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Electronic Devices 

and Circuits 




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 


Jacob Millman, Ph.D. 

Professor of Electrical Engineering 
Columbia University 

Christos C. Halkias, Ph.D. 

Associate Professor of Electrical Engineering 
Columbia University 



New York St. Louis San Francisco Diisseldorf 

London Mexico Panama Sydney Toronto 




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 


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 



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 

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 



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 






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 




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 



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 






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 





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 


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 



CONTENTS / xitt 

M. 10 





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 


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 


Field-effect Tronsistors 




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 



*» 16 



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 



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 




















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 


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 


**. 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 


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 


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 



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. 


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 


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. 



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 


being in the direction of the electric field. Thus, 


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 



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. 



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 ( = 



Sac. 7-3 

■i d- 


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 


e£ = 7tta x 


a, = — = const 


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* 


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 




= v x 


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 


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 = 

— — X — = 2 X 10 9 * 

5 X 10-* 10" 7 


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 


1.76 X 10 M / S = 1.76 X 10»°(9.46 X 10" 8 )* - 1.58 X 10« m/sec 



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. 


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* 


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 


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 



v = 5.93 X 10 6 F* 



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. 



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. 


Sec, 1-6 


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 




The minus sign shows that the electric field is directed from the region of 
higher potential to the region of lower potential. 



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 

fz = % 

x = 


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 


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; 


v y = a v t 


Oy = = 


V = W 



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 

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 


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. 



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 


- 1(1.76 X 10") (10*) (4.77 X 10"*) - 4.20 X 10* m/sec 


, . 4.20 X 10 s AmM 

a. x = 2.83 X 10-* X 0.707 = 2.00 X 10"* m = 2.00 cm 


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- 



■*kS r 


+ V d + u 


Fluorescent screen 
Fig. 1-4 Electrostatic deflection in a cathode-ray tube. 

See. 7-8 


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 


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 


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 



D = 

By inserting the known values of ay ( = eV d /dm) and v ox , this becomes 
lLV d 

D - 

2dV a 


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 

The electrostatic-deflection sensitivity of a cathode-ray tube is defined as 


Sec. 1-9 

the deflection (in meters) on the screen per volt of deflecting voltage. Thus 


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. 


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 


Horizontal- deflection 



voltage v. 


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 



Fig. 1 -6 Sweep or sawtooth voltage 
for a cathode-ray tube. 


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. 



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. 


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* 


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' 


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\ 


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, 

(1-28) can be solved for v, the true velocity of the particle. The 

Vn = 


then Eq. 

result is 

v = c 

1 - 




(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 



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 



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. 



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 


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. 


Sec. 1-12 

Sec. M3 






Fig. 1-7 Pertaining to the determination of the direc- 
tion of the force f m on a charged particle in a 
magnetic field. 


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 = *!£ 


The force in newtons on a length L m (or the force on the N conduction charges 
contained therein) is 

BIL = 


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 


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. 


N electrons 



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, 



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 * 

But it has already been pointed out that T — L/v. Then 

_ _ Nev 
J ~LA 


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 


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). 



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. 


Sec. L13 




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. 

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 = 






In these equations, e/m is in coulombs per kilogram and B in webers per square 

S*c. 1-13 


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. 



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. 


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 


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), 


R => 




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 


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 


n r IL ILeB ILB 

D ~ L * = R=^ = ^r a 


The deflection per unit magnetic field intensity, D/B, given by 

d = ih rr 



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. 


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 


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 


V = ~eB^ 

It follows from Eq. 


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 

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 


Fig. 1-12 The helical path of an 
electron introduced at an angle (not 
90°) with a constant magnetic field. 


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 


8ir 2 K~ n 2 
L*B 2 


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 


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. 


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* 


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 


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 

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 

ay m — = (1.76 X 10") (1.10 X 10*) = 1.94 X 10" m/sec» 

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 


Pig. 1-14 A problem illustrating helical electronic 
•notion of variable pitch. 


Sec. 1-17 

Fig. 1-15 The projection of the path in the 
XZ plane is a circle. 


+ 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 


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 

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 


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 


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 


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 

dv. _ 
f ' = m dl = eBv ' 

By writing for convenience 


= — and 

U = B 




the foregoing equations may be written in the form 


N oju — biVz 

dv t 

Tt = + m * 

Sec, 7-17 


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 „ 


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 


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 = - 


x = 0(1 - cos 8) z = Q(8 - sin 8) 
where u and a? are as defined in Eqs. (1-53). 




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 


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 


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 


e£ = eBv a 

Km = -g = u 


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 

w = — = 1,76 X 10" X 10"* = 1.76 X 10° rad/sec 

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 


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. 



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 


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 


of C la 

Qui = U 


See. 1. 18 

0* — Magnetic fleld 
(Into paper) 


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 


Particle orbit 

Fig. 1-20 The cyclotron principle. 

South pole 


s*. its 


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). 


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 

S»c. M« 


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 


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. 


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. 



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 




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 



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 


where the energy is in joules. Combining this expression with (2-1) produces 


W = 


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. 


Sec. 2-2 



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 = 


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 


7TWT = 



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 = - 



8fc V n 2 


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. 



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 





7.73 H 

fig. 2-1 The lower energy 
levels of atomic mercury. 



Ionization level of mercury 




M I 


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 

Et — Ei 


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. 


Sec. 2-3 



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), 




= 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. 


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 


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. 



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. 


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. 


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. 


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 


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. 



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 


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, 


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 



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 } 


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 = 


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 


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. 



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- 


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 

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 























Number | 
of | 
















TABLE 2-2 Electronic configuration in Group IVA 






ls»2s J 2p* 

laW2p B 3s s 3p 8 3di°4s*4p s 

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. 



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 


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 


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 


2 A' states | 
2N electrons 

Inner-shell atomic energy 

levels unaffected by 

crystal formation 

(4N states 
Conduction band 

Interatomic spacing, d 

(4W states 
< 4N electrons 
[ Valence band 

j, Crystal lattice 
I / spacing 


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-" 


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. 


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 


Sk. 2-1 T 

E c * 6 eV 

. Holes 



JL Valence 


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 


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. 


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. 


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. 



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 

S9C. 3-1 


Fig. 3-1 Arrangement of the sodium atoms 
in one plane of the metal. 

© © © 

D © © © d 
© © © © 
) © © © 

• • m © 


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) 


Sec. 3-2 

Sec. 3-2 



= 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). 



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 

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 


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 

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 


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 


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. 


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 


Sec. 3-3 

Fig, 3-5 The potential energy of an electron as 
a function of radial distance from an isolated 

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 


Fig. 3-6 The potential energy resulting 
from two nuclei, a and 0. 

Sec. 3-4 


Fig. 3-7 The potential-energy distribution within and at the surface of 
a metal. 


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. 



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 

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. 



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. 



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 


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 


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 



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) 


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 



y ~ h* ( 2m )'(1.60 X 10-»)l = 6.82 X 10" 


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 + e< s -^)/W 


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 


for E < E F 
for E > E F 


$k. 3 ' 5 





lx i 

r — r=300°K 


' NJ 



t N 







-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. 



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 


f Er 



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* 


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). 



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^ 


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 


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 


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). 






Sac. 3-6 

energy U set equal to zero. Under these circumstances Eq. (2-14) may be 

dx 2 h 2 


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 


^ = Ci sin ax + Ca cos ax 


h 2 


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 


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 = - 


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 


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 


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. 


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 


^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 = *^ 


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 


N(W) dW = ^ (2m)»TT» dW 


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). 



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 

T-2500K I 






Distance, x 


Fig. 3-14 Energy diagram used 
to define the work function. 

Sec. 3-5 


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). 



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 



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 


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 


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. 


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 


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 



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 


Sec. 3- 70 

Sec. 3-10 


there is an exponential decrease of current / with voltage according to the 

where V T is the "volt equivalent of temperature," defined by 

V m iT. = T 
T e 11,600 



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 


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 



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 


I m I ei +viy r 



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. 


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, 

= e -Ul, 600X0/2,700 — f — 4.2a m 0.014 


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 


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. 


Sec. 3-11 



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 

£ - 


In (r„/r k ) r 


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 = 


— = 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 ^ 


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 




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. 



