PRINCIPLES OF ELECTRONICS

PHYSICAL SCIENCE TEXTS

General Editor
SIR GRAHAM SUTTON
C.B.E., D.Sc., F.R.S
Chairman of the National Environment Council and
Director General, Meteorological O ice,
Formerly Dean of the Royal Military College of Science, Sh rivcnharn,
and Bashforth Professor of Mathematical Physics.

Applied Mathematics for Engineers and Scientists

C. G. LAMBE,
B.A.,Ph.D.

Techniques of Mathematical Analysis

C. J. TRANTER,
O.B.E., M.A., n.s¢.
A Compendium of Mathematics and Physics
DOROTHY MEYLER,
M.Sc.
and
O. G. SUTTON,
KL, C.B.E., D.Sc., F.R.S.

Electron Physics and Technology

J . T HDOSM s 0 N ,
and
E. B. CALLICK,
B.Sc., A.M.I.E.E.

Principles of Electronics
M. R. GAVIN,
M.B.E., n.s¢.
and
J. E. HOULDIN,
Ph.D., F.Inst.P.

A Course in Applied Mathematics

(Covering B.A. and B.Sc. General Degrees)
D. F. LAWDEN,
M.A.

Physics for Electrical Engineers

W. P. JOLLY,
B.Sc., A.M.l.E.E.

Differential Equations for Engineers and Scientists

C. J. TRANTER,
O.B.E., M.A., D.Sc.
and
C. G. LAMBE,
B.A.,Ph.D.

Elements of Dynamic Meteorology

A. H. GORDON,
M.Sc.

Applications of Nuclear Physics

_I. H. FREMLIN,
M.A., Ph.D.(Camb.), D.Sc., F.lnst.P.
 PRINCIPLES
OF ELECTRONICS

M. R. GAVIN
M.B.E., D.Sc., F.Inst.P., M.I.E.E.
Professor of Electronic Engineering
University College of North Wales, Bangor

and

J. E. HOULDIN
B.Eng., Ph.D., F.Inst.P., A.M.I.E.E.
Senior Lecturer
Department of Physics, Chelsea College of Science and Technology

THE ENGLISH UNIVERSITIES PRESS LTD

ST. PAUL’S HOUSE WARWICK LANE
LONDON E.C.4
 First printed 1959
Second impression 1960
Third impression 1961
Fourth impression 1963
Fifth impression 1965

Copyright © 1959
M. R. Gavin and ]. E. Houldin

Printed in Great Britain for the English Universities Press, Limited,

by Richard Clay (The Chaucer Press), Ltd., Bungay, Suffolk
 GENERAL EDITOR'S FOREWORD
by
sm GRAHAM SUTTON, C.B.E., 1>.s¢., F.R.S.
Chairman of the National Environment Council and
Director General, Meteorological O ice,
Formerly Dean of the Royal Military College of Science, Shrivenham,
and Bashforth Professor of Mathematical Physics

THE present volume is one of a number planned to extend the Physical

Science Texts beyond the Advanced or Scholarship levels of the General
Certi cate of Education. The earlier volumes in this series were prepared
as texts for class teaching or self study in the upper forms at school, or
in the rst year at the university or technical college. In this next
stage, the treatment necessarily assumes a greater degree of maturity in
the student than did the earlier volumes, but the emphasis is still on a
strongly realistic approach aimed at giving the sincere reader technical
pro ciency in his chosen subject. The material has been carefully selected
on a broad and reasonably comprehensive basis, with the object of ensur
ing that the student acquires a proper grasp of the essentials before he
begins to read more specialized texts. At the same time, due regard has
been paid to modern developments, and each volume is designed to give
the reader an integrated account of a subject up to the level of an honours
degree of any British or Commonwealth university, or the graduate
membership of a professional institution.
A course of study in science may take one of two shapes. It may
spread horizontally rather than vertically, with greater attention to the
security of the foundations than to the level attained, or it may be deliber
ately designed to reach the heights by the quickest possible route. The
tradition of scienti c education in this country has been in favour of the
former method, and despite the need to produce technologists quickly, I
am convinced that the traditional policy is still the sounder. Experience
shows that the student who has received a thorough unhurried training
in the fundamentals reaches the stage of productive or original work
very little, if at all, behind the man who has been persuaded to specialize
at a much earlier stage, and in later life there is little doubt who is the
better educated man. It is my hope that in these texts we have provided
materials for a sound general education in the physical sciences, and
that the student who works conscientiously through these books will
face more specialized studies with complete con dence.

V
 PREFACE

THE object of this book is to give a general introduction to the subject

of electronics suitable for a rst degree or diploma course in physics or
electrical engineering. Emphasis is laid on the basic principles of opera
tion of valves, transistors and other electron devices and of the circuits
in which they are used. It is intended to provide the general common
background which is essential to the physicist or the engineer prior to
specialization in any particular branch of electronics. For most of the
book the standard of mathematics required is no more than that of the
Advanced Level of the General Certi cate of Education.
After the introduction there are three chapters on the behaviour of
free electrons and electrons in matter. These provide the physical back
ground needed to explain the nature of the characteristics of the various
types of vacuum and gas valves and transistors, which are the subject
of the next two chapters. This concludes the study of electron devices.
The following thirteen chapters are concerned primarily with the use of
these devices in ampli ers, oscillators, recti ers, switches, etc. In addi
tion to the usual small signal analyses, problems are frequently discussed
generally in terms of device characteristics and load lines based on cir
cuit relations. The nal chapter deals with the subject of noise. No
attempt is made to cover speci c applications of electronics such as radio,
television, radar, computers, instrumentation, etc.
At the end of the book there is a collection of about 250 examples in
groups corresponding to the chapters of the text. These examples form
an important part of the book. Not only do they provide opportunity
for testing the student's understanding but they are also used for further
development and extension of principles. In some cases guidance is
given to the solution. Many of the examples have been taken from
recent examination papers of the Institution of Electrical Engineers and
the Institute of Physics; the authors make grateful acknowledgment to
these bodies.
Perhaps a note should be added regarding the use of symbols. In
general these conform to the recommendations of the British Standards
Institution. The small letters ia, va, U9, etc. are used, as is customary, to
denote the instantaneous values of the varying components of the anode
current, anode voltage, grid voltage, etc. Also, when these variations
are sinusoidal, heavy type 1,, V,, V: is used to denote the corresponding
complex or vector values. New symbols with capital su ixes, i4, v4
and vq, are introduced to represent total instantaneous values, including
both steady and varying components. These symbols are used to repre
sent the variables in static characteristics rather than I,,, V,, and V8. In
this way the possibility is avoided of confusion, particularly in ordinary
vii
viii PREFACE
writing, between the capital letters which are used for both vector and
total values.
In conclusion the authors wish to express their deep thanks to Mrs.
Eveline Thorp who has typed the manuscript and to Mr. K. A. White
who has drawn the diagrams.
 CONTENTS

CHAPTER 1
PAGE
Introduction 1
Electronics. Electron devices. Diode characteristics. Triode charac
teristics. Triode ampli er. Steady and varying values.

CHAPTER 2
Electron Motion 9
Electron motion. Motion in a steady electric eld. The electron volt.
Electric elds. Electron motion in a uniform electric eld. Cathode ray
tube with electrostatic de ection. Motion in a uniform magnetic eld.
Motion in crossed electric and magnetic elds—the magnetron. Cathode
ray tube with magnetic de ection. Electron optics. Magnetic lens.
Electrostatic electron optics. Electrostatic lenses.

CHAPTER 3
Electrons in Matter 27
Electrons in matter. Electrons in atoms energy levels. Electrons in
gases. Electrons in solids. Carbon and the semi conductors. Impurity
semi conductors. The p n junction. Contact potential in metals.

CHAPTER 4
Electron Emission 38
Electron emission. Types of emission. Thermionic emission—Richard
son’s equation. Thoriated tungsten. Oxide coated cathodes. Com
parison of various thermionic emitters. Mechanical form of thermionic
emitters. Secondary emission. Photo electric emission. Schottky e ect
and eld emission.

CHAPTER 5
Diode Currents 51
Flow of charge. Characteristic curves of vacuum diodes. Physics of
the planar vacuum diode—potential distributions. Planar vacuum diode
—space charge flow. Effect of space charge on electron transit time.
Space charge ow for any geometry. Effect of initial velocities of the
electrons. Space charge in magnetrons. Gas diodes. Electron col
lisions with gas atoms or molecules. Breakdown. Cold cathode dis
charge. Arc discharge. Effect of pressure on breakdown. Hot cathode
discharge. Potential distribution in hot cathode diode. Ionization
counters. Crystal diodes. The junction diode.
ix
 CONTENTS
CHAPTER 6
PAGI
Triodes, Multi electrode Valves and Transistors 70
Triodes, transistors and ampli cation. Characteristic curves of vacuum
triodes. Valve equation for small changes. Triode ratings. Physics of
the vacuum triode. Equivalent diode. E ect of space charge in a triode.
Characteristic curves of tetrodes. Secondary emission and tetrode
characteristics. E ect of space charge in tetrodes—beam tetrodes.
Pentodes. E ects of gas in triodes—thyratrons. Ionization gauge.
Transistors. Transistor parameters. Transistor equations for small
changes. Physics of the transistor.

CHAPTER 7

Voltage Ampli ers 96

Valves and their characteristics. The load line. Voltage ampli er—
small signal theory. Voltage ampli er—small a.c. signal. Valve equi
valent circuits for small signals. Voltage ampli er—inductive load.
Voltage amp1i er—capacitive load. Frequency distortion and phase
distortion. Tuned ampli ers—parallel resonant load. Ampli ers with
several stages. Automatic bias. Alternative connections of valve
ampli er. Input and output impedance of an ampli er. Limitations
of small signal theory.

CHAPTER 8

Power Ampli ers ll5

Large signals. Load line and dynamic characteristics. Power output
with resistance load. The transformer coupled load. Load resistance
for maximum power output with transformer coupling. Non linear dis
tortion. Interrnodulation. Non linear devices. Push—pull ampli ers.
Class B ampli ers. Power ampli er e ciency.

CHAPTER 9

Transistor Ampli ers 138

Transistor characteristics. Common base ampli er. Small signal theory
of common base ampli er. Common emitter ampli er. Common col
lector circuit. Transistor equivalent circuits. Biasing circuits. Tran
sistor ampli ers at high frequencies.

CHAPTER 10

Feedback 151
Feedback. Automatic bia.s. Automatic bias and signal feedback.
Cathode bias condenser. Feedback—general considerations. Effect of
feedback on non linear distortion. Effect of feedback on frequency
distortion and noise. Current and voltage feedback. Output and input
impedance of feedback ampli ers. The cathode follower. The Miller
effect. Stability with negative feedback—Nyquist diagram.
 CONTENTS xi
CHAPTER ll
PAGE
'l‘ransients in Ampli ers 169
Steady state and transients. Transients in passive circuits. Transients
in valve circuits. Ampli cation of square pulses. Large transients in
valve circuits. A.c. transients. Class C ampli er as a switch.
Transients in circuits with feedback. Some general comments on
transients.
cH A PT E R 1 2
Direct coupled Ampli ers 181
The ampli cation of d.c. changes. Direct coupling. Use of negative
feedback. Balanced or differential ampli ers. High gain ampli er with
high stability. Other methods of amplifying steady signals.

CHAPTER 13
Oscillators 190
Introduction. Negative resistance oscillators. Feedback oscillators.
Tuned anode oscillator. Class C oscillators and amplitude limitation.
Other tuned oscillators. Transistor oscillators. Feedback and negative
resistance oscillators. Triode oscillators for ultra high frequencies.

CHAPTER14
Electrons and Fields . 206
Induced currents due to moving charges. Energy considerations. The
energy equation. Transit time loading.

CHAPTERI5
Special Valves for Very High Frequencies 212
The klystron. Travelling wave tubes. Linear accelerators. Space
charge wave tubes. Cavity magnetrons.

CHAPTERIB
Recti cation 221
Simpli ed diode characteristics. A.c. supply, diode and resistance in
series—half wave recti er. Full wave recti cation. Choke input full
wave recti er. A.c. supply, diode and condenser in series. Condenser
input full wave recti er. Voltage doubling circuits. Filter circuits.
Diode peak voltmeter. Some practical considerations in recti er design.
Voltage stabilization—gas diode. Voltage stabilization—feedback.

CHAPTER 17

Modulation and Detection 240

Modulation. Amplitude modulation. Circuits for amplitude modula
tion. Frequency modulation. Circuits for frequency modulation.
Detection or demodulation. Detection of amplitude modulated waves.
Receivers. Superheterodyne reception. Automatic volume control.
Detection of frequency modulated waves. Automatic frequency control.
xii CONTENTS
CHAPTER 18
PAGI
Relaxation Oscillators and Switches 258
Relaxation oscillators. Gas diode or triode oscillator. Feedback re
laxation oscillators—the multivibrator. The transitron relaxation
oscillator. The blocking oscillator. Monostable circuits. Bistable cir
cuits. Cathode coupled trigger circuits. Counting and scaling. Decade
scaling. Amplitude control and discrimination.

CHAPTER 19
Wave Shaping 276
Wave shaping circuits. Non linear wave shaping circuits. Clamping
circuits and d.c. restoration. Linear wave shaping—differentiating and
integrating. Electronic integrating circuits.

CHAPTER 20
Noise 288
Noise. Johnson noise. Shot noise. Addition of noise voltages. Equiv
alent noise resistance. Noise factor. Other sources of noise in valve
ampli ers. Transistor noise.

Examples 294

APPENDIX I
List of Symbols 338

APPENDIX II
Useful Constants 342

APPENDIX III
Bibliography 342

Index 343
 CHAPTER 1
INTRODUCTION
1.1. Electronics
Electronics has been de ned as “ that eld of science and engineering
which deals with electron devices and their utilization ”, where electron
devices are “ devices in which electrical conduction is principally by
electrons moving through a vacuum, gas or semi conductor ”. The rst
six chapters of this book are concerned with the elucidation of this some
what cryptic de nition of electron devices. However, in these days of
radio and television everyone is familiar to some extent with some of the
devices such as valves, cathode ray tubes and photo electric cells. Yet
at the beginning of the present century no electron devices as we know
them existed. The nature of the electron itself, as a minute particle
having mass and negative electric charge, had just been established.
The Fleming diode, introduced in 1904, is usually considered to be the
prototype electron device. As the name implies, this diode had two
electrodes; one was a thin lament of wire (the cathode) which could be
heated to incandescence by means of an electric current, and the other
was a metal plate (the anode) close to the wire. The electrodes were
enclosed in an evacuated glass envelope with wire leads through the glass.
The diode acted as an electrical conductor when an e.m.f. was connected
between the anode and the cathode in such a way that the anode was
positive with respect to the cathode; when the polarity was reversed
it acted as an insulator. This asymmetric or non linear behaviour is
typical to some extent of all electron devices. The one way conduction
in diodes is utilized in many ways for the recti cation of alternating
current.
In 1906 de Forest put a third electrode in the form of a wire grid between
the cathode and anode. With this arrangement the current owing to
any electrode depends on the potentials of all three electrodes. It is
found that under some conditions the grid potential acts as an effective
control of the current to the anode without taking appreciable current
itself; the grid controls large currents and power, without consuming
much power. Thus three electrode valves or triodes can act as ampli ers
of voltage, current or power. This ability to amplify opens up innumer
able possibilities and is largely the reason for the importance of electronics
to day. After the triode, other multi electrode valves with four, ve and
more electrodes were introduced. These give additional advantages of
various kinds, but their operation generally depends on either their non
linearity or the amplifying property of a grid.
In the development of valves it was found that the presence of gas
I
2 PRINCIPLES OF ELECTRONICS [cn.
inside the envelope modi ed the properties considerably. There is a
variety of gas diodes, triodes and multi electrode valves.
From the early days of electron devices certain crystals, now called
semi conductors, were known to show non linear conduction when used
as diodes. In 1948 a major development occurred when it was found
that the semi conductor germanium could be used with three electrodes
to give ampli cation. These new devices are called transistors. One
electrode, the emitter, causes a ow of current to a second electrode, the
collector. This current can be controlled readily by the potential differ
ence between the emitter and the third electrode, the base, but the latter
takes very little current. Thus, as in the triode, ampli cation can be
obtained.
We have now had examples of electron devices with electrodes separated
by vacuum, gas and semi conductor. In all of these, as we shall see later,
the currents are due almost entirely to the movement of electrons. Cur
rents also ow in metals from the movement of electrons, but this current
ow varies linearly with the potential difference across the metal, and
such behaviour is of minor signi cance for electron devices.
For the conduction process there must be electron movement. Electron
devices differ in the manner in which the electrons are made available.
In some, the semi conductors, electrons are available in the solid at
(a) Vacuu'm Valves

Thermionic emission Secondarjlr emission

I
Photo elamission

I
Electron
I I
Photo
I
Photo cells
multipliers multipliers

Amplihcation Light
I Light
I
measurement measurement
Ampli cation and control
I

I
Diodes
.
Triodes Multi
I I
Velocity Cathode ray
electrode modulation tubes
valves valves

X ray Recti Ampli cation Klystrons Travelling

tubes cation Oscillation | wave tubes
Switching Oscillation |
Counting Ampli cation Magnetrons
Control I
Oscillation
Ampli cation

I
Oscillography Eleritron Camlera [tubes
microscopy |
Picture
transmission
1] INTRODUCTION 3
(b) Gas‘ Valves

Thermionic emission
I I
Photo emission Cold cathodes Mercury pool
I
| cathodes
Photo cells

Diodes Triodes Tetrodes Light Recti cation

| | | measurement Conversion
Recti | and control
5°?‘ 5°" Recti cation
Switching | I
Counting Diodes Triodes Multi
Control electrode
vallves

Stabilizers G M tubes Switching

| Counting
Radiation
measurement

(c) Semi conductor‘ or Crystal Valves

Dioides Tran%istors

I
Photo cells '
I
Recti ers
I
Thermistors
I I
Ampli cation Photo transistors
| | Oscillation |
Light Measurement Switching Light measurement
measurement Control Counting

ordinary temperatures. In vacuum tubes the electrons enter the vacuum

from the cathode. The emission of the electrons from the cathode
requires additional energy in some form. In thermionic emission the
extra energy is given to the electrons by heating the cathode. The
energy may also be supplied as radiation, when photo emission occurs.
Finally, the electrons may be knocked out of the cathode by impact with
other fast electrons or ions. This is called secondary emission. Radiation
may also be used to enhance the conductivity of semi conductors. In
gases, conduction electrons are produced by ionization of the gas atoms by
means of radiation or fast particles. Positive ions are produced at the
same time, and they may effect the conduction process, but their move
ment makes little contribution to the currents. In some gas valves
electrons are also produced from a cathode by thermionic or photo
emission.
1.2. Electron Devices
The tables on pages 2 and 3 indicate some of the great variety of electron
devices and their elds of use under three main headings: vacuum valves,
gas valves and semi conductor or crystal valves.
4 PRINCIPLES OF ELECTRONICS [CH.
1.3. Diode Characteristics
The properties of any particular electron device can be described con
veniently by means of characteristic curves, which show how the electrode
currents vary with the potential differences between
the electrodes.
One form of vacuum diode is shown diagrammatic
ally in Fig. 1.1. The cathode K is a hollow cylinder
which is heated indirectly by radiation from an in
sulated wire inside the cylinder. The heating current
is supplied through the leads H. The anode A is
another cylinder surrounding the cathode, and the
whole system is enclosed in a glass envelope E,
which is evacuated to a low pressure. The wire
' ¢'0_
leads to the electrodes and the heater pass through
awyt\\w a glass “ pinch " P. The characteristic curves for
such a diode are found using a circuit of the form
'0m
»
I I
shown in Fig. 1.2.a, which also shows the circuit
symbol for the diode. The heater leads H are con
Fro. 1.1
nected to a suitable electrical supply. Since there are
only two electrodes in the diode, it follows that i4 + iK = 0. Also the
potentials may be measured from the cathode as zero. Thus there need

‘A
1, I
‘I
v, _"' P
I
I
H H :—'

ix
O VA

(<1) Io)
Fro. 1.2
be only two variables iA and 1:4, and the relation between them is shown
in the characteristic curve in Fig. l.2.b. It is seen that current ows
when v4 is positive but little or no current when v4 is negative.
1.4. Triode Characteristics
A cylindrical thermionic triode is illustrated in Fig. 1.3. The grid is
a wire cage surrounding the cathode. Now there are three electrode
currents i4, iq and ix and three potential differences vgg, v6 K and v40, as
shown in Fig. 1.4, which gives the circuit symbol for a triode. However,
by Kirchhoff's Laws
54 I 50 + in = 0
and "Ax = 1140 + "ox.
 1] INTRODUCTION 6
and hence there are only four independent variables. If the potential
differences or electrode voltages are measured from the cathode as v0 and
v4, then we may take i0, i4, ‘Ug and v4 as the four variables. In many
uses of the triode ‘U0 is negative and no electrons ow to the grid, so that

I",
ll >
+
)'\.
i )‘ +0
Iii 1
I 6 Q VAK
l'|.I
.. Ii|iI
; :
14!‘!
~ I
"
....
| \\\
Fro. 1.4
.’:ZI.Z '4
I
P HH
Fro. 1.3

iq = 0. The relation between i4, U0 and v4 may then be determined

with the circuit of Fig. l.5.a. The characteristics consist of the family
of curves shown in Fig. l.5.b or c. Each curve of the rst set shows the
'0
[A] O If, HA

‘A I

e Eel
I
yea VAl

'~_1_’| 5
I" Q C
II U
O VA O VG
VAI Yea Ye 1
(<1) (0) (¢)
Fro. 1.5

variation in anode current with anode voltage for a xed value of grid
voltage. Exactly the same information is given in different form in
Fig. 1.5.c, where each curve shows the variation in anode current with
grid voltage for a xed value of anode voltage. Corresponding points
are marked B, C and D in the two diagrams.
6 PRINCIPLES OF ELECTRONICS [CH.

1.5. Triode Ampli er

In Fig. 1.6.a a resistance R is shown between the anode and the h.t.
battery E2. Now the electrode voltages are
vq = — E1 and v4 = E2 —Ri4.
In the latter equation E2 and R are constant, and the equation may
be represented by a straight line in an i4, v4 diagram as shown in
Fig. l.6.b. The straight line cuts the v4 axis where v4 = E, and the

[‘n VG

R E2 O
(A + E YG3'E‘+y‘

V __ 'c="E1
A 1 ,

E I ' . _. . _a O I\I A VA

' VP VQ E2

(0) (bl
Fro. 1.6

i4 axis where i4 = E2/R. Whatever the value of Ug, i4 and v4 must

lie somewhere on this straight line. They must also lie on the valve
characteristic curves which are drawn in the same diagram. Thus the
actual values of i4 and v4 are found at the point Q on the characteristic
corresponding to v0 = — E1. If ‘U9 changes by amount v_,, i4 and v4
must still lie on the straight line, and the new values are found at P.
The change in anode current causes a change in voltage drop across R
and a corresponding change in v4. The
R anode voltage change is found from
the diagram to be ‘UP — vq. If the
magnitude of Up — vq is greater than v,,
E, then the valve has produced voltage
B —..:E2 ampli cation; the change v, is frequently
vs called a signal. The point Q which
43
6+ gives the initial or quiescent conditions
— before the change occurs is called the
FIG 1_ 7 quiescent point or the operating point.
The change in grid voltage may be pro
duced by an alternator as shown in Fig. 1.7. Now the grid voltage and
anode current vary sinusoidally and an alternating voltage appears across
the resistance R and also between the anode and cathode.
1] INTRODUCTION '1
1.6. Steady and Varying Values
In considering the currents and voltages in a valve there are several
different conditions which have to be distinguished. Thus ‘U0 and v4 are
the actual values of the electrode voltages, however these arise. In the
quiescent state
U9 = — E1,
‘U4=‘UQ=E2— Rig
and 1.4 = ‘iq,

where iq, and vq apply to the quiescent point. When the grid voltage
is changed
va = — E1 + v..
U4 = 1)p= UQ + (Up —UQ)

and 2.4 = ip = ‘lq —j— —iQ).

In each case there is a quiescent value, a change and a total value.

Throughout this book we use capital suf xes to indicate total values
('00, v4 and i4) and small suffixes to indicate changes (v,, v, and i,,). In
the case just considered
v, = v,,
‘U¢ = ‘Up — UQ
and i, = iP — iq.
When the anode current increases the anode voltage decreases because of
the greater voltage drop across R, and obviously
‘U, =7 *' Rho

v.4 = vQ ‘I’ vs
and i4 = iq + i,,.
In the absence of a signal the total values are equal to the steady or
quiescent values.
When the signal is a small alternating voltage, say
v, = ii, sin col,
then v, = 13, sin mi = ii, sin col,
1', = i, Sin wt
and v,, = — Ri, sin cot = — ii, sin ml.
Now v@= — E1+v,,=—E1 I vgsin ml,
i4=iQ+i,,=iQ I i‘, sin wt
and v4 =vQ + v, = vq — 13,, sin col.
For sinusoidal alternating voltages and currents it is frequently con
venient to use vector or complex values of these quantities. When
this is done we use capital letters V‘, V‘, I‘. Much confusion can
8 PRINCIPLES OF ELECTRONICS [cH.1
arise in dealing with valves and valve circuits unless care is taken to
distinguish carefully between quiescent values, total values, changes
and vector values. The symbols suggested facilitate this distinction
and are equally satisfactory in print or in writing. At the same
time, the letters used indicate the nature of the quantity, voltage
or current, and the electrode associated with it. The same notation may
be easily extended to other devices. For example, i3, i, and I, represent
the total, change and vector currents respectively for the emitter of a
transistor; iR, i, and I, could represent the same currents in a resistor.
 CHAPTER 2
ELECTRON MOTION
2.1. Electron Motion
Electrons may be considered as minute particles having mass and
negative charge. As such, their movements in electric and magnetic
elds can be determined by the application of the laws of electricity and
magnetism and of mechanics. As the result of numerous experiments
the following magnitudes have been established for the charge e and the
mass m of an electron:
e = 1 60 >< 10'" coulomb, m = 9 11 X 10'" kilogram.
This mass is lm of the mass of a hydrogen atom. The above value for
m is true only when the electrons are at rest or moving with velocities
small compared with the velocity of light. According to the theory of
relativity, the mass of a particle increases as its velocity approaches the
velocity of light. The mass m at velocity u is related to the rest mass mo
by the expression
l
in 1 0 0

~/1—(‘%)'
where c is the velocity of light in free space. It may be veri ed from the
Energy Equation (see Section 2.3) that the change in mass of an electron
is negligible in devices operating below 1,000 V. The increase is about
1 per cent for electrons accelerated through a potential difference of
5,000 V. In most of this book the relativity correction is ignored.
2.2. Motion in a Steady Electric Field
The force F acting on a positive charge q in an electric eld of strength
E is, by de nition,
F=qE
and the force is in the direction of the eld. If v is the potential, then
the eld strength is related to the potential gradient by
dv
E; = '—' E‘

This equation gives the component of the eld strength in the direction
of s. The component of the force in the same direction on the charge q is

Fgiiq

9
10 PRINCIPLES OF ELECTRONICS [cl I.
For an electron with negative charge the force is given by
dv
Fiezss

and is in the direction of increasing potential. If the electron is free to

move, as it would be in a vacuum, the force does work on the electron.
In moving between two points 1 and 2 the work done is
2
[Eds = evz — evl,
1
where v, and v, are the potentials at the points 1 and 2 respectively,
This work appears as kinetic energy of the electron. If u, and u, are the
electron velocities at the two points, then
mu,” mu,”
8172 i 8'01 2 2 '

This is to be known as the Energy Equation.

In the particular case where an electron starts from rest and moves
through a potential difference v its nal velocity n depends only on v,
and is given by the relation
w_mu2
_ 2
The last two equations, which equate the gain in kinetic energy to the
loss of potential energy, show how energy may be exchanged between
electric elds and electrons, and they are of fundamental importance in
the understanding of the operation of electronic devices. This energy
exchange is considered in more detail in Chapter 14.
2.3. The Electron volt
In dealing with electrons it is convenient to introduce a unit of energy
called the electron volt. This is de ned as the amount of kinetic energy
acquired by an electron in moving through a difference of potential of
one volt. One electron volt (eV) is equal to 1 60 >< 10'" coulombs X
one volt = 1 60 >< 10'“ joules.
From the Energy Equation of Section 2.2 it follows that an electron
starting from rest and moving through a potential difference of v V
acquires a velocity given by u = \/(2 ev/m) = 5 94 >< 10°\/5m/s. If
v =1 V, then the velocity is 5 94 >< 10‘ m/s. An energy of 1 eV is
extremely small, but it imparts an enormous velocity to an electron
because of the small mass.
Although the electron volt has been de ned in terms of the electron,
it may be used to measure any quantity of energy. For example, a
proton, i.e., a hydrogen atom which has lost an electron, has a positive
charge numerically equal to e and acquires a kinetic energy of 1 eV when.
it moves through a potential difference of 1 V. Since the proton has a
2] ELECTRON MOTION ll
much greater mass than the electron, an energy of 1 eV imparts to it a
lower velocity. The Energy Equation, which applies to any charged
particle, shows that the velocity varies inversely as x/hi, i.e., 3%; or
41,; in this case. Sometimes the “ electron ” is omitted from “ electron
volt ”. For example, a beam of 100 V electrons means a beam in which
the electrons have started from rest and moved through a potential
difference of 100 V. The energy of each electron is 100 eV or 1 60 X 10'1"
joules.

2.4. Electric Fields

The user of an electron device controls the potentials of the electrodes
by making suitable connections to the electrode leads. However, if the
behaviour of electrons inside the device is to be studied the potential
distribution in the space between the electrodes must be known. In the
absence of appreciable charge in the space the potential must satisfy
Laplace's Equation, which, in cartesian co ordinates, is

é+ +§=o
82 a2 a2

When the charge density p in the space is not negligible the potential
obeys Poisson's Equation
62v 82v 32v _ p
C 962 + 6y” + 322 C co’

where co is the primary electric constant having the value 8 86 >< l0‘1*'F/m;
it is sometimes called the permittivity of free space. Laplace’s and
Poisson's Equations follow directly from Gauss’s Theorem. Thus the
problem of nding the potential consists of solving one of these two
partial differential equations using the electrode potentials as boundary
conditions. For all but a few simple geometrical arrangements of the
electrodes analytical solutions are impossible. Methods have been
developed for dealing with special cases, and approximate numerical
solutions of Laplace’s Equation can be obtained experimentally by the
use of analogues, particularly the rubber membrane, the electrolytic tank
and the resistance network. These special methods are beyond the
scope of this book, and we con ne our attention to simple idealized cases
which can be readily analysed. The results give some qualitative indica
tion of what may be expected in the more complicated practical cases.

2.5. Electron Motion in a Uniform Electric Field

A uniform electric eld, i.e., one in which the eldstrength is constant,
can be realized between the plates of a parallel plane condenser in regions
remote from the edges, provided the dimensions of the planes are large
compared with the distance between them. This case is illustrated in
 I
12 PRINCIPLES OF ELECTRONICS [cH.
Fig. 2.1, in which the y axis is chosen at right angles to the planes and
the x and z axes are parallel to the planes. The potential difference
between the two electrodes, the cathode and the anode, is v4. Obviously,
except near the edges, the potential in the space between the electrodes

W
d1 } CATHODE
Y:

I
Fro. 2.1

does not vary in the x and z directions, and Lap1ace’s Equation becomes
one dimensional, i.e.,
d2v
= 0.

Integration gives
d
;;=.4 andv=Ay+B,
where A and B are constants to be determined by the boundary condi
tions. The difference of potential between y = 0 and y = d is v4;
d = the distance between the planes. It follows that the eld strength
and the potential are given by
dv_v4 _v,,y
ay_Zandv z

These expressions con rm that the eld is uniform. If there is an

electron in the space it is acted upon by a constant force F given by
_ dv_ev4
F e‘25)_ ‘T

If the space is evacuated the electron is free to move continuously under

the in uence of this force. In air the electron would still move, but its
motion would be interrupted by impact with the air molecules. It is
assumed below that the space is evacuated. Using Newton's Second
Law of Motion
_d diy ev4
F Zz2(’"“)“’"d: 4'
2] ELECTRON MOTION 13
Hence
‘fl’ 92,
dtz _ md
and the acceleration is also uniform. By integration the velocity is
found to be
“""i y at Ll:
_dt_md
where the constant of integration has been put equal to zero on the
assumption that the electron left the cathode with zero velocity at time
t = 0. The distance moved in time t is found by a further integration
to be
__ ev4t2
Y 271'
The constant of integration is again zero. The time to travel a distance
dy is equal to dy/u, and hence the time 1 for the electron to move across
the space is found from
‘Ii

From the Energy Equation and v = my/d, it is found on integrating that

1' 2d 2d sec
_ ‘I44 (5'94: X105 \/U4) '

where ii, is the velocity of the electron at the anode. This expression
could have been found by alternative reasoning. Since the motion is
uniformly accelerated, starting from rest and nishing with velocity n4,
the average velocity is n4/2. The transit time is the distance divided
by the average velocity, i.e., 2d/u,,.

2.6. Cathode ray Tube with Electrostatic De ection

Everyone is familiar with cathode ray tubes for displaying electrical
quantities visually. The name “cathode ray " is a relic of the early
days of electron physics, when streams of electrons emanating from the
cathode of a discharge tube were called cathode rays. One common
form of cathode ray tube is shown in the diagram of Fig. 2.2. A narrow
beam of electrons is produced by an electron “ gun ” and passes between
two pairs of de ecting plates. The beam nally strikes a luminescent
screen and produces a bright spot of light. Details of the electron gun
are considered later in this chapter. If the difference in potential between
the cathode and the nal anode is 1,000 V, then a fast beam of electrons
leaves the nal anode of the gun with the electron velocities corresponding
to 1,000 eV. The de ecting plates are arranged parallel to the beam.
Normally, the rst pair, or Y plates, are horizontal and the second pair,
or X plates, are vertical. The mean potential of the de ecting plates is
also 1,000 V and the beam passes through the middle of the de ecting
14 PRINCIPLES OF ELECTRONICS [cn.
system and strikes the centre of the screen. The whole of the space
from the nal anode of the gun to the screen is at a potential of approxi
mately 1,000 V, and the electrons move with constant velocity in this
space. If a potential difference is applied to the Y plates this produces
SCREEN

ea
..............
OEFLECTING
ELECTRON GUN SYSTEM

_ _ _ _ ELECTRON BEAM SPOT

CATHODE| FIRST FINAL Y X

ANODE ANODE
GRID PLATES PLATES

Fro. 2.2

a vertical de ection of the beam and the spot is displaced on the screen.
Similarly, a potential difference across the X plates causes a horizontal
displacement of the spot. The size of these de ections can be deter
mined approximately by using the results of the foregoing paragraphs.
The passage of an electron beam through one pair of de ecting plates is
represented in Fig. 2.3. The electrons reach 0 with horizontal velocity
n4, where n4 = \/(2ev4/m), v4 being the potential of the nal anode and
also the mean potential of the de ecting plates and the screen. When

y!" UA R
>§+
..\‘*‘
_ _ _ _ _ _ _ _ _' 1::
_ .._ I._.A Q
d O ——— +——— ' ' _M
0, N I

4
>§
. \ 81‘ >4 L 1

Fro. 2.3

there is no de ecting eld the electrons move with constant velocity n4

along the path ONM. The de ecting plates are similar to the parallel
plane condenser of the previous section, and when a potential difference
is maintained across the plates the electric eld is uniform except near
the edges of the plates. In order to simplify the calculations it is assumed
that the de ecting eld is uniform over the length l of the plates and is
zero elsewhere. Under these conditions the electrons reach 0 with
velocity n4 as before. Since the de ecting eld is purely vertical, the
horizontal velocity is unaffected. However, over the length l the de ect
ing eld acts on the electrons so that, at the end of the de ecting system,
2] ELECTRON MOTION 15
they have acquired a vertical displacement NP and a vertical component
of velocity, say M7. The value of up can be found by the methods of the
previous section, and is given by
My = evdt/md

where vd = de ecting potential difference, d = distance between the

de ecting plates and t = time spent in the de ecting plates. Obviously
i=1/M4 and My = GU41/mdu4'

From P onwards the electrons are free from eld and move with velocity
corresponding to potential v4. However, they have now a vertical com
ponent of velocity equal to Up‘ and at the screen they have a vertical
de ection QR equal to upT, where T is the time to travel from the de
ecting plates to the screen. As long as
“V <“4.
T=' L/n4,
where L = horizontal distance from the de ecting plates to the screen.
Hence
QR = evd IL/mdu ,
1.6., = lL‘Ud/2d‘U4.

The total vertical de ection of the spot on the screen from the mean
position should include NP. However, in most cathode ray tubes l < L
and NP is negligible. Thus the de ection y of the spot is given approxi
mately by
y = lLv¢/2dv4.
The de ection sensitivity of the tube is de ned as y/vd. It is frequently
quoted in millimetres per volt. The sensitivity is inversely proportional
to v4. However, the brightness of the spot and the sharpness of its focus
increase with v4, so that a compromise is necessary.
In deriving the formula for y several assumptions have been made.
In particular, the de ecting eld has been assumed to be con ned to the
length of the de ecting plates. This eld must extend beyond these
plates, and as a result de ection occurs over a length greater than l.
Also, the horizontal component of the velocity over the length L is less
than 244. In this region, away from the de ecting plates, the potential
is v4. The Energy Equation relates the loss of potential energy to the
total kinetic energy so that
1711432 may”
"’ = T + '2_’
where ii” and ‘U7 are used for the horizontal and vertical components.
This equation shows that My must be less than n4. However, up is
frequently much less than '1 lg, and then an and n4 are nearly equal.
In some cathode ray tubes the de ecting plates are ared as shown in
16 PRINCIPLES OF ELECTRONICS [cH.
Fig. 2.4.b. With this arrangement the value of d varies along the plates,
and obviously it gives greater sensitivity than parallel plates with a
separation equal to the nal value of d in the ared plates. The maximum
de ection would be the same for both systems.
Mention was made in Section 2.3 of the enormous velocities which can
be given to electrons by electric elds. The mass of the electron is so
f
§
f? ?

(v) (0)
Fro. 2.4

small that its response to a change of eld is practically instantaneous

and the de ection takes very little energy from the eld. In addition,
the electrons move through the de ecting plates very rapidly. For 1,000V
electrons and de ecting plates 2 cm long the time taken is about 10‘° sec.
Thus even for an electric eld varying at a frequency of 10° c/s the
eld does not change appreciably during the passage of the electrons,
and the formula for y may be used to give the instantaneous de ection.
2.7. Motion in a Uniform Magnetic Field
VVhen an electron moves with velocity u in a magnetic eld of ux
density B the electron experiences a force F which is given by

O O O O O

F O 9 O “ O

u
' O O O O O
..¢ 6 U F

B O ‘= O == O
Fro. 2.5

O O O O O
Fro. 2.6

F = Beu sin 6, where 6 is the angle between the directions of B and u.

This force is perpendicular to the plane containing B and u as shown in Fig.
2.5. The following interesting points may be established from the equation
2] ELECTRON MOTION 17
for F: (i) there is no force on a stationary electron in a magnetic eld,
(ii) the force is greatest when the electron moves at right angles to the
magnetic eld, (iii) there is no force on an electron moving parallel to a
magnetic eld, (iv) as the force is perpendicular to the velocity, an
electron cannot gain kinetic energy from a magnetic eld; the magnetic
eld may alter the direction but not the magnitude of the velocity.
In the particular case when the eld and the velocity are at right
angles the force is Beu and is perpendicular to both B and u. This case
is represented in Fig. 2.6, where the magnetic eld is uniform and is

[Y

Y 1)’
Ucose gs’ sine
/00
/' Usina x B

y ' I W Usina,o y

2
Fro. 2.7

directed into the paper. Here we have an electron moving with velocity
u and subjected to a force Beu in a direction at right angles to u. This
is a case of uniform motion in a circle. For such a motion the force
towards the centre of the circle is mu”/r, where r is the radius. Hence
F = Ben = mu’/r and r = mu/Be.
The time t for one complete circuit is 21 . r/u,
i.e., t = Zrrm/B6
The time is thus independent of u and r; it depends only on the magnetic
eld. It may also be seen that the radius of the path varies with u.
These results have importance in many devices, such as the cyclotron,
the mass spectrograph, magnetic lenses, etc.
When the electron velocity makes an angle 6 with the magnetic eld
the path is helical, as shown in Fig. 2.7. The velocity may be resolved
into two components u cos 6 parallel to the eld and u sin 6 perpendicular
to the eld. There is no force due to the former, and the latter gives rise
to a circular motion of radius %: : sin 6. The resultant motion is due to
this circular motion round the direction of the eld and the uniform
translational motion u cos 6 along the eld. The radius of the helix may
be controlled by varying B.
18 PRINCIPLES OF ELECTRONICS [cr~r.
2.8. Motion in Crossed Electric and Magnetic Fields—The Magnetron
An important case of electron motion occurs in a combination of uni
form electric and magnetic elds. This case is illustrated in Fig. 2.8,
where a difference of potential is maintained between two parallel plates
and there is a uniform magnetic eld parallel to the plates. An arrange
ment of this type is called a planar magnetron. If a single electron starts
O O O O O
ANODE +
ea Hb ea FE" ea ea
E1 ,
ea ii; Fae ea I ‘O
CATHODE '

e ea ea ea es
Fro. 2.8
from the cathode with zero velocity it experiences a force due to the
electric eld, but the magnetic eld has no effect. The force due to
the electric eld gives an acceleration towards the anode. As a result, the
electron moves towards the anode with increasing velocity. Now,
because of its velocity, there is a force on it due to the magnetic eld, and
this force, which is always at right angles to the motion, increases as the
velocity increases. The path, therefore, is curved. If the magnetic
y Y
U, E:

E /.i_. ';;“~~_, E —¢ Bea,

I/I BCUI

0' 0

V (0) " Y (bl "

2 2
Fro. 2.9
eld is sufficiently great the electron may be turned back towards the
cathode as shown in curve (a), Fig. 2.8. Energy considerations show
that at every point of the path the electron velocity depends only on the
electric potential at that point, and after the electron is turned back its
velocity decreases. As a result, the force due to the magnetic eld
decreases, until at the cathode the electron comes to rest again, and there
2] ELECTRON MOTION 19
is no force from the magnetic eld. The electron is ready to start on
another similar path as shown by the dotted line. For a weaker magnetic
eld the curvature of the path is less and the electron may move to the
anode without being turned back as shown in Fig. 2.8, curve (b). The
actual shape of the electron path is determined as follows. The problem
is two dimensional, and the co ordinate axes are chosen as in Fig. 2.9.
The electron’s velocity at any point is resolved into two components u,
and u, parallel to the axes. These two components, along with the
magnetic eld, give rise to forces Ben, and — Ben, in the x and y directions
respectively. Due to the electric eld E there is a constant force Ee in
the y direction. From Newton's Second Law the equations of motion
for the x and y directions are then
m€T:=Beu,, and m%Z'=Ee—Beu,,
. d d E
i.e., % = mu, and £1’ = ,5 — e>u,,

where co = Be/rn. By differentiating the second of these equations and

substituting from the rst it follows that
dzu
'27” 1 1 (1)21 ‘ye

The solution of this familiar differential equation is

u,, = a cos cot + b sin (oi,
where a and b are constants to be determined. When the electron starts
from the cathode,
t=0,u,,=0,and‘%:1’=Ee/m
Using these conditions, it is found that
a = 0 and b = Ee/com.
Hence
Ee .
up = V,‘ S111 (Qt.

Now by substitution for u, in the original differential equation it follows

that
E
u, = 5:; (1 — cos cot).

By integrating these two equations for u, and u,,, and using the con
dition that x = 0 and y = 0 at t = 0, it is found that the position of the
electron at time t after leaving the cathode is given by
x = £4” (A: sin col)
and y = 52%;" (1 — cos cot).
20 PRINCIPLES OF ELECTRONICS [cl I.
These two equations represent a cycloid of the form shown in Fig. 2.8.
The greatest distance reached from the cathode occurs when y has its
maximum value. This obviously is when cos col = — 1 and the actual
distance is
ym, = 2Ee/Mm = 2Ern/B'e.
If this distance is greater than d, the anode cathode distance, then the
electron is collected by the anode. The limiting case, where the electron
just reaches the anode, occurs when d = 2Em/Bze. If v is the potential
difference between the anode and cathode, then
E = v/d and B = \/(2vm/ed”).
This value of B is known as the critical magnetic eld. For greater

IA
ANODE

CONSTANT VA

\
s
‘\
s~.
O
CRITICAL B B
Fro. 2.10 Fro. 2.11

values of B no electrons reach the anode, and for smaller values all the
electrons which leave the cathode are collected by the anode. If the
anode current of a planar magnetron is measured as the magnetic eld
is varied the current should vary as shown by the full line in Fig. 2.10.
In practice, the current cut off is more inde nite, as shown by the broken
lines in the gure. Some of the factors contributing to this inde nite
cut off could be: (i) end effects of the nite electrodes, (ii) the electrons
do not leave the cathode with zero velocity but have a velocity distribution
(see Chapter 4), (iii) electron interaction, (iv) the magnetic eld may
not be uniform throughout.
Most magnetrons in use have the electrodes as co axial cylinders with
the magnetic eld parallel to the axis. The exact analysis of this case
is dif cult, but the path of a single electron is rather similar to that of
the planar magnetron. When the radius of the cathode is not much
less than that of the anode, then the electron path is nearly cycloidal.
When the cathode radius is much less than the anode radius the path is
very nearly circular (see Exx. II). Typical electron paths for a cylindrical
magnetron are shown in Fig. 2.11. Here the numbers 1, 2 and 3 corre
spond to increasing values of the magnetic eld with a xed anode voltage.
Curve 2 occurs at the critical value of the magnetic eld. These paths
Q ELECTRON MOTION 21
could also be obtained with xed magnetic eld and decreasing anode
voltage.
In all these considerations of magnetrons so far the effect of space
charge has been neglected, i.e., attention has been given to the behaviour
of a single electron and any forces due to other electrons have been
ignored. Space charge effects complicate the problem enormously.
They are considered again in Section 5.8.
2.9. Cathode ray Tube with Magnetic De ection
The de ection of an electron in a magnetic eld is sometimes used in
cathode ray tubes as shown in the diagram of Fig. 2.12. Two pairs of
coils XX’ and YY’ are arranged outside the tube. The XX’ coils are
DEFLECTING
SYSTEM SCREEN

ELECTRON ouu
x%
_

“mt
<<eI5e)
cnuooe nnsr FINAL
lmooe moor :
‘_ ,
X
omo
FIG. 2.12
normally placed horizontally on either side of the tube, and they are
joined in series. When a current is passed through these coils a vertical
magnetic eld is produced inside the tube. This eld, which is pro

.. ".. ' _'

ea ea ea ea ea I
ea ea ea ea ea I
U "'~:jO"O """"""""" "M
O o ea ," 9
ea ea ea,/o
____ /
0 “~/' R
_ ,_, _, s> / ssorus (r)
ro cams:
v ,
;
I
Q, TO ceurae
C
Fro. 2.13
portional to the coil current, causes a horizontal force on the electron
beam and produces a horizontal displacement of the spot. The other
pair of coils YY’ give vertical displacement.
B
22 PRINCIPLES OF ELECTRONICS [CH.
The de ection sensitivity of this type of cathode ray tube is deter
mined by making simplifying assumptions similar to those made for the
electrostatic case. Over the length l of the coils it is assumed that the
magnetic eld is uniform and elsewhere zero (see Fig. 2.13). If the
electron arrives at O with velocity u it describes an arc OP of a circle of
radius r in the magnetic eld. On leaving the eld, it moves in a straight
line with uniform velocity until it strikes the screen at R. The distance
L from the coils to the screen is assumed to be much greater than l.
With this condition y/L 2 tan 6, where y is the de ection of the spot
and 6 is the angle between the horizontal line OM and the tangent to the
circular arc at P. If C is the centre of the circle of which OP is an arc,
then angle OC P is also equal to 6, and 6 = arc OP/r. The angle 6 is
usually small and tan 6='6 and arc OP =1. Thus y = Ll/r. But it
has already been shown in Section 2.7 that r = mu/Be, and hence
y = LleB/mu
gives the de ection produced by the eld B, i.e., the de ection is pro
portional to the coil current. The velocity u is found from the gun
voltage and the Energy Equation.

2.10. Electron Optics

Electron optics is that branch of electronics which deals with the pro
duction of beams of electrons such as are used in cathode ray tubes.
The motion of the electrons is controlled by means of electric or magnetic
elds, and the beams may be brought to a focus by suitable adjustment
of the elds. The whole process has much in common with the control
of light beams by means of apertures and lenses, as studied in geometrical
optics. There are “electron lenses " of various types, and these have
numerous applications, the most spectacular of which is the electron
microscope, which can give a high magni cation with greater resolution
than the optical microscope. Some simple types of electron lens are
considered in the next sections.

2.11. Magnetic Lens

The helical path of an electron moving in a uniform magnetic eld is
considered in Section 2.7, and this is the basis of magnetic lenses. In
Fig. 2.14.a a cathode acts as a source of electrons which are accelerated
by means of a high voltage anode with a small aperture. The electrons
passing through the aperture form a diverging beam. An axial magnetic
eld B is provided by means of a solenoid. The electrons which pass
normally through the aperture have no motion at right angles to B, and
their motion is unaffected. A diverging electron, making an angle 6
with the axis, describes a helix of radius %3 Z sin 6, where u is the velocity
of the electrons at the aperture. As shown in Section 2.7, all the electrons
complete one helical “ pitch” in the sa.rne time, equal to 21cm/B6. The
2] ELECTRON MOTION 23

distance travelled along the axis is then 253%: 4 cos 6, since u cos 6 is the
axial component of the velocity. If 6 is small, cos 6 is nearly unity, and
hence all the electrons have approximately the same relative positions
at I, at a distance 21rmu/Be along the axis. The paths of several electrons
are shown in Fig. 2.14.a, and it is seen that they come together at I,
where an image of the aperture is formed. This type of magnetic lens
gives unit magni cation.

it (APPaox.)
a_._ 5?. __+

ooooggggooooo w“

.....;:r Q\ Z? I

| \
1’ 0
OOOOOOOOOOOOO
/
ON:j.i;i
() M)
Fro. 2.14

A second type of magnetic lens uses a non uniform magnetic eld such
as is formed by a short coil, Fig. 2.14.b. We consider an electron travel
ling from the left parallel to the axis but above it, and entering the eld
of the coil. There is an inward radial component of the eld, and this
gives the electron a velocity component sideways out of the plane of the
paper. This velocity component is normal to the axial magnetic eld,
and gives rise to a component of force towards the axis. When the
electron passes through the coil it is in a reversed radial eld which
gradually reduces the sideways component of the velocity. As long as
there is any sideways velocity there is a force on the electron towards
the axis. The electron therefore emerges from the lens with a velocity
directed to the axis, and it ultimately intersects the axis at some point
1 ‘. Since we started with a parallel ray, F is the focus of the lens.

2.12. Electrostatic Electron Optics

The force on an electron in an electric eld is in the direction of in
creasing potential. Now consider an electron moving with velocity ul in
a region where the potential is v, and which is separated by a small planar
gap from a region where the potential is U2 (see Fig. 2.15). There is no
24 PRINCIPLES OF ELECTRONICS [C1 I.
force on the electron except in the gap, where the force is normal to the
boundary. After crossing the gap the electron moves with velocity n2.
There is no change in the electron velocity parallel to the gap, and hence
u, sin i = u, sin r,
i.e., n2/u, = sin i/sin r,
where i and r are the angles between the beam and the normal to the
boundary as shown. Using the Energy Equation, it follows that
sin i/sin r = \/ E/\/17, .
In the refraction of a light ray at the boundary of two media there is a
similar formula of the form
sin i/sin r = n2/nl,
where n, and n, are the refractive indiees of the two media. Thus x/T1

I
'1 '3
I
___ _ _ r___‘1=_
1 v~.

“Q
I
e> I
Fro. 2.15

in electron optics is analogous to n in geometrical optics. In electrostatic

elds the potential does not change discontinuously, so that the strict
optical analogy would require a medium with continuously variable
refractive index.
2.13. Electrostatic Lenses
Fig. 2.16 represents a section through two equal co axial cylinders
separated by a small gap and maintained at potentials v, and v2. The

v,+| 2)
vr>_! ''/( 2 : v2
\ \ \ I
\ \ ‘I‘I \\‘
’ '1 \ \\ \\
I I
' I I
\ \ I I
\\ \ \ I ll I’

DIVERGENT ~ _ \_~ /,/___ ’

etecrsou — iL ‘ I im ' I \

BEAM
"2 >1 "1
—> FORCE ON ELECTRON
Fro. 2.16
2] ELECTRON MOTION 25
sections through some surfaces of equal potential are also shown. Three
representative rays of a divergent electron beam are shown entering the
eld. The force acting on an electron is in the direction of increasing
potential and is perpendicular to the equipotential surfaces as shown by
the small arrows. Thus up to the gap the electron paths converge towards
the axis, and after the gap they diverge again. Although the eld is
symmetrical about the gap, the divergence is less than the convergence,
since the electrons are moving with slightly greater velocity after the gap.
In addition, the force towards the axis increases with the distance from

.3 N
{R
1
mt
§
“M

,1’! ‘Qt
\\\
’Q
0' I ‘I I

1 I

\ ‘, I
\\:\ \\
\\
\ \ \ \
\ \\\ I
\
\ I

$17.’ 1“ 0" "

"2>"I
—r FORCE ON ELECTRON
Fro. 2.17

the axis. This system therefore acts as a converging lens, and the “focal
length ” may be altered by varying the potential difference between the
cylinders. Electrostatic cylindrical lenses are sometimes constructed
with cylinders of unequal radius.
Various electrostatic lenses can be formed by means of apertures in
cliaphragms. The nature of the lens depends on the potentials of the
diaphragm and the regions on either side of it. An aperture lens is shown

HEATEs§lZI::= I=II'""IIIZI:::::::::::°§§“§======....
_L_ l_|.L_ __.
CONTROL FIRST FINAL
CATHODE ELECTRODE ANOOE ANODE
ov Iov +soov Hooov $¢REEN
Fro. 2.18

in Fig. 2.17. An aperture also serves the purpose of limiting the size of
the beam which passes through.
The electron gun in a cathode ray tube is usually a combination of
aperture and cylindrical lenses. An example is shown in Fig. 2.18,
26 PRINCIPLES OF ELECTRONICS [cH.2
where there is an aperture lens and a cylindrical lens. The intensity of
the beam or “ brightness" is adjusted by varying the voltage of the
control electrode, and the focus is controlled by means of the rst anode
voltage.

L J EMITTER
I \
o I ,
' \
I MAGNETIC
¢ CONDENSER
&\\\‘ 4 LENS

11 v r‘— SPECIMENI
' MAGNETIC
m:‘r,;:z osuecnve
, LENS
I \

!~\
'41 Hd FIRST mace
mzpm PROJECTOR
=~ <=
I I

A LENS
I|\
I \

/i‘.
s:==:!SECONOlMAGE
| PHOTOGRAPHIC PLATE
Fro. 2.19

In the electron microscope either magnetic or electrostatic lenses may

be used, and the lenses correspond to the condenser, objective and pro
jector of an optical projection microscope. One system is illustrated
in Fig. 2.19.
 CHAPTER 3

ELECTRONS IN MATTER

8.1. Electrons in Matter

In Chapter 2 the free electron is considered as a small negative charge
having a certain mass. In order to explain some observed properties of
(!l0CtI'OnS it is necessary to assume that an electron also has a magnetic
moment, which may be considered as due to the spinning of the electron
charge about an axis. This spin is signi cant only in a magnetic eld,
and the axis of spin has meaning only with reference to the direction of
the eld. The spin may be in one of two possible directions relative to
the axis. Electron spin does not concern us to any great extent in
electronics. It is introduced brie y in the present chapter, in which we
consider the energies of electrons in association with matter.
3.2. Electrons in Atoms—Energy Levels
According to modern ideas, an atom of an element consists of a positive
charge or nucleus surrounded by one or more negative charges or electrons.

ENERGY

\,\i\.l./
WV) I nomzxruou

I 3,
C I2 7
B |2 |
I SOME OF THE
AVAILABLE
mouse LEVELS

A no 2

NORMAL
Qi — —— — i uuexcnen
LEVEL
F10. 3.1

Nuclei are much more massive than electrons. In a normal isolated

atom the total negative charge of the electrons equals the positive charge
of the nucleus, and the atom as a whole is electrically neutral. The
simplest atom is that of hydrogen, which has a nucleus, called a proton,
and a single electron. In order to remove this electron from the normal
27
28 PRINCIPLES OF ELECTRONICS [cH.
atom it is found that the minimum amount of energy required is always
the same. It is concluded that the energy of the electron in the normal
atom has one unique value and is the same for all isolated hydrogen atoms.
The electron may acquire additional energy, say from impact with a fast
moving particle, and the atom is said to be excited to a higher energy
state or level. It is found, however, that only certain de nite total
energy values are possible; intermediate energy levels are not observed.
This remarkable result has been established experimentally by the study
of the radiation spectra of elements. The possible energy states for the
isolated hydrogen atom are shown in Fig. 3.1, where the normal unexcited
level is chosen arbitrarily as the zero of total energy. It is seen that
excitation of the hydrogen atom requires at least 10 2 eV of energy from
some outside source. Higher energies may excite the electron to levels
such as B (12~1 eV), C or D. Electrons do not remain in these excited
levels but return to the unexcited state in one or more steps, staying at
each excited level for a short time of about l0‘° sec. During each step
a photon of radiation is emitted in accordance with the familiar relation
hf:E2 ‘—'E1,

where f is the frequency, h is Planck's constant (6 6 >< 10'“ joule seconds)

and E1 and E, are the two levels of energy associated with the step.
When the electron is removed altogether the atom is ionized and is left
with a positive charge. The energy required to just ionize a normal atom
is the ionization energy, and for a hydrogen atom this is l3~6 eV; hydrogen
is said to have an ionization potential of 13 6 V.
In the helium atom there are two electrons. In the normal isolated
atom these two electrons have the same energy but they have opposite
spins. In lithium with three electrons two have the same energy and
opposite spins; the third electron has a greater energy. As we continue
up the periodic table to atoms with higher atomic number and more
electrons, it is found that only certain energy levels are possible and that
each level may be occupied by two electrons, which always have opposite
spin.
The atoms with greater atomic number can also be excited to higher
energy states by external means. However, as there are now several
electrons, there is greater complexity of energy levels and corresponding
spectra. In Table 3.1 are given some excitation and ionization potentials
for some of the elements commonly used in electronics; these are the
least values required, rstly, to just excite the atoms, and, secondly, to
just ionize an atom by releasing a single electron. In the latter case the
electron involved is the one of highest energy in the normal atom.
Some atoms have excitation levels which are occupied for quite long
times, l0'* sec, compared with the normal excitation levels. Such atoms
are said to be in a metastable state. In this state a further acquisition of
energy may take the atom to a higher level of the normal short period
type. The atom may then return to its unexcited state by the emission
3] ELECTRONS IN MATTER 29
of radiation. The importance of the metastable states is that they permit
excitation or ionization of atoms in two steps. Mercury has metastable

First excitation First ionization

Atom. potential. potential.
V V

Hydrogen 10 2
Helium 20~9
Neon 16 6
Sodium 21
Argon
Krypton \—lF l

Xenon
Mercury '§“!°?’T'coaoca [Quno[Qn u\1 I <?‘t°".‘°"¢."’T"€“°. PI'Q@P'@@C9
I
_ b

TABLE 3.l.— EXCITATION AND IONIZATION POTENTIALS or

VARIOUS ATOMS

states, and they play an important part in the ow of charge in mercury

vapour (see Section 5.13).
3.3. Electrons in Gases
So far we have been considering electron energy levels in isolated
atoms. We shall see in the next section that there are signi cant changes
in the energy levels when atoms are brought close together. However,
in gases the atomic spacing is so great that the changes are small, and
the ionization and excitation potentials given in Table 3.1 are also approxi
mately correct for gases.
For electrical conduction to occur in any substance there must be some
free charges available. In a gas these charges are produced by ionization.
When an atom is ionized electrical conduction may occur by movement
of the electrons or the positive ions, or both. The conductivity of the
gas, <1, is given by the expression
5 = qiPii 1 i + 59¢? e»

where q, and e are the charges of the ion and the electron respectively,
p is the number of charges per unit volume and it is the charge mobility,
i.e., the ratio of the average drift speed to the applied electric eld. On
account of the difference in mass, electrons have much greater mobility
than ions and the conductivity of gases is due mainly to the electrons
(see Section 5.15).
3.4. Electrons in Solids
In solids the atoms tend to arrange themselves in an ordered array or
crystal lattice. The nuclei are more or less xed in this array, and the
spacing between them is such that the electrons of adjacent atoms inter
mingle to some extent. The effects of intermingling are greatest with
the outermost or valency electrons which have the highest energies. By
30 PRINCIPLES OF ELECTRONICS [c1~1.
studying the X ray spectra of solids and by relating the spectral dis
tributions to the electron energies it is found that the atomic proximity
modi es the energy levels. In the isolated atoms corresponding electrons
all have the same energy, and in each atom only two electrons may have
the same value of energy. In an assembly of N atoms corresponding
electrons do not have identical energy. There are now N possible energy
states close together, and again only two electrons of opposite spin may
occupy the same state. Thus the N energy states may accommodate a
g ENERGY smzeacv |

UNOCCUPIED uuoccumeo
' LEVEL amo

u ELECTRONS HALF
— IELECTRON u susncv FILLED
sures amp

,_,
2 El E¢TP°"$ { } 2N ELECTRONS
~ gmgggv 51,755 FILLED
amp
I >
' DISTANCE THROUGH CRYSTAL

(0) ISOLATED ATOM (b) ASSEMBLY o|= u ATOM5

F10. 3.2

maximum of 2N electrons. The single energy level of the isolated atom

has become a band of energies in the solid. The individual energies
within the band are so close together that the energy band may be con
sidered to be continuous for many purposes. As an example we may
consider the metal lithium, the simplest atom which forms a solid at
ordinary temperatures. In the isolated atom we have seen that there are
three electrons, two with the same energy value and one with higher
value as shown in Fig. 3.2.a. In the solid the lower level forms a band
of 2N electrons occupying N different energy states. The higher level
forms a wider band also of N energy states; 2N electrons could be
accommodated in this band, but as there are only N electrons available
the energy band is half lled (see Fig. 3.2.b). There are also unoccupied
excitation bands in the solid corresponding to the excitation levels of the
isolated atoms. In some solids there is a gap between the occupied band
and the rst unoccupied band; in others adjacent energy bands overlap.
In addition to revealing the existence of electron energy bands in
solids, the study of X ray spectra can establish the actual distribution of
electrons amongst the available energy states within a band. In the
case of the half lled band corresponding to the valency electrons in
lithium it is found that the electron distribution follows a curve of the
form shown in Fig. 3.3. Here the ordinate ‘HE is such that nE~dE is the
3] ELECTRONS IN MATTER 31
number of electrons per unit volume with energy lying between the limits
If and E + dE. The area between the curve and the x axis thus equals
the total number of electrons per unit volume. At very low temperatures
all the lowest energy states are occupied and there is a sharp limit to the
distribution at WF, which is called the Fermi level. If the solid is given
some additional energy, say by heating, then some electrons occupy
energy states above WP, thereby
leaving unoccupied states below W1. , "E
as shown by the dotted line. There
is continuous exchange of energy
taking place between the electrons,
and the higher energies are occupied
only for a short time by individual
electrons. On the average there are \
always some electrons in the higher ° enencv (E)
levels. The Fermi level may be de W;
ned as the mean energy about which Fm, 3,3
the higher electron energies are
perturbed when the solid is heated above absolute zero.
Additional energy may be given to electrons by means of an external
electric eld. The random variations of energy and motion still occur,
but the external eld brings about an increase of average energy in a
given direction, i.e., a current ows. It is only possible to give increased
energy provided the highest occupied energy band is un lled as in
lithium. If the band is lled and there is an appreciable gap to the next
unoccupied band the solid is an insulator. If the gap is not too great,
then thermal energy may be su icient for some electrons to bridge the
gap, and then they can take part in the process of electrical conduction.
Thus in insulators the number of conduction electrons increases with
temperature, and so does the conductivity. The lowest un lled energy
band is frequently called the conduction band.
The atom beryllium has four electrons, which, in the solid, ll two
energy bands. However, the upper of these bands overlaps the next
vacant band so that beryllium is a good conductor.
We have just seen that the conductivity of insulators increases with
temperature. The reverse is tnie in conductors. Increase of tempera
ture does not affect the number of conduction electrons in a metal. The
average energy and random motion of the electrons increase, but at the
same time there is increased amplitude of vibration of the atomic nuclei
about their xed equilibrium positions. These vibrations interfere with
the energy exchanges between the electrons, and as a result the con
ductivity of a metal decreases with temperature.

3.5. Carbon and the Semi conductors

Carbon in its diamond form is of particular interest, since its crystalline
structure is similar to that of the important semi conductors, silicon and
32 PRINCIPLES OF ELECTRONICS [cH.
germanium. These three elements are tetravalent, and they form crystal
lattices in which each atom is surrounded symmetrically by four other
atoms forming a tetrahedral crystal as shown in Fig. 3.4. Each atom
shares a valency electron with each of its four neighbours, thereby form

Fro. 3.4

Q EA
e EMPTY
CONDUCTION

i }""°
euenov on E;
00000000 OVERLAPPING
0 ':¢:0:0:¢:¢:*:°::Z
00 ' ° VALENCY emos
0.0‘6_1 "°°°°¢°o:o 000
0:0:0 (FILLED)

Fro. 3.6
Fro. 3.5

ing a stable structure. The diamond lattice may be represented sym

bolically by the two dimensional diagram of Fig. 3.5, in which the circles
represent the atom cores consisting of the nuclei and inner electrons with
resultant charge + 4; the dots represent the valency electrons. In this
arrangement the electrons are seen to be bound to the atoms. The shared
electrons form co valent bonds between the atoms. In terms of energy
3] ELECTRONS IN MATTER 33
levels the valency electrons occupy lled energy bands which overlap,
and there is a gap to the next unoccupied or conduction band as shown
in Fig. 3.6. The width of the energy gap E0 is a measure of the strength
of the co valent bonds. The minimum additional energy required for
conduction to take place is E0. The differences in the electrical properties
of carbon (diamond), silicon and
germanium depend largely on their
values of E0, which are respectively Q
7, l~l and 0 7 eV. The large gap for
diamond means that at all ordinary O
temperatures there are no electrons
in the conduction band and dia
mond is a good insulator. For both
silicon and germanium, thermal
energies are su icient to bridge the
energy gap, and these materials have . Pzséllzi
some conductivity at normal room FREE
temperatures. Their conductivities ELECTRON Q
increase with temperature. When
electrons acquire suf cient energy to
Fro. 3.7
bridge the gap E0 they leave vacant
energy levels near the top of the valency band. Other electrons in that
band may therefore increase their energy by moving into the vacant
levels. Vacancies then occur at a lower energy level. These vacant
levels due to the absence of electrons are called “ positive holes ”. Thus
conduction is due to the movement of electrons with energies in the
conduction band and an equal number of positive holes in the valency
band. The conditions in the lattice are shown diagrammatically in
Fig. 3.7. Materials of the type described in this section are called
intrinsic semi conductors.

3.6. Impurity Semi conductors

The behaviour of intrinsic semi conductors can be modi ed consider
ably by the presence of small quantities of impurities, particularly when
the impurity atoms have either ve or three valency electrons. In the
former case the impurity atom may replace a germanium (or silicon) atom
in the crystal lattice, contributing four electrons for the co valent bonds
and leaving one over. As long as the impurity atoms are far apart, the
single electron behaves rather as it would in an isolated atom. The energy
of this electron is just less than that of the conduction band of the
germanium, and it may easily be excited by thermal energy into this
band and so take part in conduction. An impurity atom of this type is
known as a donor, as it gives an electron to the germanium conduction
hand. With a suitable proportion of donor atoms the conductivity may
far exceed that of the intrinsic semi conductor. The conduction is now
mainly by electrons in the conduction band, and the material is called
34 PRINCIPLES OF ELECTRONICS [CI I.
an n type semi conductor (negative carriers). Phosphorus and arsenic
are impurities which give n type conduction in germanium or silicon.
When the donor atom gives an electron to the conduction band its core
is left with a positive charge. However, this positive charge is xed in
the crystal lattice and cannot take part in the conduction process. The
n type semi conductor is illustrated diagrammatically in the energy
diagram of Fig. 3.8.a and the lattice diagram of Fig. 3.8.b. The Fermi
level at room temperature of an n type semi conductor occurs in the
energy gap but near to the bottom of the conduction band.
When an impurity atom having valency three replaces a germanium

Q
El

I cououcnou
eano QQ £6
Z,’
Q QQ

W5 ' OOOOOOQ
IMPURITY
DONOR

FILLED °
{
VALENCY
} BAND 0
“C555
ELECTRQN

G
(0) (0)
Fro. 3.8

atom in the lattice there is a positive hole available for electrons. The
energy of the vacant level is just above the lled valency band of the
germanium, and an electron may be easily excited thermally into the
vacancy, leaving a positive hole in the valency band. Electrical con
duction is now by means of positive holes, and we get p type germanium.
The impurity in this case is an acceptor, since it accepts electrons from
the valency band of the parent material. Aluminium, boron and indium
act as acceptor impurities with germanium and silicon. Fig. 3.9.a and b
illustrates the p type semi conductor. The Fermi level in this case also
occurs in the energy gap but near to the top of the valency band. Since
the energies of the conduction and valency bands are the same for both
types of material, the Fermi level for p type germanium is less than for
n type.
The conductivity of impurity semi conductors can be varied over a
wide range by varying the concentration of impurity. VVhen the tempera
ture is raised intrinsic semi conduction is increased and the proportion
of majority carriers in either n or p type material decreases. It should
3] ELECTRONS IN MATTER 35
be noted that even when conduction is due primarily to electrons or to
positive holes the net charge of a semi conductor is zero.
We may thus picture semi conductors as having free charges consisting
of electrons with energies in the conduction band and positive holes in

Ge
H
iii
Q O

cououcnou ,
{ } BAND Ge In Ge
V1,]? ACCEPTOR
o o o o o o IMPURITY HOLE
O1'0OOOOOQ FILLED
:0:< :0:0:0:0:0:0:0 vAL5N¢y
0.: 0000000 BAND
Ge
— I I 9

(e) (b)
Fro. 3.9
the valency band. The electrons and holes have random motion through
the crystal. When an electric eld is applied there is superimposed on
the random motion a drift of electrons in one direction and positive holes
in the opposite direction. The process is very similar to that occurring
in gases, where there are also negative and positive carriers, and the
conductivity 0 of a semi conductor may be expressed in the form
5 = e(Ppi I p + Pal’ n)»
where p and p. again give the density and mobility of the carriers. In
many semi conductors one or other type of carrier is predominant. The
mobility of electrons is somewhat greater than the mobility of holes.
These conditions should be compared with those in gases. One major
difference between gases and semi conductors is the method of producing
the carriers. Thermal energies are sufficient to produce ionization in
semi conductors, but collision processes are necessary in gases, where the
ionization energy is much greater.
The marked variation of conductivity with temperature in semi con
ductors is put to practical use in a number of ways for measuring and
controlling temperature. Materials prepared for this purpose are known
as thermistors; they are characterized by a large negative temperature
coefficient of resistance.
3.7. The p n Junction
Due to the presence of donor impurities in n type germanium, there
are always electrons in the conduction band. Also due to thermal energies
a few electrons are excited into the conduction band, leaving behind some
36 PRINCIPLES OF ELECTRONICS [cl I.
holes in the valency band. Thus in n type germanium there is an excess
of conduction electrons but there are some holes in the valency band.
Some electrons are always falling back into the holes as well as into the
donor levels. The whole process is a dynamic one, with some statistical
average distribution depending on temperature. Corresponding dynamic
conditions exist in p type germanium, where positive holes in the valency
band are in the majority but
p n there are always some elec
[:3 :1 trons in the conduction band.
ELECTRON { A _i_ When there is a transition
E egcy CONOUCTION
i CONOUCTION } from p to n type germanium
} \v{__:__i_____
LEVELS WI". F in a single crystal a diffusion
{ 'VALENCY’
, : } { ,VALENCY'Z } p roc ess occurs a t th e b oun d
ary. The excess of conduc
L — tion electrons in the n type
'0 3
material causes a diffusion
gradient tending to drive
i electrons across the boundary
CMRGE WWW“ from the n to the p material.
Similarly, there is a diffusion
ELEC mos rmc I of positive holes from p to
eoteutuu. n. Before diffusion occurs
{¢Q the materials are electrically
etsctaorl W *1} neutral, and both diffusion
eusecv F —— — .
LEVELS { % processes result in the jJ—
' } material becoming negatively
: charged a.nd the n material
Fm M0 positively charged as shown
in Fig. 3.10. It should be
pointed out that the charged regions at the boundary are not due to excess,
but to de ciency of carriers. Thus an electrostatic eld is set up which
tends to oppose the movement of the charges and there is a potential dis
tribution across the boundary as shown in the diagram. Equilibrium occurs
when the resultant average current of holes and electrons in both directions
is zero. In this equilibrium electrons may be excited by thermal energy
from n to p conduction levels and subsequently combine with holes in the
valency band in the ji material. At the same time minority electrons
created in the p material can fall freely across the junction, giving a ow
of negative charge in the opposite direction. Similar ow of holes may
occur in both directions across the junction, but the net current in the
equilibrium condition is zero. Equilibrium occurs when the Fermi level
is the same in both materials. Thus the equilibrium energy levels in the
p n junction are as shown in the energy diagram of Fig. 3.10. At
the same time there is the electrostatic difference of potential across
the junction as described above. As shown in Chapters 5 and 6, these
various e ects explain the behaviour of junction diodes and transistors.
3] ELECTRONS IN MATTER 37

3.8. Contact Potential in Metals

In a metal the Fermi level lies close to the top of the lled part of the
conduction band. The actual value varies from metal to metal. Like
wise, the density of conduction electrons varies with the metal. When
two metals are brought into contact there is again diffusion of electrons
METAL l METAL 2

§‘2L‘3“°"°" {W
§iU ¢‘

1
}

|
{ é %¢?

