Electronic Analog Computer Primer
~ James H. Stiee-
Bernet S. Swanson
Digitized by the Internet Archive
in 2024
https://archive.org/details/electronicanalogOO00Ostic
Electronic Analog
Computer Primer
A Blaisdell Book in the Pure and Applied Sciences
CONSULTING EDITOR
Leon Lapidus, Princeton University
Electronic Analog
Computer Primer
JAMES E. STICE
University of Arkansas
AND
BERNET S. SWANSON
Illinois Institute of Technology
BLAISDELL PUBLISHING COMPANY
A Division of Ginn and Company
NEW YORK: LONDON -: TORONTO
FIRST EDITION, 1965
Copyright © 1965, by Blaisdell Publishing Company,
A Division of Ginn and Company.
All rights reserved.
Library of Congress Catalog Card Number: 65-18916
Printed in the United States of America
»
Preface
A LARGE BODY OF LITERATURE is available on the care and feeding of
analog computers, but when we first became interested in the field it
seemed that most of this material had been written for those with
a pretty solid background in electronics. After an initial period of
study, we began trying to solve some problems on our computer,
becoming more and more successful as we acquired understanding
and experience. Later when we had developed some proficiency in
the sport we found that we had accumulated quite a file of notes. It
occurred to us that these notes might be useful to others, and this
book was accordingly written.
Our aim was to present the fundamentals of analog computation
as simply as possible so that the reader who is not well grounded in
electronics, but who has at least a nodding acquaintance with
differential equations, can understand and use analog computers.
A review of circuit theory and electronics and the application of
these subjects to analog computers is included for those who may be
interested; those who are not may skip Chapter 2. Next the opera-
tions which can be performed with analog computers are examined.
Time and magnitude scaling are then presented and illustrated by
examples. A problem is stated, scaled, and programmed to show
the application of the techniques previously studied. Finally, four
types of commercially available computers are described in detail. It
is hoped that the novice, after reading this book, will be ready to try
his hand at the game.
vi PREFACE
One cannot hope to develop facility with analog computers
simply by reading a book on the subject. Application should begin
with simple problems, and if the reader cannot think of any right off,
we have thoughtfully provided some interesting ones in Chapter 8,
these being arranged in order of increasing complexity.
We are indebted to the National Science Foundation for Grants
12473 and 17773 which made this work possible. We are also
sincerely grateful to Dr. Athanassios Costikas, formerly of the
electrical engineering faculty of the Illinois Institute of Technology
and now with the Greek Atomic Energy Commission. Dr. Costikas
critically reviewed the material on circuit theory and electronics and
gave valuable advice on technical accuracy and simplicity of presen-
tation.
We further wish to acknowledge the contributions of the follow-
ing individuals, who furnished information, illustrations, and sug-
gestions: C. R. Moores (Applied Dynamics, Inc.), Paul Williams
(Datronics Inc.), Rudolf F. Wagner (Donner Division of Systron-
Donner Corp.), L. Arthur Hoyt and Edward J. Mangold (Elec-
tronic Associates, Inc.), and Bob Scowcroft and Earl F. Broihier
(Heath Company).
JAMES E. STICE
Fayetteville, Arkansas
BERNET S. SWANSON
Chicago, Illinois
Contents
CHAPTER I
Electronic Analog Computer Primer
Computer Classification
Advantages of Analog Computers
Applications of Analog Computers
Requirements for Computing Amplifiers
CHAPTER II
Review of Circuit Theory and Amplifiers
Phase Shift
Representation of Sinusoidal Voltages and Currents
by Phasors
Impedance
Amplifiers and Gain
Amplifier with Negative Feedback
CHAPTER III
The Operational Amplifier
The Internal Amplifier
The Feedback Amplifier
CHAPTER IV
Mathematical Operations Performed with
Operational Amplifiers
Symbols for Operational Amplifiers
Change of Sign
Mn Onwnh —
|
12
i3
21
25
25
28
4]
4]
42
Vill CONTENTS
Multiplication by a Constant 42
Addition 47
Subtraction 49
Integration 49
Differentiation of a Variable 53
Multiplication of a Variable by a Variable 54
Function Generation 56
CHAPTER V
Time Scaling 59
Time Constant 60
Frequency Responses of Commonly- Used
Recording Devices 62
Determination of the Frequency of an Equation 63
Performance of the Time-Scale Change 67
The Standard Form of a Second-Order Differential
Equation with Constant Coefficients 69
CHAPTER VI
Magnitude Scaling T7
Method I. Second-Order Linear Differential
Equations with Constant Coefficients 78
Method II. Second-Order Linear Differential
Equations with Constant Coefficients 80
Method III. Nth-Order Linear Differential Equations
with Constant Coefficients — The Equal-Coefficient
Rule 82
Comparison of the Methods Given for Estimation of
the Maximum Values of the Dependent Variable
and its Derivatives 85
Performance of the Magnitude Scale Change 89
Preliminary Computer Wiring Diagram 90
Systematic Procedure for Magnitude Scaling 96
Alternate Magnitude Scaling Technique 99
CONTENTS ix
CHAPTER VII
Commercially Available Analog Computers 103
Other Analog Computers 119
CHAPTER VIII
Problems 120
Solved Problems in the Literature 133
BIBLIOGRAPHY 157
INDEX 159
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Electronic Analog
Computer Primer
I
Electronic
Analog Computer Primer
The general-purpose electronic analog computer is one of several
types of modern electronic equipment used for performing mathe-
matical computations automatically. As a result of the cost of these
various types of equipment, extensive computing facilities are
currently found only in certain kinds of industries and in the larger
colleges. Digital computers are much more numerous in these
computer centers than general-purpose electronic analog com-
puters of comparable cost. This is no doubt as it should be, since the
digital machines can be used for a greater variety of calculations,
such as bookkeeping, payroll, inventory control, records processing,
Statistical studies, and scientific and engineering design calculations.