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 


Sec, 3-13 

primary energy is increased too much, the secondary-emission ratio must pass 
through a maximum. 


1. Shockley, W. : The Nature of the Metallic State, J. Appl. Phys., vol. 10 ud 543-555 

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 

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, 

McKay, K. G.: Secondary Electron Emission, "Advances in Electronics," vol. 1, 
pp. 65-130, Academic Press Inc., New York, 1948. An extensive review. 


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. 



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 



TABLE 4-1 Comparison of thermionic emitters 

Sec. 4-1 

Type of 

A. X 10-«, 







Efficiency, t 



Gas or 



Tungsten. . . , 


tungsten . . 








50-1 , 000 


Below 750 




or gas 

t K. R. Spangenberg, "Vacuum Tubes," McGraw-Hill Book Company, New York, 


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. 



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 '*'' 

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 


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 


Sec. 42 

Sec. 43 


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. 



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. 



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 

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 


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, 

& = 


at a; =0 


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. 



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- 

energy, eV 

Fig. 4-2 The potential-energy varia- 
tions corresponding to the curves of 
Fig. 4-1. 

Sec. 4-4 


tute the current. The magnitude of the current density / in amperes per 
square meter is given by Eq. (1-38), viz., 

J = pv 


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., 




= eV 




dx 2 

, P_ 


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), 


^!Z = t = L - J 

dx 2 t V€„ ~ [2(e/m)]*e Q 

K = 

7-1 = KV-i 




*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 



dy = KV-i dx m KV-i — 


ydy = KV-* dV 


Sec. 4-4 

Sec. 4-5 


which integrates to 

%■ = 2KV* + Ci 



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 

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) 



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 -^- 


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. 



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 


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 


/ c 

/ ** 

M / 





■« d — 


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 


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. 



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 


100 V P ,V 


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. 



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, 


Fig. 4-6 An ideal-diode characteristic. 

V F 

Sec. 4-8 


an important parameter is the dynamic, incremental, or plate resistance, defined 


dV P 

r„ = 



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 



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- 


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). 



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 


Fig, 4-7 The basic diode circuit. 

-v. » 



Ri. > v » 


The Load Line From Kirehhoff's voltage law, 
vp = Vi — ipRi, 


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 

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 


Sec. 4-9 

Static curve 




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. 




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. 


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. 


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. 



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. 



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 


Covalent r Valence 

oec. o* J 


• *— * v aience 

bond. Ge /[electrons 

Je*H ■ *G 

» \ 


it i i 

Fig. 5-T Crystal structure of germanium, 
illustrated symbolically in two dimensions. 



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 




Kg. 5-3 Th e mechanism by 
w tiich a hole contributes to 
the conductivity. 






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. 






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 



Sec. 5-2 

Sac 5-3 



<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! 


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. 



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 


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 



Sec. 5-3 


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 


L.60 X 10" I9 )i = 2 ( % 





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 


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 


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 


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 


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 







E = Ec + Ey kT Nc 
F 2 2 N v 



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 = 


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 


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 


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 




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 



As indicated in Eqs. (2-15) and (2-16), the energy gap decreases linearly with 
temperature, so that 

Eq — Ego — &T 


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. 

The measured values of m 



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. 




G e /Free electron 


Fig- 5-5 Crystal lattice with a germanium 
atom displaced by a pentavalent impurity 

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 








Eg Donor energy level 



Valence band 

. . 1 


See. 5-4 





/ • / •\In 



Fig. 5-7 Crystal lattice with a germa- 
nium atom displaced by an atom of a 
trivalertt impurity. 

-' ! '" J | ■ ! 

• • • 


• 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 

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 


fi i 

Fig. 5-8 Energy-band diagram of 
p-type semiconductor. 

S«c- S-6 



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 


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 


Similarly, for a p-type semiconductor, 

tt„p p = n, a 






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. 


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 


Conduction band 

Kb f 





:« band 

5 1 



Fig. 5'9 Positions of Fermi level in (a) n-type and (b) p-type 

S«. 5-7 


can be made if we substitute n = N" D from Eq. (5-25) into Eq. (5-9). We 

tf D m N c e- iE c~ £ * VkT (5-29) 

or solving for E F , 

E F = E c - kT In ^ 


Similarly, for p-type material, from Eqs. (5-28) and (5-14) we obtain 

E F = E v + kT In 



Note that, if N A - Nd, Eqs. (5-30) and (5-31) added together (and divided 
by 2) yield Eq. (5-16). 



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 

'• - -»•% 


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 


Sec. 5*8 

diffusion and mobility are statistical thermodynamic phenomena, D and /* are 
not independent. The relationship between them is given by the Einstein 

Mp Mr. 


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. 



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 


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) 


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 




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 

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 



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 — 


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 


x + dx 


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 -~ 


Since charge must be conserved, 

eAdx~- = —eA dz — + eA dx a — dl 
dt 7-* w 



The hole current is the sum of the diffusion current (Eq. (5-32)] and the drift 
current [Eq. (5-1)], or 

I = -AeD. 



+ Apefi p & 


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), 


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, 


P - p. 

^ U * dx* 




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 

_ _ Pn - p„ 

+ A 

3 2 p„ 
dx 2 


3 (p»g) 


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 




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 




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), 


clL * dx 



Sec, 5-9 




,.,..^, 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 ! 



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)*» 


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 


d 2 P» 1 + jarr, 
dx* LJ n 


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 

dx 3 


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. 



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 


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 = 





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 = - 


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 



Sec. 5-10 


Rn = 



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 


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. 


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, 

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. 


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. 



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 


5«c. 6-1 







e © e e 

o o o 

© e e e 

o o o 

e e e e 

© © © © 

• • • 

© © © © 

• • ■ 

© © © © 


p type 

n type 

Distance from junction 






Electrostatic potential V or 
potential -energy barrier for holes 

j Distance from junction 

Potential -energy barrier for electrons 

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 


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. 



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- 


Sec. 6-2 

S«. 6-2 


Metal ohmic contacts 




Fig. 6-2 (o) A p-n junction biased in the 
reverse direction, (b) The rectifkr symbol 
is used for the p-n diode. 


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 

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. 



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 


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. 



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 


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 


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 


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), 


E Cn -E F = kT In ' 

p rom Eq. (5-31), 

Ep - E Vp = kT In £p 





Sec. 6-3 




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 — 


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 


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). 


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 



Total current J 

7 BB , electron current 
I pn , hole current 




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 


.„ 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) 


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. 


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. 



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 > 


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. 


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 — 



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. 




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 »- 


n material 

Injected or excess charge 


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 


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, 


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 



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) 


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 


AeD p p, 

( e viv T _ x) 


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) 


Finally, from Eq. (6-14), the total diode current I is the sum of I pn (Q) and 
^»p(0), or 


/ = LU^T - 1) 



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 



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- 


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 

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. 



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 = 



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. 


Sec. 6-6 





♦ 0. 





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 











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 

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 


I, inA 















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.) 







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. 


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 


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 


d(ln h) 

_ m Voo 

T "'" v TVt 


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 


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 

\I. dT) 

V - (Voo + m n V T ) 


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), 


-2.1 mV/°C 

for Ge 

for Si 


Since these data are based on 'average characteristics," it might be well for 
conservative design to assume a value of 


= -2.5 mV/°C 


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 


Sec. 6-8 



F *"""^ 








_ i 




40 60 SO 100 


a 0.05 




a C 

Reverse voltage, V 


30 60 90 120 
Reverse voltage, V 


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. 



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 


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, 



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 



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. 



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. 



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 — 



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 


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 


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 


" 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 


dx 2 

eN A 



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) 


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 



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 = 


- eN A A 


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. 




25° C 





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. 



'9- 6-15 A varactor diode under reverse 
,as - (a) Circuit symbol; (b) circuit model. 



c~4 tf—L- 

C T 





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). 



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. 

Q = f* AeP n (Q)t-~iL f dx = AeLJ>„(0) 


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) 




Sec 6-l° 


dP n (0) _ _L 


AeD P dV AeD, 



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>„ 





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 

1 - Qrh = 



^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 


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 

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. 


