bewley_tensors.pdf

TENSOR ANALYSIS

OF ELECTRIC

CIRCUITS AND MACHINES

L. V. BEWLEY PROFESSOR OF ELECTRICAL ENGINEERING

AND

DEAN OF ENGINEERING

LEHIGH UNIVERSITY

THE RONALD PRESS COMPANY . NEW YORK

Copyright 0 1961 by THE RONALD PRESS COMPANY -

All Rights Resewed No part of this book may be reproduced in any form without permission in writing from

the publisher.

Library of Congress Catalog Card Number: 61-5655

PRINTED IN THE UNITED STATES OF AMERICA

PREFACE

In this book I have attempted to present the essential principles, together with applications, of the matrix-tensor methods of analysis of electric circuits and knachines as developed over the past quarter century by Gabriel Kron and published by him in four books and numerous papers and articles. The book is designed to provide a unified consolidation of developments in this field and to serve as an introduction to the remarkable power of tensor methods in electrical engineering. In my opinion, Kron's research constitutes the most outstanding contribution to the theory of electrical engineering made in this generation and deserves to rank in importance with the work of Steinmetz. There is nothing in engineering literature to compare with Kron's generalizations as highly organized methods of attack of unlimited versatility and power. The tensor method is one of great analytical beauty which in all stages of its development exhibits a clear-cut interpretation and understanding of the underlying physical phenomena.

There are five major areas discernible in Kron's methods: (1) the mathematical disciplines of matrix and tensor analyses, with their associated concepts, definitions, rules of operation, etc.; (2) the establishment of the equations of generalized (or primitive) networks and machines; (3) the transformation of voltages, currents, impedances, velocities, and torques from the primitive system (or some selected set of connections) to other systems; (4) transformations and artifices designed to save labor, or to present a new point of view, or to simplify equations; and (5) the interpretation of the equations of performance in terms of equivalent circuits and models.

Kron recognized the possibilities of tensor analysis as the tool par excellence for the study of electrical systems, and while he added nothing new to the mathematical theory per se, he must be given full credit for having adapted tensor theory to electric circuits and machines, just as others had adapted it to the study of differential geometry, relativity theory, mechanics, elasticity, electromagnetic theory, and other mathematical and physical fields. Kron showed how the fundamental equations of electric machines could be established from any of four general methods of attack: (1) by physical considerations and elementary equations, (2) by Lagrange's equations of motion, (3) by Maxwell's equations, and (4) by the Maxwell-Lorentz equation. He conceived the all-important idea of "primitive" networks and machines in which all connections are severed and for which the equations may be established by relatively simple means, and of then passing to the actual connections by means of a "transformation tensor." If any one idea in the amazing multitude of ideas which Kron has given us deserves to be

iii

iv PREFACE

singled out and crowned as his cardinal achievement, this is it. He devised innumerable clever procedures and artifices for simplifying results, or to render them more intelligible, or for saving labor. Sometimes he became so immersed in his search for a tensor or matrix which would bring about a desired operation that he failed to notice that i t could be done by inspection or by some simple device, but these things are to be expected in pioneer work and are easily forgiven. While his intuition was uncanny, i t was too often substituted for a rigorous and rational development, and this can prove disconcerting to the reader.

For twenty years I have taught a course in the applications of tensor analysis, essentially as covered in this book, to first year graduate students a t Lehigh University. I am convinced that a course, based on this book, could be given to electrical engineering seniors, and that in the not too distant future such a course will become a standard part of undergraduate curricula, so that this powerful tool may become a part of the equipment of every electrical engineer.

I wish to express my thanks to the students of my classes, especially Joseph Teno, William Hollabaugh, and Donald Talhelm, who searched so diligently for errors, and to Mrs. Marion Stempkowski, who typed the manuscript.

L. V. BEWLEY Bethlehem, Pennsylvania

January, 1961

CONTENTS

P A R T I

Matrices, Tensors, and Circuits CHAPTER PAGE

PART I1

Machine Analysis

BASIC CONCEPTS IN ELECTRIC MACHINES . THE QUASI-HOLONOMIC GENERALIZED MACHINE . USE O F THE LAGRANGE EQUATION .

USE OF MAXWELL'S EQUATION . USE OF THE MAXWELL-LORBNTZ EQUATION . COMPARISON OF THE GENERAL SOLUTIONS

THE RAISINQ AND LOWERING OF INDICES AND GENERALIZED

PER-UNIT CONCEPTS . SMALL OSCILLATIONS AND HUNTING

v

vi CONTENTS

20 SYNCHBONOUS MACHINES . . 222

21 INDUCTION MACHINES . . 246

22 COMMUTATOR MACHINES . 268

23 INTERCONNECTED MACHINES . . 283

Part I

MATRICES, TENSORS, AND CIRCUITS

1

SOME PRELIMINARY CONSIDERATIONS

1-1. Introduction. Tensor analysis was developed by the mathe- maticians Grassmann, Riemann, Beltrami, Christoffel, Ricci, and Levi-Civita as the mathematical tool par excellence for the study of n-dimensional spaces undergoing transformations of reference frame subject to some condition of invariancy. In Einstein's hands it became the language of relativity theory, and since then it has found ready application in all branches of physics. Tensor analysis was introduced into electrical engineering by Gabriel Kron in 1935, and almost entirely through his w~itings, which include four books and numerous technical papers and articles, it has become one of the most powerful analytical tools and methods of analysis in modern engineering. Its unsurpassed supremacy as a means of generalization, its great unifying concepts, its beauty of expression, and the ease with which it may be learned and applied establish tensor analysis as perhaps the most powerful analytical tool a t the disposal of the engineer.

Since tensor analysis can do everything that can be done with vector analysis in a simpler and easier fashion and since it is not restricted to three- dimensional orthogonal coordinate systems, and in addition can handle far more difficult situations in which vector analysis is helpless, there seems little doubt that it will eventually supplant vector analysis entirely.

The study of differential invariants is concerned with "a set of differential equations" which is "invariant to" a "group of transformations." Its mathematical discipline is called tensor analysis, or the absolute calculus, and the concepts of set, form, invariancy, group property, and transformation are its keynotes. Differential geometry is its geometric interpretation; and applications to physics, dynamics, the unified field theory, and electric machinery are examples of its physical interpretation. By means of it the language and concepts of abstract geometry can be brought over en masse into physical problems with unbelievable profit and enlightenment, and thus the research which has been done in the field of abstract geometry by the greatest mathematical minds becomes available a t bargain rates to the engineer.

1-2. Networks and Notation. Consider an electrical network consisting of n branches, each branch composed of any number of resistances, inductances, and capacitances in series and each branch inductively coupled

3

4 MATRICES, TENSORS, A N D CIRCUITS [Ch. I

with all the other branches of the network. At this stage of our information 2n unknowns are associated with this network : the n currents in the branches, and the n voltages across those branches. The determination of these 2n unknowns would require 2n equations relating the voltages and currents and any constraints or relationships between currents or voltages imposed by the connections. (The specification of a voltage or of a current as a known quantity is regarded here as equivalent to an equation.) It is a remarkable fact that in any such network, no matter how complicated, n, or exactly half, of the 2n equations needed for a determination of the 2n unknowns can be written from a knowledge of the composition of the branches without regard to the way in which the branches are interconnected; and the other half can be written entirely from a diagram of the connections without regard to the composition of the separate branches. The first set of n equations is called the canonical equations, and the second set the equations of constraint. The solution of these 2n simultaneous equations determines the n voltages and the n currents of the network. These equations may be expressed in various notations, such as scalar, vector, dyadic, matrix, and tensor. Each of these notations and methods of manipulation possesses certain advantages from the point of view of compactness, associated physical concepts, ease of numerical calculation, generality, and versatility. This book will adhere to tensor notation, but the special matrix notation developed by Gabriel Kron will be employed for detailed specifications and routine manipulation. To this end :

Currents (whether branch or mesh is immaterial) will be represented with superscripts :

ia, ib, . . . , i n or, in general, by ik; (k = a, b, . . . , n)

Voltages will be represented with subscripts :

e,, e,, . . . , en or, in general, by ej; (j = a, b, . . . , n)

Impedances will be represented with subscripts :

Zaa, Zab, . . . , Znn or, in general, by Zjk; (j, k = a, b, . . . , n)

Admittances will be represented with superscripts :

Yaa, Yab, . . . , Ynn or, in general, by Yjk; (j, k = a, b, . . . , n)

The use of superscripts and subscripts is not arbitrary but, as will be developed later, is a matter of the utmost significance.

In the course of events, currents, voltages, impedances, and admittances will need to be transformed to new coordinates; for example, branch currents replaced by mesh currents with attendant changes in the voltages and impedances. These new coordinates will be designated by primed quantities a', b', . . ., n' or, in the case of several transformations in succession, by double- prime quantities a", b", . . .; and triple-prime quantities a", b", . . .; etc.

Art. 1-31 SOME PRELIMINARY CONSIDERATIONS 5

1-3. The Canonical Equations. Fig. 1-1 shows n separate impedance branches, each carrying its own current and having a voltage across it. No restrictions are placed on the nature of these impedances, and they may consist of any series combination of resistances, inductances, and capacitances. Furthermore, every impedance may be mutually (inductively) coupled with every other branch. Later on it will be shown that more sophisticated concepts of impedances permit extended definitions, but for the present a general impedance may be taken as

in which, as in operational calculus,* - - Zib 4"

for transients Fig. 1 - 1 . Branches of a network. P =

jo for steady-state alternating currents

The canonical equations relate the branch voltages and currents :

There are as many (n) of these equations as there are separate branches, and they are written quite independently of how these branches may be interconnected. In fact, a t this point it is not known how they are interconnected, or even if they are to be. This set of n simultaneous equations may be greatly condensed by writing them in summation convention :

Obviously, this much more compact notation expresses everything that Eq. 1-2 expresses.

Einstein noticed that the summation sign in Eq. 1-3 is superfluous, for the duplicate occurrence of the index k, once as a subscript and once as a superscript, suffices to give instruction for summation with respect to itself and thus permits the equation to be written simply

* Unfortunately, books on operational calculus, or Laplace transforms, are about equally divided in the adoption of p or s for the transformed equation, but the majority of the papers on electric machine theory are written in terms of p.


with the understanding that it is to be summed on k. As so often happens when some simplification in nomenclature is adopted, new concepts are born. Certainly, in the way Eq. 1 4 was arrived at, ej is a particular voltage depending on the value assigned to j, and ik is a particular current depending on k, and Zjk is a particular impedance depending on the choice of j and k. But may we not now regard Eq. 1 4 as the embodiment of all the n voltages, all the n currents, and all the n2 impedances ? If so, Eq. 1 4 vastly extends our concept of a voltage, a current, and an impedance as not single quantities but entireties associated with the whole n-branch system or network. Thus ea is only a component of ej, ic is only a component of ik , and Z,, is only a component of Zj,. On this basis Eq. 1 4 may be written in matrix notation (for convenience showing only three coordinates) :

The arrows on the matrices indicate that the rows of Z j , are to be multiplied by the column of ik. (The rules and procedures for matrix manipulation are given in the next chapter.) Notice that voltages are written as rows and currents as columns, corresponding to the subscript and superscript convention. Later on, these will be referred to as covariant and contravariant tensors of rank one.

Much of the engineering literature is already saturated with the dyadic notation

e = Z * i in which e, Z, and i, written in boldface type, have the same significance as the corresponding matrices in Eq. 1-5, and the dot product indicates the same arrow multiplication as in Eq. 1-5.

1-4. The Equations of Constraint. Equations of constraint express the conditions imposed by the interconnections of the network or the relationships which exist between the various currents and voltages. In circuit theory these conditions are usually given by Kirchhoff's first and second laws :

X (currents a t a junction or node) = 0 X (voltages around a circuit or mesh) = 0 (1-7)

But as a by-product of the idea of invariancy of power, under a transformation of coordinates in an electrical network, i t will be shown in Chapter 3 that one of the two laws of Kirchhoff (either one) is redundant. From the point of view of tensor analysis it is usually sufficient to set up relationships


among the currents (or voltages) only, although hybrid relations may sometimes have to be established. Numerous cases will be dekeloped in this book by way of illustration.

Usually, but not always, interconnection of the branches results in the need for fewer currents to describe the performance of the network. Thus mesh currents may replace branch currents, or currents flowing in one direction may be substituted for currents flowing in opposite directions, or one current at a junction may be replaced by the algebraic sum of other currents

Fig. 1-2. Examples of substitutions of variables ("transformations of coordinates").

a t that junction in accordance with Kirchhoff's law, and so on. In any event, a set of linear relationships can be formulated to express the relationship between the original (or "old") branch currents and the "new" coordina.tes. In setting up these relationships, or equations of constraint, i t is sufficient to show on the connection diagram only simple links, each consisting of a line with terminals a t its ends, since the equations of constraint are independent of the composition of the branches. Some examples of different kinds of constraints are shown in Fig. 1-2.

In Fig. 1-2a a bra,nch current ia is replaced by a branch current im' flowing in the opposite direction, and the equation for this constraint is

ia = -im'

pig. 1-2b shows a junction of three currents ia, ib, ic . Obviously, two currents ib' and ic' are sufficient to describe this situation, since

ia = ib' + ic'

ib = jb'

ic = p'


and, in general, the number of k' terms is usually less than the number of k terms. Fig. 1-2c shows a mesh composed of four links in a network. Replac- ing the branch currents ia, ib, ic, id by the loop current i", iP, t', is, i', there results

i" = i& - ii"

i b I i& - iLJ

ic = j& - iY

i d I i d - j e

Fig. 1-2d shows an ideal transformer in which the sum of the ampere-turns is balanced (negligible magnetizing current). Accordingly, the secondary current may be expressed in terms of the primary current by the turn ratio:

Another transformation is the substitution of symmetrical components (9, il, 3) for phase quantities (ia, ib, iC) by the relationships

And a final example is the substitution of direct, quadrature, and zero components ( id , iQ, z?) for phase quantities (ia, ib, ic) in the two-reaction theory of synchronous machines :

ia = i0 + id cos 6 + iQ sin 0

ib = i0 + id cos (I3 - 120") + ia sin (I3 - 120")

iC = i0 + id cos (0 + 120") + iQ sin ( 8 + 120")

There are, of course, innumerable other ways in which new coordinates may be substituted for old, but they all reduce to a set of linear equations of the form

ia = ~',a,ia' + Ci,ib' + . . + CE,~" '

These equations may be expressed in tensor notation as

The C$ coefficients may be numericallj. equal to -1 as in Fig. 1-2a, +1 as in Fig. 1-2b, f 1 as in Fig. 1-2c, or they may be complex numbers as in the


symmetrical component substitution, or sines and cosines as in the synchronous machine example. Or they may be much more complicated. It is clearly evident from Eq. 1-8 or 1-9 that

This is the transformation tensor. I t expresses the relationship between the old coordinates or variables and the new coordinates or variables. The transformation tensor is the mathematical model of the connection diagram. The study of its characteristics and behavior might be said to be the essential feature of tensor analysis itself. Much of this book will be devoted to its consideration. In order properly to develop the study of the transformation tensor, it is necessary first to explore and make available certain mathematical tools. Accordingly, the next chapter provides the essential elements on matrices. A separate chapter on the definitions and manipulative procedures of tensor analysis is also included, and the necessary ideas of this field are further developed along with their application to circuits and machines. I t will be found that they evolve quite naturally and painlessly and that no great amount of mathematical knowledge is necessary.

1-5. A Few Topological Con~iderations.~?~* I n the analysis of networks and electric machines i t is sometimes advantageous to use mesh networks and a t other times to use node networks, depending on which method of attack involves the least number of variables, the ease with which constraints may be recognized, and the known characteristics of certain elements. For example, it is usually better to handle vacuum tube circuits as node networks. I t is therefore desirable to know a t the onset how many branches, nodes, meshes, and junction pairs there are in a network. Let

number of branches (which can be counted) number of junctions or nodes (countable) number of subnetworks, which are not physically connected to one another but may be inductively coupled (countable) minimum number of independent non-redundant meshes or closed circuits in the network for which each branch will be traced a t least once minimum number of junction pairs (any two junctions in a subnetwork constitute a junction pair) for which each junction is included a t least once

Then these concepts are related by the two equations

B = M + P = M + J - S (1-12)

* Superior numbers refer to publications in which ideas discussed in this book first appeared. See the Bibliography at the end of the book.


Hence, given any network comprising S independent subnetworks, with a total number of branches B and junctions J, the minimum number of independent non-redundant meshes M is

M = B + S - J (1-13)

If M < P, the network can generally be more easily analyzed as a mesh network; while if M > P, it can generally be more easily analyzed as a node network. Eqs. 1-11 and 1-13 thus furnish simple criteria for choosing the

Fig. 1-3. A network.

type of analysis. Complicated networks may necessitate an appeal to the concepts of all-mesh or all-junction pairs or hybrids. These will be considered a t the proper time.

There is also the concept of the open mesh, which is simply any unclosed circuit traced through the network from one junction to some other junction.

As an illustration consider the network shown in Fig. 1-3. This network may be part of a much larger system from which it has been detached for purposes of analysis in much the same fashion as portions of a structure are isolated in the theory of statics and analyzed as free bodies. In this network there are

S = 3 subnetworks (I, 11, 111) B = 14 branches (1, 2, . . . , 10; 11, 12, 13; 14) J = 12junctions(A, B , C , D , E , F , G , H ; L, M , N ; Q )

Notice that in subnetwork I11 the two junctions of branch 14 coalesce into a single junction Q by virtue of the short-circuiting jumper. Alternatively, the jumper could have been regarded as an impedanceless branch.

Art. 1-61 SOME PRELIMINARY CONSIDERATIONS

Then, by Eqs. 1-11 and 1-13,

P = 1 2 - 3 = 9 junction pairs

M = 14 + 3 - 12 = 5 meshes (a, p, y, 6, a)

One choice of the fire meshes is shown as (a, p, y , 6, a), but any other five independent meshes may be selected. Similarly, any nine independent pairs of the junctions may be taken as the nine junction pairs. At each junction in a network a relationship (Kirchhoff's first law) may be written between the currents a t that junction. (J - S) such non-redundant relationships, or

TABLE 1-1

Equations

How established

Equations and

quantities remain

invariant n form upon

Example

The most simple representative

physical system

Scalar

From physical principles

Aggregations of any number of

simple systems of the same type for

a particular reference frame

with detined axes or coordinates

Matrix

First generalization

postulate

Substitution of matrix for scalar

quantities

Transformation to other reference

frames of the same type

Transformatior to other reference frames of

different types

Matrix

Second generalization

postulate

C arbitrary

Tensor

Third generalization

postulate

Transformation by means of a transformation

matrix and a law of transformation

e' = Z' . i' aik

C?, = - a aij'

Transformatior by means of a transformation tensor and a

law of transformation

e j , = Zj,k.ik'

equations of constraint, may be written for the network. Hereby any (J - S) currents may be expressed in terms of the remaining currents.

1-6. Generalization Postulates. Kron focused attention on the generality of the tensor method through the introduction of various postulates.

First generalization postulate. The method of analysis and form for an %-degree system of any complexity are the same as for a 1-degree system of the same general type, provided each quantity is replaced by an appropriate matrix.

Second generalization postulate. A matrix equation true in one reference frame remains invariant in form upon transformation to a new reference frame of the same type.


Third generalization postulate. A tensor equation true in one reference frame remains invariant in form upon transformation to a new reference frame of the same or a different type.

The procedure to be followed in passing from simple to more complex systems and interconnections is outlined in Table 1-1.

1-7. Types of Networks6 A network consisting of n coils or branches will have k meshes and ( n - k) junction pairs, the selection of which is to a certain extent arbitrary.

A network in which n voltages e, are impressed in series with each coil and in which response currents i" flow in the coils is called a mesh network. However, not all these currents are independent, and i t is sufficient to replace them by k non-redundant mesh currents ik . In setting up the k equations, the n impressed voltages e, are replaced by k mesh voltages.

A network in which n currents Ia are impressed across each coil and in which response voltages E, appear across the coils is called a junction-pair network. However, not all these voltages are independent, and it is sufficient to replace them by (n - k) non-redundant junction-pair voltages E,. In setting up the (n - k) equations, the n impressed currents Ia are replaced by (n - k) junction-pair currents.

A network in which n voltages e, are impressed in series with each coil and n currents I" are impressed across each coil, while response currents ia flow in the coils and response voltages E, appear across the coils, is called an orthogonal network. However, not all the response quantities are independent, and i t is sufficient to replace them by k non-redundant mesh currents ik and (n - k) non-redundant junction-pair voltages E,.

Whether a network should be solved as a mesh, junction-pair, or orthogonal network depends upon the relative number of meshes and junction pairs, and the types of variables it is required to determine. Some networks (rotating machines, transmission lines, magnetic circuits, multiwinding transformers) are inherently mesh networks. Other networks (vacuum tube and dielectric circuits) are inherently junction-pair networks. Each type of network has its own equation of performance and its own special methods of handling. The three types of networks, the corresponding primitive networks, the number and kind of impressed and response quantities, and the equations of performance are indicated in Table 1-2. Each of these types will be dealt with in detail in subsequent chapters; i t is unnecessary here to do more than call attention to their existence.

A coil may be regarded as two or more coils in series by introducing one or more junctions (Fig. 1-4a).

A junction may be regarded as a branch by stretching it into a zero impedance coil (Fig. 1-46).

A coil voltage may be regarded as a junction-pair voltage by inserting a junction (Pig. l 4 c ) .

Art. 1-71 SOME PRELIMINARY CONSIDERATIONS

TABLE 1-2

The Three Types of Networks

Mesh I Junction Pair

n Impressed voltoges ea n Impressed currents Ia

(e, , eb , - . ., e, I (I", r b , . .., I , )

k Response currents i k In-k) Response voltages E,

l i p , i q / ( E n , Ec , Ef I

ej = zjk i k

k Equations

I" = YYYE"

In-k) Equations

TABLE 1-3

Orthogonal Network

n Impressed voltages ea

n Impressed currents Ia

k Response currents i k

v-k) Response voltages E,

B a ea + E,= ZCB l i + I I

0 r

i a + ~ ' = y a B ( e + E ) B B

fl Equations

I Mesh I Junction Pair

Mesh Coil (or branch) Impressed voltage ei Response current i k

Impedance zjk Transformation tensor

(Kirchhoff's first law) C:, Equation of performance ef = zj,ik

Junction pair Junction (or node) Impressed current I" Response voltage EV Admittance Yuu Transformation (inverse) tensor *

(Kirchhoff's second law) C: Equation of performance I" = YY"E,

I Transformation equation P = C$ik' I Transformation equation a E, = C;E,*

MATRICES, TENSORS, AND CIRCUITS

Fig. 1-4.

TABLE 1-4

Covariance Resistance Magnetic flux

Permeance Magnetic flux density Inductance Elastance Voltage

Kinetic energy Series Short circuit

Contravariance Conductance G ~ B Dielectric flux

(or charge) Qa Reluctance pap Magnetizing force (rnrnf) ha Reciprocal inductance Capacitance Cap Dielectric flux density

(displacement) Da Potential energy V Shunt Open circuit

A r t 1-81 SOME PRELIMINARY CONSIDERATIONS 15

A junction-pair voltage may be regarded as a coil voltage by closing

through a zero impedance branch (Fig. 1 4 ) .

1-8. Duals6 There is a certain duality or parallelism in the methods of for mesh and junction-pair circuits, and a correspondence between

the quantities and concepts associated with these two methods. Quantities

having corresponding roles are called duals. The value of recognizing such

dua,ls lies in the possibility of using identical processes of reasoning and procedures in the analysis of dual systems. For the two types of networks

there is the duality shown in Table 1-3. Numerous other duals are encountered in physical systems; examples

are given in Table 1-4. The dual character of Nature is manifest everywhere, and indeed there is

no physical quantity which does not have its dual.

PROBLEMS

1-1. How many branches, junctions, junction pairs, subnetworks, and meshes does the delta-quadruple zigzag transformer of Fig. 5-8 have? Would you analyze this transformer as a mesh or junction-pair network?

1-2. Show that Eq. 1-10 follows from Eq. 1-8 or Eq. 1-9 if the CF, are independent of the currents.

1-3. Con~pile a set of duals for mechanical, optical, and thermodynamical systems.

1-4. Much of this book is concerned with developing methods and procedures for analyzing a complex system by first breaking it up into its component parts, finding the solution for each part in terms of its own terminal conditions, and then interconnecting the parts into a composite whole and obtaining the over-all solution. Consider a single-phase power system comprising a generator, sending- end transformer, transmission line, receiving-end transformer, and motor. Sup- pose that the A, B, C, D constants for each separate part of this system are known and thus that the relationships between the terminal voltages and currents for each part are given in the form

I, = CE, + DI,

Obviously, once the terminal voltages and currents for any part are known, the internal behavior of that part can be determined. For example, the voltage and current at any distance z from the receiving end of the transmission line may be found in terms of its terminal voltages and currents by the well-known formulas

E = E, cosh Z/%z + I, dm sinh d E z

I = I, cosh Z / E x + E,Z/= sinh d E x


Using subscripts 1, 2, 3, 4, 5 for the A, B, C, D constants of the generator, transformer, line, transformer, and motor respectively, find the generalized constants A,, B,, C,, Do of the complete system such that

El = AoE5 + B$5

1, = CoE, + D0I5

The philosophical concepts involved in this problem do not differ in reality from those which will be invoked repeatedly in this book.

MATRICES

2-1. n-Way Matrices. Tensor analysis, in its grand concepts and broad generalizations, may be regarded as a strategic mathematical tool. But, for handling the routine operations involved in its applications-addition, subtraction, multiplication, and inverses-matrix algebra is indispensable. This chapter is concerned with the matrix algebra necessary for our purposes. Matrix algebra was developed for the systematic solution of sets of simultaneous equations and for effecting transformations of variables.

A matrix is a rectangular (not necessarily square) array of numbers (real, complex, constant, variable, or operational) with which are associated certain rules of procedure or operation. In books on mathematics such an array is usually enclosed by brackets or parentheses:

But in electric circuit theory i t is highly desirable to know a t all stages of an analysis to just what circuit element any particular number belongs. Thus, if expressions (2-1) represent an impedance matrix, there is nothing in this notation to indicate to what branch or mesh of the circuit the impedance h might belong. I n his writings on this subject, Kron devised a-far more explicit matrix notation which tells a t a glance to what circuit element every constituent belongs.

In Kron's "box notation," voltages and currents are represented by 1-way matrices :6-8

These I-way matrices might just as well be shown as column matrices rather than as row matrices, depending on the use to which they are to be put.

17

18 MATRICES, TENSORS, A N D CIRCUITS [Ch. 2

Impedances and admittances are represented by 2-way matrices :

A 3-way matrix may be shown as a cube:

. - a b c -

Y

For manipulation purposes a 3-way matrix can be used only by breaking i t up into 2-way matrices. This can be done in many different ways, for example :

1. By taking horizontal slices along the x-axis so that

Ax,, = Aauz + Abuz + Acvz

2. By taking vertical slices along the y-axis so that

Ax,, = A,, + Ax,, + Axcz

3. By taking vertical slices along the z-axis so that

Azuz = Azua + Am, + Axvc

4. By taking slices in any other fashion with no requirement that the slices be similar or equal or symmetrical-single cells, columns, rows, double columns, and so on, can all be segregated by appropriate use of the indices.

For 4-way and higher-rank matrices no pictorial symbols are feasible (or necessary), and they are specified in terms of their 2-way slices. The way in

Art. 2-21 MATRICES 19

I t has the unique property, to be proved later, that any matrix multiplied by it remains unchanged.

A matrix having all its elements equal to zero is called a null matrix, and it has the property that any matrix multiplied by i t is reduced to zero.

which higher-rank matrices are handled will be demonstrated as they are encountered.

A 0-way matrix is a number which has no reference to any coordinate system-a simple constant of multiplication, for example.

An n-way matrix has kn cells or constituents, although in practice many of these may be unfilled or zero. The way in which a matrix is filled generally offers considerable information about the type of physical system involved and leads to systematic classifications.

Since a matrix expressed by a single symbol like e or i or Z or A represents not one quantity but a whole set of quantities, there follows the conception of set. A 2-way matrix represents a 2-way set. The idea of set is of great importance in modern mathematics.

A 2-way matrix has all its diagonal elements equal to unity and all other elements equal to zero; it is called the unit matrix or idemjactor I or Kronecker delta 6 :

a b c

2-2. Addition and Subtraction. Only matrices of the same dimensions and with the same indices may be added. Addition or subtraction then consists in adding or subtracting corresponding (same indices) components. Thus

a b c a b c a b c L=JI i2=ml A & D B ~ E C & F

1 i f j = k

= ( 0 i f j # k (2-7)

a b c a b c a b c

0

0

1

I It often happens, however, that two matrices with different indices need to be combined into a single matrix. It is then imagined that both matrices

0

1

0

a

% = I = b

1

0

c o 7

20 MATRICES, TENSORS, A N D CIRCUITS [Ch. 2 Art. 2-31 MATRICES 21

possess all indices, but that the elements of the non-present indices in each matrix are zero. For example,

a b c a b c a b c

This idea is habitually employed in adding the impedances of one network to those of another network to give the over-all impedance of the combined network.

I n matrix algebra the commutative law applies :

A + B = B + A = C (2-8) or, in tensor notation,

u v w . . . u v w . . . - uvw. . . AXVZ ... + BXVZ ... - CXVZ ... (2-9)

and the rank of c;::: is the same as the ranks of A:;:::: and PVW-.. .,,.. , that is, all have the same superior indices and the same inferior indices.

2-3. Multiplication of Matrices. Multiplication of a matrix by a -

constant simply multiplies each element of the matrix by that constant:

Notice that in the tensor notation this is obvious. Multiplication of two 1-way matrices corresponds exactly to the dot

product of vector analysis; that is, corresponding elements of the two matrices are multiplied and added. The result is a scalar or 0-way matrix. Thus

I' = s . i = l l mI= eiia + ebib + ecic (2-1 1)

In tensor notation, remembering the summation convention,

When an index, like k , occurs in a product once as a superscript and once as a subscript, it is called a dummy index and it is the instruction for summation. In a tensor product a dummy index can never occur more than twice. Dummy indices can be replaced by any other dummy indices :

since, no matter how they are designated, they mean the same thing-sum- rnation. Matrices of any rank are multiplied according to the rule

where the dummy index y vanishes in the result, and therefore the h a 1 matrix always has two less indices than the sum of the number of indices in the factor matrices of the product. In actual practice, only 1- and 2-way matrices need be multiplied:

The arrows indicate that the rows of Zjk are to be multiplied by the column of ik in accordance with the tensor notation. The sum of the ranks of Zjk and of ik is 2 + 1 = 3, and the rank of their product e j is 2 less than this, or 1.

The product of two square matrices is

the dots showing the positions vacated by the dummy indices and the arrows showing that the rows of Ajk are to be multiplied by the columns of Bk" and added:

Of particular interest is the multiplication of a matrix by the idemfactor :

22 MATRICES, TENSORS, AND CIRCUITS [Ch. 2

Of course, this is easily seen from the tensor notation

Ajk6k = A j a S k + Ajb& + . . + Ainz6: + - - + A j n S Z = A j , (2-16)

since all 6: = 0 for k # nz, while 6::; = 1.

