Electronic Devices
and Circuits
MILLMAN & HALKIAS
INTERNATIONAL STUDENT EDI"
McGRAWHILL ELECTRICAL AND
ELECTRONIC ENGINEERING SERIES
Frederick Emmons Terman, Consulting Editor
W. W. Harmon and J. G. Truxal, Associate Consulting Edno.
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Aselline • Transform Method In Linear System Analyst*
Atwater  Introduction to Microwave Theory
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Cuccio ■ Harmonics, Sidebands, and Transients In Communication Engineering
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Eastman ■ Fundamentals of Vacuum Tubes
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Kraut  Electromagnetics
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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
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Millman • Vacuumtube and Semiconductor Electronics
Millman and Hatktat • Electronic Devices ond Circuits
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Moore • Travelingwave 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
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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
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Spangenberg ■ Fundamentals of Electron Devices
Spang enberg ■ Vacuum Tubes
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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 • Alternatingcurrent and Transient Circuit Analysis
Tou  Digltol and Sampleddata Control Systems
Tou  Modem Control Theory
Trvxal ■ Automatic Feedback Control System Synthesis
Turtle ■ Electric Networks: Analysis and Synthesis
Vatdet • The Physical Theory of Transistors
Van Model • Electromagnetic Fields
Weinberg • Network Analysis ond Synthesis
Williams and Young ■ Electrical Engineering Problems
ELECTRONIC DEVICES
AND CIRCUITS
Jacob Millman, Ph.D.
Professor of Electrical Engineering
Columbia University
Christos C. Halkias, Ph.D.
Associate Professor of Electrical Engineering
Columbia University
INTERNATIONAL STUDENT EDITION
McGRAWHILL BOOK COMPANY*
New York St. Louis San Francisco Diisseldorf
London Mexico Panama Sydney Toronto
KOGAKUSHA COMPANY, LTD.
Tokyo
ELECTRONIC DEVICES AND CIRCUITS
INTERNATIONAL STUDENT EDITION
Exclusive rights by Kogokusha Co., Ltd., for manufacture
and export from Japan. This book cannot be reexported
from the country to which it it coniigned by Kogakusha
Co., Ltd., or by McGrawHill Book Company or any of iti
subsidiaries.
XI
Copyright © 1967 by McGrawHill, 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 o716934
TOSHO J'HINTINQ CO., LTD., TOKYO, JAPAN
PREFACE
This book, intended as a text for a first course in electronics for elec
trical engineering or physics students, has two primary objectives: to
present a clear, consistent picture of the internal physical behavior of
many electronic devices, and to teach the reader how to analyze and
design electronic circuits using these devices.
Only through a study of physical electronics, particularly solid
state science, can the usefulness of a device be appreciated and its
limitations be understood. From such a physical study, it is possible
to deduce the external characteristics of each device. This charac
terization allows us to exploit the device as a circuit element and to
determine its largesignal (nonlinear) behavior. A smallsignal
(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 smallsignal models, are made for a
wide variety of amplifiers.
A great deal of attention is paid to the effects of feedback on
input and output resistance, nonlinear distortion, frequency response,
and the stabilization of voltage or current gains of the various devices
and circuits studied. la order that the student appreciate the different
applications of these circuits, the basic building blocks (such as untuned
amplifiers, power amplifiers, feedback amplifiers, oscillators, and power
suppliers) are discussed in detail.
For the most part, real (commercially available) device charac
teristics are employed. In this way the reader may become familiar
with the order of magnitude of device parameters, the variability of
these parameters within a given type and with a change of temperature,
the effect of the inevitable shunt capacitances in circuits, and the effect
of input and output resistances and loading on circuit operation. These
vii
viii / PREFACE
considerations are of utmost importance to the student or the practicing engi
neer since the circuits to be designed must function properly and reliably in
the physical world, rather than under hypothetical or ideal circumstances.
There are over 600 homework problems, which will test the student's
grasp of the fundamental concepts enunciated in the book and will give him
experience in the analysis and design of electronic circuits. In almost all
numerical problems realistic parameter values and specifications have been
chosen. An answer book is available for students, and a solutions manual
may be obtained from the publisher by an instructor who has adopted the text.
This book was planned originally as a second edition of Millman's
"Vacuumtube and Semiconductor Electronics" (McGrawHill 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 solidstate devices, particularly the bipolar tran
sistor. In recognition of the growing importance of integrated circuits and
the fieldeffect 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,
siliconcontrolled rectifiers, polyphase rectifiers, tuned amplifiers, modulation,
or detection circuits. The companion volume to this book, Millman and
Taub's "Pulse, Digital, and Switching Waveforms" (McGrawHill 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 selfstudy and that the practicing engineer will find the text
useful in updating himself in this fastmoving 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 steadystate 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.
(McGrawHill Book Company, New York, 1965) in connection with Chapter
15. We express our particular appreciation to Miss S. Silverstein, adminis
trative assistant of the Electrical Engineering Department of The City College,
for her most skillful service in the preparation of the manuscript. We also
thank J. T. Millman and S. Thanos for their assistance.
Jacob Millman
Christos C. Halkias
CONTENTS
Preface
Electron Ballistics and Applications 1
11 Charged Particles 1
12 The Force on Charged Particles in an Electric Field
13 Constant Electric Field S
14 Potential 6
15 The eV Unit of Energy 7
16 Relationship between Field Intensity and Potential
17 Twodimensional Motion 8
18 Electrostatic Deflection in a Cathoderay Tube 10
19 The Cathoderay Oscilloscope 12
110 Relativistic Variation of Mass with Velocity IS
111 Force in a Magnetic Field 15
112 Current Density 16
113 Motion in a Magnetic Field 17
114 Magnetic Deflection in a Cathoderay Tube 20
115 Magnetic Focusing 21
116 Parallel Electric and Magnetic Fields 24
117 Perpendicular Electric and Magnetic Fields 26
118 The Cyclotron SI
Energy Levels and Energy Bands 36
21 The Nature of the Atom 36
22 Atomic Energy Levels S8
23 The Photon Nature of Light 40
24 Ionization <{0
25 Collisions of Electrons with Atoms 41
26 Collisions of Photons with Atoms 41
27 Metastable States 42
28 The Wave Properties of Matter 48
29 Electronic Structure of the Elements 45
210 The Energyband Theory of Crystals 47
211 Insulators, Semiconductors, and Metals 49
ix
x / CONTENTS
3
5
6
Conduction in Metals 52
31 Mobility and Conductivity 62
32 The Energy Method of Analyzing the Motion of a
Particle 54
33 The Potentialenergy Field in a Metal 57
34 Bound and Free Electrons 69
35 Energy Distribution of Electrons 60
36 The Density of States 86
37 Work Function 68
38 Thermionic Emission 69
39 Contact Potential 70
310 Energies of Emitted Electrons 71
31 1 Accelerating Fields 74
312 Highfield Emission 76
313 Secondary Emission 75
Vacuumdiode Characteristics 77
41 Cathode Materials 77
42 Commercial Cathodes 80
43 The Potential Variation between the Electrodes
44 Spacecharge Current 82
45 Factors Influencing Spacecharge Current 86
46 Diode Characteristics 87
47 An Ideal Diode versus a Thermionic Diode
48 Rating of Vacuum Diodes 89
49 The Diode as a Circuit Element 90
*» 7
80
88
8
Conduction in Semiconductors 95
51 Electrons and Holes in an Intrinsic Semiconductor 96
52 Conductivity of a Semiconductor 97
53 Carrier Concentrations in an Intrinsic Semiconductor 99
54 Donor and Acceptor Impurities 108
55 Charge Densities in a Semiconductor 105
56 Fermi Level in a Semiconductor Having Impurities 105
57 Diffusion 107
58 Carrier Lifetime 108
59 The Continuity Equation 109
510 The Hall Effect 113
Semiconductordiode Characteristics 115
61
62
63
64
65
66
67
68
115
Qualitative Theory of the pn Junction
The pn Junction as a Diode 117
Band Structure of an Opencircuited pn Junction
The Current Components in a pn Diode 12$
Quantitative Theory of the pn Diode Currents
The VoltAmpere Characteristic 127
The Temperature Dependence of pn Characteristics
Diode Resistance 1S2
9
120
124
ISO
CONTENTS / xi
69 Spacecharge, or Transition, Capacitance CV
610 Diffusion Capacitance 138
611 pn Diode Switching Times 140
612 Breakdown Diodes 148
613 The Tunnel Diode 147
614 Characteristics of a Tunnel Diode 153
134
156
166
175
Vacuumtube Characteristics 156
71 The Electrostatic Field of a Triode
72 The Electrode Currents 159
73 Commercial Triodes 161
74 Triode Characteristics 162
75 Triode Parameters 16$
76 Screengrid Tubes or Tetrodes
77 Pentodes 169
78 Beam Power Tubes 1 71
79 The Triode as a Circuit Element 173
710 Graphical Analysis of the Groundedcathode Circuit
711 The Dynamic Transfer Characteristic 178
712 Load Curve. Dynamic Load Line 179
713 Graphical Analysis of a Circuit with a Cathode
Resistor 181
714 Practical Cathodefollower Circuits 184
Vacuumtube Smallsignal Models and Applications 187
81 Variations from Quiescent Values 187
82 Voltagesource Model of a Tube 188
83 Linear Analysis of a Tube Circuit 190
84 Taylor's Series Derivation of the Equivalent Circuit
85 Currentsource Model of a Tube 196
86 A Generalized Tube Amplifier 197
87 The Thevenin's Equivalent of Any Amplifier 199
88 Looking into the Plate or Cathode of a Tube 200
89 Circuits with a Cathode Resistor 204
810 A Cascode Amplifier 207
811 Interelectrode Capacitances in a Triode 209
81 2 Input Admittance of a Triode 211
813 Interelectrode Capacitances in a Multielectrode
Tube 215
814 The Cathode Follower at High Frequencies 216
194
Transistor Characteristics 220
91 The Junction Transistor 220
92 Transistor Current Components 222
93 The Transistor as an Amplifier 225
94 Transistor Construction 226
95 Detailed Study of the Currents in a Transistor
96 The Transistor Alpha 230
97 The Commonbase Configuration 23 1
227
xlf / CONTENTS
CONTENTS / xitt
M. 10
11
12
98
99
910
911
912
913
914
915
916
917
918
The Commonemitter Configuration 234
The CE Cutoff Region 237
The CE Saturation Region 239
Largesignal, DC, and Smallsignal CE Values of Current
Gain 242
The Commoncollector Configuration 243
Graphical Analysis of the CE Configuration 244
Analytical Expressions for Transistor Characteristics £47
Analysis of Cutoff and Saturation Regions 251
Typical Transistorjunction Voltage Values 256
Transistor Switching Times 267
Maximum Voltage Rating 260
Transistor Biasing and Thermal Stabilization 263
101 The Operating Point 263
102 Bias Stability 285
103 CollectortoBase Bias 268
104 Selfbias, or Emitter Bias 271
105 Stabilization against Variations in Vbe and § for the
Selfbias Circuit 276
106 General Remarks on Collectorcurrent Stability 280
107 Bias Compensation 28S
108 Biasing Circuits for Linear Integrated Circuits 285
109 Thermistor and Sensistor Compensation 287
1010 Thermal Runaway 288
1011 Thermal Stability 290
Smallsignal Lowfrequency Transistor Models 294
111 Twoport Devices and the Hybrid Model 294
112 Transistor Hybrid Model 296
1 13 Determination of the h Parameters from the
Characteristics 298
114 Measurement of h Parameters 302
115 Conversion Formulas for the Parameters of the Three
Transistor Configurations 305
116 Analysis of a Transistor Amplifier Circuit Using h
Parameters S07
117 Comparison of Transistor Amplifier Configurations 312
118 Linear Analysis of a Transistor Circuit 316
119 The Physical Model of a CB Transistor S16
1110 A VacuumtubeTransistor Analogy 319
Low frequency Transistor Amplifier Circuits 323
121 Cascading Transistor Amplifiers 323
122 nstage Cascaded Amplifier 327
123 The Decibel 332
124 Simplified Commonemitter Hybrid Model 333
125 Simplified Calculations for the Commoncollector
Configuration 335
13
126 Simplified Calculations for the Commonbase
Configuration SS9
127 The Commonemitter Amplifier with an Emitter
Resistance 340
128 The Emitter Follower 346
129 Miller's Theorem 348
1210 High inputresistance Transistor Circuits 350
1211 The Cascode Transistor Configuration 366
1212 Difference Amplifiers 357
The Highfrequency Transistor 363
131 The Highfrequency T Model 363
132 The Commonbase Shortcircu itcurrent Frequency
Response 366
133 The Alpha Cutoff Frequency 366
134 The Commonemitter Shortcircuitcurrent Frequency
Response S68
135 The Hybridpi (n) Common emitter Transistor
Model 369
136 Hybrid pi Conductances in Terms of Lowfrequency
h Parameters 371
137 The CE Shortcircuit Current Gain Obtained with the
Hybridpi Model 376
138 Current Gain with Resistive Load S78
139 Transistor Amplifier Response, Taking Source
Resistance into Account 380
14
Fieldeffect Tronsistors
384
390
15
141 The Junction Fieldeffect Transistor
142 The Pinchoff Voltage V P 388
143 The JFET VoltAmpere Characteristics
144 The FET Smallsignal Model 392
145 The Insulatedgate FET (MOSFET) 396
146 The Commonsource Amplifier 400
147 The Commondrain Amplifier, or Source Follower
148 A Generalized FET Amplifier 403
149 Biasing the FET 406
1410 UnipolarBipolar Circuit Applications 4**
1411 The FET as a Voltagevariable Resistor (WE) 4*$
1412 The Unijunction Transistor 415
Integrated Circuits 418
151 Basic Monolithic Integrated Circuits 418
152 Epitaxial Growth 428
153 Masking and Etching 4%4
154 Diffusion of Impurities 4&5
155 Transistors for Monolithic Circuits 430
156 Monolithic Diodes 4$4
157 Integrated Resistors 436
402
xlv / CONTENTS
*» 16
17
18
158 Integrated Capacitors and Inductors 488
159 Monolithic Circuit Layout 440
1510 Integrated FieJdeffect Transistors 444
1511 Additional Isolation Methods 449
Untuned Amplifiers 450
161
162
163
164
165
166
167
168
169
1610
1611
1612
Classification of Amplifiers 460
Distortion in Amplifiers 46$
Frequency Response of an Amplifier 462
The ACcoupled Amplifier 455
Lowfrequency Response of an /eCooupled Stage 467
Highfrequency Response of a Vacuumtube 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)wfrequency Response 468
Spurious Input Voltages 472
Noise 47S
Feedback Amplifiers and Oscillators 480
171
172
173
174
175
176
177
178
179
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
Classification of Amplifiers 48O
The Feedback Concept 48S
General Characteristics of Negativefeedback
Amplifiers 488
Effect of Negative Feedback upon Output and Input
Resistances 491
Voltageseries Feedback 498
A Voltageseries Feedback Pair 602
Currentseries Feedback 604
Currentshunt Feedback 508
Voltageshunt 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 Phaseshift Oscillator 628
Resonantcircuit Oscillators 680
A General Form of Oscillator Circuit 582
Crystal Oscillators 686
Frequency Stability 687
Negative Resistance in Oscillators 588
Largesignal Amplifiers 542
181 Class A Largeaignal Amplifiers 542
182 Secondharmonic Distortion 644
183 Higherorder Harmonic Generation 546
CONTENTS / xv
184 The Transformercoupled Audio Power Amplifier
185 Power Amplifiers Using Tubes 558
186 Shift of Dynamic Load Line 556
187 Efficiency 556
188 PushPull Amplifiers 668
189 Class B Amplifiers 660
1810 Class AB Operation 564
549
**. 19
Photoelectric Devices 566
191 Photocmissivity 666
192 Photoelectric Theory 568
193 Definitions of Some Radiation Terms 571
194 Phototubes 578
195 Applications of Photodevices 575
196 Multiplier Phototubes 678
197 Photoconductivity 580
198 The Semiconductor Photodiode 588
199 Multiplejunction Photodiodes 586
1910 The Photovoltaic Effect 687
20
Rectifiers and Power Supplies 592
201 A Halfwave Rectifier 692
202 Ripple Factor 597
203 A Fullwave Rectifier 698
204 Other Fullwave Circuits 600
205 The Harmonic Components in Rectifier Circuits
206 Inductor Filters 603
207 Capacitor Filters 606
208 Approximate Analysis of Capacitor Filters 609
209 Lsection Filter 611
2010 Multiple Lsection Filter 616
201 1 11section Filter 617
2012 flsection Filter with a Resistor Replacing the
Inductor 620
2013 Summary of Filters 621
2014 Regulated Power Supplies 621
2015 Series Voltage Regulator 623
2016 Vacuumtuberegulated Power Supply 629
602
Appendix A Probable Values of General Physical
Constants 633
Appendix B Conversion Factors and Prefixes 634
Appendix C Periodic Table of the Elements 635
Appendix D Tube Characteristics 636
Problems 641
Index 745
1
ELECTRON BALLISTICS
AND APPLICATIONS
In this chapter we present the fundamental physical and mathemati
cal theory of the motion of charged particles in electric and magnetic
fields of force. In addition, we discuss a number of the more impor
tant electronic devices that depend on this theory for their operation.
The motion of a charged particle in electric and magnetic fields is
presented, starting with simple paths and proceeding to more complex
motions. First a uniform electric field is considered, and then the
analysis is given for motions in a uniform magnetic field. This dis
cussion is followed, in turn, by the motion in parallel electric and mag
netic fields and in perpendicular electric and magnetic fields.
11 CHARGED PARTICLES
The charge, or quantity, of negative electricity of the electron has
been found by numerous experiments to be 1.602 X 10  " C (coulomb).
The values of many important physical constants are given in Appen
dix A. Some idea of the number of electrons per second that repre
sents current of the usual order of magnitude is readily possible. For
example, since the charge per electron is 1.602 X 10~ 19 C, the number
of electrons per coulomb is the reciprocal of this number, or approxi
mately, 6 X 10 18 . Further, since a current of 1 A (ampere) is the flow
of 1 C/sec, then a current of only 1 pA (1 picoampere, or 10 12 A)
represents the motion of approximately 6 million electrons per second.
Yet a current of 1 pA is so small that considerable difficulty is experi
enced in attempting to measure it.
In addition to its charge, the electron possesses a definite mass.
A direct measurement of the mass of an electron cannot be made, but
the ratio e/m of the charge to the mass has been determined by a
1
2 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 72
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, onesixteenth 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 Wavemechanical 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 largescale phenomena, such as electronic trajectories in a vacuum
tube, the classical model yields accurate results. For smallscale 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.
12
THE FORCE ON CHARGED PARTICLES IN AN ELECTRIC FIELD
The force on a unit positive charge at any point in an electric field is, by definition,
the electric field intensity £ at that point. Consequently, the force on a positive
charge q in an electric field of intensity £ is given by q£, the resulting force
Sec. 73
ELECTRON BALLISTICS AND APPLICATIONS / 3
being in the direction of the electric field. Thus,
(11)
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 (meterkilogramsecond) 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. (11), must be related to the mass and the acceleration
of the particle by Newton's second law of motion. Hence
dt
(12)
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 (13)
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.
13
CONSTANT ELECTRIC FIELD
Suppose that an electron is situated between the two plates of a parallelplate
capacitor which are contained in an evacuated envelope, as illustrated in Fig.
11 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 ( =
(14)
4 / ELECTRONIC DEVICES AND CIRCUITS
Sac. 73
■i d
e*
Fig. 11 The onedimenstona) 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 onedimensional, and the electron moves along the X axis.
Newton's law applied to the X direction yields
or
e£ = 7tta x
68
a, = — = const
m
d5)
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 wellknown expressions for the velocity and displacement, viz.,
v. = tv, + aj. x = x„ + v OI t + lad*
d6)
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. (16) are no longer valid. Under these
circumstances the motion is determined by integrating the equations
dV;
dl
and
dx
dl
= v x
(17)
These are simply the definitions of the acceleration and the velocity, respec
tively. Equations (16) follow directly from Eqs. (17) by integrating the
latter equations subject to the condition of a constant acceleration.
Sec 14
ELECTRON BALLISTICS AND APPLICATIONS / 5
EXAMPLE An electron starts at rest on one plate of a planeparallel 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. 11. The magnitude of the electric field intensity is
a. 6 =
whence
— — X — = 2 X 10 9 *
5 X 10* 10" 7
V/m
a* = ^   = — = (1.76 X 10»)(2 X 10»()
at m M
= 3.52 X 10 M ( m/sec J
Upon integration, we obtain for the speed
v, = T a x dt = 1.76 X 10*V
At t = 5 X 10~ a sec, v x = 4.40 X 10* m/sec.
6. Integration of v x with respect to (, subject to the condition that x =
when t = 0, yields
x m j* Vz dt = P 1.76 X \0*H*dt = 5.87 X 10 ,9 f 3
At t m 5 X 10"" sec, x = 7.32 X 10~ 3 m = 0.732 cm,
c. To find the speed with which the electron strikes the positive plate, we
first find the time t it takes to reach that plate, or
/ x Y / 0.05 Y
[ 1 = f J  9.46 X 10'
\5.87 X 10'7 \5.87 X 10'V
Hence
1.76 X 10 M / S = 1.76 X 10»°(9.46 X 10" 8 )*  1.58 X 10« m/sec
14
POTENTIAL
The discussion to follow need not be restricted to uniform fields, but £ x may
be a function of distance. However, it is assumed that E x is not a function
t 1 /^ec = 1 microsecond = 10~»sec. 1 nsec = 1 nanosecond = 10  *sec. Conversion
factors and prefixes are given in Appendix B.
6 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 14
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* (18)
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 (19)
By virtue of Eq. (19), Eq. (18) integrates to
eV = §m(v x *  *,*) (110)
where the energy eV is expressed in joules. Equation (110) 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. (110) 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. (110) becomes
qVzA = fymA* — £wu>s*
(111)
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 lefthand side of Eq.
(111) is the rise in potential energy from A to B. The righthand side repre
sents the drop in kinetic energy from A to B. Thus Eq. (111) 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. 15
ELECTRON BALLISTICS AND APPLICATIONS / 7
It must be emphasized that Eq. (111) 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. (111) with v A — 0,
v B = v, and V B a = V, is
or
M
v = 5.93 X 10 6 F*
d12)
(113)
Thus, if an electron "falls" through a difference of only 1 V, its final speed
is 593 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. (113) 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. (111)] must be used.
15
THE eV UNIT OF ENERGY
The joule (J) is the unit of energy in the mks system. In some engineering
power problems this unit is very small, and a factor of 10 3 or 10 8 is introduced
to convert from watts (1 W = 1 J/sec) to kilowatts or megawatts, respectively.
However, in other problems, the joule is too large a unit, and a factor of 10~ 7
is introduced to convert from joules to ergs. For a discussion of the energies
involved in electronic devices, even the erg is much too large a unit. This
statement is not to be construed to mean that only minute amounts of energy
can be obtained from electron devices. It is true that each electron possesses
a tiny amount of energy, but as previously pointed out (Sec. 11), 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 electronvolt unit may be used for any type
of energy, and is not restricted to problems involving electrons.
The abbreviations MeV and BeV are used to designate 1 million and
1 billion electron volts, respectively.
8 / ELECTRONIC DEVICES AND CIRCUITS
Sec, 16
16 RELATIONSHIP BETWEEN FIELD INTENSITY AND POTENTIAL
The definition of potential is expressed mathematically by Eq. (19). 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. 11 is
given by
— V V
£ * = x^J a = ~d (114)
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. (19). We obtain
dV
ax
(115)
The minus sign shows that the electric field is directed from the region of
higher potential to the region of lower potential.
17
TWODIMENSIONAL MOTION
Suppose that an electron enters the region between the two parallel plates of a
parallelplate capacitor which are oriented as shown in Fig. 12 with an initial
velocity in the f X direction. It will again be assumed that the electric field
between the plates is uniform. Then, as chosen, the electric field £ is in the
direction of the — Y axis, no other fields existing in this region.
The motion of the particle is to be investigated, subject to the initial
conditions
fz = %
x =
(116)
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, 12 Twodimensional electronic motion
in a uniform electric field.
Sec. 17
ELECTRON BALLISTICS AND APPLICATION > / 9
zero. Hence the component of velocity in the Z direction remains constant.
Since the initial velocity in this direction is assumed to be zero, the motion
must take place entirely in one plane, the plane of the paper.
For a similar reason, the velocity along the X axis remains constant and
equal to v ox . That is,
H = Mm
from which it follows that
x = v ex t (117)
On the other hand, a constant acceleration exists along the Y direction, and
the motion is given by Eqs. (16), with the variable x replaced by y;
where
v y = a v t
e£„
Oy = =
m
V = W
md
(118)
(119)
and where the potential across the plates is V = V d . These equations indi
cate that in the region between the plates the electron is accelerated upward,
the velocity component v v varying from point to point, whereas the velocity
component v x remains unchanged in the passage of the electron between the
plates.
The path of the particle with respect to the point is readily determined
by combining Eqs. (117) and (118), the variable ( being eliminated. This
leads to the expression
^2 »«y
(120)
which shows that the particle moves in a parabolic path in the region between
the plates.
EXAMPLE Hundredvolt electrons are introduced at A into a uniform electric
field of 10* V/m, as shown in Fig. 13. 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. 13 Parabolic path of an electron in
a uniform electric field.
J^
10 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 78
Solution The path of the electrons will be a parabola, as shown by the dashed
curve in Fig. 13, 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. (113).
R, = 5.93 X 10 s s/lOO = 5.93 X 10 s m/sec
Since the speed along the X direction is constant, the distance AB = xte given by
x = (v„ cos 6)t = (5.93 X 10 fi cos 0)(4.77 X 10~») = 2.83 X 10~ 2 cos 8
Hence we first must find 8 before we can solve for x. Since the acceleration a, in
the Y direction is constant, then
y = (v sin 8)t — ^Oyt*
and y = at point B, or
v„ sin 9
*i?"i©
 1(1.76 X 10") (10*) (4.77 X 10"*)  4.20 X 10* m/sec
and
, . 4.20 X 10 s AmM
a. x = 2.83 X 10* X 0.707 = 2.00 X 10"* m = 2.00 cm
18 ELECTROSTATIC DEFLECTION IN A CATHODERAY TUBE
The essentials of a cathoderay tube for electrostatic deflection are illustrated
in Fig. 14. The hot cathode A' emits electrons whieh are accelerated toward
the anode by the potential V a . Those electrons which are not collected by
the anode pass through the tiny anode hole and strike the end of the glass
envelope. This has been coated with a material that fluoresces when bom
Anode
Cathode
■*kS r
Verticaldeflecting
plates
+ V d + u
s
Fluorescent screen
Fig. 14 Electrostatic deflection in a cathoderay tube.
See. 78
ELECTRON BALLISTICS AND APPLICATIONS / II
barded by electrons. Thus the positions where the electrons strike the screen
are made visible to the eye. The displacement D of the electrons is deter
mined by the potential V d (assumed constant) applied between the delecting
plates, as shown. The velocity v ox with which the electrons emerge from the
anode hole is given by Eq. (112), viz.,
\ m
(121)
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 straightline
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. (120). 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 fieldfree.
The straightline 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. (120) 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
(122)
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 (122) reduces to
ttjju
T
D =
By inserting the known values of ay ( = eV d /dm) and v ox , this becomes
lLV d
D 
2dV a
(123)
This result shows that the deflection on the screen of a cathoderay tube is
directly proportional to the deflecting voltage V d applied between the plates.
Consequently, a cathoderay tube may be used as a linearvoltage indicating
device.
The electrostaticdeflection sensitivity of a cathoderay tube is defined as
12 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 19
the deflection (in meters) on the screen per volt of deflecting voltage. Thus
B D IL
S = V d = 2dV a (124)
An inspection of Eq. (124) 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. (124). Typical measured values of sensitivity are 1.0 to
0.1 mm/V, corresponding to a voltage requirement of 10 to 100 V to give a
deflection of 1 cm.
U9 THE CATHODERAY OSCILLOSCOPE
An electrostatic tube has two sets of deflecting plates which are at right angles
to each other in space (as indicated in Fig. 16). These plates are referred to
as the verticaldeflection and horizontaldeflection 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. 16 is impressed across the
horizontaldeflection plates. Since this voltage is used to sweep the electron
beam across the screen, it is called a sweep voltage. The electrons are deflected
Verticaldeflection
plates
Horizontal deflection
plates
Vertical
signal
voltage v.
Horizontal
sawtooth
voltage
Electron beam
Ftg. 15 A waveform to be displayed on the screen of a
cathoderay tube is applied to the verticaldeflection plates,
and simultaneously a sawtooth voltage is applied to the hori
zontaldeflection plates.
Sec. 170
ELECTRON BALLISTICS AND APPLICATIONS / 13
Voltage
Fig. 1 6 Sweep or sawtooth voltage
for a cathoderay tube.
Time
linearly with time in the horizontal direction for a time T. Then the beam
returns to its starting point on the screen very quickly as the sawtooth voltage
rapidly falls to its initial value at the end of each period.
If a sinusoidal voltage is impressed across the verticaldeflection 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 cathoderay 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. 15. The sensitivity is greatly increased by means
of a highgain 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.
110
RELATIVISTIC VARIATION OF MASS WITH VELOCITY
The theory of relativity postulates an equivalence of mass and energy accord
ing to the relationship
W = mc* (125)
where W = total energy, J
m = mass, kg
c = velocity of light in vacuum, m/sec
t Superscript numerals are keyed to the References at the end of the chapter.
T4 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 770
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*
(126)
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'
(127)
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. (126)
and (127), the decrease in potential energy, or equivalently, the increase in
kinetic energy, is
eV = m** ( X  i\
(128)
This expression enables one to find the velocity of an electron after it has
fallen through any potential difference F. By defining the quantity v x as the
velocity that would result if the relativistic variation in mass were neglected,
i.e.,
J2eV
(128) can be solved for v, the true velocity of the particle. The
Vn =
(129)
then Eq.
result is
v = c
1 
1
"li
(130)
(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. (130). The result becomes
'*MfrW*'")
(131)
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. IW
ELECTRON BALLISTICS AND APPLICATIONS / 15
S«
if the speed of the electron is onetenth of the speed of light, Eq. (131) shows
that an error of only threeeighths 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. (130)
and the true value of v then calculated. In cases where the speed is not too
great, the simplified expression (131) may be used.
The accelerating potential in highvoltage cathoderay 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 xray tubes, the cyclotron,
and other particleaccelerating machines. Unless specifically stated otherwise,
nonrelativistic conditions are assumed in what follows.
111
FORCE IN A MAGNETIC FIELD
To investigate the force on a moving charge in a magnetic field, the well
known motor law is recalled. It has been verified by experiment that, if a
conductor of length L, carrying a current of /, is situated in a magnetic field of
intensity B, the force /„ acting on this conductor is
/.  BIL
(132)
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 (132) 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. 17. 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. 17.
If the particle under consideration is a positive ion, then I is to be taken along
the direction of its motion. This is so because the conventional direction of
the current is taken in the direction of flow of positive charge. If the current
's due to the flow of electrons, the direction of I is to be taken as opposite to
the direction of the motion of the electrons. If, therefore, a negative charge
t One weber per square meter (also called a testa) equals 10* G. A unit of more prac
tical size in most applications is the milliweber per square meter (mWb/m 1 ), which equals
10 G. Other conversion factors are given in Appendix B.
16 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 112
Sec. M3
ELECTRON BALLISTICS AND APPLICATIONS / 17
L"
o.^
90'
T
Fig. 17 Pertaining to the determination of the direc
tion of the force f m on a charged particle in a
magnetic field.
lorv*
moving with a velocity v~ is under consideration, one must first draw I anti
parallel to v~ as shown and then apply the "direction rule."
If N electrons are contained in a length L of conductor (Fig. 18) and if
it takes an electron a time T sec to travel a distance of L m in the conductor,
the total number of electrons passing through any cross section of wire in
unit time is N/T. Thus the total charge per second passing any point, which,
by definition, is the current in amperes, is
T = *!£
T
(133)
The force in newtons on a length L m (or the force on the N conduction charges
contained therein) is
BIL =
BNeL
Furthermore, since L/T is the average, or drift, speed v m/sec of the electrons,
the force per electron is
f» = eBv (134)
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. 17. f
112 CURRENT DENSITY
Before proceeding with the discussion of possible motions of charged particles
in a magnetic field, it is convenient to introduce the concept of eurrent density.
t In the crosaproduct notation of vector analysis, f m m eB x v~. For a positive ion
moving with a velocity v + , the force Is f m = ev+ X B.
m
N electrons
D
i
Fig. T8 Pertaining to the determination of the
magnitude of the force f m on a charged particle
in a magnetic field.
This concept is very useful in many later applications. By definition, the
current density, denoted by the symbol J, is the current per unit area of the
conducting medium. That is, assuming a uniform current distribution,
"i
(135)
where J is in amperes per square meter, and A is the crosssectional area (in
meters) of the conductor. This becomes, by Eq. (133),
r _ N *
J TA
But it has already been pointed out that T — L/v. Then
_ _ Nev
J ~LA
(136)
From Fig. 18 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
(137)
N
n = LA
and Eq. (136) reduces to
J = nev = pv (138)
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. 18 does not necessarily represent a wire conductor. It
may represent equally well a portion of a gaseousdischarge tube or a volume
element in the spacecharge 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. (138).
113
MOTION IN A MAGNETIC FIELD
The path of a charge particle that is moving in a magnetic field is now investi
gated. Consider an electron to be placed in the region of the magnetic
field. If the particle is at rest, /„ = and the particle remains at rest. If
the initial velocity of the particle is along the lines of the magnetic flux,
there is no force acting on the particle, in accordance with the rule associated
with Eq. (134). Hence a particle whose initial velocity has no component
normal to a uniform magnetic field will continue to move with constant speed
along the lines of flux.
18 / ELECTRONIC DEVICES AND CIRCUITS
Sec. L13
Fieldfree
region
x
X
K Magnetic
field Into
* paper
Fig. 19 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. 19. Since the force f m
is perpendicular to v and so to the motion at every instant, no work is done
on the electron. This means that its kinetic energy is not increased, and
so its speed remains unchanged. Further, since v and B are each constant
in magnitude, then f m is constant in magnitude and perpendicular to the
direction of motion of the particle. This type of force results in motion in a
circular path with constant speed. It is analogous to the problem of a mass
tied to a rope and twirled around with constant speed. The force (which
is the tension in the rope) remains constant in magnitude and is always directed
toward the center of the circle, and so is normal to the motion.
To find the radius of the circle, it is recalled that a particle moving in
a circular path with a constant speed v has an acceleration toward the center
of the circle of magnitude v 3 /R, where R is the radius of the path in meters.
Then
from which
The corresponding angular velocity in radians per second is given by
_ v_ _ eB
R m
The time in seconds for one complete revolution, called the period, is
m _ 2t __ 2irni
cd eB
For an electron, this reduces to
3.57 X 10 11
T =
B
(139)
(140)
(141)
(142)
In these equations, e/m is in coulombs per kilogram and B in webers per square
meter.
S*c. 113
ELECTRON BALLISTICS AND APPLICATIONS / T9
It is noticed that the radius of the path is directly proportional to the
speed of the particle. Further, the period and the angular velocity are inde
pendent of speed or radius. This means, of course, that fastermoving 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 magneticfocusing apparatus.
EXAMPLE Calculate the deflection of a cathoderay 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. 110).
Solution According to Eq. (113), 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. (139) 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. 110 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 cathodebeam spot in a lowvoltage cathoderay tube. If
Fig. 110 The circular path of an elec
tron in a cathoderay tube, resulting from
the earth's transverse magnetic field
(normal to the plane of the paper).
This figure is not drawn to scale.
(1120)
20 / ELECTRONIC DEVICES AND CIRCUITS
Sec. I U
the anode voltage is higher than the value used in this example, or if the tube is
not oriented normal to the field, the deflection will be less than that calculated.
In any event, this calculation indicates the advisability of carefully shielding a
cathoderay tube from stray magnetic fields.
114 MAGNETIC DEFLECTION IN A CATHODERAY TUBE
The illustrative example in Sec. 113 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. 11 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
(143)
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. (139),
mv
R =>
eB
(144)
(145)
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. 11 T Magnetic deflection in a
cathoderay tube.
S«. M5
ELECTRON BALLISTICS AND APPLICATIONS / 21
be made in assuming that the straight line MP', if projected backward, will
pass through the center 0' of the region of the magnetic field. Then
D « L tan <p « L<p
By Eqs. (143) to (145), Eq. (146) now becomes
(146)
n r IL ILeB ILB
D ~ L * = R=^ = ^r a
2m
The deflection per unit magnetic field intensity, D/B, given by
d = ih rr
(147)
WtM
is called the magneticdeflection 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. 124). If the magneticdeflection coil is driven by a sawtooth current
waveform (Fig. 16), the deflection of the beam on the face of the tube will
not be linear with time. For such wideangle deflection tubes, special linearity
correcting networks must be added.
A TV tube has two sets of magneticdeflection coils mounted around
the neck at right angles to each other, corresponding to the two sets of plates
in the oscilloscope tube of Fig. 15. Sweep currents are applied to both coils,
with the horizontal signal much higher in frequency than that of the vertical
sweep. The result is a rectangular raster of closely spaced lines which cover
the entire face of the tube and impart a uniform intensity to the screen. When
the video signal is applied to the electron gun, it modulates the intensity of
the beam and thus forms the TV picture.
'15 MAGNETIC FOCUSING
As another application of the theory developed in Sec. 113, one method of
measuring e/m is discussed. Imagine that a cathoderay tube is placed in
22 / ELECTRONIC DEVICES AND CIRCUITS
Sec. J 15
a constant longitudinal magnetic field, the axis of the tube coinciding with
the direction of the magnetic field. A magnetic field of the type here con
sidered is obtained through the use of a long solenoid, the tube being placed
within the coil. Inspection of Fig. 112 reveals the motion. The Y axis
represents the axis of the cathoderay 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.
(141) that
27T771
V = ~eB^
It follows from Eq.
(148)
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. 112), since they all will have made just one revolution.
Under these conditions an image of the anode hole will be observed on the
screen.
As the field is increased from zero, the smudge on the screen resulting
from the defocused beam will contract and will become a tiny sharp spot
(the image of the anode hole) when a critical value of the field is reached.
This critical field is that which makes the pitch of the helical path just equal
to the anodescreen distance, as discussed above. By continuing to increase
Sec. TT5
ELECTRON BALLISTICS AND APPLICATIONS / 23
Y
Fig. 112 The helical path of an
electron introduced at an angle (not
90°) with a constant magnetic field.
Electronic
path
the strength of the field beyond this critical value, the pitch of the helix
decreases, and the electrons travel through more than one complete revolution.
The electrons then strike the screen at various points, so that a defocused
spot is again visible. A magnetic field strength will ultimately be reached
at which the electrons make two complete revolutions in their path from the
anode to the screen, and once again the spot will be focused on the screen.
This process may be continued, numerous foci being obtainable. In fact, the
current rating of the solenoid is the factor that generally furnishes a practical
limitation to the order of the focus.
The foregoing considerations may be generalized in the following way:
If the screen is perpendicular to the Y axis at a distance L from the point of
emergence of the electron beam from the anode, then, for an anodecathode
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, (148) may be rearranged to read
e
m
8ir 2 K~ n 2
L*B 2
(149)
where n is an integer representing the order of the focus. It is assumed, in
this development, that eV a ~ fynvey 2 , or that the only effect of the anode
potential is to accelerate the electron along the tube axis. This implies that
the transverse velocity x oz> which is variable and unknown, is negligible in
comparison with v oy . This is a justifiable assumption.
This arrangement was suggested by Busch, and has been used 2 to measure
the ratio e/m for electrons very accurately.
A Short Focusing Coil The method described above of employing a
longitudinal magnetic field over the entire length of a commercial tube is
not too practical. Hence, in a commercial tube, a short coil is wound around
24 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 116
Rg. 113 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.
116 PARALLEL ELECTRIC AND MAGNETIC FIELDS
Consider the case where both electric and magnetic fields exist simultaneously,
the fields being in the same or in opposite directions. If the initial velocity
of the electron either is zero or is directed along the fields, the magnetic field
exerts no force on the electron, and the resultant motion depends solely upon
the electric field intensity £. In other words, the electron will move in a
direction parallel to the fields with a constant acceleration. If the fields arc
chosen as in Fig. 113, the complete motion is specified by
v v = Vey — at y = v<n) t — frl*
(150)
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. 116
ELECTRON &ALLISTICS AND APPLICATIONS / 25
EXAMPLE Given a uniform electric field of 1.10 X 10* V/m parallel to and
opposite in direction to a magnetic field of 7.50 X 10 4 Wb/ra*. An electron gun
in the XY plane directed at an angle <p = arctan f with the direction of the
electric field introduces electrons into the region of the fields with a velocity
v = 5.00 X 10 8 m/sec. Find:
a. The time for an electron to reach its maximum height above the XZ
plane
6. The position of the electron at this time
c. The velocity components of the electron at this time
Solution a. As discussed above, the path is a helix of variable pitch. The
acceleration is downward, and for the coordinate system of Fig. 114,
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 Kdirected motion. At maximum height i'„ =
and f = v ov /a. Since v 0l) = v cos <p  (5 X 10*) (0.8) = 4 X 10 8 m/sec and
e£
ay m — = (1.76 X 10") (1.10 X 10*) = 1.94 X 10" m/sec»
m
we find
f tmZSm
4 X 10 8
1.94 X 10"
= 2.06 X 10"* sec = 20.6 nsec
6. The distance traveled in the +Y direction to the position at which the
reversal occurs is
y  v ov t  iaf* = (4 X 10 8 )(2.06 X lO" 8 )  £(1.94 X 1Q»)(4.24 X 10~ 18 )
* 4.13 X 10* m = 4.13 cm
It should be kept in mind that the term reversal refers only to the Fdirected
motion, not to the direction in which the electron traverses the circular compo
nent of its path. The helical rotation is determined entirely by the quantities
B and f„. The angular velocity remains constant and equal to
a = — = (1.76 X 10") (7.50 X 10"*)  1.32 X 10« rad/sec
TO
Pig. 114 A problem illustrating helical electronic
•notion of variable pitch.
2d / ELECTRONIC DEVICES AND CIRCUITS
Z
Sec. 117
Fig. 115 The projection of the path in the
XZ plane is a circle.
180e
+ Z\ u sin \p = u.
By^use of either the relationship T = 2r/w or Eq. (142), there is obtained
T = 4,75 X 10~ 8 sec, and hence less than one revolution is made before the
reversal.
The point P' in space at which the reversal takes place is obtained by con
sidering the projection of the path in the XZ plane (since the Y coordinate U
already known). The angle 8 in Fig. 115 through which the electron has rotated
is
9  «rf  1.32 X 10» X 2.06 X 10" 8 = 2.71 rad = 155°
The radius of the circle is
fi ^, = (5 X 10»)(0.6)
to 1.32 X 10 8
From the figure it is clear that
X = R sin (180  $) = 2.27 sin 25° = 0.957 cm
Z = R + R cos (180  $) = 2.27 + 2.05 = 4.32 cm
c. The velocity is tangent to the circle, and its magnitude equals v a sin *> =
5 X 10' X 0.6 = 3 X 10" m/sec. At 9 = 155°, the velocity components are
9,  #«, cos (180  6)  8 X 10 s cos 25° = 2.71 X 10 8 m/sec
v ¥ =
f. = v„ sin (180  6) = 3 X 10« sin 25° = 1.26 X 10 8 m/sec
117 PERPENDICULAR ELECTRIC AND MAGNETIC FIELDS
The directions of the fields are shown in Fig. 116. The magnetic field is
directed along the  1' axis, and tho electric field is directed along the X
axis. The force on an electron due to the electric field is directed along the
+ X axis. Any force due to the magnetic field is always normal to B, and
Sec. M7
ELECTRON BALLISTICS AND APPLICATIONS / 27
Fig. 116 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
(151)
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. 111. Similarly, the force
due to v, is eBv t in the —X direction. Hence Newton's law, when expressed
in terms of the three components, yields
j dv x „ i,
f x = m j = e8 — eBv z
at
dv. _
f ' = m dl = eBv '
By writing for convenience
eB
m
= — and
U = B
(152)
(153)
28 / ELECTRONIC DEVICES AND CIRCUITS
the foregoing equations may be written in the form
~dl
N oju — biVz
dv t
Tt = + m *
Sec, 717
(154)
A straightforward procedure is involved in the solution of these equations.
If the first equation of (154) is differentiated and combined with the second,
we obtain
d 2 v x dv t „
(155)
This linear differential equation with constant coefficients is readily solved
for v x . Substituting this expression for v x in Eq. (154), 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
(156)
In order to find the coordinates x and z from these expressions, each equa
tion must be integrated. Thus, subject to the initial conditions x = z = 0,
4i tk»
x =  (1 — cos at) z = ut —  sin o)t
If, for convenience,
8 s at and Q = 
then
x = 0(1  cos 8) z = Q(8  sin 8)
where u and a? are as defined in Eqs. (153).
(157)
(158)
(159)
Cycloid a! Path Equations (159) 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. 117. 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 (160)
which are identical with Eqs. (159), thus proving that the path is cycloidal
as predicted.
S«. M7
ELECTRON BALLISTICS AND APPLICATIONS / 29
Fig. 117 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. 117 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. 117). This is also seen from the fact that each cusp is along
the Z axis, and hence at the same potential. Therefore the electron has gained
no energy from the electric field, and its speed must again be zero.
If an initial velocity exists that is directed parallel to the magnetic field,
the projection of the path on the XZ plane will still be a cycloid but the
particle will now have a constant velocity normal to the plane. This path
30 / ELECTRONIC DEVICES AND CIRCUITS
might be called a "cycloidal helical motion."
(159), with the addition of Eqs. (151).
Sec. 117
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. 116), and the magnetic force is eBv<„ along the — X direction. If the
net force on the electron is zero, it will continue to move along the Z axis with
the constant speed u„. This conditions is realized when
or
e£ = eBv a
Km = g = u
(161)
from Eqs. (153).
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. 116. 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. (153) and (158), it is found that
eB
w = — = 1,76 X 10" X 10"* = 1.76 X 10° rad/sec
m
8 10* tn , .
u = — = • —  = 10 6 m/sec
B 10*
Q =  =
10 6
= 5.68 X 10'* m = 0.0568 cm
1.76 X 10 s
6 = tDf = (1.76 X 10»)<2 X 10~ 8 ) = 35.2 rad
sec. trs
ELECTRON BALLISTICS AND APPLICATIONS / 31
Since there are 2ir rad/revolution, the electron goes through five complete cycles
and enters upon the sixth before it emerges from the plate. Thus
35.2 rad = lOr + 3.8 rad
Since 3.8 rad equals 218°, then Eqs. (159) 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 initialvelocity 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. 118. 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. l19a). If Q'  Q, the path is called a
common cycloid, illustrated in Fig. 117 or 1196. If Q' is less than Q, the path
is called a aviate cycloid, 6 and has blunted cusps, as indicated in Fig. l19c.
118
THE CYCLOTRON
The principles of Sec. 113 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 highenergy 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. 120. The essential elements are the "dees," the
Rolling
circle
Angular
velocity to
Fig. 1 1 8 The locus of the point P at the
end of a "spoke" of a wheel rolling on a
straight line is a trochoid.
Track of
rolling
circle'
Linear
velocity
of C la
Qui = U
32 / ELECTRONIC DEVICES AND CIRCUITS
See. 1. 18
0* — Magnetic fleld
(Into paper)
6*
fig. 11? 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 highfrequency ac potential applied to the dees.
A moving positive ion released near the center of the dees will be acceler
ated in a semicircle by the action of the magnetic field and will reappear at
point 1 at the edge of dee I. Assume that dee II is negative at this instant
with respect to dee I. Then the ion will be accelerated from point 1 to point 2
across the gap, and will gain an amount of energy corresponding to the poten
tial difference between these two points. Once the ion passes inside the metal
dee II, the electric field is zero, and the magnetic field causes it to move in the
semicircle from point 2 to point 3. If the frequency of the applied ac poten
tial is such that the potential has reversed in the time necessary for the ion to
Dees
Particle orbit
(schematic)
Fig. 120 The cyclotron principle.
South pole
Vacuum
chamber
s*. its
ELECTRON BALLISTICS AND APPLICATIONS / 33
go from point 2 to point 3, then dee I is now negative with respect to dee II,
and the ion will be accelerated across the gap from point 3 to point 4. With
the frequency of the accelerating voltage properly adjusted to this "resonance"
value, the ion continues to receive pulses of energy corresponding to this
difference of potential again and again.
Thus, after each half revolution, the ion gains energy from the electric
field, resulting, of course, in an increased velocity. The radius of each semi
circle is then larger than the preceding one, in accordance with Eq. (139),
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 5MeV 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. (139),
(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. (141),
. _ 1 _ eB
T 2rm
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 50keV 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. (141),
the time of flight is
1= SOT 
50 X 1
9.15 X 10 8
* Hi « hertz = cycles per second
m 5.47 X 10* sec = 5.47 usee
MHz = megahertz (Appendix B).
34 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 178
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 vacuumtight chamber. Further, means must be provided for introducing
the ions into the region at the center of the dees and also for removing the high
energy particles from the chamber, if desired, or for directing them against a
target.
S»c. M«
ELECTRON BALLISTICS AND APPLICATIONS / 35
hollow cylinder, since there is need for a magnetic field only transverse to the
path. This results in a great saving in weight and expense. The dees of the
cyclotron are replaced by a singlecavity 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 alternatinggradient magnetic field focusing,*
higherenergyparticle accelerators (70 BeV) have been constructed. 10
The bombardment of the elements with the highenergy 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 taggedatom," technique is used in industry,
medicine, physiology, and biology.
FM Cyclotron and Synchrotron It is shown in Sec. 110 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 highfre
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 fm 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 fm cyclotron, deuterons, a particles,
and protons have been accelerated to several hundred million electron volts. 7
It is possible to give particles energies in excess of those for which the
relativistic correction is important even if the oscillator frequency is fixed,
provided that the magnetic field is slowly increased in step with the increase
in the mass of the ions so as to maintain a constant angular velocity. Such
an instrument is called a synchrotron. The particles are injected from a gun,
which gives them a velocity approaching that of light. Since the radius of
the orbit is given by R = mv/Be and since the ratio m/B is kept constant and
v changes very little, there is not much of an increase in the orbit as the energy
of the electron increases. The vacuum chamber is built in the form of a
doughnut instead of the cyclotron pillbox. The magnet has the form of a
REFERENCES
1. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," chaps. 14
and 19, McGrawHill 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. 4763, 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, McGrawHill 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. 219, January,
1944; The Cyclotron, II, ibid., pp. 128147, February, 1944.
Livingston, M. S.: Particle Accelerators, Advan. Electron., Electrochem. Eng., vol. 1,
pp. 269316, 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. 449450,
April, 1947.
8. Livingston, M. S., J. P. Blewett, G. K. Green, and L. J. Haworth: Design Study for
a ThreeBev Proton Accelerator, Rev. Set. Tnstr., vol. 21, pp. 722, January, 1950.
9. Courant, E. D., M. S. Livingston, and H. 8. Snyder: The Strongfocusing Syn
chrotron: A New High Energy Accelerator, Phys. Rev., vol. 88, pp. 11901196,
December, 1952.
10. Livingston, M. S., and J. P. Blewett: "Particle Accelerators," chap. 15, McGraw
Hill Book Company, New York, 1962.
2 /ENERGY LEVELS AND
ENERGY BANDS
In this chapter we begin with a review of the basic atomic properties
of matter leading to discrete electronic energy levels in atoms. We
also examine some selected topics in quantum physics, such as the
wave properties of matter, the Schrodinger wave equation, and the
Pauli exclusion principle. We find that atomic energy levels are
spread into energy bands in a crystal. This band structure allows us
to distinguish between an insulator, a semiconductor, and a metal.
21
THE NATURE OF THE ATOM
In order to explain many phenomena associated with conduction in
gases, metals, and semiconductors and the emission of electrons from
the surface of a metal, it is necessary to assume that the atom has
loosely bound electrons which can be torn away from it.
Rutherford, 1 in 1911, found that the atom consists of a nucleus of
positive charge that contains nearly all the mass of the atom. Sur
rounding this central positive core are negatively charged electrons.
As a specific illustration of this atomic model, consider the hydrogen
atom. This atom consists of a positively charged nucleus (a proton)
and a single electron. The charge on the proton is positive and is
equal in magnitude to the charge on the electron. Therefore the atom
as a whole is electrically neutral. Because the proton carries practi
cally all the mass of the atom, it will remain substantially immobile,
whereas the electron will move about it in a closed orbit. The force
of attraction between the electron and the proton follows Coulomb's
law. It can be shown from classical mechanics that the resultant
closed path will be a circle or an ellipse under the action of such a
force. This motion is exactly analogous to that of the planets about
36
$•**•»
ENERGY LEVELS AND ENERGY BANDS / 37
the sun, because in both eases the force varies inversely as the square of the
distance between the particles.
Assume, therefore, that the orbit of the electron in this planetary model
f the atom is a circle, the nucleus being supposed fixed in space. It is a
simple matter to calculate its radius in terms of the total energy W of the
electron. The force of attraction between the nucleus and the electron is
e i /^ir€ r 2 , where the electronic charge e is in coulombs, the separation r between
the two particles is in meters, the force is in newtons, and e<, is the permittivity
of free space. f By Newton's second law of motion, this must be set equal
to the product of the electronic mass m in kilograms and the acceleration v ! /r
toward the nucleus, where is the speed of the electron in its circular path,
in meters per second. Then
4fir€„r 3
r
(21)
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
(22)
where the energy is in joules. Combining this expression with (21) produces
»3
W =
(23)
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. (23). Consequently, as the atom radiates
energy, the electron must move in smaller and smaller orbits, eventually fall
t The numerical value of e„ is in Appendix B.
38 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 22
S*c>22
ENERGY LEVELS AND ENERGY BANDS / 39
ing into the nucleus. Since the frequency of oscillation depends upon the size
of the circular orbit, the energy radiated would be of a gradually changing fre
quency. Such a conclusion, however, is incompatible with the sharply denned
frequencies of spectral lines.
The Bohr Atom The difficulty mentioned above was resolved by Bohr in
1913. 2 He postulated the following three fundamental laws:
1. Not all energies as given by classical mechanics are possible, but the
atom can possess only certain discrete energies. While in states correspond
ing to these discrete energies, the electron does not emit radiation, and the
electron is said to be in a stationary, or nonradiating, state.
2. In a transition from one stationary state corresponding to a definite
energy W% to another stationary state, with an associated energy W\, radi
ation will be emitted. The frequency of this radiant energy is given by
Wi  Wi
f =
(24)
where h is Planck's constant in jouleseconds, the W's are expressed in joules,
and / is in cycles per second, or hertz.
3. A stationary state is determined by the condition that the angular
momentum of the electron in this state is quantized and must be an integral
multiple of k/2w. Thus
nh
7TWT =
2tt
(25)
where n is an integer.
Combining Eqs. (21) and (25), we obtain the radii of the stationary
states (Prob. 21), and from Eq. (23) the energy level in joules of each state
is found to be
W n = 
me*
1
8fc V n 2
(26)
Then, upon making use of Eq. (24), 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.
22
ATOMIC ENERGY LEVELS
Though it is theoretically possible to calculate the various energy states of the
atoms of the simpler elements, these levels must be determined indirectly from
spectroscopic and other data for the more complicated atoms. The experi
mentally determined energylevel diagram for mercury is shown in Fig. 21.
The numbers to the left of the horizontal lines give the energy of these
levels in electron volts. The arrows represent some of the transitions that
10.39
8.86,
8.61
8.53
8.38"
7.33
7.73 H
fig. 21 The lower energy
levels of atomic mercury.
5.46
4.88
4.66
Ionization level of mercury
r~=
10140
3650
M I
J_J
Normal state of neutral mercury
have been found to exist in actual spectra, the attached numbers giving the
wavelength of the emitted radiation, expressed in angstrom units (10~ 10 m).
The light emitted in these transitions gives rise to the luminous character of
the gaseous discharge. However, all the emitted radiation need not appear
in the form of visible light, but may exist in the ultraviolet or infrared regions.
The meaning of the broken lines is explained in Sec. 27.
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. (24) may be rewritten in the form
X m
12,400
Et — Ei
(27)
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. 21. 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 4.88eV level to the zero state. The emitted radiation,
as calculated from Eq. (27), is 12,400/4.88 = 2,537 A, as indicated in the
diagram. It is primarily this line that is responsible for the ultraviolet burns
which arise from mercury discharges.
40 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 23
23
THE PHOTON NATURE OF LIGHT
The mean life of an excited state ranges from 10 T to 10 10 sec, the excited
electron returning to its previous state after the lapse of this time. 3 In this
transition, the atom must lose an amount of energy equal to the difference in
energy between the two states that it has successively occupied, this energy
appearing in the form of radiation. According to the postulates of Bohr, this
energy is emitted in the form of a photon of light, the frequency of this radi
ation being given by Eq. (24). 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 50W mercuryvapor 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. (27),
12,400
K
2,337
= 4.88 eV/photon
The total power being transformed to the 2, 537 A line is 0.05 W, or 0.05 J/sec.
Since 1 eV = 1.60 X 10~ 19 J, the power radiated is
0.05 J/sec
 3.12 x 10 17 eV/sec
1.60 X 10" J/eV '
Hence the number of photons per second is
3.12 X 10»'eV/sec ' iM . J .
. oc ... . —  6.40 X 10 18 photons/sec
4.88 eV/photon
This is an extremely large number.
24 IONIZATION
As the most loosely bound electron of an atom is given more and more energy,
it moves into stationary states which are farther and farther away from the
Sec. 26
ENERGY LEVELS AND ENERGY BANDS / 41
nucleus. When its energy is large enough to move it completely out of the field
of influence of the ion, it becomes "detached" from it. The energy required
for this process to occur is called the ionization potential and is represented as
the highest state in the energylevel diagram. From an inspection of Fig. 21,
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.
25
COLLISIONS OF ELECTRONS WITH ATOMS
The foregoing discussion has shown that, in order to excite or ionize an atom,
energy must be supplied to it. This energy may be supplied to the atom in
various ways, one of the most important of which is electron impact. Other
methods of ionization or excitation of atoms are considered below.
Suppose that an electron is accelerated by the potential applied to a dis
charge tube. When this electron collides with an atom, one of several effects
may occur. A slowly moving electron suffers an "elastic" collision, i.e., one
that entails an energy loss only as required by the laws of conservation of
energy and momentum. The direction of travel of the electron will be altered
by the collision although its energy remains substantially unchanged. This
follows from the fact that the mass of the gas molecule is large compared with
that of the electron.
If the electron possesses sufficient energy, the amount depending upon the
particular gas present, it may transfer enough of its energy to the atom to
elevate it to one of the higher quantum states. The amount of energy neces
sary for this process is the excitation, or radiation, potential of the atom. If
the impinging electron possesses a higher energy, say, an amount at least equal
to the ionization potential of the gas, it may deliver this energy to an electron
of the atom and completely remove it from the parent atom. Three charged
particles result from such an ionizing collision: two electrons and a positive ion.
It must not be presumed that the incident electron must possess an energy
corresponding exactly to the energy of a stationary state in an atom in order to
raise the atom into this level. If the bombarding electron has gained more
than the requisite energy from the electric field to raise an atom into a par
ticular energy state, the amount of energy in excess of that required for exci
tation will be retained by the incident electron as kinetic energy after the
collision. Or if the process of ionization has taken place, the excess energy
divides between the two electrons.
2 "6 COLLISIONS OF PHOTONS WITH ATOMS
Another important method by which an atom may be elevated into an excited
energy state is to have radiation fall on the gas. An atom may absorb a
photon of frequency / and thereby move from the level of energy Wi to the
higher energy level W h where tF a = TFi + kf.
42 / ElECTRONIC DEVICES AND CIRCUITS
Sec. 27
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 2,537A 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.
27 METASTABLE STATES
Stationary states may exist which can be excited by electron bombardment
but not by photoexcitation. Such levels are called metastable states. A tran
sition from a metastable level to the normal state with the emission of radiation
has a very low probability of taking place. The 4.66 and 5.46eV levels in
Fig. 21 are metastable states. The forbidden transitions are indicated by
dashed arrows on the energylevel diagram. Transitions from a higher level to
a metastable state are permitted, and several of these are shown in Fig. 21.
The mean life of a metastable state is found to be very much longer than
s*c. 28
ENERGY IEVE1S AND ENERGY BANDS / 43
the mean life of a radiating level. Representative times are 10~ 2 to 10~* sec
for metastable states and 10 7 to 10~ 10 sec for radiating levels. The long
lifetime of the metastable states arises from the fact that a transition to the
normal state with the emission of a photon is forbidden. How then can the
energy of a metastable state be expended so that the atom may return to its
normal state? One method is for the metastable atom to collide with another
molecule and give up its energy to the other molecule as kinetic energy of
translation, or potential energy of excitation. Another method is that by
which the electron in the metastable state receives additional energy by any
of the processes enumerated in the preceding sections. The metastable atom
may thereby be elevated to a higher energy state from which a transition to
the normal level can take place, or else it may be ionized. If the metastable
atom diffuses to the walls of the discharge tube or to any of the electrodes
therein, either it may expend its energy in the form of heat or the metastable
atoms might induce secondary emission.
28
THE WAVE PROPERTIES OF MATTER
In Sec. 26 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
(28)
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 standingwave pattern. In other words, a stable
orbit is one whose circumference is exactly equal to the electronic wavelength X,
44 / ELECTRONIC DEVICES AND CIRCUITS Soc. 26
or to nX, where n is an integer (but not zero). Thus
2ttt = n\=— (29)
Clearly, Eq. (29) is identical with the Bohr condition [Eq. (25)],
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
(210)
where
3z 2 x By 2
B 2
+ —
^ Bz 2
and v is the velocity of the wave, and t is time. The physical meaning of <j>
depends upon the problem under consideration. It may be one component
of electric field, the mechanical displacement, the pressure, etc., depending
upon the physical problem. We can eliminate the time variable by assuming
a solution of the form
*(*, V, *i  Mx, V, z)e*< (211)
where a = 2wf is the angular frequency. Then Eq. (210) becomes
VV+^*0 (212)
where X = v/f = the wavelength. From De Broglie's relationship [Eq. (28)],
X s h* h 2 K }
(213)
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. (213) in (212) gives the timeindependent Schrodinger equation
87r 2 m
vv + ^jF (W  W =
(214)
The $ in Eq. (214) 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*. 29
ENERGY LEVELS AND ENERGY BANDS / 45
finding the electron somewhere must be unity. Quantum mechanics makes
no attempt to locate a particle at a precise point P in space, but rather the
Schrodinger equation determines only the probability that the electron is to
be found in the neighborhood of P.
The potential energy U(z, y, z) specifies the physical problem at hand.
For the electron in the hydrogen atom, U = — e 2 /lw€„r, whereas for a crystal,
it is a complicated periodic function of space. The solution of Schrodinger's
equation, subject to the proper boundary conditions, yields the allowed total
energies W n (called characteristic values, or eigenvalues) of the particle and the
corresponding wave functions ^ n (called eigenf unctions). Except for the very
simplest potRntial functions (as in Sec. 36), 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.
(23) which were obtained from the simpler Bohr picture of the atom.
29
ELECTRONIC STRUCTURE OF THE ELEMENTS
The solution of the Schrodinger equation for hydrogen or any multielectron
atom need not have radial symmetry. The wave functions may be a function
of the azimuthal and polar angles as well as of the radial distance. It turns
out that, in the general case, four quantum numbers are required to define
the wave function. The total energy, the orbital angular momentum, the
component of this angular momentum along a fixed axis in space, and the
electron spin are quantized. The four quantum numbers are identified as
follows :
1. The principal quantum number n is an integer 1, 2, 3, . . . and deter
mines the total energy associated with a particular state. This number may
be considered to define the size of the classical elliptical orbit, and it corre
sponds to the quantum number n of the Bohr atom.
2. The orbital angular momentum quantum number I takes on the values
0, 1, 2, , . . , (n — 1). This number indicates the shape of t he classical
orbit. The magnitude of this angular momentum is s/{l)(l + 1) (h/2ir).
3. The orbital magnetic number mi may have the values 0, ± 1, ±2, ... ,
± I. This number gives the orientation of the classical orbit with respect to
an applied magnetic field. The magnitude of the component of angular
momentum along the direction of the magnetic field is mi{h/2ir).
4. EUctron spin. In order to explain certain spectroscopic and magnetic
phenomena, Uhlenbeck and Goudsmit, in 1925, found it necessary to assume
that, in addition to traversing its orbit around the nucleus, the electron must
also rotate about its own axis. This intrinsic electronic angular momentum
is called electron spin. When an electron system is subjected to a magnetic
field, the spin axis will orient itself either parallel or antiparallel to the direc
46 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 2?
S«. 210
tion of the field. The spin is thus quantized to one of two possible values.
The electronic angular momentum is given by m.(A/2ir), where the spin quan
tum number m, may assume only two values, +i or — £.
The Exclusion Principle The periodic table of the chemical elements (given
in Appendix C) may be explained by invoking a law enunciated by Pauli in
1925. He stated that no two electrons in an electronic system can have the same
set of four quantum numbers, n, I, m h and m,. This statement that no two
electrons may occupy the same quantum state is known as the Pauli exclusion
principle.
Electronic Shells All the electrons in an atom which have the same value
of n are said to belong to the same electron shell. These shells are identified
by the letters K, L, M, N, . . . , corresponding to n = 1, 2, 3, 4, ... ,
respectively. A shell is divided into subskells corresponding to different values
of I and identified as s, p, d, f, . . . , corresponding to I = 0, 1, 2, 3, . . . ,
respectively. Taking account of the exclusion principle, the distribution of
electrons in an atom among the shells and subshells is indicated in Table 21.
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 21, 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 21 Electron shells
and si
bshell:
Shell
K
1
L
2
M
3
N
4
/
s
s
1
P
3
1
P
2
d
s
1
P
2
d
Subshell
3
/
Number 
of 
2
2
6
2
6
10
2
6
10
14
electrons)
2
8
18
32
ENERGY IEVEIS AND ENERGY BANDS / 47
TABLE 22 Electronic configuration in Group IVA
Element
Atomic
number
Configuration
C
Si
Ge
Sn
6
14
32
50
ls»2s J 2p*
laW2p B 3s s 3p 8 3di°4s*4p s
Is^s^^Ss^pW^sH^d'oS^Sp*
same property is possessed by all the alkali metals (Li, Na, K, Rb, and Cs),
which accounts for the fact that these elements in the same group in the
periodic table (Appendix C) have similar chemical properties.
The innershell 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 22. 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.
210
THE ENERGYBAND THEORY OF CRYSTALS
Xray 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. (214), 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 innershell elec
trons are not affected appreciably by the presence of the neighboring atoms.
