Analog Computer Techniques
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ANALOG
COMPUTER
TECHNIQUES
CLARENCE L. JOHNSON
CAPTAIN,U.S, AIRF FORCE io
ASSISTANT PROFESSOR, DEPARTMENT OF MATHEMATICS
W.3, AIR FORCE INSTITUTE OF TES. HNOSOLOGY
Ati a ot ee BOOK Cc OM P Ad
ANALOG
COMPUTER
TECHNIQUES
Designed as an aid to those learning
to wWse electronic analog computers a)
electronic differential analyzers), this
book will make the transition period
from neophyte to experienced computer
operator easier for the engineer.
Following the introductory and gen-
eral material on electronic analog corn
putation, the volume consists of specific
techniques for the solution of difficult
or unusual problems. Wherever prac
tical, techniques and principles have
been presented in such a manner that
an individual with a minimum knowl-
edge of mathematics and electronics can
readily understand them.
The book treats in some detail the
use of diodes and differential relays in
analog computation. Special emphasis
is given to function generating tech-
niques including methods of represent
ing functions of more than one variable.
Phere is a chapter on repetitive analog
computers and a chapter describing the
principles of operation of the digital dif-
ferential analyzer type equipment.
The problems at the end of the chap-
ters will be helpful to both students and
those engaged in self study to insure
that they have understood the material
presented, A bref review of the termi
nology of differential equations and a
brief introduction to operator notation
is included in the Appendix as an aid to
the less advan el reader,
ANALOG METHODS
IN COMPUTATION
AND SIMULATION
By WALTER W. SOROKA
University of California
Here is an organized treatment of the
more important and useful methods ol
analog computation and simulation
with particular reference to enginet
ing and scientific applications.
The book describes mechanical, electro
mechanical. electrical and electronic
analog components for performing basi
mathematical operations and then
shows how such components may ito
combined into mathematical machine:
for solving sets of simultaneous linear
algebraic equations, polynomial equa
tions. and differential equations of all
kinds. The final three chapters show
how dynamical analogies, finite differ
ence networks, membrane analogies
and conducting sheets of uniform and
variable thickness may be applied to
the simulation of various physical 85
lems aril to the solution of cilferential
equations.
Treatment is fundamental and direct
The author carefully develops the theo
retical basis for each approach, thu:
providing the student with the tool
necessary for applying these methods to
actual problems.
This 1s a pioneer attempt to organize
classily and present a wide range ol
material on analog computation in text
book form. It is a valuable text and rel
erence book for all scientists and eng
neers at both the student and profes
sional levels.
CL) eee ae in|
Other McGRAW-HILL Books
ENGINEERING CYBERNETICS: The Science of Control
By H. S. Tsren, California Institute of Technology. 304 pages,
$7.50
This important hook aims to establish Engineering Cybernetic “ne ff
new branch of enginedring science. It covers, as far as possible within
the limited space, the whole field of scientific principles of control, from
the simple conventional servomechanisms to the very complex con
trolled and guided systems, Non-interacting controls of many variable
systems, control design by perturbation or with prescribed performance,
optimalizing control, noise filtering and detection, and von Neumann's
theory of error contro] are some of the important topics trea tec
PRINCIPLES OF NUMERICAL ANALYSIS
By Ausron S. Housenotper, Oak Ridge National Laboratory
International Series in Pure and Applied Mathematics, 2/4 pages,
be A a 18
A senior-graduate text which develops the mathematical principles upon
which many computing methods are based and in the light of which
they can be assessed, Directed primarily toward digital computation
the book is designed to give a unified treatment rather than a com
plete catalogue of methods. Treatment is primarily theoretical, Tech
niques for making estimates of errors are indicated wherever possible
Funetional equations as such are not discussed, but emphasis is placed
upon the methods of solving the finite systems and perhorming the
nilerpolations which ore required in the digital olution (vl finris hoorwal
MLNS
INTRODUCTION TO NUMERICAL ANALYSIS
iy F. B. Hrpepranp., Massachusetts Institute of Technology, fr
ternational Series in Pure and Applied Mathematics, S20 pages
so OM)
Designed for workers in fields of engineering, mathematics and physi
for research personnel and for courses in Numer ical Analysis offered
by colleges and universities, this book provides an introductory treat
ment of the fundamental processes of Numerical Analysis. These pro
esses are the foundation of existent techniques associated with the
effective use of modern high-speed calculating devices, Thus, the text
provides a substantial treatment of the basic operations of computation
approximation, interpolation, numerical differentiation and integration
and the numerical solution of equations.
