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INTRODUCTORY
CIRCUIT THEORY
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INTRODUCTORY
CIRCUIT THEORY
Ernst A. Guillemin
PROFESSOR OF ELECTRICAL COMMUNICATION
DEPARTMENT OF ELECTRICAL ENGINEERING
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
NEW YORK • JOHN WILEY & SONS, INC.
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LONDON • CHAPMAN & HALL, LIMITED
£ngin. library
.£45
Copyright, 1953
By
John Wiley & Son,, Inc.
All Right, Re,erved
Thi, book or any part thereof mu,t not
be reproduced in any form without the
written permi,,ion of the publi,her.
FIFTH PRINTING, SEPTEMBER, 1958
Library of Congre,, Catalog Card Number: 5311754
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Printed in the United State, of America
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To my sophomores, whose enthusiastic
cooperation has been the inspiration
for this work
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PREFACE
For the orientation of the reader of this volume, it should be pointed
out that this is the first of a contemplated sequence. The second volume
will begin with a study of the approximation properties and uses of
Fourier series in connection with circuit problems, and will lead logically
into a discussion of Fourier and Laplace transform theory, its correlation
with the classical differential equation viewpoint, and its application to
analysis and synthesis procedures. The remainder of this volume will
deal with an introduction to synthesis on a survey level, including some
conventional filter theory and the closely related topic of transmission
lines. The advanced aspects of (linear, passive, bilateral) network anal
ysis and synthesis will be the subject of one or two final volumes. Work
on the second volume has been interrupted in favor of proceeding im
mediately with the advanced part which is more urgently needed.
The present volume, as its title states, is intended to be an introductory
treatment of electric circuit theory—the text for a first course in circuits
for undergraduate students majoring in electrical engineering or for
physics students who need a good orientational background in the sub
ject. It is the result of my past five years' experience in getting our E.E.
sophomores headed in the right direction and our physics sophomores
provided with a broad orientation in circuit principles and a flexible
attitude toward their use. I feel that circuit theory (that is, linear,
passive, lumped, finite, bilaleral circuit theory—hereafter called just plain
circuit theory) is the electrical engineer's bread and butter, so to speak.
He needs to know this subject well before he can tackle any of the other
subjects in his curriculum; and it is of the utmost importance that his
first course shall provide him with a set of basic concepts and ways of
thinking that will not become obsolete throughout the rest of his under
graduate and graduate years. He should be started off with the same
basic concepts and processes of analysis that he will be using in his
doctorate research or in his professional work four or five years later. He
will not understand them so well or be able to use them with the same
facility as a sophomore, but he should never have to unlearn or discard
any of his earlier concepts later on. His thoughts as a sophomore should
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vii
viii
PREFACE
sprout from the same roots that will feed and sustain his creative thinking
throughout his professional life. In other words, his first course should
not be a "terminal" first course but the beginning of a career.
I have always held that, where the teaching of basic concepts and pro
cedures are concerned, no distinction should be made between the so
called "elementary" and the "advanced" methods. We refer to things
as being "advanced" only so long as we understand them insufficiently
well ourselves to be able to make them clear in simple terms. Once we
understand a subject fully and clearly, it is no longer difficult to make it
understandable to the beginner. And, if we do not warn the beginner
beforehand, he will not be able to distinguish when we are teaching him
the "elementary" methods and when the "advanced." Such a dis
tinction will reside only in the teacher's mind; to the student both will be
equally novel and equally clear.
I am pointing out these things because some teachers, upon perusing
the pages of this book, may consider some of the topics dealt with (as
well as the general level of the work) to be somewhat more advanced than
is ordinarily considered appropriate for sophomore or junior students.
It is important to remember in this regard that a concept is not neces
sarily more difficult for the student because it happens to be unfamiliar
to the teacher. Conceptually none of the material in this book is any
more difficult than that involved in the differential or integral calculus
which we consider quite appropriate for the sophomore level. Compared
with the oldfashioned brand of circuits course, the work is more chal
lenging, to be sure, but it is also far more interesting. To my students,
who are my most ardent and reliable critics, there is nothing drab about
this subject any longer. Their enthusiasm and morale are high, and the
future looks bright and exciting to them. This is how things should be.
Let me be a little more specific about the ways in which the intro
ductory treatment in this book differs from most. Primarily it hits
harder at the things that are more fundamental, and attempts in every
way possible to present basic ideas and principles so as to promote flex
ible thinking in terms of them and facile use of them in their application
to a wide variety of simple practical problems.
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We teachers talk much about fundamentals, but sometimes we don't
get very close to them. Take the matter of setting up equilibrium
equations for an electric circuit. The very first step is to decide upon a
suitable set of variables. They must be independent, and must be ade
quate to define the state of the network at any moment. The usual
approach to the selection of variables is to choose a set of mesh currents
or loop currents. But do we stop to consider how we can be sure that
these will be independent and adequate, or whether they are reversibly,
PREFACE
uniquely, and unambiguously related to the branch currents? No. We
take all this for granted, and we also take for granted that the student
will straighten this "obviously simple" matter out for himself. He
doesn't realize it at the time, but right here he stores up a lot of trouble
for himself that does not show until much later in his career when he
meets a slightly unorthodox situation and suddenly discovers that he
can't even get started on it.
A similar and even more confusing situation exists when we attempt to
choose a set of voltages as variables as in node analysis. This topic, even
the instructor admits, never gets across. Needless to say, I don't think
we are being very fundamental about these things. Of course, our usual
defense is to say that this is not a very important aspect of circuit theory
anyway; it's one of these advanced topics too highbrow for sophomores;
and, besides, no practical engineer ever uses it anyway. This last re
mark is really one for the book. Of course he doesn't use it. How can
he, when he doesn't understand what it's all about and never had it
explained to him or was shown its possibilities? As for the topic being
too highbrow for sophomores, this is plain nonsense (to which my sopho
mores will most vehemently attest).
So far as the practical potential of this item is concerned, let me
mention just one of a number of pertinent incidents that occurred re
cently. A group of engineers concerned with the Bonneville power
development in the Pacific Northwest were having a conference here, and
one of them described a new approach to the analysis problem which is
particularly effective for such powerdistribution networks and leads to
a systematized computational procedure that beats using the old network
analyzer all hollow. This "new" approach consists in picking an appro
priate tree and identifying the link currents with loop currents, the tree
in this instance being the distribution system and the links being the
branches formed by the sources and loads. It seems that power en
gineers also can benefit by a more fundamental approach to circuit
theory!
Another topic that is essential in getting closer and giving more em
phasis to fundamentals is the use of scale factors and the process of
normalization. We tell the student at the outset that we are going to
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restrict our discussion to linear circuits, but do we clearly impress upon
him the significance of this property or how we can capitalize on it?
Many of my graduate students, as well as many engineers in industry,
are not aware of the implications of this property and of its usefulness if
suitably exploited. In fact, the conventional procedure in teaching
circuit theory deliberately obscures this important aspect of the subject
through overemphasizing what is mistakenly regarded as a "practical"
X
PREFACE
attitude. I can well remember, in my own circuits course that I attended
as a sophomore, that the excitation in the numerical problems was
invariably 110 volts or 220 volts or some other value in current practice.
It was believed by the teacher (and still is by some today) that we must
make the student aware of such practical values of voltage; that it is an
important collateral function of an introductory circuits course to en
lighten our young men about the magnitudes of significant quantities
in current practical use.
To begin with, our students of today are not so stupid as all that.
They already know that common "house current" is supplied at 110 and
220 volts, and they even know that the frequency is 60 cycles per second
(except in some parts of Canada), and a host of other practical data
too numerous to mention. Furthermore, these factual data about
practical values should be and are far more appropriately presented in a
correlated laboratory subject. It is much more important to emphasize
that the assumption of 1 volt or 1 ampere as an excitation value is en
tirely sufficient to take care of any eventuality regarding source in
tensity. Moreover, if we do this, we achieve a certain simplification of
the numerical work, in that we have one less factor to carry through the
pertinent multiplications and divisions, and we become ever so much
more clearly aware of the implication of the linear property of networks
and of the distinctions to be made between power calculations and volt
age or current calculations, because the necessary factors by which the
solution must afterward be multiplied are different.
A similar argument may be advanced concerning the specification of
frequency. Unless there seems to be an urgent need to do otherwise, it
is far more instructive to assume 1 radian per second as the frequency
of an ac source. Through learning how he can subsequently adapt the
response thus found to any other value of excitation frequency, the stu
dent acquires a far better appreciation of the fundamental way in which
circuit behavior depends upon frequency as a parameter; and again a
very material advantage is gained with regard to the numerical com
putations. This latter item alone is more important as a practical
matter than many readers might suppose. I had occasion recently to
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set up an analysis procedure on a research project in an industrial
laboratory and neglected to suggest frequency scaling. The ensuing
calculations fairly bristled with fantastic powers of 2*r and 10, causing all
sorts of silly errors and absurd results. A program of frequency scaling
that brought all the relevant parameter values (critical frequencies and
such) into the range 110 straightened things out in a hurry. (The men
in this research group, incidentally, were trained as physicists; so the
lack of our teaching procedures to provide a sufficiently clear under
PREFACE
xi
standing of fundamentals i8 apparently not restricted to engineering
courses.)
The matter of element values lies within the framework of these same
discussions because of its intimate relation to frequency and amplitude
scaling. Most of the problems in this book involve element values
(henrys, farads, ohms) in the range 110. Here, again, critics will argue
that these values are unrealistic and may give our students mistaken
ideas concerning usual practical values. To this challenge I reply: (a)
Our students are not that dumb. (6) They have lived and will live in
the world of reality where they have ample opportunity to find out what
"really goes." (c) They are concurrently taking a coordinated laboratory
subject where they cannot help but become aware of the fact that 1
farad is a rather large capacitance, (d) It is much more important for
them to learn how, for purposes of calculation, we can so normalize our
problem as to bring the element values into a range where powers of 10
are absent or at least reduced to a minimum. In fact, it is this nor
malized problem that yields what are sometimes called "universal
curves" representing the pertinent circuit response under a wide variety
of conditions.
There are other important consequences of linearity that cannot be
overstressed such as the additive property (superposability of solutions)
and the fact that excitation and response functions as a pair may be dif
ferentiated or integrated any finite number of times without their
appropriateness one to the other being destroyed. But of utmost and
supreme importance is the proper discussion of and approach to the im
pedance concept. In this connection we cannot regard transient analysis
as an advanced topic to be dealt with later on. Transient analysis must
precede the discussion of ac steadystate response in order that the
true character of the impedance function may be recognized. Unless
this phase of an introduction to circuit theory is properly accomplished,
the student will be left with a false notion about the impedance con
cept that he will have to unlearn later on before he can acquire a mentally
clear picture of what an impedance really is and of the omnipotent role
it plays in circuit behavior. To teach the impedance concept initially in
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its conventional restricted form regarding ac steadystate response
materially impedes a later understanding of its true nature and causes a
waste of student time and effort that we cannot afford today. In this
regard I have frequently observed that many of my graduate students
have greater difficulty mastering the impedance concept than some of my
better sophomores whose mental attitude is not preconditioned by some
limited viewpoint.
Besides, the time has passed when we could regard the discussion of
xii
PREFACE
transient response of circuits as a luxury item in our E.E. curriculum.
The widespread use of electroniccontrol devices and the increased im
portance of communication links in our fastmoving modern world have
made that attitude as obsolete as the rotary converter. A discussion of
the transient behavior of circuits is a must in our present physics as well
as in our E.E. curricula at least. And it is wrong to think that it logically
belongs in a later discussion following the introductory subject. With
out an appreciation of the natural behavior of at least some simple cir
cuits it is not possible to present the impedance concept because the
natural frequencies are the quantities by which the impedance is de
termined, apart from an unimportant constant multiplier. The im
pedance is thus more intimately related to the transient behavior than to
the socalled steadystate response, although it characterizes both.
This intimate relation between the transient and steadystate behavior
of circuits is extremely important as a fundamental principle, and we
cannot claim to be hitting at fundamentals unless this item is dealt with
properly.
In close relationship with this interpretation of the impedance function
is the concept of complex frequency and its graphical representation in
the complex frequency plane. Through this means, the evaluation of an
impedance for a given applied frequency is reduced to a geometrical
problem that in many practical cases can be solved by inspection,
especially where reasonable approximations are allowable. Further ex
ploitation of these same ideas leads us, in a logical manner, to interpret
similarly the evaluation of the constants determining the transient
response, and ultimately to all of the practically useful results ordinarily
regarded as being obtainable only through use of Laplace transform
methods. Such a wealth of knowledge about circuits lies within this
conceptual framework that, without question, it may be regarded as the
foundation of circuit theory; yet the conventional "first course" in
circuits as it is now presented (with few exceptions) makes no mention
of these things.
Finally the principle of duality may be mentioned as an important
fundamental concept that should be prominent throughout the dis
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cussions comprising an introductory treatment of circuit theory. Here
the term "throughout" is used literally, since the principle of duality is
not a topic that can effectively be disposed of by a concentrated dis
cussion injected at some seemingly appropriate point, but instead is best
dealt with by touching upon it again and again, bringing out each time
some additional important aspect or application of this useful concept.
Considering the general structure of this book, it is significant to point
out that the first three chapters may be regarded as a separate unit
PREFACE
xiii
which could be used as the text for a rather solid subject in dc circuits or
resistance circuits if this seemed appropriate. Similarly, the succeeding
Chapters 4 through 8 form a closely knit unit that can be used separately.
In fact this portion of the book was written in such a way that it could
be used independently as the text for a onesemester subject, provided
the students had previously been exposed to Kirchhoff's laws and simple
resistance circuits in their physics course. If only one semester can be
devoted to circuits (as with our physics students), then this material
offers a reasonable compromise, while the availability of the discussions
in the first three chapters as collateral reading material (to be consulted
either concurrently or at any later time) serves as a stopgap in lieu of
being able to provide a really adequate foundation at this point in the
curriculum. If two semesters can be devoted to the introductory cir
cuits subject, then Chapters 1 through 9 form an appropriate text, and
Chapter 10, which rounds off and generalizes some of the previous dis
cussions, remains as a collateral reading assignment or as a reminder
that the study of circuit theory really has no ending. In any event, the
student who later goes on with advanced work in network analysis and
synthesis will need the material of Chapter 10 as a necessary background.
Thus the book may serve a dual purpose, as indeed it has served during
the period of its development, the onesemester version being appropriate
for our physics students and the twosemester one for the E.E.'s.
It is only fair to warn the potential reader that this book will prove
only moderately satisfactory as a reference work. Thus the discussion
relevant to any significant item like Thevenin's theorem, duality, the
reciprocity theorem, source transformations, etc., will not be found
nicely packaged within certain pages. Discussion of such items as well as
that pertinent to various fundamental principles are scattered through
out the book—a first presentation here, a little more there, and still more
later on. The reason for this kind of piecemeal presentation is that the
book is intended to be used as a text, and the learning process is a piece
meal procedure. We like at any stage to have some repetition of what we
already know, presented with the addition of a few new ideas, followed
by some illustrations, and then by further additions, etc. Another
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reason for this type of presentation is the dual purpose the book is meant
to serve. Any repetitiousness resulting from these objectives I hope the
reader will find pleasing rather than otherwise.
At this point I would like to make some specific comments on the
material in the various chapters and the reasons for its particular mode
of presentation. The first two chapters are the result of years of practice
and much troubled thinking about how best to present the subject of
establishing equilibrium equations for a network, and why, in spite of
xiv
PREFACE
all my efforts, there always remained so much confusion and so little con
fidence in the student's mind about this topic. At long last I think I
have found the answer to this perplexing question, and Chapters 1 and 2
embody that answer. Thus the conventional approach (and I am as
guilty as anyone of having followed it) attempts to present too much at
once and achieves only confusion. The various methods using tensor or
matrix algebra suffer from the same defect. Moreover, they fail to dis
cuss adequately the most important issue of network geometry, and in
other respects are not suitable for an introductory presentation.
The process of establishing equilibrium equations involves actually
four topics which individually require careful thought and concen
tration for clear understanding. When these are superimposed to form
one conglomerate mass, it is little wonder that nothing but misunder
standing and muddled thinking results.
The first topic is that of selecting an appropriate set of variables and
establishing the relations between these and the branch variables. It is
concerned only with the network geometry (no mention need nor should
be made at this point of Kirchhoff's laws, or the voltampere relations for
the elements, or the sources). The topic involves a number of subtleties,
and its understanding requires a reasonably good appreciation of the
principle of duality, but these matters can be clarified easily if we ex
clude at this time everything else except the purely geometrical proper
ties, as is done in Chapter 1.
Having selected variables, we are in a position to write equilibrium
equations, and so the discussion of the Kirchhoff laws and how to apply
them is the next logical topic. The third topic concerns the voltrampere
relations for the branches; and now we can combine topics 1, 2, 3 to
form the equilibrium equations in terms of the chosen variables. Finally
comes the discussion of sources, and our problem of establishing equi
librium relations is done.
Compare this with the usual procedure of writing Kirchhoff law
equations immediately in terms of loop currents. Here the four steps
outlined above are all tossed into the pot at once and stirred together.
The result is a violent case of indigestion, unless we so restrict and
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simplify the network structure as to render the end result trivial.
I might mention, incidentally, that the discussions in Chapter 1 are
rather complete, perhaps more so than might be regarded appropriate
or necessary in an introductory course. In answer to such comment I
can only say that, when I wrote the chapter, I could see no point in
deliberately stopping before I had finished what I had to say and what
I consider to be a minimum of necessary material to form a good back
ground on which to build later. To postpone the discussion of some of
PREFACE
xv
this material seemed unwise, since a subsequent continuation (perhaps
in another volume) would have to repeat parts of the earlier arguments
in order to achieve coherence in the presentation as a whole. I don't
think that the availability of more information than one cares to as
similate at the moment should pose any serious problem. Chapter 1
may profitably be read and reread several times by the student at
various stages in his educational program.
With regard to the geometrical aspects of duality, which play an im
portant part in the topic of Chapter 1, I found it convenient to invent
names for two things that to my knowledge at least had not previously
been named. Thus the dual of a cut set I have named a "tie set," and
the dual of a tree a "maze." These names seemed most appropriate to
me, and I hope the reader will find them appropriate also.
Chapter 3 is a collection of topics, all of which are directly or indirectly
concerned with expediting the process of obtaining solutions. Syste
matic elimination procedures, solution by determinants, special artifices
applicable where various types of symmetry prevail, short methods
usable with ladder structures, wyedelta transformations, source trans
formations (which are what Thevenin's and Norton's theorems
amount to), the reciprocity theorem (frequently an effective aid in ob
taining a desired result), a knowledge of how power calculations must be
made (the fact that these effects when caused by separate sources are
not additive in contrast to currents and voltages which are), the trans
formations that leave power relations invariant, the equivalence relations
pertinent to the tee, pi, bridgedtee and lattice structures—all these
things are useful when we are dealing with the business of constructing
solutions. I feel that they belong together and that it is useful to make
a first presentation of them while discussing the restricted case of re
sistance networks where there are no other complications to interfere
with their assimilation. Although here, as in Chapter 1, the treatment
may seem to be somewhat more inclusive than is essential at an in
troductory level, no serious difficulty need thereby be created, since the
relative emphasis given to various topics can always be appropriately
adjusted.
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In this chapter an attempt is made to have the various topics introduce
themselves logically rather than be forced upon the reader's attention in
a haphazard fashion. Thus, having discussed network geometry, and
having shown how a numerical set of equations may be solved by sys
tematically eliminating variables, what is more logical than for the
reader to become curious about the geometrical implications of this
elimination process? The elimination of a node potential should corre
spond geometrically to the elimination of a node, and the elimination of
xvl
PREFACE
a mesh current to the elimination of the pertinent mesh. Such a correla
tion, which is indeed possible, not only leads logically to a presentation
of wyedelta or deltawye transformations and their generalizations, but
does so with a minimum of disagreeable algebra, as contrasted with other
presentations of this item, particularly in the general starmesh case. A
particularly simple proof of the reciprocity theorem which likewise fits
in with the pattern set by the systematic elimination procedure is
achieved through showing that the symmetry of the parameter matrix
characterizing the equilibrium equations is unchanged by a typical step
in this procedure.
Chapter 4, which introduces the voltampere relations for the induc
tance and capacitance elements and shows that inductance networks and
capacitance networks are dealt with by means of the same methods
applicable to resistance networks, is primarily concerned with a dis
cussion of the unit step and impulse functions, in terms of which various
more arbitrary source functions and switching operations may con
veniently be described. In connection with the impulse function, it has
been stated that the subtleties involved in its interpretation are too
difficult for a class at the sophomore level and that the concept is too
abstract and unreal. Neither criticism is consistent with our prevailing
attitude. The limit process involved in the definition of the impulse is
precisely of the same nature as that pertinent to the formation of a de
rivative or of an integral. If the comprehension of this sort of limit
process is too much for a sophomore, then we shall also have to give up
trying to teach him the differential calculus.
As for the impulse being unreal, nothing could be further from the
truth. In our daily life we frequently see things bumping into other
things. Take a bat hitting a baseball for instance. The ball changes its
velocity from minus to plus in a wink—and that's short enough (com
pared with the time of flight of the ball) to be negligible. For all practical
purposes the ball acquires its kinetic energy of flight in no time. If we
want to be fussy about this situation and say that the nonzero time of
impact must be considered and so we really are not dealing with an im
pulse in its true sense, then to be consistent we should be equally fussy
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about the step function, because a change in value (of a force for in
stance) cannot occur instantly either; yet we no longer object to step
functions in our engineering analysis, because we have lived with this
concept longer and are used to it. Our mathematical methods of analysis
always represent an idealization of the true state of affairs, and the im
pulse function involves nothing different in this respect from all the other
mathematical concepts that we are accustomed to use.
Engineering analysis involving singularity functions of all orders is
PREFACE
xvii
becoming so common today that we can no longer neglect making our
students familiar with them at an early stage. My chief reason for in
troducing the impulse as well as the step when I first wrote this text
material was the desire to use Thevenin's and Norton's theorems with
capacitance and inductance elements in the transient state. Since these
elements involve differentiation and integration, it was clear that a
step function might have to be differentiated in the course of solving a
problem by these means. To deprive the student of this flexible way of
dealing with transient problems, I felt, was not in keeping with my basic
objectives, and so I moved the presentation of singularity functions
from the graduate curriculm into the sophomore year.
It might also be pointed out that the early introduction of these con
cepts into the study of circuit theory develops a more openminded
attitude on the part of the student toward characteristic behavior
patterns. In my student days, for example, we were told that the
current in an inductance just had to be continuous. Though this is true
in most practical situations, it is much better not to make such sweeping
assertions. It is far more instructive to show the student that a dis
continuous current can be produced in an inductance only through the
application of a voltage impulse but that physical conditions may some
times approximate this kind of excitation function.
Chapter 5 deals with the transient response of simple circuits, making
use of all the artifices mentioned above. The primary objective is to give
the student a physical understanding of transient response in first and
secondorder cases, together with a facile way of dealing with the perti
nent mathematical relationships, so that he will develop an easy and
circumspect approach to problems of this sort, rather than always use
the same mathematically ponderous and slowly moving machinery of
the "general case." In this respect I have seen some awful crimes
committed, particularly by students who have learned the Laplace
transform method. They are determined to Laplacetransform every
thing that comes their way, and they get so they can't solve the simplest
problem without this machinery. They can't write down the discharge
of a capacitor through a resistance without Laplacetransforming the
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poor thing to death. I don't want any of my students to get into a fix
like that. I want them to know their simple transients as well as they
know their own names, and Chapter 5 aims to give them the kind of
workout that can accomplish this end.
In Chapter 6 we come to la piece de resistance as the French would say.
Here we introduce the sinusoid, the notion about complex frequency,
the impedance concept, its interpretation in terms of the natural fre
quencies of the circuit, graphical portrayal of the polezero pattern in
xviii
PREFACE
the splane, evaluation of impedances through geometrical visualization
of their frequency factors, interpretation of resonance as a near coin
cidence between applied and natural frequencies, reciprocal and com
plementary impedances, magnitude and frequency scaling, vector dia
grams, and other related aspects pertinent to this general theme. Tran
sients and steady states are stirred together into a pretty intimate mix
ture, with the impedance function keeping order and clarifying all of the
pertinent interrelationships. The circuits dealt with are for the most
part still the simple ones touched upon in Chapter 5 so that the student
will have no difficulty following the mathematical steps while getting
used to the many new concepts and methods of interpretation presented
here. A few more elaborate element combinations, such as the constant
resistance networks and doubletuned circuits, are discussed toward the
end of this chapter in order to show the student how simple a matter it
is to deal with such situations in terms of the rather powerful tools which
the earlier discussions have placed at his command.
Chapter 7 introduces a formal discussion of energy and power relations.
Instead of the conventional restriction in the derivation of pertinent
quantities to inphase and outofphase components of current and
voltage, an attempt is made to develop a more physical appreciation of
these phenomena through specific attention to the stored energy func
tions and their significance in the sinusoidal steady state, along with the
role played by the dissipation function. Thus the definition of reactive
power as the product of the voltage and the quadrature component of
current leaves the student with no physical picture of what this quantity
is or why it exists and needs to be considered. When it is seen to be
proportional to the difference between the average values of the stored
energies, its significance begins to be appreciated in physical terms.
Through expressing impedances in terms of energy functions, through
their determination by these means, and through the ability thus to
perceive from a singlefrequency computation the whole course of their
behavior in a given vicinity (for instance, the determination of the im
pedance behavior in a resonance vicinity and computation of the factor
Q), the student is given a glimpse of how energy and power considerations
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may be useful in a much broader sense than merely for the computation
of energy consumption.
The object of Chapter 8 is to provide the means for dealing with more
extensive and more random circuits in the sinusoidal steady state than
the simple ones so far considered. Most important in this regard is the
consideration of mutual inductive coupling. The traditional stumbling
block involved in the treatment of random situations, namely, the
determination of algebraic signs, is overcome by a systematic approach
PREFACE
xix
which is straightforward in its use for the computation of pertinent
parameter matrices on both the loop and node bases.
In Chapter 9 the subject of transient response is generalized, first,
through consideration of the socalled ac transients and, second, through
development of the complete solution for any finite lumped network,
leading to a result that is identical in form with, but much more simply
derived than, that alternately obtainable through Laplace transform
methods supplemented by complex integration. It is in these discussions
that the concept of complex frequency is fully developed and illustrated
by a consideration of the exact coincidence between excitation and
natural frequencies (perfect resonance). It is shown how all the many
useful theorems ordinarily derived only by Fourier and Laplace trans
form methods are easily and rigorously established by inspection of the
form of the solution for the general case, and these theorems (or proper
ties as I prefer to call them) are discussed and illustrated by means of
numerous examples.
These examples were constructed by starting from assumed polezero
configurations for the desired transfer functions and synthesizing the
pertinent networks. Thus, for the first time in the history of textbooks
on transient analysis, the reader is presented with illustrative examples
involving higher than secondorder systems. He will find a multiple
order pole problem other than the hackneyed RLC circuit for the criti
cally damped case; and he will find examples that are representative of
useful response characteristics, as well as illustrative of the theoretical
analysis that precedes them.
Before the advent of synthesis it was not possible to construct really
interesting illustrative examples. If a circuit with more than two or
three meshes was assumed, the solution of a characteristic equation of
high degree was immediately involved, and the resulting random char
acter of the response obtained after much disagreeable work was hardly
representative of anything interesting. Being able to start from a pole
zero pattern and work in both directions (to a network on the one hand,
and to its transient response on the other) opens up a host of possibilities
that were not available to the textbook writer of the past. Within a
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limited space, I have made the most of this situation in working out a set
of illustrative examples for Chapter 9.
Chapter 10, as mentioned previously, supplies a certain generality and
completeness to the derivation of equilibrium equations and energy
relations that have been discussed already but have not been established
in this way. Thus, when the reader reaches this chapter, he will be
familiar with all the topics it contains except the mathematical methods
needed to state them in perfectly general and yet compact and concise
XX
PREFACE
form. The final item is a critical discussion of the principle of duality
and of the results derivable from it in the light of the broader viewpoint
just presented. The story of network theory is, of course, nowhere nearly
completed at this point, but, since one volume cannot contain all of it,
this seems to be a reasonable point at which to stop.
I should probably say something about historical notes (who did what,
when, and why) and references to source material and all that, because
with few exceptions I haven't done any of this sort of thing. As a matter
of fact, if one takes the works of Kirchhoff, Helmholtz, Cauchy, Lord
Rayleigh, and maybe a few others of similar standing and vintage, there
isn't much else that is needed to establish the background for network
theory. If a student has the inclination to "do some digging" (most of
them prefer not to) to ferret out historical facts, he will have no diffi
culty finding the bibliographical help and the encouragement from his
instructor that he needs. I do not mean to belittle the importance of
having some historical background on the evolution of science and
mathematics (and network theory), but the wherewithal to go into this
aspect of things is already available. I would rather confine my limited
energies (and heaven knows they are limited!) to making available the
things that are not now available.
One final point. In the teaching of this subject I regard it as im
portant to remind the student frequently that network theory has a
dual character (no connection with the principle of duality); it is a
Dr. JekyllMr. Hyde sort of thing; it is twofaced, if you please. There
are two aspects to this subject: the physical and the theoretical. The
physical aspects are represented by Mr. Hyde—a smooth character who
isn't what he seems to be and can't be trusted. The mathematical
aspects are represented by Dr. Jekyll—a dependable, extremely precise
individual who always responds according to established custom.
Dr. Jekyll is the network theory that we work with on paper, involving
only pure elements and only the ones specifically included. Mr. Hyde
is the network theory we meet in the laboratory or in the field. He is
always hiding parasitic elements under his jacket and pulling them out
to spoil our fun at the wrong time. We can learn all about Dr. Jekyll's
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orderly habits in a reasonable period, but Mr. Hyde will continue to fool
and confound us until the end of time. In order to be able to tackle him
at all, we must first become well acquainted with Dr. Jekyll and his
orderly ways. This book is almost wholly concerned with the latter. I
am content to leave Mr. Hyde to the boys in the laboratory.
And, speaking of the "boys in the laboratory," that is to say, the
able and cooperative staff who assist in administering this material to
our undergraduate students, I wish here to thank them one and all for
PREFACE
xxi
their many helpful suggestions and their enthusiastic cooperation
throughout the period of this "fiveyear plan." I cannot name one with
out naming them all, and I cannot name them all because I can't be sure
that I won't miss one or two. So they'll all have to remain nameless;
however, for the time being only. It won't be long before each one makes
a name for himself as some have already.
And that is all, except to wish you all a pleasant voyage—through the
pages of this book and wherever you may be going.
E. A. Guillemin
Wellasley Hills
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November 1968
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CONTENTS
Introduction 1
CHAPTER
1 Network Geometry and Network Variables 5
Art. I. The Classification of Networks S
2. The Graph of a Network 5
3. The Concept of a "Tree" 7
4. Network Variables J 8
5. The Concept of Loop Currents; Tie Sets and TieSet Schedules ./ 10
6. The Concept of NodePair Voltages; Cut Sets and CutSet Schedules 17
7. Alternative Methods of Choosing Current Variables 23
8. Alternative Methods of Choosing Voltage Variables 33
9. Duality 42
10. Concluding Remarks 51
Problems 58
2 The Equilibrium Equations 64
Art. 1. Kirchhoff's Laws 64
2. Independence among the Kirchhoff Law Equations 68
3. The Equilibrium Equations on the Loop and Node Bases 71
4. Parameter Matrices on the Loop and Node Bases 77
5. Regarding the Symmetry of Parameter Matrices 79
6. Simplified Procedures That Are Adequate in Many Practical Cases 81
7. Sources 86
8. Summary of the Procedures for Deriving Equilibrium Equations 96
9. Examples 99
Problems 105
3 Methods op Solution and Related Topics 112
Art. 1. Systematic Elimination Methods 112
2. Use of Determinants 116
3. Methods Applicable to Ladder and Other Special Network Con
figurations 121
4. Network Transformations; WyeDelta (FA) Equivalents 127
5. Thevenin's and Norton's Theorems 138
6. The Reciprocity Theorem 148
7. DrivingPoint and Transfer Functions 153—
8. Common Network Configurations and Their Equivalence Relations 161
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9. Power Relations, and Transformations under Which They Remain
Invariant 169
Problems 179
xxiv CONTENTS
CHAPTER
4 Circuit Elements and Source Functions 188
Art. 1. The VoltAmpere Relations of the Elements 188
2. Voltage and Current Sources 190
3. The Family of Singularity Functions; Some Physical Interpreta
tions 196
4. SingleElement Combinations 203
Approximate Formulas for Parameters of Simple Geometrical Configura
tion 211
Problems 218
5 Impulse and StepFunction Response of Simple Circuits 222
Art. 1. The Series RL Circuit; General Properties of the Solution 222
2. Correlation between Mathematical and Physical Aspects 230
3. Source Transformations; TheVenin's and Norton's Theorems and
Their Uses 235
4. The Dual of the Series RL Circuit 241
5. The Series RLC Circuit 243
6. The Dual of the Series RLC Circuit 251
7. Consideration of Arbitrary Initial Conditions 253
Summary Regarding the Transient Response of One, Two, and ThreeEle
ment Combinations 257
Problems 262
6 Behavior of Simple Circuits in the Sinusoidal Steady State 270
Art. 1. Why Sinusoids Play Such a Predominant Part in the Study of Elec
trical Networks 270
2. Complex Representation of Sinusoids 273
3. Elaborations upon the Impedance Concept 282
4. Interpretation of Impedance in the Complex Frequency Plane 286
5. Impedance and Admittance Functions for Simple Circuits 289
6. The Phenomenon of Resonance 297
7. Rectangular versus Polar Forms of Impedance and Admittance
Functions; an Alternative Interpretation of Resonance 301
8. Reciprocal and Complementary Impedances and Admittances 305
9. Magnitude and Frequency Scaling 309
10. Vector Diagrams 311
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11. More Elaborate Impedance Functions; Their Properties and Uses 315
Problems 325
7 Energy and Power in the Sinusoidal Steady State 340
Art. 1. Energy in the Storage Elements 340
2. Energy in the Storage Elements When Voltage and Current Are
Sinusoids 342
3. Energy and Power Relations in a Complete Circuit 343
4. Active and Reactive Power; Vector Power 348
5. RootMeanSquare, or Effective Values 352
6. Impedance or Admittance in Terms of Energy Functions 354
7. Computation of the Energy Functions for More Complex Networks 357
8. Some Illustrative Examples 358
Problems 362
CONTENTS xxv
CHAPTER
8 More General Networks in the Sinusoidal Steady State 366
Art. 1. The SteadyState Equilibrium Equations 366
2. Use of Parameter Matrices 371
3. Duality Again 373
4. Mutual Inductance and How to Deal with It 374
5. Coupling Coefficients 380
6. Forming the Equilibrium Equations When Mutual Inductances
Are Present 382
7. Computation of DrivingPoint and Transfer Impedances for Lad
der Networks 385
8. Networks Embodying Symmetry in Structure and Source Distri
bution—Polyphase Circuits 388
Problems 392
9 Additional Topics Dealing with SteadyState and Transient Be
havior of Lumped Linear Circuits 401
Art. 1. Transient Response with Alternating Excitation 401
2. Further Exploitation of the Concepts of Complex Frequency and
Impedance 412
3. Frequency and Time Domains 414
4. The Complete Solution for Any Finite LumpedConstant Network 419
5. The Derivation of Equilibrium Equations for DrivingPoint and
Transfer Situations; Reciprocity Again 426
6. Properties of the General Solution 431
7. Illustrative Examples 440
8. DrivingPoint and Transfer Functions 462
9. Arbitrary Initial Conditions 468
Problems 469
10 Generalization of Circuit Equations and Energy Relations 483
Art. 1. Use of Matrix Algebra 483
2. BranchParameter Matrices and VoltAmpere Relations 491
3. Equilibrium Equations on the Node Basis 496
4. Equilibrium Equations on the Loop Basis 499
5. Remarks and Examples 502
6. Energy Functions 510
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7. Equivalence of Kirchhoff and Lagrange Equations 520
8. Relation to Impedance Functions 522
9. Duality Once More 535
Problems 540
Index
547
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Introduction
Although the discussions in this book, and those in the ones to follow
it, are restricted to the simplest class of electrical networks, the reader
should not expect that he will find them to be either simple or restricted
as to scope and practical importance. Regarding their importance, it
may be pointed out that an understanding of the theory of this simplest
class of networks is an indispensable prerequisite to the study of all
others; and as to scope it is significant to observe that because of their
simplicity one is able to develop the theory of this class of networks to
a remarkable degree of completeness. As a result, this theory plays a
dominant role in the study and development of almost all electrical
devices and systems, and is therefore as fundamental to the intellectual
equipment of the electrical engineer as is a knowledge of mathematics
to the physicist.