The general-purpose analog computer, however, 1s particularly
suited for certain kinds of calculations. Such a computer is mainly
used for the solution of differential equations, and is therefore useful
in the study of systems which can be described by a set of simul-
taneous differential equations. This is the reason for the wide
application of analog computers in the study of automatic control
systems; the equations which describe the system may be set up on
the computer, and the resulting “‘model”’ of the real system may
then be studied to see how the real system will behave. Analog
computers are also widely used in the study of nonlinear differential
equations, a field in which they have no equal.
]
2 ELECTRONIC ANALOG COMPUTER PRIMER
x Computer Classification
Computers are much in the news these days, and a lot of the news
is so garbled and distorted that it is pretty hard for the average
reader to separate the truth from the static. Some eager but mis-
informed journalists have dubbed the computers “thinking ma-
chines,” and the resulting flood of half-truths and rumors would
make H. G. Wells and Curt Siodmak rub their paws with glee.
Accordingly, a brief discussion of several kinds of computers will
be given to help the reader differentiate between them.
Digital computers are devices which deal in numbers only and
which use addition as their basic function. A desk calculator adds
the revolutions and partial revolutions of gears and displays the
total number of revolutions of the various gears as a sum. The
mileage register of an automobile speedometer is partly digital in
its action; it counts the number of revolutions of the wheels, and by
Suitable gearing displays the total number of miles traveled. An
electronic digital computer counts electrical pulses, and multiplies
and divides by addition or subtraction. For instance, in multiplying
548 by 7357, the computer adds 7357 to itself 548 times (in con-
siderably less than a second), and can produce an answer to almost
any number of significant figures desired. Such a machine can also
make logical decisions, in that it can compare two numbers to
determine if one is larger than, smaller than, or equal to the other.
There are several types of analog computers. All of them relate
the variables and parameters in a problem to variables and param-
eters in the analog model. Examples of direct analogs include wind
tunnels (aeronautical and mechanical), network analyzers (elec-
trical), process pilot plants (chemical), and model dams and model
basins (hydraulic).
Indirect analog computers include nomographs (general), the
Slide rule (general), the general-purpose mechanical analog com-
puters such as Dr. Vannevar Bush’s Mark I calculator (the differen-
tial analyzer), and indirect electrical analog computers.
The indirect electronic analog computer is probably the most
common of the indirect type. This type uses high-gain “‘opera-
tional” amplifiers to perform mathematical operations. This is the
type of computer with which we will be concerned here, and when
ADVANTAGES OF ANALOG COMPUTERS 3
the term “analog computer” is used in the following pages, the
reader will understand that the indirect electronic analog computer
iS meant.
Recently digital and analog computers have been combined to
take advantage of the unique features of both machines, The result-
ing machine is the so-called “hybrid” computer, and the use of
these computers will no doubt increase as their capabilities become
better known.
Incidentally, the-question of whether computers can “think” has
been vigorously debated, and one of the conclusions which has
emerged from this argument seems to be that we do not really know
what “thinking” is.7 It is certain that present-day computers cannot
do anything which they have not been programmed to do. As a
matter of fact, they must be told exactly what to do in solving a
given problem, down to the smallest detail. Since they are entirely
literal, they follow their programmed instructions explicitly. If the
instructions are not detailed enough, or are incorrect, the machine
will either just sit there and look at the operator, or it will develop
electronic indigestion and belch out nonsense.