2
}cououct|ou
amo

CHARGE ‘Z?

POTENTIAL 1 : <¢2_ /Q

I
iii
I
5‘ E¢"‘°" W
eueecv ' CONDUCTION

Fro. 3.11

due to difference in concentration, and equilibrium occurs with an

electrostatic potential difference across the junction and alinement of
the Fermi levels. However, the highest energy electrons can move
equally readily in either direction across the junction. The conditions
are illustrated in Fig. 3.11. This case should be contrasted with the j>—n
junction, where thermal energy is required to move electrons from n to
/> material but they may fall freely from p to n. This fundamental differ
ence is due to the fact that conduction electrons are in the majority of the
current carriers in n type germanium but in the minority in p type. In
metals only conduction electrons are concerned.
The contact potential difference between two metals is not affected
by the insertion of any number of other metals between them. Its value
is equal to the difference in work function of the two metals (see Chapter
4). Contact potential difference is of importance in valves. For ex
ample, it may exist between the anode and cathode of a diode, and it is
additive to any externally applied potential difference.
 CHAPTER 4

ELECTRON EMISSION
4.1. Electron Emission
Many electronic devices depend for their operation on the movement
of electrons across the space between two electrodes in a vacuum. This
process involves the emission of electrons from one of the electrodes. In
this chapter we consider some of the factors governing electron emission.
In a metal many electrons are free to move in a random manner
amongst the atoms of the crystal lattice, and the electron energy dis
tribution is of the form shown in Fig. 4.1.a. This distribution assumes,
amongst other things, that each electron moves in the metal _in a region

nE l n

AT o°|<

\\AT T°K
\\ _

WP W1
Lil
(<1) (b)
Fro. 4. 1

of constant potential, and that there are no electric forces acting on the
electron on the average. Since the forces are due to all the atoms and
free electrons, then in the interior of the metal the assumption of zero
force on the average is reasonable, since each electron has a very large
number of atoms and electrons on all sides of it. Some of the electrons
near the surface have, in their random motion, velocities directing them
outwards from the metal surface. These electrons are no longer sur
rounded on all sides by charges. Fig. 4.2 represents an electron which
has moved a distance x from the surface of the metal. As the electron
has left the metal, the latter is positively charged by an amount + 0.
There is a force of attraction between the negative electron and the
positive metal. The amount of this force may be determined by the
method of images, and is equivalent to the force between two equal and
38
cu. 4] ELECTRON EMISSION 39
opposite charges separated by a distance 2x. Thus the retarding force
is given by
e2 1
F = Z? 41teo

The variation of this force with x is shown by the full line in Fig. 4.3.a.
The above expression assumes that the metal has a continuous surface
SURFACE

O O O

I
f 0" ’cn Q___
Ԥ
\
/
I ‘
I
__@__ .____‘,___,2¢
K
\\\
/1 '\'\ \\1’/C
@343‘ it 13"’; ‘

\\
'/I
O§+O O\
///’$‘\
\
6 t’
”

* | | I I +

0 o"'o ° "
Fro. 4.2

and is therefore true only for values of x large compared with the atomic
spacing. Inside the metal we have assumed that F is zero, and hence
the effective force on the electron near the surface must take a form similar
to the broken curve in Fig. 4.3.a. As an electron moves out from the
surface, work must be done against the retarding force of amount

W = /zFdx.
0

W is thus the area under the curve of F and is asymptotic to some value
W1, which is the work done against the retarding eld by an electron in

F W

'00 I I I I I II I I I I I III
_£

E1%

(0) (b)
F10. 4.3
40 PRINCIPLES OF ELECTRONICS [crr.
escaping from the surface (see Fig. 4.3.b). W1 represents a “ potential
barrier” which must be surmounted if an electron is to escape com
pletely from the metal. In order to get over the barrier an electron must
have kinetic energy satisfying the relation
Qmu” > W1,
where u is the component of velocity normal to the surface. Any electron
with a value of u less than this limiting value is brought to rest before it
has surmounted the barrier and returns to the metal. In Fig. 4.1.a only
the relatively small number of electrons with energies greater than W1
may be emitted. It should be noted that by no means all these electrons
escape. The kinetic energy must be associated with a velocity at right
angles to the surface. An electron with kinetic energy just equal to W1
E WI
etecteons
ivnicn MAY .
escape
""' "'wi"“ ' ' ' ' " ' ' _ T ' " _ _ "'3
' (wonx FUNCTION)
we
We wt
(Y) (POTENTIAL BARRIER)
'1 O
ne ° Y
Fro. 4.4

is emitted with zero velocity. Those with greater energy are emitted
with velocity ‘Mg, where
Qmugz = Qmu” — W1.
There is therefore a distribution of velocities of the emitted electrons as
shown in Fig. 4.1.b. It might be thought at rst that most electrons
would be emitted with zero velocity. It is true that there are more
electrons in the metal with energy equal to W1 than with higher energies.
However, the velocities of these electrons are distributed in all directions,
and a negligible number are moving precisely normal to the emitting
surface.
In a metal at absolute zero there are electrons with energies up to W; ,
the Fermi level. The additional energy required for emission is then

This quantity 95 is called the work function of the metal surface. It is

usually measured in electron volts. The work functions of various sur
faces lie in the range 1 to 6 eV. The conditions for electron escape from the
surface of a metal are summarized in Fig. 4.4. The energy distribution
of the electrons in the metal is shown at the left, where Fig. 4.1.a has been
4] ELECTRON EMISSION 41
turned through a right angle in order that the energies may be compared
with the potential barrier.
If a suitable electric eld is created near the surface of the emitter (or
cathode) by having a nearby electrode (or anode) at a positive potential
with respect to the emitter, then the escaping electrons are collected and
a current ows from the cathode to the anode. The electrons which
escape are amongst those with the greatest energies and the emission
current causes a loss of energy from the cathode. Emission is therefore
accompanied by a drop in temperature of the cathode. The actual drop
is small, but it can be measured. The process of evaporation of electrons
is closely analogous to the evaporation of molecules of liquid. The cool
ing of the cathode arises from the latent heat of evaporation of electrons.
If there is no collecting electrode, then the emitted electrons form a
cloud of negative charge near the cathode, creating an opposing eld
which tends to drive the electrons back. An equilibrium condition is
set up with electrons continuously entering the cloud and returning to
the cathode. This equilibrium is also very similar to that occurring
between a liquid and a vapour.
4.2. Types of Emission
At room temperatures very few electrons have sufficient energy to
enable them to escape from a solid. The additional energy may be
supplied in a yariety of ways. When it is in the form of heat we have
thermionic emission or primary emission. Altematively, the electrons
may be given the extra energy by the impact on the surface of the solid
of fast electrons or positive ions, and then we have secondary emission.
Thirdly, if the energy comes in the form of radiation there is photo
electric emission. There is another type of emission in which the elec
trons are not necessarily given additional energy but the potential
barrier is reduced by means of a strong extemal electric eld. This is
known as eld emission.
4.3. Thermionic Emission—Richarilson’s Equation
As the temperature of the cathode is increased more of the electrons
have suf cient energy to escape. The relation between the temperature
and the number of electrons emitted per second from unit area of the
cathode has been established by Richardson in the form
J8 = AT21 o/1',
where J3 = current per unit area, T = absolute temperature, and A and
b are constants. It can be shown that b is related to the work function
¢ by the equation
b = kw
where k is Boltzmann's constant. If 95 is in electron volts and k in
joules/° K, then
b = 11,600 <,6,
42 PRINCIPLES OF ELECTRONICS [CI I.
and b is said to be the temperature equivalent of the work function, in
degrees absolute. The validity of Richardson's Equation can be con
rrned by plotting log, j8/ T2 against 1/ T for a diode, in which all the
emitted current is collected at the anode for various cathode temperatures,
which may be measured with an optical pyrometer. Curves for three
commonly used thermionic cathodes are shown in Fig. 4.5. The straight

Io91J,/Ta

“O 2ooo°i< iooo°i< soo°i<

os io is 2 011163 1
O \ T
=i.
\°\‘
\ s

‘ oxioe COATED CATHODE

20

so
tnoimiteo tuncsten
4o

TUNGSTEN
FIG. 4.5

lines con rm the validity, and they may be used for approximate deter
mination of A and ¢>. It is found that A varies from 60 A cm‘? (° K)‘*
for many pure metals to 0 01 for a mixture of barium and strontium oxides.
The work function ¢ ranges from 6 eV for platinum to 1 eV for the same
oxide mixture. It is seen from Richardson's Equation and the curves
pa

A, 41 b g;:;'€i::€ Emission Emission

Substance A cm" ev’ .R “£6 density, e iciency,
(° K)" ° . K’ mA/cm’. mA/W.

Tungsten . . 60 52,000 2,500 250 4

Thoriated tungsten 3 *9’? cam 30,000 1,900 1,500 60
Barium oxide and
strontium oxide 0 01 10 12,000 1,000 300 200
on nickel 1,100
O

TABLE 4.I.——PROPERTIES or THERMIONIC EMITTERS.

of Fig. 4.5 that the emission density j,9 depends much more on b or ¢
than on A. j5 also increases rapidly with temperature, so that an im
portant factor in determining the suitability of a material as an emitter
is the maximum working temperature. Of the pure metals tungsten is
4] ELECTRON EMISSION 43
the most widely used, largely on account of its high melting point.
Values of A, b and ¢ for various materials used as emitters are summarized
in Table 4.1.

4.4. Thoriated Tungsten

A mixture of two metals may have a lower work function than either
of the pure metals alone. Thus a tungsten emitter with a small quantity
of thorium has a work function of 2 6 eV, compared with 3 4 for thorium
and 4 5 for tungsten. At the same time the thoriated tungsten may be
used at a temperature
above the melting point of ENERGY GAINED in
thorium. Thus thoriated ',;1‘,§"*, ,'1’§°,_'§,$§2°°"
tungsten has the advant W
age of a reduced work _ _ _ _ _ _ _ _ _ _ _ _ __
function and a high work /" """
. . I ,1 REDUCTION in
ing temperature. During W \£gi<_r|=un¢1 |Qn
operation the thorium 1_,§E%'P°LE
gradually evaporates from W1
the surface but more
atoms diffuse from the
W, .....
core, maintaining the con
ditions at the surface. o x
The working temperature Fro. 4.6
must be restricted to pre
vent excessive evaporation of thorium. The thorium is introduced in
the form of the oxide thoria (about 1 per cent).
It is believed that a single layer of thorium atoms is formed on the
surface of the tungsten and serves to reduce the potential barrier. The
atoms arrange themselves on the surface as dipoles with their positive
sides on the outside. An electron passing through the dipole layer has
a considerable outward acceleration. As a result, the potential barrier
is reduced after the manner of Fig. 4.6. At rst sight it appears that an
electron requires to have energy W’ for escape in order to get over the
top of the barrier. However, in the case of a narrow barrier of the type
shown in Fig. 4.6, it is possible for an electron to “ tunnel " through the
peak. Thus minimum energy needed for escape is W1. The “ tunnel
effect ” can be explained using the Principle of Uncertainty.

4.5. Oxide coated Cathodes

Oxides of any of the alkaline earth metals (calcium, strontium and
barium) have very good emission characteristics. They have to be sup
ported on a suitable metal base, which may affect the emission. The
usual combination is a base of nickel, coated with a mixture of about
equal parts of barium and strontium oxides. Oxide coated nickel
cathodes are used in practically all small thermionic valves. Although
44 PRINCIPLES OF ELECTRONICS [CH.
they have been in use and have been studied continuously for many
years, their behaviour is still not fully understood, and widely differing
values of the emission constants have been quoted.
4.6. Comparison of Various Thermionic Emitters
Pure tungsten, thoriated tungsten and oxide coatings all have their
uses as thermionic emitters. The work functions are respectively 4 5,
2 6 and 1 0 eV, and the corresponding operating temperatures are
2,500° K, 1,900° K and 1,000° K. Of course the emission from any
material increases rapidly with temperature, but the maximum tempera
ture is limited to a value that gives a reasonable life. This is usually
determined by the rate of evaporation of an essential constituent
tungsten, thorium and barium respectively in the cases under con
sideration.
Since oxide coated cathodes give their emission at much lower tempera
tures than either of the tungsten cathodes, they require much less energy
to heat them. The ratio of the emission current to the heating power,
usually measured in mA/W, is called the emission e iciency. Another
important parameter of an emitter is the emission current per unit area
at the operating temperature (see Table 4.1). The values quoted are
subject to wide variations, particularly for emitters other than pure
tungsten. As already stated, the emission densities may be increased
considerably by operation at higher temperatures, but this results in
reduced life. The gures in the table are all based on continuous opera
tion. Under certain intermittent conditions much higher emission cur
rents may be obtained, especially from oxide coatings. Increases up to
20 times have been quoted for operation in pulses of 1 as duration. Cur
rent densities of 100 A/cm” and more have been obtained from oxide
cathodes in high voltage pulsed operation, but some of this may be due
to the large electric eld causing enhanced emission.
In addition to emission density and emission efficiency, electrical and
mechanical robustness are important properties of emitters. In any
valve, no matter how careful the evacuation, there are always present
large numbers of gas molecules which may form ions by impact with
electrons when current ows. The positive ions move under the in uence
of the electric eld and nally strike the cathode. If high voltages are
used the cathode is subject to considerable bombardment and may be
damaged, particularly in the case of oxide coatings. For this reason
continuous operation with oxide cathodes is limited to less than 2,000 V.
Thoriated tungsten cathodes are also liable, to some extent, to be
damaged by ion bombardment, and their use is restricted to voltages
of about 5,000 or less. Pure tungsten cathodes may be used at still
higher voltages up to 20,000. The ability to withstand high voltage
operation may be called electrical robustness.
VVhen mechanical robustness is considered the position is reversed
and the oxide coating is distinctly superior to either of the tungsten
4] ELECTRON EMISSION 45
cathodes‘. This is particularly true when the oxide cathode is used in
the indirectly heated form which is described below. Tungsten cathodes,
which are usually in the form of wire laments, are inherently fragile.
The elds of use of the various types of cathode may be summarized
roughly as follows. On account of the high emission efficiency, oxide
coated cathodes are used whenever possible. All receiving valves, small
transmitting and amplifying valves and gas lled valves have oxide
cathodes. Pure tungsten is used in the largest valves, where high powers
or high voltages are involved, as in large transmitters, liigh frequency
industrial heating sets and X ray generators. Thoriated tungsten is
used in some medium power valves. In high voltage pulsed applications,
such as radar, oxide coated cathodes are generally preferred.

4.7. Mechanical Form of Thermionic Emitters

Practically all tungsten and some oxide coated cathodes are con
structed from wire laments. Filamentary cathodes are heated directly
by passing current through them.
There is therefore a potential drop oxioe COATING
along the cathode, and for some _/\
purposes this is undesirable. This
difficulty is eliminated in the in A
directly heated cathode, two forms kl
of which are illustrated in Fig. 4.7.
In each of these a nickel cylinder has
an oxide coating on part of its out "l AM" ¢*I"°°5
side surface, and it is heated with a OPO O O OQ
separate wire inside the cylinder.
The heater is usually of tungsten
coated with alumina to insulate it ¢YL|NQp|¢AL ¢A "1905
from the cathode. Indirectly heated Fro. 4.7
cathodes are normally used with oxide
coatings where the nickel base may be formed readily in any convenient
shape.
In some directly heated cathodes, particularly in large transmitting
valves, the magnetic eld produced by the heating current may influence
the electron flow.

4.8. Secondary Emission

When a beam of electrons or other particles strikes a surface with
sufficient energy secondary electrons may be knocked out of the surface.
The emitted electrons are called secondary to distinguish them from the
incident or primary beam. The energy of the primary electrons must
exceed some minimum value, and at rst the number of secondaries
increases with the primary energy. The ratio of the number of secondaries
to the number of primaries is called the secondary emission coefficient, 8.
46 PRINCIPLES OF ELECTRONICS [cH.
The value of 8 depends not only on the energy of the primary beam but
also on the nature of the surface and the angle of incidence of the beam.
The variation of 8 with the energy of the primaries usually follows a
curve of the form shown in Fig. 4.8. As the primary energy increases 8
rises fairly rapidly at rst as more secondaries are knocked out. At
some voltage, usually in the region of a few hundred volts, 8 reaches a

6
IO
SECONDARY
EMISSION
COEFFICIENT O
CAESIATED SILVER
6

‘ BARIUM OXIDE
2

O 200 400 6OO BOO IOOO 1200

VOLTAGE OF PRIMARY ELECTRONS
Fro. 4.8

maximum and then drops slowly. When a primary electron is moving

very fast it may penetrate some distance into the metal before striking
an electron, and this electron has less chance of escape than one released
near to the surface; hence the maximum in the curve for 8.
In a diode with a positive anode the emission of secondary electrons is
relatively unimportant, since they are immediately attracted back to the

A2 .
A1 ,, Ur
i§~“:'
\\
. I ‘\ SECONDARY
(2 I ELECTRONS
IPRIMARY
|ELEC'I’RON$
I Z =

I CATHODE

Iz'1+z',)
Fro. 4.9

positive anode. However, in a valve with two positive electrodes the

secondary electrons emitted by one of them may be collected by the other.
In Fig. 4.9 a primary beam strikes a target electrode A1 and the emitted
secondaries go to the surrounding collector A1. The effect of the
4] ELECTRON EMISSION 47
secondary emission is to reduce the target current i1. If the secondary
emission coef cient is greater than unity, then the secondary current
exceeds the primary current and i1 is negative. The total electrode
current i1 + i1 is, of course, equal to the primary current, and is unaffected
by secondary emission.
Electrodes with values of 8 > 1 may be used to give current ampli ca
tion. By use of an arrangement such as that shown in Fig. 4.10
with several secondary emitting electrodes, large ampli cations are
possible. If 8 is the coefficient for each electrode, then the current from
the primary cathode is ampli ed at the nal collector A by a factor of
8", where n is the number of secondary emitting electrodes. The primary
beam in a device of this kind is usually produced from a photo electric
cathode and the tube is called a photo multiplier. Current ampli cations
of the order of 10° may be obtained with tubes of this type. Satisfactory

LIGHT 0 O 0 I Q

I. 3 5 7 9
'\ I\ 1\ /\ I\ A

,
.' \\’ \\, \\, xxf
'\ \\f

2 4 B IO
PRIMARY .
EMITTER
Fro. 4.10

operation of multiplier tubes depends on the production of surfaces with

values of 8 greater than unity by an appreciable amount. For clean
metal surfaces 8 ranges from about 0 5 to 1 5 at the optimum voltage.
However, certain surface impurities consisting of thin layers of insulators
or semi conductors give values up to about 10 or even more. Caesium
oxide on a base of silver with a value of 8 about 6 is used in photo multi
pliers. Satisfactory operation of multipliers depends not only on having
high values of 8 but also stable values of 8, if uniform performance is to
be obtained. Since 8 varies with the electron velocities, very stable
voltage supplies are essential.
Secondary emission plays an important part in the operation of a
cathode ray tube. The phosphors which are used in the screen are
insulators, and but for secondary emission the screen would develop a
negative potential as the primary electrons are collected. The negative
potential would increase until it exceeded the cathode potential, and then
no more electrons would reach the screen. However, there is secondary
emission from the screen, and the screen potential automatically adjusts
itself to an equilibrium value nearly equal to the beam voltage.
Secondary emission occurs in many other electronic devices, and it may
modify the working conditions considerably. Its effect on the charac
48 PRINCIPLES OF ELECTRONICS [cH.
teristics of triodes and tetrodes is considered in Chapter 6. Where it is
desirable to minimize secondary emission from an electrode, its surface
is sometimes coated with carbon black, for which 8 is 0 5.

4.9. Photo electric Emission

Photo electric emission occurs when electrons are emitted from a
surface as the result of incident radiation falling on the surface. The
energy for emission is supplied to the electrons by the photons of radiation.
An essential condition for emission is that the energy of a photon must
exceed the work function of the surface, i.e.,
hf>¢
where qt is the work function. Thus the higher frequency (shorter wave
length) radiations are more effective in producing photo electric emission.
For a given surface the wavelength corresponding to hf = ¢ is called the
threshold wavelength, and it may be veri ed that it is given by
i = 12,400/¢ A,
where 95 is in electron volts. Emission is not possible with radiation of
longer wavelength. For photo emission to occur in the visible spectrum

RELATIVE
INTENSITY
IOO

so CD —

60

40

2° Cs Q Ag
0
4000 6000 3000 10000 12000
WAVELENGTH K
Fro. 4.11

with light of wavelength 6,000 A, the work function of the emitter must
be about 2 eV or less. Amongst the pure metals only caesium (1 9)
satis es this condition. The threshold wavelength is greater for certain
composite cathodes. Amongst commercial cathodes caesium antimony
(Cs Sb) and caesium oxygen silver (Cs O Ag) have threshold wave
lengths of about 7,000 and 12,000 A respectively. The latter wavelength
is well into the infra red region of the spectrum.
Another important property of a photo cathode is the variation of the
emission with wavelength or the spectral sensitivity. This depends in a
complicated manner on many factors, such as the optical properties of
 4] ELECTRON EMISSION 49
the cathode surface. If all the radiation is re ected or transmitted there
is no energy transferred to electrons. Where energy is absorbed the
emission is greatest when the absorption is mainly at the surface. The
spectral sensitivity curves for Cs—Sb and Cs O—Ag photo cathodes are
shown in Fig. 4.11. In these curves the ordinates represent the emission
current divided by the energy of radiation per unit bandwidth over a
small band of wavelengths. It is found experimentally that for radiation
of a given wavelength the current is proportional to the intensity of the
radiation.

4.10. Schottky Effect and Field Emission

An external electric eld may affect the emission of electrons from a
surface. The external eld combines algebraically with the potential
barrier, and the emission is reduced or increased, depending on whether
the eld is retarding or accelerating. The conditions are illustrated in

W W‘ EXTERNAL W‘ WI
A+ ev“ _ / '4 — — — — — — — — — — —

£>
/ In Q _

I
I’ POTENTIAL
BARRIER

I
4

0 xa x 0 i0

(<1) (b) (¢)

RETARDING FIELD ACCELERATING FIELD STRONG ACCELERATING FIELD
Fro. 4.12

Fig. 4.12. With a retarding eld, Fig. 4.l2.a, i.e., when a nearby electrode
is at i negative potential with respect to the emitter, the potential
barrier is effectively increased and only electrons with energy greater
than W4 escape. When the external eld is accelerating, Fig. 4.12.b,
the effective potential barrier is reduced to W3, and any electrons with
energy greater than this may escape. This is known as the Schottky
Effect, and explains the increase of thermionic emission in a saturated
diode when the anode voltage is increased. When a very strong extemal
eld is set up near the emitter the potential barrier is narrowed as shown
by the broken curve in Fig. 4.12.c, and now the tunnel effect may
operate and large emission may be obtained (T). Under these conditions
the many electrons with energy near to or below the Fermi level may be
emitted, and the emission is little affected by temperature. Very intense
elds are required to produce this Field Emission. Usually it is ob
tained only from sharp points on the emitter surface when the voltage
gradient is of the order of 2 to 5 X 10° V/m. Emission densities as high
as 1,000 A/cm” or more can be obtained.
50 PRINCIPLES OF ELECTRONICS [cH.4

A retarding eld of the type shown in Fig. 4.12.a may be used with a
diode to deterrnine the numbers of electrons emitted with various
velocities from zero upwards, for thermionic, photo electric or secondary
emission. The velocities are found from the Energy Equation when v
is the retarding anode potential, and the numbers are proportional to the
anode current.
 CHAPTER 5
DIODE CURRENTS
5.1. Flow of Charge
In this chapter the ow of charge in the space between the two elec
trodes of a diode is considered. The diode is the simplest possible elec
tronic tube, but even so, the evaluation of space charge ow is a com
plicated process and can be done exactly only in a limited number of
idealized cases. Some of the factors affecting the ow are: (i) the nature
of the medium between the electrodes, (ii) the size and relative positions
of the electrodes, (iii) the electrical potentials of the electrodes, (iv) the
availability of free charges in the space, (v) the physical and chemical
nature of the electrode surfaces. External in uences, such as radiation
and magnetic elds, can also affect the current through the diode. Some
or all of these factors may play a part either by deliberate action on the
part of the valve designer or the user, or sometimes accidentally. It
would be extremely di icult to try to take account of many of these
factors at one time. In this chapter certain idealized cases are con
sidered, and these may be used to give some qualitative explanation of
the measured characteristic curves of actual diodes.
Three main types of diode may be distinguished by the medium be
tween the electrodes: (i) vacuum diodes, (ii) gas lled diodes, (iii) crystal
diodes. Each type is dealt with in turn, and before attempting to explain
the physical principles representative characteristic curves are given.
5.2. Characteristic Curves of Vacuum Diodes
For a ow of current to take place there must be charges available to
act as current carriers. In a vacuum diode these charges are emitted
from the cathode by using the thermionic effect or the photo electric
effect. In the thermionic diode the
cathode is heated, and the number of [AI I
electrons emitted per second may be [f,:',1 'EER;TURE: ifs‘? EESHMGE
controlled by varying the cathode CURRENT : CURRENT
temperature. If the anode is main
tained at a constant positive potential
v4 with respect to the cathode, the
emitted electrons ow to the anode, and $71 CONSTANT
there is a current in the external circuit.
At room temperatures the emission is j
negligible. A measurable current is
rst obtained at some temperature T1, '0 T‘ T3 T2 T
as shown in Fig. 5.1. The current Fm‘ 5'1
51
52 PRINCIPLES OF ELECTRONICS [cH.
increases with temperature up to T1, after which it remains practically
constant. If the cathode temperature is kept constant at some value such
as T3, then the anode current varies with the potential difference 1 ,, after
the manner shown in Fig. 5.2.a. This curve was obtained for a diode

‘AI Iii
1, P e e
I
T constant T constant
N

° va I° va
(<1) (b)
tuncsten catnooe oxioe coateo catnooe
Fro. 5.2

with a pure tungsten cathode. Fig. 5.2.b shows the characteristic curve
for a diode with an oxide coated cathode. In both cases the current
starts to ow at a small negative anode voltage and increases steadily
as v4 is made more positive up to a point B. In the tungsten diode the
current then increases only very slightly with v4. In the oxide coated

fa

LIGHT FLUX CONSTANT

la CONSTANT

O uont FLUX O 9A
Fro. 5.3 Fro. 5.4

diode the current continues to rise beyond B but at a slower rate. The
rate of variation of the current with the voltage varies considerably.
The quantity r,,, de ned by
1 82',
fa — 8124’

is called the diode slope resistance. Its value at any point on the curve
may be found from the gradient of the tangent at that point. For
example, in Fig. 5.2.a, 1,, = MP/NM gives the slope resistance at the
point P.
5] DIODE CURRENTS 53
In the photo electric vacuum diode the electron emission is controlled
by varying the quantity of radiation falling on the cathode as shown by
the characteristic curve in Fig. 5.3. If the incident radiation is kept
constant, then the anode current varies with the voltage across the diode
as shown in Fig. 5.4. From these curves it is seen that for a given voltage
the current varies linearly with the light ux, and for a xed light ux
the current is practically independent of the anode voltage after the
initial rapid rise.

5.3. Physics oi the Planar Vacuum Diode—Potential Distributions

The rst idealized case that we consider is a thermionic diode consist
ing of two parallel conducting planes whose dimensions are large com
pared with the distance between them, and we con ne our attention to

1; 0
anooe (A)

W’ _
1!l!.!
9:0 F ' catnooe (K)
Fro. 5.5
E‘
current ow in regions remote from the edges of the planes (Fig. 5.5).
When the cathode is cold there is negligible charge in the space and the
electric potential varies linearly from zero at the cathode to v4 at the anode,
as shown in Fig. 5.6.a, where E2 is the e.m.f. of the battery connected to
the diode and v4 = E1. The slope of this curve, 2%; gives the magni
tiide of the electric eld strength, which is constant across the diode
(Fig. 5.6.b). The conditions are also illustrated in Fig. 5.6.c with equal
positive and negative charges on the anode and cathode. When the
cathode temperature is raised until there is some emission the electrons
move to the anode under the in uence of the electric eld and constitute
the current whose value is given by Richardson's Equation. However,
there are now negative charges in the space, and these cause a reduction
in potential throughout the space. The cathode and anode potentials
are maintained at their previous values by the battery, and the potential
distribution across the diode now takes the form shown in Fig. 5.7.a.
The corresponding eld strength is given in Fig. 5.7.b. The presence of
the electrons has reduced the eld strength near the cathode and increased
it near the anode. The negative charge in the space induces positive
charges on the cathode and anode. These induced charges add to those
C
V5A4_ R1
‘A
A1__
Q
E
EO
_tO
+++K
“W
i ___
__
E *2
_
( _1/ A
IIIII|II
6
_d
3
________d
F_9_7_ +++++_+
+++
+ + _+ ++
A
++++A
K
K‘
K
‘_
_
_K
__
AII

_O
VA
VH
VOOOA W
X ‘X
_X
“XX K._
K._
._
._
‘._
’_
._
_._

0
M) ._._
._(
‘_
._
I._
((“Cc_.‘I‘I1/
.c
._
_
F 5
5
5

).__

F_m
x I

( 0)

AI|I|l|l A
l :A{I1

‘._
._
‘_

F_m
X
( 0 b

MI
I‘
‘A F_ _oK
m
X M
(U 0)
6] DIODE CURRENTS 65
already in existence, and now the total positive charge on the anode is
greater than the total negative charge on the cathode (Fig. 5.7.c). The
actual charge density on the electrodes is proportional to the eld strength
just outside the surface. In considering these diagrams it must be
realized that there is a steady ow of electrons across the space all the
lime. The total number of electrons in ight at any instant remains the
same, and, although isolated charges are shown in Fig. 5.7.c, charge is
distributed throughout the whole space. As the emission is increased
farther, the potential in the space continues to drop. Also, the eld
strength at the cathode decreases and may become zero as shown in
l"ig. 5.8, or negative as in Fig. 5.9. In the latter case the force acting on
electrons in the region between the cathode and the potential minimum
at M is such that it tends to oppose the emission. Now for an electron
to reach the anode, it must be emitted from the cathode with a suf ciently
high velocity to enable it to pass through the region of the retarding eld.
lilectrons emitted with lower velocities are brought to rest between K
and M and then return to the cathode. Under these conditions many
of the electrons emitted by the cathode fail to reach the anode, and the
current density is considerably less than the value given by Richardson's
l quation. The current is limited by the retarding eld set up by the
electrons in the space and is called a space charge limited current. Where
the emission is small as in Fig. 5.7 and the eld strength at the cathode
is accelerating, all the electrons emitted from the cathode reach the anode.
Under these conditions there is a temperature limited current, since the
current is determined by the cathode temperature only. The limiting
case between temperature limitation and space charge limitation is shown
in Fig. 5.8, where the eld strength is zero at the cathode surface. This
case would occur if all the electrons were emitted from the cathode with
zero velocity. Then the density of the current would adjust itself to the
value that would just reduce the electric eld to zero at the cathode
surface. Any tendency for the current density to increase would give a
retarding eld at the cathode, and the ow would drop, since the electrons
have no initial velocity to overcome the retarding eld. On the other
hand, any tendency for the current to decrease would give an accelerating
lielcl at the cathode, and if the cathode emission is su icient the current
would immediately increase to give the equilibrium state with zero eld
at the cathode. Conditions such as those just described can be realized
if 21,4 is varied. Increase in 12,4 raises the potential across the diode.
The current density then increases to give zero eld at the cathode again.
Similarly, reduction in v4 gives a lower current density.
In the case of the photo electric diode the electron currents involved
are always very small, and they have practically no effect on the electric
lield except for very low values of '04. There is therefore no space charge
limitation to the current, which depends almost entirely on the amount
of radiation falling on the cathode and not on '04 as shown by Figs. 5.3
and 5.4. By analogy with the thermionic diode the photo electric
56 PRINCIPLES OF ELECTRONICS [Cl I.
current is sometimes called “ temperature limited ”, though strictly it
would be more correct to call it “ light limited ” Temperature limited
currents are frequently called saturated currents.

5.4. Planar Vacuum Diode—Space charge Flow

In the case of space charge limitation the current density varies with
v4, and is practically independent of the cathode temperature. In most
applications valves are used under these conditions. A quantitative
relation between the current density and v4 can be established in the
following manner for the limiting case when the electrons leave the
cathode with zero velocity. In order to ensure that space charge limita
tion is maintained it is also assumed that there is an unlimited supply of
electrons available from the cathode.
Any region with space charge must satisfy Poisson’s Equation
8% 8% 8'1: p
w+w+w".;
where v is the potential and p the charge density at the point (x, y, z).
For the parallel plane case considered in the last section the problem is
one dimensional (see Fig. 5.5). If the x axis is chosen perpendicular to
the electrodes and distances are measured from the cathode, then
Poisson's Equation becomes
@__a
dx* _
so
The Energy Equation provides an additional relation
mu’/2 = ev,
where u is the electron velocity at distance x from the cathode. This
equation assumes zero velocity at the cathode where v = 0. A third
relation connects the current per unit area ] with p and u
] = pu.
This Continuity Equation merely a irms the continuous ow of charge
analogous to the ow of liquid along a pipe. From the Energy Equation

and by substitution in the Continuity Equation

m 1/2
P=](§) Fm
Hence from Poisson's Equation
dz” _ _ l ’l 1/2 v 1/2
(?_ so 26 '
2 1/2
i.e., 3 ]; = Av“/2, where A = — éogé)
5] DIODE CURRENTS 57

The equation may be integrated by multiplying both sides by 2 (1%, giving

d

3% = 2Av'1/2 gt, i.e., = 4Av1'2.

The constant of integration is zero, since 5 : and v are both zero at x= 0.

The rst of these, g = 0, is the condition for space charge limitation

with zero initial velocities. If we now take the square root and rearrange
the terms we nd that
v"/‘ dv = 2A ‘/*dx.
A nal integration gives
vs/4 =% A1/2x_
The constant of integration is again zero. On substituting for A we nd
for the current density
4:0 2e 1/2 vi’/2
1 " aia) 3;=
If numerical values are substituted
2 33 X 10" vi‘/2
] A I » xz A/m‘.

The minus sign indicates that the positive direction of the current is from
anode to cathode in the valve. At the anode v = v4 and x = d, and
then
J X d] 20 6 Ulla’: Alma.

This equation is called the Child—Langmuir Equation or the “ three

halves power law " for space charge ow. It may easily be veri ed from
the above equations that v cz x‘/3, E oz x1/3, p a if”/3 and u a x2/3.

5.5. E ect of Space Charge on Electron Transit Time

In some uses of diodes the time of ight of an electron between the
cathode and the anode is important. This is found from the equation
fdx
1'= —,
0 u
where u is given by the Energy Equation. In Section 2 5 it is shown that
the time of ight between parallel planes in the absence of space charge
is 1 = 2d/u_,, where u4 is the electron velocity at the anode.
2;:
l~‘or space charge limitation u varies as x2/3, and so u = 14,.
With this value of u we nd the transit time 1 = 3d/144. Thus the
58 PRINCIPLES OF ELECTRONICS [cm
effect of the space charge is to increase the transit time by 50 per cent.
The numerical value for 1 with space charge can be determined from
1 3d _ sec.
5 9 >< 1051/1,,
5.6. Space charge Flow for any Geometry
The current owing in a space charge limited diode is proportional to
the three halves power of the voltage for the case of parallel plane elec
trodes. This relationship holds for a diode of any shape. Poisson's
Equation, the Energy Equation and the Continuity Equation are uni
versally true. Poisson's Equation shows that p is proportional to v
provided the potential gradient and v are zero at the cathode. The energy
equation gives u proportional to v1/2. It follows therefore from the con
tinuity equation that ] is proportional to vi’/2. The constant of pro
portionality may vary considerably with the geometry.

5.7. E ect of Initial Velocities of the Electrons

In deriving the Child Langmuir Equation for the current density under
space charge limited conditions it is assumed that the electrons all leave
the cathode with zero velocity. It is shown in Section 4.1 that the
electrons are emitted with a range of velocities. This affects the value
of the current and also, as explained in Section 5.3, there is a potential
minimum at some position between the cathode and the anode. The
potential minimum acts as a “ virtual cathode ” and the Child Langmuir
Equation is still applicable to some extent, provided the potentials and
distances are measured from the virtual cathode. The current density is
now given by an expression of the form
. a _ a/2
1 2 33 X <i10_ iii). ””> <1 + K> A/ma
where v4 and d have the same meanings as before, vu = the difference
of potential between the potential minimum and the cathode and dy is
the corresponding distance. The factor K is included to make allowance
for the electrons passing the potential minimum with some velocity.
All of these new factors increase the value of ] (v,1; is negative). The
actual values of v,1, d1, and K depend in a complicated manner on the
cathode temperature, the value of _] and the total emission given by
Richardson's Equation. The effect of initial velocities is greatest at low
values of v4. With an oxide coated cathode operated at v4 =1 V,
_] may be about double the value given by the normal Child Langmuir
Equation. Even at v_4 = 10 V, the initial velocities may increase ] by
25 per cent.

5.8. Space Charge in Magnetrons

In Chapter 2 the paths of individual electrons in magnetrons are dis
cussed ignoring the effects of other electrons in the eld. In practice,
5] DIODE CURRENTS 59
large currents may be involved, and then the space charge must modify
the behaviour considerably. The analysis of the magnetron, taking
space charge into account, is extremely dif cult. However, in the case
of the cylindrical magnetron at cut off a possible state of equilibrium
may be reached with all the electrons describing circular paths round the
cathode. The greatest density of electrons occurs near the cathode and
the density is just su icient to reduce the eld strength at the cathode
to zero, i.e., to give the necessary condition for space charge limitation.
The force on each electron due to the magnetic eld is towards the cathode.
The resultant of this force and the forces due to the electrode potentials
and the space charge distribution is just suf cient to maintain the uni
form circular motion. The presence of space charge does not affect the
cut off conditions. The critical magnetic eld is derived from the
Energy Equation and the condition of zero radial velocity at the anode,
and hence it depends only on the potential difference between the
electrodes.

5.9. Gas Diodes

When gas is introduced into a diode the ow of current may be modi
ed considerably. This can be illustrated with reference to some of the
diodes considered earlier in this chapter. For example, the 1'4, '04 charac

lk z',_l
I I
I
’ I
I
I
’I
'0

0 <
Oil Ul
>‘ '0 6 <
:51
Fro. 5.10 Fro. 5.11

teristic curve of a vacuum photo electric diode is shown by the full line
in Fig. 5.10. The current reaches its saturation value when the anode is
a few volts positive with respect to the cathode. The introduction of a
small quantity of gas into such a diode has little effect on the charac
teristic at lower voltages, but when '04 reaches about 15 V the current
begins to rise again, and continues to increase fairly rapidly with v4 as
shown by the broken line.
The thermionic diode is also affected by the presence of gas. Fig. 5.11
shows the current in a diode which has an oxide coated cathode and
contains some mercury vapour. As '04 is increased from zero the current
rises slowly, and at about 10 V has reached 1 mA. When v4 is increased
60 PRINCIPLES OF ELECTRONICS [c1~r.
slightly above 10 V the current suddenly rises at an enormous rate. In
practice, the characteristic curve of a gas lled diode is obtained with a
resistance in series to limit the current to a safe value. The circuit is
shown in Fig. 5.12. When the
current rises suddenly a glow R
appears inside the diode. The I +
size and intensity of the glow A |
increase as the current increases.
For the ow of charge in a 9A +
vacuum diode it is essential that
the cathode should be heated V
or irradiated to produce elect
rons. In the presence of gas, Fro. 5.12
however, there is no need to
energize the cathode. If the voltage across the diode is gradually in
creased from zero no measurable current is obtained at rst, but at some
voltage vs there is a sudden ow of current, as shown in Fig. 5.13; vs is
called the breakdown voltage. The value of vs depends on many factors,
such as the pressure of the gas and the geometrical arrangement of the
electrodes. In air at atmospheric pressure breakdown occurs between

[A
IA‘ i

(Lggats) ,
|

.° vs vA
Fro. 5.13 "s lg,
Fro. 5.14

parallel plane electrodes when the voltage gradient is about 3 >< 10° V/m.
At lower pressures the breakdown occurs at much lower voltages. In
order to investigate the is, vs curve, the circuit of Fig. 5.12 is again used.
A typical characteristic for a diode lled with neon at a pressure of about
1 mm Hg is shown in Fig. 5.14. Again it is found that at the breakdown
voltage glow appears and the glow increases with current. Flow of cur
rent like that represented by Fig. 5.14 is frequently described as a “ cold
cathode discharge ".
There is one feature which is common to the characteristics of all these
gas diodes. There is a marked increase in is at some value of vs. In
some cases this increase is so rapid that we say that breakdown occurs.
Full explanation of the results is even more di icult than in the case of
5] DIODE CURRENTS 61
vacuum diodes. Some of the effects of gas cannot be adequately explained
even qualitatively, and little has been achieved by way of quantitative
explanation.

5.10. Electron Collisions with Gas Atoms or Molecules

When light shines on the cathode of a photo electric diode some elec
trons are liberated and move towards the anode under the in uence of
the electric eld due to the potential difference between the cathode and
the anode. At low voltages some of these electrons are collected by the
anode. Others escape from the eld or are collected on the walls of the
container. As the voltage is increased more of the electrons are collected
and soon saturation occurs, corresponding to the horizontal part of the
characteristic in Fig. 5.10 when all the liberated electrons reach the
anode. The electrons in their ight may “ collide ” with the atoms or
molecules of the gas. There are several possibilities, depending on the
energy of the electron at the time of impact. A slow electron has an
elastic collision with conservation of the total momentum. Since the
atom is much heavier than the electron, the speed of the latter is barely
affected, but its direction of motion may be altered. If the electron
energy at the time of impact is equal to or greater than the excitation
potential of the atom, then it may use some or all of its energy to excite
the atom to a higher level, with subsequent emission of a quantum of
radiation as the atom returns to its normal state. Finally, if the electron
has su icient energy it may cause ionization of the atom, resulting in
the production of a positive ion and another electron. There are now
three charges, instead of one, free to move in the electric eld. The
positive ion moves towards the cathode, the two electrons towards the
anode and all three charges contribute to the current. When ionization
takes place there is therefore an increase in the current. This is the
explanation of the difference between the characteristics of the vacuum
and the gas lled photo cells in Fig. 5.10. At about 15 V ionization
occurs, causing the current to increase. This effect is known as “gas
ampli cation ".

5.11. Breakdown
If the voltage across the photo electric gas diode is several times the
ionization potential, then one electron liberated at the cathode by
radiation may produce ionization at a short distance from the cathode.
The positive ion moves towards the cathode and two electrons now move
towards the anode. Each of these may gain enough energy to cause
further ionization, giving two more positive ions and two electrons. Now
there are four electrons available to continue the process. It is obvious
that very large current ampli cation is possible. Another effect may
also occur due to the positive ions. It is possible for an electron to be
ejected from the cathode due to bombardment by the ions, or for an
62 PRINCIPLES OF ELECTRONICS [CH.
electron to be produced close to the cathode owing to ionization by ions,
although the latter is an inefficient process. If the ions arising from one
initial photo electron manage to produce at least one electron at the
cathode by some method or other, then this electron initiates the whole
process again, and the discharge once started maintains itself, even if the
radiation is shut off. Breakdown has occurred, and the current increases
enormously unless restricted by series resistance. Under these conditions
there is a self maintained discharge. The characteristic curve for the
gas lled photo cell has now become extended, as shown in Fig. 5.15.
The region of the characteristic from the commencement of ionization
to the breakdown is called the Townsend discharge. After breakdown
there is usually a visible glow, and this region is known as the glow
discharge. The visible glow is due to the emission of radiation by those

.
1,1 isr_
____ _ _ sac DISCHARGE
GLOW DISCHARGE C

ABNORMAL
— GLOW DISCHARGE
B __ _
uoamt
BREAKDOWN GLOW DISCHARGE
: .1 rovmseuo I A — — — ——
. DISCHARGE .

's I5, "M "s lg

Fro. 5.15 Fro. 5.16

atoms which have been excited to higher energy levels by electron

collisions and are returning to their normal state.
In practice, gas lled photo cells are never used in the glow. discharge
region, since the bombardment of the cathode by ions damages the
photo electric coating on the cathode. The limits of safe operation with
reasonable gas ampli cation are speci ed by the maker.

5.12. Cold cathode Discharge

There is always some ionization taking place in a gas. Cosmic rays,
radioactive materials or some other source of radiation cause the libera
tion of electrons. Normally these electrons move through the gas until
they recombine with positive ions. Equilibrium is set up when the rate
of ionization by external causes equals the rate of recombination of ions
and electrons. If, however, an electric eld is maintained between
electrodes in the gas some of the electrons and ions are collected at the
5] DIODE CURRENTS 63
electrodes. If the eld is increased all the charges may be collected
before there is any recombination, resulting in a saturated current. If
now the eld is increased further, ionization may occur, and obviously
the process may pass through the Townsend discharge to a glow discharge
exactly as in the case of the gas lled photo cell. The main difference is
that the initial current and saturation may occur at a very low level of
the order of 10'“ A. The actual characteristic of a cold cathode diode,
as determined by the circuit of Fig. 5.12, is therefore of the form shown
in Fig. 5.16. After breakdown occurs at vs the voltage across the diode
drops to vy. The value of vM depends on the type of gas, the pressure
and the nature of the electrodes. It varies from about 50 V to several
hundred volts for different diodes. For a given diode it is found that in
this condition of glow discharge the voltage drop across the diode is
practically constant, even though the current changes considerably, as
shown by the region AB in Fig. 5.16. The voltage vM is called the main
tenance voltage or sometimes the operating or burning voltage. The
breakdown voltage vs is sometimes called the striking or ignition voltage.
Examination of a diode working in the region AB shows a distinct glow
on the cathode surface, and the area of glow increases with the magnitude
of the current. It is believed that the current density in the cathode glow
remains constant and the change in current arises from the change in
area of cathode glow. Under these conditions the voltage drop across
the diode is practically constant. When the whole of the cathode surface
is glowing, then further increase in current causes an increase in voltage
drop across the diode, as shown by BC in Fig. 5.16. The region BC is
sometimes referred to as the abnormal glow to distinguish it from the
normal glow discharge over AB.

5.13. Arc Discharge

If the current in a glow discharge is increased through the nonnal glow
to the abnormal glow it is found that at some point such as C in Fig. 5.16
there is a sudden change in the nature of the discharge. The voltage
across the diode drops to the order of the ionization potential or even less.
The mechanism of this new discharge, called an arc, is not fully under
stood. It is probable that the metastable energy levels play an important
part in maintaining an arc discharge with a voltage drop lower than the
ionization potential. Atoms may be excited to a metastable state and
then receive additional energy to cause ionization. For example, mer
cury has an ionization potential of 10 4 V and it has a metastable level of
5 5 V. Thus if an atom of mercury collides with, say, a 6 V electron the atom
may be raised to its metastable level. If this atom is now struck by
another electron whose energy is 4 9 eV or more it may be ionized. Thus
ionization may occur with voltages considerably less than the ionization
potential. The possibility of successive stages of excitation to produce
ionization is obviously much greater with atoms which have metastable
levels of relatively long duration.
64 PRINCIPLES OF ELECTRONICS [C1 I.
In addition to the low voltage drop, an arc discharge is characterized
by an intense bright spot on the cathode which may cause evaporation
of the cathode material. Another common feature of an arc discharge
is a reduction in voltage across the diode as the current is increased, i.e.,
the arc has a negative slope resistance. The currents involved in arc
discharges may be extremely high and frequently result in destruction
of the cathode material. In many applications precautions must be
taken to ensure that an arc discharge cannot occur. However, in some
cases the high current densities are exploited, notably in the mercury arc
tube, in which a pool of mercury acts as the cathode, and in the carbon
are lamp, in which the intense bright spot is used for illumination.

5.14. E ect of Pressure on Breakdown

For ionization an electron must have suf cient energy when it collides
with the atom. The energy of an electron starting from rest at the
cathode depends on the potential difference through which it moves
before a collision. This potential difference in turn depends not only on
the voltage applied to the diode but also on the distance the electron
travels before the collision occurs. At the next collision the energy
depends on the nature of the previous collision and the distance travelled.
The important factors are obviously the applied potential difference and
the average distance between the gas atoms, i.e., the “ mean free path ”
of the atoms. The latter is known to be inversely proportional to the
pressure when the temperature is constant. At atmospheric pressure the
electrons collide with atoms before they have gained much energy from
the eld, and ionization is unlikely to occur. On the other hand, at very
low pressures, although the electrons may acquire high energies from the
eld, the chances of striking an atom are small and again little ionization
occurs. There is therefore an optimum pressure, and in the electron
devices in common use this is usually around 1 mm of Hg.

5.15. Hot cathode Discharge

The discharge in a cold cathode diode is initiated by the few electrons
produced by stray radiations. In the gas lled photo cell the electrons
are produced by radiation falling on the cathode. In both cases the
number of initial electrons is relatively small, and if breakdown is to occur
these electrons must be involved in a number of ionizations. Thus the
breakdown voltage is several times the ionization potential of the gas.
In a gas diode with a thermionic cathode there is a plentiful supply of
electrons immediately available, and it is found that there is a marked
increase in current when the voltage across the diode is of the order of
the ionization potential (see Fig. 5.11). However, the physical process
involved is different from the cold cathode case. The main cause is the
neutralization of the negative space charge by the positive ions. We
have already seen with the thermionic vacuum diode that the current
density is limited by negative space charge. Partial neutralization of
5] DIODE CURRENTS 65
the space charge by positive ions allows the current density to be in
creased at the same anode voltage. With complete neutralization, i.e.,
with equal electron and ion densities, the full emission current density ]s
could be drawn from the cathode. A negligible part of this current is
due to the extra charges produced by ionization, as may be seen from
the following considerations. The speed u of a charged particle at a
position of potential v is given by the Energy Equation
14 = \/(2qv/ml
where m and q are the mass and charge of the particle which has moved
from rest through a potential difference v without collisions. Thus for a
positive ion and an electron with equal but opposite charges
141/us = \/(mi/me) ,
where the suffices e and i denote electron and ion respectively. In this
equation it is assumed that the two charges have moved from rest through
the same potential difference but in opposite directions. The current
density due to electrons is
' ' J: = Pcue»

where p, is the density of the electrons. Similarly, the current density

due to ions is
]r = Pr“:
Taking the case of atoms of mercury,
m, = 1,840 >< 201 m,;
and hence for p, + pg = 0
], = 600 ], .
Thus the space charge can be completely neutralized while the con
tribution of the ions to the current is negligible. A region with p, + p, = 0
is called a plasma. In practice, the ow of current is controlled by the
resistance of the external circuit. It is important that the current
should be limited to less than the total cathode emission.
We have explained qualitatively the sudden rise in the characteristic
of the hot cathode mercury vapour diode given in Fig. 5.11. The actual
voltage across the diode is less than the ionization potential, so that
successive collisions and metastable energy levels must play a part in
the process.

5.16. Potential Distribution in Hot cathode Diode

In a parallel plane diode with the cathode unheated the potential varies
linearly from zero at the cathode to vs at the anode, as shown by the
broken line KQ in Fig. 5.17. When the cathode is heated and a gas dis
charge occurs the potential variation is as shown by the full line KPQ.
Practically all of the voltage drop across the diode occurs in a short range
66 PRINCIPLES OF ELECTRONICS [cr~r.
from the cathode, KP. The region P to Q, where the voltage is nearly
constant, corresponds to the plasma. In the region KP electrons move from
the cathode to the plasma under the in uence of the electric eld and ions
leave the plasma for the cathode. The electron current is much greater
than the ion current, but on account of the much lower velocity of the
ions the ion density is greater than the electron density. The region from
the plasma to the cathode behaves rather like a diode with positive ion
ow obeying a three halves power law. The distance from K to P
adjusts itself to give the ion ow needed to maintain the equilibrium
conditions for the particular value of total current through the tube, as
determined by the external circuit. The
POSITIVE voltage across KP remains constant.
9| '°" 5"5‘“" The region KP round the cathode is
]/'| PLASMA known as a positive ion sheath. The rest
|£:: ' Q of the space is lled with plasma. On
account of this, the geometrical shape of
P I ,’ the electrodes in a thermionic gas diode
,’ is much less critical than in a vacuum
/' diode. In the latter the distance be
,’ tween cathode and anode is an important
’ factor in the current ow. In the gas
_O
X\ > K diode the thickness of the positive ion
_ ,—_.;1—cn

FIG, 5,17 sheath adjusts itself automatically to suit

the value of current required, and the
rest of the space is lled with plasma. The voltage across the whole tube,
which is practically equal to the voltage across the sheath, remains con
stant even though the current varies over wide limits.
As already mentioned, it is important in using a hot cathode diode to
keep the current below the total emission of the cathode. Otherwise,
the voltage drop across the diode would rise and the increased energy of
the positive ions would cause damage to the cathode, which is usually
oxide coated.
When the anode of a hot cathode diode is negative with respect to the
cathode the device behaves like a cold cathode diode, but the reverse
breakdown voltage is much greater than the ionization potential. The
value of the breakdown voltage varies with the gas pressure, and hence
with the temperature. The latter has to be maintained within de nite
limits in a hot cathode diode.

5.17. Ionization Counters

Certain types of gas lled diode are used for the detection or counting
of atomic, nuclear or radiation particles. A common fonn of diode
counter is shown in Fig. 5.18. The cathode is a metal cylinder forming
part of the diode envelope. The anode is a thin axial wire, and the
particles to be counted pass through a thin window of mica or some
5] DIODE CURRENTS 67
light alloy. In one type of counter the anode voltage is well below the
breakdown voltage. When a fast particle penetrates the window it
causes ionization and the electrons ow towards the wire anode. The
eld is weak except near to the anode, where the electrons may acquire
suf cient energy for ionization and so give an enhanced ow of charge.
The total pulse of current owing depends on the original ionization
produced by the particle, and a diode of this type is called a proportional
counter.
If a diode of the type shown in Fig. 5.18 is used at a higher voltage,
then a fast particle may initiate a discharge which spreads along the
whole length of the diode. Used in this way, the pulse of current through
the diode is the same for all
particles with energy great cxraoor :
enough to trigger the dis
charge. As in the propor WINDOW . ANOOE
tional counter, most of the
ionization occurs near to the /4
anode and the electrons are cas |=a_uzo
removed rapidly by the eld, 1 . ,G_ 5_1s
leaving the more massive
positive ions as a space charge surrounding the anode. This space charge
reduces the eld near the anode, and the discharge ceases when the eld
is no longer suf cient to maintain it. Thus the initial particle produces a
large pulse of current which is independent of the nature of the particle.
Diodes of this type are known as Geiger counters. The rate of counting
with these diodes is limited by the time taken for the positive space
charge to disappear.
Counters may be used for photons of radiation, in which case the
ionizing process is initiated by the emission of photo electrons from the
cathode.
Electronic methods of counting the pulses of current are described in
Chapter 18.

5.18. Crystal Diodes

The properties of semi conducting crystals, such as silicon and germa
nium, which are discussed in Chapter 3, are used in certain types of diode.
These crystal diodes may be divided into two main classes—point contact
diodes and junction diodes. In the point contact diode a thin metal wire,
or “ cat's whisker ", makes contact with the surface of a slice of ger
manium or silicon crystal, as shown in Fig. 5.l9.a. When the metal
wire is made positive relative to the crystal it is found that the resistance
to current ow is much less than with the opposite polarity. A typical
characteristic curve is shown in Fig. 5.l9.b.
A semi conductor like germanium may be either n type or p type, de
pending on whether its free charges are mainly electrons or mainly holes.
68 PRINCIPLES OF ELECTRONICS [cH.
Also n and p type materials may be combined in one crystal to form a
p—n junction as shown in Fig. 5.19.0. Such a crystal junction has a
non linear characteristic of current against voltage of the form shown in
Fig. 5.l9.d. The nature of the p n junction characteristic is explained
qualitatively in the following section. The behaviour of the metal semi
conductor point contact diode is not fully understood. However, this
type of diode is of considerable practical importance as a recti er, par
ticularly at very high frequencies, on account of its low capacitance
Of the order Of 0 I p.p.F.
Other semi conductors are also found to give non linear contacts with
l.

A
WHISKER
cavsm. ° lg;
K
(1) (b)
1
ASP" K

A K
O ‘Ki:
CRYSTAL

(c) (4)
Fro. 5.19

metals. Selenium, copper oxide and copper sulphide are all used, as
well as germanium and silicon.
The crystals of silicon or germanium used in diodes have to be pre
pared most carefully to a high degree of purity, with accurately con
trolled traces of impurities, in order to give the required type of semi
conduction. The connecting leads must make low resistance contact
with the crystal. As seen in Chapter 3, semi conductors are highly sensi
tive to temperature changes. As a result, it is most important that
crystal diodes are operated within their speci ed ratings.
5.19. The Junction Diode
It is shown in Section 3.7 that across the transition region of a p n
junction there is a potential difference and distribution of electron energy
5] DIODE CURRENTS 69
levels as shown in Fig. 5.20.a. Equilibrium is brought about by the
charge concentrations and the resulting electrostatic eld, which opposes
the diffusion of the negative carriers from n to p and the positive holes
from p to n. The ow of these majority carriers is limited to those with
sufficient energy to overcome the potential barrier. There is also a con
tinuous creation by thermal energy of minority carriers, i.e., electrons in
the p region and positive holes in the n region. These minority carriers
pass readily across the barrier. In the isolated junction in equilibrium
there is equal ow of electrons in both directions across the junction.
There is also equal ow of holes, and the resultant current is zero, as
indicated by the arrows in the energy diagram in Fig. 5.20.a. Now con
_ t li; _ — +
p I1 p n p n

POTENTIAL — —/—__ ———/___ —— /—_—_

{9 . erscrnous {9 euscrnoss (9 > E‘ E¢T"°"5

— \‘>‘.~LQ}
D

i@—souzs
—x_‘ “F
4Q} {
T
O ;L_
“HOLE: Q ]
‘_\‘;1 “'
HQL55 4 Q}

(°) (0) (‘=1

Fro. 5.20

sider the connection of a small potential difference across the crystal so

that the p region is positive with respect to the n region, as in Fig. 5.20.b.
This reduces the potential barrier at the junction so that more holes move
across to the n region and electrons to the p region. The holes recombine
with free electrons in the n material. Similarly, the electrons recombine
with holes in the p material. Both of these movements give positive
current ow from p to n. This current obviously increases with the
applied potential difference. There are still minority carriers being
created in both regions, and they cause a current in the reverse direction.
However, their contribution is no greater than in the equilibrium case
with no external potential difference. When the polarity of the applied
potential difference is reversed the potential barrier is increased and it
opposes further the ow of majority carriers (see Fig. 5.20.0). The
movement of the minority carriers still occurs, and this now accounts
for most of the current. Thus with reversed polarity there is a small
saturated current. Since the number of minority carriers in both regions
depends on the temperature, the saturated reverse current in a p n
junction varies with temperature.
 CHAPTER 6

TRIODES, MULTI ELECTRODE VALVES AND TRANSISTORS

6.1. Triodes, Transistors and Ampli cation
In Chapter 5 we consider the nature of the current voltage character
istics of various types of diode—vacuum, gas and semi conducting.
These characteristics vary considerably, but they show that the cur
rent depends in some manner on the potential difference between the
cathode and the anode. In this chapter we consider in turn the charac
teristic curves of the triode and other multi electrode valves, both vacuum
and gas lled, and the transistor. Some attempt is made to give physical
explanations of the nature of these characteristics.

6.2. Characteristic Curves of Vacuum Triodes

At a xed cathode temperature the current owing to any electrode
depends on the potentials of all the electrodes. If the potentials are
measured from the cathode, which is taken as the zero of potential, then

1, "

Q
.
"6
in“: O =

.. [K 1
Fro. 6.1

the electrode currents is, is and iK are functions of vs and vs (see

Fig. 6.1) of the form
50 =f1(‘"0, '04)» 5.4 =f2(‘U0, 11.4), ix = — (5.4 + 50)
The anode current is now a function of two variables, vs and vs, and any
number of characteristic curves may be drawn. If vs is taken as the
independent variable, then a set of characteristics is obtained as shown
in Fig. 6.2, where each curve corresponds to a xed value of vs. These
is, vs curves are called grid or mutual characteristics. A second set of
curves relating is to vs and vs, but using vs as the independent variable,
is shown in Fig. 6.3, where each curve corresponds to a xed value of
vs. These are known as anode characteristics. A third way of showing
the relationship between the same variables is given in Fig. 6.4, where
vs is plotted against vs with anode current constant for each curve;
70
CH. 6] TRIODES, VALVES AND TRANSISTORS 71
these are constant current characteristics. All of these gures contain
the same amount of information and, given any one set, the other two
may be derived.

|‘A v Isl
\*\ ‘I VG
VA; l R

"Ar
"As / "or
)§4
L_____
‘A2 F
D
B _ _ _ _ _ _ _1 l
|
‘C M IE
I I V O y
"or "oz ' G . '61 6
Fro. 6.2 Fro. 6.3

The rate of variation of the anode current with the grid voltage when
the anode voltage is constant is an important quantity for any triode.
It is de ned by the relation
_ as)
gm — (3710 vs constant, /4.
" ‘ii
and is called the mutual conductance, O \
or sometimes the transconductance, of \ K\
the triode. It is given by the slope \ 5 J21
of the is, vs characteristics, and its H‘,
value varies with the point on the ‘\
curve where it is measured, particu \
\
larly when is is small. Hence in \
\
quoting the mutual conductance it is \
necessary to give the operating values \
of the anode current, anode voltage \ __
and grid voltage. In Fig. 6.2 a +yG1 ° vs
tangent has been drawn at the point
B, and then gm = CD/BC ; g,,, is usu FIG‘ 6'4
ally measured in mA/V.
The slope of the anode characteristics for constant grid voltage provides
a second important triode parameter, de ned by
_l__ 81 s
€¢—‘7,a—(a_ ;
VA )1 s constant

gs is called the anode slope conductance and rs the anode slope resistance.
It is seen that rs also varies with the value of the anode current, being
72 PRINCIPLES OF ELECTRONICS [cl 1.
greatest when is is least. Its value, at the same operating point B as
was used for g,,, above, is given in Fig. 6.3 by rs = BE/EF.
Finally, the slope of the constant current characteristics gives a measure
of the relative importance of the anode and grid voltages in controlling
the anode current, and the relation,
,, _ _ (@111)
avG is constant,
de nes the ampli cation factor [J . The minus sign is inserted, since the
changes in vs and vs are usually in opposite directions if the anode current
is to remain constant. Except when vs is low, the constant current
characteristics are parallel straight lines and n is almost independent of
the operating point over quite a wide range. The value of p. at the point
B is given in Fig. 6.4 by |J. = HK/HB.
It follows from the de nitions of [J , gm and rs that P. = g,,,r,. There is
no need to have three sets of characteristics in order to determine approxi
mate values of [J , gm and rs. They may be found from any one set by
taking the variation in any two of the variables is, vs and vs, whilst the
third is kept constant. For example, we have seen already how to nd
g,,, from the grid characteristics in Fig. 6.2. The same characteristics
may be used to deterrnine p. by using the line BC along which is is con
stant. Along BC, vs changes from vs; to vss and vs from vs; to U92,
giving 11. = (vs; — vss)/(vss — vs;) at an anode current of is;. Similarly,
along CL in the same gure, vs is constant and rs = (vs; — vss)/(iss — is;)
at about the same value of anode current. The values of |J and rs ob
tained in this way are approximate, since nite increments of the variables
are used instead of the in nitesimals implied in the formal de nitions.
The curves considered so far have given the anode current in terrns of
the grid and anode voltages. Similar curves may be drawn relating the
grid current to vs and vs, and corresponding grid current parameters
could be detennined. However, in many triode applications the opera
tion is such that very little grid current ows. It can be seen from
Fig. 6.2 that currents ow to the anode when the grid voltage is negative.
Few electrons ow to the grid under these conditions although it acts as
an effective control of the anode current. When the grid voltage is
positive some of the electrons ow to it and the total space current is
divided between the anode and the grid. Under most conditions of
operation the grid current is much less than the anode current, except
when the anode voltage is very low. The curvature at the lower ends
of the constant anode current curves in Fig. 6.4 is due to an appreciable
part of the total space current owing to the grid. This is also the cause
of the change of curvature from concave upwards to convex upwards in
the anode characteristics when vs is positive (see Fig. 6.3).
6.3. Valve Equation for Small Changes
Changes in anode current arising from small changes in both the grid
and anode voltages may be determined approximately as shown in Fig. 6.5,
6] TRIODES, VALVES AND TRANSISTORS 73
in which two grid characteristics are drawn for anode voltages of vs;
and vs; + vs. The anode current is is; when the grid and anode voltages
are vs; and vs;, and it becomes is; + is when the voltages change to
vs; + vs and vs; + vs. In the gure, BC = vs and CE = is. Since BD

is

"Ar"%

El ' “T 'iAt+i0
I | .
' ‘Kr ‘v
B ' 0
16 ~ (M
I I '
| I

"or Yer“? ° '5

I
—>y9<—

Fro. 6.5

is a grid characteristic and BD is not much different from the tangent

at B, then CD ~,g,,,vs. Also, along DE, vs is constant and hence
8is
DE = a— vs = vs/rs.
".4
Since CE = CD + DE,
in = gravy 'i" ‘U0/7:»

It is implied above that gs, and rs are constant over the range of variation.
We obtain the important Valve Equation relating the change in anode
current to changes in grid and anode voltages. This equation applies as
long as the changes are suf ciently small for the valve parameters to
remain constant. The grid or anode characteristics may be nearly
parallel straight lines over an appreciable range of vs and vs (see Figs. 6.2
and 6.3). For that range, gs, and rs are constant and the Valve Equation
can be used.
6.4. Triode Ratings
The operating conditions for triodes vary considerably with their size
and the purpose for which the valves are designed. Current may range
from a few milliamperes to 100 A, voltage from a few volts to 10 kV,
and powers from about a watt to over 100 kW. The current taken by
any valve must usually be kept well within the saturation limit. If
higher currents are required, then valves with larger cathode area must
be used. Voltage limitations usually depend on the quality of the
74 PRINCIPLES OF ELECTRONICS [cH.
insulation, and on the possibility of an arc occurring between the elec
trodes. Frequently the voltage and current are not limited by any of
these factors, but by the power capabilities of the valve. The product
vsis represents the power dissipated by
FILAMENTS the electrons striking the anode, and its
value must be restricted to keep the anode
temperature within safe limits. In most
" small valves the anode gets rid of its heat
I by radiation through the glass envelope.
onto " P,<;|_A55 The permissible anode dissipation may be
increased considerably by making the anode
part of the external envelope as shown in
Fig. 6.6. The anode may now be cooled
Q‘:;—': :f:
by natural or forced air convection, or by
4"I. £5111’
Z}.~. 7"": i i GLASS METAL . .
SEAL circulating water. In cases where electrons
ow to the grid another limitation is set
by the maximum permissible grid tempera
METAL ture. This limit may be set not only to
mssg prevent damaging the grid by over heating,
but also to avoid thermionic emission from
the grid. The maximum perrnissible grid
dissipation depends on the nature, number,
diameter and length of the grid wires, and
on the grid supports, which may conduct or
Fro. 6.6
radiate heat from the wires.
In valves used at high frequencies a
further limitation may arise from heating of the electrode leads and con
nections, due to the high frequency currents owing in them.
The various current, voltage and power ratings of any valve are
closely interrelated. In use it is desirable to ensure that a limiting rating
is not exceeded.

6.5. Physics of the Vacuum Triode

The triode characteristics described in Section 6.2 were all obtained
with the cathode temperature su iciently high to maintain space charge
limitation. If this condition is not satis ed the anode current curves
show saturation effects similar to those in diodes when temperature
limitation occurs.
In Section 5.3 it is shown that current ow in a diode is determined
by the combined effects of the elds due to the positive anode and the
negative electrons in the space. When initial electron velocities are neg
lected the equilibrium condition occurs when the eld at the cathode
surface produced by the space charge just cancels the eld due to
the anode potential. When the anode potential changes the current
density adjusts itself to restore zero eld at the cathode. Clearly the
current density is closely related to the eld at the cathode due to
6] TRIODES, VALVES AND TRANSISTORS 75
the anode potential alone. The same conditions apply in the triode,
but now the grid and anode potentials both affect the eld strength at
the cathode. If the combined effect gives a positive potential gradient
at the cathode in the absence of space charge, then when the cathode is
heated to produce adequate emission, the‘ current density assumes the
value that just reduces this potential
gradient to zero. These potential varia ANE
tions may be illustrated with reference
to a planar triode of the type shown in x< <..
.><. (<1)
section in Fig. 6.7.a. The grid consists
of a series of equally spaced wires. Each PLcarso e ‘i€_T'T_ PLanooeANE
CO
ANEOD
of the electrodes is maintained at a de
nite potential by means of an external
battery. The potential at any point in PL
GRD
the space depends on the electrode » KI (b)
potentials. For all of the diagrams, Fig.
6.7.b to e, the cathode is at zero potential ' ' \':_)( vs (swmu. mo
Q NEGATIVE )
and the anode at potential vs. Each EFFECT OF
diagram corresponds to a different value SPACE CHARGE
of vs. In b and c the potential gradient
at the cathode is positive, and when the Y VA (C)
cathode emits, current ows at such a a IX ‘
V630
density that the potential gradient be oy
EFFECT OF
comes zero as shown by the broken lines. SPACE CHARGE
In d the cathode eld is zero, as a result
of the combined effects of the positive Y ‘K (<1)
vs and negative vs, and no current ows.
This is the cut off condition. In e the ° x It; (cur oi=|=)
conditions are well beyond cut off. The
relation between vs and vs at cut off is
a straight line passing through the origin r (¢)
if emission velocities of the electrons are Y
neglected (see curve for is = 0 in Fig. ‘o BEYOND
6.4). Hence the ampli cation factor as X VG (cur OFF)
previously de ned in Section 6.2 is given Fro. 6.7
by p. = — vs/vs at cut off. These values
of vs and vs produce zero eld at the cathode in the absence of space
charge. The ampli cation factor may therefore be de ned in terms
of a purely electrostatic system with no current ow. As de ned
in this electrostatic manner, p. may be calculated from the geometry of
the electrodes. The actual calculations are beyond the scope of this
book, but some of the factors affecting pi are self evident. The further
the anode is from the grid, the less is its effect on the eld at the cathode.
Thus p. increases with the grid—anode spacing. Also, increase of the
number of grid wires per unit length, or of the wire diameter, increases
the screening of the anode from the cathode, and so increases [J .
76 PRINCIPLES OF ELECTRONICS [cl r.
It may be deduced from the electrostatic de nition of n, that
p. = Css/Css, where Css and Css are respectively the grid cathode and anode
cathode capacitances between the electrodes. These capacitances include
only the active electrodes and not any effects due to leads or other elds
outside the electrode system.

6.6. Equivalent Diode

From Sections 6.2 and 6.5 it is seen that the grid is much more effective
than the anode in controlling the current in a triode; in fact, the grid
voltage is (J. times as effective as the anode voltage. For some purposes
it is convenient to imagine that the triode is equivalent to a diode with
anode voltage equal to vs + vs/p.. This anode may be placed at any
position, but usually it is considered to coincide with the grid. It is
shown in Section 5.6 that the anode current density of any diode varies
as the 3/2 power of the anode voltage. Hence for the triode
is = ktva + vi/I1)’/2.
where k is a constant.
It follows then that
gm = iktva + vs/i )1” = iii’/° 11"“
k”/3 .
and 1/rs = % J Z41/3.
These two relationships show that gs, and 1/rs vary as the cube root of
the anode current.

