Introductory Circuit Theory by Guillemin Ernst

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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

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: 53-11754

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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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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 old-fashioned 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 power-distribution 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"

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 a-c 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 1-10 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

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 1-10. 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 a-c steady-state 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 a-c steady-state 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 electronic-control devices and the increased im-

portance of communication links in our fast-moving 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 so-called steady-state response, although it characterizes both.

This intimate relation between the transient and steady-state 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 d-c 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 one-semester 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 one-semester version being appropriate

for our physics students and the two-semester 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

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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 volt-ampere 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

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, wye-delta 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, bridged-tee 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

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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 wye-delta or delta-wye 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 star-mesh 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 volt-ampere 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 open-minded

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

second-order 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 Laplace-transform 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 Laplace-transforming 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 pole-zero pattern in

xviii

PREFACE

the s-plane, 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 double-tuned 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 in-phase and out-of-phase 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 single-frequency 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 so-called a-c 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 pole-zero

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 second-order 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

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. Jekyll-Mr. Hyde sort of thing; it is two-faced, 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 "five-year 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 Tie-Set Schedules ./ 10

6. The Concept of Node-Pair Voltages; Cut Sets and Cut-Set 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; Wye-Delta (F-A) Equivalents 127

5. Thevenin's and Norton's Theorems 138

6. The Reciprocity Theorem 148

7. Driving-Point 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 Volt-Ampere Relations of the Elements 188

2. Voltage and Current Sources 190

3. The Family of Singularity Functions; Some Physical Interpreta-

tions 196

4. Single-Element Combinations 203

Approximate Formulas for Parameters of Simple Geometrical Configura-

tion 211

Problems 218

5 Impulse and Step-Function 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 Three-Ele-

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. Root-Mean-Square, 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 Steady-State 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 Driving-Point 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 Steady-State 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 Lumped-Constant Network 419

5. The Derivation of Equilibrium Equations for Driving-Point and

Transfer Situations; Reciprocity Again 426

6. Properties of the General Solution 431

7. Illustrative Examples 440

8. Driving-Point 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. Branch-Parameter Matrices and Volt-Ampere 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 volt-ampere 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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INTRODUCTION

along the line; the behavior of a motor-generator 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 voltage-current 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 motor-generator 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 long-distance 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 volt-ampere behavior of an electric

circuit are its energy-storage and energy-dissipation properties. Energy

storage takes place in the electric and magnetic fields associated with the

network, while energy dissipation is practically ever-present 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

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 volt-ampere 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 iron-cored 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

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 so-called

two-element 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 all-important 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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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"

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

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

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 volt-ampere 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 last-mentioned 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

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 tree-branch 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 tree-branch voltages, and the link voltages. Since the tree

branches connect all of the nodes, it is clear that, if the tree-branch 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 short-circuiting 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

tree-branch voltages equal to zero forces all voltages in the network

to be zero. The tree-branch 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

tree-branch 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

tree-branch 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 tree-branch currents as appropriate superpositions of

the loop currents, thus,

(4;

= *1

- *4

= i2

- l'i

= *S

- *S

= *4

~ *3

or, being mindful of the relations 3, have

-3i

-3 a

= 32

-3i

-h

-3*

-33

(5)

These last four equations express the tree-branch 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 tree-branch 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.

THE CONCEPT OF LOOP CURRENTS

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

NETWORK GEOMETRY AND NETWORK VARIABLES

between the branch currents and the loop-current 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

(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

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 tie-set

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

i-i4-*-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 tree-branch 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 NODE-PAIR 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 tie-set 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 Node-Pair Voltages; Cut Sets and Cut-Set

Schedules

On the voltage side of the network picture, an entirely analogous

situation prevails. Here we begin by regarding the tree-branch 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 tree-branch 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 node-pair voltage, is uniquely expressible in terms of the

tree-branch voltages. The link voltages, which are a particular set

of node-pair voltages, are thus recognized to be uniquely expressible in

terms of the tree-branch 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

tree-branch voltages and hence may be regarded as an independent set.