Also, present-day computers cannot make value judgments in
subjective areas such as the social and moral fields, art, and the like
(and one wonders whether they will ever be able to do so). Much
research is under way to develop computers which are capable of
rudimentary thought, but there are no machines which are capable
of thinking at this time.
x Advantages of Analog Computers
The choice of a particular kind of computer for a specific kind of
calculation depends upon several factors, including the nature of
the problem and the degree of precision required in the solution.
The analog computer has certain features which are not found in
digital computers.
Probably the chief advantage in analog computation is that the
operator retains a “‘feel’’ for his problem. The twisting of a po-
+See, for example, Ulric Meisser’s article, “The Imitation of Man by Machine,”
Science, 139 (January 18, 1963), 193-197, and the resulting letter in Science, 140
(April 12, 1963), 212-218.
4 ELECTRONIC ANALOG COMPUTER PRIMER
tentiometer on the computer represents, in a very real sense, the
variation of a controller setting or the changing of a coefficient in
the process which is under study. Thus, the analog computer set-up
becomes a working model, or simulation, of the real physical prob-
lem, and the operator finds himself thinking of one block of com-
puter components as a control valve and another block as a heat
exchanger. Changes in settings of computer components thus be-
come meaningful in terms of the real process, and the results of
these changes can be interpreted immediately in the same terms.
The operator can “think as he goes,” and if interesting side-
avenues open up, these can be immediately explored.
Time is also capable of variation in analog computation. The
choice of time scale is limited only by the speed of the read-out
equipment being used; thus a problem may be speeded up so that it
is many times faster than the actual process, or slowed down so that
the simulated process occurs more slowly than the real process.
The actual solution time for a problem is quite short. Even quite
complicated problems rarely require minutes, and several seconds
is much more common.
Before discussing accuracy and precision, these terms require
definition. The word accuracy as used here denotes how closely a
solution conforms to fact. Precision of a solution is an indication of
the sharpness of definition. As an example, consider the value of e,
the base of natural logarithms. The value 2.718282 is more precise
than 2.718, but both values are accurate.
Analog computers generally yield results having three, or at most,
four significant figures. However, for many engineering purposes,
three significant figures will be adequate, since the original data will
be no better. Unfortunately, many people seem to think that a
Solution which contains ten significant figures is highly accurate,
even though some of the input data have only two or three signifi-
cant figures.
It is not difficult to learn how to program an analog computer.
Of course the more difficult problems require a correspondingly
greater amount of experience and knowledge on the part of the pro-
grammer, but persons with technological backgrounds can pick up
the fundamentals of the analog computing art quickly. The develop-
REQUIREMENTS FOR COMPUTING AMPLIFIERS 5
ment of greater skill is then only a matter of practice and continued
study.
x Applications of Analog Computers
Analog computers can solve ordinary linear and nonlinear
differential equations with either constant or variable coefficients,
algebraic equations, and partial differential equations. Since this is
intended only as an introduction to the field, the solution of
algebraic and partial differential equations will not be discussed
here. However, those who wish to investigate them are referred to
Jackson [1] or to Rogers and Connolly [2].
Aside from the use of the analog computer as an equation-
solving machine, it is particularly useful as a device for simulating
systems. Perhaps the widest use of the computer has been in this
field, with applications in the chemical process industries, missile
and high-speed aircraft programs, and instrument development.
Synthesis of proposed control systems or the analysis of existing
systems by means of analog simulation techniques give rapid and
reliable information about optimum control settings and sluggish or
unstable responses. It is hoped that the following discussion and the
laboratory exercises will awaken the reader to the possibilities of
the computer.
x Requirements for Computing Amplifiers
Amplifiers which are to be used for computing purposes must
meet several requirements, among which are
1. High open-loop gain.
2. High input impedance.
3. Constant closed-loop gain for all frequencies from direct
current to several thousand cycles per second.
4. A phase shift of 180 degrees between input voltage and
output voltage.
5. Linearity over a wide operating region.
6. Zero output voltage when there is no input signal.
Probably some readers are not familiar with all the terms used
above. It is our opinion that it is not necessary for the beginner to
wee ~
\
|
|
zi
6 ELECTRONIC ANALOG COMPUTER PRIMER
understand the inner workings of the machine in order to learn how
to program it for the solution of problems of moderate complexity.
On the other hand, those who wish to become adept at the art, and
who intend to use the computer for the investigation of difficult
problems, will sooner or later be obliged to learn something about
the electronics of the computer components.
