current Ip is determined by the impurity concentration (the resistivity) and the junction area. A spread of 20 percent in the value of Ip for a given tunnel-diode type is normal, but tighter-tolerance diodes are also available. For computer applications, devices with I F in the range of 1 to 100 mA are most common. However, it is possible to obtain diodes whose Ip is as small as 100 ^t A or as large as 100 A. The peak point (Vp, Ip), which is in the tunneling region, is not a very sensitive function of temperature. Commercial diodes are available 8 for which Ip and V P vary by only about 10 percent over the range —50 to + 150°C. The temperature coefficient of Ip may be positive or negative, depending upon the impurity concentration and the operating temperature, but the tempera- ture coefficient of Vp is always negative. The valley point Vy, which is affected by injection current, is quite temperature-sensitive. The value of Iy increases rapidly with temperature, and at 150°C may be two or three times its value at — 50°C. The voltages Vy and W have negative temperature coefficients of about 1.0 mV/°C, a value only about half that found for the shift in voltage with temperature of a p-n junction diode or transistor. These values apply equally well to Ge or GaAs diodes. Gallium arsenide devices show a marked reduction of the peak current if operated at high current levels in the forward injection region. However, it is found empirically 8 that negli- gible degradation results if, at room temperature, the average operating cur- rent / is kept small enough to satisfy the condition I/C < 0.5 mA/pF, where C is the junction capacitance. Tunnel diodes are found to be several orders of magnitude less sensitive to nuclear radiation than are transistors. The advantages of the tunnel diode are low cost, low noise, simplicity, high speed, environmental immunity, and low power. The disadvantages of the diode are its low output-voltage swing and the fact that it is a two-terminal device. Because of the latter feature, there is no isolation between input and output, and this leads to serious circuit-design difficulties. Hence a transistor (an essentially unilateral device) is usually preferred for frequencies below about 1 GHz (a kilomegacycle per second) or for switching times longer than several nanoseconds. The tunnel diode and transistor may be combined advantageously. 9 SEMICONDUCTOR-DIODE CHARACTERISTICS / 155 REFERENCES \. Gray, P. E., D. DeWitt, A. R. Boothroyd, and J. F. Gibbons: "Physical Electronics and Circuit Models of Transistors," vol. 2, Semiconductor Electronics Education Committee, John Wiley & Sons, Inc., New York, 1964. Shockley, W.: The Theory of p-n Junctions in Semiconductor and p-n Junction Transistors, Bell System Tech. J., vol. 28, pp. 435-489, July, 1949. Middlebrook, R. D.: "An Introduction to Junction Transistor Theory," pp. 115— 130, John Wiley & Sons, Inc., New York, 1957. 2. Middlebrook, R. D.: "An Introduction to Junction Transistor Theory," pp. 93-112, John Wiley & Sons, Inc., New York, 1957. 3. Phillips, A. B.: "Transistor Engineering," pp. 129-133, McGraw-Hill Book Com- pany, New York, 1962. A. Moll, J.: "Physics of Semiconductors," pp. 117-121, McGraw-Hill Book Company, New York, 1964. Sah, C. T., R. N. Noyce, and W. Shockley: Carrier-generation and Recombination in P-N Junctions and P-N Junction Characteristics, Proc. IRE, vol. 45, pp. 1228- 1243, September, 1957. 5. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," pp. 745- 749, McGraw-Hill Book Company, New York, 1965. 6. Corning, J. J.: "Transistor Circuit Analysis and Design," pp. 40-42, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. 7. Esaki, L.: New Phenomenon in Narrow Ge p-n Junctions, Phys, Rev., vol. 109, p. 603, 1958. Nanavati, R. P.: "Introduction to Semiconductor Electronics," chap. 12, McGraw- Hill Book Company, New York, 1963. 8. Tunnel Diode Manual, TD-30," Radio Corporation of America, Semiconductor and Materials Division, Somerville, N.J., 1963. Tunnel Diode Manual," General Electric Company, Semiconductor Products Dept., Liverpool, N.Y., 1961. •• Ref. 5, chap. 13. Sac. 7-1 VACUUM-TUBE CHARACTERISTICS / 157 7 /VACUUM-TUBE CHARACTERISTICS The triode was invented in 1906 by De Forest, 1 who inserted a third electrode, called the grid, into a vacuum diode. He discovered that current in the triode could be controlled by adjusting the grid potential with respect to the cathode. This device was found to be capable of amplifying small-signal voltages, a discovery of such great practical importance that it made possible the electronics industry. In this chapter we study the volt-ampere characteristics of triodes, tetrodes, and pentodes and define certain parameters which are useful in describing these curves. We carry through the analysis of a simple circuit containing a triode and show that such a circuit is indeed an amplifier. 7-1 THE ELECTROSTATIC FIELD OF A TRIODE Suppose that the mechanical structure of a vacuum diode is altered by inserting an electrode in the form of a wire grid structure between the cathode and the anode, thus converting the tube into a triode. A schematic arrangement of the electrodes in a triode having cylindri- cal symmetry is shown in Fig. 7-1. A study of the potential variation within a triode is very instruc- tive. For simplicity, consider a plane cathode and a parallel anode, each of infinite extent. The grid is assumed to consist of parallel equidistant wires lying in a plane parallel to the cathode. The diam- eter of the wires is small compared with the distance between wires. Such an arrangement is shown in Fig. 7-2. If we assume that the cathode is so cold that it emits no electrons, the potential at any point in the tube satisfies Laplace's equation, with boundary conditions determined by the applied electrode voltages. The results of such a 156 Anode Control grid Cathode Fig, 7-1 Schematic arrangement of the electrodes in a triode. (a) Top view; (b) side view. {The constructional details are similar to those indicated in Fig. 7-13.) V Anode (b) Cathode Control grid calculation are shown in Fig. 7-3, where equipotential surfaces are indicated for various values of grid voltage. Since the electrodes are assumed to be of infinite extent, it is only necessary to plot the equipotentials over a distance corresponding to the spacing between grid wires. Each picture is to be imagined repeated indefinitely to the right and left. It should be noted, in particular, that the grid structure does not produce an equipotential plane at the position of the grid. If it did, there could never be plate current for any value of negative grid voltage because the electrons would find themselves in a retarding field as soon as they left the cathode. (We assume, for the moment, that the cathode is heated but that the elec- trons leave with zero initial velocity.) Because of the influence of the positive plate potential, it is possible for an electron to find a path between grid wires rig. 7-2 A plane-electrode triode, showing the paths for the potential profiles given in F 'Q. 7-4. V Path between grid wires ♦J $ Path through grid wires 9 1 © a ® 158 / ELECTRONIC DEVICES AND CIRCUITS Stc, 7-} such that it does not collide with a potential-energy barrier (provided that the grid is not too highly negative) . Thus the potential variation between cathode and anode depends upon the path. The potential-vs.-distance curves (called profile presentations) corresponding to Fig. 7-3 are given in Fig. 7-4 for the two extreme conditions, a path midway between grid wires (upper curve) and a path directly through the grid wires (lower curve). If an electron finds itself in a retarding field regardless of what part of the cathode it comes from, it certainly cannot reach the anode. This situation is pictured in (a) of Figs. 7-3 and 7-1 and corresponds to conditions beyond cutoff. If we assume that all electrons leave the cathode with zero velocity, they can- not enter the shaded area in Fig. 7-3 because they encounter there a retarding field. In (6) are shown the conditions just at cutoff, where the electric field intensity at the cathode is nowhere positive. Actually, cutoff is obtained at a grid voltage slightly less than this value, so that the field at the cathode is somewhat negative and hence repels all the emitted electrons. It should be clear from a study of these figures that the current distribution is not constant along paths at different distances from the grid wires. If the grid is made suf- ficiently negative, cutoff will occur throughout the entire region. This con- fa) 100 -90 ■ -80- -70- -60- -50- -40- Anode Fig. 7-3 Equipotential contours in volts in the plane-electrode triode. (a) Grid beyond cutoff potential (V G = -25 V); (fa) grid at cutoff potential (V a = -12 V); (c) grid negative at one-half cutoff value {V G = -6 V). (From K. R. Spangenberg, "Vacuum Tubes," McGraw-Hill Book Company, New York, T948.) s^r-2 VACUUM-TUBE CHARACTERISTICS / 159 00 — _ 1 — i — rid-J I so ^ i oil i M 3 o i o .■v. 9 - a. < •ill i ■J.v i \ y ' — - V / HI i i — 1 1 1 Grid- \ 1 1 ! 1 1 i> i I < 1 > - --.- \'/ 1 1 ! Grid^JL N 1 gg — £ — > o 1 1 1 ' 1 o 1 < yj r- < 1 - - 1 40 -20 0+20 +40 +60 Distance from grid, mils (a) 40-20 + 20 + 40 + 60 Distance from grid, mils (6) -40 -20 +20 +40 +60 Distance from grid, mils <<0 Fig. 7-4 Potential profiles of a plane-electrode triode. (a) Grid at twice the cutoff value of potential; (fa) grid at the cutoff value of potential; (c) grid negative at one- half the cutoff value of potential. (From K. R. Spangenberg, "Vacuum Tubes," McGraw-Hill Book Company, New York, 1948.) dition prevails for all grid voltages more negative than that indicated in (6). If the grid voltage is made more positive than this cutoff value, then, as shown in (c), current will flow only in the region midway between the grid wires, because any electrons starting out toward a grid will be repelled. This situ- ation corresponds to the usual operating conditions of a triode voltage amplifier. It should be emphasized that these diagrams represent space-charge-free conditions. In Chap. 4 it is shown that under space-charge conditions the electric field intensity at the cathode is reduced to zero. Hence, for a hot cathode, the potential curve of Fig. 7-4c must be modified somewhat and, in particular, must have zero slope at the cathode. 7-2 THE ELECTRODE CURRENTS From the qualitative discussion already given, it follows that the plate current should depend upon the space-charge-free cathode field intensity. This elec- trostatic field, in turn, is a linear function of the grid and plate potentials. Since the grid is much closer to the cathode than the plate, a given change in Potential of the grid has a much greater effect on the field intensity at the c athode than does the same change in potential of the anode. For example, if the plate voltage is changed slightly in Fig. 7-4, it will affect the slope of the Potential curve at the cathode very little. If the grid voltage is altered the same amount, the slope will change by a very much larger amount. In view °* this discussion and the known three-halves-power law for diodes (Sec. 4-4), *t is anticipated that the plate current ip may be represented approximately 160 / ElfCTRONfC DEVICES AND CIRCUITS by the equation 2 i P = G (vo + V A' Sec. 7-2 (7-1) where v P ~ plate potential Vg ~ grid potential m = a measure of relative effectiveness of grid and plate potentials The parameter n is known as the amplification factor, and is substantially con- stant and independent of current. The exponent n is approximately equal to $. The constant G is called the perveance. The validity of Eq. (7-1) has been verified experimentally for many triodes. Grid Current Ideally, the grid electrode should control the plate cur- rent without drawing any grid current i a . In practice, it is found that if the grid is made positive with respect to the cathode, electrons will be attracted to it. For many triodes this positive-grid current increases in the range of 0.5 to 4 mA for each volt increase in positive-grid voltage. Such an increment corresponds to an effective sialic grid resistance r G m v 6 /i a of 250 Q to 2 K. Positive-grid triodes are available for power-amplifier applications. Also, in many pulse and snitching circuits 3 the grid is driven positive during a portion of the waveform (Fig. D-3). Because the electrons from the cathode are emitted with nonzero initial velocities, some of them will be collected when the grid is zero or even some- what negative with respect to the cathode. Typically, I a = 0.5 mA at V = and I a = 10 mA at V e = -0.5 V. As the magnitude of the negative-grid volt- age is increased, the grid current decreases further, then goes to zero, and may reverse in sign." This negative-grid current consists mainly of four components. First, we have gas current, consisting of positive ions (carbon dioxide, carbon monoxide, hydrogen, etc.) collected by the negative grid. The positive- ion grid current is proportional to both the pressure in the tube and plate cur- rent. When the grid voltage becomes sufficiently negative, the plate current is zero (cutoff) and no ionization takes place. Second, electrons leave the grid (and hence negative-grid current flows) because of photoelectric emission from the grid^ Third, the grid is usually operating at a temperature between 600 and 700°K, and therefore grid thermionic emission takes place. Finally, we have a component of grid current due to leakage between the grid and the other electrodes. Ordinarily, the glass stem used to support the leads and the mica pieces used to space the tube parts have a high resistance. However, sublimed materials from the cathode form films on the stem and mica surfaces which act to decrease the resistance. When the grid is negative, leakage cur- rents develop, consisting of a flow of electrons from the grid to the cathode and plate. The negative-grid current due to all sources seldom exceeds a small fraction of a microampere. Unless otherwise stated, we neglect the grid cur- rent (positive or negative) for all negative values of grid voltage. COMMERCIAL TRIODES VACUUM-rUBE CHARACTERISTICS / 161 In Sec. 4-2 the construction of commercially available cathodes is described, practical anodes are discussed in Sec. 4-8. Grids Conventional grids for vacuum tubes consist of supporting side rods on which are wound fine lateral wires. The wire size, the number of turns per inch, the grid-to-cathode spacing, and the dissipation capability of the grid structure determine the individual tube characteristics. 25 An improved grid structure, called the strap frame gridf consists of a rigid self- supporting rectangular frame that permits the use of very small lateral wire (0.3 mil = 0.0003 in. in diameter) and thus makes possible the use of a large number of lateral wires per inch. This type of construction also permits elose grid-to-cathode spacing which results in a tube with a large value of transconductance (Sec. 7-5). The Nu vis tor Another type of grid structure is employed in the manu- facture of the nuvistor-type vacuum tube shown in Fig. 7-5. This tube utilizes an all-ceramic and metal construction with cantilever-supported cylindrical electrode structure. The cylindrical-tube elements are supported by conical bases, which, in turn, rest on strong supporting pillars. This type of con- struction is mechanically rigid and of low mass, and is well suited to withstand shock and vibration. R 9- 7-5 Nuvistor triode. (Cour- tesy of Radio Corporation of America.) Cathode Grid — Heater Plate 'Indexing lugs 1*2 / ftfCTROWC DEVICES AHO CIKUITS S«r. 7-4 A. Anode B. Ceramic spacers C. Heater D. Cathode ring E. Heater buttons F. Grid G. Grid ring H. Oxide-coated cathode /. Cathode M§^9 Ceramic W/ft Titanium Fig. 7-6 Construction of a ceramic planar triode. (Courtesy of Gen- eral Electric Co.) Planar Ceramic Tube A type of electron tube construction that is extremely resistant to shock and vibration is indicated in Fig. 7-6. The close grid-to-cathode spacing (about 1.3 mils when the tube is hot) and the fine grid structure result in large values of transconductance. Noise and microphonics, as well as the danger of grid-to-cathode shorts due to loose grid wires, are minimized by a specially designed, tensioned wire grid struc- ture. The small size of the elements and the close spacing of the electrodes result in a tube which is useful at frequencies in the gigahertz region. 7-4 TRIODE CHARACTERISTICS The plate current depends upon the plate potential and the grid potential, and may be expressed mathematically by the functional relationship i P = /(i>, vq) (7-2) read "tj» is some function / of v P and v a ." This relationship is sometimes written if = ip(v P , %), the quantities in the parentheses designating the vari- ables upon which the function / (or ip) depends. If it is assumed that the grid current is zero, then under space-charge-limited conditions the approxi- mate explicit form of this function is that expressed by Eq. (7-1). By plot- ting ip versus v P and vq on a three-dimensional system of axes, a space diagram is obtained. The traces of this surface on the three coordinate planes (and on planes parallel to these) give three families of characteristic curves which are easy to visualize. Figure 7-7a shows a family of curves known as the plate characteristics, since they give the variation of the plate current with the plate potential for various values of grid potential, vg = V Q i, Vgi, etc. The effect of making the grid more negative is to shift the curves to the right without changing the VACUUM-TUBE CHARACTERISTICS / 163 Stc.7-5 fig. 7-7 (a) Plate and (b) transfer characteristic curves of a triode. Voi > Vo * > Vo * > Vo *> Vf\ > V r* > Vp *- slopes appreciably. If the grid potential is made the independent variable and if the plate voltage is held constant as a parameter, vp = V Ph Vp it etc., the family of curves known as the mviual, or transfer, characteristics, illus- trated in Fig. 7-7&, is obtained. The effect of making the plate potential less positive is to shift the curves to the right, the slopes again remaining sub- stantially unchanged. These conditions are readily evident if it is remem- bered that the sets of curves in these diagrams are plots of Eq. (7-1) with either vq or Vp maintained constant as a parameter. The simultaneous vari- ation of both the plate and the grid potentials so that the plate current remains constant, ip = Ip it Ip it etc., gives rise to a third group of curves, known as the constant-current characteristics (Prob. 7-1). The most important family of characteristics is the plate family, and these are supplied in convenient form in data books provided by the tube manu- facturers. The plate characteristics for several representative tubes are repro- duced in Appendix D. These curves are average values, and the character- istics for a specific tube may differ appreciably from these published values. The Military Specifications for Electron Tubes, MIL-E-1, give the limits of variability which may be expected in a given tube type. The volt-ampere characteristics vary with heater temperature and with aging of the tube. As with a diode, so for a multielement tube, the tempera- ture effect is found experimentally to be equivalent to a 0.1-V shift in cathode voltage (relative to the other electrodes) for each 10 percent change in heater voltage. TRIODE PARAMETERS 7-5 « the analysis of networks using tubes as circuit elements (Chap. 8), it is °und necessary to make use of the slopes of the characteristic curves of Fig. I • Hence it is convenient to introduce special symbols and names for these Entities. This is now done. Amplification Factor This factor, designated by the symbol n, is defined ^ e ratio of the change in plate voltage to the change in grid voltage for a 164 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-5 constant plate current. Mathematically, n is given by the relation (7-3) The subscript Ip denotes that the plate current remains constant in perform- ing the indicated partial differentiation. In order that ju be a positive number, the minus sign is necessary because an increasing plate voltage will require a decreasing grid potential if the current is to remain unchanged. The recipro- cal of the amplification factor is called the durchgriff, or the penetration factor, Plate Resistance The quantity (dvp/dip)v , which expresses the ratio of an increment of plate potential to the corresponding increment of plate current when the grid potential is kept constant, has units of resistance, and is known as the plate resistance of the tube, designated by the symbol r p . We note that the plate resistance is the reciprocal of the slope of the plate charac- teristics of Fig. 7-7 a. It should be recalled that the dynamic plate resistance of a diode was denned in a similar manner. The reciprocal of the plate resist- ance is called the plate conductance, g p = l/r„. Transconductance The quantity (di P /dva)v Pt which gives the ratio of an increment of plate current to the corresponding increment in grid potential for constant plate potential, has the units of conductance. This quantity is known as the plate-grid transconductance, and represents the change of current in the plate circuit for unit change in potential of the grid. The transcon- ductance is frequently referred to as the mutual conductance, and is designated by the symbol g n . The quantity g m is the slope of the mutual characteristic curves of Fig. 7-7&. Summary The triode coefficients, or parameters, which are character- istic of the tube are (dip\ \dV )y p - 9m _ (§Vp\ \dv ) Ir M plate resistance mutual conductance amplification factor (7-4) Since there is only one equation, (7-2), relating the three quantities ip, vp, and vq, the three partial derivatives cannot be independent. The interrelationship may be shown to be (Sec. 8-4) M = TpQ m (7-5) Parameter Values For a 6CG7 tube, the parameters n, r p , and g m as a function of plate current (for three particular values of plate voltage) are VACUUM-TUBE CHARACTERISTICS / 165 Each section 11 gm 3.5 5 ft; |L> 13U I 3.0 | £0 y< \y <& '■$ BT ^ 3m V pj? V f = rated value 2.0 i I iJ •a ^7" K 15(_ V_ 8 1.0 g § ISO <Y 0.5 22 28 20 24 18 *20 —16 £ 16 g a +* m at a I U * 12 < 12 E 8 10 10 Plate current, mA Fig. 7-8 The parameters u, r v , and g m for a 6CG7 triode as a function of plote current for three values of plate voltage. (Courtesy of Gen- eral Electric Co.) shown in Fig. 7-8. Note that the plate resistance varies over rather wide limits. It is very high at zero plate current and varies approximately inversely as the one-third power of the plate current (Prob. 7-3). The transconduc- tance increases with plate current from zero at zero plate current and varies directly as the one-third power of the plate current. The amplification factor is observed to remain reasonably constant over a wide range of currents, although it falls off rapidly at the low currents. The usual order of magnitudes of the tube parameters for conventional tnodes are approximately as follows : ti: from 2.5 to 100 V from 0.5 to 100 K g m : from 0.5 to 10 mA/V, or millimhos "Pecial tubes with extremely small grid-to-cathode spacing d 9k may have even ft rger values of transconductance. For example, the Western Electric type 41 6B triode with d gk = 0.018 mm has the following parameters : g m = 60 mA/V, * * 300, and r, = 5 K. Among the most commonly used triodes are those listed in Table 7-1. e se contain two triode units in one envelope, and each section has, at the ommended operating point, the parameters given in the table. Since the , current is given in milli amperes and the potentials in volts, it is con- sent to express the plate resistance in kilohms and the transconductance in 166 / ELECTRONIC DEVICES AND CIRCUITS TABLE 7-1 Some Mode parameters Sec. 7^ Triode type M r P , K ffm, mA/V 6CG7 20 55 17 100 47 7.7 5.5 7,7 62 7.2 2.6 10 2.2 1.6 6.5 12AT7 12AU7 12AX7 5965 millimhos. Note that the product of milliamperes and kilohms is volts and that the reciprocal of kilohms is millimhos or milliamperes per volt (mA/V). Approximate values of r pi ft and g m may be obtained directly from the plate characteristics. Thus, referring to the definitions in Eqs. (7-4) and to Fig. 7 -7 a, we have, at the operating point Q, AVp Aip p ~ AiZ L. = reciprocal of slope of characteristic 9m. = \Au ft ~ — Aip | Ave \y* ~ V 0i - Vat Av P I \Avpl Avq \h V02 - V c If t p were constant, the slope of the plate characteristics would every- where be constant; in other words, these curves would be parallel lines. If * were constant, the horizontal spacing of the plate characteristics would be constant. This statement assumes that the characteristics are drawn with equal increments in grid voltage (as they always are). If r p and M are con- stant, so also is g m = „/>„. Hence an important conclusion can be drawn: // over a portion of the i P -v P plane the characteristics can be approximated by parallel hues which are equidistant for equal increments in grid voltage, the param- eters n, r v , and g m can be considered constant over this region. It is shown in the next chapter that if the tube operates under this condition (tube parameters sensibly constant), the behavior of the tube as a circuit element can be obtained analytically. 7-6 SCREEN-GRID TUBES OR TETRODES In Chap, 8 it is shown that the capacitive coupling between the plate and grid of a triode may very seriously limit the use of the tube at high frequencies. In order to minimize this capacitance the screen-grid tube'-" was introduced commercially about 1928. In these tubes a fourth electrode is interposed between the grid and the anode of the triode of Fig. 7-1. This new electrode is similar in structure to the control grid, and is known as the screen grid, the SK 7-6 VACUUM-TUBE CHARACTERISTICS / 167 shield grid, or grid 2, in order to distinguish it from the grid of the triode. Because of its design and disposition, the screen grid affords very complete electrostatic shielding between the plate and the grid. This shielding is such that the grid-plate capacitance is divided by a factor of about 1,000 or more. However, the screen mesh is sufficiently coarse so that it does not interfere appreciably with the flow of electrons. Because of the shielding action of the plate by the screen grid, the electric field produced in the neighborhood of the cathode by the anode potential is practically zero. Since the total cathode or space current is determined almost wholly by the field near the cathode surface, the plate exerts little effect on the total space charge drawn from the cathode. The plate in a triode performs two distinct functions, that of controlling the total space current and that of collecting the plate current. In a tetrode, the plate only serves to collect those electrons which succeed in passing through the screen. Volt-Ampere Characteristics We have already noted that the total Bpace current remains essentially constant with variations in plate voltage provided that the control-grid and screen-grid potentials are held constant. Hence that portion of the space current which is not collected by the plate must be collected by the screen; i.e., the two currents are complementary. Where the plate current is large, the screen current must be small, and vice versa. These features can be noted in Fig. 7-9. Although the plate voltage does not affect the total space current very markedly (a slight dip does occur in the curve of total space current at the lower plate potentials), it does determine the division of the space current between the plate and the screen. At zero plate potential, none of the elec- trons has sufficient energy to reach the anode, if it is assumed that the elec- trons are liberated with zero initial velocities. Hence the plate current should be zero. As the plate voltage is increased, one should expect a rapid rise in plate current and a corresponding fall in the screen current. When the plate potential is very much larger than the screen potential, the plate current should approach the space current, and the screen current should approach sero. This asymptotic behavior is noted in Fig. 7-9, ^'9- 7-9 The currents in a tetrode. The screen po- tential is 100 V, and the 9 r 'd potential is -2 V. j Total space current f Plate ' t ' N - - •*" *1 \J \ — Screen h i — .-- 100 150 200 Plate voltage, V 168 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-6 Anode Fig, 7-10 The approximate potential pro- files in an idealized tetrode for several values of plate voltage. Two curves are shown for each plate voltage. One is for a path between grid and screen wires, and the other is for a path through the wires. Negative-resistance Region An inspection of Fig. 7-9 indicates that the plate current rises very rapidly for the first few volts, but it is then followed by a rather anomalous behavior in the region of plate potentials from a few volts to potentials somewhat lower than the screen voltage. The plate cur- rent is seen to decrease with increasing values of plate potential. That is, the tube possesses a negative plate resistance in this region. The general character of the curves of Fig. 7-9 may be described on the basis of the approximate potential-distribution diagram of Fig 7-10 This diagram should be compared with Fig. 7 A, which shows the potential profiles in a triode. The control-grid and the screen-grid voltages are held at fixed values and the plate voltage V P may be adjusted from zero to a value con- siderably in excess of the screen voltage. The kinks, or folds, that appear in the curves of Fig. 7-9 in the region where the plate potential is lower than the screen potential are caused by the liberation of secondary electrons from the plate by the impact of the primary electrons with the plate. These secondary electrons are attracted to the screen. lne screen current is increased, whereas the plate current is decreased. The number of secondary electrons liberated by this electron bombardment depends upon many factors, and may even exceed the total number of primary electrons that strike the plate and thus result in an effective negative plate current. In the region where the plate potential is higher than the potential of the screen, the secondary electrons that are liberated from the plate by the impact of the primary electrons are drawn back to the plate. In addition, some secondary electrons may also be liberated from the screen by the impact of the primary electrons on it. These secondary electrons from the screen are attracted to the plate, with the result that the plate current is slightly higher than it would be in the absence of secondary emission from the screen. furthermore, the plate current continues to increase with increasing plate potentials because the collection of these secondary electrons is more complete. At tnc same time, the screen current tends toward zero. Parameter Values In a tetrode circuit application the screen potential is almost always held at a fixed value. Hence the tetrode small-signal param- Sec 7-7 VACUUM-TUBE CHARACTERISTICS / 169 eters r p , g m , and n are defined as in Eqs. (7-4) for the triode, with the added constraint that the partial derivatives are taken at constant screen voltage. The construction and spacing of the grid and cathode are essentially the same in a tetrode as in a triode. Hence the control of the electron stream by the grid is nearly alike for both tube types. Consequently, the order of mag- nitude of g m is the same for a tetrode as for a triode. Since changes in plate voltage have very little effect upon the plate current, it follows that the plate resistance of a tetrode must be very high. Correspondingly, the amplification factor of the screen-grid tube must also be large. This statement follows from the fact that ^ measures the relative effectiveness of changes in plate and grid voltage in producing equal plate-current increments. In summary, the tetrode is characterized by the following features: a plate- grid capacitance which is only a few thousandths of that of a triode, a plate-grid transconductance which is roughly the same as that of a triode, and an amplification factor and plate resistance which are about ten times that of a triode. 7-7 PENTODES* Although the insertion of the screen grid between the control grid aria the plate serves to isolate the plate circuit from the grid circuit, nevertheless the folds in the plate characteristic arising from the effects of secondary emission limit the range of operation of the tube. This limitation results from the fact that, if the plate-voltage swing is made too large, the instantaneous plate potential may extend into the region of rapidly falling plate current, which will cause a marked distortion in the output. The negative-resistance portion of the plate characteristic curves of the tetrode may be removed or suppressed by inserting a coarse grid structure between the screen grid and the plate. Tubes equipped with this extra sup- pressor grid are known as pentodes, and were first introduced commercially in 1929. The suppressor grid must be maintained at a lower potential than the instantaneous potential reached by the plate. It is usually connected directly to the cathode, either internally in the tube or externally. Because the poten- tial of the screen is considerably above that of the suppressor grid, a retarding force prevents the secondary electrons liberated from the screen from flowing to the plate. On the other hand, the secondary electrons emitted from the plate are constrained, by the retarding field between the suppressor grid and the plate, to return to the plate. However, the electrons from the cathode that pass through the screen are not kept from reaching the plate by the pres- ence of the suppressor grid, although their velocities may be affected thereby. Volt- Ampere Characteristics The plate, screen, and total current curves as ^ function of the plate voltage are shown in Fig. 7-11 for a pentode. These 8 Wld be compared with the corresponding tetrode curves of Fig. 7-9. Note l W the kinks resulting from the effects of secondary emission are entirely 170 / ELECTRONIC DEVICES AND CIRCUITS S«. 7.7 5 1 1 1 1 4 Total space current < ] r — 1 Plate 1 { 3 O 1 L - Screen (1 - Fig. 7-11 The currents in a pentode. The suppressor is at zero voltage, the screen at 100 V, and the grid at -2 V. 50 100 150 250 300 Plate voltage, V missing in the pentode. Furthermore, the screen current no longer falls asymptotically to zero, but approaches a constant value for large plate volt- ages. This value is determined principally by the amount of space current that is intercepted by the screen-grid wires. An examination of the charac- teristics of a number of the more important voltage pentodes indicates that the screen current is ordinarily from 0.2 to 0.4 of the plate current at the recommended operating point. The total space current is seen to remain practically constant over the entire range of plate voltage, except for the very r \fV c =Q 1 1 1 V c = -0.5 V -1 .0 10 I" - < s - -1 *r 5 1 J -2.C K If s 1 2.5 r 4" 1 — 3.0- .0. I r- -A 10 I — '■ 20 g 30 inn Plate voltage, V Fig. 7-12 The plate characteristics of a 6AU6 pentode with V G2 = 150 V and V Gt = V. (Courtesy of General Electric Co.) 500 7-8 low values of potential, given in Fig. 7-12. VACUUM-TUBE CHARACTERISTICS / 171 The plate characteristics of a typical pentode are Parameter Values The plate resistance r p , plate-grid transconductance and amplification factor p of a pentode are defined exactly as for a triode (but with the suppressor and screen grid held constant) by Eqs. (7-4). Typi- cal values lie in the range from r„ = 0.1 to 2 M, g m = 0.5 to 10 mA/V, and „ = 100 to 10,000. Since the shape and disposition of the control grid and cathode are the same for triode and pentode, these tubes have comparable values of g m . The highest transconductance available is about 50 mA/V, and is obtained with a frame-grid pentode (for example, Amperex type 7788) whose grid-to-cathode spacing is extremely small (0.05 mm). The values of r p and p may be 100 times as great in the pentode as in the triode. The most important pentode parameter is the grid-plate transconduct- ance. Since g m is not a constant but depends sensitively upon the operating point, a manufacturer usually supplies curves of g m as a function of grid volt- age, with screen voltage as a parameter. Applications The pentode has displaced the tetrode (except the beam power tube discussed in Sec. 7-8) in all applications. The tetrode was dis- cussed above for historical reasons and because an understanding of this tube is necessar3 r before the pentode can be appreciated. The pentode, rather than the triode, is used in radio-frequency voltage amplifiers because the former virtually eliminates feedback from the plate to the grid. The pentode is used as a video amplifier because a triode at these high frequencies has a very large input admittance which acts as a heavy load on the preceding stage. The pentode has also found extensive application as an audio-frequency power- output tube. Finally, the pentode has been used as a constant-current device because the plate current is essentially constant, independent of the plate potential. 7-8 BEAM POWER TUBES The ideal power-tube plate characteristic has a constant current for all values °f plate voltage; it is capable of delivering large amounts of power in the plate circuit with negligible loss to the other electrodes; and it generates small distortion. These desirable properties are approached in the beam power tube, 2 s a sketch of which is given in Fig. 7-13. One feature of the design of this tube is that each spiral turn of the screen a %ned with a spiral turn of the control grid. This serves to keep the r een current small. The screen current in such tubes ranges from 0.05 to *° of the plate current, which is considerably below the range 0.2 to 0.4 for voltage pentodes. Other features are the flattened cathode, the beam-forming de plates (maintained at zero potential), and a relatively large spacing 172 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-f Be am -forming plate Cathode Grid Screen Fig. 7-13 Schematic view of the shapes and arrangements of the electrodes in a beam power tube. (Courtesy of Radio Corpora- tion of America.) between the screen and the plate. As a result of these design character- istics, the electrons flow between the grid wires toward the plate in dense sheets or beams, as indicated schematically in Fig. 7-13. The region between the screen and the plate possesses features which are somewhat analogous to those existing in the space-charge-limited diode. That is, a flow of electrons exists between two electrodes between which a difference of potential exists. There is one significant difference, however. Whereas the electrons leave the cathode of a diode with almost zero initial velocities, the electrons that pass through the screen wires in the beam tube do so with a velocity corresponding essentially to the screen potential. As described in Sec. 4-5 in connection with the effects of initial velocities on the space-charge equations, the effect of the initial velocities of the electrons in the screen- plate region will appear as a potential minimum in this region (Fig. 4-3). This minimum is shown in the approximate potential profile in Fig. 7-14, which should be compared with the corresponding figure for the tetrode (Fig. 7-10). The potential minimum produced acts as a virtual suppressor grid, since any secondary electrons emitted from either the plate or the screen will encounter a potential-energy barrier. They will be compelled to return to the electrode Fig. 7-14 Approximate potential profiles in on idealized beam power tube for two values of plate voltage. Two curves are shown for each plate voltage, one for o path between grid and screen wires, and the other for a path through the wires. Note the potential minimum in the region between the screen grid and the anode. S«.7-9 VACUUM-TUBE CHARACTERISTICS / 173 300 400 Plate voltage, V Fig. 7-15 The plate characteristics of a 616 beam tube with V G t = 250 V. (which is at a positive potential with respect to the potential minimum) from which they originate. Variable Suppressor Action The actual potential distribution in the screen-plate region will depend upon the instantaneous plate potential and the plate current (a constant screen potential being assumed), and so is not con- stant. This variable suppressor action is quite different from that which arises in a simple pentode provided only with a mechanical grid structure for supplying the retarding field. Thus, because of the beam formation, which serves to keep the screen current small, and because of the variable suppressor action, which serves to suppress secondary emission from the screen and from the plate, the ideal power-tube characteristic is closely approximated. A family of plate charac- teristics for the 6L6 is shown in Fig. 7-15. It should be noted that this tube is a tetrode when considered in terms of the number of active electrodes. At low currents, where the suppressor action of the beam is too small, the charac- teristic "kinks" of a tetrode are noticeable. 7 ' 9 THE TRIODE AS A CIRCUIT ELEMENT Even if the tube characteristics are very nonlinear, we can determine the behavior of the triode in a circuit by a graphical method. This procedure is ^sentiaUy the same as that used (Sec. 4-9) in treating the diode as a circuit element, except that the diode has two active electrodes and one character- istic curve, whereas the triode has three active elements and a family of curves. The th ree terminals are marked P (plate), K (cathode), and G (grid). A 174 / ELECTRONIC DEVICES AND CIRCUITS Soc. 7-9 Fig. 7-16 The basic circuit of q triode used as an amplifier. grounded-cathode circuit in which the triode acts as an amplifier is shown in Fig. 7-16. Before proceeding with an analysis of this circuit, it is necessary to explain the meanings of the symbols and the terminology to be used in this and subsequent analyses. The input circuit of this amplifier refers to all elements of the circuit that are connected between the grid and cathode terminals of the tube. Similarly, the output, or plate, circuit usually refers to the elements that are connected between the plate and cathode terminals. In the circuit illustrated, the out- put circuit contains a dc supply voltage in series with a load resistor R L . The input circuit consists of a dc supply voltage in series with the input voltage. The input signal may have any waveshape whatsoever, but it is usually chosen, for convenience in analysis, to be a sinusoidally varying voltage. Notation Because a variety of potentials and currents, both dc and ac, are involved simultaneously in a vacuum-tube circuit, it is necessary that a precise method of labeling be established if confusion is to be avoided. Our notation for vacuum-tube symbols is adopted from the IEEE standards 8 for semiconductor symbols, and may be summarized as follows: 1. Instantaneous values of quantities which vary with time are repre- sented by lowercase letters (i for current, v for voltage, and p for power). 2. Maximum, average (dc), and effective, or root-mean-square (rms), values are represented by the uppercase letter of the proper symbol (/ V, or P). K ' ' 3. Average (dc) values and instantaneous total values are indicated by the uppercase subscript of the proper electrode symbol (G for grid, P for plate, and K for cathode). 4. Varying component values are indicated by the lowercase subscript of the proper electrode symbol. 5. If necessary to distinguish between maximum, average, and rms values, maximum and average values may be distinguished by the additional subscript m and avg, respectively. VACUUM-TUBE CHARACTERS TICS / 175 Trtode symbols Instantaneous total value Quiescent value Instantaneous value of varying component Effective value of varying component. Amplitude of varying component Supply voltage Grid voltage with respect to cathode V* vZt Plate voltage with respect to cathode V P Current in direc- tion toward plate through the load Ip t These are positive numbers, giving the magnitude of the voltages. 6. Conventional current flow into an electrode from the external circuit is positive. 7. A single subscript is used if the reference electrode is clearly under- stood. If there is any possibility of ambiguity, the conventional double- subscript notation should be used. For example, v^ = instantaneous value of varying component of voltage drop from plate to cathode, and is positive if the plate is positive with respect to the cathode. If the cathode is grounded and all voltages are understood to be measured with respect to ground, the symbol v pk may be shortened to v p . The ground symbol is N. For example, Vp N = instantaneous value of total voltage from plate to ground. 8. The magnitude of the supply voltage is indicated by repeating the electrode subscript. Table 7-2 summarizes the notation introduced above. In the table are also listed some symbols not yet defined but which are used in later sections. This table should serve as a convenient reference until the reader is thoroughly familiar with the notation. For example, if the input^signal voltage is sinus- oidal and of the form 9 t = V tm sin tat m s/2 V, sin at then the net instantaneous grid voltage in Fig. 7-16 is v e = — Vq G + i>,= — Vqq + V tm sin mt (7-6) 7 "!0 GRAPHICAL ANALYSIS OF THE GROUNDED-CATHODE CIRCUIT Assume for the moment that no grid signal is applied in Fig. 7-16, so that "• = 0. H must not be supposed that there will be no plate current, although this might be true if the bias were very negative. In general, a definite direct 176 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-lQ Fig. 7-17 The operating point Q is located at the intersection of the load line and the plate characteristic for the bias v B = — V current will exist when the input signal is zero. The value of this current may be found graphically in the same way as that used to determine the instantaneous plate current in the diode circuit of Fig. 4-7 for a given instan- taneous input voltage. Because of the presence of the load resistor R Lf the potential that exists between the plate and the cathode will depend upon both the magnitude of the supply voltage and the magnitude of the current in the load resistor. It follows from Fig. 7-16 that Vp m Vp P — ipR! (7-7) This one equation is not sufficient to determine the current corresponding to any voltage V PP because there are two unknown quantities in this expression, Vp and ip. A second relation between these two variables is given by the plate characteristics of the triode. The straight line represented by Eq. (7-7) is plotted on the plate curves of Fig. 7-17. This line is obviously independent of the tube characteristics, for it depends only upon elements external to the tube itself. The intersection of this load line with the curve for v = - V q is called the operating point, or the quiescent point, Q. The quiescent current in the external circuit is I P , and the corresponding quiescent plate potential is V P . The simplest method of drawing the load line is to locate two points of this line and to connect these with a straightedge. One such point is the intersection with the horizontal axis, namely, i P = and v P = V PP . Another is the intersection with the vertical axis, namely, v P = and i P = V PP /R L . These are illustrated in Fig. 7-17. Sometimes this latter point falls off the printed plate characteristics supplied by the manufacturer, the current Vpp/Rl being considerably greater than the rated tube current. In such a situation any value of current, say /', that is given on the plate characteristics is chosen, and the corresponding plate voltage is found from Eq. (7-7), namely, Vp P - I'R t . EXAMPLE (a) One section of a GCG7 triode is operated at a bias of -8 V and a supply voltage of 360 V. If the load resistance is 12 K, what are the quiescent current and voltage values? (ft) If the peak-to-peak signal voltage is 12 V, what is the peak-to-peak output swing? S« 7-10 VACUUM-TUBE CHARACTERISTICS / \77 Solution a. The plate characteristics are given in Fig. D-2 (Appendix D). One point on the load line is i P = and v P = 360 V. Corresponding to v P = 0, i F = V P p/Rt = 360/12 = 30 mA, whereas the largest current in Fig. D-2 is 28 mA. Hence a second point on the load line is found by choosing i#> = 20 mA = V, and then v P m V PP - I'R L = 360 - 20 X 12 - 360 - 240 = 120 V The load line is now drawn through the pair (i P , v P ) of points (0, 360) and (20, 120) on Fig. D-2. This line is found to intersect the plate characteristic for Vq — — 8 V at I P = 9.2 mA and V p = 250 V. (The reader should check these values.) 6. For a peak swing of 6 V, the extreme values of grid voltage are — 8 + 6 = — 2 V and — 8 — 6 = —14 V. The intersection of the load line with the curve for Vg m -2 V is V P * 170 V, and with the characteristic V G = —14 V is V P = 315 V. Hence the peak-to-peak plate swing is 315 - 170 = 145 V. The output swing is 2 ^- = 12.1 times as great as the input signal. This example illustrates that the tube has functioned as a voltage amplifier. The grid base of a tube is defined as the grid-voltage swing required to take the tube from Vq = to cutoff. In the above example, since cutoff corresponding to V P = 360 V is —22 V, the grid base = 22 V. Note that the grid base depends upon the peak plate voltage. The foregoing method of finding the output current corresponding to a given input voltage is now discussed in more detail. Suppose that the grid potential is given by Eq. (7-6). The maximum and minimum values of vg will be — V G q ± V, m , which indicates that the grid swings about the point — Vgg- Consequently, the plate current and the plate voltage will then swing about the values I P and V P> respectively. The graphical construction show- R 9. 7-1 8 The output current and voltage waveforms for a 9'ven input grid signal are determined from the plate characteristics and the load line. 178 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7- J | V r , =0 Fig. 7-19 (a) A pentode amplifier, (b) The graphical construction for obtaining i Pi and cpj corresponding to Vq 2 . ing these conditions is illustrated in Fig. 7-18. For any given value of v G , the corresponding values of i P and v F are located at the intersection of the load line and the i P -v P curve corresponding to this value of v G . This construction is valid for any input waveform, and is not restricted to sinusoidal voltages. The points a', b', c', etc., of the output-current waveform correspond, respectively, to the points a", b", c", etc., of the output-voltage waveform. A Pentode Circuit The simplest amplifier circuit using a pentode ia indicated in Fig. 7-19a. The suppressor is tied to the cathode, and the screen is held at a fixed voltage V ss . The input signal is applied to the grid, and the output is taken at the plate, just as in the triode amplifier. The load line [Eq. (7-7)] expresses Kirchhoff's voltage law (KVL) at the output port, and hence is independent of the device (diode, triode, pentode, etc.). This load line is drawn in Fig. 7-1% on the pentode plate characteristics corresponding to the given screen voltage V ss . The graphical analysis is identical with that described for the triode. For example, corresponding to an instantaneous grid voltage v a = V B t, we find ip = i P % and v P = v P2 . 7-11 THE DYNAMIC TRANSFER CHARACTERISTIC The static transfer characteristic of Fig. 7-7b gives the relationship between ip and v a with the plate voltage held constant. The dynamic transfer character- istic gives the relationship between i P and v G for a given plate supply voltage V PP and a given load resistance R L . This functional relationship is obtained from the plate characteristics and the load line by the graphical construction described in the preceding section. Thus the values of i P and t' c at points 1, I Sec. 7-12 VACUUM-TUBE CHARACTERISTICS / 179 Dynamic transfer curve Fig. 7-20 The dynamic trans- fer characteristic is used to determine the output wave- shape for a given input signal. Q, and 2 in Fig. 7-20 are the same as those obtained at the corresponding points 1, Q, and 2 in Fig. 7-18. The dynamic characteristic will, in general, be curved, although often it may be approximated by a straight line. The utility of the dynamic characteristic is that it allows the output wave- form to be determined for any given input waveform. The construction should be clear from Fig. 7-20, where points a', b\ c', etc., of the output current corre- spond to points A, B, C, etc., respectively, of the input grid- voltage signal «, = v e . M2 LOAD CURVE. DYNAMIC LOAD LINE A graphical method of obtaining the operating characteristics of a triode with a distance load is given in Sec. 7-10. It is there shown that the operating region in the i P -v P plane is a straight line, called the load line. However, " the load is reactive, the work curve is no longer a straight line, but attains tn e form of an ellipse. This result follows from the fact that if the plate v oltage is sinusoidal, then (under conditions of linear operation) the plate cur- en t is also sinusoidal of the same frequency but shifted in phase with respect "° the voltage. Hence the plate current and the plate voltage are given by t-V = V vm sin ut and i p = —I pm sin (wt + 9) (7-8) ten are the parametric equations of an ellipse. If the angle 6 is zero, the 10 of these equations yields i 7 ~ Hl 180 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-12 Load curve Load line Fig. 7-21 The load line for a resistive load becomes a load curve {an ellipse, under linear operation) for a reactive load. which represents the load line for a resistance load. This load line, and also the elliptical work curve for a reactive load, are shown on the volt-ampere characteristics of Fig. 7-21. The above analysis depends upon the tube parameters p, r p , and g m being constant over the range of operation in the i P -v P plane. If these parameters are not constant, the operating curve will no longer be an ellipse. No simple analysis of the output of an amplifier with a reactive load exists under these conditions. An /i!C-coupled Load Consider the reactive load indicated in Fig. 7 -22a. Here the output is taken, not across the plate-circuit resistor R p , but rather across R g , which is isolated from the plate of the tube by means of a capacitor C. Since a capacitor cannot pass direct current, no dc voltage appears across R . The ac signal voltage developed across R g may then be applied to the input of another amplifier without affecting its bias voltage. This method of con- nection between amplifier stages is called RC coupling, and is discussed in detail in Chap. 16. Under de conditions the capacitor C acts as an open circuit. Hence the quiescent tube current and voltage are obtained as in Fig. 7-17 by drawing a static load line corresponding to the resistance R p through the point v P = Vpp, ip = 0, If we assume, as is often the case, that at the signal frequency the Static load line; slope = - l/R p Dynamic load line; elope = - \/R L R g ^ Output Vpp up (*>) Fig. 7-22 (a) An flC-coupled circuit, (b) Static and dynamic load lines for the RC- coupled circuit. VACUUM-TUBE CHARACTERISTICS / 181 Static load line Dynamic load line Fig. 7-23 (a) A transformer-coupled load, (b) Static and dynamic load lines for a transformer-coupled load. reactance of C is negligible compared with R g> then under signal conditions the effective load is again resistive. This dynamic load R L represents the parallel resistance of R p and R„ and has a value given by Rl = R,\]R = ^f^- g < R, The dynamic load line must be drawn with a slope equal to <-1/Rl through the quiescent point Q, as indicated in Fig. 7-22b. A Transformer-coupled Load For the RC-coupled circuit the ac load re- sistance is always smaller than the dc resistance. If the load is transformer- coupled to the plate, as indicated in Fig. 7-23a, the converse is true. The static load corresponds to the very small dc resistance of the transformer pri- mary, and hence is almost a vertical line, as indicated in Fig. 7-236. The dynamic load line corresponds to the much larger resistance R L reflected into the plate circuit. If the dynamic load resistance were infinite, the dynamic load line would be horizontal. Under these circumstances the output voltage would vary with signal voltage, but the output current would remain constant. Hence a circuit mtk a very large effective load acts as a constant-current device. 7 - T3 GRAPHICAL ANALYSIS OF A CIRCUIT WITH A CATHODE RESISTOR * ai *y practical circuits have a resistor R k in series with the cathode in addition (or in place of) the load resistor Rt in series with the plate. The resistor * is returned either to ground or to a negative supply — V K k, as indicated in * l g. 7-24. We consider now how to use the characteristic curves of a vacuum triode e termine such matters as range of output-voltage swing, proper bias volt- 182 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-13 Fig. 7-24 {a} A tube with both a cathode and a load resistor, (b) An alterna- tive representation of the same circuit. age, and operating point for any arbitrary input voltage Vi. In Fig, 7-24, fi, v G , vp, and ip are, respectively, the total instantaneous input voltage, grid- to-cathode voltage, plate-to-cathode voltage, and plate current. Kirchhoff's voltage law (KVL) applied to the plate circuit yields Vpp + Vkk = v P + i P {R L + Rk) Similarly, from the grid circuit, we obtain Vi = v 6 + ipRt — V KK (7-9) (7-10) Equation (7-9) is the equation of the load line corresponding to an effective voltage Vpp + Vkk and a total resistance R& + Rk. The procedure for con- structing the dynamic characteristic (plate current vs. external input voltage) is, then, the following: 1. On the plate characteristics draw the load line corresponding to the given values of Vpp -f Vkk and R L + Rk- 2. Note the current value corresponding to each point of intersection of the load line with the characteristic curves. In each case relabel the indi- vidual plate characteristics with an input voltage Vi equal to v G + ipRk — Vkk in accordance with Eq. (7-10). The procedure is illustrated in Fig. 7-25. 3. The required curve is now a plot of the plate current vs. the input voltage. For example, i r i and va are corresponding values on the graph. When cutoff occurs, there is, of course, no drop across the cathode resistor. Consequently, the externally applied voltage required to attain cutoff is inde- pendent of the size of the cathode resistor. If the tube operates within its grid base, the potential of the cathode will be slightly (perhaps a few volts) positive with respect to the grid. Hence, if the grid is grounded (v* = 0), the cathode voltage is slightly positive with respect to ground, independent of the mag- nitudes of the supply voltages or the resistances as long as the tube is within Us grid base. As the input voltage vtt increases positively, the grid-to-cathode voltage must decrease slightly in magnitude in order to supply the increased c^.7-13 Fio. 7~25 Construction for obtaining the dynamic char- acteristic of a circuit with both a cathode and a load resistor, as in Fig. 7-24. The symbolism »ej — ► f,i means that I'd is replaced by »u ■ vo\ + ip\Rk — Vkk- VACUUM-TUBE CHARACTERISTICS / 183 Load line Bin +*>n ( = «ci + i ei R* - V KK ) oca — **»a ■Vp2— — *\* H ~V„ + V KK H tube current demanded by this increased t>,-. Hence the cathode tries to follow the grid in potential. If R t . = 0, it turns out (Sec. 7-14) that the change in cathode voltage is almost exactly equal to the change in grid voltage. Hence such a circuit is called a cathode follower. The grid voltage is sometimes driven highly (perhaps several hundreds of volts) positive with respect to ground. The maximum input voltage is limited by grid current, which takes place approxi- mately where the grid-to-cathode voltage is zero. The Quiescent-point Calculation It is often desirable to find the current corresponding to a specified fixed input voltage without drawing the entire dynamic characteristic as outlined above. A very simple procedure is as follows: 1. On the plate characteristics draw the load line as in Fig. 7-25. 2. Corresponding to each value of v a for which there is a plotted plate characteristic, calculate the current for the specified value of quiescent input voltage V. In accordance with Eq. (7-10), this current is given by i P m V + Vkk — vq Rk -Ihe corresponding values of i p and % are plotted on the plate characteristics, ft s indicated by the dots in Fig. 7-26. The locus of these points is called the bias curve. 3- The intersection Q of the bias curve and the load line gives the plate Cu rrent I P corresponding to the given input voltage V. The foregoing outlined procedure is very easy to carry out. It is not ^•hy neeessary to use all values of v G , but only two adjacent values which P v e currents above and below the load line, as indicated by points A and B * ! K- 7-26. The intersection of the straight line connecting A and B with e toad line gives the desired current. In particular, it should be noted that, 184 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-U Uc VBi&s curve Fig. 7-26 The intersec- tion of the load line and the bias curve gives the quiescent point. Vpf + Vkk v f if V + Vrk is large compared with the range of values of v G within the grid base, then ip will be almost constant, and hence the curve connecting the dots in Fig. 7-26 will be approximately a horizontal straight line. Self-bias Often no negative supply is available, and self-bias is obtained from the quiescent voltage drop across Rk. For example, if the plate current and the grid-to-cathode voltage at the quiescent point are Ip and V G , respec- tively, the proper bias is obtained by choosing Rk = — V G /Ip. On the other hand, if a circuit with a definite R k is specified, the quiescent point is obtained from the construction in Fig. 7-26. For the special case under consideration, V = Vkk ~ 0, and the bias curve is obtained from i P = — v a /Rk- For a pentode, the screen current I s also passes through the cathode resists ance Rk. Hence, for proper self-bias, we must choose R k = — Va/(Ip + Is). 7-U PRACTICAL CATHODE-FOLLOWER CIRCUITS In order to see why it is sometimes advantageous to use a negative supply, consider the cathode-follower configuration of Fig. 7-27. O300V ■i(6CG7) Fig. 7-27 An example of a cathode-follower rcuit. . Sec 7-1 4 VACUUM-TUBE CHARACTERISTICS / 185 EXAMPLE Find the maximum positive and negative input voltages and the corresponding output voltages. Calculate the voltage amplification. Solution From the characteristics (Fig. D-2) and the load line it is found that the current corresponding to v = is ip = 10.4 raA. Hence the maximum output voltage is ipRk = 208 V, and since v g = 0, the maximum input voltage is also 208 V. The cutoff voltage for the 6CG7 corresponding to 300 V is found to be — 19 V. The cathode-follower input may swing from +208 to —19 V without drawing grid current or driving the tube beyond cutoff. The corresponding output swing is from +208 V to zero. Hence the amplification is 208/227 = 0.916. A more general proof that the voltage gain of a cathode follower is approximately unity (but always less than unity) is given in Sec. 8-6. In passing, we note that the corresponding input range for an amplifier using the same tube and the same supply voltage is only to — 19 V, which is far narrower than that of the cathode follower. In the preceding example the input could swing 208 V in the positive direction before drawing grid current, but could go only 19 V in the negative direction before driving the tube to cutoff. If a more symmetrical operation is desired, the tube must be properly biased. One configuration is that indi- cated in Fig. 7-24, where the bottom of R t is made negative with respect to ground, R L = 0, and the output is taken from the cathode. Two other bias- ing arrangements, indicated in Fig. 7-28a and b, do not require the use of a negative supply. In (a) the grid is held V (volts) positive with respect to ground by the use of a voltage divider across the plate supply. In (6) self- bias is used, the self-biasing voltage appearing across Ri. That is, with no input signal, the grid-to-cathode voltage is the drop across R%. The resist- ance Ri is chosen so that the quiescent voltage across R k is approximately one- half the peak-to-peak output swing. In the above example, where the total qv p & ok. €> " ? +1 * r © •Rk v° (a) (o) Fig. 7-28 Two biasing arrangements for a cathode-follower circuit. 184 / ELECTRONIC DEVICES AND CIRCUITS Sac. 7-1 4 output swing was ~200 V, the quiescent value is chosen as 100 V across the 20-K resistance. This corresponds to a quiescent plate current of 5 raA. From the plate characteristics of the 6CG7 and the 20-K load line, the grid- to-cathode voltage corresponding to 5 mA is —7 V. Hence Ri must be chosen equal to £ K = 1.4 K. REFERENCES 1. De Forest, L.: U.S. Patent 841,387, January, 1907. 2. Spangenberg, K. R.: "Vacuum Tubes," McGraw-Hill Book Company, New York, 1948. 3. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," McGraw- Hill Book Company, New York, 1965. 4. Valley, G. E., Jr., and H. Wallman: "Vacuum Tube Amplifiers," p. 418, MIT Radiation Laboratory Series, vol. 18, McGraw-Hill Book Company, New York, 1948. Natapoff, M.: Some Physical Aspects of Electron-receiving-tube Operation, Am. J. Phys., vol. 30, no. 9, pp. 621-626, September, 1962. 5. Gewartowski, J, W., and H. A. Watson: "Principles of Electron Tubes," chap. 5, D. Van Nostrand Company, Inc., Princeton, N.J., 1965. 6. Noiles, D., E. Campagna, and A. Overstrom: Performance of Frame Grid I-F Tubes, Electron. Prod., December, 1964, p. F3. 7. Pidgeon, H. A.: Theory of Multi-electrode Vacuum Tube, Bell System Tech. J., vol. 14, pp. 44-84, January, 1935. 8. Schade, 0. H.: Beam Power Tubes, Proc. IRE, vol. 26, pp. 137-181, February, 1938. 9. IEEE Standard Letter Symbols for Semiconductor Devices, IEEE Trans. Electron Devices, vol. ED-11, no. 8, pp. 392-397, August, 1964. Reich, H. J.: Standard Symbols for Electron Devices, Proc, IEEE, vol. 51, no. 2, pp. 362-363, February, 1963. VACUUM-TUBE SMALL-SIGNAL MODELS AND APPLICATIONS If the tube parameters r p , g m , and p. are reasonably constant in some region of operation, the tube behaves linearly over this range. Two linear equivalent circuits, one involving a voltage source and the other a current source, are derived in this chapter. Networks involving vacuum tubes are replaced by these linear representations and solved analytically (rather than graphically, as in the preceding chapter). The voltage gain and the input and output impedances are obtained for several amplifier configurations. 8-1 VARIATIONS FROM QUIESCENT VALUES Suppose that in Fig. 7-16, v, represents the output from a microphone and R L is the effective resistance of a loudspeaker. There is no par- ticular interest in the quiescent current, which is the current to the speaker when no one talks into the microphone. (Actually, the speaker would be transformer-coupled into the plate circuit, and the current in the secondary under quiescent conditions would be zero.) The principal interest is in the speaker output for a given microphone output. Thus the variations in current and voltage with respect to the quiescent values are most important. If the load is a resistor and not a speaker and if the output from this resistor is taken through a coupling capacitor (as in Fig. 7-22a), then, under zero input conditions, the capacitor will charge up to the quiescent voltage V P . The voltage across R g is zero under these con- ditions. If a varying grid voltage is now added to the bias, the output will again represent voltage variations about the quiescent value. It is evident that the significant quantities are the currents and voltages with respect to their quiescent values. To examine this 187 188 / ElKTRONfC DEVICES AND CIRCUITS Sec. 8-2 Fig. 8-1 The dynamic transfer characteristic is used to determine the out- put waveshape for a given input signal. matter in some detail, refer to Figs. 7-18 and 7-20. For convenience, the latter is repeated in Fig. 8-1. We see that the output current, defined by the equation i-p = ip — Ii (8-1) is simply the current variation about the quiescent-point current Ip. The output voltage v p , which is similarly defined, represents the potential vari- ations about the Q point. Consequently, if the input signal is a pure sinus- oidal wave and if the tube characteristics are equidistant lines for equal inter- vals of v e , i P will also be a sinusoidal wave. If the characteristic curves are not equidistant lines over the range 1-2 for equal intervals of v c „ the waveform of i p will differ from that of the input-signal waveform. Such a uonlinearity generates harmonics, since a nonsinusoidal periodic wave may be expressed as a Fourier series in which some of the higher-harmonic terms are appreciable. These considerations should be clear if reference is made to Figs. 7-18 and 8-1. Corresponding to Eq. (8-1), the variables v p and v e are defined by the equations V p = Vp — Vp Vg m Vg — ( — Vgg) = Vg -f- Vgg (8-2) If the symbol A is used to denote a change from the quiescent value, then Avp m v p Av G = v g Aip = i„ (8-3) 8-2 VOLTAGE-SOURCE MODEL OF A TUBE The graphical methods of the previous chapter are tedious to apply and often are very inaccurate. Certainly, if the input signal is very small, say, 0.1 V S*-*-2 VACUUM-TUBE SMAU-SIGNAL MODELS AND APPUCAT/ON5 / 189 r less, values cannot be read from the plate characteristic curves with any degree of accuracy. But for such small input signals, the parameters /u, r pt and 0m will remain substantially constant over the small operating range. Under these conditions it is possible to replace the graphical method by an analytical one. This is often called the small-signal method, but it is appli- cable even for large signals, provided only that the tube parameters are con- stant over the range of operation. The constancy of the parameters is judged by an inspection of the plate characteristics. If these are straight lines, equally spaced for equal intervals of grid bias over the operating range, the parameters are constant. Under these conditions it will be found that the tube may be replaced by a simple linear system. The resulting circuit may then be ana- lyzed by the general methods of circuit analysis. Thevenin's Theorem The small-signal equivalent circuit between the plate and cathode terminals may be obtained from Thevenin's theorem. This theorem states that any two-terminal linear network may be replaced by a gener- ator equal to the open-circuit voltage between the terminals in series with the outr put impedance seen at this port. The output impedance is that which appears between the output terminals when all independent energy sources are replaced by their internal impedances. From the definition of r, given in Eqs. (7-4) as ~ \AipJvo H this dynamic plate resistance is the output resistance between the terminals P and K. The open-circuit voltage v pts between P and K is — fiv gk . This result follows from the definition of n given in Eqs. (7-4) as /AvA _ Vp I Vpk \AVg/ip v„ Vf v gk (8-4) where use has been made of the definitions in Eqs. (8-3) and, for the sake of clarity, v p (v w ) has been replaced by v pk (v gk ) to represent the voltage drop from plate (grid) to cathode. The subscript I P in Eq. (8-4) means that the plate current is constant, and hence that variations in plate current are zero. Since h = 0, the plate is open-circuited for signal voltages. Therefore the open- circuit plate voltage is v pk = — pv ff * for a signal voltage v gk . The Small-signal Voltage-source Equivalent Circuit From Thevenin's theorem it follows that the tube may be replaced, viewed from its output terminals, by a generator — && in series with a resistor r p . This linear equiv- alent circuit is indicated in Fig. 8-2 for instantaneous voltages and currents. This diagram also includes a schematic of the tube itself in order to stress the correspondence between it and its equivalent representation. A point of the utmost importance is that no dc quantities are indicated in Fig. 8-2 because the small-signal model of the tube applies only for signal y oltages, that is, for changes about the Q point. Moreover, the equivalent tube-circuit representation is valid for any type of load, whether it be a pure 190 / ELECTRONIC DEVICES AND CIRCUITS Sec. 8-3 Fig, 8-2 (a) A tube triode (or pentode) and (fa) its voltage- source model. The quantity v„ k is to be evaluated by traversing the circuit from to K and adding all the voltage drops on the way. resistance, an impedance, or another tube. This statement is true because the above derivation was accomplished without any regard to the external circuit in which the tube is incorporated. The only restriction is that the parameters ft, r p , and g m must remain substantially constant over the operating range. If sinusoidally varying quantities are involved in the circuit— and this is usually assumed to be the case — the analysis proceeds most easily if the phasors (sinors) of elementary circuit theory are introduced. For this case of sinusoidal excitation, the tube is replaced by the equivalent circuit of Fig. 8-26, with Vgk, v pk> and i p replaced by the phasors V gk , V pk , and I p . Since in a pentode the screen voltage is held constant, then with respect to variations from the quiescent point, the small-signal model of a triode is equally valid for a pentode (or a tetrode). 8-3 LINEAR ANALYSIS OF A TUBE CIRCUIT Based on the foregoing discussion, a tube circuit may be replaced by an equivalent form which permits an analytic determination of its small-signal (ac) operation. The following simple rules should be adhered to in drawing the equivalent form of even relatively complicated amplifier circuits: 1. Draw the actual wiring diagram of the circuit neatly. 2. Mark the points G, P, and K on this circuit diagram. Locate these points as the start of the equivalent circuit. Maintain the same relative positions as in the original circuit. 