6.7. E ect of Space Charge in a Triode

In Fig. 6.7.b and c the effect of the space charge on the potential dis
tribution is shown by the broken lines. The space charge lowers the
potential appreciably between the cathode and the grid but has little
effect near the anode. The reason for this is that the electrons are moving
most rapidly at the anode. The current density ] is constant across the
valve and the Continuity Equation tells us that j = pu, where p = space
charge density and u is the electron velocity. Hence when u is greatest
p is least, and the space
P°TE"TW charge has least effect on
B C the potential. For many
yG= vA purposes, where valves are
Z: §\ operated
F
with negative grid
\
,;*’ 4/ : and positive anode, it is
\ \ perrnissible to neglect the
\ \ O\
\\

\\
/'3’
\
\
// n \\\ I‘
51ancn
effect of space charge on
CATHODE onto X
/ E moo: the P°t@"ti=11 in the grid“
anode space. However,
"'dk9 _"'i dis _'_’ when the grid potential is
1=;s_ 6,3 positive and comparable
6] TRIODES, VALVES AND TRANSISTORS 77
with the anode potential, the space charge may have considerable
effect on the conditions in the grid anode space. This may be shown
with reference to Fig. 6.8, which shows the potential distribution in a
triode with the mean grid potential equal to the anode potential, and
the grid—anode distance double the cathode grid distance. The full line
OBC shows the variation in potential across the valve in the absence of
current. A mean grid potential is assumed and the variation around the
grid wires is ignored. When the cathode emits and a space charge
limited current ows the potential between the cathode and grid drops,
as shown by the broken line ODB with zero slope at the cathode. We
may assume in the rst place that most of the electrons pass through the
grid into the grid anode space. The grid and the anode are maintained
at equal potentials vs and vs by
the external supplies. The space POTENTIAL
charge, however, depresses the
potential between grid and anode. If v
Minimum potential occurs mid way G I I I\:\\\
\ _1,/ I

between, at E. As a result, the I’ '| \\\_2

\
\
I I \\
electron velocities are least and the
space charge density is greatest at
. I
' 9 \\ ._\.L_. 0‘N‘
O
E. This concentration of space CATHODE onto anooe
charge tends to depress the poten Fro. 6.9
tial still further. With the par
ticular arrangement shown, with dss equal to twice dss, it is seen from
symmetry that the potential at E falls to zero if all the electrons leaving
the cathode pass through the grid and move on to the anode. The
potential and charge distributions for E FB and EHC are the same as
for ODB. At E the potential is zero, the electron velocities are zero
and a virtual cathode is said to be forrned there. Although the electrons
come to rest at the virtual cathode, it must be assumed that they all go
on to the anode from E. Otherwise there would be more charge on the
grid side than on the anode side of E and the symmetry would be upset.
If vs is reduced below vs, the potential at E tends to drop below zero.
This is obviously not possible, since no electrons can reach the negative
potential region. In this case a zero potential is reached, but at some
place nearer to the grid than E, as shown by curve BKL in Fig. 6.8.
Only a fraction of the electrons now pass on to the anode. Those that
return to the grid are responsible for the movement of the potential
minimum towards the grid. If the grid—anode distance is increased the
limiting case for all of the electrons to reach the anode occurs at some
value of vs greater than vs. Similarly, with smaller grid—anode distance,
equal grid and anode voltages produce a positive potential minimum as
shown by curve 1 in Fig. 6.9. The limiting case now occurs at some
lower anode voltage vs, as shown by curve 3. For anode voltages greater
than vs a positive potential minimum occurs as shown by curves 1 and 2.
Thus the presence of space charge between the grid and the anode affects
78 PRINCIPLES OF ELECTRONICS [cm
the operation of a triode in the positive grid region. The effects are most
noticeable with large grid anode distances. These considerations are of
importance in Class C operation of ampli ers and oscillators, in which
the positive grid voltage is comparable with the anode voltage. Also,
the existence of a potential minimum between a positive grid and anode
is exploited in the beam tetrode (see Section 6.10).

6.8. Characteristic Curves of Tetrodes

In the triode the anode serves two purposes. It acts as the collector
of the electron current, but it also controls the amount of current. It is
possible to separate these two functions by introducing a fourth electrode
between the grid and the anode. This additional electrode, called the
screen, consists of a grid of wires which allows most of the electrons to
pass through it. For a given value of grid voltage, the current leaving

s 1— — so —.
0 [61 g G2 +
‘I’ I '5 YA ® 3

0 9 T
. I ._ _
F10. 6.10

the cathode depends mainly on the voltage of the screen and very little
on the anode voltage. In order to distinguish between the two grids
they are called the control grid or G1 and the screen grid or Gs. In the
four electrode valve or tetrode the screen grid serves an additional pur
pose in high frequency circuits by acting as a screen between the anode
and the control grid.
In the tetrode there are possibilities for electron ow from the cathode
to three electrodes, and the current division depends on the relative values
of the electrode potentials. Many different sets of characteristics may
be obtained using the circuit of Fig. 6.10, but the most interesting are
those with the screen grid at a xed positive potential and showing the
variation in anode current with either the control grid voltage or the
anode voltage. These may be referred to as control grid characteristics
and anode characteristics. One control grid characteristic is shown in
Fig. 6.11. The slope of this curve is again de ned as the mutual con
ductance gs, of the tetrode, i.e.,
_ ais
gs, _ awn at constant vss and vs.
The full line in Fig. 6.11 shows the anode current is and the broken line
the cathode current iK. The difference between these gives the screen
6] TRIODES, VALVES AND TRANSISTORS 79
current iss ; is;, the control grid current, is negligible, since vs; is negative
for the curves shown. It may be seen from these results that iss is less
than is. In most cases it is desirable to have iss as small as possible;
the ratio of iss to is is usually about 1 to 116. The ratio depends not only

II ls j
_, I
K ‘I
I

IIA cousrmr ,' D Vs,

V62 cousrmr 1' (‘A I
E
I
I
II

I
B
I
I
II

V62 cousraur
C
o ' O ' "'
' "or ‘ix
Fro. 6.11 Fro. 6.12

on the electrode voltages but also on the number and size of the screen
grid wires, and on their alinement with the control grid.
Two anode characteristics are shown in Fig. 6.12. Each of these shows
a peculiar “kink” with considerable variation in slope. Anode slope
resistance rs may be de ned as before as
1 8
Z = 8%‘: at constant vs; and vss.

Then rs varies from a relatively low value at low anode voltages (OB)
to a negative value (BC), then it is positive again (CD) and nishes at a
fairly high value (DE). The negative resistance region BC is particu
larly interesting; over that range the current changes in the opposite
direction to the voltage. This is
equivalent to the valve acting as .
a source of power for changes in ‘I 0
anode current. It is shown in “ll:
Chapter 13 that negative resist _ H
ance may be used to maintain e F ‘A
self oscillation in a circuit. Cd
If the current to the screen is D
plotted as well as the anode cur C .
rent, it is found that the two cur B E I ‘G1’ /I
rents are complementary, varying o I T;
equally and oppositely, as shown
in Fig. 6.13. The total cathode Fro. 6.13
80 PRINCIPLES OF ELECTRONICS en.
current iK is also shown in this gure; is is found to be free from “kink”
and to be nearly constant, rising very slowly with anode voltage. In
some cases it is found that the anode current actually reverses, the
point C in Fig. 6.12 being below the vs axis. The nature of these
characteristics may be explained in terms of the emission of secondary
electrons from the anode.

6.9. Secondary Emission and Tetrode Characteristics

Electrons striking a surface with su icient energy may knock out
secondary electrons. Within limits, the higher the velocity of the im
pinging electrons, the greater is the number of secondary electrons (see
Fig. 4.8). In a triode the electrons striking the anode may cause
secondary emission. However, since the control grid is usually negative
and the anode positive the emitted electrons are pulled back to the anode
by the electric eld, and the net
y effect on the anode current is
negligible. In a tetrode, how
VG; ¢°"5TA"T V 7vs ever, the screen is at a positive
y CONSTANT " 0 potential, and the secondary elec
ON
L
4 v‘ trons emitted from the anode may
go to the screen, thereby causing a
2 "0 net reduction in anode current.
Whether a secondary electron re
: I tums to the anode or goes to the
T_____
°|( _ _é1' _ G2 A x screen depends mainly on the
Fm Us relative values of the anode and
screen voltages. This may be ex
plained by reference to Fig. 6.14, which shows how the potential distri
bution in a tetrode varies with the anode voltage, when the control grid and
screen voltages are kept constant. When vs > vss (curve 1) the potential
gradient between the screen and the anode is such that the force on an
electron in the screen—anode space is towards the anode. Under these
conditions a secondary electron usually returns to the anode. When
vs < vss (curve 2) the force on an electron is towards the screen, and
secondaries from the anode are collected by the screen. The shape of
the characteristics in Fig. 6.13 may now be explained as follows. Elec
trons emitted from the cathode pass through the negative control grid
to the screen. Most of them pass through the screen with high velocity,
and their subsequent behaviour depends on the anode voltage. When
the anode voltage is zero the electrons gradually slow down and come to
rest at the anode (if initial velocities of emission from the cathode are
neglected). They are not collected by the anode but return to the screen
and the screen current is large (b). When the anode is made slightly
positive the electrons may reach it and is increases rapidly (BC) with a
corresponding reduction in iss (bc). However, at vs equal to about 5 V,
some of the electrons reaching the anode produce secondary electrons.
6] TRIODES, VALVES AND TRANSISTORS 81
Since vs < vs s, these secondaries go to the screen and reduce is, as shown
by the reduction in the rate of increase of is (CD). As vs is increased
further, still more electrons reach the anode, but with greater energy
and giving rise to a greater proportion of secondaries. When D is reached
the rate of increase in primary electrons equals the rate of increase of
secondaries. From D to E the increase in secondaries exceeds the in
crease in primaries, and is decreases. However, at E the anode voltage
is becoming comparable with the screen voltage and some of the second
aries return to the anode. From E to F an increasing proportion of
the secondaries return to the anode. When F is reached vs exceeds vss
sufficiently to ensure that no anode secondaries go to the screen. Through
out these variations of anode voltage, the initial velocities of the second
aries assist the electrons to reach the screen. At the same time the space
charge due to primary and secondary electrons prevents some of the slow
velocity secondaries from reaching the screen, even when the screen voltage
exceeds the anode voltage. These factors account for the absence of
sudden change when vs = vss.
During the variation of is, iss varies in the opposite direction, con rm
ing that the secondary electrons leaving the anode go to the screen.
The total current is remains practically constant throughout, thus con
rming that the anode voltage has very little effect on the current leaving
the cathode. We see that the peculiar variations in the anode charac
teristics of a tetrode at low anode voltages are due to secondary emission.
At higher anode voltages when the secondary emission effects are not
noticeable the anode current varies very little with anode voltage (FH).
The secondary emission coefficient 8, i.e., the ratio of the number of
secondary to the number of primary electrons, may exceed unity. Thus
it is possible in a tetrode for the anode current to be negative.
If we de ne the ampli cation factor as in terms of
3v .
as = — é at constant is and vs;,
then as is high; as is a measure of the relative effectiveness of the anode
and control grid voltages in controlling the anode current. Just as in
the triode, as may be interpreted electrostatically as the ratio of vs to
vs; to give zero eld at the cathode in the absence of space charge. Now
there is an additional electrode, the screen, between the anode and the
cathode, and the penetration of the anode eld through to the cathode is
very small. Obviously, as for a tetrode is greater than p. for a triode.
A second ampli cation factor as may be de ned for a tetrode as
av .
as = — 8%: at constant is and vs.
as measures the relative effectiveness of the control grid and the screen
in controlling the anode current. Values of as are similar to the values
of triode ampli cation factors.
As in the case of a triode, the idea of an equivalent diode may be used
82 PRINCIPLES OF ELECTRONICS [cH.
with tetrodes. The anode voltage of the equivalent diode is given by
vs; + vs;/as 1 vs/as and the anode current is given approximately by
the expression
5.4 = K(v01 I" ‘U02/P s 1 114/I1 413/2,
where K is constant.

6.10. E ect of Space Charge in Tetrodes—Beam Tetrodes

In considering the potential distribution in the screen anode space in
a tetrode in Fig. 6.14 the effect of space charge is neglected. This
assumption may be justi ed when the anode voltage is large, since the
electrons are then moving with high velocity between the screen and the
anode and the charge density is small. However, when the anode voltage
is comparable with or less than
V the screen voltage the conditions
are similar to those described for
the grid—anode space of a triode
in Section 6.7. There it is seen
that a potential minimum may
arise due to the space charge.
Thus, instead of the potential dis
tributions given in Fig. 6.14, it is
I
possible to obtain conditions as
_

Q ' I_ '_ i
X shown in Fig. 6.15, particularly
K G; G; A
Fro. 6.15
if there is a large distance be
tween the screen and the anode.
Now the potential variation near to the anode, even at low values of anode
voltage, gives a force on electrons towards the anode. As a result the
secondary electrons return to the anode instead of going to the screen,
and the “kink” is removed from the anode characteristics. The greater

I ., . //—_:—\'§A
1...,
(0) (b) fl.
E. . ._ 1 E: : : :1
‘I I I I IW
1
I =""
IE 11 I’
\v

A 8,8, K 8,8, A \;__//

FIG. 6.16

the density of the space charge between the screen and the anode, the
more effective is the “suppression” of secondary electrons. As well as
utilizing large spacing to produce low potential minima, it is possible to
gain the same effect by concentrating the electrons into high density
beams. This is achieved in the beam tetrode, which is shown diagram
matically in Fig. 6.16. The control grid and screen wires have identical
6] TRIODES, VALVES AND TRANSISTORS 83
pitch, and they are carefully alined so that the electrons are formed into
beams as shown in the vertical section (Fig. 6.l6.a). The width of the
beams in the screen anode space is restricted by two zero potential plates
()1
vs: cousrmr V‘ I
o
"oi

I
___________L

O
I Q co Q___ I» >.._
' “A FIG. 6.18
Fro. 6.17

P as shown in the horizontal section (Fig. 6.l6.b). Anode characteristics

for a beam tetrode are shown in Fig. 6.17, in which it can be seen that the
" kinks ” have been eliminated.
6.11. Pentodes
The effects of the secondary emission from the anode on the charac
teristics of a tetrode can be eliminated by the insertion of an additional
grid between the screen and the anode, as is done in the pentode. This
additional grid is usually maintained at zero potential, i.e., cathode
potential, and then the potential distribution between the screen and the
anode is as shown in Fig. 6.18. The space between the wires of this new
grid is at a potential somewhat above zero when the screen and the anode
are at positive potentials. With this arrangement the eld outside the
anode, even at low anode voltages, is such that secondary electrons
return to the anode. The extra electrode, G3, is usually called a sup
pressor grid. The pentode
with ve electrodes has a large in
number of possible charac
teristics. The most important 0 V
are those showing the vari G1
ation of is with vs; and vs,
whilst vss and vss are constant,
the latter usually being equal
to the cathode potential. A
typical set of anode charac I/G2 cousraur
teristics for a pentode is shown V63 CONSTANT
in Fig. 6.19. These are very ,,
similar in shape to the anode Vs
characteristics of the beam Fro. 6.19
84 PRINCIPLES OF ELECTRONICS [cl I.
tetrode. The two types of valve are almost interchangeable for many
purposes. Careful comparison of the characteristics of pentodes and
beam tetrodes of similar size shows that the “knee” of the anode
characteristics is sharper and steeper with tetrodes. The pentode knee
may sometimes be sharpened by operating the suppressor at a small
positive voltage. Historically, pentodes were introduced before beam
tetrodes.
Another type of pentode characteristic which is of some interest is
shown in Fig. 6.20, which gives the variation of anode and screen currents
with suppressor voltage. Normally the suppressor is maintained at zero

‘ I
0 %_}
IA
I
| ’

' o ‘,1:U
Fro. 6.20
Fro. 6.21

potential relative to the cathode. Then the potential in the rest of the
space between the screen and the anode is maintained positive by the
combined effects of the screen, anode and suppressor potentials. If
the suppressor is made negative a small region round each suppressor
wire is at a negative potential and some of the electrons, which previously
passed through the suppressor to the anode, now return to the screen.
As the suppressor is made more negative the anode current decreases,
and becomes zero when the whole space between the suppressor wires is
reduced to negative or zero potential. It is seen from the gure that is
and iss vary equally and oppositely and that their sum is constant. One
feature of these curves is that is 2 varies in the reverse direction to vss,
an increase of vss giving a decrease in is 2, and vice versa. Thus the slope
of the curve of screen current against suppressor voltage has a negative
region (see Section 13.8).

6.12. Effects of Gas in Triodes—Thyratrons

In Chapter 5 it is shown that the presence of gas in diodes may modify
the ow of current considerably. In particular, it is seen that there is
an increase in anode current when the anode voltage approaches or
6] TRIODES, VALVES AND TRANSISTORS 85
exceeds the ionization potential of the gas. As the voltage increases
further there is a sudden and very large increase in current. Similar
effects are observed in the anode current of triodes with gas. At the
same time grid currents are found to ow even when the grid potential
is negative. The geometrical arrangement of the electrodes in gas
triodes is usually different from the vacuum triodes which have already
been considered. Fig. 6.21 shows one type of gas triode. The cathode

R1
‘A + 0 |

R2 is
Q + g VA Q + ii"
0 VG ya

T Fro. 6.22
1 _ . .I

and anode are approximately planar, but the “ grid ” is a cylindrical

shield round the anode and cathode with a central disk between them.
There is a hole in the disk. This structure differs considerably from the
grid of wires of vacuum triodes, but it is obvious that the eld at the
cathode is determined mainly by the “ grid ” potential and to a small
extent by the anode potential. It is a high (J. structure. The behaviour
of this structure may be determined with the circuit shown in Fig. 6.22.
Consider the case where the cathode is heated, the anode voltage is, say,
100 V and the grid is su iciently negative to cut off the anode current.
As the negative grid voltage is reduced gradually, a point is reached
where the anode current changes suddenly from zero to a large value.
At the same time a glow discharge appears inside the valve. It is then
found that further variation of the grid
voltage has little effect on the anode y
current, even if the grid is made more I A
negative. Once the discharge is struck, I
the grid loses control and the discharge _
may be extinguished only by reduction
of the anode voltage to a low value I
less than about 10 V. If this experi
ment is repeated with a different initial |
value of the anode voltage, it is found I
that the valve strikes at some other I
value of grid voltage. The curve in I
Fig. 6.23 gives the corresponding values P
of anode and grid voltages at which V
the valve strikes. This curve is some ‘ G
times called the control characteristic. Fro. 6.23
D
86 PRINCIPLES OF ELECTRONICS [CH.
If, on switching on, the values of vs and vs correspond to a point on or
above the full curve in the gure the valve strikes. As with the diode, it
is found that in the conducting state the voltage drop v’ across the valve
between anode and cathode remains practically constant at v’ = 10 V.
The current that ows is given by (v s — v’)/R. The greater part of the
control characteristic is a straight line. The slope of this line is called the
control ratio; its value is usually about 20. The anode current after the
discharge is struck is unaffected by the grid voltage. However, there is a
current owing to the grid itself, and this varies with the grid voltage, as
shown in Fig. 6.24. At positive and low negative grid voltages this cur
_ rent ows in the positive direction, i.e., it
Us corresponds to a ow of electrons from
_ the gas to the grid. The current in
wimpy creases very rapidly as the grid voltage
E¢LUE,§,TE'f. N is made less negative. At some small
negative grid voltage the grid current is
zero. As the grid voltage is made more
O negative a reversed current ows to the
_ '6 grid, as shown.
' The nature of these characteristics may
be explained in the following manner.
MMNLY U On switching on with the grid voltage
'°"
CURRENT
suf ciently negative there is no anode
Fro. 6.24 current. The limiting value of vs for
cut off, when the anode voltage is vs, is
vs = — vs/ii. As soon as vs is made less negative than the limiting
value, electrons leave the cathode and move towards the anode. Nonn
ally, vs is considerably greater than the ionization potential of the
gas and breakdown occurs as described in Sections 5.15 and 5.16. The
voltage between anode and cathode drops to about the ionization
potential and most of the space is lled with plasma. A positive
ion sheath forms round the cathode just as in the diode. With the
grid at a negative potential, positive ions are attracted to it from the
plasma, and so a positive ion sheath is formed round the grid. The
size of this sheath depends on the grid potential; the more negative the
grid, the greater the number of positive ions surrounding it. Equilibrium
is reached when the ion sheath just neutralizes the negative grid so that
no more ions are drawn from the plasma. Under these conditions the
main plasma is unaffected by the grid potential, and we see why the grid
voltage does not affect the ow of current to the anode. However, the
ion current to the grid varies with the grid voltage; in use, a resistor is
always connected in series with the grid supply to limit this current to a
safe value. If the grid voltage is reduced the ion current drops, and
some electrons are collected at the grid because of their random velocities.
When the grid is positive it collects electrons from the plasma and also
some positive ions, as they too have random velocities. It might appear
6] TRIODES, VALVES AND TRANSISTORS 87
that no current should ow to the grid when it is at zero potential. How
ever, the random velocities of the electrons in the plasma are greater than
the ion velocities, and more electrons are therefore collected. This is
why it is found that the total grid current is zero at a negative grid
voltage (see Fig. 6.24).
From what was said above about the control characteristic it should
be expected that its slope would give the electrostatic ampli cation factor,
de ned in terms of the ratio of vs to vs for cut off. On this basis the
straight portion of the characteristic when extrapolated should pass
through the origin. The actual extrapolation crosses the grid axis at a
slightly negative value. This is probably due to the initial velocities of
emission of the electrons from the cathode. Breakdown occurs at very
low anode currents. Hence the extra energy of a small number of elec
trons emitted with appreciable velocity may assist breakdown. The
actual characteristic cuts the vs axis near the ionization potential.
The main properties of a gas triode or thyratron are similar to those of
a relay. The breakdown corresponds to the closing of the relay. The
grid operating at a low voltage may be used to close the relay, and to
control the ow of a large current. However, once the relay is closed
the grid has no further control. Thus thyratrons have to be used in
circuits where the anode voltage drops to a low or negative value when
the relay is to be opened. When the anode voltage drops below the
ionization potential, the positive ions and electrons recombine to form
neutral molecules. This takes a de nite time, depending on the ion
mobilities. If the discharge is to cease completely so that the grid
resumes control, it is important that the anode voltage should not change
again until the process of recombination or de ionization is complete.
This factor sets a limit to the maximum frequency of operation. In the
case of mercury vapour thyratrons the frequency is limited to a few
thousand cycles per second. Hydrogen or inert gas thyratrons may be
used at higher frequencies. As in the case of gas diodes, temperature
affects the operation of thyratrons particularly when mercury vapour is
used. It is necessary to control the temperature within limits.
Some gas tubes have a fourth electrode whose potential modi es the
striking voltage of the control electrode. These gas tetrodes usually
have a high control ratio.
So far, all the considerations on gas triodes have been based on tubes
with heated cathodes. Gas diodes can operate with cold cathodes, where
the ionization is initiated by cosmic rays or other external agency. In
such tubes the discharge between two electrodes may be controlled by
the insertion of a third electrode. When a discharge starts between one
pair of electrodes it initiates a discharge to another pair. By biasing the
first pair near to the ring voltage, a small voltage change may start a
discharge to the second pair, thus giving switch or relay action as in the
thyratron. Cold cathode triodes and tetrodes are used in this way; they
have the big advantage over thyratrons that they do not require any
88 PRINCIPLES OF ELECTRONICS [ci~I.
supply for heating the cathode. Multi electrode cold cathode tubes have
been introduced for counting purposes (see Section 18.10).

6.13. Ionization Gauge

In valves which have been evacuated to a very low pressure there are
still many molecules of gas present. Indeed, in the best vacua available
the number of gas molecules usually exceeds the number of electrons even
when large currents ow. Thus there is always some ionization taking
place when the electrode potentials are sufficiently high. In valves
operated with a negative grid, the positive ions produced by ionization
ow to the grid. This positive ion current gives a measure of the density
of the gas molecules or the gas pressure. In this way a negative grid
tube may furnish an indication of its own pressure. Frequently small
triodes are incorporated in vacuum systems to indicate the pressure.
Such triodes are called ionization gauges. At low pressures the ion current
is proportional to the pressure.

6.14. Transistors
Crystal triodes or transistors are made from suitably prepared crystals
of germanium or silicon. As in the case of crystal diodes, there are two
main types—the junction transistor and the point contact transistor.

E
a
(0) (b) (¢)
Fro. 6.25

The junction transistor consists essentially of two 12 »n junctions com

bined in one crystal as shown diagrammatically in Fig. 6.25.a and b, where
it is seen that there are two main forms, depending
E C on whether the middle section is p or n type
material. In both cases the middle section is
WHISKER WI IISKER
CRYSTAL
called the base, and it is made as thin as possible
for reasons given in Section 6.17. The outside
regions are called the emitter and the collector.
B One form of 15 n ;i> junction transistor is shown
Fm 626 in Fig. 6.25.c, in which two beads of indium are
on opposite sides of a thin slice of n type ger
manium. The p type regions are formed at the boundary of the indium
and the germanium. In the point contact transistor, Fig. 6.26, the
6] TRIODES, VALVES AND TRANSISTORS 89
emitter and collector are two metal wire cat's whiskers, very close
together and making contact with the surface of a slice of germanium (or
silicon) which acts as the base. Various symbols are used to represent
transistors. Some of these are shown in Fig. 6.27.a to f; (a) and (b) are
alternative methods of representing ;t>—n—p junction transistors, and (c)

C C
3

E B ‘ if“ I Tl?» CD

E 110" I 8 I11

(0) (5) (¢) 647

P“/I'p JUNCTION TRANSISTORS I7 p 0 JUNCTION TRANSISTORS

E C E C

B POINT CONTACT TRANSISTORS B

(¢) (I)
Fro. 6.27

and (d) are the corresponding symbols for n p n junction transistors.

The two point contact symbols shown at (e) and (f) are sometimes used
for junction transistors also.
The transistor is a three terminal device, and its properties may be
speci ed in terrns of characteristic curves connecting the three currents
and three voltages which are shown in Fig. 6.28. Since
iz+¢'s I ¢'c=0
imd ‘vss+"iic+ vcs=0
there are only four independent variables. If is, is, vs s and vss are
chosen, the two voltages are measured from the base, and these variables

__ + V ac _ __
1 I 1
E _‘ T ° +c
vBE [5 ‘EB
4. __ _ ...
B
Fro. 6.28

helong to what is called a common base arrangement. In the case of the

triode there are also four independent variables is, is, vs and vs, but as
is is zero in many applications, we represent the triode characteristics
by a single equation is = f(vs, vs) and a single family of curves. In the
90 PRINCIPLES OF ELECTRONICS [C1 I.
transistor there is no such simpli cation, and all four variables must be
taken into account. One way of expressing the characteristics is
‘van = f1(iEi van)
and ig =f2(Ig, Ugg).

These are known as the hybrid characteristics. Two separate families

of curves are required to represent the complete characteristics. Typical
curves for a p n p junction type transistor are shown in Fig. 6.29.a to d;
(a) and (b) give in alternative forms families of curves for the rst of the

'59] o \"ce I "ea

| .
Y E
——r
.0 ‘E (Ii O "ca
(0) (0)
[C Io [E I ic
o 'ce
___________________,/

o IE —'__——__ _ _J
I j "cs
(¢) (0')
Fro. 6.29

above equations, and (c) and (d) do likewise for the second equation.
The characteristics are completely speci ed by one pair of families, e.g.,
(b) and (d). Important and typical features of these curves are:
(i) is and is are related linearly, being practically equal and
opposite. Actually, is is usually slightly less than is in magnitude.
Consequently is is much smaller than either is or is.
(ii) is changes considerably with vss.
(iii) is is almost independent of vss.
(iv) vss changes slowly with vss.
From (i) and (ii) we see that the emitter base voltage acts as an effective
control of the emitter current and of the collector current, but at the
6] TRIODES, VALVES AND TRANSISTORS 91
same time the base current is small. This corresponds to some extent
with conditions in a triode, where the grid acts as a control of the anode
current without taking appreciable grid current. The emitter, base and
collector are analogous to the cathode, grid and anode respectively.
Point contact transistor characteristics are similar to those of the
junction transistor, but they show greater variations from one sample to
another, and the typical features mentioned above are not so marked.
In normal use with a p n p transistor vss is a positive voltage of one

‘ VBE A VBE

° 1; ‘_’______j°7ss
'65‘ °
‘B /

fa) (0)

3. @
InuIn

* Q

O‘
In
o
— —icw

'o
(¢) (07
Fro. 6.30

volt or less, whereas vss is negative and of greater magnitude than vss.
The emitter current is positive and the collector current is negative. All
these polarities are reversed for an n p n transistor.
The common base characteristics use the four variables is, is, vss and
ess as independent. When vss and is are required they are determined
lI'O1T1 if; + ig 1 I3 = 0 and ‘U33 —j Ugg 1 ‘U33 = O, 1.6., Ugg = U95 + ‘U33.
Since is and is are approximately equal but of opposite sign, is is the
small difference between two relatively large quantities. Thus charac
teristics based on is and is do not give the most accurate overall repre
sentation. Common emitter characteristics based on is, is, vss and vs;
are frequently used. A set of such characteristics is shown in Fig. 6.30.a
92 PRINCIPLES OF ELECTRONICS [CI 1.
to d. Corresponding functional relationships may be expressed in the
form
vsis = falfs, van)
and Ig =fs(i3, ‘L'g3).

6.15. Transistor Parameters

As in the case of triodes and other electronic devices, the slopes of the
various characteristics are important parameters. For the common base
connection the hybrid parameters are de ned as follows:
av
hu = ii at constant vss,
alg

hm = $9 at constant is,
"ca
3i
hm = at T: at constant vss
dis .
and hss E at constant is .
The quantity hm has the dimensions of a resistance. It is the resistance
of the emitter to changes of the emitter base voltage when the collector
base voltage is kept constant; it has a low value. It may be considered
as a measure of the transistor input resistance when the output voltage
is constant. The quantity hs, is the conductance between collector and
base for changes in vss while the emitter current is constant. It may be
thought of as the output conductance when the input current is constant.
It has a very low value. It may be noted in passing that the is, vss
characteristics are similar in shape to the is, vs characteristics of pentodes,
and the low value of hs, corresponds to the low value of 1 /rs of a pentode.
The quantities hm and hsl are dimensionless. The fonner is a measure of
the fraction of the collector base voltage change that exists between the
emitter and base when the emitter current remains constant; hm is
usually small. Finally, hsl is the ratio of the change in collector current
to the change in emitter current when the output voltage is constant.
It is usually negative, and its value is just slightly less than unity. It
is sometimes called the collector—emitter current ampli cation factor.
Frequently has is replaced by — as, and then as, is usually positive.
Point contact transistors and certain types of junction transistor have
values of as, greater than unity.
The slopes of the common emitter characteristics may also be used as
transistor parameters. They are de ned as follows:
h avss h avari
11e = —
623
128 = avg;
—
_ alig __
h21c — a7, and IP22: — E,‘
6] TRIODES, VALVES AND TRANSISTORS 93
Then h;;, is the transistor input resistance for the common emitter con
nection; hss, is the output conductance, h;s, is a measure of the voltage
fed back from output to input; and hs;, is the current ampli cation factor
from base to collector. This quantity is usually large and positive for a
[2 n—p junction transistor; it is sometimes denoted by the symbol ass.
When it is necessary to distinguish between the h parameters for com
mon base and common emitter connections the former may be denoted
by h;;s, hm, etc. Corresponding h values for both arrangements for a
typical p—n—p junction transistor are shown in Table 6.1.

Common base. Common emitter.

Input resistance . hm, = 30 Q hm, = 1500 Q
Voltage feedback factor . hm, = 7 X 10“ hm, = 8 X 10"
Current ampli cation factor hsm = — as, = — 0 98 hm, = ass = 50
Output conductance hm, = 1 |.|.1111'lO hm, = 50 pmho

TABLE 6.1.
Mathematical relations between the h parameters may be established
(see Exx. VI). In particular, it may be shown that ass’: as,/(1 — ass)
and h;;, :h;;s/(1— ass). Since ass: 1, it follows that the common
emitter arrangement gives a considerable increase in input resistance.
The input resistance of a transistor in the common emitter connection is
much less than the input resistance of a thermionic triode operated with
negative grid. Also, in contrast with the triode, there is always some
reaction of the output on the input in the transistor due to hm.

6.16. Transistor Equations for Small Changes

In dealing with the triode in Section 6.3 it is shown that small changes
is, vs and vs in the variables is, vs and vs are related by the valve equation
is = gs,v, + vs/rs. With the transistor there are similar relations for
small changes vs;,, vs;,, is and is in the variables vss, vss, is and is. From
the equations vss = f1(is, vss) and is = fs(is , vss) it follows that
avgg . avg;
vab =——1—Z —vcb =hi
azg c+avCB v
11e'l'h1gcb

. 8' . 6' .
and is = 3% is + Ti; vs, = hslis —] hmvss.

For the junction transistor hm and hm are small, and the magnitude of
hm is nearly unity, so that we get the approximate transistor equations

1:; Z Z ‘* dais 1:; Z W‘ 1.‘.

From these equations it is seen that vss is the important voltage; it may
be called the driving voltage. A change in the driving voltage gives a
94 PRINCIPLES OF ELECTRONICS [cH.
change in 1'3 and an almost equal change in ig. Changes in 11¢; and vw
are relatively unimportant in determining the currents owing. This
simpli ed picture is useful in determining the approximate behaviour of
a transistor as long as the changes involved are small.

6.17. Physics of the Transistor

In an isolated 12 11 12 junction transistor there is a potential variation
and distribution of energy levels of the fonn illustrated in Fig. 6.31. In
the equilibrium state the Fermi energy level is constant throughout.
This state is set up in a similar manner to that described in Chapter 5 for
a single p—n junction, and involves zero resultant current of electrons and
P '7 ,0
iii_ _ r _ j —°C
°—'.
m I

POTENTlAL

{GT \ / *9}

l e) G)/—,<9}
@ > <

FIG. 6.31

holes across each junction. If the collector is joined directly to the base
and a small positive voltage is applied between emitter and base, then a
current flows just as with a p n junction diode in the forward direction.
Similarly, if the emitter is joined to the base and a negative voltage is
applied between collector and base, the current ows as with a p n
junction diode in the reverse direction, showing the same saturation of
collector current. We consider now what happens when these voltages
with the same polarity are applied simultaneously. The potential and
energy variations are shown in Fig. 6.32. The potential barrier between
emitter and base is reduced and the ow of holes across the barrier is
greatly increased. There is an increased ow of electrons from base to
emitter, but the hole density is, by design, much greater in the p region than
the electron density in the n region, so that the current may be considered
as due mainly to holes. The holes enter the n region and diffuse through
it. There is a tendency for these holes to combine with the electrons in
the region. However, if the n region is sufficiently thin a large number
of the holes reach the collector base junction and very few arrive at the
base terminal. At the collector base junction the holes fall easily into
6] TRIODES, VALVES AND TRANSISTORS 95
the collector region on account of the eld at the junction. Thus, we
see that the collector current is very nearly equal to the emitter current
and the base current is almost zero. The collector base voltage has little
effect on the current as long as its magnitude is above some minimum

P '7 P
E $ c

POTENTIAL

9}
{G) ’, _r:W
eneacv

{G9 %_=®®4
Fro. 6.32

value and the base region is su iciently thin. A small change in emitter
base voltage causes a change in emitter current. This results in an
almost equal change in collector current.
As with the junction diode, there are always some reverse currents due
to minority carriers in the different regions. The minority carriers, and
hence the reverse currents, increase with temperature. This temperature
effect is one of the chief limitations to the use of transistors.
 CHAPTER 7

VOLTAGE AMPLIFIERS
7.1. Valves and their Characteristics
In Chapters 5 and 6 the characteristics of various valves are discussed
and it is shown that the electrode currents can be determined from the
characteristics when the electrode voltages are known. In the use of
valves the electrode voltages usually depend on a number of factors, such
as the battery supplies and the circuits connected to the electrodes.
Sometimes the voltages depend on one another and on the electrode

[A [A
lG= 0+ Q Q

R VG R R
I x
El .. A E‘ {A

Y6 ='E‘ y5='E1_ RKIA

(0) (b)
Fro. 7.1

currents. In order to determine the actual electrode voltages these

factors must all be taken into account. In many applications of valves
with control grids the grid cathode voltage is negative, so that only a
negligible amount of grid current ows. Then it is frequently possible

R
1,, + I 1,, + I
. V
mi
_
. Q "
Q V
c—ni
_

it vs
_I

I/G= E;
A
._
l e
I
E2 vs
.
A E2

VA: E2 VA=E2 Rik

(0) (5)
F10. 7.2
96
CR. 7] VOLTAGE AMPLIFIERS 97
0 determine the grid cathode voltage without having to take into con
ideration the valve itself. Such is the case in the circuit of Fig. 7.l.a.
lowever, in Fig. 7.l.b the grid cathode voltage depends upon the value
»f anode current owing through the resistance RK, and hence upon the
lature of the rest of the circuit through which the anode current ows.
n this case it is not possible to write down directly the value of the grid
'oltage. Throughout this chapter it is assumed that the grid voltage is
negative and the grid current zero.
From Fig. 7.2.a the values of U0 and v4 are known directly from the
:ircuit as ‘Ug = — E1 and v4 = E2. This allows the anode current to be
leduced from the static characteristics. In Fig. 7.2.b, however, where
here is a resistance in the anode circuit, only the value of vq is known

1}, [A
52
[R VG = E1

if
° vs’ vs o vA
E2
LOAD use iA= (E; yR VALVE CHARACTERISTIC iA= l(vA)
FOR vs = —E1
(<1) (bl
Fro. 7.3

lirectly from the circuit. The determination of v4 involves the anode

zurrent through the valve, which itself requires a knowledge of v4 before
t can be determined. There are in fact two relations to be satis ed: the
/alve characteristic
2'4 =f(v4) for 22¢ = — E1,
lild the circuit condition
‘U4 ‘ = E2 '— O1’ 1.4 = (E2 U4)/R.

l"he values of £4 and v4 for the circuit of Fig. 7.2.b must satisfy these
aquations simultaneously.

7.2. The Load Line

We consider, rstly, the circuit equation
‘U4 = E2 — R214.

When 1'4 = 0, v4 = E2.

The relation between £4 and v4 is a linear one, since E2 and R are constant,
98 PRINCIPLES OF ELECTRONICS [cH.
and only one other point is required for drawing the graph of the equation.
A suitable point is
v4 = 0 and is = E2/R.
Any other point, such as
v4 = v4’ when 2'4 = (E2 — v4’)/R,
could be used. The circuit relation is represented by the straight line
in Fig. 7.3.a. This is known as the load line and R as the load resistance.
The equation
£4 = (E2 — U4)/R

is called the Load Line Equation. The slope of the load line is — 1/R.
The static characteristic corresponding to £4 = f(v_4) is shown in Fig.
1“
7.3.b. The actual values of £4 and
v4 are determined from the inter
section at Q of the curves of Fig. 7.3a.
E y =_ E and 7.3.b‘whe.n they are plotted to
1’/R G " gether as lI'l Fig. 7.4.
Now let us consider a d.c. change
v, in the grid voltage so that
9*. I I | u | I O ‘Ug = — E1 + 11,, as shown in Fig. 7.5.
0
The load line is unchanged, but there
Fol

is a new valve characteristic £4 = f(v4)

'0"" Fl N_
13' for ‘Ug = — E1 + 11,. The anode volt
FIG. 7.4
age and current are changed to the
values Up and ip, where the point P
gives the intersection of the load line and the new characteristic. Thus,
due to a change in the grid voltage, we have changes in the anode
current and voltage of is = (ip — iq) and vs = (‘UP — vq). It may be
seen that v, is, in this case, negative in value. In many cases 11,, is much
greater in magnitude than v,, so that the triode and its resistance load

[A + R IA_A ya: E1 I |/9‘vi

vG=_E1

+ _._
v Y "' .
‘I '6 i E’ " ‘}"I::.'."_"' ." O
E It . T“ its
l° .1 .0

vs= E,+v, ' ,<I

La. V 0‘QI
.m N
YA: E2“R[A ‘

= YQ + Y‘

Frc. 7.5
7] VOLTAGE AMPLIFIERS 99
act as a voltage ampli er. The voltage ampli cation is de ned as
A = '0,/vs, and the magnitude of this voltage ampli cation is often re
ferred to as the stage gain. It may be noted that this voltage ampli ca
tion has been obtained without expending any power in the grid circuit,
i.e., no current has been drawn from the source of the voltage 11,.
The point Q in Fig. 7.4 or Fig. 7.5 which gives the operating conditions
before any changes are introduced is called the quiescent point, and iq
and vq are the quiescent anode current and voltage respectively. When
R_
R to +
10+ ‘.0 " +

+ vs
+
E2
_ yo g

»z+ 3°“ ET '° E1

. II _ _ vs= E,
E1
Fro. 7.6 (0)
F10. 7.7

[A [s+ is R
5,, *
jR P vG= E,+v Q ' "Lt"; +
10 IA + vs
'
f lo Q; yG= E1 V‘ VG yQ+'5 “i E2
"

_ ...E1
O >5 vs= E,+v,
_§
15* .5"
(b) (¢‘)
Fro. 7.7

it is necessary to distinguish between the quiescent values of the anode

voltage and grid voltage, the symbols '04,; and vgq may be used. In the
present case vgq = — E1. The terminals across which the voltage
change v, is introduced are referred to as the input terminals, whilst those
across which the ampli ed voltage change occurs are the output terminals.
The input voltage change is sometimes called the signal voltage, and is given
the symbol 11,. The output voltage change is given the symbol v,,, so that
the voltage ampli cation may also be de ned as A = v,/v, (see Fig. 7.6).
What has been described above for a triode ampli er applies equally
well to a pentode ampli er. The appropriate conditions are shown in
Fig. 7.7.a, b and c.
 100 PRINCIPLES OF ELECTRONICS [c1~1.
7.3. Voltage A.mpli er—Sma1l Signal Theory
In the previous section no restrictions are placed on the size of the
changes in voltages and currents, except that the grid voltage does not
become positive. In this section only small changes are considered.
The changes are suf ciently small for the values of gm, r, and |J to remain
constant throughout the range of variation, so that the Valve Equation
derived in Section 6.3 may be used. A change '0, is made in the grid voltage
(see Fig. 7.8.a and c), and it gives rise to changes is and v,, in the anode
current and voltage. Then the Valve Equation gives 2', = g,,,v, + v,/1,.
The change in anode voltage may also be related to the change in

R R
‘
IQ +<—r—w
lA‘ YGQ
= E1
.
lQ+(a +~: ———:
._ L + E2 yL+v‘

+ V r— R + + Q L '2
o 52 V9 . 7* Vo 4 Vs

vs Yo
is Q
ET | _| _ ._
VG=—E1
O 5___ E2
v E‘A VG: E1‘? V9

('7) (b) (c)

Fro. 7.8

anode current, using the Load Line Equation £4 = (E, — v4)/R. In this
case 1', = — v,/R. The two equations for i, yield
gmvv
"° (1/R + 1/1,)
and the voltage ampli cation is
.4 5""
(1/R + 1/70)
This expression may be written in the alternative forms
A gm gm l ‘R ’
(1/R + 1/H) (G + Ea) (R + '¢)
where G = 1/R and using p. = g,,,r,,. The values of p., gm and 1,, used in
these formulae must be determined for the actual Q point in use, as shown
in Fig. 7.8.b. The above formulae all apply to small d.c. signals for any
valve ampli er. In the case of pentodes r, is very large and frequently
1, > R, so that the voltage gain is simply
A = — g,,,R.
In triodes 1', is usually comparable with R, and smaller voltage gain is
obtained than with a pentode for the same mutual conductance and load
7] VOLTAGE AMPLIFIERS 101
resistance. It is to be noted that R2}, is the change in voltage drop vi
across the load resistance, and as v, = — Ric, then v,, = — v;. The
voltage change across the valve is equal and opposite to the voltage
change across the load. This relation is obvious from Kirchhoff’s Second
Law but it is a useful check in determining the signs of the circuit
voltages.
7.4 Voltage Ampli er—Small a.c. Signal
The discussion of the ampli er has so far been in terms of a d.c. signal.
When the value of the signal is changing all the time, as with an a.c.
R
is "'

‘I’ ‘I’ Q V
Q uln col Q C. A E2
._ vs

E, _
(<1)
VG r
O

5, ________ ":1'¢.
(0)
VA L’ _
~. :10/R*%',)
I

,
5
O
(=2 ———{
__
gave

1° """""" " ' WW4)

I

° rd)
Fro. 7.9
voltage, then as long as the change is not too rapid it is found that the
values of the valve parameters corresponding to the instantaneous
values of the electrode voltages are identical with those found from the
static characteristics. This implies that the static characteristics also
102 PRINCIPLES OF ELECTRONICS [C1 I.
apply to instantaneous values of applied voltages. When the time of
transit of the electrons through the valve is comparable with the period of
the a.c. signal, this is no longer true. However, this source of error can be
ignored at frequencies below some tens of megacycles per second.
In the circuit of Fig. 7.9.a a sinusoidal alternating signal is applied in
series with the grid bias supply, and the load is again a resistance R.
The instantaneous value of the signal is v, = 23, sin wt, and thus the
instantaneous grid voltage is ‘Ug = — E1 + 13, sin col. This is represented
in Fig. 7.9.b, which also serves to de ne the polarity of the signal voltage
with respect to the bias battery at any given time. When the signal
voltage is zero, the anode current and anode voltage have their quiescent
values, iq and vq, as given by the load line construction. At any other
time the instantaneous change of grid voltage is 13, sin mt. Provided 13,
is suf ciently small, the changes of anode voltage and current at the same
instant may be found from the Valve and Load Line Equations, as in the
previous section. Thus
v,, gm ’ A sin mt = i——g'" ' sin (wt + 1:)
(1 + 1 1 + l
R 1,, R 1,
and £4 = mr? ":1iT Sin 0)‘.

Rf? + 2)
The actual anode voltage and current are
v4=vq+v,, and i4=iq+i¢.
These are shown in Fig. 7.9.c and d. It is seen from the equations above
that there is a phase difference of 180° between the a.c. components of the
anode current and anode voltage. The voltage ampli cation is again

A = 5""
(1/R + 1/n)
as for the d.c. signal, and it is independent of the frequency of the signal.
7.5. Valve Equivalent Circuits for Small Signals
In Fig. 7.l0.a is shown a valve ampli er with a load resistance R.
When a signal is applied we are usually interested only in the changes
that take place. We may therefore redraw the circuit as shown in Fig.
7.l0.b, in which the d.c. supplies E1 and E2 have been omitted and only
the varying components of voltage and current are included. It is
assumed that E1 and E, have negligible impedance to the varying cur
rents. For small signals the Valve Equation gives
in = gmvg + va/70

The circuit conditions give

U4 Z ' ' Ride
7] VOLTAGE AMPLIFIERS 103
In Fig. 7.l0.c a circuit has been drawn in which a current generator gmvs
is in parallel with a resistance rs and a load R. It may be seen that the
same two equations hold for this circuit as for the ampli er. The circuit
of Fig. 7.l0.c is known as the current generator equivalent circuit of the
ampli er. The valve is equivalent to a current generator gmvs in parallel
with a resistance of rs.
The Valve Equation for Fig. 7.l0.b can be rewritten as
707:0 = (gm7a)vg + va = I we + va»
whilst the circuit relation is still vs = — Ris. If we study the circuit
of Fig. 7.l0.d, where a voltage generator pus is in series with a resistance

R A
+

_ Om vs
I." Gm
8+
_.
7;‘ D

5+
1" ' :n>*" ,%s
K (<1) K (0)
A0 A O
IP
3*‘
G
O
9|.‘
‘:.
‘ R Q'
"I' ‘I’ 9 8*
+ D
"I +
' 7, '" Q‘
'~ v, my

K (c) K (4)
Fro. 7.10

rs and the load R, it is seen that these two equations hold also for this
circuit. The circuit of Fig. 7.10.d is known as the voltage generator
equivalent circuit of the ampli er. The valve is equivalent to a voltage
generator which has an internal series resistance of rs. These equivalent
circuits are useful in solving ampli er problems, and they are used
frequently throughout this book. It must be stressed that the equivalent
circuits are only alternative expressions of the Valve Equation, and hence
are restricted to small signals.
7.6. Voltage Ampli er—Inductive Load
With a resistance load, the ampli cation is independent of the frequency,
and there is a constant phase difference between the anode current and
1 04 WE
V‘sg
_VA
_

H
MI
UIn
)W
UI %
W
mMa H
iL I W
u _
_Q L_
)___
VG
__¢ ___ 1L

_
U/N
__MEV‘ nmzM+VM_Gw_Uu/ Q‘
u
O Hd_ nQA.“
_HR_)“_
_
+_
_r
)_
t
2
K
QV’
/l

VI __
‘vs
I
_m
B‘
'_‘|
_+_1
+_
Q(V
Q__(¢_
___
___h_PML
OE
U,
_
M
u
V0
"

3
)
G.+
V’
w’
__

V.

_
___/

PM
LA
VG
Q
*
“V
VA my
y2___72
O+IO I_lmI hI__lI +I
I
111“t n
o_ _ _ __

vs
_ _3
__

‘W
__O
GV0
‘iv
_
_ II
'' __
I_ __

3
1

_ _ _

f
f
N
R 7 1
7] VOLTAGE AMPLIFIERS 105
the anode voltage of 180°. Frequently there is some inductance or capaci
tance associated with the load, and then the ampli cation and the phase
difference vary with the signal frequency. In this section we consider an
ampli er with an inductive load with zero resistance, as shown in Fig.
7.ll.a. In this case the static load line equation is simply v4 = E2 (Fig.
7.ll.b). This gives a vertical line through E2, as seen in Fig. 7.1l.d.
Also ‘Uq = E2. If the grid voltage is given by
v@= —E1 + 13,sin mt
then we are primarily interested in the alternating components of the
anode current and the anode voltage (Fig. 7.11.0). These take the form
2', = i, sin (mt + 96) and v, = 13., sin (wt + z/1).
To nd 5,, 13,, 45 and 4,6 in terms of v, we may use the equivalent circuit with
vector voltages and currents. The constant voltage generator may be
used as shown in Fig. 7.11.e. By applying Kirchho ’s Laws and using
vector algebra we nd
L = p.V,/(1, + jcol.) and V, = — jwLL,
i.e., I, = pV,(r, — jwL)/(r,,2 + ML’).
Hence tan ¢ = (DL/7'4
and 1, = pv,/\/(13 + ML”)
Also, since V, = — jwLI,, then
~/I = ¢ + 3~/2
and ii, = p.coLi3,/\/(rag | 021.2).
Figs. 7.11.f, g, h and j summarize the relations between the total and a.c.
values of the various currents and voltages. It may be seen from the
equations that the magnitude and phase of the anode voltage vary with
the signal frequency. The stage gain is
[AI = 5,/il, = p.o>L/\/(132 + 02L’)
When COL > r,, then |A| = p. and is independent of frequency. The
variation of |A| and 51: with frequency are shown graphically in Figs.
7.ll.k and Z.

7.7. Voltage Ampli er—0apacitive Load

This case is shown in Fig. 7.l2.a, where a resistance load R is shunted
by capacitance C. With no signal the equilibrium d.c. circuit is shown
in Fig. 7.l2.b, where it is seen that the circuit is exactly like that with
resistance load only. The condenser is charged up to a voltage equal to
the potential drop across R. The quiescent point Q is found as usual in
Fig. 7.12.d, and then the appropriate values of gm, 1,, and p. may be deter
mined. If the grid signal is given by
1);‘: 1j;SiI1(0t
J_
M"
M
V__E
1_____V
Ml L“ K
[

W“ Q __ _ E
C um

Q“Qgi _”_y::
+_VA
+VV___
G R im M
2_
6%
Q_°_“ I\ Q:_:'__y

_1
_
G_____1__
Q_O
Q
+%_
___G_
+Q_)
VG“
V”
__
>“
Q
rm
LR__m_ _ o_

.1‘K
M
A.
_\ EV\
Q___
V0 ‘2
L‘C
g/IHnu/LR
Q/__
/I‘O
R 4C

_
+
V’
V,

I‘I

\
\
git I
0

M+N_ 1 ‘V8

MWJ%_ : 4
M
%
_ _

M}

VA
YO _ _ __
_ _1 H
_ _i_ 1|!t Ava

_O _ _ I _ _ mm m_)" _ m_
_ _

,
7] VOLTAGE AMPLIFIERS 107
then we assume that it gives rise to an alternating anode voltage of the
fonn
v, = 13,, sin (col | ¢).
To determine 1'2‘, and q‘> it is convenient to use the current generator equiva
lent circuit, as shown in Fig. 7.12.e. With vector notation this gives
gm“ = Io + It I, Io = ']'¢°CVa»
It Z ' ' and I Z V;/fa.

Hence V,/V, = —g,,,/(1% F g } jcoC),

i.B., V;/V; 1 1 + 3; + rl)2 + Qzcz}.

Thus tan¢=—<»>C/(%+% I)

and IAI = IV./V I =.../N/{(,1,+ ;,)=+

The variations of the voltages, the stage gain and the phase angle with
frequency and a vector diagram are shown in Fig. 7.12.f to k. The stage
gain varies from (l §——_*':‘l/rt‘) at very low frequencies to zero at very
high frequencies. The gain drops to 1/\/Q of its low frequency value,
i.e., by approximately 3 dB, when mC = 1/R + 1/1,.

7.8. Frequency Distortion and Phase Distortion

With reactive anode loads, the stage gain of an ampli er varies with
the frequency of the signal. If two signals of equal magnitude but
different frequency are applied to the input terminals the output magni
tudes are not equal. The ampli er is said to introduce frequency dis
tortion. Also, with a reactive load the phase difference between the
input and the output voltages varies with frequency, and the ampli er
may have phase distortion. When a signal consists of a combination of
several sinusoidal waves of different frequencies, frequency distortion
and phase distortion cause the output waveform to differ from the input.
When an ampli er has a resistor as its anode load there is always some
capacitance in parallel with it. This may arise from the anode—cathode
capacitance of the valve, or from stray capacitance to a metal chassis
from the anode and the components attached to it. It is dif cult to
reduce the effective capacitance across the load resistor below about
10 p.p.F, and this sets a limit to the upper frequency range of the ampli er.

‘7.9. Tuned Ampli ers—Parallel Resonant Load

An ampli er with a load consisting of inductance L, capacitance C and
resistance R, all in parallel, is shown in Fig. 7.l3.a. Such a load is purely
108 PRINCIPLES OF ELECTRONICS [CI I.
resistive and of value R at the resonant frequency when cool. = 1/QQC.
At any other frequency the load has some reactance and its impedance
is less than R. The performance of the ampli er may be analysed by

R. I’
Er

+ ‘I’ Q y .._l_.
V [A'17 +A 1
3 VG :——v C i=: E2

EYE _ ____
(°)
V
<0 I _i_ _o
V
J ma mo .—. _—

vs (b) ______ _ _ ___

\\Q¢~\¢ 00> co,
(1 V . ._. _ . ',_ y _ __,
an < co, ‘OI
>:\*'“ Q‘ 1;‘

V Fro. 7.13
(2
the method used in Section 7.7, the only difference being the extra com
ponent L in parallel with the rest of the circuit. It is found that
vs gm{_' (1/R + 1/70) '_.7'(_ QC + 1/(°L)}
V1 I (1/R + 1/Ya)” + ( QC + 1/">1 )2 0
The output voltage is seen to lead the signal voltage by an angle qt where

t . —
. (— (DC + I/col.)
.
an "‘ <1/R + 1/1.)
¢ is an angle in the third quadrant as long as 1 /(DL > 0C. At the resonant
frequency f,,(= coo/Zn),
0°C = l/cooL
and V;/V, = — gm/(l/R + l/7,).
This con rms that the ampli er behaves at its resonant frequency as if
it had a load resistance R. At frequencies below the resonant frequency
the perfonnance is similar to that of an ampli er with an inductive load.
At higher frequencies it approximates to an ampli er with a capacitive
load. The various cases are illustrated in Fig. 7.13.b, c, d and e. Such
an ampli er may be used for the selective ampli cation of a signal cover
7] VOLTAGE AMPLIFIERS 109
ing a narrow band of frequencies around fo. At a frequency fl below fo
at which
— (1)16 +1/<o1L =1/R + 1/1,,
the stage gain is 1/ \/Q of the value at resonance. The stage gain has the
same value at a frequency fa where
co2C 1/021. =1/R +1/7,.

The fraction (fa — fl)/fo is a measure of the sharpness of the resonance

curve or of the selectivity of the ampli er. When the voltage across the
tuned circuit is 1/\/§ of the resonant voltage, the corresponding power
ratio is 1/2. Thus f, — fl is the width of the resonance curve at the
“ half power points ”, i.e., when the power output of the stage is approxi
mately 3dB below the value at fo.

7.10. Ampli ers with Several Stages

So far only single valve ampli ers have been considered. Frequently,
the voltage ampli cation required is greater than can be obtained from
one valve, and then several stages have to be used. Obviously, the overall

H.T.+
o
R1 C ‘0

'4
Q R, V<51’
V01
E11 _ _ H.1:_

(<1)
[L . _. . . __._QH.T.+

11 » E _

M
+

K» "
+ Q
" Val
E, .s{—'=
" 1 0 — ——O HI
Kb)
Fro. 7.14
110 PRINCIPLES OF ELECTRONICS [cH.
stage gain is the product of the individual gains and the phase shift is the
sum of the phase shifts of each stage. In coupling the output of one
stage to the input of the next, care must be taken to ensure that the
electrode d.c. voltages are correct. In a.c. ampli ers the stages are
normally isolated from one another for d.c. by one of the two methods
shown in Fig. 7.14. The condenser coupling of Fig. 7.14.a is usual in
voltage ampli ers at low frequencies, whilst the mutual inductance
coupling is commonly used at high frequencies with tuned ampli ers,
and sometimes with low frequency power ampli ers (see Section 8.5).
In the condenser coupled ampli er the d.c. grid voltage for the second
valve is obtained from the battery E1’ through the resistance R,, which
is called a grid leak. In normal use the grid voltage does not become
positive and no d.c. current ows through R9, so that its presence does
not affect the grid bias voltage. Over the working frequency range of
the ampli er it is desirable that R,> R1 and I/(DC < R9. If both of
these conditions are satis ed, then the coupling arrangement does not
affect the stage gain obtainable from the rst valve. The voltage ampli
cation of the stage is now V‘;/V If 1 /(DC is comparable with R,, then
C and R, act as a voltage divider across R1. Thus some of the output
voltage across R1 is dropped across C and is not passed on to the grid of
the second valve, with a resulting loss of gain. There is also some
frequency and phase distortion. It may easily be shown that the
reduction in gain in the coupling circuit due to C is given by

R0/\/{R02 + (1/¢~>C)”}
and the phase change is given by the angle 56, where
tan qi = 1/mCR,.
Thus the frequency and phase distortion are greater at lower frequencies.
If 1/<oC < R, there may still be a loss of gain if R, is comparable with
R1, since they are effectively in parallel for a.c. This loss is not accom
panied by frequency or phase distortion.
It follows that resistance loaded ampli ers with condenser grid leak
coupling are all subject to frequency and phase distortion at low fre
quencies and high frequencies (see Section 7.8). In designing ampli ers
care must be taken to use components that keep these distortions within
reasonable limits over the required range of frequencies.
In the circuit with mutual inductance coupling shown in Fig. 7.14.b,
the a.c. owing in the primary inductance L1, which is part of the anode
load of the rst valve, induces an e.m.f. in the secondary L2, and this is
passed on to the grid of the next valve. If the current in L1 is i‘, sin mt
the induced e.m.f. is
v, = M %; = coMi; cos mi or V2 = jmMI1.

The value of M is related to L1 and L2 by the coefficient of coupling k,

7] VOLTAGE AMPLIFIERS 111
where M2 = k2L1L2 and 0 < k < 1. The a.c. in the secondary winding
is normally very small, since it is open circuited apart from the input
capacitance of the second valve. Hence vl = L1 lg? It follows that

1»./1». = = k~/—_'<L./La.
Thus if L2 > L1 it is possible to get additional voltage ampli cation from
the mutual inductance coupling. It is shown in Section 8.5 that the
mutual coupling also serves to give efficient operation in power ampli ers.

7.11. Automatic Bias

In the circuits considered so far in this chapter separate batteries are
used for the anode and grid supplies. In most ampli ers only one battery
is used even when there are several stages. The grid bias voltage is

R R
‘A + {A +

Vc __1__ Yo “ _._
_ + _ _ "
'— *+ E2 K Q ' ' E2

Rx RKIA _ R; CK
3 I“ _ 5. . T
(v) (0)
Fro. 7.15

obtained from the voltage drop across a resistor RK as shown in Fig.

7.15.a. In the circuit it is assumed, in the rst place, that the input
terminals A and B are joined together and that current 1'4 is owing
through the valve, the anode resistance R and the cathode resistance RK.
This current produces a voltage drop Rm} across RK. One end of RK
is joined to the cathode and the other end to the grid, and it follows that
U9 = — R51}. If the value of RK is suitably chosen, then '00 gives the
correct value of grid bias voltage for the required Q point. When an
a.c. signal is applied to the input terminals the anode current varies and
the grid bias varies. In order to keep the bias voltage steady a capacitor
CK is connected in parallel with RX as shown in Fig. 7.l5.b. Then, if
1 /coC K < RK at the frequency of operation, the a.c. voltage across RK is
much less than the d.c. voltage, and the grid bias is steady and equal to
— RA 1'Q, where iq is the quiescent anode current. The quiescent anode
voltage is now given by vq = E2 — (R + RK)iQ.
The steady voltage required for the screen of a pentode is also obtained
112 PRINCIPLES OF ELECTRONICS [cH.
from the E2 battery, and its value may be adjusted by means of a series
resistance in the lead to the screen. The arrangement is shown in Fig.
7.16. In the quiescent state the screen voltage vc 2 is given by
1'02 = E2 — Rgigg — RKiK, where 1'02, is the quiescent screen current. When
a signal is applied the screen current
varies, but the screen voltage is kept
R l constant by means of the capacitor
Rs C3, provided 1/<»C,q < R8 at the
' signal frequency.
__ A circuit for a two stage pent

V
‘
+
“rm _
"*2‘
5
J,
Es
52 ode ampli er with resistance loads
is shown in Fig. 7.17. In such a
circuit it is essential to have a d.c.
C ' conducting path through the input
" source in order to give the rst
F;G_ 7_15 valve its grid bias.

7.12. Alternative Connections of Valve Ampli er

Since the triode valve has three electrodes, there are six possible ways
in which the input could be applied between one pair of electrodes and
the output could be obtained across another pair. In the conventional
ampli er the input is applied between grid and cathode, and the output
is obtained between anode and cathode. This may be referred to as

. g .. g‘_ .
I‘
INPUT
I‘ = ourpur
O

FIG. 7.17

the common cathode connection. Two other connections are frequently

used in which the grid and the anode are the common electrodes. The
common grid ampli er is shown in Fig. 7.l8.a and tlie equivalent circuit
in Fig. 7.l8.b. From the latter it is seen that

V (u+1)=('¢+Z)LandV@=—ZL
7] VOLTAGE AMPLIFIERS 113
The voltage ampli cation is given by A = V,/V,
ie, A= Z01 + 1)/(70 + Z) = Ztgm + 1/70}/{I + Z/'a}
The corresponding formula for the conventional or common cathode
ampli er is
A = — p.Z/(7,, + Z)
so that the common grid ampli er gives slightly higher voltage gain.

Z 0I 0o
)~O

0
Q‘?~E +>§
O
+ 9,;
+“Q
Q +99
0.
_ <+<5
o<+ &
°+
QM
_ Q
Q9OQ 1»
I/V:
E1 — 1 _ 1"
K K
(0) (b)
Fro. 7.18

This type of ampli er is sometimes used at very high frequencies (see

Section 13.9).
The common anode ampli er and its equivalent circuit are shown in
Fig. 7.l9.a and b. The input is connected between grid and anode and

A
[A + | 1, +

+ we* Q
Q '~ ’~ 9 '
_
y‘ K L _ _ .'_ E2 .
G _K.... j+
yo + + bf, V

E1 R RIA V‘ Z
" 1, _O

(0) (0)
Fro. 7.19

the output is taken between anode and cathode. The voltage ampli
cation in this case may be derived in the form
A = gmz/(1 + gmz + Z/'~)
When the load Z is resistive, |A| is less than unity. Such an ampli er
has certain special properties. It is usually called a cathode follower,
and it is dealt with further in Section 10.10.
114 PRINCIPLES OF ELECTRONICS [CH.7

7.13. Input and Output Impedance of an Ampli er

W'hen a signal generator is connected to an ampli er the current ow
ing in the input circuit depends on the input impedance of the ampli er,
Z,, which is de ned to be Z, = V,/I, (see Fig. 7.20). In a conventional
ampli er the input impedance depends on the inter electrode capacitances
of the valve and the leakage resistance between the grid and cathode
electrodes. At low frequencies the input impedance can be very high.
On the output side of an ampli er
1 the valve acts as a generator feeding
o + S a load. The internal impedance of
the generator is sometimes called the
WPUT is "MP"'F'ER output impedance of the ampli er.
.. In the conventional ampli er the
output impedance is ra. Input and
FIG 7 20 output impedances are considered
further in Chapter 10. It is shown
there that the common grid ampli er has a low input impedance and the
cathode follower has high input impedance and low output impedance.

7.14. Limitations of Small Signal Theory

In most of this chapter we have limited the signals to small values,
and it may be worth while repeating and emphasizing the signi cance
of this limitation. The use of static characteristics and the load line by
themselves is subject to no limitation of amplitude. The true output
voltage may always be determined in principle for any given input voltage
as long as the characteristics are available for the range of operation. Large
signals are given more detailed consideration in the next chapter. The
small signal limitation has arisen from the use of the Valve Equation,
which relates small changes of anode current to the changes in grid and
anode voltages,
in = gmvg + 7371/70

01' In = gmV¢ + vs/7a

when the changes are sinusoidal. The changes are limited to a range
over which the valve parameters gm and r,, are constant. These restric
tions also apply to the valve equivalent circuits.
 CHAPTER s
POWER AMPLIFIERS
8.1. Large Signals
In Chapter 7 the operation of ampli ers is con ned to signal amplitudes
over which the valve parameters remain constant. Such ampli ers,
though dealing with small input and output voltages, may give large
voltage ampli cation. These are the conditions which usually exist in
the early stages of an ampli er. However, in the nal stages it frequently
happens that considerable power is required for some purpose, such as
operation of a loudspeaker, energizing a transmitting aerial or activating

R
is 1' I

'1‘ + _|_.
VA i
n E,
Ex _

(0)
‘Al Q [A

R O
....................
0 VA: YQ
o= E,
| — (,7 _______________ __ _
P _' Q j(______ ________9
I ‘/9 DYNAMIC

Vo E2 '/ VA
1

Q *\ . Q Q Q

(b) (¢)
Fro. 8.1

some control mechanism. Then both large voltage and large current.
changes are required. The use of the Load Line Equation is not limited
with regard to size of signal, and the load line is therefore the starting
point in our consideration of power ampli ers.
115
116 PRINCIPLES OF ELECTRONICS [cH.
8.2. Load Line and Dynamic Characteristics
A set of anode characteristics with a load line is shown in Fig. 8.1.b for
the valve and circuit of Fig. 8.l.a. As 1'0 is varied and E2 and R are kept
constant, the values of v4 and 2'4 always lie on the load line, and the actual
values may be found by picking the appropriate static characteristic
corresponding to "the instantaneous value of vq. This diagram is based
on the anode characteristics and gives the values of 1'4 in terms of v4, as
1'0 is varied. The same information could be given directly in terms of

R '

1, +
Q —.=
'1' '9' 7;? VA E2
V Q i =5
‘_ VG! E3

ET. 7 1 . .
(0)
‘Al “A
El | VG VA = yo
R , """""""" "\; '

_E.. ...... @ ,~,

DYNAMIC
O
Q
Q
Q

'o 6“
‘5 >*_1_">~=‘* o’ "6
Q... “\... Q... 0 Q
rn IO
_q_. _ . ._
vcq
(bl (¢)
Fro. 8.2

vq in a diagram based on the grid characteristics as shown in Fig. 8.l.a.

This diagram, which shows the variation of the anode current with grid
voltage when there is a load resistance is called the dynamic grid charac
teristic or merely the dynamic characteristic. The load line and the
dynamic characteristic give the same information, but sometimes one is
more convenient to use than the other.
Corresponding load line and dynamic characteristic for a pentode with
resistive load are given in Fig. 8.2. The pentode dynamic characteristic
coincides with the static grid characteristic, except when vq approaches
ZCTO.
8] POWER AMPLIFIERS 117
When a signal is applied to the grid of an ampli er the variation in
anode current and anode voltage may be determined from the load Line
or the dynamic characteristic as shown in Fig. 8.3. The variation in grid
voltage is con ned to a region over which the dynamic characteristic is
practically straight and the variation in anode current is a faithful replica

R
[A + I

+ * Q vA '=
V, a E2
_ ye

eIl"_ _ . _.
(0)
IA [A

1. E1.
R
Q 0 arc: — O — O I _ Q Q _ — — 19

DYNAMIC

Q" “100 3 st 5
___ E1 4

'E1 O v5 O VQ I'll N VA

O '8‘ ‘Q
O
1
‘Q
'0
_—n ~___

U U

4 4

5 5

I 1
Fro. 8.3

of the grid signal. However, if a large signal is applied as in Fig. 8.4,

then the anode current waveform differs from the signal waveform.
When this happens the anode current is said to have non linear distortion.
The output voltage, which is proportional to the anode current, suffers
similarly from non linear distortion. This type of distortion is some
times called amplitude distortion, since it increases with the amplitude of
the signal. The subject of distortion is considered in more detail later in
this chapter.
E
118 PRINCIPLES OF ELECTRONICS [cl I.
The conditions for small signals (constant gm and rd) imposed in Chapter
7 correspond in Fig. 8.1.b or 8.2.b to any range along the load line, on
either side of Q, over which the anode characteristics are parallel and cut
off equal intercepts on the load line. In Fig. 8.l.c or 8.2.0 the correspond
ing condition is the portion of the dynamic characteristic about Q which
is straight.
8.3. Power Output with Resistance Load
The power output is most easily determined from the anode character
istics and the load line. In Fig. 8.5 it is assumed that the signal is

Li, I I,

ovwwnc 2

Q || [Q l 3 st
__p__ T1""'ls |l :
I '0 I I | __‘__
| _E1 ' O ye '

. .01 I
'§=“

I f
Fro. 8.4

sinusoidal and the effects of distortion are ignored. Characteristics are

drawn for a triode and a pentode. As a result of the signal the anode
current varies sinusoidally about its mean value iq up to im, and down
to imin. The anode voltage varies similarly about vq down to vmin and
up to vmu. These values give the current and voltage variations across
the load resistance, and the power in the load is found from the product
of the r.m.s. values of the current and voltage. Obviously (ima, — imin)/2
gives the amplitude and (im, — imin)/2‘/2 the r.m.s. value of the a.c.
component of the load current, and similarly (vm, — rqm)/2\/2 gives the
r.m.s. voltage. The power output is therefore
P0 = (rm, i.,.,,,)(v.,,,, Umm)/8 = R(¢',,,,, @',,,,,,)=/s
= (um v,,.,,)*/sR.
1] POWER AMPLIFIERS 119
This expression for P0 is derived on the assumption that the grid operating
roltage vqq (= — E1) is given and there is no distortion. This means that
:9, the amplitude of the grid signal, is limited to the range on either side of
vqq over which the static characteristics make equal intercepts on the
oad line. Frequently the range of operation of a power ampli er is

lA‘
V0
R O

{A + 'Ef"I‘>'
Ei
+ + ii ._§=
V § '3 VA I. l.mox— E1 V‘
‘ v 1 r
E, ° lg: basin of
"° /2
fl‘! 0"

‘O
.léa§.
yo E2 VA

Vmin Vrnc:

(g) 0U
Q’
E 2/R O ya

. | E+£
("Fm _ | Q “El
0 ‘H El’ vs
lrnln
I | _____

Q M E= W2
Vrnln Vino:

nu
Fro. 8.5

limited to negative grid voltages, and this imposes a further limitation

on 23,, namely 13, < vgq. We have assumed above that vgq is given.
Usually in practice, E2 and R are xed and the diagram is studied for
the range of grid voltage over which the distortion is negligible; voq is
chosen at the mid point of this range. An ideal case would occur where
there is no distortion over the whole length of the load line from '00 = 0
to the v4 axis. For maximum output the anode current then varies
from zero to imu, the value for ‘U0 = 0 (see Fig. 8.6.a); the anode voltage
range is um to E2. Under these conditions
imgx Z 25¢ Z
120 PRINCIPLES OF ELECTRONICS [CH.
and the power output is
P0 = 1.Q(E2 — l'm1n)/4.

An alternative expression for the power output in this ideal case may
be obtained from the voltage equivalent circuit (Fig. 8.6.b). Its use is
justi ed, since we are assuming constant valve parameters over the
"0
_ o
‘Al
A
E2/R I . +

lmo: — / V R

... . _.
ymln E2 VA K

(<1) (b)
F10. 8.6

whole working range. As the current is cut off at '04 = E2, it follows
that the full range of grid voltage is E2/(.1, and hence
Ugq = — g = — E‘/2|! .

From the equivalent circuit

£4 = “vi/(R + 70)
and thus 5, = E,/2(R + 1,).
But 6,, = Rid, and so
P0 = '5,i,,/2 = RE,”/8(R + 11,)’.
When R is varied the power output thus has a maximum value when
R = r,,. This is in conformity with the usual relation that a generator
gives maximum power output when the load resistance equals the internal
resistance of the generator. The a.c. output power P0 is developed by
the alternating anode current owing in the load resistance. Although
this current is controlled by the grid voltage, its source is the E2 battery,
and P0 is obtained by the consumption of energy in that battery. The
total power P; taken from the battery is its voltage multiplied by the
mean current owing through it, i.e.,
P1 = E2iQ =' E2?“ = + fa).

The e iciency of power conversion from d.c. to a.c. is given by

11 = Po/P1.
i.e., 11 =. + T3).
i] POWER AMPLIFIERS 121
Thus under the condition of maximum power output, R = 1,, the e i
:iency is 12 5 per cent. Greatest e iciency occurs when R > 1,, and the
imiting value to 1; is then 25 per cent. Under this condition both the
aower output and the power input are negligibly small.
It is important to consider where all of the power taken from the
aattery is used. In addition to the a.c. component of the anode current
zhere is also a d.c. component iq, which dissipates energy in the load at
zhe rate
Riq* = Ri,,* = RE}/4(R + 1,)” = 2~qP,.
This is double the a.c. power output. The remainder of the battery
power, (1 — 31;)P1, is dissipated as heat at the anode of the valve, and

[A *

V
. .g .,Q3 E, 1

E,
(0)
iii
"0
O
[mox“"“'

1. 1. ,
1 ——
'5‘
R I "Q
|'|'\ll'\_ .1 i M

*0 vrnin E; Vina: VA
(6) (c)
Fro. 8.7

represents the rate of loss of kinetic energy of the electrons striking the
anode. This particular power loss is called the anode dissipation; it is
considered further in Section 8.11.