However, the levels of the outershell 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 outershell electrons of
^e atoms results in a band of closely spaced energy states instead of the
48 / ELECTRONIC DEVICES AND CIRCUITS
See. 270
widely separated energy levels of the isolated atom (Fig. 22). A qualitative
discussion of this energyband structure follows.
Consider a crystal consisting of N atoms of one of the elements in Table
22. 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 22 contain two s electrons and two p electrons. Hence,
if we ignore the innershell levels, then, as indicated to the extreme right in
Fig. 22a, 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
onethird 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. 22a), an atom will exert an electric force
on its neighbors. Because of this coupling between atoms, the atomicwave
functions overlap, and the crystal becomes an electronic system which must
obey the Pauli exclusion principle. Hence the 2N degenerate s states must
spread out in energy. The separation between levels is small, but since N is
very large (~10 23 cm" 8 ), the total spread between the minimum and maximum
energy may be several electron volts if the interatomic distance is decreased
sufficiently. This large number of discrete but closely spaced energy levels
is called an energy band, and is indicated schematically by the lower shaded
Isolated
atom
f 6N states ^*4
' I 2N electrons ri«i , ,
J 2N p electrons
/'{QN s tates
[Energy tfe % electrons
8 a P y\ 2 N s tates
1
2 A' states 
2N electrons
I
Innershell atomic energy
levels unaffected by
crystal formation
(4N states
electrons
Conduction band
Interatomic spacing, d
(a)
(4W states
< 4N electrons
[ Valence band
j, Crystal lattice
I / spacing
(6)
Fig. 22 Illustrating how the energy [evels of isolated atoms are split
into energy bands when these atoms are brought Into close proximity
to form a crystal.
&K. 2"
ENERGY LEVELS AND ENERGY SANDS / 49
region in Fig. 22a. The 2N states in this band are completely filled with
2/V electrons. Similarly, the upper shaded region in Fig. 22a 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, 22a but shown
in Fig. 226) 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 22 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 energyband diagram
somewhat as pictured 4 in Fig. 226. At the crystallattice 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.
2H INSULATORS, SEMICONDUCTORS, AND METALS
A very poor conductor of electricity is called an insulator; an excellent con
ductor is a metal; and a substance whose conductivity lies between these
extremes is a semiconductor, A material may be placed in one of these three
classes, depending upon its energyband structure.
Insulator The energyband structure of Fig. 226 at the normal lattice
spacing is indicated schematically in Fig. 23a. For a diamond (carbon)
crystal the region containing no quantum states is several electron volts high
(£c « 6 eV). This large forbidden band separates the filled valence region
trom the vacant conduction band. The energy which can be supplied to an
electron from an applied field is too small to carry the particle from the filled
lr *to the vacant band. Since the electron cannot acquire externally applied
energy, conduction is impossible, and hence diamond is an insulator.
Semiconductor A substance for which the width of the forbidden energy
region is relatively small (~1 eV ) is called a semiconductor. Graphite, a
50 / ELECTRONIC DEVICES AND CIRCUITS
Sk. 21 T
E c * 6 eV
. Holes
T
Conduction
band
JL Valence
band
(c)
Fig. 23 Energyband 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. 236. 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. 236, 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. 51. 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 bandgap energy of a crystal is a function of interatomic spacing
(Fig. 22), it is not surprising that Eg depends somewhat on temperature.
It has been determined experimentally that E G for silicon decreases with
ENERGY LEVELS AND ENERGY BANDS / 51
temperature at the rate of 3.60 X 10 4 eV/°K. Hence, for silicon, 5
E (T) = 1.21  3.60 X 10 4 T (215)
and at room temperature (300°K), E = 1.1 eV. Similarly, for germanium, 8
E (T) = 0.785  2.23 X 10~*r (216)
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. 23c. Under the influence of an applied electric field the electrons may
acquire additional energy and move into higher energy states. Since these
mobile electrons constitute a current, this substance is a conductor, and the
empty region is the conduction band. A metal is characterized by a band
structure containing overlapping valence and conduction bands.
REFERENCES
1. Rutherford, E.: The Scattering of a and Particles by Matter and the Structure of
the Atom, Phil. Mag., vol. 21, pp. 669688, 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. 476502. September, 1913.
3. Richtmyer, F. K., E. H. Kennard, and T. Lauritsen: "Introduction to Modern
Physics," McGrawHill 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. 2835, October. 1954.
*• Morin, F. J., and J. P. Maita: Conductivity and Hall Effect in the Intrinsic Range of
Germanium, Phys. Rev., vol. 94, pp. 15251529, June, 1954.
3/ CONDUCTION
IN METALS
In this chapter we describe the interior of a metal and present the
basic principles which characterize the movement of electrons within
the metal. The laws governing the emission of electrons from the
surface of a metal are also considered.
31
MOBILITY AND CONDUCTIVITY
In the preceding chapter we presented an energyband 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 31 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
threedimensional array of heavy, tightly bound ions permeated with
52
S9C. 31
CONDUCT/ON IN METALS / 53
Fig. 31 Arrangement of the sodium atoms
in one plane of the metal.
© © ©
D © © © d
© © © ©
) © © ©
• • m ©
0X2345
1 ' a' ' ' '
A units
a swarm of electrons that may move about quite freely. This picture is known
as the electrongas description of a metal.
According to the electrongas 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& (31)
where ^ (square meters per voltsecond) is called the mobility of the electrons.
According to the foregoing theory, a steadystate 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. 112)
J = nev = nefi& = erS (32)
54 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 32
Sec. 32
CONDUCTION IN METALS / 55
where
= net (3 _ 3)
is the conductivity of the metal in (ohmmeter)" 1 . Equation (32) 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).
32 THE ENERGY METHOD OF ANALYZING
THE MOTION OF A PARTICLE
A method is considered in Chap. 1 by which the motion of charged particles
may be analyzed. It consists in the solution of Newton's second law, in which
the forces of electric and magnetic origin are equated to the product of the
mass and the acceleration of the particle. Obviously, this method is not
applicable when the forces are as complicated as they must be in a metal.
Furthermore, it is neither possible nor desirable to consider what happens
to each individual electron.
It is necessary, therefore, to consider an alternative approach. This
method employs the law of the conservation of energy, use being made of the
potentialenergy curve corresponding to the field of force. The principles
involved may best be understood by considering specific examples of the
method.
EXAMPLE An Idealized diode consists of planeparallel 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 32a is a linear
Potential, V
'.rurgy
Potential
energy V
Total energy Jl'
Distance, x
Fig 32 (o) Potential vs. distance in a planeparallel diode, (b) The
potentialenergy barrier encountered by an electron in the retarding
field.
plot of potential vs. distance, and in Fig. 326 is indicated the corresponding
potential energy vs. distance. Since potential is the potential energy per unit
charge (Sec. 14), 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. 326 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. Potentialenergy 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. 33) 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 potentialenergy function is illus
trated graphically in Fig. 34.
fig. 33 Point F represents the mass
•8 of a mathematical pendulum
swinging in the earth's gravitational
field.
56 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 32
Fig. 34 The potential energy of
the moss m in Fig. 33 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. 34 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 totalenergy
line and the potentialenergy 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. 34
represent the potentialenergy 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 socalled "bound" elec
trons in a metal, as shown in Sec. 34.
Now consider the case when the mass has a total energy equal to W 2 ,
which is greater than the maximum of the potentialenergy 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 potentialenergy barrier, and its course is never altered, so that it
S«. 33
CONDUCT/ON IN METALS / 57
moves through an everincreasing 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 socalled
"free" electrons in a metal.
This simple but powerful energy method facilitates the discussion of the
motion of a particle in a conservative field of force, such as that found in the
body of a metal. It is also applied to many other types of problem. For
example, the method of analysis just considered is extremely useful in deter
mining whether electrons will possess sufficient energy to pass through grids
and reach the various electrodes in a vacuum tube, whether or not electrons
will be able to penetrate electron clouds in a vacuum tube, and whether charge
carriers can cross a semiconductor junction. This method is now applied to
the analysis of the motion of electrons in metals.
THE POTENTIALENERGY FIELD IN A METAL
It is desired to set up the potentialenergy field for the threedimensional 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. 14), 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 potentialenergy curve illustrated in Fig. 35. 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 potentialenergy axis. This difficulty is avoided by plotting U
58 / ELECTRONIC DEVICES AND CIRCUITS
U
Sec. 33
Fig, 35 The potential energy of an electron as
a function of radial distance from an isolated
nucleus,
along some chosen line through the crystal, say, through a row of ions. From
this graph and the method by which it is constructed it is easy to visualize
what the potential energy at any other point might be. In order to build up
this picture, consider first two adjacent ions, and neglect all others. The con
struction is shown in Fig. 36. 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. 31 and 37) 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. 37 is obtained.
According to classical electrostatics, which does not take the atomic
structure into account, the interior of a metal is an equipotential region.
The present, more accurate, picture shows that the potential energy varies
apprecmbly in the immediate neighborhoods of the nuclei and actually tends
to  « in these regions. However, the potential is approximately constant
t/=o
Fig. 36 The potential energy resulting
from two nuclei, a and 0.
Sec. 34
CONDUCTION IN METAtS / 59
Fig. 37 The potentialenergy distribution within and at the surface of
a metal.
E/0
for a very large volume of the metal, as indicated by the slowly varying por
tions of the diagram in the regions between the ions.
Consider the conditions that exist near the surface of the metal. It is
evident, according to the present point of view, that the exact position of the
"surface" cannot be defined. It is located at a small distance from the last
nucleus e in the row. It is to be noted that, since no nuclei exist to the right
of e, there can be no lowering and flattening of the potentialenergy curve
such as prevails in the region between the nuclei. This leads to a most impor
tant conclusion; A potentialmergy "hill," or "barrier," exists at the surface
of the metal.
34
BOUND AND FREE ELECTRONS
The motion of an electron in the potentialenergy field of Fig. 37 is now dis
cussed by the method given in Sec. 32. Consider an electron in the metal
that possesses a total energy corresponding to the level A in Fig. 37. 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 innershell elec
trons of an isolated atom, discussed in Sec. 29. 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. 31.
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. 37, corresponding to an energy in the conduction band. At no point
vritkin the metal is its total energy entirely converted into potential energy.
Hence, at no point is its velocity zero, and the electron travels more or less
60 / ELECTRONIC DEVICES AND CIRCUITS
Energy, eV
Sec. 35
Outside of metal
Fig. 38 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 potentialenergy 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 Potentialenergy 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 37 is redrawn in Fig. 38, 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 zeroenergy 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 potentialenergy barrier in electron volts.
35
ENERGY DISTRIBUTION OF ELECTRONS
In order to be able to escape, an electron inside the metal must possess an
amount of energy at least as great as that represented by the surface barrier
f This figure really represents potential energy, and not potential. However, the
phrase "potential barrier" is much more common in the literature than the phrase "poten
tialenergy barrier." When no confusion is likely to arise, these two expressions are used
interchangeably. Those barriers are measured in electron volts, and hence the symbol E
replaces the U of the preceding sections. It must be emphasized that one unit of B repre
sents 1.60 X lO 1 * J of energy.
S«.3.$
CONDUCTION IN METALS / 61
v It is therefore important to know what energies are possessed by the
electrons in a metal. This relationship is called the energy distribution func
tion. We here digress briefly in order to make clear what is meant by a distri
bution function.
Age Density Suppose that we were interested in the distribution in age
of the people in the United States. A sensible way to indicate this relation
ship is shown in Fig. 39, 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
(34)
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.
39. Evidently, the total population n is given by
(35)
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. (34), we may write
dn E = pb dE 06)
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. 39 The distribution
function in age of people
in the United States.
40 60
Age, years
100
62 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 35
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)
(37)
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
(38)
4ir
y ~ h* ( 2m )'(1.60 X 10»)l = 6.82 X 10"
(39)
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 jouleseconds.
The FermiDirac Function The equation for f(E) is called the Fermi
Dirac probability function, and specifies the fraction of all states at energy E
(electron volts) occupied under conditions of thermal equilibrium From
quantum statistics it is found 23 that
/(£) =
1
1 + e< s ^)/W
(310)
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 (310) becomes zero and f{E) = 1. All ouantum levels
with energies less than E F will be occupied at T = 0°K
From Eqs. (37), (38), and (310), we obtain at absolute zero temperature
»{f
for E < E F
for E > E F
(311)
$k. 3 ' 5
f{E)
*
^r=o°K
1.0
lx i
r — r=300°K
0.8
' NJ
^T=2500°K
0.6
t N
0.4
\
0.2
i\.,
CONDUCTION IN METALS / 63
T=0"K
1.0 0,6 0.2 0.2
(«)
0.6 1.0
EE r ,eV
0.2 0.4 0.6 0.8 1.0 f(E)
(&)
Fig. 310 The FermiDirac 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. (311) 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. 29). Hence not all electrons can have the
same energy even at 0°K. The application of FermiDirac statistics to the
theory of metals is due primarily to Sommerfeld. 3
A plot of the distribution in energy given by Eqs. (37) and (311) for
metallic tungsten at T = 0°K and T = 2500°K is shown in Fig. 311. 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. (310), 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. 311 Energy distribution in metallic tungsten
otOand 2500° K.
T=0°K
64 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 35
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. 311
represents the total number of free electrons (as always, per cubic meter of
the metal). Thus
or
f Er
M3
(312)
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*
(313)
Since the density n varies from metal to metal, E F will also vary among metals.
Knowing the specific gravity, the atomic weight, and the number of free elec
trons per atom, it is a simple matter to calculate n, and so E F . For most
metals the numerical value of E F is less than 10 eV.
EXAMPLE The specific gravity of tungsten is 18.8, and its atomic weight is
184.04 Assume that there are two free electrons per atom. Calculate the
numerical value of n and E F .
Solution A quantity of any substance equal to its molecular weight in grams is
a mole of that substance. Further, one mole of any substance contains the same
number of molecules as one mole of any other substance. This number is Avo
gadro's number and equals 6.02 X 10 23 molecules per mole. Thus
n = 6.02 X ]Q*a molecules x lmole x 188 JL x 2 electrons 1 atom
mole 184 g ' cm 3 atom molecule
 12.3 X io»»? J<iCtron ?  i.23 X JQ'» electrons
cm 3 m 3
since for tungsten the atomic and the molecular weights arc the same. There
fore, for tungsten,
E F = 3.64 X 10" 13 (123 X 10") ! = 8.95 eV
t The atomic weights of the elements are given in the periodic table (Appendix C).
CONDUCT/ON *N METALS / 65
THE DENSITY OF STATES
As a preliminary step in the derivation of the density function N(E) we first
show that the components of the momentum of an electron in a metal are
quantized. Consider a metal in the form of a cube, each side of which has a
length L. Assume that the interior of the metal is at a constant (zero) poten
tial but that the potentialenergy barrier (Fig 38) 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 potentialenergy diagram is given in Fig. 312a, and
two possible wave functions are indicated in Fig. 3126 and c. Clearly, this
situation is possible only if the dimension L is a halfintegral multiple of the
De Broglie wavelength X, or
r x
L = n^
(314)
where n x is a positive integer (not zero). From the De Broglie relationship
(28), X = h/p x and the x component of momentum is
_ n x h
px ~2L
(315)
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 onedimensional problem is
_ p,» _ n,'A«
2m SmL 2
(316)
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 onedimensional Schrodinger equation with the potential
Fig 312 (o) A one
dimensional problem in
which the potential U is
zero for a distance L
but rises abruptly
toward infinity at the
boundaries x = and
x = L. {b, c) Two
possible wave functions
f °r an electron in the
system described by (a).
L^
^d
(6)
(c)
66 / ELECTRONIC DEVICES AND CIRCUITS
Sac. 36
energy U set equal to zero. Under these circumstances Eq. (214) may be
written
dx 2 h 2
(317)
The general solution of this secondorder linear differential equation has two
arbitrary constants, d and C%, and in the interval < z < L is given by
where
^ = Ci sin ax + Ca cos ax
Sr*mW
h 2
(318)
(319)
Since for x = 0, ^ = 0, then C 2 ™ 0. Since for x = L, ^ = 0, sin oL = 0, or
aL = n x w (320)
where n x is an integer. Substituting from Eq, (320) into Eq. (319) and
solving for W, we again obtain the quantized energies given in Eq. (316).
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. 28,
Ci 2 L
/ V& = l/ L Ci«8in*^«fc«
or d = (2/L)» and
/2\i . u x tx
(321)
Note that n x cannot be zero since, if it were, ^ would vanish everywhere.
For n x = 1 the function ^ is plotted in Fig. 312b, and for n x = 2 the wave
function $ is as shown in Fig. 312c. 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. 28), 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 onedimensional 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 = 
(322)
This equation is a statement of the uncertainty principle, first enunciated
by Heisenberg. He postulated that, for all physical systems (not limited to
Sac. 36
CONOUCnON IN METALS / 67
electrons in a metal), there is always an uncertainty in the position and in the
momentum of a particle and that the product of these two uncertainties is of
the order of magnitude of Planck's constant h.
Quantum States in a Metal The above results may be generalized to
three dimensions. For an electron in a cube of metal, each component of
momentum is quantized. Thus
p x = n x p
p y = n v p
Pz = n t p
(323)
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. (323). These are indicated in Fig. 313, 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. 29), 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. 313 represents two electrons, one for
s — j and the other for s = — £.
We now find the energy density function N(E). Since in Fig. 313 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. 313 Momentum space.
Each dot represents three
quantum numbers, n x , n y ,
Qr >d n,. There are two
electrons per dot, corre
sponding to the two possible
values of spin.
68 / ELECTRONIC DEVICES AND CIRCUITS
lying in the shaded spherical shell of Fig. 313. This number is
r l\ 7rp 3 dp 8ttL ? p 2 dp
(?)<**■ *>(§)
5m. 37
(324)
^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 (325)
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. (324) that
N(W ) dW = *^
(326)
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. (325) in Eq. (326), we finally obtain
4?r
N(W) dW = ^ (2m)»TT» dW
(327)
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. 15), the energy density N{E) is given by
Eq. (38), with j defined in Eq. (39).
37
WORK FUNCTION
In Fig. 314, Fig. 311 has been rotated 90° counterclockwise and combined
with Fig. 38, so that the vertical axis represents energy for both sets of curves.
At 0°K it is impossible for an electron to escape from the metal because this
requires an amount of energy equal to E B , and the maximum energy possessed
by any electron is only Er. It is necessary to supply an additional amount
of energy equal to the difference between E B and E F in order to make this
escape possible. This difference, written Ew, is known as the work function
of the metal.
Ew = Eg — Ep
Enet
T2500K I
ft
eV
Outside
""^
fig
Distance, x
(328)
Fig. 314 Energy diagram used
to define the work function.
Sec. 35
CONDUCT/ON IN METALS / 69
Thus the work function of a metal represents the minimum amount of energy
that must be given to the fastestmoving 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 potentialenergy
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).
38
THERMIONIC EMISSION
The curves of Fig. 314 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. 314. 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
(329)
70 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 39
Equation (329) is called the thermionicemission, 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 thermionicemission equation has received considerable experimental
verification. 8 The graphical representation between the thermionicemission
current and the temperature is generally obtained by taking the logarithm of
Eq. (329), viz.,
log /*  2 log T = log SA  0.434 ^f
(330)
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. (329), we obtain
hh \
■ Ew\ dT
(331)
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. (329) 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.
39 CONTACT POTENTIAL
Consider two metals in contact with each other, as at the junction C in Fig.
315. 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. 315 Two metals in contact at the junction C.
S«. 310
CONDUCTION IN METALS / 71
point B just outside metal 2. The reason for the existence of the difference
of potential is easily understood. When the two metals are joined at the
boundary C, electrons will flow from the lowerworkfunction 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 energylevel diagram. To satisfy this
condition, the potentialenergy difference E A n between points A and B is given
by (Prob. 316)
Eab = Ewz ~ Ewi (332)
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 higherworkfunction metal
becomes charged negatively. In a vacuum tube the cathode is usually the
lowestworkfunction metal. If it is connected to any other electrode exter
nally by means of a wire, the effective voltage between the two electrodes is
not zero, but equals the difference in the work functions. This potential
difference is in such a direction as to repel the electrons being emitted from
the cathode. If a battery is connected between the two electrodes, the effec
tive potential is the algebraic sum of the applied voltage and the contact
potential.
310
ENERGIES OF EMITTED ELECTRONS
Since the electrons inside a metal have a distribution of energies, those which
escape from the metal will also have an energy distribution. It is easy to
demonstrate this experimentally. Thus consider a plane emitter and a plane
parallel collector. The current is measured as a function of the retarding
voltage V, (the emitter positive with respect to the collector). If all the elec
trons left the cathode with the same energy, the current would remain con
stant until a definite voltage was reached and then it would fall abruptly to
zero. For example, if they all had 2 eV energy, then, when the retarding
voltage was greater than 2 V, the electrons could not surmount the potential
barrier between cathode and anode and no particles would be collected.
Experimentally, no such sudden falling off of current is found, but instead
72 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 3 70
Sec. 310
CONDUCTION IN METALS / 73
there is an exponential decrease of current / with voltage according to the
equation
where V T is the "volt equivalent of temperature," defined by
V m iT. = T
T e 11,600
(333)
(334)
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. 15 it follows that k = 1.60 X 10" 19 &.)
The VoltAmpere Characteristic Equation (333) 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 righthand side of Eq. (329) by changing E w to E w + E T . Hence
/ = SA Th lB ' +E '> liT = I lk eWT
(335)
where use was made of Eq. (329). Since F r is numerically equal to E r , and
V T is numerically equal to kT, then
kT
(336)
Hence Eq. (333) follows from Eq. (335).
If V is the applied (accelerating) anode potential and if V is the (retard
ing) contact potential, then V r = V  V, and Eq. (333) becomes
where
I m I ei +viy r
(337)
(338)
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. 316.
The nonzero slope of this brokenline 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 planeparallel anode the
voltampere characteristic indicated in Fig. 316 is only approached in practice.
Fig. 316 To verify tke retarding
potential equation, log / is plotted
versus V.
Accelerating
Furthermore, since the effect of space charge (Chap, 4) has been completely
neglected, Eq. (333) is valid only for low values (microamperes) of current.
For larger values of /, the current varies as the threehalves power of the
plate potential (Sec. 44).
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. (333), with V r = 1, and remembering that Vr = f/11,600,
yields
= e Ul, 600X0/2,700 — f — 4.2a m 0.014
Ith
Hence only about 1.4 percent of the electrons have surfacedirected energies in
excess of 1 e V.
If the emitter is an oxidecoated 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 surfacedirected 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
(339)
F °r operating temperatures of 2700 and 1000°K, the average energies of the
emitted electrons are 0.47 and 0.17 eV, respectively.
These calculations demonstrate the validity of the assumption made in
a P 1 in the discussion of the motion of electrons in electric and magnetic
. s > Vlz , that the electrons begin their motions with very small initial veloei
■ In most applications the initial velocities are of no consequence.
74 / ElfCTRON/C DEVICES AND CIRCUITS
Sec. 311
311
ACCELERATING FIELDS
Under normal operating conditions, the field applied between the cathode and
the collecting anode is accelerating rather than retarding, and so the field aids
the electrons in overcoming the image force at the surface of the metal. This
accelerating field tends, therefore, to lower the work function of the metal,
and so results in an increased thermionic emission from the metal. It can be
shown 6 that the current / under the condition of an accelerating field of £
(volts per meter) at the surface of the emitter is
/ = Itf" ■«"»»• (340)
where I& is the zerofield 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 externalfield thermionicemission
current because of the Schottky effect.
Solution The electric field intensity (volts per meter) at any point r (meters)
in the region between the electrodes of a cylindrical capacitor, according to
classical electrostatics, is given by the formula
£ 
I
In (r„/r k ) r
(341)
where In = logarithm to the natural base t
V = plate voltage
r a ~ anode radius
rt = cathode radius
Thus the electric field intensity at the surface of the cathode is
6 =
500
— = 1 .085 X 10 s V/m
2.303 log 100 10*
It follows from Eq. (340) that
log 1  (0434)(0.44)(l.Q85 X 10°)* _ Q ^
2,500
Hence ///, A  1.20, which shows that the Schottky theory predicts a 20 percent
increase over the zerofield emission current.
S#c. 3 J3
CONDUCnON IN METALS / 75
HIGHFIELD EMISSION
312
Suppose that the accelerating field at the surface of a "cold" cathode (one for
which the thermionicemission current is negligible) is very intense. Then,
not only is the potentialenergy 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. 613).
Under these circumstances the variation of the emissioncurrent 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 highfield, coldcathode, 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 highfield 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
coldcathode effect has been used to provide several thousand amperes in an
xray tube used for highspeed radiography.
313
SECONDARY EMISSION 8
The number of secondary electrons that are emitted from a material, either a
metal or a dielectric, when subjected to electron bombardment has been found
experimentally to depend upon the following factors: the number of primary
electrons, the energy of the primary electrons, the angle of incidence of the
electrons on the material, the type of material, and the physical condition of
the surface. The yield, or secondaryemission 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
aryemission 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 lowenergy primaries, the number of secondaries that are able to over
come the surface attraction is small. As the energy of the impinging electrons
ncreases, more energetic secondaries are produced and the yield increases.
, nce > however, the depth of penetration increases with the energy of the
incident electron, the secondaries must travel a greater distance in the metal
T^fore they reach the surface. This increases the probability of collision in
e metal, with a consequent loss of energy of these secondaries. Thus, if the
76 / ELECTRONIC DEVICES AND C/RCU/TS
Sec, 313
primary energy is increased too much, the secondaryemission ratio must pass
through a maximum.
REFERENCES
1. Shockley, W. : The Nature of the Metallic State, J. Appl. Phys., vol. 10 ud 543555
1939.
2. Fermi, E.: Zur Quantelung des idealen cinatomigen Gases, Z. Physik, vol. 36 dd
902912, May, 1926.
Dirac, P. A. M.: On the Theory of Quantum Mechanics, Proc. Roy. Soc. (London)
vol. 112, pp. 661677, October, 1926.
3. Sommerfeld, A., and H. Bethe: Elektronentheorie der Metalle, in "Handbuch der
Physik," 2d ed., vol. 24, pt. 2, pp. 333622, Springer Verlag OHG t Berli n , 1933.
Darrow, K. K.: Statistical Theories of Matter, Radiation and Electricity, Bell
System Tech. J., vol. 8, pp. 672748, 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. 619627, August, 1928.
5. MUlman, J., and S. Seely; "Electronics," 2d ed., McGrawHill Book Company
New York, 1951.
6. Dushman, S.: Thermionic Emission, Rev. Mod. Phys., vol. 2, pp. 381476 October
1930.
7. Dyke, W. P., and W. W. Dolan: Field Emission, "Advances in Electronics," vol. 8,
Academic Press Inc., New York, 1956.
Fowler, R. H., and L, Nordheim: Electron Emission in Intense Electric Fields, Proc.
Roy. Soc. {London), vol. 119, pp. 173181, May, 1928.
Oppenhcimer, J. R.: On the Quantum Theory of Autoelectric Field Circuits, Proc.
Natl. Acad. Sci. U.S., vol. 14, pp. 363365, May, 1928.
8. Spangenberg, K. R.: "Vacuum Tubes," McGrawHill Book Company, New York,
1948.
McKay, K. G.: Secondary Electron Emission, "Advances in Electronics," vol. 1,
pp. 65130, Academic Press Inc., New York, 1948. An extensive review.
4 /VACUUMDIODE
CHARACTERISTICS
The properties of practical thermionic cathodes are discussed in this
chapter. In order to collect the emitted electrons, a plate or anode is
placed close to the cathode in an evacuated envelope. If an acceler
ating field is applied, it is found that the plate current increases as the
anode voltage is increased. When a large enough plate potential is
applied to collect the thermionicemission 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 voltampere characteristics are considered,
and an analysis of a circuit containing a diode is given.
41
CATHODE MATERIALS
The three most important practical emitters are pure tungsten, thori
ated tungsten, and oxidecoated cathodes. The most important prop
erties of these emitters are now discussed, and are summarized in
Table 41.
Tungsten Unlike the other cathodes discussed below, tungsten
does not have an active surface layer which can be damaged by
positiveion bombardment. Hence tungsten is used as the cathode
in highvoltage high vacuum tubes. These include xray 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 cathodeemission efficiency,
defined as the ratio of the emission current, in amperes, to the heating
77
78 / ELECTRONIC DEVICES AND CIRCUITS
TABLE 41 Comparison of thermionic emitters
Sec. 41
Type of
cathode
A. X 10«,
A/(m»)(°K')
eV
Approximate
operating
temperature,
°K
Efficiency, t
A/W
Plate
voltage,
V
Gas or
vacuum
tube
Tungsten. . . ,
Thoriated
tungsten . .
Oxidecoated
60.2
3.0
0.01
4.52
2.63
1.0
2,500
1,900
1,000
20100
501 , 000
10010,000
Above
5,000
7505,000
Below 750
Vacuum
Vacuum
Vacuum
or gas
t K. R. Spangenberg, "Vacuum Tubes," McGrawHill Book Company, New York,
1948.
power, in watts, is small. However, a copious supply of electrons can be pro
vided by operating the cathode at a sufficiently high temperature. The higher
the temperature, the greater will be the evaporation of the filament during its
operation and the sooner it will burn out. Economic considerations dictate
that the temperature of the filament be about 2500°R, which gives it a life of
approximately 2,000 hr. The melting point of tungsten is 3650°K.
Thoriated Tungsten 1 In order to obtain copious emission of electrons
at moderately low temperatures, it is necessary for the material to have a low
work function. Unfortunately, the lowworkfunction 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 lowworkfunction material, such as
thorium, on a filament of tungsten. Thoriatedtungsten 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 thoriatedtungsten emitters is the deacti
vation due to positiveion 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. Thoriatedtungsten
filaments are used in a number of moderatevoltage transmitting tubes as well
as in highpower beamtype microwave tubes.
EXAMPLE At what temperature will a thoriatedtungsten filament give 5,000
times as much emission as a pure tungsten filament at the same temperature?
The filament dimensions of the two emitters are the same.
Uc.41
VACUUMDIODE CHARACTERISTICS / 79
Solution It is required that I T w = 5,000/n. From Eq. (315) and Table 41,
Itw  (S) (3.0 X 10<)(7 ,I )€ !  63 '*''
and
I w = (5) (60.2 X 10*) (T*)t*^ lkT
Upon dividing these two equations, there results
1 T ~ w = 5 000 = — — e (<B2_2  63) ' (a  ,fl2X10 " 5T ' J
I w ' 60.2
20.1
e 21,900/r
where the value of k in electron volts per degree Kelvin given in Appendix A was
used. We can solve for T with the aid of logarithms. Thus
(0.434) (21 ,900)
T = 1900°K
 log (5,000) (20.1) = 5.00
Oxidecoated Cathodes 2 The modern oxidecoated 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. Konalmctal 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 alkalineearth 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.
Oxidecoated cathodes are subject to deactivation by positiveion bom
bardment, and so are generally used in lowvoltage tubes only. The emission
properties of an oxidecoated 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 oxidecoated cathodes
se rve satisfactorily in any circuit? It is shown in Sec. 44 that tubes usually
°perate under conditions of spacecharge limitation and not under conditions
01 temperature limitation. This statement means that the current is determined
80 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 42
Sec. 43
VACUUMDIODE CHARACTERISTICS / 81
by the plate voltage and not. by the cathode temperature. Thus, despite their
rather unpredictable emission characteristics, oxidecoated cathodes make
excellent tube elements, provided only that their thermionicemission current
never falls below that required by the circuit.
Oxidecoated cathodes are used in the greatest percentage of commercial
electron tubes. Almost all receiving tubes, many lowvoltage transmitting
tubes, and practically all gas tubes use such cathodes.
42
COMMERCIAL CATHODES
The cathodes used in thermionic tubes are sometimes directly heated filaments
in the form of a V, a W, or a straight wire, although most tubes use indirectly
heated cathodes.
The indirectly heated cathode was developed so as to minimize the hum
(Sec. 1611) 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 Konalmetal 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.
43
THE POTENTIAL VARIATION BETWFEN THE ELECTRODES
Consider a simple thermionic diode whose cathode can be heated to any desired
temperature and whose anode or plate potential is maintained at V P . It
will be assumed that the cathode is a plane equipotential surface and that the
collecting plate is also a plane parallel to it. The potential variations between
Potential, V
T 3 > T a > 7\
Fig. 41 The potential variation
between planeparallel elec
trodes for several values of
cathode temperature.
"For nonzero initial
velocities
the electrodes for various temperatures of the cathode are given in Fig. 41.
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. 41. 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. 41 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 Potentialenergy Curves Since the potential energy is equal to the
product of the potential V and the charge — e, the curves of Fig. 42 are simply
se °f Fig. 41 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 potentialenergy
arriers have already been considered in Chap. 3. On the basis of our previ
18 discussions, it is clear that only those electrons which possess an initial
er gy greater than B m , the maximum height of the barrier, can escape from
82 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 44
the cathode and reach the anode. Consequently, the assumed condition of
zero initial velocities of the emitted electrons precludes the possibility of any
electrons being emitted. As a result, the barrier will be broken down, since
the applied field will cause those electrons which produce the barrier to leave
the intereleetrode space and become part of the anode current. This auto
matic growth and collapse of the potential barrier outside the cathode may be
considered as a self regulating valve that allows a certain definite number of
electrons per second to escape from the cathode and reach the anode, for a
given value of plate voltage.
The Field Intensity at the Cathode It can be inferred from the fore
going argument that the maximum current that can be drawn from a diode
for a fixed plate voltage and any temperature whatsoever is obtained under
the condition of zero electric field at the surface of the cathode. Thus, for
optimum conditions,
& =
dx
at a; =0
(41)
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. 41, However, since the potential minimum in
Fig. 41 is usually small in comparison with the applied potential, it is neg
lected, and condition (41) is assumed to represent the true status when space
charge current is being drawn.
44
SPACECHARGE CURRENT
We shall now obtain the analytical relationship between the current and volt
age in a diode. The electrons flowing from the cathode to the anode consti
Potential
energy, eV
Fig. 42 The potentialenergy varia
tions corresponding to the curves of
Fig. 41.
Sec. 44
VACUUMDIODE CHARACTERISTICS / 83
tute the current. The magnitude of the current density / in amperes per
square meter is given by Eq. (138), viz.,
J = pv
(42)
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.,
(43)
(44)
imv*
= eV
Poisson's
equation
is
dW
dx 2
, P_
«0
where x = distance from cathode, m
V = potential, V
p = magnitude of electronic volume charge density, C/m 3
e = permittivity of free space, mks system
There results, from Eqs. (42) to (44),
where
^!Z = t = L  J
dx 2 t V€„ ~ [2(e/m)]*e Q
K =
71 = KVi
[2(e/m)h
(45)
(46)
*s a constant, independent of z.
The Solution of Eq. (45) Let y = dV/dx, and this nonlinear differential
equation may be solved by the separationofvariables method. Thus
or
Hence
dy = KVi dx m KVi —
V
ydy = KV* dV
84 / ElfCTRON/C DEVICES AND CIRCUITS
Sec. 44
Sec. 45
VACUUMDIODE CHARACTERISTICS / 85
which integrates to
%■ = 2KV* + Ci
■6
(47)
The constant of integration C\ is zero because, at the cathode, V — and
2/ = dV/dx = 0, from Eq. (41). By taking the square root of Eq. (47) there
results
and V* dV = 2KS dx
,_£_«.*
This equation integrates to
$F* = 2K*x + C 2
The constant of integration Ca is zero because V = at x = 0. Finally,
F = (1)*KV (48)
It is seen that the potential depends upon the fourthirds power of the
interelectrode spacing. For example, the curve marked T% in Fig. 41 is
expressed by the relation
V = ax* (49)
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. (49) is valid for the entire interelectrode space, including the
boundary x = d, where V = Vp.
The Threehalvespower Law The complete expression for the current
density is obtained by combining Eqs. (48) and (46). The result is
9\ m)
VI
(410)
In terms of the boundary values, this becomes, upon inserting the value of
e/m for electrons and eo = 10~ 9 /36rr,
J m 2.33 X 10" fl ^
Qr
(411)
Therefore the plate current varies as the threehalves 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 LangmuirChild law, the threekalvespower law,
or simply, the spacecharge 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 spacecharge 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. (411)
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
turelimited current.
The velocity of the electrons as a function of position between the cathode
and anode can be found from Eq. (43) with the aid of Eq. (410). Then the
charge density as a function of x can be obtained from Eq. (42). It is found
(Prob. 46) that v varies as the twothirds power of x and that p varies inversely
as the twothirds 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 planeparallel 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 { (412)
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.
45
FACTORS INFLUENCING SPACECHARGE CURRENT
Several factors modify the equations for space charge given above, particu
larly at low plate voltages. Among these factors are:
I. Filament Voltage Drop The spacecharge 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 threehalvespower equation. In
& ct, the results depend on whether the plate current is returned to the positive
°r to the negative end of the filament. Usually, the filament is heated with a
ransformer, and the plate is returned to the center tap of the secondary
binding.
86 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 45
2. Contact Potential lit every spacecharge 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 spacecharge 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 spacecharge equations because the positive ions
that are formed neutralize the electroniccharge 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, 41, which is
reproduced in Fig. 43 for convenience. This represents a potentialenergy
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. 310, 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 spacecharge 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 voltampere equation, taking into
account the energy distribution of the electrons, is somewhat involved. 7 To
Potential, V
/ •
,
.
/ t
/ o
Vp
/ c
/ **
M /
1
v^i^'
*Tu
\
■« d —
V
Fig. 43 The potential variation in a plane
parallef spacecharge diode, with the initial
velocities of the electrons taken into account.
See 46
VACUUMDtODE CHARACTERISTICS / 87
summarize, the spacecharge 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.
46
DIODE CHARACTERISTICS
The two most important factors that determine the characteristics of diodes
are thermionic emission and space charge. The first gives the temperature
saturated value, i.e., the maximum current that can be collected at a given
cathode temperature, regardless of the magnitude of the applied accelerating
potential. The second gives the spacechargelimited 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.
Largevoltage Characteristics The voltampere curves obtained
experimentally for an oxidecoated cathode are shown in Fig. 44. It should
be noted that the spacecharge 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
chargelimited and the temperaturelimited portions of the curves, but rather
a gradual transition occurs. Also, the current for the temperaturelimited
regions gradually rises with increased anode potentials (because of the Schottky
effect, Sec. 311). The shapes of these curves are determined by the factors
mentioned in the preceding section.
Lowvoltage Characteristic The diode curve does not follow Eq, (412)
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 voltampere curve near the origin is given in Fig. 45. Space charge is
negligible at these small currents, and the voltampere relationship is given
by Eq. (337), namely,
Ip = I e v r'Vr
•"ig. 44 Voltampere diode characteristics
*or various filament temperatures.
r s > Ti > T z > T t > TV
(413)
100 V P ,V
88 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 47
Fig. 45 The voltampere 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. (334)] is the volt equivalent of temperature. Note that the curve doea
not pass through the origin.
47
AN IDEAL DIODE VERSUS A THERMIONIC DIODE
An ideal diode is defined as a twoterminal 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 voltampere characteristic of an ideal diode
is shown in Fig. 46.
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 voltampere 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 smallsignal operation,
r.
Fig. 46 An idealdiode characteristic.
V F
Sec. 48
VACUUMDIODE CHARACTERISTICS I 89
an important parameter is the dynamic, incremental, or plate resistance, defined
by
dV P
r„ =
dh
(414)
This dynamic forward resistance will also be designated by R f . Of course,
if the voltampere 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 smallsignal 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
lowresistance regions) is not sharp, and may not occur at zero applied voltage.
5. As already mentioned in Sec. 45, the voltampere characteristic is not
strictly spacechargelimited, but does depend somewhat upon the filament
temperature. Experiment reveals that there is a shift in the voltage at con
stant current of about 0.1 V for a 10 percent increase in the healer voltage.
The higher the filament voltage, the more the curves shift to the left, because
the increase in the initial velocities of the electrons with increase in tempera
ture results in higher currents at a given voltage. The shift with tube replace
ment or tube aging is found in practice to be of the order of ± 0.25 V.
6. Since a diode consists of two metallic electrodes (a cathode and an
anode) separated by a dielectric (a vacuum), this device constitutes a capaci
tor. The order of magnitude of this capacitance is 5 pF. To this value must
be added the wiring capacitance introduced when the diode is inserted into a
circuit.
48
RATING OF VACUUM DIODES
The rating of a vacuum diode, i.e., the maximum current that it may normally
carry and the maximum potential difference that may be applied between the
cathode and the anode, is influenced by a number of factors.
1. The plate eurrent cannot exceed the thermionicemission current,
2. In order that the gas adsorbed by the glass walls should not be liber
ated, the temperature of the envelope must not be allowed to exceed the tem
perature to which the tube was raised in the outgassing process.
3. The most important factor limiting the rating of a tube is the allowa
ble temperature rise of the anode. When a diode is in operation, the anode
becomes heated to a rather high temperature because of the power (IpV P )
hat must be dissipated by the anode. The temperature of the anode will rise
Un til the rate at which the energy supplied to the anode just equals the rate
at which the heat is dissipated from the anode in the form of radiation. Conse
90 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 49
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 higherpower operation.
These anodes may be operated at a cherryred 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 forcedaircooling radiator fins attached to the anode.
4. The voltage limitation of a highvacuum 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, highvoltage 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 highvoltage 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 peakinversevoltage rating.
Commercial vacuum diodes are made to rectify currents at very high
voltages, up to about 200,000 V. Such units are used with xray equipment,
highvoltage cabletesting equipment, and highvoltage 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).
49
THE DIODE AS A CIRCUIT ELEMENT
The basic diode circuit of Fig. 47 consists of the tube in series with a load
resistance Rl and an inputsignal source i»,. Since the heater plays no part
in the analysis of the circuit, it has been omitted from Fig. 47, and the diode
is indicated as a twoterminal 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. 49
VACUUMDtODE CHARACTERISTICS / 91
Fig, 47 The basic diode circuit.
v. »
®
*>
Ri. > v »
T
The Load Line From Kirehhoff's voltage law,
vp = Vi — ipRi,
(415)
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. 44). In Fig. 48a is indicated the simultane
ous solution of Eq. (415) and the diode plate characteristic. The straight
line, which is represented by Eq. (415), 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 voltampere 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. 486. As y, changes, the slope of the load line does not
vary since R L is fixed. Thus, when the applied potential has the value v' i} the
corresponding current is iV This current is plotted vertically above v t at B'.
The resulting curve OB'B that is generated as v* varies is the dynamic
characteristic.
It is to be emphasized that, regardless of the shape of the static charac
teristic or the waveform of the input voltage, the resulting waveform of the
current in the output circuit can always be found graphically from the dynamic
characteristic. This construction is indicated in Fig. 49. The inputsignal
Waveform (not necessarily sinusoidal) is drawn with its time axis vertically
92 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 49
Static curve
Dynamic
(«)
<*>
Fig. 48 (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. 49 indicates that, for
negative input voltages, zero output current is obtained. If the dynamic
Output current
b / g
Fig. 49 The method of
obtaining the outputcurrent
waveform from the dynamic
curve for a given input
voltage waveform.
Sec
49
VACUUMDIODE CHARACTERISTICS / 93
characteristic is linear, the output voltage v = IpRl is an exact replica of the
input voltage Vi except that the negative portion of ty is missing. In this
application the diode acts as a clipper. If the diode polarity is reversed, the
positive portion of the input voltage is clipped. The clipping level need not
be at zero {or ground) potential. For example, if a reference battery Vr is
added in series with R L of Fig. 47 (with the negative battery terminal at
ground), signal voltages smaller than V R will be clipped. Many other wave
shaping circuits 8 employ diodes.
One of the most important applications of a diode is rectification. If the
input voltage is sinusoidal, the output consists of only positive sections (resem
bling half sinusoids). The important fact to note is that, whereas the average
value of the input is zero, the output contains a nonzero dc value. Hence
rectification, or the conversion from alternating to direct voltage, has taken
place. Practical rectifier circuits are discussed in Chap. 20. Diodes also find
extensive application in digital computers 8 and in circuits used to detect radio
frequency signals.
REFERENCES
1. Dushman, S., and J. W. Ewald: Electron Emission from Thoriated Tungsten, Pkys.
Rev., vol. 29, pp. 857870, June, 1927.
2. Blewett, J. P.: Oxide Coated Cathode Literature, 19401945, J. Appl. Phys., vol.
17, pp. 643647, August, 1946.
Eisenstein, A. S.: Oxide Coated Cathodes, "Advances in Electronics," vol. 1, pp.
164, Academic Press Inc., New York, 1948.
Hermann, G., and S. Wagner: "The Oxidecoated 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. 647654, August, 1946.
Sproull, R. L.: An Investigation of Shorttime Thermionic Emission from Oxide
coated Cathodes, Phys. Rev., vol. 67, pp. 166178, 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. 450486, December, 1913.
5. Child, C. D.: Discharge from Hot CaO, Phys. Rev., vol. 27, pp. 492511, 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. 191257,
April, 1931.
7  Fry, T. C: The Thermionic Current between Parallel Plane Electrodes; Velocities
of Emission Distributed According to Maxwell's Law, Phys. Rev., vol. 17, pp. 441
452, April, 1921.
94 / ELECTRONIC DEVICES AND CIRCUITS
Fry, T. C. : Potential Distribution between Parallel Plane Electrodes, ibid., vol. 22,
pp. 445446, 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. 419435, April, 1923.
8. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," McGraw
Hill Book Company, New York, 1965.
5 /CONDUCTION
IN SEMICONDUCTORS
In Chap. 2 we consider the energyband 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.
51 ELECTRONS AND HOLES IN AN
INTRINSIC SEMICONDUCTOR 1
From Eq. (33) 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. 51. Germanium has a total of 32 electrons in its atomic
structure, arranged in shells as indicated in Table 22. As explained
in Sec. 210, each atom in a germanium crystal contributes four valence
electrons, so that the atom is tetravalent. The inert ionic core of the
germanium atom carries a positive charge of +4 measured in units
95
96 / ELECTRONIC DEVICES AND CIRCUITS
Covalent r Valence
oec. o* J
Ge
• *— * v aience
bond. Ge /[electrons
Je*H ■ *G
» \
Ge
it i i
Fig. 5T Crystal structure of germanium,
illustrated symbolically in two dimensions.
>Ge
Ge
of the electronic charge. The binding forces between neighboring atoms
result from the fact that each of the valence electrons of a germanium atom is
shared by one of its four nearest neighbors. This electronpair, or covalent,
bond is represented in Fig. 51 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. 51 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. 52.
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. 52, and such an incomplete covalent
S*
52
CONDUCTION IN SEMICONDUCTORS / 97
Kg. 53 Th e mechanism by
w tiich a hole contributes to
the conductivity.
(a)
(6)
o
10
O
bond is called a hole. The importance of the hole is that it may serve as a
carrier of electricity comparable in effectiveness to the free electron.
The mechanism by which a hole contributes to the conductivity is quali
tatively as follows: When a bond is incomplete so that a hole exists, it is
relatively easy for a valence electron in a neighboring atom to leave its covalent
bond to fill this hole. An electron moving from a bond to fill a hole leaves a
hole in its initial position. Hence the hole effectively moves in the direction
opposite to that of the electron. This hole, in its new position, may now be
filled by an electron from another covalent bond, and the hole will correspond
ingly move one more step in the direction opposite to the motion of the elec
tron. Here we have a mechanism for the conduction of electricity which does
not involve free electrons. This phenomenon is illustrated schematically in
Fig. 53, where a circle with a dot in it represents a completed bond, and an
empty circle designates a hole. Figure 53o 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. 536 results.
If we compare this figure with Fig. 53a, 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 holeelectron
pairs, whereas other holeelectron pairs disappear as a result of recombination.
Ge*
Ge*
Ge
52
CONDUCTIVITY OF A SEMICONDUCTOR
Fig. 52 Germanium crystal with a
broken covalent bond.
With each holeelectron pair created, two chargecarrying "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. 31)
J ~ (nun + pn P )e£ = <rS
w here n — magnitude of freeelectron (negative) concentration
P = magnitude of hole (positive) concentration
f = conductivity
(51)
98 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 52
Sac 53
CONDUCTION IN SEMICONDUCTORS / 99
Hence
<r « (nfin + pju P )e (52)
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 holeelectron
pair for every 2 X 10 9 germanium atoms. With increasing temperature, the
density of holeelectron pairs increases [Eq. (521)], 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 (53)
The constants E 0Q , n n , /*„, and many other important physical quantities for
germanium and silicon are given in Table 51.
The conductivity of germanium (silicon) is found from Eq. (53) 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 microwavefrequency 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 57 Properties of germanium and silicon!
Property
Atomic number
Atomic weight
Density, g/cm 3 .
Dielectric constant (relative)
Atoms/cm 1
Ego, eV, at 0"K
Eg, eV, at 300°K
«, at 300°K, cm"'
Intrinsic resistivity at 30Q°K, Jlcm .
}i n , cm V Vsec
M P , cmVVsec
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. 17941806, December, 1955. E. M.
Conwell, Properties of Silicon and Germanium, Part II, Proc. IRE, vol.
46, no. 6, pp. 12811299, 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.
53 CARRIER CONCENTRATIONS IN AN
INTRINSIC SEMICONDUCTOR
In order to calculate the conductivity of a semiconductor from Eq. (52) it is
necessary to know the concentration of free electrons to and the concentration
of holes p. From Eqs. (36) and (37), with E in electron volts,
dn = N(E)f{E) dE (54)
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. 36 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. (38) must be generalized as follows:
N(E) = y(E  Ec)* (55)
The Fermi function f(E) is given by Eq. (310), namely,
f(E) =
I __ ^EErWiT
(56)
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. 310). 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. 54. This diagram shows the FermiDirac
distribution of Eq. (56) superimposed on the energyband diagram of a semi
conductor. At absolute zero (T = 0°K) the function is as shown in Fig. 54o.
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. 546 marked "T = 1000°K."
The concentration of electrons in the conduction band is, from Eq. (54),
n = j~ c N{E)f{E)dE
(57)
100 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 53
and
For E> E C ,E — E F y>kT and Eq. (56) reduces to
f(E) = z( E  E rW T
n = (" y(E  E c )h^ B ^' kT dE
J Ee
This integral evaluates to
where
L.60 X 10" I9 )i = 2 ( %
fc..{S«v
(58)
(■39)
(510)
In deriving this equation the value of y from Eq. (39) 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. 15.) 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, 54 FermiDirac distribution and energyband diagram for
an intrinsic semiconductor, (a) T = 0°K and (b) T = 300°K and
T m 1000°K.
Sbc. 53
CONDUCT/ON IN SEMICONDUCTORS / 101
the lattice structure. In a perfect crystal these imaginary particles respond
only to the external fields.
In conclusion, then, the effectivemass 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.
(55) J is given by
N(E)  y(E v  E)
— FM
(511)
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. (56) for f{E), we obtain
,{Ee r ) ikT
(512)
1  f(E) =
I + € IEE,)I
e {ErEyikT
where we have made use of the fact that E F — E y> kT for E < E r (Fig. 54).
Hence the number of holes per cubic meter in the valence band is
(513)
p = J* v m y(E r  E)*<i*r*iw dE
This integral evaluates to
p = N ¥tL (£r~E v MT (514)
where N v is given by Eq. (510), 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. (59) and (514) 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. (59) and (514)
N ci ~{E c E r )ikT = gffa»jrWfim
■taking the logarithm of both sides, we obtain
flic E(j f Ey — lEp
(515)
In
a
N,
kT
ence
E = Ec + Ey kT Nc
F 2 2 N v
(516)
102 / HECTRONfC DEVICES AND CIRCUITS
Sec. 54
If the effective masses of a hole and a free electron are the same, N~c = Mr,
and Eq. (516) yields
Ep =
(517)
Hence the Fermi level lies in the center of the forbidden energy band, as shown
in Fig. 54.
The Intrinsic Concentration Using Eqs. (59) and (514), we have for
the product of electronhole concentrations
np = N c N v e< E ^ E r» kT = N c Nv* B °< kT
(518)
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 (518)
is valid for either an extrinsic or intrinsic material. Hence, writing n = n, and
p — pi = n,, we have the important relationship (called the massaction law)
np = nc
(519)
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. (510), we obtain
N c  4.82 X 1Q 21
&)'
Ti
(520)
where Nc has the dimensions of a concentration (number per cubic meter).
Note that Nv is given by the righthand side of Eq. (520) with m n replaced by
m p . From Eqs. (518) to (520),
np m n t * = (2.33 X 10")
/ m n m p \l
'pi^BalkT
(521)
As indicated in Eqs. (215) and (216), the energy gap decreases linearly with
temperature, so that
Eq — Ego — &T
(522)
where E co is the magnitude of the energy gap at 0°K. Substituting this
relationship into Eq. (521) 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 51.
(523)
The measured values of m
54
DONOR AND ACCEPTOR IMPURITIES
If, to pure germanium, a small amount of impurity is added in the form of a
substance with five valence electrons, the situation pictured in Fig. 55 results.
S»C'
s4
CONDUCTION IN SEMICONDUCTORS / 103
G e /Free electron
4
Fig 55 Crystal lattice with a germanium
atom displaced by a pentavalent impurity
atom.
The impurity atoms will displace some of the germanium atoms in the crystal
lattice. Four of the five valence electrons will occupy covalent bonds, and
the fifth will be nominally unbound and will be available as a carrier of current.
The energy required to detach this fifth electron from the atom is of the order
of only 0.01 eV for Ge or 0.05 eV for Si. Suitable pentavalent impurities are
antimony, phosphorus, and arsenic. Such impurities donate excess (negative)
electron carriers, and are therefore referred to as donor, or ntype, 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. 56. 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 rctype 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 56 Energyband diagram of
n "type semiconductor.
Conduction band
0.01 eV
T
E<
,
1
8
c
N
Eg Donor energy level
4
*i
Valence band
. . 1
104 / ELECTRONIC DEVICES AND CIRCUITS
See. 54
Ge
Ge
Q«
\«/
/ • / •\In
Hole
>Ge
Fig. 57 Crystal lattice with a germa
nium atom displaced by an atom of a
trivalertt impurity.
' ! '" J  ■ !
• • •
Ge
• Ge
semiconductor, only three of the covalent bonds can be filled, and the vacancy
that exists in the fourth bond constitutes a hole. This situation is illustrated
in Fig. 57. Such impurities make available positive carriers because they
create holes which can accept electrons. These impurities are consequently
known as acceptor, or ptype impurities. The amount of impurity which must
be added to have an appreciable effect on the conductivity is very small. For
example, if a donortype 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 ptype, 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. 58. Since a very small amount of energy
is required for an electron to leave the valence band and occupy the acceptor
energy level, it follows that the holes generated in the valence band by these
electrons constitute the largest number of carriers in the semiconductor
material.
We have the important result that the doping of an intrinsic semiconductor
not only increases the conductivity, but also serves to produce a conductor in
which the electric carriers are either predominantly holes or predominantly
electrons. In an ntype semiconductor, the electrons are called the majority
carriers, and the holes are called the minority carriers. In a ptype material,
the holes are the majority carriers, and the electrons are the minority carriers.
r— ■ —
Conduction band
/ Acceptor energy level
 «^^ o.„!.v
E,
fi i
Fig. 58 Energyband diagram of
ptype semiconductor.
S«c S6
CONDUCTION IN SEMICONDUCTORS / 105
5 _ 5 CHARGE DENSITIES IN A SEMICONDUCTOR
Equation (519), namely,
np = n, 2 (519)
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 positivecharge 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 negativecharge density is N A + n. Since the semiconductor is
electrically neutral, the magnitude of the positivecharge density must equal
that of the negative concentration, or
N D +p = N A + n (524)
Consider an ntype material having N A = 0. Since the number of elec
trons is much greater than the number of holes in an ntype semiconductor
(n^>p), then Eq. (524) reduces to
n~ N D (525)
In an ntype material the freeelectron concentration is approximately equal to
the density of donor atoms.
In later applications we study the characteristics of n and ptype 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 ntype or a
ptype substance, respectively. Thus Eq. (525) is more clearly written
n n = Nd
(526)
The concentration p» of holes in the ntype semiconductor is obtained from
Eq. (519), which is now written n n p n = n< 2 . Thus
n,*
Similarly, for a ptype semiconductor,
tt„p p = n, a
**Nj
(627)
(528)
56
FERMI LEVEL IN A SEMICONDUCTOR HAVING IMPURITIES
r ° m Eqs. (51) and (52) it is seen that the electrical characteristics of a semi
conductor material depend on the concentration of free electrons and holes.
106 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 56
The expressions for n and p are given by Eqs. (59) and (514), respectively,
and these are valid for both intrinsic semiconductors and semiconductors with
impurities. The only parameter in Eqs. (59) and (514) 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 donortype
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
electronhole 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. 59a for an ntype 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 ptype material, as indicated in Fig. 596.
If for a given concentration of impurities the temperature of, say, the «type
material increases, more electronhole 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 ntype or ptype 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 ntype material
*t
Conduction band
Kb f
*
Ea
"
Valeni
:« band
5 1
(a)
f{E)
Fig. 5'9 Positions of Fermi level in (a) ntype and (b) ptype
semiconductors.
S«. 57
CONDUCTION IN SEMICONDUCTORS / 107
can be made if we substitute n = N" D from Eq. (525) into Eq. (59). We
obtain
tf D m N c e iE c~ £ * VkT (529)
or solving for E F ,
E F = E c  kT In ^
(530)
Similarly, for ptype material, from Eqs. (528) and (514) we obtain
E F = E v + kT In
Ni
(531)
Note that, if N A  Nd, Eqs. (530) and (531) added together (and divided
by 2) yield Eq. (516).
57
DIFFUSION
In addition to a conduction current, the transport of charges in a semiconductor
may be accounted for by a mechanism called diffusion, not ordinarily encoun
tered in metals. The essential features of diffusion are now discussed.
We see later that it is possible to have a nonuniform concentration of
particles in a semiconductor. Under these circumstances the concentration p
of holes varies with distance x in the semiconductor, and there exists a concen
tration gradient dp/dx in the density of carriers. The existence of a gradient
implies that, if an imaginary surface is drawn in the semiconductor, the density
of holes immediately on one side of the surface is larger than the density on
the other side. The holes are in a random motion as a result of their thermal
energy. Accordingly, holes will continue to move back and forth across this
surface. We may then expect that, in a given time interval, more holes will
cross the surface from the side of greater concentration to the side of smaller
concentration than in the reverse direction. This net transport of charge
across the surface constitutes a flow of current. It should be noted that this
net transport of charge is not the result of mutual repulsion among charges
°f like sign, but is simply the result of a statistical phenomenon. This dif
fusion is exactly analogous to that which occurs in a neutral gas if a concen
tration gradient exists in the gaseous container. The diffusion hole current
density J p (amperes per square meter) is proportional to the concentration
gradient, and is given by
'•  »•%
(532)
nere D p (square meters per second) is called the diffusion constant for holes.
similar equation exists for diffusion electroncurrent density [p is replaced
y n > and the minus sign is replaced by a plus sign in Eq. (532)]. Since both
108 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 5*8
diffusion and mobility are statistical thermodynamic phenomena, D and /* are
not independent. The relationship between them is given by the Einstein
equation
Mp Mr.
(533)
where V T = IcT/e = T/l 1,600 is defined as in Eq, (334). 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 51, on page 98.
58
CARRIER LIFETIME
In Sec. 51 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 holeelectron pairs while other boleelectron 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 ntype 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 ntype specimen, we have
p w  p w = n M  n no (534)
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
(535)
(536)
We should emphasize here that majority and minority carriers in a specific
region of a given specimen have the same lifetime t. Using Eqs. (535) and
(536), we can obtain the expressions for the rate of concentration change.
For holes, we find from Eq. (535)
d]>n _ Pn — pno _ d_
dt T ~ dt {Pn Vno)
(537)
For electrons, a similar expression with p replaced by n is valid. The quantity
p n — Pno represents the injected, or excess, carrier density. The rate of change
S«
59
CONDUCTION IN SEMICONDUCTORS / 109
f excess density is proportional to the density — an intuitively correct result.
The minus sign indicates that the change is a decrease in the case of recombi
nation and an increase when the concentration is recovering from a temporary
depletion.
The most important mechanism through which holes and electrons recom
bine is the mechanism involving recombination centers** which contribute
electronic states in the energy gap of the semiconductor material. These new
states are associated with imperfections in the crystal. Specifically, metallic
impurities in the semiconductor are capable of introducing energy states in the
forbidden gap. Recombination is affected not only by volume impurities, but
also by surface imperfections in the crystal.
Gold is extensively used as a recombination agent by semiconductor
device manufacturers. Thus the device designer can obtain desired carrier
lifetimes by introducing gold into silicon under controlled conditions. 78
59
THE CONTINUITY EQUATION
In the preceding section it is seen that if we disturb the equilibrium concen
trations of carriers in a semiconductor material, the concentration of holes or
electrons will vary with time. In the general case, however, the carrier con
centration in the body of a semiconductor is a function of both time and dis
tance. We now derive the differential equation which governs this functional
relationship. This equation is based upon the fact that charge can be neither
created nor destroyed. Consider the infinitesimal element of volume of area A
and length dx (Fig. 510) 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 —
(538)
If g is the thermal rate of generation of holeelectron pairs per unit volume,
the number of coulombs per second
Increases within the volume = eA dx g (539)
Fig, 510 Relating to the conservation of
charge.
I+dl
x + dx
110 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 59
In general, the current will vary with distance within the semiconductor. If,
as indicated in Fig. 510, 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 (540)
Because of the three effects enumerated above, the hole density must change
with time, and the total number of coulombs per second
Increases within the volume = eA dx ~
dt
Since charge must be conserved,
eAdx~ = —eA dz — + eA dx a — dl
dt 7* w
(541)
(542)
The hole current is the sum of the diffusion current (Eq. (532)] and the drift
current [Eq. (51)], or
I = AeD.
dp
dx
+ Apefi p &
(543)
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. (542),
(544)
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. (542), (543), and (544) yields the
equation of conservation of charge, or the continuity equation,
dp
Tt
P  p.
^ U * dx*
th
d{pZ)
dx
(545)
If we are considering holes in the ntype material, the subscript n is added to
p and p . Also, since p is a function of both t and *, partial derivatives should
be used. Making these changes, we have, finally,
dp n
dt
_ _ Pn  p„
+ A
3 2 p„
dx 2
My
3 (p»g)
dx
(546)
We now consider three special cases of the continuity equation.
Concentration Independent of x and with Zero Electric Field We now
derive Eqs. (535) and (537) 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 thermalequilibrium value
Sec.
59
CONDUCTION IN SEMICONDUCTORS /111
At t = the illumination is removed. How does the concentration vary
with time? The answer to this query is obtained from Eq. (546), which now
reduces to
dp* = _ Pn — Pno (547)
dt r p
in agreement with Eq. (537). The solution of this equation is
p„  P*.  (Pnc ~ pno)«r ( '' (548)
which is identical with Eq. (535). 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
(549)
(550)
(551)
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. (550) into Eq.
(549). 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 thermalequilibrium 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)*"% (552)
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. 511 and Eq. (552),
vp.\m<
clL * dx
(553)
112/ RECTRONIC DEVICES AND CIRCUITS
Sec, 59
5JO
CONDUCTION IN SEMICONDUCTORS / 113
Tfetsk
,.,..^, J.
■as*** i
p^. ■ i *
^^■■w^lv'i ;■.'■'■' •
x=0 x
)dP„\ holes
re com bine in
the distance dx
Fig. 511 Relating to the injected hole
concentration in ntype material.
dP» gives the number of injected holes which recombine in the distance
between z and x + dx. Since each hole has traveled a distance x, the total
distance traveled by \dP n \ holes is x \dP n \. Hence the total distance covered by
all the holes is J Q x \dP n \. The average distance x equals this total distance
divided by the total number P„(0) of injected holes. Hence
/;*iip n i !
PM
(554)
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)*»
(555)
where the space dependence of the injected concentration is given by P n (x).
If Eq. (555) is substituted into the continuity equation (546), the result is
or
d 2 P» 1 + jarr,
dx* LJ n
(556)
where use has been made of Eq. (551). At zero frequency the equation of
continuity is given by Eq. (5^49), which may be written in the form
d*I\
dx 3
W
A comparison of this equation with Eq. (556) 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.
510
THE HALL EFFECT 1
If a specimen (metal or semiconductor) carrying a current I is placed in a
transverse magnetic field B, an electric field £ is induced in the direction per
pendicular to both I and B. This phenomenon, known as the Hall effect, is
used to determine whether a semiconductor is n or ptype 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.
512 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 ntype, so that the current is carried by electrons, these
electrons will be forced downward toward side 1 in Fig. 512, 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
(557)
where e is the magnitude of the charge on the carrier, and a is the drift speed.
From Eq. (114), £ = V H /d, where d is the distance between surfaces 1 and 2.
From Eq. (138), J = pv = I/wd, where J is the current density, p is the
charge density, and w is the width of the specimen in the direction of the
magnetic field. Combining these relationships, we find
V H = £d = Bvd =
BJd
H
pw
(558)
If V H , B, I, and w are measured, the charge density p can be determined from
Eq. (558). 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 ptype, and p = pe,
where p is the hole concentration.
It is customary to introduce the Hall coefficient Ru defined by
Rh = 
(559)
Fi 9. 512 Pertaining to the Hall effect.
•he carriers {whether electrons or holes)
ar e subjected to a force in the negative Y
direction.
4=2=35
114 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 510
Hence
Rn =
BI
(560)
If conduction is due primarily to charges of one sign, the conductivity <t
is related to the mobility ju by Eq. (33), or
a = pft (561)
If the conductivity is measured together with the Hall coefficient, the mobility
can be determined from
(i = aRu
(562)
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. (560) remains valid provided that Rn is defined by 3tt/8p. Also,
Eq. (562) must be modified to m = (8<t/3it)Rh.
REFERENCES
1. Shockley, W.: Electrons and Holes in Semiconductors, D. Van Nostrand Company,
Inc., Princeton, N.J., reprinted February, 1963.
Gibbons, J. F.: "Semiconductor Electronics," McGrawHill 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. 170212, January,
1947.
3. Adler, R, B,, A. C. Smith, and R. L. Longini: "Introduction to Semiconductor
Physics," vol, 1, Semiconductor Electronics Education Committee, John Wiley &
Sons, Inc., New York, 1964.
4. Conwell, E. M.: Properties of Silicon and Germanium: II, Proc. IRE, vol. 46, pp.
12811300, June, 1958.
5. Shockley, W., and W. T, Read, Jr. : Statistics of the Recombination of Holes and
Electrons, Phys. Rev., vol 87, pp. 835842, September, 1952.
6. Hall, R. N.: ElectronHole Recombination in Germanium, Phys. Rev., vol. 87, p. 387,
July, 1952.