3. Replace the tube by its linear model (Fig. 8-26). 4. Transfer all circuit elements from the actual circuit to the equivalent circuit of the amplifier. Keep the relative positions of these elements intact. VACUUM-TUBE SMALL-SIGNAL MODELS AND APPLICATIONS / 191 Fig. 8-3 (a) The sche- matic diagram and (b) the equivalent circuit of a simple grounded-cathode amplifier. («) (b) "). Replace each independent dc source by its internal resistance. The ideal voltage source is replaced by a short circuit, and the ideal current source by an open circuit. A point of special importance is that, regardless of the form of the input circuit, the fictitious generator that appears in the equivalent representation of the tube is always nV gk , where V gk is the signal voltage from grid to cathode. The positive reference terminal of the generator is always at the cathode. To illustrate the application of these rules, two examples are given below. The first is a single-mesh circuit, the results being given in terms of symbols rather than numerical values. The second example is a two- mesh circuit, solved numerically. EXAMPLE Find the signal output current and voltage of the basic tube ampli- fier circuit illustrated in Fig. 8-3c. Solution According to the foregoing rules, the equivalent circuit is that of Fig. 8-36, Kirchhoff s voltage law (KVL), which requires that the sum of the voltage drops around the circuit equal zero, yields I p Rl + TpT p — iiV ek , = A glance at this circuit shows that the voltage drop from grid to cathode is Vi. Hence V gk = V it and the output current I p is Um t*Vi Rl + t p The corresponding output-voltage drop from plate to cathode is V o — Vpk — — 'pR-L The minus sign arises because the direction from P to K is opposite to the positive reference direction of the current /„. l\ = -flVjRL Rl + r p 192 / ELECTRONIC DEVICES AND CIRCUITS Sac. 8-3 Fig. 8-4 The gain of the amplifier of Fig, 8-3 as a function of the load resistance, n and r p are assumed to be constant. 6 8 10 12 14 RJr P The voltage gain, or voltage amplification, A of the tube eireuit is defined as the ratio of the output- to input-voitage drops. For the simple amplifier of Fig. S-3a, is;*. Vi = —n Rl Rl + U = — M 1 + r p /R L (8-5) The minus sign signifies a phase shift of 180° between the output and the input voltages; as the input becomes more positive, the current increases and the output becomes more negative. The magnitude of the gain increases with the load resistance and approaches a maximum value as Rl becomes much greater than r„. The general form of this variation is illustrated in Fig. 8-4. We note that the maximum possible gain is pt, although this can be obtained only if Rl = "° . Too large a value of Rl cannot be used, however, since, for a given quiescent current, this would require an impractically high power-supply voltage. Nevertheless, since | A | increases rapidly at first and then approaches n asymp- totically, a gain approaching m may be realized with a reasonable value of Rl. FotRl = r p , then, \A\ = p/2. From Eq. (7-5), g m = n/r p . The total output resistance R f at the plate, taking the load into account, is Rl in parallel with r p or R' = r„7?L/(#L + **,). Hence Eq. (8-5) may be put in the form A = -g m R' (8-6) a very compact and easily remembered expression : the voltage gain of a tube is the product of the transconductance and the total impedance between the plate and cathode. If the load Z L is reactive, R f in Eq. (8-6) must be replaced by Z', where Z' represents the parallel combination of r p and Zl- If a pentode is under consideration, then usually r p ^> Zl. Hence Z' » Zl, and A m —g m Z t (8-7) EXAMPLE Draw the equivalent circuit and find the signal plate voltage for the circuit shown in Fig. 8-5a. The tube parameters are /u = 10 and r„ = 5 K. The 1-kHz oscillator V has an rms output of 0.2 V. VACUUM-TUBE SMALL-SIGNAL MODELS AND APPLICATIONS / 193 f>0 ■ P SK Fig. 8-5 (o) Illustrative example, (b) The small- signal equivalent circuit. v„ P -$ 10K 10K (a) 10K,, +T 10K G 10K X A-AAArO-A'W ' Solution Following the rules emphasized above, the small-signal equivalent cir- cuit ia indicated in Fig. 8-56. In numerical problems we express currents in miliiamperes and resistances in kilohms. (Note that the product of milliamperes and kilohms is volts.) The reference directions for the mesh directions are com- pletely arbitrary and have been chosen clockwise. It is important to note that V B k is not equal to the input voltage. It can be found by traversing the network from the grid to the cathode and adding all the voltage drops encountered. Any path from G to K may be chosen, but the most direct one is usually taken since it involves the least amount of labor. Thus V ek - 10(7! - h) (8-8) KVL around the two indicated meshes yields 107,* + 257 1 - 207 2 = (8-9) -20/i + 257 2 - 0.2 = (8-10) If the expression for V„* is substituted into (8-9), we obtain 1007, - 1007 2 + 25/i - 20/ 2 - or U = -r&r/i = 1-0427, From this value of 7 2 and Eq. (8-10) we obtain h m 0.0331 mA and h = 0.0345 mA Also, V Qk m 10(7, - h) = 10(7, - 1.0427.) = (-0.42) (0.0331) 0.0138 mV The signal voltage drop from plate to cathode is, from mesh 1, V pk = -51,- 107 ok - -(5) (0.0331) + (10) (0.0138) - -0.028 mV Alternatively, from mesh 2, V pk = -Q.2 + 57 s = -0.2 -I- (5) (0.0345) = -0.028 mV 194 / ELECTRONIC DEVICES AND CIRCUITS Sec. 8-4 _ V c constant Fig. 8-6 If the grid voltage is constant, then Aip = (slope) (At?/.) = (di F /dVr) Vl Av P , 8-4 TAYLOR'S SERIES DERIVATION OF THE EQUIVALENT CIRCUIT It is instructive to obtain the equivalent circuit of a triode from a Taylor's series expansion of the current i P about the quiescent point Q. This deri- vation will show the limitations of this equivalent circuit and will also supply the proof that m = r p g m . If the grid voltage remains constant but the plate voltage changes by an amount Av P , then the change in current equals the rate of change of current with plate voltage times the change in plate voltage, or trip \dvpf AVp The subscript indicates the variable held constant in performing the partial differentiation. This relationship is illustrated in Fig. 8-6 and is seen to be strictly true only if the slope of the plate characteristic is constant for the assumed change in voltage. Similarly, if the plate voltage remains con- stant but the grid voltage changes by Av G , then the change in current is given by Aij \dVGjV! Av G If both the grid and plate voltages are varied, the plate-current change is the sum of the two changes indicated above, or Aip = (i£)v *»* + ($£) Vi Av G (8-11) As mentioned above, this expression is only approximate. It is, in fact, just the first two terms of the Taylor's series expansion of the function ip(vp, %). In the general case, Aip Consider the third term in this expansion dHi Av p Av G + ■ (842) Bvp ovq Since from Eqs. (7-4) the plate Sx. 8-5 Vacuum-tube small-signal models and applications / 195 resistance is given by l/r p = (di P /dv P )v B , this term equals (Av P )'< Similarly, the fourth-, fifth-, and higher-order terms in Eq. (8-12) represent derivatives of r p and g m with respect to plate and grid voltages. Small-signal Model This method of analysis is based on the assumption that the tube parameters are sensibly constant over the operating range Av P and Av G - Under these conditions a satisfactory representation of the vari- ations in plate current about the quiescent point is given by Eq, (8-11). This expression may be written in the following form, by virtue of Eqs. (7-4) : 1 Atp = — Avp + g M Av c (8-13) Using the notation of Eqs. (8-3), and remembering that g m = h/t p (see below), Eq. (8-13) becomes v p = V* - &» (8-14) This expression shows that the varying voltage v p with respect to the Q point ia made up of two components: One is a generated emf whieh is n times as large as the grid-to-cathode voltage variation v g ; the second is a signal voltage across the tube resistor r p that results from the signal load current i p through it. The result of this discussion is the circuit model shown in Fig. 8-2. It is seen from the diagram that the voltage drop v,* from plate to cathode is equal to the voltage drop in the plate resistor less the generator voltage, or *>** = ipT p — fiVak This is exactly Eq. (8-14), which verifies that Fig. 8-2 is the correct equivalent- circuit representation of the tube. Relationship between n, r p> and g m It follows from Eq. (8-13) that, if ^e plate current is constant so that Ai P = 0, then Avp Avq = gmT v *>ut since the plate current has been taken to be constant, then — Av P /Av G is bv definition [Eq. (7-3)] the amplification factor. Hence n = g m r p (8-15) 8 ' 5 CURRENT-SOURCE MODEL OF A TUBE This ne venin's equivalent circuit is used if a network is analyzed by the mesh nod. However, if a nodal analysis is made, Norton's equivalent circuit is m ° r e useful. 196 / ELECTRONIC DEVICES AND CIRCUITS Sec. 8.5 -oi -I® -02 -ol -02 (a) (ft) Fig, 8-7 As viewed from terminals 1 and 2, the Thfivenin's circuit in (a) is equivalent to the Norton's circuit in (b). Norton's Theorem The equivalent circuit between two points in a network consists of the impedance Z, seen looking back between these two terminals, in parallel with a current generator whose value is the current I which flows when the terminals are short-circuited. In other words, a voltage source V in series with an impedance Z is equivalent to a current source V/Z in parallel with an impedance Z. These equivalent circuits are indicated in Fig. S-7a and b. The Small-signal Cur rent- source Equivalent Circuit From the voltage- source representation of a tube given in Fig. 8-26 we see that the short-circuit current has a magnitude nv k/r p = g m v B k, where use is made of Eq. (8-15). The direction of the current is such that it will flow through an external load from cathode to plate. Hence the current-source equivalent circuit is as indicated in Fig. 8-8. We shall now again solve the first example in Sec. 8-3, using the Norton's equivalent representation. For convenience, the circuit of Fig. 8-3 is repeated in Fig. 8-9a. Its current-source model in Fig. 8-96 is the same as that indi- cated in Fig. 8-8, but with the addition of the load resistor R L in parallel with r p . If R' m r p R L /(r p + R L ) = the parallel combination of r p and Rl, then the output voltage is v = —iR'= —g m ViR' The voltage amplification A is a = & = - gJ r which is identical with Eq. (8-6). (8-16) I = gm <V Q) T Fig. r *< v p* g, 8-8 The current-source model of a triode- Sk 8-6 VACUUM-TUBE SMALL-SIGNAL MODELS AND APPLICATIONS / 197 V K V GC J?~ Fig. 8-9 (a) The common-cathode amplifier configuration and (b) its current-source equivalent circuit. 8-6 A GENERALIZED TUBE AMPLIFIER The circuit (Fig. 8-9) considered in the preceding section has its cathode common to the input and output circuits, and hence is called the common- cathode (or grounded-cathode) amplifier. This circuit is the one most frequently used, but two other configurations, the grounded-grid and the grounded-plate amplifiers, are also possible. The Grounded-grid Amplifier This circuit is shown in Fig. 8- 10a. As far as signal voltages are concerned, the grid is at ground potential, which accounts for the name grounded-grid amplifier. The input signal v is applied between cathode and ground, and R t is the resistance of the signal source. The output v e is taken across the plate-circuit resistor R p . Since the grid is F '9- 8-10 (a)Thegrounded- fln'd amplifier and (b) the fl'ounded-plate (cathode- f °Hower) amplifier. v ac -±- («) (b) 198 / ELECTRONIC DEVICES AND CIRCUITS Sk, 8-6 common to the input and the output circuits, this configuration is also called the common-grid amplifier. The Grounded-plate Amplifier This circuit is indicated in Fig. 8-106. The signal v is applied between grid and ground, and the output v is taken across a resistor Rk between cathode and ground. As far as signal (ac) volt- ages are concerned, the plate is at ground potential, which accounts for the name grounded-plate amplifier. For an increase in input-signal voltage v, the current i p increases, and so does the output-signal voltage v = ipR p . Conse- quently, the polarity of the output signal is the same as that for the input signal. Furthermore, as verified for a particular circuit in Sec. 7-14 and as demonstrated in general in Sec. 8-8, the magnitudes of these voltages v„ and Vi = v are almost the same (unity gain). Hence the cathode voltage follows the grid voltage closely, and this feature accounts for the name cathode fol- lower given to the circuit. The Generalized Circuit The analysis of the grounded-grid and the grounded-plate amplifiers is made by considering the generalized configuration indicated in Fig. 8-1 la. This circuit contains three independent signal sources, Vi in series with the grid, v k in series with the cathode, and v a in series with the anode. For the grounded-grid amplifier v t — v a = 0, the signal voltage is vt with a source resistance R k , and the output is v»t taken at the plate. For the cathode follower, R p = 0, v t = v a = 0, the signal voltage is v it and the output is « 2 taken at the cathode. (The signal-source impedance is unimportant since it is in series with a grid which, we assume, draws negligible current.) , (a) (b) Fig. 8-11 (a) A generalized amplifier configuration, (b) The small-signal equivalent circuit. S*8-7 VACUUM-TUBE SMALL-SIGNAL MODELS AND APPLICATIONS / 199 If the effect of the ripple voltage in the power supply Vpp is to be investi- gated, then f will be included in the circuit to represent these small voltage changes in V PP . Following the rules given in Sec. 8-3, we obtain the small-signal equivalent circuit of Fig. 8-116, from which it follows that and Vgk — v% — vt — ipRk Ml'ul- — »Jb — V a p r p + R k + R P Substituting from Eq. (8-17) in Eq. (8-18), we find . _ im/(p + 1) — Vk — Qq/Qi + 1) '* " (r„ + R,)/b + l) + *i The output voltages v ol and u s are found as follows: v»\ = — ij>Rp — V a V„2 = tpRk + v k (8-17) (8-18) (8-19) (8-20) Using the basic concepts enunciated in the following section, the physical sig- nificance of Eqs. (8-19) and (8-20) is given in Sec. 8-8. 8-7 THE THEVENIN'S EQUIVALENT OF ANY AMPLIFIER If an active device (tube or transistor) in a circuit acts as amplifier, this con- figuration is characterized by three parameters, the input impedance Z iy the output impedance Z„, and the open-circuit voltage gain A*. If these param- eters are independent of the source impedance Z s and the external load imped- ance Z L , then the Thevenin's model of the amplifier is as shown in Fig. 8-12. The external source voltage V, is applied in series with Z, to the input termi- nals marked 1 and 2. The voltage across this input port is F*. The output terminals are marked 3 and 4. Since the open-circuit voltage is the amplifier voltage gain A v times the input voltage, the Thevenin's generator is A t Vi, as indicated. Xote that A v is the unloaded voltage gain, i.e., the gain with no external load placed across the amplifier, and hence zero load current, t& = 0. The loaded gain (the amplification with the load Zl in place) is called A v . Fi 9- 8-12 TheThevenin *quivalent circuit of an Qn >Plifler. When Z L is con- "*cted to the output fermi- " als . a current l L flows in tKe load. 200 / ELECTRONIC DEVICES AND CIRCUITS The output voltage is given by V = A v Vi - I L Z Sec. 8-8 (8-21) This equation may be used to define A v and Z for a particular circuit. For example, if we find that the output voltage of an amplifier varies linearly with load current, as indicated in Eq. (8-21), the factor multiplying the input volt- age Vi is the unloaded gain A v and the factor multiplying the load current l h is the output impedance Z ol provided that these factors A v and Z B are independent of the load Z L . The following theorem offers an alternative method for finding Z B . Open-circuit Voltage-Short-circuit Current Theorems As corollaries to Thdvenin's and Norton's theorems we have the following relationships: If V represents the open-circuit voltage, / the short-circuit current, and Z (Y) the impedance (admittance) between two terminals in a network, then V Z = l I=Z = VY V = IZ = ± (8-22) The first relationship states that "the impedance between two nodes equals the open-circuit voltage divided by the short-circuit current." This method is one of the simplest for finding the output impedance Z . The last relationship of Eqs. (8-22) is often the quickest way to calculate the voltage between two points in a network. This equation states that "the voltage equals the short-circuit current divided by the admittance," The Output Impedance A third method for obtaining Z is to set the source voltage V, to zero and to drive the amplifier by an external voltage generator connected across terminals 3-4. Then the ratio of the voltage across 3-4 divided by the current delivered by the generator yields the output imped- ance Z . This same method may be used to find the input impedance if the above measurement is made at terminals 1-2 instead of 3-4. 8-8 LOOKING INTO THE PLATE OR CATHODE OF A TUBE Let us now return to the generalized amplifier of Fig. 8- 11a and find a Thevenin's equivalent circuit, first from plate to ground and then from cathode to ground. The Output from the Plate The signal v a and the resistor R p are now considered external to the amplifier. Hence, for the moment, we set v a = ® and interpret R p as the external load R L . The load current i L from plate to ground is the negative of the plate current i p . Hence, with R L = R p = 0, we Sac. 8-8 VACUUM-TUU SMALL-SIGNAL MODELS AND APPLICATIONS / 201 obtain the short-circuit load current I from Eq. (8-19) : PLVi i. " + 1 -? + v k — HVi + 0* + l)Vk T P + (M + 1)R* (8-23) The open-circuit voltage V is found as follows, using Eq. (8-19) : u + 1 V = lim (-i P Rp) = lim „ R t —*oa iJ p ~->= i }> ~t~ "'P = -fiVi + <jt* 4- l)w* + n R, + Rk (8-24) The open-circuit voltage gain A v for the signal ft is — m> and for the signal v t is +(m + !)• The output impedance Z is given by Eqs. (8-22). Thus Z = j - r, + 0* + D«* (8-25) The above results lead to the Thevenin's circuit of Fig. 8-1 3a. We conclude that, "looking into the plate" of an amplifier, we see (for small-signal operation) an equivalent circuit consisting of two generators in series, one of — m times the grid-signal voltage v it and the second (n + 1) times the cathode-signal voltage v k . These generators are in series with a resistance r v + (p + 1)jR*. Note that the voltage v k and the resistance R k in the cathode circuit are both multiplied by the same factor, n + 1. Since R P and v a were considered external to the amplifier, they have been drawn to the right of the output terminals P and N in Fig. 8-1 3a. r p (m+ iWk P -o- flUi (*+iK ^9- 8-1 3 The equivalent circuit for the generalized amplifier of Fig. 8-1 1 between W) plate and ground, (b) cathode and ground. 202 / ELECTRONIC DEVICES AND CIRCUITS Sec. 8-8 The Output from the Cathode The signal v k and the resistor ft* are con- sidered external to the amplifier. Hence, for the moment, set v * = and interpret Rk as the external load Rl in Fig. 8-1 la. The load current %l from cathode to ground equals i p . Hence, with Rl = Rk = 0, we obtain for the short-circuit load current / from Eq. (8-19) / = liVi — v a Tp + R, The open-circuit voltage V is given by V = lim i, ft*— »« Rk — li + 1 (8-26) (8-27) The open-circuit voltage gain A v for the signal Vi is m/(^ + 1) ; and for the signal v a is — 1/fri + 1). The output impedance is z- Y ~- l --j - M+ 1 (8-28) The above results lead to the Thevenin's circuit of Fig. 8-136. We conclude that, "looking into the cathode" of an amplifier, we see (for small-signal oper- ation) an equivalent circuit consisting of two generators in series, one of value m/0* + 1) times the grid signal voltage Vi, and. the second 1/(m + 1) times the plate signal voltage v a . These generators are in series with a resistance (r p + R p )/ (m+ 1). Note that the voltage v a and the resistance in the plate circuit are both divided by the same factor, n + 1 . The Ground ed-g rid Amplifier This configuration is obtained from the generalized circuit of Fig. 8-11 by setting d = Vi = 0. The equivalent circuit, obtained from Fig. 8-13a, is indicated in Fig. 8-14a. By inspection the gain is A = h = J»+\ )R * UJ? (8-29) v k R p + r p + (n + l)R k K Note that, since A is positive, there is no phase shift between output and input. If Rt — and n » 1, then the gain has almost the same value as for r p + (m + 1 )R k r P /(r+iy (//+ l)v k A/W («> (&) (c) Fig. 8-14 The Thevenin circuits for the three bosic amplifier configurations $« 8-8 VACUUM- TUBE SMALL -SIGNAl WODEIS AND APPUCATIONS / 203 a common-cathode amplifier [Eq. (8-16)]. The voltage gain is greatly reduced unless Rk is kept small compared with (R P + r p )/(n + 1), which is usually of the order of 1,000 ft or less. The output impedance of the grounded-grid amplifier will be much higher than the plate resistance if the source has appreciable resistance Rk or if an additional resistance is intentionally added in series with the cathode. On the other hand, the input impedance is quite low (Prob. 8-24). Hence a grounded- grid amplifier may be employed when a low input impedance and a high out- put impedance are desired. Such applications are infrequent. The grounded- grid amplifier is used as a tuned voltage amplifier at ultrahigh frequencies' because the grounded grid acts as a grounded electrostatic shield which pre- vents coupling between input and output circuits. The Cathode Follower This configuration is obtained from the general- ized circuit of Fig. 8-11 by setting Vk = v a = and R p = 0. The equivalent circuit is indicated in Fig. 8- 14b. By inspection the gain is A =^ = Vi M+ 1 Rk nRk g m Rk m + 1 + Rk + (m + DRk 1 + g m Rk if m » 1 (8-30) Since A is positive, there is no phase shift between grid and cathode. Note that, since the denominator is always larger than the numerator, then A never exceeds unity. However, if (pi + l)Rk » r P > then A « ^4-1 (8-31) which approaches unity. For example, for a type 6CG7 tube with m = 20, A = 0.95 (which is to be compared with the value A = 0.91 obtained graphi- cally in Sec. 7-14). The output impedance of the cathode follower is much smaller than the plate resistance. For example, if n » 1, then 2 = - ~ r -i = ± (8-32) M + 1 M 9m * 0r a ff m of 2 millimhos, the output impedance is only 500 ft, and for a higher value of transconductance, Z is even less. On the other hand, since the input sl Snal is applied to the grid, the input impedance (for negative grid voltages where the grid current is negligible) is very high (ideally infinite). A cathode .lower ' s usually employed when a high input impedance and a low output ^Pedance are desired. The high input impedance of a cathode follower makes it ideal for appli- *«ons where the loading on a signal source must be kept at a minimum. ne low output impedance permits it to support a heavy capacitive load. 204 / ELECTRONIC DEVICES AND CIRCUITS S«. 8-9 These features account for the many applications found for cathode followers. For example, the cathode follower is very often used as the input tube in oscilloscope amplifiers. It is also used where a signal must be transmitted through a short section of coaxial cable or shielded wire, with its attendant high shunt capacitance. If the output from one circuit acts as the input to another circuit and the second circuit reacts back onto the first, a cathode follower may be used as a buffer stage to eliminate this reaction. Because the cathode follower is a feedback amplifier (Sec. 17-5), it possesses great stability and linearity. Many electronic instruments take advantage of these desirable features of cathode followers. The high-fre- quency characteristics of the cathode follower are considered in Sec. 8-14. The Grounded-cathode Amplifier The equivalent circuit for this con- figuration is given in Fig. 8-36 and repeated in Fig. 8-14c, for comparison with the grounded-grid and grounded-plate amplifiers. The grounded-cathode amplifier has a high input impedance, an output impedance equal to the plate resistance, and a voltage gain which may approach the ^t of the tube (although an amplification of the order of p/2 is more common). There is a phase inversion between the plate and grid. This circuit is employed more often than the other two configurations. 8-9 CIRCUITS WITH A CATHODE RESISTOR 1 Many practical networks involve the use of a resistor in the cathode circuit. Some of the most important of these "cathode-follower-type" circuits are described in this section. The Split-load Phase Inverter This circuit appears in Fig. 8-15. A single input signal provides two output signals, v kn , which is of the same polarity as the input, and v pn , which is of opposite polarity. Further, if the o+ Fig. 8-15 The split-load phase in- verter. . Sec. 8-9 VACUUM-TUBE SMAll-SIGNAt MODELS AND APPLICATIONS / 205 -late and cathode resistors are identical, the magnitudes of the two signals must be the same, since the currents in the plate and cathode resistors are equal. The amplification \A\ = \v kn /v\ = \v pn /v\ may be written directly by comparison with either of the equivalent circuits of Fig. 8-13 (with v k = v a = 0) as \A\ = fiR g m R r„ + 0* + 2)R l + g m R (8-33) The exact result differs from that given for the cathode follower [Eq. (8-30)] only in the appearance of a factor n + 2 in place of the factor p + 1. The gain may be made to approach 1 if g m R » 1. The ratio of the plate-to-cathode signal to the input signal may then approach 2. The output impedances at the plate and at the cathode are different, the plate impedance being higher than the cathode impedance. If the capacitance from the plate to ground is greater than that from cathode to ground, it is possible to equalize the frequency response of the two outputs by adding capacitance across the cathode resistor. A phase inverter, also called a pamphase amplifier, is used to convert an input voltage v, one terminal of which is grounded, into two symmetrical output voltages (Vol = -V 6t ). The Cathode-coupled Phase Inverter This circuit, shown in Fig. 8-1 6a, serves the same purpose as the split-load phase inverter but additionally pro- vides some gain and equal output impedances. The two signals v a i and v i are of opposite polarity and are nominally of equal amplitude. The equiva- lent circuit of Fig. 8-13b may again be used to advantage to analyze the oper- ation of the cathode-coupled phase inverter. We replace each tube by its r„ + R, r B + R a F 'Q. 8-16 ( ) The cathode-coupled phase inverter and (b) its equivalent circuit from cathode to ground. 206 / ELECTRONIC DEVICES AND CIRCUITS Sec. 8-9 equivalent circuit as seen from the cathode. The resulting circuit is shown in Fig. 8-166. The signal currents flowing, respectively, out of the cathode of V\ and into the cathode of 72 are ti and i 2 . The output signals are v„i - —iiR p and Voi = iiRp- The output signals will be of equal magnitude if ii = U. This require- ment will be satisfied nominally if R k » (r, + R,)f (m + 1 ). Typically, if, say, r p = R p = 10 K and p + 1 = 20 as for a 12AU7 tube, (r, + «,)/(/* + 1) = 1 K and R k should be selected to be about 10 K if an unbalance of no more than about 10 percent is desired (Prob. 8-19). It is possible to obtain balanced outputs by choosing unequal values for the two plate load resistors. By applying Kirchhoff's voltage law to the outside loop of Fig. 8-166, we find for the plate-to-plate gain A m Vol — v o1 (ii + U)R„ nR, + R, (8-34) which is the same gain that would be provided by a single-tube grounded- cathode amplifier with plate resistor R v . If each tube carries a quiescent current of, say, 5 mA, the quiescent drop across R k is 100 V. We may require for convenience that the quiescent grid voltages be at ground potential. In the linear range of operation the grid-to- cathode voltage of a tube is usually only of the order of several volts. The voltage at the cathodes is therefore also required to be in the neighborhood of ground potential These requirements with respect to quiescent operating voltages may be satisfied by returning the cathode resistor, as in Fig. 8- 16a, to an appropriately large negative voltage (in this example, Vgo = 100 V). The Difference Amplifier Suppose that we have two signals, »i and v%, each measured with respect to ground. It is desired to generate a third signal, also to be referred to ground, which signal is to be proportional to the voltage difference vi — H- One such application would occur if it were required to convert the symmetrical signals, which appear at the plates of a paraphase amplifier, back to an unsymmetrical signal. If the voltage v in Fig. 8-16o is v 1 and if v 2 is applied between grid and ground of 72 (in place of the short circuit), this circuit is a difference amplifier. If L> + l)R k » r„, then it turns out that v i and d o2 are each proportional to v t — » 8 . The transistorized version of the difference amplifier is discussed in detail in Sec. 12-12. An Amplifier with a Constant-current Source The cathode follower, paraphase amplifier, and difference amplifier all operate with improved per- formance as the cathode resistance becomes larger. A large cathode resist- ance, however, results in a large dc voltage drop due to the quiescent tube current. Hence a device which has a small static resistance but a very large dynamic resistance may be used to advantage in the cathode circuit to replace a large ordinary resistance. An arrangement of this type is shown in the difference amplifier of Fig. 8-17. Referring to Fig. 8-13a, it appears that the 8-10 VACUUM-TUBE SMALL-SIGNAL MODELS AND APPLICATIONS / 207 fig. 8-17 Tube V3 acts as a very high dynamic resistance of value + ( M + i)R k in the cathode circuit- of tubes VI and V2. The voltage divider R is used to balance the outputs from the two plates. -A/W-i impedance seen looking into the plate of the tube 73 in the cathode circuit is rt + (i 4- /i )ig fc « nR k if R k is large. Under typical circumstances, — V G0 might be -300 V, R k = 500 K, and the cathode tube a 12AX7 with p = 100 and r, = 100 K. The effective cathode impedance of the difference amplifier would then be about 50 M. In the circuit of Fig. 8-17, high-M low-current tubes would be appropriate. Suppose, then, that the individual tubes carried only 0.1 mA of current. The total cathode current is 0.2 mA, and if an ordi- nary 50-M resistor were used, a negative supply voltage of 10,000 V would be required. This voltage is, of course, impractically high, which demonstrates the advantage of tube VZ over an ordinary 50-M resistor in this application. A large dynamic resistance is plotted as a horizontal load line (Sec. 7-12) and corresponds to a constant current. Hence the difference amplifier of Fig. 8-17 is said to be fed from a constant-current source. 8-10 A CASCODE AMPLIFIER This circuit, consisting of two triodes in series (the same current in each), is indicated in Fig. 8-18. That this circuit behaves like a pentode can be seen as follows: The load for Fl is the effective impedance looking into the cathode of 72; namely, R p = (R + r,)/fci + 1). For large values of m this m ay be very small, and to a first approximation can be considered as a short ci rcuit for signal voltage. Hence the plate potential of 71 is constant. The definition of the transconductance is ft \Ava/v P ** ei ice the signal current is Ai P = g m Av G = g m v h where vi is the signal-input 208 / ELECTRONIC DEVICES AND CIRCUITS Sec. 8-?o Fig. 8-18 The cascode amplifier. Jy-x — R + ( M + 2)r p (8-35) If (>* + 2)r p ;» iJ and if p. » 1, this is approximately ~nR At~ = —g m R which is the result obtained by the qualitative arguments given above. It is possible to apply an ac signal voltage 7 2 (in addition to the bias voltage 7') to the grid of 72. Under these circumstances 71 acts as ao voltage. The gain is A = — R&ip/Vi — —g m R, which is the expression for the gain of a pentode [Eq. (8-7) with Z L = R]. Another point of view is the following: The plate dc voltage V Pl of 71 is determined by the grid-to-ground voltage V of 72. Actually, V Fl = V - V 2 where Voi is the drop from grid to cathode of 72 and is a negative number. The value of 7 ei may vary between zero and the cutoff voltage, but this is small compared with the value of V (which may be one or several hundred volts). Hence V Pl is essentially constant, and the dc tube current is also con- stant since it is determined by V Pi and the bias V GG . Hence a curve of dc plate current vs. dc voltage from the plate of V2 to ground resembles the constant-current characteristic of a pentode. From this discussion it is clear that V takes the place of the screen voltage in a pentode. The cascode ampli- fier has the advantages over the pentode that no screen current need be sup- plied and it has the low noise of a triode. The exact expression for the amplification is found by replacing V2 by an impedance (R + r p )f(fi -\- 1) and 71 by a generator fiVi in series with an impedance r P . The result is S0C.8-11 VACUUM-TUBE SMALL-SIGNAL MODELS AND APPLICATIONS / 209 impedance of magnitude r p in the cathode of 72. The voltage gain for this signal W is A 2 = R+ 0* + 2)r T (8-36) If sinusoidal signals V\ and 7 S are applied simultaneously to both inputs, then, by the principle of superposition, the output 7 will be V = A x Vt + ArVt The quiescent operating current in a cascode amplifier is found by the method of successive approximations. The method converges very rapidly, and is best illustrated by a numerical example. EXAMPLE Find the quiescent current in the cascode amplifier of Fig, 8-18 if R - 20 K, V PP = 300 V, V - 125 V, and V aa - 4 V. The tube is a 6CG7, whose plate characteristics are given in Appendix D (Fig. D-2), Solution If V2 is not to draw grid current, then Kt must be at a higher potential than G 2 . However, it cannot be at too high a potential, or V2 will be cut off. Let us take as a first approximation V Gi = — 5 V, and hence V P i = 125 + 5 = 130 V. Corresponding to this value of V P \ and to F ffI = —4 V, the plate current Ip is found from the 6CG7 characteristics to be 4.2 mA. Hence V F2 = V PF — r P R - V n - 300 - (4.2) (20) - 130 = 86 V. For V P2 = 86 V and /,, = 4.2 mA, we find that Vat = — 2 V. The second approximation is V Pl = 125 + 2 = 127 V. Corresponding to this value of V P1 and to V Gl = -4 V, we find that I P = 4.0 mA. Hence V Pt - 300 - (4.0) (20) - 130 = 90 V. Corresponding to this 7« and to I Pi = 4.0 mA, we find Vqi « -2.1 V. The third approximation to V pi is 125 + 2.1 = 127.1 V, which is close enough to the previous value of 127 V so as not to affect the value of the current appre- ciably. Hence I P = 4.0 mA. 8-11 1NTERELECTRODE CAPACITANCES IN A TRIODE 2 e assumed in the foregoing discussions that with a negative bias the input urrent was negligible and that changes in the plate circuit were not reflected ™e grid circuit. These assumptions are not strictly true, as is now shown. d; i . ^"^ pl ate > ana * cathode elements are conductors separated by a ectnc (a vacuum), and hence, by elementary electrostatics, there exist Paeitances between pairs of electrodes. Clearly, the input current in a n ded-cathode amplifier cannot be zero because the source must supply jk . *° the grid-cathode and the grid-plate capacitances. Furthermore, ' lr mut and output circuits are no longer isolated, but there is coupling 210 / ELECTRONIC DEVICES AND CIRCUITS Sec. 8-T1 V nt = V. 1 i Fig. 8-19 The schematic and equivalent circuits of a grounded-cathode amplifier, taking into account the interelectrode capacitances. between them through the grid-plate capacitance. Although these capaci- tances are small, usually less than 10 pF, yet, at the upper audio frequencies and above, they produce appreciable loading of the input source and also cause output-to-input feedback. They must therefore be taken into account. A more complete circuit and its equivalent circuit, which includes the interelectrode capacitances, are given in Fig. 8-19. In this circuit C BP repre- sents the capacitance between the grid and the plate, C g t is the capacitance between the grid and the cathode, and C„* is the capacitance between the plate and the cathode. The inclusion of these tube capacitances (shown dashed in the schematic diagram and explicitly in the equivalent amplifier circuit of Fig. 8-196) yields results that are more precise than those resulting from the analysis of the simple circuit of Fig. 8-3. It will be noted that the same procedure outlined in Sec. 8-3 has been followed in order to obtain the equivalent circuit of the amplifier. It is evident that F B * = F„ and so pVi has been written for the emf of the Thevenin's generator in the equivalent circuit of Fig. 8-196. The Voltage Gain The output voltage between terminals P and K is easily found with the aid of the theorem of Sec. 8-5, namely, V„ = IZ, where / is the short-circuit current and Z is the impedance seen between the terminals. To find Z, the generators F, and juF, in Fig. 8-196 are (imagined) short- circuited, and we note that Z is the parallel combination of the impedances corresponding to Z L , C p k, r p , and C BP . Hence y -z Y L + Y pk + g p + Y e C8-37) where Y L = \/Z L = admittance corresponding to Z L Y p k = juC p k = admittance corresponding to Cpk Qp = 1/rp = admittance corresponding to r p Y p = jvCgp = admittance corresponding to C ap S*c 8- J 2 VACUUM-TUBE SMALL-SIGNAL MODELS AND APPLICATIONS / 211 The current in the direction from P to K in a zero-resistance wire con- necting the output terminals is —fiVi/r p = —g„V t due to the generator nV { and is ViYev <* ue to the signal V,. Hence the total short-circuit current is I - -$mVi + V t Y„ The amplification A with the load Z L in place is given by _V_ a _IZ __ I Vi Vi ViY or, from Eqs. (8-37) and (8-38), —g m + Y g (8-38) A = Y L + Yp k + g p + Y s (8-39) It is interesting to see that Eq. (8-39) reduces to the expression already developed for the case where the interelectrode capacitances are neglected. Under these conditions, Y pk = Y ap = 0, and Eq. (8-39) reduces to A m — g» -9* 9v + Y L 1/r, + 1/Z L = —g m Z' L (8-40) where Z' L is r p \\Z L . This equation is identical with Eq. (8-6). It is a simple matter to show that the error made in the calculation of the gain is very small when the interelectrode capacitances are neglected for fre- quencies covering the entire audio-frequency range. These interelectrode capacitances are seldom as large as 15 pF, which corresponds to an admittance of only about 2 micromhos at 20 kHz. Since the transconductance g m of a tnode is generally several millimhos, Y gp may be neglected in comparison with 9^. Furthermore, if g p is greater than 20 micromhos (r p < 50 K), the terms Y„ + Y pk may be neglected in comparison with g p + Y L . Under these con- ditions the gain is that given by the simple expression (8-40). Since the interelectrode capacitances have a relatively minor effect on the audio gain of an amplifier, why is it important to make note of them? The answer is to be found in the input impedance of the tube (the loading of the gge on the input circuit) and in the feedback between output and input rcuits. Also, if the amplifier is to be used beyond the audio range, say, a video (television or radar) amplifier, the capacitances may seriously affect the are now examined gain and the exact expression, Eq. (8-39), must be used. These effects INPUT ADMITTANCE OF A TRIODE 8.12 from n8 if eCt * 0n °^ ^' ^"^ revea ^ s tnat tne g^ circuit is no longer isolated the plate circuit. The input signal must supply a current U. In order 212 / ELECTRONIC DEVICES AND CIRCUITS to calculate this current, it is observed from the diagram that it - ViY* and h= V BP Y gp m (F..+ V kp )Y sp Since V kp = — ?",,* = — AV U then the total input current ia Ii = h + h = [Y, k + (1 - A) Y„)Vi From Eq. (8-41), the input admittance is given by F. = ^= 7 Bk +(l -A)Y„ Sec, 8-12 (Wl) (8-42) This expression clearly indicates that, for the triode to possess a negligible input admittance over a wide range of frequencies, the grid-cathode and the grid-plate capacitances must be negligible. Input Capacitance (Miller Effect) Consider a triode with a plate-circuit resistance R p . From the preceding section it follows that within the audio- frequency range, the gain is given by the simple expression A = — gjtl pi where R p is r P ||i2 p . In this case Eq. (8-42) becomes Yi = MC Bk + (1 + g m R p )CJ (8-43) Thus the input admittance is that arising from the presence of a capacitance from the grid to the cathode of magnitude &, where C, = C„ k + (1 + g m R P )C e (8-44) This increase in input capacitance d over the capacitance from grid to cathode C B t is known as the Miller effect. The maximum possible value of this expres- sion is C gk + (1 + ti)C 9P , which, for large values of u, may be considerably larger than any of the interelectrode capacitances. This input capacitance is important in the operation of cascaded ampli- fiers. In such a system the output from one tube is used as the input to a second tube. Hence the input impedance of the second stage acts as a shunt across the output of the first stage and R p is shunted by the capacitance &■ Since the reactance of a capacitor decreases with increasing frequencies, the resultant output impedance of the first stage will be correspondingly low for the high frequencies. This will result in a decreasing gain at the higher frequencies. EXAMPLE A triode has a plate-circuit resistance of 100 K and operates »* 20 kHz. Calculate the gain of this tube as a single stage and then as the first tube in a cascaded amplifier consisting of two identical stages. The tube parame- ters are g m - 1.6 millimhos, r„ = 44 K, » = 70, C ek = 3.0 pF, C ph = 3.8 pF, and C ep = 2.8 pF. 1 s« *-i2 VACUUM-TUBE SMALL-SIGNAL MODELS AND APPLICATIONS / 213 Solution Y Q t = jo)C ek = j2t X 2 X 10' X 3.0 X 10"" = j'3.76 X 10~ 7 mho Y& = jmCpt m J4.77 X 10" 7 mho Y„ = jo>C gp = j3.52 X 10~ 7 mho g = — = 2.27 X 10" 6 mho Y p = — = 10-* mho R p g m = 1.60 X 10- 3 mho The gain of a one-stage amplifier is given by Eq. (8-39) : -ff« + Y„ -1.60 X 10-* + i3.52 X 10~ 7 A = g, + Y P + Y pk + Y e 3.27 X 10-* + j"8.29 X 10" 7 It ia seen that the j terms (arising from the interelectrode capacitances) are negligible in comparison with the real terms. If these are neglected, then A = —48.8. This value can be checked by using Eq. (8-5), which neglects inter- electrode capacitances. Thus A = - Rl ~\~ T p -70 X 100 100 + 44 = -48.6 Since the gain is a real number, the input impedance consists of a capacitor whose value is given by Eq, (8-44) : Ci = C gk + (1 + g m R p )C sp = 3.0 + (1 + 49) (2.8) - 143 pF Consider now a two-stage amplifier, each stage consisting of a tube operating as above. The gain of the second stage is that just calculated. However, in calculating the gain of the first stage, it must be remembered that the input impedance of the second stage acts as a shunt on the output of the first stage. Thus the plate load now consists of a 100-K resistance in parallel with 143 pF. To this must be added the capacitance from plate to cathode of the first stage since this is also in shunt with the plate load. Furthermore, any stray capacitances due to wiring should be taken into account. For example, for every 1 pF capacitance between the leads going to the plate and grid of the second stage, 50 pF is effec- tively added across the load resistor of the first tube! This clearly indicates the importance of making connections with very short direct leads in high-frequency amplifiers. Let it be assumed that the input capacitance, taking into account the various factors just discussed, is 200 pF (probably a conservative figure). Then the load admittance ia Yl = —+ juCi = 10" s + j2ir X 2 X 10 4 X 200 X 10" 11 Rp = 10" s + j*2.52 X 10" s mho 214 / ELECTRONIC DEVICES AND CIRCUITS The gain is given by Eq. (8-40) : -9m -1.6 X 10"* Sac. 8-12 A = g p + Yt, 2.27 X 10~ s + 10" 6 + j2.52 X 10~» = -30.7 + J23.7 - 38.8 /143.3° Thus the effect of the capacitances has been to reduce the magnitude of the amplification from 48.8 to 38.8 and to change the phase angle between the output and input from 180 to 143.3°. If the frequency were higher, the gain would be reduced still further. For example, this circuit would be useless as a video amplifier, say, to a few megahertz, since the gain would then be less than unity. This variation of gain with fre- quency is called frequency distortion. Cascaded amplifiers and frequency dis- tortion are discussed in detail in Chap. 16. Negative Input Resistance If the plate circuit of the amplifier includes an impedance instead of a pure resistance, then A is a complex number in general and the input admittance will consist of two terms, a resistive and a reactive term. Let A be written in the general form A = Ai + jA* Then Eq. (8-42) becomes Yi = uC op Ai + MC ek + (1 - A$C„\ (8-45) (8-46) The expression indicates that the equivalent grid input circuit comprises a resistance R, in parallel with a capacitance CV For such a parallel circuit, Comparing Eqs. (8-46) and (8^7), we have 1 Rt- aCgpAi d = C„* 4- (1 - A^C (8-47) (8-48) Since no restrictions have been placed on the system, it is possible for the term At to be negative and the effective input resistance to be negative. It is interesting to note that an effective negative input resistance is possible only when the load is inductive, with the inductance in a definite range. 8 The presence of a negative resistance in a circuit can mean only that some power is being generated rather than being absorbed. Physically, this means that power is being fed back from the output circuit into the grid circuit through the coupling provided by the grid-plate capacitance. If this feed- back feature reaches an extreme stage, the system will lose its entire utility as an amplifier, becoming in fact a self-excited amplifier, or oscillator. Sec. 8-13 VACUUM-TUBE SMALl-S/GNAi MODELS AND APPIICAT/ONS / 215 g_13 INTERELECTRODE CAPACITANCES IN A MULT1ELECTRODE TUBE 2 The wiring diagram of a tetrode is given in Fig. 8-20a, and the equivalent cir- cuit taking interelectrode capacitances into account is indicated in Fig. 8-206. In drawing the equivalent circuit, the rules given in Sec. 8-3 have been appropriately extended and employed. Thus, in addition to the points K, G, and P, the screen terminal S is also marked. The circuit elements of the original circuit are included in their appropriate positions between these four points, except that all dc potentials are omitted and the tube itself is replaced by an equivalent current generator <? m F,, having an internal resistance r p , between the points K and P. The capacitances between all pairs of the four electrodes are included, the double subscript denoting the pair of electrodes under consideration. Since the screen supply must be short-circuited in the equivalent circuit, this puts the sereen at ground potential in so far as signal variations about the Q point are concerned. Usually, the screen potential is obtained from the plate supply through a screen dropping resistor. In this case a capacitor is connected from the screen to cathode. This capacitance is chosen sufficiently large so that the screen potential remains constant even though the screen current may vary. In this case, too, the screen is at signal ground potential. Thus, as indicated in the figure, this effectively shorts out C k * and puts C ek and C at in parallel. Let this parallel combination be denoted d. The capac- ity C v , now appears from plate to ground and is effectively in parallel with C,*. Let this parallel combination be denoted C 2 . From the discussion of the shielding action of the screen grid in Sec. 7-6, the capacitance between the plate and the control grid C ep has been reduced to a very small value. If this capacitance is assumed to be negligible, Fig. 8-206 may be redrawn more simply, as shown in Fig. 8-21, where Cl — Cgk + Cg Ci — Cp, -f- c t* (8-49) • 1 i O— — • • • — •X © (a) K (6) Fig. 8-20 The schematic and equivalent circuits of a tetrode con- nected as an amplifier. 216 / ELECTRONIC DEVICES AND CIRCUITS Sec. 8-14 Fig. 8-21 The ideal equivalent circuit of a tetrode. The grid-plate capacitance has been assumed equal to zero. Because of the shielding action of the screen, little error will be made if C pk is neglected in comparison with C p „ so that C% = C pt , to a good approximation. This capacitance acts aa a shunt across the plate-circuit impedance. The input admittance of the tube is seen to be simply ¥i = jod (8-50) A significant difference is seen to exist between the ideal equivalent cir- cuit of the tetrode and the complete equivalent circuit of the triode, given in Fig. 8-19. The idealization made here consists in the assumption that the grid-plate capacitance is zero rather than a very small fraction of a picofarad. The circuit of Fig. 8-21 clearly shows that under this condition the plate cir- cuit has been isolated from the grid circuit. It should be pointed out, however, that the mere substitution of a tetrode for a triode will not, in general, effect any marked difference in the amplifier response. This statement follows because the wiring and stray capacitances between circuit elements external to the tube may provide the capacitances that the tube itself seeks to eliminate. It is necessary, therefore, that the elements of the circuit be carefully arranged in order to permit short inter- connecting leads and generally neat wiring so as to reduce wiring capacitances. It is only if the capacitance between the grid and anode circuits external to the tube is small that the inherent possibilities of the tetrode can be utilized. Pentode When used as a voltage amplifier, the pentode is connected in a circuit exactly like a tetrode (Fig. 8-20), with the addition that the sup- pressor grid is connected to the cathode. Then, from an equivalent circuit analogous to that in Fig. 8-206, it follows that the equivalent circuit of a pentode is also given by Fig. 8-21. In this diagram Ct = C ek + C gt Ci = C pk + (?„ + C p3 (8-51) Ci — Cpk + Cpt + Cpi where C p z is the capacitance between the plate and grid 3 (the suppressor). When the input and output capacitances of a tube are listed by the manu- facturer, reference is being made to Ci and Ct, respectively. 8-14 THE CATHODE FOLLOWER AT HIGH FREQUENCIES Our previous discussion of cathode followers neglected the influence of the tube capacitances. These capacitances are now taken into account. 1 Sic. 8-1* VACUUM-TUBE SMALL-SIGNAL MODELS AND APPLICATIONS / 217 Voltage Gain The grounded-plate configuration of a triode, including all capacitances, is given in Fig. 8-2 2a, and its linear equivalent circuit, in Fig. 8-226. The capacitance from cathode to ground is C kn and includes the capacitance from cathode to heater if, as usual, the heater is grounded. The output voltage V„ can be found as in Sec. 8-11 from the product of the short- circuit current and the impedance between terminals K and N. We now find for the voltage gain A m V /Vi Yq " (8-52) A = Ft + g p + g m + Yt where Y k = "5~ Kk Yt — j&Ct Ct — C g k + Cpk + Ckn, Equation (8-52) may be written in the form (9m + ju>C g k)Rk A m 1 + Km + D/r, + jwCrlfo (8-53) (8-54) Assuming ^ 4- 1 w m and g m R k !» 1, ^ _ gm + jtoCgk gm + jtaCr The term juC gk in the numerator represents the effect of the coupling from input to output through C gk . If the cathode follower is driving a capacitive load Cl, the expression for A need but be modified by adding Cl to Ct. Usually, C T is much larger than C gk , and hence the decrease in gain with frequency is due principally to Ct + Cl. The frequency / 2 at which the mag- nitude of the amplification has dropped 3 dB to 0.707 of its low-frequency value is, under these circumstances, given by the condition u(C T + Cl) = g m - Typically, if the total capacitance is, say, 50 pF and g m = 3 millimhos, as for a half section of a 12AU7, then / a >w 9.5 MHz. This calculation shows that a cathode follower may be useful well into the video range. I ^ 1/ - 1 ' \\ *? ' 1 > l > o o < * ] - V Ckn (6) N or P Fig. 8-22 (a) The cathode follower, with interelectrode capacitances taken into account, and (b) its equivalent circuit. 218 / ELECTRONIC DEVICES AND CIRCUITS Sec. B-U Input Admittance An important advantage of the cathode follower over a conventional triode amplifier is that the capacitive impedance seen looking into the grid of the cathode follower is appreciably larger than the capacitive impedance looking into the amplifier. We now calculate the input admittance from Kg, 8-226. The current h = V\0'o>C op ) and U = (Vi - V )ju,C Bk = 7,(1 - A)U»C*) where A is the amplifier gain. Hence the input admittance „ m I Ji + J, is given by ti m j w C s + jaC*(l ~ A) (8-55) In general, Yi contains a resistive as well as a capacitive component. If the frequency is low enough so that A may be considered a real number, the input impedance consists of a capacitance C„ and hence Yi = jtad. From Eq. (8-55) the input capacitance is given by C,(cathode follower) ■ C sv + C gk (l - A) (8-56) On the other hand, for a grounded-cathode amplifier, we have, from Eq. (8-44), C(amplifier) = C ek + C„(l - A) (8-57) A numerical comparison is interesting. Consider a half section of a 12AU7, first as a cathode follower of nominal gain, say, equal to 0.8, and then as an amplifier of nominal gain, say, A = — 10. The capacitances are C gp = 1.5 pF, Cgk = 1.6 pF. At a frequency at which the capacitances do not yet have a marked effect on the gain, we have C, (cathode follower) = 1.5 + 0.2 X 1.6 - 1.8 pF C,(amplifier) = 1.6 + 11 X 1.5 = 18 pF The input capacitance of the amplifier is ten times that of the cathode follower. A fairer comparison may be made between the cathode follower and a conventional amplifier of equivalent gain. In this case C,(amplifier) = 1.6 + 1.8 X 1.5 = 4.3 pF which is still more than twice that for the cathode follower. Output Admittance The output impedance, or more conveniently, the output admittance Y of a cathode follower, taking interelectrode capacitances into account, is obtained by adding to the low-frequency admittance g m + g p [Eq. (8-32) j the admittance of the total shunting capacitance C T - Thus Y = g m + g p + Y T (8-58) S« 8-U VACUUM-TUBE SMALt-SlGNAL MODELS AND APPLICATIONS / 219 This result may be justified directly by applying a signal V to the output terminals and computing the current which flows through V with the grid crrounded (and Rk considered as an external load). Since g m = y.g p and assum- ing M » 1' we ma y n eglect g v compared with g m and consider that the output admittance is unaffected by the capacitance until Y T becomes large enough to be comparable with g m . The calculation made above in connection with the frequency response of the cathode follower indicates that the output impedance does not acquire an appreciable reactive component until the fre- quency exceeds several megahertz. REFERENCES 1. Valley, G. E., Jr., and H. Wallman: "Vacuum Tube Amplifiers," MIT Radiation Laboratory Series, vol. 18, chap. 11, McGraw-Hill Book Company, New York, 1948. 2. Gewartowski, J. W., and H. A. Watson: "Principles of Electron Tubes," D. Van Nostrand Company, Inc., Princeton, N.J., 1965. 3. Millman, J., and S. Seely: "Electronics," 1st ed., p. 536, McGraw-Hill Book Com- pany, New York, 1941. 9 /TRANSISTOR CHARACTERISTICS The volt-ampere characteristics of a semiconductor triode, called a transistor, are described qualitatively and also derived theoretically. Simple circuits are studied, and it is demonstrated that the transistor is capable of producing amplification. A quantitative study of the transistor as an amplifier is left for Chap. 11. 9-1 THE JUNCTION TRANSISTOR 1 A junction transistor consists of a silicon (or germanium) crystal in which a layer of n-type silicon is sandwiched between two layers of p-type silicon. Alternatively, a transistor may consist of a layer of p-type between two layers of n-type material. In the former case the transistor is referred to as a p-n-p transistor, and in the latter case, as an n-p-n transistor. The semiconductor sandwich is extremely small, and is hermetically sealed against moisture inside a metal or plastic ease. Manufacturing techniques and constructional details for several transistor types are described in Sec. 9-4. The two types of transistor are represented in Fig. 9-la. The representations employed when transistors are used as circuit elements are shown in Fig. 9-16. The three portions of a transistor are known as emitter, base, and collector. The arrow on the emitter lead specifies the direction of current flow when the emitter-base junction is biased in the forward direction. In both cases, however, the emitter, base, and collector currents, I e , 1 B , and Ic, respectively, are assumed posi- tive when the currents flow into the transistor. The symbols Vsbi V C b, and Vce are the emitter- base, collector-base, and collector-emitter voltages, respectively. (More specifically, V E b represents the voltage drop from emitter to base.) 220 Emitter Base Collector I C TRANSISTOR CHARACTERISTICS / 221 Emitter Base Collector (a) i Collector Vcb Vm p-n-p type (ft) fl -P-< Emitter ' n-p-n type Fig, 9-1 (a) A p-n-p and an n-p-n transistor. The emitter (collector) junction is J s (Jc). (b) Circuit representation of the two transistor types. Emitter ± (a) Base (b) Collector |V'*„i i r — J Space- charge width — * V *~ Effective base width — IV |V M | Emitter (p-type) Base (n-type) CO Collector (p-type) I Fig. 9-2 (a) A p-n-p transistor with biasing voltages, (b) The potential bar- riers at the junction of the unbiased transistor, (c) The potential variation through the transistor under biased conditions. As the reverse-bias collector junction voltage \V C b\ is increased, the effective base width W decreases. 222 / ELECTRONIC DEVICES AND CIRCUITS Sec, 9-2 The Potential Distribution through a Transistor We may now begin to appreciate the essential features of a transistor as an active circuit element by considering the situation depicted in Fig. 9-2a. Here a p-n-p transistor is shown with voltage sources which serve to bias the emitter-base junction in the forward direction and the collector-base junction in the reverse direction. The variation of potential through an unbiased (open-circuited) transistor shown in Fig. 9-26. The potential variation through the biased transistor indicated in Fig. 9-2c. The dashed curve applies to the case before the appL cation of external biasing voltages, and the solid curve to the case after the biasing voltages are applied. In the absence of applied voltage, the potential barriers at the junctions adjust themselves to the height F — given in Eq. (6-13) (a few tenths of a volt) — required so that no current flows across each junction. (Since the transistor may be looked upon as a p-n junction diode in series with an n-p diode, much of the theory developed in Chap. 6 for the junction diode is used in order to explain the characteristics of a transistor.) If now external potentials are applied, these voltages appear essentially across the junctions. Hence the forward biasing of the emitter-base junction lowers the emitter-base potential barrier by |7 M |, whereas the reverse biasing of the collector-base junction increases the collector-base potential barrier by \V CB \. The lowering of the emitter-base barrier permits the emitter cur- rent to increase, and holes are injected into the base region. The potential is constant across the base region (except for the small ohmic drop), and the injected holes diffuse across the n-type material to the collector-base junction. The holes which reach this junction fall down the potential barrier, and are therefore collected by the collector. 9-2 TRANSISTOR CURRENT COMPONENTS In Fig. 9-3 we show the various current components which flow across the forward-biased emitter junction and the reverse-biased collector junction. The emitter current I E consists of hole current I pE (holes crossing from emitter into base) and electron current I nB (electrons crossing from base into the emitter). The ratio of hole to electron currents, I pE /I nB , crossing the emitter junction is proportional to the ratio of the conductivity of the p material to that of the n material (Prob. 9-1). In a commercial transistor the doping of the emitter is made much larger than the doping of the base. This feature ensures (in a p-n-p transistor) that the emitter current consists almost entirely of holes, Such a situation is desired since the current which results from electrons crossing the emitter junction from base to emitter does not contribute carriers which can reach the collector. Not all the holes crossing the emitter junction J E reach the collector junction J c because some of them combine with the electrons in the «-typ e base. If I pC is the hole current at J c , there must be a bulk recombination current I pE — I pC leaving the base, as indicated in Fig. 9-3 (actually, electrons TRANSISTOR CHARACTERISTICS / 223 Fig. 9-3 Transistor current components for a forward-biased emitter junction and a reversed-biased collector [unction. enter the base region through the base lead to supply those charges which have been lost by recombination with the holes injected into the base across J E )- If the emitter were open-circuited so that I E = 0, then I pC would be zero. Under these circumstances, the base and collector would act as a reverse- biased diode, and the collector current le would equal the reverse saturation current I C o- If Ib ^ 0, then, from Fig. 9-3, we note that Ic = h pC (9-1) For a p-n-p transistor, Ico consists of holes moving across Jc from left to right (base to collector) and electrons crossing Jc in the opposite direction. Since the assumed reference direction for Ico in Fig. 9-3 is from right to left, then for a p-n-p transistor, Ico is negative. For an n-p-n transistor, Ico is positive. We now define various parameters which relate the current components discussed above. \ Emitter Efficiency 7 The emitter, or injection, efficiency 7 is defined as _ current of injected carriers at J B total emitter current *B the case of a p-n-p transistor we have 7 = pE IpE + I»l IpM I B (9-2) where I pli i s the. injected hole diffusion current at emitter junction and I, lE is the Ejected electron diffusion current at emitter junction. 224 / ELECTRONIC DEVICES AND CIRCUITS Sec. 9-2 Transport Factor |8* The transport factor 0* is defined as _ injected carrier current reaching J c injected carrier current at Je In the case of a p-n-p transistor we have P ~ -j— (9-3) Large-signal Current Gain a We define the ratio of the negative of the collector-current increment to the emitter- current change from zero (cutoff) to Ig as the large-signal current gain of a common-base transistor, or Ic — Ico (9-4) Since Ic and Is have opposite signs, then a, as defined, is always positive. Typical numerical values of a lie in the range of 0.90 to 0.995. From Eqs. (9-1) and (9-4), . IpC _ IpC IpE Ib IpE I B Using Eqs. (9-2) and (9-3), a - 0*7 (9-5) (9-6) The transistor alpha is the product of the transport factor and the emitter efficiency. This statement assumes that the collector multiplication ratio 2 a* is unity, a* is the ratio of the total current crossing Jc to the hole current (for a p-n-p transistor) arriving at the junction. For most transistors, a* = 1. The parameter a is extremely important in transistor theory, and we examine it in more detail in Sec. 9-6. It should be pointed out that a is not a constant, but varies with emitter current Is, collector voltage Vcb, and temperature. From our discussion of transistor currents we see that if the transistor is in its active region (that is, if the emitter is forward-biased and the collector is reverse-biased), the collector current is given by Eq. (9-4), or Ic ■ — olI b + 1 1 (9-7) In the active region the collector current is essentially independent of collector voltage and depends only upon the emitter current. Suppose now that we seek to generalize Eq. (9-7) so that it may apply not only when the collector junction is substantially reverse-biased, but also for any voltage across J c - To achieve this generalization we need but replace Ico by the current in a p-n diode (that consisting of the base and collector regions). This current is given by the volt-ampere relationship of Eq. (6-31), with L S*. *- 3 TRANSISTOR CHARACTERISTICS / 225 replaced by —Ico and V by V c , where the symbol V c represents the drop cross Jc fr° m tne V to the n side. The complete expression for I c for any y c and Is 1S Ic = -al s + Ico{\ ~ * Vclv *) (9-8) Note that if V c is negative and has a magnitude large compared with Vt, Eq. (9-8) reduces to Eq. (9-7). The physical interpretation of Eq, (9-8) is that the p-n junction diode current crossing the collector junction is aug- mented by the fraction a of the current I B flowing in the emitter. This relationship is derived in Sec. 9-6. 9-3 THE TRANSISTOR AS AN AMPLIFIER A load resistor R L is in series with the collector supply voltage V C c of Fig. 9-2o. A small voltage change AF,- between emitter and base causes a rela- tively large emitter-current change Al E - We define by the symbol a' that fraction of this current change which is collected and passes through R&. The change in output voltage across the load resistor AF e = a'Ri Al E may be many times the change in input voltage AF,-. Under these circumstances, the voltage amplification A = AVjAVi will be greater than unity, and the transistor acts as an amplifier. If the dynamic resistance of the emitter junc- tion is r' t , then AF,- = r' e Alg, and a'R L Me <x'R l A m A/j (9-9) From Eq. (6-41), r' t = 26/ 1 E , where I E is the quiescent emitter current in milli- amperes. For example, if r\ = 40 U, a 1 = —1, and R L = 3,000 0, A ■ —75. This calculation is oversimplified, but in essence it is correct and gives a physi- cal explanation of why the transistor acts as an amplifier. The transistor pro- vides power gain as well as voltage or current amplification. From the fore- going explanation it is clear that current in the low-resistance input circuit is transferred to the high-resistance output circuit. The word "transistor," whk-h originated as a contraction of "transfer resistor," is based upon the above physical picture of the device. The Parameter a' The parameter a' introduced above is defined as the r& tio of the change in the collector current to the change in the emitter current at constant collcctor-to-base voltage and is called the small-signal forward wart-circuit current transfer ratio, or gain. More specifically, Mi (9-10) AI B \ y c* P n the assumption that a is independent of I E > then from Eq. (9-7) it follows th at«' „ 226 / ELECTRONIC DEVICES AND CIRCUITS Sec. 9-4 9-4 TRANSISTOR CONSTRUCTION contact 3mm rT 3 mm c» Fig. 9-4 Construction of transistors, (a) Grown, (b) alloy, and (e) diffused, or epitaxial, planar types. (The dimensions are approxi- mate, and the figures are not drawn to scale. The base width is given in microns, where T p = T0~ s m = TO -3 mm.) Five basic techniques have been developed for the manufacture of diodes, transistors, and other semiconductor devices. Consequently, such devices may be classified 3 - 4 into one of the following types: grown, alloy, electro- chemical, diffusion, or epitaxial. Grown Type The n-p-n grown-junction transistor is illustrated in Fig. 9-4a. It is made by drawing a single crystal from a melt of silicon or ger- manium whose impurity concentration is changed during the crystal- drawing operation by adding n- or p-type atoms as required. Alloy Type This technique, also called the fused construction, is illus- trated in Fig. 9-46 for a p-n-p transistor. The center (base) section is a thin wafer of n-type material. Two small dots of indium are attached to opposite sides of the wafer, and the whole structure is raised for a short time to a high temperature, above the melting point of indium but below that of germanium. The indium dissolves the germanium beneath it and forms a saturation solu- tion. On cooling, the germanium in contact with the base material recrystal- lizes with enough indium concentration to change it from n type to p type. The collector is made larger than the emitter, so that the collector subtends a large angle as viewed from the emitter. Because of this geometrical arrange- ment, very little emitter current follows a diffusion path which carries it to the base rather than to the collector. Electrochemically Etched Type This technique consists in etching depressions on opposite sides of a semiconductor wafer in order to reduce the Aluminum Silicon metal izat ion dioxide Emitter J^ ^ contact QE TRANSISTOR CHARACTERISTICS / 227 thickness of this base section. The emitter and collector junctions are then formed by electroplating a suitable metal into the depression areas. This type of device, also referred to as a surface-barrier transistor, is no longer of commercial importance. Diffusion Type This technique consists in subjecting a semiconductor wafer to gaseous diffusions of both n- and p-type impurities to form both the emitter and the collector junctions. A planar silicon transistor of the diffusion type is illustrated in Fig. 9-4c. In this process (described in greater detail in Chap. 15 on integrated-circuit techniques), the base-collector junction area is determined by a diffusion mask which is photoetched just prior to the base diffusion. The emitter is then diffused on the base, and a final layer of silicon oxide is thermally grown over the entire surface. Because of the passivating action of this oxide layer, most surface problems are avoided and very low leakage currents result. There is also an improvement in the current gain at low currents and in the noise figure. Epitaxial Type The epitaxial technique (Sec. 15-2) consists in growing a very thin, high-purity, single-crystal layer of silicon or germanium on a heavily doped substrate of the same material. This augmented crystal forms the collector on which the base and emitter may be diffused (Fig. 15-116). The foregoing techniques may be combined to form a large number of methods for constructing transistors. For example, there are diffused-alloy types, grown-diffused devices, alloy-emitter-epitaxial-base transistors, etc. The special features of transistors of importance at high frequencies are discussed in Chap. 13. The volt-ampere characteristics at low frequencies of all types of junction transistors are essentially the same, and the discussion to follow applies to them all. Finally, because of its historical significance, let us mention the first type of transistor to be invented. This device consists of two sharply pointed tungsten wires pressed against a semiconductor wafer. However, the relia- bility and reproducibility of such point-contact transistors are very poor, and 48 a result these transistors are no longer of practical importance. 9-5 Th DETAILED STUDY OF THE CURRENTS IN A TRANSISTOR 18 analysis follows in many respects that given in Sec. 6-5 for the current mponents in a junction diode. From Eq. (6-14) we see that the net current J 81n S a junction equals the sum of the electron current I njt in the p side the hole current I vn in the n side, evaluated at the junction (x = 0). For P~ n ~P transistor (Fig. 9-la) electrons are injected from the base region across i errilt 'ter junction into a p region which is large compared with the diffusion Ei-n. This is precisely the condition that exists in a junction diode, and ce the expression for I np calculated previously is also valid for the transis- 228 / ELECTRONIC DEVICES AND CIRCUITS tor. From Eq. (6-25) we find that at the junction /-p(0) = AeD n nBo ( € v a iv r _ j) Sttc. 9-5 (9-11) where in Eq. (6-25) we have replaced V by V E ; we have changed n^ to n B0 because there are now two p regions and the emitter (E) is under consider- ation; we have changed L„ to L B in order to refer to the diffusion length of the minority carriers in the emitter. A summary of the symbols used follows : A = cross section of transistor, m ! e = magnitude of electronic charge, C •D* (D p ) = diffusion constant for electrons (holes), m'/sec nso (nco) = thermal-equilibrium electron concentration in the p-type mate- rial of the emitter (collector), m -3 Lb (Lc) (Lb) — diffusion length for minority carriers in the emitter (collector) (base), m Vg (Vc) = voltage drop across emitter (collector) junction; positive for a forward bias, i.e., for the p side positive with respect to the n side V T — volt equivalent of temperature [Eq. (6-34)] p„ = hole concentration in the ra-type material, m~* p ne = thermal-equilibrium value of p„ W = base width, m /jm (In P ) ~ hole (electron) current in n (p) material The Hole Current in the n-type Base Region The value of I pn is not that found in Sec. 6-5 for a diode because, in the transistor, the hole current exists in a base region of small width, whereas in a diode, the n region extends over a distance large compared with L„. The diffusion current is given, as usual, by Eq. (6-18) ; namely, * pn — AfiUp —z — where p„ is found from the continuity equation. From Eq. (5-50), - p»„ - Kic-* /£ » + K if +" L > (9-12) (9-13) where K\ and K% are constants to be determined by the boundary condition** The situation at each junction is exactly as for the diode junction, and the boundary condition is that given by Eq. (6-22), or and Pn — Pno* " T Pn = Piiot rclVf at x — at x = W (9-14) S»c. 9-5 TRANSISTOR CHARACTERISTICS / 229 The exact solution is not difficult to find (Prob. 9-3). Usually, however, the base width W is small compared with L B , and we can simplify the solution hy introducing this inequality. Since < x < W, we shall assume that x/Lb « 1> an( * ^ en tne ex P on e nt i a ls in Eq. (9-13) can be expanded into a power series. If only the first two terms are retained, this equation has the form p„ - Pno = K s + K& (9-15) where Kt and Kt are new (and, as yet, undetermined) constants. To this approximation, p„ is a linear function of distance in the base. Then, from Eqs. (9-12) and (9-15), — AeDpKt = const (9-16) This result — that the minority-carrier current is a constant throughout the base region — is readily understood because we have assumed that W « L B . Under these circumstances, little recombination can take place within the base, and hence the hole current entering the base at the emitter junction leaves the base at the collector junction unattenuated. This means that the transport factor jS* is unity. Substituting the boundary conditions (9-14) in (9-15), we easily solve for K t and then find /,«(0) = - AeD p p n W [( € VdVr - 1) - ( € VMlV T _ !)] (9-17) The Ebers-Moll Equations From Fig. 9-3 we have for the emitter current t M = I P B + InB = /,.