CONTROL-SYSTEM DYNAMICS
By Water R. Evans, North American Aviation, Inc., Downer
California, McGraw-Hill Electrical and Electronic Engineering
Series. 277 pages, $7.50
An exposition of the “Root Locus Method” invented and developed by
the author, this new volume demonstrates the techniques for determin
ing the response of linear control systems. The root locus, a tool to
factor an aeekradt polynomial, is useful in analyzing the pertinent
differential equations in feedback control systems. Developing from the
simple to the complex, each solution establishes a concept which paw
mits a simpler technique to be applied to the next, more complioated
problem.
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ANWdWOD NHOoOd8 THH-MY¥YeDSAN
ANALOG COMPUTER TECHNIQUES
Analog Computer Techniques
CLARENCE L. JOHNSON
Captain, U.S, Air Force
Assistant Professor, Department of Mathematics
U.S. Air Force Institute of Technology
McGRAW-HILL BOOK COMPANY, INC.
New York Toronto London
1956
ANALOG COMPUTER TECHNIQUES
Copyright @ 1956 by the McGraw-Hill Book Company, Inc. Printed in the
United Statea of America, All rights reserved. This book, or parts thereof,
may not be reproduced in any form without permission of the publishers.
Library of Congress Catalog Card Number 56-7560
iv
THE MAPLE PREAA COMPANT, TOME, PA,
To My Wife, Lillian
and My Daughter, Tonya Sue
PREFACE
The electronic differential analyzer, frequently referred to as the
electronic analog computer, has become an extremely important tool
to the engineer, physicist, and applied mathematician in the past
decade. Many of the rapid advances in the field of automation would
have been impossible without this tool.
The treatment of the electronic differential analyzer contained in
this book was written as an aid to the computer operator. The
presentation of the material is such that the average person with a
knowledge of Ohm's law, Kirchhoff’s laws, and a basic knowledge of
differential equations can read and understand the majority of the
material presented. In a few isolated instances Laplace-transform
notation is used to prove a statement. The results obtained, however,
are stated in such a manner that they can be applied to the setup of
computer problems without a knowledge of Laplace-transform theory.
In the various stages of development, the material contained in
this book has been used for the past two years in the computer courses
taught to advanced undergraduate and graduate students at the U.S.
Air Force Institute of Technology at Wright-Patterson Air Force
Base, Ohio.
The analog-computer course was included in the curricula because
of the relatively large number of students undertaking independent
study investigations in the fields of guidance systems and automatic
control, The use of computing devices in the level of work attempted
was virtually mandatory.
Prior to the inauguration of the computer courses, too much of the
student's effort in his thesis work was directed toward learning to use
the computer and too little toward the actual investigation of his
problem. Itis the author's belief that this situation has been improved
by the inclusion in the curricula of the material in this book.
The author further believes that the engineer, whether his specialty
he electronics, mechanics, chemistry, or aerodynamics, has an increas-
ing need to be familiar with both analog and digital computers.
Although he may never be called upon to operate a computer, he
vil
Vill PREFACE
should recognize the capability of a particular class of computers to
solve a problem with which he may be confronted.
The arrangement of the material in this book was chosen in such
a manner as to permit laboratory work to accompany the classroom
work. For this reason the material proceeds rapidly in the early
chapters into the discussion of amplitude- and time-scale factor adjust-
ment and the setup of linear systems of differential equations. Non-
linear components and function-generating techniques are also con-
sidered in the early chapters. The later chapters consider such topics
as the application of analog computers to the solution of problems
other than ordinary differential equations, a more detailed description
of computer components, checking computer results, and repetitive
analog computers. The book is concluded with a brief introduction
to the logic operation of the digital integrating differential analyzer.
A brief review of the terminology of differential equations and a
brief introduction to operator notation is included in the Appendix
as an aid to the less advanced reader.
The author is indebted to Professor C. E. Warren of The Ohio State
University for his valuable suggestions during the preparation of an
earlier form of the manuscript, and to Professor R. 'T. Harling of the
U.S. Air Force Institute of Technology and other faculty members
of The Ohio State University and the U.S. Air Force Institute of
Technology who have influenced the writing of this book.
The author wishes to acknowledge the able assistance of Mrs. Ada
Williams for the typing of the earlier manuscript and of Mrs. Frances
Borum for typing the manuscript in its present form.
CLarence L. Jonnson
CONTENTS
Preface . 4 ww we ee el
Chapter 1. Introduction
-l. Historical Development . ., te
Classification of Computing Equipme nt
l
-2.