Chapter 2 provides a review of alternating current circuit theory
and basic electronics theory which are fundamental to an under-
standing of the operational amplifiers—the heart of an analog
computer. Chapter 3 discusses the operational amplifier from the
standpoints of gain, phase shift, drift, and linearity. Those readers
who have some training in electronics should find these two chapters
helpful, especially if their background has accumulated some fiecks
of rust. The beginner may wish to skip Chapters 2 and 3 for the
time being and return to them when he finds he needs the infor-
mation.
I]
Review of Circuit Theory
and Amplifiers
* Phase Shift
In electrical circuits, it is found that when a sinusoidal voltage is
applied across a linear, passive component such as a resistor,
capacitor, or inductor, a sinusoidal current flows in the component.
The instantaneous voltage is represented by the equation
e = Em sin (wt + 6), (1)
where e is the instantaneous voltage, E,, is the maximum value of
the voltage, wt is the angular displacement in radians, and @ is the
initial phase angle in radians, measured from some arbitrary point
in time.
The instantaneous current resulting from this applied voltage is
represented by
i = Im Sin (wt + 9), (2)
where / is the instantaneous current in amperes, /,, is the maximum
value of the current, and @ is the initial phase angle in radians,
measured from the same point in time as @ in Equation (1).
Figure 2.1 shows a plot of a voltage and its associated current as a
function of time.
In Figure 2.1 the arbitrary zero point in time was chosen to be the
point at which the voltage wave crosses the horizontal axis while
7
8 REVIEW OF CIRCUIT THEORY AND AMPLIFIERS
=
Ot <->
Current and Voltage
FIGURE 2.1. Instantaneous voltage and current in a circuit component.
changing from a negative to a positive value. Then the initial phase
angle for voltage is 0°, or,
e = E,, sin (wt + 6) = En Sin ot. (3)
With respect to the same zero point in time, the current has a phase
angle of —90° (or +270°), which is —5 radians. Then
j = In sin (wt + 8) = In sin (wr ~ 2) (4)
The current is said to /ag the voltage by 90°, or the voltage Jeads the
current by 90°. There is a phase shift of 90° between the two
waveforms.
x Representation of Sinusoidal Voltages and Currents by Phasors
In alternating current (ac) circuit theory, a knowledge of in-
stantaneous values of voltages and currents is required. However,
mathematical manipulation of sine waves of different amplitudes and
phase angles often involves a formidable amount of labor if trigo-
nometric expressions are used for currents and voltages. In order to
simplify the solution of circuit problems, Steinmetz originated the
idea of representing a sinusoidally-varying quantity by a rotating
line segment called a phasor. By definition, this line segment has a
constant magnitude which is equal to the maximum value of the
sinusoidally-varying quantity. Also, the phasor is positioned so that
its vertical projection represents the instantaneous value of the sine
SINUSOIDAL VOLTAGES AND CURRENTS 9
wave at some chosen time, usually at time ¢ = 0. Finally, the line
segment rotates about the origin in a counterclockwise direction
with an angular velocity equal to that of the sinusoidal quantity
which it represents. Since the phasor concept is an important one in
ac circuit theory, it will be developed in some detail here, following
the general development in Middendorf [3].
It will become apparent presently that a phasor is a complex-
plane representation of a sine wave. To begin our development,
consider a stationary vector of magnitude E and direction @ plotted
in the complex plane (see Figure 2.2). The vector E can be repre-
sented by the equation
E = a+ jo, (5)
where a is the horizontal projection of E measured along the real
axis, and b is the vertical projection of E measured along the
imaginary axis. The symbol / precedes a quantity to be measured
along the imaginary axis. The term “‘imaginary”’ was used by the
early mathematicians, probably because it means the opposite of
“real.” This was an unfortunate choice of term, since many be-
ginning students of mathematics infer that the imaginary quantities
do not exist, although in fact, there is nothing imaginary about
them. It is helpful to look upon j as an operator which rotates the
quantity it precedes by 90° in the counterclockwise direction, Then
j? rotates the quantity by 180°, 73 by 270°, and so on. The E is
printed in boldface type to denote the fact that the voltage is a
complex quantity, having both real and imaginary parts, and here-
after all complex quantities will be so represented.
From Figure 2.2, it can be seen that
a = Ecos @ (6)
b = Esin 6. (7)
Upon combination with Equation (5),
E = Ecosé+ jE sin @ = E(cos @+/ sin @). (8)
One of the elementary relationships of complex variable theory is
e/® = cos 6+ sin 0. (9)
10 REVIEW OF CIRCUIT THEORY AND AMPLIFIERS
Imaginary Axis
——_— — «——— ee ie i Or i
| —> Real Axis
ZY
Q
Six.