«>) + /„,((» Using Eqs. (9-11), (9-17), and (9-18), we find Ib - a n (€ v » lv * - 1) + au(e v 'i v ' - 1) where «* a similar manner we can obtain Ic m a«(« v -' v r - 1) + anifiW* - 1) where we can show (Prob. 9-2) that A /D p p no a 22 = Ae I — B a n = - AeD »P» 21 W W + Dnn Co \ Lc / (9-18) (9-19) (9-20) (9-21) (9-22) We note that a n ■ On. This result may be shown 6 to be valid for a isistor possessing any geometry. Equations (9-19) and (9-21) are valid any positive or negative value of V s or Vc, and they are known as the ^ers-Moll equations. 230 / ELECTRONIC DEVICES AND CIRCUITS 9-6 THE TRANSISTOR ALPHA If V E is eliminated from Eqs. (9-19) and (9-21), the result is Ic m 5» I B + (« M - ^A ( 6 VcfVr - l) an \ on / Sec. 9. (9-23) This equation has the same form as Eq. (9-8). Hence we have, by comparison, _ _% Oil 02lOi2 Ico = On — fl22 (9-24) (9-25) (9-26) Using Eqs. (9-20) and (9-22), we obtain = 1 1 + D n n BQ W/LgD p 'p no Making use of Eq. (5-2) for the conductivity, Eq. (5-33) for the diffusion constant, and Eq. (5-19) for the concentration, Eq. (9-26) reduces to a = 1 + wl B /L*,, < 9 - 27 > where <r B We) is the conductivity of the base (emitter). We see that, in order to keep a close to unity, o E hs should be large and W/L E should be kept small The analysis of the preceding section is based upon the assumption that W/L B «1. If this restriction is removed, the solution given in Prob. 9-3 is obtained. We then find (Prob. 9-5) that 7 » 1 and 1 + (DnLBUBe/DpLspno) tanh (W/L B ) W |8* = sech ~ Lib (9-28) (9-29) If W « L B , the hyperbolic secant and the hyperbolic tangent can be expanded in powers of W/L B , and the 6rst approximations are (Prob. 9-6) W, I/? and 1 + W<Tb/LeVB LtE&B WOB LgGB (9-30) (9-31) (9-32) As the magnitude of the reverse-bias collector voltage increases, the spa^ charge width at the collector increases (Fig. 9-2) and the effective base width W 5* 9-7 TRANSISTOR CHARACTERISTICS / 231 , eases . Hence Eq. (9-32) indicates that a increases as the collector junction becomes more reverse-biased. The emitter efficiency and hence also a is a function of emitter current. TTnuation (9-30) indicates that 7 decreases at high currents where <tb increases because of the additional charges injected into the base. (This effect is called conductivity modulation.) Also, it is found that 7 decreases at very low values of Is- This effect is due to the recombination of charge carriers in the transition region at the emitter junction, 8 At low injection currents this barrier recombination current is a large fraction of the total current and hence <y must be reduced. 7 Since silicon has many recombination centers in the space-charge layer, then 7 — ► (and a — ► 0) as I E — > 0. On the other hand, a s= 0.9 for germanium at Is = because germanium can be produced rela- tively free of recombination centers. The collector reverse saturation current can be determined using Eqa. (9-25), (9-20), and (9-22). 9-7 THE COMMON-BASE CONFIGURATION If the voltages across the two junctions are known, the three transistor cur- rents can be uniquely determined using Eqs. (9-19) and (9-21). Many differ- ent families of characteristic curves can be drawn, depending upon which two parameters are chosen as the independent variables. In the ease of the tran- sistor, it turns out to be most useful to select the input current and output voltage as the independent variables. The output current and input voltage are expressed graphically in terms of these independent variables. In Fig. 9-2a, a p-n-p transistor is shown in a grounded-base configuration. This cir- cuit is also referred to as a common-base, or CB, configuration, since the base w common to the input and output circuits. For a p-n-p transistor the largest current components are due to holes. Since holes flow from the emitter to the Collector and down toward ground out of the base terminal, then, referring to he polarity conventions of Fig. 9-1, we see that J E is positive, I c is negative, ^d I B i s negative. For a forward-biased emitter junction, V E b is positive, &Q u for a reverse- biased collector junction, V C b is negative. For an n-p-n jMststor all current and voltage polarities are the negative of those for a ' n ~P transistor. We may completely describe the transistor of Fig. 9-la or & y the following two relations, which give the input voltage V EB and output Tent I c in terms of the output voltage Vcs and input current I E : V*b = MVcs, Ib) 0-33) Ic = MVcb, Ib) (9-34) 18 equation is read, "I c is some function <£ 2 of V C b and I B ") Th (Th /He relation of Eq. (9-34) is given in Fig. 9-5 for a typical p-n-p ger- Ur n transistor and is a plot of collector current Ic versus collector-to-base 232 / ELECTRONIC DEVICES AND CIRCUITS Sec. o.y Saturation A.. tlve region —\ < 40 £ « -30 I £ u -20 5 I (3 -io /., = 40mA 1 30 20 10 1 • ■ I ^~ i I Tco \) j v i Cutoff region i — i — -i 1 Fig. 9-5 Typical common- base output characteristics of a p-n-p transistor. The cutoff, active, and satura- tion regions are indicated. Note the expanded voltage scale in the saturation region. 0.25 -2 -4 -6 -8 Collector-to-base voltage drop V cs , V voltage drop V C b, with emitter current I E as a parameter. The curves of Fig. 9-5 are known as the output, or collector, static characteristics. The rela- tion of Eq. (9-33) is given in Fig. 9-6 for the same transistor, and is a plot of emitter-to-base voltage V BB versus emitter current I B , with collector-to-base voltage V C s as a parameter. This set of curves is referred to as the input, or emitter, static characteristics. We digress now in order to discuss a phenomenon known as the Early effect, 3 which is used to account for the shapes of the transistor characteristics. The Early Effect An increase in magnitude of collector voltage increases the space-charge width at the output junction diode as indicated by Eq, (6-47). From Fig. 9-2 we see that such action causes the effective base width W to decrease, a phenomenon known as the Early effect. This decrease in W has Fig. 9-6 Common-base input characteristics of a typical p-n-p germanium junction transistor. Wi o I 10 20 30 40 Emitter current I s , mA V 1 ■ n open _ 4 •> K» -■ = 0V r ~> .-10 '-20 n See 9-7 TRANSfSTOR CHARACTERISTICS / 233 consequences: First, there is less chance for recombination within the region- Hence the transport factor #*, and also a, increase with an urease in the magnitude of the collector junction voltage. Second, the haree gradient is increased within the base, and consequently, the current of inority* carriers injected across the emitter junction increases. The Input Characteristics A qualitative understanding of the form of the input and output characteristics is not difficult if we consider the fact that the transistor consists of two diodes placed in series "back to back" (with the two cathodes connected together). In the active region the input diode (emitter-to-base) is biased in the forward direction. The input characteristics j jrig. 9-6 represent simply the forward characteristic of the emitter-to-base diode for various collector voltages. A noteworthy feature of the input char- acteristics is that there exists a cutin, offset, or threshold, voltage V yt below which the emitter current is very small. In general, V y is approximately 0. 1 V for germanium transistors (Fig. 9-6) and 0.5 V for silicon. The shape of the input characteristics can be understood if we consider the fact that an increase in magnitude of collector voltage will, by the Early effect, cause the emitter current to increase, with V B b held constant. Thus the curves shift downward as \Vcb\ increases, as noted in Fig. 9-6. The curve with the collector open represents the characteristic of the forward-biased emitter diode. When the collector is shorted to the base, the emitter current increases for a given Vrb since the collector now removes minority carriers from the base, and hence the base can attract more holes from the emitter. This means that the curve with V C b = is shifted down- ward from the collector characteristic marked "Vcb open." The Output Characteristics Note, as in Fig. 9-5, that it is customary to plot along the abscissa and to the right that polarity of Vcb which reverse- biases the collector junction even if this polarity is negative. The collector- to-base diode is normally biased in the reverse direction. If Is = 0, the col- lector current is I c = Ico. For other values of I E) the output- diode reverse current is augmented by the fraction of the input-diode forward current which Caches the collector. Note also that lev is negative for a p-n-p transistor and Positive for an n-p-n transistor. Active Region In this region the collector junction is biased in the reverse direction and the emitter junction in the forward direction. Consider first that l " e emitter current is zero. Then the collector current is small and equals he reverse saturation current I C o (microamperes for germanium and nano- ani peres for silicon) of the collector junction considered as a diode. Suppose Tk W tllat a f° rwarc * emitter current I B is caused to flow in the emitter circuit. hen a fraction —aI B of this current will reach the collector, and J E is therefore ©ven by Eq. (9-7). In the active region, the collector current is essentially "^dependent of collector voltage and depends only upon the emitter current. 234 / ELECTRONIC DEVICES AND CIRCUITS S«, 9-8 S* 9-8 TRANSISTOR CHARACTERISTICS / 235 However, because of the Early effect, we note in Fig. 9-5 that there actually i 8 a small (perhaps 0.5 percent) increase in \I C \ with \V C b\. Because a is less than, but almost equal to, unity, the magnitude of the collector current i B (slightly) less than that of the emitter current. Saturation Region The region to the left of the ordinate, V CB = 0, and above the I E = characteristics, in which both emitter and collector junctions are forward-biased, is called the saturation region. We say that "bottoming" has taken place because the voltage has fallen near the bottom of the charac- teristic where V CB « 0. Actually, V CB is slightly positive (for a p-n-p tran- sistor) in this region, and this forward biasing of the collector accounts for the large change in collector current with small changes in collector voltage. For a forward bias, I c increases exponentially with voltage according to the diode relationship [Eq. (9-21)]. A forward bias means that the collector p material is made positive with respect to the base n side, and hence that hole current flows from the p side across the collector junction to the n material. This hole flow corresponds to a positive change in collector current. Hence the collector current increases rapidly, and as indicated in Fig. 9-5, I c may even become positive if the forward bias is sufficiently large. Cutoff Region The characteristic for I s = passes through the origin, but is otherwise similar to the other characteristics. This characteristic is not coincident with the voltage axis, though the separation is difficult to show because I C o is only a few nanoamperes or microamperes. The region below and to the right of the I E = characteristic, for which the emitter and col- lector junctions are both reverse-biased, is referred to as the cutoff region. The temperature characteristics of I C o are discussed in Sec. 9-9. 9-8 THE COMMON-EMITTER CONFIGURATION Most transistor circuits have the emitter, rather than the base, as the terminal common to both input and output. Such a common-emitter CE, or grounded- emitter, configuration is indicated in Fig. 9-7. In the common-emitter, as in the common-base, configuration, the input current and the output voltage Fig. 9-7 A transistor common-emitter con- figuration. The symbol Vcc is a positive number representing the magnitude of the supply voltage. ken as the independent variables, whereas the input voltage and output 8Xe en t are the dependent variables. We may write Vbs = ZiC^ca, Ib) Ic = MVcm, Ib) (9-35) (9-36) Equation (9-35) describes the family of input characteristic curves, and F (9-36) describes the family of output characteristic curves. Typical out- ut and input characteristic curves for a p-n-p junction germanium transistor are given in Figs. 9-8 and 9-9, respectively. In Fig. 9-8 the abscissa is the collector-to-emitter voltage Vce, the ordinate is the collector current I c , and the curves are given for various values of base current I B , For a fixed value of h, * ne collector current is not a very sensitive value of Vcs- However, the slopes of the curves of Fig. 9-8 are larger than in the common-base charac- teristics of Fig. 9-5. Observe also that the base current is much smaller than the emitter current. The locus of all points at which the collector dissipation is 150 mW is indi- cated in Fig. 9-8 by a solid line P c = 150 mW. This curve is the hyperbola Pc = VcbIc ~ VcbIc = constant. To the right of this curve the rated col- lector dissipation is exceeded. In Fig. 9-8 we have selected Rt = 500 Q and a supply Vcc = 10 V and have superimposed the corresponding load line on the output characteristics. The method of constructing a load line is identical with that explained in Sec. 4-9 in connection with a diode. The input Characteristics In Fig. 9-9 the abscissa is the base current Ib, the ordinate is the base-to-emitter voltage Vbb, and the curves are given for various values of collector-to-emitter voltage V C s- We observe that, with the collector shorted to the emitter and the emitter forward-biased, the input char- acteristic is essentially that of a forward-biased diode. If V BB becomes zero, fig, 9-8 Typical common-emitter Output characteristics of a p-n-p 9«rmanium junction transistor. A load line corresponding to Vcc = 10 V and R L = 500 U is super- posed. (Courtesy of Texas lns trumer,ts, Inc.) -2 -4 -6 -8 -10 Collector- emitter voltage V cg . , V 236 / ELECTRONIC DEVICES AND CIRCUITS S«. 9.j H -0.4 -as -0.2 -0.1 1 r=25°C t^OA ~~0~~ Fig. 9-9 Typical common-emitter input characteristics of the p-n-p germanium Junc- tion transistor of Fig. 9-8. o -1 -2 -3 -4 -5 Base current I B , m A then I B will be zero, since under these conditions both emitter and collector junctions will be short-circuited. For any other value of V C s, the base cur- rent for Vbb ■» is not actually zero but is too small (Sec. 9-15) to be observed in Fig. 9-9. In general, increasing \V C b\ with constant V BS causes a decrease in base width W (the Early effect) and results in a decreasing recombination base current. These considerations account for the shape of input character- istics shown in Fig. 9-9. The input characteristics for silicon transistors are similar in form to those in Fig. 9-9. The only notable difference in the case of silicon is that the curves break away from zero current in the range 0.5 to 0.6 V, rather than in the range 0.1 to 0.2 V as for germanium. The Output Characteristics This family of curves may be divided into three regions, just as was done for the CB configuration. The first of these, the active region, is discussed here, and the cutoff and saturation regions are considered in the next two sections. In the active region the collector junction is reverse-biased and the emitter junction is forward-biased. In Fig. 9-8 the active region is the area to the right of the ordinate V cs = a few tenths of a volt and above I B - 0. In this region the transistor output current responds most sensitively to an input signal. If the transistor is to be used as an amplifying device without appreci- able distortion, it must be restricted to operate in this region. The common-emitter characteristics in the active region are readily under- stood qualitatively on the basis of our earlier discussion of the common-base configuration. The base current is la - -(/c + / g ) Combining this equation with Eq. (9-7), we find Id- 1 - + ali 1 - a (9-37) (9-38) 5* 9-9 TRANSISTOR CHARACTERISTICS / 237 uation (9-7) is based on the assumption that V C b is fixed. However, if V is larger than several volts, the voltage across the collector junction is c \ larger than that across the emitter junction, and we may consider ^ U „ y CB . Hence Eq. (9-38) is valid for values of Vcs in excess of a few a1 1 8 If a were truly constant, then, according to Eq. (9-38), I c would be inde- dent oi VcE m ^ fa e curves of Fig. 9-8 would be horizontal. Assume that, because of the Early effect, a increases by only one-half of 1 percent, from 98 to 0.985, as \Vcs\ increases from a few volts to 10 V. Then the value of a/(l - «) increases from 0.98/(1 - 0.98) = 49 to 0.985/(1 - 0.985) = 66, or about 34 percent. This numerical example illustrates that a very small change (0.5 percent) in a is reflected in a very large change (34 percent) in the value of «/(l - a). It should also be clear that a slight change in a has a large effect on the common-emitter curves, and hence that common-emitter characteristics are normally subject to a wide variation even among transis- tors of a given type. This variability is caused by the fact that I B is the difference between large and nearly equal currents, I E and I c . 9-9 THE CE CUTOFF REGION We might be inclined to think that cutoff in Fig. 9-8 occurs at the intersection of the load line with the current I B = 0; however, we now find that appreci- able collector current may exist under these conditions. The common-base characteristics are described to a good approximation even to the point of cutoff by Eq. (9-7), repeated here for convenience: Ic = — oJb + 1 1 (9-39) From Fig. 9-7, if I B = 0, then I E = -Ic- Combining with Eq. (9-39), we have Ico Ic = —Is ~ 1 - SpJi (9-40) The actual collector current with collector junction reverse-biased and base open-circuited is designated by the symbol lew- Since, even in the neighbor- hood of cutoff, a may be as large as 0.9 for germanium, then Ic « 10/co at 2e ro base current. Accordingly, in order to cut off the transistor, it is not e nough to reduce I B to zero. Instead, it is necessary to reverse-bias the WBitter junction slightly. We shall define cutoff as the condition where the col- lector current is equal to the reverse saturation current I c o and the emitter cur- ^t, is zero. In Sec. 9-15 we show that a reverse-biasing voltage of the order of °- 1 V established across the emitter junction will ordinarily be adequate to cut off _ a germanium transistor. In silicon, at collector currents of the order of Ico, 11 is found 6 ' ' that a is very nearly zero because of recombination in the emitter- 238 / ELECTRONIC DEVICES AND CIRCUITS Sec. 9-9 junction transition region. Hence, even with I B = 0, we find, from Eq. (9-40), that Ic = I co = — Ib, so that the transistor is still very close to cutoff We verify in Sec. 9-15 that, in silicon, cutoff occurs at V BE « V, correspond- ing to a base short-circuited to the emitter. In summary, cutoff means that Is ~ 0, Ic = Ico, Ib = —Ic= — I co, and Vbb is a reverse voltage whose mag- nitude is of the order of 0.1 V for germanium and V for a silicon transistor. The Reverse Collector Saturation Current I CB0 The collector current in a physical transistor (a real, nonidealized, or commercial device) when the emitter current is zero is designated by the symbol I C bo. Two factors cooper- ate to make |/ Cfl0 | larger than \I C0 \. First, there exists a leakage current which flows, not through the junction, but around it and across the surfaces. The leakage current is proportional to the voltage across the junction. The second reason why \I CB0 \ exceeds \I C0 \ is that new carriers may be generated by collision in the collector-junction transition region, leading to avalanche multiplication of current and eventual breakdown. But even before break- down is approached, this multiplication component of current may attain con- siderable proportions. At 25°C, Icbo for a germanium transistor whose power dissipation is in the range of some hundreds of milliwatts is of the order of microamperes. Under similar conditions a silicon transistor has an I CB o in the range of nano- amperes. The temperature sensitivity of I CB0 in silicon is approximately the same as that of germanium. Specifically, it is found 9 that the temperature coefficient of I C bo is 8 percent/°C for germanium and 6 percent/°C for silicon. Using 7 percent as an average value and since (1.07) 10 « 2, we see that Icbo approximately doubles for every 10°C increase in temperature for both Ge and Si. However, because of the lower absolute value of I CB o in silicon, these transistors may be used up to about 2G0°C, whereas germanium transistors are limited to about 100°C. In addition to the variability of reverse saturation current with tempera- ture, there is also a wide variability of reverse current among samples of a given transistor type. For example, the specification sheet for a Texas Instru- ment type 2N337 grown diffused silicon switching transistor indicates that this type number includes units with values of I CBG extending over the tremendous range from 0.2 nA to 0.3 M. Accordingly, any particular transistor may have an I CB0 which differs very considerably from the average characteristic for the type. Circuit Considerations at Cutoff Because of temperature effects, ava- lanche multiplication, and the wide variability encountered from sample to sample of a particular transistor type, even silicon may have values of Icbo of the order of many tens of microamperes. Consider the circuit configuration of Fig. 9-10, where Vbb represents a biasing voltage intended to keep the tran- sistor cut off. We consider that the transistor is just at the point of cutoff, with I s = 0, so that I B = -I CB0 . If we require that at cutoff Vbb « -0.1 V, L S«c 9-10 TRANSISTOR CHARACTERISTICS / 239 fig. 9-10 Reverse biasing of the emitter junction to maintain the transistor in cutoff in the presence of the reverse saturation current Icbo through Us. ^rv c then the condition of cutoff requires that Vbb = ~V BB + Rshso < -0.1 V (9-41) As an extreme example consider that Rb is, say, as large as 100 K and that we want to allow for the contingency that I C bo may become as large as 100 tiA, Then V BB must be at least 10. 1 V. When I C bo is small, the magnitude of the voltage across the base-emitter junction will be 10.1 V. Hence we must use a transistor whose maximum allowable reverse base-to-emitter junction volt- age before breakdown exceeds 10 V. It is with this contingency in mind that a manufacturer supplies a rating for the reverse breakdown voltage between emitter and base, represented by the symbol BVebo- The subscript indi- cates that BVebo is measured under the condition that the collector current is zero. Breakdown voltages BVebo may be as high as some tens of volts or as low as 0.5 V. If BVbbo = 1 V, then Vbb must be chosen to have a maximum value of 1 V. For Vbb = IV and for Icbo = 0. 1 mA maximum, R B cannot exceed 9 K. For example, if R B = 8 K, then -Vbb + IcboRb = -1 + 0.8 = -0.2 V so that the transistor is indeed cut off. MO THE CE SATURATION REGION A load line has been superimposed on Fig. 9-8 corresponding to a load resistor "■t *= 500 £2 and a supply voltage of 10 V. The saturation region may be defined as the one where the collector junction (as well as the emitter junction) 18 "^ward-biased. In this region bottoming occurs, | Vce\ drops to a few tenths °f a volt, and the collector current is approximately independent of base cur- ren t, for given values of V C c and R L . Hence we may consider that the onset °' saturation takes place at the knee of the transistor curves in Fig. 9-8. Satu- ration occurs for the given load line at a base current of —0.17 mA, and at this J*° mt the collector voltage is too small to be read in Fig. 9-8. In saturation, he collector current is nominally V C c/Rl, and since R L is small, it may well e necessary to keep V C c correspondingly small in order to stay within the Wit ations imposed by the transistor on maximum current and dissipation. 240 / ELECTRONIC DEVICES AND CIRCUITS j -30 -20 « -10 J n 1 1 - 0.35mA T A = 25°C -0.30 -0.2 -0.2 5 soon II [Z -1 -0.15 — 1 1 1 ^ "rjn — 1 — 1 — ' -0.10 — 1 -0.0& 1 r Sec. 9- JO Fig. 9-11 Saturation-region com- mon-emjtter characteristics of the type 2N404 germanium transistor. A load line corresponding to Vcc = 10 Vand R L = 5000 is super- imposed. (Courtesy of Texas Instruments, Inc.) -0.1 -0.2 -0.3 -0.4 -0.5 Collector -emitter voltage V cs , V We are not able to read the collector-to-emitter saturation voltage, Vce (sat), with any precision from the plots of Fig. 9-8. We refer instead to the characteristics shown in Fig. 9-11. In these characteristics the 0- to — 0.5-V region of Fig. 9-8 has been expanded, and we have superimposed the same load line as before, corresponding to R L = 500 fi. We observe from Figs. 9-8 and 9-11 that V CB and I c no longer respond appreciably to base current I B) after the base current has attained the value —0.15 mA. At this current the transistor enters saturation. For I B = —0.15 mA, |Fcsl « 175 mV. At I B = -0.35 mA, | V CB \ has dropped to ( V CE \ ~ 100 mV. Larger magnitudes of I n will, of course, decrease \V C b\ slightly further. Saturation Resistance For a transistor operating in the saturation region, a quantity of interest is the ratio Fcs(sat)/Je- This parameter is called the common-emitter saturation resistance, variously abbreviated Res, Reus, or Resist). To specify R c $ properly, we must indicate the operating point at which it was determined. For example, from Fig. 9-11, we find that, at Ic = -20 mA and I B = -0.35 mA, R C s = -0.1/(-20 X 10~ 3 ) = 5 fl- The usefulness of R C s stems from the fact, as appears in Fig. 9-11, that to the left of the knee each of the plots, for fixed I B> may be approximated, at least roughly, by a straight line. Saturation Voltages Manufacturers specify saturation values of input and output voltages in a number of different ways, in addition to supplying characteristic curves such as Figs. 9-9 and 9-11. For example, they may specify R cs for several values of J B or they may supply curves of Fc^(sat) and F ss (sat) as functions of I B and I c . in The saturation voltage Fc^(sat) depends not only on the operating point, but also on the semiconductor material (germanium or silicon) and on the type of transistor construction. Alloy-junction and epitaxial transistors g» ve S* 9- 10 TRANSISTOR CHARACTERISTICS / 241 the lowest values for Fca(sat) (corresponding to about 1 fi saturation resist- ance), whereas grown- junction transistors yield the highest. Germanium transistors have lower values for F C s(sat) than silicon. For example, an alloy- junction Ge transistor may have, with adequate base currents, values for VciKsat) as low as tens of millivolts at collector currents which are some tens of milliamperes. Similarly, epitaxial silicon transistors may yield satu- ration voltages as low as 0.2 V with collector currents as high as an ampere. On the other hand, grown-j unction germanium transistors have saturation voltages which are several tenths of a volt, and silicon transistors of this type may have saturation voltages as high as several volts. Typical values of the temperature coefficient of the saturation voltages are ~— 2.5 mV/°C for Vss($&t) and approximately one-tenth of this value for Fas (sat) for either germanium or silicon. The temperature coefficient for VWsat) is that of a forward-biased diode [Eq. (6-39)]. In saturation the transistor consists of two forward-biased diodes back- to-back in series opposing. Hence, it is to be anticipated that the temperature-induced voltage change in one junction will be canceled by the change in the other junction. We do indeed find 10 such to be the case for F cg (sat). The DC Current Gain kn A transistor parameter of interest is the ratio I c /Ib, where Ic is the collector current and I B is the base current. This quantity is designated by do or k rB , and is known as the dc beta, the dc forward current transfer ratio, or the dc current gain. In the saturation region, the parameter h FB is a useful number and one which is usually supplied by the manufacturer when a switching transistor is involved. We know |/ c |, which is given approximately by V C c/Rl, and a knowledge of h FB tells us how much input base current (ZcA^js) will be needed to saturate the transistor. For the type 2N404, the variation of h FB with collector current at a low value of V CB is as given in Fig. 9-12. Note the 140 F| 9- 9-12 Plots of dc current gain A '« (at V cs = -0.25 V) versus col- le tfor current far three samples of * typ e 2N404 germanium transistor. (Courtesy of General Electric Com- pany.) 10 20 30 40 50 60 70 80 90100110120130 — I c ,mA 242 / ElECTRONfC DEVICES AND CIRCUITS Sec. 9-11 wide spread (a ratio of 3 : 1) in the value which may be obtained for hps even for a transistor of a particular type. Commercially available transistors have values of hps that cover the range from 10 to 150 at collector currents as small as 5 mA and as large as 30 A, Tests for Saturation It is often important to know whether or not a transistor is in saturation. We have already given two methods for making such a determination. These may be summarized as follows: 1. If Ic and Is can be determined independently from the circuit under con- sideration, the transistor is in saturation if \In\ ~> \Ic\fh-FE- 2. If Vcb is determined from the circuit configuration and if this quantity ia positive for a p-n-p transistor (or negative for an n-p-n) f the transistor is in satu- ration. Of course, the emitter j unction must be simultaneously forward- biased, but then we should not be testing for saturation if this condition were not satisfied. 9-11 LARGE-SIGNAL, DC, AND SMALL-SIGNAL CE VALUES OF CURRENT GAIN If we define by a = 1 - a and replace Ico by Icbo, then Eq. (9-38) becomes Ic = (1 + 0)1 cbo + 01 b From Eq. (9-43) we have = Ic — h Ib — ( — Icbo) (9-42) (9-43) (9-44) In Sec. 9-9 we define cutoff to mean that Is = 0, Ic = Icbo, and I B = — Icbo- Consequently, Eq. (9-44) gives the ratio of the collector-current increment to the base-current change from cutoff to I B , and hence represents the large-signal current gain of a comm-on-emitter transistor. This parameter is of primary impor- tance in connection with the biasing and stability of transistor circuits as dis- cussed in Chap, 10. In Sec. 9-10 we define the de current gain by 0d C = j- = h Ib (9-45) In that section it is noted that h FS is most useful in connection with deter- mining whether or not a transistor is in saturation. In general, the base current (and hence the collector current) is large compared with Icbo- U nder Sec 9-12 TRANSISTOR CHARACTER/ST/CS / 243 these conditions the large-signal and the dc betas are approximately equal; then hrs =* 0- The small-signal CE forward short-circuit current gain 0' is defined as the ratio of a collector- current increment Al c for a small base-current change AI B (at a given quiescent operating point, at a fixed collector-to-emitter voltage Vcb), or F d! B k« (9-46) If is independent of current, we see from Eq. (9-43) that 0' = m h FE . However, Fig. 9-12 indicates that is a function of current, and from Eq. (9-43), « + (Icbo + Ib) ~ oIb (9-47) The small-signal CE forward gain 0' is used in the analysis of amplifier cir- cuits and is designated by h ft in Chap. 11. Using 0' = ft/, and - k? B , Eq. (9-47) becomes ht e «■ 1 - (Icbo + Ib) dhp (9-48) Since k FB versus I c given in Fig. 9-12 shows a maximum, then h fe is larger than h FB for small currents (to the left of the maximum) and h fls < h PS for currents larger than that corresponding to the maximum. It should be empha- sized that Eq. (9-48) is valid in the active region only. From Fig. 9-11 we see that h f . — * in the saturation region because A/ c — * for a small increment AI B . 9-12 THE COMMON-COLLECTOR CONFIGURATION Another transistor-circuit configuration, shown in Fig. 9-13, is known as the common-collector configuration. The circuit is basically the same as the cir- cuit of Fig. 9-7, with the exception that the load resistor is in the emitter circuit rather than in the collector circuit. If we continue to specify the oper- ation of the circuit in terms of the currents which flow, the operation for the 9- 9-13 The transistor common-collector ^"figuration. 244 / aCCTRONIC DEVICES AND CIRCUITS S«e. 9-U common-collector is much the same as for the common-emitter configuration. When the base current is I C o, the emitter current will be zero, and no current will flow in the load. As the transistor is brought out of this back-biased condition by increasing the magnitude of the base current, the transistor will pass through the active region and eventually reach saturation. In this condi- tion all the supply voltage, except for a very small drop across the transistor, will appear across the load. 9-13 GRAPHICAL ANALYSIS OF THE CE CONFIGURATION It is our purpose in this section to analyze graphically the operation of the circuit of Fig. 9-14. In Fig. 9- 15a the output characteristics of a p-n-p germanium transistor and in Fig. 9-156 the corresponding input characteristics are indicated. We have selected the CE configuration because, as we see in Chap. 11, it is the most generally useful configuration. In Fig. 9-15o we have drawn a load line for a 250-fl load with Vcc = 15 V. If the input base-current signal is symmetric, the quiescent point Q is usually selected at about the center of the load line, as shown in Fig. 9-15o. We postpone until Chap. 10 our discussion on biasing of transistors. Notation At this point it is important to make a few remarks on tran- sistor symbols. The convention used to designate transistor voltages and currents is the same as that introduced for vacuum tubes in Sec. 7-9. Spe- cifically, instantaneous values of quantities which vary with time are repre- sented by lowercase letters (i for current, v for voltage, and p for power). Maximum, average (dc), and effective, or root-mean- square (rms), values are represented by the uppercase letter of the proper symbol (J, V, or P). Aver- age (dc) values and instantaneous total values are indicated by the uppercase subscript of the proper electrode symbol (B for base, C for collector, E for emitter). Varying components from some quiescent value are indicated by the lowercase subscript of the proper electrode symbol. A single subscript is used if the reference electrode is clearly understood. If there is any possi- Fig. 9-14 The CE transistor configur- ation. TRANSISTOR CHARACTERISTICS / 245 Base voltage v BEt V ""] 1 1 1'l'ITI rrm , nin .J LI I ' ' iftV "T--^7t *o -Dynamic curve ■P -^zjJ-rfF .... ,.'l I ' l f - s J^TI l£r-*Tir y***H) ^" • r VcF ::i!:::2!gj2: "i't* i -~ smm -0.15 -j-'l irToV.---- _Vcb i ;: ■ 2 -4 -6 -8 -10 -12 -14 Collector voltage v ct: ,V (a) O -100-200-800-400-600-600 Base current t B , ^ A (b) Fig. 9-15 (a) Output and [b] input characteristics of a p-n-p germanium transistor. bility of ambiguity, the conventional double-subscript notation should be used. For example, in Figs. 9- 16a to d and 9-14, we show collector and base currents and voltages in the common-emitter transistor configuration, employing the notation just described. The collector and emitter current and voltage com- ponent variations from the corresponding quiescent values are it. — ic — Ic = Aie ib = Ib — Is — Ata f e = vc — Vc = Ave Vb = vb — Vb = A-Wb The magnitude of the supply voltage is indicated by repeating the electrode subscript. This notation is summarized in Table 9-1. TABLE 9-1 Notation k|Bta ntaneouB total vaiue g"eacent value . . . Eff aa . taneoUB value of varying component e ctive value of varying component (phasor, if a ainusoid) ^PPly voltage (magnitude). Base (collector) voltage with respect to emitter vb (t>c) V B {Vc) vt <»«) n if 4 Vbb {Vcc) Base (collector) current toward electrode from external circuit t« (tc) Ib Uc) u(i<) h Uc) 246 I ELECTRONIC DEVICES AND CIRCUITS Sec. 9-7 3 Sinusoid 2t •! (6) («*) Fig. 9-16 (a, b) Collector and (c, d) base current and voltage waveforms. The Waveforms Assume a 200-^iA peak sinusoidally varying base current around the quiescent point Q, where /« = -300 juA. Then the extreme points of the base waveform are A and B, where i B = —500 /iA and — 100 nA, respectively. These points are located on the load line in Fig. 9-15a. We find ic and vce, corresponding to any given value of i B , at the intersection of the load line and the collector characteristics corresponding to this value of is- For example, at point A,i B = —500 nA, ic = —46.5 niA, and v C b - —3.4 V. The waveforms ic and vcb are plotted in Fig. 9- 16a and b, respectively. We observe that the collector current and collector voltage waveforms are not the same as the base-current waveform (the sinusoid of Fig. 9-16c) because the collector characteristics in the neighborhood of the load line in Fig. 9- 15a are not parallel lines equally spaced for equal increments in base current. This change in waveform is known as output nonlinear distortion. The base-to-emitter voltage vbb for any combination of base current and collector-to-emitter voltage can be obtained from the input characteristic curves. In Fig. 9-156 we show the dynamic operating curve drawn for the combinations of base current and collector voltage found along A-Q-B of the load line of Fig. 9-15a. The waveform v B s can be obtained from the dynaflU fl operating curve of Fig. 9-156 by reading the voltage v H b corresponding to » S*. 9-U TRANSISTOR CHARACTERISTICS / 247 jriven base current i B . We now observe that, since the dynamic curve is not a straight line, the waveform of Vb (Fig. 9-16rf) will not, in general, be the same as the waveform of i b . This change in waveform is known as input nonlinear distortion. In some cases it is more reasonable to assume that v b in Fig- 9-16d is sinusoidal, and then i b will be distorted. The above condition will be true if the sinusoidal voltage source v. driving the transistor has a small output resistance R, in comparison with the input resistance Ri of the transistor, so that the transistor input-voltage waveform is essentially the game as the source waveform. However, if R, » Ri, the variation in i B is given by % «= v t /R„ and hence the base-current waveform is also sinusoidal. From Fig. 9-156 we see that for a large sinusoidal base voltage Vt, around the point Q the base-current swing |*V| is smaller to the left of Q than to the right of Q. This input distortion tends to cancel the output distortion because, in Fig. 9- 15a, the collector-current swing |z c | for a given base-current swing is larger over the section BQ than over QA. Hence, if the amplifier is biased so that Q is near the center of the ic-ves plane, there will be less distortion if the excitation is a sinusoidal base voltage than if it is a sinusoidal base current. It should be noted here that the dynamic load curve can be approximated by a straight line over a sufficiently small line segment, and hence, if the input signal is small, there will be negligible input distortion under any conditions of operation (current-source or voltage-source driver). M4 ANALYTICAL EXPRESSIONS FOR TRANSISTOR CHARACTERISTICS The dependence of the currents in a transistor upon the junction voltages, or vice versa, may be obtained by starting with Eq. (9-8), repeated here for convenience: Ic - -a N I B - Ico(* v ° lv r - 1) (9-49) We have added the subscript N to a in order to indicate that we are using the transistor in the normal manner. We must recognize, however, that there to no essential reason which constrains us from using a transistor in an inverted fashion, that is, interchanging the roles of the emitter junction and the col- lector junction. From a practical point of view, such an arrangement might not be as effective as the normal mode of operation, but this matter does not concern us now. With this inverted mode of operation in mind, we may now Wr 'te, in correspondence with Eq. (9-49), Ib m ~ aj Ic - Ibo(* v ' ,v t ~ I) (9-50) tr , ere a r is the inverted common-base current gain, just as on in Eq. (9-49) is current gain in normal operation. I so is the emitter- junction reverse satu- on current, and V B is the voltage drop from p side to n side at the emitter c tion and is positive for a forward-biased emitter. In the literature, 248 / ELECTRONIC DEVICES AND CIRCUITS ■V c S«c. 9-U C (collector) Fig. 9-17 Defining the voltages and currents used in the Ebers-Moll equa- tions. For either a p-n-p or an n-p^n transistor, a positive value of current means that positive charge flows into the junction and a positive Vg (F c ) means that the emitter (collector) junction is forward-biased (the p side positive with respect to the n side). a R (reversed alpha) and a F (forward alpha) are sometimes used in place of m and a Ni respectively. The Base-spreading Resistance r w The symbol Vc represents the drop across the collector junction and is positive if the junction is forward-biased. The reference directions for currents and voltages are indicated in Fig. 9-17. Since Vcb represents the voltage drop from collector-to-base terminals, then Vqb differs from F c by the ohmic drops in the base and the collector materials. Recalling that the base region is very thin (Fig. 9-4), we see that the current which enters the base region across the junction area must flow through a long narrow path to reach the base terminal. The cross-sectional area for current flow in the collector (or emitter) is very much larger than in the base. Hence, usually, the ohmic drop in the base alone is of importance. This dc ohmic base resistance r»* is called the base- spreading resistance, and is indicated in Fig. 9-17. The difference between Vcb and V c is due to the ohmic drop across the body resistances of the transistor, particularly the base-spreading resistance r»«. The Ebers-Moll Model Equations (9-49) and (9-50) have a simple inter- pretation in terms of a circuit known as the Ebers-Moll model 6 This model & shown in Fig. 9-18 for a p^n-y transistor. We see that it involves two ideal diodes placed back to back with reverse saturation currents — I bo and -lco and two dependent current-controlled current sources shunting the ideal diodes. For a p-nrp transistor, both Jco and I bo are negative, so that —*c° $*. 9-U TRANSJSTOR CHARACTERISTICS / 249 and — I so are positive values, giving the magnitudes of the reverse saturation currents of the diodes. The current sources account for the minority-carrier transport across the base. An application of KCL to the collector node of Fig. 9-18 gives Ic = -a N r S + / = -a N I B + I^ctvr - 1) where the diode current I is given by Eq. (6-26). Since /„ is the magnitude of the reverse saturation, then I a = —lco' Substituting this value of I e into the preceding equation for I c yields Eq. (9-49). This model is valid for both forward and reverse static voltages applied across the transistor junctions. It should be noted that we have omitted the base-spreading resistance from Fig. 9-17 and have neglected the difference between Icbo and lco- Observe from Fig. 9-18 that the dependent current sources can be elimi- nated from this figure provided an = a r = 0. For example, by making the base width much larger than the diffusion length of minority carriers in the base, all minority carriers will recombine in the base and none will survive to reach the collector. For this case the transport factor 0*, and hence also a, will be zero. Under these conditions, transistor action ceases, and we simply have two diodes placed back to back. This discussion shows why it is impossi- ble to construct a transistor by simply connecting two separate (isolated) diodes back to back. Currents as Functions of Voltages We may use Eqs. (9-49) and (9-50) to solve explicitly for the transistor currents in terms of the junction voltages as denned in Fig. 9-17, with the result that ail CO. /„v„iv_ ,\ 1 so Ie = Ic = 1 — a^ai aiflgo ( t vcir r - 1) - ( t v,iv T _ 1) _ 1 — aitai lco ( e r,iv r _ !) ( e vciv r _ j) (9-51) (9-52) 1 — an (xi v 1 — atfai These two equations were first presented by Ebers and Moll, 6 and are identical with Eqs. (9-19) and (9-21), derived from physical principles in Sec. 9-5. In eo~ k V <~ J Fig. 9-18 The Ebers-Moll model for a p-n-p transistor. 250 / ELECTRONIC DEVICES AND CIRCUITS that section it is verified that the coefficients oliIco S«c. 9-1 4 an = 1 — ayai and Oil ■ T- 1 — ayai are equal. Hence the parameters aw, a*, Icoj and Iso are not independent, but are related by the condition ailco = owlso (9-53) Manufacturer's data sheets often provide information about ow, lco, and Iso, so that «i may be determined. For many transistors Iso lies in the range 0.51 co to I Co- Since the sum of the three currents must be zero, the base current is given by /*= -(I* + Ic) (9-54) Voltages as Functions of Currents We may solve explicitly for the junction voltages in terms of the currents from Eqs. (9-51) and (9-52), with the result that = v T in(i- lB + aiIc ) \ iso / Iso c + a^h \ lco / (9-55) (9-56) We now derive the analytic expression for the common-emitter charac- teristics of Fig. 9-8. The abscissa in this figure is the collector-to-emitter voltage Vcs = Vs — Vc for an n-p-n transistor and is Vcs = V c — V E for a p-n-p transistor (remember that V c and Vs are positive at the p side of the junction). Hence the common-emitter characteristics are found by subtract- ing Eqs. (9-55) and (9-56) and by eliminating I s by the use of Eq. (9-54). The resulting equation can be simplified provided that the following inequali- ties are valid: I B » Iso and I B » Ico/a N . After some manipulations and by the use of Eqs. (9-42) and (9-53), we obtain (except for very small values of h) where V C M = ± V T In ft'- ai Pi In . _l u PU (9-57) 1 — aj and 0N = P 1 - a Note that the + sign in Eq. (9-57) is used for an n-p-n transistor, and the — sign for a p-n-p device. For a p-n-p germanium-type transistor, at Ic = " Vcs = — Vt In (1/aj), so that the common-emitter characteristics do not po>& 8 through the origin. For a T ■ 0.78 and V T = 0.026 V, we have V C s = -6 mV S*c 9- J 5 TRANSISTOR CHARACTERISTICS / 251 pjg.9-19 The common- emitter output character- istic for a p-n-p transistor as obtained analytically. k I, f~ J 0.9/3 T / * = / , ! . , . — i 100 f r in - nfaooe 0.1S 0.2 03 04 OS -v cs ,v at room temperature. This voltage is so small that the curves of Fig. 9-8 look as if they pass through the origin, but they are actually displaced to the right by a few millivolts. If I c is increased, then Vcs rises only slightly until Ic/Ib approaches 0. For example, even for Ic/Is = 0.9/3 = 90 (for = 100), v „.- amt1n y™±*m. -0.15 V This voltage can barely be detected at the scale to which Fig. 9-8 is drawn, and hence near the origin it appears as if the curves rise vertically. However, note that Fig. 9-11 confirms that a voltage of the order of 0.2 V is required for Ic to reach 0.9 of its maximum value. The maximum value of Ic/Ib is p, and as this value of I c /Is is approached, Vcs -* - * , Hence, as Ic/Ib increases from 0.9/3 to 0, |V C u| increases from 0.15 V to infinity. A plot of the theoretical common-emitter characteristic is indicated in Fig. 9-19. We see that, at a fixed value of Vcs, the ratio Ic/Ib is a constant. Hence, for equal increments in I B , we should obtain equal incre- ments in Ic at a given Vcs- This conclusion is fairly well satisfied by the curves in Fig. 9-8. However, the Is = curve seems to be inconsistent since, for a constant Ic/Ib, this curve should coincide with the I c = axis. This discrepancy is due to the approximation made in deriving Eq. (9-57), which is not valid for I B = 0. The theoretical curve of Fig. 9-19 is much flatter than the curves of Fig. 9-8 because we have implicitly assumed that a N is truly constant. As already Pointed out, a very slight increase of <xn with V C s can account for the slopes of the common-emitter characteristic. 9-1 5 ANALYSIS OF CUTOFF AND SATURATION REGIONS ^t us now apply the equations of the preceding section to find the dc currents ^d voltages in the grounded-emitter transistor. 252 / ELECTRONIC DEVICES AND CIRCUITS Sec. 9-15 The Cutoff Region If we define cutoff as we did in Sec. 9-9 to mean zero emitter current and reverse saturation current in the collector, what emitter- junction voltage is required for cutoff? Equation (9-55) with I s = and Ic m Ico becomes (9-58) Ve = V T In f 1 - ^\ = V T In (1 - a N ) where use was made of Eq. (9-53). At 25°C, V T = 26 mV, and for a N = 0.98, V B = — 100 mV. Near cutoff we may expect that a N may be smaller than the nominal value of 0.98. With a N - 0.9 for germanium, we find that Ve = —60 mV. For silicon near cutoff, a* « 0, and from Eq. (9-58), Ve *» Vt In 1 = V. The voltage Ve is the drop from the p to the n side of the emitter junction. To find the voltage which must be applied between base and emitter terminals, we must in principle take account of the drop across the base-spreading resistance r»' in Fig. 9-17. If r& = 100 12 and Ico = 2 mA, then IcoTbb' = 0.2 mV, which is negligible. Since the emitter current is zero, the potential Ve is called the floating emitter -potential. The foregoing analysis indicates that a reverse bias of approximately 0.1 V (0 V) will cut off a germanium (silicon) transistor. It is interesting to determine what currents will flow if a larger reverse input voltage is applied. Assuming that both Ve and Vc are negative and much larger than Vt, so that the exponentials may be neglected in comparison with unity, Eqs. (9-49) and (9-50) become U - - ai I c + I bo (9-59) Ic — —onIe + h Solving these equations and using Eq. (9-53), we obtain 7 Ico{\ — a/) T Ieo(1 — <*n) Ic = — : IB = ~ 1 — asoti 1 — CtNCtI (9-60) Since (for Ge) a N « 1, Ic « Ico and Is «* 0. Using a N = 0.9 and at = 0.5, then Ic = /co(0.5O/0.55) = 0.91/co and I E = Imo(0. 10/0.55) - 0.18/ fiO and represents a very small reverse current. Using ai «» and as =* (for Si), we have that I c *■ Ico and Is « I so- Hence, increasing the magnitude of the reverse base-to-emitter bias beyond cutoff has very little effect (Fig. 9-20) on the very small transistor currents. Short-circuited Base Suppose that, instead of reverse-biasing the emitter junction, wc simply short the base to the emitter terminal. The currents which now flow are found by setting V& = and by neglecting exp (Vc/Vr) in the Ebers-Moll equations. The results are Ico Ic = 1 — asati mli and Ie = —ail ail ess (9-61) where Ices represents the collector current in the common-emitter configu- ration with a short-circuited base, If (for Ge) ow = 0.9 and m = 0.5, then Ices is about 1.8/co and I R - -0.91/ co . If (for Si) a v * and at « 0, then Sic. 9-1 S TRANSISTOR CHARACTERISTICS / 253 less m I°° an( * I* m 0* Hence, even with a short-circuited emitter junction, the transistor is virtually at cutoff (Fig. 9-20) . Open-circuited Base If instead of a shorted base we allow the base to "float," so that Ib — 0, the cutoff condition is not reached. The collector current under this condition is called Icbo, and is given by I ceo - r^- 0-62) 1 — a N It is interesting to find the emitter-junction voltage under this condition of a floating base. From Eq. (9-55), with I B - —I c , and using Eq. (9-53), a N (l — ai) = V T In [ 1 + ai{\ — an) (9-63) I For a N - 0.9 and o/ = 0.5 (for Ge), we find V B = +60 mV. For a N m 2ai *= (for Si), we have V B « V T In 3 = +28 mV. Hence an open-circuited base represents a slight forward bias. The Cutin Voltage The volt-ampere characteristic between base and emitter at constant collector-to-emitter voltage is not unlike the volt-ampere characteristic of a simple junction diode. When the emitter junction is reverse-biased, the base current is very small, being of the order of nano- amperes or microamperes for silicon and germanium, respectively. When the emitter junction is forward-biased, again, as in the simple diode, no appreciable base current flows until the emitter junction has been forward-biased to the extent where \Vbe\ > \V y \, where V 7 is called the cutin voltage. Since the col- lector current is nominally proportional to the base current, no appreciable collector current will flow until an appreciable base current flows. Therefore a plot of collector current against base-to-emitter voltage will exhibit a cutin voltage, just as does the simple diode. Such plots for Ge and Si transistors are shown in Fig. 9-20o and b. In principle, a transistor is in its active region whenever the base-to- emitter voltage is on the forward-biasing side of the cutoff voltage, which occurs at a reverse voltage of 0.1 V for germanium and V for silicon. In effeet, however, a transistor enters its active region when V B b > V y . We may estimate the cutin voltage V y in a typical case in the following ttianner: Assume that we are using a transistor as a switch, so that when the switch is on it will carry a current of 20 mA. We may then consider that the cutin point has been reached when, say, the collector current equals 1 percent of the maximum current or a collector current I c = 0.2 mA. Hence V% is the value of V B given in Eq. (9-55), with I B = -(Ic + /*) * -Ic = -0.2 mA. Assume a germanium transistor with ai = 0.5 and I so = 1 mA. Since at room temperature V T - 0.026 V, we obtain from Eq. (9-55) 7, - (0.026,(2.30) log [l + <» X 10^1-0-5) j . 0.12 V 254 / ELECTRONIC DEVICES AND CIRCUITS Sec. 9-TS T — T " •*C — *CES— -, 1 — I *c — *t ' I If, ~. /^JPrt — 1 - a N (Open -circuit base) -as -0.2 -o.i o |v T =o.i a2 v;=o.3 (a) 0.06 Active Cutoff- Cutin- regton *C — IcES^^i CES-^^CO- Silicon £ Ic = I CEO * I\ CEO ^ A CO -0.3 -0.2 -0.1 (*») Cutoff A 0.03 0.1 0.2 t Open-circuit base 0.3 0.4 V r =0.S 0.6 W*0.7 V aB ,V Cutin - Acttve region " -Saturation Fig. 9-20 Plots of collector current against base-to-emitter voltage for (a) germanium and (b) silicon transistors. (/ c is not drawn to scale.) ,. $ac- 9-15 p ig 9-21 Plot of collector current against base-to- emitter voltage for various temperatures for the type 2N337 silicon transistor. (Courtesy of Transitron Electronic Corporation.) TRANSISTOR CHARACTERISTICS / 255 111 B s 7 ti 5 4 3 , / / 180"C / / 100° 7 J 125 c l 2 1 h S9°C .y m* 1 *■ J - / ai 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Input voltage V BK , V If the switch had been called upon to carry 2 mA rather than 20 raA, a cutin voltage of 0.06 V would have been obtained. For a silicon transistor with at = 0.5 and I bo = 1 nA and operating at 20 rnA (2 mA) we obtain from Eq. (9-55) that V y = 0.6 V (0.3 V). Hence, in Fig. 9-20 the following reason- able values for the cutin voltages V y are indicated: 0.1 V for germanium and 0.5 V for silicon. Figure 9-21 shows plots, for several temperatures, of the collector current as a function of the base-to-emitter voltage at constant collector-to-emitter voltage for a typical silicon transistor. We see that a value for V y of the order of 0.5 V at room temperature is entirely reasonable. The temperature dependence results from the temperature coefficient of the emitter-junction diode. Therefore the lateral shift of the plots with change in temperature and the change with temperature of the cutin voltage V y are approximately -2.5 mV/°C [Eq. (6-39)]. The Saturation Region Let us consider the 2N404 p-n-p germanium transistor operated with I c - -20 mA, I B = —0.35, and I B = +20.35 mA. Assume the following reasonable values: Leo — —2.0 pA, I bo = —1.0 mA, and «.v - 0.99. From Eq. (9-53), «r = 0.50. From Eqs. (9-55) and (9-56), we calculate that, at room temperature, V B = (0.026) (2.30) log and V c = (0.026) (2.30) log *or a p-n-p transistor, V CB = Vc-V s = 0.11 [- [- 20.35 - (0.50) (20)' -lO" 3 -20 + 0.99(20.35) -(2)(10- s ) = 0.24 V = 0.11 V 0.24 « -0.13 V 256 / ELECTRONIC DEVICES AND CIRCUITS See. 9-16 Taking the voltage drop across rw (~100 fl) into account (Fig. 9-17), Vcb - V c - IbTw = 0.11 -h 0.035 = 0.15 V and Vbb = I B r w - Vg = -0.035 - 0.24 m 0.28 V Note that the base-spreading resistance does not enter into the calcu- lation of the collector-to-emitter voltage. For a diff used-junction transistor the voltage drop resulting from the collector-spreading resistance may be sig- nificant for saturation currents. If so, this ohmic drop can no longer be neg- lected, as we have done above. For example, if the collector resistance is 5 0, then with a collector current of 20 mA, the ohmic drop is 0.10 V, and \Vcb\ increases from 0.13 to 0.23 V. 9-16 TYPICAL TRANSISTOR-JUNCTION VOLTAGE VALUES Quite often, in making a transistor-circuit calculation, we are beset by a compli- cation when we seek to determine the transistor currents. These currents are influenced by the transistor-junction voltages. However, to determine these junction voltages, we should first have to know the very currents we seek to determine. A commonly employed and very effective procedure to overcome this problem arises from the recognition that certain of the transistor-junction voltages are ordinarily small in comparison with externally impressed voltages, the junction voltages being in the range of only tenths of volts. We may therefore start the calculation by making the first-order approximation that these junction voltages are all zero. On this basis we calculate a first-order approximation of the current. These first-order currents are now used to determine the junction voltages either from transistor characteristics or from the Ebers-Moll equations. The junction voltages so calculated are used to determine a second-order approximation of the currents, etc. As a matter of practice, it ordinarily turns out that not many orders are called for, since the successive approximations converge to a limit very rapidly. Furthermore, a precise calculation is not justifiable because of the variability from sample to sample of transistors of a given type. The required number of successive approximations may be reduced, or more importantly, the need to make successive approximations may usually be eliminated completely by recognizing that for many low- and medium- power transistors, over a wide range of operating conditions, certain transistor- junction voltages lie in a rather narrow range, and may be approximated by the entries in Table 9-2. This table lists the collector-to-emitter saturation voltage [FcaCsat)], the base-to-emitter saturation voltage [Kb* (sat) s V,], the base-to-emitter voltage in the active region [Fes (active)], at cutin [F Bif (cutin) ■ V y \, and at cutoff [F^fcutoff)]. The entries in the table are appropriate for an n-p-n transistor. For a p-n-p transistor the signs of all entries should be reversed. Observe that the total range of V B s between cutin S-c. 9-17 TRANSISTOR CHARACTERISTICS / 257 and saturation is rather small, being only 0.2 V. The voltage Vbb (active) has been located somewhat arbitrarily, but nonetheless reasonably, at the mid- point of the active region in Fig. 9-20. Of course, particular cases will depart from the estimates of Table 9-2. But it is unlikely that the larger of the numbers will be found in error by more than about 0.1 V or that the smaller entries will be wrong by more than about 0.05 V. In any event, starting a calculation with the values of Table 9-2 may well make further approximations unnecessary. TABLE 9-2 Typica 1 rir-p-n transistor- unction voltages at 25°Cf VcMisat) VWBftt) = V, Fstf(active) Vbe (cutin) ■ V y Vbe(cuU>8) Si Ge 0.3 0.1 0.7 0.3 0.8 0.2 0.5 0.1 0.0 -0.1 t The temperature variation of these voltages is discussed in Sec. 9-15. Finally, it should be noted that the values in Table 9-2 apply to the intrinsic junctions. The base terminal-to-emitter voltage includes the drop across the base-spreading resistance *v. Ordinarily, the drop r&tjt is small enough to be neglected. If, however, the transistor is driven very deeply into saturation, the base current I B may not be negligible, but we must take Vbb = V. + IbTw 9-17 TRANSISTOR SWITCHING TIMES When a transistor is used as a switch, it is usually made to operate alternately in the cutoff condition and in saturation. In the preceding sections we have computed the transistor currents and voltages in the cutoff and saturation states. We now turn our attention to the behavior of the transistor as it makes a transition from one state to the other. We consider the transistor circuit shown in Fig. 9-22a, driven by the pulse waveform shown in Fig. 9-226. fhis waveform makes transitions between the voltage levels F s and Vi. At v * the transistor is at cutoff, and at Vj. the transistor is in saturation. The •nput waveform v, is applied between base and emitter through a resistor R„ . 1Cfl "lay be included explicitly in the circuit or may represent the output "^Pedance of the source furnishing the waveform. The response of the collector current %c to the input waveform, together 'tn its time relationship to that waveform, is shown in Fig. 9-22c. The cur- nt does not immediately respond to the input signal. Instead, there is a ® ,a y, and the time that elapses during this delay, together with the time quired for the current to rise to 10 percent of its maximum (saturation) 258 / ELECTRONIC DEVICES AND CIRCUITS Sec 9.17 € | t Vy v % ic 0.9I C s o-A/W T Vcc (6) v, T f (a ^ ~7t \ . (c) 0.1I C8 M 1 \^ ' ' b i i i t., i i 1 t t H"-*GN-M U *— *OFF -*t Fig. 9-22 The pulse waveform in (b\ drives the transistor in (a) from cutoff to saturation and back again, (c) The collector-current response to the driving input pulse. value Ics ** V C c/Rl, is called the delay time t d . The current waveform has a nonzero rise time U, which is the time required for the current to rise from 10 to 90 percent of Ics- The total turn-on time £ n is the sum of the delay and rise time, *on — U -\- U. When the input signal returns to its initial state at / = T, the current again fails to respond immediately. The interval %vhich elapses between the transition of the input waveform and the time when ic has dropped to 90 percent of Ics is called the storage time t„ The storage interval is followed by the fall time t f , which is the time required for ic to fall from 90 to 10 percent of Ics- The tumoff time /off is defined as the sum of the storage and fall times, £off = t s + t f . We shall consider now the physical reasons for the existence of each of these times. The actual calculation of the time intervals (td, t r , I,, and tf) is complex, and the reader is referred to Ref. 11- Numerical values of delay time, rise time, storage time, and fall time for the Texas Instruments n-p-n epitaxial planar silicon transistor 2 N 3830 under specified conditions can be as low as id = 10 nsec, t r = 50 nsec, t, = 40 nsec, and t f = 30 nsec. The Delay Time Three factors contribute to the delay time: First, when the driving signal is applied to the transistor input, a nonzero time is required to charge up the emitter-junction transition capacitance so that the transistor may be brought from cutoff to the active region. Second, even when the transistor has been brought to the point where minority earners have begun to cross the emitter junction into the base, a time interval Jl required before these carriers can cross the base region to the collector junction Sec- 9-17 TRANSISTOR CHARACTERISTICS / 259 j^d be recorded as collector current. Finally, some time is required for the collector current to rise to 10 percent of its maximum. Rise Time and Fall Time The rise time and the fall time are due to the fact that, if a base-current step is used to saturate the transistor or return it from saturation to cutoff, the transistor collector current must traverse the active region. The collector current increases or decreases along an exponential curve whose time constant r P can be shown 11 to be given by r r = h FE (C c Rc + 1/W), where C c is the collector transition capacitance and wr is the radian frequency at which the current gain is unity (Sec. 13-7). Storage Time The failure of the transistor to respond to the trailing edge of the driving pulse for the time interval t t (indicated in Fig. 9-22c) results from the fact that a transistor in saturation has a saturation charge of excess minority carriers stored in the base. The transistor cannot respond until this saturation excess charge has been removed. The stored charge density in the base is indicated in Fig. 9-23 under various operating conditions. The concentration of minority carriers in the base region decreases linearly from p n <>e v * lv T at x = to -p w fi VclVT at x ■» W, as indicated in Fig. 9-246. In the cutoff region, both Vb and Vc are negative, and p n is almost zero everywhere. In the active region, Vs is positive and Vc negative, so that p n is large at i = and almost zero at x = W. Finally, in the saturation region, where V s and Vc are both positive, p n is large everywhere, and hence a large amount of minority-carrier charge is stored in the base. These densities are pictured in Fig. 9-23. Consider that the transistor is in its saturation region and that at i = T an input step is used to turn the transistor off, as in Fig. 9-22. Since the turnoff process cannot begin until the abnormal carrier density (the heavily shaded area of Fig. 9-23) has been removed, a relatively long storage delay time t, may elapse before the transistor responds to the turnoff signal at the input. In an extreme case this storage-time delay may be two or three times Density of minority '9- 9-23 Minority-carrier con- centration in the base for cutoff, a ctive, a n{ j saturation conditions of °Pe ration. Emitter Collector jc = x= W 260 / aecrnoN/c devices and circuits S«e. ?-T« Emitter junction Collector junction t V t 1 Pa \ \ A.** 7 * *-JC*J X- = r* Fig, 9-24 The minority-carrier density in the base region. (6) the rise or fall time through the active region. In any event, it is clear that, when transistor switches are to be used in an application where speed is at a premium, it is advantageous to restrain the transistor from entering the saturation region. 9-18 MAXIMUM VOLTAGE RATING 10 Even if the rated dissipation of a transistor is not exceeded, there is an upper limit to the maximum allowable collector-junction voltage since, at high voltages, there is the possibility of voltage breakdown in the transistor. Two types of breakdown are possible, avalanche breakdown, discussed in Sec. 6-12, and reach-through, discussed below. Avalanche Multiplication The maximum reverse-biasing voltage which may be applied before breakdown between the collector and base terminals of the transistor, under the condition that the emitter lead be open-circuited, is represented by the symbol BV C bo- This breakdown voltage is a characteristic of the transistor alone. Breakdown may occur because of avalanche multi- plication of the current Ico that crosses the collector junction. As a result of this multiplication, the current becomes MI C o, in which M is the factor by which the original current Ico is multiplied by the avalanche effect. (We neglect leakage current, which does not flow through the junction and is there- fore not subject to avalanche multiplication.) At a high enough voltage namely, BVcuo, the multiplication factor M becomes nominally infinite, and the regiou of breakdown is then attained. Here the current rises abruptly) and large changes in current accompany small changes in applied voltage. The avalanche multiplication factor depends on the voltage Vcb between collector and base. We shall consider that M m * (9-6*) 1 - {Vcb/BVcboY Equation (9-64) is employed because it is a simple expression which g lVC ^ a good empirical fit to the breakdown characteristics of many transistor typ 68 * TRANSISTOR CHARACTERISTICS / 261 Xbe parameter n is found to be in the range of about 2 to 10, and controls the sharpness of the onset of breakdown. If a current Is is caused to flow across the emitter junction, then, neglect- ing the avalanche effect, a fraction otls, where a is the common-base current gain, reaches the collector junction. Taking multiplication into account, Ic has the magnitude Malg. Consequently, it appears that, in the presence of avalanche multiplication, the transistor behaves as though its common-base current gain were Ma. An analysis 10 of avalanche breakdown for the CE configuration indicates that the coHector-to-emitter breakdown voltage with open-circuited base, desig- nated BVcso, is (9-65) BVcBO — BVcBO ■yjr — For an n-p-n germanium transistor, a reasonable value for n, determined experimentally, is 7i = 6. If we now take h?s m 50, we find that BVcxo - 0.52B Vcbo so that if BVcbo = 40 V, BVcso is about half as much, or about 20 V. Ideal- ized common-emitter characteristics extended into the breakdown region are shown in Fig. 9-25. If the base is not open-circuited, these breakdown char- acteristics are modified, the shapes of the curves being determined by the base-circuit connections. In other words, the maximum allowable collector- to-emitter voltage depends not only upon the transistor, but also upon the circuit in which it is used. Reach-through The second mechanism by which a transistor's usefulness may be terminated as the collector voltage is increased is called punch-through, or reach-through, and results from the increased width of the collector-junction transition region with increased collector-junction voltage (the Early effect). The transition region at a junction is the region of uncovered charges on both sides of the junction at the positions occupied by the impurity atoms. As the voltage applied across the junction increases, the transition region penetrates deeper into the collector and base. Because neutrality of charge must be maintained, the number of uncovered charges on each side remains '9- 9-25 Idealized common- •""Her characteristics tended into the breakdown r *9ion. CBO V CM 262 / RfCTRONlC DEVICES AND C/RCU/TS Sec. 9- ? 8 equal. Since the doping of the base is ordinarily substantially smaller than that of the collector, the penetration of the transition region into the base is larger than into the collector (Fig. 9-2c). Since the base is very thin, it is possible that, at moderate voltages, the transition region will have spread com- pletely across the base to reach the emitter junction. At this point normal transistor action ceases, since emitter and collector are effectively shorted. Punch-through differs from avalanche breakdown in that it takes place at a fixed voltage between collector and base, and is not dependent on circuit configuration. In a particular transistor, the voltage limit is determined by punch-through or breakdown, whichever occurs at the lower voltage. REFERENCES 1. Shockley, W.: The Theory of p-n Junctions in Semiconductors and p-n Junction Transistors, Bell System Tech. J., vol. 28, pp. 435-489, July, 1949. Middlebrook, R. D.: "An Introduction to Junction Transistor Theory," pp. 115- 130, John Wiley & Sons, Inc., New York, 1957. Terman, F. E.: "Electronic and Radio Engineering," 4th ed., pp. 747-760, McGraw- Hill Book Company, New York, 1955. Moll, J. L.: "Junction Transistor Electronics," Proc. IRE, vol. 43, pp. 1807-1819, December, 1955. 2. Phillips, A. B.: "Transistor Engineering," pp. 157-159, McGraw-Hill Book Com- pany, New York, 1962. 3. Ref. 2, chap. 1. 4. Texas Instruments, Inc.: J. Miller (ed.), "Transistor Circuit Design," chap. 1, McGraw-Hill Book Company, New York, 1963. 5. Ebers, J. J., and J. L. Moll: Large-signal Behavior of Junction Transistors, Proc, IRE, vol. 42, pp. 1761-1772, December, 1954. 6. Sah, C. T., R. N. Noyce, and W. Shockley : Carrier-generation and Recombination in p-n Junctions and p-n Junction Characteristics, Proc. IRE, vol. 45, pp. 1228— 1243, September, 1957. Pritchard, R. L. : Advances in the Understanding of the P-N Junction Triode, Proc. IRE, vol. 46, pp. 1130-1141, June, 1958. 7. Ref. 2, pp. 236-237. 8. Early, J. M. : Effects of Space-charge Layer Widening in Junction Transistors, Proc- IRE, vol. 40, pp. 1401-1406, November, 1952. 9. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," p. l 9 ^' McGraw-Hill Book Company, New York, 1965. 10. Ref. 9, chap. 6. 11. Ref. 9, chap. 20. "Transistor Manual," 7th ed., pp. 149-169, General Electric Co., Syracuse, N.**» 1964. TRANSISTOR BIASING AND THERMAL STABILIZATION This chapter presents methods for establishing the quiescent oper- ating point of a transistor amplifier in the active region of the charac- teristics. The operating point shifts with changes in temperature T because the transistor parameters (jS, Ico, etc.) are functions of T. A criterion is established for comparing the stability of different biasing circuits. Compensation techniques are also presented for quiescent-point stabilization. 