MATHEMATICAL MACHINES Volume II: Analog Devices By
Francis
J.
Murray
analogy demonstrates principle that diverse physical systems may often be described by the same mathematical and logical relationships. This principle is essen tial to engineering and scientific design arc! to such devices as guided missile controls, The
of
automatic airplane automatic transmissions.
pilot trainers,
pi!o?5,
ard
Three categories of mathematical ma chines which use the principle of analogy are discussed in this volume. The action prin ciples, characteristics, and use of continuous computers are presented in the first part.
These computers consist of components which represent individual mathematical opera tions. In their commercial form, these de vices are basic to the engineering design of complex machines such as airplanes and tanks, and they are incorporated into many control devices.
True analogs, discussed in the second are continuous devices wh?ch have
part,
been
utilized
by
individual
investigators to
obtain deeper insights into complex situa tions. The author examines the theory of
analogs and includes descriptions of Dimension Theory, Models, and principles of sp^fc elationships. Tha third part of the volume D-JJCM^ Me procedures and deigns which permit various 3 TiUiicji! ~rft JTrn+s to I liTjVh i 3cranced mathematical computations even though the devices themselves are simple. The instruments discussed include planim-
true
"
j
mat"
i
integrometers, and various geomet trigonometrical devices, and the author demonstrates that many of th-sie instruments appeared early in the de/esopment of mathematical machines and are the predecessors of devices now employed in continuous computers.
eters, rical
and
Aboul
the Author
Francis matics at
Murray is professor of mathe Duke University and Director of
J.
Special Projects
in
Numerical Analysis.
\m
8
MAR 24
1983
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kansas city
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MATHEMATICAL MACHINES VOLUME
II
ANALOG DEVICES
MATHEMATICAL MACHINES FRANCIS
J.
MURRAY
VOLUME
II
ANALOG DEVICES
COLUMBIA UNIVERSITY PRESS
NEW YORK,
1961
Parts of this material were previously published under the
title
The Theory of Mathematical Machines Reproduction for the
Copyright
in
whole or
in
part of this
work
is
permitted
the United States Government purposes of
1947, 1948, 1961,
Columbia University
Library of Congress Catalog Card
Press,
New York
Number: 61-7812 Manufactured in the United States of America
CONTENTS OF VOLUME
II
Part III
CONTINUOUS COMPUTERS 1.
H.
Introduction A. B.
C.
A CONTINUOUS COMPUTER AMPLIFIERS AND STABILITY PROBLEM RANGE IDEA OF
D. CONTINUOUS
COMPUTATION
55
THEOREM 56
3
J.
COMPLEX CIRCUIT THEORY AND
4
S
ILLUSTRATIONS; FILTERS Electrical
and Multipliers INTRODUCTION TO MECHANICAL
A.
Computing INTRODUCTION
69
B.
POTENTIOMETERS
69
C.
ELECTRICAL ADDITION
75
D.
CONDENSER INTEGRATION
77
COMPONENTS
7
ADDERS
8
C.
MULTIPLYING BY A CONSTANT
12 7.
D. SIMILAR TRIANGLE MULTIPLIER
RESISTANCE
15
Amplifiers A. THE BASIC NOTION OF
AN AMPLIFIER VACUUM TUBES AS AMPLIFIERS
81
B.
81
C.
FEEDBACK AMPLIFIERS
AND MECHANICAL
COMPONENTS DIVISION
17
22
D. STABILITY 3.
Cams and Gears A. CAM THEORY
26
B.
FUNCTION CAMS
27
C.
INVOLUTE GEARS AND WRAPAROUNDS
31
D.
LOG AND SQUARE CAM MULTIPLIERS BACKLASH
33
8.
E.
E.
DRIFT COMPENSATION
F.
SUMMING AMPLIFIERS
G.
INTEGRATING AMPLIFIERS
Electromechanical Components A. MOVING WIRE IN FIELD
34 B.
4.
A.
INTEGRATORS
37
B.
DIFFERENTIATORS
39
C.
MECHANICAL AMPLIFIERS
43
95
97 101
103
MECHANICAL ANALOGS OF ELECTRICAL CIRCUITS
106
WATT HOUR METER
107
D.
SYNCHRO SYSTEMS
109
9. Electrical
Circuit
88 91
C.
Mechanical Integrators, Differentiators,
and Amplifiers
5.
63
6
B.
F.
THEOREM
APPLICATIONS OF THEVENTN
2. Differentials
E.
S
i.
6.
A.
THEVENTN
3
Multiplication
A.
INTRODUCTION
112
B.
TIME DIVISION MULTIPLIER
113
MODULATION MULTIPLIER
118
A.
Theory INTRODUCTION
48
C.
B.
NOTION OF A CIRCUIT
48
D. STRAIN
C.
THE CIRCUIT EQUATIONS
50
E.
STEP MULTIPLIER
119
D.
MESH EQUATIONS
52
F.
CATHODE-RAY MULTIPLIERS
120
SOLUTION OF THE CIRCUIT PROBLEM
53
E. F.
G.
THE MESH CURRENTS AS SOLUTIONS OF
10.
GAUGE MULTIPLIER
118
Representation of Functions
DIFFERENTIAL EQUATIONS
54
A.
FUNCTION TABLE
123
THE NATURE OF THE SOLUTION
54
B.
SCOTCH YOKE AND OTHER RESOLVERS
125
CONTENTS
VI
C.
THE ELECTRICAL REPRESENTATION 127
D. POTENTIOMETER
METHODS OF
REPRESENTING A FUNCTION
128
E.
MULTI-DIODE FUNCTION GENERATOR
129
F.
CATHODE-RAY TUBE FUNCTION
G.
INTRODUCTORY DISCUSSION AND SETUP
ANALYZERS
178
THE SHANNON THEORY FOR THE SCOPE
134
OF MECHANICAL DIFFERENTIAL
MAGNETIC MEMORY METHODS
135
ANALYZERS
1
178
OF MECHANICAL DIFFERENTIAL
180
D. REFERENCES
36
REPRESENTATION OF SPECIAL
E.
187
INTRODUCTORY DISCUSSION OF ELECTROMECHANICAL DIFFERENTIAL
140
Linear Equation Solvers
ANALYZERS
187
F.
PRELIMINARY SETUP
188
A.
INTRODUCTION
144
G.
SCALING AND LOAD CONSIDERATIONS
189
B.
TWO-WAY CONTINUOUS DEVICES
144
H.
WIRING AND OUTPUT CONNECTIONS
192
C.
MANUAL ADJUSTMENT
146
I.
GOLDBERG-BROWN DEVICE
149
D. E. F.
J.
MACHINES USING THE GAUSS-SEIDEL
MACHINE
FEEDBACK
Equation Solvers A. INTRODUCTION
E.
Error Analysis for Continuous Computers A.
INTRODUCTION
B.
THE TYPES OF ERROR
199
C.
LINEARIZATION
200
D.
THE NOTION OF FREQUENCY RESPONSE 202
198
156 159
Harmonic Analyzers and Polynomial
D.
196
153
H, STABLE MULTIVARIABLE
E.
A ERROR EFFECT
F.
THE a ERROR; SENSITIVITY
165
HARMONIC ANALYSIS AND SYNTHESIS
C. FINITE
195
EQUIPMENT
AUTOMATIC MULTIVARIABLE
FEEDBACK IN THE LINEAR CASE
B.
193
COMMERCIALLY AVAILABLE
K. REFERENCES
14.
G. STABLE
IMPLICIT SYSTEMS OF DIFFERENTIAL
EQUATIONS
POSITIVE DEFINITE CASE OF ADJUSTERS 151
METHOD AND THE MURRAY-WALKER
12.
B.
GENERATOR
FUNCTIONS
11.
Equation Solvers INTRODUCTION
C.
H. FOURIER SERIES REPRESENTATION I.
A.
13. Differential
OF FUNCTIONS
HARMONIC ANALYZERS
204
EQUATIONS
165
ERROR AND NOISE
G.
THE
H.
SOLUTION OF LINEAR DIFFERENTIAL
166
FOURIER ANALYSIS
203
ft
167
EQUATIONS WITH CONSTANT
CONTINUOUS ANALYZERS AND
208
COEFFICIENTS
169
SYNTHESIZERS F.
POLYNOMIAL REPRESENTATION BY
15.
SPECIAL DEVICES
Digital
Check Solutions
A.
USE OF DIGITAL CHECK SOLUTIONS
170
B.
STABILITY OF DIGITAL
172
C.
HARMONIC ANALYZERS; ZEROS;
G.
206
THE REPRESENTATION OF THE
SOLUTIONS
COMPLEX PLANE
Part
214
THE ACCURACY OF DIGITAL CHECK SOLUTIONS
172
H. CHARACTERISTIC EQUATIONS
212
CHECK
216
IV
TRUE ANALOGS 1.
Introduction to
"True
THE CONCEPT OF
B.
ANALOG APPLICATIONS
C.
MATHEMATICAL PROBLEMS SOLVED BY
ANALOGS
223
Dimensional Analysis and Models A. INTRODUCTION
224
B.
MEASUREMENTS
C.
DIMENSIONALLY COMPLETE RELA
2.
Analogs"
A.
"ANALOG"
225
TIONS
228
228
229
CONTENTS D.
BUCKINGHAM S THEOREM
229
E.
MODELS
231
F.
APPLICATIONS
231
7.
Electromechanical Analogies A. DEFINITION
B. 3. Electrolytic
VH
Tanks and Conducting
C.
Sheets
OF MECHANICAL
SYSTEM
273
CONNECTION DIAGRAMS
273
MATHEMATICAL RELATIONS IN CONNECTION DIAGRAMS
275
276
A.
INTRODUCTION
233
D. ELECTRICAL ANALOGIES
B.
ELECTROLYTIC TANKS
233
E.
MASS-CAPACITANCE ANALOGY
277
C.
CONDUCTING SHEETS
236
F.
IDEAL TRANSFORMERS
278
G.
MASS-INDUCTANCE ANALOGY
283
236
H.
ELECTROACOUSTIC ANALOGIES
286
I.
ELECTROMECHANICAL SYSTEMS
288
J.
APPLICATIONS
292
D. REPRESENTATION OF
THE COMPLEX
PLANE E.
F.
ELIMINATION OF ERRORS DUE TO FINITE SHEETS
239
POTENTIAL FLUID FLOW
241
G. SPECIAL
FLOW PROBLEMS
243
8.
Two-Dimensional
Electromechanical
Analogies 4.
Membrane Analogies
A.
245
A.
INTRODUCTION
B.
DIFFERENTIAL EQUATION OF
5.
9.
Network Representation of
Partial
250
Differential Equations
SOURCES OF ERROR
251
A.
INTRODUCTION
B.
SCALAR-POTENTIAL EQUATION
305
C.
RECTANGULAR LATTICE
306
NETWORK REPRESENTATION METHOD OF FINITE DIFFERENCES BOUNDARY CONDITIONS AND APPLI
307
INTRODUCTION
252
D.
B.
THEORY OF ELASTICITY
252
E.
C.
PHOTOELASTIC MODEL
255
F.
D. PHOTOELASTIC EFFECT
255
Analogies Between Two-Dimensional Stress
Problems
AIRY S STRESS FUNCTION
259
B.
BOUNDARY CONDITIONS
260
309
CATIONS
310
CURL RELATIONS
311
H.
RECTANGULAR LATTICE
312
CONNECTION DIAGRAM AND NETWORK 313
ANALOG j.
317
MAXWELL S EQUATIONS
K. DERIVATION OF EQ. IV.9.C.3,
265
ANALOGIES
305
G.
I.
A.
C.
297
MODELS
Photoelasticity A.
6.
247
RUBBER SHEET MODELS
D. SOAP FILM E.
294
ELASTICITY
245
MEMBRANE C.
B.
TWO-DIMENSIONAL LUMPED-
CONSTANT SYSTEMS
4, 5
320
PartV
MATHEMATICAL INSTRUMENTS 1.
Introduction A.
MATHEMATICAL INSTRUMENTS
329
D.
TRANSFORMATIONS OF THE PLANE
E.
GENERAL THEORY AND THE CONSTRUC TION OF CURVES
2.
335
337
Transcendental Algebraic and Elementary
Operations A. FIXED PURPOSE COMPUTERS B.
C.
3.
Instruments for the Differential and
331
Integral Calculus
SLIDE RULES
332
A. DIFFERENTIATORS
PLOTTING DEVICES
333
B.
INTEGRATION AIDS
340 341
CONTENTS C.
VARIABLE SPEED DRIVE DEVICES
D. AREA-MEASURING DEVICES
342 342
Planimeters A. BASIC PRINCIPLES
5.
Integraphs A.
INTRODUCTION
353
B.
INTEGRAPH INSTRUMENTS
353
C.
THE INTEGRAPH OF ABDANK
345
B.
EXAMPLES OF LINEAR PLANIMETERS
348
C.
INTEGROMETERS
350
D.
GRAPHS IN POLAR COORDINATES
351
ABAKANOWICZ WHEEL INTEGRATORS
D. STEERING
INDEX 357
354
354
Part III
CONTINUOUS COMPUTERS
Chapter
1
INTRODUCTION
m.l.A. Idea of a Continuous Computer In a
computer individual quantities as a sequence of digits. The registers
digital
appear in
is subject to another operation. Such operation a mathematical relationship is represented in the
computer by connecting the output of the com
and combinations of these
ponent associated with the first operation to the the second. In this way, the input of that of
operations provide approximate representation
mathematical equations are translated into a
for all the operations of analysis.
setup for the computer.
digital computer performs arithmetical opera
tions
on
these digits,
possible,
up a computing device in which
however, to set
quantities are represented instance,
It is
by magnitudes. For
the value of a variable x,
a rotation, a linear dis
either
represented by
may be
A
The above
may not be quantities
may
as
represented,
which can be
unknown
interconnected.
appropriately is
necessary,
it
is
desirable that variables throughout the device be
of approxi represented by physical quantities
mately the same
sort.
of Generally speaking, there are two types
computers in which in which magnitudes: mechanical computers, and have representation, geometrical quantities quantities are represented by
electrical
Mechanical computers
computers.
would normally contain such components "differentials"
"multipliers,"
tors,"
and
as
(which are used for addition), "gear
boxes,"
"integrators."
would have analogous
"function
quantities
the computer
may
properly set up,
it
On the
well be
itself
other hand,
such that when
can immediately generate the
quantities or functions as functions
of time.
The output of such a
device appears in one of
two forms. The desired output may be certain numbers which are obtained by measurements. one Frequently the desired output is a function of variable in which case the variable, independent will
the correspond to time, and the output of
computer will be a graph of the function desired. The output appears in a device which records the value of the ordinate either at intervals or continuously.
genera
Electrical
electrical
and the
have to be adjusted by the device
until certain relations hold.
interconnection
somewhat
immediately realizable between
a computing placement, a voltage, or current. device of this type consists of components mathematical operations representing specific
Because
is
since frequently the relation desired simplified
computers components. In
ffl.LB. Amplifiers and Stability
A
continuous computer, then, consists of
confine ourselves to computers in
various components to perform mathematical
which each component has clearly specified and outputs. There are other types of inputs continuous mathematical machines which we
means of connecting them operations with some in order to represent mathematical relations.
Part III
we
will treat in Part IV. this type is Normally, a computing device of in a given each operation by having
utilized
system of mathematical equations represented by mathematical a component in the computer.
A
the equations can normally be relationship in to mean that the result of one interpreted
The operation of
these
components requires
power. On the other hand, these components tend to be inaccurate when power is transmitted
through them from input to output. Con connections between components sequently, the
must be supplemented with sources of power called amplifiers. In most cases it is desirable that the
power transmitted through a component
INTRODUCTION be negligible. The power which appears output
is
at the
and the power used
normally negligible,
at the inputs should be adequate for the function
This means that corre ing of the component. to most of the outputs, there are
sponding
which
"amplifiers"
component
and produce a quantity
this signal
with considerable power.
representing
In
the
of
case
amplifiers
output of a
will receive the
as a signal
electrical
computers,
vacuum tubes or
use
electronic devices,
equivalent
and because of called
these
this,
these
computers are frequently In a mechanical continuous computer one may "electronic."
gear boxes
for
addition,
computer, certain
ing a
power
one of the
rules
stated above.
Considerable
outputs.
Multipliers
their
and automatic function
a considerable generators, in general, require
amount of
input power even
required at their output. puter,
then,
A
if
no power
mechanical
is
com
and function generator. The gear multiplier, boxes and differentials would then be part of the interconnection
system
between
these
com
the interconnection system con
contain amplifiers. sequently would not it is not customary, a similar setup Although
can be used for a continuous
A
electrical
number of mathematical
represented
When
this
occurs the
most
For
of the
translation
facile
relations of a
problem
it
Often the
mathematical
results in
an unstable
setup.
this
reason our discussion in this part will
be concerned greatly with questions of stability. However, this stability discussion has a far
than those range of applications mentioned here. Indeed, practically every non-
computing problem has stability questions it no matter what method is used
associated with
or what devices are used for solving
example,
in
the ideas
digital
For
it.
computation one often uses
and procedures of
stability analysis
developed for continuous computers.
normally would have amplifiers
associated with the output of each integrator,
ponents, and
in turn
considered unstable. In order
a stable setup. necessary to obtain
trivial
at
required
power
problem on a continuous computer
to solve a
the other hand, integrators almost invariably are is
is
computer setup
broader
when power
this
and these deviations
the desired computation.
of these power can be transmitted to the outputs devices from the inputs without inaccuracy. On
inaccurate
The output of
may be amplified by other power sources. This of unwanted signals can swamp repeated buildup
which permit one to add, integrators, multipliers, and function generators. Gear boxes and a certain extent to
source.
the desired output
machine
differentials are exceptional to
Weak
arise.
may
source then contains stronger deviations from
following components: which permit one to multiply by a constant, differentials which are combinations of gears
the
instabilities
accidental variations occur in the signal govern
is
find
and
integration,
multiplication,
functions. representation of Since power sources are present within the
by
passive
computer. can be
relations
networks
with
fair
networks accuracy, the outputs of these passive as signals for amplifiers which feed used being
The more customary arrangement based on components in which some mathe
IH.1.C. Problem Range
The design of a complex in general, plicated structure,
device or a is
preliminary theoretical investigations. Normally the objective of physical experiments and tests is either
to
theoretical
obtain
basic
investigations
information
obtained from them. Under present-day circum stances it is impractical, with regard to both time
and expense,
to
investigate
plicated action of
In such components the input appears as a signal while the output is associated with a quantity of relatively high power.
In electronic
differential
of this type are available analyzers, components
by
One can readily give examples of devices whose
is
combined,
possibilities
constructing full-scale experimental examples.
successful functioning depends
is
the
for
or to verify results
the networks.
matical operation and amplification
com
dependent upon
include
many
many
parts.
upon
the
com
These would
electronic devices, a large variety
of airplanes and guided missiles, motors of both and the nuclear types, machines
the customary
used in manufacturing processes, automatic factories, and chemical plants. Their action can
PROBLEM RANGE
III.LC.
be described in terms of systems of equations,
possible to set
either algebraic or differential. In
yield
many instances
up continuous computers which
approximate answers to problems in partial
impractical to solve these systems of equa tions in closed explicit form. The theoretical
differential
investigation of a device in question
of such complicated devices is an essential part of present-day technology. Most of these uses, it seems fair to say, have been developed during
it is
must be
based upon some computational method. It is often possible to use continuous computers for such investigations. Many solutions to a of equations can be obtained for different
and
values
initial
different values
system
and since the Second World War. These devices
of
have also been used for control purposes for There are many years. many problems which a
sets
of the design
Thus, one can gain an excellent notion of the behavior of a proposed device parameters.
and can base an
effective
design
upon
findings
supplemented with certain experimental
investi
gations.
Even when results
purely by experimental methods, such is
experimentation time consuming.
both extremely expensive and Furthermore, in extremely
complicated devices to
same
possible to obtain the
it is
it is
practically impossible
obtain the desired information by purely
experimental
information,
When
methods.
results are required to it
is
experimental
supplement theoretical
still
theoretical investigations.
necessary
to
make
To successfully design
a device, a designer must have a thorough under standing of the basis for its action. Both theoretical
and experimental
equations.
The use of continuous computers in the design
investigations are
continuous computer can solve accurately,
more
faster,
and more economically than a human
being can.
Fire control for artillery
is
the
example, and much of present-day
classical
computing equipment was developed in its modern commercial form for this one specific purpose. Automatic computation
is
essential for
antiaircraft fire control.
There are many other control purposes for
which continuous computers are used. Auto matic pilots both for sea and for air purposes utilize
continuous computation. There are many
industrial processes
whose success depends upon mathe
the fast and accurate solution of certain
To make
matical problems.
these processes
automatic, continuous computers are frequently used.
Frequently there
is
the situation in
which the
a certain device can only be
directed for this purpose.
desired action of
adequate.
described in a rather complicated mathematical way. Here either a continuous computer must be used or, when appropriate, the principle of a
Experimental results without a theoretical framework could never be
The
designer,
therefore,
uses a continuous
computer as a means to simulate a proposed device in the laboratory.
He may
think of the
continuous
computer
incorporated
into
device, for example, temperature controls.
the
Many
computer setup as the mathematical equivalent of a working model which, however, possesses
ideas developed for continuous computing or
and which,
such devices as automatic transmissions for
far greater flexibility of adjustment also,
is
far
more convenient
for observation.
Using the continuous computer, the designer can
control purposes have been incorporated into
automobiles. lay
down
One
cannot, therefore, practically
limits for
what one would
call
con
determine the value of design parameters and of the proposed device investigate the behavior
tinuous computing. In a complex technology the
under many
applied in very
different circumstances.
By omitting
principles
of continuous
many
computing can be
forms.
or varying terms in the equations, the designer
The above discussion has tended to emphasize
can determine factors important for the success of the design as well as factors to which the
the use of continuous computers for the solution
design
is insensitive.
of Frequently the system
in this equations considered
ordinary
way
differential equations.
is
a system of
However,
it is
of ordinary differential equations. For this purpose there are a number of commercial devices
on the market. There are mechanical
differential analyzers
and two types of electronic
INTRODUCTION One of
differential analyzers.
types
these electronic
suitable for systems of ordinary dif
is
equations with constant coefficients. is not subject to this restriction.
ferential
The other type
In addition to the commercial devices there are various university installations,
including
In general, a designer can
required.
utilize
a
continuous computer much more directly than he can a digital computer. The setup is more
and the equivalent of design changes can be made by the designer himself without intermediate coding. These advantages
readily understood,
however, part and parcel of certain dis
mechanical devices which can be used for solving
are,
ordinary differential equations and electronic devices capable of giving approximate solutions
advantages associated with the limited logical of continuous computers. con flexibility tinuous computer has some specified purpose, for
to
problems in partial differential equations. Devices have also been developed for solving
A
instance, the solution of differential equations.
modern commercial
simultaneous linear equations, polynomial equa
For
one unknown, and harmonic analyzers. Harmonic analyzers are devices which will either
and are very easy to code. They cannot, however, be used for most other computational purposes and, therefore, have to
tions in
permit the evaluation of the Fourier Series of a given function or,
when
the coefficients
are
give values for the function.
known,
are, nevertheless, certain limitations in
the use of continuous computers. technological
difficulties
limited accuracy.
For
There are
which prevent the con
struction of devices having
more than a
instance,
certain
one limitation
is
the accuracy with which measurements can be made. Another is associated with the length of
time an adjustment can be maintained. How ever, this accuracy limitation can be compensated for in
many
instances by using auxiliary digital
computation.
It is also true that there are
supplemented with digital computation. However, when proper supplementary digital
many
made.
The major
other. The continuous computer can be used for general exploration purposes for which it is faster and more convenient. The
ment each
precise investi
gations of regions for which higher accuracy
is
basic
for
is
the
these
theory of electrical components and the action of computers as a whole, one must understand basic circuit theory, which is developed in Chapter 5.
are
adequate. Frequently the two types of devices, that is, the continuous and digital, can be used to supple
of the
computers. In Chapters 2, 3, and 4, the com ponents used in mechanical computers are In order to understand both the described.
Among
is
objective of the present part
theory continuous computers, both with regard to the individual components and to the use of these
presentation
devices
computer can be used for
devices
flexibility
be
problems for which the limited accuracy of these
digital
purpose,
computation is available, full use of the speed and economy of continuous computers can be
UI.l.D. Continuous Computation
There
this
have great
commercial continuous computers there
many instances
of either electrical or electro
mechanical devices.
We
describe
the
corre
sponding components and their uses in linear equation solvers and harmonic and differential analyzers.
theory for
In Chapters 14 and 15 we present a the validity of the solutions of
continuous computers as well as an analysis of their joint use with digital equipment.
Chapter 2
AND MULTIPLIERS
DIFFERENTIALS
m.2.A.
Introduction to Mechanical Components
The purpose of Chapters
and 4
2, 3,
is
to
and, of course, the remaining angle also be measured.
amount can
Practically, this permits the
describe devices which are used as components in
use of scales divided into 100,000 or
continuous computers of a mechanical nature, In general, each of these components has a
for measuring rotations far in excess of
number of inputs and one output such that some
variable.
mathematical relationship is represented. For instance, a differential is a device for representing
quantities
Two
addition.
inputs are represented by the
rotation of two shafts, and the output
sented by the rotation of a shaft which
is
is
repre
equal to
the average of the rotations of the input shafts.
possible with
parts
what
is
any other type of continuous
methods
both
However,
more
have a number of
of
representing
difficulties.
From
the computer point of view probably the most
important objection flexibility
in the
is
the difficulty of obtaining
setup of the machine. It
is
with
addition, multiplication, integration, and repre
of setup that mechanical suffer in computers comparison with electrical ones. Nevertheless, a number of systems have
sentation of a function.
been
The
operations
represented
Probably the most frequent
will
consist
of
way of represent
ing input and output is by means of rotations of a shaft. On the other hand, in mechanical devices there
is
also the possibility of representing a
regard to
set
flexibility
in
up
which computing
mechanical devices. The output into
an
electrical
signal
which
is
done by
is
transformed
is
transmitted to
another component where it is again translated back into mechanical form. Unfortunately, this
two problems of time
quantity by the translation of a rod, by a linear
in turn introduces
or angular velocity, or by a force or a torque. Each of these has been extensively used. In the
One of these is
the average time delay of a system
as a whole.
The other and more important
present chapter we
time-delay problem
stress the
use of rotations and
translations to represent mathematical quantities. It is
and
more
efficient to treat
forces
Jn
the
the use of velocities
same way that electrical on the basis of the well-
parts
of a
is
delay.
the tendency of different
computing
device
to
represent
with different time delays. Direct mechanical connections for rotating
quantities
quantities are treated,
shafts are normally
made by means of gears.
If
known
the shafts are parallel, spur gears are used.
If
analogy. Indeed, velocities and forces are
used mostly in what we refer to as true analogs a detailed treatment will be given (see Part IV)
they are at an angle, bevel gears are used. By properly shaping the teeth on these gears, the
under that heading.
rotation of one shaft can be
;
Thus, for the
moment our major
interest will
be in devices in which numerical quantities are represented by
either translations or rotations.
For extremely inexpensive devices offer easy construction
methods.
translations
On
the other
hand, they are limited in scale. Rotation has the
advantage that large amounts of rotation can be
measured by introducing a counter on the shaft which counts the total number of revolutions,
made to be an
exact
rational multiple of the rotation of the other
when
the teeth are engaged. However,
motion
is
initiated there
be a
may when
before the teeth engage, and the rotation
is
reversed there
may
when
the
slight play
the sense of
be an interval
during which the teeth are not engaged. This is in "backlash," and must be taken into account the design of mechanical computers.
of gear teeth
is
The theory
a special case of the theory of
DIFFERENTIALS
AND MULTIPLIERS
Cams are used to make a given motion a function of the rotation of a shaft. (The basis of
cams.
cams
the theory of
will
be discussed in Section
III.3.A, backlash in Section III.3.E.)
Another
difficulty
move
the pair
AA along their line of centers an
amount x and point
2(x
R
+ y).
on
the pair
BB
an amount y, then a
the chain will
It is
move an amount
clear that this
can be applied
to
of mechanical devices arises
when it is
necessary to transmit a force or torque to the output. This usually results in slippage or friction which produces
from the inputs
inaccuracies in the desired mathematical repre
To
sentation.
devices
avoid
this,
must be introduced
power amplifying form of either
in the
torque amplifiers or servos.
However, there are many applications where flexibility
is
not of
A
interest.
problem
may
involve the repeated solution of precisely the
same mathematical problem, would want what
in which case, one
referred to as a
is
"special
purpose computer." In many such cases a simple mechanical computer may be most advantageous
on
the score of reliability, inexpensiveness,
and Fig. IIL2.B.1
general sturdiness. Examples of such computers are gunsights, automatic airplane pilots, and a
any number of addends. In Fig.
variety of ship installations.
In the present chapter, in Sections III.2.B and
D,
respectively,
we
Another way
consider devices for addition
and
multiplication. In Chapter 3 representation of functions by mechanical means is discussed;
III.2.B.1,
y
is
negative. to
do
this
would be
to have the
three parallel rods with a connection joining the three in such a way that three points, one fixed on
Chapter 4 considers mechanical integration and In these chapters, quantities are by rotation of shafts or by the
differentiation.
represented either
translation of a piece.
In Section III.8.D
we
will describe the selsyn
system which permits the transmission of a shaft to position from one mechanical
component
another by an
electrical
signal.
IH.2.B. Adders
Suppose given quantities are represented by the displacement of certain rods (from fixed positions) in an apparatus. We wish to obtain a displacement corresponding to the sum of two such displacements. There is a simple initial
Fig, IH.2.B.2
arrangement by which one uses an endless chain or tape to add displacements (see Fig. III.2.B.1). The chain passes around sprocket wheels A, A, B,
B
}
and
position;
C the
distance apart
C sprocket wheels are fixed in wheels A and A are a fixed as are the If we pair B and B
The
.
each rod, remain collinear. If one outside rod displaced an is
amount x and
displaced an
amount y, then the middle rod
displaced an amount \(x
is
the other outside rod is
+ y) (see Fig. III.2.B.2).
III.2.B.
ADDERS
There are a number of ways in which the three rods can be connected so that the specified three points will remain collinear. One may have a crossbar pivoted upon the central bar with a slot
on each
side in
which a pin which
is
fixed
on the
Owing to the equality of opposite sides, BCED is a parallelogram. Hence, BD and CFare parallel.
B is the midpoint of AC and BD is one CF in length, D is the midpoint of the line segment AF and, hence, A, D, and Fare always Since
half
There
collinear.
also a
is
"lazy tong"
gram arrangement, which we There
is
a third way of accomplishing the same
Two
objective.
parallelo
will discuss later.
racks are used instead of the
outer rods, and a pinion
middle bar (see Fig.
is
III.2.B.5).
mounted on
the
Again the output
+
#x y). The gear teeth are constructed in such a fashion that the movement of this system
is
Fig. IH.2.B.3
corresponding rod
slides (see Fig, III.2.B.3).
the pivot are constrained then pins and
The
to be
collinear,
This crossbar arrangement can be replaced by
a pantagraph (see Fig.
III.2.B.4).
The
bars
AC
Fig. ffl.2.B,5
is
similar to the
strictly
rods and a wheel which
and does not IIL2.B.6).
slip
movement of a is
relative to the
The motion of
pair of
in contact with
them
rods (see Fig.
the pair of rods and
?
and the wheel can be easily specified. Let C , , 2 denote a reference position for the center of the wheel Fig. HI.2.B.4
ing
CF are rigid and equal in length with mid B and E respectively, DE = BC and points = BD CE. AC and CF are hinged at C, AC and BD at B, BD and DE at D, DE and CF at E. and
and
its
contact to the points of
two
C1} PI, and Q l refer to the correspond the circle, P2 an(* 62 ^e points fixed on
bars. Let
on the bars (see Fig. corresponding points fixed moved to a new III.2.B.7). Now if the system is in which P and Q are the new points of position contact,
we
see that since the wheel did not slip
DIFFERENTIALS
10
relative to the bars that
Q 2 Q. 22
Since
and
QP = Q2,
P2 P
AND MULTIPLIERS
= P P = Q& = X
this yields that
P2 Cl5 ,
center of the wheel
is
the average of the other
displacements,
combination
This
are collinear.
will
conveniently
add
considerable size but normally displacements of a there will be backlash between the pinion and the
two racks.
The customary method of adding rotations is by means of a differential. This device is analogous to the rack and pinion adder with the by rotations. There are two types of differentials, the bevel gear differential and the spur gear differential. translations, however, replaced
Qo
Fig. III.2,B,8 illustrates the
Fig.
Note
arrangement of gears
m.2.B.6
that the displacement of the
P
rod has
the value
where 6
/.PjCjP in radians and
r
=
QP.
Fig. HI.2.B.8
in a bevel gear differential.
the input spur gear
A
around the axle
A A 9
which
A".
gear
The
B
is
,
bevel gear A, collar
C".
B
is
and the
free to rotate
rigidly
which
entire
is
connected to the bevel
rotation of this combination
input y.
The bevel
gears
C
are
around the axle C. However, the
combination of C and C",
The rotation of the com
constitutes the input rotation
A"
applied through a gear meshing with
shaft
constitutes the
C
is
connected to the axle
perpendicular to
C
,
so that this
combination may rotate around the axis of
the axle
C".
The
output \(x + y)
Fig. HI.2.B.7
The
and the connecting
are rigidly connected but are free to rotate
bination jc,
A"
rotation of the shaft
C"
is
the
of the combination.
Geometrically, the motion of the bevel gears Similarly the displacement of the
Q
rod
is
given
= rfl. Hence, by the expression x x + y = 2QC0, i.e., the displacement of the
QQ
is
equivalent to the motion of nonslipping right circular cones or frustums of cones.
We can even
replace each gear with a disk contingent to other
IIL2.B.
ADDERS
disks, the disk representing a cross section of the
cone
to the axis (see Fig. III.2.B.9). perpendicular
These disks must rotate without
11
(see Fig. III.2.B.11).
moves so
that
P
and
Let us suppose the system Q are the new points of
tangency.
slipping.
The rotation of the the arc
P2 P
,
i.e.,
x
disk
A
can be measured by Let z
=PP = 2
P
This
x
is
measured by the arc
= z + P P.
and 6
is
around
QQ = PP C
Since the disks
2
without slipping, P2 P
+ #V
C
denote the rotation of the axis
and
.
C".
Thus
A move
= f-f = rt, where r = P&
the radian measure
of the angle
= QiQg. Hence jc z + r0. A similar discussion will show that the rotation y of 5 is
Pjd? = z
- rQ.
Hence, x
+ y = 2z.
Fig.ffl.2,B.9
let
C
and
g
Let us consider only one disk C, and
denote the center of this disk and
P
denote the points of tangency with the other
Fig,
It is clear that the
insure that if the relative to the
precisely
the
DL2JB.il
of the gear
purpose
x input
output
z,
rotates
the
same amount
y
C
is
to
an amount
input
will rotate
relative to z
but in the
This purpose can also be opposite a spur gear combination. In by accomplished direction.
the output Fig. III.2.B.12,
combination involves
the two outside disks, which
Fig, ffl.2.B.10
form a mounting
two meshing spur gears C. One of these mesh the other with the gear B; A and with the for
disks in
some
reference position for the system
Let (see Fig. III.2.B.10).
C15 PI}
and g 2 the points
fixed
Ql
denote
C
disk,
on the other
disks
the corresponding points fixed
P2
and
on the
gear A,
B mesh with input gears A and ff respectively. When the mounting for the C gears is held the A gear rotated by A, then the stationary and
B
AND MULTIPLIERS
DIFFERENTIALS
12
and in the opposite gear will turn equally
direction.
Thus, 5,
if the
and z
is
inputs
x
and}>
are applied to
the rotation of the
then the relative motion of
A
A and
C gear
mounting,
to C,
x
i.e.,
z
outer one being an plane, the the teeth are i.e., one in which
The
gear.
axle
intermediate gear
which in turn
two.
It is
is
common
the rotation of this
For
output.
gear,"
on the
we
simplicity
radius of the central gear
inside of
mounted on an
attached to an
is
revolves around the
"annular
arm which
axis of the other
arm which
shall
is
the
suppose that the
equal to the diameter
is
2r of the intermediate gear.
Let us
now consider what happens
in a
motion
of the system. For the purpose of this discussion, considered as circles. Let us suppose gears can be that the output
arm has rotated an amount z and
the central gear an
motion
relative
amount x, so of
that
two
these
x
z
is
the
z- 6).
(x
Relative to the arm, the intermediate gear will
turn an
amount
99,
which has the same arc length
gear or
rep
= 2r0.
on the
as 6 has
on the intermediate gear
Similarly,
outer gear has a rotation of an
we
central
see that the
amount
if
relative
to the output arm, rotated in opposite direction
is
must equal the z
y.
There
relative
Hence, x is
z
y
or
C
to B,
i.e.,
* + J = 2z.
y
ry>
- 4ry.
= 20.
Hence,
If y we have
y>
the total rotation of the outer gear,
=
=z
y or y
z
and
in
the
Substituting for
y.
y
equation
= 20,
we
y
obtain
another form of the differential which
consists of III.2.B. 13).
motion of
z =
which
to 0, for
Fig. HI.2.B.12
an epicycloidal gear train (see Fig. Here we have three gears in the same
Since the differential permits the simultaneous
addition of two quantities,
it
can be used in an
feed and adding machine to combine the regular the tens transmission from a lower place. This
avoids
the
the represented by This device
difficulties
additional tens transmission feed.
was used by E.
Selling in his
multiplication
machine.
in.2.C Multiplying by a Constant a constant Multiplication of a displacement by can be theoretically accomplished by means of the lever principle (see Fig. III.2.C.1).
Let us
suppose that our input and output displacements apply to bars which are constrained to move to each other, for instance, in grooves or parallel guides.
On each bar, we have a pin which slides
in a slot of a crossbar.
the slot
We
which may be fixed
also
in
have a pivot in
any position along
a line perpendicular to the two parallel bars. When the position of the pivot is determined and Fig, IH.2.B.13
one of the
parallel
bars
is
moved, the other
is
III.2.C.
displaced
congruent
a
proportional
MULTIPLYING BY A CONSTANT
amount.
For,
by
If the desired ratio
A so that fl^ =
triangles,
Then we must arms so that This arrangement even takes care of the sign, for if the pivot is not between the two parallel
is
13
r^, we must first choose B so that bjbz = r-Jr^
rjrz and
fix
the lengths of the remaining
AP =
x
and
= a%.
BP
These arrangements are not suitable for a con tinuous input of the ratio p = rjr^. However,
we can
position the pivot in Fig. III.2.C.1
by an
arrangement whose input is linear in p as shown in Fig. III.2.C.3. Let us suppose we have two such arrangements in parallel planes, with a
common
pivot which
may move
in a direction
Fig, ffl.2.C.l
bars the displacements are in the same direction, the equivalent of multiplying by a positive factor,
while the displacements are in opposite directions when the pivot is between the parallel bars.
Fig.
IE.2.C3
to the bars. We assume that the perpendicular of bars are parallel, one set above the other. pairs
Let us suppose that the lower arrangement is one and that the upper analogous to our previous
arrangement Fig.
It is
possible
HL2.C.2
to replace the crossbar with a
The
pantagraph arrangement (see Fig. III.2.C.2). better mechanical pantagraph arrangement gives have to be made in results but four adjustments
order to set up the smaller parallelogram.
is
to be used to position the pivot.
bar over the x bar to Suppose we move the upper the unit position and the one over the y bar an This will determine the position of amount p.
the
common pivot so
output y
x can be upper
will
variable.
and
that rjr%
have the value y
lower
/>.
Hence, the
= px where
and />
The separation between
the
should
be
combinations
DIFFERENTIALS
14
AND MULTIPLIERS
minimized. Normally this arrangement runs into difficulties of leverage.
The use of
a gears permits one to multiply
by one fixed ratio. One method of doing to have matched sets of demountable gears
rotation this is
Such a system
in various ratios.
commercially available
Another shift
arrangement.
is
introduced.
symbol for a IIL2.C.7.
Two
pointing toward
first
sides of the it,
box have arrows
indicating the inputs, which
used in one
the use of a clutch
and
We will define a clutch as an
arrangement to disconnect a shaft. The clutch symbols to be used are shown in Fig. IIL2.C.4, the
The most widely used shown in Fig.
differential is that
differential analyzer.
is
possibility
is
differential
indicating that
it is
Fig. HI.2.C.7
normally closed or
may appear in different positions. The connected to the middle. Fig. III.2.C.8
-\
output is is a gear
box arrangement which would give a number of ratios from relatively few
large gears.
INPUT
Fig.
IH2.C.4
Fig. IH.2.C.5
Fig.
m.2,C.6
closed in the situation described. For a pair of we shall use the notation shown in gears Fig. III.2.C.5. In the first of these the five times the rotation
upper shaft has of the lower. The ratio of
rotation of corresponding shafts will be indicated in the boxes box size. The by a number, not possibility of shifting will
by be indicated as shown
in Fig. III.2.C.6; in this example the upper shaft
capable of being gear connected to the lower in any one of three possible ratios. is
The
full
possibilities
of shift and clutch gear
arrangements can only be realized when the
Fig.
(There
is little
indicated gear
ra,ic.8
practical difficulty in arranging the
Notice that the output can be any multiple from to 99 of the input.) The mechanical advantage in this arrangement might require the use of a torque amplifier. shifts.
A slightly
SIMILAR TRIANGLE MULTIPLIER
III.2.D.
different
arrangement could yield gear ratios of
Clutches and differentials can be used together for the
to
2
Vi>
2%.! +
.01 to .99.
same purpose without
in this case the ratio
1.
of input to output
expressed with radix two. In what follows, y
can assume
Since s1
this that sk
However,
gears.
+y
15
all
values
from
to
= 3, we can conclude from
= 2k+1 -
1
(III.2.C.5)
is is
to stand for a "clutch function," i.e., its independ
ent variable
is
the value of
1
the clutch condition and
when
the clutch
is
it
has
engaged, zero
otherwise.
Fig.
midO
In closing this section, that
the variable
we wish
speed drive
to point out
which
will
be
discussed as an integrator in Section III.4.A will also
a rotation by a permit one to multiply The limitations on this will also be
constant.
discussed.
m.2.C.9
Fig.
IH.2.D. Similar Triangle Multiplier
Let us consider a single differential with two Let x be the clutches (see Fig. III.2.C.9). rotation of one side wheel of the differential,
In Section III.2.C
we
described a similar
in which the central pivot triangle multiplier
y
the negative of the other, and z the rotation of
the center wheel. Then,
x-y = 2z
x
or
= y + 2z
(III.2.C.2)
Let yj and y 2 be the clutch functions of the z and
y
clutches respectively,
X the input.
and
Then
the output
(HL2.C.3)
Thus, the possible ratios are
Now
suppose that
yield ratios 0, 1,
number.
We
.
will
from
yield ratios
.
.
,
k
-
Vi
0, 1, 2, 1
wnere s*-i
where a
Here again x
is
differentials.
is
can
an oc^
sk
= 2^ +
shown
1.
in Fig.
= y + 2z, and z = aX -1
a possible ratio from the k Hence,
x
where y
*s
differentials will
where
For consider the arrangement III.2.C.10.
3.
differentials
show that k to sk
and
= (20 +
y)X
(DI.2.C.4)
the clutch function for the
It is clear that if
y
clutch.
a can take on the values from
Fig,
m.2.D.l
was
DIFFERENTIALS
16
movable.
In the present section
similar triangle in multiplier
pivot
is fixed.
we
AND MULTIPLIERS
central
In Fig. III.2.D.1 the lines g stand
grooves and indicate that the elements contained between them (or the parts of elements)
for
can only move
parallel
to
itself.
There
describe a
which the
The output
another kind of multiplier which,
is
while only approximate and limited in range, has
been frequently used because of positive
action.
referred
to
This the
as
what
is
"links
is
its
simple
frequently
The
multiplier."
can be understood from Fig. III.2.D.3, principle where, however, the method for putting in the x
x PIVOT
input
z are rotations.
output
The y input and
not indicated.
is
strained to
The
move parallel to
a translation.
The
crosspiece
itself.
crosspiece
is
the
con
The x input is
causes the two
to have a common side c. For common side we have c = sin z, c = x(sin y), smz = x(smy), which indicates approxi
triangles pictured this
or
mately z
Fig.
= xy.
HL2.D.2
element and one input element are essentially similar but at right angles to each other. The other input element has a pivot the fixed pivot with the elements.
It
arm which
common pin
of
all
links
three
can readily be seen that the two
inputs and the output are related as in the triangle shown in Fig. III.2.D.2. By similar triangle
we
see that
i\y
= xjl
or z
= xy.
Fig. IH.2.D.4
The x input is usually entered by means of a screw or a groove cam. In Fig. III.2.D.4, for example, the groove on the
cam
positions
pivot along the y bar, according to the relative
the bar
the*
amount of
motion between cam and bar. Hence, if turned an amount y, we must turn the
is
cam an amount y
+ x.
This can be accomplished
by means of a differential which adds the two inputs x and y. For a screw cam the situation is similar.
The Fig. IH.2.D.3
differs
device utilizing a groove or screw
from the ordinary
cam
links multiplier in the
RESISTANCE
IIL2.E.
way the crosspiece or link is parallel to
itself.
AND MECHANICAL COMPONENTS
constrained to
Usually this
is
move
accomplished by
using another pivot arm similar to the output arm. The result is a parallelogram as indicated in
c(smA)
= <2(sin
C)
respectively.
17
These angles and
A are added. The actual output is #A and
A
is
+ B + C),
increased until this quantity has the
value 45.
Fig. III.2.D.5.
Fig. IH.2.D.5
Fig.ffl.2.DJ
=
The equation sinz
:csin;>
may
itself
be
very useful, especially in connection with the law
of sines in trigonometry. For the arrangement with the link in a groove, which was discussed previously,
variable
we may
arm
take the z
also as a
z)
(III.2.D.1)
on the
basis of
Eq.
Let the symbol shown in Fig.
given.
III.2.D.6 denote a links this
is
achieved by the measurement of a
One convenient way in which
multiplier
which
accuracy
is
obtainable by null measurement
methods. For example, temperature
III.2D.1 for solving a triangle in which three sides are
quantity
physical quantity. to
= x(sin y)
We can construct a device
Components
In a continuous computer a mathematical
measure a displacement or a rotation is by means of an associated resistance. Good
and get the equation
n>(sin
IH.2.E. Resistance and Mechanical
realizes
effects will
frequently cancel since only relative values are required.
In
this section
we
discuss
methods for
representing mathematical operations by dis
placements or rotations which are measured by
equation.
associated resistances. It is
possible to base multiplication of
positive quantities
principle (see Section III.5.I).
the
two
on the Wheatstone bridge If the current in
galvanometer in Fig. III.2.E.1
is 0,
then
Fig,m,2.D.6 2
Suppose three sides, a, b, c, of a triangle are of these so that angle given. Let a be the largest
A
the largest.
is
We now
acute.
Consequently,
B
and
C
are
use two links multipliers as in
Fig.
III.2.D.7.
In
this
device,
starting, say,
two
links
and
C
with
we continue
A
multipliers
for
which
= 0. will
to feed in A,
The outputs of
the
be the acute angles
= a(sin B) &(sin A)
B
and
(HL2.E.1)
G
We may consider the resistances F and Rz as G as a constant (or as a quantity by which
inputs,
we
are dividing), and
^
as ihe output.
multiply,
we have F and jR 2
rheostats
and then vary
galvanometer, Since
is
R
in the
until
i,
To
form of linear as read
on the
0.
F and R% are linear rheostats, the inputs
can be considered as the rotations which position
DIFFERENTIALS AND MULTIPLIERS
18
the contacts,
a rotation
The output can also be obtained as
servo
a linear rheostat turned by a
there
if jRx is
certain disadvantages in that
somewhat a delayed response.
The device in Fig.
servo motor.
A
motor has is
"servo motor" is
an
electrical (or hydraulic)
motor which can be controlled
to turn in either
take advantage of the use of a servo
III.2.E.2 does not, of course,
all
the possibilities inherent in
motor and a Wheatstone bridge
to force the equality of
two
There
resistances.
are other uses for this combination.
One
other obvious application
is
obtaining the
sum of a number of input
quantities.
for example, that
n are inputs,
sum
s
is
f
.
1}
.
.
,
We
desired.
t
Suppose,
and the
wish to represent the
t by resistances. Resistances are and consequently if certain of the f s may assume negative values, it will be necessary
quantities
positive,
to introduce offsets.
we
Thus, for each quantity introduce a quantity a t such that t i a{
+
always positive. If
potentiometer ri
on the
i
represented, then,
by the
we can mount a
linear
t { is
rotation of a shaft,
t
is
shaft so that the resistance
between one end and the contact
is
equal to
Fig. HI.2.E.1
direction
on a
depending
current
shown
i
For the
signal.
signal
would be the
in the galvanometer.
In the bridge
Wheatstone bridge, the
in Fig. III.2.E.1, for example, let us
G
replace the galvanometer (see Fig. III.2.E.2).
Then
R AB
by a resistance
the voltage drop
has the same sign as i. If i is positive, R: is too small, and, hence, should be increased. If the voltage,
AB,
relay circuit
is
increase J?x .
Rl
is
positive,
a voltage amplifier and a
win cause the motor to turn so
On the
too large.
other hand,
if
i
is
as to
negative,
Since the voltage drop
AB
is
negative, the relay circuit will cause the motor to
turn in the opposite direction and reduce jR lt Notice that the inputs and the output of the device illustrated in Fig. III.2.E.2 are rotations.
There
is another advantage in this device which however, not immediately obvious. The output comes from a motor in such a way that
Fig, IH.2.E.2
is,
the load
on
most devices
the inputs
is
precisely uniform.
In
for multiplication of a mechanical
t
t
+a
series,
t
.
we
If these resistances are connected in
obtain a resistance with value
load on one input depends on the value of the other input and the output. This is not sort, the
the case
other
when a
hand,
servo motor
is
used.
On
the
one must admit that using a
If this resistance is used as the R% in the Wheatstone bridge arrangement in Fig. III.2.E.2,
RESISTANCE AND MECHANICAL COMPONENTS
III.2.E.
then at equilibrium the resistance R will have a a n and consequently the value s the rotation of the motor shaft can be used as a
+
representation
+
.
of the
,
,
+
sum
19
The corresponding logarithmic equation
,
= log K! +
log z
+
s (see Fig. III.2.E.3).
We
is
x2
log
logx 3 -log); 1 -log); 2 suppose that xl9 * 2 yl9 and
will
,
available
as
shaft
logarithmic potentiometers
we
j>
are
2
By means of
rotations.
will obtain resist
ances associated with the logarithms of these
Again, the fact that our resistances
quantities.
must always be positive is taken into account. For each of the quantities xl9 x2 x^y l} and j2 ,
we determine
may vary between between
and
.01
For example, x1
the range of size.
and 10 2 while y may vary ratio of the lower and
.1
,
The
10.
upper bounds will determine the type of potenti ometer used. In order to use positive resistances
we must
also determine the quantity analogous
to the a t of our discussion dealing with the
of
representation log x1 since
for
y
jx
-log;/
10,
<
We
.
sum.
the
%=
and
1
>
Since
^
x^
.1,
be used. Similarly, 1 1, we can use c4
1 will >
=
endeavor to represent,
will then
instead of Eq.III.lE.4, log z
+
+
fli
+
(fl a
+
.
.
+
5
= (% + log *i)
log x2)
+ (0
8
(III.2.E.5)
Now <2
X
let
us consider the representation of
+ log xv In the above example, ^ = % + Iog x or log (lOxJ. Now 10
hence, log
pose that
Let us
now
consider a
method for obtaining a
z
simplicity,
which
m=
where OQ
=
n
= 3,
10
Iog10
s ,
OQ/
We
the largest available rotation.
is
x
shaft
by a
that a rotation
suitable gear
^
arrangement so
of the x shaft produces a
rotation of the potentiometer shaft such that
10
3
-l
=
I
The
resistance then between
i.e.,
one terminal and the contact z =
is
connect the shaft of such a potentiometer with
will confine ourselves to the case 2,
,
that if the contact is rotated potentiometers such an angular amount a, then the resistance between
the
in
and 10 5 and,
V
The by means of logarithmic potentiometers. For be must and always positive. 7, quantities ^
we
1
10^ varies between and 3. We sup we have available logarithmic wound
one terminal and the contact
ratio of products of input quantities,
10^
lt
between the values
varies in size
Fig. ni.2,E.3
and,
1,
hence,
(III.2.E.3) 1
+ log
Xj,
;
is
log
I
1
or
* 2 and x3 are treated in the same way.
AND MULTIPLIERS
DIFFERENTIALS
20
The procedure for the quantity y 1 is the same with one exception. We wish to represent the
- log ft
1
quantity
by a
We
resistance.
have,
of course,
This by squares, rather than by logarithms. particularly linear
when
true
combination of variable inputs
y=2^ +
(III.2.E.6)
for
varies in size
quantity
between
1
and 10 3
There are two
from
and correspondingly
input which
its logarithm varies use here also a logarithmic potenti
may
is
ways of
the square of an
be either a displacement or a
rotation.
One method
3 ometer which yields a resistance log 10
simple
relatively
which obtaining a resistance
one terminal and the contact.
(IH.2.E.9)
,
yi
We
b
i=l
jy
to 3.
as,
example,
-
= log The
is
desired to obtain a
it is
involves
K-wound
a
between ometer card (see Fig.
a
III.2.E.4).
potenti
Let us suppose,
We connect the y
shaft with the potentiometer so that contact shift
10
rotation,
3 ,
a
to
corresponds
.
ID
2
However, we use the resistance between the contact to obtain remaining terminal and the log ft. This resistance
1
is
(III.2.E.7)
The a2
+
a5
log *2
log ft are
will
+
a3
,
log
*3
4
,
now connected
be used as the resistance
-
+ log x
ax
resistances corresponding to
l9
log ft,
and
in series.
This
Rz
in the
Wheat-
stone bridge arrangement of Fig. III.2.E.2. If a g we will endeavor to obtain flj
a
=
+
.
.
.
+
,
Fig. IDL2.E.4
a resistance for simplicity in our explanation, that our contact is
(III.2.E.8)
The
size
range of the quantity
z, is
presumably
known, and so also is the size range of the which is never less than 1. Corre quantity
zW
sponding to the maximum size of a logarithmic potentiometer which
z!0
a/
representing log (z!0
).
a
is
is
The
a right triangles, with
K
is
with a leg of one along an extension of the leg of the other. The common line of these two legs is the line of motion of the contact,
and the wire
from the center point is proportional to the is proportional to length of wire which in turn
drives
connected to the
we will have Eq.
III.2.E.5.
The servo motor shaft then produces a rotation z equal to the product.
Frequently
it is
is
we choose
the area of the
K between the center point
the line
on
contact
edge at the point of contact.
desirable to use multiplication
and
the triangle perpendicular to the
at the apex one physical reasons,
then,
of
apex and
capable of
contact shaft of this potentiometer so that
For equilibrium,
made up
common
wound perpendicular to this edge. The resistance
The servo motor
a z shaft which, in turn,
to be displaced linearly.
two
the windings with a solid bar.
For
must replace
If the beginning
of the winding has a proper resistance between it and the center point, this does not introduce any error in
any other portion of the
scale.
Let
Ax
RESISTANCE AND MECHANICAL COMPONENTS
III.2.E,
be the length of one half of this solid contact Then for a contact in a neighborhood of
region.
the zero the absolute error in the resistance
(Ax)
By
is
.
displacing the contact in one direction by
an amount a and the direction
In general, one can hope that the error due to cards will not occur.
the center piece in the
K
In any case, a negligible error of the amount Ax 2 relative to the other involved can be quantities insured by proper design.
K winding in the opposite
by an amount
resistance with value (a
we can
x,
+ x)
2
obtain a
1
Similarly, if both
.
are displaced in the same direction,
-
21
nAA/VWH
we can obtain
2 a resistance (a x) This can be used to obtain a linear combination .
by means of a Wheatstone III.2.E.9
For Eq.
bridge.
can be written in the form
Fig.
There is
HL2.E.6
another practical way of obtaining
still
a square by means of resistances. This involves only a linearly wound potentiometer and, hence, does not have any of the
and we can obtain the value of y from the illustrated in
circuit
Fig. III.2.E.5.
associated
difficulties
with the center piece. Let us suppose we have a potentiometer whose ends are joined (see Fig. Let x denote the displacement of the Then the resist
IH.2.E.6).
contact from the center point.
common ends
ance between the contacts and the consists of
value R(\
two
- x),
resistances in parallel.
the total has the value R(l
or in suitable units
As 1
before, t
written in
Fig.
?"
m,2,E.5
To
- x2
.
obtain
and Eq. the form
which can be G
1
we can
- (a - xy,
One has
+ x\ and hence - *2)/2 = RQ - R^x\
the other R(\
1
(a t
III.2.E.9
+ x^ 2
and
can then be
realized as before.
linear combination represent, then, the
we need In
linear potentiometers.
Two
potenti
a^
The potentiometer P
is
the midpoint, the resistance of
R
2y and in the other
and
linear,
sponds to the displacement of
R+
its
it
if
y
corre
contact from
in one circuit
2y.
One
from our equation that for equality, F = G, we must have Eq. III.2.E.3.
is
sees then i.e.,
for
The two ometers are used for each product contacts are ganged on a common shaft so that xf corresponds to the amount of rotation of this the contact is at and x{ shaft. For a t
=
=
the midpoint of the resistances for each potenti
ometer. The quantity a { is represented by rotating one potentiometer in one direction and the other
AND MULTIPLIERS
DIFFERENTIALS
22
potentiometer a like amount in the opposite direction.
If,
the ganged contacts
then,
are
rotated an
amount x one potentiometer will have
resistance
l(a + x^, the other
{
i
1
(a {
x^
may be obtained by means of a servo and a linear potentiometer (see Fig. IIL2.E.8). The purpose of the servo is to place the contact so
voltage
that the displacement
(see Fig. III.2.E.7).
from the center of the
scale
scheme can be readily extended to
This
represent a
The
rotation or a displacement proportional to a
number of linear combinations,
variable
xt
then associated with six
is
contacts, all ganged
each a it b it and c {
ment of a
The
is
say,
on a
common
shaft while
represented by the displace
pair of potentiometers.
T
pads described in Section III.6.B are designed so that they can be used in tandem to
number of
represent the product of any
shaft
rotations.
HL2.E.8
Fig.
proportional to the input voltage. By using the voltage across the potentiometer as another is
a quotient can be obtained. The center tap on the potentiometer is used to permit the signs of the various quantities to be free. variable,
IH.2.F. Division
Since
division
multiplication
the inverse
is
we can
operation to use the servo motor idea
and any mechanical multiplier to obtain division. For instance, suppose we want to obtain z
and have will
= xjy
(III.2.F.1)
available a mechanical
multiplier which
multiply one shaft rotation by another shaft
Consider Fig.
rotation.
entered into the
III.2.F.1.
The input;; is
and the output z is produced by a servo motor whose shaft rotation also enters the multiplier. The output of the multiplier
Fig.m.2.E.7
x
differential,
multiplier,
is compared with the input x in the and the difference between the two
appears on a shaft.
In
the
foregoing
we have not
discussed
multiplying in the case where both inputs are variable and electrical in nature. Of course, one
way
in
which
this
can be done
is
by converting one electrical input into a geometrical one by a servo arrangement and then using one of the
methods
described
above.
For example,
a
rotates in
one
When
direction,
an
this difference shaft
electrical
made which produces a rotation of motor, and when the difference shaft the
opposite direction, a contact
is
contact
is
the servo rotates in
made which
produces a rotation of the servo motor in the opposite direction. tions the
irregular
However, in many applica motion which results from this
HL2.F. DIVISION
arrangement
more
is
and somewhat
objectionable,
elaborate arrangements are introduced to
allow this difference x
motor
x
For example, the
z.
23
motor which positions the
rotation of the servo
contact
on
^
is
given by
to control the servo
difference shaft
may
z
= %&
(IIL2.F.2)
y
The discussion associated with Eq. indicates
how
III.2.E.2
a Wheatstone bridge can be used
with logarithmic potentiometers to obtain an even more general result.
Another approach to the problem of division
MULTIPLIER
based on the inverse function,
is
the range of the variable
way
as not to
x
is
i.e.,
y
=-
it
is
frequently
methods of representing
this
function so that the problem of division that
referred to
If
limited in such a
contain zero,
possible to obtain
.
x
is
Function
of multiplication.
cams can be utilized for this purpose. Frequently the reciprocal
Fig. HI.2.F.1
drive
the
equally above voltage, or
on a potentiometer. The on this potentiometer may be
contact
terminal voltages
and below a
ground
certain reference
level (see Fig.
IIL2.K2). The
center of the potentiometer then corresponds to the zero position of the difference shaft.
Any
deviation from this central position will produce
obtained by a simplification of
i.e., a process by which in Eq. III.2.F.2 one utilizes constant inputs Xi and x2 An additional method for obtaining .
the reciprocal in voltage
x
is
form from a
shaft input
given in Section III.9.A.
For a limited range of x the reciprocal function can be obtained by simple approximations. For example, is
x-x
is
the Wheatstone bridge method,
if
a range centered around the value b
being dealt with,
l
CONTROL SIGNAL
pA/VWVS -E
+E or Fig. ffl.2.F.2
a voltage difference between the contact and the
can be used. In addition to the above procedures, there are
reference voltage proportional to the difference
x
x
.
This voltage difference can be used as a
control signal for the servo motor.
A Wheatstone bridge can be used with a servo motor
to obtain simultaneously the result of a
multiplication and
the resistances
a division. In Fig. IIL2.E.1, if
F., jR 2)
whose contacts
x1} xZ) and y
an(^
^
are potentiometers
are positioned
by input variables
respectively,
then the output
a
number of
specific
mechanical devices which
can be used to obtain reciprocals. The simplest of these is the invertor based on the relationship
between the tangent and cotangent of an angle. Consider Fig. III.2.F.3. This device consists of
two rods located in
parallel grooves
with two mutually perpendicular is
pivoted
at a
and an arm
slots.
The arm
point midway between the two
DIFFERENTIALS
24
AND MULTIPLIERS
Fig. IH.2.F.3
Fig. HI.2.F.4
grooves, and the distance between the grooves
taken to be two is
inserted in
Let a and
f}
is
groove, then, again,
oc
+
= 90
]8
Each rod has a pin which one of the slots on the pivoted arm.
cotangents.
denote the angles the lines through make with the line through
A linkage arrangement which willrealize the
units.
the middle of the slots the pivot which
is
perpendicular to the grooves.
+ p = 90, and^ = tan = I/tan a = A similar effect can be obtained by putting
argument can be applied,
and the above time to the
this
(see
Fig.
relationship* -y
can also be
set
up.
=
III.2.F.5)
constant
Consider a linkage with
Then a 1/x
the two rods in the same groove as shown in If a and ft are the angles between the lines through the middle of the slots and this
Fig. IIL2.F.4.
.
GAB and OBD are congruent since corresponding sides are equal and hence OB is the angle bisector of angle AOD. Since OC is The triangles
also the angle bisector of
AOD,
it
follows that
IIL2.F.
OB
lies
intersect
along OC.
OC at P.
If
we draw AD,
Since triangle
are congruent, angle
APO
it
25
DIVISION
will
AOP and POD
will be
(OB
a right angle.
b*
Subtracting
= h* + (BP) we
(III.2.F.5)
2
obtain
= (OB + J3?) 2 ~ (JBP) = (OB + 2BP)OB = OC-OB (III.2.F.6) Thus if we let OB = x and OC = y, we have a
z
2
-b*
the desired relationship. for reciprocals,
is
fixed
If this
and
is
B
is
used simply constrained
move along a line through 0. C will then move along this same line. However, if alone to
is the pivoted, Eq. IIL2.F.6 shows that C image of B relative to the inversion of B in the 6 2) 1/2 B will of radius (a 2 circle around is
-
Fig,
Also
BP
= PC,
parallelogram
since
ABCD
the
diagonals
bisect each other.
.
b within this circle. cover the ring OB Because of the properties of this inversion
m.2.F.5
>a
of the
Thus,
relative to circles
used for
many
and
straight lines, this
purposes.
can be
Chapter 3
CAMS AND GEARS
IH.3.A.
Cam Theory
The purpose of
point of contact.
chapter is to discuss of one variable by of functions representation shaft rotations. Ostensibly we want a device this
with two shafts such that when the turned an amount
first
shaft
is
the second shaft turns an
jc,
Then they must be cotangent
We will make certain
at this point.
These statements are
all
intuitively,
easily
comprehended
method of
x lamina, the y lamina
Another
is
the
a variety of
are, also,
method of cams. There
electrical devices for this
purpose. For the present cams of a special type only will
We suppose that the x and y mentioned above are parallel, and the x
be considered.
shafts
shaft drives the
Each of these
y
shaft
pieces
is
by certain metal
pieces.
planar in the sense that
it
has two plane faces perpendicular to the axis of the shaft connected to it, and the edges of the plane faces and the remaining surface consist of
of the shaft.
lines parallel to the axis
Mathe
matically such a solid is termed a "right cylinder."
The boundary of a plane "directrix." The figures in called
are called
bases
The
"bases."
is
lines
"elements."
called
face
on
The
termed the
the outer surface
distance between
When
"altitude."
is
the plane faces are
the altitude
small relative to the other dimensions,
customary
and the
to refer to this solid as a
altitude as the
"thickness"
it
in the
we continue will
the rotation of the
have to move. The
points of contact along the edge of the laminae
change, but the two directrices will remain cotangent at their point of contact. This motion involving the point of contact is
will
said to be rolling if there
no
is
relative
motion
between the laminae at the points of contact. For rolling motion the distance along the first directrix
between two points of contact Pl and ?/ on the second
equals the corresponding distance directrix.
there
may
In the more general case, however, be a relative motion between the
laminae at the point of contact in a direction tangential to the directrices. This is termed a sliding motion.
THEOREM
III.3.A.1.
Let
Q
C2
and
be two
laminae free to rotate around axes perpendicular to the common plane of their bases. They can rotate in contact provided that for each position
is
of one there
"lamina,"
of the lamina.
so that corresponding plane faces are
same plane. This
If after contact
is
Presumably there are two such laminae of equal thickness, one for each shaft. They are
mounted
but their
precise proof requires considerable mathematical care.
III.B.h).
possible,
give references for proof.
amount y =f(x). There are a number of methods for accomplishing this. One is the linkages (see also A. Svoboda, Ref.
geometrical
when
statements without proof, and,
is
a point
position of the
other
P
on
directrix
its
and a
such that the second
is tangent to the first at P. Let 0! be the angle of rotation of the lamina from some fixed position, and let
directrix
the angle of rotation of the second lamina
first 2
be
from a
possible if the axes are
reference position to the position described in
not too near each other. If the axes are not too
Theorem III.3.A.1, where it is cotangent with the
far apart, the
x
lamina
is
may
be rotated until
it
touches the y lamina. Ordinarily contact between the laminae would be just along an element.
Suppose both
directrices
have tangents
at a
first
lamina at ?. This determines
2
as a function
offli.
THEOREM
III.3.A.2.
Let
Q
and
laminae as in Theorem III.3.A.1, and
C2 let
be two
O x and
III.3.B.
refer
FUNCTION CAMS
respectively to the points
where the axes
of rotation meet the base plane.
Let n be the
2
common normal to let
A
the two directrices at P, and
be the intersection of this
common normal
will
to each other, but this corre
slip relative
sponds to a difference in the tangential com ponents of the motion of the curves at P.) Thus,
Aft is determined approximately by the equation
with Ofifr
Then
27
O^sin ft)Aft
^2 = M
Let
(ni.3.A.l)
= $PA0
=
2 P(sin
ft)A0 2
(HL3.A.2)
Then $.PAO Z =180-9, and
lt
by the law of sines we obtain
O aP(sin ft)
Proof: Consider the relation between the two directrices in the base plane (see Fig. III.3.A.1).
2
- (VKsin 9)
P(sm ft) =
2 4(sin
(ffl.3.
(180
- 9)) =
Z
A3)
A sin
(III.3.A.4)
Substituting III.3.A.5,
we
Eqs.
IIL3.A.3
and 4
in
Eq.
obtain
Dividing Eq. A.5 by sin
we
obtain
<JP,
(IE.3.A.6)
Aft"^ In the
limit, therefore,
we have
J = ~^
Fig. IIL3.A.1
IH.3.B. Function
(HI.3.A.7)
Cams
p
Normally, in obtaining a pair of cams, the is to have be a prescribed function of 2 a
object /(ft).
We shall see that one must restrict oneself
to the case
the
Fig. IH.3.A.2
mined,
Draw
AP is
the line AP, and the lines
the
common normal
Consider the triangle
and ft
P. 2
directrices.
Let ft = ^APO
0-fO^
P
>
0.
Since, as
= A.
first
THEOREM
IH.3.B.1.
Suppose the two laminae
l
lamina be rotated an amount Aft,
A0 2 be
be displaced an amount OjPAft (approximately),
and the component normal be approximately 0-f the point
P2
fixed
normal component
to the directrix will
sin ft Aft.
on the second will
be
2
the second directrix rotates an
At P
the normal
must be the same
Similarly, for directrix the
P sin ftA0 2 when amount
shall see,
described in Theorems III.3.A.1 and 2 rotate so
the corresponding rotation of the second lamina. The point Pl fixed on the first directrix, which corresponds to P initially, will let
we
we begin by considering the case in which
- $APOi (see Fig. IIL3.A.2).
Let the
and
OJ
of the two
and
where/
shapes of the cams are not uniquely deter
Aft.
component of the motion
for each of the curves if they
are to remain in contact, (In general, the curves
Fig. HI.3.B,1
CAMS AND GEARS
28
=A
that
?
lies
on
their point of contact
i.e.,
9
their line of centers.
rolling contact,
i.e.,
They
always
are then in
the point of contact?
along each directrix at the same
rate.
moves
(See Fig.
cotangent at A.
Now let ^ be the angle between
the tangent to the
first directrix
vector (Fig. III.3,B.3). this is
Let
moves along
sl
the
be the distance that first directrix
0! of the first lamina,
and
P
0^ One
(III.3.B.7)
during a rotation
be the distance
let $2
along the second directrix. Let a be the angle the common tangent makes with :
=
tan
=A
= A moves
P
calculus
such that
III.3.B.1.)
Proof;
and the radius
By elementary
can show that
Let 9? 2 be the angle between the tangent to the second directrix and the radius vector (Fig. III.3.B.4).
By
we
calculus,
tan
see that
=
Q?9
(III.3.B.8)
2
(IIL3.B.2)
(Notice
that
we have
-
negative flC/
and 2 and Theorem
III.2.B.1
Hence, by Eqs.
III.3.B.2.)
III.3.A.2
For the two
we must have
d6%
d8}
^=
q?2)
directrices to
or
\0 2 /t
d6i
Pi
= esc aO^ =
1 -
(III. 3. B. 3)
d6 1 This establishes the theorem.
We now which,
determine the shapes of two cams
when
rotating in rolling contact along
their line of centers, will satisfy the relation 2
/(0i).
Consider the second directrix as
given by a curve in polar coordinates p 2
where
jF is to
and two
2
and the distance
directrices will
and only
have
2
= ^(0^
Now
0^ = D. P
m.3 J.2
suppose we =/(0i) between 0!
be determined.
have the desired relation
Fig.
=A
in
Then
the
common
if
if
Pl
+ P2 = D
(HI.3.B.4)
where
(IH.3.B.5)
Fig. IDL3.B.3
Thus, Pl
which
= D - F(f(6J)
(HL3.B.6)
a polar coordinate equation for the first cam. If we wish to graph plt we must let 61 be increasing in a counterclockwise is
shape of the
direction,
and we must
let
2
be increasing in a
clockwise direction. (See Fig. III.3.B.2.)
However, we must also have the two directrices
in
Fig.
2
Fig. EGL3.B.4
be cotangent
FUNCTION CAMS
III.3.B.
But by the
definition of />
and by Eq.
2
III.3.B.6
Now we
29
recall that ft, the angle
0^
tangent and
is
between the
given by
Df
%
which
(HI.3.B.10)
d01
d@i
d^i
tanft
= A = Mn = AL/l)
f
Df
dpi
yields
(IIL3.B.17)
(IIL3.B.U) If tan ft
Eq. III.3.B.11 then determines F(Q parametrically, i.e., one can solve for fand 2 in terms of .
X
is
is
large,
^
-
close to a
is
.
is
no
On the other hand,
of friction between the two
F
must be
less
than
Eq, III.3.B.12
,
But then Eq. III.3.B.13 determines 61 as a function of 2 Hence Eq. B,
implies that/
^J
0.
>
F as
a function of
2
and
consequently the shape of the second cam. The shape of the first cam is determined by Eq.
and
if
with ft
the coefficient
is
and ft
small,
is
a right angle, it is obvious that the driving cam must exert a strong normal force on the driven cam in order to transmit a torque from close to
one shaft to the other.
This can readily be
,
12 and 13 determine
III.3.B.6
difficulty
(III.3.B.12)
(HL3.B.13) Since
right angle.
a strong factional contact between the
two laminae, then there close to
=
jf
If there
seen by considering the special case in which
/(0J
=k
directrices line
X
0j.
Here the two laminae have
which are
of centers.
12, i.e.
circles in
contact on their
A torque can be transmitted from
one lamina to the other only by friction. In these cases, where would tend to be close
^
(III.3.B.14)
a right angle,
to
Eq. B, 12, 1 3, and 14 will insure that the directrices can rotate so as to be at P which is on cotangent
their line of centers. Furthermore,
that this motion there
no
is
is
a rolling one,
we have
i.e.,
we define F by Eq. 2)
III.3.B.12
and
13.
^
Then,
>
with/
way
that
2
if />
2
cams can rotate
0.
=/(flj).
From Eq. B.12 we dF
derive
since
^ ;= D
%i and
Let
#2 be the two directrices obtained in the
above manner which would give the required rolling contact at the point A. We
introduce another curve rolling contact
of
#
is
with
$"
which
^ and #
2
is
at A.
relatively arbitrary, and,
to every such choice, directrices
to
move
in
The choice
corresponding
2l and ^
2
are
need only suppose that ? can move in rolling contact with at A, which is a obtained.
We
^
Df"
(HI.3.B.15)
and
$
motion in
problem of obtaining cams with;; =/(0) >
^
strong frictional motion can be obtained.
Except for one practical difficulty this would represent a solution of the
Other solutions to the problem of finding and 2 which do not require such
0,
rolling contact along their line of centers in
such a
are in rolling contact along the line of centers.
directrices
and Pl = D - F(f(9j) - / /(! +/ )
are used to shape two cams, these in
the problem of finding the shapes of the laminae which rotate according to the rule 2 =/(0j) and
slip.
Thus, given any function/^) with/
= f(0
seen
such that
would be preferable to have some solution other than those given above to it
point along the line of centers. For suppose rolls
F,
remains always at A.
Df
contact
(DUJ.16) roll
%
on #! in such a way that the point of contact it
also rolls
From the nature on
#2
,
i.e., all
together with contact at A.
of rolling
three curves
CAMS AND GEARS
30
Suppose we consider just the first lamina Ci and its theoretically associated curve #lt We
# at the point A on ^ corre = and, of course, cotangent with here. We then roll on ^. Let P be any point whose position relative to # is fixed.
place the curve sponding to 0j
@
The path l of P is a possible directrix for Q. The corresponding directrix for #2 is obtained
#
by placing
at the point
to
2
=/(0) and
of
P
obtained in
directrix for
#
2
.
rolling this
A on #2 corresponding
^ on ^
way
is
2
.
The path^ 2
the corresponding
(See Fig. III.3.B.5.)
and #2
=/(0 1), #!
a
%
will roll in contact
on
^ and P on ^
since
its
^
2
will continue to coincide
2
fixed relative to
is
position
@! and
more,
will roll in contact at A, with both of these, and P
#
Further
.
P
both nave a normal at
will
Al
and, hence, will remain cotangent. Thus, &-L and z are also two possible shapes for
through
@
#! and #2 which will rotate in the proper relation However, by properly placing P, we 2 =f(6j). can insure that the angle between the common
^
^
and 2 anci 0-f is sma11 This tangent to means that torques transmitted from one shaft to the other will correspond to forces applied in a normal direction to
The
-
^.
generality of the above indicates that one
has a wide choice for the curve *$
one can choose a
.
For example,
one can choose a
circle or
straight line.
^ and ^
In the case of gears,
These
circles are called pitch circles.
curve
#
may be
are circles.
2
In gears the
another circle called the
"tooth
circle,"
and the actual lamina has a number of
profiles
B, which
These
successively engage.
in the case of cycloidal gears are obtained
on
rolling the tooth circle
indicated above.
E.
the pitch circles as
(See also
Manufacturing Co., Ref.
Buckingham, Ref.
2 by
Brown
III.3.b,
&
Sharpe
pp.
54-60;
pp. 24-29:
III.3.C,
E.
Buckingham, Ref. III.3.d, pp. 16-26; R. Willis, Ref. III.3.1, pp. 84-87.) However, one may introduce teeth surfaces for a
#
are not circles.
2
In
fact,
cam even this
if
^ and
can be done
whenever one wishes to change the direction of the surfaces of contact. Fig. IH.3.B.5
If
then
In proving these statements, one notices that the normal to
^
at
P will
go through the point of contact between 9\ and at each instance.
#
(See
Lemma
Now we
and Corollary III.3.B.1.) and 2 so that initially the
III.3.B.1
place
^
^
point P on the two curves coincides. that
we have
their
placed the two curves
placed
<&
Now
if
2
in
=
P
is
^
corresponding to P.
pp.
pp. 87-89; Ref.
III.3.J,
LEMMA
27-30;
,
2
(See also E. Buckingham,
Ref. III.3.C, pp. 58-59; III.3,d,
# #
a tangent line and P is a point on and 2 will be the involutes of
^ and ^
E.
R.
Buckingham, Ref.
Willis,
W. A. Wilson and
Ref. J.
I.
III.3.i,
Tracey,
pp. 198-99.)
III.3.B.1.
If
one curve
rolls
other, then the instantaneous center of
around the point of contact. Proof: We assume that # 2
is
fixed
^
on the
motion
and
^
is
rolls
are cotangent at P.
without slipping, i.e., and #2 are in sliding contact. Let the origin be the point of
we
contact and the
to each of these at
@i and ^2
^ and #
and corresponding to 6 1 cotangent with them at A. Since the
position
normal
This means
#
rotate the
goes through A,
two laminae so that
on
it
x
axis the
common tangent.
Let
INVOLUTE GEARS AND WRAPAROUNDS
IIL3.C.
the equations of the curves be, respectively
yl
=
2
x
2
+ a^ +
.
.
.
have a similar motion, only upwards instead of downwards. One readily sees that the rotation
(UI.3.B.18)
which describes the motion of the curve must have
Now
if the
point
from
(0, 0)
from
(0, 0)
j/ 2)
,
#
on
to be in contact
is
then the arc length
2,
must equal the arc length to (x2 j 2 ). This follows from our
to (x^y^) ,
definition
of rolling
III.3.B.1).
We
somewhere between the
center
its
line
segments corresponding to these two displace
Ob ^J on ^
with the point (x 2
31
contact
Theorem
(see
have, therefore,
ments, which are approximately parallel but in opposite directions.
This matter
true for
is
how we
process,
every motion of this sort,
small.
no
Hence, applying the limiting
obtain that the instantaneous center
of motion of one lamina on another rolling the point of contact.
COROLLARY
IIL3.B.1,
If
S
is
is
at
the path of any
point attached to the rolling curve, then the
(HL3.B.2Q)
By
the binomial
expansion
+ y *fJ = l + W* +
...
l
Substituting the value for
at
P
passes through the point of
contact of the curves.
*
(l
normal to
y
9
(IIL3.B.21)
Proof:
P moves as if it were moving on a circle
with center at the point of contact of the two
we
obtain
curves.
fai
IH.3.C. Involute Gears and Wraparounds
( /o
Involute shapes are frequently used for gears. this case
However, in (III.3.B.22)
reasoning, for the right side of
By analogous Eq. III.3.B.19
we
<
radii rx
(IIL3.B.23)
which can be regarded
ratio.
we
2
a^x
+
.
the .
.
,
(-2[%
-
in
change
in general
order, the change of 2
circles,
r2 are in
III.3.C.1.
^ and #
whose
2,
k, in
proportion rj^
Suppose that a tape
is
wrapped
and Taylor
s
obtain
while
Thus, (6 2
Fig.
and
as defining jc2 as a function
differentiation
implicit
expansion
2
obtain
Consider the two
x^ By
#a and #
which are tangent. Instead one may use as base curves circles ^l and #2 whose radii are in the appropriate
of
one need not refer the
construction of the gear teeth to circles
bfixf +
x
that
is,
of the third order
is
...).
j,
of the second
is
Now
consider
for
small values of x, the displacement of the point (
x
to i>yd
(
x
y&
whi ch
occlirs as the
curve rolls in such a way that
its
uPP er
point of contact
Fig.m,3.C.l
x This displacement changes from the origin to z .
is
nearly perpendicular
component
will
to the
be much
It is clear
x
axis since
than larger
that the point
on
its
the
its
y
x com
moving ponent. curve which was originally at the origin must
around ^\ and then around to the other along the
^
one passing from
common tangent.
If
rotated counterclockwise, the tape will
^
is
wrap
around $\ and unwind from #2 always along the
CAMS AND GEARS
32
common
tangent. for a rotation
The amount of tape passed = 2 A6 2 and thus rjAflj.
A^ is
/-
,
38-48 ; E. Buckingham, Ref.
Ref. IIL3.g; R. Willis, 1
"r*-
(III.3.C.1)
Now, if we take a point P fixed on this tape, the locus of this point on a plane which rotates with
#
x
is
an involute
#
for
1}
the normal to this involute Fig. III.3.C.2).
is
and
at
any instant
along the tape (see
Similarly the locus of
P
in a
pp. 60-1
III.S.c,
W.
E. Buckingham, Ref. III.3.d;
11
;
Steeds, Ref.
pp. 123-27.)
III.3.I,
However, the wraparound principle can itself be used to transmit a desired motion between
One can
shafts.
inextensible cable
drums and an
consider two
which wraps around one drum
and unwinds from the
The drums can be
other.
of varying diameters, and the cable
which
spiral
is
amount of rotation
61
and
in channels
Now
around the drum.
the
if
of the two drums
2
is
which the diameters
at large relative to the rate
of the drums are changing, the cable will unwind
from the drum perpendicular to the radius of This implies that the relation the drum. rl
M
l
=
r2
A0 2
essentially correct or that
is
=^ MI
III.3.C.1,
,
holds.
Thus, given
/
Eq. (fl,),
>2
we can determine the ratio of the drum diameters at
each value of 0^
A variation
of this
the case where a tape
is
is
used which winds up on one drum and unwinds from the other. Here rx and r2 are functions of r
We make the same assumption as before that unwinds
first
#2 is an involute to #2
with normal along the tape. Thus, if we have two laminae shaped like these two involutes, they
can rotate in contact
^ = OiA = H dB l
Z
A
drum
occurs.
k of the
gear teeth
is
(m .3.C.2)
&
same with
H.
r
(IH.3.C3)
= **-*
(HI-3-C.4)
Z77
"involute
this
Z7T
gear (IIL3.C.5)
shape for
the theoretical action.
simply to set up a with a of tape between the amount system larger two base circles. Otherwise the two systems are the
by the number
that slight variations hi the distance
Oflz have no effect on The effect theoretically
J.
tape multiplied
r2
This principle is used in shaping The major advantage of
tape
-
thickness
precisely as
teeth."
of the
its
change in radius from a to the value rl9 and the increase in radius r a will be equal to the
though P were moving along a tape which unwinds from one base circle and winds around the other. is
X
drum and
first
will
the motion of P, the point of contact of these two
laminae
The
the
radius,
of revolutions of the drum. Thus,
Since
at P.
the
perpendicular Suppose, then, that a large rotation
tape Fig. IH.3.C.2
plane which rotates with
to
the
same
and
is
ratio r2 Ai.
(See also
Billings, Ref. III.3.a, pp. 160-207;
Sharpe Manufacturing Co., Ref.
Brown
III.3.b, pp.
-
1
*o-where
C
}
is-
=C
(III.3.C.6)
a constant of integration. (See also III.3.f, and R. 0. Yavne,
R. A. Harrington, Ref. Ref. III.3,k.)
IIL3.D.
LOG AND SQUARE CAM MULTIPLIERS it
Log and Square Cam
ffl.3,D.
The
Multipliers
discussion of function
cams
III.3.C permits us to mechanize
of one variable
for
>
0,
For
dQ
example, we may take y = log 6 and by the use of three such cams and a differential we multiply two positive quantities. to be operated with
as y>
customary to use a pin cam.
a
spiral.
sively
As
is
a disk with
pins
mounted on it in
the disk revolves, these pins succes
push a wheel which
is
mounted on an
axle
paraUel to the disk (see Fig. IIL3.D.3). In order
may One of them is
input (see Fig.HI.3 D.I).
Fig.
Fig.
(See also
Fry, Ref. IILS.e, Part II, pp. 9-11.)
A pin cam
in Section
any function y
which
is
M.
33
IE3.D.2
m,3,D.l
There are other types of cams besides those described
in
Sections
III.2.D,
III.3.A,
and
In Section IIL2.D we have mentioned
IIL3.B.
the groove cam. A groove cam, in general, must have the angle of rotation as the input. The
output
is
the displacement of a pin which slides
along a fixed radius of a disk.
The
output, of described by the equation in polar coordinates of the groove. Sometimes it is useful
course,
is
to have variations of this in which the slot which
guides the pin is moveable or if fixed is not radial
Logarithmic multiplication does not permit a change of sign. An alternate method of multiply ing
is
that of
"quarter squares"
based on the Fig,
formula
to
"(^
permit more than one turn of the main disk,
provision must be
Notice that Eq. III.3D.1 involves addition, subtraction,
parallel
to
itself.
made to move
This motion
is
We will return to this point.
For this equation three but only two cams are used (see
wheel to revolve. Let the pin be
sented by a cam.
Fig. III.3D.2). In the usual
form of this
device
the
the output, only the rotation of the
As a
pin passes the
little
from the center of the disk
little
wheel
not to appear in
and the operation of squaring, i.e., variable which can be repre
a function of one
differentials
inJ.D3
wheel, at
little
it
wheel.
causes the
a distance p
(see Fig. HI.3.D.4).
CAMS AND GEARS
34
The
relative
motion of disk and wheel
is
In the squaring pin cam, p
as
= k&.
Hence,
though the wheel and spiral were turning in contact with the same component of motion in a direction
which
is
the disk parallel to
dd
and the
plane of the wheel at their point of contact.
We As we have mentioned, it is necessary for the wheel to move along its axle as the disk
little
turns.
The
axle has a polygonal cross section
and
by means of prongs which are mounted on a screw which turns with 0. The linear displacement then given the pinion
is
displaced along
it
by the screw is proportional to 6. This is effective in the case of squaring cams since p is essentially equal to
Presumably in the case of pin cams
6.
with a different
spiral,
some
sort of guiding
groove on the disk itself would be necessary. Electrical methods for squaring are discussed
Fig. HI.3.D.4
in Chapter 111.10. In Section III.2.E we discussed
methods for getting square value ffl.3.E.
Backlash
Backlash
when
resistances.
is
the
amount of play between
gears
the driver reverses direction.
Sometimes backlash because
it
is
deliberately introduced
makes a device insensitive to vibrations
of less than a certain
size.
For
we have two
if
moving together, with a certain momentum the driven gear, and if the driver has a high
teeth in
frequency vibration of amplitude less than one half of the backlash, then the backlash permits the driver to
move back and
forth,
touching the
driven wheel only at one point in each cycle of the vibration. Fig. IH.3.D.5
can see
and
III.3.D.5).
r,
From
its
radius" "pitch
the contact,
we
(see
Fig.
see that
^
P dt
dt
For
eliminate backlash.
gear
made
is
to
purpose, the driven
this
each other, but which are fastened
together by a spring in such a is
half of the
on either
split gear
move
way that the
acting
pressed on both sides, one
tooth of the driver
side.
driver reverses direction,
half of the split gear
it
Consequently, it
reverses the
presses on, but
it
does not
relative to the split gear.
There (III.3.D.2)
inertia
may be split into two parts, which may move
relative to
when the
or
dd
the driven gear receives
In most cases, however, effort
this if we take
a cross section of the pins and gears at the region of contact. Let 6 be the angle of rotation of the disk; ft that of the little wheel;
Of course,
number of impulses but its moment of will smooth these out considerably. a
are
also
other
arrangements
for
eliminating backlash. In the above the pressure
III.3.E,
BACKLASH which have a is
35
W. Nieman. Again we on the driven shaft but each half split gear
is
credited to C.
free to rotate
around the driven shaft.
On each
a pin which protrudes from the face of the gears, and we believe that one passes
half, there is
through a slit on the other. On the driven shaft there is a rod perpendicular to the axis. On this rod there
which
a stud which
is
slides
up and down, but
normally pressed between the two pins on the halves of the split gear by a spring (see is
Fig. III.3.E.1).
With no load the spring presses the stud onto pins, and this pressure causes the two halves
the
of the
split
gear to press against the teeth of the
driven gear. this force is
When
the driver applies a force,
transmitted by one half of the split
gear and one pin to one side of the stud. Since this force is not balanced by a hie force from the
on the opposite side of the stud, the stud presses on its axial rod. Thus friction prevents
pin
the stud
torque
is
from moving up on the rod and the transmitted by the rod to the driven
shaft.
This arrangement has the advantage that the by the spring on the gear teeth may be
pressure
relatively light.
This
is
in quite important
many
precision devices (see Fig. III.3.E.2).
3 References for Chapter
Fig. HI.3.E.1
of the spring must be great enough to stand the pressure between wheel and driver and is itself another pressure on the faces of the gear teeth. To eliminate this, it has been proposed that the be kept apart by a wedge. split gears
Consider the following patented arrangement
a.
Kinematics for Students J. Harland Billings. Applied and Mechanical Designers. New York, D. Van Nostrand Co., 1943.
b.
Brown
c.
& Sharpe Manufacturing Co. Practical Treatise on Gearing. Providence, 1941. Gears. New E.Buckingham. Analytical Mechanics of York, McGraw-Hill Book Co., 1949.
d.
E. Buckingham. Hill
Fig. IH.3.E.2
Book
Spur Gears.
Co., 1928.
New
York, McGraw-
CAMS AND GEARS
36
e.
f
.
M. Fry, "Designing computing mechanisms," Machine
Co.,
1946. Design, Aug., 1945-Feb.,
Vol. 27.
R. A. Harrington, "Generation of functions by windup Rev. Sci. Instruments, Vol. 22, no. 9
i.
mechanisms"
g.
h.
(1951), pp. 701-2. W. Steeds. Involute Gears.
j.
New
York, Longmans,
Green, and Co., 1948. A. Svoboda. Computing Mechanisms and Linkages. Ed. by H. M. James. New York, McGraw-Hill Book
k.
R.
M.I.T.
1948.
Willis.
Radiation
Principles
Laboratory
of Mechanisms.
Series,
London,
Longmans, Green, and Co., 1870. W. A. Wilson and J. I. Tracey. Analytic Geometry, alternate ed. Boston, D. C. Heath and Co., 1937. contour cams," Product "High accuracy Vol. 19, no. 8 (Aug., 1948), pp. 134-39.
R. 0. Yavne, Engineering,
Chapter 4
MECHANICAL INTEGRATORS, DIFFERENTIATORS,
m.4.A.
A
Integrators
if
variable input
time, or
it
may
is,
of course, a function of
be considered as a function of
another variable.
We
have seen in Sections
D
and
and III.3.D methods by which we could add or multiply two such functions. The III.2.B
AND AMPLIFIERS
remaining two operations which we would like to
the
principles of the integrators
and
differen
tiators are available.
The standard method
for the integration of a
based on a principle
displacement or rotation
is
which can be
by a simple consider
illustrated
ation of a sphere rotating in contact with a disk (see
Fig. III.4.A.1).
The sphere
is
mounted
consider are integration and differentiation.
For these a relatively large number of methods are
known. For the
integration of a rotation or
displacement we have a
which can be used
variable speed drive,
as a differentiator also with
the use of a suitable servo arrangement.
There
number of instruments which
are a large
are
essentially differentiators, for
example, speed ometers or tachometers (see Section IIL8.A).
A word is
of caution, which to a certain extent
also applicable to our previous discussion,
should be inserted here concerning the objectives of Sections III.4.A and B. We are dealing with the principles which
may be used to perform the
indicated operations, and our discussions are devices. In to describe still
inadequate
practical
Sections III.4.A and B, in particular,
we
are
Fig, IH.4.A.1
forced to assume that our outputs have zero or zero loads. Consequently, to com here to the point where the the discussion plete
practically
can
principles devices,
we
be
incorporated into actual must utilize the theory of amplifiers
as given in Section III.4.C
To a certain
extent this
and Chapter
III.7,
was true of the
earlier
but in the previous cases a certain amount of load could be tolerated without devices,
introducing essential errors, no load integrators, almost
In the theory of is
permitted.
Our
reason for discussing integrators and differen tiators before amplifiers is that the theory of amplifiers
is,
in general,
more readily understood
on an
axle which intersects the axis of the
rotating disk.
and the sphere the
If the contact is
between the disk
a nonslipping one, and
if
for
moment we consider the relative positions of
the center of the sphere and disk as fixed, then a rotation Aa of the disk will cause a rotation A/?
=
f of the sphere, such that rAa A/? where t is the radius of the sphere, and where r is the distance of the center of the sphere from the axis
of the disk.
A =
Let us suppose
t
=L
Then
r Aa,
For a
brief introductory discussion, let us first
the contact between the two occurs suppose that
INTEGRATORS, DIFFERENTIATORS, AMPLIFIERS
38
Let us suppose also that there is another mechanism not shown, which moves r in
the other /*Aa (see Fig. III.4.A.2). If we suppose
such a
sphere also has this motion, and one can see that the instantaneous rotation of sphere is about an
at a point.
way
that r and a are both functions of a
that contact
is
nonslipping, then a point of the
Since the motion of r is along the of rotation of the sphere, it does not contribute to the rotation of the sphere, and r dv., or is an hence, we have dft integral of
contact at a point. Actually,
the differential
that
contact with a line of no relative motion and
move with perfect slip along
areas in which the relative motion is opposing. This gives a tendency to rotate around an axis of the disk. The perpendicular to the plane actual instantaneous rotation then must be com
variable r. axis
-
r d&.
However,
the point of contact
this
requires
plane parallel to the plane of the disk. This was, of course, on the assumption of
axis in the
pounded from
this
we have an area of
instantaneous rotation and
the instantaneous rotation about an axis parallel
plane of the disk. This can be done by considering the motion of
to the
a point at the end of a radius perpendicular to both axes of rotation. It is easily seen that the actual instantaneous axis of rotation
is
in the
plane given by the other two axes of rotation.
we have
In fact, Fig,
rotation in
IH4.A.2
the radii of the disk but with perfect nonslipping contact on the circles with the same center as the disk.
seen that the instantaneous
which we are
interested
can be con
sidered as being
made up of
each of which
a rotation around one of three
is
three components,
Of
mutually perpendicular axes. true for
any rotatory motion
course, this
is
(see Fig. III.4.A.3).
Very many variations of the above have been introduced using, say, instead of a disk and a sphere, a cone and a sphere, or two spheres, or a
cone and an
ellipsoid.
However, these are
subject to the difficulty given above
more than two elements The modern
ball
all
when no
are used.
cage variable speed drive uses
essentially four elements either the difficulty
and
is
not subject to
mentioned above
or, to
a
certain extent, to the difficulty represented by the fact that the point of contact of the disk and
sphere
is
not a point.
To understand
Fig. IH.4.A.3
this device, let
rotation of a sphere which disk,
is
us consider the
placed on a rotating
and which is being shoved by an apparatus
that does not interfere with radial line of the disk.
that the contact
is
At
its
rotation, along a
first let
us suppose
at a point.
immediately apparent that the point of contact moves on the disk a vectorial amount is
the
sum of two mutually
now consider the H. Ford variable speed
The
essential elements are a disk,
perpendicular
components, one of which has the value Ar and
two
spheres in a cage, one on the disk, the other on the
first,
and a
cylinder, with axis parallel to the
face of the disk and in contact with the
It is
which
Let us drive.
uppermost
sphere (see Fig. III.4.A.4).
Let us consider such an arrangement and see to each of the three components
what happens
of the rotation.
The component due
to the
III.4.B.
rotation of the disk
is
around an
DIFFERENTIATORS
axis of the
sphere which passes through the axis of the disk, This component is easily seen to be transmitted to the
transmits the
The upper
ball,
upper
component
to the
constructed so that there are the contact points. are not
and
if
large pressures
Presumably the
on
last eifects
negligible.
Two
sphere, in turn, cylinder,
39
w^$udv + jvdu.
Thus, ever,
integrators can be used as a multiplier.
the
limits
slipping
In general,
application
how
of
this
formula to the case where one uses the formula dv du -J = U T+V d(uv)
x
,
ax
dx
.1
dx
DDL4.B. Differentiators
A
ball cage variable speed drive can be used as a differentiator by the use of a servo hook-up.
For example,
whose
in Fig. III.4.B. 1
x
we have the input x
The quantity x is matched with a quantity^ in a differential which - y can be introduced produced x y. Now, x
the
amount of rotation of the
cylinder, a that
of the disk, and
r the displacement of the point of contact from the center of the disk, then
$=
desired.
T
which
generator for a voltage
m.4,A.4
input shaft is
is
a tachometer
into Fig.
rate
is
is
like
an
electrical
when
its
rotating (see Section III.8.A).
If
the tachometer
is
is
generated
considered to be a direct-
current generator, the output voltage can be used as a servo The signal. output of the servo motor
used to position the ball cage in the variable speed drive which produces y. Clearly any is
r
The component of the rotation which is due to r is
about an
axis changing perpendicular to the previous one but parallel to the disk. The upper axis sphere receives a rotation about a
discrepancy between the rate of will
produce a signal which
will
x and
that of y
change y in such
parallel
but in the opposite direction. If the upper sphere is in nonslipping contact with the cylinder, this rotation will cause
it
to roll
along an element of
the cylinder without transmitting any motion to the cylinder, which, of course, is the desired result.
Thus, in these two cases, as long as we have nonslipping contact, the desired effects occur.
The
situation relative to the third
component
is
not a happy one. For, as we have seen, if there is an area of contact between the disk and the sphere,
we must have
a rotation around the axis
perpendicular to the disk slipping contact.
same conditions
On
if
we
are to have non-
for the contact of cylinder
and
upper sphere, we must have no rotation about this axis. Consequently, there must be a certain
amount of slipping at the if
the device
is
to
Fig.
m.4.B.l
the other hand, under the
move
at
a way as to reduce the difference in rate between x and y. Thus, y is a measure of the desired quantity
three points of contact all.
These devices are
Thus, is
x. if
a; is
changing at a constant rate and y
initially zero,
the servo motor will change y
INTEGRATORS, DIFFERENTIATORS, AMPLIFIERS
40
x
until
y
is
The operation of such a
zero.
device tends to give a smoothed or averaged
value of
x which
desirable. frequently very
is
(See Fig. III.4.B.2.)
except for a small motion,
loaded output. If x
is
fed to the
spring-
- y is larger than the contact
motion, the spring-loaded motion will maintain contact in the second output, and equilibrium
=
This arrangement matches x occur if ;c y. and y, but it may very well give a poor relation between x and y. One would expect that y might x. To produce a match lag behind or overshoot will
of
jc
and y one should not
try to
match x and y
directly.
The above
a simple case which
represents
Most problems,
occurs in computing devices.
however, involving servomechanisms require a more complicated response than the one de
To obtain sucharesponse a theory of servomechanisms has been set up analogous scribed above.
to electrical circuits (see, for example, H. Chest
W. Mayer, Ref. III.4.b; L. A. MacColl, Ref. IIIAf; H. M. James, N. B.
nut and R.
Fig. HI.4.B.2
Purely mechanical arrangements can also be
made For
for the
same purpose. (See
instance, the difference can be fed into a
differential
load.
The
whose output has a
Nichols, and R. S. Phillips, Ref. IIIAe).
We
Fig. III.4.B.3.)
slight frictional
other input of the differential can also
x-y
be regarded as an output. On the second output of the differential we mount a contact which
moves between two contact
is
limit contacts.
made, the servo motor
If
one
is
is
if
x
y
is
one TO SERVO
made, the
reversed. This
x
device produces essentially the sign of
For
limit
will turn in
direction; if the other limit contact
direction of motion of the servo
have already discussed a number of
methods for finding rate of rotation. In addition,
- y.
Fig. IEL4.B.4
changing, the second output will
be driven to one of
its
two extreme positions
there are stroboscopic methods, which, while
very accurate, have not been adapted for lating purposes as far as
x-y
we know;
calo>
there are
methods involving the drag of viscous fluids; there are electrical methods involving charging and discharging of condensers is
the
and
;
finally there
gyroscope principle of the airplane turn
indicator.
A gyroscope is a solid body having rotational Fig. DI.4.B.3
symmetry around an axis.
owing
to the frictional load
the differential.
on the first output of
Note again that equilibrium
=
y. A corresponds to the situation in which x variation of this device can be utilized to match
x and y. In
Fig. III.4.B.4 the difference
x-y,
axis
The moment of
which
inertia
is
called the spin
around
this axis
be denoted by u. If we call the moment of inertia around any axis perpendicular to this one
will
and through the center of gravity v in the usual ,
gyroscopes v It is
is
smaller than u.
customary to regard the gyroscope as
DIFFERENTIATORS
III.4.B.
rotating with a motion which has a large around the spin axis. The
com
gyroscope is considered as being mounted in such a manner that the center of gravity remains fixed. The
ponent
spin axis vertical
is
generally
drawn with a considerable
component.
The
intersection
of the
41
The equations of motion for the gyroscope were discovered by Euler, who introduced the three angles which are known as the Eulerian
The
angles.
between the spin axis and
is
angle
between the fixed y axis and of nodes, and y between the axis of nodes
the fixed z axis,
the axis
99
plane perpendicular to the spin axis with the horizontal plane is then called the axis.of nodes.
the usual
manner of mounting
We
gimbals.
The gimbals
also consider a third axis, perpendicular to
these two. (See Fig. III.4.B.5.)
and the body-fixed x
axis.
These are related to the gyroscope in
are a set of concentric
each pivoted in the next outermost one. Let us number them from the outside in: The
rings,
outermost is
is fixed,
the next inner, or second, ring
pivoted so as to permit a rotation around a
vertical axis.
The
third ring
is
pivoted so that
gyroscope
itself is
it
axis,
and the
mounted on the third
ring with
can rotate around a horizontal
the spin axis perpendicular to the axis of rotation
of the third ring. Notice that the fixed point also the center of the mass.
is
In this gimbal mounting the axis of nodes
is
the pivot axis for the third ring. Fig. HI.4.B.5
is
Consequently,
the angle of rotation of this ring around
pivot,
is
around
its
pivot,
and y is the angle of rotation of
the gyroscope relative to
its
spin axis.
suggested that the reader prove that
(It is
the direction cosines relative to the
sin
9?,
cos y, 0,
cos
cos
0,
9),
and z 97,
of the axis of nodes are sin
99,
and of the
sin
y,
jc,
cos
axes system of the spin axis are sin
its
the angle of rotation of the second ring
third axis
sin
cos
cos
9?,
sin 0.)
We now wish to consider the motion of a rigid body
in general, with one point fixed.
Let us
consider any transformation, Fig. HI.4.B.6
y It is also
sets
of axes.
these
is
customary to introduce two other As shown in Fig. III.4.B.6, one of
fixed in space, with the z axis extending
positively upward,
to the right
The
the
and the y
x axis extending positively axis positively
toward
us.
other axes are considered as fixed in the
gyroscope. The the spin
axis.
1
z axis of this set coincides with
Consequently,
the
for this body-fixed set remain
x and} in
of a coordinate system
(IIL4.B.1)
OJ, OY,
02
into
one
OX\ OT, 02
which preserves distances and as a fixed point. If u is the vector (x, y, z\ has then u is the transform (x , /, z). If u and Y are any
two vectors and a
is
any
real
number,
the distance-preserving property yields that
axes
the plane
determined by the axis of nodes and the third axis discussed in Section III.4.A.
= g(x,y,z)
and
also that
(to)
= to
(IIIAB.3)
INTEGRATORS, DIFFERENTIATORS, AMPLIFIERS
42
(Since a straight line
between two points,
must be pre
Consequently, the transformation
served.) linear,
the shortest distance
is
collinearity
motion
total
+
=
is
= u* + u
fry)
two vectors u and
and
v
and any two numbers transformation
the
b.
Consequently, equations may be written
y
= a^x + a^y +
z
=
now
Let us vector u
3jl
+
x
it is
= Au.
u
itself, i.e.,
Z
(HI.4.B.5)
taken into a multiple of
=
+
0(sin
+
#(0,0,1)
= 8fl
+
x
x
(u
+ u^y + (u 8fS
sufficient
III.4.B.6
satisfying Eq.
A)z
(1II.4.B.6)
condition that a I
will
exist
that the
is
determinant of the coefficients be zero.
determinant
This
a cubic in A and hence must have
1
a real root Since
is
= Au
u
cos
y
sin 6 sin
y
cos
0)
9?,
+
99
these.
w = (y +
the
must
+ 0n +
lengths
latter involves
an
does not correspond to any possible
motion of a
rigid body. then assume: If a motion of a rigid
body leaves one point on a line unchanged.
fixed,
The planes perpendicular themselves,
it
leaves every point
to this fixed line are
i.e.,
each
is
Newton
s
it is
laws,
necessary to consider
w=
placed relative to an axis,
.
is
particularly is
an
axis of
it is
clear that a
torque
must be applied around the axis to effect the rate of rotation around this axis. It can be shown that in general
Newton s laws of motion become
(III.4.B.10)
dt
where
applied torque. The torque be resolved into three components
w, where
w
is
the vectorial
On
r
is
itself
can
If the vector
u
p is
moving
the other hand, from the expression for
given by Eq.
,
n
This
where there
vector u fixed in the
moving body is approximately given by u x nAoc
rotation
instead of the usual
rotated
the small interval of time Ar, the body will rotate an amount Aa about a line, with unit vector u.
x
(III.4.B.9)
point, in order to
symmetry and the fixed point is center of mass. For if there are two equal masses symmetrically
through the same angle. Let us now consider a moving body. During
The displacement of a
sin 61
For a rigid body with a fixed apply
and
cos 0)s
easy to see in the case
ut
to be
Then
one or minus one. Since the
and hence
is
which instantaneously coin cides with the spin axis, axis of nodes, and third axis. Let s, n, and f denote unit vectors along
considerations.
=u
cp,
consider a set of axes which
momentum
du
cos
(III.4.B.8)
preserved, this real root must have the values
taken into
9?,
+$
momentum p
We may
6 sin
+
y
the angular
it
an
cos 6)
be
inversion,
is
fixed in space but
2>2
The necessary and
99,
ty>
We may
-
that there
sin B sin
9?,
sin 6 cos
(
w li8 z
8tl
we suppose
sin 6 cos
ij}(
For such a u we must have
=
if
0, then
consider the condition where a
such that
is
3 ,3
Thus
angular velocity y around the spin axis, 6 around the axis of nodes and 99 is changing by the
w=
a^z
+
03,2?
w is a vector, we can add its components
vectorially.
amount 1
(III.4.B.7)
(III.4.B.4)
Since for any
x w
dt
+W
au
the
is
i.e.,
(au
a
body with a motion u* then
relative to the
=
(y
III.4.B.8,
+
$ cos
we
0)s
w
see that
+
v6n
+
vy
sin 0t
at
(IIL4.B.12)
IIL4.C.
where u
is
spin axis
and
an
axis
moment
the
v is the
MECHANICAL AMPLIFIERS
of inertia around the
moment of inertia around
There
These equations, of course, remain valid only if we permit s and n to move, It is clear that the set s, n, t revolve
letting
m.4.C. Mechanical Amplifiers
perpendicular to the spin axis.
w
obtained from
= 0,
v>
w*
with a vector w*, which
as
is
(99
cos
0, d,
y
is
a need for mechanical amplifiers as
mentioned in Section
For example, it is important that the output of an integrator should not be loaded. III.2.A.
extremely
given by Eq. IIL4.B.9 by
There are
essentially
two types of devices
which are used to produce mechanical amplifica tion. One of these is a servo system, where the
i.e.,
=
43
sin 6)
(IIL4.B.13)
Consequently we have the formulas ds
= sx T at dn
=
= (0,
w*
= ($ sin 0, 0, -y cos 0)
x w*
n
at
Q=
I
= (-0, $ cos 0, 0)
x w*
1
(III.4.B.14)
at
Eq. III.4.B.14 permits us to substitute in the equation of motion Eq. IIL4.B.10 and obtain
+
u(ip
-
cos 6
$6
sin 6)
=
Fig,
rs
input
is
the rotation
m,4,C.l
x of a
shaft.
The output
the rotation of a shaft y either driven
- i(w sin vy>
+
2
u)09
sin
20
=
r
motor or with an arrangement, such as that discussed in Section III.4.B, in which the servo controls
cos 6 v
6
device.
- (v -
=
)y0 cos
rt
(1II.4.B.15)
is
by a servo
the rate of output of still another Thus, a simplified version using the
device of Fig, HI.4.B.4
is
shownin
Fig. IH.4.C.1.
In the applications of the gyroscope as a differentiator the angle
9?
is
quantity to be measured.
kept zero, 6 being the
From
Eq. III.4.B.15,
we see that if y is kept constant r t is proportional to 0.
The angular
velocity
is y>
generally kept
constant and rather large by a governor con trolled electric motor.
A.
Ref. III.4.c;
J.
(See also H. Goldstein,
Sommerfield, Ref. IIIAh;
P. L. Tea, Ref. IIIAi;
C. Inglis, Ref. IIL4.d,
Fig, ffl.4.C2
pp. 380-401.)
=
7T/2 and y kept zero, the desirable case which the cosine terms in the r t equation are
If S
in
zero occurs.
One may
observe r t by a spring
In Fig. Ill AC. 1 the servo motor drives the
y
shaft directly rather than through a variable
speed drive.
The
variable speed drive device
load on the rotation of the y pivot or there
will
be a more elaborate arrangement to intro duce an electromagnetic torque which will keep
smoother manner but with greater time delay
may
=
and
measure
r,.
in
which the current
size
will
produce,
in
general,
than the simpler version.
more power
An
alternate
in
a
method
which yields a smoother signal is an arrangement, such as that shown in Fig. III.4.C.2, in which the
INTEGRATORS, DIFFERENTIATORS, AMPLIFIERS
44
is transmitted through A These two beams pass through B and B which are mounted on
contact voltage on the potentiometer is fed to a servo amplifier which controls the matching
beams which alone
rotation y.
separate discs
In the case of a continuously changing motion, the rotation matching device of Section III.4.B
can be used with a supplementary device to take care of the difference X - Y which is lost in the
rotates also.
the output shaft
and then to separate phototubes.
The polarizing planes for B and B are at right hence there is only one position in angles, and each quadrant at which equal amounts of light
A
enter both phototubes.
variation
from
this
shaft will favor either one position by the output
phototube or the other and hence can be used to (See also T.
control a servo.
M.
Berry, Ref.
IIIAa.)
A
general theory for servo mechanisms
is
developed by H. Chestnut and R. W. Mayer (Ref. IIIAb), L. A. MacColl (Ref. III.4JF), and
H. M. James, N. B. Nichols, and R. (Ref.
IIIAe).
This theory
is
S.
Phillips
based on the
general notion of a feedback amplifier.
The customary method
for torque amplifica
tion in the case of rotations
on a
The
rotating drum.
in Fig.
IIIACA From
is
by means of a band
principle
the lever
is
illustrated
A we
have a
band which passes once around the drum and then to B. If A is pulled the band tightens on the drum, and the lever
Fig. III.4.C3
is
The
matching process.
difference
X
Y
is
to
be put in much more slowly than Fso that it will not overshoot. As a consequence Y will lag Y is used in time. The difference behind
X-
X
to control the supplementary system
of
its
derivative
is
B is
pulled by a force which
augmented by the pull of friction between the
band and the drum. For the case
in
which the drum moves
faster
than the band, we can present the following somewhat simplified discussion of the situation.
and the sign
used to control Y. (See Fig.
III.4.C.3.)
One
exceedingly effective method of making a
servo control connection
is
based on the fact that
a polarizing disc will only permit the component
of light polarized in a certain direction to pass
through it. Thus if a beam of light passes through a disc A and then through another which makes
an angle 6 with A,
the
amount of
mitted
is proportional to cos 6. Light polarization is utilized to make a servo connection in the following manner: The
polarizing disc A is attached to the input shaft. Two beams of light obtained by mirrors from the
same source are
directed
through
Fig.
light trans
this disc.
Thus
IHAC.4
Let us consider the longitudinal tension in the
band
(see Fig. Ill AC.5). At the point where the band touches the drum on the A side, this has a
value of FQ equal to the pull from A.
as the shaft rotates, the direction of polarization
longitudinal tension in the
of the
F(o) of this angle.
"extraordinary
component"
of these
Let a
denote the angle between the radius for this point of first contact and an arbitrary radius. Then the
band
is
a function
III.4.C.
To
determine F(a),
let
MECHANICAL AMPLIFIERS
us consider the piece Aa. For this we
of band between a and a
+
+
have the two tensions, F(a) and F(a Aa), at the end of the piece we have the effect pulling ;
of the
drum on
the piece and, of course, inertial
45
component pAocz?, directed outwardly along r, and a component -pAou) along F(a) (see Fig. If
III.4.C.6).
perpendicular
Newton
s
we
take components of the forces the
to
radius
for
vector
a,
law becomes
- F(a + Aa) cos Aa + /j/cos
F(a)
-/sin~ while the components parallel to the radius vector yield Aoc
F(a
+ Aa) sin Aa
/ cos
F(y.
+ Aa) sin Aa
pAau
or
Fig.
IIL4.C5
s- + /.rin-l/(m4.C.3) 2, -J
2.
Let v denote the linear speed of the band, Let f denote the resultant of the perpendicular
effects.
forces of the tude. It
drum on the
and /its magni assume that f makes
piece,
seems reasonable to
an angle JAa with the radius vector for a. We can also assume that the effect of the drum per to f is /x/ where /z is a coefficient of pendicular
This shows that /is an infinitesimal of the order
of Aa. first
of higher than Neglecting infinitesimals
order,
III.4.C.1
F(a)
we may
and
3.
eliminate
/
between Eqs.
Hence,
- F(a + Aa) + pF(a) sin Aa
Dividing Eq. 1II.4.C.4 by the limit,
we
Aa and
passing
to
get
(I1L4.C5)
may be
In general, the mass of the band III.4.C.5 simplifying to neglected, Eq.
dF
The
= pu,F_
solution of Eq. III.4.C.6, which
is
for a single loop Consequently, in the limit.) (This will be justified
Let
We
around the drum
have the output tension
of mass of the band per p denote the amount
p
=F
^
(III
unit length.
One
Of
that the force of inertia has a easily see
drum
We
the band suppose that
for
(IIL4.C.7)
Fig. III.4.C6
can
F
a = 0is
F(a+Aa)
friction.
(III.4.C.6)
is
inelastic.
course, this
is
on
the assumption
4.C.8)
that the
revolves faster than the band. If the
band
INTEGRATORS, DIFFERENTIATORS, AMPLIFIERS
46
and drum move
together, the assumption that
component of the force of the drum on the band is pfh no longer valid.
the friction
In the type of torque amplifier which is suitable for calculating devices, provision must be made for torque amplification in either direction.
A
to
the
radial
the
pressure,
exerted by the brake
drums
resultant is
torque
proportional to
the angular difference between the shafts.
angular difference, in turn,
is
The
proportional to the
torque exerted by the input shaft on the springs.
Thus, the added torque due to drums
is
propor-
commercially available amplifier of this character is
shown
in Fig. III.4.C.7.
The
drums
rotating
are concentric with the control and driven shafts,
and the bands which are brake bands to the drums.
There
is
a set of
The control
for each direction.
are interior
drum and band shaft
can be
considered as a hollow cylinder surrounding the solid driven shaft. The brake bands are mounted
on a
steel
strip
of springlike nature. One end of
connected to a projection from the control shaft, the other end to a projection on this
strip is
the driven shaft, which passes through a slot
on
the control shaft.
Normally each band
presses against
its
corre
sponding drum a certain amount due to the action of spring. When the shafts are moving together, the corresponding torques cancel.
If,
Fig. IIL4.C.8
tional to the input torque,
and hence
this is also
a
(See also R. H. Macmillan,
torque multiplier. Ref. IIIAg; C. Inglis, Ref. IIIAg, pp. 125-28.) One simple mechanical amplifier which yields a relatively smooth value
load from a shaft
loaded
shown
7 capable of handling a
X which should be only lightly
obtained by using an integrator as
is
in Fig. III.4.C.8.
the differential
is
The output
X-
Y of
applied through gearing to the
rate input of the ball cage variable speed so that
the equation
is
obtained or
X=
7
+ kT
(III.4.C.10)
7 can be considered as lagging value for X if k is small. Fig. IIL4.C.7
however, one shaft
is
fore, pulled
away from
its
large,
this
functions
as
an
References for Chapter 4 a.
T.
M.
Berry,
Electrical
"Polarized
light
servo
Engineering, Vol. 63, no. 4 pp. 195-98.
drum, and the second
permitted to press on its drum. Since the torque exerted by a brake band is proportional
is
integrating circuit.
turned relative to the other,
then one spring is tightened on itself and the other loosened on itself. The first band is, there
is
k
If
b.
system,"
(April, 1944),
H. Chestnut and R. W. Mayer. Servomechanisms and Regulating System Design. New York, Wiley and Sons, 1951.
III.4.C.
c.
H.
Goldstein.
Classical
Mechanics.
Mass., Addison- Wesley Press, 1950. d. C.Inglis.
MECHANICAL AMPLIFIERS
Cambridge, Pages 143-84.
h.