With these remarks the primary mission of this introduction is accom
plished. The following paragraphs are intended to provide the unin
itiated reader with a bit of an idea as to what an electrical network is,
and to define the simple class of networks mentioned above. Actually
it is illusory to suppose that the reader who is totally unacquainted with
this subject will derive much benefit from an exposure to such a definition
of terms, since he will understand them clearly only after he has gained
a considerable background in network theory. On the other hand, such
remarks may provide the reader with a sufficient initial orientation to
enable him to gain a proper perspective as he progresses with the studies
that he ahead.
The relevant operating characteristics of a large proportion of all
electrical devices are adequately described through a knowledge of cur
rents and voltages as time functions at appropriately selected points or
point pairs. The significant behavior of an electronic amplifier, for
example, is characterized in terms of its voltampere relations at specified
input and output terminal pairs; the performance characteristics of a
transmission line for the distribution of electric energy or for the con
veyance of electric signals representing coded information are expressible
in terms of relative voltage and current values at appropriate points
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l
2
INTRODUCTION
along the line; the behavior of a motorgenerator set is conveniently
studied in terms of the voltage and current input to the motor relative
to the voltage and current output from the generator; the electrical
characteristics of an ordinary light bulb are adequately described in
terms of the voltagecurrent relations at its terminals.
In some of these devices, other features besides the electrical ones are
usually of interest also, as are, for example, the mechanical phenomena
involved in the operation of the motorgenerator set, or the light spec
trum emitted from the light bulb referred to above. A separation of the
nonelectrical from the purely electrical studies in such cases is, however,
usually desirable, and can always be accomplished under an appropri
ately chosen set of environmental conditions. It may additionally be
necessary to make simplifying approximations and idealizations in order
to render the electrical features of the problem manageable in reasonable
terms, but, when this is done, the resulting representation of the original
device is commonly described by the term "electric circuit" or "network."
While the electric circuit may thus be an idealized or skeletonized
representation of the electrically relevant features of some physical unit
in which these circuit characteristics are only incidental or at most
partially influential in controlling its structure and behavior, there are
important instances where the circuit is the whole device and its function
is that of a controlling unit in a larger system. The electric "wave
filters" and "corrective networks" essential to longdistance telephone
communication circuits, or the "control networks" in servo mechanisms
are examples of this sort. Here the electric circuit no longer plays an
incidental role but takes its place along with other important electro
mechanical or electronic devices as a highly significant unit or building
block essential to the successful operation of modern power, communica
tion, or control systems.
Dominant in their effect upon the voltampere behavior of an electric
circuit are its energystorage and energydissipation properties. Energy
storage takes place in the electric and magnetic fields associated with the
network, while energy dissipation is practically everpresent because of
resistance offered to the flow of electric charge through conductors.
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Three things, therefore, dominate in molding the electrical behavior of a
network: namely, the two associated fields and the dissipative character
of its various conducting paths. Although their effects are physically
superimposed throughout any actual device, the idealization referred
to above frequently permits one to assign them to separate portions of
the physical system and to regard these portions as having negligible
dimensions. Thus one speaks of certain "lumped" parts as having
resistive characteristics alone, others as having influence only upon the
INTRODUCTION
3
associated magnetic fields, and a third group related solely to the per
tinent electric fields.
These parts are spoken of as the lumped parameters or elements of a
circuit. They are of three kinds: the resistance parameter or dissipative
element, the inductance parameter which is related to the associated
magnetic fields, and the capacitance parameter appropriate to the
pertinent electric fields. Physical embodiments of these network param
eters or elements (appearing wherever their occurrence is deliberate
rather than incidental) are familiar to the reader as resistors (usually
made of metallic wire having poor conductivity), inductors such as wire'
coils, and capacitors (frequently in the form of metallic sheets or plates
separated by a thin film of insulating material). It is important to
observe that these physical embodiments are not exact representations
for the separate circuit elements which, by definition, are "pure" in the
sense that each one contains none of the other two. In any physical
resistor, for example, some inductive and capacitive effects are unavoid
able, as are resistive and capacitive effects in a physical inductor, etc.
These frequently unwanted effects present in physical resistors, induc
tors, and capacitors are commonly referred to as "parasitics." Since
any physical device with its known parasitic elements can always be
represented to a sufficient degree of approximation in terms of theo
retically pure elements, a method of circuit analysis based upon pure
elements alone is both adequate and useful.
The relationship of voltage across an element to the current through
it, which is commonly referred to as its pertinent voltampere relationship,
is in most cases a linear one (throughout reasonable operating ranges),
and the appropriate constant of proportionality is designated as the
"value" of that element.
There are devices in which the values of network elements are func
tions of the voltage across them or of the current carried by them. For
example, an ironcored coil represents an inductance element whose
value is dependent upon the coil current; an electron tube represents a
resistance which varies with the applied voltage. Such elements are
said to be nonlinear because the voltage is not linearly proportional to
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the associated current (or to the current derivative or integral, which
ever is pertinent). It is important to distinguish networks that contain
such elements from those that do not, and to recognize significant differ
ences in their response characteristics, for these differences form the
basis upon which the selection of specific types of elements is made in
the practical use of circuits.
There are some devices, linear as well as nonlinear, whose voltage or
current transmission properties depend upon their orientation with
4
INTRODUCTION
respect to the points of excitation and observation. These are spoken
of as being unilateral devices or elements; and wherever the usual ones
need to be distinguished from these, they are referred to as bilateral
elements.
Another important distinction having a bearing upon network be
havior is made according to whether the network does or does not con
tain energy sources or constraints other than those explicitly given by
the associated excitation. If it does, then one may expect at times to
get more power out than one puts into the network, or to obtain a con
tinued response even in the absence of a power input. When a network
contains such implicit energy sources and/or constraints, it is called
active; otherwise it is referred to as being passive.
The finite, lumped, linear, passive bilateral network is the simplest
regarding methods of analysis needed in a study of its behavior under
various operating conditions. To an introductory understanding of the
physical and mathematical aspects of this type of network, the discus
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sions of the present volume are directed.
CHAPTER ONE
Network Geometry
and
Network Variables
1 The Classification of Networks
Linear passive networks are distinguished from one another according
to the kinds of elements that are involved, and in the manner of their
interconnection. Thus a given network consisting of resistance elements
alone is referred to as a resistance network; and inductance or capacitance
networks are similarly denned as such in which only inductances or
capacitances are involved. Next in order of complexity are the socalled
twoelement types, more precisely the LC networks (those containing
inductance and capacitance elements but, by assumption, no resistances),
the RC networks in which inductive effects are absent, and RL networks
in which capacitive effects are absent. The RLC network then repre
sents the general case in the category of linear passive networks.
2 The Graph of a Network
Quite apart from the kinds of elements involved in a given network is
the allimportant question of network geometry that concerns itself solely
with the manner in which the various elements are grouped and inter
connected at their terminals. In order to enhance this aspect of a net
work's physical makeup, one frequently draws a schematic representation
of it in which no distinction is as yet made between kinds of elements.
Thus each element is represented merely by a line with small circles at
the ends denoting terminals. Such a graphical portrayal showing the
geometrical interconnection of elements only, is called a graph of the
given network. Figure 1 shows an example of a network as it is usually
drawn so as to distinguish the various kinds of elements [part (a)] and
how this same network appears when only its geometrical aspects are
retained [the graph of part (b)]. The numbers associated with the
various branches are added for their identification only. The terminals
of the branches (which are common to two or more branches where
these are confluent) are referred to as nodes.
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s
NETWORK GEOMETRY AND NETWORK VARIABLES
Key
( —•^AN— Resistance element
—f 0 (J 0 ^— Inductance element
1 Capacitance element
(a)
(b)
Fio. 1. A network schematic and its graph.
There are situations in which various parts of a network are only
inductively connected as in part (a) of Fig. 2 where two pairs of mutually
coupled inductances are involved. Here the corresponding graph (shown
in part (b) of Fig. 2) consists of three separate parts; and it is seen also
(a) Given network (b) Network graph
Fio. 2. The schematic and graph of a network consisting of several separate parts.
that a node may be simply the terminus of a single branch as well as the
point of confluence of several branches.
With the graph of a network there are thus associated three things or
concepts: namely, branches, nodes, and separate parts. The graph is the
skeleton of a network; it retains only
its geometrical features. It is useful
when discussing how one should best
go about characterizing the network
behavior in terms of voltages and
currents and in deciding whether a
selected set of these variables are not
only independent but also adequate
Fig. 3. The graph of Fig. 2 coa for the unique characterization of the
lesced into one part. state of a network at any moment.
In this regard it is apparent that an
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economy can be effected in situations like the one in Fig. 2 through
THE CONCEPT OF A "TREE"
7
permitting one node in each of the separate parts to become coinci
dent, thus uniting these parts, as is shown in the graph of Fig. 3.
Except for the fact that the superimposed nodes are constrained to
have the same electric potential, no restrictions are imposed upon any
of the branch voltages or currents through this modification which
reduces the total number of nodes and the number of separate parts by
equal integer values. In subsequent discussions it is thus possible
without loss in generality to consider only graphs having one separate
part.
3 The Concept of a 'Tree"
The graph of a network places in evidence a number of closed paths
upon which currents can circulate. This property of a graph (that it
contain closed paths) is obviously necessary to the existence of currents
'(a) (b) (c)
Fig. 4. A graph and two possible trees (solid lines).
in the associated network. It is a property that can be destroyed through
the removal of judiciously chosen branches.
In Fig. 4 the graph of a given network is shown in part (a), and again
in parts (b) and (c) with some of the branches represented by dotted
lines. If the dotted branches were removed, there would remain in
each of the cases shown in (b) and in (c) a graph having all of the nodes
of the original graph (a) but no closed paths. This remnant of the
original graph is called a "tree" for the reason that its structure (like
that of any tree) possesses the significant property of having no closed
paths.
More specifically, a tree is defined as any set of branches in the original
graph that is just sufficient in number to connect all of the nodes. It is
not difficult to see that this number is always nt — 1 where nt denotes
the total number of nodes. For, if we start with only the nodes drawn
and no branches, it is clear that the first added branch connects two
nodes, but thereafter one additional branch is needed for each node con
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tacted. If no more than the minimum number of nt — 1 branches are
8
NETWORK GEOMETRY AND NETWORK VARIABLES
used to connect all of the nodes, then it is likewise clear that the resulting
structure contains no closed paths, for the creation of a closed path
involves the linking of two nodes that are already contacted, and hence
involves the use of more branches than are actually needed merely to
connect all of the nodes.
For a given network graph it is possible to draw numerous trees, since
the process just described is not a unique one. Each tree, however, con
nects all of the nt nodes, and consists of
branches, which are referred to, in any given choice, as the tree branches.
The remaining branches, like the ones shown dotted in parts (b) and
(c) of Fig. 4, are called links. If there are I of these, and if the total num
ber of branches in the network graph is denoted by b, then evidently
an important fundamental relation to which we shall return in the follow
ing discussions.
4 Network Variables
The response or behavior of a network is completely known if the
currents and the voltages in all of its branches are known. The branch
currents, however, are related to the branch voltages through funda
mental equations that characterize the voltampere behavior of the
separate elements. For instance, in a resistance branch the voltage
drop (by Ohm's law) equals the current in that branch times the per
tinent branch resistance; in a capacitance branch the voltage equals the
reciprocal capacitance value times the time integral of the branch cur
rent; and in an inductance branch the voltage is given by the time
derivative of the current with the inductance as a proportionality factor.
Although the lastmentioned relations become somewhat more elaborate
when several inductances in the network are mutually coupled (as will
later be discussed in detail), their determination in no way involves the
geometrical interconnection of the elements. One can always, in a
straightforward manner, relate the branch voltages directly and re
versibly to the branch currents.
We may, therefore, regard either the branch currents alone or the
branch voltages alone as adequately characterizing the network be
havior. If the total number of branches is denoted by b, then from either
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point of view we have b quantities that play the role of unknowns or
variables in the problem of finding the network response. We shall now
show that either set of b quantities is not an independent one, but that
n = nt — 1
(1)
b=l+n
(2)
NETWORK VARIABLES
9
fewer variables suffice to characterize the network equilibrium, whether
on a current or on a voltage basis.
If in a given network a tree is selected, then the totality of b branches
is separated into two groups: the tree branches and the links. Corre
spondingly, the branch currents are separated into treebranch currents
and link currents. Since a removal or opening of the links destroys all
closed paths and hence by force renders all branch currents zero, it
becomes clear that the act of setting only the link currents equal to zero
forces all currents in the network to be zero.* The link currents alone
hold the power of life and death, so to speak, over the entire network.
Their values fix all the current values; that is, it must he possihle to
express all of the^ t.re(>hraT'f'n currants uniquely in terms of the link
currents]
The inference to be drawn from this argument is that, of the b branch
currents in a network, only I are independent; I is the smallest number of
currents in terms of which all others can be expressed uniquely. This
situation may be seen to follow from the fact that all currents become
zero when the link currents are zero. Thus it is clear that the number of
independent currents is surely not larger than I, for, if one of the tree
branch currents were claimed also to be independent, then its value
would have to remain nonzero when all the link currents are set equal
to zero, and this condition is manifestly impossible physically. It is
equally clear on the other hand that the number of independent currents
is surely not less than I, for then it would have to be possible to render
all currents in the network zero with one or more links still in place, and
this result is not possible because closed paths exist so long as some of
the links remain.
Thus, in terms of currents, it must be possible to express uniquely the
state of a network in terms of I variables alone. As will be shown later,
these variables may be any appropriate set of link currents (according
to the specific choice made for a tree), but more generally they may be
chosen in a large variety of ways so that numerous specific requirements
can be accommodated.
Analogously one may regard the branch voltages as separated into two
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groups: the treebranch voltages, and the link voltages. Since the tree
branches connect all of the nodes, it is clear that, if the treebranch volt
• In these considerations it is not necessary that we concern ourselves with the
manner in which the network is energized although some sort of excitation is implied
since all currents and voltages would otherwise be zero, regardless of whether the
links are removed or not. If the reader insists upon being specific about the nature
of the excitation, he may picture in his mind a small boy tossing coulombs into the
capacitances at random intervals.
10 NETWORK GEOMETRY AND NETWORK VARIABLES
ages are forced to be zero (through shortcircuiting the tree branches, for
example), then all the node potentials become coincident, and hence all
branch voltages are forced to be zero. Thus, the act of setting only the
treebranch voltages equal to zero forces all voltages in the network
to be zero. The treebranch voltages alone hold the power of life and
death, so to speak, over the entire network. It must be possible, there
fore, to express all of the link voltages uniquely in terms of the tree
branch voltages.
Exactlyjrt of the branch voltages in a network are independent,
namely, those pertaining to the branches of a selected tree. Surely no
larger number than this can be independent because one or more of the
link voltages would then have to be independent, and this assumption
is contradicted by the fact that all voltages become zero through short
circuiting the tree branches alone. On the other hand, no smaller num
ber than n voltages can form the controlling set, for it is physically not
possible to force all of the node potentials to coincide so long as some
treebranch voltages remain nonzero.
i ) (c)
Fio. 6. Closed paths or loops corresponding, respectively, to the three trees shown
in Fig. 5.
Each link current is thus identified with a loop current; the remaining
treebranch currents are clearly expressible as appropriate superpositions
of these loop currents, and hence are uniquely determined by the link
currents, as predicted earlier.
If the branch currents in the network graph of Fig. 5(a) are denoted
byii,i2* • • •, js, numbered to correspond to the branch numbering, and
if the loop currents of the graph of Fig. 6(a) are denoted by ii , i2, it, i4t
then we can make the identifications
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ji = ii, 32 = t2, j3 = i3, J 4 = U (3)
12 NETWORK GEOMETRY AND NETWORK VARIABLES
Through comparison of Figs. 5(a) and 6(a) one can then readily express
the remaining treebranch currents as appropriate superpositions of
the loop currents, thus,
(4;
36
= *1
 *4
= i2
 l'i
37
= *S
 *S
3a
= *4
~ *3
or, being mindful of the relations 3, have
3i
3 a
= 32
3i
37
h
3$
3*
33
(5)
These last four equations express the treebranch currents, uniquely
and unambiguously, in terms of the link currents. Thus, of the eight
branch currents in the graph of Fig. 5(a), only four are geometrically
independent. These four are appropriate to the set of links associated
with any selected tree. For the tree of Fig. 5(b), the link currents are
h,h,U For tne tree of Fi6 5(c) tliey areji, h,j5,h Here we may
write, in place of Eqs. 3,
3i = *ii h = t2, 3s = h, h = U (6)
These loop currents circulate on the contours indicated in Fig. 6(b),
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which again are found through inserting, one at a time, the branches
1, 2, 5, 7 into the tree of Fig. 5(c). The treebranch currents in this
case are expressed in terms of the loop currents by the relations
33 = i2 + U
ji  *'i  i3
ja = *2 — *i
js = ii — H — *3 — U
which are found by inspection of Figs. 5(a) and 6(b) through noting that
the currents in the tree branches result from the superposition of per
tinent loop currents.
X
THE CONCEPT OF LOOP CURRENTS
13
Through substitution of Eqs. 6 into 7, one again obtains the tree
branch currents expressed in terms of the link currents
J3 = h + fa
ji = ji — js
h = H — ji
js = ji — j2 ~ Js ~ ji
thus making evident once more the fact that only four of the eight
branch currents are geometrically independent.
The reader is cautioned against concluding that any four of the eight
branch currents may be regarded as an independent set. The branches
pertaining to a set of independent currents must be the links associated
with a tree, for it is this circumstance that assures the independence of
the currents. Thus the branch currents js, ja, jV, j$, for example, could,
not be a set of independent currents because the remaining branches
1, 2, 3, 4 do not form a tree. The concept of a tree is recognized as useful
because it yields a simple and unambiguous method of deciding whether
any selected set of branch currents is an independent one. Or one can
say that the tree concept provides a straightforward method of deter
mining a possible set of independent current variables for any given
network geometry.
Part (d) of Fig. 5 shows still another possible choice for a tree appropri
ate to the graph of part (a), and in Fig. 6(c) is shown the corresponding
set of loops. In this case one has
j* = *i, js = *2, ji = 13, js = ii (9)
and through superposition there follows that
ji = ii + *2 = ji + js
32 = n — H — U = Ji — 37 — Js
J3 = *1 — U = ji — Js
ja = — *2 — h — ii = —js — ji — js
When dealing with networks having large numbers of branches and
correspondingly elaborate geometries, one must have a less cumbersome
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and more systematic procedure for obtaining the algebraic relationships
14
NETWORK GEOMETRY AND NETWORK VARIABLES
between the branch currents and the loopcurrent variables. Thus it is
readily appreciated that the process of drawing and numbering the
reference arrows for the loops, and subsequently obtaining by inspection
the appropriate expressions for the branch currents as algebraic sums
of pertinent loop currents, can become both tedious and confusing in
situations involving complex geometries.
A systematic way of indicating the loops associated with the selection
of a particular tree is had through use of a~schedule such as 11, which
Loop
Branch No.
No.
1
2
3
4
5
6
7
8
1
1
1
1
1
1
2
1
1
1
1
3
1
1
1
4
1
1
1
(11)
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pertains to the graph of Fig. 5(a) with the tree of part (c) and hence
for the loops shown in Fig. 6(b). To interpret this schedule we note that
the first row, pertaining to loop 1, indicates that a circuit around this
loop is equivalent to traversing in the positive reference direction,
branches 1, 4, and 8, and, in the negative reference direction, branch 6.
None of the remaining branches participate in forming the contour of
loop 1, and so their corresponding spaces in the first row of the schedule
are filled in with zeros. The second row is similarly constructed, noting
that the pertinent loop contour is formed through traversing branches
2, 3, and 6 positively, and branch 8 negatively. Thus the successive
rows in this schedule indicate the confluent sets of branches that partici
THE CONCEPT OF LOOP CURRENTS
15
schedule 11 in this way yields the equations
h  ii j5 = *a
h = *a' is = —*i + t.2
J3 = *2 + U h = *4
(12)
j4 = *1 — J3 Js = *1 — *2 —
*3  *4
which are seen to agree with Eqs. 6 and 7.
The reason why this schedule has the property just mentioned may
best be seen through supposing that it is originally constructed, by
columns, according to the relationships expressed in Eqs. 12. One subse
quently can appreciate why the resulting rows of the schedule indicate
the pertinent closed paths, through noting that the nonzero elements
of a row are associated with branches traversed by the same loop cur
rent, and these collectively must form the closed path in question.
The actual construction of the schedule may thus be done in either of
two ways, viz.: by rows, according to a set of independent closed paths
(for example, those associated with a selected tree), or by columns,
according to a set of equations expressing branch currents in terms of
loop currents. If constructed by columns, the rows of the schedule
automatically indicate the closed paths upon which the associated loop
currents circulate; and, if constructed by rows from a given set of closed
paths, the columns of the resulting schedule automatically yield the
pertinent relations for the branch currents in terms of the loop currents.
This type of schedule (which for reasons given later is called a tieset
acheduie) is thus revealed to be a compact and effective means for indi
cating both the geometrical structure of the closed paths and the result
ing algebraic relations between branch currents and loop currents.
Regarding this relationship, one may initially be concerned about its
uniqueness, since there are fewer loop currents than branch currents.
Thus, if asked to solve Eqs. 12 for the loop currents in terms of branch
currents, one might be puzzled by the fact that there are more equations
than unknowns. However, the number of independent equations among
this set just equals the number of unknown loop currents (for reasons
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given in the preceding discussion), and the equations collectively form
a consistent set. Therefore the desired solution is effected through sep
arating from Eqs. 12 an independent subset and solving these. Knowing
that the equations were originally obtained through choice of the tree
of Fig. 5(c), thus designating branch currents j\, j2, js, j\ as a possible
independent set, indicates that the corresponding equations among those
given by 12 may be regarded as an independent subset. These yield the
16 NETWORK GEOMETRY AND NETWORK VARIABLES
identifications ii = ji, i2 = jv, i3 = js, {4 = jV as indicated in Eqs. 6
for this choice of tree.
It is, however, not essential that the independent subset chosen from
Eqs. 12 be this particular one. Thus, if we consider the tree of Fig. 5(d)
as a possible choice, it becomes clear that branch currents j4t js, j7, js
are an independent set. The corresponding equations separated from 12,
namely,
3* = *i  k
h = i4
js = t'i — *2 — *3 — *4
(13)
may alternatively be regarded as an appropriate independent subset.
Their solution reads
t'i = j* + j5
ii4*A (14)
*3 = J5
*4 = jl
Noting from Eqs. 8 that j4 + j5 = ji , and that j4 — j7 ~ js = J2, it
is clear that 14 agrees with the former result.
Four of the eight Eqs. 12 are independent. A simple rule for picking
four independent ones is to choose those corresponding to the link cur
rents associated with a possible tree. Any four independent ones may
be solved for the four loop currents. Substitution of these solutions
into the remaining equations then yields the previously discussed rela
tions between treebranch currents and link currents.
There should be no difficulty in understanding this situation since the
previous discussion has made it amply clear that the link currents or loop
currents are an independent set and all other branch currents are uniquely
related to these. Equations ETare consistent with this viewpoint and
contain all of the implicit and explicit relations pertinent thereto. Hence
their solution cannot fail to be unique, no matter what specific approach
one may take to gain this end.
Although a schedule like 11 may be constructed either by columns or
by rows, the usual viewpoint will be that it is constructed by rows from
an observation of those sets of confluent branches forming the pertinent
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closed paths. The latter are placed in evidence, one by one, through
imagining that all the links are opened except one, thus forcing all but
one of the link or loop currents to be zero. The existence of a single loop
THE CONCEPT OF NODEPAIR VOLTAGES 17
current energizes a set of branches forming the closed path on which
this loop current circulates. This set of branches, called a tie set, is
indicated by the elements in the pertinent row of the tieset schedule!
If the geometry of the network graph permits its mappability upon a
plane or spherical surface without crossed branches, then we may regard
any tie set as forming a boundary that divides the total network into two
portions.* Hence, if the branches in such a set are imagined to shrink
longitudinally until they reduce to a single point, the network becomes
"tied off" so to speak (as a fish net would by means of a draw string),
and the two portions bounded by the tie set become effectively separated
except for a common node. It is this interpretation of the tie set that
suggests its name.
Although there are several important variations in this procedure for
establishing an appropriate set of current variables, we shall leave these
for subsequent discussion and turn our attention now to the alternate
procedure (dual to the one just described) of formulating a set of net
work variables on a voltage basis.
6 The Concept of NodePair Voltages; Cut Sets and CutSet
Schedules
On the voltage side of the network picture, an entirely analogous
situation prevails. Here we begin by regarding the treebranch voltages
as a possible set of independent variables in terms of which the state of a
network may uniquely be expressed. Since the tree branches connect
all of the nodes, it is possible to trace a path from any node to any other
node in the network by traversing tree branches alone; and therefore it
is possible to express the difference in potential between any pair of
nodes in terms of the treebranch voltages alone. Moreover, ^the path
connecting any two nodes via tree branches is unique since the tree has
no closed loops and hence offers no alternate paths between node pairs.
Therefore, the potential difference between any two nodes, referred to
as the pertinent nodepair voltage, is uniquely expressible in terms of the
treebranch voltages. The link voltages, which are a particular set
of nodepair voltages, are thus recognized to be uniquely expressible in
terms of the treebranch voltages.
Let us illustrate these principles with the network graph of Fig. 5(a),
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and choose initially the tree given in part (b) of this same figure. If the
branch voltages are denoted by vit v2, . ••, t>s, numbered to correspond
to the given branch numbering, then the quantities t>5, v6, v7, v& are the
* For a graph not mappable on a sphere (for example one that requires a doughnut
shaped surface), some but rot all tie sets have this property. This point is discussed
further in Art. 9.
18 NETWORK GEOMETRY AND NETWORK VARIABLES
treebranch voltages and hence may be regarded as an independent set.
They may simultaneously be regarded as nodepair voltages, and, since
they are to serve as the chosen set of variables, we distinguish them
through an appropriate notation and write
ei = t>5, e2 = v6, e3 = v7, e4 = t>s (15)
This part of the procedure parallels the use of a separate notation for the
loop currents t'i, t'2, • • • when choosing variables on a current basis.
There the link currents are identified with loop currents; in Eqs. 15 the
treebranch voltages are identified with nodepair voltages.
The remaining branch voltages, namely the link voltages, are now
readily expressible in terms of the four treebranch or nodepair voltages
15. Thus, by inspection of Fig. 5(a) we have
t>i = —fa + v 6 = —ei + e2
v2 = — f6 + v7 = — e2 + e3
(16)
v3 = —v7 + t>s" ~e3 + e4
v4 = t>s + v5 = —e4 + ei
The procedure in writing these equations is to regard each link voltage
as a potential difference between the nodes terminating the pertinent
link, and to pass from one of these nodes to the other via tree branches
only, adding algebraically the several treebranch voltages encountered.
If the tree of Fig. 5(c) is chosen, the branch voltages v3, v4t va, vS
become the appropriate independent set, and we make the identifications
ei = v3, e2 = v4, e3 = v6, e4 = t>s (17)
The expressions for the link voltages in terms of these read
vi = v4 + v6 — vs = e2 + e3 — e4
v2 = — t>3 — t>6 + v 8 = — ei —63 + 64
(18)
f 5 = v 4 + vs = e2 + e4
v7 = v3 + v s = ei + e4
The results expressed in Eqs. 16 and 18 bear out the truth of a state
ment made in Art. 4 to the effect that any set of treebranch voltages
may be regarded as an independent group of variables in terms of which
the remaining branch voltages (link voltages) are uniquely expressible.
In the network graph of Fig. 5, any tree has four branches. Hence, of
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the eight branch voltages, only four are geometrically independent.
THE CONCEPT OF NODEPAIR VOLTAGES
19
These may be the ones pertinent to any selected tree; and the rest are
readily expressed in terms of them.
In dealing with more complex network geometries it becomes useful to
establish a systematic procedure for the selection of nodepair voltage
variables and for the unique expression of the branch voltages in terms
of them. The accomplishment of this end follows a pattern that is
entirely analogous (yet dual) to that described in the previous article
for the current basis. [That is to say, we seek to construct a schedule
appropriate to the voltage basis in the same way that the tieset schedule
is pertinent to the current basis. To this end we must first establish the
geometrical interpretation for a set of branches which, for the voltage
basis, plays a role analogous to that defined for the current basis by a tie
set (or confluent set of branches forming a closed loop). The latter is
placed in evidence through opening all of the links but one, so that all
loop currents are zero except one. The analogous procedure on a voltage
basis is to force all but one of the nodepair (i.e., treebranch) voltages
to be zero, which is accomplished through shortcircuiting all but one
of the tree branches. This act will in general simultaneously short
circuit some of the links, but there will in any nontrivial case be left
some links in addition to the one nonshortcircuited tree branch that
are likewise not shortcircuited and will appear to form connecting links
between the pair of nodes terminating the pertinent tree branch. This
set of branches, which is called a cut set, is the desired analogue of a tie
set, as the following detailed elaboration will clarify.
Consider again the network of Fig. 5(a) and the tree of part (b) of
this figure, together with the pertinent stipulation of nodepair voltages
as expressed by Eqs. 15. The cutset schedule appropriate' to this situ
ation reads as given in 19.
Node
Pair
No.
5
ALTERNATIVE METHODS OF CHOOSING CURRENT VARIABLES 23
but Eqs. 16 show that
» 1 + »5 = t'6
v i + v 2 + vs = v7 (22)
»1 + »2 + »3 + »5 = v3
Hence the solutions 21 again agree with the definitions 15.
Of Eqs. 20, four are independent. Not any four are independent, but
there are no more than four independent ones in this group, and there
are several different sets of four independent ones that can be found
among them. A simple rule for picking four independent ones is to
choose those corresponding to the branch voltages of a possible tree.
The solution to these yields the expressions for the e's in terms of the v's;
and substitution of these solutions for the e's into the remaining equa
tions yields the previously discussed relations between link voltages and
treebranch voltages. The cutset schedule which contains the informa
tion regarding the geometrical character of the cut sets, as well as the
algebraic relationships between the implied nodepair voltages and the
branch voltages, is thus seen to be a compact and effective mode of
expressing these things. It does for the formulation of variables on the
voltage basis what the tieset schedule does for the establishment of a
system of variables on the current basis. Continued use will be made of
both types of schedules in the following discussions.
7 Alternative Methods of Choosing Current Variables
The procedure for selecting an appropriate set of independent current
variables in a given network problem can be approached in a different
manner which may sometimes be preferred. Thus, the method given in
Art. 5, which identifies the link currents with a set of loopcurrent
l
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Fig. 8. A graph with meshes chosen as loops, and two possible trees.
24 NETWORK GEOMETRY AND NETWORK VARIABLES
variables, leaves the tie sets or closed paths upon which these currents
circulate to be determined from the choice of a tree, whereas one may
prefer to specify a set of closed paths for the loop currents at the outset.
Consider in this connection the graph of Fig. 8. In addition to provid
ing the branches with numbers and reference arrows, a set of loops have
also been chosen and designated with the circulatory arrows numbered
1, 2, 3, 4. These loops, incidentally, are referred to as meshes because
they have the appearance of the meshes in a fish net. It is a common
practice in network analysis to choose, as a set of current variables, the
currents that are assumed to circulate on the contours of these meshes.
Having made such a choice, we must know how to relate in an unam
biguous and reversible manner, the branch currents to the chosen mesh
currents.
This end is accomplished through setting down the tieset schedule
corresponding to the choice made for the closed paths defining the tie
sets. With reference to the graph of Fig. 8 one has, by inspection,
schedule 23 and the columns yield
Mesh
Branch No.
No.
1
2
3
4
5
6
7
8
9
1
1
1
2
1
1
1
1
3
1
1
1
1
4
1
1
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ALTERNATIVE METHODS OF CHOOSING CURRENT VARIABLES 25
*i = 3i
i2 = ji + 32
. . 4. . 4. . (25)
*3 = 3i + 32 + 33
U = ji + 32 + 33 + j4
Substitution into the remaining Eqs. 24, gives
35 = —ji — 32 — 33 ~ 3*
36 = ii + 32 + 33
37=31+32 (26)
3s = — ji — J2
J9 = — ji ~ 32 — 33
These express the treebranch currents in terms of the link currents.
If instead, we choose tree 2 in Fig. 8, the branches 1, 5, 8, 9 become
links. The corresponding equations in group 24, namely,
i i = Ji
U = ~js
(27)
*2 = ~3S
H = ~jg
are independent and give the expressions for the mesh currents in terms
of the link currents. With these, the remaining Eqs. 24 yield again the
treebranch currents in terms of the link currents, thus:
32 = ~ji — j$
33 = js — 3g
j* = ~js + ji (28)
i6 = —J9
37 = —js
It is readily seen that the results expressed by Eqs. 25 and 26 are con
sistent with those given by Eqs. 27 and 28. That is to say, the choice^of
a tree has nothing to do with the algebraic relations between the loop
currents and the branch currents.It merely serves as a convenient way
of establishing an independent subset among Eqs. 24. In the present
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very simple example, one can just as easily pick an independent subset
26 NETWORK GEOMETRY AND NETWORK VARIABLES
without the aid of the tree concept; however, in more complex problems
the latter can prove very useful.