8.4. The Transformer Coupled Load

The steady component of the anode current owing through the load
resistance dissipates power equal to twice the useful power output.
This wastage may be avoided if the load resistance is coupled to the
valve through a transformer as shown in Fig. 8.7.a. This arrangement
has the additional advantage that the transformer may be used for
122 PRINCIPLES OF ELECTRONICS [cn.
matching the actual load resistance R to the optimum load for the valve.
It may be shown that, for an ideal transformer, the following relation
ships are satis ed (see Fig. 8.7.b):
V1/V2 = "1/'12» I1/I2 = "2/"1 and V1/11 = R ("1/nzlz»
where n1 and n2 are the numbers of turns in the primary and secondary
windings respectively. The ratio V1/I1 is the effective resistance to a.c.
appearing across the primary terminals of the transformer. This re
sistance should equal the optimum load for the valve. If the optimum
load is R1, then the turns ratio of the transformer is chosen so that
"1/"2 \/R1/ R~
The transformer circuit then provides the valve with the optimum load
resistance for a.c. However, for d.c. the load is the d.c. resistance of the

IA‘ O VG

9* I

uo

Fro. 8.8

primary winding of the transformer. This is usually very small, and the
d.c. load resistance may be taken to be zero. In these circumstances
there is negligible power loss due to the steady anode current, and
E, = UAQ. The load line conditions are shown in Fig. 8.7.c. The power
output is again given by the equation
PO = (imax '_ imln) (vine: — vmln)/8
Frequently a power ampli er has to cope with a wide range of signal,
and then it gives maximum output when it is handling the largest signal.
Under all other conditions the output is less. The power balance with
transformer coupling is given by: power from battery = power output +
anode dissipation. The power from the battery (Eziq) is constant, so
that the anode dissipation is large when the output is small. There is
always a maximum allowable anode dissipation (W4), and in designing
an ampli er where the signal level may fall to zero, a limiting condition
is that the Q point must give
Eziq Q W4.
In Fig. 8.8 this means that the Q point must not lie in the shaded area.
8] POWER AMPLIFIERS 123
In dealing with the question of power so far, only the anode circuit
has been considered. When the grid voltage remains negative no appre
ciable current ows through the grid battery or the grid signal source.
No power is therefore consumed in the grid circuit under these conditions.
In any overall balance of power, account must be taken of the power
required to heat the cathode.

8.5. Load Resistance for Maximum Power Output with Transformer

Coupling
In Section 8.3 the power output and optimum load resistance are
determined for an ampli er with a directly coupled load resistance and
a given battery supply. With this arrangement and ignoring distortion.

1“
o '4;
51
_ F
2:, ,
I
.
¢Q l| no 28‘

B‘ D‘ C /__
_° v2 E2 v2 VA
Fro. 8.9

the value of vgq is xed at the mid point between vq = 0 and the E2
point on the v4 axis, and the maximum amplitude of the grid signal is
equal to vqq. Under these conditions it is found that the maximum
output is obtained when the load resistance is equal to 1,, and the efficiency
is then 12 5 per cent. The maximum possible e iciency is 25 per cent
and occurs with R > r,,. When transformer coupling is used between
the valve and the load, v4q is xed. Then as R varies, vgq varies and
so does the maximum signal amplitude. The conditions are quite
different from those in Section 8.3, and we are now going to see that
they give di erent values for the optimum load resistance and the e i
ciency. Again it is assumed that distortion is negligible over the whole
range of the load line (see Fig. 8.9). Maximum power output is obtained
when ima, = 2iQ and im = 0. Then P0 = Riqa/2. From the geometry
of the gure,
v2 — E2 = E2 — vi
and from AQDC, {Q = (v2 — E2)/R = (E2 — v2)/R.
Also from A FOB, v2 = 2iQr,.
124 PRINCIPLES OF ELECTRONICS [cn.
By elimination of v2 it is found that
iq = E2/(R + 21,),
and hence
P0 = RE2*/2(R + 2r,,)2.
For variation of R this has ‘a maximum value when R = 2r,,. The power
taken from the battery is E2z'Q and the e iciency of power conversion is
1; = R/2(R + 21,).
When R = 2r,,, the e iciency is 25 per cent. The maximum value of ~q
again occurs when R > r, and the limiting value is 50 per cent. These
gures con rm the improvement in performance which is obtained with
a transformer coupled load. Practically all power ampli ers use trans
former coupling.
In this section and in Section 8.3 it is assumed that linear operation is
obtained over the whole load line. In practice this is not true, and the
range of operation has to be limited to keep the distortion to a reasonable
value.

8.6. Non linear Distortion

We have seen that, with large signals, the output waveform differs
from the signal waveform. This distortion arises whenever the grid
signal covers a range over which the dynamic characteristic is not
straight. A curve may be represented by a suitable power series, and the
dynamic characteristics of Fig. 8.1.0 and 8.2.0 can have equations of the
form
1.4 = A + B00 I C1102 + D093 +

where A, B, C, D . are constants. The number of terms required

depends on the shape of the curve. For the dynamic characteristic of a
triode only the rst three terms are required to give a fairly good approxi
mation to the equation of the curve. In other words, the triode dyna
mic characteristic has approximately a square law. In the case of the
pentode the dynamic characteristic requires a cubic term also (this is
obvious from the shape of the curve which shows in exion of the kind
appropriate to a cubic law).
When a signal v, is applied to the grid of an ampli er then va = — E1 + v,.
Also 11, = v,. The equation for the dynamic characteristic expresses the
total anode current 2'4 in terms of the total grid voltage vq. A similar
equation relates the varying components 2', and v2, viz.,
1', = av, + bu,” + 021,3 +
where a, b, 0 are a new set of constants. This equation is just as
general as the previous equation, but it is more convenient for analysis.
In the case of the triode the equation is
£3 =' + bilge.
8] POWER AMPLIFIERS 125
Now let v, = 13, sin cot as shown in Fig. 8.10.a. The resulting anode
current variation is obviously not sinusoidal. On substituting in the
equation, we nd

b‘2 . b‘2
= 13 + ail, sin oat — 329 cos 2@t,

since sin” wt = (1 — cos 2o>t)/2.

Thus the signal increases the mean component of the anode current by
'% It produces a sinusoidal component ail, sin mt which is proportional
to the signal, but it also produces a new component which has twice the

I IA [A (AI

¢—qor¢1§1cn1—Q \—3 Q11Qcn—¢_—Q_0—Q 1 Qignxq

\
I
1
Q
I
I

=>Y~~w¢ P , _§
1, . F+H+M
I
IQ

° ‘~""
"""""" " ,1 H" ' "(M
I

iiiijil

I I I I I
I.
I I 13131191111111 QQZ QZQQQQQQZI

—n—nu. n—
iii. ii

'II
<>:
I
OI.
“I
I

'0
I
“I
I
0
~:
I5
(6) (¢)

(0)
(1%
~ |
II '5
It
Fro. 8.10

frequency of the signal, i.e., a second harmonic of the signal is introduced.

This effect is called harmonic distortion. It may be seen that the ampli
tude of the harmonic component equals the increase in the mean current.
This provides a simple means of checking for harmonic distortion by using
an ammeter to measure the mean anode current. A moving coil meter
is suitable.
The way in which the actual waveform is built up from its components
is shown in Fig. 8.l0.b and 0. The expression for i, may be written in
the form
.2 .
1', = Q? + ad, sin wt + % cos (2@t — 1:).
126 PRINCIPLES OF ELECTRONICS [CH.
The last two terms in this equation are represented by the curves F and
H in Fig. 8. 10.0. The positive and negative peaks, P and N, of the
actual anode current are obviously given by P = F + H + M and
N = F — H — M, where F and H are now the amplitudes of the funda
mental and harmonic components and M is the increase in the mean
component. Since H = M, these expressions lead to P + N = 2F,
P — N = 4H. Thus
WF=W—NWW+N)
gives the fractional harmonic distortion. It is usually expressed as a
percentage. In a straightforward triode ampli er operating at its full

I42 ll,
I

Q __ ____ "Q o ,
I

I |° vs I

Ԥ=":
L_aQ1111ncZ11:—

inIO.1 '11

I
Fro. 8.11
output the second harmonic distortion may amount to 5 to 10 per cent.
A similar analysis may be carried out to determine the harmonic
distortion introduced by a pentode ampli er. For a sinusoidal grid
signal we get
i, = 2213, Sin rot + bi,” Sin” mi + 013,3 Sins (oi.
The rst two terms are the same as for the triode. The third term may
be resolved using sin 3<»t = 3 sin wt — 4 sin’ mt. The anode current is
then
. b‘ 2 , , . b‘ 2
2, = 3% + (av, + it 012,3) sin wt — % cos 2wt
~s
— iii Sin 30:1.

There is now also a third harmonic, and this is often of greater amplitude
8] POWER AMPLIFIERS 127
than the second harmonic. The anode current waveform is given in
Fig. 8.11, and it shows attening at both top and bottom. This is the
effect of the third harmonic component. The unequal amplitudes are
due to the second harmonic and the change in mean current. The wave
form may be synthesized from its components as in the triode case.

8.7. Intermodulation
Non linear characteristics may give rise to another type of distortion
when the signal consists of more than one sinusoid. For example, let a
grid signal
v, = v, = 132 sin calf + 172 sin 2 >2t
be applied to a triode. The anode current is then given by
2', = a(z31 sin o>1f+ 132 sin w2t) + b(131 sin o)1i+ 132 sin 2:22)”. On expanding and
using sin“ cot = (1 — cos 22¢!)/2 and 2 sin 01¢ sin w2t = cos (<02 — w2)t —
cos (0)1 + co2)f, it is found that

bag 22;
— 17131172 COS (0)1 —|— o>2)f.
Thus the two signals have produced not only output currents proportional
to the signals but also an increase in the mean current, second harmonic
components of the signal frequencies and two new a.c. components, whose
frequencies are the sum and the difference of the signal frequencies.
This new type of distortion is called intermodulation. In some respects
it is more objectionable than harmonic distortion, particularly in audio
ampli ers. Most sounds include some harmonic components of the
fundamental frequency, and some increase of their amplitudes may be
tolerated. However, the new components produced by intermodulation
bear no harmonic relationship to the signals, and their presence is readily
noticed. Interrnodulation distortion may be emphasized when it occurs
along with frequency distortion. For example, some loudspeakers show
marked peaks in their frequency output response curves. Correspondence
of these peaks with the sum or difference intermodulation terms may
explain the peculiar sounds that are sometimes heard.
Interrnodulation distortion may also arise from the higher power terms
in the equation of the dynamic characteristic.

8.8. Non linear Devices

It should be noticed in passing that non linear distortion may arise
from any device whose output is not directly proportional to its input.
Although it has been dealt with here in terms of curved valve character
istics, the same type of analysis would apply to any non linear device.
128 PRINCIPLES OF ELECTRONICS [cl r.
A few examples are transformers or chokes with iron cores, the moving
coil of a loudspeaker and the human ear.
It is also important to note that in this chapter, where we are dealing
with ampli ers, the distortions arising from non linearity are being
emphasized. The same non linear effects are essential for many important
electronic applications, such as recti cation, modulation and frequency
changing. These are dealt with later (see Chapters 16 and 17).
8.9. Push Pull Ampli ers
When a power ampli er is to be used with reasonable freedom from
non linear distortion it is important that the operation should be con
ned to straight portions of the dynamic characteristic. When more
power is required, then larger valves should be used. The power may

In

: 5+ II.
+ 2
V01 " :
INPUT I _,_ E1 _ ourvur
0
I62 V02
1 +
.=.

‘A2
Fro. 8.12

also be increased by using a pair of valves. These can be connected in

parallel, but there is an alternative method of connection which has
considerable advantages. The circuit arrangement is shown in Fig. 8.12.
Two identical valves and two centre tapped transformers are used. The
input signal is connected to the primary winding of one transformer, and
the secondary voltage is divided equally between the grids of the two
valves. As 12,1 = v,2, then v,1 = — v,2 at any instant, and when one
grid voltage rises the other drops. The primary winding of the second
transformer is connected between the two anodes in such a way that the
anode currents ow in opposite directions in the winding, and so produce
opposing uxes. The output is therefore proportional to 2,1 — 2',2. An
ampli er connected in this way is called a push pull ampli er.
Since the two valves are identical they have the same dynamic grid
characteristic and hence
ial = avgl ‘I’ bvylz + 00913 'l"
and 2',2 = av,2 + bv,22 + 012,23 +
Since 12,1 = — v,2,
2,2 = — av,1 + bv,1” — 011,13 +
{,1 — {,3 = 200,1 + 2611,13 +
8] POWER AMPLIFIERS 129
All the even powers have disappeared, and so there are no harmonics or
intermodulation components arising from them. With triodes which
have approximately square law characteristics there is very little non
linear distortion. The push pull circuit is less effective with pentodes,
since they have an appreciable cubic term in the dynamic characteristic.
The performance of the push pull triode ampli er may be explained

HM 1,,
lot
Ft
__ I
"5 OH
o
|._ I I '_._.l___ 'III, : 3..
Q " '61
"91

g “A2 IE2‘ _
__.__...__... _ l02 F2

III I .5
_l.. o
I |. I I I I
Z"\N
l
I '0" '62 '
' " '92
___.L.__
' io1'io2
I

I
I
o

FIG. 8.13

graphically as shown in Fig. 8.13. Dynamic characteristics are drawn

separately for the two valves, and I',,1 and 21,2 are determined for the
instantaneous values of 11,1 and 12,2 arising from a sinusoidal input. The
anode currents are then analysed into fundamental, second harmonic
and mean components by the method used in Fig. 8.10. In obtaining
I',,1 — 11,2 it is seen that the harmonic and mean components disappear and
the two fundamental components add together. The performance of
push pull circuits may also be determined conveniently by drawing a
combined dynamic characteristic, as shown in Fig. 8.14. The two
130 PRINCIPLES OF ELECTRONICS [CI 1.
separate dynamic characteristics are placed together so that the anode
currents may be found from a single input signal. The resultant combined
dynamic characteristic is the sloping straight line through — E1. Anode
characteristics may be combined similarly and used with a common load
line.
When a load resistance R is connected across the output terminals of
a push pull ampli er, then the effective a.c. anode to anode resistance
presented by the transformer is 4R(n1/n2)”, where n1 is the number of
IiAl

IIIFIA2
0,} ,
I

iiii

'62 {E1 Vs: ° t

I l _
o I
|
I ' .
I 2 Q,‘ I

_ . i_ ,1.‘ >

O i 7:1

in] 3
A .
V
nit 1

' Fro. 8.14

|t
tums on each half of the primary winding and 1&2 is the total number of
tums on the secondary. Each valve has therefore an effective load
resistance of 2R(n1/n2)”.
In addition to reducing non linear distortion, the push pull ampli er
has certain other desirable properties. Since the two halves of the mean
anode current ow in opposite directions in the primary winding of the
output transformer, there is no question of magnetic saturation of the
core arising from the direct current ; thus another source of non linearity
is eliminated. Also, and for the same reason, any variations in the mean
anode current, due to hum or other variations in the power supply,
8] POWER AMPLIFIERS 131
produce no output. Finally, there are no components of the signal
current in the power supply.
The push pull circuits described so far have used transformers as
essential components. Resistance loads may also be used in push pull
circuits, and examples may be found in Chapter 12.
8.10. Glass B Ampli ers
The push pull circuit can, under certain conditions, give almost dis
tortionless output even when the operation is over ranges where the
{A1 (A1

'62 "or '02 Ye;

'
U T

1
I “E1 “E1

{A2 [A2
(0) (0)
Fro. 8.15
individual dynamic characteristics show considerable curvature. The
combined characteristic of Fig. 8.15.a illustrates this point graphically.
It is possible to proceed even further, as shown in Fig. 8.15.b, where each
valve is biased very nearly to cut off. It may be seen that there is now
appreciable curvature in the combined dynamic characteristic, and dis
tortion would occur. The bias, — E1, is too great. The greatest bias
without large distortion is obtained by projecting the straight portion of
the individual characteristic, as shown by the dotted line. The com
bined characteristic may now be redrawn with bias nearer to — E1’.
When an ampli er is biased nearly to cut off it is said to operate in the
Class B state. Single valve Class B operation is seldom used for an audio
ampli er, as it would introduce excessive distortion. An ampli er which
operates in the linear region of the valve characteristic is called a Class A
132 PRINCIPLES OF ELECTRONICS [cl I.
ampli er. Practically all the ampli ers considered so far in this book
are Class A. The main advantage of Class B ampli ers over Class A
lies in their greater e iciency. Since they are biased nearly to cut off,
the quiescent anode currents are nearly zero. When a signal is received
the mean anode currents increase. Thus very little power is consumed
except when signals are received, and the consumption increases with
signal size. This is a very desirable feature in an audio ampli er, where
signal amplitudes vary considerably. The question of ampli er efficiency
is considered in more detail in the next section.
A single valve may be used in a Class B ampli er at a high frequency

"A
)o‘

"4, 2%. 3%» ‘% t

1', ______ __ ,_
lq
. "' "' " "'
11.

PA
6,10

' O l_'I_ ' I ' T’!

Fr0.8.16
with the load tuned to parallel resonance at the frequency of the signal.
Then, although the anode current waveform is far from sinusoidal, it
may be resolved into a series of components whose frequencies are
multiples of the fundamental frequency. Since the load impedance is
negligible at all frequencies except the fundamental, the anode voltage is
practically sinusoidal.
8.11. Power Ampli er Emciency
The ef ciency of a power ampli er is given by the ratio of the useful
a.c. power in the load to the total power taken from the anode battery.
That power which is not useful a.c. power is dissipated as heat, either in
8] POWER AMPLIFIERS 133
circuit components or by electron bombardment of the anode. With a
transformer coupled load the power dissipation is practically all at the
valve anode. We are now going to examine this dissipation more closely.
In Fig. 8.16 the waveforms of v4 and I4 are shown for a Class A ampli er
with resistance load and sinusoidal signal. As shown in Section 7.4, the
a.c. components, v, and id, differ in phase by 180°. We may therefore
put for the instantaneous values of v4 and £4
£4 = iq + £0 sin wt and v4 = E, — 0,, sin col.
At an instant t an electron which leaves the cathode with zero velocity
arrives at the anode with kinetic energy given by ev4 = e(E, — vo sin wt).
On striking the anode, the electron gives up this energy as heat. Since
£4 represents the rate of arrival of electrons at the anode, 1142', gives the
rate of loss of kinetic energy, i.e., the instantaneous power dissipation.
The value varies with time and the mean power P4 dissipated at the anode
is found by averaging v4I'4 over one cycle T,
.
1.6., P4 ==
1 T ‘U414dt.
.

On taking the product v4I'4, all the tenns in sines give an average of zero
except the term in sinz cot and so
1 T . ,
,P4 = T L {Ezlq — v,,i,,(l — cos 2mt)/2}dt,

using 2 sin” wt = 1 — cos Zcol. The cosine term gives an average of zero
and then _
PA 1 Ea1Q '_ o'i°!2.

In this equation E,I'q gives the total power taken from the battery and
13°50/2 must represent the useful power output. This is con rmed when
we remember that I70/\/§ and in/V5 are the r.m.s. values of the output
voltage and current respectively. From the equation we see that the
anode dissipation decreases as the output increases. Physical explana
tion of these conditions may be given as follows. With no signal applied,
an electron in going from the negative end of the battery to the positive
end takes energy eE2 from the battery. The kinetic energy of the
electron on arriving at the anode is also equal to eE2. Thus all the energy
is dissipated as heat at the anode. When a signal is applied the electrons
arriving at the anode in the rst half cycle have kinetic energy less than
eE,; those arriving in the second half cycle have kinetic energy greater
than eE2. Each electron takes energy eE2 in going from the negative to
the positive end of the battery. Since more electrons reach the anode
during the rst half cycle than during the second, the average dissipation
at the anode is reduced on the application of a signal. The dissipation
may be reduced still further by preventing electrons reaching the anode
during the half cycle when the anode voltage is above E2. This is what
134 PRINCIPLES OF ELECTRONICS [cB.
happens in the Class B ampli er. The anode current ows only during
the negative half cycle of the anode voltage. The anode voltage and
anode current waveforms for one valve are shown in Fig. 8.17. The
anode current for one valve is far from sinusoidal, but if a tuned load is

°‘>

ITI ~_£‘ I l0——>

lo % 2% 3% 4% t

ix
9
lo

O I

PA

° I
Fro. 8.17

used the anode voltage is as shown in the gure. The anode dissipation
may be determined again from the instantaneous values,

1;, = I, sin ...: from 0 to

I'4=0from€toT
and v4 = E2 — 13° sin ml.
If we average v4i4 over a complete cycle we nd for the anode dissipation
in a Class B ampli er
P4 = E250/11: — I oio/4, using T = 2n/co;
in/1: is the mean value of 1:4 and hence E250/n gives the power taken from
the battery; 13050/4 is the output power. This equation con rms that
there is no dissipation when there is no signal. The efficiencies for Class
A and Class B ampli ers are given respectively by the formulae
8] POWER AMPLIFIERS 135
The maximum possible values for 13,, and £0 in a Class A ampli er may be
seen from Fig. 8.16 to be 130 = E2 and £0 = IQ. These give for the limit
ing ef ciency 1]4 = 50 per cent. The limiting value of 13,, in the Class B
ampli er is also E2 and then 1;, = rt/4 or about 75 per cent. These

VA‘

la
E, _ _ _

.0: . :\= a\§’_

4"!
£\‘;’.. %
‘A

'O[\I lnl I>f

PA

. on I in I |,f

Frc;.8.18

limiting ef ciencies are never reached in practice. Values of 25 and 50

per cent correspond more nearly to the highest values obtainable.
The ef ciency may be increased still further by con ning the electron
ow to a narrow pulse at the time of the minimum anode voltage, as
T LOAD T

IA IIII

+ Q,__ C —.
Iz Q E1
ET _ .
Fro. 8.19
136 PRINCIPLES OF ELECTRONICS [cH.
shown in Fig. 8.18. In the limiting case when 13,, = E,, the 'anode
dissipation would be practically zero and the e iciency would approach
100 per cent. Practical values of 70 to 80 per cent can be obtained with
this type of operation. The anode current is now far removed from a
sinusoidal waveform, and such ampli ers are used only at high frequency
with tuned anode loads. They are known as Class C ampli ers. A
suitable circuit arrangement is shown in Fig. 8.19. The LC circuit is
vA| ____
I
E, __

f
'/..=%.°%.‘z.
"0
I ‘W W W W i i 1 1

cur OFF I I I IIII I X

Q‘)
_E' __ __ __ ___

;0

FIG_ _

‘ct
"15
I

;§
.

Q I
FIG. 8.20

tuned to resonance at the signal frequency, where v, = 13, sin wt. The
load, which may be an aerial or the input circuit of another power ampli
er, is coupled to the inductance of the tuned circuit. The coupling is
adjusted to give at the resonant frequency a suitable a.c. anode load for
the ampli er. In order to achieve the Class C operation certain con
ditions are required. The grid bias must be adjusted well beyond the
cut off value for '04 = E2, and the signal voltage ii, must be large enough
for the grid voltage ‘U9 to be positive for part of the operating cycle.
Typical waveforms are shown in Fig. 8.20. The anode current pulse has
8] POWER AMPLIFIERS 137
appreciable duration, and its width is conveniently measured in terms of
the angle of ow, 6 = wt, where t is the time duration of the pulse. The
efficiency increases as 6 is reduced; however, the power output also
decreases, and some compromise is necessary. A width of about a quarter
of the period of the oscillation (6 = 1:/2) is commonly used. Since vq is
positive for part of the time, there is now some grid current, and power is
consumed from the signal source. The Class C ampli er has departed a
long way from the ampli ers which are considered at the beginning of
this chapter. It is a highly non linear device and does not readily permit
of analytical study. The adjustment of the ampli er is usually carried
out empirically.
The lines of demarcation between Class A, B and C ampli ers are not
very clear, but it is usual to distinguish the three types by the amount
of the grid bias. In Class A operation the bias is of the order of half the
cut off value, in Class B the bias is about cut off, and in Class C much
more than cut off. The elds of use overlap to some extent. Class A
ampli ers are used at any frequencies with fairly small input signals.
Class B and Class C are both used as tuned ampli ers at high frequencies.
Class B ampli cation is also possible at audio frequencies provided the
push pull circuit is used.
 CHAPTER 9

TRANSISTOR AMPLIFIERS

9.1. Transistor Characteristics

The characteristics and some of the main features of transistors are
described in Chapter 6. In some respects the transistor is analogous to
a triode, and the emitter, base and collector correspond respectively to
the cathode, grid and anode. In triode ampli ers there are three main
types of circuit which are used; these are the conventional (or common
cathode) ampli er, the common grid ampli er and the cathode follower
(or common anode) ampli er. Similarly, there are three useful types
of transistor ampli er—common emitter, common base and common
collector. All three types are considered in this chapter. One important
difference between triodes and transistors is that the latter always have
some base current, whereas in triodes, grid current is frequently negligible.

[E [c R is /c R
+ + + +
+
"es "cs v"’_ Yea Vcs

“E, I 'l'E, us,‘ _ "E,

(0) (0)
Fro. 9.1

This necessitates a slightly different procedure in establishing the perform

ance of a transistor ampli er from the characteristics and the circuits.
In the following sections we consider mainly junction type transistors.
For these we show in Section 6.14 that the main factor controlling the
transistor currents is '05 B, the driving voltage. Changes in v¢ B and vgc
are relatively unimportant. As is done with triodes and other valves,
we assume that the characteristics may be used to represent instantaneous
values of the transistor currents and voltages. Thus we neglect the
effects of diffusion rate or transit time of carriers. In practice, these
effects may become important at lower frequencies in transistors than in
vacuum valves (see Section 9.8).

9.2. Common base Ampli er

The circuit of a common base ampli er with a p—n p transistor is
shown in Fig. 9.l.a, where R is the load resistance and E1 and E2 are
batteries. Then vEB=E1 and v¢B= —E2— RI'¢. In Fig. 9.l.b a
138
CH. 9] TRANSISTOR AMPLIFIERS 139
signal v,;,, either d.c. or instantaneous a.c., is shown in the input circuit.
This is the driving voltage, and it causes a change i, in the emitter current
and an almost equal but opposite change 1', in the collector current.
Consequently, there is a voltage change across R and an equal but
opposite change vd, in the collector base voltage, i.e., vcb = — RI}. The
voltage ampli cation is given by
A = 11,1,/11,1, = — RI’,/v,b.
Thus, as long as R > [11,],/I',,|, the stage gain is greater than unity. Since

"I :31 \v¢B

o
_E _
"I 2 1,;
V 3/)’/””" ... L. L I
E '° 13* .. o 0,.
1 I \4 cs
INPUT
’I>
O_
lg 0
(0) 9
V .
CB N __ II *2 ‘E ‘on ____ E9
CL 2 1Q [E 13 IQ

3 9 ourpur OUTPUT
a

'2.>L"“' 1 F’ (C)
Ins‘; " INPUT
5 4 Fro. 9.2

V1 ‘HEY 5.1”’C o
' (6)
2',, is approximately equal to — I}, the condition becomes R > |v,¢,/I',|,
which is usually of the order of 10 to 100 ohms, so that voltage ampli cation
is readily obtained in the common base circuit. Since 11,1,/2', varies with
the signal magnitude, A also varies with the signal.
When, as in Fig. 9.1, the values of E1, E2 and R are given, the actual
operating point and voltage gain may be found accurately by the following
procedure. A start is made from ‘UE3, IE characteristics with 11¢, as
parameter (see Fig. 9.2.a). From a constant E1 line, relations are estab
lished between 1105 and ig, and they are plotted in a 22¢ B, IE diagram in
Fig. 9.2.b, in which the numbers indicate corresponding points. In order
to satisfy the input circuit conditions the value of 'u¢ B must lie some
where on this line. Its actual value is uniquely determined by the con
ditions in the output circuit. In Fig. 9.2.c I} , zI¢ B characteristics are
140 PRINCIPLES OF ELECTRONICS [C1 1.
drawn for various values of 1'5. Using the output circuit relation,
1'¢ B = E2 — RIC, we can draw the load line as shown. The output relation
between 11¢ B and 1'3 is determined, and this is also plotted in Fig. 9.2.b.
The intersection of the two curves gives the operating point Q. The
operating value of 1}; is calculated from the Load Line Equation. When
the signal 11,1, is applied the conditions change in the input circuit, but the
output relation between 1.1 5 and 1}; is unchanged. The new conditions
are found at P as shown, and the output voltage vd, is determined. In
this case 11,, and 11,1, are both positive quantities and the output voltage
is in phase with the signal. The voltage gain is given by A, = vd,/1.I,;,.
The application of the signal
11,1, changes 1'5, and this means
+ that there is a nite input re
‘, sistance for the ampli er. Its
' _ R value is given by 1', = v,b/1}.
This may be of the order of 50
Er E2 Q or less. We have already
F1G_ 93 seen that the common base
ampli er may be compared to
the common grid triode ampli er, which also has a low input resistance
and can give high voltage gain.
The common base ampli er may also give appreciable power gain.
The ratio of output to input power is v,;,1',,/v¢1',. Thus the power gain A1,
is given by
A g vai. v.I’_ r1_
P U¢(,i¢ 12,52 R

The numerical value of the power gain is normally slightly less than
the voltage gain, since [1],/1',| is less than unity. Note that the power
gain in a Class A common cathode ampli er is in nite, since 1'g is zero.
In the common grid case the power and voltage gains are equal.
The horizontal nature of the 1'1, , 11¢; characteristics is maintained down
to zero voltage. Thus the output current and output voltage in a tran
sistor power ampli er may both vary down to zero. The value of the
power output, ignoring distortion, is given by
P0 = (vmax "' vmin) (imax "" 8I

as in the triode or pentode. The transistor may therefore approach the

maximum theoretical e iciency of 25 per cent when the load resistance
is in the collector lead. If the load is transformer coupled, as shown in
Fig. 9.3, the limiting efficiency of 50 per cent may be approached. Of
course, these ef ciencies apply only to the collector circuit. Also, because
of the nature of the characteristics, it is possible to design a push pull
ampli er with negligible quiescent current. Thus a Class B transistor
audio ampli er consumes negligible power in the absence of a signal.
The graphical method in Fig. 9.2 of nding the output voltage is not
9] TRANSISTOR AMPLIFIERS 141
limited as regards signal amplitude. Provided the appropriate charac
teristics are available, the method may be used for accurate determination
of the output voltage for any given signal. Several lines for equally
spaced values of 1'55 are drawn in an 1'; , 1'05 plot in Fig. 9.4.a. Also in
this diagram we plot, from Fig. 9.4.b, the output relations between 1'5 and
‘U33 for three values of the load resistance. The intercepts on any one
output relation indicate the degree of distortion obtained with the corre
sponding load resistance. This distortion arises from the non linear
relationship between 1', and v,,,. As can be seen from Fig. 9.4.a, lines of

V . '
CB‘ ‘E _E2 ‘lc
0 .
O R2 Q Q _'
I "cs
RI
R3 R3

[E R
513%»: 2
EI*2I1»b1 ,
E1‘ ‘$01 R1
E, _
(0') (b)
Fro. 9.4

equally spaced 1'3 would give very little distortion (except where Ugg
approaches zero). A linear relationship between 1', and the signal voltage
can be realized by placing in series with the input circuit a resistance
large in comparison with the transistor input resistance.

9.3. Small signal Theory oi Common base Ampli er

If the applied signal is limited to small amplitude, the transistor
equations derived in Section 6.16 may be used, i.e.,
1'11 = hrui} + 111211111
and ic = haul} + h22o1'a>
The load line relation for changes gives 11,1, = — R1}. From these
equations it can be shown that

véb hm + R(h11bh22b — h21bh12b)

_ _ veb _ h1uh»z11R _
and 7;—Z—h1u—
142 PRINCIPLES OF ELECTRONICS [cI~r.
Using the approximate relations of Section 6.16, we may put
hm = hm = 0 and — 1121,, = ac, =' 1, and hence
A " ‘—" R111/hrrb = R/hm
and 7; '3 hub.

These results may be compared with the corresponding expressions for

a common grid pentode ampli er (see Sections 7.12 and 10.9):
A = — R(§»» + 1/'1)/(1 + R/'1) = — R81»
and r, = (1 + R/r,)/(g,,, + 1/1,) =1/g,,,.

9.4. Common emitter Ampli er

The circuit of a common emitter ampli er using a p 11 1: junction
transistor is shown in Fig. 9.5. The input circuit is as that for the common

R
+ lc +
"cs

+
Vac _.Q
a (E
Yes

E1 IE;
Frc. 9.5

base ampli er except for inversion of the terminals, but the output voltage
is obtained between the collector and the emitter. The driving voltage
vb, produces similar changes 1', and 1', as before, and the output voltage
is given by 11,, = — R1}. Thus the voltage ampli cation differs very
little from that obtained with the common base circuit. The main
difference between the two ampli ers lies in the value of the input resist
ance. In this case it is given by
T; = ‘U5;/1:5.
in \
Since |1';,| is much less than |1',|, the common lmeampli er has a higher
input resistance, usually 10 to 100 times greater. As a result, the input
circuit takes less power from the signal, and common emitter ampli ers
give higher power gain. Also, since |1',,| is greater than |1',,|, the common
emitter circuit may be said to act as a current ampli er.
The graphical determination of the operating point and the output
voltage is carried out in a similar manner to that used for the common
base. In this case the characteristics used are 1135 , 1', with 11¢ E as para
9] TRANSISTOR AMPLIFIERS 143
meter and ic, v1; E with 1'5 as parameter, together with the Load Line
Equation veg = — E2 — R1'¢. The operating point is nally found from
plots of 1'B and v¢E (see Fig. 9.6).
When small signal theory is applied to the common emitter ampli er
we may use the parameters hm, hm, I121, and h22,, which were de ned in
Section 6.15. Then
A vdt __ h21¢R

‘"11. hm 1 R(h11¢h22¢ — /121. 11121)

AU — ITd¢

hlle
_ 91¢ _ h12¢h21¢R N
and 7;’ — — hlle — — h11¢.

._E .
AVBE jay‘
o I o
is '0 ":5
._ _ .. _ E,+vb¢ [B P
V
~CE\_O_ __ _ _ _ _ _ _ _.E1. Q
INPUT

:o\5n
OUTPUT '

. I .<> _
OUTPUT ' "cs
_E1+ vb c P

INPUT

_ E1
Q

Fro. 9.6

Using the approximate relations between the common base, common

emitter parameters given in Section 6.15 it follows that
Roz“
/1z__i
hm
and 7. _r=_v ii.
‘ 1 W ace

The voltage gain is approximately the same as for the common base
ampli er, but the input resistance is considerably greater.
144 PRINCIPLES OF ELECTRONICS [CI I.

9.5. Common collector Circuit

The common collector ampli er, which is illustrated in Fig. 9.7, has
the input signal v, connected between base and collector, and the output
circuit is between emitter and collector. The transistor ampli es the

R
+ 0

ls
Yes
+ 1; g "ac
Ii Q I
c

E, E,
Frc. 9.7

driving voltage v,I, between its emitter and base. In this circuit
v,;, = v,, + v,,, = — R1', — v,. The output voltage — R1, is normally much
greater than the driving voltage v,¢,. Thus v, and —R1', are approximately
equal. The voltage ampli cation A = v,,/vb, is slightly less than
unity. The conditions are very similar to those of the cathode follower,
which is described in Sections 7.12, 7.13 and 10.10. Both ampli ers
have high input impedance and low output impedance.
It may be shown that
l—h12,,~ 1 N
A 1—h11I>/R 1 1.../R‘ 1'
E 4', 1, C E C
+ ‘D H <I> 4 I’ [0 +
I10
V00 Vcb Vet ‘I’ Vcb
_ _ 512 elm ‘ _
B B
Fro. 9.8

9.6. Transistor Equivalent Circuits

The small signal transistor equations
veb = 711111} + h12bveb and 51 = 71211151 + 112211111
may be used to establish transistor equivalent circuits. It may be veri ed
that the circuit shown in Fig. 9.8 gives the same two equations for v,,,
and 1',. In this circuit hmvd, is a voltage generator and h21,,1', a current
generator. The voltage generator has series resistance hm and the current
generator parallel conductance hm. The equivalent circuit may be
9] TRANSISTOR AMPLIFIERS 145
used to replace the transistor in actual circuits provided the changes of
currents and voltages are suf ciently small.
Just as there are many ways of expressing transistor characteristics,
so there are many different types of equivalent circuit. Some of these
are given in Exx. IX.

9.7. Biasing Circuits

Separate battery supplies have been shown so far for the input and
output circuits of transistor ampli ers. In practice, only one supply is
used whenever possible. The circuit of Fig. 9.9.11 shows one method of
R, 1, I1.

,C . 1. °
'<=
Q < _3 (AI>PIIox.)
0—| 4, g YCE TE: i E2 INPUT Rs

Vs: R
_
o ' (6) j la_ (c )
52 '
(0) O was
INPUT
.5_=.
Rs
\o
"cs

(<1)
Fro. 9.9
biasing the base of a common emitter ampli er. The battery E, is
joined to the collector through the load resistance R as usual. The same
battery is connected to the base through the resistance Re, which controls
the quiescent value of the base current and consequently the collector
current. In the collector emitter circuit there are two separate relations
between collector current and collector voltage. These are the Load
Line Equation
Ugg = — E2 —

and the Transistor Equation

ic =f4(is. 1102)
These relations are plotted in Fig. 9.9.b, and from them we get the dynamic
relation between 1'5 and vee for the output circuit; this is shown in Fig.
9.9.0. There are also two relations for the base emitter circuit:
U33 = — E2 — Rgig

and U33 =f3(i3, Ugg).

These are shown in Fig. 9.9.d, and from them we plot in Fig. 9.9.1: the
146 PRINCIPLES OF ELECTRONICS [cH.
input circuit relation between 1'1, and vee. The point Q in this gure gives
the quiescent conditions. In practice, vee is usually much smaller than
E2 in magnitude, so that
1'B z E2/RB.
Thus 1'5 is xed for a given circuit and is independent of the transistor.
This circuit is therefore said to have xed bias. With a given transistor

Ii 0
"E2 VI C__ "IT
o ' O ‘ e Z
0'
I0‘In

[B Q1 is o

Q2
I
n\'5"
_____J
(<1) ' (1)

\
O‘In

Q1 O3
In

go\"U
ls o

I (~=)
.

Q. (<1) '
FIG. 9.10

the relation between 1'e and vee at zero base current may vary appreciably
with ambient temperature, so that
7:0 '3 acbi + 7:0

where 01,1, remains almost constant but 1', is temperature dependent.

The effect of this variation on the operating point is shown in Fig. 9.l0.a
to d, in which R, RB and E2 are assumed constant. In case (c) it is seen
that the bias would be unsuitable for ampli cation.
An altemative bias circuit, which is less sensitive to these variations
in the characteristics, is shown in Fig. 9.1l.a. The base is fed from the
collector through the resistance RF. The base current now depends on
9] TRANSISTOR AMPLIFIERS 147
the value of the collector emitter voltage, which in tum depends on the
collector current. This is an automatic bias circuit which operates in
such a manner that any tendency for 1'e to change is opposed by a change
of bias. In most cases 1'3 is much less than 1'e, so that the dynamic
relation between 1'5 and vee for the output circuit can be determined in

RF l.¢"'l.g R

is lic *
O l + "ca
Vs: ‘E2
Q i. ‘ '.

(0)

E jlc Isl
2 Y1 V1
O O VCE O Ye:
[B OUTPUT OUTPUT
. V
Q

T’ R
I

11>) (=1 '

Frc. 9.11

the same manner as in the xed bias circuit (Fig. 9.11.b and c). From
the circuit we also see that

is = ("cs — 9511)/R1»
Usually |vBe| < |vee|, and then
is = 1102/RF
This load line is drawn in Fig. 9.11.0, and the quiescent point is Q. When
the transistor characteristics vary with temperature, Fig. 9.12 shows how
the automatic bias circuit behaves. The variation in the operating point
is much less than in Fig. 9.10. With the circuit of Fig. 9.11.11 a.c. variations
of collector—emitter voltage are transmitted to the base emitter circuit.
In order to prevent this a decoupling circuit is used, as shown in Fig. 9.13.
The bias resistance is divided into two parts, and Rm and C act as an
a.c. lter.
The stabilizing effect of the automatic bias circuit on the Q point
148 PRINCIPLES OF ELECTRONICS [CH.
depends on the load resistance R being suf ciently large for vee to vary
appreciably with collector current. With a transformer coupled load
the d.c. resistance is negligible. Automatic bias may be obtained in
this case with a resistance Re in the emitter lead, as SIIOVITI in Fig. 9.14.a.
Although the bias voltage now depends on the collector current, its
polarity is incorrect and it is necessary to provide a counteracting xed
bias by means of resistances R1 and R2. The Q point can now be found

I1 I1
'_E2 VI I i_ _ _f'2__ _ (L
0 ' O "cs O "cs

1.B Q1 ' [B O

Q2
I
:o\'~"“
_____J~
U

&

T
I CI VI V2 Va
“'11? O "cs ' O vce

Q1
Iso Q3

Fro. 9.12

as follows. Since 1'e> 1'8 in magnitude, the dynamic relation between

vee and 1'B can be found from the collector characteristics and the Load
Line Equation
"cs = — E2 R140.
as shown in Fig. 9.l4.b and c. The voltages across R1 and R, are of the
same order as E2, and normally E, > |ve;;|. Hence
R,;1'e =' R11, and as 1'3 ' ' — 1'e
then R111 = — Re1'e.
9] TRANSISTOR AMPLIFIERS 149
On substituting in the Load Line Equation we nd
"1 = (E2 + '”CE)/RI
From the circuit it follows that
E2 _' R111 + R2("B _ 1'1) =
and hence 1'1 = °m.."“——
‘N
++ . ==I~5’
Rn RF2 R

°—|
c E1
Fro. 9.13
The two expressions for 1'1 lead to a second relation between ve5 and 1'5,
— E2R, RIR2 .
"°” R. + R. + R. + R.“
This load line is also shown in Fig. 9.l4.c, and its intersection with the
previous ve5, 1'5 curve gives the Q point. To prevent the bias voltage

ic
°_| + $ I

I is II)
R

_ lg E2

(<7)
. "ER 1'
“C R1 2
'_2'I'_R2 ‘BE:
E. 1 1, r
I ' /R. >
° °"cIs O Vce
Q ourpur Q
1'5
ourpur E2
____
Pa

Kb) Fro. 9.14

(¢)
F
150 PRINCIPLES OF ELECTRONICS [cH.9
containing a.c. components a capacitor C5 is connected in parallel
with R5.
Although this section has dealt entirely with the common emitter
ampli er, the biasing circuits for the other transistor ampli ers are
similar in principle.
9.8. Transistor Ampli ers at High Frequencies
When transistors are operated at suf ciently high frequencies their
inherent reactances become appreciable just as in triodes or other electronic
devices. The effective capacitances across the p—11 junctions, i.e., at
the boundaries of the electrodes, must be taken into account.

is ic

O t O t

Fro. 9.15

Transistors also show effects due to the time of diffusion of the carriers
from the emitter to the collector (holes in the j> 11¢ case). This time is
not the same for all carriers owing to differences in diffusion path lengths
for individual carriers. Thus if the emitter current is a square pulse,
the collector current pulse is delayed and distorted as shown in Fig. 19.15.
For a.c. signals the effect of capacitances and of diffusion can be ex
pressed in terms of variation of magnitude and phase angle of the tran
sistor parameters. In particular, 11,, falls with increasing frequency.
 CHAPTER 10

FEEDBACK
10.1. Feedback
In conventional ampli ers the output is much greater than the input
signal which is connected between the grid and cathode of the rst valve.
A small part of this output may be transferred back to the input and put
in series with the signal across the grid and cathode terminals. The
effect on the ampli er then depends on the phase relation between the
fed back voltage and the signal. If the fed back voltage is such that
the grid cathode voltage exceeds the signal, higher output and gain are
obtained; altematively, the signal may be reduced to give the same
output as is obtained with no feedback. With these conditions it is
said that positive feedback has been introduced. If the voltage fed back
is exactly equal to and of the same phase as the grid cathode voltage,
then the signal may be reduced to zero, and the output is the same as
without feedback. Such an ampli er is called a self oscillator, as it
provides an output with no extemal signal. If the fed back voltage is
such that the grid cathode voltage is smaller than the signal, the output
voltage drops "and the gain is reduced. In this case there is negative
feedback. In all cases it may be noted that the output of the ampli er
depends only on the grid cathode voltage. The valve ampli es the
voltage appearing between its grid and cathode irrespective of how that
voltage is produced. The inherent gain of the ampli er, i.e., v,/v,, is
independent of feedback; it is v,/v, which is affected.
At rst sight it might appear that negative feedback is undesirable,
since it reduces the output for a given signal input. However, it may
also have certain very desirable features which more than offset the
loss of gain. Some of these features are: (i) greater stability against
supply variations, (ii) independence of changes in valves, (iii) reduction
in noise such as hum, (iv) reduction of frequency and phase distortion,
(v) reduction in non linear distortion, and (vi) possibility of achieving
some particular frequency response. It is the main purpose of this
chapter to study the effects of negative feedback. Positive feedback
in oscillators is dealt with in Chapter 13.

10.2. Automatic Bias

In valve ampli ers the grid bias can be obtained automatically by
putting a resistance in series with the cathode lead. We are now going to
examine further some of the effects of automatic bias. In the rst place
we consider the circuit of Fig. l0.1.a with no grid battery and with the
input terrninals short circuited. The output from such a circuit is normally
151
152 PRINCIPLES OF ELECTRONICS [cH.
taken from the terminals AB. It may be seen that ve = — R515 and
the output voltage is ve = v5 + R515. Thus the grid voltage is part of
the output voltage and we have a case of feedback. It is obvious that
this circuit helps to stabilize the anode current against changes of the
V . .

A R A.>\"" E 2
""
1, + '0 $R,, "A: E1'(R*RI<) /A
Q VG*'RKlA

+
I16 "T vo _ 1°, ' """"""""""" " '6 O
_ E
2 7 _ _
,
l°__ _ _________
Q ‘T O

‘SI
RX V1 23 :
I J . I . I_,
B "co ,° Vs ° '1 '0 E '10 ‘K
2
(0) (6) (¢)
Fro. 10.1

supply voltage E2. If, for example, E2 increases, then 1'5 increases, and
so does the grid bias, thus o setting to some extent the original change.
The method of determining the Q point is shown in Fig. l0.1.b and c.
Besides being related by the grid characterstics, 1'5 and ve satisfy the
equation ve = — R515. This is a grid circuit load line, and it is drawn
in Fig. 10.1.b. The grid characteristics are drawn as parallel straight lines
for convenience. The points where the grid load line cuts each character
istic are then transferred to the accompanying 1'5, v5 diagram (Fig.

' \vIrI I
AUTOMATIC I
_ _ J"~::::::.::::
°"‘ I'll?
' °=_ ‘~....°
>1

V62 V61 ‘O vs v E2 E2, VA

1 12 “A [A ‘ 11)
WITH Q
AUTOMATIC 2 WITH

: 1 .''''''' ’.I'...'#.‘!_'_‘°:TA§_'I'_’_';'IQL : I _P° alga

I

I °v° '°" "as

Q ‘
_
ax Q\ (C) . (d) 22"A
FIG. 10.2
im FEEDBACK :wa
l0.l.c). Between £4 and v4 there is also the anode load line relation
v4 = E2 — (R + RA )i4. This, too, is plotted on the i_4, v4 diagram.
The intersection of the two lines gives the Q point for the given values of
E2, R and RK.
If the supply voltage E2 changes, the effect may be determined as shown
in Fig. l0.2.a and b. The anode current and anode voltage both increase
when E, changes from E, to E2’. At the same time the bias voltage in
increases and z',, the change in anode current, is smaller than it would
have been if a xed battery supply had been used for bias. The change
that would have occurred with xed bias may be found from the anode
R ..

1; *

Q '~
~'¢ ml = E=
+ ,., ""

" ‘*1 E2
Rx RX [A |
T“ Q _A

(<1)
U); [Al

yG="RKl.A E2

R+RK VA: E2_ (R+RK) [A

Q
"" " [Q [Q " Q

by 1 no I | |

O‘
_ ._ 0 0 0 o"“'“ E2 A
/b) (< ‘)
FIG. 10.3

characteristic through Q1 as shown. The greater the value of R5, the

smaller the change in 1'4 for a given change in E2. This is shown in Fig.
10.2.0 and d, where the value of RK has been doubled.
For a pentode the £4, vq characteristic is practically independent of
v4, except at very low values of v4. The transfer from the £4, Ug plot to
2'4, v4 is therefore a horizontal straight line (see Fig. 10.3). The Q point
is found as before. Since the grid load line gives a horizontal line in the
£4, 04 diagram, it might appear that the cathode resistor does not con
154 PRINCIPLES OF ELECTRONICS [CI I.
tribute to stabilization of 2',, against supply voltage changes. However,
the screen electrode in a pentode corresponds to the anode of a triode,
and hence the bias resistor stabilizes £4 against changes in the screen
voltage.

10.3. Automatic Bias and Signal Feedback

The effect of automatic bias on a small signal may be determined with
reference to Fig. 10.4.a, which shows the circuit of a pentode ampli er
and the total currents and voltages at various points in the circuit. As

R R
1', + 1,, + +

+ I
. "" F °
5 Q
.. RR
0on+

R; [A
,3 0*
_ RK Rgiq
+

(<1) (b)
Fro. 10.4

we are considering the effect of the signal, we can redraw the circuit as
in Fig. l0.4.b, which shows only the varying components of the currents
and voltages. The input to this ampli er is v, and the output v,,, so
that the voltage ampli cation is given by A1 = v,/v,. It may be found
from the gure that v, = — Rid, v,, = — (R + RK)z'a and v, = v, — RK1}.
Since we are dealing with small changes, the Valve Equation can be used,
giving 2',, = gmv, + v,/r,,.
in = §m('”e — Rxia) '_ (R + RK)ia/'4.
ie. 1'. = gm”:/{1 + gmRK + (R + Rr)/'..}
A! = ' Ric/‘U 1 = _ §mR/{I "l" §mR£ + (R + RK)/'a}
With a pentode 1,, > R + RE, and then
A1 = 8mR/(1 + é’mRx)
There are several interesting points which may be made about this
circuit. Firstly, this is a case of negative feedback. The voltage
appearing between grid and cathode, when the signal is applied, is not v,
but v, reduced by R51}. The reduction is proportional to the output
voltage —Rz', ,. The fraction of the output voltage fed back to the input
is R;/R. There is also a reduction in the voltage ampli cation brought
about by the presence of RK. If the bias had been obtained from a
10] FEEDBACK 155
battery, then the voltage ampli cation would have been A = v,,/v, = — g,,,R,
as is shown in Section 7.3. We thus nd that the voltage ampli cation
with feedback is given by
A
AI = $ /T.‘
'_ T
If the inherent gain A of the ampli er is su iciently great for ARK/R > l
numerically, then the gain with feedback is A;= — R/R5. Thus the
ampli er gain depends only on the ratio of two circuit components and is
independent of the valve constants. The valve may change with age or
be replaced, or the power supply voltage may vary, but as long as |A| is
suf ciently great, the ampli er gain is unaffected. This most desirable
effect is obtained at the cost of considerable loss of gain. If, for example,
R = 20,000 Q and RK = 2,000 Q, then A; = — 10. If gm = 10 mA/V
then A = — 200; ARK/R = — 20, so that the condition for stability is
satis ed fairly well, but the gain is reduced from 200 to 10.

10.4. Cathode Bias Condenser

When automatic bias is introduced in Section 7.11 in connection with
a.c. ampli ers it is stated that the cathode resistor RK is shunted by a
condenser CK whose reactance is much less than R; at the operating

R
4;, +

V‘: Q . our E2
aslnol CK

(<1)

|'°/tl gt“
‘MK v, v,,= R (1, IQ)
(High Freq.)
v =E, Q
"‘(R*R|§)l'A
(d.c.) v

O I ‘O > £2 A
R I
Fro. 10.5
156 PRINCIPLES OF ELECTRONICS [CI I.
frequency. Under these conditions there is no appreciable a.c. voltage
drop across RK (see Fig. l0.5.a). The ampli er behaves as though it had
xed bias equal to — Rgiq, and there are no negative feedback effects
on the signal. However, the feedback still operates for d.c. changes,
and the Q point must be determined by the method described in Section
10.2. Also, there is feedback at low frequencies where 1/(DCK is not much
less than RK. The voltage ampli cation varies with frequency, as shown
in Fig. l0.5.b. The anode load line varies from 1:4 = E, — (R + RK)i_,
with d.c. to v4 = E2 — Rgiq — R2) at high frequencies (see Fig. 10.5.c).
The Q point is the same in both cases.

10.5. Feedback—General Considerations

In Fig. l0.6.a a conventional ampli er is shown diagrammatically with
input terminals v, and output terminals v,,. The ampli er may be of any
type with one or more stages. A signal v, is shown connected to the input
terminals. In Fig. 10.6.b the ampli er has part of the output fed back
to the input through a feedback network. The input to the ampli er
now consists of the signal v, and the feedback voltage (iv, in series, so
that v, = v, + B0,. The signs in this diagram de ne the polarities of
the various voltages. In the analysis of the network it is convenient to
assume a sinuosoidal a.c. signal and to use vector quantities. The phase
of V, depends on any phase changes introduced in the ampli er. The
feedback network may also introduce phase change so that a phase
angle must be assigned to (3. We de ne Ag and A to be the voltage
ampli cations for the ampli er with and without feedback where
A1 Z vg/‘V. and A 1 " V9/vi.

Ag and A are also vector quantities. Since V; = V, + [3V,,

A, = v./(v. av.) 1 _v‘;,/‘éjfvg A A/<1 EBA)

Positive or negative feedback is then de ned according as |A;/A] is greater
or less than unity
i.e., |1 — [BA] < 1 gives positive feedback
and |l — [SA] > l gives negative feedback.
These conditions are illustrated in Fig. 10.6.c and d. In these gures
OB, OD and DB are vectors representing the grid voltage, the feedback
voltage and the signal voltage respectively. Fig. 10.6.0 represents
negative feedback, since DB is greater than OB; on the other hand,
Fig. l0.6.d represents positive feedback. It is obvious that positive or
negative feedback occurs according as the point D lies inside or outside
the circle with its centre at B and its radius equal to OB.
When all the reactances in the ampli er and feedback circuits are
10] FEEDBACK 157
negligible in comparison with the resistances, then the voltages at various
points in the circuits are either in phase or differ by 180°. A vector
diagram showing negative feedback in this case is drawn in Fig. 10.6.e.

+0,5 9 AMPLIFIER gs

(0)
1
QI I
OQ
‘++
if, V +
113+

+_

aw
004
A V
p 9) FEEDBACK
uerwoax

(b)
0

“V” Q

(s) (#1

o_—"—_
Apvg
———io—
Q V9

(¢)
a
V

FIG. 10.6
D
(I)
®
G”!
In Section 10.3 we consider a simple ampli er of this type in which
[3= RK/R. However, in any ampli er reactances become effective at
some frequencies, and then the feedback conditions change. We have
one example of this in the cathode bias circuit with a condenser in Section
158 PRINCIPLES OF ELECTRONICS [C1 I.
10.4. At the higher frequencies the feedback is zero, and the ampli er
gain is independent of frequency (see Fig. 10.5.b). With d.c. and at very
low frequencies there is feedback due to RK, and the gain is constant
but at a lower level. In the intervening range the feedback and the gain
vary with frequency. At the same time the phase angle of the voltage
across RK is no longer 180° relative to V,, but varies from 180° to 90° as
the frequency rises. When the phase angle is 90°, the amplitude of B is
zero and the feedback is negligible. The vector diagram in this case is
shown in Fig. 10.6.f. As the frequency rises from zero the point D
moves round the curve D D1 D2 0.

10.6. E ect of Feedback on Non linear Distortion

When an ampli er with a curved dynamic characteristic is used with
large values of 11,, non linear distortion occurs. This distortion arises
entirely from the nature of the valve characteristic and the size of 11,.
For a given output the size of v, is xed and is given by v,/A, where A is
the inherent voltage ampli cation of the valve ampli er with resistance load.
None of these factors is affected by feedback, and so feedback does not affect
the amount of non linear distortion produced by the valve. However, the
amount of distortion in the output may be reduced by negative feedback
provided su icient signal amplitude is available. This may be explained
as follows. If v, is the fundamental output required from the valve,
then v, = v,/A. If there is no feedback 0, = v,. We assume that
under these conditions a harmonic of voltage vg is produced in the output.
Now let us apply negative feedback to the ampli er and at the same time
increase v, to give the same fundamental output '0, as before. The valve
has the sa.me '0, as before and produces the same distortion. However,
the harmonic voltage is also subject to feedback, so that the total
harmonic content in the output is due to the part produced by the valve
and the part fed back and re ampli ed. If v3’ is the total harmonic
voltage in the output with feedback, then Bug’ is fed back and ampli ed
A times. Thus
‘U3’ = ‘U3 + /1B‘Ug'.

113' = ‘UH/(1 —'

The harmonic distortion may therefore be reduced to a small value if

the feedback is negative and the magnitude of AB is suf ciently great.
It is assumed above that the reactive components of the feedback
ampli er have no effect and that the voltage ampli cation is the same
for the fundamental and the harmonic. We have also ignored the effect
of the non linearity of the ampli er on the harmonic component which is
fed back. This is obviously a second order effect.
The reduction in non linear distortion with feedback requires a
greater signal v,. However, this may be obtained from a voltage ampli
er operating at a level at which little distortion occurs.
10] FEEDBACK 159
10.7. E ect of Feedback on Frequency Distortion and Noise
If an ampli er has an excessive gain at some particular frequency, then
there is a greater voltage fed back at this frequency, and this offsets the
gain to some extent. The quantitative effect of negative feedback on
frequency distortion can be found readily from the feedback equation.
If An and A1 are the voltage ampli cation with and without feedback
at one frequency, and A12 and A2 the corresponding values at a second
frequency, then
An = A1/(1 @151) and A12 = A2(1 ' f32A2).
Au/A22 = A1(1 9252)/A2(1 f31A1)
If, with negative feedback, the magnitudes of [31A1 and [52A2 are
suf ciently great compared with unity, then, provided [51 = [32, Au = A3.
Negative feedback may improve the signal to noise ratio (S/N) of an
ampli er if the noise is produced in the ampli er. For example, if hum
is introduced in one of the stages of a multi stage ampli er, then negative
feedback may be used to reduce the amount of hum in the output.
This feedback is applied between the output and a point just before the
origin of the hum. The hum is then reduced in the same way as harmonic
distortion. The gain of the earlier stages of the ampli er is now increased
to restore the output voltage of the whole ampli er to the original level,
using the same signal amplitude as before. The value of S/N at the out
put stage is improved. This effect on noise assumes that the extra gain
in the early stages may be obtained without introducing comparable noise.
If feedback is introduced in the early stages of a high gain ampli er, S/N
may be reduced if more valves have to be used to obtain the required gain;
this increases the proportion of noise due to the valves (see Chapter 20).

10.8. Current and Voltage Feedback

In the feedback circuit with a cathode resistor the voltage fed back
to the input is proportional to the output current. This is an example

R R
10 + i” +

QC, yo +
. Q R. ,
7.‘ O I
+
'
3 *0“ 9

+ _ — 1

4————§

L _ .

Iv) (b)
Fro. 10.7
160 PRINCIPLES OF ELECTRONICS [cn.
of a single valve ampli er with current negative feedback (Fig. 10.7.a).
An alternative circuit is shown in Fig. 10.7.b. Here Bv, = v,R2/(R1 + R2),
and the fed back voltage is proportional to the output voltage. This is a
single valve ampli er with voltage negative feedback. To check the

~his 7
0
4'

Q Q
(<1)

. Q '1' Q
+ '1.
"O *
' V91
nunin $5 ‘?
Eli
(b) __ _
R

‘S*O
,
1%
Fro. 10.8

nature of the feedback in resistance loaded ampli ers it is convenient

to assume that the effects of the reactances are negligible. Then the cur
rents and voltages throughout the circuit are either in phase or in anti
phase with the signal. It should be veri ed that all the circuits in Figs.
10.7 and 10.8 give negative feedback.
10] FEEDBACK 101
The common emitter transistor ampli er in Fig. 9.14.a has current
negative feedback when the capacitor CB is omitted. There is also some
feedback inherent in the transistor itself. This is measured in terms of
I112, = avg);/avgg.

10.9. Output and Input Impedance of Feedback Ampli ers

In any generator the output obtainable from it depends on the relative
values of the generator internal resistance and the load resistance. For
maximum power output the load resistance should equal the generator
resistance. For maximum voltage output the load resistance should
be much greater than the generator resistance. For a generator with low
internal impedance the value of the load impedance may vary considerably,
with frequency or otherwise, and the change in output voltage is small
as long as the load impedance is high compared with the generator im
pedance. On the other hand, if the generator impedance is much greater
than the load impedance the output current does not vary much when the
load varies. The intemal impedance of a generator is therefore an im
portant property. A conventional valve ampli er acts as a voltage
generator of internal resistance 1,, feeding the anode load, and the

R R

1 5 '3
1 Qv:
"9 Ra
‘S 9+ ‘S0*

.
+
O
\‘

(°) (4')
FIG. 10.9

ampli er is said to have an output impedance equal to 1,. We are now

going to see the effect of negative feedback on the ampli er internal
impedance or output impedance.
When the load impedance of an ampli er or any other generator
changes, the output voltage and output current both change. For
example, if the load impedance rises the output voltage increases and the
output current decreases. If this ampli er has voltage negative feed
back, then the voltage fed back to the input increases, the effective
voltage on the grid of the rst valve decreases and the output voltage
of the ampli er drops. Thus the output voltage tends to remain constant.
162 PRINCIPLES OF ELECTRONICS [CH.
This is equivalent to saying that an ampli er with voltage feedback
has a low internal impedance. The cathode follower is an example
(see Section 10.10).
If the load impedance rises in the case of an ampli er with current
negative feedback, then the decrease in output current brings about a
reduction in feedback which offsets the change in output current.
Thus with current feedback the ampli er tends to give constant current
output, i.e., it behaves as though it has a high internal impedance.
The input impedance of an ampli er, which is de ned in Section 7.13,
is another property of considerable importance, since it determines how
much the ampli er acts as a load on the source of the signal. If a signal
V, is connected to the input terminals and a current I, ows as shown in

2* 0 , 1= K4.

+ ° ,3
*1 !
. |<
K 11>)
(0)
FIG. 10.10

Fig. 10.9.a, then the ratio V,/I, is the input impedance. For a con
ventional ampli er of the type shown, the input impedance is R,. When
no resistance is connected in the circuit, R, is the leakage resistance
across the valve insulators. The effect of the inter electrode capacitances
is considered in Section 10.11.
In all the cases of negative feedback which we have considered so far
the feedback voltage has been connected in series with the input voltage
between the grid and cathode. This always results in an increase in the
input impedance of the ampli er. In the circuit in Fig. 10.9.b, |V;| is
less than |V,|, since the feedback is negative; I, is the current taken
from the generator and I, = V,/Rg. This is less than the current V,/R,
in the circuit without feedback, so that negative feedback increases the
input impedance.
It is possible to have feedback connected in parallel with the signal
voltage, and this may result in a decrease of input impedance. The
Miller Effect, described in Section 10.11, arises from this type of feedback.
Another example of parallel voltage feedback is shown in Fig. 10.10.
10] FEEDBACK 103
The impedances Z1 Z2, Z3 and Z4 are, in general, combinations of
resistance and capacitance; Z3 is the feedback impedance. The grid
voltage is derived from the signal through Z1 and also from the
output through Z3, so that the feedback and the signal are essentially
in parallel. In this case the value of (5 cannot be written down easily,
and it is necessary to proceed from the basic circuit equations. For
simplicity it is assumed that the impedances are purely resistive. If the
inherent gain of the valve is large, then approximately

I2 ' = ' v°}R3

A150 1, = (v, v,)/R1.

When there is appreciable negative feedback V, < V, and then
I1 1 V‘/R1.

If R2 is sufficiently large
I1= — I2.

Then
_m~_R
A1 — V l — F2‘

The feedback is negative and, as in previous feedback ampli ers, the

gain can be independent of the valve parameters. The input impedance
of the ampli er is approximately equal to R1, as compared with R1 + R,
when the feedback resistance R, is removed. The output impedance is
low, as it is a case of voltage feedback. The circuit of Fig. 10.10.a has
important applications in electronic computers (see Exx. XIX).
Another example of an ampli er with low input impedance is the
common grid ampli er, which is discussed in Section 7.12; it is shown in
Fig. 10.11.a and its equivalent circuit in Fig. 10.1l.b. This ampli er

Z A A
""1+ """" 1+
E= [A Z
2 G
. , 0 f
1,, + 1; 1, ° v,
+ +
V3 3 ya v8_Q .. y,

E1 _____ 1 _ , ____ __ 1
K K
(0) (b)
FIG. 10.11
164 PRINCIPLES OF ELECTRONICS [CH.
is one with feedback in which the whole of the anode current ows through
the signal circuit. Thus the input impedance is V,/I, = (r, + Z) /(11 + 1).
When the valve is a pentode and 1,, > |Z|, then the input impedance is
r,/(p. + 1), which is approximately equal to 1/gm. Since gm usually lies
between 1 and 10 mA/V, the input impedance of a common grid ampli er
is of the order of a few hundred ohms.
A further example of parallel negative feedback is the common
emitter transistor ampli er with automatic bias, which is given in Fig.
9.1l.a. The feedback occurs through the resistance RF.

10.10. The Cathode Follower

The cathode follower is discussed in Section 7.12 as a common anode
ampli er in which the signal is connected between grid and anode and the
output is taken between cathode and anode. The circuit is shown in

(‘A + £0

1D
)‘

+
‘0
G
'5 "ii ' v. 3+
'*
1. ~ 1; ,= 1 '0x
+
‘Q7: m

Rx Rx IA Rx 1%
0 '_ 0 "
E E
(0) (6)
+ I’ K L‘ + — —oG
.111, + lg, R,
‘H1 _ Rx V, : ;"' K
f .

13’; v,,=Av, RK
"1 1 A
E,A
(¢) (4)
Fro. 10.12
Fig. 10.12.a, and in its form for small changes in Fig. 10.12.b. The
cathode follower is a special case of a negative feedback ampli er in
which the whole of the output voltage is fed back to the input. It is a
voltage negative feedback ampli er in which [3 = — 1. In Fig. 10.12.b
the valve acts as a normal ampli er of its grid voltage 11, with RK as the
load resistance. As usual, v, = Avg, where A is large. Since v, = v, 11,,
it follows that v, and v, are nearly equal, and that v, and v, are
10] FEEDBACK 165
in phase relative to their common point E. Thus the cathode voltage is
in phase with the signal and is practically equal to it; hence the name
cathode follower. Since this is a case of voltage feedback, the cathode
follower has a low output resistance.
Quantitative values for the voltage ampli cation and the output
resistance may be established quite readily. The voltage ampli cation
follows directly from the feedback equation by putting B = — 1. Then
A , = A/(1 + A). This is always less than unity when A is positive, as
it is when RK is purely resistive. The value of A may be found from the
formula derived in Section 7.3 by putting RK for R. Then

vo/"0 = A = P Rx/(Rx + 70)

The positive sign is taken in this formula, since the polarity of the output
voltage is reversed when the load is in the cathode lead instead of its
usual place in the anode lead. Also, since v, = v, 11,, we get, after
substitution and rearrangement,
p. . 1,, .
We: R a+m7a~

In this equation the rst term is an e.m.f., the second is the voltage drop
due to the current 1', owing in the load resistance RK and the third is the
voltage drop in a resistance r,/(11. + 1) with the same current flowing
through it. The term RK1}, gives the output voltage of the cathode
follower. The same equation would have been obtained for a generator
of e.m.f. 1115/(P + 1) with intemal resistance 1,,/(p. + 1) feeding a load
RK with current 1],. Thus the cathode follower acts as though it were
a generator of internal resistance 1,/(11. 1 1) and e.m.f. p.'U_,/(p. + 1), as
shown in Fig. 10.12.c. Usually 11 > 1, particularly for pentodes, and
hence the e.m.f. is approximately equal to v, and the internal resistance
is 1,/p. or 1/g,,.. Thus the output impedance of the cathode follower is of
the order of 100 to 1,000 ohms. This low output impedance means
that the cathode follower can give an output voltage which is inde
pendent of the load impedance for wide variations of the latter. When a
reactive or low impedance load is to be coupled to a voltage ampli er a
cathode follower is sometimes put between the output of the ampli er
and the load.
It may easily be shown from Fig. 10.12.11 that the input resistance of
a cathode follower is approximately equal to R,(1 + A), where R, is the
resistance between the grid and the cathode.
Since v, is nearly equal to 11,, and the input resistance of the cathode
follower is very high, whilst the load resistance may be low, it follows that
the current or the power in the load may be much greater than the current
or the power of the signal. Thus, although it cannot give voltage ampli
cation, the cathode follower may, like the conventional ampli er, be
used for current or power ampli cation.
166 PRINCIPLES OF ELECTRONICS [C1 1.

10.11. The Miller E ect

In a valve there is some capacitance between the anode and grid elec
trodes. In a conventional common cathode ampli er this provides
coupling or feedback between the output and input circuits. This is a
case where the feedback is in parallel with the signal. The phase of the
feedback depends on the phase difference between the input and output
circuits. For an ampli er with resistance load of the type shown in
Fig. 10.13.a the phase difference is very nearly 180°. When the current
through C“ is small compared to the current through the valve, the as
sumption of 180° phase difference is still justi ed. If V, is the input voltage
and |A| is the magnitude of the voltage ampli cation, then the output

C" g z ¢.,
+ C, v + _
V‘ k O y. |A|’:
— 1

(1') (0)
FIG. 10.13

voltage is —|A|V,. These two voltages have one common terminal and
the other terminals are separated by C,,, the anode—grid capacitance, as
shown in Fig. 10.13.b. There is, therefore, across (3,, a potential
difference of (|A| + 1)V,. The current I through CM is equal to
jmC,,(|A| +1)V,. Thus, when the signal V, is applied to the input
terminals, it has to supply a reactive current as though it were connected
to a capacitance of value (|A| + 1)C,,. The signal is also connected
directly across C91,, the grid cathode capacitance, so that the effective
input capacitance of the ampli er is C,;, + (|A| + 1)C,,. In a triode
C,,, and C,, may be about 10 p.p.F each. If |A| = 20, then the input
capacitance is 220 |111F. This capacitance is across the load resistance of
the previous valve, and may cause considerable frequency distortion.
In pentodes Ca, is of the order of 0 01 p.|J.F, and pentodes are therefore
much better than triodes in ampli ers which have to be used at high
frequencies. This marked effect of the feedback through the anode
grid capacitance in a valve ampli er is called the Miller effect.
The input capacitance of a cathode follower may be found by a method
similar to that used for determining the input resistance. The value is
approximately Ca, + C,;,/(1 + |A|). Thus a cathode follower ampli er
has a much smaller input capacitance than a common cathode ampli er.
10] FEEDBACK 167

10.12. Stability with Negative Feedback—Nyquist Diagram

The voltage gain of an ampli er with feedback is given by the equation,
A1= A/(1 13A)
In much of this chapter we have assumed that we are operating with
resistive loads and resistive feedback networks, and that any reactance
effects are negligible. It is quite easy to design an ampli er to satisfy
these conditions and give negative feedback over a required frequency
band, such as the audio frequencies for sound reproduction. However,
at very low frequencies, reactances such as those of the grid coupling
condensers and the cathode and screen decoupling condensers have some

|A|

NO FEEDBACK
WITH FEEDBACK

WITH
EXCESSIVE FEEDBA

>
LOG f
FIG. 10.14

effect. The magnitude of A is altered, and at the same time the phase
angle of AB is affected so that the feedback may become positive. Unless
A has then become very small, self oscillation may occur at a low
frequency. The same possibility arises at high frequencies outside the
wanted range, due to stray capacitances shunting the load resistances.
Even though self oscillation does not occur, there may be a rise in gain
of the ampli er at low and high frequencies, as shown in Fig. 10.14.
The behaviour of a feedback circuit and a criterion for its stability
may be determined by means of a Nyquist diagram. This is based on a
vector diagram of the type shown in Fig. 10.6.0 to f. The starting point
is V,, which is drawn as a unit vector OB in Fig. 10.15. The output
V, is the vector A, and BA is the feedback voltage. Since V, = V, + BV,,
the vector 1 — BA represents the signal voltage. In nonnal operation
the magnitude of BA is much greater than unity. Over the required
operating range BA is negative and in antiphase with OB, as shown by
OC ; CB then represents the signal. As the frequency departs from
the operating range, BA has a phase angle differing from OC ; OD repre
sents BA at a lower frequency and DB is the vector difference 1 — BA.
If the magnitude of DB exceeds OB, then Ag is less than A and the feed
back is negative; i.e., if D, the end of the BA vector, lies outside the unit
circle round B the feedback is negative. At the point or points where
168 PRINCIPLES OF ELECTRONICS [cH. 10
the locus of D cuts the OB line to the right of 0, the feedback voltage is in
phase with the grid voltage and with the signal voltage. For inter
sections between 0 and B there is an increase in gain but the ampli er

/6
r5 '1 , _ \\\
\
\
I \
mcneasunc C ___ ° B |
FREQUENCY /4 ' I
/
/1 /
\ \ Z /

'3 D
/2
Fro. 10.15

remains stable. If the locus of D passes through B, then the feedback

voltage equals the grid voltage, and no signal is necessary to maintain
the output, i.e., the ampli er oscillates. Also, if the point B lies inside
the locus of D the ampli er is imstable.
 CHAPTER 11
TRANSIENTS IN AMPLIFIERS
11.1. Steady State and Transients
In considering the response of an electric circuit to an applied voltage
it is necessary to distinguish two different states of the circuit. There is
the steady state, i.e., the condition existing some considerable time
after switching, and there is the transient state which occurs immediately
after switching. In the electronic circuits which have been dealt with in
this book so far, attention has been restricted to steady states. In this
chapter we consider some of the features of the transient conditions in
valve circuits. Before doing so, a brief review is made of transients in
R, L and C circuits.