2-4. Inverse of a 2-Way Matrix. The inverse of a 2-way matrix is analogous to division. Only 2-way square ma,trices have inverses. Consider a system of simultaneous equations :

If this system of equations is solved for the i k by determinants, then, according to Cramer's rule,

in which, for any k and any j,

y k j = cofactor of Z j , (2-1 8) determinant of all the Z j k

For example, if the impedance matrix is

then its inverse (the admittance) tensor is

and D is the determinant:

I zca zcb zcc 1 -zaazbczcb - zcczabzba

The steps in finding the inverse of a matrix then are:

1. Interchange rows and columns (take the transpose). 2. Replace each element by its minor. 3. Alternate the signs of each minor starting with (+) in the upper left-

hand cell. 4. Divide each new element by the determinant of the original matrix.


Notice that only a square matrix can have an inverse, for a rectangular (singular) matrix is equivalent to a square matrix having a row (or column) of zero elements, whose determinant would then be zero. This would correspond to trying to solve a set of simultaneous equations having either more variables or less variables than the number of equations.

As an example let

Z= mj 4

for which D = 30

1. Interchanging rows and columns gives the transpose:

z, = R?j 2

2. Replacing each element of the transpose by its minor: RI I

3. Alternating the signs,

4. Dividing by the determinant,

y = z-1 =

2-5. Some Definitions and Relationships. A diagonal matrix has elements only along the diagonal, all other elements being zero.

(Ajk = 0 for j # k)


A symmetrical matrix is symmetrical about the main diagonal:

A skew symmetrical matrix has equal elements of opposite sign about the main diagonal of zero elements:

Obviously, A j j = -A j j = O since only O is equal to its own negative. Any matrix can be expressed as the sum of a symmetric and a skew-

symmetric matrix by

symmetric skew-symmetric

Ajk = *(Ajk + Akj) + W j k - Akj) = * ( A + At) + * ( A - At) (2-22)

The transpose of a product of two matrices is equal to the product of the transposes of the two matrices :

The inverse of a product of two matrices is equal to the product of the inverses of the two matrices :

Dyadics and matrices do not obey all the laws of algebra. In these notations,

Distributive law: A(B f C ) = AB & AC

Associative law: ABC = A(BC) = (AB)C Commutative law: A + B = B + A

AB # BA

~ r t . 2-71 MATRICES 25

~ u t in tensor notation the commutative law with respect to products is obeyed :

AjkBkm = BkmA jk

since the indices suffice to indicate the products which are to be taken. The conjugate of a matrix A which is a function of complex variables is a

matrix A* whose elements are the conjugate complex variables; that is,

if A = f(x + jy) then A* = f(x - jy)

2-4. Differentiation. A matrix of any rank is differentiated with respect to a parameter by differentiating each of its elements with respect to that parameter :

a - a aA(e) U u w . . .

ae A:;::::@) = - ae [A(%)]:::::: = x Y z... (2-25) For example,

0 sin 6 cos 0

A matrix of any rank is differentiated with respect to a I-way matrix according to the equation

That is, each element of Az:::: is differentiated in succession with respect to each element of xS and the resultant matrix has a rank higher by one. For example,

a b c a b c

sin fl cos xp 6

a B 7 -- -sin xp

2-7. Compound Matrices6 In simple problems involving relatively few variables, the corresponding matrices have few components and generally can be handled as they are. But, in more complicated problems involving many variables, the matrices are of large dimensions, and their routine multiplication, taking of inverses, and the like, become a formidable task. To alleviate this situation matrices are subdivided, or partitioned, into smaller matrices, and rules of manipulation are establishsd for dealing with


such compound matrices. For example, the matrix equation e = Z i may be partitioned as follows.

a b c d e a b c d e P 4

in which (p , q ) are compound indices each of which embraces several individual axes of the original matrices. Thus p = a, b and q = c, d, e. The partitioning is quite arbitrary and the resulting submatrices may be either square (Z1 or Z,) or rectangular (Z , and Z, or i , and i ,).

The rules for manipulating compound matrices follow. Transpose. If A, B, C, . . . are matrices of a compound matrix, the

transpose is

For example, the transpose of the previous impedance matrix is

Products. The product of two compound matrices is taken in the same way as the product of simple matrices, particular care being taken to keep


the order of multiplication of the components in proper sequence. For example,

Note that along the arrows the compound indices ( r , s) must be the same and that R . D cannot be substituted for D R in the final product.

2-8. Solution of Matrix In the matrix representation of a system of simultaneous equations, let the matrices be split in the following fashion.

k 4 v /- - ;[ml;

= p Z9k ZPC 290 a

2 " k Z", ZU"

Hereby the set of equations

has been broken up into sets of equations :

ea = e, + es + e , iS = ik + +a + iv

where e, = Zjkik + Zjaia + Z,,iv

e, = Z,,ik + Z,ia + Z,iv

e, = ZUkik + Z,,ia + Z,,iv

in which (j, k = 1, . . . , n,) , ( p , q = n, + 1 , . . . , n,), and (u , v = n, + 1, . . . , n,). Obviously, the splitting up into parts could be carried as far as desired rather than for only three splits, as in the illustration above. Eqs. 2-28 may be solved for any set of currents, say iv from the last set:

i v = Yvue, - YZIUZUkik - YvUZUaiq (2-29)

in which YvU is the inverse of Z,,. Then, inserting this value of iV in the first two sets of Eqs. 2-28, there results

(e, - Zi,Y"ue,) = ( Z j k - ZjvY " V u k ) i k + (Z ja - Z,,Y ""Z,,)ia

i (e, - Z,,Y ""e,) = (Z, , - Z,,Y ""Z,,)ik + (2, - Z,,Y ""Z,,)iq


These equations may be rewritten

ej = Zjkik + Z~, iQ

e i = Z&ik + Z k i q

Solving for iq from the last equation,

and substituting in the other equation, there results

ey = (ej - Zj,Y'Qpe;) = (Zjk - Z ~ q Y ' a p Z ~ , ) i k = Zykik (2-31)

Thus the original set of equations (2-28) has been reduced to a single equation in terms of the current set ik , where

Then iQ follows from Eq. 2-30, and iv from Eq. 2-29. This procedure is the matrix or tensor equivalent of the ordinary step-at-a-time solution of a set of simultaneous equations.

If a set of equations is reduced one a t a time, there is no need to take the inverses of matrices. Thus, if the last of Eqs. 2-28 represents a single equation, then in Eq. 2-29 YVU = l /Zuv since Zuv would then be a number and not a matrix.

PROBLEMS

2-1. Given the three matrices a b c

i = b

Find :

Note carefully how these various operations check and cross-check one another. 2-2. Given the matrix equations (in which each member is a matrix)

MATRICES 29

Eliminate i3 and reduce to two matrix equations. Then eliminate i2 and reduce to one matrix equation. Check this first result by Eq. 2-31 in the text by eliminating i2 and i3 in one step.

2-3. Show that the inverse of a diagonal matrix is a matrix each of whose elements is the reciprocal of the corresponding element in the original matrix. Remember from this that, if a matrix can be reduced to diagonal form before an inverse must be taken, the work is enormously simplified.

2-4. Consider the five simultaneous equations represented by the numerical matrices given in Art. 2-7. (a) Eliminate the c- , d-, and e-axes one at a time. ( b ) Eliminate the c-, d-, and e-axes in one step. Compare the amount of work involved in the two processes of elimination.

2-5. Given the compound matrices

Find A . B, B . A, A t . B, A t . A. 2-6. Prove that multiplication of a matrix by a scalar is equivalent to multiply-

ing each of its elements by that scalar. (Use the rule for the addition of matrices.) 2-7. Prove that the transpose of a product of two matrices is equal to the

product of the transposes of these matrices. Extend the proof to the case of n matrices.

2-8. Prove that the commutation law holds for diagonal matrices. 2-9. Prove that the inverse of the transpose is equal to the transpose of the

inverse of a matrix. 2-10. Prove that the inverse of the product is equal to the product of the

inverses of two matrices. Extend the proof to the case of n matrices.

TRANSFORMATION OF COORDINATES

3-1. Forms. The terms of a tensor equation include different types which may be classified according to their form. The idea of form is one of the essential concepts of tensor analysis. Indeed, tensor analysis may be described as that branch of mathematics which is concerned with

a set of equations invariant to a group of transformations

and the preservation of an invariant form is of primary importance. A relationship involving the variables xT is called an algebraic form, and

one involving the differentials dxT is called a differential form. Forms often encountered are listed in Table 3-1.

TABLE 3-1

Linear Quadratic Bilinear Linear set Bilinear set Multilinear

A quadratic form can always be expressed in terms of the symmetric part (Eq. 2-22) of its 2-way matrix, for

This particular relationship is frequently used in machine theory. In a tensor equation certain forms remain invariant, that is, unchanged,

after a transformation of coordinates.

3-2. Transformati~n.~-~ The concepts employed and the steps in the formulation and solution of tensor equations are illustrated by the analysis for a mesh network :

1. Set up the canonical equations or equation of performance (Eq. 1 4 ) (a "set" of simultaneous equations).

e, = Zjkik 30

Art. 3-21 T R A N S F O R M A T I O N O F C O O R D I N A T E S 3 1

2. Establish the equations of transformation or constraint (Eq. 1-9) (also a "set").

k .k' ik = Ck.z (3-3)

3. Specify the invariancy of power (the power or energy in the network is the same regardless of the particular set of variables-currents and voltages- used in its description).

P = e.if = e .,$' . . e.Cf.,ij' = e.,ij' 3 3

.'. e,. = C:,ej ( 3 4 )

. . e . = C!'e., 3 3 (3-5)

Evidently the voltages transform in an inverse fashion from the currents, since Cr is the inverse of C$.

4. Preserve the invariancy of form in the equation of performance, in passing from the "old" variables e,, ik to the "new" variables e,., ik'.

= Zjjktik'

5. Solve for the new currents by multiplying by the inverse Yk'j' of the impedance matrix and using the Kronecker delta.

~ k ' i ' ~ ~ , - - yk'i'zjrm,im' - 6k' 'm' - - ik' (3-7) 6. Find the old currents in terms of the new (Eq. 3-3).

'k - ~k , ik ' = c k , y k ' Y e j , 2 - k k (3-8) 7. Find the old voltages (Eq. 3-2).

e . 3 = Z . 3k ik = ZZjkC;, yk'i'ej, (3-9) Thus all quantities have been found in a routine fashion by making use

of the concepts of "sets," "group of transformations," and "invariancy of form," and by using the rules for matrix multiplication and for taking inverses. The seven steps outlined above are always sufficient.

Since the currents are transformed (Eq. 3-3) directly by C,k,, whereas the voltages are transformed (Eq. 3-5) by the inverse tensor CT, currents are called contravariant vectors and are designated by upper indices, while voltages are called covariant vectors and are designated by lower indices. An impedance transforms directly as a doubly covariant tensor and therefore has two lower indices. An admittance transforms inversely as a doubly contravariant tensor and therefore has two upper indices. I n general, a mixed tensor of any rank, having both upper and lower indices, transforms with respect to its upper indices as a multiple contravariant tensor, and with respect to its lower indices as a multiple covariant tensor. It is t o be noted


that the transformation tensor CE. is itself a contravariant tensor in one of its reference frames and a covariant tensor in its other reference frame.

It is worth noting here that we did not employ Kirchhoff's laws explicitly. Instead an equation of transformation, or transformation tensor, was established which related the old and the new currents. In effect, this corresponds to an application of Kirchhoff's first law only. The second

Fig. 3-1.

Kirchhoff law is not necessary, because step 3, expressing the invariancy of power or the law of conservation of energy, replaces it.

In dyadic notation the seven steps are

e = Z . i i = C . i r e' = C t o e Z 1 = C t . Z * C i . i = C . Y ' . e r e = Z . C . Y 1 . e '

As an example, consider three branches (a, b, c) as shown in Fig. 3-la and let the voltages, currents, and impedances be

The equation of performance in terms of the "old" variables is e = Z i.

Art. 3-21 TRANSFORMATION OF COORDINATES 33

Now suppose that the coils are connected as shown in Fig. 3-lb, and let - - - new currents ia and iP be selected arbitrarily. In terms of these new currents the old currents are

a B

Thus the transformation tensor C, a singular or non-square matrix, is established. Since the determinant of a singular matrix is zero, its inverse cannot be found.

The new impedance matrix is found in two steps:

The inverse of the new impedance matrix is

Pf,-I = ;l;~;l The new voltages are

where, from the connection diagram, it is seen that e, + e , = e, - e , =

= 105, the applied voltage. The new currents then ars

34 MATRICES, TENSORS, A N D CIRCUITS

The old coil currents are

i = c . i ' = o=g - 1 - 8

and the voltages across the coils are

e = Z . i = mj [=B - 8 - 43

Thus all voltages and currents, in both the old and new reference frames, have been determined. More complicated circuits are solved by exactly the same procedure, but the matrices become larger and their manipulation more cumbersome; yet the steps are the same and they proceed in the same orderly sequence.

3-3. Group P r ~ p e r t y . ~ - ~ It is sometimes necessary or desirable t apply several transformations in succession, rather than to transform dire from the primitive system to a new system. For example, suppose that transformation tensor, together with all currents, voltages, and impedance has been set up for a network connected in a certain way, and i t is the required to make some minor change in the connections which would involv respecifying some of the currents. Instead of starting all over again, a transformation tensor can be established to show the new substi involved. A complete transformation involving any number of changes ma then be arrived a t in succession as follows :

Zay = c:zaBcBB,

z,,,~. = c$za.,.c,8: = C$C;Z~,C#C$ = c2.z 4 cB. B

za,,,,- = czrz a a"8" ~ 9 , = c<rcai~a.z c6cpkK = c>z CJ%, a a a a B B 8 8 aB B

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " , c> = Qpc>C:

This is a group property of the transformation tensors. Successive changes may be occasioned by: adoption of a new set of

currents; imposition of a constraint (such as ignoring the exciting current in a transformer) ; substitution of symmetrical components; interconnection of networks; and so on. Illustrations will be given throughout this book.

1 Art. 3-41 TRANSFORMATION OF COORDINATES 35

It will suffice at this point to give a simple example. In Fig. 3-2 first let currents ia, iB be selected as the new currents so that

u B

Then suppose it is decided to use mesh currents iU, iv. The transformation between the (a , /3) and the (u, v) sets is

Then the over-all transformation tensor is u v

and this is verified by direct comparison of the ia, ib, iC and the iu, iv currents.

3-4. Interconnection of Subnetworks. Electrical systems consist of separate parts-generators, transformers, transmission lines, motors, loads, etc.-and each part has its own set of internal currents, voltages, and impedances and can be analyzed individually in terms of the conditions a t its terminals. But the connections joining them together impose constraints. It is perfectly possible to set up an impedance matrix for the entire system, including the impedance of all the parts, and then to set up a massive transformation tensor to take care of all the interconnections, both internal and external, of the separate subnetworks or parts. But this procedure might very well become prohibitive because of the amount of work involved, and most certainly i t would not take advantage of the work previously done in arriving a t the interconnections and performance equation of each subnetwork. A much more profitable procedure takes into account only those changes imposed by the interconnections between subnetworks. In Pig. 3-3 are shown a number of subnetworks whose internal impedances and internal transformation matrices are

MATRICES, TENSORS. A N D CIRCUITS [Ch. 3

Fig. 3-3.

B F

Fig. 3-4.

Now let these subnetworks be interconnected, having a totality of coordinates

(a, B ) = an (j, k) + all (p, q) + all (u, v) + . . . (a', B') = all (j', kt) + all (p', q') + all (u', v') + - . .

Then the impedance matrix of the whole network before interconnection of the subnetworks is

Z,.,? = ZjPk' + Z,.q! + Z,',' + . . . = c;,Zjkc,", + c;,z,,c,., + C ~ ? Z , , C ~ ? + . . .

Next set up the transformation tensor C:: representing the external interconnections. The impedance matrix of the entire network is

Za"f = C"Z,.~.C~":

= c$[z,.*' + z,.,. + Z ,.,, + . - qcj: (3-11)

= c,~'[c,!z,~c$ + C$Z,C: + G'$Z,,C~, + . - .]G'g; (3-12)


Thus, if the new coordinates enjoined by the external interconnections are included in the impedance matrices of the subnetworks, Eq. 3-11 is used directly, but, if i t is first necessary to convert the impedance matrices of the subnetworks to the new coordinates, then Eq. 3-12 must be used.

EXAMPLE? Fig. 3 4 shows two subnetworks (I and 11) carrying coil currents ia, ib, iC and if, ig, ih, respectively. Switches S1 and S, are to be opened and the subnetworks connected by wires BF and AG. It is required to determine the impedance of the combined network.

The first step is to select new currents for each subnetwork, (ia', ib') for I and (if', ig ' ) for 11, and in terms of these new axes to find the new subnetwork impedances.

The combined network impedance is given by the compound matrix

The transformation matrix for the interconnection of the two subnetworks, in terms of new currents ia", ib", ig", is

38 MATRICES. TENSORS, AND CIRCUITS [Ch. 3

and the new impedance becomes

Alternatively, a transformation matrix between the original and final axes can be set up :

a" b" g" a b c f g h

and then

and

and the two methods check.

3-5. Driving Point and Transfer Impedances. A network of any complexity having internal applied voltages and any number of external leads may be reduced to an equivalent network entirely in terms of the external impressed voltages and response currents on the external leads, and therefrom the driving point and transfer impedances may be determined. Fig. 3-5 shows an enclosed n-branch network with external leads (a, 6, c, . . .). It is assumed that each external lead has an impedance (which may be equal to zero), and that the enclosed network is made up of any arrangement of impedances and impressed voltages. Let indices

( r , s ) represent the external lead axes a, b, c, . . . , (j, k) represent the internal axes inside the network, (a , /3) represent all axes, both external and internal.

Then the impedance matrix may be partitioned:


Now select new coordinates, but among them include all the active external axes. The remaining new axes may be selected as convenient. I n terms of these new axes set up a transformation matrix, remembering that the (r , s) axes are the same in the new as in the old reference frames; thus

Fig. 3-5.

Then the new impedance and voltage matrices take the form

r ' j'

e , = 1 a;., + c:,e, I c;,er + q . e j I = (3-16)

Hence

and upon elimination of the iv currents there results

40 MATRICES, CIRCUITS, A N D TENSORS [Ch. 3

which may be condensed to

The latter equation is again partitioned so as to segregate the a"-axis,

(P,!I) # a :

Now, if all voltages except e,.. = ea are put equal to zero, that is, short- circuited, and the iQ" currents are eliminated, there results

and this defines the driving point impedance at terminal a.

Fig. 3-6.

The transfer impedance between two terminals a and b is found by putting all voltages except ea equal to zero and eliminating all currents except ib so that finally there results an equation of the form

which defines the transfer impedance.

EXAMPLE.^ Find the driving point impedance a t terminal a and the transfer impedance a t b for the T-circuit of Fig. 3-6. The impedance matrix (Eq. 3-13) is

a b c f o


A transformation matrix (Eq. 3-14), which preserves the (a, b) axes is

a' b' c'

The new impedance matrix (Eq. 3-15) then is

a' b' c'

and the new voltage matrix (Eq. 3-16) is

Eliminating the ic' current by Eq. 3-18 gives

Putting e, = 0 and eliminating iv, there results e, = 3817/(11 x 46)ia', or the driving point impedance is 7.53 ohms and the current iv is

or the transfer impedance is -7.37 ohms.

M. Constraints.' An enforced relationship between two or more variables is called a constraint. Every constraint permits one variable or coordinate to be eliminated. But the suppression of a contravariant variable


automatically brings about the birth of a covariant variable, and conversely. Thus, when a current is reduced to zero by the opening of a branch, a voltage appears across the opening. Or, when a. velocity component in a given direction is reduced to zero by a barrier, a reaction force appears in that direction.

Kirchhoff's first law, stating that the sum of the currents a t a junction must be zero, is an equation of constraint, since it sets up an enforced relationship between the currents a t the junction. If there are J junctions in a network, a total of (J - 1) non-redundant relationships between the currents may be written, each of the form zik = ia + ib + a . - = 0.

In an iron core multiwinding transformer having windings with turns Nu, Nb, N,, . . . , all of which have the same* closed magnetic circuit, i t is generally sufficient to ignore the magnetizing current and to assume a balance of ampere-turns :

2 Nkik = Naia + Nbib + = 0

This relationship then constitutes an equation of constraint relating the several currents.

Equations of constraint are of the form

It is required to convert such an equation of constraint to a corresponding and equivalent equation of transformation. The procedure follows.

1. Write down the set of equations specifying the constraints. 2. Rearrange the equations of constraint so that a different dependent

variable appears on the left side in each equation of constraint, and reduce the right side to combinations involving only the remaining independent variables. There will be as many dependent variables as there are equat,ions of constraint. If indices (u, v) represent these dependent variables and ( r , s) the independent variables, then the equations take the form

3. The remaining independent variables are then written down. If indices (r, s ) represent these independent variables, they may be written

4. If indices (a , p) represent both the independent ( r , s) and dependent (u, v) variables, then

8'

* This precludes 3-phase windings on different legs of s, shell type and 5-leg core-type design.


On the other hand, if the equations of transformation are given and i t is desired to change over to corresponding equations of constraint, i t is necessary only to rearrange the transformation tensor in the form

and the equations of constraint are then

so that

B;, = (C; -a;,) = u[~; 1 -at, 1 EWLE.~ A 5-branch network with four junctions is shown in Fig. 3-7.

The constraints a t J - 1 = 3 of the junctions are

The remaining junction is redundant, since adding the three equations gives -iC + id = 0, the constraint a t Fig. 3-7. D. The matrix of the equation of constraint is

c a b c d e

Express a different current in each equation of constraint as a dependent variable ; for example,

ia = -ie' + id '


Then id and ie , the remaining currents, are the independent variables and the transformation matrix for the currents is

d' e'

3-7. Equivalence of Transformation Theory to Kirchhoff's Laws. The transformation theory presented in this chapter, particularly with respect to the notion of a primitive network as a reference frame, is to a certain extent

Fig. 3-8.

naive and philosophically unsound. Actually, the power consumed in the primitive network with given impressed voltages is in no way related to the power consumed in the actual network with the same impressed coil voltages. This can be easily verified in a simple case by comparing the powers consumed in the two networks of Fig. 3-8. The powers consumed are, respectively,

P = e , i l+ e2i2= 4 x 4 + 2 x 1 = 18

P' = (el + e2)i = 6 x 2 = 12

The powers of the two networks are not the same, and the stipulation of the invariancy of power on such a basis is unsound.

In order to justify the transformation theory based on the concepts of invariancy of power and invariancy of form, a different approach must be

Art. 3-71 TRANSFORMATION O F COORDINATES 45

made and, as might be expected, a rational approach will involve both of Kirchhoff's laws.

Consider only the network as actually connected and make no reference to the primitive or any other network. Let

ei = actual voltage impressed in series with coil j ik = actual coil currents

Fig. 3-9.

The impedance drop in the coil then is

v . = z . ik 3 3k ( s 2 7 )

Choose new currents iv and express the old currents in terms of these new currents by the transformation

ik = c;,jk' (3-28)

Then there exist certain new voltages ei. to be associated with the corresponding currents i f such that the power consumed by the network remains invariant to the transformation of coordinates. That is, ejt is dejined such that

Hence e,, = C:mi

= C?,z. i" 3 3k

= c;,zjkc;,ik'

= z . . ,ik' 3 k

The voltage ej, as determined by Eq. 3-31 will be made up of certain combinations, or algebraic sums, of the voltages v j , and these will define a closed circuit or mesh in the network; see Fig. 3-9. This closed circuit will include the corresponding zik associated with each v , = zikik as well as the


series impressed coil voltages e,. For this closed circuit Kirchhoff's second law requires that C (impressed voltages) = X (consumed voltages) or

where the C includes all of the terms yielded by Eq. 3-31. But this is identically the mesh voltage ej, of Eq. 3-31 and therefore, since in the summation around the mesh the ej occur in exactly the same way as the v,,

e,, = 2 ej = C' e 3 ' j

Consequently the same final results are obtained if in all equations ej is substituted for v j . Nevertheless, there is an unsoundness in doing so, for v, is not ej, Eq. 3-27 is not true if the substitution is made, and there is no justification for the so-called primitive network which e, = zj,ik represents.

The foregoing demonstration places transformation theory on a sound and proper basis, discards the erroneous concept regarding the primitive network, and shows the roles of Kirchhoff's first law in Eq. 3-28 and his second law in Eq. 3-33. Nevertheless, since the end results are correct if one proceeds from the primitive network notion, the above may be taken as justification for using it.

PROBLEMS

3-1. Verify all operations given in this chapter. Solve the circuit of Fig. 3-6 for all currents and voltages (both old and new) if e, = 4 and .eb = 3.

3-2. Eq. 3-4 does not follow from the previous equation simply by "canceling" ij' on each side of the equation. Show in detail the proper steps and justify them.

3-3. If the two networks in Fig. 3-10 are connected, find the total impedance

Fig. 3-1 0.

in terms of axes a, b, f, h by first transforming from the primitive networks of each subnetwork and then interconnecting; and then by verifying by transforming from the primitive network to the complete network.

3-4. Determine the driving point impedance for the network of Fig. 3-11. Find the currents in all branches.

3-5. Write the equations of constraint for Fig. 3-11 and set up its matrix B. What is the corresponding C?

TRANSFORMATION OF COORDINATES

Fig. 3-1 1 .

3-6. Without resorting to diagrams or sketches, and using only the concept of the transformation tensor, prove in a few lines that the arbitrary selection of rn independent branch currents in a network of rn meshes defines m independent meshes.

4

SYMMETRICAL COMPONENTS

41. Conjugate Tensors. When the elements of a matrix A consist of - complex numbers such as (a + jb), where j = 4-1 , the corLjugate of the matrix is denoted by A* and is found by replacing each complex number (a + j b ) in A by its conjugate (a - jb), that is, by substituting -j for j throughout :

a + j b o + j d a - j b c - j d

There follows immediately (A*)* = A

In a-c circuits where voltages and currents are expressed as complex numbers,

e = El + jE2 = E(cos 8 + j sin 8) = Ede (4-3)

i = Il + j12 = I(cos 4 + j sin 4) = I$@ (4-4)

the power is given as the product of the conjugate of the voltage and the current:

P + jQ = e* . i = EE-jeIEj@

= EI[cos (8 - 4) - j sin (8 - $)I Fig. 41.

= (E14 + E212) + j(E112 - EA) (4-5) in which

P = EI cos (8 - 4) = real power (4-6)

Q = EI sin (8 - 4) = reactive power, defined as positive for a leading current (4-7)

A resistance power loss is defined as

Art. 4-31 SYMMETRICAL COMPONENTS 49

For a network having e , i, and Z as matrices, the power is

Or, expressed in terms of the impedance,

4 2 . Transformation Formula. The transformation formulas for currents, voltages, and impedances have to be modified slightly when conjugates are being used. As before,

but the invariancy of power now becomes

Hence

or, taking the conjugates of both sides,

Substituting the impedance relationship,

The application of these transformations will be illustrated in the next two articles.

4-3. Symmetrical Components. Fortescue introduced a sequence operator which transforms the phase quantities (voltages or currents) existing in three separate phases to a set of three balanced quantities called zero, positive, and negative sequence quantities. The defining formulas are (for currents)

ia = (iO + i1 + i2)/ J3 ib = (iO + a2i1 + ai2)/ J3 (4-14) ic = (iO + ail + a2i2)/ J3

in which ia, ib, iC are the phase currents; iO, il, i2 are the zero, positive, and negative phase-sequence currents; and

50

Obviously

MATRICES, TENSORS, A N D CIRCUITS [Ch. 4

Actually, Fortescue's equations of transformation did not include the 1 / d 3 factor of Eqs. 4-14, but such a factor is necessary to preserve invariancy of power in the transformation. Eqs. 4-14 may be written in matrix form:

i = C . i' where

in which S O , S1, and S2 are the so-called zero, positive, and negative phase- sequence operators. This non-singular tensor, together with its conjugates, inverses, and transposes, provides a group of transforn~ations which are (as is easily verified)

Among these ten different tensors there are only two different matrices. Currents and voltages transform by Eqs. 4-1 1 and 4-12 as

Art. 4-31 SYMMETRICAL COMPONENTS

The impedance tensor transforms f i s t with respect to the current as

which gives the phase voltages e in terms of sequence currents it. Writing this impedance as

0 1 2

the completely transformed impedance becomes

If the mutual impedances are zero, Eq. 4-22 reduces to


If, now, the phase impedance has the special form

a b c

Z + aaZ' + aZ"

Z + aZ' + a2Z"

(4-24)

In general, there is no advantage in using symmetrical components rather than phase components unless the impedance matrix reduces, as above, to a diagonal form; for it is only then that zero, positive, and negative sequence currents cause only voltages of their own sequence; thus

When the mutual impedances are zero,

a b c

As special cases of Eq. 4-26 there are

a b c 0 1 2

Art. MI SYMMETRICAL COMPONENTS 53

A grounding impedance z, between a neutral and ground carries all three

phase currents, adding up to %hi0:

-- - I [( iO + i1 + i2) + (iO + a2i1 + ai2) + (iO + ail + a2i2)] = d 3 i 0 4 3

Hence a grounding coil has only a zero sequence impedance, and its transformation tensor is n

4-4. The Basic Networks in Symmetrical Cornponent~.~.~ I n problems where symmetrical components are employed, four different networks and reference frames are, or may be, involved. They are:

1 . The primitive actual network of n coils, made up of 3-phase coil groups. Its currents, voltages, and impedances are phase quantities and will be designated by ia, e,, .zap.

2 . The given actual network of k meshes, made up of 3-phase coil groups. Its currents, voltages, and impedances are phase quantities and will be designated by ia', e,., zarp,.

3. The primitive sequence network of n coils, made up of 3-phase coil groups. Its currents, voltages, and impedances are sequence quantities and will be designated by i d , e , ~ , z , . ~ ~ .

4. The sequence network having the same number of meshes and coils as the actual network but arranged in a different number of subnetworks. Its currents, voltages, and impedances will be designated by ism, en-, zamp-.

Other mixed networks involving both actual and sequence reference frames in different combinations may be encountered or set up for special purposes; for example, a network which will give phase voltages in terms of sequence currents.


Transformation tensors can be established between any two of the four principal networks, and formulas developed for transforming from one network to another in accordance with the following scheme (in which double- headed arrows represent reversible transformations, and single-headed arrows represent generally irreversible transformations).

Primitive Primitive Actual in+-+ C,"-ia Sequence ia"

(NetworJ " (Ne tworJ 1 (4428)

+ C;,iaP ( Actual ) i a * _ C$, iajn

Network Xetwork

The tensor C,$ is the Fortescue symmetrical component transformation, and i t expresses the relationship between the sequence currents in the primitive sequence network and the phase currents in the primitive actual network, or, inversely, by CE". This tensor is always known.

The tensor CE, is the ordinary transformation between the phase currents in the primitive actual network and the actual network. If the actual network has been established in terms of phase quantities, this tensor is easily found. However, it will generally have a singular matrix and, therefore, no inverse.

Usually, the sequence network is unknown, and tensor C$, the transformation between the primitive sequence and actual sequence networks, will have to be found by a special process to be shown. In general, it will have a singular matrix and, therefore, no inverse.

The tensor C$ is the transformation between the actual and sequence networks.