They may simultaneously be regarded as node-pair 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

tree-branch voltages are identified with node-pair voltages.

The remaining branch voltages, namely the link voltages, are now

readily expressible in terms of the four tree-branch or node-pair 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 tree-branch 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 tree-branch 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 NODE-PAIR VOLTAGES

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 node-pair 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 tie-set 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 node-pair (i.e., tree-branch) voltages

to be zero, which is accomplished through short-circuiting 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 nonshort-circuited tree branch that

are likewise not short-circuited 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 node-pair voltages

as expressed by Eqs. 15. The cut-set schedule appropriate' to this situ-

ation reads as given in 19.

Node

Pair

No.

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

tree-branch voltages. The cut-set schedule which contains the informa-

tion regarding the geometrical character of the cut sets, as well as the

algebraic relationships between the implied node-pair 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 tie-set 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 loop-current

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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 tie-set 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

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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 tree-branch 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

tree-branch 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 tie-set 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 tie-set 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 mesh-current variables.

However, it is quite possible for the I rows in a tie-set 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 tie-set schedule

reads as in 29, and hence the expressions for the branch currents in terms

Loop

Branch No.

No.

-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 right-hand 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 tie-set 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 current-variable definitions, should this be

desirable. Thus one may make use of the fact that the most general

tie-set 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

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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 tie-set

schedule); it does not place them in evidence.

The most general form a linear transformation of the tie-set 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 tie-set 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 tree-branch 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 tie-set schedule 39, which more

Loop

No.

Branch No.

-i

-h

-i

-2

-1

-l

-§

-i

-h

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compactly contains this same information. This is the schedule that is

implied by definitions 35 for the loop-current variables, which no longer

possess the geometrical interpretation of being circulatory currents.

As we shall see in the following chapter, the tie-set 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 tree-branch voltages with node-pair voltage variables. In this

process (like procedure 1 mentioned above for the choice of current vari-

ables), the algebraic definitions for the node-pair 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 node-pair

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 node-pair vol-

tages for the graph of

Fig. 8.

NETWORK GEOMETRY AND NETWORK VARIABLES

this choice (whether actually drawn or merely implied) forms a structure

that has a tree-like character.

Using the principles set forth in Art. 6, one can construct cut-set

schedule 40 appropriate to the choice of node pairs indicated in Fig. 10

Node

Pair

No.

Branch No.

Picked-

Nodes

-1

a, e

-1

b, d

-1

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ALTERNATIVE METHODS OF CHOOSING VOLTAGE VARIABLES 35

fi = —ei +62 + 63-^4-65

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 node-pair 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 node-pair

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 tree-branch 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 "node-to-datum"

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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 node-pair 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 tree-like character and hence

that such a node-to-datum set of voltages

is always an independent one.

The cut sets appropriate to this group of

node-pair voltages are particularly easy to

find since we observe that setting all but

one of the node-pair 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,

cut-set schedule 42 is thus readily obtained.

Since the node-pair 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 node-to-datum

choice of node-pair voltages

for the graph of Fig. 8.

Node

Branch No.

Picked-

No.

Nodes

-1

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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 tree-branch 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 node-pair voltage defini-

tions are derivable from the simple node-to-datum set through carrying

out linear transformations on the rows of cut-set 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

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38 NETWORK GEOMETRY AND NETWORK VARIABLES

Node

Pair

No.

Branch No.

Picked-

Nodes

-1

e,f

-1

ALTERNATIVE METHODS OF CHOOSING VOLTAGE VARIABLES 39

This result suggests that the node-pair 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 picked-up nodes corresponding to the

remaining node-pair voltages evidently remain the same as before, and

hence the rest of the cut sets are unchanged.