H. M. James, N.
B. Nichols
of Servomechanisms. f.
Co., 1947. L. A. MacColl. nisms.
New
New
and R. S. Phillips. Theory York, McGraw-Hill Book
Fundamental Theory of Servomecha York, D. Van Nostrand Co., 1945.
R. H. Macmillan.
Theory of Control for Mechanical
Engineers. Cambridge, University Press, 31-32.
Applied Mechanics for Engineers. University
Press, 1951. e.
g.
47
A.
J.
Sommerfield.
Vol.1.
New
York,
195L Pages
Lectures on Theoretical Physics,
Academic
Press,
1952.
Pages
118-61. i.
L. Tea, "Elementary theory of gyroscopes,* Franklin Institute Jn,, Vol. 214, no. 3 (Sept. 1932), P.
pp. 299-325.
5
Chapter
CIRCUIT THEORY
HI.5.A. Introduction
The macroscopic
computing devices involve
flow of electricity
is
a
well-
understood phenomenon. Within certain limita tions it can be described mathematically by linear differential
ordinary
tion electricity
is
the basis of all utilized methods
are
study of mathematical machines.
In addition to immediate electrical applica moreover, the concepts of circuit theory
equations which can be
precisely solved. Because of this, in our civiliza
many
electricity;
completely electrical. Hence, a precise under standing of circuit theory is necessary for the
tions,
have proved invaluable in
efforts
to describe
of communications beyond the power of our senses and represents the most convenient
theory was studied with the introduction of
method
analogies with certain mechanical systems, such
for the transfer
and transformation of
other
as
energy.
The phenomena of
technical interest involve
the flow of electric current via certain media in a
is
Our
with
described
good
by
accuracy
ordinary
differential equations with constant coefficients.
In the general case, the phenomena of electricity
and magnetism are governed by Maxwell s equations which are partial differential equations. However, for certain systems of great particular interest, subject to the above limitation on the rate of change, there exists a description in
functions of time,
i.e.,
of one variable alone, of
circuit
considered in circuit-theory terms.
circuit
dielectric effects. If the state
Originally
pendulums. At present, however, the relation and complex mechanical systems are
of a system does not change significantly during the time it takes light to transverse the system, these phenomena can be
field,
well.
reversed,
and
vacuum, the associated magnetic
phenomena
as
discussion begins with a description of a
and branches and the
consisting of nodes
relations
known
to hold between the currents,
potential differences,
and charges present in such
a circuit (see Section III.5.B).
Our first
objective
be to derive a system of differential equations adequate to describe the variation of this system
will
with time. These are the alternate
"nodal"
"mesh
One can show
III.5.D.
is
system
equations";
an
given in Section
that
no matter how
complex a two terminal circuit may be, it may be replaced with a far simpler one. This result, which
is
called
Thevenin
s
theorem,
is
basic in
accuracy adequate for all practical purposes. These systems can be considered as constituted
most applications (see Sections III.5.H and I). The notions of circuit theory are based on the
within each of which one
use of complex exponentials, which are discussed in Section III.5J.
of
"components"
phenomenon
specific
There
is
occurs
predominantly.
a transformation of energy which can be
specified
by giving
either a current flow or a
potential difference as a function of time.
system
is
not isolated;
duced into
it
by
electric
"generators"
or
energy is
is
intro
dissipated in
the form of heat in resistances.
Computing
An
electrical circuit consists
"nodes,"
as
shown
and
This
is
conveniently done by electrical means.
and
ances,
"branches"
in Fig. IIL5.B.1.
of various
devices involve the transfer
transformation of information.
HI.5.B. Notion of a Circuit
A
of points, called
connecting the nodes
Each branch
consists
circuit elements (inductances, resist
and voltage generators)
capacitances,
most
connected in
Most
be represented in Fig.
series.
Thus, a typical branch III.5.B.2.
may
The symbols I,
III.5.B.
NOTION OF A CIRCUIT
E and the corresponding diagram symbols represent the total inductance, resist ance, capacitance, and generator output, respec
R, C, and
for this branch.
tively,
An
amplifier will appear
49
Consider any branch B.
We denote the current
by z and the potential drop across B by e. These quantities are related to the values of L, R, in
B
C,
and
E for B and to the currents
in the other
/
branches (with their respective coefficients B) by the equation
M for
idt
Fig. IH.5.B.1
o
nnnp
SAAA-
L
R
(IIL5.B.1)
This equation is
of Eq. III.5.B.1
Fig. HI.5.B.2
known as the "voltage equation"
A derivation and explanation
branch B.
for the
Institute of
is
discussed by the Massachusetts
Technology, Electrical Engineering
Staff (Ref. IILS.d),
output branch as a generator. However, the output is normally not known, but is propor tional to some unknown current or voltage in a in
and L. Page and N.
I.
Adams
its
branch.
different
Amplifiers
can
be
most
taken care of after the machinery for theory without amplifiers has been
effectively
circuit
(Ref. III.5.e). It is
usually possible, in the simpler circuits at
least, to shield
the branches
as to render the this
is
negligible. If
done, Eq. III.5.B.1 becomes simply
developed.
idt
Because of the voltage generators and possible initial
from one another so
mutual inductance
L- + RI
differences of charges on the capacitances, exist among the nodes. Such potential
+E=
-
e
(III.5.B.2)
C
dt
potential
differences give rise to flows of current along the
branches. current
in
Both the potential difference and a branch are signed quantities
with a given direction along the branch. This direction, in turn, is associated with associated
we shall assume this to be the case in the and discuss the more following development Initially,
later. general case
We now introduce an abbreviation by defining the differential operator:
an order for the nodes which are the ends of the branch. If the direction of the branch
by
is
Z(p)
reversed
The
essential
problem
where p
in circuit analysis
=
.
The
branch
B may then
and find the potential
difference
between each pair of nodes. A current flowing in one branch of the
this
+E=e
(III.5.B.4)
phenomenon
is
known
as induction.
The operationZis referred to as the "impedance" of the branch B.
circuit
affect the currents in the other branches;
If this
takes place in the circuit, then, in order to solve we must also be given the various the
This operational notation
M of mutual inductance, one for each
pair of branches.
is
very
commonly
used when dealing with a problem involving differential equations with systems of ordinary constant coefficients. The reason is that under these circumstances one
problem,
coefficients
voltage equation for the
connected, find the current flowing
in each branch
may
(IIL5.B.3)
be written:
Zi
priate
is
+ K + -r
is:
Given the values of I, R, C, and E (in appro and given the way units) for each branch,
the circuit
Lp
Cp
the signs of both the reordering the nodes,
and current change. potential difference
=
ing p algebraically
as
numerical quantity.
is
justified
though
it
in
manipulat
were an ordinary
CIRCUIT THEORY
50
We The
ffl,5.C,
Circuit Equations
In addition to the voltage equation which we
may write for each branch of an electrical circuit, we have Kirchhoff
s
laws:
The Current Law.
The
=
that there
the branches
all
which
The
.
.
B^B^BJSJSi is .
P
the
is
sum of B,
.
.
.
.
.
common node these meshes
B&*
.
.
.
.
.
B* be a mesh such that
.
B^
for
is
and B*. The
defined as the
B*Bj+1 ...Br
(See also P.
Le
Corbeiller,
j
-
and
quantities L, R,
Mesh
are non-negative
.
mesh
III.5.C.1
Fig.
circuit:
It is
equations (one for each mesh)
evident that these equations cannot
we may
Indeed,
independent.
Bfi^ ...Br Composition of Meshes. Let Bl be a mesh, and let P be the common node of Let B^
If the
way, we can
Voltage equations (one for each branch) Nodal equations (one for each node)
f
as does, also,
a null mesh.
.
circuit.
no mutual conductance between the
is
which we may write for any
Br such that B
.
common
occurs. Consequently,
BJ and Bj+1
is
numbers.
real
B^
it
it
any other node
this
Ref. III.5.C.)
is considered to be a B^. The mesh mesh and can be omitted from a mesh in
and
null
to
Now let us consider the three sets of equations
A mesh is determined by
B^
Br
not connected in
unconnected parts.
connected with the chosen node.
a sequence of branches, and Bj+l have a node in
assumption that
along a chain of branches of the circuit is
(IIL5.C1)
the summation extending over
Mesh. Definition of a
the
from one node
analyze the circuit into separate parts, provided total current flowing
toward any one node is zero. For each node, then, we have the nodal equation: i
make
shall
possible to pass
find
all
be
a linear
dependence among the nodal equations alone: we add all the nodal equations, we get zero, for
if
each current i appears in two nodal equations, with opposite signs each time (it flows away from
one node and toward the other). In order to obtain an independent system of equations for the all
arranging
unknowns i and
e,
we
begin by
and branches
the nodes
a
in
convenient order in the following manner:
AQ
1.
Choose any node, and
2.
Consider each of the nodes directly con nected to AQ by a single branch, and label these nodes
A l9 A^
For each node A f
3.
.
.
label
.
it
.
Ar
,
may be more than
there
one branch connecting it to node A Q Choose one such branch, /, for each A j .
and call it the "returnbranch" for A f Thus, .
we have the branches #/, B z
Fig. IH.5.C.1
If there are illustrates
the composition of
B^B^BZ*. The
B-^B^B^ and
addition or removal of a null is
not considered to the
change the circuit. The Equation of a Mesh. Let B1 B^Bj+l ...Bf denote a mesh, and let ef denote the voltage drop .
across the branch
el
B
.
f
+
repeating step
fi,
+
e, +1
+
...
s
(2), this
A l playing the role of A and labeling A We also repeat nodes A r+l A r+2 ,
.
,
,
.
.
s
,
step (3) to obtain a return
.
branch for each of
.
The voltage equation for a
expresses Kirchhoff
+...
.
.
over in the original
left
time with
mesh, when permissible,
mesh which
circuit,
any nodes
we continue by
,
..., Br
second law
is
these latter nodes. circuit,
we
repeat
role of playing the
If
nodes
step (2)
A Q and ,
still
remain in the
once again with
A2
so on, until there are
no more nodes.
+
er
=
(III.5.C.2)
Since no node label
Ak
is
isolated, this process assigns
a
to each node of the given electrical
THE CIRCUIT EQUATIONS
III.5.C.
4u AZ,
^o,
we
and
circuit,
.
.
.
have,
+
n
say,
nodes:
1
A n Each of these, except A Q) has .
,
been assigned a
definite branch, its return branch,
and so we have n of these; B/, LEMMA III.5.C.1. The circuit return branches
B^B^ ...B n
5a
.
,
,
,
E^.
,
non
mesh in
this circuit
B^
.
.
B^B^
.
.
.
/
.
is
a
of return branches. Let; be
the subscript with highest value that occurs.
fc
AV the
then will have the highest subscript which occurs among the nodes in the mesh.
Now A
is
the
Jk
B^
or of
common node of either B^ The argument
B^ and jB^.
and
and we suppose the second case connects A with some node
in each case,
Then B-
holds.
k+i
ijs
with a smaller subscript, since k is the j highest is the subscript of a node in the mesh. But
As
only return branch which connects
to a
=
node .
jfy
eliminate branches until eliminated.
all
starts at the
"rising part."
to indicate a flow of current
is
The node with
.
general circuit,
a 3 which
we
arise
.
defined for
I
a#
=
<
(In general unprimed symbols
The symbols Z,
reserved for return branches.)
and
f,
,
e will be subscripted
and primed
to
correspond with the symbols for the branches to
which they belong. Thus, for example, 2 stands for the impedance in the nonreturn branch B l
1
for
the return branch
(Xfl.
1 if
the
-
=
1
3 fc
These elements are <
k
<
if
mesh
for
in
its
B/ the
m as follows
:
B^ does not contain
B/
at
all.
Bk contains
the return
descending part.
mesh
for
return branch B- in
non
return branches, while primed symbols will be
n and
mesh
the
if
of the
define a matrix with elements
frequently.
<j
a .
This terminology
around the mesh.
t
usually contain nonreturn branches. l9
the chain
subscript in the mesh, at
least
branch
B B z ,..., Bm
l9
Before proceeding to the solution
=
will refer to
B
which return branches of both parts of the mesh for B l meet is not on either part of the mesh for B
branches have been
Let us say there are m of these:
for the branch
higher node, A v) and the intermediate
nodes form the
This indicates a null mesh.
In addition to return branches, a mesh will
and
l
and the intermediate nodes are said to form
fc
This permits us to strike out from the B^Bf^ given mesh. We can continue in this way to
B
while the remaining "descending part," branches (the chain of return branches which
B^
with a lower subscript, and, hence, Bk
mesh
i
similar
is
A mesh is thus formed
of return branches which starts at the lower node,
The
A it
node
Av
a chain of return branches.
In the above
Suppose
other at
consisting of the single nonreturn branch
zero mesh,
Proof:
A u and the
branches, one starting at
toward nodes of lower subscript. Eventually these chains must meet at some node A for the 9t subscripts are decreasing,
consisting of the
contains no
51
Bk
its
contains the
rising part.
LEMMA III.5.C.2, Every mesh from the original circuit
can be composed from the nonreturn
branch meshes defined above and null meshes. Proof:
mesh B]*
From Lemma ...
BT *
is
III.5.C.1
we see that if a
not simply a null mesh,
it
lt
and 4 branch
stands for the current in the return J?/.
for any branch
Now,
we
arbitrarily take the
be that positive direction to higher subscript to the
Thus,
we have
if
a branch
the current flowing from to be positive,
than
be
A l9
and
if
from the node of
node of lower
Az
subscript.
B connecting A$to A A% to A is considered
is
lt
at a higher potential
the voltage across the branch
is
said to
Fig.m.5.C.2
positive.
For each nonreturn branch introduce a
Let
B
u
v.
<
l
mesh
connect nodes
Now
B b we
shall
now
in the following unique way.
A u and A w
where, say,
consider the two chains of return
must contain nonreturn branches. Let
E
l
be a
nonreturn branch in the mesh such that
A v and A u Let B III.5.C.2).
where
connects
.
.
.
B^
u
(see
B
l
Fig.
be the chain of
CIRCUIT THEORY
52
return branches specified above which connect
nonreturn branches.
A u and A v where B^ starts with A v B B ... Br * so that we can write
B we
.
l
occurs in
Now we Suppose A v is the common node of B and B r *. Then A u is the common node of B r * and B Br * We can introduce a null mesh into B-f l
lt
.
.
For the nonreturn branch
thus have:
l
of these
apply the voltage equations.
we may
terms of Z,
and:
/,
:
.
(ZA + E I)
+ 2>(Z/i/ + <
z j
A U9
and,
and
have the
B^
#]*...
thus,
A3s and A* 5JL3i #r *. This latter mesh has .
.
.
.
I
.
.
J-
less
than
B^
branches in
...
JJ X
5f *.
* .
.
.
5,.*
Br * r
common node
3g
.
.
.
Ah
V
,
.
.
.
associated with nonreturn branches
an
and
null
immediate
consequence of the equation of the mesh associated with the addition of two meshes can is
that
the
be written as the sum of the equations of the given meshes. Also, the equation of a null mesh
can be written
LEMMA
=
III.5.C.3.
Consequently, we have: The equation of any mesh
0.
a linear combination of the equations of the meshes for the nonreturn branches.
is
LEMMA
III.5.C.4.
The
set
of equations for
meshes for the nonreturn branches are
+ I %Z/i/ =
-1
(Our discussion up
I
/ (1
<
m)
point has ignored
to this
the possibilities of amplifiers.
If
amplifiers were
present, we could treat their output as
and
a generated
input circuit as simply an
their
output impedance. Thus, one would arrive at Eq. III.5.P.3 with certain Fs and E"s corresponding the output of amplifiers. These outputs, however, are proportional to unknown currents z or //, and we may substitute for these and z transfer the resulting terms to the left-hand side
ofEq.D.3.) In addition to the system Eq. III.5.D.3 we have Consider the node A v
the nodal equations.
This
branches it is
Proof: The voltage of a nonreturn branch occurs in one and only one such equation, a fact
which readily implies independence. Lemmas III.5.C.3 and 4 settle the independ ence question for the mesh equations. We need only consider the system of equations corre sponding to the nonreturn branch meshes.
5^,
Equations
begin by writing the mesh equations, one for each of the m meshes corresponding to the m
.
.
.
,
B
number of nonreturn
and lg
Bhi ...,B
which
for
,
h
respectively a lower node or an upper node.
With regard to return branches it is the upper node of Bv only and the lower node of return
B^, B w ^..., B w
with
-. + 1 U
If
we
V +1 U
ha
w^
r
Consequently, the nodal equation for
-1; =
Av
>
v.
is
(IH.5.D.4)
U
write the equations for
have n linear equations on
A lt
//,
.
.
... .
i
,
A w we
t
whose
n
matrix of coefficients contains zeros below the diagonal, but zero.
Mesh
.
a node for a
is
branches
We
-E,
(IH.5.D.3)
independent.
HI.5.D.
be
may
to
definitions
the
Z,i,
can be eliminated in this
meshes. It
Eq. D.2
l}
The various nonreturn
that the latter must be a null mesh. Consequently, Br * can be composed from meshes BI* .
E
^ m. We thus m + n unknowns
one nonreturn branch
way with the resulting mesh consisting only of return branches. But Lemma III.5.C.1 shows
.
Transposing the
i/.
(III.5.D.2)
written:
composed of
is
*^
and
=
/)
/ Eq. III.5.D.2 holds for 1 have a system of m equations in
Now B v *
By means
express the quantities e above in
sively,
none of whose diagonal terms are
Consequently, we can solve these succes in , // beginning with the last, for in
terms of the return branch currents
.
,
.
.
i m The elimination process can be described even i
lt
.
.
.
,
.
III.5.E.
more
we know
precisely since
branch current
SOLUTION OF CIRCUIT PROBLEM a given return
that
// occurs once
with a coefficient
-
on the diagonal and at most once above the
1
diagonal with a coefficient +1. The elimination process to obtain
consists then in
i
v
previously obtained equations for f
i
Wu
which contribute
What happens
to this
adding the for those
i^
equation.
and proper direction for
IH.5.E, Solution of the Circuit Problem
THEOREM III.5.E.1. The
differential equations obtained from Kirchhoff s laws with suitable
conditions determine the currents and
voltages as functions of time, provided the return branch k .
nonreturn branch current can be easily followed in this elimination, The term for f is introduced
m, are not zero.
into this elimination by either the equation for its upper node or its lower node. Consider the
tions
fc
The term
first case.
hand
side of the
k is
i
introduced on the
left-
equation with a minus sign. The
current
its
given in each return branch.
initial
an individual term for a
to
traced out,
53
Zk
impedances,
We now return to
Proof:
=
,
1,
.
.
non and
,
the system of equa
Eq. HI.5.D.3.
we may
Using Theorem III.5.D.1, f eliminate the in and rewrite the system
Mows:
Eq. IIL5.D.2 as
return branch current for the upper node flows into some other node which is either the lowest
node on the mesh for
on the
the branch of ik or which
rising part for the ik branch.
is
Since the
elimination consists of simple additions, the term 4 is introduced with a minus sign in every
equation for a
node on the
rising part. Thus, if corresponds to a node on the rising part, the nodal equations for contain the term will
(1II.5.E.1)
(!
Denoting the right-hand member of the equation by
we
signs,
F
l
/th
and rearranging the summation
obtain:
i-
= -~h
//
11
If
Kjick-
equation for
its
h has been introduced by
the descending part,
we
still
equation for i/ will contain
node on the mesh
for
i
k)
we have a The
k does not appear.
i
elimination
is
m)
I
(IIL5.E.2)
the lowest
cancellation,
result
<
is
Eq. III.5.E.2
on
is
have that the nodal
a^. At
(1
l
the
lower node, the sign situation
whose node
=F
flZ/a,2&
k
3
reversed, but for a current //
and
z ih +
of the
a system of m linear
differential
m
equations with constant coefficients in the
unknown mesh currents
f
f
l3
.
.
.
8>
,
m As
i
.
is
well
known, such a system may be treated as if the = Z(p) were numerical differential operators Z
To
quantities.
readily seen then to be
is
solve for
i l9
then,
unknowns by Cramer
other
s
we eliminate the
rule to obtain
an
equation:
= #1
A?i
The net
result of the elimination
is,
therefore,
current //
III.5.D.1.
may be
Each
A=
A(j?)
is
Eq. III.5.E.2 and
given by the following theorem.
THEOREM
where
branch
return
written in terms of nonreturn
branch currents as follows:
members discussion
H
the
is
combination
differential
F.
(EDL5.E.3)
the determinant of the system
appropriate linear of the right-hand
The reader
of linear
is
referred
differential
to
the
equations in
Section III.14.H.
i/=i,&
(!<;
<)
(IH.5.D.6)
fc=i
The geometrical significance of this equation is that the current in each return branch algebraic
sum
is
the
of the currents in the nonreturn
branches which have
it
in their
mesh circuit. The
equations themselves are readily written inspection from the circuit
if
down by
each basic mesh
is
Now, Eq. III.5.E.3 is again a linear differential equation with constant coefficients in the single
unknown fr
We know
that
it
will determine
^
given the necessary initial conditions if and only if the operation A(p) is not identicaUy zero, that is,
if
This
the coefficients of will
be true
if
A(p) are not the
impedances are not zero.
all zero.
nonreturn branch
CIRCUIT THEORY
54
Consider the function A(x) obtained from A(p)
by replacing the operator/) by the real variable
x.
evident the statement that A(p) vanished
It is
equivalent to the statement that
is
identically
A(x) vanished
Two conditions imposed are: (1) that the mutual inductances were all zero and (2) that there were
no
same
identically, for either statement
general the
depends entirely on the coefficients which are the same for both and A(x). Consequently, in
restrictions.
order to demonstrate the existence of a solution
parameter a as a
A(/>)
to the circuit A(JC)
problem,
we need
only to show that
does not vanish identically.
Then A(x)
is
equations Eq. IIL5.E.2 with
order
S
Let
variable x.
a replaced by
p
5"
Z-d jk
,
Let a
denote the matrix with elements a#. Then A(x)
Z S
T= S+
the determinant of the matrix
where a* l
is
is
the transpose of
is
For x
a,
real,
each
a positive real quantity. Hence, the matrix a positive definite matrix with roots all
greater than 0. Now if we show that, in addition, a* S a is non-negative definite, then T must also be positive definite
not
less
and must have roots
than the smallest root of S, This
imply that A(f)
not zero.
is
On
would
the other hand,
Z- correspond to non-negative a*S a would be non-negative, for if
as long as the quantities,
= ax, then any m dimensional vector and y a*S ax x = S yy a is non0. a*$ Hence,
x
hold without these
based on introducing a of the mutual
coefficient
basic current voltage equation
for
The previous arguments can
branch.
each be
still
the equation applied to yield
AfooOi^Hi
(UI.5.F.2)
the corresponding
matrix of order n with elements
is
is
indicated that in
is
inductance terms and also the amplifier terms in
denote the diagonal matrix of
m with elements Z^ zb
it
results
This procedure
the
the determinant of the system of
We can provide readily
amplifiers.
a discussion in which
= 0,
this
we know
then
When
of Eq. III.5.E.3.
instead
ot
reduces to the
and previous case,
that A(p,
is
not identically zero. Since
in
p and
oc)
polynomial
given value of a,
it
a,
will
a
not be identically zero in
this is also true for
pi in general,
A is
then in general, for a
a
=
1.
m.5;G. The Nature of the Solution Let us A/i
now consider the nature of Eq, III.5.E.3,
=H
lt
may have.
and consider the types of It is
constant coefficients, that "
solutions
it
a linear differential equation with
1
+
is,
+
the form
*)
-#
(ffl.5.G.l)
is
>
Consequently, A(x) does not vanish
negative. identically,
and our theorem
is
H
where we have dropped the subscript 1. Here is a known linear differential combination of the given generator outputs
proved.
function of 1H.5.F.
The Mesh Currents as Solutions of Differential
The
In view of the preceding discussion, we
and, hence, that a unique solution
when
exists
i
l5
z
i=f(t) where/(r)
2)
suitable
initial
nonreturn branch currents as functions of time,
we may
H
is
solution of an equation of the
may
Having thus found the
conditions are given.
so that
a
known
form of
Eq. III.5.G.1 consists of two parts,
Equations
conclude that A(p) does not vanish identically
of Eq. III.5.E.2
,
t.
and
is
h(t) is the
h(t) is the
(III.5.G.2)
a particular solution of Eq. III.5.G.1
complementary solution, that
is,
general solution of the corresponding
homogeneous equation:
then easily find the return branch
currents f/,
z
2
,
.
.
.
,
i^ by
means of the
relations (III.5.G.3)
i/=l^k
(III.5.F.1)
Eq. III.5.G.3
*
Then,
finally,
we may
find all the ek
and
e-
by
using the voltage equations (see Eq. IIL5.B.1-4).
The general
circuit
problem is, therefore, solved, which we have imposed.
subject to the conditions
is
the
equation which would
correspond to the same circuit stripped of its
generators.
ditions (charges
Because of certain
on
capacitances,
current can flow even in
initial
etc.),
all
con
a certain
a circuit without
THfiVENIN
IIL5.H.
generators; this
is
the current
has resistance (as
If the circuit
h(i).
actual circuits do), this
all
current will die off gradually.
For
the complementary solution h(t)
is
the
"transient"
reason
usually called
The
portion of the current.
particular solution f(t) state"
this
called the
is
"steady-
portion.
S
THEOREM
55
equivalent of a voltage generator or a current
generator whose output the
unknown
linear
proportional to one of
is
currents.
We may
still
set
up
the
system of equations on which the preceding
discussion
is
based, but
will
it
not necessarily
turn out that the real parts of the A
s
are
all
nonpositive.
The actual calculation of the complete solution of Eq. III.5.G.1 would require explicit knowl edge of the generator outputs E(t\ so that we /
H explicitly and thus be able to find
would know
We
f(t).
may, however, make some general
remarks about the transient portion
knowing the J?(f) s. The function h(t)
h(t)
m,5,H. IMvenin
homogeneous equation, Eq. HI.5.G.3. By
the
may be simplified by known as Thevenin s
generally
theorem.
THEOREM IIL5.H.1.
without
the general solution of the
Theorem
of circuit theory the use of a result
circuit is
s
Much
made up
Consider an arbitrary
of two subcircuits,
which have only two nodes in III.5.H.1).
tf and Jf
common
,
(see Fig.
We may replace Jf by a single branch
elementary theory of differential equations, we
know
that the solution of Eq. G.3 is a linear combination of terms of the form t^\ where A
a root of the algebraic equation
is
a/ +
1 a/"
+
.
.
.
+
of multiplicity greater than solution of Eq. G.3
is
q.
Jf
=
ak
(IIL5.G.4)
The most
general
then
Fig.nL5.Hl where
P (t)
A, is
v
is a root of 0-fold, Eq. III.5.G.4 and a polynomial of The degree
q-L
coefficients
Pv (t)
of these
are
the
constants, which are determined
arbitrary
from
initial
conditions.
The
generator output
E* and the impedance Z*
branch depend only on Jf
.
(Z*
impedance or rational function
function h(t) will approach zero as
t
approaches infinity if the real part of every kv is negative. This occurs in passive circuits, i.e., circuits
connecting the two nodes without altering any of the currents or potential drops of $*. The
Proof:
is
of the
a generalized
of/?.)
We begin by numbering the nodes and
branches of the entire circuit in the usual (see Fig, III.5.H.2).
way One of the common nodes
which do not have amplifiers (see also Ref. III.5.a), and since, in our
H. W. Bode,
we want
applications, after a time this to
When
the solution to depend
on the generated
voltages,
we want Ji
occur also for our devices in general. the transient approaches zero,
i.e.,
when
every Av has a negative real part, the circuit will
be said to be part,
stable.
If a A,
has a positive real
the circuit is said to be unstable. In general,
in our
applications,
this is
Fig.
U15.H.2
objectionable since the
A& The
other
currents then are determined not by the generated
is
voltages but by the
assume that the return branch
initial
conditions.
Passive circuits are stable. are introduced,
When
however, one
may
labeled
amplifiers
$
have the
branch
.
We
introduce a
B"
for
A"
is
new node
which
is
labeled
Bs
r
A"
for
A f We
A s lies
in
and a return
the only connection
56
CIRCUIT THEORY
between
and A s
A"
A
lying between
in the
We choose the branch
nodes.
and
We think of the node
.
and A s+l
8
= 0.
E"
E"
as
A"
ordering of the so that
=
Z"
One can
A
8
For meshes of
with 4,.
which do not involve
permissible for a return branch
It is
to have zero impedance.
Jf which
contain terms from a fixed path in connects
the
we
B",
$
,
then,
have equations in
form
show
readily
that the introduction of the new node and branch while for those which do involve
does not
any of the currents or potential = and drops of the original circuit. Since E"
= 0,
Z"
B",
alter
no mesh equation
f
eliminating
is
B",
A
and
A"
A
original nodal equation for
yield the
.
t
Now we assert that every current of 3C may be of the current
expressed linearly in terms
flowing in
To prove
B".
this,
we
C
between the
will
s
+C=F
(HL5.H.2)
a
and
altered,
where
the current in
,
nodal equations for
(2;ZO
we have
represents the fixed linear combination
of currents and generator outputs of
J"
in the
path consisting of return branches connecting A s to A Q Now, since each current of may be
$
.
written linearly in
we may
/",
write
i"
consider the
C Then Eqs.
=
III.5.H.1
*
+
Z*i"
(III.5.H.3)
and 2 become
JC But Eqs. III.5.H.4 and equations
we would
the circuit
Fig.m.5.H,3
E* circuit
of A\ A* and Bt
consisting
A
including
s>
For any given value of
i"
and
B",
all
containing
nodal
the
tf
Since
i".
we
electrical circuit,
be solved when
It is also
alone
is
Jf
A
s
a complete
may may
i",
and the
assertion
JC
linearly in
added node and branch. The nodal equations of
$
$ and
\\
$
Theorem
$
B\
However,
tf
,
of
and the nonreturn branch
using the elimination
III.5.D.1.
equations for involve
i"
Next,
we
$
from the
,
Jf
is
circuit.
equivalent to B*.
definition of
Z* and E*
in
Eq. IIL5.H.3 that these depend only on Jf. This proves the theorem.
only the currents
We can express all the currents
in terms of
currents of
will involve
obtained above are also those for this
Thus, relative to
We know
.
We return now to the original circuit with the for the nodes of
Fig, IH.5.H.4
is
obvious that the coefficients
depend only on Jf
of/"
of
equations
required to express any current of
term
generator output
The nodal equations
(see Fig, III.5.H.4).
Jf
Consequently, they
be solved for any value of proved.
$ and a branch B*
of
Z* and
.
equivalent
see that these
= 0.
i"
precisely the
we were to consider
we may obtain a set of Jf These
equations one would normally get for
however,
with,
of
if
(see Fig. IIL5.H.3).
linear equations for the currents of
are the
consisting
having the impedance
are
5
obtain
method of
write the
mesh
Certain of these will not
and, hence, will depend only on %. those which involve will also B"
HL5.I. Applications of Thevenin s Theorem
Thevenin
s
theorem
permits
a
standard
approach to many problems in circuit theory, which is extremely useful in the study of mathematical machines. consist of various
extremely complex
Electrical
computers
components, each involving circuits.
In general, however,
APPLICATIONS OF THfiVENIN
III.5.I.
Thevenin
s
theorem
is
applicable,
and for most
practical purposes one can replace these complex circuits by extremely simplified branches. These
branches
simplified
have
will
the
THEOREM
S
57
us suppose that the internal impedance of the be the voltage generator is a resistance r. Let let
E
generated.
Suppose we connect
a resistance
R
desired
mathematical properties, and this forms the
-V\AA-
basis for using these devices.
The
for
process
the
finding
Jf
equivalent to a circuit
is
5*
branch
straightforward (see
Fig. 111.511,2).
If
A&
no external connection measured
the
is
made from A s between
voltages
to
these Fig, ffl.5.1.3
across the output (see Fig. III.5.I.3).
/
is
The current
clearly
The
actual external
to AI, here this
E
voltage
is
the
drop from
= IR = REj(r + R)
The output power, W,
(01.513)
is
W = E l = R&l(r + Kf Fig. ffl.5 .LI III.5.I.4 is
Eq.
AQ
is
zero for
jR
=
}
(ffl.514)
and
W also W
has approaches zero as R approaches infinity. a maximum at R r, and, consequently, the
=
maximum power is E2/4r. the
assumes that the all
Thus, the smaller r
more powerful the generator values of E.
have
this
is.
is,
The above
linear-circuit theory holds for
Electronic amplifiers
property, but, nevertheless,
true in general that the smaller r
powerful the voltage generator
is,
may
not
it is still
the
more
is.
Fig.IH,5.L2 terminals as the
is
A we ,
usually referred to
obtain what
"short-circuit current"
Z*J*
By
is
"open-circuit voltage."
directly to
the
*
E*. Thus,
measuring I*,
/*,
If
we connect A,
is
referred to as
which is such
= E*
that
(III.511)
we can obtain Z*
for this
Thus, by two measurements or by their E* and Z* can be theoretical relation.
equivalents
obtained.
Our notion of a
here. appropriately reviewed
actual voltage generator internal
Kg.ffl.5JL4
also be voltage generator can
impedance;
We
note that any
must have a
otherwise,
the
certain
possible
power output would be unlimited. For simplicity
The Wheatstone Bridge. Example: Wheatstone bridge is a device used to measure The
unknown
known
resistances
resistances.
by comparing them with III.5.I.4, where G
In Fig.
58
CIRCUIT THEORY
stands for a
known. reads
galvanometer, suppose
Rl
is
un
0.
=~
X-i
|
=
fa
If we take differentials, the actual
value.
R% is adjusted until the galvanometer Under these circumstances
*
can
be considered to be
(IH.5.I.5) (III.5.I.8)
Normally, limited
however,
sensitivity
mately true. general case,
the
and Eq.
To
galvanometer IIL5.I.5
is
of
only approxi
analyze what happens in the
we can
apply Thevenin
s
If the internal
G
is z,
flowing through the galvanometer,
theorem to
I==
the circuit with the galvanometer removed.
impedance of
It
^L = +
Z*
z
*!
7,
will
be
kR
**_ E (
*
+
+
*,
the current
}
\Z*
z/Ki
+
*a
(III.5.I.9)
If in
we substitute for 7the minimum
III.5.I.9
Eq.
value of the current to which the galvanometer sensitive
and the other
is
quantities, including jR1}
which are considered to be known, we can find the
maximum
error
A#
in our
Potentiometer
Example;
measurement
Potenti
circuit.
ometers are important electrical devices, which we
some
shall consider in
circuit, as
then becomes a two-terminal network with the terminals
corresponding to the terminals of the
galvanometer. The circuit can as in
Fig. IIL5.I.5.
detail in Section III.6.B.
For the present, however, we consider one such
Fig. IIL5.I.5
now be illustrated
By Thevenin
s
theorem we
shown
ometer here
is
in
III.5.I.6.
The
Fig. potenti fed by a voltage generator, E, with
internal
The
relatively
negligible
fraction of
R between the tap and the lower node
is
x.
We now
apply Thevenin
impedance.
s
theorem to the
regard this entire circuit as a single voltage generator connected across the two right-hand nodes, with a certain impedance. Let us now calculate the characteristics E* and
Z*
of this generator.
Applying our method, we obtain the following results. R! and R% constitute a voltage division as
do
3
and R^. Thus, the open-circuit voltage is Fig.
*
=
m.5.L6
R>
(III.5.I.6)
The equivalent resistance Z* may be obtained by replacing the voltage generator
E
by a
two-terminal circuit between the tap and the lower node. The open-circuit voltage is given by voltage division:
direct
E*
= Ex
(III.5110)
connection:
The equivalent
internal
impedance
is
again obtained by ignoring the generator, and con
sequently
At
=
the extent to which
Z*
is
equivalent to parallel resistances
^
balance, theoretically, the value of is such * that 0, and we can define the error
A^ as
^
differs
from this theoretical
Z*
= Rx
+ R(l - x)
= Rx(\-x)
(III.5.U1)
Suppose a load resistance
The
the right-hand nodes. these nodes
THEOREM
59
in the primary or secondary coil, depending
on
APPLICATIONS OF THEVENIN
HI.5.I.
r is
connected across
potential drop across
S
the subscript.
(Normally one can consider the
is
seen to be
secondary current as decreasing H.)
=
Ex
flux
The magnetic e
Ir
(IIL5.I.12)
+ Rx(l - x
$
From this we
of the potentiometer to the
circuit.
see that the
amount
(See Fig. III.5.I.7.)
output
effective voltage
is
very nearly proportional x tapped, provided that the load r
magnetic
-/(#)
(III.5J.14)
In the case of an air-core transformer, linear function of
where
IJL
is
a
H, and <&
Rxn-x)
in a
where
from the equivalent
H results
force
in the core
= /*#
(III.5J.15)
depends upon geometrical considera
tions.
Modem
magnetic materials permit a broad
range of linear response in which Eq. IIL5J.15 essentially holds. (See also Bozorth, Ref. III.5.b.)
Suppose
now
a voltage
is
used to drive a
current in the primary of the transformer.
The
variation in flux in the core results in a back
voltage which
Fig. IIL5J.7
is
proportional to the
number of
We suppose that the generator generates voltage Eg and has internal impedance Zg and
turns. is
with R. It large in comparison
is
the
precisely
impedance which accounts for departure from linear output.
this
internal
a
that an impedance,
ZL
secondary (see Fig.
III.5 .1.8).
is
,
connected across the
The
resistance of
If the voltages used
Example; Transformers. do not have a direct-current component, we obtain the effect of varying the internal
may
of the impedance of a generator at the expense of a transformer. This has voltage by means
wide practical application in the output stages of
Fig. HI.5J.8
radio receivers.
A transformer, in general, consists of a core of number of coils wound magnetic material and a will confine our attention We the core. around to the case in will
which there are two
be referred to as the
other as the
"secondary."
coils.
"primary"
coil,
Approximately,
coils are in the voltages across these
as the
One coil
same
the core, which i.e.,
and
it is
is
considered to be part of
Z
ffJ
2g consists of the nontransformer impedance
and the resistance of the primary contains Similarly ZL
a
series
coil in series.
impedance for the
resistance of the secondary.
Then
ratio
E,
=2
i
g l
+
nL
^-
(III.5JJ6)
dt
A current
a flowing in a coil produces magnetic force
turns
the
i.e.,
number of turns.
Let us consider such a transformer.
turns,
the
the primary coils
H on
be measured in ampere to the number of proportional
at
may
to the current in the coil.
Thus,
we
(The actual output voltage may only be a fraction of this since one must take into account the resistance of the secondary.)
have
H=
n1 i1
-n
(ffl.5J.13)
z i,
where n is the number of turns and
z
is
the current
We
also
have
*=fiOyi-A)
(HL5.I.18)
CIRCUIT THEORY
60
We can rewrite Eq.
by the square of the turns
III.5.I.16 as
The
be part of Zg in
d Using Eq. HI.5.I.18, to obtain be rewritten as
,
Eq. 1.17 can
ratio (see Fig. III.5.I.9).
resistance of the primary series
with the
is
considered to
rest.
The practical
output impedance has a term corresponding to the resistance of the secondary in series with Z .
l
(IIL5.I.20)
In the usual operational notation
Substituting in Eq. III.5.I.19 yields
(III.5.I.22)
Dividing
by
III.5.I.22,
we
the
coefficient
of
ZL
Fig. III.5.I.9 i%
in
Eq.
Transformers are used to correct discrepancies
obtain
between the generator impedance and the load ZL We have seen above that these should be .
maximum power transfer. For one desires to use a vacuum-tube
matched for instance, if
where
amplifier with a generator (III.5.I.24)
ohms to
impedance of 2,000
drive a speaker with a
4-ohm impedance,
a transformer can be used to cut
pnp Eq. III.5.I.23 is clearly the equation associated with a voltage generator producing the voltage
(III.5.I.25)
power, since, then, the generator impedance should be small relative to the load.
Example:
Voltage
Generators and Current
In the above discussion
considered only voltage generators. (III.5.I.26)
and connected to a load impedance Zz a generator
is
transformer,
connected to
it is
its
.
Thus,
if
load through a
equivalent to a generator with
voltage given by Eq. III.5.I.25 and with internal
impedance given by Eq. design,
:
is
III.5.I.26.
In the usual
chosen large enough so that Jean be
regarded as 1 except for a correction. The effect, then, of connecting a generator through a trans
former
is
equivalent to changing the voltage
generated by the turns ratio and the impedance
the
wishes to transfer a voltage signal rather than
Generators.
with internal impedance
down
apparent impedance of the generator. Impedance matching is also available even in cases where one
we have
A
typical
branch has been described as an impedance Z with a voltage generator E in series with it.
Thevenin
s
theorem shows that
sufficiently general sense,
terminal network
made
if
Z is
type can also be described in this way. then, a situation in
taken in a
even an arbitrary twoup of branches of this
which
E
We have,
contributes to the
inhomogeneous equations, and the nodal equa tions are homogeneous. But we may introduce current generators
which make the nodal equa
inhomogeneous. A current generator is considered to be connected between two nodes. tions
APPLICATIONS OF THfiVENIN
III.5.I.
It
generates a current / which
is
considered to
flow away from one node and toward the other.
A perfect current generator produces
to a
voltage generator
= E.
provided 12
a current 7
THEOREM
S
61
E with series impedance Z
Consider an arbitrary circuit which consisting of the generator E
Proof:
contains a branch
Z
and impedance
Let us denote the
in series.
two-terminal network which constitutes the rest
Jf
of the circuit by III.5.I.10
We wish to
.
3f is concerned,
that as far as
equations will be the same
For Fig,
which
that Fig.
all
Notice
nodal and mesh
e1
if
= e%,
f
a
=
/
2.
we apply Thevenin s theorem and Jf by a voltage generator E* and series
replace
Fig. 111,5.1.10
show
equivalent to Fig. III.5.I.11.
is
1.
10
impedance Z*. But one can
also readily see that
the argument of Thevenin
theorem applies to
s
independent of the voltage between the nodes. Thus, a perfect current generator can be is
considered to have
infinite
impedance.
imperfect so that the current varies
it
by a perfect current generator with
generator with the voltage between the nodes, represent
If the
is
an impedance
in parallel.
This
is
we can
analogous to Fig.
the voltage generator situation, since a perfect
HL5.L12
voltage generator has zero internal impedance,
and an
arbitrary
voltage
generator
can be
represented as a perfect one with an impedance in series.
None
of
our
above
results
is
basically
dependent on the homogeneity of the nodal equations, and one could apply the above circuit discussions
to
circuits
having both types of Fig. ffl.5J.13
generators. However, basically the two types of
generators are equivalent.
Fig.
I.I
as well,
1
by the generator
We shown
must, in
and here also Jf can be replaced * and impedance Z*.
therefore,
Figs.
I1I.5.I.12
compare the and 13. For
circuits
Fig.
1.12
we have immediately
(E
-
eJIZ
= (e l
*)/Z*
(I1I.5J.28)
+ Z*)
(III.5.I.29)
or
^= Fig.
Indeed,
ni.5J.ll
we can now
THEOREM
III.5.I.1
an impedance
:
In Fig. 1.13
(Z*E let
i
+
Z*)/(Z
denote the current through Z.
Then prove:
A current generator /with
Z in parallel with
it is
equivalent
2
=
t
z
= (/ - yz =
f
2
z*
+
E*
(1II.5.I.30)
CIRCUIT THEORY
62
when the nodal and mesh relations last
The
are used.
or
2,2, equality yields i
=
a
(12
(III.5.I.38)
-
E*)i(Z
Z*)
-1-
(III.5.L31)
and
Zl and Z 2 are case,
=
(IZZ*
+ Z*)/(Z + Z*)
E=
present,
12. This shows that a voltage
circuit
generator can be replaced by a current generator
It is
if
if
the internal impedance
to parallel. device.
Note
is
changed from
series
this is a useful analytic
Frequently
which
also that a discussion
only one type of generator
is
treats
theoretically
One can
Example.
impedances
Zl
readily
Z2
and
show
that
if
two
are in series, they are
an impedance (Zl + Z 2). For then equivalent the current i is the same through each impedance, to
and the total voltage across e
Z
2,
=
if
two impedances is
not
problem possible
is
more convenient.
frequently
to define uniquely the potential e l
at any node. We define the potential e at A Q to be zero, and for any other node A j we choose a
chain of branches connecting A^ to
sum of
define e j to be the
AQ
,
and
the potential drops e
Since the
sum of
the
around any mesh is zero, the potential drops e j thus obtained is the same for any potential such chain of branches. The potential drop e across any branch
B is the difference between the
of the node at one end and that at the potential other.
+ Z )f
(2 l
other hand,
the
is
an alternate method of solution of the
along these branches.
adequate.
On the
Zi can be interpreted as
In circuits where mutual inductance
z*
equivalent
differential operators in the general
and hence, e
(III.5.I.32)
Comparing the expressions for \ and e 1 with and e z we see that the two circuits are 2
those for
l
J.
(III.5.I.33)
2
two impedances,
Zl and
are in parallel (see Fig. IIL5.I.14), they are
Now
consider any branch
A j and A k
nodes
voltage generator
Z, /
we may
=
/Z,
with j
B
connecting the If
B
contains a
E in series with an impedance
replace
it
with a current generator,
A k and
between Aj and
another branch
with impedance Z, i.e., the voltage generators may be removed from any branch and replaced
by a current generator in parallel. Thus, we may consider our circuit as consisting of branches which are either passive,
i.e.,
which contain only
impedances or just current generators. The current through an impedance
Fig. III.5.I.14
between nodes at potential e s and ek
ZM
is
to a single impedance Z, which may be obtained as follows: The current through Z1
equivalent
and
Z2 are,
respectively,
^ f
The
total current
i
a
=
= ejZi = e/Z ^
+
i
(III.5J.34)
This
is
considered as flowing toward the node
Let Is denote the (III.5.I.35)
2 2
^
an<
nence >
>
= Zi = Z - +
-
then become
or
(III.5137)
i
Z
-2 fc=:/Z
H
A
.
f
the generated currents
flowing toward the node A s
(III.5.I.36)
Thus,
sum of
.
The nodal equations
COMPLEX CIRCUIT THEORY; FILTERS
III.5J.
We elt
have n such equations, and these determine
.
.
,
,
en
The nodal equation
.
is
particularly
where
suitable for vacuum-tube circuits.
E
may also be pointed out that Eq. HJ.5.I.41 used to represent a system of simultaneous
It is
linear
.
ci>
l5
2,
.
.
o>
,
Z
Thus, the voltage
a).
=
we can
w represent a system symmetric, which has positive terms on
is
,
the diagonal, negative terms
such that the sum of the equation
is
and which
off,
coefficients
in
= a (cos art) +
Circuit
later,
the
many
different
= a (cos
- [sin
sin
which we
(co
frequencies. Often
+
co^i]
.
.
.
will consider
+ ~ [sin (a +
voltage generators involve terms of
minimize
it is
their effects while
the terms associated with other
ft)
-
Jf
-
sin (co
a>
Jf]
necessary
emphasizing
frequencies.
(TII.5.J.2)
frequencies.
wish to transmit information by
means of an
electric signal, for example, the voltage at a certain point in the circuit, the
while
Thus,
A ^
radio, for example, amplifies the signal associated with a narrow band of
When we
.
1
-f
cot)
to discriminate against certain of these frequen cies, i.e.,
.
Theory and (a>
applications,
.
any
Illustrations; Filters
many
+
art)
is
non-negative.
m.5J. Complex In
are small relative to the
m
voltage frequency
equations. If we use merely resistances as
the impedance
which
co
reference
63
the
unmodulated
signal
E=
(cos at) has a precise frequency, the modulated
signal
E
4(cos
varying from
cor)
co
o)
has signals of frequencies
k to
CD
greatest of the frequencies
where
4- cok
to
o)j,
co
m
co
k is the
Thus, in
.
order to convey the desired information one must have a band width of 2coA This means that .
voltage in question must be modulated.
To
convey information we must have a variation
from some of
fixed situation.
fixed
this
voltage zero
situation
It is
natural to think
having a constant
and the variation, then,
variation of the value.
as
voltage
However,
away from
this is
is
the
frequencies which
fixed zero
voltage, but a high-frequency voltage
A Q cos cot. The variation amplitude E of this reference situation may consist in either
if
one
information does
s
not involve signals of frequencies above desirable to
co fc , it is
suppress the frequencies outside this
band width. The broader the is,
which
information is conveyed by electric signals. In most cases the reference situation is not a
- % and a -f o)t
between co
must be detected, while,
this fixed zero
not the usual way by
lie
available
band of frequency
the greater the possibility of unwanted effects,
Specifically, in the case
voltages,
the
the undesired
of thermally generated
energy
is
proportional to
band width.
of fixed
varying the amplitude or the frequency. These are referred to
respectively as amplitude
tion and frequency modulation.
now
modula
We will consider
The notion of band width
carrier.
is
most
an alternating-current However, the same notion is applicable
and desirable
in the case
where one
reference
s
a direct-current voltage. In the band width extends from to cot
voltage
amplitude modulation. Here, of course, the
for a signal
clearly seen in the case of
is
this case
.
variations
in
relative to the
carrier
the amplitude
should be slow
frequency of the reference, or
Suppose the amplitude which represents the signal we wish to send can be frequency.
expressed in the form
A
=a
Designing a passive network with four (or three) external
applied to
have
one
nodes such
essentially the
band of frequencies,
is
a voltage
is
components in a specified a procedure which is well
is
known. Such a circuit is It
that, if
the output pair of nodes will pair,
called
dependent on the
a band pass filter,
fact
that
a branch
CIRCUIT THEORY
64
impedance can be designed
have varying
to
with different frequencies. Let us first consider the reaction
For the component of the voltage
effects
of
elementary two-terminal network to a voltage which contains various frequencies (see Fig.
+
sin (cot
an
a),
solving Eq. III.5J.5 yields
III.5J.1). 21
1/2
sin (cot -f
a
/)
(III.5.J.6)
where
=
tan y Fig.
E
=
e
IILSJ.l It
+
customary to consider an alternating
is
voltage or current as a complex quantity when one wishes to consider only a single frequency.
+
+
(III.5J.7)
ez
[sm
...
The
+ oj]
(co 2 r
real part of this
+ 4sinKr-|-oO]
(IIL5J.3)
If the circuit has inductance L, resistance
capacity C, the voltage equation
R
and
voltage with value e Q sin e
= R[e
sin (cor Q
R[>oH)l>s
(cor
Thus,
+
(cor
terms
of the
charge on the condenser.
Differentiating Eq. III.5.J.4
we
the
+
+
a)
+ a)] +
(cor
j sin (cor
The voltage can be represented by in
the
a)
je cos
-fa)
is
is
if e is
complex quantity
usual value for the quantity.
the
a)])]
complex
quantity
get the current
Similarly the current can be represented
equation
by
(HI.5J.5)
which assumes that the
and capacity are in
-y
series.
Now if we solve Eq. the transient terms,
III.5. J.5 for
we
i
and neglect
obtain a term for each
frequency that is present in the applied voltage E. Since the network equations are linear, this
any network made up of such elementary networks. For each additivity property generalizes to
frequency present in the circuit,
we may
solve the
system of equations which we obtain by writing other components of the various voltages as
all
zero.
In each case
we
will get
a current of the
same frequency. The
current in any individual
elementary network
the
this
is
sum of the
currents in
network thus obtained for every frequency
present.
This procedure, of course, ignores the
transients.
-rr
resistance, inductance,
(IIL5J.9)
-ir/2)]
Furthermore, our result shows that
if
Zw
is
the
complex quantity
then the equation iZ holds.
Eq. III.5J.10
generalization of
= is
Ohm s
(III.5.J.10)
readily seen to be a
law.
Another way of considering the above
is
to
a complex value for the voltage in the differential equation is used a complex value for
note that
if
the current
is
obtained.
Since, however,
the
coefficients in the differential equation are real,
the
same
derivative
relationship holds between
the real parts of these complex quantities as
COMPLEX CIRCUIT THEORY; FILTERS
III.5J.
For large values of q the value of d for which
between the quantities themselves. Thus, for a fixed frequency, the differential equation for a
Ohm
It is called the
while
/
-
+ \R*
impedance. Thus,
i
phase
=
d
essentially
and
,
if
l/2
\
~i
j
J
IHI.5J.13) is
\a>L
generally
Iq
The angle, /, is called the Note that |ZJ has a minimum for
the impedance
called the impedance. shift.
is
1
complex impedance, 2
r
=
|ZJ
}
this
is
of resistance.
=
1
%~
value is often taken as the dividing point between the region of high impedance and low
.
ffl
a generalization of resulting equation s law. Z^ is the generalization of the notion
The
-
q(x
circuit can be replaced by an given elementary equation with a complex constant Z
algebraic
65
lies
between
R and Rlm (approxi
m
otherwise. than Rl mately), and is greater One can also plot the phase angle
1
/=
LC or
Alternately,
l
aictan$fo-jf
we can
(IiI.5J.14)
)
consider a parametric plot
of the complex quantity If VQ
to
is
co,
then
gives the
co
= 27rv
and the frequency which
,
1
(ni.5J.15)
)
a straight line. If we choose our just units so that R c^ 1 unit of length, geometric
This
=
minimum impedance is r
is
(Tr\m
The behavior of |ZJ can be seen if
Z^K+jtyfo-*-
the frequency of the voltage corresponding
rather clearly
one introduces dimensionless quantities:
.* i
.6
ro "
5
!
ll coo-
-.2
The circuit parameter, q, determines the behavior
-.3
of the circuit with regard to frequency. For for a small values of |ZJ has the value R
-.5
q,
relatively large range
=
of around %
>
l
i~
Rq\(%
)\
-.8
if
-.9
then |ZJ is approximately and becomes large when % differs
-1.0
i
\
}
-
r
\ \
-1.2
1
from 1. If q(% ) appreciably because q(% |ZJ tends to be close to R, is
\
-.7
is
high.
i)
V
-.6
Thus,
1.
the percentage band width of low impedance On the other hand, if q is large,
^_r
\
-.4
<
then
!
-
-1.3
l
%~
)
-1.4
to the | squared in the expression appearing is III.5.I.11. If q is large, then % in
Eq.
power
to relatively close let
y
= +
20 and
i
j,
1
for q(%
then %
-
- r1
r i
1
)
<
L
If
Fig.
m.5 J.2
we
approximately
.
,
Hne
We
the resulting plot for q give
is
m.5 J.2).
perpendicular
If
i
is
,
to the real axis at
this
specified,
=
x
=
,
1.
10 (see Fig.
Zm can be read off as
CIRCUIT THEORY
66
minimum for a specified band.
the complex quantity which corresponds to this
each of which
value of %.
For a maximum, we would use in
The above
describes a circuit in which the
inductance and capacity are in series, and for which the impedance is a minimum for a specified
band of frequency.
the
If
inductance
a
is
impedances, each of which
a
is
series
two
maximum on
a
given band.
and
which has capacity are in parallel, an impedance
a
maximum in a specified range is
obtained. For
instance, consider the circuit of Fig.
M/V
If
III.5.J.3.
nnnnr
Fig. III.5J.4
i
The
principle of a
band pass
circuit offers
shown
a
filter
if
we
in Fig. IIL5J.4,
we
readily understood.
Thus,
can now be
consider the see that if
Z2
maximum impedance for a desired range
of frequency and that
if
Zl
and
Z3
offer
a
Fig. ffl.5. J.3
e is the total voltage across the circuit,
the current through
and
i is
then our previous dis
it,
cussion of parallel impedances (SeeEq. III.5.I.39)
shows that Fig. IH.5.J.5
J. Cp<
minimum, then (III.5J.16)
tend to be a If e
is
considered to be a complex voltage,
exp (jcot\
this indicates
a complex impedance
L_
for
Z will A
for the desired range.
number of stages can be used III.5J.5).
The objective is
effectively (see Fig.
to shunt the undesired
frequencies through the shorting circuits, while
.jR_
the low impedance of the series circuits favors
3
C
=
Z
the output across the load
maximum
Co) (III.5J.17)
the desired frequency.
which we have i- 1
2
/
14 _RM J
1/2
T
_
CL
\Cco
T
(III.5J.18) Fig. III.5J.6
In general, these circuits are considered for the case in which
R
is
small relative to Leo.
Hence,
the last factor can be considered one, and
a clearly has
maximum
The above
minimum If
for
2 o>
=
Z
briefly indicate the situation with
pedance. .
some
with relatively high output im
Thus, for a constant current output circuit which tends to shunt
one would use the yields
impedances which are a
for a connected
band of
frequencies.
we want an impedance which is a minimum two bands, we use in parallel two impedances,
for
We
special cases
the nonconstant currents but offers
little resist
ance (just that necessarily associated with the inductances) III.5J.6).
to
the
direct
current
(see
Fig.
III.5J.
A
band pass
Thus, the
circuit
filter
is
shown
in
COMPLEX CIRCUIT THEORY; FILTERS
somewhat analogous.
Fig. III.5J.10 favors the
Fig. III.5J.7 will tend
the circuit of Fig. III.5J.11 favors the higher.
to shunt frequencies for which 1/Cco
<^
&L
the condenser offers a path with impedance
For l/Co>
obtain a band pass results
inductance
series.
high in
this
range.
low frequencies, while
These can be combined as before in
for these frequencies, while the impedance of the is
67
from repeating such a combination
Another type of used
series to
Sharper discrimination
filter.
to
in
element which can be
filter
discriminate
against
a
prescribed
AAAR
Fig.
DI.5J.7
Fig.
IH.5 J.8
Fig.
HI.5.X10
Fig.
m.5. J.ll
Fig,m.5.J.9
circuit Similarly the
shown
in Fig, III.5J.8
will discriminate against frequencies for
Leo
<
1/Cco,
i.e.,
against low
which
frequencies.
A Fig. ffl.5 J.12
combination can be used to discriminate against which do not lie in a certain all frequencies
interval (see Fig. III.5 J.9).
The
first
part dis
frequency
is
based on the use of an alternatingis balanced only at a
criminates against frequencies which are higher
current bridge which
than a certain number, the second against those which are lower than another number. A
prescribed frequency.
sequence of such
filter
sections
can be used to
For example, in Fig. A and voltage between
1II.5 J.12 the open-circuit
B
is
obtain discrimination as sharp as desired.
For audio frequencies of from 20 cycles per second,
the use of inductances
inconvenient, in general,
and
(III.5J.19)
to 20,000 is
resistor capacity
combinations are preferred. Thus, the
circuit
3
of
where
Zl = R +
and
Z2 =
Rf(l
CIRCUIT THEORY
68
voltages e lt e z ,
Substituting yields
and
=
1
3+jcoC -
zero
is
= frequency /
when
co
=
1/7?
+ 4)
(,
(III.5J.21)
(III.5J.22)
.
A
Rcod
C
or
when
the
+
(L
(1H.5.J.20)
42
are
1/Zo)
= pC^ +
pCRjl, and eliminating
\llvRC.
.Re 2
(III.5J.23)
(See Eq. IIL5J.41 above.) Theny^ =;co/co will
2
=
E where
^
and
yield
(IH.5J.24)}
Z =
implies ^
=
^I//?C
=
=R+ = 0.
IjpC.
Thus
co
=
co
and E
1
The importance of the bridged-T circuit lies in it provides a "bridge effect," i.e.,
the fact that
discrimination against a fixed frequency in a
A/V^-rVVV
r
circuit
where one
This
very useful in coupling circuits involving
is
vacuum
side of the
input
is
grounded.
tubes.
References for Chapter 5 Fig. III.5J.13
a.
One well-known combination
of these ideas b.
which
used to discriminate against a fixed
is
frequency III. 5 J.I 3).
is
the
The
"bridged-T"
circuit (see Fig.
discriminated value of
co
= 2JR C, and the corresponding frequency / = \ITTRC. Let i = co/co and L^pC+ljR, co
and
is
pp. 132-34. R. M. Bozorth. Ferromagmtism,
New York, D. Van Nostrand Co., 1951. P. Le Corbeiller. Matrix Analysis of Electric Networks, Cambridge, Mass., Harvard University Press, 1950. pp. 20-26.
d.
Massachusetts
e.
Institute
Staff.
of Technology,
Electric
Circuits.
Electrical
New
York, Wiley and Sons, 1940. pp. 1-120. L. and N. I. Adams. Page Principles of Electricity. New York, D. Van Nostrand Co., 1949. pp. 333-73.
Engineering
Z
be an impedance between and The nodal for the nodes with ground. equations let
c.
is
H.W.Bode. Network Analysis and Feedback Amplifier Design. New York, D. Van Nostrand Co., 1945.
Chapter 6
ELECTRICAL COMPUTING
IDL6.A. Introduction
The
used
potentiometers
in
computing
is
machines usually consist of a resistance, in the form of a wire wound on some rigid form, and a
the reason necessary to use amplifiers, However, for this is associated with certain specific aspects
this moving contact on this wire. The position of contact depends either on the rotation of a shaft
In order to use the principles of
computing purposes
of the way
a practical
way
components are used
circuit
For
this
discusses the
way
puting.
in
electricity for
for
it
com
or,
reason the present chapter in
which
circuit
components
are used to realize certain mathematical relations.
more
rarely,
schematically
in each case, principles
a
resistance
A
and
the
in
present
B
R
connected
(see Fig. III.6.B.1).
B
T_
obtained by using one of the
described
linear displacement of a
ACH-v\AAAAA
A practical computing component, is
as
between two nodes
of amplifiers Chapter 7 is devoted to the theory and shows how practical computing components are obtained.
on the
Thus we may represent a potentiometer
bar.
chapter
Fig.m.6.B.l
properly supplemented by amplifiers. This chapter, therefore, is devoted to a discus
tion,
immediate application of condensers and induct ances for integration and differentiation.
factors
variable
are
requires
the
introduction of certain motor-driven mechanical the most modern developments components. In various devices have been introduced to avoid
mechanical processes.
We
developments in Chapter
will
some point on R, so C and B is, say, yR,
while the resistance between
A and C is (1
discuss these
at
E
- y)R.
and if no
if a voltage appears across AB, current flows out through C, then a current
Thus,
The
use of potentiometers for electrical multiplication
when both
C is located
that the resistance between
The contact
sion of the use of potentiometers for multiplica the various uses for resistances, and the
= EjR will flow through R, and a voltage, = yE, will appear across CB. Then, if the yRi i
E represents the variable x, the voltage CB represents the desired product yx. We
voltage across see,
then,
that
the ideal potentiometer
is
a
multiplier.
Potentiometers built for computing purposes moving contact
8.
are invariably wire-wound. If the
IEL6.B. Potentiometers
is
Potentiometers are important in the study of
mathematical machines because they afford an of the simplest and most common example of one
A
x
variable electric multipliers. types of a voltage, a second variable y is represented by the mechanical rotation of a shaft. represented by are the inputs to a potenti These two is
controlled by a linear displacement, then the
is usually wound on a cylindrical form, while the contact slides along an element of the
wire
cylinder.
Another
common form
has the wire
potentiometer card which tangular
is
The contact is located by angular rotation of a shaft and moves along the edge of the wire-
wound card with a
possible
is
held constant for the duration of a
computation.
rotation of 300
or
330.
magnitude
when y
a rec
bent in an arc of a circle.
quantities
ometer; the output is a voltage, having (ideally) This is particularly convenient the xy.
of wire-wound
wound on
In addition to these
common
types,
potentiometer, especially designed
there
is
a
for computing
ELECTRICAL COMPUTING
70
machines, in which the wire-wound form
The
helical in shape.
contact arm can turn through ten full revolutions,
3,600. In the United States these are
i.e.,
manu
factured under two trade names: Micropot and Helipot,
Special dials are available
number of
the
which count
turns, as well as fractions of
turn. (See also F. R. Bradley
F. E. Dole, Ref. II.6.b;
Duncan, Ref.
III.6.C.)
Potentiometers are usually
B
is
impedance Z*
the desired product yE, the load
must be taken resistance
compared
large
of
y)R
y(l
generator combination. multiplication to within
we must
values of y,
the
To
to the internal
potentiometer-
insure
accuracy in
per cent for
all
take the load impedance
Z*
.05
R when E* =
to be nearly 500 times as big as
0.
y(i-y)R
resistance between the
node
a
and R. D. McCoy, D. C.
Ref. III.6.a;
In order to obtain a result reasonably close to
is
which moves the
shaft
wound
so that the
moving contact
C and the
of proportional to y, the fraction
complete rotation of the ometers are said to be
shaft.
"linear."
Such potenti
In the ideal use of
no current flows through potentiometers, In practice, however, such a current is
these
C.
Fig. HI.6.B.3
invariably present, and the effect of this current or load may be estimated by the methods of circuit
Suppose that the potentiometer
AB
is
con
nected to a voltage generator having output
E
and negligible internal impedance. Assume also C and B are connected to an arbitrary
that
By Thevenin
circuit.
s
theorem, the latter
may
be replaced by a voltage generator with output *
and
series
impedance Z*
The
difficulty
here
not so
lies
that the actual output differs
theory as follows.
(see Fig. III.6.B.2).
output;
it lies
much
in the fact
from the desired
rather in the fact that the amount
of error varies with the setting of the potenti
This
ometer.
effect
limits
the possibility of
cascading potentiometers, since the increase in values
resistance
250,000 between
would involve a factor of
first
potentiometer and second
load for two potentiometers, and of 125,000,000
between
first
potentiometer and third load for
three potentiometers. feasible,
The ratio
and for three
it
is
for
two
is
seldom
almost always
impractical. It is
possible to correct this situation by intro
ducing amplifiers between successive potenti ometers.
Despite the name, the function of an
amplifier
here
is
not to increase the voltage
output of the potentiometer, but rather to reduce its
internal impedance:
Fig. HI.6.B.2 "amplified."
it is
the energy
discussion of amplifiers in Chapter
As we have seen the circuit to the
in Section III.5.I, the part
left
of the points
CB
of
can be
replaced by a voltage generator generating a voltage
E = yE
Z = tyO -y} drawn
voltage
as in
CB is
and with internal impedance Thus, Fig. III.6.B.2 can be
Fig. III.6.B.3.
Hence, the output
For the
which
is
We shall return to a more complete
present,
we
amplifier of the proper type
two potentiometers, as
7.
simply remark that is
if
an
inserted between
in Fig. III.6.B.4, then, the
amplifier appears as a large resistance (of the
order of millions of ohms) with regard to the on the first potentiometer, while relative to
effect
the second potentiometer, the amplifier appears as a voltage generator, generating a voltage
y-f
with negligible internal impedance (of the order
POTENTIOMETERS
IIL6.B.
of 2 or 3 ohms). This situation is
needed to handle the
is
precisely
what
discussed
difficulties
above.
same phase
71
as E, a suitable condenser has to be
shunted across AB.
Unfortunately, most auto-
much
transformers have too few coils to be of
The auto-transformer may
at first glance
to be similar to a potentiometer.
seem
This device
to alternating current, but applicable only
is
is
use for computing.
(See also
J.
B. Gibbs, Ref.
E. Karplus, Ref. III.6.e; Soroka, Ref. IIL6.g, p. 50.)
IIL6.d;
It
possible to
is
obtain a substitute for a
which
potentiometer
and W. W!
has
constant
internal
impedance with, however, a certain sacrifice in voltage. This is done by means of a device
known
as a
The T-pad illustrated in by a voltage generator with
"T-pad."
Fig. IIL6.B.6
is
fed
output E and internal impedance R. The settings
on the card-wound
SETTING
SETTING
y,
resistances
X
and
Y
are
not independent but mechanically linked, as indicated by the dotted line.
^
Now,
Fig. IH.6.B.4
possible to shape the cards for
it is
X
and Fin such a way that the internal impedance of the entire circuit above is again R, while the voltage output is lyE, where y is the common setting for
prove
these resistances,
this statement,
we redraw
<
1.
To
Fig. III.6.B.6,
Fig. IIL6.B.5
analogous to the potentiometer with an induct ance instead of a resistance. (See Fig. III.6.B.5).
However,
relative to the potentiometer,
it
has
one advantage in that the output impedance related to the generator
impedance of
square of the turns factor.
is
Fig. IH.6.B.6
E by the
Thus, normally the
output impedance would be smaller than the generator impedance unless an output voltage higher than the primary voltage is
the generator impedance of
is
itself in
Z
the circuit
(Z should include any impedance due auto-transformer
If
desired.
E for
to the
the primary circuit),
the impedance of the output of the auto-trans former is jy 2Z, where y is the ratio of the number
of turns between
C and B to
B, and the generated voltage III.5.
1
.)
There
is
that between is
A
a phase difficulty with alternat
ing-current devices: currents can readily be
pared only if they are in order to
make
in
Fig. IH.6.B.7
and
yE. (See Section
com
phase. Thus, in general,
the generated voltage have the
adding an arbitrary load
Z connected across the
right-hand terminals (see Fig. III.6.B.7).
The
circuit of Fig. III.6.B.7 is readily seen to
be equivalent to a circuit with a current generator
ELECTRICAL COMPUTING
72
(see
Fig.
The
III.6.B.8).
total
nt
Solving Eq. III.5.B.2 and 3 for
impedance in
with the current generator parallel
is
now
^^
X = 2R
_ R(R + X) 1R + X
X and
7,
we
obtain
(III.6.B.4)
(IH.6.B.1)
Y
It is
apparent
= *-R
(IIL6.B.5)
now why \yE was chosen to be
desired output voltage:
if
instead, the expression for
"six
X=
the
had been chosen
yE
X would
have been
which assumes negative values as y to 1. ranges from 1,
Since the generator impedance of such a T-pad is
the
fed,
as that of the generator by which it is of T-pads can safely be number any
same
cascaded to obtain a multiplying device capable of handling many input factors at once. For
Fig. HI.6.B.8
each T-pad,
we must
accept a 50 percent reduc
tion in voltage-range size, but this
be
may
com
pensated for at the end of a single (voltage) amplifier
The
having the same impedance.
may then be described schematically as in IIL6.B.11. (See also W. W. Soroka, Ref.
system Fig.
pp. 52-55;
III.6.g,
m.6.B.9
Fig.
and
P. K.
The disadvantage of
III.6.f.)
McElroy, Ref!
the T-pads
is
the
expense.
VOLTAGE AMPLIFIER
(MUL1ULIESBY8)
SETy 2
SETy,
Fig. IH.6.B.10
Therefore,
we can
Fig. III.6.B.11
replace the current generator
by a voltage generator, producing
E = Now, suppose
IR
= ERK2R +
f
R(R
+
2R
X)
,
can double the range of a potentiometer by the use of a double-pole, double-throw switch and an
X}.
extra resistance, (see Fig. III.6.B.12). resistance has
v (III.6.B.2)
X
Then
the circuit of Fig. III.6.B.9
to Fig. III.6.B.10.
(III.6.B.3)
+X
scale."
scales,
equivalent
is
"folding
one for the right-hand position of the
looks like Fig. III.6.B.13 tion
extra
of the
The potentiometer now has two
switch, the other for the left-hand. is
The
precisely the resistance
potentiometer P. This has the effect of the
R 2R
There are a variety of ways in which potenti may be used in computing devices. One
ometers
we have
that
SETy 3
180 and the load
is
when
The
scale
the total deflec
negligible.
The upper
POTENTIOMETERS
I1I.6.B.
scale
corresponds to the right-hand position of
The
the switch.
result
is
similar to that of a
potentiometer with twice the scale length and twice the resistance.
73
and the setting of the potentiometer by a number code with a vernier corresponding to the
in this
simple potentiometer
This circuit can be
itself.
used in the process of digital to analog conversion or vice versa.
The
controlled for this
switches
Sl5
.
.
.
,
Sr can
be
purpose by relays.
OUTPUT
Fig. IH.6.B.14
Fig. III.6.B.13
possible to fold the scale repeatedly by such means. For example, in Fig. III.6.B.14 the It is
scale
is
folded twice, and
The lowest end
^ has the resistance IP.
of the scale of the equivalent
larger potentiometer
is
switches to the right.
switch position.
obtained by setting both
Let us call
this the
The reader can
successive switch positions are rJ 2 , IJ^ and i.e.,
to pass
one switch are used,
from one position
is
/
at
can be placed on the potenti
each turn in the
scale, the corre
sponding switch can be indicated.
Ten such
switches would give a multiplication of 1,024. is,
r
X 2,
to the next, only
thrown. If only a few such switches
all scales
ometer and
7y 2
verify that
of course, possible to construct a purely
a number of such switches,
Sl9
.
.
.
,
Sr
,
are used, the position of each can be specified by
It
is
possible to use six double-pole,
also
multiplication by ten almost directly (see Fig. III.6.B.15).
The
R:
resistances
or 1. The full position of the by switches can then be indicated by a number in
the value of
the cyclic binary code (see Section ILL, App. 4)
between the I range,
a bit
m.6.B,15
double-throw switches to give an element reading
dyadic potentiometer.
When
Fig.
It
i.e.,
is
5?.
R have
The
first
the value ?,
double-pole,
double-throw switch, of course, distinguishes to .5
and
.5
to 1.0. Only
ELECTRICAL COMPUTING
74
one of the remaining switches should be thrown
of the careful calibration of each and would save
to the right.
dial
This will insert the potentiometer into the circuit in series with four of the smaller jR s in
same
order.
space.
In this case, the addition of two
rheostats in series with the potentiometer, will give
a
"vernier"
for the resistance (see
setting
Fig. III.6.B.18).
Fig.
m.6,B.16
such switching arrangements are In fact, the usual method of setting up possible. a decimal potential is based on a double-selector
Many
The output has
switch (see Fig. III.6.B.16). resistance
R
of one of the
which are
s,
Fig. IH.6.B.18
the all
There are other partly
equal.
A
more
efficient
have eleven
to
method of decimal
resistors in series
setting
is
and have a
double selector which shunts two of these by a having 2R
potentiometer III.6.B.17).
be
itself
Of
resistance
a decimal arrangement of
potentiometer, for example, the in Fig. III.6.B.19,
R. (see
Fig.
course, the potentiometer could this sort.
M. Walker
\
methods for
from a
method shown
which appeared in a paper by Ref. III.6.h). Here it is
(see
assumed that the voltages + X and X are available, and then division into voltages J, .5 J, X is made. There are six five0, -.5 J, and position switches
\
digital
obtaining the voltage division obtained
whose contacts
receive these
voltages and whose movable contacts are con
nected to resistances of values R, 5R, KIR, 50.R, 100.K, is
and
50CLR, respectively.
connected to one of the
ductances of values .017, and .0027.
Thus the point EL
five
Y= (IjR), If
E
k)
voltages
by con
.2Y, .17, .027,
where
denotes the voltage to which the
(1
ML
switch
is
connected, then the current equation for the Fig. IH.6.B.17
point
I is:
In the case of continuous resistance potenti ometers and, indeed, in other cases
also, to realize
the greatest possible accuracy in setting, the
potentiometer bridge.
When
potentiometers,
may
be
set
by a Wheatstone
one has a large number of such this would eliminate the necessity
(III.6.B.6)
ELECTRICAL ADDITION
IIL6.C,
?50R
75
>500R
>IOOR
Fig, IIL6.B.19
If
we
solve for
El
+
EL we ,
In the paper
obtain
plished .2
2
.02
4
+
.Ql
5
.002
6
voltage
(III.6.B.7)
n
M=
1,332
+
1,
fc
.5,
RL 0,
in a
dividers.
-.5, -1.
these values in Eq. III.6.B.7,
we
fc
X, where
Substituting
of thirteen such
High-precision resistors are
needed only for the resistances with values R, 5R, and 10]?. In the case in point where many such
.
the settings has certain advantages.
ing
002^)
is less
expensive
than the corresponding potentiometers would be. The use of punch cards to contain the values of
The punch
than setting potentiometers would can be removed from the values the and be, machine and put on again without a resetting. There is also a permanent record of the values
obtain
M
was accom
punch card. Thus one IBM
dividers are used, this system
-
Now Ek can be written in the form a a has values
cited, the switching
card contained the setting
+
M where
by holes
X
is
simpler
used. (1II.6.B.8)
If
ax and a 2 are permitted to assume independ the values given above, then a x + -2a 2
ently will
of .1,
assume values between
.1, .
in steps
1 .2
in particular, the values -.9, -.8, .
.
- 1 .2 and
,
.9.
Similarly, .Ia 3
+
.
.
.
,
0,
assume
.02a4 will
DI.6.C. Electrical Addition
Kirchhoffs law on the currents at a point of Section III.5.C) can be used for junction (see we In Fig. III.6.C.1, at addition. electrical
C
have
/!
+
f
- h = 02 5
values between -.09 and .09 in steps of .Oloc 5
+
will
and
,002a 6 will assume values between -.009
and .009 sion
.01,
in steps of .001. Thus, the total expres
K+
.2a 2
+ .la, + .02a4 +
.01 a 5
+
.002a 6 ]
assume values between -1.299 and 1.299
steps of .001,
analogous
and we have then a voltage
to a
potentiometer.
in
division
Kirchhoff
s
law also permits the addition of a are measured from a
number of voltages, which
common point. Suppose hi an electrical network we have a number of x, j, z, etc.,
etc. points, A, B, C,
Let
denote the potential of these points
relative to a fixed point 0.
Let us suppose
(See Fig. III.6.C.2.)
now that each of the points A,
B,
ELECTRICAL COMPUTING
76
C,
etc., is
value
R
connected through a large resistance of to a point P.
potential
w relative to
Suppose
that
P
the origin 0. Let
f
x,
has a z
2,
z
or to use first
it
as
an input for another circuit. In the suppose we use a voltmeter n
case, let us
3,
whose
resistance
is
.
We
would connect
this
A between
P
and hence there would be an
and
-
additional current of
The equation x
for
w
lw
R
to P.
then becomes
+z=
4. y
flowing from
(m
+
A)w
(III.6.C.4)
Thus, the effect of introducing the voltmeter just to
is
change the constant or proportionality
for w. Fig.
m.6,C.l If
we wish
to use the voltage
of the
other portions
we
practical cases,
In Chapter 7
we
will
w
circuit,
as
an input in in most
then,
have to use an amplifier. amplifiers used
will describe the
for this
purpose. Voltages from independent circuits can be added by connecting the circuits in a proper
For example, if we have two batteries, connected as in Fig. III.6.C.3, with potenti
fashion.
ometers across them, then the voltage z
A and A
since the points
etc.,
denote the current flowing from A,
etc.,
to P,
J?,
are
x
= x + y,
volts
above
above
these.
C,
Then
(IIL6.C1) Since,
by Kirchhoff
currents
is
zero,
s
law, the
sum
of these
we have x
Fig. IIL6.C.3
i
ground, and the point
Hence,
R
B
is
x
+y
trouble with this
or (III.6.C.3)
is
passing,
we might mention
the
bination of x, y,
z, etc.,
R
rather than a fixed value.
Of course,
left.
that a linear
In
com
can be easily obtained by
using different values of
coefficient of
the
is
y
volts
above ground. The arrangement is that an addend
can be used only once. After one such connection the circuits are
number of terms on
where n
B
volts
no longer independent. However,
the batteries can be replaced coils
by the secondary
of transformers.
When two resistances are in series, the resulting the
sum of the
two. Inductances in
in the connections
resistance
is
However, then the
series or
capacities in parallel have the
same
no interaction
in the
w depends upon these values of R. we would like either to measure w
property (provided there first case).
is
CONDENSER INTEGRATION
III.6.D.
The sum of two
resistances
may be determined
by means of a Wheatstone bridge and combination,
In Fig. III.6.C.4, the
sum
servo
of the
motions
77
and
rotations),
(ordinarily
these
motions, in turn, are represented by resistances by the use of potentiometers. The Wheatstone bridge can be used for addition, multiplication,
and subtraction by a form of com For example, the following device
division,
plementation.
shown
in Fig. III.6.C.5 will
produce a rotation A
corresponding to
(IH.6.C.5)
Now
if
the resistances
xlt * 2
,
Z
and Tare each
on potentiometers, we can introduce these variables into the device in the form of rotations, and the output
a rotation.
is
HI.6.D. Condenser Integration
Normally, in
Fig. IH.6.C.4
resistances
^
and
r 2 is desired
and
is
to be read
from the position of the contact C. The position of is
C is adjusted by a servo motor, which, in turn, controlled by the
amount of unbalance of the
bridge.
This unbalance appears as a voltage
which
power amplified
is
electrical
computers integration The use of feed
involves the use of amplifiers.
to control the motor.
back amplifiers for III.7.G.
.Section
where high frequencies are used, the
particularly
simple circuits
may
We begin with
be used.
the case where the signal to be
a
is
integrated
this purpose are discussed in In certain cases, however,
direct-current
voltage.
The
a simplest type of direct-current integrator is condenser. Let us consider the simple circuit
shown in
Fig. IH.6.D.1
If q is the
.
condenser, then the voltage
denser
is
qj
condenser.
is
da ~
9
where
C
The current
is
E
charge on the
across the con
the capacity of the
through the resistance
i
and thus we have for the input voltage:
,
at
E
= R^ + dt
R
=-l + dt
dt
C
q
(IIL6.D.1)
q
(HL6.D.2)
RC*
*SC
Fig, ffl.6.C.5
-( While
this device
the purpose,
it
is
may seem
too elaborate for
the simplest example of a this
r
technique, variables are represented by physical
J fl
technique having
(IIL6.D3)
dt
many
applications.
With
2
r}RC
e
R
^_
.j-^fr
(1II.6.D.4)
ELECTRICAL COMPUTING
78
Dividing by Ce^
RC
we
obtain
Differentiating
and dividing by R, we get
*
1 = Rdt We may
+ JL, RC
dt
solve Eq. III.6.D.8 for
i:
Now if \IRC is small, i.e., if RC is large, then we see that
we have approximately
RC 2
1
f<
RC
-
EdT =
(1IL6.D.6)
I
a
(III.6.D.9)
*ti
If
The
_
factor
since
j
1/#C
is
we can
dr
/
not particularly troublesome
E
amplify the output
=
at the time
^ and
if
IjRC
is
relatively
large, then the integral on the right represents a
.
dE time-delayed value of
E
A/VW
T
.
dt
The output voltage
derivative of a current can be obtained in
an analogous way,
at least
linear inductance.
The
theoretically,
difficulty,
from a
of course,
is
most inductances have a good deal of
that
Fig. IH.6.D.1
C
isRL The
=L
?
/!
resistance associated with them. latter
For
this purpose a mica condenser or an oil paper condenser should be used. An electro lytic condenser is not suitable because to be
However, the can be compensated for by means of a
certain circuit.
or
effective the polarity
If
we
take
R=
must be maintained, 10
megohms and C
=
10
we
get a time base of a second in which the errors due to ignoring the factors
microfarads,
,
and
e^-^ IRC is
less
than
1
percent.
Fig. HI.6.D.3
Let us suppose we have a current i flowing through a circuit. Fig. III.6.D.3 shows part of
We
this circuit.
suppose that the variations in
voltage in the part is
circuit
which
is
similar
different values of the constants differentiate
can be used to Let us put the
We will have again
voltages,
On the
negligible as far as
i
drop in
(IIL6.D.7)
we
see that the output voltage
other hand,
if i varies,
is
zero.
the voltage across
di
the
box is L
+ Ri, and across the resistance R, at
it is
C
is
steady, the voltage
=
but which has
the equation
dt
shown
i is
box equals that across R, and since the effect of the resistances R R z is to average the end
E (see Fig. III.6.D.2).
output across the resistance.
If
the
Fig, IE.6.D.2
A
concerned.
L
Ri.
The output voltage
+ Ri and dt
is
then an average of
Ri and proportional to
.
dt
CONDENSER INTEGRATION
III.6.D.
If the
signal to be integrated
current voltage, one
-
is
an
alternating-
The
net result
79
is
C combination
replaces the
byanLCcombinationasshowninFig.ni.6.D.4.
-
r
WIT
I
L
o
I
+
I
For a certain frequency
this is equivalent to
combination for direct-curL.
that the
voltage
is
proportional to
q(t),
the charge on the
For
we
(If
start
We
t(cos
A,,./. Wsm
(Ujt)
/TTT,--^
x
(IIL6.D.19)
with?
=
and
^=
then a and 6
0,
dt
^ ofthesame
,
m-f.
i
+
are both zero)
(IH.6.D.10)
2
two hnearly independent of the homogeneous equation are
and cos
sin (Ojt
OIT)
Now suppose I has a component
1 2
(^(IC)-
solutions
w
\
>
^
= ~9+L
<j(cos
/, + (*
an
Then E(0
t
f /c
Suppose
(0 and the output
is
input voltage
condenser.
/
-
J_ c
at)
^C
-
frequency as
^
Then
=
*+
use the method of 1/2
variation of
parameters in order to obtain the
jo
We let
complete solution.
q =^(sinco 1 r)
With the condition
/
I
+ ^cos^)
/(r)(siQ
+
(
\L/
} sin (
L
_ l JJ )}
JT
(III.6.D.20)
(III.6.D.11)
wnere
that
,
f
refers to the contribution
+
of
to q.
we have
M
- %(sin (IIL6.D.13) q B*(cos (o^t) -^co^sin OjO + A(cos o^O - JJ^sin otf) (IIL6.D.14) 04)
-rJKoos (^ + y)) dr = d-
1/2
-(
/(T) dr\ cos
(^ + 7)
Substituting in these values for q, q in Eq. III.6.D.10,
we
obtain
(flI.6,D.21)
Eq. III.6.D.15 plus the equation ^(sin otf)
+ 5(cos co^) = /
and the
fact that
%=
1
-
Now suppose/ (r) changes only slowly relative to (IIL6.D.16)
I
yields
the second integral in the braces can
n (j
/2
/
^ T^
be written
u/2
MOOS 2ov) A)) +
v
1/2
sn
cos
fat
- y)
7
I
I
Msin 2^7) dr) sin (o) ~ y) L
f
...18 (IIL6.D.22)
ELECTRICAL COMPUTING
80
These two integrals
will
be small when cos 2co 1 r
oscillates quickly relative to/(r), and, thus, the
only important term
is
This device can be used as an
integrating
Normally, it will be necessary to measure means of an amplifier which has a negligible q by circuit.
on the
effect
circuit.
1/2
f(r)dr
2L
References for Chapter 6 a.
F. R. Bradley and R. D. McCoy, with "Computing servo-driven potentiometers," Tele-Tech, Vol. 1 1 no. 9 ,
(III.6.D.23)
(Sept. 1952), pp. 95-97, 189, 190. b. F. E. Dole, "Potentiometers," in J. F. Blackburn, ed.,
Components Handbook, Chap.
Thus, the output voltage has a component of frequency o^ and with a phase which has shifted
90 compared with the phase of the input com + The amplitude of the output com ponent E
Hill
regarded as a slowly varying function of the time. Now if, in addition, E has contained a com
ponent of frequency (!, the
show
o>
2
somewhat removed from
computer
to
negligible, that
is, it
small highly oscillating terms.
component would consist of
New York, McGraw-
of precision servo Tele-Tech, Vol. 11, no. 11
"Characteristics
potentiometers,"
(Nov. 1952), pp. 52-54. Gibbs. Transformer Principles and Practice. 2ded.
d. J.B.
New York, McGraw-Hill Book Co., e.
f.
g.
E. Karplus, "Design of Variac Eng. Vol. 63 (1944), pp. 508-13. P.
K.
McElroy,
W. W.
1950, pp. 142-45.
transformers,"
Elect.
resistive
"Designing attenuating I.R.E. Proc., Vol. 23 (1935), pp. 213-33.
Soroka. Analog Methods in Computation and New York, McGraw-Hill Book Co., 1954.
Simulation.
that the output due to this
would be
D. C. Duncan,
networks,"
same type of argument can be used
8.
Co., 1949 M.I.T. Radiation Laboratory.
Series, Vol. 17. c.
.
ponent equals the integral of the amplitude of E+
Book
h.
R. M. Walker, "An analog computer for the solution of linear simultaneous equations," I.R.E. Proc., Vol. 37, no. 12 (1949), pp. 1467-73.
Chapter 7
AMPLIFIERS
m.7.A, The Basic Notion of an Amplifier The passive circuits described in the
previous chapter have the voltage or current relations desired for computing only when the circuits are
Computing normally requires a com bination of elementary relations but if these circuits are combined the individual isolated.
directly
relations
on voltage or
currents are distorted.
However, the use of amplifiers permits one to practically eliminate the effect of one circuit on
be negligible, and that the impedance be very Consequently, as far as the input part of
large.
the circuit
is
it
concerned,
makes
the circuit that the amplifier
this
generator
is
is
to be considered, in general,
electrical signal.
At times
is
controlled by an
the ratio of output
power to input power is important, but normally one has an input signal and power output. See
is
in series
and
is
supposed to be so small as to be negligible. Thus, as far as the output part of the circuit is concerned, a
good voltage amplifier
behaves like a voltage generator with zero internal impedance.
A
good current
amplifier
behaves like a current generator with shunt impedance. Practically,
an amplifier
infinite
of course, the input impedance of not infinite; this is normally
is
taken into account in the form of a slight correc tion. Similarly, the generator impedance of a
OUTPUT
INPUT
If
considered to be a voltage
generator, the impedance
power source which
not connected.
by an impedance and a generator.
replaced
usually
amplifier
is
other hand, relative to the output part of the circuit, the input and amplifier can also be
another and thus obtain combinations suitable
An
difference
On the
for computing.
as a
little
whether the amplifier is connected or not to this circuit, and one usually considers in this part of
voltage amplifier
is
also not entirely negligible.
Amplifiers are normally rated according to these quantities.
For
instance, in a specific case, the
input impedance of an amplifier Fig. IH.7.A.1
megohms shunted by
output generator impedance
For the present we can consider that there is no connection between the circuit on the input side of the amplifier
and the
side of the amplifier.
then,
circuit
on the output
By Thevenin
s
theorem,
we can consider the amplifier and its output
as constituting a two-terminal
network
relative
IIL7.B.
Vacuum Tubes
Most
in
amplifiers
vacuum
given as 5 and the
is
2 ohms.
as Amplifiers
computing devices use
tubes or transistors and are termed
"electronic"
reader
is
15 microfarads,
is
because of this.
We assume that the
familiar with the usual discussion of the
and multi-electrode vacuum
to the part of the circuit connected to the input.
action of triodic
Thus, as far as the input part of the circuit is concerned, the amplifier can be replaced by an
tubes.
impedance and a generator connected between the input to the amplifier and ground. It is
now a
detirable in an amplifier that the input generator
are dealing with a triode.
RCA Receiving Tube 10.) We would like to
(See also
Ref. III.7.h, p. single
vacuum tube
Manual, consider
setup as an amplifier.
For our present purpose we can assume that we
AMPLIFIERS
82
vacuum tube must be
In order to operate, the
the signal
is
connected with certain sources of energy, The heater for the cathode which produces electrons
tube.
must be connected with a source of current
to have the
since
the cold cathode will not produce electrons in
any appreciable amount. nection
We
However,
independent of the
is
rest
con
this
of the
circuit.
suppose, therefore, that the connection
made, and we do not concern
ourselves with
The
vacuum tube
.
with
generator
voltage source
is
One
equivalent to a generator.
is
terminal of the generator the other terminal
is
connected to ground;
connected by a suitable
is
if
many
internal
little
very
tubes use the same
If the current
results
The high-
desirable
it is
voltage source equivalent to a
impedance, especially
5+
it
also needs a high-
source of direct current.
impedance
B+
change in the tube, which from the above-mentioned signal voltage,
to produce a voltage change
load impedance, say,
triode
voltage
for various other reasons,
voltage
is
further.
any
Now
negative, less current flows in the
between B+ and the fixed voltage, the
RL
>
itself,
some type of
must be interposed Since
B+
is at a high change in the current flowing
plate.
through the tube and, hence, through R L now will cause a voltage change of the plate. Explicitly,
we have made
the connections of Fig. III.7.B.1.
vacuum tube. The
to the plate of the
cathode of the vacuum tube in the simplest possible case is then connected to ground. The
nongrounded high voltage terminal of the current source of current If the
B+
is
of the generator
vacuum
plate of the
"J5+."
connected to the
is
an
tube,
direct-
usually called
electric current in
the usual convention flows from the plate to the
cathode. This means, of course, that the electrons
escape from the cathode and go to the plate which has a high positive potential. The flow of this current in the tube
The
in a triode. plate
grid
is
controlled by the grid
interposed between the
is
and the cathode. The grid
is
at a voltage below that of the cathode.
example,
if
the cathode
is
said to have
For volts,
the grid will normally have a voltage of say
The
volts.
3
fact that the voltage of the grid
is
lower than the voltage of the cathode tends to reduce the current through the tube. Thus a source of negative potential called voltage
is
the
C
The
negative terminal
The
"bias"
or
C
needed to keep the grid negative
The
relative to the cathode.
voltage source
E
signal
is is
this
Presumably the
this
C voltage.
voltage, there
the tube
called
signal voltage
RL
and the
C
is
zero.
for this situation.
There
is
For any
Ea and plate Ev there is a current /which is a function of Ea and Ey Define other value of the signal voltage
voltage
)
J-I =/(,) = 0.
can be considered
voltage point and the
Given
a current 7 which flows through
when the
where /(O,
"C."
is
also a plate voltage
positive terminal of
for the circuit
C
This voltage change in the plate can affect the current in the tube also.
connected to ground.
as the result of a voltage generator interposed
between
Fig. III.7.B.1
normally kept
Now
L
let
=I
(1II.7.B.1)
Jo
grid.
signal voltage does not exceed
Thus, the
effect
of this signal
is
to
Then
change the grid voltage: if the signal is positive, the grid voltage becomes less negative and a higher current flows in the tube; conversely,
if
i,= Normally
a
/(*,,<,
+
vacuum tube
*,) is
(IIL7.B.2)
designed
and
VACUUM TUBES
III.7.B.
operated so that variables.
/
a linear function of
is
Since/(0,
)
Theoretically, the full Taylor (0,
)
RL
through
h
Ohm s
tube and
is
i
1==
}1 "g
m
.
=
+
i,
+
+ l(B -
(HI.7.B.5)
i*
-g
(III.7.B.6)
1
the transconductance of the
usually denoted
be the current flowing from ground. By nodal law
2
consider only
the first-order part of this expansion.
The constant A-^ is
/
83
law
expansion should be used. For
we
Let
plate through Z to
By
s series
practical purposes, however,
.
(IIL7.B.3)
,e,
of/around the point
its
= 0, this means that
AS AMPLIFIERS
(IIL7.B.7)
Z
The constant
We
Az
is
generally written in the
form
where
r n is
.
**
called
the
plate
resistance.
can use Eq. III.7.B.4-7 in order to obtain e p in terms of e a or to obtain / in terms of e 2
(See
also
RCA
Receiving Tube Manual, Ref. III.7.h, pp. 10-29.)
From
Eq. III.7.B.5 and making substitutions from Eq. III.7.B.4, 6, and 7, we have
Thus, "
)
o c p j ff
6wi
RL
(III.7.B.4)
i
rn 1 ,
If
we wish
another place, plate
we must make a connection to
from some other
OIL7.B.8)
the If
This second
circuit.
must have a ground terminal also. By Thevenin s theorem these connections can using circuit
o
1
to transfer this current effect to
yvvvvv^
the equilibrium value of the plate
is
EQ
potential for the
Z connection, r^\
*-/T>4-
r
then
* i
n
EQ corresponds to the situation in which ea
R.
is 0.
we
Substituting Eq. III.7.B.9 in Eq. III.7.B.8
obtain
(IH.7.B.10)
(III.7.B.11) 1
_1_
Z
Rr
we
Substituting Eq. III.7.B.11 in Eq. III.7.B.7 Fig. IIL7.B.2
obtain
be replaced by a combination of an impedance and a generator. In all practical considerations
we may ignore the generator and consider that we have an extra impedance Z connected between the plate and ground.
With the above formula
for i9
we
circuit position to analyze this
are
now in a
which can be
considered to be essentially as shown in Fig. + is a source. III.7.B.2. positive high-voltage We suppose that ground is at zero voltage. Let ^ e 9 and be the current flowing from 5+ to
+
( ffl - 7 -
B
-
12 )
z-L o
Since
is
Z
a constant) we can
thus, the effective current
and>
is
_ zf-L \R L
+i+ r,
i) ZJ
1
+^+^ RL
r,
(1U.7.B.13)
AMPLIFIERS
84
The
effective
output voltage
same as the
the
is
of Fig. III.7.B.5, where
circuit
Q
e^-^-
(III.7.B.14)
i+T + ,
.
which If
f
.
,
,s,
we
.
It
T.
_
^
.,
,
of course, similar to Eq. B.I 1 vacuum tube as an
are to regard our
^^
shown
we should replace the part of the circuit
in Fig. III.7.B.2 to the left of
A
R
convenient to introduce
RL
and
9 mQ n\
which
is
The output
in parallel.
r
Qr shown JQ
y()1
.
IS
amplifier
is
equivalent to
,
(HL7.B.16)
R
m?
$
,
6
*
by a
generator and an impedance. These, of course, should produce the apparent output voltage e of
Eq. III.7.B.14 and the output current rewrite Eq. B.14 as -,
e
=
Each term on, the current.
If
we
let
2
+A+^
right f
a
z
a
.
We can
(HI.7.B.15)
of Eq, III.7.B.15
= -gmea
,
is
a
Fig.
m.7.B.5
then Eq. B.I 5 This discussion
is,
of course, simplified. Thus,
the output capacities of the tube should also be
taken into account. The output capacity
Fig.
C
can
DL7JB.3 Fig. HI.7.B.6
be considered to be in parallel with that
R
is
shown
really given as
X where p stands for It is
RL and r^ so
r
KL
in Fig. III.7.B.6.
rv
differentiation.
customary to define the voltage gain p
as;
(IIL7.B.18) Fig. IH.7.B.4
describes the current generator circuit Fig. III.7.B.3.
shown
in
Thus, the output portion of our
vacuum-tube circuit is equivalent to the circuit in Fig. IIL7.B.4, which is, relative to Z, essentially
Tne /*
<
amplification factor (It
is
would be
lification factor as
a
= r$m
.
Therefore,
better to consider the
_ 1
-\
III.7.B.
Next we consider tube.
the is
it
Frequently
VACUUM TUBES AS vacuum
input to the
necessary to provide a
from grid to ground. This resistor is considered as shunted across the input. How
AMPLIFIERS
account, the resulting differential equation can
be written:
resistor
ever, the various input capacities in the tube are
Qi(pK where gi and
the grid-to-cathode capacity
is
clearly connected to ground.
P:
=
Pi(pK
(HI.7.B.20)
are linear differential operators
with constant coefficients.
also shunted across the input (see Fig. III.7.B.7).
For example,
85
In the engineering literature a linear operator is
The term
linear with constant coefficients.
linear
does not refer to the
way
in
which the
on derivatives, but to the way the voltage across certain impedances depends on expression depends
the current.
For many purposes the response that one can get from a single vacuum tube is not adequate. For
instance, in the simplest type of amplifier such as one finds in the audio section of most
radios one tube
used purely as a voltage
is
Fig. HI.7.B ,7
In addition,
if
Cg
and the
plate,
has the same
effect
grid
on
ppCg connected from Fig. III.7.B.8).
the
is
it is
This
capacity between the
known
that this capacity
the input as the
an impedance
grid to the ground (see
last is
termed the
"Miller
effect."
Fig. ffl.7 J.9
amplifier,
and the second tube
actual power tube,
output
is
i.e.,
is
used as the
a current generator whose
to the applied through a transformer
speaker.
However, connecting a so that the output of one
in
description,
the
above
somewhat
about simplified
a vacuum tube can be replaced by
the two circuits shown in Fig. III.7.B.9, where e a is is
the signal applied to the grid, and the generator
a current generator with output,
a load impedance suppose that between e 9 and ground.
The
is
of vacuum tubes
applied to the input
of the next presents a major difficulty. Normally the output of the plate occurs at a voltage of
Fig. IH.7.B.8
Thus,
series
Z
gme a is
.
We
connected
+100
volts while the grid of a
vacuum
tube should be 3 or 4 volts below the cathode. This
can be overcome by having for each vacuum tube. But
difficulty
separate
B+
supplies
ordinarily this
is
a very expensive solution to the be amplified does
that is to problem. If the signal
not have any direct-current component, then there are two well-known methods available for
relation
coupling stages. (UI.7.B.19)
One is the transformer coupling
which consists of
inserting the
transformer between in Fig. III.7.B.10.
5+ and
primary of a
the plate, as
shown
Since the secondary can be
should be regarded as a differential equation e and e as functions of time. In general, relating v
connected to any direct-current potential desired, one can connect one side of the secondary to the
even when cross connections are taken into
C point and the other side of the secondary to the
AMPLIFIERS
86
Transformer coupling
is
differential analyzers is direct coupling.
when impedance matching
is
directly
grid of the next tube,
In a
important, as in pulse circuits (see also Section
coupled amplifier a negative voltage 300 volts is provided. 100 to source of from
III.5.I).
Then a
used, in general,
resistive
connection
is
made between
the
and the grid of the next, and plate of one tube another resistive connection is made between this grid grid
and the negative voltage source. Thus the held at a voltage which is the average of
is
and that of the negative voltage
the voltage e y
source. Thus, if the plate voltage of the preceding
tube
is
+100
volts,
and the negative voltage if the two resistors are
106 and
is
supplied
next tube equal, the grid of the
is
held at zero
Fig, ffl.7.B.10
Another form of coupling which may be used
when no
direct-current signal
is
to be amplified
condenser coupling. For example, the plate of the output tube may be connected to the grid of is
the input tube, and the
by a resistor to the
latter, in turn,
C point.
connected
(See Fig. III.7.B.11.)
Fig. III.7.B.12
signal
at
-3
volts.
This bias voltage of
clearly replaces the
discussions.
at the
-3 volts
voltage of our previous
Obviously,
mentioned above signal
C
the
two connections
will permit a direct-current
be transferred to the grid of plate to
the next tube.
Unfortunately, the plate voltage
for the zero signal tends to drift with time, Fig.
The
m.73.11
this
and
changes the bias of the next stage.
resistor maintains the average voltage of the
grid at the signals
C
voltage while alternating-current
whose frequency
is
not too low are
transmitted through the condenser and resistor
network.
This form of coupling
used in audio
amplifiers.
It is
is
frequently
possible in audio
take advantage of grid current to amplifiers to
obtain a bias without
C when
this
kind of
coupling is used. These couplings, however, are not of major interest to us. For the reader who
wants to learn more about condenser couplings
we
refer to T. S. Gray, Applied Electronics (Ref.
III.7.C, pp. 502-29).
On
the other hand, the most
Fig. IIL7.B.13
Thus, the relation between the output of one tube and the grid of the next in a directly coupled amplifier
common form
coupling in the type of amplifier that
is
of
used in
is
as
shown in Fig.
seen that Fig. III.7.B.12 III.7.B.13,
where
III.7.B.12.
is
We have
equivalent to Fig,
CI can be considered to contain
III.7.B.
VACUUM TUBES
the capacity that arises from the Miller This circuit can be analyzed by
AS AMPLIFIERS
87
effect.
replacing the
impedances and the generator to the
left
of R 1 by
&L
TV (IIL7.B.26)
Fig. IIL7,B.14 (III.7.B.27)
bp
a voltage generator and impedances in series with it. Thus if 2 is an impedance such that
Eq. III.7.B.27 means that d* ,
,
d
we have
,
(IIL7 .B.21)
*
It
(IIL7.B.22)
should be recalled that e g is the input voltage
for the given
vacuum
tube,
voltage for the next tube.
the corresponding voltage generator will generate gme and one has the equivalent of the circuit shown in Fig. III.7.B.14. Then
and e
f
is
the input
Thus,
Z
a>
e
=
Now
B 29)
in turn, the output of the second tube
if,
Z"
+
Z"
(IH.7.B.23)
+ Rl
applied to a third with input voltage
the circuit can be regarded as a voltage
divider
7
is
-
Z i.e.,
(III
cp*+bp
If
+
e,
(III.7.B.24)
Z"
we
have, then, a series of stages, each
connected to the previous one in this simple fashion, we obtain an output voltage e which is
Eq. III.7.B.24 can also be written
we have
(III.7.B.30)
and
\Z
e"
related to the input voltage e i
two
by means of
linear differential operators with constant
coefficients
e
ea
(IIL7.B.25)
= p(p)e
Q(p)e o
where e Substituting in our values for
Z
and
Z"
given in
Eq. III.7.B.21 and 22 we obtain
and
es ,
s
(III.7.B.31)
refers to the output-generator impedance
of course,
is
the voltage at the input.
Normally we are given the input impedance Zs of our amplifier. This permits us to consider the signal
as
arising
from a current generator
producing a current 7a and ,
RL
ea
= ZJa
The generator output
voltage
R,
r,
(III.7.B.32) is
given by (IH.7.B.33)
x
c il^ + *i c A/ hr + -) r L
to
current generator.
re ar(^
Let
Z
tlie
am p^ er
as
a
stand for the output
AMPLIFIERS
=
impedance. The corresponding current generator
Now,
produces
have negative real parts, then these extra terms can be written:
and we have the
each feeding the next without cross coupling between the input and the output, the input
impedance
is
Z Z
s
for the
first
stage
and the output
for the last.
IE.7.C. Feedback Amplifiers
A
is fc
in
decay
is
such that
Ja^
have a sequence of vacuum-tube stages with
impedance
is
relation
where a
If we
g
if
negative.
all
cost&f
roots of
+
yt)
(x)
(III.7.C.3)
As time goes on these terms
and eventually e
value,
becomes
essentially equal to the particular solution
which
of frequencies present in the forcing term. This particular solution is then referred to consists
as the
the terms which
"steady-state solution";
are solutions of the
homogeneous equations
called
This corresponds to a desir
"transients."
able situation as far as an amplifier
straightforward amplifier, as described in
Q
since
we
certainly
is
are
concerned,
want the output voltage to be
use in computing
controlled by the input signal. An amplifier for which Q(p) has only roots with negative real parts is said to be stable.
output and input impedances thus obtained would not be suitable and, in addition, the time variations of com
a positive or nonzero real part, the output voltage e will contain either an increasing transient,
Section III.7.B, in which the input and output are directly associated through a chain not, in general, be of
devices.
would cause
In modern
difficulty.
customary to introduce feedback, some connection between the input and
practice i.e.,
the
Frequently
ponents
much
would
it
is
output voltages other than those given above. is to effectively vary the input and output impedances and minimize the undesirable drifts which occur in the later stages of the
This feedback
which
will
drown out
signal voltage, or
a
will introduce
an amplifier Let us
is
now
Q
(x)
=
has
the effect of the original
an additional
oscillation,
which
false signal in the output.
Such
unstable.
return to a discussion of the effect
Suppose we have an amplifier which has a relation Q(p)e P(p)e l between its and an input output voltage, input impedance Zs and an output impedance Z and suppose we of feedback.
,
amplifier.
However, in the introduction of a relation between input and output voltages there danger which is apparent as soon as recognized that the output voltage
a
In the case where some root of
is
is it
a is
governed by
,
make an
additional connection
Z
between the
input and output. We then have a situation which can be diagramed as in Fig. III.7.C.1 For .
equation with constant coefficients. For suppose the relation between the input and differential
the output voltages
is
given by the equation
The expression P (p)es can be considered as a on this differential equation. Corre
forcing term
sponding to particular
precisely the
P
(p)es
.
this
forcing term,
we can have a
solution which ordinarily contains
same frequencies
In addition to
this
particular solution,
the output voltage e will contain terms which are solutions of the equation
Fig. HI.7.C.1
as those present in let us suppose that the input signal we simplicity, wish to amplify arises from a voltage generator Z that yielding a voltage E with impedance a and
output
is
given by a current generator I
.
Then,
III.7.C.
before the connection with impedance the input voltage signal
Z
is
s
i
a
FEEDBACK AMPLIFIERS
Z is made,
and the
original
and output voltage
relation between signal
i.e.,
Z
is
f
89
equivalent
Z
to
s
and
Za
in parallel.
Therefore,
is
(HL7.C4)
Now
let
the connection
Z
be made, and
let
us
across suppose that a load impedance Z appears the output. Let e x be the new voltage of the
and input point,
Z
Z
\Z
t
+
(I11.7.U11) e
the voltage at the output
2
terminal of the amplifier.
Then
Z
Z
Z/
(IIL7.C.12)
the relation for I
by the output is For
since the current generated
determined by the input signal voltage.
+ Z,
Z
established by the
Is
ji^ Z \Z
_
(
ff
brevity let
Q(P)
Then
(IIL7.C.13) J
we can
write
= ~ Fl
Z
down
(
holds for every value of Z Eq. III.7.C.13
the nodal equations for the
nodes of e z and e lt
/! Io==
\7
i
+
We eliminate I We obtain
j*
+
7
and
!
_ I? ^ _ (
z - lz
n 2
.
Now
let
j
fl |
mi 7 C
14)
; <
i
r
^ in Eq.
Z
%
(UL7 C8)
III.7.C.7, 8,
and
9.
and
let
Z* be such
that
Then Eq. IIL7.C.13 becomes
z*
\~z/
=~
+
r+
which
is,
Z Zs
generator
Z 1
+ !+1L
X i
Z
+ ^. +
^
5
impedance that
is
an impedance Z,
cnMi that that such
1=1+1 Z
Z
z
Z*
generating
Z*.
It is clear
consists of three
from Eq.
impedances
III.7.C.15
in parallel.
This means that the original output impedance
ff
It is convenient to introduce
of course, the equation of a current /* and with shunt
now
shunted by the
trivial
andtheimpedailce ?i. If ^
impedance is
much larger than
}
(III.7.C.10)
has a generator thus, the feedback arrangement
AMPLIFIERS
9 smaller than the impedance much
The
original. is
equation for output generator voltage
:
Z+
We still need to find the input impedance of this new combination. Eq. III.7.C.16 becomes
Thus,
=
e,
JLF 7
M -A zJ
If E(l- IIL7
fld
III.7.C.9,
\
C 22
(HI.7.C.22)
for *2
-
substituted in Eq.
is
obtain
-r^-H* l+^
K M
(IIL7.C.17)
-
we
=*
+~ //
The terms involving
Z
are generally very small
or
so that Eq. III.7.C.17 becomes
1+
!
(III.7.C.18)
If
/t]8
Z
becomes very large, Eq. IIL7.C.18
is
E*
= -E
(III 7
Since
C.19)
^
is
"
ne gu g ible
+
relative to l
& we
have
Zff
using Eq.III.7.C.12,
This relationship between the input signal voltage Ea and the output generator voltage does not
depend upon the gain
p
the gain
/j.
^
is
L
large,
between input and output.
%o \ _^_
E^
of the amplifier can vary without
affecting the relation It is
Thus when
Z
\
Z
1
+ ^/5
ff
= ^L e
/
ZP
or
convenient to rewrite Eq. III.7.C.14 to the
+
E a Zp(Z
Z
)
^ "~
same approximation
=^
/*
I
as Eq. III.7.C.18,
^^ _
\Z
1
10 /
==
1
E\$
1
Z \Z
Z
i.e.,
/ tf
(IIL7.C.2Q)
the Eq. III.7.C.15 also yields to
^ 2
(i
If in the input circuit,
1
(nl ?
c 24)
^)
_|_
the amplifier and
its
connections are replaced by the input impedance
Z+then
same approxi-
E
z+ ff
=
e
+ 7 T ^7 ^
(IH.7.C.25)
I
mation: (III.7.C.21)
Comparing Eq.
formulas normally Eq. III.7.C.20 and 21 are the
Zff(Z
used.
Z
III.7.C.25
+ ZQ _
and
Z
24,
we
see that
+ +
7
c
^
IIL7.D.
STABILITY The output impedance of the generator
and, thus,
Z + =-
Eq. III.7.C.21). The input impedance to this is
amplifier
Thus, we have
III.7.C.27.
(cf.
We
and Z the
is
ff
Z
in
and Z^
in
Z
parallel;
in turn,
f,
is
impedance
is
this
-
impedance Zft
-
diminished by the feedback factor is
i-i
is
m.7.D.
output impedance. If enough feedback is used, Z+ can be considered as zero. This is the normal
Thus, we can summarize the results of the
Suppose we are Eq.
(see
fj,(p)
Z
on feedback
discussion
impedance Z from a voltage generator with impedance Za We connect a feedback impedance Z between input .
circuit
*
may
can be con
at the
beginning of our the use of
amplifiers,
lead to an unstable amplifier.
Thus, while the original relationship between output voltage and input signal was given by the equation /i\
impedance Zs and output which receives a voltage signal Ea
resulting
V (IIL7.C29) f
ff
Stability
=
III.7.C.7), with input
and output. The
Z7
.)
As we have remarked
feedback design objective.
an amplifier with gain
T Z s
T Z
Q
not completely independent of the load
impedance
a certain dependence on the
as follows. present discussion
the load impedance.
.
+ /$
(1
However, there
-
is
=1+1+1
-L
Zp + (Z
Z
Eq. C.28) where have
also
Zs
parallel (see Eq. III.7.C. 10). Essentially
input
given
now
neglect the complicated term in the
denominator of Eq.
Z/?
is
(IIL7.C.27) (cf.
We may
91
where p
is
E
(III.7.D.1)
ff
a differential operator ^(p), which
we
can suppose is stable, the amplifier with feedback has the relationship
sidered to be, relative to the output, a voltage a generator producing voltage
E* such
)* = Z /W
that
and the operator If (cf.
where Eq. IIL7.C.18),
/?
equivalent
to
Z
s
and
z,
(cf.
is
such that
Za in parallel,
z,
if
Z,
is
(1
=
+
Pp)lfy may be unstable. P and Q
-P(p)IQ(p), where for
are
polynomials, stability to the statement that equivalent
1
+
1/ftw
is
as a polynomial in p does not regarded simply a non-negative real part. For with a have root;?
z,
the feedback
write p,(p)
i.e.,
the relation, Eq. III.7.D.2 between
Eq. HI.7.C.10) then,
Eq. C.12). (If be a voltage-divider (cf.
the feedback ratio
is
defined in Eq. III.7.C.12 and
we
(IIL7.D.2)
,
E* and
Ea
can be written
is
circuit, ft is
considered to
the fraction of
which appears at the output voltage
input.)
of course, a differential equation
Eq. III.7.D.3
is,
on
Ea is
as
*, since
a given function of time, and,
we have previously pointed
out,
we
obtain the
92
AMPLIFIERS
desired result only
the roots of
if all
0P
~Q
have negative real parts. But this criterion can also be expressed in terms of /*, i.e., the necessary and sufficient
Thus, by applying different frequencies to the amplifier without feedback and comparing the steady-state response with the input,
measure
//(/?)
for various values
condition for stability of a feedback amplifier is that j(p) regarded as a rational function of the
positive imaginary axis.
complex variable p does not assume the value 1/jS for a/? with a non-negative real part, For
that
the stability of
obtain experimentally the value of
Since
Q=
0P
or
-l/ J
ju
]
=
-l/0
along the
P(P)
= P(F)
(HI.7.D.9)
0(P)
(HI.7.D.10)
is
equivalent to
-p/Q =
^
one can
=;co and
P and Q have real coefficients, it Mows
requires that Q(p) be not zero
/j>
for any such p, and, hence,
of/?
(III.7.D.4)
Thus, given an amplifier with rational gain we may introduce a feedback of amount /? /i(p), 1/0 for a complex provided /*(/?) does not equal with This real becomes, then, a /? positive part.
problem in complex function theory to determine whether a given feedback
The
ratio
is
permissible.
p(p) can normally be
rational function
considered to have a denominator
Q
degree than that of the numerator
(see Section
III.7.B).
Consequently,
->
infinity.
We
can consider the function
/j(p)
as
known.
determined experimen for imaginary values of/?, p One can tally jco. apply to the input of the original amplifier a Usually
this function
and thus p can be known along the imaginary axis. The symbol/? now denotes a complex variable,
negative half of this axis,
considered as
of higher
approaches
as/?
/*(/?)
Consequently, the values of p along the positive imaginary axis also determine those along the
is
=
sinusoidal voltage,
not an operator. The problem is: does /* 1/0 have a root in the
+
plane?
Graphic methods for determining whether a root of a given function w(p) lies in a certain region are very convenient for discussing stability. These methods are based on certain
LEMMA
this is
a complex
<-,
(w
is ITT
= Ae^
times the frequency).
will
now
establish.
C
Let
be a simple closed curve in the Let complex plane. w(p) be analytic III.7.D.1.
C and its interior and not zero on C. Let w image of C, Let ri be the number
on
voltage:
be the (III.7.D.6)
The
steady-state
output can also be represented as a complex
0,
positive real
lemmas which we
Normally one assumes that
Given
zeros of
w in the interior
of times
C goes around the origin.
(While
C is
of C, and n the
voltage:
of
number
Then n
a simple closed curve,
C
= n.
C may have
double points and wind around the origin a
number of times. where y
is
the phase
shift.
The
ratio
-=
In the /<(/?)
eo
for/? is
=jco. The
given by
=
and e a relationship between p(p)e at where, of course,;? stands
for differentiation and a particular solution of this
equation corresponding to the steady state
n
-
Fig.
III.7.D. 1
.
=
.
closed curve with interior in
jy(r) on
C
can readily be found in the form Eq. III.7.D.7.
range, say, as r
n\
example
Proof: Take any function w(p) in the complex Let C be a simple plane analytic in a region
Thus, /
See, for
case illustrated in Fig. III.7.D.1, 2, in the second n 3.) first
where r
is
from zero
(15.
= JC(T) +
Let/?
a real parameter, with a to one.
We
suppose that
goes from zero to one, the curve
C is transWe plot
versed in a counterclockwise manner. (IH.7.D.8) in the H>(/?(T))
w plane.
If we start with the value
STABILITY
III.7.D.
93
For any value of T
W PLANE
- log po + j(v -
log p
= log w - log w
)
dp Jro
(IIL7.D.15)
TV(j>)
is
determined by
= log Po +M
(IIL7.D.16)
where, of course, the logarithm the condition
logw Since \v(p)
is
analytic
on
the region
we have
Consequently,
poles.
do, it
has no
for the residue
integral
^dp^lm j where
is
C on
the
(IIL7.D.17)
the
number of zeros inside the contour
p
plane and
C
transversed in a
is
counterclockwise direction. (See K. Knopp, Ref.
IHJ.d and zero
hand, Fig.