10-1 THE OPERATING POINT From our discussion of transistor characteristics in Sees. 9-8 to 9-10, it is clear that the transistor functions most linearly when it is con- strained to operate in its active region. To establish an operating point in this region it is necessary to provide appropriate direct poten- tials and currents, using external sources. Once an operating point Q is established, such as the one shown in Fig. 9- 15a, time- varying excur- sions of the input signal (base current, for example) should cause an output signal (collector voltage or collector current) of the same wave- form. If the output signal is not a faithful reproduction of the input signal, for example, if it is clipped on one side, the operating point is unsatisfactory and should be relocated on the collector characteristics. The question now naturally arises as to how to choose the operating point. In Fig. 10-1 we show a common-emitter circuit (the capacitors have negligible reactance at the lowest frequency of operation of this circuit). Figure 10-2 gives the output characteristics of the transistor used in Fig. 10-1. Note that even if we are free to choose R c , Rl, Rb, and Vcc, we may not operate the transistor everywhere in the active region because the various transistor ratings limit the range of useful 263 2M / aecreoNJC devices and circuits S»e. 70.J Fig. 10-1 The fixed-bias circuit. Signal input, v, operation. These ratings (listed in the manufacturer's specification sheets) are maximum collector dissipation P c (max), maximum collector voltage Fc(max), maximum collector current J c (max), and maximum emitter-to-base voltage V« B (max). Figure 10-2 shows three of these bounds on typical col- lector characteristics. V c V cc Vc(niax) V c e^ Fig. 10-2 Common-emitter collector characteristics; ac and dc load lines. TRANSISTOR BIASING AND THERMAL STABILIZATION / 265 The DC and AC Load Lines Let us suppose that we can select R e so that the dc load line is as drawn in Fig. 10-2. If R L = » and if the input signal (base current) is large and symmetrical, we must locate the operating point Qi ft t the center of the load line. In this way the collector voltage and current may vary approximately symmetrically around the quiescent values Vc and j c respectively. If Ri, ^ », however, an ac load line (Sec. 7-12) correspond- ing to a load of R' L = Rl\\R{ must be drawn through the operating point Qi. This ac load line is indicated in Fig. 10-2, where we observe that the input signal may swing a maximum of approximately 40 nA around Q\ because, if the base current decreases by more than 40 ^A, the transistor is driven off. If a larger input swing is available, then in order to avoid cutoff during a part of the cycle, the quiescent point must be located at a higher current. For example, by simple trial and error we locate Qt on the dc load line such that a line with a slope corresponding to the ac resistance R' L and drawn through Qi gives as large an output as possible without too much distortion. In Fig. 10-2 the choice of Qj allows an input peak current swing of about 6*»A. The Fixed- bias Circuit The point Q% can be established by noting the required current /as in Fig. 10-2 and choosing the resistance fit in Fig. 10-1 so that the base current is equal to I si. Therefore T Vcc — V B B t Ib = — m — = /bs (10-1) The voltage V# E across the forward-biased emitter junction is (Table 0-2, page 257) approximately 0.2 V for a germanium transistor and 0.6 V for a silicon transistor in the active region. Since Vcc is usually much larger than Vbb, we have Ib « Vcc R> (10-2) The current Ib is constant, and the network of Fig. 10-1 is called the fixed-bias circuit. In summary, we see that the selection of an operating point v depends upon a number of factors. Among these factors are the ac and dc loads on the stage, the available power supply, the maximum transistor stings, the peak signal excursions to be handled by the stage, and the toler- able distortion. l0 "2 BIAS STABILITY 111 the preceding section we examined the problem of selecting an operating P°mt Q on the load line of the transistor. We now consider some of the Problems of maintaining the operating point stable. Let us refer to the biasing circuit of Fig. 10-1. In this circuit the base 266 / ElECTRON/C DEVICES AND CJRCUJTS la Sec. JO-2 fig, 10-3 Graphs showing the collector characteristics for two transistors of the same type. The dashed characteristics are for a transistor whose is much larger than that of the transistor represented by the solid curves, current I s is kept constant since I B « V C c/Rb. Let us assume that the tran- sistor of Fig. 10-1 is replaced by another of the same type. In spite of the tremendous strides that have been made in the technology of the manufacture of semiconductor devices, transistors of a particular type still come out of pro- duction with a wide spread in the values of some parameters. For example, Fig. 9-12 shows a range of h FB » of about 3 to 1. To provide information about this variability, a transistor data sheet, in tabulating parameter values, often provides columns headed minimum, typical, and maximum. In Sec. 9-8 we see that the spacing of the output characteristics will increase or decrease (for equal changes in I B ) as increases or decreases. In Fig. 10-3 we have assumed that is greater for the replacement transistor of Fig. 10-1, and since I B is maintained constant at I B % by the external biasing circuit, it follows that the operating point will move to Q 2 . This new oper- ating point may be completely unsatisfactory. Specifically, it is possible for the transistor to find itself in the saturation region. We now conclude that maintaining I H constant will not provide operating-point stability as changes. On the contrary, I 1{ should be allowed to change so as to maintain I c and Van constant as changes. Thermal Instability A second very important cause for bia.s instability is a variation in temperature. In Sec. 9-9 we note that the reverse saturation current 7 C of changes greatly with temperature. Specifically, Ico doubles for every 10°C rise in temperature. This fact may cause considerable practical difficulty in using a transistor as a circuit element. For example, the collector current I c causes the collector-junction temperature to rise, which in turn increases I C q. As a result of this growth of I C o, Ic will increase [Eq. (9-43)1, which may further increase the junction temperature, and consequently Ico- f Throughout this chapter Icbo is abbreviated Ico (Sec. 9-9). s«. JO-2 TRANSISTOR BIASING AND THERMAl STABIUZATJON / 267 It i s possible for this succession of events to become cumulative, so that the atings of the transistor are exceeded and the device burns out. Even if the drastic state of affairs described above does not take place, it is sible f or a transistor which was biased in the active region to find itself in the saturation region as a result of this operating-point instability (Sec. 10-10). To see how this may happen, we note that if I B = 0, then, from Eq. (9-38), j _, [ c0 /(l — a). As the temperature increases, Ico increases, and even if W e assume that a remains constant (actually it also increases), it is clear that the j B = line in the CE output characteristics will move upward. The characteristics for other values of I B will also move upward by the same amount (provided that remains constant), and consequently the operating point will move if I B is forced to remain constant. In Fig. 10-4 we show the output characteristics of the 2N708 transistor at temperatures of +25 and -fl00°C. This transistor, used in the circuit of Fig. 10-1 with V C c = 10 V, R t = 250 12, Rt = 24 K, operates at Q with I B = (10 - 0.6)/24 ** 0.4 mA. Hence it would find itself almost in saturation at a temperature of -l-100 o C even though it would be biased in the middle of its active region at +25°C. The Stability Factor S From our discussion so far we see that in biasing a transistor in the active region we should strive to maintain the operating point stable by keeping I c and V C s constant. The techniques normally used to do so may be classified in. two categories: (1) stabilization techniques and (2) compensation techniques. Stabilization techniques refer to the use of resis- tive biasing circuits which allow I B to vary so as to keep I c relatively constant with variations in I C o, 0, and V BB - Compensation techniques refer to the use of temperature-sensitive devices such as diodes, transistors, thermistors, etc., e 30 To T = 25°C OS 1 0.6 / £4- / Q 0.2 \ N h = 0mA k < 40 1 4* 30 20 0.6 T= 100° C OS| « CL4_ oJ 0.2 OJ_ /, = 0mA 'N 2 4 6 8 10 02468 10 Collector voltage V CK , V Collector voltage V C£ , V (a) (*>) fig. 10-4 Diffused silicon planar 2N708 n-p-n transistor output CE characteristics for (a) 25°C and (b) 100°C (Courtesy of Falrchild Semiconductor.) 268 / ELECTRONIC DEVICES AND CIRCUITS $*c. 10- j S#- J 0-3 TRANSISTOR BIASING AND THERMAL STABILIZATION / 269 which provide compensating voltages and currents to maintain the operating point constant. A number of stabilization and compensation circuits are pre- sented in the sections that follow. In order to compare these biasing circuits we define a stability factor S as the rate of change of collector current with respect to the reverse saturation current, keeping and V BB constant, or S - dlco Alt Ale (10-3) The larger the value of S, the more likely the circuit is to exhibit thermal instability.! S as defined here cannot be smaller than unity. Other stability factors may also be defined, for example, Ql c /d0 and dI c /dV BB . As we show in Sec. 10-5, however, bias circuits which provide stabilization of I c with respect to Ico will also perform satisfactorily for transistors which have large variations of and V BS with temperature. In the active region the basic relationship between Ic and I B is given by Eq. (9-43), repeated here for convenience: Ic = (1 + 0)lco + &B (10-4) If we differentiate Eq. (10-4) with respect to I c and consider constant with Ic, we obtain 1 - 1 +P-L.R dI * or S = 1+0 1 - mis/dlc) (10-5) (10-6) In order to calculate the factor S for any biasing arrangement, it is only neces- sary to find the relationship between I B and I c and to use Eq. (10-6). For the fixed-bias circuit of Fig. 10-1, I B is independent of Ic [Eq. (10-2)]. Hence the stability factor S of the fixed-bias circuit is <8~ _+ I (10-7) For = 50, £ = 51, which means that Ic increases 51 times as fast as Ico- Such a large value of S makes thermal runaway a definite possibility with this circuit. In the following sections bias-stabilization techniques are presented which reduce the value of S, and hence make Ic more independent of Ico- 10-3 COLLECTOR-TO-BASE BIAS An improvement in stability is obtained if the resistor R b in Fig. 10-1 is returned to the collector junction rather than to the battery terminal. Such t In this sense, 8 should more properly be called an instability factor. connection is indicated in Fig. 10-5o. The physical reason that this circuit is an improvement over that in Fig. 10-1 is not difficult to find. If Ic tends to increase (either because of a rise in temperature or because the transistor has been replaced by another of larger 0), then Vcs decreases. Hence I B also decreases; and as a consequence of this lowered bias current, the collector current is not allowed to increase as much as it would have if fixed bias had been used. We now calculate the stability factor S. From KVL applied to the cir- cuit of Fig. 10-5a, - Vcc + (Ib + Ic)Rc + I B R» + V BS = or In = Vcc - IcRc - Vi Re + Rb (10-8) (10-9) Since V B s is almost independent of collector current (V BB = 0.6 V for Si and 0.2 V for Ge), then from Eq. (10-9) we obtain ■:Ub_ di c R c Re + Rb Substituting Eq. (10-10) in Eq. (10-6), we obtain 0+ 1 S = 1 + 0Rc/{Rc + Rb) (10-10) (10-11) This value is smaller than 0+1, which is obtained for the fixed-bias circuit, and hence an improvement in stability is obtained. Stabilization with Changes in It is important to determine how well the circuit of Fig. 10-5 will stabilize the operating point against variations in 0. r — * — i o VW B <«) 9VC (&) Fig. 10-5 (a) A coll ector-to- base bias circuit, (b) A method of avoiding ac degeneration. 270 / ELECTRONIC DEVICES AND CIRCUITS Sec. I0-3 From Eqs. (10-1) and (10-8) we obtain, after some manipulation, and with » 1, a PiVcc - V BB + (R e + R b )Ico\ (10-12) Ic « ffi* + Rh To make Ic insensitive to we must have 0R c »R b (10-13) The inequality of Eq. (10-13) cannot be realized in all practical circuits. However, note that even if R c is so small that R c = R b /0, the sensitivity to variations in is half what it would be if fixed bias (I B constant) were used. EXAMPLE The transistor in Fig. 10-5 ia a silicon-type 2X708 with /S = 50 V C c = 10 V, and R t = 250 & It is desired that, the quiescent point be approxi- mately at the middle of the load line. Find R b and calculate 5. The output characteristics are shown in Fig. 10-4. Solution Since we may neglect I b compared with F r in R c , we may draw a load line corresponding to 10 V and 250 fi. From the load line shown in Fig. 10-4, we choose the operating point at 1 B = 0.4 mA, I c = 21 mA, and V C k = 4.6 V (at a temperature of +25°C). From Fig. 10-5 we have Rb = Vcs - Vbe __ 4.6 - 0.6 Ib 0.4 = 10 K The stability factor S can now be calculated using Eq, (10-11), or 51 S = 1 + 50 X 0.25/10.25 = 23 which is about half the value found for the circuit of Fig. 10-1. We should note here that the numerical values of R t and R b of this example do not satisfy Eq. (10-13) since 0R C = 12.5 K whereas R b = 10 IC. We should then expect I c to vary with variations in 0, but to a smaller extent than if fixed bias were used. Analysis of the Col lector- to- Base Bias Circuit If the component values are specified, the quiescent point is found as follows: Corresponding to value of I B given on the collector curves, the collector voltage Vcs = I s R b + V BB is calculated. The locus of these corresponding points V CB and I B plotted on the common-emitter characteristics is called the bias curve. The intersec- tion of the load line and the bias curve gives the quiescent point. Alterna- tively, if the collector characteristics can be represented analytically by Eq- (10-4), I c is found directly from Eq. (10-12). A Method for Decreasing Signal-gain Feedback The increased sta- bility of the circuit in Fig. 10-5a over that in Fig. 10-1 is due to the feedback fro" 1 S<* |0-4 TRANSISTOR BIASING AND THERMAL STABILIZATION / 271 the output (collector) terminal to the input (base) terminal via R b . Feedback amplifiers are studied in detail in Chap. 17. The ac voltage gain of such an mplifier is less than it would be if there were no feedback. Thus, if the signal voltage causes an increase in the base current, i c tends to increase, Vcb decreases, and the component of base current coming from R b decreases. Hence the ne t change in base current is less than it would have been if Rb were connected to a fixed potential rather than to the collector terminal. This signal-gain degeneration may be avoided by splitting R b into two parts and connecting the junction of these resistors to ground through a capacitor C, as indicated in Fig. 10-56. At the frequencies under consideration, the reactance of C must be negligible. Note that if the output impedance of the signal source is small compared with the input resistance of the transistor, then the capacitance C is not needed, because any feedback current in R b is bypassed to ground through the signal impedanee and does not contribute to the base current. 10-4 SELF-BIAS, OR EMITTER BIAS If the load resistance R c is very small, as, for example, in a transformer- coupled circuit, then from Eq. (10-11) we see that there is no improvement in stabilization in the collector-to-base bias circuit over the fixed-bias circuit. A circuit which can be used even if there is zero dc resistance in series with the collector terminal is the self-biasing configuration of Fig. 10-6a. The current in the resistance R e in the emitter lead causes a voltage drop which is in the direction to reverse-bias the emitter junction. Since this junction must be forward-biased, the base voltage is obtained from the supply through the RiR 2 network. Note that if R b = J2i||iK*— • 0, then the base-to-ground voltage V BN is independent of Ico- Under these circumstances we may verify Fig. 10-6 (a) A self-biasing circuit, (b) Simplification of the base circuit in (a) by the use of Tbevenin's theorem. 272 / ELECTRONIC DEVICES AND CIRCUITS [Eq. (10-17)] that S - dlc/dlco-* 1. For best stability R t and R t must be kept as small as possible. The physical reason for an improvement in stability with Rb ^ is iq. following: If Ic tends to increase, say, because Ico has risen as a result of an elevated temperature, the current in R t increases. As a consequence of the increase in voltage drop across R„ the base current is decreased. Hence Ic will increase less than it would have had there been no self-biasing resistor R The Stabilization Factor S We now find the analytical expression for the stabilization factor S. Since such a calculation is made under dc or no- signal conditions, the network of Fig. 10-6o contains three independent loops. If the circuit to the left between the base B and ground N terminals in Fig, 10-6a is replaced by its Thevenin equivalent, the two-mesh circuit of Fig. 10-66 is obtained, where V = Rz -\- R\ Rb = RiRi Ri + Ri (10-14) Obviously, Rt, is the effective resistance seen looking back from the base terminal. Kirchhoff's voltage law around the base circuit yields V = I B R> + V BX + (Js + Ic)R. (10-15) If we consider V B s to be independent of I c , we can differentiate Eq. (10-15) to obtain dl* die R. Re + Rt> Substituting Eq. (10-16) in Eq. (10-6) results in S = 1 + 1 + 0RJ(R t + BO - (1 + fi) 1 + R+/R. I + + Rt/R. (10-16) (10-17) Note that S varies between 1 for small R b /R, and 1 + for R b /R t ^> « . Equa- tion (10-17) is plotted in Fig. 10-7 for various values of 0. It can be seen that, for a fixed Rb/R„ S increases with increasing 0. (Therefore stability decreases with increasing 0.) Also note that S is essentially independent of for small S. The smaller the value of Rb, the better the stabilization. We have already noted that even if R b approaches zero, the value of S cannot be reduced below unity. Hence Ic always increases more than Ico- As Rb is reduced while the Q point is held fixed, the current drawn in the RiR? network from the supply Vcc increases. Also, if R t is increased while Rb is held constant, then to operate at the same quiescent currents, the magnitude of Vcc must be increased. 1° either case a loss of power (decreased efficiency) is the disadvantage which accompanies the improvement in stability. In order to avoid the loss of ac (signal) gain because of the feedback caused by R e (Sec. 12-7), this resistance is often bypassed by a large capacitan^ 10-4 100 10 TRANSISTOR BIASING AND THERMAL STABIUZATION / 273 r- B = 100 80 60 40 30 20 10 5 1.0 10 1,000 Rt Fig. 10-7 Stability factor S {Eq. (10-17}] versus R b /R* for the self-bias circuit of Fig. 10-66, with as a parameter. (Courtesy of L. P. Hunter, "Handbook of Semiconductor Electronics," McGraw-Hill Book Company, New York, 1962.) (> 10 mF), so that its reactance at the frequencies under consideration is very small. EXAMPLE Assume that a silicon transistor with jS = 50, Vbb = 0.6 V, Vcc = 22.5 V, and R t = 5,6 K is used in Fig. IO-60. It is desired to establish a Q point at V C s = 12 V, T c - 1.5 raA, and a stability factor S < 3. Find R„ R u and R*. Solution The current in R. ia J c + la ~ Ic Hence, from the collector circuit of Ftg. 10-66, we have Vcc - Vcs 22.5 - 12 R t + R t = rcc rc * = _ if = 7.0 K Ic 1.5 or R t = 7.0-5.6 = 1.4 K From Eq. (10-17) we can solve for R b /R t : 3= 51 1 + R " /R * 51 + Rt/R, We find R b /R t = 2.12 and R b = 2.12 X 1.4 = 2.96 K. If R b < 2.96 K, then 5<3. The base current I B is given by p 50 274 / ELECTRONIC DEVICES AND CIRCUITS We can solve for Ri and R 2 from Eqs. (10-14). We find Ri — Rb Rt — RtV Sec. IO.4 a 0-1 s) V ~" Vce - V From Eqs. (10-15) and (10-18) we obtain V = (0.030) (2.96) + 0.6 + (0.030 + 1.5) (1.4) = 2.83 V 2.83 _ 23.6 X 2.83 _ 22.5 - 2.83 Analysis of the Self-bias Circuit If the circuit component values in Fig. 10-6a are specified, the quiescent point is found as follows ; Kirchhoff s voltage law around the collector circuit yields - Vcc + Ic(R e + R.) + IbR* + Vce = (10-19) If the drop in R e due to In is neglected compared with that due to /<?, then this relationship between Ic and Vcb is a straight line whose slope corresponds to R c + R t and whose intercept at Ic = is V C g = Vcc This load line is drawn on the collector characteristics. If Ic from Eq. (10-19) is substituted in Eq. (10-15), a relationship between Is and Vcb results. Corresponding to each value of Is given on the collector curves, Vcb is calculated and the bias curve is plotted. The intersection of the load line and the bias curve gives the quiescent point. EXAMPLE A silicon transistor whose common-emitter output characteristics are shown in Jig. 10-86 is used in the circuit of Fig. 10-6a, with V C c = 22.5 V, R c = 5.6 K, R t =» 1 K, R 2 = 10 K, and #, = 90 K. For this transistor, /J - 55. (a) Find the Q point, (6) Calculate S. Solution a. From Eqs. (10-14) we have V - "» X 22 ■» - 2.25 V 100 "i, 10 X 90 100 - 9.0 K The equivalent circuit is shown in Fig. 10-8a. The load line corresponding t0 a total resistance of 6.6 K and a supply of 22.5 V is drawn on the collector charac- teristics of Fig. 10-86. Kirchhoff'* voltage law applied to the collector and base circuits, respectively, yields (with V be = 0.6) (10-20) (10-21) -22.5 + 6.6/c + I a + Vcs = 0.6 - 2.25 + Ic + 10.07* = Eliminating 7c from these two equations, we find Vcb = 65.07* +11.6 jl See 10-4 TRANSISTOR BIASING AND THERMAL STABILIZATION / 275 9.0K V CF . -±- 22.5 V i*** v i,+icjsi B 3 - 4 ■3 2 I»* 160 ji n — in? L22— Load line I r-~ "f 40 ' Bias curve — |*" -3 ^— r — l (a) 4 8 12 16 20 I 24 Collector-to-emitter voltage V C s . V 22 - 5 (6) Fig. 10-8 (a) An illustrative example, (b) The intersection of the load line and the bias curve determines the Q point. Values of Vce corresponding to I s = 20, 40, and 60 fiA are obtained from this equation and are plotted in Fig. 10-86. We see that the intersection of the bias curve and the load line occurs at Vcs = 13.3 V, I c = 1.4 mA, and from the bias- curve equation, 7 fl = 26 pA. In many cases transistor characteristics are not available but is known. Then the calculation of the Q point may be carried out as follows: In the active region and for base currents large compared with the reverse saturation current (7« » Too), it follows from Eq. (10-4) that 7c - 01 B (10-22) This equation can now be used in place of the collector characteristics. Since = 55 for the transistor used in this example, substituting I B = 7 c /55 in Eq. (10-21) for the base circuit yields -1.65 + 7c + H7c = 7c = 1.40 mA and . 7 C 1.40 . rtc _ . Ib = 77 = -r- mA - - 25.5 pA 55 55 These values are very close to those found from the characteristics. The collector-to-emitter voltage can be found from Eq. (10-20) and the known values of 7 a and 7 C : -22.5 + 6.6 X 1.40 + 0.026 + Vcs = or Vce - 13.2 V 6. From Eq. (10-17), S = _ 56 (i±^ = 276 / ELECTRONIC DEVICES AND CIRCUITS Sec. TO.j This value is about one-sixth of the stabilization factor for the fixed-bias circuit, which indicates that a great improvement in stability can result if self-bias is used. In the colleetor-to-base bias circuit the value of Rb is determined from the desired quiescent base current, and no control is exercised over the stabiliza- tion factor S. However, in the self- bias circuit, I B and S may be specified independently because these requirements can be satisfied by the proper choice of R e and Rb. For this reason, and because generally lower values of 5 are obtained with the self-bias arrangement, this circuit is more popular than that of Fig. 10-5a. For the sake of simplicity the resistor R 2 is sometimes omitted from Fig. 10-6a. In such a circuit R\ is determined by I B but S cannot be specified as a design parameter. The value of S is calculated from Eq. (10-17), with J2 t replaced by Ri. 10-5 STABILIZATION AGAINST VARIATIONS IN Vbe AND FOR THE SELF-BIAS CIRCUIT In the preceding sections we examine in detail a number of bias circuits which provide stabilization of I c against variations in Ico- There remain to be con- sidered two other sources of instability in Ic, those due to the variation of Vbe and jS with temperature and with manufacturing tolerances in the pro- duction of transistors. We shall neglect the effect of the change of Vce with temperature, because this variation is very small (Sec. 9-10) and because we assume that the transistor operates in the active region, where I c is approxi- mately independent of Vce. However, the variation of Vbe with temperature has a very important effect on bias stability. For a silicon transistor, Vbe w about 0.6 V at room temperature, and for a germanium transistor, it is about 0.2 V. As the temperature increases, \V S e\ decreases at the rate of 2.5 mV/°C for both germanium and silicon transistors (Sec. 9-10). The Transfer Characteristic The output current Ic is plotted in Fig- 10-9 as a function of input voltage for the germanium transistor, type 2N1631. This transfer characteristic for a silicon transistor is given in Fig. 9-21. Each curve shifts to the left at the rate of 2.5 mV/°C (at constant Ic) for increasing temperature. We now examine in detail the effect of the shift in transfer characteristics and the variation of and Ico with temperature. If Fq- (10-15), obtained by applying KVL around the base circuit of the self-bi» s circuit of Fig. 10-6b, is combined with Eq. (10-4), which represents the collector characteristics in the active region, the result is V BB = V + {R b + R.) ^4^ Ico R b + fl.(l + fl) (10-23) Sk. 10-5 TRANSISTOR BIASING AND THERMAl STABILIZATION / 277 6 Fig- 10-9 Transfer characteristic for the 2N1631 germanium p-n-p alloy-type transistor at Vce = -9 V and T A = 25°C. {Courtesy of Radio Corp. of America.) ■ n 100 150 200 Vbe, mV Equation (10-23) represents a load line in the Ic-V a e plane, and is indi- cated in Fig. 10-10. The intercept on the V bb axis is V + V, where r = (R b + r<) £±i ico (Rb + R t )Ico (10-24) since » 1. If at T - Ti (Ti), I c = I CO i (I cot) and = 0i (fit), then V[ m (Rb -f Re)Icoi and V' t « (R b + Re)Icoz- Hence the intercept of the load line on the Vbe axis is a function of temperature because Ico increases with T. The slope of the load line is Rt, + R.(l + |8) and hence |er| increases with T because /J increases with T. The transfer char- acteristic for T = Ti > T t shifts to the left of the corresponding curve for F 'Q. 10-10 Illustrating that the col- lector current varies with tempera- te T because V BK , Ico, and chonge with T, 278 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1 0-5 Sec- 10-5 TRANSISTOR BIASING AND THERMAL STABILIZATION / 279 T — Ti because Vbe (at constant Ic) varies with T as indicated above. The intersection of the load line with the transfer characteristic gives the collector current Ic. We see that J C i > la because Ico, 0, and Vbe all vary with temperature. The Stability Factor 8' The variation of Ic with Vbs is given by the stability factor S', defined by 8' = dV BS where both I C o and are considered constant. From Eq. (10-23) we find -0 S (10-25) S' = R b + R.(l + 0) R b + R e + 1 (10-26) where we made use of Eq. (10-17). We now see from Eq. (10-26) that as we reduce S toward unity, we minimize the change of Ic with respect to both Vbs and Ico- The Stability Factor S" The variation of Ic with respect to is given by the stability factor S", defined by on _ ai c S =W where both Ico and Vbb are considered constant. From Eq. (10-23), 0(V + V - Vbe) Ic - R b + R t (l + 0) (10-27) (10-28) where, from Eq. (10-24), V may be taken to be independent of 0. We obtain, after differentiation and some algebraic manipulation, on _ Mc IcS d0 0(1 + 0) (10-29) A difficulty arises in the use of S" which is not present with S and S\ The change in collector current due to a change in is Ale = 8" A0 - IcS 0(1 + 0) A0 (10-30) where A0 = 2 — 0i may represent a large change in #. Hence it is not cle* r whether to use is 2 , or perhaps some average value of /S in the expressions fo r S" and S. This difficulty is avoided if S" is obtained by taking finite differ- ences rather than by evaluating a derivative. Thus 8" m Ig — la 02 — 01 Ale A0 (10-31) L From Eq. (10-28), we have let = 0* Rb + g.(l + gl) Id 01 Rb + fl.(l + 02) Subtracting unity from both sides of Eq. (10-32) yields Rb + Re hi _ 1 - (§1 _ \\ lev \0i ) R b + Re(l + 02) or IC1S2 " A0 0,(1 + 2 ) (10-32) (10-33) (10-34) where Si is the value of the stabilizing factor S when = 0s as given by Eq. (10-17). Note that this equation reduces to Eq. (10-29) as A0 - 2 — 0i — ► 0. It is clear from Eq. (10-29) that minimizing S also minimizes S". This means that the ratio Rh/R* must be small. From Eq. (10-26) it is seen that, in order to keep 8' small, a large Rb or R„ is required. Hence, in all cases, it is desirable to use as large an emitter resistance R e as practical, and a compromise will usually be necessary for the selection of Rb. In the examples given previously, illustrating how to design a bias net- work, the stability factor S was arbitrarily chosen. Equation (10-34) is of prime importance because it allows us to determine the maximum value of 8 allowed for a given spread of 0. This variation in may be due to any cause, such as a temperature change, a transistor replacement, etc. EXAMPLE Transistor type 2N335, used in the circuit of Fig. l0-6a, may have any value of between 36 and 90 at a temperature of 25°C, and the leakage cur- rent Ico has negligible effect on Ic at room temperature. Find R e , R\, and R 2 subject to the following specifications: R r = 4 K, Fee = 20 V; the nominal bias point is to be at V c e = 10 V, I c = 2 mA; and I c should be in the range 1.75 to 2.25 mA as varies from 36 to 90. Solution From the collector circuit (with I c » I B ), Ic 2 Hence R e m 5 - 4 = 1 K. From Eq. (10-34) we can solve for S 2 . Hence, with Ale = 0.5 mA, la = 1.75 mA, 0, * 36, 2 = 90, and A0 = 54, we obtain 0.5 1 .75 S s 54 36 1 + 90 Si = 17.3 280 / ELECTRONIC DEVICES AND CIRCUITS Sec. 10.6 Substituting S 2 = 17.3, R e = 1 K, and jS* = 90 in Eq. (10-17) yields (17.3)(91 + Ri) = 91(1 + R b ) or R b = 20.1 K From Eq. (10-23), with I c = 1.75 mA, = 36, R b = 20.1 K, R e = L K, Fas = 0.6 V, and Ico = 0, we obtain + R b + g.(l + 0) u / 20.1 + 37 \ FromEqs. (10-18), Vcc 20 R l = R b -£Z = 20.1 X — = 119 K F 3.38 iij — J?iF 119 X 3.38 Vcc - V 20 - 3.38 = 24.2 K 10-6 GENERAL REMARKS ON COLLECTOR-CURRENT STABILITY 1 Stability factors were defined in the preceding sections, which considered the change in collector current with respect to Ico, Vbb, and 0. These stability factors are repeated here for convenience : S = Ale, S f = AV B o„ _ Ale (10-35) Each differential quotient (partial derivative) is calculated with all other parameters maintained constant. If we desire to obtain the total ehange in collector current over a specified temperature range, we can do so by expressing this change as the sum of the individual changes due to the three stability factors. Specifically, by taking the total differential of I c — f(Ico, V B e, 0), we obtain . t die A T , die > it i die . „ OiCO OV BE Op - S Alco + S' AV BB + S" A0 (10-36) If Ale is known, the corresponding change in Vcb can be obtained from the dc load line. We now examine in detail the order of magnitude of the terms of Eq- (10-36) for both silicon and germanium transistors over their entire range of temperature operation as specified by transistor manufacturers. This range usually is —65 to +75°C for germanium transistors and —65 to +175 (■> for silicon transistors. Tables 10-1 and 10-2 show typical parameters of silicon and germanlu^ 1 S«. T0-<5 TRANSISTOR BIASING AND THERMAL STABILIZATION / 281 TABIC 7 0- T Typical silicon transistor parameters TABLE 10-2 Typical germanium transistor parameters transistors, each having the same (55) at room temperature. For Si, I C o is much smaller than for Ge. Note that Ico doubles approximately every 10°C and \V B e\ decreases by approximately 2.5 mV/°C. EXAMPLE For the self-bias circuit of Fig. 10-6a, R t = 4.7 K, R b = 7.75 K, and Rb/Re = 1.65. The collector supply voltage and R c are adjusted to establish a collector current of 1.5 mA. a. Determine the variation of I c in the temperature range of —65 to + 175°C when the silicon transistor of Table 10-1 is used. h. Repeat (a) for the range —65 to +75°C when the germanium transistor of Table 10-2 is used. Solution a. Since R t , R bl and are known, the stability factor S can be deter- mined at -f 25°C from Eq, (10-17) : S(25°C) - (1 + 0)(l + Rt/R.) (56) (2.65) = 2.57 1 + + R b /R, 56 + 1.65 Similarly, $' at +25°C can be determined from Eq. (10-26) : S'(25°C) - -S R b + R t l+0 ~(S)©— " '-*" The values of S and 5' are valid for either a silicon or a germanium transistor operating in the circuit of Fig. 10-6a. Since the stability factor S" contains both Pi and jSj, it must be determined individually for each transistor at each new temperature, using Eq. (10-34). Hence, for the silicon transistor at -|-175°C, we have, using Eq. (10-17), Si( + 175°C) = (1 + 02 1 + R b /R t = (101) (2.65) I +0z + Rh/R, 101 + 1.65 - 2.61 282 / ELECTRONIC DEVICES AND CIRCUITS Sec. 10-6 Then 5"( + 175°C) - IciSj _ (1.5) (2.62) 0,(1 + j8 2 ) (55)(101) = 0.71 X 10-» mA Similarly, Sl( _65-C) - 2fifi« . 2.49 26 + 1.65 S"(-65°C) = (l ' 5)(2 - 49) = 2.61 X 10-* mA (55) (26) We are now in a position to calculate the change in 7 C , using Eq. (10-36) and Table 10-1. A/ C ( + 175°C) = 5 Mco + S' AV BB + 5" AjS = (2.57)(33 X 10" 3 ) + (-0.203) (-0.375) + (0.71 X 10-»)(45) - 0.085 + 0.077 + 0.032 = 0.194 mA and at -65°C, A/c(-65°C) = (2.57) (-10-*) - (0.203) (0.18) + (2.61 X Hr 3 )(-30) = - 0.036 - 0.078 - -0.114 mA Therefore, for the silicon transistor, the collector current will be approximately 1.69 mA at +175°C and 1.39 mA at -65°C. 6. To calculate the change in collector current using the germanium tran- sistor, we must compute S" at +75 and — 65°C. S!(+75 .Q - yWy* - 2.60 S"(+75°C) = 91 + 1.65 = (1.5) (2.60) /S»(l + &) (55)(91) = 0.78 X 10~ a mA Similarly, S , ( _ 65 . C ) = ^^ = 2.45 21 + 1.65 S"(-65°C) = (L5)(2 ' 45) - 3.18 X 10"» mA (55)(21) Hence the change in collector current is A/ C (+75°C) « (2.57)(31 X 1G" 3 ) + (-0.203) (-0.10) + (0.78 X 10-*)(3S) = 0.080 + 0.020 + 0.027 = 0.127 mA and at -65°C, A/ C (-65°C) = (2.57)(-10" J ) + ( + 0.203) (0.1 8) + (3.18 X 10~ s )(-35) = -0.002 - 0.036 - 0.111 = -0.149 mA Therefore, for the germanium transistor, the collector current will be approxi- mately 1.63 mA at +75°C and 1.35 mA at -65°C. Sec. 10-7 TRANSISTOR BIASING AND THERMAL STABIIIZATION / 283 practical Considerations The foregoing example illustrates the supe- riority of silicon over germanium transistors because, approximately, the same change in collector current is obtained for a much higher temperature change in the silicon transistor. In the above example, with & = 2.57, the current change at the extremes of temperature is only about 10 percent. Hence this circuit could be used at temperatures in excess of 75°C for germanium and 175°C for silicon. If S is larger, the current instability is greater. For example, in Prob. 10-19, we find for R t = 1 K and S = 7.70 that the collector current varies about 30 percent at -65°C and +75°C (Ge) or + 175°C (Si). These numerical values illustrate why a germanium transistor is seldom used above 75°C, and a silicon device above 175°C. The importance of keeping S small is clear. The change in collector current that can be tolerated in any specific application depends on design requirements, such as peak signal voltage required across R c . We should also point out that the tolerance in bias resistors and supply voltages must be taken into account, in addition to the variation of 0, ho, and V be- Out discussion of stability and the results obtained are independent of R tl and hence they remain valid for R c = 0. If the output is taken across R e , such a circuit is called an emitter follower (discussed in detail in Sec. 12-8). If we have a direct-coupled emitter follower driven from an ideal voltage source, then #6 = and S is at its lowest possible value, namely, 5=1. It is clear that a circuit with R b = can be used to a higher temperature than a similar circuit with Rb ^ 0. In the above example the increase in collector current from 25 to 75°C for a germanium transistor is 0.08 mA due to Ico and 0.02 mA due to Vbb- Hence, for Ge, the effect of ho has the dominant influence on the collector current. On the other hand, the increase in h for a silicon transistor over the range from 25 to 175°C due to ho is approximately the same as that due to V BE . However, if the temperature range is restricted somewhat, say, from 25 to 145°C, then M c - 0.01 mA due to ho and A/ c = 0.06 mA due to Vbb- These numbers are computed as follows: If T mai is reduced from 175 to 145°C, or by 30°, then ho is divided by 2^ 10 = 2* = 8. Hence S AI C0 = 0.085/8 = 0.01 mA. Also, AV B s is increased by (30) (2.5) = 75 mV, °r AVbb goes from -0.375 to -0.30 V and S' AV BE = <-0.2)(-0.30) = 0.06 "lA. Hence, for Si, the effect of Vbb has the dominant influence on the col- lector current. 1 0-7 BIAS COMPENSATION' T he collector-to-base bias circuit of Fig. 10-oa and the self-bias circuit of Fig. !0-6a are used to limit the variation in the operating collector current h caused by variations in ho, Vami and /3. These circuits are examples of feedback amplifiers, which are studied in Chap. 17, where it is found that a 284 / ELECTRONIC DEVICES AND CIRCUITS Sot. 10.7 Fig. 10-11 Stabilization by means of self- bias and diode-compensation techniques. consequence of feedback is to reduce drastically the amplification of the signal. If this loss in signal gain is intolerable in a particular application, it is often possible to use compensating techniques to reduce the drift of the operating point. Very often both stabilization and compensation techniques are used to provide maximum bias and thermal stabilization. Diode Compensation for V BB A circuit utilizing the self-bias stabiliza- tion technique and diode compensation is shown in Fig. 10-11. The diode is kept biased in the forward direction by the source V D t> and resistance R*. If the diode is of the same material and type as the transistor, the voltage V across the diode will have the same temperature coefficient ( — 2.5 mV/°C) as the base-to-emitter voltage Vbb- If we write KVL around the base circuit of Fig. 10-11, then Eq. (10-28) becomes Ic = &[V ~ (Vbb - V.)] + (Rt + R t )(fl + Pico R b + fl.(l + J8) (10-37) Since V B b tracks V a with respect to temperature, it is clear from Eq. (10-37) that Ic will be insensitive to variations in Vbb. In practice, the compensation of V B b as explained above is not exact, but it is sufficiently effective to take care of a great part of transistor drift due to variations in Vbb- Diode Compensation for Ico We demonstrate in Sec. 10-6 that change 8 of Vbb with temperature contribute significantly to changes in collector current of silicon transistors. On the other hand, for germanium transistors, changes in Ico with temperature play the more important role in collector- current stability. The diode compensation circuit shown in Fig. 10-12 offera stabilization against variations in Ico, and is therefore useful for stabilizing germanium transistors. If the diode and the transistor are of the same type and material, the reverse saturation current I of the diode will increase with temperature a* 10-8 TRANSISTOR BIASING AND THERMAL STABILIZATION / 285 fm, 10-12 Diode compensation for a germanium transistor. '!>*. .P7 m „ the same rate as the transistor collector saturation current Ico- From Fig. 10-12 we have / - Vcc ~ Vi Ri = const Since the diode is reverse-biased by an amount Vbb "* 0.2 V for germanium devices, it follows that the current through D is I e . The base current is I B = I - I . Substituting this expression for I B in Eq. (1CM), we obtain I c = 01- fih + (1 + 0)1 co (10-38) We see from Eq. (10-38) that if p » 1 and if h of D and Ico of Q track each other over the desired temperature range, then I c remains essentially constant. 10-8 BIASING CIRCUITS FOR LINEAR INTEGRATED CIRCUITS 2 In Chap. 15 we study the fabrication techniques employed to construct integrated circuits. These circuits consist of transistors, diodes, resistors, and capacitors, all made with silicon and silicon oxides in one piece of crystal or chip. One of the most basic problems encountered in linear integrated circuits is bias stabilization of a common-emitter amplifier. The self-bias circuit of Fig, 10-(ia is impractical because the bypass capacitor required across R, is much too large (usually in excess of 10 n¥) to be fabricated with present-day integrated-circuit technology. This technology offers specific advantages, w hich are exploited in the biasing circuits of Fig. 10-13a and o. The special features are (1) close matching of active and passive devices over a wide temperature range; (2) excellent thermal coupling, since the whole circuit is fabricated on a very tiny chip of crystal material (approximately 90 mils 8t iuare); and (3) the active components made with this technology are no m ore expensive than the passive components. Hence transistors or diodes c an be used economically in place of resistors. The biasing technique shown in Fig. 10-13a uses transistor Ql connected 88 a diode across the base-to-emitter junction of transistor Q% whose collector 286 / ELECTRONIC DEVICES AND CIRCUITS (a) (o) Fig. 10-13 Biasing techniques for linear integrated circuits. current is to be temperature-stabilized. The collector current of Ql is given by VcC — V BE Ici = Ri — I B\ ~ 1 1 For Vbb « V C c and (I B i + /«) « hi, Eq. (10-39) becomes r V CC . Ici « -5— = const Hi (10-39) (10-40) If transistors Ql and Q2 are identical and have the same Vbb, their collector currents will be equal. Hence I c % = Ici = const. Even if the two transistors are not identical, experiments 2 have shown that this biasing scheme gives collector-current matching between the biasing and operating transistors typi- cally better than 5 percent and is stable over a wide temperature range. The circuit of Fig. 10-1 3a is modified as indicated in Fig. 10- 13b so that the transistors are driven by equal base currents rather than the same base voltage. Since the collector current in the active region varies linearly with Ib, but exponentially with V b e, improved matching of collector currents results. The resistors R% and Rz are fabricated in an identical manner, so that #3 = R2. Since the two bases are driven from a common voltage node through equal resistances, then Ibi = Im = Ib, and the collector currents are well matched for identically constructed transistors. From Fig. 10-13&, the collector current of Ql is given by If cc — Vbb Ici = Ri - ( 2 + si) Under the assumptions that V B b « Vcc, (10-41) becomes Vcc Ri (10-41) and (2 + Ri/Ri)I B « Vcc/Ru E* Ici = Ici = Sec. 10-9 TRANSISTOR BIASING AND THERMAL STABILIZATION / 287 If R e = ^Ri, then V C e = V C c ~ hiR* m V C c/2, which means that the amplifier will be biased at one-half the supply voltage Vcc, independent of the supply voltage as well as temperature, and dependent only on the matching f components within the integrated circuit. An evaluation of the effects of mismatch in this circuit on bias stability is given in Ref. 2. THERMISTOR AND SENSISTOR COMPENSATION 1 There is a method of transistor compensation which involves the use of tem- perature-sensitive resistive elements rather than diodes or transistors. The thermistor (Sec. 5-2) has a negative temperature coefficient, its resistance decreasing exponentially with increasing T. The circuit of Fig. 10-14 uses a thermistor R T to minimize the increase in collector current due to changes in I co, Vbe, or with T. As T rises, R T decreases, and the current fed through R T into Re increases. Since the voltage drop across R e is in the direction to reverse-bias the transistor, the temperature sensitivity of Rr acts so as to tend to compensate the increase in Ic due to T. An alternative configuration using thermistor compensation is to move R T from its position in Fig. 10-14 and place it across R%. As T increases, the drop across R T decreases, and hence the forward-biasing base voltage is reduced. This behavior will tend to offset the increase in collector current with temperature. Instead of a thermistor, it is possible to use a temperature-sensitive resistor with a positive temperature coefficient such as a metal, or the sensistor (manufactured by Texas Instruments). The sensistor has a temperature coefficient of resistance which is +0.7 percent/°C (over the range from -60 to -fl50°C). A heavily doped semiconductor can exhibit a positive tem- perature coefficient of resistance, for under these conditions the material acquires metallic properties and the resistance decreases because of the decrease of carrier mobility with temperature. In the circuit of Fig. 10-14 (with R r -V cc ng. 10-14 Thermistor compensation of *he increase in l e with T. <■', o 288 / ELECTRONIC DEVICES AND CIRCUITS Sec. 10- JO removed), temperature compensation may be obtained by placing a sensistor either in parallel with Ri or in parallel with (or in place of) R f . Why? In practice it is often necessary to use silicon resistors and carbon resistors in series or parallel combinations to form the proper shaping network. 3 The characteristics required to eliminate the temperature effects can be determined experimentally as follows: A variable resistance is substituted for the shaping network and is adjusted to maintain constant collector current as the operating temperature changes. The resistance vs. temperature can then be plotted to indicate the required characteristics of the shaping network. The problem now is reduced to that of synthesizing a network with this measured tem- perature characteristic by using thermistors or sensistors padded with tem- perature-insensitive resistors. 10-10 THERMAL RUNAWAY The maximum average power Fornax) which a transistor can dissipate depends upon the transistor construction and may lie in the range from a few milliwatts to 200 W. This maximum power is limited by the tempera- ture that the collector-to-base junction can withstand. For silicon transis- tors this temperature is in the range 150 to 225° C, and for germanium it is between 60 and 100°C The junction temperature may rise either because the ambient temperature rises or because of self-heating. The maximum power dissipation is usually specified for the transistor enclosure (case) or ambient temperature of 25°C. The problem of self-heating, which is men- tioned in Sec, 10-2, results from the power dissipated at the collector junction. As a consequence of the junction power dissipation, the junction temperature rises, and this in turn increases the collector current, with a subsequent increase in power dissipation. If this phenomenon, referred to as thermal runaway, continues, it may result in permanently damaging the transistor. Thermal Resistance It is found experimentally that the steady-state temperature rise at the collector junction is proportional to the power dissi- pated at the junction, or AT = Tj - T A = OP D (10-42) where Tj and Ta are the junction and ambient temperatures, respectively, in degrees centigrade, and P D is the power in watts dissipated at the collector junction. The constant of proportionality is called the thermal resistance. Its value depends on the size of the transistor, on convection or radiation to the surroundings, on forced-air cooling (if used), and on the thermal connection of the device to a metal chassis or to a heat sink. Typical values for various transistor designs vary from 0.2°C/W for a high-power transistor with an efficient heat sink to 1000°C/W for a low-power transistor in free air. The maximum collector power P € allowed for safe operation is specified Sec IO-T0 TRANSISTOR BIASING AND THERMAL STABILIZATION / 289 Pc.W pig. 10-15 Power-temperature derating curve for a germanium power transistor. 150 120 1 \. i X i X^ 90 60 30 ( ) 20 40 60 80 100 Case temperature, •c at 25°C. For ambient temperatures above this value, Pc must be decreased, and at the extreme temperature at which the transistor may operate, Pc is reduced to zero. A typical power-temperature derating curve, supplied in a manufacturer's specification sheet, is indicated in Fig. 10-15. Operating-point Considerations The effects of self-heating may be appreciated by referring to Fig. 10-16, which shows three constant-power hyperbolas and a dc load line tangent to one of them. It can be shown (Prob. 10-26) that the point of tangency C bisects the load line AB. Consider that the quiescent point is above the point of tangency, say at Qi. If now the collector current increases, the result is a lower collector dissipation because Qi moves along the load line in the direction away from the 300- W toward the 100-W parabola. The opposite is true if the quiescent point is below the point of tangency, such as at Q 2 . We can conclude that if Vce is less than Fcc/2, the quiescent point lies in a safe region, where an increase in collector current, results in a decreased dissipation. If, on the other hand, the operating ,500W '9. 10-16 Concerning transis- tor self-heating. The dashed lo °d tine corresponds to a er y small dc resistance. 290 / ElECTRONJC DEVICES AND CIRCUITS Sec, 7 0-| | Sec. 10-W TRANSISTOR BIASING ANO THERMAL STABILIZATION / 291 point is located so that Vce > V C cl% the self-heating results in even more collector dissipation, and the effect is cumulative. It is not always possible to select an operating point which satisfies the restriction Vcs < Wcc- For example, if the load R L is transformer-coupled to the collector, as in Fig. 10-17, then R c represents the small primary dc resistance, and hence the load line is almost vertical, as indicated by the dashed line in Fig. 10-16. Clearly Vce can be less than Wee only for exces- sively large collector currents. Hence thermal runaway can easily occur with a transformer-coupled load or with a power amplifier which has small collector and emitter resistances. For such circuits it is particularly important to take precautions to keep the stability factors (discussed in the preceding sections) so small as to maintain essentially constant collector current. The Condition for Thermal Stability We now obtain the restrictions to be met if thermal runaway is to be avoided. The required condition is that the rate at which heat is released at the collector junction must not exceed the rate at which the heat can be dissipated; that is, (10-43) dPc . BPd df~ Jf~ If we differentiate Eq. (10-42) with respect to Tj and substitute in Eq. (10-43), we obtain dP c dTj (10^4) This condition must be satisfied to prevent thermal runaway. By suitable circuit design it is possible to ensure that the transistor cannot run away below a specified ambient temperature or even under any conditions. Such an analysis is made in the next section. 10-11 THERMAL STABILITY Let us refer to Fig. 10-6a and assume that the transistor is biased in the active region. The power generated at the collector junction with no signal is Pc = IcVcb « IcVce ( l0A ® If we assume that the quiescent collector and emitter currents are essentially equal, Eq. (10-45) becomes Pc = IcVcc - /c«(fl. + 4) ( 1(M6) Equation (10-44), the condition to avoid thermal runaway, can be rewritten as follows: dPcdlc ^ I ( l0 -47) dl c dTj The first partial derivative of Eq. (10-47) can be obtained from Eq. (10-46) : dP c dh = Vcc — 2Ic(Re + Re) (10-48) The second partial derivative in Eq. (10-47) gives the rate at which collector current increases with temperature. From our discussion in this chapter we fcnow that junction temperature affects collector current by affecting Ico, Vbe, and /S. Hence we have, from Eq. (10-36), dl c dTj = S dlco ~df~ + s ,dV dTj (10-49) Since for any given transistor the derivatives in Eq. (10-49) are known, the designer is required to satisfy Eq. (10-47) by the proper selection of S, S' y S", and 0. In some practical problems the effect of Ieo dominates, and we present an analysis of the thermal- runaway problem for this case. From Eqs. (10-47) and (10-49), dP C l die (s dIco \ < I (10-50) In Sec. 6-7 it is noted that the reverse saturation current for either silicon or germanium increases about 7 percent/°C, or dlco ar7 = 0.07/e O Substituting Eqs. (10-48) and (10-51) in Eq. (10-50) results in [V cc ~ 2I c (R e + i2 e )](S)(0.07/ CO ) < g (10-51) (10-52) Equation (10-52) remains valid for a p-n-p transistor provided that Ic (and Ico) are understood to represent the magnitude of the current. Remembering that 0, S, and I C o are positive, we see that the inequality (10-52) is always satisfied provided that the quantity in the brackets is negative, or provided that Vcc Ic> (10-53) 2(fl. + R B ) Since V CB - V C c - Ic(R, + Re), thenEq. (10-53) implies that Vce < V C cl% and this checks with our previous conclusion from Fig. 10-16. If the inequality of Eq. (10-53) is not satisfied and V C e > V C c/2, then from Eq. (10-48) we see that dP c /dI c i s positive, and the designer must ensure that Eq. (10-50) will De satisfied, or else thermal runaway will occur. EXAMPLE Find the value of required for the transistor of the example on page 274 in order for the circuit to be thermally stable. Assume that / c « = 1 nA at 25°C. 292 / ELECTRONIC DEVICES AND CIRCUITS See. JO- J j Solution Since V C c/2 = 11.25 V and V C e = 13-3 V, the circuit is not inherently stable, because V CB > -kVcc Substituting in Eq. (10-52), we obtain {22.5 - 2 X 1.4 X (5.6 + 1.0)1(8.61) (0.07 X 10-") < i 4.0 X 8.61 X 0.07 X 10-» < ^ or < 4.1 X 10 s °C/W The upper bound on the value of is so high that no transistor can violate it, and therefore this circuit will always be safe from thermal runaway. This example illustrates that amplifier circuits operated at low current and designed with low values of stability factor (S < 10) are very rarely susceptible to thermal runaway. In contrast, power amplifiers operate at high power levels. In addition, in such circuits R e is a small resistance for power efficiency, and this results in a high stability factor S. As a result, thermal runaway in power stages is a major consideration, and the designer must guard against it. EXAMPLE Figure 10-17 shows a power amplifier using a jhn-p germanium transistor with /S = 100 and Ico = — 5 mA. The quiescent collector current is T c = — 1 A. Find (a) the value of resistor R b ; (b) the largest value of that can result in a thermally stable circuit. Solution a. The collector current is given by Eq. (10-4), or Ic = Wb + (1 + Wco *» 0(1 a + Ico) and 1 - 5 X 10-' X 100 . : . [ B = A = — 5 mA 100 -v„-=-40V Fig. 10-17 Power amplifier with a trans- former-coupled load. u, o— * &■ 70-11 TRANSISTOR BIASING AND THERMAL STABILIZATION / 293 If we neglect V B e, we have 5 X 10-»A, = 40-5 or R b = 7,000 Q b. Since \V CS \ = 40 - 15 = 25 > i|V«?| = 20 V, the circuit of Fig. 10-17 is not inherently stable. The stability factor S is obtained from Eq. (10-17). 1 + 7,000/5 S = 101 - 94.3 101 + 7,000/5 Substituting in Eq. (10-52), we obtain (40 - 2 X 1 X 15) (94.3) (0.07 X 5 X 10~ a ) < - vt or < 3.03°C/W REFERENCES 1. Hunter, L. P.: "Handbook of Semiconductor Electronics," McGraw-Hill Book Com- pany, New York, 1962. "Transistor Manual," 7th ed., General Electric Co., Syracuse, N.Y., 1964. "Motorola Power Transistor Handbook," Phoenix, Ariz., 1961. 2. Widlar, R. L: Some Circuit Design Techniques for Linear Integrated Circuits, IEEE Trans. Circuit Theory, vol. CT-12, no. 4, pp. 586-590, December, 1965. 3. Konjian, E., and J. S. Schaffner: Shaping of the Characteristics of Temperature- sensitive Elements, Commun. and Electron., vol. 14, pp. 396-400, September, 1954. SMALL-SIGNAL LOW-FREQUENCY TRANSISTOR MODELS In Chap. 9 we are primarily interested in the static characteristics of a transistor. In the active region the transistor operates with reasonable linearity, and we now inquire into small-signal linear models which represent the operation of the transistor in this active region. The parameters introduced in the models presented here are interpreted in terms of the external volt-ampere characteristics of the transistor. Methods for measuring these parameters are also given. Finally, a detailed study of the transistor amplifier in its various configurations is made. 11-1 TWO-PORT DEVICES AND THE HYBRID MODEL The terminal behavior of a large class of two-port devices is specified by two voltages and two currents. The box in Fig. 11-1 represents such a two-port network. We may select two of the four quantities as the independent variables and express the remaining two in terms of the chosen independent variables. It should be noted that, in general, we are not free to select the independent variables arbitrarily. For example, if the two-port device is an ideal transformer, we cannot pick the two voltages vi and v 2 as the independent variables because their ratio is a constant equal to the transformer turns ratio. If thc current t'i and the voltage w 2 are independent and if the two-port ' linear, we-may write v 1 = huii + hiiVi U*" ' The quantities h n , h n , ft 2 i, and A 22 are called the h, or hybrid, parameter because they are not all alike dimensionally. Let us assume that t»w 294 $•' U-I SMALL-SIGNAL LOW-FREQUENCY TRANSISTOR MODELS / 295 Fig network H-1 A two-port Input port Output port ft - e no reactive elements within the two-port network. Then, from Eqs. (11-1) and (11-2), the h parameters are defined as follows: A,, m — I = input resistance with output short-circuited (ohms). I, |m = h - Vl \ X>1 l» = o fctt 3! hit sb H I »2 k = = fraction of output voltage at input with input open- circuited, or more simply, reverse-open-circuit voltage amplification (dimensionless). = negative of current transfer ratio (or current gain) with output short-circuited. (Note that the current into a load across the output port would be the nega- tive of H.) This parameter is usually referred to, simply, as the short-circuit current gain (dimensionless). = output conductance with input open-circuited (mhos). Notation The following convenient alternative subscript notation is recommended by the IEEE Standards: 1 i «■ 11 = input / = 21 = forward transfer o = 22 = output f as 12 = reverse transfer In the case of transistors, another subscript (b, e, or c) is added to designate the type of configuration. For example, hit, = h ub = input resistance in common-base configuration h S e - %%u = short-circuit forward current gain in common-emitter circuit Since the device described by Eqs. (11-1) and (11-2) is assumed to include no reactive elements, the four parameters An, hi 2 , km, and h i% are real numbers, ^nd the voltages and currents v h i> 2 , and i h U are functions of time. However, lf reactive elements had been included in the device, the excitation would be c °nsidered to be sinusoidal, the h parameters would in general be functions of frequency, and the voltages and currents would be represented by phasors ^i. V 2) and I h h. The Model We may now use the four h parameters to construct a ma thematical model of the device of Fig. 11-1. The hybrid circuit for any 296 / ELECTRONIC DEVICES AND CIRCUITS i\ (in ohms) ft n *i ^ (in raA/V) r Fig. IT -2 The hybrid model for the two-port network of Fig. 11-1. The parameters h n and An are dimenslonless. «L device characterised by Eqs. (11-1) and (11-2) is indicated in Fig. 11-2. We can verify that the model of Fig. 11-2 satisfies Eqs. (11-1) and (11-2) by writing Kirchhoffs voltage and current laws for the input and output ports, respectively. 11-2 TRANSISTOR HYBRID MODEL The basic assumption in arriving at a transistor linear model or equivalent circuit is the same as that used in the case of a vacuum tube: the variations about the quiescent point are assumed small, so that the transistor parameters can be considered constant over the signal excursion. Many transistor models have been proposed, each one having its particular advantages and disadvantages. The transistor model presented i n this chapter, and exploited in the next chapter, is given in terms of the h parameters, which are real numbers at audio frequencies, are easy to measure, can also be obtained from the transistor static characteristic curves, and are particularly convenient to use in circuit analysis and design. Furthermore, a set of h parameters i.s specified for many transistors by the manufacturers. To see how we can derive a hybrid model for a transistor, let us consider the common-emitter connection shown in Fig. 11-3. The variables is, ic, Vb, and v c represent total instantaneous currents and voltages. From our discussion in Chap. 9 of transistor voltages and currents, we see that we may select the current i R and voltage v c as independent variables. Since vb 1S some function /i of i B and v c and since ic is another function f t of in and vc, Fig. 11-3 A simple common-emitter con- nection. L Soc. 11-2 we may write Vb = fi(iB, Vc) ic = Mis, vc) SMALL-SIGNAL LOW-FREQUENCY TRANSISTOR MODELS / 297 (11-3) (11-4) Making a Taylor's series expansion of Eqs. (11-3) and (11-4) around the quies- cent point I S) V c , similar to that of Eq. (8-12), and neglecting higher-order terms, we obtain AVB = *Zl I At* + & dis \Yc dv c V Av< (11-5) (11-6) The partial derivatives are taken, keeping the collector voltage or the base current constant, as indicated by the subscript attached to the derivative. The quantities Av B) &t>c, Mb, and Ai c represent the small-signal (incre- mental) base and collector voltages and currents. According to the notation in Table 9-1, we represent them with the symbols v b , v t , i b , and v We may now write Eqs. (11-5) and (11-6) in the following form: where and v b = h it) ib + K»Vc ic = h/etb + k oe v e A - a /l = dvB I di 8 dlB Wc h = ^ll = — dii di B Ikc h ^ 9/i = Bvb I " dvc dv c l/» Bfi die I dv c dVc u» (11-7) (11-8) (11-9) (11-10) The partial derivatives of Eqs. (11-9) and (11-10) define the A parameters for the common-emitter connection. In the next section we show that the above partial derivatives can be obtained from the transistor characteristic curves and that they are real numbers. We now observe that Eqs. (11-7) and (11-8) are of exactly the same form as Eqs. (11-1) and (11-2). Hence the model of Fig. 11-2 can be used to represent a transistor. The Three Transistor Configurations The common-emitter (CE), com- mon-collector (CC), and common- base (CB) configurations, their hybrid Models, and the terminel v-i equations are summarized in Table 11-1. We should note here that, for any one of the three different transistor connections, the input and output voltages have a common terminal. Moreover, we note from Kirchhoff's current law that ib + i. + ic - (11-11) 298 / ELECTRONIC DEVICES AND CIRCUITS TABLE 11-1 Transistor configurations and their hybrid models s «. Ij-3 The circuits and equations in Table 11-1 are valid for either an «-/>-" or p-n-p transistor and are independent of the type of load or method of biasing. 11-3 DETERMINATION OF THE h PARAMETERS FROM THE CHARACTERISTICS 2 Equations (11-3) and (11-4) give the form of the functional relationships ft* the common-emitter connection of total instantaneous collector current aa d base voltage in terms of two variables, namely, base current and collector voltage. Such functional relationships are represented in Chap. 9 by famili eS S*- IJ-3 SMALL-SIGNAL LOW-FREQUENCY TRANSISTOR MODELS / 299 f characteristic curves. Two families of curves are usually specified for transistors. The output characteristic cumes depict the relationship between iiitput current and voltage, with input current as the parameter. Figures 9-5 and 9-8 show typical output characteristic curves for the common-base and common-emitter transistor configurations. The input characteristics depict the relationship between input voltage and current with output voltage as the parameter. Typical input characteristic curves for the common-base and common-emitter transistor connections are shown in Figs. 9-6 and 9-9. If the input and output characteristics of a particular connection are given, the h parameters can be determined graphically. The Parameter h fe For a common-emitter connection the characteristics arc shown in Fig. 11-4. From the definition of A /e given in Eq. (11-10) and from Fig. 1 l-4a, we have die Bin k ft = Ate 1 Lis We IC2 — 1C\ iB2 — 1-Bl Ul-12) The current increments are taken around the quiescent point Q, which corre- sponds to the base current is = In and to the collector voltage v C s = V c (a vertical line in Fig. Ll-4a). The parameter A/ e is the most important small-signal parameter of the transistor. This common-emitter current transfer ratio, or CE alpha, is also written a,, or lV, and called the small-signal beta of the transistor. The rela- tionship between j3' = A/„ and the large-signal beta, « h FE , is given in Eq, (9-47). Wca ■ v c ^•g. H-4 Characteristic curves of a common-emitter transistor, (a) CE output characteristics— determination of h/i ar »d fewl (M CE input characteristics— determi nation of hit and A„. 300 / ELECTRONIC DEVICES AND CIRCUITS The Parameter h h From Eq, (11-10), die Ate I Av c Us s <*- 11. j (11-18) The value of h ot at the quiescent point Q is given by the slope of the output characteristic curve at that point. This slope can be evaluated by drawing the line AB in Fig, ll-4o tangent to the characteristic curve at the point Q, The Parameter ft™ From Eq. (11-9), , _ dv B ^ Av B I Olf} &l B \Vc (11-14) Hence the slope of the appropriate input characteristic at the quiescent point Q gives h*. In Fig. 11-46, ft,, is given by the slope of the line EF, which is drawn tangent to the characteristic curves at the point Q. The Parameter ft„ dv B 6.vb Finally, from Eq. (11-9), v B i - v B \ , _ vug _^ "^g _ dvc At) c u* Va — Vci (11-15) A vertical line on the input characteristics of Fig. 11-46 represents constant base current. The parameter h„ can now be obtained as the change in base voltage, v B i — v S i, divided by the change in collector voltage, v C i — Pes, for a constant base current I a , at the quiescent point Q. Since h re ~ 10~ 4 , then Av B <K Av c , and hence the above method, although correct in principle, is very inaccurate in practice. The procedure outlined here for the determination of the common-emitter h parameters may also be used to obtain the common-base and common- collector h parameters from the appropriate input and output characteristic curves. Hybrid-parameter Variations From the discussion in this section we have seen that once a quiescent point Q is specified, the h parameters can be obtained from the slopes and spacing between curves at this point. Since the characteristic curves are not in general straight lines, equally spaced for equal changes in I B (Fig. ll-4o) or V C b (Fig. 11-46), it is clear that the values of the h parameters depend upon the position of the quiescent point on the curves. Moreover, from our discussion in Chap. 9, we know that the shape and actual numerical values of the characteristic curves depend on the junction tem- perature. Hence the h parameters also will depend on temperature. Mow transistor specification sheets include curves of the variation of the h parameters with the quiescent point and temperature. Such curves are shown for a typi 06 * silicon p-n-p transistor in Fig. ll-5a and 6. These curves are plotted with respect to the values of a specific operating point, say — 5 V collector-to-eraitter voltage and — I mA collector current. The variation in h parameters as shown Sec n-3 50 I 20 1 ,0 1 5 9 * I > 1.0 J 0-5 I I 0.2 £ o.i jS 0,06 f 0.02 y o.oi -1 1 V _l 1 1 5.0V 1 7^25'C £* *rr h^- ** hfr £ K SMALL-SIGNAL LOW-FREQUENCY TRANSISTOR MODELS I 301 H II W 2.0 % IB a 1.5 "3 1.0 8 2 0.4 g -0.1-0.2-0.5-1.0-2 -5-10-20 I Collector current Ic , mA r>r. 3 «•? 1 = 1kHz - 1.0mA ho. h~ h n * n t' *J ion Be N 150 200 (a) Junction temperature T f , * C (6) Fig. 11-5 Variation of common-emitter h parameters (a) with collector current normalized to unity at V C r = -5.0 V and I c = -1.0 mA for the type 2N996 diffused-silicon planar epitaxial transistor; (b) with junction temperature, normal- ized to unity at T, = 25°C. (Courtesy of Fairchild Semiconductor.) in Fig. ll-5o is for a constant junction temperature of 25°C and a frequency of 1 kHz. Manufacturers usually also provide curves of h parameters versus V C K, although this variation with Vcs is often not significant. Specifically, h /t is more sensitive to Ic than to Vcs- Most transistors exhibit a well-defined maximum in the value of h fe as a function of collector or emitter current. Such a maximum in the variation of ft/, with emitter current and temperature is shown in Fig. 1 1-6 for an n-p-n double-diffused silicon mesa transistor. Fi 9- 11-6 Variation of h fr w 'th emitter current for the type 2N1 573 silicon mesa * r ansistor. (Courtesy of T*xas Instruments, Inc.) 180 150 120 k f* Q0 V cr . = 5V 150" C il25« C 75"C T A = 25° C 60 30 ^ -55 e C O -5 -10 -15 Emitter current I t , mA 302 / ELECTRONIC DEVICES AND CIRCUITS TABLE 11-2 Typical /(-parameter values for a transistor (at I B = 1.3 raA) Soc. TJ. Parameter ftu = hi kit = h T h tl = kj h%i = k 9 1/A. CE 1,100 n 2.5 X 10- 50 25*A/V 40K CC 1,100 Q —1 -51 25 M/V 40 K CB 21.6 R 2.9 X 10-* -0.98 0.49 jiA/V 2.04 M Table 1 1-2 shows values of ft parameters for the three different transistor configurations of a typical junction transistor. 11 -4 MEASUREMENT OF ft PARAMETERS 3 Based on the definitions given in Sees. 11-1 and 11-2, simple experiments may be carried out for the direct measurement of the hybrid parameters. Consider the circuit of Fig. 11-7. The desired quiescent conditions are obtained from adjustable supplies Vcc, Vbb, and the resistor Rt. The imped- ance of the tank circuit (~500 K) at the audio frequency (1 kHz) at which the measurements are made is large compared with the transistor input resistance Ri. The value of #1 (1 M) is large compared with R i} and the reactances of d, C 2 , and Cz are negligible at the frequency of the sinusoidal generator V B . Note that we now use capital letters to represent phasor rms voltages and currents, Hence, Av B , Ai'b, Ave, and Aic of the preceding section are replaced by Vb, h, V c , and I c , respectively. We may consider the signal-input current to be 7 6 = V,/R\. Since Rl is generally 50 U, we may consider the transistor output port as short-circuited to the signal. JJ,(1M) 1 VV\ — t "Tank circuit Fig. 11-7 Circuit for measuring ft, e and hj c . Sec M-4 The value of ft« is given by Eq. (11-14) : Vt I VtRx SMALL-SIGNAL LOW-FREQUENCY TRANSISTOR MODELS / 303 h- t = h lv.=o V. (11-16) Hence the input resistance ft,-, may be calculated from the two measured voltages V t and V&. For the parameter ft/„ we have from Eq. (11-12) fife ■= T VJli h\v*-o V,R L (11-17) since h = V g /R L - Thus ft/, is obtained from the two measured voltages V. and V t . The circuit of Fig. 11-8 may be used to measure h re and k ot . The signal is now applied to the collector circuit using a transformer. Because the impedance of the tank circuit is large compared with R i} the base circuit may be considered effectively open-circuited as far as the signal is concerned. We then obtain from Eq. (11-15) h -1*1 The output conductance is defined by Eq. (11-13): h - U flat = TT V e lA-0 R L V e (11-18) (11-19) Hence h„ is obtained from the measured voltages V and V e . In measuring V„, V b , and F e it is necessary to ground one side of the volt- meter to avoid stray pickup. This can be done by using a high input resistance voltmeter with one side connected, through a capacitor, to point ,A } or to the base or to the collector, and with the other side of the meter grounded. ng. H-8 Circuit for meos- ur 'n8 kr, and h at . 304 / ELECTRONIC DEVICES AND CIRCUITS S«. lj-4 TABLE 7 J -3 Approximate conversion formulas for transistor parameters (numerical values are for a typical transistor Q) Symbol Common emitter Common collector Common base T equivalent circuit h« 1,100 Q hid Art 1 +h /b 1—0 h„ 2.5 X 10-* 1 - A re t AitAoj 1+A„ ** (1 - o)r. A,. 50 -a +**)* A/t 1 +*/» a 1 -o h„ 25 M A/V A«t 1 + A/t 1 (1 - o)r. ha A* A, e A/e 21.6 a r. + (1 - o)r 4 Art huh,, l+h,. K ' *.-»£=--l A/ e 2.9 X 10"* A/* h f . 1 +A/c A/. -0.98 1 +A/. — a h i A<n Am ~A/ e 1 1 + fc/. 0.49 aiA/V 1 r« h ie M 1,100 Q h& l — a 1 + A/» K t 1 - At, - If 1 1 r. 1 ~~ (1 - a)r t h tc -a + A / .)t -51 1 1 1 +A/» 1 - a A« A«t 25 ^A/V A,* 1 +A/i 1 (1 - a)r t _ a h/t 1 +h f . 1 + A/« — A/i 0.980 n 1 + */. . A/ e . Aoe 1 A,* 2.04 M r. In Ao# 1-6™ hoc Act 10 fl n a,. - J= a + A A )t A,, Ar* Aoi 590 n t Exact. 5*. "-5 SMAU-S/GNAt tOW-F«fQU£NCy TRANSISTOR MOORS / 305 CONVERSION FORMULAS FOR THE PARAMETERS OF THE THREE TRANSISTOR CONFIGURATIONS 4 Very often it is necessary to convert from one set of transistor parameters to another set. Some transistor manufacturers specify all four common- emitter h parameters; others specify h /e , h ib , h*, and h rb . In Table 11-3 we give approximate conversion formulas between the CE, CC, and CB h parame- ter. For completeness, we also include the T-model parameters, although we postpone until Sec, 11-9 the discussion of the T model. Exact formulas are given in Ref. 4, but are seldom required. Those conversions marked with a dagger in Table 11-3 are exact. The conversion formulas can be obtained using the definitions of the parameters involved and Kirchhoff 's laws. The general procedure is illustrated in the following examples. EXAMPLE Find, in terms of the CB A parameters, (a) h„ and (6) Ai,. Solution a. The CB A-parameter circuit of Fig. ll-9a is redrawn in Fig. 11-96 as a CE configuration. The latter corresponds in every detail to the former, except that the emitter terminal E is made common to the input and output ports. By definition, h « Z* I If h = 0, then I e = — / e , and the current / in h^ in Fig. 11-96 is / = (1 + h f0 )I,. Since A,* represents a conductance, I = hgbVit = (1 + h/t,)I t Applying KVL to the output mesh of Fig. 11-96, *#T, -(- hrtVo, + Vu + V tt = <«) (6) '8- It -O ( a j jj, e £g hybrid model, (b) The circuit in {a) redrawn in a CE configu- r atio n . 306 / ELECTRONIC DEVICES AND CIRCUITS Combining the last two equations yields h&hob S»e. 11.$ 1 + hjt or V Hence k r< = I + Vu - k^Vte + V ta + V„ = -(1 +h fb ) hibhub + (1 — Art)(l + ^) hahab — (1 + h/bjhrt V et kith* + (1 - A rt )(l + A/i) This is an exact expression. The simpler approximate formula is obtained by noting that, for the typical values given in Table 11-2, k Tb « 1 Hence and hobhit, « 1 + A/t kibha -h. 1 +h /b which is the formula given in Table 11-3. 6. By definition, h -Ih\ If we connect terminals C and E together in Fig. 11-96, we obtain Fig. H-10. From the latter figure we see that V cb = -Ti. Applying KVL to the left-hand mesh, we have Vu + A*/. + fcrfcF* = Combining these two equations yields I, . J L^J* Fb( Fig. 11-10 Relating to the calcula- tions of h ir in terms of the CB h S« ' T6 SMALL-SIGNAl LOW -FREQUENCY TRANSISTOR MODELS / 307 Applying KCL to node B, we obtain h + I. + hfl,I t - hobVb. = or h = (i + m ^ : ^ ^ + ^n. nit Hence Vb, h- hie = ~r- = h kjl* + (1 - /U)(l + A,*) This is the exact expression. Jf we make use of the same inequalities as in part a, namely, h* « 1 and A^aA,* « 1 + A/&, the above equation reduces to l. hi. * l + Jfc* which is the formula given in Table 11-3. 