7. Collins, C. B., R. O, Carlson, and C. J. Gallagher: Properties of Golddoped Silicon,
Phys. Rev., vol. 105, pp. 11681173, February, 1957.
8. Bemski, G.: Recombination Properties of Gold in Silicon, Phys, Rev,, vol. Ill, pp
15151518, September, 1958.
SEMICONDUCTORDIODE
CHARACTERISTICS
In this chapter we demonstrate that if a junction is formed between
a sample of ptype and one of ntype semiconductor, this combination
possesses the properties of a rectifier. The voltampere 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.
61
QUALITATIVE THEORY OF THE pn JUNCTION 1
If donor impurities are introduced into one side and acceptors into the
other side of a single crystal of a semiconductor, say, germanium, a
pn junction is formed. Such a system is illustrated in Fig. 6la.
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 ptype carriers to the left of the junction and only
fttype carriers to the right. Because there is a density gradient across
the junction, holes will diffuse to the right across the junction, and
electrons to the left.
As a result of the displacement of these charges, an electric field
will appear across the junction. Equilibrium will be established when
the field becomes large enough to restrain the process of diffusion.
The general shape of the charge distribution may be as illustrated in
115
116 / ELECTRONIC DEVICES AND CIRCUITS
5«c. 61
Acceptor
Hole
(«>
Junction
4
Donor
ion
e © e e
o o o
© e e e
o o o
e e e e
© © © ©
• • •
© © © ©
• • ■
© © © ©
Electron
p type
n type
Distance from junction
ib)
•HE/
(c)
U>
(e)
Electrostatic potential V or
potential energy barrier for holes
j Distance from junction
Potential energy barrier for electrons
i
Distance from junction
Fig. 61 A schematic diagram of a pn junction, including the
charge density, electric field intensity, and potentialenergy
barriers at the junction. (Not drawn to scale.)
Sec. 62
SEMICONDUCTORDIODE CHARACTERISTICS / 117
Fig 61& 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 ptype germanium have
disappeared as a result of combination with electrons which have diffused
across the junction. Similarly, the neutralizing electrons in the ntype 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 spacecharge 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. 61 c. Note that this curve is the integral of the density func
tion p in Fig. 61&. The electrostaticpotential variation in the depletion
region is shown in Fig. 6ld, and is the negative integral of the function 8
of Fig. 6lc. This variation constitutes a potentialenergy barrier against the
further diffusion of holes across the barrier. The form of the potentialenergy
barrier against the flow of electrons from the n side across the junction is
shown in Fig. 6le. It is similar to that shown in Fig. 6ld, 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 opencircuited 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. (68)] in terms of the donor and acceptor
concentrations. The numerical value for V a is of the order of magnitude of a
few tenths of a volt.
6 " 2 THE pn JUNCTION AS A DIODE
Th
e essential electrical characteristic of a pn 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. 62, a battery is shown connected across the
ttiinals of a pn junction. The negative terminal of the battery is con
118/ ELECTRONIC DEVICES AND CIRCUITS
Sec. 62
S«. 62
SEMICONDUCTORDIODE CHARACTERISTICS / 119
Metal ohmic contacts
<^jy
V
(a)
Hp
Fig. 62 (o) A pn junction biased in the
reverse direction, (b) The rectifkr symbol
is used for the pn diode.
V
(6)
nected to the p side of the junction, and the positive terminal to the n side.
The polarity of connection is such as to cause both the holes in the p type and
the electrons in the n type to move away from the junction. Consequently,
the region of negativecharge density is spread to the left of the junction (Fig.
616), and the positivechargedensity 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 ntype germanium. And there are very few holes in the ntype side.
Hence, nominally, zero current results. Actually, a small current does flow
because a small number of holeelectron pairs are generated throughout the
crystal as a result of thermal energy. The holes so formed in the ntype ger
manium will wander over to the junction. A similar remark applies to the
electrons thermally generated in the ptype 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. (628)],
and hence the back resistance of a crystal diode decreases with increasing
temperature.
The mechanism of conduction in the reverse direction may be described
alternatively in the following way: When no voltage is applied to the pn
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. 62, the
height of the potentialenergy 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. 62 is called the reverse, or
blocking, bias.
Forward Bias An external voltage applied with the polarity shown in
Fig. 63 (opposite to that indicated in Fig. 62) is called a forward bias. An
ideal pn 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 potentialenergy barrier at the junction
Fig. 63 (a) A pn junction biased in the
forward direction, (b) The rectifier sym
bol is used for the pn diode.
/Metal contacts.
V
(a)
V
(b)
will be disturbed. Hence, for a forward bias, the holes cross the junction
from the p type to the n type, and the electrons cross the junction in the
opposite direction. These majority carriers can then travel around the closed
circuit, and a relatively large current will flow.
Ohmic Contacts 1 In Fig. 62 (63) we show an external reverse (forward)
bias applied to a pn 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. 62 and 63 we indicate metal contacts
with which the homogeneous ptype and ntype materials are provided. We
thus see that we have introduced two metalsemiconductor 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. 62 and 63 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 metalsemiconductor contacts are lowresistance 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 pn junction.
The Shortcircuited and Opencircuited pn Junction If the voltage V
^ Fig. 6_2 or 63 were set equal to zero, the pn 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 opencircuit 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 pn bar. The semiconductor bar, therefore, would have to cool off.
ear ly, under thermal equilibrium the simultaneous heating of the metal and
120 / ELECTRONIC DEVICES AND CIRCUITS
See. 63
cooling of the bar is impossible, and we conclude that I = 0. Since under
shortcircuit conditions the sum of the voltages around the closed loop must
be zero, the junction potential V must be exactly compensated by the
metaltosemiconductor 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.
63 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 pn junction. We conclude that, as the for
ward voltage V becomes comparable with V , the current through a real pn
diode will be governed by the ohmiccontact resistances and the crystal bulk
resistance. Thus the voltampere characteristic becomes approximately a
straight line.
63
BAND STRUCTURE OF AN OPENCIRCUITED pn JUNCTION
As in the previous section, we here consider that a pn junction is formed by
placing p and ntype 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. 56 it is verified that the Fermi level E F is closer to
the conduction band edge E Cn in the ntype 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 energyband diagram for
a pn 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
(61)
This energy E g represents the potential energy of the electrons at the junction,
as is indicated in Fig. 61 e.
Sac. 63
SEMICONDUCTORDIODE CHARACTERISTICS / 121
Fig. 64 Band diagram for a pn junction under opencircuit condi
tions. This sketch corresponds to Fig. 61 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. 64 we see that
Ep — Ey. = t%Eq — Ei
and
Ecn — Ep = \Ea — E%
Adding these two equations, we obtain
E„ = E x + E 2 = Eo  (E c «  Ep)  (E F  E Vp )
From Eqs. (518) and (519),
E s = kT In NcNv
From Eq. (530),
nr
E Cn E F = kT In '
p rom Eq. (531),
Ep  E Vp = kT In £p
(62)
(63)
(64)
(65)
(66)
(67)
122 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 63
S9C
6*A
SEMICONDUCTORDIODE CHARACTERISTICS / 123
Substituting from Eqs. (65), (66), and (67) in Eq. (64) yields
£, = fer f m ^_ ln ^_ ln £A
= kTlJ^^XA = kTln?*4± (68)
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. (526), (527),
and (528) in Eq. (68). We find
E = kT In ^ = kT In —
(69)
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. 61 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. (532)] must equal the hole drift current [Eq. (32)], or
eDp dx = epLpp& (6 ~ 10)
The Einstein relation [Eq. (533)] is
^ = V T (611)
where the volt equivalent of temperature V T is defined by Eq. (334). Substi
tuting Eq. (611) in Eq. (610) and remembering the relationship (115)
between field intensity and potential, we obtain
dp _ &dx _ dV
p Vt Vt
(612)
If this equation is integrated between limits which extend across the junction
(Fig. 6ld) 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. (613) is equivalent to Eq. (69).
(613)
Fig. 65 The hole and
electroncurrent compo
nents vs. distance in a pn
junction diode. The space
charge region at the Junc
tion is assumed to be
negligibly small.
Ipp, hole current
I npt electron current
(6)
jc=0
Total current J
7 BB , electron current
I pn , hole current
Distance
64
THE CURRENT COMPONENTS IN A pn DIODE
In Sec. 62 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. (550)]. Since the diffusion current of minority carriers is
proportional to the concentration gradient [Eq. (532)], this current must also
vary exponentially with distance. There are two minority currents, /„* and
In P , and these are indicated in Fig. 65. 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
(614)
.„ 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. 65. 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)
(615)
This current is plotted as a function of distance in Fig. 65, as is also the
corresponding electron current /„„ in the n material. This figure is drawn for
a n unsym metrically doped diode, so that /,,„ ^ l np .
Note that deep into the p side the current is a drift (conduction) current
ipp of holes sustained by the small electric field in the semiconductor. As the
t If the letters p and n both appear in a symbol, the first letter refers to the type of
carrier, and the second to the type of material.
124 / ELECTRONIC DEVICES AND CIRCUITS
See. 65
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 forwardbiased pn 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 pn 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. 65.
65
QUANTITATIVE THEORY OF THE pn DIODE CURRENTS
We now derive the expression for the total current as a function of the applied
voltage (the voltampere characteristic). In the discussion to follow we neg
lect the depletionlayer 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 thermalequilibrium value p no and, as indicated in Eq.
(552), is given by
y«{x) = Pno + Pn(0)r* ,L >
(616)
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.
(617)
These several holeconcentration components are indicated in Fig. 66, which
shows the exponential decrease of the density p n (x) with distance x into the
n material.
From Eq. (532) the diffusion hole current in the n side is given by
■* «n —
■**.£
(618)
Taking the derivative of Eq. (616) and substituting in Eq. (618), we obtain
I P n(x)  AeD P P »W € *IL t (649)
Li p
This equation verifies that the hole current decreases exponentially with dis
tance. The dependence of I pn upon applied voltage is contained implicitly in
the factor P»(0) because the injected concentration is a function of voltage.
We now find the dependence of P„(0) upon V.
Sec
65
SEMICONDUCTORDIODE CHARACTERISTICS / 125
I Concentration, p„
Rg. 66 Defining the
several components of hole
concentration in the n side
f a forwardbiased diode.
The diagram is not drawn
to scale since p„(0) » p n »
M0>
n material
Injected or excess charge
Distance
The Law of the Junction If the hole concentrations at the edges of the
spacecharge 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
(620)
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 lowlevel injection, we may to a good approximation again equate the
magnitudes of the diffusion and drift currents. Starting with Eqs. (610) and
(612) and integrating over the depletion layer, Eq. (620) is obtained.
If we apply Eq. (620) to the case of an opencircuited pn junction, then
Pp ■ Pr°, Pn = p»», and V B = V* Substituting these values in Eq. (620),
it reduces to Eq. (613), 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. 66. At the edge of the depletion layer, x = 0, p»  Pn(0). The
Boltzmann relation (620) is, for this case,
(621)
Pvo = p.(0)«<*.™^
Combining this equation with Eq. (613), 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
(622)
126 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 65
greater than the thermalequilibrium value p„„. A similar law, valid for elec
trons, is obtained by interchanging p and n in Eq. (622).
The hole concentration f\,(0) injected into the n side at the junction is
obtained by substituting Eq. (622) in Eq. (617), yielding
Pn(0) = p«,(e vlv r  1)
(623)
The Forward Currents The hole current 7 pn (0) crossing the junction
into the n side is given by Eq. (619), with x «=• 0. Using Eq. (623) for
.Pti(O), we obtain
V(0)
AeD p p,
( e viv T _ x)
(624)
The electron current I np (0) crossing the junction into the p side is obtained
from Eq, (624) by interchanging n and p, or
AeD n n.
/»p(0) =
L n
**1 ( € VIV T _ J)
(625)
Finally, from Eq. (614), the total diode current I is the sum of I pn (Q) and
^»p(0), or
where
/ = LU^T  1)
(626)
(627)
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. 69).
The Reverse Saturation Current In the foregoing discussion a positive
value of V indicates a forward bias, The derivation of Eq. (626) is equally
valid if V is negative, signifying an applied reversebias 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. (527), (528), and (627), we obtain
Io  Ae \LpW D +l
where n? is given by Eq. (523),
n, 2 = AeThZootkr = A T*e r °° lr r
(628)
(629)
where V GO is a voltage which is numerically equal to the forbiddengap energy
Eqq in electron volts, and Vr is the volt equivalent of temperature [Eq.
(334)]. For germanium the diffusion constants D p and D n vary approxi
SEMICONDUCTORDIODE CHARACTERISTICS / 127
mately 3 inversely proportional to T. Hence the temperature dependence
of L is
I. = JttSfVWS* (630)
where K\ is a constant independent of temperature.
Throughout this section we have neglected carrier generation and recombi
nation in the spacecharge 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 transitionlayer chargegeneration 3  4
current, which is given approximately by
I = J ( € viiv r  1) (631)
where 57 « 2 for small (rated) currents and j? « 1 for large currents. Also,
/„ is now found to be proportional to n, instead of n*. Hence, if Kt is a
constant,
I = K t T l *c v «>i* v T (632)
The practical implications of these diode equations are given in the
following sections.
66
THE VOLTAMPERE CHARACTERISTIC
The discussion of the preceding section indicates that, for a pn junction, the
current / is related to the voltage V by the equation
I = I (€ v ^ v r  1) (633)
A positive value of I means that current flows from the p to the n side. The
diode is forwardbiased 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. (334), repeated here for convenience:
V T =
(634)
11,600
At room temperature (T = 300°K), V T = 0.026 V = 26 mV.
The form of the voltampere characteristic described by Eq. (633) is
shown in Fig. 67a. When the voltage V is positive and several times V T ,
the unity in the parentheses of Eq. (633) 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 reversebiased and F is
several times V T , I m —U. The reverse current is therefore constant, inde
pendent of the applied reverse bias. Consequently, h is referred to as the
reverse saturation current.
128 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 66
LmA
e
5
4
3
2
1
♦ 0.
0.5
1.0
A
v,v
(b)
Fig. 67 (a) The voltampere characteristic of an ideal pn diode, (fa) The
voltampere 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. 67a 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. 676, to use two different current scales.
The voltampere 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. 676 indicates that, at a reverse
biasing voltage V Zi the diode characteristic exhibits an abrupt and marked
departure from Eq. (633). At this critical voltage a large reverse current
flows, and the diode is said to be in the breakdown region, discussed in Sec. 612.
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 voltampere characteristics is
brought out in Fig. 68. Here are plotted the forward characteristics at room
temperature of a generalpurpose 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. 68 is that there
exists a cutin, offset, breakpoint, 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. 68 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 silicondiode characteristic is offset about
0.4 V with respect to the break in the germaniumdiode characteristic. The
Sec. 66
SEMICONDUCTORDIODE CHARACTERISTICS / 129
/,mA
100
SO
60
40
20
/
J
Ge/
Si/
0.2 0.4 0.6 0.8 1.0 V,V
Fig. 68 The forward voltampere characteristics of a
germanium (1N270) and a silicon (1N3605) diode at
25°C.
reason for this difference is to be found, in part, in the fact that the reverse
saturation current in a germanium diode is normally larger by a factor of
about 1,000 than the reverse saturation current in a silicon diode of com
parable ratings. Thus, if T is in the range of microamperes for a germanium
diode, I will be in the range of nanoamperes for a silicon diode.
Since t) = 2 for small currents in silicon, the current increases as t v!2V r for
the first several tenths of a volt and increases as e vlv r only at higher voltages.
This initial smaller dependence of the current on voltage accounts for the
further delay in the rise of the silicon characteristic.
Logarithmic Characteristic It is instructive to examine the family
of curves for the silicon diodes shown in Fig. 69. 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. (633), 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. 69, 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 pn junction. At high currents the externally
T30 / ELECTRONIC DEVICES AND CIRCUITS
I, inA
1,000
500
100
50
10
5
1
0.5
0.1
0.05
A
50°
C
A
l*€
55
°C
Sec. 67
Fig. 69 Voltampere
characteristics at three
different temperatures for
a silicon diode (planar
epitaxial passivated types
1N36G5, 1N3606, 1N3608,
and1N3609). The shaded
area indicates 25°C limits
of controlled conductance.
Note that the vertical scale
is logarithmic and encom
passes a current range of
50,000. (Courtesy of
General Electric Company.)
0.2
0.4
0.6
0.8
1.0
v,v
impressed voltage is called upon principally to establish an electric field to
overcome the ohmic resistance of the semiconductor material. Therefore, at
high currents, the diode behaves more like a resistor than a diode, and the
current increases linearly rather than exponentially with applied voltage.
67 THE TEMPERATURE DEPENDENCE OF pn CHARACTERISTICS
Let us inquire into the diode voltage variation with temperature at fixed
current. This variation may be calculated from Eq. (633), 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. (630) and (632),
given approximately by
L = KT m <r v °oWT
(635)
where if is a constant and eVgo (e is the magnitude of the electronic charge)
is the forbiddengap 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. (635), we find
LdT
d(ln h)
dT
_ m Voo
T "'" v TVt
(636)
At room temperature, we deduce from Eq. (636) 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
SEMICONDUCTORDIODE CHARACTERISTICS / 131
is that, in a physical diode, there is a component of the reverse saturation
current due to leakage over the surface that is not taken into account in Eq.
(635) 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. (633), dropping the unity in comparison with the exponential,
we find, for constant /,
d I = v_
dT T ' T
{ldh\
\I. dT)
V  (Voo + m n V T )
(637)
where use has been made of Eq. (636). 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. (637),
dV
dT
2.1 mV/°C
2.3mV/°C
for Ge
for Si
(638)
Since these data are based on 'average characteristics," it might be well for
conservative design to assume a value of
dV
dT
= 2.5 mV/°C
(639)
for either Ge or Si at room temperature. Note from Eq. (637) that \dV/dT\
decreases with increasing T.
The temperature dependence of forward voltage is given in Eq. (637) as
the difference between two terms. The positive term V/T on the righthand
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. 69 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. 610a and 6. Observe the very pronounced dependence of current on
Verse voltage, a result which is not consistent with our expectation of a con
ant saturated reverse current. This increase in /„ results from leakage across
le surface of the diode, and also from the additional fact that new current
rr iers may be generated by collision in the transition region at the junction.
1 the other hand, there are man}" commercially available diodes, both ger
an nun and silicon, that do exhibit a fairly constant reverse current with
132 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 68
1.0
0.1
F *"""^
W
r
p^*"'*
55°
C
25
°C
_ i
5"
c
2
40 60 SO 100
o.r.
a 0.05
150"C
100"C
25
a C
Reverse voltage, V
(a)
30 60 90 120
Reverse voltage, V
150
Fig. 610 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. 610a 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. 610 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.
68
DIODE RESISTANCE
The static resistance R of a diode is denned as the ratio V/I of the voltage
to the current. At any point on the voltampere characteristic of the diode
(Fig. 67), 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
SEMICONDUCTORDIODE CHARACTERISTICS / 133
ca l values for a silicon planar epitaxial diode are Vr = 0.8 V at /^ = 10 mA
(corresponding to Rf = 80 fl) and Ir = 0.1 mA at Vr = 50 V (corresponding
to Rb= 500 M).
For smallsignal 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. (633) that the dynamic conductance g = 1/r is
_ dJ I*rWT
9 ~ dV V V T
I + I,
nVr
(640)
For a reverse bias greater than a few tenths of a volt (so that F/^Tr 2> l)j
g is extremely small and r is very large. On the other hand, for a forward
bias greater than a few tenths of a volt, 7 » I 0) and r is given approximately by
rfVr
I
(641)
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 largesignal 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. 611. 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
9 611 The piecewise linear character
nation of a semiconductor diode.
Slope,
134 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 69
significance even for this largesignal 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. 68 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. 612, V r = V z , and
Rf is the dynamic resistance in the breakdown region.
69
SPACECHARGE, OR TRANSITION, CAPACITANCE 1 C 7
As mentioned in Sec. 61, a reverse bias causes majority carriers to move away
from the junction, thereby uncovering more immobile charges. Hence the
thickness of the spacecharge layer at the junction increases with reverse volt
age. This increase in uncovered charge with applied voltage may be con
sidered a capacitive effect. We may define an incremental capacitance Ct by
Ct —
dQ
dV
(642)
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
tr dV
(643)
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, spacecharge, barrier, or depletionregion, 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. (642) 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 ntype 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*. 69
SEMICONDUCTORDIODE CHARACTERISTICS / 135
P type
n type
Charge density
Pig, 612 The charge density and
potential variation at a fusion
pn junction (W « 10 4 cm).
x = G
junction. Figure 612 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
(644)
" Na « Ar B; then W p » W „. For simplicity, we neglect W n and assume that
the entire barrier potential Vb appears across the uncovered acceptor ions.
*ne relationship between potential and charge density is given by Poisson's
equation,
d*V
dx 2
eN A
6
(645)
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. 612, and £ = — dV/dx = at x = 0. Also, since the zero
Potential is arbitrary, we choose V = at x = 0. Integrating Eq. (645)
T36 / ELECTRONIC DEVICES AND CIRCUITS
subject to these boundary conditions yields
V = eN * xi
At x = W p « W , V  V B , the barrier height. Thus
V B = ^ W*
Sec. 69
(646)
(647)
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. (642), is
Ct =
dQ
dV
 eN A A
dW
dV
From Eq. (647), \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 parallelplate 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. (647) W represents the total spacecharge width, and 1/N A is replaced
by 1/N A + 1/JW Equation (649) 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 ptype and then ntype
impurities. For such a grown junction the charge density varies gradually
(almost linearly), as indicated in Fig. 613. If an analysis similar to that
Charge density
Fig. 6T3 The chargedensity variation at a
grown pn junction.
SEMICONDUCTORDIODE CHARACTERISTICS / T37
4.U
3.2
2.4
1.6
25° C
■■■■
1N914
1N916
0.8
Fig. 6M Typical barriercapaci
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. (649) is found to be valid
where W equals the total width of the spacecharge 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 spacecharge width W, and hence the smaller
the capacitance C T  The variation is illustrated for two typical diodes in Fig.
614. Similarly, for an increase in forward bias (V positive), W decreases and
C T increases.
The voltage variable capacitance of a pn 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 selfbalancing
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. 615.
ve
to
'9 615 A varactor diode under reverse
,as  (a) Circuit symbol; (b) circuit model.
Rr
AAA
c~4 tf—L
C T
AAAr
R.
0>)
138 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 610
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 reversebiased 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 reversebiased diode will flow through the capacitor
(Fig. 6156).
610
DIFFUSION CAPACITANCE
For a forward bias a capacitance which is much larger than that considered in
the preceding section comes into play. The origin of this capacitance is now
discussed. If the bias is in the forward direction, the potential barrier at the
junction is lowered and holes from the p side enter the n side. Similarly,
electrons from the n side move into the p side. This process of minority
carrier injection is discussed in Sec. 65, where we see that the excess hole
density falls off exponentially with distance, as indicated in Fig. 66. The
shaded area under this curve is proportional to the injected charge. As
explained in Sec. 69, 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. 66 multiplied by the diode cross section A and the electronic charge e.
Hence
Q = f* AeP n (Q)t~iL f dx = AeLJ>„(0)
and
f>  d ® at
dP n (0)
dV —* dV
The hole current / is given by l pR (x) in Eq. (619), with a; = 0, or
, = AeD v P n {u)
(650)
(651)
(652)
Sec 6l°
and
dP n (0) _ _L
dV
dl
AeD P dV AeD,
SEMICONDUCTORDIODE CHARACTERISTICS / 139
(653)
where g = dl/dV is the diode conductance given in Eq. (640). Combining
Eqs. (651) and (653) yields
Since from Eq. (551) the mean lifetime for holes t p = t is given by
7 Z>„
(654)
(655)
(656)
(657)
then
C D = rg
From Eq. (641), 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. (656) 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. 630).
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. (656),
rC D = r (658)
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.
Chargecontrol Description of a Diode From Eqs. (650), (652), and
(655),
1  Qrh =
(659)
Thi
^his very impjrtant equation states that the diode current (which consists of
«oles crossing the junction from the p to the n side) is proportional to the
140 / ELECTRONIC DEVICES AND CIRCUITS
Sac. 6 J J
stored charge Q of excess minority carriers. The factor of proportionality is
the reciprocal of the decay time constant (the mean lifetime t) of the minority
carriers. Thus, in the steady state, the current I supplies minority carriers
at the rate at which these carriers are disappearing because of the process of
recombination.
The chargecontrol characterization of a diode describes the device in
terms of the current I and the stored charge Q, whereas the equivalentcircuit
characterization uses the current J and the junction voltage V. One immedi
ately apparent advantage of this chargecontrol 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 reversebiased. The diode is
forwardbiased if Q is positive and reversebiased if Q is negative.
611 pn DIODE SWITCHING TIMES
When a diode is driven from the reversed condition to the forward state or in
the opposite direction, the diode response is accompanied by a transient, and
an interval of time elapses before the diode recovers to its steady state. The
forward recovery time t fT is the time difference between the 10 percent point
of the diode voltage and the time when this voltage reaches and remains within
10 percent of its final value. It turns out 6 that t /T does not usually constitute
a serious practical problem, and hence we here consider only the more impor
tant situation of reverse recovery,
Diode Reverse Recovery Time When an external voltage forward biases
a pn junction, the steadystate density of minority carriers is as shown in
Fig. 616o (compare with Fig. 66). The number of minority carriers is very
Junction
Junction
P type
n type
Fig. 616 Minoritycarrier density distribution as a function of the distance x
from a junction, (a) A forwardbiased junction; (b) a reversebiased junction.
The injected, or excess, hole (electron) density is p„  p na {n„  n po ).
Sec. oM
SEMICONDUCTORDIODE CHARACTERISTICS / T41
large. These minority carriers have, in each case, been supplied from the other
side of the junction, where, being majority carriers, they are in plentiful supply.
When an external voltage reversebiases the junction, the steadystate
density of minority carriers is as shown in Fig. 6166. Far from the junction
the minority carriers are equal to their thermalequilibrium values p n <> and n po ,
as is also the situation in Fig. 616o. 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 steadystate reversevoltage value. For the current
cannot attain its steadystate value until the minoritycarrier distribution,
which at the moment of voltage reversal had the form in Fig. 6 16a, reduces
to the distribution in Fig. 6166. Until such time as the injected, or excess,
minoritycarrier 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. 617.
We consider that the voltage in Fig. 6176 is applied to the dioderesistor
circuit in Fig. 6I7a. For a long time, and up to the time hi the voltage
Vi = Vr has been in the direction to forwardbias 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. 617c, the injected minoritycarrier 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 storedminority 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 reversebiased junction has
barged through R L to the voltage  V R .
Manufacturers normally specify the reverse recovery time of a diode t„
142 / ELECTRONIC DEVICES AND CIRCUITS
•H—
Rl
(a)
/„ =s
V.
Forwardi .
, storage, t,
bias
Sec. 61 1
h
t
\
(b)
(c)
(d)
(e)
Fig. 617 The waveform in (b) is applied to the diode circuit in (a); (c) the
excess carrier density at the junction; (d) the diode current; (e) the diode
voltage.
in a typical operating condition in terms of the current waveform of Fig.
617<f. The time t„ is the interval from the current reversal at ( = (i until
the diode has recovered to a specified extent in terms either of the diode cur
rent or of the diode resistance. If the specified value of R L is larger than
several hundred ohms, ordinarily the manufacturers will specify the capaci
tance Cl shunting Ri in the measuring circuit which is used to determine t„*
Thus we find, for the Fairchild IN 3071, that with I F = 30 mA and I K = 30 mA,
the time required for the reverse current to fall to 1.0 mA is 50 nsec. Again
we find, for the same diode, that with I? = 30 mA, — V R = — 35 V, R L » 2 K,
and C L = 10 pF (/« = 35/2 = 17.5 mA), the time required for the
diode to recover to the extent that its resistance becomes 400 K is t„ = 400 nsec.
Sac.
612
SEMICONDUCTORDIODE CHARACTERISTICS / 143
Commercial switchingtype 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.
612
BREAKDOWN DIODES 8
The reversevoltage 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 voltagereference or constant voltage devices. Such
diodes are known as avalanche, breakdown, or Zener diodes. They are used
characteristically in the manner indicated in Fig. 6186. 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 voltampere curve.
The upper limit on diode current is determined by the powerdissipation rating
of the diode.
Two mechanisms of diode breakdown for increasing reverse voltage are
recognized. In one mechanism, the thermally generated electrons and holes
acquire sufficient energy from the applied potential to produce new carriers
by removing valence electrons from their bonds. These new carriers, in turn,
produce additional carriers again through the process of disrupting bonds.
I
Vz
;
1
Izk V
1 («)
h
I
—fc
R
(6)
l 9 61 8 ( a ) The voltampere characteristic of an avalanche, or Zener, diode.
lfa ) A circuit in which such a diode is used to regulate the voltage across Rl
q 9atnst changes dve to variations in load current and supply voltage.
144 / ELECTRONIC DEVICES AND CIRCUITS
Sac. 612
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, 619a and 6, is typical of what may be expected generally.
In Fig. 01 9c the temperature coefficient oT the reference voltage is plotted as
a function of the operating current through the diode for various different
diodes whose reference voltage at 5 mA is specified. The temperature coef
ficient is given as percentage change in reference voltage per centigrade degree
10 15 20 25 30 35 40 45 50 35
Iz.mA
(a)
(6)
Fig. 619 Temperature coefficients for a number of Zener diodes having different
operating voltages (a) as a function of operating current, (b) as a function of
operating voltage. The voltage V z is measured at I z = 5 mA {from 25 to 100°C).
(Courtesy of Pacific Semiconductors, Inc.}
0.10
0.08
07
06
05
0.04
0.01
0.02
0.03
0.04
0.05
0 06
07
i
/
I
I
J
0.08
5 t
r
l
V z @5
J i
niA.V
5
SfMrCONDUCTORD/ODE CHARACTERISTICS / 145
change in diode temperature. In Fig. 6196 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. 619a 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 voltampere 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 voltampere 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 erdissipation rating and various voltages is shown in Fig. 620. 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) 200V Zener diode
yj* atin g at 1.2 mA has an r of 1,500 0. Finally, we observe that, to the left
. e minimum, at low Zener voltages, the dynamic resistance rapidly becomes
R 18° ^ e ' Some manufacturers specify the minimum current Izk (Fig.
th i k^ ow which the diode should not be used. Since this current is on
K&ee of the curve, where the dynamic resistance is large, then for currents
146 / ELECTRONIC DEVICES AND CIRCUITS
S«. 612
no
100
yn
60
70
80
50
40
3 Cl
20
10
f)
r ~
»
I z = 1mA
6J
1 >
f\ J^
20
XjS^j^j^Ssdg^
10 12 14 IG 18 20 22 24 26 28 30 32
v z ,v
Fig, 620 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 60Hz
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 crosssectional area of the diode, highpower 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. 68, the voltampere characteristic of a forwardbiased 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 forwardbiased 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. 620, 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*. 6n
SEMICONDUCTORDIODE CHARACTERISTICS / 147
diode must operate over a range of currents. Under these circumstances
temperaturecompensated avalanche diodes find application. Such diodes
consist of a reversebiased Zener diode with a positive temperature coefficient,
combined in a single package with a forwardbiased diode whose temperature
coefficient is negative. As an example, the Transitron SV3176 silicon 8V
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 highvoltage reference is required, it is usually advantageous
(except of course with respect to economy) to use two or more diodes in
series rather than a single diode. This combination will allow higher volt
age, higher dissipation, lower temperature coefficient, and lower dynamic
resistance.
613
THE TUNNEL DIODE
A pn junction diode of the type discussed in Sec. 61 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. 621 Voltampere characteristic of a tunnel diode.
148 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 613
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 voltampere characteristic, depicted
in Fig. 621.
The Tunneling Phenomenon The width of the junction barrier varies
inversely as the square root of impurity concentration [Eq. (647)] and there
fore is reduced from 5 microns to less than 100 A (10 9 cm). This thickness is
only about onefiftieth the wavelength of visible light. Classically, a particle
must have an energy at least equal to the height of a potentialenergy 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 quantummechanical behavior is referred to as tunneling,
and hence these highimpuritydensity pn junction devices are called tunnel
diodes. This same tunneling effect is responsible for highfield emission of
electrons from a cold metal and for radioactive emissions.
We explain the tunneling effect by considering the following onedimen
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 potentialenergy barrier of height U„ > W, and as indicated in Fig. 622a,
the potential energy remains constant in region 2 for x > 0.
Region 1 The Schrodinger equation (214),
dx* T h 2 w
(660)
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. 28
the product of ^ and its complex conjugate ^* is interpreted as giving the
probability of finding an electron between x and x + dx (in a onedimensional
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. 622 (a) A potential
energy step of height U .
The electronic energy is
W < U . (b) A potential
energy hill of height U
and depth d may be pene
trated by the electron
provided that d is small
enough.
w
x a x = d
m
$*c. 61 *
SEMICONDUCTORDIODE CHARACTERISTICS / 149
Region 2 The Schrodinger equation for x > is
dhp 8jt%i
dx*
h*
(U e WW =
Since U > W, this equation has a solution of the form
if, = A£U***mlk*HU.W)]lz _ J^gmtm,
where A is a constant and
si
8ir*m(U  W)
* _ h r i f
4vl2m(U  IF) J
(661)
(662)
(663)
The solution of Eq. (661) 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 *'
(664)
From Eq. (664) we see that an electron can penetrate a potentialenergy
barrier and that this probability decreases exponentially with distance into the
barrier region. If, as in Fig. 6~22&, the potentialenergy 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. 32).
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 pn 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 energyband picture when the
ln npurity concentration is very high. In Fig. 64, 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. (66),
Ef = E c  kT In %£
150 / ELECTRONIC DEVICES AND CIRCUITS
Sac. 61 3
Spacecharge
region
n side*
(«)
Valence
band
(6)
Fig, 623 Energy bands in a heavily doped pn diode (a} under opencircuited
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. 310.)
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. 64. 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 ntype 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. (67)]. A comparison of Eqs. (65) and (68)
indicates that E > Eq, so that the contact difference of potential energy E,
now exceeds the forbiddenenergygap voltage E G . Hence, under opencircuit
conditions, the band structure of a heavily doped pn junction must be as
pictured in Fig, 623o. 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 opencircuited diode.
S*c.6I3
SEMICONDUCTORDIODE CHARACTERISTICS / 151
The VoltAmpere Characteristic With the aid of the energyband picture
of Fig. 623 and the concept of quantummechanical tunneling, the tunnel
diode characteristic of Fig. 621 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 reversebias voltage is
applied, we know from Sec. 62 that the height of the barrier is increased
above the opencircuit value E . Hence the Tiside levels must shift down
ward with respect to the pside levels, as indicated in Fig. 623b. We now
observe that there are some energy states (the heavily shaded region) in the
valence band of the p side which lie at the same level as allowed empty states
in the conduction band of the n side. Hence these electrons will tunnel from
the p to the n side, giving rise to a reverse diode current. As the magnitude
(a)
(b)
E v 
K,
fee
(c)
id)
24 The energyband pictures in a heavily doped pn diode for a forward
* As the bias is increased, the band structure changes progressively from
la) to (d).
152 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 613
S«r 6M
SEMICONDUCTORDIODE CHARACTERISTICS / 153
of the reverse bias increases, the heavily shaded area grows in size, causing the
reverse current to increase, as shown by section 1 of Fig. 625.
Consider now that a forward bias is applied to the diode so that the
potential barrier is decreased below E . Hence the nside levels must shift
upward with respect to those on the p side, and the energyband picture for
this situation is indicated in Fig. 624a. 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. 625.
As the forward bias is increased further, the condition shown in Fig.
6246 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. 625.
If still more forward bias is applied, the situation in Fig. 624c is obtained,
and the tunneling current decreases, giving rise to section 3 in Fig. 625.
Finally, at an even larger forward bias, the band structure of Fig. 624d 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 quantummechanical current described above, the
regular pn junction injection current is also being collected. This current is
given by Eq. (631) and is indicated by the dashed section 4 of Fig. 625.
The curve in Fig. 6256 is the sum of the solid and dashed curves of Fig.
625a, and this resultant is the tunnel diode characteristic of Fig. 621.
Fig. 625 (a) The tunneling current is shown solid. The injection current is the
dashed curve. The sum of these two gives the tunneldiode voltampere charac
teristic of Fig. 621, which is reproduced in (b) for convenience.
6 _!4 CHARACTERISTICS OF A TUNNEL DIODE 8
From Fig. 621 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 negativeresistance 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 socalled 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. 626a.
The smallsignal model for operation in the negativeresistance region is indi
cated in Fig. 6266. 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 highfrequency (microwave)
oscillator.
The most common commercially available tunnel diodes are made from
germanium or gallium arsenide. It is difficult to manufacture a silicon tunnel
diode with a high ratio of peaktovalley current Ip/Iv. Table 61 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
9 626 (a) Symbol for a tunnel
'ode ; (fc>) smallsignal model In
* negativeresistance region.
o — VWTTW^
is. L,
c
JL
(a)
(b)
154 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 61 4
TABLE 61
parameters
Typical tunneldiode
Ip/Iy .
V F ,V
V V ,V
Ge
8
0.055
0.35
0.50
GaAfl
15
0.15
0.50
1.10
Si
3.5
0.065
0.42
0.70
has the highest ratio Ip/Iy and the largest voltage swing V p — V P «s 1.0 V as
against 0.45 V for germanium.
The peak current Ip is determined by the impurity concentration (the
resistivity) and the junction area. A spread of 20 percent in the value of Ip
for a given tunneldiode type is normal, but tightertolerance 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 temperaturesensitive. 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 pn 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 outputvoltage swing and the fact that it is a twoterminal
device. Because of the latter feature, there is no isolation between input and
output, and this leads to serious circuitdesign 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
SEMICONDUCTORDIODE CHARACTERISTICS / 155
REFERENCES
\. Gray, P. E., D. DeWitt, A. R. Boothroyd, and J. F. Gibbons: "Physical Electronics
and Circuit Models of Transistors," vol. 2, Semiconductor Electronics Education
Committee, John Wiley & Sons, Inc., New York, 1964.
Shockley, W.: The Theory of pn Junctions in Semiconductor and pn Junction
Transistors, Bell System Tech. J., vol. 28, pp. 435489, 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. 93112,
John Wiley & Sons, Inc., New York, 1957.
3. Phillips, A. B.: "Transistor Engineering," pp. 129133, McGrawHill Book Com
pany, New York, 1962.
A. Moll, J.: "Physics of Semiconductors," pp. 117121, McGrawHill Book Company,
New York, 1964.
Sah, C. T., R. N. Noyce, and W. Shockley: Carriergeneration and Recombination
in PN Junctions and PN 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, McGrawHill Book Company, New York, 1965.
6. Corning, J. J.: "Transistor Circuit Analysis and Design," pp. 4042, PrenticeHall,
Inc., Englewood Cliffs, N.J., 1965.
7. Esaki, L.: New Phenomenon in Narrow Ge pn 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, TD30," 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. 71
VACUUMTUBE CHARACTERISTICS / 157
7 /VACUUMTUBE
CHARACTERISTICS
The triode was invented in 1906 by De Forest, 1 who inserted a third
electrode, called the grid, into a vacuum diode. He discovered that
current in the triode could be controlled by adjusting the grid potential
with respect to the cathode. This device was found to be capable of
amplifying smallsignal voltages, a discovery of such great practical
importance that it made possible the electronics industry.
In this chapter we study the voltampere characteristics of triodes,
tetrodes, and pentodes and define certain parameters which are useful
in describing these curves. We carry through the analysis of a simple
circuit containing a triode and show that such a circuit is indeed an
amplifier.
71
THE ELECTROSTATIC FIELD OF A TRIODE
Suppose that the mechanical structure of a vacuum diode is altered
by inserting an electrode in the form of a wire grid structure between
the cathode and the anode, thus converting the tube into a triode.
A schematic arrangement of the electrodes in a triode having cylindri
cal symmetry is shown in Fig. 71.
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. 72. If we assume that the
cathode is so cold that it emits no electrons, the potential at any point
in the tube satisfies Laplace's equation, with boundary conditions
determined by the applied electrode voltages. The results of such a
156
Anode
Control grid
Cathode
Fig, 71 Schematic arrangement
of the electrodes in a triode. (a)
Top view; (b) side view. {The
constructional details are similar
to those indicated in Fig. 713.)
V
Anode
(b)
Cathode
Control grid
calculation are shown in Fig. 73, 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. 72 A planeelectrode triode, showing
the paths for the potential profiles given in
F 'Q. 74.
V
Path between grid wires
♦J
$
Path through grid wires
9
1 ©
a
®
158 / ELECTRONIC DEVICES AND CIRCUITS
Stc, 7}
such that it does not collide with a potentialenergy barrier (provided that the
grid is not too highly negative) . Thus the potential variation between cathode
and anode depends upon the path. The potentialvs.distance curves (called
profile presentations) corresponding to Fig. 73 are given in Fig. 74 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. 73 and 71 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. 73 because they encounter there a retarding
field. In (6) are shown the conditions just at cutoff, where the electric field
intensity at the cathode is nowhere positive. Actually, cutoff is obtained at a
grid voltage slightly less than this value, so that the field at the cathode is
somewhat negative and hence repels all the emitted electrons. It should be
clear from a study of these figures that the current distribution is not constant
along paths at different distances from the grid wires. If the grid is made suf
ficiently negative, cutoff will occur throughout the entire region. This con
fa)
100
90 ■
80
70
60
50
40
Anode
Fig. 73 Equipotential contours in volts in the planeelectrode triode. (a) Grid
beyond cutoff potential (V G = 25 V); (fa) grid at cutoff potential (V a = 12 V);
(c) grid negative at onehalf cutoff value {V G = 6 V). (From K. R. Spangenberg,
"Vacuum Tubes," McGrawHill Book Company, New York, T948.)
s^r2
VACUUMTUBE CHARACTERISTICS / 159
00
—
_
1
— i —
ridJ
I
so
^
i
oil
i
M
3
o
i
o
.■v.
9
 a.
<
•ill
i
■J.v
i
\ y
'
—

V /
HI
i
i
—
1 1
1
Grid
\
1
1
!
1
1
i>
i
I
<
1 >

.
\'/
1 1
!
Grid^JL
N
1
gg
—
£
— >
o
1
1
1 '
1
o
1
<
yj
r
<
1


1
40 20 0+20 +40 +60
Distance from grid, mils
(a)
4020 + 20 + 40 + 60
Distance from grid, mils
(6)
40 20 +20 +40 +60
Distance from grid, mils
<<0
Fig. 74 Potential profiles of a planeelectrode 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,"
McGrawHill 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 spacechargefree
conditions. In Chap. 4 it is shown that under spacecharge conditions the
electric field intensity at the cathode is reduced to zero. Hence, for a hot
cathode, the potential curve of Fig. 74c must be modified somewhat and, in
particular, must have zero slope at the cathode.
72
THE ELECTRODE CURRENTS
From the qualitative discussion already given, it follows that the plate current
should depend upon the spacechargefree 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. 74, 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 threehalvespower law for diodes (Sec. 44),
*t is anticipated that the plate current ip may be represented approximately
160 / ElfCTRONfC DEVICES AND CIRCUITS
by the equation 2
i P = G (vo + V A'
Sec. 72
(71)
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. (71) 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 positivegrid current increases in the range of
0.5 to 4 mA for each volt increase in positivegrid voltage. Such an increment
corresponds to an effective sialic grid resistance r G m v 6 /i a of 250 Q to 2 K.
Positivegrid triodes are available for poweramplifier applications. Also, in
many pulse and snitching circuits 3 the grid is driven positive during a portion
of the waveform (Fig. D3).
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 negativegrid volt
age is increased, the grid current decreases further, then goes to zero, and
may reverse in sign." This negativegrid current consists mainly of four
components.
First, we have gas current, consisting of positive ions (carbon dioxide,
carbon monoxide, hydrogen, etc.) collected by the negative grid. The positive
ion grid current is proportional to both the pressure in the tube and plate cur
rent. When the grid voltage becomes sufficiently negative, the plate current is
zero (cutoff) and no ionization takes place. Second, electrons leave the grid
(and hence negativegrid 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 negativegrid current due to all sources seldom exceeds a small
fraction of a microampere. Unless otherwise stated, we neglect the grid cur
rent (positive or negative) for all negative values of grid voltage.
COMMERCIAL TRIODES
VACUUMrUBE CHARACTERISTICS / 161
In Sec. 42 the construction of commercially available cathodes is described,
practical anodes are discussed in Sec. 48.
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 gridtocathode 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 gridtocathode spacing which results in a tube with a large value of
transconductance (Sec. 75).
The Nu vis tor Another type of grid structure is employed in the manu
facture of the nuvistortype vacuum tube shown in Fig. 75. This tube utilizes
an allceramic and metal construction with cantileversupported cylindrical
electrode structure. The cylindricaltube 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 75 Nuvistor triode. (Cour
tesy of Radio Corporation of
America.)
Cathode
Grid
— Heater
Plate
'Indexing
lugs
1*2 / ftfCTROWC DEVICES AHO CIKUITS
S«r. 74
A. Anode
B. Ceramic spacers
C. Heater
D. Cathode ring
E. Heater buttons
F. Grid
G. Grid ring
H. Oxidecoated cathode
/. Cathode
M§^9 Ceramic
W/ft Titanium
Fig. 76 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. 76. The close
gridtocathode 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 gridtocathode 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.
74
TRIODE CHARACTERISTICS
The plate current depends upon the plate potential and the grid potential, and
may be expressed mathematically by the functional relationship
i P = /(i>, vq)
(72)
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 spacechargelimited conditions the approxi
mate explicit form of this function is that expressed by Eq. (71). By plot
ting ip versus v P and vq on a threedimensional 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 77a 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
VACUUMTUBE CHARACTERISTICS / 163
Stc.75
fig. 77 (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. 77&, 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. (71) 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 constantcurrent characteristics (Prob. 71).
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, MILE1, give the limits of
variability which may be expected in a given tube type.
The voltampere 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 0.1V shift in cathode
voltage (relative to the other electrodes) for each 10 percent change in heater
voltage.
TRIODE PARAMETERS
75
« the analysis of networks using tubes as circuit elements (Chap. 8), it is
°und necessary to make use of the slopes of the characteristic curves of Fig.
I • Hence it is convenient to introduce special symbols and names for these
Entities. This is now done.
Amplification Factor This factor, designated by the symbol n, is defined
^ e ratio of the change in plate voltage to the change in grid voltage for a
164 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 75
constant plate current. Mathematically, n is given by the relation
(73)
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. 77 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 plategrid 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. 77&.
Summary The triode coefficients, or parameters, which are character
istic of the tube are
(dip\
\dV )y p  9m
_ (§Vp\
\dv ) Ir M
plate resistance
mutual conductance
amplification factor
(74)
Since there is only one equation, (72), relating the three quantities ip, vp, and
vq, the three partial derivatives cannot be independent. The interrelationship
may be shown to be (Sec. 84)
M = TpQ m (75)
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
VACUUMTUBE CHARACTERISTICS / 165
Each section
11
gm
3.5
5
ft;
L>
13U
I
3.0

£0
y<
\y
<&
'■$
BT
^ 3m
V
pj?
V f = rated value
2.0 i
I
iJ
•a
^7"
K
15(_
V_
8
1.0 g
§
ISO
<Y
0.5
22 28
20 24
18 *20
—16 £ 16
g a
+* m
at a
I U * 12
< 12 E 8
10
10
Plate current, mA
Fig. 78 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. 78. Note that the plate resistance varies over rather wide
limits. It is very high at zero plate current and varies approximately inversely
as the onethird power of the plate current (Prob. 73). The transconduc
tance increases with plate current from zero at zero plate current and varies
directly as the onethird 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 gridtocathode 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 71.
e se contain two triode units in one envelope, and each section has, at the
ommended operating point, the parameters given in the table. Since the
, current is given in milli amperes and the potentials in volts, it is con
sent to express the plate resistance in kilohms and the transconductance in
166 / ELECTRONIC DEVICES AND CIRCUITS
TABLE 71 Some Mode parameters
Sec. 7^
Triode type
M
r P , K
ffm, mA/V
6CG7
20
55
17
100
47
7.7
5.5
7,7
62
7.2
2.6
10
2.2
1.6
6.5
12AT7
12AU7
12AX7
5965
millimhos. Note that the product of milliamperes and kilohms is volts and
that the reciprocal of kilohms is millimhos or milliamperes per volt (mA/V).
Approximate values of r pi ft and g m may be obtained directly from the
plate characteristics. Thus, referring to the definitions in Eqs. (74) and
to Fig. 7 7 a, we have, at the operating point Q,
AVp
Aip
p ~ AiZ L. = reciprocal of slope of characteristic
9m. =
\Au
ft ~ —
Aip 
Ave \y* ~ V 0i  Vat
Av P I \Avpl
Avq \h
V02  V c
If t p were constant, the slope of the plate characteristics would every
where be constant; in other words, these curves would be parallel lines. If
* were constant, the horizontal spacing of the plate characteristics would be
constant. This statement assumes that the characteristics are drawn with
equal increments in grid voltage (as they always are). If r p and M are con
stant, so also is g m = „/>„. Hence an important conclusion can be drawn:
// over a portion of the i P v P plane the characteristics can be approximated by
parallel hues which are equidistant for equal increments in grid voltage, the param
eters n, r v , and g m can be considered constant over this region. It is shown in the
next chapter that if the tube operates under this condition (tube parameters
sensibly constant), the behavior of the tube as a circuit element can be obtained
analytically.
76 SCREENGRID TUBES OR TETRODES
In Chap, 8 it is shown that the capacitive coupling between the plate and grid
of a triode may very seriously limit the use of the tube at high frequencies.
In order to minimize this capacitance the screengrid 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. 71. This new electrode
is similar in structure to the control grid, and is known as the screen grid, the
SK
76
VACUUMTUBE CHARACTERISTICS / 167
shield grid, or grid 2, in order to distinguish it from the grid of the triode.
Because of its design and disposition, the screen grid affords very complete
electrostatic shielding between the plate and the grid. This shielding is such
that the gridplate 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.
VoltAmpere Characteristics We have already noted that the total
Bpace current remains essentially constant with variations in plate voltage
provided that the controlgrid and screengrid 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. 79.
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. 79,
^'9 79 The currents in a
tetrode. The screen po
tential is 100 V, and the
9 r 'd potential is 2 V.
j Total space current
f
Plate
'
t '
N  
•*" *1
\J
\
—
Screen
h i —
.
100 150 200
Plate voltage, V
168 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 76
Anode
Fig, 710 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.
Negativeresistance Region An inspection of Fig. 79 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. 79 may be described on the
basis of the approximate potentialdistribution diagram of Fig 710 This
diagram should be compared with Fig. 7 A, which shows the potential profiles
in a triode. The controlgrid and the screengrid 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. 79 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 smallsignal param
Sec 77
VACUUMTUBE CHARACTERISTICS / 169
eters r p , g m , and n are defined as in Eqs. (74) 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 screengrid 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 platecurrent 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
plategrid 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.
77
PENTODES*
Although the insertion of the screen grid between the control grid aria the
plate serves to isolate the plate circuit from the grid circuit, nevertheless the
folds in the plate characteristic arising from the effects of secondary emission
limit the range of operation of the tube. This limitation results from the fact
that, if the platevoltage 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 negativeresistance 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. 711 for a pentode. These
8 Wld be compared with the corresponding tetrode curves of Fig. 79. Note
l W the kinks resulting from the effects of secondary emission are entirely
170 / ELECTRONIC DEVICES AND CIRCUITS
S«. 7.7
5
1 1 1 1
4
Total space current
<
] r
—
1
Plate
1
{
3
O
1
L

Screen
(1

Fig. 711 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 screengrid wires. An examination of the charac
teristics of a number of the more important voltage pentodes indicates that
the screen current is ordinarily from 0.2 to 0.4 of the plate current at the
recommended operating point. The total space current is seen to remain
practically constant over the entire range of plate voltage, except for the very
r
\fV c =Q
1 1 1
V c = 0.5 V
1
.0
10
I" 
<
s 
1
*r
5
1 J
2.C
K If
s 1
2.5
r
4"
1
— 3.0
.0.
I
r
A
10
I
— '■
20
g
30
inn
Plate voltage, V
Fig. 712 The plate characteristics of a 6AU6 pentode with V G2 = 150 V
and V Gt = V. (Courtesy of General Electric Co.)
500
78
low values of potential,
given in Fig. 712.
VACUUMTUBE CHARACTERISTICS / 171
The plate characteristics of a typical pentode are
Parameter Values The plate resistance r p , plategrid 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. (74). 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 framegrid pentode (for example, Amperex type 7788) whose
gridtocathode 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 gridplate 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. 78) 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 radiofrequency 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 audiofrequency power
output tube. Finally, the pentode has been used as a constantcurrent device
because the plate current is essentially constant, independent of the plate
potential.
78
BEAM POWER TUBES
The ideal powertube 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. 713.
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 beamforming
de plates (maintained at zero potential), and a relatively large spacing
172 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 7f
Be am forming
plate
Cathode
Grid
Screen
Fig. 713 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. 713.
The region between the screen and the plate possesses features which are
somewhat analogous to those existing in the spacechargelimited 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. 45 in connection with the effects of initial velocities on the spacecharge
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. 43). This
minimum is shown in the approximate potential profile in Fig. 714, which
should be compared with the corresponding figure for the tetrode (Fig. 710).
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 potentialenergy barrier. They will be compelled to return to the electrode
Fig. 714 Approximate potential profiles
in on idealized beam power tube for two
values of plate voltage. Two curves are
shown for each plate voltage, one for o
path between grid and screen wires, and
the other for a path through the wires.
Note the potential minimum in the region
between the screen grid and the anode.
S«.79
VACUUMTUBE CHARACTERISTICS / 173
300 400
Plate voltage, V
Fig. 715 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
screenplate 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
powertube characteristic is closely approximated. A family of plate charac
teristics for the 6L6 is shown in Fig. 715. It should be noted that this tube
is a tetrode when considered in terms of the number of active electrodes. At
low currents, where the suppressor action of the beam is too small, the charac
teristic "kinks" of a tetrode are noticeable.
7 ' 9 THE TRIODE AS A CIRCUIT ELEMENT
Even if the tube characteristics are very nonlinear, we can determine the
behavior of the triode in a circuit by a graphical method. This procedure is
^sentiaUy the same as that used (Sec. 49) in treating the diode as a circuit
element, except that the diode has two active electrodes and one character
istic curve, whereas the triode has three active elements and a family of curves.
The th ree terminals are marked P (plate), K (cathode), and G (grid). A
174 / ELECTRONIC DEVICES AND CIRCUITS
Soc. 79
Fig. 716 The basic circuit of q
triode used as an amplifier.
groundedcathode circuit in which the triode acts as an amplifier is shown in
Fig. 716. 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 vacuumtube circuit, it is necessary that a
precise method of labeling be established if confusion is to be avoided. Our
notation for vacuumtube 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 rootmeansquare (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.
VACUUMTUBE CHARACTERS TICS / 175
Trtode symbols
Instantaneous total value
Quiescent value
Instantaneous value of varying
component
Effective value of varying component.
Amplitude of varying component
Supply voltage
Grid voltage
with respect
to cathode
V*
vZt
Plate voltage
with respect
to cathode
V P
Current in direc
tion toward plate
through the load
Ip
t These are positive numbers, giving the magnitude of the voltages.
6. Conventional current flow into an electrode from the external circuit is
positive.
7. A single subscript is used if the reference electrode is clearly under
stood. If there is any possibility of ambiguity, the conventional double
subscript notation should be used. For example, v^ = instantaneous value
of varying component of voltage drop from plate to cathode, and is positive
if the plate is positive with respect to the cathode. If the cathode is grounded
and all voltages are understood to be measured with respect to ground, the
symbol v pk may be shortened to v p . The ground symbol is N. For example,
Vp N = instantaneous value of total voltage from plate to ground.
8. The magnitude of the supply voltage is indicated by repeating the
electrode subscript.
Table 72 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. 716 is
v e = — Vq G + i>,= — Vqq + V tm sin mt (76)
7 "!0 GRAPHICAL ANALYSIS OF THE
GROUNDEDCATHODE CIRCUIT
Assume for the moment that no grid signal is applied in Fig. 716, so that
"• = 0. H must not be supposed that there will be no plate current, although
this might be true if the bias were very negative. In general, a definite direct
176 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 7lQ
Fig. 717 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. 47 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. 716 that
Vp m Vp P — ipR!
(77)
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. (77) is
plotted on the plate curves of Fig. 717. 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. 717. 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. (77), 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 peaktopeak signal voltage is 12 V, what
is the peaktopeak output swing?
S«
710
VACUUMTUBE CHARACTERISTICS / \77
Solution a. The plate characteristics are given in Fig. D2 (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. D2 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. D2. 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 peaktopeak 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 gridvoltage 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. (76). 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. 71 8 The output current
and voltage waveforms for a
9'ven input grid signal are
determined from the plate
characteristics and the load
line.
178 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 7 J 
V r , =0
Fig. 719 (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. 718. 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 outputcurrent waveform correspond, respectively,
to the points a", b", c", etc., of the outputvoltage waveform.
A Pentode Circuit The simplest amplifier circuit using a pentode ia
indicated in Fig. 719a. 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. (77)] 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. 71% 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 .
711
THE DYNAMIC TRANSFER CHARACTERISTIC
The static transfer characteristic of Fig. 77b gives the relationship between ip
and v a with the plate voltage held constant. The dynamic transfer character
istic gives the relationship between i P and v G for a given plate supply voltage
V PP and a given load resistance R L . This functional relationship is obtained
from the plate characteristics and the load line by the graphical construction
described in the preceding section. Thus the values of i P and t' c at points 1,
I
Sec. 712
VACUUMTUBE CHARACTERISTICS / 179
Dynamic
transfer curve
Fig. 720 The dynamic trans
fer characteristic is used to
determine the output wave
shape for a given input
signal.
Q, and 2 in Fig. 720 are the same as those obtained at the corresponding
points 1, Q, and 2 in Fig. 718. 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. 720, where points a', b\ c', etc., of the output current corre
spond to points A, B, C, etc., respectively, of the input grid voltage signal
«, = v e .
M2
LOAD CURVE. DYNAMIC LOAD LINE
A graphical method of obtaining the operating characteristics of a triode with
a distance load is given in Sec. 710. 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
tV = V vm sin ut and i p = —I pm sin (wt + 9)
(78)
ten are the parametric equations of an ellipse. If the angle 6 is zero, the
10 of these equations yields
i 7 ~ Hl
180 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 712
Load curve
Load line
Fig. 721 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 voltampere
characteristics of Fig. 721.
The above analysis depends upon the tube parameters p, r p , and g m being
constant over the range of operation in the i P v P plane. If these parameters
are not constant, the operating curve will no longer be an ellipse. No simple
analysis of the output of an amplifier with a reactive load exists under these
conditions.
An /i!Ccoupled Load Consider the reactive load indicated in Fig. 7 22a.
Here the output is taken, not across the platecircuit 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. 717 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. 722 (a) An flCcoupled circuit, (b) Static and dynamic load lines for the RC
coupled circuit.
VACUUMTUBE CHARACTERISTICS / 181
Static load line
Dynamic
load line
Fig. 723 (a) A transformercoupled load, (b) Static and dynamic
load lines for a transformercoupled 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. 722b.
A Transformercoupled Load For the RCcoupled 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. 723a, 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. 7236. 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 constantcurrent device.
7  T3 GRAPHICAL ANALYSIS OF A CIRCUIT
WITH A CATHODE RESISTOR
* ai *y practical circuits have a resistor R k in series with the cathode in addition
(or in place of) the load resistor Rt in series with the plate. The resistor
* is returned either to ground or to a negative supply — V K k, as indicated in
* l g. 724.
We consider now how to use the characteristic curves of a vacuum triode
e termine such matters as range of outputvoltage swing, proper bias volt
182 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 713
Fig. 724 {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, 724,
fi, v G , vp, and ip are, respectively, the total instantaneous input voltage, grid
tocathode voltage, platetocathode 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
(79)
(710)
Equation (79) 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. (710). The procedure is illustrated in Fig. 725.
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 gridtocathode
voltage must decrease slightly in magnitude in order to supply the increased
c^.713
Fio. 7~25 Construction for
obtaining the dynamic char
acteristic of a circuit with
both a cathode and a load
resistor, as in Fig. 724. The
symbolism »ej — ► f,i means
that I'd is replaced by
»u ■ vo\ + ip\Rk — Vkk
VACUUMTUBE CHARACTERISTICS / 183
Load line
Bin +*>n ( = «ci + i ei R*  V KK )
oca — **»a
■Vp2— — *\* H
~V„ + V KK H
tube current demanded by this increased t>,. Hence the cathode tries to follow
the grid in potential. If R t . = 0, it turns out (Sec. 714) 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 gridtocathode voltage is zero.
The Quiescentpoint Calculation It is often desirable to find the current
corresponding to a specified fixed input voltage without drawing the entire
dynamic characteristic as outlined above. A very simple procedure is as
follows:
1. On the plate characteristics draw the load line as in Fig. 725.
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. (710), this current is given by
i P m
V + Vkk — vq
Rk
Ihe corresponding values of i p and % are plotted on the plate characteristics,
ft s indicated by the dots in Fig. 726. 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 726. The intersection of the straight line connecting A and B with
e toad line gives the desired current. In particular, it should be noted that,
184 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 7U
Uc VBi&s curve
Fig. 726 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. 726 will be approximately a horizontal straight line.
Selfbias Often no negative supply is available, and selfbias is obtained
from the quiescent voltage drop across Rk. For example, if the plate current
and the gridtocathode 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. 726. 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 selfbias, we must choose R k = — Va/(Ip + Is).
7U PRACTICAL CATHODEFOLLOWER CIRCUITS
In order to see why it is sometimes advantageous to use a negative supply,
consider the cathodefollower configuration of Fig. 727.
O300V
■i(6CG7)
Fig. 727 An example of a cathodefollower
rcuit.
.
Sec 71 4
VACUUMTUBE CHARACTERISTICS / 185
EXAMPLE Find the maximum positive and negative input voltages and the
corresponding output voltages. Calculate the voltage amplification.
Solution From the characteristics (Fig. D2) 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 cathodefollower 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. 86.
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. 724, 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. 728a 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 selfbiasing voltage appearing across Ri. That is, with no
input signal, the gridtocathode 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 peaktopeak output swing. In the above example, where the total
qv p
&
ok.
€>
" ? +1 *
r ©
•Rk v°
(a) (o)
Fig. 728 Two biasing arrangements for a cathodefollower circuit.
184 / ELECTRONIC DEVICES AND CIRCUITS
Sac. 71 4
output swing was ~200 V, the quiescent value is chosen as 100 V across the
20K resistance. This corresponds to a quiescent plate current of 5 raA.
From the plate characteristics of the 6CG7 and the 20K load line, the grid
tocathode voltage corresponding to 5 mA is —7 V. Hence Ri must be chosen
equal to £ K = 1.4 K.
REFERENCES
1. De Forest, L.: U.S. Patent 841,387, January, 1907.
2. Spangenberg, K. R.: "Vacuum Tubes," McGrawHill Book Company, New York,
1948.
3. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," McGraw
Hill Book Company, New York, 1965.
4. Valley, G. E., Jr., and H. Wallman: "Vacuum Tube Amplifiers," p. 418, MIT
Radiation Laboratory Series, vol. 18, McGrawHill Book Company, New York,
1948.
Natapoff, M.: Some Physical Aspects of Electronreceivingtube Operation, Am. J.
Phys., vol. 30, no. 9, pp. 621626, 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 IF
Tubes, Electron. Prod., December, 1964, p. F3.
7. Pidgeon, H. A.: Theory of Multielectrode Vacuum Tube, Bell System Tech. J.,
vol. 14, pp. 4484, January, 1935.
8. Schade, 0. H.: Beam Power Tubes, Proc. IRE, vol. 26, pp. 137181, February, 1938.
9. IEEE Standard Letter Symbols for Semiconductor Devices, IEEE Trans. Electron
Devices, vol. ED11, no. 8, pp. 392397, August, 1964.
Reich, H. J.: Standard Symbols for Electron Devices, Proc, IEEE, vol. 51, no. 2,
pp. 362363, February, 1963.
VACUUMTUBE SMALLSIGNAL
MODELS AND APPLICATIONS
If the tube parameters r p , g m , and p. are reasonably constant in some
region of operation, the tube behaves linearly over this range. Two
linear equivalent circuits, one involving a voltage source and the other
a current source, are derived in this chapter. Networks involving
vacuum tubes are replaced by these linear representations and solved
analytically (rather than graphically, as in the preceding chapter).
The voltage gain and the input and output impedances are obtained for
several amplifier configurations.
81
VARIATIONS FROM QUIESCENT VALUES
Suppose that in Fig. 716, 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 transformercoupled 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. 722a),
then, under zero input conditions, the capacitor will charge up to the
quiescent voltage V P . The voltage across R g is zero under these con
ditions. If a varying grid voltage is now added to the bias, the output
will again represent voltage variations about the quiescent value.
It is evident that the significant quantities are the currents and
voltages with respect to their quiescent values. To examine this
187
188 / ElKTRONfC DEVICES AND CIRCUITS
Sec. 82
Fig. 81 The dynamic
transfer characteristic is
used to determine the out
put waveshape for a given
input signal.
matter in some detail, refer to Figs. 718 and 720. For convenience, the
latter is repeated in Fig. 81. We see that the output current, defined by the
equation
ip = ip — Ii
(81)
is simply the current variation about the quiescentpoint 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 12 for equal intervals of v c „ the waveform
of i p will differ from that of the inputsignal waveform. Such a uonlinearity
generates harmonics, since a nonsinusoidal periodic wave may be expressed as a
Fourier series in which some of the higherharmonic terms are appreciable.
These considerations should be clear if reference is made to Figs. 718 and 81.
Corresponding to Eq. (81), the variables v p and v e are defined by the
equations
V p = Vp — Vp Vg m Vg — ( — Vgg) = Vg f Vgg (82)
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„ (83)
82
VOLTAGESOURCE MODEL OF A TUBE
The graphical methods of the previous chapter are tedious to apply and often
are very inaccurate. Certainly, if the input signal is very small, say, 0.1 V
S**2
VACUUMTUBE SMAUSIGNAL MODELS AND APPUCAT/ON5 / 189
r less, values cannot be read from the plate characteristic curves with any
degree of accuracy. But for such small input signals, the parameters /u, r pt
and 0m will remain substantially constant over the small operating range.
Under these conditions it is possible to replace the graphical method by an
analytical one. This is often called the smallsignal 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 smallsignal equivalent circuit between the
plate and cathode terminals may be obtained from Thevenin's theorem. This
theorem states that any twoterminal linear network may be replaced by a gener
ator equal to the opencircuit 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. (74) as
~ \AipJvo H
this dynamic plate resistance is the output resistance between the terminals
P and K. The opencircuit voltage v pts between P and K is — fiv gk . This
result follows from the definition of n given in Eqs. (74) as
/AvA _ Vp I Vpk
\AVg/ip v„ Vf v gk
(84)
where use has been made of the definitions in Eqs. (83) 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. (84) means that the plate
current is constant, and hence that variations in plate current are zero. Since
h = 0, the plate is opencircuited for signal voltages. Therefore the open
circuit plate voltage is v pk = — pv ff * for a signal voltage v gk .