TO for which w(p(r)) pQ
>
m.7.D.l
i=-
0,
E.
J.
Townsend, Ref.
we
see
from Eq. III.7D.17,
when we have made
we can choose
p
% and
= po and = 9?
such that
j
Therefore, n (III.7.D.13)
,
Now a
the other
and 14 that
15,
we have
= limj
^)
(IIL7.D.18)
we have proved
Thus,
.
On
the circuit completely and
= j(9/ =n
IIIJ.j.)
~ 0.
a point where w(p)
is
the
lemma.
Wecan generalize Lemma III.7.D.1 as follows:
See Fig. III.7.D.2.
LEMMA
III.7.D.2.
Let
C
be a simple closed
curve in the complex plane. Let w(p) be analytic on C and its interior; w(p) does not have the of C. Let value a on C. Let C be the w
image
f
n a be the
number of times w assumes
in the interior of
C and na
C goes around the point Proof:
Fig.m.7.D.2
w(p)
- a.
inside C.
Then
as
we vary
Now, p and and p is
will
T,
we can
w
express
will vary continuously
come back
but single valued,
to 9?
its
as pe
j
since original value,
will return to
.
around C,
w
= 2rto Geometrically, n
has
wound
itself
is
-
the
number
is
Then na
is
the
Then n
III.7.D.1
na
.
to v(p)
=
number of zeros of
v
Furthermore, v(p) going around the to w(p) go^g around a, since
equivalent
= v(p) + a.
Thus na
is
also the
number of
some value
(IIL7.D.14)
the boundary of C, then the angle
Note: If vi^) assumes the value a k times on
of times the curve
around the origin in a counter
clockwise direction.
v^)
a.
Apply Lemma
the value a
number of times
times v(p) goes around the origin. The above na . lemma applied to v shows na
9/ such that
origin
the
2nir by an amount
9?
will
change
+ kir in the circuit.
Now let us return to
the
problem of establish an amplifier to it, of
or lack of ing the stability,
AMPLIFIERS
94
which feedback has been added. ratio
and
is
given,
sufficient
we have
condition
amplifier be stable
value
1/0 for a
for a
p
that
is
p
If the feedback
seen that the necessary
feedback
will
tion
the
with positive real part,
in the right-hand half of the
i.e.,
complex
(j,(p) along contour of Fig. III.7.D.3. Starting at the origin
C
when
and only
tour, if
when
it
occurs.
p assumes
the
if
is
1/0
1/0 within this image
equivalent to having
contour for
within the image of
Thus, instability for the feedback
the contour.
P PLANE
it
III.7.D.2 states that
1/0 within the given semicircular con
value
is
For
clockwise.
transversed
is
us ignore this possibility since one
let
easily interpret
Lemma the
be mapped on the exterior of image of C
If this happens, the
be transversed in a counterclockwise direc
simplicity,
can
plane.
Let us consider the values of
C may
the image of C.
does not assume the
that
p
interior of
all
large
sufficiently
semicircular
contours. It
convenient to
is
semicircle
when
R
is
the radius
let
R
of the
In this process, a part of the sufficiently large, only
approach
infinity.
contour image near the origin varies, and in the limit the // image of the positive half of the
imaginary axis joined to the negative half at the origin.
p image
The
assumes on the portion of the
of the
two images.
/*
plane with
p
this positive real part are enclosed in
tion
of the
values which
These,
combina then,
are
forbidden values for the quantity -1/0. Normally one can infer the enclosed values
from a plot of axis, since the
Fig. III.7.D,3
and increasing
co,
we can
part of the contour that
is
as p
-^
on the
//(/co)
becomes
If the contour
oo.
//.
Eventually as
small, since ^(p) is
axis.
large, this plot
->
large enough, p(p)
circular arc will be small, and, hence,
confined to origin.
is
with a large value of
& increases,
along the positive imaginary
the plot p(jco) along
on the imaginary
If the gain for direct current will start
fj,
other half yields just a mirror
some small neighborhood of the other hand, if we plot \L along the
On the
negative segment of the contour on the imaginary with p 0, we obtain the mirror axis, starting
image of the curve obtained from the positive segment. These two plots are joined correspond = 0, and they also have a connection ing to /* corresponding to the p image of the circular segment in a small neighborhood of the origin.
Notice that
we now transverse the contour C in a
clockwise direction.
If
/*
This
image.
plot
of
p along
the
positive
imaginary axis is termed the "Nyquist diagram." In most cases of interest here, this plot has a special character starting with a large value of
for
p
= 0.
expresses the
This
plot
in
modulus of p
polar
/*
coordinates
as a function of
C will of C in a
the angle of phase shift between input and output
has poles, then the
closed can be readily estimated if the value of
Thus the image of
wind around the image of clockwise direction.
Fig. III.7.D.4
=
the interior
signals.
(See Fig. III.7.D.4.)
Clearly, the region
DRIFT COMPENSATION
III.7.E.
= 90 and 180. For our is known for \p\ immediate purposes these values are the most
Nevertheless, the design of ampli in general, is a art in which
significant. fiers,
factors
complex must be considered and the
of
curve
this
We
is
entire
a positive real quantity
imaginary one. the
along
reason regulated power supplies are desirable to regulate the main source
this
is
of power also.
shape
introduce
which
real
or
axis
{I
is
a purely
or
either
is
Consequently, -I//?
negative
For
used. It
can also Temperature changes
along
in
variations
There have been
shall deal with situations in
either
Variations occur also in plate and bias voltages.
many
significant.
95
many
which minimize the
Korn and
G. A.
components.
passive
efforts to
of
effect
develop circuits (See also
drift.
M. Korn,
T.
Ref. III.7.e,
pp. 190-223.)
the
imaginary one. In the first case, we would like ft to be as large as possible to standardize perform ance of our feedback amplifier independently of the original gain. But is
the modulus of
have
/5
<
contour.
p
if
ft
is
such that
at the 180
point,
- I/ft
we must
ft so that -I//? remains outside the
Thus, the modulus of
at the 180
/A
maximum
point determines the
real feedback
permissible.
A similar consideration holds for an imaginary feedback.
Since the complete contour
we
are
symmetric around the real axis, we can suppose that ft=jb where b is positive. considering
is
-1 //?=;/&
Consequently,
a
has
positive
Here again, if b Q is such that the modulus of p at the 90 point, we
imaginary part. is
1/6
must have b
<
bQ
if
1//5
is
to remain outside
the contour. Thus, the modulus of
determines the
maximum
/*
at the 90
imaginary feedback.
Further reference to the Nyquist diagram given in H.
W. Bode
and H. Nyquist
is
L
F. E.
Terman and
If the gain is
discussed in E. Peter
A.
M.
J.
Fig.ffl.7.E.l
Obtaining the
(Ref. IHJ.f).
diagram experimentally son, J. G. Kreer, and
and
is
(Ref. IILV.a, pp. 151-57)
Ware
(Ref. III.7.g)
Pettit (Ref, IIL7.i,
compensate manually
possible to
it is
for drift stabilization.
One
method is to use a double triode for the first stage with a
pp. 31 1-379).
not too large,
common cathode (see
Fig. III.7.E.1).
One
the double triode corresponds to the grid of
HI,7.E. Drift Compensation
As we have mentioned coupled amplifier drift.
The
is
normal input for the
equilibrium
directly
summing
direct-current
feedback.
above,
subject to
a
con vary owing to several causes, and,
to
The
current in tubes will tend
first
signal
to
the
At
output reference will change.
output.
One
serious cause
filament voltages. desirable
to
It
stabilize
of is
drift
is
variation in
therefore
the
frequently
filament
voltage.
this is the
other grid
is
manually
the beginning of operation
output voltage. with zero signal input
produces changes
i.e.,
for any drift of the adjusted to compensate
will change. Since sequently, the plate voltages in the bias voltages, the
this
stage,
to the amplifier before point or input
ea
is
that
this is adjusted to zero
If e is the voltage
on the first
on the second, the second
input,
and
a grid causes
so that the com change in the cathode voltage to a single stage with input bination is
equivalent
AMPLIFIERS
96
ea
ea .
Thus,
if
there
is
a
drift
of amount k of
summing point, which will cause an apparent false input signal of amount k, we can com
the
ea
= k.
Drift
^. We can
be denoted
will
consider
an extra output voltage K^
Thus, the actual output voltage
is
is
frequently pensate by letting measured by the amount of adjustment voltage ea necessary to bring the output voltage back to
the reference value for no signal.
amplifier
the effect of drift as
It
could also
be measured by the amount of variation
We
define
Za
as equivalent to Z,
Z
and
ff ,
Z
s
in
parallel:
!=!+!+!
in the
ZQ
output voltage for the reference value.
Then
Za
Z
Z
the nodal equation for
*,
=
If the gain of the
K
Q
+
(HI.7.E.2)
s
yields
e
(1U.7.E.3)
narrow-band amplifier
is
^2
,
then
and
thus,
Suppose then a relation in the form
and eliminate
e
a
III.7.E.3.
by Eq.
We then have
N.B.D.C AMPLIFIER
(IIL7
E 6)
Fig. HI.7.E.2
In manual adjustment,
drift is
compensated
(IIL7.E.7)
for before computing and perhaps at intervals
work
during a requires
Automatic compensation to be made at a rate
slow relative to the rate in which the signal changing. In commercial feedback amplifiers
that is
day.
some adjustment
is
For large values of
/*,
Z = -* +
is,
1
,
o
for electronic computers, the voltage of the
of course,
-
this
rX
-
measured and amplified by a summing point very-narrow-band direct-current amplifier, and the output is applied as an adjust voltage. This is
method
due to E. A. Goldberg (see Ref. This requires, of course, two inputs, one
is
III.T.b).
the usual voltage,
point, the other
summing and these are normally the two
Eq. III.7.E.8 can be interpreted in two ways. we can see that the second term //l5 IJL
If we let
=
shows that a feedback amplifier
compensate for any additional signal
grids of a
output.
double triode as shown in Fig. III.7.E.1. In effect, then, we have a circuit which consists
The amount of
given essentially by
/*
this
,
of the usual directly coupled amplifier and an
also the factor
reduced by feedback, this
In Fig. III.7.E.2 we have denoted the two input Q and A. Their voltages are, respec
points by eq
and ea
.
The gain of the
original
normally
K
Q
in
compensation
i.e., /tfi.
Since this
its is
is
ii
additional narrow-band direct-current amplifier.
tively,
will
an adjust
improvement.
by which the desired But
if
III.7.E.6 for the case in IJL
is
is
we now which
signal
is
not a percentage
//
consider Eq,
= /^(l
// 2 ).
extremely large for direct-current and for
III.7.F.
SUMMING AMPLIFIERS
very slowly varying quantities. Thus, relative to
we have an
drift
extra
compensating reduction due to the large factor 1 This is a // 2 .
If
percentage improvement factor.
we measure
by the amount k of adjustment voltage, normal feedback has no percentage effect, but drift
the additional amplification
fa has a
1
direct
effect.
The
contact.
vibrating,
grid of the tube
is
alternately connected to the direct-current signal and to ground or a negative reference value by
the
III.7.E.4). Thus, the input a square wave whose height is
chopper (see Fig.
to the
the
is
amplifier
desired
direct-current
(See also A.
HI.7.E.5).
Tarpley, and
The use of an
97
W.
J.
signal
Williams,
(see Jr.,
Fig.
R. E.
R. Clark, Ref. III.7.L) This can
auxiliary amplifier for drift
compensation illustrates one interesting aspect of feedback amplifiers. An unwanted signal at any intermediate stage in the amplifier is reduced by, essentially, the gain preceding
back
ratio
Suppose we
/?.
as consisting of
one
after the
two
amplifiers,
unwanted
JI
times the feed
it
consider an amplifier
A.C.
^AMPLIFIER NEC. 6S BIAS
one before and
signal (see Fig. IIL7.E.3).
Fig.
HL7.E.4
be amplified by a normal alternating-current amplifier with condenser coupling and the result
However, the frequency range of
rectified.
which may be amplified by this device, must be small compared with the frequency of signal,
the vibrator so that this device
is
a narrow-band
It can be effectively pass direct-current amplifier.
used in a computer for
drift
compensation.
Fig. HI.7.E.3
fraction of output fed Suppose jS is as usual the back to the input, i.e., /? is the ratio of total input
impedance
Z
fl
to
NE6. BIAS
Zq + Z where Z is the feedback
connection impedance,
Now
let
the amplifier Fig.
before the unwanted signal have gain fa and the amplifier
Let e
after the
and
e
unwanted
signal
have gain
fa.
be the unwanted signal and the
Then the corresponding corresponding output. we have and is , /te input voltage (IH.7.E.9)
The problem of drift is not simply one of effect. Drift compensating for the total output introduces
variations
certain stages being at
drift
this
KB. One method of avoiding
Korn, Ref.
purpose.
result in
levels
with
Consequently, design to the greatest extent
(See also G. A.
III.7.e,
voltages
may
Many circuits have been
In effect then, e appears decreased by the factor
developed for
Korn and T. M.
pp. 231-248.)
the difficulty with
a direct-current amplifier
is
to change the
to an alternating one, given direct-current signal and then rectify the result. amplify the latter, means of a "chopper," or This can be done
by
reference
poor operating
of gain.
should minimize possible.
drift in
in
which throughout the amplifier resultant loss
(III.7.E.10)
IHJ.E.5
III.7.F.
Summing
In electronic are represented
Amplifiers
differential analyzers, quantities
by
voltages.
Each quantity
is
a voltage thought of as being produced by
98
AMPLIFIERS of
generator
internal
negligible
Consequently,
when an
impedance,
is
operation
performed,
we must
consider the output as arising from a voltage generator of negligible impedance. Let us consider the operation of addition, or more precisely the operation of a linear
X
voltage generators which produce voltages c i i to the summing point through resistances *R*. (See Fig. III.7.F.3.)
c,x
forming
combination. voltages
Suppose we have, say, three and J3 We will introduce an
J J
,
2,
l5
Fig. HI.7.F.1
amplifier
and a
network which
resistance
will
produce a voltage with value X Jx + #2^2 + a z 3 Let us suppose that originally the voltages and 3 are produced by generators of l9 2 <2
J X X .
J
,
negligible internal impedance, one terminal of
X
is grounded. The voltage 1 is connected by a resistance network to the summing point Q
which
of the
summing
amplifier.
X
This network and the
l equivalent to a voltage generator and a series impedance. Since the only
voltage generator for
is
voltage generator in this network
Fig.
is
Fig.
m.7.F.3
Fig.
HL7.F.4
the original
Now,
HL7.F.2
let
R
be equivalent to
Rlt R t
,
and R%
in
parallel: this
one,
equivalent voltage
produce a voltage which i.e.,
is
generator
the original voltage generator and
work.
Fig. III.7.F.1
generator
qJ
x
and a
is
must
proportional to its
Jb net
equivalent to a voltage
resistance
R1
in series (see
By Thevenin
s
theorem
turn, be replaced
this
combination can, in
by a single voltage generator
which produces a voltage
Fig. III.7.F.2).
Consequently,
if
we connect each one of these
voltage generators by networks
ming point,
the result
is
JV, to the
sum
equivalent to connecting
(HI.7.F.2)
SUMMING AMPLIFIERS
III.7.F.
which has a
series
impedance
R
D
1
(see
Fig.
The
IIL7.F.4).
Let us
now
and feedback
introduce an amplifier with gain
R
resistance
99
from the output
Fig.III.7.F,5), (See also Section III.7.C.)
p
(see
We do
quantities
are usually referred to as
R
and would have values
factors"
"scale
t
like, say, 1, 2, 4,
or 10, with the quantity 10 appearing at most once.
In differential analyzers there
is
a certain
freedom in choosing connections so that ft can vary between J and ^. The gain \n of a com mercial differential analyzer
order of 10 value of
6
fift
is
The
the usually at least 50,000. Thus,
+ pity
factor pftl(l
frequently of the
is
ignoring drift compensation.
,
to within
is 1
an error smaller
than the computing tolerances in the machine, The output impedance of the amplifier is given Fig. HI.7.F.5
not use any grid
III.7.C.21,
by Eq.
Z*
resistance, and, consequently,
Z is infinite (see Fig. IH.7.C.1), Za = R Ea = X (as given by Eq. III.7.F.2), and, hence, Z
(IU.7.F.7)
,
s
is
the output impedance of the amplifier
without feedback. P
= -^ R+R
(III.7.F.3)
x * = XRlj4_\
(m7F4)
R \l+pftl
lift 1(1
can suppose that p
+
fifty
as
Normally last
this is the
is
is
In
1.
essentially
the output of the amplifier
is
so
large
that
this case, then,
equivalent
to that
output stage a cathode follower, Z may be
is
stage.
On
low as 200 ohms.
the other hand, a
could lead to
resistance-coupled output stage
However,
obvious from Eq. III.7.F.7 that
it is
with sufficient feedback the output impedance
can be reduced to a fraction of an ohm. input side of the amplifier
we know
to the equivalent input generator resistance
of a voltage generator producing a voltage
the point
R,
X
f
t-9
and its series
has an apparent
Q
III.7.C.28.
ground given by Eq. effect of load impedance, the Neglecting the
\Y At I
apparent input impedance
is
(III.7.F.5) \J?
discuss
the
(IIL7.F.8)
3
impedance
output
we must look generator
of
this
at If
B = P
R is
then
+R negligible.
(Eq. III.7.F.3)
where
R
is
determined by
1.1+1+1 R K RI
1
megohm, and
of the order of
R
R,
On the
that relative
resistance to
A
To
output
If the
ohms. impedances as high as the order of 5,000
III.7.C.18
We
impedance of the of the amplifier
of the amplifier Eq. III.7.C.12). The output a voltage X* which we can obtain from Eq.
(cf. is
-^-
=
We
1
/t
=
10
ohm. Normally
6 ,
then
R+
is
this also is
can consider the summing point
Q
to be grounded, relative to the input circuits.
is
With regard to stability, our feedback factor ft now real and of the order of magnitude of J.
t
a margin of Normally we would require
(Eq. III.7.F.1), and, hence,
safety
so that our feedback amplifier should be stable
even for
ft
=
1
.
Thus, the gain
(III.7.F.6)
must be reduced
means
to
1
p of the amplifier
at the 180
point.
This
that the original amplifier gain, which was
AMPLIFIERS
100
6
10 for
co
= 0, must now be reduced to
1
at the
180 point (see also Section III.7.D). The theory of amplifiers shows that if one wishes to reduce the gain in this fashion from a very high gain at direct-current to a low gain at the 180
but generator which produces a voltage with a resistance x(l x)P in series (see Fig. III.7.F.7). (See also Section III.5.L) This extra Xx>
resistance
- x)P can be thought of as added
jc(l
point,
one must use a very-wide-band amplifier. (See also H. W. Bode, Ref. IILV.a, pp. 454-58.) This
why wide-band
is
must be used
amplifiers
in
electronic computers.
Our output
voltage
is
given by
Eq. IIL7.F.5). To obtain a desired voltage combination a%X2 a^X3 the networks
Fig. III.7.F.7
(cf.
a^ +
N
t
+
must be chosen so
to the scale-factor resistance
R
t
that
.
the total contribution to the final
R
Consequently,
sum
voltage
is
x\X
t
As
under
circumstances
certain
the
required
D ratios
a
t
correspond to the scale factors
One has a number of
resistances
R
t
-.
R
In practice R t may be as small as 10 5 ohms. be 2 104 and, of course, x(l x) may
-
P may
be as large as
t
associated
from the
differ
may
with the input of these amplifiers which can be used for this purpose. Consequently, the resist
ance network consists of the resistance
R
t
factor so that c i
set
and
1.
Normally more accurately the quantity provided it does not exceed 1. The
the larger c t
can be
between
is
is,
setting of c t by a factor
R +
the
-
*(3
{
x)P
Ri
in
X
with the voltage generator it If, however, the quantity has to be specified by a number of digits, we begin by choosing a scale series
of*
Consequently, the setting
.25.
which
The
from
differs
setting
by as much as 5 percent.
1
x may be found by an experimental
method which measures ming
amplifier
standard value, say zero. This cases.
is
done
Another
difficulty
1,
sum
the output of the
which
in the case in
and
all
X
i
has a
the other Jf s are
as standard practice in certain is
possibility
to eliminate this
Rt
always less than the by having a factor .25P. Then one adds by
desired result in series
?[.25
an adjustable resistance with value This can be readily done by
- *(1 - x)].
using a rheostat with a properly calibrated In fact, if
y
=x-}
the resistance to be added
Fig. ffl.7.F.6
dial.
(III.7.F.10)
is
quantity c i must be entered by means of a poten tiometer, (See Fig, HL7.F.6.)
The combination
of a voltage generator with output
X
and a
potentiometer with total resistance P, which set at the value x,
is
is
equivalent to a voltage
Note that while
it is
desirable to keep
P
as large
as possible in order to minimize the load amplifiers,
the larger
P
is,
the
more
on the
accurately
the above compensation has to be made.
INTEGRATING AMPLIFIERS
III.7.G,
should also be pointed out that the desired can be obtained by a switching arrange
It
ratio c i
ment involving resistances able
Z
which the feedback impedance condenser,
that of a
is
i.e.,
rather than the adjust
For
potentiometer.
101
we
simplicity,
(IIL1G.1)
will
fC
consider the case in which the scale factors are
and we
one input,
(We
The desired ratio fl x is expressed as a decimal
X=
all 1,
X
lf
will also consider only
and a 3 are
fraction o^ocs, where a1} a 2 ,
Each digit, in turn,
set
in the
up
expressed in biquinary form
is
^
The input network form
digits.
for the voltage
shown
as
J
ea
is
the voltage to be integrated,
R a we have
has the value
,
by
and
Za
Eq. III.7.C.18 for
the generated output voltage:
W
+
then
is
x
use the terminology of Section III.7.C.) If
P CR *
in Fig. III.7.F.8.
Thus, the derivative of values for .1
or
1
C are
1
*
- J.
is
Possible
CRa
or 10 microfarads.
megohm. The feedback
ratio
pCR
Rc may be is
ff
(III.7.G.3)
+ pCR
ff
Fig, III.7.F.8
If
Each
resistance r
is
connected either to
ground by the switches
in such a
fy
r yi
,
,
fy
Pt
.05, and* .005, respectively.
r
,
7a
,
and
r^
.002, respectively,
resistances
The
have conductances
.1, .01,
and and
with
a
voltage
r yi
r
,
Vi
,
and
.02,
.2,
and
r^
have
a^
series
and
the resistances in
The required
kept small.
resistors vary with the
are
high
only
for
load must be
include
high
on
is
the in
amplifiers
this
The
is,
there
that the loop does not
reason
same both
is
1/0
co,
and, thus, normally
it is
use practical to
feedback
summing
difference
"operational"
amplifiers
and
between the two
is
the feedback impedance used. purely in
An
for hold
also
amplifier
integrating
provision
and
reset.
must have
The hold provision
to disconnect the input of the permits one from the rest of the circuit so that the
voltage output
.
Reset
Integrating Amplifiers in electronic differential analyzers
based on the use of feedback amplifiers in
is
of the amplifier remains the same.
used before computation begins on an
electronic differential analyzer,
to insure that
This
Integration
For
.
the
conductances
associated with the most significant digit a x
m.7,G.
1
1
amplifier
tolerances
conductance values and the
the stability condition
X
maximum
values of
reasonably close to
integrators.
the load on the generator l parallel. However, varies considerably with the setting percentage wise, and, hence, the
relatively high
basically
circuit
all
For
Since
be equivalent to a
generator
(III.7.G.4)
fore,
.001, respectively.
to impedance equivalent
~j(o,
o>CR,
resistances
the resistances are either used or grounded, this circuit is readily seen to
p
that the
way
and the output is l to the desired ratio a^ The resist proportional and r have relative conductances ances .5,
let
X
conductance between
total
we
X or to
is
will
and
its
purpose
is
the proper initial value.
done by an additional feedback connec
tion which period.
E* has
is
The
connected only during the reset
details
of the hold and reset circuits
be given in Section IIL13.H.
AMPLIFIERS
102
f.
References for Chapter 7 a.
H. W. Bode. Network Analysis and Feedback Amplifier Design. New York, D. Van Nostrand Company,
g.
A. Goldberg,
"Stabilization
of the wide-band direct gain,"
RCA
e.
Bell
System
and
experiment,"
I.R.E. Proc., Vol. 22
RCA Receiving Tube Manual Tech. Series RC15, Tube of America, Harri Department, Radio Corporation son, N.J.
John
Dover Publications, 1945. pp. 129-34. G. A. Korn and T. M. Korn. Electronic Analog Book Computers. 2d ed. New York, McGraw-Hill Co., 1956.
h.
Rev.,
Applied Electronics. New York, Wiley and Sons, 1954. Principles of Electrical Engineering Series, 2d ed., Technology Press M.I.T. d. K.Knopp. Theory of Functions. Part One. New York, T. S. Gray.
theory,"
(Oct. 1934), pp. 1191-1210.
current amplifiers for zero and Vol. 11 (1950), pp. 296-300. c.
"Regeneration
TechnicaUournal, Vol. 11 (Jan. 1932), pp. 126-47. E. Peterson, J. G. Kreer, and L. A. Ware, "Regenera tion theory
1945. b. E.
H. Nyquist,
i.
F. E.
Terman and G. M.
ments.
2d
ed.
New
Pettit.
Electronic
York, McGraw-Hill
Measure
Book
Co.,
1952. j.
E.
J.
New k.
A.
J.
Townsend. Functions of a Complex Variable. York, Henry Holt and Co., 1915. Chapter 7. Williams, R. E. Tarpley, and W. R. Clark, "D-C
for zero and gain," A.I.E.E. Trans., amplifier stabilized Part 1, Vol. 67 (1948), pp. 47-57.
Chapter 8
ELECTROMECHANICAL COMPONENTS
m.8.A. Moving Wire
magnet. Within the poles
in Field
The usual speedometer
is
an instrument
for
measuring the rate of rotation of a shaft. If a is moving in an electromagnetic field, a
wire
voltage
is
induced which
is
proportional to the
rate of increase of flux within the loop
formed by
the wire (see In order to be Fig. III.8.A.1).
more
iron core
we
ignore.
dotted loop which
lies in
is
uniform
a plane. If
is
the
from a fixed angle of rotation of the loop of flux then maximum position
an armature whose armature we have
in fact
may be
it
supposed to indicate that the armature
way around
a coil which loops the armature
number of times. The
are the
commutator
off brushes simplified
solid portions of circle
m,8.A,2
Fig.
us assume that the field
and that the wire loop
is
the wire goes all the
Fig. IH.8.A.1
let
is
this
mounted various wire loops which are repre sented in the figure by a solid radial line and a
a
specific,
On
bars.
which are
A
and
B are the take
fixed in space.
(We have
the following explanation by doubling
the brushes.
The
actual windings used vary in
ways depending on the voltage and current characteristics desired and the number of different
= A cos B
(IH.8.A.1)
= fc0=-k4sine
(IH.8.A.2)
and poles, fl
dt
The
can a principle of direct-current generator
be described by reference to Fig. III.8.A.2. Here a permanent field is given by the two poles of a
but those aspects of the situation which
are of interest to us are precisely the same.) It is clear that the coils
minimum
flux across
to the field, parallel
near the A brushes have
them
since they are almost
and those near the B brushes
have maximum flux through them. Thus the flux
ELECTROMECHANICAL COMPONENTS
104
is
increasing for those coils
and IV, decreasing Thus, in
marked
This device
rise.
pole the contacts
A
is
and
I
II
III.
The fact that the output is a
shaft rotation
is
very
useful for computation with potentiometers.
A to B we have
An alternating-current generator differs from a
a generator with one
direct-current generator in that the rotating coils
four paths going from
all
a voltage
we have marked
for those
and the other the
5
s.
are in series (except
where
it is
desired to increase
depends upon the position of the coils as well as the rate of rotation. Thus, the This voltage
voltage rise
is
rise
X
not simply proportional to the rate
of rotation,
,
OUTPUT
but contains a varying com-
dt
ponent. In Fig. III.8.A.2, however, it is clear that the geometric situation repeats itself every 45, and we could make this repetition occur at
oflt)
360
by
taking n coils instead of
8.
Fig. IH.8.A.4
the current at the expense of the voltage) electrically slip rings
and
the position of the coils relative to the
is
fixed.
In Fig. III.8.A.5, the solid
radial lines represent wire loops, the dotted lines
simply represent connections.
The two
circles
are slip rings.
m.8.A.3
Fig.
Thus the tiator.
direct-current generator
Notice that the
of relatively is
little
difficulties
importance
if all
the sign of the derivative, as
is
a differen
mentioned are that
is
desired
when the rotation
of the difference shaft in Fig. III.4.B.3 is con sidered. generator could be used instead of
A
the friction arrangement to control the relay (see Fig. III.8.A.3).
somewhat
The
characteristics
would be
different; the generator might permit
a certain accumulation of difference
were slow,
i.e.,
if
x
y
if
the rate
Fig. IH.8.A.5
discussed in Section
III.4.B is not too great in absolute value.
A
motor generator set can be used as an integrator whose output x is a shaft rotation.
The
In Fig. III.8.A.5 the total flux through coils 1,
2,
integrand, f(t\
is
represented by an elec
signal input which
is compared with the generator output. The difference/- x is used as a servo signal to drive x (see Fig. III.8.A.4).
trical
and
3
is
= A cos 6 + A cos (0 + 45) + A cos (0 + 90)
where 6
is
the
(III.8.A.3)
amount of rotation from the
position shown. (The other circuit is analogous.)
MOVING WIRE
III.8.A.
Consequently the voltage
IN FIELD
Considering the
rise is
the result that if
=
=
+
-A[sm 6
at
+
sin (0
elements,
we
get
a current through the it,
we would
get a
torque instead of a voltage rise. This, of course, is the of the electric motor. In the principle
+ 90)]0
usual ammeter,
- -4sin 0(1 + cos 45 + cos 90) -f cos 0(1
sum of such
we put
conductor, instead of rotating
+ 45)
sin (0
105
this
is
the
method used to
measure currents. The current produces a torque which is proportional to it. This torque is
+ sin 45 + sin 90)]
measured by observing
how
far
a spring
is
displaced,
cos 0(1
=
-A(l
+ V 2 /2)[sin + cos 0]0
=
-A(\
+
Now
if 6 is
V
)[sin (6
+ 45)]0
V2/2)]0
(III.8.A.4)
we
relatively constant,
+
see that the
output is an alternating current which is modu lated both in magnitude and frequency by 6. This can be used in a
number of ways. The
above discussion generalizes to a number of coils. It
be well at
may
this point to indicate the
relationship between generators
and motors. In
Fig. III.8.A.6, let us consider the action of the field
magnetic
Then
element.
which
is
H on an element
Let
the wire.
m
the voltage rise e
e
is
is
a vector
given by the equation
e
i.e.,
dl of length of
denote the motion of the
= mxH
(III.8.A.5)
m and H and has
perpendicular to both
size
mH sin where
On
6
is
dl
the angle between
the other hand, if
m and H.
we had
a current
i
H
is the same, then there through the wire and would be a force f on the wire element
Fig,
f=Hxi
IDL8JL6
(III.8.A.6)
To sum up, a generator is a device by which an If
we had two such elements relative
placed
conductor,
to
we would
the
axis
symmetrically
of the rotating
angular velocity voltage. is
get a torque
is
A motor
converted into a proportional a device by which a current
is
converted into a torque. These correspond to
the interaction of electromagnetism and motion.
T=
2r/sin0
(III.8.A.7)
The corresponding electrostatic interaction is one in
where
r
is
the radius of the conducting loop.
which a voltage produces a
motion produces a
current.
force,
and a
ELECTROMECHANICAL COMPONENTS
106
IDL8.B. Mechanical Analogs of Electrical Cir
(The
cuits
in acceleration, the second
In Part
IV we will discuss
mechanical and IV. 7), It
systems electrical
is
the analogy between
electrical elements (see
Chapter
often convenient to study mechanical
by analogy with the corresponding circuits, and in the case of mixed
systems where conversion occurs between elec trical and mechanical this permits a quantities
uniform consideration of the
full
system. There
an analogy between electrical and hydraulic systems. The techniques developed for also
is
term in Eq. III.8.B.2
first
overcome the viscous
AW-
nnnnr
friction,
torque used to
and the
third term
introduction to the subject of mechanical and electrical
analogy, which will be treated in detail
IV
in Part
also
method of handling a combined
the
electrical It is
Chapter IV.7). However, it does mechanical equivalent of filters and
(see
illustrate the
and mechanical system.
readily seen that this analogy extends to
mechanical
filters
electrical ones.
CONDENSER
the torque used
corresponds to the torque of the spring.) The above discussion is intended merely as an
more complicated o
is
the
is
Hence, we can have
circuits.
and smoothers as well as
In general, the frequencies are
smaller in the mechanical case. Unfortunately, in
unknown
general, the resistance coefficients are
and highly varying, which
is
a
difficulty
with
designing mechanical circuits. There is another way in which the analogy can be drawn which is particularly suitable for FRICTION
Here the notions of
electromagnetic devices.
BRUSH
VISCOUS FRICTION
force, mass, velocity,
and the compliance of a
spring correspond to the electrical notions of
Fig. IH.8.B.1
current, capacity, voltage,
have useful mechanical analogs. circuits of Section III.5J have
electrical circuits
The
filter
and inductance. Thus,
the impulse equation
important mechanical equivalents which are used for the analogous mechanical process of smooth ing.
Let us consider Fig. III.8.B.1. The second
combination of flywheel, viscous
friction, drag,
and spring
is
we assume
that the total charge passing into the
circuit is
analogous to the
The torque
circuit
is
is
compared with
applied to the mechanical
Ha i
dt
the inductance, the viscous friction to resistance,
is
kF where
jc is
the voltage equation
- ej
(III.8.B.4)
=x
(III.8.B.5)
the displacement with the inductance
relation
is
r
proportional to the angle turned is analogous to the condenser. All this is immediately apparent
we compare
C(e 2
to
analogous
and the spring whose torque
=
and the compliance relation for a spring
analogous to the applied voltage, the
moment of inertia of the flywheel is analogous
if
that for the flow of current into
a condenser
electric circuit if
analogous to the rotation of the input
shaft.
(IH.8.B.3)
i
To
illustrate this,
resistance
R
t
=
\edt
let
(III.8.B.6)
us consider a coil of
pivoted to turn so that
it
cuts a
constant magnetic field (see Fig. IIL8.B.2).
us suppose that the coil has
with the torque equation
and
~
~
dt
dt
Let
moment of inertia /
is subject to springs which tend to keep it in a position 0. (The Arsonval movement of a course, is, good example.)
D
IIL8.C.
kFa
Let k be defined by the equation
That is,
F
s
is
the force exerted
WATT HOUR METER
by the spring
=
107
Power considerations
6.
when
that if
= Ri
(III.8.B.15)
the resistance voltage drop
in the coil, then the
e"
amount
the coil has been turned an
show
also
0. is
total
is
voltage drop
"equivalent
III.8.B.4, where,
+
e"
circuit"
e
Consequently, the
.
for our coil that of Fig.
of course, R, C, and
I have
the
This means that as far
values indicated above.
as the electrical circuit connected to the coil
concerned,
we can
same
the
get
effect
is
by
A
instead of the coil. substituting this circuit
shunt resistance will indicate the
effect
of viscous
friction.
Fig.
If there
is
a current
field will exert
proportional to
HL8.B.2 i
a force
FH
on the
coil
which
is
i.
hFH If
in the coil, the magnetic
FH
=
i
(III.8.B.7)
does work there must also be a counter Fig. HI.8.B.3
dB electromotive force e
must equal
ie
and the power
FH r
,
Since (III.8.B.8)
we must have (III.8.B.9)
hJt~ The equations of motion Fig. IIL8.B.4
m.8,C. Watt Hour Meter
become
A
watt hour meter
Essentially
h
r dt
kr
this is
used as an integrator.
is
an electric motor with a
which permits type of load
it
special
to represent
or r*.
IEdt
r/dt
Jti
kr
Thus, the current voltage relationship asthatofFig.ffl.8J.3,i.e., r fc
-
"
dt
L
is
the
same
Consider a motor whose
the current / in the
edt
field is
an electromagnet and, hence, coil.
obtained from
to proportional current / for the
is
The
load to be measured goes through this coil while the armature current is obtained through a resistance shunted across the load and, hence,
provided 2
,/t
to E.
proportional that the torque
is
From
Section III.8.A
proportional
to IE.
we
is
see
ELECTROMECHANICAL COMPONENTS
108
Let
be the rate of rotation of the armature.
co
m
Let
moment
denote the
of inertia of the
and the associated rotating
armature
Then we have
parts.
the torque equation
=
I
+
Lco
is
(IIL8.C1)
the load torque.
Solving for
we
co,
E=
current
i.e.,
Since there
a weighted average
is
the
function - IE over the
a current in the armature of
we have a
counter torque, i.e., the generator begins to act like a motor. In turning
we do work against this
the armature
III.8.C.3
now
is
the generator
write
this
Eq.
Then
(III.8.C.12)
R~
^t
and
(III.8.C.11)
E (IE)dr
m
Let us ignore the transient in Eq. III.8.C.2,
Ce~
/ceo
across a resistance R.
~
J-oo
potential
is
i
ft
the term
E
Let us apply
obtain
given by the
Let us turn a generator with
:
The generated
(o.
dt
where Leo
is
Theoretically the load torque Leo could be
obtained as follows a velocity
=m
T
Thus, the value of the integral rotation of the shaft.
from
interval
of the oo to
work
torque, and the energy dissipated in the resist
is
ance. This counter torque is proportional to the current and hence equal to Leo.
t.
LJ
Changing the variable of integration, we have
L *
A -r/JS -Hr/JS
f
,^
J.^m where IE is taken
L
at
\L
^
f
customary to
dr
(III.8.C.4) v
+ T in the integration, and
t
=
dr
It is
\
/
refer to this
1
(III.8.C.5)
average as a time
L delay,
and indeed if-
is
sufficiently large
and IE
does not change too rapidly, Eq, IIL8.C.4 be approximated as to
where TO
from
= IE(t - T
)
Fig.
(III.8.C.6)
dissipating generator,
the value such that the total weight co to TO is \. For this we have is
~r 1
-
=
f
4
L
em
m
~
IIL8.C1
may
We
must then hook our meter motor to a
remembering to keep m accomplished in the meter by introducing an aluminum disk between two poles ^ a ma The meter motor turns the disk. low.
This
is
net>
r
fa
_
e
T
m
(IH.8.C.7)
The magnetic
located
field is
center of the disk.
The
or
up a voltage TO
=
7L
log, 2
=
,7-
L
(IIL8.C8)
on
in the disk
Since
rise
side of the
from the center
this side,
to the edge of and, hence, a current flows
whose energy
is
dissipated
by the
resistance of the disk.
-
current
Such
see that
1-82=
will
IE(t-rQ )dt
is
(III.8.C.9)
dt
we
the disk
on one
rotation of the disk sets
which are called eddy currents, any conductor rotating in a
currents,
occur in
magnetic (III.8.C.10)
In Fig. III.8.C.1 the indicated by dotted lines.
electrical
field.
They
are very objectionable in
machinery, where they introduce a
SYNCHRO SYSTEMS
III.8.D.
109
The other
the
dissipating torque proportional to the speed.
be transmitted
To minimize
which reproduces the rotation. "synchro These units differ only in minor respects. Both
machines
is
these
losses
the
iron
in
such
laminated so that the resistance to
these currents
is
up.
unit
is
In
units are similar in appearance to motors.
as high as possible.
due to three sym each having a coil. metrically placed pole pieces, The rotor has a single coil and is an electro each case there
HI.8.D. Synchro Systems
A
synchro system is a system set up to reproduce a rotation at a distance. The connec tion between the two points is purely electrical.
The importance of
set
is
motor"
the system in a mechanical
is
a
field
magnet.
However, the
units are intended to function as
transformers in which the rotor coil
primary and the
field coils are
If the signal generator rotor
is
the
the secondaries.
and the motor rotor
are similarly placed relative to the field pieces,
then the voltages induced in the pair of field coils are equal and opposite, and no current will flow in
any of the three
field circuits.
If,
however, the
rotors are not similarly placed, the voltages will
not cancel, and current will flow. This will set up a magnetic field which will act on both rotors,
up a torque tending to cause the rotors to be similarly placed. We can indicate the direction of the magnetic setting
field
by the following considerations.
Let us
suppose that the rotor of the motor has been removed. The effect of the induction due to the transformer rotor current which sets
of course, to induce a
is,
up
a magnetic field in the
generator field poles opposite to the field of the rotor. Let us look at this first in the transformer
or signal generator. Of course, there are three each with a magnetic field. But it is poles,
field
clear that the resulting magnetic field
one that induced
to the
it.
is
opposite
However, the current
that flows in the generator poles also flows in the
motor poles and
Fig, HI.8.D.1
there
computing device
lies
mechanical units
to
positions and
in the fact that
have
also that
nections between units.
between units
it
it
arbitrary
permits
permits relative
flexible
con
If the only connection
by rotating shafts, there are major alignment and layout problems which must be solved for each setup. A synchro system is
can be used to eliminate these problems. which The type of synchro system, simplest
in
which the rotation
The
is
to
we
sets
up a
precisely similar field
neglect resistance
effect
and other
of the motor rotor
is,
losses.
of course,
However, actually the current that are present must set up a
analogous. flows
when both rotors
field that
corresponds to the vector difference
between the two magnetic
fields,
one of which
is
the field of the generator rotor, the opposite to
other opposite to the motor rotor
The
shown in Fig. III.8.D.1, involves two parts or One is a "synchro signal generator" or "units." "synchro transformer,"
if
effect
field.
of this can be readily obtained. Let
us suppose that the generator rotor magnetic field and the motor field are as indicated. Let 5
H
denote the vector denoting the generator rotor
magnetic
field,
H
TO
the corresponding vector for
ELECTROMECHANICAL COMPONENTS
110
the motor;
the field then
torque exerted by this
Hm
is
field
-
H5). The k(E m on the electromagnet
is
shown
in Fig. III. 8. D. 3 has three coils
and three
to the field pole pieces. precisely analogous The use of the differential selsyn permits one
faces
given by
to specify the difference between the input
and
the output rather than just insure equality. If we
This torque, of course, to bring
H m and H
s
is
in the direction tending
in coincidence,
and
consider the effect of a selsyn generator alone,
it
its size is
proportional to the sine of the angle between them. (See Fig. IIL8.D.2.) The generator rotor
has precisely the opposite torque exerted on it, but presumably the input determines its position.
Note that this means that the input must do work if
is
any load on the output.
Torque
amplification, however,
there
is
possible
one uses an additional motor on the output
The
rotor
is
the previous parallel to
if
shaft.
motor rotor
and so are the corresponding pole pieces. The current in the new motor field pole coils is
Fig, IE.8.D.3
controlled to be a multiple of the current in the original field coils.
This
is
done by magnetic
means.
is
the differential selsyn. This unit
is
in appearance to the other selsyn units, field
clear,
field set
Another selsyn unit of considerable import ance
is
pole pieces are the same.
similar
and the
But the rotor
is
from
Fig. III.8.D.4, that the magnetic
up by the rotor of the
differential
selsyn
a replica of that of the generator rotor relative
to the field coils. differential is
Hence,
if
the rotor of the
positioned so that the rotor coils
face the pole pieces, the field coils of the dif ferential are affected in the
the generator. differential will
the
field
coils
same way
as those of
Also rotating the rotor of the change the induced currents in to
those corresponding
to
a
same angle in the field of the Thus, if a is the rotation of the
rotation of the generator.
generator rotor,
ft
that of the
motor
that of the differential rotor, for Fig.
HL8.D.2
rotor,
and
no torque we
must have *
Fig. HI.8.D.4
+ =
ft
(III.8.D.1)
IIL8.D.
The
differential selsyn
is
SYNCHRO SYSTEMS
particularly useful in
where two quantities are to be added to yield a third and the three are far apart in
rotor
is.
111
(There are, of course, two positions of apart.) This can be used to
situations
zero voltage 180
space.
control an alternating-current servo. selsyn system or its equivalent is valuable in
A
Another selsyn device of wide application is the selsyn control transformer. If a selsyn motor
freedom in connecting various
has no voltage impressed upon
the later version of the differential analyzer at
power source, the field
itself will
rotor from the
induce a voltage.
zero only when the receptor rotor perpendicular to the position corresponding to
This voltage is
its
is
the generator, and its phase will show on which side of the perpendicular position the receptor
calculating devices because
allows for great
it
M.I.T., a selsyn system (based
units.
on
Thus, in
capacity,
how
than induction) was used to connect the units, and, of course, this gave one the same ever, rather
freedom
in-
the interconnecting that one
have with a purely
electrical setup.
would
Chapter 9
ELECTRICAL MULTIPLICATION
IEL9.A. Introduction
We
a digital one, multiply digitally and reconvert. (See, for
multiplying an electrical voltage by a constant
and obtaining another voltage (see Section III.6.B). For instance, the given voltage may be applied across a potentiometer, and the poten
tiometer
is
We may
set
according to the constant factor.
also use a transformer to multiply a
voltage by a constant, provided the variable input is
an alternating-current voltage and the turns
ratio of the coils corresponds to the constant (see Section III.5.I).
A machine using transformers for this purpose is
discussed by R. R.
On
the
M. Mallock
transformers
in
this
(Ref. III.9,r).
machine the
primary has a fixed number of turns. There however,
many
series to
These can be connected in
obtain a wide range of secondary turn (See also Ref. 111.9.x.)
values.
we
frequently have the situation where the values of
the two variables are given by the two voltages desired. It
is
quite a
common
practice to use one voltage to control a servo in
way
that the rotation of a shaft
is
propor
tional to the variable (see Section III.4.E). If this shaft drives the contact
on a potentiometer and
the input voltage
we can
instance,
is applied to the potentiometer, the get product as the voltage at the
contact.
means of a representation of the square function or of the log function. These are functions of one variable;
we will
discuss
methods of representing
comparatively arbitrary functions of one variable
by purely electrical methods in Chapter III. 10, Another method of representing the product of
two functions, recently developed, is the time division multiplier, which is discussed in Section III.9.B.
another possibility
Still
which
will
eliminate the difficulty inherent in the response of the servo system is the use of a mechanical in
which one variable
represented by a force
is
rather than affects the
by a displacement. If this force value of a resistance, one can use this
also for multiplication.
This will be described in
For
electrical
computing
devices, in general,
we can obtain a procedure
for a given purpose
by considering the corresponding mechanical system and replacing the elementary operators
on addition, their
and multiplication by equivalents and by replacing
integration,
electrical
multiplier
and servo
and amplifier are
in Fig. III.9.A.1.
the difficulty of the shaft position of the servo tending to lag behind the voltage it is supposed to
a very small value.
difficulties
instance, the use of
a
to obtain division in a
mechanical system (see also Section III.2.F) has an electrical equivalent in which a multiplier, differencer,
This can introduce serious
For
servos by amplifiers.
However, the method described above for multiplying one voltage by another is subject to
represent.
For
can be obtained by
multiplication
Section III.9.D.
In continuous computing devices, however,
and their product is
these electrical multipliers can be based.
system
are,
secondaries with turn numbers
in multiples of 10.
such a
example Ref. III.9.a, 1, m.) There are a number of principles on which
have previously discussed methods for
gain, the difference
utilized, as
shown
If amplifier II has
x
adequate x can be maintained at
This principle of amplifying a difference is many ways. Suppose jc is a shaft rotation
used in
in the computation. For this reason many efforts have been made to develop purely electrical
input and
multipliers or to convert the continuous signal to
sponding to
we wish y
to obtain a voltage corre
= - We use an amplifier whose .
TIME DIVISION MULTIPLIER
III.9.B.
/
and whose input is the difference output is between xy and a reference voltage correspond
ing to unity (see Fig. III.9.A.2).
x
positions the contact
When
applicable,
on
The
shaft input
the potentiometer.
approximation methods
the reciprocal are desirable in electrical
for
com
modulate the width of a square wave, and the other variable y gives the amplitude of the square wave. The product xy then can be obtained by
an averaging process. There have been a number of time division
Second World
multipliers developed since the
War
puters (see also Section III.2.F).
113
Baum and
R. V.
(see
An
III.9.d).
E. Flater
C. D. Merrill, Ref.
Argentinian device
and K. Franz
discussed by
is
division multiplier credited to L.
time
division
on the
Variations
(Ref. III.9.aa, pp. 67-68).
time
G. Walters
W. W. Soroka
described by
is
(Ref. III.9.ac)
A
(Ref. III.91).
have been presented
principle
recently (see Ref. IH.9.s,ab).
However, one such device completely described in the literature
is
due to E. A. Goldberg (Ref.
III.9.gorh).
A
voltage at a point in the circuit
represent a square
wave
is
said to
alternately assumes two values in a regular manner. Suppose, for
Fig.m.9.A.l
if it
example, that the two values are -f e and
There are certain
electrical
methods by which
products can be obtained directly, but these are of limited usefulness. For example, one can use
a
"mixer"
III.9.w).
tube such as the
6SA7
(see Ref,
Mixer vacuum tubes have two control
and the output is essentially the product of the signals on these two grids. There are feedback procedures using various frequencies
grids,
which can be
utilized to
improve the accuracy,
that for a time interval
Now, suppose
voltage assumes the value -f
I2
time
repeated.
*
assumes
it
,
e
Tx
and then for a
and the process
,
.
the
is
The graph of this voltage as a function
of the time consists of alternate bars at heights
+e
and
e
indicated
and widths
before
that
T:
in
and
order
T2 We .
to
have
represent
information by a voltage, the voltage must be
modulated, reference
that
must be a
there
is,
fixed-
and the information
situation,
is
conveyed by the departure from this fixedreference situation. In the case of a square wave
two ways readily available by which a indicated. One method depends on the width of the intervals T and T2 The
there are
number can be
-
other
and Fig.
is
T2
the height of the square are varied,
modulation.
m.9.A,2
we have
wave
e
.
If 7i
pulse width ratio
For example, a variable x can be
represented as follows: Let
but normally the voltage range available for the input is regarded as too limited. (See also Ref.
I2 and
)
(HL9.B.1)
let
IIL9.e, pp. 668-74.) :
HI.9.B.
One
Time fast
division
T(l
method of multiplication with two
variables represented
by voltages
method. Here one variable
7i
(III.9.B.2)
Tz
(HI.9.B.3)
and
Division Multiplier
is jc is
the time
made to
It is also
- x) =
to modulate the height e possible
of the square wave.
Suppose then
we have
,
a
ELECTRICAL MULTIPLICATION
114
square wave pulse width ratio modulated as in the above with x and height ratio modulated with Its
y.
shown
is
graph
in Fig. III.9.B.1,
The
wave form described above which
the
ratio
by
modulated by x and height This output
y.
is
component of this voltage can be obtained by averaging this square wave function
voltage xy.
and
and condenser combination
proportional to xy.
For the averaging in
-
/
2
?i
we may use a
circuit
In
III.9.B.3.
Fig.
relative to
width
is
modulated
then averaged to yield a
direct-current
is
ratio
resistor
circuit as indicated
this circuit
RC
small
is
the total time interval over
which the average
to be taken,
is
and
large
the input voltage E, relative to 2T, the period for
which E(f)
is
Let
to be averaged.
is
a periodic function,
it
co
= 2irj2T.
Since
can be expressed as
a Fourier Series (Ux)T
ao
+2a
fc(
cos
k^O
+
-y
(III.9.B.4)
EO
-OE
y/\/wR
Fig, IH.9.B.1
Thus a time Fig.
division multiplier
IIL9.B.2.
voltage,
is
modulation.
first
The input
jc,
is
shown
in
presumably a
applied to a pulse width ratio
This
is
Fig. III.9.B.3
a device which produces a
square wave which is pulse width ratio modulated
by x but with a constant amplitude, The output of this pulse width ratio modulator is used to
The
relation
between
E
in Section III.6.D (see
KG
and
E has been obtained
Eq. III.6.D.5) as
Jfc.
+E
1
-;1)/C]
exp-[(f2
(IIL9.B.5)
r
where E^
is
the value of
interval of integration. f
2
*i
is
E at l9 the start of the We are assuming that t
large relative to
the last term involving
/
RC. Consequently..
in Eq. III.9.B.5
can be
neglected, and
(III.9.B.6)
Fig.
m.9,B,2 is
control an electronic switch.
The
electronic
switch has two inputs, y and -y, and the voltage
y when y when it
output
and
is
the controlling input
is
obtained.
We may then
substitute
IH.9.B.6 and integrate term by term.
of integrating the constant term
E into The
Eq.
result
is
positive
is negative. Consequently, the of this electronic switch will be precisely output
(III.9.B.7)
TIME DIVISION MULTIPLIER
III.9.B.
If
we
we may,
neglect the exponential term, as
we see that E
When we
or
has the same constant term aQ as E.
any of the remaining terms, for instance, b k sinka)t, we obtain an
say,
115
integrate
1
1,000,000
2T
ITT
~
160,000
(III.9.B.12)
Eq. III.9.B.12 is the repetition frequency, i.e., 160 kilocycles, which can be depended on to give
expression in the form
excellent results
using a simple averaging
circuit.
fc
we
provided
terms
neglect
exp
involving
There are a wide variety of switching circuits one of which has been designed specifically for this
Ref. type of service (see E. A. Goldberg,
Switching circuits are described in Puckle (Ref. IIL9.V, pp. 52-88).
III.9.g, h).
= 2irkRCI2T
RCkcD
and
since
RC is
(III.9.B.9)
0.
S.
the expres large relative to 2!T,
sion Eq. III.9.B.9
large relative to
is
1,
SQUARE WAVE INPUT
and the
b k are approximately k
sn (III.9.B.10)
RCkoo
fe
sn Fig.
If
we
let t
circuit
=
we
1^
when E is
the input voltage
E
m.9,B.4
see that the effect of this is
which has # as
an
to yield
A
has three inputs switching circuit normally
term, and the remaining terms are divided by a
and one output. One input is a square wave, the other two inputs are voltages X and J2 (see Kg-
factor which
III.9.B.4).
output voltage
RCo)
is
at least
RCa>
its
in size.
constant
Thus,
if
the only important term in the large,
is
Now for
our square wave which is output and width ratio height ratio modulated by y is OQ.
modulated by
x,
aQ
is
proportional
The purpose of the switching
This
to xy.
the desired output voltage. is, of course, In the devices using such time division multipliers,
the time
t
is
circuit is to
V such that when the produce an output voltage V = Jls square wave is at the positive value, s.w.
QJ
the independent variable, t must corre ? : 2
and, thus, the averaging time
in the independent spond to the smallest change t which has mathematical significance.
variable
may mean that we wanted RC small
This
the factor exp
(f a
[
*
^
2
about
is
relative to
-
f
a
.01.
Now
to
make
^
tJlRC] small. This
2
D/"
2T factor
Kg. m.9,B.5
will
if (f a fJ/JRCis about certainly be accomplished 10. Thus RC should be about .001. We want
and when the square wave value,
to be large 5 relative to
1
since this
is
the
terms are by which the nonconstant
reduced. For example,
let
us take
V=
=
which has an output, which
negative 1,000
(III.9.B.11)
Ordinarily
is
at the negative
this is
accomplished
circuit
""
2T
8
.
(see Fig. IIL9.B.5) by means of a simpler which has input, say X, and a square wave, and
feed
Xl
is
X when the square
and when it has the positive value, value. If we have two such circuits, and
wave has "
J
its
into
one and control with the original
ELECTRICAL MULTIPLICATION
116
J
square wave and
2
into the other with the
square wave inverted and add the outputs, we obtain the desired result (see Fig. III.9.B.6). The values
Xl and J
2
can both be positive if we add a
bias voltage to the input of the final amplifier
.
diode has no effect on the
X
circuit, and the voltage controls the triode as a cathode follower.
When the switching voltage is negative, will
this
approximately negative voltage the
the diode
conduct, and the plate of the diode will be at
is
negative
voltage.
If
this
sufficient to cut off the triode,
voltage output will be zero.
The pulse width ratio modulator is generally based on the use of a multivibrator circuit. A multivibrator circuit consists triodes with a resistive
normally of two
and with the
plate load
grid of each capacitatively coupled with the other (see Fig. III.9.B.8).
Ci and
C2 are
bias
voltages
Fig. IIL9.B.6
The simplest type of switching circuit in Fig. III.9.B.7.
One has a
resistance
is
shown
R
which
connected to the plate of a diode and also to the grid of the tube. The cathode of the diode is is
Fig. IH.9.B.8
normally in the conducting range. If the symmetric, and the gain of the tubes is
circuit is
adequate,
the output of this circuit is a square wave. By varying the bias Ci, the length of the positive part of the square wave can be lengthened relative to the negative part. The higher the bias Cl9 the shorter the time the tube D is nonconducting, and, hence, the shorter the time the output square wave is positive. Varying the bias C2 has the
opposite effect. Changing one bias tends to change the period of the square wave also. Thus, this circuit can be considered as one which has as
input a direct-current voltage and as its output a square wave for which the ratio of positive to
Fig. IH.9.B.7
connected by a
resistor to
ground and
square wave switching voltage. ing voltage is
high
is
positive, the
relative to
its
its
also to the
When the switch
cathode of the diode
plate and, consequently, the
negative
part
depends
on the
voltage (see Fig. III.9.B.9).
direct-current
This device can be
used as part of a pulse width ratio modulator as
shown
in
Fig. III.9.B.10.
TIME DIVISION MULTIPLIER
IIL9.B.
This circuit works as follows:
The
an averager for a pulse width
multi-
vibrator produces a square controls an electronic
wave output which switch. The electronic
switch has constant input voltages, eQ and
-e
.
117
signal as
shown
ratio
in Fig. III.9.B.1
The feedback arrangement x is such that relatively
modulated
1.
insures that the
output
is
a constant, and thus x
average of x
is
the
negative of an
.
Fig. HI.9.B.9
1
The output x is a pulse width signal. The signals x and -x
ratio
modulated
f
integrating amplifier
current
together feed an
whose output
is
a
direct-
voltage which controls the multivibrator
- x ) dt remains essen f
in such a tially
way
that
constant.
If
J (x
x exceeds x
f
we wish
,
Fig,
m.9.B.ll
to
For an arrangement of this type involving the use of two extra
amplifiers the repetition time of
the multivibrator need only be large relative to the smallest significant time interval. It is also
to
possible to use a diode limiting circuit
amplitude modulate the pulse width ratio
modulated signal The diode limiter can be used to
produce output voltages with a specified
maximum or minimum, For example, the circuit
Fig. III.9.B.10
increase the time in which the output
This
is
is
positive.
accomplished by using the output of the
integrating amplifier
as the
C2
vibrator (see Fig, III.9.B.8).
bias in the multi
This produces a
pulse width ratio modulated signal correspond
x with constant amplitude switching voltage can also control a number
ing to the value
The
.
Fig.
HL9.B.12
of electronic switches whose inputs correspond to
quantities
y and whose output
is
the
product
xy. In such a setup a number of products xylt
xy 2 etc., ,
can be obtained with the same factor
x.
The integrating amplifier idea can also be used as
of Fig. III.9.B.12 produces an output whose
maximum and minimum
have absolute value y,
but whose intermediate values are the same as the input.
Owing
to the action
of the diode, the
ELECTRICAL MULTIPLICATION
118
summing
amplifier
equal to the
Al
The
of y.
amplifier
is
Al
and produces an input for a
inverts this input
similar circuit.
an input which
receives
minimum
The output of
original input clipped of! at
+y
Az
is
and
then the
The input
modulated
ratio
is
then the pulse width
signal corresponding
factor x, and, in this case, the
A 2 may be
summing
to
In
(Ref. III.9.p).
McCool
s
paper two
earlier
references are cited (see R. Price, Ref. III.9.U,
and M.
y.
This circuit can be used in a time division multiplier.
A
frequency modulation frequency modulation. W. A. McCool multiplier has been developed by
J.
Somerville, Ref. IIL9.z).
Another modulation multiplier is presented a paper by B. N. Locanthi (Ref. IIL9.o).
in
the
amplifier
UI.9.D. Strain Gauge Multiplier
Another device for obtaining the product of
replaced by an integrator.
two variables represented by voltages is based on the strain gauge. Here one variable is trans
in,9.C. Modulation Multiplier
In general, when a signal can be represented by
formed into a
force,
Hooke
since
or,
which
law
s
A
some modulation method other than amplitude
a displacement applies, to
modulation, one has a device called a
force applied to a strain gauge will vary the
which
is
"detector,"
capable of producing a voltage propor
tional both to the modulation
and the amplitude
of the signal received. In Section III.9.B we had a pulse time
electrical resistance
is
associated with
small.
it,
and a
variable resistance can be used as a factor in multiplication.
modu
There are a number of ways a force can be
modulated
associated with a change in electrical resistance.
by another variable. The time averaging circuits mentioned are detectors for this type of modula
Thus, for a coat of conducting particles on an
In general, given any method of modulation other than amplitude modulation, a multiplier
conductor will increase the resistance since the
can be constructed and provided with a detector with output proportional both to the amplitude
subjected to a force or to a tension, a conducting
lated signal which
was
also amplitude
tion.
and
to the modulation.
elastic
nonconductor, stretching the
contact between particles
lessened.
non
When
wire will increase in length and decrease in crosssection,
Thus, Fig, III.9.C.1 is the general diagram for this type of multiplier.
is
elastic
and both
effects increase the resistance.
These principles are used as the basis of
strain
gauges, and, in general, yield the resistance as a
monotonically increasing function of the force
M.
applied (see also X
Hetenyi, Ref. III.9.k).
Relatively, the change in resistance
MODULATED SIGNAL
is
small,
and, consequently, to use this principle in a multi
one must introduce
plier,
it
in
an
alternating-
current form. However, this does permit the use
of alternating-current amplifiers and has, there fore, certain
compensations.
A simplified strain gauge multiplier then would Fig.
be described as follows. There are two inputs.
HL9.C.1
One of In general, to control the modulation of the
modulated
signal
producer,
combination of detector amplifiers.
one needs some
circuits
and feedback
It is also desirable, if possible, to
control the;; amplitude modulation by feedback and detector methods.
One method
of modulation for which tech
niques of detection are very well developed
is
these
an alternating-current voltage
is
representing the variable x.
The other is a direct-
current voltage representing the variable y, and
the
output
is
an alternating-current voltage
representing the product xy. are connected
Two
strain gauges
mechanically so that they are
same force. Presumably then, they have the same resistance. The force applied to subject to the
these
two gauges
is
given by a solenoid. Thus,
IH.9.E.
the force
is
STEP MULTIPLIER
determined by the current in the
solenoid.
119
of the strain gauge with the other terminal
grounded. Hence,
This can be used as part of the multiplier
shown
in
Fig. III.9.D.1.
A
constant voltage
+
*.
IIIU
w-i
(See also A. C. Hall, Ref.
-
D 2)
and W. W.
III.9.J,
Soroka, Ref. III.9.aa, pp. 70-71.)
A
multiplier
making use of the
torque on an armature
fact that the
proportional to the product of the current in the armature windings * and field windings of a which "dynamometer, is
analogous to an electric motor, is described by G. A. Korn and T. M. Korn (Ref. IIL9.n,
is
pp. 256-57).
m.9.E. Step Multiplier
The variable resistance method of representing a factor in multiplication can be utilized in ways. factor
One
many
method of representing a
ingenious
by a variable resistance consists in express
ing the quantity in binary form electronic
by means of an and introducing a con
counter,
ductance into a
circuit,
stage in the counter as
accordingly as a specific or 1 If the conductances .
have values proportional to the stage of the electronic counter, the desired result can be
Fig. HI.9.D.1
obtained. is
applied across the resistance of one gauge and
a resistance
RQ
in series.
If R(t) is the resistance
of the gauge, the voltage
Such a multiplier has been developed for the U.S.
Navy
Devices
Special
Typhoon and has been
Center
carried out
Project
by R.C.A.
described by E. A. Goldberg (Ref. III.9.i). The following is a simplified version of the
It is
R(t)
p
appears at one terminal of the gauge if the other terminal is grounded. This voltage is compared
with x, and the difference
is
applied to an
principles involved in a multiplier
of
First such a device takes a
x and
forms
it
counter.
into a value
voltage
this
type.
trans
represented by a binary
(How this is to be done will be indicated
Then
the value of each stage of the
amplifier which, in turn, drives the current in the
later.)
solenoid so that R(t) varies in such a
binary counter must be used to control a corre
minimize
this difference.
way
as to
Thus,
sponding
we c
x
(III.9.D.1)
relay.
For
simplicity
we suppose
that
are dealing with a two-pole relay for each
if for the place. Thus,
purpose of explanation we jc can be represented by
suppose that the variable (This
is
an alternating-current
voltage.)
The
other variable y
is
voltage which
applied across another resistance
with value
is
RQ
represented by a direct-current
and the resistance of the second
gauge. The output z
is
taken from one terminal
we have
for each a; a relay -
which
activated or not accordingly as a3
-
is 1
is
or
either 0.
ELECTRICAL MULTIPLICATION
120
In Fig. IIL9JB.1, is
&$, that is
point PJ
grounded.
is,
if
clear that the voltage
it is
the ;th relay
P
j
activated, the
is
connected to y. Otherwise, Pj is The effect of the amplifier is, of
course, to produce a voltage z such that the
point
has essentially
Q
nodal equation for
ftj
,
V
Q
w
8? |
2
2
of different polarity on its two inputs. The electronic switch is controlled in such a way that if
the reading in the binary counter
is
less
than
jc,
pulses of a polarity which will cause the binary
current
yields
.fey.
*
The
voltage.
control an electronic switch which receives pulses
2
3
=0
.z i
i
K
R^
R, (III.9.E.2)
or
(III.9.E.3)
This device has been simplified to the extent
on y depends on the
that the load
voltage
y
factor x. If the
generated directly by means of a
is
feedback amplifier, this may not be a serious objection, but by means of a somewhat more complicated the
T
Typhoon
circuit this
has been overcome in
Calculator.
Fig. IDL9.E.2
counter to increase counter.
its
value go to the binary
On the other hand, if the reading of the
binary counter applied to
it
is
greater than x, a signal
which causes
it
to count
is
down.
m,9.F. Cathode-Ray Multipliers Special purpose cathode-ray tubes have been constructed for multipliers. One such tube, credited to A. Sommerville (Ref. III.9.y) has a
normal electron gun with horizontal and deflection plates.
vertical
This gun, however, instead of
producing a concentrated beam, produces a uniform density of electrons across a square on the face. Behind the face are four collector plates positioned so that if the electron square
Fig.
HL9.E.1
A voltage
on the horizontal In the Typhoon Calculator the reading of the binary counter is also translated into a conduct ance in order to permit a comparison of the value of the binary counter with the x voltage
(see Fig. III.9.E.2).
The
difference
is
not deflected, the electrons divide into four equal x parts going to each deflection plate.
is
used to
y on
deflection plates
and a voltage
the vertical deflection plates will deflect the
center of the If the
square of electrons to the point xy.
original square has side la, then the area of
the electron
beam which
receiving plates
is
falls
on the various
- x)(a + y), (a + x)(a + y), (a
IIL9.F,
+ x)(a - y\ and (a - x)(a - y).
(a
the
first
third
and
is
last
The sum of
of these minus the second and
The
4xy.
CATHODE-RAY MULTIPLIERS
electrons
falling
deflection plate produce a current
on each
which when
combined by the proper summation technique will produce a voltage proportional to xy. The
121
so that the spot remains always along the vertical
edge of the mask.
M, Korn,
T.
MacNee,
Ref.
III.9.q,
and
III.9.aa, p. 70,
and R. E.
(See also G. A.
Korn and
Ref. IIL9.n, pp. 257-59), A. B.
J.
and W. W. Soroka, Ref. A. Miller, A.
S. Soltes,
Scott, Ref. III.9.t.)
The pulse width
ratio
modulation which
is
used for a time division multiplier can be obtained by means of a suitable mask on a
A
cathode-ray tube.
sawtooth voltage
the vertical deflection
of the variables x.
linear
voltage corresponds to one
A
mask
sequence of vertical wedges cell.
sweep voltage or
applied horizontally while
is
is
in the shape of a used with a photo
(See Fig. III.9.R2.) For zero deflection the
beam will
cross the middle of these
wedges, and,
consequently, the photocell will be off and
same fraction of the
on the
Raising the beam will increase the amount of time the beam is on in time.
proportion to the amount of vertical deflection and, hence, will produce the desired pulse width ratio modulation.
Fig, III.9.F.1
above device
described
is
device credited to E.
m.9.b,
A
of
area
circular
J.
by W. W. Soroka
who
(Ref. III.9.aa, pp. 68-69),
also describes a
Angelo,
electrons.
Jr.,
based on a
(See
also
Ref.
c.)
somewhat
different device is credited to
A. B. MacNee (Ref.
III.9.q) (see Fig. III.9.F.1).
This device has both horizontal and vertical deflection
an
plates
and also provision for producing one of the
axial magnetic field proportional to
input variables x.
Now,
if
a voltage y
to the vertical deflection plates, the
is
applied
beam of
electrons will acquire a vertical velocity
ponent
The to
x
magnetic field which is proportional produce a horizontal deflection of the
axial
will
beam proportional to is
com
proportional to y.
xy.
applied to the horizontal deflection plate. This
maintained in such a way that the actual horizontal deflection is zero and corrective voltage z
is
proportional to xy. A photo arrangement and a mask on the left-hand side of the tube are used to control the voltage z
consequently z cell
is
Fig. III.9JF.2
A corrective voltage z A complete time division multiplication, then, will
be obtained
if
the intensity of the
beam can
be modulated by another variable y. Theoreti cally this can be done by measuring the peak-to-
peak voltage of the output and controlling the intensity of the
beam
so that this has the desired
ELECTRICAL MULTIPLICATION
122
However, it might be more desirable to diode clipping procedures to control the amplitude modulation.
and Automation, Vol.
value.
29, no.
(June,
1956), pp.
1109-17.
utilize
n.
G. A. Korn and T. M. Korn. Electronic Analog Computers. New York, McGraw-Hill Book Co., 1956, pp. 251-84.
o. B.
References for Chapter 9 a.
"Analog-digital
conversion
techniques,"
c.
"A
time division
electronic differential a multiplier for general purpose I.R.E. Proc., Vol. 39, no. 3 (1951), p. 306. analyzer," e.
p.
Book
Co., 1949.
M.I.T. Radiation Lab. Series,
New q.
"Electronic
r.
g.
41 (Sept. 1952), pp. 570-73. E. A. Goldberg, "A high accuracy time-division multiplier,"
in Project Cyclone,
Symposium
Navy Bureau
New
s.
i.
computer,"
j.
pp. 120-24. A. C. Hall,
k.
M.
1.
v.
high accuracy time-division Review, Vol. 13, no. 3 (1952),
x.
missile multiplier in guided Electronics, Vol. 24, no. 8 (1951),
y.
"A
"An
electrical
analyzer,"
calculating
muHiplier,"
Electronics, Vol. 29, no. 8 (Aug.,
A. Miller, A. S. Soltes, and R. E. Scott, "Wide band analog function multiplier," Electronics, Vol. 28, no. 2 (Feb., 1955), pp. 160-63. J.
"An
FM-AM
range,"
multiplier of high accuracy for
M.LT. Research Laboratory
Electronics Technical Report, no. 213 (Oct. 4, 1951). 0. S. Puckle. Time Bases. New York, John Wiley
1946.
RCA Receiving Tube Manual.
Radio Corporation of America, Tube Dept., Harrison, N.J. F. L. Ryder, "Linear algebraic computation by multiwinding transformers," Franklin Inst. Jn., Vol. 259, no. 5 (May, 1955).
"Step
for generalized analogue computer A.I.E.E. Trans., Vol. 69, Part I flight simulation," (1950), pp. 308-20. "A
Handbook of Experimental New York, John Wiley & Sons,
Hetenyi.
Stress
"A beam-type tube that multiplies," National Electronics Conf. Proc., Vol. 6 (1950),
A, Somerville,
z.
pp. 145-54. ~ M. J. Somerville, "An electronic multiplier," Vol. 24, no. 288 (Feb., 1952), Electronic Engineering,
aa.
pp. 78-80. W. W. Soroka. Analog- Methods in Computation and Simulation. New York, McGraw-Hill Book Co.,
1957,
pp. 160-93. M. L. Klein, F. K. Williams, and H. C. Morgan, speed digital conversion," Instruments and
1954. ab. R. L.
Van Allen, "Four-quadrant multiplication with
transistors
"High
ac.
1302.
m. M. L, Klein, F. K. Williams, and H. C. Morgan, analog-digital
converters,"
Instruments
and magnetic
cores,"
A.I.E.E.
Trans.,
(Nov., 1955), pp. 643-48. L. G. Walters, "A study of the series-motor relay Ph.D. Thesis, University of servomechanism,"
Vol. 74, Part
Automation, Vol. 29, no. 7 (July, 1956), pp. 1297-
"Practical
Mallock,
Royal Soc. (London) Proc., Series A, Vol. A841 (May, 1933), pp. 457-83. R. A. Meyers and H. B. Davies, "Triangular wave
& Sons, w.
RCA
M.
R.
and wide
1952,
pp. 265-74. E. A. Goldberg,
Analysis.
R.
u. R. Price,
1952.
pp. 215-23. h. E. A. Goldberg, multiplier,"
t.
2,
28-May
electronic differential
1956), pp. 182-85.
2,
of Aeronautics, April
York, Reeves Instrument Corp. Part
"An
2, 1952,
140, no.
II on
Simulation and Computing Techniques, under sponsor Devices Center and U.S. ship of U.S. Navy Special
analog
machine"
multiplier using
in duration and amplitude," pulses modulated Reoista Telegrafica Electronica (Buenos Aires), Vol.
electronic
York, Reeves Instrument Corp. Part
pp. 225-37. A. B. MacNee,
analog
E. Plater and K. Franz,
AM-FM
in Project Cyclone,
I.R.E. Proc., Vol. 37, no. 11 (1949), pp. 1315-24.
Vol. 19. f.
"An
Symposium II on Simulation and Computing Techniques, under sponsor Devices Center and U.S. ship of U.S. Navy Special Navy Bureau of Aeronautics, April 28-May 2, 1952.
B. Chance, V. Hughes, E. MacNichol, D. Sayre, and F. C. Williams. Waveforms. New York, McGrawHill
W. A. McCool, multiplier,"
Research, Vol. 4, no. 4 (1953), p. 2. E. J. Angelo, Jr. "An Electron-Beam Tube for Analogue Multiplication." Sc.D. Thesis, M.I.T.,
Dept. of Elec. Eng., 1952. d. R. V. Baum and C. D. Morrill,
high speed multiplier for analogue Electronic Engineering, Vol. 69 (Aug.,
1950), p. 717.
M.I.T.,
Dept. of Elec. Bug., M.LT. Photo Service, Cambridge, Mass. (1956), pp. 5.1-5.95. b. "Analogue multiplier tube," M.I.T., Reports on
A
N. Locanthi,
computers,"
California,
I
Los Angeles,
1951.
Chapter 10
REPRESENTATION OF FUNCTIONS
m.lO.A. Function Table
We
begin by describing mechanical means of
representing a function.
function cams
we
(For a discussion of
operates, the variable x changes, and the operator turns the handwheel so as to maintain
the
pointer
on the graph. This
also provides
y
an input to the machine. Function tables are widely used in mechanical differential analyzers as
refer to Section III.3.B.)
(see also Section III.13.B).
Normally they are arranged so that a pen can be substituted for the pointer. Then if both the x and y inputs are also
obtained from the machine, the graph of the is obtained as a machine output.
function f(x)
(See also E. Janssen, Ref. IILlO.r and V. Bush, Ref. IILlO.e.)
One
variation of the above which
convenient
is
is
quite
to use a cylindrical surface, instead
of a plane, as shown in Fig. IIL10.A.2. The y is along an element of the cylinder, but the
axis
leadscrew can now be fixed, and changes in x accomplished by rotating the cylinder (see also V. Bush and S. H. Caldwell, Ref. IILlO.f).
y
Fig.
m.lO.A.l
A function table is a device by which a relation y =f(x) is
function
f(x)
into
given in
a mechanical
differential
The function/ The variable x corre graphical form. (See Fig,
analyzer. is
(III.10.A.1)
maintained manually in order to introduce the
III. 10. A.I.)
sponds to an output of the
differential analyzer
This output signal controls (see Fig. a screw arrangement of the function table on III. 10. A.I).
which a pointer
is
mounted.
The
pointer
moved by
the screw arrangement so that
abscissa
x.
is
The ordinate of
the pointer
Fig. IE.10.A.2
is
its
is
determined by another screw arrangement which As the machine is driven by a handwheel.
Automatic
curve
followers
have
been
the human operator in developed which replace the y position of the point in the following
function table.
One automatic curve
follower
REPRESENTATION OF FUNCTIONS
124
utilizes
is
a small beam of light to scan the plotted assume that the part below the curve
We
curve.
black and that the part above the curve
is
white as shown in Fig. III.10.A.3. The beam of light is sent from the carriage onto x. If it hits
no light
is
the value of y
is
black,
made
to increase.
too low, the value of y
Similarly, if the
is
used.
If a
direct-current
component is desired, the output of the pentode must be directly coupled to an
amplifier,
and normally feedback
necessary to prevent excessive
to the grid drift.
On
is
the
Since
reflected in the photocell.
now
unless an alternating light source
-OB+ is
beam of light
white, the photocell receives light which
hits
indicates that the value of
y
is
too high.
The
control circuit then tends to decrease the value
of y.
Equilibrium can be obtained by having
the spot half
on and
Alter
half off the curve,
natively, instead of a spot
we may have a
short
Fig, III.10.A.4
other hand, if an alternating light source is used, the signal output of the photocell is alternating, and a regular condenser can be used to
coupling connect the pentode to the amplifier.
After
amplification the alternating-current signal can
Fig. III.10.A.3
line
of light, with the photocell balanced so that
equilibrium corresponds to the line being half in the white region and half in the black.
The
control circuit of the automatic curve
follower cell is
the is
is
shown
in
Fig. III.10.AA The photo a essentially conductance which varies with
amount of light forming on
in series with
a
it.
Ordinarily
large resistance,
voltage across the large resistance of the amount of illumination
is
it
a function
falling
on the
photocell.
The voltage as a
Fig. III.10.A.5
and the
be linearly detected and used as a servo error signal to control the servos
ordinates.
across the large resistance signal to the grid of a pentode and
is
used
curve or
is
con
motor,
sequently amplified. The output in these applica tions has to involve a direct-current
component
by
a
The beam of
which position the light which follows the
mask may be
positioned by a servo
or, in
one instance,
cathode-ray
III.10.F.)
tube.
it
may
(See
be positioned also
Section
SCOTCH YOKE AND OTHER RESOLVERS
IH.10.B.
Consider
another
device
similar
to
the
placed below the curve
is
Conducting paper
An
(see Fig. III.10.A.5).
electrical contact
is
used here instead of the photocell. Its vertical position is controlled by a servo which will decrease the ordinate
if
no contact
the conducting paper and increase
is
made
it if
a slot in
lies in
of A
automatic curve follower discussed above.
is
A
r(sin a).
125
so that the vertical displacement
A rack-and-pinion arrangement
converts the linear motion of
Scotch yoke
This
harmonic analyzers
A
into a rotation.
been used in
has
and
also
many
fire-control
in
with
contact
is
made. (For descriptions of automatic curve followers M. S. Blackett and F. C. Williams, Ref.
see P.
III.lO.c,
G.
S.
and H.
L. Hazen,
Brown, Ref,
J.
J.
Jaeger,
and
III.lO.p.)
IILIO.B. Scotch Yoke and Other Resolvers
A large number of devices has been developed to
represent
and
sines
mechanical device
is
consists essentially of
cosines.
the
two
One
purely
Scotch yoke. parts,
This
one of which
is
Fig. EI.10.B.2
The
apparatus.
represented by
from a
cosine of an angle
the
same device
if
a
is
can be
measured
different reference position.
There
is
a special case of the hypocycloid
which permits one to obtain a cosine (and
by the use of gears without using is perhaps most apparatus. The device
sine)
sliding
readily
understood by considering the general equation which is the path of a point of the hypocycloid,
on the circumference of a
which
circle
rolls
inside another circle (see Fig. III.10.B.2).
Let r that
R
denote the radius of the fixed
of the rolling
circle,
and
circle,
us choose a
let
Cartesian coordinate system in a convenient
Let
fashion.
g
between the two Let
Fig. IH.10.B.1
is
mounted is
so that
the piece
vertically.
A
it
has only linear motion. This
which can
slide
other piece consists of an
arm
in Fig. III.10.B.1,
The
denote the point of contact when it is on the x axis.
circles
denote the center of the iked
circle
also the origin of the coordinate system
the center of the rolling circle
point
is
at
Q
.
Also
when
at this time, let
?
which
and A
the contact
denote the
the point whose motion generates the position of
mounted on a
shaft
which rotates an amount a
hypocycloid,
around a iked
axis.
A pin mounted on this arm
clockwise.
and
let
Q
- 2o^o
measured
REPRESENTATION OF FUNCTIONS
126
When
the inner circle has rolled so that the
let A^ point of contact has moved from 2 to be the new position of the center of the rolling 2i>
circle
and P1 the new position of the point which
These are the parametric equations of a straight
= 0. When through the origin, k$ + k x = back and P forth moves R 2r each point along
line
2
a fixed radius in a harmonic motion.
Thus,
^ = o, x = 2a cos 6 and y = 0. If % = x = and y = -2a cos (0 + rr/2) = 2a sin 0.
if TT,
This readily permits one to construct a device
produce a linear harmonic motion from a uniform rotating motion (see Fig. III.10.B.3). to
The spur gear
B
mounted within the
is
fixed
C which rotates around amount 0. On the opposite
annular gear A on an arm the central axis an side of
B
from the rotating arm C, we have a
P
on the circumference of the pitch circle pivot of B, which is connected to a sliding rod. The radius of the pitch circle of
A
is
circumstances
P
will
= if
= Q^OQ
measured clockwise and
= 2i^A
(p 1
Under
move according 2fl
cos
is
these
to the law
(III.10.B.6)
Fig. IIL10.B.3
generates the hypocycloid. Then,
where a
la,
the radius of the pitch circle of B.
A sliding rod perpendicular to this one will move according to the law
(counter
clockwise), then tp
= (Rj r)d +
(III.10.B.1)
0>
and the equation of the hypocycloid
is
readily
If
P
is
not on the pitch
circle
b from the displaced an amount
of
B
but
is
center, the pivot
seen to be
x
= =
-
(R
-
(R
r)
cos
r)cos
-
+
r
+
Ik rcos I-
cos (
0)
-
r
\
+
(I1L10.B.2)
y
-
-
(R
r) sin
-
/R
-
V
r
-
r sin
r
\
-
+
/
(1II.10.B.3)
Now,
if we let R = la,r =
x
a,
we have
= fl(cos + cos (0 + = 2a cos ^ /2 cos (0 +
0>
))
9?
/2)
Fig. III.10.B.4
y
= = =
a (sin
sin (0
la sin
-/c 2 cos (0
+
/2 cos (0
+
9>o/2)
+
<
(1II.10.B.5)
will
have an
Let
?
contact
elliptic
motion
be the position of is
at
g
in
P
(see Fig. III.10.B.4).
on the x
Fig. III.10.B.2.
axis
Then
when
after
C
III.10.C.
has rotated the amount will
6,
127
ELECTRICAL REPRESENTATION
the coordinates of
Nonlinear potentiometers are frequently used
P
be given by:
to
and cosines
sines
represent
Section
(see
III.10.D).
we have produced
In the above discussion
Devices to represent the sine or cosine of an angle are generally referred to
as
resolvers.
Frequently both sine and cosine are needed. There is a well-known device for this purpose which supposes that the angle a is represented by
a rotation and whose output consists of two alternating voltages corresponding to sin a and cos a.
This device
is
similar to the selsyn transformer
In appearance
(see Section III.8.C).
to
an
electric
motor.
However,
it is it
similar
behaves
a transformer. The principle of an essentially as follows: Suppose
electrically like
the device
is
alternating magnetic field
is
maintained between
y = sin x,
(see
III.10.B.5).
Fig.
Let a
=
a shaft rotation.
is
are a wide variety of applications in
independent variable to
it,
is
time and x
is
There
which the
proportional
e.g.,
=
x
at
(III.10.B.10)
As we have pointed out
in Section III.8.A,
a
in a uniform magnetic plane loop of wire rotating field about an axis perpendicular to the field will have induced in it a sinusoidal voltage if the
voltage is
uniform. (If not, the angular velocity
is
a factor in this voltage.)
Thus, generators
produce alternating voltages
two pole pieces. A coil is placed in this field and mounted on an axis so that it can rotate through an angle a
where x
their rates of rotation are uniform.
provided quantity
o>
is
proportional
The
to the angular velocity
of the shaft.
However, another method of producing an is by means of an electronic oscillating voltage circuit.
The circuits used vary with the frequency
desired, but the objective in each case
a critically
damped
circuit,
i.e.,
W. A.
Amateur
s
Edson, Ref.
Handbook,
III. 10. j,
tial
analyzer.
variable
y
=
equation y
cos
is
choice of the
III.10.C. Fig,
m.lO.B.5
a position correspond to the coil
is
parallel
flux intercepted
cos
a,
by the
coil is
The amount of proportional
and, hence, the induced voltage
proportional
to cos
<x.
is
to
also
A coil mounted at 90
to
this one would have an induced voltage propor fields tional to sin a. In practice, two alternating
in different phases are used.
2
_y
In this case the independent
The
produced initial
y
= sin o)t
or
in accordance with the
conditions.
Electrical Representation of
Func
tions
A common situation in continuous computers
such that the axis of
to the field.
by representing the on a differen
+ co =
the time, and either
ait is
and Radio
Ref. lll.10.ac.) Sines and
cosines can also be obtained differential
to obtain
nor decreases. (See
oscillation neither increases
also
is
one in which the
a function of a variable x, which representing form of a voltage, as a voltage the in is present also. There are a variety of ways of doing this. is
In one of these the variable x shaft rotation and, then,
is
is
converted into a
used with a special
to represent the function. type of potentiometer a function of method a is There representing
REPRESENTATION OF FUNCTIONS
128
which is based on the use of diodes, and there are other methods which use a mask and a cathode-
On
occasion a simpler device can be used to
some
represent
functions
by
using
poten
represent a
tiometers with taps. These taps can be connected
function electrically using some form of magnetic storage or to represent it by means of a Fourier
by resistances of various values to one terminal and thus obtain the equivalent of a variable
series.
resistance
ray tube.
It
also possible to
is
IIL10.D, Potentiometer Methods of Representing a Function
In
many
provision
is
made
to
commerical
devices
change a variable from an
a shaft rotation by a servo done, the rotating shaft can be
setup.
is
sometimes
is
even
effective
simpler to use a
is
enough to affect the (See also G. A. Korn
large
linearity of the position.
andT.M. Korn,
electrical voltage to
If this
is
load resistance between the contact and one terminal which
well-known
An
potentiometer.
procedure which
It is
Ref. III.lO.s.)
possible to represent the function
1
x 2 as
means of a linear potentiometer. very useful in multiplication. If one
a resistance by
used to position a contact on a special type of potentiometer in order to represent a function,
This
There are two types of potentiometers which are used. In one type the amount of rotation of
tiometer to one point A, then the resistance
the contact
ance
is
is
connects the two terminals of the linear poten
between the contact
B and these terminals can be
proportional to a, but the resist
is
not a linear function of the contact Ordinarily the resistance wire
position.
rectangular card which
on a
then bent into
is
a
into the cylindrical shape and inserted poten tiometer. The wire is bared along one edge of
the card, and the this
moving contact moves over
edge in the potentiometer.
In these special-function potentiometers the card on which the resistance wire is wound is not Fig.
rectangular in shape. straight,
and
this is
The other edge
is
One
lengthwise edge
used as the contact edge.
at a variable distance
from the
straight edge so that the width of the card
function
where a
(a)>
is
is
a
a variable measured
along the straight edge. (See also I. A. Green J. V. Holdam, Jr., and D. MacRae, Jr.,
wood, Jr., Ref. If
III.
10.m, pp. 106-111.)
regarded as a function of the variable x, which
corresponds to the amount of displacement from the center point as
resistance wire to
shown
in Fig. III.10.D.1.
Suppose the total resistance in the linear poten tiometer
is
2R. Then the resistance between the
contact and one terminal can be designated as
x)R, and the resistance between the contact
(1
we connect one end of the
HL10.D.1
is
and the other terminal
is (1
+
x)R.
The
resist
a fixed terminal of the potentiometer, and this end corresponds to, say, a 0, then the amount
ance, then, between the connected terminals
of resistance between the contact when
and
=
a and
this
terminal
it is
in
proportional to
the contact
and
B consists of (1
this is readily
x)R and (1 + x)R, seen to be J(l x z)R.
the length of the resistance wire between these
Another type of special potentiometer has been developed for use with the REAC device.
two
Here a
position
terminals.
mately J g/(a)
is
This length, in turn, da., i.e.,
proportional
is
approxi
to
linear resistance
is
used in the poten
tiometer, but contact with this linear resistance
is
obtained by means of a wire mounted on a rotating cylinder (see Fig. III.10.D.2). is
and
this will
which
is
permit us to obtain a resistance
a given function of
a.
wrapped around the
that
its
The wire
cylinder in such a
way
displacement along an element of the
cylinder corresponds to the function /(a), where
MULTI-DIODE FUNCTION GENERATOR
IIL10.E.
a
a variable measured on a directrix of the
is
based on diodes are often used to represent functions.
cylinder.
an
We
element of the cylinder. The variable a corre the cylinder, sponds to the amount of rotation of the linear resistance element lies and at a
diode
The
129
linear resistance element lies along
Fig.
first is
consider a simple case, in which one
used.
III.10.E.1.
We have the circuit shown in Ea is the voltage corresponding
=
to the y along that element which corresponds
on the
axis
cylinder.
amount
rotated an
When
the
cylinder
EO-IOO
is
the resistance element will
a,
be along the cylinder element corresponding to a. Contact will be made with the contact wire in
such a way that the resistance between one terminal and the contact is a prescribed function of
It
a.
should be mentioned that ordinarily the
O OUTPUT
o-xAAAALINEAR RESISTANCE ELEMENT
WAAA/V
Fig. ra.lO.E.l
CONTACT WIRE
Fig. ffl.lO.D.2
thus obtained is applied to the output voltage in such a way that the load an of amplifier input
on
The x
is
this
negligible. potentiometer drum is obtained by a servo setup
rotation of the
similar to that used in the servo multiplier,
the
x input to
this type
also voltage. (See
of function generator
W. W.
and is
a
to a position
ffllO.E. Multi-diode Function Generator
other characteristics of a servo in an electrical
Thus,
an
electrical
device
it is
with
input
y =f(x) is
ordinary
the input voltage. resistances
would
represented
The
best
known
of these
conducting.
For x
<
Ea
the output voltage
is
is
-x
(ffl.10.Rl)
(IIL10.E.2)
A circuit involving yield
only linear
"non-linear"
is
.
and output
functions of the input voltage. It is necessary, functions are to be therefore, if more general to introduce
on the potentiometer. The output
Ea This is this circuit is x when x voltage of as if the diode were not there since it is not
desirable to have
that the output voltage voltages such
where x
m.lO.E.2
>
avoid the delay and
It is highly desirable to
differential analyzer.
Fig.
Soroka, Ref. III.10.ah.)
a diode, and
devices. circuits
Thus, for x
<
Ea
the cathode
is
more
negative
than the plate, and, hence, the current is flowing the diode, and the resistance of the
through diode Fig.
is
negligible
III.10.E.2
relative to that
represents
of
^ and
R%.
the output of the
REPRESENTATION OF FUNCTIONS
130
circuit.
Thus,
it
would seem
as if the output
using two diodes
By
one can describe a
function with two changes of slope, and with k consists
of a straight
drawn from
line
0, -) R l + Ay \
of slope
to the point (Ea ,
straight line of slope
from
The sharp
there.
angle
Ea
)
is
1
continuing
supposedly due
slope.
We first consider the case in which the change of slope
and then a
k changes of
diodes one can introduce
an increase
is
x
as
The
increases.
idea to
be discussed here can be illustrated by consider ing the case in which three diodes are used.
Our
We
have
shown
circuit is
in Fig. III.10.E.4.
three potentiometers which can be adjusted to yield three voltages,
Ea Eb E
tiometers in practice.
c,
,
,
R^ R&
A* l5
resistors,
and we have three
which are also poten
We suppose Ea
<E
<
l)
E
c
.
The system then will have four ranges of output y with different slopes. cuts off
We
assume that the diode
The ranges
sharply.
Ea
y<Ea ,
^y<E b ,
E
E,^y<E ct
For y
diodes
all
<E
a
are:
and
conduct,
if
the
resistances of the potentiometers are negligible,
then
VA
Ag
A^
A
AU/
K
Ajj
Fig. III.10.E.3
Ag
(III.10.E.3)
Thus, in
this
region y has slope
R (III.10.E.4)
A
A!
A2
This certainly holds until y
Ag
= Ea or
(III.10.E.5)
Or
A Fig.
I\2
(III.10.E.6)
m.lO.E.4 (E b
R
R
to the fact that the diode suddenly stops con
shown
really looks as
it
Such a smooth
transition is
-E RZ
=
Ev Actually, ducting at (Ea9 Ea), i.e., when x the break-off of the flow of electrons is a gradual rather than a sudden process. Hence, the output is
A|j.
a)
(g.-J RZ (IIL10.E.7)
For
Ea
<
y
<
E
b
the relation becomes:
in Fig. III.10.E.3.
more
general, for representing functions.
desirable, in
A*3
(UL10.E.8)
MULTI-DIODE FUNCTION GENERATOR
III.10.E.
and, hence, the slope for this interval
of
is
-
Similarly, for
6
first
:
-1.1.1 +-+R <
y
<
R to R% will be determined by the slope of the R to A\ by the
second segment, and the ratio of
-
m*
131
RZ
(III.10.E.9)
gi
segment. Thus, this graphical process will ve us tne ree and c correbt voltages Ea
^
E
E
,
spending to the values of y at which the break
RS
E we c
have the slope
J.
R
and
for
c
<,
y
m4 = From readily
1
Eq. IH.10.E.3 to Eq. III.10.E.11, we obtain a process
for
representing
suppose that three diodes
are
adequate and
proceed as follows. Let y =f(x) be a function with an increasing slope on a certain interval.
We
graph the function and choose three inter mediate values of y in such a way that the line
determined by these three values
polygonal and the end value constitute a good approxi
As shown
mation to the function. IIL10.E.5, we have four line
points of the graph have been taken,
R ,
^2
segments, then,
I, II,
(see Eqs. III.10.E.9
R
actual value of
tiometers for
method
Marshall,
To
circuit
chosen so that
is
fl ,
and
Eft
E
c
.
a
In this
new
will determine the ratio
.
see B, 0.
reverse
shown
in Fig. IIL10.E.6.
y
<
E
y
>
a
,
plate
and
We
see that if
the diode has no
a9
and, since the output if
the
Thus, we consider the
is
unloaded, y
the diode
is
circuit
= x.
conducting,
with output
this output voltage corresponds to a polyg onal line function with two slopes, the first 1,
,
we
the second less than
decrease in slope
we
1.
Therefore, to obtain a
use a diode with plate and
call
cathode reversed from previous connection. In
Eq. III.10.E.10
the general case we can use the above to represent
scale situation, if
ws
= x2
Thus
We choose a scale representing preceding one. the function so that the segment IV has slope 1.
y
monotonic function with
we
slope,
and we have a voltage divider
w4 =
are
Jr., III.lO.v.)
represent
However,
the slope of the third segment n
we
of the poten (For a discussion of
for representing
the output voltage effect,
HL10.K5
The
E.4).
justified in neglecting the resistance
cathode of the diode.
Fig.
and
-KS
decreasing
7-
,
^1
R and
Fig.
IV, such that each has a larger slope than the
and the n
above computation will determine the ratios
this
in
m.lO.E.6
Fig.
a
We
function which has an increasing slope.
III,
O OUTPUT
(HI.lO.E.ll)
Similarly the ratio
a monotonically increasing function in which the slope
increases
or
decreases.
Consider,
for
REPRESENTATION OF FUNCTIONS
132
shown
instance, the circuit
We
suppose
then
Ea and E
Ea c
<E^<
E
c
in Fig. III.10.E.7.
Then
.
and y
are conducting,
if is
y
<
Ea
with slope
,
R
given by
(III.10.E.18)
R
i+i+1 p p Ao Ap Ao
R1 (III.10.E.13)
Finally for
<
c
j we have
=-+
(III.10.E.19)
and I
R (III.10.E.2Q)
1
1+ R
The
m3
<
AO
m1
slopes are, therefore, related;
w4
<m
2
Such a function would appear as
.
>
in
III.10.E.8.
Fig.
to represent a function Suppose we want
y=f(x) /* 2 ,
Fig.
fa,
line with slopes given a polygonal
and
// 4
which
are, say,
// 1}
proportional to
m.lO.E.7
(neglecting the resistances of the diodes).
This
linear function has slope
R (III.10.E.14)
For
Ea
the
diode
first
does
not
conduct, and, hence,
J1 + \A
1)
=
*
A
AJJ/
+
*?
A3
(IIL10.E.15) Fig. HI.10.E.8
This function has slope J.
1+ * On
the other hand, 1
,
1
are conducting, and
1
(IH.10.E.16)
<
<
slopes
z
of Eq. III.10.E.14, m 2 of Eq. of Eq. IIL10.E.18, and m 4 of Eq.
III.10.E.20, respectively.
^3
Eb y we have
if
ml m.io.E.16, m
the
Jl
7
E&
two diodes
v *
_
can choose a new
,
,
.
,
holds where A
,
graph has slopes
m^
j
,
.
is
an in u on lUlilU.Ju.Zl
in, *^/
i
,
We
scale, 7, so that the result
x
a scale factor.
m m2 m3 lt
= /*^
,
,
and
(1 <j^4)
XT
Now
.
the Y,x
m4 where (III.10.E.22)
MULTI-DIODE FUNCTION GENERATOR
IIL10.E.
Ea
to voltages
Letting
with
133
and with cathode
k>j
connected to the output are conducting, and those with k and with plate connected to
3)
<j
output are conducting. Let
and
IjRj,.
1
=-
v
(IIL10.E.24)
h
that
denote the
Ea
<
Ea
we have a
.
sum
of
For y such
linear function
with slope
we have
III.10.E.14,
Eqs.
16,
18,
and
20,
respectively, as
-v
2
for these conducting devices.
m,
+ Wi =
+ toUi +
-to
(HI.10.E.25)
v
IIR
=
(III.10.E.29)
introduce represented by this diode device. If we a scale factor h, the original function represented
has slope
v
p s such
that
m =
v
(III.10.E.30)
hto
f
We can also write Eq. Mowing way. Let S^j) denote the sum of i/ b where R k and connected to a is a resistance with k III.10.E.29 in the
If the determinant
-1
to
>j
-100
diode whose cathode
and
-1
S 2 (j)
p,
diodes with k
u.
output. Then
not
0,
we can
R u2
,
and w 3
ul
.
in addition, v
If,
we can
which
E
a}
^ (IIL10.E.31)
where S^y) decreases with; and
y=fto. We can now infer the general situation readily. line function
<
is
we can represent the given function
If we use n diodes,
y
l
^3 then
< .
= --,
R
=
RZ positive,
= -,
solve for v
and with plate connected to
Ea
for
R
1
is
connected to the output, sum of those
are the corresponding <j
-1
is
represent a polygonal
is
monotonically increasing under 1 and has n , slopes p^ / 2 /yi certain circumstances. The n intermediate output
+
,
.
.
.
S 2f ; )
increases
with/.
Consider to
y
>
E
aj
,
now i.e.,
the change in
}
from y
One and
the actual one. slope, not
term, S^j) or
m
Ea
.
only one
S 2(j), in the denominator changes,
so that the change in the denominator
We
<
the theoretical sharp change in
is
1/1^.
can write
will be voltages at which the slope changes
We set
written in increasing order
Ea ^
n potentiometers so that
their output voltages
.
.
.
,
v
values. If at E the slope correspond to these aj a diode with cathode introduce we increases,
__ _
voltage
For;
E^
and for y
to
correspond
On the
<
and cathode
its
to
diode
not. This
is
is
if
seen to
= Ea
the slope decreases at
..
E aj
,
connected to the output y plate
E a}
fl
,
a
S ay)
u
For
m n+1 we have
= 1 + II - = 1 + IkRS
^n+l
(ffl.lO.E.33)
where SJ
is
the
sum
over
all
the j for which
of the diode is connected to the output. plate If we introduce the scale factor h, as in Eq. III.10.E.30, Eqs. III.10.E.32
and 33 become
through R,.
Consider the situation in which the output;; is Those diodes connected and E
between
(
conducting,
readily
an increase in slope at y
other hand,
the diode has
this
aj
Eaj it is
>
E
= R*
connected to the output y and plate connected contact with through Rj to the potentiometer
=
^
J--L=^
(III.10.E.34)
REPRESENTATION OF FUNCTIONS
134
and
IEL10.F. Cathode-Ray Tube Function Generator 1
*
~~
y
i
*
/
;
__
/
in
rr
T
c
oc\
or
an
-L-J^-I)-,
W^il
\M
36)
is
cathode-ray tube which can be used to position a beam of light in accordance with certain voltages. (See Fig. III.10.F.1.) Such a tube consists of an
Wn /
J.1
If the left-hand side
o
i
A cathode-ray tube of the flying spot type can be used to represent a function in a number of ways. A flying spot tube is an (oscilloscopic-
negative, then
we can
not
obtain a scale factor h to permit representation by n diodes. On the other hand, if a positive scale factor
available, then each u s
is
deter
is
mined, and the given function is representable. Now, if, however, the n diodes are to be used
ELECTROSTATIC DEFLECTION PLATES
to represent a polygonal line with only n different slopes,
p lt
.
slope
,
/j n ,
we can continue
this function
segment with
line
is satisfied
for
h.
we would like to mention briefly
Finally in
.
so that Eq. III.10.E.36
// B+1
a positive
.
by introducing a
arbitrarily
Ea
which the voltage breakpoints
can be obtained for representation.
Since
abscissas.
We we
maximum
,
a
.
.
.
,
E^
accuracy in the
will find the
corresponding
are dealing with a polygonal
approximation, the error over an interval of length h the
that
proportional to
is
error
in
z .
y"h
various
This suggests
intervals
Fig. IIL10.F.1
way
can
be
1/2
and dividing the x equalized by plotting |/ interval into subintervals for which the areas
electron
gun
beam of
a narrow
to produce
electrons, electrostatic deflection plates to deflect
the
beam of electrons, a glass envelope
enclosing
and the electron
the electrostatic deflection plates
a gun, and a fluorescent screen consisting of
white fluorescing phosphor.
(See
RCA
also
Ref. III.10.ad.) Receiving Tube Manual,
|
under
this
curve
are
equal.
distinct intervals of length A x
Thus, for two
There are two pairs of deflection plates. Each flat pieces of metal. pair consists of two parallel
we have
In the customary position of the tube, one pair of
and
/z
2,
plates
approximately
is
mounted
individual plates
(IIL10.E.37)
mounted
(HI.10.E.38)
these plates. first
This implies that the
two
maximum
error
on
these
intervals are equal. Frequently this criterion
alone
adequate to obtain the approximate equality of the maximum errors on the interval to within practical limits. If necessary, a further is
adjustment can be made by contracting those intervals
on which the maximum
error
is
larger
than the average and expanding those in which the error is less than the average until the equality of the errors
is
obtained.
Meissinger, Ref. III.lO.w.)
(See also
H. F.
The
A
so that the plane
corresponding
is
plane
beam
passes between difference in voltage between the electron
pair of plates produces
directed
of the
the others are so
vertical;
the
that
horizontal.
and
is
horizontally.
an
This
electrostatic field will
proportional horizontal velocity
produce
a
component and, beam. In
thus, a corresponding deflection of the
a similar manner the other pair of plates will
produce a vertical velocity component of the electron beam. The place where the beam hits the fluorescent screen
is
displaced horizontally or
vertically in a proportional
plates
manner
also.
The
which produce a horizontal deflection are
called the horizontal deflection plates.
An electromagnetic field can be used instead of
III.10.G.
the electrostatic
The
field.
MAGNETIC MEMORY METHODS
deflection then
perpendicular to the direction of the the direction of the
is
beam and In the
135
mask-shaped electrodes within the tubes have been used instead of an exterior How target.
case of electrostatic deflection the tangent of the
can only be used for a fixed function. (See also A. C. Munster, Ref. IILlO.y, and
angle of deflection in any one direction is propor deflecting voltage. In the case of
Ref. III.10.af.)
electromagnetic
field.
tional to the
ever, these
H. W. Schultz,
J.
and E.
F. Calvert,
L
Buell,
deflection
the electromagnetic angle itself is proportional to the strength of the magnetic field, and, hence, to the current in the deflection
The
coils.
electromagnetic field
produced by
is
coils exterior to the
normally
cathode-ray
tube. intense
beam of
electrons
hitting the
phosphorescent screen produces a brilliant spot of light. In certain examples (see E. J. Hancock,
G. A. Korn and T. M. Korn, A. B. MacNee, Ref. Ill.lO.t;
Ref. IILlO.n; III.lO.s;
D. M. McKay, Ref. III.lO.u; C. N. Pederson, A. A. Gerlach, and R. E. Zenner, Ref. III.10.aa; D. E. Sunstein, Ref. is
made
III.10.aj) the
beam of light
to follow along the top of a
face of the tube in the
follower (see Section
way
mask on the
described for a curve
III. 10. A).
However, a flying spot tube can also be used to represent a function by pulse width ratio modula In this case the flying spot has a rapid vertical oscillation of constant height and is tion.
moved
horizontally
in
accordance with the
A
mask in the shape of the function placed in front of the screen. If this mask is
variable x. is
cut out in such a
way
that
it
corresponds to
points above the graph of the function, then the flying spot will
tion an
show during each vertical
amount of time proportional
ordinate y.
A
oscilla
to
the
photocell can then be placed in
such a way that it will be illuminated by a fraction of the time corresponding to the ordinate y. Thus, the output voltage associated with this photocell will be a pulse width ratio modulated
In connection with the
new
digital electronic
computers, "memories" of various sorts have been introduced. These, of course, can be used
continuous form. They have the advantage that a function can be produced in a
computer
automatically used
One must, of
Since this signal
modulated,
it
is
now
pulse width ratio
can be readily used as a factor in
multiplication in a time division multiplier.
time division multiplier actually uses
One
this device
with the function corresponding to one factor. feedback arrangement from a photocell to the vertical deflection plates can be used to center
A
the spot on the top of the mask. In certain cases,
later.
course, take into account the
magnetization of the tape and the resulting playback of the signal will not be a fact that the
linear function of the original signal.
therefore,
it is
necessary to use
In general,
some method of
modulation other than amplitude. (See also Sections III.5J and III.9.B.) Alternately, one can convert the analogue signal to a digital signal and use any digital storage procedure. To use result again one must convert back to continuous form.
the
Frequency modulation has been used
effec
with a magnetic tape recorder to represent a function. signal of fixed amplitude but with
tively
A
varying frequency can be used to represent a
manner as to permit both automatic recording and playback. The function
function in such a
appears originally as a voltage function of the This voltage is used to control the
time.
frequency of an oscillator and the output of the oscillator recorded with a tape recorder. In
playback the output of the tape reading head fed to a frequency meter
whose output
is
is
a
voltage.
signal with modulation representing the function y(x).
Memory Methods
to hold a function table in either a digital or
The
Ref.
IEL10.G. Magnetic
is
Another popular method of modulation used It can be used pulse width ratio modulation.
in a perfectly analogous
way
to the frequency
modulation system described above for recording and using a function.
The magnetic-type memory can be used devices represent a function for use in
to
whose
output appears on an oscillograph. For instance, a magnetic wire memory consists of a wire, which
REPRESENTATION OF FUNCTIONS
136
is
a rapidly through reading head, which
moving
can show that
if
responds to the variations in the magnetic state
To maintain a periodic function, we
of the wire.
use a
wire which will enter the reading head one period
circuits
later.
signal
Such a function could be depicted on an If
oscillograph.
with time, as we
we can impress
we wish in
may
there will be co \^^>
a phase shift of 90 for the no) component. In order to synchronize the outputs, we
which corresponds to the value of the function on another point on the
can impress the
R^ = n
common
may
oscillator for all the integrating
tuned to a frequency corresponding to co, of a type which is provided this oscillator
to modify the function an adjuster-type device,
the modified value
on
the wire.
If the function does not vary in the problem,
it
could be placed on a closed loop of wire which runs through the machine. An acoustical delay in a similar way.
memory can be used
line
also F. E. Brooks,
W. H. Coombs,
A.
Ill.lO.d;
A. E. Hastings and
m.lO.H. Fourier
J.
(See
and H. W. Smith, Ref.
Jr.,
E.
Ref.
IILlO.i;
Meade, Ref.
III.lO.o.)
Fig. HI.10.H.1
produces terms corresponding to the various A relaxation oscillator is one higher harmonics. that
Series Representation
is
characterized by the property that there
in each cycle a period in
Another way of representing a function is by means of the Fourier series. There must be a
method sin
for representing a constant function
nx and cos nx
(this
supposes that the interval
for the independent variable
method
and
is
TT
to +TT)
and a
for taking a linear combination of the
In the case of a voltage, tuned output of these. of representing the methods offer circuits
slowly
number of such circuits
are given
by 0.
S.
is
A
then rapidly discharged.
charged,
is
which a condenser
Puckle
(Ref. III.10.ab, pp. 56-64), but the principle
can
by means of any gaseous tube having a breakdown characteristic. Thus, in the circuit shown in Fig. III.10.H.2 if the input be simply
is
illustrated
a fixed high voltage, the condenser will slowly
trigonometric functions.
Let us briefly discuss the possibilities for such We consider a function on the interval
a device.
-TT
<
x
<
TT.
to continuity
With certain reservations
and
differentiability
can be represented by a
/(x)
=
relative
such a function
series
sn
a
Fig. IH.10.H.2
(III.10.H.1)
In Section III.10.B
we saw how
where a n voltage a n sin nx, input and x
=
cot,
where
convenient frequency.
be used to produce
CD
An
is
to
produce a
a direct-current
corresponds to some analogous circuit can
b n sin nx.
If
we apply
this
which introduces a phase output to a circuit obtain b n cos nx. we of 90, change
A circuit containing a condenser and a resistor if
we
circuit illustrated in Fig. III.10.H.1,
we
can be used to change the phase. Thus,
have the
charge until the breakdown potential
is
reached,
will discharge. Unfortunately, the current
then
it
will
continue even with a
unless the plate
is
made
much
lower voltage
to negative with respect
(This assumes that the cathode
the cathode.
is
emitting electrons.) This can be accomplished by a capacity in the introducing a resistance and cathode circuit. Thus, in the circuit shown in
two condensers Cx and C 2 C2 reaches the potential at which the
the Fig. III.10.H.3
charge
until
"trigger grid"
of the gaseous triode will
initiate
a
IIL10.H.
Then
discharge.
C2
tube and
C
FOURIER SERIES REPRESENTATION
will
x discharge through the through R%. This discharge passes
partly through
R^ and
C3
partly into
L
actually
plate.
(/,$)
=
permits the cathode to be to the
relative
positive
and
space the inner product of two vectors/ (x) is given by the formula
g(x)
until the
cathode becomes positive relative to the
(The inductance
137
Pre
plate.)
|
(Note:
(IIL10.H.2)
/WJ(*)
J- v (
Two
the complex conjugate of g.)
is
sumably, the output should be amplified and a condenser coupling should be used to the
zero.
integrating circuit instead of the transformer
spond to vectors along a
coupling.
since
functions are orthogonal if their inner product
The functions
1,
sin x, set
is
and cosx corre
of coordinate axes
they are mutually orthogonal.
(See also
R. V. Churchill, Ref. IILlO.h, pp. 34-52 and
The
P. Franklin, Ref. IILlO.k, pp. 48-49.)
coordinate axes forms a
no
there
is
finite
sum
"complete"
vector orthogonal to
all
set
of
set since
of them.
A
(III.10.H.3)
an approximation to /if the integral
is
a
is
Fig. ffl.lO.H.3
The output of
a
the circuits for each term of
Eq. III.10.H.1 are averaged by means of ances.
The
result
resist
a periodic function of the
is
,
minimum when
a n , and b n
n
corresponds to function
is
periodic
tinuous derivative,
used
<
x
^
TT.
<
TT/W
t
<
TT/W
If the original
N).
However, in order to get a series representation not only in which the series approximation cr v approximates/, but where
7
v approximates/
/
we must
then
and, say, A approximates minimize the integral cr
time in which the interval
regarded as a function of
(=!,...,
/",
and continuous with con representation can be
this
effectively.
On the
other hand,
if it is
desired to represent
a function just for the interval n x IT, even if the function itself is continuous, the <
representation
may have
<
To
a
obtain
discussion
preceding one in this case,
which the inner product
in
analogous
to
the
we must use a space
W
is
discontinuities at the r
end
points, either in itself or its derivatives.
this case, neither is the representation
In
However,
it is
differentiation
possible
which permit term-by-term situation
may
differentiation.
be described as follows:
The It
is
a vector in an infinite dimensional function space
M. H.
Stone, Ref. III.10.ai).
That there
(III.10.H.4)
In this
shown
in
is
such a space with the requisite
two independent variables is a thesis of the author s (Ref. III.10.z).
properties
new terms
customary to consider the orthogonal series representation of a function as representing it as
(see also
+fg")dx
by means
apply.
to introduce
<
j-t
of trigonometric function uniformly convergent
nor can term-by-term
(/,*)=( (fi+f S
for
However, the discussion can be readily
simplified
to one variable or expanded to any number.
One can sin
readily
show
that the functions
nx and cos nx are orthogonal
also.
1,
in this space
Thus, they also determine coordinate axes.
But this
set is incomplete, i.e., there are functions
REPRESENTATION OF FUNCTIONS
138
which are orthogonal to every one of these. Let us find
Let 9 denote any of the functions
and cos
nx,
the space
We
also have
functions /with this property.
all
to suppose /is orthogonal
and
W,
>
J
sin/a,
1,
J_
in
99
^ ff
-,
/
i.e.,
=0
-/"
(IIL10.H.14)
J-ir
+/
(/
|
=
&x
+/"$
(We
clearly sufficient.
(We
have assumed that/ has two further derivatives. fourth
continuous
a
has
/
suppose
f
)
(
The assumption of f r tne
Then
derivative.)
=
These conditions are
(HI.10.H.5)
-ff
=
the existence and continuity
ner derivatives of /can be justified,
but a discussion would be somewhat lengthy.) BY the usual considerations of ordinary linear
+ /y + /y) dx
y; ^
["
m
J-v
differential equations,
^
solutions of
it
can be shown that
all
e iuation
(ii)
f-/ +/ )?& If
we
(HI.10.H.6)
are given
(III.10.H.7)
f=Ae* smfa
let
n+1
l
(- w)]
[/"(x)-/
where
/ff
=
(/-/"+
/
(ir)
)^ ^ x
( IIL
10
-
H 8)
oo of the right-hand side
->
since the right-hand expression coefficient
the
for
function
is
is
(/-/"
+f
of
be
zero
a Fourier
and y 2 are constants
y lt
B,
The successive derivatives of/are easily shown
-
to
as n
A,
integration.
J ~*
The limit
the expression
VJ B
= smnx
)
by
v^
=
/
2
Ae
+
sin ftx
yl
+
77/6)
(iv) ).
This clearly ; implies F
/=/ Similarly, if
we
(-*)
_
an.io.H.9)
v
let /"
y
= cos nx
= ^e^
(III.10.H.10) "
"
we
+
yx
+ w/3)
v
weobtain
If
sin (Jx -
use the facts we have just established,
we
/
=
Ae
T
*
sm
&+
ft
+ */
sn
+Be
If we substitute the value given
for each of the above
=
Since
1,
sin nx, cos nx,
a complete system of function orthogorthogonal functions, the only is of them all to onal zero, and, hence,
and n
1, 2,
.
M=
.
.
are
(-TT
<
x
^ TT)
(III.10.H.12)
fwf" 9
^
we
by Eq. III. 10.H. 1 8 becomes
find that Eq. III.10.H.13
sin (Vi
+ 5?r 6 ~ ^ )
/
.
sin
^-1 s n (^ + 57T/6) - By sin (y - 57r/6) =
_^
^2
- 5^/6)
i
a
(III.10.H.20)
III.10.H.
mi
"
FOURIER SERIES REPRESENTATION j
t.
ereas + *) = -sm
sin (
a.)
It
f
i.
(We have used
.
,
the fact that
+r
Thus, smce (y
can be shown that Eq. III.10.H.29 .. satisfies the alues of ^ and a. Letus .
.
conditions for/for :/fo all i) is
never zero,
take .,
^ =
R 29
Eq. ffl JQ
^ ^ ^^ m -
f
the form
A sin fa +
577/6)
= B sin (ya -
/=
677/6)
(IIL10.H.21)
By a
139
use of the trigonometric identity for the
differences of sign, Eq. IIL10.H.17
and 19
wnere
=
fi
+/
cos a
8
a
sin
(III.10.H.30)
-
v 2
e
sin (Jx
5?r/6)
Vg
yield
T
+
-
v
/2
=
*
sin (lx
2
cos (Jx
+ 577/6) 577/6)
*
Be
+
^ + W6)
COS
(HL10.H.32)
(ffl.lO.H.22)
We
Eq. III.10.H.14 then becomes
can readily rind the length of
/
in the
new
Thus, using the conditions Eqs. space. ffl.io.H.12, 13, and 14, we have .
2
(ffl.10.R23)
We now use the
/!
= (/-/) =
2
+
d/i
i/T +
I/
""
fact that
cos (a
+
IT)
n ff
=
=
r
(/
-cos a
+y
and by rearranging and dividing by y
l
~/
")/
+
/Tj ^
ff
we
+
obtain
f
y _y* + /
(l c)
)/ Jx
J-,
A cos (ft +
577/6)
= B cos (y - 577/6)
= [(/ (*) ^/X/W -/HO)]
2
(I1I.10.H.24) If
we
we
square Eqs. III.10.H.21
and24andadd
_ ,-
^
?