4-5. Changing C,"$$:"" to C ~ ~ ~ : ~ ~ ~ . . If the currents of the actual network are coil currents, then the transformation

in which the (r, s ) independent variables and the (u, v) dependent variables may be replaced by an equation of constraint,

0 = B,U.ia' or 0 = B,Uia since S' = s (4-30)

where Ba is found from C:, by subtracting the Kronecker delta from the independent variables (in accordance with Eq. 3-26). Converting to sequence currents,

0 = Bija = B:c,"-~~" = B:iaX = (~21;'" + &$'") (473l) where

S v 8" 2)" 8" v"


Now, expressing the dependent currents iv" in terms of the independent currents in" = 8:: is-, there results

Finally, changing is" to is", the transformation tensor becomes

= } independent variables sy -(D:c;. - p . ) - l ( D : C ~ - C:) } dependent variables

Fig. 4-2.

The procedure then is:

1. Arrange C,$ as a compound tensor separating the independent (r, S)

and dependent (u, v) variables. 2. Arrange CEn also as a compound tensor separating the independent

and dependent variables to agree with the first step. 3. Find C$ from Eq. 4-34. The sequence network may be found there-

from by inspection. 4. Find em- and ZaYI from eae and Za-,,".


5. Use (7,"- = C;"C,"- to establish the sequence network from the primitive actual network e, and Zap.

6. Determine actual phase currents from i"' = c > ~ .

EXAMPLE. In Fig. 4-2 is shown a %phase grounded neutral generator supplying an unbalanced load grounded through an impedance. The generator impedances are specified in terms of sequence quantities, but the load and grounding impedances are given in terms of phase values. The given data are

The combined impedance and voltage matrices referred to sequence axes are

The transformation tensor between the actual network and its primitive, and its rearrangement separating the independent and dependent variables is


The corresponding compound tensor for the Fortescue transformation is

Hence by Eq. 4-34, upon performing the several operations,

The sequence network currents are then found in the usual manner by

The sequence network impedance and voltage then are

1; 2; . 2

It is clear from the Zanrflm and earn tensors that the equivalent circuit of the sequence network is as shown in Fig. 4-3, and this may be set up on an a-c calculating board.

After the sequence currents have been found, the phase currents are

1; zafmPm = c $ . z ~ , ~ ~ " 8" =

2;

2, + Z1 + 2 2 + 3 2 , 2, + + 3 2 ,

Z o + 2 + 3 2 , 1 2, + 2, + 2 2 + 3 2 ,


The sequence voltages are

The actual phase currents are

which may be checked by ia = C:-C$iaW. The voltages across each coil are

which may be checked by ea = &".Za,.~$:ip*.

PROBLEMS

4-1. Write the conjugate of

z + j y 3 - j 4 cos 6 + j sin 6

/- 1200 ( -A2 A L ~

4-2. Confirm Eqs. 4-18 to 4-26 inclusive, together with all the special cases of Eq. 4-26.

4-3. For the example of Fig. 4-2 identify in detail each step in the analysis with corresponding Eqs. 4-29 to 4-34 inclusive, defining the procedure.

SYMMETRICAL C O M P O N E N T S 59

4-4. In Fig. 4 4 a generator with a neutral impedance supplies a load. The impedances and voltages are

Calculate the sequence and actual short-circuit currents and the voltage differences across each phase of the generator and of the load for the following conditions: (a) line-to-ground fault (switch S1 closed); (b) line-to-line fault (switch S4 closed); (c) double line-to-ground fault (switches S1 and Sp closed); (d) three- phase fault (switches S1, S2, S3 closed).

Fig. 4-4.

4-5. Repeat the preceding problem when the load is a balanced delta-connected impedance with z in each side of the delta.

MULTIWINDING TRANSFORMER CIRCUITS21-4e

5-1. Basic Def in i t ion~.~ l -~~ Multiwinding transformer circuits are unique among static networks in that ignoring the magnetizing currents reduces the number of simultaneous equations by one for each complete magnetic circuit, and this in turn leads to the concepts of "leakage impedances" and of voltage differences rather than absolute voltages in the canonical equations.

Consider two windings a and b linked by a common magnetic circuit, and having turns Nu and Nb, respectively. Select a winding of No turns as the "reference winding." Let

Nu turns of winding a n =-= Nb n, = - a No turns of winding 0 No

zaa = Raa + jwLaa = self-impedance of winding a

zab = zba = jwMab = mutual impedance between windings a and b

Then, ignoring the magnetizing current, and writing the voltage equations

Multiplying Eq. 5-2 by (lln,) and Eq. 5-3 by (lln,), and subtracting, and dividing by naia,

= leakage impedance between windings a and b referred to reference winding 0 (5-4

60

Art. 5-11 MULTlWlNDlNG TRANSFORMER CIRCUITS 61

It is evident from the construction of Eq. 5 4 that the voltage and current terms involved are of the form

- ea - - % No = (volts per turn)% na Na . N ia (ampere turns in winding) nuta = Q- =

No No that is, it is as though both windings had No turns.

Now consider a multiwinding transformer with (n + 1) windings numbered from 0 to n inclusive, and let winding 0 be selected as the reference winding. Let indices ( j , k) range from 1 to n inclusive. Then, ignoring the magnetizing currents,

.k O = i" + nkik or i0 = -nkz (5-5) and

e. 3 = z. 30 i0 + zikik = (zjk - nkzj0)ik (5-6)

eo = zooiO + zokik = (zOk - nkzm)ik (5-7)

Subtracting n j times Eq. 5-7 from Eq. 5-6 and rearranging,

Comparing the terms in parentheses in Eq. 5-8 with the definition of a leakage impedance as given by Eq. 5 4 , i t is seen that these terms are leakage impedances, and thus Eq. 5-8 may be rewritten

in which 8 , is the branch impedance matrix. This same equation may also be arrived a t by using matrices and trans-

formation tensors as follows: Let indices (u, p ) range from 0 to n inclusive. Thus Eqs. 5-5, 5-6, and 5-7 are replaced by

e, = za8iS (5-11) in which

0 1 2 ... n . I

. . . 0 zoo 0 1 ! 0 I 20.

(5-1 2) =

n zno znl Z,P . . . Z,"

7


The magnetizing current is eliminated by the transformation

ia = Citk where

1 2 . . . n

The new voltage then is

and the new impedance tensor becomes

Eqs. 5-15 and 5-16 agree with the f i s t line of Eq. 5-8. The advantage of using the final specification of C: in Eq. 5-14 rather than the entire matrix expression is evident.

5-2. Windings in Series. I n Fig.

;a1 : 6' 5-1 is shown a transformer having I I - - - primary windings numbered from 0 to

m, of total turns N in series carrying current iO, and secondary windings numbered ( m + 1) to n of total turns N'

I 1 ' ' 1 I in series carrying current in. Let indices . -

(j, k) range from 1 to m, and indices (r , s) range from (m + 1) to n.

Select new currents ia' and ib' flowing through the primary and secondary

I I I I windings respectively, and put I

N = N 0 + N 1 + . - . + N m

= total primary turns

N' = Nm+, + . . + N , N N ' = total secondary turns

Fig. 5-1. n = NIN'

= turn ratio

Art. 5-21 MULTlWlNDlNG TRANSFORMER CIRCUITS

The balance of ampere-turns requires that

Nia' + N'ib' = 0 or ib ' - -N ;a1 = -nial N '

The transformation tensor then is a'

Ci, = 1 for all (j, k)

c r - - a' - n for a11 (7, 8 )

The voltage is then transformed as

e,, = C;,(e, - n,eo) = Ci.(ej - njeo) + C;,(e, - VO)

= (el + e, + . . + em) - (nl + n, + . . + nrn)eo

- n(ern+, + . . . + en) + n(nrn+, + . . . + nn)eo

where E = (eo + el + . + em) is the total primary voltage and E' = (em+, + . . + en) is the total secondary voltage.

The impedance matrix before transformation is

= zjk f Zjs + Zrk + (5-19) and i t transforms to

z,.,. = C,",Cj.Z,, = C;.Cpjk + c ; * c ~ , Z j s + c;,c;,z,, + c:,c:,zrs = (Z,, + Z1, + . . + Zlrn + . . + Zmm)

- n(Z1(,+,) + . . . 21, + ' ' ' + Zmn)

- n(Z(m+l)l + ' ' ' + Z(rn+~)m f ' ' ' Znm)

+ n2(Z,m+l)(m+l) + . . . + Z(rn+l)n + . . . + Znn) (5-20)

In the case of a single primary winding No and two secondary windings Nl and N, in series (m = 0, n = 2) , the impedance reduces to

Za,B, = n2(Zll + 2Zl2 + Z2,)

This formula is much used in transformer design for the calculation of reactances.


5-3. Windings in Parallel. Pig. 5-2 shows two independent windings 0 and 1, and ( n - 1) windings 2 to n in parallel with a common load impedance Z(,+l,(n+l, . Let indices (j, k ) and ( a , p) both range from 1 to n inclusive. Then the current in+' flowing in the load impedance Z(n+l,(n+l, is eliminated by the connection tensor whose components are

c; = 6; ct+l = (6: - g,"+ l ) (5422)

where g;+l = 1 for all a. The "new" voltages are

The transformed impedance becomes

4

Fig. 5-2. Windings in parallel.

or, rearranged as a matrix, putting Z(n+l,(n+l, = Z ,

1 2 . . . n

The new equation is then e, = Zapip

which can be solved for the current iB upon taking the inverse YPa of the impedance

= YBae,

Art. 5-41 MULT lWlNDlNG TRANSFORMER CIRCUITS 65

For example, if there are four windings ( n = 3 ) and the turns of the second and third windings are equal, n2 = n3, the equations become

el - nleo = Zllil + Z12i2 + Z13i3

-n2eo = Z2,i1 + ( Z 2 2 + Z ) i 2 + ( Z 2 3 + Z ) i 3

-n2eo = Z3,i1 + ( Z S 2 + Z ) i 2 + ( Z 3 3 + Z ) i 3 (5-26)

5-4. Coupling Windings. Coupling windings are used to reduce reactances and stray flux, and to transfer ampere-turns. A multiwinding transformer with a number of coupling windings is shown in Fig. 5-3. It

Fig. 5-3. Coupling windings.

is convenient to number each pair of coupling windings by an odd and an even number, (2k + 1) and 2k, respectively. The transformation tensor which eliminates the currents in the odd-numbered coupling windings (2k + 1) has components for k = 1, 2 , . . . , n as follows :

Since e,, = e2,+,, the "new" voltages are

If the turns are equal for each pair of coupling windings, the only surviving voltage term is

(el - nleo) (5-29) The "new" impedance is


For example, a transformer with two pairs of coupling windings, and equal turns on each winding of a pair, would be represented by the equations

5-5. Load Ratio Control Circuit. The connection diagram of a load ratio control transformer in the off- ratio position is shown in Fig. 5 4 . The transformation tensor for eliminating the common winding current

r i1 is

Fig. 5-4. Load rat io control transformer. -u The branch impedance matrix of Eq. 5-9 is

1 2 3

The new impedance then is

2

The voltage, after putting E = el f e2 = el + e3, is

Art. 5-61 MULTlWlNDlNG TRANSFORMER CIRCUITS

The admittance matrix is the inverse of Zap or

The currents then are

In simple problems of this type the tensor method does not compare favorably with ordinary methods of solution. I t is easier and simpler to identify the circuit constraints and insert them direct,ly into the canonical equation^.^^

5-6. Forked Autotransformer. A forked autotransformer is shown in Fig. 5-5. The ampere-turn balance and the current constraints require that

Therefore, since i2 = 4 3 ,

The transformation tensor, which eliminates the current in the common winding, is

3 n

The old and new impedance matrices then are

1 3

( 5 4 1 )

3 1 0 0 ~ 1 Fig. 5-5. Forke-d autotransformer.


The new voltage is

which is, of course, simply the line voltage referred to the load side.

E7. Group of Transformers. The connection diagram for a three- winding transformer, I, with regulating units consisting of an exciting

Fig. 5-6. Group of transformers.

transformer, 11, and a series transformer, 111, is shown in Fig. 5-6. The transformation tensor and primitive impedance matrix are

Art. 5-81 M U L T l W l N D l N G TRANSFORMER CIRCUITS 69

The new impedance and voltage are

= -(n,n;n; + nz)eo + n;n;(el - eb) + n;(e; - e;)

= -(nln;n; + n,)e, ( 5 4 6 ) since

el = eh and e; = eg

5-8. Wye-Delta-Zigzag Transformers. The connection diagram for a wye-delta-zigzag transformer with a load on the zigzag winding is shown

Fig. 5-7. Wye-delta-zigzag.

in Fig. 5-7. There are four windings on each magnetic circuit, but each "electrical" leg involves, through the zigzag, two "magnetic" phases (legs). As this is typical of all phase-shifting connections, some care is required in setting up the transformation tensor. Let (1, a , a2) be the three roots of unity, so that

i l = -i2" - - -a2i2 or i 2 = - a i l = & 4

The transformation matrix and its conjugate for the "magnetic" phase, therefore, are

3 4 3 4

* and C; =


The new voltage is 3 4

2 3 4 * Ch(e, - njeo) = 'm 1 el -lnleo 1 e2 - n2eo 1 e3 - n3eo 1 el - a2e2

3

4

3 4

- - e3 - n3e0 (nl - a2n2)e, ( 5 4 8 )

The primitive impedance is 1 2 3 4

1 2 1 1 2 . 2 ' 2 1 3 I I

and the new impedance becomes 3

Fig. 5-8. Delta-quadruple-zigzag transformer connections.

5-9. Delta-Quadruple-Zigzag Transformer. Quadruple-zigzag transformers are used for power supply to 12-anode mercury arc rectifiers. The diagram of connections under short-circuit conditions is shown in Pig. 5-8.

Art. E-91 M U L T l W l N D l N G TRANSFORMER CIRCUITS 71

Let the windings be numbered as shown in Fig. 5-8. For simplicity assume perfect interleaving of the windings so that there are only three branch impedances :

In terms of these impedances the primitive impedance matrix is

The transformation matrix, in terms of the 120" phase-sequence operators (1, a, a2), is

1 2 3 4


The new voltage matrix then is

since nl = n2 = n, = n4 = n, n5 = n , = n , = n8 = n', and from Fig. 5-8 it is clear that el - ae, = 0, e2 - a2e8 = 0, e4 - ae, = 0, e3 - a2e5 = 0. The new impedance matrix is

1 2 3 4 I

1 z - az" - a%" + z' I z - az" - az" + a%' z - az" - az" + a%' z - az" - a%" + z'

and since the new voltages are el = e4 and e2 = e,, while in the impedance matrix the coefficients of i1 and i2 are the same, respectively, as those for i4 and i3, only axes 1 and 2 need be retained, so that

where the 2 is required to compensate for the ignored currents i3 and i4.

1 An. 5-91 MULT lWlNDlNG TRANSFORMER CIRCUITS

I The admittance matrix is

a ( l - a)[zn t - z"(n + an') + z'na] ( a - l ) [ z n T a - z"(an + n') + z'n] - - 6(zz' - z " ~ ) eo

(5-59) Hence

- z"a2(an + n') + z'na2] eo 6(zz1 - zV2)

The impedance as measured from the primary terminals is

(5-61) The ratio of currents is

i1 zn' - z"(an' + n) + z'na - - - = AEje i2 -znr + z"a2(an + n') - z'na2

Thus il and i2 are of unequal magnitude and out of phase by an angle 0 if both resistance and reactance are present. If the resistances are zero, then A = 1, but 6 may have any value.

It is interesting to check the preceding equations by the use of compound matrices. Let the various matrices be written in terms of the following indices: (j, k) = 1 , . . . , 8; (r , s) = 1, . . . , 4; ( p , q) = 5 , . . . , 8. Then


and, carrying out these operations, Zap is confirmed. Similarly, for the voltage,

PROBLEMS

5-1. Verify and carry out in detail the calculations for each case given in this chapter. In this regard it is highly instructive to show the various transformation tensors, such as Eqs. 5-22 and 5-27, in matrix form.

5-2. Set up the transformation n~atrices and the performance equations for the stub delta connection of Fig. 5-9.

Fig. 5-9. Stub delta.

5-3. Set up the transformation matrices and the performance equations for the extended delta connection of Fig. 5-10.

Fig. 5-10. Extended delta.

5-4. Set up the transformation matrices and the performance equations for the inscribed delta connection of Fig. 5-11.

Fig. 5-1 1 . Inscribed delta.

MULTIVELOCITY TRAVELING WAVES22945.47

61. Differential Equations of a Line. A multiconductor transmission system having n incoming lines and n outgoing lines connected through a transition network is shown in Fig. 6-1. Incident voltage and current waves (e,, i') approach the transition point and give rise to reflected waves (ei, i'3 and transmitted waves (ef, inT). It is required to set up the equations for these waves.

I waves waves waves

I Fig. 6-1. Multiconductor transmission system.

The electrostatic charges &Z accompanying the incident waves, expressed in terms of Maxwell's potential coefficients p,,, are given by

1 from which

where KT8 is the inverse of the matrix psT. These coefficients are calculated for parallel cylindrical conductors in the presence of ground by including the images of the conductors in the ground surface.

The magnetic flux linkages are given in terms of the inductance coefficients LTS by

q$ = LTsiS (6-3)

1 The currents flowing to ground and between conductors, including the effects of both leakage and corona, are given by Grse,. And, ha l ly , the resistance drops in the conductors due to the currents are RTsiS.

I 75


Hence the differential equations for the multiconductor transmission system become (putting p = a/at in the Heaviside sense)

Eliminating iq, there results

in which 6: is the Kronecker delta and

J," = (Lmp + Rm)(KqSp + Gas) (6-7)

Now, if the losses are ignored (R,, = 0, GrS = 0), then Eq. 6-6 is satisfied by the traveling wave function

in which a closed index (ci) is used to suspend temporarily the summation convention with respect to cc. Substitution of Eq. 6-8 in Eq. 6-6 gives

Since this equation must be satisfied fok waves of the same velocity, and since p2f(a, = v&f& and (a2/ax2)f(,) = f&,

(Lr,$'9& - 6:)~:' = 0 (6-10)

In order that this set of equations have other than trivial solutions the determinant of the coefficients must vanish; that is,

and this condition then determines the velocities q,,. To each root of Eq. 6-1 1 there correspond n values of a(,"), and any (n - 1)

of them may be determined in terms of one value taken arbitrarily. Let this one arbitrary value be taken as a?) = 1. Solving Eq. 6-10,

where s f 1, r f 1, and (bi81 is the determinant Ic;'I with row s = 1 and column r = 1 deleted, and A: is the cofactor of b: in 1 bi81.

Art. 6-21 MULTIVELOCITY TRAVELING WAVES 77

The complete solution then becomes

in which Yra = KrSa;vca,

Eqs. 6-13 and 6-14 show that there are on each of the n conductors n pairs of forward- and backward-traveling waves and each of these has its own particular velocity v(,, .

In the event that the zero-potential plane for both the voltage and current images is the ground surface, and, if there is no corona, all the L,, and KTS are related so that in Eq. 6-11

where v is the velocity of light. Then i t may be shown22 that the expansion of the determinant (Eq. 6-11) gives v(,, = v and all velocities are thus equal to the velocity of light, and the sum of waves of equal velocity becomes

a,"ja(x - vt) = f,(x - vt)

so that Eqs. 6-13 and 6-14 reduce to

e, =f,@ - 4 + FAX f vt) (6-1 7)

ir = YTS[fS(x - vt) - Fs(x + vt)] (6-18)

Eqs. 6-17 and 6-18 are the traveling wave equations for a multiconductor single-velocity transmission line, in which f(x - vt) represents a forward wave and P(x + vt) a backward wave. In practice, the backward-traveling waves develop as reflections of the forward waves a t a transition point.

6-2. Transition Points. In the general case of Fig. 6-1 any number of incoming lines terminate a t a transition point consisting of an interconnected network and any number of outgoing lines. When incident waves on the incoming lines reach the transition point, currents will flow in the network, transmitted waves will move out on the outgoing lines, and reflected waves will start back on the incoming lines. In temor notation, let

z,, = surge impedance of the incoming lines yTS = surge admittance of the incoming lines (inverse of z,,) zuv = surge impedance of the outgoing lines yuv = surge admittance of the outgoing lines Zj, = branch impedance of the transition network CE, = transformation tensor specifying the total interconnections of the

network and outgoing lines


Then

ZaB = ( Z j k + z,,) = impedance of network and outgoing lines before interconnection

Za,B, = C$C$,ZaB = impedance after interconnection of the network and outgoing lines

Now ZaTB, may include branches other than those connected to the incoming lines. The open-circuit branches will have been eliminated by Ci,, but the grounded branches other than those connected to the incoming lines will have to be eliminated by the substitutions

S !l

Zayp = z,, + 2, + z,, + 2, = (6-19)

E, = Z,p + Z,Za (6-20)

The impedance matrix has been partitioned in Eq. 6-19 so as to separate the axes (r , s) connected to the incoming lines from the axes (p, q) not connected to incoming lines. From Eqs. 6-20 and 6-21, upon eliminating the short-circuit currents 1 9 , there results

where E, and I 8 are the total voltages and currents on the incoming lines at the transition point.

Now let (e,, i r ) and (ei, i") be the incident and reflected waves, respectively, on the incoming lines. Then the resultant voltages E, and currents I' a t the transition point are the sum of the incident and reflected waves, or


First the impedance matrix for the transition network and the outgoing lines is set up :

i k u '19

zoo ju - 211

Fig. 6-2.

In the second step a transformation matrix is set up, including among its new axes all the branches connected to the incoming lines, that is, the (T , S) axes a and b,

i k u v

Then the new impedance matrix is i r + i f , = (6-24) so that

e, + ek = Z;,(iS + i t s ) = ZisySt(et - e',) (6-25)

The total voltage E, and the total current I+ follow from Eq. 6-23, the remaining network currents from Eq. 6-21, the network voltages from Eq. ' (

agpl Za,s. = Cg,ZasC$, = b'

6-20, and so on; thus all system quantities may be determined.

EXAMPLE. As an example of the application of the general equations p ( cP -2.1 -%a RZ + 2.2 previously derived, consider the 2-conductor transmission system shown in Pig. 6-2 in which incident waves e,, e, impinge on a transition network Rl, In this new impedance matrix the (a', b') axes are connected to the incom- R, and give rise to reflected waves e;, e; and transmitted waves e:, e,". ing lines, but the c'-axis is not, and therefore i t must be eliminated by splitting

80 MATRICES. TENSORS, A N D CIRCUITS [Ch. 6

the matrix as specified by Eq. 6-19 and then using Eq. 6-22. Splitting according to Eq. 6-19,

27, = afml b' zva=;;cl a' b' C'

Z, = q - 5 q - l - Z z a Z, = c' R, + z,, y a p = cr - n R, + z2a

and Eq. 6-22 becomes El

E, B= (R1-B R, + 222 m)d

The admittance matrix of the incoming lines is

y.' = ( ~ ~ $ 1 5

6-25 relatine. the reflected and incident voltage waves then is


Since the equations are becoming unwieldy in algebraic form, i t will serve our purpose just as well to pass to numerical values for illustrating the procedure. Let

1 y a p = -

500

and Eq. 6-26 becomes

Solving for the reflected voltages, gives

The total voltages on the incoming side of the transition point are

The reflected current waves are

The total currents on the incoming side are


The impedance of the network and outgoing lines is

Hence the currents are

The branch currents are

And, finally, the transmitted voltage waves are

Thus all incident, reflected, and transmitted voltage and current waves and all network currents have been de$ermined in a routine fashion.

PROBLEMS

6-1. Confirm all equations in the chapter, and verify in detail every step in the derivations.

6-2. Why must Eq. 6-11 hold for non-trivial solutions? 6-3. If Eq. 6-15 holds, show that all v(,) are equal to the velocity of light. 6-4. Numerous examples and solutions of transmission line networks will be

found in texts such as the author's Traveling Waves on Transmission Lines.= These examples can be assigned as problems in tensor analysis and the results checked against those in the reference text. Actually, the methods of tensor analysis do not present any advantage over the solutions by simpler methods, but of course the problems provide exercises in the tensor method.

6-5. Write out the set of Eqs. 6-10 and show the determinant (Eq. 6-11) for three conductors (r, s = 1, 2, 3). Then solve for the a?) as in Eq. 6-12 and write out the set of Eqs. 6-13 and 6-14.

7

JUNCTION-PAIR NETWORKS6

The previous chapters treated mesh networks. The essential steps in the analysis of mesh networks in earlier chapters were :

1. Setting up an equation of performance, e = z . i , for the primitive mesh network.

2. Establishing the equation of transformation, or equations of constraint (corresponding to Kirchhoff's first law), relating the "old" currents to the "new" currents by a transformation tensor.

3. Finding the equation of transformation for voltages by invoking the invariancy of power.

4. Finding the equation of transformation for impedances by preserving the invariancy of form.

5. Determining the new currents, using the inverse of the impedance. 6. Finding the old currents with the aid of the transformation tensor. 7. Finding the coil voltages.

The analysis of junction-pair networks follows the same operations, step for step, upon the substitution of the corresponding dual quantities. For a comparison of three methods of analysis see Table 8-1, p. 102.

Consider n separate coils having admittances YUv (including both self and mutual values), and across each coil let a current I" be impressed, flowing into one junction and out of the other. As a consequence, a response voltage E,appears across the coil. The equation of performance for all the coils is

Iu = YUvED (7-1) If the coils of Fig. 7-1 are now inter-

connected in some fashion so that the new network consists of J junctions (or nodes), then P = (J - 1) independent junction pairs may be established. Any two junctions may be selected as a junction pair; the minimum number of non-redundant junction pairs is (J - 1) and every junction is included a t least once. Let the "new" voltages between

83

Fig. 7-1. The primitive junction- ppir network.

84 MATRICES, TENSORS, A N D CIRCUITS

the junction pairs be designated by E,,. Kirchhoff's second law permits (J - 1) linear relationships to be set up between the voltages E, of the primitive network (Fig. 7-1) and the voltages E,, of the new network. These relationships define a transformation tensor C c , so that

E , = CE'E,,

Stipulating the invariancy of power between the primitive and new networks, there is

P = E,IU = E,,IU'

Substituting Eq. 7-2 in Eq. 7-3,

C: 'E , , I~ = E,P' from which

I,' = c;'p Substifuting Eqs. 7-1 and 7-2 in succession in Eq. 7-4,

I,' = c;' p " ~ , = c;' y U v c ; ' ~ , , = ~ " ' v ' E , , in which

p ' v ' c;' y,vc;'

is the transformed admittance. Eq. 7-5 may be solved for the "new" voltages in terms of the junction-pair

currents by taking the inverse of the new admittance tensor:

E,, = ( p '" ' ) -q* ' = ,,,,I"'

The voltages across the individual coils follow from Eq. 7-2, and the currents flowing in these coils from Eq. 7-1. Thus all currents and voltages may be determined in routine fashion.

7-1. The Selection of Junction Pairs. In the analysis of a junction-pair network of J junctions i t is mandatory to select a minimum of (J - 1) independent junction pairs, even though currents may be impressed acros a fewer number of junction pairs, because unknown response voltages occ across all junction pairs and must be recognized in setting up the trans formation tensor. This is the counterpart of selecting as many independen meshes as there are unknown currents in a mesh network.

After the (J - 1) junction pairs have been selected, the transformatio tensor is established by assuming new voltages E,, for these junction and then equating the old voltages E , across each coil to the sum of jun pair voltages around closed circuits for which the coil junctions are termin Alternatively, equations of constraint may be set up for short-cir junction pairs in terms of the new junction-pair voltages. The proc will be clarified in the examples to follow.

EXAMPLE.^ In Fig. 7-2 is shown a network having six coils and fo junctions, and therefore (J - 1) = 3 junction pairs. Currents Id and I f ar impressed across two of the coils.

Art. 7-11 JUNCTION-PAIR NETWORKS 85 Select, arbitrarily, three junction pairs such that every junction is included

a t least once, and designate the new junction-pair voltages by Ea., E , , E,.. The voltage between any pair of junctions, that is, across any coil, is readily found as the sum of the voltages around closed circuits through the two coil

Ed'

( a ) Actual network ( b ) Coil voltages ( c ) Junction-pair voltages

Fig. 7-2.

junctions, as shown in Fig. 7-2c. A direct comparison between the coil voltages (Fig. 7-2b) and the junction-pair voltages (Fig. 7-2c) defines the transformation tensor.

E, = Ed' - E,? - E,,' . Ed = - E d' d

E, = E,, + E,? e E, = E,. - Ed.

f 1~ - 1

Let the primitive coil admittance tensor be


The new admittance matrix is

YU'V' = c:' Y U V ~ " ' 0

The old and new currents are

When these values of I"' are inserted on the diagram of connections there results the distribution shown in Fig. 7-3, from which i t is seen that the

original currents Id and P are correctly given by the superposition of the new junction-pair currents. If the inverse of t,he admittance tensor i s

then the junction-pair voltages follow from Eq. 7-7, the coil voltages from Eq. 7-2, and the coil currents from Eq. 7-1.

Fig. 7-3. Superposition of junction- EXAMPLE OF CONSTRAINTS.~ A net-

pair voltages. work in which certain junction pairs are short-circuited is shown in Fig.

7-4a. The voltages across individual coils are shown in Fig. 7 4 b . Since, there are four junctions, there are three junction pairs, and these may be selected arbitrarily so as to include each junction a t least once, as shown in Fig. 7 4 c , where new junction-pair voltages Em,, E,,, and E,, are shown.

There are two ways in which the transformation tensor may be set up: either by introducing equations of constraint specifying the short circuits, or by writing the tensor from inspection of the diagram of connections.

Using first the method of constraints, let as many new junction-pair voltages E,., E,,, E,., Em,, En, be selected as there are coils. In terms of these voltages, as seen from Fig. 7-4c, the short-circuited junction pairs

Art. 7-11 JUNCTION-PAIR NETWORKS 87

furnish two equations of constraint, permitting En. and E,. to be eliminated; thus

En, = 0

-E, . + Ea, + E,, + Em. = 0 :. E,. = E,'+ Er# + Em,

( a ) Actual network (b) Coil voltages ( c ) Junction-pair voltages

Fig. 7 4 . Circuits with constraints.

Expressing the "old" coil voltages in terms of the remaining variables Egt, Erj, Em,, there results

q' r' m'

Es = Ear + Em*

E , = E,, = Ea, + Em, + E,.

But the same tensor may be written directly from Fig. 7 4 by observing from the connection diagram that En. = 0 and - E , + E,. + Em, + E , =

0 ; hence

ED = Em# E , = E,. + Em, + E,.

Let the primitive admittance tensor be

MATRICES. TENSORS, AND CIRCUITS [Ch. 7

The impressed currents are I g and I t , and therefore the new junction-pair currents are

a' r' m'

.. - ...- - -

coordinates, it may be desirable for some reason or other to restate them in terms of a different set of junction pairs. Of course, there must be as many junction pairs in the new set as there were in the old, and each set of junction pairs must include each junction in the network a t least once. The procedure in setting up the appropriate transformation tensor to change from one set of variables to another consists simply in writing the equations which express

For example, in Fig. 7-5 are shown two sets of junction-pair voltages, (Ea,, Eb,, Ec,, Ed,, Ee,) and (Ea-, E,", R,, Ed.., E,"), for a network with six junctions. It is clear from the diagram that

a" b" c" d" e" !