In approaching the establishment of a set of current variables through
making at the outset a choice of closed paths, a difficulty arises in that
the independence of these paths is in general not assured. A necessary
(though not sufficient) condition is that all branches must participate in
forming these paths, for, if one or more of the branches were not traversed
by loop currents, then the currents in these branches in addition to the
loop currents would appear to be independent. Actually, the loop cur
rents chosen in this manner could not be independent since altogether
there can be only I independent currents.
A sufficient (though not necessary) procedure to insure the inde
pendence of the closed paths (tie sets) is to select them successively in
i such a way that each additional path involves at least nrift braiich that
1 is not part of any of the previously selected paths. This statement fol
lows from the fact that the paths or tie sets form an independent set
if the I rows in the associated tieset schedule are independent: that is,
if it is not possible to express any row in this schedule as a linear com
bination of the other rows. If, as we write down the successive rows in
this schedule, each new row involves a branch that has not appeared in
any of the previous rows, that row can surely not
be formed from a linear combination of those already
chosen, and hence must be independent of them.
A glance at schedule 23 shows that this principle
is met. Thus, construction of the first row involves
only branches 1 and 2. The second row introduces
the additional branches 3, 7, 8; the third row
adds branches 4, 6, and 9, and the last row involves
the previously unused branch 5. It is not difficult
to convince oneself that, if one designates only meshes
as closed paths (which is, of course, possible only in a
graph that is mappable on a plane or sphere), then
the rows in the associated tieset schedule can always
be written in such a sequence that the principle just
described will be met. This simple choice in a plane
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mappable graph, therefore, always assures the in
dependence of the closed paths and hence does the
same for the implied meshcurrent variables.
However, it is quite possible for the I rows in a tieset schedule to be
independent while not fulfilling the property just pointed out. Thus,
as already stated above, this property of the rows is a sufficient though
not necessary condition to insure their independence. When closed
Fig. 9. A modified
choice of loops for
the graph of Fig. 8
that turns out not
to form an inde
pendent set.
ALTERNATIVE METHODS OF CHOOSING CURRENT VARIABLES 27
paths are chosen in a more general manner, as they sometimes may be, it
is not always evident at the outset whether the choice made is acceptable.
To illustrate this point, let us reconsider the network graph of Fig. 8
with the choice of closed paths shown in Fig. 9. The tieset schedule
reads as in 29, and hence the expressions for the branch currents in terms
Loop
Branch No.
No.
1
2
3
4
5
6
7
8
9
1
1
1
1
1
2
1
1
1
1
1
3
1
1
1
1
4
1
1
1
1
1
1
(29)
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1
of the loop currents are
ji = *i + h j4 = — i* k = t'i + i2 + U
k = *.2 k = ~*3  U is = *i ~ *2  U (30)
k = *i + *3 k = *2 + h + U k = ~*2  i3  U
as may also be verified by inspection of Fig. 9.
To investigate the independence of the chosen loops, we observe that
the choice of tree 1 in Fig. 8 indicates that the branch currents ji, j2, k, k
form an independent set. Hence the first four of Eqs. 30 should be
independent. They obviously are not, since the righthand members
of the second and fourth equations are identical except for a change in
algebraic sign. Hence the loops indicated in Fig. 9 are not an inde
28 NETWORK GEOMETRY AND NETWORK VARIABLES
independent set of loops (or tie sets) is in general not a matter that is
evident by inspection, although one has a straightforward procedure for
checking a given selection. Namely, the chosen set of loops are inde
pendent if the I rows of the associated tieset schedule are independent;
and they are, if it is possible to find in this schedule a subset of I inde
pendent columns (i.e., I independent equations among a set like 30).
The simplest procedure for making this check among the columns is to
pick those columns corresponding to the links of any chosen tree. These
must be independent if the I rows of the schedule are to be independent.
They are if the pertinent equations (like the first four of 30 in the test
discussed in the previous paragraph) have unique solutions. Usually
one can readily see by inspection whether or not such solutions exist.
An elegant algebraic method is to see if the determinant of these equa
tions is nonzero. Thus the nonvanishing of the determinant formed
from the subset of columns corresponding to the links of a chosen tree
suffices to prove the independence of an arbitrarily selected set of closed
paths.
In the case of graphs having many branches this method may prove
tedious, and so it is useful to be aware of alternative procedures for
arriving at more general currentvariable definitions, should this be
desirable. Thus one may make use of the fact that the most general
tieset schedule is obtainable through successive elementary transforma
tions of the rows of any given one, and that such transformations leave
the independence of the rows invariant. We may, for example, start
with a schedule like 23 that is based upon a choice of meshes so that its
rows are surely independent. Suppose we construct a new first row
through adding to the elements of the present one the respective elements
of the second row. The new schedule is then as shown in 31.
Loop
No.
Branch No.
1
2
3
4
6
7
8
9
1
1
1
1
1
2
1
1
1
1
3
1
1
1
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5
ALTERNATIVE METHODS OF CHOOSING CURRENT VARIABLES 29
Loops 2, 3, 4 are still the meshes 2, 3, 4 of Fig. 8. However, loop 1
is now the combined contour of meshes 1 and 2, as a comparison of the
first row of the new schedule with the graph of Fig. 8 reveals. If we
modify this new schedule further by constructing a new second row with
elements equal to the sum of the respective ones of the present rows
2, 3, and 4, there results another schedule that implies a loop 2 with the
combined contours of meshes 2, 3, and 4. It should thus be clear that
more general loops or tie sets are readily formed through combining
linearly a set of existing simple ones. So long as only one new row is
constructed from the combination of rows in a given schedule, and if the
pertinent old row is a constituent part of this combination, the procedure
cannot destroy the independence of a given set of rows.
Each new schedule has the property that its columns correctly yield
the expressions for the branch currents in terms of the implied new loop
currents. That is to say, since transformation of the schedule through
making linear row combinations implies a revision in the choice of loops,
it likewise implies a revision in the algebraic definitions of the loop cur
rents. Nevertheless the relations expressing the branch currents in
terms of these new loop currents is still given by the coefficients in the
columns of the schedule. For example, we would get for schedule 31
the relations
ii = i'i j4 = i'z + i\ h = t'i + i'2
h = i'2 js = ~i\ js = t'i  i'2 (32)
J3 = *''i  i'2 + i'z j& = i'z jg = i'z
where primes are used on the i's to distinguish them from those in Eqs.
24 which are pertinent to schedule 23.
Comparison of Eqs. 24 and 32 reveals the transformation in the loop
currents implied by the transformation of schedule 23 to the form 31,
namely:
ii = t'i
t'2 = t'i + i'2
. ., (33)
*3 = t 3
i4 = i'4
This result is at first sight somewhat unexpected. Thus the transforma
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tion from schedule 23 to schedule 31 implies leaving the contours for the
loop currents i2, iz, U the same as in the graph of Fig. 8, but changes the
contour for loop current *i. Offhand we would expect the algebraic
30 NETWORK GEOMETRY AND NETWORK VARIABLES
definition for t'i to change and those for i2,%3, and i4 to remain the same.
Instead we see from Eqs. 33 that t'i, ^3, and %4 are unchanged while tj
changes. Nevertheless Eqs. 32 are correct, as we can readily verify
through sketching in Fig. 8 the altered contour for loop 1 and expressing
the branch currents as linear superpositions of the loop currents, noting
this altered path for t'i. The results expressed by Eqs. 33, therefore, are
surely correct also.
The mental confusion temporarily created by this result disappears
if we concentrate our attention upon schedule 23 and Eqs. 24 and ask
ourselves: What change in relations 24 will bring about the addition of
row 2 to row 1 in schedule 23 and leave rows 2, 3, and 4 unchanged?
The answer is that we must replace the symbol i2 by t'i + i2, for then
every element in row 2 will also appear in row 1, in addition to the ele
ments that are already in row 1, and nothing else will change. The lesson
to be learned from this example is that we should not expect a simple and
obvious connection between the contours chosen for loop currents and
the algebraic definitions for these currents, nor should we expect to be
able to correlate by inspection changes in the chosen contours (tie sets)
and corresponding transformations in the loop currents until experience
with these matters has given us an adequate insight into the rather
subtle relationships implied by such transformations.
The reason for our being misled in the first place is that we are too
prone to regard the choice of contours for loop currents as equivalent to
their definition in terms of the branch currents, whereas in reality the
fixing of these contours merely implies the algebraic relationships be
tween the loop currents and branch currents (through fixing the tieset
schedule); it does not place them in evidence.
The most general form a linear transformation of the tieset schedule
may take is indicated through writing in place of 33
t'i = ani\ + ai2i'2 H h otiii'i
*a = «2i*'i + a22^2 + • • • + a2[i'i
(34)
it = + az2*'2 + • • • + aid'1
in which the as are any real numbers. If t'i • • • ii are an independent
set of current variables, then t'i • • • t'j will be independent if Eqs. 34
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are independent; that is, if they possess unique solutions (which they
ALTERNATIVE METHODS OF CHOOSING CURRENT VARIABLES 31
will if their determinant is nonzero). In general the currents i\ • • • i'i
will no longer have the significance of circulatory currents or loop cur
rents, although for convenience they may still be referred to by that
name. They will turn out to be some linear combinations of the branch
currents.
If such a very general set of definitions for the loop currents is desired,
one can approach the construction of an appropriate tieset schedule
directly from this point of view, which we will illustrate for the network
graph of Fig. 8. Thus let us suppose that one wishes to introduce current
variables which are the following linear combinations of the branch
currents:
*1 = ~ji + 32  33 + 3*  3j9
*2 = k + 273 + k  js
(35)
*a = 3i + h + 3s + h + J9
*4 = 32 + 2ji + j6 + ja
The first step is to rewrite these expressions in terms of I (in this case
four) branch currents. To do this we may follow the usual scheme of
picking a tree and finding the relations for the treebranch currents in
terms of the link currents. For tree 1 of Fig. 8, these are given by Eqs.
26. Their use transforms Eqs. 35 into
t, = 2k + 4j2 + 2j3 + lj4
H = 2i, + 3j2 + 373 + Q74
(36)
13 = Q/i  \j2  lj3  ljt
U = Oji + lj'2 + lis + 2j4
having the solutions
3i = Oii + \i2 + 3i3 + §*4
h — 3*1 — 2*2 — 2*3 — 1*4
(37)
h =  hi + 2*2  2*3 + 0*4
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ji = Oii + 0t2 + lt3 + 1*4
32 NETWORK GEOMETRY AND NETWORK VARIABLES
Using Eqs. 26 again we have the additional relations
is = Oii ~ 5*2  2i3  §u
h = Oii + £i2 + 1*3 + \U
h = K + 0i2 + \ i3 + hi (38)
Js =  5*1 + °*2  §*3  5*4
J9 = Oii  \i2  lis  h*
The results in Eqs. 37 and 38 yield tieset schedule 39, which more
Loop
No.
Branch No.
1
2
3
4
5
6
7
8
9
1
h
i
1
h
2
i
i
i
i
i
3
3
!
i
1
2
i
a
1
2
4
§
l
1
§
i
J
i
h
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compactly contains this same information. This is the schedule that is
implied by definitions 35 for the loopcurrent variables, which no longer
possess the geometrical interpretation of being circulatory currents.
As we shall see in the following chapter, the tieset schedule plays an
important role in the formulation of the equilibrium equations appropri
ate to the chosen definitions for the current variables. The present dis
cussions, therefore, provide the basis for accommodating such a choice,
regardless of its generality or mode of inception. Thus we have shown
that the process of selecting an appropriate set of current variables can
take one of essentially three different forms:
1. The approach through choice of a tree and identification of the
ALTERNATIVE METHODS OF CHOOSING VOLTAGE VARIABLES 33
the meshes of a mappable network), but no facile control is had regarding
the associated algebraic definitions of the loop currents.
3. The approach through making an initial and arbitrarily general
choice for the algebraic definitions of the current variables (like those
given by Eqs. 35). In this case the variables no longer possess the simple
geometrical significance of circulatory currents. This approach will
probably seldom be used, and is given largely for the sake of its theo
retical interest.
8 Alternative Methods of Choosing Voltage Variables
When voltages are chosen as variables, we similarly have three possible
variations which the form of the approach may take. The first, which is
discussed in Art. 6, proceeds through choice of a tree and the identifica
tion of treebranch voltages with nodepair voltage variables. In this
process (like procedure 1 mentioned above for the choice of current vari
ables), the algebraic definitions for the nodepair voltages are as simple
as they can be, but little or no direct control can be exercised over the
geometrical distribution of node pairs. A second form of procedure,
which permits a forthright choice of node pairs
at the outset, and a third, in which the process
is initiated through an arbitrarily general choice
for the algebraic definitions of the voltage vari
ables, are now presented in detail.
To illustrate how a designation of nodepair
voltage variables may be approached through
the initial selection of an appropriate set of
node pairs, let us consider the network of Fig. 8.
In Fig. 10 are indicated the nodes of this net
work, lettered a, b, • • f for ease of reference,
and a system of lines with arrowheads intended
to indicate a choice of node pairs and reference
directions for the voltage variables ei, e2, e5.
These arrows are not to be confused with
branches of the network; yet, if we momentarily
think of them as such, we notice that the structure in Fig. 10 has the
characteristics of a tree, for it connects all of the nodes, and involves the
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smallest number of branches needed to accomplish this end. Hence
this choice for the variables ei • • • e5 is an appropriate one since the
variables surely form an independent set, and their number equals the
number of branches in any tree associated with a network having
these nodes. In making a forthright choice of node pairs it is
sufficient to see to it that the system of reference arrows accompanying
Fig. 10. A possible
choice of nodepair vol
tages for the graph of
Fig. 8.
34
NETWORK GEOMETRY AND NETWORK VARIABLES
this choice (whether actually drawn or merely implied) forms a structure
that has a treelike character.
Using the principles set forth in Art. 6, one can construct cutset
schedule 40 appropriate to the choice of node pairs indicated in Fig. 10
Node
Pair
No.
Branch No.
Picked
1
2
3
4
5
6
7
8
9
Up
Nodes
1
1
1
1
d
2
1
1
1
a
3
1
1
1
1
1
1
a, e
4
1
1
1
1
1
1
b, d
5
1
1
1
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ALTERNATIVE METHODS OF CHOOSING VOLTAGE VARIABLES 35
fi = —ei +62 + 63^465
t>6 = —64 — 65
= 61 + e2 + e3  e4 — e5
v7 = e2  e3 + e4 + es
t>3 = 63 + 64 + e5
v& = 61 + e3 — e4  e5 (41)
f4 = es
vg  —c3 + es
The correctness of these may readily be checked with reference to Figs. 8
and 10, remembering again that the v's are drops and the e's are rises.
For example, vi is the voltage drop from node a to node d. If we pass
from a to d via the system of nodepair voltage arrows in Fig. 10, we
observe that we first traverse the arrows for e2 and e3 counterfluently,
and then the arrows for e5, e^, and ei confluently. Since confluence indi
cates a rise in voltage, the terms for ci, e4, and e5 are negative. There
should be no difficulty in thus verifying the remaining equations in set 41.
One could have written Eqs. 41 from inspection of Figs. 8 and 10 to
start with and thus constructed schedule 40 by columns, whence the
rows would automatically yield the cut sets. This part of the procedure
is thus seen to be the same as with the alternate approach given in Art. 6.
So is the matter regarding the solution of Eqs. 41 for the nodepair
voltages in terms of the branch voltages. One selects any five independ
ent equations from this group and solves them. Again the selection of a
tree in the associated network graph (such as tree 1 or 2 in Fig. 8) is a
quick and sure way to spot an independent subset among Eqs. 41, and
the remaining ones will then yield the appropriate expressions for the
link voltages in terms of treebranch voltages, as discussed previously.
In this method of approach to the problem of defining an appropriate
set of independent voltage variables, a rather common procedure is to
choose the potential of one arbitrarily selected node as a reference and
designate as variables the potentials ei . • • en of the remaining nodes with
respect to this reference. Thus, one node serves as a datum or reference,
and the node pairs defining the variables ei • • • en all have this datum
node in common. The quantities ei • • • en in this arrangement are
spoken of as node potentials and are referred to as a "nodetodatum"
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set of voltage variables.
The rather simplified choice of node pairs implied in this specialized
procedure is in a sense the parallel of choosing meshes for loops in the
specification of current variables. This theme is elaborated upon in
Art. 9 where the dual character of the loop and node procedures is stressed
and the implications of this duality are partially evaluated.
36 NETWORK GEOMETRY AND NETWORK VARIABLES
The equivalent of Fig. 10 for a choice of nodepair voltages of this sort
is shown in Fig. 11, pertinent to the network graph of Fig. 8. Again, for
the moment regarding the arrows in this diagram as branches, we see
that it has treelike character and hence
that such a nodetodatum set of voltages
is always an independent one.
The cut sets appropriate to this group of
nodepair voltages are particularly easy to
find since we observe that setting all but
one of the nodepair voltages equal to zero
causes all of the nodes to coincide at the
datum except the one at the tip end of
the nonzero voltage. Hence the branches
divergent from this single node form the
pertinent cut set. With reference to Fig. 8,
cutset schedule 42 is thus readily obtained.
Since the nodepair voltages are the po
tentials of the separate nodes with respect
to a common datum, each branch voltage drop is given by the
difference of two node potentials, namely those associated with
the nodes terminating the pertinent branch. If the latter touches the
datum node, then its voltage drop is given by a single node potential
Fio. 11. A nodetodatum
choice of nodepair voltages
for the graph of Fig. 8.
Node
Branch No.
Picked
No.
Up
Nodes
1
2
3
4
6
7
8
9
1
1
1
1
/
2
1
1
1
e
3
1
1
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5
ALTERNATIVE METHODS OF CHOOSING VOLTAGE VARIABLES 37
potentials are thus formed either by inspection of Figs. 8 and 11 or
from the columns of schedule 42 to be
fi = e3 + e5 f4 = ei IV = e4  e5
v2 = —e3 + e5 t>5 = —Cj t>s = e2 — e3 (43)
t>3 " —e2 + c4 = —e4 Ug = ei — e2
The node potentials in terms of the branch voltages are found from
these by the usual process of selecting from these equations a subset of
five independent ones. According to tree 1 of Fig. 8, the last five are
such a subset. Their solution yields
ei = vs
c2 = v5 — vg
63 = —t>«  vS  vg (44)
e4 = v6
and the remaining equations in set 43 then give the following expressions
for the link voltages in terms of the treebranch voltages
vi = v5  t>6  v7 + t>s + vg
v2 = v 5 — t>6 — v7 + l>s + vg
(45)
v3 = v5 — v6 + vg
v4 = l>5
It is interesting to observe how more general nodepair voltage defini
tions are derivable from the simple nodetodatum set through carrying
out linear transformations on the rows of cutset schedule 42. Thus,
suppose we form from this one a new schedule through adding the ele
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ments of the second row of 42 to the respective ones of the first row,
1
e
3
1
1
1
d
4
1
1
1
b
5
1
1
1
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38 NETWORK GEOMETRY AND NETWORK VARIABLES
Node
Pair
No.
Branch No.
Picked
Up
Nodes
1
2
3
4
5
6
7
8
9
1
1
1
1
1
e,f
2
1
1
ALTERNATIVE METHODS OF CHOOSING VOLTAGE VARIABLES 39
This result suggests that the nodepair voltage diagram has changed
from the form shown in Fig. 11 to that shown in Fig. 12, since the poten
tial of node e with respect to the datum (which in Fig. 11 is ez) now is
equal to the sum of e\ and e\. We note further that, when e\ is the
only nonzero voltage, nodes e and / coincide at the tip end of e'i; so the
associated cut set is found through picking up these two nodes, as is also
indicated in schedule 46. The pickedup nodes corresponding to the
remaining nodepair voltages evidently remain the same as before, and
hence the rest of the cut sets are unchanged.
Fig. 12. Revision in the nodepair volt Fio. 13. The graph of Fig. 8
age definitions of Fig. 11 corresponding with node designations as
to a transformation of cutset schedule given in Figs. 10, 11, and 12.
42 to form 46.
Other simple transformations in schedule 46 may similarly be inter
preted. For example, if row 3 is added to row 4, the pickedup nodes
for cut set 4 become d and b, which in Fig. 12 implies that the tail end
of e'3 shifts from the datum to node b, and we will find that now 63 ** e"s
+ e"4 where the double prime refers to the latest revision of the set of
nodepair voltages (the rest of the e's remain as in Eqs. 49 with double
primes on the righthand quantities).
One soon discovers upon carrying out additional row combinations
in schedules 42 or 46 that it is by no means always possible to associate
a nodepair voltage diagram like the ones in Figs. 10, 11, or 12 with the
resulting nodepair voltages, for the reason that some of these are likely
no longer to be simply potential differences between node pairs but
instead are more general linear combinations of the branch voltages.
The same is true if one constructs a cutset schedule (as is also a pos
sible procedure) by making arbitrary choices for the pickedup nodes.
To illustrate such a method we may consider again the graph of Fig. 8
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which is redrawn in Fig. 13 with the nodes lettered as in Figs. 10, 11,
40
NETWORK GEOMETRY AND NETWORK VARIABLES
and 12. Cutset schedule 50 is constructed by simply making an arbi
trary choice for the pickedup nodes relating to the pertinent cut sets.
The term "node pair" here retains only a nominal significance since we
are not at all assured that the implied voltage variables are potential
Node
Pair
No.
Branch No.
Picked
1
2
3
4
5
6
7
8
9
Up
Nodes
1
1
1
a, d
2
1
1
c,f
3
1
1
1
1
1
a, b, c
4
1
1
1
1
d, e
5
1
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ALTERNATIVE METHODS OF CHOOSING VOLTAGE VARIABLES 41
graph of Fig. 13 or 8. Picking tree 1 in Fig. 8 designates v5, v6, v7, vg, vg
as independent treebranch voltages and hence stipulates that the last
five equations in set 51 should be independent. It is readily seen that
they are, for they yield the solutions
ei = f7
e2 = t'6
*3 = »5 — "7 + H (52)
e* = v6  vg
e5 = ~v7 + vS
We may conclude that the cut sets in schedule 50 are independent, and
Eqs. 52 tell us what the implied voltage variables are in terms of the
branch voltages. The first two are simple potential differences between
nodes, but the remaining three are not. There is no reason why the
selected voltage variables have to be potential differences between
nodes. So long as they form an independent set, and we know the
algebraic relations between them and the branch voltages, they are
appropriate.
Lastly let us consider for the same network of Fig. 8 the following set
of independent linear combinations of the branch voltages as a starting
point:
«i =
—»i + v3 + vi + 2v6 + 2t,7 + 5vs + 5v9
e2 =
—»2 + »3 + «4 — v 5 + v6 + v 7 + 4»s + 4»9
e3 =
—»2 + 2«3  vs + vo + 3»s + 2f9
e* =
— vi + 2v3  v5 + v6 — v7 + 2vs + vg
e5 =
v3  v5 + v6
(53)
Through use of Eqs. 45 one can eliminate all but five of the branch volt
ages and get definitions 53 into the form
«\ = »5 + 2v6 + 3v7 + 4fs + 5t'9
«2 = v 6 + 2v7 + 3t's + 4»9
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e3 = » 7 + 2i>s + 3t'9 (54)
e* = vs + 2v9
ee = Vg
42 NETWORK GEOMETRY AND NETWORK VARIABLES
The solutions to these equations together with Eqs. 45 yield the complete
set of relations for the branch voltages in terms of ei • • • eg, thus
=
ei
 3e2 + 2e3
~ «5
t'3

ei
 3e2 + 2e3
 e5
»S
ei
 3e2 + 3e3
 ei + «s
v*

ei
 2e2 + e3
vs
=
e\
 2e2 + e3
»e

e2
 2e3 + e4

e3
 2«4 + e5
v»
=
2e5
v9

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ej
These results may be summarized in cutset schedule 56. Thus we see
Node
Pair
No.
Branch No.
1
2
3
4
5
6
7
8
9
1
1
1
1
1
1
2
3
3
3
2
2
1
DUALITY
43
to these two methods of approach. We wish now to call specific attention
to this aspect of our problem so that we may gain the circumspection
that later will enable us to make effective use of its implications. In a
word, this usefulness stems from the fact that two situations which, on
a current and voltage basis respectively, are entirely analogous, have
identical behavior patterns except for an interchange of the roles played
by voltage and current, while physically and geometrically they are
distinctly different. Not only can one recognize an obvious economy
in computational effort resulting from this fact since the analysis of only
one of two networks so related yields the behavior of both, but one can
sense as well that an understanding of these ideas may lead to other
important and practically useful applications, as indeed the later dis
cussions of our subject substantiate.
A careful review of the previous articles in this chapter shows that
essentially the same sequence of ideas and procedures characterizes
both the loop and the node methods, but with an interchange in pairs
of the principal quantities and concepts involved. Since the latter are
thus revealed to play a dual role, they are referred to as dual quantities
and concepts. First among such dual quantities are current and voltage;
and first among the dual concepts involved are meshes and nodes or
loops and node pairs. Since a zero current implies an open circuit and a
zero voltage a short circuit, these two physical constraints are seen to
be duals. The identification of loop currents with link currents and of
nodepair voltages with treebranch voltages shows that the links and
the tree branches likewise are dual quantities. The accompanying table
gives a more complete list of such pairs.
Dual Quantities or Concepts
Current
Branch current
Mesh or loop
Number of loops (Z)
Loop current
Mesh current
Link
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Link current
Treebranch current
Tie set
Short circuit
Parallel paths
Voltage
Branch voltage
Node or node pair
Number of node pairs (n)
Nodepair voltage
Node potential
Tree branch
Treebranch voltage
Link voltage
Cut set
Open circuit
Series paths
It should be emphasized that duality is strictly a mutual relationship.
There is no reason why any pair of quantities in the table cannot be
interchanged, although each column as written associates those quan
44
NETWORK GEOMETRY AND NETWORK VARIABLES
tities and concepts that are pertinent to one of the two procedures
commonly referred to as the loop and node methods of analysis.
Two network graphs are said to be duals if the characterization of one
on the loop basis leads to results identical in form with those obtained
for the characterization of the other on the node basis. Both graphs will
have the same number of branches, but the number of tree branches in
one equals the number of links in the other; or the number of inde
pendent node pairs in one equals the number of independent loops in
the other. More specifically, the equations relating the branch currents
and loop currents for one network are identical in form to the equations
relating the branch voltages and the nodepair voltages for the other,
so that these sets of equations become interchanged if the letters i and j
are replaced, respectively, by e and v, and vice versa. For appropriately
chosen elements in the branches of the associated dual networks, the
electrical behavior of one of these is obtained from that of the other
simply through an interchange in the identities of voltage and current.
Apart from the usefulness that will be had from later applications of
these ideas, a detailed consideration of the underlying principles is
advantageous at this time because of their correlative value with respect
to the foregoing discussions of this chapter.
Geometrically, two graphs are dual if the relationship between
branches and node pairs in one is identical with the relationship between
branches and loops in the other. The detailed aspects involved in such
a mutual relationship are best seen from actual examples. To this end,
consider the pair of graphs in Fig. 14. Suppose the one in part (a) is
given, and we are to construct its dual as shown in part (b). At the
outset we observe that the graph of part (a) has seven meshes and five
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independent node pairs (a total of six nodes). Hence the dual graph
DUALITY
45
must have seven independent node pairs (a total of eight nodes) and five
meshes. The total number of branches must be the same in both graphs.
In proceeding with the construction of the dual of (a), one may begin
by setting down eight small circles as nodes—one for each mesh in the
graph of part (a), and an extra one that can play the part of a datum
node if we wish to regard it as such, although any or none of the eight
nodes needs to be considered in this light. We next assign each of these
seven nodes to one of the seven meshes in the given graph, as is indi
cated in Fig. 14 through the letters a, b, • • ., g. The procedure so far
implies that we are considering as tie sets those confluent branches in
graph (a) that form the contours of meshes and as cut sets those branches
in the dual graph that are stretched in the process of picking up single
nodes. At least, this implication is true of the nodes a, • • •, g that are
assigned to specific meshes; the cut set pertaining to the remaining
unassigned node will correspond to a tie set in graph (a) that will reveal
itself as we now proceed to carry out the process of making all tie sets
in the given graph identical to all the cut sets in its dual.
Initially let us disregard reference arrows entirely; these will be added
as a final step. To begin with mesh a, we observe that it specifies a tie
set consisting of branches 1, 6, 7; therefore the cut set formed through
picking up node a in the dual graph must involve branches 1, 6, 7, and
so these are the branches confluent in node a. Similarly the branches
7, 10 form the tie set for mesh b, and therefore these branches are con
fluent in node b of the dual graph; and so forth. The actual process of
drawing the dual graph is best begun by inserting only those branches
that are common to any two tie sets and hence must be common to the
respective cut sets. That is to say, we note that any branches that are
common to two meshes in the given graph must be common to the two
corresponding nodes in the dual graph and hence are branches that form
direct connecting links between such node pairs. For example, branch 7
is common to meshes a and b, and hence branch 7 in the dual graph
connects nodes a and b; similarly branch 10 links nodes b and c; branch
11 links nodes c and d; and so forth.
In this way we readily insert branches 7, 10, 11, 8, 12, 9, and then note
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that the remaining branches 1, 2, 3, 4, 5, 6 in the original graph form a
tie set that must be identical with the cut set of the dual graph that is
associated with the remaining unassigned node. Hence these branches,
which have one terminus in an assigned node, are the ones that must be
confluent in the remaining node. The latter is thus seen to be assignable
to the loop formed by the periphery of the given graph. In a sense we
may regard this periphery as a "reference loop" corresponding to the
originally unassigned node playing the role of a "reference node,"
46 NETWORK GEOMETRY AND NETWORK VARIABLES
although the following discussion will show that this view is a rather
specialized one and need not be considered unless it seems desirable
to do so.
Now, as to reference arrows on the branches of the dual graph we note,
for example, that the traversal of mesh a in a clockwise direction is
confluent with the reference arrow of branches 1 and 6, and counterfluent
with the reference arrow of branch 7. Hence on the dual graph we
attach reference arrows to branches 1 and 6 that are divergent from node
a, and provide branch 7 with an arrow that is convergent upon this node.
That is to say, we correlate clock
wise traversal of the meshes with
divergence from the respective nodes,
and then assign branch arrows in the
dual graph that agree or disagree
with this direction, according to
whether the corresponding branch
arrows in the given graph agree or
disagree with the clockwise direction
for each corresponding mesh. We
could, of course, choose a consistent
counterclockwise traversal of the
meshes, or in the dual graph choose
convergence as a corresponding direc
tion. Such a switch will merely re
verse all reference arrows in the dual graph (which we can do anyway),
but we must in any case be consistent and stick to the same chosen
convention throughout the process of assigning branch reference arrows.
This is done in the construction of the graph of Fig. 14(b), as the reader
may readily verify by inspection.
Being mindful of the fact that duality is in all respects a mutual rela
tionship, we now expect to find that the graph (a) of Fig. 14 is related to
the graph (b) in the same detailed manner that (b), through the process
of construction just described, is related to (a). Thus we expect the
meshes of (b) to correspond to nodes in (a) as do the meshes of (a) to
the nodes in (b). However, we find upon inspection that such is not
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consistently the case. For example, the mesh in graph (b) having its
contour formed by the consecutively traversed branches 1, 7, 10, 12, 9, 4
corresponds in graph (a) not to a single node, but instead is seen to be
the dual of the group of three nodes situated at the vertexes of the
triangle formed by the branches 2, 3, 11, since the act of simultaneously
picking up these nodes reveals the same group of branches 1, 7, 10, 12,
9, 4 in graph (a) to be a cut set.
Fig. 15. A graph topologically
equivalent to that in Fig. 14b.
DUALITY
47
This apparent inconsistency is easily resolved through consideration
of a slight variation in the construction of the dual of graph (a) as shown
in Fig. 15. Here all meshes correspond to the nodes of graph (a) in Fig.
14 in the same way that the meshes of graph (a) correspond to nodes in
the graph of Fig. 15, as the reader should carefully verify. The addi
tionalprinciple observed in the construction of the graph of Fig. 15 is
that the sequence of branches about any node is chosen to be identical
with that of the similarly numbered branches around the respective
mesh, assuming a consistent clockwise (or counterclockwise) direction
of circuitation around meshes and around nodes. For example, the
branches taken in clockwise order around mesh a of the graph of Fig.
14(a) are numbers 1, 7, 6; around node a in the graph of Fig. 15 this
sequence of branches corresponds to counterclockwise rotation. Corre
spondingly, the clockwise sequence of branches around mesh c in Fig.
14(a) is 10, 11, 12, and this is the counterclockwise sequence of the
corresponding branches around node c in Fig. 15. This correspondence
in the sequence of branches is seen to hold for all meshes and their corre
sponding nodes not only between meshes in Fig. 14(a) and nodes in Fig.
15 but also between the meshes in Fig. 15 and their corresponding nodes
in Fig. 14(a). The duality between these two graphs is indeed complete
in every respect.*
So far as the relationships between branch currents and loop currents
or between branch voltages and nodepair voltages are concerned, how
ever, these must be the same for the graph of Fig. 14(b) as they are for
the graph of Fig. 15, since both involve fundamentally the same geo
metrical relationship between nodes and branches, as a comparison
readily reveals. For this reason it is not essential in the construction of a
dual graph to preserve branchnumber sequences around meshes and
nodes as just described unless one wishes for some other reason to make
meshes in the dual graph again correspond to single nodes in the original
graph. From the standpoint of their electrical behavior, the networks
whose graphs are given by Figs. 14(b) and 15 are entirely identical.
These graphs are, therefore, referred to as being topologically t equivalent,
and either one may be regarded as the dual of Fig. 14(a), or the latter
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as the dual of either of the networks of Figs. 14(b) and 15.
An additional interesting example of dual graphs is shown in Fig. 16.
The meshes a, b, c, . . . in the graph of part (a) correspond to similarly
* The correlation of clockwise rotation in one graph with counterclockwise rota
tion in its dual is an arbitrary choice. One can as well choose clockwise rotation in
both, the significant point being that a consistent pattern is adhered to.
t The mathematical subject dealing with the properties of linear graphs is known
as topoiogy.
48 NETWORK GEOMETRY AND NETWORK VARIABLES
lettered nodes in the graph of part (b); and, conversely, the meshes in
graph (b) correspond to nodes in part (a). It will also be observed that
the sequences of branches around meshes and around corresponding
nodes agree; and it is interesting to note in this special case that, although
both graphs have the form of a wheel, the spokes in one are the rim seg
"12 6^
(a) (b)
Fio. 16. A pair of dual graphs.
ments of the other. It is further useful to recognize that these graphs
may be redrawn as shown in Fig. 17, where they take the form of so
called ladder configurations with "feedback" between their input and
output ends. Removing link 16 in the graph of Fig. 16(a) corresponds
to shortcircuiting link 16 in the dual graph of part (b), since open and
16
1*
10
11
12
13
14
15
(a) Datum
161
10
11
12
13
14
151
(b) Datum
Fig. 17. The dual graphs of Fig. 16 redrawn in the form of unbalanced ladder
networks.
shortcircuit constraints are dual concepts (as previously mentioned).