11.2. Transients in Passive Circuits

Some of the fundamental properties of R, L and C circuits may readily
be established from energy considerations. VVhen a current i flows in a
coil with inductance L, then energy of amount §Li2 is stored in the mag
netic eld surrounding the coil. Similarly, when a condenser of capaci
tance C has a potential difference v across it, energy of amount 1}Cv2 is
stored in the electric eld of the condenser. In a resistor R there is no
storage of energy. When a current 1' ows in the resistor, energy is
dissipated as heat of amount Rial, where t is the length of time the current
ows. By the nature of energy it is impossible to take it at an in nite
rate from a nite supply. Thus, when a battery is connected through a
switch to an inductance the current cannot rise instantaneously. It must
be zero immediately after closing the switch, and a de nite time must
elapse before it reaches its nal or steady state with energy }Li”.
Similarly, it is not possible to change the voltage across a condenser
instantaneously. There is no such limitation with a resistor, and the
current rises immediately to its steady state value E/R, where E is the
e.m.f. of the battery. This involves no energy consumption until some
time has elapsed. Inductance or capacitance is therefore responsible
for the transient conditions in electric circuits. The impossibility of
sudden change of current in an inductance implies that immediately
after the sudden change in applied voltage the inductance acts as an
open circuit to the change. Similarly, the impossibility of sudden
change of voltage across a condenser implies that immediately after the
initiation of the transient a condenser acts as a short circuit to the change.
These considerations are useful in determining the initial conditions in
transient changes.
169
170 [C1 I

E
,' S
. >3
1 O
15 ,, 1 (b)
.. [
SWITCH c1.oseo A1’ t=o lo " " " 7' " "
/
(0) l'qt1'1/<)“"7
/ I
I

° T11 C '
Fro. 11.1

v
E

1° 5

R :15 ° (11 t
[EL _ ¢ _ Eli‘. _ '1
I
("’
|+ [G 1
E(1 la) r’ I /
F‘ _*i@|
| |
I I
—iu I
'C | .° _ 1
I 5 (F)
R '6
E
(<11 \
\
E/¢ "\
41 — 1 —>f
,0 —l
(¢)
F10. 11.2
11] TRANSIENTS IN AMPLIFIERS 171
The mathematical laws of transient currents and voltages are derived
in numerous text books, and it is found that they follow exponential
laws. For example, in the case of a coil with inductance L and resistance
R connected suddenly to a steady supply of e.m.f. E (see Fig. 11.1.a),
we have the relation Lg + Ri = E. Solution of this equation shows
that the current 2' after time t follows the law
1:= — E"/T),

where iq is the steady state current equal to E/R, and T = L/R. T is

called the time constant ; it is the time for the current to reach (1 — 1/e)
of its nal value (i.e., approximately % of iq). The variation with time
of v, the voltage across the inductance, and 1', the current through it,
are shown in Fig. 11.l.b and 0.
v
E

1' 5
+1
,
_
E
[
O (0) "
(<1) E/R

o T‘
(¢)
Fro. 11.3

When a battery of e.m.f. E is connected suddenly to a condenser C

through a resistance R, as shown in Fig. 11.2.a and b, the voltage '0, across
the condenser follows a similar law to the current in the inductance, and
. d .
v, = vq(1 — e"/7') andz = C7129 (see Fig. 11.2.0),

where vq = E, the nal value of 11,. The time constant T equals RC

in this case. If the same condenser after being charged were then dis
charged through R (Fig. 11.2.d), the condenser voltage would decay
exponentially to zero according to the law
‘U, 1 E8 ‘IT

which is shown in Fig 11.2.0. The corresponding case of sudden con

nection of a supply to a pure resistance is shown in Fig. l1.3.a, b and 0.
As mentioned already, there is no transient delay in this case.
172 PRINCIPLES OF ELECTRONICS [cH.
In circuits with inductance and capacitance there are two separate
energy stores, and it is possible, under certain conditions, for the energy
to be transferred backwards and forwards from one store to the other.
An example of a charged condenser suddenly connected to a coil is shown
in Fig. 11.4.a. At the instant of switching, all the energy is stored in the
condenser with no current in the inductance. The condenser then dis
charges through the inductance, and a current grows until the condenser

'0
I
4%
S 1 ___
+ Kr 0 _ Q & _ _ _ __

VC C LR Q I t

swncu c1.oseo AT t o / '

(<1) (0)
F10. 11.4

is completely discharged. At this instant the current has its maximum

value, and all the stored energy is in the inductance. The current then
decreases and the condenser charges again but with reversed polarity.
The charging continues until the current is zero and so on. Charge
oscillates backwards and forwards round the circuit. While the current
is owing energy is dissipated in the circuit resistance, so that the amount
of stored energy decreases and ultimately the oscillation dies away.
The variation of current and voltage are shown in Fig. 11.4.b. The oscil
latory current varies sinusoidally with steadily decreasing amplitude.
The actual law is 1' = /1e"°“ sin cot, where or = R/2L and co Q If
the circuit resistance is large the energy may be dissipated so rapidly
that no oscillation takes place. The condition for oscillation is R < 2J

11.3. Transients in Valve Circuits

The anode—cathode path of a valve may be represented by a resistance
1,, at least for small changes of the currents and voltages. Also, there
are capacitances between the electrodes. Thus the valve makes con
tributions to the resistance and capacitance of the circuits associated
with it, and so must affect the transient behaviour for sudden changes
either of the supply voltages or of a signal. The behaviour of the valve
circuit depends to some extent on the place where the change occurs.
 11] TRANSIENTS IN AMPLIFIERS 173
Firstly, we consider the circuit in Fig. 1l.5.a, which shows a valve
ampli er with RC coupling to another valve. Inter electrode capacitances
are ignored at present. It is assumed that a sudden change occurs in the
supply voltage at the point P as shown by the step function in the
diagram. This affects the anode voltage of the rst valve V1, but does

HI
,8 PHI*' H.T.+
.
o g R;
c |_o 0 R,
v1 v2 V‘ c _ V
'5 7.‘ G2
R2 yea O t . R O

G 2 V’ 51
E‘ "" ' """
(<1) (b)
1.9, ,2.._'3ss»'s__,"/'r.,,;'/1
wens R=R +'iR.L 9 R1'5+R1rv+R1R2
2 '3"'R1
F10. 11.5

not affect its grid voltage. If the changes are small, then any anode
current or voltage changes are related by 1', = v,/r,. The valve is there
fore replaced by r,,, as shown in Fig. 1l.5.b. The effect of the sudden
voltage change on the ampli er depends on the voltage 11,, appearing at the
grid of V2. On account of the coupling condenser C, a transient occurs.
Ultimately the steady state gives an increase of charge across C and no
H.T.+
s o
R, C
[A [C I O £4

V1 V2 C
9'1 + "02
BE‘
"1
T
E °'R2 v
1T_
.. _To_
f
o 1 "= R
V,
° 1

(°) (5)
T=RC 12 v — — —I ‘R2917: ) "'/T
¢ = v / 1:4/T
vmene R=R, + t 92 R16 *’R2G *R1R2 ‘

Fro. 11.6
 CH

[A * LR
O
at "' v, E2
O (E VG

E1
1A1 (0) _
E1+vs

1, ________ P ..E‘

1," 3L Q
O
AIYAL VR VP VQ E2 VA
I

(0)
'71
r, L
o R I lye """ "
,, t
.'_ . o 1

V5 C
O l

E +e
' v. 5,
,“ (<1) L
[P . _ _ — . _ — — —_ _ T =

I
1.=.er*..<1 ~"’*>
f

VA‘ (C )

'5
"P
,0 = 1121. (1.. £1 ,"/1}
1+5‘R R
V11

O
(I)
F10. 11.7
ll] TRANSIENTS IN AMPLIFIERS 175
permanent effect on 11,2. While C is charging there is a current through
R2, and 12,2 varies as shown by the exponential waveform in the diagram.
As a second example of transients in valve circuits we consider the
case of a sudden small change in grid voltage in the rst valve of a two
stage RC coupled ampli er as shown in Fig. ll.6.a. After its initial
sudden change the grid voltage remains constant, and the transient
conditions are determined by the rest of the circuit. As the changes are
small, the equivalent circuit may be used as shown in Fig. ll.6.b, from
which the transient change in ‘Ugg may be determined. The actual shape
and magnitude are indicated in the gure. The value of 1,, is, of course,
determined from the 1'4, v4 characteristic at the operating point.
Finally, we consider an ampli er with an inductive load (Fig. ll.7.a),
and we assume a sudden grid voltage rise as shown. Since the current
through an inductance cannot be changed instantaneously, it follows
that the anode voltage must drop suddenly in order to maintain constant
anode current. Thereafter the anode current changes gradually to a
new steady value. The variations may be determined as in Fig. ll.7.b
from the valve characteristics and the load line corresponding to the re
sistance R. The initial quiescent point is Q. The sudden application of
v, results in an instantaneous anode voltage change from vq to ‘U3 at
constant £4. The anode current then rises from iq to ip, whilst the anode
voltage rises from ‘U3 to ‘Up along the — E1 + v, grid characteristic. If
the changes are small the transient may be analysed by using the equiv
alent circuit of Fig. ll.7.c. The nature and magnitude of the transient
changes are shown in Fig. ll.7.d, e and f.

11.4. Ampli cation of Square Pulses

An RC coupled ampli er is incapable of amplifying faithfully a single
step signal. Frequently it happens that an ampli er is required for a
at topped pulse of the type shown in Fig. 1l.8.a. The circuit of Fig.
ll.6.a could give at the grid of the second valve a signal of the type shown
in Fig. ll.8.b, which has little resemblance to the original signal. The
nature of the waveform produced by the ampli er varies with the time
constant of the valve and circuit. For the case in Fig. ll.8.b the time
constant is much less than the width of the pulse. For a time constant
of the order of the pulse width the ampli ed signal would be as shown in
Fig. ll.8.c. With a time constant much longer than the pulse width the
ampli ed waveform would be as shown in Fig. ll.8.d, and would be
almost the same shape as the original pulse. The conditions required
for longer time constant are the same as those for extending the low
frequency range of the ampli er. Thus good low frequency response is
necessary for the faithful reproduction of the at top of a pulse.
In dealing with transients in valve circuits so far, we have neglected
inter electrode capacitances or stray capacitances. In the case of an
ampli er with a resistance load the stray capacitance from the anode,
C,9, is effectively across the load resistance as shown in Fig. ll.9.a. When
 [cu

'91 ‘ ‘ (0)

M (bl

‘$2 (C)

‘$2 (d)

FIG. 11.8

1.: E1".
7;...... 5
E‘
1', ‘O

.‘9 Q
: : _
11$

,0 V’ V VA

“ (<=)° E=
§

=é ’~ cf}

1+6 » ‘z HMO %€ O t
(v) (b)
Fro. 11.9
ll] TRANSIENTS IN AMPLIFIERS 177
a small square pulse is applied to the grid of such an ampli er the tran
sient behaviour is found from Fig. ll.9.b or c. The effect of C3 is to
prevent the sudden rise and fall at the beginning and end of the anode
voltage pulse, as shown in the gure. The anode current varies along the
path QRP in Fig. ll.9.c. For an ampli er to give rapid rise and fall to
the sides of a pulse it must obviously have good high frequency response.

1.
R
+—1_
E,
gs
V0
*5
,3: o
o +c$

r" *t<r"
.\

E; — _ .

(0) (9)
Fro. 11.10

In some ampli ers an inductance is used in series with the load resistance
to extend the high frequency range of the ampli er (Fig. ll.l0.a). The
load circuit now has L, C and R, and hence oscillation is possible. When a
square pulse is applied to the input of such an ampli er the output voltage
may have oscillations as shown in Fig. ll.l0.b.

11.5. Large Transients in Valve Circuits

In most of this chapter we have assumed that the transients have been
suf ciently small to justify the use of the valve equivalent circuit. Fre
quently large transients occur, and then mathematical analysis is ex
tremely dif cult, as the valve behaviour may be highly non linear over
the range of the transient. The perfonnance may usually be determined
qualitatively on the basis of the principles discussed in this chapter by
using anode characteristics and a load line.

11.6. A.c. Transients

We have considered transients arising from a step change in a steady
voltage, and we have seen that the resultant behaviour can be determined
in terms of the initial conditions, the nal steady state and the transient
state. Similar effects are obtained on introducing any sudden change.
The nature of the transient part is the same in all cases, and any differences
arise from the different initial conditions and nal steady state. In the
178 PRINCIPLES OF ELECTRONICS [C1 I.
case of a.c. supplies the initial conditions vary with the time of switching;
the magnitude of the resulting transient is affected. However, the dura
tion of the transient is the same as in switching d.c. supplies.

11.7. Glass 0 Ampli er as a Switch

A sudden change in a circuit with both L and C may give an oscillatory
current which gradually dies away as the energy is dissipated in the circuit
resistance. In Fig. ll.ll.a an oscillatory circuit is shown connected to a
power supply E2 through a switch S. Let us assume that the condenser

+ I

DI
C
_ =
E,
s4 |
(0)
"c
I
+
Vc : “\‘
II I
I
I
5 , '“‘

v+
s
e
'1' |
*= Q I, ’ ""'
'
" /’ smrca ctosso

(b) (¢)
Fro. 11.11

is charged initially to a potential difference equal to E2 and that the switch

is open. The condenser discharges through the inductor, and after one
cycle the voltage across C is slightly less than E2, on account of the energy
dissipated in R during the cycle. If now the switch S is closed for a short
time the condenser voltage is restored to E2, and the switch may be
opened again (see Fig. ll.ll.c). A second oscillatory cycle occurs and,
provided S is always closed at the right time each cycle, the battery
restores the ‘energy dissipated; the oscillation therefore continues with
out loss of amplitude. Provided the energy lost each cycle is only a small
fraction of the energy stored, i.e., provided the circuit has a high Q, the
oscillation is practically sinusoidal. The rate at which a condenser
charges depends on the resistance of the charging circuit. If the resistance
of the condenser battery switch circuit is negligible, then the switch need
be closed only momentarily.
ll] TRANSIENTS IN AMPLIFIERS 179
The valve in a Class C ampli er behaves very much like the switch
in Fig. ll.ll.a. Once every cycle the valve passes a pulse of current to
restore the initial charge amplitude to the condenser C (Fig. l1.1l.b).
Since the valve has some resistance, it must conduct for a de nite part
of the cycle in order to recharge the condenser, and also there is some
voltage drop across the valve, so that the condenser voltage amplitude is
less than E2. In the switch circuit we postulated that S should be closed
at the right time every cycle. With the valve this is achieved by con
necting to the grid a signal of the same frequency as the oscillation.
It may be noted in passing that the grid signal does not supply any power
to the oscillatory circuit. It merely serves as a suitable timing device
to ensure that the losses in R are made good from the anode supply. In
practice, R arises mainly from a coupled load resistance. The resistive
components of the circuits are made as small as possible. The operation
of the Class C ampli er as a switch supplying power to the oscillatory
circuit once every cycle is closely analogous to the operation of a clock
pendulum, which is given an impulse once every cycle by means of the
escapement. The energy is supplied from the potential energy stored in
weights or in a coiled spring. Another analogy is found in the child's
swing, which is kept going by giving it a push at the right time each cycle.
It may be seen that it is not essential to supply energy to these oscillators
each cycle. Every other cycle, or one cycle in three, is enough, provided
the Q is suf ciently high. When a valve is supplied with a grid signal of
frequencyf and the anode circuit oscillates at 2f, 3f or higher, then we have
a frequency multiplying Class C ampli er.

11.8. Transients in Circuits with Feedback

In Section 11.3 we considered the response of valve ampli ers to sudden
changes of grid voltage. It was assumed that after the initial change
the grid voltage remained constant and that the transient behaviour de
pended on the rest of the valve and the circuit. In an ampli er with
feedback this assumption is unjusti ed, as the grid voltage continues
to change with the transient changes in the output circuit. The problem
is then much more complicated. Some consideration is given to the case
of an ampli er with positive feedback in Chapter 13.

11.9. Some General Comments on Transients

We consider above transient effects in circuits with R, L or C in various
combinations. In particular, in circuits which are purely resistive,
changes in current and voltage occur instantaneously. However, when a
resistor carries current there must be a magnetic eld around it, so that the
resistor has some inductance as well as resistance. Also, when a potential
difference exists across a resistor there is an electric eld around it, so
that it must also have capacitance. The inductance and capacitance,
which are distributed along the resistor, are small. Similarly, inductors
180 PRINCIPLES OF ELECTRONICS [cH.ll
have distributed resistance and capacitance and condensers have in
ductance and resistance. The time constants arising from these distri
buted circuit components are small, usually much less than 10'“ sec.
For longer time constants than this the circuits can be represented by
simple combinations of R, L and C. For shorter time constants it may be
necessary to consider each circuit component in terms of its distributed
properties, rather like a transmission line.
In circuits with very short time constants the transit time of the
electrons across the inter electrode spaces may not be negligible. In
some transistor circuits diffusion or reactive effects may be important
at quite low frequencies.
 CHAPTER 12
DIRECT COUPLED AMPLIFIERS
12.1. The Ampli cation oi d.c. Changes
A common requirement in electronics is the ampli cation of a steady
current or voltage, or of the mean component of a changing current or
voltage. When several stages of ampli cation are required, then the
capacitance or mutual inductance couplings previously described cannot
be used, since neither of these couplings is capable of transmitting a d.c.
change. A direct connection must be made between the anode of one
valve and the grid of the next valve. If a d.c. signal is applied to the
input of a direct coupled ampli er, then the output is also a d.c. change.
However, changes in the grid or anode supplies or in the heater supplies
also give d.c. changes in output, and hence are indistinguishable from
signals. This effect, sometimes referred to as zero drift, constitutes a
serious limitation in direct coupled ampli ers, and special circuits are
used to minimize the effects of supply changes. Similar changes in
supply voltages are of much less account in a.c. ampli ers with capacitance
or mutual inductance coupling, since they do not affect the output, except
in so far as the valve constants, 1,, and gm, vary with the supply changes.
Direct coupled ampli ers are also capable of amplifying a.c. signals.
However, when an ampli er is required for a.c. signals only it is usual to
avoid direct coupling, except perhaps when the frequency of the signals is
very low.
12.2. Direct Coupling
A simple two stage direct coupled ampli er is shown in Fig. 12.1.
The direct connection from the anode of the rst valve to the grid of the
second valve is made through a battery E12. The purpose of this

R‘ E12 R2 I
in * in *
Vl V2 __
+ + f‘ E2
V‘ VA1 VA:

'01 Ycz

EJl_ _.1—. L—,._.

Fro. 12.1
G 181
182 PRINCIPLES OF ELECTRONICS [ca.
battery is to ensure that the second valve has the correct operating
grid bias. From the circuit it may be seen that UG2 = v41 — E12 and
v41 = E2 — R1z'41. Since 1102 has a small negative value, E12 is ap
proximately equal to v41. Normally, v41 is an appreciable fraction of E2,
so that the battery E12 is fairly large. When a small signal '0, is applied
to the input of the ampli er the ampli ed output of the rst valve is
passed directly to the grid of the second valve, and hence
1 1
W2 = val = gmlvs/(E +

The signal output of the ampli er is Uag, where

1 1 1 1
U02 = gm1gm2va/(E + 11;) (F2 + Q)’

If the valves are identical pentodes with equal load resistances R, then the
voltage ampli cation is A = (g,,,R)”. With an a.c. signal the output is in

. . 5
‘Ar *' 1' ‘A2 I

4’ +
"1 Q5 I/M
"2
+ vs VA? E2

' V51 R2 G2

E1 IT . “ "" " I“ r 0

I I "E1.
'62 = VA: " R11
(RF R2)‘ = E12 *' "Ar
Fro. 12.2
phase with the signal. If a small change en should occur in the grid
battery supply Eu, this would be ampli ed and give an output change
of —(g,,,R)2en (assuming pentodes). A variation in E12 of amount em
would give a change in output of g,,,Re12. A change of amount ea in the
main h.t. supply E2 would give an output change with pentodes of
(1 — Rg,,,)e2. Normally Rg,,,> l, and then the change in output is
—g,,,Re2. On account of these variations in output it is essential to
use highly stable supplies with this circuit. Also, the need for the fairly
large battery E12 has certain disadvantages. Not only is it an extra
cost, but its capacitance is in parallel with the load resistance R1, and so
limits the high frequency and transient response of the ampli er.
An alternative method of obtaining direct coupling is shown in Fig.
12.2. Here the anode of the rst valve is joined to the grid of the second
12] DIRECT COUPLED AMPLIFIERS 133
valve through a resistor R1. The appropriate quiescent grid voltage for
V2 is obtained from an additional supply E12 through the resistor R2.
Ilere
. 1 UAI ‘_ R11’,

where z(R1 + R2) = E12 + v41

(it is assumed that U92 is negative so that V2 has no grid current).
It follows that
v RIEI2 .

"’ R1+R2 R1+R2

Since 121,2 is a small negative voltage, then
E12 "" R2'”A1/R1»
and this is one design equation for the circuit. When a signal is applied

IA; + R

R vs T
R 1,, 1+ ,, VA3 E= +'+
v2 “E _ '1‘ _Er I’
[A1 ++ op "A2
"°3 _ E2:
II, + vs VA!
'01
_ _ E12
Err
Fro. 12.3

to the grid of V1 only part of the resulting change in anode voltage,

11,1, is applied to the grid of V2, since R1 and R2 act as a potential
divider. Thus
"92 = Ravel/(R1 "l" R2)
(It is assumed that the internal resistance of E12 is negligible.) It is
therefore desirable to make R1/R2 small to give high stage gain. From
the rst design equation this means large E12, and obviously there must
be some compromise. This circuit is sometimes called the “third rail
circuit”. In most other ampli ers there are two h.t. “rails”, the h.t.
positive and h.t. negative lines. In Fig. 12.2 there is an additional h.t.
rail to the grid of V2.
A third type of direct coupling is shown in Fig. 12.3, where each anode
is connected straight to the next grid. This circuit involves the cathodes
of the valves being at different potentials, which may give rise to cathode
heater insulation problems unless separate heater supplies are used for each
valve.
184 PRINCIPLES OF ELECTRONICS [cH.

12.3. Use of Negative Feedback

In Chapter 10 it is shown that the use of negative feedback can render
ampli ers insensitive to changes in supply voltage, and it is therefore
worth considering the introduction of feedback to direct coupled ampli
ers. A two stage ampli er with current feedback applied to each
stage is shown in Fig. 12.4. The cathode resistors RK provide the feed
back and at the same time give automatic bias. This bias may be too

[A1 "' I *'

V1
. + Q + Q TE,
v‘ VG.‘ _ '2‘ '5’ "oz
+ H‘ ‘ 2 7+''''' '
E Rxliu Rx Ruin Rx
rr _ _ _ _

E12
Vs: "'Rx [A2 ’ ‘bi’ R11
(R99 2) 4. ’ Er2*"or
Fro. 12.4

great, particularly in the rst valve, and then an additional battery E11
is required. In each of the stages the feedback fraction B is equal to
R;/R. Provided each stage has suf ciently high inherent gain, the gain
of each with feedback is 1/B. The overall gain of the ampli er is then
(R/RX)’; this assumes that R1/R2 is small. The gain of the ampli er is
thus independent of changes in supply voltage. However, this is far
from the whole story, since changes in supply voltages still give changes in
output. For example, a change in E11 is indistinguishable from a signal
v,. Also, it may be shown that a change e12 in E12 gives an output change
of
B...__R1 2
RK R1+R212

and a change in E2 of amount e2 gives an output voltage change of

R R2 e.

Thus, although feedback renders the ampli er gain independent of

supply changes, it is still necessary to have highly stable supplies to
prevent zero drift.
12] DIRECT COUPLED AMPLIFIERS 185

12.4. Balanced or Di erential Ampli ers

The effects of power supply changes on zero drift can be considerably
reduced by the use of certain balanced or differential circuits. These
usually employ twice the number of valves that would be used in a
normal ampli er, and the connections are arranged so that the changes
produced in the output circuit by the extra valves cancel those in the

" + I

g V2 E2
03+ O ‘S
|
R11 v
1| :

Rx
v,’

G =
I

R B

(<1)
P R

R, RI
(6)
FIG. 12.5

normal ampli er. An example of such a circuit is shown in Fig. l2.5.a.

V1 and V2 form a third rail, two stage ampli er with feedback
similar to the one considered in the last section. The output of this
ampli er v, depends on the signal applied to the input of V1 and also
186 PRINCIPLES OF ELECTRONICS [crr.
on any changes in the supply voltages. V1’ and V2’ form a second and
similar ampli er, whose output v2’ depends only on changes in the supply
voltages. There is no signal applied to this ampli er. The output v of
the whole system is taken between the points A and B, and so v = v, — v,,'.
The valves and components of the two ampli ers are identical, so that
with no signal v,, and v,’ are equal and v is zero. Now any changes in the
power supplies cause equal changes in v, and 0,,’ so that v is still zero.
A signal applied to V1 gives a change in vo, but does not affect v,,',
and hence v changes by the same amount as v,,. With the same assump
tions as in the previous section, the overall gain of the ampli er is there
fore (R/R3)”. It may be noted in passing that the valves V2 and V2’,

"cal
C Vce C tvco

T E '0 :
o °'l|' '05‘ o t
1| _ t
E D °.U
(*7) | (b)
V1
Rs
+
"Q E |
R9

§ V1’
(¢)
FIG. 12.6

together with their anode loads, form a Wheatstone bridge network, as

shown in Fig. l2.5.b, where R, and R,’ represent the equivalent d.c.
resistances of V2 and V2’. Changes in supply voltages cause equal
changes in R, and R,’ so that the bridge remains balanced. A d.c.
signal, however, alters the value of R, only, and the balance is upset.
The amount of unbalance is then a measure of the signal.
If the signal is omitted from Fig. l2.5.a the circuit is completely sym
metrical, and by comparison with the circuits of Section 9.9 it may be
seen that this is a push pull circuit with resistive components. In
push pull circuits changes in the supply voltages do not appear in the
12] DIRECT COUPLED AMPLIFIERS 187
output. In the use of the ampli er described above the signal is con
nected asymmetrically between one grid C and the earth point E. Such
a signal is said to be unbalanced. One terminal of the signal is assumed
to remain constant at earth potential, and the other terminal varies posi
tively and negatively with respect to earth, as shown in Fig. 12.6.a.
Sometimes the signal is balanced with respect to earth, i.e., the two
terminals both vary, but by equal and opposite amounts measured from
earth potential, as shown in Fig. l2.6.b. Such a signal would be con
nected to the ampli er symmetrically, as shown in Fig. 12.6.0, and this
is the normal input connection for a push pull ampli er.

12.5. High gain Ampli er with High Stability

We see above that stability against power supply changes can be ob
tained by the use of a second ampli er giving a differential action. At
the same time stage gain stability is achieved by the use of current

1..
C V1
R
1 1+ YK "I

'= "0 ~"'——"‘~

‘A1+lA2 Rx E2

E E1
R9 _ R
Q v2
1., _ _
B
Fro. 12.7

feedback separately on each valve. Even greater stability against

supply changes can be realized by using a common bias resistor for V1
and V1’. As shown below, this gives feedback against supply changes
but no feedback on the signal. Thus high signal gain with high stability
to supply changes is obtained. The circuit is shown in Fig. 12.7. The
anode currents of both valves ow through the common cathode resistor
RK, giving a bias voltage vg. If need be, this may be offset by a supply
E1 in order to bring the valves to a suitable operating point. If there
is an increase in supply voltage E2 this causes an increase in the anode
currents of both valves, and so an increase in vg. This increase in bias
reduces the anode current, and so opposes the original change. Thus, in
188 PRINCIPLES OF ELECTRONICS [cB.
addition to the differential action obtained by identical valves and com
ponents, there is also a feedback against supply changes, and this ampli
er is extremely stable against such changes.
When a signal is applied which makes C positive with respect to E
the anode current in V1 rises, and so increases vg. This reduces to
some extent the effect of the signal on the anode current to V1, and
at the same time reduces the anode current to V2. The valve currents
thus vary in opposite senses. The variations are very nearly equal and
opposite, as is now shown. For pentodes and small signals
in = gmvgl and 11.2 = gmva.
where ‘U21 = ‘U; — RK('i¢1 1 and ‘U213 = — Rg(I:21 1 1.112).

Hence 12,; = — Rgg,,,(v,1 + 11,2), i.e., 11,2 = i 11,1.

If R5 gm > 1 then 1:22 = — 12,1 = — v,/2. The input signal is thus shared
equally by the two valves, but in opposite senses. The output voltage

I
R
<so|ue"rmes)
ZERO
O
'1'

FIG. 12.8. —

of each valve is g,,,Rv,/2 and the total output across the terminals A
and B is g,,,Rv,. There is therefore no feedback on the signal.
The circuit of Fig. 12.7 has several forms and it is known by a variety
of names, such as a cathode coupled ampli er and a long tailed pair.
One variation of the circuit is shown in Fig. 12.8, in which the output is
taken from V2. This output may be connected to the grid of another
valve in a second cathode coupled ampli er. It may be noted that the
output voltage of a cathode coupled ampli er is in phase with the signal
voltage.

12.6. Other Methods oiAmr>liTYi118 Steady Signals

Throughout this chapter we have come across the problem of zero
drift in direct coupled ampli ers —a dif culty which does not occur to
anything like the same extent in a.c. ampli ers. Other methods are
sometimes used for the ampli cation of steady signals. For example, the
12] DIRECT COUPLED AMPLIFIERS 189
steady signal may be connected to the terminals of an interrupter, i.e., a
mechanical vibrator which interrupts the steady signal, thereby producing
an alternating signal which may be ampli ed by a conventional a.c.
ampli er.
The magnetic ampli er is another means of amplifying d.c. by using the
d.c. to produce an a.c. The d.c. is passed through one winding of a
saturable iron cored inductor. The magnetization of the core varies
with the magnitude of the d.c. A second winding is fed with a.c.,
and the inductance of this winding varies with the magnetization. Thus
the d.c. change is converted into an a.c. change, which may be ampli ed
further and nally recti ed to restore a steady signal.
 CHAPTER 13
OSCILLATORS
13.1. Introduction
Electronic circuits are used as generators of oscillations at frequencies
from 1 to 1011 c/s. To cover this enormous range, valves or transistors
and circuits of great variety are used. In considering the operating
principles it is convenient to divide oscillators into two main classes,
negative resistance and feedback. In the former an electronic device
acts, between two terminals, as a negative resistance which cancels the
normal ohmic resistance of a tuned circuit connected across the terminals.
A feedback oscillator can be realized from any device capable of power
ampli cation. Sufficient output power is fed back to the input so that
the ampli er supplies its own input signal. This chapter deals with both
classes of oscillator, but only with those producing sinusoidal oscillations
and using negative grid valves and transistors. Special valves for very
high frequencies are considered in Chapter 15, and non sinusoidal oscil
lators in Chapter 18. In all oscillators the ultimate source of energy is a
d.c. power supply. The valves and circuits serve to convert some of
the d.c. energy into a.c. energy.

13.2. Negative Resistance Oscillators.

When a condenser is discharged through an inductance the current is
oscillatory provided that the resistive damping of the circuit is not too
great (see Section 11.2). The amplitude of the current decreases ex
ponentially, and ultimately becomes negligible when all the original energy
stored in the condenser has been dissipated in the resistance. In the
circuit in Fig. 13.1 a capacitance is connected in parallel with an inductance
L and a resistance R. When the switch is closed an oscillatory current
ows and the rate of decay of the oscillation
S depends on R; the larger the value of R, the
*\ slower the rate of decay. At any instant the
Cl Lé $2 rate of dissipation of energy in R is 1:2/R, where
v is the instantaneous value of the voltage
across C. When R is in nite there is no loss
F111 13_1 of energy, and once the current starts it con
tinues inde nitely with constant amplitude. If
R is negative the resistance acts as a source of energy and the amplitude
of the oscillation steadily increases.
Under certain conditions, a tetrode valve behaves as though there is a
negative resistance between the anode and the cathode. An anode
190
CH. 13] OSCILLATORS 191
characteristic for a tetrode is shown in Fig. 13.2.a. Over the region BC
the anode slope resistance r, is negative. An arrangement for using this
negative resistance to generate oscillations is shown in Fig. 13.2.b, where

L
1, 1
I/5 3 E3 ‘A
B Y
er n—E r g R
73
"A
52
"b I'|‘l U .5"
QV
" ' o.1" m
_ __
U
><
(<1) W
.2 ‘

* R‘
'3
(¢) (<1)
is
1.
Q o 2

o >§

o '°

(¢)

I
Fro. 13.2

a parallel tuned circuit is used as the anode load. As far as alternating cur
rents are concerned, the valve is in parallel with the tuned circuit (Fig.
13.2.0), and the resultant resistance across L and C is R’ = R1,,/(R + 1,)
(Fig. 13.2.11). If the quiescent point is chosen near the middle of BC, then
r, is negative, and the sign of R’ depends on the relative sizes of R and 1,.
192 PRINCIPLES OF ELECTRONICS [CH.
If R and 1,, are numerically equal R’ is in nite, and an oscillatory current
once started does not decay. In practice, it is extremely di icult to
arrange that R and 1,, are exactly equal. The circuit and voltages are
adjusted so that 1,, is negative and numerically less than R and then R’ is
negative. The amplitude of oscillation therefore increases. However,
it may soon reach a value where the anode voltage goes beyond the range
BC; 1,, is then positive and no longer acts as a source of power. In
operation the anode voltage variation is just enough to give r, an average
value over one cycle equal to R. The anode voltage and current varia
tions are shown in Fig. 13.2.12. Although the anode current is not sinu
soidal, it produces voltage across the load only at the fundamental
frequency, since the load impedance is small at frequencies away from the
resonant value, f,,, where 2nf,, = 1/\/F. The initial adjustment of this
oscillator is more than enough to produce oscillation, and the amplitude
is limited by the valve characteristics. This is an example of a practice
which is common in valve oscillators.
In Section 5.13 it is shown that a gas diode may produce an arc dis
charge which has a negative resistance. In the early days of radio, arc
oscillators were frequently used in transmitters. Diodes and other
vacuum tubes may have the property of negative resistance at certain
very high frequencies, when the electron transit time is greater than a
period of the oscillation. The tetrode negative resistance oscillator is
usually called a dynatron oscillator.

13.3. Feedback Oscillators

The vast majority of electronic oscillators are ampli ers which provide
their own input. In the general feedback circuit of Fig. 10.6.b, which is
discussed in Sections 10.5 and 10.12, it is shown that any ampli er oscil
lates when 1 — [BA is zero, where BA is the feedback factor. This means
that the feedback voltage is exactly equal in magnitude and phase to
the grid voltage. Oscillation may also occur if the feedback voltage is
greater than 11,, but again the phases must be identical. In general,
there are, therefore, two separate conditions to be satis ed in feedback
oscillators. The phase condition determines the frequency of oscillation.
The other condition is concerned with the feedback magnitude.
The principle of the feedback oscillator may be illustrated with re
ference to a single stage resistance loaded ampli er of the type shown in
Fig. 13.3.a. In this circuit the output voltage 12,, is greater than the signal
voltage, and differs from it in phase by 180°, provided the capacitances
of the valve and circuit elements are negligible. If this ampli er is to
provide its own input there must be a further change of phase of 180°
in the feedback circuit. This can be achieved with a phase shifting
network of the form illustrated in Fig. 13.3.b. The phase difference
between the voltages v1 and 1:2 varies from 90° to zero, as the frequency is
raised from zero to a very high value. At the same time the phase differ
ence between v, and v1 ranges from 270° to zero. If this network is put in
13] OSCILLATORS 193
parallel with the anode load R1, then at some frequency, f,,, the voltage
1:, is in phase with 12,. If R > R1 there is approximately 180° phase
shift through the ampli er and a further 180° in the R, C network.
The magnitude of the voltage 11,, is considerably less than 1:1,. Provided
that the voltage gain of the ampli er at frequency f, is greater than 11,/v,,,

C C C
+ . .1. +"'

,1, Q
'1.
'9 R1 '1 R '2 R R '0
1 1 0 1 1 1

(0) (0)

R1 c c c

9129'

(<1‘)
FIG. 13.8

then 1:, is greater than and in phase with 11,. Hence, on joining the out
put of the phase shifting network to the input of the ampli er, as shown
in Fig. 13.3.0, the circuit oscillates at frequency f,,.

13.4. Tuned anode Oscillator

A common form of feedback oscillator is shown in Fig. 13.4, where a
tuned anode load is used, and the feedback to the grid is through the
mutual inductance M. In order to determine the conditions for oscil
lation certain assumptions are made. Firstly, small a.c. changes are
assumed so that the valve equivalent circuit may be used with vector
currents and voltages. Also, it is assumed that no grid current ows so
that the mutual inductance has no effect on the current in L. The
equivalent circuit for the valve and load is given in Fig. l3.4.b. If this
ampli er provides its own input, then the voltage across the grid winding
of M must be V; as shown. If Z is the vector impedance of the LC
circuit then
z = (R +j1.>L)/(1 0>”LC + ;'R<.>c).
194 PRINCIPLES OF ELECTRONICS [CH.
The following equations follow directly from the network
{N3 = I,(r,, + Z), (R + jc >L)I| = L,/j<»C,
L = 11+ L, and V, =jcoMI|.
Then L = I1(1 }—j¢oCR — co2LC),
i.e., I; = (1 + j<oCR — <..*LC)V,/j¢.>M
and p.V¢ = (1 + j<oCR — <»2LC)V¢(r,, + Z)/j<»M.
On substituting for Z it is found that
jcoMp. = (1 | j<oCR — <o2LC)f, + R +j<oL.

I.
M M I1 1:
*5 c '?l§ c
. E2 ! i .
G IQ? P.
E, 'v, W‘

|< |<
(v) Fro. 13.4
(0)
This equation must be satis ed if the circuit is to provide its own input.
The real and imaginary parts must be equal, and hence
M = L/u + CR/gm (using g,,, = p./r,,)
and co2LC = 1 + R/r,,.
Both of these equations are necessary conditions for oscillation. The
rst gives the value of M for given values of L, C, R, [.1 and gm. The
second equation gives the angular frequency of oscillation
(02 = Zlz. (1 + R/7“).

Normally 1, > R and then (02 =' 1/LC, the familiar value of the resonant
frequency of the LC circuit when resistances are neglected. This analysis
shows how self oscillation can be obtained from a tuned anode ampli er
working under Class A conditions with small a.c. amplitudes and feeding
back to the grid a voltage which just maintains these conditions.

13.5. Class 0 Oscillators and Amplitude Limitation

In practice, it is extremely di icult to operate stably under the con
ditions described in the last paragraph. The feedback is usually greater
13] OSCILLATORS 195
than the value required to maintain Class A operation. Then the ampli
tude of the oscillation increases until it is limited by the valve character
istics in much the same manner as with the negative resistance oscillator
described in Section 13.2. Large amplitudes of anode and grid voltages
are built up, and the analysis based on small signals no longer applies,
except as an indication of the starting conditions. The frequency of
oscillation is still given approximately by (.02 = 1/LC, but the M condition
becomes
M > L/p. + CR/gm.
However, this applies only in the early stages of the build up of oscillation,
since gm, and to some extent p., vary over the cycle as the amplitudes grow.

ll ('1

(A _
DYNAMIC IA
’.

1’ (5
I
1
I
5:2)
‘ l Io’ — . ‘*I"
O
"E1 cur o|=|= ~ "0
I *

FIG. 13.6
II

When the grid voltage goes into the positive region, grid current ows.
The oscillator is usually adjusted to operate under the conditions de
scribed for Class C ampli ers in Chapter 8. This means that the grid bias
is beyond cut off and the grid is driven well into the positive region. The
behaviour of such an oscillator can be determined to some extent if the
dynamic grid characteristic is known into the positive region. Such
a characteristic is shown in Fig. 13.5. When the grid is positive the anode
current continues to rise at rst, but subsequently the curve may tum
over. This is largely due to the increasing current taken by the grid
(shown by broken line). In this region gm and (J. both drop. Thus at the
peak of the grid voltage the condition for maintenance of oscillation is not
satis ed. The amplitudes settle to equilibrium values in which the average
196 PRINCIPLES OF ELECTRONICS [Cl !.
values of g,,, and p. over a cycle satisfy the M condition. It may be noted
that g,,, is zero over an appreciable part of the cycle.
A self oscillator cannot operate under Class C conditions with xed
grid bias. On switching on, the anode current would be cut off, and no
oscillation could build up. The bias is usually obtained automatically
by utilizing the ow of grid current through a grid leak R,, as shown in
Fig. 13.6.a. There is a condenser C across R, such that 1/(DC < R,.
Then, only the mean grid current ows through R,, and the bias is equal

.. g
R:

(<1)
l [6

? [A is
_Q’lG i

O ' " ' f

i ’ VG O

I lg

f
(0)
Fro. 13.0
to R, times the mean current, i.e., —R,i@. With this arrangement there
is no bias on switching on. The bias gradually builds up until the
equilibrium condition is reached, rather after the manner shown in Fig.
13.6.b.
The equilibrium condition for a Class C oscillator with grid leak bias
adjusts itself automatically so that, during the portion of the cycle when
the valve is conducting, it passes enough energy from the h.t. supply to
13] OSCILLATORS 197
the oscillatory circuit to make up the loss of energy during the whole
cycle. If for any reason the amplitude of oscillation decreases slightly,
then it is essential that the bias decreases also, otherwise the circuit losses
are not made up and the amplitude of the oscillation decreases further
and ultimately falls to zero. The rate at which the bias voltage may be
changed depends on the time constant of R, and C, and there is therefore
a limit to the value of R,C. The actual value is related to the rate at which
oscillations decay in the resonant circuit. When oscillation ceases in this
manner the valve remains cut off until C is discharged suf ciently. Then
the valve passes current and ampli es once more. The oscillation
builds up again with steadily increasing amplitude and bias. If the bias
lags behind too much, it increases beyond the equilibrium value and oscilla
tion again decreases. Thus when R,C is too great, intermittent oscilla
tion may be obtained. The effect is known as squegging.
The operating conditions for a Class C oscillator are the same as
those for a Class C ampli er. The power output is rather lower, since
the oscillator has to supply its own grid power.

13.6. Other Timed Oscillators

In the tuned anode oscillator of Fig. 13.4 the grid and anode a.c. voltages
are very nearly in antiphase. The grid voltage V; differs in phase by
90° from I1, and I1 differs from V, by 90°, except for the small effect of
R. The vector diagram is shown in Fig. 13.7. In this diagram V, is
drawn rst and then I1, 90° behind V‘. The voltage vectors for RI1 and
j< >LI1 are drawn next, in phase with and 90° ahead of I1 respectively.
The resultant gives the voltage across the load, which is —V,. V, and V;
are very nearly in antiphase, but the
small departure from 180° is essential Vv
to give the correct feedback. For V _
many purposes it is convenient to Fl:/“MIA
ignore the effects of resistances of RI! Va
the oscillatory circuits, and then V,
and V, are directly in antiphase.
This is a useful approximate rule
for applying to other tuned oscil
lators. Then the rough criterion _
of suitability of a circuit as an oscil In
lator is that the grid and anode
F10. 13.7
voltage should be in antiphase, and
there should be ready means of adjusting the size of the grid voltage to
ensure that it is suf cient.
A common triode oscillator, known as a Hartley circuit, is shown in
Fig. l3.8.a. Only the essential a.c. connections are given. In this
circuit V, = jmL,I| and V, = — jcoL2I|, ignoring mutual inductance
between L1 and L2. (The oscillatory current I1 is large compared with
198 PRINCIPLES OF ELECTRONICS [cH.
any current in the other leads, and we may assume that the same current
I1 ows in L2 and L2.) Thus V,/V, = — L2/L1 and the size of V2 may be
varied by moving the tapping point on the inductance. This is a simple
oscillator and easy to adjust. The frequency is determined by L and C,
where L is equal to L2 + L2. A circuit diagram showing the d.c. con

C» C.

1: j L
+1' I L‘ C
0 P CB _

v L’ Q
0 l L 1. 1. I C”
v, ' 1* 2 C’
!___ _

(0) (0)
Frc. 13.8

nections as well is given in Fig. l3.8.b. The condensers C5, of large

capacitance, are for d.c. separation; the inductances Ch are of high value
to act as chokes at the frequency of oscillation; R, and C, provide the
grid bias.
Another commonly used oscillator, known as the Colpitts’ circuit, is
shown in Fig. 13.9. Here V;/V. = — C2/C2.

A
7| T" — '_ °
:r'"' I
E§
. o'T§>
v,‘ ‘ml
°
+
%»
Va
1
Q
Ow 1
0
'
E]
"4 '
vi
: ' —— — :1. 1 0 ~ —— 7
Frc. 13.9 K
Fro. 13.10

In Fig. 13.10 a generalized triode oscillator circuit is drawn showing

reactances between each pair of the electrodes. Circuit and valve re
sistances are neglected, and inter electrode capacitances are included in
X2, X2 and X2. The three reactances form a closed oscillatory circuit.
Obviously they cannot all be inductances or all capacitances, as there
would then be no tuned circuit. The same current ows through all
13] OSCILLATORS 199
the reactances. Since V, and V, must be of opposite sign for oscillation,
it follows that X2 and X2 must both be inductances or both capacitances.
Thus the possible arrangements of the reactances are as shown in the table.

x, x, X,
I L C L
11 c ‘ L c j

It may be seen that the Hartley circuit conforms to type I and the Colpitts
circuit to type II. This generalized circuit may be used to interpret any
tuned oscillator in which there are no mutual couplings.
Another circuit, known as a tuned anode, tuned grid oscillator, is
shown in Fig. 13.11. This case conforms to the generalized circuit if the
anode—grid capacitance is included to form
X2. This is a type I oscillator with X1 and
X2 both inductive. Thus the two tuned
circuits, including the other inter electrode
capacitances, must both be tuned to a .1.'1
C22 L.
frequency above the frequency of oscil I
lation.
In the generalized oscillator circuit of
Fig. 13.10 the current owing round the
circuit depends on the values of the three
reactances. If any one reactance is very
high the current is small and the grid volt FIG, 13,11
age may not be su icient to give oscillation.
Oscillation in the circuit of Fig. 13.1 1 occurs more readily at high frequencies
when the reactance of C,2 is small. In pentodes C,2 is very small and there
is much less likelihood of oscillation. Hence the circuit of Fig. 13.11 with
a pentode may be used as a stable ampli er; sometimes it is necessary
to place a screen between the input and output circuits.

13.7. Transistor Oscillators

Transistor ampli ers with suitable feedback may act as oscillators,
and most of the triode circuits described in Sections 13.4 and 13.6 have
corresponding transistor versions. The circuit in Fig. 13.12 shows a
tuned collector oscillator in which a common emitter arrangement is
used. The main oscillatory circuit is LIC1, and the feedback is obtained
through the mutual inductance between L2 and L2; the resistance R
provides the correct operating base voltage, and C is a d.c. blocking
condenser. The similarity to the tuned anode oscillator is obvious.
In considering the latter in Section 13.4 we assumed that the resistance
between grid and cathode was in nite. In the transistor case the input
200 PRINCIPLES OF ELECTRONICS [cn.
impedance with the common emitter circuit is a few hundred ohms, and
this factor constitutes a major difference between valve and transistor
oscillators. In Section 13.6 we estab
L2 L2 _ lished some general properties of triode
L oscillators on the assumption that only
2 C‘ R reactances between the electrodes need
be considered. Now the resistance be
M tween base and emitter may be com
parable with the reactance, and the
Q: 9 conditions for oscillation are rather
more complicated. One result is that,
_ _ _ _ in order to obtain the correct phase of
2 2G_ 23_22 the feedback, the frequency of oscil
lation may differ appreciably from the
resonant frequency of the LC circuit. Also, the input impedance acts
as a damping load on the oscillatory circuit.
Transistor versions of the Hartley and Colpitts oscillators are shown in
Fig. 13.13.a and b. These circuits should be compared with Fig. l3.8.a
and 13.9.

I" i
~
l_T‘
~
OI!

(0) (5)
Fro. 13.13

13.8. Feedback and Negative Resistance Oscillators

So far we have distinguished between feedback and negative resistance
oscillators. Actually, it is always possible to deal with a feedback
oscillator in tenns of negative resistance. For example, consider the
two stage, direct coupled transistor ampli er of Fig. 13.l4.a, in which
the output is fed back to the input; R2 and R2 are the coupling resistances,
and R is the load of the second stage. If the relation between v2 and i2
is measured it is found that there is a negative resistance portion of the
curve, as shown in Fig. l3.14.b. Thus if a parallel tuned circuit is con
nected with a suitable biasing voltage, as shown in Fig. 13.14.c, oscillation
can occur. A triode circuit with similar properties is shown in Fig. 13.15.
Another interesting oscillator which may be discussed in terms of feed
back or negative resistance is the transitron, which uses a pentode valve
in a rather unusual way. In Section 6.11 we discuss the variation of
rs] 201
[1 R1 R2 [2
+
R .. Er
"1
'1

E:
(<1) (b)
R2 R2

E2 52
(¢)
Fro. 13.14

I3 Q3

T . 1
Fro. 13.15

“ca

C Q '
PT

0
ET _ |1 I. TE.
O "cs I _
.1"
(<1) (b)
Fro. 13.16
202 PRINCIPLES OF ELECTRONICS [cH.
screen current with suppressor voltage, when all the other voltages are
constant. The characteristic takes the form of Fig. 13.l6.a, and as '00;
rises, 2'02 falls. If G2 is used as a control grid and G2 as an output elec
trode, then the output current is in antiphase with the signal voltage.
Hence, with a resistance load, the output voltage is in phase with the
signal. This means that direct feedback from the screen to the suppressor
gives the necessary phase condition for oscillation. The transitron
oscillator, based on this principle, is shown in Fig. 13.16.b; the con
denser C provides the feedback. The oscillator can also be analysed in
terms of negative resistance. For constant v4 and v01,
1'02 =f(vaa. 1102),
and for small changes
dim dim
iv‘! = 3;?’ ‘"03 ‘l’ E v = €2a'”0<3 + 902/72

From Fig. 13.l6.a it is seen that g22 can be negative in value; r2 is positive.
When G2 and G2 are joined together changes in v03 equal changes in ‘U02.
Then
Is‘ = (gas 'l‘ 1/'2)vs'
or V2: 1 7.
Is‘ E23 + 1/'2
As long as g22 is negative and numerically greater than 1/r2, the dynamic
resistance r is negative. Hence, with a suitable parallel tuned circuit
between the screen grid and the cathode, oscillation can be obtained.

is 1c

RE l.1 + RC
E
r '[
l~=
..
:| E2

(0)
"1
I
o —~.42

Ra Re
53 ____ __ _
E21 I IE3 jg:

(b) (¢)
Fro. 13.17
rs] OSCILLATORS 203
The transitron oscillator is sometimes referred to as a negative mutual
conductance (or transconductance) oscillator.
Transistors can be used in a variety of ways to give negative resistance
oscillators. For example, if a transistor has 01¢, greater than unity this
can lead to a negative resistance characteristic. With a point contact
transistor in the circuit of Fig. 13.l7.a the relation between v2 and 2'2 is as
shown in Fig. l3.17.b. If a parallel tuned circuit is connected in the base
lead with suitable bias, as in Fig. l3.17.c, oscillations occur when the
magnitude of the negative resistance is less than the parallel resonant
resistance of the tuned circuit. This circuit can also be considered to

[E [2 .
+
Rs
Ra v2

Er [B _

(<1)
E2/RC v2 2
lr
O

Dm II O

7U 0 OI“
R
E2 E2
(0) (c)
FIG. 13.18

give positive feedback through the resistance of the tuned circuit, since
the changes in collector current are greater in magnitude than changes in
emitter current.
All the negative resistance oscillators discussed above have used a
parallel resonant circuit as the frequency determining element. Con
tinuous oscillations may be obtained in a suitable series tuned circuit,
and an example is given in Fig. 13.18, using a point contact transistor.
Oscillation occurs in this case, provided the magnitude of the negative
resistance exceeds R, the series resistance of the tuned circuit. The
quiescent point is determined by RC and E2. The base resistance R3
can be considered to give positive feedback. It should be noted that,
in this case, the negative resistance portion of the characteristic exists
uniquely for a de nite range of current, whereas in the previous cases of
negative resistance the controlling factor is the voltage range.
204 PRINCIPLES OF ELECTRONICS [CH.

13.9. Triode Oscillators tor Ultra high Frequencies

As the frequency of operation is increased the inductance of the elec
trode leads and the inter electrode capacitances become increasingly
important. With ordinary valves it is impossible to make a simple
Hartley or Colpitts oscillator, since the connections from the valve to the
tuned circuits have appreciable reactance. Some high frequency oscil

O ~Z
Q 9.
lv

r___
0
......|._..
_ O.. Q
' Cr»
_
QD
_
_
O ck so O

9*

Ch : C, E1

'I (0) (0)

FIG. 13.19

lators use the anode—cathode and grid cathode capacitances as C2 and

C2 in a Colpitts circuit, and then there is no need for the connection from
the common point to the cathode. The ratio of V2 to V, equals—C,1,/C21.
This modi ed Colpitts circuit is shown in Fig. 13.19. The anode and
cathode supplies are fed through chokes. Circuits of this type may
operate quite successfully at high frequencies, but they do not conform to
any of the simple oscillators which have already been described, since
there are always stray and indeterminate
couplings between different parts of the cir
cuits. Special triodes have been designed
which do permit of fairly simple and control
lable operation even at frequencies of 4,000 Mc/s
and higher. Such a triode is shown diagram
r—_——\ matically in Fig. 13.20. This valve has a
K planar electrode system, and the leads to the
2 !|| anode and the grid are in the form of copper
disks. The cathode lead is essentially a metal
B tube. This valve is constructed in such a
way that it forms an integral part of the ex
temal circuit, which, instead of using lumped
H
inductances and capacitances, employs short
Fro. 13.20
sections of high frequency transmission lines.
The arrangement is illustrated in Fig. 13.21. The triode plugs into a
circuit consisting of three co axial tubes, each electrode being attached
to one tube with the anode outermost and the cathode innermost. There
13] OSCILLATORS 205
are then two co axial transmission lines attached to the electrodes.
The grid tube is part of both lines. Its outside surface is the inner
conductor of the anode—grid line, and its inside surface is the outer
conductor of the grid cathode line. Although one conductor is shared
by these two circuits, the high frequency currents are completely isolated
from one another, as they ow only on the surfaces of the conductors.
The two co axial transmission lines are short circuited by means of sliding
plungers, P2 and P2, whose positions may be varied for tuning purposes.
The lengths of short co axial lines act as inductances or capacitances,
depending on their length relative to the wavelength of the oscillation.
When the length lies between 0 and 1/4 the reactance is inductive and

INSULATOR

Am/_ l u _:l R _
Gr '
_I ‘ _ |N$ULATQR
_::]‘: _ _ I _ _ _ _/ INSULATOR
I1
1 ‘i1 " |

Fro. 13.21

when it lies between 7./4 and J./2 the reactance is capacitive. Both of these
relationships are unaffected by the addition of any integral number of half
wavelengths to the length of the transmission line. It may be seen
from Fig. 13.21 that the valve and its capacitances are attached to the
two transmission lines in such a way that they conform to the simple
generalized circuit of Fig. 13.10; X2 is the cathode—grid transmission
line with C2,, in parallel, X2 is the anode—grid line with C2, in parallel and
X2 is simply C21. Since X2 is capacitive, then X2 must be capacitive
and X2 inductive. The transmission lines are adjusted to give the re
quired reactances for oscillation.
The reactances of X2, X2 and X2 have all to be suf ciently small to give
oscillation. In some cases C21 has to be increased if oscillation is to occur
in the circuitof Fig. 13.21. On the other hand, if C21, is very small there is
no oscillation, and this circuit may be used as a stable ampli er. The
signal is connected to the cathode—grid line, and the output is taken
from the anode—grid line. This is an example of a common grid ampli er.
It is frequently but inaptly called a grounded grid ampli er.
Common anode circuits have also been used with transmission lines
as oscillators for very high frequencies. The common cathode oscillator,
which is another name for the tuned anode, tuned grid circuit of Fig. 13.11,
is not suitable for operation at the highest frequencies.
 CHAPTER 14
ELECTRONS AND FIELDS
14.1. Induced Currents due to Moving Charges
In this chapter we investigate further the interchange of energy between
moving electrons and electric elds with a view to explaining the operation
of some of the special valves which are used for the generation of alternating
currents at ultra high frequencies. Consider the movement of a negative

d =1 :2
' —_ +92 K

Fro. 14.1

charge between two large parallel planes (Fig. 14.1), under the in uence of
the steady eld produced by the battery E2. When the charge —q is in
the space between the two planes it induces positive charges +q2 and
+q2 in the planes, such that
qr + Q2 = q~
When the charge leaves K,
qr = q and 92 = 0
and when it reaches A,
q2=0 and q2=q.
As the charge moves across the space the positive charge is continuously
transferred from K to A. The transfer must take place through the
battery and the total transfer is +q. When the charge moves a distance
dx in the space we assume that charge dq2 is transferred from K to A.
The electric eld strength in the space is —E2/d, and the force on the
charge is qE2/d. The work done in moving the charge the distance
dx is qE2dx/d. This energy is obtained from the battery by the transfer
of dq2 through it in the direction of the arrow. Hence
qE2dx/d = E2dq2 or dq2 = qdx/d.
206
CH. 14] ELECTRONS AND FIELDS 207
The current owing in the external circuit is
dq dx
—‘ = qu/d, where u =
dt dt

Thus the movement of the charge induces a current in the external

circuit, and the magnitude of the current depends on the velocity of the
charge. The induced current starts when the charge leaves K and ends
when it reaches A. We thus see that the motion of a charge to or from an
electrode produces a current to that electrode in the external circuit.
When there are many moving charges the external current is the sum of all
the induced currents.
Similar considerations may be applied to the movement of a charge

.__C—l 2

<5 = —E2
~v |
|< E2
_Tll_ Fro. 14.2

from the cathode to the anode of a triode as illustrated in Fig. 14.2. As

the charge moves from the cathode to the grid an induced current ows
through the grid battery in the direction of arrow 1. After the charge
passes through the grid the induced current ows through the grid and
anode batteries as shown by arrow 2. If the triode has a high ampli ca
tion factor p. there is very little charge induced on the anode when the
moving charge is in the cathode grid space, and similarly there is neg
ligible induction on the cathode after the charge passes the grid. Of
course, the values of E2 and E2 have to be such that —E2 + E2/p. is
positive in order to make the charge move from the cathode. The values
of the induced currents again depend on the charge velocity. The direc
tions of current ow in Fig. 14.1 and 14.2 show that in both cases the
current in the E2 battery is in a direction to take energy from the battery.
The motion of the charge between K and G in Fig. 14.2 causes a charging
current in the grid battery, i.e., energy is given to the battery. The
subsequent movement from G to A gives a discharging current, and an
equal quantity of energy is taken from the battery. We have here an
example of how current may ow to an electrode even when it collects
no charge from the space. Further consideration of the energy conditions
in the batteries shows that energy is given to a battery when the negative
charge moves towards a negative electrode. Energy is taken from the
208 PRINCIPLES OF ELECTRONICS [CI I.
batteries when the negative charge moves away from a negative electrode
or towards a positive electrode.

14.2. Energy Considerations

In the triode in the previous paragraph there are two separate sources
of the steady eld, namely the grid and anode batteries. The energy
relationships between the moving charge and the sources may be stated
in an alternative form. When charge moves against, or is retarded by,
the eld of force produced by a source it gives energy to that source.
When a charge moves with, or is accelerated
tr“ by, the force produced by a source, it takes
energy from the source. In the triode ex
Q7 ample both sources are steady. However,
the same energy relationship applies when
one or more of the sources is varying. This
.|o _ _ i,’ may be con rmed by reference to Class A,
B and C ampli ers in Section 8.11. There
vs it is shown how more energy can be trans
‘ 0 ' _ ' ' "f ferred to the a.c. circuit by arranging for a
greater proportion of the electrons to move
towards the anode during the negative half
cycle of the a.c. anode voltage. We may
consider the case of the Class A ampli er in
[A more detail in terms of the ideas of the
present chapter. The relative waveforms of
anode voltage, grid voltage and anode cur
O 1 rent are shown in Fig. 14.3 for a Class A
Fm 222 ampli er with a resistive load. The elec
trons which ow from the cathode to the
grid during the rst half cycle move against the alternating grid force
(the altemating grid voltage is negative). These electrons therefore give
energy to the source of the grid a.c. eld. As these same electrons move
from the grid to the anode, they are moving with the alternating grid
force (they are being accelerated by the negative a.c. grid voltage), and
so take back the energy which they have already given to the grid. At
the same time they are moving towards the anode whilst the alternating
anode force is accelerating, and so they take energy from the anode a.c.
source. During the second half cycle the electrons rst take energy
from the a.c. grid source and then return it. They now move against the
a.c. anode eld of force, and so give energy to it. More electrons ow
during the second half cycle, and on balance the electrons give more
energy to the a.c. anode source than they take from it. The resultant
energy exchange with the grid source is zero.
We have considered above each alternating eld of force separately and
ignored the d.c. elds whilst considering the exchange between the elec
14] ELECTRONS AND FIELDS 209
trons and each separate eld. There is, of course, energy exchange going
on continuously between the electrons and all the sources of eld in each
region. The d.c. sources supply the energy, some of which is transferred
to the a.c. circuits, and the whole process goes on simultaneously. In
Chapter 15 there are examples where the electrons move rst in the d.c.
eld to acquire energy, some of which they then give up in an a.c. eld.
For energy conversion from d.c. to a.c. the electrons should be given high
velocities by the d.c. source and then slowed down in the retarding part
of the a.c. eld. The slower the electrons are nally, the greater the
e iciency of energy conversion.
When an ampli er has a resistive load the anode voltage is in anti
phase with the anode current, and the concentration of the electrons at
the negative half cycle of the anode voltage is an essential feature of the

INDUCED CURRENTS INDUCED CURRENTS

.__ .
.
I I lzaassr.
A 2__ ___
(¢)

B
BEAM OF
ELECTRONS
Fro. 14.4

operation. In most ampli ers and oscillators the load is resistive (fre
quently parallel resonant) and the conditions for energy conversion arise
automatically in the adjustment of the circuits. A common example at
high frequencies of this type of operation is found in space resonators, in
which the electrons move between two electrodes which are integral parts
of the circuit. Examples are shown in Fig. 14.4.a and b. In the former
there is a section of a space resonator in which two parallel plates A and B
are joined by a doughnut shaped conductor. This resonator, which is
sometimes called a rhumbatron, is an extension of the “ lumped " circuit
in Fig. 14.4.b, where two plates act as a capacitor and are tuned to re
sonance by a number of inductors in parallel. In such a space resonator
the plates A and B are usually grids permitting the passage of a beam of
electrons normally. The beam is modulated in density so that one com
plete cycle of modulation passes through the resonator during the natural
resonant period. Induced currents ow round the doughnut and build
up the a.c. eld across AB with the retarding half cycles coinciding
with the maximum density of electrons. A second type of space re
210 PRINCIPLES OF ELECTRONICS [CI I.
sonator is shown in section in Fig. 14.4.0. In this case the beam of electrons
builds up the a.c. eld across the gap G.

14.3. The Energy Equation

When an electron moves in a steady eld between two points at poten
tials v2 and v2 the gain in kinetic energy equals the loss of potential
energy. This is expressed in the Energy Equation
eivz _ vi) = ‘imluzz _ urz)»
u2 and u2 being the velocities at the two points. In the case of a diode
with steady potential difference E2 between anode and cathode, the energy
equation becomes eE2 = imuf, where u,2 is the electron velocity at the
anode and it is assumed that the electron leaves the cathode with zero
velocity. If the diode has an alternating potential difference as well,
then the total voltage between anode and cathode is v4 = E2 + 13 sin col.
Provided the electron transit time is negligible compared with the dura
tion of one cycle of the eld, then each electron traverses the diode in a
steady eld, and the kinetic energy on arrival at the anode is given by
imuf = e(E2 + 23 sin cot).

14.4. Transit Time Loading

When the electron transit time is not negligible the electric eld changes
during the electron’s ight, and the kinetic energy at the anode is not given
by the simple energy equation. The curve in Fig. 14.5 shows the variation
in anode voltage with time in a parallel plane diode. At any instant the
force on an electron is ev_1/d if the effects of space charge are neglected.
An electron which leaves the cathode at time t2 and arrives at t2 experiences

.
I/sin (Of
"1 V2
+ 2,2
A Q ——+4 ———
ll l

0'1 '
E2 1

I
° '5 7| L2 6\ — ‘:5 _ _ :5 _ _ _ J
Fro. 14.5

a decreasing force during its transit. At the anode its kinetic energy
exceeds what it would have been if it had moved throughout in a steady
force given by ev2/d, the value when it reaches the anode. Such an
electron arrives at the anode with kinetic energy greater than the loss of
potential energy. Similarly, an electron crossing between t2 and t2 while
the force is increasing arrives at the anode with kinetic energy less than the
14] ELECTRONS AND FIELDS 211
loss of potential energy. The excess or de ciency of the kinetic energy
over the potential energy is obtained from the alternating eld. To nd
the net energy exchange between the electrons and the eld an average
must be taken over a complete cycle. These conclusions apply whether
the current owing is temperature limited or space charge limited. In
the latter case the greatest number of electrons leave the cathode when
v4 is maximum at t2, i.e., when the force on electrons is beginning to
decrease. Also, the smallest number leave at t2 when the force is starting
to increase. Thus, on balance, there is an excess of kinetic energy gained
at the expense of the a.c. eld, and the effect of the nite electron transit
time is to put an additional load on the a.c. generator. When the electron
transit time is very large the energy exchange becomes complicated.
Similar considerations may be applied to the grid circuit of a tetrode or
pentode ampli er. Usually the transit time between the grid and screen
is much less than that between cathode and grid, on account of the high
screen voltage. Transit time effects may therefore be neglected between
the grid and the screen when they begin to be appreciable in the cathode
grid space. The energy exchanges in the latter space are practically the
same as for the diode considered above. Thus in the pentode at high
frequencies the electron transit time causes a load on the grid signal,
even though no electrons are collected by the grid. This effect is fre
quently called input damping due to electron transit time. Transit time
damping also occurs in a triode. However, the conditions are complicated
by the varying anode voltage. When the anode voltage is low transit
time effects in the grid anode space must also be taken into account.
 CHAPTER 15

SPECIAL VALVES FOR VERY HIGH FREQUENCIES

15.1. The Klystron

In Chapter 14 it is shown that electrons can be used for transferring
energy from a d.c. source to an a.c. circuit. The electrons are given high
velocity by a steady eld, and are slowed down in an alternating eld by
making them traverse the latter whilst it is in its retarding phase. Any
electrons which move through the alternating eld during its accelerating
phase decrease the eld and reduce the e iciency of power conversion from
_ _. . ._ ._ I

COLLECTOR '
Z
carcnsn I
OUTPUT

onu=r
space

auucusn! I

T”
INPUT

euacrnou 2: =,_ _ _ ,
cuu

CATHOOE
Fro. 15.1

the d.c. supply to the a.c. circuit. In some electronic devices, such as
diodes, triodes and pentodes, the electrons move simultaneously in the
steady and alternating elds. In others, the two elds are separated.
The klystron ampli er is a good example of the latter. The essential
parts of one type of klystron are an electron gun to produce a beam of
high velocity electrons and two space resonators of the type described in
Section 14.2 and having the same resonant frequency. The high fre
quency elds are con ned to the spaces inside the two resonators. The
arrangement is shown in Fig. 15.1. The nal anode of the gun and the
212
C1 1.15] SPECIAL VALVES FOR HIGH FREQUENCIES 213
two resonators are at the same steady potential. The electrons emerge
from the gun with high velocity and enter the rst resonator. This
resonator is energized at its resonant frequency from some signal source,
so that there is a small alternating eld across the resonator gap. The
electrons which traverse the gap during the half cycle that the eld
is accelerating emerge with slightly increased velocity. During the other
half cycle the electrons leave the resonator with reduced velocity. The
electron beam is said to be velocity modulated by the resonator. In the
drift space between the two resonators the faster electrons overtake the
slower ones and there is electron concentration into bunches by the time
the second resonator is reached. The bunches are repeated each alternat

DISTANCE ‘

ACCELERAT ING
FORCE
__. — —>f

Fro. 15.2

ing cycle. In passing through the second resonator the bunches of charge
induce pulses of current round the resonator and a voltage builds up across
the gap at the resonant frequency. The phase of this voltage automatic
ally gives a retarding eld while the bunches are passing through, and the
electrons emerge from the second resonator with reduced velocity, having
given up their energy to the resonator. An equilibrium condition is
reached when the rate of energy removal by the output coupling equals
the rate at which energy is extracted from the beam. The electrons are
nally collected by a separate electrode, whose voltage is just su icient to
collect all the electrons.
In Section 14.4 it is shown that energy losses occur in the grid circuit
of a conventional ampli er when the electron transit time is comparable
with the high frequency period. Similar losses occur in klystron reson
ators. However, as the electrons traverse these resonators with high
H
214 PRINCIPLES OF ELECTRONICS [CH.
velocity, the electron transit time is much less than in the grid circuit of a
pentode where the voltages are low. Hence in klystrons transit time
limitations occur at considerably higher frequencies.
If space charge effects are neglected the formation of electron bunches
in the drift space of a klystron may be illustrated graphically as in Fig.
15.2. The horizontal axis represents time, and the vertical axis represents
distance along the drift tube from the rst resonator, or buncher, as it is
frequently called. On the time axis is shown one cycle of the buncher
voltage. The electrons emerge from the buncher with modulated
velocities and then move in the eld free drift space. The subsequent
distance travelled by an electron along the drift space may be found from
a straight line whose slope represents the velocity. The greatest slope
occurs when the resonator voltage is at its positive maximum (line 1)

REFLECTOR / \

assouaron m

ELECTRON
GUN

1 _ i Q Z o

Fro. 15.3

and the least at the negative maximum (line 2). The lines starting from
the buncher are equally spaced in time, representing the uniform beam
density leaving the buncher. It may be seen from this diagram that at a
distance along the drift space there are regions of high and low charge
density. Also, the separation in time of the bunches is one period.
The second resonator, or catcher, is placed where appreciable bunching has
occurred.
Considerable voltage and power ampli cation can be obtained from a
resonator klystron. If coupling is introduced between the output and
input resonators the ampli er may provide its own input and so become a
self oscillator.
In the re ex klystron which is shown diagrammatically in Fig. 15.3
only one resonator is used and the electron beam is re ected back so that
it traverses the resonator twice. In this way the one resonator acts as
both buncher and catcher. Provided the phase of the bunches is correct,
self oscillation may be obtained. The phase may be controlled by adjust
ment of the d.c. voltages of the resonator and the re ector.
15] SPECIAL VALVES FOR HIGH FREQUENCIES 215

15.2. Travelling wave Tubes

For the transfer of energy from moving electrons to an a.c. circuit the
electrons must be bunched or density modulated, and the bunches must
move in an alternating eld during the retarding half cycle. Alternating
elds can exist in the form of travelling electromagnetic waves as in a
transmission line or a waveguide. If a stream of electrons could be

._Wi@“@@@@@b .1 {_L_'._' @
(<1) (0)
Fro. 15.4

made to move with the same velocity as the travelling wave, then it would
be possible for continuous exchange of energy between the stream and the
wave. For a net gain of energy to the wave it would be necessary to
bunch the stream and for the bunches to move in the retarding regions of
the wave. This is the principle of operation of certain types of high
frequency valves, including travelling wave tubes, space charge wave
tubes and cavity magnetrons.
Waves on a transmission line or in a normal wave guide travel with
velocities of the order of the velocity of light. Electron beams with such

,\=y&’/Axsy Ki“/2,. /(Q WAVE TRAVEL

2 VOLTAGE 2

\ .. AXIA FORCE
\ *I \ ’ ‘~/'/on ELECTRONS
(0) I \
\ “L II I \ *1
Ti ‘O0 ‘
I _¢
\ _.\._

Fro. 15.5

velocities are not practicable. If the waves are to travel in step with
the electrons the wave velocity must be reduced. This can be done
in several ways, two of which are shown in Fig. 15.4.a and b. In the
former a helix of wire is enclosed inside a conducting tube. This is like
a co axial transmission line with a coiled inner conductor. The velocity
of a wave on a transmission line is equal to 1/V (LC), where L and C are
the inductance and capacitance per unit length. For the helical line,
216 PRINCIPLES OF ELECTRONICS '[CH.
C is much the same as it would be for a line with a rod inner conductor
whose diameter equalled that of the helix. However, L is very much
greater and the wave velocity along the axis is considerably reduced.
The wave is guided round the wire rather than along the axis, and the
velocity reduction is about equal to the axial length divided by the wire
length. Fig. 15.4.b shows a cylindrical wave guide with diaphragms at
intervals along the length. The axial wave velocity in such a guide is
much less than the free space velocity.
In Fig. 15.5.a the instantaneous eld is shown along the helix. The
lines and arrows show the direction of the force on electrons, i.e., in the
opposite direction to the electric eld strength. The voltage and force
/
I
I

omacruou or "rnavst . ' ggggw

.9.
— it > J ,

,5 \ § §

\ \
wave vouace 1
\ \
\

Fro. 15.6 ‘\

distribution at the same instant are shown in Fig. 15.5.b. If now a beam
of electrons travels along with a velocity equal to the wave velocity (i.e.,
the relative velocity of the electrons and the wave is zero) some are
accelerated and others retarded, and bunches gradually form at regions A.
At A the axial force is zero, and the bunched electrons then travel along
in step with the wave, but there is no further exchange of energy, since the
bunches are in regions of zero eld. If the beam velocity is slightly
greater than the wave velocity, bunches still form, but they move to
regions to the right of A, where the force is retarding. As long as these
conditions are maintained the electrons give up some of their energy to the
wave, whose amplitude increases. The relationship between the beam
and the wave along the tube is illustrated in Fig. 15.6. Initially the
beam is uniform, but it gradually becomes more bunched, and the wave
amplitude increases at the same time. Since energy is transferred to the
wave at the expense of the kinetic energy of the electrons, the electron
velocities ultimately approach the wave velocity and the bunches move
to the positions of zero eld. No further increase in amplitude is then
obtained.
15] SPECIAL VALVES FOR HIGH FREQUENCIES 217
The relation between the electrons and the eld in a travelling wave
tube may be illustrated by the movement of traf c along an undulating
road, where the undulations correspond to the variation in eld along
the helix. The excess velocity of the electrons over the wave velocity
corresponds to the mean speed of the traf c relative to the road. Be
cause of the undulations there are also uctuations in the traf c speed.
Vehicles slow down when going up hill and speed up on the down gradients.
Thus concentrations of traf c or bunches occur on the rises and there is
relatively little traf c on the descents. Similar behaviour is obtained
with the electrons and the bunches, though it must be remembered that,
as the electrons have negative charge, they concentrate in the regions
where the potential is decreasing. Also the size of the “ hills ” increases
steadily as the electrons give up energy to the wave. The traf c analogy
may also be used to illustrate the behaviour of the electrons when their

MAGNETIC FIELD COIL

:
|—rV//W////3|
’bN%®'bbbb%b®'\
' l//////////A r
INPUT OUTPUT
Fro. 15.7

velocity equals the wave velocity. The vehicles would have no mean
forward velocity relative to the road, and they would all concentrate in
the valleys, which correspond to the regions A in Fig. 15.5.a.
In a travelling wave ampli er the wave is injected into the helix with a
wave guide coupling as shown in Fig. 15.7. The output is taken from the
helix with a similar coupling arrangement. This circuit may be used for
self oscillation by feeding back some of the output to the input circuit.
There may be some re ected wave travelling backwards from the output
end of the helix to the input. If the amplitude of the re ected wave is
su iciently great this may cause self oscillation. Usually it is necessary
to introduce some attenuation in the helix to prevent oscillation in this
manner. The length of the helix in a travelling wave tube may be 20 cm
or more. An axial magnetic focusing eld is used to keep the beam inside
the helix.
One feature which distinguishes travelling wave ampli ers from other
high frequency ampli ers is the absence of resonant circuits. A wide
range of frequencies may be covered without tuning. This feature is
valuable in any system which requires a wide frequency bandwidth.
Klystron ampli ers, which use sharply tuned resonators, are limited to
narrow bandwidths.
218 PRINCIPLES OF ELECTRONICS [cH.