E,, = Ea- - E," 4 ---

b' d' - - -Ea" - LC,- + hd" + he" 1- E - - E d - - E," e '

e' -

7-3. Interconnection of Networks.6 When two or more networks a d to be interconnected to form a combined network, two procedures are possible.

1. The primitive admittance and current tensors for the individual coib of the entire combined network may be set up, the junction pairs selected, a transformation tensor established, and the analysis completed. This

0---

7-2. Change of Variables.6 After the equationsof performance for a network h n v ~ heen established in terms of a particular set of junction-pair

the old junction-pair voltages in terms of the new:

E," = C$Eut

E,, = -Ea" + E," + Ed" + E,"

E,? = Ea-- E,- uq or CU' = c' + E," - Ed" - E," ---

- - - - , "

Art. 7-31 JUNCTION-PAIR NETWORKS 89

procedure ignores the identity of subnetworks already established and simply treats the final combined network as derived directly from its over-all primitive.

2. The admittance and current tensors of each Ebl

subnetwork as already established are combined to form the compound tensors

ydv, = ydv' + y;W + y;'vl + . . . 1

I,' = 1;' + I,"' + 1;' + . . . Each tensor is assumed to poseess all the nates for the combined network but has zero elements except for the specific coordinates of its own Eel

subnetwork. Then a transformation tensor is set Fig. 7-5. Substitution up to express the interconnection, and the com- of variables. bined admittance is calculated from

EXAMPLE.^ Consider the interconnection of the networks of Fig. 7-2 and Fig. 7 4 , as shown in Fig. 7-6. The compound admittance is the sum of the - -

subnetwork admittances, as previously found.

Select new junction-pair voltages E,, E,, Ed. so that a" 6" d"

h 1 I 1

90 MATRICES, TENSORS, AND CIRCUITS

The current tensor for the combined network is

[Ch. 7

I I Fig. 7-6. Interconnection of two subnetworks.

and the admittance matrix follows in the usual fashion as

PROBLEMS

7-1. The use of the principle of the invariancy of power under a transformation of coordinates, or for a change in the connections of a network, was justified for a mesh network in Art. 3-7. In a similar fashion prove that the use of the principle is justified for a junction-pair network.

7-2. Superimpose the new current IU' on Fig. 7-4 and show that they combine to give the actual currents at the original junctions.

7-3. Confirm all calculations indicated in the chapter. 7-4. Prepare a tabular form of three columns to illustrate the correspondence

(or duality) between the analysis of mesh and junction-pair networks. In the first column list the items: impressed variables, response variables, circuit parameters, equation of performance, equation of transformation and transformation matrix, invariancy of power, transformation of the response quantities, invariancy of form, transformation of the circuit parameters, new response variables, old response variables, old coil variables. I n the second column show the corresponding steps in the analysis of mesh networks. In the third column show the corresponding steps in the analysis of junction-pair networks.

7-5. Consider a network in the form of a hexagon with six junctions having a coil between each pair of junctions (15 in all). Let junction-pair currents be

JUNCTION-PAIR NETWORKS 9 1

applied between junctions 1 and 2 and between junctions 3 and 5. Make a complete junction-pair analysis of this network, finding all response voltages.

7-6. Solve the network of Fig. 7-7 and indicate on the diagram all coil voltages and currents. The mutual admittances are all zero.

Fig. 7-7.

8

COMPLETE NETWORKS6

& I . The Variables of Complete Networks.6 Every network of n coils comprises m meshes and (n - m) junction pairs, and in general there is a voltage applied in series with each coil and a current applied in shunt across each coil. Then m response currents ik' flow in the meshes, and (n - m) response voltages E , appear across the junction pairs. Therefore there are

n applied series voltages ek

n applied shunt currents IU

m mesh currents i k '

(n - m) junction-pair voltages E,.

In pure mesh networks there are n known impressed series coil voltages e,, the shunt currents Iu are zero, the junction-pair voltages Eu, are of no concern, and the m mesh currents ik' are related to the coil currents ik by a singular transformation tensor. I t is sufficient to solve for m unknown mesh currents ik' in terms of m impressed mesh voltages ek, from the equation of performance,

e., = z., ,ik' 3 3 k

A voltage in an impedanceless branch (z = 0) is treated as a voltage in series with a zero impedance coil, and each such voltage adds anadditionalreference axis, or mesh, to the network.

In pure junction-pair networks there are n known impressed shunt coil currents I U , the series voltages ek are zero, the mesh currents ik' are of no. concern, and the (n - m) junction-pair voltages E,. are related to the coil shunt voltages Eu by a singular transformation tensor. It is sufficient to solve for (n - m ) unknown junction-pair voltages E,, in terms of (n - m) impressed junction-pair currents I,' from the equation of performance,

A current across an admittanceless coil ( z = co) is treated as a current in shunt with a zero admittance coil, and each such current adds an additional reference axis, or junction pair, to the network.

92

Art. 8-21 COMPLETE NETWORKS 93

In complete networks (or what Kron calls "orthogonal networks"), series applied coil voltages ek and shunt applied coil currents 1" coexist, and consequently m currents ik' flow in the meshes and (n - m) voltages E,. appear across the junction pairs. No restrictions are placed on which voltages or currents are known or which are to be solved for, but there are a total of n unknowns: for example, the m mesh currents and the (n - m) junction- pair voltages. Under these conditions a non-singular transformation tensor can be established.

When a branch with a mesh current ia (a contravariant variable) is open- circuited, that current is reduced to zero and in its place appears a junction- pair voltage E, (a covariant variable). Or, conversely, when a junction-pair

Fig. 8-1. Replacement of open and closed meshes by apparent coils.

voltage Ea is short-circuited, that voltage is reduced to zero and in its place appears a mesh current ia. Thus the total number of variables is not changed by the imposition of constraints; there is only a substitution of one type of variable for another. If, however, one type of variable assumes a known value along an axis, there will appear in that same axis the dual type of variable as an unknown quantity.

An open mesh across which appears an unknown voltage Ea may be regarded as closed through an impedanceless apparent coil through which flows a zero current ia. Similarly, a closed mesh through which flows an unknown current ia may be regarded as opened across an admittanceless apparent coil in shunt with which is a zero voltage Ea. These possibilities are indicated in Fig. 8-1.

8-2. The Three Types of Terminals. In any network, three types of adjacent-terminals should be recognized :

1. Terminals between which there is a coil of impedance zkk in series with an impressed voltage e, carrying a coil current itc) and bridged by an


impressed shunt current Ik and a voltage E,. Such a general element is shown in Fig. 8-2a. It is to be noted that the current through the terminals is

From the point of view of the remainder of the network, this is the current contributed by the series-shunt coil element, and it is the current which is to be transformed. Evidently, for each such local circuit,

ej + E j = zjkifc, = zjk(ik + Ik) (8-2) 2. Terminals between which there is an admittanceless branch (that is,

no coil but an open circuit), and across which there is impressed a known junction-pair current iv and a response voltage E, as shown in Fig. 8-2b.

( a ) Coil element ( b ) Known current ( c ) Known voltage in an across an open impedanceless branch branch

Fig. 8-2. The three types of adjacent-terminal pairs.

The purpose of designating the known current iv rather than P, in spite of the fact that it is a junction-pair current, is to reduce the complete network to an all-mesh network, as will be shown. Closing the circuit through an external Ev introduces an additional mesh in the network.

3. Terminals between which there is an impedanceless branch, and through which there is impressed a known series voltage E, and a response current iU, as shown in Fig. 8-2c. The purpose of designating the known voltage E, rather than e,, in spite of the fact that it is a branch voltage, is to avoid the introduction of apparent (z = 0) coils in the network.

8-3. Procedure in Analysis of Complete Networks. The procedure in setting up the impedance and transformation tensors and the equations of performance for a complete network, regarded as an all-mesh network, follows.


1. Close all (n - m) junction pairs through impedanceless (z = 0) apparent coils. The (n - m) arbitrarily selected junction pairs must include all admittanceless branches across which known currents are impressed and also all impedanceless branches having known impressed voltages; that is, the selected junction pairs in the network must include all the apparent coils. In order to avoid the inclusion of apparent coils in the primitive impedance, all known impressed quantities associated with these apparent coils must be considered known response quantities iv and E,. The following nomenclature is adhered to:

known voltages in series with coils are denoted by ek

known voltages in series with impedanceless branches (apparent coils, z = 0) are denoted by E u

known currents impressed across real coils, and not assumed as junction-pair currents, are denoted by Ik

known currents impressed across admittanceless branches (apparent coils, z = a) are denoted by iv

2. Select arbitrarily TI. new mesh currents ik to include the (n - m) currents iV:

across admittanceless coils they are usually known

around the meshes they are usually unknown

3. Set up the impedance matrix for the "all-mesh" network. Let (j, k) coordinates represent the original n coils, and (u, v) coordinates the (n - m) impedanceless junction-pair apparent coils. The impedance matrix then takes the form, and has the dimensions, shown:


4. Set up the transformation tensor in the usual fashion, selecting arbitrarily n new currents represented by coordinates (j', k') but including the currents in the added meshes. It takes the following form :

\ at kg

When the junction-pair axes are not required, the corresponding k' columns may be ignored and Ci, reduces to the singular transformation matrix of an ordinary mesh network. In this compound matrix, the n original branch currents as given by Eq. 8-1 are related to the n new currents by

(Note that Ci. is established as a relationship involving total currents through the terminals of a coil, and not coil currents; they differ-by the impressed shunt current Ik).

The added mesh currents iv are to be preserved by the new axes, and therefore the (n - m) junction-pair currents are related to the new currents

by iv = h;,ik' (8-6)

5. Currents impressed across real coils, and not assumed as junction-pair currents, are designated by Ik.

6. The new impedance is

za.B. = Cf.C$,zaB = (C;, + 6;4)(~:, + 6:,)z,, = c;,C:.zjk (8-7)

7. The current transformation is (using Eq. 8 4 )

8. The voltage transformation is (using Eq. 8 4 )

e,. = C,b.e, = (Ci, + 6:.)ek = C:.ek = e,. (8-9)

9. The coil shunt currents Ik, being components of the total currents ik, and if not assumed as junction-pair currents, transform in the same way as ik by the inverse of Ci,:

~ k ' = c k ' p (8-10)


10. The total currents in the n new axes are

ik' + Ik' (8-11) 11. The equation of performance in the new coordinates is (by Eqs. 8-7

and 8-9 and expression 8-1 1)

eat + Eat = ~ , , ~ . ( i ~ ' + I B ' ) ( s 1 2 ) I n this equation there is a total of n unknowns. In order to solve for the

( a 1 Actual network ( b ) Equivalent "all-mesh'lnetwork

Fig. 8-3. Complete network of five coils.

unknowns it is necessary to partition the voltage equation along the mesh (j', k') and junction-pair (u', v') axes as follows:

EXAMPLE.^ The following example of the analysis of a complete network is adapted from Chapter XVI of Kron's Tensor Analysis of Networks, and i t illustrates the procedure for handling such networks.

In the actual network of Fig. 8-3a there are

n = 5 coils (zaa, %bbt Zcc, Zdd, ~ ~ f )

m = 2 meshes (n - m) = 3 junction pairs (to be selected, but to include q and r)

Note that only actual (non-zero impedance) coils are counted, while apparent (zero or infinite impedance) coils are to be regarded as junction pairs.

n = 5 known series coil voltages (e,, eb, e, = 0, ed = 0, ef) n = 5 known shunt coil currents (Ia = 0, Ib = 0, IC, Id = 0, If = 0) 1 known junction-pair current (iQ) 1 known junction-pair voltage (E,)

This leaves one additional junction pair to be chosen.


In Fig. 8-3b, the (n - nz = 3 ) junction pairs p', q', r' have been chosen to include the admittanceless branch q and the impedanceless branch r , and these junction pairs are shown as closed through impedanceless apparent coils so as to convert the original network to an "all-mesh'' network of five meshes and mesh currents (a', b', p', q', r'). Between these five new mesh currents and the primitive terminal currents a non-singular transformation can be established. There are then the following matrices.

-

ek = I e,, I eb 1 0 I 0 I ef I or n = 5 known series impressed voltages

a b c (1 f

E,,. -1 o o I ED, 1 Ed 1 E,. I or n = 5 new junction-pair voltages, of which three are known (a', b', T')

J - and two are unknown (p', q') known known

a,' b' p' q' r'

The non-singular part of the transformation tensor is established by comparison of coil terminal currents with the new mesh currents; thus

or n = 5 known shunt impressed currents 0

a '


Note that when the junction-pair axes (p', q', r') are ncit active, the corresponding columns of C;. are eliminated, and the remaining singular part of the transformation matrix C;, is identical with that which would be set up for a simple mesh network of two meshes (a', b').

The junction-pair currents transform into themselves :

I0

two are known (p', q') and three - unknown (a', b', r') known

ib'

The new impedance matrix is 0

The new voltages, all of which are known, are

0

The new mesh currents corresponding to the impressed shunt currents Ik are found upon taking the inverse of C;,.

and all these currents are known.

id if or n = 5 new mesh currents, of which


Finally, there are the voltage equations along the mesh (af and b') and junction-pair (p', q', r f ) axes:

e,. + 0 = z,,,,ia' + z,.,,ib' + zatQ.(iQ' + IQ') + ~ ~ . ~ , ( i " + I?') e,, + 0 = zb.,.ia' + zb.,ib' + zbrqr(iq' + Ia') + zbJir' + IT')

e,, + E,. = z,,,,za ' ' + ~ , , ~ , i ~ ' + ~ , . ~ . ( i ~ ' + la') + z,,,,(ir' + I") eq. + E,. = zQparia' + zQ,,ib' + + IQ') + zQ,,.,(i7' + IT') e,, + E,, = z,,,,ia' + zrTb,ib' + z,,Jia' + IQ') + zr,,,(ir' + I")

In these five equations containing 14 variables there are

9 knowns (eat, e,,, E,,, iQ', lQ', I"", e,,, eQt, eT.)

5 unknowns (E,,, E,,, in', ib', iT')

and therefore there is a sufficient number of equations to solve for the unknowns.

8-4. The Conversion of Networks6 Any n-coil network may be used as a reference frame for the same n coils interconnected into a different network. The procedure follows.

1. Convert each network to an "all-mesh" network by closing its junction- pairs through apparent coils. The mesh currents of the reference network are designated by ik', and those of the new network by ik".

1 2. Express each coil current in terms of the mesh currents in both the reference network and the new network:

ik = c,k,ik' = ~ , k , , j k " (8-14) I 3. Rearrange these equations so that only one reference current appears I on the left side of each equation (thereby defining the required transformn tion

II tensor)

i k f - c k r c k jk" = ckTj lc" - k k" P" (8-15)

I Thus all n-coil networks made up of the same impedances are related

through appropriate transformation tensors. Therefore, when the equations of performance are known in one reference frame, they may be converted to any other reference frame within the group.

EXAMFLE.~ The following example, adapted from Kron's Tensor Analysis of Networks, illustrates this procedure. In Fig. 8 4 , five coils are shown interconnected in two different ways, having three and two junction pairs, '

respectively. Equating the coil currents in terms of the currents in both the reference

and derived networks, there results p = ia' - - ia" i b " - i d - if"

ib = i n ' + i ~ ' + i ~ ' + jr' - - - ic"

ic = i b ' - p' - ia" - i d = - i b ' - - i b " + i c " + id''

if = i a ' + ib' + i"' - - - i b "


Rearranging the equations so that only one reference axis appears on the left side defines the following transformation matrix :

( a Reference network ( b ) Derived network

Fig. 8-4. Conversion of "all-mesh" networks.

The new impedance and impressed voltage matrices then are

These matrices could have been found also by setting up the transformation tensor Ci. between the new network and the primitive coil currents.

Art. 841 COMPLETE NETWORKS

. '4

m II I I I I II I I I I II I1 I1 11


8-5. "All Junction-Pair" Networks.6 A complete network also may be reduced to an "all junction-pair" network and its equation of performance established as a current equation:

The procedure in setting up this equation parallels that for an "all-mesh" network upon the interchange of dual quantities and the opening of each mesh for the insertion of an apparent ( Y = 0) coil.

8-6. Comparison of the Three Types of Networks. Every network comprises n coils, m meshes, and (n - m) junction pairs and has (n) known impressed voltages ek in series with the coils, (n) known impressed currents Ik in shunt with the coils, and a number of apparent (zero or infinite impedance) coils. In general (m) response currents ik' flow in the meshes and (n - m) response voltages E,. appear across the junction pairs. Then

1. If m < (n - m), Iu = 0, and Eut is of no concern, t,he network is preferably solved as a simple mesh network.

2. If m > (n - m), ek = 0, and ik' is of no concern, the network is preferably solved as a simple junction-pair network.

3. If both e, and Iu are impressed, the network may be solved either as an "all-mesh" or an "all junction-pair" complete (orthogonal) network.

A comparison of the several steps involved is given in Table 8-1.

PROBLEMS

8-1. The use of the principle of the invariancy of power under a transformation of coordinates, or for a change in the connections of a network, was justified for a mesh network in Art. 3-7. Justification for the use of the principle for junction-pair networks was assigned as Problem 7-1. What is the justification for the use of the principle in the case of complete networks?

8-2. A thorough understanding of the underlying concepts and philosophy of complete networks is absolutely necessary to the analysis of such networks. I t is believed that these matters have been put on a sound basis in the text and that by Eqs. 8-13 all the essential ideas have been presented in a logical order. On a large piece of paper show a closed contour within which are representative types of adjacent-terminal pairs as shown in Fig. 8-2. Alongside the diagram write the definitions and procedures to be followed. And, finally, show the impedance and transformation matrices, and equations. This "bird's-eye view" is complete when it clearly defines and explains every necessary concept of complete networks.

8-3. Check every step in the numerical examples in the text. 84. Reduce a complete network to an "all junction-pair" network and

establish Eq. 8-16 as its equation of performance.

THE ELEMENTS OF TENSOR A N A L Y S I P 4

The previous chapters have been devoted to underlying principles in the application of matrices and tensors to electric circuit analysis, but actually only the barest notions of tensor algebra have been introduced. However, electric machine theory will require the most advanced concepts of tensor analysis, and, while many of these concepts can be examined as they occur in the development of machine theory, i t is, nevertheless, desirable to present now the elements of tensor analysis as briefly as is consistent with an understanding and appreciation of them.

It has been mentioned that the essence of tensor analysis is bound up in the ideas of: set, form, invariancy, group, and traqformation. And there are, of course, the associated notations, conventions, definitions, and rules.

9-1. Sets. The totality of the variables required for the description of a geometrical or physical system comprises a set and in tensor notation is represented by a single symbol:

xr E?E (x", xb, . . . , xn) (y-1)

Here x is called the base variable and refers to the type of variable. For example, geometric coordinates, currents, voltages, velocities, and accelerations are all different types of variables and need to be designated by different base letters such as x, i, e, v, a.

The superscript index (not exponent) r may be interpreted either as indi- cating any (but not a particular) one of the several variables xa, xb, etc., or alternatively (and this is the preferred point of view) as the whole set of values. When it becomes desirable, or necessary, to specify a particular variable, a capital letter is used for the index, for example xR, or else a specific element like xb is written.

Indices may appear either as superscripts, as in x" or as subscripts, as in y,, depending on the nature of the variables. Variables which transform in a direct fashion are called wntravariant variables, and their indices are attached as superscripts. Variables which transform in an inverse fashion are called covariant, and their indices are attached as subscripts. In a given system one type of variable may be selected arbitrarily either as contravariant or covariant .

105


The idea of set is not restricted to single variables like xT or y,, but can be extended to tensors of any rank or valence; see Table 9-1.

TABLE 9-1 I

Valence (or Rank)

Contravariant Covariant -- I Total

1

1

2

2

2 2

2

3 4

Tensors

Current, velocity, coordinate

Voltage, force, gradient Impedance Admittance Rotation Kronecker delta Transformation tensor Affine connection Riemann curvature

A tensor of valence 2, such as a,, where ( r , s ) range from 1 to n, constitutes a set of n2 elements. A tensor of valence 3, such as A:4, constitutes a set of n3 elements.

Furthermore, functions of variables, or equations, comprise sets. For example (using the summation convention),

is a linear set of n terms, while

is a bilinear set of n2 terms.

9-2. The Summation Convention. The summation convention has already been defined and used in previous chapters. Whenever in a tensor product the same index occurs twice--once as a contravariant and once as a covariant index-summation with respect thereto is implied. Such an index is called a dummy index. An index which is not repeatedis called a free index.

Art. 9 4 THE ELEMENTS O F TENSOR ANALYSIS 107

When several pairs of dummy indices occur in a product, summation is implied with respect to each pair of dummy indices. Examples are

It is evident that the letter used to represent the repeated or dummy index is immaterial. I n practice, in order to assemble terms, i t is often necessary to substitute letters for the dummy indices occurring in different terms. For example,

9-3. Invariance. Invariance refers to the constancy of some entity, or form, which does not change under a transformation of coordinates. There are two types of invariancy of special importance in tensor analysis :

1 . An ent i ty , like energy W or power P, or the distance d between two points, which remains unchanged under all permissible transformations of coordinates.

2 . A mathematical f o r m , like the formula for stored magnetic energy (a quadratic form) LTSiT is , or the equation of voltage (a linear form) e, =

zTSiS, which preserves the same form in all reference frames.

It often happens that the tensor equation representing the most general case will contain terms not present in simpler reference frames. The invariant form is thus considered to be that corresponding to the most general case, but some of its terms are zero or vanish for certain reference frames.

. . . . 9-4. Group Theory. A set of elements, A, B, C, having a rule of combination (denoted by . , but not restricted to multiplication) is defined as a group if the following four conditions apply:

1. The combinations A . A and A . B are also members (the "group property").

2. The associative l a w holds: (A . B ) . C = A . ( B . C ) . 3. There exists a u n i t element I such that combination between it and any

member of the group leaves the member unchanged. 4. Every member has an inverse A-l, also a member of the group, such that

A . A-1 = I .


If only the f i s t three conditions are satisfied, the set is called a semigroup. If all elements of the group belong to a second group, the set is called a

subgroup of the second group. If a group repeats itself, it is called a cyclic group. If a group has a finite number of members, it is called a Jinite group; if i t

has an infinite number of members, it is called an inJinite group. For examples see Table 9-2.

Group

Set

Rule of combination

Unit element

Inverse

Condition 1

Condition 2

Condition 3

Condition 4

TABLE 9-2

All positive and negative integers

. . . , - 2 , - 1 , 0 , 1 , 2 , 3 ,...

Addition (+)

0

(2) + (-4 = 0

(2) + (&Y) = (2 * Y) ( x + y ) + z = X + ( ~ + d

x + O = x

(2) + (-4 = 0

lotation by multiples of 90'

y, = 90°, y, = 180°, y3 = 270°, y, = 360"

Addition (+)

y, = 360"

Yl + Y3 = Yz + Yz = Ya YI + 7, = y3, etc.

(71 + Y,) + Y, = yl + (Y, + Y,) YE + y4 = YZ. etc.

Y1 + Y3 = Yz + Yz = Ya

The rotation group is cyclic if it is agreed that any angle plus 360' cannot be distinguished from the angle. (This could not be tolerated in some investigations; for example, in determining the work done in taking e magnetic pole around a current many times.) The 90" rotation group is also a subgroup of a rotation group having any angle 8 as its modulus.

The transformation matrices which substitute one set of variables for another set in a network form a Jinite group, since only a finite number of substitutions is possible (i.e., replacing branch currents with mesh currents).

The transformation matrices which change one network into another also form a jinite group (i.e., reconnecting the elements of a network).

The transformation matrices which change one network into another with a different number of coils form a semigroup, since such matrices do not have an inverse (i.e., reconnecting with a different number of coils).

The transformation matrices which introduce hypothetical currents as variables form an inJinite group (i.e., substitution of symmetrical components).

The transformation matrices which substitute variables are a subgroup of the transformation matrices changing the interconnections of a network, which in turn are a subgroup of the group of connection matrices for the generalized machine.

9-5. Transformation. A set of coordinates xS representing a description of some configuration may be transformed into another set xa by a linear transformation,

xa = c;xs ( 9 4 )

in which the coefficients Cf are independent of the variables.

Art. 9-71 THE ELEMENTS OF TENSOR ANALYSIS 109

There are two possible interpretations of a transformation of this type:

1. The passive interpretation regards an identical physical configuration of an entity alternatively represented in two different reference frames. It is simply a case of expressing the same entity in two related systems of coordinates by a substitution of variables. (For example, the temperature distribution in a flat plate may be expressed in terms of rectangular or polar coordinates.)

2 . The active interpretation regards two different physical configurations of an entity as expressed in the same reference frame. (For example, the temperature distribution on two identical flat plates may be linearly related in terms of the same coordinate system.)

The passive interpretation is used in cases where a mere substitution of variables occurs, such as substitution of mesh currents for branch currents or of one set of branch currents for another, a permutation of variables, or an imposition of con~t~raints.

The active interpretation is adopted when a given network is to be recon- nected into an entirely different network.

The transformation (Eq. 9 4 ) is reversible if the equations can be solved for the independent variables in terms of the dependent variables.

Solving Eq. 9 4 by Cramer's rule,

in which ICfI is the determinant of the transformation matrix and A: is the cofactor of C: in the determinant. Obviously, this solution does not exist if the determinant vanishes, that is, if its matrix is singular. In other words, a singular matrix has no inverse and the transformation is irreversible.

9-6. Successive Transformation and Group Property. A set of variables x7 may be transformed to x7', and this in turn to xr", and so on, thus

etc. (9-6)

The successive products of transformation matrices possess the group property.

9-7. Definition of a Tensor. A set of numbers or functions which remain invariant under a group of linear transformations is called a tensor. If the CL are independent of the variables, then

l I0 MATRICES, TENSORS, A N D CIRCUITS [Ch. 9

and therefore the definition of a tensor requires that

9-8. lnvariancy of Form. Suppose that . the variables transform linearly according to the equation

xT = c;xa (9-9)

What is the law of transformation for the coefficients in the equation

a,xT = b (9-10)

if its form is to remain invariant? Substituting Eq. 9-9 in Eq. 9-10,

ar(Cixa) = (aTC;)xa = aaxa = b (9-1 1) That is,

aa = arc: or a, =C;a, (9-12)

Thus the coefficients transform in an inverse fashion from the variables, and this leads to the following distinctions : contravariant tensors (like xT) transform directly according to Eq. 9-9; covariant tensors (like a,) transform inversely according to Eq. 9-12; hence the use of upper indices for contravariant and lower indices for covariant tensors.

Quadratic forms transform as follows :

b 7 s ~ ~ ~ ~ = b T S ( C ; ~ a ) ( C ! ~ p ) = ( b T S c ~ C ! ) ~ a ~ / 3 = b a B ~ a ~ , 9 (9-14)

A bilinear form transforms as follows :

a y y , = a: (Cy) (C!yB) = ( a : c ~ c f ) x a y B = a!xayB (9-15)

In general, any multilinear form transforms as follows:

1 : : xrxsxt . . . ymyny, . - = am"""' C a C ~ C Y , - - - CTC~txPzuxryaypyy . . . r s t - . m n P a r

that is BY. .. = Ca CBCY . . . C T C S C ~ . . . amn"' ' ' ape, . . . m n p P a r , s t . . . (9-1 6 )

Therefore, if it is known that a quantity is a mixed tensor of multiple valence, i t is transformed by as many direct transformation tensors as i t has upper indices, and by as many inverse transformation tensors as i t has lower indices, and the transformation tensors are prescribed automatically by the position and type of the indices.

A constant is a tensor of valence 0, since it has no indices, and it transforms into itself.

9-9. Addition of Tensors. Tensors may be added or subtracted only if their base letters are of the same type, and each has the same free indices similarly placed (equal valence). Thus

e, + e; + e: = E, (9-17)

z,, 4 .is + z:s = zrs (9-1 8 )

Art. 9-1 I] THE ELEMENTS OF TENSOR ANALYSIS I I I

But e, and iT cannot be added because they have base letters of different types (a voltage and a current) and one has a covariant and the other a contravariant index.

The sum of several tensors transforms in the same way as its constituent members :

aaB baB = C ~ C B ~ T S C ~ C B ~ T S 7 s 7 s

= C:Cf(ar" fT" = c y ! d r s = dab (9-19)

9-10. Contraction. If a pair of contravariant and covariant free indices in a tensor are made identical, they become dummy indices implying summation, and the operation is called contraction. For example,

st ." a ~ n t " ' - r l t - ' r 2 t " . ~ 3 t . . . r t . . . (amno... ) s = n = r n n p . . . - amlp + a m Z p . .. + a m 3 p . . . + . . = bms. .. (9-20)

Contraction is seen to reduce the valence of a tensor by 2, since the sum of the terms is a tensor with 2 less indices than the original tensor.

9-1 1. Multiplication. In multiplication of a tensor by a constant, each element of the tensor is multiplied by this constant (just as in multiplication of a matrix by a constant), and the resulting tensor has the same valence and is the same type as the original tensor:

kars = bn (9-21) In the general (or outer) product of two tensors, each element of one tensor

is multiplied in turn by each element of the other tensor (as in the product of two matrices), yielding a tensor of valence equal to the sum of t,he valences of the factor tensors :

arb: = (9-22)

Transforming the left side of Eq. 9-22 with form invariant,

a;b,P = C " , ~ a ~ C ~ C ~ b ~ = Ca m~ CnGPCSdmr r u n s = diP, (9-23)

showing that the indices in the product are the same and in the same position as the indices in the factor terms.

When multiplication and a double contraction are combined on like terms of the same valence, the resulting operation is called the inner product or composition. It is defined by (putting r = n and s = m)

a: . b: = arb; = d (9-24)

For tensors of higher valence, repeated contraction of the product results finally in an invariant, as shown below, using the dot for inner product and the ( ) for contraction:

((a:: - b',:)) = ((amn . ,bm,)) " = ( ( d 3 = ( a 3 = ( f3 = g (9-25) A chain product consists of multiplication and a single contraction, such as


The expansion of Eq. 9-26 is the same as for the product of two matrices, each of rank 2.

9-12. Multiplication of aTensor by I t s Inverse. Ifthe set of equations

is solved for the variable xa, there is

in which C: is the inverse of CL. Substituting Eq. 9-28 in Eq. 9-29, there must be

2" = c a c T x 8 = p x S = xa 8 B (9-30)

:. c;c; = s; (9-31)

It is also possible to prove this relationship from the definition of the element of an inverse as

CT = cofactor of Cz (9-32)

" determinant of the Cz

Then, from the theory of determinants,

This is true for the product of any tensor of valence 2 by its inverse. By virtue of Eq. 9-31, the equivalent valence of the indices on the two

sides of a tensor equation is easily demonstrated. Consider the equation

and contract by putting y = a on both sides:

a;, = CkC,"C;arD = bgCiarD = CFaFm (9-35)

showing that a , m, and n are dummy indices and there is only one free index, b, on each side of the equation.