The Smallsignal Voltagesource 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. 82 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. 82 because the smallsignal model of the tube applies only for signal
y oltages, that is, for changes about the Q point. Moreover, the equivalent
tubecircuit representation is valid for any type of load, whether it be a pure
190 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 83
Fig, 82 (a) A tube triode (or pentode) and (fa) its voltage
source model. The quantity v„ k is to be evaluated by
traversing the circuit from to K and adding all the voltage
drops on the way.
resistance, an impedance, or another tube. This statement is true because
the above derivation was accomplished without any regard to the external
circuit in which the tube is incorporated. The only restriction is that the
parameters ft, r p , and g m must remain substantially constant over the operating
range.
If sinusoidally varying quantities are involved in the circuit— and this is
usually assumed to be the case — the analysis proceeds most easily if the
phasors (sinors) of elementary circuit theory are introduced. For this case
of sinusoidal excitation, the tube is replaced by the equivalent circuit of Fig.
826, 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 smallsignal model of a triode is
equally valid for a pentode (or a tetrode).
83
LINEAR ANALYSIS OF A TUBE CIRCUIT
Based on the foregoing discussion, a tube circuit may be replaced by an
equivalent form which permits an analytic determination of its smallsignal
(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. 826).
4. Transfer all circuit elements from the actual circuit to the equivalent
circuit of the amplifier. Keep the relative positions of these elements intact.
VACUUMTUBE SMALLSIGNAL MODELS AND APPLICATIONS / 191
Fig. 83 (a) The sche
matic diagram and (b) the
equivalent circuit of a
simple groundedcathode
amplifier.
(«)
(b)
"). Replace each independent dc source by its internal resistance. The
ideal voltage source is replaced by a short circuit, and the ideal current source
by an open circuit.
A point of special importance is that, regardless of the form of the input circuit,
the fictitious generator that appears in the equivalent representation of the
tube is always nV gk , where V gk is the signal voltage from grid to cathode. The
positive reference terminal of the generator is always at the cathode.
To illustrate the application of these rules, two examples are given below.
The first is a singlemesh 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. 83c.
Solution According to the foregoing rules, the equivalent circuit is that of Fig.
836, Kirchhoff s voltage law (KVL), which requires that the sum of the voltage
drops around the circuit equal zero, yields
I p Rl + TpT p — iiV ek , =
A glance at this circuit shows that the voltage drop from grid to cathode is Vi.
Hence V gk = V it and the output current I p is
Um
t*Vi
Rl + t p
The corresponding outputvoltage drop from plate to cathode is
V o — Vpk — — 'pRL
The minus sign arises because the direction from P to K is opposite to the positive
reference direction of the current /„.
l\ =
flVjRL
Rl + r p
192 / ELECTRONIC DEVICES AND CIRCUITS
Sac. 83
Fig. 84 The gain of the amplifier of Fig, 83 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 inputvoitage drops. For the simple amplifier
of Fig. S3a,
is;*.
Vi
= —n
Rl
Rl + U
= — M
1 + r p /R L
(85)
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. 84. 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 powersupply 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. (75), 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. (85) may be put in the form
A = g m R'
(86)
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. (86) 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
(87)
EXAMPLE Draw the equivalent circuit and find the signal plate voltage for
the circuit shown in Fig. 85a. The tube parameters are /u = 10 and r„ = 5 K.
The 1kHz oscillator V has an rms output of 0.2 V.
VACUUMTUBE SMALLSIGNAL MODELS AND APPLICATIONS / 193
f>0 ■ P
SK
Fig. 85 (o) Illustrative
example, (b) The small
signal equivalent circuit.
v„ P $
10K 10K
(a)
10K,,
+T 10K G 10K
X AAAArOA'W '
Solution Following the rules emphasized above, the smallsignal equivalent cir
cuit ia indicated in Fig. 856. 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) (88)
KVL around the two indicated meshes yields
107,* + 257 1  207 2 = (89)
20/i + 257 2  0.2 = (810)
If the expression for V„* is substituted into (89), we obtain
1007,  1007 2 + 25/i  20/ 2 
or
U = r&r/i = 10427,
From this value of 7 2 and Eq. (810) we obtain
h m 0.0331 mA and h = 0.0345 mA
Also,
V Qk m 10(7,  h) = 10(7,  1.0427.) = (0.42) (0.0331) 0.0138 mV
The signal voltage drop from plate to cathode is, from mesh 1,
V pk = 51, 107 ok  (5) (0.0331) + (10) (0.0138)  0.028 mV
Alternatively, from mesh 2,
V pk = Q.2 + 57 s = 0.2 I (5) (0.0345) = 0.028 mV
194 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 84
_
V c constant
Fig. 86 If the grid voltage is constant, then
Aip = (slope) (At?/.) = (di F /dVr) Vl Av P ,
84
TAYLOR'S SERIES DERIVATION OF THE EQUIVALENT CIRCUIT
It is instructive to obtain the equivalent circuit of a triode from a Taylor's
series expansion of the current i P about the quiescent point Q. This deri
vation will show the limitations of this equivalent circuit and will also supply
the proof that m = r p g m .
If the grid voltage remains constant but the plate voltage changes by an
amount Av P , then the change in current equals the rate of change of current
with plate voltage times the change in plate voltage, or
trip
\dvpf
AVp
The subscript indicates the variable held constant in performing the
partial differentiation. This relationship is illustrated in Fig. 86 and is seen
to be strictly true only if the slope of the plate characteristic is constant for
the assumed change in voltage. Similarly, if the plate voltage remains con
stant but the grid voltage changes by Av G , then the change in current is given by
Aij
\dVGjV!
Av G
If both the grid and plate voltages are varied, the platecurrent change is
the sum of the two changes indicated above, or
Aip
= (i£)v *»* + ($£)
Vi
Av G
(811)
As mentioned above, this expression is only approximate. It is, in fact,
just the first two terms of the Taylor's series expansion of the function
ip(vp, %). In the general case,
Aip
Consider the third term in this expansion
dHi
Av p Av G +
■ (842)
Bvp ovq
Since from Eqs. (74) the plate
Sx. 85
Vacuumtube smallsignal 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 higherorder terms in Eq. (812) represent
derivatives of r p and g m with respect to plate and grid voltages.
Smallsignal 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, (811). This
expression may be written in the following form, by virtue of Eqs. (74) :
1
Atp = — Avp + g M Av c
(813)
Using the notation of Eqs. (83), and remembering that g m = h/t p (see
below), Eq. (813) becomes
v p = V*  &» (814)
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 gridtocathode 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. 82. 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. (814), which verifies that Fig. 82 is the correct equivalent
circuit representation of the tube.
Relationship between n, r p> and g m It follows from Eq. (813) that, if
^e plate current is constant so that Ai P = 0, then
Avp
Avq
= gmT v
*>ut since the plate current has been taken to be constant, then — Av P /Av G is
bv definition [Eq. (73)] the amplification factor. Hence
n = g m r p
(815)
8 ' 5 CURRENTSOURCE MODEL OF A TUBE
This
ne venin's equivalent circuit is used if a network is analyzed by the mesh
nod. However, if a nodal analysis is made, Norton's equivalent circuit is
m ° r e useful.
196 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 8.5
oi
I®
02
ol
02
(a)
(ft)
Fig, 87 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 shortcircuited. 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. S7a and b.
The Smallsignal Cur rent source Equivalent Circuit From the voltage
source representation of a tube given in Fig. 826 we see that the shortcircuit
current has a magnitude nv k/r p = g m v B k, where use is made of Eq. (815).
The direction of the current is such that it will flow through an external load
from cathode to plate. Hence the currentsource equivalent circuit is as
indicated in Fig. 88.
We shall now again solve the first example in Sec. 83, using the Norton's
equivalent representation. For convenience, the circuit of Fig. 83 is repeated
in Fig. 89a. Its currentsource model in Fig. 896 is the same as that indi
cated in Fig. 88, 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. (86).
(816)
I = gm <V
Q)
T
Fig.
r *< v p*
g, 88 The currentsource model of a triode
Sk 86
VACUUMTUBE SMALLSIGNAL MODELS AND APPLICATIONS / 197
V K
V GC J?~
Fig. 89 (a) The commoncathode amplifier configuration
and (b) its currentsource equivalent circuit.
86
A GENERALIZED TUBE AMPLIFIER
The circuit (Fig. 89) considered in the preceding section has its cathode
common to the input and output circuits, and hence is called the common
cathode (or groundedcathode) amplifier. This circuit is the one most frequently
used, but two other configurations, the groundedgrid and the groundedplate
amplifiers, are also possible.
The Groundedgrid 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 groundedgrid 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 platecircuit resistor R p . Since the grid is
F '9 810 (a)Thegrounded
fln'd amplifier and (b) the
fl'oundedplate (cathode
f °Hower) amplifier.
v ac ±
(«)
(b)
198 / ELECTRONIC DEVICES AND CIRCUITS
Sk, 86
common to the input and the output circuits, this configuration is also called
the commongrid amplifier.
The Groundedplate Amplifier This circuit is indicated in Fig. 8106.
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 groundedplate amplifier. For an increase in inputsignal voltage v, the
current i p increases, and so does the outputsignal 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. 714 and as
demonstrated in general in Sec. 88, 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 groundedgrid and the
groundedplate amplifiers is made by considering the generalized configuration
indicated in Fig. 81 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 groundedgrid 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 signalsource impedance is unimportant
since it is in series with a grid which, we assume, draws negligible current.)
,
(a)
(b)
Fig. 811 (a) A generalized amplifier configuration, (b) The
smallsignal equivalent circuit.
S*87
VACUUMTUBE SMALLSIGNAL MODELS AND APPLICATIONS / 199
If the effect of the ripple voltage in the power supply Vpp is to be investi
gated, then f will be included in the circuit to represent these small voltage
changes in V PP .
Following the rules given in Sec. 83, we obtain the smallsignal equivalent
circuit of Fig. 8116, from which it follows that
and
Vgk — v% — vt — ipRk
Ml'ul — »Jb — V a
p r p + R k + R P
Substituting from Eq. (817) in Eq. (818), 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
(817)
(818)
(819)
(820)
Using the basic concepts enunciated in the following section, the physical sig
nificance of Eqs. (819) and (820) is given in Sec. 88.
87
THE THEVENIN'S EQUIVALENT OF ANY AMPLIFIER
If an active device (tube or transistor) in a circuit acts as amplifier, this con
figuration is characterized by three parameters, the input impedance Z iy the
output impedance Z„, and the opencircuit 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. 812.
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 opencircuit 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 9 812 TheThevenin
*quivalent circuit of an
Qn >Plifler. When Z L is con
"*cted to the output fermi
" als . a current l L flows in
tKe load.
200 / ELECTRONIC DEVICES AND CIRCUITS
The output voltage is given by
V = A v Vi  I L Z
Sec. 88
(821)
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. (821), 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 .
Opencircuit VoltageShortcircuit Current Theorems As corollaries to
Thdvenin's and Norton's theorems we have the following relationships: If V
represents the opencircuit voltage, / the shortcircuit current, and Z (Y) the
impedance (admittance) between two terminals in a network, then
V
Z = l
I=Z = VY
V = IZ = ±
(822)
The first relationship states that "the impedance between two nodes equals
the opencircuit voltage divided by the shortcircuit current." This method
is one of the simplest for finding the output impedance Z .
The last relationship of Eqs. (822) is often the quickest way to calculate
the voltage between two points in a network. This equation states that "the
voltage equals the shortcircuit 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 34. Then the ratio of the voltage across
34 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 12 instead of 34.
88 LOOKING INTO THE PLATE OR CATHODE OF A TUBE
Let us now return to the generalized amplifier of Fig. 8 11a and find a
Thevenin's equivalent circuit, first from plate to ground and then from
cathode to ground.
The Output from the Plate The signal v a and the resistor R p are now
considered external to the amplifier. Hence, for the moment, we set v a = ®
and interpret R p as the external load R L . The load current i L from plate to
ground is the negative of the plate current i p . Hence, with R L = R p = 0, we
Sac. 88
VACUUMTUU SMALLSIGNAL MODELS AND APPLICATIONS / 201
obtain the shortcircuit load current I from Eq. (819) :
PLVi
i. " + 1
? + v k
— HVi + 0* + l)Vk
T P + (M + 1)R*
(823)
The opencircuit voltage V is found as follows, using Eq. (819) :
u + 1
V = lim (i P Rp) = lim „
R t —*oa iJ p ~>= i }> ~t~ "'P
= fiVi + <jt* 4 l)w*
+ n
R,
+ Rk
(824)
The opencircuit 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. (822). Thus
Z = j  r, + 0* + D«*
(825)
The above results lead to the Thevenin's circuit of Fig. 81 3a. We conclude
that, "looking into the plate" of an amplifier, we see (for smallsignal operation)
an equivalent circuit consisting of two generators in series, one of — m times the
gridsignal voltage v it and the second (n + 1) times the cathodesignal 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. 81 3a.
r p (m+ iWk P
o
flUi
(*+iK
^9 81 3 The equivalent circuit for the generalized amplifier of Fig. 81 1 between
W) plate and ground, (b) cathode and ground.
202 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 88
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. 81 la. The load current %l from
cathode to ground equals i p . Hence, with Rl = Rk = 0, we obtain for the
shortcircuit load current / from Eq. (819)
/ =
liVi — v a
Tp + R,
The opencircuit voltage V is given by
V = lim i,
ft*— »«
Rk —
li + 1
(826)
(827)
The opencircuit 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
(828)
The above results lead to the Thevenin's circuit of Fig. 8136. We conclude
that, "looking into the cathode" of an amplifier, we see (for smallsignal 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 edg rid Amplifier This configuration is obtained from the
generalized circuit of Fig. 811 by setting d = Vi = 0. The equivalent circuit,
obtained from Fig. 813a, is indicated in Fig. 814a. By inspection the gain is
A = h = J»+\ )R * UJ? (829)
v k R p + r p + (n + l)R k K
Note that, since A is positive, there is no phase shift between output and
input. If Rt — and n » 1, then the gain has almost the same value as for
r p + (m + 1 )R k
r P /(r+iy
(//+ l)v k
A/W
(«>
(&)
(c)
Fig. 814 The Thevenin circuits for the three bosic amplifier configurations
$« 88
VACUUM TUBE SMALL SIGNAl WODEIS AND APPUCATIONS / 203
a commoncathode amplifier [Eq. (816)]. 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 groundedgrid 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. 824). 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. 811 by setting Vk = v a = and R p = 0. The equivalent
circuit is indicated in Fig. 8 14b. By inspection the gain is
A =^ =
Vi
M+ 1
Rk
nRk
g m Rk
m + 1
+ Rk
+ (m + DRk 1 + g m Rk
if m » 1
(830)
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 «
^41
(831)
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. 714).
The output impedance of the cathode follower is much smaller than the
plate resistance. For example, if n » 1, then
2 = 
~ r i = ± (832)
M + 1 M 9m
* 0r a ff m of 2 millimhos, the output impedance is only 500 ft, and for a higher
value of transconductance, Z is even less. On the other hand, since the input
sl Snal is applied to the grid, the input impedance (for negative grid voltages
where the grid current is negligible) is very high (ideally infinite). A cathode
.lower ' s usually employed when a high input impedance and a low output
^Pedance are desired.
The high input impedance of a cathode follower makes it ideal for appli
*«ons where the loading on a signal source must be kept at a minimum.
ne low output impedance permits it to support a heavy capacitive load.
204 / ELECTRONIC DEVICES AND CIRCUITS
S«. 89
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. 175), it
possesses great stability and linearity. Many electronic instruments take
advantage of these desirable features of cathode followers. The highfre
quency characteristics of the cathode follower are considered in Sec. 814.
The Groundedcathode Amplifier The equivalent circuit for this con
figuration is given in Fig. 836 and repeated in Fig. 814c, for comparison with
the groundedgrid and groundedplate amplifiers. The groundedcathode
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.
89
CIRCUITS WITH A CATHODE RESISTOR 1
Many practical networks involve the use of a resistor in the cathode circuit.
Some of the most important of these "cathodefollowertype" circuits are
described in this section.
The Splitload Phase Inverter This circuit appears in Fig. 815. A
single input signal provides two output signals, v kn , which is of the same
polarity as the input, and v pn , which is of opposite polarity. Further, if the
o+
Fig. 815 The splitload phase in
verter.
.
Sec. 89
VACUUMTUBE SMAllSIGNAt MODELS AND APPLICATIONS / 205
late and cathode resistors are identical, the magnitudes of the two signals
must be the same, since the currents in the plate and cathode resistors are
equal. The amplification \A\ = \v kn /v\ = \v pn /v\ may be written directly by
comparison with either of the equivalent circuits of Fig. 813 (with v k = v a = 0)
as
\A\ =
fiR
g m R
r„ + 0* + 2)R l + g m R
(833)
The exact result differs from that given for the cathode follower [Eq. (830)]
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 platetocathode
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 Cathodecoupled Phase Inverter This circuit, shown in Fig. 81 6a,
serves the same purpose as the splitload 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. 813b may again be used to advantage to analyze the oper
ation of the cathodecoupled phase inverter. We replace each tube by its
r„ + R,
r B + R a
F 'Q. 816 ( ) The cathodecoupled phase inverter and (b) its equivalent circuit
from cathode to ground.
206 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 89
equivalent circuit as seen from the cathode. The resulting circuit is shown in
Fig. 8166. 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. 819). 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. 8166,
we find for the platetoplate gain
A m
Vol — v o1
(ii + U)R„
nR,
+ R,
(834)
which is the same gain that would be provided by a singletube 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 gridto
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. 816o 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. 1212.
An Amplifier with a Constantcurrent 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. 817. Referring to Fig. 813a, it appears that the
810
VACUUMTUBE SMALLSIGNAL MODELS AND APPLICATIONS / 207
fig. 817 Tube V3 acts as a very high
dynamic resistance of value
+ ( M + i)R k in the cathode circuit
of tubes VI and V2. The voltage
divider R is used to balance the outputs
from the two plates.
A/Wi
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. 817, highM lowcurrent
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 50M 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 50M resistor in this application.
A large dynamic resistance is plotted as a horizontal load line (Sec. 712)
and corresponds to a constant current. Hence the difference amplifier of Fig.
817 is said to be fed from a constantcurrent source.
810
A CASCODE AMPLIFIER
This circuit, consisting of two triodes in series (the same current in each),
is indicated in Fig. 818. That this circuit behaves like a pentode can be
seen as follows: The load for Fl is the effective impedance looking into the
cathode of 72; namely, R p = (R + r,)/fci + 1). For large values of m this
m ay be very small, and to a first approximation can be considered as a short
ci rcuit for signal voltage. Hence the plate potential of 71 is constant. The
definition of the transconductance is
ft
\Ava/v P
** ei ice the signal current is Ai P = g m Av G = g m v h where vi is the signalinput
208 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 8?o
Fig. 818 The cascode amplifier.
Jyx —
R + ( M + 2)r p
(835)
If (>* + 2)r p ;» iJ and if p. » 1, this is approximately
~nR
At~
= —g m R
which is the result obtained by the qualitative arguments given above.
It is possible to apply an ac signal voltage 7 2 (in addition to the bias
voltage 7') to the grid of 72. Under these circumstances 71 acts as ao
voltage. The gain is A = — R&ip/Vi — —g m R, which is the expression for
the gain of a pentode [Eq. (87) with Z L = R].
Another point of view is the following: The plate dc voltage V Pl of 71
is determined by the gridtoground 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
constantcurrent characteristic of a pentode. From this discussion it is clear
that V takes the place of the screen voltage in a pentode. The cascode ampli
fier has the advantages over the pentode that no screen current need be sup
plied and it has the low noise of a triode.
The exact expression for the amplification is found by replacing V2 by
an impedance (R + r p )f(fi \ 1) and 71 by a generator fiVi in series with an
impedance r P . The result is
S0C.811
VACUUMTUBE SMALLSIGNAL MODELS AND APPLICATIONS / 209
impedance of magnitude r p in the cathode of 72. The voltage gain for this
signal W is
A 2 =
R+ 0* + 2)r T
(836)
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, 818 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. D2),
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.
811
1NTERELECTRODE CAPACITANCES IN A TRIODE 2
e assumed in the foregoing discussions that with a negative bias the input
urrent was negligible and that changes in the plate circuit were not reflected
™e grid circuit. These assumptions are not strictly true, as is now shown.
d; i . ^"^ pl ate > ana * cathode elements are conductors separated by a
ectnc (a vacuum), and hence, by elementary electrostatics, there exist
Paeitances between pairs of electrodes. Clearly, the input current in a
n dedcathode amplifier cannot be zero because the source must supply
jk . *° the gridcathode and the gridplate capacitances. Furthermore,
' lr mut and output circuits are no longer isolated, but there is coupling
210 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 8T1
V nt = V.
1 i
Fig. 819 The schematic and equivalent circuits of a groundedcathode
amplifier, taking into account the interelectrode capacitances.
between them through the gridplate 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 outputtoinput 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. 819. 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. 8196) yields results that are more precise than those resulting
from the analysis of the simple circuit of Fig. 83. It will be noted that the
same procedure outlined in Sec. 83 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. 8196.
The Voltage Gain The output voltage between terminals P and K is
easily found with the aid of the theorem of Sec. 85, namely, V„ = IZ, where /
is the shortcircuit current and Z is the impedance seen between the terminals.
To find Z, the generators F, and juF, in Fig. 8196 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
C837)
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
VACUUMTUBE SMALLSIGNAL MODELS AND APPLICATIONS / 211
The current in the direction from P to K in a zeroresistance 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 shortcircuit 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. (837) and (838),
—g m + Y g
(838)
A =
Y L + Yp k + g p + Y s
(839)
It is interesting to see that Eq. (839) 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. (839) reduces to
A m
— g»
9*
9v + Y L 1/r, + 1/Z L
= —g m Z' L
(840)
where Z' L is r p \\Z L . This equation is identical with Eq. (86).
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 audiofrequency 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 (840).
Since the interelectrode capacitances have a relatively minor effect on the
audio gain of an amplifier, why is it important to make note of them? The
answer is to be found in the input impedance of the tube (the loading of the
gge on the input circuit) and in the feedback between output and input
rcuits. Also, if the amplifier is to be used beyond the audio range, say,
a video (television or radar) amplifier, the capacitances may seriously affect
the
are now examined
gain and the exact expression, Eq. (839), must be used. These effects
INPUT ADMITTANCE OF A TRIODE
8.12
from n8 if eCt * 0n °^ ^' ^"^ revea ^ s tnat tne g^ circuit is no longer isolated
the plate circuit. The input signal must supply a current U. In order
212 / ELECTRONIC DEVICES AND CIRCUITS
to calculate this current, it is observed from the diagram that
it  ViY*
and
h= V BP Y gp m (F..+ V kp )Y sp
Since V kp = — ?",,* = — AV U then the total input current ia
Ii = h + h = [Y, k + (1  A) Y„)Vi
From Eq. (841), the input admittance is given by
F. = ^= 7 Bk +(l A)Y„
Sec, 812
(Wl)
(842)
This expression clearly indicates that, for the triode to possess a negligible
input admittance over a wide range of frequencies, the gridcathode and the
gridplate capacitances must be negligible.
Input Capacitance (Miller Effect) Consider a triode with a platecircuit
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. (842) becomes
Yi = MC Bk + (1 + g m R p )CJ
(843)
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
(844)
This increase in input capacitance d over the capacitance from grid to cathode
C B t is known as the Miller effect. The maximum possible value of this expres
sion is C gk + (1 + ti)C 9P , which, for large values of u, may be considerably
larger than any of the interelectrode capacitances.
This input capacitance is important in the operation of cascaded ampli
fiers. In such a system the output from one tube is used as the input to a
second tube. Hence the input impedance of the second stage acts as a shunt
across the output of the first stage and R p is shunted by the capacitance &■
Since the reactance of a capacitor decreases with increasing frequencies, the
resultant output impedance of the first stage will be correspondingly low for
the high frequencies. This will result in a decreasing gain at the higher
frequencies.
EXAMPLE A triode has a platecircuit resistance of 100 K and operates »*
20 kHz. Calculate the gain of this tube as a single stage and then as the first
tube in a cascaded amplifier consisting of two identical stages. The tube parame
ters are g m  1.6 millimhos, r„ = 44 K, » = 70, C ek = 3.0 pF, C ph = 3.8 pF, and
C ep = 2.8 pF.
1
s« *i2
VACUUMTUBE SMALLSIGNAL MODELS AND APPLICATIONS / 213
Solution
Y Q t = jo)C ek = j2t X 2 X 10' X 3.0 X 10"" = j'3.76 X 10~ 7 mho
Y& = jmCpt m J4.77 X 10" 7 mho
Y„ = jo>C gp = j3.52 X 10~ 7 mho
g = — = 2.27 X 10" 6 mho
Y p = — = 10* mho
R p
g m = 1.60 X 10 3 mho
The gain of a onestage amplifier is given by Eq. (839) :
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. (85), 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, (844) :
Ci = C gk + (1 + g m R p )C sp = 3.0 + (1 + 49) (2.8)  143 pF
Consider now a twostage 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 100K 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 highfrequency
amplifiers. Let it be assumed that the input capacitance, taking into account
the various factors just discussed, is 200 pF (probably a conservative figure).
Then the load admittance ia
Yl = —+ juCi = 10" s + j2ir X 2 X 10 4 X 200 X 10" 11
Rp
= 10" s + j*2.52 X 10" s mho
214 / ELECTRONIC DEVICES AND CIRCUITS
The gain is given by Eq. (840) :
9m 1.6 X 10"*
Sac. 812
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. (842) becomes
Yi = uC op Ai + MC ek + (1  A$C„\
(845)
(846)
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. (846) and (8^7), we have
1
Rt
aCgpAi
d = C„* 4 (1  A^C
(847)
(848)
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 gridplate capacitance. If this feed
back feature reaches an extreme stage, the system will lose its entire utility
as an amplifier, becoming in fact a selfexcited amplifier, or oscillator.
Sec. 813 VACUUMTUBE SMALlS/GNAi MODELS AND APPIICAT/ONS / 215
g_13 INTERELECTRODE CAPACITANCES
IN A MULT1ELECTRODE TUBE 2
The wiring diagram of a tetrode is given in Fig. 820a, and the equivalent cir
cuit taking interelectrode capacitances into account is indicated in Fig. 8206.
In drawing the equivalent circuit, the rules given in Sec. 83 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 shortcircuited 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. 76, 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. 8206 may be redrawn more
simply, as shown in Fig. 821, where
Cl — Cgk + Cg
Ci — Cp, f c
t*
(849)
• 1 i O— — • • • —
•X ©
(a)
K
(6)
Fig. 820 The schematic and equivalent circuits of a tetrode con
nected as an amplifier.
216 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 814
Fig. 821 The ideal equivalent circuit of a
tetrode. The gridplate 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 platecircuit impedance. The
input admittance of the tube is seen to be simply
¥i = jod (850)
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. 819. The idealization made here consists in the assumption that the
gridplate capacitance is zero rather than a very small fraction of a picofarad.
The circuit of Fig. 821 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. 820), with the addition that the sup
pressor grid is connected to the cathode. Then, from an equivalent circuit
analogous to that in Fig. 8206, it follows that the equivalent circuit of a
pentode is also given by Fig. 821. In this diagram
Ct = C ek + C gt Ci = C pk + (?„ + C p3 (851)
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.
814
THE CATHODE FOLLOWER AT HIGH FREQUENCIES
Our previous discussion of cathode followers neglected the influence of the
tube capacitances. These capacitances are now taken into account.
1
Sic. 81*
VACUUMTUBE SMALLSIGNAL MODELS AND APPLICATIONS / 217
Voltage Gain The groundedplate configuration of a triode, including
all capacitances, is given in Fig. 82 2a, and its linear equivalent circuit, in
Fig. 8226. 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. 811 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 " (852)
A =
Ft + g p + g m + Yt
where
Y k = "5~
Kk
Yt — j&Ct Ct — C g k + Cpk + Ckn,
Equation (852) may be written in the form
(9m + ju>C g k)Rk
A m
1 + Km + D/r, + jwCrlfo
(853)
(854)
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 lowfrequency
value is, under these circumstances, given by the condition u(C T + Cl) = g m 
Typically, if the total capacitance is, say, 50 pF and g m = 3 millimhos, as for
a half section of a 12AU7, then / a >w 9.5 MHz. This calculation shows that a
cathode follower may be useful well into the video range.
I ^
1/ 
1
' \\ *? ' 1
> l
> o o <
* ]
 V
Ckn
(6)
N or P
Fig. 822 (a) The cathode follower, with interelectrode capacitances taken into
account, and (b) its equivalent circuit.
218 / ELECTRONIC DEVICES AND CIRCUITS
Sec. BU
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, 8226. 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)
(855)
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. (855) the input capacitance is given by
C,(cathode follower) ■ C sv + C gk (l  A) (856)
On the other hand, for a groundedcathode amplifier, we have, from Eq. (844),
C(amplifier) = C ek + C„(l  A) (857)
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 lowfrequency admittance g m + g p
[Eq. (832) j the admittance of the total shunting capacitance C T  Thus
Y = g m + g p + Y T
(858)
S«
8U
VACUUMTUBE SMALtSlGNAL MODELS AND APPLICATIONS / 219
This result may be justified directly by applying a signal V to the output
terminals and computing the current which flows through V with the grid
crrounded (and Rk considered as an external load). Since g m = y.g p and assum
ing M » 1' we ma y n eglect g v compared with g m and consider that the output
admittance is unaffected by the capacitance until Y T becomes large enough
to be comparable with g m . The calculation made above in connection with
the frequency response of the cathode follower indicates that the output
impedance does not acquire an appreciable reactive component until the fre
quency exceeds several megahertz.
REFERENCES
1. Valley, G. E., Jr., and H. Wallman: "Vacuum Tube Amplifiers," MIT Radiation
Laboratory Series, vol. 18, chap. 11, McGrawHill 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, McGrawHill Book Com
pany, New York, 1941.
9 /TRANSISTOR
CHARACTERISTICS
The voltampere 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.
91
THE JUNCTION TRANSISTOR 1
A junction transistor consists of a silicon (or germanium) crystal in
which a layer of ntype silicon is sandwiched between two layers of
ptype silicon. Alternatively, a transistor may consist of a layer of
ptype between two layers of ntype material. In the former case
the transistor is referred to as a pnp transistor, and in the latter case,
as an npn 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. 94.
The two types of transistor are represented in Fig. 9la. The
representations employed when transistors are used as circuit elements
are shown in Fig. 916. 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 emitterbase 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, collectorbase, and collectoremitter
voltages, respectively. (More specifically, V E b represents the voltage
drop from emitter to base.)
220
Emitter Base Collector
I C
TRANSISTOR CHARACTERISTICS / 221
Emitter Base Collector
(a)
i Collector
Vcb Vm
pnp type (ft) fl P<
Emitter '
npn type
Fig, 91 (a) A pnp and an npn transistor. The emitter
(collector) junction is J s (Jc). (b) Circuit representation of the
two transistor types.
Emitter
±
(a)
Base
(b)
Collector
V'*„i i
r — J Space
charge
width — * V *~
Effective
base width
— IV
V M 
Emitter
(ptype)
Base
(ntype)
CO
Collector
(ptype)
I
Fig. 92 (a) A pnp 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 reversebias collector
junction voltage \V C b\ is increased, the effective base width W decreases.
222 / ELECTRONIC DEVICES AND CIRCUITS
Sec, 92
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. 92a. Here a pnp transistor
is shown with voltage sources which serve to bias the emitterbase junction
in the forward direction and the collectorbase junction in the reverse direction.
The variation of potential through an unbiased (opencircuited) transistor
shown in Fig. 926. The potential variation through the biased transistor
indicated in Fig. 92c. 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.
(613) (a few tenths of a volt) — required so that no current flows across each
junction. (Since the transistor may be looked upon as a pn junction diode
in series with an np 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 emitterbase junction lowers
the emitterbase potential barrier by 7 M , whereas the reverse biasing of
the collectorbase junction increases the collectorbase potential barrier by
\V CB \. The lowering of the emitterbase 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 ntype material to the collectorbase junction.
The holes which reach this junction fall down the potential barrier, and are
therefore collected by the collector.
92
TRANSISTOR CURRENT COMPONENTS
In Fig. 93 we show the various current components which flow across the
forwardbiased emitter junction and the reversebiased 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. 91). In a commercial transistor the doping
of the emitter is made much larger than the doping of the base. This feature
ensures (in a pnp 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. 93 (actually, electrons
TRANSISTOR CHARACTERISTICS / 223
Fig. 93 Transistor current components for a forwardbiased emitter junction and a
reversedbiased 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 opencircuited 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. 93, we note that
Ic = h
pC
(91)
For a pnp 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. 93 is from right to left, then for
a pnp transistor, Ico is negative. For an npn 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 pnp transistor we have
7 =
pE
IpE + I»l
IpM
I B
(92)
where I pli i s the. injected hole diffusion current at emitter junction and I, lE is
the
Ejected electron diffusion current at emitter junction.
224 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 92
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 pnp transistor we have
P ~ j—
(93)
Largesignal Current Gain a We define the ratio of the negative of the
collectorcurrent increment to the emitter current change from zero (cutoff)
to Ig as the largesignal current gain of a commonbase transistor, or
Ic — Ico
(94)
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. (91) and (94),
. IpC _ IpC IpE
Ib IpE I B
Using Eqs. (92) and (93),
a  0*7
(95)
(96)
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 pnp 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. 96. 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 forwardbiased and the collector is
reversebiased), the collector current is given by Eq. (94), or
Ic ■ — olI b + 1 1
(97)
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. (97) so that it may apply not only when the
collector junction is substantially reversebiased, but also for any voltage
across J c  To achieve this generalization we need but replace Ico by the
current in a pn diode (that consisting of the base and collector regions).
This current is given by the voltampere relationship of Eq. (631), with L
S*. * 3
TRANSISTOR CHARACTERISTICS / 225
replaced by —Ico and V by V c , where the symbol V c represents the drop
cross Jc fr° m tne V to the n side. The complete expression for I c for any
y c and Is 1S
Ic = al s + Ico{\ ~ * Vclv *) (98)
Note that if V c is negative and has a magnitude large compared with Vt,
Eq. (98) reduces to Eq. (97). The physical interpretation of Eq, (98) is
that the pn 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. 96.
93
THE TRANSISTOR AS AN AMPLIFIER
A load resistor R L is in series with the collector supply voltage V C c of Fig.
92o. A small voltage change AF, between emitter and base causes a rela
tively large emittercurrent change Al E  We define by the symbol a' that
fraction of this current change which is collected and passes through R&. The
change in output voltage across the load resistor AF e = a'Ri Al E may be
many times the change in input voltage AF,. Under these circumstances,
the voltage amplification A = AVjAVi will be greater than unity, and the
transistor acts as an amplifier. If the dynamic resistance of the emitter junc
tion is r' t , then AF, = r' e Alg, and
a'R L Me <x'R l
A m
A/j
(99)
From Eq. (641), 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 lowresistance input circuit is
transferred to the highresistance output circuit. The word "transistor,"
whkh 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 collcctortobase voltage and is called the smallsignal forward
wartcircuit current transfer ratio, or gain. More specifically,
Mi
(910)
AI B \ y c*
P n the assumption that a is independent of I E > then from Eq. (97) it follows
th at«' „
226 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 94
94
TRANSISTOR CONSTRUCTION
contact
3mm
rT
3 mm c»
Fig. 94 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 npn grownjunction transistor is illustrated in Fig.
94a. 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 ptype atoms as required.
Alloy Type This technique, also called the fused construction, is illus
trated in Fig. 946 for a pnp transistor. The center (base) section is a thin
wafer of ntype material. Two small dots of indium are attached to opposite
sides of the wafer, and the whole structure is raised for a short time to a high
temperature, above the melting point of indium but below that of germanium.
The indium dissolves the germanium beneath it and forms a saturation solu
tion. On cooling, the germanium in contact with the base material recrystal
lizes with enough indium concentration to change it from n type to p type.
The collector is made larger than the emitter, so that the collector subtends
a large angle as viewed from the emitter. Because of this geometrical arrange
ment, very little emitter current follows a diffusion path which carries it to
the base rather than to the collector.
Electrochemically Etched Type This technique consists in etching
depressions on opposite sides of a semiconductor wafer in order to reduce the
Aluminum
Silicon metal izat ion
dioxide Emitter J^ ^
contact QE
TRANSISTOR CHARACTERISTICS / 227
thickness of this base section. The emitter and collector junctions are then
formed by electroplating a suitable metal into the depression areas. This
type of device, also referred to as a surfacebarrier transistor, is no longer of
commercial importance.
Diffusion Type This technique consists in subjecting a semiconductor
wafer to gaseous diffusions of both n and ptype impurities to form both the
emitter and the collector junctions. A planar silicon transistor of the diffusion
type is illustrated in Fig. 94c. In this process (described in greater detail in
Chap. 15 on integratedcircuit techniques), the basecollector 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. 152) consists in growing
a very thin, highpurity, singlecrystal 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. 15116).
The foregoing techniques may be combined to form a large number of
methods for constructing transistors. For example, there are diffusedalloy
types, growndiffused devices, alloyemitterepitaxialbase transistors, etc. The
special features of transistors of importance at high frequencies are discussed
in Chap. 13. The voltampere 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 pointcontact transistors are very poor, and
48 a result these transistors are no longer of practical importance.
95
Th
DETAILED STUDY OF THE CURRENTS IN A TRANSISTOR
18 analysis follows in many respects that given in Sec. 65 for the current
mponents in a junction diode. From Eq. (614) 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. 9la) electrons are injected from the base region across
i errilt 'ter junction into a p region which is large compared with the diffusion
Ein. This is precisely the condition that exists in a junction diode, and
ce the expression for I np calculated previously is also valid for the transis
228 / ELECTRONIC DEVICES AND CIRCUITS
tor. From Eq. (625) we find that at the junction
/p(0) =
AeD n nBo
( € v a iv r _ j)
Sttc. 95
(911)
where in Eq. (625) 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) = thermalequilibrium electron concentration in the ptype mate
rial of the emitter (collector), m 3
Lb (Lc) (Lb) — diffusion length for minority carriers in the emitter (collector)
(base), m
Vg (Vc) = voltage drop across emitter (collector) junction; positive for a
forward bias, i.e., for the p side positive with respect to the n
side
V T — volt equivalent of temperature [Eq. (634)]
p„ = hole concentration in the ratype material, m~*
p ne = thermalequilibrium value of p„
W = base width, m
/jm (In P ) ~ hole (electron) current in n (p) material
The Hole Current in the ntype Base Region The value of I pn is not
that found in Sec. 65 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. (618) ; namely,
* pn — AfiUp —z —
where p„ is found from the continuity equation. From Eq. (550),
 p»„  Kic* /£ » + K if +" L >
(912)
(913)
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. (622), or
and
Pn — Pno* " T
Pn = Piiot
rclVf
at x —
at x = W
(914)
S»c. 95
TRANSISTOR CHARACTERISTICS / 229
The exact solution is not difficult to find (Prob. 93). 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. (913) can be expanded into a
power series. If only the first two terms are retained, this equation has the
form
p„  Pno = K s + K& (915)
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. (912) and (915),
— AeDpKt = const
(916)
This result — that the minoritycarrier 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 (914) in
(915), we easily solve for K t and then find
/,«(0) = 
AeD p p n
W
[( € VdVr  1)  ( € VMlV T _ !)]
(917)
The EbersMoll Equations From Fig. 93 we have for the emitter current
t M = I P B + InB = /,.«>) + /„,((»
Using Eqs. (911), (917), and (918), we find
Ib  a n (€ v » lv *  1) + au(e v 'i v '  1)
where
«* a similar manner we can obtain
Ic m a«(« v ' v r  1) + anifiW*  1)
where we can show (Prob. 92) that
A /D p p no
a 22 = Ae I — B
a n =  AeD »P»
21 W
W
+
Dnn Co \
Lc /
(918)
(919)
(920)
(921)
(922)
We note that a n ■ On. This result may be shown 6 to be valid for a
isistor possessing any geometry. Equations (919) and (921) are valid
any positive or negative value of V s or Vc, and they are known as the
^ersMoll equations.
230 / ELECTRONIC DEVICES AND CIRCUITS
96 THE TRANSISTOR ALPHA
If V E is eliminated from Eqs. (919) and (921), the result is
Ic m 5» I B + (« M  ^A ( 6 VcfVr  l)
an \ on /
Sec. 9.
(923)
This equation has the same form as Eq. (98). Hence we have, by
comparison,
_ _%
Oil
02lOi2
Ico =
On
— fl22
(924)
(925)
(926)
Using Eqs. (920) and (922), we obtain
= 1
1 + D n n BQ W/LgD p 'p no
Making use of Eq. (52) for the conductivity, Eq. (533) for the diffusion
constant, and Eq. (519) for the concentration, Eq. (926) 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. 93
is obtained. We then find (Prob. 95) that
7 »
1
and
1 + (DnLBUBe/DpLspno) tanh (W/L B )
W
8* = sech ~
Lib
(928)
(929)
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. 96)
W,
I/?
and
1 + W<Tb/LeVB
LtE&B
WOB
LgGB
(930)
(931)
(932)
As the magnitude of the reversebias collector voltage increases, the spa^
charge width at the collector increases (Fig. 92) and the effective base width W
5*
97
TRANSISTOR CHARACTERISTICS / 231
, eases . Hence Eq. (932) indicates that a increases as the collector junction
becomes more reversebiased.
The emitter efficiency and hence also a is a function of emitter current.
TTnuation (930) 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
spacecharge 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.
(925), (920), and (922).
97
THE COMMONBASE CONFIGURATION
If the voltages across the two junctions are known, the three transistor cur
rents can be uniquely determined using Eqs. (919) and (921). 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.
92a, a pnp transistor is shown in a groundedbase configuration. This cir
cuit is also referred to as a commonbase, or CB, configuration, since the base
w common to the input and output circuits. For a pnp 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. 91, we see that J E is positive, I c is negative,
^d I B i s negative. For a forwardbiased emitter junction, V E b is positive,
&Q u for a reverse biased collector junction, V C b is negative. For an npn
jMststor all current and voltage polarities are the negative of those for a
' n ~P transistor. We may completely describe the transistor of Fig. 9la 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) 033)
Ic = MVcb, Ib) (934)
18 equation is read, "I c is some function <£ 2 of V C b and I B ")
Th
(Th
/He relation of Eq. (934) is given in Fig. 95 for a typical pnp ger
Ur n transistor and is a plot of collector current Ic versus collectortobase
232 / ELECTRONIC DEVICES AND CIRCUITS
Sec. o.y
Saturation
A..
tlve region
—\
< 40
£
« 30
I
£
u 20
5
I
(3 io
/.,
= 40mA
1
30
20
10
1
• ■
I
^~
i
I
Tco
\)
j
v
i
Cutoff region
i — i — i 1
Fig. 95 Typical common
base output characteristics
of a pnp transistor. The
cutoff, active, and satura
tion regions are indicated.
Note the expanded voltage
scale in the saturation
region.
0.25 2 4 6 8
Collectortobase voltage drop V cs , V
voltage drop V C b, with emitter current I E as a parameter. The curves of
Fig. 95 are known as the output, or collector, static characteristics. The rela
tion of Eq. (933) is given in Fig. 96 for the same transistor, and is a plot of
emittertobase voltage V BB versus emitter current I B , with collectortobase
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 spacecharge width at the output junction diode as indicated by Eq, (647).
From Fig. 92 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. 96 Commonbase input
characteristics of a typical
pnp germanium junction
transistor.
Wi
o I
10 20 30 40
Emitter current I s , mA
V
1
■ n open _
4
•>
K» ■
= 0V
r ~>
.10
'20
n
See
97
TRANSfSTOR CHARACTERISTICS / 233
consequences: First, there is less chance for recombination within the
region Hence the transport factor #*, and also a, increase with an
urease in the magnitude of the collector junction voltage. Second, the
haree gradient is increased within the base, and consequently, the current of
inority* carriers injected across the emitter junction increases.
The Input Characteristics A qualitative understanding of the form of
the input and output characteristics is not difficult if we consider the fact that
the transistor consists of two diodes placed in series "back to back" (with the
two cathodes connected together). In the active region the input diode
(emittertobase) is biased in the forward direction. The input characteristics
j jrig. 96 represent simply the forward characteristic of the emittertobase
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. 96) 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. 96.
The curve with the collector open represents the characteristic of the
forwardbiased 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. 95, 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
tobase 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 inputdiode forward current which
Caches the collector. Note also that lev is negative for a pnp transistor and
Positive for an npn 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. (97). In the active region, the collector current is essentially
"^dependent of collector voltage and depends only upon the emitter current.
234 / ELECTRONIC DEVICES AND CIRCUITS
S«, 98
S*
98
TRANSISTOR CHARACTERISTICS / 235
However, because of the Early effect, we note in Fig. 95 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 forwardbiased, 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 pnp 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. (921)]. 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. 95, 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 reversebiased, is referred to as the cutoff region. The
temperature characteristics of I C o are discussed in Sec. 99.
98
THE COMMONEMITTER CONFIGURATION
Most transistor circuits have the emitter, rather than the base, as the terminal
common to both input and output. Such a commonemitter CE, or grounded
emitter, configuration is indicated in Fig. 97. In the commonemitter, as in
the commonbase, configuration, the input current and the output voltage
Fig. 97 A transistor commonemitter 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)
(935)
(936)
Equation (935) describes the family of input characteristic curves, and
F (936) describes the family of output characteristic curves. Typical out
ut and input characteristic curves for a pnp junction germanium transistor
are given in Figs. 98 and 99, respectively. In Fig. 98 the abscissa is the
collectortoemitter 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. 98 are larger than in the commonbase charac
teristics of Fig. 95. 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. 98 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. 98 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. 49 in connection with a diode.
The input Characteristics In Fig. 99 the abscissa is the base current Ib,
the ordinate is the basetoemitter voltage Vbb, and the curves are given for
various values of collectortoemitter voltage V C s We observe that, with the
collector shorted to the emitter and the emitter forwardbiased, the input char
acteristic is essentially that of a forwardbiased diode. If V BB becomes zero,
fig, 98 Typical commonemitter
Output characteristics of a pnp
9«rmanium junction transistor. A
load line corresponding to Vcc =
10 V and R L = 500 U is super
posed. (Courtesy of Texas
lns trumer,ts, Inc.)
2 4 6 8 10
Collector emitter voltage V cg . , V
236 / ELECTRONIC DEVICES AND CIRCUITS
S«. 9.j
H 0.4
as
0.2
0.1
1
r=25°C
t^OA
~~0~~
Fig. 99 Typical commonemitter input
characteristics of the pnp germanium Junc
tion transistor of Fig. 98.
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 shortcircuited. For any other value of V C s, the base cur
rent for Vbb ■» is not actually zero but is too small (Sec. 915) to be observed
in Fig. 99. 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. 99.
The input characteristics for silicon transistors are similar in form to those
in Fig. 99. 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 reversebiased and the emitter
junction is forwardbiased. In Fig. 98 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 commonemitter characteristics in the active region are readily under
stood qualitatively on the basis of our earlier discussion of the commonbase
configuration. The base current is
la  (/c + / g )
Combining this equation with Eq. (97), we find
Id
1 
+
ali
1  a
(937)
(938)
5*
99
TRANSISTOR CHARACTERISTICS / 237
uation (97) 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. (938) is valid for values of Vcs in excess of a few
a1 1 8
If a were truly constant, then, according to Eq. (938), I c would be inde
dent oi VcE m ^ fa e curves of Fig. 98 would be horizontal. Assume that,
because of the Early effect, a increases by only onehalf 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 commonemitter curves, and hence that commonemitter
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 .
99
THE CE CUTOFF REGION
We might be inclined to think that cutoff in Fig. 98 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 commonbase
characteristics are described to a good approximation even to the point of
cutoff by Eq. (97), repeated here for convenience:
Ic = — oJb + 1 1
(939)
From Fig. 97, if I B = 0, then I E = Ic Combining with Eq. (939), we
have
Ico
Ic = —Is ~
1 
SpJi
(940)
The actual collector current with collector junction reversebiased and base
opencircuited 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 reversebias 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. 915 we show that a reversebiasing voltage of the order of
° 1 V established across the emitter junction will ordinarily be adequate to cut
off _ a germanium transistor. In silicon, at collector currents of the order of Ico,
11 is found 6 ' ' that a is very nearly zero because of recombination in the emitter
238 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 99
junction transition region. Hence, even with I B = 0, we find, from Eq.
(940), that Ic = I co = — Ib, so that the transistor is still very close to cutoff
We verify in Sec. 915 that, in silicon, cutoff occurs at V BE « V, correspond
ing to a base shortcircuited 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 collectorjunction 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. 910, where Vbb represents a biasing voltage intended to keep the tran
sistor cut off. We consider that the transistor is just at the point of cutoff,
with I s = 0, so that I B = I CB0 . If we require that at cutoff Vbb « 0.1 V,
L
S«c
910
TRANSISTOR CHARACTERISTICS / 239
fig. 910 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
(941)
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 baseemitter junction will be 10.1 V. Hence we must use
a transistor whose maximum allowable reverse basetoemitter junction volt
age before breakdown exceeds 10 V. It is with this contingency in mind that
a manufacturer supplies a rating for the reverse breakdown voltage between
emitter and base, represented by the symbol BVebo The subscript indi
cates that BVebo is measured under the condition that the collector current is
zero. Breakdown voltages BVebo may be as high as some tens of volts or as
low as 0.5 V. If BVbbo = 1 V, then Vbb must be chosen to have a maximum
value of 1 V. For Vbb = IV and for Icbo = 0. 1 mA maximum, R B cannot
exceed 9 K. For example, if R B = 8 K, then
Vbb + IcboRb = 1 + 0.8 = 0.2 V
so that the transistor is indeed cut off.
MO
THE CE SATURATION REGION
A load line has been superimposed on Fig. 98 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 "^wardbiased. 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. 98. 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. 98. In saturation,
he collector current is nominally V C c/Rl, and since R L is small, it may well
e necessary to keep V C c correspondingly small in order to stay within the
Wit
ations imposed by the transistor on maximum current and dissipation.
240 / ELECTRONIC DEVICES AND CIRCUITS
j 30
20
« 10
J
n
1 1
 0.35mA
T A = 25°C
0.30
0.2
0.2
5
soon II
[Z
1
0.15
— 1 1 1
^
"rjn
— 1 — 1 — '
0.10
— 1
0.0&
1 r
Sec. 9 JO
Fig. 911 Saturationregion com
monemjtter 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 collectortoemitter saturation voltage,
Vce (sat), with any precision from the plots of Fig. 98. We refer instead
to the characteristics shown in Fig. 911. In these characteristics the 0 to
— 0.5V region of Fig. 98 has been expanded, and we have superimposed the
same load line as before, corresponding to R L = 500 fi. We observe from
Figs. 98 and 911 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 commonemitter 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. 911, 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. 911, 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. 99 and 911. 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. Alloyjunction and epitaxial transistors g» ve
S* 9 10
TRANSISTOR CHARACTERISTICS / 241
the lowest values for Fca(sat) (corresponding to about 1 fi saturation resist
ance), whereas grown junction transistors yield the highest. Germanium
transistors have lower values for F C s(sat) than silicon. For example, an
alloy junction Ge transistor may have, with adequate base currents, values
for VciKsat) as low as tens of millivolts at collector currents which are some
tens of milliamperes. Similarly, epitaxial silicon transistors may yield satu
ration voltages as low as 0.2 V with collector currents as high as an ampere.
On the other hand, grownj 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 onetenth of this value
for Fas (sat) for either germanium or silicon. The temperature coefficient for
VWsat) is that of a forwardbiased diode [Eq. (639)]. In saturation the
transistor consists of two forwardbiased diodes back toback in series opposing.
Hence, it is to be anticipated that the temperatureinduced 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. 912. Note the
140
F 9 912 Plots of dc current gain
A '« (at V cs = 0.25 V) versus col
le tfor current far three samples of
* typ e 2N404 germanium transistor.
(Courtesy of General Electric Com
pany.)
10 20 30 40 50 60 70 80 90100110120130
— I c ,mA
242 / ElECTRONfC DEVICES AND CIRCUITS
Sec. 911
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\fhFE
2. If Vcb is determined from the circuit configuration and if this quantity ia
positive for a pnp transistor (or negative for an npn) f the transistor is in satu
ration. Of course, the emitter j unction must be simultaneously forward biased,
but then we should not be testing for saturation if this condition were not
satisfied.
911 LARGESIGNAL, DC, AND SMALLSIGNAL
CE VALUES OF CURRENT GAIN
If we define by
a
=
1  a
and replace Ico by Icbo, then Eq. (938) becomes
Ic = (1 + 0)1 cbo + 01 b
From Eq. (943) we have
=
Ic — h
Ib — ( — Icbo)
(942)
(943)
(944)
In Sec. 99 we define cutoff to mean that Is = 0, Ic = Icbo, and I B = — Icbo
Consequently, Eq. (944) gives the ratio of the collectorcurrent increment to
the basecurrent change from cutoff to I B , and hence represents the largesignal
current gain of a commonemitter 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. 910 we define the de current gain by
0d C = j = h
Ib
(945)
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 912
TRANSISTOR CHARACTER/ST/CS / 243
these conditions the largesignal and the dc betas are approximately equal;
then hrs =* 0
The smallsignal CE forward shortcircuit current gain 0' is defined as the
ratio of a collector current increment Al c for a small basecurrent change AI B
(at a given quiescent operating point, at a fixed collectortoemitter voltage
Vcb), or
F d! B k«
(946)
If is independent of current, we see from Eq. (943) that 0' = m h FE .
However, Fig. 912 indicates that is a function of current, and from Eq.
(943),
« + (Icbo + Ib) ~
oIb
(947)
The smallsignal 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. (947) becomes
ht e «■
1  (Icbo + Ib)
dhp
(948)
Since k FB versus I c given in Fig. 912 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. (948) is valid in the active region only. From Fig. 911 we see
that h f . — * in the saturation region because A/ c — * for a small increment AI B .
912 THE COMMONCOLLECTOR CONFIGURATION
Another transistorcircuit configuration, shown in Fig. 913, is known as the
commoncollector configuration. The circuit is basically the same as the cir
cuit of Fig. 97, 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
9 913 The transistor commoncollector
^"figuration.
244 / aCCTRONIC DEVICES AND CIRCUITS
S«e. 9U
commoncollector is much the same as for the commonemitter 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 backbiased
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.
913 GRAPHICAL ANALYSIS OF THE CE CONFIGURATION
It is our purpose in this section to analyze graphically the operation of the
circuit of Fig. 914. In Fig. 9 15a the output characteristics of a pnp
germanium transistor and in Fig. 9156 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. 915o we have drawn a load line for a 250fl load with Vcc = 15 V.
If the input basecurrent signal is symmetric, the quiescent point Q is usually
selected at about the center of the load line, as shown in Fig. 915o. 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. 79. 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 rootmean 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. 914 The CE transistor configur
ation.
TRANSISTOR CHARACTERISTICS / 245
Base voltage v BEt V
""] 1 1 1'l'ITI
rrm ,
nin
.J LI I ' ' iftV
"T^7t *o
Dynamic curve
■P ^zjJrfF
.... ,.'l I ' l f  s
J^TI l£r*Tir
y***H) ^"
• r VcF
::i!:::2!gj2:
"i't* i
~ smm
0.15 j'l
irToV.
_Vcb i ;:
■ 2 4 6 8 10 12 14
Collector voltage v ct: ,V
(a)
O 100200800400600600
Base current t B , ^ A
(b)
Fig. 915 (a) Output and [b] input characteristics of a pnp germanium transistor.
bility of ambiguity, the conventional doublesubscript notation should be used.
For example, in Figs. 9 16a to d and 914, we show collector and base currents
and voltages in the commonemitter 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 = AWb
The magnitude of the supply voltage is indicated by repeating the electrode
subscript. This notation is summarized in Table 91.
TABLE 91 Notation
kBta ntaneouB total vaiue
g"eacent value . . .
Eff aa . taneoUB value of varying component
e ctive value of varying component (phasor, if
a ainusoid)
^PPly voltage (magnitude).
Base (collector)
voltage with
respect to emitter
vb (t>c)
V B {Vc)
vt <»«)
n if 4
Vbb {Vcc)
Base (collector)
current toward
electrode from
external circuit
t« (tc)
Ib Uc)
u(i<)
h Uc)
246 I ELECTRONIC DEVICES AND CIRCUITS
Sec. 97 3
Sinusoid
2t •!
(6) («*)
Fig. 916 (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. 915a. 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 basecurrent waveform (the sinusoid of Fig. 916c) 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 basetoemitter voltage vbb for any combination of base current and
collectortoemitter voltage can be obtained from the input characteristic
curves. In Fig. 9156 we show the dynamic operating curve drawn for the
combinations of base current and collector voltage found along AQB of the
load line of Fig. 915a. The waveform v B s can be obtained from the dynaflU fl
operating curve of Fig. 9156 by reading the voltage v H b corresponding to »
S*. 9U
TRANSISTOR CHARACTERISTICS / 247
jriven base current i B . We now observe that, since the dynamic curve is not
a straight line, the waveform of Vb (Fig. 916rf) 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 916d 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 inputvoltage 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 basecurrent waveform is also sinusoidal.
From Fig. 9156 we see that for a large sinusoidal base voltage Vt, around the
point Q the basecurrent 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 collectorcurrent swing z c  for a given basecurrent 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 icves 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 (currentsource or voltagesource driver).
M4 ANALYTICAL EXPRESSIONS FOR
TRANSISTOR CHARACTERISTICS
The dependence of the currents in a transistor upon the junction voltages, or
vice versa, may be obtained by starting with Eq. (98), repeated here for
convenience:
Ic  a N I B  Ico(* v ° lv r  1) (949)
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. (949),
Ib m ~ aj Ic  Ibo(* v ' ,v t ~ I) (950)
tr
, ere a r is the inverted commonbase current gain, just as on in Eq. (949) 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 forwardbiased emitter. In the literature,
248 / ELECTRONIC DEVICES AND CIRCUITS
■V c
S«c. 9U
C
(collector)
Fig. 917 Defining the voltages and currents used in the EbersMoll equa
tions. For either a pnp or an np^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 forwardbiased (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 Basespreading Resistance r w The symbol Vc represents the drop
across the collector junction and is positive if the junction is forwardbiased.
The reference directions for currents and voltages are indicated in Fig. 917.
Since Vcb represents the voltage drop from collectortobase 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. 94), 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 crosssectional 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. 917. The difference between Vcb and V c is due to the ohmic drop
across the body resistances of the transistor, particularly the basespreading
resistance r»«.
The EbersMoll Model Equations (949) and (950) have a simple inter
pretation in terms of a circuit known as the EbersMoll model 6 This model &
shown in Fig. 918 for a p^ny transistor. We see that it involves two ideal
diodes placed back to back with reverse saturation currents — I bo and lco
and two dependent currentcontrolled current sources shunting the ideal
diodes. For a pnrp transistor, both Jco and I bo are negative, so that —*c°
$*. 9U
TRANSJSTOR CHARACTERISTICS / 249
and — I so are positive values, giving the magnitudes of the reverse saturation
currents of the diodes. The current sources account for the minoritycarrier
transport across the base. An application of KCL to the collector node of
Fig. 918 gives
Ic = a N r S + / = a N I B + I^ctvr  1)
where the diode current I is given by Eq. (626). 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. (949).
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
basespreading resistance from Fig. 917 and have neglected the difference
between Icbo and lco
Observe from Fig. 918 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. (949) and (950)
to solve explicitly for the transistor currents in terms of the junction voltages
as denned in Fig. 917, with the result that
ail CO. /„v„iv_ ,\ 1 so
Ie =
Ic =
1 — a^ai
aiflgo
( t vcir r  1) 
( t v,iv T _ 1) _
1 — aitai
lco
( e r,iv r _ !)
( e vciv r _ j)
(951)
(952)
1 — an (xi v 1 — atfai
These two equations were first presented by Ebers and Moll, 6 and are identical
with Eqs. (919) and (921), derived from physical principles in Sec. 95. In
eo~
k V <~ J
Fig. 918 The EbersMoll model for a pnp transistor.
250 / ELECTRONIC DEVICES AND CIRCUITS
that section it is verified that the coefficients
oliIco
S«c. 91 4
an =
1 — ayai
and
Oil ■ T
1 — ayai
are equal. Hence the parameters aw, a*, Icoj and Iso are not independent,
but are related by the condition
ailco = owlso
(953)
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)
(954)
Voltages as Functions of Currents We may solve explicitly for the
junction voltages in terms of the currents from Eqs. (951) and (952), with
the result that
= v T in(i lB + aiIc )
\ iso /
Iso
c + a^h
\ lco /
(955)
(956)
We now derive the analytic expression for the commonemitter charac
teristics of Fig. 98. The abscissa in this figure is the collectortoemitter
voltage Vcs = Vs — Vc for an npn transistor and is Vcs = V c — V E for a
pnp transistor (remember that V c and Vs are positive at the p side of the
junction). Hence the commonemitter characteristics are found by subtract
ing Eqs. (955) and (956) and by eliminating I s by the use of Eq. (954).
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. (942) and (953), we obtain (except for very small values of h)
where
V C M = ± V T In
ft'
ai Pi In
. _l u
PU
(957)
1 — aj
and
0N = P
1  a
Note that the + sign in Eq. (957) is used for an npn transistor, and the
— sign for a pnp device. For a pnp germaniumtype transistor, at Ic = "
Vcs = — Vt In (1/aj), so that the commonemitter characteristics do not po>& 8
through the origin. For a T ■ 0.78 and V T = 0.026 V, we have V C s = 6 mV
S*c 9 J 5
TRANSISTOR CHARACTERISTICS / 251
pjg.919 The common
emitter output character
istic for a pnp transistor
as obtained analytically.
k
I,
f~ J
0.9/3
T
/ * =
/ , ! . , . — i
100
f r in  nfaooe
0.1S 0.2 03 04
OS
v cs ,v
at room temperature. This voltage is so small that the curves of Fig. 98
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. 98 is drawn,
and hence near the origin it appears as if the curves rise vertically. However,
note that Fig. 911 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 commonemitter characteristic is
indicated in Fig. 919. 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. 98. 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. (957), which is
not valid for I B = 0.
The theoretical curve of Fig. 919 is much flatter than the curves of Fig.
98 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 commonemitter characteristic.
91 5 ANALYSIS OF CUTOFF AND SATURATION REGIONS
^t us now apply the equations of the preceding section to find the dc currents
^d voltages in the groundedemitter transistor.
252 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 915
The Cutoff Region If we define cutoff as we did in Sec. 99 to mean zero
emitter current and reverse saturation current in the collector, what emitter
junction voltage is required for cutoff? Equation (955) with I s = and
Ic m Ico becomes
(958)
Ve = V T In f 1  ^\ = V T In (1  a N )
where use was made of Eq. (953). 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. (958),
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 basespreading resistance r»' in Fig. 917. 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. (949) and
(950) become
U   ai I c + I bo (959)
Ic — —onIe + h
Solving these equations and using Eq. (953), we obtain
7 Ico{\ — a/) T Ieo(1 — <*n)
Ic = — : IB = ~
1 — asoti
1 — CtNCtI
(960)
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 basetoemitter bias beyond cutoff has very little effect (Fig. 920) on
the very small transistor currents.
Shortcircuited Base Suppose that, instead of reversebiasing 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 EbersMoll equations. The results are
Ico
Ic =
1 — asati
mli
and
Ie = —ail
ail ess
(961)
where Ices represents the collector current in the commonemitter configu
ration with a shortcircuited 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. 91 S
TRANSISTOR CHARACTERISTICS / 253
less m I°° an( * I* m 0* Hence, even with a shortcircuited emitter junction,
the transistor is virtually at cutoff (Fig. 920) .
Opencircuited 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^ 062)
1 — a N
It is interesting to find the emitterjunction voltage under this condition of a
floating base. From Eq. (955), with I B  —I c , and using Eq. (953),
a N (l — ai)
= V T In [
1 +
ai{\ — an)
(963)
I
For a N  0.9 and o/ = 0.5 (for Ge), we find V B = +60 mV. For a N m
2ai *= (for Si), we have V B « V T In 3 = +28 mV. Hence an opencircuited
base represents a slight forward bias.
The Cutin Voltage The voltampere characteristic between base and
emitter at constant collectortoemitter voltage is not unlike the voltampere
characteristic of a simple junction diode. When the emitter junction is
reversebiased, 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 forwardbiased, again, as in the simple diode, no appreciable
base current flows until the emitter junction has been forwardbiased 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 basetoemitter voltage will exhibit a cutin
voltage, just as does the simple diode. Such plots for Ge and Si transistors
are shown in Fig. 920o and b.
In principle, a transistor is in its active region whenever the baseto
emitter voltage is on the forwardbiasing 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. (955), 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. (955)
7,  (0.026,(2.30) log [l + <» X 10^105) j . 0.12 V
254 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 9TS
T — T "
•*C — *CES— ,
1 — I
*c — *t
' I
If, ~. /^JPrt —
1  a N
(Open circuit base)
as 0.2 o.i o v T =o.i a2 v;=o.3
(a) 0.06
Active
Cutoff
Cutin
regton
*C — IcES^^i
CES^^CO
Silicon
£
Ic = I CEO * I\
CEO ^ A CO
0.3 0.2 0.1
(*»)
Cutoff A
0.03 0.1 0.2
t
Opencircuit
base
0.3 0.4 V r =0.S 0.6 W*0.7 V aB ,V
Cutin 
Acttve
region "
Saturation
Fig. 920 Plots of collector current against basetoemitter voltage for
(a) germanium and (b) silicon transistors. (/ c is not drawn to scale.)
,.
$ac 915
p ig 921 Plot of collector
current against baseto
emitter voltage for various
temperatures for the type
2N337 silicon transistor.
(Courtesy of Transitron
Electronic Corporation.)
TRANSISTOR CHARACTERISTICS / 255
111
B
s
7
ti
5
4
3
,
/
/
180"C
/
/
100°
7
J
125
c l
2
1
h
S9°C
.y
m* 1 *■
J

/
ai 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Input voltage V BK , V
If the switch had been called upon to carry 2 mA rather than 20 raA, a cutin
voltage of 0.06 V would have been obtained. For a silicon transistor with
at = 0.5 and I bo = 1 nA and operating at 20 rnA (2 mA) we obtain from
Eq. (955) that V y = 0.6 V (0.3 V). Hence, in Fig. 920 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 921 shows plots, for several temperatures, of the collector current
as a function of the basetoemitter voltage at constant collectortoemitter
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 emitterjunction
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. (639)].