FiGurE 2.2. Complex plane representation of vector E.
By using this relationship, Equation (8) becomes
E = £e. (10)
Upon referring to Figure 2.2, the magnitude of E is, by the Pythago-
rean Theorem:
E = yva2 + b?. (11)
From elementary trigonometry, the angle @ is defined to be
b
= wes. pani
@ = tan! (12)
The vector E is now defined by Equation (10), with magnitude and
direction given by Equations (11) and (12).
The phasor we are seeking is not a vector; it is a rotating line
segment, with an angular velocity of w radians per second. Thus, if
the vector can be caused to rotate with the desired velocity, then the
phasor is obtained. The rotation can be included by multiplying
Equation (10) by a rotational operator having an angle which in-
creases with time. Then the phasor is:
E = Eeseiet = Eeiott) — (13)
= Elcos (wt + 6) + j sin (wt + 6)]. (14)
Now the object of all this manipulation is to be able to express the
instantaneous value of a sine wave at any instant of time. The
SINUSOIDAL VOLTAGES AND CURRENTS li
imaginary portion of the phasor is the sinusoidal function desired,
and represents a sinusoidal voltage having a maximum value E,,,
an angular velocity w, and a phase angle of @ radians at time ¢ = 0.
This is the quantity described by Equation (1), or
e = E,, sin (wt + 6). (1)
Then the imaginary part of the phasor (Equation 14) represents the
sinusoidal voltage of Equation (1). Figure 2.3 shows this sine wave,
together with the corresponding phasor representation.
The figure shows that, at any instant of time, the vertical projection
of the phasor is equal to the instantaneous value of the sinusoidally-
varying voltage which it represents. For example, at ¢ = 0, the
instantaneous value of the voltage sine wave is
e = Ep sin (wt + 6) = Em sin 8. (15)
At the same instant of time, since the length of the phasor is E,,,, the
vertical projection of the phasor is E,, sin 6, which agrees with
Equation (15). It should be realized that the phasor is not equal to
the sine wave, but it permits expression of the necessary information
about the sine wave for many types of calculations.
At the beginning of this section, the phasor was defined to have a
length equal to the maximum value of the sinusoidally-varying
quantity. However, an ammeter or voltmeter does not indicate the
maximum value, but an effective value called the root mean square,
or rms value. (The reader who is not familiar with rms values
should consult any elementary text on ac circuits.) In order to make
e = E» Sin (wt+8@)
FiGuRE 2.3. Sine wave of voltage represented by a phasor
and by the trigonometric function.
12 REVIEW OF CIRCUIT THEORY AND AMPLIFIERS
the phasor concept more useful, the length of the phasor has, by
agreement, been set equal to the rms value rather than the maximum
value. The rms value of a voltage sine wave is equal to E,,/+/2. In
the rest of the text, rms values are presented by capital letters, as E.
The notation which has been agreed upon for representation of a
sinusoidal voltage as a phasor is (polar form)
E = E/6, (16)
where E is the rms value of the voltage, and @ is the initial phase
angle. Similarly, a current phasor is represented by
I = J/8. (17)
The phasor of Equation (16) can also be expressed in rectangular
form as E = E, + jE,. The expression of currents and voltages as
phasors allows one to perform the following mathematical opera-
tions on sinusoidal quantities with far less labor than would be the
case if the quantities were expressed trigonometrically: addition,
subtraction, multiplication, division, raising to powers, extraction
of roots, and obtaining the logarithm. The reader who is not
familiar with the algebra of complex numbers will find a review of
the subject, and electrical engineering examples, in any good text on
elementary ac circuit theory.
x Impedance
In direct current (dc) circuit theory, the ratio of voltage to current
in a branch is called resistance, and is given by Ohm’s law,
E
where R is the resistance in ohms, E is the de voltage drop across the
resistance, and / is the de current through the resistance.
In ac circuit theory, the ratio of the phasor voltage across a
branch to the phasor current through the branch is defined to be the
impedance, Z,
E/0
I/
aad
(19)
o
AMPLIFIERS AND GAIN 13
Since the impedance is the ratio of two complex numbers, it is itself
a complex number. Then it can be represented in rectangular
form by
Z= R-+ jx, (20)
where Z is the complex value of the impedance in ohms, R is the
resistance in ohms (real component of impedance), and YX is the
reactance in ohms (imaginary component of impedance). Reactance
may be of two types, inductive or capacitive. For a pure inductor,
X = wl, where L is the inductance in henrys. For a pure capacitor,
X = —1/wC, where C is the capacitance in farads.