+
get
^=,B
2
(IIL10.H.25)
or
A= A
Reasoning similar to that of Eq, HI.10.H.33 can
B
(III.
10.H.26)
B implies *
+
/i
5W6
=
.
ya
...
^ use(
i
to
+
577/6
,
(ffl.10.IL27)
=a=y
2
577/6
+
Eq.
Substituting
H
we
16
III.10.H.27
28
or
in
Eq.
obtain
1 2
sinftx
=
(ffl.10.H34)
+ a - 577/6) +
i/2
5
l!
= CV3
(IIL10.H.35)
^
^
Vw(l
+
2
-^ ^at F(x)
577/6)
(IIL10.H.29)
4 1/2
+
)
If
.
we
apply the usual
method of obtaining the Fourier coefficients, we a ^uncti n F(x) can
the interval
_
e
= CV3
2
In the space w? the function j length the functions sin nx and cos nx have lengths
77
(III.10.H.28)
III 10
t
(/i,/2 )
= (say) a
^ = -B implies ft
t
,
5?7/6
/il |2
and
^OVf ^
77
<
x
<,
TT,
^ represented on
by a
series
=
i
(IIL10.H.36)
REPRESENTATION OF FUNCTIONS
140
where
=
a
^i
=
discussed these in Section IIL10.B. Other special functions of great interest are: log x, exp x,
-J
F(x)Jx
(IIL10.H.37)
x 1/2
,
+
Fft
+ Ffildx (III.10.H.38)
x2
,
course, the
methods which we have previously for representing functions can be
various fi
(l/CV3)|
Of
and the error function.
discussed
applied in these specific cases, but
methods which
special
we
also have
are applicable to
In addition, there
individual functions.
the is
a
and very important problem con
fascinating
cerning the representation of noise functions.
a
In
number
characteristics
of instances,
have been
vacuum-tube
utilized to
represent
either the logarithmic or exponential function
(HI.10.H.40)
This representation has the property that the term by term derivatives converge to the corre sponding derivatives of F (at least in the mean).
SQUARE WAVE GENERATOR
/
Thus, the problem of representing functions in this manner can be referred to the problem of representing /i and
/2
T"
Fig. e
2
sin Jx,
e
2
e first
2
c
HL10J.1
sn
cos Jx,
(see T. S.
The
T
These are linear
.
combinations
Gray and H.
B. Frey, Ref. III.10.1;
R. H. Miffler and G. F. Kinney, Ref. IIUO.x; cos Jx
two of these are two
pendent solutions of the
F. C. linearly inde
differential
equation
Snowden and H.
T. Page, Ref. III.10.ag;
and B. Chance, F. C. Williams, C. Yang, J. Busser, and J. Higgins, Ref. IH.lO.g). It is a mixer tube, 6BE6, to represent possible to use
x2
(see
R.
M. Walker,
Ref. IH.10.aJc).
It is also
an oscillographic representa tion of any positive power of x and also to represent polynomials. This is based on the fact possible to obtain
and the second
pair
are solutions for
that under certain circumstances a condenser will
charge exponentially. We indicate briefly the manner in which a power of x can be represented
Functions of this type which satisfy differential equations with constant coefficients are readily obtainable from equipment of the
REAC
and
on an oscilloscope tube. Suppose we have a voltage source which produces a square wave voltage (see Fig.
III. 10.1.1).
Alternately,
-eQ for
the
Philbrick types.
voltage will be
ni.10.1. Representation of Special Functions
Suppose this voltage is fed through resistances and R% and condensers Cx and C 2 Let the be the horizontal voltage across the condenser
+
and
equal intervals.
^
There are a number of great
practical
.
special functions of
importance which occur
fre
The most important are sinx and cosx, but we have undoubtedly quently in problems.
Q
voltage x for the oscilloscope, across the condenser
At the
instant in
and
let
the voltage
C2 be the vertical voltage/.
which the square wave becomes
III.10.I.
there
SPECIAL FUNCTIONS
be
zero charge on both positive condensers. During the time in which the square
wave
will
positive, the voltage
is
x = exp
-(
(IIL10.I.1)
is
in
positive.
We
Fig. III. 10.1.2.
2
t
are illustrated in
f(E)
can imagine the
rectifier in
(IIL10.I.2)
2)
from the beginning of
the time
Two
very useful for our present discussion.
y = exp -W C where
which / depends upon the previous history of would not be
the device as well as E, but these
possible such functions
the voltage
instant
in
141
the
which the square wave becomes
Now if
=
(I1I.10.L3)
Fig. ffl.10.L3
the function represented will be
If,
(III.10.I.4)
instead of using a single voltage y, a linear
combination of the voltages y nomial can be represented.
The
is
considerable
Bennett and A.
G. James
(see
R.
Fulton, Ref. IILlO.b).
S.
Thermistors
is
R.
A
R. R. Bennett (Ref.
have
represent x* (see P. Rudnick Ref. III.10.ae).
functions has
attention
reference for noise studies IILlO.a).
used, a poly
The above graphs and the functions/(/) are obtained by plotting / against E for different fixed values of the latter. We (see Fig. III.10.I.3).
assume that / depends only on E. This would that the above graphs are valid even when
mean
representation of noise
received
with an ammeter and a voltage applied
series
n y = x
been
used
to
E is
not fixed but a function of the time.
Let us suppose now that a voltage in the form e e1 sin wt is applied. Let us apply Taylor s theorem with the remainder to/() around the
+
point eQ)
with x
i.e.,
a
=
e1 sin wt.
Then
and V. Anderson,
For logarithms we
refer to
W.
.
W + W sm
ff
J
j
(Ref. IILlO.q).
.
,
,
ef
.
x
eif
+ ,
wt
flff
/
,
(e )
.
o
sm -
wt
A rectifier, in general, can be used to represent a square function for a limited range of voltage input. The input signal is an alternating current, the output signal
is
a
direct current proportional
to the square of the amplitude of the input.
+
^ W
sin
3
wf
+
3!
where el
is
/
a
V)
4
sin w*
4!
a function of e
elt
Q>
and
/.
An
ordinary direct-current ammeter will yield the average value of 7, provided w is not too
small
To
find this average value,
we
integrate /
from, say, to ITTJW and multiply by w/27r. The 3 average of sin wt and sin wt is of course zero.
Hence the
result is
41
where
Fig.
m.10.1.2
A rectifier can be defined as a device in which We regard/ (e a function, /(), of the applied voltage in such a way that/( E) does not equal f(E). Sometimes rectifying devices are used the current /
is
and
).
Of course,
^ but we will get a
H depends upon e
square output if e^H is on we negligible compared with/ (e )/2. Later will discuss methods of improving this situation. ff
REPRESENTATION OF FUNCTIONS
142
A
"full-wave"
rectifier
we have
similar result, provided
well as
+E!
can be used to give a
-E
voltage
(see Fig. III.10.I.4).
This
is
as
them suppose we operate
rectifiers;
relative to
It is clear that if
voltage.
P. M. S. Blackett and F. C. Williams, "An automatic curve follower for use with the differential analyzer,"
Camb.
true
even for direct current. Let us suppose we wish to square the value of a current z. We have two similar
c.
Phil. Soc. Proc., Vol. 35, Part 3 (1939), pp.
494-505. d.
F. E. Brooks, Jr.,
and H. W. Smith,
"A
computer for
correlation functions," -Key. ScL Instr., Vol. 23 (1952),
e.
pp. 121-26. V. Bush, "The differential analyzer: for solving differential
the voltage f.
g.
i.
A new machine
Franklin
Inst. Jn.,
Vol. 212, no. 4 (1931), pp. 447-88. V. Bush and S. H. Caldwell, "A new type of differ ential analyzer," Franklin Inst. Jn., Vol. 240, no. 4 (1945), pp. 255-326. B. Chance, F. C. Williams, C. Yang, J.
h.
equations,"
J.
Busser,
and
quarter-square multiplier using a parabolic characteristic," Rev. ScL Instr.,
Higgins,
"A
segmented Vol. 22 (1951), pp. 683-88. R. V. Churchill. Fourier Series and Boundary Value Problems. New York, McGraw-Hill Book Co., 1941. A. W. H. Coombs, "Memory systems in electronic computers,"
Communications and Electronics (Lon
don), Vol. 2, no. 3 (March, 1955), pp. 60-64. j.
W. A. Edson. Vacuum-Tube John Wiley
Fig,
k. P. Franklin. Fourier
HL10JL4
Hill 1.
across the rectifier p />
2
is
is e l9
here.)
through ft
Consequently,
m.
current
the
Book
T. S.
is
Oscillators.
New York,
1953.
Methods.
New York, McGraw-
Co., 1949.
Gray and H. B.
rithmic range of 10 pp. 117-18.
that across the rectifier
-elt (We neglect the current drawn by the
rectifiers
& Sons,
9 ,"
Frey, "Acorn diode has loga Rev. ScL Instr., Vol. 22 (1951),
A. Greenwood, Jr., J. V. Holdam, Jr., and D. MacRae, Jr. Electronic Instruments. New York, McGraw-Hill Book Co., 1948. M.I.T. Radiation
I.
Laboratory Series, Vol. 21. Hancock, "Photoformer design and perform National Electronics Conf. Proc., Vol. 7 ance,"
n. E. J.
2!
(1951), pp, 228-34. o.
/"(OK
while that through p 2
A. E. Hastings and
J.
computing correlation
4!
3!
E. Meade,
functions,"
Vol. 23 (1952), pp. 347-49. H. L. Hazen, J. J. Jaeger, p.
is
automatic curve
follower,"
"A
device for
Rev. Sci.
Instr.,
and G. S. Brown, "An Rev. ScL Instr., Vol. 7,
no. 9 (1936), pp. 353-57. W. G. James. Logarithms in Instrumentation. q.
Oak
Ridge, Tenn., Oak Ridge National Laboratory, 1949. U.S. Atomic Energy Commission, ORNL-413. r.
4!
3!
The ammeter measures
the
sum of
these
1947.
two in s.
which the odd powers
cancel;
essentially a square.
The
of course, this
difficulty
is
t.
circuit is
References for Chapter 10
G. A. Korn and T. M. Korn.
Electronic Analog York, McGraw-Hill Book Co., 1956. Pages 284-344. A. B. MacNee, "An electronic differential analyzer,"
Computers.
with this
matching the rectifiers and in general the need for a rather elaborate biasing arrangement.
E.Janssen. The Differential Analyzer of the University of California. Los Angeles, University of California,
u.
v.
New
I.R.E. Proc., Vol. 37, no. 11 (1949), pp. 1315-24. "A high-speed electronic function
D. M. MacKay,
Nature, Vol. 159, no. 4038 (1947), generator," 406-7. pp. B. 0. Marshall, Jr., "An analogue multiplier," Nature, Vol. 167, no. 4236 (1951), pp. 29-30.
a.
R. R, Bennett, "Analog computing applied to noise studies," LR.E. Proc., Vol. 41, no. 10 (1953), pp.
w. H. F. Meissinger,
1509-13. b.
S. Fulton, "The generation and measurement of low frequency random noise," Journ.
R. R. Bennett and A.
Appl. Phys., Vol. 22 (Sept., 1951), pp. 1187-91.
"An
electronic circuit for the
generation of functions of several variables," I.R.E. National Convention, 1955, Record. x.
R. H. Muller and G.
F. Kinney, "A photoelectric colorimeter with logarithmic response," Opt. Soc. Amer. Jn., Vol. 25, no. 10 (1935), pp. 342-46.
ffl.10.1.
y.
z.
A. C. Minister,
"The
monoformer,"
SPECIAL FUNCTIONS
Radio Electronic
Eng., Vol. 15, no. 4 (1950), pp. 8A-9A. F. J. Murray, "Linear transformations
af.
precise electronic function Electronics Conf. Proc., Vol.
generator,"
7
Time Bases.
National
(1951), pp. 216-
New York,
John Wiley
ah.
&
Sons, 1951. ac.
ad.
ae.
Radio Amateur s Handbook, 33d ed. West Hartford, Conn. Headquarters Staff of the American Radio
Relay League, 1956. Pages 72-73, 140-43. RCA Receiving Tube Manual Radio Corporation of America, Tube Department, Harrison, NJ. Tech. Series RC15. P. Rudnick and V. Anderson, thermistor bridge
H.
W.
Schultz,
J.
F. Calvert,
and E. L.
Buell,
"The
Snowden and H. T. Page, "An electronic circuit which extracts anti-logarithms directly," Rev. Sci. Instr., Vol. 21, no. 2 (1950), pp. 179-81. W. W. Soroka. Analog Methods in Computation and Simulation. New York, McGraw-Hill Book Co., 1954.
ai.
"A
Vol. 24, no. 5 (1953),
ag. F. C.
227. ab. O. S. Puckle.
Instr.,
photoformer in Anacom calculations," National Electronics Conf. Proc., Vol. 5 (1949), pp. 40-47.
N. Pederson, A. A. Gerlach, and R. E. Zenner,
"A
Rev. ScL
pp. 360-61.
between
Hilbert spaces and the application of this theory to linear partial differential equations," A.M.S. Trans Vol. 37 (1935), pp. 301-38. aa. C.
correlator,"
143
Pages 93-96. Stone. Linear Transformations in Hilbert Space. New York, The American Mathematical
M. H.
Society, 1932. aj.
D. E. Sunstein,
Pages 1-23. "Photoelectric
waveform generator/
Electronics, Vol. 22 (Feb., 1949), pp. 100-4. ak. R. M. Walker, "An analogue computer for the solu
tion of linear simultaneous
equations,"
Vol. 37, no. 12 (1949), pp. 1467-73.
I.R.E. Proc.,
Chapter
11
LINEAR EQUATION SOLVERS
m,ll.B. Two-Way Continuous Devices
m.ll,A. Introduction
The
solution, of simultaneous systems
of linear
considerable algebraic equations has aroused As we have discussed earlier, a interest.
we
In this section
discuss devices
which have
components which are not unidirectional.
It is
clear that a system of equations
tremendous variety of digital methods have been In addition, proposed for solving these systems.
many
have been constructed for the same
We They
objective.
will study these in the present chapter.
are associated with a
number of
interesting mathematical ideas,
rather
and they
into the give considerable insight
also
theory of
mathematical machines. There are a number of design questions involved, and stability critical
is
a
problem.
The continuous computers
that
we
deal with
between input and output; in the other type one directional flow of information is presupposed in each component so that we have definite inputs definite outputs.
Devices of the Section
III. 11. 8.
The
type will be discussed in other type can be further
divided into machines which require manual
adjustment and those which do not. In Section III.ll.C we will discuss those simple machines
which require manual adjustment and there
is
no
machines
in
special provision for stability.
use
the
Gauss-Seidel
which These
method
for
solving simultaneous linear equations, which will discuss.
chapter
we
In the remaining sections of
we
this
will discuss processes for obtaining
Both in the case of automatic adjusting machines and hand adjusting machines, this stability.
discussion yields an introduction to the general
problem of the with
many
stability
variables.
of a continuous machine
We
also refer to
Soroka (Ref. IILll.m, pp. 97-126).
W. W.
=^
(III.ll.B.l)
can be realized by gear boxes and differentials in such a fashion that when we set in the coefficients a i} and turn the 6/s to the proper value, then the x/s will be deter theoretically the values of
mined
at the proper value. Such a device is, however, extremely expensive, and the multi of parts can readily be seen to decrease the plicity
accuracy.
The
effect
It is
clear that the
of gear backlash alone here.
number of
We
kept as small as possible.
parts should be
wish 10 mention
two devices which have been constructed which, it
certainly seems,
have a minimum number of
parts.
One device is
that of J. B. Wilbur developed in
Each equation
is represented by a tape, and each variable by the sine of the angle of rotation of a shaft. The part of the device
1934.
first
3=1
would be considerable
can be divided into two types: the first type consists of computers which use components in which it is not necessary to make a distinction
and
f
purpose continuous computers
special
corresponding to a^Xf
is
illustrated
in
Fig.
III.ll.B.l.
The tape corresponding to
the fth equation in over such a fashion that the passes pulleys is shortened in length along the line corre sponding to the equation by an amount a^. This tape is part of an endless chain adder (see
tape
also Section III.2.B).
For each equation we have a device
like this
for each variable. Thus, the total shortening of
the tape
is
2%x Now if we can permit this to -.
;
3
equal b it then we will have realized the equation. later version of this device is described by
A
J.
B.Wilbur
(see Ref. III.ll.o).
TWO-WAY CONTINUOUS DEVICES
IIL11.B.
More
and
effective
definitely
more expensive is
around the
M. Mallock
we
the well-known machine of R. R. (see
Ref.
Here each unknown
Ill.ll.g),
equation
circuit
145
corresponding to the /th
find
is
-*
represented by the flux in a transformer, and
The
9
~n
Fig. IIL11.B.2.
On
in
dt
dt
_.n
__
/) l
dt
dt
circuit is indicated in
the constant transformer, besides the equation
we
coils,
also have a
power
coil
or primary and
another measuring coil of a fixed turns.
There
each
of
is
the
number of
also a similar measuring coil
transformers.
variable
on
Each
measuring coil is connected across a circuit which is essentially an alternating-current volt meter, so that the flux change in each transformer is
measured.
the
power
equation the
Fig. ffl.ll.B.l
An alternating current is applied to This induces voltages in the
coil.
circuits,
various
which, in turn, induce flux in
variable
transformers.
Except
perhaps for a brief period, the above equations
each equation is represented by a closed circuit consisting of a number of coils, one around each variable transformer
and one around the trans
former corresponding to the constant term. We have then a coil for each coefficient %, and the
number of turns on each
are satisfied.
The power
is
adjusted until the
constant measuring circuit indicates the value
which case the other measurable
in
1,
circuits
indicate the values of the variables.
proportional to
coil is
Actually each transformer has a large number of different coils, whose number of turns are
decimal fractions of a fixed number of turns.
The machine
up by connecting in series whose total number
is set
these different coils into coils
of turns are proportional to a ti Thus, on the transformer corresponding to the variable xf , coils are obtained corresponding to the coef .
%,
ficients fly,
.
term transformer to b lt
.
.
.
,
bj.
.
.
,
a nj
and on the constant-
,
coils are
The
coils
obtained proportional
a a , a i2
.
,
.
.
,
ain and
^
are connected corresponding to the ith equation in series. If J?3
-
is
the total flux in the
the voltage across the coil a
xf transformer, then is
Fig.HLll.B.2
The
and presumably we can electronic methods for the
coils are in series,
compensate
by
resistance loss in each circuit.
Hence,
if
we go
It is clear that
by
the torques
an equation can be represented shaft. Here the coefficients
on a
are the distances
from the
axis
and the variables
LINEAR EQUATION SOLVERS
146
by the
represented
The
forces.
forces can be
equalized between different shafts by hydraulic
German
(There was a
methods,
device of this
nature about half a century ago.)
Another device for solving mechanical means
is
linear
that of T. E.
An
9
If
this
electronic constant-current sources are available
is
inadequate,
based on the high plate impedance of a pentode
At each point A
.
current.
the
a network in a number of ways. of n simple network made up ...
desired
equations by
A real symmetric matrix can be represented by
A l9
source and a approximated by a high-voltage resistance which can be set for large adjustable
W, Schuman
(seeRef.III.11.1).
A<
impedance so that variations of the external connections have no effect. This situation can be
Consider a
+
1
points, >
{
(i
1)
4
Zy
which
is
Then
the current equation for the zth node
.
becomes
even higher
effectively
unbiased cathode resistor
is
when an
used (see 0.
S.
Puckle,Ref.III.ll.k, pp. 17-18).
we
have a current generator which generates a nodes is current Suppose that each pair of connected by a conductor of conductance Ytj the reciprocal of the impedance
which becomes
For direct-current itself
signals the plate current
can be used, but for other frequencies, a
transformer coupling would be used.
The use of inductances of their expense.
number of nodes, so voltage
and
its
is
not desirable because
However, by doubling the that for each
x both a
it is
possible to
negative appears,
t
use only capacities, as discussed in the paper of
+ ...-7^ = I,
A. (IIL1LB.3)
Many and S. Meiboom (Ref.
Now it has been proposed to use this as a method of solving the system of linear equations. a inx n
-
fl
Obviously,
i3
and b-s are
s
we can
=
conductances
Y
ti
fl
tf
and j tv
= aH
.
correspond to the
s
a matrix
is
are clearly restricted.
On the other hand,
if
=
the
Suppose we have n voltage
These voltages are to represent
x1? ...,#
quantities
in addition,
coefficients
can be realized for
input currents of a specified frequency,
In this case, the 6/s must be realized as current generators having a specified current output,
i.e.,
they must be obtained from constant current
one uses a normal adjustable current source with a measured output, one generators.
would have
If
to adjust each source until all the //s
coincide with the i/s.
However,
stant current sources are
current generator
is
relatively
possible.
one with
A
con
constant-
infinite internal
in
Eq.
we have n
Suppose works and a constant
voltage.
/
.
t
III. 11. C.I.
resistance net
Each of
resistance networks is intended to
of the above
o>.
(II1.11.C.1)
whose output can be adjusted to any
desired value.
these restrictions disappear, and theoretically any
!,...,*)
can be readily realized by means of voltages and
generators 7
problem of
A linear expression such as
potentiometers.
one uses reactances
matrix with real
to obtain
IE.11.C. Manual Adjustment
(i
one would prefer to use passive impedances. If resistors are used, the Ff /s are all positive and fly s
is
the same.
Because of stability questions,
.
Meiboom
solve a system of equations, but the
consider the voltages e i as the
The
unknowns x?
real
S.
or only
objective of the
the characteristic roots of a matrix rather than to
realizing
where the
Many and
paper by A.
III.l l.h)
The
resistors to realize a matrix.
these
represent
one
in the sense that if the voltages
mentioned are connected to
this
network, some
point in this circuit has a voltage whose value represents
/
.
4
Under certain special circumstances
this device
can be used to solve the system of n linear algebraic equations,
/ f
= 0.
One starts with some
4
0)
(
values of the voltages x *\
xf,
applies these voltages to the
first
equation
and adjusts the voltage x% until the
first
equation,
initial
One
,
-
MANUAL ADJUSTMENT
III. 1 l.C.
/!
= 0, is satisfied.
Let x f denote the new value (
Thus,
of the voltage x r One then applies the voltages 0) 0) 1} to tae resistance network for / 2
4 4 ,
and
>
adjusts the voltage
satisfies
/
2
adjust the
= 0.
x 2 to a value x 21} which (
One proceeds in this x n to voltages x 3 x4 ,
.
,
.
.
,
fashion to
all
with the
and
first,
cycles again through all the
that at any instant one satisfied
given equation
is
x
values of the
would have only
out of the all
adjusted,
are disturbed. However,
n, for
when a
other equations
may turn out that the
it
which we have considered
s
converge when the cycle is repeated indefinitely. If the x s converge, each linear expression will
The only value
also converge.
sion can converge to is
is
difficulties
of the x
the linear expres
zero since in each cycle
zero once. Consequently,
s,
here
:
the other
x
s
do
one is to insure convergence is
associated with the rate of
We will now
convergence.
if the
a solution. There are
converge, they converge to
two
discuss
both of these
(III.11.C6)
j,,
if
and only
if
the xf*
are obtained iteratively
,
Now if the
solution to the system
by
consider the case in which there
solution so that
we have
quantities
is
ys such
a
Let us return to the quantities
for
initial
any arbitrary
value of z (0)
solution,
the
#s
given
we need only consider the case in which and the corresponding z
are zero,
are
s
by Eq.III.ll.C7. very convenient to consider (zf\
It is
.
.
.
,
zj^)
u) and to components of a vector z (k ~ l} and z (b) by express the relation between z means of a matrix. Let A denote the matrix (%)
as the n
of the coefficients of the original linear expres u Let A denote the matrix of elements sion.
A
above the diagonal. Let
l
denote the matrix of
elements below the diagonal. Let
D
denote the
Eq. III.11.C.7
becomes
that
deter
Thus, to
.
establish convergence if there exists a unique
(D
xf which we have
The Gauss-Seidel process
jc/s
converge to the solution for any initial value of the x (0) s is that the z w s will converge to zero
will
(see also
defined above.
unique, a
is
condition that the
sufficient
matrix of diagonal elements.
questions.
We
ys and
_
jc
converge to zero
converge to
necessary and
one equation
It
$
=
Having
Ordinarily one would expect from the above
each
the z
(
>
the variables, one starts over again
variables.
process
let
satisfy the
corresponding equation in each case. adjusted
we
z
4
>
if
147
If
we
+
A^ = -A
H. HoteUing, Ref.
u
z
(k
~ (IIL11.C.8)
Thus
III.ll.c).
let
mines these quantities according to the equations
we
see that the necessary
and
sufficient condition
for convergence of the Gauss-Seidel
every
We can rewrite Eq.
= *
initial
vectors z (0) ,
ffl.ll.C.2 as
value
T
(Q]
necessary and
Th *\ (
,
if
we
III.11.C.3,
subtract
we
.
.
.
2
1,
this
equation from Eq.
obtain
is
that
T should be less than
in absolute value.
not every
A
will
lead to a
convergent Gauss-Seidel process.
In
can readily give an example.
we
system
(III.11.C5)
for
converge to zero. The
condition for this
root of every characteristic
&*-
method
(0) that for every vector z , the
sufficient
Clearly, then,
and
is
If
fact,
one
take the
LINEAR EQUATION SOLVERS
148
and
start
with a
set
of values
with i
(a, b)
^ 0, we
b), (-b, -b\ b\ (b, b\ (b, and from here on the cycle repeats,
get successively (a, b, b),(b, b),
(
T will
In general,
characteristic root Eq, III.l 1 .C. 16 holds for any 1 of #, and a bound on the A for which Eq. C.16
holds
In order to study the rate of convergence of the Gauss-Seidel process, it is more effective
roots.
to consider the matrix
T*T where T*
the
is
will transpose of T, although only overestimates be obtained in general. The transformation
H = r*Tdoes describe directly the way in which the size of the vector Tz varies from that of the _. vector z. Thus,
is
One
not have real characteristic
a bound for H.
also, then,
type of sufficiency argument
based on
is
the relative sizes of the assumptions concerning
D
diagonal terms which appear in
and the nonu
in A and A diagonal terms a if which appear Now the easiest way to bound a matrix is by .
l
noting that the /
^a^ W
^
2
bound
_
D
,
and. hence,
,
,
,
than or equal to
less
is
For simplicity,
.
T
,
has bound
us suppose a u
let
=
1
,
Let
1.
i>j
H has
if
Thus,
a bound C,
2
2
|rz|
C|z|
<
and,
1/2
furthermore, the greatest characteristic value of
H
bound
precisely this
is
C
that
C
<
1,
<
C.
we know
this will clearly yield
=
(III.11.C.17)
than the bound
ofA and A\
we know
that
c^VI and
if
Thus,
ku
Thus,
c
(in 11
A:,
and ku
is
is
not
not
less
less
ii)
+ A )%\ = l
I(D
a number of
is
Au
be the transpose of
transpose of A,.
Au
and
A
1
-
ana the
Then |
=
xA"
Let
Au
\%\
l
\A %\
(Z)
..
.
.
+ A ^ D + A i^\
>
(1
^
.
f
IIULC16
\
on
F
(m.n.c.19)
for ^
^
indicates that
H^ A (D + A^ U
(III.11.C.13)
Apply A
>
Similarly
^
Suppose A is the largest characteristic root of and 7 the corresponding characteristic vector,
A,
-
l
+ A %\
\%
sufficient conditions
which can be used to estimate characteristic roots of H. Let
AU and AU
an overestimate for the
rate of convergence.
There
l
than the bound of
u
Now
^
is
not zero, and hence,
to both sides of Eq. m.ll.C.13.
= WMJ[(D + A&D + ^
v
A1/2
1
-
\\r e
(III.11.C22)
k,
)]-U>
(III. 1 1 .C. 14)
= y.
<-^-
Z
recan that a sufficient condition for Gauss-
Se j de l convergence
is
that A
<
1.
For
this
it is
sufficient that
--
(III.11.C15)
1
Now
jD
+
thus there
It is
^[j
is
and
D+^
z
have inverses, and
1
(III.11.C.23)
z
or
a vector % such that
convenient then to consider
<
fc
fe
i/2
+
k
<
1
(III 11
C 24)
\i/2i
(nL1LC25) J
IIL1LD.
GOLDBERG-BROWN DEVICE
(This applies to the case where a i{
=
1.)
We
also have that
149
Xj. This can be done by spanning the output of the amplifier across ground or in the case of the usual electronic differential analyzer equip
ment by an inverting (III.11.C.26)
So that
if
perfect in
we have
2
a i?
tt
i
<
+ is >
,
This
therefore,
condition, for
sufficient
summing i.e.,
con
what
+
each d ti
of the
Gauss-Seidel
when
process
linear
expres 1 potentiometers and n
+
1
A
potentiometer is set for and connected to the amplifier which resistors.
corresponds to the variable
vergence
that they are
For each
follows.
we have n
sion
If inverting
amplifier.
we suppose
amplifiers are used,
xj9
taking the sign of
Another potentiometer
dy into account.
is
connected across the constant-voltage source and
There are two commercially available com puters which use the above form of the GaussSeidel process.
One
is
manufactured by the
Consolidated Engineering Company of Pasadena (see C. E. Berry, D. E. Wilcox, S. M, Rock, and
W. Washburn,
H.
Ref.
III. 11. a).
The other
is
manufactured by the Phillips Petroleum Com pany (see T. D. Morgan and F. W. Crawford, Ref.III.lLi).
b{
set at
contacts If
the
The summing
.
on
of
resistance
negligible relative to the
the
potentiometers
is
summing resistors, then
point will be at a voltage e f as given
this fixed
Now suppose the voltage
above.
connect the
resistors
these potentiometers to a fixed point.
input to the zth amplifier,
and
e t is .
let
used as an
x { denote the
=
1 ^, where output of this amplifier. Then e t G denotes the gain of the amplifier. Thus, G"
HI.11.D. Goldberg-Brown Device
Another type of continuous solver
is
W. Brown (see
G.
linear
equation
that invented by E. A. Goldberg and Ref. IILll.b). This
feedback machine which
is
not always yield a solution to a given system of simultaneous linear equations, but the system can always be modified
make
to
If
number of
operational amplifiers
tiometers
available, a
differential analyzer
2
will
the device applicable.
is
or the x/s satisfy the relation
a direct
an adequate and poten
commercial electronic
such as a
REAC
or
Good
E
(
aa
~
denoting
^Oxi - W
Now we device.
wish to discuss the
It is
essentially
of
stability
an electronic
below. satisfy a stability criterion given
we know the output specified if the x of voltages amplifiers t (t) as functions of the time. III.l
be Suppose the system we wish to solve can
These voltages are governed by Eq. 1 it is understood that is a
1.D.4 provided
G"
function of/?, the differential operator.
Thus,
n Eq. D.4 can be considered as a system of differential equations on the n unknowns xj(t).
written:
This system
Let
is
linear with constant coefficients.
a unique solution xi
If there is
/
original system
denote the discrepancy in the zth equation when are substituted in
it.
We
= 0, = z
=xw
of the
a
special
t
!,...,
solution to the differential equations Eq.
given by
x
t
=x
-
;
(0) ,
where
E is,
original
of alternating current with a frequency
these amplifiers must produce voltages x, and
cycle,
is
(In the
output of the given voltage source.
In order to provide for the sign of the coefficients
.
0.4
of course, the
source and suppose we have a constant-voltage n amplifiers, each corresponding to a variable xf
a&
this
circuit
whose behavior is
year can be set up in standard fashion to act as a device of this type, provided the amplifiers
the values *!,...,*
output of the fixed-voltage
the
generator.
Goldberg-Brown machine
E
was an 1
kilo
but we can think of it as a constant.)
LINEAR EQUATION SOLVERS
150
The general solution is obtained by considering the
2 (a - WP)" 1.D.5
Eq.
III.l
the
form x t
exists
is
solved by assuming solutions in
k/
A
1
*.
solution in this
of
determinant vanishes,
1%
form
negative.
holds are
1
We want conditions on the
of equations has a stable matrix, the device will be stable. This means that G(rj) is such that for y
which the
for
??
all
that Condition
G such that when the given system
(III.11.D.5)
amplifier gain
those values
for
Now
)**
^ are
(%) such
usually called stable.
=
1
The
Condition 2:
Matrices
homogeneous equation
i.e.,
=
1
W?)"
consider the roots A l5
!
.
.
(III.11.D.6)
.
,
l%- VI
An (III.11.D.7)
Since the original system of linear equations
nonsingular by hypothesis, no A ;
-
= 0.
If
r\
is
is
a
solution of Eq. III.11.D.6, G(f\}~^ has one of the
values Al5
,
.
.
=
(jfyj-i
An,
,
or
jt ft
Fig. ra.ll.D,2
i.e.,
G(rf)
=A
-1
with a positive real part,
(III.
Jb
Therefore, a necessary and sufficient condition that all solutions r\ of is Eq. have negative real parts. If all solutions of this equation have negative real for
stability
III.11.D.8
solution will parts, the general t
->
oo,
and only the
approach zero as
solution which special
Since G(r/)
G(rf).
1/1,-
oo, this
r\
question
can be answered by considering the Nyquist diagram for G(rj) as a contour in the complex
diagram Contour
the
is
I
be recalled that the Nyquist in the G(rj) plane of the
will
It
plane.
map
of Fig.
III.l I.D.I in
is
we have
If
positive values
The
the ^ plane.
direct-current
Nyquist diagram
G(rf)
and
will
look
the
amplifiers,
will
from
start
large
like Fig. III.l l.D.2.
exterior of curve in Fig. III.l l.D.2 will
not be assumed by
PLANE
not assumed by
is
-+
as
->
G(rj)
for the
??
s
in
the
positive half of the complex plane (see also
Section IIL7.D).
Thus,
it is
desired that none of the
be inside the contour.
unknowns and K-|
<
l,|^.|
Now if the Condition
part
m.ll.D.l
desired will be apparent. This condition depends
on the A1?
.
.
.
,
Am
.
To go
assumptions on these A
s.
further
we must make
There are two such
conditions of practical interest: Condition 1: The real parts of the negative.
A
s
Hence,
a {i
l/l^-l
have negative
is satisfied,
should
such that
all ;>
l/.
real parts,
then these I
s lie
i.e., if
in the
portion of the complex plane for which the real
CONTOUR I
Fig.
1
coefficients
I/A,-
For a system with n
^
are
all
is
negative, outside a circle with radius 1/n.
Thus, in terms of amplifier design it is desired that at the point ?15 where a phase shift of 90 is obtained, the amplifier have gain l/. If this <
holds,
the 1/A X ,
.
.
.
,
l/A n
are all outside the
Nyquist contour (see Fig. III. 1 1 .D. 1). G(rj) does not assume these values for any r\ with a positive real
part.
stable,
Hence, the
and we
circuit
of
this device is
will obtain the answer.
III.11.E.
On
the other hand,
Condition 2
if
and negative and from the origin. Thus a real
is
I/A,
POSITIVE DEFINITE CASE OF ADJUSTERS
away
for stability
is
is satisfied,
is
and
further than 1/w
is
A sufficient condition
that the gain be less than Ijn
a phase shift of 180 E. A. Goldberg and G.
151
obtained.
W. Brown,
when
= 11
(See also
(!**&& /k\fc
i
i
Ref. Ill.ll.b.)
Notice, then, that our device will be adequate
which
to solve systems of linear equations either Condition
ki
satisfy
or 2 provided the amplifiers
1
where
the corresponding conditions given above. These conditions are readily verified from the
y
= fc
satisfy
Nyquist diagram Systems which
satisfy
On
arise in practice.
Condition
1
frequently
the other hand,
if
we
are
of linear equations, given an arbitrary system we can transform the system into one which satisfies
(IIL1LD.14)
(see also Section III.7.D).
A characteristic vector is not zero by hypothesis, and zero.
A
is
nonsingular, {y,
.
.
.
,
A
a
RCA
described and
The machine
Princeton.
in
Laboratories
is
is
device of this type was constructed by W. Brown at the
E. A. Goldberg and G.
combinations of the given equations.
not
is
y n}
Hence, Eq. III.11.D.14 shows that A
positive quantity.
Condition 2 by taking appropriate linear
Suppose our original system
since
is
discussed in Ref. stability
its
Ill.ll.b.
We
A
can obtain an equivalent system
further
discussion
stability
is
given
in
G. A. Kora (Ref. IILll.d) and G. A. Kora and
LA.Zadeh(Ref.IILll.e). determinant
the
provided III.11.D.10
may
0,
\c i
Eq.
also be written
HI.11.E. Positive Definite Case of Adjusters
The Gauss-Seidel method can be guaranteed to work if the matrix of the system of equations is
i\k
positive
we
definite as
shall
prove in
this
section.
If
A
denotes the matrix of the original system, the matrix of the equivalent
cA Eq. IIL11.D.9,
is
system, Eq. D.10.
Any nonsingular matrix c may
be used for is
this purpose. In particular, if
nonsingular,wihch corresponds
that there
is
A itself
to the property
a unique solution to the system
we Eq. D.9, then
may
take
C
= A*,
the trans
of A. The matrix of the new posed conjugate
system A*A elements cntS9
is
positive
A*A
If
definite.
suppose
As
in
<x
.
i3
Thus
A*A
has real characteristic roots AI}
A
is
* any such root and 1}
sponding
.
is .
.
.
.
.
.
,
xn
the corre
characteristic vector.
(for
z
=
!,...,)
definite
coefficients
of
for convergence |Ty|
<
Now
|y|.
Ay y =
if
is
U ,
and a
sufficient
that for
for every y,
and only
if
matrix.
are
real.
we can introduce
HI.ll.C, 1
A
y
condition
every vector y, *
AY y
= 0.
>
Let y
0,
and
= Ty.
Consider the difference
has
symmetric and A n Suppose ,
a positive
T(j) + A^~ A
formulas:
%=
is
the
Section
where y*
Clearly
A
Suppose
We
(IIL11.D.13)
is
obtained from y by the successive
LINEAR EQUATION SOLVERS
152
Consider
=
Hence,
for
=
;
.
.
1,
.
,
and
,
this
is
equivalent
Now
hold
y*...,y
and regarding yl as so that p is a minimum.
the variable, choose y
The
minimum
condition for a
j
n fixed,
A
Since
that
is
^=
=
or
(III.11.E.3)
,
dy l
=
y
.
.
.
n)
,
(III.11.E.12)
= 0,
is
quadratic
2
_L a
(lll.ll.E.13)
^ 0,
if z
.
(III.1LE.4)
we hold JV^s* relative to y 2
We
.
(0)
, . u n this result we will now show that .
...
and form y
(1)
(0)
,y
(2)
,y
.
,
.
.
,
y
(fc)
^
^n
and positive
^
>
^
}
since
^
(fc)
5
= By
->
is
0.
(III.11.E.14)
z w _^ o
nonsingularj
Furthermore,
- BTB~\
(t)
z<*>
and only
if
Let
z (1) ,
will
The sequence
z (0)
,
app roach zero for every choice of z if a| t the characteristic roots of S are
(III.11.E.5)
andsimilarl?
if
= BTB~W~u. (o)
.
if
,
.
,
and only than
1
S and
A
less
in absolute value.
= ^iy*
+
y*
[D(y
-
y*)
(y
-
Let z be any characteristic vector of
y*)]
(III.11.E.6) is
A
positive definite,
definite matrix B.
positive
= Ty
Let
definite matrices.
obtain
y
A
we take (k ~u
if
we e^ a sequence of vectors such that y (k) 0. This will justify the Gauss-Seidel method for
z<*>
?
.,
,
->
v *)
(y
y = ^{^i*, 72*
Since
to
^-i^ __ z *j ^-i^ _ z *j = D(y - y*) (y y*) ^ any y
y
The
contrary
= ETErh * 2.
With
Ay
= 0.
means y
z
z*
...,
^y
1,
implies
Thus,
hypothesis.
in yls
Similarly,
=
Consequently,
= yf, and since p
Clearly, then ft
minimize
i
nonsingular, this
is
statement
=
o (for
=
5y
2
for a
Let *
=
Sz
^% z*
Qur previous
V==Z
Z
=5
tne corresponding characteristic root.
=
fa
Let
(III.11.E.15)
nULL10
shows
^
(IIL1LE7)
Now z
= By* = BTy == BTB
*
1
!
(IIL11.E.8)
>
z*
z*
=
Uz
l
-
Uzl
2
Eq.III.ll.E.6 becomes
52 J
52 J
J
,
*
.
J
.
-r
and, hence,
/
|A|
<
Thus, z
1.
= z*
z*
+
l
[DB~
(z
- z*)]
B"\z
- z*)
For z*
z 7=0,
= z implies
singular, this
y minimizes
z
y*
means y* ja
^z
is
y lt
.
.
.
,
yn
w
->
.
if
one replaces
by an equivalent system, one can obtain a system in which the
matrix
is
positive definite.
ID.) However,
replacement (lll.ll.li.il)
= 5y, and since B not = y. But if y* = y, then
for each variable
7
the given system of equations
III. 1
prove: it
0,
and the Gauss Seidel method conv e rges.
(III. 1 I.E. 10)
Now we
->
.
j
z
)
(IIL11.E.16)
D(u _ ^; ^ u _ j; ^
,
^e kave geen prev ousiy that 2
z
UI (z
and
it is
(See also Section
conceivable that this
may slow up the convergence rate, it may not be desirable to do this
therefore,
unless absolutely necessary.
The following Theorem for positive definiteness.
is
a well-known test
MACHINES USING GAUSS-SEIDEL METHOD
III.ll.F.
THEOREM form
is
A
III. 11. E.I.
positive definite if
u
symmetric quadratic
and only
if
v n_
combined conditions
the
Thus,
153
%>0
and
are equivalent to
>
,
flu. >0
>0
(III.11.E.17)
The proof
/:
n =
clearly true for
can write
It is
n
for
A:
=
1,
.
.
equivalent to
.
and we know that these are
n,
,
p
>
0.
We show it for n.
1.
as defined above, following
/*
Eq. m.ll.E.2h
n.
Let us suppose that the
1.
result holds in the case
We
by induction on
is
Machines Using the Gauss-Seidel
ffl.ll,F.
Method and the Murray-Walker Machine
the form
It is,
of course, relatively simple to construct
devices to solve simultaneous linear equations
For x 2
,
.
.
.
xn
,
given, Eq. III.11.E.18
for all values of
xt
if
and only
flu
and
if
is
positive
(III.1LE.19)
>
in
\2
%A-i
%0ij>%-
i
(III.11.E.21)
hypothesis of our induction,
*2
values of
,
.
.
.
,
xn
is
Vi
>
f r
equivalent to the
is
circuit
possible
equations in Fig. is
all
based on direct-current methods,
tiometer tiometer
u
fl
such
for
III. 11. F.I.
x, gives
,
.
,
n.
Now if we take the determinant
The
six-pole switch
a voltage
x.
We
A poten
%
as its output gives add au x and a12y and -bt
to realize the voltage anx .
a device for two
&e
across
voltage output.
= 2,
multiplication
The double-pole switch and poten
marked
>0
k
The
a double-throw switch, which determines the
equation.
inequality
for
and a gang
realized at a time,
by a potentiometer method and the addition is by means of the addition of voltages. We show a
>
3=2 i=2
By the
also
is
-
=2
one equation
switch changes the equation.
Eq. III.11.E.20 can be written
(see
Section III.ll.C) are of this type. One of these (see T. D. Morgan and F. W. Crawford, Ref. III.11.6.i) is
(III.11.E.20) >(!%**)
V n-l
based on the Gauss-Seidel method, and the
abovementioned commercial devices
we have
+ a12y -
measured by the voltmeter. Alternatively, we have seen
b t which
how
is
a linear
combination can be realized as a resistance (see Section III.2.E). Using linear potentiometers, we flu,
,
%
and multiply each column except the first by and then proceed by subtracting multiples of the first
in
column from the
which the
first
others to obtain the
row is
1, 0,
.
.
.
,
0,
form
we can show
that
mount two potentiometers for each coefficient on the variable shaft. This, of course, can be in a
number of ways, but
done
for simplicity let us
suppose that the resistance portion of the poten Let us
tiometer turns with the variable shaft.
begin with each variable shaft in the zero Then each coefficient is entered by position.
contacts displacing the
from the center position a
the contacts going in proportionate amount,
(HI.11.E.22)
opposite directions
on
associated with the
same
the
two potentiometers
coefficients.
This also
enter the sign of each coefficient. permits one to These contacts are now fixed in space. If, then,
LINEAR EQUATION SOLVERS
154
the
x
shaft
is
rotated,
we
see
from Section III.2.E
that one potentiometer will have resistance
2
and the other
R 12 (see Fig.
III.
1 1
,F.2).
resistances of the
and
all
For a given equation
first kind
all
the
are connected in series
the second kind in another series,
resistance corresponding to the constant term
connected to one or the other of these
depending on
The two
its
A is
series,
sign.
used as two branches of a
series are
Wheatstone bridge. we have
When
equality
is
obtained,
This circuit has the disadvantage that two potentiometers are used for each coefficient, each
must be entered twice or some
coefficient
mechanical
arrangement to accomplish this purpose has to be used. However, it has the advantage that only one voltage is used and the value of this voltage does not enter into the
In
calculation, this
voltage
purposes.
may
This permits one to use sensitive
The
galvanometers. far
a potentiometer across be used for volume-control
fact,
simpler
switching arrangement
and the device can be
augmented so
as to
is
readily
produce the value of each
equation.
An S.
article
by C. E. Berry, D. E. Wilcox,
M. Rock, and H. W. Washburn
(Ref. IILll.a)
describes a device for solving simultaneous linear
equations by the Gauss-Seidel method. nating current multiplication
is is
and the addition
used for convenience.
by
Alter
The
successive potentiometers
by the resistance-averaging method. The coefficients are set by a Wheatstone bridge method.
is
Fig. m.ll.F.2
III.11.F.
MACHINES USING GAUSS-SEIDEL METHOD
The devices given above can be used only when the equations permit one to apply the GaussSeidel method, In general, this would require a transformation
preliminary
Section IIL11.D.
indicated
as
However, F.
J,
in
Murray has
155
F. J. Murray had previously constructed a crude model of such a device for the four-
equation
case.
the
However,
can be
idea
explained readily in the case of two equations.
A schematic
pointed out a method of constructing devices of
The
given in Fig. III.11.F.3.
is
diagram
x and y, produce an and by an averaging process
variable boxes,
alternating voltage,
the voltages proportional to
a l2 y
-
and
are produced
is
(t
a scale variable which
is
frequently very useful in fitting the device to the
problem and for volume-control purposes). The signals, l and 2 are alternating voltages ,
and hence may be amplified by means of an audio-frequency amplifier suitable for the frequency used. The amplified signal is applied to the plates of a diode, and by square law rectification
is
obtained and read on the meter.
The Fig.
HL11.F.3
entering
Rg
shown
are the output grid
Each
is
of the final amplifying stage joined to the plate condenser. These resistances are a
by R. M. Walker (see Ref. IILll.n). This machine was for twelve equations in twelve the
resistances
resistances of the amplifying stages.
an adjusting type which are applicable to any system. One such machine has been constructed
unknowns and
a direct current proportional to
utilized a
punch-card method for
coefficients.
(See
also
Section
III.6.B.)
by
blocking
for the normal necessary since they supply a path
by thermal emission in The voltage drop generated by this current would introduce excessive bias voltage on
direct current generated
the diode.
the diode plates if the battery were not provided.
A
smoothing
circuit
is
with the
associated
microammeter.
The machine produces
The variables
directly
are used in rotation to minimize p,
whose value appears on a meter.
This
is
equivalent to applying the Gauss-Seidel method to the system
J-
Fig.
=
(i
=
1,
.
.
.
,
n)
(III.11.F.4)
Each
variable
Fig. III. 11.
whose matrix
when one
sets
is
b{
as positive definite,
=
one
sees
in the expression for p.
FA
m.ll.F.4
box has
shown
the circuit
The power
is
in
obtained from a
step-down transformer across the transformers are used in the model.
line. It
Bell
would be
LINEAR EQUATION SOLVERS
156
better if a single transformer having secondaries for
each
and
variable
were
constants
the
The double-pole switch determines and the 400-ohm wire-wound poten
available.
the sign of x
tiometer values
P2
determines the
4x4
for the
resistance
volume
R
of 3,000
control,
Pl9
of
size
model.)
ohms and
x.
(We
give
The constant the
0.1-megohm
are used to equalize the load
on the transformer with different x
It is
settings.
not necessary that this be done with great accuracy, and one adjustment when the value of
x is approximately known is
all
that
is
x
to
one for +jc.
as well as
device this
is
voltage in a balanced position relative to ground so that one terminal
other
is
is
as far
below ground as the
above.
It is seen then that the use of alternating current in this device has three advantages. One
of these
the ease of positioning the variable
is
circuits relative to is
ground. The second advantage
the use of simple audio-frequency amplifiers.
The
third
is
the ease of squaring.
The model gave
necessary,
percent of the largest unknown.
be balanced by increasing the resistance in the shunt Pl5 R.)
expected from
is
to
and
crude,
out
1
The model was
better results could hardly have been it.
However,
it
should be pointed
connection with devices
in
about
results accurate to
(The larger value of x, the less is the total resistance of the potentiometer P 2 and its load. This
In the present
accomplished by locating the x
for
solving
a very matter to the results simple improve by an simultaneous linear equations that
it
is
1
Suppose, for example, x/ ,. an approximation of the answer. Let Ax t
iterative process.
xn
(1)
is
.
.
,
-
be defined by the equation
x^xp+Ax;
(III.11.F.8)
The equation Fig.
m.ll.F.5
iX,x,
=
fc,
(IU.11.F.9)
then becomes
The
single-pole single-throw switch
con
is
n
JL/
venient for testing purposes.
The coefficient boxes with values for the 4 model are
illustrated in Fig.
III.11.F.5.
double-pole double-throw switch is
set
x 4 The
Now,
according
The potentiometer P is 10,000 ohm wire wound, and R[i$ a resistance matched
to the sign a tf
to P.
R%
is
with the
can be
megohm and must be matched
if
the
the Ax/s, solve for
A6/s have only one tenth the value
we may
of the 6/s,
i.e.,
introduce a scale factor for
multiply the equation by 10 and
10Axl5 10Ax 2
,
.
.
.
,
10Ax w
.
Thus, as
of the other coefficient boxes. This
long as the accuracy is adequate to reduce the constants by a factor of 10 at each stage, we may
done by taking twice
conveniently obtain any accuracy desired. Since
easily
megohm
.
0.5
2 s
i=i
3=1
resistors,
combining them
in
as
many 0.25
%
evaluating each, and then
it is
the proper total pairs with
the reiterative process, the labor in each stage
not necessary to reset the coefficients
mainly one of calculating the errors
resistance.
The groundings in the various coefficient boxes
A^
in is
and
resetting the constants to these values.
locate the variable circuits relative to ground
and hence permit one to add by averaging the voltages fl u jc, a l2 y, and b^t through the matched resistances
Rz
.
desirable since
is
Adding by averaging voltages it permits one to obtain the
different equations simultaneously.
But
if
we
have negative coefficients and wish to average voltages, we must have a voltage corresponding
HI.11.G. Stable Automatic Multivariable
Feedback in the Linear Case It is
the
possible to usethe
sum of
same idea of minimizing
the squares of the error to provide
an automatic adjustment of the variables which will this
work
in
every case.
We
will
question in detail for there are a
consider
number
III.1LG.
STABLE AUTOMATIC MULTIVARIABLE FEEDBACK
of interesting variations,
and
discussion
trie
157
linear combinations. pair of devices for forming
directions.
The coefficient inputs can be mechanically linked
Consider again a system of simultaneous linear
so that a shaft will determine the values of single
generalizes
equations.
in
many
Of course, we must minimize
a certain coefficient in both boxes.
formed
-I
,
we may
integrate
Having
this to obtain
v%i x,
Now we
obtain
i.e., jc is
dx that 7-
in such a continuously changed
dt
We
We
way
du
= -A -A fai
show
that such a device will converge
recall that
asymptotically
For we
to the correct answer.
have
is
a positive definite quadratic form in
e
C
Let
inverse.
v is
1 bound of N-
.
.
.
.
,
positive definite,
dt
en .
we have a
Associated with the quadratic form v
matrix N. Since
l5
N has
Thus,
an
denote the reciprocal of the
Then is
In any calculating device, the x functions of the time,
more negative than -AC. Hence,
the device at
t
log
/i
must be
s
and, hence, p must be also.
if
we
start
= 0, -log ft
<
-AC*
(IIL11.G.10)
and we may conclude that
Thus,
So we see
t-idx
dt
is
Thus,
that, in general,
a device with the
will block diagram shown in Fig. IH.l I.G.I
the inner product of the gradient
dt
vector \
,, ...,d 3v
the rate vector
to.!
dxA
/&! dx,
dear
^ reach a
.
solution
we must minimize
/<,
i.e.,
we wantf _C dt
to be negative. dXj
One method of doing
-.
this is to take
=
Then Fig. ffl.lLG.l
a position of converge to
One method of realizing
the equations
solution
is
this device.
a linear to reah ze the e/s and then form is desired is a
combination of these, Thus, what
= 0,
when
the
unique.
Notice that
is
p
we have
An
dt
established the stability of
for instance arbitrary feedback,
LINEAR EQUATION SOLVERS
158
not stable except for special matrices
A
~ (a
A
as one can readily see by solving this system of
complete feedback arrangement is always relatively expensive even if the auxiliary feedback
differential
is
is
)
i}
equations.
The idea used here For
equations.
not confined to linear
is
instance, if
we
are engaged in
solving a system of equations
s
(the/
feedback circuit
is
sensitivity
around zero, while
relatively forge percentage errors can be tolerated.
(III.11.G.12)
we may
not as accurate as the original equations. It is from the above that what is desired in the
clear
may depend upon parameters not shown) let ^ f and form t
This permits a number of compromises between a completely automatic setup and a minimum of device parts. The device described
was obtained
in Section III.ll.F
such a compromise, not as an or improve existing devices.
for
TT
~ 2 oXj
and then feed
down
in the
the gradient,
x
to obtain
In
it,
/
only
The operator determines the
calculated.
and proceed as before
originally as
effort to generalize
each
in
i
and
succession
is
sign of
varies
x-l
v*i
accordingly.
= 2*i~ i
s
(IIL11.G.13)
Another compromise
is
readily obtainable.
One can use a relatively inexpensive combination
OX]
so that the machine goes
of resistances with highly amplified versions of to obtain f
the i.e.,
... at
0;^
and
Of course,
the accuracy of the device depends
upn the accuracy with which the
visually represent this to the operator who then choose the method of varying the a/s. If the matrix A is positive definite, we
will
=
equations
dx>
can use the feedback are realized.
-~
= -e
Since
On the other hand, we are permitted
a large range in percentage accuracy as far as realizing
we
see that
dx
dt
concerned. For instance, as long as
is
-
and
dt
(III.11.G.20)
fy are opposite in sign, and there
3
8
>
0,
is
a constant
Since
A is positive definite,
for (III.11.G.16)
dt will
always negative
such that
same type of exponential decay
and, in fact, the
we
is
have the
dx.
result
fi
can be established as
in the above.
Another procedure for trying to obtain feed back without the introduction of additional circuits
involving
proposed by
the
coefficients
W. A. McCool
has
been
(see Ref. Ill.ll.f).
This process consists in successively introducing
which
will
imply as in the
above that ~
the equations so that at that stage in which k is stable in the equations are present, the
system
first
m (HI.11.G.18)
k unknowns,
constant.
The k
the other
+
1
unknowns being held
equation
is
introduced to
III.11.H.
+
feedback to the k
unknown with
1
which
integration rate
STABLE MULTIVARIABLE FEEDBACK
is
a variable
until stability adjusted
obtained.
is
For
have three suppose that we
The
unknowns.
in three
equations
first
two
with gain up using an amplifier to feed back to xl and x 2
equations are set
.
)
Now if a is very small, the roots p of this equation are near to the roots of Eq. IIL11.G.23 with
exception,
instance,
159
which is near
therefore, if
a
cc0
If oj 33
33 .
small enough,
is
The
negative real parts.
all
is
one
negative,
roots will have
form of Eq.
precise
III.11.G.22 does not enter the argument, and,
hence,
we can
to the general readily generalize
case.
The above
(III.11.G.21)
and second equations is i.e., to obtain x1 and x 2 Eq. III.11.G.21 amplified can be written the error in the
first
.
probably about the simplest
is
method of adjusting the feedback to obtain One can show that, in general, some stability. type of adjustment
is
F. necessary (see also
J.
Murray, Ref. IILll.j).
m.ll.H. Stable Multivariable Feedback In view of the fact that
many
questions
can be approximately answered by analysis (III.11.G.22) If jc s
is
a constant, these equations are
stable,
i.e.,
we discuss in the present section approximations, the possibility of using a general feedback device for problems involving functions.
the rootsp of
For
=
all
have negative
experimentally
real parts.
(IIL11.G.23)
Suppose
it is
suppose that
definiteness, let us
we have a
second-order differential equation
found
,
dx
that the introduction of the third
equation will yield instability
if
xs
the same manner. Alternatively,
# 3 by
of
finite
integrating
at
any
fed back in
is
we may
rate desired,
obtain
to be solved in an interval a
<
x
<
b,
with
boundary conditions
i.e.,
(IIL11.G.24)
where a
We
is
are
=
a parameter.
integrating amplifiers
are available as well as
We then let
other components of an electronic differential III.11.G.24 is then readily realized
(III.11.H.4)
analyzer. Eq.
with variable
a.
be written Eq. G.24 can
The solution to (III.11.G.25)
for the new Thus, the characteristic equation
system
(ni.n.H.3)
electronic assuming that standard
is
makes
= p
c-p (III.11.G.26)
function.
a function}? which
which by introducing a by
smaller value to
=
is
we have an approximation or this function, we can improve
If
even a starting function
value of
a 31 a
the problem
0.
\L
/*.
We
must, then, find
gives
how
a
the
an increment dy to the depends on
Actually
it is
desirable to express this
in terms of dy. The procedures of dependence us to do this. the calculus of variations permit (In
the
interesting
special
case
in
which
LINEAR EQUATION SOLVERS
160
/
d z v\
dv
This
F pc,y,~,-4 dx
\
Euler
the
is
is
clearly
the
norm squared of
the trans
for
equation
formation
dx*/
minimizing an integral
Tfy /, f]
=
[F(x, y, y
,
/),
/), H(y, y
G(j;,
)]
(III.11.H.7)
between the spaces mentioned. Let us try to find the vectorial increment
6
f
we may
F 2 dx.)
substitute this integral for
1
by
(dy,
,
by"),
which
will
minimize p.
In the
Jo,
At
this
function
we
point,
introduce for the
=f(x) a
y
unknown
combination
linear
finite
usual notation of the calculus of variations,
we
have
=
which is supposed to approximate/and for are which o (x) and supposed to approxi a(x),
(FFy by
+ FF
>
dy
V
+ FF
,,
y
dy")
dx
a"(x)
mate/ ^)
and/"(x),
in Section III.10.H
We have seen
respectively.
how such
constructed for the interval
IT
a a could be <
x
<
n
in the
form
Let us introduce K(x)
= FFy dx and
integrate
Jd
= a(x)
the
first
term under the integral sign by
FV
- K(x) dy + FFy
parts.
dy"]
dx
We have two alternative procedures at this point. One of these
is
to realize a as a function of the
time as indicated in Section III.10.H, and then (III.11.H.9)
obtain
p by
The other
applying the necessary operations.
possibility
at this
point
is
to express
Now let
p
as a function of the coefficients in the above
We
a and realize this function. expression for
The
net
effect,
however, in each case
to
is
produce p as a function of the coefficients. We may then try to minimize p as in Section III.l l.F or G. If we have a feedback device, the
partial
derivatives
of
p
coefficients as outputs of a part
insert this constant in the coefficient of
under the integral
sign.
The net
result
is
dy the
following:
we must have
relative
to
the
of the device.
In connection with the feedback, the following operational considerations are worth noting.
We
can consider our problem as concerning an operator from an infinite dimensional vector space consisting of
form [F(x),
\y,
y
, y"]
triples
to a function
is
obtained from the usual
Let us consider
let
,-*(*)
space with elements
A, B] consisting of the functions F(x) on
obvious manner.
Next
to eliminate
of functions in the
the interval and with two extra dimensions.
norm
The purpose of this maneuver was the dy at x = 4 terms.
L 2 norm
The in
an
+
(IIL11.H.11)
and again integrate the first term of Eq. III. 1 1 .H. 10 under the integral sign by parts. The result is
III.ll.H.
STABLE MULTIVARIABLE FEEDBACK
We then let 2
.
,
and
x
to
is
purpose
C = (HH + L)] x=b
161
suppose that the
also
will
V
.._,,. under
,,.
insert this constant
the
.
,
mtegrals,gn.
Wetohave
boundary does not contain to an h
^
and for the
simplify the formulas,
same reason we
line
^
wlM
segments parallel
J non
4
ntial
restriction.
-
C2)
Let us suppose that our problem
is
to obtain a
solution of the equation
F(x, y,
on the region
=
z, p, 5, r, s, t)
(IIL11.H.15)
S, where
Let
2
p
a z
= a~
dx
and
now evident that if we let dy be any negative
It is
multiple of
subject to the fc
2
F
-IT ^a (IIL11.H.14)
<5/j
be negative. Under suitable continuity
variation in
variation in dy
is
We
have continuous partial derivatives apparent variables.
consider the equivalent of the
discussion.
imply that the actual be negative provided the
will
//
to
We
above
let
will
this
restrictions,
is
relative to the various
/a
will
boundary condition
- a) (x
(M
utilized
no matter what method
is
used to represent the functions. In the case of a linear F, the expression for V// is the well-known expression T*Ty. The basic ideas generalize with to problems in two
F2 dS J
+j
G 2 ds (HL11.H.18)
s
small.
The formulas thus obtained could be to plan a feedback,
=J
or
more
Again we must establish the dependence of du on a variation in the function z in order that a given approximate z can be improved by adding a dz
which lowers the value of
We
p.
then have the
usual notation little
difficulty
variables,
To
a problem of a region S have we Suppose
illustrate this, let us briefly consider
the following sort:
whose boundary consists of a rectifiable curve which has a continuously turning tangent except for at most a finite number of points. For in the formulas, we will suppose that S simplicity can be described in both of the following ways: and g? 2 (4 (1) There are two functions, q^(x)
defined on the interval a
<
x
^
b such that
S is
(ffl.ll.H.19)
Now if we integrate the last term of the spatial integral
by
parts,
we
the set of points (x,y) with a<x
obtain
<
t
t
^(y) and y 2 such that
defined
on an interval
c
<
(j)>
S
consists of points (x 9 y) for
y
<
d
l\(FF 6t)dydx=j(FF
which
c Jand ^(y) VaOO* These restric y tions are by no means essential. Their sole
(ffi.ll.H.20)
"
<
<
dq)dx
<x<
dy
LINEAR EQUATION SOLVERS
162
Now consider
L et
Let us choose a fixed point
WW-Wfl
(FF,)iS =
and
let
fc have the
P on
valued
at
P
the
boundary
Thencmthe
.
boundary we have
(tpdx
+ tqdy)
u
Jp,ds\ds Let
we have
P
f
Ki0 = j his
integration,
(GG, Po
of course, extends along a
the boundary whose arc length o te branc branch of
/
C
(K2(x,
y)
<Ss)
dS
(III.11.H.22)
Let
r
is s.
denote
J v
(?,* + My)
Thus, Eq.IttllJH.19 becomes
(ffl.H.H.27)
Then
(GG,fe)dj Js
j
p ds d3t)
?
= j^ + f j
We now integral sign
the
integrate
by
first
t
,
i/S
term under the
=LK40 +
parts, letting r
We
S
1
f
f f
Jp
(te)
d
^5
-[z
U*
nNow
JB
obtain
ds
K^p
Hence, /
^\
f *
Jj? I
4
3
dsl
(III.11.H.31)
(IIL11.H.24)
Thus,
III.11.H.31
Eq.
substituting
163
STABLE MULTIVARIABLE FEEDBACK
IIL11.H,
in
Now,
Eq.
for
z
=
1, 2, let
III.11.H.24
+ (FFJ 7 r dx dx
(IIL11.H.35)
f
+
dp(GG, ds
+ FF
r
dy
-K^dx- K,dx)
Then
JB
dq(GG y ds
+J
+ FF dx-K 1 dy-K^dy) t
(IIL1LH
-
\
32)
We now let
and
let
result
us integrate the
first
term by
parts,
The
^
-
is
\
^a
[FF.^ + ^F.
dx
HJdpl Jv*?>i
+ Ki = UJ
(H^
&
-
,
ds
(X,
= PJ
- K, - ^) ^]
Jo
(Hj
dy dx
_
Jv-Vi
(ffl.ll.BL36)
(III.11.H.33) Similarly,
we
Similarly,
let
if
we
put
for
i
=
1,
2
-^
and obtain
FF8 + K1
H-^-K.-^W.
+ K - K5 -K ),] dS 6
a
(III.11.H.37)
we obtain
/
1
dp[GG 9
ds
+ FF
r
P.H.H.38)
LINEAR EQUATION SOLVERS
164
Substituting Eq.
we
IIL1LH.38 for Eq. HI.11.H.34,
integral, p, itself is clearly a quadratic function
in a, and, hence, a
obtain
a
=
can be chosen
a which minimizes
of the function z
/*.
~
therefore, z
is,
at the value
The improved value This
V/u.
<x
improved value could then be used as a function
One may under
z for another improvement step.
from the
certain circumstances be able to infer
(H.- H^dx
]ip\
Jy =
>a
l
result
of a sequence of n such steps the accuracy
of the
last
approximation.
References for Chapter 11 (III.11.H.39)
We
P
have chosen a point ,7 ). Let
a.
S. M. Rock, and H. W. computer for solving linear simul
C. E. Beny, D. E. Wilcox,
Washburn,
with coordinates
taneous
"A
equations,"
Journ.
no. 4 (1946), pp. 262-72. b. E. A. and G. W.
Goldberg
AppL
Vol.
Phys.,
Brown,
"An
17,
electronic
simultaneous equation solver," JWn, AppL Phys., Vol. 19, no. 4 (1948), pp. 339-45. c.
H. Hotelling, "Some new methods in matrix calcula Am. Math. Stat., Vol. 14, no. 1 (1943), pp.
tion,"
1-34. d.
G. A. Korn,
"Stabilization
of simultaneous
equation
I.R.E. Proc., Vol. 37, no. 9 (1949), pp.
solvers,"
1000-2.
-HZ and
let
S(x,y) denote the
Swith??
^x,
I
(iii.ii.H.4i)
of points
set
(rj,
e.
in
DC
siderations.
dS
R.
Circuit
Stability
Con
Washington, D.C., 1949. Naval Research
M.
R.
machine,"
(III.11.H.42)
one readily shows that
Stabili
Lab., Report No. 3533, g.
Now
on
of
Linear Algebraic Equations: 5
"Discussion
zation of simultaneous equation solvers, by G. A. Korn," I.R.E. Proc. Vol. 38, no. 5 (1950), p. 514. W. A. McCool. Simultaneous Analog Solution t
f.
Let
G. A. Korn and L. A. Zadeh,
Mallock,
"An
A841 (May, A. Many and S. Meiboom, Vol. 140, no.
h.
electrical
calculating
Roy. Soc. (London) Proc., Series A, 1933), pp. 457-83. "An
electrical
network for
determining the eigenvalues and eigenvectors of a real symmetric matrix," Rev. Sci. Instr., Vol. 18,
and i.
-- = tf
i
#2 when y
=
#
=
no. 11 (1947), pp. 831-36. T. D. and F. W.
Morgan
Crawford,
"Time
saving
computing instruments designed for spectroscopic Oil Gas Journ., Vol. 43 analysis," 26, 1944),
<
(Aug.
pp. 100-5.
3V/* -r1-
= #4
3
when x
j.
k.
and
C. F.J.J Murray, "Linear equation solvers," Quart. Appl Math., Vol. 7, no. 3 (1948), pp. 263-74. 0. S.Puckle. Time Bases. New York, John Wiley &
Sons, 1951. 1.
T. E.
W. Schuman,
method
"The
principles of
a mechanical
for
calculating regression equations and multiple correlation coefficients and for the solution of
Thus,
if
our increment dz
-V//, we
see
that
dp
is
is
simultaneous linear Philos. Ma?., Series 7 equations," Vol. 29, no. 194 (March, 258-73. 1940),
proportional to
negative
by Eq.
pp.
m.
III.11.H.39. n.
Thus, given a function z which
an approxi mate solution to our we can problem, improve z by adding ccV/j where V/j is given III.11.H.42 and
a
is
is
sufficiently
by Eq. small. The
W. W.
Soroka.
Analog Methods in Computation and Simulation. New York, McGraw-Hill Book Co.; 1954.
R.
M. Walker,
"An
analog computer for the solution
of linear simultaneous 37, no. 12 (1949),
o.
equations,"
I.R.E. Proc.
Vol
pp. 1467-73. J. B. Wilbur, "The mechanical solution of simul taneous equations," Frank. Inst. Jn., Vol. 222 no 6 (December, 1936), pp. 715-24.
Chapter 12
HARMONIC ANALYZERS AND POLYNOMIAL EQUATION SOLVERS
TC.12.A. Introduction
There
Two
problems of considerable interest in the development of mathematical machines are the polynomial equation in one These problems are related rather closely by De Moivre s theorem. De Moivre s
ficients are
of a
theorem z
n
=
r
that
states
if
z
=
n
We
(cosw0+7smn0).
considering
various
the
r(cos0 will
harmonic
+; sin0),
,
"synthesis,"
A
ways to
solve
unknown and characteristic
equation
of
a
one
methods of Solving the
matrix
is
an
to be found.
when
the
called a
analyzers
of solving important special case of the problem a polynomial equation.
Harmonic
is
given
A
in
is
harmonic analyzer. device which sums the series is called a is
harmonic synthesizer.
polynomial equations
the complex plane. representing
given and the function
begin by
These can be used in certain
are related to various
the problem in which the coef
is
device which finds the coefficients
function
A large number of devices have been set up for
which are concerned with the Fourier representa tion of functions.
"analysis"
,
fl
solution
with
associated
the problem of
is
where /is given and the Fourier coefficients n b n need to be found; the other problem,
Fourier representation of a function and the
unknown.
two problems
are
Fourier series: one
They can be roughly
these purposes.
is
classified
two
In the first type the function f(t) types. as a function of a continuous represented
into
variable in
some manner and the
sines
and
cosines are also represented as functions of this variable.
Then
the problem
is
evaluating
in analyzers have been developed
various commercially available forms in Europe. (See F. A. Willers, Ref. III.12.aa, and
zur Capellen, Ref.
W. Meyer
III.12.r.)
HL12.B. Harmonic Analysis and Synthesis
Both Fourier
series
In this method,
if
all
that
is
desired
is
the
numerical value of the coefficients, they can be obtained successively. Similarly the Fourier transform problem involves the evaluation of
and the Fourier transform
are widely used in applied mathematics. The Fourier series of a function in the interval <>
f(x)
x C
^ 277 is given by fl
Many devices of this type have been constructed;
+ a l(COS *
later. they will be discussed
In the other type of device the values of f(f) sines and cosines are considered only
and of the
where
at a discrete set of points.
basis for these devices "finite
(III.12.B.3)
Fourier
shall give this
analysis."
of a function.
The mathematical
usually referred to as
In Section III.12.C
we
mathematical discussion and refer
to devices based
devices based
is
on
it.
We
shall return later to
on the continuous representation
HARMONIC ANALYZERS; POLYNOMIAL EQUATION SOLVERS
166
HL12.C.
Finite Fourier Analysis
Let us consider
now
by the
the simplest numerical
=
approximation to an integral. Let x^, p^n for In and y v -/(*,). The points x 9 1, p divide the interval x ITT into In equal
=
.
.
.
for
sin(?ffJfl)<
we
^r
have for p
,
= 1 + 21 J + cos ^ =1 cosgfl)
--
<
From the obvious approximation integral we obtain
subintervals.
for the
formula
addition
Substituting Eq. III.12.C.7 in Eq. III.12.C.5,
= 1 + sin
(w
__
i)0
_
sjn
nfl
ig
+
cos nO
Smt
(ffl.12.Cl)
(HL12.Q8)
= (!/")
2
l,...,ii)
Now
4
(Hi:i2.C2)
(HL12.C.3) 31=1
(Note that 6 B
=
= sin (p -
n0
sin
for
= pa and sinp- = 0.)
nx f
=
(III.12.C.10)
r>
Now form the expression
+ irW cos nx + r
1
&
sin
+ 2? [(cospc.Xcosgjc,) 0=1
+ (cos TI^XCOS wc
=
nxr)
r)
Thus
"f />
+ cos nfl =
nfl
^
.
M
0,
and
(III.12.C.11)
^=
r,
to Eq. III.12.C.4 Eqs. IIL12.C11 and 12 we obtain
substitute
= y =/(Xf
nxr)
r
)
(III.12.C13) In (III
12
C 4)
tller
V and ^
W
if
rds>
We
calculate the coefficients
,
n
Am = 1 +
63,
by the above approximation formulas,
get a trigonometric polynomial Values For
^
?
(
>
reasons this
+ cos jifo - x ) f
(III.12.C,5)
A*. Let
= x9 - x
^ r we have
=
r
wliere
For p
p
r)
= (l/2n
We now evaluate
If
when we go back
Consequently,
]
6 n sin
+ cosn(*,-x
0.
from Eq. III.12.C5 we have
and
+ (sin roOCsin nx
-cos
on^
for
Now r
(IIL12.C.6)
not zero, and we have
show
which takes convergence
in general, a desirable procedure situati n in which is continuous
is,
tlie
y
^ aboj
that
if
=
fl
we
formal calculations essentially
consider the system of equations
A + 2,
~
,
o
fl
(III.12.C7)
(IIL12.C.14)
IIL12.D.
as a system of In equations
%
0i,
,
a n &!,...,
Iv
>
1?
on the In
HARMONIC ANALYZERS
quantities
then the system
167
IV.12.D. Harmonic Analyzers It
from the above discussion that
clear
is
what
desired
is
a device to produce linear
is
combinations (III.12.C.15)
cos gx,
te=s
~
s
t
"
(III.12.C16)
,
L
c
Such
linear combinations
can be obtained by
""
"
A
using the components previously described. (III.12.C17)
has
Bi
number of
constructed,
.
,
large
matrix which
Eq. III.12.C.14.
is
inverse to the matrix of
devices of this type have been
with
beginning
analyzer credited
harmonic
the
Lord Kelvin
to
(see
Ref.
Consequently, each of these III.12.Z).
is
systems
if
nonsingular.
This has the very important consequence that a device is constructed to the various yield
combinations
(IIL12.C.18)
then
it
can be used
coefficients of
either to obtain the Fourier
a given function
or,
given the
coefficients, to obtain the value
Fourier
do vary
can readily be taken The summation from 1 to In can be
slightly,
care of.)
of the
(The formulas
function at the specified points.
but
Fig.m.l2,D.l
this
evaluated by sums from
1
For
to n.
instance,
Let us consider the harmonic analyzer of
A. A. Michelson and III.12.S).
J
aq
3=71+1
sin qx = 2 a q+ n(sin ((n
arrangement
5=1
prrfn
2a
For
this
is
S.
W.
we have
such that
Stratton (see Ref.
Fig.
if
we
by a continuous variable
III.12JXL This replace
x,
a
the input
we get the sums
(IIL12.D.3)
( c=l
P.12.D.4)
By adding are readily adapted Naturally these formulas
whose Fourier
arithmetical or
This fact
also H.Lipson and punched card machines. (See C. A. Beevers, Ref. III.12.rn, and H. Shimizu,
function.
to calculations based
P.
J.
Elsey,
on ordinary
and D. McLachlan,
Jr.,
Ref. III.12.y.)
these to
is
a
,
we
t
t
f
.
evaluate the function
coefficients are the rf/s
and
e
a
s.
used to draw a complete graph of the
The
addition and the multiplication
can be done in various standard ways and the sine
and cosine obtained as
in Section III.10.B.
HARMONIC ANALYZERS; POLYNOMIAL EQUATION SOLVERS
168
In the Michelson-Stratton instrument, the
x
input pirjn or
by q
is
an
is
angle.
obtained by a gear
The
multiplication
The
ratio.
cosine
is
Suppose we have a cylinder which can around its axis. (SeeFig.III.12.DA) On one side of the cylinder, we have two bands.
ideas.
rotate
These bands are fastened cylinder, is
wrapped around the cylinder and
partly
down
extends
vertically to the
Each band and
spring.
On
input.
one end to the
at
end to springs. The band
at the other
corresponding
spring corresponds to
the opposite side of the cylinder
an
is
a
band and spring
similar arrangement of single
for the output.
We
are suppose that the two input springs
We suppose that extended an amount lv By
these are normally
similar.
HL12.D.2
Fig.
force exerted
obtained by an eccentric.
Hooke
s
law, the
by each of these springs
is
k^
Notice that in Fig.
HL12JX2
=
co
r(cos a)
+
/(cos
/?)
2 2\l/2
/
= r(cosa) + I
I
+
2
-
/ll-(sin a)-2
r(cos a)
+
{(l
- (sin
]
1/2 2
a)
-
^)
l]
+ r(cos a)
Fig.
(III.12.D.5)
Thus, the percentage error which results when we consider the other end of the eccentric to have a
harmonic motion
The
about 100r/2/.
is
multiplication
by the constant dq
where
^
the force necessary to extend the
is
spring a unit length.
is
obtained by means of a simple similar triangle
the cylinder
is
in
moments shows
Now
suppose
input springs
The
that 2
z.
amounts x
moment
2
be the corre Since
we move
=
fc /
2 2.
the other end of the
down amounts x and y respectively.
The input
-z
springs will be extended
and y
equation
and the output spring be extended an
is fixed will
will
- z,
still
respectively.
The
be
addition that
it is
is used is very a mechanical counterpart to
voltage averaging which we have previously Let us describe it in the case of two
discussed.
addends since
/
1 1
cylinder will rotate
amount Fig. ni.l2,D.3
the
/
equilibrium, a consideration of
whose lower end
interesting since
Let k z and
sponding quantities for the output spring.
(see Fig. 111.12.0,3).
The method of
m,12.D.4
this case contains the essential
(III.12.D.6)
or
CONTINUOUS ANALYZERS AND SYNTHESIZERS
III.12.E.
should be clear how any number of inputs can be introduced into such a device. The output is It
then a linear displacement. When the machine is used as a synthesizer,
169
With such an approximation how both can be handled simultaneously. of a function ever, a continuous these problems.
If>
representation
i.e.,
graph a function y whose Fourier coefficients are given, the output appears as a displacement
to
is
used, one generally finds different devices for
the two purposes.
of a pencil above a horizontal line which corre sponds to the x axis. This pencil presses against
a piece of paper on a vertical drawing board. above, in this case we have a
As mentioned
continuous input x and the drawing board is with this input. continuously displaced to the left
Thus while the
pencil remains
vertical plane, the
drawing board
graph of
in
the same
y appears on
the Fig. ffl.!2.E,l
(see Fig. III.12.D.5).
A
harmonic analyzer can be readily con from the following four components
structed
(see Fig. III.12.E.1):
Function generator
A resolver to represent sin nx or cos nx A multiplier An integrator
XAXIS If
Fig.m.l2.D.5
A modern development by
S. L.
of the above
one has a ball-cage-type integrator which
permits
is
one to represent
given
Brown (see Ref. IILlld). The schematic is
diagram
the same as Fig. IIL12.D.1, thepr/n is
an angle, the multiplication by q is again so that/^/n appears as an again by a gear box,
input
one can vary Fig.
III.12.E.1 as in Fig. III.12.E.1
A large number of harmonic analyzers have been constructed in this way.
angle.
The cosines and sines of this angle are obtained on the x axis of a line by taking projections an angle a with x axis. makes which segment
TmVis
readily accomplished
(see Section HI.10.B).
The
by a Scotch yoke by dq
multiplication
of the segment. obtained by varying the length chain. endless an of means Addition is
is
by
are described Other analyzers and synthesizers D. C. Miller, and Ref. IIL12.k, W. F. Kranz, by
Ref. III.12.t.
m.llE.
We
Fig.ffl.llE.2
Continuous Analyzers and Synthesizers
two have mentioned that there are
putational problems
involved in Fourier
com series,
and synthesis, and we have discussed analysis for devices which use a finite approximation
The harmonic
analyzers
of the above type are
obtain the coefficients one simple because they to have the at a time. If, however, one wants
continuous representation of afunction by
means
HARMONIC ANALYZERS; POLYNOMIAL EQUATION SOLVERS
170
of
its
Fourier
series,
tion of each sine
sum
we must have
a representa
and cosine which appears
in the
and H. W. Smith Hastings and
(Ref.
Meade
E.
J.
and A. E.
IIL12.c)
(Ref. III.12.i).
(see Fig. III.12.E.3).
In a typical device of this sort the resolver for
producing
sine
and cosine
is
an
electrical
generator which produces sincof and coswf provided the shaft of the generator is driven at a rate of co/277 revolutions per unit time.
The
shafts
IEL12.F. Polynomial Representation by
Harmonic
Devices Analyzers; Zeros; Special
one has a harmonic synthesizer, one can and imaginary parts of the
If
represent the real
values of a polynomial as follows: Let
=
z
re
(III.12.F.1)
Then P(z)
=
+
=a +
+
o
.
+
.
.
...
We can suppose
s
29)
ft
+
fl
+
a ra r (sinn0)]
(III.12.F.2)
convenient that
%a
as
is
r (cOS 710)] a
n
.
ly
.
.
,
a n are real. (The complex case can be taken care
jw
of bya slight extension of our present discussion.) Then
Fig.
z)
m,12.E.3
a
of these generators are driven from the x shaft
through gear boxes to produce the appropriate The output voltages of the resolvers
+
voltages.
can be multiplied by a constant factor by means of a potentiometer and added by any standard method.
Mechanical devices of
also possible.
this
type
are
Various timing devices can be
connected to the x shaft
One might mention
also at this point devices
Many of these devices are analogous to the continuous harmonic analyzers except for the oo fact that the integration will have to be from some approximation. Photoelectric means for obtaining Fourier
to oo or
transforms are discussed by R. Fiirth and R.
W.
Devices for obtaining correlation functions similar to those of the principles
work on
A
mechanical continuous harmonic synthesizers. is discussed by H. R. Seiwell (Ref.
device
devices for obtaining correla
by
and imaginary parts of P(z\
and they can be obtained from the synthesizer provided
we
use as coefficients
a^.
/ are represented by the expression as indicated, and these can be used for various purposes connected with the polynomial. We give a tentative block diagram for such a device in Fig.III.llF.l.
Such a device may be
F. E. Brooks,
Jr.,
realized in
many
ways.
For example, in a mechanical device we generate the powers of r
each
Pringle (see Ref. Ill.llg).
tion functions are discussed
these are real
(I
There are a variety of devices in which St and
also.
for obtaining the Fourier transform of a function.
III. 12.x). Electrical
Then
n
a n r (sin n0)
r,
in turn,
by repeated multiplication. Then is multiplied by the constant and
multiplied by the radial input of the resolver. The quantity 6 is also entered in the form of a rotation. Then by the use of gears, multiples of
are introduced into the angular inputs for the resolvers.
The
Cartesian
outputs
of
resolvers will give the terms c/^cosfcfl)
these
and
IIL12.F.
POLYNOMIALS; ZEROS; SPECIAL DEVICES
fl/(sin k&), and by addition one can obtain the desired result.
(For mechanical multiplication
by a constant,
see Section III.2.C; for resolvers,
see Section IIL10.B).
However, electromechanical methods can be used,
Here the powers of
r are
also
so that this effect can be ignored.
-^
~T
Notice then
no gears are used, but
that in this device
*)
171
just
.
_I_
\
obtained as
voltages determined by potentiometers on a
common
shaft provided suitable amplifiers are
utilized.
The resolver part of such a device could consist of sine and cosine potentiometers linked by shafts or we could use having the correct gear ratio, some form of generator resolvers in which the field is
ak r
determined by the direct-current voltages
k .
Alternatively,
sine
and cosine poten
tiometers can be used from a
common
shaft
to introduce the constants although it is necessary a later. The full device can be described as
g SHAFT
j
s
shown
in Fig. III.12.F.2.
An individual box can
be described as shown in Fig. It
should be remembered that amplifiers in
electromechanical devices normally reverse the the output. sign of III.12.F.3,
two
In most cases, as in Fig.
amplifiers
Fig.m.l2.F3
III.12.F.3.
are used in succession
the usual electromechanical components of a differential analyzer, It is also
possible
representation
in
to
have a purely
which
electrical
multiplication
is
HARMONIC ANALYZERS; POLYNOMIAL EQUATION SOLVERS
172
obtained by electronic and the sine multipliers,
and cosine functions are generated by These devices then have the
and
are r and B
oscillators.
effect that
inputs
the outputs are $(P(z)) and
J(P(z)) (see Fig. III.12.F.4). These devices can be used for various purposes. For instance, one
might use them that z
have
this is to
moves over
and
6
vary so
the complex plane in an
This
increasing spiral.
r
is
obtained by letting r
W. Bubb,
S. Fifer (Ref. III.12.b), F.
Jr. (Ref.
Ill.llf), B. 0. Marshall, Jr. (Ref. III.12.q),
M. G. Scherberg and
and
Riordan (Ref.
F.
J.
III.12.W).
= 0.
to locate the roots of P(z)
One way of doing
Examples of electromechanical polynomial equation solvers are given by L. Bauer and
Examples of
electrical
solvers are discussed in
H. Glubrecht (Ref.
polynomial equation
W. Bader
III.12.h),
(Ref. III.12.a),
and L. lofgren
(Ref. m.!2.n).
Examples of mechanical polynomial equation solvers are given
by
S.
Brown and
L.
L. L.
Wheeler (Ref. IIL12.e). IH.12.G. The Representation of the Plane
In Part Four to
Fig, IH.12.F.4
it
will
be shown that
Complex
it is
possible
represent the complex logarithm of a rational
function of a complex variable by an electrical
vary far more slowly than set
6.
An
arrangement is and are up which indicates when both
^
/
zero and records the corresponding values of r and 6. It has been proposed to use this method to represent the roots of a polynomial
As
cathode-ray tube.
beam
ray
is
and
r
on a
vary, the cathode-
positioned according to the real and
imaginary parts of
z.
However,
its
intensity
/ are zero.
There
are,
Suppose, for instance, one held vary.
not treat
r fixed
and
it
The
device, though, is
components, and thus we do computer, but as a
as a continuous
true "analog,"
which
will
be discussed under the
heading of analogs. IH.12.H. Characteristic Equations
There
however, other methods by which such a device can be used to locate the zeros.
in
including solving
a polynomial equation. not constructed of
is
normally kept below cutoff except when both 9t
and
analog based on a plane conductor. This can be used for a variety of purposes,
let 6
There are many ways of realizing devices this can be done very effectively.
is,
of course, an enormous
number of
problems in which one wishes to find the characteristic roots A of a matrix. If the matrix
A
has elements a iit the problem is to find the values of A for which the determinant
which
Suppose as 6 goes from of 3t and
J
to
2?r,
we use the values
to plot the values of P(z) in the
complex plane.
=
This can be done, of course,
with standard recording equipment with the available from a continuous accuracy
maximum
computing
somewhat
The
device, less
or
it
may be done
accuracy on a cathode-ray tube.
plot of the values of P(z) will consist of a
closed curve normally having n loops. The this curve encircles the origin is,
number of times
of course, the number of zeros of P(z) in the circle of radius r. Notice that r will be the
modulus of a root when the origin.
this
(III.12.H.1)
with
plot passes through
is a polynomial of the w th degree, one evaluates the coefficients, it can be
vanishes. This
and
if
solved by any device for finding the roots of the
However,
equation.
the
evaluation
of the
of the characteristic equation is a complicated matter which requires considerable coefficients
analysis if
E. Saibel
it is
to be
and W.
J.
done
efficiently.
(See also
Berger, Ref. III.12.V.)
CHARACTERISTIC EQUATIONS
III.12.H.
There
however, a certain
is,
with
difficulty
evaluating the roots of a polynomial equation when the coefficients are given.
for
Consider,
example,
the
Tchebycheff
polynomial
a linear combination of
is
these,
Now suppose the system is stable, i.e., the real parts
of the characteristic roots
positive.
T n+l cos (n(arc cos
TM (x) -
and every solution
173
If
we
not
are
I,
consider a particular function in
the solution
x))
(III.12.H.2)
(III.12.H.3)
-2
(III.12.H.7) (III.12.H.4)
we
see that if the
ar
are negative or zero, the
s
terms in which ar
is
more negative
in this
expansion will become negligible first. If there are only one or two terms in which a r has its
maximum
value, then eventually the solution
practically of only these terms. The decay of the maxima of these factors with time is a measure of a and the periodic part indicates
will consist
.
(See also A. S. Householder, Ref. IIL12.J,
C. Lanczos, Ref.
and
III.12.1.)
Tw say, of the order of 20.
Suppose n is large in
of the coefficients of the highest powers present will exceed one. Consequently, a scale
Many
would have
to be chosen with
On
greater than one.
of the polynomial be
<,
2~ n+1
~2
maximum
itself
for
-1
6
Our
scale has
10"
value
the other hand, the value
.
^
x
<
would
1
maximum
On such a scale the difference maximum value of the polynomial
and zero would be
than about one part in a million and undetectable by most methods. This less
example has been used by Goldstine and von
Neumann to
evaluate the accuracy requirements
for digital methods for
soMng
used to find the characteristic roots of a stable matrix for which a
least negative.
is
Further
more, the result can be improved by a repetition using a new set of initial conditions determined
by the solution
first
Note
obtained.
that the
magnitudes of the solutions with least decay can be made to yield the information
relative
exceeding one.
between the
Thus, a machine for solving the system of differential equations, Eq. III.12.H.5, can be
characteristic
concerning the characteristic vectors associated with the roots for which a is least negative. Actually, since the purpose of a repetition
to
conditions multiples of the values of the
xf which
occur after a long tune in the
solution.
Essentially this
equations.
is
emphasize the terms which decay least, we can begin our second solution by taking as initial
is
continuing the
first first
solution
however, certain methods which
with a change of scale. There is a mathematical device which can be
permit one to find the characteristic vectors of a matrix directly. For instance, one may consider
used to vary the convergence in the above process
There
are,
or indeed to handle the unstable case when
the system of differential equations necessary. Suppose x^t),
(IIL12.H.5) for
some
arbitrary initial conditions clt
.
.
.
,
of Eq. III.12.H.5, and
let
.
.
.
,
yjf)
x n (t) is a solution
= 4*xtf).
Then
cn ,
=
This is a homogeneous system jc/0) Cj. of equations, and if Af is a characteristic root of r is the corre the matrix and if X[ \ ,
i.e.,
.
.
.
X%>
sponding characteristic vector of the matrix, a solution of the system
is
given by
(HL12.H.8)
HARMONIC ANALYZERS; POLYNOMIAL EQUATION SOLVERS
174
Thus, the j
-
satisfy
3
an equation which
differs
from Eq, H.5 by the addition of a to the diagonal The addition, then, of a to the coefficients. diagonal terms in the equation is equivalent to adding a to the real part of each root. If we are
a given an unstable equation, we can add for the a to yield stability negative adequate Actually from the computational point of
result.
view, the
most
desirable situation
is
one in which
Under
2 which has the remaining charac
Choose 72-2 6 35
.
.
.
,
en
= 3,
for ;
determine /2 the
.
,
.
.
,
e2
2
.
s
2
x
i
>f
have been determined, the
e s
=
vectors
and e y 2 = = %, / = V* and / = d^. (When n so that e /,
^=
Let
n. .
.
and
independent
linearly
such that ej .
=
Let el
roots of A.
teristic
we can find a matrix
these assumptions
of order n
/
s
are
determined.) For an arbitrary vector x
the least negative real part of the root is zero. This can also be obtained by the introduction of
a suitable a along the diagonal. Thus, the above process will yield, one or two
characteristic values of
in general,
a matrix A.
We can by solving a homogeneous equation with determinant zero find the corresponding charac the teristic vectors, or our process for finding characteristic roots
and Let
9? 2
A!
,
characteristic vectors for
[THEOREM
III.12.H.1. is
If
A
is
Thus,
if A is
s.
We
n then
,
if
A
is
suppose that
^
\A
y2
cp z
A2 ^2
(III.12.H.14)
= 0. A/| affect
the transpose of A,
and
= Vi
A% =
that is
the
where aik
=
Let
/,..
order matrix of the a^.
transposed conjugate of A. If A is a characteristic root of A, A is a characteristic root of A*.]
On
.
obtain from Eq. III.12.H.11
a characteristic
- A/I = 0. Taking conjugates shows - 17| = or |,4* - A/j = where A* We
we
Ae l
we have the determinant
the determinant. Thus,
zero.
.
Similarly,
a characteristic
Transposing rows and columns does not
\A
.
=
one for A*.
Proof: For any matrix A,
\A
3,
respectively.
root of A, I
root, then
= ;
A*,
the transposed conjugate of A, corresponding to ,
if
t
denote the corresponding vectors for A.
\ and A 2
note that
we have obtained
and A 2 which are not equal. Let
yt and y 2 be
coordinate system determined by the e first
vectors also. yield the
may
for definiteness that
Suppose two roots,
obtain the matrix Eq. III.12.H.11 can be used to of the transformation corresponding to A in the
\A
- A/I =
(A x
5
denote the n
Then
-2
the determinant
- A)(A - A)|B - A/I 2
are not
(III.12.H.15)
the other hand,
Thus, the characteristic roots of B are the n
remaining characteristic roots of A.
4*y,2)
this
that ^2)
(III.
Since A x
^A
Similarly that
2
2,
this is ip
only possible
= 0, We
choose
if
(p l
^
process will yield
On
l*%
and
it
roots.
should be pointed out
is
and
The
difficult cases are
the quantity is not zero but differs
so
2
^2
i=-
0.
the type of condition which
ten ds to introduce difficulty in an actual machine process.
=
all
^ ^^
we assumed
Unfortunately this
12.H.9)
-ip 2
the other hand,
2
Repeating
those in which
from zero only
by an amount comparable with the the machine.
If
^ ^=
Jordan normal form with a
0, 1
tolerances in
the matrix has a
below the diagonal.
III.12.H.
CHARACTERISTIC EQUATIONS
The above method of
locating the character roots can be used with regular commercial
istic
differential
analyzer equipment.
Also, special
devices have been constructed for this See, for example, L. A. Lusternik
Prokhorov
A
inductance
and AQ.
is
connected between each node
Let (-1) 1/2 =/.
Then
the
current
equations become
purpose.
and A. M.
L
175
j(a u
-
a>
(see Ref, IILllo).
system for using a passive network for
-
finding the characteristic roots of a matrix has
l/Lco>2
-f
.
.
.
been developed by A.
Many and S. Meiboom Their method is based on
(see Ref. III.12.p).
obtaining a passive network which resonates, i.e., appears to have zero impedance at the characteristic
elements,
(III.12.H.17)
frequencies.
network
This
consists
of reactive
purely inductances and capacitances. For
i.e.,
If
we
divide these
see that
reasons of expense, capacitances are preferred, and off-diagonal elements are represented
we have
by ;co and
let
1
=
2
1/Lco
,
we
represented the system
purely
by condensers. However,
in order to
do
voltage nodes in the network are doubled.
has 2n
The
+
1
nodes,
A_ n A_ n+l
relative to
,
be
circuit is to
x corresponds t
A
,
jt
f
An
.
.
.
9
A Q9 A l9
A_
voltage,
,
is
4.
that
way
voltage
t
to
a
e
i i
+
a nz e z
^ and A_
{
has
equation (IIL12.H.16)
*<
also the equation obtained 1.
If
a it
.
.
.
+ (0 B - A)en = IJjco (HI.12.H.18)
(Note that the quantities for each individual
+ **.=
multiplying both sides by
+
be
unknown
the
i.e.,
.
.
,
One
is
% which are adjusted
problem are now completely
represented by capacities.)
Now we
and
realized
{
.
in such a
to a voltage e i of
d*l+... is
up
A$ or ground, the
the negative of the
voltage
set
the
this,
can
and obtain
solve, say, for el
by
=I
positive,
we connect A and A^ by a conductance of this value and we do the same with A^ and A_jt On
(III.12.H.19)
t
the other hand,
with at
A
that
For
if
a^
is
negative,
we connect A
A_ and we connect A^ with A and A_ we locate sources which
.
s
i9
i
/^
i9
and
/_,-
t
Finally are such
are always negatives of each other.
suitable current sources, for example, trans
former output sources, this can be accomplished by using two terminals of the source. These transformers can
all
be driven by a voltage source
obtain the characteristic roots of a real
symmetric matrix, in the manner used by
and Meiboom, the matrix
frequencies for which A
= IjLco
roots of the matrix. Here A
we have
/B . In
= 0.
must be positive, but
already insured this
when we added a
constant to the diagonal terms in order to obtain a matrix which can be realized by capacitative
A
= (a
)
i}
is
Many
realized
With actual matrices of
finite
g, the voltage
lumped at the resonant points and the current decreases to a minimum. But this is not response
used.
is
At resonance
the input currents and the
sharply detected by an oscilloscope.
is
,
yields A(A)
diagonal elements of the matrix (see below), but
term
.
2
voltages are in phase,
to the characteristic simply added no difficulty. In addition, an
roots and offers
.
These are the resonance frequencies of the matrix, and the values of A are the characteristic
by a capacitator network as indicated above. It may be necessary to add a constant term to the this
.
the theoretical case of no resistance, there are
elements.
with variable frequency.
To
where /is a linear combination of /i,
The
difficulties
and these points can be
with finding the characteristic
roots of a matrix are often associated with the
HARMONIC ANALYZERS; POLYNOMIAL EQUATION SOLVERS
176
situation in
possibilities
12 References for Chapter
which roots coincide or approxi There are discussions of these
mately coincide.
a.
Although these
in the literature.
discussions are mainly for digital methods, the
M.
(See also
J.
R. Hestenes, and
There are
Karush, Ref,
digital processes
One
indicate the error in the analog process.
c.
roots do not coincide) the
Newton
(if
process to
New
"perturbation process"
has been
developed which is, in general, applicable to
this
Pages 3 1-36. F. E. Brooks,
S.
L.
Brown,
"A
Inst. Jn., Vol. 228,
pp. 675-94. S. L. Brown
and L. L. Wheeler, method for graphical solution of
no. 6 (1939),
mechanical
"A
polynomials,"
W. Bubb, Jr., "Circuit for generating polynomials and finding their zeros," I.R.E. Proc., Vol. 39, no. 12
F.
R. Furth and R.
W.
fourier
"A
Pringle, photoelectric Series 7, Vol. 37, no. Philos.
Mag.,
h.
A. Glubrecht,
"Elektrisches
ungen Hoheren
-A
Vol. 2, no. i,
=
computer for Vol. 23, no. 3
264
(1946), pp. 1-13.
the equation
1
Rechengerat fur Gleich-
Zeit.
Grades,"
Angew. Physik,
filr
(1950), pp. 1-8.
A. E. Hastings and
E. Meade,
J.
computing correlation
(III.12.H.20)
flffl
"A
Instr.,
mechanical harmonic synthesizer-
Franklin
transformer,"
moment
Rev. Sci.
functions,"
(1951), pp. 1556-61. g.
situation.
Consider for a
and H. W. Smith,
Jr.,
March
Special Devices Center,
York, Reeves Instrument Corp.
Franklin Inst. Jn., Vol. 231, no. 3 (1941) pp. 223-43. f.
obtain the roots more accurately.
In physics a
Navy
15-16, 1951.
analyzer,"
e.
One can then use
solution of polynomial
"The
(1952), pp. 121-26. d.
roots of the characteristic equation
are given approximately.
S. Fifer,
of the U.S.
correlation
can, of course, consider the analog process as one
by which the
Bauer and
on the REAC," in Project Cyclone under sponsorship Symposium Ion REAC Techniques,
which can be used
an approximate solution obtained by to analog methods. These also can be used
Zeit.
Wege,"
equations
III.12.ii.)
to improve
von Polynomgleichungen auf Angew. Math. Mech. Vol.
"Auflosung
30 (1950), pp. 289-91. b. L.
B. Rosser, C. Lanczos,
W.
Bader,
Elektrishchen
theoretical difficulty applies to analog processes
as well.
W.
functions,"
device for
"A
Rev. Sci. Instr.,
Vol. 23, no. 7 (1952), pp. 347-49. OOI
j.
A.
Householder. Principles of Numerical Analysis, York, McGraw-Hill Book Co., 1953. Pages 197-200, 223-25. S.
New Eq. III.12.H.20 can be considered as determining A as a function of #12 %, %, 31 c 32 if we regard ,
k. F.
W. Kranz,
,
analyzer,"
A1? A 2 and A 3 as fixed. These a ,
be
small,
fly
=
We
and A
is
s are supposed to one of three functions which at
have respectively the values
expand
Eq. III.12.H.20
/1
1?
A 23
pp. 245-62. L C. Lanczos,
1952.
and obtain
mechanical
synthesizer
Jn., Vol. 204, no.
"Introduction,"
and
2 (1927),
in Tables of Chebyshev
Sn(x) and Cn(x). Washington,
D.C., National Bureau of Standards, Applied Math.
Polynomials, ^3-
"A
Franklin Inst.
Series 9.
m. H. Lipson and C. A. Beevers, "An improved numeri cal method of two-dimensional fourier synthesis for crystals,"
Phys. Soc. Proc., Vol. 48, Part
5,
no. 268
(Sept. 1936), pp. 772-80. n. L.
Lofgren,
algebraic
"Analog
equations,"
computers for the roots of I.R.E. Proc., Vol. 41, no. 7
(1953), pp. 907-13. o.
(III.12.H.21)
If the fly are small, the function which at fly
has the value Ax
is
=
L A. Lusternik and A.
M. Prokhorov, "The deter mination of eigenvalues and eigenfunctions of certain operators by means of a recurrent circuit," Rendus (Doklady) de
I
Comptes Academie des Sciences de
rU.R.S.S., Vol. 55, no. 7 (1947), pp. 575-78.
given by p.
Many and S. Meiboom,
"An
electrical
network for
determining the eigenvalues and eigenvectors of a real symmetric matrix," Rev. Sci. Instr., Vol. 18, no. 11
(III.12.H.22)
q.
Eq. III.12.H.22 generalizes to determinants of nth order and permits a
(1947), pp. 831-36. B. 0. Marshall, Jr.,
roots of
"The
polynomials,"
electronic
isograph for
Jour. Appl. Phys., Vol. 21,
no. 4 (1950), pp. 307-12.
relatively straightforward
way of improving an approximate solution to a characteristic value problem.
A.
r.
W. Meyer zurCapellen. Leipzig,
Akademische
Pages 273-88.
Mathematische Instrumente. Verlagsgesellschaft,
1944.
s.
A. A. Michelson and
monic t.
W. Stratton, "A new har Jour. ScL, Series 4, Vol. 5.
S.
Am.
analyzer,"
no. 25 (1898), pp. 1-13. D. C. Miller, "The henrici harmonic analyzer and devices for extending and facilitating its use," Franklin Inst. Jn., Vol. 182, no. 3 (1916), pp. 285-322.
M. R. Hestenes, and W. of close eigenvalues of a real
u. J. B. Rosser, C. Lanczos,
Karush,
"The
separation
National Bureau of Standards
symmetric matrix," Jour, of Research, Vol.
47, no.
4
expansions,"
7,
w.
7,
no. 41 (1953), pp.
x
H R
.
Seiwell
"A
new mechanical
+
autocorrelator,
to. Sd. Instr., Vol. 21, no. 5 (1950), pp. 481-84.^ P. J. Elsey, and D. McLachlan, Jr., y. H. Shimizu, fourier machine for synthesizing two-dimensional structures, Rev. series in the determination of crystal
A
Set. Instr., Vol. 21,
z
W
no. 9 (1950), pp. 779-83. tor Kelvin), "On an instrument
Thomson (Lord
two
of the product of given calculating the integral Soc. (London) Proc., Vol. 27 (1878), functions,"
(Oct. 1951), pp.
Roy.
E. Saibel and teristic
M.T.A.C., Vol.
61-65.
291-97. v.
177
CHARACTERISTIC EQUATIONS
III.12.H.
W.
equation
J.
Berger,
of a square
"On
finding the charac M.T.A.C., Vol.
matrix,"
no. 44 (1953), pp. 228-36. F. Riordan, Scherberg and J.
M. G.
computation
of
polynomial
and
pp. 371-73.
aa
F
A.
Instmmente, "Analogue
trigonometric
194-235.
,
,
Willers.
Mathematische
Berlin,
Maschinen
Akademie-Verlag,
1951,
met pp.
Chapter 13
DIFFERENTIAL EQUATION SOLVERS
IH.13.A, Introduction
solved.
There are two types of continuous computers which have been developed for solving simul
following:
One of
taneous differential equations.
these
is
the mechanical differential analyzer generally
based
on a
disk-type
integrator
in
quantities are represented
by rotations of
Addition in such a device
is
also Section III.2.B),
constant
is
by
which shafts.
differentials (see
by
and multiplication by a
gears (see also Section III.2.C),
The components This
Integrators. logically.
is
It consists
some form of
to
are, in general,
the
be considered as a unit
of a disk integrator plus
amplification
and
for the output,
may even contain some provision for trans ones for lating mechanical signals into electrical it
transmission to other parts of the device. Differentials,
or
adders.
These,
of course,
provide us with a shaft rotation equivalent to the
obtained by a Multiplication and division can be
sum of
combination of integrators, as we shall see, and very general classes of equations can be realized
Here again the unit may be more complex than the
on
contain provision for changing the answer into
these devices.
The
other type
of differential analyzer
"electromechanical."
is
In electromechanical dif
an
the shaft rotations of two given inputs.
mechanical
simple
electrical signal, or
it
differential.
It
may
may indeed be based on
the so-called selsyn differential.
Logically
it is
shaft rotation
simply an adder. Gear boxes. These provide us with a shaft rotation which is a constant multiple of an input
variable
shaft rotation (see Section III.2.C).
ferential
analyzers
quantities
are
normally
represented by voltages, although conversions to
may also appear. The independent There are normally the time. versions of use only these which simplified integrators based on feedback amplifiers and "linear"
can
is
elements such as potentiometers. These
be used
differential
to
solve
simultaneous
linear
equations with constant coefficients.
However, even in these simpler machines, nonlinearities expressed by "diode limiters" can be represented.
involve
some
The more complicated provision
for
devices
multiplying
two
varying quantities and also for representing and more complex equations can be
functions, solved.
may
Function tables. Multipliers
(See Section III. 10. A.) similar units.
These are not
since the corresponding result can be obtained
a combination of integrators and adders. ever, these units frequently
easier to use
making
and
make
by
How
the machine
also extend its capacity
by
integrators available for other purposes.
In addition to the above components one must have some method of connecting these
components together to correspond to given
EI.13.B. Introductory Discussion and Setup of
Mechanical Differential Analyzers typical
consists
and
absolutely necessary for a differential analyzer
problems.
A
The output
have a torque amplifier.
mechanical
differential
the interconnection
analyzer
of a collection of components and
some method of connecting
In a purely mechanical differential
analyzer such as those that were first constructed,
these together in
accordance with the problem that has to be
was made bymeans of shafts.
We describe a typical layout. On a long table large
number of
The various are
shafts are
mounted
integrating units
mounted along
a
lengthwise.
and function
tables
the side of this table with
SETUP OF MECHANICAL ANALYZERS
III.13.B.
input and output shafts perpendicular to the mechanical length of the table. In the simplest
179
Thus, the procedure for setting up a problem can be described as a process involving a number
layout gear boxes and differentials are mounted
of steps.
on
problem and express it in terms of the variables U, V, y, Y, and Z, For these variables we
the table.
In setting up a problem each quantity com puted, as well as the independent variable x, is assigned a shaft which runs lengthwise on tie table. This assignment also an requires
First
we have
take the given mathematical
have the equations
assign
ment of a scale factor between the rotation of the shaft and the quantity represented. Thus, if n integrators are used,
we
dx
dx
n shafts for the
(III.13.B.2)
output quantities y l9 y n If r function tables are to be used, we have r shafts for the output Zls Zr There is a shaft for the quantities
where the fis are functions whose graph is available. In addition we have the relations Eq.
,
.
.
.
.
.
.
.
.
x.
In addition,
if
gears and
are used, these will have output
differentials l5
,
,
independent variable
shafts
.
Qs
,
.
Each of
these shafts
is,
of
course, driven by the output shafts of the unit
involved, and the setup process would begin with introducing bevel gear or spur gear connections
The remaining quantities. connections must be used to specify the inputs Uj for each of these
and Vj
to the integrating units
to the function tables.
V
jt
Vj,
Y
and
and x
and the inputs 7f
Each of these
must be obtained from the j,-,Z,-,
it
by means of gear boxes and Thus, each such quantity must be
shafts
differentials.
given as a linear combination of the
Fig. ffl,13.B.l
quantities,
/s andZ s.
IIL13.B.1 between variables U, V, and the
first
step
is
to express the
of Eq. III.13.B.2.
The next
Thus,
step
to provide a
is
setup diagram which will indicate the necessary
connections which will have to be
Thus,
j?.
problem in the form
An
machine.
made on
the
assignment of shafts and scale
factors for each variable
is
This setup
required.
diagram is normally set up on a prescribed form which indicates the position of the lengthwise shafts and the integrating and function table *=i
j=i
units.
A typical simplified diagram
is
shown
in
Fig. III.13.B.1.
=
i;
*fl>
+
^+i^+iftA. 3=1
fc
The longitudinal
=l
initially
(III.13.B.1)
The
fl
i0 ,
6 f0 and ci0 represent an ,
initial setting
of
the reference marks for each of these shafts.
Furthermore,
the
individual
function units have to be set
integrating and
up
in accordance
with the initial values specified by these constants. The remaining expression for each of these variables
must be mechanized by means of the
gear boxes and
differentials.
shafts occur in sets
which are
disconnected but which can be connected
along any line in some simple fashion. The boxes underneath the lines correspond to the One set of integrating and function units. longitudinal bars are driven variable
x
unit.
The coder
tion between these units
by the independen-
indicates the connect
and the longitudinal
bars by drawing lines such as those Fig.
IH.13.B.2.
may be
The gear boxes and
indicated
III.13.B.2.
in
the
space
shown
in
differentials
above
Fig.
DIFFERENTIAL EQUATION SOLVERS
180
Let us consider a very simple problem in which are requested to solve the following system of
Normally the
new
we
in the
differential equations:
the output.
original quantities are represented
certain of these will constitute
setup;
Some
of the function tables
may be
output devices or there may be counters on certain shafts which are recorded either by means
dx
of an electrical remote control setup or photo graphically (see also Section III.10.G). In each
a Z2 y,
Tx
(III.13.B.3)
case
it is
necessary to record the x-shaft position
simultaneously with the dependent variable.
When
the diagram has been constructed, the
connections indicated on the diagram,
e.g., Fig.
machine. The
III.13.B.3, are introduced into the
computation proceeds as the x unit drives the x shaft. The rate at which the x shaft is driven can be chosen for
maximum accuracy.
x shaft can be stopped
at
Also, the
any instant in order to
introduce discontinuities, such as step functions.
The x
Fig. IIL13.B.2
this
We let Zl =fL(x)
and
Z =/a(*). 2
unit
x
on certain other
functions
possibilities
Our
differential
=7 =7 =x 1
a
of these variables.
from the
follow
independent variable (IU.13.B.4)
equation, Eq. III.13.B.3,
is
variables to control the
unit so that discontinuities can be introduced
into
y1= ya
have a predetermined stop for even possible to put limit
It is
purpose.
switches
Then
may
is
fact
All these that
the
not time but a shaft
rotation under the control of the operator.
When function tables are to be used, the choice of scale factors for the input and output for these
clearly equivalent to Eq. IIL13.B.2.
devices is normally the major influence in deter mining the scale factors for the full setup. As far as practical, one endeavors to obtain a scale
which corresponds to a large fraction of the and output-shaft
possible range for the inputrotations for these units.
A similar consideration
applies to the input of a recording unit. f.tx)
scale factor
used should produce a
available range for the input quantity.
W. W.
The
maximum (See also
Soroka, Ref. III.13.am, pp. 159-90.)
ffll3.C. The Shannon Theory for the Scope of
Mechanical Differential Analyzers
The Fig. IH.13.B.3
It
should be clear from Fig. III.13.B.3
how
Eq. III.13.B.1 can be represented on such a diagram by means of gear boxes and differentials. It is also
necessary in Fig. III.13.B.3 to indicate
connections
to
the
various
differential
analyzer
is
essentially
a
combination of integrators, gear boxes, and differentials which is used to solve systems of
output
devices.
ordinary equations. The problem of taking a general system of differential equations differential
and transforming it into a form which can be represented on a mechanical differential
analyzer
(see Eqs. III.13.B.1, 2) has
been solved by C. E.
IIL13.C.
Shannon
THE SHANNON THEORY
This process applies
(Ref. III.13.ak).
not only to the mechanical differential analyzer but also to digital differential analyzers such as the
CRC 105.
Maddida and
by the
we
we have n
If
and
integrators
W
let
dWu _n U
.
.
1,
we have
\
UM ~ a m + % *
"f
2
dx
/
c
V*
the
,
(IH.13.C4)
= Ak
%
=
akl&2
,
The U-s and V-s must be
n.
,
= AM
(,,,.,3.0.,)
1/, s,
F/s, and
Since they must be uniquely deter
x.
must be
it
mined,
\
(dVkl
HT"/
dx
\dxl
linear combinations of the other
Wi and
this
Let
dx
=
do
V,Q
equations
i
If we
V^ the angle
input, of the zth generator, then
for
respectively.
"T~~
denote the
i
the linear input, and
C/i,
zn&pq
get
Function generators
are not considered in the present section.
output,
pairs kl
181
possible to solve for these
then
explicitly:
Ukl
Then
does not depend upon
Vkl does not Vsk for every
/,
depend upon k, and, indeed, Ukl = pair of / and s with each less than
we
=U =V
wk
let
kl
system of equations
is
sk
n.
Then
.
Suppose the above
such that
^ =w fe
3=1
k
dx
Thus, the solve
and
also
differential analyzer permits us to
differential
any
system in the above form
differential
shall
differential
If
we
have to discuss the equivalence of a process of
d
"expanding"
the system. If we
that if we eliminate the
we
The
^-A A dx
elimination process
may
It
clear that if
is
we
this is
is
a system,
appear at
may
To show this, first
,
(IIL13.C.7)
dx
= 2 A kvqw r Now if we suppose that
where Pkg
3?
one of the equations in the form of Eq. III.13.C.7
dw form
in the
=
this
1,
system
dx i.e.,
is
clearly
Eqs.
remarkably inclusive although
not readily apparent.
lP^ flf
is
and 2,
take what
=^ +
solve the
original system.
for
<
1
Pt q
involve an
expanded system, we obtain solutions of the
The above form
+r ^^jfcjxz^j) ?A w
ft
get the
we have expanded the
integration and a choice of the constant of
III. 1 3. C. 1
by means of the others
get the system
add an unknown
new unknown we
original system back, then
integration.
(Wpq)
equations and also
to the system and an equation in such a fashion
system.
n equations and
differentiate the last
eliminate the
the above type.
We
(III.13.C6)
equation system whose are a subset of the variables of a set of
any
unknowns
\t~
let
equivalent to the system
us
to be a special
2 Let us suppose that we have n integrators, 2 w so that i for the 1, equation
dx
dt
case.
.
.
.
=Vl dx
Now we
wish
where
PM
and the
P^
s
are arbitrary hnear
,
to
(DI.13.C3)
\dxJ
replace
combinations of the
w s and
x.
We
have thus
proven the following theorem:
THEOREM III. 1 3.C. 1 Any system of differential .
the
subscripts
equations
form of Eq. IIL13.C.8 can be a system in the form of Eq.
in the
expanded into
DIFFERENTIAL EQUATION SOLVERS
182
III.13.C4 and hence can be
set in the differential
Consider
Conversely, every system in the form
analyzer.
of Eq. C.4 subject to the condition Eq. C5, can
be contracted to the form of Eq. C.8 the l/ s and F s).
dx
Notice that the equations in the form of Eq,
The enlarged set consisting of Eqs. III.13.C11 and 12 clearly has the desired properties.
III.13.C.6 give the exact connections for the analyzer.
W
The
kl
w
integrators, the
^
s
ft
are linear combinations of
The upper set of equations shows how
these.
Hence:
outputs of the
are the
integrators are to be connected,
i.e.,
dx
3=0
(eliminate
the
THEOREM III.13.C2. Every
set
Eq. III.13.C10
can be expanded into a set Eq. III.13.C.8 and hence can be solved by the differential analyzer.
our Suppose in Eq. III.13.C10, we replace
their inputs,
while the lower set shows what linear combina tions
Wj-
are to be taken.
Again we can remark
more first
general than
it
3=1
is
we
this,
^=f
lemma:
Let
.+T
Tax
To show
appears.
establish the general
LEMMA III.13.C.1.
that Eq. III.13.C.8
polynomials P^ by rational functions and obtain the system
\
dx
(
-
Theoretically,
R..^?
dx
3=0
at
least,
J?
w
=
p -p *"
(III.13.C13)
dx there
no
is
in
loss
the denominators generality in assuming that all
..... -)
/
same and,
are the
(III.13.C9)
thus, that Eq. III.13.C13
can
be written
denote a system of differential equations in which
Pw s are polynomials in x and of the mth or lower degree where m
the
wlt >
.
1.
.
.
,
wn
We wish to expand this to a system in the form We introduce the variables
the system Eq. III.13.C9 can be expanded into a
system of the same sort in which the degree of the
P
s is
m-
or lower.
1
of Eq. III.13.C10. !