Fig. 12. Revision in the node-pair volt- Fio. 13. The graph of Fig. 8

age definitions of Fig. 11 corresponding with node designations as

to a transformation of cut-set 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 picked-up 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

node-pair voltages (the rest of the e's remain as in Eqs. 49 with double

primes on the right-hand 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 node-pair voltage diagram like the ones in Figs. 10, 11, or 12 with the

resulting node-pair 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 cut-set schedule (as is also a pos-

sible procedure) by making arbitrary choices for the picked-up 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,

NETWORK GEOMETRY AND NETWORK VARIABLES

and 12. Cut-set schedule 50 is constructed by simply making an arbi-

trary choice for the picked-up 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-

Nodes

-1

a, d

c,f

a, b, c

-1

d, e

-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 tree-branch 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

- 3e2 + 2e3

~ «5

t'3

- 3e2 + 2e3

- e5

»S

- 3e2 + 3e3

- ei + «s

- 2e2 + e3

»e

- 2e3 + e4

- 2«4 + e5

v»

-2e5

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These results may be summarized in cut-set schedule 56. Thus we see

Node

Pair

No.

Branch No.

-3

-2

DUALITY

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

node-pair voltages with tree-branch 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

Tree-branch current

Tie set

Short circuit

Parallel paths

Voltage

Branch voltage

Node or node pair

Number of node pairs (n)

Node-pair voltage

Node potential

Tree branch

Tree-branch 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-

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 node-pair 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

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

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-

tional-principle 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 node-pair 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 branch-number 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 short-circuiting link 16 in the dual graph of part (b), since open- and

(a) Datum

161

151

(b) Datum

Fig. 17. The dual graphs of Fig. 16 redrawn in the form of unbalanced ladder

networks.

short-circuit 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

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 node-pair voltages with tree-branch 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

node-pair 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,

NETWORK GEOMETRY AND NETWORK VARIABLES

»io. •"■ "i6 are respectively identified with node-pair voltages e\,

e2, • • ., eg. For the tree-branch 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 node-to-datum 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 tying-off 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

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 node-to-datum

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 tree-branch

* 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 tree-branch 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

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

tree-like 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 node-pair 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 node-pair voltage (potential

difference between two nodes) equals the algebraic sum of tree-branch

voltages along the path connecting this node pair. The difference

(im ~ in), which might be called a mesh-pair current, is the real dual of a

node-pair 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 node-potential differences with tree-branch voltages.

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 node-to-datum 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 node-to-datum choice of node-pair

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 tree-branch voltages read

ei = vi

e2 = v2 + vi

e3 = — vi — i's - t'6 (57)

e4 =

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—»5 — »6

CONCLUDING REMARKS

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,

i2 -

is ~

-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 mesh-current 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 node-pair 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 =

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

- *i =

t'a

= ii

t'a

= t'5

= -ja

- is =

-h

~ *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 node-pair 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 node-pair voltage diagram [like part (b) of

Fig. 20] corresponding to a chosen loop-current 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

Since for cut sets picked at random there does not necessarily corre-

spond a set of "node-pair 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

Fig. 20. A revised choice of loop currents in the graph of Fig. 19(a) and the corres-

ponding revision in the node-pair voltage definitions for the dual graph.

differences between nodes and i's that are mesh-current 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 tie-set 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 tie-set schedule.

2. With reference to the graph of Prob. 1, de-

termine whether each of the accompanying tie-set

schedules defines an independent set of loop currents. If so, express the loop cur-

Loop

No.

Branch No.

-1

Loop

No.

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Branch No.

PROBLEMS

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 tie-set schedule and its associated graph, trace the

Loop

Branch No.

No.

-1

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60 NETWORK GEOMETRY AND NETWORK VARIABLES

7, 8, 9, 10, 11, 12; identifying, in each case, the node-pair voltages with the tree-

branch voltages. For each choice of tree, express the link voltages in terms of the

tree-branch or node-pair 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 cut-set schedule for the node-to-

datum voltages, ei, . • ., e.i. Express these in terms of

1 2 each of the two sets of tree-branch 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 cut-set schedule,

and express the node-pair 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 node-to-datum voltages of Prob. 6, and show that the cut-set

schedule written there is now the appropriate tie-set 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 tree-branch voltages found in

Prob. 6.

9. For the dual graph of Prob. 8 define loop-current variables (mesh-pair currents)

that are dual to each of the independent sets of node-pair voltages specified in Prob. 7.