In graph 17(b) this alteration identifies the first node on the left with the
datum, thus in effect paralleling branches 1 and 9 at the left and branches
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8 and 15 on the right. Such ladder configurations are much used in
DUALITY
49
practice, and it is therefore well to know that the dual of a ladder is
again a ladder with the essential difference that its series branches corre
spond to shunt branches in the given ladder, and vice versa.
It is helpful, in the process of constructing a dual graph, to visualize
the given one as mapped upon the surface of a sphere instead of on a
plane. If this is done, then the periphery appears as an ordinary mesh
when viewed from the opposite side of the sphere. For example, if the
graph of Fig. 16(a) is imagined to consist of an elastic net and is stretched
over the surface of a sphere until the periphery contracts upon the oppo
site hemisphere, and if one now views the sphere from the opposite side
so as to look directly at this hemisphere, then the periphery no longer
appears to be fundamentally different in character from an ordinary
mesh, for it now appears as a simple opening in the net, like all the other
meshes. Thus the branches 9, 10, 11, 12, 13, 14, 15, 16 forming the
contour of this mesh appear more logically to correspond to the simi
larly numbered group of branches in the dual graph 16(b) emanating
from the central node which, like all the other nodes, now corresponds
to a simple mesh in the given graph.
When, in the choice of network variables, one identifies loop currents
with link currents and nodepair voltages with treebranch voltages, it
will be recalled that each tie set consists of one link and a number of tree
branches, while each cut set consists of one tree branch and a number of
links. Since the tie sets of a given graph correspond to cut sets in the
dual graph, one recognizes that the tree branches in one of these graphs
are links in the other. That is to say, corresponding trees in dual
graphs involve complementary sets of branches. In Fig. 16, for example,
if one chooses the branches 1, 2, 3, 4, 5, 6, 7, 8 in graph (a) as forming a
tree, then the corresponding tree in graph (b) is formed by the branches
9, 10, 11, 12, 13, 14, 15, 16. Or, if in graph (a) we choose branches 1, 2,
3, 4, 12, 13, 14, 15 as forming a tree, then in graph (b) the corresponding
tree is formed by branches 5, 6, 7, 8, 9, 10, 11, 16.
It should now be clear, according to the discussion in the preceding
articles, that, if in a given graph we pick a tree and choose the comple
mentary set of branches as forming a tree in the dual graph, then the
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resulting equations between branch currents and loop currents in one
of these graphs becomes identical (except for a replacement of the letters
j and i, respectively, by v and e) with those relating branch voltages and
nodepair voltages in the dual graph. In the graphs of Fig. 16, for
example, we may choose branches 1 to 8 inclusive as the tree of graph (a)
and branches 9 to 16 inclusive as the tree of graph (b). Then in graph
(a), the branch currents jg, j\0, • • •, jia are respectively identified with
loop currents ii, i2, • • •, is> while in graph (b) the branch voltages vg,
50
NETWORK GEOMETRY AND NETWORK VARIABLES
»io. •"■ "i6 are respectively identified with nodepair voltages e\,
e2, • • ., eg. For the treebranch currents in graph (a) we then have, for
example, j2 = *'i + i2 = jt + Jio;j3 = *2 + *3 = Jio + jn,
etc.; while for the link voltages in graph (b) we have correspondingly
»2 = — »g + vio = —e\ + e2; t'3 = —t'10 + »n = e2 + e3, etc. The
reader may complete these equations as an exercise, and repeat the
process for several other trees as well as for the graphs of Fig. 14.
It should likewise be clear that similar results for a pair of dual graphs
and their current and voltage variables are obtained if for one graph
one chooses meshes as loops and in the other the corresponding nodes as
a nodetodatum set of node pairs. In this case it may be desirable to
regard the unassigned node as a datum and the corresponding peripheral
mesh as playing the role of a datum mesh. Since more general choices
of loops or of node pairs may be expressed as linear combinations of these
simple ones, it is seen that the parallelism between the current and
voltage relations of dual networks holds in all cases, regardless of the
approach taken in formulating defining relations for network variables.
It is important, however, to note a restriction with regard to the exist
ence of a dual graph. This restriction may most easily be understood
through recognizing that all possible choices of tie sets in a given net
work must correspond to cut sets in its dual, and vice versa. In this
connection, visualize the given graph as some net covering the surface
of a sphere, and a tie set as any confluent group of branches forming a
closed path. As mentioned at the close of Art. 5, let us think of inserting
a draw string along this path and then tying off, as we might if the
sphere were an inflated balloon. We would thus virtually create two
balloons, fastened one to the other only at a single point where the con
tracted tie set has become a common node for the two subgraphs formed
by the nets covering these balloons. Whether we thus regard the tie
set as contracted or left in its original form upon the sphere, its primary
characteristic so far as the present argument is concerned lies in the fact
that it forms a boundary along which the given network is divided into
two parts, and correspondingly the totality of meshes is divided into
two groups.
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In the dual graph these correspond to two groups of nodes. If we
think of grasping one of these node groups in each of our two hands and
pulling them apart, the stretched branches place in evidence the cut set
corresponding to the tie set of the original graph. The act of cutting
this set of branches is dual to the tyingoff process described above, since
by this means the dual graph is separated into two parts which are,
respectively, dual to the two subgraphs created by contracting the tie set.
Duality between the original graph and its dual demands that to
every creatable cut set in one of these there must correspond in the other
CONCLUDING REMARKS
51
a tie set with the property just described. It should be clear that this
requirement cannot be met if either network is not mappable upon a
sphere but requires the surface of some multiply connected space like
that occupied by a doughnut or a pretzel. For example, if the mapping
of a graph requires the surface of a doughnut, then it is clear that a
closed path passing through the hole is not a tie set because the doughnut
is not separated into two parts through the contraction of this path.
The surface of a simply connected region like that of a sphere is the only
surface on which all closed paths are tie sets. There is obviously no
corresponding restriction on the existence of cut sets, since we can visu
alize grasping complementary groups of nodes in our two hands and,
through cutting the stretched branches, separating the graph into two
parts regardless of whether the geometry permits its being mapped
upon a sphere or not.
Thus, mappability upon a sphere is revealed as a necessary condition
that a tie set in the original graph shall correspond to every possible cut
set in its dual, and hence the latter is constructible only if the graph of
the given network is so mappable.*
10 Concluding Remarks
As expressed in the opening paragraphs of the previous article, the
object in discussing the subject of duality is twofold. First, duality is a
means of recognizing the analytical equivalence of pairs of physically
dissimilar networks; so far as mappable networks are concerned, it
essentially reduces by a factor of two the totality of distinct network
configurations that can occur. Second, and no less useful, is the result
that the principle of duality gives us two geometrically different ways of
interpreting a given situation; if one of these proves difficult to compre
hend, the other frequently turns out to be far simpler. This characteris
tic of the two geometrical interpretations of dual situations to reinforce
the mental process of comprehending the significance of either one we
wish now to present through a few typical examples.
Suppose, for a given mappable graph, we consider a nodetodatum
set of voltage variables. That is to say, we pick a datum node, and
choose as variables the potentials of the remaining nodes with respect
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to this datum. If we now wish to obtain algebraic expressions for these
node voltages in terms of a like number of independent branch voltages,
the simplest procedure is to select a tree and recognize that each node
potential is then uniquely given by an algebraic sum of treebranch
* Further detailed discussion of these as well as all foregoing principles presented
in this chapter are given throughout the succeeding chapters dealing with their appli
cation. A general method for the construction of dual networks and the evaluation
of their properties is given in the last article of Ch. 10.
52 NETWORK GEOMETRY AND NETWORK VARIABLES
voltages, since the path from any node to the datum via tree branches
is a unique one. The geometrical picture involved and the pertinent
algebraic procedure are simple and easily comprehensible.
Contrast with this the completely dual situation. For a given map
pable graph, we consider the mesh currents as a set of appropriate
variables, and ask for the algebraic expressions for these in terms of a like
number of independent branch currents. Since the latter may be re
garded as the currents in a set of links associated with a chosen tree,
the initial step in the procedure is clearly the same as in the previous
situation. At this point, however, the lucidity of the picture is suddenly
lacking, for we do not appear to have a procedure for expressing each
mesh current as an algebraic sum of link currents that has a geometric
clarity and straightforwardness comparable to the process of expressing
node potentials in terms of treebranch voltages, and yet we feel certain
that there must exist a picture of equivalent clarity since to every
mappable situation there exists a dual which possesses all of the same
features and with the same degree of lucidity. Our failure to find the
mesh situation as lucid as the one involving node potentials must be
due to our inability to construct in our minds the completely dual
geometry. Once we achieve the latter, our initial objective will easily
be gained, and our understanding of network geometry will corre
spondingly be enhanced.
It turns out that our failure to recognize the dual geometry stems from
an initial misconception of what is meant by a mesh. Since we use the
term mesh to connote a particular kind of loop, namely the simplest
closed path that one can trace, we establish in our minds the view that
the term mesh refers to the contour (the associated tie set) instead of
the thing that it should refer to, namely the space surrounded by that
contour 1 A mesh is an opening—not the boundary of that opening.
This opening is the dual of a node—the point of confluence of branches.
A tree consists of nodes connected by tree branches. The dual of a
tree branch is a link. Therefore the dual of a tree should be something
that consists of spaces (meshes) connected by links. If we add to the
mental picture created by these thoughts the fact that traversing a
branch longitudinally and crossing it at right angles are geometrically
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dual operations (since a branch voltage is found through a longitudinal
summation process while a branch current is given by a summation over
the cross section), we arrive without further difficulty at the geometrical
entity that must be recognized as the dual of a tree. It is the space
surrounding the tree.
This space is subdivided into sections by the links. Each of these
sections is a mesh; and one passes from mesh to mesh by crossing the
links, just as in the tree one passes from node to node by following along
CONCLUDING REMARKS
53
the tree branches. Figure 18 shows in part (a) a graph in the form of a
rectangular grid and in part (b) a possible tree with the links included
as dotted lines. The space surrounding the tree, and dual to it, is best
described by the word maze as used to denote a familiar kind of picture
puzzle where one is asked to trace a continuous path from one point in
this space to another without crossing any of the barriers formed by the
treelike structure.
Such a path connecting meshes m and n is shown dotted in part (b) of
the figure. It is clear that the path leading from one mesh to any other
is unique, just as is the path from one node to another along the tree
(a) (b)
Fio. 18. A graph; a possible tree and its dual which is interpreted as a maze.
branches. In passing along a path such as the one leading from mesh m
to mesh n, one crosses a particular set of links. These links characterize
this path just as a set of confluent tree branches characterize the path
from one node to another in a given tree.
Having recognized these dual processes, we now realize that we have
not been entirely accurate in the foregoing discussions where we refer to
a loop current as being dual to a nodepair voltage. The latter is the
difference between two node potentials, and its dual is, therefore, the
difference between two mesh currents, like the currents in meshes m and
n in Fig. 18(b). The difference (im — in) is algebraically given by the
summation of those link currents (with due attention to sign) charac
terizing the path from m to n, just as a nodepair voltage (potential
difference between two nodes) equals the algebraic sum of treebranch
voltages along the path connecting this node pair. The difference
(im ~ in), which might be called a meshpair current, is the real dual of a
nodepair voltage. With the addition of the maze concept to our
interpretation of network geometry, we have acquired a geometrical
picture for the clarification of the algebraic connection between mesh
current differences and link currents that is as lucid as the familiar one
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used to connect nodepotential differences with treebranch voltages.
54
NETWORK GEOMETRY AND NETWORK VARIABLES
These matters are further clarified through more specific examples. In
Fig. 19 is shown a simple network graph [part (a)], its dual [part (b)], and
a schematic indicating a choice of nodetodatum voltages characterizing
the dual graph [part (c)]. In the graph of part (a) the tree branches
are the solid lines, and the links (branches 1, 2, 4, 5, 6) are shown dotted.
In the dual graph of part (b), these same branches (1, 2, 4, 5, 6) form the
tree, and the rest are links. The datum node surrounds the whole dual
(a) (b) (c)
Fig. 19. A network graph (a), its dual (b), and a nodetodatum choice of nodepair
voltages (c) corresponding to the mesh currents in (a). The tree branches (solid) in
(a) become links (dotted) in (b) and vice versa.
graph. Mesh currents ii, i2, • • , 15 are chosen to characterize the graph
(a), while correspondingly the node potentials e\, e2, . • •, e5 characterize
the dual graph (b).
Starting with the dual graph, it is evident that the expressions for the
e's in terms of the treebranch voltages read
ei = vi
e2 = v2 + vi
e3 = — vi — i's  t'6 (57)
e4 =
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—»5 — »6
CONCLUDING REMARKS
55
Analogously, the mesh currents in terms of the link currents in graph
(a) must be given by
ii = ji
*2 = h + ji
H = Ji  h ~ h (58)
U = —js — je
is = ja
One can verify these last results either by expressing the link currents
as superpositions of the loop currents in the following manner,
32

i2 
*i
h

u
i3
h

is ~
it
k

is
(59)
and solving for the i's, or by noting that each mesh current (like a node
potential) is the difference between the current circulating on the contour
of that mesh and the datum mesh current, which is visualized as circu
lating on the periphery of the entire graph. In this sense the datum
mesh is the entire space outside the graph, just as the datum node in the
dual graph surrounds it. Following the pattern set in Fig. 18(b) for
expressing meshcurrent differences in terms of link currents, one readily
establishes Eqs. 58 as representing the situation depicted in graph (a)
of Fig. 19, and simultaneously recognizes how the algebraic signs in these
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equations are related to the reference arrows involved.
Consider now the same networks, but with an altered choice for the
voltage and current variables. In Fig. 20(a) are shown the paths for
the new loop currents. The dual graph is not repeated in this figure, but
part (b) shows the diagram for the choice of nodepair voltages in the
dual graph that correspond to the new loop currents in graph (a). All
variables corresponding to this revised choice are distinguished by primes.
So far as the voltage picture is concerned, one has little difficulty in
56 NETWORK GEOMETRY AND NETWORK VARIABLES
recognizing that one now has
fii
= e2 
Ci =
02
e'a
"= ei =
t>i
e'a
= e5 =
f«
✓4
= e< 
c5 =
e'a
= c3 
c4 =
~f4
and so, by analogy, the corresponding relations for the loop currents in
terms of the link currents of the graph in part (a) of Fig. 20 must be
t'i
=H
 *i =
h
t'a
= ii
= ii
t'a
= t'5
= ja
i\
=U
 is =
h
t\
~ *4 =
h
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= *3
These can readily be verified through the usual procedure of writing
expressions for the link currents in terms of the loop currents and solving.
It is more interesting, however, to establish them entirely by analogy to
the dual voltage situation, for we learn in this way more about the
manner in which the loop currents are related to the link currents.
Thus a loop current like i'3, for example, surrounds three meshes, and
correspondingly the nodepair voltage e'3 contributes to the potentials
of the three nodes 0, p, q [Fig. 20(b)]. In forming the cut set associated
with e'3 we would pick up nodes 0, p, q, whereas in forming the tie set
associated with i'3 we may say that we "pick up" the meshes whose
combined contour places that tie set in evidence.
Having established the fact that picking up meshes is dual to picking
up nodes, and recognizing that loop currents, as contrasted to mesh
currents, circulate on the resulting contours of groups of meshes, we are
in a position to sketch the nodepair voltage diagram [like part (b) of
Fig. 20] corresponding to a chosen loopcurrent diagram [like part (a)
of Fig. 20], provided one exists, and, by analogy to the dual voltage
equations, obtain directly the pertinent relations for the loop currents.
CONCLUDING REMARKS
57
Since for cut sets picked at random there does not necessarily corre
spond a set of "nodepair voltages" that are simple potential differences
between pairs of nodes, it is analogously true that for loops (i.e. tie sets)
picked at random there does not necessarily correspond a set of "mesh
pair currents" that are simple differences between currents in pairs of
meshes. In the example of Fig. 20, pertinent to Eqs. 60 and 61, the
conditions are chosen so that one does obtain e's that are potential
1
o9
Fig. 20. A revised choice of loop currents in the graph of Fig. 19(a) and the corres
ponding revision in the nodepair voltage definitions for the dual graph.
differences between nodes and i's that are meshcurrent differences, but,
when loops are picked at random, it is in general no longer possible to
give any simple geometrical interpretation to the implied current rela
tionships, just as on the voltage side of the picture a straightforward
interpretation fails when cut sets are chosen at random.
Wherever simple relationships do exist, the principle of duality is
distinctly helpful in clarifying them. For example, in comparing parts
(a) of Figs. 19 and 20, one might be tempted to conclude offhand that
i'i = i2, or i's = i3 because the contours on which these pairs of currents
circulate are the same. As pointed out in Art. 7, it is fallacious to imply
that there is any direct relation between the contours chosen for loop
currents and their algebraic expressions in terms of link currents.
Equations 61 show that the above offhand conclusions are false. Use
of the duality principle, as in the preceding discussion, shows why they
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are false.
58 NETWORK GEOMETRY AND NETWORK VARIABLES
PROBLEMS
1. For the graph shown, pick the indicated meshes as loops and write a corre
sponding tieset schedule. Select an independent set of columns as those pertinent
to the links of a chosen tree, and from the corre
sponding equations find expressions for the mesh
currents ti, it, is, 14 in terms of branch currents.
Do this specifically for (a) the tree composed of
branches 1, 2, 3, 6; (b) the tree composed of
branches 5, 6, 7, 8; and show that the two sets of
relations for the t's in terms of j's are equivalent.
For the tree defined under (b) show that the mesh
currents are link currents.
Pick the link currents 4, 5, 7, 8 as loop currents.
Find the corresponding set of closed paths, and
construct an appropriate tieset schedule.
2. With reference to the graph of Prob. 1, de
termine whether each of the accompanying tieset
schedules defines an independent set of loop currents. If so, express the loop cur
Loop
No.
Branch No.
1
2
3
4
5
6
7
8
1
1
1
1
1
1
2
1
1
1
1
1
3
1
1
1
1
1
4
1
1
1
1
1
Loop
No.
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Branch No.
1
PROBLEMS
59
rente in terms of the currents in links 1, 2, 3, 4. In each case, trace the closed paths
traversed by the loop currents.
Express the currents in branches 5, 6, 7, 8 in terms of the link currents 1, 2, 3, 4.
3. Given the accompanying tieset schedule and its associated graph, trace the
Loop
Branch No.
No.
1
2
3
4
5
6
7
8
9
10
11
12
1
1
1
1
1
1
1
2
1
1
1
1
1
1
3
1
1
1
1
1
1
4
1
1
1
1
1
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60 NETWORK GEOMETRY AND NETWORK VARIABLES
7, 8, 9, 10, 11, 12; identifying, in each case, the nodepair voltages with the tree
branch voltages. For each choice of tree, express the link voltages in terms of the
treebranch or nodepair voltages.
6. For the graph of Prob. 3 and the designation of nodes shown in the accompany
ing sketch, choose 0 as the datum node, and write a cutset schedule for the nodeto
datum voltages, ei, . • ., e.i. Express these in terms of
1 2 each of the two sets of treebranch voltages specified in
Prob. 5.
7. With reference to the graph of Prob. 3 and the node
designation given in Prob. 6, determine which of the
following sets of node pairs are independent, and for
each of the latter construct a pertinent cutset schedule,
and express the nodepair voltages in terms of the branch
voltages of tree (a) in Prob. 5: (a) 02, 04, 13, 17, 26, 35,
57; (b) 02, 06, 13, 15, 24, 46, 57; (c) 02, 06, 13, 15, 24,
36, 37.
8. Construct the dual of the graph in Prob. 3, giving
Prob. 6. the appropriate numbering and reference arrows for all
branches. On this dual graph indicate a set of mesh
currents dual to the nodetodatum voltages of Prob. 6, and show that the cutset
schedule written there is now the appropriate tieset schedule. Show further that
the relations for the mesh currents in terms of link currents are identical in form with
the expressions for the node potentials in terms of treebranch voltages found in
Prob. 6.
9. For the dual graph of Prob. 8 define loopcurrent variables (meshpair currents)
that are dual to each of the independent sets of nodepair voltages specified in Prob. 7.
Show in each case that the appropriate tieset schedule is identical with the pertinent
cutset schedule of Prob. 7, and thus find the relations between the loop currents and
the link currents dual to the branch voltages in tree (a) of Prob. 5. For each set of
independent loop currents (making use of the appropriate tieset schedule) find the
associated set of closed paths and trace these in the dual graph.
10. Through making appropriate linear combinations, show that any set of lin
early independent rows is reducible to the particular set shown here, in which ele
1xxxx
01xxx•••
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0 0 1 x x •••
0 0 0 1 x .••
ments marked x may have any finite values (including zero). If necessary, some
column interchanges (corresponding to changes in branch numbering) are, of course,
permitted at any stage in the transformations. Thus show that, if the I rows of a
tieset schedule are independent, it must always be possible to find at least one set
of I independent columns.
11. If the links corresponding to the chosen tree of any given graph are numbered
1, 2, . • •, I, and loop currents are defined as n = ji, it = jt, • • •, ii — ji, show that
the first I columns of the tieset schedule represent a matrix having l's on its prin
PROBLEMS
61
16
15
14
13*
121
*7
*8
A10
5
cipal diagonal (upper left to lower right) and all other elements zero (called a unit
matrix). Compare this situation with that in the previous problem.
12. Draw a regular pentagon with branches numbered 1 to 5 and additional
branches 6 to 10 so that each vertex (node) is connected with every other one. For
any appropriate cutset schedule prove that any four of the
columns 1 to 5 or 6 to 10 are independent.
13. With reference to the graph of the preceding problem,
consider any appropriate tieset schedule, and prove that any
six of its columns including either 1 to 5 or 6 to 10 are inde
pendent.
14. Consider a graph in which a branch connects every node
with every other node. Determine the number n of inde
pendent node pairs and the number I of independent loops in
terms of the number of total nodes nt. Compute the number
of equilibrium equations needed for this graph on the loop and
node bases for the cases nt = 2, 3, 4, 5, 10, 50, 100, and tab
ulate the results.
15. Consider a threedimensional graph in the form of a uni
form cubical grid with n, nodes on a side and n,' total nodes.
Show that the number of independent loops is I = 2(n,« — 1)
— 3(n,2 — 1). Make a table showing the numbers n and I
for n,  2, 3, 4, 5, 10, 100.
16. Consider the graph shown here, and choose a tree consist
ing of the branches 6 to 16 inclusive. Let the loop currents be
the link currents t* = j* for k — 1 • • • 5, and construct the per
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tinent tieset schedule. Define a second set of loop currents as
those circulating in the clockwise direction around the boundaries of the meshes
a, b, c, d, e, and construct a second tieset schedule appropriate to this choice of
current variables.
Referring to the respective rows of the first schedule by the numerals 1 • • • 5 and
to those of the second by the letters a . . • e, express the rows (tie sets) of each
schedule as the appropriate linear combinations of rows in the other schedule. For
example:
a = 1 — 2; 6 = 2 — 3; etc. and 5 — e; 4 = d + e; etc.
These are the topological relationships between the two sets of closed paths involved
in the definition of loop currents. Now find the algebraic relationships between the
set of loop currents ii • . • % and the set ia • • • i,; that is to say, express the t'i • • • it
in terms of ta • • • i,, and vice versa. Compare the topological and the algebraic
relationships thus found, and note carefully the distinction that must be made be
tween them.
17. With reference to the situation in Prob. 16, suppose we introduce some new
loop currents as the meshcurrent differences given by the algebraic relationships
11
Prob. 16.
*& — toj *s = *c — ta; k — u — *z> = *, — tc
IE = 19 is numerically negative; and, since its algebraic sign in Eq. 2 is plus,
we see that this term involves an arithmetic subtraction. In branch 10,
on the other hand, the actual drop in altitude may be contrary to the
arrow direction so that vi0 has a negative value. The corresponding
term in Eq. 2 becomes numerically positive, as is appropriate since we
actually experience a drop in altitude when we encounter branch 10 in
traversing the circuit to which Eq. 2 applies.
The Kirchhoff voltage law thus expresses the simply understandable
fact that the algebraic sum of voltage drops in any confluent set of
branches forming a closed circuit or loop must equal zero. Symbolically
this fact may be expressed by writing
where the Greek capital sigma is interpreted as a summation sign and
the quantities ±v which are summed are voltage drops, with due regard
to the possible agreement or disagreement of their pertinent reference
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arrows with the (arbitrary) direction of traversal around the loop, thus
indicating the choice of the plus or minus sign respectively.
It is interesting to observe an important property of equations of this
type with reference to a given network geometry such as that shown in
t>2 + v3 + t>12 + t>19 + f20  vi7 — vi0 — v4 = 0
(2)
2±t > = 0
(3)
KIRCHHOFF'S LAWS
67
Fig. 1. Suppose we write voltagelaw equations for the upper lefthand
corner mesh and its righthand neighbor, thus:
»i + H ~ H = 0
(4)
v2 + v5 — v7 — vt = 0
Addition of these two equations gives
» i + v2 + t/5 — t'7 — t'6 = 0 (5)
which we recognize as an equation pertinent to the closed loop which is
the periphery of the two meshes combined. The reason for this result is
that branch 4, which is common to both meshes, injects the terms +vi
and — »4 respectively into the two Eqs. 4, and hence cancels out in their
addition.
It is immediately clear that such cancelation of voltage terms will
take place in the summation of any group of equations relating to meshes
for which these terms correspond to branches common to the group of
meshes. Suppose we write separate equations for the meshes imme
diately below those to which Eqs. 4 refer, thus:
«6 + via — vi3 + v9 = 0
(6)
»7 + »n  fi4  »10 = 0
Adding Eqs. 4 and 6, we have
»i + »2 + »5 + ^11 ~ »i4 — vi3 + v9 = 0 (7)
This equation is pertinent to the periphery of the block of four upper
lefthand meshes in the graph of Fig. 1. If all the equations for the sep
arate meshes in this graph are added, one obtains Eq. 1 relating to the
periphery of the whole graph. The student should try this as an exercise.
We now turn our attention to an analogous law in terms of branch cur
rents: the socalled Kirchhoff current law. The electric current in a branch
is the time rate at which charge flows through that branch. Unless the
algebraic sum of currents for a group of branches confluent in the same
node is zero, electric charge will be either created or destroyed at that
node. Kirchhoff's current law, which in essence expresses the principle
of the conservation of charge, states therefore that an algebraic summa
tion of branch currents confluent in the same node must equal zero.
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Symbolically this fact is expressed by writing (as in Eq. 3):
S±i = 0 (8)
As illustrations of this law suppose we write equations of this sort
for nodes a and h and the one immediately to the right of h in Fig. 1.
68
THE EQUILIBRIUM EQUATIONS
These read
3i + J2 + h
ji + 36 — J9
(9)
~3* + 37 + iio — jo
Each equation states that the net current diverging from a pertinent
node equals zero.
Now suppose we add the three Eqs. 9. This gives
Branch currents ji, j4t and ja cancel out in the process of addition.
Reference to the graph of Fig. 1 reveals that these branches are common
to the group of three nodes in question, while the branches to which
the remaining currents in Eq. 10 refer terminate only in one of these
nodes.
An interesting interpretation may be given the resulting Eq. 10.
If we regard the portion of the graph of Fig. 1 formed by branches 1, 4,
and 6 alone (referred to as a subgraph of the entire network) as enclosed
in a box, then Eq. 10 expresses the fact that the algebraic sum of currents
divergent from this box equals zero. In other words, the current law
applies to the box containing a subgraph the same as it does to a single
node. That is to say, it is not possible for electric charge to pile up or
diminish within a box containing a lumped network any more than it is
possible for charge to pile up or diminish at a single node. This fact
follows directly from the current law applied to a group of nodes, as shown
above, and yet students usually have difficulty recognizing the truth
of this result. They somehow feel that in a box there is more room for
charge to pile up, and so it may perhaps do this, whereas at a single
node it is clear that the charge would have to jump off into space if more
entered than left the node in any time interval. The above analysis
shows, however, that what holds for a simple node must hold also for a
box full of network.
2 Independence among the Kirchhoff Law Equations
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Equilibrium equations are a set of relations that uniquely determine
the state of a network at any moment. They may be written in terms of
any appropriately chosen variables; the uniqueness requirement de
mands, however, that the number of independent equations shall equal
the number of independent variables involved. We have seen earlier
that the state of a network is expressible either in terms of I = b — nt + 1
32 + 37 + 3io ~ 3g = 0
(10)
INDEPENDENCE AMONG THE KIRCHHOFF EQUATIONS
69
independent currents (for example, the loop currents) or in terms of
n = ni — 1 = 6 — I independent voltages (for example the nodepair
voltages). On a current basis we shall, therefore, require exactly I
independent equations; and on a voltage basis exactly n independent
equations will be needed.
For these equations we turn our attention to the Kirchhoff laws. It
is essential to determine how many independent equations of each type
(the voltagelaw and the currentlaw types) may be written for any
given network geometry. Consider first the voltagelaw equations, and
assume that these have been written for all of the nine meshes of the
network graph in Fig. 1. Incidentally, this graph has 20 branches and
a total of 12 nodes (6 = 20, nt = 12). Hence I  20  12 + 1 = 9,
which just equals the number of meshes. Any tree in this network
involves n = 11 branches. There are 9 links, and hence there are
9 geometrically independent loop currents.
From what has been pointed out in the previous article, it is clear
that a voltagelaw equation written for any other loop enclosing a group
of meshes in Fig. 1 may be formed by adding together the separate
equations for the pertinent meshes. Such additional voltagelaw
equations clearly are not independent. The inference is that one can
always write exactly / independent equations of the voltagelaw type.
This conclusion is supported by the following reasoning. Suppose, for
any network geometry, a tree is chosen, and the link currents are identi
fied with loop currents. For the correspondingly determined loops a set
of voltagelaw equations are written. These equations are surely inde
pendent, for the link voltages appear separately, one in each equation,
so that it certainly is not possible to express any equation as a linear
combination of the others. Each of these equations could be used to
express one link voltage in terms of treebranch voltages. This fact
incidentally substantiates what was said earlier with regard to the
treebranch voltages being an independent set and the link voltages
being expressible uniquely in terms of them (see Art. 6, Ch. 1).
Now any other closed loop for which a voltagelaw equation could be
written must traverse one or more links since the tree branches alone
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can form no closed paths. If in this equation the previous expressions
for the pertinent link voltages are substituted, the resultant equation
must reduce to the trivial identity 0 = 0, since no nontrivial relation
can exist among treebranch voltages alone (the treebranch voltages
are independent and hence are not expressible in terms of each other).
It follows, therefore, that the voltagelaw equation written for the
additional closed loop expresses no independent result. There are indeed
exactly I independent voltagelaw equations.
70
THE EQUILIBRIUM EQUATIONS
a
*9
2 6 X)
kif
14
20
Let us turn our attention now to the Kirchhoff currentlaw equations
and see how many of these may be independent. Referring again to the
graph of Fig. 1, suppose we begin writing equations for several nodes
adjacent to each other. If we examine these equations carefully, we
observe that each contains at least one term that does not appear in the
others. For example, if we consider the equations written for nodes a
and h, it is clear that the terms involving j2 and j4 do not appear in the
equation for node h, and that the j6 and is terms in the equation for
node h do not appear in the one for
node a. If we also write an equation
for the node immediately to the right
of h, this one contains terms with
jj and ji0 which are not contained
in either of the equations for nodes
a or h. Such sets of equations are
surely independent, for it is mani
festly not possible to express any
one as a linear combination of the
others so long as each has terms that
the others do not contain.
As we proceed to write current
law equations for additional nodes
in the graph of Fig. 1, the state of
affairs just described continues to hold
true until equations have been written for all but one of the nodes. The
inference is that exactly n = nt — 1 independent equations of the
currentlaw type can always be written. This conclusion is supported
by the following reasoning.
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 Suppose, for any network geometry, a tree is chosen, and the tree
branch voltages are identified with nodepair voltages. For the corre
spondingly determined node pairs, a set of Kirchhoff currentlaw equa
tions are written. The set of branches taking part in the equation for
any node pair is the pertinent cut set, just as the group of branches in
volved in the voltagelaw equation for any loop is the tie set for that loop.
TThe cut set pertinent to the node pair defined by any tree branch
evidently involves that tree branch in addition to those links having one
of their ends terminating upon the pickedup nodesJ(see Art. 8, Ch. 1).
Figure 2 illustrates the choice of a tree for the network graph of Fig. 1,
and, with respect to the node pair/, e joined by branch 20, indicates by
dotted lines the links that take part in the pertinent cut set. Since the
treebranch voltage v20 is identified with the respective nodepair
voltage, the latter has its reference arrow pointing from / to e. That
Fio. 2. A tree for the graph of Fig. 1.
The cutset pertinent to node pair
fe consists of tree branch 20 and the
links shown dotted.
EQUATIONS ON THE LOOP AND NODE BASES
71
is to say, the pickedup nodes are e, q, I, b, c, d. Hence the pertinent
currentlaw equation reads
J20  ju  h  32  0 (11)
Schedules like 40, 42, 46 in Art. 8 of Ch. 1 are helpful in writing the
currentlaw equations for a chosen set of node pairs, for the elements in
the rows of such a schedule are the coefficients appropriate to these
equations.
Suppose that currentlaw equations like 11 are written for all of the
node pairs corresponding to the n tree branches. These equations are
surely independent, for the treebranch currents appear separately, one
in each equation, so that it certainly is not possible to express any equa
tion as a linear combination of the others. Each of these equations
could be used to express one treebranch current in terms of the link
currents. This fact incidentally substantiates what was said earlier
with regard to the link currents being an independent set and the tree
branch currents being expressible uniquely in terms of them (see Art. 5,
Ch. 1).
Now any other cut set pertinent to a node pair for which a current
law equation could be written would have to involve one or more tree
branches, since the tree connects all of the nodes, and therefore no node
exists that has not at least one tree branch touching it. If in such an
additional currentlaw equation one substitutes the expressions already
obtained for the pertinent treebranch currents, the resultant equation
must reduce to the trivial identity 0 = 0, since no nontrivial relation
can exist among link currents alone (the link currents are independent
and hence are not expressible in terms of each other). It follows, there
fore, that the currentlaw equation written for any additional node pair
expresses no independent result. There are indeed exactly n independent
currentlaw equations.