15.3. Linear Accelerators

When a travelling wave tube is used with the electron velocity slightly
less than the wave velocity, the bunches form in the accelerating parts
of the wave and the electron energies increase at the expense of the wave.
This principle is used in the linear accelerator for giving extremely high
velocities to electrons. If the wave velocity is gradually increased, say
by opening out the helix, the electron velocity may be increased until it
approaches close to the velocity of light.

15.4. Space charge wave Tubes

If a beam of electrons is density modulated as, say, in a long drift tube
of a klystron ampli er, there is a series of bunches along the tube with
low density regions between them. The axial electric forces set up by this
charge distribution are similar to those shown in Fig. l5.5.a. Such a
beam is equivalent to a slow travelling wave, and can be used to exchange
energy with a second beam of electrons owing along with the rst and
with a slightly different velocity. Double beam arrangements of this type
are called space charge wave tubes, or double stream ampli ers.

15.5. Cavity Magnetrons

In Fig. 15.4 two arrangements are shown for producing slow travelling
electromagnetic waves. Another possible arrangement is shown in Fig.
15.8, in which the wave travels round the slot and hole cavities and the

Q) ..........¢ FIELD ~10 ......

_ _ ' __'.. L L K
DIRECTION OF WAVE TRAVEL
Fro. 15.8

horizontal velocity is considerably less than the free space velocity.

Each slot and hole acts rather like a resonant circuit. The electric eld
is greatest near the slot and the magnetic eld within the hole. The
lines with the arrows show a possible instantaneous distribution of the
electric force on electrons, and the correspondence between this and
Fig. 15.5 is obvious. If a horizontal beam of electrons travels in the space
between A and K with the correct velocity, this arrangement produces
bunching and energy exchange between electrons and eld.
A horizontal ow of electrons may be maintained in this structure if it is
used as a planar magnetron of the type described in Section 2.8, with K
as the cathode, A as the anode and a magnetic eld into the plane of the
15] SPECIAL VALVES FOR HIGH FREQUENCIES 219
paper. With a steady electric eld, the electrons describe cycloidal
paths provided space charge effects are neglected. The nature of the
path depends on the relative sizes of the electric and magnetic elds.
In the absence of a high frequency eld and with the magnetic eld much
greater than the cut off value, an electron leaving the cathode with zero
velocity describes a path as shown in the curve in Fig. 15.9.a and just
returns to the cathode with zero velocity. There are electrons describing
similar paths all along the cathode, and there is therefore a horizontal

® MAGNETIC FIELD INTO PAPER

: (<1)

an ca
* 2,, rn ca T
+_____ A +

9 ii

___.|____ . (<=)
EB U U :0:
DIRECTION OF
TRAVEL OF WAVE
Fro. 15.9

ow of charge. If now an electromagnetic wave is introduced at the left

and travels along the system with a velocity equal to the mean horizontal
velocity of the electrons, energy interchange takes place between the elec
trons and the wave. An instantaneous distribution of electric eld is
shown in Fig. 15.9.b, where the arrows again show the direction of the
force on electrons. Electrons in regions D have the horizontal com
ponents of their velocities increased, whilst those at B lose velocity.
The effect of the increase in velocity is to increase the force (Beu) due to
the magnetic eld, and these electrons are driven back to the cathode
with appreciable velocity and are removed from the eld. Electrons
with reduced velocity have a reduced force due to the magnetic eld, and
are therefore brought to rest before they reach the cathode. They then
move on again under the in uence of electric eld (steady and varying).
As the mean horizontal velocity of the electrons equals the wave velocity_.
the electrons continue in the retarding high frequency eld and describe
220 PRINCIPLES OF ELECTRONICS [cH. 15
a path as shown, ultimately reaching the anode. From the nature of the
path it is seen that the mean velocity of the electrons remains constant
during the passage from cathode to anode. At rst sight it might appear
that they have given no energy to the wave. However, although they
have maintained their mean kinetic energy, they have lost potential
energy in moving from the zero potential cathode to the positive anode,
and this energy has been given to the wave. Thus all the unfavourable
electrons which leave the cathode at D are quickly removed, and those
from B continue to move along with the travelling wave, giving up energy
to it as they go.
This planar magnetron acts as a travelling wave ampli er. Most
cavity magnetrons are used as self oscillators, and this is achieved by
bending the planar structure into cylindrical form, as shown in Fig. 15.10,
the output end being joined to the input. For this arrangement to work
it is essential for there to be an exact number of waves round the circuit.

® OUTPUT

MAGNETIC FIELD
INTO PAPER

'1]

5
§Q$\\~
~\\}“‘\

Fro. 15.10

With the eight cavity magnetron it is possible to have one, two, three or
four complete waves. With four waves there is a phase difference of 1:
radians across each cavity, as shown by the signs in the gure; this is
known as the 1: mode of operation. There are four regions for favourable
electrons, and the electron distribution forms an axle and spoke arrange
ment as shown shaded in the gure. The eld and the electrons travel
together round the system in the direction of the arrow. The favourable
electrons in a magnetron can give up most of their potential energy
(equal to ev,1) to the high frequency circuit. The unfavourable electrons
are in the eld for only a very short time. The magnetron may therefore
operate as a generator of high ef ciency. Conversion e iciency from d.c.
to a.c. of 70 per cent can be obtained.
The main eld of use of the special valves described in this chapter is for
frequencies from about 1,000 to 100,000 Mc/s.
 CHAPTER re
RECTIFICATION
18.1. Simpli ed Diode Characteristics
The applications of diodes, whether vacuum, gas or semi conductor,
are based on the asymmetrical features of the characteristics. It may be
seen from Fig. 16.1 that these characteristics all show similar general
trends. When v4 is positive the resistance is very much lower than when

, ‘A IA ‘A
1,, '

' ‘A

0 "A O VA O ‘Ia
VACUUM DIODE SEMICONDUCTOR GAS DIODE
DIODE
FIG. 16.1

v4 is negative. When v4 is zero the current is either zero or small. In

order to simplify the analysis of circuits using diodes, the characteristics
can be assumed to have the ideal form shown in Fig. 16.2, in which 1'4 = 0
when v4 is negative, and £4 = 124/R2 when v4 is positive.

[AA

('2 _ _ _ _ __

o _ DIS. _. — _
"VA

Fro. 16.2

The rst application which is to be considered is that of the recti ca

tion of an alternating current to produce a direct current. The frequency
of the a.c. supply may be 50 c/s, but the application is not limited to this
frequency.
221
222 PRINCIPLES OF ELECTRONICS [cl I.

16.2. A.c. Supply, Diode and Resistance in Ser'ies—Hali wave Recti er

The circuit arrangement is shown in Fig. 16.3, where a transformer is
used to provide a voltage, v22 = 23 sin mt, and also to act as a low resistance
return path for the diode current. The resistance R represents the load
+"A
mi

.9 I .
1
>“'
+A X9“
“' +
MAINS
SUPPLY l I V12 R "R=R ‘A
i ‘ii 1

2
I/22 I: 9810 = VA+l/9

Fro. 16.3

'12 ___
I. A

'0 1 r 11' 2?‘

2 2

9" ' " on

' o
/\ IWO
1' _T‘

. >1
o 1' I; j

‘A ___
A
_"_
R+ R0
o ' 1' TI
Fro. 16.4

which has to be supplied with d.c. When terminal 1 is positive with

respect to terminal 2 current ows through the diode and the resistance.
On assuming the ideal diode characteristic, it is found that the current is
given by
£4 = 1%‘, sin mt from t = 0 to T/2, T to 3T[2, etc.
16] RECTIFICATION 223
At the same time the voltages across the load and the diode are
A

vR=—Ri—sinmt

and v4 = sin (oi.

0

When terminal 1 is negative with respect to terminal 2 no current ows

through the diode, so that
2'4 = 0 from t = T/2 to T, 3T/2 to 2T, etc.,
whilst vg = 0 and v4 = 13 sin cot.
The variations of current and voltage are shown graphically in Fig. 16.4.

‘A 9 . 1
.\<\\ =1
_r..\\\— , (0;
I 2 '//4////1
O
2;: Q 1:
\
.\\\\ Q. 7::
_r.\\\\\i 2 (5)
O '//I////I
I

° o
Fro. 16.5

The current through the resistance and the voltage vR across it are both
cyclic but unidirectional. Each may be separated into a steady com
ponent and a cyclic component. The steady component of the current is
the average value of 2'4, and the cyclic component has zero average value
over one cycle. The two added together give the actual 2'4. The average
value of the current is

a4 = T1 / T 44.1:
_ = ‘U/1t(R
_ + 12,).
0

The cyclic component i is shown in Fig. 16.5.a along with i4. The areas
1 and 3 together equal area 2, since the cyclic component has zero average
value. The voltages are shown similarly in Fig. 16.5.b; the steady
component of the voltage is
17,; = R54 = R13/n(R + R2).
From Fig. 16.4 it is seen that the maximum voltage across the diode
224 PRINCIPLES OF ELECTRONICS [CI ‘I.
occurs when it is not passing current and the anode is negative. This
voltage is equal to the peak value of the supply voltage, and is called the
peak inverse voltage.
The above circuit has produced a d.c. supply with a superimposed cyclic
component, which is usually called a “ ripple ”. For some purposes this
ripple is undesirable, and various methods of reducing it are described later
in this chapter. Since current is drawn from the supply only during
alternate half cycles, this circuit is known as a half wave recti er. A
more e icient arrangement, which is described in the next section, gives
full wave recti cation by using a second diode during those half cycles
when the rst diode is inoperative.

16.3. Full wave Recti cation

The circuit of the full wave recti er is shown in Fig. 16.6. Two identical

I
0 _ —
T 4A1 +
1 '12 _ v2.2. V1 VA1
_ { ii? _

SUPPLY o 0 _ — _
T R lAr*/A2
1 '22 I V2 YA2
A2 _____ +
3
V22=VA2+ ya =p$II'\ ; “V23 "I/42+ ya 9310

'12: R([A1+1A2)
Fro. 16.6

valves are used and v22 = v22 = ii sin (oi. From the circuit it is found that
v22 = v41 + U3 = 11 sin (oi and —v22 = v44 I U3 = — 13 sin wt.
From t = 0 to T/2, terminal 1 is positive with respect to terminal 2 and
diode 1 passes current. At the same time terminal 3 is negative with
respect to terminal 2 and diode 2 is non conducting. Thus for this
period of time
1'41 = L S111 cot, 1'43 = 0,
R + R2

va = l'— sin col,

R+R2

t'1=' —SlI1toi
" R I R2
and v44 = — ii (1 1 sin cot.
o
16] RECTIFICATION 225
From I = T/2 to T, diode 2 conducts and diode 1 is non conducting, and
hence
1:42 = — L Slfl oat, 4'41 = 0,
R + R2
A

U3=—lLS1HQt
R I R2 ’
R .
U41‘ = ' + o) S111 (Ct

and v42 = — R%°l'F sin mi.

o
During this time interval sin mt is negative, and hence £44 and vg are
positive. This may be veri ed readily from Fig. 16.7. Now U3 consists

,3 '2:
Q>

\ vs
O
N I O I
I0 I 70 !

141 .
' '2:

o t o t

7A1
VA2
O 1 O r
29

'* 1..
O I
O _
YR 2| i :2 O I

° "'1 Y
0 0
Fro. 16.7
226 PRINCIPLES OF ELECTRONICS [CH.
of a unidirectional voltage in half sine wave pulses, and the average
value is
13;; = 2R1?/n(R + R0).
This mean voltage is double the value obtained from the half wave
recti er, but, for the full wave ciruit, the total transformer secondary
voltage is also doubled. It may be seen that the frequency of the ripple
is twice the supply frequency. The maximum voltage across each diode
again occurs during the non conducting half cycles. Usually R0 < R,
and hence the peak inverse voltage is nearly equal to 213. With the same
approximation 133 = 2'5/1r.
16.4. Choke input Full wave Recti er
When an inductance is used in series with the load the recti er be
haviour is modi ed in certain respects. The circuit, known as a choke

1 (A1

SUPPLY 0 0 0 ~ ~ 4

E v2
‘A2
FIG. 16.8

input recti er, is shown in Fig. 16.8. Provided R0 is suf ciently low, the
voltage between terminals 4 and 2 is a series of half sine waves of ampli
tude zi as shown in Fig. l6.9.a. As before, 1140 may be separated into its
steady component 17 and a ripple component v (Fig. l6.9.b); again
17 '= ' 213/1:.
The steady voltage causes a steady current 5 through the load R, where
5 = '5/(R + R1)
and R1 is the resistance of the choke. The d.c. voltage across the load is
therefore 17R/ (R + R1), and it is seen that R1 should be small compared
with R. The cyclic voltage v gives rise to a cyclic current 1', where

v= (R+R1)i+L‘%
If L is su iciently large
dz’
v— La t

and hence
1
i=Z](:vdt,
16] RECTIFICATION 227
i.e., 1' may be determined from the area under the curve of v against t.
This is done in Fig. l6.9.c using Fig. l6.9.b. At the point B, i is zero and
then increases in the negative direction to point C. Here v goes positive
and i becomes less negative, until at D, 2' is zero and area 1 equals area 2.
The current now goes positive, passes through a maximum value at E and

“Q :> (<1)
O t

sg0= 7+ v

O
' ll. 0»)

[A1+iA2 _ (C)
'1 ._|__. _ R.
B Q.._ __
vs
O C F I
In — T
(d)
"5
O

IA: N E‘ " (e)

O I

"R
: , T (I)
I I t
O i — —Z—>

Frc 169

drops to zero again at F. The cyclic current adds to the steady current E,
giving the total current, which is supplied in alternate half cycles by the
two diodes as indicated in Fig. l6.9.d and e. The ripple voltage across
the load is iR, and this may be made very small with large L. Thus one
advantage of the choke input recti er is considerable reduction of ripple.
The individual diode currents consist essentially of rectangular pulses of
228 PRINCIPLES OF ELECTRONICS [cl I.
maximum height little in excess of the mean load current 5. This is
another feature in favour of the choke input circuit in comparison with
some other recti ers. For satisfactory operation of this circuit the choke
must have high inductance but low resistance. Also it must have high
inductance with some d.c. owing in it. To meet these requirements the
choke usually has an iron core but with an appreciable air gap.

16.5. A.c. Supply, Diode and Condenser in Series

The circuit for this case is shown in Fig. 16.10. It is assumed that
v10 = 13 sin mt and that the condenser C is uncharged at the time of
switching on. When terminal 1 is positive with respect to terminal 2,
current ows through the diode and charges up the condenser to voltage

4 VA _

1 A K {A
4. __‘ +

SUPPLY I V12 C Iyc

S 2
. d
"12=YA+Yc » Vc"q/C» IA’ q/dt
FXG. 16.10

vc. During the negative half cycle no current ows and the condenser
retains its charge, provided its insulation resistance is in nite. When
the next positive half cycle occurs the diode anode is positive with respect
to the cathode for only part of the time on account of the voltage vc.
The charging process increases vc during each positive half cycle until the
condenser is charged to a voltage equal to 13, when no further current ows.
This process may be analysed as follows using Fig. 16.11. At all times
um = v4 + ‘Ug = ii sin mt.
At t = 0, '04 is just about to become positive and current 1'4 ows through
the diode of amount
2'4 = (13 sin cot — vc)/R0 as long as 13 sin wt > vc.
While the current ows vc increases, since the condenser charge

q = fgdt and vq = q/C.

o
At time £1, v4 is zero and vc = vm, where 12¢; = 6 sin coil. At t1, 1'4 also
is zero, and it remains zero as long as v4 is negative, i.e., until t0 where
wt, = 31: — coil. The condenser voltage remains constant at 11¢ 1 during
the time t1 to t0. After £0, 1:4 is again positive, diode current ows and the
condenser charges to a voltage Ugg at time t0, where '5 sin wt, = v¢ 3.
1e] RECTIFICATION 229
Current again ceases from t0 to t0, ows from t0 to t0 and so on. An equili
brium condition is reached in which:
(a) the condenser is charged to a steady voltage equal to '5 (this is
the recti ed or d.c. voltage, and C is called the reservoir condenser),
(b) the potential difference between anode and cathode of the diode
is cyclic and equal to —13(l — sin wt), i.e., it varies from zero to
223, thus giving a peak inverse voltage of 26, and
(c) there is no diode current at any time.
The equilibrium condition is approached more quickly with smaller
C or smaller R0. Maximum diode current occurs during the rst cycle,

"12 ‘
VA

"12

o 1, :0 :0 '

Vc __ ___ _ _§‘
an Q»
UI
Q |"cs "c
_ 0 I . 0 . _>t

I _;§=' I
—r ___l_‘*!._
O‘
—r
2_ ".1.
lit
t
o
Fro. 16.11
and its value increases with the capacitance of the reservoir condenser.
This circuit has applications for small recti ers, voltage doubling and
peak voltage measurement.

16.6. Condenser input Full wave Recti er

The steady voltage across the reservoir condenser may be used for
supplying d.c. to a load resistance by connecting the latter across C.
Usually a full wave circuit is used, and this involves a centre tapped
transformer and two diodes, as shown in Fig. 16.12. With this arrange
ment a current 1'3 ows in the load where £3 = Ug/R. The condenser is
being discharged continuously by R, and it is being charged only during
those periods when the anode—cathode voltage of either diode is positive.
230 PRINCIPLES OF ELECTRONICS [CH.
In analysing this system it is assumed that the equilibrium condition
has been reached. At time t1 (Fig. 16.13) when v41 becomes positive,
V1 passes current and charges C. During the interval tl to t0 the diode
supplies current to both C and R. Then
. . . . . d
141 = Zg + an, where 13 = 11¢ /R and zg = C

1 [A1
_ +

f
Vc
ii}
+ V11 VAI
suppur 2 . Z
C lc lA1+lA2
. T "1
3 R in V2 If
‘A2
"12 = 'A1* 'c ' '2: = "A2 " "c
FIG. 16.12

From t0 to t0, £41 = £42 = 0 and z'¢ = — in. The condenser is now being
discharged. If at time £0, vc = v0, then at time r after t0,
‘Ug = v0e"/3°.
If 1 /RC is small
Ug =' v0(l — 1/RC).

At time t0, Ug = v0 =1’ v0 (1 —"’—RC.—'”)

The greatest possible value of t0 — t0 is 1/2f, where f is the supply fre

quency, hence the above approximation is justi ed if RC > 1/2f. The
waveforms of the various currents and voltages are shown in Fig. 16.13.
It is seen that there is a ripple current which can be reduced by increasing
the capacitance of the reservoir condenser. The average voltage across
the load is 17¢ and the average load current is 17¢/R. The recharging of
the condenser by the two diodes takes place during the intervals of time
equal to t0 — t1. The shorter the value of t0 — t1, the nearer 17¢ is to ii,
but the larger is the maximum current taken by each diode to produce the
average load current. The value of t0 — t1 may be shortened by in
creasing CR. Thus for a given load resistance the greater the value of the
reservoir condenser, the lower the ripple, but the greater the maximum
diode current. The condenser input recti er may require a maximum
diode current several times the mean load current. In this respect the
choke input recti er has a distinct advantage, since its maximum diode
current is approximately equal to the load current. On the other hand,
the condenser input circuit gives a recti ed voltage equal to 13 from a
16] RECTIFICATION 231
given transformer, which is about 50 per cent more than the value 213/1:
obtained from the choke input circuit.
The average recti ed voltage with the condenser input recti er is given
by
t —t
V0 Q’ ('02 + vs)/2 = '02 (1 "

As rough approximations '00 = 13 and t0 — t0 =

HCHCC g i ' )'

Also 13/R = ER = 50, the recti ed current, and so

v0 — vC ~
This shows that the recti ed voltage decreases with the current to the
load. This variation of the d.c. voltage with the load current is called the

I
V“ '12
V‘ 71 ~?
I '.3 '.4 v—— QB

'0 . ‘ t
VA:
v0 _
V2 Va
'o ; r, :0 ' “t
In I
.0
' . >(

%no fl

Fo. _,,(

{A1 : _

In [MAX
'<> v1 ' v21 v1 v2 '
F10 161's
232 PRINCIPLES OF ELECTRONICS [cH.
regulation of the recti er. In this case the regulation may be improved
by increasing C. In practice, the regulation is also affected by voltage
drop in the resistance of the transformer winding and in the diodes.
The regulation of the choke input lter depends on similar resistances,
including the resistance of the choke. However, in that case there is no
drop in voltage comparable to that due to the discharging of the reservoir
condenser. As a result, the choke input recti er has better regulation.
The condenser input full wave recti er is used mainly for small power
supplies of a few hundred volts and 200 mA or less. Larger supplies use
the choke input circuit.

16.7. Voltage doubling Circuits

It is shown in Section 16.5 that the voltage across the diode in a diode
condenser rectifying circuit varies from 0 to 26, with the greatest value

1 3

supwur I C' V1 V2
é C2
2 4
F10. 16.14

occurring when the anode is negative with respect to the cathode. If a

second diode and condenser are connected across the rst diode with the
correct polarity, as shown in Fig. 16.14, then this diode passes current
until the condenser C0 is charged to a steady voltage equal to 26, thus
giving double the recti ed voltage of the single circuit. The peak inverse

0| <>

‘‘+
1 0) so J

+
~
Q_ +.
29 '1
_.
2§
.+ _.

SUPPLY

'29’ "'25 +26‘

2 I? '
49
o~ Q)
_|+
5+ ' ' * " '1
Fro. 16.15
16] RECTIFICATION 233
voltage across the second diode is 211, and occurs when the anode is nega
tive with respect to the cathode. A third diode and condenser may be
connected across the second diode, and this condenser is also charged to a
voltage 213. The process may be continued as often as required, as illus

SUPPLY +
2i>_
* T.
FIG. 16.16

trated in Fig. 16.15, where voltages up to 611 can be obtained from six
diodes and condensers.
An alternative voltage doubling circuit is shown in Fig. 16.16. Here
two half wave recti ers use a common transformer, and the voltage
across the two reservoir condensers in series is 213.

16.8. Filter Circuits

Recti er circuits produce a current in the load which consists of a steady
part and a superimposed ripple. The amount of ripple is usually ex
pressed in terms of the ratio of the r.m.s. ripple current through the load

LORL

c, R c, c R
R >>1/ooC, R» 1/<.oC <.oL>>1/coC
Fro. 16.17 Fro. 16.18

to the mean current (or the r.m.s. ripple voltage across the load to the
mean voltage).
In the choke input recti er the ripple current through the load may be
reduced by connecting a condenser C1 in parallel with it. The cyclic
ripple current passes mainly through C1, provided ‘% < R, where the
1
ripple is assumed to be sinusoidal and of frequency f = (1)/211'. The ripple
current through the load is reduced in the ratio 1 /coC1R (see Fig. 16.17).
Further reduction in ripple may be ‘achieved by the connection of a
“ lter circuit " between the recti er output terminals and the load.
The lter circuit consists of one or more inductors and capacitors. A
single L, C lter is shown in Fig. 16.18, where col. > 1/(DC and 5'6. < R.
234 PRINCIPLES OF ELECTRONICS [Cl I.
In this circuit the condenser C1 is either the parallel condenser used above
in the choke input circuit or else the reservoir condenser in the condenser
input circuit. Using the above conditions, the magnitude of the ripple
voltage across the load is reduced to 1/co2LC of its value across C1. Any
resistance R1, associated with the inductor causes a reduction in steady
voltage across the load. The higher the supply frequency, the smaller
the values of L and C needed for ltering. Also, the ltering or smooth
ing of a full wave recti er is easier than a half wave recti er, since the
ripple has twice the frequency in the former case.

16.9. Diode Peak Voltmeter

The series diode—condenser circuit is used widely, and particularly at
high frequencies, as an a.c. voltmeter for reading the peak value of any
applied voltage. The circuit in Fig. l6.19.a has an equilibrium state
when the condenser is charged to a steady voltage vc equal to ii, and the

1 1 1
+
VA
Vu';eln0)f — T; I
0 <3 vs , C v e.s.v. 2 c R
Y"P(sln(0('1) (9! 5"/R
V¢';'

(0) (0) (c)

1: +| ,1>_|¢ 1 P1 = +
C R
R C1 ";§Y

2 2 4 _

(d) (¢)
Fro. 16.19

measurement of the peak voltage reduces to the measurement of a

steady voltage. Thus, if an electrostatic voltmeter is placed in parallel
with C, as shown in Fig. 16.l9.b, it reads the peak value of the applied
voltage. As an electrostatic voltmeter has itself a capacitance, it can act
as C. An alternative method of measuring the voltage across C is shown
in Fig. 16.19.c, which uses a voltmeter of high resistance R. The current
taken by this voltmeter must be negligible, and this means a large value
of CR, as shown in Section 16.6. For this circuit to work the source of
the a.c. voltage must have a d.c. path, otherwise there is no circuit for the
voltmeter current. The d.c. path must also have a resistance much less
16] RECTIFICATION 235
than R to avoid reduction in the voltage indicated by the voltmeter.
A circuit which avoids the need for the supply to have a d.c. path is shown
in Fig. l6.l9.d. It is based on the equilibrium voltage v4 across the diode.
For a sinusoidal applied voltage this is shown in Section 16.5 to be given
by 1/A = — 13' + 23 sin wt. The d.c. voltmeter, consisting of a high re
sistance and a galvanometer, is connected across the diode, and it measures
the steady value of the voltage, irrespective of whether or not there is a
d.c. path between terminals 1 and 2. To protect the galvanometer from
the current arising from the cyclic component of v4, it may be shunted
by a condenser as shown. The form of this circuit which is usually used
in practice is drawn in Fig. 16.l9.e, where a lter R,C1 is inserted between
the diode and the voltmeter. Across terminals 3 and 4 there is a steady
voltage 13 and a cyclic voltage of magnitude 13/{1 + R1%>”C1’}1/2, which
can be made very small. It might appear at rst sight that the resistance
R is not necessary in this circuit. However, even with terminals 1 and 2
shorted there is some diode current due to initial velocities of the electrons.
This current requires a path back to the cathode, and it produces a
voltage drop across the resistance of the path, thereby giving the diode a
negative anode voltage for its quiescent condition. Without R the re
sistance of the path depends on leakages and is inde nite. The purpose
of R is therefore to stabilize the quiescent voltage across the diode. The
value of R is normally many megohms. The steady voltage across C1
is usually measured with a d.c. ampli er; the actual peak value of the
a.c. voltage is the difference between this steady voltage and the quiescent
voltage.
Any good voltmeter should take a negligible current from the supply
being measured. The impedance at high frequencies of the diode peak
voltmeter of Fig. 16.l9.e is determined almost entirely by the effective
shunt capacitance C, between terminals 1 and 2, and this should be as
small as possible. It depends mainly on the capacitance of the diode
and stray capacitance from the condenser C to terminal 2. When the
voltmeter is used at very high frequencies the leads from the circuit
under test to terminals 1 and 2 should be as short as possible. Any
inductance in series with the input capacitance C; across terminals 1
and 2 effectively changes the input impedance to _1'(—l/(DC; + <oL);
the voltage across the capacitance C; is greater than the voltage to be
measured, as LC, is a series circuit. At the resonant frequency of this
circuit the voltmeter reading may be many times the applied voltage.
In using a diode voltmeter care must be taken when the magnitude of
the applied voltage varies with time. The rate at which equilibrium is
reached depends largely on the product CR. In order to measure rapid
changes in amplitude CR must be small in comparison with the duration
of the changes. However, CR must be large in comparison with the
alternating period if the measured voltages are to approximate to the
peak values. Thus there must be some compromise in the choice of C
when a high frequency voltage of varying amplitude is to be measured.
236 PRINCIPLES OF ELECTRONICS [CH.

16.10. Some Practical Considerations in Recti er Design

In the design of recti ers there are several important practical considera
tions, some of which have already been mentioned. In the choice of a
suitable diode there are two main factors: (a) the maximum current in
the forward direction, and (b) the maximum peak inverse voltage. The
maximum permissible values are speci ed by the valve maker. These
factors become increasingly important in larger recti ers. When large
currents are required it is obviously desirable to have the mean recti ed
current nearly equal to the maximum allowed current. The choke input
recti er is therefore preferred in such cases. The peak current demand
on switching on also favours the choke input circuit. V1/hen the current
taken from a recti er varies, then an important factor is the regulation,
i.e., the change in the output voltage when the load current varies between
its minimum and maximum values. The ratio of the change in output
voltage to the corresponding change in load current is called the internal
resistance of the supply. For good regulation the intemal resistance
should be low. In any recti er the resistance of chokes, transformer wind
ings and R0, the diode forward resistance, all contribute to the internal
resistance. Thermionic gas diodes, mercury arc diodes and crystal diodes
have very low values of R0, and are therefore used in recti ers when high
currents are required. In addition to having stability of the output
voltage against load changes, it is sometimes important to have stability
of output voltage against changes in the mains voltage. Both types of
stability may be achieved with special circuit arrangements, examples
of which are described in the next two sections. When thermionic diodes
are used in recti ers care must be taken with the insulation of the windings
of the heater transformers. This is particularly necessary in voltage
doubling circuits.

16.11. Voltage Stabilization—Gas Diode

A cold cathode gas diode can maintain a discharge in which the voltage
across the diode remains almost constant while the current varies over
considerable limits. A typical characteristic is shown in Fig. l6.20.a.
Over the current range from £0000 to z'000,, the voltage across the diode varies
from vmm to um“. The average value of v4 over the range is the main
tenance voltage '04,; vs is the striking voltage. This diode may be used
to provide a stable voltage equal to v4, across a varying load R1, with
the circuit shown in Fig. 16.20.b, in which E is the supply voltage. The
arrangement provides stability against changes in the load over quite an
appreciable range. The value of R is chosen so that 2'4 equals its maxi
mum value imu, when RL is in nite, i.e., zero load current. Then
U4 Z E "— Rimuo

But v4 = ' ‘Uy

and hence R = (E — v44)/£00,,
16] RECTIFICATION 237
When there is some load current 1'1,, the diode current adjusts itself to
give
1'4 + 1'1. = imu.
and the voltage across the load is still Uy approximately. Over the current
range 1,010 to 1,00, the diode acts as a current reservoir; as the load current
increases, the diode current decreases by a corresponding amount.

‘Al

"MAX'"""""""'
1 R
;~.

D 1,1‘ +,5 In
1'...~
. . t
_ _ 1 » VA
Q.l'{1
___9
"Mm <_¢ 4 MAX ‘
(R1 (0)
FIG. 16.20

This circuit also gives a stable voltage v4, across a xed load R1, for an
appreciable range of variation of supply voltage E.
If 1'1, = ‘Uy/R1,,
then 1'( = 1'1, + 1'4) may vary from
1'1, +1000 to 1'1, +1',00,.
The supply voltage may therefore vary from
vu + R(¢'z 1 imm) to vi! + R(1'z + ism),
and the load voltage remains nearly constant at vM.
In all uses of the circuit shown in Fig. l6.20.b it is essential to ensure
that the discharge strikes initially. This means that L must be
greater than vs when the circuit is switched on.
Gas diodes with values of v4, of 50 V and upwards are available, but one
of the disadvantages of this type of stabilizer is that the voltage cannot
be varied once a particular diode is chosen. Another limitation is that
the maximum load current is determined by the diode. An alternative
stabilizer, which gives greater freedom in these respects and at the same
time gives greater stability, is described in Section 16.12. Crystal diodes
are available with characteristics which make them suitable as stabilizers
for low voltages of the order of 5 to 10 V.
238 PRINCIPLES OF ELECTRONICS [cr I.

16.12. Voltage Stabi1ization—Feedback

By using the principle of negative feedback a power supply can be
constructed with low intemal resistance. One suitable circuit is shown
in Fig. 16.21. The load R1, is fed from the supply E through a triode
V1 in a cathode follower circuit. Part of the output voltage, across the
resistance R0, is connected to the grid of a pentode ampli er. The
actual grid cathode voltage of the pentode V2 is the difference between
the voltage across R0 and the voltage across a gas diode V3, which acts as
a diode stabilizer. The output of the pentode ampli er is direct coupled
to the grid of the triode.
The operation of this circuit is somewhat as follows. If, for any reason,
|.°
Ra

. V'
(L + V2
"y "‘v00 E
—
*' vr
R1 '62
RL VL _ '

*== V3

FIG. 16.21

the value of v1, tends to rise, then the voltage across R0 rises. The
reference voltage across the diode V3 remains constant over a range of
variation of current through it. Hence the grid cathode voltage of V2
becomes less negative, its anode current increases and its anode voltage
decreases. This means that the grid voltage of V1 becomes more negative
and the triode passes less current, thus offsetting to some extent the
original increase in v 1,. In effect, the pentode provides ampli ed negative
feedback between the input and output circuits of the triode cathode
follower, thus keeping the output voltage nearly constant. Suppose
that the supply voltage changes by amount e and gives rise to changes
v02 and v02 in the grid and anode voltages of V2. Then
‘U02 = 8 — gm2‘UggR3.
This is the input voltage to Vl and,
provided R ;,g,,,1 > l,
the output voltage of the cathode follower equals the input voltage, and
hence v; = e — g,,,2v0 R0.
But v02 = R0v;/(R1 + R0) and

so v;=e/(l+gm2R0.%)
1 2
16] RECTIFICATION 239
The factor g,,,2R, is the gain of the pentode ampli er, and the fraction
R2/(R1 + R2) need not differ greatly from unity, so that
R
"‘ = ‘/(g"‘*R‘*' R——.
13 R2)
gives a measure of the stability of this circuit against changes in the supply
voltage. It has been assumed above that the voltage across the gas diode
remained constant. Actually, the voltage does change by a small amount
in the same direction as the change across the load, so that the control
is reduced somewhat. Where very high stability is required the gas
diode is sometimes replaced by an h.t. battery. Altematively, some
improvement may be obtained if the resistance R4 to the diode is fed from
the stabilized side of the supply. The magnitude of the output voltage
may be varied by adjusting the value of R2/(R1 + R2). When this fraction
equals unity the output voltage approximately equals the maintenance
voltage of the diode. VVhen the fraction is decreased the output voltage
rises, but for small values of the fraction the stability is reduced.
 CHAPTER 17

MODULATION AND DETECTION

17.1. Modulation
If the output of an oscillator is connected to an aerial, some of the
output is radiated into space as an electromagnetic wave. A small part
of the radiation may be intercepted by a second aerial which is connected
to a receiver. In order to convey information over this communication
system some characteristic of the original oscillation must be varied in
time in accordance with the information. The sinusoidal output of the
oscillator may be varied in amplitude, giving amplitude modulation;
alternatively, the phase angle may be varied, in which case there is either
frequency or phase modulation. There are also several ways of using
pulses, which are varied by the information, and which themselves
control the output of a sinusoidal oscillator, thus producing pulse modu
lation.
Modulation techniques are also used in other applications of electronics,
such as instrumentation.

17.2. Amplitude Modulation

The main sinusoidal oscillation is known as the carrier. In amplitude
modulation the amplitude of the carrier is detennined by the instantaneous
value of the information, or modulation.

represents the carrier voltage wave and the modulating signal is given by

then the modulated carrier is given by

e = é, {I + fé1'f(t)} sin mt.

C

In the case when the modulating signal is a single sine wave,

e,,, = é,,, cos coml, the modulated carrier wave is
e = é,{l + m cos w,,,t} sin 0,1,
where m = é,,,/é,,. This quantity m is called the modulation factor or the
depth of modulation, and it should not exceed unity, otherwise distortion
is introduced into the system (see Fig. 17.1). Both experiment and
analysis show that the modulated wave may be represented by three
240
C1 1. 17] MODULATION AND DETECTION 241
separate sinusoidal components, the carrier and two side bands of angular
frequency (0¢ — com and w, + com. Since
e = é,,{l + m cos comi} sin w,,t
méc . mé,
then e = é, sin 0),! + 2 sin (0),, + w,,,)t + 3 sin (0),, — <.>,,,)t.

If this modulated voltage is produced across a resistance whose value is

the same at the carrier and the side band frequencies, the power in each

6
:4“.
' ‘ 1 O 1

2t
0‘
_.. (0) “mu ‘ (5.)MO0Ul.A1’lON
Q _ — — Q3 1
‘\

A ll
|ll \

Y j
' / f0 II lo—cI 3 Q)c
I \ \ /
,,Q>
0Q) _ ___ A ’/ A \ I
\\ mtg Q‘ " ° I’
.. .. v \ _‘
\ \

2 \‘ ' \

I I
1
’ m<1 \ ' In) 1 \

\ / \\

1 \
I s
(c) uooutxreo cannu :9 "‘ (d) oven uoouurao
| cnmea
Fro. 17.1

side band is m”/4 of the carrier power, with a maximum value of 1/4 when
m is unity.
For a more complicated modulating signal the modulated wave is
e = é,,{l + mf(t)} sin Q;
and f(t) is de ned so that it has a maximum value of unity and m can
have any value up to unity. For distortionless amplitude modulation
the envelope of the modulated wave has the same wave shape as the
modulating signal as shown in Fig. l7.l.b and c.

17.3. Circuits for Amplitude Modulation

It is not usual to attempt to modulate the amplitude of a self oscillator,
as this produces some unwanted frequency modulation as well. Ampli
tude modulation is effected in the power ampli er stages following the
oscillator. At low power levels amplitude modulation may be carried
242 PRINCIPLES OF ELECTRONICS [CI I.
out using a device with a non linear current voltage characteristic such as
a diode. The carrier and modulating voltages are connected in series
with the diode as shown in Fig. 17.2.a. The dynamic characteristic of the
diode can be expressed as a power series
i,,=av+bv2+cv3+.
where the applied voltage is given by
v = é, sin <.>,,t + é,,, cos <.>,,,t.
These equations are similar to those used in Section 8.7, where intermodu
lation in a triode is discussed. If we consider only the rst two terms in
the power series we nd that the current contains components of angular
c, (A
+ + + +
¢¢ Q Gc Q
4. V 4. V

*Q *
(Q) (6)
Fro. 17.2
frequency coc, com, 20¢, 2m,,,, 0)¢ — com andw, + com. The components in
0),, 0),, — <.>,,, and <0, + com represent the carrier and side bands. The out
put voltage of these components is proportional to
(15,, Sin 0),; + bé,,é,,, Sin (co, —{ com)! | bécém Sin (co, — co,,,)l.
The depth of modulation is given by
m bé,,é,,,. 2bé,,,
—= ac: 16 a m=—
2 é " a
If the cubic term of the power series is included the output current also
includes components of angular frequency, <0, + 2<o,,, and 0),, — 2w,,,.
These components introduce unwanted side bands whose amplitudes are
proportional to ém”. Since m is proportional to é,,,, it is important to
keep the modulation depth small to avoid distortion. In addition to the
carrier and side bands the diode current contains components at frequen
cies ¢.>,,,, 2w,,,, 3w,,,, 20)., 3co,,, etc. In practice the diode load is arranged
to have appreciable impedance only over the range covering the carrier
and side bands. One way of achieving this is shown in Fig. 17.2.b, where
the parallel resonant circuit is tuned to the carrier frequency.
Amplitude modulation may also be produced by connecting the carrier
and modulation voltages in series with the grid and cathode of a triode or
other ampli er and operating over a curved region of the dynamic grid
characteristic.
In a pentode, amplitude modulation can be achieved by applying the
cairier voltage to the control grid and the modulating voltage to the
17] MODULATION AND DETECTION 243
suppressor grid, as shown in Fig. 17.3. In this case the dynamic character
istic may be represented approximately by the expression
‘ia = £1(‘U91 + 12093) + b(U91 + k‘U93)2

and the square term gives rise to side bands as before.

In high power transmitters it is more usual to modulate the anode of
the nal stage of a power ampli er operating under Class C conditions.

+ T‘T...

FIG. 17.3

OUTPUT

l AMPLIFIER
Q e.
I_ 1" _ T C»

Q g MODULATOR E;
Q,” T E2

T
Fic. 17.4

The circuit is shown in Fig. 17.4. In a Class C ampli er the amplitude

of the sinusoidal component of the anode voltage is approximately
proportional to the quiescent value of the anode voltage, i.e., to E2’.
Thus, if the latter is varied at the modulation frequency so that
E2’ = E2(l + m cos <.>,,,t),
the alternating component of the anode voltage is
e, = é,(l + m cos <.>,,,t) sin mci.
Negative feedback can be used to reduce distortion in the modulation
process. Some of the modulated carrier is demodulated in a linear
244 PRINCIPLES OF ELECTRONICS [cH.
detector (see Section 17.7) and the output is fed to the input of the modu
lating ampli er. The action is similar to that described in Section 10.6
for the reduction of harmonic distortion.

17.4. Frequency Modulation

When a carrier wave is frequency modulated the instantaneous devia
tion of the frequency from the carrier is proportional to the instantaneous
value of the modulating signal, so that
co; = ac + /lém COS amt.
The high frequency wave completes one cycle of frequency variation
during one cycle of the modulation, as shown in Fig. 17 .5.a and b.
In the unmodulated carrier, é,, sin a,t, the total phase angle is 6, where

0,8

(0) O r

(b) F.M.

9|

(c) RM: ' '

Frc. 17.5

0 = a,t. The angular frequency a,, is seen to be the time rate of change
of 6. Similarly, the instantaneous frequency in the frequency modulated
wave is de ned as the rate of change of the instantaneous phase angle 6,
where 6= a 5». sin amt.
mm

Thus the frequency modulated voltage wave is given by

e = 2, sin {act + 18"’ sin amt}

l
17] MODULATION AND DETECTION 245
When a carrier wave is phase modulated the phase angle varies accord
ing to the modulation so that
e = é, sin {a,,t + Bém cos amt}.
Thus, in addition to the normal linear increase of the total phase angle
with time, there is a varying component of the angle which is proportional
to the modulating signal (see Fig. 17.5.c). The instantaneous angular
frequency in this case is given by

and the frequency deviation is proportional to the frequency of the

modulating signal. This means that, in a phase modulated wave, the
carrier is frequency modulated by a signal whose amplitude varies pro
portionately with the frequency of the signal. Obviously, frequency and
phase modulation are closely related.

17.5. Circuits for Frequency Modulation

An oscillator can be frequency modulated if it has, in parallel with its
resonant LC circuit, a small susceptance whose magnitude varies with the
modulating signal. If the susceptance is capacitive, Cm(t), then the re
sonant frequency is
. . 1 CM‘)
f‘ 2m/L(C + c,..(z)) f°/'\/1 + T’
where fa =
1
21r\/LC.
\rVh€n Cm(t) < C,

then f, =fo {I — C (fl

Thus if Cm(t) is proportional to the instantaneous value of the modulating

signal the oscillator is frequency modulated. As long as the output of the
oscillator remains constant over the range, then there is no amplitude
modulation. A device giving a susceptance with the required properties
for Cm(t) is the reactance valve, shown in Fig. l7.6.a. The essential features
of this circuit are a pentode valve with a capacitor C connected between
its anode and control grid, and a resistor R between the control grid
and cathode, as shown in Fig. 17.6.b. In the former diagram all the
other capacitors have negligible reactance at high frequency. The choke
Ch has a very high reactance and merely serves the purpose of feeding the
d.c. supply to the anode of the pentode. The pentode is connected across
the tuned circuit of the oscillator so that there is an alternating voltage
v, between the anode and the cathode. Provided 1/aC > R, the voltage
across R, i.e. '09, leads the anode voltage by nearly 90° and its magnitude
is z3,,aCR. The anode current of a pentode is nearly independent of the
I
246 PRINCIPLES OF ELECTRONICS [CH.
anode voltage and i,, Q gmvg. Thus the anode current has a magnitude
gmaC R13, and it also leads the anode voltage by nearly 90°. The total
current taken from the source of v,, is, therefore, of magnitude
(gmaC R + aC)'5,, and the current leads va by 90°. The effective susceptance
across v,, is (gmR + l)aC and is equivalent to a capacitance (gmR + 1)C.
If gmR > 1, then the reactance valve behaves as a capacitance of gmRC
across the tuned circuit. In some pentodes there is almost a linear re
lationship between mutual conductance and grid voltage. Then if the

L " _ ‘O I, "+
C Ch C t

gq TO Q
El OSCILLATOR "P "a
R R
O O

WE’
(0) (0)

lr I I
C

5 K
'1
(¢)
FIG. 17.6

grid voltage is varied at modulation frequency through a transformer, as

shown in Fig. 17.6.a, the capacitance gmRC is proportional to the modulat
ing signal, and this is the required condition for frequency modulation.
Using the equivalent circuit of Fig. 17.6.0 and the conditions aCR < 1
and gmR > 1, it can be con rmed that there is an effective capacitance
gmRC across v,,. It can also be shown that there is a parallel conductance
Gm given by
0,, = ...=c=R=g,,, +
The conductance is small and is usually neglected. However, as it varies
with gm and frequency, it may cause some amplitude modulation.
Some alternative reactance valve circuits are given in Exx. XVII.

17.6. Detection or Demodulation

The useful information in a modulated carrier wave is contained in the
modulation. The extraction of the information is achieved in the process
17] MODULATION AND DETECTION 247
known as detection or demodulation. In the case of amplitude modula
tion the envelope of the modulated wave has to be extracted, whereas
with frequency modulation a voltage has to be produced proportional to
the instantaneous frequency deviation from the carrier frequency.
Before detection, the modulated wave is usually ampli ed by means of a
tuned high frequency ampli er. In the detection process use is made of
the non linear characteristics of diodes, triodes or other electronic devices.
For small signals detection may be dealt with in terms of the curvature
of the characteristics. If the received signal is ampli ed suf ciently
before detection the electron device is best represented as a switch. In
this case the principle of detection is very similar to that of recti cation,
which is discussed in Chapter 16.
17.7. Detection of Amplitude modulated Waves
A thermionic or a crystal diode is frequently employed as the non
linear device. The modulated carrier voltage
e = é,(1 + m cos amt) sin act
is applied in series with the diode and a load resistance R, as shown in
. ._.__.."
9* ‘ . __VA_."
' :
lg lg . (A

* (v)
R —Il FILTER

(a) 0
1. _
lg — V°=R (A

RESISTANCE \\ \ I, / _

RESISTANCE
INFINITE
"~ ~ ' ;'
Iv,
"""".16.
''
.
O O O
O v, I t
| >

~\ (<1) (*1
\\\ I I

' (¢)
I

I I’ \
I I

Fro. 17.7
Fig. 17.7.a. The diode characteristic is assumed to be ideal, and then
the average value of the diode current over one cycle at the carrier
frequency is found by the method used in Section 16.2. It is found that
5 é,,(1 + m cos amt)_
A 1r(R + R0)
248 PRINCIPLES OF ELECTRONICS [cl I.
It is assumed here that am < ac, so that cos amt does not change ap
preciably during one cycle of the carrier frequency. In addition to the
mean anode current there is also the carrier component (see Fig. 17.7.0
and d). This component can be removed by means of a suitable lter
circuit, shown in Fig. 17.7.b. Then the output voltage consists of a

+vA
(————@

0
+

+ V VA: G V
G C o o

(<1)
VA
I I
$,(1+m) O
_ .. _ _ _. V,
f § _ _‘ ‘

... C LARGE ‘ ~
C

(6)
VA
I

I’ _
T vO
\
I
I
\
1 \ \
Q Csuatusn

(¢)
Fro. 17.8

steady part proportional to the carrier amplitude and a part, rim cos amt,
proportional to the original modulating signal, as shown in Fig. 17.7.e.
Since the amplitude rim varies as m, the modulation depth, this is called
linear detection. The linearity, of course, depends essentially on the
assumed ideal characteristics of the diode.
The detection ef ciency 1; is de ned as 13m/mé, and is given by
n = R/"(R + R0)
When R > R0 the ef ciency has a maximum value of 1 /rt. The maximum
efficiency can be increased by connecting a capacitor C across the load
17] MODULATION AND DETECTION 249
resistance, as illustrated in Fig. 17.8.a. If this condenser is made very
large the output of the detector is a steady voltage and, as shown in
Section 16.5, it is equal to the maximum value reached by the applied
voltage, i.e., (1 + m)é,, (see Fig. 17.8.b). However, if the capacitance is
reduced in value the detector output is able to follow changes at modula
tion frequencies (Fig. 17.8.0.). In order to give improved ef ciency the
time constant CR must be large compared with the period of the carrier,

+
0 + v,
0
C

R, C‘ _ (b)
I V1
R v,

t:,*~*@~—~ *5 (¢)
.
Fro. 17.9

but to follow the modulation envelope CR must be small compared with

the period of the highest modulation frequency.
Several linear detector circuits are shown in Fig. 17.9. The rst,
Fig. 17.9.a, has a lter R1, C1, and a variable resistor R2 to control the
size of the output; C2 provides d.c. separation between R2 and the recti er
circuit. Fig. 17.9.b shows parallel connection of a diode detector with
L, C ltering. Finally, Fig. 17.9.0 is an example of a push pull detector.
The assumption of the ideal characteristic for the diode detector
is justi ed only if the signal is large. For small signals the dynamic diode
characteristic is represented by a power series
1', = av I bv” +
where v = é,,(1 + m cos amt) sin a,t.
The output voltage across the load resistance includes high frequency
components which may be ltered out as before. The square terrn in the
power series gives rise to low frequency components in the output voltage
of value
bmzé “R
vm = bmé,2R cos amt + — 4% cos 2amt.
250 PRINCIPLES OF ELECTRONICS [c1~r.
The rst term is the required modulating signal, and the second represents
second harmonic distortion. If other modulation frequencies are
present there are also inter modulation terms. Thus small signal de
tection is always accompanied by appreciable distortion, and normally
this method is not used.
Detection can be achieved with triode or other valves by using the
non linear variation of anode current with grid voltage when the valve is
biased near to cut off. Alternatively, the non linear grid current grid
voltage relation near to zero grid bias may be used.
17.8. Receivers
For satisfactory radio reception the signal applied to a linear detector
must be of the order of a few volts. The received radio signal is not likely
to be more than a few millivolts, and may be much less. Considerable
AERIAL

fc /0 AUDIO
DETECWR AMPLIFIER

FIG. 17.10

ampli cation is therefore necessary. The signal covers a band of fre

quencies which includes the carrier and the side bands. To avoid
frequency distortion, the whole of this bandwidth should be ampli ed
uniformly. At the same time it is necessary to discriminate against
signals arising from other transmitters whose carrier frequencies are
adjacent to the wanted signal. For example, in a sound broadcasting
system with amplitude modulation, the side bands may cover a range on
either side of the carrier of about 5,000 c/s. The carrier frequencies
of the transmitters are separated by 9,000 c/s. Thus the receiver must
satisfy fairly stringent requirements of bandwidth and frequency selection.
Also it must have considerable voltage ampli cation prior to the de
tector. Added to all this is the requirement that the receiver must
be capable of selecting any one of many signals of widely differing radio
frequencies. One simple type of receiver is shown schematically in
Fig. 17.10. This includes a high frequency ampli er, a detector and an
audio ampli er. With this kind of receiver it is virtually impossible to
satisfy simply the above requirements of high frequency ampli cation,
bandwidth and selectivity over a wide tuning range.
17.9. Superheterodyne Reception
The difficulty of maintaining adequate and constant performance with
wide tuning is overcome to a large extent in the superheterodyne receiver,
17] MODULATION AND DETECTION 251
whose essential features are represented in the diagram in Fig. 17.11.
The main tuning element of the receiver is in the local oscillator, whose
frequency is adjusted so that it differs from the signal frequency by an
amount equal to the intermediate frequency. The signal voltage and
local oscillator voltage are both applied to a “ mixer" or frequency
changer, which is a non linear device producing output current com
ponents whose frequencies are the sum and difference of the signal and
local oscillator frequencies. The output circuit of the mixer selects the

AERIAL

1
= ’= .::.':ss.2z. ’= ’~ °"
g~\
I. <=¢~=»»
/0
LOCAL
OSCILLATOR

Fro. 17.11

difference frequency, which is also the interrnediate frequency. The

selected output contains all the components of the modulated wave,
with the intermediate frequency replacing the canier frequency. The
signal is now ampli ed at intermediate frequency, and the modulation
is extracted by using a linear detector, as before. When a different
carrier frequency is to be received the local oscillator is tuned so that the
difference between its frequency and the wanted ca.rrier is again the inter
mediate frequency. Thus gain, selectivity and bandwidth are all achieved
almost entirely at the xed intennediate frequency. In some super

+ ___ _
¢° ¢I°—¢I
+ OR
¢¢ I]. ¢I¢ OI

F10. 17.12

heterodyne receivers there is a single stage high frequency ampli er

before the mixer to give additional selectivity.
Crystal or thermionic diodes are frequently used as the non linear
devices in the mixer stage. The signal voltage and local oscillator voltage
are connected in series with the diode and the load circuit, which is
tuned to an angular frequency of either mo — a, or 0),, — coo, as shown in
Fig. 17.12. The oscillator voltage is usually much greater than the signal
252 PRINCIPLES OF ELECTRONICS [cH.
voltage. The operation of this circuit is similar to the circuit of Fig.
17.2. The diode current contains components at frequencies coo, 2am
ac, 2ac, ac — coo, ac + coo, etc. The impedance of the load is negligible
at all angular frequencies except ac — ao, and hence the output voltage
of the mixer is at this frequency. Only the carrier frequency has been
considered in the above analysis. However, the side band components
are treated similarly, and the tuned load circuit has a bandwidth su icient
to accept the side bands. Thus the modulation of the carrier wave is
transferred unchanged to the intermediate frequency wave.

: 1 F.
Q '1 ourwur

Iii ' L .
1 _
0|| —

sucmn. l I I

OSCILLATOR

Fro. 17.13

Mixing may also be achieved with specially designed multi grid valves,
much in the same way as the pentode is used for producing amplitude
modulation, as described in Section 17.3. One form of multi grid mixer
is shown in Fig. 17.13. The valve has ve grids, and it serves the triple
function of mixer, local oscillator and rst intermediate frequency
ampli er. The cathode and rst two grids act as a triode Hartley
oscillator. The signal is connected to the third grid. The fourth and
fth grids act like the screen and suppressor grids of a pentode ampli er,
and the anode circuit is tuned to the intermediate frequency. The oscil
lator operates in the Class C condition so that the anode current consists of
a series of pulses at angular frequency mo. When the small signal ec is
applied to the third grid the anode current varies linearly according to the
relation i = gmec. However, owing to the action of the oscillator grid,
gm varies at oscillator frequency and may be represented by the series
gm = go ‘I’ gr Sin wot + E2 Sin 2‘°ot "I"
The conditions are similar to those for the diode mixer and output is
obtained at the intermediate frequency.

17.10. Automatic Volume Control

In a communication system it is sometimes found that the received
signals vary slowly in magnitude due to changes in the propagation
17] MODULATION AND DETECTION 253
properties of the medium between the transmitter and the receiver.
These slow variations are known as fading, and they are detected in the
receiver along with the wanted modulations. The fading usually occurs

'|
A.V.C. s‘ R cl C
use
R1
I A.V.C. HIGH FREQUENCY
FILTER FILTER

Fro. 17.14

at a rate much slower than the lowest modulation frequencies, and it is

possible to introduce suitable compensation with a circuit arrangement
of the type shown in Fig. 17.14. The detector circuit is the same as the
one shown in Fig. 17.9.a, but it has an additional lter circuit C3, R3
which has a time constant of about 0~1 sec. Thus across C3 there is a
voltage which remains steady except for the slow variations due to fading.
This voltage can be used as a grid bias voltage for the preceding amplifying
valves, varying the gain in such a way that the effect of fading is reduced
considerably. Special amplifying valves are used in the automatic
volume control circuit. By varying the pitch of the control grid wires
along the length of the system a valve is produced whose mutual con
ductance varies over a wider range of grid bias than in a nonnal pentode.
These valves are known as variable it pentodes.

17.11. Detection ofF1 equency modulated Waves

The detection of a frequency modulated signal requires a circuit whose
output is proportional to the instantaneous frequency difference between
the modulated and unmodulated carrier. This can be achieved simply
by adjusting a resonant circuit so that the carrier frequency is in the sloping
part of the resonance curve, as shown in Fig. 17.15. Satisfactory opera
tion requires the range of frequency deviation to be within the straight
portion of the resonance curve. This is possible only for very small
deviations, and this circuit is not often used.
254 ICH

AMPLITUDE I

AMPLITUDE
VARIATION
11111111 11

O
_ ‘_
_|
__._I__;__
/
FREQUENCY
VARIATION

Fro. 17.15

V1
C 0
. R
Li R
Q :L‘
I uuii .
t

V2
R

(<1)

I’
Q‘
U‘.
H

V1
+ 4. °
Q _
+ ' yo
92 ° '2 C3
+ O

ii}
V2
+702”

(0)
Fro. 17.16
17] MODULATION AND DETECTION 255
A more useful form of detector is shown in Fig. 17.16.a, in which 01,
the frequency modulated signal, is connected across the primary section
of a tuned high frequency transformer. The windings are tuned to
resonate at the carrier frequency. The secondary inductance is centre
tapped so that equal voltages are developed across the two halves.
The high potential end of the primary circuit is joined to the centre tap
through a condenser C, whose reactance is small at high frequencies.
The two diodes and their load circuits C2, R are identical. The common
point of the load circuits is joined to the centre tap on L2 through an
inductance L2. The reactance of L2 is large compared with the reactance
of L2; L1 and L2 are effectively in parallel so that the voltage across
L 2 is 02. For sinusoidal voltages it can be shown that

2E2 rw C1,

E1 . 1 C2
R2 + ] (COL: ' ' E)

where M is the mutual inductance between L2 and L2, and R2 is the series
resistance of the secondary winding. This assumes that the circulating
current in the primary is large in comparison with the current taken from
the generator 02. At the carrier frequency acL2 = 0716 and E2 and E1
¢ 2
differ in phase by 1:/2. At a frequency ac + 8a near to resonance it
can be veri ed, that the phase difference between E2 and E1 is 1:/2 + 9'»,
where tan ¢ '=28aL2/R2. When 8a is small qb is therefore proportional
to the difference between the actual frequency of the signal and the carrier
frequency.
For determining the voltages applied to the diodes the circuit can be
redrawn as shown in Fig. 17.16.b. Then

V1=E2+E1 and V,=—E2+E1.

These voltages are recti ed separately by the diodes operating as linear

detectors. The voltage v2 across the load is proportional to the difference
between the outputs of the diodes, i.e.,

"0 = KIIE2 + Erl I E2 + E1|}

If 02 = él sin at then 02 = é2 sin (at + g —I 95).

The numbers of turns on the windings are chosen so that él = é2 and then

. 11: <6
IE2 + Ell = 281 C05 (4 'I" Q)

. .
and |—E2 + E1] = 201 S111 (21: + 95Q)
256 PRINCIPLES OF ELECTRONICS [CH.
Hence we nd that
4K‘ . .
vo = 7; sin (5 =1 Keqlx/2.

As 45 is proportional to 8a, then the output voltage is proportional to the

instantaneous difference between the modulated frequency and the carrier
frequency, which is the required condition. Since the output of this

‘J
§

§ 4;

ac "" " 1.“ ac ‘~\_

L“
51 2,0 __;.+_4>_ '1? ____..$__
Q‘
.1“
1.4)’
<6 S‘40/
9:16;
~$x ‘*6.»;‘ +1
—'11
IIIcc———n'_
6‘
13’
','{_<,\
\®x

_ _ ‘:‘_
(<1) (b) (¢)
@
~>I.<

| (4)
FIG. 17.17

circuit is proportional to the phase angle 43, it is called a phase dis

criminator. As it gives an output proportional to the frequency difference
8a, it is also a frequency discriminator. The conditions in the circuit are
shown diagrammatically in Fig. 17.17.
The output of the discriminator is proportional to the signal amplitude
as well as to the frequency deviation. Thus amplitude variations due to
interference or noise may produce unwanted output. It is usual to
eliminate these amplitude variations before the discriminator by means of
an amplitude limiter, which consists of a cut off triode clipping circuit,
similar to the one described in Section 19.2. This ensures constant
amplitude without affecting frequency variations. Thus any amplitude
modulated interference is removed.
17] MODULATION AND DETECTION 257

17.12. Automatic Frequency Control

_A frequency discriminator produces an output voltage proportional
to the difference between the frequency of a signal and some xed fre
quency. If the output voltage is used to control the bias of a reactance
valve the frequency of an oscillator may be kept automatically at a value
that gives, on the average, zero output at the detector.
 CHAPTER 18

RELAXATION OSCILLATORS AND SWITCHES

18.1. Relaxation Oscillators

Various types of valve oscillator for producing sinusoidal voltages
are discussed in Chapter 13. We now consider relaxation oscillators
which produce non sinusoidal waves. There are two distinctive features
of these oscillators. Firstly, they depend on valves acting as voltage
sensitive switches, and secondly, the frequency of oscillation is determined
by the time constants of the circuits. The switching action may cause a
sudden change or relaxation of the conditions and is responsible for the

vcl

E2 """""""""""" ;;a'
Ir 0"
II
I/’
V5 _ _ _..

R vM _ I II II . .. ..
+ E2
5"cC _ _ J_ _ _

O
an g

{If fa
5+"
iO——————4J

(0) (b)
Fro. 18.1

non sinusoidal waveform. The principle may be explained with re

ference to the circuit of Fig. l8.1.a, which shows a resistance and con
denser in series with a battery. Across the condenser is a rather special
type of switch S. This switch has the property of closing automatically
when the condenser has charged up to some speci c voltage vs. The
condenser then discharges rapidly down to some lower voltage vM, when
the switch automatically opens. The condenser charges again and the
process is repeated, giving an alternating voltage across the condenser
of the shape shown in Fig. l8.l.b. The period of the relaxation oscillation
is To, which is determined primarily by the time T1 taken for the con
denser C to charge from a voltage vM to vs. This is given by the equation
='T*'”° = (E2 vs)/(E2 W)
258
cH.l8] RELAXATION OSCILLATORS AND SWITCHES 259
The time T2 of discharge of the condenser from voltage vs down to v1; is
determined by the series resistance of the condenser plus the resistance
of the switch. Usually T, < T1, so that the period of oscillation is
nearly equal to T1, which depends on the time constant RC. It is shown
in Section 18.2 how certain types of valve may act as the switch S.
In Fig. l8.l.b the condenser is assumed to be uncharged at time t = 0.
If the switch is not present, the condenser ultimately charges up to the
battery voltage E2. The initial part of the rise of voltage is nearly linear,
but, as the condenser voltage approaches E2, the rate of rise of voltage
decreases. For linear voltage rise from vM to vs it is necessary for E, to
be considerably greater than vs. The waveform, usually called a “ saw
tooth ”, is used for the time base of a Qithode ray tube. The period of
oscillation can be altered by changing R or C, and also by changing
either vs or vy or both.

18.2. Gas Diode or Triode Oscillator

A cold cathode gas diode may be used as the switch S, as shown in
Fig. 18.2. The voltages vs and vg correspond to the striking and main
tenance voltages respectively (see
Section 5.12). This is a simple R1 R
type of oscillator whose frequency +
is altered by changing the time con
stant RC. The_ amplitude of the 5 bf, E2
oscillation is constant and equal to
vs — vy. As long as this is small ..
compared to E2, the condenser volt Fro. 18.2
age rises reasonably linearly with
time. A small resistance R1 is included to limit the diode current
during the discharge of the condenser. The actual value of R1 is de
termined from the maximum permissible diode current.
The hot cathode gas triode or thyratron may also be used as the switch

R1 R
+ +

4. g VA C V¢

Fro. 18.3

in a saw tooth generator with the circuit shown in Fig. 18.3. The
thyratron has the advantage that the striking voltage vs can be changed
by altering the value of the grid bias E1. The greater the magnitude
of the negative bias, the higher the striking voltage. However, as shown
in Section 6.12, the maintenance voltage vy is approximately equal
260 PRINCIPLES OF ELECTRONICS [C1 I.
to the ionization potential of the gas and is almost independent of the
grid bias. Thus it is possible to control the amplitude of oscillation
by varying the bias, but this is accompanied by a change of frequency.
The frequency can also be changed by varying the time constant, and this
does not alter the amplitude.
A re nement of this circuit is the use of a pentode in place of the charg
ing resistance, as shown in Fig. 18.4. The pentode has the property of

$ C v ' 1,1
, + [3 __ '02

., e . ~
‘ii’
L
11..
I
_ _ _ |
O o VA
' mm E2"% 52"»:
Fro. 18.4

passing an anode current which is almost independent of the applied

anode voltage, as long as the voltages on all the grids are kept constant,
and the anode voltage is greater than a certain minimum, vmqs. The
constant charging current produces a linear rise of voltage across C.
The anode current, and hence the frequency of oscillation, can be altered
by changing the screen voltage.

18.3. Feedback Relaxation 0scillators—The Multivibrator

In dealing with feedback oscillators in Chapter 13 it is shown that two
conditions are necessary for oscillation. The phase of the feedback
voltage must be exactly correct, and its magnitude must be suf ciently
great. If the output of a two stage resistance—capacitance coupled
ampli er is connected back to the input, as shown in Fig. l8.5.a, the mag
nitude of the feedback may be much more than enough for oscillation.
On account of the coupling condensers the phase of the feedback is exactly
l 1 . . . .
correct when 0 E; 3 C2 _ 0, i.e., at in nite frequency. In practice,
because of lead inductances and valve capacitances, this would probably
mean some fairly high frequency. However, it is found that this circuit
gives rise to a relaxation oscillation in which each valve conducts in turn
while the other is cut off; the frequency is determined by the time con
stants of the circuits. The mechanism of the oscillation may be explained
18] RELAXATION OSCILLATORS AND SWITCHES 261
roughly as follows. Because of the positive feedback, any small change
of voltage produces large and sudden changes throughout the circuit.
This results in one of the valves being driven beyond cut off and ampli
cation ceases. The grid bias of this valve now increases at a rate de

in:
R1 +"ci— R2 +"ga—
lu * C1 [A2 *¢
. 3 0
V1 Q I 1 v2 Q ()
+ vs + v.~ E3
“A1 VA:
"ca R92 9c: I

[A1 lo:
ovumnc ron
. V¢1"° Q E2R3
41 H srxrnc
Q 9 "A1 Q y "ca

E2“
V
. C
M
d
= so¢ /R... :2
I E "7 R‘ l
G

O
y
3“ 2

.
E2“"co"'1)

9 . I
b b Y ""
("1*I'z'Yc<>)
vi "___ 0 co c d c d
" 5"" t "YE: o e
VA2 0 Q vG2 b d b d t
O ¢ C

VCO 0 q

c d c
A b
O I
(d)
Fro. 18.5
termined by the circuit time constant. As the bias passes through the
cut off value ampli cation occurs again, resulting in large and sudden
changes in the opposite direction, and ending in the other valve being cut
off. The whole process is repetitive and the anode voltage waveforms
are nearly rectangular (see Fig. l8.5.d). We now attempt detailed
262 PRINCIPLES OF ELECTRONICS [cH.
explanation of these waveforms. Let it be assumed that at a certain
instant V1 is conducting with vm = 0 and is; =i1, and V2 is cut
off with vs; = vss and £42 = 0. Under these conditions
v41 = E1 — vs OR;/RG2 — Rz'1 = v1 (see Fig. l8.5.b)
and vs; = E2.
The corresponding points are marked a in Fig. 18.5.d, which shows the
variations with time of the anode and grid voltages of the two valves.
At the same instant the voltages across the condensers C1 and C1 are
Ug1=U41—Ugg=‘U1—‘U(; 9
and Ugg=‘U42—‘Ug1=E2.

Now let vs; increase slightly for any reason so that anode current just
starts to ow in V2. Then v42 becomes less than E1. As the voltage
across C1 cannot change instantaneously, then vs; goes negative by the
same amount. This reduces 2'41 so that vs; becomes greater than v1.
The voltage across C1 cannot change abruptly so that vss increases further.
This means that in increases, v42 falls and vm becomes even more nega
tive. The whole process is cumulative giving a sudden avalanche of change
in which V1 becomes cut off, whilst V2 conducts and its grid is driven
positive. Then
I41 = 0 and ‘U41 = E2 — R1i31.

The end point of the avalanche may be detennined by considering the

grid current taken by V2. If R92 is large so that ‘U02/R93 is negligible
compared with iss, then, as £41 = 0,
igg = 1:31 and U41 = E2 — R11:gg.

The avalanche has taken place instantaneously without change of con

denser voltages so that vsl must have changed from its initial value of
v1 by an amount equal to the change in v02. Thus after the avalanche,
i.e., at b in the diagrams
"41 = v1 + ‘"02 — ‘vco
so that vs, = (E2 + vso — v1) — R1is2. This equation may be represented
by a load line in a vss, 2'02 plot as shown in Fig. l8.5.c. In this diagram
is shown also the dynamic characteristic of 2'02 against vss for V2
operating with battery voltage E1 and anode load R1. The point Q in
this gure gives the end of the avalanche where
vs, = v1 and vs; = v1 + v1 — vs;,.
If £42 is the anode current of V2 at b then
vs; = E1 — Rziss = v3 (see Fig. 18.5.d)
and vsl = v3 — E1.
The condenser C1 now begins to discharge from E1 and condenser C1
begins to charge from v1 — vss. As V2 is taking grid current, the
charging of C1 is determined mainly by the time constant R1C1, as long
18] RELAXATION OSCILLATORS AND SWITCHES 263
as vs;/is, < R1 and R11. Thus C1 charges up rapidly to E2 whilst vss
falls to zero. The anode current of V2 falls somewhat and v42 rises
to v1. There is a consequent change of vs; of amount v1 — vs, although
vm still remains beyond cut off. The corresponding positions on the
waveforms are marked c. The next part of the waveform arises from
condenser C1 discharging through R1 and R111 in series, with a time constant
C1(R1 + R,1). During this discharge everything else is quiescent until

'1

O I

'2

O I

R, C1
R1.

R,» R1
Fro. 18.6

vs; reaches vs;, at points d. V1 now begins to take current and a

second avalanche occurs but in the reverse direction, giving an abrupt
transfer of current from V2 to V1. The second avalanche is limited
as before, and the whole process is repetitive with waveforms shown in
Fig. l8.5.d. In drawing these diagrams it has been assumed that the
circuit is symmetrical with identical valves and R1 = R1, C1 = C1
and R11 = R92. This relaxation oscillator is known as a free running
multivibrator.
264 PRINCIPLES OF ELECTRONICS [CH
A circuit for a transistor multivibrator is shown in Fig. 18.6. The
principle of operation is similar to the one described above, and the wave
forms are summarized in the diagram.
To some extent the multivibrator may be likened to a limiting case of
a squegging oscillator of the type described in Section 13.5. The valves
attempt to build up an oscillation at a very high frequency in agreement
with the phase relation mentioned at the beginning of this section.
However, the oscillation builds up only part of one quarter cycle before
the bias condition of one valve causes the oscillation to cease. The bias
now changes slowly until conditions for oscillation are again correct,
when a further quarter cycle occurs before the oscillation stops once more.
18.4. The Transitron Relaxation Oscillator
Another relaxation oscillator using feedback may be based on the
transitron circuit which is described in Section 13.8. When such a
circuit has a resistive load (R in Fig. 18.7) the output voltage v,1 is in

,. vs I R
'93
"ca '62
E3 _ _ IE2

Fro. 18.7
phase with the input voltage v13. Thus direct feedback between screen
and suppressor may give the required phase conditions, and the mag
nitude of the feedback may be more than is necessary for oscillation.
In order to separate the d.c. supplies, the feedback is introduced by means
of a capacitance C as shown in Fig. 18.8.a. The presence of C gives the

"ca

R;
+ ° a "l
,_ +
__ "ca
"ca V02 E
2
EFF". ___
O _—| __>r

(<1) (b)
Fro. 18.8
18] RELAXATION OSCILLATORS AND SWITCHES 265
correct phase condition for oscillation only at very high frequency. It
may be seen that the conditions are similar to those in the multivibrator,
and again relaxation oscillations are obtained. Sudden large changes
occur in vss and vs;, with resulting waveforms of the type shown in
Fig. 18.8.b. The limits to the changes are set by the regions of the
characteristics where the suppressor voltage no longer affects the screen
current and ampli cation ceases. The recovery time after each sudden
change depends on the time constant C (R1 + R1).