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 



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 


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„ 





-/„ =s 


Forwardi . 

, storage, t, 

Sec. 6-1 1 








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 

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. 




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. 



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. 





Izk V 

1 («) 






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. 


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 



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 06 







5 t 


V z @5 

J i 



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 


S«. 6-12 




3 Cl 



r ~ 


I z = 1mA 


1 > 

f\ J^ 



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 


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 



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. 


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 


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 


x a x = d 


$*c. 6-1 * 


Region 2 The Schrodinger equation for x > is 
dhp 8jt%i 



(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 


8ir*m(U - W) 

* _ h r i f 

4vl2m(U - IF) J 




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 *' 


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 %£ 


Sac. 6-1 3 


n side-* 




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. 



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 



E v - 





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). 


Sec. 6-13 

S«r- 6-M 


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. 


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) 

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, 





Sec. 6-1 4 

TABLE 6-1 


Typical tunnel-diode 

Ip/Iy . 
V F ,V 
V V ,V 











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 



\. 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 



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 



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 


Control grid 


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.) 




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. 


Path between grid wires 



Path through grid wires 


1 © 




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- 


-90 ■ 


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.) 







— i — 













- a. 





\ y 




V / 





1 1 












1 > 




1 1 







— > 




1 ' 











40 -20 0+20 +40 +60 
Distance from grid, mils 


40-20 + 20 + 40 + 60 
Distance from grid, mils 


-40 -20 +20 +40 +60 
Distance from grid, mils 


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. 



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 

by the equation 2 

i P = G (vo + V A' 

Sec. 7-2 


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 

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. 



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 



— Heater 





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. 



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) 


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 



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 



« 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 


Sec. 7-5 

constant plate current. Mathematically, n is given by the relation 


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 


\dV )y p - 9m 

_ (§Vp\ 

\dv ) Ir M 

plate resistance 

mutual conductance 

amplification factor 


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 

Each section 
















^ 3m 



V f = rated value 

2.0 i 







1.0 g 





22 28 

20 24 

18 *20 

—16 £ 16 

g a 

+* m 

at a 
I U * 12 

< 12 E 8 


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 

TABLE 7-1 Some Mode parameters 

Sec. 7^ 

Triode type 


r P , K 

ffm, mA/V 











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, 



p ~ AiZ L. = reciprocal of slope of characteristic 

9m. = 


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 


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 




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 




t ' 

N - - 

•*" *1 





h i — 


100 150 200 

Plate voltage, V 


Sec. 7-6 


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 


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. 



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 


S«. 7.7 


1 1 1 1 


Total space current 


] r 













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 


\fV c =Q 

1 1 1 
V c = -0.5 V 




I" - 


s - 




1 J 


K If 
s 1 





— 3.0- 







— '■ 





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.) 



low values of potential, 
given in Fig. 7-12. 


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 



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 


Sec. 7-f 

Be am -forming 




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. 



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. 


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 


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. 


Trtode symbols 

Instantaneous total value 

Quiescent value 

Instantaneous value of varying 


Effective value of varying component. 

Amplitude of varying component 

Supply voltage 

Grid voltage 

with respect 

to cathode 



Plate voltage 
with respect 

to cathode 

V P 

Current in direc- 
tion toward plate 
through the load 


t These are positive numbers, giving the magnitude of the voltages. 

6. Conventional current flow into an electrode from the external circuit is 

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) 



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 


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! 


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? 




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 


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 . 



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, 


Sec. 7-12 


transfer curve 

Fig. 7-20 The dynamic trans- 
fer characteristic is used to 
determine the output wave- 
shape for a given input 

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 . 



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) 


ten are the parametric equations of an ellipse. If the angle 6 is zero, the 
10 of these equations yields 

i 7 ~ Hl 


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 

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. 

Static load line 

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. 



* 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- 


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 



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 


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- 


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 


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 

-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, 


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). 


In order to see why it is sometimes advantageous to use a negative supply, 
consider the cathode-follower configuration of Fig. 7-27. 



Fig. 7-27 An example of a cathode-follower 


Sec 7-1 4 


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 




" ? +1 * 

r © 

•Rk v° 

(a) (o) 

Fig. 7-28 Two biasing arrangements for a cathode-follower circuit. 


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. 


1. De Forest, L.: U.S. Patent 841,387, January, 1907. 

2. Spangenberg, K. R.: "Vacuum Tubes," McGraw-Hill Book Company, New York, 


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, 

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. 


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. 



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 



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 

i-p = ip — Ii 


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 

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) 



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 



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 


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 


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 

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). 



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. 


Fig. 8-3 (a) The sche- 
matic diagram and (b) the 
equivalent circuit of a 
simple grounded-cathode 



"). 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 



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\ = 


Rl + r p 


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, 



= —n 


Rl + U 

= — M 

1 + r p /R L 


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' 


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 


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. 


f>0 ■ P 


Fig. 8-5 (o) Illustrative 
example, (b) The small- 
signal equivalent circuit. 

v„ P -$ 

10K 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 - 

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 

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 


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 , 



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 




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 



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 


= (i£)v *»* + ($£) 


Av G 


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, 


Consider the third term in this expansion 


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) : 


Atp = — Avp + g M Av c 


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 


= 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 



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. 


Sec. 8.5 








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). 


I = gm <V 




r *< v p* 

g, 8-8 The current-source model of a triode- 

Sk 8-6 

V K 

V GC J?~ 

Fig. 8-9 (a) The common-cathode amplifier configuration 
and (b) its current-source equivalent circuit. 



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 -±- 




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.) 




Fig. 8-11 (a) A generalized amplifier configuration, (b) The 
small-signal equivalent circuit. 



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 


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 




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. 



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. 


The output voltage is given by 

V = A v Vi - I L Z 

Sec. 8-8 


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 

Z = l 

I=Z = VY 

V = IZ = ± 


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. 


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 


obtain the short-circuit load current I from Eq. (8-19) : 


i. " + 1 

-? + v k 

— HVi + 0* + l)Vk 
T P + (M + 1)R* 


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 


+ Rk 


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«* 


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 



^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. 


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 



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 


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 





Fig. 8-14 The Thevenin circuits for the three bosic amplifier configurations 

$« 8-8 


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 =^ = 


M+ 1 



g m Rk 

m + 1 

+ Rk 

+ (m + DRk 1 + g m Rk 

if m » 1 


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 « 



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. 


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. 



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 


Fig. 8-15 The split-load phase in- 


Sec. 8-9 


-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) 


\A\ = 


g m R 

r„ + 0* + 2)R l + g m R 


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. 


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„ 


+ R, 


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 



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. 


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. 



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 


\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 


Sec. 8-?o 

Fig. 8-18 The cascode amplifier. 

Jy-x — 

R + ( M + 2)r p 


If (>* + 2)r p ;» iJ and if p. » 1, this is approximately 


= —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 



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 


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. 



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 


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 


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 


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 


A = 

Y L + Yp k + g p + Y s 


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» 


9v + Y L 1/r, + 1/Z L 

= —g m Z' L 


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 


are now examined 

gain and the exact expression, Eq. (8-39), must be used. These effects 



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 


to calculate this current, it is observed from the diagram that 

it - ViY* 


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 



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 


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 


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 

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. 


s« *-i2 



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 

= 10" s + j*2.52 X 10" s mho 


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„\ 



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 




d = C„* 4- (1 - A^C 



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. 




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 



• 1 i O— — • • • — 

•X © 



Fig. 8-20 The schematic and equivalent circuits of a tetrode con- 
nected as an amplifier. 


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. 



Our previous discussion of cathode followers neglected the influence of the 
tube capacitances. These capacitances are now taken into account. 


Sic. 8-1* 


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 


Y k = "5~ 

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 



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 

> l 

> o o < 

* ] 
- V 



N or P 

Fig. 8-22 (a) The cathode follower, with interelectrode capacitances taken into 
account, and (b) its equivalent circuit. 


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) 


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 





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. 


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. 


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. 



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.) 


Emitter Base Collector 
I C 


Emitter Base Collector 


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. 






|V'*„i i 

r — J Space- 

width — * V *~ 

base width 
— IV 

|V M | 






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. 


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. 