Or consider the double contraction of Eq. 9-23 by writing (putting p = f i and a = a )

((azb:)) = a;b{ = ((CkCFCyzarb:))

= CkCzC;~;arbb: = bb",b,"arb:

= aFb: (9-36)

Or consider the single contraction in the chain product ( p = /?) :

(sib:) = ( C k C ~ C ~ C ~ a ~ b ~ ) = ~ & C z ~ ; C ! a z b : = CkCzG:arb: = CkCzaFb' (9-37)

The Kronecker delta is itself a tensor, as an examination of its transformation shows :

CzCibl= C;Ci = 6; (9-38)

Art. 9-14] THE ELEMENTS OF TENSOR ANALYSIS 113

9-13. The Quotient Law. In tensor analysis, an equation as a whole may be a tensor, but i t does not follow that every factor among its terms is a tensor. The quotient law provides a criterion for testing such factors. Suppose that

a(rst)ba = dT (9-39)

The right side is a tensor dr of valence 1, and the left side contains the tensor bst of valence 2. What, then, is the nature of the object a(&) ? Assuming that the form of the equation remains invariant under transformation, there results

a(a,5'y)bsY = da = Czar = Czbsta(rst) = C,"CjC:bsYa(rst) ( 9 4 0 ) Then

a(a,!ly) = C;C;C:a(rst) ( 9 4 1 )

and therefore a(&) is a tensor of valence 3 , contravariant in r and covariant in s and t :

a(&) = a:, and Eq. 9-39 can be rewritten

(9-42)

aT bst = d7 st ( 9 4 3 ) The indices now balance on both sides of the equation. This requi rement that in every product in a tensor equation the indices mnst balance with the indices in all the other terms of the equation-furnishes a quick and infallible means of identifying an unknown factor by inspection.

9-14. General Functional Transformation. Up to now only linear transformations have been considered, and tensors have been defined only with respect to such transformations.

A general functional transformation is one which relates the variables in two different reference frames by an arbitrary set of functions as

Suppose that the transformation is reversible, so that

and that the functions possess derivatives to any order required. Then

in which the CF and Cr, are now no longer constants, as in the case of linear transformations, but functions of the variables. Thus a linear transformation is merely a special case of the general functional transformation. But the differentials transform linearly and are integrable. Therefore the necessary and sufficient condition that the transformation between the differentials


be reversible simply means that the determinant of the coefficients (the Jacobian) does not vanish, as was demonstrated in the case of linear transformations. That is,

It is easily seen from the rule for the product of two determinants that

Icj . Ic;l = lc:c;l = la$ = 1 (9-50)

The extended definition of a tensor now becomes :

A function of any order (valence) is a tensor with respect to a general functional transformation if it transform in accordance with a linear transformation between the old and new differentials.

Thus, if a& is a third-order function of the variables x*, it is a tensor of valence 3 if its law of transformation is

A tensor of valence 1 transforms as . axa %a = - i'

axr If a linear form remains invariant under transformation,

eaia = eTiT and, upon substituting for ia,

Hence

If a quadratic form remains invariant under transformation, then

9-15. Weighted Tensors. If transformation (9-51) is multiplied on the right side by a power M of the Jacobian, it is said to be a tensor of weight M . In general,

is a tensor of weight M.

Art. 9-17] THE ELEMENTS OF TENSOR ANALYSIS 115

9-16. Derivatives of a Tensor. The derivative of a tensor is not a tensor. Let

Taking the derivative, and remembering that Ck is a function of the variables,

If xn is an electric charge and t is time, then in = dxn/dt, and

This is not a tensor transformation because an additional term, other than Ck, appears on the right side of the equation.

9-17. Rectilinear Reference Frames. A point in a plane is uniquely determined by two numbers, a point in ordinary space by three numbers, and a point in n-dimensional space by n numbers. These numbers are the coordinates of the space in question. The coordinates may be rectangular (orthogonal or not) or general curvilinear. Familiar examples of different coordinate systems (or reference frames) are: rectangular, polar, spherical, and cylindrical. In every case, the point is determined by the intersection of a set of parametric surfaces. It is even possible to use different units of measurement along each axis ; for example, in a rectangular coordinate system the units of measurement along the x-, y-, z-axes, respectively, may be inches, feet, and centimeters. As will be shown, there is a remarkable entity called the metric tensor which automatically takes care of the different yardsticks as well as providing the correlation between coordinate systems.

Let yT = (ya, yb, ye) be the Cartesian orthogonal coordinates of a point in space. Then the equat.ion

where the C: are constants, defines a linear transformation to new variables x* - (xl, x2, s), and these variables may be looked on as new coordinates for the point in a rectilinear reference frame. The transformation is reversible, so that

If PI and P2 are two points having rectilinear coordinates xi and x;, respectively, and if y; and y7, are their corresponding Cartesian coordinates,

1 16 MATRICES, TENSORS, A N D CIRCUITS [Ch. 9

then the square of the distance between these two points is given by

s2= (=I2 = (ylf - Y ; ) ~ + (Y; - Y;12 + (Y; - ~3~ = (Y; - Y;)(Y; - Y;) + (Y; - Y;)(Y; - Y;) + (Y: - Y%YE - ~3 = C;(x; - X;)C;(X; - x[ ) + C:(x: - x;)c;(x{ - 2;)

+ C:(xq - x;)c;(xf - x i ) B B = ( C p ; + c:c; + C;C;)(x; - xi)(xl - x2)

= c;C;(x; - x;)(x! - 2;)

= g&; - x ; ) ( g - 4 (9-58) in which

g,, = ctc; + cp; + cy; (9-59)

In particular, if one point is the origin,

s2 = g xax, (9-60)

If the two points are infinitely close, (xy - x i ) = dxa and

Now s or ds is an invariant, and therefore the right side of Eq. 9-60 or 9-61 is an invariant and, since xu is a tensor, it follows from the quotient law of tensors and Eq. 9-60 that

gmp = gBa is a symmetric double covariant tensor of valence 2. It is called the metric tensor and is of fundamental importance.

As an example of these concepts, consider a point P in a plane whose Cartesian coordinates are (ya, yb) and whose rectilinear coordinates referred to axes a t fixed angles 8 and 4 from ya are x1 and x2. From Fig. 9-1 it is clear that

ya = x1 cos 8 + x2 COS 4 yb = x1 sin 6 + x2 sin 4

Then Cy = cos 8, Cg = cos 4, C! = sin 8, ~ b , = sin 4, and therefore

gll = cos2 8 + sin2 8 = 1

g2, = g12 = cos 8 cos 4 + sin 6 sin 4 = cos (4 - 8)

g22 = cos2 4 + sin2 4 = 1

The distance of the point from the origin then is

But even an orthogonal coordinate system may attract a metric tensor with other than unit elements if the units of measurement are not the same

Art. 9-1 91 T H E ELEMENTS O F TENSOR ANALYSIS 117

for the different axes. For example, suppose the units of measurement along the axes are in the ratio 1 : h: k. Then, in terms of the unit measurement, the distance of a point from the origin would be

When the unit measurements are the same for all axes, gTs is 1 for r = s and 0 for r # s.

Fig. 9-1. Cartesian and rectilinear axes. Fig. 9-2.

9-18. The Magnitude of a Vector. Similarly to Eq. 9-60, the magnitude of a contravariant vector A T is defined as

A = ( g T , ~ r ~ y ) " (9-63)

and the magnitude of a covariant vector is defined as

B = (~ 'T~B,))"

in which gT8 is the inverse of grs; that is,

gTdm" gTrngms = s: A unit vector is one of unit magnitude. If aT and bT are unit vectors,

9-19. Angle Between Two Directions. In Fig. 9-2 let A and B be two points unit distance from the origin represented by the unit vectors aT and bT. Then

- - - - - AB" 0 A 2 + 0 B 2 - 2 0 A . OB cos 8 = 2(1 - cos 6) (9-68)

But A% = (aT - bT) and hence, by Eqs. 9-58 and 9-66,

l I8 MATRICES, TENSORS, A N D CIRCUITS [Ch. 9 Art. 9-21] THE ELEMENTS OF TENSOR ANALYSIS 119

A comparison of Eqs. 9-68 and 9-69 shows that

cos I9 = g,,aTbs

If a' and b7 are perpendicular, 0 = 90' and therefore the condition for two directions being orthogonal is

0 = g,,a'bs (9-7 1 )

In general, for two vectors AT and BT of magnitudes A and B, respectively, the unit vectors are a' = AT/A and bT = BT/B and there is

A2 = grsA'AS

B2 = gTsBTBS (9-72)

cos 19 = gTSaTb" gT,ATBS/AB

The dot product familiar in vector analysis follows as

AB cos I9 = gTsATB"

9-20. Associated Tensors: Raising and Lowering of Indices. If the contravariant vector An is multiplied by g,, and contracted, there results the covariant vector

A, = grmArn (9-73)

This process is called lowering the index. If, now, An is multiplied by grn and contracted,

gTmAm = gTmgmnAn = 6',An = AT (9-74)

and this is called raising the index. Thus, lowering an index and then raising i t restores the original vector.

Two vectors with the same base letter, but with covariant and contravariant indices, are called associated vectors and are said to represent the covariant and contravariant components of the same vector.

Any number of indices may be raised or lowered by repeated applications of the metric tensor. When an index is thus changed, a dot is left to show the position from which it was moved. Examples are

ATs = gT"Ams (9-75)

A,! = gSmATm (9-76) A 7 = gSmA'"t (9-77)

Indices may be raised and lowered simultaneously by employing a sufficient number of metric tensors; for example,

..I: - g k u ~ m n " &.I - rm sn ' P I (9-78)

In a product of tensors, wherever a dummy index occurs it can be raised in one tensor and lowered in the other without changing the value of the product :

ATsB, = gTmAmSgTnBn = 6,mtlmsBn = AmsBm = A,BT (9-79)

The interchange of the dummy indices may be made automatically without the necessity of showing the metric tensors explicitly; for example,

Similarly, the free index may be raised or lowered simultaneously on both sides of an equation to yield an equivalent equation. For example, if

then

and so

9-21. Tensor Fields. The general tensor transformation of Eq. 9-52 is true in general for one point only, because the coefficients axa/axk are functions of Z and therefore of xu, and consequently, if the relationship holds for one point, there is no assurance that it will hold for any other point. Thus the tensor is localized at a point, and the various algebraic operations which were developed for linear transformations apply in general a t a point.

If, however, A: : : : and A: : : : are functions of xk and satisfy Eq. 9-52 a t every point in space where the functions are defined, the aggregate of all these tensors is called a tensor ,field.

PROBLEMS

9-1. Give examples of the following: h i t e group, infinite group, cyclic group, semigroup, subgroup.

9-2. Write the set of equations (9-4) for three variables x8 = (xa , xb, x C ) and xa = (xl, x2, z3) and solve for the xS in terms of the xa in accordance with Eq. 9-5.

9-3. Give an example of Eq. 9-6 showing all the C. 9 4 . Give the proof of Eq. 9-33. 9-5. In the equation e, = z(r, s)iS, if e, and is are known to be tensors; prove

that z(r, s ) must be a tensor. 9-6. Give the proof of Eq. 9-50. 9-7. The relationships between certain curvilinear (x l , x2 , x3) and rectangular

(y l , y2, y3) coordinates for some typical systems are given in the following table. Tabulate the metric tensors for each case.

Spherical Cylindrical Parabolic

Elliptic

xl sin x2 cos x3 X1 C 0 8 XZ

xlx2 cos x3

xl sin x2 sin x3 x1 cos x* x1 sin xB ~ 1 x 2 sin x3 f

TENSOR

10-1 . The Christoffel Symbols. The ordinary and partial derivatives of conventional calculus are only special cases of much more general processes encountered in tensor analysis; for this reason tensor analysis is sometimes called the absolute calculus.

Consider a vector-field of parallel vectors of equal magnitude whose positions are functions of a parameter t , and in terms of Cartesian, yr, and curvilinear, xr, coordinates let this vector-field be represented by Y r ( t ) and Xr( t ) , respectively. Since the components of these parallel vectors are equal in Cartesian coordinates, they are constant with respect to t and therefore

But

and its derivative therefore is (using Eq. 10-l),

Now, according to Eq. 9-59 the metric tensor is

and upon differentiating with respect to x p there results

By a cyclic interchange of the indices in Eq. 10-5 there are

Art. 10-21 TENSOR DIFFERENTIATION 121

Adding Eqs. 10-6 and 10-7, subtracting Eq. 10-5, and dividing by 2 there results

The left-hand side of this equation is called the Christoflel symbol of the first kind and is written

The Christofel symbol of the second kind is defined as

These three-index symbols are not tensors, as will be seen when their equations of transformation are derived. They are symmetrical in m , n , as is easily seen since the metric tensor gmn is itself symmetric. They vanish in a rectilinear reference system. They are of enormous importance in tensor analysis, and they possess numerous interesting properties.

10-2. Intrinsic and Covariant Derivatives. Return now to Eq. 10-3 and multiply it by grP(aya/ax") :

But aye aye - gm, by ~ q . 1 0 4 and gmdw = 6; axm axD

also

aye ( ' ] by Eqs. 10-8,10-9,lO-10 qr* 7 = rnn ax ax ax Hence Eq. 10-1 1 becomes

Any vector satisfying Eq. 10-12 is thus a parallel vector-field. But X r is a function of the variables xr, so

d x r axT dxn -- dt axn dt

and Eq. 10-12 then becomes


Since this must be true for all values of dxn/dt, evidently a parallel vector- field satisfies the equation

Now let X, be any covariant vector along a curve and let A' be an arbitrary contravariant parallel vector-field, satisfying Eq. 10-15, along that curve. Then (ATX,) is an invariant and its derivative is

Since the left side of this equation is an invariant, the right side must also be an invariant. Interchanging dummy indices r, m, there is the'invariant

But AT is a vector, hence by the quotient law the quantity

is a covariant vector called the intrinsic derivative of X, with respect to t and is denoted by 6XT/6t.

In exactly the same way the intrinsic derivative of a contravariant vector Xr is found to be

Putting

in identity (10-19),

and, since dxs/dt is a tensor, by the quotient law the expression in square brackets is also a tensor, contravariant in r and covariant in s. It is called. the covariant derivative of XT and is denoted by

In a similar way i t may be shown that

The commas in identities (10-21) and (10-22) denote differentiation.


By comparison with Eq. 10-15, it is seen that the condition for a vector- field to be parallel is that its covariant derivative vanish, Xrs = 0 or X,,, = 0.

10-3. Derivatives of Tensors of Higher Valence. Let Xi, be a tensor of valence 3 which is a function of a parameter t. If it is multiplied by three arbitrary parallel vector-fields A,, BS, Ct, and the consequent invariant (a function oft) is differentiated with respect to t, there results another invariant,

The intrinsic derivatives, identity (10-18) or (10-19), for each of the arbitrary parallel vectors is equal to zero; hence by the quotient law Eq. 10-23 gives

which is the intrinsic derivative for Xi,. I n a similar fashion the intrinsic derivatives for tensors of any valence are easily derived. The construction is clear from Eq. 10-24, and the intrinsic derivative for higher valence tensors may be written by inspection. Substituting

ax:, ax:, axk -- at axL at

in Eq. 10-24, and separating out dxk/dt, there remains a set of terms which by the quotient law must be a t,ensor and is called the covariant derivative,

In a Cartesian coordinate system, the metric tensor components are all constant; the Christoffel symbols are therefore zero; and the intrinsic derivative reduces to the ordinary derivative, the covariant derivative reduces to the partial derivative.

The same rules which apply to the ordinary and partial derivatives of sums and products apply to intrinsic and covariant derivatives, for any tensor relationship which is true in a particular reference system must be true in every reference system. It is therefore only necessary to substitute intrinsic or covariant differentials for ordinary or partial differentials to pass from rectangular to curvilinear coordinate systems. Some examples are given in Table 10-1.

MATRICES, TENSORS, AND CIRCUITS [Ch. 10

TABLE 10-1

Rectangular coordinates

Let C: = A: + B:

dC,' dA: dB,' : .dt=dt+dt

Let C;, = A k B ;

dc;, - d A & -- d B ; - B,"1+

" at d t

Kronecker delta 6; = constant

as: -- ax. - O

Curvilinear coordinates

10-4. Divergence of a Vector. In a rectangular system the divergence of a contravariant vector AT is defined as

In a curvilinear system it must then be

div AT = A', = gTsAs,, (10-28)

10-5. Gradient. If a covariant vector A , is derivable from an invariant 4 (a potential function), it is called the gradient of 4, and in rectangular and curvilinear coordinates i t is

(a4 a4 8 6 ) grad 4 = - , - , - = 4,, = A , ayl ay2 ay3 in which, for a scalar, the covariant derivative is equal to the partial derivative

10-6. The Laplacian. The divergence of the gradient of an invariant is called the Laplacian. By Eqs. 10-28 and 10-29, i t is


In rectangular coordinates it reduces to

10-7. The Curl of a Vector. In curvilinear coordinates the curl of a covariant vector A , is defined as

curl A , = P t A t , , (10-32)

in which there appears an operator defined by

erst = qdlgmnl where

1 0 where any two of the indices are equal

p t = { + 1 when r s t is an even permutation of 1 2 3 (10-33)

1 - 1 when r s t is an odd permutation of 1 2 3

In rectangular coordinates this reduces to

If A, is derived from a potential (Eq. 10-29), then A,,, = A,,, and therefore

curl grad 4 = 0 (10-35)

10-8. The Curvature Tensor. The curvature tensor is of importance in electric machine theory when oscillating reference axes are encountered. It is a generalized second partial derivative of a vector.

The covariant derivative of a vector X , is given by identity (10-22). Regarding X,,, as a covariant tensor of valence 2: there follows from Eq. 10-26 (leaving out the contravariant r index and then substituting r for s, s for t , and t for k)


By exchanging dummy indices this equation may be rearranged:

Interchanging indices s and t , subtracting, and remembering that the Christoffel symbols are symmetric, there results

In a rectangular coordinate system Eq. 10-38 reduces to zero, since

Since the left side of Eq. 10-38 is zero for rectangular coordinates, i t must be zero for all coordinate systems, and therefore, in all reference frames,

R",,, = 0 (10-41

The left side of Eq. 10-38 is a tensor and therefore the right side is also a tensor, and, by the quotient law, since X , is a tensor, R!,, is also a tensor. It is called the Riemann-Christoffel tensor. I t depends only on the metric tensors gTs and their derivatives of the first and second orders.

The associated curvature tensor is

Substituting Eq. 1 0 4 3 in Eq. 10-39 and remembering Eq. 10-10, there finally results

or, substituting for the Christoffel symbols,


From this expression i t is evident that

That is, i t is skew-symmetric in p, r and also in s, t .

10-9. Holonomic and Non-holonomic Systems. I n a holonomic ace there are as many canonical equations, n, relating the variables as ere are n unknown variables xu, and the transformation tensor is

I n a non-holonomic space there are 2n unknowns ( n old variables, and n new differentials), and they are determined from n canonical equations and n equations of transformation. The transformation tensor, which is a function of the differentials, is

10-10. Transformation of the Christoffel Symbol. The Christoffel symbol of the first kind is defined by Eq. 10-9. I ts transformation is now sought. Let the metric tensor transform as

ga,fi, = gafiC;,C$ (1 0 4 9 )

in which the transformation tensor is a function of the coordinates

Also, for brevity put

Taking the partial derivative of Eq. 1 0 4 9 with respect to d'

and, by a cyclic interchange of the indices,

agyTa* agya acy, ac:. - -c;,cj,c;. + gya - c:,cj. + gyac:* c;, (10-S4) ab' ad ax@

1 28 MATRICES, TENSORS, A N D CIRCUITS [Ch. 10

Add Eqs. 10-53 and 10-54 and subtract Eq. 10-52, divide each side by 2, and then to the right side add and subtract

The resulting equation becomes

The term on the left is recognized as the holonomic Christoffel symbol of the first kind in the new coordinate system. The first term on the right is recognized as the holonomic Christoffel symbol in the old coordinates multiplied by the three transformation tensors. The remaining terms may now be combined by vertical pairs, and a t the same time the old g4 transformed to new gatr. There results

But, in these combinations,

Also, since the derivative of any of the expressions in (10-57) is zero, there is, for example, -

co'cB - go; .-. C" , - acj. = - aca '

8 @ ' - 8 2 C j , (10-58) axa axa

Art. 10-101 TENSOR DIFFERENTIATION

Then by Eqs. 10-57 and 10-58, Eq. 10-56 may be expressed as

The last three terms are called the non-holonomic objects; they are written

" * Y , a:) na.8.,y, = -(act c:.c;. 2 ad a x a

It is skew-symmetric in its first two indices, and it is not a tensor. It is evident from Eq. 10-59 that the holonomic Christoffel symbol is not

a tensor, since its transformation involves terms other than the transformation tensors.

If the non-holonomic objects are moved to the left sid; of the equation, there results an identity called the afine connection:

In a holonomic reference frame, where Eq. 1047 applies, the non-holonomic obiects vanish and

I From Eq. 10-61 it is seen that the affine connection in holonomic space

I is the same as the Christoffel symbol whose law of transformation is

But, in non-holonomic space, the non-holonomic objects appear on the left.

SPECIAL A N D USEFUL ARTIFICES

A number of helpful artifices are used in circuit and machine analysis. Rather than introduce them as they occur in the development of machine theory, it seems better to assemble them here in a single chapter so that they can be more conveniently referred to.

I I - I. T h e "Short-Ci rcuit The~rern ."e-~ This is simply the procedure which has frequently been used in this book for the simultaneous elimination of several rows and columns in a matrix or tensor equation. It consists of an appropriate partitioning of the matrix equation so that the variables not needed can be eliminated. Thus, if there are n variables and those up to m are to be eliminated, the partitioned equations are

in which indices (j, k) range from 1 to m and indices (u. v) from (nz + 1) to n . Solving Eq. 11-1 for the currents ik,

Substituting Eq. 11-3 in Eq. 11-2 and rearranging,

and substituting Eq. 11-5 in Eq. 11-3,

By Eq. 11-5 any set of currents may be solved for, and then if the remaining set of currents is required, Eq. 11-6 may be used.

It is often preferable, in eliminating variables by the procedure above, to do so one at a time, for then the inverse matrices are much easier to com- pute.

If, as often happens, the set of voltages e, = 0 (as in short-circuit problems), then the corresponding terms drop out of Eqs. 11-5 and 11-6 and the calculations are greatly simplified.

130

Art. 1 1-31 SPECIAL A N D USEFUL ARTIFICES 13 1

11-2. T h e "Short-Circuit" M a t r i ~ . ~ - ~ If only a single winding is short-circuited, say winding R, then in Eq. 11-1 put j = R and k = R and ej = 0, and let indices (u, v) include all the remaining indices except R. Then

0 = zRRiR + z&iV from which

ZRR and, substituting in Eq. 11-2,

This expression follows also from Eq. 11-4 upon substituting e j = 0, j =

k = R, and ykj = (zRR)-l = l /zRn Thus a new matrix, with row R and column R deleted, and having elements

zu RZR, = ZUV - - (11-10)

ZRR defines the retained axes.

The advantage of this method over the previous method is that the inverse of a matrix need not be calculated. The disadvantage is that the procedure must be repeated for every axis eliminated.

11-3. Lauder's Rule.45 A. H. Lauder has given an interesting and practical rule for the elimination of reference axes. To establish his rule, let A7:3 be the cofactor of z,, in the determinant D = IzIkl, so that

Eq. 11-4 may then be rewritten

But the numerators in this equation may Be recognized as the expansion of the following "bordered determinants,"* so that the equation becomes

As long as the determinants do not become unwieldy, this is a fast and simple procedure.

When the voltages (el . . e,,) are all zero, as for amortisseur windings or ground wires, the first term, e l , drops out.

* See NcConnel1.l p. 19.

MATRICES. TENSORS, A N D CIRCUITS [Ch. I I

Fig. 1 1 - 1

As a simple example take the case of Fig. 11-1 in which there is a generator on winding 3 and loads on 1 and 2. The impedance matrix, partitioned to permit elimination of the short-circuited axes, is

213222231 - 213232221 + 223211232 - 223212231 = (z33 -

211222 - '12221

Or the same result may be obtained from Lauder's rule (Eq. 11-13) :

Art. I I d ] SPECIAL A N D USEFUL ARTIFICES 133

Finally, for el = e2 = 0, let the result be confirmed by eliminating one axis a t a time (by Eq. 11-10). First put R = 1, so that

Applying Eq. 11-10 again to eliminate axis 2:

which checks with the results found by the other methods.

1 1-4. Elimination of Complex number^.^-^ Complex numbers may be dispensed with in favor of real numbers a t the price of doubling the number of rows and columns in a matrix equation.

The voltage drop in an impedance is

Arranging in matrix form, with the first row or column referring to real quantities and the second row or column to imaginary values, gives

in which (1) and (2) now refer to real and imaginary components, respectively. For example, the complex matrices of a polyphase induction motor are

134 MATRICES, TENSORS, A N D CIRCUITS [Ch. I I

11-5. The Polyphase Matrix. In balanced polyphase circuits having symmetrical polyphase windings and balanced voltages and currents, only one phase needs to be dealt with explicitly, since the variables in any other phase are the same except for a phase displacement. Let there be P phases and N coordinates per phase. The time displacement between adjacent phases then is 2.rrlP or 2(x - 1)n-/P between phase xand thereference phase 1. Then, if indices (r, s) apply to the reference phase and indices (m, n) to all phases, any current is

in = E-32;r(z-l)/P 8 n (*-1pfsis = PsniS (11-15)

in which P," is the polyphase matrix ~ - ~ ~ " ( ~ - ' ) ~ ~ 6 ~ ~ - , , ~ + ~ . The voltage equation then is

em = Zmnin = ZmnP:iS (11-16)

For the reference phase (x = l ) , = 6; and in = i s , em = e,, and

Z,, = Z,,. In rotating machine analysis often P = 4 and N = 2; thens

11-6. Substitution for the T ime Derivative. The impedances of static networks and machines usually involve the time derivative and its inverse :

d 1 p = - and - =St dt

at P

For example, in an R, L, C series circuit, the impedance is

In addition, in the impedance functions of machine theory, the mechanical angular velocities of the moving conductors, p0, moving reference axes, pel,, and moving magnetic paths, PO,, are encountered.

The solutions for the transient case are found by treating the impedances in the Heaviside operator, or Laplace transform, sense in the usual way.

In steady-state a-c cases the instantaneous value of a current may be expressed

i = I cos hmt = Re Iejhmt (11-19)

in which w = 2n-f is the synchronous (or base) value, and h is the fraction of the synchronous value a t which the current is varying (as, for example, during hunting or a t subsynchronous speed). Likewise, the mechanical

Art. I 1-81 SPECIAL A N D USEFUL ARTIFICES 135

velocities may be fractions v, v,, v2 of synchronous speed. Then (dispensing with Re)

pi = jhWIEjhWt (11-20)

1 . 1 - = - l E i h o t (11-21) P jho

Or, in general, = EEj(hmt+a) = Z( P, PO, ~ 0 1 , P02)i

= Z(jho, v o , v lo, v20 ) I~ jhwt ( 1 1-22)

and, canceling the exponentials, the a-c relationship

E =- Z(jhw, v o , v,o, v2w)I ( 1 1-23) results. For example, if

Z(p) = R + LP + Mp0

and i = Idhmt and p0 = vw, then

11-7. Product Frequencies. The product of two sinusoidally varying quantities of different frequencies yields two terms whose frequencies are the sums and differences of the parent frequencies. Thus, if

y, = $(A cos o,t - B sin w,t)

y, = J ~ ( c cos w2t - D sin o p t ) then

yly2 = [(AC - BD) cos (o, + 02)t - (BC + AD) sin (ol + w2)t]

+ [(AC + BD) cos (w, - w2)t - (BC - AD) sin (o, - 02) t ]

(1 1-25)

The coefficients in the product above are given correctly by the products of the following complex numbers :

Y , = ( A - jB),l Y: = ( A + jBIm1 ( 1 1-26) Y2 =( C - jDL2 y2* = (C + jQoZ

Y1Y2 = [(AC - BD) - j (AD + BC)1!m1+m2, (11-27)

Y,Y,*= [(AC + BD) - j(BC - AD)1(,1-m2) ( 1 1-28)

11-8. Cri ter ia of Stability. When a disturbance superimposed on a dynamic system induces an oscillation, the oscillation may die out as a result of the damping effects of the losses in the system, or may be sustained indefinitely, or may cumulatively increase in amplitude. In the latter case the system reaches its limit of stability and falls out of synchronism.

136 MATRICES, TENSORS, A N D CIRCUITS [Ch. I I

In an electrodynamic system, such as an electric machine, the equation of small oscillations takes the form

Ae, = Z,, Ais ( 1 1-29)

in which the determinant of is a function of the time derivative p of the form

IT,^( = anpn + an-lpn-l + . . + alp + a0 (1 1-30)

When equated to zero, this is the characteristic equation of the system, and its roots define the characteristic values which determine the nature of the transient. I n general, roots will occur which are pairs of complex numbers of the type ( a f j o ) , and they give rise to terms in the solution of the form

eat sin ( o t + /3) (11-31)

When a is negative, the oscillation dies out; when a is zero, the oscillation is sustained; and, when a is positive, the oscillation increases in amplitude. When (11-30) is a polynomial of high degree, it is difficult, or time consuming, to determine the roots. A number of criteria have been developed for ready determination of the nature of the roots without actually solving for them.

Routh's criterion applies when the a coefficients are all real numbers. The procedure in applying this criterion follows.

1. Arrange the coefficients in pairs :

2. Cross-multiply the even coefficient, a,, in the first column by each of the odd coefficients in the other columns; and also multiply the odd coefficient, a,, in the first column by the even coefficients in the other columns, and form the combinations bl = a4a3 - a5a2, b2 = a4al - a5ao, etc., and make the new arrangement

a4 a2 a0

bl b2

3. Cross-multiply aga,in to form

c1 = bla2 - a4b2 c2 = blao

and make the new arrangement

b, b2

Cl c2

4. Continue to cross-multiply and make new arrangements until no more products can be formed.

5 . If all the coefficients are positive, the system is stable. If any of the coefficients is negative, the system is unstable.

Art. 11-81 SPECIAL A N D USEFUL ARTIFICES

Hurwitz's criterion for stability requires that

1. All the coefficients a be positive.

2. The n determinants

formed by letting k be 1 , 2, . . . , n in succession be positive.

Part I1

MACHINE ANALYSIS

BASIC CONCEPTS IN ELECTRIC MACHINES

The complete analysis of any rotating electrical machine leads to the development of the following equations, but not necessarily in this order.

1 . The equation of mmf, including the effects of the various field excita- tions, armature reactions, and currents induced in short-circuited (amortisseur) windings; and taking into account the way in which the several windings are connected (number of turns, coil pitch, skew of conductors, distribution, and connections).

2 . The equation of $ux, in terms of the mmf's and permeances of the magnetic paths; or, alternatively, in terms of the currents and the self- and mutual inductances of the windings.

3. The equation of voltage, showing the way in which voltage is induced in the windings by variation of the fluxes, movement of the conductors, and movement of the magnetic paths.

4. The impedance equation, relating the voltages and the currents through generalized impedance functions (motional impedance; synchronous, transient, and subtransient impedances in direct and quadrature axes; etc.)

5 . The equation of torque, showing the force due to interactions between the currents and the flux densities.

6 . The equation of motion, combining the equation of voltage and the equation of torque into a composite expression which may be derived from energy considerations or general laws (Lagrange equation, Hamiltonian principle, Boltzmann-Hamel equation, Maxwell-Lorentz equation, etc.)