Proof: For simplicity in our discussion, let us add the dependent variable w x. Our system is
dx
dx
Then
=Q
and
v
=
2
This
l/^.
leads
the
to
expanded system
then equivalent to ,
dx
dx
3=0
dx
dx
(IIL13.C.10)
To prove the lemma, we append the variables = w^j (i and j = 0, n) by adding the
Zy
(n +
.
.
.
,
2
I)
do
equations
which
in the
polynomial in
dx
dx
is
THEOREM
each of which
is
but with linear
in the
P
s.
we
system, Eq. C.10,
form of Eq. III.13.C.10
Returning to the original see that in every
monomial
of degree 2 or larger, we may substitute for a a z{j This process will lower every product of polynomial degree greater than 1 in the set.
w^
.
form of Eq. C.10,
w
,
.
.
.
,
wm
.
since
Hence,
we
is
a
have:
Every system in the form Eq. III.13.C13 can be expanded into the III.13.C3.
form Eq. III.13.C10, and, hence,
set
up
in the
differential analyzer.
(In the latest differential analyzers,
servo feedback
mechanism on the
one has a
integrators
that permits one to use the angle input as the
THE SHANNON THEORY
IIL13.C.
output. Thus if (/and Fare inputs and PFis the
we have
output,
However, this does not exhaust the possibilities for the differential analyzer.
/i= =0 dx
dx
where
or
/
dW ~^F
III. 13. C.I 9
l_/dV (IIL13.C15)
U\dx
dx
This permits a more
use of integrators
efficient
fractions are present,)
However, our system Eq. III.13.C.13 can be
^
rational in the
is
We may
w
For
s.
clear
it is
and, on the other hand, every system Eq. C.13 equivalent to a system in the form Eq. C.16 as
is
dW
explicitly for the
-~
.
dx
III.13.C.16 for a system of
ordinary differential equations is, of course, a There is, of course, a well-known
process by which one can take equations of higher order than the first and express them in the form of Eq. C.16.
only one further extension. Let w n and system of equations on wlt
we have
if
Thus,
.
,
a
J
*>
\
73
*
,
7
dx dx
\
*
z
5
T
f>5
j._,
dx*
.
.
,
(HI.13.G20)
be constructed by the use
of the
rational
which are themselves solutions of algebraic differential equations. Such a system can be expanded into the form of Eq. IIL13.C.19. Let F(X) be a nonrational function occurring
I
s,
functions used
Thus,
V
in
with an argument 2 which is a rational function of the w s. Of course, some of the
the/
we
have nonrational
may themselves
But then we
functions in their arguments.
equation
T. k
=
ft
go further until
dx
obtain a
operations and by functions of one variable
familiar one.
differential
which
solve for these highest deriva
We will make
their derivatives
The form of Eq.
w n and
,
and then by introducing more
explicitly
are special cases of the system Eq. IIL13.C.13
by solving
.
variables as in the above example,
the
see
.
In general, the system Eq.
can be expanded into a system in the
that the equations of the system Eq. HI.13.C.16
one can
.
system in the form Eq. III.13.C.16.
dx
where
(IU.13.G19)
wb
linear in the derivatives of highest order
tives
(1II.13.C.16)
!,..,,)
formEq.III.13.C.16. Let us differentiate the system Eq. IIL13.C.19. The result will be a set of equations which are
appear.
expressed in a simpler form:
=
(i
Consider a system
a polynomial in
is
their derivatives.
when
183
we meet an
see that there
may
F such as described. always an
is
F
as
described.
(III.13.G17)
we may H>
2
=
introduce
dy
unknowns w
=
x,
Wj
= y,
= d^y -^ and setup the system
F may
have a number zl9 ...
=
may
use the equations
dFldzA
= ~
dz\dx)
z
dz \dxl
%
w >
r
. ,
.
.
.
an d tne i f n>
dy?
d*F ,
dZl dz^
dx
dx
dz\dx*!
dF to express -
,
form of Eq. C.16
if
/is
rational.
derivatives.
y>
,, az 1
.
rationally in terms
determined by an equation
in the
dy
tf/AV
fFtdtf dx
is
zr of rational
-
Wj.
dx
wMch
t
arguments in different places in the system of y. We equations. Let us consider only F(zL)
of
Now F is
DIFFERENTIAL EQUATION SOLVERS
184
Our If
by hypothesis.
we
substitute for z1?
F
}
etc.,
the values given above,
If in the
derivatives,
become an
y and
s,
W
WK
outputs,
2)
Ws,
be
W^ with dx
we
substitute 7,
dx
have
will
dW,
F(z-j).
The above
will
their
expanded our system to one which does not contain
it
adequate to introduce four integrators, with
original system, wherever
we now
F(ZJ) occurs,
will
g
w
equation on the
algebraic
discussion above shows that
,
dz1
=
dz_
dx is
process
repeated for F(z 2),
.
.
.
dx
,
dW, It
clear
is
that the above process
continued until
all
can be
dx
dx
the nonrational functions are
=y
removed.
Thus we have
THEOREM
dx
established:
Every system of dif form of Eq, IIL13.C.20
III.13.C.4.
ferential equations in the
and
U s which
three
(III.13.C.25)
dx
Fs
are also
can be expanded into a system of the form of Eq. III.13.C.4 and, hence, can be differential
The above interesting in
discussion
theoretical
that
it
gives us
process for finding the system Eqs.
2 which we must have
if
(z=)(73=
is
very
we
a step-by-step III. 1 3. C.I
and
are to use the dif
analyzer to solve a system of differential
form of Eq.
equations, say, in the
It is clear that the essential
differential
system
the
but the major reason for
itself,
it is
considering
ferential
set in
analyzer.
analyzer are
a system in
into
III.13.C.1
first
(Each
x
W has
- XQ.
It
Since
(ffl.l3.C26)
W=
at
from the above how the
integration can be entered in
this
any
type.)
we have the equations
III.13.C.20.
.
steps in using a
2
and 2 making a careful note of how
dx
the constants of integration are concerned in the setting
up
{dx
dx
expand a given the form of Eqs. to
expansion process and then
+ Zo
set so as to yield
clear
is
constants of
problem of
been
jf4
dWt
the
analyzer.
Let us take a few examples. Suppose we have
(III.13.C.27)
a system
dx dy
=x+y .
dx
dz z
dx
dy the connections of the differential
dx
analyzer are
completely indicated.
dz (III. 1 3 . C. 23)
ft i s
c^ear
We
wish to find the solution which at x
=
provided for each
This system rewrite
is
in the
form Eq.
III.13.C.8.
We
it
mind i.e.,
dy ~ dx
=x
dx
.
{-
dx
y
-- z~ dz
dy
dx
dx
=, dx
(III.13.C.24)
dx
^at tae c o ns^nts of integration will
no
introduce
difficulty
we know /.
It is
in
the
the value of
just this that
in our successive
i
for
x
case,,
=*
we must keep rn
expansions of the systenfl,
whenever a new variable
initial
general
W
is
introduced,
iits
value must be determined.
Let us consider another
example:
Kl=x(sinj;)
(III.13.C.28)
III.13.C
This step
is is
z
=
form of Eq, III.13.C20. Our first expand this to a system in the form of
To
in the to
To do
III.13.C19.
Eq.
THE SHANNON THEORY
introduce
we
this,
an Eq. IIL13.C.10 system, we
return to
.
w
=
185
_
1
}
introduce
and Eq. IIL13.C34 becomes
sinj for which we have
~+
z
^
=
(III.13.C29)
dy dz__
Now,
if
we
eliminate the derivatives of i relative
V
dx~
by means of the equivalent of Eq.
to y,
we
III.13.C21,
obtain
du
= \.,(wxu +
,
,
wzj
dx dx \dx
2
dx \dx
/
2
fa
\dxl
/
=
,,2^2
+
dx
(III.13.C.30)
with
,
d
M- =xz
_
^
,
~ = ~^ ~ 2
mr n r in
l y\*
^
(III.13.C31)
(HL13.C.35) v
\ dx )
fa
\dxt
Eqs.
IIL13.C30and31areintheformofEq.
EqJIIJ3.C.16 it is Eq. C.31. Hence
desirable to differentiate only
V
=x- + z
(III.13.C32)
dx
To
q
=
zw
s
r
=
w 2 and ,
the
in
is
reduce
it,
s
/
\
21
= zu.
=
jz
2
/
\ 4~1
/
\
dx
"
dxl
We
dx
i-(x- + z)!
^
(III.13.C33)
,(dy\\ dx
= dy
introduce u
dz
,
Jx
o
Ttr
= -r We
^
.
.
1
then have
dv
Jw-
=
-r
du
dx
dx
the Eq. C.16 system
+ Ww^
^-u
^=w
^
dx"dx
dx
^=
=Z
+ 7x
7x
v
}V
dx
dx
T dx
dw
dr
j
= 2w T T dx dx
<
du
1
==
,
,
+z r( 2w xt)
.
tf
)
(
IIL11C34)
^
=z
^
form
of
we introduce^
dx
and our system becomes r
III.13.C35
Eq.
III.13.C10.
then becomes
.1Z T
form of
to the
However, to revert
III.13.C19,
+ "^
Eq.
= uw,
Eq. III.13.C35
DIFFERENTIAL EQUATION SOLVERS
186
Finally
and j
we introduce
= su,
f=
= Fp and obtain dy -i dx
=u
dz
=v
dx
dw
dw
= KWi +
i
+
+
(W
=M dx
\dx/l""dx
,
dx
dx
dx
d/i
= s du
dx
dx
dx
*\dx/\
= p dx dx
dx .
ds
dx
=,+/ dx dx
dx (ffl.13.C37) "dx"
Eq. IH.13.C.37
is
in the form Eq. III.13.C8
and
;an be expanded as in our above example to the
orm Eq.
III.13.C.4.
least lealing with the
For the arrangement
number of
integrators
~dx
we
:
ntroduce "dx"
"dx"
"dT
dx,
K/o
+ 9o)
ELECTROMECHANICAL DIFFERENTIAL ANALYZERS
IH.13.E.
=
w
dx
=
dv
Maginniss, Ref. III.13.J;
dx,
HI.13.V;
later version
of
this
was also
installed at the
University of California at Los Angeles (see Ref. m.l3.i; L. L. Grandi and D. Lebell,
dw
dV,
Heffron, Ref.
H. P. Kuehni and H. A. Peterson,
Ref. IIU3.x; F. L Maginniss, Ref. III.13.ab; H. A. Peterson and C. Concordia, Ref. III.13.af).
dw v
A
d^
W. G.
187
dx
Ref. IH.13.r).
Small-scale computers in the United States were built for the Institute of Actuaries (see
5
\dx
R.
E.
Beard,
amplifiers,
G.
dx,
and
Ref. at
III.13.c), using torque Yale (see R. E. Meyerott and
Breit, Ref. IH.13.ad).
German machines which used
=
selsyn trans
mission systems are discussed by R. Sauer and
"? dx
"3
H. Poesch (see Ref. IIL13.aj) and A. Walther (see Ref. III.13.as).
A
survey of British differential analyzers
discussed in
J.
Crank
(see Ref. III.13.k).
is
We
note several articles on devices based on the M.I.T. differential analyzer and produced at the University of Manchester (Ref. III.13.m;
dx
dx
^o = 7 p* \
dx
/
(III.13.C38)
dx/
\
D. R.
and D. R. Hartree and
A. K. Nuttall, Ref. HI.13.t). D. R. Hartree also
= j2-
6
dx
Hartree, Ref. IIU3.s;
dx/
machine which was a developed a Meccano low-cost differential analyzer at the University
HI.13.D. References
The
of
notion
of Manchester (see D. R. Hartree and A. Porter,
a
mechanical
differential
Ref. III.13.U).
Devices similar to the Meccano
Lord Kelvin (see analyzer was proposed by W. Thomson, Ref. IIL13.an, ao). However, the
machine were developed
of this device required the practical development introduction of torque amplifiers, and this was
Ref. III.13.z)
done
at
1925-30 (see
M.I.T. in the period
V. Bush, Ref.
III.13.f, g;
W. W.
Soroka, Ref.
III.13.am, pp. 160-65).
at
Later on devices of this type were constructed School (see including the Moore
I.
Travis, Ref. III.13.aq;
many places,
M.
see also
Vallarta,
these using
an
electrical
method
for
interconnection between units was constructed at M.I.T. (see V.
Bush and
S.
and R.
and
Massey,
at
Cambridge
V. Wilkes, and
at J.
Queen
s
J.
(see J. E.
B. Bratt,
University, Belfast
Wylie, R. A. Buckingham,
Sullivan, Ref. HI.13.ac).
Some
of the companies
making
ball-cage
variable speed drives for differential analyzers are the Ford Instrument Long Island
Company,
City,
New
York, and the Libroscope Company,
Glendale, California. discussion of the components used in the
mechanical differential analyzers design of Macon Fry (see Ref. III.13.o). given by
is
IH.13.E. Introductory Discussion of Electro
General Electric introduced
a mechanical
differential analyzer in which the output of each
was connected purely by polarized which produced the
the servo system light with
torque amplification (see
S.
M.
H. Caldwell,
Ref. III.13.h).
integrator
H.
(see
A
Ref. III.13.ar).
One of
Lennard-Jones,
A. C. Cook and F.
J.
mechanical Differential Analyzers
The electromechanical
differential analyzers
are a post-war development based essentially on of the electronic firethe
computing procedures
control computer developed by Bell Laboratories
DIFFERENTIAL EQUATION SOLVERS
188
The
during the war,
critical
development was
quarter-square
method using
A number
triodes.
that of the feedback or operational amplifier
of units for function representation purposes are
which can be
available.
utilized in either
a summer or an
integrator (see also Sections III.7.F and G).
An
assembly of such operational amplifiers with the proper feedback networks and linear (tenturn) potentiometers will constitute a differential
In addition to the above differential equation solvers there are also
special-purpose devices
which are normally wired up for the solution of only one system of differential equations. Air
analyzer capable of solving systems of linear
plane trainers, airplane automatic pilots, and
equations with constant coefficients.
various other control systems are of this type.
In a somewhat more elaborate installation one
These can be, of course, based on the principles of the mechanical or the electromechanical
differential
would have function
generators, resolvers,
and
In addition, output recorders must
multipliers.
be available. In the more elaborate installations there are function generators of the III.10.D),
drum type (see Section
cathode-ray type, photoformer (see
Section III.10.F), or the diode function genera
For multipliers one
tors (see Section III.10.E).
has a servo multiplier division
(see Section III.2.E),
time
multiplier (see Section III.9.B), or a
quarter-square multiplier (see Section III.2.E),
which uses squaring.
many
differential
a
diode
For
function
generator
for
fundamental
The
motor.
principle.
integrating element velocity
of the
an
is
The
electric
output shaft
is
measured by a tachometer, and the resulting signal is compared with an electrical signal which represents the desired integrand.
The
difference
used as a servo signal through a servo amplifier to drive the motor. Calculations are
is
by means of potentiometers linear)
mounted on
(linear
performed and non
these integrator shafts.
division (see Section III.9.A),
types of output equipment are available
m,13.F. Preliminary Setup
(see also Section III, 13.H).
The coding of an electromechanical differential
The computing devices are usually assembled on a set of racks with power supplies. The recorders
The
analyzers, but frequently they are
based on a somewhat different
are
on
generally separate chasses. various individual components are perma
nently wired to a patch bay which
is
analyzer if
the
is
relatively simple
and straightforward
given system of differential equations gives
explicitly the highest derivatives
functions.
For
of the
unknown
we
are given
instance, suppose
centrally
mounted.
This patch bay consists of an array of hubs which can be interconnected by patch cords. Recently there has been a tendency to introduce problem boards which can be mounted
on the patch bay
in a manner analogous to the punched card machine plugboards. The problem is wired up either on the patch bay or on the problem board.
We
can construct a coding diagram for
problem as follows
:
this
We use the symbols given in
Fig. III.13.F.1.
In addition to the electromechanical analyzers is a purely electronic continuous differential
there
analyzer available from the Philbrick Company,
INTEGRATOR
SUMMER
MUmpLER
POTENT]OMETER
of Cambridge, Massachusetts. The integrating
and summing
amplifiers are similar to but less
elaborate than the others. feature of this
solution process
equipment is
The is
the fact that the
repeated thirty times a second,
and the output functions
are
Fig. DI.13.F.1
distinguishing
displayed on a
cathode-ray oscilloscope; multiplication
is
by a
We
start
with two points which correspond to
nodes at which we are
suppose the highest derivatives
present in the form of voltages,
We connect these to
i.e., 5c
and j>.
integrators in series so that
SCALING AND LOAD CONSIDERATIONS
IIL13.G,
our diagram, Fig. III.13.F.2, contains nodal points for each of the unknown quantities which appear on the right-hand side of the given equations, and we also introduce a function generator dependent on the time
189
While
this
IIL13.F.2)
is
a perhaps the easiest to follow for
it is
a frequently desirable to introduce
beginner,
of
type
diagram
Fig.
(e.g.,
standard form on which each component of the
will
equipment is indicated with its output and inputs.
produce /(r), the remaining quantity on the
One might begin the planning for the problem by
right-hand side of the equations. The variable
a rough diagram such as that shown in Fig.
is
t
which
t
introduced by integrating a constant voltage.
We may now make the multiplier and summation
and then pass
III.13.F.2, If
7
we suppose integrators,
tiometers, tors,
4
to the standard form.
that total equipment consists of 7
20 poten amplifiers, and 2 function genera
summing
multipliers,
we can suppose
a standard form, as
shown
made up indicating various pieces of equipment. One procedure setting up a diagram would be first to make in Fig. III.13.F.3, can be
the for
the
diagram in the form of Fig. IIL13.F.2, label the components, and then introduce corresponding connections on the standard form.
The resulting
may be
confusing in
set
complicated
many
practical
desirable
of
lines
situations,
and,
it
hence,
is
introduce an abbreviated wiring
to
diagram in which the connections are indicated by labeling inputs. Since the origin of the various derivatives of a variable will clearly be an integrating
eliminates
labeling
amplifier,
many
lines.
long
in
For
above problem, Eq. III.13.F.1 can be Fig. III.13.F.4.
cases
these
instance, the set
up
as in
(Certain potentiometers have
been omitted.)
The convention we have mentioned concern Fig. DI.13.F.2
ing Fig.
We
connections indicated by the equations.
must
x,
x,
x
}
y,
and
y are
and
a, b, c, d,
of the amplifiers
simplifies
Further conventions can be
introduced for the same purpose.
In
order
precisely,
to
we now
indicate
the
label each
machine
setup
component accord
that ing to the specific machine component to use. Thus, since
these are
now
labeled
we have
we
six integrators,
/, 2, 3, 4, 5, 6.
We
have
used three summing amplifiers, three poten tiometers, two multipliers, and one function generator,
and
these,
in
turn,
enumerated to specify the machine initial
should setup.
be
The
conditions are normally set on poten
tiometers permanently wired to the correspond
ing integrator.
ffl.l3.G. Scaling and
Load Considerations
In the above section
respectively.
want
output
III.13.F.4.
also indicate the initial-conditions values
which for e,
the
we have assumed that the
were equivalent to voltage generators amplifiers which had such low generator impedance that loading could be neglected.
However, a great
deal of commercial equipment
is
such that the
number of potentiometers that can be placed on the output without loading effects is limited. If a greater
load
is
needed, either a
"booster"
circuit
must be introduced or additional output must be obtained by inverting amplifiers or by a parallel ing process.
For example,
if
the load
on the
which yields x, is too output of the integrator, jc may be fed into two integrators each of great,
DIFFERENTIAL EQUATION SOLVERS
190
I/
\5/ \l/
\
Fig. EQ.13.F.3
-x
V x
,5xy
p-cn
i-i
r^r
T
-
v v hri ..
7
-.5xy -x
v Fig.
m.l3.F.4
-y
III.13.G.
which
will
A
x.
produce
variation of the
discussion of the
amplifier circuits which results
from the introduction of a is
SCALING AND LOAD CONSIDERATIONS
resistance
"booster"
G. A. Korn and T. M. Korn
in
(see Ref.
III.13.W, p. 250).
Another
scaled so that
it is
smaller than
is
represented by a voltage which
necessary.
analog computer
191
is
For accuracy
meaningful only by
in
an
reference
to the full available scale. Thus, the preliminary
computations on the machine for scale purposes
which must be taken
effect
account in setting up the machine occurs when
must check so that no quantity ever exceeds the allowable limit, and also so that the scale or
varying gains for the summing amplifiers are
scales are not too small.
used.
Normally the input impedance is
operational amplifier
megohm, and
1
on
effect
the
the
loading
into
for
an
some standard value,
say
has a negligible loading
this
output of a potentiometer. In effect is
readily
operational amplifier
is
computed
fact,
since the
designed so that one
may
consider the input of the amplifier as consisting of this resistance connected to ground. This 1
-megohm
gain of
4
is
input impedance corresponds to a
in the
1
amplifier.
If,
however, a gain of
desired, then a resistance of .25
megohms
is
used in the input circuit. Connected to ground, such an impedance would not have a
negligible
on a potentiometer of total 25,000 ohms. The effect of this load effect
ever,
be computed
resistance
how
can,
(see Section III.7.B)
and a
Frequently a good deal of information available
by a
method.
If,
available
may find a proper trial-and-error
however, the type of information
inadequate from the scaling point of
one should
set
This would
may have mean that
up an equation corresponding
scale
range
for
the
of
output
each
This process can begin by assuming scale values for each dependent variable.
Ordinarily
one would apply the same scale to each of the derivatives of this variable, including the highest is
to be computed. To determine this a given variable, one should consider
It is
accuracy and scale factor for each quantity
tion relationship will preserve the scale.
become
the magnitude available larger than
from the output of
the
corresponding
ponent, the component involved would
com fail to
certain hypotheses as to
how
these will vary.
convenient to have the same scale for
all
We also
have scale values for the functions of
x which appear.
The other
quantities
in the
setup are obtained by computation from these.
The various components can be ordered so the
Thus, normally a voltage scale range of -100 to 100 volts is available, but if a somewhat larger
the output of integrators or functions of
is
represented, this representation
is
first set
inputs.
The second
set
sidered will have inputs
which
or the original inputs.
that every ing quantities must be introduced so
these various components
within the
allowable range.
cannot always be done completely
One must make
This
in advance.
a preliminary estimate of the
each pertinent quantity, and one must check so that the computation remains within range by actually carrying it out on the machine. must not be On the other hand, the sizes of
quantity
x
as
of components con
first set
is
that
of components considered will have
extremely inaccurate. Thus, a scale for represent
output
the
derivatives of a given variable since the integra
function and would give an incorrect result.
quantity
and
the initial values for each of the derivatives
be obtained experimentally. When one has set up the diagram for a problem, one must consider the question of
to
such
component.
scale for
represented on the diagram. If a quantity were
to
every component in the device and estimate a
which
obtained. This higher value can also
case one
to be analyzed in detail.
the correct voltage division from the poten is
is
this
straightforward
view, then the proposed computation
higher setting for the potentiometer used so that
tiometer
and in
quantities,
scaling
is
concerning the sizes of the desired
arise
may be
succession beginning with the
from the
The outputs of scaled in
first set.
When
one reaches the point where one computes the one may have to introduce highest derivatives, extra potentiometers or amplifiers to get the
prescribed
scale for the highest derivative of a
dependent variable.
It
should be emphasized
that if the voltages which represent a quantity
DIFFERENTIAL EQUATION SOLVERS
192
are too small, one has accuracy difficulties as
= xy
Letting z
serious as running off-scale.
We now illustrate our We assume
scaling process for Eq.
that the range of the
III.13.F.1.
machine
is
-100
volts to 100 volts.
We
want
relative to the other
multiplier
The potentiometer whose output
similar.
have the following range:
f(t) to
The discussion
is
is
to
represent 0.65j will have a voltage output
V* Therefore,
/*
is
-25
volts
/*
100 volts where
the voltage proportional to/.
25 volts
^
1,
r Suppose the
We
have
and
initial
= 0.65/ = (25)(0.65)y
amplifier should be
x*
f
(III.13.G.1)
= 25x volts
(25X0.65)0
summer. The
into the
first
summer which
input for the
by a factor of
The
4. is
scale
for
x and y could
v*
is
fact that the
with a voltage which the size of x
y*
is
the best
we can
only J the voltage used
and y be monitored. by
It is desirable
a factor of 2 each
before going into the multiplier, or one set
of computing elements consists of
factor of 4,
if
do.
product xy appears
components suggests that
that these be amplified
The first
it
identical.
for the output of other
y*
multiplies
situation relative to the
the hypotheses concerning the sizes of
However, the initial
only J of the
into an
the given quantities, this the
is
25/
we must introduce this term
Under
conceivably be:
term
+
desired value, so that
equation for y
Thus,
(III.13.G.8)
The way the setup has been given, one would feed
values are
and we hypothesize that these variables do not vary above 4. Initially
(III.13.G.7)
Now it is desired that the output of a summing
by a
possible. This process will involve
the multiplier which yields xy, the multiplier
the introduction of two operational amplifiers,
which
but normally the increase in accuracy
The second
summing
A
and the two potentiometers. of components consists of the two
yields xy, set
amplifiers
which produce x and
y.
multiplier normally has a voltage relation
is
worth
this,
For a further discussion of
scale factors see
G. A. Korn and T. M. Korn (Ref.III.13.w, pp. 30-32, 58-61).
Z*7* (III.13.G.5)
100
HL13.H. Wiring and Output Connections
where Z*, X*, and 7* are expressed
For the xy
multiplier our voltage inputs are Jt*
and, hence,
in volts.
= x25 and
7*
- y25
When the precise setup has been determined as described above, the machine can be set up in accordance with the above connection diagrams.
The input and output of the various components are
all
connected to an array of hubs (called the
which appears in the front of the "patch bay") machine, and the required wiring connections are
made by means
electrical
Above
of
"patch
cords"
which are
connections with jacks on each end.
this
patch bay corresponding to each
IH.13,1.
integrator,
one
and a
finds a potentiometer
switch to permit the entry of the
initial
the output of this integrator.
IMPLICIT SYSTEMS
sign
193
The
solution of Eq. III.13.H.1
and
after
Other poten
a while
will
it
tiometers for multiplying by a constant are also available. Their input
in the patch bay.
and output appear
at
state
The
or
is initial
third
reset.
in
one of three
The second
The
hold.
is
is
the extra connection to the output has a negligible effect
the
because of the low generator impedance of
is
computing.
machine
switch
If the
amplifier.
One
position,
state of the
of the other two positions,
If the switch is in either
states.
(OI.13.H.3)
Q
hubs
connections for the output recorders. After the power has been turned on, the differential analyzer
be
=y y
also necessary to introduce
It is
is
value for
is
the input to the integrator
in
the hold
is
zero, and,
thus, the output remains constant.
is
The
determined by a relay system which, in turn, determines the state of each integrating amplifier.
above
differential
is
-for
electromechanical
the
For the Philbrick
analyzer.
type,
in
which the computation process is repeated thirty times a second, the timing is under the control of a central timing component. three-valued
is
voltage
alternates
and a negative value. available both directly and in integrated
between a This
step
This produces a
which
positive, zero,
form, This voltage
is
fed to various integrators
and function generating units so that the desired outputs are
first
produced with time increasing
corresponding to the independent variable After a computation period the increasing. reversal of the sign of the voltage restores all quantities
to their initial state before the next Initial
computation period begins.
Fig. IIL13.H.1
conditions or
constants of integration are introduced
A single three-position switch controls the com when the power is
puter is
on.
When the computer
in the initial or reset state, each integrating
amplifier
is
disconnected from the rest of the
computer, and initial
given
its
output voltage held at the
voltage value.
When
the switch
is changed to computing, the integrator normal connection. At hold the integrators
position is
in
are
disconnected,
but the
condenser charge
remains unchanged. This permits the introduc tion of discontinuities in the input functions
when
m.13.1.
Systems
Implicit
of
Differential
Equations
Our previous
discussion of the use of electro
mechanical differential analyzers assumed that the system of equations was explicitly solved for the highest derivatives of the unknowns. Suppose
our unknowns are jlr ,y n , and the highest derivative of y i is of the r^th order. We have .
,
.
supposed that the system of equations
necessary.
j^ - F
situation for a single integrator can be
described by Fig. III.3.H.1. initial
integrator
is
When
the switch
is
or reset position, the input to the the
sum of the
values,
j
,
set
on
the potentiometer and the output value, y. Thus
is
in the
(III.13.I.1)
t
where the F-s are functions of less
than
rk
y$
where j
is
.
But the given system of differential equations simply be relations between the specified
may
derivatives such that
or
step
form
The
at the
by
functions.
it is
solve for impractical to
the highest derivatives. There
is
one case where
a standard coding for a system can be obtained
DIFFERENTIAL EQUATION SOLVERS
194
must determine y in a
even though the equations cannot be put in the form of Eq. III.13.I.1. Unfortunately, there is
It is clear
no reason
defer any discussion of Chapter III. 14, where the error theory developed by K. S. Miller and the author is indicated. However, there is another aspect of
and
to believe that the result will be stable
to our problem. yield a solution
we can
solve the zth equation for
y^
w = F? y
Suppose
i) :
(III.13.I.2)
may
is
we
if
are to obtain a solution.
desirable to
stability to
(
appropriately discussed
here.
<
depend Eq.
upon^ and the lower derivatives of jv
III. 13.1.2
can be coded for a standard com
mercial electromechanical differential analyzer in
a manner similar to Eq. III.13.I.1. We introduce for each equation of the set Eq. 1.2 a node whose voltage corresponds to
y^\ The various lower
l
derivatives
$*~
\
.
.
.
,y*p
obtained
are
by
repeated integration, using integrating amplifiers.
We
It
that this equation
manner
the situation that can be
F*
does not depend on y ?*\ although it t r it and it may also contain y^ with ;
where
stable
It is
clear that
no
such as
heuristic procedure
two feedback
given by either of the above
methods can work unless the given system of equations
=
F, (
(i
=
l,2,...,n) n)
(
determine y { l \y ^\ other derivatives.
.
.
.
,jj[
This
is
(III.13.I.7)
as functions of the to
equivalent
the
statement that the Jacobian
also introduce functional inputs, and, con
sequently,
we have nodes
for each input for
F*
^
t
i. use the nodes for ffi for j Indeed one can construct this coding even when
Ff
depends upon $*\
the
system
must be regarded as
determining these highest derivatives implicitly. We have then a feedback system which may or
not be
may If,
at least,
one can obtain an equiva which the highest
lent system of equations for
order derivatives do not exceed the given rf in order,
and
one case the order
in at least
is less.
For, suppose
stable.
however, the equations
Then, by a well-known result in the theory of
F+ = are not solved explicitly, there
using the error in the
fth
(III.13.I.3) is
The lower
equation as the input of is considered to be
derivatives
functions, there exists a functional relation
a technique of
an amplifier whose output
$*\
Now if this Jacobian is zero, we can show that theoretically,
In any case as long as derivatives of the highest orders are used on the right-hand side of the equations,
(III. 13.1.8)
^
provided we
are
obtained as
H=
(III.13.I.10)
a consequence of Eq. which does not involve y^\ y$*\
which
III. 13.1.7
is
.
.
.
,
jj
and E. R. Hedrick, Ref.
n) .
and (See
above, and the various derivatives are used as
also E. Goursat
inputs to expressions Ff.
Consequently, if ^ is the gain of the amplifiers used to produce the
pp. 52-58.)
highest order derivatives, the system actually set
set Eq. 1.7 by Eq. III.13.I.10. For each;, let st denote the order of the highest derivative of ys
can be obtained by replacing one equation of the
present in H, and
Suppose, for instance, our given equation
F
= ? + ?-y + y+ /(O = o
is
(III.13.L5)
our system cpnsists of only one equation on one unknown, y. If an amplifier with large gain
i.e.,
p
is
used,
we may
ffl.lS.q,
A system equivalent to Eq. IIL13.I.7
u
;>
1.
u
min
Let j\ be such that
differentiate will
let
Then
s f ).
(r^
u = r
f
;>
sf
.
We
Eq. 1,10 u times. The derivative^
be present in the result in the form
realize
Since the coefficient of (III.13.I.6)
III.13.I.1
yfd
is
can be used to express
not zero, Eq. in terms of
y^
COMMERCIALLY AVAILABLE EQUIPMENT
IIL13.J.
and lower
yW
the
(
y ?^
replace
which
is
We
derivatives.
new system of
in the
can then
195
equations,
but these appear one can linearly. Consequently, +l] solve the system and Eq. 1.13 for the
expression
express
yf>
equivalent to Eq.
1.7,
by
obtained from Eq. III.13.L11.
this
We now
have a
To
it
form Eq.
in the standard
new system of Eq.
solve the
III. 1 3.1.
we
1.13,
L
need,
system equivalent to Eq. 1.7 in which the highest derivative of y j is less than rf in order, and the
of course, the additional information concerning
highest derivative of the other y } does not exceed
obtained above.
r 3 in order.
Now
new system has a or we can repeat the
either this
nonvanishing Jacobian, above with further reduction in the highest order of the derivatives of some y. Thus, we can continue until our system is clearly contradictory,
we
or for
get a system with nonvanishing Jacobian
This means, of
the highest derivatives.
1 the initial values of thej/f *,
It
rfl) .
.
.
,
j4
which we
also be possible to
may
obtain a linear system without differentiating every equation of Eq. 111.13.1.12. ffl.13. J.
Commercially Available Equipment
The commercial development of electro mechanical differential analyzers has occurred since the war. A number of companies manu
course, that theoretically one can always express
facture differential analyzers of this type.
the system in the form used previously in
of these companies are the following:
this
Beckman
chapter. If the system
is
and the Jacobian
not zero,
Computer Corporation of America New York 7, N.Y.
we can readily solve
for the highest derivatives to obtain the form
Eq.
Electronics Associates, Incorporated
III.13.I.9.
If,
however, the
Long Branch, New
in solving a system of difficulty
Jacobian equations with nonvanishing
is
due to
Goodyear
Akron
the fact that the highest derivatives appear in the there is one way of in a nonlinear at the expense of increasing solving the system now describe this method. its order.
We
f.
=
New
(HI.13.L12)
The does determine the
initial
values of the highest
derivatives v (ri) /!
at the initial value
r
M
J
>
V /n
at least to within a discrete is
III.13.I.6 determines instance, the system of Eq. three of one values, and we be to y, in general, a other information is available to make
suppose
choice.
available,
When
this
we may
additional informa
differentiate the system
once and obtain a new system Eq. IIL13.I.13
1 in
N.Y.
12,
17) tf
1
which the highest order
York, N.Y.
Associates,
Goodyear
by
Aircraft,
Electronic
and
now
is
justified
Reeves
common
in regarding these
as examples of commercially available differential analyzers.
The
basic element in this equipment
a drift stabilized operational amplifier with In the case of Reeves equipment high loop gain. is
this
amplifier
is
of either an normally part or
inverter, summing integrating amplifier, phase In the case of the Goodyear Aircraft amplifier. and also Electronic Associates feed
Corporation units so back networks are available as plug-in serve either an the same that
integrating
derivatives are
produced
equipment
features so that one
For available to make a precise determination.
is
Instrumatic Corporation
Instrument Corporation have certain
of alternatives, and additional information
tion
Ohio
Reeves Instrument Corporation
equations
specific
15,
George A. Philbrick Researches, Incorporated Boston 10, Massachusetts
that the given system of
One may assume
Jersey
Aircraft Corporation
Mid Century New York
way,
equation
set
Instruments, Inc.
Richmond, California
linear in the highest derivatives, is
Some
amplifier
or
may
summing purpose.
These
inte
occur in unit groups of twenty for Reeves grators of fourteen for and unit
and Goodyear
Electronic Associates.
groups
DIFFERENTIAL EQUATION SOLVERS
196
Another unit group consists of four servounits mounted in common cabinet.
mechanism
Each of these companies also makes a variety of function generators. The most common of these based on a diode function generator. Another common type is based on a servo-driven non
is
linear potentiometer element.
tions contain a
Normally installa number of resolvers of this type.
Vivian (Ref. III.13.ae).
machine
a
describes
An anonymous
article
by M.I.T.
(Ref.
built
IILlS.a).
French electromechanical differential analyzers
by F. H. Raymond (Ref. and B. A. SokolorT (Ref. III.13.al). are discussed
A
Canadian computer
is
III.13.ai)
discussed by
J.
G.
function table which can be utilized either with
Bayly (Ref. m.l3.b). Other electromechanical differential analyzers and their applications are given in: R. L. Garwin
suitable servo drives for output purposes or to
(Ref. III.13.p);
introduce a specified plotted function. Photoformer function generators have also been
Rajko Tomovich (Ref.
There
is
also available equipment similar to a
D. H. Pickens
(Ref. III.13.ag);
III.13.ap);
B. B.
Young
(Ref. in.13.at).
introduced in this equipment.
Normal output equipment
of
consists
recording oscillographs mounted on a
six
References for Chapter 13
common
The equipment comes mounted in rack cabinets. The front panel of these chassis.
relay
racks contains the patch board connections and
which
also the ten-turn helical potentiometers
a.
b. J.
c.
are used. Electronic Associates has introduced a special device
which permits the
and with a
central location
digital setting
is
discussed
in
Bromberg and R. D. McCoy S.
Frost (Ref. III.13.n).
A
given in the two Cyclone
articles
of each
and
detailed analysis
is
Symposium Reports
(Ref. III.13.ah).
The IDA,
duced by the Computer Corporation of America discussed by S. Bosworth (Ref. IILlS.d).
In
the
M. Korn,
V. Bush,
differential
analyzer:
A new machine
(1945), pp. 255-326.
Cook and F. J. Maginniss, "More differential analyzer applications," Gen. Elec. Rev., Vol. 52, no. 8 (1949), pp. 14-20. A. C.
k. J.
1.
Crank.
The
Differential
Analyzer.
London,
Longmans, Green & Co., 1947. A. A. Currie, "The general
purpose analog
puter,"
Ref. HI.13.W, pp. 395-96, 399-400).
"The
California, University of, Dept. of Engineering. The Differential Analyzer of the University of California. Los Angeles, 1947.
Project Typhoon.
T.
"A
i.
Radio
Smaller devices have been constructed by Boeing and Curtiss Wright (see G. A. Korn and
193-227.
new analog computer," Electronics, Bosworth, Vol. 24 (1951), pp. 21 6-24. E. Bromberg and R. D. McCoy, "Calculating machines new tools for the Product
V. Bush and S. H. Caldwell, "A new type of differ ential Franklin Inst. Jn., Vol. 240, no. 4 analyzer,"
j.
has
Instr.,
h.
(Ref. III.13.1).
Corporation of produced a large differential analyzer for the United States Navy known as addition,
computer,"
solving differential equations," Franklin Inst. Jn., Vol. 212, no. 4 (1931), pp. 447-88. V. Bush, "Instrumental Amer. Math. Soc. g. analysis," Bull., Vol. 42 (1936), pp. 649-69.
The
America
analog
Vol. 21, no. 3 (1950), pp. 228-31. R. E. Beard, "The construction of a small-scale
for
Bell Laboratories differential analyzer is discussed in articles by E. Lakatos (Ref. IH.lS.y)
and by A. A. Currie
(1951),
Rev. ScL
Engineering, Vol. 22 (1951), pp. 85-88. f.
integro-dirTerential analyzer, pro
is
24
designer,"
by E.
(Ref. Ill.lS.e)
"An
Jn., Vol. 71 (1942), pp.
e.
REAC
G. Bayly,
d. S.
potentiometer.
The
Instruments, Vol.
differential analyzer and its application to the calculation of actuarial functions," Inst. of Actuaries
setting of all the
potentiometers used in the installation from a
computer,"
"Analog
p. 772.
relay
com
Bell Lab. Record, Vol. 29, no. 3 (1951), pp.
101-8.
m.
"The
Differential
Analyzer,"
160
Engineer, Vol.
(1935), pp. 56-58, 82-84. n. S.
Frost,
"Compact
analogue
computer,"
Elec
tronics, Vol. 21, no. 7 (1948), pp. 116-22.
o.
IH.13.K. References Descriptions of electromechanical differential analyzers at M.I.T. are given by A. B. (Ref. III.13.aa)
Macnee
and D. W. Peaceman and
J.
E.
Macon
Fry. Designing Computing Mechanisms. Reprinted from Machine Design, Vols. 17 and 18. Part I: "Basic elements," 1-6. 1945,
Aug.
Part II:
pp.
and dividing," Sept., 1945, "Multiplying pp. 7-14. Part III: "Cam mechanisms," Oct., 1945, pp. 15-21. Part IV: "Integration," Nov., 1945,
REFERENCES
III.13.K.
Part V:
pp. 22-26.
"Differential
equations 1945, pp. 27-30. Part "Servomechanisms," Jan., 1946, pp. 31-35. VII: Feb., 1946, "Stepping followups,"
and
ae.
VI:
Dec.,
differentiation,"
af.
R. L. Garwin,
"A
differential
analyzer for the Rev. Sci. Instruments, Vol.
equation,"
21 (1950), pp. 411-16. E. Goursat and E. R. Hedrick. Course in Mathe q. matical Analysis, Vol. 1. New York, Ginn & Co.,
Pages 52-58.
1904.
L. L. Grandi and D. Lebell,
"Analogue
computers
solve complex problems," Elect. West., Vol. (1950), pp. 70-72. s.
D. R. Hartree, differential
t.
"The
equations,"
its
5,
applications
"The
differential
in electrical engineer
(1935), pp. 51-71. v. W. G. Heffron, "Operation and applications of the differential analyzer," Product Engineering, Vol. 23, no. 4 (1952), pp. 164-70.
M. Korn. Electronic Analog D-C Analog Computers. New York, McGraw-Hill Book Co, 1952.
w. G. A. Korn and T.
P.
ential
analyzer,"
"A
Elec. Eng., Vol. 63
new
E. Lennard-Jones,
design
of a
M.
V. Wilkes, and differential
small
Phil. Soc. Proc., Vol.
A. B. Macnee,
J.
ao.
W. Thomson (Lord
et
realisation
d une
Operateur Mathedes Tele (OME)," Annales dite
Kelvin), "Mechanical integra
24 (1875-76), pp. 271-75. (Lord Kelvin), "Mechanical integra tion of the linear differential equations of the second order with variable coefficients," Roy. Soc. (London)
"A
35 (1939), pp. 485-
electronic differential high speed
Massey, "A
construction and
operation,"
Irish
its
aq ar.
universal unit for the electrical
Franklin Inst. Jn., Vol. 254
(1952) pp. 143-51. eliminates brain I. Travis, "Differential analyzer Mach. Des., Vol. 7 (1935), pp. 15-18.
M.
fag,"
Review of S. Rossland, "Mechanische von differentialgleichungen" [in Natur729-35], Math. Vol. 27 (1939),
Vallarta,
pp. wissenschaften, 127. Reviews, Vol. 1, no. 4 (1940), p. as.
A. Walther,
Differential
"Losungen gewohnlicher IPM-OTT," gleichungen mit der Integrieranlage Math. Mech. Vol. 29, no. 1/2 (1949), Zeit.
Angew.
differential
and selsyn analyzer with ball carriage integrators Rev. Sci. Instr., Vol. 20 (1949), pp.
coupling,"
"A
analyzer,"
Integration
Acad. Proc., Vol.
45A, no. 1 (Oct., 1938), pp. 1-21. ad. R. E. Meyerott and G. Breit, "Small
24 (1875-76), pp. 269-71.
Rajko Tomovich, differential
and R.
A.
Buckingham, Wylie, R. small scale differential analyser J.
W. Thomson
Proc., Vol. ap.
54-59.
874-76.
"Principe
tion of the general linear differential equation of any order with variable coefficients," Roy. Soc. (London)
I.R.E. Proc., Vol. 37 (1948), pp. 1315-24.
analyzer,"
Sullivan,
machine for
1954. an.
B. Bratt,
analyzer,"
ab F. J. Maginniss, "Differential analyzer applications," Gen. Elec. Rev., Vol. 48, no. 5 (1945), pp. S.
"Integrating
communications, Vol. 5, no. 4 (1950), pp. 143-59. am. W. W. Soroka. Analog Methods in Computation and Simulation. New York, McGraw-Hill Book Co.,
93.
H.
(1950), pp. 2-20.
Proc., Vol.
Cambridge
ac.
1
mathematique
matique Electronique
109-14.
aa.
(1941), pp. 337-54. A. Sokoloff,
B.
machine
(1944), pp.
the analog solving with y. E. Lakatos, Bell Lab. Record, Vol. 29, no. 3, pp. computer,"
"The
no.
analyzer,"
al.
differ
"Problem
J.
5,
R. Sauer and H. Poesch,
differential equations," Engineer s solving ordinary no. 4 (1944), pp. 94-96. Digest, Vol. 4, ak. C. E. Shannon, "Mathematical theory of the differ M.I.T. Jn. Math. Phys., Vol. 20 ential
221-28.
z.
algebriques,"
munications, Vol.
.
Kuehni and H. A. Peterson,
1 and 2. Corp., 1952. Parts F. H. Raymond, "Sur un type general de machines Annales des Telecom
mathematiques aj.
/., Vol. 83 (1938), pp. 643-47. D. R. Hartree and A. Porter, "The construction and a model differential analyzer," Man operation of chester Lit. and Phil. Soc. Mem. and Proc., Vol. 79
H.
28-May
ai.
I.E.E.
Computers
D. H. Pickens, "The electrical analog computer," Product Engineering, MQ\. 24, no. 5 (1953), pp. 176-85. ah. Project Cyclone. Symposium I on REAC Techniques, under sponsorship of U.S. Navy Special Devices Center and U.S. Navy Bureau of Aeronautics, April New York, Reeves Instrument 2, 1952.
no. 6
ing,"
x.
for use
ag.
mechanical integration of Math. Gaz., Vol. 22 (1938),
pp. 342-64. D. R. Hartree and A. K. Nuttall, analyzer and
u.
"Analyzers
in
Schrodinger
r.
pp. 106-7. H. A. Peterson and C. Concordia,
Gen. Elec. engineering and scientific problems," Rev., Vol. 48, no. 9 (1945), pp. 29-37.
36-39. p.
D. W. Peaceman and J. E. Vivian, "Bantam differ Chem. Eng., Vol. 57, No. 8 (1950), analyzer,"
ential
Part pp.
197
at
pp. 37-38. Bruce B. Young, "Advanced time scale analog Franklin Inst. Jn., Vol. 253, no. 2 (1952), computer,"
pp. 169-72.
Chapter 14
ERROR ANALYSIS FOR CONTINUOUS COMPUTERS
HI.14.A. Introduction
A
continuous
The must
computer
obtain
a
theoretical situation has
the
in
explained
case
linear
Winson
been thoroughly with constant
who
of representation of a solution of a system
coefficients
equations
introduced certain modifications into the original
F
4
=
(III.14.A.1)
theory,
J.
by
which we
(see Ref. III. 14.1)
will follow to
a considerable
Of course, many continuous computers deal with
There has been a number of practical applications of Winson s work to operational
systems of equations which do not involve derivatives, but in our present discussion it is
that there will be also a practical application of
extent.
III. 14. A.I is
advisable to assume that Eq.
of n
differential
functions
y lt
derivative of
Section
.
.
y}
III. 13.1,
.
,
yn
such
present
is
a
set
on n unknown
equations
that
the
As
yf*\
we may regard
highest
indicated in
the system Eq.
^ (
as functions A.I as determining yfi\ y of t and the lower order derivatives of y l9 yn .
.
.
,
.
.
.
,
.
One can hope
amplifiers with excellent results.
an
extended
version
applicable to
modern
of a theory multipliers
which
is
and function
generators.
An
error theory can be used to
knows
correction provided one
If,
however, one has
statistical estimate,
an error theory
original cause of the error.
merely a
compute a
precisely the
For the system Eq. A.I consisting of a single differential equation which is linear and has
yields merely probabilities or overestimates of the
constant coefficients, A. B. Macnee has given an
question which arises from the possibility of A
error theory (see
Ref.
III.Kd), and F.
H.
Raymond has investigated the error in a linear system with reference to a special type of computer
K.
S. Miller
and the author. The basis of is
this
the general existence theory for
ordinary differential equations which involves a discussion of properties of solutions (see Ref.
IH.Ke,
an important
There
are,
however, other ways in which an
error theory which describes the effect of errors
instance,
one
can
obtain
utilized.
specifications
For on
But perhaps the most important aspect of error theory stems from the circumstances under which differential analyzers individual components.
are normally used.
Suppose, for instance, one tentative design for
g).
stability
errors.
on the actual solution can be
(see Ref. III.Ki).
In the present chapter we will present an intro duction to the general error theory developed by
development
error. In this case there is
some
is
concerned with a
device.
By
the use of
However, the possibility that the machine may
physical principles one obtains a mathematical
be governed by a system of higher order than the system means that one has the
description in terms of differential equations.
represented
One then attempts
additional problem of comparing two systems, one of which is of higher order than the other.
device
This introduces certain phenomena which we /I errors. There is no classical equivalent for
call
this
in the usual existence theories.
technique to cover below.
this situation is
A
new
introduced
to obtain the behavior of the
by solving the system of differential equations on a differential analyzer. The given system of differential equations
precisely
known due
is
generally not
to uncertainty concerning
the physical assumptions
made and
the freedom of design possibilities. objective
of the
simulation
is
to
also
due to
The main show the
THE TYPES OF ERROR
III.14.B,
existence of certain behaviors for the device and
199
to determine approximately design parameters
Discrepancies of this type will be termed A errors. We use this terminology in the following
which
sense.
We introduce
G (y
a1 ,...,a m ,A1 ,...,Ar
will
yield the desired result.
Thus, the
uncertainty in the given equations may be more than that introduced by the machine itself. An error theory should indicate the effect of these differences not only between the given
mathe
matical problem and the machine solution, but
i
matical problem.
It is
and the mathe
extremely desirable to
know whether these differences produce behavior differences.
Of course, no general mathematical theory can The
yield this information in specific cases.
mathematics, at most, provides a framework within which the individual case can be con
The purpose of the present chapter is
sidered.
indicate
this
To
general framework.
one needs additional
results
specific
estimates and also computation.
It
to
obtain specific
is
)-0
are zero, the equations of the system Eq. B.2
become Eq. A.I, and
if
possible. Thus,
this
may require more computation than
analysis
original problem.
the error problem
is
A,
have certain
certain circumstances only one a parameter
be used, in others
many A
may be
s
a
many
For
used.
may
Similarly one or
s.
instance, a simple
equation in the form
y
may
=
-0.65y
(HL14.B.3)
be realized as
the
y
not surprising since basically more difficult than
This
and
The choice of the parameters a and A is not unique and can be utilized to emphasize any appropriate aspect of the error problem. Under
highly
type of error
if o^
nonzero values, then Eq. B.2 becomes Eq. B.I. Furthermore, if all the A s are zero, then Eq. B.2 is of the same order as Eq. A.l. specified
desirable that the machines themselves be used
for the latter
(IIL14.B.2)
which we consider as intermediate between Eq. A.I and Eq. B.I in the sense that if all the oc and A,-
also between the actual situation
?)
}
new system
a
= (-0.65
-f Oo)y
(HL14.B.4)
is
where a
due to an inaccuracy in the poten
is
tiometer setting.
the problem of computation.
Or,
owing to the dynamic Eq. B.3 may be
characteristics of the computer,
m.RB.
The Types of Error
realized as
The use of a continuous computer
=
0, system Eq. III.14.A.1, F{ solving a system of equations
G,
and
equivalent
The
i.e.,
y B.I, are
we do not
take
There are three ways in which the machine solution differs from the desired solution of Eq. B.I
may differ from that of
in the effort to Eq. A.1 because of inaccuracies
which does Eq. A.I. Such a discrepancy not raise the order of Eq. B.I above that of Eq. realize
A. 1
will
be called an a
error.
+
ly
=
(-0.65
+ a)j;
(IH.14.B.6)
discrepancies
errors in setup into account.
The system Eq.
analysis applies to
the intermediate system
Eq. III.14.B.1 as a
between the systems, Eqs. A.I and
A.I.
Our mathematical
constants.
(IIL14.B.1)
III. 14. A.I.
supposedly unavoidable,
to
In Eq. B.5, OQ and AQ presumably are small
=
of utilizing the solution
solution for Eq.
is
to solve the
On the other hand,
the dynamic response of the analyzer B.I be of usually requires that the system Eq. A.I. higher order than the given system Eq. differential
where A and oc
,
<x
range between
and A and
and
respectively.
However, in addition to the discrepancies between the systems Eq. A.I and Eq. B.I, the machine solution obtained may differ because of other causes.
Suppose we have assigned
initial
conditions for the system Eq. B.I which corre to the desired solution of Eq. A.I. The
spond initial
values of higher derivatives of the solution
be obtained by repeated differentiation. with Eq. (See also the discussion associated
may
III.13.I.13.) differ
The
actual
machine solution may
from the desired machine solution owing
ERROR ANALYSIS FOR CONTINUOUS COMPUTERS
200
to a discrepancy in the
initial
conditions or to
various perturbations which
may occur during the course of generating the solution. This between the actual machine solution
difference
and the desired one
The
be called a
will
error.
ft
addition
The machine must the
to
with
associated
Disturbances in
also utilize, in
information
stored
setup,
output of integrators. information produce
the
this stored
errors.
indicate the general effect of these three
types of errors in this chapter. The usual theory of differential equations shows that if a system of
equations depends upon aparameter analytically, then under certain circumstances the solution
How
depends analytically upon this parameter,
ever, this theory requires that the order of the
system does not change.
Thus,
this
theory
is
present, and, as
Since, however,
when no
one can show,
it
we must consider the
situation in
we must
provide for
present,
frequently utilized in practice for
purposes and, hence, is of considerable interest in itself. We will use it to study the A
We
error theory.
will
not develop a mathe
matically precise and detailed theory since the latter is available in the
and the author (Refs.
Winson
papers of K.
III.14.e, h)
Instead
(Ref. III. 14.1).
S. Miller
and in the thesis
we
will try
to indicate certain factual aspects of the situation
which cases.
will
permit an investigation in individual
Practically,
linearization
is
we
Let us
first
will
indicate
justified in general,
phenomena occur when
it
a solution of Eq.
is
is
valid for a range
a
t
<
<
This
b.
= 0.
also a solution of Eq, III.14.C.1 for a
The
classical
existence theory for systems of
ordinary differential equations
is
applicable here
and shows that for a range of a around zero, we also have a solution of Eq. C,l. is one point which should be made here connection with the dependence of solutions
There in
upon a parameter a which appears in the differential equations. The theory which yields results in this
connection
based on a process of
is
repeated integration relative to the independent variable. (See E. L. Ince, Ref. III. 14.c, pp. 62-75 F.
Murray and K.
J.
Thus,
48-93.)
Miller, Ref, III.14.h,
S,
cannot be
that
the
and what
even
when
a
is still
valid
these conditions are true only piece-
This permits the application of the theory than is normally
much broader manner
supposed. (See F. J. Murray, Ref. III.14,g.) Let us return to our system Eq. III.14.C.2 on the variables
y lt
.
.
.
,
and
vn,
which
/,
that the highest order derivatives yfi\
such
is
y^ (
.
,
.
,
}
are determined as functions of the lower order derivatives
and
tion process
parameter
oc
/,
We now consider the lineariza
system by means of a which we introduce below. This for this
willpermit us to apply the above existence theory. Let zl9 ...,z n denote a particular solution valid for the interval a
point in this interval, and
t
<,
let
<
Let t* be a
b.
z^,
.
.
.
,
zn
*
denote
the values of the z s at this point. If every function t is analytic in all its variables at this
F
point, justified.
thehypotheses or derivatives of a
although
"analyticity"
certain order, the basic construction
in
many
J.
in addition, is
wise.
In the present section we discuss the process of linearization for a system of differential equations.
of
Suppose, then, that there
pp.
HI.14.C. Linearization
is
are of
{
III.14.C.2 with specified initial conditions which,
also
the possibility of increasing the order.
This process
F
order.
normally require
which A errors are
and
l
same
error under these circumstances.
applies to a
(III.14.C.2)
{
;
a type of error directly applicable to the
A error
is
F = = 0. We suppose that H
when a
and A errors correspond to discrepancies
oc
equations.
We
reduces to a system
the
in the realization of the system of differential
/?
which depends on a parameter a and which
we may expand
in
a Taylor
s series
and
obtain
indicate the nature of the results of
the usual theoretical discussion of a system of differential equations having
a parameter. Let us
v v L L
?Fj*_ ( Z 2 (IA( J
__
consider a system of differential equations
%=
f
(III.14.C.1)
ot
=
(ffl.14.C3)
LINEARIZATION
HI.14.C.
201
where yf } denotes the variable ys and where the
where the
asterisk denotes that the
within each interval.
sion
corresponding expres
R
evaluated at t* and
is
dependent on
is
t
A) quadratic and higher powers of the(zj and (t - t*). We now consider a system
z^*)
coefficients
obtaining the existence of a solution of Eq. IIL14.C.5 with specified initial conditions.
there
is
a parameter which in Eq. III.14.C.3
is
For
1.
<x
=
some neighborhood
for
We
assume a
let r*
Eq. III.14.C.4 holds
1,
t
d.
<
t*\
of values
set
t
...
l9
t
t
m
In order to obtain a result which
is
significant
for the usual processes of linearization
consider Eq.
=
t*
t
III.14.C.4
For present
lf
for
/,
^
/
<
we
will
M
for
t
theoretical purposes one
might consider the aj]
dF-*
=
as functions of
-r-^-
solution of
Eq. III.14.C.4 or 3 which is equivalent
Eq. IIL14.C.2, differs from the solution of
C.5 because of the dependence on the parameter a. This dependence can be assumed to be analytic. If an adequate set of points, ? *m h as b een chosen, the systems Eqs. Eq.
.
these functions at
.
t
.
,
and
lt
this also
to the
applies
then not depend on zf \ These quantities would a, and we would obtain a theoretically simpler }
result.
However,
if
one wishes to apply
this
the theory to the usual linearization process, z n and their derivatives which values of zls .
.
.
differ
on the
the
greatly
same
in the
are the values of the
4j and the
solutions
zf**
at
<
t
<
termed
?
For
"linearization."
fixed initial
t
the solutions of the two systems differ merely in the introduction of a parameter
should have
little
oc
and normally
effect
qualitative
.
conditions
let
z,
=
z,M
(IIL14.C.6)
denote solutions of Eq. C.4. Theoretically know that there is a range of a around a
study on each
interval
t
<
t
<
l
t
l+l
before
for
which zf
suppose that Zj
is
this
analytic
defined for a
is
in
<
a range includes
oc
at
oc
=
0.
we
set
a
t
<,
b.
1
We
= 0, and that
Then
(III.14.C.7)
= 0, Eq. III.14.C.4 becomes
and of Eq. C.2 for a = 1. wish, therefore, to study the relation between
f*)
=
-
a Eq. C.5 for these
two
Let
of
sets
that Eq. C.7 -
III.14.C.7 constitute a solution of
The zf of Eq.
We
the next. passing to If
we
=
on a
fc)
0$ and zj as dependent we have made a complete theoretical t
since the
dependence of the solution of Eq. III.14.C.4 upon a is analytic. For prescribed initial
t
= a are
*
and that
con
Furthermore, one
initial conditions.
consider the
and has intervals,
and for the circumstances specified above,
can show that they are analytic in a. From the formal point of view, however, they require that
we
"general"
linear
is
which are constant on
coefficients
ditions
which
l
These values of zs and the derivatives may be obtained by an induction on / under certain since the values at general hypotheses
two systems.
The process of passing from the 5,
have
shall explore
t
obtained by
C.4 over the interval a integrating Eq,
We
further the relation between these
Eq. III.14.C.2 to
do not
specified region if they
initial conditions.
,
one must substitute
the given
3
i>
C.5 and 2 should have solutions which
is
determined by substituting the values of the z n and the derivatives of given solution zl5 ti
suppose that this range these circumstances a
Under
1.
<
<
We
a.
=
*o>
such that the corresponding d intervals match up and fill out the t interval a t b.
= 0,
a
includes a
to
has the value
the system Eq. C.4 with
is
also a solution of Eq. C.4 defined for
some range of
where a
and af are constant is, of course, no
There
difficulty in
Since Eq. C.5
(III.14.C.4)
dj$
z;,(f, ;
is
a)
The
z.
critical
convergent for
= -^
.
a
assumption
=
Since the
is
1.
z-
and
their
doc
(IIL14.C.5)
derivatives
depend
analytically
on the
oc s,
and
ERROR ANALYSIS FOR CONTINUOUS COMPUTERS
202
this is also true for the
ag and
term. This yields values of the zs and z* which can be used in an inductive solution for Eq. C.8
zf\ we may
the
differentiate Eq. C.4 with respect to a. Since differentiation with respect to t and a are inter
we changeable,
each
for
obtain
to obtain the v it
The values of the
interval
t
initial
v j at?
=
f j
are used both as
conditions for the interval
and also as derivatives of the zf f.(f }
a)
=
The
(III.14.C8)
%
l
t
l+l
similar.
is
In general, then, the linearization process can be referred to the solution of systems of equations
W
= d$
is an / and t depend on its derivative relative and expression involving to a and the derivatives of the af and the zf
where the ojg
situation relative to
t
relative to a.
^
}
The
relative to a.
interest
form of
exact
from the theoretical point
W
{
is
not of
of view. If our
to obtain a basis for the A error only object were assume as indicated above could we discussion,
that
W<
= Ri +
oc
However, to provide
.
dec
from the integration of the
over previous t intervals. system of equations will yield a coefficient III.14.C.8 for a Eq.
=
The above
intervals.
reasoning shows that
type of linear system
this
holds not only for the
first approximation z/f, 0) that similar systems hold Eq. III.14.C.7, but
in
for the usual linearization process other terms are involved, but it may be pointed out that these, in turn, arise
(IIL14.C.10)
which are constants over
for the other differential coefficients as well, and, thus, theoretically the only important point to be settled is
whether Eq. C.7 converges for
oc
=
1.
IH.14.D. The Notion of Frequency Response
*\
of a in Eq. IIL14.C.7 provided one knows the conditions for the vt,
initial
at
Normally the convergence of Eq. III.14.C.7 a = 1 cannot be settled on theoretical
for
doc
f
= Z& z/0, a)
= a,
for all values of
and
oc,
oc)
at
t
and hence
we can
Similarly,
=
obtain uft, a)
We may
analogous process.
III.14.C.8 with respect to interval
t
< l
l
t
t<
But
at
t
=a
i?,-
= 0.
oc
= a.
the given initial values
which can be used as the basis for a computa tional investigation of the sort upon which reliance is normally put in practice. This discus
g*
-
However, we can give a discussion
grounds.
by an
differentiate
sion can be refined to the point where yield
Eq.
a and obtain for each
t
merely indicate the
l+l
results of theoretical interest
W
i
(HL14.C.9)
St (t, a) depends on R it the partials of the to a, the partials of the a s and with { respect but not on z* s with respect to oc, and on the
it
initial
shall
can be used to show that
S much
are small and the
Around the
=
how
will
it
but we
value of
t
t
smaller.
one may linearize as
above by introducing the parameter a. A t interval is chosen so that the expansion relative
a has a convenient number of terms which are
where
to
R
determined by the procedure indicated above. This is assisted by the fact that the initial con
W\
the w
The
s.
also zero
the
vf
by
initial
the
values of the
]
uf
at
t
= a are
same argument used above for
.
ditions for the Wj,
Sit etc.,
are zero. Presumably,
then, one makes a
careful study of the solutions
on
?
this
t
interval
<
t
<
^ by means
of the
expansion relative to a. This study should provide accurate values of the unknown
can be obtained Theoretically similar results for the higher derivatives of zj with respect to a.
power
The
a sequence of systems, Eqs. etc,, which can be solved
functions zs at
= 0.
can be solved by successively solving systems
Thus, the given system Eq. III.14.A.1 has been referred to linear systems with constant coef
with constant coefficients and a linear forcing
ficients,
result
III.14.C.5,
is
9,
8,
inductively at
cc
Thus, the system Eq. C.5
series
repeated
for
and
^ so that the above process can be
an
interval
it is
^
<
/
<,
t%.
solutions of the latter that
one
HI.14.E. A
uses as a basis for the
ERROR EFFECT
approximate discussion.
Solutions for homogeneous systems with con stant
are
coefficients
linear
exponential functions,
exp
where I
(If)
i.e.,
combinations of
functions in the form
a complex number.
is
It is
203
equivalent to the treatment given in R. A. Frazer,
W.
Duncan, and A. R. Collar (Ref. IIL14.b, and E. L Ince (Ref. III.Ke, pp.
J.
pp. 160-62)
but modified
148-50),
slightly
so
as
be
to
appropriate for our present purposes.
natural, then, to try to express solutions of a
system of differential equations as linear com binations of such exponential functions with
much more
expansion of comparable com The A l5 hr which appear in the .
plexity.
.
.
,
exponential functions can be considered as the
complex frequencies present the solution
is
in the solution,
f\
highest derivative of
system
Eq.
if is
device. Clearly, then,
we
the
real
part of an
used as input to the can assess the accuracy
of such a component when the input
form
SQ exp (A we
and
4 r),
if
this
is
say
that
.
The
.
.
,
as
x%*>
a system
$= on xl9
.
.
.
,
xn
(i
in
=
l,...,n)
(HI.14.E.2)
which the order of the highest
derivatives $!,...,$ exceed, respectively, rlt
The machine
rn .
in general,
will,
.
.
.
,
have more
is
the original system of equations, for example, a
accuracy
the
-
con response throughout the computer can be sidered to be the frequency response of the com It is
rs
regarded
degrees of freedom than the original system. In
t
puter.
may be
in the
component has adequate frequency response. The range of values of A for which one has adequate frequency adequate
t
introduced on a machine, one
is
system
Now, in general, it is possible to determine how an electrical or mechanical computing com respond
(III.14.E.1)
x being of order
III.14.E.1 .
this
will
!,...,)
in terms of the determining x p\ x$*\ other quantities (see also Section IIL13.I). When
realizes
exponential function
=
on n unknowns xl9 ...,x n with, of course, the
(1IL14.D.1)
i
ponent
=
(
i.e.,
approximately
2c,exp(V)
Effect
Suppose we are trying to solve a system
useful approximation than a
series
s
Taylor
X Error
Such an expression may
constant coefficients.
be a
IH.14.E.
that the computer clearly necessary
have a frequency range which includes the ^ which enter into the approximation for the solution if the computer is to handle a given system of equations. The notion of frequency response is imme to linear systems with constant
of product does not introduce a new degree freedom. But a multiplier will, in general, not ?
output precisely linked with its input, and, consequently, a degree of freedom will be
have
its
introduced.
Most computing elements will new degrees of freedom.
generally introduce
The system Eq.
III.14.E.2
is
supposed to be an
III.14.E.1. approximation of the system Eq. the are derivatives introduced, Although higher
on system Eq. E.2 should depend only slightly these higher derivatives.
The
partial
derivatives
diately applicable coefficients.
In the more general case experience
has indicated that often one can apply these to nonlinear systems more frequency notions series or polynomial than effectively
power
latter indeed are a special approximations. (The motivates much of the This case.) limiting
should be relatively small.
Now let us consider the process Eqs.
III.14.E.1
2.
of linearizing
For defmiteness suppose
we have obtained
/)
following discussion.
For the convenience of those who
for the linearized version of Eq. E.I
solution of a system of linear equations precise with constant coefficients we discuss this problem is
(IIL14.E.3)
are not
familiar in detail with the processes used for a
in Section IIL14.H. This discussion
and
essentially
for Eq, E.2.
and
ERROR ANALYSIS FOR CONTINUOUS COMPUTERS
204
To
obtain a transition between Eqs. III.14.E.3 we introduce the following system
4
and
depending on a parameter
a positive
real
to
part corresponds
unstable
even if the original setups for the machine. Thus, system of equations, Eq. III.14.E.1 is well behaved in the sense that all characteristic roots
A:
of the system Eq. IIL14.E.3 correspond to stable Tc
=/,
behavior, the machine realization, Eq. III.14.E.2
%-/)
+
(HU4.E.5)
may
=
This system has the property such that at A 1 it it is corresponds to Eq. Eq. E,3 and at I E.4. Furthermore, the coefficients of the various
which appear multiplied by I should
derivatives
be small.
all
=
At A
we
have as
will
many
constants of
E.5 as for the integration for the system Eq.
order of the original system Eq. E.I.
we
If
It
nevertheless be unstable. also true that the extra terms in the
is
deteiminant have produced variations in the roots of Eq. III.14.E.4 which correspond to the roots of Eq. III.14.E.3. These variations have been termed by J. Winson a errors. However, if
the computer
should be
is
well designed, these variations
Unfortunately, in most cases
slight.
information
not
is
consider the determinant of the system Eq. E.3,
extent of this effect.
which, of course,
ever,
supposed to depend on;?, as in Section IIL14.H, we will have as many roots of A(p) At A
= =
is
as there are constants of integration. 1
and intermediate values of A we
many more
have
original roots,
root for A
^
the roots for A
i.e.,
will
For each one of the
roots.
= 0, there
is
a
which depends continuously on A
and approaches
the given zero value as A
A root for A ^ 0,
->
0.
which does not approach one
of the roots for A
=
0,
new mode
represents a
which does not correspond to anything original system of equations, Eq. E.I.
root will be termed an extraneous
in the
Such a
root.
In
J.
available
An
Winson has
this,
how
of linear elements which
in the case
is
concerning the
exception to
investigated thoroughly (see Ref.
III. 14.1).
The A
error
presents
two problems:
the
problem of obtaining a stable setup for the equations; and the problem of designing com ponents.
The
variation of the roots of Eq. E.4
from certain corresponding roots of Eq. E.3, i.e., the a errors, is dependent on the design of the components.
In a well-designed machine this
on the
variation should have a negligible effect solution. (See also
Ref. III.14.C,
K. S. Miller and F. J. Murray,
f.)
general, extraneous roots are large in absolute
magnitude, since they are the extra roots of a polynomial equation which appear when small terms of degree higher than the original degree are introduced into the polynomial.
An
exponential term in the form e^\ where A
an extraneous
is
something which will
root,
will
contribute
either
will disappear rapidly, or
it
contribute something which will destroy
completely the validity of the solution. If A has a negative real part of large size, the term $* will diminish rapidly. real
part which
is
extraneous term,
On the
other hand,
positive
and large
e*
completely dominate
*,
will
if
A has a
in size, this
the value obtained from the solution, and hence, the solution
is
useless.
An
which would require further but
we
The
The a Error;
Sensitivity Equations
In a discussion of a and
is
not
necessary to take into account A errors.
In
f$
errors
practice, instead of applying the
it
a and
f}
error
F =
0, we apply theory to the original system t it to the machine 0, in which the system G^ rise of order corresponding to the A errors has already taken place. This means that the errors
=
are obtained
on the
basis of two comparisons: a comparison of the original correct system of equations with an idealized system
the
first is
which has the A errors but no a or second
is
a comparison of
/?
errors
this idealized
with the actual machine system having
a,
;
the
system /?,
and
A errors.
investigation exists,
Consider the system
of an extraneous solution with
G
will ignore
possibility
intermediate case,
IH.14.F.
it.
i
=
(IIL14.F.1)
IIL14.F.
on the variables
xl5
t,
.
,
.
,
xn
.
,
.
.
.
.
,
(III.14.F.2)
i
how probably be of little value. It is possible, ever, by means of the theory to compute as many terms of the expansion of Eq. F.5 as desired. We how
begin by showing
to compute theoretically
the expressions
E
{
contains
all
the variables of G;, and, in
addition, has certain parameters
we
.
of order fy
is
E =
If
x **\ (
x^\
,
where the highest derivative of xs Consider also a system
where
205
ERROR; SENSITIVITY EQUATIONS
oc
<x
.
l5
.
.
,
aM
dx l dxz
-
then by the existence theory for exists a
III.14.F.1,
ordinary differential equations there solution
of
on the parameters a 1}
analytically
x. = x fa a
.
l5
.
.
.
a M)
.
,
depends
.
a,
,
M
amounts
.
ft,
III.14.g).
.
The
.
,
(III.14.F.3)
oc
s
we
solution
therefore, can be
both on the
F.
shown
to
and on the
J.
depend
either
or
oc
ft.
t
.. ,y,
.
.
the system of equations Eq. III.14.F.2, satisfy
.
.)
we
Taking
derivatives of each equation of the system partial
we
Eq. F.2 with respect to y,
Murray, Ref.
are interested
/Ts,
where y stands for
obtain
N
number of times by
finite
ftv (see
dy
substitute these functions in Eq. F.2.
This solution, Eq. III.14.F.3, can be perturbed in
a number of ways a
a/
By
Since x^t, .. ,,y,. ..),... ,x n (t
which
III.14.F.2
Eq.
9
are given a solution of the system Eq.
ay
ax, ay
dy
in,
(i=l,...,n)
analytically
(III.14.F.6)
Let
i.e.,
y
=
(III.14.F.7)
dy for
some region around the point a a
= 0, ft = 0.
Then r
cannot be Unfortunately, the size of this region
is
We
assume that the region to include the values which enough large
foreseen, in general.
actually appear
We
as errors in the machine.
of Eq. IIL14.F.4 around expand the functions the point a p
= 0, ft = 0, in a Taylor
x/f, &!,...,
ocjtf ,
ft,
j,ao
.
.
.
,
s series
= ?L
(IIL14.F.8)
dy since integration relative to
changeable.
Eq.
We
III.14.F.7
y and
t
niter-
is
see then that the partials
andjf
of Eq. m.!4.F.8
yf
of
satisfy
the linear system of differential equations
fa)
0=1 90L (IIL14.F.9)
In our present simplified discussion we assume we are interested in the solution of Eq.
that
III.14.F.9
when a
= 0.
Setting
a
= =
yields
(IIL14.F.10) (III.14.F.5)
If the effect of the error
solution, in general,
sequently, it
is
is
is
useless.
the large, then,
We assume, con
that the effect of the error is small,
to consider only the necessary
first
and
and
more
second degree terms in Eq. III.14.F.5. If would than these terms are required, the solution
We
can, therefore, find the
or a /? parameter for the /s.
if
we know
The following
/s
for either an
a
the initial conditions initial
conditions are
at the appropriate assuming computation begins the a errors and is an a. At t time t
=
=
y have no effect, and the solution given III.14.F.4
is
still
equal
to
its
initial
in Eq.
value.
ERROR ANALYSIS FOR CONTINUOUS COMPUTERS
206
-
Consequently,
j>
=
=
for all;
3
initial cases for the
1,
.
.
.
The
n.
,
errors are more complicated,
/?
and we defer discussing them. parameter y considered there by y x , with respect to
F.9
Eq,
parameter y 2
we
If
.
we
dif
another
linear
differential
the
systems
of
problem
second partial derivative, w jt evaluating reduces to finding an adequate number of linearly the
independent solutions of the system
let
=
z
According to theorems for
Eq. III.14.F.10.
Returning to Eq. III.14.F.9 and indicating the
ferentiate
on the w s Eq. III.14.F.15 is a linear system with precisely the same homogeneous part as
(III.14.F.11)
v f EL.
and and certain associated
We omit the
integrations.
explicit solutions.
t
=
now
we were
to try to
evaluate the higher partial derivatives,
we would
It is
then noting that differentiation with respect to and y 9 p 1, 2 is interchangeable, we obtain
apparent that
if
come upon exactly this same linear homogeneous system of ordinary differential equations.
an,
We
consequently, that the problem of
see,
evaluating the effect of the a and
y
y -
4-
errors after
ft
the initial conditions have been specified
z fti) y }a -^
is
a
<*i)
matter of solving equations of the system Eq. III.14.F.16. This system, Eq. F.16, can be justly considered
the
as
system for the
sensitivity
machine solution. Note that Eq. F.16 has in it already the perturbation due to the A error. If the A error does not introduce instability,
the a error effect
v + 2,
is
by the A error into the
,
and
if
small, then the perturbation
equations
sensitivity
is
small.
=
(III.14.F.13)
IH.14.G. The
p Error and Noise
Our previous Before xf*
and
partials
we
calculate the second
Xj with respect to y1?
by
IIL14.F.10,
the
solving first
sequently, the
for y 1
/s and
partials
y a we ,
linear
of the
find all first
system
and then for y 2
.
Eq.
Con
the z s in Eq. III.14.F.11
are to be considered as
known
functions of
t
so
that Eq. F.I 3 can be abbreviated in the form n
Sj
Orr
3rr
n
y y 5i wW) + y w .j.T(0 +y AAaxf) Aac/ (t)
(III.14.F.14)
process
we
we set a
refer to our
Eq. III.14.F.5.
We
ft
errors as well as
a errors provided some minor modifications are Since
introduced.
the
differential
themselves are not affected by
derivatives,
III.14.F.13
is
modification
is
the
homogeneous injy and and similarly the last term of Eq.
system Eq. IIL14.F.9 its
equations errors,
]8
is
omitted.
The only important
the determination of the initial
i
l
In Eq. III.14.F.14
discussion which specifies the
sensitivity equations applies to
= ^ = 0, and for this
remarks above following
obtain the equation
conditions for the /Ts.
The p
errors
correspond to jumps in the
quantities stored in the computer.
For
instance,
suppose the variable x1 appears in our equation to the third order so that xf\ xf\ x i\ xx appear (
in the equations.
We know that we can consider
our system of equations as determining the highest derivatives in terms of the remaining
ax (IIL14.F.15)
derivatives
and the
variables themselves.
(See
III.14.G.
ft
ERROR AND NOISE
is one Consequently, x of the variables determined by the system of
also Section
while
equations
(
f
III. 13.1.)
x *\ x \ (
(
%
are
variables
207
by introducing a noise
the noise can be described
generator into the system which appears in the Ordinarily an
differential equations.
oc
obtained by integration. These lower order derivatives would be stored in the machine in the
be
form of charges on condensers or
rotation of shafts. If any of these quantities are
parameter in front of it This description, however,
disturbed momentarily, the computer will con
describing the situation where the noise
tinue as
if
the disturbed value were the correct
new
value. In
its
solve the
same
The
as the total
continuation the computer will
equation as before. effect of the perturbation is the same as a
change of
differential
initial
conditions so that
jumped from one solution to another. For example, suppose in the case above a perturbation occurs stored value of
at time
we have described
^
in the
x^\ Then
introduced
describe
to
this
error can
effect.
We
introduce a noise generator term with an a
not adequate for is part of
is
the storage devices which contain the integrated
In
quantities.
we must introduce a
this case
mechanism based on the above description of the
We
errors.
ft
can divide the entire run into a
number of
large
relatively small
time intervals
At,-,
and for each stored element we can introduce
a
error at the end of the time interval corre
ft
in that stored sponding to the noise generated element during that time interval. Thus, the effect of the noise generator in a storage element
number of ft errors each equivalent to a large of which can be regarded as a chance variable.
is
We
can regard the machine solution for
t
^
>
as a solution of the machine equations Eq,
The contribution
III.14.F.1 with the values of the corresponding
individual variable can be given
except for ft
x^
total
error in any by the Taylor s
In practical
conditions
series
expansion on these
which has values dp(r) + ft. can be specified by stating
cases
we
the linear terms usually consider only
ft
where the
functions in every case as the
A
the
to
initial
error, in general,
stored quantity has been perturbed exactly which and the time at which it occurred. This fact will
I
^ eh
ft
errors.
is
coefficient
a known
opt
function of the time while
a chance variable
is
ftj
error.
In the above example, for instance, the
whose character depends on the actual chance variable which appears in the storage device. In
initial
conditions are given at the time
other examples
specify
the initial conditions for each kind of
and except
x^
for
all
this error are zero at
^
partials
t
=
/J
t
lt
with respect to
while
it
the chance variable
=1
(HI.14.G.2)
may
to
go
our procedure permits us
ft,
further. It is reasonable to
each stored quantity x
and the higher at
are
t
with respect to
ft
This will generalize to any
ft
partials
=^
of
x^
error.
Notice the symmetry between the /Ps and the a s inasmuch as the a error corresponds to
be necessary to go to
second order or higher degree terms. If we suppose a specific method of generating
assume that for
the expected
number
n
of noise perturbations per unit interval of time is of the time, and that when such a
independent
the amplitude distribution perturbation occurs, of the time with mean zero. is also
Let
cr
independent denote the variance of the amplitude
distribution.
Both n and
may depend on u, in
solutions of an inhomogeneous sensitivity system with zero initial conditions while a ft error
which case we write
involves the solution of a
chance variables involved are independent for different values of u. In the usual noise theory
with nonzero
initial
homogeneous system
conditions.
discussion of a and ft Ordinarily the above to handle the problem of errors will be
adequate
noise in a computing device. Noise can appear in one of two forms. In one case in a
computer
(see S.
0. Rice, Ref.
0ti
O
and
cr
0u .
III.14.J, k)
However, the
one shows that
under these circumstances one can divide the interval from f to t into subintervals in such a
way
that
the possibility
of more than one
ERROR ANALYSIS FOR CONTINUOUS COMPUTERS
208
in an interval may be perturbation occurring In the zth interval, say, a perturbation
neglected.
This will choose a time // in the
occurred at
r/.
zth interval.
If
be
chosen .
u (f )
=
no perturbation occurred,
For a given
0.
of an occurrence
may
the
in
arbitrarily
t
<
r_.
^
<
We
form then the polynomials
and
interval the probability
the probability of precisely ?
Af
<
A? (where, Actually
1).
k occurrences
in
and write the original system, Eq. III.14.H.1,
form
the
24(pto=/(0
(IIL14.H.3)
i*i
Mfc
j
in
n
an
is
/j.
with constant coefficients,
the differential operator/? were a number.
may
interval
be taken as
of course, Ar is so small that
interval
f/
differential equations
the system of equations can be manipulated as if
It is
convenient to introduce the matrix
-OL-
(HL14.G.3) k\
according to certain results in noise theory (see S.
0. Rice, Ref.
In particular, the
III.14.J, k).
of a perturbation at time
occurrence
is
t-
or successive occur independent of previous rences. Consequently, the variance of /?(*/) is
(
(We
assume
shall
linear noise effect
We
determinant of A(p) suppose that the
zero. In matrix notation Eq. III.14.H.3
seen to be readily
% A0
<7
is
is
is
not
becomes
1/2
(III.14.G.4)
independent of
The
u.)
then a chance variable
n
A(p)y=f these "Manipulating
e-
U
"noise"),
equations"
to multiplying both sides
equivalent (the presuperscript n refers to
whose elements are if f/is
Thus,
(III.14.H.4)
also
of course,
is,
by a matrix
polynomials in p.
any such matrix, any solution of Eq.
III.14.H.4 will be a solution of u*
Since the
and limit
/J
large
are
in
Op
now considered to be independent
number, we
may
theorem and obtain the
apply the central
result that
n
ef
is
zero and variance ^,
rftO
2
^
(We assume for
a
normally distributed chance variable with mean
this that
S.
Miller and F.
J.
/is
If the determinant of
required.)
constant, the inverse of U,
U~\
with polynomial elements.
Now
(IU.14.G.6)
denotes the cofactor of
Op
K.
(III.14.H.5) as differentiate as
the adjugate of U, the matrix
(Again in the unconditionally stable case we can obtain upper bounds for this expression.) (See also
=
U(p)A(p)y
Murray, Ref.
III.
He.)
u^
is
all
where U^
Since the determinant of [/is k, the inverse of is
U
equal to a constant multiple of the adjugate of
U which is a matrix with polynomial elements. if U is such that the determinant of U is k,
Thus
then every solution of Eq. III.14.H.5 solution of Eq. H.4. equivalent.
constant
equations with constant coefficients
a
are polynomials in/?.
IH.14.H, Solution of Linear Differential
differential
k,
elements of
(U^
Equations with Constant Coefficients
Consider the system of linear
is
U(p)
also a matrix
But we can construct a
determinant
where
is
also a
Thus, Eq. H.4 and 5 are
such
that
U
UA
with a
=
D(p)
,..
d
(
(p)
...
J=U=0 The usual method
for solving these
in certain
However, modification of the
respects.
elementary methods will lead to procedure.
Since
we
are
is
indefinite
a
D(p)
=
df\p)
definite
dealing with linear
(III.14.H.6)
SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS
HI.14.H.
This could be done by steps as in the proof of the theorem given below.
THEOREM
HI.14.H.1.
Let
B
Multiplying Eq.
an n x n
be
III.
Vi(p)^i(p)
14.H.9 by
$?(/?),
209
we
obtain
+
polynomial matrix
(III.14.H.10)
We ...
UP)
.-.
b nn (p)l
also have the relation
(III.14.H.11)
Thus,
MP)
we
let
...
There
exists
0,
a matrix ... ...
1
o
\
O/
WI(P) such that
||I7|
c
=
1
UB
and
(IIL14.H.12)
= C where ...
u (p)
(III.14.H.13)
...
Denoting by d
C
i.e.,
is
C
c nz(P)
l(P)
M
a matrix having
ilc
"-
all its
and by row and kih column of
the Kronecker delta
the element in the zth
wjJJ
U
i9
we have
^nn(P)/
superdiagonal
terms zero. Proof:
We begin by constructing a
b
[
$
=
=
0. (Note we B UjB, where 6$ denote the element in the rth row and jth [1]
such that let
U^p) with
(IIL14.H.14)
Let
column of B [k] .)
(IIL14.H.15)
There are three possible forms of U^p):
where
and b% n = 0, then U^p) is the which interchanges the first matrix permutation 1. and second rows and ||t/i(p)|| = 2. If
b ln
3. If
^
b ln and
consider
= (b\]\p)) =
#"
0,
b zn are unequal to zero,
the greatest
b ln (p) and b 2n (p). Thus,
common
[,w
= 2 (5.^
.(p)
fcJ
g6 w j
|2
(m.!4.H.16)
+ (fc/(p) - 1) ^(p)
we
factor of
+
(%(P)
+
^I(P) 5f2^i;
"~
~"
^2^2/P)
(^
(P)
^iA/P))
we have p)
(III.14.H.17)
(III.14.H.7) If
i is
neither
1
nor
2,
then
(III.14.H.8)
where b n
(p)
which are exist
and b n
"(p)
6g
are two polynomials in p
relatively prime.
Therefore, there i.e.,
two polynomials in p, ^(p), ^(p), such that
^i(p)^;(p)+y2(p^;(p)
=
i
(HI.14.H.9)
If
only the i
is
first
either
1
=
b
(III.14.H.18)
tti
and second rows are
or 2 and
are equal to zero, then
/ is
affected.
such that by and b%
b$ is
still
zero.
ERROR ANALYSIS FOR CONTINUOUS COMPUTERS
210
If
is 1,
i
and;
then
is n,
in
which
superdiagonal elements with the
all
exception of
b$~
l]
are zero,
(III.14.H.19)
If iis 2
and /
(Note;
C/s ,
1
<
s
1
(III.14.H.25)
m-
<
constructed similarly to
then
is n,
and such that
1,
C/1}
is
or
either
it is
a matrix
a product of
two matrices one of which permutes two rows of Bte-V
wniie
the other
is
of a type similar to
Uv
sometimes necessary in order to ensure at least one nonzero entry in the final diagonal matrix C.)
This
(III.14.H.20)
Hence, we have that
C/i(p)||
||
=
constructed a matrix ^(p) such 1
and a matrix
B \p)=U1(p)B(p) [1
U%(p) with determinant
B(p) such that
matrix
struct t/2
Since
(p)
common is
Actually B(p)
[1]
(/?)
by a matrix 0.
(p).
we must find
the
the greatest
4l~
1]
ujp)
-
of
4^
common
is
and
UmB
[m ~ 1 ^
is
similar
to
Ufy) by
common factor of ^~ 1]
so constructed that
=
1
(III.14.H.26)
a matrix having
all
super-
diagonal elements equal to zero.
To con
as in the case of i71
factor, 0(pj,
Um(p)
IIC/JI
in order to obtain a
to eliminate 4*1,
construct
= &!H(p)
g-6 =
we proceed
we want
greatest
1
Now
considering the greatest
and
such that
We now want to premultiply5
is
and
- UB
6gJ.
factor of
is
(III.14.H.27)
the matrix that satisfies the theorem.
Furthermore,
B
Premultiplying
[l]
(p)
by a matrix U^p)
equivalent to premultiplying
U2 C/!
matrix
for
by the
B by
is
the product
associativity
(III.14.H.28)
of matrix
The diagonalization process
multiplication
is
discussed in
E. L. Ince (see Ref. IIL14.C, pp. 148-50)
R. A. Frazer, (III.14.H.21)
We have
constructed
!WI
=
1
C/j
and
C/2
(1<J<2)
such that
(III.RH.22)
Hence,
IIWHItf The
2 lll|tfill
=
l
elimination proceeds until
(see Ref.
W.
J.
and
in
Duncan, and A. R. Collar
III.Kb, pp. 160-62).
Given the system Eq. III.14.H.3 by the above theorem we can find an equivalent system in the diagonal form
(ffl.14.EL23)
we have
zeroed
but the diagonal element in the last column. We then start on the next to the last column and all
then zero the elements
There are
init
above the last column.
= m elements of B that we
(III.14.H.29)
The determinant of Eq.
have to eliminate. Let us assume that we have obtained a matrix
4 (III.14.H.24)
n)
(p)
III.14.H.9,
III.14.H.
SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS
the product of the d ( and equals the f(p) determinant of the original system Eq.III.14.H.3.
is
211
A. R. Collar (Ref. IILKb, pp. 156-201), and E. L. Ince (Ref. III.Kc, pp. 133-56).
The system Eq. IIL14.H.29 can be solved by considering the individual equations successively. In general, then, when we have solved the first 1 xj_1 and have , equations for xit obtained these solutions with suitable constants
j
of integration,
we can
is
-
}
xj-i
d. We
can then assign
r
xf
-J,
we may
.
.
.
Xj.
,
The is
assign
total
rx -f
.
.
number of constants .
+ rn
which
is
d.
A. B. Macnee,
.
.
instance, since
*! alone,
each variable
r 2 is
that
we can
it is
is
fixed.
the
The
significance
^
any individual variable, we diagonalize in a manner that makes it the first variable. in
G. Doetsch (Ref. III.Ka, W, J. Duncan, and A. R. Frazer, pp. 321-39),
ficients
we
refer to
Error Analysis for Air Development
F.
1956.
mathematical basis and F. J. Murray, an error analysis of differential analyzers," Jn.
J.
on
"A
136-63. Phys., Vol. 32, no. 2-3 (1953), pp. "Mathematical error analysis for con
Murray,
tinuous
computers,"
in Project
H
Cyclone Symposium Techniques, under
and Computing
Simulation
Devices Center and sponsorship of U.S. Navy Special the U.S. Navy Bureau of Aeronautics, April 28-May 2, 1952, h.
i.
New
York, Reeves Instrument Corp., 1952.
Pages 139-146. F. J. Murray and K.
S. Miller. Existence
Theorems for
New
New York, Ordinary Differential Equations. York University Press, 1954. Pages 48-93. "Sur un F. H. type general des machines Raymond,
mathematiques tions,
Vol.
5,
algebriques,"
no.
1 (1950),
Annales Telecommunica
pp. 2-20.
S. 0. Rice, "Mathematical analysis of random noise," Bell Sys. Tech. Jn., Vol. 23, no. 3 (1944), pp. 282-332. k. S. 0. Rice, "Mathematical analysis of random noise," j.
discussion of solutions of systems of
linear differential equations with constant coef
on the accuracy of
I.R.E. Proc., Vol. 40,
Wright
S. Miller
Math g.
of
which we would obtain by taking xs first. In e r 2 we obtained by does not eclua^ general, jR 2 to get the maximum free taking x1 first. In order
For a
K. for
that can be
the above assignment of assigned to x 2 after values has been made to x^. Let jR,,- denote the r
dom
Sept. f.
higher
amount of freedom
Analyzers.
New
Center, Report 54-250 (Part 14). Published by the Advisory Board on Simulation, University of Chicago,
r assign initially x initial values, derivatives are all
this equation.
limitations
analyzers,"
no. 3 (1952), pp. 303-8. K. S. Miller and F. J. Murray. Differential
For
^ satisfies an equation involving
but no more since determined by
e.
.
associated with
"Some
electronic differential
the
order of the original system of equations. When an order has been assigned to the x n9 the number of constants variables xlt ,
1943.
J.
Cambridge University Press, 1952. E. L. Ince. Ordinary Differential Equations. York, Dover Publications, 1944.
specifically
to
W.
c.
f
constants in integrating theyth equation, or more we can assign arbitrary values initially
Anwendung der Laplace
New York, Dover Publications,
Duncan, and A. R. Collar. Elementary Matrices and Some Applications to Dynamics and Differential Equations. Cambridge,
i
the degree of
Theorie und
G. Doetsch.
Transformation. b. R. A. Frazer,
substitute these values for
n tne y th equation which now becomes an equation in x s alone. Its degree, r it
Xi,
a.
.
.
.
References for Chapter 14
1.
Bell Sys. Tech. Jn., Vol. 24, no. 1 (1945), pp. 46-156. J. Winson, "The error analysis of electronic analogue
computation linear
differential
equations with con
Unpublished Ph.D. dissertation, Columbia University, New York, 1954.
stant
coefficients,"
Chapter 15
DIGITAL CHECK SOLUTIONS
HI.15.A. Use of Digital Check Solutions is
Normally analog equipment the behavior of a complex system having components. The system variety of parameters, and behavior of the system
parameters are
sequence digital computers, which use a
used to study
many
may depend upon
a
in order to study the
many combinations
of
Thus, the study of the
tried.
became
it
memory,
practical
drum
obtain such
to
check solutions automatically. Formerly, these solutions were obtained by laborious hand digital
methods and were subject themselves to con siderable
solution
error.
Obtaining
involve an
may
a
check
digital
amount of effort almost
problem may involve finding many solutions
equivalent to solving the full system problem
corresponding to different parameter values. The advantage of the analog computer lies in the fact
the analog,
that such a study can be
made
cheaply and at
low cost per solution if many solutions of similar systems are to be obtained. Normally
relatively
most of the work involved in obtaining a solution is in the initial setup of the problem. Once the computer has been set up, parameters are readily varied, and the equivalent of major changes are easily
made by
the
person
having ultimate
responsibility for the design.
If the system
is
solution
and the machine time for one
may
digital
be the equivalent of the time for
hundreds of analog solutions. Nevertheless, the two processes complement each other nicely. It is
quite reasonable and practical to determine the
size
of the error
From
this
made
in the digital solution.
possible to obtain a reasonably
it is
good idea of the error
in the analog results. Frequently the objective of the whole study is to narrow the range of parameters to within
The analog computing equipment
certain limits.
reasonably large, a certain
on
is
excellent for this purpose
and can be used then
amount of preliminary study is desirable. For instance, we should try to establish scale factors
to determine the desired parameter values. Since
important to determine the desired type of behavior and also objection able behavior in some way before the problem
may be
for every variable.
is
It is
put on the machine.
An important part of this
preliminary study is obtaining a digital check solution for the problem. This digital solution is obtained by numerically integrating the system
the solution corresponding to these parameters
obtain
of great it
also
numerical
interest, it
information
An
may be
desirable to
digitally, particularly
given
is
alternative procedure
when
the
reasonably
to attempt experimental checks on the original system.
precise.
Many
studies
digital study,
go through repeated phases of analog study, experimental study,
over a few units of the independent variable. The original purpose of such a check solution is to
represents efforts to obtain
obtain a solution which could be utilized in
the system based
and
is
repetitions.
The analog and on
digital
work
an understanding of
scientific
knowledge.
How
checking the setup of the analog computer. The corresponding problem would be set up on the
experimental investigations, and in most practical
analog computer and solved by recording as
cases the basic data are not
many
variables
as
possible.
between the analog and the
Discrepancies
digital solution
could
ever,
such an understanding
is
always clarified by
known
well
enough
for adequate theoretical studies.
The
digital
check solutions are obtained by
be used to detect errors in the analog or
numerical methods.
conceivably in the original digital solution. With the arrival of the smaller automatic
Ref. Hl.lS.g;
J.
(See also
W.
E. Milne,
B. Scarborough, Ref. III.15.k;
E. T. Whittaker and G. Robinson, Ref.
III. 151)
USE OF DIGITAL CHECK SOLUTIONS
IIL15.A.
A
method
typical numerical
Milne
is
W.
that of
We illustrate
(see Ref. IH.15.f, h).
it
E.
with
an example.
Suppose we wish differential
solve
to
system of
the
and obtaining If
we
in
t
it is
similarly
g^
,
take a polynomial
R
9
M
.
,
.
of the p
M
.
g^.
,
1
degree
which agrees with/at f r _ / n 1} Mfl reasonable to assume that this polynomial ,
,
.
.
.
,
with /approximately on between ^_ x and t n We can
will continue to agree
equations
the
interval
.
y=f(t>y,z)
integrate the polynomial over this interval and
= g(t,y,z)
*
213
(IIL15.A.1)
obtain an approximation;
where the functions/and g can be evaluated by a digital calculator to a desired accuracy. III. 15, A.I
change Eq.
We first
R(t)df.
f(t,
y>
z)dt
to a system of integral (III.15.A.4)
equations.
This
an
is
"open"
process described
or extrapolated step.
The
and we can compute
is linear,
coefficients -(0)
=
+
ZQ
g(t, y,
\
Z )dt
(III.15.A.2)
Jt*
The system Eq. step process
in
(0)
is solved by a step-bywhich the values of y and z are
III.15.A.2
calculated for equally spaced values of
t.
denote the difference between successive
t
(o)
suchthat
Let h values.
Suppose now that we have
(Note: The discussion preceding Eq. HI.15.A.4 holds for
g as well as /.) Having obtained this open approximation, we can refine it by a closed
z
Z
and wish
z fr
l>
Q>
9
to continue to
approximation obtained as follows. We take our open values /Jf and z and substitute them
z n-fr z n-l tn ,
(
y n and z n ,
.
The
into
and g
/
respectively,
|X
tn-v+l,
f f(t,y,z)dt
\
and now we pass a
polynomial which agrees with /on
integral equation, Eq. A.2, is equivalent to
Vl
and on
t
n has the
^n-2>+2,
(0>
value/n
,
*n-l
where/^ is the value
obtained by substituting the open values.
(IIU5.A.3)
g(t,y,z)dt
can integrate
and
t
this
polynomial
also between
n in order to approximate Eq.
We t
n_I
A.4and there
We may proceed to do this in a number of steps
by obtain a closed value y^. The above discuss
which are roughly equivalent to the abovementioned method of Milne. We suppose that
ion for/and y also holds for g and z. The values yj* and z % are given by constants (
g are well behaved i.e., they of some reasonably high order.
the functions /and
have derivatives
As
is
well
known,
this
the solutions implies that
also have derivatives of at least the
same
which are independent of/,
we
substitute functions of
t
/ and g become functions of t suppose that we have done calculated solutions.
of/ and g
for
t l9
.
.
.
,
for y
= /n-1 v
this
v /n
and z then
alone.
g, or h 9 as follows:
order. (1)
If
-d)
-d)
-d)
i
Let us
with
the
We have, therefore, values fixed n _ v We can take a t
of /for ?_ integer;? and consider the values
Z
W=
These values, to obtain tively,
z tt-1
-fc
(III.15.A.6)
(
y^ and z $ can be used, over again closed values y& and zf respec
new
and
+ h 2 c?ig B
this
t
an process can be iterated until
DIGITAL CHECK SOLUTIONS
214
agreement
same
as c
for
j
For
reached.
is
( }
<;
all
(
i
0, c }
}
the
is
W.
(See also
1.
-/>+
>
E.
Most of the new old ones
quantities
by a simple
shift
are obtained
from the
of subscript while
yn
Milne, Ref. IILlS.f, h.) H. Salzer has a table of
and z n are obtained by a process such as that
these constants for the various values
discussed above.
Ref.
III.
As an example of
15.j.)
give for the case
p
dp.
(See
the values
we
=4
C^fi^ = -ttrf?8 = tt rf?4=
This method, however,
more
these
Murray, Ref. III.151) Notice that the first p values of y and z must be obtained
taking the open value
in z,
as the Taylor Series.
in
obtained by
is
(
j ^.
Consider, then, the example described above
J.
by some method such
illustrate
methods by an example
direct
which we suppose simply that y n
(See also F.
We
are available in the special case. "ft
(IIL15.A.7)
a
is
somewhat cumbersome way of establishing in this case, and more direct methods stability
which we have two dependent variables y and and suppose that we have in each computed
quantity
As we have remarked
y s an error ^y
Check Solutions
ffl.l5.B. Stability of Digital
above, the important
thing with regard to digital check solutions
accuracy of the solution.
is
the
In a digital process,
such as that described above, at each step we introduce a truncation error, which can be
y n on
described as the error in that
y n^
is
perfect.
the assumption
This means that
we con
sider the error in the solution as being
a succession of ft (See Chapter
We
errors,
III. 14,
assume the
one
at
due to
each step point.
an error
Z} It is
e,.z
the essential point in stability studies that the
of immediate errors
is not important; the whether errors will tend to grow. question past Thus, we can suppose that the errors mentioned
effect
is
here are due to past errors, and in the
find
we
propagation of this error
ny
from
t n_ k
y
for
k
=
1,
.
are interested
i.e., .
.
we wish
?J p.
We
to
then
have the equations
in particular, Section E.)
digital
machine
starts at z n_ l3
=
t nt and j n_i, ^_i, goes along a solution until / at that a makes error ft corresponding to point
the truncation error. ft
error theory shows
Our mechanism, then, how we can compute
final error in the solution ft
error at each step.
enough, the in these
ft
provided
If these
final error
ft
we know
X /(*-*,
J>
n -fe
+
-*?, Zn-fc
+ *n-ifeZ)
n for
zn
the
+ e nz =
z n _!
+
n _jZ
+ h j c.j. fc=i
the
X
g(f nHfc , y n . k
+
e n . k y,
z n .k
+
n _ k z)
errors are small (III.15.B.1)
can be regarded as linear
We
errors.
linearize
Eq. IIL15.B.1 and suppose that
Normally, there are two situations which arise in connection with these systems of differential equations.
ation that
and
We is
must plan in advance a comput and has adequate accuracy,
stable
after the
is
computation
finished
we must
estimate the accuracy of the numerical result.
Our
discussion of stability (see Chapter II.8)
is
for the range of variables considered.
Thus,
applicable to the present situationif we regard the quantities that are being cyclically
j
J n-ls
computed
^n
as
+ (III.15.B.3)
III.15.B.
Eq. III.15.B.3
a
is
we
system of difference
linear
equations as can be
STABILITY
shown
Eq. IH.15.B.K
for convenience,
if,
215
the result of
is
let
dz
dy *n-*Z
With Eq.
=
n -*
III.15.B.4, system
=
+
u n _!
/iB1 (Sc_jW
Eq. B.3 becomes
n . i ) -f
/iJ3
2
(Sc_
t; fc
fill 15
Stability in this case
is
(III.15.B.15)
(HI.15.B.4)
k
vn
diagonalizing the
matrix
dy
n -u
Returning to Eq. IIL15.B.13 and the various
n _ fc)
components u n and n ,we
B
equation, Eq. B.I 3, becomes:
t?
5)
see that our difference
equivalent to the rate of
growth of the solution of the difference equations, Since Eq. B.5
Eq. III.15.B.5.
we can
replace the pair u n
is
a linear system,
and v n by any pair of
Let
linear combinations.
The variables now Un
=
fll
h 1
un vn
B
n
+
4, h 2
fc
(y4 1
-J-
1
=
and each of the
(Notice that the above construction
fc
2
t;
n _ fc)
general. I
ati ori
B
7)
-
is
No matter what linear method of is
we can reduce
used,
the
}
2
( IIL15 -
w
separated,
individually.
nrri^Btt
n
- w n _ + /iEc_ w n _ -f 4 = v n ~ 1 hie ~ k(B u n ~ k 4- B x
are
equations of Eq. III.15.B.16 can be considered
flat?,/
quite
extrap-
stability
problem to a consideration of a system such as Eq. B.16 in which \ and 1 2 are roots of the characteristic equation of the matrix
(IH.15.B.8)
{>}
/?f
T wn
dz
dy
(III.15.B.9)
= w,^ dy
Introduce a change of variable
The procedure also generalizes (III.15.B.11) f
where the components of w n
are
{u n
,
vn
}.
Substituting Eq. III.15.B.11 into Eq. III.15.B.10,
readily to the case
O f more than two unknowns where
we would
have a more general Jacobian matrix and, say, Al9
r roots:
Of
we obtain
.
course,
.., Ar
we have made assumptions
con-
the relative constancy of the partial
or
wn
certain extent, the linearization process of Section
=
IIL14.C, above,
<-i
(III.15.B.13)
Ordinarily,
and T,
T has
distinct characteristic roots
distinct characteristic vectors so that ,
.
choose a matrix
,* t 1T C such that tf-TS ,
5
,
.
j.
diagonal,
i
ex pansio
^r
Sr IIL14
we can
is
f
f
is
^ ^.
justified in
..,,... denvatves are
partial
1
(III
15
B
a 14)
y s ^em
r
offrequency response for
^ nonlinear equations (see Section
s
HI.14.D)
that the supposing r
constant over reasonable
intervals that the notion
-
=
applicable to every system,
? , a 9 must be lun
case where one
-
STS~ l
is
but to be of practical value, the number of terms
is
applicable.
DIGITAL CHECK SOLUTIONS
216
Our
with the
deals stability question, then,
of the equation
stability
maximum In
(IH.15.B.17)
by H.
always of the same size. Then stability requires that an error should not grow in size. The solutions of this system are obtained
by
first
considering special solutions in the form
The
III. 1 5. B. 17.
III.15.B.18
Eq,
substituting
result
in
Eq.
an equation on 6
is
itself has
changes as the problem progresses, is obtained by expressing
then a stability criterion n
Q
in the appropriate scale for
Of course, it is not always
t
n.
possible to determine
completely the stability of a computing procedure advance since Ax and A 2 which are introduced
in
,
for
and
Ref. IILlS.d).
Jr. (see
Gray,
a negative real part.
may not be known in advance the whole problem. In many cases, however, III.l 5.B.14,
by Eq.
(III.15.B.18)
J.
of interest, one has that h
many cases If the scale
Let us suppose for simplicity that the variables for which we are trying to solve y n and z n remain
Such mappings have been
real part.
investigated
we have
which
situations in
provided h
is
sufficiently
Frequently, a
to take a tentative value of
is
practical procedure
appears
stability
small.
(III.15JB.19)
proceed with the computation and test by various methods for both the accuracy and the
6*-* from Eq. III.15.B.19 and
discussion of this type of procedure which will
h,
c_
n
=
fc
fc
We
stability of the solution.
We can factor out
which 15
2,
will .
.
.
in
have, P
,
= solutions
p
general,
which ate functions of h^. The on u
is
=
S^
give a
or not.
m.!5.C. The Accuracy of Digital Check Solutions
The
general solutions to Eq. B.17
wn
now
be valid over intervals of a certain length is stable irrespective of whether the integration
obtain fc
will
basis for the various
integration
B
(III.15.B.21)
a linear combination of the corresponding 1, then clearly
by
methods used for
various means
approximation to the functions integrated.
a polynomial
is
which are to be
Suppose we have a system
solutions. If all the 0, are less than
u n will approach a fixed scale
as
n increases. For the case of
we would
the
call
integration stable for this system
for both values of
and g do not have
A/z.
Thus,
if this
if
/
which vary
too rapidly with respect to y and z, a relatively simple method can be used to determine the stability of the proposed integration method. The 6 S just obtained as functions of Aft are
complex numbers. interested in
\B }9
it
Since is
we
are
at
happened
the functions
partial derivatives
T
method of
Our
^o
ft
desirable to investigate
(
n
~
at these
t
is
log
\6 3 \.
If log
desired result obtains.
|0^|
In
is
.
.
,
zr at points /,
To
/
+
/z,
.
.
and consequently also the values
points.
.
,
of,
We now take a polynomial Q(t) at the points
+ (n-p + l)h, + (n - l)h, where
+
r
p
+
f
(H
p)h,
(- p + 2)ft,...,
is
j
some
fixed integer.
find
log 6 j and, in particular, the real part of this
which
%
.
which agrees with/^
f
ordinarily
+
we have
step-by-step procedure assumes that
the values of zl9
nh)
-
negative, then the
many
instances
it is
ffi)dt
highly desirable to study for a given integration process log 3 as a complex function of A/z. More
(ffl.15.C2)
-
precisely,
of log
0,-
what one usually
desires
is
a mapping has a 0^
for the value of; such that log
we
integrate Q(f) between (n
result
is
V)h
and
nh.
The
a certain linear combination of the
ACCURACY
III.15.C.
values t
6
+
of/; at
(n
!
-
+(
\)h,
(_ 2
+
,
Since J
...,
)h,
p)h:
217
0, the coefficient
>
of
to Eq. III.15.C.7
not zero, and, hence, there
is
such an S. let
is
(n-L
I)!
...
(III.i5.C8)
and
-p)h)] tfO
= *W-rtO
for
=L
(IIL15.C.3)
To
evaluate the
N W accuracy of this process
consider a function F(r) which has
tinuous derivatives.
of
F
at
f
-
-f
(
p
+
us
let
con-
1
~^P
Suppose we know the value
%
...
4-
rn
Zer
- v)h (n
^
=
^ ~ m
S
^
skow
that
^
(
^
^>
*+11
by the above of the
p
-
1
Let
process.
(-!)*
F
+
)
Eq IIU5
at
C7
*
p
-
1
distinct
^ induction one can /
,
in the interval
Thus,
= F^V)-
Hence, we have shown that
g(0 be a polynomial
degree which coincides with
^%
=
/(/) has
and
W has a zero,
~(p~l)h
(IIL15.C4)
"
~(P
dlstmct zeros in the interval
this interva1
and
wish to evaluate
has
(III.i5.C9)
~A
there
if
^ is determined by
a value f such
is
^
--fi
Wewillhave and
+ (B +
,,
Since the coefficients c
depend on r
+
(n
q,
.
.
.
- j)fc)]
Vl
,
(()
A - %F(0) + ^(-/j
do not
=
we may consider the case in which = 0. Our problem is then to find 1)A r,
the accuracy with
2(0
,
(n
fA
"
^
^t^
J
"^
.
R(Q ^
which the expression
+
(P
=p
{(g
1) !
+ A).
A= + V/HP -
W]
(IH-15.C6)
*
f
approximates
^) A
A
.
,
, N
given the function F(t]
with
f+
this,
we must clearly find the accuracy with which
I
To
continuous derivatives.
2(0 approximates
F(t)
LeU be any value
on
evaluate
^ ^ A. /
=
Let
0
.
.
Let us suppose that our A interval
we can regard
that
^ml ^ _ this
<
t
^_
-, (J ,
^
so small
as constant
L
<
integration procedure
,..
e
l]h
F (M} (t)
is
on the
the error
h
is
+1)
"
x ^
be a constant such that
o
(p
+
1
1)-
(III.15.C13)
^
= /? ..
.
/
^\r^
+ (?-*)*) i
The dependence of e on h
is
readily determined.
Letus produce a new variable
I)!
(HI.15.C.7)
xft
=?
(III.15.C.14)
DIGITAL CHECK SOLUTIONS
218
we
Substituting in the integral in Eq. III.15.C13,
obtain
+
.
.
.)
can be shown to involve A to be a power of
at least
+
p
>
on
z i depends only
2 while
/z
p+1 .
Thus, to the lowest power of h we can regard BJZ, as in the
=
h
+
f *(*
v+l
form
Jo
where
+
(p
l)l
- h* +1A 9
(III.15.C.15)
=
A
-
(p
+
D!
Consequently,
Now
let
(III.15.C.22)
us return to the integration of the
differential equations
of Eq. III.15.C.1. If we had
the precise values of/i at
+
?
(n
t
=
t
+ (n
.
l)h,
.
.
,
p)h the open integration procedure t
described above would introduce an error in the
value of z i of
made
and we have found that
at each step,
y+l proportional to h
it is
.
Now suppose we have some fixed interval over which we must integrate the system of equations of Eq. III.15.C.1. Let us suppose that our error
z^tf^h^A,
eo
Thus, under these simplifying assumptions we have obtained the expression for the p error
(III.15.C17)
wherebyEq.IIL15.C2andl2,
is
we may apply a purely This means that in con
small enough so that
linear error theory.
sidering the dependence of the solution
on
these
p errors, we need to consider only the first partial derivatives. Then the total error in the solution (IIL15.C18)
Now if we wish to manner
the
+ (n
t
are
that the values of zl5 z 2 ,
the
correct,
+
f
computed values
- 2)A,
(n
+
f
-
(
are incorrect to a certain extent. fl
,
.
.
,
zr
l)h are correct, but if these values
quantities at
the
we
indicated in Section III.14.B,
must suppose at
apply the /terror theory in
error at
?
+
/z
in z
is
be a linear combination of the p errors given
will
approximately by Eq. III.15.C21. Suppose now we take a sequence of intervals which are
approaching zero. The individual f} n errors will approach zero, but the total number of them
of these
will increase in
3)A,
finite
The value
for
then
proportion to \\h for any fixed
length of interval. Consequently, the total error in the solution will approach zero in
proportion to
R
For an open integration method based on - 1 order, i.e., an polynomials of the p open
method using p
integration
constants, c
,
.
.
.
,
tne total error in the solution will approach zero with the p\h power of the step length h.
where/;* denotes the computed values of zl5 the f
.
.
,
of substituting the
zr These are not on .
solution which has the given values at
+ (n -
and
result .
18
we
l)/z.
Thus
if
we
use Eq. III.15.C17
This
linear error
are
obtain
permits
one very effective method of on an interval for which a
estimating the error
is
theory
applicable.
Two
solutions
computed, one with a given value of
other with a value half as large.
The
h,
the
error in the
second solution should be (2~*) times the error in (III.15.C20)
A
the
first.
If
p
is
large, say 5,
complete error theory should, of course, handle the last term in Eq. III.15.C20. A
between the solutions
procedure for doing
solution
and
F.
this is described
by P. Brock
Murray (see Ref. IILlS.b, pp. 99-109). However, we will point out that ^(/J -/,*) J.
the
first
solution, is
is
then the difference
essentially the error in
and the error in the second
p approximately 2~ times
This procedure
this.
dependent on an assumption of linear error theory. However, there will be a is
ACCURACY
HI.15.C.
length / such that the above estimates of the error
can be used to
justify the
error theory for tion
r
<
t
assumption of a
<
f
+
/.
this is
However,
process
that
not
is
many is
are
proceeding
simply by open integration methods,
We
same value
the
(p
2)/z,
p
-
1
how
find
let ?(r)
degree which has
as F(f) at the points h, 0,
Ay
is
positive,
By
is
negative,
Thus, suppose that our integration procedure one for which in each step we obtain a set of values z ( [\ ..., z
"open"
closed integration the equivalent problem to the one stated in connection with Eq, III.15.C.6 is
the following: Let F(t) be as before, and
we can effectively use
these circumstances
(
}
for the
unknown
can,
however, apply a similar evaluation to the ft error obtained in the case of a closed procedure using the same kind of assumptions, In the case of a
be a polynomial of the
given by
the fact that while
acceptable in
we
an open and closed integration
(IIL15.C.26)
Under
practical instances.
The above assumes
error in
ft
is
one has no
reason to believe that the linear theory applicable.
while the
This justifica
essentially the statement that
is
linear
219
h,
.
.
.
functions zl9 ...,zr and a set of z (f,
.
this
.
.
for them.
,
If the
"closed"
"true
values
result"
z<J>
is
step
solution zl5
.
.
.
,
zr which pass through the z
values obtained in the previous step, then z z (f differ from this true solution by j8 nz
(<
ft^ (
zf
9
of
considered to be the values of the
Since
respectively.
we can compute
we have both
}
>
and and
(
z f and
the difference
,
closely
(III.15.C27)
and obtain from them an expression for the errors
C,. 1 F((p-2)h)] f
(III.15.C23)
F(t) dt.
approximates
By making
the
same
Jo
assumptions, error
A,
A
we get the equivalent result that
and
the
is
(IIU5.C.24)
For
we use
procedure involving an open Eq. III.15.C29 will give us an approximate expression for the truncation error which we make at each step in our development.
Thus,
if
and a closed
B
= (p
+
1)!
If we are using this
a
step,
purpose,
an automatic
we can
computer for compute the
digital
readily
conse expression Eq. C.29 at each step, and, (IIL15.C.25) quently,
The assumptions made
are of such a nature
is as good an estimate of the error which corresponds to the first step of
that Eq. III.15.C.24
the closed procedure as
verged
it is
for the final con
result of the closed procedure.
Actually
the improvement obtained in pushing through the convergence process for the closed integration is
a matter of higher powers of
The expression Eq.
estimate of the
If this truncation error indicates that
using too large an interval,
we
are
then there are
procedures which will permit us to shorten the interval. Conversely, if our error expression indicates that our error
that our total sequently, will
h.
III.15.C.24 can be regarded
we can always have an
truncation error.
be
\B 9
\
large, is,
then
is
too small and, con
amount of computation
we can lengthen
the interval.
in general, smaller than [XJ, and,
in the closed
an open and consequently, a procedure involving
same assumptions procedure. Making, then, the as before, we can say that the ft error in an open
a closed step will be more accurate than a an open step only. This procedure involving means, however, that for each step twice as much
as the error after the
integration process
is
first
step
III.15,C.21 given by Eq.
DIGITAL CHECK SOLUTIONS
220
computation has to be done in the open and closed case than in the open case alone. Approxi mately the same amount of computation would
be done for an open procedure and step length /z/2 as for an open and closed procedure with step length h. Thus, in comparing the merits of
the
two procedures, we should use
the situation
which the same amount of computation is done in each case. In the open and closed
in
procedure the error for a single step
is
propor
By while the error for two steps in the open procedure with half the interval is propor tional to with the same constant of tional to
A^
In
order
approximating the unknown function by a poly many types of problems the function
nomial. In to
be integrated can be more
effectively
mated by a sum of exponentials.
approxi
Such an
approximation permits an integration procedure with constants which do not depend upon the independent variable
In fact, one can
t.
that approximation by polynomials
show
a limiting
is
case or a special case of the approximations
by sums of exponentials. Dr. Paul Brock and the author have extended certain of the methods of numerical analysis to the case in which sums of exponentials are used instead of polynomials.
the
This applies, in particular, to the numerical
accuracy of the two possible procedures we must compare \B P with AJ2*. We list our results for
integration procedures discussed in the present section (see P. Brock and F. J. Ref.
proportionality.
to
compare
\
=
/
Murray, and R, E. Greenwood, Ref. III.15.e).