=. Problems Solvable on the Electronic Analog Computer
4. Major Components
-5, Summary
wheres 2. The Linear Computer Components
. Introduction . .
22 The Operational Amplifers
. Potentiometers , ;
24. Computer Design Differences
2-6. Concluding Remarks .
Chapter 3. Time- and Amplitude-scale Factors
d-1, Factors Influencing the Choice of Time Scale .
1-2. Determination of Approximate Problem Frequencies .
dd, Approximate Frequencies of Higher-order Systeme
t4. The Influence of Recorder Characteristics on the Choice of Time
Scale . re
‘#6. Performing the Time-acale Change
4-6, Choice of Amplitude-scale Factor
7. Approximating the Magnitude of the Variables of A Problem
‘8, Summary
Chapter 4. The Synthesis of Servomechanism Systems
lL. Introdwetion
2. The Block-diagram Notation. .
4, Betup of a Simple Servomechanism System
|, Setup of a Transfer Function
f, Setup of the Factored Form of a Tranafe r Function ;
i Surnmary
Chapter 6, Multiplying and Resolving Servos
fel. Introduction
fed. Borvomultipliors
OO. Division. . 2. 2. 1 6 ee ltl lll
ied, Bquare Root 2. . 4 6 te thle
Vil
Se oe bo
12
14
17
20
22
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25
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32
42
45
45
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66
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65
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74
Sf.
§-6.
CONTENTS
Resolving Servos . . os.
Choice of Rectangular or Polar Coordinates
Chapter 6, Additional Computer Techniques
(1.
6-2.
6-3.
6-4.
6-5. A
6-6.
6-7.
Introduction
Analytic-function Generation,
Generalized Integration
ximate Differentiation .
Sean ans plifer Circuit Representation of 1 a Second-order System
Computer Instability . a er ee ee
Concluding Remarks .
Chapter 7. The Representation of Nonlinear Phenomena
7-1.
7-2.
7-3.
7-4.
7-5.
7-6.
7-7.
7-8,
Introduction . ‘
Differential Relays and Diodes—General ok
Applications of Relays and Diodes to Simple Limiting
Other Diode and Relay Circuits . Lo.
Limiting the Output of an Integrator
Representation of a Unit Impulse
The Repetitive Unit Impulse.
Approximation of a True Time Leg.
Chapter 8. Multipliers and Function Generators
8-1.
8-2.
a.
8-4.
8-5.
ef.
8-7.
8-8.
8-9.
8-10.
8-11.
&-12.
Introduction.
The Time-division Multiplier
Quarter-square Multipliers.
The Crossed-fields Multiplier.
Other Multiplying Devices
Function-generating Equipment .
The Input Table _ .
The Photoformer
Diode-type Arbitrary-function- generating Equipment
Tapped Potentiometers ces
Resistive Materials as Function Generators
Conclusions,
Chapter 9. Miscellaneous Applications of the Electronic Analog Computer
§-1.
9-2.
9-3.
9-4.
9-5.
0-6.
9-7.
9-8.
Introduction ;
Simultaneous Algebraic Equations ;
Partial Differential Equations (Introduction) .
The Solution of Eigenvalue Problems . .
Replacing Partial Derivatives by Finite Differences ;
Computers as Curve-fitting Devices.
The Roots of Polynomial Equations
Conclusions. — ss 8
Chapter 10. Analog Computer Components and Computer Control
10-1.
10-2,
10-3,
Introduction.
Stabilized D-C Amplifiers. kon ee
Inherent Erroraof D-C Amplifiers. . «© «© «+ :
89
89
07
08
100
105
107
107
109
113
120
123
126
127
136
137
142
143
144
144
146
151
153
157
159
163
166
166
168
168
171
175
176
181
184
184
180
CONTENTS
10-4. Overload Warning Systeme
10-5. Boost ..
10-6, Operate-Reaet Systems
10-7. Potentiometer-setting Systems
10-8. Automatic Programming .
Chapter 11. The Checking of Computer Results
ll-1. Introduction ..
11-2, Cheeking Problem Preparation .
11-3. Partial System Checking .
li-4. Checking Problem Stability .
11-5. Specialized Checking Procedures
11-6. The Role of the Digital Integrating Differential Analyser | in n Prob-
lem Checking
11-7. A Dynamic Check of Computer Operation.