3 The Equilibrium Equations on the Loop and Node Bases
Having established the fact that the state of a network can be charac
terized uniquely either in terms of a set of I loop currents or in terms of
a set of n nodepair voltages, and having recognized that the numbers of
independent Kirchhoff voltagelaw and currentlaw equations are I and n
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respectively, the conclusion is imminent that the equilibrium condition
for a network can be expressed in either of two ways: (a) through a set
of I voltagelaw equations in which the loop currents are the variables,
or (b) through a set of n currentlaw equations in which the nodepair
voltages are the variables. These procedures, which are referred to
72
THE EQUILIBRIUM EQUATIONS
respectively as the loop and node methods of expressing network equilib
rium, are now discussed in further detail.
Consider first the loop method. The voltagelaw equations, like Eq. 1,
p. 65, involve the branchvoltage drops. If these equations are to be
written with the loop currents as variables, we must find some way of
expressing the branch voltages in terms of the loop currents. These
expressions are obtained in two successive steps.
The branch voltages are related to the branch currents by the volt
ampere equations pertaining to the kinds of elements (inductance,
resistance, or capacitance) that the branches represent; and the branch
currents in turn are related to the loop currents in the manner shown in
Ch. 1. Detailed consideration of the relations between branch currents
and branch voltages is restricted at present to networks involving re
sistances only. Appropriate extensions to include the consideration of
inductance and capacitance elements will follow in the later chapters.
Let the resistances of branches 1, 2, 3, • • • be denoted by ti, r2, r3) etc.
Then the relations between all the branch voltages and all the branch
currents are expressed by
The complete procedure for setting up the equilibrium equations on
the loop basis will be illustrated for the network graph shown in Fig. 3.
Part (a) is the complete graph, and part (b) is a chosen tree. Branches
1, 2, 6 are links, and the link currents ji, j2, • • •, ja are identified
respectively with the loop currents t'i, t'2, • • t'6.
The following tieset schedule is readily constructed from an inspection
of the resulting closed paths pertinent to these six loop currents [as the
reader should check through placing the links 1,2, • • •, 6, one at a time,
into the tree of Fig. 3(b)]. The Kirchhoff voltagelaw equations written
for these same loops are immediately obtained through use of the coef
vk = Tkjk for fc = 1, 2, • • •, b
(12)
(a) (b)
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Fio. 3. A nonmappable graph (a), and a possible tree (b).
3
1
1
1
4
1
1
1
5
1
1
1
1
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EQUATIONS ON THE LOOP AND NODE BASES
73
Loop
Branch No.
No.
1
2
3
4
5
6
7
8
9
10
1
1
1
1
1
1
2
1
1
1
74
THE EQUILIBRIUM EQUATIONS
then vi = 2ji, v2 = j2, i'3 = 5/5, v+ = 37'4 volts, and so forth. Use of
Eqs. 15 then gives
6(n + u  i6)
lO(ii + i,  U  is + *«) (17)
Pi
—
Z*,
1)7
»a
=
5t3
va
0*

v9
=
4i5
9(t'i  *a + is)
The desired loop equilibrium equations are obtained through sub
stituting these values for the v's into Eqs. 14. After proper arrangement
of the results, one finds
35i,
 18t2
 17t3 + 16*4
+ 27i5
 24*6
=0
lWi
+ 19t2
+ Si3  10it
 18iB
+ 18*6
=0
m
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+ 8*2
+ 22i3 + 0*4
 17i5
+ 8i6
=0
16ii
 10t2
+ 0t3 + 19t4
+ lOtg
 16i6
=0
27i!
 18i2
 17t3 + 10l4
+ 31i5
 18t6
=0
24h
+ 18i2
+ 8*3  I614
 18t5
+ 24i6
=0
(18)
Considering next the node method of writing equilibrium equations we
observe first that the currentlaw equations, like Eq. 11 above, involve
the branch currents. If these equations are to be written with the
nodepair voltages as variables, we must express the branch currents
in terms of the nodepair voltages. To do this, we note that the branch
currents are related to the branch voltages through Eqs. 12, and the
branch voltages in turn are related to the nodepair voltages in the man
ner shown in Ch. 1. Equations 12 are now more appropriately written
in the form
jk = gkvk for fc = 1, 2, b (19)
EQUATIONS ON THE LOOP AND NODE BASES
75
following cutset schedule is then readily constructed from an inspection
of Fig. 3, noting the pickedup nodes pertinent to these four node pairs.
Node
Pair
No.
Branch No.
Picked
1
2
3
4
5
6
7
8
9
10
Up
Nodes
1
1
1
1
1
a
2
1'
r
1J
1,.
c, d, e
3
1
1
1
i
1
1
a, b, c
4
1
1
i
1
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1*
a, b, c, d
(20)
The Kirchhoff currentlaw equations corresponding to this choice of
76
THE EQUILIBRIUM EQUATIONS
0.5(ei + e2 — e3  e4)
e2 + e3
ja = 0.143(ci  e2 + e3)
j7 = 0.167e,
Use of Eqs. 22 then gives
h = 0.2(e3 + e4)
= 0.1e2 (23)
/» = 0.333(ei + e2) j9 = 0.125e3
j5 = 0.25(e2 — e3 — e4) jio = 0.111e4
The desired node equilibrium equations are obtained through substitut
ing these values for the j's into Eqs. 21. After proper arrangement, the
results read
1.142ei  0.976e2 + 0.643e3 + 0.500e4 = 0
0.976e, + 2.326e2  1.893e3  0.750e4 = 0
(24)
0.643ei  1.893e2 + 2.218e3 + 0.950e4 = 0
0.500ei  0.750e2 + 0.950e3 + 1.061e4 = 0
In summary it is well to observe that the procedure for setting up
equilibrium equations involves, for either the loop or node method,
essentially three sets of relations:
(a) The Kirchhoff equations in terms of pertinent branch quantities.
(b) The relations between branch voltages and branch currents.
(c) The branch .quantities in terms of the desired variables.
The coefficients in the rows and in the columns of the appropriate tieset
or cutset schedule supply the means for writing the relations (a) and (c)
respectively. The relations (b), in the form of either Eqs. 12 or Eqs. 19,
are straightforward in any case.
The desired equilibrium equations are obtained through substituting
relations (c) into (b), and the resulting ones into (a). In the loop method,
the branch quantities in the voltagelaw equations (a) are voltages while
the branch quantities in (c) are currents. In the node method, the
branch quantities in the currentlaw equations (a) are currents while the
branch quantities in (c) are voltages. The relations (b) are needed in
either case to facilitate the substitution of (c) into (a); that is to say,
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this substitution requires first a conversion from branch currents to
branch voltages or vice versa. It is this conversion that is supplied by
the relations (b) which depend upon the circuit elements (resistances or
conductances in the above example).
The tieset or cutset schedule is thus seen to play a dominant role in
either method since it summarizes in compact and readily usable form
all pertinent relations except those determined by the element values.
PARAMETER MATRICES
77
The rows of a tieset schedule define an independent set of closed paths,
and hence provide a convenient means for obtaining an independent set
of Kirchhoff voltagelaw equations. Any row of a cutset schedule, on
the other hand, represents all of the branches terminating in the sub
graph associated with one or more nodes. Since the algebraic sum of
currents in such a set of branches must equal zero, the rows of a cutset
schedule are seen to provide a convenient means for obtaining an inde
pendent set of Kirchhoff currentlaw equations.
The columns of these same schedules provide the pertinent relations
through which the desired variables are introduced. They are useful not
only in the process of obtaining the appropriate equilibrium equations,
but also in subsequently enabling one to compute any of the branch
quantities from known values of the variables.
In situations where the geometry is particularly simple, and where
correspondingly straightforward definitions for the variables are ap
propriate, one may, after acquiring some experience, employ a more
direct procedure for obtaining equilibrium equations (as given in Art. 6)
which dispenses with the use of schedules.
4 Parameter Matrices on the Loop and Node Bases
It should be observed that the final equilibrium Eqs. 18 and 24 are
written in an orderly form in that the variable ii (resp. ei) appears in the
first column, the variable i2 (resp. e2) in the second column, and so forth.
Taking this arrangement for granted, it becomes evident that the essen
tial information conveyed by Eqs. 18, for example, is contained with
equal definiteness but with increased compactness in the array of
coefficients
35
18
17
16
27
24
18
19
10
18
18
17
8
22
17
8
16
10
19
10
16
27
18
17
10
31
18
24
18
8
16
18
24.
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8
known as the loopresistance parameter matrix. Equilibrium Eqs. 24 are
similarly characterized by the following nodeconductance parameter
matrix.
r 1.142
0.976
0.643
78
THE EQUILIBRIUM EQUATIONS
The term matrix is a name given to a rectangular array of coefficients
as exemplified by forms 25 and 26. As will be discussed in later chapters,
one can manipulate sets of simultaneous algebraic equations like those
given by 18 and 24 in a facile manner through use of a set of symbolic
operations known as the rules of matrix algebra. These matters need
not concern us at the moment, however, since the matrix concept is at
present introduced only to achieve two objectives that can be grasped
without any knowledge of matrix algebra whatever, namely: (a) to recog
nize that all of the essential information given by the sets of Eqs. 18 and
24 is more compactly and hence more effectively placed in evidence
through the rectangular arrays 25 and 26; (b) to make available a greatly
abbreviated method of designating loop or nodeparameter values in
numerical examples.
The second of these objectives may better be understood through
calling attention first to a common symbolic form in which equations
like 18 are written, namely thus:
»nii + ri2i2 H h rnii = 0
r2iii + r22i2 H r r2iii = 0
(27)
rzit'i + n2i2 I h Wi = 0
Here each coefficient is denoted by a symbol like rn, r12, and so forth.
The corresponding matrix reads
[R] =
721
ri2
rM
r2i
(28)
irii ri2 ••• mi
The general coefficient in this matrix is denoted by r,* in which the
indexes s and k can independently assume any integer values from 1 to I.
Observe that the first index denotes the row position, and the second one
denotes the column position of the coefficient with respect to array 28.
Analogously, a set of node equations like 24 would symbolically be
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written
ffnei + gi2e2 \ h ginen = 0
021^1 + 022^2 H h ?2nen = 0
(29)
gmei + gn2e2 H h gnnen = 0
SYMMETRY OF PARAMETER MATRICES
79
with the matrix
011 012
9in
[G] =
021 923
02n
(30)
Identification of loop Eqs. 27 in analytic form, with the specific
numerical Eqs. 18 would necessitate (without use of the parameter
matrix concept) writing
which is clearly an arduous and spaceconsuming task compared with
writing down the numerical matrix 25. Use of the matrix concept takes
advantage of the fact that the row and column position of a number
identifies it as a specific r,k value; it is no longer necessary to write
identifying equations like those given by 31. Similar remarks apply to
the numerical identification of parameters on the node basis and the
usefulness of the corresponding parametermatrix notation.
5 Regarding the Symmetry of Parameter Matrices
The parameter matrices 25 and 26 given above have an important
and interesting property in common which is described as their sym
metry. For example, in matrix 25 we note that ri8 = r2i = —18,
T\3 =* r3i = —17, ri4 = r41 = 16, and so forth. More specifically,
matrix 25 is said to possess symmetry about its principal diagonal, the
latter being represented by the elements rn = 35, r22 = 19, r33 = 22,
etc. on the diagonal extending from the upper left to the lower right
hand corner of the array. Elements symmetrically located above and
below this diagonal are equal. Symbolically this symmetrical property
is expressed by the equation
Similar remarks apply to the nodeconductance matrix 26.
This symmetry of the parameter matrix is neither accidental nor
inherent in the physical property of linear networks. It is the result
of having followed a deliberate procedure in the derivation of equilibrium
equations that need by no means always be adhered to.
In order to understand the nature of this procedure, let us recall first
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that the process of deriving equilibrium equations involves predom
rn = 35, ri2 = 18, r13 = 17,
(31)
T,k = Tk,
(32)
80
THE EQUILIBRIUM EQUATIONS
inantly the two sets of relations designated in the summary in Art. 3
as (a) the Kirchhofflaw equations and (c) the defining equations for the
chosen variables. [The circuit element relations (b) are needed in carry
ing out the substitution of (c) into (a) but are not pertinent to the present
argument.] On the loop basis the variables are loop currents, and the
Kirchhoff equations are of the voltagelaw type; on the node basis the
variables are nodepair voltages, and the Kirchhoff equations are of the
currentlaw type.
The choice of a set of loopcurrent variables involves the fixing of a
set of loops or closed paths (tie sets), either through the choice of a tree
and the identification of link currents with loop currents or through the
forthright selection of a set of geometrically independent loops. The
writing of Kirchhoff voltagelaw equations also necessitates the selection
of a set of geometrically independent loops, but this set need not be the
same as that pertaining to the definition of the chosen loop currents. If
the same loops are used in the definition of loop currents and in the
writing of the voltagelaw equations, then the resulting parameter
matrices become symmetrical, but if separate choices are made for the
closed paths denning loop currents and those for which the voltage
law equations are written, then the parameter matrices will not become
symmetrical.
Thus a more general procedure for obtaining the loop equilibrium
equations involves the use of two tieset schedules. One of these per
tains to the definition of a set of loopcurrent variables (as discussed in
Art. 5, Ch. 1); the tie sets in the other one serve merely as a basis for
writing the voltagelaw equations. Instead of using the rows and col
umns of the same schedule for obtaining relations (a) and (c) respectively
in the summary referred to above, one uses the rows of one schedule and
the columns of another. The reader should illustrate these matters for
himself by carrying through this revised procedure for the numerical
example given above and noting the detailed changes that occur.
Analogously, on the node basis, one must choose a set of geometrically
independent node pairs and their associated cut sets for the definition of
nodepair voltage variables, and again for the writing of the Kirchhoff
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currentlaw equations. The second selection of node pairs and associated
cut sets need not be the same as the first, but, if they are (as in the
numerical example leading to Eqs. 24), then the resulting parameter
matrix becomes symmetrical.
Thus a more general procedure for obtaining the node equilibrium
equations involves the use of two cutset schedules. One of these per
tains to the definition of a set of nodepair voltage variables (as dis
cussed in Art. 6, Ch. 1); the cut sets in the other one are utilized in writing
SIMPLIFIED PROCEDURES
81
currentlaw equations. Instead of using the rows and columns of the
same schedule, one uses the rows of one schedule and the columns of
another.
The significant point in these thoughts is that the choice of variables,
whether current or voltage, need have no relation to the process of writ
ing Kirchhofflaw equations. It is merely necessary that the latter be an
independent set; the variables in terms of which they are ultimately
expressed, may be chosen with complete freedom.
When the same tie sets are used for voltagelaw equations and loop
current definitions, or the same cut sets are used for currentlaw equa
tions and nodepair voltage definitions, then we say that the choice of /
variables is consistent with the Kirchhofflaw equations. It is this con ,
sistency that leads to symmetrical parameter matrices.*
The question of symmetry in the parameter matrices is important
primarily in that one should recognize the deliberateness in the achieve
ment of this result and not (as is quite common) become confused into
thinking that it is an inherent property of linear passive bilateral net
works to be characterized by symmetrical parameter matrices. We
shall, to be sure, follow the usual procedure that leads to symmetry, not
only because it obviates two choices being made for a set of loops or
node pairs, but also because symmetrical equations are easier to solve,
and because a number of interesting network properties are more readily
demonstrated. So in the end we follow the customary procedure, but
with an added sense of perspective that comes from a deeper under
standing of the principles involved.
6 Simplified Procedures That Are Adequate in Many Practical
Cases
We have given the preceding very general approach to the matter of
forming the equilibrium equations of networks because, through it as a
background, we are now in a position to understand far more adequately
and with greater mental satisfaction the following rather restricted but
practically very useful procedures applicable to many geometrical net
work configurations dealt with in practice. Thus, in many situations
* These matters were first pointed out by the author at an informal roundtable
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conference on network analysis and synthesis sponsored by the AIEE at its midwinter
convention in 1938. The discussions (supplemented by a distribution of pertinent
mimeographed notes) included derivation of the general loop and node equilibrium
equations for bilateral networks in symmetrical or dissymmetrical form and the con
sequent possibility of obtaining symmetrical matrices for networks containing uni
lateral elements through an appropriate definition of variables. During the past 15
years the presentation of this material was continually simplified through classroom
use.
82
THE EQUILIBRIUM EQUATIONS
encountered in engineering work, the network geometry is such that the
graph may be drawn on a plane surface without having any branches
cross each other. As mentioned in Art. 9, Ch. 1 such a network is spoken
of as being "mappable on a plane," or more briefly as a mappable net
work. The network whose graph is shown in Fig. 3 is not of the map
pable variety, but the one given by the graph in Fig. 1 is.
When the equilibrium equations for a mappable network (such as
that shown in Fig. 1) are to be written on the loop basis, it is possible
to choose as a geometrically independent set of closed loops the meshes
of this network graph (as pointed out in Art. 7 of Ch. 1). A simple
example of this sort is shown in Fig. 4 in which the meshes are indicated
by circulatory arrows. The corresponding voltagelaw equations are
(33)
vi — vi = 0
v2 — t'e = 0
»3 — v 6 = 0
» 4 + 1'5 + v6 = 0
The branch currents in terms of the loop currents are seen to be given by
Ji = *i j* = H ~ *i
h = *2 j& = U ~ *2 (34)
h = i3 k = n — t3
Suppose the branch resistance values are
ri = 5, r2 = 10, r3 = 4, r4 = 2, r5 = 10, r6 = 5 (35)
Equations 34 multiplied respectively by these values yield the corre
sponding v's by means of which Eqs. 33 become expressed in terms of
the loop currents. After proper arrangement this substitution yields
7*'i + 0i2 + 0i3  2u = 0
Ot'i + 20i2 + 0i3  10t4 = 0
Oii + 0i2 + 9i3  5n = 0
2ii  10i2  5i3 + 17i4 = 0
with the symmetrical matrix
(36)
[R] =
.7
2"
20
10
9
5
.2
10
5
17.
(37)
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SIMPLIFIED PROCEDURES
83
A simple physical interpretation may be given to these equations by
reference to Fig. 5 in which the same network as in Fig. 4 is redrawn with
the branch numbering and reference arrows left off but with the branch
resistances and their values indicated. The terra 7*i in the first of Eqs.
36 may be interpreted as the voltage drop caused in mesh 1 by loop
current t'i since the total resistance on the contour of this mesh is 7 ohms;
the rest of the terms in this equation represent additional voltage drops
caused in mesh 1 by the loop currents i2, 13, U, respectively. Since no
part of the contour of mesh 1 is traversed by the currents i2 and i3, these
Fig. 4. A mappable network graph Fig. 5. The resistance network whose
in which the meshes are chosen as graph is shown in Fig. 4. Element
loops. values are in ohms.
can cause no voltage drop in mesh 1; hence the coefficients of their terms
in the first of Eqs. 36 are zero. The term —2i4 takes account of the fact
that loop current ii, in traversing the 2ohm resistance, contributes to
the voltage drop in mesh 1 and that this contribution is negative with
respect to the loop reference arrow in mesh 1.
The second of Eqs. 36 similarly expresses the fact that the algebraic
sum of voltage drops caused in mesh 2 by the various loop currents
equals zero. Only those terms have nonzero coefficients whose associated
loop currents traverse at least part of the contour of mesh 2. The value
of any nonzero coefficient equals the ohmic value of the total or partial
mesh 2 resistance traversed by the pertinent loop current, and its alge
braic sign is plus or minus, according to whether the reference direction
for this loop current agrees or disagrees, respectively, with the reference
arrow for mesh 2. Analogous remarks apply to the rest of Eqs. 36.
With this interpretation in mind, one can write the loopresistance
matrix 37 directly. Thus the coefficients on the principal diagonal are,
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respectively, the total resistance values on the contours of meshes 1, 2,
84
THE EQUILIBRIUM EQUATIONS
3, • • •. The remaining coefficients are resistances of branches common
to a pair of meshes, with their algebraic signs plus or minus according
to the confluence or counterfluence of the respective mesh arrows in the
pertinent common branch. Specifically, a term r,k in value equals the
resistance of the branch common to meshes s and k; its algebraic sign is
plus if the mesh arrows have the same direction in this common branch;
it is minus if they have opposite directions.
In a mappable network, with the meshes chosen as loops and the loop
reference arrows consistently clockwise (or consistently counterclock
wise), the algebraic signs of all nondiagonal terms in the loopresistance
matrix are negat*ve. It is obvious that this procedure for the derivation
of loop equilibrium equations yields a symmetrical parameter matrix
(r,k = Tk,) since a branch common to meshes s and fc, whose value deter
mines the coefficient r,k, is at the same time common to meshes k and s.
This simplified procedure for writing down the loop equilibrium equa
tions directly (having made a choice for the loops and loop currents)
does not, of course, require mappability of the network, but it is not
difficult to appreciate that it soon loses its simplicity and directness
when the network geometry becomes random. For, in a random case
it may become difficult to continue to speak of meshes as simplified
versions of loops; moreover, their choice is certainly no longer straight
forward nor is the designation of loop reference arrows as simple to indi
cate. Any given branch may be common to more than two meshes;
the pertinent loop reference arrows may traverse such a branch in
random directions, so that the nondiagonal coefficients in the parameter
matrix will no longer be consistently negative. Although the simplified
procedure may still be usable in some moderately complex nonmappable
cases, one will find the more general procedure described earlier prefer
able when arbitrary network geometries are encountered.
An analogous simplified procedure appropriate to relatively simple
geometries may be found for the determination of node equilibrium
equations. In this simplified procedure the nodepair voltage variables
are chosen as a nodetodatum set, as described in Art. 8 of Ch. 1. That
is, they are defined as the potentials of the various single nodes with
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respect to a common (arbitrarily selected) datum node, as illustrated in
Ch. 1 by Fig. 11 for the network graph of Fig. 8. The cut sets (which
determine the Kirchhoff currentlaw equations) are then all given by
the groups of branches divergent from the single nodes for which the
pertinent node potentials are defined.
With regard to the network of Fig. 4 one may choose the bottom node
as the datum or reference and define the potentials of nodes 1 and 2
respectively as the voltage variables ei and e2. Noting that the pertinent
SIMPLIFIED PROCEDURES
85
cut sets are the branches divergent from these nodes, the currentlaw
equations consistent with this selection of nodepair voltages are seen
to read
ji + j*  ja  h = 0
(38)
3i ~ 3* + 32 + 3s  0
The branch voltages in terms of the node potentials are, by inspection
of Fig. 4,
vi = ei — e2 V4 = ei — e2
v2 = e2 v 5 = e2 (39)
v3 = ~ei t>6 = ei
The branch conductances corresponding to the resistance values 35 are
9i = 0.2, 02 = 0.1, g3 = 0.25, ff4 = 0.5, g5 = 0.1, g6 = 0.2
(40)
Equations 39 multiplied respectively by these values yield the corre
sponding fa in terms of the node potentials. Their substitution into
Eqs. 38 results in the desired equilibrium equations, which read
1.15ei  0.70e2 = 0
(41)
0.70ei + 0.90e2 = 0
with the symmetrical nodeconductance matrix
T 1.15 0.701
[O = (42)
[0.70 0.90J
A simple physical interpretation may be given to the node equilibrium
Eqs. 41 that parallels the interpretation given above for the loop equa
tions. Thus the first term in the first of Eqs. 41 represents the current
that is caused to diverge from node 1 by the potential ei acting alone
(that is, while e2 = 0); the second term in this equation represents the
current that is caused to diverge from node 1 by the potential e2 acting
alone (that is, while e: = 0). Since a positive e2 acting alone causes
current to converge upon node 1 (instead of causing a divergence of
current), the term with e2 is numerically negative. The amount of cur
rent that ei alone causes to diverge from node 1 evidently equals the
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value of ei times the total conductance between node 1 and datum when
e2 = 0 (that is, when node 2 coincides with the datum). This total
conductance clearly is the sum of the conductances of the various
branches divergent from node 1; with reference to Fig. 5 (in which the
86
THE EQUILIBRIUM EQUATIONS
given parameter values are resistances) this total conductance is 1/5
+ 1/2 + 1/5 + 1/4 = 1.15, thus accounting for the coefficient of the
term with ei in the first of Eqs. 41.
The current that e2 alone causes to diverge from node 1 can traverse
only the branches connecting node 1 directly with node 2 (these are the
2ohm and 5ohm branches in Fig. 5), and the value of this current is
evidently given in magnitude by the product of e2 and the net con
ductance of these combined branches. In the present example the per
tinent conductance is 1/2 + 1/5 = 0.70 mho, thus accounting for the
value of the coefficient in the second term of the first of Eqs. 41 (the
reason for its negative sign has already been explained). A similar
interpretation is readily given to the second of Eqs. 41.
Thus these equations or their conductance matrix 42 could be written
down directly by inspection of Fig. 5, especially if the branchresistance
values are alternately given as branchconductance values expressed
in mhos. The elements on the principal diagonal of [G] are, respectively,
the total conductance values (sums of branch conductances) divergent
from nodes 1, 2, • • • (in a more general case there will be more than two
nodes). The nondiagonal elements of [G] all have negative algebraic
signs, for the argument given above in the detailed explanation of Eqs. 41
clearly applies unaltered to all cases in which the nodepair voltage
variables are chosen as a nodetodatum set. In magnitude, the non
diagonal elements in [G] equal the net conductance values (sums of
branch conductances), for those branches directly connecting the per
tinent node pairs. More specifically, the element g,k in [6] equals the
negative sum of the conductances of the various branches directly con
necting nodes s and k. If these nodes are not directly connected by any
branches, then the pertinent g,k value is zero. Note that the consistent
negativeness of the nondiagonal terms follows directly from the tacit
assumption that any node potential is regarded as positive when it is
higher than that of the datum node. This situation parallels the con
sistent negativeness of the nondiagonal terms in the [R] matrix obtained
on the loop basis for a mappable network in which all the mesh reference
arrows are chosen consistently clockwise (or consistently counterclock
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wise), whence in any common branch they are counterfluent.
7 Sources
When currents and their accompanying voltage drops exist in a re
sistive network, energy is being dissipated. Since at every instant the
rate of energy supply must equal its rate of dissipation, there can be
no voltages or currents in a purely resistive or in any "lossy" network
unless there are present one or more sources of energy.
SOURCES
87
Until now the role played by sources has not been introduced into the
network picture and indeed their presence has nothing whatever to do
with the topics discussed so far. Sources were purposely left out of
consideration for this reason, since their inclusion would merely have
detracted from the effectiveness of the discussion. Now, however, it
is time to recognize the significance of sources, their characteristics, and
how we are to determine their effect upon the equilibrium equations.
Their most important effect, as already stated, is that without them
there would be no response. This fact may clearly be seen, for example,
from the loop equilibrium Eqs. 36 for the network of Fig. 5. Since these
four equations involving the four unknowns i2, iz, u are independent,
and all of the righthand members are zero, we know according to the
rules of algebra that none but the trivial solution t'i = *2 = iz = U = 0
exists. That is to say, in the absence of excitation (which, as we shall
see, causes the righthand members of the equations to be nonzero)
the network remains "dead as a doornail."
It was pointed out in the introduction that an electrical network as we
think of it in connection with our present discussions is almost always an
artificial representation of some physical system in terms of idealized
quantities which we call the circuit elements or parameters (the resist
ance, inductance, and capacitance elements). We justify such an arti
ficial representation through noting (a) that it can be so chosen as to
simulate functionally (and to any desired degree of accuracy) the actual
system at any selected points of interest, and (b) that such an idealiza
tion is essential in reducing the analysis procedure to a relatively simple
and easily understandable form.
Regarding the sources through which the network becomes energized
or through which the physical system derives its motive power, a con
sistent degree of idealization is necessary. That is to say, the sources,
like the circuit elements, are represented in an idealized fashion. We
shall see that actual energy sources may thus be simulated through such
idealized sources in combination with idealized circuit elements. For
the moment we focus our attention upon the idealized sources themselves.
Although the physical function of a source is to supply energy to the
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system, we shall for the time being find it more expedient to characterize
a source as an element capable of providing a fixed amount of voltage
or a fixed amount of current at a certain point. Actually it provides
both voltage and current, and hence an amount of power equal to their
product, but it is analytically essential and practically more realistic
to suppose that either the voltage or the current of the source is known
or fixed. We could, of course, postulate a source for which both the
voltage and the current are fixed, but such sources would not prove useful
88
THE EQUILIBRIUM EQUATIONS
in the simulation of physical systems, and we must at all times be mind
ful of the utility of our methods of analysis.
When we say that the voltage or the current of a source is fixed, we do
not necessarily mean that it is a constant, but rather that its value
or sequence of values as a continuous function of the time are independent
of all other voltages and currents in the entire network. Most important
in this connection is the nondependence upon the source's own voltage,
if it is a current, or upon its own current if it is a voltage. Thus a so
called idealized voltage source provides at a given terminal pair a voltage
function that is independent of the current at that terminal pair; and an
idealized current source provides a current function that is independent
of the voltage at the pertinent terminal pair.
By way of contrast, it is useful to compare the idealized source as just
defined with an ordinary passive resistance or other circuit element.
In the latter, the voltage and current at the terminals are related in a
definite way which we call the "voltampere relationship" for that ele
ment. For example, in a resistance the voltage is proportional to the
current, the constant of proportionality being what we call the value
of the element in ohms. At the terminals of an ideal voltage source, on
the other hand, the voltage is whatever we assume it to be, and it cannot
depart one jot from this specification, regardless of the current it is
called upon to deliver on account of the conditions imposed by its
environment. An extreme situation arises if the environment is a short
circuit, for then the source is called upon to deliver an infinite current;
yet it does so unflinchingly and without its terminal voltage departing
in the slightest from its assigned value. It is, of course, not sensible to
place an ideal voltage source in such a situation, for it then is called
upon to furnish infinite power. The ideal voltage source is idle when
its environment is an open circuit, for then the associated current be
comes zero.
Similarly, at the terminals of an ideal current source the current is
whatever we assume it to be, and it cannot depart from this specification,
regardless of the voltage it is called upon to produce on account of the
conditions imposed by its environment. An extreme situation arises
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in this case if the environment turns out to be an open circuit, for then
the source must produce an infinite voltage at its terminals since the
terminal current, by definition, cannot depart from its specified value.
Like the shortcircuited voltage source, it is called upon to deliver in
finite power, and hence it is not realistic to place an ideal current source
in an opencircuit environment. This type of source is idle when short
circuited, since the associated voltage is then zero.
In the discussion of Kirchhoff's voltage law we found it useful to think
of voltage as analogous to altitude in a mountainous terrain. The
SOURCES
89
potentials of various points in the network with respect to a common
reference or datum are thought of as being analogous to the altitudes of
various points in a mountainous terrain with respect to sea level as a
common reference. Instead of an actual mountainous terrain, suppose
we visualize a miniature replica constructed by hanging up a large
rubber sheet and suspending from it various weights attached at random
places. Since altitude is the analogue of voltage, the problem of finding
the altitude of various locations on the sheet (above, say, the floor as a
common reference) is analogous to determining the potentials of various
nodes in an electrical network with reference to a datum node.
Suppose first that we consider the electrical network to have no sources
of excitation; all node potentials are zero. The analogous situation in
volving the rubber sheet would be to have it lying flat on the floor.
To apply a voltage excitation to the network may be regarded as causing
certain of its node potentials to be given fixed values. Analogously,
certain points in the rubber sheet are raised above the floor to fixed
positions and clamped there. As a result, the various nodes in the
electrical network whose potentials are not arbitrarily fixed, assume
potentials that are consistent with the applied excitation and the char
acteristics of the network. Analogously, the freely movable portions
of the rubber sheet assume positions above the floor level that are
consistent with the way in which the sheet is supported at the points
where it is clamped (analogous to excitation of the electrical network)
and the structural characteristics of the sheet with its system of attached
weights.
It is interesting to note from the description of these two analogous
situations that electrical excitation by means of voltage sources may be
thought of as arbitrarily fixing or clamping the voltage at a certain
point or points. A voltage source is thus regarded as an applied con
straint, like nailing the rubber sheet to the wall at some point.
Ideal current sources when used to excite an electric network may
likewise be regarded as applied constraints. In any passive network
the currents and voltages in its various parts are in general free to assume
an array of values subject only to certain interrelationships dictated by
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the structure of that network, but, without any excitation, all voltages
and currents remain zero. If we now give to some of these voltages and
currents arbitrary nonzero values, we take away their freedom, for they
can no longer assume any values except the specified ones, but the re
maining voltages and currents, whose values are not pegged, now move
into positions that are compatible with the network characteristics inter
relating all voltages and currents, and with the fixed values of those
chosen to play the role of excitation quantities. As more of the voltages
and currents are clamped or fixed through the application of sources,
90
THE EQUILIBRIUM EQUATIONS
fewer remain free to adjust themselves to compatible values. Finally,
if all voltages and currents were constrained by applied sources, there
would be no network problem left, for everything would be known
beforehand. In the commonest situation, only a single voltage or cur
rent variable is constrained through an applied source; determination
of the compatible values of all the others constitutes the network problem.
Various ways in which sources are schematically represented in circuit
diagrams are shown in Fig. 6. Parts (a), (b), and (c) are representations
of voltage sources, whereas part (d) shows the representation for a cur
rent source. Specifically (a) and (b) are common ways of indicating
(a) (b) (c) (d)
Fio. 6. Schematic representations for sources, (a) A constant voltage (batiery), (b)
a constant voltage (dc generator), (c) arbitrary voltage function, (d) arbitrary cur
rent function.
constantvoltage sources, also called "direct current" or "dc" voltage
sources. The schematic (a) simulates a battery, for example, a dry cell
in which the zinc electrode (thin line) is negative and the carbon elec
trode (thick line) is the positive terminal. The dc source shown in (b)
is drawn to resemble the commutator and brushes of a generator. The
symbolic representation in (c) is intended to be more general in that the
wavy line inside the circle indicates that e,(t) may be any function of
time (not necessarily a sinusoid, although there is an established prac
tice in using this symbol as the representation for a sinusoidal generator).
It should be particularly noted that e,(l) in the symbolic representation
of part (c) may be any time function and, in particular, may also be
used to denote a constantvoltage source (dc source).
Part (d) of Fig. 6 shows the schematic representation for a current
source in which i,(t) is any time function and hence may be used to denote
a constant or dc source as well as any other.
In all of these source representations it will be noted that a reference
arrow is included. This arrow does not imply that the source voltage or
current is assumed to act in the indicated direction but only that, if it
should at any moment have this direction, it will at that moment be
regarded as a positive quantity. The reference arrow establishes a
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means for telling when the quantity e,(t) or is positive and when it is
SOURCES
91
negative. A source voltage is said to "act in the direction of the refer
ence arrow" when it is a voltage rise in this direction. The + and —
signs of parts (a) and (b) of Fig. 6 further clarify this statement. In
most of the following work the representations shown in parts (c) and
(d) will be used.