The Saturation Region Let us consider the 2N404 pnp 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. (953), «r = 0.50. From Eqs. (955) and (956), we
calculate that, at room temperature,
V B = (0.026) (2.30) log
and
V c = (0.026) (2.30) log
*or a pnp transistor,
V CB = VcV s = 0.11
[
[
20.35  (0.50) (20)'
lO" 3
20 + 0.99(20.35)
(2)(10 s )
= 0.24 V
= 0.11 V
0.24 « 0.13 V
256 / ELECTRONIC DEVICES AND CIRCUITS See. 916
Taking the voltage drop across rw (~100 fl) into account (Fig. 917),
Vcb  V c  IbTw = 0.11 h 0.035 = 0.15 V
and
Vbb = I B r w  Vg = 0.035  0.24 m 0.28 V
Note that the basespreading resistance does not enter into the calcu
lation of the collectortoemitter voltage. For a diff usedjunction transistor
the voltage drop resulting from the collectorspreading 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.
916
TYPICAL TRANSISTORJUNCTION VOLTAGE VALUES
Quite often, in making a transistorcircuit calculation, we are beset by a compli
cation when we seek to determine the transistor currents. These currents are
influenced by the transistorjunction 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 transistorjunction
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 firstorder approximation that
these junction voltages are all zero. On this basis we calculate a firstorder
approximation of the current. These firstorder currents are now used to
determine the junction voltages either from transistor characteristics or from
the EbersMoll equations. The junction voltages so calculated are used to
determine a secondorder 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 92. This table lists the collectortoemitter saturation
voltage [FcaCsat)], the basetoemitter saturation voltage [Kb* (sat) s V,],
the basetoemitter 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 npn transistor. For a pnp transistor the signs of all
entries should be reversed. Observe that the total range of V B s between cutin
Sc. 917
TRANSISTOR CHARACTERISTICS / 257
and saturation is rather small, being only 0.2 V. The voltage Vbb (active) has
been located somewhat arbitrarily, but nonetheless reasonably, at the mid
point of the active region in Fig. 920.
Of course, particular cases will depart from the estimates of Table 92.
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 92
may well make further approximations unnecessary.
TABLE 92 Typica
1 rirpn transistor
unction voltages at 25°Cf
VcMisat)
VWBftt) = V,
Fstf(active)
Vbe (cutin) ■ V y
Vbe(cuU>8)
Si
Ge
0.3
0.1
0.7
0.3
0.8
0.2
0.5
0.1
0.0
0.1
t The temperature variation of these voltages is discussed in Sec. 915.
Finally, it should be noted that the values in Table 92 apply to the
intrinsic junctions. The base terminaltoemitter voltage includes the drop
across the basespreading 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
917
TRANSISTOR SWITCHING TIMES
When a transistor is used as a switch, it is usually made to operate alternately
in the cutoff condition and in saturation. In the preceding sections we have
computed the transistor currents and voltages in the cutoff and saturation
states. We now turn our attention to the behavior of the transistor as it
makes a transition from one state to the other. We consider the transistor
circuit shown in Fig. 922a, driven by the pulse waveform shown in Fig. 9226.
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. 922c. The cur
nt does not immediately respond to the input signal. Instead, there is a
® ,a y, and the time that elapses during this delay, together with the time
quired for the current to rise to 10 percent of its maximum (saturation)
258 / ELECTRONIC DEVICES AND CIRCUITS
Sec 9.17
€
 t
Vy
v %
ic
0.9I C s
oA/W
T Vcc
(6)
v,
T
f
(a
^
~7t
\ .
(c)
0.1I C8
M
1 \^ '
'
b
i i
i
t.,
i i
1 t
t
H"*GNM U *— *OFF *t
Fig. 922 The pulse waveform in (b\ drives the transistor in (a) from cutoff to
saturation and back again, (c) The collectorcurrent response to the driving input
pulse.
value Ics ** V C c/Rl, is called the delay time t d . The current waveform has
a nonzero rise time U, which is the time required for the current to rise from
10 to 90 percent of Ics The total turnon 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 npn 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 emitterjunction transition capacitance so that the
transistor may be brought from cutoff to the active region. Second, even
when the transistor has been brought to the point where minority earners
have begun to cross the emitter junction into the base, a time interval Jl
required before these carriers can cross the base region to the collector junction
Sec
917
TRANSISTOR CHARACTERISTICS / 259
j^d be recorded as collector current. Finally, some time is required for the
collector current to rise to 10 percent of its maximum.
Rise Time and Fall Time The rise time and the fall time are due to the fact
that, if a basecurrent 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. 137).
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. 922c)
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. 923 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. 9246. 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 minoritycarrier charge is stored in the base. These densities are pictured
in Fig. 923.
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. 922. Since the
turnoff process cannot begin until the abnormal carrier density (the heavily
shaded area of Fig. 923) 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 storagetime delay may be two or three times
Density of
minority
'9 923 Minoritycarrier con
centration in the base for cutoff,
a ctive, a n{ j saturation conditions of
°Pe ration.
Emitter
Collector
jc =
x= W
260 / aecrnoN/c devices and circuits
S«e. ?T«
Emitter
junction
Collector
junction
t
V
t
1
Pa
\
\
A.** 7 *
*JC*J
X
= r*
Fig, 924 The minoritycarrier density in the
base region.
(6)
the rise or fall time through the active region. In any event, it is clear that,
when transistor switches are to be used in an application where speed is at
a premium, it is advantageous to restrain the transistor from entering the
saturation region.
918
MAXIMUM VOLTAGE RATING 10
Even if the rated dissipation of a transistor is not exceeded, there is an upper
limit to the maximum allowable collectorjunction 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. 612,
and reachthrough, discussed below.
Avalanche Multiplication The maximum reversebiasing voltage which
may be applied before breakdown between the collector and base terminals
of the transistor, under the condition that the emitter lead be opencircuited, 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 * (96*)
1  {Vcb/BVcboY
Equation (964) is employed because it is a simple expression which g lVC ^
a good empirical fit to the breakdown characteristics of many transistor typ 68 *
TRANSISTOR CHARACTERISTICS / 261
Xbe parameter n is found to be in the range of about 2 to 10, and controls
the sharpness of the onset of breakdown.
If a current Is is caused to flow across the emitter junction, then, neglect
ing the avalanche effect, a fraction otls, where a is the commonbase 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 commonbase
current gain were Ma.
An analysis 10 of avalanche breakdown for the CE configuration indicates
that the coHectortoemitter breakdown voltage with opencircuited base, desig
nated BVcso, is
(965)
BVcBO — BVcBO ■yjr —
For an npn 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 commonemitter characteristics extended into the breakdown region are
shown in Fig. 925. If the base is not opencircuited, these breakdown char
acteristics are modified, the shapes of the curves being determined by the
basecircuit connections. In other words, the maximum allowable collector
toemitter voltage depends not only upon the transistor, but also upon the
circuit in which it is used.
Reachthrough The second mechanism by which a transistor's usefulness
may be terminated as the collector voltage is increased is called punchthrough,
or reachthrough, and results from the increased width of the collectorjunction
transition region with increased collectorjunction 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
'9 925 Idealized common
•""Her characteristics
tended into the breakdown
r *9ion.
CBO V CM
262 / RfCTRONlC DEVICES AND C/RCU/TS
Sec. 9 ? 8
equal. Since the doping of the base is ordinarily substantially smaller than
that of the collector, the penetration of the transition region into the base is
larger than into the collector (Fig. 92c). 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.
Punchthrough 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
punchthrough or breakdown, whichever occurs at the lower voltage.
REFERENCES
1. Shockley, W.: The Theory of pn Junctions in Semiconductors and pn Junction
Transistors, Bell System Tech. J., vol. 28, pp. 435489, 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. 747760, McGraw
Hill Book Company, New York, 1955.
Moll, J. L.: "Junction Transistor Electronics," Proc. IRE, vol. 43, pp. 18071819,
December, 1955.
2. Phillips, A. B.: "Transistor Engineering," pp. 157159, McGrawHill Book Com
pany, New York, 1962.
3. Ref. 2, chap. 1.
4. Texas Instruments, Inc.: J. Miller (ed.), "Transistor Circuit Design," chap. 1,
McGrawHill Book Company, New York, 1963.
5. Ebers, J. J., and J. L. Moll: Largesignal Behavior of Junction Transistors, Proc,
IRE, vol. 42, pp. 17611772, December, 1954.
6. Sah, C. T., R. N. Noyce, and W. Shockley : Carriergeneration and Recombination
in pn Junctions and pn Junction Characteristics, Proc. IRE, vol. 45, pp. 1228—
1243, September, 1957.
Pritchard, R. L. : Advances in the Understanding of the PN Junction Triode, Proc.
IRE, vol. 46, pp. 11301141, June, 1958.
7. Ref. 2, pp. 236237.
8. Early, J. M. : Effects of Spacecharge Layer Widening in Junction Transistors, Proc
IRE, vol. 40, pp. 14011406, November, 1952.
9. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," p. l 9 ^'
McGrawHill Book Company, New York, 1965.
10. Ref. 9, chap. 6.
11. Ref. 9, chap. 20.
"Transistor Manual," 7th ed., pp. 149169, General Electric Co., Syracuse, N.**»
1964.
TRANSISTOR BIASING AND
THERMAL STABILIZATION
This chapter presents methods for establishing the quiescent oper
ating point of a transistor amplifier in the active region of the charac
teristics. The operating point shifts with changes in temperature T
because the transistor parameters (jS, Ico, etc.) are functions of T.
A criterion is established for comparing the stability of different
biasing circuits. Compensation techniques are also presented for
quiescentpoint stabilization.
101
THE OPERATING POINT
From our discussion of transistor characteristics in Sees. 98 to 910,
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. 101 we show a commonemitter circuit (the capacitors
have negligible reactance at the lowest frequency of operation of this
circuit). Figure 102 gives the output characteristics of the transistor
used in Fig. 101. Note that even if we are free to choose R c , Rl, Rb,
and Vcc, we may not operate the transistor everywhere in the active
region because the various transistor ratings limit the range of useful
263
2M / aecreoNJC devices and circuits
S»e. 70.J
Fig. 101 The fixedbias circuit.
Signal
input, v,
operation. These ratings (listed in the manufacturer's specification sheets)
are maximum collector dissipation P c (max), maximum collector voltage
Fc(max), maximum collector current J c (max), and maximum emittertobase
voltage V« B (max). Figure 102 shows three of these bounds on typical col
lector characteristics.
V c V cc Vc(niax) V c e^
Fig. 102 Commonemitter collector characteristics; ac and dc load lines.
TRANSISTOR BIASING AND THERMAL STABILIZATION / 265
The DC and AC Load Lines Let us suppose that we can select R e so that
the dc load line is as drawn in Fig. 102. 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. 712) 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. 102, 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. 102 the choice of Qj allows an input peak current swing of about
6*»A.
The Fixed bias Circuit The point Q% can be established by noting the
required current /as in Fig. 102 and choosing the resistance fit in Fig. 101
so that the base current is equal to I si. Therefore
T Vcc — V B B t
Ib = — m — = /bs
(101)
The voltage V# E across the forwardbiased emitter junction is (Table 02,
page 257) approximately 0.2 V for a germanium transistor and 0.6 V for a
silicon transistor in the active region. Since Vcc is usually much larger than
Vbb, we have
Ib «
Vcc
R>
(102)
The current Ib is constant, and the network of Fig. 101 is called the
fixedbias circuit. In summary, we see that the selection of an operating point
v depends upon a number of factors. Among these factors are the ac and
dc loads on the stage, the available power supply, the maximum transistor
stings, the peak signal excursions to be handled by the stage, and the toler
able distortion.
l0 "2 BIAS STABILITY
111 the preceding section we examined the problem of selecting an operating
P°mt Q on the load line of the transistor. We now consider some of the
Problems of maintaining the operating point stable.
Let us refer to the biasing circuit of Fig. 101. In this circuit the base
266 / ElECTRON/C DEVICES AND CJRCUJTS
la
Sec. JO2
fig, 103 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. 101 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. 912 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. 98 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. 103 we have assumed that is greater for the replacement transistor of
Fig. 101, 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 operatingpoint 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. 99 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 collectorjunction temperature to rise, which in turn
increases I C q. As a result of this growth of I C o, Ic will increase [Eq. (943)1,
which may further increase the junction temperature, and consequently Ico
f Throughout this chapter Icbo is abbreviated Ico (Sec. 99).
s«.
JO2
TRANSISTOR BIASING AND THERMAl STABIUZATJON / 267
It i s possible for this succession of events to become cumulative, so that the
atings of the transistor are exceeded and the device burns out.
Even if the drastic state of affairs described above does not take place, it is
sible f or a transistor which was biased in the active region to find itself in
the saturation region as a result of this operatingpoint instability (Sec. 1010).
To see how this may happen, we note that if I B = 0, then, from Eq. (938),
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. 104 we show the
output characteristics of the 2N708 transistor at temperatures of +25 and
fl00°C. This transistor, used in the circuit of Fig. 101 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 l100 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 temperaturesensitive devices such as diodes, transistors, thermistors, etc.,
e 30
To
T = 25°C
OS
1
0.6
/
£4
/
Q
0.2
\
N
h
= 0mA
k
< 40
1
4*
30
20
0.6
T= 100° C
OS
«
CL4_
oJ
0.2
OJ_
/, = 0mA
'N
2 4 6 8 10 02468 10
Collector voltage V CK , V Collector voltage V C£ , V
(a) (*>)
fig. 104 Diffused silicon planar 2N708 npn transistor output CE characteristics
for (a) 25°C and (b) 100°C (Courtesy of Falrchild Semiconductor.)
268 / ELECTRONIC DEVICES AND CIRCUITS
$*c. 10 j
S#
J 03
TRANSISTOR BIASING AND THERMAL STABILIZATION / 269
which provide compensating voltages and currents to maintain the operating
point constant. A number of stabilization and compensation circuits are pre
sented in the sections that follow. In order to compare these biasing circuits
we define a stability factor S as the rate of change of collector current with
respect to the reverse saturation current, keeping and V BB constant, or
S 
dlco
Alt
Ale
(103)
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. 105, 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. (943), repeated here for
convenience:
Ic = (1 + 0)lco + &B
(104)
If we differentiate Eq. (104) with respect to I c and consider constant with
Ic, we obtain
1  1 +PL.R dI *
or
S =
1+0
1  mis/dlc)
(105)
(106)
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. (106). For the
fixedbias circuit of Fig. 101, I B is independent of Ic [Eq. (102)]. Hence the
stability factor S of the fixedbias circuit is
<8~ _+ I (107)
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 biasstabilization techniques are
presented which reduce the value of S, and hence make Ic more independent
of Ico
103
COLLECTORTOBASE BIAS
An improvement in stability is obtained if the resistor R b in Fig. 101 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. 105o. The physical reason that this circuit
is an improvement over that in Fig. 101 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. 105a,
 Vcc + (Ib + Ic)Rc + I B R» + V BS =
or
In =
Vcc  IcRc  Vi
Re + Rb
(108)
(109)
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. (109) we obtain
■:Ub_
di c
R c
Re + Rb
Substituting Eq. (1010) in Eq. (106), we obtain
0+ 1
S =
1 + 0Rc/{Rc + Rb)
(1010)
(1011)
This value is smaller than 0+1, which is obtained for the fixedbias circuit,
and hence an improvement in stability is obtained.
Stabilization with Changes in It is important to determine how well
the circuit of Fig. 105 will stabilize the operating point against variations in 0.
r — * — i
o VW
B
<«)
9VC
(&)
Fig. 105 (a) A coll ectorto base bias circuit, (b) A method of
avoiding ac degeneration.
270 / ELECTRONIC DEVICES AND CIRCUITS
Sec. I03
From Eqs. (101) and (108) we obtain, after some manipulation, and with
» 1, a
PiVcc  V BB + (R e + R b )Ico\
(1012)
Ic «
ffi* + Rh
To make Ic insensitive to we must have
0R c »R b (1013)
The inequality of Eq. (1013) 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. 105 ia a silicontype 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. 104.
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. 104,
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. 105 we have
Rb =
Vcs  Vbe __ 4.6  0.6
Ib 0.4
= 10 K
The stability factor S can now be calculated using Eq, (1011), or
51
S =
1 + 50 X 0.25/10.25
= 23
which is about half the value found for the circuit of Fig. 101. We should note
here that the numerical values of R t and R b of this example do not satisfy Eq.
(1013) 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 commonemitter 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
(104), I c is found directly from Eq. (1012).
A Method for Decreasing Signalgain Feedback The increased sta
bility of the circuit in Fig. 105a over that in Fig. 101 is due to the feedback fro" 1
S<*
04
TRANSISTOR BIASING AND THERMAL STABILIZATION / 271
the output (collector) terminal to the input (base) terminal via R b . Feedback
amplifiers are studied in detail in Chap. 17. The ac voltage gain of such an
mplifier is less than it would be if there were no feedback. Thus, if the signal
voltage causes an increase in the base current, i c tends to increase, Vcb decreases,
and the component of base current coming from R b decreases. Hence the
ne t change in base current is less than it would have been if Rb were connected
to a fixed potential rather than to the collector terminal. This signalgain
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. 1056. 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.
104
SELFBIAS, OR EMITTER BIAS
If the load resistance R c is very small, as, for example, in a transformer
coupled circuit, then from Eq. (1011) we see that there is no improvement
in stabilization in the collectortobase bias circuit over the fixedbias circuit.
A circuit which can be used even if there is zero dc resistance in series with
the collector terminal is the selfbiasing configuration of Fig. 106a. The
current in the resistance R e in the emitter lead causes a voltage drop which
is in the direction to reversebias the emitter junction. Since this junction
must be forwardbiased, the base voltage is obtained from the supply through
the RiR 2 network. Note that if R b = J2iiK*— • 0, then the basetoground
voltage V BN is independent of Ico Under these circumstances we may verify
Fig. 106 (a) A selfbiasing circuit, (b) Simplification of the base
circuit in (a) by the use of Tbevenin's theorem.
272 / ELECTRONIC DEVICES AND CIRCUITS
[Eq. (1017)] 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 selfbiasing 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. 106o contains three independent loops.
If the circuit to the left between the base B and ground N terminals in Fig,
106a is replaced by its Thevenin equivalent, the twomesh circuit of Fig.
1066 is obtained, where
V =
Rz \ R\
Rb =
RiRi
Ri + Ri
(1014)
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.
(1015)
If we consider V B s to be independent of I c , we can differentiate Eq. (1015)
to obtain
dl*
die
R.
Re + Rt>
Substituting Eq. (1016) in Eq. (106) results in
S =
1 +
1 + 0RJ(R t + BO
 (1 + fi)
1 + R+/R.
I + + Rt/R.
(1016)
(1017)
Note that S varies between 1 for small R b /R, and 1 + for R b /R t ^> « . Equa
tion (1017) is plotted in Fig. 107 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. 127), this resistance is often bypassed by a large capacitan^
104
100
10
TRANSISTOR BIASING AND THERMAL STABIUZATION / 273
r B = 100
80
60
40
30
20
10
5
1.0
10
1,000
Rt
Fig. 107 Stability factor S {Eq. (1017}] versus R b /R* for the selfbias
circuit of Fig. 1066, with as a parameter. (Courtesy of L. P.
Hunter, "Handbook of Semiconductor Electronics," McGrawHill
Book Company, New York, 1962.)
(> 10 mF), so that its reactance at the frequencies under consideration is very
small.
EXAMPLE Assume that a silicon transistor with jS = 50, Vbb = 0.6 V, Vcc =
22.5 V, and R t = 5,6 K is used in Fig. IO60. 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. 1066, we have
Vcc  Vcs 22.5  12
R t + R t = rcc rc * = _ if = 7.0 K
Ic 1.5
or
R t = 7.05.6 = 1.4 K
From Eq. (1017) we can solve for R b /R t :
3= 51 1 + R " /R *
51 + Rt/R,
We find R b /R t = 2.12 and R b = 2.12 X 1.4 = 2.96 K. If R b < 2.96 K, then
5<3.
The base current I B is given by
p 50
274 / ELECTRONIC DEVICES AND CIRCUITS
We can solve for Ri and R 2 from Eqs. (1014). We find
Ri — Rb
Rt —
RtV
Sec. IO.4
a 01 s)
V ~" Vce  V
From Eqs. (1015) and (1018) we obtain
V = (0.030) (2.96) + 0.6 + (0.030 + 1.5) (1.4) = 2.83 V
2.83
_ 23.6 X 2.83 _
22.5  2.83
Analysis of the Selfbias Circuit If the circuit component values in
Fig. 106a 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 =
(1019)
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. (1019) is substituted
in Eq. (1015), 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 commonemitter output characteristics
are shown in Jig. 1086 is used in the circuit of Fig. 106a, 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. (1014) we have
V  "» X 22 ■»  2.25 V
100
"i,
10 X 90
100
 9.0 K
The equivalent circuit is shown in Fig. 108a. 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. 1086. Kirchhoff'* voltage law applied to the collector and base
circuits, respectively, yields (with V be = 0.6)
(1020)
(1021)
22.5 + 6.6/c + I a + Vcs =
0.6  2.25 + Ic + 10.07* =
Eliminating 7c from these two equations, we find
Vcb = 65.07* +11.6
jl
See
104
TRANSISTOR BIASING AND THERMAL STABILIZATION / 275
9.0K
V CF . ± 22.5 V
i*** v i,+icjsi
B 3  4
■3 2
I»*
160
ji n
— in?
L22—
Load line I
r~ "f 40
'
Bias
curve — *"
3
^—
r
—
l
(a)
4 8 12 16 20 I 24
Collectortoemitter voltage V C s . V 22  5
(6)
Fig. 108 (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. 1086. 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. (104) that
7c  01 B (1022)
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.
(1021) for the base circuit yields
1.65 + 7c + H7c =
7c = 1.40 mA
and
. 7 C 1.40 . rtc _ .
Ib = 77 = r mA   25.5 pA
55 55
These values are very close to those found from the characteristics.
The collectortoemitter voltage can be found from Eq. (1020) and the known
values of 7 a and 7 C :
22.5 + 6.6 X 1.40 + 0.026 + Vcs =
or
Vce  13.2 V
6. From Eq. (1017),
S =
_ 56 (i±^ =
276 / ELECTRONIC DEVICES AND CIRCUITS
Sec. TO.j
This value is about onesixth of the stabilization factor for the fixedbias circuit,
which indicates that a great improvement in stability can result if selfbias is
used.
In the colleetortobase 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 selfbias arrangement, this circuit is more popular than that
of Fig. 105a.
For the sake of simplicity the resistor R 2 is sometimes omitted from Fig.
106a. 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. (1017), with J2 t
replaced by Ri.
105 STABILIZATION AGAINST VARIATIONS IN Vbe AND
FOR THE SELFBIAS CIRCUIT
In the preceding sections we examine in detail a number of bias circuits which
provide stabilization of I c against variations in Ico There remain to be con
sidered two other sources of instability in Ic, those due to the variation of
Vbe and jS with temperature and with manufacturing tolerances in the pro
duction of transistors. We shall neglect the effect of the change of Vce with
temperature, because this variation is very small (Sec. 910) 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. 910).
The Transfer Characteristic The output current Ic is plotted in Fig
109 as a function of input voltage for the germanium transistor, type 2N1631.
This transfer characteristic for a silicon transistor is given in Fig. 921. 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
(1015), obtained by applying KVL around the base circuit of the selfbi» s
circuit of Fig. 106b, is combined with Eq. (104), 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)
(1023)
Sk.
105
TRANSISTOR BIASING AND THERMAl STABILIZATION / 277
6
Fig 109 Transfer characteristic
for the 2N1631 germanium pnp
alloytype transistor at Vce =
9 V and T A = 25°C. {Courtesy
of Radio Corp. of America.)
■
n
100 150 200
Vbe, mV
Equation (1023) represents a load line in the IcV a e plane, and is indi
cated in Fig. 1010. The intercept on the V bb axis is V + V, where
r = (R b + r<) £±i ico
(Rb + R t )Ico
(1024)
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. 1010 Illustrating that the col
lector
current varies with tempera
te T because V BK , Ico, and
chonge with T,
278 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 1 05
Sec 105
TRANSISTOR BIASING AND THERMAL STABILIZATION / 279
T — Ti because Vbe (at constant Ic) varies with T as indicated above. The
intersection of the load line with the transfer characteristic gives the collector
current Ic. We see that J C i > la because Ico, 0, and Vbe all vary with
temperature.
The Stability Factor 8' The variation of Ic with Vbs is given by the
stability factor S', defined by
8' =
dV BS
where both I C o and are considered constant. From Eq. (1023) we find
0 S
(1025)
S' =
R b + R.(l + 0)
R b + R e + 1
(1026)
where we made use of Eq. (1017). We now see from Eq. (1026) 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. (1023),
0(V + V  Vbe)
Ic 
R b + R t (l + 0)
(1027)
(1028)
where, from Eq. (1024), V may be taken to be independent of 0. We obtain,
after differentiation and some algebraic manipulation,
on _ Mc
IcS
d0 0(1 + 0)
(1029)
A difficulty arises in the use of S" which is not present with S and S\ The
change in collector current due to a change in is
Ale = 8" A0 
IcS
0(1 + 0)
A0
(1030)
where A0 = 2 — 0i may represent a large change in #. Hence it is not cle* r
whether to use is 2 , or perhaps some average value of /S in the expressions fo r
S" and S. This difficulty is avoided if S" is obtained by taking finite differ
ences rather than by evaluating a derivative. Thus
8" m
Ig — la
02 — 01
Ale
A0
(1031)
L
From Eq. (1028), we have
let = 0* Rb + g.(l + gl)
Id 01 Rb + fl.(l + 02)
Subtracting unity from both sides of Eq. (1032) yields
Rb + Re
hi _ 1  (§1 _ \\
lev \0i )
R b + Re(l + 02)
or
IC1S2
" A0 0,(1 + 2 )
(1032)
(1033)
(1034)
where Si is the value of the stabilizing factor S when = 0s as given by Eq.
(1017). Note that this equation reduces to Eq. (1029) as A0  2 — 0i — ► 0.
It is clear from Eq. (1029) that minimizing S also minimizes S". This means
that the ratio Rh/R* must be small. From Eq. (1026) 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 (1034) 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. l06a, 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. (1034) we can solve for S 2 . Hence, with Ale = 0.5 mA, la =
1.75 mA, 0, * 36, 2 = 90, and A0 = 54, we obtain
0.5 1 .75 S s
54 36 1 + 90
Si = 17.3
280 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 10.6
Substituting S 2 = 17.3, R e = 1 K, and jS* = 90 in Eq. (1017) yields
(17.3)(91 + Ri) = 91(1 + R b )
or
R b = 20.1 K
From Eq. (1023), 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. (1018),
Vcc 20
R l = R b £Z = 20.1 X — = 119 K
F 3.38
iij —
J?iF
119 X 3.38
Vcc  V 20  3.38
= 24.2 K
106 GENERAL REMARKS ON COLLECTORCURRENT STABILITY 1
Stability factors were defined in the preceding sections, which considered the
change in collector current with respect to Ico, Vbb, and 0. These stability
factors are repeated here for convenience :
S =
Ale,
S f =
AV B
o„ _ Ale
(1035)
Each differential quotient (partial derivative) is calculated with all other
parameters maintained constant.
If we desire to obtain the total ehange in collector current over a specified
temperature range, we can do so by expressing this change as the sum of the
individual changes due to the three stability factors. Specifically, by taking
the total differential of I c — f(Ico, V B e, 0), we obtain
. t die A T , die > it i die . „
OiCO OV BE Op
 S Alco + S' AV BB + S" A0
(1036)
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
(1036) 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 101 and 102 show typical parameters of silicon and germanlu^ 1
S«. T0<5
TRANSISTOR BIASING AND THERMAL STABILIZATION / 281
TABIC 7 0 T Typical silicon transistor parameters
TABLE 102 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 selfbias circuit of Fig. 106a, 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 101 is used.
h. Repeat (a) for the range —65 to +75°C when the germanium transistor
of Table 102 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, (1017) :
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. (1026) :
S'(25°C) 
S
R b + R t l+0
~(S)©— " '*"
The values of S and 5' are valid for either a silicon or a germanium transistor
operating in the circuit of Fig. 106a. Since the stability factor S" contains both
Pi and jSj, it must be determined individually for each transistor at each new
temperature, using Eq. (1034). Hence, for the silicon transistor at 175°C,
we have, using Eq. (1017),
Si( + 175°C) = (1 + 02
1 + R b /R t = (101) (2.65)
I +0z + Rh/R, 101 + 1.65
 2.61
282 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 106
Then
5"( + 175°C) 
IciSj _ (1.5) (2.62)
0,(1 + j8 2 ) (55)(101)
= 0.71 X 10» mA
Similarly,
Sl( _65C)  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. (1036) and
Table 101.
A/ C ( + 175°C) = 5 Mco + S' AV BB + 5" AjS
= (2.57)(33 X 10" 3 ) + (0.203) (0.375) + (0.71 X 10»)(45)
 0.085 + 0.077 + 0.032 = 0.194 mA
and at 65°C,
A/c(65°C) = (2.57) (10*)  (0.203) (0.18) + (2.61 X Hr 3 )(30)
=  0.036  0.078  0.114 mA
Therefore, for the silicon transistor, the collector current will be approximately
1.69 mA at +175°C and 1.39 mA at 65°C.
6. To calculate the change in collector current using the germanium tran
sistor, we must compute S" at +75 and — 65°C.
S!(+75 .Q  yWy*  2.60
S"(+75°C) =
91 + 1.65
= (1.5) (2.60)
/S»(l + &) (55)(91)
= 0.78 X 10~ a mA
Similarly,
S , ( _ 65 . C ) = ^^ = 2.45
21 + 1.65
S"(65°C) = (L5)(2 ' 45)  3.18 X 10"» mA
(55)(21)
Hence the change in collector current is
A/ C (+75°C) « (2.57)(31 X 1G" 3 ) + (0.203) (0.10) + (0.78 X 10*)(3S)
= 0.080 + 0.020 + 0.027 = 0.127 mA
and at 65°C,
A/ C (65°C) = (2.57)(10" J ) + ( + 0.203) (0.1 8) + (3.18 X 10~ s )(35)
= 0.002  0.036  0.111 = 0.149 mA
Therefore, for the germanium transistor, the collector current will be approxi
mately 1.63 mA at +75°C and 1.35 mA at 65°C.
Sec. 107
TRANSISTOR BIASING AND THERMAL STABIIIZATION / 283
practical Considerations The foregoing example illustrates the supe
riority of silicon over germanium transistors because, approximately, the same
change in collector current is obtained for a much higher temperature change
in the silicon transistor. In the above example, with & = 2.57, the current
change at the extremes of temperature is only about 10 percent. Hence this
circuit could be used at temperatures in excess of 75°C for germanium and
175°C for silicon. If S is larger, the current instability is greater. For
example, in Prob. 1019, 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. 128).
If we have a directcoupled emitter follower driven from an ideal voltage source,
then #6 = and S is at its lowest possible value, namely, 5=1. It is clear
that a circuit with R b = can be used to a higher temperature than a similar
circuit with Rb ^ 0.
In the above example the increase in collector current from 25 to 75°C
for a germanium transistor is 0.08 mA due to Ico and 0.02 mA due to Vbb
Hence, for Ge, the effect of ho has the dominant influence on the collector
current. On the other hand, the increase in h for a silicon transistor over
the range from 25 to 175°C due to ho is approximately the same as that due
to V BE . However, if the temperature range is restricted somewhat, say, from
25 to 145°C, then M c  0.01 mA due to ho and A/ c = 0.06 mA due to
Vbb These numbers are computed as follows: If T mai is reduced from
175 to 145°C, or by 30°, then ho is divided by 2^ 10 = 2* = 8. Hence
S AI C0 = 0.085/8 = 0.01 mA. Also, AV B s is increased by (30) (2.5) = 75 mV,
°r AVbb goes from 0.375 to 0.30 V and S' AV BE = <0.2)(0.30) = 0.06
"lA. Hence, for Si, the effect of Vbb has the dominant influence on the col
lector current.
1 07 BIAS COMPENSATION'
T he collectortobase bias circuit of Fig. 10oa and the selfbias circuit of Fig.
!06a are used to limit the variation in the operating collector current h
caused by variations in ho, Vami and /3. These circuits are examples of
feedback amplifiers, which are studied in Chap. 17, where it is found that a
284 / ELECTRONIC DEVICES AND CIRCUITS
Sot. 10.7
Fig. 1011 Stabilization by means of self
bias and diodecompensation 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 selfbias stabiliza
tion technique and diode compensation is shown in Fig. 1011. 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 basetoemitter voltage Vbb If we write KVL around the base circuit
of Fig. 1011, then Eq. (1028) becomes
Ic =
&[V ~ (Vbb  V.)] + (Rt + R t )(fl + Pico
R b + fl.(l + J8)
(1037)
Since V B b tracks V a with respect to temperature, it is clear from Eq. (1037)
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. 106 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. 1012 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*
108
TRANSISTOR BIASING AND THERMAL STABILIZATION / 285
fm, 1012 Diode compensation for a germanium
transistor.
'!>*.
.P7
m „
the same rate as the transistor collector saturation current Ico From Fig.
1012 we have
/ 
Vcc ~ Vi
Ri
= const
Since the diode is reversebiased 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
(1038)
We see from Eq. (1038) 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.
108
BIASING CIRCUITS FOR LINEAR INTEGRATED CIRCUITS 2
In Chap. 15 we study the fabrication techniques employed to construct
integrated circuits. These circuits consist of transistors, diodes, resistors, and
capacitors, all made with silicon and silicon oxides in one piece of crystal or
chip. One of the most basic problems encountered in linear integrated circuits
is bias stabilization of a commonemitter amplifier. The selfbias 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 presentday
integratedcircuit technology. This technology offers specific advantages,
w hich are exploited in the biasing circuits of Fig. 1013a 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. 1013a uses transistor Ql connected
88 a diode across the basetoemitter junction of transistor Q% whose collector
286 / ELECTRONIC DEVICES AND CIRCUITS
(a) (o)
Fig. 1013 Biasing techniques for linear integrated circuits.
current is to be temperaturestabilized. The collector current of Ql is given by
VcC — V BE
Ici =
Ri
— I B\ ~ 1 1
For Vbb « V C c and (I B i + /«) « hi, Eq. (1039) becomes
r V CC .
Ici « 5— = const
Hi
(1039)
(1040)
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
collectorcurrent 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. 101 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. 1013&, the collector current of Ql is given by
If cc — Vbb
Ici =
Ri
 ( 2 + si)
Under the assumptions that V B b « Vcc,
(1041) becomes
Vcc
Ri
(1041)
and (2 + Ri/Ri)I B « Vcc/Ru E*
Ici = Ici =
Sec. 109
TRANSISTOR BIASING AND THERMAL STABILIZATION / 287
If R e = ^Ri, then V C e = V C c ~ hiR* m V C c/2, which means that the
amplifier will be biased at onehalf the supply voltage Vcc, independent of
the supply voltage as well as temperature, and dependent only on the matching
f components within the integrated circuit. An evaluation of the effects
of mismatch in this circuit on bias stability is given in Ref. 2.
THERMISTOR AND SENSISTOR COMPENSATION 1
There is a method of transistor compensation which involves the use of tem
peraturesensitive resistive elements rather than diodes or transistors. The
thermistor (Sec. 52) has a negative temperature coefficient, its resistance
decreasing exponentially with increasing T. The circuit of Fig. 1014 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
reversebias 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. 1014 and place it across R%. As T increases,
the drop across R T decreases, and hence the forwardbiasing 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 temperaturesensitive
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. 1014 (with R r
V cc
ng. 1014 Thermistor compensation of
*he increase in l e with T.
<■', o
288 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 10 JO
removed), temperature compensation may be obtained by placing a sensistor
either in parallel with Ri or in parallel with (or in place of) R f . Why?
In practice it is often necessary to use silicon resistors and carbon resistors
in series or parallel combinations to form the proper shaping network. 3 The
characteristics required to eliminate the temperature effects can be determined
experimentally as follows: A variable resistance is substituted for the shaping
network and is adjusted to maintain constant collector current as the operating
temperature changes. The resistance vs. temperature can then be plotted
to indicate the required characteristics of the shaping network. The problem
now is reduced to that of synthesizing a network with this measured tem
perature characteristic by using thermistors or sensistors padded with tem
peratureinsensitive resistors.
1010
THERMAL RUNAWAY
The maximum average power Fornax) which a transistor can dissipate
depends upon the transistor construction and may lie in the range from a
few milliwatts to 200 W. This maximum power is limited by the tempera
ture that the collectortobase 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 selfheating. The maximum
power dissipation is usually specified for the transistor enclosure (case) or
ambient temperature of 25°C. The problem of selfheating, which is men
tioned in Sec, 102, 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 steadystate
temperature rise at the collector junction is proportional to the power dissi
pated at the junction, or
AT = Tj  T A = OP D
(1042)
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 forcedair 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 highpower transistor with an
efficient heat sink to 1000°C/W for a lowpower transistor in free air.
The maximum collector power P € allowed for safe operation is specified
Sec
IOT0
TRANSISTOR BIASING AND THERMAL STABILIZATION / 289
Pc.W
pig. 1015 Powertemperature
derating curve for a germanium
power transistor.
150
120
1 \.
i X
i X^
90
60
30
(
) 20 40 60
80
100
Case temperature,
•c
at 25°C. For ambient temperatures above this value, Pc must be decreased,
and at the extreme temperature at which the transistor may operate, Pc is
reduced to zero. A typical powertemperature derating curve, supplied in a
manufacturer's specification sheet, is indicated in Fig. 1015.
Operatingpoint Considerations The effects of selfheating may be
appreciated by referring to Fig. 1016, which shows three constantpower
hyperbolas and a dc load line tangent to one of them. It can be shown (Prob.
1026) 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 100W parabola. The opposite is true if the quiescent point is below the
point of tangency, such as at Q 2 . We can conclude that if Vce is less than
Fcc/2, the quiescent point lies in a safe region, where an increase in collector
current, results in a decreased dissipation. If, on the other hand, the operating
,500W
'9. 1016 Concerning transis
tor selfheating. The dashed
lo °d tine corresponds to a
er y small dc resistance.
290 / ElECTRONJC DEVICES AND CIRCUITS
Sec, 7 0 
Sec. 10W
TRANSISTOR BIASING ANO THERMAL STABILIZATION / 291
point is located so that Vce > V C cl% the selfheating 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 transformercoupled
to the collector, as in Fig. 1017, 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. 1016. Clearly Vce can be less than Wee only for exces
sively large collector currents. Hence thermal runaway can easily occur with
a transformercoupled 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,
(1043)
dPc . BPd
df~ Jf~
If we differentiate Eq. (1042) with respect to Tj and substitute in Eq. (1043),
we obtain
dP c
dTj
(10^4)
This condition must be satisfied to prevent thermal runaway. By suitable circuit
design it is possible to ensure that the transistor cannot run away below a
specified ambient temperature or even under any conditions. Such an analysis
is made in the next section.
1011 THERMAL STABILITY
Let us refer to Fig. 106a 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. (1045) becomes
Pc = IcVcc  /c«(fl. + 4) ( 1(M6)
Equation (1044), the condition to avoid thermal runaway, can be rewritten
as follows:
dPcdlc ^ I ( l0 47)
dl c dTj
The first partial derivative of Eq. (1047) can be obtained from Eq. (1046) :
dP c
dh
= Vcc — 2Ic(Re + Re)
(1048)
The second partial derivative in Eq. (1047) 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. (1036),
dl c
dTj
= S
dlco
~df~
+ s
,dV
dTj
(1049)
Since for any given transistor the derivatives in Eq. (1049) are known, the
designer is required to satisfy Eq. (1047) 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.
(1047) and (1049),
dP C l
die
(s dIco \ < I
(1050)
In Sec. 67 it is noted that the reverse saturation current for either silicon or
germanium increases about 7 percent/°C, or
dlco
ar7
= 0.07/e O
Substituting Eqs. (1048) and (1051) in Eq. (1050) results in
[V cc ~ 2I c (R e + i2 e )](S)(0.07/ CO ) < g
(1051)
(1052)
Equation (1052) remains valid for a pnp 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 (1052) is always
satisfied provided that the quantity in the brackets is negative, or provided
that
Vcc
Ic>
(1053)
2(fl. + R B )
Since V CB  V C c  Ic(R, + Re), thenEq. (1053) implies that Vce < V C cl%
and this checks with our previous conclusion from Fig. 1016. If the inequality
of Eq. (1053) is not satisfied and V C e > V C c/2, then from Eq. (1048) we see
that dP c /dI c i s positive, and the designer must ensure that Eq. (1050) will
De satisfied, or else thermal runaway will occur.
EXAMPLE Find the value of required for the transistor of the example on
page 274 in order for the circuit to be thermally stable. Assume that / c « = 1 nA
at 25°C.
292 / ELECTRONIC DEVICES AND CIRCUITS
See. JO J j
Solution Since V C c/2 = 11.25 V and V C e = 133 V, the circuit is not inherently
stable, because V CB > kVcc Substituting in Eq. (1052), we obtain
{22.5  2 X 1.4 X (5.6 + 1.0)1(8.61) (0.07 X 10") < i
4.0 X 8.61 X 0.07 X 10» < ^
or
< 4.1 X 10 s °C/W
The upper bound on the value of is so high that no transistor can violate it,
and therefore this circuit will always be safe from thermal runaway.
This example illustrates that amplifier circuits operated at low current
and designed with low values of stability factor (S < 10) are very rarely
susceptible to thermal runaway. In contrast, power amplifiers operate at
high power levels. In addition, in such circuits R e is a small resistance for
power efficiency, and this results in a high stability factor S. As a result,
thermal runaway in power stages is a major consideration, and the designer
must guard against it.
EXAMPLE Figure 1017 shows a power amplifier using a jhnp 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. (104), or
Ic = Wb + (1 + Wco *» 0(1 a + Ico)
and
1  5 X 10' X 100 . : .
[ B = A = — 5 mA
100
v„=40V
Fig. 1017 Power amplifier with a trans
formercoupled load.
u, o— *
&■
7011
TRANSISTOR BIASING AND THERMAL STABILIZATION / 293
If we neglect V B e, we have
5 X 10»A, = 405 or R b = 7,000 Q
b. Since \V CS \ = 40  15 = 25 > iV«? = 20 V, the circuit of Fig. 1017
is not inherently stable. The stability factor S is obtained from Eq. (1017).
1 + 7,000/5
S = 101
 94.3
101 + 7,000/5
Substituting in Eq. (1052), we obtain
(40  2 X 1 X 15) (94.3) (0.07 X 5 X 10~ a ) < 
vt
or
< 3.03°C/W
REFERENCES
1. Hunter, L. P.: "Handbook of Semiconductor Electronics," McGrawHill 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. CT12, no. 4, pp. 586590, December, 1965.
3. Konjian, E., and J. S. Schaffner: Shaping of the Characteristics of Temperature
sensitive Elements, Commun. and Electron., vol. 14, pp. 396400, September, 1954.
SMALLSIGNAL LOWFREQUENCY
TRANSISTOR MODELS
In Chap. 9 we are primarily interested in the static characteristics
of a transistor. In the active region the transistor operates with
reasonable linearity, and we now inquire into smallsignal 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 voltampere 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.
111
TWOPORT DEVICES AND THE HYBRID MODEL
The terminal behavior of a large class of twoport devices is specified
by two voltages and two currents. The box in Fig. 111 represents
such a twoport 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 twoport 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 twoport '
linear, wemay write
v 1 = huii + hiiVi U*" '
The quantities h n , h n , ft 2 i, and A 22 are called the h, or hybrid, parameter
because they are not all alike dimensionally. Let us assume that t»w
294
$•'
UI
SMALLSIGNAL LOWFREQUENCY TRANSISTOR MODELS / 295
Fig
network
H1 A twoport
Input
port
Output
port
ft  e no reactive elements within the twoport network. Then, from Eqs.
(111) and (112), the h parameters are defined as follows:
A,, m — I = input resistance with output shortcircuited (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, reverseopencircuit voltage
amplification (dimensionless).
= negative of current transfer ratio (or current gain)
with output shortcircuited. (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 shortcircuit current gain (dimensionless).
= output conductance with input opencircuited (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 commonbase configuration
h S e  %%u = shortcircuit forward current gain in commonemitter
circuit
Since the device described by Eqs. (111) and (112) 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. 111. The hybrid circuit for any
296 / ELECTRONIC DEVICES AND CIRCUITS
i\ (in ohms)
ft n *i ^ (in
raA/V)
r
Fig. IT 2 The hybrid model for
the twoport network of Fig. 111.
The parameters h n and An are
dimenslonless.
«L
device characterised by Eqs. (111) and (112) is indicated in Fig. 112. We
can verify that the model of Fig. 112 satisfies Eqs. (111) and (112) by
writing Kirchhoffs voltage and current laws for the input and output ports,
respectively.
112
TRANSISTOR HYBRID MODEL
The basic assumption in arriving at a transistor linear model or equivalent
circuit is the same as that used in the case of a vacuum tube: the variations
about the quiescent point are assumed small, so that the transistor parameters
can be considered constant over the signal excursion.
Many transistor models have been proposed, each one having its particular
advantages and disadvantages. The transistor model presented i n this chapter,
and exploited in the next chapter, is given in terms of the h parameters, which
are real numbers at audio frequencies, are easy to measure, can also be obtained
from the transistor static characteristic curves, and are particularly convenient
to use in circuit analysis and design. Furthermore, a set of h parameters i.s
specified for many transistors by the manufacturers.
To see how we can derive a hybrid model for a transistor, let us consider
the commonemitter connection shown in Fig. 113. 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. 113 A simple commonemitter con
nection.
L
Soc. 112
we may write
Vb = fi(iB, Vc)
ic = Mis, vc)
SMALLSIGNAL LOWFREQUENCY TRANSISTOR MODELS / 297
(113)
(114)
Making a Taylor's series expansion of Eqs. (113) and (114) around the quies
cent point I S) V c , similar to that of Eq. (812), and neglecting higherorder
terms, we obtain
AVB = *Zl I At* + &
dis \Yc
dv c V
Av<
(115)
(116)
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 smallsignal (incre
mental) base and collector voltages and currents. According to the notation
in Table 91, we represent them with the symbols v b , v t , i b , and v We may
now write Eqs. (115) and (116) in the following form:
where
and
v b = h it) ib + K»Vc
ic = h/etb + k oe v e
A  a /l = dvB I
di 8 dlB Wc
h = ^ll = —
dii
di B Ikc
h ^ 9/i = Bvb I
" dvc dv c l/»
Bfi die I
dv c dVc u»
(117)
(118)
(119)
(1110)
The partial derivatives of Eqs. (119) and (1110) define the A parameters
for the commonemitter 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. (117)
and (118) are of exactly the same form as Eqs. (111) and (112). Hence the
model of Fig. 112 can be used to represent a transistor.
The Three Transistor Configurations The commonemitter (CE), com
moncollector (CC), and common base (CB) configurations, their hybrid
Models, and the terminel vi equations are summarized in Table 111. 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 
(1111)
298 / ELECTRONIC DEVICES AND CIRCUITS
TABLE 111 Transistor configurations and their hybrid models
s «. Ij3
The circuits and equations in Table 111 are valid for either an «/>"
or pnp transistor and are independent of the type of load or method of
biasing.
113 DETERMINATION OF THE h PARAMETERS
FROM THE CHARACTERISTICS 2
Equations (113) and (114) give the form of the functional relationships ft*
the commonemitter connection of total instantaneous collector current aa d
base voltage in terms of two variables, namely, base current and collector
voltage. Such functional relationships are represented in Chap. 9 by famili eS
S*
IJ3
SMALLSIGNAL LOWFREQUENCY TRANSISTOR MODELS / 299
f characteristic curves. Two families of curves are usually specified for
transistors. The output characteristic cumes depict the relationship between
iiitput current and voltage, with input current as the parameter. Figures
95 and 98 show typical output characteristic curves for the commonbase
and commonemitter 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 commonbase
and commonemitter transistor connections are shown in Figs. 96 and 99.
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 commonemitter connection the characteristics
arc shown in Fig. 114. From the definition of A /e given in Eq. (1110) and
from Fig. 1 l4a, we have
die
Bin
k ft =
Ate 1
Lis We
IC2 — 1C\
iB2 — 1Bl
Ul12)
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. Ll4a).
The parameter A/ e is the most important smallsignal parameter of the
transistor. This commonemitter current transfer ratio, or CE alpha, is also
written a,, or lV, and called the smallsignal beta of the transistor. The rela
tionship between j3' = A/„ and the largesignal beta, « h FE , is given in Eq,
(947).
Wca ■ v c
^•g. H4 Characteristic curves of a commonemitter transistor, (a) CE output
characteristics— determination of h/i ar »d fewl (M CE input characteristics—
determi nation of hit and A„.
300 / ELECTRONIC DEVICES AND CIRCUITS
The Parameter h
h
From Eq, (1110),
die
Ate I
Av c Us
s <* 11. j
(1118)
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, ll4o tangent to the characteristic curve at the point Q,
The Parameter ft™ From Eq. (119),
, _ dv B ^ Av B I
Olf} &l B \Vc
(1114)
Hence the slope of the appropriate input characteristic at the quiescent
point Q gives h*. In Fig. 1146, 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. (119),
v B i  v B \
, _ vug _^ "^g _
dvc At) c u* Va — Vci
(1115)
A vertical line on the input characteristics of Fig. 1146 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 commonemitter
h parameters may also be used to obtain the commonbase and common
collector h parameters from the appropriate input and output characteristic
curves.
Hybridparameter 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. ll4o) or V C b (Fig. 1146), 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 pnp transistor in Fig. ll5a and 6. These curves are plotted with
respect to the values of a specific operating point, say — 5 V collectortoeraitter
voltage and — I mA collector current. The variation in h parameters as shown
Sec
n3
50
I 20
1 ,0
1 5
9
* I
> 1.0
J 05
I
I 0.2
£ o.i
jS 0,06
f 0.02
y o.oi
1
1 V
_l
1 1
5.0V
1
7^25'C
£*
*rr
h^
**
hfr
£
K
SMALLSIGNAL LOWFREQUENCY TRANSISTOR MODELS I 301
H
II
W 2.0
%
IB
a 1.5
"3
1.0
8
2 0.4
g 0.10.20.51.02 51020
I Collector current Ic , mA
r>r.
3 «•?
1
= 1kHz
 1.0mA
ho.
h~
h n *
n t'
*J
ion
Be
N
150 200
(a)
Junction temperature T f , * C
(6)
Fig. 115 Variation of commonemitter 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
diffusedsilicon planar epitaxial transistor; (b) with junction temperature, normal
ized to unity at T, = 25°C. (Courtesy of Fairchild Semiconductor.)
in Fig. ll5o 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 welldefined
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 16 for an npn doublediffused silicon mesa transistor.
Fi 9 116 Variation of h fr
w 'th emitter current for the
type 2N1 573 silicon mesa
* r ansistor. (Courtesy of
T*xas Instruments, Inc.)
180
150
120
k f* Q0
V cr . = 5V
150" C
il25«
C
75"C
T A
= 25°
C
60
30
^ 55 e C
O
5 10 15
Emitter current I t , mA
302 / ELECTRONIC DEVICES AND CIRCUITS
TABLE 112 Typical /(parameter values for a
transistor (at I B = 1.3 raA)
Soc. TJ.
Parameter
ftu = hi
kit = h T
h tl = kj
h%i = k 9
1/A.
CE
1,100 n
2.5 X 10
50
25*A/V
40K
CC
1,100 Q
—1
51
25 M/V
40 K
CB
21.6 R
2.9 X 10*
0.98
0.49 jiA/V
2.04 M
Table 1 12 shows values of ft parameters for the three different transistor
configurations of a typical junction transistor.
11 4
MEASUREMENT OF ft PARAMETERS 3
Based on the definitions given in Sees. 111 and 112, simple experiments
may be carried out for the direct measurement of the hybrid parameters.
Consider the circuit of Fig. 117. 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 signalinput current
to be 7 6 = V,/R\. Since Rl is generally 50 U, we may consider the transistor
output port as shortcircuited to the signal.
JJ,(1M)
1 VV\ — t
"Tank circuit
Fig. 117 Circuit for measuring ft, e and hj c .
Sec M4
The value of ft« is given by Eq. (1114) :
Vt I VtRx
SMALLSIGNAL LOWFREQUENCY TRANSISTOR MODELS / 303
h t =
h lv.=o
V.
(1116)
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. (1112)
fife ■= T
VJli
h\v*o V,R L
(1117)
since h = V g /R L  Thus ft/, is obtained from the two measured voltages V.
and V t .
The circuit of Fig. 118 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 opencircuited as far as the signal is concerned.
We then obtain from Eq. (1115)
h 1*1
The output conductance is defined by Eq. (1113):
h  U
flat = TT
V e lA0 R L V e
(1118)
(1119)
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. H8 Circuit for meos
ur 'n8 kr, and h at .
304 / ELECTRONIC DEVICES AND CIRCUITS
S«. lj4
TABLE 7 J 3 Approximate conversion formulas for transistor parameters
(numerical values are for a typical transistor Q)
Symbol
Common
emitter
Common
collector
Common
base
T equivalent
circuit
h«
1,100 Q
hid
Art
1 +h /b
1—0
h„
2.5 X 10*
1  A re t
AitAoj
1+A„ **
(1  o)r.
A,.
50
a +**)*
A/t
1 +*/»
a
1 o
h„
25 M A/V
A«t
1 + A/t
1
(1  o)r.
ha
A*
A, e
A/e
21.6 a
r. + (1  o)r 4
Art
huh,,
l+h,. K '
*.»£=l
A/ e
2.9 X 10"*
A/*
h f .
1 +A/c
A/.
0.98
1 +A/.
— a
h i
A<n
Am
~A/ e
1
1 + fc/.
0.49 aiA/V
1 r«
h ie
M
1,100 Q
h&
l — a
1 + A/»
K t
1  At,  If
1
1
r.
1 ~~ (1  a)r t
h tc
a + A / .)t
51
1
1
1 +A/»
1  a
A«
A«t
25 ^A/V
A,*
1 +A/i
1
(1  a)r t _
a
h/t
1 +h f .
1 + A/«
— A/i
0.980
n
1 + */. .
A/ e .
Aoe
1
A,*
2.04 M
r.
In
Ao#
16™
hoc
Act
10 fl
n
a,.  J= a + A A )t
A,,
Ar*
Aoi
590 n
t Exact.
5*. "5
SMAUS/GNAt tOWF«fQU£NCy TRANSISTOR MOORS / 305
CONVERSION FORMULAS FOR THE PARAMETERS
OF THE THREE TRANSISTOR CONFIGURATIONS 4
Very often it is necessary to convert from one set of transistor parameters
to another set. Some transistor manufacturers specify all four common
emitter h parameters; others specify h /e , h ib , h*, and h rb . In Table 113 we
give approximate conversion formulas between the CE, CC, and CB h parame
ter. For completeness, we also include the Tmodel parameters, although
we postpone until Sec, 119 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 113 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 Aparameter circuit of Fig. ll9a is redrawn in Fig. 1196
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. 1196 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. 1196,
*#T, ( hrtVo, + Vu + V tt =
<«)
(6)
'8 It O ( a j jj, e £g hybrid model, (b) The circuit in {a) redrawn in a CE configu
r atio n .
306 / ELECTRONIC DEVICES AND CIRCUITS
Combining the last two equations yields
h&hob
S»e. 11.$
1 + hjt
or
V
Hence
k r< = I +
Vu  k^Vte + V ta + V„ =
(1 +h fb )
hibhub + (1 — Art)(l + ^)
hahab — (1 + h/bjhrt
V et kith* + (1  A rt )(l + A/i)
This is an exact expression. The simpler approximate formula is obtained by
noting that, for the typical values given in Table 112,
k Tb « 1
Hence
and hobhit, « 1 + A/t
kibha
h.
1 +h /b
which is the formula given in Table 113.
6. By definition,
h Ih\
If we connect terminals C and E together in Fig. 1196, we obtain Fig. H10.
From the latter figure we see that
V cb = Ti.
Applying KVL to the lefthand mesh, we have
Vu + A*/. + fcrfcF* =
Combining these two equations yields
I, . J L^J* Fb(
Fig. 1110 Relating to the calcula
tions of h ir in terms of the CB h
S« ' T6
SMALLSIGNAl LOW FREQUENCY TRANSISTOR MODELS / 307
Applying KCL to node B, we obtain
h + I. + hfl,I t  hobVb. =
or
h = (i + m ^ : ^ ^ + ^n.
nit
Hence
Vb, h
hie = ~r =
h kjl* + (1  /U)(l + A,*)
This is the exact expression. Jf we make use of the same inequalities as in part
a, namely, h* « 1 and A^aA,* « 1 + A/&, the above equation reduces to
l.
hi. *
l + Jfc*
which is the formula given in Table 113.
116 ANALYSIS OF A TRANSISTOR AMPLIFIER
CIRCUIT USING k PARAMETERS
To form a transistor amplifier it is only necessary to connect an external load
and signal source as indicated in Fig. 1111 and to bias the transistor properly.
The twoport active network of Fig. 1111 represents a transistor in any one
of the three possible configurations. In Fig. 1112 we treat the general case
(connection not specified) by replacing the transistor with its smallsignal
hybrid model. The circuit used in Fig. 1112 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. 1112, 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 1111 A basic amplifier
tir cuit.
i
308 / ELECTRONIC DEVICES AND CIRCUITS
Sec. J?<J
Fig. 1112 The transistor in
Fig. 1111 is replaced by its
Aparameter 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. 1112, we have
l % = k f h + KV,
Substituting Vt = —IiZl hi Eq. (1121), we obtain
is _ ft/
/i 1 + hoZt
At
(1120)
(1121)
(1122)
The Input Impedance Z, The resistance R, in Figs. 1111 and 1112
represents the signalsource 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. 1112, we have
V l  KJi + hrV*
Hence
„ hilt + KV* V,
Li — f = Hi r fir y~
Substituting
Vi = —IsZl — AiIiZl
in Eq. (1125), we obtain
Zi = hi f hrAtZh = hi —
hfh r
Y L + h
(1123)
(1124)
(1125)
(1126)
(1127)
where use has been made of Eq. (1122) 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 ( U28)
A v =
Vt
s« " 6
SMALLSIGNAL LOWFREQUENCY TRANSISTOR MODELS / 309
From Eq. (1126) we have
AjUZl AiZl
A v =
Vi
Zi
(1129)
The Output Admittance Y e For the transistor in Figs. 1111 and 1112,
Y 9 is denned as
(1130)
(1131)
(1132)
(1133)
Substituting the expression for I\JV% from Eq. (1133) in Eq, (1131), we
obtain
h/hr
Y B
It
v z
with 7. =
From Eq.
(1121),
Y
1 7l
= hf V i
+ h
From Fig.
1112, with V. = 0,
RJi + hili
+ KV 2 =
or
fi
h r
v t
s;
•f fi.
(1134)
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. ll13o,
V,Zi
(1135)
Vi =
Th,
en
A v . =
Z< + R,
A v Zi
AtZ L
Zi f R, Zi + R,
(1136)
Wher,
""ere u se has been made of Eq. (1 129). Note that, if R, m 0, then Ay, m A v .
^ce A v is the voltage gain for an ideal voltage source (one with zero internal
stance). In practice, the quantity Av, is more meaningful than Av since,
310 / ELECTRONIC DEVICES AND CIRCUITS
Set. 1L 6
(a)
(6)
Fig. 1113 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. 11136, then this overall current gain Aj, is defined by
U
A u =
/.
7TT. mA 'Z
(1137)
From Fig. 11136,
7 I,Rt
1 1 —
and hence
Zi 4 R,
AtR,
A Im = """ (1138)
U Zi + R t
Note that if R, = <*> , then An = Ai. Hence At is the current gain for an
ideal current source (one with infinite source resistance).
Independent of the transistor characteristics, the voltage and current
gains, taking source impedance into account, are related by
Z L
A Vs = A u
R,
(1139)
This relationship is obtained by dividing Eq. (1136) by Eq. (1138), '■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. 1111 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
(1140)
A  S  
Ap ~ P, ~
V*Ii A A A 1^ L
vrr AvAt ~ At &
S*c. >>6
SMALLSIGNAL LOWFREQUENCY TRANSISTOR MODELS / 311
TABLE 7 74 Smallsignal analysis of
a transistor amplifier
A, = 
1 + hoZt,
Zi = ki + h T AiZi, = hi —
kfh T
hT+Yl
Av 
AiZl
Zi
Y Q = K
hfh.
Av, =
h{ + R, Z e
AvZi AiZh Ai,Zl
Zi + R. Zi + R, R t
AtR.
Zi + R.
Summary The important formulas derived above are summarized for
ready reference in Table 114. 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 hybridII model of Sec. 135).
EXAMPLE The transistor of Fig. 1111 is connected as a commonemitter
amplifier, and the h parameters are those given in Table 112. If R L = R, =
1,000 12, find the various gains and the input and output impedances.
Solution In making the smallsignal 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 114 and the A parameters from Table 112,
At m 
50
= 48.8
1 + k a Jti, 1 I 25 X 10' X 10*
Ri  ht. + h T ,A t R L = 1,100  2.5 X 10"« X 48.8 X 10 a  1,088 &
A v =
AjRi
Ri
48.8 X 10*
1.088 X 10 s
m 44.8
. A v Ri ,,„ 1,088 nnrv
A Yt m * = 44.8 X r = 23.3
Au~
Ri + R*
AjR t
2,08K
48.8 X 10*
Ri + Rt 2.088 X 10*
= 23.3
1
312 / ELECTRONIC DEVICES AND CIRCUITS
Sac. II.7
Note that, since R 3 — Rt, then Ay, = A t ,.
T.  &^  ^ = 25 X 10 
2,100
= 19.0 X 10~ 6 raho
= 19.0 jiA/V
1 10 8
Z = — = — Q = 52.6 K
Y 19.0
Finally, the power gain is given by
A p = A V A, = 44.8 X 48.8 = 2,190
117
COMPARISON OF TRANSISTOR AMPLIFIER CONFIGURATIONS
From Table 11^1 the values of current gain, voltage gain, input impedance,
and output impedance are calculated as a function of load and source imped
ances. These are plotted in Figs. 1 114 to 1 117 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 115.
A, (CB)
/cc
iV(CC)
Or
1.0
0.8
0.8
A,(CE)
\ CB
0.98
^5sCC
CE*^
\CB
40
30
0.4
02
20
10
^— 0.20
10*
10 s
10*
10»
10*
10 1
R L ,a
Fig. 1114 The current gain At of the typical transistor of Table
112 as a function of its load resistance.
117
SMALLSIGNAL LOWFREQUENCY TRANSISTOR MODELS / 313
The CE Configuration From the curves and Table 115, it is observed
that only the commonemitter 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 zeroimpedance voltage source be used, but also that R L be many times
larger than the output impedance. Normally, however, so large a value of
R L is not feasible. Suppose, for example, that a manufacturer specifies a
maximum collector voltage of, say, 30 V. Then we should not be inclined
to use a collector supply voltage in excess of this maximum voltage, since in
such a case the collector voltage would be exceeded if the transistor were
driven to cutoff. Suppose, further, that the transistor is designed to carry a
collector current of, say, 5 mA when biased in the middle of its active region.
Then the load resistor should be selected to have a resistance of about ^ = 3 K.
We compute for the CE configuration a voltage gain under load of A? ■ — 129
(for R, = 0). Of course, the load resistance may be smaller than 3 K, as,
>MCC)
A„(CB)
or
A V (CE)
3300
3000
1.0
/cc
2500
0.8
2000
0.6
/CB or CE
1500
0.4
'0.31
1000
0.2
0.45
500
*^ 1
Iff 1
10 3
10 s
10 s
10 1
R Lt a
Fig. 1115 The voltage gain of the typical transistor of Table
112 as a function of its load resistance.
314 / ELECTRONIC DEVICES AND CIRCUITS
Sec. J J .7
. n
2.03M
10'
10"
10*
10*
cc/
.,600/
CE
10*
1,100
600
503
21.6
CB
10
10 s
10 s
10*
10»
io' J?j.,n
Fig. 1116 The input resistance of the typical transistor
of Table 112 as a function of its load resistance.
TABLE 115 Asymptotic values of transistor gains and resistances
(for numerical values of h parameters see Table 112)
Quantity
(A/.)™** (B*,  0, R, m «)
Ri (Rl = 0)
Ri (Rl — ™ )
(Avx)«**(flji = *,B. = 0)
B„ (B. • 0)
B (B, = »)
A parameter
expression
A,
A
Aift — ArA/
CE
50
1,100 n
eoo a
•3,330
73.3 K
40 K
15 X 10*
cc
51
l.ioo n
2.04 M
21.6 fi
40 K
51.0
CB
0.98
21.6 Q
600 a
3,330
73.5 K
2.04 M
2.94 X 1°'
S*c N7
SMALL SIGNAL LOWFREQUENCY TRANSISTOR MODELS / 315
f or example, when a transistor is used to drive another transistor. Or in
gome applications a higher value of Rl may be acceptable, although load
resistances in excess of 10 K are unusual.
The CB Configuration For the commonbase 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 highimpedance load, or as a noninverting amplifier with a voltage gain
greater than unity. It is also used as a constantcurrent source (for example,
as a sweep circuit to charge a capacitor linearly 11 ).
The CC Configuration For the commoncollector 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 highimpedance source and
a lowimpedance 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 116,
where the various quantities are calculated for Rl = 3 K and for the k parame
ters in Table 1 12,
R B ,tl
10 s
w
Iff
Iff
CB
2.04 M
105 K
1/
CE
73 K
40 K
33.2 K
CC>
21.70
10 ]
10 s
]0*
10 s
10*
Iff
R t ,a
Fig. 1117 The output resistance of the typical transistor of
Table 112 as a function of its source resistance.
316 / ELECTRONIC DEVICES AND CIRCUITS
TABLE 116 Comparison of transistor configurations
Sec. ?T8
Quantity
CE
CC
CB
Aj
Av
Ri (Rl = 3 K)
R e (R.  3 K)
High (46.5)
High (131)
Medium (1,065 tt)
Medium high (45.5 K)
High (47.5)
Low (0.99)
High (144 K)
Low (80.5 ft)
Low (0.98)
High (131)
Low (22.5 a)
High (1.72 M)
118
LINEAR ANALYSIS OF A TRANSISTOR CIRCUIT
There are many transistor circuits which do not consist simply of the CE, CB,
or CC configurations discussed above. For example, a CE amplifier may have
a feedback resistor from collector to base, as in Fig. 105, or it may have an
emitter resistor, as in Fig. 106. Furthermore, a circuit may consist of several
transistors which are interconnected in some manner. An analytic determina
tion of the smallsignal 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 Aparameter model (Table 111).
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 114.
11?