11-6 ANALYSIS OF A TRANSISTOR AMPLIFIER CIRCUIT USING k PARAMETERS To form a transistor amplifier it is only necessary to connect an external load and signal source as indicated in Fig. 11-11 and to bias the transistor properly. The two-port active network of Fig. 11-11 represents a transistor in any one of the three possible configurations. In Fig. 11-12 we treat the general case (connection not specified) by replacing the transistor with its small-signal hybrid model. The circuit used in Fig. 11-12 is valid for any type of load whether it be a pure resistance, an impedance, or another transistor. This is true because the transistor hybrid model was derived without any regard to the external circuit in which the transistor is incorporated. The only restriction is the requirement that the h parameters remain substantially constant over the operating range. Assuming sinusoidally varying voltages and currents, we can proceed w 'th the analysis of the circuit of Fig. 11-12, using the phasor (sinor) notation to represent the sinusoidally varying quantities. The quantities of interest ar e the current gain, the input impedance, the voltage gain, and the output impedance. ■ 9- 11-11 A basic amplifier tir cuit. i 308 / ELECTRONIC DEVICES AND CIRCUITS Sec. J?-<J Fig. 11-12 The transistor in Fig. 11-11 is replaced by its A-parameter model. The Current Gain, or Current Amplification, A T For the transistor amplifier stage, Aj is defined as the ratio of output to input currents, or From the circuit of Fig. 11-12, we have l % = k f h + KV, Substituting Vt = —IiZl hi Eq. (11-21), we obtain is _ ft/ /i 1 + hoZt At (11-20) (11-21) (11-22) The Input Impedance Z, The resistance R, in Figs. 11-11 and 11-12 represents the signal-source resistance. The impedance we see looking into the amplifier input terminals (1, 1') is the amplifier input impedance Z it or _ Vi Zi = -s- j i From the input circuit of Fig. 11-12, we have V l - KJi + hrV* Hence „ hilt + KV* V, Li — f = Hi -r fir -y~ Substituting Vi = —IsZl — AiIiZl in Eq. (11-25), we obtain Zi = hi -f- hrAtZh = hi — hfh r Y L + h (11-23) (11-24) (11-25) (11-26) (11-27) where use has been made of Eq. (11-22) and the fact that the load admittance is Yl = 1/Zl. Note that the input impedance is a function of the load impedance. The Voltage Gain, or Voltage Amplification, A v The ratio of outpu voltage Vt to input voltage Vi gives the voltage gain of the transistor, or V t ( U-28) A v = Vt s«- "- 6 SMALL-SIGNAL LOW-FREQUENCY TRANSISTOR MODELS / 309 From Eq. (11-26) we have AjUZl AiZl A v = Vi Zi (11-29) The Output Admittance Y e For the transistor in Figs. 11-11 and 11-12, Y 9 is denned as (11-30) (11-31) (11-32) (11-33) Substituting the expression for I\JV% from Eq. (11-33) in Eq, (11-31), we obtain h/hr Y B It v z with 7. = From Eq. (11-21), Y 1 7l = hf V i + h From Fig. 11-12, with V. = 0, RJi + hili + KV 2 = or fi h r v t s; •f fi. (11-34) fk + R, Note that the output admittance is a function of the source resistance. If the source impedance is resistive, as we have assumed, then Y is real (a conductance). In the above definition of Y = 1/Z„, we have considered the load Z h external to the amplifier. If the output impedance of the amplifier stage with Z L included is desired, this loaded impedance can be calculated as the parallel combination of Z L and Z . The Voltage Amplification Av», Taking into Account the Resistance R t of the Source This overall voltage gain An is defined by A v, = ?p — — yt = Ay tt V, V, V. AV V. From the equivalent input circuit of the amplifier, shown in Fig. ll-13o, V,Zi (11-35) Vi = Th, en A v . = Z< + R, A v Zi AtZ L Zi -f- R, Zi + R, (11-36) Wher, ""ere u se has been made of Eq. (1 1-29). Note that, if R, m 0, then Ay, m A v . ^ce A v is the voltage gain for an ideal voltage source (one with zero internal stance). In practice, the quantity Av, is more meaningful than Av since, 310 / ELECTRONIC DEVICES AND CIRCUITS Set. 1L 6 (a) (6) Fig. 11-13 Input circuit of a transistor amplifier using (a) a Thevenin's equivalent for the source and (b) a Norton's equivalent for the source. usually, the source resistance has an appreciable effect on the overall voltage amplification. For example, if Zi is resistive and equal in magnitude to R„ then A v , = 0.5Av. The Current Amplification At,, Taking into Account the Source Resistance R, If the input source is a current generator /, in parallel with a resistance R„ as indicated in Fig. 11-136, then this overall current gain Aj, is defined by -U A u = /. 7TT. mA 'Z (11-37) From Fig. 11-136, 7 I,Rt 1 1 — and hence Zi -4- R, AtR, A Im = """ (11-38) U Zi + R t Note that if R, = <*> , then An = Ai. Hence At is the current gain for an ideal current source (one with infinite source resistance). Independent of the transistor characteristics, the voltage and current gains, taking source impedance into account, are related by Z L A Vs = A u R, (11-39) This relationship is obtained by dividing Eq. (11-36) by Eq. (11-38), '■M' i is valid provided that the current and voltage generators have the same source resistance R„ The Operating Power Gain A p The average power delivered to the load Z L in Fig. 11-11 is P 2 - | V»| |/ L [ cos 6, where 6 is the phase angle between V 2 and I L . Assume that Z L is resistive. Then, since the h parameters are real at low frequencies, the power delivered to the load is P 2 — V%Il = " \ Since the input power is Pi = VJi, the operating power gain A p of the transis * is defined as (11-40) A - S - - Ap ~ P, ~ V*Ii A A A 1^ L vrr AvAt ~ At & S*c. >>-6 SMALL-SIGNAL LOW-FREQUENCY TRANSISTOR MODELS / 311 TABLE 7 7-4 Small-signal analysis of a transistor amplifier A, = - 1 + hoZt, Zi = ki + h T AiZi, = hi — kfh T hT+Yl Av - AiZl Zi Y Q = K hfh. Av, = h{ + R, Z e AvZi AiZh Ai,Zl Zi + R. Zi + R, R t AtR. Zi + R. Summary The important formulas derived above are summarized for ready reference in Table 11-4. Note that the expressions for Ay. Av*, and At, do not contain the hybrid parameters, and hence are valid regardless of what equivalent circuit we use for the transistor. In particular, these expres- sions are valid at high frequencies, where the A parameters are functions of frequency or where we may prefer to use another model for the transistor (for example, the hybrid-II model of Sec. 13-5). EXAMPLE The transistor of Fig. 11-11 is connected as a common-emitter amplifier, and the h parameters are those given in Table 11-2. If R L = R, = 1,000 12, find the various gains and the input and output impedances. Solution In making the small-signal analysis of this circuit it is convenient, first, to calculate A It then obtain R t from .4;, and A v from both these quantities. 1 sing the expressions in Table 11-4 and the A parameters from Table 11-2, At m - 50 = -48.8 1 + k a Jti, 1 -I- 25 X 10-' X 10* Ri - ht. + h T ,A t R L = 1,100 - 2.5 X 10"« X 48.8 X 10 a - 1,088 & A v = AjRi Ri -48.8 X 10* 1.088 X 10 s m -44.8 . A v Ri ,,„ 1,088 nnrv A Yt m *- = -44.8 X r = -23.3 Au~ Ri + R* AjR t 2,08K -48.8 X 10* Ri + Rt 2.088 X 10* = -23.3 1 312 / ELECTRONIC DEVICES AND CIRCUITS Sac. II.7 Note that, since R 3 — Rt, then Ay, = A t ,. T. - &^ - -^- = 25 X 10- - 2,100 = 19.0 X 10~ 6 raho = 19.0 jiA/V 1 10 8 Z = — = — Q = 52.6 K Y 19.0 Finally, the power gain is given by A p = A V A, = 44.8 X 48.8 = 2,190 11-7 COMPARISON OF TRANSISTOR AMPLIFIER CONFIGURATIONS From Table 11^1 the values of current gain, voltage gain, input impedance, and output impedance are calculated as a function of load and source imped- ances. These are plotted in Figs. 1 1-14 to 1 1-17 for each of the three configura- tions. A study of the shapes and relative amplitudes of these curves is instruc- tive. The asymptotic end points of these plots (for R L or R, equal to zero or infinity) are indicated in Table 11-5. A, (CB) /cc iV(CC) Or 1.0 0.8 0.8 --A,(CE) \ CB 0.98 ^5sCC CE*^ \CB 40 30 0.4 02 20 10 ^— -0.20 10* 10 s 10* 10» 10* 10 1 R L ,a Fig. 11-14 The current gain At of the typical transistor of Table 11-2 as a function of its load resistance. 11-7 SMALL-SIGNAL LOW-FREQUENCY TRANSISTOR MODELS / 313 The CE Configuration From the curves and Table 11-5, it is observed that only the common-emitter stage is capable of both a voltage gain and a current gain greater than unity. This configuration is the most versatile and useful of the three connections. Note that Ri and R vary least with R L and R t , respectively, for the CE circuit. Also observe that the magnitudes of R< and R B lie between those for the CB and CC configurations. To realize a gain nominally equal to (Air,) BB would require not only that a zero-impedance voltage source be used, but also that R L be many times larger than the output impedance. Normally, however, so large a value of R L is not feasible. Suppose, for example, that a manufacturer specifies a maximum collector voltage of, say, 30 V. Then we should not be inclined to use a collector supply voltage in excess of this maximum voltage, since in such a case the collector voltage would be exceeded if the transistor were driven to cutoff. Suppose, further, that the transistor is designed to carry a collector current of, say, 5 mA when biased in the middle of its active region. Then the load resistor should be selected to have a resistance of about -^ = 3 K. We compute for the CE configuration a voltage gain under load of A? ■ — 129 (for R, = 0). Of course, the load resistance may be smaller than 3 K, as, >MCC) A„(CB) or -A V (CE) 3300 3000 1.0 /cc 2500 0.8 2000 0.6 /CB or CE 1500 0.4 '0.31 1000 0.2 0.45 500 *^ 1 Iff 1 10 3 10 s 10 s 10 1 R Lt a Fig. 11-15 The voltage gain of the typical transistor of Table 11-2 as a function of its load resistance. 314 / ELECTRONIC DEVICES AND CIRCUITS Sec. J J .7 . n 2.03M 10' 10" 10* 10* cc/ .,600/ CE 10* 1,100 600 503 21.6 CB 10 10 s 10 s 10* 10» io' J?j.,n Fig. 11-16 The input resistance of the typical transistor of Table 11-2 as a function of its load resistance. TABLE 11-5 Asymptotic values of transistor gains and resistances (for numerical values of h parameters see Table 11-2) Quantity (A/.)™** (B*, - 0, R, m «) Ri (Rl = 0) Ri (Rl — ™ ) (Avx)«**(flji = *,B. = 0) B„ (B. • 0) B (B, = ») A- parameter expression -A, A Aift — ArA/ CE -50 1,100 n eoo a •3,330 73.3 K 40 K 15 X 10* cc 51 l.ioo n 2.04 M 21.6 fi 40 K 51.0 CB 0.98 21.6 Q 600 a 3,330 73.5 K 2.04 M 2.94 X 1°' S*c N-7 SMALL -SIGNAL LOW-FREQUENCY TRANSISTOR MODELS / 315 f or example, when a transistor is used to drive another transistor. Or in gome applications a higher value of Rl may be acceptable, although load resistances in excess of 10 K are unusual. The CB Configuration For the common-base stage, Ai is less than unity, Av is high (approximately equal to that of the CE stage), Ri is the lowest, and R B is the highest of the three configurations. The CB stage has few applica- tions. It is sometimes used to match a very low impedance source, to drive a high-impedance load, or as a noninverting amplifier with a voltage gain greater than unity. It is also used as a constant-current source (for example, as a sweep circuit to charge a capacitor linearly 11 ). The CC Configuration For the common-collector stage, A r is high (approximately equal to that of the CE stage), Av is less than unity, Ri is the highest, and R a is the lowest of the three configurations. This circuit finds wide appli cation as a buffer stage between a high-impedance source and a low-impedance load. This use is analogous to that of the cathode follower, and this transistor circuit is called an emitter follower. Summary The foregoing characteristics are summarized in Table 11-6, where the various quantities are calculated for Rl = 3 K and for the k parame- ters in Table 1 1-2, R B ,tl 10 s w Iff Iff CB 2.04 M 105 K 1/ CE 73 K 40 K 33.2 K CC> 21.70 10 ] 10 s ]0* 10 s 10* Iff R t ,a Fig. 11-17 The output resistance of the typical transistor of Table 11-2 as a function of its source resistance. 316 / ELECTRONIC DEVICES AND CIRCUITS TABLE 11-6 Comparison of transistor configurations Sec. ?T-8 Quantity CE CC CB Aj Av Ri (Rl = 3 K) R e (R. - 3 K) High (-46.5) High (-131) Medium (1,065 tt) Medium high (45.5 K) High (47.5) Low (0.99) High (144 K) Low (80.5 ft) Low (0.98) High (131) Low (22.5 a) High (1.72 M) 11-8 LINEAR ANALYSIS OF A TRANSISTOR CIRCUIT There are many transistor circuits which do not consist simply of the CE, CB, or CC configurations discussed above. For example, a CE amplifier may have a feedback resistor from collector to base, as in Fig. 10-5, or it may have an emitter resistor, as in Fig. 10-6. Furthermore, a circuit may consist of several transistors which are interconnected in some manner. An analytic determina- tion of the small-signal behavior of even relatively complicated amplifier cir- cuits may be made by following these simple rules: 1. Draw the actual wiring diagram of the circuit neatly. 2. Mark the points B (base), C (collector), and E (emitter) on this circuit diagram. Locate these points as the start of the equivalent circuit. Maintain the same relative positions as in the original circuit. 3. Replace each transistor by its A-parameter model (Table 11-1). 4. Transfer all circuit elements from the actual circuit to the equivalent circuit of the amplifier. Keep the relative positions of these elements intact. 5. Replace each independent dc source by its internal resistance. The ideal voltage source is replaced by a short circuit, and the ideal current source by an open circuit. 6. Solve the resultant linear circuit for mesh or branch currents and node voltages by applying Kirchhoff s current and voltage laws (KCL and KVL). It should be emphasized that it is not necessary to use the foregoing general approach for a circuit consisting of a cascade of CE, CB, and/or CC stages. Such configurations are analyzed very simply in Chap. 12 by direct applications of the formulas in Table 11-4. 11-? THE PHYSICAL MODEL OF A CB TRANSISTOR The circuit designer finds the small-signal model of the transistor described by the hybrid parameters very convenient for circuit analysis. As indicated in Sec. 11-1, these h parameters characterize a general two-port network. When this model is applied to a specific transistor, the values of the hybrid parameters are measured experimentally (Sec. 11-4) by the user or by the Sec. 71-9 SMALL-SIGNAL LOW-FREQUENCY TRANSISTOR MODELS / 317 manufacturer. The device designer, on the other hand, prefers to use a model containing circuit parameters whose values can be determined directly from the physical properties of the transistor. We now attempt to obtain such a small-signal equivalent circuit which brings into evidence the physical mecha- nisms taking place within the device. To be specific, consider the grounded-base configuration. Looking into the emitter, we see a forward-biased diode. Hence, between input terminals E and B', there is a dynamic resistance r' et obtained as the slope of the (forward- biased) emitter-j unction volt-ampere characteristic, Looking back into the output terminals C and B f , we see a back-biased diode. Hence, between these terminals, there is a dynamic resistance r c obtained as the slope of the (reverse-biased) collector-junction volt-ampere characteristic. From the physical behavior of a transistor as discussed in Chap. 9, we know that the collector current is proportional to the emitter current. Hence a current generator od» is added across r' c , resulting in the equivalent circuit of Fig. 11-18. The Early Feedback Generator The equivalent circuit of Fig. 11-18 is unrealistic because it indicates a lack of dependence of emitter current on collector voltage. Actually, there is some such small dependence, and the physical reason for this relationship is not hard to find. As indicated in Sec. 9-7, an increase in the magnitude of the collector voltage effectively narrows the base width W, a phenomenon known as the Early effect* The minority-carrier current in the base in the active region is proportional to the slope of the injected minority-carrier density curve. From Fig. 9-23 we see that this slope increases as W decreases. Hence the emitter current injected into the base increases with reverse collector voltage. This effect of collector voltage Vd,> on emitter current may be taken into account by including a voltage source fiv e b> in series with r' e , as indicated in Fig. 11-19. A little thought should convince the reader that the polarity shown for generator Miw is consistent with the physical explanation just given. The Base-spreading Resistance To complete the equivalent circuit of Fig. 11-18, we must take into account the ohmic resistances of the three transistor regions. Since the base section is very thin, the base current passes through a region of extremely small cross section. Hence this resistance rw, called the base-spreading resistance, is large, and may be of the order of several Emitter E Collector C F '9- 11-18 A simplified physical ""odel of a CB transistor. f^ V,b -I r.' ai,(T) Base B' n + 318 / ELECTRONIC DEVICES AND CIRCUITS V,b C.6 <'.(♦) Web 1 Tb iz U& Soc. I ] .p Fig. 11-19 A more com- plete physical model of a CB transistor than that indicated in Fig. 11-18. hundred ohms. On the other hand, the collector and emitter ohmic resistances are only a few ohms, and may usually be neglected. If the external connection to the base is designated by B, then between the fictitious internal base node B' and B we must place a resistance r w , as indicated in Fig. 11-19. If the base-spreading resistance could be neglected so that B and B' coincided, the circuit of Fig. 11-19 would be identical with the hybrid model of Fig. 11-2, with r' e = k& n = Art, a, ** —h» and r' c = j— hob The T Model The circuit of Fig. 11-19 contains elements each of which has been identified with the physics of the transistor. However, this circuit, which includes a dependent voltage generator, a dependent current generator, and three resistors, is fairly complicated to use in circuit analysis. By means of network transformations it is possible to eliminate the voltage generator and obtain the simpler T model of Fig. 11-20. This new circuit should be considered in conjunction with Table 11-7. This tables gives the transforma- tion equations and, in addition, specifies typical values of the parameters in each of the circuits. The derivation of the equations of transformation is an entirely straightforward matter. It is necessary only to find ^ as a function of i t and *,. (and also to determine w rt as a function of t, and i e ) for both circuits and to require that the corresponding equations be identical TABLE 1 1-7 Typical parameter values and the equations of transformation between the circuits of Figs. 11-19 and 11-20 Parameter in Fig. 11-1*1 Transformation equations Parameter in Fig. 11-20 t, = 40 n - 5 X 10-* r'„ - 2 M a m 0.98 r, = r' e — (I ~ a)fir' c K = (*i r. - (1 - tt)r e a — ft a = 1 — » r, = 20 a K - 1 K r c = 2M a = 0.98 Sec TI-TO SMALL-SIGNAL LOW-FREQUENCY TRANSISTOR MODELS / 319 Fig.' 11 -20 The T model of a CB tran- sistor. The transformed circuit, we observe, accounts for the effect of the collector circuit on the emitter circuit essentially through the resistor r 6 rather than through the generator fiv cb >. Note from Table 11-7 that r c « r e , a « a, and r , m rJ2. The resistor r b in the base leg is given by n m r' b + rfe, where r' b and rw are resistances of comparable magnitudes. The circuit components in the T model cannot be interpreted directly in terms of the physical mechanisms in the transistor. Values for these elements are difficult to obtain experimentally. And, finally, the analysis of a circuit is somewhat simpler in terms of the h parameters than through the use of the T equivalent circuit. For these three reasons the T model is not used in this text. It is included here because of its historical significance and because we refer to this circuit when we discuss the transistor at high fre- quencies {Sec. 13-1). The relationships between the hybrid parameters and those in the T equivalent circuit are given in Table 11-3. 11-10 A VACUUM-TUBE— TRANSISTOR ANALOGY 7 It is possible to draw a very rough analogy between a transistor and a vacuum tube. In this analogy the base, emitter, and collector of a transistor are identified, respectively, with the grid, cathode, and plate of a vacuum tube. Correspondingly, the grounded-base, grounded-emitter, and grounded-collector configurations are identified, respectively, with the grounded-grid, grounded- cathode, and grounded-plate (cathode-follower) vacuum-tube circuits, as in *'ig. H-21. Consider, for example, the circuits of Fig. ll-21a. For the tube circuit, w e find that, in the normal amplifier region, |/ t | = \I P \. In the transistor circuit, in the active region, we find that \I t \ « |/ e |, the difference between J« and ] c being of the order of 2 percent. In both the transistor and tube cir emts of Fig. ll-21a, we find that the input impedance is low because of the * ar ge current at low voltage which must be furnished by the driving generator. Also, both circuits are capable of considerable voltage gain without inverting th e input signal. The transistor configuration of Fig. 11-216 has a higher input imped- 320 / ELECTRONIC DEVICES AND CIRCUITS ?¥] o ■ T T V ri («> (*) (c) Fig, 11-21 Analogous transistor and vacuum-tube circuits, (a) Grounded base and grounded grid, (b) Common emitter and common cathode, (c) Emitter fol- lower and cathode follower. ance than the CB circuit. As a voltage amplifier, a large gain with polarity inversion is possible. In all these respects the grounded-emittar configuration is analogous to the grounded-cathodc vacuum-tube amplifier stage. In Fig. ll-21c, the grounded-collector (emitter-follower) configuration is compared with the grounded-plate (cathode-follower) circuit. In the emitter- follower circuit the input current is relatively small, and the voltage difference between base and emitter is essentially the small voltage drop aeross the for- ward-biased emitter junction when operating in the active region. Hence we may expect the input voltage and the output voltage, as in a cathode follower, to be nominally the same. The emitter follower, as the cathode follower, provides a voltage gain slightly less than unity without polarity inversion. The emitter follower may also be expected to handle an input signal comparable in size with the collector supply voltage. The input^current swing from cutoff to saturation is the same for grounded-emitter and grounded-collector opera- tion, but in the grounded-collector operation the input-voltage swing is larger. The cutoff region of the transistor corresponds to the region in the vacuum tube where the grid voltage is larger than the cutoff bias. The active region of the transistor corresponds to the region in which the tube operates as a linear amplifier. This region covers not only the region within the grid base, but also the region of positive grid voltages, where the tube operates linearly- S« U-IO SMAU-SIGNAl lOW-FRfQl/fNCV TRANSISTOR MODELS / 321 The saturation region of the transistor corresponds to the tube region where the grid is so positive and the plate voltage is so low that the plate current is almost independent of grid voltage (Fig. D-3). The transistor base takes cur- rent at all points in its active region, whereas in the tube the grid draws appreciable current only when it is positive. The analogy may be improved by assuming that cutoff occurs in the tube at zero grid bias; i.e., the grid base is zero. Also, the volt-ampere transistor characteristics are shaped more like pentode curves than like triode characteristics. It need hardly be emphasized that the analogies drawn above are far from exact. On several occasions we have already noted that a transistor is a more complicated device than a vacuum tube. In the former the current is due to charge carriers of both signs moving in a solid, whereas in the latter the cur- rent is carried by electrons in a vacuum. There is nothing in a vacuum tube corresponding to minority-carrier storage in a transistor. The low-frequency input impedance of a grounded-cathode or cathode-follower circuit is infinite, whereas a transistor has a relatively low input impedance in all three con- figurations. The low-frequency equivalent circuit of a tube contains only two parameters, m and r p (or g m and r p ) l whereas four parameters, hu, hn, /i 22 , and An, are required in the corresponding transistor small-signal equivalent circuit. The analogies are principally useful as mnemonic aids. For example, we may note that the most generally useful tube circuit is the grounded-cathode circuit. We may then expect from our analogy that the grounded-emitter configuration will occupy the same preferred position in the transistor con- figurations. This anticipated result is borne out in practice. Tubes versus Transistors The semiconductor device has replaced the vacuum tube in many applications because the bipolar transistor possesses the following advantages over the tube: 1. The transistor has no filament, and hence requires no standby power or heating time. 2. It is smaller and lighter than a tube. 3. It has longer life and hence greater reliability. 4. It may operate with low voltages and power dissipation. 5- It is mechanically more rugged and cannot be microphonic. 6- It is a more ideal switch. 7. The transistor is readily adapted to microminiaturization, as described m Chap. 15 on Integrated Circuits. 8- Because there are two types of transistors (n-p-n and v-n-p) t some cir- s resigns are possible which have no tube counterparts. Wherever space, weight, or power is at a premium, the circuits are tran- orized. Digital computers (large-scale or special-purpose), hearing aids, ■ctronic circuits for space vehicles, and portable equipment fall into this °gory. The future of extremely complex systems lies in the direction of cfominiaturization, using transistor technology. However, there are appli- 322 / ELECTRONIC DEVICES AND CIRCUITS Sec. U-JQ cations where the tube will continue to be used because of the following dis- advantages of the semiconductor triode: 1. The transistor characteristics are very temperature-sensitive. 2. The transistor is damaged by nuclear radiation. 3. It is easily damaged by transient overloads. 4. The maximum output power (100 W) is lower than from a tube (300 kW). 5. The upper frequency response (1 GHz) is lower than from a tube (10 GHz). 6. It is difficult to obtain voltage swings in excess of about 100 V. 7. Under some operating conditions transistors are noisier than tubes. 8. The spread in the characteristics of a given type of transistor is often very great. Systems involving high voltage, high power, or high frequencies (and par- ticularly those requiring several of these characteristics simultaneously) use tubes. Such applications include communications transmitters, radar indi- cators, oscilloscopes, and test equipment. Systems which must operate under unusual environments of temperature or nuclear radiation use tubes. Also, systems designed some time ago, and still operative, use tubes. Such equip- ment is often in production today because it is not economically feasible to redesign the system using semiconductor devices. REFERENCES 1 . IRE Standards on Semiconductor Symbols, Proc. IRE, vol. 44, pp. 935-937, July, 1956. 2. "Transistor Manual," 7th ed., General Electric Co., pp. 52-55, Syracuse, N.Y., 1964. 3. Ref. 2, pp. 477-482. 4. Electronics Reference Sheet, Electronics, Apr. 1, 1957, p. 190. 5. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," pp. 528- 532, McGraw-Hill Book Company, New York, 1965. 6. Early, J. M.: Effects of Space-charge Layer Widening in Junction Transistors, Proc. IRE, vol. 40, pp. 1401-1406, November, 1952. 7. Giacoletto, L. J.: Junction Transistor Equivalent Circuits and Vacuum-tube Analogy, Proc. IRE, vol. 40, pp. 1490-1493, November, 1952. Dosse, J.: "The Transistor," pp. 104-123, D. Van Nostrand Company, Inc., Prin ce * ton, N.J., 1964. 12 /LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS In the preceding chapter we consider the small-signal analysis of a single stage of amplification. Very often, in practice, a number of stages are used in cascade to amplify a signal from a source, such as a phonograph pickup, to a level which is suitable for the operation of another transducer, such as a loudspeaker. In this chapter we con- sider the problem of cascading a number of transistor amplifier stages. In addition, various special transistor circuits of practical importance are examined in detail. Also, simplified approximate methods of solu- tion are presented. All transistor circuits in this chapter are examined at low frequencies, where the transistor internal capacitances may be neglected. 12-1 CASCADING TRANSISTOR AMPLIFIERS 1 When the amplification of a single transistor is not sufficient for a particular purpose, or when the input or output impedance is not of the correct magnitude for the intended application, two or more stages may be connected in cascade; i.e., the output of a given stage is con- nected to the input of the next stage, as shown in Fig. 12-1. In the circuit of Fig. 12-2a the first stage is connected common-emitter, and the second is a common-collector stage. Figure 12-2o shows the small- signal circuit of the two-stage amplifier, with the biasing arrangements omitted for simplicity. In order to analyze a circuit such as the one of Fig. 12-2, we make use of the general expressions for Ar, Z t , Av, and Y a from Table 11-4. It is necessary that we have available the h parameters for the specific transistors used in the circuit. The /i-parameter values for a specific transistor are usually obtained from the manufacturer's data sheet. 323 324 / ELECTRONIC DEVICES AND CIRCUITS Sac. 12- J '/- :*, Fig. 12-1 Two cascaded stages. Since most vendors specify the common-emitter h parameters, it may be necessary (depending on whether a certain stage is CE, CC, or CB) to con- vert them with the aid of Table 11-3 to the appropriate CC or CB values. In addition, the k parameters must be corrected for the operating bias con- ditions (Fig. 11-5). EXAMPLE Shown in Fig. 12-2 is a two-stage amplifier circuit in a CE-CC con- figuration. The transistor parameters at the corresponding quiescent points are h it = 2 K h ic = 2 K h ft = 50 k fe = -51 h„ = 6 X 10" 4 A rc = 1 A M = 25 vA/V ho, = 25 fiA/Y Find the input and output impedances and individual, as well as overall, voltage and current gains. Solution We note that, in a cascade of stages, the collector resistance of one stage is shunted by the input impedance of the next stage. Hence it is advantageous to start the analysis with the last stage. In addition, it is convenient (as already noted in Sec. 11-6) to compute, first, the current gain, then the input impedance and the voltage gain. Finally, the output impedance may be calculated if desired by starting this analysis with the first stage and proceeding toward the output stage. The second stage. From Table 11-4, with R L = R*2, the current gain of the last stage is An - — -— - = -A, 51 1 + 25 X 10"« X 5 X 10 J - 45.3 ifti 1 + hgcReZ The input impedance fl, 2 is Ra = ku + h rt A I2 R e2 = 2 + 45.3 X 5 = 228.5 K Note the high input impedance of the CC stage. The voltage gain of the second stage is . V R tl 45.3 X 5 Set- 12-1 1 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 325 «+ V cc o-V, (a) Fig. 12-2 (a) Common-emitter— common-collector amplifier, (b) Small-signal circuit of the CE-CC amplifier. (The component values refer to the example in Sec. 12-1.) The first stage. We observe that the net load resistance Rn of this stage is the parallel combination of R tt and R i2 (written in symbolic form, Rli = Rt\\\Ra), or R el R it 5 X 228.5 Rli = Re\ + Ril 233.5 = 4.9 K Hence -A, -50 An- -- = I bl 1 + h et Rn 1 + 25 X 10" B X 4.9 X 10 s = -44.5 The input impedance of the first stage, which is also the input impedance of the cascaded amplifier, is given by Ru = kt, + K c A n Ru = 2 - 6 X 10~* X 44.5 X 4.9 = 1.87 K The voltage gain of the first stage is V % A n Ru -44. 5 X 4.9 Atrt = — = = = — llo.o r, Ra 1.87 326 / ELECTRONIC DEVICES AND CIRCUITS >ec. J2.| The output admittance of the first transistor is, from Eq. (11-39) or Table iy 1'.. - h„ - kfjlr hu + R, = 15 nA/V = 2o X 10"« - = 15 X 10-s mkft 2 X 10 3 + 1 X 10 3 nno Hence 1 10* R ol = — = — 12 = 66.7 K Y Bl 15 The output impedance of the first stage, taking R Bl into account, is /J c il|ft,[, or ^ i = RdRoi _ 5 X 66.7 Rcl + Rax ~ 5 + 66.7 = 4.65 K The output resistance of the last stage. The effective source resistance R',. for the second transistor Q2 is R ol \\R ti . Thus R'^ = R' ol = 4.65 K, and Y . = h — I az 'tor A/cAt A.- B + ^,2 T = 25 X 10-" - (-5I)(1) 2 X 10 3 + 4.65 X 10 3 = 7.70 X 10-' A/V Hence R oi = 1/F 2 = 130 O, where R o2 is the output impedance of transistor Q2 under open-circuit conditions. The output impedance R' g of the amplifier, taking R ei into account, is R B 2\\R e i, or Hi* Ro2Rt2 130 X 5.000 - 127 £2 K„2 + fl e:! 130 + 5,000 The overall current and voltage gains. The total current gain of both sta^ Ai = lc2 hi lb2 /cl hx hx From Fig 12-3, we have lit RcX hx fl, s + RcX Hence 4#» A/jA/i — R c i = 45.3 > 1 ^ bi A Art r- An l*i Rn + Rex 228.5 + 5 -43.2 (12-1) (12-2) C, o- I*x Fig. 12-3 Relating to the calculation of overall current gain. N o— For the voltage gain of the amplifier, we have LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 327 7, V 2 Vi = AviAx (12-4) or Av = 0.99 X (-116.6) m -115 The voltage gain can also be obtained from Ar - A.& - -43.2 X £ - -115 The overall voltage gain, taking the source impedance into account, is given by Av ' = vr Av R~x~fR. = -us* *: 8 ! i m - 75 - 3 1.87 + 1 Table 12-1 summarizes the results obtained in the solution of this problem. TABLE J2-7 Results of the examp e on page 324 Transistor Q2 CC Transistor Ql CE Both stages CE-CC At Ri Ay K 45.3 228.5 K 0.99 127 -44.5 1.87 K -116,6 4.65 K -43.2 1.87 K -115 127 fi 1 12-2 n-STAGE CASCADED AMPLIFIER The function of a low-level amplifier is to raise a weak signalto a usable level, perhaps from the range of microvolts to several volts. This is usually done by cascading several transistors in the common-emitter connection. A typi- cal two-stage cascaded CE audio amplifier with biasing arrangements and coupling capacitors included is shown in Fig. 12-4. We now examine in detail the small-signal operation of an amplifier con- sisting of n cascaded common-emitter stages, as shown- in Fig. 12-5. The biasing arrangements and coupling capacitors have been omitted for simplicity. The Voltage Gain We observe from Fig. 12-5 that the resultant voltage gain is given by the product of the individual voltage gains of each stage. A "is statement is verified as follows: = F 2 _ output voltage of first stage = A / 6i Vl = y~j — input voltage of first stage 328 / ELECTRONIC DEVICES AND CIRCUITS 12V Soc J 2.} (22K) >(6K) (5mF) jQI 2N338 *,; (16K) >(700n) ,Q2 2N338 (3.3K) <R tl ^ (50uF , S(G.2K)< R J_(50 >(1K)T ( ^ ' f > ' 2 T>F) 1(4700) >F) 1 — i Fig. 12-4 Practical two-stage CE audio amplifier. (Courtesy of Texas Instruments, Inc.) where A i is the magnitude of the voltage gain of the first stage, and tfi is the phase angle between output and input voltage of this stage. Similarly, 4 _ Vk+\ m output voltage of fcth stage _ . . Vk ~ V k input voltage of kth stage " k/ — The resultant voltage gain is defined as j. m V^ _ output voltage of nth stage _ . V\ input voltage of first stage ~ — Since V, V, V 2 V 3 F_, v n it follows from these expressions that Ay = AvxAvz • ■ ' Ay n = A t As • ■ ■ A n /8i + $ t + ■ - ■ or A = AlAl • - • An S = 01 + $ 2 + + 8* = (12-5) (12-6) ■ + 6 n (12-7) The magnitude of the voltage gain equals the product of the magnitudes of tht voltage gains of each stage. Also, the resultant phase shift of a multistage amph' fier equals the sum of the phase skifts introduced by each stage. The voltage gain of the Ath stage is, from Table 11-4, AikRhk A V k = Rik (12-8) where Ri k is the effective load at the collector of the ftth transistor. ™ e quantities in Eq. (12-8) are evaluated by starting with the last stage and p 1 **" S<* 12-2 IOW-EREQUENCV TRANSISTOR AMPLIFIER CIRCUITS / 329 ding to the first. Thus the current gain and the input impedance of the C |h stage are given in Table 11-4, respectively, as At* = — h fe 1 + h ot Rln Rin = hie + hreAl„Rl (12-9) w here Rl« = R™- The effective load R L , n -i on the (n - l)st stage is R Cin ^Rin (12-10) Now the amplification Ar, n -i of the next to the last stage is obtained from Eq (12-9) by replacing R L « by R L .n-i- The input impedance of the (n - l)st stage is obtained by replacing n by n - 1 in Eq. (12-9). Proceeding in this manner, .we can calculate the base-to-collector current gains of every stage, including the first. From Eq. (12-8) we then obtain the voltage gain of each stage. The Current Gain Without first finding the voltage amplification of each stage as indicated above, we can obtain the resultant voltage gain from v ~ Al W l cl = li F *cl< V St c 2 Q2 -o— — o- /*-■ h„ Rc2< Vi *«..-.' (12-11) /«, = h R t \ -R« (a) *• G) 7 ' < R - *■>■ ►*i *E t (b) (c) 1 *9- 12-5 (o) n transistor CE stages in cascade, (e) The &th stage, (c) The '•"onsistor input stage driven from a current source. 330 / ELECTRONIC DEVICES AND CIRCUITS See. 1 2- 2 where Ai is the current gain of the n-stage amplifier. Since Ai is defined aa the ratio of the output current I of the last stage to the input (base) current hi of the first stage, /, u hi hi (1242) where I cn s /„ is the collector current of the nth stage. We now obtain expressions from which to calculate Ai in terms of the circuit parameters. Since 1* hi lib hi 1 1 h—1 In h—2 in— 1 then where Ai = AiiA' n ■ ■ ■ A'i^A'^ d hi An = - — - - 42 A' = Ibl lbi A. h-1 (12-13) (12-U) Note that An is the base-to-collector current gain of the first stage, and A' lk is the collector-to-collector current gain of the kth stage (Jfc = 2, 3, . . . , n). We now obtain the relationship between the collector-to-collector current gain A Ik = It/h-i and the base-to-collector current amplification h A n = ~~ *bk where I ek m Ik is the collector current and hk is the base current of the fcth stage. From Fig. 12-56, hk — — I*_l Hence Alt — Rc.k-] + Rik h h hk AiicRe.k—l h-i hk h-i Rc,k-i + Rit (12-15) (12-16) The base-to-collector current gain A Tk is found by starting with the output stage and proceeding to the fcth stage, as indicated above in connection with Eqs. (12-9) and (12-10). The collector-to-collector gains are then found froffl Eq, (12-16), and the current gain of the n-stage amplifier, from Eq. (12-13). If the input stage of Fig. 12-5a is driven from a current source, as indicated in Fig. 12-5c, the overall current gain is given by Ai t = Ai R, R, + Rh (12-17) Input and Output Impedances The input resistance of the amplifier is obtained, as indicated above, by starting with the last stage and proceeding toward the first stage. The output impedance of each transistor stage and of the overall ampler Sec 12-2 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 331 • g calculated starting with the first stage and using Eq. (11-34). The output 'edance R' ok of the fcth stage is the parallel combination of the output j mpe dancc R„k of transistor QK and R ck . The effective source impedance of the (fc + !) st sta S e is also R *- of Power Gain The total power gain of the n-stage amplifier is output power VJ a . , A P = - — , = - «-r~ ■ AvAt input power V ihi Ren A P = (A,) 2 RiX (12-18) (12-19) Choice of the Transistor Configuration in a Cascade It is important to note that the previous calculations of input and output impedances and voltage and current gains are applicable for any connection of the cascaded stages. The discussion has assumed that all stages are CE. However, they could be CC, CB, or combinations of all three possible connections. Consider the following question : Which of the three possible connections must be used in cascade if maximum voltage gain is to be realized? For the intermediate stages, the common-collector connection is not used because the voltage gain of such a stage is less than unity. Hence it is not possible (without a transformer) to increase the overall voltage amplification by cascading com- mon-collector stages. Grounded- base SC-coupled stages also are seldom cascaded because the voltage gain of such an arrangement is approximately the same as that of the out- put stage alone. This statement may be verified as follows : The voltage gain of a stage equals its current gain times the effective load resistance Rl divided by the input resistance R t . The effective load resistance Rl is the parallel combination of the actual collector resistance R c and (except for the last stage) the input resistance Ri of the following stage. This parallel combination is certainly less than R it and hence, for identical stages, the effective load resistance is less than R t . The maximum current gain is h Jb , which is less than unity (but approximately equal to unity). Hence the voltage gain of any stage (except the last, or output, stage) is less than unity. (This analysis * not strictly correct because the R { is a function of the effective load resistance and hence will vary somewhat from stage to stage.) Since the short-circuit current gain h /e of a common-emitter stage is m "ch greater than unity, it is possible to increase the voltage amplification by fading such stages. We may now state that in a cascade the intermediate tr< insistors should be connected in a common-emitter configuration. The choice of the input stage may be decided by criteria other than the Maximization of voltage gain. For example, the amplitude or the frequency f es Ponse of the transducer V, may depend upon the impedance into which 11 operates. Some transducers require essentially open-circuit or short-circuit 332 / ELECTRONIC DEVICES AND CIRCUITS Sec 12-3 operation. In many cases the common-collector or common-base stage [» used at the input because of impedance considerations, even at the expense of voltage or current gain. Noise is another important consideration which may determine the selection of a particular configuration of the input stage. 12-3 THE DECIBEL In many problems it is found very convenient to compare two powers on a logarithmic rather than on a linear scale. The unit of this logarithmic scale is called the decibel (abbreviated dB). The number N of decibels by which the power P 2 exceeds the power Pi is defined by AT = 10 log 1J * 1 (12-20) It should be noted that the specification of a certain power in decibels is meaningless unless a standard reference level is implied or is stated specifically. A negative value of N means that the power P 2 is less than the reference power Pi. If the input and output impedances of an amplifier are equal resistances, then P 2 = Vf/R and Pi = Vf/R, where V 2 and Fi are the output and input voltage drops. Under this condition, Eq. (12-20) reduces to N = 20 log ^ = 20 log Ay (12-21) where A v is the magnitude of the voltage gain of the unit. The input and output resistances are not equal, in general However, this expression is adopted as a convenient definition of the decibel voltage gain of an amplifier, regardless of the magnitudes of the input and output resistances. That is, if the voltage amplification is 10, its decibel voltage gain is 20; if the voltage amplification is 100, the decibel voltage gain is 40; etc. If there is the possi- bility of confusion between voltage and power gain, the designation dBV can be used for decibel voltage gain. The logarithm of the magnitude of the expression for voltage gain in Eq. (12-7) is given by log A v = log Ai + log A t + + lOg An (12-22) By comparing this result with Eq. (12-21), which defines the decibel voltage gain, it is seen that the overall decibel voltage gain of a multistage amplifier « the sum of Ike decibel voltage gains of the individual stages. The foregoing considerations are independent of the type of interst:»g e coupling and are valid for both transistor and vacuum-tube amplifiers. Ho*' ever, it must be emphasized that, in calculating the gain of one stage, the loading effect of the next stage must be taken into account. Sec 12-4 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 333 l2 _ 4 SIMPLIFIED COMMON-EMITTER HYBRID MODEL 2 In the preceding chapter, and also in Sec. 12-1, we carried out detailed calcula- tions of current gain, voltage gain, input, and output impedances, of illustrative transistor amplifier circuits. In most practical cases it is appropriate to obtain approximate values of Ai, Av, A P , Ri, and R<> rather than to carry out the more lengthy exact calculations. We are justified in making such approximations because the h parameters themselves usually vary widely for the same type of transistor. Also a better "physical f ee ling" for the behavior of a transistor circuit can be obtained from a simple approximate solution than from a more laborious exact calculation. Since the common-emitter connection is in general the most useful, we first concentrate our attention on the CE ft-parameter model shown in Fig. 12-6a. How can we modify this model so as to make the analysis simple without, greatly sacrificing accuracy? Since l/h oe in parallel with R h is approximately equal to R L if UK, » Rl* then h ot may be neglected in Fig. 12-0a provided that k et R L «\- Moreover, if we omit h Be from this figure, the collector current I c is given by I t = h f J b . Under these circumstances the magnitude of the voltage of the generator in the emitter circuit is K*\V,\ - KJ c Rl - kjifjtj* Since h rt h it ~ 0.01, this voltage may be neglected in comparison with the h ie l b drop across *,-„ provided that R L is not too large. We therefore conclude that if the load resistance R L is small, it is possible to neglect the parameters K. and h oe in the circuit of Fig. 12-6a and to obtain the approximate equivalent o — j — VW- ft*, R. < + I 1 V„ h„V e Q (j)h fe *-^r* F '9. 12-6 (a) Exact CE hybrid model; (b) ap- proximate CE model. 334 / ELECTRONIC DEVICES AND CIRCUITS Sec. I2.4 Fig. 12-7 Approximate hybrid model which may be used for afl three con- figurations, CE, CC, or CB. circuit of Fig. 12-66. We are essentially making the assumption here that the transistor operates under short-circuit conditions. In subsequent discus. sion we investigate the error introduced in our calculations because of the nonzero load resistance. Specifically, we show that if h„R L < I the error in calculating A r , R h A v , and R' B for the CE connection is less than 10 percent. Generalized Approximate Model The simplified hybrid circuit of Fig 12-7 which we used in Fig. 12-66 for the CE circuit may also be used for the CO (or the CB) connection by simply grounding the appropriate terminal, lne signal is connected between the input terminal and ground, and the load is placed between the output terminal and ground. We examine in detad in the following sections the errors introduced in our calculations by using the simplified model of Fig. 12-7 for the analysis of the CC and CB connections. In summary, we claim that two of the four h parameters, k u and hf„ are sufficient for the approximate analysis of low-frequency transistor circuits, provided the load resistance R L is no larger than 0.1/A M . For the value of k ae given in Table 1 1-2, R L must be less than 4 K. The approximate circuit is always valid when CE transistors are operated in cascade because the low input impedance of a CE stage shunts the output of the previous stage so that the effective load resistance R' L satisfies the condition hjft' h < 0.1. We now justify the validity of the proposed simplification for the CB configuration. Current Gain From Table 1 1-4 the CE current gain is given by A r = — ~ hfe 1 + h oe R L Hence we immediately see that the approximation (Fig. 12-66) Ar «* —hf t (12-23) (12-24) overestimates the magnitude of the current gain by less than 10 percent if h ot R L < 0.1. Input Impedance From Table 11-4 the input resistance is given by Ri = h u + hreArRt (12-25) 12-5 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 335 (12-26) ff hich may be put in the form Using the typical A-parameter values in Table 11-2, we find h Te h; e /hi t h oe ~ 0.5. From Eq. (12-23), we see that \A r \ < h ft . Hence, if KMl < 0.1, it follows from Eq. (12-26) that the approximation obtained from Fig. 12-66, namely, Ri — ~r~ ** *<* lb overestimates the input resistance by less than 5 percent. Voltage Gain From Table 11-4 the voltage gain is given by (12-27) Av - At Ri h/iRi, (12-28) Ri hie If we take the logarithm of this equation and then the differential, we obtain dAv _ dAr dRj (12-29) A Y " IT " ~Rl From the preceding discussion the maximum errors for KMl < 0.1 are ^ = +0.1 At and -n~ l = +0.05 Hence, the maximum error in voltage gain is 5 percent, and the magnitude of Ay is overestimated by this amount. Output Impedance The simplified circuit of Fig. 12-66 has infinite out- put resistance because, with V, = and an external voltage source applied at the output, we find h = 0, and hence I c = 0. However, the true value depends upon the source resistance R, and lies between 40 and 80 K (Fig. 11-17). For a maximum load resistance of Ri = 4 K, the output resistance of the stage, taking R L into account, is 4 K, if the simplified model is used, and the parallel combination of 4 K with 40 K (under the worst case), if the exact solution is used. Hence, using the approximate model leads to a value °f output resistance under load which is too large, but by no more than 10 Percent. The approximate solution for the CE configuration is summarized in the frst column of Table 12-2. 12 ' 5 SIMPLIFIED CALCULATIONS FOR THE COMMON-COLLECTOR CONFIGURATION *We 12-8 shows the simplified circuit of Fig. 12-7 with the collector grounded With respect to the signal) and a load Rl connected between emitter and ©"c-und. 33d / ELECTRONIC DEVICES AND CIRCUITS Sec 72. j Fig, 12-8 Simplified hybrid model for the CC circuit. Current Gain From Fig. 12-8 we see that At i - g w 14. ^ From Tables 11-4 and 11-3, the exact expression for Ai is — A /c 1 + A /B A, m 1 + h oe R L 1 + h oe R L (12-30) (12-31) Comparing these two equations, we conclude that when the simplified equivalent, circuit of Fig. 12-8 is used, the current gain is overestimated by less than 10 percent if h ot R L < 0.1. Input Resistance From Fig. 12-8, we obtain Rt = T b = hit + (1 + hft)RL (12-32) Note that ft » h u « 1 K even if R L is as small as 0.5 K, because h /t » 1. The expression for ft is, from Tables 11^ and 11-3, ft = h ic + k rc AiR L = h it 4- AiR L (12-33) where we have neglected h n (~2.5 X 10"*) compared with unity, and hence have written h TC = 1 - h re = 1. If we substitute from Eq. (12-30) in (12-33), we obtain Eq. (12-32). However, we have just concluded that Eq. (12-30) gives too high a value of Ai by at most 10 percent. Hence it follows that Hi, as calculated from Eq. (12-32) or Fig. 12-8, is also ovemstimated by less than 10 percent. Voltage Gain If Eq. (12-29) is used for the voltage gain, it follows fro* the same arguments as used in the CE case that there will be very little error in the value of A v . An alternative proof is now given. The voltage gain of the emitter- follower is close to unity, and we obtain an expression for its devi- ation from unity. Using Eq. (12-33), 1 _ a v = 1 _ a iRl = Ri ~ A T R L ' hu ft ft Ri il-j-34) SfC j 2-5 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 337 TABLE 12-2 Summary of approximate equations for A oe (ft -f Rj,) <0.1f CE CE with R t CC CB A, -h f . -A/. 1 +h /t L h/. " ~ 1 + *,. Ri hi. hi. + (1 + h f .)R t hu + (1 + A/,) fix. A, hit 1 + k„ Ay h/.Ri, hi. h/ t Rz, rT hi. Rl hi. R„ 00 at R, + hi. 1 +h f . SO K R L R L R \\Rl R L t (Ri)cB is an underestimation by less than 10 percent. All other quan- tities except R 9 are too large in magnitude by less than 10 percent. This expression is nearly exact since the only approximation made is that hn, = 1 — A M is replaced by unity. If, for example, ft = 10A,a, then Ay — 0.9. If, however, we use an approximate value of Ri which is 10 percent too high, then hu/Ri = tt = 0.09 and Av = 0.91. Hence the approximate calculation for A v gives a value which is only 1 percent too high. Output Impedance In Fig. 12-8 the open-circuit output voltage is V, and the short-circuit output current is r M /, X t vr (1 +h;.)V, Hence the output admittance of the transistor alone is, from Eq. (8-22), (12-35) Y L * + h f V, hit + Rt From Tables 11-4 and 11-3, the expression for Y„ is Y« = h M — hrji fclt-rc hic + R. Kt + 1 + h f€ kit + % (12-36) *^ven if we choose an abnormally large value of source resistance, say *■ = 100 K, then (using the typical A-parameter values in Table 11-2) we jhd that the second term in Eq. (12-36) is large (500 jiA/V) compared with h e first term (25 /1A/V). Hence the value of the approximate output admit- [* nc e given by Eq. (12-35) is smaller than the value given by Eq. (12-36) y less than 5 percent. The output resistance R„ of the transistor, calculated 338 / ELECTRONIC DEVICES AND CIRCUITS from the simplified model, namely, hi. + R, R„ = 1 + h fr Sec, 1 2-5 (12-37) is an overestimation by less than 5 percent. The output resistance Rl of the stage, taking the load into account, is R a in parallel with R L . The approximate solution for the CC configuration is summarized in the third column of Table 12-2. EXAMPLE Carry out the calculations for the two-stage amplifier of Fig, ]2-2 using the simplified model of Fig. 12-7. Solution First note that, since h„R L = 25 X 10" 6 X 5 X 10 3 = 0.125, which is slightly larger than 0.1, we may expect errors in our approximation somewhat larger than 10 percent. For the CC output stage we have, from Table 12-2, An = 1 + h fe - 51 Rn = h it +(l+ h St )R L = 2 + (51) (5) = 257 K Avt = AjtRt (51)(5) Rn 257 or alternatively, = 0.992 An = 1 - ~ = 1 - -?- = 0.992 jSij 257 For the CE input stage, we find, from Table 12-2, An = -h /e = -50 Rn =A i( = 2K The effective load on the first stage, its voltage gain, and output impedance are Rn = Rci + Ra (5) (257) 262 - 4.9 K _ AnRti -(50) (4.9) Av\ = — ■ = —123 Ru 2 R l = Rel = 5 K Since R' ol is the effective source impedance for Q2, then, from Table 12-2, D h ie + R, 2,000 + 5,000 il„2 = — R»-) = l+A/. R02R1.2 R»2 + Ri 51 (137) (5,000) 5,137 = 137 a = 134 Q S*. I*-* LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 339 Finally, the overall voltage and current gains of the cascade are Av = AviAvt = (-123) (0.992) m -122 R c i Ai — AnAit = (-50) (51) Rci + Riz Alternatively, Ay may be computed from A A Rn 48.7 X 5 A v = A; — = = -122 Rn 2 \5 + 257/ 48.7 Table 12-3 summarizes this solution, and should be compared with the exact values in Table 12-1. We find that the maximum errors are just slightly above 10 percent, as anticipated. It should also be noted that all the approxi- mate values are numerically too large, as predicted. TABLE 72-3 Approximate results of the example on page 338 12-6 SIMPLIFIED CALCULATIONS FOR THE COMMON-BASE CONFIGURATION »pire 12-9 shows the simplified circuit of Fig. 12-7 with the base grounded n a a load resistor R L connected between collector and ground. Following Procedures exactly analogous to those explained in Sees. 12-4 and 12-5 for the * and CC configurations, respectively, the approximate formulas given in yourth column of Table 12-2 may be obtained. Note that ft is too small y ess than 10 percent, whereas A Jt A v , and R[ are too large by no more than 1U Percent. e II. ? Fig 1 -j _ R, "• '*■? Simplified hybrid model Tor rhu rn - - me CB circuit e=- h f , I b K 1 1 O 1. R, B -+r R» 340 / ELECTRONIC DEVICES AND CIRCUITS Sac, 12-7 12-7 THE COMMON-EMITTER AMPLIFIER WITH AN EMITTER RESISTANCE Very often a transistor amplifier consists of a number of CE stages in cascade. Since the voltage gain of the amplifier is equal to the product of the voltage gains of each stage, it becomes important to stabilize the voltage amplification of each stage. By stabilization of voltage or current gain, we mean that the amplification becomes essentially independent of the k parameters of the tran- sistor. From our discussion in Sec. 1 1-3, we know that the transistor param- eters depend on temperature, aging, and the operating point. Moreover, these parameters vary widely from device to device even for the same type of transistor. The necessity for voltage stabilization is seen from the following example; Two commercially built six- stage amplifiers are to be compared. If each stage of the first has a gain which is only 10 percent below that of the second, the overall amplification of the latter is (0.9) 6 = 0.53 (or about one-half that of the former). And this value may be below the required specification. A simple and effective way to obtain voltage-gain stabilization is to add an emitter resistor R t to a CE stage, as indicated in the circuit of Fig. 12-10. This stabilization is a result of the feedback provided by the emitter resistor. The general concept of feedback is discussed in Chap. 17. We show in this section that the presence of R e has the following effects on the amplifier performance, in addition to the beneficial effect on bias stability discussed in Sec. 10-4: It leaves the current gain ^4/ essentially unchanged; it increases the input impedance by (1 + h/ t )R,; it increases the output impedance; and under the condition (1 + hf,)R t » h it) it stabilizes the («) (6) Fig. 12-10 (a) Common-emitter amplifier with an emitter resistor. The base biasing network {RiR t of Fig. 12-1 3a J is not indicated, (b) Approximate small-signal equivalent circuit. S«. '2-7 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 341 voltage gain, which becomes essentially equal to —R L /R t (and thus is inde- pendent of the transistor). The Approximate Solution An approximate analysis of the circuit of Fig. 12- 10a can be made using the simplified model of Fig. 12-7 as shown in Fig. 12-I0b. The current gain is, from Fig. 12-106, <■! j~ f = — «/« lb lb (12-38) The current gain equals the short-circuit value, and is unaffected by the addition of R e . The input resistance, as obtained from inspection of Fig. 12-106, is V* Ri = -r- = hu + (1 + k/ t )R, (12-39) The input resistance is augmented by (1 + k fc )R t) and may be very much larger than k ie . For example, if R, = 1 K and h f , = 50, then (1 + h,.)R, = 51 K » hu « 1 K Hence an emitter resistance greatly increases the input resistance. The voltage gain is Ay = AiRt — h/tRi (12-40) Ri hi, + (1 + h /t )R t Clearly, the addition of an emitter resistance greatly reduces the voltage amplification. This reduction in gain is often a reasonable price to pay for the improvement in stability. We note that, if (1 + h ft )R, » h it , and since hf, » 1, then a ~ "hf* Rl _ —Rl ,.„.., Ar ~ T+hf-.T.- -RT (1JM1) Subject to the above approximations, A v is completely stable (if stable resistances are used for Rt and R t ) t since it is independent of all transistor Parameters. The output resistance of the transistor alone (with R L considered exter- n& l) is infinite for the approximate circuit of Fig. 12-106, just as it was for the C E amplifier of Sec. 12-4 with R e = 0. Hence the output impedance of the s1 *ge, including the load, is R L - Looking into the Base, Collector, and Emitter of a Transistor On the b *s's of Eq. (12-39), we draw the equivalent circuit of Fig. 12-1 la from hl ch to calculate the base current with the signal source applied. This net- 0r k is the equivalent circuit "looking into the base." From it we obtain h = V. R< + Ai, + (1 + h t .)R, (12-42) 342 / ELECTRONIC DEVICES AND CIRCUITS Sec. J 2-7 (7) ^Yi 3 J (6) ZL ir— =±=- JV !, = (! + £,,)/(, I W\* o i- - !_=. (c) Fig. 12-11 (a) Equivalent circuit "looking into the base" of Fig. 12-10. This circuit gives (approximately) the correct base current, (b) Equivalent circuit "looking into the collector" of Fig. 12-10. This circuit gives (approximately) the correct collec- tor current, (e) Equivalent circuit "looking into the emitter" of Fig. 12-10. This circuit gives (approximately) the correct emitter voltage V t and the correct emitter and base currents. Since the output voltage at the collector is ~h fr V t R L V m — I C R L = —k/JJii, = R, + h t . + (1 + h f .)R t (12-43) and since the output impedance is infinite, the Norton's equivalent output circuit is as given in Fig, 12-116. This network "looking into the collector" gives the correct collector voltage. This equivalent circuit emphasizes that (subject to our approximations) the transistor behaves like an ideal current source and that the collector current is ft/, times the base current. From Fig. 12-106 and Eq. (12-42) we find the emitter- to-ground voltage to be V,R e V m = V t = (1 + h /e )I b R, = (R. + A*)/(l + */.) + £« (12-44) This same expression may be obtained from Fig. 12- lie, which therefore represents the equivalent circuit "looking into the emitter." Validity of the Approximations For the CE case, with R. = 0, th e approximate equivalent circuit of Fig. 12-7 is valid if h ot R L < 0.1. What i s Sec 12-7 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 343 the corresponding restriction for the circuit with R f y^ 0? We can answer this question and, at the same time, obtain an exact solution, if desired, by proceeding as indicated in Fig. 12-12. The exact value of the current gain f Fig. 12- 12a (which is the same as that of Fig. 12- 10a) is At = —I e /h- The two amplifiers of Fig. 12-12a and 6 are equivalent in the sense that the base and collector currents are the same in the two circuits. This fact can be verified by writing the KVL equations for the two loops of each of the amplifiers. The effective load impedance R' L is, from Fig. 12-126, R'l~ '■l + ^t^R, At (12-45) We know from the above approximate solution that Ai » — ft/ e , and since ft /e » 1, then R' L = R L + R»- Since in Fig. 12-126 the emitter is grounded and the collector resistance is R' L , the approximate two-parameter (ft,*« and ft/.) circuit is valid, provided that KJt' L = h oe (R L + R 4 ) < 0.1 (12-46) This condition means that the sum of Ri and R t is no more than a few thousand ohms, say 4 K for l/ft , = 40 K. Furthermore, R t is usually several times smaller than Rl in order to have an appreciable voltage gain [Eq. (12-41)]. The approximate solution for the CE amplifier with an emitter resistor R, is summarized in the second column of Table 12-2. *ig. 12-12 (a) Transistor ampli- fier stage with unbypassed emitter resistor R t , (b) Small- signal equivalent circuit. U(1-A,)I> K -*rv e Wv- 344 / ELECTRONIC DEVICES AND CIRCUITS Sec. 12-7 The Exact Solution If the above inequality (12^6) is not satisfied for a particular amplifier, an exact solution can readily be obtained by referring to Fig. 12-126 and to Table 11-4. For example, the current gain is At = -h f* -h, l + KM'l l + K, Ul + ^3— «.) (12-47) From this equation we can solve explicitly for At, and we obtain fl at Re — hje At = 1 + K.{Rl + R.) (12-48) If the inequality (12-46) is satisfied, then h ,R, « h/ a} and the exact expression (12-48) reduces to At W — A/« in agreement with Eq, (12-38). The exact expression for the input resistance is, from Fig. 12-126 and Table 12-2, Bt + ^r - (1 ~ A Z )R. -T- Ik, + h Tt AiR' L (12-49) where R' L is given by Eq. (12-45). Usually, the third term on the right-hand side can be neglected, compared with, the other two terms. The exact expres- sion for the voltage amplification is Ay = A t Ri Rt (12-50) where the exact values for Ai and Ri from Eqs. (12-48) and (12-49) must be used. The exact expression for the output impedance (with Rl considered external to the amplifier) is found, as outlined in Prob. 12-14, to be R = 1 (1 + h f ,)R, + (R, + hi,) (I + h,R t ) h oi R, + R, -f hi, — KJift/K, Note that, if R. » R, + h it and h<Jl, « 1, then d 1 1 + h f< K* h^ (12-51) (12-52) where the conversion formula (Table 1 1-3) from the CE to the CB h parameters is used. Since I /hob *» 2 M, we see that the addition of an emitter resistor greatly increases the output resistance of a CE stage. This statement is true even if R, is of the same order of magnitude as R, and h ie . For example, for R, — R, = 1 K, and using the ^-parameter values in Table 11-3, we rind from Eq. (12-51) that R Q = 817 K, which is at least ten times the output resistance for an amplifier with R, — (Fig. 11-17). Sec. 12-8 LOW-FREQUENCV TRANSISTOR AMPLIFIER CIRCUITS / 345 12-8 THE EMITTER FOLLOWER Figure 12- 13a is the circuit diagram of a common-collector transistor amplifier. This configuration is called the emitter follower, and is similar to the cathode follower in its operation, although there are a number of important differences worth noting. First, this amplifier has a voltage gain which is very close to unity (much closer to unity for typical loads than the cathode follower). Second, the voltage drop across the emitter resistor (from emitter to ground) may be either positive or negative, depending on whether an n-p-n or a p~n-p transistor is used. In the case of the cathode follower, the drop across the cathode resistor is always positive. Third, the input resistance of the emitter follower, although high (tens or hundreds of kilohms), is low compared with that of a cathode follower. Fourth, the output resistance of the emitter follower is much lower (perhaps by a factor of 10) than that of a cathode follower. In the discussion on cascading transistor stages in Sec. 12-2, we note that the common-collector stage is not used as an intermediate stage, but rather the most common use for the emitter follower is as a circuit which performs the function of impedance transformation over a wide range of frequencies with voltage gain close to unity. In addition, the emitter follower increases the power level of the signal. The input circuit of Fig. 12-13a includes the biasing resistors R h R 2 , and the blocking capacitor C. This circuit may be simplified by the use of Theve- nin's theorem. Let R' = R1WR2. If, at the lowest frequency under considera- tion, the reactance of C is small compared with is!, + R', we may neglect the effect of this capacitor. The equivalent input circuit is then indicated in Fig. 12-136, where R b - R t \\R' R' - fl,l|/2, and F fl = V.R' R, + R' (12-53) If the input resistance of the amplifier is Ri = Vi/I b , the input resistance Rt, taking the bleeder into account, is R' t = R'\\Ri, The impedance which the source V, sees is R" = R, + R+ The voltage Vt at the input terminals of the amplifier is 1 r, + r: (12-54) The circuit of Fig. 12-136 is examined in some detail in Sec. 12-5, where *e obtain approximate, as well as exact, expressions for At, Ri, Av, and R a . *"e approximate formulas are given in the third column of Table 12-2, with iL replaced by R e , and R, replaced by R b . The approximate equivalent Wteuita looking into the base and emitter are given in Fig. 12- 11a and c, respec- ively, where V, is replaced by F„. For exact expressions for A 1, Ri, Av, and Y B , the reader is referred to Eqs. (12-31), (12-33), (12-34), and (12-36), respectively. 346 / ELECTRONIC DEVICES AND CIRCUITS Sec. 12-8 Pig. 12-13 (a) The circuit of an emitter follower, in- cluding the biasing resistors Ri and R 2 . (fa) The input circuit is replaced by its Thevenin's equivalent. Extreme Values of Hi and Ay It is interesting to calculate Ay for the largest load for which the approximate equivalent circuit is valid, namely, Rl = 4 K (for l/h, e = 40 K). From Eqs, (12-32) and (12-34) and Table 11-2, Ri = 1.1 + (51) (4) = 205 K ^r-J-Sff 1 -*- 00054 = 0.9946 205 If a t-riode is used in a cathode-follower configuration, the maximum ga 111 obtained for infinite load resistance is m/(m +1)- A value of p — 200 would be required to obtain Ay = 0.995. Since such a large value of p is difficult to obtain with a triode, we see that an emitter follower can give a value of Av much closer to unity than can be obtained with a cathode follower (provided that the emitter follower is driven from a very low impedance; Eq. ( 12-58) J- Let us now calculate Ri and Av for an infinite load resistance. Of course, we must now use the exact formulas, Eqs. (12-31) and (12-33), rather than the Sec. 12-8 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 347 1 approximations, Eqs. (12-30) and (12-32). With R L m fl,-> », 1 + kf t 1 + h/ e Ar - 1 ■+• KeRt h ot R t Ri = h it + AjR e ~ h ie + 1 + h f , 1 + hu J_ k b (12-55) (12-56) where use has been made of the transformation from the CE to the CB h parameters in Table 1 1-3. We have proved that, even if the emitter resistance is infinite, the input resistance of an emitter follower is finite and equals I /hob ia 2M. This result is evident from an inspection of Fig. 12-136, where we see that, with i? e — > w , the input resistance is the resistance between base and collector. However, by definition, ft<* is the admittance between collector and base, with zero emitter current (A%— * =o), and therefore Ri = I /hob. The input resistance #■, taking the bleeder RiRi into account, will be much smaller than a megohm. Methods for increasing the input resistance of a transistor circuit are given in Sec, 12-10. Fori?*— ► », Eq. (12-34) becomes 1 - A v «fe kiji (12-57) Ri 1 + h f . If we use the fe-parameter values in Table 11-2, we find A v = 1 - 5.4 X 10- 4 - 0.99946 This value is probably somewhat optimistic (too close to unity) because, for a large value of R B) and hence a small value of transistor current, h is will be larger and h fe smaller than the nominal values in Table 11-4. The voltage gain A v = V /Vi gives the amplification between the output and the input to the base. The overall gain Av„ taking the signal-source impedance into account, gives the amplification between the output and the signal source V.. Thus V R'i V V- A = U — ° * — * Av '- V, ViV t - Av R. + R'< (12-58) where use has been made of Eq. (12-54). Hence, in order for Ay, to be very close to unity, it is required that Ay be very nearly unity and, in addition, that R, be extremely small compared with R[. This latter condition may be difficult to satisfy in practice (Sec. 12-10). The Effect of a Collector-circuit Resistor It is important to investigate 'he effect of the presence in the collector circuit of a resistance R e in Fig. 12-13. °uch a resistance is frequently added in the circuit to protect the transistor a 8ainst an accidental short circuit across R e or a large input- voltage swing. From Fig. 12-12a we see that the relationship between the CE current 348 / ELECTRONIC DEVICES AND CIRCUITS Sec. 12-9 gain An (designated simply Ai in the figure) and the CC current gain A Ie is A Ie - 1 - Au (12-59) where A Ie = - y and 4*. ~ -? Substituting Eq. (12-48) in Eq. (12-59) with Rl replaced by R e , we obtain the exact expression An = 1 + k oe R e + kf„ 1 + h oe (R<: + Re) (12-60) The value of Ri is obtained from Eq. (12-49), with Ai replaced by An and Rl by R e . The voltage gain of the emitter follower with R c present in the collector circuit is obtained as follows : A V„ _ A R* Ay _ Vi - An^ (12-61) Subject to the restriction k e(Rc + R*) « 1, the approximate formulas given in the third column of Table 12-2 are valid, and the protection resistor R e has no effect on the small-signal operation of the emitter follower. 12-9 MILLER'S THEOREM We digress briefly to discuss a theorem which is used in the next section and also in connection with several other topics in this book. Consider an arbitrary circuit configuration with JV distinct nodes, 1, 2, 3, . . . , N, as indicated in Fig. 12-14a. Let the node voltages be Vi, V%, F 3 , . . . , V&, where Fat — z,= l-K z 2 = z;k k-\ (a) (6) Fig. 12-14 Pertaining to Miller's theorem. By definition, K = PVVj. The networks in (a) and [b) have identical node voltages. Note that /i = — h. S«. '2-9 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 349 and JV is the reference or ground node. Nodes 1 and 2 (referred to as iVi and $%) are interconnected with an impedance Z' . We postulate that we know the ratio F s |Fi- Designate the ratio F a /Fi by K, which in the sinusoidal steady state will be a complex number and, more generally, will be a function of the Laplace transform variable s. We shall now show that the current A drawn from iV\ through Z' ean be obtained by disconnecting terminal 1 from Z 1 and by bridging an impedance Z'/(l — K) from N\ to ground, as indicated in Fig. 12-146. The current I\ is given by h = Vx - F 2 Fi(l - K) Vi F, Z' Z' £'/(! - K) z- L (12-62) Therefore, if Z x = Z'/(l — K) were shunted across terminals N\-N, the cur- rent 1 1 drawn from N t would be the same as that from the original circuit. Hence, KCL applied at JVi leads to the same expression in terms of the node voltages for the two configurations (Fig. 12-14a and b). In a similar way, it may be established that the correct current 7 2 drawn from JV 2 may be calculated by removing Z' and by connecting between JV a and ground an impedance Z 2 , given by Z, m Z f Z'K L-l/K K - 1 (12-63) Since identical nodal equations (KCL) are obtained from the configurations of Fig. 12- 14a and b, then these two networks are equivalent. It must be emphasized that this theorem will be useful in making calculations only if it is possible to find the value of K by some independent means. Let us apply the above theorem to the grounded-cathode stage, taking interelectrode capacitances into account. Terminal JV is the cathode (Fig. 8-19), whereas nodes 1 and 2 are the grid and plate, respectively. Then Z' represents the capacitive reactance between grid and plate, or Z' = —j/<*)C 0]> , and K represents the voltage gain between input and output. If R p = plate- circuit resistance, r p = plate resistance, and R' v = R v \\r v , then, in the mid- band region, K « — g m R' p . Shunting the input terminals of the amplifier is an effective impedance Z\, as in Fig. 12-146, given by Z, m Z' -J 1 - K (12-64) «C»(1 + g.R' P ) Clearly, Z x is the reactance of a capacitance whose value is C = C op (l + g m R p ). The total input capacitance C\ of the stage is C augmented by the direct capacitance C a k between grid and cathode, or Ci = C B k + C ap (l 4- g m R p ) (12-65) This result agrees with Eq. (8-44), first derived by Miller. 