It should not be overlooked that the representations in Fig. 6 are for
ideal sources. Thus the voltage between the terminals in the sketch
of part (c) is always e,(t) no matter what is placed across them. Like
wise the current issuing from the terminals in the sketch of part (d) is
always i,(t) no matter what the external circuit may be. An actual
physical voltage source may, to a first approximation, be represented
by placing a resistance in series with the ideal one so that the terminal
voltage decreases as the source current increases. A physical current
source may similarly be represented to a first approximation through
the ideal one of part (d) with a resistance in parallel with the terminals,
thus taking account of the fact that the net current issuing from the
terminals of the combination depends upon the terminal voltage, and
decreases as this voltage increases. These matters will further be
elaborated upon in the applications to come later on.
It is common among students that they have more difficulty visualiz
ing or grasping the significance of current sources than they do in the
understanding of voltage sources. A contributing reason for this
difficulty is that voltage sources are more commonly experienced. Thus
our power systems that supply electricity to our homes and factories
are essentially voltage sources in that they have the property of being
idle when opencircuited. Sources that are basically of the current
variety are far less common. One such source is the photoelectric cell
which emits charge proportional to the intensity of the impinging light
and hence is definitely a current source; it clearly is idle when short
circuited because it then delivers no energy. Another device that is
commonly regarded as a current source is the pentode vacuum tube.
Its plate current is very nearly proportional to its grid excitation under
normal operating conditions, and hence, for purposes of circuit analysis,
it is appropriate to consider it as being essentially a current source.
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In any case it can with very good accuracy be regarded as an ideal cur
rent source in parallel with a resistance.
Whether actual sources are more correctly to be regarded as voltage
sources or as current sources is, however, a rather pointless argument
since we shall soon see that either representation (in combination with
an appropriate arrangement of passive circuit elements) is always pos
sible no matter what the actual source really is. Again we must be
reminded that circuit theory makes no claim to be dealing with actual
92
THE EQUILIBRIUM EQUATIONS
things. In fact it very definitely deals only with fictitious things, but
in such a way that actual things can thereby be represented. "Like all
other methods of analysis, circuit theory is merely the means to an end;
it lays no claim to being the real thing.
Now as to determining how source quantities enter into the equilib
rium equations for a given network, we first make the rather general
observation that the insertion of sources into a given passive network
is done in either of two ways. One of these is to insert the source into
(a) / (b)
Fia. 7. Network graph involving voltage source (constraint) in parallel with a branch
(a), and the equivalent revised graph (b) showing disposition of voltage source.
the gap formed by cutting a branch (as with a pliers); the other is to
connect the source terminals to a selected node pair (as with a soldering
iron). These two methods will be distinguished as the "pliers method"
and the "solderingiron method" respectively. We shall now show that
one may consider the pliers method restricted to the insertion of voltage
sources and the solderingiron method to the insertion of current sources.
That is to say, the connection of a voltage source across a node pair or
the insertion of a current source in series with a branch implies a revision
of the network geometry, with the end result that voltage sources again
appear only in series with branches and current sources appear only in
parallel with branches (or across node pairs).
For example, in part (a) of Fig. 7 is shown a graph in which a voltage
source e, appears in parallel with branch 6 of some network, and in part
(b) of this figure is shown the resultant change in the network geometry
and source arrangement which this situation reduces to. Thus, in
considering the given arrangement in part (a), one should first observe
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that branch 6 is rendered trivial by having e, placed in parallel with it
SOURCES
93
since the value of v6 is thus forced to be equal to e, and hence (along
with j6) is no longer an unknown. That is to say, the determination of
the current in branch 6 is rendered trivially simple and independent of
what happens in the rest of the network. Therefore we can remove
branch 6 from our thoughts and from the rest of the graph so that e,
alone appears as a connecting link between nodes a and b. Next we ob
serve that the potentials of nodes c, d, f, relative to that of node a are
precisely the same in the arrangement of part (b) in Fig. 7 as they are
in part (a). For example, the potential of node c with respect to that
of node a is (e, — vs) as is evident by inspection of either part (a) or
part (b) of this figure. Similarly the potential of node d with respect
to that of node a is seen to be (e, + 1^7) in the arrangement of part (a)
or of part (b). It thus becomes clear that the branch voltages and cur
rents in the graph of part (b) must be the same as in the graph of part
(a), except for the omission of the trivial branch 6.
We may conclude that placing a voltage source across a node pair
has the same effect upon the network geometry as placing a short circuit
across that node pair. Comparing graphs (a) and (b) in Fig. 7, we see,
for example, that the voltage source e, in graph (a) effectively unites
nodes a and b in that graph, thus eliminating branch 6, and yielding
the revised graph (b). The effect of the voltage source so far as this
revised graph is concerned is taken into account through placing iden
tical voltage sources in series with all branches confluent in the original
node b. We can alternately place the identical voltage sources in series
with the branches originally confluent in node a: that is, in branches
4 and 5 instead of 7, 8, and 9.
It is useful in this connection to regard a voltage source as though it
were a sort of generalized short circuit, which indeed it is. Thus, by a
short circuit we imply a link or branch for which the potential difference
between its terminals is zero independent of the branch current, while
for a voltage source the potential difference is e, independent of the
branch current. For e, = 0, the short circuit is identical with the voltage
source. Or we may say that a dead voltage source is a short circuit.
The preceding discussion shows that the effect of a voltage source upon
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the network geometry is the same as that of an applied shortcircuit
constraint.
Analogously, part (a) of Fig. 8 depicts a situation in which a current
source i, appears in series with branch 4 of some network, and part (b)
shows the resultant change in geometry and source arrangement which
is thereby implied. With reference to the given situation in part (a)
it is at once evident that branch 4 becomes trivial since its current is
identical with the source current and hence is known. It is also evident
94
THE EQUILIBRIUM EQUATIONS
that the effect of the current source i, upon the rest of the network is
the same as though there had been no branch linking nodes a and b
through which the source is applied. We can, therefore, regard the cur
rent source to be bridged across the node pair ab in a modified graph
in which branch 4 is absent.
A further step that results in having all current sources in parallel
with branches may be carried out as shown in part (b) of Fig. 8. The
equivalence of the four identical current sources i, bridged across
Fig. 8. Network graph involving current source (constraint) in series with a branch
(a), and the equivalent revised graph (b) showing disposition of current source.
branches 11, 9, 8, 7, with a single source i, bridged across the node pair
ab is evident by inspection since the same amount of source current
still leaves node a and enters node b, while no net source current enters
or leaves the nodes /, g, and h.
We may conclude that inserting a current source in series with a
branch has the same effect upon the network geometry as does the open
circuiting or the removal of that branch. In this altered network the
source appears bridged across the node pair originally linked by the
removed branch, or in the form of several identical sources bridged
across a confluent set of branches joining this node pair.
According to these results we may regard a current source as a gen
eralized open circuit. By an open circuit we understand a branch for
which the current is zero independent of the branch voltage; and by a
current source we understand a branch for which the current is t, inde
pendent of the branch voltage. For i, = 0, the current source is identical
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with an open circuit; the latter may be regarded as a dead current source.
SOURCES
95
In summary we may say that, so long as voltage sources appear only
in series with branches, and current sources are associated only in parallel
with branches or across node pairs, their presence does not disturb the
network geometry in the sense that all matters pertaining to that
geometry remain unaltered, such as the numbers of independent voltages
and currents uniquely characterizing the state of the network, or their
algebraic relations to the branch currents and voltages. In a sense, the
opencircuit character of a current source and the shortcircuit character
of a voltage source become evident
here as they do in the reasoning of ^*
the immediately preceding para a b
voltage source in parallel with Fig. 9. Passive branch with associated
question becomes trivial and can
be removed, leaving in its place an open circuit if the inserted source
is a current, and a short circuit if the inserted source is a voltage.
After this revision in the geometry is carried out, the source appears
either as a current in parallel with a branch (or with several branches)
or as a voltage in series with a branch (or with several branches).
These two source arrangements alone, therefore, are all that need to
be considered in the following discussion.
Thus we may regard any branch in a network to have the structure
shown in Fig. 9. Here the link ab represents the passive branch with
out its associated voltage and current sources; that is to say, when the
sources are zero (as they usually are for most of the branches in a net
work), then the branch reduces to this link ab alone. However, we
shall take the attitude at this point that any or all of the branches in a
network may turn out to have the associated sources shown in Fig. 9.
The network is thus regarded as a geometrical configuration of active
instead of passive branches. This turn of events changes nothing with
regard to all that has been said previously except the relations between
branch voltages and branch currents [designated as the relations (b)
in the summary of Art. 3 regarding the formulation of equilibrium
equations].
Since vk and jk denote the net voltage drop and the net current in
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branch fc, the voltage drop and current in the passive link ab (noting
the reference arrows in Fig. 9) are (vk + e,k) and (jk + i,k) respectively.
On the other hand, we see that
the network geometry is affected
whenever a current source is placed
in series with a branch or a
graphs.
one. In both cases the branch in
current and voltage source.
96
THE EQUILIBRIUM EQUATIONS
These are the quantities that are related by the passive circuit element
which the branch represents. If the functional relationship between
voltage drop and current in the passive link is formally denoted by
v = z(j) or j = y(v), we have, for the general active branch of Fig. 9,
0>* + e,k) = z(jk + i,k) or (j* + *',*) = y (t>* + e,*) (43)
In a resistance branch, the notation z(j) reduces simply to a multi
plication of the current j by the branch resistance, and y(v) denotes a
multiplication of the voltage drop v by the branch conductance. In
capacitive or inductive branches the symbols z(j) and y(v) also involve
time differentiation or integration, as will be discussed in detail later
on when circuits involving these elements are considered. For the
moment it will suffice to visualize the significance of Eqs. 43 with regard
to resistance elements alone.
It may be mentioned, with reference to the arrangement in Fig. 9,
that the same results are obtained if the current source i,k is assumed to
be in parallel with the passive link ab alone rather than with the series
combination of this link and the voltage source e,*. If i,k = 0, the link
is activated by a series voltage source alone; if e,k = 0, one has the
representation of a passive branch activated by a current source alone.
For e,k = i,k = 0, the arrangement reduces to the usual passive branch.
Thus the voltampere relations 43 are sufficiently general to take care
of any functional dependence between net branch voltages and currents
that can arise in the present discussions.
The method of including the effect of sources in the derivation of
equilibrium equations is now easily stated. Namely, one proceeds
precisely as described in the previous articles for the unactivated net
work except that the relations between branch voltages and branch
currents are considered in the form of Eqs. 43, so as to take account of
the presence of any voltage or current sources. This statement applies
alike to the determination of equilibrium equations on the loop or the
node basis. Thus, regardless of the nature and distribution of sources
throughout the network, the procedure remains straightforward and is
essentially the same as for the unexcited network.
8 Summary of the Procedures for Deriving Equilibrium
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Equations
At this point it is effective to bring together in compact symbolic
form the steps involved in setting up equilibrium equations. Thus we
have on the loop basis:
(a) The Kirchhoff voltagelaw equations in terms of branch voltages:
2±t>* = 0
(44)
SUMMARY OF THE PROCEDURES
97
(b) The relations between branch voltages and branch currents
(Eqs. 43):
vk = e.k + z(jk + i,k) (45)
(c) The branch currents in terms of the loop currents:
ju = 2±ir (46)
The rows of a tieset schedule (like 13, for example) place in evidence
the Kirchhoff Eqs. 44, while the columns of this schedule yield the branch
currents in terms of the loop currents, Eqs. 46. The expressions for the
vk's in terms of the jVs, Eqs. 45, are obtained from a knowledge of the
circuit parameters and the associated voltage and current sources, as
illustrated in Fig. 9.
The desired equilibrium equations are the Kirchhoff Eqs. 44 expressed
in terms of the loop currents. One accomplishes this end through sub
stituting the jk's given by Eqs. 46 into Eqs. 45, and the resulting
expressions for vk into Eqs. 44. Noting that the linearity of the network
permits one to write z(jk + i,k) = z(jk) + z(i,k), the result of this
substitution among Eqs. 44, 45, 46 leads to
Z±z(2±tr) = 2±[e.*  z(i,k)] = etl (47)
Interpretation of this formidable looking result is aided by pointing
out that z(2±ir) represents the passive voltage drop in any branch k
due to the superposition of loop currents ir in that branch, and that the
lefthand side of Eq. 47 is the algebraic summation of such passive
branch voltage drops around a typical closed loop I. The righthand side,
which is abbreviated by the symbol e,i, is the net apparent source voltage
acting in the same loop. It is given by an algebraic summation of the
voltage sources present in the branches comprising this closed contour
(tie set) and the additional voltages induced in these branches by current
sources that may simultaneously be associated with them. The latter
voltages, which are represented by the term — z(i,k), must depend upon
the circuit parameter relations in the same way as do the passive voltage
drops caused by the loop currents, except that their algebraic signs are
reversed because they are rises.
Thus the resulting equilibrium Eqs. 47 state the logical fact that the
net passive voltage drop on any closed contour must equal the net active
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voltage rise on that contour. If we imagine that the loops are deter
mined through selecting a tree and identifying the link currents with
loop currents, then we can interpret the source voltages e,i as equivalent
link voltages in the sense that, if actual voltage sources having these
values are placed in the links and all original current and voltage sources
are removed, the resulting loop currents remain the same. Or we can
98
THE EQUILIBRIUM EQUATIONS
say that, if the negatives of the voltages e,i are placed in the links, then
the effect of all other sources becomes neutralized, and the resulting
network response is zero; that is, the loop currents or link currents are
zero, the same as they would be if all links were opened.
Hence we have a physical interpretation of the e,i in that they may be
regarded as the negatives of the voltages appearing across gaps formed
by opening all the links. In many situations to which the simplified
procedure discussed in Art. 6 is relevant, this physical interpretation
of the net excitation quantities e,i suffices for their determination by
inspection of the given network.
An entirely analogous procedure and corresponding process of physical
interpretation applies to the derivation of equilibrium equations on the
node basis. Here one has
(a) The Kirchhoff currentlaw equations in terms of branch currents:
2±j* = 0 (48)
(b) The relations between the branch currents and branch voltages
(Eqs. 43):
jk = i,k + y(vk + e,k) (49)
(c) The branch voltages in terms of the nodepair voltages:
vk = 2±er (50)
The rows of a cutset schedule (like 20, for example) place in evidence
the Kirchhoff Eqs. 48, while the columns of this schedule yield the
branch voltages in terms of the nodepair voltages, Eqs. 50. The ex
pressions for the jk's in terms of the vk's, Eqs. 49, are obtained from a
knowledge of the circuit parameters and the associated voltage and
current sources, as illustrated in Fig. 9.
The desired equilibrium equations are the Kirchhoff Eqs. 48 expressed
in terms of the nodepair voltages. One obtains this end by substituting
the v^s given by Eqs. 50 into Eqs. 49, and the resulting expressions for
jk into Eqs. 48. Noting that the linearity of the network permits one
to write y(vk + e,k) = y(vk) + y(e,k), the result of this substitution
among Eqs. 48, 49, 50 leads to
Z±2/(Z±er) = 2± [i.k  y(e,k)] = i.n (51)
Interpretation of this formidable looking result is aided through
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recognizing that ?/(2±er) represents the passive current in any branch k
due to the algebraic sum of nodepair voltages er acting upon it, and
hence the lefthand side of Eq. 51 is the summation of such branch cur
rents in all branches of a typical cut set; for example, the set of branches
EXAMPLES
99
divergent from a given node n if the nodepair voltages are chosen as a
nodetodatum set.
The righthand side of Eq. 51, which is abbreviated by the symbol i,n,
is the net apparent source current for this cut set, for example, it is the
net apparent source current entering node n in a nodetodatum situa
tion. The net source current is given by an algebraic summation of the
current sources associated with the branches comprising the pertinent
cut set and the additional currents induced in these branches by voltage
sources that may simultaneously be acting in them. The latter currents,
which are represented by the term — y(e,k), must depend upon the cir
cuitparameter relations in the same way as do the passive currents
caused by the nodepair voltages except that their algebraic signs are
reversed because they represent a flow of charge into the cut set rather
than out of it.
Thus the resulting equilibrium Eqs. 51 state the logical fact that the
net current in the several branches of a cut set must equal the total source
current feeding this cut set. If we imagine that the cut sets have been
determined through selecting a tree and identifying the treebranch
voltages with nodepair voltages, then we can interpret the source cur
rents i,n as equivalent sources bridged across the tree branches in the
sense that, if actual current sources having these values are placed in
parallel with the tree branches and all original current and voltage
sources are removed, the resulting nodepair voltages remain the same.
Or we can say that, if the negatives of the currents i,n are placed across
the tree branches, then the effect of all other sources becomes neutralized,
and the resulting network response is zero; that is, the nodepair voltages
or treebranch voltages are zero, the same as they would be if all tree
branches were shortcircuited.
Hence we have a physical interpretation of the i,n in that they may be
regarded as the negatives of the currents appearing in short circuits
placed across all the tree branches. In a nodetodatum choice of node
pairs, the i,n may be regarded as the negatives of the currents appearing
in a set of short circuits placed across these node pairs, and a nodeto
datum set of current sources having these values can be used in place
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of the original voltage and current sources in computing the desired
network response. In many situations to which the simplified procedure
discussed in Art. 6 is relevant, this physical interpretation of the net
excitation quantities i,n suffices for their determination by inspection
of the given network.
9 Examples
The complete procedure for setting up equilibrium equations will now
be illustrated for several specific examples: Consider first the resistance
100
THE EQUILIBRIUM EQUATIONS
network of Fig. 10. The element values in part (a) are in ohms, and the
source values are i, = 10 amperes, e, = 5 volts (both constant). In
Fia. 10. A resistance network (element values in ohms) and its graph showing the
'choice of meshes as loops.
part (b) of the same figure is shown the graph with its branch numbering
and a choice of meshes to define loop currents.
Loop
Branch No.
No.
1
2
3
4
5
6
1
1
1
2
1
1
3
1
1
4
1
1
1
(52)
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The tieset schedule corresponding to this choice is given in 52. The
rows give us the voltagelaw equations:
vi — vi = 0
v2 — v5 = 0
v3 — v6 = 0
v4 + » 5 + v 6 = 0
(53)
EXAMPLES
101
and the columns yield the branch currents in terms of the loop currents,
thus:
Ji
= *i
3i =
ii +14
h
= i2
J5 =
*2 + *4
33
= i3
36 =
~i3 + U
These correspond respectively to Eqs. 44 and 46 in the above summary.
With regard to Eqs. 45 relating branch voltages to branch currents,
we observe that, if we associate the current source with branch 5 (we
could alternately associate it with branch 2), then all branches except
1 and 5 are passive and no special comment is needed for them. The
net voltage drop in branch 1 is vi = — e, f ji, and the net current in
the arrow direction in branch 5 is j6 = i, + (v5/2), the term (»5/2)
being the current in the 2ohm resistance which is the passive part of this
branch. Noting the source values given above, the relations expressing
net branchvoltage drops in terms of net branch currents read
fi =■ h  5 »4 = 2/4
»2 = 32 vs = 2js  20 (55)
v3 = j3 v 6 = 2j6
The relations involving the active branches are seen to contain terms
that are independent of current.
The desired equilibrium equations are found through substitution of
Eqs. 54 into 55, and the resulting expressions for the v's into the voltage
law equations 53. After proper arrangement this gives
3ii + 0*2 + 0i3  2U = 5
Ot'i + 3i2 + 0i3  2n = 20
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(56)
Oii + 0*2 + 3i3  2i4 = 0
2t'i  2i2  2i3 + 6t4 = 20
These are readily solved for the loop currents. One finds
*, = 5, i2 = 10/3, *3 = 10/3, i4 = 5 (57)
whence substitution into Eqs. 54 yields all the branch currents
3i  5, j2 = 10/3, j3 = 10/3, u = 0, is = 25/3, j6 = 5/3 (58)
The value of js is the net current in branch 5. That in the passive part
of this branch is smaller than j5 by the value of the source current, and
hence is (25/3)  10 = 5/3.
102
THE EQUILIBRIUM EQUATIONS
Now let us solve the network given in Fig. 10 by the node method,
choosing as nodepair voltages the potentials of nodes a and b respec
tively, with the bottom node as a reference. The appropriate cutset
schedule is 59. The rows give us the currentlaw equations,
Node
Pair
No.
Branch No.
1
2
3
4
5
6
1
1
1
1
1
2
1
1
1
1
(59)
ji + h ~ h + h  0
32 + h  is + ie = 0
(60)
and the columns yield the branch voltages in terms of the nodepair
voltages, thus:
e2
ei  e2
(61)
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ei — e2
These correspond respectively to Eqs. 48 and 50 in the above summary.
Regarding Eqs. 49 relating the branch currents to the branch voltages,
we note as before that 3\ = vi + e, and js = i, + 0.5»5, so that the com
plete set of these equations reads
Ji
3a
33
vi + 5
H
3*
3s
3t
0.5«4
0.5f5 + 10
0.5«6
(62)
which are simply the inverse of Eqs. 55.
The desired equilibrium equations are found through substitution of
Eqs. 61 into 62, and the resulting expressions for the j's into the current
law equations 60. After proper arrangement one finds
3ei — 1.5e2 = —5
1.5ei + 3e2 = 10
The solution is readily found to be
(63)
EXAMPLES
103
ei =0, e2 = 10/3 (64)
and the branch voltages are then computed from Eqs. 61 to be
„, = 0, t>2 = 10/3, t>3 = 10/3, v4 = 0, v5= 10/3, v8 = 10/3 (65)
With regard to branch 1 it must be remembered that the value of vi is
for the total branch, including the voltage source. The drop in the
passive part, therefore, is 5 volts.
As a second example we shall consider the network graph shown in
Fig. 11(a). The sources in series with the branches are voltages having
(a) (b)
Fio. 11. Graph of a resistance network (a) with branch conductance values given by
Eqs. 69. Choice of nodepair voltage variables is indicated in (b).
the values indicated. Since for this graph b = 10, n = 3, and I = 7, it
will be advantageous to choose the node method. A geometrical specifi
cation of nodepair voltages is shown in part (b) of the same figure. In
cutset schedule 66 pertaining to this choice of node pairs a last column
Node
Pair
No.
Branch No.
Picked
Up
1
2
3
4
5
6
7
8
9
10
Nodes
1
1
1
1
1
1
1
1
a, c
2
1
1
1
1
.0
1
a, c, d
3
1
1
1
1
1
b, c, d
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1
104 THE EQUILIBRIUM EQUATIONS
indicating the corresponding "pickedup" nodes is added to facilitate
understanding its construction.
According to the rows of this schedule one obtains the Kirchhoff
currentlaw equations,
fa + fa ~ fa  fa + J5 + fa  fa fa =0
fa + fa ~ fa ~ fa +fa =0 (67)
fa  fa + fa + fa + ji0 = 0
while the columns yield the following relations for the branch voltages
in terms of the nodepair voltages:
vi = ei + e2 — e3 v6  ei
v2 = e\ + e2 — e3 v7 = —ei + e3
t'3 = —ei — e2 va = — ei + e3 (68)
f4 = —1\ — e2 v$ = e2
v6 = ei viq = e3
The branches are again considered to be resistive. Let us assume for
their conductances the following values in mhos:
gi = 2, g2 = 2, g3 = 1, gt = 3, g5 = 4,
(69)
!76 = 5, g7 = 1, 0s = 3, gg = 2, gi0 = 6
The relations expressing the branch currents in terms of the net branch
voltage drops are then readily found by noting the appropriate expression
for the drop in the passive part of each branch and multiplying this by
the corresponding conductance. For example, the voltage drop in the
passive part of branch 1 is »i + 10, in branch 3 it is t'3 + 2, in branch 5
it is t'5 — 8, and so forth. Thus we see that
(70)
Substitution of the «'s from Eqs. 68 into Eqs. 70 and the resulting
expressions for the fs into Eqs. 67 gives the desired equilibrium equa
fa
= 2vi + 20
fa
= 5v6
fa
= 2v2
fa
fa
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= v7 + 6
= v3 + 2
fa
= 3«s
fa
= 3t'4  12
fa
= 2vg
fa
= 4t'5  32
fan
= 6t,io  30
PROBLEMS
105
tions. After proper arrangement these read
21ei + 8e2  8e3 = 8
8d + 10e2  4e3 = 30 (71)
8ei  4e2 + 14e3 = 44
Their solution yields
ei = 3.49, e2 = 4.22, e3 = 3.93 (72)
from which the net branchvoltage drops may readily be computed using
Eqs. 68, and the branch currents are then found from Eqs. 70.
PROBLEMS
1. Regarding the independence of Kirchhoff voltagelaw equations, it might be
supposed that, if the number of equations equals I = b — n, and if collectively they
involve all of the branch voltages, then they must form an independent set. Show
that this conclusion is false by constructing a counter example. Thus, with regard
to the accompanying graph, consider equations written for the combined contours of
meshes 1 and 2, 2 and 3, 3 and 4, 4 and 1. Although all branch voltages are involved,
show that these equations do not form an independent set.
r.
2
3^
4
5
6
7
8
9
V
Prob. 1. Prob. 3.
2. Prove or disprove the statement: "The number of independent Kirchhoff
voltagelaw equations equals the smallest number of closed paths that traverse all
of the branches."
3. With reference to the graph shown, determine whether a set of voltagelaw
equations written for the following combined mesh contours is an independent one:
(1+2+3), (4 + 5 + 6), (7 + 8 + 9), (1+4 + 7), (2 + 5 + 8),
(3+6 + 9), (1+2 + 4 + 5), (2 + 3 + 5 + 6), (5+6+8 + 9)
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4. Construct the dual to Prob. 1, and solve it.
5. In Prob. 1 show that voltagelaw equations written for the following combined
mesh contours do form an independent set (1 + 2 + 3), (2 + 3 + 4), (3 + 4 + 1),
(4 + 1 + 2).
Is the following set independent: (1  2), (2  3), (3  4), (4  1)?
106
THE EQUILIBRIUM EQUATIONS
6. In a 5mesh mappable network, are voltagelaw equations written for the fol
lowing mesh combinations independent:
(1+2), (2+3), (3+4), (4 + 5), (5 + 1)?
or (1  2), (2  3), (3  4), (4  5), (5  1)?
7. Translate Prob. 6 into its dual. Make appropriate sketches and answer the
pertinent questions involved.
8. Prove that voltagelaw equations written for the mesh contours in a mappable
network always form an independent set by constructing the dual situation and
carrying out the corresponding proof. In which situation is the proof more readily
obvious?
9. Consider the graph of Prob. 1, Ch. 1, and choose branches 5, 6, 7, 8 as consti
tuting a tree. For the meshes, which become the closed paths upon which the link
currents circulate, write Kirchhoff voltagelaw' equations, and use these to express
the link voltages in terms of the treebranch voltages. Now write a voltagelaw
equation for an additional closed path, say, for the mesh combination (1+2 — 3)
or any other one. In this equation substitute the expressions for the link voltages
obtained above, and note that it reduces to the trivial identity 0=0.
10. Construct the dual to the situation described in Prob. 9, and thus give an
illustrative example showing that no more than n Kirchhoff currentlaw equations
are independent.
11. In the sketch below, the series source is a voltage, and the parallel one is a
current. Numerical values are in volts and amperes. The passive element is a re
sistance of 3 ohms, as indicated.
v
Using the superposition principle which allows us to add separate effects, treating
each as though the others did not exist, and remembering that a nonexistent current
is an open circuit, demonstrate the correctness of each of the following relations
f = 5 + 3X2 + 3Xj = 3j + ll
and check them, using Eq. 43. Thus show that the given active branch is replace
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able by either of the following ones:
PROBLEMS
107
12. Using the ideas brought out in the preceding problem, reduce the following to
(a) an equivalent single passive element with a series voltage source, (b) an equiva
lent single passive element with a parallel current source.
Prob. 12.
13. Apply the statement of Prob. 12 to the following:
Prob. 13.
14. Apply the statement of Prob. 12 to the arrangement of sources and passive
elements shown below.
2
9
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Prob. 14.
108
THE EQUILIBRIUM EQUATIONS
15. In the following circuit the central source is a current. The other sources
are understood to be voltages or currents, according to their series or parallel associ
ation with the pertinent passive element. Element values are in ohms. Through
appropriate manipulation, reduce this problem to one involving a single loop current,
and, after finding its value, obtain the four currents ii, t2, is, u in terms of this one.
16. In the pertinent graph, the branch numbers may be regarded as also indicating
branch conductance values in mhos. Construct two cutset schedules, one for the
choice of nodepair voltages, ei = «i, e2 = t% «s = »s, and the other one for the
pickedup nodes, ac, bc, d.
Using the first schedule for the definition of variables and the second one for the
determination of the Kirchhoff currentlaw equations, obtain the equilibrium equa
tions (having a nonsymmetrical parameter matrix), and solve. Alternately obtain
symmetrical equilibrium equations through use of the first schedule alone. Solve
these, and check the previous solutions.
17. Construct the complete dual to Prob. 16 and solve.
18. Consider the 2, 4, 5ohm branches as forming a tree.
(a) Find equivalent voltage sources in the links alone. Set up loop equations, and
solve.
(b) Find an equivalent set of current sources across the tree branches alone. Set
up node equations, and solve. Obtain all currents and voltages in the passive
branches by each method and check.
Find the equivalent voltage sources in (a), first, by replacing the —4volt and 2volt
sources in the tree branches by respectively equal sources in the links and combining
these with the other linkvoltage sources and converted current sources; second, by
opening all the links and noting the net voltages across the gaps thus formed (the
desired linkvoltage sources are the negatives of these). Check the results found by
these two methods. Similarly in part (b) find the desired equivalent current sources,
first, through conversion of voltage to current sources and then replacing current
sources across links by equal ones across tree branches and combining these with
other sources across these branches; second, by shortcircuiting all the tree branches
and noting the net currents in these short circuits (the desired current sources are
the negatives of these). Again check the results found by the two methods.
Prob. 15.
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Prob. 16.
PROBLEMS
109
Note carefully that the sources in (a) yield the correct loop currents but that the
voltages across the tree branches, which are now purely passive, are not the actual
net treebranch voltages. Hence, if we convert the voltage sources in (a) to equiva
lent current sources and transfer these across the tree branches, we should not expect
to check the current sources found in part (b). Similarly, we cannot expect from
the results of (b) to find those of (a) through source transformation methods alone.
Discuss this aspect of the problem.
Prob. 18.
19. The sketch below shows the graph of a network consisting of seven 1ohm
branches and a 1volt source. Find the values of the node potentials ei, ej, ej with
respect to that of the common node at 0. Although any valid method is acceptable,
it is suggested that you use the technique of source transformations in order to avoid
leriving and solving a set of algebraic equations.
Prob. 19. Prob. 20.
20. (a) In the network shown consider branches 1, 3, and 4 as forming a tree.
Identify the link currents with the loop currents, and write a tieset schedule for the
network. Write down explicitly the three sets of equations: (1) Kirchhoff's voltage
law equations, (2) the appropriate voltampere relations for the branches, (3) the
branch currents in terms of the loop currents. Substitute (3) into (2) and then (2)
into (1) to obtain the equilibrium equations on a loop basis.
(b) Write down this last set of equations directly, using mesh currents as variables
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and the simplified procedure discussed in Art. 6.
no
THE EQUILIBRIUM EQUATIONS
21. (a) For the network of Prob. 20 consider the nodepair voltages from a, 6,
and c to ground as an independent set. Write a cutset schedule for them. Then
obtain the three sets of equations: (1) Kirchhoff's currentlaw equations, (2) the
appropriate voltampere relations for the branches, (3) the branch voltage drops
in terms of the nodepair voltages. By substitution of (3) into (2), and then these
into (1), obtain the equilibrium equations on a node basis.
(b) Write down this last set of equations directly, using the same nodepair volt
ages as variables and the simplified procedure discussed in Art. 6.
22. Choosing the link currents 1, 2, 3, 4, 11, as variables, repeat parts (a) and (b)
of Prob. 20 for the network shown here. Branches 1 through 10 are 2ohm resist
ances. Branch 11 is a 1ohm resistance in parallel with a 1ampere current source.
23. When a branch with its associated sources as shown in Fig. 9 becomes degen
erate through having its passive resistance assume an infinite value, then its voltage
source is trivial, and its current is constrained by the associated current source to
the value jk = — i,k. One way of dealing with this situation is to revise the network
geometry and dispose of the current source as shown in Fig. 8. Show, however, that
one may alternately meet this situation by treating this branch in the normal man
ner. Thus on a node basis this type of degeneracy creates no problem since terms
in the Kirchhoff currentlaw equations involving the current jk = —i,k simply be
come known quantities and are transposed to the righthand sides. On a loop basis,
show that one can construct the tieset schedule so that its first I — 1 rows do not
involve this branch, thus identifying loop current tj with the known branch current
and rendering the first I — 1 of the loop equations sufficient for the determination
of all unknowns. As an illustration, treat the following circuit in this manner. Let
the branch numbers equal resistance values in ohms.
1
3
Prob. 22.
1
Prob. 23.
24. When a branch with its associated sources as shown in Fig. 9 becomes degen
erate through having its passive resistance assume a zero value, then its current
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source is trivial, and its voltage is constrained by the associated voltage source to
PROBLEMS
111
the value t■* — — e,*. One way of dealing with this situation is to revise the network
geometry and dispose of the voltage source in the manner shown in Fig. 7. Show,
however, that one may alternately meet this situation by treating this branch in the
normal manner. Thus on a loop basis this type of degeneracy creates no problem
since terms in the Kirchhoff voltagelaw equations involving the voltage t,* — — e,*
simply become known quantities and are transposed to the righthand sides. On a
node basis, show that one can construct the cutset schedule so that its first n — 1
rows do not involve this branch, thus identifying nodepair voltage en with the known
branch voltage and rendering the first n — 1 of the node equations sufficient for the
determination of all unknowns. As an illustration, treat the following circuit in this
manner. Let the branch numbers equal conductance values in mhos.
Prob. 24.
25. For the circuit shown in the accompanying sketch, assume the branch num
bers to indicate also the resistance values in ohms, and let i, be one ampere. Choosing
branches 1, 2, 3, 4, 5 as links, find a set of linkvoltage sources equivalent to the given
current source as being the negatives of the voltages appearing at gaps cut simul
taneously into all links. With these replacing the current source i,, write down by
Prob. 25.
inspection the equilibrium equations on a mesh basis using the simplified procedure
given in Art. 6 and inserting the net source voltages around meshes as the right
hand members. Alternately obtain these same equations using the procedure de
scribed in Prob. 23 in which the current source is treated as a normal branch, and
check.
Now replace i, by identical sources in parallel with branches 3 and 4; convert to
voltage sources in series with these branches, and again write mesh equations. Will
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these yield the same mesh currents as above? Explain in detail.
CHAPTER THREE
Methods of Solution
and
Related Topics
1 Systematic Elimination Methods
Having written the equilibrium equations for a given network, the
next task is to carry through their solution. Here one may proceed in
several ways, the proper choice depending largely upon the objective
for which the analysis is done. Thus, one may be interested merely in
the numerical solution to a specific situation, or in a more general solu
tion in which some or all of the network parameters enter symbolically.