7. The equation of oscillations, expressing the effects of oscillating motion of conductors, oscillating reference axes, and perturbations in the currents, fluxes, and voltages.

8. The equation of power, showing the power inputs, outputs, and losses.

These equations may be developed from several different points of view, and by employing quite different concepts, and their forms may be quite different, subject to widely different physical interpretations, yet yield identical numerical results.

It is the purpose of the remainder of this book to examine the ways in which electric machine theory may be developed from the most general and comprehensive points of view, to reconcile the results, and to show how the

141

142 MACHINE ANALYSIS [Ch. 12

general solutions may be applied to special cases. To this end, certain simplifying assumptions need to be made in the interest of economy of time and space. In particular, the analysis will be idealized to the extent that i t will be assumed that:

1. There is no saturation in the magnetic paths. 2. Hysteresis and eddy current losses in the iron may be ignored or else

included in fictitious circuits. 3. Stray losses in the adjoining structures or frame may be ignored. 4. Space harmonics in the distribution of flux do not exist or may be

ignored. 5. Time harmonics in the voltages, currents, and fluxes do not exist

under steady-state conditions or may be ignored. 6. The effects of armature reaction and of field flux distribution are

sinusoidally distributed in the air gap space. (This, of course, is equivalent to assumption 4, but it is restated in this form for emphasis. This assumption is essentially true in a properly designed machine in which the windings and magnetic paths are properly distributed.)

7. The winding reduction factors, such as the skew, pitch, distribution, and connection coefficients, are incorporated in the "effective" turns of a winding and need not be shown explicitly.

8. The windings on a magnetic structure (the stator or rotor) are sym- metrically placed with respect to a reference axis, and are balanced (that is, each group of windings or phase is like the others).

12-1. The Fundamental Voltage E q u a t i ~ n . ~ ~ . ~ l The form of the equation of voltage and its physical interpretation depends on the choice of the arbitrary reference axis. In order to obtain as general and versatile an expression as possible, and one from which the process of induction in any machine follows as a special case, i t will be assumed that with respect to the reference axis :

1. The flux density, sinusoidally distributed in space, is varying in time in any arbitrary manner.

2. The flux density distribution is moving. 3. The winding is moving.

Fig. 12-1 shows a coil of N turns, axial length I , and coil pitch p centered a t xo(t), a function of time t, with respect to an arbitrary reference axis, and moving with respect to an arbitrary reference axis with a velocity dxo/dt. This coil is linked with a sinusoidal flux density distribution of wavelength 27 given by

/I = B sin (z - y) (12-1) T

and whose amplitude B(t), and displacement y(t) from the reference axis are, in general, functions of time t.

Art. 12-11 BASIC CONCEPTS I N ELECTRIC MACHINES 143

The total flux linked with the coil a t any instant then is

in which K , = sin (pn/2) is the pitch coefficient. The average value of a sinusoid is 2/3r times its amplitude, and the area in this case is 71, so that Eq. 12-2 may be expressed in terms of the flux @ = (2/3r)dB:

in which @ = @(t ) , xo = xo(t), and y = y(t) are all functions of time.

Fig. 12-1.

By Faraday's law the voltage induced in the coil is

This equation shows that a voltage may be induced in the coil in three different ways :

1. By variation of the flux (the so-called transformer component), with both the coil and flux stationary in position, d@/dt.

2. By movement of the conductors (the so-called "cutting action"), with the flux momentarily fixed and unvarying, dxo/dt.

3. By movement of the flux, with its amplitude and the position of the coil momentarily fixed, dyldt.

It is obvious from Eq. 1 2 4 that, if the reference axis is taken as

fixed to the flux wave, then dy/dt = 0, and there is no component of voltage induced by movement of the flux;

fixed on the coil, then dxo/dt = 0, and there is no component of voltage due to cutting action.

1 44 MACHINE ANALYSIS [Ch. I 2

Thus the appearance or vanishing of a component of voltage can often be made to depend only on the arbitrary selection of the reference axi~,~Os~l and consequently the physical interpretation must change with the selection of the axis.

Put 8 = r x 0 / r and p = dldt, and take the flux density distribution as stationary with respect to the reference axis so that dyldt = 0. Then Eq. 1 2 4 reduces to

in which the effective turns N are now assumed to include the effects of coil pitch, skew, distribution, and connection of coils in a group.

12-2. Direct and Quadrature Axes. In machine analysis i t is often advisable, and sometimes necessary (as in the case of salient pole machines),

Quadrature axis ___)

u Fig. 12-2. Direct and quadrature axes and coil designation.

to select a pair of axes centered on the field poles (or main axis of magnetization in the machine) and in quadrature (half a pole pitch) therefrom. These two orthogonal axes are called the direct (d) and quadrature (q) axes, respectively.

Fig. 12-2 shows a symmetrical machine to which are attached direct and quadrature axes, labeled d and q, respectively. The stator (or stationary) coils are designated by s l , 82, etc., counting from the air gap outward. The rotor (or moving) coils are designated by r l , r2, etc., counting from the air gap inward. To fix ideas, brushes are shown on the rotor windings in both


axes. I t is clear that, if a current flows in a winding through the brushes in the direct axis, the resulting axis of magnetization is also in the direct axis. For a single stator layer and a single rotor layer let the currents be id', ids, iQr, iqs and the self- and mutual inductances be L,,, Lds, La,, Lqs, Md, Ma. This simplifying variation of tensor notation avoids a complicated ranking of subscripts and superscripts. There is, of course, no mutual inductance between windings in quadrature (90" apart in space), and for simplicity i t is assumed that the mutual inductance in an axis is the same for all windings. Then the resultant fluxes linked with the rotor in the two axes are

The positions of these two fluxes with respect to the reference axis are indicated in Fig. 12-3, and by comparison with Fig. 12-1 the following identifications are easily made :

y = -90" for q5dT

y = 0 for 4a, 8 = 0 for coil (or brush) on d-axis

8 = 90" for coil (or brush) on q-axis

p8 = velocity of rotation from d- to q-axis

Inserting Eqs. 12-6 and 12-7 in Eq. 12-5, with due regard to Eqs. 12-8, the voltages are found to be

- - - i r ) ( ~ ~ ~ i ~ ' + Mdids) sin (0 + 90) + (Ld,idT + Mjds)p8 cos (0 + 90)]

= -[p(Lq,iqT + lWqiqs) sin (0 + 0) + (L,,iaT + 1M,iaJ)p8 cos (0 + O)]

.'. ed, = -Li)(LdTidT + Mdids) + (LqTiqT + MqiqS)p8]

= -Li)4dT + + g T ~ e l (12-9)

e,, (due to 4 d T )

= -@(Ld,idr + Mdids) sin (90 + 90) + (Ld,idT + Mdids)p8 cos (90 + go)]

= --[p(Lq,iQT + MaiaS) sin (90 + 0) + (LaTiar + M,iaS)p8 cos (90 + o)]

.'. ear = -[p(Lq,iqT + MaiaS) - (LdridT + Mdids)p]0

= -[PA' - 4dTp81 (12-10)

1 46 MACHINE ANALYSIS [Ch. 12

The first term in Eq. 12-9 is an induced voltage due to the variation of the flux. The second term is a generated voltage due to the movement (PO) of the rotating conductors cutting the flux. The corresponding (space) vector diagram shown in Fig. 1 2 4 follows directly from Eqs. 12-6, 7, 9, and 10.

Fig. 12-3. Fig. 12-4. Components of induced and generated voltages.

These equations may be arranged as matrices (taking for simplicity a single stator layer and a single rotor layer of windings), and including the resistances of the windings :

I Y 4 J. +

ind. ind. gen. gen. (12-11)

Thus the voltage equation takes the matrix form

ds dr Pr P

but now the impedance Zap has been generalized to include not only resistances and inductances but also the effects of rotation.

12-3. The Fundamental Torque Equation. The force on a current- carrying conductor of length 1 in a magnetic field of density and a t an angle 1 therewith, by the law of Biot-Savart is

0

-Mope

-%P

-'US - L 0 8 ~

P = Bli sin 1 (12-12)

+ gene + gen. ind. t( ind. 7

0

- L W P ~

-rv - lor^

-Map

-Tds - Ld#

-Mdp

&PO

0


- " d ~

-rv - Ldrp

L d r ~ e

0

Hence the torque on the coil of Fig. 12-1, if the air gap radius is R, is

But 27rR/r = P, the number of poles, and, as before, (2/7r)71B is the flux per pole. Therefore Eq. 12-13 becomes, putting 8 = (7rx0/r) and absorbing the pitch factor K, in t,he "effective" turns I?,

The machine torques on the rotor for the several fluxes (Eqs. 12-6 and 12-7) and rotor currents (per pair of poles and in terms of flux linkages) then are: for &,, y = -90°, and iqT, 8 = 90°,

-- z ( + Maids) cos (90 + 90) = --iW(LdTidr + 1%faids) (12-15) f - '" L id'

for #,,, y = 0, and idT, 0 = 0,

fi = idT(LgTiqr + Mgigs) cos (0 - 0) = idT(Lqriqr + ~ & f ~ j ~ ~ ) (12-16)

for y = -90, and id', 8 = 0,

f3 = idT(Ldjdr + Naids) cos (0 + 90) = 0 (12-17)

for #,,, y = 0, and igT, 8 = 90,

f4 = igT(Lq,ig' + .&fgigs) cos (90 - 0) = 0 (12-18)

Thus, as could have been anticipated, no torque is developed between fluxes and currents in the same axis. There are, of course, torques on the stator also, but these do not produce power and are ignored.

The torque equations may be arranged in matrix form:

Although the torque is an invariant (tensor of zero valence), nevertheless i t may be considered a vector in the direction of the machine shaft if a suitable index is attached. The same index may be used with angular velocity vectors. This artifice will permit the consolidation of the torque and voltage equations into a single equation of motion, as will be shown in subsequent chapters.


12-4. The Energy Relationships. The stored magnetic energy in a system with self- and mutual inductance Lab is

W , = +La,i"ip (12-20)

The electric energy dissipated in the resistance is

W e = Ra,iai@ (12-21)

The kinetic energy stored in the rotating parts of moments of inertia Jab and angular velocities o" is

Wo = &JaSoad (12-22)

The mechanical energy dissipated by friction will be taken as

Wf = raPw"oB (12-23)

These quadratic functions will be made use of in subsequent work.

PROBLEMS

12-1. Confirm all equations in the chapter and verify in detail every step in the derivations.

12-2. Identify the separate terms of Eq. 12-4 in the following mechines and circumstances :

Machine I Transformer D-c generator Syn. generator Syn. generator Induction motor Induction motor Induction motor

Voltage .Reference Axes on Desired

Core Winding Field Armature Field Armature Armature Armature Stator Rotor Rotor Rotor Rotating field Rotor

THE QUASI-HOLONOMIC GENERALIZED MACHINE

Of the three methods of analysis which will be discussed in detail in Chapter 16, the approach which is the most easily set up and which best lends itself to a simple physical picture is through an application of the Maxwell- Lorentz equations in a quasi-holonomic reference frame. The Maxwell- Lorentz equations are the extension of Maxwell's equations (which are true only for stationary media) to moving electrodynamic media. In fundamental machine theory they are Eqs. 1 2 4 and 12-5, which are simply Faraday's law extended to the case of moving conductors in a moving and varying field. The quasi-holonomic reference frame uses as coordinates the direct and quadrature (d and q) axes attached to the machine field. The corresponding generalized (or primitive) machine has multilayer windings on both stator and rotor, as indicated in Fig. 12-2. The resulting equation of voltage is of the type given by Eq. 12-11, and the equation of torque is of the type given by Eq. 12-19. These developments will now be described in greater detail.

13-1. The Generalized Machine (Primitive Machine of the First Kind).7@923~~~ The generalized machine with quasi-holonomic reference axes consists of a cylindrical stator with salient poles and a cylindrical rotor, each with any number of concentric layers of windings. In each winding layer on the stator there are separate coils centered on the direct and quadrature axes. In each winding layer on the rotor there is a pair of brushes (hypothetical perhaps) centered on the direct axis and another pair of brushes centered on the quadrature axis. Currents id flow in the direct axis coils or brushes, and currents iq flow in the quadrature axis coils or brushes.

It is assumed that the windings are so designed (coil skew, pitch, and distribution) that the mmf produced by a current through a brush axis is sinusoidally distributed in space and centered on the brush axis. The significance of this assumption is that the mmf along any brush axis may then be resolved into components along orthogonal directions upon multiplying by cos a: and sin a:, where a: is the angle of brush shift. Likewise, a voltage existing along a brush axis may be resolved into components. Only in this way may voltages, currents, and mmf's be conceived of as space vectors resolvable into direct and quadrature components; see Pig. 13-1.

149

M A C H I N E ANALYSIS

L i d

Fig. 13-1. Current and voltage components as space vectors.

[Ch. 13

Fig. 13-2. Different representations for a rotor winding layer. (The brush is the axis of magnetization.)

jdrl

( a ) ( b )

Fig. 13-3. The generalized machine with two stator and two rotor winding layers.

Art. 13-21 T H E GENERALIZED M A C H I N E 151

A rotor winding layer may be represented by any of the diagrams in Fig. 13-2, but in the interest of simplicity the last one will be used hereafter. Brushes are shown on both the direct and the quadrature axes. Actual brushes are used only on commutating machines, but hypothetical brushes are imagined to permit visualization of the axes along which voltages and currents are defined.

A generalized (or primitive) machine with two layers of windings on both the stator and the rotor is illustrated in alternative forms in Fig. 13-3. The case of a machine with more layers is obvious. The direct and quadrature axes are as shown, and rotation is clockwise (c.w.). Equations will be set up originally for a generator. The generalized machine described here is identical, as a matter of fact, with the machine envisaged by the two-reaction theory of synchronous machines as developed by Blondel, Doherty and Nickle, and by Park. But Kron extended the concept to include a large variety of machines-synchronous, induction, and commutating.

13-2. The Equation of Voltage. There is no loss of generality in restricting the analysis to one winding layer on the stator and one on the rotor, for additional layers merely add additional terms of an obvious nature to the analysis. If needed, the equations for a multilayer machine may be written by inspection and without difficulty. The voltage equation for this case has already been derived in Eq. 12-11, and the various generated and induced voltage components, shown in Fig. 1 2 4 . But in commutating machines the proportionality factors (inductances) between currents and generated voltages due to rotation are somewhat different from the proportionality factors between currents and induced voltages due to transformer action, because the assumption of pure sinusoidal distribution in space is not exact. For this reason, in the interests of accuracy, a distinction needs to be made in the inductance coefficients of Eq. 12-11 by priming those associated with generated voltages. Eq. 12-11 may then be rewritten:

This equation was set up for a generator with rotating armature, C.W.

rotation, and q-axis ahead of the d-axis. It is now necessary to see what changes need to be made for a motor instead of a generator, if the direction of rotation be reversed, if the field instead of the armature is the rotating element, or if the q-axis is behind the d-axis.

152 MACHINE ANALYSIS [Ch. 13 Art. 13-21 THE GENERALIZED MACHINE 153

In the case of a motor, either the voltages are reversed (that is, impressed voltages are substituted for output voltages) or the currents are reversed (for given terminal voltages the currents flow in the opposite direction). Thus the equation of voltage for a motor becomes

The modifications in the basic impedance tensor for a generator (Eq. 13-l), occasioned by changing the direction of rotation p8, the position of the d- and q-axes, from rotating armature to rotating field, or from generator to

TABLE 13-1 Changes Effected by Reversal of Direction of Rotation,

Axes, or Currents and, therefore, all that is necessary is to change the signs of all terms in the

Member

Field / Armature

Axes Change in Zap

of Eq. 13-1 -

'2 -

Right

-

Right

No change. Stationary Rotating C.W.

Change terms. signs of pi

--

Change signs o f p ~ terms.

Stationary

Stationary

--

Stationary

If the direction of rotation is reversed, i t is only necessary to substitute -p8 for p8 in the impedance matrix.

If the q-axis is reversed, then the direction of rotation is reversed and i t is necessary to change the sign of p8 in the impedance matrix. Thus Eq.

Rotating C.C.W.

Rotating C.W. Left

Rotating C.W. Change signs of ;o( terms.

13-3 would become de d r qr 'P

Down Right

(134)

ds

d+ - .

Zmotor - qr

q*

Change signs of p( terms; see Eq 13-5 for a motor.

If the field is the rotating element and is rotating c.w., then relative to the field the armature is rotating counterclockwise (c.c.w.), so that in Eq.

Right

0

-M;pe

Map

T,. + L,,P Rotating

Change signs of all terms; see Eq. 13-3.

0

- L ; d

r , + LavP

Map

rd8 + L d 8 ~

MdP

M;Pe

0 Stationary C.W.

I-- Any of above

M d ~

r , + LdVP

L k p e

0

13-1 for a generator, or in Eq. 13-3 for a motor, i t is necessary to substitute -p8 for p8. For example,

motor, are given in Table 13-1. For example, what changes are necessary in Eq. 13-1 to adapt i t to a revolving field motor, C.C.W. rotation, and q-axis to the left ?

1. Change signs of p0 terms for a revolving field. ( -) 2. Change signs of p e terms for C.C.W. rotation. ( + I 3. Change signs of pe terms for q-axis to left. ( - ) 4. Change signs of all terms to convert to a motor. ( - )

where now df and qf have replaced ds and qs for the field, and da and qa have replaced dr and qr for the armature. The net result then is: change signs of all terms except the p8 terms.


The impedance matrices for machines with more than one winding layer on the stator and on the rotor, as well as for the case of zero sequence currents, will be introduced under appropriate machine headings. For example, a synchronous generator may have several amortisseur windings in both the direct and the quadrature axes, and zero sequence currents may flow in addition to the d- and q-component currents.

For certain reasons it is sometimes advantageous to rearrange the rows and columns of the impedance ma,trix. Where this is indicated in the analysis of particular machines, the changes will be made without apology.

13-3. Components of the Impedance T e n ~ o r . ~ A 2 ~ J ~ The impedance matrix of the generalized machine may be broken down into three component matrices :

1. The resistance matrix, comprising all resistance terms. 2. The inductance matrix, comprising all terms having p as a coefficient. 3. The torque matrix, comprising all terms having p0 as a coefficient.

Thus Eq. 13-3 becomes

so that the impedance tensor may be written

z a p = Rap f Lapp f Gap~e (13-7)

The voltage equation for a motor then is

Art. 13-31 THE GENERALIZED MACHINE

in which the $us-linkage vector is

ds LdsidS + Mdid7

dr Mdida + Ld7idT +a = Lapis =

qr R Lqliqp + Mqigb

qsI Mqiqr + LqsigE 1 and the cross-flux vector is

The torque equation becomes, as seen from Eq. 12-18,

However, since ya has zero elements in its ds- and qs-axes, i t must be multiplied only by rotor currents in Eq. 13-12, so that if indices (r, m) refer to the rotor and (s) to the stator,

T = iaG ip aS

= ir(Grm + Grs)(im + is)

= irGr,jm + irGrsis (1 3-1 3) in which

ds dr qr qs ds qs dr qr

drml dr rl dr rl Qap = -- - -

qr - M : / -hir qr - M & 9. --Ad,

= Q,, + Q,, (13-14)

The reaction torque of a salient pole machine is

But, if the air gap is smooth (non-salient poles), Lir = LA,. and the reaction torque reduces to zero. The only torque in the smooth air gap machine is then

Tsmooth = irGrsis = idrlviiqS - iqrMP (13-16)


In the case of non-commutating machines, the primes may be omitted in the torque matrix, and i t is then seen that this matrix may be obtained &om the inductance matrix by the following changes:

1. The dr row of Gap is the same as the qr column of L,,. 2. The qr row of Gd is the negative of the dr column of L@.

This is equivalent to rotating the pertinent columns of the L,, matrix counterclockwise 90" so that

c a p = Y;% (13-17) where

Similarly,

The tensor y z is the rotation tensor. It rotates a vector with which i t is multiplied by 90' in space. It possesses the group property, as shown in Table 9-2 on p. 108.

It will be shown directly (in Eq. 13-32) that the transformation from stationary ( d , q) to moving (a , 6 ) reference frames is effected by a transformation tensor,

C I = (13-20) q sin 0 , cos 0 ,

in which 8, is the angular displacement, as a function of time t , of the moving axes with respect to the stationary axes. The angle of the moving axes is not necessarily the same as the angular displacement, 8, of the rotor. The derivative of CE, is

* ac;, -- a01

(13-21)

Art. 13-41 THE GENERALIZED MACHINE 157

Multiplication of a vector in the moving reference frame (a , b)-say the current vector ia'-by Eq. 13-20 expresses this vector in the stationary reference frame ( d , q) without changing its direction or magnitude; thus

ia = C,".ia' = ( ia cos 8, - i b sin 8,)d + ( ia sin 8, + i b cos 8,)q

= idd + iqq (13-22)

On the other hand, multiplication of the vector by Eq. 13-21 gives

Comparison of Eqs. 13-22 and 13-23 clearly shows that the q component of ia has become the negative d component of ia, and the d component of ia has become the q component of iu; that is, the vector ia has been rotated in the positive direction of 8, by 90" (Fig. 1 3 4 ) .

If, now, Eq. 13-21 is multiplied by the inverse of Eq. 13-20, there results

and multiplication by this matrix will also rotate a vector by positive 90'. The inverse of Eq. 13-24 is

which is the matrix of Eq. 13-18 and rotates a vector with which it is multiplied in the negative direction by 90".

The matrices (13-24) and (13-25) are elements of the group

13-4. Transformation of the Primitive Machine.698s23924 There are a number of routine transformations to which the primitive machine is subjected in passing to particular machines in its group. They are assembled in this chapter for ready reference. Each such transformation attracts a transformation tensor C:, to the voltages, currents, and impedances of the

158 M A C H I N E ANALYSIS [Ch. 13

equation of performance so that ia = cz,ia' (13-27)

ea, = C:,e, (13-28) = C ; , ~ . Z = ~ (13-29)

13-5. Turn-Ratio Transformation. The constants of the primitive machine may have been calculated, or determined experimentally, for certain turns on each winding (say, equal turns for all windings). The connections

Fig. 13-4. Fig. 13-5. Stationary and moving reference axes.

of the windings of an actual machine will then require that the turn ratios be taken into account. If the windings are (a, b, e, d, . . . ) and the turns are in the ratio ( 1 : n, : n, : n, : - - - ), the appropriate transformation tensor is

a' b c' d'

13-6. Moving Reference Axes. The primitive machine has stationary d- and q-axes. Let a new set of orthogonal moving axes, (a, b) , be chosen and let the displacement of the new axes from the stationary axes be 8,(t), a function of time t. Since the voltages, currents, and fluxes in the ideal generalized machine are assumed to be sinusoidally distributed in space, a component in any direction is simply the projection of the space vector on that direction. Thus in Fig. 13-5 the current ia has components ( id , 67 on the stationary axes and components (ia, ib) on the moving axes, and these' are related by

id = ia cos 8, - i b sin 8, (13-31) iQ = ia sin 8, + i b cos 8,

Art. 13-81 T H E GENERALIZED M A C H I N E

The transformation tensor therefore is b .:, =

(13-32) q sin 0, cos 0,

The current vector ia in the stationary reference frame has the same magnitude and direction as the current vector ia' in the moving reference frame, but of course the components are different in the two frames. Indeed, the components ( id , iq) are constant if ia is constant, whereas the components ( ia, i b ) change with time in accordance with the relationships

ia = ia cos (u - 01) (13-33)

i b = i a sin (U - el) (13-34)

13-7. Slip Rings. Slip rings are connected to fixed points on the revolving armature. If the armature is rotating with an angular displacement 0, a function of time, the projections of the current vector ia on the d- and q-axes give the transformation tensor :

a

This may be regarded as a special case of moving reference axes (Eq. 13-32), in which the a-axis is attached to the slip ring connection and rotates with it, so that O1 = 8. If there is an additional set of slip ring connections in quadrature, then the transformation tensor is identical with Eq. 13-32 (upon putting O1 = 8) .

13-8. Brush Shift. The brushes of the primitive machine are on the d- and q-axes. I n an actual machine the brushes may be shifted by some angle a or /3. Since the fundamental assumption of sinusoidally distributed currents and voltages was put down a t the outset, the component of current in any axis is simply the projection of the space current vector on that axis. Thus in Fig. 13-7 the current im through brushes shifted a t an angle a from the d-axis has components

if = i m cos u

i: = i" sin u

and the current i n through brushes at an angle (90' + /3) has components

i,d = - in sin /I a - n , - i cos

The total direct and currents therefore are

id = i m cos u - i n sin

iQ = i m sin u + i n cos /I

160 MACHINE ANALYSIS

and the transformation matrix is m " n

[Ch. 13

- sin

If the pairs of brushes are a t right angles, ,!? = a. If there are brushes only on the a-axis, the n-column may be deleted. If there are several pairs of brushes a t angles a,, a,, a,, . . . , the trans-

formation matrix is

(13-37) sin a, sin cc,

Fig. 13-6. Slip rings. Fig. 13-7. Brush shifts.

13-9. Displacement of a Stator Coil. The current of a stator coil displaced by an angle a from the d-axis (Fig. 13-8) has components given by the transformation matrix

8

C:, = (13-38)

13-10. Interconnection of Coils. Stator and rotor coils may be interconnected in any fashion. New currents may be selected and the transformation matrix set up in the usual way. For example, in Fig. 13-9

Art. 13-12] THE GENERALIZED MACHINE 161

13-1 1. Successive Transformations. Any number of transformations may be made in succession-turn ratios, brush shifts, coil displacements,

Fig. 13-8. Displaced stator coil. Fig. 13-9.

interconnections, etc.-and the effects of each may be considered separately; or an over-all transformation tensor may be established in accordance with the group property :

q," = c:,c$?$ . . - (1340)

13-12. Steady-State Calculations. Under steady-state a-c conditions, certain substitutions are made in the impedance matrix of the generalized machine. In particular, iff is the fundamental machine frequency, w = 2rf, and

actual speed v =

synchronous speed then

d jw replaces p = - (1342)

dt

and vw replaces pB

p L = j w L = jX

If the a-c currents are given as rms valnes, the torque, in synchronous watts, is

T,., = ~ . e (~LGJ) (1345)

and the torque, in pound-feet, is

33,000 (No. of poles) T = Tm (13-47)

4rf60 x 746


13-13. The Equation of P o ~ e r . ~ 9 ~ . ~ ~ $ ~ ~ The electric power input to the machine (motor) follows from Eq. 13-8 as

di fl P = eaia = Rafliaifl + LaBia - + pOGa8iaip (1348)

dt The stored magnetic energy is

W , = &La,iaip ( 1349)

Since the inductances are constant in a quasi-holonomic system, and symmetric (Lao = LBa), the rate of change of the stored magnetic energy is

Then the terms of Eq. 1 3 4 8 are identified as

RaSiaiS = resistance losses

dip LaSia- = rate of change of stored magnetic energy

dt

PBG,BiaiB = (speed) (torque) = mechanical power output

e,ia = electric power input

13-14. Classification of Machines. There are a great variety of machines, both a-c and d-c, which permit analysis starting with the quasi- holonomic generalized machine (the primitive machine of the first kind with stationary reference axes). A number of such machines are considered in the following chapters, but primarily from the point of view of showing how tensor methods may be used in establishing the machine equations, and not with any intention of exploring their detailed performance characteristics. The derivations for some machines will be carried further in a particular direction than for other machines, because repetit,ion of an obvious procedure would demonstrate nothing new. In general, however, a complete analysis of any machine would include the following steps (but not necessarily in this order) :

1. A brief description of the particular machine, including its salient features, purpose, advantages, and uses.

2. The diagram of connection and its transformation matrix. 3. The R,,, L,,, G,,, Z,,, em, and in of its primitive machine. 4. The transformed L,,,,, G,,,,, Z,,,,, em,, in' corresponding to its diagram

of connections and transformation matrix. 5. The transient analysis based on stationary axes and the derivation of the

equations of performance (voltages, currents, torques). 6. The steady-state Z,,,,, G,,,,, em,, in'. 7. The equivalent circuit of the cross-jield theory.

Art. 13-14] THE GENERALIZED MACHINE 163

8. The steady-state analysis based on stationary axes, and the derivation of the equations of performance (voltages, currents, torques).

9. The elimination of short-circuited axes. 10. The elimination of axes by the polyphase mairix. 11. Transformation to moving axes. 12. Transient analysis based on moving axes. 13. Steady -state analysis based on moving axes. 14. Transformation by symmetrical components (revolving field theory). 15. Equivalent circuit of revolving .field theory. 16. The motional impedance matrix. 17. Oscillations and hunting. 18. Interconnection of machines with other machines and networks.

All the machines discussed in succeeding chapters develop from a primitive generalized machine with stationary axes having two layers of winding in both the stator and the rotor and representation in both the direct and quadrature axes; but for any particular machine certain layers of winding and certain axes may be missing. In such cases the corresponding elements of the general impedance and torque matrices are deleted. The impedance matrix, Z,,,, for the double-layer generalized machine is (for a motor)

ds2 ds 1 drl dr2 qr2

I

I?. L ~ P + 1 M.11~ 1 M.1.P

--,

M.11~ s.1 + 1 MqsP L . 8 1 ~ 1

I-

The inductance matrix, L,,, comprises all terms in the impedance matrix having p as a coefficient, and the torque matrix, G,,, comprises all terms in

[Ch. 13 Art. 13-14] THE GENERALIZED MACHINE 165

other combinations of stator and rotor windings, brush and slip ring arrangements, salient poles, and polyphase coils are possible and have been employed. Rotating brushes open up additional possibilities, and rotating magnetic poles or teeth, with or without windings, still others.20


the impedance matrix having p% as a coefficient; thus

ds2 dsl drl dr2 qr2 qrl qsl qs2

TABLE 13-2

drl Winding layers

Stationary axes Special cam

or combinatioi

of

Machine Stator Stator

Generalized Salient pole

synchronous Round-rotor

synchronous Reactor motor Salient pole

synchronous Single-phase

alternator Slip ring

induction motor Asymmetrical

induction motor Squirrel-cage

induction motor Single-phase

induction motor Shaded pole motor Compound-wound

d-c motor Polyphase

commutator motor Leblanc advancer scherbius advancer DBri motor Repulsion motor squirrel-cage

repulsion motor Fynn-Wcichsel motor lchrage motor lynchronous

converter Frequency converter

ds 1 drl dr2 or2 arl as1 us2

ds2 --

cis 1 I I

In general, voltages and currents may be assumed to exist in all axes, although in any particular case many of them are zero.

Any winding (axis) may be considered non-existent either by deleting the entire row and column in the general matrices given above, or by putting the corresponding current equal to zero in all equations and ignoring the corresponding voltage.

Some machines may be regarded as special cases (or simplified editions) of other machines, and some are combinations of others. In any event they may be classified by groups as shown in Table 13-2 (in which S, I , and C refer to synchronous, induction, and commutator, respectively). Numerous

PROBLEMS


13-2. If the q-axis is reversed, the text states that the direction of rotation is reversed and it is then necessary to change the sign of p0 in the impedance matrix. Show that this is consistent with changing the signs of the q-axis voltage8 and currents.


13-3. A commutating machine is equipped with two pairs of brushes rotating at constant speeds in opposite directions. The pairs of brushes are connected in series. What is the transformation tensor for this arrangement?