11-8 Accuracy and Precision Soe
Chapter 12. Repetitive Analog Computers
12-1. Introduction
12-2, Advantages and Disad: vantages of Rapatitive Computers
12-3. Philbrick Computer coe
12-4. Initial Conditions .
12-6. Problem Setup. . .
12-6. Trends in the Design of Repetitive Computers
Chapter 13. The Digital Integrating Differential Analyzer
13-1. Historical Development
13-2, Design Features of the Newer Digital Integrating Differential
Analyzers.
13-3. Operation of the Digital Integrating Differential Analyser
13-4. The Number System . tok
1-6, Communication between Integrators
1-6. The Coding Diagram .
13-7, Conclusions.
Appendix
A-1. The Terminology of Differential Equations
A-2. Time Constants and Natural Frequencies .
A-3. Operator Notation .
A-4, Conditions for Stability
A-§, The Equivalence of jw and the Differential Operator "
Inaleg .
194
195
195
198
199
208
210
213
214
216
216
217
222
227
229
2o1
233
233
235
oT
242
242
244
247
251
252
255
257
CHAPTER 1
INTRODUCTION
1-1. Historical Development. In the period of years immediately
following World War II, there has been developed a tool which has
lal much influence upon the methods of analysis employed by engi-
neers. This tool is the electronic analog computer. The credit for its
early development, both in this country and in England, is not easily
placed because of wartime security measures,
If any two men can be credited with the first. published use of oper-
wlional amplifiers as computer components, they are C. A. Lovell and
I), B. Parkinson of Bell Telephone Laboratories. Their use of oper-
ntional amplifiers was in the computer of the M9 antiaircraft-gun
director built by Western Electric Company.* J. B. Russell of
Columbia University noted the circuits utilized in the M9 com-
puter and brought them to the attention of Ragazzini, Randall, and
ussell,'} who proceeded to build the first general-purpose electronic
inalog computer under contract with the National Defense Research
Committee. This work led to the publication of the first article, in
May, 1947, describing the operational amplifier as a computer com-
ponent, G, A. Philbrick is credited by some with having independently
pioneered the use of high-gain d-c amplifiers as computer components
in unpublished work conducted prior to World War II.
In 1947, the Reeves Instrument Corporation, under a Navy con-
tract, developed a computer which was the forerunner of the present-
ny REAC, At about the same time, many others began the inde-
pendent development of analog computers. No attempt will be made
hore to assign credit for the very rapid developments made since the
publication of Ragazzini's first paper, The list of individual contribu-
tora would be quite large and would require a major research effort to
oneure proper recognition for all concerned. Due credit should also
lo given to those men responsible for the earlier development of the
* Instruction booklet prepared by the Bell Telephone Laboratories for the
Weetern Electric MO antinireraft-gun director,
| Neforences denoted by superscripts appear at the end of each chapter.
1
E
2 ANALOG COMPUTER TECHNIQUES
d-c amplifier. Without their efforts, d-c analog computers would not
be possible today.
The main purpose of this brief historical review has been to impress
upon the reader the relatively short time between the construction of
the first d-c analog computer and its general acceptance as a tool by
engineers throughout the country. Ragazzini’s paper was published
in 1947, and by 1949 several computing facilities had sprung into
existence using equipment similar to that described in the paper.
It is the rapid development of the field of analog computation that
has inspired the preparation of this book. The development of new
equipments and techniques has left a gap in the literature such that
each individual when learning to use electronic analog computers must,
to a certain extent, travel the path of learning the hard way. There is
considerable literature available, in the form of papers, that will guide
the computer operator in the application of special techniques to par-
ticular problems, but nowhere does there exist a treatment of the use
of the analog computer sufficiently simple to serve as introductory
material yet sufficiently complete to be useful to the more advanced
machine operator. It is the purpose of the author to attempt to
shorten the period of transition from neophyte to accomplished analog-
computer operator for the engineer learning to use this new tool.
1-2. Classification of Computing Equipment. ['or many years the
engineer has had available computing devices of various types to aid
him in his work. The classification of the various instruments avail-
able should help to clarify exactly the type of equipment to be dis-
cussed extensively in the following pages.
Computing equipment may be divided into two main classifications:
analog and digital. The analog computer operates by representing the
variables of a problem by physical quantities easily generated or controlled
such as shaft rotaitons or electrical voltages. The representation is such
as to give continuous correspondence to the variables of the problem
being studied. A digital computer counts and obeys logic rules exactly.