18.5. The Blocking Oscillator

Frequently it is required to produce voltage pulses whose width is much
less than the separation between the pulses. It is possible to do this
with a multivibrator by proper choice of the circuit components. How
ever, a more suitable arrangement is the blocking oscillator, which is

t|l _
OUT
E=
C
O
+ g VA E2 v
R V5 G
O
v I
CUTOFF __ ___________ ....

Yo .. ..

(<1) (b)
Fro. 18.9

shown in Fig. 18.9.a. In Section 18.3 the multivibrator is likened to a

very high frequency squegging oscillator which has excessive feedback
and which builds up only a fraction of one cycle of the high frequency
oscillation before the valve is cut off. The blocking oscillator is similar
to some extent, but it uses a single valve with mutual inductance feed
back, usually through an iron cored transformer. The detailed action
of the oscillator is quite complicated, as it depends not only on the non
linear characteristics of the triode but also to some extent on the mag
netic saturation of the transformer. The sequence of operations may be
followed by assuming that the grid of the triode is initially beyond cut off
266 PRINCIPLES OF ELECTRONICS [cl I.
but is becoming less negative. W'hen the grid rises above the cut off
value anode current begins to ow and an induced voltage, proportional
to the rate of change of anode current, makes the grid less negative, thus
increasing the anode current even further. The action of the feedback
is to give a sharply decreasing anode voltage and rising grid voltage, as
shown in Fig. 18.9.b. Two factors bring this feedback action to a stop.
Firstly, when the grid becomes positive there is effectively across the
transformer a low resistance damping the oscillation, and secondly, the
mutual conductance of the triode is decreasing. The grid voltage then
falls slowly for a short time until it again becomes slightly negative, whilst
the anode goes more positive. The effect of removing the load from the
transformer together with the falling anode current is to produce a rapid
feedback action in the opposite direction, which drives the grid beyond
cut off and charges the condenser C to a voltage vo. The condenser now
discharges through the resistance R. When the grid voltage again reaches
the cut off value the whole process is repeated. The length of the pulse
depends upon the characteristics of the transforrner and the value of C,
and is usually much shorter than the time interval of repetition. The
latter is deterrnined mainly by the time constant of the RC circuit, so
that the frequency varies approximately linearly with the reciprocal
of R.
18.6. Monostable Circuits
The circuits described in this section have one stable and one unstable
state. When a suitable signal is applied the circuit goes from one state
to the other, and then returns to the stable condition after a time de
termined by the constants of the circuit. The circuit produces an output

"so R v¢I
O I "E2 """"" “"

B 1
O—— | Q C_ E2
SIGNAL
PULSE
0
0
E1 V“ (

(v) ' W
Fro. 18.10

voltage of xed magnitude and waveform for input signals which may
vary in amplitude and shape over wide limits.
A monostable thyratron circuit is shown in Fig. 18.10. The stable
state occurs when the anode voltage is below the striking value, and the
condenser C is charged to a voltage E1. When a positive pulse is applied
18] RELAXATION OSCILLATORS AND SWITCHES 267
to the grid the thyratron strikes and rapidly discharges the condenser
down to the maintenance voltage of the thyratron when the discharge is
extinguished. The condenser then recharges to a voltage E1 through
resistance R, returning the circuit to its stable condition, in which it
remains until it is triggered again by another signal.
In the multivibrator described in Section 18.3 the operation is closely
connected with the charging and discharging of the coupling capacitances
between the stages of the two stage resistance loaded ampli er. We
know that such ampli ers may have direct coupling between the stages,
and it is interesting to consider the effect of this on the behaviour of the
multivibrator. In the circuit shown in Fig. 18.11 one of the coupling
condensers is replaced by a battery E1. We assume that the circuit has
initially both valves operating with zero grid voltage, and we nd the
. . 1,0

"eo R R E2
° r +
e v E v2 <3
on .‘ ‘ 1, =E. ° '
sucmu. v v °
PULSE R, v61 A‘ vs; *2
D
Q. .._
_ __ _.

Fro. 18.11

effect of a small change in one of them. For example, let the rst grid
voltage become slightly negative. This change is ampli ed by the two
stages and is fed back to the rst valve, making its grid much more
negative. An avalanche occurs and the valve is driven beyond cut off.
At the same time the second valve passes a large current and vs, drops to a
low value. The condenser C then discharges at a rate depending on
C (R + R1), just as in the free running multivibrator, and the negative
grid voltage on the rst valve decreases until current starts to ow.
This initiates another avalanche which makes the rst grid positive
and the second grid negative. At the same time the rst anode voltage
drops to a low value. Now the actual value of the second grid voltage
is given by vs; = vs; — E1. It is possible therefore, by suitable choice
of E1, for vs, to cut off the current in V2 after the second avalanche.
There is now no mechanism in the circuit to change this condition. Thus,
this circuit has one unstable condition with V1 cut off and one
stable condition with V2 cut off; hence the name monostable multi
vibrator, or univibrator. VVhen in its stable state, the second valve
may be rendered conducting by the application of a positive signal to its
grid. This circuit has many applications as a switch or relay, whose
operation is controlled by an external signal or trigger of the correct
polarity. After operation the relay is automatically reset. When re
268 PRINCIPLES OF ELECTRONICS [cH.
quired to operate on receiving a negative signal the latter is connected to
the rst grid.
Practical univibrator circuits do not have battery coupling. One of
the alternative methods of direct coupling is used, as described in Chapter
12. Two circuits using a third rail and cathode coupling are shown in
Fig. 18.12.a and b respectively. An interesting point in the circuit of
Fig. 18.12.b is that the grid resistance R, is returned to the h.t. positive

_
Ir ._.__
(0)

5 Q "1'
7 ? V3 l

(b)
FIG. 18.12

rail instead of h.t. negative. Following an unstable avalanche making

vs; very negative, the grid voltage gradually becomes less negative until
the stable avalanche takes place. When R, is returned to h.t. negative
the cut off value of vs; occurs near the end of the discharge of C when the
voltage is varying slowly, and the time of occurrence is rather uncertain.
When R, is returned to h.t. positive the cut off voltage occurs whilst the
discharge rate is still changing rapidly and the time of occurrence is more
certain. This arrangement may also be used with advantage in the
free running multivibrator, particularly when R, is large.
The blocking oscillator circuit can be adapted for monostable working
18] RELAXATION OSCILLATORS AND SWITCHES 269
by using a grid battery which cuts off the anode current in the stable
condition, as shown in Fig. 18.13. When a positive pulse is applied to the
grid, then one cycle of oscillation is induced just as in the free running

'eo

OB Ra

OI air
sncmt C E3
PULSE
, .
R ,0
o E1 _
FIG. 18.13

circuit. At the end of the pulse of anode current the condenser C is

charged to a voltage somewhat greater than E1 and then discharges to its
original state. The short pulse of output voltage can be made nearly
rectangular in shape by adjusting the value of resistance R1.

18.7. Bistable Ci1"cuits

Some circuits possess two separate stable states and can be transferred
from one to the other by suitable input signals. However, they can exist
inde nitely in any one of the two states when required. The circuit of
Fig. 18.14 shows a bistable thyratron circuit. In one stable condition the

V VFO t VAC

1:01 + |.€ O
B __ E;
O O—I E mése E1
t '1‘
PULSE
o "*
D E1 _. V“ ‘I 1 I
O T ' 'o m€.)se 91%? _ T‘

Fro. 18.14

anode voltage is less than the striking voltage. The application of a

positive pulse to the grid causes the thyratron to strike, giving the second
stable condition, which is unaffected by any further positive pulses to the
grid. The new condition remains until a negative pulse is applied to the
anode, extinguishing the discharge and retuming the circuit to its original
270 PRINCIPLES OF ELECTRONICS [cm
stable state, which in turn is unaffected by further negative pulses to the
anode.
The substitution of direct coupling for one of the coupling condensers
in the free running multivibrator gives a circuit with one stable and one
unstable state. When the second coupling condenser is also replaced by
direct coupling, as in Fig. 18.l5.a or b, there are two stable states. When

Q Q =
(0)

2El‘
2
I
(6)
FIG. 18.15

this circuit receives a suitable triggering signal at one grid it changes

from one stable state to another. It returns to its original state when a
signal of reversed polarity is applied to the same grid. This circuit
may be used as a switch or relay when it is desired that the action be con
trolled by two separate signals.

18.8. Oathode coupled Trigger Circuit

A direct coupled multivibrator can have two stable states. If a
common cathode resistor is used for coupling, then bistable operation
18] RELAXATION OSCILLATORS AND SWITCHES 271

...Q “* . V2 Q"
Vci
‘I * __
"ca yo E;
Q ) Q

v1 * . 1
Rx V5 §R2IV2 I

Q " T _ . i. E“ A

Fro. 18.16

"01 I 1 '1 A I 1
Q >
' 4—' |:::::f.\

> i
'0 0
,5 '\

v2‘ E E
1_ E2 — ..

0 1 — 71 O ,
' |

.1, . :

O 91

Fro. 18.17
is still obtained, but the existence of either state depends on whether the
signal voltage is above or below a threshold value. The circuit is there
fore able to discriminate between the amplitudes of signals. It is shown
in Fig. 18.16, and it is sometimes called the Schmitt Trigger circuit.
One stable state occurs with zero input voltage and with V1 cut off.
Its grid cathode voltage is equal to vB, which is due to the anode current
of V2 owing through the cathode resistor Rs. Under this condition
272 PRINCIPLES OF ELECTRONICS [CI—1.
vss is slightly negative. This voltage is determined by vs and the potential
divider R, R1, R2. If v1 is now increased to approach vs, V1 begins to
pass current and causes a fall in vsl. As a result vss becomes more
negative and vs decreases, thus increasing the current in V1. There
is a sudden transfer of current from V2 to V1 until the second valve is
cut off, giving the second stable state, which is una ected by further
increase of v1, though vs now varies with v1. When v1 is reduced the
circuit remains in its second stable state until V2 starts to take
current, when there is a sudden transfer back to the original stable state.
The diagrams in Fig. 18.17 show the voltage variations. The threshold
voltage depends to some extent on whether v1 is increasing or decreasing,
but the “ backlash” Av1 can be made small compared with v1. The
capacitance C1 is included to avoid the delay in transfer of rapid changes
in vs; to vss arising from the stray capacitance across R2.

18.9. Counting and Scaling

Some of the circuits described in this chapter are used for counting
electronically, as in digital computing machines or in counting the pulses
produced by radiation particles in ionization counters. In any counting
process the principle of scaling is used. In ordinary arithmetical count

'eo B ournur
'1 —

" 7.‘

D
O

Fro. 18.18

ing we use the decimal scale of 10. Any particular number is expressed
in powers of 10, e.g., 971 is equal to 9 X 102 + 7 >< 101 + 1 >< 10°.
Electronically it is convenient to use the binary scale of 2, and this can
be done by means of a number of bistable multivibrator circuits. In
Section 18.7 it is shown that a bistable circuit requires two signals to take
the circuit through one complete cycle of operation and that two signal
pulses produce only one output pulse from the anode of one of the valves.
If this output pulse is passed on to a second bistable circuit, then four
18] RELAXATION OSCILLATORS AND SWITCHES 273
signal pulses are required to produce one output pulse from an anode of the
second pair of valves. Each pair of valves gives a division by 2 of the
number of signal pulses and so a counting system in the scale of 2 can be
produced with a series of bistable circuits in tandem. If there are n
circuits, pulses can be counted up to 2". One basic bistable circuit for
binary scaling is shown in Fig. 18.18. The bistable multivibrator of
valves V3 and V4 is the same as that of Fig. 18.l5.b except that the con
densers C are added to speed up the response to rapid changes. The
diodes V1 and V2 are used to feed the signal pulses to V3 and V4. Some
times a bistable circuit has the signals connected directly to both valves
through isolating condensers. This may work all right, but the applica
tion of a large negative pulse to both valves simultaneously may give rise
to undesirable effects. The purpose of the diodes in Fig. 18.18 is to ensure
that the signal pulse affects only one of the valves V3 or V4 at a time.
If the circuit is quiescent with V3 passing current and V4 cut off, then the
anode of V3, is at a lower potential than the anode of V4. Thus the
negative voltage across V1 is less than the negative voltage across V2.
When the negative signal pulse is applied to the common cathode of V1
and V2, V1 conducts but not V2, and the triggering signal is applied to
V4 but not to V3. In the other stable state the reverse situation exists.
This is sometimes referred to as a steering circuit.

18.10. Decade
An electronic decade counter can be produced using four bistable
multivibrators suitably connected (see Exx. XVIII). An interesting cold
cathode gas valve, called a Dekatron, has been specially designed for decade
counting. Fig. 18.19 shows diagrammatically one type of Dekatron,
in which there is a cylindrical anode, surrounded by nine cathodes“ a ”, ten
cathodes “ b ”, ten cathodes " c ” and one cathode “ d ”. All the cathodes
of one type are connected together, and the “ d ” cathode is normally at

I — 1*) '" 1 ZZ
~12/21 .s.._:§
0 0 ' E '
O b 9

c
0 .

b .

<>¢__ _1
°’ *1 t

Fro. 18.19
274 PRINCIPLES OF ELECTRONICS [cn.
the same potential as the “ a ” cathodes. Each “ a ” cathode has a “ b ”
and a “ c " electrode on either side of it. Under quiescent conditions the
“ b " and “ c ” cathodes are at the same negative potential with respect
to the anode and the “ a " cathodes are more negative still. Thus a local
glow discharge is set up between one “ a ” cathode and the anode. A
signal pulse now makes the “ b " cathodes more negative than the “ a ”
cathodes and the discharge moves to the adjacent “b” electrode. A
short time later the same pulse is arranged to make the “ c ” cathodes
even more negative so that the discharge moves on one place to a “ c ”
electrode. The “ b " and “ c ” potentials now return to their quiescent
values and the discharge moves to the next “ a ” cathode. The signal
pulse has thus advanced the discharge in a clockwise direction by one
“ a ” cathode. The single “ d ” cathode functions similarly to the “ a "
cathodes, but, although it is at the same potential as the “ a ” electrodes,
it is connected separately through a resistance. When the discharge
reaches the “ d " electrode there is a pulse of current through the resist

INPUT PULSE B
yao O

O t R I
10¢ O ——
1 toes OUTPUT _
Qq O

c R2
D
FIG. 18.20

ance. Thus for every ten signal pulses there is one output pulse, which
may, after suitable ampli cation, be passed on to another similar Deka
tron. One output pulse is obtained from the second Dekatron for every
100 signal pulses. Provision is usually made for zero setting by switching
a large negative pulse to the “ d ” electrode so that the discharge moves
to that electrode. Normally the gas discharge can be seen through the
end of the Dekatron, and a scale marked 0 to 9 gives a visible indication
of the count.
A typical circuit for use with a Dekatron is shown in Fig. 18.20. The
potential divider R1, R2 ensures that about half of the input pulse is
applied to the “ b ” cathodes. At a short time later, determined by the
RC circuit, the full pulse is applied to the “ c ” cathodes.

18.11. Amplitude Control and Discrimination

For satisfactory operation of the counting circuits described in Sections
18.9 and 18.10 it is desirable to have pulses of constant amplitude. This
18] RELAXATION OSCILLATORS AND SWITCHES 275
can be achieved by using the signal pulse to operate a monostable multi
vibrator (see Section 18.6). As long as the signal pulse is su icient to
operate the multivibrator the latter produces an output pulse of constant
amplitude and shape.
Sometimes it is desired to count only pulses which are greater than a
certain amplitude. This can be done readily with a cathode coupled
trigger circuit (see Section 18.8). Frequently, counting circuits embrace a
combination of trigger, switching, shaping and sca.ling circuits.
 CHAPTER 19

WAVE SHAPING

19.1. Wave shaping Circuits

In the operation of certain electronic circuits it is essential for the signal
to have some particular waveform. For example, pulses may be re
quired with at top, steep sides, suitable duration or suitable magnitudes.
To achieve these requirements the signal may have to pass through a
wave shaping circuit. There are two main classes of such circuits. In
the rst, use is made of the non linear characteristics of valves for changing
the shape of a sinusoidal or other type of signal. In the second class,
the transient properties of linear circuits are exploited.

19.2. Non linear Wave shaping Circuits

The asymmetric conductivity of thermionic or crystal diodes may be
used for wave shaping. For example, if a sinusoidal voltage is applied to a
resistance and a diode in series, as shown in Fig. 19.1, the voltage across
the resistance consists of a series of half sine waves. Provided the series
resistance is much greater than the diode forward resistance, the voltage

KI

+vp a
0
R + 4. O I
+

V‘ 6 VA Yo

_ VAI
1I __'°~
Re
‘ 119+". ,
RESISTANCE
lo c
R, vs
nesusrmce VA
INFINITE
"R 9, R
/I ‘H R2
O I

FIG. 19.1
276
CH. 19] WAVE SHAPING 277
across the diode also consists very nearly of half sines. The diode has
clipped or limited the waveform. When, as in Fig. 19.2, there is a
battery in the series circuit another clipped waveform is obtained. By

'2
_ ¢
o
E1 ''' " ' “'’ “' “ ’ ' ' °'

_ _ (_1R
35>+ 11.!“
O
"5
o
+vA
ii}

+
21+ (<1)
_ El R Y.
ll 1'0
R
+
+ S.

" '° rs)

Va‘

E1
. O. __ I

A
V8
I ._ _

Fro. 19.2
using two diodes in the circuit of Fig. 19.3 both halves of the sine wave
are limited.
Wave clipping is frequently achieved by means of the non linear grid
characteristic of a triode. With a high resistance in series with the input
K
273 PRINCIPLES OF ELECTRONICS [CH

"0

+ 1 I _ t

I II I
If
+® ..
‘Q
.. E
+
yo (
J,“
I"
r .O _—————' —_

In
)
Q‘ 1/
I "' E3 0‘ —E2) O_____ __ ¢\

Fro. 19.3

signal the grid and cathode can act together like the diode in the previous
paragraph. VVhen the grid is driven positive the grid cathode voltage
is limited, and the resulting output voltage takes the forrn shown in
Fig. 19.4. When the grid voltage is positive the cathode current is
shared between the grid and the anode. As the voltage rises an increasing
share of the current goes to the grid, and the anode current passes through

1| [A l
DYNAMIC
CHAIACTERISTIC

>'§.

K.
10 5*
O O M
+ 0‘$
P __I'l‘I ...;.6*
.

'3
M"
‘I’ +A
v2
~ E
3
" o _v2
J‘
'0
_ _ _ 'A.Ea RlA

Fro. 19.4

a maximum value as seen in Fig. 19.5. Thus wave clipping can arise even
without a high resistance in series with the grid signal. If, at the same
time, the signal is sufficiently large to give cut off over part of the cycle
the other half of the waveform is also limited, as shown in the diagram.
By suitable adjustment of the grid bias cut off clipping is easily achieved.
A two stage ampli er can give cut off clipping in each stage. Because of
the phase reversal in a resistance loaded ampli er both peaks are limited
as shown in Fig. 19.6.
19] WAVE SHAPING 279
A junction transistor ampli er of the form illustrated in Fig. 19.7 can
be used for shaping a signal waveform. The transistor ampli es linearly
only for a limited range of signal. Cut off occurs when the base is positive

R
ls I
51
+ "A
*3 vs ‘E’

I [A 11’
DYNAMIC
CI IARACTERISTIC

is
— O ya O
I
F.

O 0‘

m
yA.Ea RIA

Fro. 19.5

with respect to the emitter. Also, with a large enough signal, particularly
when the load resistance is high, the collector voltage is almost zero over
part of the negative peak. Thus both halves of the signal are limited in
the manner shown.

19.3. Clamping Circuits and d.c. Restoration

In the diode clipping circuit of Fig. 19.1 the output voltage across the
diode does not rise appreciably above zero. The output voltage is said to
be clamped at zero. Similarly, vs in Fig. 19.2.a and v2 in Fig. 19.2.b are
clamped at zero and E1 respectively. In the triode circuit of Fig. 19.4
the grid voltage is clamped at zero.
A clamping circuit is frequently used to restore the d.c. component
of a pulse which has been passed through an a.c. ampli er or through
any other circuit which does not pass d.c. In Fig. 19.8.a is shown a
280 crr

VA!
E1 """" "'

_.>f
O

VA2
E; _

..o._ — >(

+ ‘I’

, OQ VAI "A2 E2
'~ —T _ — A

Fro. 19.6

+
"cs
I
'c

+
E:

Fro. 19.7
19] WAVE SHAPING 281
signal pulse which has a d.c. component. If this is passed through a
transformer or an a.c. ampli er it is possible to reproduce the shape of the
pulse without distortion, but the d.c. component disappears and the average

ye

(<1)
O z
T C
V + 1
I

(0) " R '°

t _ _
'0
(¢)
v0

(d)
O I

FIG. 19.8

value of the output is zero as shown in Fig. 19.8.b. The d.c. component
of the pulse can be restored by using the circuit of Fig. 19.8.c, in which
the time constant RC is large compared with one complete cycle of the
pulse. During the negative portion of the pulse the diode conducts and

'0

C I
+ + °

V1 R yo

Fro. 19.9

charges condenser C. This continues until at no point of the cycle does the
total voltage v2 fall below zero, as shown in Fig. 19.8.d. Whatever the
mean level v1 of the pulse, this circuit automatically produces positive
output pulses with zero level at the base. The circuit in Fig. 19.9 gives a
negative output pulse which just rises to zero.
282 PRINCIPLES OF ELECTRONICS [cl I.

19.4. Linear Wave Shaping—Di erentiating and Integrating

In Chapter 11 we consider the transient response of various circuits
possessing resistance, capacitance and inductance. When the time con
stants of the circuits are very small or very large the response may be
used to modify certain wave forins. Consider the circuit of Fig. 19.10.a,
I Y
+£
c 1
+ +

'= R v, (<1)
Y. v,

o 1 o 1

'1, v,

o 3 o s

,1 (b) (¢)

O t y°

v. ° '

° ' (¢)
(4)
Fro. 19.10

in which a voltage v, varying in time is connected in series with a resistor

R and a condenser C. The circuit equations are
. dvs
‘ = C 2;’
dv,
‘U0 3 E

and v, = v, + v,.
19] WAVE SHAPING 283
If RC is very small in comparison with the time required for any signal
change, then v, '= vs and
dv,
v, RC W
Thus the output voltage is approximately equal to the differential co
efficient of the signal voltage. The response of this circuit to three different
signals is shown in Fig. l9.10.b, c and d. The sharp spikes in Fig. 19.10.d
in response to the step signals are really limiting values of the exponential
waveforms of Fig. l9.10.e. The smaller the value of RC, the more
nearly does the actual response correspond to Fig. 19.l0.d. Another

R
+ 'I'

K yo

Fro. 19.11
differentiating circuit may be achieved with resistance and inductance
connected as shown in Fig. 19.11. Provided that the time constant
L/R is suf ciently small it follows that
Ldv, .
vo.'=. '
Rdt
When the positions of the components in Fig. 19.10.a or Fig. 19.11 are
interchanged then the circuits give a response corresponding to the
mathematical process of integration, but now it is necessary for the time

R ‘I
n
4‘

V; V0

_ _. 0 O t

(0)

'._
H R
+
:1,
AA ,
(0) Fro. 19.12
Kc)
284 PRINCIPLES OF ELECTRONICS [CI 1.
constants. to be large in comparison with the time of duration of signal
changes. In Fig. l9.12.a
I . l
'Us=C.[Zdt=~1?€[URdt

1
and v,=vs+1 etfvsdt.

Then, provided RC is sufficiently great, v, '= vs and

v, =' F16 . I v,dt.

In the LR circuit of Fig. 19.12.b it may be shown similarly that

v,’ =%[v,dt

provided that L/R is sufficiently large. These integrating circuits give

an output of triangular waveform in response to a square wave signal,
as shown in Fig. 19.12.c.

19.5. Electronic Integrating Circuits

An interesting electronic integrator can be realized with a single
pentode valve and by exploiting the Miller Effect. The circuit is shown
in Fig. 19.13.a, in which there is a capacitance C between the anode and
control grid of a pentode ampli er. I t is shown in Section 10.11 that
the effective capacitance between grid and cathode in this circuit is

's

I
O I

E1
R1 C I *
+
gq
QQ
‘
V __i__
A
_
'“
E1
'
vs "c vs

(01 . ,
as
FIG. 19.13
19] WAVE SHAPING 285
(A + 1)C, where A is the voltage gain of the ampli er and, in this case,
A = g,,,R2. The time constant of the grid circuit is R1C(1+ g,,,R2),
which is much greater than R1C. When a step signal is applied to the
grid circuit it is effectively integrated, the grid voltage rises linearly and
the ampli ed output voltage drops linearly. If R1 is large the input

'6:

R2 .0 ...:

c ‘
R1 i “' + '= VA

‘I V ' ' ' ' ' * ' ' "'

nvcail
T T 1 .
()
0 O (0) 1 I

FIG. 19.14

voltage is clamped at vs = 0, so that the linear variations cease abruptly.

The waveforms are shown in Fig. 19.13.b. The Miller integrator is usually
controlled on the suppressor grid. The suppressor is biased negatively
so that the anode current is zero and the cathode current ows to the
VA“

R1

g =

I
FIG. 19.15

screen. A positive pulse is applied to C3, so that anode current starts to

ow, and there is a sudden drop in anode voltage which is passed on to
G1, through C, and the integrating action occurs as the grid voltage
returns to its quiescent value. The circuit and waveforms are shown in
Fig. 19.14.a and b. This circuit may be used for producing a single
stroke time base for a cathode ray tube. A continuous time base can
286 PRINCIPLES OF ELECTRONICS [CH.
be achieved by combining this integrator with the transitron relaxation
oscillator described in Section 18.4. Successive sweeps are initiated by
each transitron avalanche, but the ow of anode current is controlled by

‘I’
E‘ Q+ vs
VA R
vs VG
_ _ _ E2

(0)
'6
O I
51
cur o|=|=

VA
E2 ____ _ :;_;::":="'==

0 ——ii>(

(0)
FIG. 19.16

the integrator. The circuit and waveforms are shown in Fig. 19.15;
the integrating resistor R1 is returned to h.t.+, but this does not affect
the principle of operation. The circuit gives, from a single valve, a wave
forrn which is highly linear and which is of considerable amplitude.

'19 V3 5
I
I

O
,1‘
.2."
FIG. 19.17
19] WAVE SHAPING 287
Another electronic integrating circuit is illustrated in Fig. 19.l6.a,
where a triode has an anode load consisting of resistance R and capacitance
C in parallel. A square wave voltage is applied to the grid, and the bias is
chosen so that the grid voltage varies between zero and beyond cut off,
as shown in Fig. l9.l6.b. Whilst the valve current is cut off the con
denser C charges through R from E2. When the grid voltage changes to
zero the anode slope resistance is effectively in parallel with C and, pro
vided ra < R, the condenser discharges rapidly to a voltage vo = E2 Rio,
where 1'0 is the anode current at zero grid voltage. The total range
of anode voltage variation is limited to a small fraction of the possible
range, and then the output voltage varies linearly with time. A common
adaptation of this circuit is shown in Fig. 19.17, where the output is

V,‘

*
+
a
vs
VA
_._ VA‘

Y. O 1

FIG. 19.18

applied to a cathode follower, which in turn feeds back to a tapping

point on the load resistor of the triode ampli er. The cause of non
linearity in a condenser charging circuit is the decrease in charging
current as the condenser voltage opposes the supply voltage. If the
supply voltage is [increased appropriately, then the charging current may
be kept constant. This is what is attempted in the circuit of Fig. 19.17.
As the condenser voltage rises so does the output voltage of the cathode
follower, and this contributes to the charging of the condenser C. Thus
the charging current is maintained very nearly constant and the circuit
gives a high degree of linearity of output voltage against time. This
circuit, usually called a Bootstrap circuit, is used as a single stroke or
repetitive time base.
An electronic differentiating circuit and its waveforms are shown in
Fig. 19.18. This circuit and also the Miller integrator can be considered
as special cases of the parallel voltage feedback ampli er analysed in
Section 10.9. (See Exx. XIX.)
 CHAPTER 20

NOISE
20.1. Noise
Before the information in a small signal can be used the signal fre
quently has to be ampli ed considerably. Any output from the ampli er
other than that due to the signal introduces extraneous information and
is classed as noise. Noise may arise from many sources, and some of these
have been mentioned in other chapters. Variations in the output may
arise from the a.c. sources used to heat the valve cathodes or to produce
the electrode supply voltages; this type of noise is usually referred to
as mains hum. Other noises come from sources outside the ampli er
and its associated equipment. A common example is radiation from
sparks in motor car ignition systems or in the commutators of electric
motors. Interference of this type may be reduced by taking suitable
precautions in the offending equipment. There is one source of noise
which is inevitable in all electrical apparatus, including ampli ers.
This noise arises from the nite size and random movements of electrons.
As a result of these random movements an electric current does not have a
perfectly steady value but varies randomly about an average value.
The variations are very small, but they set a limit to the size of the signal
which may be usefully ampli ed. When the output voltage due to the
signal is less than that due to the noise no more information can be obtained
from the signal by further ampli cation.
Noise due to random electronic uctuations is sometimes called funda
mental noise. It may arise from resistances or valves, when it is referred
to as Johnson or shot noise respectively.

20.2. Johnson Noise

In a conductor the electrons move through the crystal lattice and they
continually exchange energy with the thermal vibrations of the ions in
the lattice. As a result, there are random variations in the charge density
of electrons, and hence variations in potential, through the conductor.
When the conductor is part of an ampli er circuit these potential variations
are ampli ed just as any other voltage across the conductor. It has been
shown experimentally by Johnson that the mean square value of the
electronic voltage uctuations across an open circuited resistance R is
given by
Z5 = 4kTRB
where k is Boltzmann's constant, T is the absolute temperature and B
is the measured frequency bandwidth of the equipment which is used to
288
cl r. 20] NOISE 2so
observe the effect. It is noted that the mean square noise voltage per unit
bandwidth is independent of frequency. The formula is applicable to
the effective resistance of any circuit, however complicated.

20.3. Shot Noise

Consider the ow of electrons from the cathode to the anode of a diode.
As shown in Section 14.1, each electron during its ight induces on the
anode a charge which varies from zero up to +e. The change of charge
with time constitutes a pulse of current in the external circuit. The
total current is the sum of the pulses due to all the electrons in transit
at any instant. When the current is temperature limited all the electrons
reach the anode. Due to the random motion of the electrons in the
cathode material, the number and velocity of the electrons emitted in
equal time intervals uctuate in a random manner about average values.
Thus the anode current uctuates also, and produces a noise voltage
across the load in the diode circuit. This uctuation is called shot noise.
It has been shown experimentally that the mean square noise current in a
temperature limited diode passing a current 1'4 is given by
F = 281248,

where B is again the bandwidth of the equipment which is used to observe

the effect. The mean square current per unit bandwidth is independent
of frequency.
In the temperature limited diode each electron is assumed to move
independently of the others. In the space charge limited diode this is
no longer the case, and it is found that the effect of space charge is to
“ smooth out ” the uctuations, giving a considerable reduction in the
noise current. The formula is now
‘F2 = 261248172,

where F2 is the space charge reduction factor which is determined experi

mentally; it may be as low as 0 03, and is about 0 1 on the average.
The noise current in a space charge limited triode is given by the
same formula, as long as the grid is at a negative potential and collects
no electrons. In valves where the cathode current is collected by more
than one electrode there is an increase in shot noise. This arises since
the fraction in which the current divides between the electrodes uctuates
in accordance with the random transverse components of the electron
velocities. In the case of a pentode with anode and screen currents the
uctuating anode current is given by
2 _ . F21.‘ +
$2 — 2€Z4B{ }°

The value of F* for a pentode is about 0 1 as in a triode. However, igg

may be as much as 0 2 of the total current, and a pentode thus has more
noise than a triode with the same anode current. When the cathode
290 PRINCIPLES OF ELECTRONICS [CH.
current is temperature limited it is seen that the noise current is unaffected
by the partition of current between screen and anode.

20.4. Addition of Noise Voltages

Noise arises from unrelated random events, and noise from one source
can be treated independently of noise from a different source. The
resultant effect is then obtained by the direct addition of the appropriate

Rt

_ Rt
=2 ’ ‘*0 '= :
1’ 2¢z,e|='
O2 =4

(<1) (b)
FIG. 20.1

mean square values. As an example we may consider the diode with

load resistance shown in Fig. 20.l.a. The diode shot noise may be
represented by a current generator supplying its current in parallel with
the diode slope resistance 1,. The noise in the load resistance R1, may be
represented by a voltage generator in series with R1,. The resultant
circuit is shown in Fig. 20.1.b. Due to shot noise the mean square noise
voltage between anode and cathode is

= *2 = “R”
where R is equivalent to R1, and 1 ,, in parallel. Due to resistance noise
the corresponding mean square noise voltage is
_ ‘ear,’ EER2
U22'—(RL_*_’ “)2 RL2'

The resultant noise voltage is therefore given by

F2 = B? + 17;” = R2675 + ‘Z2/R1,“) = R2B(2ei4F2 + 4kT/R1,).

20.5. Equivalent Noise Resistance

In Fig. 20.2 a triode is shown with load resistance RL. It may be
veri ed, in most practical cases, that the noise voltage across R L due to
its own resistance noise is much less than the noise voltage across R1, due
to the shot noise current of the valve. The latter may be evaluated from
20] NOISE 291
Fig. 20.2.b exactly as in the previous section for the diode, and it is found
that the output voltage is
Y2 = FR” = 2ei4BF’R’.
For some purposes it is convenient to express this noise voltage in terms
of an equivalent noise voltage from a resistance between grid and cathode

R: RLra
RL RL+r0 RL

F RL RE

5?
£2'2¢iABF2

(0) (b) (c)

Fro. 20.2
and to assume a noiseless valve. If RE is this resistance, then it produces
an input noise voltage of
Z17 = 4kTBRE.
After ampli cation this gives an output voltage of

v7 _ I5'}?1 2 _ 4g,,,2kTBRER2 .
__ + _
(Rn H)
The noise voltages are equivalent if

Thus the shot noise of a valve may be represented by an equivalent noise

resistance between its grid and cathode, as shown in Fig. 20.2.0. This
is convenient for many purposes, since RE can be determined by measure
ment, and then the noise performance of the valve may be compared
with noise from any resistance in the actual input circuit. The formula
for RE shows that a low noise valve has a high gm at low 2'4.

20.6. Noise Factor

Any signal is always associated with a source of some internal resistance.
For example, the aerial of a receiver has a radiation resistance. In the
case of a signal generator there is the intemal resistance of the generator.
The resistance is a source of noise, and from the source there is therefore a
292 PRINCIPLES OF ELECTRONICS [cl I.
de nite signal to noise ratio. This may be expressed as a signal to noise
power ratio When the signal is applied to the input of an
ampli er as shown in Fig. 20.3 it is ampli ed giving an output voltage
v,_,. However, in the ampli er there is some noise arising from the
effective parallel resistance of the input circuit, and there is also some shot
noise from the rst valve. If this valve has appreciable gain the noise
of subsequent circuits and valves is negligible. In any case, it is obvious

“ = e'* v (0)

R, vi. Q _2

— :3.
Q
Ys:

Frc. 20.3
that the ampli er has reduced the signal to noise ratio. If v,,,, is the noise
voltage in the output of the ampli er, then the quantity N de ned by

N = U“:/van:
is called the noise factor of the receiver. It is usually expressed in
decibels. An ideal receiver producing no noise would have a noise
factor of 0 dB. The various signal and noise voltages in the circuit are
shown in Fig. 20.3.b. In this gure R, is the resistance of the signal circuit,
Rd is the resistance of the input circuit of the ampli er, v,,d is the noise
voltage associated with Rd, and v,,,, is the noise voltage equivalent to the
valve shot noise. In using the concept of equivalent noise resistance
it must be realized that RE has no physical existence, and it must not
be included in the circuit in determining the actual voltages applied
between grid and cathode.
20] NOI SE 293
20.7. Other Sources of Noise in Valve Ampli ers
The presence of small quantities of gas in valves may increase valve
noise appreciably. Electrons ionize the gas atoms, producing more
electrons and positive ions. The former ow to the anode and produce
additional uctuations, whilst the ions ow to the grid and cause uctua
tions in grid current.
In some valves it is found that the mean square noise current per unit
bandwidth increases as the mean frequency is reduced below about
4 kc/s. This icker effect, as it is called, is most notable with oxide
coated cathodes and especially under temperature limited conditions,
though it is still present when the current is space charge limited.
Finally, mention may be made of noise arising from mechanical vibra
tion of the valve system, particularly in the early stages of a high gain
ampli er. This may cause variation in the valve currents which may be
ampli ed and appear in the output. This type of noise can be minimized
by using valves specially designed to reduce “ microphony ’ ’, as it is called.
20.8. Transistor Noise
Transistors and crystal diodes produce Johnson noise like any other
resistance. When they are passing steady currents additional noise is
produced. Unlike shot noise in valves, transistor noise is not constant
over the frequency band. It is greater at lower frequencies. Although,
in some cases, transistors are inferior to vacuum valves as far as noise is
concemed, inherently solid state valves have a distinct advantage over
the thermionic valve. The latter requires a high temperature cathode
as the electronic source. There is no such temperature requirement in
crystal devices; indeed they may be operated at low temperatures where
electron random movements are much reduced.
 EXAMPLES

EXAMPLES II
1. Find the percentage increase in mass of an electron accelerated
through a p.d. of 5,000 V.
(1 per cent.)
2. Calculate the transit time of an electron in the de ecting plates of a
cathode ray tube if the plates are 2 cm long and the nal anode voltage is
1,000.
(1 1 X 10’° sec.)
3. A cathode ray tube has de ecting plates 2 cm long, 0 5 cm apart and
20 cm distant from the screen. If a de ecting p.d. of 40 V produces a
spot de ection of 2 cm, calculate the approximate value of the nal anode
voltage. Point out any assumptions which you make.
(800 V.)
4. What is the shortest time it would take for a proton starting from
rest to move between two points differing in potential by 1 V and separated
by a distance of 1 cm? What factors might increase this time?
(1 4 p.S.)
5. How would you show experimentally that electrons acquire energy
in moving from a cathode to an anode under the in uence of a p.d.?
6. The potential distribution between co axial cylinders is given by the
formulae v = k log(r/r1) and k = v4/log(r,/11), where 11 and 1, are the radii
of the cylinders and v4 is the potential difference between the cylinders.
Draw a graph showing the variation of v in a cylindrical magnetron with
anode radius of 1 cm and cathode radius of 0 025 cm. Hence show that
with a magnetic eld parallel to the axis the path of an electron in a
magnetron with a thin wire cathode is very nearly circular.
7. A cylindrical magnetron has a lamentary cathode 0 2 mm in dia
meter and the anode diameter is 2 cm. If v4 = 1,000 V, calculate the
approximate value of the magnetic ux density for an electron just to
reach the anode. Explain any assumptions that you make.
(0 022 Wb/ma.)
8. If 101° electrons per second pass steadily along a 100V electron
beam, nd the beam current and the power dissipated at the collector.
(1 6 mA, 0 16 W.)
9. An electron with energy 400 eV moves in a uniform magnetic eld
of ux density 0 001 Wb/m”, the eld and the velocity being mutually
perpendicular. Calculate the radius of the electron path.
(6 8 cm.)
10. A 1,000V electron moves in a uniform magnetic eld of ux density
294
 EXAMPLES 295
0 01 Wb/m”, the electron velocity making an angle of 5° with the eld.
Calculate the path of the electron.
(Helix of radius 0 93 mm and pitch 6 7 cm.)
11. Discuss critically some of the differences in construction and use
of cathode ray tubes with electrostatic and magnetic de ection.
12. Determine the path of an electron which enters a uniform magnetic
eld of ux density B with a velocity v at right angles to the eld. How
is the path modi ed when the velocity and the eld are inclined at an
angle 6? Mention brie y one important practical example of each of
these cases. [I.E.E., II, 1954.]
13. With the aid of a sketch show the essential parts of a cathode ray
tube with electrostatic de ection. Derive an approximate expression
for the de ection sensitivity in terms of the de ecting voltage and the
nal anode voltage.
When there is no de ecting voltage how does the electron velocity vary
between the nal anode and the screen? (Give reasons.)
[I.E.E., I1, October 1956.]
14. The de ecting plates of a cathode ray tube are 3 cm long and 0 5 cm
apart. The distance from the centre of the plates to the uorescent
screen is 20 cm. A de ecting potential difference of 100 V produces a
spot de ection of 4 cm. Calculate the nal anode voltage. Derive any
formula that you use, and explain any assumptions you make in the
derivation. What is the velocity of the electrons leaving the nal
anode? (1500 Y, 2 3 X 107 m/s.) [I.E.E., II, October 1957.]
15. An electron of charge e and mass m is projected with velocity v into
a uniform magnetic eld of ux density B. If the direction of projection
is normal to the eld, determine the path of the electron.
Describe with the appropriate theory any experiment for the determina
tion of e/m for an electron. [I.E.E., II, April 1956.]
16. A cathode ray tube has de ecting coils which produce a uniform
eld of 6 >< 10'“ Wb/m2 when the coil current is 1 A (d.c.). This eld
extends an axial distance of 2 5 cm and its centre is 25 cm from the screen.
If an alternating current of 0 25 A (r.m.s.) produces a trace 15 cm long on
the screen, what is the nal anode voltage of the tube? Prove any form
ulae used relating to electron ballistics. (2750 V.) [I.E.E., III, 1954.]
17. A high vacuum diode has a cylindrical anode of diameter 1 cm.
The cathode, of very small diameter, is on the axis of the cylinder. The
anode is maintained at a positive potential of 800 V relative to the cathode.
VVhat value of uniform axial magnetic eld is required just to cause the
anode current to be zero? Derive any necessary formulae and state
clearly any assumptions made.
(1 35 >< 10'“ Wb/m2.) [I.E.E., III, October 1956.]
18. A cathode ray oscillograph has a nal anode voltage of +2 0 kV
with respect to the cathode. Calculate the beam velocity.
Parallel de ecting plates are provided, 1 5 cm long and 0 5 cm apart,
their centre being 50 cm from the screen: (a) nd the de ection sensitivity
296 PRINCIPLES OF ELECTRONICS
in volts applied to the de ecting plates per millimetre de ection at the
screen ; (b) nd the density of a magnetic cross eld, extending over 5 cm
of the beam path and distant 40 cm from the screen, that will give a
de ection at the screen of 1 cm. Prove all the formulae used.
(2 7 X 107 m/s, 2 8V, 7 6 >< 10" Wb/m“.) [I.E.E., III, April 1956.]
19. Derive an expression for the electric eld strength in the annular
space bounded by two concentric cylinders when there is a potential
difference between them.
An electron is injected with a certain velocity and at a certain radius
into the evacuated space between the cylinders in a tangential direction.
Determine the relation that must exist between electron velocity, cylinder
radii and potential difference if the electron is to follow a concentric
circular orbit. Calculate the potential difference required to give a
circular orbit if the electron velocity is 10" m/sec and the relevant cylinder
radii are 2 cm and 6 cm, respectively.
(The ratio of charge to mass of the electron is e/m = 1 76 >< 1011
coulomb/kg.) (630 V.) [I.E.E., III, April 1957.]
20. An electron moves with velocity 2 >< 10" m/sec mid way between
and in a plane parallel to the electrodes of a planar magnetron. Calculate
the p.d. between the electrodes. If the distance between cathode and
anode is 0 5 cm, calculate the magnetic flux density. Indicate clearly
any assumptions that you make.
(1,100 V, 1 1 >< 10" Wb/m'.)
21. Explain how an electron beam can be focused by: (a) a magnetic
eld; (b) an electric eld. Describe brie y one application of each
method, indicating the way in which the eld is provided and its approxi
mate magnitude.
The anode and cathode of a vacuum diode are parallel plates 1 cm apart.
The cathode is at zero potential and the potential of the anode is given by
V = sin 21: ft volt, where f = 50 Mc/s. At time t == 0 an electron is at
rest near the cathode. Describe its subsequent motion and nd: (i) its
velocity at time t — 2 . 10" sec; (ii) its maximum velocity.
(0, 1 1 X 105 m/s.) [I. of P., 1957.]
EXAMPLES III
1. Compare the mechanism of electrical conduction in gases and semi
conductors.
2. What is the evidence for the existence of electron energy levels in
matter?
3. Explain the difference between n type and p type semi conductors
and discuss the conditions at the junction between 15 and n type german
rum.
4. Discuss the important differences and similarities between diamond
and silicon.
5. Discuss the equilibrium potential, charge and energy conditions at
the junction of: (a) two di erent metals; (b) two different semi con
ductors.
 EXAMPLES 297
6. Give an account of the electron theory of electrical conduction in
solids. Explain the differences in the variation of conductivity with
temperature in conductors, insulators and semi conductors.
[I.E.E., II, April 1955.]
7. Write an account of the conduction of electricity in solids, referring
particularly to the factors determining the electrical resistivity of metals,
semi conductors and insulators.
Explain what is meant by the work function of a surface, and describe
brie y its importance in (a) thermionic emission, (b) metal recti ers.
[I. of P., 1957.]
8. Write an account of the mechanism of the conduction of electricity
in solids. [I. of P., 1954.]
9. If the conductivity of a semi conductor is given by
0 = o,,a'4/7'
where T is the absolute temperature, show that the change in conductivity
caused by a small change in temperature is Ao/T“ times the change in
temperature.
(Note. If we write A = ev/k, then av is related to the value of the
energy gap between the top of the valency band and the bottom of the
conduction band, i.e., E0.)

EXAMPLES IV
1. De ne the work function of a metal and show how it is related to the
Fermi level.
2. Discuss the relative advantages and disadvantages of the more
commonly used thermionic emitters.
3. How could you demonstrate experimentally that electrons are
emitted from a hot cathode with a distribution of velocities?
4. Describe brie y the phenomenon of photo emission.
The work function of the cathode of a photo cell is 3 5 electron volts.
What is the maximum velocity of the emitted electrons when the cell is
irradiated with light of frequency 4 X 101‘ c/s? How could the maximum
velocity of emission be determined experimentally?
(2 1 X 10° m/s) [I.E.E., II, October 1955.]
5. Explain the meaning of the various symbols in Richardson's emis
sion equation I = A T25‘/*7’.
Describe and compare the main features of the various types of thermi
onic cathode which a.re in general use. [I.E.E., II, April 1956.]
6. In what way does the current from a vacuum photocell vary with
the intensity of the incident radiation? How is the variation affected by
the presence of gas in the cell?
The work function of barium is 2 5 eV. Would barium be suitable as a
cathode in a photocell for violet light of wavelength 4,300 A? (Give
reasons.) [I.E.E., II, April 1957.]
298 PRINCIPLES OF ELECTRONICS
7. How does the thermionic emission from a valve cathode depend
upon: (a) the nature of the cathode surface; (b) the heater power?
Describe how the cathodes of modem valves are designed so as to
minimize the heater power required for a given emission.
By what percentage will the emission from a tungsten lament at 2,400° C
be changed by a change in temperature of 10° C? (8 1.) [I. of P., 1953.]
8. Explain what is meant by thermionic emission and describe how you
would investigate its variation with temperature for a particular surface.
Explain brie y why, although the three common types of thermionic
cathode have widely different emission e iciencies, all three are never
theless in general commercial use.
By how many electron volts must the work function of a surface change
in order to reduce the emission from that surface at 2,400° C by 10 per
cent? (+ 0 025.) [I. of P., 1955.]
9. Explain what is meant by “ secondary emission ” and describe how
you would measure the secondary emission properties of a surface.
Discuss the importance of secondary emission in: (a) triodes and pen
todes; (b) cathode ray tubes; (c) photomultiplier tubes.
[I. of P., 1955.]
10. Describe asuitable model by means of which the emission of electrons
from a metal surface may be described. State the condition under which
thermionic emission, eld emission and photo electric emission will take
place, and draw attention to common features and to differences in the
three processes. ~[I. of P., 1956.]
11. Explain what is meant by: (a) secondary emission; (b) photo
electric emission.
Describe brie y the principles of operation of an electron multiplier
photocell. Indicate suitable materials for the various component parts
of the device and discuss its advantages and disadvantages compared
with a vacuum photocell followed by a high gain ampli er.
[I. of P., 1956.]
12. When monochromatic radiation of wavelength 2,000 A falls upon
a nickel plate the latter acquires a positive charge. The wavelength is
increased, and at a wavelength of 3,400 A the effect ceases, however
intense the beam may be. Explain this, calculate the maximum velocity
of the electrons emitted in the rst case and describe, with a diagram and
a circuit diagram, the construction and use of a practical photocell based
on this effect. (9 5 X 105 m/s.) [I. of P., 1952.]

Examrrrzs V
1. A parallel plane diode is operated at an anode voltage of 10 V.
Calculate the velocity of an electron half way between the cathode and
anode when: (a) the current is space charge limited, and (b) temperature
limited. (Ignore initial velocities of the electrons.)
(1 2 X 10° m/sec, 1 3 X 10° m/sec.)
 EXAMPLES 299
2. A p n junction and a junction between two dissimilar metals both
give a contact potential difference, but only the p n junction can act as a
recti er. Explain this.
3. With the aid of potential distribution diagrams distinguish between
temperature limited and space charge limited current in a planar diode.
State the Child—Langmuir formula for the space charge limited current
density and explain how the formula is modi ed when the initial velocities
of the electrons are taken into account.
4. The distance between the cathode and anode of a planar diode is d
and the anode potential is v4 relative to the cathode. At what distance
from the cathode is the potential equal to v4/2 when a space charge
lirnited current ows?
(0 6 d.)
5. The anode current of a particular themrionic diode is given by
£4 = i,eK’4 when v4 is negative.
A resistance R is connected directly between the anode and the cathode.
Calculate the voltage across the diode when R = 1,000 MQ, k = 11 V'1
and i, = 60p.A.
(U4 = — 1'0 V.)
(The load line relation gives £4 = — v4/R.)
6. The relation between current and voltage for a junction diode is
given approximately by
i= i,(r :4" 1).

Draw the characteristic when i, = 1 p.A and A = e/kT between v = — 1V

and v = 0 5 V. (e and k are the electronic charge and Boltzmann's
constant, and T is the absolute temperature.)
7. VVhen the value of v is su iciently positive the characteristic of the
junction diode is approximately
i = i, e4’.

Show that under these conditions the a.c. conductance of the diode is
proportional to the current.
8. The voltage current characteristic of a diode valve is given for
positive values of v4 by
1:4 = 2 X U48/2 X 10 3 A

and for negative values of v4 by £4 = 0.

A voltage, v, = 4 cos mt is applied to the diode. Plot the variation
of current through the diode for values of wt from 0 to 2n.
A d.c. voltage of 3 V is applied to the diode in series with a small alter
nating voltage. Show that the ratio of the amplitude of the alternating
300 PRINCIPLES OF ELECTRONICS
current through the diode to the amplitude of the alternating voltage is
about 5 2 X 10*’ A/V.
9. A parallel plane diode has an anode cathode clearance of 1 cm and a
cathode area of 16 cm’. Assuming space charge limitation and an anode
voltage of 100 V, nd:

the current density;

the total current;
the number of electrons in the anode cathode space;
35: /93$, the average density of electrons.
/ \;E s\/ \I'\

((a) 2 3 mA/cm’; (b) 37 mA; (c) 1 2 X 10°; (d) 7 4 X 10" cm'*.)

(Note. The density of free electrons in metals is of the order of
10“ cm'3.)
10. Describe the formation of a space charge in a thermionic valve and
explain: (a) why this modi es the distribution of electric eld between the
electrodes, and (b) its effect on the anode current.
11. Explain how electric current can be carried between cold electrodes
in a gas at low pressure and describe how and why the nature of the dis
charge varies with the pressure in the gas. Why is such a discharge often
accompanied by the emission of light and what factors determine the
intensity and colour of the light emitted?
Mention two applications of gas discharge devices and indicate brie y,
with circuit diagrams, how they are used. [I. of P., 1953.]
12. Discuss brie y any two of the following: (a) grid current; (b)
space charge; (c) electron temperature. [I. of P., 1953.]
13. Explain, with reference to a typical diode valve: (a) cathode
emission; (b) space charge; (c) saturation.
Calculate the space charge density at (i) the anode, and (ii) the cathode,
of a plane parallel diode the plates of which are 5 mm apart, when the
potential difference between them is 300 volts. Assume the current
through the diode to be limited by space charge. What is the signi cance
of the calculated value of the space charge density at the cathode?
(4 7 X 10" c/m2, in nite.) [I. of P., 1954.]
14. Give a simple explanation in tenns of energy levels of the difference
between conductors, intrinsic semi conductors and impurity semi con
ductors of the p type and n type. Using either p type or n type as your
example, account for the rectifying action exhibited by an impurity semi
conductor in contact with a metal. 1/Vhat are the respective elds of
application of point contact and junction recti ers?
[I.E.E., III, April 1956.]
15. A plane diode has an anode cathode spacing of s metres. A sinu
soidal voltage of maximum value V,,, at a frequency co/211: is applied
between anode and cathode. Show that the distance travelled by an
 EXAMPLES 301
electron released with zero initial velocity from the cathode at time t, is, in
the absence of space charge:

x = 3%? "E S [(0)1 — onto) COS mi, + Sin col, — Sin col].

If the anode—cathode spacing of such a diode is 0 5 mm, and the applied

voltage has maximum value of 100 V at a frequency of 1,000 Mc/s, how
long will it take an electron to reach the anode, assuming it to be released
from the cathode with zero initial velocity at the commencement of a
cycle? (2 5 X 10'1° sec.) [I.E.E., III, April 1956.]
16. Sketch a graph showing how the current between two cold electrodes
in an inert gas at low pressure varies as the voltage between the electrodes
is increased.
Explain (giving skeleton circuit diagrams) how certain parts of the
curve may be applied in the following ways:
(i) Detection of charged particles.
ii) Voltage stabilization.
I'\

(iii) High power recti cation.

[I.E.E. III, October 1956.]

EXAMPLES VI
1. The anode current of a triode is given by the equation 1'4 = f(v@, v4)
When no and v4 change to ‘U0 + 8120 and 04 + 811,4 then 1'4 becomes
1'4 + 814. Tayl0'r's Theorem for functions of two variables shows that
. 8' 8' 621' 26’ 1'
824 = 8% 8'09 + 5:73 8'04 + (8U0)2 + 5 8114809

82'
+ (8'UA)2} + »

Use this to show that

814 = g,,,8v@ + 8124/1',
o
o
" o
0
8 .x° =_,0
O

15’
Q
5:15_
e_ao Q
14o L.

it < 7"
_

(MA)
‘A {go §°
1/’ I '\v

" ' ‘ /' 1 "

Q‘ I // /E’

A J 4 J . J}. J‘
0 10° 20° $00 49° $00 09° 70° 00° 90°

VA (V)
Fro. VI.i

provided that either 8110 and 8124 are small or the characteristics are free
from curvature.
302 PRINCIPLES OF ELECTRONICS

2. Use the previous example to prove that p. = <3 72:5 )1. = g,,,r,,.
0 .1
3. Using the 1'4, v4 characteristics of Fig VI. i, draw 1'4, ‘U0 and v4, U9
characteristics for values of v4 from 0 to 500 V and 00 from 0 to — 50 V.
From the three sets of characteristics determine p., gm and 1', for
(i) 09 = — 10 V, 1'4 =60 mA; (ii) va = — 30V,1'4 =20mA; and
(iii) ‘U0 = — 50 V, 1'4 = 10 mA. Verify in each case that p. '= ' g,,,r,,.
4. Draw typical static characteristics for a screen grid tetrode and a
pentode and explain in detail the reasons for their shapes.
5. Explain why the control grid current of a thyratron varies from
negative to positive values as the grid voltage is made less negative.
6. Draw common base hybrid characteristics for a transistor from the
following data:

1;(mA 0 75 0 75
vq4(V) — — —50 00
.
v"(m
Y) ea
II 5 0I 1 127 to
Il 131
1¢(mA) _ ’?'°‘¢."’T‘o8
o¢ _ ?'°*:°’?‘ c°8
oQ _ ?‘°?"." soooo~1 — 0 75 — ?‘°':°?4o~1anO! — 0 74

1'4(mA) 0 so 0 so 0 50 0 25
v¢ 4(V) 50 20 00 00
v,,(mV ) 115 117 119 co co es
1'¢(mA) 05 0 4s 0 40 ?°‘¢?‘? onon
1»: O1 ?°°'9?tocwca
O1 0 24
Find the values of the hybrid parameters at 1'4 = 1 mA, 11¢, = — 4 V and
1:3 =0'5ITlA,‘Ug3 = —2V.
7. From the common base transistor equation, changes in electrode
currents and voltages are related by 8044 = h11b81'4 + hm 80¢; and
81}; = hm81'4 + h.m8v¢ B. Using the conditions 1'4 + 1'B + 1}; = 0 and
v¢ B = 11¢; + 04 B, show that
— 3vBE(1 hm) = h11b(35s 1 350) 1 hrrrsvcr:
and 35c(1 + h zro) = — h 211353 + h22b(8vCE — 311511)
Hence, by keeping 80¢); constant, i.e., 8v¢4 = 0, show that
ha — h 211 + h21bh12b — hzzohna
e 1 + h21o — hm — h211h121 + hwhno
_ —h 211 + h 2112121 — h mhrro and hm = 1%

i.e., ac), = Oi and hm ¢ * L13

1 — 01,, 1 — aw
since hm, and hwhm, are both much less than 1 (see Table I, Chapter 6).
8. By setting 81'4 = 0 in the previous example show that
hm — hm — h zrzélrzb + hush 221 and hm = héga

l.6., h1g4‘: — h1g1, + Q and hag, =

 EXAMPLES 303
9. Discuss the physical reasons why the parameters hm, ha, and ha for
the common emitter connections are approximately equal to the corre
sponding common base values multiplied by the factor 1/(1 — 014,).
10. De ne h parameters for the common collector connection using
1'4 and ‘U33 as the independent variables. Show that
~ hm ~
hllc '—' —' ll

1 h
h21c Q 1 1;; and h22c Q 1 _2_2ba“

ll. To a rst approximation the emitter current of a p n p transistor

is given by
14 = 1,: '1" when 114 4 rs positive.
I O A O Q O

Show that the rate of change of v4 B with 1'4 is inversely proportional

110 ig.
12. Explain how and why the grid voltage control action in a gas
lled relay valve differs from that in a high vacuum triode. What is
meant by the control ratio of a gas lled relay and how does it depend
upon: (a) the nature and pressure of the gas lling; (b) the anode voltage?
Why is it usual to include a resistor in series with the grid of a gas lled
relay? [I. of P., 1954.]
13. Describe the action of a triode valve and explain how the properties
of the materials with which it is made, and the size and spacing of the
electrodes, determine: (a) the maximum current and voltage which can
be handled; (b) the highest frequency signal which can usefully be
handled; (c) the maximum voltage and power ampli cations; (d) the
useful life of the valve.
Describe brie y how you would determine the anode characteristics of
a pentode valve. [I. of P., 1955.]
14. Explain how the potential applied to the suppressor grid of a
pentode valve determines the division of the cathode current between the
anode and the screen grid.
What are the advantages of the pentode over: (a) a triode; (b) a
tetrode? Discuss the types of applications for which each of these three
types of valve is most suitable. [I. of P., 1956.]
15. Describe the construction of a low power pentode valve. Sketch
the following static characteristic curves and explain their shapes with
the aid of potential distribution diagrams:
(') anode current/anode voltage;
j

(ii) anode current/suppressor grid voltage;

(iii) screen grid current/suppressor grid voltage.
Sketch also characteristic (i) for a beam tetrode, and explain the
differences. [I.E.E., III, April 1957.]
304 PRINCIPLES OF ELECTRONICS

EXAMPLES VII
1. A single stage ampli er has an anode load of 20 kQ. Calculate the
ampli er gain when the valve is: (a) a triode with gm = 2 mA/V and
1', = 10 kQ, and (b) a pentode with g,,, = 2 mA/V.
(13, 40.)
2. The ampli er of Example 1 is coupled to another stage through a
capacitor of 0 005 p.F and a grid leak of 1 MQ. If the total effective cap
acitance across the load is 50 ;111F, calculate the frequencies at which the
gain has dropped by 3 db with the triode and the pentode.
(32 c/s, 480 kc/s; 32 c/s, 160 kc/s)
3. A pentode ampli er has a load consisting of an inductor of 250 11H, a
capacitor of 0 0001 11F and a resistor of 20 k all in parallel. If the pen
tode has gm = 5 mA/V, calculate the frequencies at which the ampli er
has: (a) maximum gain, and (b) gain of 3 db below the maximum. What
is the maximum gain?
(l 01 Mc/s, 0 97 Mc/s, 1 05 Mc/s, 100.)
4. For freedom from phase distortion in an ampli er show that the

10kQ 4Ou11F

Q o our E 3'
+ +
v w

FIG. VII.i

phase angle delay produced by the ampli er should be proportional to the

frequency.
5. For the circuit shown in Fig. VII.i calculate the “ middle frequency "
gain (i.e., the value of v9,/11,1 when the reactances are negligible) when the
rst valve is a triode with g,,, = 5 1nA/V and r, = 10 kQ. Also calculate
the two frequencies at which the gain has dropped to half the middle
frequency value and the corresponding phase shifts.
(25, 18 c/s, 1 4 Mc/s, 240°, 120°.)
6. A pentode ampli er has an anode load consisting of a resistance R in
parallel with a capacitance C. Show that the complex gain A at an
angular frequency <0 is given by
A '_~_. .... L
. (.0
1 +JE
where <0, = 1/RC, i.e., Q, is the “ half power angular frequency ”.
 EXAMPLES 305
7. If an inductance L is put in series with the load resistance R of
Example 6, show that
. 0)
’ J 1 + JQ (32)
A ;.
co 2 0)
g"R ‘"Q(z;) WI.)
whereQ = w,L/R.
8. Use Example 7 to draw curves showing how the magnitude of the
gain varies with frequency when Q = 0, 0 25, 0 5 and 1 0.
(Note. Ampli ers for wide frequency bands often use inductance
compensation to extend the frequency range. A value of about 0 5 is
usually chosen for Q.)
9. Show that the voltage gain of a cathode follower is given by the
formula
A =g,,,R/(1 +g,,,R + R/r,,)
where R is the cathode load resistance.
10. A pentode with gm = 5 mA/V has a signal of 0 1 V connected be
tween its grid and earth. The output is taken across a resistance of 1,000 Q
connected between the cathode and earth. There is an h.t. supply
between earth and the anode to give the required operating conditions.
Calculate the output voltage.
(O 083 V.)
(Note. When a pentode is used in a cathode follower, then, in order to
keep 1', large, the total voltage between screen grid and cathode must remain
unchanged on application of the signal.)
ll. An automatic bias circuit uses cathode resistance and capacitance
of 400 Q and 20 p.F. If the valve current is (10 + 5 sin 21: ft) mA calculate
the magnitudes of the direct and alternating voltages across the bias
circuit when f = 10, 50 and 1,000 c/s.
(4 V, 1 8 V; 4 V, 0 7 V; 4 V, 0 04 V.)
12. Show that the input impedance of a common grid ampli er is

A R
1

1 + B

_ K
—

(0)
V‘

_
_'i"_

= RL

(0)
L
F10. VII.ii
306 PRINCIPLES OF ELECTRONICS
equal to (Z + r,,)/(pr + 1) where Z is the load impedance between anode
and grid.
13. In circuit of Fig. VII.ii.a show that V; = V, Z2/(Z1 + Z2).
Hence, if Z1 is a resistance R and Z2 a capacitance C, show that the
impedance between anode and cathode is equivalent to that shown in
Fig VII.ii.b, where R’ = 1/gm and L = CR/gm.
14. Two identical triode valves are connected in series, the anode of
one being connected to the cathode of the other. The two in series are
connected across a steady h.t. voltage supply. The valve which is on the
negative side of the h.t. supply is provided with an alternating voltage V1
connected between its grid and its cathode. The other valve has its grid
connected to h.t. negative through a very large capacitance and to its
own cathode through a very large resistance. Prove that the alternating
component of anode current is given by
§».(1 + 1 )V1/(2 + 1 )
Also show that the alternating voltage across the second valve, is
P V1/(2 + 9)
l5. We may write the characteristics of a triode valve when used in a
common cathode circuit as
1'0 = f(v@, v4) and 1'4 = F(v@,v4),
so that for small changes
1} = 8111 11¢ l 8121'". 1 and ia = 8211 111 l 822.1%
Show that for the conventional negative grid triode
gm = 0, gm = 0
gm = gm and gm = 1/r,.
16. We may write the characteristics of a triode valve when used in a
common grid circuit as
ix =f(vr0.‘".40) and 5.1 = F(vsa,'".10),
so that for small changes
it = 81111111 + 812% and 51 = 8111": 1 1' 8221011
Show that for the negative grid triode
8211 = 8111 a d 8221 = 8121
whilst gm, = gm + 1/r, and gm, = — 1/1,.
The input resistance of the system is given by r, = 0; 9/1'4. Show that
when the output is short circuited so that 0,, = 0 then
n_ 1 1 _
8111 (gm +1/'1)
 EXAMPLES 307
17. We may write the characteristics of a triode valve when used in a
common anode circuit as
50 =f(1104. vim) 9 Rd ix = F (1104, 1111.1)
so that for small changes
1} = En {"91 1 8120"“ and 51 = £211 ‘"98 1 3226/18
Show that for the conventional negative grid triode
gm = 0. £124 = 0 821 1 = — gm, and $22,, = gm 1 1/711
The effective internal resistance of the system is given by ro = 01,/1'), when
the input terminals are shorted. Show that ro = g% = 1/(gm + 1/r,,).
2a
18. A common cathode pentode ampli er has an anode load consisting
of resistance R2 in series with resistance R1. Across R1 is a condenser C1.
Show that the gain at low frequencies is times greater than at
2
high frequencies.
(This can be used to give bass boost in a valve ampli er.)
19. Discuss in a qualitative manner the variation in stage gain of a
triode common cathode ampli er as the load resistance is altered, keeping
the high tension constant.
20. Explain why, over a large range of anode voltage, the anode current
in a pentode valve is almost independent of the anode voltage. Explain
also how it is possible to increase the anode slope resistance of a pentode
without affecting its mutual conductance, and how this enables a large
voltage ampli cation to be obtained with a pentode valve.
An amplifying valve has a grid leak of 1 MQ. Neglecting all other forms
of low frequency attenuation calculate the minimum capacitance of the
coupling condenser in order that the ampli cation at a frequency of 30 c/s
shall be within 3 dB of that at mid frequencies. Assume the load re
sistance of the previous valve to be small compared with that of the grid
leak. (0 0053 |J.F.) [I. of P., 1954.]
21. Determine the expressions for the frequency response and phase
characteristics of a conventional RC coupled ampli er stage. The effects
of anode and screen grid decoupling and of the cathode by pass capacitor
are to be neglected.
Explain how the high frequency response of such a stage may be
extended. [I.E.E., III, October 1956.]
(See Examples 7 and 8 above.)

EXAMPLES VIII
1. A valve used as an audio frequency power ampli er takes a quiescent
current of 30 mA from an anode supply of 200 V. When a sinusoidal
signal is applied to the grid the anode voltage varies from 40 to 360 V and
the anode current from 50 to 10 mA. Calculate: (a) the power output;
308 PRINCIPLES OF ELECTRONICS
(b) the ampli er efficiency; and (c) the turns ratio of the output trans
former if the valve is to feed maximum power to a load of 20 Q.
(l 6 W, 27 per cent, 20.)
2. Explain the difference between the d.c. and a.c. load lines in a
transformer coupled power ampli er.
A triode, whose characteristics are given in Fig VI.i, is to be used to
supply power to a resistance load of 5 Q using a supply voltage of 440 V.
The anode dissipation must not exceed 20 W. Choose a suitable operating
point and load line and determine: (a) maximum power output; (b) grid
driving voltage; (c) transformer ratio; (d) percentage distortion.
3. Explain why a moving coil meter is suitable but a moving iron or a
thermal meter is unsuitable for indicating distortion in a triode ampli er.
4. By considering anode characteristics, or otherwise, explain why a
pentode gives higher gain and greater output than a triode of comparable
size.
5. For the ideal transformer coupled ampli er represented by Fig. 8.9
show that the amplitude of the grid signal for maximum output power is
E, R + 1,,
I R + 270 .

6. With the notation of Section 8.6 show that the fundamental and
second harmonic output powers are given respectively by the expressions
(P + N)”R/8 and (P — N)’R/32, where R is the load resistance.
7. A push pull ampli er has a sinusoidal signal of value é sin mt supplied
to each grid. The output transformer has n, turns on each half of the
primary and n2 secondary turns. Show that the power output is
P 27l2RL(|J.é)2
° (1.. + 2n”Rr)’
where R1, is the load resistance, n = n1/n, and the operation is assumed
to be linear.
8. Describe, with circuit diagrams, the principles of operation of choke
capacity coupled, transformer coupled and resistance capacity coupled
ampli ers. How do the characteristics of each type vary with frequency?
In a low frequency ampli er the input voltage is applied across a 600 Q
resistor in parallel with the grid and cathode of a triode: the valve has an
ampli cation factor of 20 and an anode slope resistance of 12,000 Q. In
the anode circuit there is a 15 : l transformer supplying a resistive load
of 60 Q. Calculate the overall gain in decibels. (7dB.) [I. of P., 1952.]
9. Describe brie y the push pull method of ampli cation and list its
advantages over the single valve method.
The anode current/anode voltage characteristics of a triode are given
in the table below. Two such valves are to be used in push pull in the
output stage of an audio frequency ampli er feeding a non reactive load
of 500 Q via a transformer. Each half of the primary winding of the
transformer has 332 turns and the secondary winding has 100 turns. If
 EXAMPLES 309
the available h.t. supply is 400 V and the peak a.c. input signal is
18 V determine: (a) the grid bias required; (b) the maximum power
output; (0) the overall voltage gain.

Grid volts 0 —6 — 12 — 18 — 24
if
Anode volts
120 17 5
180 42 8
240 80 20
300 128 42 10
360 1 80 80 24
420 129 50 9
480 178 88 28 2
540 138 57 12
600 180 100 32

( 18 v, 4 7 w, 12 2.) [I. of P., 1953.]

10. Explain carefully the purposes and properties of the second and
third grids in a pentode valve.
How would you use a pentode valve or valves: (a) in an ampli er
stage; (b) in a push pull output stage?
In each case draw a typical circuit showing the use of the valve, and
indicate the characteristics required of the valve and the circuit.
[I. of P., 1955.]
ll. The anode current/anode voltage characteristics of a triode valve
are given below. I‘ The valve is transformer coupled to a resistive load of
13 3 Q, the ratio of primary to secondary turns being 20, and is used with
an h.t. supply of 400 V. The grid bias voltage is — 36 V, and a sinusoi
dal signal voltage of amplitude 30 V is applied to the grid. Calculate:
(a) the percentage of second harmonic in the output voltage; (b) the
fundamental output power; and (c) the e iciency. The resistance of the
windings of the transformer may be neglected.

Anode current, mA
Grid voltage —6 — 18 — 36 — 54 — 66
Anode voltage
50 5
100 i 35
150 90 5
200 150 25
250 68
300 1 25 7
350 25
400 55
450 100 10
500 25
550 50 11
600 25
650 50

(7 5, 4 2 W, 19 per cent.) [I. of P., 1956.]

I.
310 PRINCIPLES OF ELECTRONICS
12. Explain the principle and the advantages of push pull operation
in a valve ampli er. Each of the triodes used in a push pull Class A
power ampli er has a mutual conductance of 6 mA/V and an anode
slope resistance of 1,500 Q. The combined anode current of 120 mA is
fed from a 350 V d.c. supply to the centre tap of a primary winding of an
iron cored transformer, the secondary winding of which is connected to a
non reactive resistor of 2 5 Q resistance. The ratio of the number of
primary turns to secondary turns is 40. Estimate the anode power
e iciency of the stage when a sinusoidal signal of 20 V peak is applied to
each valve. (Assume ideal linear characteristics for the valves.)
(12 6 per cent.) [I. of P., 1957.]
13. Show that, if two frequencies fl and fa are applied simultaneously to
a non linear impedance, component voltages having frequenciesfl :1; fa are
present in the output.
Discuss the importance of this result in telecommunication practice.
[I.E.E., III, April 1956.1
14. A non linear device has the current/voltage relation
i=a+bv+cv”+dv°.
Show that the third order term leads to cross modulation between two
amplitude modulated sinusoidal signal voltages.
A carbon microphone, when subjected to a sinusoidal sound wave of
frequency co, has a resistance given by
1o0(1 + 0 2 sin ...¢) o.
Find the percentage second harmonic distortion current in its output.
(10.) [I.E.E., III, October 1956.]
15. Explain what is meant by the term non linear distortion when
applied to an ampli er.
The dynamic £4/vq characteristic of a triode valve with a resistive load
may be represented by
1:4 = A + BUG + C1192.

Deduce the ratio of the second harmonic component in the output to the
fundamental. '
If the valve has a steady no signal anode current of 60 rnA and the
application of a sinusoidal grid voltage causes the anode current to vary
between 105 and 25 mA, calculate the percentage second harmonic in the
output current. (6 3.) [I.E.E., III, April 1957.]