Equation (19) defines impedance for sinusoidal voltages and
currents only, since phasors apply only to sinusoidal quantities.
However, the concept of impedance may be generalized to include
voltage and current waveforms which are arbitrary functions of
time. This involves writing the differential equation of the circuit
involved and then taking the Laplace transform of this differential
equation. The ratio of the voltage transform to the current trans-
form is the transform of the impedance (called the Z transform),
and the inverse transform of the Z transform is the generalized
impedance of the circuit. The method is applicable to any wave-
form whatever. Since a knowledge of Laplace transform techniques
is not presupposed here, the method will not be enlarged upon, but
the interested reader is referred to van Valkenburg [4] for very
readable discussions of the application of Laplace transforms to
electrical circuits and the use of the Z transform.
To summarize, impedance is the ratio of the voltage across a
branch to the current through the branch. For sinusoidal voltages
and currents the impedance is the ratio of phasor voltage to phasor
current, and for nonsinusoidal voltages and currents, the im-
pedance is the inverse Laplace transform of the Z transform.
Impedance is a complex quantity, having both magnitude and direc-
tion, and in complex notation (rectangular coordinates) it consists
of a real part (resistance) and an imaginary part (reactance).
» Amplifiers and Gain
In general, an amplifier is a device which produces an output that
is a magnified form of the input. Only electronic amplifiers will be
14 REVIEW OF CIRCUIT THEORY AND AMPLIFIERS
| eee Ih = ip
Vacuum Triode +
Rr —(1, Rr)
1 ache Ey = ep -
E, =, = Ey
Ece 4. |
oa |
|| :
FiGureE 2.4. Simple triode amplifier, no-signal condition.
discussed here, and the development given follows the material in
Gray’s Applied Electronics [5].
Figure 2.4 shows a simple triode amplifier with a resistance load
under quiescent conditions (no signal applied to the triode grid). In
the figure, E., is the grid bias supply voltage (direct current) which
always maintains the grid at a potential lower than the cathode
potential, so that no current flows in the grid circuit. E,, is the plate
(anode) voltage supply (also direct current) which maintains the
plate at a higher potential than that of the cathode. The voltage
drop across the tube (difference between the plate potential and the
cathode potential) is E,. The voltage drop across load resistor Rz is
I,Rzr, with polarity defined as shown in the diagram. The current
flowing in the plate circuit is /,, and the direction of this current is
as Shown. This is the “‘conventional”’ current, which flows in a direc-
tion opposite to the flow of electrons. Since the electron flow in the
tube is from cathode to plate, the direction of the flow of conven-
tional current is from plate to cathode.
It is necessary to define the symbols for the various currents and
voltages in the amplifier of Figure 2.4 in order to discuss the opera-
tion clearly. The following symbols-and definitions are those
adopted by the Institute of Radio Engineers [6].
I, = value of the current through the external circuit toward
the plate, when there is no time-varying component of
grid voltage.
AMPLIFIERS AND GAIN 15
Cp
|
value of the voltage rise from cathode to plate, when
there is no time-varying component of grid voltage.
value of the voltage rise from cathode to grid, when there
is no time-varying component of grid voltage.
instantaneous total current through the external circuit
toward the plate.
instantaneous total voltage rise from cathode to plate.
instantaneous total voltage rise from cathode to grid.
instantaneous value of the time-varying component of
current through the external circuit toward the plate.
instantaneous value of the time-varying component of
the grid-signal voltage.
instantaneous value of the time-varying component of
the voltage rise from cathode to plate.
For the simple triode amplifier of Figure 2.4, a little study will
show that the following relationships exist under no-signal con-
ditions:
e, = 0 (no grid signal applied)
ep =
lp =
Cc = Lc
= Es
I = I.
When a time-varying signal is applied to the grid, the above
relationships change. Figure 2.5 shows the same amplifier with a
varying grid signal applied. In Figure 2.5 all of the voltages and
currents now contain a varying component in addition to the no-
16 REVIEW OF CIRCUIT THEORY AND AMPLIFIERS
= in +1p
+
+ Ry S —(ipt+ bb) Rr
+ T - a Dees
© a = Exp
= Eee Pa
= | i He
41
FiGureE 2.5. Simple triode amplifier with grid signal applied.
signal (steady-state) component, as a result of the application of the
grid signal e,. The relationships become:
Cg = es
@- = Eve + Cg
ey = Es a Cp
@€p = —1)Rx
ij ec Tesh ss,
Consider that the amplitude of the grid signal e, is small, so that
the operation of the amplifier is linear. Also, the frequency of e, is
low enough for the effect of tube interelectrode capacitances to be
negligible. Then it can be shown [7] that the circuit of Figure 2.5 is
lp
mV AVAYAYA :
| ee
Ps “ +
A) €g © Meg Rr ep
FiGuRE 2.6. Voltage-source equivalent circuit for
simple triode amplifier with resistance load.