A major difference between analog and digital computers is the accu-
racy attainable. Accuracy of a digital computer can be extended by
simply carrying more significant figures, whereas the accuracy of an
analog device is limited primarily by the accuracy of individual com-
ponents and of the measuring device.
There are many types of analog computers in use today. These
may be subdivided into two broad categories: general-purpose and
special-purpose, Figure 1-1 shows a few of the analog computers and
the class to which each belongs, The special-purpose computers will
not be considered further in this work, The general-purpose com-
INTRODUCTION 3
puters can be further subdivided into two classes: the direct- or
physical-analog computers, and the indirect- or mathematical-analog
computers.
The network analyzer, which belongs in the direct-analog class, is the
oldest of the general-purpose analog computers. Its operation is based
upon the analogy of the mechanical behavior of dashpots, springs, and
masses to the corresponding electrical behavior of resistors, condensers,
and inductors.
There are several reasons for the network analyzer’s not having
achieved the widespread use that the electronic analog computer has
Analog
| |
General Special
purpose purpose
| Planimeters
Direct Indirect Computing bombsights
Gun directors.
Link trainers
Scale models
Equivalent D-c electronic Wind tunnels
circuits analog computers Polynomial evaluators
Network Mechanical ;
analyzers differential F-86D flight simulator
analyzers
Digital integrating
differential
analyzers
Fra. 1-1, Classification of analog-computing equipment.
realized. First, the cost of a large analyzer installation is in the same
order of magnitude as that of the large-scale digital computer and the
mechanical differential analyzer ($100,000 to $500,000). Second,
although the equivalent electrical networks for mechanical systems
may be easily derived, the actual simulation is much more difficult
bocause of the absence of perfect components. The resistance associ-
ated with inductors, the leakage of capacitors, and the inductance of
reaiators all introduce errors into the system. It is the consideration
of such factors that makes the simulation of many systems more diffi-
cult than on the electronic analog computer.
Despite its disadvantages, there are computational tasks for which
ihe physical-analog computer remains unexcelled. An example is the
analysis of electrical power-distribution systems, Several of the major
4 ANALOG COMPUTER TECHNIQUES
network analyzers in the country are kept busy a large portion of the
time on this one type of problem. Other problems such as the dynamic
analysis of structural problems, e¢.g., an aircraft wing, can be handled
well on this type of computer. New network analyzers will probably
continue to be built by power companies, aircraft: manufacturers, and
universities to supplement the capacity of existing facilities.
The mathematical-analog computers include the mechanical differ-
ential analyzers, the digital integrating differential analyzers,* and the
electronic differential analyzers. The mechanical differential analyzers
are capable of accuracies up to five significant figures and usually con-
sist of ball-and-disk integrators having mechanical or electromechani-
cal couplings. They are at present in a very unenviable position of
being squeezed by the accuracy of the digital computer on one hand
and the speed and ease of operation of the electronic differential ana-
lyzer on the other hand, Since the cost of a mechanical differential
analyzer 1s of the same order of magnitude as that of the large-scale
digital computer, it is very doubtful whether another large computer
of this type will ever be constructed.
The digital integrating differential analyzer is a newcomer in the
computation field. The existence of such devices is widely known,
but few people, other than those actively engaged in their operation,
understand their basic principles of operation, An attempt will be
made in Chap. 13 to familiarize the reader with this type of equip-
ment in order that he may be more aware of its capabilities and
limitations.
The electronic differential analyzer is the device that will be treated
most extensively in the subsequent pages. Other than in Chap. 13,
the major effort will be to present the limitations and capabilities of
this class of equipment. Any future reference to analog computers will
mean specifically the electronic differential analyzer.
1-3. Problems Solvable on the Electronic Analog Computer. There
is a considerable overlap of fields of usefulness of analog and digital
computers. In general, a large-scale digital computer can do any job
which can be accomplished on an analog computer, but many prob-
lems can be handled adequately on an analog computer far more
rapidly and easily than by any other means, The correct choice of
*The digital integrating differential analyzer ia, by the strictest definition, a
digital computer. The technique of problem preparation ia, however, very similar
to that used for the other mathematical analog computers, For that reason the
author has arbitrarily listed the computer as an analog device, Bome authorities
prefer to list the digital integrating differential analyzer in a category sparate
from other computers, i-c., aa a digital analog compuler,
INTRODUCTION 5
computer is important to the economical solution of problems. Fae-
tors to consider in making the choice are the accuracy required and
the nature of the problem. In general, if more than four-significant-
figure accuracy is required, the electronic analog computer cannot
satisfy the requirements. Few analog computer laboratories can do
that well.