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 


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 



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 = 


IpE + I»l 

I B 


where I pli i s the. injected hole diffusion current at emitter junction and I, lE is 


Ejected electron diffusion current at emitter junction. 


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— 


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 


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 



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 


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 


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. 



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 



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, 



AI B \ y c* 

P n the assumption that a is independent of I E > then from Eq. (9-7) it follows 

th at«' „ 


Sec. 9-4 






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 

Silicon metal izat ion 

dioxide Emitter J^ ^ 

contact QE 


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. 



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- 


tor. From Eq. (6-25) we find that at the junction 

/-p(0) = 

AeD n nBo 

( € v a iv r _ j) 

Sttc. 9-5 


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 
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 > 



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 


Pn — Pno* " T 

Pn = Piiot 


at x — 
at x = W 


S»c. 9-5 


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 

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 


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 

[( € VdVr - 1) - ( € VMlV T _ !)] 


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) 

«* 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 



Dnn Co \ 
Lc / 




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. 



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. 


This equation has the same form as Eq. (9-8). Hence we have, by 

_ _% 



Ico = 


— fl22 




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 + (DnLBUBe/DpLspno) tanh (W/L B ) 


|8* = sech ~ 




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) 




1 + W<Tb/LeVB 






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 




, 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). 



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 ") 


/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 


Sec. o.y 



tlve region 


< 40 


« -30 



u -20 


(3 -io 


= 40mA 






• ■ 










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 

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 

o I 

10 20 30 40 

Emitter current I s , mA 


■ n open _ 



K» -■ 

= 0V 

r ~> 







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. 


S«, 9-8 




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. 



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) 


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 


S«. 9.j 

H -0.4 







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 


1 - 



1 - a 





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 . 



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 


From Fig. 9-7, if I B = 0, then I E = -Ic- Combining with Eq. (9-39), we 


Ic = —Is ~ 

1 - 



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- 


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, 





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 


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. 



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 


ations imposed by the transistor on maximum current and dissipation. 


j -30 


« -10 



1 1 

- 0.35mA 

T A = 25°C 




soon II 




— 1 1 1 



— 1 — 1 — ' 


— 1 


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 


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 


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- 

10 20 30 40 50 60 70 80 90100110120130 
— I c ,mA 


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 



If we define by 


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) 




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 


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 


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« 


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. 

« + (Icbo + Ib) ~ 


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) 



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 . 


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 


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. 


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- 

Base voltage v BEt V 

""] 1 1 1'l'ITI 

rrm , 


.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't* i 

-~ smm 

-0.15 -j-'l 


_Vcb i ;: 

■ 2 -4 -6 -8 -10 -12 -14 
Collector voltage v ct: ,V 


O -100-200-800-400-600-600 
Base current t B , ^ A 


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) 

h Uc) 


Sec. 9-7 3 


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 


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). 



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 

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) 


, 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, 


■V c 

S«c. 9-U 



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 


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 

( t vcir r - 1) - 

( t v,iv T _ 1) _ 

1 — aitai 

( e r,iv r _ !) 
( e vciv r _ j) 



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 


k V <~ J 

Fig. 9-18 The Ebers-Moll model for a p-n-p transistor. 


that section it is verified that the coefficients 

S«c. 9-1 4 

an = 

1 — ayai 


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 


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) 


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 / 


c + a^h 

\ lco / 



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) 


V C M = ± V T In 


ai Pi In 

. _l u 



1 — aj 


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 


pjg.9-19 The common- 
emitter output character- 
istic for a p-n-p transistor 
as obtained analytically. 



f~ J 



/ * = 

/ , ! . , . — i 


f r in - nfaooe 

0.1S 0.2 03 04 


-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. 


^t us now apply the equations of the preceding section to find the dc currents 
^d voltages in the grounded-emitter transistor. 


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 


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 


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 


Ic = 

1 — asati 



Ie = —ail 

ail ess 


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 


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) 



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 


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 





*C — IcES^^i 




Ic = I CEO * I\ 


-0.3 -0.2 -0.1 


Cutoff A 

0.03 0.1 0.2 



0.3 0.4 V r =0.S 0.6 W*0.7 V aB ,V 

Cutin - 

region " 


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.) 


















c l 






m* 1 *■ 




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 


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 


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 


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. 



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 


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 


VWBftt) = V, 


Vbe (cutin) ■ V y 









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 



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) 


Sec 9.17 


| t 


v % 

0.9I C s 


T Vcc 








\ . 


0.1I C8 


1 \^ ' 



i i 


i i 

1 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 

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 




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 

'9- 9-23 Minority-carrier con- 
centration in the base for cutoff, 
a ctive, a n{ j saturation conditions of 
°Pe ration. 



jc = 

x= W 

260 / aecrnoN/c devices and circuits 

S«e. ?-T« 









A.** 7 * 



= r* 

Fig, 9-24 The minority-carrier density in the 
base region. 


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. 



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 * 


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 


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. 



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. 


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.**» 


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. 



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 


2M / aecreoNJC devices and circuits 

S»e. 70.J 

Fig. 10-1 The fixed-bias circuit. 

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. 


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 

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 


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 « 



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. 


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 



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). 




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 


T = 25°C 












= 0mA 


< 40 






T= 100° C 







/, = 0mA 


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.) 


$*c. 10- j 


J 0-3 


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 - 





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 

Ic = (1 + 0)lco + &B 


If we differentiate Eq. (10-4) with respect to I c and consider constant with 
Ic, we obtain 

1 - 1 +P-L.R dI * 


S = 


1 - mis/dlc) 


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- 



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 = 


In = 

Vcc - IcRc - Vi 

Re + Rb 


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 

di c 

R c 

Re + Rb 

Substituting Eq. (10-10) in Eq. (10-6), we obtain 
0+ 1 

S = 

1 + 0Rc/{Rc + Rb) 



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 





Fig. 10-5 (a) A coll ector-to- base bias circuit, (b) A method of 
avoiding ac degeneration. 


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\ 


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 


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 




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. 



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. 


[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 = 

Ri + Ri 


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. 


If we consider V B s to be independent of I c , we can differentiate Eq. (10-15) 
to obtain 




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. 



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^ 





r- B = 100 







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 

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 


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 

The base current I B is given by 

p 50 


We can solve for Ri and R 2 from Eqs. (10-14). We find 

Ri — Rb 

Rt — 


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 


_ 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 = 


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 


10 X 90 

- 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) 



-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 






V CF . -±- 22.5 V 

i*** v i,+icjsi 

B 3 - 4 

■3 2 



ji n 

— in? 


Load line I 

r-~ "f 40 



curve — |*" 







4 8 12 16 20 I 24 

Collector-to-emitter voltage V C s . V 22 - 5 


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 


. 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 = 

Vce - 13.2 V 

6. From Eq. (10-17), 

S = 

_ 56 (i±^ = 


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 

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. 



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) 





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.) 



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 


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- 


current varies with tempera- 

te T because V BK , Ico, and 
chonge with T, 


Sec. 1 0-5 

Sec- 10-5 


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 

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 


S' = 

R b + R.(l + 0) 

R b + R e + 1 


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) 



where, from Eq. (10-24), V may be taken to be independent of 0. We obtain, 
after differentiation and some algebraic manipulation, 

on _ Mc 


d0 0(1 + 0) 


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 - 


0(1 + 0) 



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 




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) 



" A0 0,(1 + 2 ) 




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 


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 ) 

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 — 


119 X 3.38 

Vcc - V 20 - 3.38 

= 24.2 K 


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 = 


S f = 


o„ _ Ale 


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 . „ 


- S Alco + S' AV BB + S" A0 


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 


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) - 


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 


Sec. 10-6 


5"( + 175°C) - 

IciSj _ (1.5) (2.62) 
0,(1 + j8 2 ) (55)(101) 

= 0.71 X 10-» mA 


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 


S , ( _ 65 . C ) = ^^ = 2.45 
21 + 1.65 

S"(-65°C) = (L5)(2 ' 45) - 3.18 X 10"» mA 

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 


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. 


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 


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) 


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* 



fm, 10-12 Diode compensation for a germanium 



m „ 

the same rate as the transistor collector saturation current Ico- From Fig. 
10-12 we have 

/ - 

Vcc ~ Vi 


= 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 


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. 