Show in each case that the appropriate tie-set schedule is identical with the pertinent

cut-set 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 tie-set 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

tie-set 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 tie-set schedule represent a matrix having l's on its prin-

PROBLEMS

13*

121

A10

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 cut-set 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 tie-set 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 three-dimensional 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 tie-set 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 tie-set 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 mesh-current differences given by the algebraic relationships

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

Fig. 1. Suppose we write voltage-law equations for the upper left-hand

corner mesh and its right-hand 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

left-hand 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 so-called 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.

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

independent currents (for example, the loop currents) or in terms of

n = ni — 1 = 6 — I independent voltages (for example the node-pair

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 voltage-law and the current-law types) may be written for any

given network geometry. Consider first the voltage-law 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 voltage-law 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 voltage-law

equations clearly are not independent. The inference is that one can

always write exactly / independent equations of the voltage-law 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 voltage-law 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 tree-branch voltages. This fact

incidentally substantiates what was said earlier with regard to the

tree-branch 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 voltage-law 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 tree-branch voltages alone (the tree-branch voltages

are independent and hence are not expressible in terms of each other).

It follows, therefore, that the voltage-law equation written for the

additional closed loop expresses no independent result. There are indeed

exactly I independent voltage-law equations.

THE EQUILIBRIUM EQUATIONS

2 6 X)

kif

Let us turn our attention now to the Kirchhoff current-law 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

current-law 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 node-pair voltages. For the corre-

spondingly determined node pairs, a set of Kirchhoff current-law 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 voltage-law 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 picked-up 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

tree-branch voltage v20 is identified with the respective node-pair

voltage, the latter has its reference arrow pointing from / to e. That

Fio. 2. A tree for the graph of Fig. 1.

The cut-set pertinent to node pair

f-e consists of tree branch 20 and the

links shown dotted.

EQUATIONS ON THE LOOP AND NODE BASES

is to say, the picked-up nodes are e, q, I, b, c, d. Hence the pertinent

current-law equation reads

J20 - ju - h - 32 - 0 (11)

Schedules like 40, 42, 46 in Art. 8 of Ch. 1 are helpful in writing the

current-law 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 current-law equations like 11 are written for all of the

node pairs corresponding to the n tree branches. These equations are

surely independent, for the tree-branch 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 tree-branch 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 current-law equation one substitutes the expressions already

obtained for the pertinent tree-branch 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 current-law equation written for any additional node pair

expresses no independent result. There are indeed exactly n independent

current-law 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 node-pair voltages, and having recognized that the numbers of

independent Kirchhoff voltage-law and current-law 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 voltage-law equations in which the loop currents are the variables,

or (b) through a set of n current-law equations in which the node-pair

voltages are the variables. These procedures, which are referred to

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 voltage-law equations, like Eq. 1,

p. 65, involve the branch-voltage 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 tie-set 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 voltage-law 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).

-1

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EQUATIONS ON THE LOOP AND NODE BASES

Loop

Branch No.

No.

-1

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)

—

Z*,

1)7

»a

5t3

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

-lWi

+ 19t2

+ Si3 - 10it

- 18iB

+ 18*6

-m

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+ 8*2

+ 22i3 + 0*4

- 17i5

+ 8i6

16ii

- 10t2

+ 0t3 + 19t4

+ lOtg

- 16i6

27i!

- 18i2

- 17t3 + 10l4

+ 31i5

- 18t6

-24h

+ 18i2

+ 8*3 - I614

- 18t5

+ 24i6

(18)

Considering next the node method of writing equilibrium equations we

observe first that the current-law equations, like Eq. 11 above, involve

the branch currents. If these equations are to be written with the

node-pair voltages as variables, we must express the branch currents

in terms of the node-pair 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 node-pair 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

following cut-set schedule is then readily constructed from an inspection

of Fig. 3, noting the picked-up nodes pertinent to these four node pairs.

Node

Pair

No.

Branch No.