The type of problems best adapted to solution on an electronic
analog computer are those involving systems of simultaneous differ-
ential equations, linear or nonlinear, with constant or nonconstant
cocficients. Fortunately, the complexity of problem setup is increased
only slightly for nonlinear problems and problems involving non-
constant coefhcients. Some problems, other than those which belong
in the category of ordinary differential equations, can be satisfactorily
handled on an analog computer. These will be discussed in later
chapters.
Analog-computer results are normally presented graphically as a
continuous plot of the variable quantities. In many instances this
method of presentation is most convenient for engineering use. The
graphical representation of results has another important aspect. It
is relatively easy for the engineer, as he operates the analog computer,
to visualize the results as the actual dynamic response of the physical
svatem under investigation. Thinking of the analog computer as just
a mathematical device is to be discouraged as much as possible. The
usefulness of the computer will be greatly enhanced if the operator
views it as a tool to help him think in terms of the physical system.
1-4. Major Components. The major types of components of an
electronic analog computer are relatively few in number. First, and
the most important components, are d-c amplifiers or operational
amplifiers. It is these amplifiers that become summers and integrators
upon the addition of proper feedback and input impedances. Second,
il is necessary to be able to set coefficients in a problem. As will be
shown later, this may be accomplished either by the use of potenti-
ometers or by adjusting the ratio of feedback and input impedances
applied to the operational amplifiers. Third, it is necessary to have
a set of controls capable of starting and stopping the computation.
In addition, it is desirable to have a control position to perform the
funetion of holding the problem solution at any point in the solution.
‘These control operations are performed by a system of relays. Usu-
ally provision is made in the control system for the automatic appli-
eation of the initial conditions of the problem while the computer
controls are in the nearer position, A knowledge of the actual oper-
ation of the operate-reset relays is unimportant for the solution of
"
i] ANALOG COMPUTER TECHNIQUES
simple routine problems but is very important to the operator as the
problem complexity increases.
If problems other than those involving linear differential equations
with constant coefficients are to be solved, units capable of multiply-
ing variable quantities must be provided. As problem complexity
increases, there arises more and more frequently the need for arbitrary-
function-generating equipment capable of generating functions not
easily represented mathematically. Similarly, more complex problems
often require the representation of nonlinear phenomena. This is
accomplished by the introduction of diodes or relays into the com-
puting circuits.
The basic components mentioned in the preceding paragraphs,
together with suitable recording equipment, make up the major por-
tion of the equipment used in the solution of problems on analog com-
puters. Each of these components will be treated extensively in suc-
ceeding chapters.
1-5. Summary. It is desirable to emphasize to the beginner that
the understanding of the manner in which an analog computer per-
forms its operations is extremely important. This is true, perhaps
even toa greater extent than for any other computational aid. With-
out an understanding of the actual operation and limitations of the
equipment the operator can never rise above the level of knob twister.
An effort will be made throughout the remainder of this book to present
the fundamental principles of analog computation in a manner that
will allow the reader to attain that understanding easily.
REFERENCE
1, Ragazzini, J. R., R. H. Randall, and F. A. Russell: Analysia of Problems in
Dynamics by Electronic Circuita, Proc. [RE, May, 1947, pp. 444-452,
CHAPTER 2
THE LINEAR COMPUTER COMPONENTS
2-1. Introduction. Theoretically, there are two logical philosophies
along which an analog computer for use in solving differential equa-
tions may be developed, The first of these might be based upon
repeated differentiation, and the second upon a process of repeated
integration. From mathematical considerations, both systems are
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Fie, 2-1, Symbolism frequently used to represent the computer components neces-
sary to solve linear differential equations with constant coefficients, All voltages
e and ¢, referred to in the diagram are varying d-c voltages measured with reapect
to a common ground. The components shown are (a) an integrating amplifier,
(b) a summing or inverting amplifier, and (c) a potentiometer,
adequate. From an engineering approach, however, the process of
differentiation has a serious drawback, Differentiation is a ‘noise’’-
amplifying process, and all electronic gear, to a greater or lesser extent,
produces random noise. This noise, however slight, results in a much
higher noise level if differentiation is used. The second possibility,
that of repeated integration, offers no such difficulty; the noise is
amoothed, since integration is an averaging process. IJtepeated inte-
gration ia the basis of the electronic differential analyzers as we know
them today.