3 Hence the trans- formation indicated in Fig. 12-14 is referred to as Miller's theorem. 350 / ELECTRONIC DEVICES AND CIRCUITS Sec. 12-10 (a) (b) Fig, 12-15 (a) Darlington pair. Some vendors package this device as a single composite transistor with only three external leads, (b) The Darlington circuit drawn as two cascaded CC stages. 12-10 HIGH-INPUT-RESISTANCE TRANSISTOR CIRCUITS 4 In some applications the need arises for an amplifier with a high input imped- ance. For input resistances smaller than about 500 K, the emitter follower discussed in Sec, 12-8 is satisfactory. In order to achieve larger input imped- ances, the circuit shown in Fig. 12-15a, called the Darlingtan connection, is uscd.f Note that two transistors form a composite pair, the input resistance of the second transistor constituting the emitter load for the first. More specifically, the Darlington circuit consists of two cascaded emitter followers with infinite emitter resistance in the first stage, as shown in Fig. 12-15o. The Darlington composite emitter follower will be analyzed by referring to Fig. 12-16, Assuming that hJR. < 0.1 and h f Ji e y>h ie> we have, from Table 12-2, for the current gain and the input impedance of the second stage, Ru « (1 + h ft )R t (12-66) Since the effective load for transistor Ql is R i2 , which usually does not meet the requirement KJtn < 0.1, we must use the exact expression of Eq. (12-31) for the current gain of the first transistor: 1 +h f . 1 +'M1 +A/.)/2 ( A = — = * + ft/« /,' 1 + h ae Ri2 and since h o0 R t < 0.1, we have 1 ~\~ h ae hf,,R e t For many applications the field-effect transistor (Chap. 14) with its extremely high input impedance would be preferred to the Darlington pair. (12-67) (12-68) Sec. 12-10 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 351 or The overall current gain for Fig. 12-16 is At = ~r = -f y = AnAii Al m J!±_M!_ 1 + h oe hf C R ( Similarly, for the input resistance of Ql, we must use Eq. (12-33) Bit - hu + AnRit (1 + h/.) l R, 1 + k Jif t Rt (12-69) (12-70) This equation for the input resistance of the Darlington circuit is valid for KJt t < 0-1, and should be compared with the input resistance of the single- stage emitter follower given by Eq, (12-32). If R s = 4 K, and using the h parameters of Table 11-2, we obtain Rn = 205 K for the emitter follower and R A = 1.73 M for the Darlington circuit. We also find Ai = 427, which is much higher than the current gain of the emitter follower ( = 51). The voltage gain of the Darlington circuit is close to unity, but its devi- ation from unity is slightly greater than that of the emitter follower. This result should be obvious because Fig. 12-16 represents two emitter followers in cascade (and the product of two numbers, each less than unity, is smaller than either number). If we make use of Eq. (12-34), we obtain 1 — Avt = hie Ril 1 A - *■'« 1 — Avi — -5- hi. (12-71) AnRiZ where A V2 = V /V 2 and Avi = VifVi. Finally, we have, for A v = V,/Vt, A v = Av.Av, ~\\ - g-j (l - ~^J - 1 - j^fa - Jfa and since AuRn » Ru, expression (12-72) becomes 1 - h± Ri2 (12-72) (12-73) Fig. 12-16 Darlington emitter 'Oilower, 352 / ELECTRONIC DEVICES AND CIRCUITS Sec. ?2-20 This result indicates that the voltage gain of the Darlington circuit used as an emitter follower is essentially the same as the voltage gain of the emitter follower consisting of transistor Q2 alone, but very slightly smaller. The output resistance R oi of Ql is, from Eq, (12-35), p R. + h ie and hence the output resistance of the second transistor Q2 is, approximately, R* + hi e , , (12-74) R»9 «* 1 + hfe R, + hi e 1 + h H* + hie (1 + hjeY ^ l+hje We can now conclude from the foregoing discussion, and specifically from Eqs. (12-69), (12-70), (12-73), and (12-74), that the Darlington emitter follower has a higher current gain, a higher input resistance, a voltage gain less close to unity, and a lower output resistance than does a single-stage emitter follower. Practical Considerations We have assumed in the above computations that the h parameters of Ql and Q2 are identical. In reality, this is usually not the case, because the h parameters depend on the quiescent conditions of Ql and Q2. Since the emitter current of Ql is the base current of Q2, the quiescent current of the first stage is much smaller than that of the second. Hence h f , may be much smaller for Ql than for Q2, and h ie may be much larger for Ql than for Q2 (Fig. 11-5). In order to have reasonable operating current in the first transistor, the second may have to be a power stage. A second major drawback of the Darlington transistor pair is that the leakage current of the first transistor is amplified by the second, and hence the overall leakage current may be high. For these two reasons, a Darlington connection of three or more transis- tors is usually impractical. The composite transistor pair of Fig. 12- 15a can, of course, be used as a common-emitter amplifier. The advantage of this pair would be a very high overall h ft , nominally equal to the product of the CE short-circuit current gains of the two transistors. In fact, Darlington integrated transistor pairs are commercially available with h /t as high as 30,000. If the condition h oe R* « 1 is not satisfied, an exact analysis of the Darling- ton circuit must be made. We may proceed as in Sec. 12-1, using the CC h parameters of each stage, or we may derive the k parameters of the com- posite pair in terms of the parameters h' and h" of Ql and Q2, respectively- The Biasing Problem In discussing the Darlington transistor pair, we have emphasized its value in providing high-input impedance. However, we have oversimplified the problem by disregarding the effect of the biasing arrangement used in the circuit. Figure 12- 13a shows a typical biasing net- work (resistors R t and R 2 ). The input resistance R' t of the stage of the 12-10 LOW -FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 353 emitter follower of Fig. 12-13a consists of Ri\\R', where R' m Ri\\Rt. Assume that the input circuit is modified as in Fig. 12-17 by the addition of K 3 but ff j t h C" = (that is, for the moment, ignore the presence of C), Now R' is increased to Rz + Rt |iR*. However, since Ri is usually much greater than R', jt is seen that R^ «* R', which may be a few hundred kilohms at most. To overcome the decrease in the input resistance due to the biasing net- work, the input circuit of Fig. 12-17 is modified by the addition of C" between the emitter and the junction of R\ and R*. The capacitance C" is chosen large enough to act as a short circuit at the lowest frequency under consideration. Hence the bottom of Rz is effectively connected to the output (the emitter), whereas the top of Rz is at the input (the base). Since the input voltage is Vi and the output voltage is V - AyV it the circuit of Fig. 12-14 and Miller's theorem can be used to calculate the current drawn by Rz from the input signal. We can then see that the biasing arrangement Ri, Ri, and Rz represents an effective input resistance of Rm — St 1 - A, (12-75) Since, for an emitter follower, Ay approaches unity, then R^t becomes extremely large. For example, with Av = 0.995 and Rz = 100 K, we find R e(i = 20 M. Note that the quiescent base current passes through Rz, and hence that a few hundred kilohms is probably an upper limit for Rz. The above effect, when Av— * +1, is called bootstrapping. The term arises from the fact that, if one end of the resistor R 3 changes in voltage, the other end of R 3 moves through the same potential difference; it is as if Rz were "pulling itself up by its bootstraps." The input resistance of the CC amplifier as given by Eq. (12-34) is Ri — A«/(l — A v ). Since this expression is of the form of Eq. (12-75), here is an example of bootstrapping of the resistance hu which appears between base and emitter. In making calculations of A/, Ri, and Ay, we should, in principle, take into account that the emitter follower is loaded, not only by R„ and /JiHfls, but also by R 3 . The extent to which R 3 loads the emitter follower is calcu- fig. 12-17 Trie boot- >rop principle increases ^e effective value of Rz. 354 / ELECTRONIC DEVICES AND CIRCUITS Sec. I2-IQ lated as follows: The emitter end of #3 is at a voltage Av times as large aa the base end of Rz. From Fig. 12-14, illustrating Miller's theorem, the effec- tive resistance seen looking from the emitter to ground is not Ri but, exagger- ated by the Miller effect, is A vRz RsM — Ay — 1 (12-76) Since Av is positive and slightly less than unity, then R 3 m is a (negative) resist- ance of large magnitude. Since R 3 m is paralleled with the appreciably smaller resistors R, and Ri\\Rt, the effect of R 3 wall usually be quite negligible. Bootstrapped Darlington Circuit We find in the preceding section that even neglecting the effect of the resistors R x , R^ and #3 and assuming infinite emitter resistance, the maximum input resistance is limited to l//u K 2M. Since l/£<* is the resistance between base and collector, the input resistance Fig. 12-18 (a) The boot- strapped Darlington cir- cuit, (b) The equivalent circuit. +o-j^AAA O — o VVW^ I R t — RciWRti ' <P _£ I A/,2 lb (ft) 5«c. 12-17 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 355 can be greatly increased by bootstrapping the Darlington circuit through the addition of C„ between the first collector Ci and the second emitter #2, as indi- cated in Fig. 12- 18a. Note that the collector resistor R„i is essential because, without it, Rut would be shorted to ground. If the input signal changes by y i} then E 2 changes by A v Vi and (assuming that the reactance of C is negligible) the collector changes by the same amount. Hence 1/hob is now effectively increased to l/(h ob )(l — Av) ~ 400 M, for a voltage gain of 0.995. An expression for the input resistance Ri of the bootstrapped Darlington pair can be obtained using the equivalent circuit of Fig. 12-186. The effective resistance R t between Ei and ground is R e — R e i\\R e t. If KM* < 0.1, then Q2 may be represented by the approximate A-parameter model. However, the exact hybrid model as indicated in Fig, 12-186 must be used for Q2. Since l/h M i » hi e %, then h oel may be omitted from this figure. Solving for Vi/hi, we obtain (Prob. 12-21) Ri * hfelhfeiRe (12-77) This equation shows that the input resistance of the bootstrapped Darlington emitter follower is essentially equal to the product of the short-circuit eurrent gains and the effective emitter resistance. If k/ t i = ft/«j = 50 and R t = 4 K, then Ri «* 10 M. If transistors with current gains of the order of magnitude of 100 instead of 50 were used, an input resistance of 40 M would be obtained. The biasing arrangement of Fig. 12-17 would also be used in the circuit of Fig. 12-18. Hence, the input resistance taking into account the bootstrap- ping both at the base and at the collector of Ql would be RestWhftih/^R,, where flat is given in Eq. (12-75). 12-11 THE CASCODE TRANSISTOR CONFIGURATION 6 The cascode transistor configuration shown in Fig. 12-19 consists of a CE "age in series with a CB stage (the collector current of Ql equals the emitter current of Q2). This circuit should be compared with the vacuum-tube triode cascode amplifier discussed in Sec. 8-10. In the case of the tube cascode connection, two triodes are used in a series circuit and the combination behaves like a pentode. In the circuit shown in Fig. 12-19 transistors Ql and Q2 in cascode act like a single CE transistor with negligible internal feedback (negli- ® Die K e ) and very small output conductance for an open-circuited input. Derivation of Parameter Values To verify the above statement let us . nipute the k parameters of the Q1-Q2 combination. From our discussion lnS ec n-1 and Fig. 12-19, 1 B °*ever, if y 2 = 0, then the load of Ql consists of h ib2 , which, from Table 11-3, 356 / ELECTRONIC DEVICES AND CIRCUITS Sec. 12. j j Fig. 12-19 The cascode configura- tion. (Supply voltages are not indicated.) is about 20 8. Hence transistor Q\ is effectively short-circuited, and ftu m h ia (12-78) Similarly, we have for the short-circuit current gain *» -Jllr.-. ** ZT?\r~* " - h ' J " b " h " (l2 - 79) since —hfb — a « 1. The output conductance with input open-circuited is given by If 1 1 = 0, the output resistance of Ql is equal to 1/A M « 40 K. Hence the equivalent source resistance for transistor Q2 is 40 K. From Fig. 11-17 we see that, for the CB connection, the output resistance R„ with R, = 40 K is essentially the same as that for R t = °° , so that R a — l/hc*. Therefore ha = -mr « hob is Finally, for the reverse open-circuit voltage amplification, we have hrJlrb (12-80) (12-81) Equation (12-81) is valid under the assumption that the output resistance of Ql (which is l/h M m 40 K) represents an open-circuited emitter for QZ. Summary Using the h parameters of the typical transistor of Table 11-* and Eqs. (12-78) to (12-81), we find h = h u " 1,100 8 » hi. k f = hii = 0.98 X 50 = 49 « h ft K = h n = 0.49 j*A/V m k* K m hu = 7.25 X 10~ a « KXb (12-82) S* 12-12 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 357 Kote that the input resistance and current gain (with the output short- Vcuited) are nominally equal to the corresponding parameter values for a ngle CE stage. The output resistance (with the input open-circuited) is oproximately equal to the CB value of 2 M, which is much higher than the pE value of 40 K. The reverse open-circuit amplification parameter h r is ver v much smaller for the cascode connection than for a single CE stage. In view of the foregoing discussion, it should be clear that the simplified model mven in Fig. 12-7 is a better approximation for the cascode circuit than for a single transistor. As a matter of fact, calculations based upon this hybrid model will result in less than 10 percent error if the load resistance Rl satisfies the inequality H-aRl < 0.1 or for R L less than about 200 K. The small value of h r for the cascode transistor pair makes this circuit particularly useful in tuned-amplifier design. The reduction in the "internal feedback" of the compound device reduces the probability of oscillation and results in improved stability of the circuit. 12-12 DIFFERENCE AMPLIFIERS 8 The function of a difference, or differential, amplifier is, in general, to amplify the difference between two signals. The need for differential amplifiers arises in many physical measurements, in medical electronics, and in direct-coupled amplifier applications. Figure 12-20 represents a linear active device with two input signals Pi, vi and one output signal v 0} each measured with respect to ground. In an ideal differential amplifier the output signal v a should be given by v = Ad(vi — v%) (12-83) where A d is the gain of the differential amplifier. Thus it is seen that any signal which is common to both inputs will have no effect on the output voltage. However, a practical differential amplifier cannot be described by Eq. (12-83) since, in general, the output depends not only upon the difference tignal v d of the two signals, but also upon the average level, called the common- Wode signal v e , where v d = V\ — v<i and v c = i(«i + w») (12-84) For example, if one signal is +50 mV and the second is —50 pf, the output w 'll not be exactly the same as if v t = 1,050 fiV and v 2 = 950 pM, even though ™e difference v d = 100 fiV is the same in the two cases. The Common-mode Rejection Ratio The foregoing statements are now clarified, and a figure of merit for a difference amplifier is introduced. The '9- 12-20 The output is a [ineor function of 1 Q nd v 2 . For an ideal differential ampli- V ». - A,(*. - ir,). 358 / ELECTRONIC DEVICES AND CIRCUITS Sec. I2.j j output of Fig. 12-20 can be expressed as a linear combination of the two input voltages v B = AiV! + A2V2 (12-85) where A 1 (A 2 ) is the voltage amplification from input 1 (2) to the output under the condition that input 2 (1) is grounded. From Eqs. (12-84), i>i = v e + §& and v 2 m v e — %v d If these equations are substituted in Eq. (12-85), we obtain v ~ A d Vd + A c v c where (12-86) A d = %(Ai — At) and A e = A 1 + A- (12-87) (12-88) The voltage gain for the difference signal is A d , and that for the common-mode signal is A c . We can measure A d directly by setting Vi = — » s = 0.5 V, so that Vd = 1 V and v c - 0. Under these conditions the measured output voltage v gives the gain A d for the difference signal [Eq. (12-87)]. Similarly, if we set vj. = y 2 = 1 V, then v d = 0, v c = 1, and v e = A e . The output voltage now is a direct measurement of the common-mode gain A e . Clearly, we should like to have A d large, whereas, ideally, A e should equal zero. A quantity called the common-mode rejection ratio, which serves as a figure of merit for a difference amplifier, is P= 2: ( 12 - 89 > From Eqs. (12-87) and (12-89) we obtain an expression for the output in the following form : / 1 .. \ (12-90) v = A d v d (i + is) \ pVdJ From this equation we see that the amplifier should be designed so that p is large compared with the ratio of the common-mode signal to the difference signal. For example, if p = 1,000, v e = 1 mV, and v d = 1 pV, the second term in Eq. (12-90) is equal to the first term. Hence, for an amplifier with a common-mode rejection ratio of 1,000, a 1-pV difference of potential between the two inputs gives the same output as a 1-mV signal applied with the same polarity to both inputs. EXAMPLE (a) Consider the situation referred to above where the first set oi signals is », = +50 pV and v t = -50 pV and the second set is t>i = 1,050 M v and v 2 = 950 pV. If the common-mode rejection ratio is 100, calculate the per- centage difference in output voltage obtained for the two sets of input signal- (6) Repeat part a if p = 10,000. Solution a. In the first case, v d = 100 pV and v e = 0, so that, from Eq. (12-9°)' v Q = 100A d pV. Sec 12-12 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 359 In the second ease, v d = 100 pV, the same value as in part a, but now v e = 4(1,050 + 950) - 1,000 jtV, so that, from Eq. (12-90), ». - 100^ fc*9- 100A d (l + *&) pV These two measurements differ by 10 percent. b. For p = 10,000, the second set of signals results in an output Vt = 100A d (l + 10 X 10"") pV whereas the first set of signals gives an output v c = 100/lj pV. Hence the two measurements now differ by only 0.1 percent. The Emitter-coupled Difference Amplifier The circuit of Fig. 12-21 is an excellent difference amplifier if the emitter resistance R e is large. This statement can be justified as follows: If V t i = V s » = V t , then from Eq. (12-87), we have V d = V $l — F« 2 = and V e = A C V„ However, if R, m °°, then because of the symmetry of Fig. 12-21, we obtain J e i = 7,2 = 0. Since hi « /e2 f then l ti *= I a, and it follows that V = 0. Hence the common- mode gain A c becomes zero, and the common-mode rejection ratio is infinite for R t = « and a symmetrical circuit. We now analyze the emitter-coupled circuit for a finite value of R e . A e can be evaluated by setting V ti = F j2 = V, and making use of the symmetry of Fig. 12-21. This circuit can be bisected as in Fig. 12-22a. An analysis of this circuit (Prob. 12-28), using Eqs. (12-48) to (12-50) and neglecting the term in fe„ in Eq. (12-49), yields V e (2h oe R. - h f .)R. A c = V. 2R.(1 + h ft ) + {R. + hi,)(2h Jt. + 1) (12-91) provided that h et R e « 1. Similarly, the difference mode gain A d can be obtained by setting V.i = — V t % = V./2. From the symmetry of Fig. 12-21, we see that, if V,i = — V t2 , then the emitter of each transistor is grounded for -v e v ei = v a '8- 12-21 Symmetrical emitter- c °upled difference amplifier. 360 / ELECTRONIC DEVICES AND CIRCUITS Sec. J 2, 1? °V„ oK„ Fig. 12-22 Equivalent circuit for a symmetrical difference amplifier used to determine (a) the com- mon-mode gain A e and (M the difference gain A d . («) (6) small-signal operation. Under these conditions the circuit of Fig. 12-226 can be used to obtain Ad. Hence A d = ^ = ± 1 hr.R c V 8 2 /?, + fa (12-92) provided h Q Jt c <<C 1. The common-mode rejection ratio can now be obtained using Eqs. (12-91) and (12-92). From Eq. (12-91) it is seen that the common-mode rejection ratio increases with R t as predicted above. There are, however, practical limitations on the magnitude of R e because of the quiescent dc voltage drop across it; the emitter supply Vbs must become larger as R„ is increased in order to maintain the quiescent current at its proper value. If the operating currents of the tran- sistors are allowed to decrease, this will lead to higher h it values and lower values of h /e . This can be seen from Fig. 11-5. Both of these effects will tend to decrease the common-mode rejection ratio. Difference Amplifier Supplied with a Constant Current Frequently, » n practice, R e is replaced by a transistor circuit, as in Fig. 12-23, in which Ru R-l, and R% can be adjusted to give the same quiescent conditions for Ql and Q2 as the original circuit of Fig. 12-21. This modified circuit of Fig. 12-23 presents a very high effective emitter resistance R e for the two transistors Ql and Q2, Since R, is also the effective resistance looking into the col- lector of transistor Q3, it is given by Eq. (12-51). In Sec. 12-7 it is verified that R e will be hundreds of kilohms even if ff 3 is as small as 1 K. We now verify that transistor Q3 acts as an approximately constant current source, subject to the conditions that the base current and the base- to-emitter voltage of Q3 are negligible. The voltage across R 2 (and hence also across R 3 ) is V E bR»/(Ri + R z ). Hence the emitter current I B = ls\ + l * % in Fig. 12-23 is given by I E = h = Ri{Ri + R z ) (12-93) S^ 12-12 LOW-FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 361 Since this current is independent of the signal voltages V.i and F. a , then 03 acts to supply the difference amplifier consisting of Ql and Q2 with the constant current I s . Consider that Ql and Q2 are identical and that Q3 is a true const an t- gurrent source. Under these circumstances we can demonstrate that the common-mode gain is zero. Assume that V,i = V.t = V„ so that from the e ymmetry of the circuit, the collector current hi (the increase over the quiescent value for V, = 0) in Ql equals the current Id in Q2. However, since the total current increase hi + I C 2 — if Ig = constant, then J £ i = 7 e3 = and Practical Considerations Since the h parameters vary with the quiescent current, the common-mode rejection ratio depends upon the Q point. The values of hf e and l/h ee should be as large as possible, and h ie as small as possible. A reasonable set of values might be h /e = 100, h it = 2 K, l/k ot — 100 K, and k rt = 2.5 X 10~<. For R 3 = 27 K, R. = 1 K, and R x R % /(Ri + R t ) = 1 K, we find from Eqs. (12-51), (12-91), and (12-92) that R, = 9.95 M and p = 338,000. More elaborate transistor configurations giving higher values of p are found in the literature. fl For the analysis of nonsymmetrical differ- ential circuits the reader is referred to Ref. 6. In some applications the choice of V tl and V ti as the input voltages is not realistic because the resistances R,i and R t z represent the output impedances of the voltage generators V,i and F, 2 . In such a case we use as input voltages the base-to-ground voltages V&i and Vb* of Ql and Q2, respectively. The differential amplifier is often used in dc applications. It is difficult to design dc amplifiers using transistors because of drift due to variations of V> V BS> and Icbo with temperature. A shift in any of these quantities changes the output voltage and cannot be distinguished from a change in input-signal voltage. Using the techniques of integrated circuits (Chap. 15), it is possible -v c F 'B. 12-23 Differential omplifier with constont- Cu "*rent stage in the emit- ter circuit. Nominally, 362 / ELECTRONIC DEVICES AND CIRCUITS S«c. !2. ? j to construct a difference amplifier with Ql and Q2 having almost identical properties. Under these conditions any parameter changes due to tempera- ture will cancel and will not vary the output. A number of manufacturers t sell devices designed specifically for difference-amplifier applications. These consist of two high-gain n-p-n silicon planar transistors in the same hermeti- cally sealed enclosure. The manufacturer guarantees that for equality f collector currents the maximum difference in base voltages is 5 mV, that the base- voltage differential at fixed collector current will not exceed 10 juV/°C and that h fe of one transistor will not differ from h ft of the other by more thao 10 percent. It has been found 7 that a substantial reduction in thermal drift is obtained if the two transistors are operated with equal V B s instead of equal collector current. Difference amplifiers may be cascaded in order to obtain larger amplifi, cations for the difference signal and also better common-mode rejection. Out- puts V a i and V o2 are taken from each collector (Fig. 12-23) and are coupled directly to the two bases, respectively, of the next stage. Finally, the differential amplifier may be used as an emitter-coupled phase inverter. For this application the signal is applied to one base, whereas the second base is not excited (but is, of course, properly biased). The output voltages taken from the collectors are equal in magnitude and 180° out of phase. REFERENCES 1. Coblenz, A., and H. L. Owens: Cascading Transistor Amplifier Stages, Electronics, vol. 27, pp. 158-161, January, 1954. 2. Dion, D. F,: Common Emitter Transistor Amplifiers, Proc, IRE, vol. 46, p. 920, May, 1958. 3. Miller, J. M.: Dependence of the Input Impedance of a Three-electrode Vacuum Tube upon the Load in the Plate Circuit, Nail. Bur. Std. (U.S.) Res Papers vol. 15, no. 351, pp. 367-385, 1919. 4. Levirie, I.: High Input Impedance Transistor Circuits, Electronics, vol. 33, pp. 50-54, September, 1960. 5. James, J. R.: Analysis of the Transistor Cascode Configuration, Electron. EnQ-> vol. 32, pp. 44-48, 1960. 6. Slaughter, D. W.: The Emitter-coupled Differential Amplifier. IRE Trans. Circuit Theory, vol. CT-3, pp. 51-53, 1956. Middlebrook, R. D.: Differential Amplifiers, John Wiley & Sons Inc New York, 1963. 7. Hoffait, A. H., and R. D. Thornton: Limitations of Transistor DC Amplifi"* 9 ' Proc. IEEE, vol. 52, no. 2, pp. 179-184, February, 1964. t Fairchild Semiconductor Corporation, Sprague Electric Co., Texas Instruments, Ii» c "» and Motorola, Inc. 13 /THE HIGH-FREQUENCY TRANSISTOR At low frequencies it is assumed that the transistor responds instantly to changes of input voltage or current. Actually, such is not the case because the mechanism of the transport of charge carriers from emitter to collector is essentially one of diffusion. Hence, in order to find out how the transistor behaves at high frequencies, it is necessary to examine this diffusion mechanism in more detail. Such an analysis 1 is complicated, and the resulting equations are suggestive of those encountered in connection with a lossy transmission line. This result could have been anticipated in view of the fact that some time delay must be involved in the transport of charge across the base region by the diffusion process. A model based upon the transmission-line equations would be quite accurate, but unfortunately, the resulting equivalent circuit is too complicated to be of practical use. Hence it is necessary to make approximations. Of course, the cruder the approximation, the simpler the circuit becomes. It is therefore a matter of engineering judgment to decide when we have a reasonable compromise between accuracy and simplicity. 13-1 THE HIGH-FREQUENCY T MODEL .. Experience shows that, as a first reasonable approximation, the dif- fusion phenomenon can be taken into account by modifying the basic common-base T model of Fig. 11-19 as follows: The collector resis- tor r' c is shunted by a capacitor C e , and the emitter resistor r e is shunted by a capacitor C e , as indicated in Fig. 13-1. Also, the dependent cur- rent generator is made proportional to the current i\ in r fl and not to the emitter current i t . The low-frequency alpha is designated by a„. If an input current step is applied, then initially this current is 363 364 / ELECTRONIC DEVICES AND CIRCUITS Sw, J 3. Fig. 13-1 Transistor T model at high fre- quencies. bypassed by C t and t'i remains zero. Hence the output current starts at zero and rises slowly with time. Such a response is roughly what we expect because of the diffusion process. A better approximation is to replace C, and r, by a lumped transmission line consisting of resistance-capacitance sections, but as already emphasized, such an equivalent circuit is too complicated to be useful. The physical significance of C, is not difficult to find. It represents the sum of the diffusion capacitance C D . and the transition capacitance CV, across the emitter junction, C t = Cd* + Ct,- The diffusion capacitance is directly proportional to the quiescent emitter current. Usually, C D * » Ct, (except for very small values of emitter current), and hence C, is approximately equal to the diffusion capacitance Co*. Since the collector j unction is reverse-biased, the collector diffusion capacitance C De is negligible, so that C c is essentially equal to the collector transition capacitance C Tc . Usually, C a is at least 30 times as large as C c , The High-frequency Alpha We shall assume that the input excitation is sinusoidal of frequency / = u/2ir. Then, using capital letters for phasor currents, we have, from Fig. 13-1, h or where liK = It* 1/K+joC. Um l+jf/fa 1 2-KT'jC, (13-1) (13-2) It is possible to consider the current generator to be proportional to the emitter current (rather than the current through r t ) provided that we alio* the proportionality factor a to be a complex function of frequency. Thus, if we write o<Ji = ctl. (13-3) 5* 13-2 then, from E( l- t 13 " 1 ). a„ i+tf/U THE HIGH-FREQUENCY TRANSISTOR / 365 (13-4) The magnitude of the complex or high-frequency alpha a is a at zero fre- quency and falls to 0.707a o at / = /„. This frequency /„ is called the alpha cutoff frequency. The diffusion equation leads to a solution for a equal to the hyperbolic secant of a complex quantity. If this expression is expanded into a power series in the variable f/f a and only the first two terms are retained, Eq. U 3 " 4 ) is obtained (Prob. 13-1). Hence Eq. (13-4) and the equivalent circuit of Fig. 13-1 are valid at frequencies which are appreciably less than /«, (up to perhaps f a /2). General-purpose transistors have values of f a in the range of hundreds of kilohertz. High-frequency transistors may have alpha cutoff frequencies in the tens, hundreds, or even thousands of megahertz. Since a = -h th the symbol / A /& is sometimes used for /„. The Approximate CB T Model If the load resistance R L is small, the output voltage v*, and hence v A >, will be small. Since fx « 10~ 4 , we can neg- lect the Early generator iti>*'. Under these circumstances the network of Fig. 13-1 reduces to the circuit of Fig. 13-2, which is known as the approximate CB high-frequency model. The order of magnitudes of the parameters in Fig. 13-2 are t, « 20 Si r»- « 100 Q r|«lM C e « 1-50 pF and C. m 30-10,000 pF 13-2 THE COMMON-BASE SHORT-CIRCUIT^CURRENT FREQUENCY RESPONSE Consider a transistor in the eommon-base configuration excited by a sinusoidal current I, of frequency /. What is the frequency dependence of the load cur- rent l L under short-circuited conditions? If terminals C and B are connected together in Fig. 13-2, then rw, r' £ , and C t are placed in parallel. Since r e » r»., o„»i I, r@~ i f, 9- 13-2 The approximate high- •-Ayv/^—. — o — <-AA/V^' o fre quency T model. c. ' r bH C c '«t 366 / ELECTRONIC DEVICES AND CIRCUITS s «*- 13-3 we may omit r e . Usually, rtb'C e « r t C e , and under these circumstances, the response is determined by the larger time constant r' e C e . Hence we shall also omit C e from Fig. 13-2. With these simplifications, I L — aj h or from Eqg, (13-3) and (13-4), the common-base short-circuit current gain is given by A. = l^ — a °^ 1 e 1* i + if//. The magnitude of a and its phase angle 8 are given by <*o vi + (f/f a y e = f — arctan ~ (13-5) (13-6) Iff=U,a= ac/y/S, and 20 log \a/a.\ = -20 log y/2 = -3 dB. Hence the alpha cutoff frequency f a is called the Z-dB frequency of the CB short-circuit current gain. Equation (13-6) also predicts that a has undergone a 45° phase shift in comparison with its low-frequency value. This calculated amplitude response is in close agreement with experiment, but the phase-shift calculation may well be far off. The reason for the discrepancy is that our lumped-circuit equivalent representation of the transistor is simply not accurate enough. It is found, empirically, that the discrepancy between calculation and experiment can be very substantially reduced by introducing an ' 'excess- phase" factor 2 in the expression for a, so that Eq. (13-5) becomes i+iCf//-) ^—jmflfa (13-7) In this equation m is an adjustable parameter that ranges from about 0.2 for a diffusion transistor to about unity for a drift transistor. Diffusion tran- sistors are transistors in which the base doping is uniform, so that minority carriers cross the base entirely through diffusion. In drift transistors the doping is nonuniform, and an electric field exists in the base that causes a drift of minority carriers which adds to the diffusion current. 13-3 THE ALPHA CUTOFF FREQUENCY Obviously, for high-frequency applications we want /„ to be very large. 1° order to construct a transistor with a definite value of f a , it is necessary t< know all the parameters upon which f a depends. As a first step towar obtaining the desired equation for f a , an expression for the emitter capaci" tance will be obtained. The Diffusion Capacitance Refer to Fig. 13-3, which represents the injected hole concentration vs. distance in the base region of a p-n-p transi tor. The base width W is assumed to be small compared with the diff us 10 S# 13-3 THE HIGH-FREQUENCY TRANSISTOR / 367 length La °f the minority carriers. Since the collector is reverse-biased, the injected charge concentration P at the collector junction is essentially zero (Fig- 9-24). If W <£L B , then P varies almost linearly from the value P(0) a t the emitter to zero at the collector, as indicated in Fig. 13-3. The stored base charge Qb is the average concentration P(0)/2 times the volume of the base WA (where A is the base cross-sectional area) times the electronic charge e; that is, Qb = %P(0)AWe The diffusion current is [from Eq. (5-32)] a r> dP ' „ P(0) I m ~AeD B ar = AeD B -^- where Db is the diffusion constant for minority carriers in the base, bining Eqs. (13-8) and (13-9), IW % (13-8) (13-9) Com- (13-10) The emitter diffusion capacitance Cn, is given by the rate of change of Qb with respect to emitter voltage V, or Cd« = dQi W* dl dV 2D B dV W 2 1 2D s r' t (13-11) where r[ = dV/dl is the emitter-junction incremental resistance. From Eq. (6-41) and neglecting junction recombination, r, = Vt/Ik, where Vr = fcT/e, & ~ Boltzmann's constant in J/°K, T = absolute temperature, and e = elec- tronic charge [Eq. (3-34)]. Hence Cue — 2D B V T (13-12) which indicates that the diffusion capacitance is proportional to the emitter bias current I s . Since Db varies 3 approximately inversely with T t and Vt is pro- portional to T, then Cd* is almost independent of temperature. Except for Very small values of I Ei the diffusion capacitance is much greater than the transition capacitance CV, and hence C e = Co* + Ct» * Cd. Emitter Collector "8- 13-3 Minority-carrier charge distribution in the p ' ) b Q Se region. * = x = W 368 / aECTRONfC DEVICES AND CIRCUITS s «- T3.4 Dependence of /„ upon Base Width or Transit Time From Eqs. (13-2) and (13-11), and since C, » Cz>., then (13-13) This equation indicates that the alpha cutoff frequency varies inversely as the square of the base thickness W. For a p-n-p germanium transistor with W = 1 mil = 2.54 X 10~ 3 cm - 25.4 microns Eq. (13-13) predicts an /. = 2.3 MHz. An interesting interpretation of w a is now obtained. By combining Eqs. (13-10) and (13-13), I = Q B V e (13-14) If Ib is the base transit time (the number of seconds it takes a carrier to cross the base), then in time t B an amount of charge equal to the base charge Q B reaches the collector. The resulting current is i = 9* (13-15) From Eqs. (13-14) and (13-15) we have that w a = 1/fe, or that the alpha cutoff (angular) frequency is the reciprocal of the base transit time. 13-4 THE COMMON-EMITTER SHORT-CIRCUIT-CURRENT FREQUENCY RESPONSE The T model of Fig. 13-2 is applicable in the CE configuration if E is grounded, the signal is applied to B, and the load is placed between C and E. The CE short-circuit current gain A it is obtained by shorting the collector terminal C to E as indicated in Fig. 13-4. Since r e » r. and C. » C c , we may omit the parallel elements r' e and C c> and then I L = a„Ii — al„ But from KCL, «0 il = (XI t r€h O W\/ O 'H-VS/V-* ■="•■ i r' Fig. 13-4 The T circuit in the CE configura- tion under short-circuit conditions. $*. 13- 5 THE HIGH-FREQUENCY TRANSISTOR / 369 j L = I b + I t , so that 1.(1 — a) = —I b . Finally, -a(u) A - ? L - *!> _ " " h ~ ~h " 1 a(w) Using Eq. (13-4), A ie may be put in the form -A where and jAi* ft = i+jy/A «o 1 - a. U = /«(i — <*•) (13-16) (13-17) (13-18) (13-19) At zero frequency the CE short-circuit current amplification is ft *= A /e and the corresponding CB parameter is a m -h /h . Hence Eq. (13-18) is con- sistent with the conversion in Table 11-3. The CE 3-dB frequency, or the beta cutoff frequency, is/* (also designated /*/. or/„). From Eqs. (13-18) and (13-19) h — h/Jff = Ctofa (13-20) 1 Since a, is close to unity, the high-frequency response for the CE configuration is much worse than that for the CB circuit. However, the amplification for the CE configuration is much greater than that for the CB circuit. Note that the so-called short-circuit-current gain-bandwidth product (amplification times 3-dB frequency) is the same for both configurations. '3-5 THE HYBRID-PI (II) COMMON-EMITTER TRANSISTOR MODEL 4 fn Chap. 11 it is emphasized that the common-emitter circuit is the most important practical configuration. Hence we now seek a CE model which will be valid at high frequencies. The circuit of Fig. 13-1 can be used in he CE configuration, but it is too complicated to be useful for analysis. On the other hand, the model of Kg. 13-4 (with a load R L between C and E 'nstead of the short circuit) is fairly simple but inaccurate (except for small Ues of Rl) because it neglects the Early generator. A circuit, called the hybrid-XL, or Giacoletto, model, which does not have the °ve defects, is indicated in Fig. 13-5. Analyses of circuits using this model e not too difficult and give results which are in excellent agreement with Perirnent at all frequencies for which the transistor gives reasonable amplifica- n - Furthermore, the resistive components in this circuit can be obtained ec- 13-6) from the low-frequency h parameters. All parameters (resistances capacitances) in the model are assumed to be independent of frequency. 370 ELECTRONIC DEVICES AND CIRCUITS B r *» f rfi See. T3.5 Fig. 13-5 The hybrid-n model for a transistor in the CE configuration. They may vary with the quiescent operating point, but under given conditions are reasonably constant for small-signal swings. Discussion of Circuit Components The internal node B' is not physically accessible. The ohmic base-spreading resistance rw is represented as si lumped parameter between the external base terminal and B'. For small changes in the voltage IV* across the emitter junction, the excess minority-earner concentration injected into the base is proportional to and therefore the resulting small-signal collector current, with the co!l< shorted to the emitter, is proportional to V be . This effect accounts for the current generator g m V b -„ in Fig. 13-5. The increase in minority carriers in the base results in increase bination base current, and this effect is taken into account by insert :r conductance g b > t between B' and E, The excess-minority-carrier storag the base is accounted for by the diffusion capacitance C, connected beti B' and E (Sec. 13-3). The Early effect (Sec. 9-7) indicates that the varying voltage : collector-to-emitter junction results in base-width modulation. A chant the effective base width causes the emitter (and hence collector) curren change because the slope of the minority-carrier distribution in changes. This feedback effect between output and input is taket by connecting g b > c between B' and C. The conductance between C and E 1 Finally, the collector-junction barrier capacitance is included in C t . S^^H times it is necessary to split the collector-barrier capacitance in ' and connect one capacitance between C and B f and another betweei B. The last component is known as the overlap-diode capacitance. itlK Hybrid-pi Parameter Values Typical magnitudes for the elemei the hybrid-pi model for a germanium transistor at room temperature and io r Ic = 1.3 niA are g m = 50 mA/V r w = 100 fi tv« = 1 K r b . c = 4 M r ce - 80 K C t = 3 pF That these values arc reasonable is justified in the following section. C, = 100 pF THE HIGH-FREQUENCY TRANSISTOR 371 13-6 HYBRID-PI CONDUCTANCES IN TERMS OF LOW-FREQUENCY h PARAMETERS ty'c now demonstrate that all the resistive components in the hybrid-pi model cft n he obtained from the k parameters in the CE configuration. These h parameters are supplied by the manufacturers or can be easily measured (Ch: Transistor Transconductance g m Figure 13-6 shows a p-n-p transistor in the CE configuration with the collector aborted to the emitter for time-varying signals. In the active region the collector current is given by Eq. (9-7), ;ed here for convenience, with ay = a„: IC = I CO — Ct e Is The transconductance g m is denned by die 1 SI B gm ■ dT B'B l^ c « dV S 'E a ° dl dis_ E (13-21) i above we have assumed that ay is independent of V E , For a p-n-p transistor Vg = — Vb>s as shown in Fig. 13-6. If the emitter diode resistance w t\ (Fig. 13-2), then r, = dV s /dI B , and hence 9m = — r' To evaluate r t , note from Eq. (9-19), with V c ~ —V C c, that Ib = a U € v * lv T — an — aii (13-22) (13-23) At <utoff, V E is very negative and Ig ™ — an — an. Since the cutoff current * very small, we neglect it in Eq. (13-23). Hence and : an* 91 E aV E a U i v * ,Vr Is V, V (13-24) 8- 13-d Pertaining to the derivation of B r w>* o J WV 372 / ELECTRONIC DEVICES AND CIRCUITS Substituting Eq. (13-24) in Eq. (13-22), we obtain ObIe Ico — Ic Qm = V, S«. I3. 4 (13-25) For a p-n-p transistor I c is negative. For an n-p-n transistor I c is positive but the foregoing analysis (with V s = + V b >b) leads to g m = (I c — Ico)/V T . Hence, for either type of transistor g m is positive. Since |/<?| » \Ico\, then g m is given by 9* V T (13-26) where, from Eq. (3-34), V T = T/ 11,600. Note that g m is directly proportional to current and inversely proportional to temperature. At room temperature _ |/c|(mA) ff ~ 26 (13-27) For Ic = 1.3 mA, g m = 0.05 mho = 50 mA/V. For I c = 10 mA, g m « 400 mA/V. These values are much larger than the transconductances obtained with tubes. The Input Conductance g Vu In Fig. 13-7o we show the hybrid-pi model valid at low frequencies, where all capacitances are negligible. Figure 13-76 represents the same transistor, using the A-parameter equivalent circuit. From the component values given in Sec. 13-5, we see that ?v e » r b '„ Hence I b flows into r h > e and W, » I b r b >,. The short-circuit collector current is given by I« = gmVb'M « gvJhfi'. Fig. 13-7 (a) The hybrid-pi model at low frequencies,- (b) the A-pararn- eter model at low frequencies. Sec. 13-6 THE HIGH-FREQUENCY TRANSISTOR / 373 The short-circuit current gain h f , is defined by J. I h/ e — hlVm — gmn't or - hit — V*Y r . or (13-28) Jfote that, over the range of currents for which h /e remains fairly constant, f h ,, is directly proportional to temperature and inversely proportional to current. Observe in Fig. ll-5o that at both very low and very high currents, h fB decreases. Since g m = a a /r t and h fe » ^/(l — a*), then n>, may be expressed in terms of the T-model emitter resistor r t as r _ */« _ < Tb ' r ~ Z — t g m 1 — a (13-29) The Feedback Conductance gv. With the input open-circuited, h re is defined as the reverse voltage gain, or from Fig. 13- 7a with lb = 0, ?v, (13-30) or JV,(1 — k rt ) = h T jTi>c Since K» <K 1, then to a good approximation Tb'» = hrtTb'c or g b 'e = h Tt g bta (13-31) Since h n » 10~«, Eq. (13-31) verifies that r b , c » tv,. It is found that h„ is quite insensitive to current and temperature. There- fore r b > c has the same dependence upon \I C \ and T as does r 6 ',. The Base-spreading Resistance rw The input resistance with the out- put shorted is hu. Under these conditions r b >„ is in parallel with 7v c . Using Eq. (13-31), we have »v,]|»v e m r h >„ and hence hie = Tbb 1 + rt*. (13-32) or r»' = hi, — fb't (13-33) Incidentally, note from Eqs. (13-28) and (13-32) that the short-circuit input "npedance A,-, varies with current and temperature in the following manner: l 1 hf e V T h ie = rw + -Vt (13-34) The Output Conductance g ee With the input open-circuited, this con- ductance is defined as h oe . For I b «■ 0, we have L = + Tb'c + r b '. + g m V b (13-35) 374 / ELECTRONIC DEVICES AND CIRCUITS Sec. ?3-* With h = 0. we have, from Eq, (13-30), W« — h rt V e „ and from Eq (13-36), we find Ho* — T7 I T T 9mnrc ' e* ' et i o c (13-3ft) where we made use of the fact that rv„ S> r»«,. If we substitute Eqs. (13-28) and (13-31) in Eq. (13-36), we have h ot — get + 06V + ff6'J»/ e ff« = h ae — (1 + h ft )9b'e (13-37) iiis equation may be put in the form [using Eqs. (13-29) and (13-31)] 0« « & oe — 0mA r (13-38) Summary If the CE ft parameters at low frequencies are known at a given collector current Ic, the conductances or resistances in the hybrid-11 circuit are calculable from the following five equations in the order given: 9m = TV, = fkb' v T hi 5h ' e = /v: hf, — r !r (13-39) Tb'e h rt or ffi'e = ry ?« = A„f - (1 + A/«)ff6'c « — For the typical h parameters in Table 11-2. at Ic = 1.3 mA and room tempera- ture, we obtain the component values listed on page 370. The Hybrid-pi Capacitances The collector-junction capacil C c = CVc is the measured CB output capacitance with the input open (Is - 0)» and is usually specified by manufacturers as Cot. Since in the active regioO the collector junction is reverse-biased, then C* is a transition capacita and hence, varies as Fes - ", where n is ^ or ^ for an abrupt or gradual junction, respectively (Sec. 6-9). Since C = Cfe represents, principally, the diffusion capacitance the emitter junction, it is directly proportional to the current and is apprO»" mateiy independent of temperature (Sec. 13-3). Experimentally, C, is deter- mined from a measurement of the frequency jV at which the GE short- circu 1 Sec A 13-7 THE HIGH-FREQUENCY TRANSISTOR I 375 Fig. 13-8 The hybrid-II circuit for a single transistor with a resistive load R r .. jurrent gain drops to unity. We verify in Sec. 13-7 that C «* 9m 2rfr (13-40) Reasonable values for these capacitances are - - 3 pF C t = 100 pF 13-7 THE CE SHORT-CIRCUIT CURRENT GAIN OBTAINED WITH THE HYBRID-PI MODEL Consider a single-stage CE transistor amplifier, or the last stage of a cascade. The load ft L on this stage is the collector-circuit resistor, so that Ii c — Rl- In this section we assume that R L m 0, whereas the circuit with Rl f* is analyzed in the next section. To obtain the frequency response of the tran- sistor amplifier, we use the hybrid-II model of Fig. 13-5, which is repeated for convenience in Fig. 13-8. Representative values of the circuit components we .specified on page 370 for a transistor intended for use at high frequencies. We use these values as a guide in making simplifying assumptions. The approximate equivalent circuit from which to calculate the short- Wrcuit current gain is shown in Fig. 13-9. A current source furnishes a sinusoidal input current of magnitude /;, and the load current is /j,. We have Neglected 06v which should appear across terminals B'C, because 0&< c « g B - e . And of course g et disappears, because it is in shunt with a short circuit. An af Ulitional approximation is involved, in that we have neglected the current l '9- 13-9 Approximate equivalent Clrcuit for the calculation of the ■tort-circuit CE current gain. 8v HY. ^ ^ C e + C c J> -o—J I" 376 / ELECTRONIC DEVICES AND CIRCUITS S»c. 13.7 delivered directly to the output through g b > e and C c . We see shortly that this approximation is justified. The load current is II = — g^V**, where (13-41) Vb '' g b >. + MC + C.) The current amplification under short-circuited conditions is a = l± = ~ g" * h J^+MC + CJ Using the results given in Eqs. (13-39) , we have — h ft Ai = 1+X//M (13-42) (13-43) where the frequency at which the CE short-circuit current gain falls by 3 dB is given by /,- U<y I 0. 2r(C, + C e ) h f , 2rr(C, + C.) (13-44) The frequency range up to ft is referred to as the bandwidth of the circuit. Note that the value of Ai at w = is — h ft , in agreement with the definition of — h f e as the low-frequency short-circuit CE current gain. The expression for ft obtained in Sec. 13-4 from the high-frequency T model is essentially the same as that given in Eq. (13-44). (See also Prob. 13-12.) Since, for a single-time-constant circuit, the 3-dB frequency ft is given by ft = 1/2-jtRC, where R is the resistance in parallel with the capacitance, we could have written ft by inspection as J& 2rtv.(C. + C.) in agreement with Eq. (13-44). The Parameter ft We introduce now/r, which is defined as the frequency at which the short-circuit common-emitter current gain attains unit magnitude. Since h /t » 1, we have, from Eqs. (13-43) and (13-44), that ft is given by St • h ie ft = since C„ » C, Aim 9m _ Qm 2ir(C. + C t ) ~ 2*C t Hence, from Eq. (13-43), l+JA/.(f/M (13-45) (13-46) The parameter ft is an important high-frequency characteristic of a transistor. Like other transistor parameters, its value depends on the operating condition 8 of the device. Typically, the dependence of ft on collector current is *® shown in Fig. 13-10. $*. 13-7 THE HIGH-FREQUENCY TRANSISTOR / 377 fig, 13-10 Variation of f T with collector current. /V. MHz 400 - 300 - 200 V„-5V r=25°C 1 10 100 I c (log scale), mA Since ft "* hf t ft, this parameter may be given a second interpretation. It represents the short-circuit current-gain-bandwidth product; that is, for the CE configuration with the output shorted, ft is the product of the low-frequency current gain and the upper 3-dB frequency. For our typical transistor (page 370), ft = 80 MHz and ft = 1.6 MHz. It is to be noted from Eq. (13-45) that there is a sense in which gain may be sacrificed for bandwidth, and vice versa. Thus, if two transistors are available with equal ft, the transistor with lower h fe will have a correspondingly larger bandwidth. In Fig. 13-11, Ai expressed in decibels (i.e., 20 log \Ai\) is plotted against frequency on a logarithmic frequency scale. When/ «/^, Ai « — h fe , and Ai (dB) approaches asymptotically the horizontal line Ai (dB) = 20 log h fe . When f»ft, \Ai\ * h/Jf/f = ft/f, so that At (dB) = 20 log A- - 20 log /. Accordingly, Ai (dB) = dB at/ = ft. And for/»/s, the plot approaches J 4,(dB)=201og|j4,| 20 log V 6dB/octave = 20dB/decade log L log/r log/ Fig. 13-11 The short-circuit CE current gain vs. frequency (plotted on a log-log scale). 378 / ELECTRONIC DEVICES AND CIRCUITS Sec. 13-3 as an asymptote a straight line passing through the point (/r, 0) and having a slope which causes a decrease in A, (dB) of dB per octave, or 20 dB p er decade. The intersection of the two asymptotes occurs at the "corner" frequency / = /$, where A< is down by 3 dB. Earlier we neglected the current delivered directly to the output through g b - c and C c . Now we may see that this approximation is justified. Consider, say, the current through C e . The magnitude of this current is uC.l whereas the current due to the controlled generator is g m Vb>*- The ratio of currents is mC e /g m . At the highest frequency of interest fr, we have, from Eq. (13-45), using the typical values of Fig. 13-8, g m g m C. + C c 0.03 In a similar way the current delivered to the output through g b - c may be shown to be negligible. The frequency /r is often inconveniently high to allow a direct experimental determination of f T . However, a procedure is available which allows a measurement of /r at an appreciably lower frequency. We note from Eq. (13-43) that, for/ »/ 3 , we may neglect the unity in the denominator and write \Ai\f » fifth - h from Eq. (13^5). Accordingly, at some particular fre- quency /i (say /i is five or ten times f fi ), we measure the gain \An\. The parameter f T may be calculated now from f T = /i|A;i|. In the case of our typical transistor, for which f T = 80 MHz and fr = 1.6 MHz, the frequency /i may be /i = 5 X 1.6 = 8.0 MHz, a much more convenient frequency than 80 MHz. The experimentally determined value of /r is used to calculate the value of C e in the hybrid-II circuit. From Eq. (13-45), C = gm 2-KJT (13-473 From Eqs. (13-20) and (13-45), f T » M# = «»/«■ H e nce [t is expected that f a and f T should be almost equal. Experimentally, it is found that in diffusion transistors /„ ~ 1.2/ r , whereas in drift transistors f a ~ 2/ r . These values may be accounted for if the excess-phase factor for a in Eq. (13-7) « taken into consideration. 13-8 CURRENT GAIN WITH RESISTIVE LOAD To minimize the complications which result when the load resistor Rl in *' l &' 13-8 is not zero, we find it convenient to deal with the parallel combination of g b > c and C c , using Miller's theorem of Sec. 12-9. We identify F 6 <« with Vi in Fig. 12-14 and V ce with 7 8 . On this basis the circuit of Fig. 13-8 may b 6 replaced by the circuit of Fig. 13-12a. Here K m VJV b > e . This circuit « still rather complicated because it has two independent time constants, on THE HIGH-FREQUENCY TRANSISTOR ' 379 fig. 13-12 (a) Approximate equivalent circuit for calculation of response of a transistor amplifier stage with a resistive toad; (b) further simplification of the equivalent circuit. associated with the input circuit and one associated with the output. We now show that in a practical situation the output time constant is negligible in comparison with the input time constant, and may be ignored. Let us therefore delete the output capacitance C e (K - l)/K, consider the resultant Circuit, and then show that the reintroduction of the output capacitance makes no significant change in the performance of the circuit. Since K = V c ,/V b ; is (approximately) the voltage gain, we normally have |jK| » 1. Hence g b - c (K - 1)/K « g Ve . Since g b > e <5C g» fon m4r. M and r et = 80 K), we may omit g b > c from Fig. 13-12a. In a wideband amplifier, Rl seldom exceeds 2 K. The conductance g c * may be neglected compared with Ri, and the output circuit consists of the current generator g m V b >, feeding the load Rl, as indicated in Fig. 13-126. Even if the above approximations were not valid for some particular transistor or load, the analysis to follow fe still valid provided that Rl is interpreted as the parallel combination of the collector-circuit resistor, r ce and rv*. By inspection of Pig. 13-126, K = VJV V . - -g m Rz. Forg m = 50 mA/V **id R L = 2,000 Si, K = —100. For this ax mum value of K, conduct- anc e g b 'c(l — K) =* 0.025 mA/V is negligible compared with g b >« « 1 mA/V. Hence the circuit of Fig. 13-12a is reduced to that shown in Fig. 13-126. The load resistance Rl has been restricted to a maximum value of 2 K because, at values of Rl much above 2,000 ft, the capacitance C c (l + g*ML) becomes excessively large and the bandpass correspondingly small. Now let us return to the capacitance C e (K — 1)/K ^ C c , which we Neglected above. For R L - 2,000 Q t RlC c = 2 X 10 3 X 3 X 10" 1Z = 6 X lO- 8 sec - 6 nsec 380 / ELECTRONIC DEVICES AND CIRCUITS The input time constant is S«. )3-9 ui> tune u u lis ta.ii v is n».[C. + C.(l + gJt L )] = 10'(100 + 3 X 101)10-" sec = 403 nsec It is therefore apparent that the bandpass of the amplifier will be determined by the time constant of the input circuit and that, in the useful frequency range of the stage, the capacitance C e will not make itself felt in the output circuit. Of course, if the transistor works into a highly capacitive load this capacitance will have to be taken into account, and it then might happen that the output time constant will predominate. The circuit of Fig. 13-126 is different from the circuit of Fig. 13-9 only in that a load R L has been included and that C c has been augmented by g„R L C . To the accuracy of our approximations, the low-frequency current gain Ai„ uuder load is the same as the low-frequency gain A* with output shorted. Therefore Alo = —hf« However, the 3-dB frequency is now / 2 (rather than //j), where gv where 13-9 n 2irr b >£ 2ttC C = C. + C.(l + gM TRANSISTOR AMPLIFIER RESPONSE, TAKING SOURCE RESISTANCE INTO ACCOUNT (13-48) (13-49) In the preceding discussions we assumed that the transistor stage was driven from an ideal current source, that is, a source of infinite resistance. We now remove that restriction and consider that the source has a resistive impedance R,. We may represent the source by its Norton's equivalent, as in Fig- 13-1 3a, or by its Thevenin's equivalent, as in Fig. 13-13&. At low frequencies (and with R t = » ) the current gain is A Io = I L /Ii = —g^V Vt /g Vt V v % — —A/* from Eq. (13-28). Therefore the low-frequency current gain, taking the load ■"-CD •*l Fig. 13-13 (o) A transistor is driven by a generator of resistance R, which is represented by its Norton's equivalent circuit, (fa) The generator is represented by its Thevenin's equivalent. $tc 1 3-9 THE HIGH-FREQUENCY TRANSISTOR / 381 and source impedances into account, is j. lL __ Jl Ii _ i Rt __ — kfeRt At* J - jr* j - ft/. ^ _j_ ^ + ^ ~ Rt + hu (13-50) since A« = r tb> + rv* Note that Ai, is independent of Rl. The 3-dB fre- quency is determined by the time constant consisting of C and the equivalent resistance R shunted across C. Accordingly, fr- 1 2ttRC (13-51) where C is given by Eq. (13-49), and R is the parallel combination of R t + r& and tb'; namely, n s {R. + rw}ry, R, + h it (13-52) From Eq, (11-39) we have that the voltage gain Ar» at low frequency, taking load and source impedances into account, is Rl _ — hftRt R, R, -f- hit (13-53) Note that A Yso increases linearly with Rl. The 3-dB frequency for voltage gain Ay is also given by Eq. (13-51). Note that /s increases as the load resistance is decreased because C is a linear function of Rl- At Rl ■= 0, the 3-dB frequency is finite (unlike the vacuum-tube amplifier, which has infinite bandpass for zero plate-circuit resistance; Sec. 16-6) and from Eq. (13-47) is given by J _ h h QmR ft* R L = (13-54) 2ir/e(C. + C t ) g m R gy.R For R, = 0, this quantity is of the order of /r/5 » 10//j, and for R, = 1 K (and Rl = 0), ft ** /r/25 w 2f#. Of course, for Rl = 0, the voltage gain is *ero. In practice, when Rl ?* 0, much lower 3-dB frequencies than those indicated above will be obtained. The equality in 3-dB frequencies for current and voltage gains applies only in the case of a fixed source resistance. The voltage gain Ay (for the case of an ideal voltage source) and the current gain At (for the case of an 'deal current source) do not have the same value of fo. In the former case, "• = 0, and in the latter case, R, = «*. Equation (13-51) applies in both cases provided that, for Av, we use R — Rv, where, from Eq. (13-52) with ft. - 0, R v = ***»« m &&1 (13-55) Tbb' + Tb' e hit and for A T we use R = R h where, from Eq. (13-52) with R, = « , Rt = n,, (13-56) h 't>ce R Y «i? /( the 3-dB frequency f 2V for an ideal voltage source is higher ban fa for an ideal current source. 382 / ELECTRONIC DEVICES AND CIRCUITS 7 3-9 S<K 13-9 THE HIGH-FREQUENCY TRANSISTOR / 383 The Gain-Bandwidth Product This product is found in Prob. 13-18 to b Rl Jt Rl \A It J t \ = ■2wC R t + r». 1 + %tfrC c Ri, R t + r u - St R, 1 + 2irf T C e RL R, + r«> ( 13-58) The quantities / 2 , At»o, and j4v«,, which characterize the transistor stage, depend on both Rt and R„ The form of this dependence, as well as the ordfl of magnitude of these quantities, may be seen in Fig. 13-14. Here /•■ has plotted as a function of R L , up to R L ■= 2,000 U, for several values of R,. The topmost / 2 curve in Fig. 13-14 for R t = corresponds to ideal-voltage-source drive. The current gain is zero, and the voltage gain ranges from ze: Rl = to 90.9 at Rl = 2,000 0. Note that a source impedance of only 100 Q reduces the bandwidth by a factor of about 1.8. The bottom curve has R t — w and corresponds to the ideal current source. The voltage gain is zero for all R L if R, = « . For any Rl the bandwidth is highest for lowest R„ In the case of a vacuum-tube stage of amplification, the gain-bandwidth product is a useful number (Sec. 16-6). For a transistor amplifier con lA-.o/il.MHz 300 300 100 ft , MHz 16 19 14 w ./.■ 8 -^R, = 50 _J 11 Vsi .=0 4 50q 90.9 36011 ron"* 87.0 2. 1 7 83.3 4.17 t «-£■=■*• 74. 1 9.25 50 500 1,000 1,500 2,000 J?/.,Q Fig. 13-14 Bandwidth/., as a function of R L , with source resistance as a parameter, for an amplifier consisting of one CE transistor whose param- eters are given in Sec. 13-5, Also, the gain-bandwidth product for a 50-11 source is plotted. The tabulated values of |.4k..| correspond to Rl = 2,000 fi and to the values of R. on the curves. The values of |/li„| ore inde- pendent of R L . 1 j n c of a single stage, however, the gain-bandwidth product is ordinarily not a il parameter; it is not independent of R, and Rl and varies widely with both- The currcnt-gain-bandwidth product decreases with increasing R !t and increases with increasing R„. The voltage-sain -bandwidth product inert increasing Rl and decreases with increasing R a . Even if we know the gain-bandwidth product at a particular R, and Rl, we cannot use the product to determine the improvement, say, in bandwidth corresponding to a sacrifice in gain. For if we change the gain by changing R M or R L or both, generally, the gain-bandwidth product will no longer be the same as it had been. Summary The high-frequency response of a transistor amplifier is obtained by applying Eqs. (13-49) to (13-53). We now show that only four independent transistor device parameters appear in these equations. Hence these four (A,,, fa, h ic , And C e = £*) are usually specified by manufacturers of high-frequency transistor.- From the operating current Ic and the temperature T, the transconduct- ance is obtained [Eqs. (13-39)] as g m p PM/F* and is independent of the par- ticular device under consideration. Knowing g m we can find, from Eqs. (13-39) and (13-40), TW — hit — TV* l"* "* ^T Wt = If R, and R L are given, then all quantities in Eqs. (13-49) to (13-53) are known. We have therefore verified that the frequency response may be determined from the four parameters hj, t f T , hu, and C* REFERENCES 1. Phillips, A. B.: "Transistor Engineering," chaps. 13 and 14, McGraw-Hill Book Company, New York, 1962. Pritchard, R. L.: Electric-network Representations of Transistors: A Survey, IRE Trans, Circuit Theory, vol. CT-3, no. 1, pp. 5-21, March, 1956. Searle, C. L., A. R. Boothroyd, E. J. Angelo, Jr., P. E. Gray, and D. 0. Pederson: "Elementary Circuit Properties of Transistors," vol. 3, Semiconductor Electronics Education Committee, John Wiley & Sons, Inc., New York, 1964. 2 - Thomas, D. E., and J. L. Moll: Junction Transistor Short-circuit Current Gain and Phase Determination, Proc. IRE. vol. 46, no. 6, pp. 1177-1184, June, 1958. 3 - Phillips, A. B.: "Transistor Engineering," pp. 129-130, McGraw-Hill Book Com- pany, New York, 1962. *• Giacoletto, L. J.: Study of p-n-p Alloy Junction Transistors from DC through Medium Frequencies, RCA Rev., vol, 15, no. 4, pp. 506-562, December, 1954. Searle, C. L., A. B. Boothroyd, E. J. Angelo, Jr., P. E. Gray, and D. 0. Pederson: "Elementary Circuit Properties of Transistors," vol. 3, chap. 3, Semiconductor Electronics Education Committee, John Wiley & Sons, Inc., New York, 1964. 14 /FIELD-EFFECT TRANSISTORS The field-effect transistor 1 is a semiconductor device which depends for its operation on the control of current by an electric field. There are two types of field-effect transistors, the junction field-effect transis- tor (abbreviated JFET, or simply FET) and the insulated-gate field- effect transistor (IGFET), more commonly called the metal-ox idc-semir conductor {MOS) transistor (.MOST or MOSFET). The principles on which these devices operate, as well as the differences in their characteristics, are examined in this chapter. Representative circuits making use of FET transistors are also presented. The FET enjoys several advantages over the conventional transistor: 1. Its operation depends upon the flow of majority carriers only. It is therefore a unipolar (one type of carrier) device. The vacuum tube is another example of a unipolar device. The conventional tran- sistor is a bipolar device. 2. It is relatively immune to radiation. - It exhibits a high input resistance, typically many meg- ohi 4. It is less noisy than a tube or a bipolar transistor. 5. It exhibits no offset voltage at zero drain current, and hence makes an excellent signal chopper. 2 6. It has thermal stability (Sec. 14-4). The main disadvantage of the FET is its relatively small gain- bandwidth product in comparison with that which can be obtained with a conventional transistor. 384 Sec. M-1 FIELD-EFFECT TRANSISTORS / 3B5 14-1 THE JUNCTION FIELD-EFFECT TRANSISTOR I The structure of an n-channel field-effect transistor is shown in Fig. 14-1. Ohmic contacts are made to the two ends of a semiconductor bar of n-type material (if p-type silicon is used, the device is referred to as a p-channel FET). Current is caused to flow along the length of the bar because of the voltage supply connected between the ends. This current consists of majority carriers which in this ease are electrons. The following FET notation is standard. Source The source S is the terminal through which the majority carriers enter the bar. Conventional current entering the bar at S is designated by la. Drain The drain D is the terminal through which the majority carriers leave the bar. Conventional current entering the bar at D is designated by Id- The drain-to-source voltage is called Vns, and is positive if D is more positive than S. Gate On both sides of the n-type bar of Fig. 14-1, heavily doped (p + ) regions of acceptor impurities have been formed by alloying, by diffusion, or by any other procedure available for creating p-n junctions. These impurity regions are called the gate G. Between the gate and source a voltage Vaa is applied in the direction to reverse- bias the p-n junction. Conventional cur- rent entering the bar at G is designated Iq. Channel The region in Fig. 14-1 of n-type material between the two gate regions is the channel through which the majority carriers move from source to drain. FET Operation It is necessary to recall that on the two sides of the transition region of a reverse-biased p-n junction there are space-charge regions (Sec. 6-9). The current carriers have diffused across the junction, leaving only uncovered positive ions on the n side and negative ions on the p side. The electric lines of field intensity which now originate on the positive ions and terminate on the negative ions are precisely the source of the voltage drop across the junction. As the reverse bias across the junction increases, so also does the thickness of the region of immobile uncovered charges. The con- ductivity of this region is nominally zero because of the unavailability of cur- rent carriers. Hence we see that the effective width of the channel in Fig. 14-1 w iH become progressively decreased with increasing reverse bias. Accordingly, '°r a fixed drain-to-source voltage, the drain current will be a function of the Averse-biasing voltage across the gate junction. The term field effect is used to describe this device because the mechanism of current control is the effect °f the extension, with increasing reverse bias, of the field associated with the e £ion of uncovered charges. FET Static Characteristics The circuit, symbol, and polarity conventions f or a FET are indicated in Fig. 14-2. The direction of the arrow at the gate 3&6 ELECTRONIC OF/ICES AND CIRCUITS Sec. J4.fi Depletion region D Drain Fig. T4-1 The basic structure of an n -channel field -effect tran- sistor. The normal polarities of the drain-to-source and gate-to- source supply voltages are shown. In a p-channel FET the volt- ages would be reversed. of the junction FET in Fig. 14-2 indicates the direction in which gate current would flow if the gate junction wore forward-biased. The common-source drain characteristics for a typical n-ehannel FET shown in Fig. 14-3 give Id against Vdb, with Voa as a parameter. To see qualitatively why the charac- teristics have the form shown, consider, say, the case for which Vos = 0. For Io = 0, the channel between the gate j unctions is entirely open. In resp io a small applied voltage V DS) the n-type bar acts as a simple semiconductor resistor, and the current I D increases linearly with Vd$. With increasing cur- rent, the ohmic voltage drop between the source and the channel region reverse- biases the junction, and the conducting portion of the channel begins to con- strict. Because of the ohmic drop along the length of the channel itself, the Fig. 14-2 Circuit symbol for an ^-channel FET. (For a p-channe) FET the arrow at the gate junc- tion points in the opposite direc- tion.) For an n-channel FET, Id and V m; are positive and Vos |S negative. For a p-channel FET, to and P'ti.s are negative and Vg* iS positive. Vcc,^- *! V n Sec U-1 FIELD-EFFECT TRANSISTORS , 387 + 0.5 !_ - -0.5 - 1 / / ^-~"*' -1.0 -2 1.5 - 2.5 1 -3.oi -3.5 . -4.0 i it 15 20 Drain -source voltage V !ls lL 3H Fig. 14-3 Common-source drain characteristics of an n-channel field-effect transistor. (Courtesy Texas Instruments, Inc.) constriction is not uniform, but is more pronounced at distances farther from the source, as indicated in Fig. 14-1. Eventually, a voltage V ds is reached at which the channel is "pinched off." This is the voltage, not. too sharply defined in Fig. 14-3, where the current Id begins to level off and approach a constant value. It is, of course, in principle not possible for tin:: channel to close completely and thereby reduce the current Id to zero. For if such, indeed, could be the case, the ohmic drop required to provide the necessary hack bias would itself be lacking. Note that each characteristic curve has an ohmic region for small values of Vdb, where Id is proportional io V DS . Each slfio has a constant-current region for large values of Vna, where Id responds Wy slightly to V oa . If now a gate voltage Vos » applied in the direction to provide additional Averse bias, pinch-off will occur for smaller value ..■:[. and the maxi- mum drain current will be smaller. This feature is brought out in Fig. 14-3. bat a plot for a silicon FET is given even for Voa = +0.5 V, which is in wie direction of forward bias. We note from Table 9-1 that, actually, the Kate current will be very small, because at this gate voltage the Si junction is P a rely at the cutin voltage V-,. The similarity between the FET character- istics and those of a pentode tube need hardly be belabored. The maximum voltage that can be applied between any two terminals of the FET is the lowest voltage that will cause avalanche breakdown (Sec. 6-12) ^oss the gate junction. From Fig. 14-3 it is seen that avalanche occurs at a lower value of | V D s\ when the gate is reverse-biased than for V G s = 0. This 388 / ELECTRONIC DEVICES AND CIRCUITS s «. ]4. a is caused by the fact that the reverse-bias gate voltage adds to the drain volt- age, and hence increases the effective voltage across the gate junction. We note from Fig. 14-2 that the n-ehannel FET requires zero or negative gate bias and positive drain voltage, and it is therefore similar to a vacuum tube. The p-channel FET which requires opposite voltage polarities behaves like a vacuum tube in which the cathode emits positive ions instead of elec- trons. Either end of the channel may be used as a source. We can remem- ber supply polarities by using the channel type, p or n, to designate the polarity of the source side of the drain supply. The field-effect transistor existed as a laboratory device from 1952 to 1962. The reason why no large- scale production and use of this device took place is that semiconductor-device technology only recently reached the degree of refinement required for the production of a thin, lightly doped layer between two more heavily doped layers of opposite type. A Practical FET Structure The structure shown in Fig. 14-1 is not prac- tical because of the difficulties involved in diffusing impurities into both sides of a semiconductor wafer. Figure 14-4 shows a single-en ded-geometry junc- tion FET where diffusion is from one side only. The substrate is of p-type material onto which an n-type channel is epitaxially grown (Sec. 15-2). A p-type gate is then diffused into the n-type channel. The substrate which may function as a second gate is of relatively low resistivity material. The diffused gate is also of very low resistivity material, allowing the depletion region to spread mostly into the n-type channel. 