The latter type of problem is actually equivalent to the simultaneous
study of an infinity of specific numerical situations and consequently
presents greater algebraic difficulties which can be overcome only through
the use of correspondingly more general methods of analysis. An effec
tive tool for dealing with such problems is given in the next article. For
the moment we shall concern ourselves with the less difficult task of
solving a specific numerical case.
Suppose we choose as an example the Eqs. 24 appropriate to the net
work of Fig. 3 in Ch. 2, with arbitrary nonzero righthand members, thus:
1.142ei  0.976e2 + 0.643e3 + 0.500e4 = 1
0.976ei + 2.326e2  1.893e3  0.750e4 = 2
0.643ei  1.893e2 + 2.218e3 + 0.950e4 = 3
0.500e!  0.750e2 + 0.950e3 + 1.061e4 = 4
The straightforward method of solving a set of simultaneous equations
like these consists in systematically eliminating variables until an equa
tion with a single unknown is obtained. After its value is found, an
equation involving this and one other variable is used to compute the
value of a second unknown, and so forth. Unless the entire process is
systematized, however, a considerable amount of lost motion may result.
The following procedure is an effective one.
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112
SYSTEMATIC ELIMINATION METHODS
113
It is clear that only the numerical coefficients enter into the computa
tional procedure. Therefore it is sensible to omit writing the symbols
ei, e2, • • • altogether and consider only numerical matrix 2. We now
1.142  0.976 0.643 0.500 1.000
0.976 2.326 1.893  0.750 2.000
0.643 1.893 2.218 0.950 3.000
L 0.500  0.750 0.950 1.061 4.000
(2)
contemplate the detailed manner in which one may carry out the follow
ing plan in terms of the Eqs. 1: First, we undertake to eliminate ei from
all but the first of these equations; this step leaves us with three equa
tions involving e2, e3, e4. From all but the first of these, we now eliminate
e2, so that we have two equations with e3 and e4. From one of these we
eliminate e3 and have a single equation in e4.
Note at this stage that we also have an equation involving e3 and e4,
one involving ea, e3, and e4, and the first of the original equations in
volving all four unknowns. We can, therefore, readily solve these equa
tions in sequence and obtain all the unknowns without further difficulty.
Specifically, we solve first the equation in e4 alone. Next, the one in
volving e3 and e4 is solved for e3. Then, with e3 and e4 known, the equa
tion involving e2, e3, and e4 yields the value of e2, and the first of the
original equations, lastly, is used to find ei.
With reference to matrix 2, the process of eliminating ei from all
but the first of Eqs. 1 is evidently equivalent to an elimination of the
second, third, and fourth elements in the first column. This end is
accomplished by operating directly upon the rows of matrix 2 as one
would upon the corresponding Eqs. 1. Thus, if we add to the elements
of the second row the respective asmultiplied elements of the first row,
with a = 0.976/1.142, the result reads
0.000 1.492 1.343  0.323 2.855
(3)
which we regard as a new second row. Similarly, a new third row is
formed by adding to the elements of the present third row the respective
amultiplied elements of the first row with a = —0.643/1.142, yielding
(4)
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0.000 1.343 1.856 0.668 2.437
Finally a new fourth row is analogously formed with a = —0.500/1.142,
giving
0.000  0.323 0.668 0.842 3.562 (5)
114
METHODS OF SOLUTION AND RELATED TOPICS
These steps are summarized by observing that the original matrix 2
has thus been transformed into the following equivalent one:
1.142
0.976
0.643
0.500
1.000"
0.000
1.492
1.343
0.323
2.855
0.000
1.343
1.856
0.668
2.437
.0.000
0.323
0.668
0.842
3.562.
(6)
If we were to write down the equations corresponding to this matrix, it
would become clear that the numerical operations just carried out are
equivalent to the elimination of ei from the last three of the original
Eqs. 1.
We now proceed to eliminate from the last two equations corre
sponding to matrix 6. To this end we add to the elements of the third
row of this matrix the respective amultiplied elements of the second
row with a = 1.344/1.492, obtaining the new third row:
0.000 0.000 0.646 0.378 5.008 (7)
Next, multiplying the elements of the second row in 6 by a = 0.323/1.492
and adding to the respective elements of the fourth row gives
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0.000 0.000 0.378 0.772 4.180 (8)
The original matrix now has assumed the form
1.142
0.976
0.643
0.500
1.000"
0.000
1.492
1.343
0.323
2.855
0.000
0.000
0.646
0.378
5.008
0.000
0.000
0.378
0.772
4.180.
(9)
corresponding to a set of equations in which ei does not appear in the
second, while ei and e2 do not appear in the third and fourth.
We now carry out a step equivalent to eliminating e3 from the last
of the set of equations represented by the matrix 9 by adding the amulti
plied elements of the third row to the respective ones of the fourth row,
with a = —0.378/0.646, giving a final fourth row that reads
0.000 0.000 0.000 0.551 1.251 (10)
and the following final form for the matrix:
SYSTEMATIC ELIMINATION METHODS
115
1.142
0.976
0.643
0.500
1.000"
0.000
1.492
1.343
0.323
2.855
0.000
0.000
0.646
0.378
5.008
.0.000
0.000
0.000
0.551:"
1.251.
(11)
The last row represents the equation
0.551e4 = 1.251 (12)
from which e4 = 2.27 (13)
The third row in matrix 11 implies the equation
0.646e3 + 0.378e4 = 5.008 (14)
which, through use of the value 13 for e4, becomes
0.646e3 = 5.008  0.857 = 4.150 (15)
and hence yields e3 = 6.42 (16)
From the second row in matrix 11 we next have the equation
1.492e2  1.343e3  0.323e4 = 2.855 (17)
or, in view of the values 13 and 16,
1.492e2 = 2.855 + 8.629 + 0.732 = 12.216 (18)
from which e2 = 8.189 (19)
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Finally the first of Eqs. 1, corresponding to the first row in matrix 11,
together with the values for e2, e3, and e4 already found, becomes
1.142ei  7.992 + 4.130 + 1.135 = 1 (20)
and thus
1.142c, = 3.727 or = 3.264 (21)
The basic process in this systematic elimination method is the trans
formation of the original matrix 2 into the socalled triangular form 11,
whence the unknowns are obtained through an obvious recursion process
which begins with the computation of the last of the unknowns • • • e4
and successively yields all the others. It may readily be seen that this
computational procedure involves a minimum of lost motion and hence
is the best method to apply in any numerical example.
If some terms in the given equations already have zero coefficients, it
may be necessary first to rearrange the equations in order that the
elimination method be applicable in precisely the form described above
while enabling one to take advantage of the simplifications implied by
116
METHODS OF SOLUTION AND RELATED TOPICS
such missing terms. These modifications in procedure, however, the
reader can readily supply for himself as he carries out actual examples,
and further pertinent discussion of them will not be given here.
2 Use of Determinants
Although the determinant method of solving simultaneous algebraic
equations may be used in numerical examples, the amount of computa
tion involved is usually greater than in the systematic elimination process
just described. It does, on the other hand, afford a means for expressing
the solutions in a compact symbolic form that enables one to study their
functional properties and thus deduce with little effort a number of
important and useful general network characteristics, some of which
will be pointed out in the latter part of this chapter. Our immediate
objective is to discuss briefly some of the more important algebraic
properties of determinants.
The socalled determinant of the system of equations
an^i + 012^2 H h ainZn = yi
021X1 + a22x2 \ h a2nxn = 2/2
(22)
Oni^i + an2X2  h annxn = yn
is written in the form
A=
aii a12 "' . °in
0,21 022 • • • 02n
ani On2 '"' ann
(23)
In appearance it is much like the corresponding matrix (differing only
in that the array of coefficients is enclosed between vertical lines instead
of square brackets), but in its algebraic significance it is entirely different
from the matrix in that it is a function of its elements and has a value
corresponding to the values of these elements as does any function of
several variables. The elements are the coefficients a,k in Eqs. 22. For
n equations the determinant has n2 elements and is said to be of order n.
The determinant is a particular kind of function of many variables
that was created by mathematicians for the sole purpose of its being
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useful in the solution of simultaneous equations. Hence it was given
those properties that turn out to serve best this objective. These may
USE OF DETERMINANTS
117
be summarized in the following three statements:
The value of a determinant is unchanged if the elements of
any row (or column) are added to the respective ones of an
other row (or column).
The value of a determinant is multiplied by k if all the ele
ments of any row or column are multiplied by k.
The value of a determinant is unity if the elements on the
principal diagonal are unity and all others are zero.
The last statement may be written in the form
10 0 ••• 0
0 10 ••• 0
(24)
(25)
(26)
00
(27)
Through combining the properties 24 and 25 it follows that the value
of A remains unchanged if the fcmultiplied elements of any row (or
column) are added to the respective ones of another row (or column).
Since k may be numerically negative, this statement includes the sub
traction as well as the addition of respective elements. It also follows
from these properties that a determinant has the value zero (a) if the
elements of any row or column are all zero, or (b) if the elements of any
two rows (or columns) are respectively equal or proportional, for a row
or column of zeros implies k = 0, and a condition of equal or propor
tional rows (or columns) immediately leads to a row (or column) of
zeros through appropriate manipulations of the sort just mentioned.
The value of a numerical determinant may readily be found through
use of these properties since, by means of them, one can consecutively
reduce to zero all but the diagonal elements (after the fashion that
matrix 2 in the previous article is transformed to form 11). Once the
determinant has this diagonal form, properties 25 and 26 show that the
value equals the product of the diagonal elements. In fact it can be
shown that the determinant in triangular form has this same property;
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that is,
an a12 a13 '" • ain
A=
a22 a23
0 a33
2/22 = jjj' 2/12 = r^r (104)
where the determinant of Eqs. 101 is
I z I = ZnZ22  Z122 (105)
and the symmetry condition z12 = Z2i holds. The determinants  y \
and I z , of course, have reciprocal values; that is,
2/ = M1 (106)
as is clear, incidentally, from a comparison of relations 102 and 104.
These results contain the interesting and useful relationship expressed
by
J/1iZn = 2/22z22 (107)
A simple example will illustrate the unusual character of this result.
With reference to the network of
Fig. 32 in which the element values j.0 WW t Q2*
are in ohms, we note that zn is the
resistance of the series combination
of the two branches, while yw is the ^8
conductance of the 2ohm branch
alone. Hence
lo I 02
2/11 = 1/2, Zii = 10 (108) FiQ 32. a simple example of a dis
. , , 1 ., j » ... symmetrical two terminalpair network
At the opposite end of this two / .
. . .for which the property expressed by
terminal pair we observe that z22 is Eq. 107 is illustrated,
given by the 8ohm resistance alone
while 2/22 is the conductance of the two branches in parallel. Thus we
have
2/22 = 5/8, z22 = 8 (109)
Physically and numerically the pairs of quantities 108 and 109 seem
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unrelated; yet it is clear that they do fulfill the condition expressed by
Eq. 107, as indeed they must, since this relationship holds for any two
terminalpair network.
160
METHODS OF SOLUTION AND RELATED TOPICS
Returning to Fig. 31 again, it is often useful to express the quantities
ei, t'i in terms of e2, iz or vice versa. Such relations are readily obtained
through an appropriate manipulation of Eqs. 100 or 101. It is cus
tomary to write them as
ei = Ae2 — Bt2
. r, (110>
Xi — Ce2 — J.JT.2
It is a simple matter to determine the coefficients A, B, C, D, called
the general circuit parameters, in terms of the y's or the z's. The fol
lowing relations are selfexplanatory, and make use of Eqs. 100, 101,
and Eqs. 102 and 104 relating the y's and z's.
A = (eA = zii=—^ (Hi)
\e2/ 0, the current function i,(t) ap
proaches a step of the value I, and e(/dt, n being the number of turns in the coil and ct>
the flux linking it. Since by definition L = nct>/i, we note that a current
of the value 1/L corresponds to a flux linkage ncj) of unity. The state
ment in the second sentence of this paragraph may now be made more
precise: A unit current impulse applied to a capacitance instantaneously
places unit charge (1 coulomb) in that capacitance; a unit voltage im
pulse applied to an inductance instantaneously places unit flux linkage
(1 weberturn) in that inductance.
These two statements, one about a current impulse and the other
about a voltage impulse, are identical except for an interchange of
quantities in the pairs: e and i, C and L, charge and flux linkage. Or
we may say that only one statement is made, and that this one remains
true upon interchange of the dual quantities in the pairs mentioned.
Here again we have an example of the principle of duality which we
shall elaborate further as our discussions continue.
The sudden introduction of electric charge into a capacitance repre
sents the sudden addition of a finite amount of energy to the system of
which that capacitance is a part. Q coulombs in C farads represents an
energy of Q2/2C joules, which may alternatively be written CEc2/2 if
Ec = Q/C denotes the voltage produced in the capacitance by the
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charge Q. Similarly, the sudden introduction of flux linkage into an
inductance represents the addition of energy to the network of which
that inductance is a part. nct> weberturns in L henrys represents an
energy of (nct>)2/2L joules, which may alternatively be written LIl2/2
if II = rut>/L denotes the current produced in the inductance by the
flux linkage nct>.
A remark somewhat apart from the present topic but nevertheless
appropriate at this point is to the effect that some readers may not like
196 CIRCUIT ELEMENTS AND SOURCE FUNCTIONS
the statement about flux linkage producing current. They may feel
that it is quite the other way about, that current produces flux linkage.
While it is true that teachers of electricity and magnetism have con
sistently presented the situation in this way for as long as the subject
has been taught, there is actually more reason based upon physical
interpretation (if "physical" interpretation of such purely mathematical
fictions as electric and magnetic fields makes any sense at all) to adhere
to the view that the electromagnetic field produces voltage and current
rather than that the reverse is true. For purposes of analysis it does
not matter one jot how we interpret the mathematical relationships.
It is best to take a very flexible view of such things and be ready to
accept either interpretation, whichever is consistent with the tenor of
reasoning at the moment.
To summarize the statements about current or voltage impulses and
the energy they impart to network elements we may say: A unit current
impulse applied to a capacitance of C farads establishes instantly a
charge of 1 coulomb and inserts 1/2C joules of energy; a unit voltage
impulse applied to an inductance of L henrys establishes instantly a
flux linkage of 1 weberturn (hence a current of 1/L amperes) and in
serts 1/2L joules of energy.
3 The Family of Singularity Functions; Some Physical
Interpretations
The impulse and step functions introduced in the previous article
are found to be practically useful because many actual excitation func
tions can be represented in terms of them. In this regard, the step
function is probably the most widely known of the two, for it has been
discussed and employed in the literature on circuit theory for many
years, having been introduced through the writings of Oliver Heaviside
during the latter part of the nineteenth century.
In order to appreciate the usefulness of such a concept as the step
function, consider the commonly occurring situation pictured in part
(a) of Fig. 7 where some passive network (shown by the box) is assumed
to be connected to a battery with the constant value of E volts through
the switch S. The problem usually is to study the network response
that takes place following closure of the switch, with the assumption
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that rest conditions obtain before this time.
If the principal interest is in the network response long after the
instant of switch closure, then the excitation function is regarded as a
constant voltage having the value E. However, if the interest Lies
chiefly in the behavior of the network immediately following the switch
closure, then it is obviously not appropriate to regard the excitation as
THE FAMILY OF SINGULARITY FUNCTIONS 197
a constant, for it is the discontinuity in this function occurring at the
switching instant that is its outstanding characteristic. That is to say,
it is the sudden change in the excitation from the zero value prior to the
switching instant to its nonzero constant value afterward that charac
terizes the nature of the network response near this time instant.
(a) (b)
Fig. 7. Application of a constant voltage E through closure of switch 5 at the
instant ( = that is to say, they are node flux linkages.
Denoting the branch currents as usual byji, 32, • • . we have accord
ing to the relations between current and flux linkage in an induct
ance element (as given in Eq. 20)
ii = fa
h = 2(0i — fa)
h = 3 fit, the result for this case is immediately obtained from
Eq. 90, thus:
i(t) =Vc/Lu0tX eai (91)
Again it may be pointed out that a variety of modifications of the
circuit arrangement of Fig. 21 may be carried out without affecting the
nature of the response, through
use of one or more of the source
transformations shown in Figs. 12,
13, and 14. Thus, if the capaci
tance C in Fig. 21 is split into
Ci + C2 = C and the transforma
tion of Fig. 14 applied to i,(c)
and Ci in parallel to convert this
Fio. 24. Circuit arrangement yielding combination into a voltage source
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the same t( 0 is fixed by the
value of the current for / = 0 (Eq. 14 is the result for i(0) = l/L).
Similarly the behavior of the series RC circuit for t > 0 is fixed by the
value of the charge for t = 0 (in Eq. 47 the capacitance voltage is given
for t > 0 when the initial charge is 1 coulomb). The series RLC circuit
has a definite behavior for t > 0 in terms of known values of charge and
current at t = 0 (as will be shown in further detail presently).
That is to say, the state of the network at t = 0 is adequately de
scribed by the values of the initial charges and currents; it is not neces
sary to know how these values came about! Although a given set of values
may have come about as a result of many completely different behavior
patterns before the initial instant, the behavior that this set determines
for t > 0 can have only one pattern because the solution to the pertinent
differential equation of equilibrium involves as many integration con
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stants as there are independent initial charges and currents, so that the
latter determine these constants uniquely, and nothing else can exert an
influence on the resulting solution. This point is now further illustrated
for the series RLC circuit discussed in Art. 5.
Let us return to the point in the discussion of the RLC circuit where
the formal solutions for current and charge (Eqs. 57 and 58) are ob
tained from the differential Eq. 52. The reason we refer to these as
* See Eq. 36 for a specific example of this sort.
CONSIDERATION OF ARBITRARY INITIAL CONDITIONS 255
"formal" solutions is that they are not yet explicit relations for the
current and charge but merely represent these quantities in functional
form because the integration constants Ai and A2 appearing in them are
as yet not fixed. Since only two unknown constants are involved, two
special conditions suffice to render the formal solutions explicit.
For these conditions we may choose two arbitrarily specified values of
the current at selected instants of time, or two values of the charge at
chosen instants; or, what is more commonly done, we may specify values
for the current and the charge at t = 0. We call these values the "initial
conditions" since they determine the state of the network at t — 0.
Thus, through considering Eqs. 57 and 58 for t = 0, we have for the
determination of the integration constants (in place of Eqs. 59)
Ai + A2 = *(0), — + — = 9(0) (99)
Pi P2
Solving these we have
Pii(0)  PiP29(0) t p2i{0)  p2Pi?(0)
Ai = 1 A2 = (100)
Pi — P2 P2  Pi
Since the characteristic values pi and p2 are conjugate complex, we
see that the A's are conjugate complex. By Eqs. 74 and 75 we find
Pi =  a + jud = j(ud + jot) = juoe'*
P2 = —a — jud = — i(wd — ja) = —ju0e~3
= ud/uo. These relations are useful in that they are the representation
for any unfinished business that the RLC circuit may find itself in the
process of carrying out if at t = 0 a fresh excitation is applied to it.
Thus, for the completion of this unfinished business, one writes Eqs. 106
and 107 with t(0) and g(0) equal to the appropriate values (these must
be part of the given data), and then adds the response due to the fresh
excitation (for tacitly assumed rest conditions) to obtain the complete
behavior for t > 0.
It is interesting also to note that Eqs. 106 and 107 give the response
of the series RLC circuit for a number of special excitation functions.
For example, if we want the response of this circuit to an applied unit
voltage impulse, we observe according to the discussion given earlier
that this excitation instantly establishes a current in the inductance of
the value 1/L. Hence we need merely consider Eqs. 106 and 107 for
i(0) = 1/L and 17(0) = 0 to have the response appropriate to this
excitation. In Art. 5 it is shown (through consideration of Figs. 20 and
21) that an applied unit step voltage is equivalent to starting from an
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initial capacitance charge q(0) = — C coulombs. Hence Eqs. 106 and
107 for t'(0) = 0 and q(0) = — C yield results appropriate to this case,
as may be verified by comparison with Eq. 77.
These considerations lead us to recognize that the process of taking
arbitrary initial conditions into account in a transientnetwoik problem
may be done in an alternate way. Thus the existence of a current in an
SUMMARY REGARDING THE TRANSIENT RESPONSE 257
inductance at t — 0 is equivalent to inserting a voltage impulse (of
appropriate value and occurring at t = 0) in series with this inductance,
while the existence of a charge in a capacitance at t = 0 is equivalent
to bridging a current impulse (of appropriate value and occurring at
t = 0) across this capacitance. In other words, any set of arbitrary
initial currents and charges may be replaced by an appropriate set of
voltage and current impulse sources connected into the network. Super
position of their individually produced responses and that due to some
specific excitation, all computed for initial rest conditions, yields the
desired net response.
Thus it may be seen that a discussion of network response that tacitly
considers only initial rest conditions is nevertheless sufficient to deal
with problems involving arbitrary initial conditions.
SUMMARY REGARDING THE TRANSIENT RESPONSE
OF ONE, TWO, AND THREEELEMENT COMBINATIONS
A Single Elements
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Voltage Source Current Source
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258 IMPULSE AND STEPFUNCTION RESPONSE
SUMMARY REGARDING THE TRANSIENT RESPONSE 259
C Two Elements—R, C
CO
grot
(A)
1
J
ifl, ic same as
in part (A)
oo »—l 1
i(t)\(A)
J
ic
\(A)
e(0\E "C
oI
£ amen
iWj7 * c=j=
eit, ec same as
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in part (A)
260 IMPULSE AND STEPFUNCTION RESPONSE
D Two Elements—L, C u0 = l/VZc
e(t)\ (A)
00
i(t)\(A)
eW £cos«„
1 has the opposite effect. *
10 Vector Diagrams
Although the Ohm's law relation linking complex voltage and current
amplitudes through the impedance of the circuit is so simple that it
hardly needs any further clarification to be fully understood, yet it
may in some cases be found additionally helpful to give to this relation
its corresponding graphical interpretation. Such representation is par
ticularly useful when a given problem involves more voltages and cur
Fio. 22. Vector diagrams illustrating relative magnitude and phaseangle relation
ships between complex voltage and current amplitudes. In (a) the current lags
while in (b) it leads the voltage which is chosen as the reference vector.
rents (those in other branches of the network), for it lends circumspec
tion and unity to the sum total of voltampere relations involved and
enables one more readily to recognize significant special amplitude and
phase relationships and the circuit conditions for which they arise.
Figure 22 shows such a graphical representation—called a vector
diagram—for the simplest case in which only one voltage vector E and
current vector / are involved. Part (a) of the figure represents a situa
tion in which the angle of the impedance Z is positive (specifically
6 = +30°), while in part (b) of the same figure the angle of the imped
ance is assumed negative (specifically 6 — —60°). In the first case the
current vector lags the voltage vector; in the second it leads.
The relative lengths of the voltage and current vectors in these dia
grams are completely arbitrary, for, although they are related through
the magnitude of the impedance Z, the scales determining length may
be chosen independently for voltage and current. Thus, suppose the
* Although frequency scaling has no effect upon the amplitude of an impedance or
admittance function since it involves only the independent variable (the frequency
8), a closer study reveals that the time function characterizing the transient response
of the pertinent network not only has its independent variable (the time t) affected,
but its amplitude becomes multiplied by a constant also. These matters are discussed
in Art. 6 of Ch. 9 and are summarized there in the statement 132.
(•)
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(b)
312 SIMPLE CIRCUITS IN THE SINUSOIDAL STEADY STATE
voltage magnitude were 1 and the impedance magnitude 10, so that
the current magnitude becomes 0.1. If we choose a voltage scale of
2 inches per volt and a current scale of 20 inches per ampere, the vectors
E and / have equal length; whereas, if we change to a current scale of
10 inches per ampere, the J vector has half the length of the E vector.
Unless the diagram contains several voltage or current vectors, relative
lengths have not much meaning, but relative angles have.
Elaborating upon this theme, we may say that a choice of scales for
the quantities E and / fixes a scale for the associated impedance Z
(which may or may not appear on the same
diagram). Or a choice of scales for J and Z
fixes that for E; while a choice of scales for E
and Z fixes that for /. For example, a choice of
10 volts per inch and 2 amperes per inch implies
a scale of 5 ohms per inch if the scaled length
Fig. 23. The diagram of (;n inches) of a voltage vector divided by the
Fig. 22(a) redrawn with scaled { h (Jn inches) Qf & vectof ig
the current chosen as the . , , ., . , .... ,
reference vector. to yie1d the appropriate length in inches for
the associated impedance vector. For these
scales, a voltage vector 2.5 inches long represents 25 volts; a current
vector 2.0 inches long represents 4 amperes; the length of the associ
ated impedance vector is 2.5/2.0 = 1.25 inches and represents 5 X 1.25
= 6.25 = 25/4 ohms.
One may tacitly assume for Z the scale of 1 ohm per inch, whence it
follows that the scales for E and / become equal; that is, the number of
volts per inch equals the number of amperes per inch. This tacit con
dition need, however, not always apply; and in fact it may in many
problems be difficult to accommodate.
Observe, with regard to angles, that we specifically use the term
relative angles. Thus the diagram of Fig. 22(a) could just as well be
drawn as shown in Fig. 23, or in any one of an infinite number of addi
tional possible angular orientations. The one significant fact which
this simple diagram portrays is that the current lags the voltage by
0 radians.
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Since the angular orientation of the diagram as a whole is thus per
fectly arbitrary, one is free to choose that orientation which seems to
be appropriate to the physical conditions of the problem. For example,
if the source is a voltage, then it is customary to choose the angle of E
as being zero; if the source is a current, the angle of / is usually taken
to be zero. In the first of these choices the vector E serves as phase
reference for the diagram; in the second choice the vector / becomes
the phase reference. Whichever vector is chosen to have zero angle is
VECTOR DIAGRAMS
313
designated as the reference vector. Although several different voltages
and currents may be involved in a given problem, it is clear that the
angle of only one voltage or of one current vector may arbitrarily be set
equal to zero.
When the impedance is represented in rectangular form, the volt
ampere relation may be separated into a sum of terms corresponding
to the resistive and the reactive components of Z, as in
E = IZ = IR(w) +jIX(.w) (161)
The separate vector components of E represented by the terms IR(u)
and IX(ui)—called the resistance drop and reactance drop respectively—
may be indicated in the corresponding
vector diagram. If this is done for the
situation depicted in Fig. 22(a), the re
sult has the appearance shown in Fig. 24.
Observe that the vector IR(u) must have
the same angular orientation as the vec
tor /, since R(ui) is merely a positive real
number. We express this fact by stating
that the IR drop is in phase with the
vector J. The voltage component given
by jIX(u), on the other hand, clearly is
ir/2 radians in advance of I; that is, it leads the vector / by 90°. This
fact is alternatively expressed by stating that the IX drop is in quad
rature with the vector /, although this terminology is a bit ambiguous
since quadrature merely implies a rightangle relationship without re
gard to lead or lag.
Observe that the resistive and reactive components of E vectorially
add to yield E. The lengths of these component vectors are fixed, for
a given impedance angle 6, as soon as a length for the vector E is chosen.
The vector I in the diagram must coincide in direction (must be in
phase) with the IR vector; its length (as already mentioned) is arbitrary.
In dealing with certain problems it may be convenient or useful to
decompose the current vector I into components that are respectively
in phase with E and in quadrature with E; or it may be expedient to
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subdivide the components of Z into subcomponents. A common exam
ple of the latter procedure arises in dealing with the series RLC circuit
for an impressed voltage E. Here
E
Fio. 24. The diagram of Fig.
22(a) with the resistive and re
active components of the volt
age drop added according to
their definition in Eq. 161.
Z = R+juL + —
(162)
314 SIMPLE CIRCUITS IN THE SINUSOIDAL STEADY STATE
which may be written
Z = R +jXL+jXc
(163)
with Xl and Xc as given in Eq. 147.
In drawing the vector diagram for this ex
\jIXL ample, it is effective to choose the current as
phase reference, notwithstanding the fact that
the voltage may be the source function. The re
sulting diagram, shown in Fig. 25, is drawn for
a condition in which the capacitive reactance
IR ^ J Xc predominates so that the net voltage E
^ i lags the current / (the latter leads the voltage).
Note that the net reactance drop IX is small
compared with either component IXl or IXc, so
that even the total voltage E, which includes
(vectorially) the IR drop, is smaller than either
reactive component drop. If Xl + Xc = 0, we
have the resonant condition for which E = IR
alone. The vector diagram makes more evident
the fact that, at resonance, one may have volt
ages across the inductance and capacitance ele
ments separately that can be many times larger
than the net applied voltage. For this reason
it is important that caution be exercised
when experimenting with resonance in the
laboratory unless the power source used is small
enough so as to preclude the possibility of
dangerous shock due to accidental contact with
the apparatus.
Another example that illustrates the circum
spection afforded through use of a vector diagram, is the circuit sche
matically shown in Fig. 26, which consists of the three impedances
Zi, Z2, Z3 in series. Suppose we write
each in its rectangular form as o— zx — z2 — z3 —o
B
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Fig. 25. Vector dia
gram for the series
RLC circuit showing
the capacitive and in
ductive reactance
drops as well as the
net reactance and the
resistance drops.
Note that the net re
actance drop is smaller
than either of its com
ponents (at resonance
it is zero).
Z1 = +jX,
%2 ~ R2 + 3X2
Z% = R3 + jX3
(164)
Fig. 26. Schematic representation
of a circuit involving three arbi
Figure 27 shows the vector diagram trary impcdanccs in series,
in which the current is chosen as
phase reference and the impedances Zi and Z2 are assumed to be in
MORE ELABORATE IMPEDANCE FUNCTIONS 315
ductively reactive (Xi > 0 and X2 > 0), while Z3 is assumed to be
capacitively reactive (X3 < 0). The diagram shows all three impedance
drops (that is, voltages across the separate impedances) broken down
into resistive and reactive components, as
well as their vector sum which equals
the net voltage E. The circumspection
which this diagram affords relative to
magnitudes and phase relationships of all
voltages with respect to the common
current / cannot be had in equal meas
ure from the purely analytic relationship
involved. It is this property of the vector
diagram that justifies its use.
Although these remarks have been
made with specific reference to the im
pedance as parameter linking E and /,
it is evident that one may equally well
carry through the graphical procedure in
terms of the reciprocal parameter Y.
Thus, if the impedances of Fig. 26 were connected in parallel, such
a "switch" to an admittance basis would be indicated. The details
of this situation would then be exactly analogous to the ones given
above with the roles of E and / interchanged, R's replaced by G's, and
X's by B's.
IS,
Fio. 27. The vector diagram
associated with the circuit of
Fig. 26 showing all resistance
and reactance drops as well as
the net current and voltage vec
tors.
11 More Elaborate Impedance Functions; Their Properties and
Uses
As pointed out in Eq. 37, Art. 2 of Ch. 5, the differential equation
linking current i(t) in some part of a network with voltage e(l) at the
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same point or at any other point is always of the form
an— + ani —
dtn dln'
di
t H h — + O0*
dt
cTe
de
+ bmi~—r + ...+b1T + b0e (165)
dt dt
in which an • • • a0 and bm • . • b0 are real constants. They are all posi
tive if e(t) and refer to the same point in the network; otherwise
either some a's or some b's may be negative according to whether
or e(t) is the excitation function.*
* These matters are fully elaborated upon in Arts. 4 and 5 of Ch. 9.
316 SIMPLE CIRCUITS IN THE SINUSOIDAL STEADY STATE
For an excitation of the form e", the particular integral yielding the
steadystate response must have the same form. Hence for the steady
state solution to the differential Eq. 165 it is appropriate to substitute
e(t) = Ee" and t(0 = Ie'1 (160)
with the result
(ansn + an^s"1 + . • • + ais + a0)/e" =
(bmsm + bm—lsm1 + . • • + b1S + bo)Ee" (167)
After canceling the common factor e'i, one has
E anSn + a»—1Sn1+...+ a1S + a0 P(S)
 = Z(s) = ; = (168)
/ bmsm + bmism1 +...+biS + b0 Q(s)
If the polynomials P(s) and Q(s) are factored in terms of their zeros,
the impedance Z(s) assumes the form
, «(—.)t.^(.^)
(s  s2)(s  S4) • • • (a  s2m)
in which H = an/bm is a positive real constant.
If the excitation is e(l), the transient (forcefree) part of the solution
is determined by Eq. 165 with e = 0. Assuming for the solution to
this homogeneous differential equation the expression
i0(0 = Aept (170)
leads through direct substitution to
P(p).Aept = 0 (171)
whence a nontrivial solution (A j£ 0) demands
P(p) = anpn + anjpn1 + • • • + aiP + do = 0 (172)
This is the characteristic equation determining the complex natural
frequencies associated with the transient current. We observe that
they are the critical frequencies Si, s3, • • • s2n—1 appearing in the numera
tor of the impedance 169. The complete response (transient plus steady
state) is thus given by
i(t) = Aie'" + A3e'" + .••+ A2n^e'^1 + — e" (173)
Z(s)
If the excitation is i(t), the transient (forcefree) part of the solution
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is determined by Eq. 165 with i = 0. Assuming for the solution to this
MORE ELABORATE IMPEDANCE FUNCTIONS
317.
homogeneous differential equation the expression
eo(0 = Be"
(174)
yields
Q(p)Bept = 0
and a nontrivial solution (B ^ 0) demands
Q(P) = bmpm + 6^ip"1 + • • •+ 6iP + b0 = 0
(175)
This is the characteristic equation determining the complex natural
frequencies associated with the transient voltage. We observe that
they are the critical frequencies s2, S4, . . •, s2m appearing in the denom
inator of the impedance 169. The complete response (transient plus
steady state) is in this case given by
The transient amplitudes Ai • • • A2ni in Eq. 173 and Bi . • . B2m
in Eq. 176 are determined from the known state of the network at the
time the excitation is applied and the demands made by the steady
state response function at that same instant, the discrepancies between
these two factors being the quantities upon which the sizes of these
amplitudes depend. The details of their determination do not interest
us at the moment.* It is significant to point out however that, for a
nonzero initial state, the results 173 and 176 are meaningful, even when
the respective excitation functions are zero. When e(  A , show from a geometrical construction that  B + A 
\B\ \A\.
4. Consider the complex numbers
A  «i + io2
B = 61 +jbt
(a) Show that Re [A ± B]  Re [A] ± Re [B) and Im [4 ± B)  Im [A] ± Im [B].
(b) Show that Re [AB] = Re [A] Re [B]  Im [A] Im (B) and Im [AB\ = Re [A]
Im [B] + Im [.4] Re [B]. A'oie. Re [AB] * Re [A] Re [BJ.
(c) Find Re [Ae''j where A is real. What does this result become when A =
(d) Find Re [(3 +j4)(0.2 + j0.2)].
5. Given the complex impedance
Z  R +jX ohms
(a) Find the impedance in the polar form Z =  Z \/e. (b) Express it in the form
Z  Ae'\ (c) Find Z" and Z1/n.