W
W
1
.12500
.25000
2
.02778
.02778
3
.01172
.01042
analysis for the numerical solution of a test system of
4
.00436
.00528
in
5
.00172
.00313
!,.. .,5:
P
III.15.b, c }
References for Chapter 15 a.
P.
Brock and F.
differential
Thus, in the cases for
p
=
1,
2, 4,
and
There
is
a certain auxiliary
h as small as possible
since this tends to decrease
steps were used normally, but at every tenth step an open and closed step was used so that the truncation error could be
estimated by Eq. IIL15.C.28. The result of the closed integration was not used to continue the
P.
Brock and F. J. Murray, sums in step by step
same values of the derivatives
"The
integration
use of exponential
II,"
MTAC,
Vol.
d.
H.
J.
e.
Gray,
Jr., "Numerical
methods
in
digital real
Quart. Appi Math., Vol. 12. no. 2 (1954), pp. 133-40. R. E. Greenwood, "Numerical integration of linear sums of exponential functions," Ann. Math. simulation,"
Stat.,
Vol. 20, no. 4 (1949), pp. 608-11. f. W. E. Milne, "Numerical integration of ordinary differential equations," Amer. Math. Monthly, Vol. 33 no. 9 (1926), pp. 455-60.. W. E. Milne. Numerical Solutions g. of Differential Equations. New York, Pages 53-71. h.
W.
E. Milne,
"On
John Wiley
the numerical
&
Sons, 1953.
integration of certain
differential
at the
equations of the second Math. Monthly, Vol. 40, no. 6 (1933),
upper end of an interval which are used for
the closed integration are also used for the open integration on the next interval. Consequently, the derivatives were computed only once at each step. In the specific cases considered, it seemed
evident that this was the most
i.
j.
k.
principles for numerical analysis.
mental approach
in this
procedure consists in
J.
Murray,
"Planning
order,"
Amer.
pp. 322-27.
and error considerations for system of differential
equations on a sequence calculator," MTAC Vol 4 no. 31 (1950), pp. 133^4. H. E. Salzer, "Table of coefficients for
repeated 7
integration with differences," Phihs. Ma?., Ser Vol. 38, no. 280 (1947), pp. 331-38.
The above procedureis based on the
customary The funda
F.
the numerical solutions of a
efficient solution
procedure.
6,
no. 39 (1952), pp. 138-50.
solution but an open step was used in the tenth stepalso. Thus, the
equations on the IBM sequence calculator,"
Project
time
which open integration
and error
c.
advantage in taking
the errors due to the various assumptions we have made. The author has used a procedure in
"Planning
b.
5 the
open procedure is at least comparable accuracy with the open and closed procedure.
Murray,
New York, Reeves Cyclone Report. Instrument Corp., 1950. P. Brock and F. J. Murray, "The use of exponential sums in step by step integration," MTAC, Vol. 6, no. 38 (1952), pp. 63-78.
single step
in
J.
1.
J.B.Scarborough. Numerical Mathematical Analysis. Baltimore, Johns Hopkins Press, 1950. Pages 135-308. E. T. Whittaker and G. Robinson. The Calculus Observations:
A
of
Treatise on Numerical Mathematics.
London, Blackie and Son, 1944. Pages 132-63.
Part
IV
TRUE ANALOGS
1
Chapter
INTRODUCTION TO
IV.1.A. The Concept of
Our
"Analog"
information will permit us in cases to describe in mathematical language
many
scientific
a natural system which we wish to study. Often this mathematical description has an implicit character,
i.e., it
involves
unknown
functions in
such a way that we cannot readily solve for the unknown functions or quantities. Under these circumstances, very frequently an effort
is
made
up an analogous system which satisfies same mathematical relations but practically the
to set
whose behavior can be
readily investigated.
In the present volume, we distinguish a special type of analog device in which individual mathe matical operations are realized by specific com of these components, one can ponents. In each
between inputs and output. Such a device can be justly called a "computer"
clearly distinguish
and,
when
are represented
the, quantities
magnitudes, a
"continuous
devices were treated in Part
computer."
III.
There are devices of a somewhat class in
by These
different
which numerical information can be
obtained by measuring magnitudes. In these, mathematical relations, as distinguished from operations, are realized, is
made
signal
and
in general
no
effort
to establish a unidirectional flow of
from inputs
one has an description,
to output.
original system,
and an
"analog"
to
is
made
description.
investigation"
which the same
called
of the original system.
cases, the use of
the setting
This
necessary, in the
description, scientific
"analog
In some
"dimensional analysis"
up of an analog in the form
permits
of a model
without complete knowledge of the intermediate mathematical description.
instance, to
first
since
the
latter
But often the mathematics set
up
the
represents
expression of the natural laws involved. is
such that one can
correct principles for the correspondence
without explicit reference to the mathematics. This is convenient in many instances and extremely useful to persons
who
com
readily
prehend the behavior of the analog. This behavior may be known, even without a physical realization of the analog.
Some people
process quite correctly, but call
it
use this
"reasoning
by analogy," a term which also applies to an incorrect logical process involving extrap olation.
The notion of an analog based on an
inter
mathematical description is general include as, a special case, certain to enough Con applications of continuous computers.
mediate
tinuous computers are often used as analogs in the sense described above. On the other hand,
an analog is used as a computing device when one obtains numerical information from it. But it will
be convenient for our purposes to
above as follows:
without reference to the intermediate mathe matical
it is
base the correspondence on the mathematical
Thus, normally
on the comparison system and the analog is
manner,
a mathematical
to proceed
between the original
To set up the analogy between an original system and an analog in a sound scientific
terms
mathematical description applies. Since the behavior of the analog is readily observed, every effort
ANALOGS"
"TRUE
"computer"
and
"analog"
restrict the
as indicated
A computer is constructed of
at any one instant components, each of which realizes one mathematical operation and for
which there are
clearly distinguished inputs
and
the analog, one has mathematical output. In relations and no effort for a unidirectional flow
of signal.
It is
a continuous computing device in
the sense that numerical information
by measurement, but we "computer."
shall
is
not
obtained call
The present part of the book
it
a
will
be concerned with analogs in this sense. (Of course, this distinction is never perfectly clear cut in practice.)
INTRODUCTION TO
224
It
should be
clear, then, that
notion of an analog in a
one can
sound manner. Analogs have been
on
of
electrical circuits, the flow
set
and
scientific
set
up
the
logically
produce many such mathematical concepts in the future.
up based
electricity
in
ANALOGS"
"TRUE
Analogs
offer fascinating illustrations
of the
and
other
mathematics
between
interplay
Mathematics
has
been
electrolytes, the transmission of light through
sciences.
and a host of other phenomena involving mechanical and elastic properties. Persons
enriched by concepts derived from analogs.
familiar with the behavior of certain types of
matics have had strikingly unusual applications,
systems can
in
which some analog
in
the condition of the special relativity theory,
matter,
such analogs for technical purposes very effectively. This means that their study of the original given problem is made in utilize
terms of concepts associated with the analog system rather than concepts associated with the intermediate mathematics.
The relation between an original system and a mathematical description is based on an abstrac tion process which recognizes the applicability of
the other hand, concepts developed in
is
almost
these
every
case,
a converse process.
The mathematical
precise.
description of
The setup
of
not
an
number of
original system usually represents a
approximations.
are
processes
In
the
ordinarily involves a simplification of the
The
matics
For
system.
this
purpose,
many
it
are abstracted and developed in
itself,
general natural laws.
Because of simply
the
this,
applied mathematics
application
is
not
of pure mathematics
developed for itself alone. The two are joined in such a manner that the development of each
would be
far less if the other
were not present.
differences in emphasis, should not
subtle concepts
instance,
order to permit the expression of the most
analog
objective
For
Concepts introduced into mathematics by analogs, as well as those which arise in mathe
mathe
of the mathematical description should contain the best possible scientific analysis of the original
and
involved.
The
matical description and an approximate physical realization of this simplification.
is
On
mathe
constancy of the speed of light is treated as if were the imposition of a metric on the space.
general natural laws to a particular situation. Setting up an analog
frequently
surface distinctions between the two, the
be permitted
to obscure the fundamental unity of mathematics
and the unity of mathematics with science
as a
whole.
IV.1.B. Analog Applications
ever, practical considerations frequently force
It may seem that in design problems one has an option of a purely experimental approach, or a theoretical approach, or a compromise between
one to ignore much of
the two extremes.
distinctions are available, representing past
experience summarized in natural laws.
setting
up
the
When this is the case, it is inadvisable to
analog. rely
this in
How
on analog reasoning alone; one must keep
the mathematical description in mind. interesting to note that
It is
of mathematical
analysis
are
tangent,
and normal
most concepts expressed
clearly
refer
geometrical analogs of analytic situations.
new and
in
to
When
interesting concepts are introduced into
mathematics by applications, they are usually associated with an analog. Striking examples of this are furnished by the notions of vectors and tensors,
and one can expect
However, in most practical information is of little
empirical
on a thorough and understanding, experimentation both time consuming and expensive. Thus, in
practical
value unless based
theoretical
geometrical terminology, although the concepts are defined so as to have numerical significance. Thus, the terms point, set, neighborhood, distance, curve,
instances,
that analogs will
is
design problems one would normally prefer a predominantly theoretical investigation with a minimum of experimental verification. When
theory permits a mathematical formulation, this theoretical investigation can
putation, either ever,
be based on
by computers or analogs.
com
How
even when sucha mathematical formulation
not practical, dimensional analysis may permit a theoretical investigation based on models,
is
rather than a investigation.
more expensive full-scale
empirical
THE MATHEMATICAL PROBLEMS SOLVED
IV.l.C.
From
a theoretical study one desires two types
of information: behavior of the
information.
qualitative
proposed device, and
more
a
permit
The general
(1)
precise
At
design,
present,
design
based on either continuous or
(2)
to
quantitative studies
digital
are
computa
Continuous computation, using either
tion.
computers or analogs,
when
it
applicable;
is
generally less expensive
gives the desired qualitative
behavior more quickly and permits the designer a greater flexibility in considering his problem,
The
significance of each aspect of the
tional result
is
computa
apparent to the person with design
and he may make suggestions which are equivalent to major design changes and
225
building components, and machine parts can be solved by the highly developed elasticity analogs.
have been
useful in the Analogs study of heat flow and the transmission of high-frequency radio waves. Impedance functions of electrical
networks have been studied with electrolytic tanks. The design of acoustical devices and
shock absorbers has been aided by
electric
network analogs. Control computations
in general involve continuous computers rather than analogs and, therefore, are not discussed in this part of the
book.
responsibility,
have the
almost immediately. On the basis of continuous computation, the given
IV.l.C. Mathematical Problems Solved by
effect assessed
mathematical description
is
often simplified by
The various analogs
are classified in this
in accordance with the
establishing that certain portions of the equations
used. (The table of contents for Part
have
list
These are
little effect.
all
consequences of
the fact that, in these forms of computation, the
ultimate
designer
may
actually
computing device or be very with
its
operate
the
closely associated
operation.
book
types of analog devices
IV forms a
of these analog types.)
Mathematically, considered as
we may
classify the
problems
Mows:
1. Boundary-value problems in two or three dimensions in which an unknown function, is
With regard to
qualitative behavior, therefore,
continuous computation has
many advantages at
on a
defined
and
9? satisfies
region,
Laplace
present. However, the accuracy of the numerical results is seldom ascertained. It may be that the
accuracy of the numerical result is as high as desired or as high as the accuracy of the data justifies; the analog can then be justified for
computational purposes, as distinguished from qualitative studies. But normally, if numerical accuracy is essential, one should supplement such continuous computation by digital procedures. as
Analogs,
are
generally
distinguished
from
special-purpose
computers,
devices
and
its
boundary, ^,
equation on
conditions.
21,
(IV.1.C1)
on 38
certain auxiliary conditions
boundary
called
The usual boundary
conditions for these problems
fall
into three
categories:
a)
specified
on the boundary
b) -r- specified
on
on
and c)
studies include the use of scale
s
and
VV =
individually have limited ranges of application.
Analog
91,
specified
the
boundary
on a piece of the boundary and
dtp
-^ on the remainder.
models
on
in designing dams, bridges, harbor works, ships,
and
airplanes.
Vacuum
tubes
may
be designed
Two- and
three-dimensional problems involving
of electrolytic tank studies or studies. The stability of an airplane
equation and any of these boundary conditions have been solved using electrolytic
wing against flutter is often investigated by means of electric networks, and flow patterns,
tanks (see Chapter IV.3) and electric networks
on the
basis
membrane
and
drag functions are established by electrolytic tanks or networks. Complex problems in the strength of structures,
approximate
lift,
Laplace
s
(see Chapter IV.9). Two-dimensional problems with boundary conditions specifying (p on the boundary have been solved using membranes (see
Chapter IV.4).
INTRODUCTION TO
226
A
2.
second type of boundary-value problem
involves Poisson
s
The boundary
is
uniquely determined by Eq. IV.1.C5. These may be computed for any simply-
conditions
equation,
connected
VV = constant those for Laplace
problems
involving multiply connected regions, available
conditions are the same as
information usually defines only the tangential
s
derivative of grad
equation. s
interior
Chapter IV.4).
It
be generalized to the
may
on the boundary, plus the integrals around all the
bounding curves of expressions involving normal and tangential derivatives of V 2
/>
situation
of contour
values
constant and y specified on the equation with boundary have been solved with membranes (see
The
for
However,
region.
(IV.1.C.2)
Two-dimensional problems involving Poisson
3.
ANALOGS"
"TRUE
the
assumed that these conditions plus Eq.
is
IV.1.C.5 determine y (see Section IV.6.B).
There are a number of analogs available for partial derivatives of 99, or
scalar potential equation,
computing the second div (a
grad
=
+
b
+
(IV.1.C.3)
b"
where
time
also
Here, a
variable.
enters
an
as
a tensor, and
is
the stresses, in problems where they cannot be directly calculated or
ot
independent b , and
a, b,
A
6.
are all given functions of the spatial variables.
=
*
q>
specified at each point of the
boundary
type of problem involves
relationships between
curl"
-
fifth
b"
In general, if a is a positive definite matrix, and b and b are positive, or (a grad 9?) n must be
measured. These analogies
are discussed in Chapter IV.6.
curlcp
+
c
c <]>
two vectors, 9
+
"cross-
and<|>.
c"
dt
as a
function of the tune, and the values of 99 through
$
out
initial
at
t
=
must be
1
values of
given.
If b
= 0,
the ot
are not needed.
Two- and
three-dimensional problems involv
ing Ep. IV. 1 .C.3 have been solved using electrical
networks 4.
We
(see
wave equation
in
c,
c
and d are
d,
,
tensors,
c"
and
d"
vectors, all given functions of the coordinates.
Chapter IV.9).
shall consider
where
Maxwell
problems involving the
field
two dimensions,
s
equations for the electromagnetic
and the equations describing incompressible,
source-free vortex fluid flow are specializations
ofEq.IV.LC.6.
The boundary solutions of these
with
(p
specified as a function of time
on the
boundary and as a function of the coordinates at an initial time / These have been solved using
are
not
clear;
conditions in this
conditions
determining unique equations in all their generality the
of boundary be limited to special
discussion
book
will
.
membranes 5.
A
number of two-dimensional
problems give differential
cases.
(see Section IV.4.C).
rise
to
If elasticity
a fourth-order partial
equation on Airy
s
stress function,
c,
c
,
and d
d,
are all
matrices, then the values of
9
positive definite
and<J>
determined by their values at an
are uniquely
initial time, r
,
and the tangential components of either 9 as functions of time on the boundary. (See also or<]>
J.
where/(x, y) depends upon the given body force and temperature distribution. The components of stress are linearly related to the second partial derivatives of If
both
d(p
are defined
on the boundary,
The equations
that
arise
from general
problems are discussed in detail in Section IV. 5.B General elasticity problems have
elasticity
.
been
(p.
and
A. Stratton, Ref. IV.9.m, pp. 486-88.) 7.
solved
with
photoelastic
Chapter IV.5) and with Section IV.8.B).
electric
models
(see
networks (see
IV.l.C.
8.
We
also consider the
polynomial
algebraic
THE MATHEMATICAL PROBLEMS SOLVED problem of solving Given a
equations.
function
rational function
9.
where a t are constants and z
an
The
an
electrolytic
to plot the value of 9 at any point
tank
is
used
on the complex
plane (see Section IV.3.D).
variable.
%
227
zeros of y(z)
may
is a complex be found with
electrolytic tank.
Conversely, given the poles and zeros of a
Finally,
we
consider the solution of second-
order linear differential equations with constant those that arise in coefficients, specifically
mechanics and acoustics.
means of IV.7, 8).
electric
They are solved by network analogs (see Chapter
Chapter 2
DIMENSIONAL ANALYSIS AND MODELS
IV.2,A. Introduction
average time of one rotation of the earth with
There are many physical problems, particularly in the field of fluid mechanics,
whose solution
is
by dimensional analysis and the use of models. For example, suppose an
greatly facilitated
engineer
is
interested in
computing the drag and
functions of an airplane wing of given design.
lift
Theoretically, these functions could be
mathe
matically derived from basic physical principles.
In practice, however, the partial differential equations involved become too complicated for a
The
purely analytic solution.
model
scale
studies as
engineer turns to
an aid to computation.
Since the functions are nonlinear, a dimensional analysis of the problem
is essential.
This will
reduce the number of variables in his problem
and put them into dimensionless form, independ ent of the scale of his test models. Model studies are frequently used for airplanes, ships,
dams, and harbors.
hydraulic machines,
it is
computations, equations into dimensionless form.
digital all
In
also desirable to put
respect to the sun.
There are several systems of units in which mass, length, and time are primary quantities. force,
Alternatively,
ture
be considered
may
additional
comparison with another of the same kind but is a mathematical function of a number of primary quantities. For example, the magnitude of a velocity
is
obtained by dividing the magnitude of is a
a length by the magnitude of a time. If mass
primary quantity, the measurement of force based on Newton s second law,/= Ma.
arbitrarily established.
the mass of a platinum cylinder that at the International Bureau of
is
y w primary
,
.
.
.
,
zr, primary quantities comparable
Suppose
a function of these primary
is
(p
=
one meter, which
is
the
distance between two scratches on a platinumiridium bar, also kept in Sevres. is
one mean solar second
1/86,400 part of a
mean
.
.
.
,
xm
,
ft,
.
.
.
,
y n z l5
.
,
solar day, the
.
.
,
zf)
(IV.2.B.1)
To
deposited
at Sevres, France. is
g^fo,
is
Weights and
is
.
to a third standard, say time.
which
.
.
comparable to a second standard, say
length; zl9
in the m.k.s. system of units, the
Time. The unit
y l9
quantities:
primary quantities are the following: 1. Mass. The unit is one kilogram, which
3.
is
Let us consider the dimensionality of second ary quantities. Let (p be a secondary quantity. Let *!,..., x m be primary quantities
be directly compared with standards that are
The unit
as
secondary or derived quantities. The magnitude of a secondary quantity is not obtained by direct
quantities
Length.
primary
comparable
In any system of dimensions, certain measured quantities are regarded as primary. These may
2.
are
known
All other measured quantities are
IV.2.B. Measurements
Measures
time
quantities.
to one standard, say mass;
For example,
and
length,
fundamentals of a number of other systems. In certain cases, an electrical quantity or a tempera
be useful in dimensional analysis,
be
must
the mathematically homogeneous primary quantities comparable to one standard,
that
in
all
is,
,
ax m by l9 ,
.
.
.
,
n, ,
.
cz l5 .
.
,
.
.
yw
.
,
czr)
z lt
.
.
.
(IV.2.B.2)
BUCKINGHAM S THEOREM
IV.2.D.
If a secondary quantity meets this requirement,
we may
say that
has dimensions.
it
IV.2.D. Buckingham
In dimen
sional notation, the dimensionality of
229
Theorem
s
xl9 ...,xm be a
Let
is
of
set
m
physical
whose dimensions are based on
quantities,
fundamental
r
units.
Suppose a dimensionally complete relationship between them:
exists
For instance,
our fundamental quantities are
if
mass, length, and time, the dimensionality of force
is
given by
Then
=
[F]
less
2
many secondary
must be borne
reducible to a
is
"dimension-
^,...,0 = quantities
are determined directly with a single measure it
equation
[M][L][T]-
In actual practice,
ment. However,
this
form":
in
mind
where
rr
l9
the j/s
.
.
.
,
ir
n are
dimensionless products of
and
that
=m-
n
they are basically functions of primary quantities. For example, a pressure gauge is really measuring
(IV.2.D.2)
r
(IV.2.D.3)
Proof:
xm
Solve Eq. IV.2.D.1 explicitly for
a force divided by an area, and hence pressure
Xm
has the dimensionality
:
=
or
xw
IV.2.C. Dimensionally Complete Relations physics are basic
The fundamental laws of
between physical vari empirical relationships describe as ables, written in a form which we "dimensionally
.
.
l9
.
,
x n)
If
complete."
= 0, where x
l9
...
9
the
xn
are
law
is
measur
able physical quantities, this relationship must hold regardless of the choice of standard on
which the
size of units for is
quantities
length
is
when
valid
Any
based.
length
measured
physical
particular
if it is valid
in meters,
is
it
must
It
also complete.
particular problem
is
Suppose we change one of the fundamental new units each quantity,
units, say, length. In the
x it can be
expressed as
original value
and a
much
is
the
new
unit
are used.
Therefore,
(IV.2.D.5) so that
com
(IV.2.D.6)
From
and D.6 Eq. IV.2.D.4
we
get
derived from
often happens that a
cannot in practice be equations governing derived from fundamental laws. Dimensional as
c^x t where xt
the ratio of the
of length to the old. Since it is dimensionally IV.2.D.4 is valid, whatever units complete, Eq.
(IV.2.D.7)
it
analysis gleans
is
complex that the precise
so
(IV.2.D.4)
be problem may
logically
any relationship is
=
when
in feet.
these fundamental laws are dimensionally
them
...,x m . 1)
also be
described by equations derivable mathematically from the fundamental laws of physics. Since
plete,
(x 1)
primary and secondary
Thus,
measured
-
information as possible
Taking the and setting a
partial
=
1,
derivative with respect to
we
a9
get
from a knowledge of the variables involved in the required relationship, subject that
it
(IV.2.D.8)
to the condition
must be dimensionally complete.
The theorem discussed in the following section, originally
stated
by
Buckingham
(see
dimensional analysis. IV.2.c), is the heart of
But
Ref.
dx s
DIMENSIONAL ANALYSIS AND MODELS
230
For
so that
an
=
(IV.2.D.10)
spherical
a
i
or
of
example
Buckingham
of
incompressible
D
1 1)
^e kdy ^ e is
we form
in terms
solve.
Let
in
the
some distance ahead of
=
(IV.2.D.18) fluid
and
of viscosity of the
fluid.
density
of
the
Let us consider the dimensions of the variables,
L For
immersed
fluid.
/>,/*)
coefficient
dynamic
fi
(IV.2.D.12)
and
= mass />
dx m
is
smooth
v-
?(/,M, where
the differential equations
a
Then the drag force/ on the body of the form represented by an equation
This first-order partial differential equation may be solved by the method of Lagrange as follows:
of
application
body of diameter d that
stream
velocity of the stream at
(IV 2
the
consider
theorem,
s
of fundamental dimensions of
(length),
and
M (mass),
T (time).
instance,
dx m
(
dx
d*
_
t\
M-MP
(
\XJJ-M\ X
(IV.2.D.13) .
Then,
a m log x f
=a
-
3
+ constant
log x m
(IV.2.D.14)
Since we have five variables and three dimensions, the equation
or
must be reducible
two independent dimensionless (IV.2.D.15)
*j
x
are
t
h e pressure
to
one involving These
variables.
coefiicient,
*"m.
The usual theory
for linear partial differential
equations of the first order shows that any solution of Eq. IV.2D.11 is a function of these constants, bp
i.e.,
in the
form
%,..., ^.3) =
(IV.2.D.19)
and the Reynold
s
number,
R=
(IV.2.D.16)
-
(IV.2.D.20)
Thus, to be invariant under change of unit of length,
99
yi>-->y
must depend simply on
variables
The equation
is
(f(P, jR)
y^^
(IV.2.D.17)
P
xm
The
variables
yl9
.
.
.
}
y m^
are independent of
the dimension of length.
by one.
We
end up with a
set
of
dimensionless variables equal in number to the original number of variables less the number of
fundamental dimensions
effectively present.
=
which a graph can
experimental
Thus, any primary dimension that is effectively present can be used to reduce the number of variables
for
restricted to
form
reducible to the
m-i where
data.
-
(IV.2.D.21)
y(R) easily
This
(IV.2.D.22)
be drawn from
reasoning
spherical bodies.
is
not
It is valid for
a
body of any shape, an airplane wing, for example, For more extensive discussions of dimensional analysis,
consult the
following texts:
P.
W.
Bridgman, Ref. IV.2.b; H. L. Langhaar, Ref. IV.lk; G. Murphy, Ref. IV.2.m.
APPLICATIONS
IV.2.F.
IV.2.E. Models
An
Ships.
who
engineer
plans a model study begins
with a dimensional analysis of his problem. We have seen that he can reduce his variables to a set
of dimensionless products, Within the limits of he tries to build his model with practicability, these products identical to those of the prototype.
In fluid mechanics, the most common variables are force (/),
length
(/),
velocity
mass
(v),
coefficient of viscosity
(//),
acceleration of gravity (g), speed of sound
(c),
density
(p),
dynamic
231
Drag forces and wake patterns of naval boat hulls, and some commercial
vessels, flying
vessels are investigated
by towing models with a
power-driven carriage that runs on a track above
a canal in which the model
A. B. Murray, B. V. Korvinand F. V. Lewis, Ref. IV.2.n; Kroukovsky, H. E. Rossell and L. B. Chapman, Ref. IV,2.q; F. H. Todd, Ref. IV.2.r.)
Hydraulic Engineering.
can independent dimensionless products which
great importance in this
and
be formed from these variables:
s
Reynold
number
Pressure coefficient
R= F
vlp
W.
Ref. IV.2.a;
=
E.
P, Creager, J.
W.
Harris, Ref. IV.2.g;
Wisler,
teristics
W=-
number
a In an ideal model,
all
would
these products
have the same value for the model as for the
this
impose
Therefore,
complete
some of
and
and
J.
H. W. King, C. 0.
G. Woodburn, Ref.
Model
IV.2.J.)
studies are fre
of the normal modes of vibration of
have been particu proposed structures. Models in the design of suspension larly important (See also J. B. Wilbur and C. H.
bridges.
Norris, Ref. IV.2.S.)
usually not feasible to
it is
prototype, However,
Justin,
the total stresses, quently used to determine stress distribution, critical loads, and charac
M =c
s
D.
Hinds, Ref. IV.2.d; C. V. Davis, Ref. IV.le;
Structural Analysis.
Weber
studies are of
The designs of
and harbor control and improvement is planned on the basis of model studies. (See also J. Allen,
F=
Mach s number
Model field.
most major dams are checked, before construc tion, by model tests. Much of the work on river
J.
Froude number
Occasionally
Ref. IV.2.h;
are the
The following
surface tension (a).
floats.
models are also used. (See self-propelled ship also Ref. IV.2.1; G. Hughes and J. F. Allan,
similarity
in
practice.
the dimensionless variables,
2 References for Chapter
which are believed to have secondary influences or which effect the
manner,
are
phenomenon
allowed
to
deviate
in a
known
from
correct values.
AHen. Scale Models in Hydraulic Engineering. London, Longmans, Green & Co., 1947. P. W.Bridgman. Dimensional Analysis. New Haven,
a. J.
their b.
Yale University Press, 1931. E. Buckingham, "On physically similar systems," no. 4 (1914), p. 345. Phys. Rev., Vol. IV, d. W. P. Creager, J.D. Justin, and J. Hinds. Engineering New York, John Wiley Sons, Design for Dams. c.
IV.2.F. Applications
The
the extent of following examples indicate
&
1945. Vol.
the field in which models have proven useful. e.
Wind-tunnel testing provides im such as lift, drag, in airplane design, portant data and moment coefficients. There are also freeAirplanes.
f.
which the performance of observed. (See also H. L.
g.
Ref, IV.li; Dryden, Ref. IV.2.f; G. G. Kayten, H. C. Pavian, Ref. IV.2.o; Alan Pope, Ref.
h.
flight
wind tunnels
models flying
IV.2.p.)
is
in
1,
Chap.
4.
C.V.Davis. Handbook of Applied Hydraulics. New York, McGraw-Hill Book Co., Inc., 1952. Sec. 24. H. J,. Dryden, "The design of low turbulence wind for Aeronautics, Technical tunnels," Nat. Adv. Comm, no. 940 (1949). Report, E. W. Harris, "Hydraulic
models,"
University of
Washington, Engineering Experiment Station, Bull, no. 112(1944).
G. Hughes and J. F. Allan, "Turbulence stimulation on ship models," Soc. of Naval Architects and Marine Vol. 59 (1951), pp. 281-314. Engineers, Trans.,
DIMENSIONAL ANALYSIS AND MODELS
232
i.
G. G. Kayten, control
"Analysis
of wind-tunnel stability and
terms of flying qualities of full-scale Nat. Adv. Comm. for Aeronautics,
tests in
airplanes,"
Technical Report, no. 825 (1945). j.
k.
1.
m. n.
H. W. King, C. 0. Wisler, and J. G. Woodburn. Hydraulics. New York, John Wiley & Sons, 1948. Chap. 10. H. L. Langhaar, Dimensional Analysis and Theory of Models. New York, John Wiley & Sons, 1951. "M.I.T. ship model towing tank," Mar. Eng. & Shipg. Rev., Vol. 56, no. 7 (July, 1951), pp. 48-50, 64. Glenn Murphy. Similitude in Engineering. York, Ronald Press Co., 1950.
Lewis,
"Self-Propulsion
tests
p.
q.
r.
New
A. B. Murray, B. V. Korvin-Kroukovsky, and E. V. with small models," Soc.
&
Marine Engineers, Trans., Vol. 59 (1951), pp. 129-67. o. H. C. Pavian. Experimental Aerodynamics. New York, Pitman Pub. Corp., 1940. of Naval Architects
s.
Alan Pope. Wind-Tunnel Testing. New York, John Wiley & Sons, 1947. H. E, Rossell and L. B. Chapman. Principles of Naval Architecture. New York, Soc. of Naval Architects & Marine Engineers, 1939. Vol. 2. F. H. Todd, "Fundamentals of ship model testing," Soc. of Naval Architects & Marine Engineers, Trans., Vol. 59 (1951), pp. 850-96. B. Wilbur and C. H. Norris,
J.
analysis,"
"Structural
Handbook of Experimental
New York,
John Wiley
& Sons,
Stress
1950.
model
Analysis.
Chap.
15.
Chapter 3
ELECTROLYTIC TANKS AND CONDUCTING SHEETS
IV.3.A. Introduction
An equation of very great importance branches of physics and is engineering
equation for a potential function
This equation
is
many
1.
Electric
Laplace
on the
applicable, for example, to
E = V^
E
is
(IV.3.A.2)
rest
boundary for
intensity
prescribed,
or
be
may
of the boundary.
In problems in which the geometry of the
(IV.3.A.1)
electric field
is
on one piece of the boundary and
prescribed
potential in a charge-free electro
The
or
9?
3
following scalar potentials:
static field,
either
s
9?.
VV = the
in
is
simple, there are classical
solving the Laplacian (see also A.
Ref. IV.3.t).
feld,
the
boundary
is
methods
Sommer-
However, when the shape of
it is frequently convenient to resort to methods of com analog the puting the value of throughout given
complicated,
(See, for example,
Chapter
J.
A. Stratton, Ref. IV.3.U,
region.
magnetic
potential in a current-free
slightly
The
3.)
2. Scalar
The magnetic
field.
field
intensity H
electric-potential function e(x,y,z) in a
conducting material constitutes a useful
representation of a function
is
which
satisfies the
Laplacian. Boundary conditions of prescribed
H = -Vp
~
(IV.3.A.3)
(p
dq>
or
Velocity potential in a field of potential fluid flow. The velocity v is
are simulated
3.
v (See also H. 4.
(IV.3.A.4)
-V
Ref. IV.3.i, pp. 17-20.)
Lamb,
in a region of steady-state
Temperature
heat conduction.
Ref. IV.3.m, p. 29.)
satisfied
by the
two dimensions, is also and imaginary parts of an
real
analytic function of a J.
complex variable
(see also
F. Ritt, Ref. IV.3.p, p. 69). This idea will be
distribution
in
the
Then
material
is
given problem.
The conducting material most commonly used is
in
voltage
determined by means of probes. Equipotentials are plotted to provide a graphical solution to the
W. H. McAdams,
(See also
The Laplacian,
applied to analogously shaped boundaries. the
=
by voltages or currents
a dilute
salt
solution,
but metalized
or
paper has also been successfully In the following sections, the details of
graphitized utilized.
some will
electrolytic tanks
and conducting sheets
be discussed.
further developed in Section IV.3.D.
In
all
there
the
arise
problems."
applications
many The
mentioned above,
so-called
IV.3.B. Electrolytic Tanks
Most
"boundary-value
solution of such a problem
electrolytic tanks are limited in their
application to two-dimensional problems. Later
we
some
consists of finding, out of all possible solutions
in
of Eq. IV.3.A.1, that particular one which satisfies a given set of boundary conditions. A
specialized three-dimensional tanks, but
problem would be that of finding a y which satisfies the Laplacian through out a region bounded by a given surface on which typical
function
this
section
by assuming
shall
discuss
highly
we
start
that our problems are limited to a
plane,
A shallow bath filled with dilute electrolyte is required.
The
electrolyte is considered to
have
ELECTROLYTIC TANKS AND CONDUCTING SHEETS
234
negligible thickness.
It
is
which analogous to the unknown function satisfies the Laplacian. There must be a provision
for the simulation of boundary conditions for the
there
or
(p
must be a
its
This voltage or
device for measuring the voltage
at the V
its
gradient corresponds to
y or
corresponding point on the analogous
such as shape
is
constructed of insulating material, or cement-lined wood.
slate, glass,
is
Its
appropriate to the problems for which
For problems involving parallel fields, such as fluid flow around an airplane wing, the
it is
used.
tank
is
For problems involving
rectangular.
radial fields, the tank
each problem tank.
may
is
circular.
In
some
cases,
require a specially shaped
For example, heat flow through various
geometric configurations (square edge, square
and square prism) has been studied in similarly shaped glass models filled with electrolyte by Langmuir,
corner, plane edge, plane comer,
Adams, and Meikle (see Ref. IV.31). In general, rectangular electrolytic tanks have dimensions of several feet
on each
side.
have pointed out that there are two types of boundary conditions likely to be encountered in boundary-value problems. In the
first, 9? is
a
prescribed constant on given curves. This con dition is simulated in an tank
by
electrolytic
carefully
cleaned,
shaped metal
analogously
electrodes raised to the corresponding voltage.
In the second case,
where ~ on
is
that
so
potentiometer,
prescribed, a
null
indicator
is
connected
the
in
points
the
same potential as the
may be
a
cathode-ray
whose unbalance
amplified.
potential gradients, the probe
may be
replaced by a dipper consisting of a pair of needles mounted rigidly approximately 10
diameters
apart
vertical axis
but free to rotate
about a
midway between them. By balancing
out the voltage drop between the two needles at is possible to measure approxi mately the potential gradient in both magnitude
various angles,
and
it
direction.
The voltage probe
usually connected by
is
mechanical linkage to a needle that makes a mark on a sheet of drawing paper corresponding to the position of the probe on the surface of the electrolyte.
curves through
By drawing
all
the
points on the paper which correspond to single voltages, one can plot a representation of an
of the
map
field
under considera
tion. If the alternative, voltage gradient,
probe is
similarly used, the field or flux lines, orthogonal to the equipotentials, are
plotted.
Automatic equipment for tracing the equi potentials has been designed. The probe is con strained to move along an equipotentiai by an electronic servomechanism,
the probe
is
transferred
drawing board.
(See
IV.S.o.)
mated by feeding uniform
distribution along
current through large
may be
movable contact of
oscillograph or a bridge circuit
corresponding current density is maintained at the electrode surface. Parallel fields are approxi
is
movable contact may be accurately determined.
The
equipotential
We
It
across a null indicator to the
To determine
plane.
The tank
mm.
diameter of 0.1-0.5
electrolyte having the
gradient at any point of the electrolyte.
some electrode
the voltage of
to
measured by a needle probe. The probe, made either of copper plate, silver or platinum, has a
And
or -r- over prescribed curves. on
voltage of any point of the electrolyte
relative
a
9(39
prescribed
The
considered to be a
conducting sheet analogous to the two dimen sional region of the problem, with voltage
A
J.
and the motion of
by a rigid arm to a K. Mickelsen, Ref.
rapid method of obtaining the potential
any
straight line in the elec
of a rectangular tank with dimensions very much larger than the region of interest. Alternating current with a
trolyte consists of feeding the voltage
frequency between 400 and 1,000 cycles per second is used to minimize polarization errors
one pair of plates of a cathode-ray oscillograph.
The other
due to decomposition of the
linear time-base
metal
strips at opposite edges
electrode surfaces.
electrolyte
at
probe moving
in a straight line
electrode, rectified
and
between a
and the base
amplified, directly into
pair of plates
is
fed a direct-current
potential synchronized through
a system of gears with the movement of the
ELECTROLYTIC TANKS
IV.3.B.
probe.
Then the potential distribution
along the
may be viewed and photographed as
straight line
a trace on the face of the
cathode-ray tube. (See
L. Jacob, Ref. IV.S.g.)
Department
This
of
and voltage up to 100
is
is
diagramed
the largest tank in the
Electrical
the
at
Analogy
Sorbonne, (See also L. Malavard, Ref. IV.S.n.) The tank is made of slate, measuring 2,0 by 1.5m. An electric dipper made of platinum wire,
m
volts.
Tap water is used as
the electrolyte.
The tank provides 0.1
A typical electrolytic tank setup in Fig. IV.3.B.1.
235
mm
in the
results with a
equipotentials
potentials with a precision of
The
over-all
measurements Ref.
is
1
part in 10,000.
accuracy of electrolytic tank discussed by P. A. Einstein (see
The
IV.3.d).
precision of
and measures the
errors
inaccuracies, polarization,
due to mechanical and the
distortion of
the liquid surface due to surface tension are
considered, with the conclusion that, with careful
work, accuracy up to 0.1-0.2 percent should be obtainable.
There have been a number of extensions of the
methods
just described that permit the study of
parts of three-dimensional systems.
For example, the equipotential
map
of the
electrostatic field of the
plane of symmetry of an electrode system such as that in a vacuum tube
or electron lens (See V. K.
is determined by tank methods. Zworykin and G. A. Morton, Ref.
IV.3.W.) From this map,the electron trajectories, which are a prime factor in tube design, can be calculated.
Most
electrode configurations of interest have
mirror symmetry. The equipotential surfaces cross the plane of symmetry at right angles.
Therefore, the potential distribution would be
Fig.IV.3.B.l
with diameter from 0.1 to 0.2 the electrolyte. vertically into
mm,
plunges
The probe P
is
connected across a cathode-ray oscillograph with amplifier to the tiometer, which
is
movable contact of a poten connected in parallel with the The oscillograph has a
supplying electrodes E. straight-line trace
when
the dipper and movable
contact of the potentiometer are at the same potential.
parallel
A T-shaped carriage moves along two
rails
along the sides of the tank, while a
rod moves perpendicular to these
rails
long arm of the carriage. The probe of this rod while an
is
along the at
one end
electric inscriber at the other
end moves over a drawing board and transfers the points of equal potential onto a sheet of
unchanged if an insulating plane on the plane of symmetry were substituted for half the electrode system. In the electrolytic model,the free surface
of the electrolyte simulates
An
cides with the plane of symmetry. This
is
rods.
Potentials proportional to those in the proto
type are applied to the model electrodes. Then the potential at any point in the is electrolyte
the potential of the proportional to analogous in the original electrode system. The point distribution over the free surface is potential
by means of a probe connected by to a mapping pencil. The pantograph linkage
400 to 500 alternating at 1,000 cycles, although
equipotential
may
model
made of sheet metal and supported on insulating
plotted
cycles
insulating plane.
along the plane of symmetry, is immersed in the tank so that the surface of the electrolyte coin
drawing paper, thus plotting the equipotentials. The current supplied to the tank is usually be used. Power varies up to 30 watts
this
exact scale model of half the system, cut
sections
lines
of the
plotted represent the inter equipotential
surfaces
with
ELECTROLYTIC TANKS AND CONDUCTING SHEETS
236
the plane of symmetry of the electrode system.
A special adaptation
of an electrolytic tank has
been used for automatic plotting of electron Ref, (See also D. B. Langmuir,
these problems,
it is
possible
to limit the investi
a small segment of the total volume, gation to such a small central angle that the one
having
curvature of
outer boundary
its
trajectories.
is
essentially
in
one plane.
IV.3.J.)
The
Institute of Hydraulic Research,
At the Iowa
State University of Iowa, electrolytic tanks have been used to simulate three-dimensional fluid
Ref. IV.3.f; (See also P. C. Hubbard, Rouse and M. M. Hassan, Ref. IV.3.q.)
electrolyte
confined between an inclined
is
two terminal
sheet of plate glass, flexible
of plastic which
strip
the
boundary
profile.
plates, is
and a
curved to
A series of fixed
flow.
represent
H.
electrodes are accurately placed along the inner
face of the flexible strip.
ment
is
shown
The schematic arrange
in Fig. IV.3.B.2.
Probes for real three-dimensional tanks have
been studied. They are made of wires insulated over all but the tips, so that they may be plunged deep into the J.
(See also S. Softky
electrolyte.
Jungerman, Ref.
and
IV.3.S.)
IV.3.C. Conducting Sheets
For two-dimensional boundary-value prob some experimenters have found it simpler
lems,
and
less
use sheet metal, metalized expensive to
as their conducting paper, or graphitized paper material instead of an electrolytic solution.
A portable made
by
"analog
New
Schenectady,
field
at
plotter" is,
General
the
Electric
The
York.
present,
Company, conducting
material consists of graphite-impregnated paper.
An
connected through a exploratory stylus, microammeter to a potentiometer, can be used to plot the equipotentials directly on the con
Fig.IV.3.B.2
ducting
Note, in Eq. IV.3.A.4, that in potential flow, velocity
v
fluid
Conducting
paper.
inserted with silver paint
boundaries
are
and fed from a low-
voltage direct-current source operated from 115
= \
volts, alternating current.
where y
satisfies
sponds to
the Laplacian. Thus,
Ve corresponds
to
v.
if
Two
e corre
types of
problems have been studied. The first is the effect of an obstruction of given shape in a field of uniform
parallel
flow.
The
obstacles
are
represented by models of nonconducting lucite, with permanently mounted electrodes replacing the usual probes.
The second
is
is
with the velocity distribution along
the boundary, from which boundary pressures
are calculated.
Kayan has used uniform metalized paper com
plicated
heat
boundary shapes
flow
Ref.
(see
for two-dimensional
IV.3.h).
To
represent
materials of different heat conductivities in one structure, square
paper model,
meshes are cut in parts of the
yielding
sections
of
different
electrical conductivities.
concerned with flow confined by
a surface of revolution. In this case, the principal
concern
C. F.
cut into the desired shape to represent
Due
to the axial
symmetry of
IV.3.D. Representation of the Complex Plane Electrolytic tanks
wide
application
complex plane.
and conducting sheets find as
representations
of
the
REPRESENTATION OF COMPLEX PLANE
IV.3.D.
Consider a planar conductor and introduce into the plane a Cartesian coordinate system, Let e(x, y) be the and J and J the com voltage,
x
v
ponents of the two-dimensional current-density vector (with dimensions of current per unit a
if
Then,
length).
the
is
two-dimensional
which
is
237
a constant of the and to the electrolyte,
three-dimensional current density. con Specific ductivity is defined as the amount of current that crosses a unit surface under unit
Our two-dimensional
voltage gradient.
conductivity
is
then equal
to the specific conductivity multiplied
by the
conductivity (with dimensions of current divided
electrolyte depth. Likewise, our two-dimensional
by
current density
voltage),
current *
2
ox
Now
-a
a
e satisfies the
o
0x
Then there
exists
an
by the depth of
Now let us assume that equal currents poles and
o
are fed
into the electrolyte at points corresponding to
Laplacian, so that
~+~ =
equal to the three-dimensional multiplied
electrolyte.
"i
oy
is
density
(IV
-
drawn out
at points corresponding to
m2)
ay "electric
current
function,"
f(x,y), such that
y lf=J ox
^=-J
l
a
(IV.3.D.3)
to
the
oy Fig. IV.3.D.1
(Eq.
IV.3.D.2
is
equivalent
cross
derivative condition for the existence of/)
zeros in a tank corresponding to the entire
Thus, plex plane.
de
=
dx
i
a/
tion of the logarithm of a rational function
y(x +jy). This fact
tfdy (IV.3.D.4)
and
e
dx (See also
functions.
(7
J.
tion
is
(IV-3.D.5)
an analytic function of a complex variable in the region or corresponding to the conducting tank of a shallow sheet. is
electrolytic
uniform depth can be used as a physical realiza tion of a region of the complex plane. (See, for S. G. Hooker, and example, K. N. E. Bradford, R. V. Southwell, Ref. IV.3.C.)
and two-dimensional conno
of finite depth, assuming ductivity to a tank vertical flow
of current.
by
Frame,
A modem example of its applica by A. R. Boothroyd, Makar (see Ref. IV.3.a).
source of strength 7 at a point
E. C.
However, we must
relate these quantities to the specific conductivity,
Q
with z
=^
and a corresponding sink at infinity. Let P(z) be any other point, at a distance s from 2, and
let
J8 be the component of
current-density vector at
vector
2?
P
total
Q
the
directed along the
(see Fig. IV.3.D.1).
Since the radial flow from
outward flow across a
and radius
s
= PQ =
To simplify this and subsequent discussions, we have applied the concepts of two-dimensional current-density vector
also J. S.
Let us consider the potential distribution in a conducting sheet of infinite extent with a current
Consequently,
Thus,
and
discussed
Cherry, and R.
F. Ritt, Ref. IV.3.p, p. 69.)
the basis of a process for
(see Ref. IV.3.1,
Ref. IV. 3. e).
and -/are conjugate
is
the solution of algebraic equations developed
Lucas dy
com
We then have an effective representa
Q
is
circle
\z-
ct
uniform, the
with center at is
\
(IV.3JD.6)
g utj
^ Eqjy.3.D.l, & (IV.3.D.7) 3s
ELECTROLYTIC TANKS AND CONDUCTING SHEETS
238
Therefore,
the interior of the circle will have the
potential,
S=-,
(IV.3.D.8)
same current and voltage distribution as before, no matter what happens outside the circle.
we can
Alternatively,
specify
the total
that
This equation holds on every radius originating at Q, and on a radius this partial derivative is
amount of current flowing from the
a equivalent to an ordinary derivative. Thus, for
the source at the origin,
fixed direction,
of an extended sink. Thus, the circular conductor
circular
conductor be equal to the current from
yields the equivalent
(IV.3.D.9) ITTO
The flow
=
is
uniform
in every direction, and, at
sheet into the
making the equivalent
of a sink
at infinity for the
it. If the source is not at part of the plane within the origin but at a distance from the origin which
small relative to the radius of the outer con
is
has a iixed value eQ independent of the direction (e is determined only up to a constant).
ductor, the sink at infinity
Thus, for every direction, we have
ductor will approximate the point at infinity for a
s
e
1,
with only a small error.
number of sources and
=
e
=
*o-;r-tog*
*o-
Itra
^g
|z
-
cj
2-77(7
Now log
\z
Ci\
function to e
is
Thus,
the
its
at the source point and connected to the outer conductor.
located
Now
center.
The
dlt
,
.
.
dm
27TCT
,
The
infinity.
(IV.3.D.11)
other
and m equal sinks at points with a sink of strength (n m) at
points .
the
suppose there are n equal sources at
conjugate
is
-c,)
sinks near
each individual source or sink
electric current for
the real part of the complex
c f).
log (z
approximated
may be supplied by a generator with one terminal (IV.3.D.10)
function
is still
Thus, one outer con
any point
potentials are additive, so that, at
z,
and
-/ = C
-
bg ( Z
-
Ci )
(IV.3.D.12)
(z-cl)...(z-c n)
C
is a complex constant. a source and 7 is a positive quantity. If considered a sink of equal strength at a finite
where
2 we
.
27TCT
(7
is
point,
we would
(IV.3.D.14) 1
Thus, e(z)
obtain an analogous result:
+j -/(z) is an exact representation
of the logarithm of a rational function with zeros e
+ j -f =
C
+ A. log (z -
a
Ci )
(IV.3.D.13)
ITTU
at the points
q,
.
.
.
The
The above
derivation refers to a conducting
sheet, infinite in extent, with a source at a finite
point and a sink of equal strength at infinity. For a physical realization of this condition, let us
=
a single source with c{ 0. The current from this source flows outward from the consider
first
origin uniformly in every direction. If we take a
e
on
this circle
is
constant.
ness of potentials,
if
we
cn
d^
.
.
.
,
dm and poles
at the
points
.
potential e
is
directly
measurable with a
probe and/may be approximately determined by integration, using a set of measurements with a double-needle dipper (see Ref. IV.3.a). Conversely, given an arbitrary rational func tion, one its
large circle with center at the origin, the potential
,
J.
can use an
electrolytic
tank to determine
poles and zeros by the Lucas method (see also S. Frame, Ref. IV.3.e, and Lucas, Ref. IV.3.1).
Consider the polynomial
Owing to the unique
introduce a conductor
with the same shape and maintain it at the proper
(IV.3.D.15) i=0
ERRORS DUE TO FINITE SHEET
IV.3.E.
We has
choose an arbitrary function,
m+
known
1
real roots, c 1}
= (z - CjXz -
c 2)
.
.
,
.
.
),
which
ct
=z-
jy
cit
we have by
Eq. IV.3.D.17
.
-
(z
Then, since x
239
1
de.de
c m+1 )
,
^^t
.
6,-
,
N
i^i z
(IV.3.D.16)
Let
(IV.3.D.20) (IV.3.D.17)
x method
where the residues
k = -~~ are real,
/i
equal to
is
and the
proven.
rational functions
are fed into the
2770^
0,
ay
Impedance functions of electrical networks are
vfe) Currents
= 0,
if
Thus,
~ =~=
The scale conducting sheet at the points c t factor k has the dimensions of voltage. As
Therefore, the
of the complex frequency.
method
just discussed is very
-.
before,
we can suppose
that the infinite con
ducting sheet has been approximated by a circular sheet of large radius with a conducting
rim corresponding to the point at infinity. We shall show that the roots of #(z), which are
useful in finding the poles and zeros of impedance functions. This application
are the points at
which
and
Be
both vanish. These points
may be determined
dy
by exploring an
electrolytic
tank with a double-
carefully discussed (see Ref.
IV.3.a).
IV.3.E. Elimination of Errors It
the roots of
is
by Boothroyd, Cherry, and Makar
Due to Finite Sheet
inconvenient and inefficient to use a
is
rimmed
circular sheet large
enough so that one
has a good approximation of an infinite con Various other methods of ducting sheet. are used. eliminating boundary errors
needle probe.
Our
previous
potential
argument
function e
on
shows
the sheet
is
that
the
given by
Ino n+i
1
Z - PLANE
Z-PLANE
Fig, IV.3.E.1
(IV.3.D.18)
Then,
if
z
is
the conjugate of
In one method, two circular sheets of con
z,
ducting material with equal radii are joined on their outer rims.
One
circular sheet corresponds
around the origin in the complex and sinks plane and contains all the sources to a circle
(x-
located at finite points.
The other corresponds
to an image of the rest of the plane with the point at infinity
mapped onto
the center of the circle.
Consider two circular conducting sheets of equal radius contains
all
R
(see Fig. IV.3.E.1).
the sources
and corresponds to a (IV.3.D.19)
the z-plane (z
= re*
9 ).
and
circle
The
first
sinks at finite points
around the origin
in
The other has one source
ELECTROLYTIC TANKS AND CONDUCTING SHEETS
240
or sink at
each
Inside
Thus, on the
center and corresponds to a circle
its
around the origin
the
circle,
continuous and
=
in the z -plane (z
function
potential
circle in Fig. IV.3.E.2,
r e
the Laplacian every-
satisfies
rims,
the boundary potentials and the
components
=
\z
Jlr J Therefore, as z
121
(IV
3
E
13)
(IV.3.E.14)
r+R 30
define a function
^
to
qual
(IV.3.E.2)
,
lQ
E
inside outside? and equal to and ^ on the circk The func ti on e(z) has
^
= ~^, =J
3
fa
^^
NQW we may
^
= R,
\
~^ R
= lim~
Jim
the current densities are equal and opposite. \z\
jr
r-*R 3$
of the current-density tangential components of vectors match, while the radial
When
^i2
?fi_r
^R
where except at the sources or sinks. Since the two circular sheets are connected on their
(TV is
(IV.3.E.3)
d>
and z approach
r (see Section
e e IV.3.D), the partial derivatives of l and a
approach
such that
limits,
a
=
-lim^ -+R OT
limfOr r+R
(IV.3.E.4)
r
Fig.IV.3.E,2
^ = Hm^ 2
r
(IV.3.E.5)
^R
continuous partial derivatives everywhere except
.
at sources
(IV.3.E.6)
still
e (z
)
=g
/R*\ a
(--I
= g (z)
This
is
that, if
circle
and outside the
that
construct functions
We must
circle.
the Laplacian
on the
-/^z) for
R
it satisfies
r
<,
and
ff
an anticonformal map of the
and the second
and it satisfies the Laplacian
sinks,
1
interior of
the second circle onto the exterior of the circle,
prove
We
(IV.3.E.7)
s
\zl
and
inside the circle
onto the
first,
first
such
arbitrai
r>R
We
as follows:
7 P oint on the circle and cal1
i
a
|
for
2 (z) -/ a
A(z)
f
=
~J
z
2
lde
f
;^^ + j
z
de r
*
select
an
z* Then,
(IV3 R15)
r
z
a"
Now,
R 2 3e 2
Then, as
r
wnere tne P atn
approaches^, the partial derivatives
approach the following
limits
where the path of integration
r-^j?
30
is
entirely outside
;
On r
entirely inside
is
tne circle-
M *! = -M%L r~*R dr
f integration
-^R OT
/-*B 30
the circle, ,
(IV.3.E.10) -*
f f-\
~/iW
^
fi
f f-\
I
"~/2\^J
I
p
-K
where the path of integration
Jfi ^*"
is
/TV ^ T7 1 1\ / V .j.Jti.l ^1
on the
J
circle.
POTENTIAL FLUID FLOW
IV.3.F,
The
functions/!
constants.
On
and/2 are uniquely defined up to
the
*[ z\
f
f \
/jy 3
z
v Wl lim
= lim W2
r^R or
r-+R dr
Then we may
i$\
made, which, with
symmetry a strip for the
substitute
infinite plane.
(IV.3.E.19)
(IV.3.E.20)
<0,2)
define the function /(z) with
continuous derivatives equal to circle, /2 outside, and equal to/x
From Eqs.
circle.
g
is
considerations, permits one to
i
= ^\
2g
logarithmic transformation certain
circle,
241
IV.3.E.15, 16,
/x
inside the
(0,1)
and/2 on the and 17, we may {2,0}
conclude that -/is everywhere conjugate to
(0,0)
(-1,0)
(2,0)
(1,0)
Z-PLANE
e,
(7
so that
we have
e(z)+j-f(z),
constructed a function
analytic
tp(z)
=
over the entire plane
(7
and sinks. Its real part, e(z) 9 is except at sources then the unique solution of the Laplacian with the prescribed boundary conditions. The second method of eliminating boundary errors
limited to problems
is
which involve
about the x-axis. symmetry of sources and sinks
With
this
;c-axis.
flows across the symmetry no current
Therefore, a thin insulating wall placed
distribution along the x-axis leaves the potential
in the upper half-plane unchanged. Now, by a conformal transformation, the upper half-plane
can be mapped Let
onto the interior of a
I
=-
(0,0) -
PLANE
circle.
Fig. IV.3.E.3
IV.3.F. Potential Fluid
2+]
Electrolytic
This transformation maps the upper half of the onto the interior of a circle, radius J,
z-plane
center j\ in the
mapped
-plane.
The point
at infinity
is
onto the origin (see Fig. IV.3.E.3). This
circle is easily represented
by an
electrolytic
tank.
Sources or sinks on the real axis in the z-plane sources or sinks of half be
may
represented
by
circular boundary of the tank. strength on the
Two
tanks
principles
constructed
discussed
Flow
(IV.3.E.21)
according
to
the
above are described by
tanks have been used to solve
fluid flow, i.e., where the problems in potential is determined by a poten of a fluid local velocity tial
This method
function.
is
discussed
by G.
(Ref. IV.3.v).
I.
in
Taylor and C. F. Sharman
Modern apparatus and
mental work at the Sorbonne
L Malavard (Ref. Potential flow
finite sheets is by R. E. Scott (Ref. method in which a a describes He IV.3.r).
velocity
rigid
normal
at infinity in
an
experi
described by
characterized as follows:
Makar (see Ref. IV.3.a). Boothroyd, Cherry, and Another method of eliminating error due to
normal to a
is
IV.3.n). is
Neither sources nor vortices
discussed
most popular
is
where it is often used in England and France, The theory of the analogy aerodynamic studies.
exist.
bounding surface
to all other infinite
The is
velocity
zero.
The
bounding surfaces or
region
is
defined.
ELECTROLYTIC TANKS AND CONDUCTING SHEETS
242
Analogy
Mathematically (three dimensional), Fluid
V
(pT)
-
Electric
Quantity
9
where v
is
At a
At
rigid
is
bounding
surface,
Eq. IV.3.F.1 and
n
is
f
f
-e -J.
defined.
we conclude
2,
e
U U
the density.
conditions are:
other bounding surfaces, v
From
w
(IV.3.F.2)
the velocity vector, p
The boundary
that l
there exists a scalar potential function (velocity potential)
99
B
Analogy
(1Y.3.F.1)
v -
V x
A
Quantity Electric Quantity
->.
a
and a vector potential function
(stream function) u } such that
Y
=
(IV.3.F.3)
-V
As an pv
= -V
x
u
(IV.3.F.4)
sider If
Eq.IV.3.F.3
is
regarded as asystem of partial
on
differential equations
9?,
then Eq. IV.3.F.2
is
the
the integrability condition for this system. If as a system of differential
equations situation
on
which
in
way
let
us con
problem of uniform flow in the
^-direction past a solid cylindrical cross-sectional
Eq. IV.3.F.4 is regarded
of the
illustration
boundary conditions are simulated,
example
the three components of u, the
body of any an airplane wing, for
shape
(see Fig. IV.3.F.1).
ANALOGY A
ANALOGY B
is similar.
In the case of a flow parallel to the x y plane 0, and Eq. IV.3.F.4 has a solution u for which only the z component y is not zero, and t
vg
=
4 become Eq. IV.3.F.3 and
Fig. IV.3.F.1
^ = -pv
a
ox It is clear that
Mathematically, x
(IV.3.F.6)
distance
this
means
vx
Eq. IV.3.F.5 and 6 are anal
If
fluid quantities
and
electrical quantities.
represent
lines
of
equal
At the
surface of the body,
B
of equal electric the stream lines while the potential represent lines of electric flow represent lines of equal velocity potential.
the
lines
vn
-
(IV.3.F.8)
velocity
while lines of flow of electric current potential, lines. stream represent
In Analogy
some
(IV.3.F.7)
Analogy A is applied, lines of equal electric
potential
at
= constant
ogous to Eq. IV.3.D.1 and 3 on the electrical e There are two useful analogies quantities and/.
between
that,
from the body,
where v n
is
the normal component of velocity.
Analogously, in Analogy
from the
A
at
some distance
obstacle,
JT
= constant (IV.3.F.9)
SPECIAL
IV.3.G.
At
the surface,
the normal
is
density, so that the
(IV.3.F.10)
component of current
body
is
represented by a
perfect insulator of identical shape, and the fluid
flow at a distance
is
represented by uniform
current in the x direction,
Jy
J
t
=
(IV.3.F.12)
the tangential
density so that the
component of current
body
is
represented by a
perfect conductor of identical shape, and the fluid flow at a distance
is
represented by uniform
current in the negative
y
direction.
uniform
the
Practically,
electric
a
Current
is
field
tank
rectangular approximated by dimensions large compared to the obstacle.
Analogously, the velocity of an ion in a given is electrolyte J
= -c
v(
where vt
-
is
Ve
{
(IV.3.G.2)
the velocity of the ion, c i
of the ion and electrolyte, e
is
is
a constant
the potential.
allow the motion of ions to be observed
constant
the surface,
is
pressure.
electrolyte has conductivity sufficiently small to
(IV.3.F.11)
where Jt
the
is
p
Therefore, an electrolytic tank in which the
In Analogy B at some distance.
At
243
constant of the fluid and medium,
=
Jn where Jn
FLOW PROBLEMS
size
fed into the tank
suitable
model
is
a
for the flow of fluid through a
homogeneous field. The substratum is repre sented by an agar gelatin solution on a glass and the fluid to be recovered, by colorless plate; zinc
ammonium ions from the chloride salt. The
input wells are simulated
ammonium
blue copper
by tubes containing
chloride in agar gelatin
solution with the anodes at the top.
The output
wells are tubes containing the zinc
ammonium
is
chloride solution with the cathodes at the top.
with
The progress of the blue copper ammonium ions
of the
by means
due to the
electric field
is
photographed atvarious
intervals in order to study the flooding process.
of large metal strips at opposite edges. Stream can be directly plotted and the slope of the
lines lift
References for Chapter 3
curve and the angle of zero incidence can be a.
directly measured.
IV.3.G. Special
of allied properties response, transient response, and networks," Inst. Elec. Eng., Proc,, Vol. 96, Part I
Mow Problems
In some cases, direct analogy is
(1949), pp. 163-77.
made between
of fluid through a porous medium. In the petroleum industry, for
problems
H. G. Botset,
c.
K. N. E. Bradford,
example,
flooding.
models are used to study recovery H. G. Botset, Ref. IV.3.b). Two
When
engineer knows
these
methods are
applied, the
electrical
d. P.
tank,"
Royal
A. Einstein,
Soc.,
"Factors
electrolytic plotting
Series
London, Proc.,
A,
of limiting the accuracy Brit. Jn. Appl Physics,
tanks,"
Vol. 2 (1951), pp. 45-55. e. J.
S.
Frame,
"Machines
for
solving
algebraic
M.T.A.C, Vol. I (1945), pp. 346-48. G. Hubbard, "Applications of the electrical
equations," f.
P.
g.
Instruments, Vol. 20 (1949), pp. 802-6. L. Jacob, "The field in an electron-optical immersion
the injection and production producing wells and rates used. He is interested in a method of
analogy in
objective,"
the
G. Hooker, and R. V. South with the aid of an
Vol. 159 (1937), p. 315.
the location of his injection and
of the flood through investigating the progress
S.
well, "Conformal transformation
(see
methods of recovery of petroleum products are fields and water gas recycling in condensate
"The electrolytic model and its applica tion to the study of recovery problems," A.I.M.E., Vol. 165 (1946), pp. 15-25. PetroleumDivision, Trans.,
b.
the motion of ions in an electrolyte and the flow
electrolytic
A. R. Boothroyd, E. C. Cherry, and R. Makar, "An tank for the measurement of steady-state electrolytic
field.
mechanics
fluid
Soc.,
Phys.
research,"
Rev.
Sci.
London, Proc., Vol. 63B
(1950), pp. 75-83.
The flow of a homogeneous
viscous fluid
medium through a homogeneous permeable d approximately governed by Arcy
s
for
Kayan, "An electrical geometrical analogue Vol. 67 (1945), complex heat flow," A.S.M.E., Trans.,
h. C. F.
is
pp.
law. i.
H.
713-18.
Lamb,
Hydrodynamics.
New
York,
Dover
Publications, 1945. j.
where
Y,
is
the velocity of the fluid, cf
is
a
D. B. Langmuir, trajectories,"
"An
automatic plotter for electron
R.C.A. Rev., Vol. 11 (1950), pp. 143-54.
ELECTROLYTIC TANKS AND CONDUCTING SHEETS
244
k.
1.
Langmuir, E. Q. Adams, and F. S, Meikle, "Flow of heat through furnace walls," Am. Electrochemical Soc., Trans., Vol. 24 (1913), pp. 53-84.
I.
F. Lucas,
moyen de
"Resolution 1
electricite,"
r.
Sci., Paris,
Comptes
of
Report, s.
Rendus, Vol. 106 (1888), pp. 645-48.
n. L.
New
York, t.
use of rheo-electrical analogies in certain aerodynamical Aeronautical problems," Royal
Malavard,
"The
Soc., Jn., Vol. 51 (1947), pp. 739-56. o. J.
K.
Mickelsen,
"Automatic
equipment
Crown
Vol. 71 (1949), pp. 213-16.
in
8, 1950).
Jungerman, three
"Electrolytic
tank
Rev.
Sci.
dimensions,"
Chap.
IV.
and
Press, 1947. Page 69. Hunter Rouse and M. M. Hassan, "Cavitation free inlets and contractions," Mechanical Engineering,
J.
Instruments, Vol. 23 (1952), pp. 306-7. Arnold Sommerfeld. Partial Differential Equations in II, Physics. New York, Academic Press, 1949.
u. J.
techniques for field mapping," G.E, Rev., Vol. 52, no. 11 (Nov., 1949), pp. 19-23. p. J. F. Ritt. Theory of Functions. New York, King s q.
No. 137 (June
Softky and
S.
measurements
m. W. H. McAdams. Heat Transmission. McGraw-Hill Book Co., Inc., 1942.
analog device for solving the
"An
network synthesis," approximation problem M.I.T., Research Laboratory of Electronics, Technical
immediate des equations au
Acad. de
R. E. Scott,
v.
A. Stratton. Electromagnetic Theory.
McGraw-Hill Book Co., Inc., 1941. G. I. Taylor and C. F. Sharman, method for solving problems of flow
"A
New
York,
mechanical
in
compressible fluids," Royal Soc., London, Proc., Series A, Vol. 121 (1928), pp. 194-217. w. V. K. Zworykin and G. A. Morton. Television. York, John Wiley Sons, 1940. Chap. 3.
&
New
Chapter 4
MEMBRANE ANALOGIES
IV.4.A. Introduction
value of
Membrane analogies provide another method for the solution of Laplace s equation in two
the two-dimensional region.
be shown that
It will
over
s
dy*
vertical deflection
equation,
3V +
^=
a*
3/
2
entire surface
is
a constant tension
T
and subjected to a constant
small vertical pressure from below, then whatever deflections may be imposed at specific points, the
2
dx*
its
on the boundary of
t
membrane is stretched
if a
horizontally so that there
dimensions,
or of Poisson
as a function of
y
mately const.
satisfies
(IV.4.A.2)
z of
*
*! + = 2 dx*^dy
or of the wave equation,
where p
is
all
other points approxi
the equation
_* + *! T^Tdf
(IV.4.A.4) ^
the mass per unit area.
Thus, the vertical deflection z of a membrane constitutes a useful representation of the In problems involving Eq, IV.4.A.1 and 2, the are satisfied in a prescribed twoequations dimensional region bounded by a number of
The
closed curves.
solution
mined by the value of
on
or
its
is
uniquely
normal
Membrane
derivative,
analogs
may
dn be used for those problems in which 99 is defined on the boundary, but there is considerable technical
difficulty i
.
problems involving
in
using
membranes
for
.
the wave In problems involving Eq. IV.4.A.3,
a cylindrical region in bounded by the x~y plane and a x-y-t space surfaces perpen closed number of cylindrical
equation
is
satisfied in
dicular to the x-y plane.
The
of
this
projection to the two-
is analogous region on the x-y plane dimensional region of the preceding paragraph. The solution of the wave equation is uniquely on the boundary of value of
determined by the
vertical
boundary represented by in
corresponding
deflections
positions
on
the
imposed membrane. Membrane analogies have been used for problems in electron optics, electromagnetic
conduction theory, heat
and
stress analysis.
There are two types of membrane models and soap films. commonly used, rubber sheets
Rubber
sheet models are
most widely used
in
are electron optics, where electron trajectories
fy
y
deter
q>
the boundary.
in Laplace s, Poisson s or the wave on the given the values of with equation,
potential
y, t) the cylindrical region. In other words, at every of value initial the 9 is determined by and the of the two-dimensional region point
a simulated by the paths of steel balls rolling on its rubber sheet deformed so that the height of surface
is
proportional
under consideration.
to the electrostatic field
They are
and
cavities.
Soap
films are widely used in
of cylindrical bars
studies of torsion or flexure
and have been
also used for
fields in guides
models of the electromagnetic
applied
to
studies
of heat
conduction. IV.4.B. Differential Equation of
Membrane
We
define a
flexible
membrane
as follows:
It is
thickness which body of negligible
a
may
MEMBRANE ANALOGIES
246
be considered the physical geometrical surface. to
its
It
realization
surface only. If any curve
drawn across the membrane
membrane
into
two
of a
transmits tension tangent is
imagined to be
surface dividing the
then the two parts parts,
on each other normal to
exert tensile force
the
derive
the differential equation
vertical displacement of points
on the
tension
T is
+
T(/ 2 flj
where 6 dA
is
for this
/!
a)
-
Td
dA
(IVAB.2)
the increase in area.
Then the work done over
the entire
membrane
is
WT =
of the
where 6A
assumptions:
The
is,
T dA
(IV.4.B.3)
surface of
a horizontal membrane, we make the following
1.
dWT =
surface
dividing curve.
To
The work done against tension small rectangle,
constant and uniform in
is
the increase in the area of the
membrane. Thus, the potential energy due to tension
may
be written: all directions.
2.
The motion
surface
is
of any point
P.E.
on the membrane
= TA
This, of course, corresponds to a choice of the
purely vertical.
additive constant.
Now, we pressure
P
in
work done against
consider the
any deformation. Here, (IV.4.B.4)
where d Fis the change in volume enclosed by the
membrane and
the
deformation. But, Fig. IV.4.B.1
Now we
6V
an expression for the Lagrangian function of a vibrating membrane.
kinetic energy
is
obtained by integrating
reference
plane
in
the
the deflections are vertical,
=J \dzdxdy
(IV.4.B.5)
A
shall derive
(The Lagrangian is equal to the kinetic energy minus the potential energy.)
The
if
where dz is the change in vertical deflection at the point (x,y).
Then, the potential energy due to pressure and tension
may be
expressed
one half the density times the square of the velocity over the
membrane 1
K.E.=
P.E.
surface.
= TA- P
J
[z
dx dy
ff (IV.4.B.1)
2 J J\pzfdxdy
A where
the mass per unit area of the dz and z t
membrane,
=
.
ot
The
(IV.4.B.6)
is />
is
potential energy
it
a
Lagrangian,
and
/
2,
with area, dA 9
under uniform tension T(see Fig. IV.4.B.1). Let it be stretched by amounts 6^ and 2 <5/
.
we have
L = K.E.-P.E. yz?-T(i
into
a given shape. Let us first consider the work done in any deformation against tension. Consider a small /
for the
a constant plus the
work done on the membrane in deforming
rectangle of dimensions
Then
+ z* + z*)W + Pz)dxdy
A (IV.4.B.7)
Now we
apply Hamilton
s
principle (see also E. T. Whittaker, Ref. IV.4.k, pp. 245-46):
Ldt
d I
=
(IV.4.B.8)
RUBBER SHEET MODELS
IV.4.C.
where
f
and ^ are any
f* J
P
1
P
1
Ldt=\
8\ *
fixed values of l
to
fields
surfaces of prescribed potential. the necessary that the potential throughout as a function of two co field be
bounded by
\{pz (6z) t t
JJ(
Jt n
to
determine electron paths in electrostatic
fff
6Ldt=\
commonly used
sheet models are
Rubber
But
t.
247
It
is
expressible
and that the electron path be confined
ordinates
to the plane of these coordinates.
A
(IV.4.B.9)
that the Integrating by parts and assuming
membrane boundary,
we
as well as
f
and
f
is
1?
fixed,
is
rubber membrane, stretched over a frame,
down over a model of the electrode is made in such a way that its plane
pressed
system which
view corresponds to the geometrical configura tion of the electrodes in the x-y plane while the
get
pZtt
_T
is
height
each
the negative voltage on proportional to Then the contours of the
electrode.
membrane surface correspond to an equipotential
map
-pj (IV.4.B.10)
VT^
(d -I-
A small sphere
is
at a point corresponding to the electron placed
Since this integral vanishes for any variation dz,
T
of the electrode system,
). ^aa; I
source and allowed to roll on the rubber.
The
horizontal projection of its path is then a map of an electron trajectory in the electrode system
under investigation. Let us consider the path of an electron in a
+
U-
static
field.
potential
electron source
(IV.4.B.11)
a further assumption, that zx small, that
and zy are everywhere
neglect
that the
is,
We everywhere small. we z and z and of second powers y x
of the all
is
conservative system, where the potential energy of time, the time integral of the
is
slope
assume that the
located at the origin of the
emits electrons of zero velocity. x-y plane and We apply the principle of least action. In a
-2
Now we make
We
membrane
independent
two
a path between points an extremum (see also E. T.
kinetic energy over fixed in space
is
is
Whittaker, Ref. IV.4.k, pp. 247-48).
,
Thus,
derive the following approximate expression:
^+^=
-^--
=Q where
IV.4.C. Rubber Sheet Models
In this section
we
shall discuss
used at zero pressure, so stationary,
that,
membranes if
they
If they vibrate,
4.^=0 their vertical
and
e
is
the mass of the electron, v
final times,
points
the Laplacian, mately governed by
M
the electron, velocity of
are
their vertical deflection is approxi
^!?
(IV.4.C.3)
(IV.4.B.12)
P
?
and
f
and pi are tbe
x
is
the
are the initial
initial
and final
of the path.
is set equal If the electrostatic potential e(x, y)
to zero at the electron source, field is static, at
P
,
on the any point
then, since the
electron path,
(IV.4.C.1)
deflection
is
^
approximately governed by
^
^ ^^
or
1/2
z
M r+
2 = .! dy*
T3f
(IV.4.C.2)
(IY.4.C.5)
MEMBRANE ANALOGIES
248
z
Since
Then Eq. IV.4.C.3 becomes (IV.4.C.6)
=
(IVAC.7)
structed
Eq. IV.4.C.7 leads to the differential equation of the motion of an electron in a field of potential e(x,y).
to
e.
let
us consider the motion of a rolling
We again use
M
6
is
is
is
the
moment
the linear velocity of the
Q1
are the initial
and
final
and
points of the
and
stretching
The
electrode models are
(aluminum or
made
lead) or
metal
either of soft
wood, cut
to the correct
height to represent the potential, and shaped to
with the
electrode
shape,
Where
necessary, clamps are used to insure contact
between the membrane and the entire electrode surface. Steel ball bearings are held at the model cathode by a small electromagnet and released
desired by cutting off the current. Models can be set up either for visual observa
set
-v R
0)
where
achieved by ruling
when
path.
We
is
squares on the rubber before
conform (IV ACS)
the angular velocity of the ball,
ball, a) is
and
proportional
=
the mass of the ball, Ib
of inertia of the ball, v
QQ
is
the principle of least action,
!
where
either rectangular or
feet wide,
Constant tension
round.
maintaining their shape after stretching.
sphere on a surface whose height
is
stretching surgical rubber over a
by
frame several
Now
Eq. IV.4.C.15
e,
path of the electron. In practice, rubber sheet models are con
or
:
proportional to
is
analogous to Eq. IV.4.C.7, and the horizontal the path of the ball represents the projection of
R is
tion
or
for
observation
Then
the radius of the sphere.
table with a
(IV.4.C.10)
desired, the
plate-glass top
When
at the origin
of the x-y plane
is
zero,
cold arc.
visual
up on a
is
set
is
illuminated
which
a permanent record
motion of the
ball
is
is
photographed source, such as a 60-cycle
using a pulsating light
But, assuming that the starting energy of the ball
model
desired, the
is
from beneath.
=Q
Where
photographing.
(IV.4.C.9)
The path appears
as a dotted line, with
the spacing of the dots giving a measure of velocity.
(IV.4.C11)
The rubber
sheet
model has been further ,
adapted to problems in which radio frequency
or v
= const. (-z)
1/2
(IV.4.C.12)
and Eq. IVAC.10 becomes CQi
d\
(~ z)
1/2
ds
voltages are comparable in magnitude to direct-
current voltages, such as in the designing of
microwave power tubes
=
(IVAC.13)
or
(see J.
W.
Clark, Ref.
IVAb). Radio frequency voltages are simulated by causing the appropriate electrode models oscillate
up and down.
The frequency
to
scale
factor relating the radio frequency with the
dx
oscillation rate in the
model turns out to be
(IVAC.14)
We have already assumed that the slope of the membrane neglected.
is
so
Then
small
that
I
may be
sufficiently
large
oscillation to
The rubber
to
allow
be practically sheet
the
mechanical
realized.
model can
also be used to
estimate the capacity of an electrode system (see, for example, J. H. Fremlin and J. Walker, Ref. IV.4.c).
Membrane
analogies are useful in the study of
the electromagnetic
(see, for example, E. C. Cherry, Ref.
IVAa).
Plane sections of the electromagnetic inside
waveguides and
like
systems
on the membrane measured per unit length (y} be the z-comto the x-axis, and letf
guides and cavities
fields in
249
RUBBER SHEET MODELS
IV.4.C.
parallel
field
cavities,
the membrane measured ponent of the force on
having certain boundaries, with either
TE or TM
membrane
modes
similar to
clear that, approximately,
are governed
by equations of
vibrations
the
governing
elastic
membranes having analogous boundaries.
Let us
by
consider the electromagnetic
first
field
The
perpendicular
electric field is
dx
dt
parallel
is
any
in
fa
so that
the plates, and the magnetic perpendicular to to the plates. This constitutes a field is
TE mode
(IVAC.21)
a
r
homogeneous isotropic dielectric enclosed two infinite parallel conducting sheets, to the z-axis.
dy (IV.4.C.22)
which the direction of propagation
direction in the plane parallel to the plates.
We set H = Q,Ex = Ey = s
become equations
^_^ dx
dx
dt 0.
2
Then Maxwell
^
=e
IV.4.C.21 in Eq. IV.4.C.19, Substituting Eq.
dy
az+ai^p a* dy
an.
p
It is clear that
dt
(IV.4.C.17) *
we
find
(IV.4.C16)
dt
(ivAc.23) dt
and 20, are Eq. IVAC.23, 22,
IVAC.16,
analogous to Eq.
17,
and
18,
where
the following quantities correspond:
d_Hy
dx
is
it
are everywhere small,
slopes
stretched
those
in a
If the
to the j-axis. per unit length parallel
dt
Membrane
Electromagnetic Quantities
Quantities
and the wave equation
ax
=
+
2
dt*
and p are the
where
(IVAC.18)
p
a/
f
H.
(v}
and magnetic
electric
inductive capacities of the dielectric.
We
have
already
pointed
deflections of a vibrating
under uniform tension
that
out
membrane
T
are
the
One important
stretched
approximately
H ,f y
governed by
3x
2
+
?-?-
=
a/
-?
T at 2
(IVAC.19)
?** c/jt
where
with respect to time,
+
^=1^ vy a
*
Consider an brane.
Let/
(x)
Let us pursue
that
of interest here.
Bx = pHx corresponds to - -/
(x)
r
v z is vertical velocity.
obvious.
and j-axes.
i-iYti ^/~i^i\-. it.i that (from Eq. IVAC.21) so
The analogy between Eq. IVAC.18 and 20 quite
H
and/
One more analogy is
(IV.4.C.20)
be noted in
a vector. Both are directed parallel to the z-axis and are measured per unit length parallel to the x-
or, differentiating
restriction should
x and the application of this analogy. Unlike (1/) (x) are not x and y components of
it
is
B x is
A
.
represented by
-
dy
a bit further.
infinite stretched
elastic
mem
be the z-component of the force
dz
By is
JU
az
represented by
ox
We note
=-
^
MEMBRANE ANALOGIES
250
This discussion has been limited to an field,
we note
finite case,
infinite
To proceed to the
or an infinite membrane.
that either mechanical or
the
cross section
s
If the
from
from the ends,
different
boundary shapes can be
built up.
IVAD. Soap
sides
of the
membrane
is
and
equal,
it
tion of the
membrane obeys
the second, there
is
the Laplacian.
greater constant pressure
In
??-axes
be
that,
shown
it is
its
ends,
at a sufficient distance
any cross section
(see Ref.
IVAg,
i)
according to the Saint-Venant theory,
a + a__* where
9?
,.,
a stress function of f and
is
*-
was
demonstrated in Section IV.4.B that the deflec
and
-
bar is twisted by couples applied at
then, in
Film Models
Two types of soap film models are used for analogy solution of torsion and heat conduction problems. In the first, the pressure on the two
of centroids of
line
and the
parallel to the principal axes of the cross section.
electromagnetic waves can be superimposed so that the interference patterns given by reflections
The analogy is maintained no matter how complex the wave pattern becomes.
with the
-axis coincide
the bar
*
|
are shear stresses) (see also
r^
Section IV.5.B), 6
(IV.4.D.3)
d
drj
(where T# and
such that
r\
is
the angle of twist,
on
one side of the membrane than on the other.
Membrane
measured with a
are
deflections
membrane
pointed micrometer head and
slopes
+
2(1
(where
are measured optically.
E is Young s
v)
modulus and
Poisson
v is
s
ratio).
We
have shown, in Eq. IV.4.A.4,
stationary
that, for
a
membrane,
S
^ + ^ = _^T
dx*
i
J.
(IVAD.l)
have been used for the solution of
was pointed out steady state heat con
Miles, Ref. IV. 4.1).
in Section IV.3.A that, in
a constant,
is
<^.=
(IVAD.4)
Jc,.
alternative expression for torsion
can be
obtained by setting
v
It
duction, the temperature distribution
by the Laplacian,
on each boundary curve
dy*
heat conduction problems (see also L. H. Wilson
and A.
that
of any cross section y
An films
Soap
shown
It is also (
where
F is
=
9
-F
any function of f and
(IV.4.D.5) r\
such that
is
governed For two-dimensional heat
-~ + -~ = -2p6
(IV.4.D.6)
conduction problems involving a single isotropic conducting material, the zero-pressure membrane
analogy
is
therefore applicable.
deflection of the
to
tional
If the vertical
Then Eq. IV.4.D.3 becomes
membrane boundaries is propor
g+
the
temperature of the analogous boundaries of the prototype, the deflection at any
point of the membrane
is
proportional to the
_0
(IV.4.D.7)
with the boundary condition,
temperature of the corresponding point in the
(IV.4.D.8)
prototype.
Both
zero-pressure and pressure soap films are used to solve problems in the torsion and flexure of cylindrical bars (see also R. D. Mindlin and
Thus torsion problems in cylindrical bars may be solved by the solution of either Poisson s or Laplace s equation with appropriate boundary
M. G.
conditions.
and
J.
Salvadori, Ref.
IVAg;
S.
Timoshenko
N. Goodier, Ref. IV Ai),
The membrane analogy has been
successfully applied to these problems
Consider a cylindrical bar whose transverse dimensions are small compared to its Let length.
by allow
ing the deflection z to represent either of the
or
functions
%
with membrane coordinates
SOURCES OF ERROR
IV.4.E.
corresponding to
(x, y)
in
(|, rj).
Eq. IV.4.D.2, the pressure
If
z represents
membrane
is
9
used,
with PIT representing 2^6. Boundary conditions are simple, with constant deflection over each
boundary. If z represents y in Eq. IV.4.D.8, the zero-pressure membrane may be used, but the
boundary must be its
profile
may
W. Clark, "A dynamic electron trajectory I.R.E., Proc., Vol. 38 (1950), pp. 521-24.
b. J.
c.
e.
g.
Membrane
h.
How i.
important to keep the approximations involved clearly in mind. We have assumed
are
it is
commonly
used.
A
detailed discussion of
errors in rubber sheet models has been written
G. B. Walker (Ref.
by
IV.4.J).
24 (1947), pp. 50-51. motion of an electron in
Jn. Sci. Instruments, Vol.
J.
A. Kleynen,
"The
fields,"
Philips Tech.
R. E. B. Makinson,
"A
mechanical analogy for
J.
G. McGivern and H. L. Supper,
"A
M. G. Salvadori. Handbook of Experimental Stress York, John Wiley & Sons, 1950. R. D. Mindlin and
membrane Franklin
"Analogies,"
Analysis.
Chap.
assumptions is fulfilled in practice. In fact, slopes of the order of magnitude of 30 percent
tracer,"
stretched rubber
analogy supplementing photo-elasticity," Inst., Jn., Vol. 217 (1934), pp. 491-504.
mem
uniform tension and slopes so small that their squares may be neglected. Neither of these
"The
transverse electric waves in a guide of rectangular section," Jn. Sci. Instruments, Vol. 24 (1947), pp. 189-90.
brane constitutes a representation of the potential in Poisson s, Laplace s, or the wave equation.
ever,
Walker,
Rev., Vol. 2 (1937), pp. 338-45.
f.
approximate solution of these equations.
H.
J.
two-dimensional electrostatic
represent Eq. D.8.
analogies have proven useful in the
H. Fremlin and
sheet,"
carefully constructed so that
have seen that the deflection of a
J.
d. P.
IV.4.E. Sources of Error
We
251
in
New
16.
R. Pierce, "Electron multiplier design," Bell Lab. Rec., Vol. 16 (1938), pp. 305-9. S. Timoshenko and J. N. Goodier. Theory of J.
Elasticity.
2d ed.
New York, McGraw-Hill Book Co.,
1951.
j.
Chap. 11. G. B. Walker, "Factors influencing the design of a rubber model," Inst. Elec. Eng., Proc., Vol. 96, Part II (1949), pp. 319-24. A Treatise T. Whittaker.
k. E.
on
the
Dynamics of Particles and Rigid Bodies.
Analytical
New
York,
Dover Publications, 1944. Pages 245-48. 1.
L.
H. Wilson and A.
J.
Miles,
"Application
of the
membrane analogy to the solution of heat conduction Jn. Appl Phys., Vol. 21 (1950), pp. 532-35. problems," m. L. E. Zachrisson, "On the membrane analogy of torsion and
its use in a simple apparatus," Royal Inst. Technology, Stockholm, Trans., No. 44 (1951). n. V. K. Zworykin and G. A. Morton. Television. New
References for Chapter 4 a.
E. C. Cherry,
"The
analogies between the vibrations
of elastic membranes and the electromagnetic guides and cavities," Part III (1949), pp. 346-60.
fields in
Inst. Elec. Eng., Proc., Vol. 96,
o.
York, John Wiley and Sons, 1940. Pages 83-90. V. K. Zworykin and J. A. Rajchman, The electro static
electron
multiplier,"
(1939). pp. 558-66.
I.R.E., Proc.,
Vol. 27
Chapter 5
PHOTOELASTICITY
IV.5.B. Theory of Elasticity
IV.5.A. Introduction photoelastic method
The of
constitutes a
information
obtaining
about
means
the
stress
too
complex for The method is based on the analytical methods. fact that transparent materials under stress in
distribution
structures
colored
exhibit
when examined
patterns
in
STRESS If
any solid body fixed
in
space
is
subjected to
external forces, internal forces between the parts
of the body are set up. internal forces
is
The
intensity of these
defined in terms of stress.
Consider an arbitrary surface
(5
through the
The discovery of this effect is polarized light. credited to Sir David Brewster (1816). By 1900 the relation between the optical effect and the in
distribution
stress
was well
the material
understood, and engineering applications were to
beginning
be
found.
Many
improving the accuracy of the
In recent years, the development of
developed.
new
techniques
method were
photoelastic
materials
plastic
and the
invention of Polaroid have
made the photoelastic
method reasonable and
simple,
and a very
effective tool in stress analysis.
In essence, the method consists of fashioning a scale
model of a
made of model
is
structure
under consideration
which are to be applied to the prototype and examined in a field of polarized light. A series of brilliant
bands of different colors
alternate
bright
in white light or
and dark bands in mono
chromatic light are observed. Correct inter pretation of these optical effects yields much information about the distribution of stresses in the model and hence in the prototype.
The method
is
well
developed
the following sections, principles of the
student
who
we
method
two-
is
shall
explain the basic
for
two dimensions.
interested in pursuing the
subject further will find a very large literature to
which he can
refer.
body and a point IV.5.B.1). limit, as
body of
P on
this surface (see Fig.
We define the stress
on
S
at
P as
the
the area approaches zero, of the force
on
an area of S surrounding P, divided by that area. In general, this stress is a vector inclined at an arbitrary angle to the surface S.
This vector
is
two components, a normal perpendicular to 6 and a shear stress
usually resolved into stress
for
dimensional problems, and, in recent years, has been extended to three-dimensional studies. In
The
Fig. IV.5.B.1
a suitable transparent material The loaded with forces analogous to those
tangent to It is
(5.
assumed that the
completely specified by a
stress at stress
any point matrix
P
is
(IV.5.B.1)
IV.5.B.
The
on any surface
stress
(3
calculated from this matrix.
at
P
is
Now we
253
assume that
displacements are
all
small and adopt a linear definition of strain.
Normal
strain:
m,
/,
equal to o
through P may be Let h be a unit
6 at P, with components n. Then the stress on G
vector perpendicular to or direction cosines
THEORY OF ELASTICITY
dw
dv
du
h,
*
T"
dy
It
(o
h),,
(o
h\
(o
h) 2
=
9
+
l
Shear strain:
= rw + oy m + rw n = rzx + rzy m + a n l
l
dv
z
?xz
TW = T W
(See A. E. H. Love, Ref. IV.S.g, p. 78, and
7yz
In a rigorous analysis of general displacements of the is necessary to include higher powers
partial
,
the
of o,
2,
=
a-h 2
=
a h3
=
and
(T
3
2
h2
(IV.5.B.4)
H. Love, Ref. IV.S.g, p. 60). However, methods of photoelasticity apply only to
IV.5.B.6) are adequate for our purpose.
HOOKE
h
3 3
are the characteristic roots
S
=
(IV.5.B.5)
MacLane, Ref.
cr
l5
cr
2,
and
cr
3
are
strain.
When
components of the as
Now isotropic
IV.5.a, p. 306.)
roots
and
by relations
the displace
component
known and S. (See also G. Birkhoff
stress
ments are small, these relations are approxi each mately linear. The relations expressing of strain as a linear function of all of the six
The
LAW
Elastic materials are characterized
between
roots of the equation
i.e.,
derivatives in the definition of strain (see
small displacements so that the equations (Eq.
ojbi
<7
r~ 5z
also A. E.
and h 3 such that
a hx
dv f~
ay
it
T"
5z
dw "I
S.
three mutually perpendicular characteristic unit
cr
,
r
"T
_
(IV.5.B.3)
Timoshenko and J. N. Goodier,Ref. IV.S.j, p. 4.) Now, since a is a symmetric matrix, there are
where alt
_ ~ 9x
T M = TW
vectors h 1} h 2 ,
du
,
(IV.5.B.2)
can be proven that T W = T W!
Toz
known
as the
Hooke
s
stress
matrix are
law.
methods apply only to media. For isotropic materials,
photoelastic elastic
Hooke s law is reduced to the following relations
:
The planes and h 3 are known as
stresses at the point P. principal
perpendicular
to h^ h 2
,
at the point P. On the principal principal planes the shear stresses are equal to zero, and
planes,
the planes. the stress vectors are perpendicular to stress at a point is completely the Clearly, h and the defined by the unit vectors h 1? h 2 and 3 ,
stresses
principal
a it
2
and
cr
3
.
2(1
+
)
STRAIN
When is
an
elastic
body
is
subjected
deformed so that each point
displaced
from
its
to stress,
in the
original position,
body
it
is
We resolve
these displacements into components w,
v,
and w
to the x- t J-, and z-axes, respectively. parallel
(IV.5.B.7)
PHOTOELASTICITY
254
where
E
ratio.
(See also A. E. H. Love, Ref. IV.5.g ? pp.
is
s
Young
modulus and
Timoshenko and
102-4, S.
v is Poisson
s
N. Goodier, Ref.
J.
Timoshenko and
(See also S.
The above
equations, Eq. IV.5.B.6, 7, 8
constitute 21 equations
on
15
These components are related
u, v,
and w of Eq. B.6, we may consider Eq. B.7,
and 9 as 15 equations on 12 unknowns, the 6 components of stress and the 6 components of 8,
Eq. B.6 can then be
by the three differential equations of equilibrium.
strain.
A consideration of the equilibrium offerees on a
solved for the displacements, u,
small cube inside an elastic body readily yields
Eq. B.7
the three equations:
in Eq. B.9,
If these are solved
may be
will yield 9
fog
i
feoa
d>
X=Q
i
|
dx
^ + ^ + ^+7=0 dx
oy
fa**
3rw
,
%GZ
,
ox
_ uA
Z are, respectively, the x, y, and
Timoshenko and J. N. Goodier, Ref.
IV.5.J, pp. 228-29.)
Eq. IV.5.B.6
and used to
may be
solved for the
substitute for the stress
and the
com
sufficient conditions that
result,
components.
A third possibility consists of using Eq.
B.6 to
substitute for the strains in Eq. B.7, solving for
the stresses
and
substituting the results in Eq.
B.8, yielding 3 equations
The necessary and
with Eq. B.8
together with Eq. B.9, will yield 9 equations on the 6 strain
components of body force per unit volume, including inertial force if the body is in motion.
the displacements
result, together
ponents in Eq. B.8
z
(See also S.
and the
equations on the 6 stress components.
Alternatively, Eq. B.7 stresses
dz
cy
where J, 7, and
7
,
~j- z,
[-
and w. Finally,
pp. 230-32). (IV.5.B.8)
dz
j-
v,
used to substitute for the strains
For a further discussion of these 9 equations, see S. Timoshenko and J. N, Goodier (Ref. IV.5.J,
dz
dy
and 9
unknowns. Since
those of Eq. B.9 are sufficient for the existence of
In Eq, IV.5.B.1 we specified the stress at a point in terms of nine components, six of which are independent.
N. Goodier, Ref,
pp. 48-50.)
IV.5.j,pp.7-9.)
PARTIAL DIFFERENTIAL EQUATIONS
J.
A. E. H. Love, Ref. IV.S.g, IV.5.J, pp. 229-30;
u, v,
and
on the
3 displacements
w, such as
and w be determined by
u, v,
may be
expressed as six second
order partial differential equations between the strain
These are known as the
components.
conditions of compatibility.
= 9/
(IV.5.B.10)
dx*
with corresponding also S.
/ dx d
2
2
dz
2
x
__d_l_
tyyz
dx\
w. (See
dx dz
Any problem in elasticity theory of small displacements consists of finding the solution to one of the sets of differential equations discussed above, subject to given boundary conditions for
,
d rxy
fyxz
dx
dy \ dx
and
N. Goodier, Ref.
IV.5.J, pp. 233-34.)
f
dy
the
problem.
generally the
dx dz
J.
dy dz
2
dydz
expressions for v
Timoshenko and
dy
The boundary conditions
known
are
forces or
displacements applied to the boundary of the elastic body. For
dz
multiply connected bodies, additional conditions involving discontinuities or multivaluedness of
dx dy
dz
\
dx
dy
dz
displacements are required. These are discussed (IV.5.B.9)
in Section IV.6.B.
IV.5.D.
PHOTOELASTIC EFFECT
The above
partial differential equations have constant coefficients, and linear problems in
have
elasticity
been
handled
effectively
by
dimensional system with holes, or any three-
dimensional system, Poisson
some
boundary conditions are complicated,
influence
often
convenient to solve the equations by analogy
methods.
However, in any two-
1036-38).
IV.5.C, pp.
considering exponential functions or Laplace or Fourier transforms. However, when the it is
255
influence
been
on the
may have
ratio v
It
that
demonstrated
experimentally
(see, for
very small
is
s
stress distribution.
has this
M. M.
example,
Chapter 6). For for Filon has given a method greater accuracy, correction on a twothe
Frocht, Ref. IV.5.f, Vol.
2,
making
necessary dimensional system with holes (see E. G. Coker
IV.5.C. Photoelastic Model
In
photoelastic
material
is
the
investigations,
stress
and L. N. G. Filon, Ref. IV.5.b, pp. 521-24).
model fashioned of photoelastic
distribution in a
studied and
is
assumed to be the same
IV.5.D. Photoelastic Effect
The phenomenon of
as the stress distribution in a metal prototype.
This assumption requires some consideration, in view of the fact that the two systems are governed
by
differential equations
with different constants
light
is
due to
field
vector perpendicular to each other and to if, on a given
Now
E and v. We assume that our model is made accurately
the electric vectors all lie in light ray,
to scale and that the load distribution
geometri
plane,
However,
and the direction of the
is
that of the prototype. cally similar to
Young
s
modulus
E
and Poisson
necessarily different in the
been proven
that,
s
ratio v are
two systems.
It
two-dimensional
in
the stress distribution
has
of
independent systems, these constants so long as the elastic limit is not
exceeded and the materials are homogeneous, isotropic,
and
free
from body
that the region of study
is
forces, provided
simply connected (see
E. G. Coker and L. N. G. Filon, Ref. IV.5.b, pp. 516-18). Under
the direction of propagation.
as
in
both model and proto
called
is
a
single
"plane polarized"
known
electric vector is
direction of vibration.
Most
The pulsation of the
termed the vibration.
crystalline materials,
such as mica and
Iceland spar have the property of resolving a
beam of
into
light
two components, whose
directions of vibration are perpendicular to each other.
The two components are transmitted
different velocities.
This phenomenon
at
known
is
as double refraction.
Almost
these conditions, the stresses at
corresponding points
its
that light ray
electric vector is
stress
is
electro
a ray of light magnetic waves. At any point on and a magnetic vector field an electric are there
all
transparent
noncrystalline
materials, such as glass, celluloid,
and
bakelite,
to their applied type are directly proportional to their and loads inversely proportional
exhibit double refraction while they are
dimensions.
effect disappears.
stress.
elastic
When effect"
the stress
This
is is
removed,
known
under
this optical
as the
"photo-
and follows the following laws:
If the direction of propagation of the incident
where a and a m are model, s and s m are faces of prototype
thicknesses,
M. M.
P
and
prototype linear dimensions of the
Pn
1.
are applied loads.
stress
The
light
polarized (See
principal 1,
p. 363.)
in which the system whatever so that the sufficiently small
2.
distribution
E
(see T.
is
J.
independent of
Young
s
modulus
Dolan and W. M. Murray, Ref.
stress axes
is
split
into
two
components two
in the direction of the other stress axes.
The index of refraction for each component is on the intensity of the linearly dependent
deformations are
stresses are linearly related to the loads, the stress
one of the principal
then:
and model, d and dn are
Frocht, Ref. IV.5.f, Vol.
In any
and
stresses in
to light is parallel
principal
Let
stresses.
be the index of refraction of the
unstressed material;
let
nit n^ and w 3 be the
PHOTOELASTICITY
256
indices of refractionfor the
components vibrating h1} h 2 and
to the characteristic vectors parallel
,
device for determining, besides 1
o!
-
cr
2,
^
h2
the velocity in the material, v { .)
photoelastic model, an analyzer
,
and thus and
at
whose
axis
perpendicular to that of the polarizer,
Then,
hj_
each point. Our polariscope consists of a source of monochromatic light, a polarizer, a
index of refraction, n i} is respectively. (The the ratio of the velocity of light in vacuum, c, to
h3
t%
the directions of principal stress
is
set
and a
screen for observation (see Fig. IV.5.D.1).
C2
(
(IV.5.D.1)
where
Q and
known
C2
are constants of the material,
as the stress optical coefficients.
Now
consider a thin plate of photoelastic h is its own plane so that 3
material, loaded in
that
to
perpendicular
plane.
If
light
falls
SOURCE
Fig. IV.5.D.1
normally on the plane,
we
1
2
plate,
a[cosa][cosajt]
then at each point on the
have, from Eq. IV.5.D.1,
= (Q ~
C*)(ffl
~
0a)
= ^Gfj ~
(J
2)
(IV.5.D.2)
where
C=
constant;
C2 is the relative
C-
^ and
cr
2
stress
optical
The two components transmitted on the two 2 principal planes take different times, ^ and t to ,
pass through the plate. If d is the thickness of the plate,
= dlv = n l dlci = n dlc Hence, ^ proportional to % therefore, to ^ ti
t
1
J
2 is
(7
Now
let
2
Fig. IV.5.D.3
are the principal stresses at
the given point.
z
Ih 2
a[sina][co$(i)fj
Monochromatic
light
of wave length
amplitude of vibration and
2
vibration 2,
and,
.
us consider a simple experimental
/I
passes
through the polarizer and emerges vibrating in one plane (see Fig. IV.5.D.2). The emergent vibration is described by 0[cos M], where a is the
is
co
= ITTCJL
The
parallel to the axis of the polarizer.
On enteringthe model, the vibration is resolved into
two components
parallel to the principal
stress axes (see Fig. IV.5.D.3).
If a
is
the angle between
hx and the
axis of the
then the two component vibrations are
polarizer,
There
is
a dark spot on the screen at any point
for which the amplitude
two types of dark
expressed a [cos alfcos L JL
a)t\ J parallel
to hx
a[sin
o)t] parallel
to h 2
<x][cos
257
PHOTOELASTIC EFFECT
IV.5.D.
fringes
Thus there are
zero.
is
which appear:
.
Isoclmics:
a
On leaving the model, the two components are
These are the
out of phase (see Fig. IV.5.D.4),
=
loci
or
?r/2
of constant stress direction.
Isochromatics:
where
We
tTj
m
is
an
recall 2
.
integer.
that
t
l
-t
l
is
proportional
to
Then we have a dark spot whenever
or whenever
a[coso][c<(t-t,j|
(IV.5.D.6)
for where /is Infringe constant of the material of thickness the is d of wave length I and
a[sin a][cos
light
Fig. IV.5.D.4
The component a [cos a] [cos a[sin a] [cos
The
the plate.
vibrations are
- ^)] parallel to hx - )] parallel to h co(t
a)(t
t
2
z
to the polarizer in analyzer acts similarly
the sense that, for any vibration, the component axis is transmitted and the along a specified not.
The
is analyzer perpendicular component axis is set perpendicular to the polarizer axis (see
Fig. IV.5.D.5).
The analyzer transmits vibrations
then corresponding to each of the above terms.
The total vibration transmitted by the
analyzer
is
a[cos
<x][sin <x][cos
o)(t
tj
- cos a(t -
fj)]
(IV.5.D.3)
which
is
equal to
(IV.5.D.4)
The amplitude of the
vibration
is
Fig. IV.5.D.5
thus
The (1V.5JD.5)
isochromatics
difference
are
between principal
loci
of
stresses.
constant
PHOTOELASTICITY
258
The isoclinics correspond to all points at which
From
the differential equations of elasticity,
it
the principal stress directions coincide with the
may be shown that in the absence of nonconstant
axes of the polarizer and the analyzer. Thus, by
body
model with
rotating the
to the polarirespect
also
forces,
M. M.
+
al
<7
2 satisfies
the Laplacian (see
Frocht, Ref. IV.S.f, Vol. 2, pp. 9-10).
angles a can be
Thus any of the methods of solving the Laplacian
The isochromatics form a pattern of successive
those which use the membrane, electrolytic tank,
m is
or conducting sheet, as well as numerical or
scope, the isoclinics for
all
discussed in other sections of this book, such as
determined.
fringes.
To determine
determined It
first
o1
between the
isoclinics
model is
2
at
any point,
merely by counting
of course,
is,
<;
necessary
distinguish
and isochromatics.
may be used to determine M. M. Frocht, Ref. IV.S.f,
analytical methods,
fringes.
to
When
^+
2
(see also
Vol. 2, Chapter 8).
rotated, the position of the isoclinics
R. D. Mindlin, in an excellent article reviewing
changes, but that of the isochromatics does not. Also, if white light is substituted for the mono
the whole subject of photoelasticity, surveys the
various
chromatic, the isochromatics become bands of
individually (see Ref. IV.S.i, pp. 273-82).
the
color, while the isoclinics remain black.
white light
is
determining
preferable to
isoclinics
When
isoclinics.
alone are desired,
the isochromatics
possible to eliminate the
it is
by using
Thus
monochromatic for
circularly,
instead of plane,
methods of determining
The preceding
sections
o^
constitute
and
2
a brief
introduction to the principles of two-dimensional
In recent years,
photoelasticity.
many
refine
ments to the methods have been developed, and these methods have been extended to threedimensional problems.
polarized light.
We
have shown how the photoelastic method value of ^ - 2 at any point on a yields the The desired result is the values of ol and plane. <;
<7
2
References for Chapter 5
themselves. In a great many cases, the value of
on the boundary is all that is required. any unloaded boundary, the only nonzero component of stress is parallel to the boundary, the stress
On
so that
2
= 0,
and the photoelastic method
a.
To compute the
values of
^ and
it is
2
at interior d.
use auxiliary information.
There are several to
commonly
used
methods based
determine
entirely
Some
the
cr
isoclinics (see
M. M.
2
plus information
from the g.
Chapters 7, 8, 9). Other methods use data obtained from other
h.
point.
This lateral extension
measured (see
proportional to o1 -f #2 at that
may
be accurately
optically, mechanically, or electrically
M. M.
Frocht, Ref. IV.S.f, Vol.2, Chapter 7).
photoelasticity,"
M. M.
Frocht,
"The
growth and present state of Appl Mech. Revs.,
photoelasticity,"
M.M. Frocht, Photoelasticity. New York, John Wiley Sons. Vol.
A, E. H. Love.
(1941); Vol. 2 (1948).
1
A Treatise on the Mathematical Theory
of Elasticity. New York, Dover Publications, 1944. R. D. Mindlin, "Optical aspect of three-dimensional Franklin
/., Vol. 233 (1942),
Inst.,
pp. 349-64.
.
i.
the relative increase of thickness of a thin plate at is
"Three-dimensional
photoelasticity,"
than photoelastic sources to determine cr 2 From Hooke s law (Eq. IV.5.B.7), we see that
any point
D. C. Drucker,
&
1,
^+
J.
Vol. 5 (1952), pp. 337-40.
data, the
Frocht, Ref. IV.S.f, Vol.
London, Cambridge University Press, 1931. Dolan and W. M. Murray. "Photoelasticity," in Handbook of Experimental Stress Analysis. New York, John Wiley & Sons, 1950. Chapter 17, Part I. T.
three-dimensional
f.
values of o^
Modem
Handbook of Experimental Stress Analysis. New York, John Wiley & Sons, 1950. Chapter 17, Part
are graphical
on photoelastic
Survey of
in
e.
individual principal stresses.
A
MacLane.
elasticity,
c.
necessary to
methods
S.
Algebra. New York, MacMillan, 1944. b. E. G. Coker and L. N. G. Filon. A Treatise on Photo-
yields the required stress value.
points or at loaded boundaries,
G. Birkhoff and
R. D. Mindlin, review of the photoelastic method of stress analysis," Jn. Appl Phys., Vol. 10 (1939), "A
pp. 222-41, 273-94. y.
S.
Timoshenko and J. N. Goodier. Theory of Elasticity, McGraw-Hill Book Co., 1951.
New York,
NOTE: References
b, c, d, e,
and
fall contain extensive
bibliographies for further reference.
Chapter 6
ANALOGIES BETWEEN TWO-DIMENSIONAL STRESS PROBLEMS
IV.6.A. Airy s Stress Function cylindrical
There are several types of two-dimensional stress problems which are very difficult to investigate
by
direct observation,
model
study, or
photoelastic methods, but which are analogous to problems conveniently handled by one of these methods. In this chapter, we shall survey some of these analogies, It will be demonstrated that the stress distribution in a cylinder with boundary conditions uniform along its axis is
a thin
analogous to the
stress distribution in
plate, that stresses
due to certain body forces are due to boundary loads, and
analogous to
stresses
that steady-state thermal stresses are analogous to dislocation stresses, These last two analogies
were developed by M. A. Biot
The
(see Ref. IV.6.a, b).
slab analogy, developed
(see Ref, IV.6.i),
between the
by K. Wieghardt stress in
a
slice
body loaded with forces perpendicular and uniform over its
to the axis of the cylinder
length. If a body is in a state of plane stress, the z components of stress, a Z9 rzx and rzy , are equal to zero. These are the conditions satisfied by a free ,
slice,
a thin cylindrical
slice in
which the
parallel
bounding planes are unstressed and free to warp. The loading is symmetrically distributed so that
A
the middle plane remains plane.
thin plate
loaded with boundary forces parallel to the plane of the plate approximately satisfies these conditions.
A
constrained slice or a free slice may be simply connected, bounded by one closed curve, or it may have holes in it, in which case the boundary consists of several closed curves.
Most two-dimensional problems which
of
arise
material with loads applied in its own plane and the curvature of a slab loaded perpendicular to its
in
be explained. An extensive treatment plane of the entire subject, with a number of additional
along the axis of the cylinder, or constrained
will
may be found in a paper by R. D. Mindlin and M. G. Salvadori (Ref. IV.6.f).
references
systems may be classified in two major categories, those involving
Two-dimensional
plane strain
elastic
and those involving plane
order to define these terms, stress,
stress.
we use the concepts of
and displacement as defined
strain,
In
in
a body
is
applications
structures with
slices.
involve
cylindrical
boundary conditions uniform
On the other hand, direct measurements of
stresses,
two-dimensional model studies, and
to photoelastic studies are far simpler thin plates or free slices.
It will
make on
be shown in
Section IV.6.C that the two types of slices are
analogous. For two-dimensional problems,
known
to define a function
it is
convenient
as Airy s stress
function, such that
Section IV.5.B. If
technical
in a state of plane strain, the
the z-axis are equal to displacements w parallel to and the displacements u and v parallel to the
zero,
x- andj-axes are functions of x and y only. These are the conditions satisfied by a constrained slice,
a
slice
of a cylindrical body bounded by rigid These conditions are approxi satisfied far from the ends of a long
parallel planes.
mately
ay
(IV.6.A.1)
TWO-DIMENSIONAL STRESS PROBLEMS
260
where
V is the body force potential, $y
=
T~
dV
such that
~"^
IV.6.A.1,
we have our
X and 7 are
Plane strain (constrained slice) :
,.
(IV.6.A.2)
Stress
the x and
7 components
of
^^- V F --
=-
V4
components w
*
^ and T
OT
for any function
Eq. IV.6.A.1
which
pkm
satisfy
the satisfy
^ ^ ^ Ay j_
j_
"
1
"
V>
^r
v*
(IV.6.A.3)
vy
Hooke
s
(Eq.IV.5.B.9)maybecombinedwithEq.IV.6.A.l
^n
P lane
-
Let
and a is the
is
T
the
X=
^r
Y
=
<1
+
^~ and ~^
I
yw
=
1
+ vm
(IV.6.B.1)
dx
-
=
dy -7 ^s
v
,
f
+ Tw m
thermal expansion:
2
[(1
The boundary of a
* and y components of
ox l
dn
-^K-
(IV.6. A.7)
,
Plane strain (constrained slice):
= .
2
Then, at any point on a
stress.
the tempera
from a reference temperature
coefficient of
- Ea V
_
boundary,
law, with an added term for thermal
T
V2 K
of one or more closed curves in a
expansion, assumes the following forms for our
where
v)
x and Y be
Hooke
categories,
-
the boundary.
slice consists
boundary
two
_
,
_
fourth-order partial differential equations on 9.
ture difference
(1V.6. A.6)
(
-(1
~ on
_
to yield the equations of elasticity expressed as
=T
V2
g
9 and
law (Eq. IV.5.B.7) and the compatibility relations
s
a r
Boundary by J conditions are expressed P y defining * o
for two-dimensional problems,
Now,
*
IV.6.B. Boundary Conditions
P/T
2a+?2i + y =
v
~
n
3y
"jk"
^ =
equations of equilibrium:
17
2
~
unit volume.
body force per
and the
fourth-order equations:
1
where
5
conditions (Eq, IV.5.B.9) in Eq. compatibility
= -7
T
and
IV.6.A.4
Substituting Eq.
are
TXV
normal and tangential
derivatives,
\
= rZX = Xw = 7ZX = ^ = ^ = r (^ + a - E^ (IV.6. A.4)
respectively
I
.
TV2
y)
_ ~
P/a/?e 5/rejj (free slice):
,
T/
,
^ = ;;(^-^) + oc0 1,
e y
=
,=
,
i
a
E
( v
-i
~ ^^ + a ^
-|(
A
fl
,
+
<0
+
ds \dyf
e
F=--^2 8x87*
ax
2
/ds
BOUNDARY CONDITIONS
IV.6.B.
Then, integrating along the
fth
boundary
curve,
and
let
one of the edges formed by
slightly displaced
lation
261
and
this cut
relative to the other
rotation.
Then
let
the
by
be
trans
two edges be same
joined by the insertion of a thin wedge of the
OX
material as that constituting the original
=
-Z
|(^-F/)^ + ft
(IV.6.B.3)
The new slice formed This process
where a 4 and
are constants of
ft-
is
will
called a
slice.
stress. usually be under
"dislocation."
integration.
Let
4(s)=-
(f-Vm)ds
= (X-Vl)ds
(IV.6.B.4)
Then, integrating along (^ again,
(IV.6.B.5)
Also 99?!
dffdy
d
dnJ^
dxdn
dydn (IV.6.B.6)
These are used in the
boundary conditions
on
7. They involve three arbitrary Eq. IV.6.A.6 and constants, oc,, ft-, and y,, for each boundary
curve.
a
set ao;c
_|_
For a simply connected
=
= y = 0, ft
fi^y
_[-
y
we may
addition
to the stress function
change the values partial
slice,
the
since
9?
of
of the stresses, which are second
derivatives of 99.
for a slice with
However,
n holes, three conditions on
99
are required for
each of the n internal boundary curves in order to We now focus our and determine a,,
y,.
ft,
u and v and
on the linear displacements the angular displacement co on each boundary lldv du\ T attention
curve, where it is
co
= -k- -
In
many
cases
j-j-
convenient to derive three conditions on the
value of
Fig.
IV.6.B.1
does not dislocation consists of a small Suppose a given horizontal translation a it a vertical translation b it
and a small rotation
^ about an axis at the origin
of coordinates (see Fig. IV.6.B.1).
Consider a boundary (,- which intersects the and of cut. Let Pi and ?/ be the intersections
^
two edges formed by the cut. We integrate fromP { to P/, assuming du, do, and da around (, that u, v, and co are continuous and single-valued the
at the cut. everywhere except
on each boundary curve from the
that u, v, and co are continuous and assumption also J. H. Michell, Ref. IV.6.e). single-valued (see we shall provide for our in derivation, However, dislocations (see also
(TV.6.B.7)
A. E. H. Love, Ref, IV.6.d,
pp. 221-28).
connected Suppose a multiply
slice is
made
connected by a system of barriers. Now simply cut be applied at a barrier, let an actual physical
where (xi9 y ) are the coordinates of Pt l
.
TWO-DIMENSIONAL STRESS PROBLEMS
262
Thus
we have reasoned with
far,
u>
v,
and
co
^
,
~~
continuous and single-valued everywhere except at a distinct physical cut, where they are dis-
^ \jx
2
32
22
/
vv
continuous. However, the dislocation is assumed
Jf
__
j
j
cut edges are joined very small, and after the two the same material, of of a the insertion wedge by to locate the distinct
it is
impossible physically cut. The same state of strain would exist
and
were
co
and
continuous
throughout the
slice
(IV.6.B 12)
9/
\9x9j; /
g^
j
We have used
co
=-
1
-
^
if
v,
,
many-valued
i nterms
of strains,
with
r
f
Jco-c,
2 9x
J(
j *
f
du
=a -
3x7
2 9y
t
(IV.6.B.13)
Then
(IV.6.B.8)
Adopting this latter point of view, we express du, dv, and doi in terms of strains. Using Hooke s
a
.
=
+
Thus, Eq. IV.6.B.8 becomes conditions on
-
-
y
2 ox
law, the strains are expressed as stresses, and,
the stresses are expressed in terms of finally,
y
r
(p.
n ~
+
I
J
7w
dx
1
oy/
~y
A ~ (
^e
g
~j^ ~~~r
on
the internal boundary curves.
(IV.6.B.14)
We have Now we assume that the strains are continuous .
*
_
j
f
y..
\
flu
=
I fi II
u
j
rfx
Jw^c
JG.
^u
i
--
and
9y
integrating
(IV 6 -
by
parts,
B 9)
A
"
(
2%
A
(L. ^
Then.
we have
-if 2
But, integrating by parts,
= yi \
on everywhere y
singleo valued
-|
"
3,
\
9x (IV.6.B.15)
, s
ffV.6J.lQ)
that
a^-x+
,
Then
(IV.6.B.16)
But Eq. IV.6.B.16 \
J
ydco
= \
J
constrained
yi^-dx^ \
9x
9; 9j;
Hooke
s
slices.
is
vah d for both
Now we use the
law relations to express
it
free
and
appropriate in terms of
BOUNDARY CONDITIONS
IV.6.B.
For a constrained
stresses.
slice,
we
use Eq,
IV.6.A.4, and obtain:
263
Rearranging terms, vn _
_ +a
r
39
pa
y \a dy L3*
J /!((!_ Ll J
^_
K1
fe
~T 3
+ ,)^ 3x
r
J^ + W
,
_ fe -
L9x
5
rfx
Now we substitute for the stresses in terms of
9?
using Eq.IV.6.A.l:
Ea^{
= -1 \
x
\ ,.