THE PHYSICAL MODEL OF A CB TRANSISTOR
The circuit designer finds the smallsignal model of the transistor described
by the hybrid parameters very convenient for circuit analysis. As indicated
in Sec. 111, these h parameters characterize a general twoport network.
When this model is applied to a specific transistor, the values of the hybrid
parameters are measured experimentally (Sec. 114) by the user or by the
Sec.
719
SMALLSIGNAL LOWFREQUENCY TRANSISTOR MODELS / 317
manufacturer. The device designer, on the other hand, prefers to use a model
containing circuit parameters whose values can be determined directly from
the physical properties of the transistor. We now attempt to obtain such a
smallsignal equivalent circuit which brings into evidence the physical mecha
nisms taking place within the device.
To be specific, consider the groundedbase configuration. Looking into
the emitter, we see a forwardbiased diode. Hence, between input terminals
E and B', there is a dynamic resistance r' et obtained as the slope of the (forward
biased) emitterj unction voltampere characteristic, Looking back into the
output terminals C and B f , we see a backbiased diode. Hence, between
these terminals, there is a dynamic resistance r c obtained as the slope of the
(reversebiased) collectorjunction voltampere 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.
1118.
The Early Feedback Generator The equivalent circuit of Fig. 1118 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. 97, an increase in the magnitude of the collector voltage effectively
narrows the base width W, a phenomenon known as the Early effect* The
minoritycarrier current in the base in the active region is proportional to
the slope of the injected minoritycarrier density curve. From Fig. 923 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. 1119. A little
thought should convince the reader that the polarity shown for generator
Miw is consistent with the physical explanation just given.
The Basespreading Resistance To complete the equivalent circuit of
Fig. 1118, 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 basespreading resistance, is large, and may be of the order of several
Emitter E
Collector C
F '9 1118 A simplified physical
""odel of a CB transistor.
f^
V,b
I
r.' ai,(T)
Base B'
n
+
318 / ELECTRONIC DEVICES AND CIRCUITS
V,b
C.6
<'.(♦)
Web
1 Tb
iz
U&
Soc. I ] .p
Fig. 1119 A more com
plete physical model
of a CB transistor than
that indicated in Fig.
1118.
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. 1119.
If the basespreading resistance could be neglected so that B and B'
coincided, the circuit of Fig. 1119 would be identical with the hybrid model
of Fig. 112, with
r' e = k& n = Art, a, ** —h» and r' c = j—
hob
The T Model The circuit of Fig. 1119 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. 1120. This new circuit should be
considered in conjunction with Table 117. 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 17 Typical parameter values and the equations of
transformation between the circuits of Figs. 1119 and 1120
Parameter in Fig. 111*1
Transformation equations
Parameter in Fig. 1120
t, = 40
n  5 X 10*
r'„  2 M
a m 0.98
r, = r' e — (I ~ a)fir' c
K = (*i
r.  (1  tt)r e
a — ft
a =
1 — »
r, = 20 a
K  1 K
r c = 2M
a = 0.98
Sec
TITO
SMALLSIGNAL LOWFREQUENCY TRANSISTOR MODELS / 319
Fig.' 11 20 The T model of a CB tran
sistor.
The transformed circuit, we observe, accounts for the effect of the collector
circuit on the emitter circuit essentially through the resistor r 6 rather than
through the generator fiv cb >. Note from Table 117 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. 131). The relationships between the hybrid parameters and
those in the T equivalent circuit are given in Table 113.
1110
A VACUUMTUBE— TRANSISTOR ANALOGY 7
It is possible to draw a very rough analogy between a transistor and a vacuum
tube. In this analogy the base, emitter, and collector of a transistor are
identified, respectively, with the grid, cathode, and plate of a vacuum tube.
Correspondingly, the groundedbase, groundedemitter, and groundedcollector
configurations are identified, respectively, with the groundedgrid, grounded
cathode, and groundedplate (cathodefollower) vacuumtube circuits, as in
*'ig. H21.
Consider, for example, the circuits of Fig. ll21a. 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. ll21a, 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. 11216 has a higher input imped
320 / ELECTRONIC DEVICES AND CIRCUITS
?¥]
o ■
T
T V ri
(«>
(*)
(c)
Fig, 1121 Analogous transistor and vacuumtube 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 groundedemittar configuration
is analogous to the groundedcathodc vacuumtube amplifier stage.
In Fig. ll21c, the groundedcollector (emitterfollower) configuration is
compared with the groundedplate (cathodefollower) 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
wardbiased 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 groundedemitter and groundedcollector opera
tion, but in the groundedcollector operation the inputvoltage swing is larger.
The cutoff region of the transistor corresponds to the region in the vacuum
tube where the grid voltage is larger than the cutoff bias. The active region
of the transistor corresponds to the region in which the tube operates as a
linear amplifier. This region covers not only the region within the grid base,
but also the region of positive grid voltages, where the tube operates linearly
S«
UIO
SMAUSIGNAl lOWFRfQl/fNCV TRANSISTOR MODELS / 321
The saturation region of the transistor corresponds to the tube region where
the grid is so positive and the plate voltage is so low that the plate current is
almost independent of grid voltage (Fig. D3). 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 voltampere 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 minoritycarrier storage in a transistor. The lowfrequency
input impedance of a groundedcathode or cathodefollower circuit is infinite,
whereas a transistor has a relatively low input impedance in all three con
figurations. The lowfrequency 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 smallsignal equivalent circuit.
The analogies are principally useful as mnemonic aids. For example, we
may note that the most generally useful tube circuit is the groundedcathode
circuit. We may then expect from our analogy that the groundedemitter
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 (npn and vnp) 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 (largescale or specialpurpose), hearing aids,
■ctronic circuits for space vehicles, and portable equipment fall into this
°gory. The future of extremely complex systems lies in the direction of
cfominiaturization, using transistor technology. However, there are appli
322 / ELECTRONIC DEVICES AND CIRCUITS
Sec. UJQ
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 temperaturesensitive.
2. The transistor is damaged by nuclear radiation.
3. It is easily damaged by transient overloads.
4. The maximum output power (100 W) is lower than from a tube
(300 kW).
5. The upper frequency response (1 GHz) is lower than from a tube
(10 GHz).
6. It is difficult to obtain voltage swings in excess of about 100 V.
7. Under some operating conditions transistors are noisier than tubes.
8. The spread in the characteristics of a given type of transistor is often
very great.
Systems involving high voltage, high power, or high frequencies (and par
ticularly those requiring several of these characteristics simultaneously) use
tubes. Such applications include communications transmitters, radar indi
cators, oscilloscopes, and test equipment. Systems which must operate under
unusual environments of temperature or nuclear radiation use tubes. Also,
systems designed some time ago, and still operative, use tubes. Such equip
ment is often in production today because it is not economically feasible to
redesign the system using semiconductor devices.
REFERENCES
1 . IRE Standards on Semiconductor Symbols, Proc. IRE, vol. 44, pp. 935937, July,
1956.
2. "Transistor Manual," 7th ed., General Electric Co., pp. 5255, Syracuse, N.Y., 1964.
3. Ref. 2, pp. 477482.
4. Electronics Reference Sheet, Electronics, Apr. 1, 1957, p. 190.
5. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," pp. 528
532, McGrawHill Book Company, New York, 1965.
6. Early, J. M.: Effects of Spacecharge Layer Widening in Junction Transistors, Proc.
IRE, vol. 40, pp. 14011406, November, 1952.
7. Giacoletto, L. J.: Junction Transistor Equivalent Circuits and Vacuumtube
Analogy, Proc. IRE, vol. 40, pp. 14901493, November, 1952.
Dosse, J.: "The Transistor," pp. 104123, D. Van Nostrand Company, Inc., Prin ce *
ton, N.J., 1964.
12 /LOWFREQUENCY TRANSISTOR
AMPLIFIER CIRCUITS
In the preceding chapter we consider the smallsignal analysis of a
single stage of amplification. Very often, in practice, a number of
stages are used in cascade to amplify a signal from a source, such as a
phonograph pickup, to a level which is suitable for the operation of
another transducer, such as a loudspeaker. In this chapter we con
sider the problem of cascading a number of transistor amplifier stages.
In addition, various special transistor circuits of practical importance
are examined in detail. Also, simplified approximate methods of solu
tion are presented. All transistor circuits in this chapter are examined
at low frequencies, where the transistor internal capacitances may be
neglected.
121
CASCADING TRANSISTOR AMPLIFIERS 1
When the amplification of a single transistor is not sufficient for a
particular purpose, or when the input or output impedance is not of
the correct magnitude for the intended application, two or more stages
may be connected in cascade; i.e., the output of a given stage is con
nected to the input of the next stage, as shown in Fig. 121. In the
circuit of Fig. 122a the first stage is connected commonemitter, and
the second is a commoncollector stage. Figure 122o shows the small
signal circuit of the twostage amplifier, with the biasing arrangements
omitted for simplicity.
In order to analyze a circuit such as the one of Fig. 122, we make
use of the general expressions for Ar, Z t , Av, and Y a from Table 114.
It is necessary that we have available the h parameters for the specific
transistors used in the circuit. The /iparameter values for a specific
transistor are usually obtained from the manufacturer's data sheet.
323
324 / ELECTRONIC DEVICES AND CIRCUITS
Sac. 12 J
'/
:*,
Fig. 121 Two cascaded stages.
Since most vendors specify the commonemitter 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 113 to the appropriate CC or CB values.
In addition, the k parameters must be corrected for the operating bias con
ditions (Fig. 115).
EXAMPLE Shown in Fig. 122 is a twostage amplifier circuit in a CECC 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. 116) to compute, first, the current gain, then the input impedance
and the voltage gain. Finally, the output impedance may be calculated if desired
by starting this analysis with the first stage and proceeding toward the output
stage.
The second stage. From Table 114, with R L = R*2, the current gain of the
last stage is
An  — —  =
A,
51
1 + 25 X 10"« X 5 X 10 J
 45.3
ifti 1 + hgcReZ
The input impedance fl, 2 is
Ra = ku + h rt A I2 R e2 = 2 + 45.3 X 5 = 228.5 K
Note the high input impedance of the CC stage. The voltage gain of the second
stage is
. V R tl 45.3 X 5
Set
121
1
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 325
«+ V cc
oV,
(a)
Fig. 122 (a) Commonemitter— commoncollector amplifier, (b)
Smallsignal circuit of the CECC amplifier. (The component values
refer to the example in Sec. 121.)
The first stage. We observe that the net load resistance Rn of this stage is
the parallel combination of R tt and R i2 (written in symbolic form, Rli = Rt\\\Ra),
or
R el R it 5 X 228.5
Rli =
Re\ + Ril
233.5
= 4.9 K
Hence
A,
50
An  =
I bl 1 + h et Rn 1 + 25 X 10" B X 4.9 X 10 s
= 44.5
The input impedance of the first stage, which is also the input impedance of the
cascaded amplifier, is given by
Ru = kt, + K c A n Ru = 2  6 X 10~* X 44.5 X 4.9 = 1.87 K
The voltage gain of the first stage is
V % A n Ru 44. 5 X 4.9
Atrt = — = = = — llo.o
r, Ra 1.87
326 / ELECTRONIC DEVICES AND CIRCUITS
>ec. J2.
The output admittance of the first transistor is, from Eq. (1139) or Table iy
1'..  h„ 
kfjlr
hu + R,
= 15 nA/V
= 2o X 10"«  = 15 X 10s mkft
2 X 10 3 + 1 X 10 3 nno
Hence
1 10*
R ol = — = — 12 = 66.7 K
Y Bl 15
The output impedance of the first stage, taking R Bl into account, is /J c ilft,[, or
^ i =
RdRoi _ 5 X 66.7
Rcl + Rax ~ 5 + 66.7
= 4.65 K
The output resistance of the last stage. The effective source resistance R',.
for the second transistor Q2 is R ol \\R ti . Thus R'^ = R' ol = 4.65 K, and
Y . = h —
I az 'tor
A/cAt
A. B + ^,2
T = 25 X 10" 
(5I)(1)
2 X 10 3 + 4.65 X 10 3
= 7.70 X 10' A/V
Hence R oi = 1/F 2 = 130 O, where R o2 is the output impedance of transistor Q2
under opencircuit conditions. The output impedance R' g of the amplifier, taking
R ei into account, is R B 2\\R e i, or
Hi*
Ro2Rt2
130 X 5.000
 127 £2
K„2 + fl e:! 130 + 5,000
The overall current and voltage gains. The total current gain of both sta^
Ai =
lc2 hi
lb2 /cl
hx
hx
From Fig
123,
we
have
lit
RcX
hx
fl, s
+ RcX
Hence
4#»
A/jA/i —
R c i
= 45.3 >
1 ^ bi A
Art r An
l*i
Rn + Rex
228.5 + 5
43.2
(121)
(122)
C, o
I*x
Fig. 123 Relating to the calculation of
overall current gain.
N o—
For the voltage gain of the amplifier, we have
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 327
7, V 2 Vi
= AviAx
(124)
or
Av = 0.99 X (116.6) m 115
The voltage gain can also be obtained from
Ar  A.&  43.2 X £  115
The overall voltage gain, taking the source impedance into account, is given by
Av ' = vr Av R~x~fR.
= us* *: 8 ! i m  75  3
1.87 + 1
Table 121 summarizes the results obtained in the solution of this problem.
TABLE
J27 Results of the examp
e on page 324
Transistor Q2
CC
Transistor Ql
CE
Both stages
CECC
At
Ri
Ay
K
45.3
228.5 K
0.99
127
44.5
1.87 K
116,6
4.65 K
43.2
1.87 K
115
127 fi
1
122 nSTAGE CASCADED AMPLIFIER
The function of a lowlevel 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 commonemitter connection. A typi
cal twostage cascaded CE audio amplifier with biasing arrangements and
coupling capacitors included is shown in Fig. 124.
We now examine in detail the smallsignal operation of an amplifier con
sisting of n cascaded commonemitter stages, as shown in Fig. 125. The
biasing arrangements and coupling capacitors have been omitted for simplicity.
The Voltage Gain We observe from Fig. 125 that the resultant voltage
gain is given by the product of the individual voltage gains of each stage.
A "is statement is verified as follows:
= F 2 _ output voltage of first stage = A / 6i
Vl = y~j — input voltage of first stage
328 / ELECTRONIC DEVICES AND CIRCUITS
12V
Soc J 2.}
(22K) >(6K)
(5mF)
jQI
2N338
*,;
(16K) >(700n)
,Q2
2N338
(3.3K) <R tl ^ (50uF , S(G.2K)< R J_(50
>(1K)T ( ^ ' f > ' 2 T>F)
1(4700)
>F)
1 — i
Fig. 124 Practical twostage CE audio amplifier. (Courtesy of
Texas Instruments, Inc.)
where A i is the magnitude of the voltage gain of the first stage, and tfi is the
phase angle between output and input voltage of this stage. Similarly,
4 _ Vk+\ m output voltage of fcth stage _ . .
Vk ~ V k input voltage of kth stage " k/ —
The resultant voltage gain is defined as
j. m V^ _ output voltage of nth stage _ .
V\ input voltage of first stage ~ —
Since
V, V, V 2 V 3 F_, v n
it follows from these expressions that
Ay = AvxAvz • ■ ' Ay n
= A t As • ■ ■ A n /8i + $ t + ■  ■
or
A = AlAl •  • An S = 01 + $ 2 +
+ 8* =
(125)
(126)
■ + 6 n (127)
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 114,
AikRhk
A V k =
Rik
(128)
where Ri k is the effective load at the collector of the ftth transistor. ™ e
quantities in Eq. (128) are evaluated by starting with the last stage and p 1 **"
S<*
122
IOWEREQUENCV TRANSISTOR AMPLIFIER CIRCUITS / 329
ding to the first. Thus the current gain and the input impedance of the
C h stage are given in Table 114, respectively, as
At* =
— h fe
1 + h ot Rln
Rin = hie + hreAl„Rl
(129)
w here Rl« = R™ The effective load R L , n i on the (n  l)st stage is
R Cin ^Rin (1210)
Now the amplification Ar, n i of the next to the last stage is obtained from
Eq (129) by replacing R L « by R L .ni The input impedance of the (n  l)st
stage is obtained by replacing n by n  1 in Eq. (129). Proceeding in this
manner, .we can calculate the basetocollector current gains of every stage,
including the first. From Eq. (128) we then obtain the voltage gain of each
stage.
The Current Gain Without first finding the voltage amplification of each
stage as indicated above, we can obtain the resultant voltage gain from
v ~ Al W
l cl = li
F
*cl< V
St c 2
Q2
o— — o
/*■ h„
Rc2< Vi *«...'
(1211)
/«, = h
R t \ R«
(a)
*• G) 7 ' < R  *■>■
►*i
*E t
(b)
(c)
1
*9 125 (o) n transistor CE stages in cascade, (e) The &th stage, (c) The
'•"onsistor input stage driven from a current source.
330 / ELECTRONIC DEVICES AND CIRCUITS
See. 1 2 2
where Ai is the current gain of the nstage amplifier. Since Ai is defined aa
the ratio of the output current I of the last stage to the input (base) current
hi of the first stage,
/,
u
hi
hi
(1242)
where I cn s /„ is the collector current of the nth stage. We now obtain
expressions from which to calculate Ai in terms of the circuit parameters.
Since
1*
hi
lib
hi 1 1
h—1 In
h—2 in— 1
then
where
Ai = AiiA' n ■ ■ ■ A'i^A'^
d
hi
An =  —   42 A' =
Ibl lbi
A.
h1
(1213)
(12U)
Note that An is the basetocollector current gain of the first stage, and A' lk is
the collectortocollector current gain of the kth stage (Jfc = 2, 3, . . . , n).
We now obtain the relationship between the collectortocollector current
gain A Ik = It/hi and the basetocollector current amplification
h
A n = ~~
*bk
where I ek m Ik is the collector current and hk is the base current of the fcth
stage. From Fig. 1256,
hk — — I*_l
Hence
Alt —
Rc.k] + Rik
h h hk AiicRe.k—l
hi hk hi Rc,ki + Rit
(1215)
(1216)
The basetocollector 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. (129) and (1210). The collectortocollector gains are then found froffl
Eq, (1216), and the current gain of the nstage amplifier, from Eq. (1213).
If the input stage of Fig. 125a is driven from a current source, as indicated
in Fig. 125c, the overall current gain is given by
Ai t = Ai
R,
R, + Rh
(1217)
Input and Output Impedances The input resistance of the amplifier is
obtained, as indicated above, by starting with the last stage and proceeding
toward the first stage.
The output impedance of each transistor stage and of the overall ampler
Sec
122
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 331
• g calculated starting with the first stage and using Eq. (1134). The output
'edance R' ok of the fcth stage is the parallel combination of the output
j mpe dancc R„k of transistor QK and R ck . The effective source impedance of
the (fc + !) st sta S e is also R *
of
Power Gain The total power gain of the nstage amplifier is
output power VJ a . ,
A P =  — , =  «r~ ■ AvAt
input power V ihi
Ren
A P = (A,) 2
RiX
(1218)
(1219)
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 commoncollector 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
moncollector stages.
Grounded base SCcoupled 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 shortcircuit current gain h /e of a commonemitter 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 commonemitter 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 opencircuit or shortcircuit
332 / ELECTRONIC DEVICES AND CIRCUITS
Sec 123
operation. In many cases the commoncollector or commonbase 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.
123
THE DECIBEL
In many problems it is found very convenient to compare two powers on a
logarithmic rather than on a linear scale. The unit of this logarithmic scale
is called the decibel (abbreviated dB). The number N of decibels by which
the power P 2 exceeds the power Pi is defined by
AT = 10 log 1J
* 1
(1220)
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. (1220) reduces to
N = 20 log ^ = 20 log Ay
(1221)
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. (127) is given by
log A v = log Ai + log A t +
+ lOg An
(1222)
By comparing this result with Eq. (1221), 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 vacuumtube amplifiers. Ho*'
ever, it must be emphasized that, in calculating the gain of one stage, the
loading effect of the next stage must be taken into account.
Sec
124
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 333
l2 _ 4 SIMPLIFIED COMMONEMITTER HYBRID MODEL 2
In the preceding chapter, and also in Sec. 121, 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 commonemitter connection is in general the
most useful, we first concentrate our attention on the CE ftparameter model
shown in Fig. 126a. 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.
120a 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. 126a and to obtain the approximate equivalent
o — j — VW
ft*,
R. < + I 1
V„ h„V e Q (j)h fe
*^r*
F '9. 126 (a) Exact CE
hybrid model; (b) ap
proximate CE model.
334 / ELECTRONIC DEVICES AND CIRCUITS
Sec. I2.4
Fig. 127 Approximate hybrid model
which may be used for afl three con
figurations, CE, CC, or CB.
circuit of Fig. 1266. We are essentially making the assumption here that
the transistor operates under shortcircuit 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
127 which we used in Fig. 1266 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. 127 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 lowfrequency 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 12, R L must be less than 4 K. The approximate
circuit is always valid when CE transistors are operated in cascade because
the low input impedance of a CE stage shunts the output of the previous
stage so that the effective load resistance R' L satisfies the condition hjft' h < 0.1.
We now justify the validity of the proposed simplification for the CB
configuration.
Current Gain From Table 1 14 the CE current gain is given by
A r = — ~ hfe
1 + h oe R L
Hence we immediately see that the approximation (Fig. 1266)
Ar «* —hf t
(1223)
(1224)
overestimates the magnitude of the current gain by less than 10 percent if
h ot R L < 0.1.
Input Impedance From Table 114 the input resistance is given by
Ri = h u + hreArRt (1225)
125
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 335
(1226)
ff hich may be put in the form
Using the typical Aparameter values in Table 112, we find h Te h; e /hi t h oe ~ 0.5.
From Eq. (1223), we see that \A r \ < h ft . Hence, if KMl < 0.1, it follows
from Eq. (1226) that the approximation obtained from Fig. 1266, namely,
Ri — ~r~ ** *<*
lb
overestimates the input resistance by less than 5 percent.
Voltage Gain From Table 114 the voltage gain is given by
(1227)
Av  At
Ri
h/iRi,
(1228)
Ri hie
If we take the logarithm of this equation and then the differential, we obtain
dAv _ dAr dRj (1229)
A Y " IT " ~Rl
From the preceding discussion the maximum errors for KMl < 0.1 are
^ = +0.1
At
and n~ l = +0.05
Hence, the maximum error in voltage gain is 5 percent, and the magnitude
of Ay is overestimated by this amount.
Output Impedance The simplified circuit of Fig. 1266 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.
1117). For a maximum load resistance of Ri = 4 K, the output resistance
of the stage, taking R L into account, is 4 K, if the simplified model is used,
and the parallel combination of 4 K with 40 K (under the worst case), if the
exact solution is used. Hence, using the approximate model leads to a value
°f output resistance under load which is too large, but by no more than 10
Percent.
The approximate solution for the CE configuration is summarized in the
frst column of Table 122.
12 ' 5 SIMPLIFIED CALCULATIONS FOR THE
COMMONCOLLECTOR CONFIGURATION
*We 128 shows the simplified circuit of Fig. 127 with the collector grounded
With respect to the signal) and a load Rl connected between emitter and
©"cund.
33d / ELECTRONIC DEVICES AND CIRCUITS
Sec 72. j
Fig, 128 Simplified hybrid
model for the CC circuit.
Current Gain From Fig. 128 we see that
At i  g w 14. ^
From Tables 114 and 113, the exact expression for Ai is
— A /c 1 + A /B
A, m
1 + h oe R L 1 + h oe R L
(1230)
(1231)
Comparing these two equations, we conclude that when the simplified
equivalent, circuit of Fig. 128 is used, the current gain is overestimated by
less than 10 percent if h ot R L < 0.1.
Input Resistance From Fig. 128, we obtain
Rt = T b = hit + (1 + hft)RL
(1232)
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 113,
ft = h ic + k rc AiR L = h it 4 AiR L
(1233)
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. (1230) in (1233),
we obtain Eq. (1232). However, we have just concluded that Eq. (1230)
gives too high a value of Ai by at most 10 percent. Hence it follows that Hi,
as calculated from Eq. (1232) or Fig. 128, is also ovemstimated by less than
10 percent.
Voltage Gain If Eq. (1229) 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. (1233),
1 _ a v = 1 _ a iRl = Ri ~ A T R L ' hu
ft
ft
Ri
ilj34)
SfC j 25 LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 337
TABLE 122 Summary of approximate equations for A oe (ft f Rj,) <0.1f
CE
CE with R t
CC
CB
A,
h f .
A/.
1 +h /t
L h/.
" ~ 1 + *,.
Ri
hi.
hi. + (1 + h f .)R t
hu + (1 + A/,) fix.
A, hit
1 + k„
Ay
h/.Ri,
hi.
h/ t Rz,
rT
hi.
Rl
hi.
R„
00
at
R, + hi.
1 +h f .
SO
K
R L
R L
R \\Rl
R L
t (Ri)cB is an underestimation by less than 10 percent. All other quan
tities except R 9 are too large in magnitude by less than 10 percent.
This expression is nearly exact since the only approximation made is that
hn, = 1 — A M is replaced by unity. If, for example, ft = 10A,a, then Ay — 0.9.
If, however, we use an approximate value of Ri which is 10 percent too high,
then hu/Ri = tt = 0.09 and Av = 0.91. Hence the approximate calculation
for A v gives a value which is only 1 percent too high.
Output Impedance In Fig. 128 the opencircuit output voltage is V,
and the shortcircuit output current is
r M /, X t vr (1 +h;.)V,
Hence the output admittance of the transistor alone is, from Eq. (822),
(1235)
Y L * + h f
V, hit + Rt
From Tables 114 and 113, the expression for Y„ is
Y« = h M —
hrji
fcltrc
hic + R.
Kt +
1 + h f€
kit + %
(1236)
*^ven if we choose an abnormally large value of source resistance, say
*■ = 100 K, then (using the typical Aparameter values in Table 112) we
jhd that the second term in Eq. (1236) 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. (1235) is smaller than the value given by Eq. (1236)
y less than 5 percent. The output resistance R„ of the transistor, calculated
338 / ELECTRONIC DEVICES AND CIRCUITS
from the simplified model, namely,
hi. + R,
R„ =
1 + h fr
Sec, 1 25
(1237)
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 122.
EXAMPLE Carry out the calculations for the twostage amplifier of Fig, ]22
using the simplified model of Fig. 127.
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 122,
An = 1 + h fe  51
Rn = h it +(l+ h St )R L = 2 + (51) (5) = 257 K
Avt =
AjtRt
(51)(5)
Rn 257
or alternatively,
= 0.992
An = 1  ~ = 1  ? = 0.992
jSij 257
For the CE input stage, we find, from Table 122,
An = h /e = 50 Rn =A i( = 2K
The effective load on the first stage, its voltage gain, and output impedance are
Rn =
Rci + Ra
(5) (257)
262
 4.9 K
_ AnRti (50) (4.9)
Av\ = — ■ = —123
Ru 2
R l = Rel = 5 K
Since R' ol is the effective source impedance for Q2, then, from Table 122,
D h ie + R, 2,000 + 5,000
il„2 = —
R») =
l+A/.
R02R1.2
R»2 + Ri
51
(137) (5,000)
5,137
= 137 a
= 134 Q
S*. I** LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 339
Finally, the overall voltage and current gains of the cascade are
Av = AviAvt = (123) (0.992) m 122
R c i
Ai — AnAit
= (50) (51)
Rci + Riz
Alternatively, Ay may be computed from
A A Rn 48.7 X 5
A v = A; — = = 122
Rn 2
\5 + 257/
48.7
Table 123 summarizes this solution, and should be compared with the
exact values in Table 121. 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 723 Approximate results of the
example on page 338
126
SIMPLIFIED CALCULATIONS FOR THE
COMMONBASE CONFIGURATION
»pire 129 shows the simplified circuit of Fig. 127 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. 124 and 125 for the
* and CC configurations, respectively, the approximate formulas given in
yourth column of Table 122 may be obtained. Note that ft is too small
y ess than 10 percent, whereas A Jt A v , and R[ are too large by no more than
1U Percent.
e II.
?
Fig 1 j _ R,
"• '*■? Simplified hybrid model
Tor rhu rn  
me CB circuit
e=
h f , I b
K
1 1 O 1.
R, B +r
R»
340 / ELECTRONIC DEVICES AND CIRCUITS Sac, 127
127 THE COMMONEMITTER AMPLIFIER
WITH AN EMITTER RESISTANCE
Very often a transistor amplifier consists of a number of CE stages in cascade.
Since the voltage gain of the amplifier is equal to the product of the voltage
gains of each stage, it becomes important to stabilize the voltage amplification
of each stage. By stabilization of voltage or current gain, we mean that the
amplification becomes essentially independent of the k parameters of the tran
sistor. From our discussion in Sec. 1 13, we know that the transistor param
eters depend on temperature, aging, and the operating point. Moreover, these
parameters vary widely from device to device even for the same type of
transistor.
The necessity for voltage stabilization is seen from the following example;
Two commercially built six stage amplifiers are to be compared. If each stage
of the first has a gain which is only 10 percent below that of the second, the
overall amplification of the latter is (0.9) 6 = 0.53 (or about onehalf that of
the former). And this value may be below the required specification. A
simple and effective way to obtain voltagegain stabilization is to add an
emitter resistor R t to a CE stage, as indicated in the circuit of Fig. 1210.
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. 104: It leaves the current gain ^4/ essentially
unchanged; it increases the input impedance by (1 + h/ t )R,; it increases the
output impedance; and under the condition (1 + hf,)R t » h it) it stabilizes the
(«)
(6)
Fig. 1210 (a) Commonemitter amplifier with an emitter resistor.
The base biasing network {RiR t of Fig. 121 3a J is not indicated,
(b) Approximate smallsignal equivalent circuit.
S«. '27
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 341
voltage gain, which becomes essentially equal to —R L /R t (and thus is inde
pendent of the transistor).
The Approximate Solution An approximate analysis of the circuit of
Fig. 12 10a can be made using the simplified model of Fig. 127 as shown in
Fig. 12I0b.
The current gain is, from Fig. 12106,
<■! j~ f = — «/«
lb lb
(1238)
The current gain equals the shortcircuit value, and is unaffected by the
addition of R e .
The input resistance, as obtained from inspection of Fig. 12106, is
V*
Ri = r = hu + (1 + k/ t )R,
(1239)
The input resistance is augmented by (1 + k fc )R t) and may be very much
larger than k ie . For example, if R, = 1 K and h f , = 50, then
(1 + h,.)R, = 51 K » hu « 1 K
Hence an emitter resistance greatly increases the input resistance.
The voltage gain is
Ay =
AiRt
— h/tRi
(1240)
Ri hi, + (1 + h /t )R t
Clearly, the addition of an emitter resistance greatly reduces the voltage
amplification. This reduction in gain is often a reasonable price to pay for the
improvement in stability. We note that, if (1 + h ft )R, » h it , and since
hf, » 1, then
a ~ "hf* Rl _ —Rl ,.„..,
Ar ~ T+hf.T. RT (1JM1)
Subject to the above approximations, A v is completely stable (if stable
resistances are used for Rt and R t ) t since it is independent of all transistor
Parameters.
The output resistance of the transistor alone (with R L considered exter
n& l) is infinite for the approximate circuit of Fig. 12106, just as it was for the
C E amplifier of Sec. 124 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. (1239), we draw the equivalent circuit of Fig. 121 la from
hl ch to calculate the base current with the signal source applied. This net
0r k is the equivalent circuit "looking into the base." From it we obtain
h =
V.
R< + Ai, + (1 + h t .)R,
(1242)
342 / ELECTRONIC DEVICES AND CIRCUITS
Sec. J 27
(7) ^Yi 3 J
(6)
ZL
ir—
=±= JV
!, = (! + £,,)/(,
I W\* o i 
!_=.
(c)
Fig. 1211 (a) Equivalent circuit "looking into the base" of
Fig. 1210. This circuit gives (approximately) the correct base
current, (b) Equivalent circuit "looking into the collector" of
Fig. 1210. This circuit gives (approximately) the correct collec
tor current, (e) Equivalent circuit "looking into the emitter"
of Fig. 1210. 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
(1243)
and since the output impedance is infinite, the Norton's equivalent output
circuit is as given in Fig, 12116. 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. 12106 and Eq. (1242) we find the emitter toground voltage
to be
V,R e
V m = V t = (1 + h /e )I b R, =
(R. + A*)/(l + */.) + £«
(1244)
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. 127 is valid if h ot R L < 0.1. What i s
Sec 127
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 343
the corresponding restriction for the circuit with R f y^ 0? We can answer
this question and, at the same time, obtain an exact solution, if desired, by
proceeding as indicated in Fig. 1212. 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. 1212a 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. 12126,
R'l~
'■l + ^t^R,
At
(1245)
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. 12126 the emitter is grounded
and the collector resistance is R' L , the approximate twoparameter (ft,*« and ft/.)
circuit is valid, provided that
KJt' L = h oe (R L + R 4 ) < 0.1
(1246)
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. (1241)].
The approximate solution for the CE amplifier with an emitter resistor
R, is summarized in the second column of Table 122.
*ig. 1212 (a) Transistor ampli
fier stage with unbypassed
emitter resistor R t , (b) Small
signal equivalent circuit.
U(1A,)I>
K
*rv e
Wv
344 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 127
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. 12126 and to Table 114. For example, the current gain is
At =
h
f*
h,
l + KM'l
l + K, Ul + ^3— «.)
(1247)
From this equation we can solve explicitly for At, and we obtain
fl at Re — hje
At =
1 + K.{Rl + R.)
(1248)
If the inequality (1246) is satisfied, then h ,R, « h/ a} and the exact expression
(1248) reduces to At W — A/« in agreement with Eq, (1238).
The exact expression for the input resistance is, from Fig. 12126 and
Table 122,
Bt + ^r  (1 ~ A Z )R. T Ik, + h Tt AiR' L
(1249)
where R' L is given by Eq. (1245). Usually, the third term on the righthand
side can be neglected, compared with, the other two terms. The exact expres
sion for the voltage amplification is
Ay =
A t Ri
Rt
(1250)
where the exact values for Ai and Ri from Eqs. (1248) and (1249) must be
used.
The exact expression for the output impedance (with Rl considered
external to the amplifier) is found, as outlined in Prob. 1214, to be
R =
1 (1 + h f ,)R, + (R, + hi,) (I + h,R t )
h oi R, + R, f hi, — KJift/K,
Note that, if R. » R, + h it and h<Jl, « 1, then
d 1 1 + h f<
K*
h^
(1251)
(1252)
where the conversion formula (Table 1 13) 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 113, we rind
from Eq. (1251) that R Q = 817 K, which is at least ten times the output
resistance for an amplifier with R, — (Fig. 1117).
Sec. 128
LOWFREQUENCV TRANSISTOR AMPLIFIER CIRCUITS / 345
128
THE EMITTER FOLLOWER
Figure 12 13a is the circuit diagram of a commoncollector 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 npn or a p~np
transistor is used. In the case of the cathode follower, the drop across the
cathode resistor is always positive. Third, the input resistance of the emitter
follower, although high (tens or hundreds of kilohms), is low compared with
that of a cathode follower. Fourth, the output resistance of the emitter
follower is much lower (perhaps by a factor of 10) than that of a cathode
follower.
In the discussion on cascading transistor stages in Sec. 122, we note that
the commoncollector 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. 1213a 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. 12136, where
R b  R t \\R' R'  fl,l/2,
and
F fl =
V.R'
R, + R'
(1253)
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:
(1254)
The circuit of Fig. 12136 is examined in some detail in Sec. 125, 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 122, 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. (1231), (1233), (1234), and (1236), respectively.
346 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 128
Pig. 1213 (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, (1232) and (1234) and Table 112,
Ri = 1.1 + (51) (4) = 205 K
^rJSff 1 * 00054 = 0.9946
205
If a triode is used in a cathodefollower 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. ( 1258) J
Let us now calculate Ri and Av for an infinite load resistance. Of course,
we must now use the exact formulas, Eqs. (1231) and (1233), rather than the
Sec. 128
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 347
1
approximations, Eqs. (1230) and (1232). With R L m fl,> »,
1 + kf t 1 + h/ e
Ar 
1 ■+• KeRt h ot R t
Ri = h it + AjR e ~ h ie +
1 + h f , 1 + hu
J_
k b
(1255)
(1256)
where use has been made of the transformation from the CE to the CB h
parameters in Table 1 13. 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. 12136, 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, 1210.
Fori?*— ► », Eq. (1234) becomes
1  A v «fe
kiji
(1257)
Ri 1 + h f .
If we use the feparameter values in Table 112, 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 114.
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 signalsource
impedance into account, gives the amplification between the output and the
signal source V.. Thus
V
R'i
V V
A = U — ° * — *
Av ' V, ViV t  Av R. + R'<
(1258)
where use has been made of Eq. (1254). 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. 1210).
The Effect of a Collectorcircuit Resistor It is important to investigate
'he effect of the presence in the collector circuit of a resistance R e in Fig. 1213.
°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. 1212a we see that the relationship between the CE current
348 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 129
gain An (designated simply Ai in the figure) and the CC current gain A Ie is
A Ie  1  Au (1259)
where
A Ie =  y and
4*. ~ ?
Substituting Eq. (1248) in Eq. (1259) with Rl replaced by R e , we obtain
the exact expression
An =
1 + k oe R e + kf„
1 + h oe (R<: + Re)
(1260)
The value of Ri is obtained from Eq. (1249), 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^
(1261)
Subject to the restriction k e(Rc + R*) « 1, the approximate formulas
given in the third column of Table 122 are valid, and the protection resistor
R e has no effect on the smallsignal operation of the emitter follower.
129
MILLER'S THEOREM
We digress briefly to discuss a theorem which is used in the next section and
also in connection with several other topics in this book. Consider an arbitrary
circuit configuration with JV distinct nodes, 1, 2, 3, . . . , N, as indicated in
Fig. 1214a. Let the node voltages be Vi, V%, F 3 , . . . , V&, where Fat —
z,=
lK
z 2 =
z;k
k\
(a)
(6)
Fig. 1214 Pertaining to Miller's theorem. By definition, K = PVVj. The
networks in (a) and [b) have identical node voltages. Note that /i = — h.
S«. '29
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 349
and JV is the reference or ground node. Nodes 1 and 2 (referred to as iVi and
$%) are interconnected with an impedance Z' . We postulate that we know
the ratio F s Fi Designate the ratio F a /Fi by K, which in the sinusoidal
steady state will be a complex number and, more generally, will be a function
of the Laplace transform variable s. We shall now show that the current A
drawn from iV\ through Z' ean be obtained by disconnecting terminal 1 from
Z 1 and by bridging an impedance Z'/(l — K) from N\ to ground, as indicated
in Fig. 12146.
The current I\ is given by
h =
Vx  F 2 Fi(l  K) Vi
F,
Z' Z' £'/(!  K)
z L
(1262)
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. 1214a and b).
In a similar way, it may be established that the correct current 7 2 drawn
from JV 2 may be calculated by removing Z' and by connecting between JV a and
ground an impedance Z 2 , given by
Z, m
Z f
Z'K
Ll/K K  1
(1263)
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 groundedcathode stage, taking
interelectrode capacitances into account. Terminal JV is the cathode (Fig.
819), 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. 12146, given by
Z, m
Z'
J
1  K
(1264)
«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 )
(1265)
This result agrees with Eq. (844), first derived by Miller. 3 Hence the trans
formation indicated in Fig. 1214 is referred to as Miller's theorem.
350 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 1210
(a)
(b)
Fig, 1215 (a) Darlington pair. Some vendors package this
device as a single composite transistor with only three external
leads, (b) The Darlington circuit drawn as two cascaded CC
stages.
1210
HIGHINPUTRESISTANCE TRANSISTOR CIRCUITS 4
In some applications the need arises for an amplifier with a high input imped
ance. For input resistances smaller than about 500 K, the emitter follower
discussed in Sec, 128 is satisfactory. In order to achieve larger input imped
ances, the circuit shown in Fig. 1215a, 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. 1215o.
The Darlington composite emitter follower will be analyzed by referring
to Fig. 1216, Assuming that hJR. < 0.1 and h f Ji e y>h ie> we have, from
Table 122, for the current gain and the input impedance of the second stage,
Ru « (1 + h ft )R t
(1266)
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.
(1231) 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 fieldeffect transistor (Chap. 14) with its extremely high
input impedance would be preferred to the Darlington pair.
(1267)
(1268)
Sec. 1210
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 351
or
The overall current gain for Fig. 1216 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. (1233)
Bit  hu + AnRit
(1 + h/.) l R,
1 + k Jif t Rt
(1269)
(1270)
This equation for the input resistance of the Darlington circuit is valid for
KJt t < 01, and should be compared with the input resistance of the single
stage emitter follower given by Eq, (1232). If R s = 4 K, and using the h
parameters of Table 112, 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. 1216 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. (1234), we obtain
1 — Avt =
hie
Ril
1 A  *■'«
1 — Avi — 5
hi.
(1271)
AnRiZ
where A V2 = V /V 2 and Avi = VifVi. Finally, we have, for A v = V,/Vt,
A v = Av.Av, ~\\  gj (l  ~^J  1  j^fa  Jfa
and since AuRn » Ru, expression (1272) becomes
1 
h±
Ri2
(1272)
(1273)
Fig. 1216 Darlington emitter
'Oilower,
352 / ELECTRONIC DEVICES AND CIRCUITS
Sec. ?220
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, (1235),
p R. + h ie
and hence the output resistance of the second transistor Q2 is, approximately,
R* + hi e , ,
(1274)
R»9 «*
1 + hfe
R, + hi e
1 + h
H*
+
hie
(1 + hjeY ^ l+hje
We can now conclude from the foregoing discussion, and specifically from
Eqs. (1269), (1270), (1273), and (1274), 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 singlestage 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. 115). 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
commonemitter amplifier. The advantage of this pair would be a very high
overall h ft , nominally equal to the product of the CE shortcircuit 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. 121, 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 highinput 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
1210
LOW FREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 353
emitter follower of Fig. 1213a consists of Ri\\R', where R' m Ri\\Rt. Assume
that the input circuit is modified as in Fig. 1217 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. 1217 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. 1214 and Miller's
theorem can be used to calculate the current drawn by Rz from the input signal.
We can then see that the biasing arrangement Ri, Ri, and Rz represents an
effective input resistance of
Rm —
St
1  A,
(1275)
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. (1234) is Ri — A«/(l — A v ). Since this expression is of the
form of Eq. (1275), 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. 1217 Trie boot
>rop principle increases
^e effective value of Rz.
354 / ELECTRONIC DEVICES AND CIRCUITS
Sec. I2IQ
lated as follows: The emitter end of #3 is at a voltage Av times as large aa
the base end of Rz. From Fig. 1214, 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
(1276)
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. 1218 (a) The boot
strapped Darlington cir
cuit, (b) The equivalent
circuit.
+oj^AAA O — o VVW^
I
R t — RciWRti '
<P
_£
I A/,2 lb
(ft)
5«c.
1217
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 355
can be greatly increased by bootstrapping the Darlington circuit through the
addition of C„ between the first collector Ci and the second emitter #2, as indi
cated in Fig. 12 18a. Note that the collector resistor R„i is essential because,
without it, Rut would be shorted to ground. If the input signal changes by
y i} then E 2 changes by A v Vi and (assuming that the reactance of C is
negligible) the collector changes by the same amount. Hence 1/hob is now
effectively increased to l/(h ob )(l — Av) ~ 400 M, for a voltage gain of 0.995.
An expression for the input resistance Ri of the bootstrapped Darlington
pair can be obtained using the equivalent circuit of Fig. 12186. 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 Aparameter model. However,
the exact hybrid model as indicated in Fig, 12186 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. 1221)
Ri * hfelhfeiRe
(1277)
This equation shows that the input resistance of the bootstrapped Darlington
emitter follower is essentially equal to the product of the shortcircuit 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. 1217 would also be used in the circuit
of Fig. 1218. 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. (1275).
1211 THE CASCODE TRANSISTOR CONFIGURATION 6
The cascode transistor configuration shown in Fig. 1219 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 vacuumtube triode
cascode amplifier discussed in Sec. 810. 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. 1219 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 opencircuited input.
Derivation of Parameter Values To verify the above statement let us
. nipute the k parameters of the Q1Q2 combination. From our discussion
lnS ec n1 and Fig. 1219,
1
B
°*ever, if y 2 = 0, then the load of Ql consists of h ib2 , which, from Table 113,
356 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 12. j j
Fig. 1219 The cascode configura
tion. (Supply voltages are not
indicated.)
is about 20 8. Hence transistor Q\ is effectively shortcircuited, and
ftu m h ia (1278)
Similarly, we have for the shortcircuit current gain
*» Jllr.. ** ZT?\r~* "  h ' J " b " h " (l2  79)
since —hfb — a « 1.
The output conductance with input opencircuited 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. 1117
we see that, for the CB connection, the output resistance R„ with R, = 40
K is essentially the same as that for R t = °° , so that R a — l/hc*. Therefore
ha = mr « hob
is
Finally, for the reverse opencircuit voltage amplification, we have
hrJlrb
(1280)
(1281)
Equation (1281) is valid under the assumption that the output resistance
of Ql (which is l/h M m 40 K) represents an opencircuited emitter for QZ.
Summary Using the h parameters of the typical transistor of Table 11*
and Eqs. (1278) to (1281), 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
(1282)
S*
1212
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 357
Kote that the input resistance and current gain (with the output short
Vcuited) are nominally equal to the corresponding parameter values for a
ngle CE stage. The output resistance (with the input opencircuited) is
oproximately equal to the CB value of 2 M, which is much higher than the
pE value of 40 K. The reverse opencircuit 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. 127 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 HaRl < 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 tunedamplifier design. The reduction in the "internal
feedback" of the compound device reduces the probability of oscillation and
results in improved stability of the circuit.
1212
DIFFERENCE AMPLIFIERS 8
The function of a difference, or differential, amplifier is, in general, to amplify
the difference between two signals. The need for differential amplifiers arises
in many physical measurements, in medical electronics, and in directcoupled
amplifier applications.
Figure 1220 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%) (1283)
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. (1283) 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») (1284)
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 Commonmode Rejection Ratio The foregoing statements are now
clarified, and a figure of merit for a difference amplifier is introduced. The
'9 1220 The output is a [ineor function of
1 Q nd v 2 . For an ideal differential ampli
V ».  A,(*.  ir,).
358 / ELECTRONIC DEVICES AND CIRCUITS
Sec. I2.j j
output of Fig. 1220 can be expressed as a linear combination of the two
input voltages
v B = AiV! + A2V2 (1285)
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. (1284),
i>i = v e + §& and v 2 m v e — %v d
If these equations are substituted in Eq. (1285), we obtain
v ~ A d Vd + A c v c
where
(1286)
A d = %(Ai — At)
and
A e = A 1 + A
(1287)
(1288)
The voltage gain for the difference signal is A d , and that for the commonmode
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. (1287)]. 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 commonmode gain A e .
Clearly, we should like to have A d large, whereas, ideally, A e should
equal zero. A quantity called the commonmode rejection ratio, which serves
as a figure of merit for a difference amplifier, is
P= 2: ( 12  89 >
From Eqs. (1287) and (1289) we obtain an expression for the output in the
following form :
/ 1 .. \
(1290)
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 commonmode 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. (1290) is equal to the first term. Hence, for an amplifier with
a commonmode rejection ratio of 1,000, a 1pV difference of potential between
the two inputs gives the same output as a 1mV 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 commonmode 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. (129°)'
v Q = 100A d pV.
Sec
1212
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 359
In the second ease, v d = 100 pV, the same value as in part a, but now v e =
4(1,050 + 950)  1,000 jtV, so that, from Eq. (1290),
».  100^
fc*9
100A d (l + *&) pV
These two measurements differ by 10 percent.
b. For p = 10,000, the second set of signals results in an output
Vt = 100A d (l + 10 X 10"") pV
whereas the first set of signals gives an output v c = 100/lj pV. Hence the two
measurements now differ by only 0.1 percent.
The Emittercoupled Difference Amplifier The circuit of Fig. 1221 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.
(1287), 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. 1221, 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 commonmode rejection ratio is infinite
for R t = « and a symmetrical circuit.
We now analyze the emittercoupled 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. 1221. This circuit can be bisected as in Fig. 1222a. An analysis of
this circuit (Prob. 1228), using Eqs. (1248) to (1250) and neglecting the
term in fe„ in Eq. (1249), yields
V e (2h oe R.  h f .)R.
A c =
V. 2R.(1 + h ft ) + {R. + hi,)(2h Jt. + 1)
(1291)
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. 1221,
we see that, if V,i = — V t2 , then the emitter of each transistor is grounded for
v e
v ei = v a
'8 1221 Symmetrical emitter
c °upled difference amplifier.
360 / ELECTRONIC DEVICES AND CIRCUITS
Sec. J 2,
1?
°V„
oK„
Fig. 1222 Equivalent
circuit for a symmetrical
difference amplifier used
to determine (a) the com
monmode gain A e and (M
the difference gain A d .
(«)
(6)
smallsignal operation. Under these conditions the circuit of Fig. 12226 can
be used to obtain Ad. Hence
A d = ^ = ±
1 hr.R c
V 8 2 /?, + fa
(1292)
provided h Q Jt c <<C 1.
The commonmode rejection ratio can now be obtained using Eqs. (1291)
and (1292).
From Eq. (1291) it is seen that the commonmode 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. 115. Both of these effects will tend
to decrease the commonmode rejection ratio.
Difference Amplifier Supplied with a Constant Current Frequently, » n
practice, R e is replaced by a transistor circuit, as in Fig. 1223, in which Ru
Rl, and R% can be adjusted to give the same quiescent conditions for Ql
and Q2 as the original circuit of Fig. 1221. This modified circuit of Fig.
1223 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. (1251). In Sec. 127 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
toemitter 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. 1223 is given by
I E = h =
Ri{Ri + R z )
(1293)
S^
1212
LOWFREQUENCY TRANSISTOR AMPLIFIER CIRCUITS / 361
Since this current is independent of the signal voltages V.i and F. a , then
03 acts to supply the difference amplifier consisting of Ql and Q2 with the
constant current I s .
Consider that Ql and Q2 are identical and that Q3 is a true const an t
gurrent source. Under these circumstances we can demonstrate that the
commonmode 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 commonmode 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. (1251), (1291), and (1292) 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 basetoground 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 inputsignal
voltage. Using the techniques of integrated circuits (Chap. 15), it is possible
v c
F 'B. 1223 Differential
omplifier with constont
Cu "*rent stage in the emit
ter circuit. Nominally,
362 / ELECTRONIC DEVICES AND CIRCUITS
S«c. !2. ? j
to construct a difference amplifier with Ql and Q2 having almost identical
properties. Under these conditions any parameter changes due to tempera
ture will cancel and will not vary the output. A number of manufacturers t
sell devices designed specifically for differenceamplifier applications. These
consist of two highgain npn 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 commonmode rejection. Out
puts V a i and V o2 are taken from each collector (Fig. 1223) and are coupled
directly to the two bases, respectively, of the next stage.
Finally, the differential amplifier may be used as an emittercoupled phase
inverter. For this application the signal is applied to one base, whereas the
second base is not excited (but is, of course, properly biased). The output
voltages taken from the collectors are equal in magnitude and 180° out of phase.
REFERENCES
1. Coblenz, A., and H. L. Owens: Cascading Transistor Amplifier Stages, Electronics,
vol. 27, pp. 158161, 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 Threeelectrode Vacuum
Tube upon the Load in the Plate Circuit, Nail. Bur. Std. (U.S.) Res Papers vol. 15,
no. 351, pp. 367385, 1919.
4. Levirie, I.: High Input Impedance Transistor Circuits, Electronics, vol. 33, pp. 5054,
September, 1960.
5. James, J. R.: Analysis of the Transistor Cascode Configuration, Electron. EnQ>
vol. 32, pp. 4448, 1960.
6. Slaughter, D. W.: The Emittercoupled Differential Amplifier. IRE Trans. Circuit
Theory, vol. CT3, pp. 5153, 1956.
Middlebrook, R. D.: Differential Amplifiers, John Wiley & Sons Inc New York,
1963.
7. Hoffait, A. H., and R. D. Thornton: Limitations of Transistor DC Amplifi"* 9 '
Proc. IEEE, vol. 52, no. 2, pp. 179184, February, 1964.
t Fairchild Semiconductor Corporation, Sprague Electric Co., Texas Instruments, Ii» c "»
and Motorola, Inc.
13 /THE HIGHFREQUENCY
TRANSISTOR
At low frequencies it is assumed that the transistor responds instantly
to changes of input voltage or current. Actually, such is not the case
because the mechanism of the transport of charge carriers from emitter
to collector is essentially one of diffusion. Hence, in order to find out
how the transistor behaves at high frequencies, it is necessary to
examine this diffusion mechanism in more detail. Such an analysis 1
is complicated, and the resulting equations are suggestive of those
encountered in connection with a lossy transmission line. This result
could have been anticipated in view of the fact that some time delay
must be involved in the transport of charge across the base region
by the diffusion process. A model based upon the transmissionline
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.
131
THE HIGHFREQUENCY T MODEL
..
Experience shows that, as a first reasonable approximation, the dif
fusion phenomenon can be taken into account by modifying the basic
commonbase T model of Fig. 1119 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. 131. 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 lowfrequency alpha is designated by a„.
If an input current step is applied, then initially this current is
363
364 / ELECTRONIC DEVICES AND CIRCUITS
Sw, J 3.
Fig. 131 Transistor T
model at high fre
quencies.
bypassed by C t and t'i remains zero. Hence the output current starts at
zero and rises slowly with time. Such a response is roughly what we expect
because of the diffusion process. A better approximation is to replace C, and
r, by a lumped transmission line consisting of resistancecapacitance 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 reversebiased,
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 Highfrequency 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. 131,
h
or
where
liK =
It*
1/K+joC.
Um
l+jf/fa
1
2KT'jC,
(131)
(132)
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.
(133)
5*
132
then, from E( l t 13 " 1 ).
a„
i+tf/U
THE HIGHFREQUENCY TRANSISTOR / 365
(134)
The magnitude of the complex or highfrequency 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. 131). Hence Eq. (134) and the equivalent
circuit of Fig. 131 are valid at frequencies which are appreciably less than /«,
(up to perhaps f a /2). Generalpurpose transistors have values of f a in the
range of hundreds of kilohertz. Highfrequency 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.
131 reduces to the circuit of Fig. 132, which is known as the approximate
CB highfrequency model. The order of magnitudes of the parameters in Fig.
132 are
t, « 20 Si r» « 100 Q r«lM
C e « 150 pF and C. m 3010,000 pF
132 THE COMMONBASE SHORTCIRCUIT^CURRENT
FREQUENCY RESPONSE
Consider a transistor in the eommonbase configuration excited by a sinusoidal
current I, of frequency /. What is the frequency dependence of the load cur
rent l L under shortcircuited conditions? If terminals C and B are connected
together in Fig. 132, then rw, r' £ , and C t are placed in parallel. Since r e » r».,
o„»i
I,
r@~ i
f, 9 132 The approximate high
•Ayv/^—. — o — <AA/V^' o
fre
quency T model.
c.
' r bH
C c
'«t
366 / ELECTRONIC DEVICES AND CIRCUITS
s «* 133
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. 132. With these simplifications, I L — aj h or from Eqg,
(133) and (134), the commonbase shortcircuit current gain is given by
A. = l^ — a °^
1 e 1*
i + if//.
The magnitude of a and its phase angle 8 are given by
<*o
vi + (f/f a y
e =
f
— arctan ~
(135)
(136)
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 ZdB frequency of the CB shortcircuit
current gain. Equation (136) also predicts that a has undergone a 45° phase
shift in comparison with its lowfrequency value. This calculated amplitude
response is in close agreement with experiment, but the phaseshift calculation
may well be far off.
The reason for the discrepancy is that our lumpedcircuit 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. (135) becomes
i+iCf//)
^—jmflfa
(137)
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.
133 THE ALPHA CUTOFF FREQUENCY
Obviously, for highfrequency 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. 133, which represents the
injected hole concentration vs. distance in the base region of a pnp transi
tor. The base width W is assumed to be small compared with the diff us 10
S#
133
THE HIGHFREQUENCY TRANSISTOR / 367
length La °f the minority carriers. Since the collector is reversebiased, the
injected charge concentration P at the collector junction is essentially zero
(Fig 924). 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. 133. The stored
base charge Qb is the average concentration P(0)/2 times the volume of the
base WA (where A is the base crosssectional area) times the electronic
charge e; that is,
Qb = %P(0)AWe
The diffusion current is [from Eq. (532)]
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. (138) and (139),
IW %
(138)
(139)
Com
(1310)
The emitter diffusion capacitance Cn, is given by the rate of change of Qb
with respect to emitter voltage V, or
Cd« =
dQi
W* dl
dV 2D B dV
W 2 1
2D s r' t
(1311)
where r[ = dV/dl is the emitterjunction incremental resistance. From Eq.
(641) 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. (334)]. Hence
Cue —
2D B V T
(1312)
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
"8 133 Minoritycarrier charge distribution in the p ' )
b Q
Se region.
* =
x = W
368 / aECTRONfC DEVICES AND CIRCUITS
s « T3.4
Dependence of /„ upon Base Width or Transit Time From Eqs. (132)
and (1311), and since C, » Cz>., then
(1313)
This equation indicates that the alpha cutoff frequency varies inversely as the
square of the base thickness W. For a pnp germanium transistor with
W = 1 mil = 2.54 X 10~ 3 cm  25.4 microns
Eq. (1313) predicts an /. = 2.3 MHz.
An interesting interpretation of w a is now obtained. By combining Eqs.
(1310) and (1313),
I = Q B V e
(1314)
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*
(1315)
From Eqs. (1314) and (1315) we have that w a = 1/fe, or that the alpha
cutoff (angular) frequency is the reciprocal of the base transit time.
134 THE COMMONEMITTER SHORTCIRCUITCURRENT
FREQUENCY RESPONSE
The T model of Fig. 132 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 shortcircuit current gain A it is obtained by shorting the collector terminal
C to E as indicated in Fig. 134. Since r e » r. and C. » C c , we may omit
the parallel elements r' e and C c> and then I L = a„Ii — al„ But from KCL,
«0 il = (XI t
r€h
O W\/ O 'HVS/V*
■="•■ i r'
Fig. 134 The T circuit in the CE configura
tion under shortcircuit conditions.
$*. 13 5
THE HIGHFREQUENCY TRANSISTOR / 369
j L = I b + I t , so that 1.(1 — a) = —I b . Finally,
a(u)
A  ? L  *!> _
" " h ~ ~h " 1
a(w)
Using Eq. (134), A ie may be put in the form
A
where
and
jAi*
ft =
i+jy/A
«o
1 
a.
U = /«(i — <*•)
(1316)
(1317)
(1318)
(1319)
At zero frequency the CE shortcircuit current amplification is ft *= A /e and
the corresponding CB parameter is a m h /h . Hence Eq. (1318) is con
sistent with the conversion in Table 113.
The CE 3dB frequency, or the beta cutoff frequency, is/* (also designated
/*/. or/„). From Eqs. (1318) and (1319)
h — h/Jff = Ctofa
(1320)
1
Since a, is close to unity, the highfrequency 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 socalled shortcircuitcurrent gainbandwidth product (amplification
times 3dB frequency) is the same for both configurations.
'35 THE HYBRIDPI (II) COMMONEMITTER TRANSISTOR MODEL 4
fn Chap. 11 it is emphasized that the commonemitter 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. 131 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. 134 (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 hybridXL, or Giacoletto, model, which does not have the
°ve defects, is indicated in Fig. 135. 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 136) from the lowfrequency h parameters. All parameters (resistances
capacitances) in the model are assumed to be independent of frequency.
370 ELECTRONIC DEVICES AND CIRCUITS
B r *»
f rfi
See. T3.5
Fig. 135 The hybridn
model for a transistor
in the CE configuration.
They may vary with the quiescent operating point, but under given
conditions are reasonably constant for smallsignal swings.
Discussion of Circuit Components The internal node B' is not physically
accessible. The ohmic basespreading 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
minorityearner concentration injected into the base is proportional to
and therefore the resulting smallsignal 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. 135.
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 excessminoritycarrier storag
the base is accounted for by the diffusion capacitance C, connected beti
B' and E (Sec. 133).
The Early effect (Sec. 97) indicates that the varying voltage :
collectortoemitter junction results in basewidth modulation. A chant
the effective base width causes the emitter (and hence collector) curren
change because the slope of the minoritycarrier 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 collectorjunction barrier capacitance is included in C t . S^^H
times it is necessary to split the collectorbarrier capacitance in '
and connect one capacitance between C and B f and another betweei
B. The last component is known as the overlapdiode capacitance.
itlK
Hybridpi Parameter Values Typical magnitudes for the elemei
the hybridpi model for a germanium transistor at room temperature and io r
Ic = 1.3 niA are
g m = 50 mA/V r w = 100 fi tv« = 1 K
r b . c = 4 M r ce  80 K C t = 3 pF
That these values arc reasonable is justified in the following section.
C, = 100 pF
THE HIGHFREQUENCY TRANSISTOR 371
136
HYBRIDPI CONDUCTANCES IN TERMS OF
LOWFREQUENCY h PARAMETERS
ty'c now demonstrate that all the resistive components in the hybridpi model
cft n he obtained from the k parameters in the CE configuration. These h
parameters are supplied by the manufacturers or can be easily measured
(Ch:
Transistor Transconductance g m Figure 136 shows a pnp transistor in
the CE configuration with the collector aborted to the emitter for timevarying
signals. In the active region the collector current is given by Eq. (97),
;ed here for convenience, with ay = a„:
IC = I CO — Ct e Is
The transconductance g m is denned by
die 1 SI B
gm ■
dT B'B l^ c «
dV S 'E a ° dl
dis_
E
(1321)
i above we have assumed that ay is independent of V E , For a pnp
transistor Vg = — Vb>s as shown in Fig. 136. If the emitter diode resistance
w t\ (Fig. 132), then r, = dV s /dI B , and hence
9m = —
r'
To evaluate r t , note from Eq. (919), with V c ~ —V C c, that
Ib = a U € v * lv T — an — aii
(1322)
(1323)
At <utoff, V E is very negative and Ig ™ — an — an. Since the cutoff current
* very small, we neglect it in Eq. (1323). Hence
and
: an*
91 E
aV E
a U i v * ,Vr Is
V,
V
(1324)
8 13d Pertaining to the derivation of
B r w>*
o J WV
372 / ELECTRONIC DEVICES AND CIRCUITS
Substituting Eq. (1324) in Eq. (1322), we obtain
ObIe Ico — Ic
Qm =
V,
S«. I3. 4
(1325)
For a pnp transistor I c is negative. For an npn transistor I c is positive
but the foregoing analysis (with V s = + V b >b) leads to g m = (I c — Ico)/V T .
Hence, for either type of transistor g m is positive. Since /<? » \Ico\, then
g m is given by
9*
V T
(1326)
where, from Eq. (334), 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
(1327)
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. 137o we show the hybridpi model
valid at low frequencies, where all capacitances are negligible. Figure 1376
represents the same transistor, using the Aparameter equivalent circuit.
From the component values given in Sec. 135, we see that ?v e » r b '„
Hence I b flows into r h > e and W, » I b r b >,. The shortcircuit collector current
is given by
I« = gmVb'M « gvJhfi'.
Fig. 137 (a) The hybridpi model
at low frequencies, (b) the Apararn
eter model at low frequencies.
Sec. 136
THE HIGHFREQUENCY TRANSISTOR / 373
The shortcircuit current gain h f , is defined by
J. I
h/ e —
hlVm
— gmn't
or
 hit — V*Y r .
or
(1328)
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. ll5o 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 Tmodel emitter resistor r t as
r _ */« _ <
Tb ' r ~ Z — t
g m 1 — a
(1329)
The Feedback Conductance gv. With the input opencircuited, h re is
defined as the reverse voltage gain, or from Fig. 13 7a with lb = 0,
?v,
(1330)
or
JV,(1 — k rt ) = h T jTi>c
Since K» <K 1, then to a good approximation
Tb'» = hrtTb'c or g b 'e = h Tt g bta (1331)
Since h n » 10~«, Eq. (1331) 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 Basespreading 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. (1331), we have »v,]»v e m r h >„ and hence
hie = Tbb 1 + rt*.
(1332)
or
r»' = hi, — fb't (1333)
Incidentally, note from Eqs. (1328) and (1332) that the shortcircuit input
"npedance A,, varies with current and temperature in the following manner:
l 1 hf e V T
h ie = rw + Vt
(1334)
The Output Conductance g ee With the input opencircuited, this con
ductance is defined as h oe . For I b «■ 0, we have
L =
+
Tb'c + r b '.
+ g m V b
(1335)
374 / ELECTRONIC DEVICES AND CIRCUITS
Sec. ?3*
With h = 0. we have, from Eq, (1330), W« — h rt V e „ and from Eq
(1336), we find
Ho* — T7 I T T 9mnrc
' e* ' et i o c
(133ft)
where we made use of the fact that rv„ S> r»«,. If we substitute Eqs. (1328)
and (1331) in Eq. (1336), we have
h ot — get + 06V + ff6'J»/ e
ff« = h ae — (1 + h ft )9b'e
(1337)
iiis equation may be put in the form [using Eqs. (1329) and
(1331)]
0« « & oe — 0mA r
(1338)
Summary If the CE ft parameters at low frequencies are known at a
given collector current Ic, the conductances or resistances in the hybrid11
circuit are calculable from the following five equations in the order given:
9m =
TV, =
fkb'
v T
hi
5h ' e = /v:
hf, — r !r
(1339)
Tb'e
h rt
or
ffi'e =
ry
?« = A„f  (1 + A/«)ff6'c « —
For the typical h parameters in Table 112. at Ic = 1.3 mA and room tempera
ture, we obtain the component values listed on page 370.
The Hybridpi Capacitances The collectorjunction 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 reversebiased, then C* is a transition capacita
and hence, varies as Fes  ", where n is ^ or ^ for an abrupt or gradual junction,
respectively (Sec. 69).
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. 133). Experimentally, C, is deter
mined from a measurement of the frequency jV at which the GE short circu 1
Sec
A
137
THE HIGHFREQUENCY TRANSISTOR I 375
Fig. 138 The hybridII circuit for a single transistor with a
resistive load R r ..
jurrent gain drops to unity. We verify in Sec. 137 that
C «*
9m
2rfr
(1340)
Reasonable values for these capacitances are
  3 pF C t = 100 pF
137 THE CE SHORTCIRCUIT CURRENT GAIN OBTAINED
WITH THE HYBRIDPI MODEL
Consider a singlestage CE transistor amplifier, or the last stage of a cascade.
The load ft L on this stage is the collectorcircuit 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 hybridII model of Fig. 135, which is repeated for
convenience in Fig. 138. 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. 139. A current source furnishes a
sinusoidal input current of magnitude /;, and the load current is /j,. We have
Neglected 06v which should appear across terminals B'C, because 0&< c « g B  e .