Picked-

Nodes

-1

-1'

-1J

1,.

c, d, e

-1

-i

a, b, c

-1

-i

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a, b, c, d

(20)

The Kirchhoff current-law equations corresponding to this choice of

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.

The coefficients in the rows and in the columns of the appropriate tie-set

or cut-set 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 voltage-law equations (a) are voltages while

the branch quantities in (c) are currents. In the node method, the

branch quantities in the current-law 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 tie-set or cut-set 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

The rows of a tie-set schedule define an independent set of closed paths,

and hence provide a convenient means for obtaining an independent set

of Kirchhoff voltage-law equations. Any row of a cut-set 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 cut-set

schedule are seen to provide a convenient means for obtaining an inde-

pendent set of Kirchhoff current-law 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

-18

-17

-24

-10

-18

-17

-10

-16

-18

-17

-18

-16

-18

24.

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known as the loop-resistance parameter matrix. Equilibrium Eqs. 24 are

similarly characterized by the following node-conductance parameter

matrix.

r 1.142

-0.976

0.643

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 node-parameter 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

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

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 space-consuming 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 parameter-matrix 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 node-conductance 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)

THE EQUILIBRIUM EQUATIONS

inantly the two sets of relations designated in the summary in Art. 3

as (a) the Kirchhoff-law 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 voltage-law type; on the node basis the

variables are node-pair voltages, and the Kirchhoff equations are of the

current-law type.

The choice of a set of loop-current 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 voltage-law 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 voltage-law 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 tie-set schedules. One of these per-

tains to the definition of a set of loop-current variables (as discussed in

Art. 5, Ch. 1); the tie sets in the other one serve merely as a basis for

writing the voltage-law 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

node-pair voltage variables, and again for the writing of the Kirchhoff

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current-law 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 cut-set schedules. One of these per-

tains to the definition of a set of node-pair voltage variables (as dis-

cussed in Art. 6, Ch. 1); the cut sets in the other one are utilized in writing

SIMPLIFIED PROCEDURES

current-law 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 Kirchhoff-law 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 voltage-law equations and loop-

current definitions, or the same cut sets are used for current-law equa-

tions and node-pair voltage definitions, then we say that the choice of /

variables is consistent with the Kirchhoff-law 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 round-table

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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.

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 voltage-law 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 — t-3

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] =

-2"

-10

-5

.-2

-10

-5

17.

(37)

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SIMPLIFIED PROCEDURES

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 2-ohm 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 loop-resistance

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,

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 loop-resistance

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 node-pair voltage variables

are chosen as a node-to-datum 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 current-law 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

cut sets are the branches divergent from these nodes, the current-law

equations consistent with this selection of node-pair 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 node-conductance 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

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

2-ohm and 5-ohm 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 branch-resistance

values are alternately given as branch-conductance 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 node-pair voltage

variables are chosen as a node-to-datum 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

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 right-hand 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 right-hand 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

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 "volt-ampere 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 short-circuited 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 open-circuit 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

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,

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 (d-c generator), (c) arbitrary voltage function, (d) arbitrary cur-

rent function.

constant-voltage sources, also called "direct current" or "d-c" 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 d-c 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 constant-voltage source (d-c 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 d-c 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

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 open-circuited. 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

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 "soldering-iron method" respectively. We shall now show that

one may consider the pliers method restricted to the insertion of voltage

sources and the soldering-iron 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

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 short-circuit

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

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 a-b 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

a-b 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

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

open-circuit character of a current source and the short-circuit 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 a-b 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 a-b 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 a-b (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.

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 a-b 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 volt-ampere 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 voltage-law equations in terms of branch voltages:

2±t>* = 0

(44)

SUMMARY OF THE PROCEDURES

(b) The relations between branch voltages and branch currents

(Eqs. 43):

vk = -e.k + z(jk + i,k) (45)

ju = 2±ir (46)

The rows of a tie-set 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

left-hand side of Eq. 47 is the algebraic summation of such passive

branch voltage drops around a typical closed loop I. The right-hand 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

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 current-law 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)

vk = 2±er (50)

The rows of a cut-set 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 node-pair 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 node-pair 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 node-pair voltages er acting upon it, and

hence the left-hand 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

divergent from a given node n if the node-pair voltages are chosen as a

node-to-datum set.