EXAMPLES IX
l. Using the transistor of Example VI.6, nd the operating point for a
common base circuit with v1.; B = l 10 mV and a load resistance and a battery
of — 5 V between the collector and the base, when the load resistance is:
(a) 2 5 kQ; (b) 5 kQ; and (0) 10 kQ. Find the output voltage for signal
changes of 10 mV, 20 mV and 30 mV, when the load resistance is 5 kQ.
Explain how the output could be made to follow the input more linearly.
 EXAMPLES 311
2. Determine the approximate value of the power gain in Example l.
3. The current gain of an ampli er A, is de ned as
A_ Current change in load _
' Corresponding current change in the input circuit
Show that for small signals the following formulae give the current gain
for common base, common emitter and common collector ampli ers
respectively
A‘ hero
1 + h22oR1,
h21e
A‘ 1 + h22¢R1.
_ h21¢ Q, 1
3.11
d A‘ 1 + h‘Z2cRL 1 "' ace
4. The output impedance of a common base ampli er is de ned as
r,, = vd,/ic when no signal is applied to the input. If the input circuit
consists of a resistance R between the emitter and base show that
' R + hllb _
0 hrrohezb — h21bh12b + Rhea
5. Write down an expression for the output impedance of a common
emitter ampli er.
6. Show that the voltage gain of the common collector ampli er
shown in Fig. IXA is given by
A ‘_' h21cRL Q 1
hm + R1.(h11¢h 22¢ — h12ch‘Z1c) '

3
v.
_ RL _
Fro. IX.i

7. Draw an equivalent circuit for a transistor based on the equations

vbe = hrreio + h12evce and 1} = hzreio + h22¢‘"¢¢
8. Common base transistor characteristics may be expressed in the
form
‘UE3 and U03 = F(’ig, 1:5’).
Use these expressions to show that, for small changes, 12,, and vcb may be
put in the forms
11¢» = 71101} + 712$} and 11¢» = 72101} + 722$}.
where rm, rm, 121;, and 122,, are resistances, whose values may be found
from the characteristics.
312 PRINCIPLES OF ELECTRONICS
9. Show that the resistance parameters rm, rm, rm and rm are related
to the h parameters by the following equations:
h —h h h
711» nbhmhw mhm, '12:» = biz. 7210 = I $3 and '22» =
(Start with the transistor equations using h parameters and use the de
nitions of the r parameters. See Examples VI, I and 2.)
10. Use Example 8 to show that Fig. IX.ii, iii and iv are suitable
equivalent circuits for a transistor,
WIICIC 7, = 7115 — 7125 = hub + h21b)h12ba
h22b
hm 1 — hm 1
1'o=1'12o=—1'=1'22b—1'1o ""
hazel’ C 2 h22b 71220’
C
1'21» — 1'12» ‘
hm + hm ,__,
’ C .
1'22» — 1'12» 1 — hm C‘
E 1', l, C
+ +
run '22:
'06 4 f2la['+ vtb

O ___ '12:‘:
__ $ __ _ _

B
Fro. IX.ii
oi,

e 1, r. u 1, c
+ ° ° +

'0 Q "co

.. . O _
O

B
Frc.IX.iii

E 'rra"'r2¢ '22o"'12a+ _ [C ¢
'2‘
lo . I
y (60 626) lo
“ rrzo V“

B
FIG. IX.iv
 EXAMPLES 313
The circuits of Fig. IX.iii and iv are known respectively as the current
generator and voltage generator T equivalent circuits.
ll. Find the values of re, rb, 1,, and at for the transistor of Example VI. 6.
12. Using the T equivalent circuit, show that for a common base
transistor" ampli er with load resistance R1, and resistance R between E
and B
‘val R1.(<17¢ + 1'0)
A
v.» RL(7e + 1») + nr. + an + n>n(l — '1)
~ GR],
n+nu Q’
. R C I —
r, = vd,/z, = r, + rb I€’L++rr(b + 7:) z 1, + r,(l — a)

_ . _ 75(7), + Clfc) ~ _ C1757}

and 1,, vcb/1., r,,+r, R+re+rb_r. R+re+rb

13. The output of a common emitter ampli er is connected to a parallel

combination of resistance R2 and condenser C2 which is in series with the
input of a second ampli er of input resistance R1. The collector load of
the rst ampli er is a resistance R3. Assuming that
vbe = hrreio 3 Rd 1} = Olebib,
nd the ratio of voltage across the input of the second ampli er to that at
the rst ampli er.
(Such a circuit can be used to give treble boost in a low frequency
ampli er.)
Ii °‘¢b R1Ra(l +j‘°C2R2)
hue .7R2(R1 + Ra)‘°C2 + (R1 + R2 + Ra)
14. A common emitter ampli er of input resistance R1 is fed from a
signal generator of resistance R2. Across the input of the ampli er is
connected a series circuit of resistance R3 in series with a condenser C3.
Show that the gain of the ampli er varies with frequency according to
A Cc Rr(Ra + 1/7.‘°Ca) _
(RrR2 + R2Ra + R1Ra) + (R1 + R2)/7" °Ca
(This circuit can be used to give a bass boost in a low frequency ampli er.)
15. Let ico be the collector current for a given transistor when the
emitter current is zero. If an is assumed constant, then
50 = — dis + ico
Hence show that the collector current is also given by
. . 1 .
¢c=é‘1B+lf°‘¢c0

(This shows how much more serious can be the effects of temperature in
a common emitter than in a common base circuit, particularly where
314 PRINCIPLES OF ELECTRONICS
at == 1 and the base current is determined mainly by the external circuit:
1}; O varies considerably with temperature.)
16. A p n ;b transistor is connected in a common emitter circuit. A
by passed resistance R1, is included in series with the emitter lead to the
positive terminal of the battery E1. The base bias is obtained from a
potentiometer chain R1 to the negative and R1 to the positive terminal
of the battery. The base is connected to the centre point. If we assume
that the resistance R1, is such that the d.c. voltage across it is large
compared with vBE, show that when there is zero resistance in the collector
lead
1 '2. <>(R1R2 + R=R1 + R3R1) — <=R2E=_
C 12112,(1 2) + 12,12, + 12,121
17. In the previous example with R1 = 22 kQ, R1 = 4 7 l :0 and
R3 = 1 kQ, nd the rate at which the collector current changes with
changes in igo.
With R1 unchanged but R1 equal to zero and R2 removed so that £3
is constant, nd the rate at which the collector current now changes with
igo I01‘ d =
(5, 50.)
18. Explain how a metal semi conductor contact can act as a recti er.
In a thermionic triode currents at a low power level in the grid circuit
can be used to control larger currents in the lower impedance anode
circuit. Describe brie y how the action of a transistor may be explained
in similar terms, and point out the principal points of similarity and
difference between the two devices. [I. of P., 1956.]
19. Assume that a transistor has ideal characteristics such that 1}; is
independent of veg and is proportional to 1'3. If two such transistors
operate in a Class A push pull ampli er using a 12 V supply and giving
10 W output in a load resistance of 15 Q, nd the collector current, the
collector dissipation and the transformer turns ratio.
(0 83 A, 10 W, 0 7 + 0 7 : 1.)
20. If, in the previous example, the operation is in Class B nd the
peak collector current.
(1 7 A.)
EXAMPLES X
1. Explain brie y in words why negative feedback makes the gain of an
ampli er independent of variations in supply voltage.
A single stage ampli er without feedback has a voltage gain of 10. A
second ampli er, operated from the same power supply, has two stages each
with a gain of 10, but there is negative feedback reducing the overall gain
of the ampli er to 10. Calculate the percentage feedback. As a result
of supply voltage variations the gain of the rst ampli er drops to 9.
What is now the gain of the second ampli er?
(9 per cent, 9 8.)
2. An ampli er, in the absence of feedback, has a gain which is liable
 EXAMPLES 315
to fall by 40 per cent of its rated value as a result of uncontrollable varia
tions of supply voltages. If, by the application of negative feedback, an
ampli er is to be produced with a rated gain of 100 and with the require
ment that the gain shall never fall below 99, determine the required
initial gain of the ampli er in the absence of feedback.
(6,600.)
3. Show that the output impedance of a common grid ampli er is
1,, + ((1. + l)Z,, where Z, is the impedance between grid and cathode.
4. Show that the output conductance of the circuit in Fig. 10.10 is
approximately equal to gm/2 (assuming R2 is very large and R1 = R3).
5. At suf ciently high frequencies the inductance of the electrode leads

1.,
I
%
+ C I 1°
"_ L‘

Z +1,
Fro. X.i

may be appreciable. In Fig. X.i L1 represents the inductance of the

cathode lead and C is the grid cathode capacitance of a pentode. Show
that the input admittance is given by the expression
r g,,,...=L,,c ;'...c(2..=L.c 1)_
V, (gm(.v)L1 )2 + (0.)2LgC — 1)2

If the operating frequency is not too high, show that the effect of the
lead inductance is to introduce a conductance between grid and cathode
of amount g,,,m*L;,C.
6. With the aid of circuit diagrams explain current negative feedback.
Discuss qualitatively its effect on the input and output impedance of an
ampli er. ‘
Calculate the voltage gain of a triode cathode follower stage in which
gm = 2 5 mA/V, 1,, = 10,000 Q and the cathode load resistor is 5,000 Q.
Derive any formula that you use.
(0 9.)
7. Discuss in detail the purpose, operation, and design of a cathode
follower stage.
8. Discuss the effects of current and voltage negative feedback on:
(a) the gain, input impedance and output impedance of an ampli er, and
(b) the distortion produced by an ampli er.
If the gain of an ampli er without feedback is 90 dB, what must be
316 PRINCIPLES OF ELECTRONICS
the attenuation in the feedback loop if, with feedback, the gain is reduced
to 60 dB? (60 dB.) [I. of P., 1952.]
9. Explain, with reference to the equivalent circuit, the inherent
disadvantages of a triode valve having a large ampli cation factor, when
used as an ampli er at high frequencies. Why is a tetrode superior for
this purpose?
Calculate the mutual conductance of a pentode valve used in a single
stage I.F. ampli er operating at 465 kc/s given that the voltage gain
between grid and anode is 58 6 dB and that the anode tuned circuit
consists of a 200 11oF condenser in parallel with an inductor having a Q
of 100. (5 mA/V.) [I. of P., 1953.]
10. Describe what is meant by feedback and explain its effect with
reference to: (a) a cathode follower circuit; (b) the Miller effect.
A triode having an a.c. impedance of 10,000 Q and an ampli cation
factor of 30 is used in a cathode follower circuit with a cathode load
resistance of 200 Q. Calculate the voltage gain of the circuit and the
effective intemal impedance. (0 37, 320 Q.) [I. of P., 1953.]
ll. Explain in physical terms the effects of current and voltage negative
feedback in any system for the ampli cation of electrical signals. Con
sider the effects of both types of feedback on: (a) the ampli cation;
(b) the effective internal resistance; (c) distortion.
Two identical triode valves are connected in parallel. The resistor in
the anode circuit of the combination has a value of 25,000 Q, and the
value of the cathode bias resistor is 1,000 Q. The grid bias is arranged so
that the anode slope resistance of each valve is 10,000 Q and the ampli ca
tion factor is 30. Calculate how much of the cathode resistor must be
by passed to alternating current in order to increase the output impedance
of the combination to 25,000 Q. (355 Q.) [I. of P., 1954.]
12. Explain in words the effects of current and voltage negative feed
back in any system for the ampli cation of electrical signals. Refer
particularly to: (a) input and output impedance ; (b) frequency response;
(c) distortion.
In a single valve pentode ampli er the load resistance is 65,000 Q,
gm = 1 75 mA/V, the impedance of the valve is 0 95 MQ, and the anode
current is 3 15 mA. Draw the equivalent a.c. circuit and from it deduce
the gain of the ampli er. Calculate the values of: (a) the cathode
resistor necessary to reduce the gain to 25; (b) the bias resistor required
for a grid bias of — 2 2 V. Sketch the complete circuit and indicate
suitable values for components in the grid and cathode portion of the
circuit. (2000 Q, 700 Q.) [I. of P., 1955.]
13. The anode resistor of a triode valve circuit has a resistance of
50,000 Q, and so also has the resistor connecting the cathode to the nega
tive terminal of the h.t. supply and earth. The grid is so biased that the
anode slope resistance of the valve is 10,000 Q and the ampli cation factor
is 25. Calculate the approximate value of the internal resistance of
(a) the anode circuit, (b) the cathode circuit, when each is used to drive a
 EXAMPLES 317
subsequent stage of ampli cation. The input voltage of the triode is
applied between grid and earth. (1 3 MQ, 2 3 KQ.) [I. of P., 1956.]
14. Draw the equivalent circuit, including inter electrode capacitances,
for a simple triode ampli er feeding a resistive load, and derive an expres
sion for the equivalent input impedance. Hence explain why such a
circuit is unsuitable for the ampli cation of high frequency signals. Why
is a pentode more suitable than a triode for this purpose?
What is the effect, on the properties of the ampli er, of increasing the
effective grid anode capacitance by connecting a condenser between grid
and anode? Indicate two applications of this type of circuit.
[I. of P., 1957.]
15. In a single valve ampli er a fraction (3 of the output voltage is
fed back to the input as negative feedback, and negative current feed
back is also applied through a resistance R1. The anode slope resistance
of the valve is R, and its ampli cation factor is (1.. Calculate from rst
principles the output impedance of the ampli er.

mwur
2 R2 oureur

Ra
O . ___

R1 zmn. R2=4ooo, R1 =1sm.

Fro. X.ii

Explain brie y why the non linear distortion of the ampli er is reduced
by the use of negative feedback.
Calculate the input impedance, output impedance and voltage gain of
the cathode follower stage shown in Fig. X.ii.
(Note. Neglect the effect of condenser.) [I. of P., 1957.]
16. Derive an expression for the shunt resistive component of the input
impedance of an earthed cathode ampli er valve due to the inductance
of its cathode connection. Calculate the input resistance for a pentode
valve having the following properties:
Capacitance between control grid and cathode = 5 p.p.F
Inductance of cathode connection = 0 03 p.H
Mutual conductance = 8 mA/V
The frequency is 100 Mc/s.
(See Example 5 above.) (2000 Q.) [I.E.E., III, April 1957.]
M
318 PRINCIPLES OF ELECTRONICS
17. A triode ampli er has a resistance RK between the cathode and earth.
The anode load is a resistance R. The input is applied between the grid
and earth, whilst the output is taken between the anode and earth. A
by pass condenser C is in parallel with the resistance RK. Show that the
ratio of the voltage ampli cation at high frequencies to that for d.c.
signals is given by 1 + RKgm, if 1/1,, is assumed negligible.
If R = 20 kQ, RX = 100 Q, gm = 10 mA/V, and re = co, nd the value
of C such that the stage gain at 5 kc/s is 1 5 times the d.c. stage gain.
(0'54 |.1.F.)
(Note. This type of circuit can be used to give a treble boost in an
audio ampli er.)
18. A pentode ampli er has an anode load resistance R1, and the signal
is applied in series with a resistance R1 between the grid and cathode.
The output voltage is taken between anode and cathode. A feedback
path Z3, consisting of resistance R, in series with a condenser C, is con
nected between anode and grid. Show that at any frequency
l
A_ _(gm' z;)Rr

1 +%8<R1+ R. + R.R.g...>
if 1/r, is assumed negligible.
Find the ratio of high frequency gain to d.c. gain, when R1 = 20 kQ,
R1 = 1,000 Q, R, = 100 kQ and gm = 10 mA/V.
(0 3.)
(This type of circuit can be used to give a bass boost in an audio am
pli er.)
19. A transistor ampli er has a resistance R1 between collector and
earth, and a parallel combination of resistance R, and capacitance C,
between emitter and earth. The input is applied between base and
earth, and the output taken between collector and earth. Show that the
voltage gain is given by
R
— deb R1/{hue + deb

if we assume that vb, = hneib and 1'. = a¢;,i;,.

(This feedback circuit can be used to give treble boost.)

EXAMPLES XI
1. Use static characteristics and load lines to show the transient
variation of v4 and £4 in the rst valve of the ampli er in Fig. ll.6.a.
(Through the Q point draw a load line corresponding to the load
resistance and the grid leak in parallel.)
2. Show that the initial value and the time constant of 11,, in Fig. ll.5.b
are respectively v_,R,r,,/(R1/,1 + R911, + R;,R,) and RC, where R1, = load
resistance, R, = grid leak and R = R, + r,,R1,/(r, + R1,).
 EXAMPLES 319
3. For the circuit of Fig. ll.9.a show that the output voltage due to a
sudden change e in grid voltage is A (1 — e"/T), where A = — p.6R/( R + 1,)
and T = Rr1,C,/(R + 1,1).
4. The anode load of a triode ampli er consists of resistance R and
inductance L in parallel. If the grid voltage is changed suddenly by a
small amount e, show that the anode current is iq + g,,,e(l — As"/T),
where A = R/(R + 1,) and T = L(R + r,,)/R1,. Show on anode
characteristics how the anode current changes.
5. In the previous example show that the variation in anode voltage is
given by 0,, = g,,,eR'e‘R"/L, where R’ = Rr,/(R + 1,).
6. A diode is switched in series with a battery E and an uncharged
condenser C. The condenser has a parallel leakage resistance R. Using
the characteristic of the diode, indicate qualitatively the pulse of current
through the diode, and the pulse of voltage across the condenser.
(The steady state condition is given by the cross over of the diode
characteristic and the load line E = v4 | Rid ; the initial diode current
is given by 11,1 = E.)
Show that the steady state condenser voltage is equal to E if the diode
current for v4 = 0 is E/R.
7. A diode is switched in series with a battery E and an inductance L of
negligible resistance. Using the characteristic of the diode, indicate
qualitatively the pulse of current through the diode.
8. A thermionic diode has a resistance R between anode and cathode.
This parallel combination is suddenly switched in series with an uncharged
condenser and a battery. Indicate qualitatively the change with time
of the voltage across the diode, and show that the voltage across the
condenser can nally exceed the battery voltage.
9. A common cathode triode ampli er has an anode load of resistance
R in series with inductance L. The d.c. grid voltage is suddenly changed
by a small amount v,. Show that
11° = ___'_g2!)_ '_{1 + Q 1._ 1/T},
1 1 R
(12 + 7.)
where T = L/(r, + R).
10. A pentode ampli er (1,, = co) has a resistance R in series with an
h.t. battery E, to earth. From the cathode to earth is a parallel com
bination of resistance RK and condenser C. The voltage between the
grid and earth is changed suddenly by a small amount v,. Show that
_ gmv. _ _
'° T (1 +£1.12‘12)“ +g"‘R"e W’
_ CR;
where T —————(1
+ gmRK)

ll. A common cathode triode ampli er has a resistance load of R1.

It is coupled to a second ampli er through a condenser C in series with a
320 PRINCIPLES OF ELECTRONICS
resistance R. Show that when the voltage on the grid is suddenly
changed by a small amount v, that the voltage across the resistance R is

1 g"iv' 1 ‘M’
(E+E+F)
__ R11,
Wh€re T —— + }°

l2. A common cathode triode ampli er has the primary of a trans

former as the anode load (inductance L1, negligible resistance). If M is
the mutual inductance between primary and secondary, show that, when
the grid voltage changes by a small amount 0,, the secondary voltage is
1% 8 1/T
L1 ’
where T = L1/1,.
(Note. In Examples 9 to 12 nd qualitatively, using the valve charac
teristics, the changes in currents and voltages in the cases where these
changes are not small.)
13. A common cathode pentode ampli er (1,, = oo) has an anode load
composed of C, R and L in parallel. Show that if the grid voltage
changes by a small amount v, and T = 2CR > V LC,

then v. = gmv.1‘/é e"/T sin 72:6 t.

14. A common cathode triode ampli er has an anode load resistance R1

and a capacitance C between anode and grid. Show that, if the grid
voltage changes by a small amount v_,,
1
U11 Z — g,,1R1v_,{l . HT},

where T = CR1.
15. A common cathode pentode ampli er (1, = co) has an anode load
composed of resistance R in parallel with a capacitance C and with the
primary L of a transformer. The secondary of the transformer (mutual
inductance M) is in series with the d.c. grid supply between the grid and
cathode. If the grid voltage changes by a small amount v,, show that
L . 1
v, = —g,,,v_,\/Ce"/T srn E t,

where 5. 216 g"iM} < 1 /\/I.—C‘.

16. The table on next page gives some data on the static character
istics of a photo electric cell:
 EXAMPLES 321

Anode cathode Amount of

potential 0 10 25 250 light falling on
di erence, V cathode, lumens

0 0 75 08 09 0 02
Anode
current, I 0 15 16 18 0 04
114 I
0 30 32 36 0 08

(a) What type of cell is this, and for what applications is it particularly
suited?
(b) The cell is used in the circuit shown in Fig. XI.i. The light falling
on the cathode is mechanically chopped into “ square ” pulses such that

+25OV
R1 mo

O O1|1F
ceu. R2 mo
O
Fro. XI.i
it is constant at 0 08 lumen for 0 02 sec and is zero for 0 02 sec. Assuming
that steady conditions have been reached, what are the maximum and
minimum values of pulse height developed across R2?
(1 31 V, 1 31 V.) [I.E.E., III, April 1956.]
17. For applications involving transient phenomena it is often necessary
to use an ampli er whose gain is constant over a wide frequency range,
and whose phase shift changes in proportion to the frequency. Explain
simply why these characteristics are necessary, and show why it is dif cult
in practice to obtain them simultaneously with high gain.
[I.E.E., III, 1954.]
18. A single stage ampli er is to use a pentode having an anode slope
resistance of 1 M!) and a mutual conductance of 2 mA/V. The ampli er
is to feed a circuit of resistance 0 5 MQ and shunt capacitance 12 |J.|J.F
through a coupling capacitor. The gain is to be uniform, within 3 dB,
from 20 c/s to 100 kc/s. If the anode—earth capacitance plus stray
capacitance is 8 11oF, what load resistance is required, what middle
frequency gain can be achieved and what is the minimum value of coupling
capacitor required?
If the input to this ampli er consists of rectangular pulses, estimate:
(i) The rise time (to 90 per cent) of the output pulses.
(ii) The greatest pulse length that can be handled if the decay of
the output pulse is not to exceed 10 per cent.
(l04,000 Q, 160, 0 016 11F, 3 7 p.s, 22 ms.) [I.E.E., III, 1956.]
322 PRINCIPLES OF ELECTRONICS
19. A valve having an ampli cation factor of 5 and a mutual conduct
ance of 5 mA/V feeds a load circuit of inductance 25 henry and resistance
1,000 Q. Negative feedback is applied by a non bypassed resistor of
1,000 Q in the cathode circuit. By what value will the anode current
increase if a positive step signal of 35 V is applied to the grid circuit, and
how long will it take to reach 90 per cent of its nal value?
(21 9 mA, 7 2 ms.) [I.E.E., III, 1954.]

EXAMPLES XII
1. In the direct coupled ampli er of Fig. 12.1 show that a small change
of amount e2 in the battery voltage E2 would give a change in output
voltage of
1 _ _g,,,
R 1 + R/r2
—T_T_ 6*’
12 + 7,
when both valves are identical and have the same load resistance R.
2. For the ampli er in Fig. 12.4 show that the change in output arising
from a change e12 in the supply E12 is approximately equal to
RR1e12 _
Rx(R1 + R2)
For a change e2 in E2 show that the output changes by
RR2e2 _
RK(R1 + R1)
3. Explain why the performance of the circuit in Fig. 12.7 may be
improved by replacing RK by a suitably biased pentode.
4. Draw the circuit diagram of a cathode coupled ampli er stage and
explain its operation. Show how such a circuit can be used as a difference
ampli er and derive an expression for the gain in this case.
5. What is meant by “ drift ” in connection with direct coupled ampli
ers, and why is it so serious in high gain d.c. ampli ers, whereas it is not
usually troublesome in equally high gain systems using capacitive or
transformer inter stage coupling? Mention the main causes of drift, and
explain how their effects can be minimized.
[I.E.E., III, October 1956.]
6. The diagram of Fig. XII.i shows the circuit of a d.c. valve voltmeter.
Describe the manner of operation of the circuit and explain the function
of each component. What are the disadvantages of this type of circuit
for this purpose?
E — 100 V, R1 — 6,500 Q, R2 — 3,500 Q, R2 — 500 Q, R1 — 9,500 Q.
Meter resistance 300 Q. Valve (J. — 19, gm — 2 1 mA/V.
 EXAMPLES 323
Find: (a) the anode cmrent when the meter reads zero; (b) the d.c.
input required to produce a current of 1 mA through the meter.
(3 5 mA, 0 85 V.) [I. of P., 1955.]
7. Two identical triodes, with identical anode resistors but with a
common cathode resistor, are connected across an h.t. supply. Analyse

Q a
R1

=5

R2
1

Fro. XII.i
the action of this circuit and explain how the anode voltages vary when a
sinusoidal voltage is applied to one grid, the potential of the other grid
remaining constant. Why is it desirable that the resistance in the cathode
circuit should be as great as possible?
Indicate possible uses for this type of circuit. [I. of P., 1956.]
8. Discuss the. di iculties which are encountered in the construction and
operation of a direct current ampli er, and explain how these may be
minimized.
Give an example of a situation in which it is possible to modify appar
atus which would normally yield a direct current response so that an
alternating current output can be obtained. [I. of P., 1957.]

EXAMPLES XIII
1. Sketch a typical anode current/anode voltage curve for a tetrode
when the grid and screen voltages are xed. Explain fully the nature of
the curve. Show how a tetrode may be used as an oscillator.
2. In the vector diagram of Fig. 13.7 add vectors representing L and
p.V;, and explain the conditions to be satis ed for oscillation.
3. In the phase shift oscillator of Fig. 13.3 show that the frequency of
oscillation is given by the formula

= 2 1 TR%/é (Assume R1 < R.)

4. In the phase shift oscillator of Fig. 13.3 show that, for oscillation,
the ampli er voltage gain must be at least 29.
5. Draw a phase shift transistor oscillator corresponding to the triode
oscillator in Fig. 13.3.
6. It is shown in Section 13.6 that both circuits of a tuned anode
324 PRINCIPLES OF ELECTRONICS
tuned grid oscillator must be tuned to resonant frequencies above the
frequency of oscillation. Use this result to explain why this type of
oscillator is not suitable for operation at very high frequencies.
7. When resistive components are taken into account the generalized
triode circuit of Fig. 13.10 takes the form shown in Fig. XIII .i, where the

'
' g + ° @

_
FIG. XIII. FIG. XIII.ii

Y's are admittances which may be expressed in the form Y = G 1 jB,

where G is a conductance and B is a susceptance. Using the equivalent
circuit of Fig. XIII.ii and assuming that an oscillator is an ampli er
which provides its own input, show that the condition for oscillation is
Y1Yg + YQY3 + Y3Y1 + Yggm =

(Nate. The anode slope conductance of the valve g2 is included in P.)

8. By separating the Y's into their real and imaginary parts show that
the condition for oscillation of the last example reduces to two separate
and necessary conditions
(G10: '1' G26: '1' 6:161) (B132 + B2Ba '1‘ BaB1) + 5 1162 = 0
and G1(Ba '1‘ Ba) + G2(Ba '1“ B1) + Ga(B1 "1' B2) + 81 132 = 0
Use these two equations and the knowledge that all the conductances
are positive to show that B1 and B2 must have the same sign and B2 the
opposite sign.
9. If the circuit resistances in Example 8 are negligible (i.e., all the
conductances are zero except g2 and gm) show that the condition for
oscillation becomes B2/B1 > 1/11.
10. Derive the condition for sustained oscillation, and the exact value
of the oscillation frequency, in a circuit consisting of a coil (L, R), a
capacitance (C), and a negative resistance (— r), all connected in parallel.
11. Compare brie y, with reference to the electric eld in the valves, the
action of: (a) a tetrode; (b) a pentode; (c) a beam tetrode.
Show that sustained oscillations may be obtained with any device for
which the voltage current characteristic has a negative slope and describe
brie y an oscillator based on this principle. [I. of P., 1952.]
12. Describe two possible circuit arrangements for the maintenance of
continuous oscillations in an oscillatory circuit by a triode valve, and
 EXAMPLES 325
derive the necessary initial conditions for the continuous increase of
oscillatory amplitude in one of them.
How is the amplitude of oscillation limited by the characteristics of the
valve? [I. of P., 1954.]
13. It can be shown that all single valve feedback oscillators can be
represented by the basic circuit of Fig. XIII.iii (a.c. conditions only).

FIG. XIII.iii

For oscillation to be possible either of the following sets of conditions

must be ful lled:
Z1 and Z2 inductive ; Z3 capacitive.
Z1 and Z2 capacitive; Z; inductive.
Illustrate each of these criteria with a circuit diagram (showing supply
connections) of a practical oscillator arrangement. What properties of
the valve limit the highest frequency at which such circuits will operate,
and which of the two basic arrangements is preferable when the highest
possible frequency is required? [I.E.E., III, April 1956.]
14. State the conditions for self oscillation of a valve ampli er circuit,
and explain the operation of the oscillator shown in Fig. XIII.iv.

H.T.+
Sk
O O13? |°kQ

s e xv 10kQ

Fro. XIII.iv

Assuming valves of identical characteristics and high slope resistance,

determine the frequency of oscillation and the least value of mutual
conductance for the valves. The effects of C and R are to be neglected.
(1 30 kc /s, 0 32 mA/V.) [I.E.E., III, October 1956.]
15. A direct coupled ampli er consists of three identical resistance
326 PRINCIPLES OF ELECTRONICS
loaded stages in cascade, and has an overall gain frequency characteristic
(relative to zero frequency) as follows:

Frequency, kc/s 0 1 10 50 100 200 400 800 1,600

Change in gain, dB 0 0 36 9 21 37 54 72 90

Voltage gain at zero frequency = 8,000. A fraction of the output

voltage is tapped off the load resistance of the last stage and returned
without modi cation to the input circuit of the rst stage, so that the
feedback at low frequencies is negative. It is found that oscillation
occurs if the voltage feedback fraction exceeds a certain value. Deter
mine this value, and the frequency of oscillation.
(10'°, 88 k/cs.) [I.E.E., III, April 1956.]
16. In terms of the quantities shown in the skeleton circuit diagram
(Fig. XIII.v), deduce expressions for: (a) the minimum mutual inductance

L0

$12
L @ {9,

FIG. XIII.v

required for oscillation to commence, and (b) the frequency of oscillation.

What method might be used, in a practical oscillator, to ensure reasonable
amplitude stability of the output? [I.E.E., III, October 1956.]
17. Describe with the aid of a circuit diagram a valve oscillator suitable
for generating frequencies in the audio frequency range. Give a physical
explanation of the generating process and derive an expression for the
conditions determining the inception of oscillations.
[I.E.E., III, October 1956.]
18. Draw a circuit diagram for a tuned anode valve oscillator and
explain its action. Sketch a vector diagram for the equivalent circuit
and derive an expression in tenns of the circuit constants for the least
value of the mutual inductance necessary for oscillations to commence.
[I.E.E., III, April 1957.)
19. The a.c. circuit of a transistor is equivalent to a resistance R;
between emitter and earth, and a resistance RC between collector and
earth. If we assume that for small signals
12,2 = hubi, and 1', = — 01221,,
 E XAM PLE S 327
show that the a.c. resistance between the base and earth is
hllb + R1. _
1 01,,

(If the transistor has a value of 01¢, greater than unity, then the a.c.
resistance is negative.)
20. The same transistor is in an a.c. circuit which can be represented by
a resistance RB between base and earth, whilst there is a resistance R1;
between collector and earth. Show that if

ace>(1

the a.c. resistance between emitter and earth is negative.

21. Two identical triodes have their cathodes connected together, and
at the frequency under consideration the anode of each triode is effectively
connected to the grid of the other triode. Show that the value of the
a.c. resistance between the two anodes is
__L_
(1/'4 _ gm)
if grid current is negligible.
(NOIC that 7:21 = — 1:22.)

EXAMPLES XIV
1. Show that the movement of a charge between two planar electrodes
under the in uence of a uniform eld produces a saw tooth pulse of current
in the external circuit.
2. Indicate approximately the effect of space charge on the shape of
the current pulse in the external circuit in Example 1.
3. The constant current leaving the cathode of a temperature limited
planar diode is 1'2 and the anode voltage is v11 + 6 sin wt. If d is the anode
cathode distance show that at time t the electron velocity and displacement
are given by
d A

7: = 16%; (t — to) — 1% uz (cos cot — cos wto)

A I

and x = 2% (t — to)” + 2 o%d (t — to) cos oaio — 1 0 ; ‘i—1d (sin cot sin oaio),

where it is assumed that the electron leaves the cathode at time to with
zero velocity, and space charge effects are neglected.
Using the expression 1}m(%)2 for the kinetic energy and Ly for the
potential energy, nd, in terms of to and T, the difference between the
K.E. and P.E. at the anode of an electron which reaches the anode at
time t = to | T, where T is the transit time.
328 PRINCIPLES OF ELECTRONICS

Since Z9%£° is the number of electrons leaving the cathode in time dt,,
the integral
21.
P = 51/0 7 (K.E.
(1) (010
P.E.) dz,
gives the additional power consumed by the electrons at the expense of
the h.f. eld. If v2 > 5, so that T is the same for all electrons, show that
P £2132 {2 — 2 cos wT — (OT sin wT}_
v2 (coT)*
4. If the additional source of power consumed by the electrons in the
previous example is represented by a conductance g in parallel with the
diode show that

5 = 2(2 2 cos or or sin (OT)/((0T)2,

0

where go = 1'11/vo.
Draw a graph of g/go against mT and hence explain why a diode can
be used as a negative resistance oscillator at very high frequencies.

EXAMPLES XV
1. Discuss the interchange of energy between electric elds and moving
charges.
Explain qualitatively the growth of the amplitude of the wave along
the helix of a travelling wave tube.
2. The mean voltage of a klystron resonator is v2 and the bunching
voltage is 01 sin wt. If 171 < vo, show that the velocities of the electrons
leaving the buncher are given approximately by the equation
_ 111 .
u _ u11(1 + 270 srn cot),

where uo is the velocity corresponding to v2.

3. Show that the distance—time diagram for an electron in the resonator
re ector space of a planar re ex klystron is parabolic, and hence show
how bunching occurs.
4. Explain why the collector in a double resonator klystron is some
times operated at a voltage below the resonator voltage.
5. On a diagram of a planar magnetron similar to Fig. 15.8 show the
electron distribution.
6. Explain brie y how oscillations are maintained in: (a) a Hartley
oscillator; (b) a magnetron; (c) a multivibrator ; (d) a dynatron oscillator;
and (e) a resistance capacity oscillator.
A triode with a mutual conductance of 1 5 mA/V is to be used as an
oscillator. The parallel tuned grid circuit has an equivalent series
 EXAMPLES 329
resistance of 25 Q and a total capacitance of 200 p.p.F. Calculate the
minimum value of the mutual inductance between the untuned anode
coil and the grid coil in order that oscillations may just be maintained.
(3 3 p.H.) [I. of P., 1957.]
7. A diode has a cylindrical anode of radius a metres and a cathode of
radius b metres (a > b). A constant potential difference of V volts is
maintained between them, and the axis of the diode lies along a uniform
magnetic eld of ux density B webers per square centimetre.
Calculate the value of B which just allows electrons leaving the cathode
radially with zero initial velocity to reach the anode.
Explain brie y the basic principles of the operation of the cavity
magnetron oscillator. [I.E.E., III, October 1956.]
8. Show how velocity modulation of an electron beam can produce
bunching in a drift space. Show the application of this effect to: (i) an
ampli er, and (ii) an oscillator suitable for very high frequencies.
Upon what factors does the operating frequency of these devices depend?
[I.E.E., III, April 1957.]

EXAMPLES XVI
1. A diode has the following values of v_.1 and 1'4:

v4, V 0 5 10 15 20 25 30 35 40 60
1'4, mA 0 4 10 20 32 50 72 96 120 240

Find graphically the anode current when a sinusoidal alternating

voltage of peak value 50 V is applied to the diode.
2. In the recti er circuit of Fig. 16.3 show that the d.c. power output
in the load is i1‘”R/1:2(R1, + R)”, the power dissipated in the diode is
NR2/4(R2 + R)’ and the efficiency (i.e., d.c. power in load/power taken
from source) is
__4;__.
1=“(l + R»/R)
3. Show that in a condenser input half wave recti er the voltage across
the load is given approximately by the expression 17¢ = ii — 5/2fC, where
the symbols have the same meanings as in Section 16.6.
4. Compare the relative merits of condenser input and choke input
recti ers.
5. Draw a diagram similar to Fig. 16.11 to show how the voltage Ugg
builds up across the condenser C2 in the voltage doubling circuit of Fig.
16.14.
6. Explain how the thyratron circuit shown in Fig. XVI.i may be used
to control a unidirectional current through the load R. Draw waveforms
of the anode voltage and anode current.
7. A certain gas diode has a striking voltage of 105 V and over the
3M) PRINCIPLES OF ELECTRONICS
voltage range 72 to 78 V the diode current varies from 5 to 45 mA. Show
that this diode may be used to supply a load current from 0 to 40 mA
whilst the voltage remains constant at 75 V 3); 4 per cent.

V = V slab) I

FIG. XVI.i

A diode with similar characteristics is used to maintain a stable voltage

of 75 V across a resistance of 5,000 Q using a nominal 200 V d.c. supply,
with the circuit shown in Fig. l6.20.b. Calculate the value of R and show
that the d.c. supply voltage may vary by 30 per cent for a 4 per cent
voltage variation across the load. What is the minimum value of the
supply voltage for switching on?
(3,120 Q, 170 V.)
8. A half wave diode recti er feeds a load of resistance 10,000 Q,
across which is connected a 1 (.LF capacitor. The supply is sinusoidal, of
frequency 50 c/s, and the diode has zero resistance when conducting and
in nite resistance when non conducting. Determine the instants in the
cycle at which diode conduction commences and ceases.
(280° and 380° from peak) [I.E.E., III, October 1956.]
9. Explain the action of the two diode voltmeter circuits in

+ 9 + H
v1 I/1 — '2

Frc. XVI.ii

Fig. XVI.ii and indicate the type of application for which each is suitable.
If the output voltage from these circuits is to be ampli ed before measure
ment, sketch suitable direct current ampli ers for use with each type of
circuit. [I. of P., 1952.]
10. Compare the actions of a thyratron and a discharge tube voltage
stabilizer.
Sketch: (a) a circuit using a thyratron to control the opening and
closing of a relay in response to an external signal, and (b) a circuit using
 EXAMPLES 331
a discharge tube voltage stabilizer. Comment on the performance of the
voltage stabilizer and indicate brie y how any residual voltage uctua
tions might be eliminated. [I. of P., 1952.]
11. Explain, with circuit diagrams, the action of two types of full
wave recti er circuit suitable for supplying an X ray tube at 200 kV
from a 50 c/s supply. Describe the essential features of the components
used and explain brie y how the voltage across the X ray tube could be
measured.
If the X ray tube current is 10 mA, what must be the capacity of the
smoothing condensers if the ripple is not to exceed 0 5 per cent?
(0 1 p.F.) [I. of P., 1953.]
12. The circuit diagram of a series parallel voltage stabilizer circuit
is given in Fig. XVI.iii. V1 and V2 are identical triodes for which

v1

B R,
mvur v2 Q oureur
VOLT AGE V0 LTAGE

va R’
Fro. XVI.iii

R2 = 40,000 Q and (J. = 20. V3 is a gas discharge tube. R1 = R2 and

R2 = 1 MQ. Calculate, approximately, the change in the output voltage
if the input voltage changes by 10 V. (0 04 V.) [I. of P., 1956.]

EXAMPLES XVII
1. Explain what is meant by an amplitude modulated wave and show
that it may be represented by a carrier and side bands. VVhat is the
signi cance of this for communication purposes?
A transmitter radiates a power of 1,000 W when fed with a carrier
amplitude modulated to a depth of 50 per cent. Calculate the power
in each side band.
(56 W.)
2. Describe and explain the main features of a superheterodyne receiver
for the reception of amplitude modulated broadcast signals. Describe
in more detail the action of the second detector.
3. A superheterodyne receiver has a calibrated r.f. dial which indicates
the frequency of the received signal. A certain station is received strongly
at 100 Mc/s on the dial and weakly at 76 Mc/s. Explain this and detennine
332 PRINCIPLES OF ELECTRONICS
the actual frequency of the station and the intermediate frequency of the
receiver.
(100 Mc/s, 12 Mc/s.)
4. The receiver in Example 3 receives the same station very weakly
when the dial indicates 82, 84, 92 and 94 Mc/s. Show how these results
arise from the harmonics of the station and the local oscillator.
5. A second station is received on the same receiver when the dial is
set to 106 or 94 Mc/s. Explain this and nd the actual frequency of the
station.
(224 Mc/s.)
6. Explain how a pentode valve may be used as a variable reactance.
7. Draw the circuit of a variable reactance valve in which the grid
bias may be varied at audio frequency. The phase shift potential divider
consists of a capacitor of 30 pF connected between grid and negative h.t.
and a 50 kQ resistor.
Show that the impedance between anode and cathode of the valve is
equivalent to an inductance shunted by a resistance, and nd the values
of these components for mean values of g,,, = 2 mA/V and f = 1,000 c/s.
Explain how the circuit may be used for frequency modulation of an
oscillator.
What steps could be taken to avoid amplitude modulation?
(8 5 H, 500 Q.) [I.E.E., III, April 1956.]
8. Show that the processes of modulation, demodulation and frequency
changing are essentially the same.
The dynamic characteristic of a triode with a 10,000 ohm resistive load
is represented by
I2 = 2 5 (V2,, + 5) + 0 2(V2;, + 5)” milliamperes,
where V2; is the potential difference between grid and cathode in volts.
The valve is operated with a xed bias of — 3 V, and sinusoidal signals of
amplitudes 1 V and 0 5 V at frequencies of 2 kc/s and 5 kc/s, respectively,
are applied simultaneously to the grid circuit. Determine the amplitudes
and frequencies of the various components of voltage across the load.
(59 3,33, 16 5, 1, 0 25, 1, 1 V; 0, 2,5,4, 10, 7,3 k/cs.) [I.E.E., III, 1954.]
9. Explain the operation of a diode recti er used for the demodulation
of a modulated radio signal.
Show that for an unmodulated signal the load presented to the preceding
valve by the diode circuit is a resistance of half the value of the series load
resistor in the diode circuit. Assume the diode to act as a perfect recti er
and to have a large condenser in parallel with the load resistor.
[I. of P., 1957.]

EXAMPLES XVIII
1. Describe in detail the operation of a free running multivibrator.
What general considerations govern whether a trigger circuit will be free
running, mono stable or bi stable? Give examples of each type of circuit.
 P30

_m_
flq
I1
.‘
.
.‘e
_____Qé mi‘?
~®‘Ow‘1”‘®?_
_>
T

My)1_ =_ _ _
1I

3:5“
6:“

_PUW UCG IU_ _ ,W

334 PRINCIPLES OF ELECTRONICS
2. Use Sections 6.9, 18.1 and 18.4 to show how a transitron relaxation
oscillator may be used as a switch across a charging condenser for the
production of a saw tooth waveform with the circuit shown in Fig.
XVIII.i.

%
FIG. XVIII.i

3. Valves 1 to 6 in the circuit shown in Fig. XVIII.ii form three bistable

binaries which count up to seven in the normal manner. The output of
the rst binary is also arranged to switch one stage of the output bistable
circuit. Show that the whole circuit gives one negative output pulse for
every ten input pulses.
4. Analyse the transistor multivibrator circuit of Fig. 18.6, assuming
that the collector current is a linear function of the base current but is
almost independent of the collector emitter voltage.

EXAMPLES XIX
1. Show that a pentode valve used with a high resistance load acts as
a clamping device at approximately zero voltage. Hence show that the

OUTPUT
0

0 |
ggg |—o

INPUT 1 ‘ ‘ INPUT 2

0 O
FIG. XIX.i

circuit of Fig. XIX.i acts as a coincidence counter, i.e., a device which

operates only when two negative signal pulses occur simultaneously.
 EXAMPLES 335
2. Explain how the circuit of Fig. XIX.ii can also act as a coincidence
counter for negative pulses.
3. Explain how the circuits of Examples 1 or 2 may be adapted to act
as “ gates ”, i.e., circuits which will count only for speci ed time intervals.
4. The Operational Ampli er. The circuit in Fig. XIX.iii shows an

<>—l
IE
moor 1 ' mvur 2

Fro. XIX.ii
I
I.

FIG. XIX.iii

E
1'1,

OOQ
+ +00
+0“

cm
M

I..9
+'9. QD:0 I,,l,
FIG. XIX.iv

ampli er with a network z, in series with the input signal and a feedback
impedance network z). Show that for a sinusoidal signal

E0 =1 _ 51
Z1 E1
when the inherent gain of the ampli er is very large.
336 PRINCIPLES OF ELECTRONICS
When z, is a resistance R, and 2; is due to a capacitance C1 show that for
any signal
1
<» 1,2,1,
e 2 — —— e, dt.

Also, when z, is due to a capacitance C1 and 2; = R: show that

dc,
e2 z —R;C, '
dt
(Note. These ampli ers are able to carry out respectively the mathe
matical operations of integration and differentiation. They are used in
differential analysers and analogue computers.)
5. Show that the operational ampli er in Fig. XIX.iv can be used to
can'y out the mathematical operation of addition.

EXAMPLES XX
1. Write an account of the causes and effects of ampli er noise with
particular reference to resistance noise, shot noise, equivalent noise
resistance and noise factor.
2. A signal generator has an open circuit output e.m.f. e, and its output
impedance is resistive and of value R. Show that the ratio of signal to
noise across the generator terminals on open circuit is ea/4k TBR. When
a resistance R1 is connected across the output terminals show that the
signal to noise ratio is e2R1/4kTBR(R + R1) and the noise factor is
R1/(R + R1)
Hence show that in a receiver with a rst stage of zero noise resistance
and high gain the noise factor is 3 dB when the receiver input resistance,
assumed not to be due to feedback, is matched to the generator.
3. Give a full account of how noise may be generated in thermionic
ampli ers, and indicate why noise considerations are important in:
(a) carrier telephony repeaters; (b) television ampli ers.
(I.E.E., III, April 1957.]
4. Available power is de ned as the maximum power which may be
obtained in a load from a source, and it is achieved by matching the load
to the source. Show that the maximum available noise power from any
resistance is kTB.
5. Show that in a receiver with power gain G and an available noise
power output of PN per unit band width, the noise factor is P1, /kTG
when the receiver is matched to a generator at temperature T.
6. A receiver has its input matched by a resistance R, across which is
connected a temperature limited diode. The available output noise
power of the receiver is doubled when the diode current is increased from
zero to 1'4. Show that the noise factor of the receiver is ez'4R/2kT.
(Note. This method is used for the experimental determination of
receiver noise factor.)
 EXAMPLES 337
7. A wide band ampli er is to consist of a number of identical stages,
each giving a voltage gain of 10 times, and is to respond uniformly from
direct current to 10 Mc/s. The signal source resistance is 3,000 Q, the
input resistance is 2,000 Q and the equivalent noise resistance of the valves
used is 1,500 Q, referred to the grid. The nal stage can develop 30 V
(r.m.s.) output. How many stages are required if, at full gain in the
absence of signal, full output is to be provided by noise? The temperature
of the equipment is 27° C.
The mean square noise voltage developed by a resistance R Q is:
3 = 4 kTBR,
where k = 1 37 >< 10'” joule per degree; T = absolute temperature;
B = effective band width, c/s.
(7.) [I.E.E., III, April 1957.]
8. A 75 Q resistive signal source is coupled to the rst stage of an
ampli er by a network giving an impedance step up of 100 times and an
e ective band width of 5 Mc/s. Valve and circuit damping amount to
5,000 Q, and the noise resistance of the valve, referred to its grid, is 2,000 Q.
Assuming a unifonn temperature of 17° C, what is the r.m.s. noise voltage
referred to the grid of the valve?
You may assume that the “ available power ” from a source of
noise is kTB watts, where k = 1 37 >< 10"” joule per degree absolute;
T = temperature in degrees absolute; B = effective band width in cycles
per second. (20 .11V.) [I.E.E., III, 1954.]
9. A common cathode ampli er is the rst stage of a receiver. If a
resistance R is across the input terminals the noise output of the receiver
is found to be twice the noise output when the input terminals are short
circuited. Show that the noise resistance of the ampli er is equal to R.
(Assume that there is no change in band width.)
 APPENDIX I
LIST OF SYMBOLS
THE list below contains certain symbols which are used frequently in
the book. Some symbols are used with more than one meaning, but
where this occurs the context makes it clear which meaning is intended.
Symbols in heavy type denote complex or vector quantities. The
rationalized M.K.S. unit in which each quantity is measured is also
given.
A or A = voltage ampli cation.
Ag or A1 = voltage ampli cation with feedback.
A, = power ampli cation.
|A| = stage gain.
61 . .
01,11, = 5.9 at constant ow in a transistor = h21,.
B
81 . .
a,,, = — at constant U33 in a transistor = — hm.
E
B = magnetic ux density, Wb/m2.
B = frequency bandwidth, c/s.
[3 or (3 = fraction of output voltage fed back in series with
the input.
C = capacitance, F.
C22, C2,, C2, . . = inter electrode capacitances, F.
d = distance between parallel planes, m.
d,,2, d2, = interelectrode clearances, m.
8 = secondary emission coefficient.
E = energy, W.
E1, E2, etc. = e.m.f. of battery supplies, V.
E = electric eld strength, V/m.
= efficiency = P0/P4.
= force, N.
'~. ,'7'l.s = frequency, c/s.
g,,, = 3%‘ at constant v4 = mutual conductance or trans
o
conductance of a triode, A/V.
g,,, = 5% at constant v4, 1102, 110;», = mutual conductance
or transconductance of a pentode, A/V.
G1 refers to control grid in multi electrode valves.
G2 refers to screen grid in multi electrode valves.
G3 refers to suppressor grid in multi electrode valves.
338
 LIST OF SYMBOLS 339
3v
h11 or hm = 3 1.2 at constant veg in common base transistor, Q.
1:
avsn at constant 1'g in common base transistor.
h12 or hm = ——
av”
81 . .
h21 or hm = at constant veg m common base transistor
11
: i aw.

81 . . .
h:2 or hm = i
av“ at constant 1g in common base transistor, mho.
3v . .
h11, = 87 BE at constant veg, in common emitter tran
B
sistor, Q.
av . . . .
hm, = avingat constant 1g, in common emitter transistor.
c1:
81 . .
hm, = at constant veg, in common emitter tran
B
sistor = 012,.
61' . . . .
hgg, = a—vc— at constant 1g, in common emitter transistor,
as
mho.
1' = current, A.
1'4 = total anode current, A.
1'2 = varying component of anode current, A.
I2 = vector value of sinusoidal component of anode
current, A.
ie = steady value of anode current in absence of a
signal, A.
1,1 = mean value of total anode current, A.
1,, = peakvalueofvaryingcomponent of anode current, A
ie = total grid current, A.
1'2 = varying component of grid current, A.
I2 = vector value of sinusoidal component of grid
current, A.
ie = mean value of total grid current, A.
1‘, = peak value of varying component of grid current, A.
1'g = total cathode current, A.
1'g, 1'g, 1'e = totalemitter, base, collectorcurrent,respectively, A.
1}, 1'2, 1', = varying component of emitter, base, collector
current respectively, A.
I2, I2, L, = vector value of sinusoidal component of emitter,
base, collector current respectively, A.
] = current density, A/ma.
jg = total emission current density, A/m2.
l = effective length of de ecting system of a cathode
ray tube, m.
340 PRINCIPLES OF ELECTRONICS
L = inductance, H.
L = length from de ecting system to screen of a
cathode ray tube, m.
7. = wavelength, m.
m = mass, kg.
m or m, = mass of electron, kg.
mi = mass of ion, kg.
(1 = charge mobility = ratio of drift speed to applied
electric eld, m2 s'1 V'1.
|J = — 3 3‘ 2 at constant 1'4 = triode ampli cation factor.

I *8 = — at constant 1,1 and vg = tetrode ampli ca

0
tion factor.
Po = power output, W.
P4 = anode power dissipation, W.
q = charge, C.
R = resistance, Q.
1' = radius, m.

31
7a = anode slope resistance where 1/r2 = “
= charge density, C/ma.
= conductivity, Q‘1 m'1.
= time, sec.
= alternating period, sec.
= absolute temperature, ° K.
= electron transit time, sec.
= phase angle, rad.
= work function, eV.
= velocity, m/s.
"~:1~*e<2~le. _a 18 = potential, V.
"4 = total anode voltage measured from cathode, V.
Va = varying component of anode voltage, V.
V. = vector value of sinusoidal component of anode
voltage.
vge (sometimes ve) = steady or quiescent value of anode voltage in
absence of signal, V.
17.4 = mean value of total anode voltage, V.
15¢ = peak value ofvarying component ofanode voltage,V.
"0 = total grid voltage, V.
"0 = varying component of grid voltage, V.
V2 = vector value of sinusoidal component of grid
voltage, V.
Ugq = steady or quiescent value of grid voltage in
absence of signal, V.
I70 = mean value of total grid voltage, V.
 LIST OF SYMBOLS 341
62 = peak value of varying component of grid voltage, V.
vel, U02, U93, . . = total voltages of G1, G2, G3 . . ., V.
vel, ‘U92 (sometimes) = total voltages of the control grids of two different
valves, V.
v21, v22, v23, . . = varying componentsof voltagesof G1,G2,G3. . ., V.
v21, v22 (sometimes) = varying components of two different voltages
applied to control grid, or, varying components
of voltages of the control grids of two different
valves, V.
v, = signal voltage, V.
ve = total output voltage, V.
v, = varying component of output voltage, V.
vg = value of potential minimum in space charge, V.
vg = maintenance or operating or burning voltage in
gas discharge.
ve = breakdown or striking voltage in gas discharge.
All the above voltages are measured from the cathode.
vgg, vge, veg = total transistor voltages between electrodes indi
cated, V.
‘U25, v22, vc, = varying components of transistor voltages between
electrodes indicated, V.
V2,, V2,, V2, = vector value of sinusoidal component of transistor
voltages between electrodes indicated, V.
W4 = maximum permissible anode dissipation, W.
Wg = Fermi energy level, eV.
0) = angular frequency = 21tf, rad/s.
X = reactance, Q.
= impedance, Q.
NN = vector impedance, Q.
 APPENDIX rr
USEFUL CONSTANTS
= velocity of light in free space = 3 00 X 10° m/s.
= electronic charge = 1 60 X 10'" C.
= primary electric constant = 8 86 X 10'" F/m.
= 2 72.
= Planck's constant = 6 62 X 10'“ J s.
= Boltzmann's constant = 1 38 X 10'” _]/° K.
§Zv § n§'<\<'s = electronic rest mass = 9 11 X 10'" kg.

APPENDIX rrr
BIBLIOGRAPHY
THE following list gives a selection of books for further reading:
General
Gray, T. S. Applied Electronics, 2nd edition (Wiley, 1954).
Parker, P. Electronics (Arnold, 1950).
Tennan, F. E. Electronic and Radio Engineering, 4th edition (McGraw
Hill, 1955).
Williams, E. Thermionic Valve Circuits, 3rd edition (Pitman, 1955).
Valves
Beck, A. H. W. Thermionic Valves (Cambridge University Press, 1953).
Harman, W. W. Fundamentals of Electronic Motion (McGraw Hill,
1953)
Spangenberg, K. Vacuum Tubes (McGraw Hill, 1948).
Transistors
Evans, ]. Fundamental Principles of Transistors (Heywood, 1957).
Lo, A. W., Endres, R. 0., Zawels, _]., Waldhauer, F. D., and Cheng, C.
Transistor Electronics (Prentice Hall, 1955).

342
 INDEX

Abnormal glow, 63 Aperture lens, 25

Accelerator, linear, 218 Arc discharge, 63
Acceptor atom, 34 mercury, 63, 192
Aerial, 240, 291 Automatic bias, 111, 146, 154, 194
Ampli cation factor, 72, 76, 81, 92, 93, frequency control, 257
302 volume control, 253
gas, 61 Available power, 336
power, 115, 165
voltage, 96, 99, 138, 181 Balanced ampli er, 185
Ampli er, audio, 96, 115, 138, 250 input, 186
balanced, 185 Band, conduction, 31
bandwidth, 109, 305 energy, 30
cathode coupled, 187, 268 valency, 32
cathode follower, 113, 164, 162, 238, 287 Bandwidth, 109, 288, 305
Class A, 131, 208 Barium oxide, 42, 46
Class B, 131, 140, 208 Barrier, potential, 40, 43, 49, 69, 94
Class C, 78, 136, 178, 195, 208 Beam, electron, 22
common anode, 112, 164 tetrode, 78, 82
common base, 138, 311 Beryllium, 31
common cathode, 112 Bias, automatic, 111, 146, 154, 194
common collector, 144, 311 Binary scaling, 272
common emitter, 142, 164, 311 Bistable circuits, 269
common grid, 112, 163, 205, 305 Blocking oscillator, 265
differential, 185 Boltzmann’s constant, 288, 342
direct coupled, 181, 200 Bombardment, 44, 61
distortion in, 107, 117, 124, 141, 158, Bond, co valent, 32
159, 175 Bootstrap circuit, 287
double stream, 218 Boron, 34
e iciency of power, 120, 132, 140 Breakdown in gas, 60, 61
feedback in, 151, 179, 184 voltage, 63
grounded grid, 205 Bridge recti er, 233
high frequency, 107, 132, 136, 150, Brightness control, 26
205, 212, 215, 218, 250 Buncher, 212
intermediate frequency, 251
klystron, 212 Caesium oxide, 47, 48
magnetic, 189 Calcium oxide, 43
noise, 159, 288, 336 Capacitance, input, 114, 161, 166
operational, 163, 335 inter electrode, 76, 107, 175, 199, 204
power, 115 Capacitive load, 105, 175
push pull, 128, 186 Carbon, 31, 48
radio frequency, 107, 132, 136, 150, Carrier, 240
205, 212, 215, 218, 250 modulated, 240
stage gain, 99, 105 Carriers, majority, 34, 69
transients in, 169 minority, 36, 69, 95
transistor, 138, 310 Catcher, 212
travelling wave, 215, 218 Cathode bias, 111, 154
tuned, 107, 132, 136 cold, 60, 62, 87
voltage, 6, 96, 138, 181 follower, 113, 114, 162, 164, 238, 287
wide band, 305 indirectly heated, 45
Amplitude distortion, 117 mercury pool, 64
modulation, 240, 247 oxide coated, 43, 52
Analogues, 11 photo electric, 47, 48, 53
Anode characteristics, 70, 78, 83, 116 thermionic, 42
dissipation, 74, 121, 133, 208 thoriated tungsten, 43
modulation, 243 tungsten, 42
slope resistance, 52, 71, 76 virtual, 55, 58, 77
343
344 INDEX
Cathode coupled ampli er, 188, 268 Di erentiating, 282, 287, 336
trigger circuit, 270 Di usion, 36, 94, 150
Cathode ray tube, 13, 21, 47 Diode clipper, 277
Cat's whisker, 67, 89 cold cathode, 62, 236, 259
Cavity magnetron, 215, 218, 220 crystal, 51, 67, 221, 236, 237
Characteristic, anode, 4, 70, 78, 83, 116 gas lled, 51, 59, 221, 236, 238
common base, 89 hot cathode, 64
common emitter, 91 junction, 67
combined, 129, 131 limiter, 279
constant current, 71 mercury vapour, 65
control, 85 mixer, 251
diode, 4, 51, 59, 62, 68, 221 photo electric, 52, 55, 59
dynamic, 116, 124, 195, 242 point contact, 67
grid, 5, 70, 78, 152 semiconductor, 51, 67, 221
hybrid, 90 slope conductance, 252
mutual, 70 resistance, 52
pentode, 83, 116, 124 thermionic, 4, 51
tetrode, 78 vacuum, 4, 51, 221, 247, 251
transistor, 89, 138 voltmeter, peak, 234
triode, 4, 70 Direct coupled ampli er, 181, 200
Choke input recti er, 226, 236 Discharge, abnormal glow, 63
Clamping, 279 arc, 63, 192
Class A ampli er, 131, 208 cold cathode, 60, 62
Class B ampli er, 131, 140, 208 glow, 62, 63
Class C ampli er, 78, 136, 178, 195, 208 hot cathode, 64
Clipping, 276 self maintained, 62
Coincidence counter, 334 Townsend, 62
Colpitts oscillator, 198, 200, 204 Discriminator, 256, 274
Communications, 240 Dissipation, anode, 74, 121, 133, 208
Computer, 163, 336 Distortion, amplitude, 117, 124, 158
Condenser input recti er, 228, 236 frequency, 107, 110, 127, 151, 159, 166
Contact potential, 37 harmonic, 125, 250
Control ratio, 86 intermodulation, 127
Copper oxide recti er, 68 non linear, 117, 124, 127, 141, 151, 158
Counter, coincidence, 334 phase, 107, 110, 151, 304
Geiger, 67 Distribution, electron energy, 30, 38
ionization, 66 potential, 53, 65, 75, 82
proportional, 67 velocity, 40, 58
Counting, 67, 272 Donor atom, 33
Critical magnetic eld, 20, 59 Driving voltage, 93, 139, 142
Crystal structure, 29, 31 Dynamic characteristic, 116, 124, 195,
valves, 3, 67, 88 242
Cut o in magnetrons, 20, 59, 218 Dynatron oscillator, 192
in thyratrons, 85
in triodes, 75 E iciency, detection, 248
Cyclotron, 17 emission, 44
power ampli er, 120, 132, 140
D.c. ampli er, 100, 181 recti er, 329
restorer, 279 Electron beam, 22
Damping, input, 210, 315 charge, 9, 342
transit time, 210 energy, 10, 27, 30, 38
Decade scaling, 273, 334 gun, 13, 21, 25, 212
Decoupling circuit, 111, 147, 233, 248 lens, 22
de Forest, 1 mass, 9, 342
Dekatron, 273 microscope, 26
Demodulation, 246 multiplier, 47
Detection, 246, 247, 255 spin, 27
linear, 248 Electron volt, 10
Detector, second, 251 Electrostatic lens, 24
Device, non linear, 127 Emission, 38
Diamond structure, 31 eld, 41, 49
Di erential ampli er, 185 photo electric, 41, 48
analyser, 336 primary, 41
 INDEX 345
Emission secondary, 41, 45, 80 Hot cathode diode, 64
thermionic, 41 discharge, 64
Energy band, 30 Hum, 159
conversion, 10, 132, 206, 212 Hybrid characteristics, 90
Equation, 10 parameters, 92
gap, 33
level, 27 Ignition voltage, 63
state, 28, 30 Impedance, input, 114, 161, 210, 315
Equivalent circuits, 102, 144, 165, 312 internal, 114, 161, 165, 236, 291
diode, 76, 81 output, 114, 161
noise resistance, 290 Impurity atom, 33
Excitation energy, 28 Indirectly heated cathode, 45
potential, 28, 61 Indium, 34, 88
Induced current, 206
Fading, 253 Inductive load, 103, 175
Feedback, 151, 184, 190 Initial velocity of electron, 40, 54, 58
current, 159 Input capacitance, 114, 161, 166
negative, 151, 184, 238, 243 damping, cathode inductance, 315
parallel, 162, 166 transit time, 211
positive, 151, 167, 190 impedance, 114, 161, 210, 315
stability with, 167 Insulator, 33
voltage, 159, 163 Integration, 282, 284, 336
Fermi level, 31, 37, 40, 94 Inter electrode capacitance, 76, 107, 175,
Field emission, 41, 49 199, 204
Filter circuit, 233, 248 Intermediate frequency ampli er, 251
Fleming, 1 Intermodulation, 127
Flicker effect, 293 Internal impedance, 114, 161, 165, 236,
Focusing, 22, 26 291
Frequency changer, 251 Intrinsic semiconductor, 33
discriminator, 256 Inverse voltage, peak, 224, 236
distortion, 107, 110, 127, 151, 159, 166 Ionization, 27
modulation, 240, 244, 255 counter, 66
multiplier, 179 ' gauge, 88
Full wave recti er, 224, 226, 229 potential, 28, 61, 86
Gain, power, 1, 140, 142 Junction diode, 67
stage, 99, 105 p n, 35, 68, 88
voltage, 1, 96, 138, 181 transistor, 88
Gas ampli cation, 61
pressure, 64 Kinetic energy, l0
tetrode, 87 Klystron, 212, 328
Gases, collisions in, 61
conduction in, 29 Laplace’s Equation, ll
electrons in, 29 Lens, aperture, 25
Gas lled diode, 51, 59, 221, 236, 238 cylindrical, 25
triode, 84 electron, 22
Gate, 335 electrostatic, 24
Geiger counter, 67 magnetic, 22
Germanium, 32, 67, 88 Limiter, amplitude, 257, 276, 279
Glow discharge, 62, 63 Linear accelerator, 218
Grid bias, 111, 151 detection, 248
characteristics, 5, 70, 78, 152 Load, capacitive, 105, 175
emission, 74 inductive, 103, 175
leak, 110 line, 97, 116, 138
Grid circuit load line, 152 Equation, 98
Grid leak bias, 196 grid circuit, 152
Grounded grid ampli er, 205 optimum, 122
parallel resonant, 107, 132, 136, 202
Half wave recti er, 222 resistance, 98
Harmonic distortion, 125, 250 transformer coupled, 121
Hartley oscillator, 197, 200, 204, 252 tuned, 107, 132, 136
High frequency ampli er, 107, 250 Local oscillator, 251
Hole, positive, 33 Long tailed pair, 188
346 INDEX
Magnetic ampli er, 189 Non linear characteristic, 68, 221, 242
eld, 16 247, 276
critical, 20, 59 device, 127
lens, 22 distortion, 117, 124, 127,141,151 158
moment, 27 Normal glow, 63
Ma8 netron i cavitY , 215, 218, 220 Nucleus of atom, 27
cut off, 20, 59, 218 Nyquist diagram, 167
cylindrical, 20, 58, 220, 294, 296
planar, 18, 218
space charge in, 21, 58 Operating point, 6, 99
Maintenance voltage, 63, 236, 259 Operational ampli er, 163, 335
Majority carriers, 34, 69 Optics, electron, 22
Mass, electron, 9, 342 Optimum load resistance, 122
spectrograph, 17 Oscillation, 172, 178
Maximum power output, 119, 120, 123 Oscillator, 151, 190, 258
Mean free path, 64 bias, 196
Mercury, 29, 59, 63, 65, 87 blocking, 265
Mercury arc tube, 64 monostable, 266
Mercury vapour diode, 65 Class C, 178, 196
thyratron, 87 Colpitts, 198, 200, 204
Metastable energy, 28, 63 dynatron, 192
Microphony, 293 feedback, 151, 190, 192, 200
Miller effect, 162, 166, 284 Hartley, 197, 200, 204, 252
Minority carriers, 36, 69, 95 Klystron, 212
Mixer, 251 local, 251
Mobility, 29, 35 magnetron, 218
Modulation, amplitude, 240, 247 negative resistance, 190, 200, 328
phase shift, 192, 323
density, 209, 215 relaxation, 258
depth, 240 squegging, 197, 264
factor, 240
frequency, 240, 244, 255 transistor, 199, 200, 203
phase, 240, 244 transitron, 200, 264
pulse, 240 tuned anode, 193, 197
velocity, 213 tuned grid, 199
Monostable circuits, 266 tuned collector, 199
Multi electrode valve, 70 ultra high frequency, 204
Multi stage ampli er, 109 Output conductance, transistor, 92
Multiplier, electron, 47 impedance, 114, 161, 316
frequency, 179 Oxide coated cathode, 43, 62
Multivibrator, bistable, 269
free running, 260, 266 p type semiconductor, 34, 88
monostable, 266 p n junction, 35, 68, 88
transistor, 264 p~n" ¢ transistor, 88, 138
Mutual characteristic, 70 Parallel feedback, 162, 166
conductance, 71, 76, 78 resonant load, 107, 132, 136
tuned circuit, 107, 132, 136
n type semiconductor, 34, 88 Parameters, hybrid, 92
n 12 4: transistor, 89 transistor, 92
Negative feedback, 151, 154, 184, 238, triode, 71
243 Passive circuits, 169
Negative mutual conductance, 203 Peak inverse voltage, 224, 236
resistance, 64, 79, 190, 192, 200, 328 Pentode, 83
oscillator, 190, 200, 328 characteristic, 83, 116, 124
transconductance, 203 variable p, 253
Noise diode, 336 Phase discriminator, 255
factor, 291 distortion, 107, 110, 151, 304
Johnson, 288 modulation, 240, 244
resistance, equivalent, 290 Phase shift oscillator, 192, 323
shot 288 Photo cell, 53, 59
space charge limited current, 289 Photo electric cathode, 47, 48, 52
temperature limited current, 289 emission, 41, 48
transistor, 293 Photo multiplier, 47
valve, 159, 289 Photon, 28, 48
 INDEX 347
Planar magnetron, 18, 218 Schottky effect, 49
Planck's constant, 28, 342 Screen grid, 78
Plasma, 65, 66 Second detector, 251
Point contact diode, 67 Secondary electron, 41, 45
transistor, 88, 203 emission, 41, 45, 80
Poisson's Equation, 11, 56 coef cient, 45, 81
Positive feedback, 151, 167, 190 Selectivity of ampli er, 109, 250
hole, 33 Selenium, 68
ion, 28, 29, 44, 61, 65 Self maintained discharge, 62
bombardment, 41, 44, 61 Self oscillation, 151, 167, 214
sheath, 66, 86 Self oscillator, 151
Potential barrier, 40, 43, 49, 69, 94 Semiconductor, 31
distribution, 53, 65, 75, 82 Side bands, 241
Power ampli er, 115 Signal to noise ratio, 159, 292
efficiency, 120, 132, 140 Silicon, 31, 67, 88
ampli cation, 165 Single stroke time base, 285, 287
available, 336 Slope resistance, diode, 52
gain, 140, 142 Smoothing circuits, 233
Pressure, gas, effects of, 64 Solids, electrons in, 29
Primary electron, 45 Space charge, 51
Proportional counter, 67 density, 11, 56, 57
Proton, 27 flow, 56, 58
Pulse ampli cation, 175 on potential, e ect of, 53, 75, 76, 82
generators, 265, 274, 276 Space charge limited current, 53, 21 1, 289
modulation, 240 Space charge wave tube, 215, 218
Push pull ampli er, 128, 186 Spectral sensitivity, 49
detector, 249 Spin, electron, 27
Squegging, 197
Quiescent point, 6, 99 oscillator, 264
Stability with feedback, 167
Radio frequency ampli er, 107, 250 Stabilized power supplies, 236, 238, 330,
Reactance valve, 245, 257, 332 331
Receiver, 250 Stage gain, 99, 105
Recombination, 36, 62, 67, 69, 87, 94 Steering circuit, 273
Recti cation, 221 Stored energy, 169
Recti er, choke input, 226, 236 Striking voltage, 63, 236, 259
condenser input, 228, 236 Superheterodyne receiver, 250, 331
full wave, 224, 226, 229 Suppressor grid, 83
half wave, 222 Switch, voltage sensitive, 258
Re ex klystron, 214, 328
Regulation, voltage, 232, 236 Temperature, e ect on transistor, 95, 146
Relaxation oscillators, 258 Temperature limited current, 51, 55, 211,
Relay, thyratron, 87 289
Reservoir condenser, 229 Tetrode, 78
Resistance, anode slope, 52, 71, 76, 79 ampli cation factor, 81
equivalent noise, 290 beam, 78, 82
input, 140, 142 characteristic, 78
transistor, 92 cold cathode, 87
internal, 236, 291 gas, 87
load, 98 Thermionic emission, 41
negative, 64, 79, 190, 192, 200, 328 Thermistor, 35
Resonance curve, sharpness of, 109 Third rail circuit, 183, 268
Resonator, 209, 212, 218 Thoriated tungsten cathode, 43
Restorer, d.c., 279 Three halves power law, 57, 66
Rhumbatron, 209 Threshold wavelength, 48
Richardson's equation, 41, 42, 53 Thyratron, 84, 259
Ripple, 224 as relay, 87
bistable, 269
Saturated current, 56, 63, 69, 94 characteristic, 85
Sawtooth generator, 259, 284, 287, 334 cut off, 84
Scaling, binary, 272 inert gas, 87
decade, 273, 334 mercury vapour, 87
Schmitt trigger circuit, 271 monostable, 266
348 INDEX
Time base, 259, 284, 287 Tuned ampli er, 107, 132, 136
Time constant, 171, 175, 197, 258 Tuned anode oscillator, 193, 197
Townsend discharge, 62 tuned grid oscillator, 199
Transconductance, 71, 203 Tuned collector oscillator, 199
Transformer coupling, 121 load, 107, 132, 136
Transients, 169 Tungsten cathode, 42
Transistor, 2, 36, 70, 88, 94 Tunnel e ect, 43, 49
ampli ers, 138, 310
biasing circuits, 145 Ultra high frequency oscillator, 204
characteristics, 89, 138 Unbalanced signal, 187
effect of temperature on, 95, 146 Unidirectional current, 223
equations, 93 Univibrator, 267
equivalent circuits, 144, 311, 312
input resistance, 92 Vacuum diode, 4, 51, 221, 247, 251
junction, 88 Valency electron, 29
multivibrator, 264 Valve Equation, 72
n p n, 88 noise, 159, 289
noise, 293 Variable p pentode, 253
oscillators, 199, 200, 203 Velocity distribution, 40, 58
output conductance, 92 modulation, 213
p =n—p, 88, 138 Virtual cathode, 58, 77
parameters, 92 Voltage ampli cation, 96, 99, 138, 181
point contact, 88, 203 feedback, 159, 163
Transit time, 13, 57, 102, 210, 213, gain, 96, 138, 181
328 regulation, 232, 236
damping, 210, 213, 328 stabilization, 236
Transitron oscillator, 200, 264 Voltage doubling circuit, 232
Transmission line, 204 Voltage sensitive switch, 258, 267
Travelling wave tube, 215
Trigger circuits, 266, 269, 270, 275 Waveguide, 217
Triode, 4, 70 Wavelength, threshold, 48
ampli er, 6, 99 Wave shaping circuits, 276
characteristics, 4, 70 Work function, 37, 40
cut off, 75
oscillator, 192, 204, 259 X rays, 30
parameters, 71, 72, 301
ratings, 73 Zero drift 181