And of course g et disappears, because it is in shunt with a short circuit. An
af Ulitional approximation is involved, in that we have neglected the current
l
'9 139 Approximate equivalent
Clrcuit for the calculation of the
■tortcircuit CE current gain.
8v
HY.
^ ^ C e + C c
J>
o—J
I"
376 / ELECTRONIC DEVICES AND CIRCUITS
S»c. 13.7
delivered directly to the output through g b > e and C c . We see shortly that this
approximation is justified.
The load current is II = — g^V**, where
(1341)
Vb '' g b >. + MC + C.)
The current amplification under shortcircuited conditions is
a = l± = ~ g"
* h J^+MC + CJ
Using the results given in Eqs. (1339) , we have
— h ft
Ai =
1+X//M
(1342)
(1343)
where the frequency at which the CE shortcircuit current gain falls by 3
dB is given by
/,
U<y
I
0.
2r(C, + C e ) h f , 2rr(C, + C.)
(1344)
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 lowfrequency shortcircuit CE current gain. The expression
for ft obtained in Sec. 134 from the highfrequency T model is essentially
the same as that given in Eq. (1344). (See also Prob. 1312.)
Since, for a singletimeconstant circuit, the 3dB frequency ft is given
by ft = 1/2jtRC, 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. (1344).
The Parameter ft We introduce now/r, which is defined as the frequency
at which the shortcircuit commonemitter current gain attains unit magnitude.
Since h /t » 1, we have, from Eqs. (1343) and (1344), that ft is given by
St • h ie ft =
since C„ » C,
Aim
9m _ Qm
2ir(C. + C t ) ~ 2*C t
Hence, from Eq. (1343),
l+JA/.(f/M
(1345)
(1346)
The parameter ft is an important highfrequency 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. 1310.
$*. 137
THE HIGHFREQUENCY TRANSISTOR / 377
fig, 1310 Variation of f T with
collector current.
/V. MHz
400 
300 
200
V„5V
r=25°C
1 10 100
I c (log scale), mA
Since ft "* hf t ft, this parameter may be given a second interpretation.
It represents the shortcircuit currentgainbandwidth product; that is, for the
CE configuration with the output shorted, ft is the product of the lowfrequency
current gain and the upper 3dB frequency. For our typical transistor (page
370), ft = 80 MHz and ft = 1.6 MHz. It is to be noted from Eq. (1345)
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. 1311, 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)=201ogj4,
20 log V
6dB/octave = 20dB/decade
log L
log/r log/
Fig. 1311 The shortcircuit CE current gain vs. frequency (plotted
on a loglog scale).
378 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 133
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. (1345), using the typical values of Fig. 138,
g m
g m
C. + C c
0.03
In a similar way the current delivered to the output through g b  c may be shown
to be negligible.
The frequency /r is often inconveniently high to allow a direct experimental
determination of f T . However, a procedure is available which allows a
measurement of /r at an appreciably lower frequency. We note from Eq.
(1343) 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 = /iA;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 hybridII circuit. From Eq. (1345),
C =
gm
2KJT
(13473
From Eqs. (1320) and (1345), 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 excessphase factor for a in Eq. (137) «
taken into consideration.
138 CURRENT GAIN WITH RESISTIVE LOAD
To minimize the complications which result when the load resistor Rl in *' l &'
138 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. 129. We identify F 6 <« with Vi
in Fig. 1214 and V ce with 7 8 . On this basis the circuit of Fig. 138 may b 6
replaced by the circuit of Fig. 1312a. Here K m VJV b > e . This circuit «
still rather complicated because it has two independent time constants, on
THE HIGHFREQUENCY TRANSISTOR ' 379
fig. 1312 (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. 1312a. 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. 13126. 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 collectorcircuit resistor, r ce and rv*.
By inspection of Pig. 13126, 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. 1312a is reduced to that shown in Fig. 13126.
The load resistance Rl has been restricted to a maximum value of 2 K
because, at values of Rl much above 2,000 ft, the capacitance C c (l + g*ML)
becomes excessively large and the bandpass correspondingly small.
Now let us return to the capacitance C e (K — 1)/K ^ C c , which we
Neglected above. For R L  2,000 Q t
RlC c = 2 X 10 3 X 3 X 10" 1Z = 6 X lO 8 sec  6 nsec
380 / ELECTRONIC DEVICES AND CIRCUITS
The input time constant is
S«.
)39
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. 13126 is different from the circuit of Fig. 139 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 lowfrequency current gain Ai„
uuder load is the same as the lowfrequency gain A* with output shorted.
Therefore
Alo = —hf«
However, the 3dB frequency is now / 2 (rather than //j), where
gv
where
139
n 2irr b >£ 2ttC
C = C. + C.(l + gM
TRANSISTOR AMPLIFIER RESPONSE, TAKING
SOURCE RESISTANCE INTO ACCOUNT
(1348)
(1349)
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
131 3a, or by its Thevenin's equivalent, as in Fig. 1313&. 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. (1328). Therefore the lowfrequency current gain, taking the load
■"CD
•*l
Fig. 1313 (o) A transistor is driven by a generator of resistance R, which is
represented by its Norton's equivalent circuit, (fa) The generator is represented
by its Thevenin's equivalent.
$tc
1 39
THE HIGHFREQUENCY TRANSISTOR / 381
and source impedances into account, is
j. lL __ Jl Ii _ i Rt __ — kfeRt
At* J  jr* j  ft/. ^ _j_ ^ + ^ ~ Rt + hu
(1350)
since A« = r tb> + rv* Note that Ai, is independent of Rl. The 3dB fre
quency is determined by the time constant consisting of C and the equivalent
resistance R shunted across C. Accordingly,
fr
1
2ttRC
(1351)
where C is given by Eq. (1349), and R is the parallel combination of R t + r&
and tb'; namely,
n s {R. + rw}ry,
R, + h it
(1352)
From Eq, (1139) we have that the voltage gain Ar» at low frequency,
taking load and source impedances into account, is
Rl _ — hftRt
R, R, f hit
(1353)
Note that A Yso increases linearly with Rl. The 3dB frequency for voltage
gain Ay is also given by Eq. (1351). Note that /s increases as the load
resistance is decreased because C is a linear function of Rl At Rl ■= 0, the
3dB frequency is finite (unlike the vacuumtube amplifier, which has infinite
bandpass for zero platecircuit resistance; Sec. 166) and from Eq. (1347) is
given by
J _ h h
QmR
ft*
R L =
(1354)
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 3dB frequencies than those
indicated above will be obtained.
The equality in 3dB 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 (1351) applies in both
cases provided that, for Av, we use R — Rv, where, from Eq. (1352) with
ft.  0,
R v = ***»« m &&1 (1355)
Tbb' + Tb' e hit
and for A T we use R = R h where, from Eq. (1352) with R, = « ,
Rt = n,, (1356)
h 't>ce R Y «i? /( the 3dB frequency f 2V for an ideal voltage source is higher
ban fa for an ideal current source.
382 / ELECTRONIC DEVICES AND CIRCUITS
7 39
S<K
139
THE HIGHFREQUENCY TRANSISTOR / 383
The GainBandwidth Product This product is found in Prob. 1318 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«>
( 1358)
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. 1314. 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. 1314 for R t = corresponds to idealvoltagesource
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 vacuumtube stage of amplification, the gainbandwidth
product is a useful number (Sec. 166). For a transistor amplifier con
lA.o/il.MHz
300
300
100
ft , MHz
16
19
14 w
./.■
8
^R, = 50
_J
11
Vsi
.=0
4
50q
90.9
36011
ron"*
87.0 2. 1 7
83.3 4.17
t
«£■=■*•
74. 1 9.25
50
500 1,000 1,500 2,000 J?/.,Q
Fig. 1314 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. 135, Also, the gainbandwidth product for a 5011
source is plotted. The tabulated values of .4k.. correspond to Rl =
2,000 fi and to the values of R. on the curves. The values of /li„ ore inde
pendent of R L .
1
j n c of a single stage, however, the gainbandwidth product is ordinarily not a
il parameter; it is not independent of R, and Rl and varies widely with
both The currcntgainbandwidth product decreases with increasing R !t and
increases with increasing R„. The voltagesain bandwidth product inert
increasing Rl and decreases with increasing R a . Even if we know the
gainbandwidth 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 gainbandwidth product will no longer be the same as it had been.
Summary The highfrequency response of a transistor amplifier is
obtained by applying Eqs. (1349) to (1353). 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
highfrequency transistor.
From the operating current Ic and the temperature T, the transconduct
ance is obtained [Eqs. (1339)] as g m p PM/F* and is independent of the par
ticular device under consideration. Knowing g m we can find, from Eqs.
(1339) and (1340),
TW — hit — TV* l"* "* ^T
Wt =
If R, and R L are given, then all quantities in Eqs. (1349) to (1353) are known.
We have therefore verified that the frequency response may be determined
from the four parameters hj, t f T , hu, and C*
REFERENCES
1. Phillips, A. B.: "Transistor Engineering," chaps. 13 and 14, McGrawHill Book
Company, New York, 1962.
Pritchard, R. L.: Electricnetwork Representations of Transistors: A Survey, IRE
Trans, Circuit Theory, vol. CT3, no. 1, pp. 521, 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 Shortcircuit Current Gain and
Phase Determination, Proc. IRE. vol. 46, no. 6, pp. 11771184, June, 1958.
3  Phillips, A. B.: "Transistor Engineering," pp. 129130, McGrawHill Book Com
pany, New York, 1962.
*• Giacoletto, L. J.: Study of pnp Alloy Junction Transistors from DC through
Medium Frequencies, RCA Rev., vol, 15, no. 4, pp. 506562, December, 1954.
Searle, C. L., A. B. Boothroyd, E. J. Angelo, Jr., P. E. Gray, and D. 0. Pederson:
"Elementary Circuit Properties of Transistors," vol. 3, chap. 3, Semiconductor
Electronics Education Committee, John Wiley & Sons, Inc., New York, 1964.
14 /FIELDEFFECT
TRANSISTORS
The fieldeffect 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 fieldeffect transistors, the junction fieldeffect transis
tor (abbreviated JFET, or simply FET) and the insulatedgate field
effect transistor (IGFET), more commonly called the metalox idcsemir
conductor {MOS) transistor (.MOST or MOSFET).
The principles on which these devices operate, as well as the
differences in their characteristics, are examined in this chapter.
Representative circuits making use of FET transistors are also
presented.
The FET enjoys several advantages over the conventional
transistor:
1. Its operation depends upon the flow of majority carriers only.
It is therefore a unipolar (one type of carrier) device. The vacuum
tube is another example of a unipolar device. The conventional tran
sistor is a bipolar device.
2. It is relatively immune to radiation.
 It exhibits a high input resistance, typically many meg
ohi
4. It is less noisy than a tube or a bipolar transistor.
5. It exhibits no offset voltage at zero drain current, and hence
makes an excellent signal chopper. 2
6. It has thermal stability (Sec. 144).
The main disadvantage of the FET is its relatively small gain
bandwidth product in comparison with that which can be obtained
with a conventional transistor.
384
Sec.
M1
FIELDEFFECT TRANSISTORS / 3B5
141
THE JUNCTION FIELDEFFECT TRANSISTOR
I
The structure of an nchannel fieldeffect transistor is shown in Fig. 141.
Ohmic contacts are made to the two ends of a semiconductor bar of ntype
material (if ptype silicon is used, the device is referred to as a pchannel 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 draintosource voltage is called Vns, and is positive if D is more positive
than S.
Gate On both sides of the ntype bar of Fig. 141, heavily doped (p + )
regions of acceptor impurities have been formed by alloying, by diffusion, or
by any other procedure available for creating pn 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 pn junction. Conventional cur
rent entering the bar at G is designated Iq.
Channel The region in Fig. 141 of ntype 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 reversebiased pn junction there are spacecharge regions
(Sec. 69). 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. 141
w iH become progressively decreased with increasing reverse bias. Accordingly,
'°r a fixed draintosource voltage, the drain current will be a function of the
Aversebiasing 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. 142. The direction of the arrow at the gate
3&6 ELECTRONIC OF/ICES AND CIRCUITS
Sec. J4.fi
Depletion region
D
Drain
Fig. T41 The basic structure of an n channel field effect tran
sistor. The normal polarities of the draintosource and gateto
source supply voltages are shown. In a pchannel FET the volt
ages would be reversed.
of the junction FET in Fig. 142 indicates the direction in which gate current
would flow if the gate junction wore forwardbiased. The commonsource
drain characteristics for a typical nehannel FET shown in Fig. 143 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 ntype 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. 142 Circuit symbol for an
^channel FET. (For a pchanne)
FET the arrow at the gate junc
tion points in the opposite direc
tion.) For an nchannel FET, Id
and V m; are positive and Vos S
negative. For a pchannel FET, to
and P'ti.s are negative and Vg* iS
positive.
Vcc,^
*! V n
Sec
U1
FIELDEFFECT TRANSISTORS , 387
+ 0.5 !_ 
0.5

1 / /
^~"*'
1.0
2
1.5
 2.5 1
3.oi
3.5
.
4.0
i it 15 20
Drain source voltage V !ls
lL
3H
Fig. 143 Commonsource drain characteristics of an
nchannel fieldeffect transistor. (Courtesy Texas
Instruments, Inc.)
constriction is not uniform, but is more pronounced at distances farther from
the source, as indicated in Fig. 141. Eventually, a voltage V ds is reached
at which the channel is "pinched off." This is the voltage, not. too sharply
defined in Fig. 143, 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 constantcurrent 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, pinchoff will occur for smaller value ..■:[. and the maxi
mum drain current will be smaller. This feature is brought out in Fig. 143.
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 91 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. 612)
^oss the gate junction. From Fig. 143 it is seen that avalanche occurs at
a lower value of  V D s\ when the gate is reversebiased than for V G s = 0. This
388 / ELECTRONIC DEVICES AND CIRCUITS
s «. ]4. a
is caused by the fact that the reversebias gate voltage adds to the drain volt
age, and hence increases the effective voltage across the gate junction.
We note from Fig. 142 that the nehannel FET requires zero or negative
gate bias and positive drain voltage, and it is therefore similar to a vacuum
tube. The pchannel 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 fieldeffect 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 semiconductordevice
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. 141 is not prac
tical because of the difficulties involved in diffusing impurities into both sides
of a semiconductor wafer. Figure 144 shows a singleen dedgeometry junc
tion FET where diffusion is from one side only. The substrate is of ptype
material onto which an ntype channel is epitaxially grown (Sec. 152). A
ptype gate is then diffused into the ntype 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 ntype channel.
142
THE PINCHOFF VOLTAGE V M
We derive an expression for the gate reverse voltage Vp that removes all the
free charge from the channel using the physical model described in the pre
ceding section. This analysis was first made by Shockley, 1 using the structure
of Fig. 141. In this device a slab of ntype semiconductor is sandwiched
between two layers of ptype material, forming two pn junctions.
Assume that the ptype region is doped with N A acceptors per cubic meter,
that the ntype region is doped with No donors per cubic meter, and that the
Di a: ii
Gate
Source
Channel
Fig. 144 Singleendedgeometry
junction FET.
S*
142
FIELDEFFECT TRANSISTORS / 389
junt'tJ° n formed is abrupt. The assumption of an abrupt junction is the same
£9 that made in Sec. 69 and Fig. 612, and is chosen for simplicity. More
over, if Na 2> N D , we see from Eq. (644) that W p « W n) and using Eq. (647),
ffC have, for the spacecharge width, W n (x) = W{x) at a distance x along the
channel in Fig. 141 :
W(x) =ab(x) = \^W.  7(»)l}*
(141)
where e — dielectric constant of channel material
e = magnitude of electronic charge
V„ = junction contact potential at x (Fig. 6ld)
V(x) = applied potential across spacecharge 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. 141)
If the drain current is zero, b{x) and V(x) are independent of x and
b{x) = b. If in Eq. (141) we substitute b(x) = b = and solve for V, on the
assumption that \V„\ << \V\, we obtain the pinchoff 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
(142)
If we substitute V G s for V and a — 6 for x in Eq. (646), we obtain, using
Eq. (142),
V 0S = (l^V,
(143)
The voltage V G s in Eq. (143) represents the reverse bias across the gate
junction and is independent of distance along the channel if I D = 0.
EXAMPLE For an wchannel silicon FET with o = 3 X 10~* cm and N D  10 1S
electrons/cm 3 , find (c) the pinchoff voltage and (fa) the channel halfwidth for
Vos  %V P and l a = 0.
Solution a. The relative dielectric constant of silicon is given in Table 51 as
12, and hence e = 12e„. Using the value of e and e„ from Appendixes A and B,
^e have, from Eq. (142), 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. (143) for b, we obtain for V gs — \Yp
6ol ( ~ j  (3 X 10^*) [1  (m = 0.87 X 10" cm
Hence the channel width has been reduced to about onethird its value for Vos ~ 0.
390 / ELECTRONIC DEVICES AND CIRCUITS
Sec.
"3
U3
THE JFET VOLTAMPERE CHARACTERISTICS
Assume, first, that a small voltage V&s is applied between drain an
'ting small drain current Id will then have no appreciable enV
the channel profile. Under these c consider the
channel cross section A to be constant throughout its length. Hence A ~ ^H
when be channel width corresponding to zero drain current as give
Eq. (143) for a specified Vas, and w is the channel dimension perpendicular to
the b direction, as indicated in Fig. 141.
Bince no current {Lows in the depletion region, then, using Ohm's law
[Eq. (51)], 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. (143) in Eq. (144), we have
2aweN D n„
Id =
I
»[>mh
The on* Resistance rj(o.v) Equation (14.").) describes the voltai
characteristics of Fig. 143 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. (145), with Vas = 0.
r<t( on) = 7
I
2aweN Diu
For the alues given in the illustrative example in this section
with L/to = 1. we find that r rf (o!v) = 3.3 K. For the dimensions and con
tralion used in commercially available I'LTs and MOSFETs (Sec. i ;
values of r d (0N) ranging from about 100 to to 100 K arc measured. Th«
parameter is important in switching applications where the FET
heavily ox. The bipolar transistor has the advantage over the fieldc
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. 914), whereas the I
characteristics pass through the origin, Id = and V D s = 0.
The Pinchoff 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 reversebiased 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. 141. If the convergence of the depl 6 "
tiou region is gradual, the previous onedimensional analysis is valid 1 in
thin slice of the channel of thickness Ax and at a distance x from the source
Sec
143
FIELDEFFECT TRANSISTORS I 391
Subject to this condition of the "gradual" channel, the current may be written
pection of Fig. 141 as
I D = 2b(x)weN Dfin S x (147)
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 pinchoff (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 ntype 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 pinchoff,
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. (147) we now see that both Ip and b remain constant, thus explain
ing the constantcurrent portion of the VI characteristic of Fig, 143.
What happens* if Vds is increased beyond pinchoff, 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.
145, the velocitylimited region U increases with V DS , whereas 5 remains at
a fixed value.
The Region before Pinchoff We have verified that the FET behaves as
an ohmic resistance for small V D a and as a constantcurrent device for large
*ds An analysis giving the shape of the voltampere 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. 145 After pinchoff,
as ' , is increased, then I J
'""eases but 5 and I D re
main essentially constant.
"' a nd Gi are tied to
other.)
+ ll~ or
Depletion region
I 1 ? Gl
> —
26(.r) ^Cs^is.
H7
^w*
^SM^lm
f*
I, x .1
ll *
'1 +
392 / afCTRONJC DEVICES AND CIRCUITS
U4
varies with 6«~* for larger values of & x (before pinchoff). 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 pinchoff (also called the constantcurrent
pentode, or currentsaturation 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
(148)
This simple parabolic approximation gives an excellent fit, with the experi
mentally determined transfer characteristics for FETs made by the diffusion
process.
Cutoff Consider a FET operating at a fixed value of Vds in the pentode
region. As V s is increased in the direction to reversebias the gate junction,
the conducting channel will narrow. When Vos = Vp, the channel width is
reduced to zero, and from Eq. (147), 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 gatetosource 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.
144
THE FET SMALLSIGNAL MODEL
The linear smallsignal 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 timevarying
and dc currents and voltages as used in Sees. 79 and 913 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)
(149)
The Transconductance g n and Drain Resistance r$ We now proceed ft 8
in Sec. 8^. If both the gate and drain voltages are varied, the change i°
i
S*
144
REIDEFFECT TRANSISTORS / 393
drain current is given approximately by the first two terms in the Taylor's
series expansion of Eq. (149), or
a ■ d»z> I . . diD I A
Md = ~£T~ „ Av OS + T L &V DS
OVqs l v *a OVds \ v °*
(1410)
jn the smallsignal notation of Sec. 81, Atj> = id, Avqs = f ff „ and Avds = v<u,
so that Eq. (1410) becomes
id  gmVf H — Vd.
where
g m =
Bid I
At'p j
— I
V„ \Vbs
(1411)
(1412)
is the mutual conductance, or transconductance. It is also often designated
by y/t or g f , and called the (commonsource) forward tran&admittance. The
second parameter rd in Eq. (1411) is the drain (or output) resistance, and is
defined by
dVps I _ &VPS I _ Vdt I
Bio \ v <>* Aio Was id l^o*
Td =
(1413)
The reciprocal of r& is the drain conductance g^. It is also designated by y ot
and g , and called the (commonsource) output conductance.
The parameters g m and r* are completely analogous to the vacuumtube
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. 84, we verify that ft, u, and g m are related by
(1414)
M = Tdg„
(1415)
A circuit for measuring g m is given in Fig. 146a. It follows from Eq.
(1412) that (if \V t \ « V DD> so that Vds = const)
9m " Vi
Vt/R*
Vr
V x Rd
(1416)
Similarly, the circuit of Fig. 1466 allows r d to be measured. From Eq. (1413)
it follows that
" I d V./Rd V,
(1417)
An expression for g m is obtained by applying the definition of Eq. (1412)
10 Eq. (14^8). The result is
ffm = *» ^1  ^J
(1418)
394 ELECTRONIC DEVICES AND CIRCUITS
Sac, l4^f
144
FIELDEFFECT TRANSISTORS / 395
Oscillator
— , Oscillator ^
&
©
!f
(a)
(6)
Fig. 146 Test circuits for measuring (a) p m and (b) r rf . The rms volt
ages Vj and T, are measured with ac highimpedance voltmeters.
where g„<. is the value of g m for V f ,.s = 0, and is given by
— 2 / 1
<?mo =
r
Siaee/j»s* and V> area! opposil positive. Thi ship,
com has beei experimentally. 1 Since f/™.
vith tin circuit of Fig. 146a = 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. 147 for the 2b
;.: v and th< i FET [with 17 V) The lii
relationship predicted by Eq. (1418
C 100
so
■
f~
10V
l kHz
^■^^^
^•^
\
X T\
f&
^»
*
Fig, 147 Tronsconductance g„ versus
gate voltage for types 2N3277 and
2N3278 FETs. (Courtesy of Fairchild
Semiconductor Company.)
1.0 2.0 3.0 4.0
Gate voltage V GS . V
5.0
l.h
: *
1 J
„
50 tOO 150
Ambient temperature T Al "C
0.8
0.6
50
(«)
50 100
Ambient temperature T A , "C
<6)
Fig. 148 (a) Normalized tronsconductance g„, versus ambient temperature T A
and (b) normalized drain resistance r„ versus 7\ (for the 2N3277 and the 2N3278
FETs with Vbb = 10 V, Yea = V, and /  I kHz). (Courtesy of Fairchild
Semiconductor.)
Temperature Dependence Curves of g m and r d versus temperature are
given in Fig. 148a 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 majoritycarrier current decreases with temperature (unlike the
bipolar transistor whoso minoritycarrier current increases with temperature),
the me phenomenon of thermal runaway (Sec. 1010) is not encount
ritb fieldeffect transistors.
The FET Model We note thai Kq. (1411) is identical with Eq. (813)
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 88 is valid for the FET.
This model is repeated in Fig. 140, with the appropriate change of notation.
In this figure we have also included I deh exist between pairs
°f nodes, i corresponding to the highfrequency triode model of Fig. 819). The
Btacitor C vt represents the barrier capacitance between gate and source, and
"••149 Small signal FET
•ftodel
Gate G o • 1( »
Source So *■
♦ o Drain D
5=* u «i(p < r " SC
i OS
396 / aECTRONIC DEVICES AND CIRCUITS
s « J4.J
TABLE M.I
Range of parameter values for a FET
Parameter
JFET
MOSFETf
ffm
0.110mA/V
. 120 mA/V or more
U
0.11 M
150 K
C*
0,11 pF
0.11 pF
"an "ad
110 pF
110 pF
Tgi
>10* a
>io i ° a
T B d
>io fl a
>10'*fi
t Discussed in Sec. 145.
C e d is the barrier capacitance between gate and drain. The element C&
represents the draintosource capacitance of the channel.
The order of magnitudes of the parameters in the model for a diff used
junction FET is given in Table 141. Since the gate junction is reverse
biased, the gatesource resistance r a , and the gatedrain resistance r gd are
extremely large, and hence have not been included in the model of Fig. 149.
U5
THE INSULATEDGATE FET (MOSFET)
In preceding sections we developed the voltampere characteristics and small
signal properties of the junction fieldeffect transistor. We now turn our
attention to the insulatedgate FET, or metaloxidesemiconductor FET,'
which promises to be of even greater commercial importance than the junction
FET.
The rtchannel MOSFET consists of a lightly doped ptype substrate into
which two highly doped n + regions are diffused, as shown in Fig. 1410. 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 gatemetal area is overlaid on the oxide,
Source Gate(+) Drain
, Aluminum
Fig. 1410 Channel enhancement
in a MOSFET. (Courtesy of
Motorola Semiconductor products,
Inc.)
See
US
FIELDEFFECT TRANSISTORS / 397
covering the entire channel region. Simultaneously, metal contacts are made
*o the drain and source, as shown in Fig. 1410. 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 parallelplate capacitor.
The insulating layer of silicon dioxide is the reason why this device is called
the insulatedgate fieldeffect 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. 1410 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, 1410. The nega
tive charge of electrons which are minority carriers in the ptype 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 ntype 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
enhancementtype MOS.
The voltampere drain characteristics of an «channel enhancementmode
MOSFET are given in Fig. 141 la, and its transfer curve, in Fig. 14llb. 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 gatesource 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. 1411 (a) The drain characteristics, and (fa) the transfer curve (for Vds =
") of an nchannel enhancementtype MOSFET.
398 ELECTRONIC DEVICES AND CIRCUITS
Dili'.
3iO.. channel
Source i Aluminum
t
P (substrain
Enhancement
Fig. 1412 (a) A depletiontype MOSFET. (b) Channel depletion with the appli
cation of a negative gate voltage. (Courtesy of Motorola Semiconductor
Products.'lnc.)
The Depletion MOSFET A second type of MOSFET can be made
the basic structure of Fig. 1410, an n channel is diffused between the source
and the drain, as shown in Fig. M12o. With this device an appreeiab;
current loss flows for zero gatetosource 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. 14126 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 pinchoff
occurring in a JFET at the drain end of the channel (Fig. 141). As a
•. the veil ampere characteristics of the depletionmode 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 rctype channel. In this
manner the conductivity of I he channel increases and the current rises above
Jdss The voitampere characteristics of this device are indicated in J' 1 ?*
1413n, and the transfer curve is given in Fig. 14136. The deplete
enhancement rt ending to Vos negative and positive, n
should be noted. The manufacturer sometimes indicates the galesourct ! I'utojf
voltage Vgs(orr), at which F D is reduced to some specified negligible val
recommended Vos This gate voltage corresponds to the pinchoff vol tag*
V P of a JFET.
The foregoing discussion is applicable in principle also to the /)chan' ie
FIEIDEFFECT TRANSISTORS 399
l B ,mA
Depletion « —
J D (on) = 6
5
4
3
— * Enhancement
^ """l
.. : — . 1 .
t  3  2
V& (OFF)
Vm.V
Fig. 1413 (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
voltampere characteristics of Figs. 1111 and 1413 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. 1414 Some
the symbol of Fig. 142 for the JFET is also used for the MOSFET,
Rtfa the understanding that Gt is internally connected to *S.
Smallsignal MOSFET Circuit Model m If the small bulk resistances of
the source and drain are neglected, the smallsignal equivalent circuit of the
FET between terminals G (= GV), >'. and D is identical with that given in
Fig 149 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 141 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 141 that the input resistance r s , and the feedback resistance r 9i are
v *ry much larger for the MOSFET than for the JFET.
Drain D
OD
6,
o
6 Source S
OS
6 S
(a) (6) (C)
Fig, 1414 Three circuit symbols for a pchannel MOSFET.
400 / HECTRONJC DEVICES AND CIRCUITS Sec. l4
If the substrate terminal G* is not connected to the source, the model of
Fig. 149 must be generalized as follows: Between node ff 2 and S, a diode t}\
is added to represent the pn junction between the substrate and the source
Similarly, a second diode D2 is included between (? s and D to account for the
pn junction formed by the substrate and the drain.
146
THE COMMONSOURCE AMPLIFIER
The three basic JFET or MOSFET configurations are the commonsource
(CS), commondrain (CD), and commongate (CG). The configurations are
shown in Fig. 1415 for a pchannel JFET. Unless specifically stated other
wise, the circuits discussed throughout this chapter apply equally well to
JFETs or MOSFETs.
Voltage Gain The circuit of Fig. 1416o is the basic CS amplifier con
figuration. If the FET is replaced by the circuit model of Fig. 149, we obtain
the circuit of Fig. 14166, which is equivalent to that of Fig. 8196 for a CK
triode amplifier with interelectrode capacitances taken into account. [In Fig.
819 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. 85).] Hence the voltage gain
Av = VJVi for the CS amplifier as given by Eq. (839), which is repeated
here, using FET notation,
Ay =
g m + y
ad
Y l 4 Yd, + Qd + Y 9d
(1420)
where Y L = 1/Z L = admittance corresponding to Zi
Yd* = juCd, = admittance corresponding to Cd,
§d = l/rd = conductance corresponding to Td
Ygd ■« juCgd ~ admittance corresponding to C td
At low frequencies the FET capacitances can be neglected.
Under these con
v^y Output
Input
Output
(<*>
Input
Input
£1
— o
Output
(c)
Fit 1415 The three FET configurations: (a) CS, (b) CD, and (c)
CG.
1
Sec.
U6
FIEIDEFFECT TRANSISTORS / 401
ditions, Yd, = Y a d = 0, and Eq. (1420) reduces to
Qm QmZt.
A v =
Y L +
1 4 g d Z L
= ~9mZ' L
(1421)
where Z' L m r d \\Z L . This equation is identical with Eq. (840).
Input Admittance An inspection of Fig. 14166 reveals that the gate cir
cuit is not isolated from the drain circuit. Since Figs. 14166 and 819 are
identical, the input admittance is given by Eq. (842), or
Y<= Y st +{\~ A v )Y g d (1422)
This expression indicates that for a fieldeffect transistor to possess negligible
input admittance over a wide range of frequencies, the gatesource and gate
drain capacitances must be negligible. Also, as explained in Sec. 812, 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 draincircuit
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. (1422) becomes
5 m d  C„ + (1 + fJKJC*
(1423)
This increase in input capacitance d over the capacitance from gate to source
is caused by the familiar (Sec. 812) Miller effect.
This input capacitance is important in the operation of cascaded ampli
fiers, as is discussed in Sec. 812 in connection with vacuum tubes.
Output Resistance For the commonsource amplifier of Fig. 1416a, the
rrxrr
gwVi
I
s £ V a
i
(a)
(6)
F '9 1416 {a) The commonsource amplifier circuit; (b) smallsignal equiva
le "t circuit of CS amplifier.
402 / afCTRONIC DEWCES AND CIRCUITS
Sec. ?4.y
output resistance R e is given by the parallel combination of r d and Rd, or
TdRd
R a =
Td + Rd
(1424)
Equation (1424) is valid at low frequencies, where the effect of the capacitors
in Fig. 14166 is negligible, and with a resistive load, Zc = Rd.
EXAMPLE A MOSFET has a draincircuit 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. 812. The solution of this example is therefore the same as that given
in Sec. 812. Hence
(Ay).
= 48,6
and
(4v)«„t.u,. = 38.8/143.3'
Sec
ue
FIELDEFFECT TRANSISTORS / 403
jfote that the amplification is positive and has a value less than unity. If
g „R, » 1, then Av « gj(g m + g d ) = n/(u + 1).
Input Admittance The source follower offers the important advantage of
lower input capacitance than the CS amplifier. The input admittance Yi is
given by Eq. (855), or
Yi « juC ed + juCM ~ Av) (1427)
Output Admittance The output admittance Y , with R t considered
external to the amplifier, is given by Eq. (858), or
Y = g m + g d + jtaC T
At low frequencies the output resistance R is
1 1
Ra =
g m + gd g*
(1428)
(1429)
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. 88).
147 THE COMMONDRAIN AMPLIFIER, OR SOURCE FOLLOWER
The CDamplifier connection shown in Fig. 1417 is analogous to the cathode
follower discussed in Sec. 814. The voltage gain of this circuit is given by
Eq. (853), or in FET notation,
(ffm + j(*C g ,)R t
A v =
(1425)
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 (1426)
v 1 + (g m + g d )R t
V„
V,o )\
Fig. 1417 Sourcefollower circuit.
■o V a
148
A GENERALIZED FET AMPLIFIER
The analysis of the CS amplifier with a source resistance R„ the CG con
figuration, and the CD circuit at low frequencies is made by considering the
generalized configuration in Fig. 1418. 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 signalsource 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 .
88
The Output from the Drain From the analysis given in Sees. 86 and
we obtain the Thevenin's equivalent circuit from drain to ground (Fig.
19a) and from source to ground (Fig. 14196). From the former circuit
e conclude that ''looking into the drain" of the FET we see (for smallsignal
^Peration) an equivalent circuit consisting of two generators in series, one of —ft
€s the gatesignal voltage v,* and the second (^ + 1) times the sourcesignal
0e y. and the resistance r d + (ft + 1)R.. Note that the voltage v, and
resistance in the source lead are both multiplied by the same factor, /* + 1 •
tow
th
404 / ELECTRONIC DEVICES AND CIRCUITS
Fig. 1418 A generalized FET amplifier.
o v e3
The CS Amplifier with an Unbypassed Source Resistance From Fig,
1419a, with v, = v a = 0, we obtain for the voltage gain, Ay = foi/»»,
— flRd —{JmRd
Ay =
(1430)
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. (1421), 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* (1431)
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 reversebiased.
The CG Amplifier From Fig. 1419a, 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 (1432)
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. (1431), 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. 14196 :
(1433)
* ■ (ttt) "*■
S«c. H9
FIEIDEFFECT TRANSISTORS / 405
l
The commongate amplifier with its low input resistance and high output
resistance has few applications. The CG circuit at high frequencies is con
sidered in Prob. 1411.
The Output from the Source From Fig. 14196 we conclude that "looking
into the source" of the FET we see (for smallsignal operation) an equivalent
circuit consisting of two generators in series, one of value m/(m + 1) times the
gatesignal voltage $ and the second t/(jt + 1) times the drainsignal 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. 14196, 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. (1426), obtained by setting w =
into the highfrequency formula for Ay. If R d 9* 0, then A 7 in Eq. (1434) 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. 14196,
R =
U
(1435)
P + 1 g m + gd
which agrees with Eq. (1429). The output impedance &„ taking R, into
account, is R' = R \\R,.
149
BIASING THE FET
The selection of an appropriate operating point (I D , V GS , Vz> s ) for a FET
amplifier stage is determined by considerations similar to those given to tubes
r d + R„
s
—o
'flfc.
X
(a)
(6)
9 1419 The equivalent circuits for the generalized amplifier of Fig. 1418
00 king into" (a) the drain and (b) the source. Note that n = r lt g m .
406 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 1 4.9
Ftg. 1420 Source selfbias circuit.
and transistors, as discussed in Sec. 713 and Chap. 10. These considerations
are outputvoltage 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 fieldeffect devices.
Source Self bias The configuration shown in Fig. 1420 is the same as
that, considered in connection with the biasing of vacuum tubes. It can be
used to bias junction FET devices or depletionmode MOS transistors. For
a specified drain current I D , the corresponding gatetosource voltage Vos can
be obtained either using Eq. (148) 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. 1420 utilizes an nchannel FET for which
V, = 2.0 V, gmo = 1.60 mA/V, and /o« = 1.65 mA. It is desired to bias the
circuit at 1„ = 0.8 mA, using Vdd = 24 V. Assume r d » R d . Find (a) F«*
(6) g m , (c) A%, (<0 Rd, such that the voltage gain is at least 20 dB, with R, bypassed
with a very large capacitance C,.
Solving,
Solution a. Using Eq
V QS = 0.62 V.
(148), we have 0.8 = 1.65(1 + V as /2.0y
b. Equation (1418) now yields
A _ ^\ = LH mA /V
?, = 160
e , Rt m _ Z« = ^ = 0.77 K = 770 «
/d 0.8
d. Since 20 dB corresponds to a voltage gain of 10, then Av  gmR<i > l0, °
R<t >
10
1.11
= 9K
1
Sec. M9
FIELD EFFECT TRANSISTORS / 407
Biasing for Zero Current Drift 11 Figure 1421 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 fieldeffect transistor for zero draincurrent
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 majoritycarrier 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 gatetochannel
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. 67.
Since a change in gate voltage A Fes causes a change in drain current of
Qn AFos, then the condition for zero drift is
oi
0.007/i, = O.OO220 n
9*
= 0.314 V
(1436)
(1437)
If we substitute Eqs. (148), (1418), and (1419) in Eq. (1436), we obtain
\Vp\  \V as \ = 0.63 V (1438)
Equation (1438) 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. (148), (1418), and (1438),
'91421 Transfer characteristics
f °r an ?tchannel FET as a function
of temperature T.
V B
408 / ELECTRONIC DEVICES AND CIRCUITS
and
Id = h
9m
/0.63\ a
0.63
W7\
Sec. ?49
(1439)
(1440)
Equations (1439) and (1440) 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. (1438) to (1440) are measured at T = 25°C
EXAMPLE It is desired to bias the amplifier stage of the previous example for
zero draincurrent 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. (1439),
To = 1.65  1 m 0.165 mA = 165 jiA
pfy
6. From Eq. (1438),
Vm  1.37 V
c. Since Fes = —IdR,
1.37
B. =
K = 8.3 K
0.165
d. From Eq. (1440), we have
ft. = 1.60
/0.63\
\~) m
0.50 mA/V
Hence Ay « ff m fl* = 0.50 X 10 = 5.0.
We thus see that zero drift has been obtained at the expense of g m and voltage
gain, which are now onehalf 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 nchanne
PET may appear as in Fig. 1422a, 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. 1422a. 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. U9
h
( + )
less f max)
l oss (rnin)
^ \ V 3 /
j Bias line
h
\ ~ x — jSt^^
h
FIELDEFFECT TRANSISTORS / 409
lo < + )
Bias line
W(min)
fr(max) V as
V o0
(+) _o— •<)
(a)
(6)
(+)■
Fig. 1422 Maximum and minimum transfer curves for an nchannel 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. 14226, 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*
(1441)
Such a bias relationship may be obtained by adding a fixed bias to the gate
in addition to the source selfbias, as indicated in Fig. 1423a. A circuit
requiring only one power supply and which can satisfy Eq. (1441) is shown
^=c.
(a)
(6)
Fig. 1423 (a) Biasing a FET with a fixedbias Vera in addition to
selfbias through R,. (b) A single powersupply configuration
which is equivalent to the circuit in (a).
410 / ELECTRONIC DEVICES AND CIRCUITS
Sec. I4»9
in Fig. 14236.
V QG —
For this circuit
RiV DC
Ri 4 R2
R m
R1R2
R 1 \ Ri
We have assumed that the gate current is negligible. It is also possible for
Vqq to fall in the reversebiased region so that the line in Fig. 14226 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. 14236. For this nchan
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. 1424. 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. =
40
1.1 0.9
= 20 K
Then, from the first point and Eq. (1441), we find
Vgg  IdR. = (0.9) (20) = 18 V
Fig. 1424 Extreme trans
fer curves for the 2N3484
field effect transistor.
(Courtesy of Union Car
bide Corporation.)
Sac. M70
FIELDEFFECT TRANSISTORS / 411
6. If R, = 3.3 K, we see from the curves that Jx>(min) = 0,4 mA and /e(max)
= 1.2 mA. The minimum current is far below the specified value of 0.8 mA.
Biasing the Enhancement MOSFET The selfbias technique of Fig. 1420
cannot be used to establish an operating point for the enhancementtype
MOSFET because the voltage drop across R a is in a direction to reversebias
the gate, and a forward gate bias is required. The circuit of Fig. 1425o 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. 14256 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.
129), Rf corresponds to an equivalent resistance Ri = R//(l — Av) shunting
the amplifier input.
Finally, note that the circuit of Fig. 14236 could also be used with the
enhancement MOSFET, but the dc stability introduced in Fig. 1425 through
the feedback resistor R/ would then be missing.
1410
UNIPOLARBIPOLAR CIRCUIT APPLICATIONS 12
The main advantages of the unipolar transistor, or FET, are the very high input
impedance, no offset voltage, and low noise. For these reasons a FET is most
useful in a lowlevel highinputimpedance circuit, such as a signal chopper or
the first stage of a unipolarbipolar cascade combination. In this section
we consider the advantages of some representative FETbipolar transistor
or FETFET combinations.
Source Follower with Constantcurrent Supply Consider the source fol
lower of Fig. 1417, 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. (1426), R, > 49 K, provided g m » g 4 .
v„
F '9 1425 (o) Drain
'°gate bias circuit for
enhancementmode
M °S transistors; (b)
""proved version of (a)
„
v B
412 / ELECTRONIC DEVICES AND CIRCUITS
Sec. W.Jo
QV„
QV n
(a)
(6)
Fig. 1426 A source follower with (a) a bipolar transistor and (b) o
FBT constantcurrent supply.
It is clear that the drain supply must exceed 49 V. Since most FETs have low
breakdown voltages, it might be impractical to obtain Av > 0.98 with this
circuit.
This difficulty is circumvented in the configuration of Fig. 1426a, which
shows a source follower with the constantcurrent supply circuit discussed in
Sec. 1212. Here the effective source resistance of Ql is the output impedance
of Q2, whose value is given by Eq. (1251). 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. 14266 makes use of the high dynamic resist
ance R', = r d + (m + !)#« in the source circuit of Ql.
9v n
~v n
(a) (6)
Fig. 1427 Bootstrap circuits for very high input impedance.
MH
FIELDEFFECT TRANSISTORS / 413
Vn
Fig. 1428 Directcoupled cascode
circuit.
Bootstrap FET Circuits for Very High Input Impedance The input resist
ance in the circuits of Fig. 1426 is essentially 5iiZa. If very high input
impedance is desired, the bootstrap principle discussed in Sec. 1210 must be
invoked. The circuits of Fig. 1427 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. 1427a, the output circuit is an
emitter follower, and a voltage gain close to unity is possible. In Fig. 14276,
the output is taken from the collector circuit of Q2, and hence this circuit is
a lownoise highinputimpedance amplifier with Av — v /V{ > 1. Expres
sions for Ay and also for v t /vi are given in Prob. 1430.
The Cascode Amplifier Circuit This configuration is a version of the
cascode circuit discussed in Sees. 810 and 1211. In Fig. 1428 a common
source FET drives a commonbase 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 raintosou 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 highfrequency amplification.
1411
THE FET AS A VOLTAGEVARIABLE RESISTOR 13 (WR)
,
y* most linear applications of fieldeffect transistors the device is operated
l ° the constantcurrent portion of its output characteristics. We now consider
*ET transistor operation in the region before pinchoff, where Vds is small.
in this region the FET is useful as a voltagecontrolled resistor; i.e., the drain
wsource resistance is controlled by the bias voltage Vgs In such an applica
414 / ELECTRONIC DEVICES AND CIRCUITS
r„,K
too
80
60
40
r*
l
H_
If
T
J,9N^91^ 
LH
"J
T
Sec. MJj
(a)
(6)
2.0 3.0
Fig. 1429 (a) FET lowlevel drain characteristics for 2N3278.
(b) Smallsignal FET resistance variation with applied gate voltage.
(Courtesy of Fairchild Semiconductor Company.)
tion the FET is also referred to as a voltagevariable resistor (WR) or voltage
dependent resistor (VDR).
Figure 1429a shows the lowlevel bidirectional characteristics of a FET.
The slope of these characteristics gives r d as a function of Vqs. Figure 1429o
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. 143 we
derive Eq. (145), which gives the draintosource conductance g d = Id/Vds
for small values of Vds From this equation we have
w = 0*[i (tj)*]
(1442)
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 pinchoff
voltage V P . Variation of r d with Vqs is plotted in Fig. 14296 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
(1443)
where r = drain resistance at zero gate bias
K = a constant, dependent upon FET type
Vos = gatetosource 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
?4I2
FIELDEFFECT TRANSISTORS / 415
pjg. 1430 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. 1430. The signal is taken at a highlevel point, rectified, and filtered to
produce a dc voltage proportional to the outputsignal 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. 14296. We thus may cause the gain
of transistor Ql to decrease as the outputsignal level increases. The dc bias
conditions of Ql are not affected by Q2 since Q2 is isolated from Ql by means
of capacitor C*.
1412
THE UNIJUNCTION TRANSISTOR
Another device whose construction is similar to that of the FET is indicated
in Fig. 1431. A bar of highresistivity ntype 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 3mil aluminum wire, called the emitter E,
Base, B
ntype Si bar
Al rod
sO"
p~n junction
(a)
J32
> Ohmic
/ contacts
LI 31
(b)
Fig. 1431 Unijunction transistor, (a) Constructional details; (b)
circuit symbol.
416 / HfCTRONJC DEVICES AND CIRCUITS
Sac. ?4.j 2
S 10
T^=2S C
V M = 30V
1
1
L20
^10
Sc 5
*S2 =
Fig. 1432 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 doublebase diode, but is now com
mercially available under the designation unijunction transistor (UJT). The
standard symbol for this device is shown in Fig. 14316. 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 reversebiased,
whereas the useful behavior of the UJT occurs when the emitter is forward
biased.
As usually employed, a fixed interbase potential Vbb is applied between
B\ and B2. The most important characteristic of the UJT is that of the input
diode between E and Bh If 52 is opencircuited so that I B z = 0, then the
input voltampere relationship is that of the usual pn junction diode as given
by Eq. (631). In Fig. 1432 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 negativeresistance characteristic. A
qualitative explanation of the physical origin of the negative resistance is
given in Ref. 14. The principal application of the UJT is as a switch which
allows the rapid discharge of a capacitor (Ref. 13).
REFERENCES
1. Shockley, W.: A Unipolar Fieldeffect 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. 11491189, November, 1955.
FIELDEFFECT TRANSISTORS / 417
Wallmark, J. T., and H. Johnson: "Fieldeffect Transistors," PrenticeHall, Inc.
Englewood Cliffs, N.J., 1966.
Sevin, L. J.: "Fieldeffect Transistors," McGrawHill Book Company, New York
1965.
2. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," sec. 1720,
McGrawHill Book Company, New York, 1965.
3. Wallmark, J. T., and H. Johnson: "Fieldeffect Transistors," p. 115, PrenticeHall,
Inc., Englewood Cliffs, N.J., 1966.
4. Sevin, L. J.: "Fieldeffect Transistors," pp. 1317, McGrawHill 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. ED13, no. 6, pp, 531532
June, 1966.
6. Ref. 4, p. 21.
7. Ref. 4, p. 23.
8. Ref. 4, p. 34.
9. Ref. 3, pp. 187215.
10. Ref. 3, pp. 256259.
11. Hoerai, J. A., and B. Weir: Conditions for a Temperature Compensated Silicon
Field Effect Transistor, Proc. IEEE, vol. 51, pp. 10581059, July, 1963.
Evans, L. L.: Biasing FETs for Zero dc Drift, Electrotechnol, August, 1964, po.
9396.
12. Gosling, W.: A Drift Compensated FETBipolar Hybrid Amplifier, Proc. IEEE,
vol. 53, pp. 323324, March, 1965.
'3. Bilotti, A.: Operation of a MOS Transistor as a Voltage Variable Resistor, Proc.
IEEE, vol. 54, pp. 10931094, August, 1966.
'*■ Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," sees.
123 and 1313, McGrawHill Book Company, New York, 1965.
15 /INTEGRATED
CIRCUITS
An integrated circuit consists of a singlecrystal 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.
151
BASIC MONOLITHIC INTEGRATED CIRCUITS 12
We now examine in some detail the various techniques and processes
required to obtain the circuit of Fig. 15 la in an integrated form, W
shown in Fig. 1515. This configuration is called a monolithic inte
grated circuit because it is formed on a single silicon chip. The wor
"monolithic" is derived from the Greek monos, meaning "single," » in
lithos, meaning "stone." Thus a monolithic circuit is built into
single stone, or single crystal.
In this section we describe qualitatively a complete epita xlfl
diffused fabrication process for integrated circuits. In subsequeo
sections we examine in more detail the epitaxial, photographic, ft
diffusion processes involved. The circuit of Fig. 15 lo is chosen
discussion because it contains typical components: a resistor, dio«
41 S
S9C
151
INTEGRATED CIRCUITS / A\9
Q 3
+«
(a)
Resistor
Diode junctions
Transistor
Aluminum
metalization
Silicon dioxide
Collector
contact n*
n* Emitter
Base
Collector
Fig. 151 (a) A circuit containing a resistor, two diodes, and a tran
sistor, (b) Crosssectional 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. 152 and described
below.
Step 1. Epitaxial Growth An retype epitaxial layer, typically 25
microns thick, is grown onto a ptype substrate which has a resistivity of typi
cally 10 ficm, corresponding to N A = 1.4 X 10 15 atoms/cm 3 . The epitaxial
Process described in Sec. 152 indicates that the resistivity of the ntype epi
taxial layer can be chosen independently of that of the substrate. Values of
•fom 0.1 to 0.5 Ucm are chosen for the ntype layer. In contrast to the situa
tion depicted in Fig. 152a, the epitaxial process is used with discrete transistors
10 obtain a thin highresistivity layer on a lowresistivity 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. 152a. 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. 1526 the wafer is shown with the
ide removed in four different places on the surface. This removal is accom
420 / ELECTRONIC DEVICES AND CIRCUITS
$»c. 1 5. 
S«'
151
INTEGRATED CIRCUITS / 421
Silicon dioxide
(a)
i Isolation islands
Sidewall C
Bottom C,
•Resistor
xnesisior .
Anode of diode /Base
ULSU
(c)
Cathodes of diodes n
Wx
w>
rr«5
\w,\\v\w,\
I
/•^■H
P*
,
^__
u
p
n
Emitter n"
Fig. 152 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, 153.
The remaining SiOz serves as a mask for the diffusion of acceptor impurities
/jn this case, boron). The wafer is now subjected to the socalled isolation
diffusion, which takes place at the temperature and for the time interval
required for the ptype impurities to penetrate the ntype epitaxial layer and
jeach the ptype substrate. We thus leave the shaded ntype regions in Fig.
l52?>. These sections are called isolation islands, or isolated regions, because
they are separated by two backtoback pn 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 ptype sub
strate must always be held at a negative potential with respect to the isolation
islands in order that the pn junctions be reversebiased. If these diodes were
to become forwardbiased 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 ptype substrate. The reason for this
higher density is to prevent the depletion region of the reversebiased isolation
tosubstrate junction from extending into p + type material (Sec. 69) 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 ptype substrate, which capacitance can affect the oper
ation of the circuit. Since Ct, is an undesirable byproduct 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 ntype region to the substrate (Fig. 1526) 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. 152) and hence varies inversely as the square root of the voltage V
Between the isolation region and the substrate (Sec. 69). 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 ntype layer. The total capaci
ance is of the order of a few picofarads.
Step 3, Base Diffusion During this process a new layer of oxide is
ormed over the wafer, and the photolithographic process is used again to create
the
cliff
Pattern of openings shown in Fig. 152c. The ptype impurities (boron) are
used through these openings. In this way are formed the transistor base
^'°us as well as resistors, the anode of diodes, and junction capacitors (if
y ■ It is important to control the depth of this diffusion so that it is shallow
^~ does not penetrate to the substrate. The resistivity of the base layer will
e ral! y be much higher than that of the isolation regions.
Sec. Fs_
422 / ELECTRONIC DEVICES AND CIRCUITS
Step 4. Emitter Diffusion A layer of oxide is again formed over the
entire surface, and the masking and etching processes are used again to ope n
windows in the ptype regions, as shown in Fig. 152d. Through these open,
ings are diffused ntype 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. 152d) 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 ptype impurity in silicon, and
a large concentration of phosphorus prevents the formation of a pn junction
when the aluminum is alloyed to form an ohmic contact. 4
Step 5. Aluminum Metalization All pn 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. 152e,
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. 152e between resistors, diodes, and transistors.
In production a large number (several hundred) of identical circuits such
as that of Fig. 15la are manufactured simultaneously on a single wafer.
After the metalization process has been completed, the wafer is scribed with
a diamondtipped 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
1mil aluminum or gold wire from the terminal pad on the circuit to th«
package lead (Fig. 1526).
Summary In this section the epitaxialdiffused 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 tripledin uS8 ^
A L
INTEGRATED CIRCUITS / 423
process and the d iff usedcollector process. 1 The method just described, how
ever, is in more general use because of a number of inherent advantages. l
152
EPITAXIAL GROWTH 1
The epitaxial process produces a thin film of singlecrystal silicon from the
gas phase upon an existing crystal wafer of the same material. The epitaxial
layer may be either ptype or ntype. The growth of an epitaxial film with
impurity atoms of boron being trapped in the growing film is shown in Fig. 153.
The basic chemical reaction used to describe the epitaxial growth of pure
silicon is the hydrogen reduction of silicon tetrachloride:
1200°C
SiCU + 2H 2 » Si + 4HC1
(151)
Since it is required to produce epitaxial films of specific impurity concen
trations, it is necessary to introduce impurities such as phospbine for ntype
doping or biborane for ptype doping into the silicon tetrachlo ridehydro gen
gas stream. An apparatus for the production of an epitaxial layer is shown in
Fig. 154. In this system a long cylindrical quartz tube is encircled by a
radiofrequency 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 pn junction similar to the junction shown in
Fig. 612.
F '9l53 The epitaxial
9'owth of an epitaxial film
lowing impurity (boron)
°toms being trapped in the
flawing film. (Courtesy of
M °toroIa, Inc. 1 )
• Gas phase
®®®®®'®®1 Epu _
® ®®@@®@ J
© © © © © © ©
® ® © © © © ®
© @®®©@®
fsT) CsT) fsT) (sT) (sT) (s?) (sT)
Substrate
424 / ELECTRONIC DEVICES AND CIRCUITS
Induction coll
Sec. 15.
Outlet
Silicon wafers
Graphite boat
Fig. 154 A diagram
matic representation of a
system for production
growth of silicon epi
taxial films. (Courtesy
of Motorola, Inc. 1 )
153
MASKING AND ETCHING 1
The monolithic technique described in See. 151 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. 155. 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 blackand
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. 155a. 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. 1556.
Mask
Photoresist
Si0 2 " /
Silicon chip
Ultraviolet
Polymerized
photoresist
/ I \
'S10 2
■Silicon chip
(«)
(6)
Fig. 155 Photoetching technique, (o) Masking and exposure to
ultraviolet radiation, (b) The photoresist after development.
S#
154
INTEGRATED CIRCUITS / 425
The emulsion which was not removed in development is now fixed, or cured,
e that it becomes resistant to the corrosive etches used next. The chip is
immersed in an etching solution of hydrofluoric acid, which removes the oxide
from the areas through which dopants are to be diffused. Those portions of
the SiOs which are protected by the photoresist are unaffected by the acid.
After etching and diffusion of impurities, the resist mask is removed (stripped)
with a chemical solvent (hot H2SO4) and by means of a mechanical abrasion
process.
154
DIFFUSION OF IMPURITIES 6
The most important process in the fabrication of integrated circuits is the
diffusion of impurities into the silicon chip. We now examine the basic theory
connected with this process. The solution to the diffusion equation will give
the effect of temperature and time on the diffusion distribution.
The Diffusion Law The continuity equation derived in Sec. 59 for
charged particles is equally valid for neutral atoms. Since diffusion does not
involve electronhole recombination or generation (r P = «) and since no
electric field is present (£ = 0), Eq. (546) now reduces to
d_N
at
= D
em
(152)
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 ntype impurities, such as phosphorus, these atoms will diffuse into
the silicon crystal, and their distribution will be as shown in Fig. 156a. 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. (152) is solved and the above boundary conditions are applied,
N(x, t) = N t
0
erf
nnr.
erfc
(153)
2 y/DiJ " 2 ^/Dt
"ere erfc y means the errorfunction complement of y, and the error function
of V is defined by
erf *^/° VX,< * X
(154)
426 / ELECTRONIC DEVICES AND CIRCUITS
S «. 1 4^
Fig. 156 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. 157.
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. (152), we find
N(x, t) 
Q
t'UDt
(155)
Equation (155) is known as the Gaussian distribution, and is plotted in
Fig. 1566 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. (153) and (155) 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 ntype
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. (153) is less than the
diffusant's solid solubility. The solid solubility is defined as the maximum
concentration N„ of the element which can be dissolved in the solid silico°
S«c
\S4
INTEGRATED CIRCUITS / 427
1.0
5x 10"'
10"'
5x 10r»
1L1'
pig, 1 57 The complemen
tary error function plotted
>> 5x10"'
o
E
on semilogarithmic paper.
io J
5x 10 1
10"*
Sx Mr 1
. [— ^
10" 1
at a given temperature. Figure 158 shows solid solubilities of some impurity
elements. It can be seen that since for phosphorus the solid solubility is
approximately 10 S1 atoms/cm 3 , and for pure silicon we have 5 X 10 !2 atoms/
cm s , the maximum concentration of phosphorus in silicon is 2 percent. For
most of the other impurity elements the solubility is a small fraction of 1
percent.
Diffusion Coefficients Temperature affects the diffusion process because
Wgher temperatures give more energy, and thus higher velocities, to the dif
Fi 9. 158 Solid solubili "
•■m of some impurity £
demerits in silicon. 
'* f ter Trumbore/ I
Curtesy of Motorola, J
 IIIIU
Slli,
" "'Ml
a J
1100 /
I
1
\3 rrj
>y
ill "' 1
toy
Auinr
j_p5.
Jjt
Ve
u if
, «]ttl
 IJ
\ llllii
 fcu
1
\ II
kj Silicon
sb jjj
J t
500 1
□
1
1
10" 10 JI 10™ 10" 10" 10" 10" 10"
10" 10" 10"
Atoms/cm 3
428 / ELECTRONIC DEVICES AND CIRCUITS
Sec J 5 . 4
Temperature, "C
1300 1200 1100 1000
900
8 io
s io" 
io
1
1
i
t
i
—
i Aluminum
kGailiurr. "
^_ Boron and
v phosphorus
Antimony
' \
Fig, 159 Diffusion coefficients q s
a function of temperature for some
impurity elements in silicon. (After
Fuller and Ditzenberger, 6 courtesy
of Motorola, Inc. 1 )
0.60 0.65 0.70 0.75 0.80
1000/7/ (Tin'K)
0.85
fusant atoms. It is clear that the diffusion coefficient is a function of tempera
ture, as shown in Fig. 159. 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, (153) and (155) appears in the product Dt, an increase in either
diffusion constant or diffusion time has the same effect on diffusant density.
Note from Fig. 159 that the diffusion coefficients, for the same tempera
ture, of the ntype impurities (antimony and arsenic) are lower than the
coefficients for the ptype impurities (gallium and aluminum), but that phos
phorus («type) and boron (ptype) have the same diffusion coefficients.
Typical Diffusion Apparatus Reasonable diffusion times require high
diffusion temperatures (~1000°C). Therefore a hightemperature 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 1510 shows the
apparatus used for POCH diffusion. In this apparatus a carrier gas (mixture
of nitrogen and oxygen) bubbles through the liquiddiffusant source an d
carries the diffusant atoms to the silicon wafers. Using this process, we obtain
the complementaryerrorf unction distribution of Eq. (153). A twosWP
procedure is used to obtain the Gaussian distribution. The first step involve*
predeposition, carried out at about 900°C, followed by drivein at abou
1100°C.
S.c 154
INTEGRATED CIRCUITS / 429
Quartz diffusion tube
' — ■
poootC
Silicon wafers
S^l
■ Furnace
Liquid POC1
Thermostated bathn\
input
Fig. 1510 Schematic representation of typical apparatus for POCIs diffusion.
(Courtesy of Motorola, Inc. 1 }
EXAMPLE A uniformly doped ntype silicon substrate of 0.5 Sicm resistivity
is subjected to a boron diffusion with constant surface concentration of 5 X 10"
cm 3 . It is desired to form a pn 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. 156a. 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. (52) :
1
= 0.96 X 10 IB cm""
tine (0.5) (1,300) (1.60 X 10" 19 )
where all distances are expressed in centimeters and the mobility (t n for silicon
is taken from Table 51, 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. 157 that y = 2.2. Hence
2.2 = ''_ =, 27 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. 159 at T = 1130°C.
430 / ELECTRONIC DEVICES AND CIRCUITS
'55
155
TRANSISTORS FOR MONOLITHIC CIRCUITS 17
A planar transistor made for monolithic integrated circuits, using epitaxy and
diffusion, is shown in Fig. 151 la. Here the collector is electrically separated
from the substrate by the reversebiased 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. 151 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 collectorcurrent path and thus increases the collector resistance
and Fcfi(sat). All these undesirable effects are absent from the discrete
epitaxial transistor shown in Fig. 15116. 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. 1212).
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 1512 showsa typical
impurity profile for a monolithic integrated circuit transistor. The back
Emitter contact
Base contact
Collector contact
nepitaxial collector
p substrate
ptype isolation
diffusion
(b)
Emitter contact
Base contact
Fig. ? 511 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, 1513,1
^Collector contact
Sec
155
INTEGRATED CIRCUITS / 431
3 x, v
Collector — *■
Fig, 1512 A typical impurity profile in a monolithic
integrated transistor. [Note that N(x) is plotted on a
logarithmic scale.]
ground, or epitaxialcollector, concentration N S c is shown as a dashed line in
Fig. 1512. The base diffusion of ptype 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 emittertobase junction is usually treated as a step junction, whereas the
hasetocollector junction is considered a graded junction.
EXAMPLE (a) Obtain the equations for the inpurity profiles in Fig. 1512.
(6) If the phosphorus diffusion is conducted at 1100°C, how long should be
allowed for this diffusion?
Solution a. The base diffusion specifications are exactly those given in the
example on page 429, where we find (with x expressed in microns) that
y « 2.2 =
2.7
avS
or
/ — 2.7
2 V Dt = — = 1.23 microns
2.2
432 / ELECTRONIC DEVICES AND CIRCUITS
Hence the boron profile, given by Eq. (153), is
x
Sec.
'55
N* = 5 X 10" erfc
1.23
The emitter junction is formed at x = 2 microns, and the boron concentration
here is
N B  5 X 10" erfc  5 X 10" X 2 X 10~»
1.23
= 1.0 X 10" cm"'
The phosphorus concentration N B is given by
iVp = 10" erfc — %=*.
2VDt
At a: = 2, JV,  AT B = 1.0 X 10", so that
erfc
2 VDt
1.0 X 10"
10"
= 1.0 X 10"*
From Fig. 157, 2/(2 Voi) = 2.7 and 2 Vfli = 0.74 micron. Hence the
phosphorus profile is given by
N r = 10" erfc —
0.74
o. From Fig. 159, 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 smallgeometry 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. 1513. The
emitter rectangle measures 1 by 1.5 mils, and is diffused into a 2.5 by 4.0mi 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 lateraldiffusion distance will also be 1 mil The
dashed rectangle in Fig. 1513 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 0.1iicm collectors is given in Table 151.
Stc
155
INTEGRATED CIRCUITS / 433
Indicates
contacts
Emitter diffusion
Base diffusion
Isolation diffusion
Fig. 1513 A typical doublebase stripe geometry of an integrated
ctrcuit transistor. Dimensions are in mils. (For a side view of the
transistor see Fig. 1511.) (Courtesy of Motorola Monitor.)
TABLE 151 Characteristics for 1  by 1 .5miI double
base stripe monolithic transistors 3
Transistor parameter
BVcbq,V
BVtmhV
BVceo, V
Ct. (forward bias), pF
C T . at 0.5 V, pF
C T . at 5 V, pF
A;rjat 10 mA
Res, «
Vc*(sat) at 5 mA, V. ,
VsE&t 10 mA, V
/r at 5 V, 5 mA, MHz
t Golddoped.
0.5 ncm 0.1 ficmt
55
25
7
5.5
23
14
6
10
1.5
2.5
0.7
1.5
50
50
75
15
0,5
0.26
0.85
0.85
440
520
434 / ELECTRONIC DEVICES AND CIRCUITS
Se c. fs^
Fig. 1514 Utilization of
"buried" n + layer to
reduce collector series
resistance.
Buried Layer 1 We noted above that the integrated transistor, because of
the top collector contact, has a higher collector series resistance than a similar
discretetype 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 ntypc epitaxial collector, as shown in
Fig. 1514. The buriedlayer structure can be obtained by diffusing the n +
layer into the substrate before the ntype 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 npn 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. 159 shows that
n type impurities have smaller values of diffusion constant D than ptype
impurities, the collector is usually ntype. In addition, the solid solubility
of some ntype impurities is higher than that of any ptype impurity, thus
allowing heavier doping of the n + type emitter and other n + regions.
156
MONOLITHIC DIODES'
The diodes utilized in integrated circuits are made by using transistor struc
tures in one of five possible connections (Prob. 159). The three most popular
diode structures are shown in Fig. 1515. They are obtained from a transistor
Rg. 1515 Cross section
of various diode struc
tures, (a) Emitterbose
diode with collector
shorted to base; (b)
emitterbase diode wi*
collector open; and \ c >
collectorbase diode 1°
emitter diffusion).
(a)
(b)
$c<~
T56
Anode
1
Common
cathode
3
O
*f
Anode
2
INTEGRATED CIRCUITS / 435
Common
anode
3
9
Cathode
Cathode
lYftViNVW
» V
WWWVVj
p substrate
i
(a)
Fig. 1516 Diode pairs, (a) Commoncathode pair, and (b) common
anode pair, using collectorbase diodes.
structure by using (a) the emitterbase diode, with the collector shortcircuited
to the base; (6) the emitterbase diode, with the collector open; and (c) the
collectorbase diode, with the emitter opencircuited (or not fabricated at all).
The choice of the diode type used depends upon the application and circuit
performance desired. Collectorbase diodes have the higher collectorbase
voltage breakdown rating of the collector junction (~12 V minimum), and
they are suitable for commoncathode diode arrays diffused within a single
isolation island, as shown in Fig. 1516o. Commonanode arrays can also be
made with the collector base diffusion, as shown in Fig. 15166. A sepa
rate isolation is required for each diode, and the anodes are connected by
oietalization.