The right-hand 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 node-to-datum 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-

cuit-parameter relations in the same way as do the passive currents

caused by the node-pair 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 tree-branch

voltages with node-pair 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 node-pair 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 node-pair voltages

or tree-branch voltages are zero, the same as they would be if all tree

branches were short-circuited.

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 node-to-datum 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 node-to-

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

(52)

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The tie-set schedule corresponding to this choice is given in 52. The

rows give us the voltage-law 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:

= *i

3i =

-ii +14

= i2

J5 =

-*2 + *4

= 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 2-ohm resistance which is the passive part of this

branch. Noting the source values given above, the relations expressing

net branch-voltage 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 node-pair voltages the potentials of nodes a and b respec-

tively, with the bottom node as a reference. The appropriate cut-set

schedule is 59. The rows give us the current-law equations,

Node

Pair

No.

Branch No.

-1

(59)

-ji + h ~ h + h - 0

-32 + h - is + ie = 0

(60)

and the columns yield the branch voltages in terms of the node-pair

voltages, thus:

e-2

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

vi + 5

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 node-pair 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 node-pair voltages is shown in part (b) of the same figure. In

cut-set schedule 66 pertaining to this choice of node pairs a last column

Node

Pair

No.

Branch No.

Picked-

Nodes

-1

a, c

-1

a, c, d

-1

b, c, d

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104 THE EQUILIBRIUM EQUATIONS

indicating the corresponding "picked-up" nodes is added to facilitate

understanding its construction.

According to the rows of this schedule one obtains the Kirchhoff

current-law 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 node-pair 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-

= 2vi + 20

= 5v6

= 2v2

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= v7 + 6

= v3 + 2

= 3«s

= 3t'4 - 12

= 2vg

= 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 branch-voltage 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 voltage-law 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.

Prob. 1. Prob. 3.

2. Prove or disprove the statement: "The number of independent Kirchhoff

voltage-law 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 voltage-law

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 voltage-law 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 5-mesh mappable network, are voltage-law 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 voltage-law 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 voltage-law' equations, and use these to express

the link voltages in terms of the tree-branch voltages. Now write a voltage-law

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 current-law 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.

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.

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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 cut-set schedules, one for the

choice of node-pair voltages, ei = «i, e2 = t% «s = »s, and the other one for the

picked-up nodes, a-c, b-c, d.

Using the first schedule for the definition of variables and the second one for the

determination of the Kirchhoff current-law 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-, 5-ohm 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 —4-volt and 2-volt

sources in the tree branches by respectively equal sources in the links and combining

these with the other link-voltage sources and converted current sources; second, by

opening all the links and noting the net voltages across the gaps thus formed (the

desired link-voltage 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 short-circuiting 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 tree-branch 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 1-ohm

branches and a 1-volt 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 tie-set schedule for the

network. Write down explicitly the three sets of equations: (1) Kirchhoff's voltage-

law equations, (2) the appropriate volt-ampere 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.

THE EQUILIBRIUM EQUATIONS

21. (a) For the network of Prob. 20 consider the node-pair voltages from a, 6,

and c to ground as an independent set. Write a cut-set schedule for them. Then

obtain the three sets of equations: (1) Kirchhoff's current-law equations, (2) the

appropriate volt-ampere relations for the branches, (3) the branch voltage drops

in terms of the node-pair 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 node-pair 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 2-ohm resist-

ances. Branch 11 is a 1-ohm resistance in parallel with a 1-ampere 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 current-law equations involving the current jk = —i,k simply be-

come known quantities and are transposed to the right-hand sides. On a loop basis,

show that one can construct the tie-set 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.

Prob. 22.

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 voltage-law equations involving the voltage t,* — — e,*

simply become known quantities and are transposed to the right-hand sides. On a

node basis, show that one can construct the cut-set schedule so that its first n — 1

rows do not involve this branch, thus identifying node-pair 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 link-voltage 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

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