7
bt ANALOG COMPUTER TECHNIQUES
In order to solve a system of linear differential equations with con-
stant coefficients, the following types of equipment must be available:
1. Devices capable of performing the process of integration
9, Devices capable of summing several quantities
3. Devices capable of multiplying a quantity by a constant
4, Devices capable of multiplying by the constant —1
The symbolism of Fig. 2-1 may be adopted to represent the above
devices, Note that the ability to perform sign inversions and to sum
two or more quantities has been given to the integrating amplifier and
to the summing or inverting amplifier. Assuming that the above
devices are available, they may be interconnected in such a manner
as to produce the solution of any linear differential equation with con-
stant coefficients. The method of connection is illustrated by the
following example.
Example 2-1
a
OF ay SF + ase = f(b) (2-1)
Equation (2-1) may be rewritten in the form
dr dr
dt = ~Oi — gt + fit) (2-2)
Then, assuming that d*z/df* is a known quantity, it may be integrated to give
dx/dt and this in turn may be integrated to produce z. From Eq. (2-2) above, it
a
d*x _de . (a2) ar Ne a ~agx tit)
de dt * a, 2
dz
+
Fie. 2-2. Computer circuit showing the component interconnections necessary to
aolve Eig. (2-1).
can be seen that, having z and dz/dt, it is only necessary to multiply each by an
appropriate constant and sum them together with f(f) in order to get dx jdt, the
quantity that was assumed to have been known, The block diagram of Fig. 2-2
illustrates the method.
First examination of the block diagram may lead one to believe that he is reach-
ing down and lifting himself by the bootstraps. This ia not the case, however, as
the system operates much as any closed-loop servo system. The rate of change of
THE LINEAR COMPUTER COMPONENTS 9
a quantity depends on the magnitude of the quantity and the time history of itself
and ite derivatives. It is very important that one reason out the flow of informa-
tion in the above closed-loop system.
2-2. The Operational Amplifiers. In the above illustrative exam-
ple, the magnitude of the constants a, and a; must be less than unity,
ca | K,
" Ke Tu &,* - ft >. Kyejdt
tq ——————__ f,..
ne :
(6)
Fig, 2-3. More complete symbolism representing (a) the integrating amplifiers-
(}) summing amplifers, The symbols show the gain associated with each input
of the amplifiers,
since a potentiometer is only able to multiply by a constant less than
unity. This need not be the case in general, however, as the integra-
tors and summing amplifiers have the ability to multiply by constants
other than unity. The transfer functions of an integrating amplifier
and of a summing amplifier are,
therefore, more completely repre- ty
sented as in Fig. 2-3.
Until now, it has been assumed i High-gain
that the processes of integration « 4 | Se amplifier &s
and summation can be performed. =A
It might be well at this point to Fic, 2-4. Block diagram of an opera-
show one means by which this may tional amplifier showing the high-gain
, id A J detailed bl t d-c amplifier and the feedback and in-
e one. more detal . oc put impedances zy anc 2,.
diagram of the above operational
amplifiers is given in Fig. 2-4. In the figure — A represents the gain of
the amplifier, and z, and 2; are the feedback and input impedances,
respectively. If the amplifier is assumed to draw no current at its input
grid, then from Fig. 2-4, Eqs. (2-3) to (2-6) can be written
hi ta (2-3)
i= — “ (2-4)
pe ot % (2-5)
ot
¢ = —Ae, (2-6)
10 ANALOG COMPUTER TECHNIQUES
From Eqs. (2-3) to (2-5),
a zy
Substituting Eq. (2-6) in Eq. (2-7) gives
e+ eo/A _ _ Go/A + & (2-8)
mi zy
Simplifying Eq. (2-8) gives
@ Aa
ej zp +i + Az, (2-9)
Multiplying numerator and denominator by 1/ A and factoring 2 from
the denominator gives the final form of the transfer function
@ ow _ 2-10
ei z 1 + 1/A(z;/2% + 1) 2-20)
If A is sufficiently high, a good approximation for Eq. (2-10) is
fe 1
ei Fi (2-11)
An alternate and simpler derivation of Eq. (2-11) can be made by
assuming that the voltage e, is equal to zero in Fig. 2-4. This assump-
tion is valid to a good approximation providing the gain A of the
amplifier is sufficiently high. Suppose the amplifier output ¢ is
restricted to remain within some finite region of voltage, usually
+100 volts. Since e, = — Ae, it is apparent that e, is approximately
equal to zero for A >> 1. Further, since the amplifier can be assumed
to draw no current,
ty = Ta
and hHeEe- == -—
1 z, zy
Rearranging and solving for ¢,//e; again gives
fe, — Fi .
as (2-11)
The derivation of this important result by the latter method has
the disadvantage of not emphasizing the nature of the approximations
made in the derivation. For the present, however, it is not necessary
to discuss the second-order effects of the operational amplifier, ‘These
effects are, of course, important in the design of the amplifiers and in
attempting to understand the limitations of the equipment, They
THE LINEAR COMPUTER COMPONENTS ll
are, however, less important to the beginner learning to use the analog
computer.