The emitterbase diffusion is very popular for the fabrication of diodes
Provided that the reversevoltage requirement of the circuit does not exceed
the lower baseemitter breakdown voltage (^7 V). Commonanode arrays
°an easily be made with the emitterbase diffusion by using a multipleemitter
transistor within a single isolation area, as shown in Fig. 1517. The collector
^9 1517 A multiple
fitter npn transistor.
™) Schematic, (b) mono
lttl 'c surface pattern.
me base is connected
to ,h e collector, the
[ esu "isa mu tipe
"""ode diode structure
*ith
Q common anode.
(a)
436 / ELECTRONIC DEVICES AND CIRCUITS
Sec. ?S.»
10
S 6
■§ 4
r±
t t J
{a)j (b)l (cy
11/
—J J/
J^Z
f/
~//
Jl?
2
Fig. 1518 Typical diode voltampere
characteristics for the three diode types
of Fig. 1515. (a) Baseemitter (collector
shorted to base); (b) baseemitter (col
lector open); (c) collectorbase (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, 151 is
constructed in this manner, with the collector floating (open).
Diode Characteristics The forward voltampere characteristics of the
three diode types discussed above are shown in Fig. 1518. It will be observed
that the diodeconnected transistor (emitterbase 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, onethird to onefourth
that of the collectorbase diode.
157
INTEGRATED RESISTORS 1
A resistor in a monolithic integrated circuit is very often obtained by utilizing
the bulk resistivity of one of the diffused areas. The ptype base diffusion
is most commonly used, although the ntype 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. 1519, 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
(156)
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
1512 are 200 fi/square and 2.2 Si/ square, respectively. .
The construction of a basediffused resistor is shown in Fig. 151 an
repeated in Fig. 1520o. A top view of this resistor is shown in Fig 1&"
S*c. 1S7
INTEGRATED CIRCUITS / 437
Fig. 1519 Pertaining to sheet
resistance, ohms per square.
The resistance value may be computed from
ff  pl  p *
K — — lis —
yw w
(157)
where I and w are the length and width of the diffused area, as shown in
the top view. For example, a basediffusedresistor 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 diffusedresistor design are
stripe length and stripe width. Stripe widths of less than one mil (0.001 in.)
are not normally used because a linewidth variation of 0.0001 in. due to
mask drawing error or mask misalignment or photographicresolution error
can result in 10 percent resistortolerance 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. 1520 A monolithic resistor, (o)
Crosssectional view; (b) top view.
1 o
k^^m,kwww
/] isolation region
P substrate
(«)
i
(6)
438 / ELECTRONIC DEVICES AND CIRCUITS
R
AAV
bt
T
1
2
o p layer
C
Sec. I5.J
Fig. 1521 The equivalent circuit
of a diffused resistor.
A/VV
A/W
o n isolation region
o p substrate
to 30 K for a basediffused resistor and 10 U to 1 K for emitterdiffused
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. 109) 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. 1521,
where the parasitic capacitances of the baseisolation (Ci) and isolationsub
strate (Cj) junctions are included. In addition, it can be seen that a parasitic
pnp transistor exists, with the substrate as collector, the isolation ntype
region as base, and the resistor ptype material as the emitter. Since the
collector is reversebiased, 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 ntype 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.
Thinfilm Resistors 1 A technique of vapor thinfilm 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 sheetresistance values for nichrome thinfib* 1
resistors are 40 to 400 Q/square, resulting in resistance values from about 20
Q to 50 K.
158
INTEGRATED CAPACITORS AND INDUCTORS 1 2
Capacitors in integrated circuits may be obtained by utilizing the transit 100
capacitance of a reversebiased pn junction or by a thinfilm technique.
Sec. J 58
Al metal izat ion
INTEGRATED CIRCUITS / 439
C a ^0.2pF/mil J
Substrate
(4)
Fig. 1522 (a) Junction monolithic capacitor, (b) Equivalent circuit. (Courtesy of
Motorola, Inc.)
Junction Capacitors A crosssectional view of a junction capacitor is
shown in Fig. 1522a. The capacitor is formed by the reversebiased junction
3 1, which separates the epitaxial ntype layer from the upper ptype diffusion
area. An additional junction J\ appears between the ntype 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. 15226, 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.
(649). The series resistance R (10 to 50 £1) represents the resistance of the
ntype 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 reversebiased. It should also be pointed out that the junction
capacitor Ct is polarized since the pn junction J \ must always be re verse biased.
Thinfilm Capacitors A metaloxidesemiconductor (MOS) nonpolarized
capacitor is indicated in Fig. 1523a. This structure is a parallelplate capa
Al metalizaton
C=0.25pF/mil 2
R =
5 ion B
■)\ T 1 — Vv\ o
C 1
4/
J >?i
»r»C i
ptype substrate
(b)
9 1523 A MOS capacitor, (a) The structure and (b) the equivalent circuit.
440 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 159
TABLE 752 Integrated capacitor parameters
Characteristic
Diff usedjunction
capacitor
Thinfilm MOS
Capacitance, pF/mil*
0.2
2 X 10«
406
520
kV~*
±20
0.250.4
2 X 10*
800
50200
Tolerance, percent , .'.' i 
±20
citor with SiO s as the dielectric. A surface thin film of metal (aluminum) is
the top plate. The bottom plate consists of the heavily doped n + region that
is formed during the emitter diffusion. A typical value for capacitance 8 is
0.4 pF/mil* for an oxide thickness of 500 A, and the capacitance varies inversely
with the thickness.
The equivalent circuit of the MOS capacitor is shown in Fig. 15236,
where Ci denotes the parasitic capacitance J t of the collectorsubstrate junc
tion, and R is the small series resistance of the n+ region. Table 152 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 thinfilm
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.
159 MONOLITHIC CIRCUIT LAYOUT 1 '
In this section we describe how to transform the discrete circuit of Fig. 1524a
into the layout of the monolithic circuit shown in Fig. 1525. Circuits involv
ing diodes and transistors, connected as in Fig. 1524o, 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 collectorpotential 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«. 1S9
INTEGRATED CIRCUITS / 441
5. In layout, allow an isolation border equal to twice the epitaxial thick
ness to allow for underdiffusion.
6. Use 1mil widths for diffused emitter regions and jmil widths for base
contacts and spacings, and for collector contacts and spacings.
7. For resistors, use widest possible designs consistent with diesize
limitations.
8. Always optimize the layout arrangement to maintain the smallest
possible die size, and if necessary, compromise pin connections to achieve this.
9. Determine component geometries from the performance requirements
of the circuit.
10. Keep all metalizing runs as short and as wide as possible, particularly
at the emitter and collector output connections of the saturating transistor.
Pin Connections The circuit of Fig. 1524a is redrawn in Fig. 15246,
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 powersupply 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. 15246) to cross over each
other. This crossover cannot be made directly because it will result in electric
contact between two parts of the circuit. Since all resistors are protected by
the SiOj layer, any resistor may be used as a crossover region. In other words,
©
+ 6.5V(
kputs
®«
©<
®
1K«
Dl
T*~
4000
D2
D3
D4
H
D5
w
Substrate
6.5V
5.6K
06.5V6
© @ © i
GND
F '9 1524 (a) A DTL gate, (b) The schematic redrawn to indicate the 10 external
c °nnections arranged in the sequence in which they will be brought out to the
"eader pins. The isolation regions are shown in heavy outline.
442 / ELECTRONIC DEVICES AND CIRCUITS
Sec. T59
•" ■■' ' "■ '■■'
— — Indicates isolation region ig^ssa Indicates metalization
Fig. 1525 Monolithic design layout for the circuit of Fig. 1524. (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. 15246 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. 157.
Sec 159
INTEGRATED CIRCUITS / 443
In order to determine the number of isolation regions required for the
diodes, it is necessary first to establish which kind of diode will be fabricated.
In this case, because of the low forward drop shown in Fig. 1518, it was
decided to make the commonanode diodes of the emitterbase 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 commonanode diodes.
Finally, the remaining diode is fabricated as an emitterbase diode, with the
collector opencircuited, and thus it requires a separate isolation island.
The Fabrication Sequence The final monolithic layout is determined by
a trialanderror process, having as its objective the smallest possible die size.
This layout is shown in Fig. 1525. The reader should identify the four iso
lation islands, the three resistors, the live diodes, and the transistor. It is
Ffg. 1526 Monolithic
fabrication sequence for
the circuit of Fig. 1524.
(Courtesy of Motorola
Monitor, Phoenix, Ariz.)
Metalization
Flat package assembly
444 / ElECTRONIC DEVICES AND CIRCUITS
Sec. 75 10
interesting to note that the 5.6K resistor has been achieved with a 2milwide
1.8K resistor in series with a 1milwide 3.8K 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. 1525, 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. 1526.
Largescale Integration (LSI) The monolithic circuit layout shown in
Fig. 1525 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 integratedcircuit 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. Largescale integration
(LSI) represents the process of fabricated largecomponentdensity chips
which represent complete subsystems or equipment components. A packaged
LSI slice 2j in. square with 32 leads on each side is pictured in Ref. 12.
1510
INTEGRATED FIELDEFFECT TRANSISTORS 113
The MOSFET is discussed in detail in Chap. 14. In this section we point out
the advantages of this device as an integratedcircuit active element (Fig
1527).
Size Reduction The MOS integrated transistor typically occupies only
5 percent of the surface required by an epitaxial doublediffused transistor ifl
a conventional integrated circuit. The doublebase stripe 1 by 1.5mil 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 enhancementtype fieldeffect transistor. In this step
(Fig. 1527 a) two heavily doped ntype regions are diffused into a lightly
doped ptype 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. 15TO
INTEGRATED CIRCUITS / 445
Source Drain
S I)
S
sa
9 S /
* Mrliilization
W o Hate (
WJ
a substrate
}> substrate
•n
l>
Fig. 1527 An nchannel insulatedgate 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.
15276.
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 crossoverdiffused regions (with R s =* 80100 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
pn junctions are reversebiased during the operation of the circuit.
The MOS as a Resistor for Integrated Circuits In our discussion of
diffused resistors in Sec. 157, 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. 1528, where the eate and drain are tied
together and a fixed voltage Vdd is applied between drain and ground. A
<*V m
Fig. 1528 The MOS as a resistor.
90
®
«?s
1 *
446 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 15. j,
Thevenin's equivalent circuit looking into the source is obtained in Sec
148. From Eq. (1435) 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.
1511
ADDITIONAL ISOLATION METHODS
Electrical isolation between the different elements of a monolithic integrated
circuit is accomplished by means of a diffusion which yields backto back
pn junctions, as indicated in Sec. 151. With the application of bias voltage
to the substrate, these junctions represent reversebiased diodes with a very
high back resistance, thus providing adequate dc isolation. But since each
pn 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 diodeisolation 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
pnp and npn transistors within the same silicon substrate. It is also simple
to have both fast and chargestorage diodes and also both high and lowfre
quency transistors in the same chip through selective gold diffusion — a process
prohibited by conventional techniques because of the rapid rate at which gold
diffuses through silicon unless impeded by a physical barrier such as a dielectric
layer.
One isolation method employing silicon dioxide as the isolating material
is the EPIC process, 12 developed by Motorola, Inc. This EPIC isolation
method reduces parasitic capacitance by a factor of 10 or more. In addition,
the insulating oxide precludes the need for a reverse bias between substrate
and circuit elements. Breakdown voltage between circuit elements and sub
strate is in excess of 1,000 V, in contrast to the 20 V across an isolation j unction
Beam Leads The beamlead concept 16 of Bell Telephone Laboratories
was primarily developed to batchfabricate semiconductor devices and bite
15JI
INTEGRATED CIRCUITS / 447
•rated circuits. This technique consists in depositing an array of thick (of the
order of 1 mil) contacts on the surface of a slice of standard monolithic circuit,
ft nd then removing the excess semiconductor from under the contacts, thereby
separating the individual devices and leaving them with semirigid beam leads
cantilevered beyond the semiconductor. The contacts serve not only as elec
trical leads, but also as the structural support for the devices; hence the name
beam leads. Chips of beamlead circuits are mounted directly by leads, with
out 1mil aluminum or gold wires.
Isolation within integrated circuits may be accomplished by the beam
.
Isolation area
C ommoD e mltte r
beam lead
Load res is tor
beam lead
Load resistor
Semiconductor
wafer
Common collector
beam lead
Baseresistor beam
intraconnectlon
(one of four)
Input resistor (one of four)
Inputresistor
beam lead
(one of four)
Isolation area
Semiconductor wafers
Fig, 1529 The beamlead isolation technique, (a) Photomicro
graph of logic circuit connected in a header, (b) The underside of
the same circuit, with the various elements identified. (Courtesy of
Bell Telephone Laboratories.)
448 / ELECTRONIC DEVICES AND CIRCUITS
$«. 7511
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 metaloveroxide overlay 6
This is much lower than the junction capacitance incurred with pn junction
isolated monolithic circuits.
It should be pointed out that the dielectric and beamlead isolation
techniques involve additional process steps, and thus higher costs and possible
reduction in yield of the manufacturing process.
Figure 1529 shows photomicrographs of two different views of a logic
circuit made using the beamlead 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 beamlead circuits.
REFERENCES
1. Motorola, Inc. (R. M. Warner, Jr., and J. N. Fordcmwalt, eds.): "Integrated
Circuits," McGrawHill Book Company, New York, 1965.
2. Phillips, A. B.; Monolithic Integrated Circuits, IEEE Spectrum, vol. 1, no. 6,
pp. 83101, June, 1964.
3. Jahnke, E., and F. Emde: "Tables of Functions," Dover Publications New York,
1945,
4. Hunter, L. P.: "Handbook of Semiconductor Electronics," 2d ed„ sec. 8, McGraw
Hill Book Company, New York, 1962.
5. Fuller, C. S., and J. A. Ditzenberger: Diffusion of Donor and Acceptor Elements in
Silicon, /. Appl. Phys., vol. 27, pp. 544553, 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. 205234, January, 1960.
INTEGRATED CIRCUITS / 449
7. King, D., and L. Stern: Designing Monolithic Integrated Circuits, Semicond. Prod.
Solid State TechnoL, March, 1965.
8. "Custom Microcircuit Design Handbook," Fairchild Semiconductor, Mountain
View, Calif., 1963.
9. Phillips, A. B.: Designing Digital Monolithic Integrated Circuits, Motorola
Monitor, vol. 2, no. 2, pp. 1827, 1964.
10. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," pp. 330
334, McGrawHill Book Company, New York, 1965.
11. Baker, O. R.: Aspects of Large Scale Integration, 1967 IEEE Intern. Conv. Dig.,
pp. 376377, March, 1967.
12. Weber. S.: LSI: The Technologies Converge, Electronics, vol. 40, no. 4, pp. 124127,
February, 1967.
13. Farina, E. D., and D. Trotter: MOS Integrated Circuits, Electronics, vol. 38, no. 20,
pp. 8495, 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.: Beamlead Technology, BeU System Tech. J., February, 1966
pp. 233253.
16/UNTUNED
AMPLIFIERS
Frequently the need arises for amplifying a signal with a minimum
of distortion. Under these circumstances the active devices involved
must operate linearly. In the analysis of such circuits the first step is
the replacement of the actual circuit by a linear model. Thereafter
it becomes a matter of circuit analysis to determine the distortion
produced by the transmission characteristics of the linear network.
The frequency range of the amplifiers discussed in this chapter
extends from a few cycles per second (hertz), or possibly from zero,
up to some tens of megahertz. The original impetus for the study
of such wideband amplifiers was supplied because they were needed
to amplify the pulses occurring in a television signal. Therefore such
amplifiers are often referred to as video amplifiers. Basic amplifier
circuits are discussed here. Modifications of these configurations to
extend the frequency range of these amplifiers are considered in Ref. L
In this chapter, then, we consider the following problem : Given
a lowlevel input waveform which is not necessarily sinusoidal but
may contain frequency components from a few hertz to a few mega
hertz, how can this voltage signal be amplified with a minimum of
distortion?
We also discuss many topics associated with the general problem
of amplification, such as the classification of amplifiers, hum and noise
in amplifiers, etc.
161
CLASSIFICATION OF AMPLIFIERS
Amplifiers are described in many ways, according to their frequency
range, the method of operation, the ultimate use, the type of lo* d '
the method of interstage coupling, etc. The frequency classification
$K. 161
UNTUNED AMPLIFIERS / 451
450
includes dc (from zero frequency), audio (20 Hz to 20 kHz), video or pulse
(up to a few megahertz), radiofrequency (a few kilohertz to hundreds of
megahertz), and ultrahighfrequency (hundreds or thousands of megahertz)
amplifiers.
The position of the quiescent point and the extent of the characteristic
that is being used determine the method of operation. Whether the transistor
or tube is operated as a Class A, Class AB, Class B, or Class C amplifier is
determined from the following definitions.
Class A A Class A amplifier is one in which the operating point and the
input signal are such that the current in the output circuit (in the collector,
plate, or drain electrode) flows at all times, A Class A amplifier operates
essentially over a linear portion of its characteristic.
Class B A Class B amplifier is one in which the operating point is at an
extreme end of its characteristic, so that the quiescent power is very small.
Hence either the quiescent current or the quiescent voltage is approximately
aero. If the signal voltage is sinusoidal, amplification takes place for only
onehalf a cycle. For example, if the quiescent outputcircuit current is zero,
this current will remain zero for onehalf 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 onehalf 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 onehalf of an
input sinusoidal signal cycle.
In the case of a vacuumtube 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 generalpurpose amplifiers. In general, the load
of an amplifier is an impedance. The two most important special cases are
the idealized resistive load and the tuned circuit operating near its resonant
frequency.
Class AB and Class B operation are used with untuned power amplifiers
'^hap. 18), whereas Class C operation is used with tuned radio frequency
^ttplifiers. Many important waveshaping functions may be performed by
'ass B or C overdriven amplifiers. This chapter considers only the untuned
*udio or video voltage amplifier with a resistive load operated in Class A.
452 / HECTRONtC DEVICES AUD CIRCUITS
Sec. 1<J,j
162
DISTORTION IN AMPLIFIERS
The application of a sinusoidal signal to the input of an ideal Class A amplifier
will result in a sinusoidal output wave. Generally, the output waveform ig
not an exact replica of the inputsignal 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. 182 and 183. 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 frequencyresponse 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 phaseshift 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.
163 FREQUENCY RESPONSE OF AN AMPLIFIER
A criterion which may be used to compare one amplifier with another wi
respect to fidelity of reproduction of the input signal is suggested by *
following considerations: Any arbitrary waveform of engineering importan
may be resolved into a Fourier spectrum. If the waveform is periodic,
spectrum will consist of a series of sines and cosines whose frequencies are
integral multiples of a fundamental frequency. The fundamental frequen J
is the reciprocal of the time which must elapse before the waveform rep«*
itself. If the waveform is not periodic, the fundamental period extends i
sense from a time — « to a time + °° ■ The fundamental frequency is c
Sec
163
UNTUNED AMPLIFIERS / 453
jjjfjnitcsimally small; the frequencies of successive terms in the Fourier series
differ by an infinitesimal amount rather than by a finite amount; and the
foil I'ii' i' series becomes instead a Fourier integral. In either case the spectrum
includes terms whose frequencies extend, in the general case, from zero fre
«iency to infinity.
Fidelity Considerations Consider a sinusoidal signal of angular fre
quency w represented by V m sin (wt + <£). If the voltage gain of the amplifier
)jas a magnitude A and if the signal suffers a phase lag B, then the output
will be
AV m sin (wt + — B) = AV m sin
[»H) + *]
Therefore, if the amplification A is independent of frequency and if the phase
shift 9 is proportional to frequency (or is zero), then the amplifier will preserve the
form of the input signal, although the signal will be delayed in time by an amount
D = 6/u>.
This discussion suggests that the extent to which an amplifier's amplitude
response is not uniform, and its time delay is not constant with frequency,
may perve as a measure of the lack of fidelity to be anticipated in it. In
prim i pie, it is really not necessary to specify both amplitude and delay response
rince, for most practical circuits, the two are related and, one having been
id, the other is uniquely determined. However, in particular cases, it
may well be that either the timedelay or amplitude response is the more
sensitive indicator of frequency distortion.
Lowfrequency Response Video amplifiers of either the transistor or tube
variety are almost invariably of the 2?Ccoupled 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 (lowfrequency) region, below the midband, an
am pliiier stage behaves (See. 165) like the simple highpass circuit of Fig. 161
°f time constant n = RiCi. From this circuit we find that
V„ =
ViRx
V,
Ri — jfwCi 1 — j/wRiCt
(161)
Th
ne voltage gain at low frequencies Ai is defined as the ratio of the output
'* ^1 A highpass RC circuit may be used to calcu
e the lowfrequency response of an amplifier.
f
Vt
L
IH
[R,
T
J
454 / ELECTRONIC DEVICES AND CIRCUITS
l Li Y 1 1 1
(<»
(6)
Fig. 162 [a\ A lowp Qss
BC circuit may be used t e
calculate the highfre
quency response of an
amplifier, (b) The Norton's
equivalent of the circuit i n
(a), where / = Vi/R t .
voltage F to the input voltage F,, or
1
where
A  V '
h =
2tt/liCi
The magnitude \Ai\ and the phase lag 6\ of the gain are given by
H.i =
Vi + (/i//) 2
$i — — arctan —
/
At the frequency f = f h A t = 1/V2 = 0.707, whereas in the midband
region (/» /i), A\ — * 1. Hence f x is that frequency at which the gain has
fallen to 0.707 times its midband value A Q . From Eq. (1221) 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 ZdB frequency. From Eq. (163) we see
that /i is that frequency for which the resistance R i equals the capacitive
reactance 1/2tt/iCi.
Highfrequency Response In the third (highfrequency) region, above
the midband, the amplifier stage behaves (Sec. 166) like the simple lowpa* 8
circuit of Fig. 162, with a time constant r% = R 2 Cz. Proceeding as above,
we obtain for the magnitude \Az\ and the phase lag 62 of the gain
where
it,
*■
1
VTTWW
f
h = arctan 7
h
(165)
(166)
Since at/ = / s the gain is reduced to l/\/2 times its midband value, then;'
is called the upper 3dB 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 lowfrequency range is to be *
in Fig. 163.
UNTUNED AMPLIFIERS / 455
Fig. 163 A loglog plot of the amplitude frequencyresponse characteristic of
an fi!Ccoupled amplifier.
Bandwidth The frequency range from fi to f 2 is called the bandwidth of
the amplifier stage. We may anticipate in a general way that a signal, all
of whose Fourier components of appreciable amplitude lie well within the
^ge/i to/i, will pass through the stage without excessive distortion. This
criterion must be applied, however, with caution. 2
l6 *4 THE tfCCOUPLED AMPLIFIER
* cascaded arrangement of commoncathode (CK) vacuumtube stages is
*°wn in Fig. 164a, of commonemitter (CE) transistor stages in Fig. 1646,
^d of commonsource (CS) FET stages in Fig. 164e. The output Y\ of one
7*86 is coupled to the input X% of the next stage via a blocking capacitor
* which is used to keep the dc component of the output voltage at Fi from
lining the input X 2 . The resistor R a is the grid (gate) leak, and the plate
Rector) (drain) circuit resistor is R p (R c ) (R d ). The cathode resistor R k , the
r ce resistor R„ the emitter resistor R t> the screen resistor R K , and the resis
^ °i and R2 are used to establish the bias. The bypass capacitors, used
. Prevent loss of amplification due to negative feedback (Chap. 17), are C*
*j/* e cathode, C, in the emitter, C, in the source, and C, e in the screen circuit.
Present are interelectrode capacitances in the case of a tube, and junction
456 / ELECTRONIC DEVICES AND CIRCUITS
S * ld.4
From C
preceding 0*~
stage
(a)
From d
preceding O *■ (
stage
(c)
From C,
preceding O  ( O
stage
(b)
Fig, T64 A cascade of (a) commoncathode (CK) pentode stages; (b) common
emitter (CE) transistor stages; (c) commonsource (CS) FET stages.
capacitances if a transistor is used. These are taken into account when *•
consider the highfrequency 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 smallsignal models are used throughout this chap**'"
S*
Pfl
165
165 A schematic representa
UNTUNED AMPLIFIERS / 457
tion of «'™er a tube, FET, or transis
tor stage. Biasing arrangements
and suppfy voltages are not indi
cated.
'Mi
165
LOWFREQUENCY RESPONSE OF AN ftCCOUPLED STAGE
The effect of the bypass capacitors C k> C t , and C, on the lowfrequency charac
teristics is discussed in Sec. 1610. 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. 164 may be represented sche
matically as in Fig. 165. The resistor Rb represents the gridleak 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 lowfrequency 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. 166a. 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. 112; 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
timeconstant highpass circuit of Fig. 1666 results. Hence, from Eq. (163),
ft
o—
Q)
Mt
Ri
<«) *
^g. 1 66 o) The lowfrequency model of an flCcoupled 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& — jBiKs,
K = R e , and Ri « h it . Also, R = Ri\\R b and R'„  R B \\R V .
458 / ELECTRONIC DEVICES AND CIRCUITS
the lower 3dB frequency is
1
/*
2tt(R'„ + #;.)<?(,
$<*. T^
(167)
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 vacuumtube 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 3dB frequency of not more than 10 H»
for an flCcoupled 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. (167) we have
fl = 2t(R' + R'jC b ~ 10
or
C*> s r
" Q2.8(K + R<)
Since R' 4 = 1 M and R' < R v = \ K, then R'„ + R' t « 1 M and d > 0.016
mF.
&. From Eq. (1134) we find for a transistor R > \/h et = 40 K, and hence
R' e « R c « 1 K. If we assume that R b » fl; = I K, then J2^ « 1 K. Henw
1
Ci, >  F = 8.0 uF
(62.8) (2 X 10 3 ) M
Note that because the input impedance of a transistor is much smaller
than that of a FET or a tube, a coupling capacitor is required with the
transistor which is 500 times larger than that required with the FET or tube.
Fortunately, it is possible to obtain physically small electrolytic capacitors
having such high capacitance values at the low voltages at which transistors
operate.
166 HIGHFREQUENCY RESPONSE OF A VACUUMTUBE STAGE
For frequencies above the midband range we may neglect the reactance of the
large series capacitance C b . However, we must now include in Fig. 164 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 highfrequency model of Fig. 167 can be drawn. In order to keep &*
S*
166
UNTUNED AMPLIFIERS / 439
f a 167 The highfrequency model of
ftCco 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. 811 and 813). 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. 1626), the
amplifier stage at high frequencies behaves like a singletimeconstant low
pass circuit, where Ct — C and R% = R = r p \\R v \\R a .
Hence, from Eq. (166), the upper 3dB frequency ft is given by
A
i
2vRC 2irRJJ
(168)
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
(169)
GainBandwidth Product The upper 3dB 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 3dB 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. (168) and (169) we have, since
C  Ci + C„
F  \A e \f* =
th
(1610)
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 gainbandwidth
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
"torit F may be interpreted as giving the maximum possible bandwidth obtain
&lD to with a given tube if R p is adjusted for unity gain. For video pentodes
*toh as the 6AK5, 6BH6, 6AU6, 6BC5, and 6CL6, values of g m ranging from
to 1 1 millimhos (mA/V,) and values of Co + d from 7 to 20 pF are obtain
JJfe. The value of F for all these tubes lies between 80 and 120 MHz, with
e 6AK5 having the largest value.
460 / ELECTRONIC DEVICES AND CIRCUITS
S«c.
>«*
An amplifier with a gain of unity is not very useful. Hence let us assum
that 4„ is at least 2. Then f 2 = F/\A e \ = 60 MHz for the 6AK5 tube, i
a practical circuit, the inevitable extra stray capacitance might easily redu
the bandwidth by a factor of 2. Hence we may probably take a bandwidth
of 30 MHz as a reasonable estimate of a practical upper limit for an uncom.
pensated tube amplifier using lumped parameters. If the desired gain is \n
instead of 2, the maximum 3dB 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!
tocathode 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 3dB 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 cathoderay tube).
The equivalent circuit of a FET is the same as that of a triode (Fig.
149). Hence the input capacitance of an internal stage may be very large
because of the Miller effect (Sec. 812). This shunting capacitance limits
the bandwidth of a FET.
167
CASCADED CE TRANSISTOR STAGES
The highfrequency analysis of a singlestage CE transistor amplifier, or the
last stage of a cascade, is given in detail in Sees. 137 and 138. 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. 168. We omit from
this diagram all supply voltages and components, such as coupling capacitor*.
Fig. 168 An infinite cascade of CE stages. The dashed, shaded rectangle (blocW
encloses one stage.
$K.
167
UNTUNED AMPLIFIERS / 461
which serve only to establish proper bias and do not affect the highfrequency
response. The collectorcircuit 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. 1646 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 collectorcircuit resistance. A complete
g t a ge from collector to collector is included in the shaded block. We define the
current gain of the stage to be An m It/1%, Each stage behaves like a current
generator of impedance R. = R e delivering current to the following stage.
We define the voltage gain to be A v = V a /Vi. Since we have specified Vi as
the voltage precisely at the stage input, then A v is the gain for an ideal volt
age source. We now prove that At, m Av for an infinite cascade of similar
stages.
In a long chain of stages the input impedance Z t between base and emitter
of each stage is identical. Let Z[ represent Z» in parallel with R c . Accord
ingly, Z'i = Vi/Ii = Vi/h, so that h/h = An =V % /V l = A v in this special
case.
We now calculate this gain A Tt = A v = A. For this purpose Fig. 160
shows the circuit details of the stage in the shaded block in Fig. 168. 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 169
is obtained from Fig. 1312a. The elements involving g b >c have been omitted
since, as demonstrated in Sec. 138, their omission introduces little error.
The gain A e = I%fl\ at low frequencies is given by Eq. (1350), except
with R m replaced by R e , and we have
j. _ — h/,R e
R e + hft
(1611)
To calculate the bandwidth we must evaluate K. From Fig. 169 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 zerofrequency value of K. We show
c 2
^AA
B' 3
~iT
R c < r b .<
c.
A 1
E
c e aK)
^'9 169 The equivalent circuit of the enclosed stage of Fig. 168 (K = V^/V*,).
462 / FircTRONJC DEVICES AND CIRCUITS
s «. 16.*
below that the response obtained experimentally is somewhat better than that
predicted by this analysis, and hence that we are erring in the conservative
direction. At zero frequency, K = K B = g m R L , in which Rl is the *
tive load on the transistor from C to E and consists of R e in parallel with
r »* + n't = hi,. Therefore
Rl =
RJij
(1612)
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 singletimeconstant
circuit. Hence we can calculate the 3dB 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 3dB frequency is
1
/t
2t/bc
(1614)
(1615)
This halfpower 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 gainbandwidth 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. (1612).
(1616)
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. (1612) and also reduces R in Eq. (1614) 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. (1615). It may be that a decrease
in gain is more than compensated for by an increase in /,. To investigate
1
J 67
UNTUNED AMPLIFIERS i 463
this point we differentiate the gainbandwidth 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 gainbandwidth product is obtained
is designated by (R e ) ovt and is given by
{flclopt —
K
y/x 1
with
hr.C.
C, + C c r»'
(1617)
(1618)
In Fig. 1610 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. (1617)] 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.
1310).
Note in Fig. 1610 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. 139),
the gainbandwidth product takes on some importance as a figure of merit.
/a, MHz \AJ lAJsl , MHz
I!)
M
41)
\AJ 2 \
\A B \
M
UAM
20
TV
10
IA 4 *
1
r 1
>
500
1,000
1,500
2,000
R c , n
Fig. 1610 Gain \A„\, bandwidth / 2 , and gainbandwidth product \A„f,\
as a function of R e for one stage of a CE cascade. The transistor
parameters are given in Sec. 135.
464 / ELECTRONIC DEVICES AND CIRCUITS
Sec. Td>
For our typical transistor, f T = 80 MHz, whereas the constant value of 4«/J
in Fig. 1610 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 hybridII 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 gainbandwidth 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 <
(1619)
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. (1611). Then the bandwidth which will be obtained is found
from Eq. (1615). On the other hand, if the desired bandwidth is specified,
then/ 2 substituted into Eq. (1615) 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. (1612). 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. 1610 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. 139 for a single stage apply, provided that the load
Rl is taken as the collectorcircuit resistance in parallel with the input resistance
of the second stage:
R c h it
Rl = *
^e + hi.
For the last stage in a cascade use the formulas for a single stage, with A
Sec
168
UNTUNED AMPLIFIERS / 465
equal to the collectorcircuit resistance R e of the preceding stage and with Rl
equal to the R e of the last stage.
168
STEP RESPONSE OF AN AMPLIFIER
\n alternative criterion of amplifier fidelity is the response of the amplifier
to a particular input waveform. Of all possible available waveforms, the most
generally useful is the step voltage. In terms of a circuit's response to a step,
the response to an arbitrary waveform may be written in the form of the
superposition integral. Another feature which recommends the step voltage
is the fact that this waveform is one which permits small distortions to stand
out clearly. Additionally, from an experimental viewpoint, we note that
excellent pulse (a short step) and squarewave (a repeated step) generators
are available commercially.
As long as an amplifier can be represented by a singletimeconstant 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 highfrequency response.
Similarly, there is a close relationship between the lowfrequency response
and the distortion of the flat portion of the step. We should, of course,
expect such a relationship, since the highfrequency response measures essen
tially the ability of the amplifier to respond faithfully to rapid variations in
fflgnal, whereas the lowfrequency 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 lowpass circuit of Fig. 162 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 steadystate value V, as shown in Fig. 1611. The output
'9.1611 Stepvoltage response
of *e lowpass RC circuit. The
Se f ime t r is indicated.
.
466 I ELECTRONIC DEVICES AND CIRCUITS
is given by
Sec. ]fi.j
v„ = 7(1  e" R ' c ») (1620)
The time required for v B to reach onetenth of its final value is readily found
to be O.IR2C2, and the time to reach ninetenths 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. 1611, The time U is an indication of how fast the
amplifier can respond to a discontinuity in the input voltage. We have, using
Eq. (166),
2.2 0.35
t T — li.Z/f2t'2 —
2tt/ 2
(1621)
Note that the rise time is inversely proportional to the upper 3dB 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 highpass circuit
of Fig. 161, the output is
v = Fe" fi . c .
(1622)
For times t which are small compared with the time constant R1C1, the response
is given by
F ( x «k)
(1623)
From Fig. 1612 we see that the output is tilted, and the percent tilt or aag
in time h is given by
P =
V
X 100 =
R\C)
X 100%
(1624)
It is found 6 that this same expression is valid for the tilt of each half cycle
of a symmetrical square wave of peaktopeak 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. (163), we may express P in the form
r X 100 = j^r x 100 = ^ X 100%
%RlCl
(1625)
Fig. 1612 The response t>„, when
a step v % is applied to a highp° sS
RC circuit, exhibits a tilt.
S«. J 69
UNTUNED AMPLIFIERS I 467
pjote that the tilt is directly proportional to the lower 3dB frequency. If
w e wish to pass a 50Hz square wave with less than 10 percent sag, then fi
must n °t exceed 1.6 Hz.
Squarewave Testing An important experimental procedure (called
squarewave testing) is to observe with an oscilloscope the output of an amplifier
excited by a squarewave 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 squarewave frequencies, to
examine individually the highfrequency and lowfrequency distortion. For
example, consider an amplifier which has a highfrequency time constant of
1 Msec and a lowfrequency 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 steadystate
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 frequencyresponse 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 steadystateresponse circumstances where transient cal
culations are extremely cumbersome. Finally, it happens occasionally that
a Q amplifier is required whose characteristics are specified on a frequency
basis, the principal emphasis being to amplify a sine wave.
T *~9 BANDPASS OF CASCADED STAGES
*he upper 3dB frequency for n cascaded stages is/a <n> and equals the frequency
[° r which the overall voltage gain falls to l/V 2 " (3 dB) of its midband value.
*hus/,w is calculated from
Vi + </2 ( »V/ a ) 2 J
\/2
468 / ELECTRONIC DEVICES AND CIRCUITS
to be
f.OO . .
J ~ = V2" R  1
Sec. 76f0
(1626)
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 10kHz stages give a resultant upper 3dB frequency of 54
kHz, etc.
If the lower 3dB frequency for n cascaded stages is /i (n \ then correspond
ing to Eq. (1626) we find
h
V2 1 '"  1
(1627)
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 loglog 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. 163. Hence
every time the frequency / doubles (which, by definition, is one octave), the
response drops by 6 dB. For an ?istage 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
outputsignal rise time t T is given (to within 10 percent) by
tr « 1.1 vV + W + W +
+ *r
(1628)
If, upon application of a voltage step, one PCcoupling 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.
1610 EFFECT OF AN EMITTER (OR A CATHODE) BYPASS
CAPACITOR ON LOWFREQUENCY RESPONSE
If an emitter resistor R e is used for selfbias 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. 1646, It is shown below that the effect of this capacitor is *°
affect adversely the lowfrequency response.
Consider the single stage of Fig. 161 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. 127 is valid. The equivalent circuit subjec
$K.
1610
UNTUNED AMPLIFIERS / 469
Fig. 1613 (a) An amplifier with a bypassed emitter resistor; (b) the lowfrequency
simplified Aparameter model of the circuit in (a).
to these assumptions is shown in Fig. 16136. The blocking capacitor C b is
omitted from Fig. 16136; its effect is considered in Sec, 165.
The output voltage V is given by
Vo m Ithf.R* = 
V,h /t R c
where
R, + k ie + Z' t
z;  (1 + h f .)
R.
1 + jtaC.R,
(1629)
(1630)
Substituting Eq. (1630) in Eq. (1629) and solving for the voltage gain A v,
we find
7. h f ,R, 1 ( foCiR,
A *T.= 
R + R , 1 . „ R t R
1 +j(>>C,
R + R'
where
R a R. + K and IP m ( + h f ,)R,
The midband gain A is obtained as n — » « , or
* _ h/gR c _ —hf e R e
Ao __
Hence
Where
/.
R R, 4* h ie
1 1 +■?/ //»
1 + R'/R) + jf/f P
1
, = 1 + R'/R
jp —
(1631)
(1632)
(1633)
(1634)
(1635)
°te that f determines the zero and f p the pole of the gain A v /A e . Since
dually R'/R » l, then f p » / , so that the pole and zero are widely separated.
470 / aECTRONJC DEVICES AND CIRCUITS
See. 1610
For example, assuming R, = 0, R, = 1 K, C* = 100 nF, h fe = 50, h it = 1.1 ^
and tf e = 2 K, we find /«  1.6 Hz and /, = 76 Hz.
A plot of 20 log \A v/A \ versus log / is indicated in Fig. 1614. The piece
wise linear curve shown dashed indicates the asymptotic behavior of the fre
quency response. This dashed characteristic is constant at — 20 log (1 + R'/R)
for /</<,; it increases linearly at 6 dB/octave for f e < f < /„, and remains at
dB for / > /„. Remembering that f p ;$> /„ and using Eqs. (1634) and
(1635), the magnitude of A v /A becomes, for / = f p ,
Av
1
UIU
1 + R'/R vT+T
l
Hence / = f p is that frequency at which the gain has dropped 3 dB. Thus
the lower 3dB frequency /j is approximately equal to f p . If the condition
fp » fa is not satisfied, then /i ?* /,,. As a matter of fact, a 3dB frequency
may not exist (Prob. 1629).
Squarewave Response Since the network in Fig. 1613 is a singletime
constant circuit, the percentage tilt to a square wave is given by Eq, (1625), 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%
(1636)
(1637)
<si*c
/
/
10
V
//
20
fi
30
20
log
("
R'\ /'
33.5
•^
..^..6
76^
i i
1.0 f„
/plOO
1,000 /, Hz
Fig. 1614 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. 1610
UNTUNED AMPLIFIERS / 471
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
obtain
(51)(100)
c* =
(2)(50)(1,100)(10)
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. 1615 must be used. An
analysis of this circuit (Prob. 1630) yields
Av _ 1 1+j///.
A e "
where
1+17ifcl+tf//,
A = —g m Ri, f e =
(1638)
2irCkRk
, m 1 + QmRh
zirCkRk
(1639)
These equations are analogous to Eqs. (1634) and (1635), and the frequency
response is of the form indicated in Fig. 1614. If g m R k » 1, the pole and
zero frequencies are widely separated, and hence /i « f p . Then, from Eq.
(1625), it follows that the percentage tilt to a square wave of frequency / is
P = ^ X 100 =
1 + 9mRk
2CkRkf
X 100
2C k f
X100%
(1640)
Note that for g m R k » 1, P is independent of R k . If g m for a pentode is 5 mA/V
(onetenth that of a transistor), then for no more than a 10 percent output tilt
with a 50Hz squarewave 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. 1615 The equivalent circuit of a pen
°de stage with a cathode impedance.
472 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 76.li
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 3dB 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
3dB frequency for the stage.
1611
SPURIOUS INPUT VOLTAGES
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 often being very difficult to determine.
The undesired, or parasitic, oscillation may occur with any type of circuit,
such as audio, video, or radiofrequency amplifier, oscillator, modulator, pulse
waveform generating circuits, etc. Parasitic oscillations are particularly prev
alent with circuits in which (physically) large tubes are used, tubes or transis
tors are operated in parallel or pushpull, and in power stages. 7 The frequency
of oscillation may be in the audio range, but is usually much higher and often is
so high (hundreds of megahertz) that its presence cannot be detected with an
oscilloscope.
Parasitic oscillations can usually be eliminated by a change in circuit
parameters, a rearrangement of wiring, some additional bypassing or shield
ing, a change of tube or transistor, the use of a radiofrequency inductor in the
output circuit, radiofrequency chokes in series with filament leads, etc. A
small resistance (50 to 1,000 fi) placed in series with a grid and as close to the
grid terminal as possible is often very effective in reducing highfrequency
oscillations in a tube.
Hum Even if an amplifier is not oscillating, undesirable output voltage 8
may be present in a vacuumtube amplifier in the form of hum from the use
of ac heated filaments. 8 There are several sources of this hum:
1. The magnetic field produced by the filament current will deflect tfl
electron stream. During some portion of each half cycle the electrons ma?
S«c. 1612
UNTUNED AMPLIFIERS / 473
be deflected to such an extent as to miss the plate. A 120Hz hum results
(if a 60Hz power source is used).
2. Effective capacitances exist between each side of the heater and the
grid. If these capacitances are not equal, an effective 60Hz voltage is
impressed upon the grid.
3. The heatercathode insulation is not infinite. If selfbias is used,
leakage will take place from the heater through this insulation resistance in
series with the cathode impedance Z k , The voltage across Z k appears as hum.
4. The heating and cooling of the cathode, because the heating power is
periodic, introduces a 120Hz hum. This would not be true if the plate cur
rent were strictly spacechargelimited since it would then be independent of
the temperature of the emitter. However, some parts of the cathode are at
a low enough temperature so that some temperaturelimited current exists.
Furthermore, the effect of the initial velocities is a function of temperature.
Hum from the above sources can be eliminated completely only by using
dc heating power.
In addition to hum that is inherent in the heater construction, some hum
may appear from pickup resulting from the stray magnetic fields of the power
transformer or from the fields produced by the heater current in the connecting
leads. The effect of the former is negligible with properly shielded trans
formers, and that of the latter may be reduced if the heater leads are twisted.
There may also be electrostatic pickup from the ac line. Finally, there is the
possibility of pickup of radiofrequency signals radiated through space. These
spurious voltages can often be eliminated by proper shielding or bypassing.
Some of these sources of hum cause difficulties with transistors as well as
with tubes. It should be emphasized that hum troubles are usually of
importance only in the first stage of a highgain amplifier, for the small spu
rious voltages introduced in this stage are amplified by all succeeding stages.
Microphonics The spurious output voltages caused by the vibrations of
the electrodes arising from mechanical or acoustical jarring of the tube are
called microphonics. Some tubes are much more microphonic than others of
presumably identical construction, and this source of trouble can often be
eliminated by changing tubes. In many cases, it is necessary to mount the
tubes in rubber or in special supports. In addition, special tubes are avail
able in which the microphonic effect, and also the heater hum effects outlined
above, have been minimized. A transistor is, of course, completely non
j^crophonic because there can be no mechanical motion between the emitter,
ba se, and collector.
16 ~12 NOISE
It *
is found that there is an inherent limit to the amplification obtainable
0tt » an amplifier even after the abovementioned sources of hum have been
474 / ELECTRONIC DEVICES AND CIRCUITS
Sec. 1 6 J J
eliminated. Under these conditions, the output of the amplifier, when there
is no impressed input signal, is called amplifier noise. 9 If, therefore, only a
very small voltage is available, such as a weak radio, television, radar, etc.
signal, it may be impossible to distinguish the signal from the background
noise. The term noise arises from the fact that with no input, the output
of an audio amplifier with the gain control set at a maximum is an audible
hiss, or crackle. In the case of a video amplifier the term snow is often used
in place of noise because of the snowlike appearance on a TV screen when
the set is tuned to a weak station. The various noise sources in an amplifier
are now considered.
Thermal, or Johnson, Noise The electrons in a conductor possess varying
amounts of energy by virtue of the temperature of the conductor. The slight
fluctuations in energy about the values specified by the most probable distribu
tion are very small, but they are sufficient to produce small noise potentials
within a conductor. These random fluctuations produced by the thermal
agitation of the electrons are called the thermal, or Johnson, noise. The rms
value of the thermal resistance noise voltage V n over a frequency range
/2 — /i is given by the expression
V n * m AkTRB (1611)
where k — Boltzmann constant, J/°K
T = resistor temperature, °K
R = resistance, fi
B = ji — fi = bandwidth, Hz
It should be observed that the same noise power exists in a given bandwidth
regardless of the center frequency. Such a distribution, which gives the same
noise per unit bandwidth anywhere in the spectrum, is called white noise.
If the conductor under consideration is the input resistor to an ideal
(noiseless) amplifier, the input noise voltage to the amplifier is given by Eq.
(1641). An idea of the order of magnitude of the voltage involved is obtained
by calculating the noise voltage generated in a 1M resistance at room tem
perature over a 10kHz bandpass. Equation (1641) yields for V n the value
13 mV. Clearly, if the bandpass of an amplifier is wider, the input resistance
must be smaller, if excessive noise is to be avoided. Thus, if the amplifier
considered is 10 MHz wide, its input resistance cannot exceed 1,000 fi, if t,ie
fluctuation noise is not to exceed that of the 10kHz audio amplifier.
It is obvious that the bandpass of an amplifier should be kept as low tf
possible (without introducing excessive frequency distortion) because the noi#
power is directly proportional to the bandwidth. The noise output squared
from the amplifier due to R, only is given by Eq. (1641) provided that the
value of W is multiplied by \A Vo \ 2 and that the noise bandwidth B is defined
by
S«. 1«I2
UNTUNED AMPLIFIERS / 475
where Avo is the midband value of the voltage gain Av(f). We thus see that
the noise bandwidth given by Eq. (1642) may be different from the amplifier
voltage gain bandwidth.
Shot Noise Among the various possible sources of noise in a tube, one of
the most important is the shot effect. Normally, one assumes that the current
in a tube under dc conditions is a constant at every instant. Actually,
however, the current from the cathode to the anode consists of a stream of
individual electrons, and it is only the time average flow which is constant.
These fluctuations in the number of electrons emitted constitute the shot noise.
If the cathode emission is temperaturelimited, the rms noise current /„ in
a diode is given by the expression
I»> = 2eI P B
(1643)
where e = electronic charge, C
Ip = emission current, A
B = bandwidth, Hz
If the load resistor is R, a noise voltage of magnitude I n R will appear across
the load. Temperaturelimited diodes are used as constantcurrent whitenoise
generators for test purposes.
If the tube is spacechargelimited, the irregularities in emission are
decreased, and the platecurrent fluctuation is much less in a spacecharge
limited tube than in one which is temperaturelimited. This fact is explained
qualitatively by the automaticvalve action of the spacecharge cloud in the
neighborhood of the cathode, as discussed in Sec. 71. The spacecharge
limited noise power is of the order of 10 percent of the temperaturelimited value.
Other Noise Sources in Tubes In addition to Johnson and shot noise,
there are the following physical mechanisms for the generation of extraneous
signals in a tube: gas noise, caused by the random ionization of the few mole
cules remaining in the tube; secondaryemission noise, arising from the random
variations of secondary emission from the grid and plate ; flicker noise, caused
by the spontaneous emission of particles from an oxidecoated cathode, an
effect particularly noticeable at low frequencies; and induced grid noise, result
lf ig from the random nature of the electron stream near the grid. In addition
to these, we also have in a pentode partition noise, which arises from the random
fluctuation in the current division between the screen and the plate. Because
of this partition effect, a pentode may be much noisier (perhaps by a factor
°f 10) than a triode. Hence the input stage to a highgain amplifier is usually
a triode. We should note that it is the input stage whose noise must be kept
extremely low, because any noise generated in this tube is amplified by all
the following stages.
Noise Figure A noise figure NF has been introduced in order to be able
10 specify quantitatively how noisy a circuit is. By definition, NF is the ratio
476 / ELECTRONIC DEVICES AND CIRCUITS
Sec. J 6? J
of the noise power output of the circuit under consideration to the noise power
output which would be obtained in the same bandwidth if the only source
of noise were the thermal noise in the internal resistance R, of the signal source.
Thus the noise figure is a quantity which compares the noise in an actual
amplifier with that in an ideal (noiseless) amplifier. Usually, NF is expressed
in decibels.
We define the following symbols:
Spi (Syi) = signal power (voltage) input
Npi (N Yi ) = noise power (voltage) input due to R,
Sp e (Svo) ~ signal power (voltage) output
N P o (Nvo) = noise power (voltage) output due to R t and any noise
sources within the active device
From Eq. (1641), N ri = V n = (4&TR.B)*.
From the definition of noise figure
total noise power output = lfl . N Po (iq_aa\
noise power output due to R t A P N P j
NF m 10 log
where the power gain of the active device is A P = Sp /Spi. Hence,
NF  10 log ^#? = 10 log Spi/Npi
Sntfi
S P ,/N
Pa
(1645)
The quotient Sp/Np is called the signaltonoise power ratio. The noise
figure is the input signaltonoise power ratio divided by the output signalto
noise power ratio. Expressed in decibels, the noise figure is given by the
input signaltonoise power ratio in decibels minus the output signaltonoise
power ratio in decibels. Since the signal and noise appear across the same
load, Eq. (1645) takes the form
NF = 20 log
Sn/Nv
S
Si
« 20 log £^20 log £2
Sve/Ny, 6 N Yi 6 N Vo
where Sv/Ny is called the signaltonoise voltage ratio.
(1646)
Measurement of Noise Figure A very simple method 10 for measuring
the noise figure of an active device Q is indicated in Fig. 1616. An audio
sinusoidal generator V, with source resistance R t is connected to the input of
Fig. 1616 A system used to measure the noise figure of an active
device Q.
Sec 16} 2
UNTUNED AMPLIFIERS / 477
q. The active device is cascaded with a lownoise amplifier and a filter, and
the output of this system is measured on a true mis reading voltmeter M.
The experimental procedure for determining NF is as follows :
1. Measure R t and calculate N yi = V n from Eq. (1641). The bandwidth
B is set by the filter.
2. Adjust the audio signal voltage so that it is ten times the noise voltage:
V, «■ 10F„ or Svi — lOiVV,. Measure the output voltage with M. For such a
large signaltonoise ratio {Svi/Nvt m 20 dB) we may neglect the noise and
assume that the voltmeter reading gives the signal output voltage Sy .
3. Set V s ■» and measure the output voltage JW« with M.
4. From Eq. (1646) the noise figure is given by
NF = 20  20 log «S
Nvo
(1647)
where Sv<, and Nv* are the meter readings obtained in measurements 2 and 3,
respectively.
The lownoise amplifier is required only if the noise output of Q is too
low to be detected with M. It should be pointed out that the amplifier
filter combination does not affect NF (for a given B) since the ratio Sve/N Y o
is used in Eq. (1647).
The accuracy of the method described is based on the assumption that
the output signal and noise can be measured separately. This is not strictly
true since the noise cannot be turned off while measuring the output signal.
It is found 10 that for a 20dB input signaltonoise ratio, transistor noise
figures may be measured up to 10 dB with less than 0.5 dB error. The larger
the Svi/Nn t the smaller is the error in this measurement. Usually the output
signal voltage is monitored on an oscilloscope to make certain that the system
operates linearly so that no clipping takes place and no 60Hz hum is present.
If a filter with a very narrow bandwidth (a few hertz) is used, the fore
going measurement gives the spot, singlefrequency, or incremental noise
figure. On the other hand, if the filter bandwidth is large (from ft = 10 Hz
to Si = 10 kHz), then the circuit of Fig. 1616 gives the broadband or inte
grated noise figure. Other methods of measuring NF are available, ll  ls but
these have the disadvantage of requiring a calibrated noise generator.
Transistor Noise 13 In addition to thermal noise in a transistor, there is
ft oise due to the random motion of the carriers crossing the emitter and collector
Junctions and to the random recombination of holes and electrons in the base.
Ihere is also a partition effect arising from the random fluctuation in the
division of current between the collector and base. It is found that a transistor
does not generate white noise, except over a midband region. Also, the amount
01 noise generated depends upon the quiescent conditions and the source
re si stance. Hence, in specifying the noise in a transistor, the center frequency,
the operating point, and R, must be given.
478 / ELECTRONIC DEVICES AND CIRCUITS
Sec. T6F2
.a
o
Z 2.0
T
1 1
5.0 V
^ Bandwidth = 15,7 kHz
T
^/ C = 10M
100 s
10
m 8.0
•a
of
I 6 "
$4.0

Z
2,0
'
P«
i
i i "
5.0V
\
^j
C =20 M A
R,=
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1
;
\
X
r
!1 I III
J ( . = 0.5 mA, R t =
l.OK
H s
*%„
..
1.0 5.0 10 50 100
Source resistance R„ K
(a)
l.OK IOK 100K l.OM
Frequency, Hz
(b)
10M
Fig. 1617 Noise figure of a 2N3964 transistor, (a) Broadband NF as a function
of source resistance; (b) spot NF as a function of frequency. (Courtesy
of Fairchild Semiconductor Corp.)
Figure 16I7a and b show the noise figure vs. source resistance and fre
quency for the 2N3964 diffused planar resistor. There are three distinct
regions in Fig. 16166. At low frequencies the noise varies approximately
as 1/f, and is called excess or flicker noise. The source of this noise is not
clearly understood, but is thought to be caused by the recombination and
generation of carriers on the surface of the crystal. In intermediate frequencies
the noise is independent of frequency. This white noise is caused by the
bulk resistance of the semiconductor material and the statistical variation of
the currents (shot noise). The third region in Fig. 1617& is characterized
by an increase of the noise figure with frequency, and is essentially caused
by a decrease in power gain with frequency. 12
FET Noise 14 The fieldeffect transistor exhibits excellent noise charac
teristics. The main sources of noise in the FET are the thermal noise of the
conducting channel, the shot noise caused by the gate leakage current, and the
1// noise caused by surface effects. The FET is also superior, from a noise
point of view, to a vacuum tube of comparable transconductance. 12
The noise figure vs. frequency for the 2N2497 FET transistor is shown
Fig. 1618 The spot noise figure
for a 2N2497 FET. (Courtesy of
Texas Instruments, Inc.)
12
g 10
« 8
u
IS 8
J A.
111 "
v D , 
1 Hill
5V
1 D =  1mA
ff«=lM
T A =
t\
5
= C
ill!
*
Z
2
'
I
0.01
kHz
S«e. 16 1 2
UNTUNED AMPLIFIERS / 479
in Fig. 1618. It should be pointed out that, unlike the bipolar transistor,
the noise figure of the FET is essentially independent of the quiescent point
(I D and V D s).
REFERENCES
1. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," chap. 5,
McGrawHill Book Company, New York, 1965.
2. Ref. 1, sec. 44.
3. Ref. 1, sec. 412.
4. Ref. 1, sec. 58,
5. Johannes, V.: "Transient Response of Transistor Video Amplifiers," sec. 1, Uni
versity Microfilms, Ann Arbor, Mich., 1961,
6. Ref, 1, sec. 21.
7. Terman, F. E.: "Electronic and Radio Engineering," 4th ed., pp. 503506,
McGrawHill Book Company, New York, 1955.
8. RCA Appl. Note 88.
9. Van der Ziel, A. : "Noise," PrenticeHall, Inc., Englewood Cliffs, N.J., 1954.
Ref. 7, pp. 434442, 79679S.
Seely, S.: "Radio Electronics," pp. 143149, McGrawHill Book Company, New
York, 1956.
Valley, G. E., Jr., and H. WaUman (eds.): "Vacuum Tube Amplifiers," MIT
Radiation Laboratory Series, vol. 18, pp. 496720, McGrawHill Book Company,
New York, 1948.
10. Miller, J, R. (ed.): "Solidstate Communications," Texas Instruments Electronic
Series, pp. 194197, McGrawHill Book Company, New York, 1966.
11. Terman, F. E., and J. M. Pettit: "Electronic Measurements," pp. 362379,
McGrawHill Book Company, New York, 1952.
12. Dosse, J.: "The Transistor," 4th ed., pp. 144152, D. Van Nostrand Company,
Inc., Princeton, N.J., 1964.
13. Thornton, R. D., et al.: "Characteristics and Limitations of Transistors," Semi
conductor Electronics Education Committee, vol, 4, John Wiley & Sons, Inc., New
York, 1966.
U. Van der Ziel, A.: Thermal Noise in Field Effect Transistors, Proc. IRE, vol. 50,
pp. 18081812, August, 1962.
Van der Ziel, A.: "Electronics," chap. 23, Allyn and Bacon, Inc., Boston, 1966.
Sevin, L. J.: "Fieldeffect Transistors," pp. 4650, McGrawHill Book Company,
New York, 1965.
/FEEDBACK AMPLIFIERS
AND OSCILLATORS
In this chapter we introduce the concept of feedback and show how to
modify the characteristics of an amplifier by combining a portion of
the output signal with the external signal. Many advantages are to
be gained from the use of negative (degenerative) feedback, and these
are studied. It is possible for the feedback to be positive (regener
ative), and the circuit may then oscillate. Examples of feedback
amplifier and oscillator circuits are given.
171
CLASSIFICATION OF AMPLIFIERS
Before proceeding with the concept of feedback, it is useful to classify
amplifiers into four broad categories, 1 as either voltage, current, trans
conductance, or transresistance amplifiers. This classification is based
on the magnitudes of the input and output impedances of an amplifier
relative to the source and load impedances, respectively.
Voltage Amplifier Figure 17la shows a Thevenin's equivalent
circuit of a twoport network which represents an amplifier. If the
amplifier input resistance /?, is large compared with the source resistance
R„ then V { = V a . If the external load resistance R L is large compared
with the output resistance R of the amplifier, then V = A„V\ « 4.^
This amplifier provides a voltage output proportional to the voltage
input, and the proportionality factor is independent of the magnitudes
of the source and toad resistances. Such a circuit is called a voltage
amplifier. An ideal voltage amplifier must have infinite input resist
ance Ri and zero output resistance R„. The symbol A, in Fig. I 7 " 10
represents V /Vi with R L = w, and hence represents the opencircuit
voltage amplification, or gain.
480
$*. 177
FEEDBACK AMPLIFIERS AND OSCILLATORS / 48T
R «R
Fig. 171 (a) Thevenin's equivalent circuit of a voltage amplifier, (fa) A simple
vacuumtube voltage amplifier. For this circuit R t = R ff , R = R p \\r p> and
A, = —QmR«
A practical circuit which approximates the ideal voltage amplifier is the
simple triodetube voltage amplifier shown in Fig. 1716. Note that the open
circuit voltage gain is computed with R L = «s , but with R p in place.
Current Amplifier An ideal current amplifier 1 is defined as an amplifier
which provides an output current proportional to the signal current, and the
proportionality factor is independent of R, and R L . An ideal current amplifier
must have zero input resistance Ri and infinite output resistance R B . In
practice, the amplifier has low input resistance and high output resistance.
It drives a lowresistance load {R » R L ), and is driven by a highresistance
source (fi, « R,). Figure 172a shows Norton's equivalent circuit of a cur
rent amplifier. Note that A< = I L /I U with R L = 0, representing the short
circuit current amplification, or gain. We see that if R { « R„ I { « j„ and if
«b » R L , I L sa AJi s= A J,. Hence the output current is proportional to the
signal current. The characteristics of the four ideal amplifier types are sum
marized in Table 171.
A practical circuit which approximates the ideal current amplifier is the
'A&LE 171 Ideal amplifier characteristics
I
Parameter
Amplifier type
Voltage
Current
Tranaconductan ce
Tran Bt esistan ce
it..
CO
F.  AxV.
171
00
II = AJ,
172
00
II  G m V.
173
V. m R m I.
174
*.. .
f&nsfer characteristic . .
* l 8ure
**—
482 / ELECTRONIC DEVICES AND CIRCUITS
Sec. J 7. j
V,
Rl j (\) S*
(a)
(o)
Fig. 172 (a) Norton's equivalent circuit of a current amplifier, (b) A simple
commonemitter transistor current amplifier. For this circuit Rt, y> Ri « A»„
A ( = ~h/ t , R 9 « R c , assuming that A ,(fl e /2i) < 0.1.
simple commonemitter transistor amplifier of Fig. 1726. The amplifier of
Fig. 1726 can be considered as a voltage amplifier if R t <3C A« and Rl » R»
In that case the amplifier should be represented by its Thevenin's equivalent
circuits at the input and the output ports.
Transconductance Amplifier The ideal transconductance amplifier 1
supplies an output current which is proportional to the signal voltage, inde
pendently of the magnitudes of R, and Rl. This amplifier must have an
infinite input resistance Ri and infinite output resistance R . A practical
transconductance amplifier has a large input resistance (Ri » R t ) and hence
must be driven by a lowresistance source. It presents a high output resist
ance (R e ^> Rl) and hence drives a lowresistance load. The equivalent cir
cuit of a transconductance amplifier is shown in Fig. 173a.
A practical transconductance amplifier using a pentode is shown in Fig.
1736. From the circuit of Fig. 17~3a we have that V t ~ V, if R< » R,. Also.
if R » R L , then I L ~ G m V t « G m V„ Hence the output current is propor
tional to the inputsignal voltage. The proportionality factor is G m = II/'*
with Rt = 0, and represents the shortcircuit mutual or transfer conductance
Note that the voltage and current amplifiers of Figs. 171 and 172 may also be
considered as imperfect transconductance amplifiers.
Trans resistance Amplifier Finally, in Fig. 174a, we show the equiva*
lent circuit of an amplifier which ideally supplies an output voltage V a in P 1 " "
portion to the signal current I, independently of R, and Rl. This amplin er
called a transresistance amplifier. For a practical transresistance amplifier *
must have /?, « R, and R„ <K Rl. Hence the input and output resistances a
low relative to the source and load resistances. From Fig. 174a we see thu it
R t » R it h « /„ and if R « R L , V « /? m 7, « R K I t . Note that R m * V»' U
S* 172
FEEDBACK AMPLIFIERS AND OSCILLATORS / 483
Fig. 173 (a) A transconductance amplifier is represented by a Thevenin's
equivalent in its input circuit and a Norton's equivalent in its output circuit, (b)
A pentode transconductance amplifier. For this circuit Ri = R g , R„ » R p » R Lt
G m = —gm. A FET also approximates a transconductance amplifier provided that
R, = Rd\\r* » Rl.
with Rl = *>. In other words, R m is the opencircuit mutual or transfer
resistance.
The commonemitter circuit of Fig. 1726 may be considered as a trans
resistance amplifier if Rl » R e . In that case we convert the output current
source into a voltage source, as indicated in Fig. 1746.
172
THE FEEDBACK CONCEPT 2
In the preceding section we summarize the properties of four basic amplifier
types. In each one of these circuits we may sample the output voltage or
R,«R,
R < .«R L
'8 174 (a) A transresistance amplifier is represented by a Norton's equivalent
n ,T s input circuit and a Thevenin's equivalent in its output circuit, (b) Equivalent
r cuit of a commonemitter transistor transresistance amplifier. For this circuit
*' * fit* R„  h /4 R„ R m R e « Rl, assuming that MA%i#i) < 0.1.
484 / ELECTRONIC DEVICES AND CIRCUITS
Sac. J7.j
Basic amplifier
Forward transfer gain
A
Feedback amplifier
Feedback
network
Reverse
transmission
£
Fig. 175
amplifier.
Representation of any singleloop feedback connection around a basic
The transfer gain A may represent A*, Ai, G m , or R m .
current by means of a suitable sampling network and apply this signal *o the
input through a feedback twoport network, as shown in Fig. 175. At the
input the feedback signal is combined with the external (source) signal through
a mixer network and is fed into the amplifier proper.
Feedback Network This block in Fig. 175 is usually a passive twoport
network which may contain resistors, capacitors, and inductors. Very often
it is simply a resistive configuration.
Sampling Network Several sampling blocks are shown in Fig. 176. In
Fig. 176a the output voltage is sampled by connecting the feedback network
in shunt across the output. In this case it is desirable that the input imped
ance of the feedback network be much greater than R L so as not to load the
output of the amplifier. Another feedback connection which samples the out
Baalc
ampliHer
A
Basic
amplifier
A
m — i
>J
I
i
i
i
L?
1
%
1
Feedback
network
.. _
Feedback
network
(a) (b)
Fig. 176 Feedback connections at the output of a basic amplifier, sampling ni«
output (a) voltage and (b) current.
i
Sec
172
FEEDBACK AMPLIFIERS AND OSCILLATORS / 485
put current is shown in Fig. 176&, where the feedback network is connected
in series with the output. Here the input impedance of the feedback network
should be much smaller than Rl in order not to reduce the current gain apprecia
bly (without feedback). Other sampling networks are possible.
Mixing Network Various mixing blocks are shown in Fig. 177. Figure
177 a and 6 show the simple and very common series input and shunt input
connections, respectively. Figure 177c shows a mixing network consisting of
a single transistor, and in Fig. 177 d we indicate a differential input connection.
Feedback may be classified as either positive or negative. In the former
case any increase in the output signal results in a feedback signal into the input
in such a way as to increase further the magnitude of the output signal. When
the feedback results in a decrease in the magnitude of the output signal, the
amplifier is said to have negative feedback.
§>
r ~"t7""!
' ~X CC
rfl
kg)
,
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1 1
D
Nonln verting
amplifier

(c) (d)
B
'9 177 Feedback connections at the input of a basic amplifier, (a, c, d) Series
Redback, (fa) Shunt feedback. In (c) and (d) the gain for V, may not be the
SQnr *e as the gain for V f .
486 / ELECTRONIC DEVICES AND CIRCUITS
See. I7.j
Transfer Ratio or Gain The symbol A in Fig. 175 represents the ratio of
the output signal to the input signal of the basic amplifier. For an ideal
amplifier the trans