14-2 THE PINCH-OFF VOLTAGE V M We derive an expression for the gate reverse voltage Vp that removes all the free charge from the channel using the physical model described in the pre- ceding section. This analysis was first made by Shockley, 1 using the structure of Fig. 14-1. In this device a slab of n-type semiconductor is sandwiched between two layers of p-type material, forming two p-n junctions. Assume that the p-type region is doped with N A acceptors per cubic meter, that the n-type region is doped with No donors per cubic meter, and that the Di a: ii Gate Source Channel Fig. 14-4 Single-ended-geometry junction FET. S* 14-2 FIELD-EFFECT TRANSISTORS / 389 junt'tJ° n formed is abrupt. The assumption of an abrupt junction is the same £9 that made in Sec. 6-9 and Fig. 6-12, and is chosen for simplicity. More- over, if Na 2> N D , we see from Eq. (6-44) that W p « W n) and using Eq. (6-47), ffC have, for the space-charge width, W n (x) = W{x) at a distance x along the channel in Fig. 14-1 : W(x) =a-b(x) = \^W. - 7(»)l}* (14-1) where e — dielectric constant of channel material e = magnitude of electronic charge V„ = junction contact potential at x (Fig. 6-ld) V(x) = applied potential across space-charge region at x and is a negative number for an applied reverse bias a - b(x) — penetration W(x) of depletion region into channel at a point x along channel (Fig. 14-1) If the drain current is zero, b{x) and V(x) are independent of x and b{x) = b. If in Eq. (14-1) we substitute b(x) = b = and solve for V, on the assumption that \V„\ << \V\, we obtain the pinch-off voltage V P , the diode reverse voltage that removes all the free charge from the channel. Hence , T/ I eN D \ v r\ - ~2T a (14-2) If we substitute V G s for V and a — 6 for x in Eq. (6-46), we obtain, using Eq. (14-2), V 0S = (l-^V, (14-3) The voltage V G s in Eq. (14-3) represents the reverse bias across the gate junction and is independent of distance along the channel if I D = 0. EXAMPLE For an w-channel silicon FET with o = 3 X 10~* cm and N D - 10 1S electrons/cm 3 , find (c) the pinch-off voltage and (fa) the channel half-width for Vos - %V P and l a = 0. Solution a. The relative dielectric constant of silicon is given in Table 5-1 as 12, and hence e = 12e„. Using the value of e and e„ from Appendixes A and B, ^e have, from Eq. (14-2), expressed in mks units, T/ 1.60 X 10~ 19 X 10 !l X (3 X lO" 8 )* 2 X 12 X (36x X 10 8 )-' b. Solving Eq. (14-3) for b, we obtain for V gs — \Yp 6-ol- ( ~ j - (3 X 10^*) [1 - (m = 0.87 X 10-" cm Hence the channel width has been reduced to about one-third its value for Vos ~ 0. 390 / ELECTRONIC DEVICES AND CIRCUITS Sec. "-3 U-3 THE JFET VOLT-AMPERE CHARACTERISTICS Assume, first, that a small voltage V&s is applied between drain an 'ting small drain current Id will then have no appreciable enV the channel profile. Under these c consider the channel cross section A to be constant throughout its length. Hence A ~ ^H when be channel width corresponding to zero drain current as give Eq. (14-3) for a specified Vas, and w is the channel dimension perpendicular to the b direction, as indicated in Fig. 14-1. Bince no current {Lows in the depletion region, then, using Ohm's law [Eq. (5-1)], we obtain for the drain current Id = AeN D nn& = 2bweN D fi n —^ where L is the length of the channel. Substituting b from Eq. (14-3) in Eq. (14-4), we have 2aweN D n„ Id = I »[>-mh The on* Resistance rj(o.v) Equation (14-.").) describes the volt-ai characteristics of Fig. 14-3 for very small Vdb, and it suggests that under these conditions the FET behaves like an nhmic resistance whose value is >• mined by Vcs- The ratio Vos/Id igin is called the ox drain re ance r d {os). For a JFET we obtain from Eq. (14-5), with Vas = 0. r<t( on) = 7 I 2aweN Diu For the alues given in the illustrative example in this section with L/to = 1. we find that r rf (o!v) = 3.3 K. For the dimensions and con tralion used in commercially available I'LTs and MOSFETs (Sec. i ; values of r d (0N) ranging from about 100 to to 100 K arc measured. Th« parameter is important in switching applications where the FET heavily ox. The bipolar transistor has the advantage over the field-c device in that Hcs is usually only a few ohms, and hence is much smaller than r d (ox). However, a bipolar transistor has the disadvantage for choppy applications 2 of possessing an offset voltage (Sec. 9-14), whereas the I characteristics pass through the origin, Id = and V D s = 0. The Pinch-off Region We now consider the situation where an el' field S x appears along the x axis. If a substantial drain current Id flows, ,nC drain end of the gate is more reverse-biased than the source end, and h<' I,ce the boundaries of the depletion region are not parallel to the center channel, but converge as shown in Fig. 14-1. If the convergence of the depl 6 " tiou region is gradual, the previous one-dimensional analysis is valid 1 in thin slice of the channel of thickness Ax and at a distance x from the source- Sec 14-3 FIELD-EFFECT TRANSISTORS I 391 Subject to this condition of the "gradual" channel, the current may be written pection of Fig. 14-1 as I D = 2b(x)weN Dfin S x (14-7) As \'ds increases, S, and Id increase, whereas b{x) decreases because the channel narrows and hence the current density J = /jq/26(jc)w increases. We n ow see that complete pinch-off (6 = D> cannot take place because, if it did, J would become infinite, which is a physically impossible condition. If J W ere to increase without limit, then, from 1 .aso would 6*, provided that m« remains constant. It is found experimentally, 3 * however, that the mobility is a function of electric field intensity and remains constant only for £* < 10 3 V/crn in n-type silicon. For moderate fields, 10* to 10 4 V/>m, the mobility is approximately inversely proportional to the square root of the applied field. For still higher fields, such as are encountered at pinch-off, p. is inversely proportional to 8*. In this region the drift velocity of the electrons (v x = /i„C x ) remains constant, and Ohm's law is no longer valid. From Eq. (14-7) we now see that both Ip and b remain constant, thus explain- ing the constant-current portion of the V-I characteristic of Fig, 14-3. What happens* if Vds is increased beyond pinch-off, with Vas held constant'? As explained above, the minimum channel width 6 min = 8 has a small nonzero constant value. This minimum width occurs at the drain end of the bar. As V DS is increased, this increment in potential causes an increase in 6* in an adjacent channel section toward the source. Referring to Fig. 14-5, the velocity-limited region U increases with V DS , whereas 5 remains at a fixed value. The Region before Pinch-off We have verified that the FET behaves as an ohmic resistance for small V D a and as a constant-current device for large *ds- An analysis giving the shape of the volt-ampere characteristic between these two extremes is complicated. It has already been mentioned that in this region the mobility is at first independent of electric field and then m Fi 9. 14-5 After pinch-off, as ' ,- is increased, then I J '""eases but 5 and I D re- main essentially constant. "' a nd Gi are tied to- other.) + ll~ or Depletion region I 1 ? Gl > — 26(.r) ^Cs^is. H7 ^w* ^SM^lm f* I, x .1 ll * -'1 + 392 / afCTRONJC DEVICES AND CIRCUITS U-4 varies with 6«~* for larger values of & x (before pinch-off). Taking this rela- tionship into account, it is possible* -6 to obtain an expression for I D as a function of Vj>$ and V G s which agrees quite well with experimentally deter- mined curves. The Transfer Character isHc In amplifier applications the FET is almost always used in the region beyond pinch-off (also called the constant-current pentode, or current-saturation region). Let the saturation drain current be designated by Ids, and its value with the gate shorted to the source (Vq S = 0) by loss- It has been found* that the transfer characteristic, giving the rela- tionship between Ids and Vos, can be approximated by the parabola Ids = 1 1 o - w (14-8) This simple parabolic approximation gives an excellent fit, with the experi- mentally determined transfer characteristics for FETs made by the diffusion process. Cutoff Consider a FET operating at a fixed value of Vds in the pentode region. As V s is increased in the direction to reverse-bias the gate junction, the conducting channel will narrow. When Vos = Vp, the channel width is reduced to zero, and from Eq. (14-7), Ids = 0. With a physical device some leakage current /d(off) still flows even under the cutoff condition \Vos\ > \Vp\. A manufacturer usually specifies a maximum value of J d (off) at a given value of Vos and Vds- Typically, a value of a few nanoamperes may be expected for / d (off) for a silicon FET. The gate reverse current, also called the gate cutoff current, designated by lass, gives the gate-to-source current, with the drain shorted to the source for \Vas\ > \Vp\. Typically, loss is of the order of a few nanoamperes for a silicon device. 14-4 THE FET SMALL-SIGNAL MODEL The linear small-signal equivalent circuit for the FET can be obtained in ft manner analogous to that used to derive the corresponding model for a vacuum tube or a transistor. We employ the same notation in labeling time-varying and dc currents and voltages as used in Sees. 7-9 and 9-13 for the vacuum tube and transistor. We can formally express the drain current %d as a func- tion / of the gate voltage vq& and drain voltage v D s by Id — f(vas, Vds) (14-9) The Transconductance g n and Drain Resistance r$ We now proceed ft 8 in Sec. 8^. If both the gate and drain voltages are varied, the change i° i S* 14-4 REID-EFFECT TRANSISTORS / 393 drain current is given approximately by the first two terms in the Taylor's series expansion of Eq. (14-9), or a ■ d»z> I . . diD I A Md = ~£T~ „ Av OS + T- L &V DS OVqs l v *a OVds \ v °* (14-10) jn the small-signal notation of Sec. 8-1, Atj> = id, Avqs = f ff „ and Avds = v<u, so that Eq. (14-10) becomes id - gmVf H — Vd. where g m = Bid I At'p j — I V„ \Vbs (14-11) (14-12) is the mutual conductance, or transconductance. It is also often designated by y/t or g f , and called the (common-source) forward tran&admittance. The second parameter rd in Eq. (14-11) is the drain (or output) resistance, and is defined by dVps I _ &VPS I _ Vdt I Bio \ v <>* Aio Was id l^o* Td = (14-13) The reciprocal of r& is the drain conductance g^. It is also designated by y ot and g , and called the (common-source) output conductance. The parameters g m and r* are completely analogous to the vacuum-tube parameter g m and r p . An amplification factor /i for a FET may be denned, just as it is for a tube, by _ dvps I _ At?gg | _ v^, I dvos \ l ° Avas Vd v , \io Proceeding as in Sec. 8-4, we verify that ft, u, and g m are related by (14-14) M = Tdg„ (14-15) A circuit for measuring g m is given in Fig. 14-6a. It follows from Eq. (14-12) that (if \V t \ « V DD> so that Vds = const) 9m " Vi Vt/R* Vr V x Rd (14-16) Similarly, the circuit of Fig. 14-66 allows r d to be measured. From Eq. (14-13) it follows that " I d V./Rd V, (14-17) An expression for g m is obtained by applying the definition of Eq. (14-12) 10 Eq. (14^8). The result is ffm = *» ^1 - -^J (14-18) 394 ELECTRONIC DEVICES AND CIRCUITS Sac, l4^f 14-4 FIELD-EFFECT TRANSISTORS / 395 Oscillator — -, Oscillator ^ & © !f- (a) (6) Fig. 14-6 Test circuits for measuring (a) p m and (b) r rf . The rms volt- ages Vj and T, are measured with ac high-impedance voltmeters. where g„<. is the value of g m for V f ,.s = 0, and is given by — 2 / 1 <?mo = r Siaee/j»s* and V> area! opposil positive. Thi ship, com has beei experimentally. 1 Since f/™. vith tin- circuit of Fig. 14-6a = 0, and /W ■■■m '« read on a de milliarnmel d in the drain 1 i same circuit (wit I gab methdd for obtain? i The dependei upon Vea is indicated in Fig. 14-7 for the 2b ;.: v and th< i FET [with 1-7 V) The lii relationship predicted by Eq. (14-18 C 100 so ■ f~ -10V l kHz ^■^^^ ^•^ \ X T\ f& ^» * Fig, 14-7 Tronsconductance g„ versus gate voltage for types 2N3277 and 2N3278 FETs. (Courtesy of Fairchild Semiconductor Company.) 1.0 2.0 3.0 4.0 Gate voltage V GS . V 5.0 l.h : * 1 J „- 50 tOO 150 Ambient temperature T Al "C 0.8 0.6 -50 («) 50 100 Ambient temperature T A , "C <6) Fig. 14-8 (a) Normalized tronsconductance g„, versus ambient temperature T A and (b) normalized drain resistance r„ versus 7\ (for the 2N3277 and the 2N3278 FETs with Vbb = -10 V, Yea = V, and / - I kHz). (Courtesy of Fairchild Semiconductor.) Temperature Dependence Curves of g m and r d versus temperature are given in Fig. 14-8a and I. The drain current Ids has the same temperature variation as does g m . The principal reason for itive temperature coefficient of Ids is that the mobility decreases with increasing temperature. 8 Since this majority-carrier current decreases with temperature (unlike the bipolar transistor whoso minority-carrier current increases with temperature), the me phenomenon of thermal runaway (Sec. 10-10) is not encount- ritb field-effect transistors. The FET Model We note thai Kq. (14-11) is identical with Eq. (8-13) tor the triode provided that k icaUujdf) is replaced by • ^mtrce), that p (plate) J8 replaced by d {drain-), am both identified as gate (instead of grid). ■bee the smull- signal tube equivalent circuit of Fig 8-8 is valid for the FET. This model is repeated in Fig. 14-0, with the appropriate change of notation. In this figure we have also included I deh exist between pairs °f nodes, i corresponding to the high-frequency triode model of Fig. 8-19). The Btacitor C vt represents the barrier capacitance between gate and source, and "••14-9 Small -signal FET •ftodel Gate G o • 1( » Source So *■ ♦ o Drain D 5=*- u «i(p < r " SC i OS 396 / aECTRONIC DEVICES AND CIRCUITS s «- J4.J TABLE M.I Range of parameter values for a FET Parameter JFET MOSFETf ffm 0.1-10mA/V . 1-20 mA/V or more U 0.1-1 M 1-50 K C* 0,1-1 pF 0.1-1 pF "an "ad 1-10 pF 1-10 pF Tgi >10* a >io i ° a T B d >io fl a >10'*fi t Discussed in Sec. 14-5. C e d is the barrier capacitance between gate and drain. The element C& represents the drain-to-source capacitance of the channel. The order of magnitudes of the parameters in the model for a diff used- junction FET is given in Table 14-1. Since the gate junction is reverse- biased, the gate-source resistance r a , and the gate-drain resistance r gd are extremely large, and hence have not been included in the model of Fig. 14-9. U-5 THE INSULATED-GATE FET (MOSFET) In preceding sections we developed the volt-ampere characteristics and small- signal properties of the junction field-effect transistor. We now turn our attention to the insulated-gate FET, or metal-oxide-semiconductor FET,' which promises to be of even greater commercial importance than the junction FET. The rt-channel MOSFET consists of a lightly doped p-type substrate into which two highly doped n + regions are diffused, as shown in Fig. 14-10. These n + sections, which will act as the source and drain, are separated by about 1 mil. A thin layer of insulating silicon dioxide (Si0 2 ) is grown over the surface of the structure, and holes are cut into the oxide layer, allowing contact with the source and drain. Then the gate-metal area is overlaid on the oxide, Source Gate(+) Drain , Aluminum Fig. 14-10 Channel enhancement in a MOSFET. (Courtesy of Motorola Semiconductor products, Inc.) See US FIELD-EFFECT TRANSISTORS / 397 covering the entire channel region. Simultaneously, metal contacts are made *o the drain and source, as shown in Fig. 14-10. The contact to the metal ver the channel area is the gate terminal. The metal area of the gate, in conjunction with the insulating dielectric oxide layer and the semiconductor channel, forms a parallel-plate capacitor. The insulating layer of silicon dioxide is the reason why this device is called the insulated-gate field-effect transistor. This layer results in an extremely high input resistance (10 10 to 10" £2) for the MOSFET. The Enhancement MOSFET If we ground the substrate for the structure of Fig. 14-10 and apply a positive voltage at the gate, an electric field will be directed perpendicularly through the oxide. This field will end on "induced" negative charges on the semiconductor site, as shown in Fig, 14-10. The nega- tive charge of electrons which are minority carriers in the p-type substrate forms an "inversion layer." As the positive voltage on the gate increases, the induced negative charge in the semiconductor increases. The region beneath the oxide now has n-type carriers, the conductivity increases, and current flows from source to drain through the induced channel. Thus the drain cur- rent is "enhanced" by the positive gate voltage, and sueh a device is called an enhancement-type MOS. The volt-ampere drain characteristics of an «-channel enhancement-mode MOSFET are given in Fig. 14-1 la, and its transfer curve, in Fig. 14-llb. The current Idss at Vgs < is very small, being of the order of a few nanoamperes. As V as is made positive, the current Id increases slowly at first, and then much more rapidly with an increase in Vgs- The manufacturer sometimes indicates the gate-source threshold voltage Vasr at which In reaches some defined small value, say 10 uA. A current Id(on), corresponding approxi- mately to the maximum value given on the drain characteristics, and the value of Vgs needed to obtain this current are also usually given on the manu- facturer's specification sheets. h. mA ■9. 14-11 (a) The drain characteristics, and (fa) the transfer curve (for Vds = ") of an n-channel enhancement-type MOSFET. 398 ELECTRONIC DEVICES AND CIRCUITS Dili'. 3iO.. channel Source i Aluminum t P (substrain Enhancement Fig. 14-12 (a) A depletion-type MOSFET. (b) Channel depletion with the appli- cation of a negative gate voltage. (Courtesy of Motorola Semiconductor Products.'lnc.) The Depletion MOSFET A second type of MOSFET can be made the basic structure of Fig. 14-10, an n channel is diffused between the source and the drain, as shown in Fig. M-12o. With this device an appreeiab; current loss flows for zero gate-to-source voltage, V GS ~ 0. If the ga age is made negative, positive charges arc induced in the channel through the SiOj of the gate capacitor. Since the current in a FET is due i majoriflj carriers (electrons for an Retype material), the induced positive charge; the channel less conductive, and the drain current drops as Vas is nun: negative. The redistribution of charge in the channel causes an etl depletion of majority carriers, which accounts for the designation d< MOSFET. Note in Fig. 14-126 that, because of the voltage drop due drain current, the channel region nearest the drain is more depleted than w the volume near t.lic source. This phenomenon is analogous to that of pinch-off occurring in a JFET at the drain end of the channel (Fig. 14-1). As a •. the veil -ampere characteristics of the depletion-mode MOS and the JFET are quite similar. A MOSFET of the depletion type just described may also be op> i in an enhancement mode. It is only necessary to apply a positive ga age so that negative charges are induced into the rc-type channel. In this manner the conductivity of I he channel increases and the current rises above Jdss- The voit-ampere characteristics of this device are indicated in J' 1 ?* 14-13n, and the transfer curve is given in Fig. 14-136. The deplete enhancement rt ending to Vos negative and positive, n should be noted. The manufacturer sometimes indicates the gale-sourct ! I'utojf voltage Vgs(orr), at which F D is reduced to some specified negligible val recommended Vos- This gate voltage corresponds to the pinch-off vol tag* V P of a JFET. The foregoing discussion is applicable in principle also to the /)-chan' ie FIEIDEFFECT TRANSISTORS 399 l B ,mA Depletion -« — J D (on) = 6 5 4 3 — *- Enhancement ^ -"""l .. : — . 1 . t - 3 - 2 V& (OFF) Vm.V Fig. 14-13 (a) The drain characteristics and (b) the transfer curve (for Vaa = 10 V) for an ^-channel MOSFET which may be used in either the enhancement or the depletion mode. FET. For such a device the signs of all currents and voltages in the volt-ampere characteristics of Figs. 11-11 and 14-13 must be reversed. Circuit Symbols It is possible to bring out the connection to the sub- ex lernally so as to have a tetrode device. Most Ah ►SFETSj however, are triodes, with the substrate internally connected to the source. The circuit iscd by several manufacturers arc indicated in Fig. 14-14 Some- the symbol of Fig. 14-2 for the JFET is also used for the MOSFET, Rtfa the understanding that Gt is internally connected to *S. Small-signal MOSFET Circuit Model m If the small bulk resistances of the source and drain are neglected, the small-signal equivalent circuit of the FET between terminals G (= GV), >'. and D is identical with that given in Fig 14-9 for the JFET. The transconductance g m and the interelectrode capacitances have comparable values for the two types of devices. However, as noted in Table 14-1 on page 396, the drain resi pi the MOSFET is very much smaller than that of the JFET The magnitude of r d for a :omparable with the plate resistance of a triode. whereas u for a •FF.T has a value approximating the r, of a pentode. It should also be noted 111 Tabic 14-1 that the input resistance r s , and the feedback resistance r 9i are v *ry much larger for the MOSFET than for the JFET. Drain D OD 6, o- 6 Source S OS 6 S (a) (6) (C) Fig, 14-14 Three circuit symbols for a p-channel MOSFET. 400 / HECTRONJC DEVICES AND CIRCUITS Sec. l4 If the substrate terminal G* is not connected to the source, the model of Fig. 14-9 must be generalized as follows: Between node ff 2 and S, a diode t}\ is added to represent the p-n junction between the substrate and the source Similarly, a second diode D2 is included between (? s and D to account for the p-n junction formed by the substrate and the drain. 14-6 THE COMMON-SOURCE AMPLIFIER The three basic JFET or MOSFET configurations are the common-source (CS), common-drain (CD), and common-gate (CG). The configurations are shown in Fig. 14-15 for a p-channel JFET. Unless specifically stated other- wise, the circuits discussed throughout this chapter apply equally well to JFETs or MOSFETs. Voltage Gain The circuit of Fig. 14-16o is the basic CS amplifier con- figuration. If the FET is replaced by the circuit model of Fig. 14-9, we obtain the circuit of Fig. 14-166, which is equivalent to that of Fig. 8-196 for a CK triode amplifier with interelectrode capacitances taken into account. [In Fig. 8-19 the voltage source ttV t in series with r„ may be transformed into a current source /iVi/r r = g m Vi, in parallel with r p (Sec. 8-5).] Hence the voltage gain Av = VJVi for the CS amplifier as given by Eq. (8-39), which is repeated here, using FET notation, Ay = -g m + y ad Y l 4- Yd, + Qd + Y 9d (14-20) where Y L = 1/Z L = admittance corresponding to Zi Yd* = juCd, = admittance corresponding to Cd, §d = l/rd = conductance corresponding to Td Ygd ■« juCgd ~ admittance corresponding to C td At low frequencies the FET capacitances can be neglected. Under these con- v^y Output Input Output (<*> Input Input £1 — o Output (c) Fit- 14-15 The three FET configurations: (a) CS, (b) CD, and (c) CG. 1 Sec. U-6 FIEID-EFFECT TRANSISTORS / 401 ditions, Yd, = Y a d = 0, and Eq. (14-20) reduces to Qm QmZt. A v = Y L + 1 4- g d Z L = ~9mZ' L (14-21) where Z' L m r d \\Z L . This equation is identical with Eq. (8-40). Input Admittance An inspection of Fig. 14-166 reveals that the gate cir- cuit is not isolated from the drain circuit. Since Figs. 14-166 and 8-19 are identical, the input admittance is given by Eq. (8-42), or Y<= Y st +{\~ A v )Y g d (14-22) This expression indicates that for a field-effect transistor to possess negligible input admittance over a wide range of frequencies, the gate-source and gate- drain capacitances must be negligible. Also, as explained in Sec. 8-12, it is possible for the input resistance to be negative for an inductive load, and the circuit may oscillate. Input Capacitance (Miller Effect) Consider a FET with a drain-circuit resistance R d . From the previous discussion it follows that within the audio- frequency range, the gain is given by the simple expression A v = -g m R' d , where R' d is R d \\r d . In this ease, Eq. (14-22) becomes 5 m d - C„ + (1 + fJKJC* (14-23) This increase in input capacitance d over the capacitance from gate to source is caused by the familiar (Sec. 8-12) Miller effect. This input capacitance is important in the operation of cascaded ampli- fiers, as is discussed in Sec. 8-12 in connection with vacuum tubes. Output Resistance For the common-source amplifier of Fig. 14-16a, the rrxrr gwVi I s £ V a i- (a) (6) F '9- 14-16 {a) The common-source amplifier circuit; (b) small-signal equiva- le "t circuit of CS amplifier. 402 / afCTRONIC DEWCES AND CIRCUITS Sec. ?4.y output resistance R e is given by the parallel combination of r d and Rd, or TdRd R a = Td + Rd (14-24) Equation (14-24) is valid at low frequencies, where the effect of the capacitors in Fig. 14-166 is negligible, and with a resistive load, Zc = Rd. EXAMPLE A MOSFET has a drain-circuit resistance R d of 100 K and operates at 20 kHz. Calculate the voltage gain of this device as a single stage, and then as the first transistor in a cascaded amplifier consisting of two identical stages. The MOSFET parameters are g m = 1.6 mA/V, r d = 44 K, C 9 , = 3.0 pF, C d , - 3.8 pF, and C ad = 2.8 pF. Solution The numerical values of the circuit parameters for this particular MOSFET are identical with the parameter values of the triode used in the exam- ple of Sec. 8-12. The solution of this example is therefore the same as that given in Sec. 8-12. Hence (Ay). = -48,6 and (4v)«„t.u,. = 38.8/143.3' Sec u-e FIELD-EFFECT TRANSISTORS / 403 jfote that the amplification is positive and has a value less than unity. If g „R, » 1, then Av « gj(g m + g d ) = n/(u + 1). Input Admittance The source follower offers the important advantage of lower input capacitance than the CS amplifier. The input admittance Yi is given by Eq. (8-55), or Yi « juC ed + juCM ~ Av) (14-27) Output Admittance The output admittance Y , with R t considered external to the amplifier, is given by Eq. (8-58), or Y = g m + g d + jtaC T At low frequencies the output resistance R is 1 1 Ra = g m + gd g* (14-28) (14-29) since g m » g d . For g m = 2 mA/V, then R = 500 G. The source follower is used for the same applications as the cathode follower, those requiring high input impedance and low output impedance (Sec. 8-8). 14-7 THE COMMON-DRAIN AMPLIFIER, OR SOURCE FOLLOWER The CD-amplifier connection shown in Fig. 14-17 is analogous to the cathode follower discussed in Sec. 8-14. The voltage gain of this circuit is given by Eq. (8-53), or in FET notation, (ffm + j(*C g ,)R t A v = (14-25) l + (gm + g d + jvC T )R. where C T = C a , + C d , + C tn , and C.„ represents the capacitance from source to ground. At low frequencies the gain reduces to Ay m g» g ' vp (14-26) v 1 + (g m + g d )R t -V„ V,o )\- Fig. 14-17 Source-follower circuit. ■o V a 14-8 A GENERALIZED FET AMPLIFIER The analysis of the CS amplifier with a source resistance R„ the CG con- figuration, and the CD circuit at low frequencies is made by considering the generalized configuration in Fig. 14-18. This circuit contains three independ- ent signal sources, t>, in series with the gate, v, in series with the source, and » a in series with the drain. For the CS amplifier, v, « v a = 0, and the output •s ».i taken at the drain. For the CG circuit, t\ = v a = 0, the signal is i>, wit h a source resistance R t) and the output is v \. For the source follower, R d = 0, v * ~ v a — 0, the signal voltage is i\, and the output is v z taken at the source, line signal-source resistance is unimportant since it is in series with a gate *hich draws negligible current.) If the effect of the ripple voltage in the Power supply Vdd is to be investigated, v a will be included in the circuit to ^present these small changes in V DD . 8-8 The Output from the Drain From the analysis given in Sees. 8-6 and we obtain the Thevenin's equivalent circuit from drain to ground (Fig. 19a) and from source to ground (Fig. 14-196). From the former circuit e conclude that ''looking into the drain" of the FET we see (for small-signal ^Peration) an equivalent circuit consisting of two generators in series, one of —ft €s the gate-signal voltage v,* and the second (^ + 1) times the source-signal 0e y. and the resistance r d + (ft + 1)R.. Note that the voltage v, and resistance in the source lead are both multiplied by the same factor, /* + 1 • tow th 404 / ELECTRONIC DEVICES AND CIRCUITS Fig. 14-18 A generalized FET amplifier. o v e3 The CS Amplifier with an Unbypassed Source Resistance From Fig, 14-19a, with v, = v a = 0, we obtain for the voltage gain, Ay = foi/»», — flRd —{JmRd Ay = (14-30) r d + ( M + 1)J2. + Kd 1 + ffm#. 4- ffd(ie. + Rd) Note that, for R t = 0, this result reduces to that given in Eq. (14-21), with Zi replaced by R d . The minus sign indicates a 180° phase shift between input and output. The resistance R„, looking into the drain, is increased by 0* + l)R> from its value r d for R, = 0. The net output resistance R' os taking R d into account, is K = [n+ b+\)R.]\\R* (14-31) We observe that the addition of R„ reduces the voltage gain and increases the output impedance. The input impedance is in excess of 100 M since the gate junction is reverse-biased. The CG Amplifier From Fig. 14-19a, with v { = v a = 0, we obtain for the voltage gain, Ay = v e i/v„ A = (m + l)Rd m (gm + g d )Rd (14-32) V r d + (ft + 1)R, + R d 1 + g m R t + g d {R. + Rd) Since Ay is a positive number, there is no phase shift between input and output. Also, since g m y> g d) the magnitude of the amplification is appro* 1 " mately the same as for the CS amplifier with R, ^ 0. The output resistance R' is given by Eq. (14-31), and unless R § is Q ul small, R'„ will be much larger than r d \\R d . The input impedance R t between source and ground is obtained by inspection of Fig. 14-196 : (14-33) * ■ (ttt) "*■ S«c. H-9 FIEID-EFFECT TRANSISTORS / 405 l The common-gate amplifier with its low input resistance and high output resistance has few applications. The CG circuit at high frequencies is con- sidered in Prob. 14-11. The Output from the Source From Fig. 14-196 we conclude that "looking into the source" of the FET we see (for small-signal operation) an equivalent circuit consisting of two generators in series, one of value m/(m + 1) times the gate-signal voltage $ and the second t/(jt + 1) times the drain-signal voltage v a and a resistance (r d + R d )/(fi + 1). Note that the voltage v a and the resistance in the drain circuit are both divided by the same factor, fi 4- 1. The CD Amplifier The voltage gain A r of the source follower is obtained, by inspection, from Fig. 14-196, with v, = v a = and R d = 0: ~ * nfiv + 1) + R, ~ 1 + (g m + 9d )R t (14 ' 34) Note that this expression agrees with Eq. (14-26), obtained by setting w = into the high-frequency formula for Ay. If R d 9* 0, then A 7 in Eq. (14-34) is modified only by the addition of the term g d R d to the denominator. The output impedance R of the source follower at low frequencies (with R d = and with R t considered external to the amplifier) is, from Fig. 14-196, R = U (14-35) P + 1 g m + gd which agrees with Eq. (14-29). The output impedance &„ taking R, into account, is R' = R \\R,. 14-9 BIASING THE FET The selection of an appropriate operating point (I D , V GS , Vz> s ) for a FET amplifier stage is determined by considerations similar to those given to tubes r d + R„ s —o- 'flfc. X (a) (6) 9- 14-19 The equivalent circuits for the generalized amplifier of Fig. 14-18 00 king into" (a) the drain and (b) the source. Note that n = r lt g m . 406 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1 4.9 Ftg. 14-20 Source self-bias circuit. and transistors, as discussed in Sec. 7-13 and Chap. 10. These considerations are output-voltage swing, distortion, power dissipation, voltage gain, and drift of drain current. In most cases it is not possible to satisfy all desired speci- fications simultaneously. In this section we examine several biasing circuits for field-effect devices. Source Self -bias The configuration shown in Fig. 14-20 is the same as that, considered in connection with the biasing of vacuum tubes. It can be used to bias junction FET devices or depletion-mode MOS transistors. For a specified drain current I D , the corresponding gate-to-source voltage Vos can be obtained either using Eq. (14-8) or from the plotted drain or transfer characteristics. Since the gate current is negligible, the source resistance R. can be found as the ratio of Vas to the desired Id- EXAMPLE The amplifier of Fig. 14-20 utilizes an n-channel FET for which V, = -2.0 V, gmo = 1.60 mA/V, and /o« = 1.65 mA. It is desired to bias the circuit at 1„ = 0.8 mA, using Vdd = 24 V. Assume r d » R d . Find (a) F«* (6) g m , (c) A%, (<0 Rd, such that the voltage gain is at least 20 dB, with R, bypassed with a very large capacitance C,. Solving, Solution a. Using Eq V QS = -0.62 V. (14-8), we have 0.8 = 1.65(1 + V as /2.0y b. Equation (14-18) now yields A _ |^\ = LH mA /V ?, = 1-60 e , Rt m _ Z« = ^ = 0.77 K = 770 « /d 0.8 d. Since 20 dB corresponds to a voltage gain of 10, then Av - gmR<i > l0, ° R<t > 10 1.11 = 9K 1 Sec. M-9 FIELD -EFFECT TRANSISTORS / 407 Biasing for Zero Current Drift 11 Figure 14-21 shows the dependence of the transfer characteristics on temperature. Observe from this figure that there exists a value of V GS for which I D = I Q does not change with temperature T. It is therefore possible to bias a field-effect transistor for zero drain-current drift. An explanation of this effect is possible if we note that two factors affect the variation of drain current with T, The first factor is the decrease of majority-carrier mobility with temperature. As T increases, the lattice ions vibrate more vigorously, and hence the carriers cannot move as freely in the crystalline structure. Thus, for a given field strength, their velocity is decreased, and this reduces the current. It has been found 12 that the reduc- tion in Id is 0.7 percent/°C. The second factor is the decrease of the width of the gate-to-channel barrier with increasing temperature. This allows Id to increase, and it has been found that the increase in I D is equivalent to a change of 2.2 mV/°C in \Vas\- This is a similar phenomenon to that which gives a bipolar transistor a change of | Vbe\ of 2.5 mV/°C, as discussed in Sec. 6-7. Since a change in gate voltage A Fes causes a change in drain current of Qn AFos, then the condition for zero drift is oi- 0.007|/i,| = O.OO220 n 9* = 0.314 V (14-36) (14-37) If we substitute Eqs. (14-8), (14-18), and (14-19) in Eq. (14-36), we obtain \Vp\ - \V as \ = 0.63 V (14-38) Equation (14-38) gives the value of Vos for zero drift if V P is known. If Vr - 0.63 V, V GS = and I D = loss- From Eqs. (14-8), (14-18), and (14-38), '9-14-21 Transfer characteristics f °r an ?t-channel FET as a function of temperature T. -V B 408 / ELECTRONIC DEVICES AND CIRCUITS and Id = h 9m /0.63\ a 0.63 W7\ Sec. ?4-9 (14-39) (14-40) Equations (14-39) and (14-40) can be used to specify the drain current and transconductance for zero drift of Id with T. The parameters V P> Idss, and gmo in Eqs. (14-38) to (14-40) are measured at T = 25°C EXAMPLE It is desired to bias the amplifier stage of the previous example for zero drain-current drift. If R<i = 10 K, find (a) I B for zero drift, (6) V QS> (c) R., (d) the voltage gain, with R. bypassed with a very large capacitance C,. Solution a. From Eq. (14-39), To = 1.65 | 1 m 0.165 mA = 165 jiA pfy- 6. From Eq. (14-38), Vm - -1.37 V c. Since Fes = —IdR, 1.37 B. = K = 8.3 K 0.165 d. From Eq. (14-40), we have ft. = 1.60 /0.63\ \~) m 0.50 mA/V Hence Ay « ff m fl* = 0.50 X 10 = 5.0. We thus see that zero drift has been obtained at the expense of g m and voltage gain, which are now one-half their values in the previous example. Biasing against Device Variation FET manufacturers usually supply information on the maximum and minimum values of Idss and Vp at room temperature. They also supply data to correct these quantities for tempera- ture variations. The transfer characteristics for a given type of n-channe PET may appear as in Fig. 14-22a, where the top and bottom curves are for extreme values of temperature and device variation. Assume that, on tl basis of considerations previously discussed, it is necessary to bias the device a a drain current which will not drift outside of Id = I a and I D — Is- Tn< the bias line Vgs = ~ IdR, must intersect the transfer characteristics betwee the points A and B, as indicated in Fig. 14-22a. The slope of the bias h is determined by the source resistance R,. For any transfer character^ between the two extremes indicated, the current Iq is such that I A < Iq ^ as desired. ~ Sec. U-9 h ( + ) less f max) l oss (rnin) ^ \ V 3 / j- Bias line h \ ~ x — jSt^^ h FIELD-EFFECT TRANSISTORS / 409 lo < + ) Bias line W(min) fr(max) V as V o0 (+) -_o— •-<-) (a) (6) (+)■ Fig. 14-22 Maximum and minimum transfer curves for an n-channel FET. The drain current must lie between I A and I B . The bias line can be drawn through the origin for the current limits indicated in (a), but this is not possible for the currents specified in (b). Consider the physical situation indicated in Fig. 14-226, where a line drawn to pass between points A and B does not pass through the origin. This bias line satisfies the equation Vgs = Vgg — IdR* (14-41) Such a bias relationship may be obtained by adding a fixed bias to the gate in addition to the source self-bias, as indicated in Fig. 14-23a. A circuit requiring only one power supply and which can satisfy Eq. (14-41) is shown ^=c. (a) (6) Fig. 14-23 (a) Biasing a FET with a fixed-bias Vera in addition to self-bias through R,. (b) A single power-supply configuration which is equivalent to the circuit in (a). 410 / ELECTRONIC DEVICES AND CIRCUITS Sec. I4»9 in Fig. 14-236. V QG — For this circuit RiV DC Ri 4- R2 R m R1R2 R 1 -\- Ri We have assumed that the gate current is negligible. It is also possible for Vqq to fall in the reverse-biased region so that the line in Fig. 14-226 intersects the axis of abscissa to the right of the origin. Under these circumstances two separate supply voltages must be used. EXAMPLE FET 2K3684 is used in the circuit of Fig. 14-236. For this n-chan- nel device the manufacturer specifies T'p(min) = —2 V, F f (max) = — 5 V, /css(min) = 1.6 mA, and /cssfmax) = 7.05 mA. The extreme transfer curves are plotted in Fig. 14-24. It is desired to bias the circuit so that /a(min) = 0.8 mA --= J A and / a (max) = 1.2 mA = I B for V OD - 24 V. Find (a) V Ga and R„ (6) the range of possible, values in Id if R, = 3.3 K and Vao = 0. Solution a. The bias line will lie between A and B as indicated if it is drawn to pass through the two points Yea = 0, Id = 0.9 mA, and V aa = —4 V, //, = 1.1 mA. The slope of this line determines R„ or R. = 4-0 1.1 -0.9 = 20 K Then, from the first point and Eq. (14-41), we find Vgg - IdR. = (0.9) (20) = 18 V Fig. 14-24 Extreme trans- fer curves for the 2N3484 field -effect transistor. (Courtesy of Union Car- bide Corporation.) Sac. M-70 FIELD-EFFECT TRANSISTORS / 411 6. If R, = 3.3 K, we see from the curves that Jx>(min) = 0,4 mA and /e(max) = 1.2 mA. The minimum current is far below the specified value of 0.8 mA. Biasing the Enhancement MOSFET The self-bias technique of Fig. 14-20 cannot be used to establish an operating point for the enhancement-type MOSFET because the voltage drop across R a is in a direction to reverse-bias the gate, and a forward gate bias is required. The circuit of Fig. 14-25o can be used, and for this case we have Vg s = V B s, since no current flows through R f . If for reasons of linearity in device operation or maximum output voltage it is desired that V G s 7* V DS , then the circuit of Fig. 14-256 is suitable. We note that Vos = [Ri/(Ri + R/)]Vds- Both circuits discussed here offer the advantages of dc stabilization through the feedback introduced with R/. However, the input impedance is reduced because, by Miller's theorem (Sec. 12-9), Rf corresponds to an equivalent resistance Ri = R//(l — Av) shunting the amplifier input. Finally, note that the circuit of Fig. 14-236 could also be used with the enhancement MOSFET, but the dc stability introduced in Fig. 14-25 through the feedback resistor R/ would then be missing. 14-10 UNIPOLAR-BIPOLAR CIRCUIT APPLICATIONS 12 The main advantages of the unipolar transistor, or FET, are the very high input impedance, no offset voltage, and low noise. For these reasons a FET is most useful in a low-level high-input-impedance circuit, such as a signal chopper or the first stage of a unipolar-bipolar cascade combination. In this section we consider the advantages of some representative FET-bipolar transistor or FET-FET combinations. Source Follower with Constant-current Supply Consider the source fol- lower of Fig. 14-17, where the g m of the FET is 1 mA/V at I D = 1 mA. In order to have A v > 0.98, then, by Eq. (14-26), R, > 49 K, provided g m » g 4 . -v„ F '9- 14-25 (o) Drain- '°-gate bias circuit for enhancement-mode M °S transistors; (b) ""proved version of (a) „ -v B 412 / ELECTRONIC DEVICES AND CIRCUITS Sec. W.Jo Q-V„ Q-V n (a) (6) Fig. 14-26 A source follower with (a) a bipolar transistor and (b) o FBT constant-current supply. It is clear that the drain supply must exceed 49 V. Since most FETs have low breakdown voltages, it might be impractical to obtain Av > 0.98 with this circuit. This difficulty is circumvented in the configuration of Fig. 14-26a, which shows a source follower with the constant-current supply circuit discussed in Sec. 12-12. Here the effective source resistance of Ql is the output impedance of Q2, whose value is given by Eq. (12-51). Since this dynamic source resist- ance is very high, then Av approaches the maximum value of m/(m + 1)- Simi- larly, the source follower of Fig. 14-266 makes use of the high dynamic resist- ance R', = r d + (m + !)#« in the source circuit of Ql. 9-v n ~v n (a) (6) Fig. 14-27 Bootstrap circuits for very high input impedance. M-H FIELD-EFFECT TRANSISTORS / 413 -Vn Fig. 14-28 Direct-coupled cascode circuit. Bootstrap FET Circuits for Very High Input Impedance The input resist- ance in the circuits of Fig. 14-26 is essentially 5i||iZa. If very high input impedance is desired, the bootstrap principle discussed in Sec. 12-10 must be invoked. The circuits of Fig. 14-27 employ a FET source follower with a bootstrapped bias network which allows input impedances on the order of tens of megohms to be obtained. In Fig. 14-27a, the output circuit is an emitter follower, and a voltage gain close to unity is possible. In Fig. 14-276, the output is taken from the collector circuit of Q2, and hence this circuit is a low-noise high-input-impedance amplifier with Av — v /V{ > 1. Expres- sions for Ay and also for v t /vi are given in Prob. 14-30. The Cascode Amplifier Circuit This configuration is a version of the cascode circuit discussed in Sees. 8-10 and 12-11. In Fig. 14-28 a common- source FET drives a common-base bipolar transistor. The FET is biased at high I Df thus giving high values of g m . The advantage of this circuit is that the drain voltage Vdd can be high since the FET d rain-to-sou rcc voltage < V. A large supply Vdd allows the resistance Rl to be high, thus giving a large voltage gain and output swing. The cascode amplifier offers good isolation between output and input and iB useful for high-frequency amplification. 14-11 THE FET AS A VOLTAGE-VARIABLE RESISTOR 13 (WR) , y*- most linear applications of field-effect transistors the device is operated l ° the constant-current portion of its output characteristics. We now consider *ET transistor operation in the region before pinch-off, where Vds is small. in this region the FET is useful as a voltage-controlled resistor; i.e., the drain- w-source resistance is controlled by the bias voltage Vgs- In such an applica- 414 / ELECTRONIC DEVICES AND CIRCUITS r„,K too 80 60 40 r* l H_ If T J-,9N^91^ - L-H "J T Sec. M-Jj (a) (6) 2.0 3.0 Fig. 14-29 (a) FET low-level drain characteristics for 2N3278. (b) Small-signal FET resistance variation with applied gate voltage. (Courtesy of Fairchild Semiconductor Company.) tion the FET is also referred to as a voltage-variable resistor (WR) or voltage- dependent resistor (VDR). Figure 14-29a shows the low-level bidirectional characteristics of a FET. The slope of these characteristics gives r d as a function of Vqs. Figure 14-29o has been extended into the third quadrant to give an idea of device linearity around Vds = 0. In our treatment of the junction FET characteristics in Sec. 14-3 we derive Eq. (14-5), which gives the drain-to-source conductance g d = Id/Vds for small values of Vds- From this equation we have w = 0*[i -(tj)*] (14-42) where g do is the value of the drain conductance when the bias is zero. In Ref. 4 it is shown that g d „ is equal to the value of the FET transconductance g m measured for Vqs = and for a drain voltage Vds higher than the pinch-off voltage V P . Variation of r d with Vqs is plotted in Fig. 14-296 for the 2N3277 and 2 N 3278 FETs. The variation of r d with V os can be closely approximated by the empirical expression r d = 1 - KVas (14-43) where r = drain resistance at zero gate bias K = a constant, dependent upon FET type Vos = gate-to-source voltage Applications of the WR Since the FET operated as described above acts like a variable passive resistor, it finds applications in many areas where ?4-I2 FIELD-EFFECT TRANSISTORS / 415 pjg. 14-30 AGC amplifier using the FET as a voltage- yariable resistor. this property is useful. The WR, for example, can be used to vary the voltage gain of a multistage amplifier A as the signal level is increased. This action is called AGC, or automatic gain control. A typical arrangement is shown in Fig. 14-30. The signal is taken at a high-level point, rectified, and filtered to produce a dc voltage proportional to the output-signal level. This voltage is applied to the gate of Q2, thus causing the ac resistance between the drain and source to change, as shown in Fig. 14-296. We thus may cause the gain of transistor Ql to decrease as the output-signal level increases. The dc bias conditions of Ql are not affected by Q2 since Q2 is isolated from Ql by means of capacitor C*. 14-12 THE UNIJUNCTION TRANSISTOR Another device whose construction is similar to that of the FET is indicated in Fig. 14-31. A bar of high-resistivity n-type silicon of typical dimensions 8 X 10 X 35 mils, called the base B, has attached to it at opposite ends two ohmic contacts, 51 and B2. A 3-mil aluminum wire, called the emitter E, Base, B n-type Si bar Al rod sO" p~n junction- (a) J32 > Ohmic / contacts LI 31 (b) Fig. 14-31 Unijunction transistor, (a) Constructional details; (b) circuit symbol. 416 / HfCTRONJC DEVICES AND CIRCUITS Sac. ?4.j 2 S 10 T^=2S C V M = 30V 1 1 L-20 ^10 Sc 5 *S2 = Fig. 14-32 Unijunction input character- istics for types 2N489 to 2N494. (Cour- tesy of General Electric Company.) 2 4 6 8 10 12 14 16 Emitter current I £ ,mA is alloyed to the base to form a p~n rectifying junction. This device was originally described in the literature as the double-base diode, but is now com- mercially available under the designation unijunction transistor (UJT). The standard symbol for this device is shown in Fig. 14-316. Note that the emitter arrow is inclined and points toward Bi whereas the ohmic contacts B\ and B2 are brought out at right angles to the line which represents the base. The principal constructional difference between the FET and the UJT is that the gate surface of the former is much larger than the emitter junction of the latter. The main operational difference between the two devices is that the FET is normally operated with the gate junction reverse-biased, whereas the useful behavior of the UJT occurs when the emitter is forward- biased. As usually employed, a fixed interbase potential Vbb is applied between B\ and B2. The most important characteristic of the UJT is that of the input diode between E and Bh If 52 is open-circuited so that I B z = 0, then the input volt-ampere relationship is that of the usual p-n junction diode as given by Eq. (6-31). In Fig. 14-32 the input current- voltage characteristics are plotted for I B t = and also for fixed values of interbase voltage Vbb- Eacn of the latter curves is seen to have a negative-resistance characteristic. A qualitative explanation of the physical origin of the negative resistance is given in Ref. 14. The principal application of the UJT is as a switch which allows the rapid discharge of a capacitor (Ref. 13). REFERENCES 1. Shockley, W.: A Unipolar Field-effect Transistor, Proc. IRE, vol. 40, pp. 1365- 1376, November, 1952. Dacey, G. C, and I. M. Rosa: The Field Effect Transistor, Bell System Tech. J-> vol. 34, pp. 1149-1189, November, 1955. FIELD-EFFECT TRANSISTORS / 417 Wallmark, J. T., and H. Johnson: "Field-effect Transistors," Prentice-Hall, Inc. Englewood Cliffs, N.J., 1966. Sevin, L. J.: "Field-effect Transistors," McGraw-Hill Book Company, New York 1965. 2. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," sec. 17-20, McGraw-Hill Book Company, New York, 1965. 3. Wallmark, J. T., and H. Johnson: "Field-effect Transistors," p. 115, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. 4. Sevin, L. J.: "Field-effect Transistors," pp. 13-17, McGraw-Hill Book Company, New York, 1965. 5. Halladay, H. E., and A. Van Der Ziel: DC Characteristics of Junction Gate Field- effect Transistors, IEEE Trans. Electron Devices, vol. ED-13, no. 6, pp, 531-532 June, 1966. 6. Ref. 4, p. 21. 7. Ref. 4, p. 23. 8. Ref. 4, p. 34. 9. Ref. 3, pp. 187-215. 10. Ref. 3, pp. 256-259. 11. Hoerai, J. A., and B. Weir: Conditions for a Temperature Compensated Silicon Field Effect Transistor, Proc. IEEE, vol. 51, pp. 1058-1059, July, 1963. Evans, L. L.: Biasing FETs for Zero dc Drift, Electrotechnol, August, 1964, po. 93-96. 12. Gosling, W.: A Drift Compensated FET-Bipolar Hybrid Amplifier, Proc. IEEE, vol. 53, pp. 323-324, March, 1965. '3. Bilotti, A.: Operation of a MOS Transistor as a Voltage Variable Resistor, Proc. IEEE, vol. 54, pp. 1093-1094, August, 1966. '*■ Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," sees. 12-3 and 13-13, McGraw-Hill Book Company, New York, 1965. 15 /INTEGRATED CIRCUITS An integrated circuit consists of a single-crystal chip of silicon, typi- cally 50 by 50 mils in cross section, containing both active and passive elements and their interconnections. Such circuits are produced by the same processes used to fabricate individual transistors and diodes. These processes include epitaxial growth, masked impurity diffusion, oxide growth, and oxide etching, using photolithography for pattern definition. A method of batch processing is used which offers excel- lent repeatability and is adaptable to the production of large numbers of integrated circuits at low cost. The main benefits derived from this technology are high reliability, size reduction, and low cost, as compared with the use of discrete components interconnected by con- ventional techniques. In this chapter we describe the basic processes involved in fabricating an integrated circuit. 15-1 BASIC MONOLITHIC INTEGRATED CIRCUITS 12 We now examine in some detail the various techniques and processes required to obtain the circuit of Fig. 15- la in an integrated form, W shown in Fig. 15-15. This configuration is called a monolithic inte- grated circuit because it is formed on a single silicon chip. The wor "monolithic" is derived from the Greek monos, meaning "single," » in lithos, meaning "stone." Thus a monolithic circuit is built into single stone, or single crystal. In this section we describe qualitatively a complete epita xlfl diffused fabrication process for integrated circuits. In subsequeo sections we examine in more detail the epitaxial, photographic, ft diffusion processes involved. The circuit of Fig. 15- lo is chosen discussion because it contains typical components: a resistor, dio« 41 S S9C- 15-1 INTEGRATED CIRCUITS / A\9 Q 3 -+«- (a) Resistor Diode junctions Transistor Aluminum metalization Silicon dioxide Collector contact n* n* Emitter Base Collector Fig. 15-1 (a) A circuit containing a resistor, two diodes, and a tran- sistor, (b) Cross-sectional view of the circuit in (a) when trans- formed into a monolithic form. (After Phillips. 1 ) and a transistor. These elements (and also capacitors, with small values of capacitances) are the components encountered in integrated circuits. The monolithic circuit is formed by the steps indicated in Fig. 15-2 and described below. Step 1. Epitaxial Growth An retype epitaxial layer, typically 25 microns thick, is grown onto a p-type substrate which has a resistivity of typi- cally 10 fi-cm, corresponding to N A = 1.4 X 10 15 atoms/cm 3 . The epitaxial Process described in Sec. 15-2 indicates that the resistivity of the n-type epi- taxial layer can be chosen independently of that of the substrate. Values of •fom 0.1 to 0.5 U-cm are chosen for the n-type layer. In contrast to the situa- tion depicted in Fig. 15-2a, the epitaxial process is used with discrete transistors 10 obtain a thin high-resistivity layer on a low-resistivity substrate of the same Polarity, After polishing and cleaning, a thin layer (0.5 micron = 5,000 A) °J oxide, Si0 2 , is formed over the entire wafer, as shown in Fig. 15-2a. The yj is grown by exposing the epitaxial layer to an oxygen atmosphere while b ®ing heated to about 1000°C. Silicon dioxide has the fundamental property Preventing the diffusion of impurities through it. Use of this property is ma de in the following steps. Step 2. Isolation Diffusion In Fig. 15-26 the wafer is shown with the ide removed in four different places on the surface. This removal is accom- 420 / ELECTRONIC DEVICES AND CIRCUITS $»c. 1 5. | S«'- 15-1 INTEGRATED CIRCUITS / 421 Silicon dioxide (a) i Isolation islands Sidewall C- Bottom C, •Resistor xnesisior . Anode of diode /Base ULSU (c) Cathodes of diodes n Wx w> rr«5 \w,\\v\w,\ I /•^■H P* ,| ^__ u p n Emitter n" Fig. 15-2 The steps involved in fabricating a monolithic circuit (not drawn to scale), (a) Epitaxial growth; (b) isolation diffusion; (c) base diffusion; (d) emitter diffusion; (e) aluminum metalization. jished by means of a photolithographic etching process described in Sec, 15-3. The remaining SiOz serves as a mask for the diffusion of acceptor impurities /jn this case, boron). The wafer is now subjected to the so-called isolation diffusion, which takes place at the temperature and for the time interval required for the p-type impurities to penetrate the n-type epitaxial layer and jeach the p-type substrate. We thus leave the shaded n-type regions in Fig. l5-2?>. These sections are called isolation islands, or isolated regions, because they are separated by two back-to-back p-n junctions. Their purpose is to ft ]|ow electrical isolation between different circuit components. For example, it will become apparent later in this section that a different isolation region must be used for the collector of each separate transistor. The p-type sub- strate must always be held at a negative potential with respect to the isolation islands in order that the p-n junctions be reverse-biased. If these diodes were to become forward-biased in an operating circuit, then, of course, the isolation would be lost. It should be noted that the concentration of acceptor atoms (Na «* 5 X MP cm -3 ) in the region between isolation islands will generally be much higher (and hence indicated as p + ) than in the p-type substrate. The reason for this higher density is to prevent the depletion region of the reverse-biased isolation- to-substrate junction from extending into p + -type material (Sec. 6-9) and possi- bly connecting two isolation islands. Parasitic Capacitance It is now important to consider that these isola- tion regions, or junctions, are connected by a significant barrier, or transition capacitance Ctm, to the p-type substrate, which capacitance can affect the oper- ation of the circuit. Since Ct, is an undesirable by-product of the isolation process, it is called the parasitic capacitance. The parasitic capacitance is the sum of two components, the capacitance Ci from the bottom of the n-type region to the substrate (Fig. 15-26) and C* from the side walls of the isolation islands to the p + region. The bottom com- ponent, Ci, results from an essentially step junction due to the epitaxial growth (Sec. 15-2) and hence varies inversely as the square root of the voltage V Between the isolation region and the substrate (Sec. 6-9). The sidewall capaci- tance C 2 is associated with a diffused graded junction, and it varies as V""'. •"or this component the junction area is equal to the perimeter of the isolation re giun times the thickness y of the epitaxial n-type layer. The total capaci- ance is of the order of a few picofarads. Step 3, Base Diffusion During this process a new layer of oxide is ormed over the wafer, and the photolithographic process is used again to create the cliff Pattern of openings shown in Fig. 15-2c. The p-type impurities (boron) are used through these openings. In this way are formed the transistor base ^'°us as well as resistors, the anode of diodes, and junction capacitors (if y ■ It is important to control the depth of this diffusion so that it is shallow ^~ does not penetrate to the substrate. The resistivity of the base layer will e ral! y be much higher than that of the isolation regions. Sec. Fs_| 422 / ELECTRONIC DEVICES AND CIRCUITS Step 4. Emitter Diffusion A layer of oxide is again formed over the entire surface, and the masking and etching processes are used again to ope n windows in the p-type regions, as shown in Fig. 15-2d. Through these open, ings are diffused n-type impurities (phosphorus) for the formation of transistor emitters, the cathode regions for diodes, and junction capacitors. Additional windows (such as W t and W 2 in Fig. 15-2d) are often made into the n regions to which a lead is to be connected, using aluminum as the ohmic contact, or interconnecting metal. During the diffusion of phosphorus a heavy concentration (called n + ) is formed at the points where contact with aluminum is to be made. Aluminum is a p-type impurity in silicon, and a large concentration of phosphorus prevents the formation of a p-n junction when the aluminum is alloyed to form an ohmic contact. 4 Step 5. Aluminum Metalization All p-n junctions and resistors for the circuit of Fig. 15- lo have been formed in the previous steps. It is now neces- sary to interconnect the various components of the integrated circuit as dic- tated by the desired circuit. In order to make these connections, a fourth set of windows is opened into a newly formed Si0 2 layer, as shown in Fig. 15-2e, at the points where contact is to be made. The interconnections are made first, using vacuum deposition of a thin even coating of aluminum over the entire wafer. The photoresist technique is now applied to etch away all undesired aluminum areas, leaving the desired pattern of interconnection* shown in Fig. 15-2e between resistors, diodes, and transistors. In production a large number (several hundred) of identical circuits such as that of Fig. 15-la are manufactured simultaneously on a single wafer. After the metalization process has been completed, the wafer is scribed with a diamond-tipped tool and separated into individual chips. Each chip is then mounted on a ceramic wafer and is attached to a suitable header. The package leads are connected to the integrated circuit by stitch bonding 1 of ft 1-mil aluminum or gold wire from the terminal pad on the circuit to th« package lead (Fig. 15-26). Summary In this section the epitaxial-diffused method of fabricating integrated circuits is described. We have encountered the following pn 1. Epitaxy 2. Silicon dioxide growth 3. Photoetching 4. Diffusion 5. Vacuum evaporation of aluminum Using these techniques, it is possible to produce the following elements on w same chip: transistors, diodes, resistors, capacitors, and aluminum interco nections. Other techniques have been used also, such as the triple-din uS8 ^ A L INTEGRATED CIRCUITS / 423 process and the d iff used-collector process. 1 The method just described, how- ever, is in more general use because of a number of inherent advantages. l 15-2 EPITAXIAL GROWTH 1 The epitaxial process produces a thin film of single-crystal silicon from the gas phase upon an existing crystal wafer of the same material. The epitaxial layer may be either p-type or n-type. The growth of an epitaxial film with impurity atoms of boron being trapped in the growing film is shown in Fig. 15-3. The basic chemical reaction used to describe the epitaxial growth of pure silicon is the hydrogen reduction of silicon tetrachloride: 1200°C SiCU + 2H 2 » Si + 4HC1 (15-1) Since it is required to produce epitaxial films of specific impurity concen- trations, it is necessary to introduce impurities such as phospbine for n-type doping or biborane for p-type doping into the silicon tetrachlo ride-hydro gen gas stream. An apparatus for the production of an epitaxial layer is shown in Fig. 15-4. In this system a long cylindrical quartz tube is encircled by a radio-frequency induction coil, The silicon wafers are placed on a rectangular graphite rod called a boat The boat is inserted in the reaction chamber, and the graphite is heated inductively to about 1200°C. At the input of the renction chamber a control console permits the introduction of various gases required for the growth of appropriate epitaxial layers. Thus it is possible to form an almost abrupt step p-n junction similar to the junction shown in Fig. 6-12. F '9-l5-3 The epitaxial 9'owth of an epitaxial film lowing impurity (boron) °toms being trapped in the flawing film. (Courtesy of M °toroIa, Inc. 1 ) • Gas phase ®®®®®'®®1 Epu _ ® ®®@@®@ J © © © © © © © ® ® © © © © ® © @®®©@® fsT) CsT) fsT) (sT) (sT) (s?) (sT) Substrate 424 / ELECTRONIC DEVICES AND CIRCUITS Induction coll Sec. 15. Outlet Silicon wafers Graphite boat Fig. 15-4 A diagram- matic representation of a system for production growth of silicon epi- taxial films. (Courtesy of Motorola, Inc. 1 ) 15-3 MASKING AND ETCHING 1 The monolithic technique described in See. 15-1 requires the selective removal of the Si0 2 to form openings through which impurities may be diffused. The photoetching method used for this removal is illustrated in Fig. 15-5. During the photolithographic process the wafer is coated with a uniform film of a photo- sensitive emulsion (such as the Kodak photoresist KPR). A large black-and- white layout of the desired pattern of openings is made and then reduced photographically. This negative, or stencil, of the required dimensions is placed as a mask over the photoresist, as shown in Fig. 15-5a. By expos- ing the KPR to ultraviolet light through the mask, the photoresist becomes polymerized under the transparent regions of the stencil. The mask is now removed, and the wafer is "developed" by using a chemical (such as tri- chloroethylene) which dissolves the unexposed (unpolymerized) portions of the photoresist film and leaves the surface pattern as shown in Fig. 15-56. Mask Photoresist- Si0 2 -" / Silicon chip Ultraviolet Polymerized photoresist / I \ 'S10 2 ■Silicon chip («) (6) Fig. 15-5 Photoetching technique, (o) Masking and exposure to ultraviolet radiation, (b) The photoresist after development. S# 15-4 INTEGRATED CIRCUITS / 425 The emulsion which was not removed in development is now fixed, or cured, e that it becomes resistant to the corrosive etches used next. The chip is immersed in an etching solution of hydrofluoric acid, which removes the oxide from the areas through which dopants are to be diffused. Those portions of the SiOs which are protected by the photoresist are unaffected by the acid. After etching and diffusion of impurities, the resist mask is removed (stripped) with a chemical solvent (hot H2SO4) and by means of a mechanical abrasion process. 15-4 DIFFUSION OF IMPURITIES 6 The most important process in the fabrication of integrated circuits is the diffusion of impurities into the silicon chip. We now examine the basic theory connected with this process. The solution to the diffusion equation will give the effect of temperature and time on the diffusion distribution. The Diffusion Law The continuity equation derived in Sec. 5-9 for charged particles is equally valid for neutral atoms. Since diffusion does not involve electron-hole recombination or generation (r P = «) and since no electric field is present (£ = 0), Eq. (5-46) now reduces to d_N at = D em (15-2) where JV is the particle concentration in atoms per unit volume as a function of distance x from the surface and time /, and D is the diffusion constant in area per unit time. This diffusion equation is also called Fiek's second law. The Complementary Error Function If an intrinsic silicon wafer is exposed to a volume of gas having a uniform concentration N atoms per unit volume of n-type impurities, such as phosphorus, these atoms will diffuse into the silicon crystal, and their distribution will be as shown in Fig. 15-6a. If the diffusion is allowed to proceed for extremely long times, the silicon will become uniformly doped with N phosphorus atoms per unit volume. The basic assumptions made here are that the surface concentration of impurity a toms remains at N B for all diffusion times and that N(x) = at t = for *> 0. If Eq. (15-2) is solved and the above boundary conditions are applied, N(x, t) = N t 0- erf n-nr. erfc (15-3) 2 y/DiJ " 2 ^/Dt "ere erfc y means the error-function complement of y, and the error function of V is defined by erf *^/° VX,< * X (154) 426 / ELECTRONIC DEVICES AND CIRCUITS S «. 1 4^ Fig. 15-6 The concentration N as a function of distance x into a silicon chip for two values ti and tt of the diffusion time, (a) The surface concentration is held constant at JV a per unit volume, {fa} The total number of atoms on the surface is held constant at Q per unit area. and is tabulated in Ref. 3. Fig. 15-7. The function erfc y = 1 — erf y is plotted in The Gaussian Distribution If a specific number Q of impurity atoms per unit area are deposited on one face of the wafer and then if the material is heated, the impurity atoms will again diffuse into the silicon. When the boundary conditions fj N(x) dx = Q for all times and N(x) = at t - for x > are applied to Eq. (15-2), we find N(x, t) - Q t'UDt (15-5) Equation (15-5) is known as the Gaussian distribution, and is plotted in Fig. 15-66 for two times. It is noted from the figure that as time increases, the surface concentration decreases. The area under each curve is the same, however, since this area represents the total amount of impurity being diffused, and this is a constant amount Q. Note that in Eqs. (15-3) and (15-5) time t and the diffusion constant D appear only as a product DL Solid Solubility 1 - 6 The designer of integrated circuits may wish to produce a specific diffusion profile (say the complementary error function of an n-type impurity). In deciding which of the available impurities (such as phosphorus, arsenic, antimony) can be used, it is necessary to know if the number of atoms per unit volume required by the specific profile of Eq. (15-3) is less than the diffusant's solid solubility. The solid solubility is defined as the maximum concentration N„ of the element which can be dissolved in the solid silico° S«c \S-4 INTEGRATED CIRCUITS / 427 1.0 5x 10"' 10"' 5x 10r» 1L1-' pig, 1 5-7 The complemen- tary error function plotted >> 5x10"' o E on semilogarithmic paper. io- J 5x 10- 1 10"* Sx Mr 1 . [— ^ 10" 1 at a given temperature. Figure 15-8 shows solid solubilities of some impurity elements. It can be seen that since for phosphorus the solid solubility is approximately 10 S1 atoms/cm 3 , and for pure silicon we have 5 X 10 !2 atoms/ cm s , the maximum concentration of phosphorus in silicon is 2 percent. For most of the other impurity elements the solubility is a small fraction of 1 percent. Diffusion Coefficients Temperature affects the diffusion process because Wgher temperatures give more energy, and thus higher velocities, to the dif- Fi 9. 15-8 Solid solubili- " •■m of some impurity £ demerits in silicon. | '* f ter Trumbore/ I Curtesy of Motorola, J - IIIIU-- --Slli, " "'Ml a J 1100 / I 1 \3 rrj >y ill "' 1 toy Auinr j_p5.- Jjt Ve u if , «]ttl| - IJ \ llllii - fcu 1 \ II kj Silicon sb jjj J t 500 1 □ 1 1 10" 10 JI 10™ 10" 10" 10" 10" 10" 10" 10" 10" Atoms/cm 3 428 / ELECTRONIC DEVICES AND CIRCUITS Sec J 5 . 4 Temperature, "C 1300 1200 1100 1000 900 8 io- s io-" - io- 1 1 i t i — i Aluminum k-Gailiurr. " ^_ Boron and v phosphorus Antimony ' \ Fig, 15-9 Diffusion coefficients q s a function of temperature for some impurity elements in silicon. (After Fuller and Ditzenberger, 6 courtesy of Motorola, Inc. 1 ) 0.60 0.65 0.70 0.75 0.80 1000/7/ (Tin'K) 0.85 fusant atoms. It is clear that the diffusion coefficient is a function of tempera- ture, as shown in Fig. 15-9. From this figure it can be deduced that the dif- fusion coefficient could be doubled for a few degrees increase in temperature. This critical dependence of D on temperature has forced the development of accurately controlled diffusion furnaces, where temperatures in the range of 1000 to 1300°C can be held to a tolerance of ± 0.5°C or better. Since time t in Eqs, (15-3) and (15-5) appears in the product Dt, an increase in either diffusion constant or diffusion time has the same effect on diffusant density. Note from Fig. 15-9 that the diffusion coefficients, for the same tempera- ture, of the n-type impurities (antimony and arsenic) are lower than the coefficients for the p-type impurities (gallium and aluminum), but that phos- phorus («-type) and boron (p-type) have the same diffusion coefficients. Typical Diffusion Apparatus Reasonable diffusion times require high diffusion temperatures (~1000°C). Therefore a high-temperature diffusion furnace, having a closely controlled temperature over the length (20 in.) o» the hot zone of the furnace, is standard equipment in a facility for the fabrica- tion of integrated circuits. Impurity sources used in connection with diffusion furnaces can be gases, liquids, or solids. For example, POCla, which is a liquid, is often used as a source of phosphorus. Figure 15-10 shows the apparatus used for POCH diffusion. In this apparatus a carrier gas (mixture of nitrogen and oxygen) bubbles through the liquid-diffusant source an d carries the diffusant atoms to the silicon wafers. Using this process, we obtain the complementary-error-f unction distribution of Eq. (15-3). A two-sWP procedure is used to obtain the Gaussian distribution. The first step involve* predeposition, carried out at about 900°C, followed by drive-in at abou 1100°C. S.c 15-4 INTEGRATED CIRCUITS / 429 Quartz diffusion tube ' — ■- poootC Silicon wafers S^l ■ Furnace Liquid POC1 Thermostated bath-n\ input Fig. 15-10 Schematic representation of typical apparatus for POCIs diffusion. (Courtesy of Motorola, Inc. 1 } EXAMPLE A uniformly doped n-type silicon substrate of 0.5 Si-cm resistivity is subjected to a boron diffusion with constant surface concentration of 5 X 10" cm -3 . It is desired to form a p-n junction at a depth of 2.7 microns. At what temperature should this diffusion be carried out if it is to be completed in 2 hr? Solution The concentration JV of boron is high at the surface and falls off with distance into the silicon, as indicated in Fig. 15-6a. At that distance x = Xj at which N equals the concentration n of the doped silicon wafer, the net impurity density is zero. For x < x it the net impurity density is positive, and for x > x,, it is negative. Hence sj represents the distance from the surface at which a junction is formed. We first find n from Eq. (5-2) : 1 = 0.96 X 10 IB cm"" tine (0.5) (1,300) (1.60 X 10" 19 ) where all distances are expressed in centimeters and the mobility (t n for silicon is taken from Table 5-1, on page 98. The junction is formed when N = n. For N n 0.96 X 10" . Q(> x, in _, erfc y = — = — = = 1.92 X 10 * N N a 5 X 10 18 we find from Fig. 15-7 that y = 2.2. Hence 2.2 = ''_ =, 2-7 X 10-* 2 VDt 2 VD X 2 X 3,600 Solving for D, we obtain D = 5.2 X 10 _1S cm s /sec. This value of diffusion con- stant for boron is obtained from Fig. 15-9 at T = 1130°C. 430 / ELECTRONIC DEVICES AND CIRCUITS '5-5 15-5 TRANSISTORS FOR MONOLITHIC CIRCUITS 17 A planar transistor made for monolithic integrated circuits, using epitaxy and diffusion, is shown in Fig. 15-1 la. Here the collector is electrically separated from the substrate by the reverse-biased isolation diodes. Since the anode of the isolation diode covers the back of the entire wafer, it is necessary t make the collector contact on the top, as shown in Fig. 15-1 la. It is now clear that the isolation diode of the integrated transistor has two undesirable effects: it adds a parasitic shunt capacitance to the collector and a leakage current path. In addition, the necessity for a top connection for the collector increases the collector-current path and thus increases the collector resistance and Fcfi(sat). All these undesirable effects are absent from the discrete epitaxial transistor shown in Fig. 15-116. What is then the advantage of the monolithic transistor? A significant improvement in performance arises from the fact that integrated transistors are located physically close together and their electrical characteristics are closely matched. For example, inte- grated transistors spaced within 30 mils (0.03 in.) have Vbb matching of better than 5 mV with less than 10 mV/°C drift and an k F s match of ±10 percent. These matched transistors make excellent difference amplifiers (Sec. 12-12). The electrical characteristics of a transistor depend on the size and geometry of the transistor, doping levels, diffusion schedules, and the basic silicon material. Of all these factors the size and geometry offer the greatest flexibility for design. The doping levels and diffusion schedules are determined by the standard processing schedule used for the desired transistors in the integrated circuit. Impurity Profiles for Integrated Transistors 1 Figure 15-12 showsa typical impurity profile for a monolithic integrated circuit transistor. The back- Emitter contact Base contact Collector contact n-epitaxial collector p substrate p-type isolation diffusion (b) Emitter contact Base contact Fig. ? 