13-4. Verify, from fis t principles, the voltage in the qr2-axis in the matrix of Eq. 13-51.

13-5. Why is the conjugate of iu taken in Eq. 13-45, and why the conjugate of GBor in Eq. 13-46?

14

USE OF THE LAGRANGE EQUATION

14-1. The Plan of Attack. Before proceeding with the analyses of particular machines, and in order to have available the tools provided by all methods of attack, three general methods will be developed in the next few chapters and the necessary relat,ionships established. The steps are

somewhat analogous to those encountered in military operations :

Strategic Objectives

The derivation of the equations of performance (voltage, torque, motion, small oscillations, and power) for holonomic, non-holonomic, and quasi-holonomic reference systems

Major Tactics

1. Lagrange's equation of motion for holonomic systems Transformation to non-holonomic systems Establishment of the Boltzmann-Hamel equation Transformation to quasi-holonomic systems

2. Maxwell's equation of voltage for a holonomic system Independent equation of torque Combination to form the equation of motion Transformation to non-holonomic systems Transformation to quasi-holonomic systems

3. Maxwell-Lorentz equation of voltage for a quasi-holonomic system Establishment of the generalized machine Independent equation of torque Combination to form the equation of motion Transformation to non-holonomic systems Transformation to holonomic systems

Organization and Equipment

The laws, symbols, operations, and concepts of dyadics, matrices, vector analysis, and tensors

Minor Tactics

The analyses of particular maqhines, to include

Performance equations Equivalent circuits Vector and circle diagrams


Each of these points of view has certain advantages and each has a different physical interpretation, but all of them give the same numerical results. By each method there will be derived:

1. The equation of voltage. 2. The equation of torque. 3. The equation of motion (combination of voltage and torque).

In each method of attack these equations will be derived and transformed to three reference systems :

1. Holonomic reference system, in which the axes are fixed to the conductors, and there are

n unknown variables xa

n equations of motion

Lagrange's equation of motion and Maxwell's equation of voltage for stationary circuits apply to a holonomic reference frame.

2. Non-holonomic reference system, in which the axes are free and moving, and there are

n old variables xa 2n unknowns

n new differentials ia' = dxd/dt

n equations of motion 2n equations

n equations of transformation

3. Quasi-holonomic system, which is a special non-holonomic system which can be treated as a holonomic system, and there are

k variables identical in old and new axes n unknowns

(n - k ) new differentials

n equations of motion

The three methods of analysis are

1. Lagrange's equation, based on energy considerations, is used to establish the equation of motion in a holonomic reference frame. The equation of voltage and the equation of torque are then special cases of the equation of motion. These equations may be transformed to non-holonomic and quasi-holonomic reference frames in accordance with the following scheme:

Art. 14-11 USE OF T H E L A G R A N G E E Q U A T I O N 169

Motion

Voltage --

The transformations may proceed from 1 to 4 and 1 to 7 with 5 and 6 as special cases of 4, and 8 and 9 as special cases of 7 ; or alternatively from 2 to 5 and 2 to 8, and 3 to 6 and 3 to 9.

2. Maxwell's equation for stationary circuits (Faraday's law) is used to establish the equation of voltage in a holonomic reference frame. An independent equation of torque is established and combined tensorily with the equation of voltage to establish the equation of motion. These equations may be transformed to non-holonomic and quasi-holonomic reference frames in accordance with the following scheme :

Holonomic

Lagrange 1

Special case 2

Holonomic Non-holonomic

Torque 1 Special case 3

Quasi-holonomic

Voltage Maxwell 1

Independent 2

Non-holonomic

Boltzmann-Hamel 4

5

The transformations may proceed from 1 to 4 and 1 to 7, from 2 to 5 and 2 to 8 ; then 4 and 5 may be combined to give 6, and 7 and 8 may be combined to give 9 ; or alternatively from 3 to 6 with 4 and 5 as special cases, and from 3 to 9 with 7 and 8 as special cases.

3. The Maxwell-Lorentz equation (Faraday's law extended to moving circuits) is used to establish the equation of voltage in a quasi-holonomic reference frame. An independent equation of torque is established and combined tensorily with the equation of voltage to establish the eqwation of motion. These equations may be transformed to non-holonomic and holo-

4 --

5

Motion

Quasi-holonomic

7

8

6

nomic reference frames in accordance with the following scheme

9

Combine 1 and 2 3 9

Holonomic

7

The transformation may proceed from 1 to 4 and 1 to 7, from 2 to 5 and 2 to 8; then 4 and 5 may be combined to give 6, and 7 and 8 may be combined to give 9; or alternatively from 3 to 6 with 4 and 5 as special cases, and from 3 to 9 with 7 and 8 as special cases.

Non-holonomic Quasi-holonomic

4 I Maxwell-Lorentz 1 - -- --

Torque 1 I 5

Motion -1- 6 1

Each method of procedure yields various interesting by-products, such as the Christoffel symbols, the affine connection, the torsion tensors, the field tensor, the torque tensor, the rotation matrices, and the non-holonomic objects.

Independent 2 --

Combined 3


14-2. The Idealized Model. In Fig. 14-1 is shown a machine having stationary (d, q) axes, a rotor rotating a t speed p8 = is, a magnetic pole (without any winding) rotating a t speed p8, = iP, and a moving axis a rotating at speed pel = i t The indices (p, T , s) represent mechanical coordinates, so that xP, xT, xS are particular values of xu, and the velocities ip, iT, is are particular values of iu. In such a machine the inductances L,, are not constant but are functions of 8, = xv. If the moving reference axis is fixed

Fig. 14-1. Generalized machine with moving conductors, pole, and axis.

to the rotor, then 8, = 8; or, if the moving pole is fixed to the rotor, then 8, = 8; but, in general, 8, 8,, 8, are all different and are assumed to be orthogonal to each other and to all electric variables xm.

14-3. The Lagrangian Equation of Moti0n.~498 The energy in a magnetic field and the energy in a rotating body are given by the quadratic forms

Wm = iL,,imP (12-20)

W o = ~ J u u ~ U ~ u (12-22)

The energy losses in an electric circuit and in a rotating mechanical system are also given by quadratic forms :

We = Rmnimin (12-21)

W, = R U u ~ U ~ u (1 2-23)

Since Eqs. 12-20 and 12-22 have identical forms, they may be combined to give the total kinetic energy in the electromechanical system, and similarly the energy loss equations may be combined; thus

T = 1 ,aa8iai8 = kinetic energy function (14-1)

F = :R,,+"ia = Rayleigh dissipation function (14-2)

Art. 14-41 USE O F T H E LAGRANGE E Q U A T I O N 171

in which indices (a, /?) include the electric coordinates (m, n) and the mechanical coordinates (u, v), and where now ia represents an electric current if u = m, n, or an angular velocity if u = u, v. Likewise xu will represent an electric charge, xm, or an angular displacement, xu, and ea will represent a voltage, em, or a torque, e,. With this understanding, Lagrange's equation is

Lagrange's equation is valid only in a holonomic reference system in which the coordinate frame is rigidly attached to the conductors.

From Eqs. 14-1 and 14-2, assuming that aa8 is a symmetrical function of the coordinates xu (actually, only of the mechanical coordinates xu), and not a function of the currents or velocities ia (that is, no saturation),

Substituting these values in Lagrange's equations yields the equation of motion :

In this equation the first term includes the electric resistance drops and mechanical friction torque; the second term includes the flux linkage induced (variational or transformer) voltage drops and the torques of angular acceleration; and the final term includes the generated (motional or cutting action) voltage drops.

14-4. Significance of the Holonomic Christoffel Symb01.~~?8 The Christoffel symbol appearing in Eq. 1 4 4 includes all the electric and mechanical coordinates. However, as will be shown, most of its elements are zero, and only three orthogonal "slices" of its representative cube have any


significance. The inductance coefficients a,, = a,, = Lmn = f (xu) are functions only of the mechanical angle, xu = e2. The mechanical moments of inertia a,, = J,,, are constants, and valid only for u = v (there is no such thing as a "mutual" moment of inertia). Also, the Christoffel symbol is symmetric, so that

[UP, Y] = [ h 71 (14-5) and

Since the aWp are either constants (moments of inertia) a,,, or functions of the mechanical coordinates only (inductances) a,,, it is clear that

must include a t least one mechanical index, u. Furthermore, since there can be no hybrids between the moment of inertia and an inductance (that is, no terms like a,,), it follows that when u = u,

when P =

and when

It follows that the equation of voltage is

din em = Rmnin + a,, - + [un, m]i"in + [ku, m]iki"

,at

Expression of Eq. 14-8c follows upon substituting the scalar mechanical angles 8 and e2 for the mechanical coordinate xu.

Art. 14-51 USE OF T H E LAGRANGE E Q U A T I O N 173

The equation of torque is di "

e, = RuViv + a,, - + [mn, u]imin (14-9a) at

div 1 aamn imin = R,,iv + a,,, - - - -

at 2 axTL

It will be convenient a t times to put

aamn N = - = motion matrix mn (14-10) ae The mechanical coordinate may be taken along the shaft of the rotating

machine, u - s. Then it is seen that the Christoffel symbol has zero elements

d q a b s - B

Fig. 14-2.

except for the three parts given in Eqs. 14-8a and 14-9a and shown as the shaded parts of the cube in Fig. 14-2, in which (d, q) are the direct and quadrature axes of the stator, (a, b) the rotating axes of the rotor, and s the mechanical axis of the shaft. Moreover, many of the elements in each of these slices are zero, and consequently the equations are not so complicated as might be supposed.

14-5. The Non-holonomic Equations.24,8 It is required to transform from a holonomic system in which there are n unknown variables xu and n equations of motion to a non-holonomic system in which there are 2n unknowns (the n old variables xa and n new differentials ia') and 2n equations (the n equations of motion and the n equations of transformation). The


reference axes in the non-holonomic system are no longer fixed to the conductors but are in any position and may move a t any velocity. Let the transformation tensor be C:,--a function of the old coordinates. Then

ia = cz,ia' and, transforming Eq. 14-4,

and, by identity Eq. 10-61,

did' = R ,.,. ia' + - + ([af/?', y'] - Qp.,.,,. + Q,,a.,fl. - Qatfl,,,,)ia'is'

at

By comparison with Eqs. 1 4 4 , and with Eq. 14-3 from which it came, the last expression may be written

in which

Eq. 14-13 is therefore the generalization of Lagrange's equation to non- holonomic reference systems. It is called the Roltzmann-Hamel equation and is of great importance in analytical dynam'ics.

The equation of voltage i n a non-holonomic system follows from Eq. 14-12 upon substituting y' = m' for the free index and using the same arguments which preceded Eq. 14-8. Then

din' em, = Rm,,.in' + a,,,, - + [u'n', m']in'iu' + [k'u', m']ik'i"'

dt

din' aamtn, .n,.ur a c ~ = Ilm,,,in' + a,.,. - + -- z z + CK, -

at axu axu'

This equation could have been found directly by transforming Eq. 14-8.

Art. 14-63 USE OF THE LAGRANGE EQUATION 175

The equation of torque in a non-holonomic system follows from Eq. 14-12 upon substituting y' = u' for the free index; then

div' 8 ~ : ' .n, .mp Q, = RutV,iv' + a,,,, - + [m'n', u']im'in' - C=,GE. - ak,,,% z

at axu

This equation could have been found directly upon transforming Eq. 14-9.

14-6. The Quasi-holonornic equation^.^^.^ I t is required to transform from the holonomic system in which there are n unknown variables x" and n equations of motion to a quasi-holonomic system in which there are n unknowns [the k mechanical variables xu" = xu which are identical in the old and new coordinates, and the (n - k) new electric variables xm"] and n equations of motion which are functions of the old mechanical coordinates xu and the new electric variables. The transformation tensor is a function only of the mechanical coordinates xu. The quasi-holonomic system is thus a special case of a non-holonomic system in which certain coordinates do not transform; for this reason it may be treated, in a sense, as though it were holonomic.

Let the transformation tensor be

Then C:,, = f ( xU) and C;,. = a:,, (14-18)

im = CK,,~"" (14-19)

i u = c;,,iu" = ,j;,,iu" = iu"

By virtue of Eq. 14-18,

and therefore the non-holonomic objects in Eq. 14-12 may be considered as tensors :

This tensor is skew-symmetric in its first two indices. The quasi-holonomic equation of motion follows from Eq. 14-12 as

This equation defines the torsion tensor, skew-symmetric in its first two indices :

Ta,.B,,Y,, = Sat,B,,,,, + SBrTyrra,, - Sv,ja3y (14-24)


Since i t occurs in the equations of the other methods of attack, a discussion of its characteristics is postponed to Chapter 16.

The eqmtion of voltage in a qwi-holonomic system follows from Eq. 14-23 upon substituting y" = m" for the free index:

din'' aam..n.. in"iu"

.,,l .u" em,, = ~ ~ , , , , , i " " + am,,,,., - + - + T u n z

at axuw

or, substituting the scalar mechanical angle 0 for xu" and iu' = deldt and observing that

the voltage equation becomes

d em,, = ~ , , , ~ , , i " " + - (am,,n,.in") + Gm.,n,,in'~e

dt

in which G,",. is the torque matrix of Eq. 13-17 :

The eqmtion of torque in a qwi-holonomic system follows from Eq. 14-23 upon substitution of y" = u" for the free index:

If the scalars B and 8, are substituted for xu., and pe and p8, for iv', the -torque (now a scalar) becomes

1 aLrn,,,,,, .mrp .,,,, T = Rpe + JP2e - ~ , , , , , , i ~ ' ; ~ " - - - 2 2 (1629) 2 ae,

in which the first term on the right is the frictional torque, the second term the acceleration torque, the third term the electric torque, and the last term the torque due to moving magnetic paths.

ch. 141 U S E OF THE LAGRANGE EQUATION ~n

PROBLEMS

14-1. Confirm all the equations in the chapter and verify in detail every step in the derivations.

14-2. Identify the three slices in the matrix cube of Fig. 14-2 with the equations.

14-3. Substituting Eqs. 14-14 and 14-15 in Eq. 1 6 1 3 , show that Eq. 14-12 results.

14-4. This chapter is concerned with the development of the equations of motion, voltage, and torque for holonomic, non-holonomic, and quasi-holonomic systems, starting with the Lagrangian equation. Prepare a comprehensive chart on a large piece of paper showing the appropriate equations in each box of the chart, in accordance with the following scheme:

Torque 1 (14-go, b, C) 1 (14-28), (14-29)

14-5. Obtain the quasi-holonomic equations of motion, voltage, and torque by transforming directly from the corresponding holonomic equations.

Eq. of

Motion

Non-holonomic

( 1 1 2 ) (14-13) (14-14), (14-15)

Holonomic

(14-11, (14-21, (14-3), (14-4)

Quasi-holonomic

(14-18), (14-19), (14-21), (14-23), (14-24)

15

U S E OF MAXWELL'S EQUATION

15-1. Equation of V0ltage.7~~ Maxwell's equation of voltage (Faraday's law) applies to stationary circuits, that is, to a holonomic reference frame in which the axes are fixed to the conductors. If the inductances are functions only of the mechanical coordinates xu and not of the currents (no saturation), the equution of voltage in a holonomic system is

Since the only mechanical coordinates are the angular displacements of the rotating armature and magnetic paths, xu = 8, 8, and iu = p8, PO,, so that

in which the motion matrix is defined by

Equations 15-1 and 15-2 are seen to be identical with Eqs. 1 P 8 a to 14&, derived Gom other considerations, and the definition of Eq. 15-3 is the same as for Eq. 14-10.

15-2. Equation of Torque.7~8 The electromagnetic energy stored in the' field is

We = QLmnimin (15-4) I78

Art. 1 5-41 USE O F MAXWELL'S E Q U A T I O N 179

and the rate a t which this is varying with respect to 13 and 8, is the electromagnetic torque :

This torque, plus the mechanical torque T applied to the shaft, is consumed by friction (assumed proportional to the velocity) and the inertial torque, so that

1 aLmn imp 1 aLmn .m.n T = Rpe+Jp28--- 2 2 (1 5-6) 2 ae 2 ae,

Let the torque be considered as a vector, e,, along the shaft with a mechanical index u, and let the angular displacements 0 and 8, be considered as a vector, xu, along the shaft, and their derivatives p8 and p8, as a velocity vector, iu. Then Eq. 15-6 may be rewritten

15-3. The Combined E q u a t i ~ n . ~ ~ ~ As a final step, let aaB represent an inductance Lmn when electric indices (m, n) are employed, and a polar moment of inertia Juv when mechanical indices (u, v) are employed. With this understanding Eqs. 15-1 and 15-7 may be combined into a single tensor equation :

Obviously, when the free index is electric, y = m, this equation reduces to Eq. 15-1, since a,, is not a function of the electric coordinates. And, clearly, when the free index is mechanical, y = u, this equation reduces to Eq. 15-7, since a,, = Juv is a constant. Eq. 15-8 may be rewritten

dim e, = RYaia + a,, - +

dt

dia = R,,ia + a,, - + [ap, y]i"is

at

This equation is identical with Eq. 1 4 4 . Thus Maxwell's equation (really, Faraday's law) yields equations identical with those given by Lagrange's equation. Transformation to other systems-non-holonomic and quasi- holonomic-would therefore be the same.

15-4. Transformation of the Equation of V ~ l t a g e . ~ ? ~ Let the equation of voltage be transformed directly to the quasi-holonomic system.


In the new system the mechanical coordinates are the same as in the holonomic system, xu" = xu(8, O,), and the transformation tensor C$ = f(xu) is a function only of the mechanical coordinates. Then

Substituting in Eq. 15-lb, there results the equation of voltage in the quasi- holonomic system :

din" a = Rm,,,.,in" + L,,,,., - + Cz,, - (LknX~) iu" in"

at axu"

.d d$m" = Rm,,n.in" + - (Lmttn,,in") + ym,,pO = Rm,.,,in'' + - + ym,,p6

dt dt (15-11)

The PO, term vanishes if the magnetic paths do not move. In these equations

is recognized as the rotation matrix y ~ k of Eq. 13-18, and consequently, by the definitions of the torque matrix Gmn given in Eq. 13-17, and of the cross- flux ym given in Eq. 13-19, there have been substituted

Art. 15-51 USE OF MAXWELL'S E Q U A T I O N 181

15-5. The Transformation of the Equation of Torque. The equation of torque (15-7) may be transformed directly from rotating (holonomic) to stationary (quasi-holonomic) axes upon substituting Eq. 15-10.

Substituting f3 and 6, for xu", and p6 and p6, for iv" (and since Lmmn. is a function of 8, only), then by Eq. 15-12 the scalar form of the torque equation becomes

.,,,. ,,, 1 aLm..,.. T = Rp6 + Jp20 - Gm,rnr,.t .t - - - i ""in" (15-15) 2 ae,

There is no need to show the transformation of the equation of motion because it would be exactly like that carried out in Chapter 14.

The equations developed in this chapter, starting with Maxwell's equation, can be further discussed in terms of their physical significance and of the construction of certain matrices and tensors, but, as these same ideas are presented more explicitly in the next chapter, consideration of them is deferred.


PROBLEMS


15-2. What is the torque equation for a, machine having a salient pole rotor without windings?

15-3. What is the electric torque on a salient pole machine whose inductance varies as

L = +(Ld + L,) + 4(Ld - L,) cos 28 + AL cos so2

15-4. Show a side-by-side comparison, including all essential steps, of the derivation of Eqs. 14-4 and 15-9. In your opinion, which method of approach is shorter and simpler? Is one approach more "general" than the other?

15-5. What cardinal assumption was made with respect to the inductances in writing Eq. 15-4? (Recall that all subsequent equations are valid only to the extent that this cardinal assumption holds.)

USE O F THE MAXWELL-LORENTZ EQUATION

16-1. The Equation of Voltage.7.8 The generalized machine described in Chapter 13 has stationary d- and q-axes with respect to which the rotor windings are moving. As shown in Chapter 13, there are two components of voltage : the induced voltage due to the variation of the flux in each axis, and the generated voltage due to the cutting of these fluxes by the rotating conductors. A coordinate frame of this type is called quasi-holonomic. There are n unknowns : the k mechanical variables xu = (8 , 13,) representing the angular displacements of the rotor and the magnetic paths, and the (n - k) electric currents im. There are (n - 1) equations of voltage and one equation of torque.

The equation of voltage for a motor (Eq. 13-10) is given by the Maxwell- Lorentz equivalent of Faraday's law:

din aLm, , w ' u = Rmnin + L,, - + - i p6, + T,,,,% % (16-1)

at ae, In this equation the term

is due to the movement of magnetic paths (Fig. 14-l), such as might be caused by the rotation of a magnetic mass other than the field poles. For the sake of generality, the angular posit~on 8, and velocity p0, of this mass have been taken as different from the rotor position O and velocity p0. In most com- mercial machines the magnetic paths, except for the effect of the rotor teeth,

I83


are stationary, and L,,, # f(f3,) but is a constant, and the term (16-2) vanishes.

The term T,,,,i21 = GmnpH (16-3)

defines one component of the torsion tensor T,!,,, which was encountered in Eq. 14-24. The torque matrix G,, was given in Eqs. 13-6 and 13-14 for a machine with one stator and one rotor winding layer.

Eq. 16-1 exhibits four distinct voltage drops in the quasi-holonomic machine :

1. The resistance drops Rmnin 2. The induced (transformer action) voltages Lmnpin 3. The generated (cutting action) voltages GmninpB

aLmn .n 4. The generated (moving flux) voltages

802 - 2 p 2

The last voltage is usually zero, although it is possible to build a machine (a revolving iron pole without windings) which will develop such a voltage.

16-2. The Equation of The stored magnetic energy is

and, if the inductances Lmn are functions of 0, only, the rate a t which the stored magnetic energy is changing due to the movement of the magnetic path is

and this results in equal and opposite torques on the moving magnetic mass and the rotor.

In addition, there is the torque due to the interaction between the rotor currents and the stator cross-fluxes, as given in Eq. 13-12 by

The sum of the applied torque T, the torque (Eq. 16-5) due to varying magnetic paths, and the torque (Eq. 16-6) due to the interaction between the rotor currents and the cross-fluxes is consumed by the frictional torque and the inertial torque, so that the equation of torque is

1 a'mn .m.n T = R p e + J p 2 8 - y n i n - - - 2 2 2 ae,

1 aLmn .m.n = Rp8 + Jp28 - G,,,i"in - -- 2 r 2 ae,

Let the torque be considered as a vector e,, along the shaft with a mechanical index u, and let the angular displacen~ents of the rotor, 13, and of the magnetic

Art. 16-31 T H E M A X W E L L - L O R E N T Z E Q U A T I O N 185

path, O,, be considered as components of a mechanical vector xu. Then the corresponding velocity is pxu = i u , and Eq. 16-7 may be written

div 1 aamn ;mjn e,, = R,,,iU + J,, - + Tm,,,i'"in - - --

at 2 a ~ ' ~ in which

a,, -- L,,,, the inductance considered as a component of the metric J,, - J , the moment of inertia considered as a tensor of valence 2

(16-8)

tensor

R,, r R, the frictional resistance considered as a tensor of valence 2 TnLn,, = --Gm,, the negative torque matrix considered as a tensor of valence 3

Thus each term of Eq. 16-7 has been endowed with an extra index.

16-3. The Equation of M o t i ~ n . ~ . ~ If, now, indices (a, 8, y) are considered to include all electric indices (m, n , k) and all mechanical indices (u, v, w), the equation of voltage (16-1) and the equation of torque (16-8) may be combined into a single tensor equation:

dia an,, 1 asap ., .B e, = R,,ia + a,, - + - iais - - -- 2 L + Tasyiais

at ax8 2 ax?

dia = R,,ia + a,, - + [ x p , y]iais + F,,ia ( 16-9)

dt

in which a new entity, the field tensor, is defined:

Upon substitution of y = m for the free index, the term aa,g/axm vanishes because the a,l, are not functions of the electric coordinates, and

since the inductances Lmn are functions only of 8,. Ordinarily this term vanishes because the Lmn are constant. Finally, if the torsion tensor is restricted to only two components in the quasi-holonomic reference frame, then

Tasy = Tunnt, Tmnu f Gmn, -Gmn (16-12)

and Eq. 16-9 reduces to Eq. 16-1 for y = m. And, upon substituting y = u for the free index, Eq. 16-9 reduces to Eq. 16-8.


The Christoffel symbol has only three components in the quasi-holonomic reference frame :

[up, yl = [un, ml, [nu, ml, Cmn, ul

The pictorial representations of Eqs. 16-12 and 16-13 are shown in Fig. 16-1. The bulk of these cubes have zero elements, and even the slabs which are present have many zero elements, as will be seen when particular machines are analyzed.

Fig. 16-1. Components of the torsion tensor and of the Christoffel symbol in the quasi-holonomic reference frame.

164. Transformation to Non-holonomic Frarne~.~ss The transformation tensor C$ is a function of the mechanical variables xu and, more specifically, of the angular rotation 8, of the reference axes (in general, different from either the conductor parameter 8 or the magnetic path parameter 8,).

c : . = ~ ( x u ) = f(el) (16-14)

Then Eq. 16-9, the equation of motion, transforms as

dim' = ~ , . , , i ~ ' + a,.,. - + I'a.S,,,.ia'iS' (16-15)

dt

Art. 16-41 THE MAXWELL-LORENTZ EQUATION 187

in which the afine connection is defined : ac:,

ra .S . , y t = c : , c ; q . [ ~ r p , + c; ,u ,~ - + T ~ , ~ , , , (16-16) axst

The first two terms of Eq. 16-16 are the transformation of the holonomic Christoffel symbol.

Eq. 16-15 may be regarded as the most general equation of motion in non-holonomic reference frames. I t gives the same numerical results as Eq. 14-12. In a holonomic reference frame the affine connection becomes the Christoffel symbol.

The equation of voltage in non-holonomic systems may be found as a special case of Eq. 16-15 upon substituting y = m for the free index and properly identifying the resulting terms. It will prove more instructive, however, to transform directly the quasi-holonomic equation of voltage (Eq. 16-1) :

In the development of this equation these points were taken into account: the L,, as defined in the quasi-holonomic frame are functions only of xu = O,, the coordinate of the magnetic path; and the CE, are functions only of xu = el, the coordinate of the rotating axes, and therefore

Then the Christoffel matrix is defined; it is also reducible from the second term on the right of Eq. 16-16:


where Vmn = 0 in the quasi-holonomic system. However, V,, is inserted on the right side of Eq. 16-18 in order to show its law of transformation. This is easily verified by transforming Eq. 16-17 with C$ and associating with C $ V ~ . ~ , ~ : ~ ~ " ~ B ~ the term

contributed from

The torque matrix Gmn is related to the inductance matrix L,, in the quasi-holonomic reference frame by the rotation matrix, Eq. 13-25 or 15-12:

Transforming to a rotating reference frame, as in Eq. 16-17, there results

Comparing Eqs. 16-20 and 16-18, it is seen that

vmtn. = G,.,. Eq. 16-17 therefore becomes

Eq. 16-22 exhibits four different voltages, in addition to the resistance drops RrnTn,in' (see Table 16-1).

TABLE 16-1

I Considering (instantaneously)

Type of Voltage I 1 Reference Axes

I Induced ~,,,.~.~i"' 1 Stationary

aL,,,, .,,, I Generated - z pB, I Stationary

3% I

Generated V,,,,,.in'pB, Generated C7,,,,in'p0 Stationary

Stationary I Moving i

Moving Stationary

Art. 16-61 THE MAXWELL-LORENTZ EQUATION I89

16-5. Equation of Voltage in Holonomic In a holonornic frame the reference axes are attached to the moving conductors so that pel = PO, and Eq. 16-22 becomes

which defines the motion matrix,

This agrees with the definition arrived a t in Eq. 15-3 by a different approach. Eq. 16-23 may then be further condensed to

which is the Maxwell equation used as a starting point in Chapter 15.

16-6. Explicit Reference System. Eqs. 16-17 and 16-22 are associated with an implicit stationary reference system (the quasi-holonomic) to which the velocities pe, pel, and pO2 are referred. If there are no moving magnetic paths, so that the p8, terms may be ignored, Eq. 16-22 may be written


where p8' = p(8 - 8,) is now the velocity of the moving conductors with respect to the reference axes. If the reference system is holonomic, 8' = 0 , the final term vanishes. But in all non-holonomic systems the voltage equation is now the same as in the quasi-holonomic system.

16-7. Transformation of the Equation of Torque to a Holonomic Transforming the quasi-holonomic equation of torque Eq. 16-8,

and remembering that = 6:.,

But a,, is a function of 8, only, and c, is a function of 8, only and not of 8,. Therefore

Also, by Eqs. 16-12 and 16-20 and by making use of the fact that am, = a, , is a symmetric tensor,

.7nll . . m,, .,,,, ac:,, .mrr T,,,,.,,,.z zn" = -Gmvn,.z z = -c;, -

881 am,' '

- - 1 aam-,.. im"in" 2 ae,

Substituting Eq. 16-28 and identity (16-29) in Eq. 16-27,

did' 1 aam-,.. .,.,. 1 aamrTn.. imr,in,, e , , = Ru..v,.iv" + J,,,,., - - - - z an" (16-30) at 2 ae, 2 ae,

Eq. 16-31 agrees with Eq. 14-9c, since 19, = 8 for holonomic frames.

Ch. 161 THE MAXWELL-LORENTZ EQUATION 191

PROBLEMS


16-2. Why is there no term involving O1 in Eq. 16-l? 16-3. In a quasi-holonomic reference frame why are the L,, functions of

O2 only? 164. Why are there only two components of the torsion tensor Tab? and

only three components of the Christoffel symbol [ap, y ] in the quasi-holonomic frame?

16-5. Justify the equivalence indicated in Fig. 16-lb. 16-6. Obtain Eq. 16-17 as a special case of Eq. 1616. 16-7. In this chapter the equations of motion, voltage, and torque were

developed for holonomic, non-holonomic, and quasi-holonomic systems, starting with the Maxwell-Lorentz equation in the quasi-holonomic system. Prepare a comprehensive chart on a large piece of paper showing the appropriate equations in each box of the chart in accordance with the following scheme:

I Eq. of / Aolonomic 1 Non-holonomio 1 Quasi-holonomic I I Mobion 1 1 ( 1 - 1 ) ( 1 1 5 ) ( 1 - 1 )

I

Voltage / (16-231, (16-24), (16-171, (16-IS), I; (16-25) i (16-22), (16-26) (lo-1), (16-2), (163)

Torque 1 (16-27), (16-30), , (16-31) i I

16-8. In the chart of Problem 16-7 fill in the blank spaces (a ) by combining the voltage and torque equations to give the equation of motion for the holonomic frame, and ( b ) by finding the equation of torque as a special case of the equation of motion for the non-holonomic frame.

Art. 17-51 COMPARISON OF SOLUTIONS 193

COMPARISON O F THE GENERAL SOLUTIONS

Three general methods for establishing the general equations of rotating machines were given in Chapters 14, 15, and 16. Although each of these methods yields the complete solutions, they are in somewhat different forms and subject to different physical interpretations. In this chapter they will be compared and their implications discussed. This comparison involves three equations (motion, voltage, and torque), three reference frames (holonomic, non-holonomic, and quasi-holonomic) and three methods of attack

Equation Method Frame

Motion + Lagrange- Holonomic

Voltage - Maxwell

Torque

/ Nan)-h!lonomic t. 4

Maxwell-Lorsntr 4- Quasi - holonomic - Energy

Fig. 17-1.