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 


(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 = 


— I B\ ~ 1 1 

For Vbb « V C c and (I B i + /«) « hi, Eq. (10-39) becomes 

r V CC . 

Ici « -5— = const 



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 = 


- ( 2 + si) 

Under the assumptions that V B b « Vcc, 
(10-41) becomes 


and (2 + Ri/Ri)I B « Vcc/Ru E* 

Ici = Ici = 

Sec. 10-9 


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. 


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 


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. 



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 


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 





pig. 10-15 Power-temperature 
derating curve for a germanium 
power transistor. 


1 \. 

i X 

i X^ 





) 20 40 60 



Case temperature, 


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 


'9. 10-16 Concerning transis- 
tor self-heating. The dashed 
lo °d tine corresponds to a 

er y small dc resistance. 


Sec, 7 0-| | 

Sec. 10-W 


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, 


dPc . BPd 

df~ Jf~ 

If we differentiate Eq. (10-42) with respect to Tj and substitute in Eq. (10-43), 
we obtain 
dP c 



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. 


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 


= Vcc — 2Ic(Re + Re) 


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 


= S 



+ s 




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 

(s dIco \ < I 


In Sec. 6-7 it is noted that the reverse saturation current for either silicon or 
germanium increases about 7 percent/°C, or 



= 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 



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 




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. 


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-» < ^ 


< 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) 


1 - 5 X 10-' X 100 . : . 

[ B = A = — 5 mA 



Fig. 10-17 Power amplifier with a trans- 
former-coupled load. 

u, o— * 




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 ) < - 



< 3.03°C/W 


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. 


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. 



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 






H-1 A two-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 

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 


i\ (in ohms) 

ft n *i ^ (in 



Fig. IT -2 The hybrid model for 
the two-port network of Fig. 11-1. 
The parameters h n and An are 



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, 



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- 


Soc. 11-2 

we may write 

Vb = fi(iB, Vc) 

ic = Mis, vc) 




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 



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: 



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 = — 


di B Ikc 

h ^ 9/i = Bvb I 
" dvc dv c l/» 

Bfi die I 
dv c dVc u» 



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 - 



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 



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 




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 



k ft = 

Ate 1 
Lis We 

IC2 — 1C\ 
iB2 — 1-Bl 


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, 

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„. 


The Parameter h 

From Eq, (11-10), 


Ate I 

Av c Us 

s <*- 11. j 


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 


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 


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 

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 



I 20 

1 ,0 

1 5 


* I 

> 1.0 

J 0-5 


I 0.2 

£ o.i 

jS 0,06 
f 0.02 

y o.oi 


1 V 


1 1 














W 2.0 



a 1.5 



2 0.4 

g -0.1-0.2-0.5-1.0-2 -5-10-20 
I Collector current Ic , mA 


3 «•? 

= 1kHz 
- 1.0mA 



h n * 

n t' 





150 200 


Junction temperature T f , * C 

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.) 




k f* Q0 

V cr . = 5V 

150" C 




T A 

= 25° 




^ -55 e C 


-5 -10 -15 

Emitter current I t , mA 


TABLE 11-2 Typical /(-parameter values for a 
transistor (at I B = 1.3 raA) 

Soc. TJ. 


ftu = hi 

kit = h T 

h tl = kj 

h%i = k 9 



1,100 n 

2.5 X 10- 



1,100 Q 



25 M/V 
40 K 


21.6 R 
2.9 X 10-* 
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 


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. 


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 


h- t = 

h lv.=o 



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 


h\v*-o V,R L 


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 



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 . 


S«. lj-4 

TABLE 7 J -3 Approximate conversion formulas for transistor parameters 
(numerical values are for a typical transistor Q) 





T equivalent 


1,100 Q 


1 +h /b 



2.5 X 10-* 

1 - A re t 

1+A„ ** 

(1 - o)r. 



-a +**)* 


1 +*/» 


1 -o 


25 M A/V 


1 + A/t 


(1 - o)r. 



A, e 

21.6 a 

r. + (1 - o)r 4 


l+h,. K ' 


A/ e 

2.9 X 10"* 


h f . 

1 +A/c 


1 +A/. 

— a 

h i 



~A/ e 


1 + fc/. 

0.49 aiA/V 

1 r« 

h ie 


1,100 Q 


l — a 

1 + A/» 

K t 

1 - At, - If 



1 ~~ (1 - a)r t 

h tc 

-a + A / .)t 




1 +A/» 

1 - a 



25 ^A/V 


1 +A/i 


(1 - a)r t _ 



1 +h f . 

1 + A/« 

— A/i 



1 + */. . 

A/ e . 



2.04 M 






10 fl 


a,. - J= a + A A )t 




590 n 

t Exact. 

5*. "-5 



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 = 



'8- It -O ( a j jj, e £g hybrid model, (b) The circuit in {a) redrawn in a CE configu- 
r atio n . 


Combining the last two equations yields 

S»e. 11.$ 

1 + hjt 



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 

and hobhit, « 1 + A/t 



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 


Applying KCL to node B, we obtain 

h + I. + hfl,I t - hobVb. = 


h = (i + m ^ : ^ ^ + ^n. 


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 


hi. * 

l + Jfc* 

which is the formula given in Table 11-3. 



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. 



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 





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* 


„ hilt + KV* V, 

Li — f = Hi -r fir -y~ 


Vi = —IsZl — AiIiZl 
in Eq. (11-25), we obtain 

Zi = hi -f- hrAtZh = hi — 

hfh r 

Y L + h 




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 = 


s«- "- 6 


From Eq. (11-26) we have 

AjUZl AiZl 

A v = 




The Output Admittance Y e For the transistor in Figs. 11-11 and 11-12, 
Y 9 is denned as 




Substituting the expression for I\JV% from Eq. (11-33) in Eq, (11-31), we 


Y B 


v z 

with 7. = 

From Eq. 



1 7l 

= hf V i 

+ h 

From Fig. 

11-12, with V. = 0, 

RJi + hili 

+ KV 2 = 



h r 

v t 


•f fi. 


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, 


Vi = 



A v . = 

Z< + R, 
A v Zi 

AtZ L 

Zi -f- R, Zi + R, 



""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, 


Set. 1L 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 


A u = 


7TT. mA 'Z 


From Fig. 11-136, 
7 I,Rt 

1 1 — 
and hence 

Zi -4- R, 

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 



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 


A - S - - 

Ap ~ P, ~ 

V*Ii A A A 1^ L 

vrr AvAt ~ At & 

S*c. >>-6 


TABLE 7 7-4 Small-signal analysis of 
a transistor amplifier 

A, = - 

1 + hoZt, 

Zi = ki + h T AiZi, = hi — 

kfh T 

Av - 


Y Q = K 


Av, = 

h{ + R, Z e 
AvZi AiZh Ai,Zl 

Zi + R. Zi + R, R t 

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 - 


= -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 = 


-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 


Ri + R* 
AjR t 


-48.8 X 10* 

Ri + Rt 2.088 X 10* 

= -23.3 



Sac. II.7 

Note that, since R 3 — Rt, then Ay, = A t ,. 
T. - &^ - -^- = 25 X 10- - 


= 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 



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) 





\ CB 








^— -0.20 


10 s 




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. 



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, 



-A V (CE) 








/CB or CE 








*^ 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. 


Sec. J J .7 

. n 














10 s 

10 s 



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) 


(A/.)™** (B*, - 0, R, m «) 

Ri (Rl = 0) 

Ri (Rl — ™ ) 

(Avx)«**(flji = *,B. = 0) 

B„ (B. • 0) 

B (B, = ») 

A- parameter 



Aift — ArA/ 



1,100 n 

eoo a 


73.3 K 

40 K 

15 X 10* 



l.ioo n 

2.04 M 

21.6 fi 

40 K 




21.6 Q 

600 a 


73.5 K 

2.04 M 

2.94 X 1°' 

S*c N-7 


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 





2.04 M 

105 K 



73 K 

40 K 

33.2 K 



10 ] 

10 s 


10 s 



R t ,a 

Fig. 11-17 The output resistance of the typical transistor of 
Table 11-2 as a function of its source resistance. 