5-11 Comparison of cross sections of (a) a monolithic integrated cir- cuit transistor with (fa) Q discrete planar epitaxial transistor. [For a top view of the transistor in (a) see Fig, 15-13,1 ^Collector contact Sec 15-5 INTEGRATED CIRCUITS / 431 3 x, v Collector — *■ Fig, 15-12 A typical impurity profile in a monolithic integrated transistor. [Note that N(x) is plotted on a logarithmic scale.] ground, or epitaxial-collector, concentration N S c is shown as a dashed line in Fig. 15-12. The base diffusion of p-type impurities (boron) starts with a surface concentration of 5 X 10 18 atoms/cm 3 , and is diffused to a depth of 2.7 microns, where the collector junction is formed. The emitter diffusion (phosphorus) starts from a much higher surface concentration (close to the solid solubility) of about 10" atoms/cm s , and is diffused to a depth of 2 microns, where the emitter junction is formed. This junction corresponds to the intersection of the base and emitter distribution of impurities. We Q ow see that the base thickness for this monolithic transistor is 0.7 micron. The emitter-to-base junction is usually treated as a step junction, whereas the hase-to-collector junction is considered a graded junction. EXAMPLE (a) Obtain the equations for the inpurity profiles in Fig. 15-12. (6) If the phosphorus diffusion is conducted at 1100°C, how long should be allowed for this diffusion? Solution a. The base diffusion specifications are exactly those given in the example on page 429, where we find (with x expressed in microns) that y « 2.2 = 2.7 avS or / — 2.7 2 V Dt = — = 1.23 microns 2.2 432 / ELECTRONIC DEVICES AND CIRCUITS Hence the boron profile, given by Eq. (15-3), is x Sec. '5-5 N* = 5 X 10" erfc 1.23 The emitter junction is formed at x = 2 microns, and the boron concentration here is N B - 5 X 10" erfc - 5 X 10" X 2 X 10~» 1.23 = 1.0 X 10" cm"' The phosphorus concentration N B is given by iVp = 10" erfc — %=*. 2VDt At a: = 2, JV, - AT B = 1.0 X 10", so that erfc 2 VDt 1.0 X 10" 10" = 1.0 X 10"* From Fig. 15-7, 2/(2 Voi) = 2.7 and 2 Vfli = 0.74 micron. Hence the phosphorus profile is given by N r = 10" erfc — 0.74 o. From Fig. 15-9, at T = 1100°C, Z) = 3.8 X 10"" cmVsec. Solving for t from 2 \/Di = 0.74 micron, we obtain (0.37 X 10"*) 1 1 = -^ r^r- = 3.600 sec = 60 min 3.8 X 10-" Monolithic Transistor Layout 1 - 2 The physical size of a transistor deter- mines the parasitic isolation capacitance as well as the junction capacitance. It is therefore necessary to use small-geometry transistors if the integrated circuit is designed to operate at high frequencies or high switching speeds. The geometry of a typical monolithic transistor is shown in Fig. 15-13. The emitter rectangle measures 1 by 1.5 mils, and is diffused into a 2.5- by 4.0-mi 1 base region. Contact to the base is made through two metalized stripes on either side of the emitter. The rectangular metalized area forms the ohrnic contact to the collector region. The rectangular collector contact of this transistor reduces the saturation resistance. The substrate in this structure is located about 1 mil below the surface. Since diffusion proceeds in three dimensions, it is clear that the lateral-diffusion distance will also be 1 mil- The dashed rectangle in Fig. 15-13 represents the substrate area and is 6.5 by » mils. A summary of the electrical properties 2 of this transistor for both the 0.5- and the 0.1-ii-cm collectors is given in Table 15-1. Stc- 15-5 INTEGRATED CIRCUITS / 433 Indicates contacts Emitter diffusion Base diffusion Isolation diffusion Fig. 15-13 A typical double-base stripe geometry of an integrated- ctrcuit transistor. Dimensions are in mils. (For a side view of the transistor see Fig. 15-11.) (Courtesy of Motorola Monitor.) TABLE 15-1 Characteristics for 1 - by 1 .5-miI double- base stripe monolithic transistors 3 Transistor parameter BVcbq,V BVtmhV BVceo, V Ct. (forward bias), pF C T . at 0.5 V, pF C T . at 5 V, pF A;rjat 10 mA Res, « Vc*(sat) at 5 mA, V. , VsE&t 10 mA, V /r at 5 V, 5 mA, MHz t Gold-doped. 0.5 n-cm 0.1 fi-cmt 55 25 7 5.5 23 14 6 10 1.5 2.5 0.7 1.5 50 50 75 15 0,5 0.26 0.85 0.85 440 520 434 / ELECTRONIC DEVICES AND CIRCUITS Se c. fs^ Fig. 15-14 Utilization of "buried" n + layer to reduce collector series resistance. Buried Layer 1 We noted above that the integrated transistor, because of the top collector contact, has a higher collector series resistance than a similar discrete-type transistor. One common method of reducing the collector series resistance is by means of a heavily doped n + "buried" layer sandwiched between the p~type substrate and the n-typc epitaxial collector, as shown in Fig. 15-14. The buried-layer structure can be obtained by diffusing the n + layer into the substrate before the n-type epitaxial collector is grown or by selectively growing the n + -type layer, using masked epitaxial techniques. We are now in a position to appreciate one of the reasons why the inte- grated transistor is usually of the n-p-n type. Since the collector region is subjected to heating during the base and emitter diffusions, it is necessary that the diffusion coefficient of the collector impurities be as small as possible, to avoid movement of the collector junction. Since Fig. 15-9 shows that n- type impurities have smaller values of diffusion constant D than p-type impurities, the collector is usually n-type. In addition, the solid solubility of some n-type impurities is higher than that of any p-type impurity, thus allowing heavier doping of the n + -type emitter and other n + regions. 15-6 MONOLITHIC DIODES' The diodes utilized in integrated circuits are made by using transistor struc- tures in one of five possible connections (Prob. 15-9). The three most popular diode structures are shown in Fig. 15-15. They are obtained from a transistor Rg. 15-15 Cross section of various diode struc- tures, (a) Emitter-bose diode with collector shorted to base; (b) emitter-base diode wi* collector open; and \ c > collector-base diode 1° emitter diffusion). (a) (b) $c<~- T5-6 Anode 1 Common cathode 3 O *f Anode 2 INTEGRATED CIRCUITS / 435 Common anode 3 9 Cathode Cathode lYftViNVW |» V WWWVVj p substrate i (a) Fig. 15-16 Diode pairs, (a) Common-cathode pair, and (b) common- anode pair, using collector-base diodes. structure by using (a) the emitter-base diode, with the collector short-circuited to the base; (6) the emitter-base diode, with the collector open; and (c) the collector-base diode, with the emitter open-circuited (or not fabricated at all). The choice of the diode type used depends upon the application and circuit performance desired. Collector-base diodes have the higher collector-base voltage- breakdown rating of the collector junction (~12 V minimum), and they are suitable for common-cathode diode arrays diffused within a single isolation island, as shown in Fig. 15-16o. Common-anode arrays can also be made with the collector- base diffusion, as shown in Fig. 15-166. A sepa- rate isolation is required for each diode, and the anodes are connected by oietalization. The emitter-base diffusion is very popular for the fabrication of diodes Provided that the reverse-voltage requirement of the circuit does not exceed the lower base-emitter breakdown voltage (^7 V). Common-anode arrays °an easily be made with the emitter-base diffusion by using a multiple-emitter transistor within a single isolation area, as shown in Fig. 15-17. The collector ^9- 15-17 A multiple- fitter n-p-n transistor. ™) Schematic, (b) mono- lttl 'c surface pattern. me base is connected to ,h e collector, the [ esu "isa mu |tip|e- """ode diode structure *ith Q common anode. (a) 436 / ELECTRONIC DEVICES AND CIRCUITS Sec. ?S.» 10 S 6 ■§ 4 r± t t J- {a)j (b)l (cy 11/ —J J--/ J^-Z- f/ ~// Jl? 2 Fig. 15-18 Typical diode volt-ampere characteristics for the three diode types of Fig. 15-15. (a) Base-emitter (collector shorted to base); (b) base-emitter (col- lector open); (c) collector-base (emitter open). (Courtesy of Fairchild Semiconductor. 8 ) 0.4 0.8 1.2 1.6 Forward voltage, V may be either open or shorted to the baae. The diode pair in Fig, 15-1 is constructed in this manner, with the collector floating (open). Diode Characteristics The forward volt-ampere characteristics of the three diode types discussed above are shown in Fig. 15-18. It will be observed that the diode-connected transistor (emitter-base diode with collector shorted to the base) provides the highest conduction for a given forward voltage. The reverse recovery time for this diode is also smaller, one-third to one-fourth that of the collector-base diode. 15-7 INTEGRATED RESISTORS 1 A resistor in a monolithic integrated circuit is very often obtained by utilizing the bulk resistivity of one of the diffused areas. The p-type base diffusion is most commonly used, although the n-type emitter diffusion is also employed- Since these diffusion layers are very thin, it is convenient to define a quantity known as the sheet resistance R3. Sheet Resistance If, in Fig. 15-19, the width w equals the length /, * e have a square I by I of material with resistivity p, thickness y, and cros sectional area A = ly. The resistance of this conductor (in ohms per square/ p _ pl _ P its — i ly y (15-6) Note that R 3 is independent of the size of the square. Typically, the sh resistance of the base and emitter diffusions whose profiles are given in 15-12 are 200 fi/square and 2.2 Si/ square, respectively. . The construction of a base-diffused resistor is shown in Fig. 15-1 an repeated in Fig. 15-20o. A top view of this resistor is shown in Fig- 1&" S*c. 1S-7 INTEGRATED CIRCUITS / 437 Fig. 15-19 Pertaining to sheet resistance, ohms per square. The resistance value may be computed from ff - pl - p * K — — lis — yw w (15-7) where I and w are the length and width of the diffused area, as shown in the top view. For example, a base-diffused-resistor stripe 1 mil wide and 10 mils long contains 10 (1 by 1 mil) squares, and its value is 10 X 200 = 2,000 Si. Empirical 1,1 corrections for the end contacts are usually included in calculations of R. Resistance Values Since the sheet resistance of the base and emitter diffusions is fixed, the only variables available for diffused-resistor design are stripe length and stripe width. Stripe widths of less than one mil (0.001 in.) are not normally used because a line-width variation of 0.0001 in. due to mask drawing error or mask misalignment or photographic-resolution error can result in 10 percent resistor-tolerance error. The range of values obtainable with diffused resistors is limited by the aizc of the area required by the resistor. Practical range of resistance is 20 Si Pig. 15-20 A monolithic resistor, (o) Cross-sectional view; (b) top view. 1 o k^^m,kwww /] isolation region P substrate («) i (6) 438 / ELECTRONIC DEVICES AND CIRCUITS R AAV bt- T 1 2 -o p layer C Sec. I5.J Fig. 15-21 The equivalent circuit of a diffused resistor. A/VV A/W -o n isolation region -o p substrate to 30 K for a base-diffused resistor and 10 U to 1 K for emitter-diffused resistors. The tolerance which results from profile variations and surface geometry errors 1 is as high as ± 10 percent of the nominal value at 25°C, with ratio tolerance of ± 3 percent. For this reason the design of integrated circuits should, if possible, emphasize resistance ratios rather than absolute values. The temperature coefficient for these heavily doped resistors is posi- tive (for the same reason that gives a positive coefficient to the silicon sensistor, discussed in Sec. 10-9) and is +0.06 percent/°C from -55 to 0°C and +0.20 percent/°C from to 125°C. Equivalent Circuit A model of the diffused resistor is shown in Fig. 15-21, where the parasitic capacitances of the base-isolation (Ci) and isolation-sub- strate (Cj) junctions are included. In addition, it can be seen that a parasitic p-n-p transistor exists, with the substrate as collector, the isolation n-type region as base, and the resistor p-type material as the emitter. Since the collector is reverse-biased, it is also necessary that the emitter be reverse- biased in order to keep the parasitic transistor at cutoff. This condition is maintained by placing all resistors in the same isolation region and connecting the n-type isolation region surrounding the resistors to the most positive voltage present in the circuit. Typical values of h/ t for this parasitic transistor range from 0.5 to 5. Thin-film Resistors 1 A technique of vapor thin-film deposition can also be used to fabricate resistors for integrated circuits. The metal (usually nichrome, NiCr) film is deposited on the silicon dioxide layer, and masked etching is used to produce the desired geometry. The metal resistor is then covered by an insulating layer, and apertures for the ohmic contacts are opened through this insulating layer. Typical sheet-resistance values for nichrome thin-fib* 1 resistors are 40 to 400 Q/square, resulting in resistance values from about 20 Q to 50 K. 15-8 INTEGRATED CAPACITORS AND INDUCTORS 1 2 Capacitors in integrated circuits may be obtained by utilizing the transit 100 capacitance of a reverse-biased p-n junction or by a thin-film technique. Sec. J 5-8 Al metal izat ion INTEGRATED CIRCUITS / 439 C a ^0.2pF/mil J Substrate (4) Fig. 15-22 (a) Junction monolithic capacitor, (b) Equivalent circuit. (Courtesy of Motorola, Inc.) Junction Capacitors A cross-sectional view of a junction capacitor is shown in Fig. 15-22a. The capacitor is formed by the reverse-biased junction 3 1, which separates the epitaxial n-type layer from the upper p-type diffusion area. An additional junction J\ appears between the n-type epitaxial plane and the substrate, and a parasitic capacitance Ci is associated with this reverse- biased junction. The equivalent circuit of the junction capacitor is shown in Fig. 15-226, where the desired capacitance Ci should be as large as possible relative to Ci. The value of Cj depends on the junction area and impurity concentration. Since this junction is essentially abrupt, Ci is given by Eq. (6-49). The series resistance R (10 to 50 £1) represents the resistance of the n-type layer. It is clear that the substrate must be at the most negative voltage so as to minimize Ci and isolate the capacitor from other elements by keeping junction A reverse-biased. It should also be pointed out that the junction capacitor Ct is polarized since the p-n junction J \ must always be re verse- biased. Thin-film Capacitors A metal-oxide-semiconductor (MOS) nonpolarized capacitor is indicated in Fig. 15-23a. This structure is a parallel-plate capa- Al metalizaton C=0.25pF/mil 2 R = 5- ion B ■)\ T 1 — Vv\ o C 1 -4-/ J >?i »-r»C i p-type substrate (b) 9- 15-23 A MOS capacitor, (a) The structure and (b) the equivalent circuit. 440 / ELECTRONIC DEVICES AND CIRCUITS Sec. 15-9 TABLE 75-2 Integrated capacitor parameters Characteristic Diff used-junction capacitor Thin-film MOS Capacitance, pF/mil* 0.2 2 X 10« 406 5-20 kV~* ±20 0.25-0.4 2 X 10* 800 50-200 Tolerance, percent , .'.' i - ±20 citor with SiO s as the dielectric. A surface thin film of metal (aluminum) is the top plate. The bottom plate consists of the heavily doped n + region that is formed during the emitter diffusion. A typical value for capacitance 8 is 0.4 pF/mil* for an oxide thickness of 500 A, and the capacitance varies inversely with the thickness. The equivalent circuit of the MOS capacitor is shown in Fig. 15-236, where Ci denotes the parasitic capacitance J t of the collector-substrate junc- tion, and R is the small series resistance of the n+ region. Table 15-2 lists the range of possible values for the parameters of junction and MOS capacitors. Inductors No practical inductance values have been obtained at the present time (1967) on silicon substrates using semiconductor or thin-film techniques. Therefore their use is avoided in circuit design wherever possible. If an inductor is required, a discrete component is connected externally to the integrated circuit. 15-9 MONOLITHIC CIRCUIT LAYOUT 1 -' In this section we describe how to transform the discrete circuit of Fig. 15-24a into the layout of the monolithic circuit shown in Fig. 15-25. Circuits involv- ing diodes and transistors, connected as in Fig. 15-24o, are called diode- transiBtor (DTL) logic gates. 1C Design Rules for Monolithic Layout The following 10 reasonable design rules are Btated by Phillips : 9 1. Redraw the schematic to satisfy the required pin connection with the minimum number of crossovers. 2. Determine the number of isolation islands from collector-potential con- siderations, and reduce the areas as much as possible. 3. Place all resistors having fixed potentials at one end in the same is lation island, and return that isolation island to the most positive potent* in the circuit. 4. Connect the substrate to the most negative potential of the circuit S«. 1S-9 INTEGRATED CIRCUITS / 441 5. In layout, allow an isolation border equal to twice the epitaxial thick- ness to allow for underdiffusion. 6. Use 1-mil widths for diffused emitter regions and -j-mil widths for base contacts and spacings, and for collector contacts and spacings. 7. For resistors, use widest possible designs consistent with die-size limitations. 8. Always optimize the layout arrangement to maintain the smallest possible die size, and if necessary, compromise pin connections to achieve this. 9. Determine component geometries from the performance requirements of the circuit. 10. Keep all metalizing runs as short and as wide as possible, particularly at the emitter and collector output connections of the saturating transistor. Pin Connections The circuit of Fig. 15-24a is redrawn in Fig. 15-246, with the external leads labeled 1, 2, 3, . . . , 10 and arranged in the order in which they are connected to the header pins. The diagram reveals that the power-supply pins are grouped together, and also that the inputs are on adjacent pins. In general, the external connections are determined by the system in which the circuits are used. Crossovers Very often the layout of a monolithic circuit requires two conducting paths (such as leads 5 and 6 in Fig. 15-246) to cross over each other. This crossover cannot be made directly because it will result in electric contact between two parts of the circuit. Since all resistors are protected by the SiOj layer, any resistor may be used as a crossover region. In other words, © + 6.5V( kputs ®« ©< ® 1K« Dl -T*~ 4000 D2 D3 D4 -H- D5 -w- Substrate -6.5V 5.6K 0-6.5V6 © @ © i GND F '9- 15-24 (a) A DTL gate, (b) The schematic redrawn to indicate the 10 external c °nnections arranged in the sequence in which they will be brought out to the "eader pins. The isolation regions are shown in heavy outline. 442 / ELECTRONIC DEVICES AND CIRCUITS Sec. T5-9 •" ■■' ' "■ '■■' — — Indicates isolation region ig^ssa Indicates metalization Fig. 15-25 Monolithic design layout for the circuit of Fig. 15-24. (Cour- tesy of Motorola Monitor, Phoenix, Ariz.) if aluminum metalization is run over a resistor, no electric contact will take place between the resistor and the aluminum. Sometimes the layout is so complex that additional crossover points may be required. A diffused structure which allows a crossover is also possible. This type of crossover should be avoided if at all possible because it requires a separate isolation region and it introduces undesired series resistance of the diffused region into the connection. Isolation Islands The number of isolation islands is determined next- Since the transistor collector requires one isolation region, the heavy rectangle has been drawn in Fig. 15-246 around the transistor. It is shown connected to the output pin 2 because this isolation island also forms the transistor col- lector. Next, all resistors are placed in the same isolation island, and tn island is then connected to the most positive voltage in the circuit, for reason 5 discussed in Sec. 15-7. Sec 15-9 INTEGRATED CIRCUITS / 443 In order to determine the number of isolation regions required for the diodes, it is necessary first to establish which kind of diode will be fabricated. In this case, because of the low forward drop shown in Fig. 15-18, it was decided to make the common-anode diodes of the emitter-base type with the collector shorted to the base. Since the "collector" is at the "base" potential, it is required to have a single isolation island for the four common-anode diodes. Finally, the remaining diode is fabricated as an emitter-base diode, with the collector open-circuited, and thus it requires a separate isolation island. The Fabrication Sequence The final monolithic layout is determined by a trial-and-error process, having as its objective the smallest possible die size. This layout is shown in Fig. 15-25. The reader should identify the four iso- lation islands, the three resistors, the live diodes, and the transistor. It is Ffg. 15-26 Monolithic fabrication sequence for the circuit of Fig. 15-24. (Courtesy of Motorola Monitor, Phoenix, Ariz.) Metalization Flat package assembly 444 / ElECTRONIC DEVICES AND CIRCUITS Sec. 75- 10 interesting to note that the 5.6-K resistor has been achieved with a 2-mil-wide 1.8-K resistor in series with a 1-mil-wide 3.8-K resistor. In order to con- serve space, the resistor was folded back on itself. In addition, two metalizing crossovers ran over this resistor. From a layout such as shown in Fig. 15-25, the manufacturer produces the masks required for the fabrication of the monolithic integrated circuit. The production sequence which involves isolation, base, and emitter diffusions, preohmic etch, aluminum metalization, and the flat package assembly is shown in Fig. 15-26. Large-scale Integration (LSI) The monolithic circuit layout shown in Fig. 15-25 contains one transistor, five diodes, and three resistors for a total of nine circuit elements. This number of elements per chip, or the component density, is determined primarily by cost considerations. Even if it were possible to fabricate and interconnect several hundred components per chip, the manufacturing cost per component would not necessarily decrease. The reason is that beyond a certain component density the cost per component increases again owing to circuit complexity, which tends to reduce the yield. At any given stage in the development of integrated-circuit techniques, there exists an optimum number of components per chip which will produce minimum cost per component. 11 In 1962, 10 components per circuit (chip) represented the optimum. In 1967 the optimum number is about 70. It is predicted 1 * that by 1970 the optimum number will exceed 1,000. Large-scale integration (LSI) represents the process of fabricated large-component-density chips which represent complete subsystems or equipment components. A packaged LSI slice 2-j in. square with 32 leads on each side is pictured in Ref. 12. 15-10 INTEGRATED FIELD-EFFECT TRANSISTORS 113 The MOSFET is discussed in detail in Chap. 14. In this section we point out the advantages of this device as an integrated-circuit active element (Fig- 15-27). Size Reduction The MOS integrated transistor typically occupies only 5 percent of the surface required by an epitaxial double-diffused transistor ifl a conventional integrated circuit. The double-base stripe 1- by 1.5-mil emitter integrated transistor normally requires about 10 X 9.5 mils of chip area, whereas the MOS requires 5 square mils. Simple Fabrication Process Only one diffusion step is required to fabricate the MOS enhancement-type field-effect transistor. In this step (Fig. 15-27 a) two heavily doped n-type regions are diffused into a lightly doped p-type substrate to form the drain and source. An insulating lay er of oxide is grown, and holes are etched for the metal electrodes for the source S,c. 15-TO INTEGRATED CIRCUITS / 445 Source Drain S I) S sa 9 S / * Mrliilization W o Hate ( WJ a substrate }> substrate •n l> Fig. 15-27 An n-channel insulated-gate FET of the enhancement- mode type, (a) The source and drain are diffused into the sub- strate, (b) The completed device. and drain. The metal for these contacts, as well as for the gate electrode, is then evaporated at the same time to complete the device shown in Fig. 15-276. Crossovers and Isolation Islands The crossovers between components of integrated MOS circuits are diffused at the same time as the source and drain. The resistive effects of crossover-diffused regions (with R s =* 80-100 Ji/ square) are negligible since these regions are in series with large- value load resistors of the order of 100 K normally used with FETs. Another important advantage is that no isolation regions are needed between MOS transistors because the p-n junctions are reverse-biased during the operation of the circuit. The MOS as a Resistor for Integrated Circuits In our discussion of diffused resistors in Sec. 15-7, we show that 30 K is about the maximum resistance value possible (in 1967). Larger values may be obtained by using a MOS structure as shown in Fig. 15-28, where the eate and drain are tied together and a fixed voltage Vdd is applied between drain and ground. A <*V m Fig. 15-28 The MOS as a resistor. 90 ® «?s 1 * 446 / ELECTRONIC DEVICES AND CIRCUITS Sec. 15. j, Thevenin's equivalent circuit looking into the source is obtained in Sec 14-8. From Eq. (14-35) we find that the impedance seen looking into the source is approximately equal to l/g m , assuming negligible drain conductance gd- If, for example, g m - 10 j*A/V, we have R = l/g m = 100 K. This value of effective resistance requires approximately 5 square mils of active area as compared with 300 square mils of chip area to yield a diffused resistance of value 20 K. 15-11 ADDITIONAL ISOLATION METHODS Electrical isolation between the different elements of a monolithic integrated circuit is accomplished by means of a diffusion which yields back-to- back p-n junctions, as indicated in Sec. 15-1. With the application of bias voltage to the substrate, these junctions represent reverse-biased diodes with a very high back resistance, thus providing adequate dc isolation. But since each p-n junction is also a capacitance, there remains that inevitable capacitive coupling between components and the substrate. These parasitic distributed capacitances thus limit monolithic integrated circuits to frequencies somewhat below those at which corresponding discrete circuits can operate. Additional methods for achieving better isolation, and therefore improved frequency response, have been developed, and are discussed in this section. Dielectric Isolation In this process 1,14 the diode-isolation concept is discarded completely. Instead, isolation, both electrical and physical, is achieved by means of a layer of solid dielectric which completely surrounds and separates the components from each other and from the common sub- strate. This passive layer can be silicon dioxide, silicon monoxide, ruby, or possibly a glazed ceramic substrate which is made thick enough so that its associated capacitance is negligible. In a dielectrie isolated integrated circuit it is possible to fabricate readily p-n-p and n-p-n transistors within the same silicon substrate. It is also simple to have both fast and charge-storage diodes and also both high- and low-fre- quency transistors in the same chip through selective gold diffusion — a process prohibited by conventional techniques because of the rapid rate at which gold diffuses through silicon unless impeded by a physical barrier such as a dielectric layer. One isolation method employing silicon dioxide as the isolating material is the EPIC process, 12 developed by Motorola, Inc. This EPIC isolation method reduces parasitic capacitance by a factor of 10 or more. In addition, the insulating oxide precludes the need for a reverse bias between substrate and circuit elements. Breakdown voltage between circuit elements and sub- strate is in excess of 1,000 V, in contrast to the 20 V across an isolation j unction- Beam Leads The beam-lead concept 16 of Bell Telephone Laboratories was primarily developed to batch-fabricate semiconductor devices and bite- 15-JI INTEGRATED CIRCUITS / 447 •rated circuits. This technique consists in depositing an array of thick (of the order of 1 mil) contacts on the surface of a slice of standard monolithic circuit, ft nd then removing the excess semiconductor from under the contacts, thereby separating the individual devices and leaving them with semirigid beam leads cantilevered beyond the semiconductor. The contacts serve not only as elec- trical leads, but also as the structural support for the devices; hence the name beam leads. Chips of beam-lead circuits are mounted directly by leads, with- out 1-mil aluminum or gold wires. Isolation within integrated circuits may be accomplished by the beam- . Isolation area -C ommoD -e mltte r beam lead Load -res is tor beam lead Load resistor Semiconductor wafer Common- collector beam lead Base-resistor beam intraconnectlon (one of four) Input resistor (one of four) Input-resistor beam lead (one of four) Isolation area Semiconductor wafers Fig, 15-29 The beam-lead isolation technique, (a) Photomicro- graph of logic circuit connected in a header, (b) The underside of the same circuit, with the various elements identified. (Courtesy of Bell Telephone Laboratories.) 448 / ELECTRONIC DEVICES AND CIRCUITS $«. 75-11 lead structure. By etching away the unwanted silicon from under the beam leads which connect the devices on an integrated chip, isolated pads of silicon may be attained, interconnected by the beam leads. The only capacity/ coupling between elements is then through the small metal-over-oxide overlay 6 This is much lower than the junction capacitance incurred with p-n junction- isolated monolithic circuits. It should be pointed out that the dielectric and beam-lead isolation techniques involve additional process steps, and thus higher costs and possible reduction in yield of the manufacturing process. Figure 15-29 shows photomicrographs of two different views of a logic circuit made using the beam-lead technique. The top photo shows the logic circuit connected in a header. The bottom photo shows the underside of the same circuit with the various elements identified. This device is made using conventional planar techniques to form the transistor and resistor regions. Electrical isolation is accomplished by removing all unwanted material between components. The beam leads then remain to support and intraconnect the isolated components. Hybrid Circuits 1 The hybrid circuit as opposed to the monolithic circuit consists of several component parts (transistors, diodes, resistors, capacitors, or complete monolithic circuits), all attached to the same ceramic substrate and employing wire bonding to achieve the interconnections. In these circuits electrical isolation is provided by the physical separation of the component parts, and in this respect hybrid circuits resemble beam-lead circuits. REFERENCES 1. Motorola, Inc. (R. M. Warner, Jr., and J. N. Fordcmwalt, eds.): "Integrated Circuits," McGraw-Hill Book Company, New York, 1965. 2. Phillips, A. B.; Monolithic Integrated Circuits, IEEE Spectrum, vol. 1, no. 6, pp. 83-101, June, 1964. 3. Jahnke, E., and F. Emde: "Tables of Functions," Dover Publications New York, 1945, 4. Hunter, L. P.: "Handbook of Semiconductor Electronics," 2d ed„ sec. 8, McGraw- Hill Book Company, New York, 1962. 5. Fuller, C. S., and J. A. Ditzenberger: Diffusion of Donor and Acceptor Elements in Silicon, /. Appl. Phys., vol. 27, pp. 544-553, May, 1956. Barrer, P. M. : "Diffusion in and through Solids," Cambridge University Press, London, 1951. 6. Trumbore, F. A. : Solid Solubilities of Impurity Elements in Germanium and Silicon, BeU System Tech. J,, vol. 39, pp. 205-234, January, 1960. INTEGRATED CIRCUITS / 449 7. King, D., and L. Stern: Designing Monolithic Integrated Circuits, Semicond. Prod. Solid State TechnoL, March, 1965. 8. "Custom Microcircuit Design Handbook," Fairchild Semiconductor, Mountain View, Calif., 1963. 9. Phillips, A. B.: Designing Digital Monolithic Integrated Circuits, Motorola Monitor, vol. 2, no. 2, pp. 18-27, 1964. 10. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," pp. 330- 334, McGraw-Hill Book Company, New York, 1965. 11. Baker, O. R.: Aspects of Large Scale Integration, 1967 IEEE Intern. Conv. Dig., pp. 376-377, March, 1967. 12. Weber. S.: LSI: The Technologies Converge, Electronics, vol. 40, no. 4, pp. 124-127, February, 1967. 13. Farina, E. D., and D. Trotter: MOS Integrated Circuits, Electronics, vol. 38, no. 20, pp. 84-95, October, 1965. K Epic Process Isolates Integrated Circuit Elements with Silicon Dioxide, Electro- Technol. (New York), July, 1964, p. 136. 15. Lepselter, M. P., et al.: Beam Leads and Integrated Circuits, Proc. IEEE, vol. 53, p. 405, April, 1965. Lepselter, M. P.: Beam-lead Technology, BeU System Tech. J., February, 1966 pp. 233-253. 16/UNTUNED AMPLIFIERS Frequently the need arises for amplifying a signal with a minimum of distortion. Under these circumstances the active devices involved must operate linearly. In the analysis of such circuits the first step is the replacement of the actual circuit by a linear model. Thereafter it becomes a matter of circuit analysis to determine the distortion produced by the transmission characteristics of the linear network. The frequency range of the amplifiers discussed in this chapter extends from a few cycles per second (hertz), or possibly from zero, up to some tens of megahertz. The original impetus for the study of such wideband amplifiers was supplied because they were needed to amplify the pulses occurring in a television signal. Therefore such amplifiers are often referred to as video amplifiers. Basic amplifier circuits are discussed here. Modifications of these configurations to extend the frequency range of these amplifiers are considered in Ref. L In this chapter, then, we consider the following problem : Given a low-level input waveform which is not necessarily sinusoidal but may contain frequency components from a few hertz to a few mega- hertz, how can this voltage signal be amplified with a minimum of distortion? We also discuss many topics associated with the general problem of amplification, such as the classification of amplifiers, hum and noise in amplifiers, etc. 16-1 CLASSIFICATION OF AMPLIFIERS Amplifiers are described in many ways, according to their frequency range, the method of operation, the ultimate use, the type of lo* d ' the method of interstage coupling, etc. The frequency classification $K. 16-1 UNTUNED AMPLIFIERS / 451 450 includes dc (from zero frequency), audio (20 Hz to 20 kHz), video or pulse (up to a few megahertz), radio-frequency (a few kilohertz to hundreds of megahertz), and ultrahigh-frequency (hundreds or thousands of megahertz) amplifiers. The position of the quiescent point and the extent of the characteristic that is being used determine the method of operation. Whether the transistor or tube is operated as a Class A, Class AB, Class B, or Class C amplifier is determined from the following definitions. Class A A Class A amplifier is one in which the operating point and the input signal are such that the current in the output circuit (in the collector, plate, or drain electrode) flows at all times, A Class A amplifier operates essentially over a linear portion of its characteristic. Class B A Class B amplifier is one in which the operating point is at an extreme end of its characteristic, so that the quiescent power is very small. Hence either the quiescent current or the quiescent voltage is approximately aero. If the signal voltage is sinusoidal, amplification takes place for only one-half a cycle. For example, if the quiescent output-circuit current is zero, this current will remain zero for one-half a cycle. Class AB A Class AB amplifier is one operating between the two extremes defined for Class A and Class B. Hence the output signal is zero for part but less than one-half of an input sinusoidal signal cycle. Class C A Class C amplifier is one in which the operating point is chosen so that the output current (or voltage) is zero for more than one-half of an input sinusoidal signal cycle. In the case of a vacuum-tube amplifier the suffix I may be added to the letter or letters of the class identification to denote that grid current does not flow during any part of the input cycle. The suffix 2 may be added to denote that grid current does flow during some part of the input cycle. Amplifier Applications The classification according to use includes voltage, power, current, or general-purpose amplifiers. In general, the load of an amplifier is an impedance. The two most important special cases are the idealized resistive load and the tuned circuit operating near its resonant frequency. Class AB and Class B operation are used with untuned power amplifiers '^hap. 18), whereas Class C operation is used with tuned radio- frequency ^ttplifiers. Many important waveshaping functions may be performed by 'ass B or C overdriven amplifiers. This chapter considers only the untuned *udio or video voltage amplifier with a resistive load operated in Class A. 452 / HECTRONtC DEVICES AUD CIRCUITS Sec. 1<J,j 16-2 DISTORTION IN AMPLIFIERS The application of a sinusoidal signal to the input of an ideal Class A amplifier will result in a sinusoidal output wave. Generally, the output waveform ig not an exact replica of the input-signal waveform because of various types of distortion that may arise, either from the inherent nonlinearity in the char- acteristics of the transistors or tubes or from the influence of the associated circuit. The types of distortion that may exist either separately or simultane- ously are called nonlinear distortion, frequency distortion, and delay distortion. Nonlinear Distortion This type of distortion results from the production of new frequencies in the output which are not present in the input signal. These new frequencies, or harmonics, result from the existence of a nonlinear dynamic curve for the active device; they are considered in some detail in Sees. 18-2 and 18-3. This distortion is sometimes referred to as "amplitude distortion. " Frequency Distortion This type of distortion exists when the signal components of different frequencies are amplified differently. In either a transistor or a tube this distortion may be caused by the internal device capacitances, or it may arise because the associated circuit (for example, the coupling components or the load) is reactive. Under these circumstances, the gain A is a complex number whose magnitude and phase angle depend upon the frequency of the impressed signal. A plot of gain (magnitude) vs. frequency of an amplifier is called the amplitude frequency-response characteristic. If this plot is not a horizontal straight line over the range of frequencies under consideration, the circuit is said to exhibit frequency distortion over this range. Delay Distortion This distortion, also called phase-shift distortion, results from unequal phase shifts of signals of different frequencies. This distortion is due to the fact that the phase angle of the complex gain A depends upon the frequency. 16-3 FREQUENCY RESPONSE OF AN AMPLIFIER A criterion which may be used to compare one amplifier with another wi respect to fidelity of reproduction of the input signal is suggested by * following considerations: Any arbitrary waveform of engineering importan may be resolved into a Fourier spectrum. If the waveform is periodic, spectrum will consist of a series of sines and cosines whose frequencies are integral multiples of a fundamental frequency. The fundamental frequen J is the reciprocal of the time which must elapse before the waveform rep«* itself. If the waveform is not periodic, the fundamental period extends i sense from a time — « to a time + °° ■ The fundamental frequency is c Sec 16-3 UNTUNED AMPLIFIERS / 453 jjjfjnitcsimally small; the frequencies of successive terms in the Fourier series differ by an infinitesimal amount rather than by a finite amount; and the foil I'ii' i' series becomes instead a Fourier integral. In either case the spectrum includes terms whose frequencies extend, in the general case, from zero fre- «iency to infinity. Fidelity Considerations Consider a sinusoidal signal of angular fre- quency w represented by V m sin (wt + <£). If the voltage gain of the amplifier )jas a magnitude A and if the signal suffers a phase lag B, then the output will be AV m sin (wt + — B) = AV m sin [»H) + *] Therefore, if the amplification A is independent of frequency and if the phase shift 9 is proportional to frequency (or is zero), then the amplifier will preserve the form of the input signal, although the signal will be delayed in time by an amount D = 6/u>. This discussion suggests that the extent to which an amplifier's amplitude response is not uniform, and its time delay is not constant with frequency, may perve as a measure of the lack of fidelity to be anticipated in it. In prim -i pie, it is really not necessary to specify both amplitude and delay response rince, for most practical circuits, the two are related and, one having been id, the other is uniquely determined. However, in particular cases, it may well be that either the time-delay or amplitude response is the more sensitive indicator of frequency distortion. Low-frequency Response Video amplifiers of either the transistor or tube variety are almost invariably of the 2?C-coupled type. For such a stage the frequency characteristics may be divided into three regions: There is a range, c&lled the midband frequencies, over which the amplification is reasonably constant and equal to A a and over which the delay is also quite constant. *°r the present discussion we assume that the midband gain is normalized to Un ity, A = 1. In the second (low-frequency) region, below the midband, an am pliiier stage behaves (See. 16-5) like the simple high-pass circuit of Fig. 16-1 °f time constant n = RiCi. From this circuit we find that V„ = ViRx V, Ri — jfwCi 1 — j/wRiCt (16-1) Th ne voltage gain at low frequencies Ai is defined as the ratio of the output '* ^-1 A high-pass RC circuit may be used to calcu- e the low-frequency response of an amplifier. f Vt -L -IH [R, T J- 454 / ELECTRONIC DEVICES AND CIRCUITS -l Li- Y 1 1 1- (<» (6) Fig. 16-2 [a\ A low-p Qss BC circuit may be used t e calculate the high-fre- quency response of an amplifier, (b) The Norton's equivalent of the circuit i n (a), where / = Vi/R t . voltage F to the input voltage F,-, or 1 where A - V '- h = 2tt/liCi The magnitude \Ai\ and the phase lag 6\ of the gain are given by H.i = Vi + (/i//) 2 $i — — arctan — / At the frequency f = f h A t = 1/V2 = 0.707, whereas in the midband region (/» /i), A\ — * 1. Hence f x is that frequency at which the gain has fallen to 0.707 times its midband value A Q . From Eq. (12-21) this drop in sig- nal level corresponds to a decibel reduction of 20 log(l/\/2), or 3 dB. Accord- ingly, /i is referred to as the lower Z-dB frequency. From Eq. (16-3) we see that /i is that frequency for which the resistance R i equals the capacitive reactance 1/2tt/iCi. High-frequency Response In the third (high-frequency) region, above the midband, the amplifier stage behaves (Sec. 16-6) like the simple low-pa* 8 circuit of Fig. 16-2, with a time constant r% = R 2 Cz. Proceeding as above, we obtain for the magnitude \Az\ and the phase lag 62 of the gain where |it,|- *■ 1 VTTWW f h = arctan 7- h (16-5) (16-6) Since at/ = / s the gain is reduced to l/\/2 times its midband value, then;' is called the upper 3-dB frequency. It also represents that frequency for whic the resistance R 2 equals the capacitive reactance 1/2jt/ 2 C 2 . In the ahoi expressions 6 X and a represent the angle by which the output lags the inp u neglecting the initial 180° phase shift through the amplifier. The frequ eD ^ dependence of the gains in the high- and low-frequency range is to be * in Fig. 16-3. UNTUNED AMPLIFIERS / 455 Fig. 16-3 A log-log plot of the amplitude frequency-response characteristic of an fi!C-coupled amplifier. Bandwidth The frequency range from fi to f 2 is called the bandwidth of the amplifier stage. We may anticipate in a general way that a signal, all of whose Fourier components of appreciable amplitude lie well within the ^ge/i to/i, will pass through the stage without excessive distortion. This criterion must be applied, however, with caution. 2 l6 *4 THE tfC-COUPLED AMPLIFIER * cascaded arrangement of common-cathode (CK) vacuum-tube stages is *°wn in Fig. 16-4a, of common-emitter (CE) transistor stages in Fig. 16-46, ^d of common-source (CS) FET stages in Fig. 16-4e. The output Y\ of one 7*86 is coupled to the input X% of the next stage via a blocking capacitor * which is used to keep the dc component of the output voltage at Fi from lining the input X 2 . The resistor R a is the grid (gate) leak, and the plate Rector) (drain) circuit resistor is R p (R c ) (R d ). The cathode resistor R k , the r ce resistor R„ the emitter resistor R t> the screen resistor R K , and the resis- ^ °i and R2 are used to establish the bias. The bypass capacitors, used . Prevent loss of amplification due to negative feedback (Chap. 17), are C* *j/* e cathode, C, in the emitter, C, in the source, and C, e in the screen circuit. Present are interelectrode capacitances in the case of a tube, and junction 456 / ELECTRONIC DEVICES AND CIRCUITS S * ld.4 From C preceding 0-*~-| stage (a) From d preceding O *■ |( stage (c) From C, preceding O - |( O stage (b) Fig, T6-4 A cascade of (a) common-cathode (CK) pentode stages; (b) common- emitter (CE) transistor stages; (c) common-source (CS) FET stages. capacitances if a transistor is used. These are taken into account when *• consider the high-frequency response, which is limited by their presence, any practical mechanical arrangement of the amplifier components there #• also capacitances associated with tube sockets and the proximity to the chass» of components (for example, the body of C 6 ) and signal leads. These str*? capacitances are also considered later. We assume that the active de* 1 * 5 * operates linearly, so that small-signal models are used throughout this chap**'" S* Pfl- 16-5 16-5 A schematic representa- UNTUNED AMPLIFIERS / 457 tion of «'™er a tube, FET, or transis- tor stage. Biasing arrangements and suppfy voltages are not indi- cated. 'Mi 16-5 LOW-FREQUENCY RESPONSE OF AN ftC-COUPLED STAGE The effect of the bypass capacitors C k> C t , and C, on the low-frequency charac- teristics is discussed in Sec. 16-10. For the present we assume that these capacitances are arbitrarily large and act as ac short circuits across Rk, R», and R., respectively. The effect of C, e is considered in Ref. 3. A single inter- mediate stage of any of the cascades in Fig. 16-4 may be represented sche- matically as in Fig. 16-5. The resistor Rb represents the grid-leak resistor for a tube or the gate resistor R s for a FET, and equals Ri in parallel with R 2 if a transistor stage is under consideration. The resistor R„ represents R p for a tube, R c for a transistor, or R d for a FET, and R { represents the input resistance of the following stage. The low-frequency equivalent circuit is obtained by neglecting all shunt- ing capacitances and all junction capacitances, by replacing amplifier A 1 by its Norton's equivalent, as indicated in Fig. 16-6a. For a vacuum tube or field- effect transistor, Ri = *> ; the output impedance is R a = r p (r d ) [the plate (drain) resistance]; and J = g m Vi (transconductance times grid or gate signal voltage). For a transistor these quantities may be expressed in terms of the CE hybrid parameters as in Sec. 11-2; Ri « h it (for small values of R e ), fi« - I /ho, (for a current drive), and I = h fe h, where h is the base signal current. Let R' Q represent R„ in parallel with R v , and let R\ be Ri in parallel With R b . Then, replacing I and R'„ by the Thevenin's equivalent, the single- time-constant high-pass circuit of Fig. 16-66 results. Hence, from Eq. (16-3), ft -o— Q) Mt Ri <«) * ^g. 1 6-6 |o) The low-frequency model of an flC-coupled amplifier; (b) an equiva- lent representation. For a tube or FET: I = g m Vi, R g = r„ {u) t R v = R v (Ra), *h — R t , and Rf = «. For a transistor: / = hfJo, R, «• l/h a „ R& — jBi||Ks, K = R e , and Ri « h it . Also, R- = Ri\\R b and R'„ - R B \\R V . 458 / ELECTRONIC DEVICES AND CIRCUITS the lower 3-dB frequency is 1 /*- 2tt(R'„ + #;.)<?(, $<*. T^ (16-7) This result is easy to remember since the time constant equals C b multiplied by the sum of the effective resistances R' a to the left of the blocking capacitor and R' t to the right of Ci. For a vacuum-tube amplifier, #' = /?„ » fl Since R' < R p because R'„ is R p in parallel with R e , then R\ = R a » R' e ar ^j /i ~ l/2irCbR a . This same expression is valid for a FET. EXAMPLE It is desired to have a low 3-dB frequency of not more than 10 H» for an flC-coupled amplifier for which R y = 1 K. What minimum value of coupling capacitance is required if (a) vacuum tubes or FETs with R B = 1 M are used; (6) transistors with R f = I K and 1/A„ = 40 K are used? Solution a. From Eq. (16-7) we have fl = 2t(R' + R'jC b ~ 10 or C*> s r " Q2.8(K + R<) Since R' 4 = 1 M and R' < R v = \ K, then R'„ + R' t « 1 M and d > 0.016 mF. &. From Eq. (11-34) we find for a transistor R > \/h et = 40 K, and hence R' e « R c « 1 K. If we assume that R b » fl; = I K, then J2^ « 1 K. Henw 1 Ci, > - F = 8.0 uF (62.8) (2 X 10 3 ) M Note that because the input impedance of a transistor is much smaller than that of a FET or a tube, a coupling capacitor is required with the transistor which is 500 times larger than that required with the FET or tube. Fortunately, it is possible to obtain physically small electrolytic capacitors having such high capacitance values at the low voltages at which transistors operate. 16-6 HIGH-FREQUENCY RESPONSE OF A VACUUM-TUBE STAGE For frequencies above the midband range we may neglect the reactance of the large series capacitance C b . However, we must now include in Fig. 16-4 the output capacitance C from Y 1 to ground and the input capacitance & f r ° m X 2 to ground. To these capacitances must also be added the stray cap» cl " tance to ground. If the sum of all these shunt capacitances is called C, t n ® n the high-frequency model of Fig. 16-7 can be drawn. In order to keep &* S* 16-6 UNTUNED AMPLIFIERS / 439 f- a 16-7 The high-frequency model of ftC-co up led stage using a pentode. input capacitance C< as small as possible, a pentode, rather than a triode, is aged for the tube (Sees. 8-11 and 8-13). Hence r p is of the order of magnitude of a megohm, as is also R B1 whereas R v is at most a few kilohms. Therefore the parallel combination R of these three resistors can be approximated by R p without introducing appreciable error. As predicted above (Fig. 16-26), the amplifier stage at high frequencies behaves like a single-time-constant low- pass circuit, where Ct — C and R% = R = r p \\R v \\R a . Hence, from Eq. (16-6), the upper 3-dB frequency ft is given by A- i 2vRC 2irRJJ (16-8) la the midband region, where the shunting effect of C can be neglected (X.^iJp), the output voltage is V„ = — g m RVi, and hence the midband gain A„ = VJVi (for R v <K r p and R p « R 9 ) is given by A e = — g m R ~ —g m R P (16-9) Gain-Bandwidth Product The upper 3-dB frequency of the amplifier may be improved by reducing the product R P C. Every attempt should be made to reduce C by careful mechanical arrangement to decrease the shunt capaci- tance. The upper 3-dB frequency may also be increased by reducing R p , but this reduces simultaneously the nominal amplifier gain. A figure of merit F which is very useful in comparing tube types is obtained by computing the product of A and fa in the limiting case where stray capacitance is considered to have been reduced to zero. From Eqs. (16-8) and (16-9) we have, since C - Ci + C„ F - \A e \f* = th (16-10) 2w(C + Ci) Since /, » f h the bandwidth f% — f\ •* f% and \A \f 2 = F is called the gain- wttdwidth product It should be noted that ft varies inversely with plate- ^cuit resistance, whereas A is proportional to R p , so that the gain-bandwidth Product is a constant independent of R P . It is possible to reduce R p to such a low value that a midband gain |A„| = 1 is obtained. Hence the figure of "tor-it F may be interpreted as giving the maximum possible bandwidth obtain- &lD to with a given tube if R p is adjusted for unity gain. For video pentodes *toh as the 6AK5, 6BH6, 6AU6, 6BC5, and 6CL6, values of g m ranging from to 1 1 millimhos (mA/V,) and values of Co + d from 7 to 20 pF are obtain- JJfe. The value of F for all these tubes lies between 80 and 120 MHz, with e 6AK5 having the largest value. 460 / ELECTRONIC DEVICES AND CIRCUITS S«c. >«* An amplifier with a gain of unity is not very useful. Hence let us assum that |4„| is at least 2. Then f 2 = F/\A e \ = 60 MHz for the 6AK5 tube, i a practical circuit, the inevitable extra stray capacitance might easily redu the bandwidth by a factor of 2. Hence we may probably take a bandwidth of 30 MHz as a reasonable estimate of a practical upper limit for an uncom. pensated tube amplifier using lumped parameters. If the desired gain is \n instead of 2, the maximum 3-dB frequency is about 6 MHz. The highest transconductance available in tubes is about 50 millimhos and is obtained with frame grid pentodes having very close (0.05 mm) grid! to-cathode spacing. For example, the Amperex type 7788 pentode has g m = 50 mA/V and C a + C, « 20 pF, corresponding to F = 400 MHz. With this tube a 3-dB frequency of about 20 MHz is possible with a gain of 10. If more bandwidth is needed, distributed amplifiers are used. 1 The foregoing discussion is valid for any stage of a tube amplifier, includ- ing the output stage. For this last stage, d, representing the input capaci- tance to the following stage, is missing, and its place is taken by any shunt capacitance of the device being driven (Bay a cathode-ray tube). The equivalent circuit of a FET is the same as that of a triode (Fig. 14-9). Hence the input capacitance of an internal stage may be very large because of the Miller effect (Sec. 8-12). This shunting capacitance limits the bandwidth of a FET. 16-7 CASCADED CE TRANSISTOR STAGES The high-frequency analysis of a single-stage CE transistor amplifier, or the last stage of a cascade, is given in detail in Sees. 13-7 and 13-8. Since the input impedance of a transistor cannot be represented by a parallel resistance- capacitance combination, the analysis of an internal stage differs from thai of the final stage. We consider now the operation of one transistor amplifier stage in a cas- cade of many stages. Such a cascade is shown in Fig. 16-8. We omit from this diagram all supply voltages and components, such as coupling capacitor*. Fig. 16-8 An infinite cascade of CE stages. The dashed, shaded rectangle (blocW encloses one stage. $K. 16-7 UNTUNED AMPLIFIERS / 461 which serve only to establish proper bias and do not affect the high-frequency response. The collector-circuit resistor R e is included, however, since this resistor has an effect on both the gain and frequency response. The base- biasing resistors R\ and R* in Fig. 16-46 are assumed to be large compared with Re- H this condition is not satisfied, the symbol R^ represents the par- allel combination of R it R it and the collector-circuit resistance. A complete g t a ge from collector to collector is included in the shaded block. We define the current gain of the stage to be An m It/1%, Each stage behaves like a current generator of impedance R. = R e delivering current to the following stage. We define the voltage gain to be A v = V a /Vi. Since we have specified Vi as the voltage precisely at the stage input, then A v is the gain for an ideal volt- age source. We now prove that At, m Av for an infinite cascade of similar stages. In a long chain of stages the input impedance Z t between base and emitter of each stage is identical. Let Z[ represent Z» in parallel with R c . Accord- ingly, Z'i = Vi/Ii = Vi/h, so that h/h = An =V % /V l = A v in this special case. We now calculate this gain A Tt = A v = A. For this purpose Fig. 16-0 shows the circuit details of the stage in the shaded block in Fig. 16-8. Also shown is the input portion of the next stage, so that we may take account of its loading effect on the stage of interest. The symbol K used in the expression C e (l - K) for one of the capacitors is K m V e ,/V b > t . Figure 16-9 is obtained from Fig. 13-12a. The elements involving g b >c have been omitted since, as demonstrated in Sec. 13-8, their omission introduces little error. The gain A e = I%fl\ at low frequencies is given by Eq. (13-50), except with R m replaced by R e , and we have j. _ — h/,R e R e + hft (16-11) To calculate the bandwidth we must evaluate K. From Fig. 16-9 we obtain for K an unwieldy expression. Since If is a function of frequency, the element marked C c {\ — K) is not a true capacitor, but rather is a com- plex network. Thus, in order to proceed with a simple solution which will give reasonable accuracy, we use the zero-frequency value of K. We show c 2 ^AA B' 3 ~iT R c < r b .< c. A 1 E c e a-K) ^'9- 16-9 The equivalent circuit of the enclosed stage of Fig. 16-8 (K = V^/V*,). 462 / FircTRONJC DEVICES AND CIRCUITS s «. 16.* below that the response obtained experimentally is somewhat better than that predicted by this analysis, and hence that we are erring in the conservative direction. At zero frequency, K = K B = -g m R L , in which Rl is the * tive load on the transistor from C to E and consists of R e in parallel with r »* + n't = hi,. Therefore Rl = RJij (16-12) Re + h^ and the total capacitance C from B\ to E is C = C. + C.(l + ? J! t ) (lfi _ The gain is A = /,//, = -g m V b , t /U, where 7 6 , e = F t - 2 . represents the volt- age across C. Instead of calculating V v . directly from the input network of * lg ' .\ 6 " 9 ' we a « ain make the observation that this is a single-time-constant circuit. Hence we can calculate the 3-dB frequency /, by inspection. Since the capacitance C is charged through a resistance R consisting of r b . t in parallel witn Kc + r»«, or R m (Re + ivjry. -fie + A,> the 3-dB frequency is 1 /t- 2t/bc (16-14) (16-15) This half-power frequency is the same for the current gain and voltage gain. In using the approximation K = K = - gm R L> we are making a conserva- tive error, since K a is the maximum magnitude of K and is attained only at zero frequency. Using K. leads to the largeat value of shunt capacitance C, and consequently to an overly low estimate of the bandwidth /,. From the equations above the gain-bandwidth product is found to be \Aoh\ = £ _ 9* «■ & 2irC R e + tw 1 + 2wfTC e R L R e + r* where R h depends upon R e , as indicated in Eq. (16-12). (16-16) Gam and Bandwidth Considerations Our only adjustable parameter is «„ and we now discuss its selection. At one extreme, if we set R e = 0, we should simply shunt all output current away from the following transistor. As a matter of fact, it seems initially not unreasonable to set R e arbitrarily nigh so as to avoid this shunting effect. However, as we reduce R c and thereby lose gam, a compensating advantage appears. A reduction of & reduces R L m Eq. (16-12) and also reduces R in Eq. (16-14) The reduction in fti reduces C - C. + C.(l+ g m R L ), and this reduction, together with the reduction in R, increases / 2 , as is seen in Eq. (16-15). It may be that a decrease in gain is more than compensated for by an increase in /,. To investigate 1 J 6-7 UNTUNED AMPLIFIERS i 463 this point we differentiate the gain-bandwidth product \A fi\ with respect to jy f Setting the derivative equal to zero, we find that a maximum does occur. The value of R e for which this optimum gain-bandwidth product is obtained is designated by (R e ) ovt and is given by {flclopt — K y/x- 1 with hr.C. C, + C c r»' (16-17) (16-18) In Fig. 16-10 we have plotted the gain, the bandwidth, and the gain- bandwidth product. The maximum which is apparent [at R e = 360 SI, as found from Eq. (16-17)] is not particularly pronounced. 5 Nevertheless, there is enough of a falling off at values of R e above or below (# e ) np t so that it may be worthwhile to operate near the maximum. It is important to bias the transistor so that at the quiescent point a large value of /r is obtained (Fig. 13-10). Note in Fig. 16-10 that l^o/sl remains roughly constant for values of R e in the neighborhood of (R c )o P t or for larger values of R c . Hence, for a cascade of stages (as distinct from the single stage considered in Sec. 13-9), the gain-bandwidth product takes on some importance as a figure of merit. /a, MHz \AJ lAJsl , MHz I!) M 41) \AJ 2 \ \A B \ M UAM 20 TV 10 IA 4 * 1 r 1 > 500 1,000 1,500 2,000 R c , n Fig. 16-10 Gain \A„\, bandwidth / 2 , and gain-bandwidth product \A„f,\ as a function of R e for one stage of a CE cascade. The transistor parameters are given in Sec. 13-5. 464 / ELECTRONIC DEVICES AND CIRCUITS Sec. Td-> For our typical transistor, f T = 80 MHz, whereas the constant value of |4«/J in Fig. 16-10 is approximately 40 MHz, or Q.5/ T . A good general rule'ij choosing a transistor as a broadband amplifier is to assume A e f 2 « 0.6/ T This conclusion is based upon calculations on more than twenty transistors for which the hybrid-II parameters were known. These had values of f T ranging from 700 kHz to 700 MHz. In each case (R c ) 0ftt was found and the value of AoU at this optimum resistance was calculated. All values of gain- bandwidth product were in the range between 0.4 and 0.8/r. The values of AJi were also calculated for several values of R e besides (# e )o P t, and it was confirmed that the gain-bandwidth product remained constant over a wide range of values of R e . It must be remembered that bandwidth cannot be exchanged for gain at low values of gain because AJt is not constant for small values of R e or A* The maximum value of ft, which occurs at R e = (and A e - 0), is given by //a St _ frhii g m rc ft/,r w < (16-19) The design of the amplifier represents, as usual, a compromise between gain and bandwidth. If A is specified, the load R e which must be used is found from Eq. (16-11). Then the bandwidth which will be obtained is found from Eq. (16-15). On the other hand, if the desired bandwidth is specified, then/ 2 substituted into Eq. (16-15) will not allow a direct calculation of R t . The reason for the difficulty is that R depends upon R c and that C=C. + C.(I + g m R L ) is also a function of R e through R L , as given in Eq. (16-12). Under these circumstances an arbitrary value of R c> say 1,000 12, is chosen, and f t is cal- culated. If this value is larger (smaller) than the desired value of f it the next approximation to R e must be larger (smaller) than 1,000 ft. By plotting /i versus R t , the desired value of R c can be found by interpolation. The approximations which we have made in this analysis are valid if Rh i» less than 2,000 SI. Since R L is the parallel combination of R c and A* « 1, 100 Q, there are no restrictions on the magnitude of R c . As R e -> « , R L = h it and A = -h f ,. The asymptotic limits in Fig. 16-10 are found to be |ii.| - 50, h = 0.59 MHz, and \A„U\ = 29.5 MHz for £.-»«. The First and Final Stages The results obtained above for an internal stage of a cascade are not valid for the first or last stage. For the first stage the equations in Sec. 13-9 for a single stage apply, provided that the load Rl is taken as the collector-circuit resistance in parallel with the input resistance of the second stage: R c h it Rl = * ^e + hi. For the last stage in a cascade use the formulas for a single stage, with A Sec- 16-8 UNTUNED AMPLIFIERS / 465 equal to the collector-circuit resistance R e of the preceding stage and with Rl equal to the R e of the last stage. 16-8 STEP RESPONSE OF AN AMPLIFIER \n alternative criterion of amplifier fidelity is the response of the amplifier to a particular input waveform. Of all possible available waveforms, the most generally useful is the step voltage. In terms of a circuit's response to a step, the response to an arbitrary waveform may be written in the form of the superposition integral. Another feature which recommends the step voltage is the fact that this waveform is one which permits small distortions to stand out clearly. Additionally, from an experimental viewpoint, we note that excellent pulse (a short step) and square-wave (a repeated step) generators are available commercially. As long as an amplifier can be represented by a single-time-constant circuit, the correlation between its frequency response and the output waveshape for a step input is that given below. Quite generally, even for more complicated amplifier circuits, there continues to be an intimate relationship between the distortion of the leading edge of a step and the high-frequency response. Similarly, there is a close relationship between the low-frequency response and the distortion of the flat portion of the step. We should, of course, expect such a relationship, since the high-frequency response measures essen- tially the ability of the amplifier to respond faithfully to rapid variations in fflgnal, whereas the low-frequency response measures the fidelity of the amplifier for slowly varying signals. An important feature of a step is that it is a combination of the most abrupt voltage change possible and of the slowest possible voltage variation. Rise Time The response of the low-pass circuit of Fig. 16-2 to a step ffi put of amplitude V is exponential with a time constant RzC*. Since the ^pacitor voltage cannot change instantaneously, the output starts from zero to d rises toward the steady-state value V, as shown in Fig. 16-11. The output '9.16-11 Step-voltage response of *e low-pass RC circuit. The Se f ime t r is indicated. . 466 I ELECTRONIC DEVICES AND CIRCUITS is given by Sec. ]fi.j v„ = 7(1 - e-" R ' c ») (16-20) The time required for v B to reach one-tenth of its final value is readily found to be O.IR2C2, and the time to reach nine-tenths its final value is 2.3fl a Cj. The difference between these two values is called the rise time t r of the circuit and is shown in Fig. 16-11, The time U is an indication of how fast the amplifier can respond to a discontinuity in the input voltage. We have, using Eq. (16-6), 2.2 0.35 t T — li.Z/f2t'2 — 2tt/ 2 (16-21) Note that the rise time is inversely proportional to the upper 3-dB frequency. For an amplifier with 1 MHz bandpass, t, = 0.35 ^sec. Tilt or Sag If a step of amplitude V is impressed on the high-pass circuit of Fig. 16-1, the output is v = Fe-" fi . c . (16-22) For times t which are small compared with the time constant R1C1, the response is given by F ( x -«k) (16-23) From Fig. 16-12 we see that the output is tilted, and the percent tilt or aag in time h is given by P = V X 100 = R\C) X 100% (16-24) It is found 6 that this same expression is valid for the tilt of each half cycle of a symmetrical square wave of peak-to-peak value V and period T provided that we set h = T/2, If / = l/T is the frequency of the square wave, then, using Eq. (16-3), we may express P in the form r X 100 = -j^r x 100 = ^ X 100% %RlCl (16-25) Fig. 16-12 The response t>„, when a step v % is applied to a high-p° sS RC circuit, exhibits a tilt. S«. J 6-9 UNTUNED AMPLIFIERS I 467 pjote that the tilt is directly proportional to the lower 3-dB frequency. If w e wish to pass a 50-Hz square wave with less than 10 percent sag, then fi must n °t exceed 1.6 Hz. Square-wave Testing An important experimental procedure (called square-wave testing) is to observe with an oscilloscope the output of an amplifier excited by a square-wave generator. It is possible to improve the response of an amplifier by adding to it eertain circuit elements, 1 which then must be adjusted with precision. It is a great convenience to be able to adjust these elements and to see simultaneously the effect of such an adjustment on the amplifier output waveform. The alternative is to take data, after each succes- sive adjustment, from which to plot the amplitude and phase responses. Aside from the extra time consumed in this latter procedure, we have the problem that it is usually not obvious which of the attainable amplitude and phase responses corresponds to optimum fidelity. On the other hand, the step response gives immediately useful information. It is possible, by judicious selection of two square-wave frequencies, to examine individually the high-frequency and low-frequency distortion. For example, consider an amplifier which has a high-frequency time constant of 1 Msec and a low-frequency time constant of 0.1 sec. A square wave of half period equal to several microseconds, on an appropriately fast oscilloscope sweep, will display the rounding of the leading edge of the waveform and will not display the tilt. At the other extreme, a square wave of half period approximately 0.01 sec on an appropriately slow sweep will display the tilt, and not the distortion of the leading edge. It should not be inferred from the above comparison between steady-state and transient response that the phase and amplitude responses are of no importance at all in the study of amplifiers. The frequency characteristics are useful for the following reasons: In the first place, much more is known generally about the analysis and synthesis of circuits in the frequency domain than in the time domain, and for this reason the design of coupling networks •s often done on a frequency-response basis. Second, it is often possible to arrive at least at a qualitative understanding of the properties of a circuit from a study of the steady-state-response circumstances where transient cal- culations are extremely cumbersome. Finally, it happens occasionally that a Q amplifier is required whose characteristics are specified on a frequency basis, the principal emphasis being to amplify a sine wave. T *~9 BANDPASS OF CASCADED STAGES *he upper 3-dB frequency for n cascaded stages is/a <n> and equals the frequency [° r which the overall voltage gain falls to l/V 2 " (3 dB) of its midband value. *hus/,w is calculated from Vi + </2 ( »V/ a ) 2 J \/2 468 / ELECTRONIC DEVICES AND CIRCUITS to be f.OO . . J ~ = V2" R - 1 Sec. 76-f0 (16-26) For example, for n — 2, fz m /fi = 0.64. Hence two cascade stages, each with a bandwidth f% = 10 kHz, have an overall bandwidth of 6.4 kHz. Similarly three cascaded 10-kHz stages give a resultant upper 3-dB frequency of 54 kHz, etc. If the lower 3-dB frequency for n cascaded stages is /i (n \ then correspond- ing to Eq. (16-26) we find h V2 1 '" - 1 (16-27) We see that a cascade of stages has a lower / 2 and a higher f% than a single stage, resulting in a shrinkage in bandwidth. If the amplitude response for a single stage is plotted on log-log paper the resulting graph will approach a straight line whose slope is 6 dB/octave both at the low and at the high frequencies, as indicated in Fig. 16-3. Hence every time the frequency / doubles (which, by definition, is one octave), the response drops by 6 dB. For an ?i-stage amplifier it follows that the amplitude response falls Qn dB/octave, or, equivalently, 20n dB/decade. Step Response If the rise time of the individual cascaded stages is Ui, tri, . . . , t m and if the input waveform rise time is („,, it is found that the output-signal rise time t T is given (to within 10 percent) by tr « 1.1 vV + W + W + + *r (16-28) If, upon application of a voltage step, one PC-coupling circuit produces a tilt of Pi percent and if a second stage gives a tilt of Pj percent, the effect of cascading these two circuits is to produce a tilt of Pi + P% percent. This result applies only if the individual tilts and the combined tilt are small enough so that in each case the response falls approximately linearly with time. 16-10 EFFECT OF AN EMITTER (OR A CATHODE) BYPASS CAPACITOR ON LOW-FREQUENCY RESPONSE If an emitter resistor R e is used for self-bias in an amplifier and if it is desired to avoid the degeneration, and hence the loss of gain due to P„ we niig° t attempt to bypass this resistor with a very large capacitance C t . The circuit is indicated in Fig. 16-46, It is shown below that the effect of this capacitor is *° affect adversely the low-frequency response. Consider the single stage of Fig. 16-1 3a. To simplify the analysis * assume that Ri\\R 2 y> R* and that the load R e is small enough so that the simplified hybrid model of Fig. 12-7 is valid. The equivalent circuit subjec $K. 16-10 UNTUNED AMPLIFIERS / 469 Fig. 16-13 (a) An amplifier with a bypassed emitter resistor; (b) the low-frequency simplified A-parameter model of the circuit in (a). to these assumptions is shown in Fig. 16-136. The blocking capacitor C b is omitted from Fig. 16-136; its effect is considered in Sec, 16-5. The output voltage V is given by Vo m -Ithf.R* = - V,h /t R c where R, + k ie + Z' t z; - (1 + h f .) R. 1 + jtaC.R, (16-29) (16-30) Substituting Eq. (16-30) in Eq. (16-29) and solving for the voltage gain A v, we find 7. h f ,R, 1 -(- foCiR, A *-T.= - R + R , 1 . „ R t R 1 +j(>>C, R + R' where R a R. + K and IP m (| + h f ,)R, The midband gain A is obtained as n — » « , or * _ h/gR c _ —hf e R e Ao __ Hence Where /.- R R, 4* h ie 1 1 +■?/ //» 1 + R'/R) + jf/f P 1 , = 1 + R'/R jp — (16-31) (16-32) (16-33) (16-34) (16-35) °te that f determines the zero and f p the pole of the gain A v /A e . Since dually R'/R » l, then f p » / , so that the pole and zero are widely separated. 470 / aECTRONJC DEVICES AND CIRCUITS See. 16-10 For example, assuming R, = 0, R, = 1 K, C* = 100 nF, h fe = 50, h it = 1.1 ^ and tf e = 2 K, we find /« - 1.6 Hz and /, = 76 Hz. A plot of 20 log \A v/A \ versus log / is indicated in Fig. 16-14. The piece- wise linear curve shown dashed indicates the asymptotic behavior of the fre- quency response. This dashed characteristic is constant at — 20 log (1 + R'/R) for /</<,; it increases linearly at 6 dB/octave for f e < f < /„, and remains at dB for / > /„. Remembering that f p ;$> /„ and using Eqs. (16-34) and (16-35), the magnitude of A v /A becomes, for / = f p , Av 1 UIU 1 + R'/R vT+T l Hence / = f p is that frequency at which the gain has dropped 3 dB. Thus the lower 3-dB frequency /j is approximately equal to f p . If the condition fp » fa is not satisfied, then /i ?* /,,. As a matter of fact, a 3-dB frequency may not exist (Prob. 16-29). Square-wave Response Since the network in Fig. 16-13 is a single-time- constant circuit, the percentage tilt to a square wave is given by Eq, (16-25), or P * 5& X 100 = 1 tiCR R X 10 ° Since R'/R » 1, R' X 100 P m l + h /t 2fC t RR, 2f{C,){R t + h it ) X 100% (16-36) (16-37) <si*c / / -10 V // -20