6. Given a current I = a + jb amperes in an impedance Z  R + jX ohms, find
the voltage drop E in the direction of the current. Express the result in (a) rectangu
lar form, (b) polar form, (c) exponential form, (d) as an instantaneous cosine function
assuming t(3 = 20°, • • •, ct>i = 60°. You are to find appropriate contents for the
boxes labeled ii, z2, etc., not exceeding twoelement combinations in complexity.
Assume w = 1 radian per second to start with, and later convert your design to the
frequency a = 2r X 60, and again to u = 2r X 1000. If there exist other solutions
of no greater complexity, state what they are. Draw a vector diagram showing
E, Ii, /j, • . ., h and the resultant current Iq. Obtain an exact analytic expression
for the latter. If the phase angles involved are
replaced by lag angles, what changes in the cir
cuits are needed?
41. The element values in the circuits shown
in the accompanying sketches are in ohms and
henrys. Determine the impedance Z in each
case as a function of the complex frequency S,
and put it into the normal form of a quotient
of frequency factors. Make a sketch of the s
plane, showing the critical frequencies and some
point s = ju on the j axis. By inspection of
this diagram, what are  Z \ and 6 (the angle of
Z) for « — 1 and w = 2? At what u value
is 8 largest, and what is this largest value?
Sketch  Z  and 6 versus u. If a unit step cur
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rent is applied to each circuit, find the resultant
transient voltage at the input terminals. If the
frequency scale is stretched so that the point
u — 1 becomes u — 1000 (and all other points are changed in the same ratio), what
do the circuit element values become in ohms and henrys? What is the effect upon
the critical frequencies of Zl What is the effect upon the transient response ob
tained above?
42. The element values in the circuits shown in the sketches are in ohms and
henrys. Determine the expression for the impedance Z(s) in each case, and put it
into the normal form of a quotient of frequency factors. Plot the critical frequencies
in the » plane appropriate to each Z. Do the results suggest anything of interest
o VWV
i
A/WV—i
i
t
Prob. 41.
336 SIMPLE CIRCUITS IN THE SINUSOIDAL STEADY STATE
or possible practical value? If the frequency scale is stretched so that u
comes 0> = 106, what do the element values become?
?VWV—I
3/8
1 be
O1
— 00V —1
3/2 1
1/2
Z.^1\
! »5
f1
O1
(a)
(4
(b)
Prob. 42.
43. Find networks *hat are dual to those given in Prob. 41; that is, ones that will
have reciprocal Z values. If a unit voltage step is applied to either one of these,
what is the resultant transient current (using the results found in Prob. 41, of course)?
Find the new element values corresponding
lf"j2 to a stretch of the frequency scale that shifts
u = 1 to a  1000.
44. Find the networks that are dual to
those given in Prob. 42; that is, those that
will have reciprocal Z values. Write their
Z functions as a quotient of frequency
factors. Find the revised element values in
these networks corresponding to a stretch
of the frequency scale that shifts u — 1 to
u = 10s. How do the critical frequencies
change?
45. For the circuits of Prob. 41 find net
works which when placed respectively in
series with each given network will yield
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a resultant impedance equal to unity at
all frequencies; that is to say, find (by the
method discussed in the text) those net
works having complementary impedances.
Prob. 47. May these networks be used interchangeably
to form constantresistance resultants?
46. Find networks that are complementary to those given in Prob. 42.
47. The element values in the networks shown at the left are in ohms, henrys, and
If
7— 1 l/vf
PROBLEMS
337
farads. Find the expressions for Z\(s) and Zi(s) as quotients of frequency factors.
For each impedance, sketch the locations of its critical frequencies in the s plane.
Form Z\ + Z% How are these impedances related? Compute the real and imagi
nary parts of these impedances, and sketch (neatly) versus « for the range 0 < w < 3,
putting both real parts on one sheet and both imaginary parts on another.
48. For the networks of Prob. 47 determine the transfer impedances Zn = Et/I\
in the form of quotients of frequency factors, and sketch the critical frequencies in
the s plane. Compute the squared magnitude of Zyi(jw) in each case, and sketch
versus w on the same sheet for range 0 < w < 3. Compare with the realpart plots
of Prob. 47.
49. Obtain the duals of the networks given in Prob. 47, and combine these so as
to yield a constantresistance combination. Using the results of Prob. 48, what are
the transfer admittances Kb = Ii/Ei for the dual networks, and what are their
squared magnitudes as functions of u? Does the constantresistance combination
suggest any practical application? How would you revise this resultant network
corresponding to a stretch of the frequency scale that puts the point u = 1 at w =
27 r X 1000?
50. The admittance of a series RLC circuit has the form
Y(S) =
(s   S2)
with si = —0.1 +jl, Si = —0.1 — jl. If you drew the resonance curve for this
circuit, what would be the resonance frequency and the width of the curve at its
halfpower points? What is the Q of the circuit? What are the values of its param
eters in ohms, henrys, and farads? How do these parameter values change if the
frequency scale is stretched by a factor 10,000 (so as to make the resonance fre
quency 10,000 times higher)? How do the critical frequencies change? How does the
width of the resonance curve at the halfpower points change, and what is the effect
upon Q? How do the parameter values change if Y is to become 1000 times larger
(at all frequencies)? Does this change have any effect upon the shape of the reso
nance curve or upon 0? Returning to the original situation, suppose the real parts
only of the critical frequencies are changed from —0.1 to —0.01, what are (a) the
resonance frequency, (b) the width of the resonance
curve at the halfpower points, (c) the Q of the cir
cuit, (d) the parameter values?
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51. For the circuit shown in the sketch, show that
the admittance is given by the expression
C(s  Si)(s  si)
Y(s) =
(»  «s)
and determine the critical frequencies si, «2, «s in terms Prob. 51.
of the parameters R, L, C, G. If si = 0.1 +/10;
»2 = —0.1 — 7IO; S3 = —0.1, what are the values of R, L, and G relative to C? If
the circuit is driven by a current source and a resonance curve is taken for the volt
age, what are the resonance frequency and the width at the halfpower points?
What is the value of Q? What is the magnitude of the impedance 5 per cent above
or below resonance relative to its value at resonance? Suppose the values of R and
G are changed to R' and G', keeping the quantity (ft'/L) + (G'/O  (R/L) + (G/C),
what is the net effect upon the impedance or admittance? If the Q of the circuit is
large, is this net effect significant so far as the resonance behavior is concerned?
338 SIMPLE CIRCUITS IN THE SINUSOIDAL STEADY STATE
52. A circuit of the sort shown in Prob. 51 but with G = 0 is to be designed to
have an impedance with a maximum absolute value of 100,000 ohms at a frequency
of 1.5 X 106 cycles per second. At frequencies 10 per cent above and below reso
nance, the impedance magnitude should be not more than onetenth of 1 per cent
of its resonance value. What are the appropriate parameter values? What is the
Q of this circuit? Suppose the data are changed by requiring that the impedance
magnitude need not be smaller than 1 per cent of its resonance value at 10 per cent
above or below resonance, what then are the answers to the above questions?
1 0.1
Ohms, henrys, farads
Prob. 53.
53. In the circuit shown, the current and voltage sources are
*,(v  Tav) (71)
Hence one can express the admittance as
2Pav+j4w(Fav Tav)
YM = ^rji ('2)
and the impedance as
E.
2Fav + jMT.v  Vav)
Z(u) = pyp (73)
If the functions Pav, Vav, Tav in Eq. 72 are assumed to be evaluated
for E, — 1 volt, then the admittance Y(u) is expressed explicitly in
terms of these power and energy functions. A similar interpretation
may be given the impedance expression 73 on the tacit assumption that
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T'av, Tav, Vav are evaluated for I = 1 ampere. In connection with the
simple RLC circuit, these results are only of nominal interest since the
conventional expressions for Y(u) and Z(w) in this case are even more
compact than the relations 72 and 73, and so it is only the novelty of
IMPEDANCE OR ADMITTANCE IN TERMS OF ENERGY FUNCTIONS 355
seeing these functions expressed in terms of power and energy that
makes them interesting. It is significant to mention that Eqs. 72 and
73 are found to apply as well to linear passive networks of arbitrary
complexity,* the expressions for Pav, Tav, and Fav being correspond
ingly more elaborate.
In terms of these results one may see again that a condition of reso
nance implies Tav = Vav. That is to say, when the average energies
stored by the electric and magnetic fields are equal, the impedance or
admittance at the driving point reduces to a real quantity; the system
is in resonance. Conversely, whenever the drivingpoint impedance or
admittance has a zero imaginary part, then one may conclude that the
average electric and magnetic stored energies are equal; the power
factor is unity, and the reactive power is zero.
Since the quantities Pav, Tav, Vav are implicit functions of the fre
quency w, the expressions 72 and 73 are not useful in the study of Y(u)
or Z(w) as functions of w except in some very special circumstances. A
case in point is the consideration of the behavior of Z(w) in the vicinity
of a resonance frequency. In the simple RLC circuit considered here,
Eq. 64 shows that Tav/ / 2 is a constant. In more elaborate circuits
one finds that the current ratios throughout the network are almost
constant over any frequency range near a resonance point, and hence
that Tav/12, which depends only upon the current distribution, is in
general almost constant in the vicinity of resonance.
Since Fav must equal for w = w0, Eq. 63 shows that we can write
for this vicinity f
Fav^7VWo7o,2 (74)
where w0 is the resonance frequency in question, and thus have in place
of Eq. 73
W.''^ —^ (75,
For values of w near w0, one may use the approximation
w2  w02 (w + w0)(w  wo) _ ,
= 2(w  w0) (76)
CO CO
* See Art. 8, Ch. 10.
f For the simple RLC circuit treated here, this expression, as well as the one given
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by Eq. 75, is of course exact, but in more general situations these are approximate
relations which are, however, very nearly correct throughout any pronounced reso
nance vicinity.
356 ENERGY AND POWER IN THE SINUSOIDAL STEADY STATE
and thus obtain for the impedance Z(u) the following explicit function
valid near w0
2Pav + j8Tav(u  u0)
Z(u) £S , 'I' = « + 3X (77)
whence
R = rjy2> X\T\i(w—Wo) (78)
The expression for R (which is exact) checks with Eq. 52 as, of course,
it should; the expression for X, through use of Eq. 64, checks with the
approximate expression for the reactance of the RLC circuit given by
Eq. 153 in Ch. 6. Again the significant feature about this result is that
one finds it to apply generally for all lowloss networks.
As pointed out in Ch. 6 and illustrated there in Fig. 19, the half
power points on the associated resonance curve lie where X = ±/? or,
using Eqs. 78, where
(«  wo) = ±P.v/47.v (79)
Hence the radianfrequency increment w between the halfpower fre
quencies (width of the resonance curve) becomes
to = Pav/27/ov (80)
and the Q of the circuit is found to be expressible as
Q = u0/W = 2w07'.v/P.v (81)
The behavior of T vs. time shown in Fig. 1 (applying to the simple
RLC circuit) is found to be representative of any lowloss system near
resonance. Thus, 2Tav = TVcak, and Eq. 81 can be written
Pav T()Pov
or
27r7,pcak 2irVpcak
Q =: ^r = ^ —r (83)
loss per cycle loss per cycle
since at or near resonance the stored energy merely swaps back and
forth between the electric and magnetic fields and so the peak value of
this energy is the same whether expressed electrically or magnetically.
The loss per cycle clearly equals the average rate of loss (Pav) times
the period t0 = 2r/«o.
Thus the factor Q which characterizes the critical behavior of a low
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loss network near resonance may be computed entirely on an energy
ENERGY FUNCTIONS FOR MORE COMPLEX NETWORKS 357
basis. Not only is the result 83 useful because it provides an inde
pendent approach to the computation of this important figure of merit
(an approach that is found to be usable in situations where parameter
calculations are difficult or not feasible) but also because it provides an
excellent basis for describing what is meant by a "lowloss" or "highQ"
system. Namely, it is one in which the loss per cycle is small compared
with the peak value of the total stored energy. In order to obtain a
circuit with an extremely sharp resonance curve, one must strive to
obtain as large an energy storage as possible relative to the associated
loss per cycle.
7 Computation of the Energy Functions for More Complex
Networks
When the network under consideration has several inductive and
capacitive branches, the expressions for the total instantaneous stored
energies T and V are obtained through simply summing the relations
40 and 41 over all pertinent branches.* Symbolically we may indicate
this procedure by writing
T = 7 £ Lk\ Ik 2 + \ Re[^'2"' E W] (84)
4*4k
and
V  A Z S* Ik 2  Refcr*"' £ SkI^ (85)
4oT k 4« k
In Eq. 84, Ik denotes the vector current in an inductive branch having
the inductance L*, and the summation extends over all inductive
branches in the network. In Eq. 85, Ik denotes the vector current in
a capacitive branch having elastance (reciprocal capacitance) Sk, and
the summation extends over all capacitive branches in the network.
The first terms in Eqs. 84 and 85 are Tav and Fav respectively for
the total network. Note that the sums yielding these quantities involve
the squared absolute values of the branch currents, while the second
terms in Eqs. 84 and 85, which are doublefrequency sinusoids, involve
the squared complex values of the branch currents. The sums in these
terms, therefore, involve complex addition (not merely the addition of
absolute values), and it is the angle of the resultant complex number
that determines the time phase of the pertinent sinusoid. Since the
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sum of a set of complex values has a resultant magnitude that is always
less than or at most equal to the sum of the absolute values of this set
* Mutual coupling between inductive branches is here assumed to be absent. A
treatment not subject to this restriction is given in Art. 6 of Ch. 10.
358 ENERGY AND POWER IN THE SINUSOIDAL STEADY STATE
of complex numbers, it is clear that the amplitude of the sinusoids,
component of either T or V is in general less than Txv or Vav respec
tively, and can equal this constant component only if all squared branch
currents are in phase, a condition that exists in all lossless networks (for
a single sinusoidal excitation) and is nearly attained in lowloss net
works operating at or near a resonance frequency.
In computing V it is sometimes more convenient to do so in terms of
the branch voltages instead of the branch currents. Since such a branch
voltage is related to its current by the expression
Ek = SJk/jo, (86)
giving
SkIk2/*2 = CkEk2 (87)
we see that Eq. 85 may be written
V = i £ Ck\ Ek\2 + i Re W2"ty£CkEk2] (8S)
k
which looks like Eq. 84 except for an interchange of E with / and C
with L, as we might have predicted through use of the principle of
duality.
8 Some Illustrative Examples
In order to show how these results are applied to a specific circuit,
consider the one in Fig. 5. If we denote by Ek the voltage drop in a
11
Henrys, farads, ohms
Fio. 5. Circuit to which the computations 89 are pertinent.
branch in which the current is Ik, and assume E4 = 1 volt, then the
following sequence of calculations for an assumed w = 1 radian per
second are selfexplanatory
h = U = 1 + jO, E3 = jl
E2 = E3 + E4 = l h = jE2 = 1 +jl (89)
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h = h + h = A Ei = jh = 1, E0 = £i + E2 = jl
SOME ILLUSTRATIVE EXAMPLES" 359
From these values we readily have
/i2 = i
h2 = A \h\2 = 2 (90)
h2 = 1,  h 2 = 1
and so Eqs. 84 and 85 yield
T = \ + 0 cos 2t, V = %  % sin 21 (91)
Since P„ = ^ /4 2 we then have
Pav = 5 watt, Tav = Fav =  joule (92)
We see that the circuit is evidently not a lowloss system, for the stored
energies are not large compared with the loss. Although the circuit is
in resonance, Tpeak = K is not equal to Fpeak = 1. There is no point
in computing a Q since it would have little meaning anyway.
It is interesting to find the impedance from the energy functions
according to Eq. 73, thus,
1 + ji X 0
Z = J = 1 (93)
which checks with Z = E0/I\ according to the values 89.
Now suppose we change the value of the resistance in Fig. 5 to l/10th
ohm. The computations 89 then become
Is = h = 10 + io, E3 = jlO
E2 = 1 +jl0, I2 = 10+jl (94)
h = Jl, Ei = 1, E0=jl0
and in place of the results 90 we have
A2 = l, \h\2 = l
h2 = 99  j20,  h 2 = 101 (95)
h2 = 100, \h\2 = ioo
The magnetic and electric stored energy functions, according to Eqs.
84 and 85, become
T = 25.25 + 24.75 cos 2t, V = 25.25  25.25 cos (2<  11.5°) (96)
So now
Tav = Fav = 25.25 joules, Pav = 5 watts (97)
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which looks a bit more like the results for a lowloss system should.
360 ENERGY AND POWER IN THE SINUSOIDAL STEADY STATE
We observe from Eqs. 96 also that
Tpeak = 50.0, 7Peak = 50.5 (9S
The loss per cycle equals t0 X Pav = 2ir X 5 = 10ir joules, and so the
factor Q, computed from either Eqs. 81 or 83, yields
2i r X 50
7
10jt
= 10
(99)
Although the resonance is not extremely sharp, it is well defined.
The input impedance, according to Eq. 73, for this case becomes
10 + j4(0)
1
10
(100)
while from the values 94 we get Z = E0/Ii = 10, thus substantiating
again the equivalence of these relationships.
Suppose now we restore the resistance to the value of 1 ohm, but add
two more reactive branches as shown in Fig. 6. This procedure should
1/2
o—npm^
Henrys, farads, ohms
Fig. 6. Circuit to which the computations 101 are pertinent.
increase the stored energy relative to the loss, and hence yield a sharper
resonance. In order to maintain resonance at w = 1 radian per second,
the first inductance now needs to be 1/2 henry as shown. This result
is easily arrived at through first computing the currents in all of the
other branches, following the pattern used above, and then noting the
value of the first inductance needed to make T*v = Fav.
The sequence of calculations appropriate to this circuit, assuming
E6 = 1 volt, read
Ia = /5 = 1 + jO, E5 = jl
E4^l+jl, /4=l+il, /3=jl
E3 = 1,
h = l.+jl,
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Eajl, J2=l (101)
Ei = (J) j(\)
SOME ILLUSTRATIVE EXAMPLES
361
From these we get
h2 =
1,
I/.
I2 = l
/42 =
/
I2 = 2
h2 =
1,
1/3
I2 = l
h2 =
1,
I/.
I2 = l
h2 =
A
/,
I2 = 2
(102)
and Eqs. 84 and 85 then give
T =  + \ sin 2t
3 VI (103)
V = cos (2<  63.5°)
44
Since Pav = 1/2 watt, and the loss per cycle is 3.14 joules, it is clear
that this situation, although somewhat better than the one in the first
example above, is still not a lowloss case. Thus Tpeak = 1 is only
moderately equal to Fpeak = 1.31. If we compute a Q at all, it is better
to use Eq. 81, which gives
e_^_£x(i)_, (104)
.* ar ( 2 /
For the input impedance we have, using Eq. 73,
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1 + j'4 X 0 1
Z = —— =  (105)
22
while from the values 101 we get
Z = 2. = II — =  (106)
h 1+Jl 2
These examples show that it is a straightforward matter to compute
TKv, Vav, Pav from a given current distribution. Since the latter or its
equivalent must in any event be determined in the course of an imped
ance computation, it turns out that it is no more tedious to find the
impedance in terms of the energy functions than in the normal manner.
The result in terms of energy functions contains more information.
For example, if Tav Vav but Tav — Fav is small compared with
either Tav or Vav, then we can conclude that the frequency considered
is near a pronounced resonance, especially if Pav is small compared
with either Tav or Fav. Through making a single computation at a
362 ENERGY AND POWER IN THE SINUSOIDAL STEADY STATE
resonance frequency, we are able to perceive the entire character of the
resonance curve, which is much more than the value of Z at resonance
can tell us. To compute Q in the normal manner, we must compute
many values of Z near resonance and plot a curve. In terms of energy
considerations we get the same information from a single calculation
made at the resonance frequency.
Another way of expressing these thoughts is to call attention to
the fact that, when we calculate the impedance of the series RLC
circuit through noting the values of the resistance R, the inductive
reactance Lw, the capacitive reactance — 1/Cu, and the net reactance
X = Lu — (l/Cw), their relative magnitudes not only enable us to
see whether the frequency in question is at or near resonance but they
also determine the character of the resonance curve. All this informa
tion is ours for the trouble of making a calculation at only one fre
quency. In more elaborate circuits such as those shown in Figs. 5 and
6, we are not in a position to get this much per unit of computing effort
unless we avail ourselves of the technique of expressing impedance in
terms of energy functions, for this scheme virtually reduces the im
pedance of any circuit to the basic form that it has for the series RLC
circuit.
PROBLEMS
1. Given e,(t)  100cos377 '31, hi, etc., experi
mentally, each determination is concerned with one pair of coils only
and can wholly ignore the presence of the others (except to see to it
that they remain opencircuited during the experiment so that there
will be no other nonzero (di/dt)'s except the one specifically intended
to be nonzero). For this reason the determination of the mutualinduct
ance coefficients for a large group of coils is every bit as simple and
straightforward as it is for just two coils, because one considers only
two coils at a time and the others are meanwhile ignored.
For a chosen set of reference arrows on the coils, as shown in Fig. 4,
the set of self and mutualinductance coefficients is completely fixed
as to both sign and magnitude. Specifically, if t'i, i2, i3, i4 are the coil
currents and vi, v2, v3, v4 are the voltage drops, both with regard to the
same set of reference arrows, then we can relate these currents and
voltages through the equations
dii
dia
~dt
di3
du
01
= hi
~kl
+ h2
+ h3
+ Im
m
0a
= hi
dii
~dt
+ I22
dia
~dt
+ h3
di3
Tt
+ h*
dh
~dT
dii
dia
H
di3
dh
1tt
t>s
"hi
Hi
+ I32
+ ^33
m
+ h*
dii
dt
di2
di3
dh
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m
MUTUAL INDUCTANCE AND HOW TO DEAL WITH IT
377
Equations 30 may formally be integrated with respect to time,
yielding
Jvi dt = lnii + l12i2 + h3H + hiU
v2 dt = £2ili + ^22l2 + ^23*3 + ^24l4
(31)
I v3dt = l31ii + l32i2 + h3i3 + hiU
Jvt dt = hiii + li2i2 + li3is +
The quantities involved here are flux linkages (since their time deriva
tives are voltages). These equations may be solved for the coil currents
in terms of the flux linkages by any algebraic process applying to the
solution of simultaneous linear equations (such as the determinant
method), yielding
*1 = 7HlAi + 712^2 + 713^3 + 714^4
*2 = 721^1 + 722^2 + 723^3 + 724^4
is = 731^1 + 732^2 + 733^3 + 734^4
ii = 741^1 + 742^2 + 743^3 + 744^4
(32)
in which the flux linkages are denoted by
ik =Jvkdt
(33)
and the y,k denote the numerical coefficients found in the process of
solving Eqs. 31 for the i*'s. For example, if the determinant of the
coefficients in 31 is denoted by
A=
11
Zi4
lti • • • In
and its cofactors by A,*, then by Cramer's rule
y.k = A*,/A
(34)
(35)
Whether the student completely understands the details of solving
simultaneous equations is at the moment of little importance. The
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point in writing these things down here is rather to be able to call atten
378 MORE GENERAL NETWORKS IN THE SINUSOIDAL STEADY STATE
tion to the fact that one can (through welldefined algebraic methods)
express the currents in the set of mutually coupled coils (Fig. 4) in
terms of their voltage drops (specifically in terms of the voltage inte
grals) as straightforwardly as one can express the voltage drops in these
coils in terms of their currents (specifically in terms of the current
derivatives). The latter is done in Eqs. 30, the former in Eqs. 32. In
Eqs. 30 the coefficients are the self and mutual inductances for the
given group of coils; in Eqs. 32 the coefficients are the self and mutual
reciprocal inductances for the same group of coils. The latter coeffi
cients are related to the former in a manner expressed by Eqs. 34 and
—"ct>
hc
>
)
>
(
H
Fig. 5. Relevant to the determination of the algebraic sign of a mutual inductance.
35, namely, as are the coefficients in inverse sets of simultaneous linear
equations. Thus, while the reciprocal inductance coefficients y,k are
not simply the respective reciprocals of the inductance coefficients l,k,
they are nevertheless related in a onetoone rational algebraic manner,
which, once understood, is simple and straightforward in its application
(although tedious if the number of coefficients is large).
Before continuing with the discussion of how the present relations
are used in the process of setting up equilibrium equations when a group
of mutually coupled coils such as those in Fig. 4 is imbedded in a given
network, a number of additional remarks may be in order with regard
to the determination of algebraic signs for mutual inductances in situa
tions where the relative directions of coil windings and mutual mag
netic fields are indicated schematically. A situation of this sort is
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shown in Fig. 5. Here the preferred path taken by the magnetic field
is indicated as a closed rectangular core structure (which may be the
iron core of a transformer), and the windings of the coils are drawn in
such a manner that one recognizes the directions in which they encircle
the core.
If a battery is applied to the lefthand winding so as to make the
indicated terminal positive, current in this winding increases in the
arrow direction, and, according to the righthand screw rule, the flux ct>
MUTUAL INDUCTANCE AND HOW TO DEAL WITH IT 379
in the core increases in the direction shown by its arrow. By the rule
for induced voltages (which is a lefthand screw rule because of Lenz's
law), we see that the increasing core flux ct> induces a voltage in the
righthand winding so as to make the bottom terminal plus with respect
to the top. If we place a reference arrow on the righthand winding as
indicated, we note that the induced voltage there is a voltage rise or a
negative drop. Hence, for the reference arrows shown, the mutual
inductance is seen to be numerically negative; it becomes positive,
however, if the reference arrow on either winding (not both) is reversed.
We may say in this example that the plusmarked ends of the two
windings are corresponding ends in the sense that they will always be
come plus together or minus to
gether when a voltage is induced
in one winding by a changing
current in the other, regardless of
which winding is doing the in
ducing. Since the marked ends
may become negative as well as
positive, the plus sign might be „ „ n . .... ,
r ' f ° . ° Fig. 6. Relevant to the sign determina
regarded as inappropriate. For tion of a set of three mutual inductances,
this reason many writers (and ap
paratus manufacturers) prefer to mark corresponding winding ends
simply with dots instead of plus signs, and this is a widely accepted
practice.
Note, however, that this scheme of relative polarity marking cannot
always be used without modification when more than two windings are
associated with the same magnetic structure, as the following discussion
of the example in Fig. 6 will show. If we assume the top terminal in
winding 1 to be positive with respect to the bottom one, current enters
this coil and increases in the arrow direction, thus producing a flux that
increases upward in the core of winding 1 and downward in the cores
of windings 2 and 3. From their winding directions relative to their
cores, one deduces that it is the bottom ends of coils 2 and 3 that become
positive. Hence we would place a dot at the top of coil 1, and corre
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sponding dots at the bottoms of coils 2 and 3. If we now move the
source from coil 1 to coil 2 and make the bottom terminal (the dot
marked one) positive, we see that flux increases downward in the core
of coil 2 and hence upward in the cores of coils 1 and 3. Thus the top
terminals of both of these coils become positive. For coil 1 this terminal
is the dotmarked one, but for coil 3 it isn't. Therefore, it becomes
clear that dotmarked terminals can in general indicate relative polarities
correctly only for a specific pair of coils. One would have to use a dif
380 MORE GENERAL NETWORKS IN THE SINUSOIDAL STEADY STATE
ferent set of dots for the pair of coils 2 and 3 from those that are already
placed upon these coils in pairing them separately with coil 1.
While the method of marking relative polarities of mutually coupled
coils by means of dots is thus seen to become prohibitively confusing
where many coupled coils are involved, the determination of a set of
self and mutualinductance coefficients consistent with assumed refer
ence arrows remains simple and unambiguous, as already explained. In
the example of Fig. 6 we clearly find all three mutualinductance coeffi
cients li2, hz, I23 numerically negative. Once these are known, the volt
ampere relations for the group of coils is unambiguously written down
as is done in Eqs. 30 or 32.
5 Coupling Coefficients
Suppose we consider the simple case of just two mutually coupled
coils, and let the associated inductance coefficients be denoted by £n,
£22, Ii2 — hi. The voltampere relations read
vi = £n (dii/dt) + li2 (di2/dt)
t>2 = hi (dii/dl) + h2 (diz/dt)
(36)
If we multiply these equations respectively by ii and i2 and add, we
have
dii . di2 , dii . dt'2
•Vi + t>2*2 = hih — + h2H — + hiiz — + £22*2 — (37)
dt dt dt di
which we may alternatively write as
viii + t>2*2 = dT/dt (38)
with
2T = £nl'i2 + ii2*i*2 *t" 121*2*1 ~T" £22*22
or
T  \Qnii2 + 2li2Hi2 + h2k2) (39)
Equation 38 states a simply understandable physical fact, namely,
that the" instantaneous power absorbed by the pair of coils (viii + t>2i2)
is equal to the time rate of change of the energy T stored in the asso
ciated magnetic fields, the latter being given by expression 39. Alge
braically this expression is homogeneous and quadratic in the current
variables ii and *2 (known as a quadratic form). Physically it is clear
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that T must be positive no matter what values (positive or negative)
the currents ii and i2 may have. Mathematicians have found that this
requirement on T imposes conditions on the coefficients l,k. Specifically
one can show that, if 39 is to be a positive definite quadratic form, it
COUPLING COEFFICIENTS
381
is necessary and sufficient that ln > 0, l22 > 0, and in addition
lnh2 ~ ha2 > 0 (40)
which can be written
h2Vhiha < 1 (41)
Since the quantity
fcli2/V^^ (42)
is defined as the coupling coefficient for the pair of coils in question, the
requirement that the associated stored energy be positive for all values
of the coil currents leads to the condition
 k  < 1 (43)
The limiting condition expressed by \k \ = 1, which is approachable
but never attainable in a pair of physical coils, is spoken of as a condi
tion of perfect coupling or close coupling. Physically it represents a
situation in which all the flux links all of the windings of both coils. If
the coupling coefficient k (Eq. 42) is derived from the standpoint of
flux linkages, condition 43 is arrived at on the basis that the state of
perfect coupling is manifestly an upper limit. A difficulty with this
method of deriving condition 43 is that it does not lend itself to gen
eralization while the method based upon stored energy is readily ex
tended to any number of coupled coils.
A logical extension of the reasoning leading from Eq. 36 to Eq. 39
shows that the stored energy is in general expressible as *
2T = Inii2 + Z12V2 H— •+ hniiin
+ ?2il2^1 + ^22*22 + ' • 1 + hnhin
(44)
+ Iniinii + h2ini2 H h Innin
Since the selfinductances In, h2, etc. are positive in any case, the con
ditions assuring T positive are expressed by stating that the determi
nant
lu Ii2 • •' 'in
I21 I22 • . • hn
I Ini ln2 "'' ^nn
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* See Art. 6; Ch. 10.
382 MORE GENERAL NETWORKS IN THE SINUSOIDAL STEADY STATE
and all minors formed through cancelation of the first row and column,
the first two rows and columns, the first three rows and columns, etc.
(called the principal minors) are positive. Although it is not the pur
pose of the present discussion to go deeply into matters of this sort, it
is nevertheless useful to point out (wherever this can easily be done!
what methods are available for extending our considerations to more
elaborate situations.
6 Forming the Equilibrium Equations When Mutual
Inductances Are Present
The procedure is most easily presented in terms of a specific example.
For this purpose consider the network of Fig. 7, for which the ecjui
Fig. 7. A circuit for which the equilibrium equations are to be found on the loop
basis. Numerical element values are in ohms and darafs. The coupled coils are
characterized by the self and mutualinductance values in matrix 46.
librium is to be formulated on the loop basis. So far as the resistance
and elastance parameter matrices are concerned, there is no new prob
lem presented here. Hence we need concern ourselves only with the
formation of the inductance parameter matrix.
In this regard we are given the three mutually coupled coils Lu L2, L3,
which, for the reference arrows indicated, shall be characterized by the
self and mutualinductance matrix.
[l.k] =
2 1 2
.1 3 2
22 5
(46)
That is to say, the selfinductance of hi is 2 henrys, the mutual between
it and L2 is — 1 henry, and so forth. If the voltage drops in these coils
are denoted by vi, v2, v3, then, since the corresponding currents are
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respectively ii, (ii — i2), and i2, we have
FORMING THE EQUILIBRIUM EQUATIONS
383
di, d di2 dit di2
t'i = 2 lfa  *2) + 2— = — + 3 —
dt dt dt dt dt
dii d dio dii dio
v2 1 — + 3  (t\  i2)  2 — = 2 5— (47)
dt dt dt dt dt
dii d dio du dio
t* = 2 2(i,  ta) + 5— = 0 — + 7 —
dt dt dt dt dt
The total inductive voltage drop around loop 1 is vi + v2, and that
around loop 2 is — v2 + v3. From Eq. 47 this gives
dii di2
vi + v2 = 3 2—.
dt dt
du di2
v2 + v3 = 2 — + 12— (48)
dt dt
whence the loop inductance matrix is seen to be
(49)
The fact that L12 must be equal to L21 serves as a partial check on the
numerical work.
Now let us consider a simple example on the node basis. Let the
network be that shown in Fig. 8. Here only the method of finding the
*2
=1
7z
6
)
The instantaneous power delivered by this source is
Pi = eiH =  Eih  cos ut cos (ut + ct>)
= [cos ct> + cos (2ut + ct>)]
(84)
392 MORE GENERAL NETWORKS IN THE SINUSOIDAL STEADY STATE
In the other two phases the voltages and currents are given by the
expressions 83 advanced (or retarded) by 120° and 240° respectively;
or we can say that the quantity (ul) in 83 is replaced by (ut ± 120°)
and (ut ± 240°) respectively. The corresponding expressions for in
stantaneous power in these phases are, therefore, the same as pi in
Eq. 84, except that (w + cos (2ut + ct>) + cos (2w< + ct> ± 240°)
+ cos (2ut + 4> db 480°)]
Since the last three terms in this expression cancel, we have simply
3 E1h 
Pi + P2 + P3 = cos 4> (86)
The important part about this result is that the pulsating components
in the several phases neutralize each other, so that the net instantaneous
power is composed of the steady component alone. It is simply three
times the average active power per phase.
In threephase rotating machinery this feature results in a steady
torque rather than one containing a pulsating component. The prac
tical advantage thus gained is significant.
PROBLEMS
1. Two inductances are characterized by the matrix
4 31
■[J 1]
Find the value of the net inductance when they are connected in the ways shown in
the diagrams (a) through (d).
(a)o—nflPP * IPPP^o (b)o—rWtP > « —0
i—rnftr^*!
i
(d) 0
Find the impedance function Z(s), and sketch a network, giving element values,
having this impedance.
26. For the circuit shown in the following sketch find
= E*/h
as a quotient of polynomials in s, and sketch the corresponding polezero configura
tion in the s plane. Determine the analytic expression for the instantaneous output
voltage et(f) if the input current ii(fl is a unit step.
Ohms, henrys, farads
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Prob. 26.
478 ADDITIONAL TOPICS STEADYSTATE AND TRANSIENT
27. A given network has the transfer impedance
ZuM = io/(S + io)
When the input current !i(