AMPLIFIERS AND GAIN 17
equivalent, for purposes of analysis, to the circuit of Figure 2.6.
Figure 2.6 is called the voltage-source equivalent circuit of the
amplifier shown in Figure 2.5.
In Figure 2.6 the triode has been replaced by a resistance r, (the
tube plate resistance) in series with an ideal voltage source ye, (with
polarity as shown), where uw is the tube amplification factor. No
steady-state (dc) voltages or currents appear in the equivalent cir-
cuit, since it applies only to incremental changes in voltage and
current caused by incremental changes in the grid signal, e,.
The voltage gain of the circuit in Figure 2.6 will now be derived.
The term gain is the amount of amplification obtained from an
amplifier, and voltage gain is defined to be the ratio of the in-
Sstantaneous value of the varying component of the output voltage
rise to the corresponding instantaneous value of the varying com-
ponent of the input grid-signal voltage rise. The symbol for gain
is A.
By Kirchhoff’s voltage law, the algebraic sum of the instantaneous
values of voltage drops around any closed path of a circuit is
equal to zero. By applying this law to the plate circuit of Figure 2.6,
and summing in a counterclockwise direction we obtain
ipRr + iprp — weg = 0 (21)
ip(Ri + rp) = neg. (22)
Also, because of the direction of current 7,, and the defined direction
of polarity for voltage e,,
i, = orn (24)
By substituting Equation (24) into Equation (22):
—E(Ri + rp) = ner. (25)
L
Upon rearranging,
i ae Pisin os
a. "Rao Gain = A. (26)
18 REVIEW OF CIRCUIT THEORY AND AMPLIFIERS
Equation (26) applies to any input signal-voltage waveform;
e, need be neither sinusoidal nor periodic. If the input is a sinusoidal
voltage, then the equation may be written in other forms. For
instance,
E Pi. uRy
E, = Rt ss at p
In a simple triode amplifier such as the one shown in Figure
2.5, suppose that a sinusoidal input signal e, is applied. As the
signal voltage becomes more positive, the triode grid becomes
more positive, and the tube current increases. This causes the
voltage drop across the load resistor, which is equal to i,Rz,
to increase also. However, the output voltage is e, = —i,Rz, so
that the output voltage decreases as i, increases. Thus, as e, goes
positive, ep goes negative, and vice versa. Then there is a phase
shift of 180° between e, and ep.
Equation (27) is the gain expression for a siraple triode amplifier
with a sinusoidal input voltage and a pure resistance load. The
amplification factor for the tube, uw, is a real number, and so are
the values of resistances Ry and r,. Then the gain is a negative
real number, having a magnitude of uRz/(Rx. + r,), and a phase
shift of 180° between input and output voltages.
If the amplifier has a load which is a complex impedance rather
than the pure resistance shown in Figure 2.5, then the gain is no
longer a real number, but becomes a complex number. If the
complex impedance load is designated by Zz, then Equation (27)
becomes
= Voltage Gain = A. (27)
Li + Pp
Suppose, for the purpose of illustration, that the load is a pure
resistance in parallel with a pure capacitance, as shown in Figure 2.7.
In the circuit shown, both branches have the same voltage drop,
V, across them. Also, by Kirchhoff’s current law, the sum of the
branch currents I; and I; must be equal to the current I entering
the branch, or
E
I=I,+h. (29)
AMPLIFIERS AND GAIN 19
FiGureE 2.7. Load consisting of resistor and capacitor in parallel.
The impedance of the capacitor is
1
foc =D ~ Jae (30)
The impedance of the resistor is
Zr= R+ j0. (31)
Since, in general, V = IZ,, then
I; = je = isis = “=p 4 = JVoC. (32)
wl
Similarly,
Ve ¥
I, = 7 (33)
The combination of Equations (29), (32), and (33) yields
l=[,+ kb =jVwoC+ ' = vi + jC | (34)
V V R
Z=-= m= (35)
I v(3 + jwC) 1 + jwRC
R
By substituting Equation (35) into Equation (28) we obtain
(TF joRc)
be ;
+ Pp
1 + jwRC
20 REVIEW OF CIRCUIT THEORY AND AMPLIFIERS
_ Equation (36) will be more meaningful if it is put into the form
A=R-+ )X.
