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