* (IV.6.B.19)
or, expressed in terms of ,
9
3
3
3
(!-,_---* g?
normal and tangential
derivatives,
f
flj
rf-^ + 1
1
"")
v
JE
(IV.6.B.20)
.dV ^ ox
Now,
, 39"]
since the stresses are continuous
^
9xJ
single-valued,
by parts and
we may
write:
=(IV.6.B.18)
and
integrate the last integral
4
i^Us 3s\3x/
(IV.6.B.21)
TWO-DIMENSIONAL STRESS PROBLEMS
264
Substituting the
But, from Eq. IV.6.B.2,
constrained
Y-Vm
:
Substituting in Eq. IV.6.B.20,
f
r
Jfr.L
9
9
dn
ds
(IV.6.B.22)
we finally obtain
f
_
f(l
Hooke
s
law relations for a
slice
(Eq. IV.6.A.4),
+
3TW
v)
1
:
1
2
+
J
1
Vd(lJfi
.
as
E
ox r
/*
L_ v
1
J^
L-
x
on
\ds
if-[7-7m]ds
dv
f
L
to
respect
= b + x^i, we obtain t
+ ^(V
["y|-(VV)
^^
with
reasoning
=
c li
(IV.6.B.23)
-vh.
By analogous
Using Eq. IV.6.A.1 to express ct in terms of 9?,
c/sJ
--i~ 1
u.jj.j6 /^
\
^/i~i
"\/\
on
ds
l-2y
^L.
_
37
f
l-vk
\y \-
\-
ds
t
3Kl
.
["
I-/
%
,
Ids
3J (IV.6.B.28)
Collecting terms,
Finally,
%
=
we
I
do)
I
do) /(,
consider the
\
K..
first
,
ax
Lox
+
of Eq. IV.6.B.8,
doo
9v dy
-J
/
ro^
+ (l + v)aj
^ dxI
[^^"g-
j
\d (IV.6.B.29)
(IV.6.B.25)
Expressed in terms of line integrals,
Expressed in terms of strains, f
f(
e. on
1
-
-
f 3fl l-2vf 3F, T"*-^ a~* 3n l-rJ l-J(,ai E(
-;
(IV.6.B.30)
IV.6.C
It is
and
ANALOGIES
generally assumed that Eq. IV.6.B.23, 24,
30, together with Eq. IV.6.B.5
Boundary Conditions:
and 6 con
stitute boundary conditions sufficient to determine a unique solution y to Eq. IV.6.A.6 through-
out a multiply connected constrained
To
is
(B
_A
ds
}
+ fty +
+
(IV.6.B.5)
_
followed, but the
slice,
^
the
Hooke
= 41 + fyn + a,/ + ftm
(IV.6.B.6)
dnJ
s
relations of Eq. IV.6. A. 5 are substituted into
Eq. IV.6.B.16 and 26, yielding:
f
,
slice.
to Eq. IV.6.B.23, 24, and 30 for a free
law
= (*
obtain the boundary conditions analogous
identical procedure
265
,= ~
(Y-Vm)ds
[y|-
=
,a
{
U - -x -1, Us
-(l-r),f
J(S;1
3sJ
9n
(
y
-- x
.
\ds
[Y-Vm]ds
_
(IV.6.B.31)
cc
f
L 3? _ x ^1
r
r
j5
"
1
f
-6, -(!-)
r
dV
JL{y-f 3s
_ _L_
3F
+ x-j-
i
F-
Fm Us
(IV.6.B.23)
J
l-rJ(jl
3
(IV.6.B.32)
J
(^ i
f "
39
a j
- -i-
.
ftj*
1 (V
IV.6.C. Analogies J
differential equations
and boundary conditions
sufficient to determine Airy
multiply
s stress
function for a
connected free or constrained
f
I-
^
2
^5
U
1
F/
1-Jdl
(IV.6.B.33)
In Sections IV.6. A and B, we derived the partial
f
(TV.6.B.24)
J
-"
n
- \^iv[ dV ^ I
^__ T
slice.
Constrained Slice:
^
Ex "
90
f
Y^
J
"^
*
(IV.6.B.30)
Free Slice:
^=
-(1
(IV.6.A.7)
TWO-DIMENSIONAL STRESS PROBLEMS
266
However, one can very simple physical systems. use the assumption that they determine
Boundary Conditions:
effectively
It serves to justify
a unique solution.
- A-m) ds
a number
(IV.6.B.5)
between systems whose elastic are difficult or impossible to determine properties and systems whose elastic properties are directly
(IV.6.B.6)
measurable.
of analogies
Note that the right-hand side of each of the and MichelFs conditions is
differential equations
where
a function of x and
y
from the given
calculable
distri body forces, boundary loads, temperature and slice the dislocations bution, and throughout
The right sides of the load along the boundaries. relations are calculable except for the
constants
a,.,
ft,
unknown
and y t which are assumed deter -,
mined by MichelTs conditions. The analogies which we shall discuss below are justified by two
that
showing
systems
analogous
have
their equations. right-hand sides in Therefore, the stress function cp is identical for the
identical
_
-Ea
f
a0
r
two systems, and they have
del,
IL\y--x-\ds on
identical
stress
distributions.
os-l
First
we
consider the analogy between a free
and a constrained
-(!
slice.
Suppose our problem
consists of finding the stress distribution in a
constrained slice of given shape, given constants (IV.6.B.31) v,
E,
and
and given body
oc,
boundary and dislocations.
forces,
loads, temperature distribution,
Direct measurement of stress in a constrained slice is
extremely difficult.
stress in
a free
On the other hand, the
slice is relatively
easy to determine.
An
examination of the equations and boundary conditions for the Airy stress function in a free
^ ^_L U+ x-
r f -Ea
T
J(j;l
os
s
and constrained
dn
slice indicates
analogous. If their shapes, (IV.6.B.32)
that the
body forces,
loads, temperature distributions,
two are
boundary
and dislocations
are identical and the appropriate relations hold
between their
i
elastic constants,
distributions are identical.
f
Jcr,9n
constants /,
E
,
the constrained
-4 We "load
31, 32,
As
-ds
a
are
equations can be
and
known as MichelPs matter,
and a of a free slice analogous to slice of constants v, E, and a.
Comparing Eq.
6 as the 1
IV.6. A.6 with Eq. IV.6. A.7,
these
conditions. differential
for very directly integrated only
-/=
1-
while Eq. IV.6.B.23, 24, 30 or
practical
stress
we
have
shall refer to Eq. IV.6.B.5 relations,"
and 33
(IV.6.B.33)
then their
Let us determine the
(IV.6.C.1)
and Eoc
(IV.6.C.2)
IV.6.C.
ANALOGIES The above analogy is useful in itself and
Comparing Eq. IV.6.B.23 with IV.6.B.31,
=
(IV.6.C.3)
v
1
is
also
an intermediate step in the analogies
as
applied
--
267
below.
The second analogy we consider is the one between body forces and boundary loads and forces are such that dislocations. If the
Thus, our required conditions are:
body
v
(IV 6
v
C 4)
v
1
(e.g.,
and the they are gravitational), the then determining constant,
if
is
temperature
slice are:
equations for a constrained
F
F = -= ~
(IV.6.C.5)
2
VV =
v
1
a
V2 7 =
= o(l + v)
(IV.6.C.6)
-
f.
=
tp
in the equations for a Substituting these values
(jy
(IV.6.C.10)
+a +
Ajn) ds
oc
t
S
are
no
dislocations, Eq. IV.6.C.3
not
is
the conditions reduce to required and
+ ft
(IV.6.C. 1 1)
free shce we obtain the equations for a constrained
^f there
fty
*>o
<
= ~J^
AJ
+Bm
A
= -\(Y-Vm)ds
t
-f a,J
+
ftm (IV.6.C12)
*
where v
r,
V= ~TT
V
i
JQ
r,,
EOC
,
In the special case that
V2 F = V2 =
the forces are gravitational
the temperature distribution slice
is
simply
connected,
is
steady),
the
=
Al
+ Bm
(for
and
T
f
J^i
1^
}
(Jf-
slices
-
1
J
2
^ ! -v 1
rfs
J
ds
= J^i_ _
are
VI) ds
_ x 1 (V 2
dn
and the
determining
and constrained
ns for both free equations
J
Bi=
V
L
example, when
S
.
i
%
_
-,
L -- x L
9
ds
^s J
"r\
(IV.6.C.9)
SnJ
where
A= _
f Jo
(F
_
Vm)
- J- f - J
ds
1
w>*
f
[I
-
W]
&
(IV.6.C.14)
5i
l (VWs w = ^._L^J ^ ds 1-v 1-JE,3B 2
J E .3n^
Thus y, and hence the stresses, are independent
(IV.6.C.15)
,
of
TWO-DIMENSIONAL STRESS PROBLEMS
268
boundary loads and
dislocations, so
that the
the same, right-hand sides of the equations remain as follows.
We set
X Y
Assume
the
body forces
^X-Vl = Y-Vm
(IV.6.C.16) (IV.6.C.17)
J=0.
V in
no dislocations need
A
)[ L,
.
on
-J
W Y
This
precisely the pressure in a fluid of
been suggested that an analog subjected to fluid pressure may be studied by the methods of photo-
A
n
M. A.
third analogy
is
dislocation stresses.
Eq. IV.6.C.16 and 17 indicate that in the analog
a boundary pressure of magnitude V must be added to the boundary loads on the prototype.
Suppose the temperature
= 0)
2
and
7=0,
slice
is
steady (V
Y=
0.
Then the determining equations
VV =
of Eq. IV.6.C.18, 19,
sponding dislocations must be applied to the by a method to be discussed a bit later.
^1
=0,1
X=
0,
are:
(IV.6.C.22)
^o^ + Ay +
y,
+ Am
(IV.6.C.23)
(IV.6.C24)
dnJc.
analog,
these conditions, the stress function
Biot, Ref. IV.6.a).
one between thermal and
distribution in a multiply connected constrained
and 20 represent integrals of couples or forces on boundaries. If these do not vanish, then corre
Under
is
elasticity (see also C
(IV.6.C.20)
The right-hand members
of
surface pressure
density p at a distance y below the surface. It has
(IY.6.C.19)
.
Now
to be considered.
model of the body made of material
magnitude py should be applied in order to simulate the stresses in the prototype due to body forces.
t,
have
V=py.
Eq. IV.6.C.18, 19, and 20,
of negligible density.
os
We
Then
clear that the line integrals all vanish, so that
consider a
(IV.6.C.18)
to the vertical with
bottom of the body.
F=-/>,
Substituting for it is
parallel
j>-axis
the origin at the
in the
analog is, at all corresponding points, equal to 9? in the to Eq. I V.6. A. 1 prototype. Referring ,
we see that the forces
may
prototype with body be very simply calculated from the
stresses in the
stresses in the
Ea
SB
analog as follows: (IV.6.C.25)
Gy
The preceding
= a; +
V
discussion
(IV.6.C21)
referred
to
f ^| /L as
--* Ea
Eh
the
2
analogy between two constrained
slices.
By
f
dd
T
i-vJ^L
ds
combining this with the first analogy, it is possible to
construct a free
slice
dnl (IV.6.C.26)
without body forces
analogous to a constrained slice with body forces. An interesting application of this analogy has
been developed by M. A. Biot
He showed
,
+ x df\\ds ,
\y
f J
-3
Ect
on
\-r
Eoc
,
-ds
(see Ref. IV.6.a).
that the gravity forces acting
(IV.6.C.27)
upon a
body of uniform density p may be represented in an analog by the pressure on a model immersed upside down in a fluid of density p.
A uniform-temperature analog of identical shape and material has identical determining equations
if its dislocations are fixed
so that the
IV.6.C.
ANALOGIES
of Eq. IV.6.C.25, 26, and 27 right-hand sides
remain the same. That b-
t
and
269
fourth-order equation:
the dislocations a/,
is,
4
(IV.6.C.31)
^
must be such that
c-
where do
r
!
del
J>7--
en
Eh*
D=
TaasJJ
x
-
5
12(1
(IV.6.C.28)
where h as the
is
- *)
2
the thickness of the slab.
"flexural
D
is
known
of the slab.
rigidity"
Now suppose we wish to study the stresses in a free or constrained simply connected
(IV.6.C.29)
(IV.6.C.30)
slice
of
and given body forces, temperature given shape We may distribution, and boundary loads. construct a slab of identical shape and subject
it
to a transverse load distribution adjusted so that If these
conditions are satisfied,
the Airy
-
is
function and the stresses are identical in model
and prototype. Here again, the with
this
or first
analogy
may be combined
one so that the thermal
constrained
slice
identical with the right side of Eq. IV.6.A.6
This makes Eq. IV.6.C.31 on
7.
with the fourth-order equation on
Now on the slice we have boundary conditions
stresses in a
be studied by applying
may
=
dislocations to a uniform-temperature free-slice
Am)
I
ds
Jo
(IV.6.C.32)
analog.
may be
Dislocations
connected free
slice as
applied
follows
an inner boundary curve
Both
curve. relative
identical
9?.
sides of the
to a multiply
A slit is cut from
:
to the outer
slit
boundary
of the two clamped edges
position
is
of screws. E. E. Weibel (Ref. adjusted by means M. A. Biot (Ref.
A
where
are clamped and the
and
B are calculated from given
The corresponding conditions on the
data.
slab
may
be imposed by adjusting the deflection and
normal slope
at the slab
boundary to equal
and
of IV.6.h) 5 following the ideas
at corresponding points
to
investigate IV.6.b), has designed an instrument thermal stresses in a constrained slice. Appro
priate
the
dislocations are applied to a constant-
temperature
free-slice
elastic material,
analog
made of
is
in Chapter IV.5.
known as the
"slab
s
analogy"
the similarity of Eq. IV.6.A.6, differential equations
7,
on Airy
stress function.
and
equal to
slice.
slab, the deflection J
at the corresponding point of the
However, our
interest lies
not in finding 9,
but in determining the stresses a^ ay , and TW In Eq. IV.6.A.1, we see that we need the second .
There is one more important analogy based on the equations determining Airy
on the boundary of
slice.
Then, at any point of the
photo-
which may then be studied by
methods discussed
It is
dn
arises
from
the fourth order
s stress
function,
and the fourth-order equation on the deflections or slab, arising from the approxi of a thin
partial v
derivatives
,, 2
,
djr
and -z-J"
in the slab analog Correspondingly,
oxoy we must
determine the quantities
plate
mate theory of thin plates. (See also A. E. H. Love, Ref. IV.6,d, pp. 487-89.) If a thin plate is subjected to
Z(x
to
its
(IV.6.C.33)
a force distribution
middle plane, and
J
t y) perpendicular the displacement perpendicular to this plane, then I approximately satisfies the following
is
dxdy
TWO-DIMENSIONAL STRESS PROBLEMS
270
and
If
ox
-r-
are small, then
and K V are the
/c,
oy
curvatures of the slab on planes through the x and axes, respectively, perpendicular to the x-y
y
Light from source reflected
P.
an angle perpendicular to the x-y plane, forming with the x axis, /c e can also be expressed approxi ,
relative to the
as the second derivative of
mately
distance s along the intersection with the x-y -
Since
plane. r
3s 2
d
__
2
The
S is collimated by lens A and
slab surface 6.
lens B, at distance a
from the
distance b between
Now if the
The curvature along any other plane
plane.
by the
slab surface
beam of light is
parallel,
is
an image at is measured.
and
B
plane,
the reflected
where/is the
b=f,
If the slab surface is
focal length of the lens.
convex and
P is
and
focused by
It is
slab, to
considered to have a circular cross
section of large radius
r,
then, as
we
shall see
= cos 6, -4 = sin 6, and
ds
ds
9s \dx ds
dy ds/
UdxY.
3
~
2
^.^
c
dxdy_
2
dx \ds!
dxdydsds
3/Vds
then
-
2* w [sin 0][cos 0]
+
a
* y [sin 0]
(IV.6.C.34)
This equation determines KXV in terms of KX *, These quantities may be measured by ,
and K e
.
optical
M. G.
methods
(see also
Salvadori,
Ref.
R. D. Mindlin and IV,6.f, -pp.
784-89)
providing a simple method of determining the
Fig.
IV.6.C1
stresses.
Slabs
may
be constructed of polished metal,
such as brass, and several methods have been used
by means of the The first method, used in 1908
to determine the curvatures reflection of light.
K. Wieghardt, Ref.
(see
IV.6.i),
below, the light reflected from the surface
approximately described as
if it
S
at
point source
from
lens B.
below the slab
Then b
consisted of
determining the slopes of the surface point by
is
V>
An
instrument to
(IV.6.C.35)
measure the radius of
(see J.
K. Robertson, Ref. IV.6.g, pp. 84-85) so
that
-
of a light source with a collimating lens which throws a parallel beam of light on the (see Fig. IV.6.C.1).
a
We now discuss
The
the
assumption that the light
B has an apparent
which reaches the lens
the image from the second lens
course, that the slab cross section
reflected
is
measured and
used to calculate the radius of curvature of the surface.
(IV.6.C.36)
*-/
by the surface onto another lens which focuses it at a point. The distance of is
the
/
tially
curved metal surface
+a
graphically.
curvature of a surface directly was developed by E. Einsporn (see Ref. IV.6.c). It consists essen
beam
/and
usual equation for a lens yields
telescope.
and K^ were then computed
,
a distance x
greater than
point by measuring the angles of incidence and reflection of a beam of light with a
KX K
may be
originated at a
S
.
We assume that the beam is narrow,
Suppose the axes of lenses
A
circular.
is
and
source and, of
B intersect at
a point, g, on the slab surface (see Fig. IV.6.C.2).
IV.6.C.
Let JR be any other point near Q on the slab cross section, such that light in the beam falls on R.
The
center of the incident
falling
on
R
or
2r sin
beam of light makes
an angle a with the normal to the slab surface at
and the ray
makes an angle a
271
ANALOGIES
+
i-, /
+J
x
Q
sin2e
(IV.6.C38)
2/J
\
e
with the normal at R. Of course, these rays are
If
is
small and
<x
is
not close to -
,
then,
approximately,
&J9) terms of the measured distance b, Expressed in
=
_.-
(IV.6.C.40)
fl
The determination of- in three different planes r
perpendicular
to the slab surface yields KX ,
KV and ,
stresses
KXV and, by analogy, the corresponding in a slice.
A refinement of this instrument, developed by and minimum
Martinelli, measures the maximum
curvatures at any point, as well as their orienta tions, directly (see Ref. IV.6.f, pp. 787-89).
The
slab
analogy
may
also be applied to
slices. Eq. IV.6.B.23, 24, and multiply connected 30 have their analogs in the approximate theory
M. G.
of thin plates (see R. D. Mindlin and
Fig. IV.6.C.2
Salvadori, Ref. IV.6.f, pp. 776-82). parallel.
At each
the angle of reflection
point,
is
of incidence. Thus, the angles equal to the angle between the incident and reflected rays are equal
2a
to
at
Q
and to
at rays intersect
S
2oc
at
+
2e at R.
The
reflected
.
is
small,
x
is
independent
of e up to a
so that S is the approximation, source of any ray in the beam. apparent intersect drawn to Q and Clearly, the radii
first-order
R
at
;
an angle
-
/_S
e
and the chord :
2r|~sin-l 2J L
is
principles
usually applied
an angle of 2e.
We shall now calculate x or S Q as a function of We shall show the radius of curvature r, a, and that, if
All the above analogies were presented in their forms in order to emphasize the basic simplest involved. In practice, scale factors are
QR
equal to
has length
/_QRO
and
methods of dimensional analysis
stresses of
(see
The
Chapter
effect of this scaling IV.2) readily indicate the
In general,
the measured quantities.
it is
on
only
a simple calculation to necessary to perform means of which obtain the conversion factor
by
stresses in the prototype
may be
calculated
from
measured in the analog. quantities
References for Chapter
ROor^-)-(y.+ )oi--cL--. a.
Then, by the law of sines,
M. A.
"Distributed
Biot,
gravity
6
and temperature
two-dimensional elasticity replaced by loading in
A.S.M.E., boundary pressures and dislocations," Vol. 57 (1935), pp. Trans, in Jn. Appl Mech.,
2r
3
to the dimensions
the prototype in constructing a model.
sin 2e
sin 2e b.
A41-A45. M. A. Biot,
. "A
thermal stress
(IV.6.C.37)
pp.
54M9.
general property
distribution,"
Phil
of two-dimensional
Ma^
Vol. 19 (1935),
TWO-DIMENSIONAL STRESS PROBLEMS
272
c.
d.
E. Einsporn, "Ebenheit," Zeitschrift fur Instrumentenkunde, Vol. 57 (1937), pp. 265-85.
A. E. H. Love. of
e.
Elasticity.
A Treatise on the Mathematical Theory New York, Dover Publications, 1944.
H. Michell, "On the direct determination of stress an elastic solid, with application to the theory of London Math. Soc., Proc., Vol. 31 (1899), plates," pp. 100-24.
R. D. Mindlin and M. G. Salvadori,
J.
K. Robertson.
New h.
i.
in cylinders by the 5th Int. Cong. Appl. Mech., Proc., 1938, pp. 213-20. K. Wieghardt, "Uber ein neues Verfahren, verwickelte
E. E. Weibel,
"Thermal stresses
method,"
Spannungsverteilungen in elastischen
"Analogies,"
in
New York,
Wege zu
Kb rpern
auf
Mitteilungen tiber Forschungsarbeiten an den Gebiete des Ingenieurwesens, Vol. 49(1908), pp. 15-30.
experimentellem
Handbook of Experimental Stress Analysis. John Wiley & Sons, 1950. Pages 751-89.
Introduction to Physical Optics,
York, Van Nostrand, 1935.
photoelastic
J.
in
f.
g.
finden,"
Chapter 7
ELECTROMECHANICAL ANALOGIES
IV.7.A. Definition of Mechanical System
When circuit theory was in its infancy, electrical systems were conceived in analogy to mechanical or hydraulic systems. The term, electromotive force, for example, is directly derived
concept of a mechanical force.
from the
The flow of
current in a wire was thought of as analogous to the flow of fluid in a pipe.
Today, knowledge of the theory of
electricity
has progressed so far beyond the theory of it has become useful to reverse
mechanics that the analogy.
and
The
solution of
acoustical problems
electrical analogies are
is
mechanical
many
facilitated if their
determined so that
circuit
theory can be applied. There are many problems to
which purely
theoretical solutions are too
time consuming or too difficult and in which mechanical experimentation is both difficult and expensive. The analogous
hand,
may
construct.
circuits,
be very much simpler and cheaper to Mechanical experimentation is likely
to require the construction of a
every new problem. However,
of
on the other
electrical analogy,
it is
new model
for the
for
methods
possible to set
up a
applied to the propagation of sound in horns, mechanical phonographs, and mechanical wave filters. Many types of vibration absorbers have
been studied in this way. Furthermore, the study of transducers, which convert electrical energy into mechanical all-electric
simplified if their
is
energy,
analogs
may be
determined.
Let us consider a mechanical system with We assume that four
motion in one dimension.
types of forces exist in this system. 1.
Inertial forces proportional to the accelera
tion of masses. 2.
Dissipative forces proportional to velocity. forces proportional to displacement, Forces generated by some outside agency.
3. Elastic
4.
We make the following further approximating assumptions:
The system may be broken up points, connected to each other and
into
1.
axis 2.
mass
to a fixed
by springs and dashpots. All elastic forces are lumped into the springs
which obey Hooke
s
law,
i.e.,
they transmit forces
proportional to their contraction. 3.
All dissipative forces are
lumped
into ideal
such that calculating board with fixed elements,
dashpots, which transmit forces proportional to
for a wide variety of appropriate connections
their rate of contraction.
problems can be rapidly set up. A wide variety of mechanical and acoustical
of a piston whose motion
be
approximated by lumpedsystems may constant systems, which have analogous electrical circuits consisting of resistance, inductance,
capacitance.
It
has become
and
common practice to
(A real dashpot consists is
damped by a fluid, a
hydraulic door check, for example. idealization
of this
device
to
We
use an
represent
all
dissipative mechanisms.)
Although no actual mechanical system exactly these assumptions, they hold approxi
fulfills
the analyze this type of mechanical system by methods of electrical network theory. Many of
mately in many transient or vibrational problems, small motions are involved. especially if only
the most useful concepts of network theory, such as Thevenin s theorem (see Section III.5.H), and
FV77.B. Connection Diagrams
modes
of
impedances
vibration
associated
(see Section III.5J)
with
may
be
given directly
and acoustical vibrating applied to mechanical network For theory has been example, systems.
The motion of this mechanical system can be analyzed in a manner analogous to the way an electrical network has been analyzed previously.
Connection diagrams, analogous to
electrical
ELECTROMECHANICAL ANALOGIES
274
Gl9
Gq
network diagrams, are drawn indicating the distribution of forces and velocities through the
contraction of
mechanical system. The
of the point of higher index minus the velocity
differential
equations of
tion"
we mean
.
.
.
,
of contrac
"rate
(by
the velocity of the terminal mass
Fl ,...,Fn Pn Fl5 Vn Pn Gn ...,Ga and G/,
the system are set up in a form analogous to those
terminal mass point of lower index);
Many of the concepts which are so useful in circuit theory, such as
be the forces impressed on Pl5 be the velocities impressed on P1}
damping, fundamental modes, and the impedance operator have analogs in the mechanical system.
connecting
The methods of complex
diagram, with a connection for each mechanical
of an electrical network.
so
analysis,
well
developed for electrical circuits, can be utilized.
The
details of this
method
be developed
will
below.
At
By drawing
P
we
point,
indicate that there are
l9
.
.
P w we
,
.
,
,
.
.
.
.
.
,
.
,
.
<7
.
,
.
.
.
,
obtain a connection
element.
As an example system (see
this
.
.
us consider the following
let
Two
IV.7.B.1):
Fig.
blocks
of
two
In one,
to be considered. possible analogies
analogous to force, voltage to velocity, capacitance to mass, conductance to dashpot constant, and inverse inductance to elastic
current
is
Current generators represent force
constant.
generators; current amplifiers, force amplifiers;
and transformers represent analogy, voltage
is
In the other
levers.
analogous to force, current to
inductance to mass, resistance to dash-
velocity,
pot constant, and inverse capacitance to constant.
elastic
Voltage generators represent force
generators; voltage amplifiers, force amplifiers;
and again transformers represent levers.
We now proceed with the analysis of the mechanical system. Let P be the fixed axis; Plt ...,P n be the mass
points, enumerated in order of increasing
distance
from the fixed
masses
at
M
axis;
?!,...,?;
i?/,
Fig. IV.7.B.1
...,Mn be the
l9
...
be
vn
9
their
material of mass
M
and
l
M
2
slide
on a horizontal
velocities.
Block 1 is connected to a vertical wall by
surface.
By Newton
s
second law, every mass exerts an
inertial force
//
^
and to block 2 by a spring a spring of constant of constant z There is factional resistance of
K
=
=
!,...,)
M,pt>/
(IV.7.B.1)
dashpot constant surface
friction.
the fixed axis (p represents differentiation).
to block 2.
connections, (//,
...,G n with
the fixed axis,
through which //, fn are transmitted. Let Gl9 Gq be the spring and dashpot .
.
.
(noninertial)
.
,
.
:
,
connections;
forces transmitted through
conventions directed
we have
/x Gl3
,
.
.
.
.
.
.
,
,/a be the
Gq
adopted, positive force
t^,
.
.
.
,
be the
i>
fl
2
between block
1
slides
and the without
A varying horizontal force F2 is applied Its
connection diagram
in Fig. IV.7.B.2.
Gl9 Gz
inertial connections.
,
and
G3
G/ and G2
connections associated with
M
is
illustrated
are the
non
are the inertial
and
We follow these rules in making
M
2.
a connection
diagram:
(with the is
from a mass point of higher index toward
one of lower index);
D
on which it slides. Block 2
with positive force and velocity directed toward
We think of the mass points as having inertial
.
rates of
1.
We draw the fixed axis, P
the motion of
measured.
all
,
relative to
which
other parts of the system
is
IV.7.C.
We
2.
P
.
.
.
insert a point for
Pn
We make
.
MATHEMATICAL RELATIONS
each mass point,
the
approximating assumption that any distributed mass in the l5
,
original system
is
275
mass is lumped with the masses of the end points. 6. If two connections are drawn in the parallel, velocity of their terminals
concentrated at these points.
is
the same.
two connections are drawn
7. If
in series, the
same force is transmitted through both. 8.
An
impressed force,
F
is
i9
represented by
symbol F, Fig. IV.7.B.3 in a connection between P the mass the point 3 to which it is applied and -
fixed axis. 9.
An impressed velocity, V^ is represented by V (see Fig. IV.7.B.3) inserted in the
symbol
connection of the mass point at which
inertial is
it
applied.
IV.7.C. Mathematical Relations in Connection
Diagrams
Now, Fig. IV.7,6.2
to return to the general system.
We
obtain the relationship between force and velocity in the connections.
We
3.
connect the mass points,
to the fixed axis,
GI,
.
.
.
Gn
,
P
,
by
Px
,
Fig. IV.7.B.3). inertial forces
.
.
,P n
In the inertial connections,
,
inertial connections,
f ,
.
represented by symbol
M
(see
These connections represent the
//
-
-
(?!,
=
!,...,)
(IV.7.C.1)
In the noninertial connections,
due to the motion of the masses
with respect to the fixed axis. 4. We connect the mass points to each other
and
(/
Vs)
or
f
=-
j
to the fixed axis by noninertial connections, .
.
.
forces.
Gq representing
,
elastic
and
We assume that any distributed elastic or
forces in the system are concentrated dissipative
in these connections.
The
elastic
or spring con
K
nections are represented by symbol or dashpot connections by dissipative
(IV.7.C.2)
dissipative
and the
D
9
Fig.
D =
where
5
constant,
K = s
spring
If a force Fj is impressed on a point Pit let v/ be the velocity across the FJ connection. Since the FJ connection is in parallel with the inertial
connection
IV.7.B.3.
-3-
dashpot
constant.
(?/, clearly
At this point we introduce the very useful concept of the hindrance of a mechanical system, Ka differential operator A. , Dj -f
= M$ +
which acts upon velocity giving force analogously to the
Any
real
elastic
element has mass and
transmits dissipative as well as elastic forces. To represent the element completely, we should
draw a
a dashpot in spring and
operates
on current to
give
a major point of voltage. difference between the methods of calculation of Note,
Fig. IV.7.B.3
5.
way impedance
parallel.
The
however,
impedance and hindrance.
Where
impedance of elements connected in added,
it is
the
total
series is
the hindrance of elements connected
in parallel that
is
added.
(In the literature,
ELECTROMECHANICAL ANALOGIES
276
hindrance
is
pedance."
Thus, in our mechanical system every non-
called "mechanical im-
generally
We
new term
the
introduce
and impressed force connection together with the inertial connections of its terminals to
to
inertial
eliminate confusion,)
the fixed axis forms a loop
In the example cited above, we have
M^
=
>
fl
K
/lAl,
^D 2%
/2
=
/,
MX
K =^
/3
We
must relate the quantities in our connections.
%= + ajlc
1
if
is
j
if Pj is
respectively,
fi
...,)
of
are the vei oc i t ies
its
or
ya
,
v
_
,
v
(IV 7
C
8)
the terminal of higher index on Ok
not on
where (a^ was defined above. The argument th eor ven y ( see Section III.5.C) can gj
Gk s
on a mass point
-F =\
=
r
^^
m
sum
third law, the is
of
now be used to show that tnis is one
A
/,
=
i
complete
set
of lo P s but of course there are other complete
zero.
>
>
sets
ft
1,
terminal mass points of lower and higher index
:
Now, from Newton forces acting
/*
^/ and
where
Ps is the terminal of lower index on Gk
= -1 ifP
= a^
(#)
=
=
- v! =
vf we
our general system, completely describe
define a matrix
;
(TV.7.C3) 3
P
P
To
% ~%
+
*
*
i
1,
.
.
.
,
>
of loops.
for we write the equation For our example, ^ r
M\ n)
fc=i
every loop: (IV.7.C.4)
we have
In our example,
^
/
Vl
Vl
=
f ~~ F+f-^2+73
J2
In our general system, velocity relations for
all
we have
geometrical
the closed loops in our
v2
v1
(e)
=
connection diagram. 6
(IV.7.C.9) (f)
-v=o
(g)
where
(^,
and
(cn),
are
(^J
^, q or ^ m =
(IV.7.C.6)
00
connection
(i)
1 if i.e., G* G or Fm is in thejth loop; b& cn or ^ m = if Gt G or ^m is not in the /th loop. We note that there are numerous ways of
matrices,
,
zs
Z5
t;
i
u2
(j)
-f t;/
1>
-
+
_
-
s
t?
=
2
=0
a
+ % - t;/ =
.
,
,
3
where (a
.
>
(c),
and
(d) are specializations
of
z,
selecting q independent loops
as there are
many ways
meshes in an
electrical circuit.
(see Section III.5.C) a
among
these, just
mesh
In circuit theory
^
IV 1C1 2 4 and -
>
,
electrical is
selected corre-
spending to each nonreturn branch. There is a rather obvious analogy between the two types of systems in which inertial connections correspond to return branches and noninertial connections to
nonreturn branches.
Electrical Analogies
of selecting independent
anal 8 ies
two P ossible
6 su est g
We may
-
let
current be
analo ous to force and Yolta e to velocity or
may let volta g e be anal to velocity,,
fi
Now
=M ^ ,,
,
we
g us to force and current tf we examine E 1 C J and 2 .
,.
fi=D
^
,
f^
= X.-
IV.7.E.
we note
that the
MASS-CAPACITANCE ANALOGY
analogous relations in a
circuit
branch are
is
correctly drawn,
it is
277
not necessary to write
down
the differential equations at all. The con nection diagram may be directly translated into **
an electrical
1$
circuit,
following the rules described
below.
or
(IV.7.D.1)
IV.7.E. Mass-Capacitance Analogy
Thus, analogous to mass we have capacitance or inductance. Analogous to dashpot constant we have conductance or resistance, and analogous to spring constant we have inverse inductance or inverse capacitance,
It will
be shown below that,
for either analogy, Kirchhoff
s
nodal and mesh
laws are exactly analogous to Eq. IY.7.C.4 and Table IV.7.D.1 summarizes the analogies.
The mass-capacitance analogy is also known as the inverse, electromagnetic, or mobility analogy.
To apply nection
it,
we can
directly translate the
con
diagram discussed above into an electrical ground corresponds to the we have a current
Electrical
circuit.
fixed axis, and, in addition,
node for every mass
Each connection has
point.
analogous branch, with a capacitance propor tional to the mass in each inertial connection, a
its
6.
conductance proportional to the constant of each dashpot, an inductance proportional to the constant of each spring, a current generated
Table IV.7.D.1 Mechanical System
Mass- Capacitance
Mass-Inductance
proportional to each generated force, and a
Analogy
Analogy
voltage generated proportional to each impressed
MassM
Capacitance
Force/
Current
Velocity v
D
Dashpot constant Spring constant
K
C
Inductance
L
Voltage e
Current
Conductance \IR
Resistance
Inverse
Susceptance 1/C
Admittance
i
tion,
R
is
proportional
and the voltage at a node is proportional
the velocity of the corresponding mass point. electrical
Impedance Z
Y
current in a branch
to the force through the corresponding connec
inductance 1/L
Hindrance^
The
velocity. Voltage e
i
mesh corresponds
to
An
to every mechanical
loop. Return branches correspond to our inertial
connections while nonreturn branches corre
spond to our nomnertial and force generator Elastic energy
i/f 2
(spring)
Kinetic energy
connections.
K
%Mv
(mass)
/fc),
defined in Section
Eq. IV.7.C. 1 ,
2, 4,
and 6, we have
the relations between voltage
and
current,
Analogous 2
I
Dissipated power
i/
RP
t
Force generator
Current generator
Voltage generator
Force amplifier
Current amplifier
Voltage amplifier
11
to
R?
R
(dashpot)
D
The matrix (a
III.5.C, corresponds to (aik).
2
= C/X/-,) 0=1,
,)
R
=~ 1
e
or
l
=
1
(IV.7.E1)
By an analogy, we mean that we can electrical circuit containing, for
set
up an
the nodal equations for the network,
each element of
the mechanical system, an analogous electrical element. The analogous circuit and elements are chosen in such a way that the differential
(IV.7.E.2)
and the mesh equations,
equations that govern the mechanical system may
be immediately changed into those governing the electrical circuit
quantities
.
by the substitution of analogous
If the mechanical
connection diagram
(IV.7.E.3)
ELECTROMECHANICAL ANALOGIES
278
If we apply the mass-capacitance analogy to our example, we derive the network shown in
to
Fig. IV.7.E.1.
system.
current are made
compute
on an analogous circuit, in order
velocity
and force
in the
mechanical
Obviously, the constants e (1) and K (I)
occupy a position of major importance in this For purely mechanical application.
latter
/m\
i
v
systems, e (1)
and K (I) may be
that the electrical
quantities
chosen so
arbitrarily
may
fall
in a range
convenient to measure.
When we study
electromechanical systems,
shall see that in the
we
analogous circuit for an
electromechanical converter, the values of e (1) and
K (1 are no longer arbitrary, but are determined by j
the constants of the converter.
IV.7.F. Ideal Transformers
At this
point,
we digress from the development
of electromechanical
analogies in order to discuss
which play a very important role in many types of electrical network analogies (see also A. Bloch, Ref. IV.7.b). (For a discussion
ideal transformers,
Fig.IV.7.E.l
Eq. IV.7.C.1,
and
IV.7.E.1, 2,
CM and
2, 4,
and
3 if the
8 are identical with Eq.
proportionality
factors
K (I) are introduced as follows:
/
=
c (1)
/
(IV.7.E.4)
of magnetic circuits, see L. Page and N.
common
type
essentially
of several
Then it Mows
= K(1) v
of ideal transformer coils
is
perfect (see Fig. IV.7.F.1). If a voltage
is
(IV.7.E.6)
K/i\
1 = 32
/e
Fig.
IW7.F1
out that this analogy has two major
areas of use. It can be used as a
way of thinking
in which the mathematical analysis of mechanical
problems is facilitated by the use of mathematical methods developed for circuit theory. On the
device, in
consists
wound on an iron core,
(IV.7.E.5)
= JilD
other hand,
Adams,
that
Xx
We point
I.
and most
first
so arranged that the magnetic coupling between
them e
The
Ref. IV.7.n, pp. 425-29.)
it
can also be used as a computational
which measurements of voltage and
applied across one of the coils, a proportional voltage
is
induced across
all
In an ideal transformer, are
no energy
losses
the others.
it is
assumed that there
and no leakage of
that the self-inductance of the large that
we may
consider
flux,
windings
it infinite.
and
is
so
IV.7.F.
The
voltage across each coil
the time derivative of the
of turns on the
is
IDEAL TRANSFORMERS
proportional to
and to the number
flux,
;th
$
is flux,
and
coil,
nt
.
(IV 7
"
3
F
1)}
I,
*
^
?
noc* e
...,Zm are
1?
branches
2)
flow through impedances
^e
nto
IV 7 F
^
e
vo * ta
*1} ...,e m , where
e
drops
e^Zfr
in the
voltage across the/th
and N,
Z
(see Fig
^
Z
1}
across
number of turns
the
is
et is the
(See L. Page
it
^
dt
where
ve
Currents
e, = - n
m
have a node with
Suppose connected to
coil,
279
coil.
pm
V
Ref. IV.7.n,
Adams,
pp. 319-20.)
Now since there is no same for
all
leakage of flux, the coils and
3=1= The
=*
W2
H!
is
the
(IV.7.F.2)
W2
voltage drops across the coils are propor number of turns.
tional to the
Furthermore, we have assumed very large self-inductance for each coil. This is
means
that n, Fig. IV.7.F.2
very large, and therefore, from Eq. IV.7.F.1,
iffy
must always be vanishingly
remains finite, -
small.
It
follows
can never
that
appreciably from zero.
differ
Now may be expressed
:
(IV.7.F.3)
J
where ^
is
uA
the current in the ;th
coil, \i is
the
A is the cross-sectional permeability of the core, area of the core. The integral is taken around the core.
(See also L. Page and N.
I.
Adams,
Ref.
IV.7.n, p. 427.)
Fig. IV.7.F.3
The denominator
we may
is
not very large, so that
write approximately
We insert m (IV.7.F.4)
Now we may In the special case where all the coils have the same number of turns, Eq. IV.7.F.2 and 4 reduce
e1
=e = 2
...
= *,.
= ...,
(IV.7.F.5)
(seeFig.IV.7.F.4).
have formed a new network in which the
branch
OPl has
zero impedance.
linked to the separate circuit, (IV.7.F.6)
3=1
any of these
short-circuit
branches (say Zj) and detach it from the network
We
to
of a nodal-type transformer
coils
m branches (see Fig. IV.7.F3).
into the
transformer linkage only. the
new network is
In the
We
Zt is part of a
main
currents z2
through
constraints as a circuit node.
branch through the impedanceless
2}
.
.
.
,
by
the old. equivalent to
new network
This special case is called a "nodal-type" same transformer, because it implements the
Z
circuit
prove that
shall
Zm
5
.
into the node,
.
.
,
and
i
m
f
flow flows
OP^ The
ELECTROMECHANICAL ANALOGIES
280
current z/ flows in the separate circuit containing
Zx fix
Across
.
.
,
.
,
Z
.
l5
.
.
Z m)
,
e w where e/
,
The voltage drops
The nodal equation
there are voltage drops
for
is
n
= Z,i/.
+ J f/ =
i
(IV.7.F.7)
=2
across the transformer coils f
are all equal, and are called e t
But the transformer equation
.
is
pm
(IV.7.F.8)
Therefore, (IV.7.F.9)
Now,
if
we
f
replace
equations for Pj_ and
i
by
in the
z/
in the nodal
new circuit, we have
prime equations equivalent to the original ones. Thus, the nodal equations for the new circuit plus the transformer equation are equivalent to the
nodal equations of the original primed and the i The system of equations for equation z/
=
Fig. IV.7.F.4
We
show
shall
equations for the
that the
the
mesh and nodal
new network can be
trans
formed into equations for the old, so that the solutions for the old network are also solutions
The mesh equations can be
divided into three
for the
new network
which do not involve the point
P1?
the
mesh equation
this last,
In the transformer just described,
we might say
that the cores of the individual coils
magnetic
circuit in
form a
which they are connected in
Now we consider the dual of this case, in
which the cores of the individual
coils are
con
of duality will be precisely defined later on.
are identical
with mesh equations of the original. 2. If a mesh of the new network involves
not
.
can be solved ignoring
nected in parallel (see Fig. IV.7.F.5). The concept
types.
The mesh equations
circuit
clearly the circuits are equivalent.
series.
for the new.
1.
and
new
will contain
but
+e
r
t
and
and
these will cancel to yield an e/, equation obtained from the analogous equation in the original circuit 3.
Now
by priming.
consider a typical
mesh
in the
new
network containing OPl9 e.g., a mesh involving PiOPz The mesh equation contains a pair of .
voltages e t Z-L,
+^
we may
have e2
=Z
z
2 2
these values,
-
But, for the
write ej .
If
= we
e
mesh containing and we
= -Z^ f
replace e t
and
e%
by
we have an equation which may be
Fig. IV.7.F.5
obtained from a mesh equation of the original by priming.
Thus
the set of
mesh equations correspond
under the operation of priming.
Nodal equations nodes original.
and
?
1}
for the
new circuit,
except for
are the prime equivalents of the
Each coil is wound on its own core. There are two magnetic junctions, each one made up of one end from every core. In the ideal case, it is assumed that no
flux leaks out of the circuit.
In this case, the
sum
of
all
the magnetic fluxes
converging on any one of the magnetic junctions
IDEAL TRANSFORMERS
IV.7.F.
must be zero, yth
SO,.
But, ef
coil.
281
= 0, where $, is the flux in the
=
-w,
-~
Therefore,
.
at
2^ = Now
(IV.7J.10)
us
consider a closed loop in the magnetic circuit through the yth and Jkth coils. We have the relation; let
(IV.7.F.11)
As before, we assume that 0, and
=
n k ik
^
are close to
zero and the integrals are not very so that J large, n.i.
Fig.
(IV.7J.12)
W.7.F.7
,
,
detach any of these Now we *****? with its transformer coil from the (say Zx) along ,
network
(see Fig. IV.7.F.8),
same number of turns,
If all the coils have the
then
(IV.7.F.13) 3=1
=
ft This
...
*,=
=
(IV.7J.14)
>...)
called a
mesh-type ideal transformer, implements the same constraints as a circuit mesh. is
because
it
we have a mesh
Suppose
branches, with
m
nodes
Currents
IV.7.F.6).
z
P
l9
containing
...
l,...,^
9
Pm
m
(see Fig.
flow through
Fig.
We have
W.7.F.8
formed a new network
in
which the
Zl is part of
branch P/a has
infinite impedance. circuit by a separate circuit linked to the main
The new network
transformer linkage only.
to the old. equivalent In the new network, currents
Z through impedances 2 the in Z flows through l ,
Fig. IV.7 J.6
impedances
1}
.
.
.
,
Zm
across impedance Zj ;
=
1,
.
.
.
,
.
is
The equal
.
.
,
Zm
2
z"
.
,
.
.
The
separate
. , z
m flow
current z/
circuit.
The
to across impedance Z, equal voltage drop, e/ 9 The . , w. voltage drop i, is
=
-
Z
.
is
voltage drop to
^
.
.
Z^, where
^
m.
We connect m coils of a mesh-type transformer IV.7.F.7). in parallel with thembranches (see Fig.
all
and are called equal
We
shall
transfonner coilsare
/.
once more prove equivalence by
ELECTROMECHANICAL ANALOGIES
282
showing that the mesh and nodal equations for the new network can be transformed into
fulcrum; v s the velocity at a distance
equations for the old.
fulcrum.
The mesh equation not involve
P^
for
equation for mesh P-f^
.
.
.
we
all
.
.
.
=
and, since the lever
=^=Z
Pm
,
.
.
.
,
Pm
contain
giving equations
obtained from analogous equations in the original circuit
by priming. The nodal equation for
yields the
By for
f/,
P:
and making
r
contains this
i
m
+
i/.
substitution
prime equivalent of the old Pl equation, new nodal equation
corresponds to the old, and
we have
completed the proof of equivalence. It is clear that for
former there
is
no
difference
between the two
we have
first
type
of transformer with
connected to ground,
if
its
we postulate that n i9
to
/,.
In a real electrical realize
circuit, it is
an ideal transformer.
impossible to
Any
real trans
the windings in the core.
and to
There
hysteresis
is
also
and eddy currents flux leakage, and
some
the self inductance of the windings
is finite.
The
and leakage reactance of the windings may be thought of as small resistances and resistance
The
may be considered as current flowing in parallel with one of the a resistance through core losses
The magnetizing current necessary to produce the flux in the core, with coils of finite inductance, may be considered a current flowing through an inductance, also in parallel with one of the ideal coils (the primary of a two-winding
2
1^-j-li-
(IV.7.F.19)
1
of the use of ideal transformers
in electromechanical let us discuss the analogies,
mass-capacitance analogue of a massless rigid lever which rotates on a fixed axis without friction (see Fig. IV.7.F.9).
coils
ideal coils.
(IV.7.R18)
As an example
so that these relations are analogous to
those for the
inductances in series with ideal windings.
a two-winding ideal trans
types discussed above, and
h
(IV.7.F.21)
former has energy losses due to the resistance of
the same reasoning, the
P2
= - =...
the number of turns in the/th coil, be proportional
The nodal equations for P 3 f/ and these cancel +f/ and
if
..
In our mass-capacitance analogy, current is is analogous to analogous to force and voltage velocity,
nodes except
all
old nodal equations,
But
= - =.
rigid,
2
5!
are clearly prime equivalents of the
=
is
(IV.7.F.17)
x iV
for
(IV.7.F.20)
(IV.7.F.16)
replace e by Z^ in the mesh equations meshes containing P-f^ we have prime
,
from the
Then, for equilibrium of torques,
we have
The nodal equations l5
from the
/,
Fig. IV.7.F.9
is
equations equivalent to the original ones.
P
-
3
A
(IV.7.F.15)
=
If
The
P m is
But the transformer equation
for
/
any mesh which does
=
e
,
the primed equivalent of the
is
analogous equation for the old network.
Therefore,
be the force at a distance
-
transformer).
known
as the
The sum "exciting
of these two currents current"
is
of the trans
former.
For transformers
in electrical analogies, the
exciting current, the winding resistance,
and the
leakage reactance must be kept small so that they will
not
introduce
appreciable
errors.
The
IV.7.G.
California Institute of
computer contains
Technology
MASS-INDUCTANCE ANALOGY electric
analog
specially constructed trans
formers with a ratio of
exciting
leakage impedance of 1400.
impedance to
(See E. L. Harder
and G. D. McCann, Ref. IV.7.g.) A method worked out by Mallock its
exciting current
losses (see R. R.
M. Mallock,
and impedance
Ref. IVJ.h).
rf
also
analogy,
known
represents velocity. Eq. IV.7.C.4 relating the forces in connections at mass a
converging point must be represented by KirchhofFs mesh relating the voltages in
a closed
relating the velocities in a mechanical
Thus, we are led to a consideration of the concept of network duality between two networks,
which we define as follows
correspondence
We
call
Each element of one network has a counter
a.
part in the other. b. The currents through the elements of one network and the voltages across their counter parts in the other network are proportional to
each other.
A planar network
corre
network dual to a nonplanar network (see H. Whitney, Ref. IV.7.p). However, any network must have an element dual network (see H. W.
all
Ref.
IV.7.C,
pp.
branches of one
from the a
direct
correspondence of the connection diagram with the corresponding electrical circuit. In the masscircuit
analogous
is
q
this case
magnetic
their
deduced that the currents through the
network are counterparts in the other
other. proportional to each 2
iV*i
=
q of which are independent,
relations,
t
exactly q
+
1
relations.
(IV.7.G.1)
is
planar, there
is
a
method of accomplishing
this dual translation.
We may look upon our connection diagram as a
map in which q
+
q
+
We
way
1
countries.
that the entire
1
loops are the boundaries of select
plane
is
our loops in such a up into non-
divided
overlapping countries. Then
we
select
one point
For every connection of the original diagram, we draw a crossing branch. This branch contains an elec in every country to serve as a node.
trical
*/
diagram
rather simple geometric
electrical, it is
branches of one network and the voltages across
able to select
r
circuits.
of dual networks are
1
we must be
of loop equations (Eq. IV.7.C.6),
whose analogs can serve as nodal equations. That is, every vk and every v must appear in
discussed in the preceding section, are good
examples of dual networks, in
+
set
If the connection
network dual to the connection diagram. The nodal and mesh-type transformers,
is
mass-capacitance
In order to find the mass-inductance analog of
which form a closed mesh. is
its
analog (see also A. Bloch, Ref. IV.7.b).
a connection diagram,
The mass-capacitance analogy
The mass-
196-99).
inductance analog of a mechanical system
network which radiate from a particular junction point have counterparts in the other network
inductance analogy, the
one that can be drawn in a
always the element dual of
sponding branch in the other.
For each network,
is
plane so that no connections intersect except at nodes. It has been proven that there exists no
Bode,
easily
concept element
this
:
Each branch of one network has a
pair
wish to
concept of duality as a between two assemblies of
elements with relation only to the behavior of the
mesh. Eq.
loop must be represented by KirchhofFs nodal law relating the currents at a node.
If a
may
introduce also another
in the
analogy.
In this analogy, voltage represents force and
2.
2
proportionality.
elements specified and not to their connection
current
1.
are currents and
duality defined:
literature as the direct or electrostatic
IV.7.C.6
e"
proceed with a discussion of the
mass-inductance
law
and // and e/ and
arrangement.
IV.7.G. Mass-Inductance Analogy
We now
i-
voltages in and across corresponding elements of the two networks, and /c x are constants of
In electrical circuit theory, one uses
amplifiers to automatically compensate a trans
former for
where
283
element analogous, according to Table
IV.7.D.1 , to the mechanical element in the crossed
ELECTROMECHANICAL ANALOGIES
284
several
prototype.
branch crosses each
inertial
same system. Bloch (Ref. IV.7.b) describes methods of constructing the dual of a
the
to connection, with an impedance proportional An inductance the hindrance of its
ideal transformers. nonplanar network, using The dual of a planar network is constructed by method described above, with an the
connection, a resist
ance branch crosses each dashpot connection, branch crosses each spring con and a
geometrical inductance branch crossing each capacitance
capacitance
force nection, a voltage generator crosses each
branch, a resistance crossing each conductance,
a
generator
voltage
generator, and a
of the
coil
dual type
crossing
each
current
assembled on a transformer crossing
each transformer
coil.
To
construct the dual of a nonplanar network,
we convert the original network into another with the
same elements and the same performance, but
with a planar circuit diagram and additional constraints ideal nodal or mesh-
implemented by
type transformers. is
The dual of this new network
constructed by the routine method discussed
above. B
Fig. IV.7.G.1
generator,
and a
current generator crosses each
An inertial
velocity generator.
taining a velocity generator
is
connection con
represented,
in this
one by two branches in parallel, the other a current and an inductance containing analogy,
generator.
Let us perform
this process
of Section IV.7.B above
The
lines
solid
on our example
(see Fig.
represent
IV.7.G.1)
connection diagram, while the
:
mechanical
the
dashed
lines
electrical circuit. represent the analogous
We
have selected (d),
(b), (c),
(e),
five
and
loops corresponding to
(i)
The
of Eq. IV.7.C.9.
translation can be performed in this
manner only
if the
system is planar. In certain cases the connection diagram cannot
be drawn in a plane without intersections that do not coincide with mass points. Fig. IV.7.G.2 is an
example of such a system: This system has no dual
mesh
for every
mass
For nonplanar convenient to electrical
For
as
which there
a
We it
is
probably most
obtain the mass-capacitance
analog and then construct
we have
is
point.
systems,
first
Fig.IV.7.G.2 in
pointed
its
dual.
out above, the mass-
inductance analog of a mechanical system
is
the
element dual of the mass-capacitance analog of
have already considered two methods of
using ideal transformers to detach a branch from
a network.
By detaching any crossed branches, we convert our original nonplanar network into a number of planar networks linked by trans formers.
Bloch describes several other methods
of
IV.7.G.
/
networks. converting nonplanar into planar interest
among these is
the
Of
where (afl!
method of the fictitious
at all cross-overs (see Fig. IV.7.G.3).
) is
to a connection matrix analogous
(%), and the nodal equations n
in which short circuits are introjunction point
duced
285
MASS-INDUCTANCE ANALOGY
^
^ 2 ftsV + IjV + m ^
z
By
a
=
**
m
(IV.7.G.5)
(/=!,..., 4) where
(ft,
),
and
(y,/),
,,-.
,
(O
ft
I?ra
are analogous to
3
Eq. IV.7.C.1, G.2,
4,
3,
proportionality
and 6 are identical with Eq.
2, 4,
and 5
as before,
if,
factors,
i
(a)
we introduce
and K (2)
:
(IV.7.G.6)
=
Then Fig. IV.7.G.3
into real nodes, the converting the cross-overs
network
is
made
and
The primary transformer are inserted
planar.
secondary of an two branches derived from one of the initial ideal 1-1
(IV.7.G.7)
in the
is made crossed branches (BD). Thus the current
equal
in those
two branches (Za and ZJ.
-i
c
It
branches follows that the current in the other two also be equal, and the current must and Zj) (Zj is unchanged have now established methods for con-
distribution
We
structing
the mass-inductance analogy of any
one-dimensional mechanical
proceed
network
to
system,
Z
*(
=
A K
^
as in this analogy for our example, Completing g P we have Fig. & IV.7.G.4. "
rig.iv./.u.i,
and we
consider explicitly the analogous
= -^ K
e
h~ ~~l J__||_J-
w
relations.
and 6, we now Analogous to Eq. IV.7.C.1, 2, 4, and voltage: have the relations between current
the total current flowing (Here, // represents inductance and the current the
through
Fig. IV.7.G.4.
generator.) massless, rigid lever discussed previously second as its mass-inductance analog, the
The or
(IV 7
G 3)
nas >
type
of the of ideal transformer in which the cores
individual coils are connected in parallel.
We recall that the equations of the lever are the
mesh equations
for the
network: (IV.7.F.20)
Hi
(IV.7.G.4)
(IV.7.F.21)
ELECTROMECHANICAL ANALOGIES
286
while the equations of the second type of ideal
system
transformer are
The
2^ = nJ!
=
ng f a
These are
=
clear
is
is
proportional to the rate of increase of fluid flow,
with the inertance the constant of proportionality
=
.
.
n,i,
=
.
.
(IV.7.F.12)
.
if
analogous
clearly
proportional to It
.
(IV.7.F.10)
ns
i
concentrated into inertance elements.
is
pressure drop across such an element
-
is
The inertance of any real
acoustical element
.
may
set
Hj
l jt
both these analogies are
that
applicable to one-dimensional rotational systems if
torque,
and moment of
angular velocity,
inertia are substituted throughout the reasoning
for force, velocity, and mass, respectively.
Fig.
IV.7.H. Electroacoustic Analogies
be calculated from
In the field of acoustics, frequent use is made of the analogy between fluid flow and electrical current flow (see also
H.
F.
Olson, Ref.
W.
Mason, Ref. IV.71; IV.7.m). Sound waves in P.
often
are
acoustical
devices
analyzed by
the methods of circuit theory. Exact
approximately
acoustical analysis involves a study of three-
dimensional fluid flow,
methods of
many
we
describe a
method of
similar to the connection diagrams for mechanical
it
by the square of the cross-
sectional area. b,
D
Dissipatance. This
is
represented by symbol
All energy dissipated in the
(Fig. IV.7.H.1).
primary causes of dissipation.
assumed that
It is
the pressure drop across a dissipative element
is
proportional to the rate of flow of fluid through the element. The dissipatance is the constant of
A
proportionality.
symbol for a
narrow
dissipative
slit
is
element.
used as a
The
dis
sipatance of a real acoustic element due to
systems.
In these acoustical systems, attention
is
focused viscosity
pressure, or force per unit area, and rate of
flow, defined as the velocity of the fluid multiplied
by the
contains, divided
system is assumed lost in the dissipative elements.
drawing flow diagrams for acoustical systems
on
approximately equal to the mass of fluid
Viscous resistance and heat conduction are the
sound waves come so close to
shall
of the cylinder, the inertance parallel to the axis is
in a great
However,
being plane waves that one-dimensional analysis yields useful results. By making a number of
assumptions,
its geometry and the density of the fluid. For a cylindrical element with flow
which precludes the
circuit analysis.
cases, the
IV.7.H1
may be calculated from its geometry and
the viscosity of the fluid.
For a narrow
slit,
cross-sectional area.
We make the following assumptions: 1.
The flow of
fluid
is
one-dimensional
throughout the system. 2.
where
/*
is
All elements of the system are so small
For a
w is
viscosity,
Visits width, /is
its
the thickness of the
slit,
length.
circular hole,
compared to the wave length of the sound waves set
up that
the rate of flow
may be
D=^
considered
3.
(IV.7.H.2)
Trr
constant throughout each element.
The system may be approximated by a set of
where
r is
the radius (see also
W.
P.
Mason,
massless, dissipation-free tubes connecting ideal
Ref.IV.7.k,p.ll6).
elements of the following types: a. Inertance. This is represented by symbol I
compressance. The first is storage compressance,
Fig. IV.7.H.1). All the effective fluid
mass
in the
c.
There are two types of
Compressance.
represented
by
s
(Fig.
IV.7.H.1).
These
ELECTROACOUSTIC ANALOGIES
IV.7.H.
elements represent the storage units ofthe system, in which the of the energy due to
compression
fluid in
rigid receptacles is stored. It is
that the
change in volume in these elements under
assumed
compression is very small compared with total volume. The difference between the
their
pressure storage element and theatmosphericpressure proportional to the decrease in volume of the
in a is
fluid contained. is
The constant of
current to rate of flow, inductance to inertance, dissipatance, and inverse capaci tance to compressance. However, in some cases, the dual inertance-capacitance analogy may be used. resistance to
Let us in
set up rules for drawing a flow diagram, which acoustic elements of the type defined
above are connected by ideal tubes.
proportionality 1.
the
287
We
draw a reference
compressance.
The compressance of an
ideal element
pressure
is
is
atmospheric.
tube, in
which the
All connections to the
atmosphere in the original system are connected
K=~
to this tube.
OV.7.H.3)
where p is the density of the fluid, v is the velocity of sound in the fluid, V is the volume of the element.
(See also
W.
P.
Ref. IV.7.k,
Mason,
pp. 103-5; H. F. Olson, Ref. IVJ.m, pp. 18-19,) The second type of compressance is
compressance,
KD in Fig. IV.7.H.
1
.
pressure
is
diaphragm
circular
diaphragm
Storage compressance
entry to the storage
This
is
volume
justified
is
connected from the
chamber to the reference tube.
by the
in this element
fact that the is
change in
proportional to the
difference in pressure between the element
and
the atmosphere.
approximately
4.
K=
represented by two ideal elements
series.
3.
example, inertance and
for
properties,
connected in
The compressance of a stretched is
these
We postulate
proportional to the volume dis
or
dissipatance,
analogous to those of the original system. An element of the original system having any two of
is
placed, the compressance being the constant of proportionality.
of inertance,
dissipatance,
such that the pressure drop across the is
Elements
diaphragm
diaphragms, whose displacement under
ideal
2.
diaphragm compressance are inserted in tubes which are connected so that the junctions are
-
Applied pressure
is
indicated by a pressure
P in Fig. IV.7.H.1, connected from the generator, (IV.7.H.4)
4 777*
reference tube to the inlet where the pressure
is
applied.
T is
where radius.
on the diaphragm,
the tension
(See
W.
P.
Mason, Ref.
r is its
IV.7.k, pp.
We shall
apply these rules to a simple example
(see Fig. IV.7.H.2).
163-66.)
Any
real
diaphragm
also has inertance
must be considered a separate element
which in our
idealized system. If these assumptions are approximated by
acoustical system,
an
we may draw a flow diagram
for the system similar to the connection diagram
drawn for a mechanical system. The flow diagram indicates the distribution of pressure
flow
and
The
the
system. throughout may be described by ordinary
system
differential
to those of an equations in a form analogous electrical network.
The most useful analogy
acoustical systems
is
for
the inertance-inductance
analogy in which voltage
is
Fig. IV.7.H.2
rate of
idealized
analogous to pressure,
A pressure P is applied to a tube of inertance It whose end
is
covered by a diaphragm of
com
K
pressance lt On the other side of the diaphragm whose is a z storage chamber of compressance
K
other end
is
separated from the atmosphere by a
ELECTROMECHANICAL ANALOGIES
288
sheet of silk cloth,
The
combined dissipatance
holes in the cloth have a
jD 3
and inertance /3
,
Following the above rules, we draw the flow diagram as shown in Fig. IV.7.H.3, We observe
inductance proportional to inertance, resistance proportional to dissipatance, and inverse capaci tance If the proportional to compressance.
pressure,
all
circuit will
proportional to applied
is
generated voltage
and currents
the voltages
rates of flow in the acoustical system.
To prove
we need only note the analogy between laws
this,
and 4 and Kirchhoff
3
in the
be proportional to the pressures and
IV.7.H.5,
6,
s
laws and between
Eq.
and 7 and the voltage relations in an
electric circuit.
The electrical analog of our illustrative Fig.IV.7,H,3
E that the
following laws apply to any correctly
drawn flow diagram; If
1.
same
two elements are
drop across both 3.
in series, they carry the
is
K
2
in
L
and I 3 are proportional to Capacitances C1 and C2 are
and 73
.
inversely proportional to compressances
rate of fluid flow.
2. If
proportional to the applied pressure P.
is
Inductances inertances /j
two elements are
.
Resistance R 2
is
4.
path
KI and
proportional to dissipatance
parallel, the pressure
the same.
The total rate of fluid flow into
any junction
pw^-i
zero.
is
example
appears in Fig. IV.7.H.4. The generated voltage
The net pressure drop around any closed is
zero.
qf is the pressure drop across an element and u f is the rate of fluid flow the 5. If
element,
through
we have the following relationships,
Fig. IV.7.H.4
derived from
the definitions of the elements: Inertance:
qf
= Ijpuj
(1V.7.H.5)
The inertance-capacitance analog may be constructed by forming the dual of the above, by the method discussed in Section IV.7.G.
Dissipatance: IV.7.I. Electromechanical Systems
(IV.7.H.6)
The methods of electromechanical
Compressance:
wide (IV.7.H.7)
systems, such as servomechanisms ducers.
where /
Z)
p indicates
J5
and Kf are the element
differentiation with
constants,
and
respect to time.
We shall call the differential operator,
an
Any
electrical circuit, a purely
electrical
diagram for the
up an
tion
established all the conditions
electrical circuit with the
transducer
We
electrical
can draw a circuit
part
and a connection
diagram for the mechanical part of the system.
necessary to form the electrical analogy. set
A
energy, energy, into another form, such
as mechanical work.
we have
mechanical system,
relating the two,
a device which converts one form of
such as
Clearly,
and trans
electromechanical system contains
and a transducer is
the acoustical impedance.
analogy find
application in the study of electromechanical
We may
same connec
arrangement as the flow diagram, with
The transducer
relates
a connection in
mechanical diagram to a branch
the
in the electrical
circuit.
Now suppose the electromechanical
analogy to
IV.7.L
ELECTROMECHANICAL SYSTEMS
289
For the
In general, electromagnetic transducers are
we then have one electrical purpose of analysis, network to study instead of an electrical network related to a mechanical system, which is often
governed by the following relations: Let Eel be the applied voltage across the
the mechanical diagram
is
constructed.
electrical
branch;
customary, in the design of
branch;
Zel
transducer systems for acoustical devices, for
branch,
when
very complex.
make
to
example,
It is
calculations
all
the
for
mechanical and electromechanical parts on the basis of electrical network theory. (See also
W.
P.
Mason,
Ref. IV.T.k;
gories,
1
.
el9
the current in the electrical
=
produced
0; /,
mechanical connection of the mechanical
;
u,
the rate of contraction
and
connection;
A
hindrance in the mechanical connection
9
the
when
H. F. Olson, Ref,
Then
IV.7.m.)
Most
>
i
the impedance in the electrical in the the force v
transducers
fall
two general
into
based on the nature of the above
Electromagnetic transducers,
cate
(IV.7.U)
relation:
which generate
force through a connection proportional to the
where a
is
a constant of the transducer.
see proof of these relations,
W.
(For
Mason, Ref.
P.
IV.7.k, p. 190.)
current through a branch. transducers, which generate
2. Electrostatic
force through a connection proportional to the
a branch. voltage across
Because of
this
proportionality,
the mass-
to the mechanical
is
applied capacitance analogy while the masspart of electromagnetic systems, is
inductance analogy
applied
to the mechanical
Some authors,
electrostatic systems. part of as Mason, draw the appropriate
but retain the mechanical
circuits
electrical
quantities.
such
analogous
A mass,
is
for example, symbolized or an inductance of a given
by a capacitance
The constant of
or pounds. between force and current or proportionality means of an is introduced
number of grams force and voltage
by
mechanical and ideal transformer linking the electrical parts of the system.
certain
systems,
particularly
advantages,
because
makes
it
This method has
the line
in
complex
clear.
seek to avoid In our treatment, however, we and confusion between connection diagrams our we Therefore, apply analogous circuits. derive one we that a such way analogies in
system.
circuit
The
for
electromechanical
each
are constants of proportionality the and or /c and
introduced through
(1)
(1)
(2)
%>,
discussed previously constants of proportionality
between mechanical quantities
As an example,
and their electrical
consider a
transducer (see Fig. IV.7J.1).
Eq.IVJllwith a
of separation
between the mechanical and electrical parts
electrical
Fig.IV.7.Ll
where /is
=
(IV.7.I.2)
B is the flux density of the polarizing field
the length of the conductor;
Listheinductanceofthecoil;
where
governed by
Bl
M
is
Z = 6l
and^
the mass of the coil and
constant of the coil
s
;
If, where
= Mp + -
9
K is the spring
suspension.
an electromagnetic Since the generated force in to current, we select is transducer proportional to apply to the the mass-capacitance analogy linked mechanical part of an electromagnetically electromechanical system.
analogs.
moving cojUype It is
ELECTROMECHANICAL ANALOGIES
290
Let i me represent II A.
represent
e
/,
m
Then, by
represent
v,
Eq. IV.7.E.4,
and 5,
Zme
and
6
At
seems worth while to indicate
this point, it
how the transformer method would be used. We that the
back to Eq, IV.7.I.1. Recalling mechanical quantities are directly applied to the refer
e
6{1)
/ ,
^ -fiil
we see that v will analogous circuit diagram, as a voltage, but with dimensions of appear a current but with dimensions of velocity, / as
In this case, since we are dealing with con version of electrical energy to mechanical energy, it is essential that both kinds of power be
measured
in the
same
units or that
(IV.7.I.4)
Fig. IV.7.I.2
so that
(IV.7.I.5)
From
Eq, IV.7.I.1,
we now have
r (IV.7.I.6)
Fig, IV.7.I.3
or
e me
We
=
set
0e (1) Z me i e
(1)
=-
s
+ Z me me i
j
and
finally
we
derive the
fl
relations
"
"
"""
Fig.
W.7.L4
(iv.7.1.8)
force,
These are the equations, of the four-terminal network in Fig. IV.7.I.2, the mass-capacitance analog of an electromagnetic transducer. In the case of the moving-coil converter
rt
and
A
as
Then a
is
the constant of the transformer linking
the electrical and mechanical parts.
shown
1
9)
The
circuit
in Fig. IV.7.I.4 satisfies Eq. IV.7.I.1.
Electrostatic
= Lp (IV 7
an admittance but with dimen
sions of hindrance (or \\A as an impedance).
(including piezoelectric)
trans
ducers are governed by relations as follows
Let
Eel
electrical
:
be the voltage applied across the branch,
i
el
be the current through
Z the impedance of the electrical branch when v = 0; /, the force through the electrical branch;
so that the analogous circuit Fig. IV.7.I.3.
is
as
shown
in
el9
the mechanical connection;
v,
the rate of con
traction of the mechanical connection;
and A,
IV.7.L
ELECTROMECHANICAL SYSTEMS
the hindrance of the mechanical connection when z
= 0.
As
in
291
Eq. IV.7.I.5 "
Then,
1
(IV.7.I.13)
From Eq.
(IV.7.I.10)
where a
W.
P.
a constant of the transducer.
is
IV.7.I.10
we have
(See (IV.7.I.14)
Mason, Ref. IV.7.k pp. 193-95.) }
or
(IV.7.L15)
Finally,
we
set
=
e (2)
-
and
arrive at the
a relations
Fig. IV.7.L5
These are the equations of the network of Fig. the mass-inductance analog of an
IV 716 -
>
Anexampleofaneiectrostatictransducerisan electrostatic system, in which a force results from the repulsion of charges on
which
For
is
movable
two
plates,
electrostatic transducer,
one of
(see Fig. IV.7.I.5).
this system,
m
(IV.7.I.U)
!
a
l
I
M.
Fig.IV.7.L6
2
47TX
where
is
the polarizing voltage;
effective area of the plates ;
between the
S
is
the
For the example
cited,
and x is the separation **&
plates.
~
1
^
Since the generated force in an electrostatic
transducer
is
proportional
to voltage,
we
(IV.7.I.17)
select
the mass-inductance analogy to apply to the mechanical portion of an electrostatically linked
an(j
electromechanical system.
IV.7.L7.
Let e me represent A.
/,
i
me represent
v,
and
we
the analogous
Zm
represent
Then
(IV.7J.12)
Fig,IV.7.L7
circuit
of Fig.
ELECTROMECHANICAL ANALOGIES
292
To
v
will
Eq. IV.7.I.10. In this representation, a voltage, A as an appear as a current, / as of a trans impedance, and a as the constant former linking the electrical and mechanical parts.
One may draw a connection diagram symboliz
use the transformer method, we study
The
circuit
shown
in Fig.
IV.7.I.8 satisfies
and a flow diagram ing the mechanical part the acoustical part of the system. symbolizing These cannot be connected since force through a connection
a
Eq. IV.7J.10.
The reader should bear
in
mind
that both the
mechanical portions purely electrical and purely of electromechanical systems are likely to be
is
related to pressure at a point, while
is related to rate of flow through velocity at a point
The mass-inductance and
tube.
inertance-
inductance analogs of these diagrams are con structed and connected. If the same constant is
used to relate voltage to force and voltage to pressure velocity
piston
and the same constant
and current to
may be
relates current to
rate of flow, then the
ideal trans represented by an
former with a winding ratio equal to the area of Alternatively, the voltage-pressure
the piston.
may be made
equal to the voltage-force constant times the area of the piston, and the
constant
constant
current-rate-of-flow Fig. IV.7.L8
the
highly complex.
A real simplification
is
Note
may be
principles
involved apply also to mechanoacoustical trans mechanoacoustical transducer is a ducers.
A
device which relates the force through a mechanical connection to the pressure at one end
of an acoustical tube. Vibrating membranes and of mechanoacoustical trans plates are examples
we have
selected analogies that relate
both force and pressure to voltage and velocity
and the
a mechanoacoustical trans
that, since
ducer relates force to pressure and velocity to rate of flow,
This section has been devoted to electro
but
transformer
the
effected
analysis are so well developed.
transducers,
Then
piston.
eliminated.
by reducing these complex systems to purely electrical circuits, for which the methods of
mechanical
equal the
may
divided by the area of current-velocity constant
On the
rate of flow to current.
other hand,
we might also have used the mass-capacitance and However, care
inertance-capacitance analogies.
must be exercised
to avoid trying to connect a
mass-capacitance
analog
an
with
inertance-
inductance analog. IV.7.J. Applications
ducers.
system which consists of a mechanical part, an acoustical part, and a piston
Consider
a
connecting the two. nection
is
A
force through a con
applied to one side of the piston
producing pressure at one end of a tube on the other
side.
The
pressure, q,
is,
of course, equal
The principles of electromechanical analogy have been utilized in the construction of a number
vania,
and the California
have computers transients
to the force,/, divided by the area of the piston, S.
Likewise, the rate of flow of the the tube
is
fluid, u,
through
equal to the mechanical velocity,
at
Institute
referreid to
of Technology
both as
and as
"mechanical
"electric
analog These machines are used to study
analyzers"
computers."
vibration t?,
The Westinghouse
of computing machines.
Electric Corporation, East Pittsburgh, Pennsyl
problems,
all
sorts
of mechanical
problems involving transient forces or torques, as well as servomechanism problems.
the piston times the area of the piston.
Special low-loss electrical units of inductance,
capacitance,
(IV.7J.18)
and
resistance are arranged
on a
board in such a way
that,
by
possible to connect
them
into a wide variety of
inserting jacks,
it is
IV.7J.
APPLICATIONS
For any given problem, they are connected to form either the mass-inductance or
networks.
mass-capacitance analog of the given mechanical Then voltages or currents exactly system.
analogous
in amplitude
and time variation
e.
f.
of synchronizing switches, these disturbances be applied repeatedly, so that the solution
g.
systems: Mechanical-electrical analogies,"//!. Appl. Phys., Vol. 9 (1938), pp. 373-87. B. "Electromechanical and electroGehlshoj,
h.
R.
R.
M.
157-70. b.
c.
London, Proc., Vol. 58 (1946), pp. 677-94. H. W. Bode. Network Analysis and Feedback
New York, Van Nostrand,
d.
1949.
Amplifier Design. H. E. Criner, G. D. McCann, and C. E. Warren, "A new device for the solution of transient-vibration
problems by the method of electrical-mechanical in Jn. Appl. Mech., analogy," A.S.M.E., Trans., Vol. 67 (1945), pp. A135-A141.
electrical
London,
Proc.,
calculating Vol. 140
G. D. McCann and H. E. Criner,
"Mechanical
problems,"
A.I.E.E.,
Trans.,
Vol.
(1946), pp. 91-96. k. W. P. Mason. Electromechanical Transducers
Wave Filters. 2d 1.
A. Bloch, "On methods for the construction of networks dual to non-planar networks," Phys. Soc.,
"An
Soc.,
problems solved electrically," Westinghouse Engineer, Vol. 6 (1946), pp. 49-56. G. D. McCann, S. W. Herwald, and H. S. Kirschbaum, "Electrical analogy methods applied to servo-
mechanism analogies and their use for the analysis of mechanical and electromechanical Inst. Elec. systems," Eng., Jn., Vol. 92 (1945), pp. "Electromechanical
Mallock,
Royal
(1933), pp. 457-83. i.
j.
A. Bloch,
scale
"A
machine,"
References for Chapter 7 a.
Academy of Technical Sciences,
Copenhagen, Scientific Paper, No. 1 (1947). E. L. Harder and G. D. McCann, large
general purpose electric analog computer," A.I.E.E., Trans., Vol. 67, Part I (1948), pp. 664-73.
may
can be reproduced on a cathode-ray oscilloscope. The traces can be measured or recorded.
F, A. Firestone, "The mobility method of computing the vibrations of linear mechanical and acoustical
acoustical analogies,"
to the
impressed mechanical disturbances are fed into the appropriate nodes of the network. By means
293
J.
Miles,
ed.
New York, Van Nostrand,
"Applications
electrical analogies,
65
and
1948.
and limitations of mechanical-
new and
old,"
Acoust. Soc. Am.,
/., Vol. 14 (1934), pp. 183-92. m. H. F. Olson. Dynamical Analogies.
New York, Van
Nostrand, 1943. n..
L. Page and N.
New o.
G.
Adams.
Principles of Electricity.
York, Van Nostrand, 1931. Thaler and R. G. Brown.
J.
New
Servomechanism
York, McGraw-Hill, 1953. Chap. 3. Hassler Whitney, "Non-separable and planargraphs," Nat. Acad. ScL, Proc., Vol. 17 (1931), pp. 125-27.
Analysis. p.
I.
Chapter 8
TWO-DIMENSIONAL ELECTROMECHANICAL ANALOGIES
We
IV.8.A. Two-Dimensional Lumped-Constant
shall
now
for this system.
Systems
The methods of electromechanical analogy were developed
in the preceding chapter for
one-dimensional systems. These same methods may be extended to apply to systems with motion in two or more dimensions. In this section we
construct a connection diagram
Our first step consists of drawing
two diagrams, one for
vertical connections
and
the other for horizontal connections (see Fig.
IV.8.A.2X
shall consider two-dimensional lumped-constant
systems, such as cross sections of cylindrical structures in which ideal mass and elastic elements
are rigidly connected (see also V. Bush, Ref. IV.S.a).
The frame of reference will be an inertial
axis system.
the
method
In Section IV.8.B, to
we
shall extend
FIXED AXIS
an approximate analysis of con
VERTICAL
tinuous elastic structures.
Let us assume that we have a two-dimensional system of mass points joined to each other by horizontal,
vertical,
and slanting
elastic
con
nections (see Fig. IV. 8. A.I).
HORIZONTAL Fig. IV.8.A.2
We
note that the inertial connections appear The horizontal and vertical
in both diagrams.
Fig. IV.8.A.1
inertial
As
a
first
approximation, we
shall
assume that
each connection can transmit forces parallel to That is, horizontal connections only.
itself
transmit only horizontal forces, vertical con nections only vertical forces, and slanting
connections only slanting forces.
Later on,
we
shall indicate the possibility of treating shear
forces, but at this point they are neglected.
connections for one mass are drawn as
they were independent, but the constant is used for both.
if
same mass
Now it is necessary to take care of the slanting connections. If a connection
with the horizontal, and then
/ has
it
makes an
angle, a,
transmits a force, /,
horizontal and vertical components
/.=/
.
(IV8A1)
IV.8.A.
The
rate of contraction,
connection,
of
its
is
equal to the
LUMPED-CONSTANT SYSTEMS
of the slanting
v,
sum of the projections
horizontal and vertical components on the
connection. That v
where
A
295
a constant characteristic of the
is
In addition,
transformer.
contraction of the
if v 1 is
the rate of
connection and u 2 the rate
first
of contraction of the second,
is,
= v x cos a +
v
y
sin
a
(IV.8. A.2)
v1
=
-4t>
2
(IV.8.A.4)
We
must introduce a separate connection Before diagram for the slanting member. must introduce the proceeding, however, we notion of a mechanical transformer, defined analogously
to
an
transformer
electrical
follows (see also Section IV.7.F)
A
mechanical transformer
Making use of the concept of a mechanical
^
the slanting
member. Obviously, the
slanting
as
:
is
a theoretical
device which appears in a connection diagram as
two connections
(see Fig. IV. 8, A. 3).
Fig. IV.8.A.4
connection contains a spring transmitting a force
/ with a rate of contraction v. It must interact with both the horizontal and vertical connection
Fig. TV.8.A.3
If/!
is
the force in the
first
connection and /2
8. A. I and 2. These diagrams according to Eq. IV. e ffects can be represented by means of two
the force in the second, the transformer causes
mechanical
these forces to be related by the equation
each in
(IV.8.A.3)
FIXED AXIS
M.
HORIZONTAL Fig. IV.8.A.5
first coil
of
with the spring to form a closed
loop (see Fig. IV.8.A.4).
@MtU
-
transformers, with the
series
VERTICAL
TWO-DIMENSIONAL ELECTROMECHANICAL ANALOGIES
296
The constant of
a,
linked with the vertical must be sin
a.
in
all
Now
the transformer linked with
the horizontal diagram must be cos
three branches of the loop
is
let
us
extend
the
electromechanical
while that
forces analogy to shear forces as well as parallel
The
through the connections. For the present,
force
let
us
the same, so
that Eq. IV.8.A.1
is satisfied by the definition of a mechanical transformer. The transformer con
nections in the loop have rates of contraction
a and
vx cos
vy sin a, so that Eq. IV.8.A.2
loop equation, and
is
the
clear that the slanting
it is
member is represented by the loop.
We
can
now draw
diagram for the system
the complete connection (see Fig. IV.8.A.5).
This
diagram constitutes a complete symbolic repre sentation of the differential equations of motion
%
IV.8.A.7
assume that we have a system of mass points horizontal and vertical elastic joined only by connections (see Fig. IV.8.A.8).
Now and
each member transmits both horizontal
vertical forces.
Let us study the forces
through any member, between P1 and P2
say,
the
connection
.
Fig. IV.8.A.6
of the mechanical system.
The equilibrium of
and horizontal components of force at each mass point is represented by the connections
vertical
joined to the point. The geometrical constraints are represented by the loops in the diagram, around which the sum of rates of contraction
must vanish. Newton law are the
s
second law and Hooke
relationships
s
Fig. IV.8.A.8
between force and rate
of contraction in individual connections.
As
in the previous case,
we
have:
We
can derive the mass-capacitance and mass-inductance analogs of the system from the
=
f**
^i>
v
(IV.8.A.5)
the routine methods
connection diagram by discussed for the one-dimensional case. Current transformers are used in the
first
case
I
-v y
is
the vertical contraction of the
P connection.
is
indicated in
Fig. IV. 8. A. 6 and the mass-inductance analog in Fig. IV.8.A.7.
=
and voltage
transformers in the second.
The mass-capacitance analog
where dy
We also have a shear force:
= ^D
(B
(IV.8.A.6)
ELASTICITY
IV.8.B.
where dx
=
- v is the relative horizontal x
ment of
the terminal mass points shear constant of the connection.
297
the diagram represents these factors displace
and S12
is
the
and therefore
determines the system. Notice that
we have two
we
diagrams, vertical and horizontal, because
have an equation of motion for each component of the velocity.
By combining the principles of this analogy with those of the preceding one, which admitted slanting connections but no shear forces, we can represent rigidly connected systems containing slanting
members.
we must consider systems of rigid bodies embodying both rotational and translational motion. Any motion of a rigid body Finally,
consists of a translation of the center of gravity
plus a rotation of the
body about
centroid.
its
may be drawn
Therefore, connection diagrams
independently for the translational motion and for the rotational motion. The translational
VERTICAL
diagrams are drawn according to the principles discussed in the preceding sections, with any rigid
body
treated as if
concentrated
the
at
its
entire
In
centroid.
diagrams, the fixed axis
is
mass were rotational
selected to coincide
=
0. We have a with some angular position torque through each connection and a rate of
to the difference in angular contraction equal Each angular velocity between the end points.
body has an inertial connection containing an element for its moment of inertia.
rigid
IV.8.B. Elasticity
The HORIZONTAL
of
principle
representing
mechanical
can be equations by connection diagrams extended to include elasticity problems (see also
Fig. IV.8.A.9
G. Kron, Ref. IV.S.b). For the sake of simplicity
When we draw
the connection diagram (see
connection Fig. IV.8.A.9) every
both horizontal and parallel force spring
vertical (e.g.,
must appear
in
as either a
diagrams ) or a shear spring
Klz
in presentation, our discussion here will be limited to the two-dimensional case, isotropic media, linear stress-strain relations.
is
(e.g, Sia).
Here
again,
that the connection point out
diagram affords a complete symbolic representa tion of the differential equations of motion of the
an inertial frame of refer
system when referred ence. In elementary mechanics to
it is
the motion of a system of particles
determined by
the forces acting
proven
is
completely
on each
constraints. It plus the geometrical
that
is
particle
clear that
and
essentials
of the method are grasped, the generalization to three dimensions
we
Once the
perfectly straightforward.
In Section IV.S.B the concepts of stress, strain,
and displacement were
defined.
Hooke s law and
the partial differential equations of elasticity were
three-dimensional problems. presented for In the two-dimensional case which we shall consider here,
we
are concerned with three
com
a a and r^ three components ponents of stress, v y ,
of strain,
#
<,
and y^, and two components of
ANALOGIES
ELECTROMECHANICAL YV V/ TWO-DIMENSIONAL
TUX 298
JL
.L<
A.LTJ
L
-
u v . !, law Hookes
,
and
"*-
.
^^ ^ ^ ^ is
Since the material
assumes
^
displacement, the following form:
homogeneous
^
=?S-tl)r B
where
Young
is
Now let us consider
]
=
(IV.8.B.4) /ce v
(IV.8.B.2)
modulus and
s
and subject
uniform and
IV.5.B.6 defin ition of strain, Eq.
from the
(IV.8.B.1)
r-
^
is
Poisson
s
a small, thin, rectangular
isotropic, piece of homogeneous, with sides of dimensions h and *,
IV.8.B.1.
elastic material
B B a 5 and ,
Suppose
in Fig. B, indicated to uniform tensile (normal)
stress.
it
is
The
t
subjected
L
J
-L v[B 2]
Fig. IV.8.B.2
Then Hooke
s
becomes, for law, Eq. IV.8.B.1,
this rectangle,
(IV.8.B.5)
"
Now
"
L
-
2J
J*
E
Eh"
consider a similar rectangular piece
Fig. IV.8.B.1
,
shear stress
horizontal stress
is
af and the
vertical stress is
vertical forces
There are horizontal and
_ ^ ;~
the rectangle
,
v
.
through
XV
On there
,
;
(IV.8.B.3)
T
(IV.8.B.6)
a*
line through the rectangle any horizontal is
vertical a horizontal force, and on any
line there is
a vertical force. (See Fig. IV.8.B.3.)
&v
Now
=
^
(IV.8.B.7)
under the action of these forces the
rectangle
is
deformed
only normal rectangle,
but
(see Fig. IV.8.B.2).
forces are involved, its
of
same dimensions, with elastic material of the here to uniform sides BI, BZ, 3 B^ subjected
it
Since
and
remains a
dimensions are changed. The to a
are subjected upper and lower boundaries and the constant vertical
side
(IV.8.B.8)
Under
the the action of this shear force,
displacement,
boundariesaresubjectedtoaconstanthorizontal and v[Bk ] be the constant Let u displacement. horizontal
and
[B,]
vertical
displacements
of the
rectangle .
to a lower boundaries are subjected upper and
constant horizontal displacement,
appropriatesides(thebracketnotationreferstoa not to characteristic of a geometrical object, and a function of a numerical quantity).
[*], and the
side
boundanes
:
[*4
]
aad
to a
.uljejj and [*, constant vertical displacement, v[B, ]
].
IV.8.B.
ELASTICITY
299
diagram in which
all
If
forces are represented.
and density is dimensions h and k
the distribution of stress, strain,
continuous,
we may
select
sufficiently small that, in
any rectangle of these
dimensions, the stresses, displacements, or density
may, without appreciable
error,
be assumed
constant.
We suppose the material has been divided into a lattice of rectangles of dimensions h and k.
Now, the
for analysis,
we need rectangles
over which
considered constant
may be
displacements
and ones corresponding to the force rectangles described above. However, these rectangles will
Fig.IV.8.B,3
not coincide.
In
we
fact,
will
have a
lattice
of
rectangles of constant horizontal displacement (u rectangles),
a
of rectangles of constant a lattice of
lattice
vertical displacement (v rectangles),
normal force
rectangles,
and a
lattice
of shear-
force rectangles.
r
^r^IL UW *^*T 1
,
Fig. IV.8.B.4
Since the strain rectangle,
IV.5.B.6,
is
L
constant throughout the
the definition of shear strain, Eq.
becomes Fig. IV.8.B.5
Let us find the relation between these (IV.8.B.9)
so that,
from Hooke
s
law, Eq. IV.8.B.2,
of rectangles.
we Suppose
force lattice.
Let
four sides of a
B l9 typical
2
start >
B*
lattices
with the normal
and
rectangle
54
&
(see
tne Fig.
IV.8.B.5).
|
In order to apply Eq. IV.8.B.5, B l and 53 should
+ v) ~-~r~k g 1
2(1
2(1
+ v)
be center lines of u rectangles and 52 and J54 center
1 j
"~
Thus the u (IV.8.B.10)
Now let us consider a sheet of isotropic elastic material,
subject
shear stresses.
to
nommiform normal and
Our aim
is
to
lines oft; rectangles.
Jfc
draw a connection
obtained rectangles are essentially
from the normal-force rectangles by a \h horizontal translation and the v rectangles by a \k
vertical translation.
On the
other hand,
we must
consider also the
TWO-DIMENSIONAL ELECTROMECHANICAL ANALOGIES
300
shear-force
Hooke
of rectangles.
lattice
Since
the
law relations for normal and shear
s
forces are
completely independent of each other, the normal-force and shear-force lattices need not coincide. Let JB/,
2
,
5 3 and B
this lattice
with a typical u rectangle,
start
A lt A* A z A ,
(see Fig. IV.8.B.8).
On its sides, there are horizontal normal forces /aPJ and fx [A$] and an d
be the sides of
,
a typical rectangle from
Suppose we with sides
&wM
horizontal shear forces
These are the horizontal
(see Fig.
IV.8.B.6).
In order to apply # Eq. IV.8.B.10, J?/ and 3 should be center lines of v rectangles and 52 and 4 center lines
J?
The u force
of u rectangles.
rectangles are obtained
rectangles
by a \k
from the shear-
vertical translation
and
the v rectangles by a \h horizontal translation.
I
Fig. IV.8.B.7
forces through the constant-force rectangles that
interlock our
u rectangle.
We have an analogous interpretation in terms of the v rectangles.
Thus, forces
we have two
lattices,
one for horizontal
and displacements, and one for
vertical
Fig.IV,8.B.6
Combining these two
make our
sets
of relations,
we can
four lattice-of-rectangles division (see
Fig. IV.8.B.7).
The
solid lines enclose force rectangles
and
the broken lines enclose displacement rectangles.
We can describe this process in precise mathe matical terms by assigning a coordinate system to
our
lattice.
Suppose the origin
lies at
the center
of a normal-force rectangle. Then the rectangles for each variable have the following centers,
where
m and
are integers
f0fv
:
gwgvx u
:
v
:
Fig. IV.8.B.8 :
mh nk
forces
9
fa
4-
%(
(m
+
})A,
+ *
(n
+ J)fc
We have set up our lattice structure by starting with force rectangles.
There
is
(see Fig. IV.8.B.9).
In
rectangles, solid lines enclose/^ rectangles, dotted
nk
lines
mh,
and displacements
the horizontal diagram, broken lines enclose u
a dual inter
pretation that should be pointed out now.
enclose
g^
rectangles.
In
the
vertical
diagrams, broken lines enclose v rectangles, solid lines enclose dotted lines enclose fy rectangles,
^rectangles.
Now let
us consider the equilibrium offerees
IV.8.B.
ELASTICITY
301
h,2k)
rxytamjsm
KKxxixtr,
fffn;
VERTICAL
Fig. IV.8.B.9
on a u rectangle.
We have
this
mass point. Then we can represent Eq. 1 and 12 by a connection diagram. (See
IV.8.B.1
Fig. IV.8.B.10.)
We where fx
is
the horizontal inertial force through
the rectangle and
Fx is any horizontal body force
impressed on the
rectangle.
can apply
rectangles,
this
same reasoning to the
and we derive a
diagram with
its
mass points located by a \h
By Newton s second law,
= Mp u = P hkp*u 2
/; where
M
is
(IV.8.B.12)
the mass of the rectangle and p
is
density.
We note that the horizontal component of the velocity of any point
is
the time derivative of u at y|/>hk
that point.
Now let us substitute for our u rectangle a mass its center. We
located at point having mass phk
think of the forces in Eq. IV.8.B.1
1
as acting
on
v
vertical connection
Fig.
IV.O.10
TWO-DIMENSIONAL ELECTROMECHANICAL ANALOGIES
302
horizontal translation and a \k vertical transla tion
from the mass points
diagram
in the horizontal
It
as two con appears in a connection diagram is the force through the first If
nections.
/!
connection
(see Fig. IV.8.B.11).
5
/2 the force through the second, d^ the
4
2h,2W
the expansion of the expansion of the first, and the mutual then the second, spring implements
following relations:
/= where
A is
(
IV
-
8
-
B
-
13 )
~-Af
a constant.
This device is analogous to a mutual inductance
& in
an
electric circuit.
We
that
recall
previously
we
defined
a
mechanical transformer that causes the force and contraction of one connection to be proportional to the force
and contraction,
second.
IV.8.B.5
Eq. IV.8.B.5, 8, and 10 to find the From Eq. connection elements.
we
discover that the relative displace
ment between
Contrast this with our mutual spring
that causes force through one connection to be
We refer to necessary
of a
W.8.B.13
Fig.
Fig. IV.8.B.11
respectively,
proportional to the contraction of the second, and vice versa.
Now let us consider the first of Eq.
B.5
u points horizontally adjacent
The
left-hand side
is
the expansion of an
On the right side,
connection.
the
first
term
fv
may
be represented by the expansion of a spring of constant
-jfc
and
horizontal
onfx but also onfyi vertical
diagrams
carrying a force fx and the second
term by the contraction of one connection of a mutual spring of constant vjE, whose second
Fig. IV.8.B.12
depends not only
K
so that our
must
connection carries a force fy (see Fig. IV.8.B.13). The second of Eq. IV.8.B.5,
be v
1
k
interrelated.
We postulate a theoretical device which we call a
"mutual
spring,"
symbolized in Fig. IV.8.B.12.
may
likewise
be
symbolized
by
analogous
IV.8.B.
ELASTICITY
303
elements in the^J, connection, the mutual spring 1
k
-an ordinary spring of constant
in series with
.
Eh
Now we
consider the shear connections.
K
=
IV.8.B.8, ftw
Eq.
h
7, may be symbolized by a K
mechanical transformer of ratio hfe linking the
gw and gm
connections,
Now we
rewrite
Eq.
IV.8.B.10:
Fig. IV.8.B.14
(IV.8.B.14)
The
first
term on the right side
may
be
represented by the relative displacement across a
*ear spring of constant force
gw and ,
arrying shear
the second term
by the contraction
Fig. IV.8.B.15
a
6
c
spring
spring
constant
=
7
= 2(14-*)* = 7
if
shear spring constant
e
mechanical transformer ratio
/
mass
-^
constant
=
1* ^7
mutual spring constant
=
v
u
^
v
=
phk
points in horizontal diagram points in vertical diagram
=T
K
TWO-DIMENSIONAL ELECTROMECHANICAL ANALOGIES
304
Fig. IV.8.B.16
Lx Ly
LW
M
inductance proportional to T,T
T
ideal transformer winding ratio r
Ik
C
capacitance proportional
inductance proportional to -^r 2(1
inductance proportional to
+v)k ^
h
whose second connection has
,
k expansion v[B3
]
v[B].
This
second con
nection of the mechanical transformer constitutes the entire
We
can
gyx
connection.
now draw
diagram as shown will is
!,
i
xyt
4, Iv
to
phk and v
tofx ,fv ,gxv
impressed currents proportional to
Fx and Fv
case
three
contains
types
of
displacement
and four types offeree parallele pipeds, one for normal forces and three for shear one for gxs and gzx forces, one for gm and g parallelepipeds
w
andonefor
w
and
w
,
.
(See Fig. IV.8.B.14.)
the complete connection
in Fig, IV.8.B.15.
References for Chapter 8
The reader
readily find that the mass-capacitance analog
the network of Fig. IV.8.B.16.
a.
V. Bush,
b.
G. Kron, "Equivalent circuits of the elastic field," A.S.M.E., Trans., in Jn. Appl Mech., Vol. 66 (1944),
The three-dimensional connection diagram and c.
lattice for the
three-dimensional
analysis by analogies," Franklin Inst, Jn., Vol. 217 (1934), pp. 289-329. "Structural
pp. A149-A161.
analogous network can be constructed according to principles identical with the preceding ones.
The rectangular
is ,
to
voltages proportional and ivai currents proportional
-=,
of one connection of our mechanical transformer of constant -
and ev
r v
mutual inductance proportional to
eu
R. D. Mindlin and M. G. Salvador! "Analogies," in Handbook of Experimental Stress Analysis. New York, John Wiley and Sons, 1950. Chap. 16.
Chapter 9
NETWORK REPRESENTATION OF PARTIAL DIFFERENTIAL EQUATIONS
IV.9.A. Introduction
where a is a tensor and a, b, b and 9
In recent years there has been a good deal of
work on and
analogs of partial such as the scalar-potential
circuit
electric
differential equations,
vector-potential field equations.
As
in the
of the spatial variables.
b"
are functions
Specializations of this
equation apply to incompressible fluid flow (see
H. Lamb, Ref.
IV.9.e, pp. 17-20), electrostatic
and magnetic flux distribution
(see J.
A. Stratton,
case of electrical analogs of mechanical systems,
Ref. IV.9.m, Chap. Ill, IV), conduction of heat
W. H. McAdams,
these networks are useful as an aid in numerical
(see
calculations, or they can be physically constructed
of shafts (see A. E. H. Love, Ref. IV.9.g, p. 311),
and thus used
and many other physical phenomena.
to solve the equations.
In general, the problem to be solved will involve
a number of unknown functions of the independ ent variables. In the case of a high-order system, the lower partial derivatives of the unknowns are
unknowns
usually treated as
represent a form of certain information which can
also be represented
A
duced into the
integral relations.
by
intro interlocking lattices are
number of
region.
Usually such a
lattice is
parallel
The information given by mutually orthogonal. the partial differential equations will correspond, in
general,
to
integral
relations
involving
individual cells of a lattice and their associated faces
and
edges.
In order to obtain a
number of unknowns,
v
finite
=
the rate at which our physical quantity flows. the effect of introducing the
Mathematically, is
change our partial
to
Consider an arbitrary, closed, simply con nected region of space, $, bounded by a surface S.
It
can be shown (see A.
P. Wills, Ref. IV.9.n,
pp. 96-98) that
These approxi (IV.9.B.3)
where n is a unit outward normal to S. Therefore,
analogs,
unknowns
are represented
by voltages or relations are
currents and the approximating or nodal equations. represented by voltage, mesh,
Eq. IV.9.B.1 integral form: -
-
JJ
(a
v)
is
n dS
scalartion in physics and engineering is the for the potential 9?, potential equation
= bpy + b + V
p?
to
+
the following
6>
+
b")
dV
91
A partial differential equation of wide applica
grad ?)
equivalent
=
IV,9.B. Scalar-Potential Equation
div (a
differential
to a equation from one second-order equation number of first-order equations.
relations can be symbolized by connection
in turn, be translated into diagrams, which can, in which the values of network electrical
the
(TV.9.B.2)
grad
which we may think of as a velocity. Then, if we think of a as a density tensor, the vector a v is
these integral relations are
replaced by approximations.
mating
vector
function v
obtained by dividing the region with sets of surfaces, the sets being
essentially
The scalar-potential equation refers to the flow of some physical quantity, such as fluid, flux, or heat. If (p is the scalar potential, we define a
partial
equations, in general,
differential
pointwise
The
also.
Ref. IV.9.h, p. 29), torsion
(IV.9.B.1)
(IV.9.B.4)
where v
= grad
9?.
the net represents is which surface the S, outward flow across
The
left-hand
integral
PARTIAL DIFFERENTIAL EQUATIONS
306
equated to the volume integral over the region of a linear form in 99 and its time derivative.
R
must apply Let the
to each one of these parallelepipeds.
1, 2,
and
3 directions
coincide with the
edges of a parallelepiped of dimensions h\ h\ and
IV.9.C. Rectangular Lattice
We
3 /z
method of drawing a con nection diagram and an analogous network to represent Eq. IV.9.B.1. The method we develop are seeking a
here applies,
explicitly,
system only, but
to
can
is
faces
Blt ...,B6
as indicated in
Fig.IV.9.C.l.
be adapted to any
reference
Our
frame.
the following:
We take our original region and divide it "small"
bounded by
a Cartesian coordinate
easily
curvilinear
orthogonal
procedure
it
,
into
rectangular parallelepipeds. Eq. IV.9.B.4
between the average values of gives us a relation grad
over the faces and averages of
and py
over the volume of the parallelepipeds, explicitly
Eq. IV.9.C.1 below. Thus, we have replaced our original problem by a number of relations
between average values. follows.
We
values of
This means
must
now
We
proceed further as
as our
unknowns
the
at the center of all parallelepipeds.
We
take
we now have a finite set of unknowns.
translate our average conditions into
conditions on these
finite
done as follows.
Given the value of
number of
points,
unknowns. This can be
we can
9?
at a
which approximates in the region around these points and whose coefficients are linear in the y>
values of 9? at the given points. If we have enough points
and
Fig. IV.9.C.1
find a polynomial
the points are not too far apart, this
polynomial will approximate to within a given accuracy, and we can, therefore, average this
Let (a vjjiy, etc., be average values of the components of a v over the indicated faces and $ $]$], etc., be the average values of the variables over the
For
this
volume of the parallelepiped. becomes
parallelepiped, Eq. IV.9.B.4
polynomial approximation and use the result in place of the corresponding (p averages. If we substitute these
approximate averages into the we now have a
given equation, Eq. IV.9.C.1,
system of ordinary differential equations on the values of cp at the specified points. Since we have
{(a-v)3 [B6]-(a-Y) 3 [B5]}/W
one equation for each parallelepiped and one we have a system of
(IV.9.C.1)
point for each parallelepiped,
ordinary differential equations with as
unknowns
as
equations.
We
will
many
obtain an
Eq. IV.9.B.2, v becomes
= grad
or a v
=a
grad
analogy between this system of equations and the system of nodal equations for an electrical network.
We now
describe this process in detail.
suppose that
we can
We
subdivide our space into
equal rectangular parallelepipeds. Eq. IV.9.B.4
(IV.9.C.2)
NETWORK REPRESENTATION
IV.9.D.
where \a H
\
[Bk ]
is
the average value of a ji
ox*/
\
over the face
Note
~
307
the following expressions:
ox*
Bk
.
that Eq. IV.9.C.1
involves first-order
(IV.9.C.3)
differences, while Eq. IV.9.C.2 involves first-order
derivatives.
We want to associate a coordinate system with our rectangular lattice. Suppose the origin
Then
center of a parallelepiped. all
1 (ft
1 /*
3
nW,
,
fl
the centers of
V
are The variables bpy, b y, and averaged over the volume of these parallel (a
epipeds.
on the other hand,
v),-
OV.9.C.5)
A3 ), where n\ n\ and n3
are integers. all
=1
must have co
parallelepipeds in the lattice
ordinates
(IV.9.C.4) 3
the
is
is
averaged
over a face whose center has coordinates as follows:
where
is
the value of
93
at
P,..
^
[y], ^([;],
and (j^;, 5J are numerical factors which can be calculated directly from the form of the poly nomial approximation assumed for (p, the n and the given surrounding lattice points selected, functions b
b,
t
and a.
IV.9.D. Network Representation 2
\ (n
(t-Y),:
2
+
|)h
Substituting Eq. IV.9.C.3, 4,
n*h*}
,
1
1
(a
v) 3
:
/!
(n
,
n
2
2 /i
3 ,
3
+
(rc
differential
J)/i }
t
as the independent variable.
at the
cp
/z
in Eq.
2,
unknowns with
Now let us consider the point values of 3 3 l We must lattice points (n h\ n*h\
and 5
we now have a system of ordinary a large number of equations in
IV.9.C.1 and
relate
).
\
these values with the averages of Eq. IV.9.C.1
and
2,
(b
ml
(bpy^l and
\a
*^i
In Section IV.9.K, we shall show that these
can be approximated by combination of the values of cp
a linear
averages
at
lattice
points.
We
neighborhood of any as a poly be can expressed 9? nomial function of the coordinates, with the origin
that, in the
assume
lattice point
P
,
taken at the
We
lattice point.
the values of polynomial through
(p
center of the parallelepiped, plus a lattice
surrounding
(6pl],
for
points,
example,
is
P
.
s
pass the
at
,
(a-v) m [ (IV.9.D.2)
average,
obtained
There is one such equation for every lattice is a linear relation between Each
by the
polynomial, multiplied by of the spatial variables b over the given function the volume. The and
integrating
where
the
number of
The
then
P
this
equation
point.
1
,
parallelepiped
other averages,
dividing
by
and
^
are
and/?
and pep
at the point
at that point
obtained by appropriate manipulation be same polynomial. In this manner, it will be can by the that shown approximated averages
differences
between
at neighboring points.
These equations can be represented by a connection diagram.
of the
and
and
of
(p
We may think of the value
at a lattice point in the
same way that we
think of the voltage to ground at a node in an in electrical circuit or the velocity of a mass point
PARTIAL DIFFERENTIAL EQUATIONS
308
mechanical connection diagram Then
a
Section IV.7.B). part
(
of a voltage drop
(see
plays
also
the
across a branch or a rate of
contraction of a connection. Eq. IV.9.D.1 plays the part of a nodal equation in which the
operators on
or on
Each term of this equation may be thought of through a branch or the force
as the current
connection through a connection. A complete connection between P contain a should diagram
and each
of
its
neighboring
lattice
points,
are like
(
electrical admittance or mechanical hindrance.
As a consider
specific
the
example of
simplest
averages, Eq. IV.9.C.3, 4, a, b,
and b vary
this process, let us
approximation to the and 5. We assume that
slowly enough that they
may be
considered constants within each parallelepiped and that a is a diagonal matrix. We further
assume that
9?
may be approximated by
a linear
function of the coordinates.
Let
P
?!,...,
be the center of our parallelepiped and be neighboring points located as
P6
indicated in Fig. IV.9.D.1.
In
this case it is
shown
in Section
IV.9.K that
Fig. IV.9.D.1
Pl3
.
.
.
,
P6
,
each containing an element analogous
to electrical conductance or mechanical resistance
(IV.9.D.3)
~
of magnitude
plus three connections to ground
to take care ofbp, b
}
and
b".
The electrical network analog of the connection diagram
is
given in Fig. IV.9.D.2, where e
is
proportional to
fl
are proportional to
ljRlt
h
Then Eq.
C
is
proportional to b
RQ
is
proportional to b
l
^
j,
h*
h
z
IV.9.D.1 becomes
IQ is proportional to It
b"
should be clear to the reader that one- or
two-dimensional problems allow a direct simplifi cation of the above network.
The
analogous network has been fully developed for this simplest case. This network is
+
J>VM
+
b"
(IV.9.D.4)
widely used.
It is
customarily obtained by a
IV.9.E.
METHOD OF
FINITE DIFFERENCES
finite difference discussion (see
Section IV.9.E).
Consider Eq.IV.9.B.l:
If
Eq. IV.9.D.1 contains more terms,
clear to the reader that
introduced.
It is
it
should be
likely that these
branches
may
contain elements of negative impedance, which are difficult to realize physically, but offer no difficulty
when
the network
is
used purely as a
original
lattice
=
v
We write
to interesting point out the geometrical
duality between the
v)
bpcp 4- b
+
b"
where
i
= grad y
these relations in terms of Cartesian
components
method of reasoning. It is
div (a
more branches must be
309
:
.
and the
a*
2
a*
3
where
(IV.9.E.2)
Now we interest
once again assume that the region of subdivided into a lattice of
may be
rectangular parallelepipeds of dimensions h\ A
and h
z .
We
focus attention
on
2 ,
the differences
between values of y at the centers of adjacent
We
parallelepipeds.
approximate the partial
derivatives of Eq. IV.9.E.1
and 2 by expressions
involving these finite differences.
The
and most obvious procedure
simplest
consists of replacing the partial derivatives
simple difference quotients. Eq. IV.9.E.2
by
may be
approximated by
Fig.
W.9.D.2
connection diagram derived from it. In the and were quantities b original lattice,
,
403.40])
b"
bpq>,
associated with the volume of a parallelepiped.
In the connection diagram, they are associated with the dual of the three-dimensional unit, with a point. In the original system, the components of a v were associated with a two-dimensional face.
connection
In the
associated
with
diagram,
one-dimensional
a
they line
are
1,
4Q]. 4Q3)
2
or
ft
connection.
IV.9.E.
There
Method is
an
of Finite Differences
approach to approxi differential equation by a
alternative
%(4o],4o],4o] + f 0,4o], 40]
+
3 ft
)
mating our partial differential equations that system of ordinary
[o],4o])
should be mentioned.
(IV.9.E.3)
PARTIAL DIFFERENTIAL EQUATIONS
310
IV.9.F. Boundary Conditions and Applications
AndEq.IV.9JE.lby
We
boundary conditions for
shall discuss the
problems involving the scalar potential equation in which a is a definite matrix and b and positive
V
are
which 6
equations
Laplace
s
and Poisson
s
into these categories.
~ does not
or
the solution
of
= 0.
dcp
= 0,
b
If
fall
we consider those
First,
nonnegative,
cases in
is
enter the equation,
uniquely determined by the value
dw on the boundary or the value of -~ on the
99
on
boundary. Some problems combine these two forms, with 9 given on one part of the boundary (a
9o9
^3^40] +
on the remainder.
and on
quite clear
It is
how
specified
To
nodes corresponding to
on the boundary are
fc
a V[0], * [0],
fixed at a
simulate
lattice
is
specified, the lattice points
should be
half the length of a parallelepiped edge displaced from the If the coordinate
x*[0])
boundary.
(IV.9.E.4)
where (^[0], * 2 [0], ^[0]) are the coordinates of
resulting
network analog
is
accurate
to
It
may
be
can be shown (see also G. Boole,
D
size h.
= |log(l+A)
The preceding approximation
many
this series,
but
terms as desired.
structed
it is
a\ are fed into the If b is
proportional
corresponding nodes.
not equal to zero, or
potential
equation, then,
uses only the
Networks can be con
with branches for the higher order
may be calculated currents fed in to com
in
addition to
boundary conditions discussed above,
the
must be
defined throughout the region at an initial time
/
in order to determine a
unique solution. In a network analog, the voltage of every node is adjusted to a specified
possible to use as
enters the scalar ot
In
(IV.9.E.5)
differences or correction terms
and small correction
boundary,
specified
v v tnen tne currents
initial value.
many problems
potential equation,
term in
the
d
Tl
h
first
system
on
finite
Ref. IV.9.a) that a derivative can be operator expanded in a series involving successively higher order differences of a function over a lattice of
Z)
~
that,
2
approximations
obtained by recourse to the calculus of differences.
d
identical with
the network of Fig. IV.9.D.2.
More
such
is
T~
a lattice point.
The
points
specified voltage.
d(p
Where
+
can be
either condition
simulated in a network analog.
a,
the
involving i,
A
,
and
V
scalar
are not
continuous functions of the coordinates through out the region in which a solution is
sought.
There
may be
surfaces at
these quantities in
an
is
which one or more of
discontinuous.
electrostatic
problem
For example, two
involving
materials of different dielectric constant, a
is
pensate for them.
discontinuous at the interface between the two
uses the latter
materials.
G. Liebmann (Ref. IV.9.f) method.
IV.9.G.
At
CURL RELATIONS
the interface, one must have conditions
relating both
9 and the normal derivative or
For
equivalent across the surface.
instance,
IV.9.G. Curl Relations
its
A number of the partial differential equations
cp
of physics are specializations of the following
may be continuous and (a grad 9) n have a jump given by a specified function (see also J. A. Stratton, Ref. IV.9.m, 34-37).
itself
relations
pp.
The
curl
representation of the interface will, in
To
represent a
jump
easier to
-
in (a
current generators are used.
It
grad
is
n,
frequently
represent integrals associated with a
of an interface given using polynomial approximations for 99 than to represent the side
between two vectors,
=c p4>
=
-(d
curli{>
general, involve the introduction of additional
nodes.
311
where
c,
c
,
d,
9?
^+
+
d -9
and
div (pv)
=
curl v
F
where
p
is
density
on the
T
is
vorticity
0,
we
appropriate side of the surface. The normal derivative can be obtained from the polynomial
is
well
curty
which the adapted to the solution of problems in boundaries are planar, so that the above geo metrical conditions can be fulfilled. For nonis often desirable to use planar boundaries, it other lattice structures, for example, cylindrical
or spherical.
The methods used
are readily
curvilinear co adaptable to any orthogonal ordinate system.
There have been a number of computing
machines designed potential equation
specially
in
If
F
are arranged
is
known
=
-pY
As a second example,
for
one-
we
consider Maxwell
s
field in a region equations for the electromagnetic
in
which the relations between
between
H
and
B
D
and
E
(See also
are linear.
J.
and A.
Stratton, Ref. IV.9.m, pp. 1-11.)
to solve the scalar-
on a calculating board in such a way
(IV.9.G.4)
function of the coordinates,
Ref, IV.9.d.)
curl
one or two dimensions with
circuits
(IV.9.G.3)
find a vector potential
have a special case of Eq. IV.9.G.1 and 2. (See also H. Lamb, Ref. IV.9.e, p. 5,202; G. Kron,
various boundary conditions. Electrical elements
that appropriate
=
Since div (pv)
function fy such that
approximation as in the Section IV.9.K. rectangular lattice here discussed
are
=
corresponding to an extrapolation from the
The
and
no sources or sinks where
velocity
at certain discrete points
c"
(IV.9.G.2)
As an example, we mention the equations of vortex motion of an incompressible fluid, with
is
are tensors,
v
values of
(IV.9.G.1)
c"
+
v{>.
vectors, all functions of the spatial variables.
curved surfaces appropriate to the immediate problem. One uses a polynomial approximation to
c
]?9
problem in finite differences. The given problem can be expressed in terms of these integrals as in Section IV.9.C. These integrals can be taken over
+
9 and
curl
E + pB
=
H - pD - J =
(IV.9.G.5)
where
and two-
dimensional flow problems can be constructed by the insertion of jacks in the proper positions. Ref. IV.9.f; G. D. (See also G. Liebmann, IV.9.1 V. Paschkis Ref. H. C. McCann and Wilts, ;
and H. D. Baker,
Ref. IV.9.J;
S.
C. Redshaw,
In addition, the network method of solving differential equations finds
tion as a
We
will consider the
wide applica
method of computation using
digital
It serves also as
its
inherent interest.
a further illustration of the ideas
discussed earlier in this chapter and provides an
one further develop opportunity to introduce in analog networks. This development
ment computers.
network for Eq. IV.9.G.1
and 2 not only because of
Ref. IV.9.L)
partial
This example will be more fully discussed later on.
PARTIAL DIFFERENTIAL EQUATIONS
312
both nodal and voltage relations to in one network.
utilizes
realize
two equations
We proceed to construct a network representa tion of our partial differential equations.
As
the case of the scalar potential equation, learn
in
Consider a face, B, in the lattice for Eq. IV.9.G.7 parallel to the 1-3 plane, with edges C4 as shown in Fig. IV.9.H.1. For this Ci, .
.
.
,
Eq. 1V.9.G.7 becomes
face,
we
most by transforming Eq. IV.9.G.1 and 2
into their integral forms.
curl9-nJS
We note that
=
(IV.9.H.1)
(IV.9.G.6)
J
where ^[CJ,
etc.,
are the average values of
l9
;
where the left-hand
integral is taken over
any
6 bounded by a contour (L (See also A. P. Wills, Ref. IV.9.n, pp. 97-98.) The positive direction along is the direction in which a surface
etc.,
etc.,
over the indicated edges and (c are averages over the face.
(
right-hand screw positive n If
we
surfaces (
,
we
is
turned to
move
it
in the
direction.
integrate Eq. IV.9.G.1 <S
and
and 2 over any (E and
bounded by contours
obtain the relations
f 9-
j
(IV.9.G.7)
-
ds ==
-
(d
p
+
d
-9
-f
d")
udS
/(
We (IV.9.G.8)
Now we
all
the faces
Eq. IV.9.G.8. The components are averaged over edges and related to the
of require two rectangular lattices, one
for Eq. IV.9.G.7
have analogous relations for
in the lattice for
IV.9.H, Rectangular Lattice
and one for Eq. IV.9.G.8. Each
i\>
average values of the components of d 9, etc., r over faces. For example, let B be a face from this lattice (see Fig. IV.9.H.2).
Eq. IV.9.G.8
For
this face,
is
(IV.9.H.2)
Now it is desirable to relate our average values to
point values at lattice points.
The methods of
the preceding section can be used to approximate the average values by a linear combination of the values of the variables at any specified lattice Fig. IV.9.H.1
points.
We lattice
consists of equal rectangular parallele
pipeds of dimensions h\ h\
must apply
3 /z
.
Eq. IV.9.G J or 8
to each face of these
parallelepipeds.
study Eq. IV.9.H.1 and
lattice structure, for
2.
In our
first
Eq. H.I, the components of are averaged over edges while the components of c and c are averaged over faces.
9
-p*\>
t|>
CONNECTION DIAGRAM; NETWORK ANALOG
IV.9.I.
Therefore, the
lattice points for ^, 2 and 993 should be the centers of to the 1, 2, edges parallel
and
The
3 directions.
lattice
points for
and y 8 should be the
centers of faces
dicular to the
and
1,
2,
consider the second
3
y l9 ^ 2
,
perpen-
Now
directions.
lattice structure, for
For simplicity, we assume that the components
,
Eq. H.2.