—uR
a= (TR art oR BH
Upon rationalization of the denominator of Equation (37):
see HR (R + Pp) = JoRCry
(R + rp) + joRCr, (R + rp) — joRCr,
_ —eR(R + Pp) + juwR*Cry (38)
(R + rp)? + w2R2C2r/?
™ —pR(R + rp) 4:55 pwR2Cr > .
~ (R47)? + wR? | /(R + rp)? + w2R2Cery
Equation (38) expresses the voltage gain of the simple triode
amplifier as a complex number (rectangular form). For sinusoidal
signals, the term w in the equations is equal to 2xf, where f is the
frequency of the signal in cycles per second.t Equation (38) shows
that the voltage gain decreases as the frequency of the grid signal
becomes large. Also, since the real part of the gain term is negative,
while the imaginary part is positive, the phase angle is in the
second quadrant. Thus, as w increases from zero to infinity, the
phase varies from — 180° to —270°, or from +180° to +90°.
The value of yu, the tube amplification factor, is also variable.
Tube characteristics vary during warming-up, under changes in
ambient temperature, and with normal aging. Further, two tubes
of precisely the same make and type will not have identical charac-
teristics, so that gain changes result when tubes are replaced.
The material in the last several pages has been presented to
demonstrate the fact that the voltage gain and the phase shift of
the simple triode amplifier are not constants, but are subject to
several sources of variation. It will now be shown that the use of
feedback will make the gain a constant for the over-all feedback
amplifier over a considerable range of frequencies.
}For nonsinusoidal signals, the generalized impedance transform must be used,
and the equivalent expression to Equation (38) for nonsinusoidal signals will not
be developed here.
AMPLIFIER WITH NEGATIVE FEEDBACK 21
a4
lg
— Z, —— fl A,
i Baeble Fe im tf
ej Cg A Cr €o
Ss s ay
FIGURE 2.8. Feedback amplifier.
x Amplifier with Negative Feedback
In the preceding section, it was shown that the gain of a simple
triode amplifier with a complex impedance load varies with the
frequency. Also, the phase shift is not 180° at all frequencies.
In order to remedy this situation, use is made of negative, or de-
generative, feedback. A feedback amplifier is depicted in Figure 2.8.
The rectangle marked A in Figure 2.8 is an amplifier such as we
have been discussing with gain A (a complex number). The input
signal e; is fed through impedance Z; to the grid of the input tube
of amplifier A. The output voltage e, (which is identical with eo)
acts on the input signal through impedance Z,; to produce the
resultant grid signal ey.
The voltage feedback through impedance Z,; is of opposite
polarity to the voltage e,, whence the name negative feedback. The
use of negative feedback makes the gain of the feedback amplifier
somewhat independent of the gain of the internal amplifier, and
if the gain of the internal amplifier is made high enough, the gain
and phase angle of the feedback amplifier become constants over a
range of frequencies. In order that this point may be well under-
stood, the gain expression for the feedback amplifier of Figure 2.8
will be derived.
Assume that e; is positive at the moment being considered.
Then the direction of the current / is as shown, since e, is less than
e;, and current always flows from a higher to a lower potential.
Also, e, is more positive than e, (remember that if e, is positive,
then e, is positive; e, is negative, then e, is at a higher potential
22 REVIEW OF CIRCUIT THEORY AND AMPLIFIERS
than e,). Therefore, current iy flows in the direction shown. The
direction of grid current i, is unknown at this time, and will be
assumed as shown.
By Kirchhoff’s current law, the algebraic sum of all currents ata
node must equal zero. Then
Upon using complex values for current, Equation (39) becomes
We will now assume that the grid current i, is small enough to be
ignored. This is an important assumption, and depends upon the
fact that the input impedance of amplifier A is large enough
(several megohms) for the small grid voltage e, to cause only a
negligible grid current to flow. Then,
I=Iy. (41)
In order to make this development entirely general, complex
quantities will be used throughout. In the input circuit, since
voltage drop is equal to the product of current times impedance,
E; — E, = IZ; (42)
E-E,
[ = Z. (43)
Similarly, for the feedback circuit,
= Ee BPs E,
I, = Z; (44)
Since E, is identical to Eo, Equation (44) may be written
E, — Eo
Substitution of Equations (43) and (45) into Equation (41) yields,
E; — E, _ E, — Eo
A ae (46)
Since the generalized gain