TABLE 11-6 Comparison of transistor configurations 

Sec. ?T-8 






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) 



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. 



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 




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. 

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. 




r.' ai,(T) 

Base B' 








1 Tb 



Soc. I ] .p 

Fig. 11-19 A more com- 
plete physical model 
of a CB transistor than 
that indicated in Fig. 

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— 


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 




Fig.' 11 -20 The T model of a CB tran- 

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. 



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- 



o ■ 


T V ri 




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- 




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- 


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. 


1 . IRE Standards on Semiconductor Symbols, Proc. IRE, vol. 44, pp. 935-937, July, 

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. 


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 



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. 


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 

The second stage. From Table 11-4, with R L = R*2, the current gain of the 
last stage is 

An - — -— - = 



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 





«+ V cc 



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), 

R el R it 5 X 228.5 

Rli = 

Re\ + Ril 


= 4.9 K 




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 


>ec. J2.| 

The output admittance of the first transistor is, from Eq. (11-39) or Table iy 

1'.. - h„ - 


hu + R, 
= 15 nA/V 

= 2o X 10"« - = 15 X 10-s mkft 

2 X 10 3 + 1 X 10 3 nno 


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.- B + ^,2 

T = 25 X 10-" - 


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 



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 


From Fig 







fl, s 

+ RcX 



A/jA/i — 

R c i 

= 45.3 > 

1 ^ bi A 

Art r- An 

Rn + Rex 

228.5 + 5 




C, o- 


Fig. 12-3 Relating to the calculation of 
overall current gain. 

N o— 

For the voltage gain of the amplifier, we have 


7, V 2 Vi 

= AviAx 



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. 


J2-7 Results of the examp 

e on page 324 

Transistor Q2 

Transistor Ql 

Both stages 




228.5 K 


1.87 K 
4.65 K 

1.87 K 

127 fi 



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 



Soc J 2.} 

(22K) >(6K) 



(16K) >(700n) 


(3.3K) <R tl ^ (50uF , S(G.2K)< R J_(50 

>(1K)T ( ^ ' f > ' 2 T>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 ~ — 


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 + ■ - ■ 

A = AlAl • - • An S = 01 + $ 2 + 

+ 8* = 



■ + 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, 


A V k = 



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 **" 




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 


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 

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 


*cl< V 

St c 2 


-o— — o- 

/*-■ h„ 

Rc2< Vi *«..-.' 


/«, = h 

R t \ -R« 


*• G) 7 ' < R - *■>■ 


*E t 




*9- 12-5 (o) n transistor CE stages in cascade, (e) The &th stage, (c) The 
'•"onsistor input stage driven from a current source. 


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, 






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. 




hi 1 1 

h—1 In 

h—2 in— 1 



Ai = AiiA' n ■ ■ ■ A'i^A'^ 



An = - — - - 42 A' = 

Ibl lbi 





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 


A n = ~~ 


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 


Alt — 

Rc.k-] + Rik 
h h hk AiicRe.k—l 

h-i hk h-i Rc,k-i + Rit 


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, + Rh 


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 




• 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 *- 


Power Gain The total power gain of the n-stage amplifier is 

output power VJ a . , 

A P = - — , = - «-r~ ■ AvAt 

input power V ihi 


A P = (A,) 2 



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 


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. 



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 


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 


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 


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. 





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- 


R. < + I 1 

V„ h„V e Q (j)h fe 


F '9. 12-6 (a) Exact CE 
hybrid model; (b) ap- 
proximate CE model. 


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 

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 



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) 




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~ ** *<* 

overestimates the input resistance by less than 5 percent. 

Voltage Gain From Table 11-4 the voltage gain is given by 


Av - At 




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 

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 

The approximate solution for the CE configuration is summarized in the 
frst column of Table 12-2. 



*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 



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 



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 


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 


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 






TABLE 12-2 Summary of approximate equations for A oe (ft -f Rj,) <0.1f 


CE with R t 




-h f . 


1 +h /t 

L h/. 

" ~ 1 + *,. 



hi. + (1 + h f .)R t 

hu + (1 + A/,) fix. 

A, hit 

1 + k„ 




h/ t Rz, 








R, + hi. 
1 +h f . 



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), 


Y L * + h f 

V, hit + Rt 

From Tables 11-4 and 11-3, the expression for Y„ is 

Y« = h M — 



hic + R. 

Kt + 

1 + h f€ 
kit + % 


*^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 


from the simplified model, namely, 
hi. + R, 

R„ = 

1 + h fr 

Sec, 1 2-5 


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 = 



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) 

- 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»-) = 


R»2 + Ri 


(137) (5,000) 

= 137 a 
= 134 Q 


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/ 


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 



»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 


h f , I b 


1 1 O 1. 

R, B -+r 





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 

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 



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 


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 


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 

Ri = -r- = hu + (1 + k/ t )R, 


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 = 


— h/tRi 


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 

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 = 


R< + Ai, + (1 + h t .)R, 



Sec. J 2-7 

(7) ^Yi 3 J 




=±=- JV 

!, = (! + £,,)/(, 

I W\* o i- - 



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 


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 + */.) + £« 


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 


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, 


'■l + ^t^R, 



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 


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. 



-*rv e 



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 = 




l + KM'l 

l + K, Ul + ^3— «.) 


From this equation we can solve explicitly for At, and we obtain 

fl at Re — hje 

At = 

1 + K.{Rl + R.) 


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 


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 


where the exact values for Ai and Ri from Eqs. (12-48) and (12-49) must be 

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< 





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 




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 

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, 


F fl = 

R, + R' 


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: 


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. 


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 

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 



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 


k b 


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 



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 V- 

A = U — ° * — * 

Av '- V, ViV t - Av R. + R'< 


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 


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) 


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) 


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^ 


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. 



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 2 = 




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 


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 


Z' Z' £'/(! - K) 

z- L 


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 


L-l/K K - 1 


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 



1 - K 


«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 ) 


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. 


Sec. 12-10 



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 



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 


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. 



Sec. 12-10 



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 



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 = 


1 A - *■'« 

1 — Avi — -5- 




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 - 





Fig. 12-16 Darlington emitter 



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 , , 


R»9 «* 

1 + hfe 

R, + hi e 

1 + h 




(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 



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 — 


1 - A, 


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. 


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 


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 

+o-j^AAA O — o VVW^ 


R t — RciWRti ' 



I A/,2 lb 





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 


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). 


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, 



°*ever, if y 2 = 0, then the load of Ql consists of h ib2 , which, from Table 11-3, 


Sec. 12. j j 

Fig. 12-19 The cascode configura- 
tion. (Supply voltages are not 

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 

Finally, for the reverse open-circuit voltage amplification, we have 




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 





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. 



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,). 


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 


A d = %(Ai — At) 


A e = A 1 + A- 


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 .. \ 


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. 




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^ 


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) 


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. 


Sec. J 2, 




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 . 



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 


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 ) 





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, 


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. 


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, 

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. 


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. 




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 



Sw, J 3. 

Fig. 13-1 Transistor T 
model at high fre- 

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, 




liK = 









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. 




then, from E( l- t 13 " 1 ). 





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 



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»., 



r@~ i 

f, 9- 13-2 The approximate high- 

•-Ayv/^—. — o — <-AA/V^' o 


quency T model. 


' r bH 

C c 



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 


vi + (f/f a y 

e = 


— arctan ~ 



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 




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. 


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 




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 % 





The emitter diffusion capacitance Cn, is given by the rate of change of Qb 
with respect to emitter voltage V, or 

Cd« = 


W* dl 

dV 2D B dV 

W 2 1 
2D s r' t 


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 


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 


s «- T3.4 

Dependence of /„ upon Base Width or Transit Time From Eqs. (13-2) 
and (13-11), and since C, » Cz>., then 


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 


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* 


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. 



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 


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 


j L = I b + I t , so that 1.(1 — a) = —I b . Finally, 


A - ? L - *!> _ 
" " h ~ ~h " 1 


Using Eq. (13-4), A ie may be put in the form 




ft = 



1 - 


U = /«(i — <*•) 





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 



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. 


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. 


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. 