In later derivations of transfer functions of computer circuits, the
derivations will frequently depend upon the approximation
Bo
ts i A =O
l'or most analog-computing equipment the approximation is very good.
The value of amplifier gain ranges from 2 * 10° for some repetitive
equipment to over 10* for at least one of the better commercially avail-
able computers.
In Eq. (2-11), if zy and z; are both 1-megohm resistors,
Ce _ Fr
ej zy
—] (2-12)
Similarly, if 2; is 0.1 megohm and z, is 1.0 megohm, the transfer func-
tion is
e& ew 1X10 |
a 4 O1X10° —
Kquations (2-12) and (2-13) show that, to change the gain of a sum-
ming amplifier, it is only necessary to vary the size of the input resistor.
(Normally, the feedback resistor is held constant at 1 megohm.)
If zy is a condenser, then*
10 (2-13)
1 1
1 = 0 = 56 (2-14)
cx I/pC_
and rim = R ~~ pRO (2-15)
if = 1% 10 farad and R = 1 & 10° ohma,
fo 1 __i1
« 10°X10p pp 10)
or é = —Je dt (2-17)
Similarly, it may be noted that the gain of an integrator can be varied
by changing the size of the input resistor, just as in the case of the
summing amplifier.
A differentiator could be as easily formed by letting z; be a con-
denser and z,; be a resistor: then
&, gy R
— a =— —
e4 a0 pl
* The equivalence of jw and the operator p is demonstrated in Bee. A-5. A
brief introduction to operator notation is included in Bec, A-d.
—pRC (2-18)
12 ANALOG COMPUTER TECHNIQUES
Differentiation is seldom used in the solution of problems on analog
computers, as the noise amplification produced by differentiation is
very undesirable. At times it is preferable to rewrite a set of differ-
ential equations completely rather than to differentiate. If differenti-
ation cannot be avoided (a very rare situation), then an approximate
differentiation may be used to keep the noise at a usably low level.
A circuit for producing approximate derivatives will be discussed in a
later section.
So far, the transfer function of the integrator and of the summing
amplifier have been developed without showing that each has the
iy
—
fa iy iy
=—-r
; iy
Fy
4 Th
-A &s
* i
*
*
Fa
Ca Fa
Fia, 2-5, Block diagram of an amplifier having a feedback impedance z; and several
input impedances z,, %,.. . , % connected in parallel to the grid input of the
amplifier,
ability to sum several functions at its input. This may be most easily
demonstrated by considering the block diagram of Fig. 2-5.
In Fig. 2-5 the single input has been replaced by several inputs
a & ».., x. Since the gain of the amplifier is very large, e, may
be set equal to zero, since
lle
0 (2-19)
Ba
a=— 7
The current drawn by the amplifier may be neglected as before;
therefore
te Fito bin = Oe (2-20)
Replacing i., i, . . . , i, by equivalent expressions, Eq. (2-20) becomes
fo ee. Ee _ oo -
. + rs T + :. zy (2-21)
me — Fe, He... Hy oR e -!
or e, = 2. fa os &, :. En zy » x (2-22)
2-3. Potentiometers. Potentiometers are frequently used in analog-
computer setups to perform multiplication by a constant loss than 1,
THE LINEAR COMPUTER COMPONENTS 13
Very often these potentiometers are 10-turn helical wire-wound types
of high resolution and excellent linearity (usually from 1 to 0.05 per
cent of full scale). By means of a vernier dial, parameters of the
problems may be accurately and conveniently set. A loading correc-
tion will have to be applied, in most cases, to compensate for the load-
ing on the potentiometer,
The potentiometers most commonly used in analog computation
vary in total resistance from 10,000 to 100,000 ohms. The lower limit
e Ry
NAG
. “ m1 e] ry High-gain “
. ad ¢z | amplifier
(a) = (5)
Fie. 2-6. Block diagram of a potentiometer when used as the input to an amplifier:
(a) schematic representation; (b) detailed circuit connections; (c) equivalent. cir-
oult representation from which the p