313
of
9 and fy can, over each parallelepiped face and
edge, be approximated
coordinates, that matrices,
d
,
and
d"
c
c,
by
linear functions of the
and d are diagonal
d,
,
and that the components of c, c
may all be approximated by
over any face.
,
c",
d,
constants
Then
(IV.9.H.3)
]
= d \[V]
and yjj ] are the values of (p t and y f (p t [j\ and ?/, the centers of the edges C, and C/.
where at?,-
Furthermore, (/%)[()] and ^ t-[0], (/^)[0 ] and are the values of pip t and y f p^ and (p it at -
,
P and ?
the centers of the faces
B and B
.
Then Eq. IV.9.H.1 becomes Fig. IV.9.H.3
Here the
(plt
2,
and
3
lattice
points
lie
on the
points
y x y% and
centers* of faces, while the
,
centers of edges.
lattices are interlocked,
on the
lie
% lattice
Thus, the two
with the edges of each
(IV.9.H.4)
Eq,IV,9.H.2 becomes
one passing through the centers of the perpen dicular faces of the other, as in Fig, IV.9.H.3.
Now we points.
can assign coordinates to our
Let the origin
parallelepiped
lie
in the first lattice.
lattice
vertex of a
(IV.9.H.5)
Then the lattice
be written for the
on the
the following coordinates: points have
Analogous equations may
other faces of both parallelepipeds.
IV.9.L Connection Diagram and
Network Analog
We 1
(B
+ *)**,(
are
diagram.
now
We
ready to draw our connection have a separate diagram for each that
we have a
is, diagram plane in our first lattice, to represent Eq. IV.9.H.4 between each of the
whose lattice points following sets of variables
The
and 2 averages of Eq. IV.9.H.1
approximated by
linear
combinations of the
values of the variables at their
own lattice points
by methods exactly analogous to in Section IV.9.K.
may
be
those developed
are coplanar:
PARTIAL DIFFERENTIAL EQUATIONS
314
^ and
Y>
Eq. IV.9.H.4 and its counterparts for the nodal equations of these diagrams 3 are
points
Here
The
points serve as nodes, while the
lattice y>,.
lattice
.
s
become connections. note
again,
between the original
the
We see that h*y z may be thought of as the voltage to
1 ground at the nodes, h 3
vertical branches,
/z
^ as the current in the
as the negative of the
3
current in the horizontal branches, the right-hand duality
geometrical
lattice structure
side of the equation as 22
connection diagrams. Each of these connection diagrams lies in a two-dimensional space and is
dual to a two-dimensional section of the
a current to ground at the
L1/.3
and the nodes,
(c /?
n
+
c
22 )
as
an admittance to
ground, c 2 "/W as a generated current.
first
A
node or point in the con nection diagram corresponds to a face in the lattice, while a connection or line in the con
lattice structure.
nection diagram corresponds to an edge in the lattice (see Fig. IV.9.I.1).
I
Fig. IV.9.L1
The
Fig. IV.9.I.2
solid lines represent the lattice structure,
while the broken lines represent the connection
diagram.
which
it
Each node corresponds to the face of is the center. Each broken line corre
sponds to the
solid line
which
Specifically, Eq. IV.9.H.4
it
crosses.
may
be represented
Now
each s appears in two relations like Eq. IV.9.H.4 with opposite signs, and is therefore represented in two of these connection diagrams. Consequently, these connection diagrams must
be interrelated as shown in Fig. IV.9.L3. Each point must be represented by two con
^
byFig.IV.9.1.2.
lattice
P
,
.
.
.
,
?4 are lattice points for y2
& and g
2
are lattice points for
2s and
4
are lattice points for
Rewriting Eq. IV.9.H.4,
^
z
nections, each carrying the j quantity h
same
current-like
with opposite signs.
In the network analog, the quantity hty, is represented by the voltage to ground of the es .
The quantity
appropriate
node,
represented
by equal and opposite currents, k in two branches parallel to the x
= i,-
(i)
and x l
i i{l]
The
directions.
positive current flow
sign convention
hffy
is
makes
from the node correspond
ing to a lattice point of higher coordinate to the node of lower coordinate. For hz is represented
by
i
example,
=
(p z
2 f^ currents in to the xl and jc 8 directions,
a
^
,
branches parallel respectively. This suggests ideal transformers
CONNECTION DIAGRAM; NETWORK ANALOG
IV.9.I.
linking these paired branches.
networks of Eq. IV.9.H.4 are
by
Thus, the three
tied to
each other
branch.
315
Then the voltage across the transformer, - Jfyjl ]. The ], represents #^[2 ]
e$] - ejl
ideal transformers as in Fig. IV.9.I.4.
Now realized
us see
let
how
Eq. IV.9.H.5
be
may
by these same networks. Thus
we
far
Fig. IV.9 J.4
P3
;
P4
L* p
+R
branch must contain an impedance z
to represent
^m
cP -
^
Fig.IV.9.L3 a generator of a voltage
have not specified the impedances in the branches connecting
ip f
We
nodes.
shall
show
that
3
#% and A ^ at nodes.
3
p
,
d
,
and
d".
We have made
the voltages, el and e3 , equivalent to
We rewrite Eq. IV.9.H, 5.
(h
The
d,
left-hand
J/Wi
3
^
^
o
(IV.9.L2)
2
of
representing
r^v^X
V )[0] + side
and + d u -r^ &
(see Fig. IV.9.L5).
it is
to represent possible to adjust these impedances
the components of
2
,,,,hw
p
this
equation
is
-
e3 [3 ]. e represented by the voltage drop s [4 ] This is equal to the voltage drop across the ideal transformer coil plus the voltage drop across the the voltage impedance in the connection plus in the connection. Thus, in order that
generated
p;
Fig. IV.9.I.5
-
P4 branch the voltage equation for the P3 transformer the ideal may represent Eq. IV.9.I.2, P2 coil must be the only element in the P/
-
We can treat the analogous relations on ^ and draw our complete network similarly and now analog as shown in Fig. IV.9.I.6.
3
PARTIAL DIFFERENTIAL EQUATIONS
316
The
following quantities correspond:
Quantity
Proportional to
Symbol
Voltage to ground at nodes
Current in branches
_##
^hW
Wll
Inductance in branches
,722
A1
h
Voltage generated in branches
1
h*
rf/w, i,
Capacitance to ground
m
w
,n
Resistance in branches
A2
^l>
^25
2
w,
AV
^
dz
,
^
3
^3
Conductance to ground
c*
-,
JW If
Current generated from node to
A?
4
ground
special case
where
c,
c
,
d,
and d are
positive
definite matrices, the solution is uniquely deter
mined by theinitial values of 9 and
fy
asfunctions
of position and the tangential components of either or fy on the boundary as functions of
9
time (see also
J.
A. Stratton, Ref. IV.9.m, pp.
The uniqueness proof rests on a computation of n dS, taken over ) JJ(9 x
486-88). (Note:
4>
the
are the differences boundary, where9 and between two sets of solutions of Eq. IV.9.G.1 and <]>
2.
With the given boundary must vanish, so that
conditions, this
integral
=
~
cur l
j j j
Fig.IV.9.1.6
taken over the volume, must vanish too. stituting
The boundary conditions sufficient to determine a unique solution to Eq. IV.9.G.1 and 2 in their most form are not clear. In the general
curl<J>
from Eq. IV.9.G.1 and
under the given conditions, only
when
proven.)
9 =
i|>
= 0,
this
2,
we
)
dV
Sub
see that,
integral vanishes
and uniqueness
is
MAXWELL S EQUATIONS
IV.9J.
Let us consider the
The
ditions.
analogous network con
where
currents and
voltages are initially adjusted to values analogous to the given initial
Now
values.
suppose one bounding surface
normal to the x so that either
on
this
tangential
The lattice is arranged
direction. 2
fall
1
and
or <%
y 2 and
conditions.
ponents of
are
lattice
constitute If the
points
the
tangential
the
given,
t|>
%
depending upon which
surface,
components
boundary
is
and
#V 2
ponents of
//,
constant,
a
is
the
/z is
9
and
993
tional to
2 /2
An
2
9=
H,
or
q9 3
cp
two
=E
com
(IV.9J.5)
voltages of the
Then c c
corresponding to the
c"
=
e
or
c
= =
d
=
d"
=
=
(A
=
c
=
c"
(IV.9J.6)
d=~e d = -0
d=ji
z
3 /z
in
We can let
points are fixed at values propor
and
and 2
IV.9.G.1
alternative ways.
- from the preceding case, and the
lattice
the dielectric
these equations constitute a
of Eq.
case
special
.
currents in the branches
is
conductivity.
hl displaced by
e
the magnetic inductive capacity,
It is clear that
If the 3 tangential com are given, the lattice structure is
#V
and a are constants,
given
corresponding nodes are fixed at values propor tional to
,
317
.
analogous procedure
is
possible in any
d"
=
locally orthogonal coordinate system, but the
general case
much more complex and
is
seem to be considered IV.9 J. Maxwell
s
does not
in the literature.
ways.
is
equations (see also
J.
s
left-hand case
is
discussed here.
=H
Equations
Maxwell
to
The
electromagnetic
field
A. Stratton, Ref. lV.9.m,
pp. 1-11; G. Kron, Ref. IV.9.c).
=
currents in the branches are proportional to the
H
and voltages components of the magnetic field at the nodes are to the proportional components of the electric field E. We have inductances proportional to the components of
curlE
where
E
and
intensities,
+ pB-0
(IV.9J.1)
curlH-^D-J^O (IV.9J.2) H are electric and magnetic field
D
is
electric
displacement,
B
is
magnetic induction, J is current density. In any material, isotropic or not, in which the relations between
D and E, B and H, and J and E
are linear,
D = e-E
-
H J=o-E
B
-
[i
(IV.9J.3)
,
with ideal transformer
in series
Capacitances proportional to the components of in parallel with conductances proportional to the components of a connect the nodes to
Then
the charges
tional to the fluxes
ground.
on the condensers are propor
components of D, the magnetic
through the inductors are proportional to
and the currents through the conductors are proportional to the components of J (see Fig. IV.9 J.I).
the components of B,
Many
electromagnetic
sional problems.
u,,
electric,
In the
or TE, waves,
rela
direction,
and
H
is
tions reduce to
E
field
problems
are
transverse-magnetic, or (IV.9J.4)
to a single direction
first is
type, transverse-
limited to a single
directed in a plane perpen
dicular to that direction.
= //H
|JL
coils in the branches.
reduced, by virtue of symmetry, to two-dimen
and a are symmetric tensors. In homogeneous, isotropic media these
where
The
right-hand analogy may be developed by similar and ^ E, we find that reasoning. If 9
A straightforward application of the foregoing analogy
Our network becomes specialized in one of two
In the second type,
TM,
waves,
H
is
limited
and E is directed in a perpen
dicular plane. Thus, for these problems,
we may
PARTIAL DIFFERENTIAL EQUATIONS
318
to
Proportional
Symbol
Quantity
nodes Voltage at Current in branches
#HI U /*
Inductance in branches
Capacitance
C
to ground
l5
C2
,
^V
TT
C3
Conductance to ground
condensers Charge on
fo
Current through conductances
M
,
(3)
(j 3 e3
*
^
**>
*** ~
*
-
.
*** ~
and 2 may be specialized: Eq. IV.9J.1
TE waves:
(IV.9J.7)
TM waves:
For
TE
waves,
specialization
we choose our
in which
9=H
left-hand
= E so
and 4>
to the
of the assume that the only nonzero components
that the voltage at nodes
field vectors are as follows:
components to 2 need to be considered are those analogous one branch to and we need
TE waves:
TM waves:
4 r
_
#
2
#i>
H
s
is
which of E. Then the only nodes
only
lattice points
_ ^i
P
^
proportional
rpnr^wit each H, and
*
H*
lattice
MAXWELL S EQUATIONS
IV.9.J.
network 1-3
is
319
analogous to the connections in the
plane, without ideal transformers (see Fig
IV.9J.2).
Network
Proportional to
Quantity
*
For
TM
we choose
waves,
the right-hand
= E, ^ = H. specialization, 9
It will
the reader to see that
is
Fig. IV.9. J.3
be left to Fig,
the
network, in which the voltage at the nodes proportional to
#
IV.9J.2
analogous is
and the currents in the
2
branches are proportional to Network Quantity
El and
3
.
Proportional to
*W
u*W
R3
L3
:c 2
Note that, in problems that do not involve any flow of current (o 0), both these networks
Fig.IV,9J.3
=
contain only capacitors and inductors and are
constructed at Stanford University, based
identical
ideas of Gabriel
in
structure,
although the network
elements represent different quantities.
Two
isotropic media,
where a
field
= 0,
problems
have been
on the
1L Spangenburg,
G. Walters, and F. Schott, Ref.
network analyzers for the solution of
two-dimensional electromagnetic in
Kron(see
also
IV.9.1).
One
is
in designed for problems having axial symmetry,
which cylindrical coordinates are used, the other for Cartesian coordinates.
320
PARTIAL DIFFERENTIAL EQUATIONS
IV.9.K, Derivation of Eq. IV.9.C3,
4,
5
btP^lj], x
anadjacentparallelepiped
2 3 tangular parallelepipeds of dimensions h\ A A
2
[j], x*[j]).
We select su ch points ?!,..., Pr We sna11 snow that the averages of Eq. IV.9.C.1
Given a region subdivided into equal reo .
,
and 2
may
be approximated as follows
:
(IV.9.C3.)
(IV.9.C.4)
(IV.9.C.5)
where p[0] value of
the value of
is
at
P
A
99
P
at
and ^[ ; ] is the
..
3
A[j\ and G mlc [j, B ] may be calculated by methods described below. These will not depend on (p. lj\
t
}
directly
Let us
mating
consider the process of approxi
first
9?
by means of a polynomial. Given a
center point for a parallelepiped,
obtain a Fig, IV,9.C.l
this
P we
point and also agrees with
99
will
,
polynomial which has the value of
at
on a number of
neighboring center points.
A
typical parallelepiped has faces
indicated in
and
Fig. IV.9.C.1,
Blt
...
t
B6
Given Eq. IV.9.C.1
2,
THEOREM IV 9
LL
Assume
that the a
PP roxi
mating polynomial can be written in the form
(IV.9.K.1)
where
is
monomial
the value of y
in
atP0?
iv is 2
(^-^[0]),
(x
a nontrivial
-j
2
[0]),
and
where where
o^, ft,
y{ are
positive integers,
and the
are constants as yet undetermined.
Pl3
neighboring center points,
+
A if the P Alternately the determinant
raw+l-g
1?
.
.
.
given below .
.
.
Pn
,
are
,Pn is
c,
Select n ,
so that
not zero.
prescribed, the
monomials w* must be chosen to make A not zero. (IV.9.C.2)
Then
should be recalled that these equations are the integral form of the (It
(IV.9.K.2)
scalar-potential equation
[Eq. IV.9.B.1] for the parallelepiped S.) *
where
Let the center point of the parallelepiped be r<) ]>
* 2 [0], ^[0]), and
let
the center
point of
is
^-[j]
a constant which can be calculated
from the values of w 1
P19
.
.
.
,
?.
n ,
.
.
.
,
\v
at the
points
IV.9.K.
For each
Proof:
x 2 [j],
x*[j]),
DERIVATION OF EQ. lattice
neighboring
we
write
the value of
is
99
at
THEOREM IV.9.K.2.
point,
321
If
an equation
] where ?[/]
IV.9.C.3, 4, 5
(IV.9.K.3)
P
(b
i9
=
ypffop)]
+
4 1 [/Mfl
"*
wllere
^
v<
;
^
parallelepiped,
(IV.9.K.4)
We
have n such equations (Eq. IV.9.K.3). if the determinant is not equal to zero, solve
them
the avera e of S
&V over the y pq is the average of V over the
parallelepiped,
Therefore,
we can
(IV.9.K.7) is
u
for the c t-.
(IV.9.K.5)
where 4[;]
=
D.r /I
Proof:
-,
We multiply Eq. IV.9.K.6 by b
(IV.9.K.8)
w s [2]
A=
We
can average Eq. IV.9.K.8 term by term by
integrating over the parallelepiped
and dividing
by the volume.
and D,[/]
is
the cofactor of
wj/l
in this deter-
minant
9?
We
have shown
may
be approximated by the polynomial
9
=
rfo]
that, in the specified region, /J
~
9L JX
w }|
-
J
where
+2
in which the coefficients are linear functions of
the values of
at lattice points.
Care must be
(& V)[9f]
=
(i) so neighboring points or of the monomials w A may not vanish. Note that
^
j
|
Uw
l z h h*h J J J
exercised in the selection of the constellation of ,
that the determinant
A depends relative to
P only upon the positions of 1}
P
.
Therefore,
of neighboring points point, the
same
is
.
.
.
,
Pn
a given constellation selected for one lattice if
relative configuration
may be
used for any lattice point in the region. From our polynomial approximation, Eq. IV.9.K.6.
we can now
calculate the approximate
averages, Eq. IV.9.C.3, 4,
and
5.
and the theorem
is
proven. (Note that A
be directly calculated.)
THEOREM iv.y.^.j.
n
[j]
may
PARTIAL DIFFERENTIAL EQUATIONS
322
where
then
~l
^M ^
w^ere s Now we set
where n
The
/:
quantities 4[;]
and the theorem
+21 4D ]X?D
.As
*~ 1
3
]
~1
is
~ ?[]X (IV 9
K
noimal 11)
we
illustrations,
IV.9.C.3, 4,
and
5,
Multiplying by b and averaging term by term,
=
for
determine
Eq.
two simple poly-
assumed as approximations to 9?. many cases, b, b and a vary slowly enough with the coordinates that, in any one paralleleforces
f
,
PP^
6rsn(pv)rol r L J L J
shall
explicitly
In
(b p(r p)l%\ l J
.
proven.
:
n
n (P9>)[0]
t
differentiate
Eq. IV.9.K.6 with respect to time, thus
=
B
and w* do not
depend on time. Therefore, we may
PV
area of the face
the^
ma^
be aPP roximated by constants.
This simplifies the determination of Eq.IV.9.C.3,
+ 22 ^MK^D] - ?[0])MPQ i=i 3=1
(IV.9.K.12)
Now we
let
24L/K^P] =
A[j] and the
=i
theorem
is
4) and 5 for any P articular Polynomial approximation, but is not theoretically necessary. It may t>
e undesirable to regard these as constants
one
is tr
cells
as possible.
To determine
proven.
THEOREM IV.9.K.4.
form
If
f
(p
+2J t=
-
b
n
n
=
di[j]((plj\
-
f
^[0])w
b,
when
with as few
simplest possible explicit
our approximations, let us assume that and a are constant and that is a linear f
function of the coordinates.
W(IV 9
K 13) In this case,
BJ
We
it is
"^^lM = 2
Proof:
,
the
lattice
3=1
l
wkere
res ectto P
yin g to construct a
1
differentiate
\
clear that, for all
(fcV)[1tt]
/,
=
C/A /
Eq. IV.9.K.6 with
(6w )[C|
=
Therefore
**>
*
Cfl
-
^
]
=
and (iv.9.K.i4)
,
(1V9
K1
face B,, Multiplying by a* and averaging over the
Now
to determine
mfc [jfy],
we
refer to
IV.9.K,
DERIVATION OF EQ. To
323
IV.9.C.3, 4, 5
determine q,
.
.
.
,
c6 ,
we
start
by
selecting
the six lattice points indicated in Fig. IV.9.D.1.
P3
:
P4
:
P6
:
Substituting
these values in Eq. IV.9.K.20
obtain the specific form of Eq. IV.9.K.3
we
:
Fig. IV.9.D.1
Let
Pl9
.
.
.
.
.
.
,
P6
.
coordinates,
,
it is
be the values of y
Since
y
is
at
a linear function of the
clear that
(IV.9.K.21) Solving,
If
(IV.9.K.19)
Betted approximations
can usually be obtained
(IV.9.K.20)
PARTIAL DIFFERENTIAL EQUATIONS
,
JJ*t
A
the
in
surrounding
region
P
Then, IV.s.R-o, Eq. polynomial approximation,
If
our
V
is
^
is.
^^^
term of Eq. not a constant, each J( and then
^ ^ ^.^ Likewise)
if
Ms
fey
a constant,
(1V.9.K.25)
or
=1 -
-
2
?[0])}(x
([4]
2h
- *2 t])
(j
=
1 ..... 6)
for
Eq. To determine the specific expression IV.9.K.23 IV9C5 we must differentiate Eq. coordinate, with respect to the appropriate
3
the appropriate multiply by
componen
ssion e the resultant expr
awras
face.
priate
+ J_ {(^[l]
treated as a constant matrix,
9 [0])
T
f a In the special case where
we
find
2 2ft
3 2/!
(IV.9.K.26)
+ (?[6]
- vtO])K - *
2
3
for the other faces.
P>])
(IV 9
Now
we can
V
Eq.
matrixaisdiagonalora, = Oif;!.Thisgreatly
our approximating
calculate
4, averages, Eq.IV.9.C.3, is a constant, or If
with similar expressions the In most cases of practical application,
K 23)
and is
simplifies
5:
a v .
=
.
a approximated by
References for Chapter
IV.9.C.3:
(f ,)PQ
ot the expressions for the components
b
b>[0]
+
-
^ {(rfll
"Eectrc
+
^C\\\ 4- r^fSl (9L^J ?[OJ)
-
d
(PFOD ^L^JJ
-
-
+ (^[6] ,
/
9(0])
- dPD rt]) + fo[?l (?[5]
^
-
<m
-
)
flui d
e H. .
(j
=
,-.
.
1,
,
6)
5.,
JOIHIL Aero.
ff HyMywnics.
5.
Lamb.
f f
S Z.
o g
A
Vol. 12,
York, Dover
New
of partial Q Liebmann, Wi th a Distance network fqua
Appl
b
=[j]
fields,"
Publications, 1945.
or
A
flow
2
1
HV9K24 IV 9 K (
rmYt
r
of circuit of the field equations 32 (1944), pp 289-99. I.R-E, Pr.c, Vol. and mcomcircuits of compressible
"Equivalent 4
pressible
+ (?[4] -
Vol. 67 (1948),
EL^valent Maxwell
-
m
Electrical Engineering,
equations,"
V[])
crcu
9
"Solution
j-.n*i
diflerenuai
analogue,"
Vol.
1
(1950), pp.
Bnt.
92-103
Phys., Treatise on the Mathematical llteory Publiations, 1944.
EH Love.
J aaftfty.
A
New
York, Dover
IV.9.K.
h.
i.
W. H. McAdams. Heat
DERIVATION OF EQ.
Transmission.
McGraw-Hill, 1942. G. D. McCann and C. H. Wilts,
New
York,
k. S. C.
"Application
of
1.
Vol. 37 electromagnetic field problems," I.R.E., Proc., (1949), pp. 724-29, 866-73.
Mech., Vol. 71 (1949), pp. 247-58. V. Paschkis and H. D. Baker, method for deter mining unsteady-state heat transfer by means of an
m.
A.S.M.E., Trans., Vol. 64 (1942),
n.
"A
electrical
analogy,"
pp. 105-12.
Redshaw, "An electrical potential analyzer," Mech. Eng., Proc., Vol. 159 (1948), pp. 55-62. K. Spangenburg, G. Walters, and F. Schott, of "Electrical network analyzers for the solution Inst.
electric-analog computers to heat-transfer and fluidflow problems," A.S.M.E., Trans., in Journ. Appl
j.
325
IV.9.C.3, 4, 5
J.
A. Stratton. Electromagnetic Theory.
McGraw-Hill, 1941. A. P. Wills. Vector Analysis. Hall, 1947.
New
New
York,
York, Prentice-
Part
V
MATHEMATICAL INSTRUMENTS
1
Chapter
INTRODUCTION
Vl.A. Mathematical
Instruments
one
Maps, graphs, and design drawings represent of methods by which situations typical examples are set
up on paper in order to permit logical The construction of these representa
analysis.
tions
and the procedures
for their use involve
or, at
most, a few mathematical operations.
Instruments of this type have been developed for centuries, and are often precursors of the com
ponents used in more complex
modem standard
computing devices. Frequently the mathematical used in their theory involved and the ingenuity
much
many mathematical operations. For centuries various instruments have been developed to
construction and design are of
in a carry out these mathematical operations
which make them the most economical or most
quick and convenient manner. Recently, the problem of handling large amounts of observational data has led to the use
convenient method for handling situations for
of
obtaining
inexpensiveness,
numerical values from empirical graphs or for obtained from graphing numerical information
Mathematical
similar
digital
Often
instruments
either
for
communication and computing systems. desirable to perform a specific mathe
it is
interest.
Very often they have specific practical advantages
which they were designed. In advantages
may
instruments plotting
and ease of operation. operations
have
The
for
which
such
constructed
include
manual inputs or
electrical
been
devices with
signals.
these specific cases,
consist of small size, portability,
resulting
plots
Most of these
can
be
either
devices assume
matical operation on the data as it is received. Instruments using various electrical signals have
cartesian or polar.
been devised for special operations, for example,
coordinates holds between the inputs. However, curves such as there are devices which
computing It is
Fourier transforms.
frequently
convenient to have instruments
which can operate directly on data in a given form
and produce
the result in a desired form.
only one
or,
operations
are involved,
at
When
a few mathematical
most,
it is
frequently possible
an to produce inexpensively instruments having such for of accuracy purposes. adequate range
On the other hand, if the necessary mathematical procedure
is
more complex
the
most
effective
be to translate the given basic procedure may suit information from the given form to a form and devices able for use in standard computing utilize
some
special
instrument for
this trans
The more complex the mathematical the more difficult and expensive the procedure, instrument will be when the form
lation.
special purpose
of input and output data
We
is
specified.
devices consider, then, in this part
operate on data
in a specified
which
form and perform
that the desired functional relation between the
plot special evolutes or ellipses or the conical sections in
have been
of
multipliers Many types rule constructed, but at present only the slide
general.
is
used as such an instrument, although, of course, the desk calculators are lineal descendants of earlier instruments.
For a given graph, one may use instruments to measure the length of a curve, the area under it to produce
the integral, or
derivative. is
In
many
its
the slope to obtain
technical procedures, there
a need for the integral
of/ or/, and ingenious
to devices for modifying a graphical integrator These devised. been have obtain these integrals
modifications are simple additional mechanisms of a type which are also used to obtain greater device for scale range, and, hence, accuracy. called a is curve a within area obtaining the devices for obtaining integrals of/, planimeter; etc., are called integrometers.
A
/,
INTRODUCTION
330
Many of these computing devices are based on the steering principle of the front wheel of a tricycle, and much more elaborate devices called "integraphs,"
using this principle,
have been
constructed in the past which permits one to obtain derivative or integral curves very directly.
Another principle often used formerly was that of the disk-type variable speed drive, and, eventually, these devices were elaborated into
harmonic analyzers and the modern mechanical differential analyzers.
It is
often desirable to
computer with a graphical output. application
is
A
frequent
that of obtaining the Fourier trans
form of a given
signal,
or of
"smoothing"
effect of noise. signal to minimize the The devices described in this part are used
the
more
widely in Europe than in the United States. For this reason we do not describe details of individual devices,
but rather give the principles on which Those who wish
the devices are constructed. specific
information are referred to References
V.I. a, b, c,
and
d.
obtain integrals of the products of pairs of functions, as, for instance, Fourier coefficients References for Chapter 1
and Fourier transforms. There are a number of
ways this can be accomplished using the variable speed drive principle. In a number of modern applications, the input signal is in the form of a voltage, and often electronic differential analyzer
components can
be effectively combined into a fixed purpose
A. Galle. Mathematische Instrumente. Leipzig, B. G. Teubner, 1912. b. W. Meyer Zur Capellen. Mathematische Instrumente. a.
Akademische Verlagsgeschellschaft, 1944. H. de Morin. Les Appareils d Integration. Paris, Leipzig,
c.
Gauthier d.
F.
A.
Villars, 1913.
Willers.
Mathematische
Maschinen
Instrumente. Berlin, Akademie-Verlag, 1951.
und
Chapter 2
ALGEBRAIC AND ELEMENTARY TRANSCENDENTAL OPERATIONS
V.2.A. Fixed Purpose Computers
There are many cases where available com to accomplish ponents can be readily combined specific purposes.
The range of
electronic
and
mechanical analog components presently avail able is such that the mathematical aspects of such devices are readily taken care of special purpose (see Part III).
Normally the practical
difficulties
are associated with translating the input into a
form to which the components Moreover,
this
are applicable.
at present type of computer
is
rather expensive and would be justified only by associated with human effort. saving the expense If the given input is in a graphical form which
can be
set
in a function input table, then both
up and mechanical components can be and the obvious consideration of availa
electronic utilized,
even graphical is the only one. However, bility information may require a change of scale, for
which pantograph instruments are available (see Section V.2.C below). The graph may also be circular arc paper, i.e., the usual recorder given on the ordinate is measured along paper on which
circular arcs while the abscissa
is
measured by the
intersection of these arcs with a fixed axis.
most function
tables, this
may
For
a trans
require
which transforms the rotation corre lating device to a linear
displacement. sponding to the ordinate This rotation can be readily obtained by a linear or the transformation can be
potentiometer, a rack and pinion. This performed by
is
clearly
an appropriate func equivalent to constructing tion table for a graph of this type, the However, the speed of obtaining
data
may
require
original
that one use photographic
by a person. However, photographic records may be in graphical form. If the form is "negative," the record appears as a white line
if
i.e.,
on a
black background or a transparent line on an one can read such a record
opaque background,
by scanning parallel to the y axis with and a photoelectric cell. The result coded pulse
signal,
i.e.,
a light ray is
a time-
the value of the signal
is
the duration of time between the start of the scan
and the activation of the photoelectric
motion of the rotating
slit
light ray
or by a
cell.
The
can be obtained by a spot"
"flying
cathode-ray
tube whose time of scan can be very effectively
For various integration purposes, such a time-coded pulse reading device can be used to convert a curve on an ordinary photo
controlled.
into graphic negative area under the curve
the
area
"mask"
is
form, in which the
transparent
or white and
above of a contrasting character.
Simultaneous with the reading by the scanning beam we can also have a flying spot scan of
light
an undeveloped
film, similar to the reading
scan
but with the ray on until the read pulse occurs, at which point the ray is blanked until the start of the next scan. Thus the ray of light hits the for those parts of the scan below undeveloped film the curve and not above. For cathode-ray tube spots,
this is readily
accomplished.
intensity of magnetiza not a dependable function of the input The majority of applications of this
For magnetic tape, the tion
is
signal
medium depends on
digital
or
"on-off"
records.
However, with good tape movement apparatus, can be used as a modulation and frequency decoded by frequency modulation techniques
records or magnetic tapes. Photographic records of meters, which, be
which yield direct-current signals. Magnetic tape is more convenient to handle than photographic
be interpreted frame by frame normally, have to
films
may
actually
pictures
and
is
used in Fourier transform devices.
ALGEBRAIC AND ELEMENTARY TRANSCENDENTAL OPERATIONS
332
value of the dividend
V.2.B. Slide Rules
have
devices
Many
been
with
its
for
developed
immediate multiplication, but only the
slide rule,
The decimal point must slide scale. determined either by rule or by a simple mental be
the
The
has survived. logarithmic scale,
basic slide rule consists of two logarithmic scales
and has the advantage of mechanical and ease of use.
the dividend
(x^) or
foxJlO) and reading the value on the frame scale corresponding to either 1 or 10 on divided by 10
simplicity
computation. The cursor permits the alignment of a point on
any
scale
on the
slide
with a point on any scale on
Thus any frame
the frame.
can be used
scale
with any slide scale. It is usual to provide scales with the direction of increase reversed to assist division,
and
scales displaced
to facilitate multiplication also provided in
which the
by
by an amount log TT Other scales are IT.
scale
numbers,
is
one
on
this
linear scale unit
half the unit of the primary scale so that x, are represented
from
1
to 100.
The quantity x on the primary scale, say, of the frame corresponds by means of the cursor to
common forms
There are two
of the slide rule,
linear slide rule consists of three parts, a
The
frame which contains a number of
which contains associated glass
slide
scales
scales,
scales,
a slide
and a cursor or
which permits one to line on the frame and slide in a desired with a
line,
quantity
#2 on The
frame.
this smaller unit scale
distance
scale has value log
=
d from the
1
on the
end of
x on the primary
scale
this
and
2
log x on this secondary scale. This convenient table of square roots on the slide rule
2 log x
on
often very useful for finding initial approxi mations for the more accurate extraction of a
the frame. The basic scale on the frame, and the
square root by an iterative procedure on a desk
up
manner. The
slide
corresponding
is
between the scale unit
is 1
log
and
(1
indicates a
If the total length of
end) to the point marked If
x.
x1 and J 2
are
and log
Cube root
scales
may
scales, linear
log scales,
also be provided.
mark xl on
(see Fig. V.2.B.1).
xz mark on
xl mark on
FRAME
x on
SLIDE
If the
mark of the scale on the slide scale is now
the jci*2
calculator.
two numbers
10, the cursor is set to
on the frame
to coincide with the
the
10.
slide,
is
taken as a unit of length, the distance
from the left end the scale
and
1
in a straight groove
on the
scale
number x between the scale
moves
set
the fixed scale,
the slide scale will correspond to
mark on
the fixed scale if
exceed 10. Otherwise the x 2
logX,
x^ does not
mark on the
sliding
be beyond the scale of the frame, one then sets the 10 mark on the that indicating slide to correspond to the xl mark on the frame scale will
(see Fig. Y.2.B.2).
scale has value
1,
Since the total length of the
this
corresponds to a displace
ment by 1 to the left of the x2 mark, and, the *2 mark is now at a distance log x^z log
(x^/lO) from
the
1
hence, 1
mark on the frame scale.
Division can be obtained by setting the slide scale value of the divisor (x^ to the frame scale
Fig.V.2.B.2
Another form of the
slide rule is the circular
slide rule.
For the log scale, the circumference of
a circle
assigned the value
is
1,
and a point
is
marked L The arclength between 1 and a point marked x measured clockwise then has value log x and multiplication corresponds angles.
The advantage of this
to the addition of device,
compared
V.2.C.
to the linear slide rule,
Square and cube
Two more
going off scale
is
more compact
in
on a
is
the cylindrical slide rule which
mounted on
is
mounted on
and can
also
the fixed
move parallel
be rotated around it
by an angle which assumes one of a discrete
The fundamental logarithm
set
scale
of is
divided into k equal parts and inscribed on k equally spaced
The
perpendicular
The mark
to
on a
for the point to be plotted
on a piece mounted on
second lead
this
screw.
This type of device can be readily set up for to position It is also possible
manual operation.
lead screws by servo motors.
A
linear
poten
tiometer can be used to measure the total rotation
of a linear scale of length k
and produce a feedback signal for comparison a mechanical input must purposes. However,
effect is that
The is
counter. is
this sliding piece
in order to give the ordinate, again
first
elements of each of these cylinders.
times that of a scale length used on the cylinder element.
by
sliding
by a counter giving the value of the abscissa in a linear scale. Finally, another lead screw can be
the
cylinder can
motion corre
linear
arrangement on a lead screw, the total rotation of which can be measured this
mounting
on a
The second
The
sponding to the abscissa can be obtained
scale
principle.
333
this relation.
One
"folded scale"
to the fixed one
values.
rule.
can be readily provided.
scales
cylinder which
cylinder.
slide
maintain
fixed cylinder, the second scale
based on a
moving
is
elaborate slide rule devices have
been used. One
is set
that
diameter than the linear
over-all
is
is
Also, the device
eliminated.
PLOTTING DEVICES
motion of the movable
linear
twice the element scale length, and
cylinder rotations are integral multiples of 360/fc.
its
The
have a fixed position and, hence, both ordinate and abscissa must be determined, say, by a lead screw in a fixed position. If each of these perpen
scale relative to the other is displacement of one obtained by rotating through one of these given of the motion and then for the
dicular lead screws carry a slot, they will properly
a linear displacement corresponding
ways: determined by a lead screw which is mounted on a table which can rotate to represent the angular
larger part
angles
performing
to a fraction less than
does not go off
by a factor of
7 times the k
The
scale.
fc.
full scale.
precision
is
One
increased
Another elaborate device uses
on
logarithmic scales Cappelen, Ref. V.2.d).
The logarithmic
steel
tapes (see
Meyer Zu has the
How advantage of portability and cheapness. must one and is in care ever, required,
setting
obtain the position of the decimal point in the
For straight multiplication, the desk calculator has
is
if
required, accuracy obvious advantages in both entry and read out,
but for taking roots and other specific logarithmic operations involving exponentials, is
coordinate.
better if the accuracy
is
the slide rule
acceptable.
V.2.C. Plotting Devices of a function y=f(x) in
cartesian coordinates, one
pair
may
use effectively a
of ruled edges with scales which are at right
other. One edge must be parallel angles to each to the x axis, and the combination of rulers may
be mounted so as to
slide in
a groove in order to
The
table is fixed
and the lead
bevel gear connection to an axle perpendicular to at the origin. However, of the the this axle
graph
must be rotated an amount which is the
sum of the
desired rotation of the lead screw
the angle through which the table
The customary
oscillograph
is
and
rotated.
is
obtained by
moving the paper to vary the abscissa
and rotating
a pen arm to represent the ordinate. The main reason for this arrangement is, of course, its
mechanical simplicity.
The
exact location of a point
photograph a point.
For obtaining a plot
(2)
screw for the radial coordinate can be driven by a
plane
slide rule generally
result independently.
a pin to plot the point. Polar graphing can be accomplished in at least can be two (1) The radial coordinate
position
A
is
on a graph or
an analogous problem to graphing an
similar apparatus is used with
operator varying is obtained. point
the coordinates until the correct
The various forms of graphing
above can be used in this fashion. apparatus given This application is important in astronomy and in gunnery assessment.
device for measuring an ordinate which forms a part of an integrating device is described
A
ALGEBRAIC AND ELEMENTARY TRANSCENDENTAL OPERATIONS
334
by A. Galle,
[Ref. V.2.c, pp. 67-68].
Now let us suppose that we have a wheel which
The co
ordinates are measured by a roller wheel principle
which
of great importance in the theory of instruments. The principle would be described is
as follows. resting
Suppose we have a wheel of radius
r
on a paper with its plane perpendicular to
the plane of the paper (see Fig. V.2.C. 1). Suppose,
is
in contact with the
initially
parallel
x
to the
constrained to
x
axis with its axle
Suppose the wheel is a circle with center on the
axis.
move
in
x axis in such a way that the axle is always parallel x axis. For instance, one might have a bar
to the
center (see Fig. V.2.C.2). pivoted at the circle
The other end of
this
bar has a pivot in which
a [/-shaped yoke, which holds the axle of rod is rigidly attached to the yoke the wheel. there
is
A
which extends perpendicular to the axle of the wheel. This rod slips freely through a collar attached to another collar which
which
is
slides
on a rod along the
rigidly
are
collars
x
axis
the
x
axis.
axis.
Since the two
mutually perpendicular,
extension of the yoke the
jc
is
and the wheel
the
rod
always perpendicular to axle
is
always parallel to
Fig. V.2.C.1
then, that
amount
we shove the wheel across the paper an makes an angle a
s in a direction which
with the axle of the wheel. During this displace ment the axle remains parallel to its original position.
The component of
parallel to the axle will induce
wheel turns
freely,
the displacement
no rotation.
there will be
perpendicular to the axle.
no
If the
slipping
Consequently, the
wheel will turn through an angle y such that
ry
= s sin a.
This formula generalizes readily to the case in which the wheel is displaced so that the point of contact moves along an arc C in such a fashion that a, the angle between the tangent and the
a Riemann integrable function of the arc length. For instance, if a is continuous except axle, is
possibly at a
finite
a right and left limit, it satisfies this condition. Since a is Riemann integrable, one can show that sin
a
Thus
is
if
also a
we
Fig.V.2.C.2
number of points where it has
Riemann
integrable function of
s.
consider the motion as a limit of
polygonal motions on sets of chords, we obtain that the wheel will turn through an angle such that
If the wheel
is
= y, where r
evident
=
sin
a ds
is
is
the radius of the wheel. This
from the above
shads
counter
=
moved from its original position
r
y>
r
is
of contact on the x axis to a point (x,y\ the wheel will turn such that through an angle
(see
Fig.
integral
V.2.C.3).
formula since Thus,
if
a
attached to measure the revolution of
such a wheel, the ordinate of a point can be
measured.
TRANSFORMATIONS OF THE PLANE
V.2.D.
335
V.2.D, Transformations of the Plane
The
sin
integral
a ds can be obtained by
We
Jc
means of a sphere and sphere
if
roller in contact
the axis of the roller
is
with the
parallel
to the
map
describe here various instruments which
a part of the xy plane on to
some
similar
a specified rule. These region according devices can, of course, be used as parts of drawing to
board instruments. In Section IIL2.F a device for inverting the circle was described. In plane relative to a given this case, if (r, 0) are the polar coordinates of the coordinates the point has
image
original point, (r
= a\
,0) where rr
The usual pantagraph will expand a part of a map a ring on a larger ring. Thus, in
plane and X-AXIS
Fig. V.2JD.1,
ABCD
is
a parallelogram formed
Fig. V.2.C.3
the sphere has (see Fig. V.2.C.4). Suppose a point of contact with the base plane which moves on a curve C. Then oc is the angle between
x axis
the tangent to
C and
desired integral
is
the axis of the roller.
proportional
rotation of the roller.
to the
The
amount of
It is desirable to
permit the
Fig. V.2.D.1
by
links pivoted at the vertices,
extension of
DC. The point A
and
Q
is fixed,
is
on an
and
P is
= CQIQD = (say) determined so that CP\DA
fc.
P is on the line AQ, device determines a mapping in which Q
Because of the latter relation,
and the
corresponds
kAP
radius and, thus, a ring of inner
and outer radius
AB + BP is uniformly expanded
on to a ring with inner radius outer radius Fig. V.2.C.4
to the rule/ stein
provided
measure
to the
oty
x
axis.
Its
rotation
is
then a
A roller parallel to the y axis will
abscissa. give the
DQ +
DQ
AD
and
AD.
A device which transforms the plane according
Means roller to displace itself along its axis. to insure that the roller remains must be parallel
AQ = AB - BP
with P. For this mapping
again
and
=y,x =
fodsgivenbyF.FreudenHere
P. Calcaterra (see Ref. V.2.b).
ABCD
is
a hinged parallelogram, but one mounts A and C so
instead of fixing A,
that they slide in a groove
on the y
axis (see
ALGEBRAIC AND ELEMENTARY TRANSCENDENTAL OPERATIONS
336
Fig. V.2.D.2). Thus, the diagonal
the
y
axis,
AC lies
is
on the extension of CB.
Q
image point by these means or similar positioning methods.
(say) 8 is
a point
mounted on the cross bar EF with pivots at E and
F=4
and PF=Fg. Let
parallel to the ^ axis.
One readily sees that
One must have I7F be
independently generate the y or x coordinate of a point and produce the desired position of the
= L ACB = i/ DC P
along
and
= = /. [7/2, and, thus, since PF = FQ,
One
particularly important application is the
new y coordinate
case in which the
specified function of y.
is
/
apparatus to produce multiplied by a constant, and
relatively simple
power of y
to be a
possible to set
It is
up
as a
one
if
wishes to elaborate the setup by the use of linear
polynomials of7 can be constructed. In the instruments described in this part,
differentials,
certain
trigonometric relations are effectively used to produce powers or basic polynomials
from which powers or other polynomials can be constructed. The principle can be illustrated by the
following relatively simple example. In Fig. V.2.D.3, we have a linked parallelo
gram
which
in
AB is constrained to
x
that
UF
is
perpendicular to
coordinates x, y, and let
Then,
clearly,
Pg.
Let
P
g have coordinates *
/=j, x=-PCsin0,
(CF+Fg)sin0,
have
and, hence,
*
=
x
,/. 7
=
-foe, for
In the reference given, the above two devices are combined to produce a quite arbitrary change
of scale. In drawing board instruments, roller-type wheels with broad threads are used to maintain
motion in one
axis,
and the vertex
along the rod AP, a rod of length a,
AD in such a way that
P AD
The
is
a right angle.
is
= 2a.
/_EP A
x
y
then
axis,
EP A
= 2a.
If
to
insure
E is the projection of P
P AE
is
90
-2<x,
that
on the
and, hence,
Thus,
= a cos 2a =
a(l
- 2 sin
Clearly, then, if one has
a given
objective of this
is
parallelogram arrangement
2
= a - 2y ja 2
a)
an apparatus for rotating wa, one can
rigid member by an amount
produce
y
=
a cos
woe
or
/ = a sin woe
x axis can be used to insure that two pins have the same;; coordinates. Alternately, a sliding groove parallel to the;; axis can assure that two pins have
For even
the same abscissa.
sin HOC is
This means that one can
C slides
whose length is a. P A mounted permanently on
direction.
A sliding groove which remains parallel to the
along the
Fig. V.2.D.3
X-AXIS
Fig. V.2.D.2
lie
72,
cos
a can be expressed
order polynomial in sin
a,
such a polynomial.
as
while, for
an nth
odd
n,
V.2.E,
Of
course, gears can be used to produce the
desired rotation, but linkage arrangements can also produce such integral multiple rotations,
CURVES
337
position
y can be moved
is x,
x
to a point
f ,
/
according to the relation:
Consider, for example, Fig. V.2.D.4, in which the
-
x
=
y
= (x - a) sin
(x
fl
)
- (y -
cos
b) sin 8
(V.2.E.1)
b) cos 6
These relationships describe a translation by a vector {
Thus
b} followed by a rotation.
a,
three parameters describe the possible motions of limit such a lamina. Restraints can be put on to
two degrees. Thus if a point specified by requiring the lamina to pivot about this point, two coordinates are specified. the freedom to one or
Fig. V.2.D.4
line
is
segments other than the sides of a are equal.
The
indicated
are
relations
angular
readily
obtained either by isosceles triangles or by the fact that
an
alternate
interior
exterior angle
is
sum
the
Clearly,
angles.
of the
one
can
produce such multiple integrations by a linkage arrangement in which the vertices are permitted
That
is
to say,
two
one relation
is
specified.
Such a lamina with one restraint represents the simplest type of transformation device. We have one
restraint,
to slide along the sides of the angle a. It is interesting to notice that if
a is
specified
x
=
a cos
a,
and one can produce:
/=
F(a, b , 0)
by
the abscissa rather than by the ordinate, one has
=
(V.2.E.2)
and two equations, Eq. V.2.E.1, corresponding x y whose
to the coordinates of the point P,
reference position
a cos n arcos xja
is x,
y.
is
the nth Tchebychef polynomial
constant.
up
to a
This could be used as the basis for
devices similar to the harmonic analyzer in which,
however,
Tchebychef polynomials
are
used
instead of sines or cosines.
V.2.E. General Theory and the Construction of
Curves
we were
concerned
with special transformations of the plane on to means of an instrument. However, one itself
by
can consider certain aspects of these instruments
The computational processes in these
machines are obtained by using a combination of each of which can be considered as a
pieces
lamina whose base moves in a plane
parallel
to a
A
lamina moving
freely in
a plane
has three
to the possibili degrees of freedom corresponding ties for a rigid displacement in the plane. Thus, a
point whose coordinates
in a specified reference
not zero.
a, b,
and B
The image point
if the
is
also
degree of freedom
is
the
available,
possible
images of a point is a curve. In general, the transforming devices will consist of a number of pieces, say n. If one has 1
restraints, positioning
one point
will
give
two more equations, so that the positions of
all
However, the Jacobian
are determined.
pieces
of this system of 3n equations relative to the 3n must not be zero; otherwise
position parameters
one would have a locking
situation.
A transformation 3n
1
restraints.
of the plane, then requires 2 restraints If one has 3
-
and varies the remaining degree of freedom, each
on the apparatus
point
given plane.
is
given by a system of equations similar to Eq. V.2.E.1 with a different value of x, y. If one
3
In our previous discussion
in general.
Jacobian
9
These three equations
can be considered to determine
which
and the
relations are given,
motion has one degree of freedom. If the lamina has a linear slit which contains a fixed pin, only
curve.
draw
An
1
will
move on a
definite
Thus, such an apparatus can be used to
curves.
example
is
W. R. Crawford
given by a device described
by
(see Ref. V,2.a) for obtaining
ALGEBRAIC AND ELEMENTARY TRANSCENDENTAL OPERATIONS
338
The
conic sections.
device (see Fig. V.2.E.1)
mechanizes the relationship
coordinate,
section, a
r
= +
1
In
e cos
device a geometrical straight line
this
0,
and
distance r\
is
by a rigid rod, and geometrical relations, such as the coincidence of two lines or a pre-
realized
pen
and the length FP is the polar if one wishes to draw the conic
is
mounted
at
P also.
The geometrical relations are readily seen. Thus /_OFA = 6, and, consequently, AF = 2
FC 2 = BF
-
b
Thus, p
BCP
yields the relation
+ a cos 0) = c
FP, or r(b
2
and, hence,
+ a cos
= c^jb and e = ajb.
If
b
is fixed,
a and c
can be adjusted to give the desired value of/? and e. The angle 6 can range from a little more than
minus ninety degrees to a little less than plus ninety. These limits are set by the interference of
A and F. In addition to geometrical restraints in the usual
sense,
restraints.
one can also have
Thus,
if a
differential
piece has a sharp wheel, the
point corresponding to the mid-axle of the wheel
can move instantaneously only in the direction of the sharp edge of the wheel. This is a differen
Fig, V.2.E.1
scribed angle between them,
is
obtained by using
sliding sleeves attached to each other.
the lines Fig. V.2.E.1, at P,
FP and PC
tial
relationship involving
a function
M of the
Thus, in
are coincident
owing to the sliding sleeves which can turn
On the other hand, BCP is maintained as a right angle by
freely relative to each other.
the angle
means of two
fixed
sleeves
whose axes are
In this apparatus the points
perpendicular,
and
F are fixed,
rod
AP to
and there
is
a sleeve at
which the fixed arm
such a manner that
FC is
FC is
F on the
attached in
perpendicular to AP.
at C can pivot OA rotates freely around and carries the sleeve AB on the rod AP. The sleeve AB can rotate freely around A] this sleeve is attached to another sleeve at B which slides on the rod BC. The position of C on FC determines the length c; that of A on OA the length a\ and that of B on AB the length b.
The combination of
relative
to FC.
These lengths,
sleeves
The arm
a, b,
and
c,
can be adjusted to
yield different conic sections.
The
extension of OF,
is
at F.
FD,
The angle
position
and parameters dy
is
the reference
DFP
is
the polar
= M dx.
If the
apparatus permits only one degree of freedom, these
restraints
differential
radius for the polar coordinate axis system. The origin
Fig. V.2.E.2
dom
are
are
equivalent
equations, but
permitted,
a
if
to
ordinary
two degrees of free
much more complex
situation arises involving partial derivatives integrability conditions.
and
CURVES
V.2.E.
An example of a device subject to the differen type of restraint is given by that for drawing the evolute to an arbitrary curve. rod has a tial
A
wheel mounted
339
Devices of considerable complexity can be constructed by combining the various trans formations or restraints discussed in the present
one end, A, with axle perpen dicular to rod length AB. Another wheel is
the more com chapter. It is also true that most of into a resolved can be of this sort devices plicated
mounted on a
sequence of pieces which produce individual transformations of the type described above.
at
sleeve at P.
around the rod
AP
This wheel revolves
so that instantaneously
always moves perpendicular to AP. That is
the normal to the locus of P. Thus,
if the
is,
P
AP
wheel
A is steered so as to be tangent to a given curve (, P will describe the evolute to (L (In the next
References for Chapter 2
at
chapter we will discuss methods of insuring that the plane of the wheel A projects onto a tangent (L)
A pen mounted on P will draw the required
evolute.
If the curve
(
is
to be a
circle,
a.
R. Crawford,
general conic 210. p. b.
R
"The
mechanical construction of the 162 (1936), Engineer, Vol.
section,"
Freudenstein
instrument
one can
for
and
P.
Calcaterra, "Tracer-type in two mutually
changing scales
Sci. perpendicular directions," Rev. no. 9 (1955), pp. 866-69.
at replace the wheel A by a rigid member pivoted the center of the circle. The last arrangement is
c.
the shape of particularly valuable for drawing
d.
gear teeth (see Fig. V.2.E.2).
W.
Instr.,
Vol. 26,
A. Galle. Mathematische Instrumente. Leipzig, E.G. Teubner, 1912. W. Meyer Zur Capellen. Mathematische Instrumente, 1944. Leipzig, AJkademishce Verlagsgeschellschaft,
3
Chapter
INSTRUMENTS FOR THE DIFFERENTIAL AND INTEGRAL CALCULUS
V.3.A. Differentiators prism.
In the previous section we have indicated a method of following a curve by a wheel whose plane projects onto the tangent to the curve. The use of a sharp-edged wheel insures that the motion of the wheel
along the curve, but the wheel must be steered to maintain the tangential relationship. is
A point P on the lower face is
by the intersection of two
lightly
indicated
etched
lines.
If
not parallel to the tangent to the curve at P, the curve will appear to consist of the prism edge e
is
two mutually displaced curves. Thus the slope of the curve can be obtained by rotating the prism until the curves
appear to meet.
Instead of the sharp-edged wheel, one might use two rollers with a broad thread to maintain the desired instantaneous motion, with the axle of the rollers projecting curve.
The
common
on the normal of the
objective of the latter arrangement
would maintain
this situation
with the midpoint
of the axle over the curve.
However, one must be able to judge whether the proper normal relationship
is
A
present.
mirror whose plane is normal to a curve will give a reflection of the curve that appears to continue the curve with a continuously turning tangent, but if the mirror is not normal to the curve, the reflection will
make an angle with the
original.
A glass prism also offers
a remarkably simple and effective manner for determining the slope of
a curve. Consider a prism with an isosceles base
Fig. V.3.A.2
Let us consider the appearance of straight lines
through the two upper faces of the prism when viewed directly from above. Under these circum stances, the plane determined by a line of sight normal to the face at the point of intersection is perpendicular to the edge of the prism between
the upper faces.
The laws of refraction permit us
to consider only this plane (see Fig. V.3.A.2). Let the intersection of this with the bottom
plane
face be the
x
direction with
origin under the
upper edge of the prism, as in Fig. V.3.A.3. If
g2
an actual point on the curve in the lower face plane, the path of a light ray from Q 2 to the
is
Fig,
observer will be Q^FG.
V.3.A1
However, the apparent be Q^. If a is the angle FG makes with the normal to the face at F, and is /5 position of
and
let
e denote the
equal angles a of V.3.A.1).
The
common
side of the
two
base triangle (see Fig, rectangular face of the prism this
corresponding to the edge e is laid on the curve, latter is viewed from above through the
the
g2 will
angle between
Q 2 F and
the normal, the law
of refraction requires : sin
a (V.3.A.1)
and the
sin/5
V.3.B.
INTEGRATION AIDS
341
* Then:
when If
m=
we
0,
are
dealing with a curve rather than
straight lines, the
(V.3A2)
x
AP
axis
can be
etched on the bottom face of the
images of the curve around correct
position
is
The proper determined by with
a
tangent
?
and
axis is
jc
P".
position for such a prism
screw or a
similar
This produces the angle the desired
may
be
reduction oc
whose
slope m, and various
m
geometrical methods for able,
Two
appear, and the
turning the mounting arrangement
worm
is
P
carefully
prism.
one in which the
tangent to both images at
apparatus.
is
are avail obtaining Willers (Ref. V.3.e, pp. 174-76) describes
an apparatus manufactured by the firm A. Ott for
Now AO is along the x axis, and the
Thus
origin.
if
x2
= -Q
corresponds to is
Z
the abscissa of
the actual position of the point, and x1 is that of the apparent position, and then: a
+ x = (a + x^l + tan a tan u) 2
Now
let
we
First,
drawing the derivative curve from a given
=
-Q-fl
AO =
a,
(V.3. A.3)
us consider a line in the base
plane.
introduce a coordinate system with
origin at P and;; axis under the upper edge of the prism (see Fig. V.3.A.4). Let us consider a line
through
P
If
Let
(actual).
position of the
AC = b,
CD
denote the actual
line,
and
AP
= a,
the
equation of the
Fig. V.3.A.4
actual line can be written:
J 2 -f b
Let
= m(x + 2
a)
(V.3.A.4)
x^ denote the apparent position of the point as viewed
(x%y^
Now
prism.
through the
since the
perpendicular to the y
left
face of the
plane of the light rays
axis,
j2
=y
lt
is
Eq. V.3.A.3
yields the equation for the apparent position of
the line:
yl
V.3.B. Integration Aids If
the
one can guide a wheel to
along a curve, registered
counter will yield the length of the curve.
on a
One
may also want to
obtain the area enclosed by the curve or draw the integral of a function whose
graph
+ b=m(l + tan a tan w)(^ + a)
roll
amount of turn of the wheel as
these
is
given. There are instruments for each of
purposes which we will consider
the present section, however,
we
later.
In
will consider
(V.3.A.5)
simple aids to integration.
Thus, the apparent slope m^
m1 =
(1
is
given by:
+ tan a tan u)m
(V.3. A.6)
Perhaps the simplest of such devices of glass ruled into squares.
One places
is
a piece
the glass
DIFFERENTIAL
342
AND INTEGRAL CALCULUS
on the area to be measured and counts the
to obtain the area originally used as a planimeter
number of squares which
enclosed by a curve, having been superseded by
lie
wholly within the
area and estimates the remaining area around the
the mechanically simpler polar planimeters.
A
boundary.
The "harp planimeter" is designed to assist one in
variable
speed drive can be utilized to
evaluate the integral
forming a sum:
f
ydx
J(L
One has a
number of threads strung
large
on a frame. To
parallel
given curve
we
find the area
in
under a
so that the threads lay the frame
are perpendicular to the
x
axis.
A
compass is used to measure the ordinates of the points on the
midway between two
curve
and counter arrangement on the compass
or by laying the ordinates off along a straight line. If in the latter case the total length
we may length
is
use a fixed length
/
excessive, this
we may shift back by /. Thus, k coordinates have a sum which exceeds
we take another compass
back
is
and whenever
exceeded,
if the first /,
/,
from the
final
or ruler and measure
point of the sum.
We may
then continue to lay off the ordinates using the point obtained by shifting back, as the starting
The shifting must be taken into
point.
considera
tion in the final answer.
Instead of the threads and compass, a glass slide
may
be
fitted in the
frame and used to
measure the ordinates. Marks along the side of the slide parallel to the x axis can be used to show
where the ordinates are to be taken. The
moves parallel to is
on the
that the ordinate of the point traced
mark
The displacement of the
slide
is
the linear
or rate input of the variable speed drive, abscissa
This can be done in a
which
and the
the rotary or disk input.
is
number of ways. We rimmed wheels
a carriage on broad
may have
rolls across the
paper parallel to the x
axis.
The rotation of the wheels then yields the abscissa. The ordinate
is
entered by means of an extension
of variable length which remains parallel to thej axis.
A
large variety of such devices can be
found
A. Galle and H. de Morin (Ref. V.3.a and
d).
Historically these devices are important since
they led to the development of the variable speed drive
itself.
These variable speed drives have one advantage in that they permit one to obtain the integral of the product of
two functions
in a reasonably This follows from the
straightforward fashion. fact that
one can use the output of one such
device as the rotatory or disk input of another.
slide
the y axis until the proper
curve.
attached to the variable speed drive in such a way
The sum of
threads.
the ordinates of these points can be obtained by a ratchet
from a graph in an obvious fashion. One has a pointer which traces the curve, and this is
we wish
Thus,
if
we
up a
to obtain the
integral
measures the ordinate.
Of course, an ordinary adding machine can be used in conjunction with a formula for numerical integration for obtaining areas. This is particu larly valuable in the case in which the function
set
pair of integrators to represent
2=
given in the form of a table rather than a graph. For a discussion of numerical integration, the reader
is
f(x)dy J
XQ
is
referred to C. Jordan, Ref. V.3.C, pp.
-c
g(x)dx
and this is readily realized by a
simple mechanical
512-27.
differential
V.3.C. Variable Speed Drive Devices
V.3.D. Area-Measuring Devices
We have considered the variable speed drive as a
component
analyzer.
in
the
However,
mechanical
this
differential
type of device was
One is
old
(V.3.C.2)
analyzer setup.
method
to cut out a
for the evaluation of an area
replica
from some material of
uniform density and thickness and to weigh the
AREA-MEASURING DEVICES
V.3.D.
A modern method is by making a mask,
result.
from which
the desired area
measuring the
is
total illumination
which the linear
g(x), throuj-h
cut out, and then
which passes
visible
from
light source
343
this
point.
which
The
light
source
is
actual length of the
visible is
is
through such a mask by means of a phototube.
With
less delicate
photosensitive methods, the be concentrated by means of a lens onto the tube itself, but, in general, it is entire light
may
disperse the light in a cavity and to determine the general level of illumination. Of
course, in the latter case, the entire light should be
concentrated
so that
first
it
will enter the
cavity
through a small aperture. Since the output of a phototube
answer is obtained
intensity
is
in general
if
we
ignore the slight variation in
due to distance, we see that the illumina
on the outer cylinder at any point on the x plane is the same and proportional to g(x). The
tion
f(x) mask, of course, permits a fraction of proportional
to/0)
to
this
pass.
not a linear
function of the illumination falling on
One
Consequently,
to
preferable
it,
the
by a bridge method.
has, besides the above, a duplicate arrange
ment
associated with another phototube with an
adjustable shutter instead of the mask.
One
until the output of the two adjusts the shutter
tubes are identical.
Then
the opening of the
shutter will have the same area as the hole in the
mask. This method of integration "cinema
H.
integraph"
is
used in the
L Hazen and G. S. Brown (Ref. V.3.b) which
also gives a historical account of the development
The cinema integraph evaluate quantities in the form
of the instrument. designed to
Fig. V.3.D.1
described in the paper of
For the general case
in
vary in sign, allowance
which f(x) and g(x)
is
made
is
possibilities
and g_
in
signs.
Let g+
and
(V.3.D.1)
for the four
max
\g(x), 0]
= max [-#(*), 0]; /+ and/_ are defined
there are similarly. In the device,
f(xy)g(x)dx
=
whose balance
two phototubes
indicates the result.
The integrals
similar quantities.
method is used to ingenious obtain the product. Let us consider the case of
The following
two
functions
positive
V.3.D.1).
Masks
but with different x linear,
scales.
common
and g(x)
f-g.dx
(see Fig.
The source of light
is
and f(x) are around concentric cylinders whose
and the masks
wrapped
/ (x)
are cut out for each function
axis
is
be on the inner
for g(x)
the linear light source; g(x) will cylinder.
The
ordinates on the
mask run along the elements of the cylinder while to the elements. the x axis is perpendicular
Now we
consider a value of x. Corresponding to x, have a plane containing the linear light source. Now let us consider a point P on the outer
The mask on the inner this plane. cylinder in is such that there is an opening of height cylinder
are obtained as indicated above, but the illumina first two goes to one phototube, the second tube. A to two other the of that
tion
from the
also enters the biasing light
first
tube so that
it is
to balance the arrangement by always possible illumination on the second tube. The positive shutter is controlled by a servo motor.
balancing
DIFFERENTIAL AND INTEGRAL CALCULUS
344
Of
course, one
may
readily rotate the outer
cylinder to obtain the integrals of Eq. V.3.D.I.
When
one
is
through
this
will give a
g(x)
mask and
the
mask
for f(x)
measure of
dealing with a situation in which
The above, of course, refers to positive /(jc) and One may add a constant to/(f) to obtain
g(x).
the positive result, but relative to g(x) one find
it
more convenient
to
may
separately integrate
g+ and g_. This procedure is effective when dealing with Fourier coefficients or Fourier
relative to
transforms or for integrals in the form of Eq. V.3.D.I. References for Chapter 3 Fig. V.3.D.2
a.
b.
used repeatedly, one can use a simpler procedure in which /(x) can be repre sented by a plane mask corresponding to its
the function g(x)
is
graph. The function g(x) is to be represented by the density of lines parallel to the y axis. (See Fig. V.3.D.2.) If these lines are transparent
mask and
integral,"
Frankl. Inst. Jn., Vol. 230 (1940), pp. 19-44,
183-205.
C. Jordan. Calculus of Finite Differences. Budapest, Eggenberger, 1939. Pages 512-27. d. H. de Morin. Les d Paris, c.
Appareils
Gauthier
on a
interspaces are not, the illumination
A. Galle. Mathematische Instrumente. Leipzig, B. G. Teubner, 1912. H. L. Hazen and G. S. Brown, "The cinema integraph: A machine for evaluating a parametric product
e.
F.
A.
Integration.
Villars, 1913.
WUlers.
Instrumente. Berlin,
Mathematische
Akademie
Maschinen
Verlag, 1951,
und
Chapter 4
PLANIMETERS
V.4.A. Basic Principles
A
Although
defined as a device for is planimeter the area under a curve. Modern
measuring
instruments made for this purpose are remarkably the mechanical point of view because simple from of the ingenious mathematical theory on which based. In this section we give the basic they are geometrical
discussions,
it is
we
not customary in mathematical in traversing a suppose that
will
the arc length is increasing simple closed curve, as the point moves in a clockwise direction. Now let us consider a simple closed rectifiable
curve (see Fig. V.4.A.2) and
on
it.
successive vertices
constructions.
let
us take n points
These can be chosen so that they are the of a polygon whose area
the area enclosed by the rectifiable approximates curve.
Y-AXIS
The area of the polygon has been proven to be given by
If we let
Eq.VAA.l.
where these appear in
this
formula,
we
A = lSJBl (y,Ax,-x,Aft) X-AXIS
If
we pass
to the limit,
we
dx-xdy) a formula which, of course, first
polygon.
polygon
area of a obtain a formula for the
is
well
(V.4.A.3)
known but
derive in order to establish the sign.
that the seen, for instance,
It is easily
shown
which we
(V.4.A.2)
obtain
Fig. V.4.A.1
We
obtain
in Fig. V.4.A.1 has area
75*6
In general,
2A
=
*
we
see that the
1
SjVi (ft*m
-
Vm) + ?n*i ~ x ^i
r * ,^
, po,
Fig. V.4.A.2
formula
a line segment consider the area swept out by This is an essential a
M -i
(V.4.A.1)
of length
/movingin
plane.
^T;::rw4;"
PLANIMETERS
346
that the motion
we suppose
is
continuous and smooth. Thus,
that each point describes an arc with
a continuously turning tangent. sufficient for this if just
line
segment move in
two
this
We first discuss the
(It
would be
distinct points
on the
manner.)
question of the definition
if
had moved upwards, the area
the line segment
would be
negative.
the
second
In
the
case
situation
not
is
immediately clear since in general, points not in the two triangles shown may be covered in the
X be the point on the
Let
motion.
of this area and, in particular, the matter of sign. Suppose we have a directed line segment QP
the initial
and
intersect.
Suppose
which moves to a new position QP (see Fig. V.4.A.3). We suppose that this motion is small,
on the it is
initial
X
segment.
Now A
X will is
the initial segment, and
the
be
image of B,
PXP
QA BA
or
Then B\
in
AP
In the
.
moving
to
its
total angle
while in the second case,
smaller rotation
is
would
it
PXQ. Now
AP
the
moves onto
common
QB
points.
is
if
A
is
on XP,
f
is
XQ
on
moves onto
common
Thus,
possible.
we may suppose B
motion
1
of
generality
the former.
either
i.e.,
QB
or
supposed to be small, hence, only the
is
segment have no
BP
essential
turn through an angle equal to
motion
segment,
on
would turn through a
position
equal to
on
initial
no
the line segment
first case,
new
is
A B when
be called
either
we assume
lost if
segment
the image of a point
is
considered to be on the
X=A = B. will
plane where
final position of the line
Hence, the line
.
XP = A P likewise,
points;
r
and these >
QX
QB
and these two also have no
Now we
suppose that the is not greatly
so slight that the area
by using the line segments AA or BB as boundaries instead of the actual paths. But then
altered
it is
Fig. V.4.A.3
clear that
we can regard the area swept out by
AP and QB as
in the
first case.
Since only a small motion
which
view of the fact that any motion of the postulated sort can be considered justifiable in
is
as the consequence of a number of small motions.
There are of these, the
QP;
two
essentially
new position
in the second case,
the area
is
QP
it
readily defined.
Q
and
first
final
position of the
segment enclose an area which
as negligible.
is
given by
]
is
small
Let us then arbitrarily assign to
it
-[,P r[-Q ]
(ydx
-x
W
(V.4.A.5)
For the area swept out by
specifies the sign
of the area.
An area such as that shown will be positive, while
AP and QB, we have,
of course, the previous formulas. If we add these expressions for the area,
we
find that
have exactly the same formula for kA. case,
it is
parts,
(V.4.A.4)
(ydx-xdy)
JAA JWA BA >/-/-
three
This formula clearly
= BA
the value
case,
the formula
QP
considered, the
and hence the area swept out can be considered
The point P describes an arc QQ, and these
together with the line
In one
is
A A = XA
does not intersect
does. In the
an arc PP and the point initial
possibilities.
actual displacement
f\
latter is
clear that the area
and
77*.
is
we
In this
broken up into two
The former
is
positive, the
negative.
Let us take now a motion which is not restricted in size and, as
we have
suggested, consider
it
as a
V.4.A.
number of
small motions to each of which the
A4
above formula for find that
Q
BASIC PRINCIPLES
Q
if
is
is
applicable.
the path of P
and
We
then
But is
it is
evident that the constant of integration
zero and, thus,
C2 the path of
A
then If the fixed
(ydx-xdy) QP
347
=
(VAA.ll)
length returns to
its
original position
we have:
without making a complete revolution
(V.4.A.6) If the large
returns to
A
motion
(ydx
sinarfs
such that the line segment
original position then
its
\
is
xdy)
(y
\
Let us
dx
now consider three
(VAA.12)
points, Q,
Plt P2
,
on
the fixed length (see Fig. V.4.A.5). Let us suppose
x dy)
Jc z
Jc?i
(VAA.7)
We
have described the integrating wheel
Section V.2.C. and the fact that
sin
where
C
it
ads
(V.4.A.8)
the curve traced out
is
contact of the wheel and a
is
in
registers
by point of
the angle between
the axle of the wheel and ds.
This expression
is
also associated with the area
the swept out by a fixed length along the axle of wheel. Consider a differential dA of the area (see Fig.
Let
V.4.A.4).
9?
denote
the
clockwise
Fig. V.4.A.5
that the fixed length
moves and returns to
Pl
its
out a path Q. original position. a path g, however, is constrained to move on either a line
way
that P!
cases for
Pl
Let
/!
Al
its
such a
original
lies
between
We will Q and P2 The other .
are treated in an entirely analogous
manner so we Let
circle in
has returned to
it
no area has been enclosed.
position,
suppose
segment or an arc of a
when
that
traces
not consider them further.
will
denote the length
QP
lt
/
2
the length Pi?8
denote the area enclosed by
Q.
(If
.
Pl
circumscribes the area in the usual sense a
number of times
Fig. V.4.A.4
in traversing
Q,
then
Al
is
a
the area as usually understood.) multiple of
rotation between the original position and the
present position.
It is readily
We first notice that by
Eq.
VAA.12,
seen that J
2
sin
a ds
=
area swept out by
P^
=
/J sinews
U
(VAA.13)
(V.4.A.9)
Thus,
the fixed length
if
position to
moves from one
2 * c^ Jc
another and C is the path of P we have
Thus the
A
=
I
I
sin
integral
a ds
is
independent of
Jo
r sin
a ds
+ J/V +
fc
(VAA.10)
the p OS jt i on O f the point
P on the line.
PLANIMETERS
348
On the
l
l\
si sin
other hand,
if /
= 2?2 we have
= area swept out by QP = A%
a ds
2
Jc (V.4.A.14)
The
last
two formulas
planimeters.
so that
it
are the basis of
many
The operator moves a pointer
at
integrating wheel
at
is
some other point
P1
The
circumscribes the desired area,
P along
The planimeters are classified point Q moves along a straight
the fixed length. as linear if the line or polar if
If
Q moves along the arc
one mounts the
fixed length
of a
circle.
Fig. V.4.B.2
on a carriage
with broad wheels or on a track in such a way that the fixed length
move
is
pivoted at g, then
in a straight line
and one has a
Q
will
linear
planimeter. If one connects the fixed length at
Q
by means of a hinge to an arm which itself is pivoted at a fixed point, we obtain a polar planimeter. It is
sible
desirable that the curve
C and the permis
2 be such that if we take any circle / = P 2 with center on a 2 point of C,
path of
of radius then this
circle will intersect the
path of
Q at only
one point. This will insure that when the operator returns the point
? to
traversing the curve,
Q
its
Fig. V.4.B.3
original position after
will return to its
original
and a horizontal wheel driven by the worm gear
position.
(see Fig. V.4.B.3).
V.4.B, Examples of Linear Planimeters
An
example of a linear planimeter is given in Fig. V.4.B.1, a polar planimeter in Fig. V.4.B.2. In
many
examples given in the references (see
Ref. V.4.a, b,
of a
and
c)
the counting device consists
worm gear mounted on the axle of the wheel
Thus, a small fraction of a
turn of the latter wheel corresponds to a full turn
of the actual integrating wheel,
Fractions of a
turn of the integrating wheel are obtained from a scale
mounted on
mounted
at
it.
Normally some device
If the linear
motion of
Q is obtained by a roller
rather than just by a guide roller
yields
is
P to help trace the given curve.
a
differential
in the integration.
rail,
the rotation of the
dx which can be used
Furthermore, by means of
one can amplify this dx so as to obtain a result with higher precision and, also,
suitable
gears,
one can minimize the amount of movement corresponding to the motion of the integrating wheel in the direction of its axle, which is not
supposed to register. Let us consider a plan view Fig. V.4.B.4 of such a roller device. The pointP follows the curve, and the two rollers
^ and
moves along the x
j? 2
axis.
insure that the point If our
Q
integration point
V.4.B.
is
x
6, then a axis
is
and the
the angle between the desired
integral
f
PQ
and the
is
the line.
For if we
349
B represent the foot of the A on to QW, and the
let
perpendicular from
a>
angular velocity of disk, the velocity of the disk
f
\ydx=*PQ i
EXAMPLES OF LINEAR PLANIMETERS
sinaJjc J
(V.4.B.1)
Here, of course, ds has become dx. The motion of the roller R2 is communicated to the
by gears
Fig, V.4.B.4
Fig.V.4.B,6
WisAWco (see Fig. V.4.B.5). If is the angle BA W then the component of this velocity along at
t
QW
is
(co*
ftAWv
= AB,
and
this last is
independent of W. Thus, the quantity registered by the counter on is the dt integral of
the integral wheel
ABk dx
kAQ
ABa>
sin
C The
Fig. V.4JB.5
the point fP (see Fig. V.4.B.5).
Now
the
component of motion of the disk Q Wh the same for any point along
along the line
I
sin
a dx
instantaneous component which does not
register is
disk with center at A, and the large horizontal is in contact with this disk at wheel integrating
~
a dx and, thus we obtain
BW,
and
if
the distance
QW
is
properly chosen, this can be kept relatively small.
The
rollers
can be replaced by gears moving on
racks.
A similar development of the polar planimeter is
possible (see Fig.
VAB.6). The rotation of the
PLANIMETERS
350
arm
QO
around
can be multiplied by gears so % is a multiple of the
that the rotation of the disk
motion ds of Q. That is to rate of
St,
CD
dt
= k ds.
say, If
if
W
o>
is
the angular
the point of
is
contact of the integrating wheel with
/_WQO
is
a, the angle
between
PQ
$1
then
Thus
the frame Fig. V.4.C.1,
in
which contains
the integrator rotates through twice the slope
angle a of PQ, and because of
its initial
orienta
tion will register the integral of sin (90
cos 2a.
2a)
=
Thus we obtain
and the f
=
tangent to the path of Q.
2
(l-2sin
ot)dx
J
VAC. The
Integrometers linear planimeter can
be elaborated to
etc.
Let us consider
produce ly* dx, $fdx,
= \dx-2\ sin
2
a dx
again Fig. V.4.B.4. In the device illustrated, we relation between PQ and as
QW
have shown the
If the curve described is closed, the first integral
when
vanish
will
curve
the
is
completely
circumscribed.
using an appropriate gear ratio, one can
By
QW arm to
cause the
reference position
If its
swing through 3 a.
perpendicular to QP, one
is
can obtain
f sin
f
3a da
J
f
3 sin a
4
da.
This will yield It is also
sin
3
a
da.
J
J
J/ dx provided one has
possible to set
We
to obtain ]y 112 dx.
dx. j}>
up a device of this sort must use a different
X-AXIS
Fig. V.4.C.1
a simple rigid connection which maintains as a right angle and hence
AQW = a.
were straight rather than a right
li
PQW AQW
angle, then
we
would obtain the integral of cos a instead of sin
<x.
However, we can replace the rigid connection between PQ and by a gear connection so that angle AQB represents 2a or 3a or na. This
Fig. V.4.C.2
QW
now
permits us to obtain the integral of certain
polynomials in y and perhaps more usefully the integrals
coordinate system to represent the curve so that the direction of the axis is reversed and y Q runs
along j
,
We jy*dx
or\y*dx
=
1
in the
also take
(V A.C.I) position
for
new system
g?= the
1.
(see Fig. V.4.C.2).
Then
setup
if
(i.e.,
the reference
a
=
0),
has
V.4.D.
WOP =
and
135
}in
(450
QW
_ 5b x =
\
swings through
sin (i(90
2/
J
=
U_
-
a))
i/ 2
cos (90
be
for the the polar coordinate inverse function on the the when slot of the W, point
fa
-a))
around a
=
equation
given curve,
r
we
the
slot, provided integrate shaping closed curve (see Fig. V .4.0.2)1*,
|ot,
willregister:
oudntegrator &
Jsin
if
351
GRAPHS IN POLAR COORDINATES
1/ 2
Astern
dx
is
taken as the origin of an
r,
9
of polar coordinates.
V.4.D. Graphs in Polar Coordinates If
and
r =/(0) one has polar coordinate graph a use can one relatively dti, wishes to obtain
JY
simple
arrangement
of an integrating
wheel
Fig.V.4.D.3
and F(r), obtain the relation between 0(r)
To
curve r =/(0) us consider a point on the the tangent to the draw and (see Fig. V.4.D.3) the line AP perpen curve at the point P, and also a neighboring point on dicular to OP. Let P be the ds. Let BP stand for
let
-
thecurve,andletPP
of the edge of the wheel at P. projection
Then
Fig.V.4.D.l
If
PC is perpendicular to BP, PC is the projection
and since of the axle of the wheel,
Let
0-Z.^PP r dti
,
then
= AP = PP
1
cos
jff
= ds cos ft
Fig. V.4.D.2
r^^rrpotrrr-" LTili.f.fc<^* rate of turn
For
show r
is
li!rf>l
.
we in explaining the pnncrple, simplicity as on the curve wheel the integrating
displaced
V.4.D.I. Actually
to permit
it
would he
sin
J
a ds
-
f
,
tf J^cos
-
s
$
the curve to be more readily
aid. followed by some optical to slot permits one the of Varying the shape in used where F is a function obtain
fW
*.-y
f
obviously rd6.
= f(e), in Fig.
TS^r^^r,*
iB
-
si
^
PLANIMETERS
352
The if
last integral is in the
the curve
value of
r,
form
J dH(r) so that
connects two points with the same the last integral is zero. This is true (
for closed curves. In any case, if 0(r) is specified it is simply a function of the values of r at the two
F(r)
= r j2a
r
2a cos
2
and we
(this
would
yield
areas),
then
the polar equation of the slot, have a circular slit. This is equivalent to a cp
is
form of the polar planimeter.
end points, which could be readily obtained from
4 References for Chapter
a table of H(r). Thus we can evaluate a.
A. Galle. Mathematische Instrumente. Leipzig, B. G. Teubner, 1912.
b.
if F(r)
$(/)
cos
when
<(r>.
F(r)
This can be used to determine is
given.
For example,
if
c.
W. Meyer Zur Capellen. Mathematische Instrumente. 1944. Leipzig, Akademische Verlagsgeschellschaft, Mathematische. Maschinen und Willers, F. A. Instrumente. Berlin, Akademie-Verlag, 1951.
Chapter 5
INTEGRAPHS
V.5.A. Introduction
In the present chapter we wish to describe An integraph is an briefly the integraph. instrument used to draw the graph of a function for which the derivative is given. In certain
modern developments, instrument for solving
The
principle
this
has
become an
that of the steering wheel
let
theoretical device (Fig. V.5.A.1).
suppose that the turning moments are counter some other manner.
acted in
essentially
set
up an
integrating device, then,
we must
steer this tricycle so that the line of the steering
us introduce a
wheel always makes an angle a with the x axis such that tan a =/(*), where/(j) is the function
This
whose
on a
describe the situation precisely,
is
moves, while the component of F perpendicular to the axle will cause the tricycle to move. We
To
differential equations,
of the integraph
counteracted by a friction force at the point of contact of the wheel and the plane on which it
tricycle,
is
To
similar
integral
is
desired.
V.5.B. Integraph Instruments
The above
principle
is
a applied in
ways which can be roughly
classified
number of under two
Fig. V.5.A.1
been to a tricycle except that the rear wheels have a pair of spherical ball bearings in replaced by sockets.
Now
if
applied
a force
F
which
to this tricycle in
is
not too great
is
a direction not perpen
the case of dicular to the front wheel, then as in
V.2.C the integrating wheel described in Section which to a in move path above, the tricycle will of F the front wheel is tangent. The component is to the axle of the wheel
which
is
parallel
headings.
One type is represented by the Conradi by A. (Me (Ref. V.S.b, We present a diagram, Fig. V.5.B.1,
instrument, described pp. 158-59).
looking at
it
from above.
INTEGRAPHS
354
The frame of the device consists of two parallel which are mounted on wheels
rails r,
frame moves
parallel to the
so that the
u and
Morin, Ref. V.5.c, pp. 136-41).
x axis. There are two
which move along r. One of these carriages, u, has an extension with
carriages,
displacement of the cylinder (see also H. de
v,
a pointer, P, which follows the given curve. On carriage we have a pivoted collar through
In Fig. V.5.C.1 the point derivative curve /(x).
rides a bar
of frame.
The
s.
other end of this bar
we
is
is
Since this fixed extension has length to the
x
f(x), the ordinate of the given curve,
y
see that the line of the bar s always has
slope
s.
On
s
we have another
contains a line
CD
s.
slide
but we show
carriage,
to
P
/>
one and the distance of the carriage u axis
given
a
is
which is on a fixed extension
pivoted at the point
traces the
maintained at a fixed distance before the carriage so that the slope of the wheel Wis essentially/ (jc).
this
which
P
Note that the point
side
is
a
which
which remains perpendicular
AB determines the
The
the wheel W.
ABCD,
direction of
carriage v has a pivot
which the mounting for arrangement the wheel slope as s,
actually
(it
part of a parallelogram
is
whose other
CD
q
as a collar)
it
W turns.
W always
upon
From
this
has the same
and the motion of Wis also the motion
W
of the carriage
i\ Since the direction of always has slope/(x), the point of contact of the wheel with the basic plane moves on a curve which is an
integral of/(jr).
The
essentially, since the
pencil
F traces this motion
upward displacement merely
changes the constant of integration. V.5.C. The Integraph of Abdank Abakanowicz
Notice that the arrangement of the bar s and is such that the slope of
the slide q in Fig. V.5.B.1 the line
This
is
CD
is
determined but not
essential since the
on the wheel Wi$ relative to the
its
position.
only permitted restraint
the determination of
its
slope
x axis.
There is one other solution, credited to Abdank
Abakanowicz,
in
which the desired connection
The mechanical arrangement this is
omitted from the
based on Fig. V.5.B. 1 would do. The front wheels of the carriage turn the cylinder d (which can be toothed like a gear), whose rotation is communi cated to the cylinder C. the wheel
W
9
Owing
to the
C rides up or down
in the
between the given curve and the slope of the wheel can be obtained (see Fig. V.5.C.1).
pressing against
Instead of having the steering wheel on the
Of course,
original plane,
we permit
it
to ride
on a cylinder
which is free to displace itself parallel to
for accomplishing diagram but a contrivance
as
a
it
turns.
C
will
on the carriage and record the motion of C.
the integral curve can wind around
the;; axis.
on the frame, and motion of wheel and
combined
so the desired relative is
C
number of times.
V.5.D. Steering Wheel Integrators
cylinder
carriage
A pencil F fixed
This cylinder turns at the same rate as the tracing point covers the x axis. However, the wheel is fixed
slope of
obtained by the
A number of steering wheel integrators can be equations.
into a device for
solving differential
A modern example of this is given by
V.5.D.
the device described by D.
This involves two
355
STEERING WHEEL INTEGRATORS
M. Myers (Ref. V.S.d).
integrating
wheels and
is
suitable for solving differential equations
where b and
c are constants
but a and d
may be dz
the variables *, functions of any one of
The two integraph wheels
z,
or
.
are connected to
realize the pair of equations:
(d-by-
Fig.
cz)
similar triangle principles.
dx
readily multiply by I/a.
of a by means
*L. dx
differential.
The connection
to
the integraph wheels
the parallelogram arrangement essentially
Conradi integraph. Consequently,
V.5.D1
z
and y
tongs"
(See Fig. V.5.D.I.)
is
References for Chapter
of the
=
"lazy
This permits one to
Addition is accomplished form of a linear
a.
^
B. Abdank-Abakanowicz. B. G. Teubner, 1889.
5
Die Integraphen. Leipzig, .
.
b.
Instrumente. Leipzig, B. A-Galle. Mathematische
c
H^Morm!
U
in the device.
are present as linear displacements to enter a as a function This permits one operator x from a one of these variables or of
Les Appareils ^integration.
*^ *
of either
graph. Similarly
The
d can be
multiplications
entered.
involved are based on
of the second order, 209-22. Vol. 16, no. 6 (1939), pp.
differential equations Inst.,
Paris,
Jour. Sci.
INDEX FOR VOLUME Abakanowicz, Abdank, 354 Abstraction, 224
II
Backlash, 7, 34, 144 Ball cage variable speed drive, 38, 187
Accuracy, 214, 216, 225 Accuracy of electrolytic tank measurements, 235 Acoustical analysis, 286
Band Band
Acoustical devices, 286 Acoustical systems, 273
Battery voltage adder, 76 Beam intensity control, 121
Adders, 178 Addition, 168
Beckman
Addition by resistance, 18 Addition of currents, 76
Bell Laboratories differential analyzer, 196
63, 66, 67
Bases, 26
Instruments, Inc., 195
Berry et al machine, 154 f errors, 199
f
Airplanes, 231 s stress
filter,
width, 63
Bell Laboratories, 187
Addition of voltages, 75 Airplane trainers, 188
Airy
pass
function, 226, 259, 266, 269
a errors, 199 7 a 204 ,
Alternating current bridge filter, 67 Alternating current generator, 104
factor, 89,
Bevel gear Bias, 82
95
differential, 10
Bias voltage, 82 Binary counter resistance multiplier, 119 Body force potential, 260
Body forces,
267, 268
Boeing, 189 Booster circuit, 189
Alternating current integration, 79 Altitude, 26
Boundary d,-, 261 Boundary conditions, 226,
Ammeter, 105 Amplification factor, 84
233, 234, 242, 250, 254, 255,
260, 265, 269, 310, 312, 316 Boundary errors, 239, 241
81 Amplifiers, 3, 37, 49, 52,
Amplifiers with potentiometers, 70 Amplitude modulation, 63
Analogies, 265
Boundary loads, 266, 267, 268 Boundary pressure, 268 Boundary stress, 260 Boundary value problems, 225,
and electrical Analogies between fluid quantities 242 quantities,
Bracket notation, 298 Branch impedance, 53
Analogous circuits, 273 Analogous network, 306
Branches, 48, 50
Analogs, 172, 223, 224, 225, 271 Analogy, 277
British differential analyzers, 187
Analog Analog
applications, 224 field
plotter,
Analogy between
236
Bridged-T
fluid flow
and
circuit,
233, 234, 245
68
S. L, harmonic analyzer, 169 Buckingham s theorem, 229
Brown, electrical current flow,
286
Analogy
table,
C voltage, 82 Cam shape, 29 Cam theory, 26
318
Analytic function, 237 Analyzer, 169, 257 Angle multiples, 350
Cams,
8
Antiaircraft fire control, 5
248 Capacity of an electrode system, Capstan principle, 45
Anticonformal map, 240 224 Applied mathematics, Arc length, 345
Cascading potentiometers, 70 Cathode, 82 Cathode-ray multipliers, 120
Area, 341 Area of polygon, 345
Area swept out by a Area-measuring
line segment,
devices,
345
342
134 Cathode-ray tube function generator, 188 Cathode-ray type function generator, Cavities, 245, 249
Astronomy, 333 Automatic adjusting machines, 144
Central timing component, 193
Automatic airplane pilots, 8 Automatic curve followers, 123, 125 Automatic drift compensation, 96
Chain, 51 Change of scale, 331, 336 Characteristic equation, 172
Automatic
pilots,
5
Automatic plotting of electron trajectories, 236 Automatic sequence digital computers, 212
Automatic transmissions, 5 Auto-transformer, 71
Characteristic equation of a matrix, 165 Characteristic root difficulties, 175
Characteristic roots by differential analyzers, 175 Characteristic roots by differential equations, 173 Characteristic roots A of a matrix, 172
INDEX
358
Characteristic vectors, 173, 253
Cos
Checking setup of analog computer, 212
Cosines, 168
amplifier, 97
Chopper direct current Cinema integraph, 343
x, 140
Coupling
CRC
circuits,
68
105, 181
216
Circle inverter, 25
Criterion,
Circuit analysis, 49 Circuit diagram for a transducer, 288
Cross-curl relationships, 226 Cube root scales, 332
Circuit elements, 48
Curl relations, 311 Current, 49
Circuit theory, 274, 278, 286 Circuit theory analogies, 48
Current density, 317 Current density vector, 237
Circuits, 48
Current generators, 60, 311 Cursor, 332 Curtiss Wright, 196
Circular arc paper, 331 Circular slide rule, 332 Circularly polarized light, 258
Closed loops, 276 Clutch and differential
Cycloidal gears, 30
Cyclone Symposium Reports, 196
multiplier, 15
Coding diagram, 188 Coding of an electromechanical
differential analyzer,
188 Coefficient boxes, 156
Commercially available Compatibility, 254
differential analyzers, 195
Complex circuit theory, 63 Complex impedance, 65 Complex logarithm, 172 Complex plane, 236 Complex values for current and
Dependence on parameters, 200 Derived quantities, 228
Descending part, 51 Design drawings, 329 voltages, 64
Components, 48, 223 Composition of meshes, 50 Compressance, 286 Computation, auxiliary digital, 6 Computer, 223 Computer Corporation of America, 195 Condenser coupling, 86 integration, 77
Conducting material, 233 Conducting paper automatic curve follower, 125 Conducting sheets, 233 Conduction of heat, 305 Conductivity, 237, 317 Conic sections, 338
Connection diagram, 273, 274, 276, 283, 294, 302, 305, 306, 307, 308, 309, 313, 314
Conradi instrument, 353 Constant current generator, 146 Constants for step-by-step integration, 214 Constants of integration, 184, 193, 211 slice,
Design parameters, 199 Design problems, 224 Design responsibility, 225 Design studies, 225 Designer, 225 Designing components, 204 Determinant of the system, 53 Device for drawing evolute, 339 Diagonalization process, 210
Diagram, 179, 180 Diaphragm compressance, 287 Dielectric, 317
Conditions of compatibility, 254 Conducting boundaries, 236
Constrained
Damping, 274 D Arcy slaw, 119 De Moivre s theorem, 165 Decimal point, 332 A, A(p), 53, 54, 204, 320 Density tensor, 305
Compatibility relations, 260 Complex analysis, 274
Condenser
A/z,
259, 266, 269
Difference quotients, 309 Differential, 7, 10 Differential analyzer, 123, 188, 198 Differential analyzer for characteristic roots, 175 Differential analyzer resolver, 127 Differential equation of membrane, 245 Differential equations, 178, 277, 338, 355 Differential equations of equilibrium, Differential equations of motion, 297
Differential Differential
254
equations of the system, 274 equations systems, 198
Differential restraints, 338 Differential selsyn, 110
Construction of curves, 337
Differentials, 3, 178
Continuous computer, 3, 6, 223, 225 Continuous harmonic analyzer, 169 Contour integrals, 226 Contours for integration, 93 Control computations, 225 Control of manufacturing processes, 5 Control purposes, use of continuous
Differentiator, direct current generator, 104
Convergence assumption, 201 Conversion factor, 271
Digital setting of potentiometers, 74, 196
Correction, 198 Correlation functions, 170
Dimensionally complete, 229
computer, 5
Differentiators, 340
Digital check solutions, 212 Digital computation, 225, 228 Digital computers, 311 Digital methods for voltage division, 74 Digital records, 331
Dimensional analysis, 223, 228, 229, 271
Diode function generator,
129, 188, 196
INDEX Diode Diode
359
limiters, 178
Electromagnetic analogy, 277
limiting circuit, 117
Electromagnetic analogy for membrane, 249 Electromagnetic fields, 245, 249
Direct analogy, 283 Direct coupling, 86
Electromagnetic transducers, 289 Electromagnetic waves, 255 Electromechanical analogies, 282, 288
Direct current differentiation, 78 Direct-current generator, 103 Direct-current integrator, 77 Direction of propagation, 255
Electromechanical analyzers, 188 Electromechanical differential analyzers, 178, 187, 196 Electromechanical polynomial equation solvers, 172 Electromechanical systems, 288
Directly coupled amplifier, 86 Directrices, 29
Disk input, 342 Disk integrator,
Electron optics, 245 38, 178
Electron trajectories, 245, 247 Electronic circuit resolver, 127
Disk-type variable speed drive, 330 Dislocation stresses, 268
Electronic computers, 4 Electronic continuous differential
Dislocations, 261, 266, 267, 268, 269
Displacements, 298 Dissipatance, 286
Electronic fire-control
Division, 332
Electrostatic
Division by reciprocals, 23 Division by servos, 22
Electrostatic fields, 235, 247 Electrostatic flux distribution, 305
Double
Electrostatic
analyzer, 4, 5, 188, 330
computer, 187
Electronics Associates, Inc., 195
refraction, 255
analogy, 283
problem, 310
Double-needle dipper, 238
Electrostatic transducer, 289, 290, 291
Drag forces, 230, 231 Drag functions, 225, 228 Drawing board instruments, 336
Elements, 26
126 Elliptic motion, Empirical information, 224 Endless chain adder, 8
Drift, causes of, 95
Drift compensation, 95 Drift stabilized operational amplifier, 195
Drum memory, 212 Drum type function
generator, 188
Equation
Dual of a nonplanar network, 284 Dynamometer, 119
Young s modulus, Eddy currents, 108 E,
254, 298
on
Elasticity analogs,
of
function, 226
254
coefficients,
Equipotential map, 234, 236 Equipotentials, 234,
235
Equivalent amplifier,
s stress
equilibrium,
Equations with constant
Equivalence
of voltage and current generators, 61 vacuum tube, 85
circuits for
procedure,
Error for open integration, 218 Error function, 140 Error in solution, 212
297
225
Electric analog computers, 292
digital
Error parameter, 199 Error problem, 199
Electric current function, 237 Electric displacement, 317
Error theory, 198,214,218 Error types, 199
Electric field E, 317 Electric field vector, 255 Electric motor, 105
Errors, 200, 204, 205, 206, 207, 218
network analogs, 225 Electric networks, 226
Errors in
Electric
membrane
analogies,
Evolute, 339
233
282 Exciting current, x, 140
Electrical analogies, 276
Exp
Electrical circuit, 277
Experimental approach,
Electrical interconnection, 187
Electrical network, 274, 306
network analogs, 305, 308 172
Electrical polynomial equation solvers, of a real symmetric matrix, 146 Electrical
representation
Electrical representation of functions, 127
Electrode system, 235 233 Electrolyte,
233 Electrolytic tanks, 225, 227,
224
224 Experimental verification, 255 Exponential function, 140, 333 Exponential operators, Extraneous roots, 204
Electrical computers, 3
Electroacoustic analogies, 286
251
Errors in rubber sheet models, 251
Electric potential, Electrical addition, 75
Electrical
208
84 Equivalent output generator, Error estimation procedure, 218 219 Error for closed
91
Elaborate slide rule devices, 333 Elastic constants, 266 Elasticity, 226, 252,
on Airy
Equations
Effect of feedback, 88 Effect of feedback
5 Engineering design, e,317 50 Equation of a mesh,
213 Extrapolated step,
Feedback, 88, 161
Feedback amplifier, 188 Feedback for differential analyzers, 194 Feedback solution of ordinary differential equations, 159
Feedback solution of
partial
differential equations, 161
INDEX
360 Filter sections, 67
Gravity forces, 268
Finite differences, 309, 311 Finite Fourier analysis, 165, 166
Gray, H. J., Jr., 216 Green s theorem, 305 Grid, 82
Fire control,
5,
125
Groove cam, 33
Fixed axis, 274, 276 Fixed purpose computers, 331
Guides, 245
Flexural rigidity, 269
Gunnery assessment, 333
Flow diagram for an Flow problems, 243
acoustical system, 287
Gunsights, 8
Gyroscope, 40
Fluid flow, 234 Fluid mechanics, 228, 231 Flutter, 225
Gyroscope differentiator, 41, 43 Gyroscope rate of turn indicator, 41
Flux, 279
Hamilton
Flying spot cathode-ray, 331 Folded scale principle, 333
Hand
Folding the Forces, 273
scale,
Harmonic Harmonic
72
Harp
Fourier representation of function, 165 Fourier series, 165
slice,
French
259, 260, 265
differential analyzers, 63,
196
Gas
recycling, 243
Gauss-Seidel convergence, 148, 151 Gauss-Seidel process, 144, 147, 155
Gear boxes, 144, 178 Gear connections, 7 Gear sets, 14 Gear teeth, 7 Gear train differential, 12 Generalized gradients, 161
Generator, function, 189
Generator impedance for a potentiometer, 70 Generator impedance for feedback, 89 Generator impedance of a T-pad, 72 Geometrical analogs, 224 Geometrical duality, 309, 314 Glass prism, 340
analyzers, 6, 125, 165, 169, 330 synthesis, 165
planimeter, 342
s stress
Higher order approximations, 323 Hindrance, 275, 284 Hold, 193
Hold and
Hooke
Frequency modulation, 331 Frequency modulation multiplier, 118 Frequency response, 202, 215 Froude number, 231 Function cams, 27 Function generators, 3, 4, 188, 196 Function input table, 331 Function space, 137 Function table, manual, 123 Function tables, 178, 180 Fundamental modes, 274
246
Helical potentiometers, 70 Helipot, 70
Fourier series function representation, 136 Fourier transform devices, 331
Fourier transforms, 165, 170, 255, 330, 334 Fourth-order partial differential equation on Airy function, 226, 260, 269
principle,
Heat conduction, 245 Heat conduction problems, 250 Height modulation of square wave, 113
Ford, H., 38 Fourier coefficients, 344
Free
s
adjusting machines, 144
s
reset, 101
law, 253, 260, 298
Horizontal connections, 294 Horizontal stress, 298
Hydraulic engineering, 231 Hypocycloidal resolver, 125 Ida, 196
Ideal diaphragms, 287 Ideal transformer, 278, 284, 315, 317
Impedance, 65, 275
Impedance Impedance Impedance Impedance
function, 239
matching, 60, 86 operator, 274
through a transformer, 60 Implicit systems of differential equations, 193 Impressed force connection, 276 Incompressible fluid flow, 305 Independent loops, 276 Index of refraction, 255, 256, 341 Inductance, 48
Inductance differentiator, 78 Inertance, 286
Inertance-capacitance analogy, 287 Inertance-inductance analogy, 287 Inertial connections, 276, 294 Initial position, 193 Initial state,
193
Initial values, 193, 199, 211,
Input impedance for an amplifier, 81
Glass ruled into squares, 341 Glass slide, 342
Inputs, 223
Goldberg-Brown linear equation solver, 149 Goodyear Aircraft Corporation, 195
Integral of the product of Integrals of powers, 350
Gradient generalization, 159
Graph, 329, 342, 351 Graphical integrator, 329 Graphical output, 330
226
Inner product, 137
Integral multiple rotations, 337
two functions, 342
Integraph, 353 Integrating amplifier, 101, 188 Integrating wheels, 334, 348, 351, 355
Integration aids, 341
INDEX
361
Integrators, 3, 4, 178, 188, 342
Low-cost
Integrometers, 329, 350 Interface, 311
Lusternik-Prokhorov device, 175
Internal impedance, 277 Internal impedance of a voltage generator, 57
Machine
Inverting device, 335 Involute gear teeth, 32
Maddida, 181 Magnetic
circuit, 280,
Involute gears, 31 Ion analogies, 243
Magnetic
field
Ion motion, 243
Magnetic
field resolver,
Isochromatics, 257
Magnetic
field vector,
187
setup, 189
Mach s number,
Magnetic
231
281
H, 317
field in
synchro units, 109 127
255
Magnetic flux distribution, 305
257
Isoclinics,
differential analyzer,
59
253 Isotropic elastic media,
Magnetic flux
253 Isotropic materials, Iterative procedure for characteristic values, 174
Magnetic force H, 59 Magnetic induction, 317
<,
317 Magnetic inductive capacity,
Jacobian, 194, 337 Joint use of digital
K-wound
and continuous computers, 6
Magnetic tapes, 331 223 Magnitudes,
20
potentiometer,
Mallock,R.R.M.,112, 145
Kinetic energy, 246 Kirchhoff s laws, 50, 277
Mallock linear equation
246 Lagrangian function, 55
Many-Meiboom
A errors, 199
Lamina,
Mask form,
267 s equation, 225, 233, 245, 258 Laplacian, 240, 247, 250,
Laplace
296, Mass-capacitance analogy, 106, 277, 282, 283, 289, 304
299
Mass-inductance analogy, 106, 283, 285, 289, Mathematical analysis, 278 223, 224 Mathematical
screw, 333
Length, 228 of curve, 341
Mathematical instruments, 329 225 Mathematical
Lift functions, 228 integrals,
problems,
264
relations, 223 Mathematical relations in connection diagrams, 275 Mathematical theory, 345
Mathematical
Linear, 85
144
Linear algebraic equations, Linear difference equations for stability, 215 188 Linear differential
Mathematics, 224
equations,
Linear differential equations with constant 208 Linear harmonic motion, 126
planimeter,
Linear-circuit theory, 57
Linearization, 200
337
Load considerations, 189 Load impedance, 82 Load relations, 266 scales,
332
x, 140
Logarithm,
Logarithm Logarithm
141
20 of a rational function, 237, 238
multiplier,
Logarithmic Logarithmic
scale,
Matrix of a ib 51
Matrix representation, 146
Maximum and minimum curvatures, s
332
transformation, 241
271
311, 317 equations, 226, 249,
McCool feedback method,
Linear potentiometers, 70 Linear slide rule, 332
Linkage arrangements, Links multiplier, 16
coefficients,
Maxwell
Linear log scales, 332 Linear operator, 85 348 Linear
Log Log
2%
description,
Length Line
331
Mass,.228
Lattice structure, 314
Lead
device, 175
Maps, 329 Mask, 343
26, 337
255 Laplace transform,
Lattice,
solver, 145
Manual adjustment, 144, 146 Manual drift compensation, 95 Manual operation, 333 Many and Meiboom, 146
230 Lagrange method, A,,
135 Magnetic memory function generators, Magnetic potential, 233
158
Mean, 208 Measure of a displacement or a rotation by 333 Measuring an ordinate,
resistance, 17
Meccano, 187 Mechanical amplifiers, 43 Mechanical analogs of electrical circuits, Mechanical computers, 3 Mechanical connection diagram, 277
Mechanical differential analyzers,
106
5, 123, 178, 187, 330,
Mechanical multiplier, 13 Mechanical phonographs, 273
Mechanical polynomial equation solvers, Mechanical system, 273, 277 Mechanical transformer, 295, 302
292 Mechanical transients analyzers, Mechanical wave, 273 Mechanoacoustical transducer, 292
172
342
INDEX
362
Membrane,
Noise theory, 207, 208 Nonlinear potentiometer
225, 245
Membrane analogies, 245 Membrane equation, 245 Membrane models, 245 Membrane slopes, 249
Nonrectangular lattice, Nonreturn branch, 51
Mesh, 50
Mesh Mesh
Mesh-type
48, 50, 52, 277, 280
ideal transformer, 281
Metalized paper, 236 Method of the fictitious junction point, 285 Mica condenser, 78
MichelTs conditions, 266 167 Michelson-Stratton harmonic analyzer, Micropot,
Miller,
K.
Instrumatic Corporation,
S.,
Normal
derivative, 263
Normal
force lattice, 299
304
Normal
forces,
Normal
relationship,
Normal
strain,
253
Normal
stress,
252
340
Null mesh, 50
Numerical integration, 342 150 Nyquist diagram, 94,
70
Mid Century
311
Nonreturn branch current, 53
currents, 54
equations,
resolvers, 127
283, 284 Nonplanar network, 284 Nonplanar systems,
195
198 Oil condenser, 78
Miller effect, 85
On-off records, 331
Milne method, 213
closed step, 220
Open and
Mirror, 340
113
Open
integration procedure,
m.k.s. system of units, 228
Open
step,
Mixer tube
multiplier,
Model
Open-circuit voltage,
277
Mobility analogy,
Optical
Modulation, 63
Oscillograph,
multiplier,
Monochromatic
light,
Ordinary
Output,
118
Morgan-Crawford Motor generator integrator,
104, 188
type transducer, Moving wire in field, 103 coil
resolver, 127
by
Pantograph,
constant, 168
112 logarithm function,
Multiplication by
functions, 20, 112 Multiplication by square of a displacement by a constant, 12 Multiplication 39 Multiplier using integrators, 178, 188, 203 Multipliers, 3, 4,
connected bodies, 254 Multiply connected regions, 226 Multiply connected slice, 261, 265 Multiply Multivibrator circuit, 116
Perturbations, 200
Philbrick
s
equations,
283
Photoelastic effect, 255
317
Photoelastic models, 226, 255, 256
Photoelastic stress analysis, 252 Photoelasticity,
Network
duality,
Network
312 representation, 307,
Network
289 theory, 273, 48, 50, 52, 62, 277, 280
Noise function, 141
Company, 188
Photocell curve follower, 124
313, 314, 316, 319
Noise generator, 207
306
Parameters, 204, 212
Phase angle, 65 Phase shift, 65
Network analogy for Maxwell Network analyzers, 319
Nodal-type transformer, Nodes, 48, 50
8
lattice,
176 Perturbation method for characteristic roots,
Needle probe, 234 331 Negative forms,
Nodal equations,
rod adder,
Parallelepiped
175 Passive network for characteristic roots,
302
Network analogy,
Parallel
Patch bay, 188, 192 Patchcords, 188, 192
Murray-Walker machine, 153 Mutual inductance, 49, 302 spring,
335
331 Pantograph instruments, 78 Paper condenser, Parallel fields, 234
311 Partial differential equations, 161, 228, 254, 305, 309,
Murray, F. J, 155
Mutual
192 Output connections,
332
Multiplication,
Multiplication
84 Output capacity 188 Output cathode ray oscilloscope,
188 Output recorders,
299
lattices,
333
223
180 Output devices, 81 Output generator impedance, of an amplifier, 81 Output generator
289
Moving
Multiple
differential equations, 159
Q
256
linear equation solver, 153
Moving wire
188
measurement of curvature, 270
Models, 224, 228, 231 Modes of vibration, 273
Modulation
57
Operational amplifier,
electrodes, 235
218
213
279
268
Photoelectric cell, 331
Photoelectric Fourier transformers, 170
Photoformer function generators, 188, 196 331 Photographic records, 248 Photographing models,
Phototube, 343 Piezoelectric transducers, 290
INDEX
363
Pin cam, 33 Planar flow, 242
Rack and pinion
Planar network, 283 Plane mask, 344
Rate of angular contraction, 297 Rate of contraction, 295
Plane polarized light, 255 Plane strain, 259, 260
Reac, 196
Plane
Radial
stress, 259,
adder, 9
234
fields,
Reac potentiometer function generator, 128 Reactance network for characteristic roots, 175
260
Planimeter, 329, 342, 345, 348
Reasoning by analogy, 223
Plate, 82
112 Reciprocals, 23,
Plate resistance, 83
24 Reciprocals by linkage,
333 Plotting devices,
Reciprocals by Recorders, 188
Point at
infinity,
239
equation, 226, 245, 250 254, 255, 298
Poisson
s
Poisson
s ratio,
Polar coordinate graph, 333, 351 Polar planimeters, 342 Polariscope,
256
light,
252
Polarized light servo signal, 44 Polarized light servo system, 187
Refraction, 340
Regulated power supplies, 95 Relaxation oscillator, 136 7 Representation by shaft rotation, 7 Representation by translation, 7 Representation by velocities, 146 Representation of linear expression, of the complex plane, 172, 236 Representation Reset, 193
Polarizer, 256, 257
Poles, 237, 238
Polynomial,
Rectangular lattices, 311, 312 Reeves Instrument Corporation, 195
7 Representation by forces,
Polarization, 235
Polarized
mechanical means, 24
238
Polynomial approximation, 307, 311 in one unknown, 165 Polynomial equation
Polynomial equation
solvers,
172
Residues, 239 Resistance, 48
Resistance addition, 76
Positive definiteness, 153
Resistance averaging, 76 Resistance measuring procedures, 17 Resistance multipliers, 17
Potential difference, 49
Resistance network, 146
Potential energy, 246 Potential equation, 233
Resistance-capacity
Potential fluid flow, 236, 241
Restraints, 337, 339
Potential gradients, 234 Potentiometer, 58, 69
Return branch, 50 Return branch current, 53
Potentiometer function generators, 128
Reynold
Potentiometer range, 72 Potentiometer setting, 191
26 Right cylinder,
Polynomial equations, 227 Polynomial representation, 140, 170
s
number, 230, 231 242
Potentiometer setting by Wheatstone bridge, 74 Potentiometers with amplifiers, 70
Rigid bounding surface, Roller wheel principle, 334
Rollers with broad thread, 340
Powers, 336 Precision, 235
29 Rolling contact, 26 Rolling motion,
Pressure, 229
Rolling sphere,
Pressure coefficient, 230, 231
Roots, 239 Rotor, 109
Primary
quantities,
Principal planes,
228
Rubber
253
stress axes, 255, Principal
Problem boards, 188
sheet models, 245, 247, 248
Scalar potential, 233, 242 Scalar potential equation, 226, 305, 310 Scale factors, 212, 271 Scale
Product, 343
Propagated
248
256
257 Principal stresses, 253, 255, 256, Probe, 235, 236, 238
Project Typhoon,
67
filters,
Resolvers, 125, 127, 170, 188
196
error, 214, 218
278 Proportionality factors, Prototype, 255 Pulse width ratio modulation, 114, 121, 135 Punched card machine Fourier computation, 167
Punched card setting of potentiometers, 75 Punched cards for coefficients, 155 Qualitative behavior, 225
Quarter square multiplier, 188 Quarter squares, 33
model
studies, 225,
228
Scale values, 191 Scales,
332
191 Scaling, 189, Schumann linear equation solver, 146
Scotch yoke resolver, 125 227 Second-order linear differential equations, 228 Secondary quantities, Self-inductance, 279 111 Selsyn control transformer, 187 Selsyn transmission systems, Sensitivity equations, Sensitivity system,
204
206
INDEX
364 vacuum
Series connection of
Stable automatic multivariable feedback, 156
tubes, 85
Series representation of a function 139
Starting values, 214
Servo amplifier, 43 Servo differentiator, 39
Steady-state solution, 88
Servo feedback mechanism, 182 Servo function generator, 127
Steering wheel, 353
Servo motor, 18, 333 Servo multiplier, 22, 112, 188 Servo-driven nonlinear potentiometer, 196
Steering wheel principle, 330 Step error estimate, 219, 220
Step multiplier, 119
Servo-mechanism units, 196 Servo-mechanism problems, 292
Storage compressance, 287
Steel ball bearings, 248
Steering wheel integrators, 354
Step-by-step integration, 213
Servos, 8
Setup diagram, 179, 189 Setup of mechanical differential analyzers, 178 Shaft assignment, 179 Shaft interconnection, 178
Shannon theory
for differential analyzers, 180
Shift
Stress
243
297
components, 260
Stress function, 261, 268 Stress matrix,
252
Stresses, 271
Structural analysis, 231
Sum
300
and clutch gear
lines,
Stress distribution, 255, 259, 266
Shear forces, 303, 304 Shear spring, 303 Shear strain, 253 Shear stress, 252 lattice,
Stream
Stress, 252,
Sharp-edged wheel, 338, 340 Shear connections, 303
Shear-force
Strain, 253, 297 Strain gauge multiplier, 118 Stream function, 242
of exponentials, 220
Summer, 188
multipliers, 14
Summing
amplifier, 97, 188
Ships, 231 Short-circuit current, 57
Surface pressure, 268 Surface tension, 235
Side boundaries, 298 Similar triangle multiplier, 15
Switching arrangements for potentiometers, 73 Switching circuits, 115
Similar triangle principles, 355
Symmetric matrix representation, 146
Sin x, 140 Sines, 168
Symmetry of
Single step open procedure, 220
Sink at
infinity,
stress matrix,
253
Synchro motor, 109
Synchro signal generator, 109 Synchro system, 109
238
Sinks, 239
Synchro transformer, 109
Slab analogy, 259
Synthesizers, 169
Slide rule, 332
System of ordinary
differential
equations, 309
Sliding motion, 26 Slip rings, 104 Slope, 340, 341
T-pad, 71 Table, 342
Small displacements, 253, 254 Smoothing, 330 Soap film models, 245, 250
Tandem connection
Tachometer, 188
Solution of algebraic equations, 237
Sound waves, 286 Special purpose character of continuous computers, 6 Special purpose computers, 8 Special purpose differential analyzers, 188 Specific conductivity, 237
Speedometer, 103 Split gear, 34
tubes, 85
Tchebychef polynomials, 173, 337 TE waves, 317
Temperature distribution, 266 Tensile stress, 298 Tension, 246 Term-by-term differentiation, 137 Theory for servo mechanisms, 44 Theory of thin plates, 269 Thermal expansion, 260 Thermal stresses, 268, 269
gear differential, 10, 11
Spur Square cam multipliers, 33 Square function, 141 Square roots on the slide rule, 332 Squares by linear potentiometers, 21 Squares by resistances, 21
Thevenin s theorem, 55, 56, 273 Thickness of the lamina, 26 Three-dimensional fluid flow, 236
Squaring, 188
Time, 228
Stability, 3, 4, 144, 157, 173, 204, 214, 216,
Goldberg-Brown
Stability for feedback amplifiers, 91
Stable amplifier, 88
vacuum
Tapped potentiometer function generator, 128
Sources, 239
Stability charts, 216 Stability criterion for
of
Tangential derivatives, 263 Tank, 234, 235, 243
225
device, 149
Time Time
delay, 7 division multiplier, 112, 113, 121, 188
Time-coded pulse waves, 317 Tooth circle, 30
TM
signal, 331
INDEX Torque Torque Torque
amplification, 110 amplifier, 8, 14, 44, 46, 187 linear equation solvers, 145
365
Vector potential function, 422 Velocity, 305 Velocity potential, 233, 242
Torsion, 245 Torsion of shafts, 305
Velocity vector, 242 Vertical connections, 294
Torsion problems, 250 Total error, 218
Vertical stress, 298
Transconductance, 83 Transducer, 288
Very-narrow-band direct-current amplifier, 96 Viscosity, 286
Transformation of network by ideal transformers, 280 Transformations, 339
Voltage, 234, 237 Voltage division by digital methods, 74
Transformations of the plane, 335 Transformer, 59, 296
Voltage equations, 50 Voltage generator, 48, 60 Voltage gradient, 234
Vertical velocity, 249
Transformer coupling, 85 Transformer method, 292 Transients, 88
Voltage stabilization, 95 Vortex motion of an incompressible
Translating device, 331 Transverse electric waves, 317
Wake patterns,
Transverse load distribution, 269 Transverse magnetic waves, 317
Water flooding, 243 Watt hour meter integrator, 107
Triangle solver, 17 Truncation error, 214, 219
Wave
Two-dimensional Two-dimensional Two-dimensional Two-dimensional Two-dimensional
Two-winding
Typhoon
conductivity, 237
current-density vector, 237 elasticity
problems, 297
electromechanical analogies, 294
model
studies,
259
ideal transformer, 282
Calculator, 120
fluid,
311
231
equation, 226, 245
Waveguides, 249 Weber s number, 231 Wheatstone bridge, 17, 57, 77, 154 Wheatstone bridge multiplier divider, 23
Wide-band amplifiers, 100 Wilbur linear equation solver, 144 Wind-tunnel testing, 231 Winson, J., 198, 204 Wiring, 192
Working model using a continuous computer, Wraparound, 32
Uniform flow, 242 Uniform temperature, 268 Uniform-temperature Uniqueness, 316 Units, 228
free-slice,
x unit, 179 jet,
ideal transformers,
Vacuum
140
x2 140
Unity of mathematics, 224 Unstable amplifier, 88
Use of
269
,
282
tube amplifier, 81
Variable speed drive, 15, 37 Variance, 208
Young
s
modulus E, 254, 255, 298
Zeros, 237, 238 Zeros of a polynomial, 172
5
MATHEMATICAL MACHINES Volume
I:
Computers
Digital
Contents I
DESK CALCULATORS
AND
PUNCHED CARD MACHINES Registers
and Counters
Accumulators Multiplication
and Other
Operations
Survey of Individual Machines Electrical Counters and Accumulators
Punched Card Machines Specific
Punched Card Machines
Sequence Calculators
II
AUTOMATIC SEQUENCE DIGITAL CALCULATORS Machines for Automatic Sequence Computation Spatial and Temporal Organization of a Computer Logical Organization of a
Computer Instruction Systems,
Codes, and
Checking Boolean Analysis of Computers Circuit Elements
and Detailed
Boolean Design
Setup
of
Computation,
ming, Coding, and Layout Errors
Mathematical Methods Survey of Computers
Program
Memory