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 




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 


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 




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 = — 


To evaluate r t , note from Eq. (9-19), with V c ~ —V C c, that 
Ib = a U € v * lv T — an — aii 



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 


: an* 

91 E 

aV E 

a U i v * ,Vr Is 




8- 13-d Pertaining to the derivation of 

B r w>* 

o J WV 


Substituting Eq. (13-24) in Eq. (13-22), we obtain 
ObIe Ico — Ic 

Qm = 


S«. I3. 4 


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 


V T 


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 


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 short-circuit current gain h f , is defined by 
J. I 

h/ e — 


— gmn't 


- hit — V*Y r . 



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 


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, 




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*. 



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 


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 



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 


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 


iiis equation may be put in the form [using Eqs. (13-29) and 

0« « & oe — 0mA r 


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, = 


v T 


5h ' e = /v: 

hf, — r !r 



h rt 


ffi'e = 


?« = 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 





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 «* 




Reasonable values for these capacitances are 
- - 3 pF C t = 100 pF 



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 


'9- 13-9 Approximate equivalent 
Clrcuit for the calculation of the 
■tort-circuit CE current gain. 



^ ^ C e + C c 





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 


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 = 




where the frequency at which the CE short-circuit current gain falls by 3 
dB is given by 





2r(C, + C e ) h f , 2rr(C, + C.) 


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, 

9m _ Qm 

2ir(C. + C t ) ~ 2*C t 

Hence, from Eq. (13-43), 




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 


fig, 13-10 Variation of f T with 
collector current. 

/V. MHz 
400 - 

300 - 



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). 


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 


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 = 




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. 


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 


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 


The input time constant is 



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. 

Alo = —hf« 

However, the 3-dB frequency is now / 2 (rather than //j), where 




n 2irr b >£ 2ttC 

C = C. + C.(l + gM 



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 



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. 


1 3-9 


and source impedances into account, is 

j. lL __ Jl Ii _ i Rt __ — kfeRt 

At* J - jr* j - ft/. ^ _j_ ^ + ^ ~ Rt + hu 


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, 





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 


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 


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 



R L = 


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. 


7 3-9 




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 





ft , MHz 



14 w 



-^R, = 50 










87.0 2. 1 7 
83.3 4.17 



74. 1 9.25 

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 . 


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* 


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. 


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 

The FET enjoys several advantages over the conventional 

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- 


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. 








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 



Depletion region 



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 


*! V n 




+ 0.5 !_ - 



1 / / 





- 2.5 1 





i it 15 20 

Drain -source voltage V !ls 



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 


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. 



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 




Fig. 14-4 Single-ended-geometry 
junction FET. 




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}* 


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 


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, 


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. 






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 = 



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 


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- 




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- 

+ ll~ or 

Depletion region 

I 1 ? Gl 

> — 

26(.r) ^Cs^is. 





I, x .1 

ll * 

-'1 + 



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 


This simple parabolic approximation gives an excellent fit, with the experi- 
mentally determined transfer characteristics for FETs made by the diffusion 

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. 



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) 


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° 





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 °* 


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. 


g m = 

Bid I 

At'p j 

— I 

V„ \Vbs 



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 = 


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 


M = Tdg„ 


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 


V x Rd 


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, 


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 



Sac, l4^f 




— -, Oscillator ^ 






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 = 


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 




l kHz 




X T\ 




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 



: * 

1 J 


50 tOO 150 

Ambient temperature T Al "C 




50 100 

Ambient temperature T A , "C 


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 

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 


Gate G o • 1( » 

Source So *■ 

♦ o Drain D 

5=*- u «i(p < r " SC 

i OS 


s «- J4.J 


Range of parameter values for a FET 






. 1-20 mA/V or more 


0.1-1 M 

1-50 K 


0,1-1 pF 

0.1-1 pF 

"an "ad 

1-10 pF 

1-10 pF 


>10* a 

>io i ° a 

T B d 

>io fl a 


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. 



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 

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, 




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. 



3iO.. channel 

Source i Aluminum 


P (substrain 


Fig. 14-12 (a) A depletion-type MOSFET. (b) Channel depletion with the appli- 
cation of a negative gate voltage. (Courtesy of Motorola Semiconductor 

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 


l B ,mA 

Depletion -« — 

J D (on) = 6 

— *- Enhancement 

^ -"""l 

.. : — . 1 . 

t - 3 - 2 

V& (OFF) 


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 




6 Source S 


6 S 

(a) (6) (C) 

Fig, 14-14 Three circuit symbols for a p-channel MOSFET. 


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. 



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 

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 


Y l 4- Yd, + Qd + Y 9d 


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 







— o 



Fit- 14-15 The three FET configurations: (a) CS, (b) CD, and (c) 





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 


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* 


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 




s £ V a 




F '9- 14-16 {a) The common-source amplifier circuit; (b) small-signal equiva- 
le "t circuit of CS amplifier. 


Sec. ?4.y 

output resistance R e is given by the parallel combination of r d and Rd, or 

R a = 

Td + Rd 


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 


= -48,6 


(4v)«„t.u,. = 38.8/143.3' 




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* 



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). 


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 = 


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,o )\- 

Fig. 14-17 Source-follower circuit. 

■o V a 



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 . 


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 • 




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 = 


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 : 


* ■ (ttt) "*■ 

S«c. H-9 



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 = 



P + 1 g m + gd 

which agrees with Eq. (14-29). The output impedance &„ taking R, into 
account, is R' = R \\R,. 



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„ 






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 . 


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,. 


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 > 



= 9K 


Sec. M-9 


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 


0.007|/i,| = O.OO220 n 


= 0.314 V 



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 



Id = h 


/0.63\ a 



Sec. ?4-9 


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 


6. From Eq. (14-38), 
Vm - -1.37 V 
c. Since Fes = —IdR, 

B. = 

K = 8.3 K 

d. From Eq. (14-40), we have 

ft. = 1.60 

\~) 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 


( + ) 

less f max) 

l oss (rnin) 

^ \ V 3 / 

j- Bias line 


\ ~ x — jSt^^ 


lo < + ) 

Bias line 


fr(max) V as 

V o0 

(+) -_o— •-<-) 




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* 


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 




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). 


Sec. I4»9 

in Fig. 14-236. 

V QG — 

For this circuit 


Ri 4- R2 

R m 


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. = 

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 


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. 



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 . 


F '9- 14-25 (o) Drain- 
'°-gate bias circuit for 
M °S transistors; (b) 
""proved version of (a) 


-v B 


Sec. W.Jo 


Q-V n 



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 

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. 




Fig. 14-28 Direct-coupled cascode 

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. 




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- 











J-,9N^91^ - 




Sec. M-Jj 



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)*] 


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 


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 



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*. 



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 


p~n junction- 



> Ohmic 
/ contacts 

LI 31 


Fig. 14-31 Unijunction transistor, (a) Constructional details; (b) 
circuit symbol. 


Sac. ?4.j 2 

S 10 

T^=2S C 

V M = 30V 




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- 

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). 


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. 


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 


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. 

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. 


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. 



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 




Q 3 




Diode junctions 


Silicon dioxide 
contact n* 

n* Emitter 



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 

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- 


$»c. 1 5. | 




Silicon dioxide 

i Isolation islands 

Sidewall C- 

Bottom C, 


xnesisior . 

Anode of diode /Base 



Cathodes of diodes 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 


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_| 


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 


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 



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: 


SiCU + 2H 2 » Si + 4HC1 


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) 



Induction coll 

Sec. 15. 


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 ) 



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. 



Si0 2 -" / 
Silicon chip 



/ I \ 

'S10 2 
■Silicon chip 



Fig. 15-5 Photoetching technique, (o) Masking and exposure to 
ultraviolet radiation, (b) The photoresist after development. 




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 



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 



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 






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 



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) - 




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° 





5x 10"' 


5x 10r» 


pig, 1 5-7 The complemen- 
tary error function plotted 

>> 5x10"' 



on semilogarithmic paper. 

io- J 

5x 10- 1 


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 

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-- 


" "'Ml 

a J 

1100 / 



\3 rrj 


ill "' 1 






u if 

, «]ttl| 

- IJ 

\ llllii 

- fcu 


\ II 

kj Silicon 

sb jjj 

J t 

500 1 




10" 10 JI 10™ 10" 10" 10" 10" 10" 

10" 10" 10" 
Atoms/cm 3 


Sec J 5 . 4 

Temperature, "C 
1300 1200 1100 1000 


8 io- 

s io-" - 








i Aluminum 

k-Gailiurr. " 

^_ Boron and 

v phosphorus 


' \ 

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) 


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 

S.c 15-4 


Quartz diffusion tube 

' — ■- 


Silicon wafers 


■ Furnace 

Liquid POC1 
Thermostated bath-n\ 


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) : 


= 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. 





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 


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 




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 
2 V Dt = — = 1.23 microns 


Hence the boron profile, given by Eq. (15-3), is 



N* = 5 X 10" erfc 


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~» 


= 1.0 X 10" cm"' 

The phosphorus concentration N B is given by 

iVp = 10" erfc — %=*. 

At a: = 2, JV, - AT B = 1.0 X 10", so that 


2 VDt 

1.0 X 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 — 

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. 





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 



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 
























Se c. fs^ 

Fig. 15-14 Utilization of 
"buried" n + layer to 
reduce collector series 

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. 



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). 

















|» V 


p substrate 



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 

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 

Q common anode. 



Sec. ?S.» 


S 6 

■§ 4 


t t J- 

{a)j (b)l (cy 


—J J--/ 






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. 



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 


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 


Fig. 15-19 Pertaining to sheet 
resistance, ohms per square. 

The resistance value may be computed from 
ff - pl - p * 

K — — lis — 

yw w 


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 


/] isolation region 

P substrate 











-o p layer 


Sec. I5.J 

Fig. 15-21 The equivalent circuit 
of a diffused resistor. 



-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. 



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 


C a ^0.2pF/mil J 



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 


J >?i 

»-r»C i 

p-type substrate 


9- 15-23 A MOS capacitor, (a) The structure and (b) the equivalent circuit. 


Sec. 15-9 

TABLE 75-2 Integrated capacitor parameters 


Diff used-junction 

Thin-film MOS 

Capacitance, pF/mil* 


2 X 10« 



2 X 10* 



Tolerance, percent , .'.' i - 


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. 


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 


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 

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( 
















© @ © i 


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. 


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 


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.) 


Flat package assembly 


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. 



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- 

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 


Source Drain 
S I) 



9 S / 

* Mrliilization 
W o Hate ( 


a substrate 

}> substrate 



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. 

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. 




1 * 


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. 



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 

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- 



•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 


Common- collector 
beam lead 

Base-resistor beam 


(one of four) 

Input resistor (one of four) 

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.) 


$«. 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. 


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, 

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. 


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. 



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 

We also discuss many topics associated with the general problem 
of amplification, such as the classification of amplifiers, hum and noise 
in amplifiers, etc. 



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 



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) 

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 

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. 


Sec. 1<J,j 



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. 


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 




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„ = 



Ri — jfwCi 1 — j/wRiCt 



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. 








-l Li- Y 1 1 1- 



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 


A - V '- 

h = 


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 







h = arctan 7- 



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. 


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 


* 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 


S * ld.4 

From C 

preceding 0-*~-| 


From d 

preceding O *■ |( 


From C, 

preceding O - |( O 



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**'" 




16-5 A schematic representa- 


tion of «'™er a tube, FET, or transis- 
tor stage. Biasing arrangements 
and suppfy voltages are not indi- 




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), 






<«) * 

^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 . 


the lower 3-dB frequency is 


2tt(R'„ + #;.)<?(, 

$<*. T^ 


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 

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 

&. 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 


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 


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 &* 




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 



2vRC 2irRJJ 


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 


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* = 



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. 




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. 



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. 




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 

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 

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 


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 


B' 3 


R c < r b .< 


A 1 


c e a-K) 

^'9- 16-9 The equivalent circuit of the enclosed stage of Fig. 16-8 (K = V^/V*,). 


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 = 



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 





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). 


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 


J 6-7 


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 — 


y/x- 1 



C, + C c r»' 


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. 

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 




\AJ 2 \ 
\A B \ 






IA 4 * 


r 1 






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. 


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 < 


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 




equal to the collector-circuit resistance R e of the preceding stage and with Rl 
equal to the R e of the last stage. 



\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. 



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 


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 . 


For times t which are small compared with the time constant R1C1, the response 
is given by 

F ( x -«k) 


From Fig. 16-12 we see that the output is tilted, and the percent tilt or aag 
in time h is given by 

P = 


X 100 = 


X 100% 


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% 



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 


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. 


*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 



to be 

f.OO . . 

J ~ = V2" R - 1 

Sec. 76-f0 


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 


V2 1 '" - 1 


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 


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. 



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 




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 


R, + k ie + Z' t 

z; - (1 + h f .) 


1 + jtaC.R, 



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' 


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 __ 




R R, 4* h ie 

1 1 +■?/ //» 
1 + R'/R) + jf/f P 


, = 1 + R'/R 

jp — 




°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. 


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 , 




1 + R'/R vT+T 


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% 















R'\ /' 





i i 

1.0 f„ 


1,000 /, Hz 

Fig. 16-14 The frequency response of on amplifier with a bypassed 
emitter resistor. The numerical values correspond to the component 
values given at the top of this page. 

Sec. 16-10 


Let us calculate the size of C x so that we may reproduce a 50- Hz square wave 
with a tilt of less than 10 percent. Using the parameters given above, we 


c* = 


F = 4,600 M F 

Such a large value of capacitance is impractical, and it must be concluded 
that if very small tilts are to be obtained for very low frequency signals, the 
emitter resistor must be left unbypassed. The flatness will then be obtained 
at the sacrifice of gain because of the degeneration caused by R t . If the loss in 
amplification cannot be tolerated, R t cannot be used. 

A Tube or FET Stage If the active device is a pentode (with r p y>R L + R k ) 
instead of a transistor, the equivalent circuit of Fig. 16-15 must be used. An 
analysis of this circuit (Prob. 16-30) yields 

Av _ 1 1+j///. 

A e " 



A = —g m Ri, f e = 



, m 1 + QmRh 


These equations are analogous to Eqs. (16-34) and (16-35), and the frequency 
response is of the form indicated in Fig. 16-14. If g m R k » 1, the pole and 
zero frequencies are widely separated, and hence /i « f p . Then, from Eq. 
(16-25), it follows that the percentage tilt to a square wave of frequency / is 

P = ^ X 100 = 

1 + 9mRk 


X 100 

2C k f 



Note that for g m R k » 1, P is independent of R k . If g m for a pentode is 5 mA/V 
(one-tenth that of a transistor), then for no more than a 10 percent output tilt 
with a 50-Hz square-wave input, the capacitor C* must be at least 

C k = 

5 X 10-" X 100 
2 X 50 X 10 

F = 500 mF 

The analysis of a FET stage (with r d » Rl + R t ) is identical with that for 
a pentode, except that C* and R k must be replaced by C, and R„ respectively. 

'9. 16-15 The equivalent circuit of a pen- 
°de stage with a cathode impedance. 



Practical Considerations Electrolytic capacitors are often used as 
emitter, cathode, or source bypass capacitors because they offer the greatest 
capacitance per unit volume. It is important to note that these capacitors 
have a series resistance which arises from the conductive losses in the electro- 
lyte. This resistance, typically 1 to 20 &, must be taken into account in com- 
puting the midband gain of the stage. 

If in a given stage both C» and the coupling capacitor C& are present, we 
can assume, first, C, to be infinite and compute the lower 3-dB frequency due 
to Cb alone. We then calculate /i due to C t by assuming C& to be infinite. 
If the two cutoff frequencies are significantly different (by a factor of more 
than four or five times), the higher of the two is approximately the lower 
3-dB frequency for the stage. 



It often happens that, with no apparent input signal to an amplifier, an output 
voltage of considerable magnitude may be obtained. The amplifier may be 
oscillating because some part of the output is inadvertently being fed back 
into the input. 

Parasitic Oscillations Feedback may occur through the interelectrode 
capacitance from input to output of the active device, through lead induct- 
ances, stray wiring, etc., the exact path ofte