(Lagrange, Maxwell, and Maxwell-Lorentz). It will probably be most illuminating to examine each equation and each method for each reference frame in succession, and to number the equations as in the previous chapters. The interrelationships are shown in Fig. 17-1. In this chapter, unprimed, primed, and double-primed quantities refer to holonomic, non-holonomic, and quasi-holonomic reference frames, respectively.

Holonomic Reference Frame

17-1. Equation of Motion. The equations as developed in Eqs. 1 4 4 , 15-9, and 16-15 all reduce to

dia e, = R,,iU + a,, - + [up, y]i"is (17-1)

at

for a holonomic system, since the affine connection as given by (10-61) and Eq. 16-16 for a non-holonomic system becomes the Christoffel symbol in e holonomic system. Thus all three methods of attack yield the same equation in the holonomic frame, and there is no distinction.

192

17-2. Equation of Voltage. The equation of voltage as developed in Eqs. 14-8, 15-1, and 16-25 by any of the three methods reduces to

Thus all three methods of attack yield the same equation of voltage, and there is no distinction. This is Faraday's law. The inductances may be functions of the rotor displacement 8 and of the magnetic path displacement 8,.

17-3. Equation of Torque. The equations of torque as developed in Eqs. 14-9, 15-7, and 16-30 have the identical form :

1 aamn imp T = Rye + J p 2 e - 1 aaR^ imin - - - 2 ae 2 ae,

Thus all three methods yield the same equation of torque and there is no distinction. The inductances may be functions of the rotor displacement 8 and of the magnetic path displacement 8,.

In the holonomic frame, Eqs. 17-1, 17-2, 17-3, and 1 7 4 are identical for all three methods.

Quasi-holonomic Reference System

17-4. The Equation of Motion. Both the Lagrange equation and the Maxwell-Lorentz equation lead to an identical form (Eqs. 14-23 and 16-9), for the quasi-holonomic equation of motion. Also, since the Maxwell equation in the holonomic frame is the same as that from the Lagrange method, all three methods may be said to give the same quasi-holonomic equation of motion : dia"

e,,, = R,.,a,.ia" + a,..,, - + [ ~ " b " , y"]ia"ifl" + Ta,.s,.y,.i""ifl" dt

&a" = R,,,,,,.ia" + a,,,,., - + [ul',Y"' r"]i""ifl" + Fy,,,,,iu'' (17-5)

dt 17-5. The Equation of Voltage. The Lagrangian method led to the

equation

1 94 MACHINE ANALYSIS

The Maxwell method led to the equation

[Ch. 17

din'' aarnptn,, em.. = Rmnn,,in" + am,,,,,, - +- in'p02 + y,-pe (15-11)

dt a02 The Maxwell-Lorentz derivation started from

d+m. em. = Rm.,,.,in" + - + y m 4 d = . . - dt

(16-1)

and in its development embraced Eqs. 14-25 and 15-11. Therefore all three methods yield identical equations of voltage, which

take on a variety of forms as shown in Eq. 16-1.

17-6. The Equation of Torque. The Lagrangian method led to the equation

1 aL,.,. .,-.,st T = RpO + Jp20 - Gm,,n,.im"in" - - - 2 z (14-29) 2 ae,

The Maxwell method led to an identical equation (15-15). The Maxwell-Lorentz method established an independent equation of

torque (Eq. 16-7) of identical form. Thus all three methods yield the same equation of torque.

Non-holonomic Reference Frame

17-7. Equation of Motion. Each method of attack gives the solution in the form

but the affine connection has a different definition for one of the methods than for the other two:

Lagrange : ra,p,,y, = [ a l l l , y l ] - Rpry. ,a . + Qy,a, ,p , - Ra,p,,yt

Maxwell : derivable from the same holonomic equation as in the Lagrange method and therefore the same as (14-12)

Maxwell-Lorentz : I'.r,pt,yr = C$C$Z$[~",!?", y"] + Ta,B,yr

However, since the non-holonomic object is skew-symmetric in its first two indices,

!&p.,yia'iB' = 0 (17-7)

Art. 17-81 COMPARISON OF SOLUTIONS

Also, for the same reason, and upon interchanging dummy indices,

as in the final expression of Eq. 14-12. But the CE, included in the non-holonomic objects of Eq. 14-12 refer to

the transformation from the holonomic frame and are not the same as the C:: included in Eq. 16-16, which refer to the transformation from the quasi- holonomic frame. Consequently, it is not possible to make a direct comparison between Eqs. 14-12 and 16-16, and the proof that they give identical results must rest on the argument:

1. The holonomic equation (14-4) transforms to the quasi-holonomic equation (14-23), which is identical with Eq. 16-9.

Holonomic Non- holonomic Quasi- holonomic

Eq. 14-4

Eq. 14- 12

- Eq 14-213 Eq. 16-15 - Eq. 16-9 \

Fig. 17-2.

2. The holonomic equation ( 1 4 4 ) transforms to the non-holonomic equation (14-12), and the quasi-holonomic equation (16-9) transforms to the non-holonomic equation (16-15).

3. Therefore Eqs. 14-12 and 16-15, although different in form, must give identical results.

This can be diagrammed as illustrated in Fig. 17-2.

17-8. Equation of Voltage. The equations of voltage developed in Eqs. 1 6 1 6 and 16-17 from the holonomic and quasi-holonomic frames, respectively, have the forms

din' aamtn, .n, .u, ~ C Z .nt .u, em, = Rmjn.in' + a,,,, - + - 2 '8 + CE. 3aktn.t 1 (14-16)

at axu' din' ac::'

em. = Rm.,.in' + a,.,. - + CE:'am,,n.. - inke l dt 801

aamTnP +- in$& + Gm,,.inpO (16-17) 802

Here, again, the E, is different from the E: and a direct comparison between Eqs. 1 6 1 6 and 16-17 is not possible. However, by letting zu' in


Eq. 14-16 take on the particular values 8, €4, 8, and defining G,.,. in Eq. 16-17 as in Eq. 16-20, the two equations become

din' aa,,,, em, = Rm,,.in' + a,.,. - + - in 'p8

at ae

Thus both equations present the same type of terms. By the same argument as for the equations of motion, the two equations must give identical results.

17-9. Equation of Torque. The equations developed in Eqs. 14-17 and 16-27 have the forms

div' 1 aarn,,, .,, .n, ack,' .,, .n. e,. = R,.,.iV' + a ,,,, - - - - 2 2 - C: , -

at 2 axu' axu,akpm," (14-17)

dig' 1 aarn,,, .,, .n, .,, .,, e,, = R,,,,iU' + a,,, , - - - - 2 z + T,,,. , ,z 2 at 2 axu'

But, by the definition of the torsion tensor (Eq. 14-24), and in terms of the non-holonomic objects (Eq. 14-22),

Thus Eqs. 1617 and 16-27 are identical in form and give the same results. Or, as shown in Chapter 16,

t7-10. The Components of Voltage. A comparison of the equations of voltage for stationary axes (Eq. 16-1) and moving axes (Eq. 16-17) brings to light two different physical interpretations, although both equations give identical numerical results. The different components of voltage, other than resistance drops, as measured by an observer on stationary (quasi-holonomic) axes and an observer on moving (non-holonomic) axes, and the corresponding physical interpretations, are given in Ta.ble 17-1.

Art. 17-10] COMPARISON O F SOLUTIONS 197

Each observer measures exactly the same total voltage, but the components into which i t is divided are quite different for the two observers. Indeed, the moving observer measures a component of voltage Vmtn . in~O1 which the stationary observer does not see a t all. If the magnetic paths (iron masses) move a t the same speed as the rotor, p8, = p0, the two voltage components involving p02 and p8 cannot be separated by either observer.

TABLE 17-1

II)

d Q

3 2 Axes in which

p measured:

8 2 Consider: Ref. axes

) Conductors Magnetic path1 Currents

measured :

1 Magnetic pathi Currents

Voltages measured by observer

Induced

Lrnnpin

Stator and rotor axes

Stationary Stationary Stationary Varying

pi"'


Stationary Stationary Stationary Varying

Generated


Moving Stationary Stationary Constant

Generated Generated

Rotor axes rotor axes

Stationary Moving Stationary Constant

Stationary Stationary Moving Constant

-

a,.,.in'pO

Rotor axes

Stationary Moving Stationary Constant

aL,.,. - a,*


Stationary Stationary Moving Corstant

And, if the magnetic paths, rotor, and reference axes all move a t the same speed, pB2 = pel = p8, both observers measure a single generated voltage, but neither is able to separate i t into its parts. The four voltage components of the moving observer and the three voltage components of the stationary observer add up to exactly the same value, but the way in which this total voltage is distributed among the components is quite different for the two observers, and i t can be changed a t will merely by changing the arbitrary speed, pel, of the reference axes. Whether a voltage is induced by a varying flux, or generated by a moving flux, or generated by moving conductors, or generated by moving reference axes is to a certain extent a matter of the selection of the reference frame, and one type of voltage when viewed from one frame is an entirely different type of voltage when viewed from a different frame.


17-1 1. Recapitulation of the Basic Concepts. The different concepts introduced by the three methods of analysis used in deriving the equations of voltage, torque, and motion are recapitulated here in Table 17-2 for ready reference, and classified with respect to their origin, equation of definition,

H Q and law of transformation. The = or = means that the relationship is true only in a holonomic or quasi-holonomic system, respectively.

17-12. The Two Preferred Generalized Machines. In the great majority of practical cases, the analysis of a machine is the simplest and most clear-cut if referred to either a holonomic or quasi-holonomic reference

( a ) Quasi - holonomic ( b ) Holonomic

.Fig. 17-3. The two preferred generalized machines.

frame. There are a few cases, however, in which more general non-holonomic frames are of advantage, either because of construction details or because of special operating conditions in a machine. But, for most cases, one or the other of the two generalized machines shown in Fig. 17-3 can be adapted. The equations of either may be converted to those of the other by the transformation tensor, and i t *is immaterial which machine is used as a starting point. The conditions affecting a choice will be made clear in Chapters 20 through 23.

The transformation matrix between the two machines and the inductance matrix (Eq. 13-6) of the quasi-holonomic machine are (for 8, = 8):

L,, =

Art. 17-1 21 COMPARISON OF SOLUTIONS

The inductance of the holonomic machine therefore is

b -Mdsint9 -LDs in28 L , - LD cos 28 M , cos 8

qs 0 M , sin 8 M, cos 8 1 Lo, I

where Ls = (L,, + L,,)/2 and LD = (L,, - L,,)/2. The impedance matrix is

where p = dldt refers to both the 8 terms and to the currents. For example,

ds n b PS

Z,,,,ib' = - LDp(sin 28 ib') = - LD(2ib' cos 26pO + sin 20pib')

a

h

!la

Eq. 17-13 may be verified by the general transformation formula (in which p refers only to the currents),

That Eqs. 17-13 and 17-14 give identical results is easily shown by expanding as follows (see Eqs. 16-23, 16-24, 16-25) :

(17-13)

r , ~ + pLd,

M d p cos 8

= (R,,,. + ~ , , , , p + Gm.,,p8 + vrn.,,pe)in'

= Rm,,,in' + p(L,,,tin') (17-15) Therefore

~ , p cos o I I - ~ , p sin o 1 o

r. + p(L , + LD cos 28) / - LDp sin 28 filQp sin 0 1-

C,m.C~.Zmn + LrnnC,m. Z m t n = [ m + P

p refers only to in' p refers to 8 and in'

- M d p sin 0 I -L,p sin 28 I r, + p(L , - L, cos 20) I MQp cos 8 --

0 ~ MQp sin 8 &fop cos 8 + L ~ P 1

MACHINE ANALYSIS [Ch. 17

PROBLEMS


17-2. Check every entry in Table 17-2 against the chapter and equation in which it first appeared.

17-3. Prove that the a f i e connection as given in Eq. 1 6 1 6 reduces to the Christoffel symbol in a holonomic system.

174. The voltage in the brush axis of any machine contains a "generated voltage" term which depends upon the velocity p6 of conductors$xed in position. Discuss the philosophical implications of endowing a stationary conductor with a velocity. There is more to this than meets the eye.20

THE RAISING A N D LOWERING OF INDICES A N D GENERALIZED PER-UNIT CONCEPTS

18-1. The Raising and Lowering of in dice^.^ The metric tensor g,, was defined in Eq. 9-59, and its inverse gap in Eq. 9-65, such that the invariant infinitesimal distance between two points in general curvilinear space is given by

= gas dxb dxB = gas dx, dxs = dxs dxs (9-6 1 )

It was also shown that the metric tensor has the property of raising or lowering indices; thus

g,As = A, (9-73)

I n an electric machine the stored magnetic and kinetic energy is given by

and by analogy with Eq. 9-61 the asp, which include both inductances L,, and moments of inertia J,,, may be adopted as the metric tensor with the property of raising and lowering indices.

The definition of the flux linkage (Eq. 13-11) may be extended to include mechanical as well as electric variables (that is, 4, represents both flux linkage and momentum) by

4, = aasis = i , (18-1)

This relationship shows that the covariant form of the current ia is the $uz linkagf 4,. The energy may then be written in the following equivalent forms :

2T = a,,ibifi = i,ia = aaB4,4, = 4,y = 4,ia (1 8-2)

38-2. The Generalized Per-Unit Equation of M o t i ~ n . ~ The equation ,of motion in its most general (non-holonomic) form is

Let i t be multiplied by the inverse of the metric tensor:


in which

The terms of Eq. l P 1 2 represent voltages and torques, but the terms of Eq. 18-3 represent rates of change of currents and accelerations. The latter equation greatly simplifies calculations, since the inductance and inertia coefficients have disappeared, leaving the coefficient of piY simply unity. Moreover, as will be shown, the resistance coefficients have been replaced by reciprocal time constants, and the aEne connection is made up of inductance ratios only,

18-3. Conversion of the Equation of V ~ l t a g e . ~ The equation of voltage for the quasi-holonomic generalized machine, if there are no moving magnetic paths, is

Substituting in = ank$k

and remembering that the a,, are constant, and that amnank = d:,

18-4. The Mixed Design constant^.^ The a,,, R,,, and G,, for the simple (quasi-holonomic) generalized machine were given as

ds dr qr qs da dr qr qe

Art. 1841 THE RAISING A N D LOWERING OF INDICES

The inverse of the metric tensor a,, is

--

in which appear

L: = (LsLp - = short-circuit inductance (18-8) LT

mutual inductance il = = leakage coeflcient

self-inductance

The mixed tensors R: and G.2 are

in which appear the decrement factors

resistance 6 =

short-circuit inductance

and


in which G: is seen to be the same as the rotation matrix y: upon comparison with Eq. 13-18. Consequently, by Eqs. 18-12 and 13-19,

and Eq. 18-6 may be rewritten

The mixed impedance tensor, for pB = constant, follows from Eq. 18-14 as

This impedance can be expressed in terms of other design constants, and further simplified, if it is transformed by

ds dr qr Pa

ds' dr' qr' qs'

ds

z: = R: + + aipe =

!IT

whereupon the new voltage, flux, and impedance matrices become

Art. 1841 THE RAISING A N D LOWERING OF INDICES 207

in which appear the coupling coeficients in place of the leakage coefficients:

(18-15)

+ P -

!

A comparison of the impedance matrices Z,,, 22, and 22; discloses the following :

Z,, has 9 design constants: r,,, r,,, r,,, L,,, L,,, L,,, L,,, Md, M,, and 4 zero components, and the coefficients of p and p8 are not unity.

22 has 8 design constants: A,,, Ad,, A,,, A,,, d,,, d,,, d,,, d,,, and 6 zero components, and the coefficients of p and p8 are unity.

22: has 6 design constants: l ; ld , q,, dds , Bar, d,,, 6,,, and 6 zero components, and the coefficients of p and p8 are unity.

-&,6,*

s,, + p

-PO

Moreover, the design constants in 22 and 22: are all ratios which have the following considerable advantages over actual resistances and inductances :

1. These ratios are of the same order of magnitude for machines of all sizes, and consequently in design or performance calculations there is less chance of making a decimal error.

2. These ratios are not nearly so sensitive to the effects of saturation as are the inductances themselves.

3. Within reasonable design limits, the decrement factors depend only on the copper in the windings, and the coupling factors only on the iron in the magnetic circuits, so that the effects of design changes are more easily anticipated and determined.

4. Graphical performance curves are more naturally and simply plotted for a whole line of machines in terms of 6 and as parameters.

5. A unified graphical analysis (vector and circle diagrams) can be established corresponding to the concepts of Eq. 18-19 and using only the 6 and q as constants.

PO

+ P

-

PROBLEMS

-&Jar

6,. + P


18-2. Are the Christoffel symbols shown in Eq. 18-4 non-holonomic or holonomic?

18-3. In the matrix of Eq. 18-7 show that the following mutual terms are equal.

- A d ~ / L A ~ = - A d s / L A ~

-'w/Lis = -'qs/L&


184. Complete the following tabulation:

( Impedance 1 Design Constants I Zem Components Coefficients of p and p0 I I ~ o t unity I

18-5. Are the corresponding mutual terms of Eq. 18-19 equal?

SMALL OSCILLATIONS A N D HUNTING

19-1. The Equation of Small Oscil la ti on^.^.^ The general equation of motion, applicable to machines with either stationary or moving reference axes, is

dip ea = RaSiS + a,, - + r,,,,iSiY (19-1)

dt

In general, the design constants Rap and a@, as well as the transformation matrices C:. included in the affine connection rSr,a, are functions of the mechanical coordinates xu.

Suppose, now, that for a machine or a group of machines there suddenly is superimposed on a condition of steady motion or of uniform acceleration abrupt changes in the applied torques and voltages, Ae,. As a consequence, corresponding changes hia occur in the currents and velocities of the rotor. If the reference axes are attached to the rotor, or move a t a speed depending on the applied frequency, they too will oscillate about their instantaneous mean position.

The first-order variation of any quantity (ia, e,, Rap, aa8, rS,,,) in Eq. 19-1 is of the form

The first-order terms in the Taylor series expansion of Eq. 19-1, that is, the first dqrivatives, become

sea d Aib Ae, + - axs AxS = RaSAiS + a,, - dt + r,,,,(iflAiY + hi%,)

d Aib = RaSAib + a,, - + rsY,,(isAiY + iYAiS)

dt


In this equation, Ae, represents a suddenly applied external voltage or torque ; (aea/ax@) Ax@ represents the variation in voltage or torque brought about by the oscillation of the reference axes. The f i t four terms on the right represent the changes due to the oscillating currents and velocities. The last three terms on the right represent the changes due to the oscillating reference axes.

In a quasi-holonomic reference frame (stationary axes), the design parameters remain constant and the last three terms of Eq. 19-3 vanish.

In a holonomic reference frame, the affine connection becomes the Christof- fel symbol, which is symmetrical in its &st two indices, so that

rSy,a(iDAiY + AiW) [By, a](i@AiY + AiSiY) = 2[&, a]iYAiS

and the equation of small oscillations becomes

A number of special cases arise, depending on the reference frame and the underlying assumptions.

19-2. Stationary Non-oscillating Axes. In this case put Axa = 0 in Eq. 19-3; i t reduces to

The equation of small voltages follows upon substituting electric indices for the free indices, u = m, so that

dAin he, = RmnAin + a,, - + l',,,,AiniU + l',,,,inAiU

dt

(or, if the magnetic paths do not move)

dAin = RmnAin + a,, - + ARp,iu + FIm,,Aiu

dt

(and, if there is only one machine)

d hin = RmnAin + a,, - + GmnAinptl + Gmnin(ApO)

dt

= RmnAin + - dA4m + Aymy,pB + V,(AP~) at

Art. 19-31 SMALL OSCILLATIONS A N D H U N T I N G 21 1

The equation of small torques follows from Eq. 19-5 upon substituting mechanical indices for the free index, u = u, so that

d hiv Ae, = RuVAiv'+ a,, - + l?,,,,Ainim + rmn,,inAim

dt

(or, if the magnetic paths do not move)

-dAiv = R,,AiV + a,, - + AFd,im + F,,Aim

dt

(and, if there is only one machine)

Eq. 19-5 defines the motional impedance:

(or, separating the electric and mechanical axes, neglecting R, and if the magnetic paths do not move)

The changes in the currents and velocities are

The motional impedance transforms from the quasi-holonomic system in accordance with the law

19-3. Classification of Comp~nents.~ For the affine connection substitute (Eq. 1P23)

r p y , a ' [ P Y ~ a] + Tpya (19-13)


in Eq. 19-3, which then becomes

[Ch. 19

8% d Aip he, + - AxS = Rap hiS+ a,, - + ([By, LX] + TSy,)(iSAiY + hi%?) axS at

If for the free index is substituted electric ( a = m) and mechanical ( a = u ) indices, respectively, and if the magnetic paths do not move or oscillate, the equations of small voltage and torque result:

aem ~ , , , i ~ ~ i ~ Ae, + - Ax" = axu +oscillating conductors+ 1

+ [nu, m]iYAin + TunmiYAin RmnAin + a,, - dt

oscillating currents-

["-. i n Ax" + -- aamn din Axu + [nu, m]inAiu I axtL axu at 1

oscillating reference axes-

(19-15) and (neglecting R,,)

dAiv Tmn,jmAin + Tmn"Aimin

conductors oscillating currents oscill. ref. axes

In thkse equations the terms which are due to oscillating conductors, oscillating currents, and oscillating reference axes have been segregated or grouped together in that order inside braces. The oscillation of the impressed voltage is due to the oscillation of the reference axes.

19-4. Transformation of the Equation of O~cillations.~ For the generalized machine with stationary axes the equation of oscillations is (from Eq. 19-8)

Ae, = 9'aSAiS (19-17)

Let the currents and voltages be transformed as

ib = Cj,i@' and e, = C,"'e,, (19-18)

Art. 19-41 SMALL OSCILLATIONS A N D H U N T I N G

Substituting Eq. 19-18 in Eq. 19-17,

Multiplying both sides by C:., there results, since C Z , ~ ' = 65,

Ae,, + C,",AC,Y'ey, = C,",9,&C$,AiS' + AC$ifl') (19-19) But, since

A(C;C:') = C,".ACz' + C:'AC:, = 0 i t follows that

C,",AC,Y'e,, = -C,Y'AC:,e,, = - AC:.e, (1 9-20)

Since the mechanical coordinates are not transformed,

and therefore Eq. 19-19 becomes

ac:, ae, , + Z,, --; Ci.iS'AxY' - -Ax7 ax? ax?' The last term of Eq. 19-22 was added to allow for the possibility that the voltage is a function of the coordinates and changes even though there is no change in the applied terminal voltage.

If Air is split into electric current changes Aim' and mechanical velocity changes Aiu', and dummy indices are changed, Eq. 19-22 becomes

in which Axw' has been substituted as the variable in place of hiu' = p AxzL' by transferring the p to the last (geometric) column of sap. If p = dldt


refers only to hia, this equation takes the form (since only the a,, terms of Tap have p coefficients and C;,Z,,iy' = e,)

Equation 19-23 may be expressed in the form

Ae,. = C t , T d B ~ i , h i n ' (oscillating currents)

+ c,Q.T,&!~~Ax"' (oscillating conductors)

+ C:.TaBAC$iV' (oscill. ref. axes, currents oscillating)

+ AC,",Z,,C~iY' (oscill. ref. axes, voltages oscillating)

ae,. -- Axv ' (forced oscill. of impressed voltages) (19-25) axv3 From Eq. 19-9

3 4 , = Zmn + ( rmn , , im + F u n ) + FTVp + Jug2

= Rmn + Lmnp + Gmnpe + rmn,,im + Fbn + PIm3,p + JUvp2 (19-26)

Substituting this in Eq. 19-25 and rearranging the terms, there results the classification

oscill. steady- change in torque due to change in voltage, state voltage oscill. of currents oscill. conductors k

change in torque, torque due to oscill. ref. axes oscill. rotor mass

i r ' forced oscillations of currents f

+ C:,Rm,Ct,Ain' + C:,Lm,Cl,pAin' + C:.Lm,pC~.Ain' + Cz,Omnp0C,".Ain' resistance drop, inductance drop, voltage due to voltage due to

c oscill. currents currents varying moving ref. axes moving conductors I E forced oscillations of reference axes f

+ C~,R,,AC:,i'" + Cz,LmnAC,",pin' + ~,7 ,L, , (pAq, ) i" ' + C ~ , O m n p 0 A C ~ , i d 3 ( ' resistance drop currents varying moving ref. axes moving conductom g m + AC:.RmnC:,in' + AC:,L,,,,C;,pin' + AC:.L,,(pC;,)in' + A C ~ , O , , , , p O ~ , i d

resistance drop currents varying moving ref. axes moving conductors I

Art. 19-51 SMALL OSCILLATIONS A N D H U N T I N G 215

19-5. The Generalized Machine (Stationary Axes).7*8 The motional impedance for a single machine with stationary (quasi-holonomic) reference axes was given in Eq. 19-10. Evidently, this matrix consists of the transient impedance matrix Zmn bordered by the matrices Gmnin, -(Gmn + G,,)i*, and Jp. For the simple generalized machine (by Eqs. 13-3 and 13-14),

By Eq. 19-10, the motional impedance is

The steady-state currents in the bordering row and column are those existing a t the instant of disturbance and are found from the solution of the transient impedance equation,

em = Zmnin or in = (Zmn)-le, = Ynmem (19-29)

The velocitypfl is the constant speed vo existing at the instant of disturbance.


For any machine with stationary reference axes,

For any machine with holonomic reference axes,

In either case, the steady-state currents in the bordering row and column must be replaced by

i m = C;,im' (19-32)

so that only the new currents im' for the particular machine will appear in its motional impedance.

19-6. Invariant Form of the Equation of Small oscillation^.^^ The equation of small oscillations (19-3) was obtained by taking the ordinary derivative of the equation of motion (19-1). But in this form no single term of Eq. 19-3 is a tensor, nor can any combination of the terms be found which is a tensor. Moreover, the transformation to other reference frames becomes complicated and involved. It is therefore desirable to set up the equation of small oscillations in an invariant form such that all its terms are tensors and transformations may be effected by simple tensor transformations.

The intrinsic derivative for a holonomic reference frame was defined in Eq. 10-19 in terms of the Christoffel symbols. This definition is extended to nonholonomic reference frames upon replacing each Christoffel symbol

(!d with the corresponding affine connection FtY . The two are equal in a

holonomic frame. In general then, from Eqs. 10-19 and 10-24, the absolute or intrinsic derivatives and differentials are defined as

dip - dip --- dxy dip - + r:,i" - = - + rS imiy 6t dt dt dt by

It is left as an exercise to show that the absolute differential of the metric tensor is zero, Baas = 0.

The equation of motion (19-1) may be rewritten in terms of the intrinsic derivative as

6ip = R,$ + a,, -

6t

Art. 19-61 SMALL OSCILLATIONS A N D H U N T I N G 217

The equation of small orjcillations (19-3) is then equivalent to

3% de, + - d 2 = d(RaBip) + d a2

(19-36)

But, as was pointed out, this is not an invariant equation. However, by replacing the ordinary differentials with absolute (intrinsic) differentials, there results an invariant form for the equation of small oscillations :

Now, remembering that the intrinsic differential of a product follows the same rules as ordinary differentials,

since 6aap = 0. The (6/6t)6is terms were added and subtracted arbitrarily for reasons which will soon become apparent.

Substituting Eq. 19-34 in Eq. 19-38, th2re results

in which the quantity in parentheses defines a new "resistance tensor of rank 3" designated Rvpa.

Applying the definition (Eq. 19-33) to the terms of Eq. 19-39, there results


The difference between Eqs. 1 9 4 1 and 1 9 4 2 is

= ~,;;&'i& dxY (1943)

in which there has been defined a new tensor, KGifl, called the "generalized Riemann-Christoffel curvature tensor." Its covariant form is

a U S K " . @ = K & Y A &Y .la (1944)

Substituting Eq. 1 9 4 3 in Eq. 19-39 and then substituting Eqs. 19-39 and 19-40 in Eq. 19-37, there finally results as the invariant form of the equation of small oscillations

Now in rotating electric machines the affine connection is a function only of the mechanical variables, and therefore i t can be differentiated only with respect to the mechanical variables, xu = 8.

The curvature tensor (Eq. 1943) is skew-symmetric in its f i s t two indices as can be seen by interchanging E and y in its definition; that is,

Moreover, must always be a mechanical index, u, as can be seen by writing out the affine connections in full, according to identity 10-61, remembering that the metric tensor amp is a function only of the mechanical variables. Furthermore, either E or y must be a mechanical index for the same reason.

From the observations above i t is found that the only curvature tensors which need to be calculated are

The equation of small voltages then is (putting u = m)

The equa,tion of small torques then is (putting u = u)

6% 6 de, + - dxu = Ruv 6iV + a,, - 6iv + RyuuiS dxY f Keyu,iEiu dzy (1948) 6x" 6t

Invariant forms will not be discussed further here.

Art. 19-71 SMALL OSCILLATIONS A N D HUNTING 219

19-7. Sinusoidal disturbance^.^.^ Suppose that the disturbance is a sinusoidal variation ho, where h is its ratio to normal frequency. Such a disturbance may be due to a variation of shaft torque in a motor driving a reciprocating compressor, or to the variation of terminal voltage on a motor connected to a power source containing harmonics.

Where the currents in the bordering row and column are constant (this case includes all d-c machines, synchronous machines with stationary axes, and induction or commutator machines whose reference axes move with the

TABLE 19-1

Oscillating

Fundamental, w

em, im

rotating air gap field), the current and velocity changes Aia are all of hunting frequency h o corresponding to the torque and voltage changes Ae, of the same frequency. Then in the motional impedance matrix the following changes are made :

1. p = jho where

Hunting, o h

oscillation frequency h =

fundamental frequency

Double, w ( l f h)

2. Multiply all terms in the bordering column by w and compensatingly divide the velocity change Ape by o so that

A @ = ~ 2 ) 0 = ~ v = speed variation w o synchronous (or average) speed

3. Multiply all terms in the bordering row by w so that the torque change AT is measured in synchronous watts :

w AT = A ( o T ) = AT,,,,.,

In those cases where the steady-state torques and velocities are constant and the voltages and currents are of fundamental frequency, the torque and velocity changes will be of two different frequencies, m(1 f h) (see Table 19-1). Therefore the motional jmpedance equation must be solved by separating the electrical and mechanical quantities:


From Eq. 1949, the current changes are

hin = (Zrnn)-l (he , - Gmnin hi") = Ynm(Aem - ym Ape) (19-50)

and therefore the equation of small currents is the same as the current equation if the voltage is replaced by the sum of the applied voltage variation he, and a generated voltage ( - y , Ape) due to the oscillation of the conductors. The frequency of the flux y, is w, and that of the oscillation Ape is h o , and there are therefore two product frequencies, (h f l ) o , of the current. Each frequency component of current must be calculated separately by replacing p in Zmn first by p = (h + l ) w and then by p = (h - 1 ) ~ . Then for each hin as found from Eq. 19-50 the change in torque is calculated by Eq. 1949 . But in the equation of small torque the current im is of frequency o and its product with Ain of frequencies (h f 1)w yields resultant frequencies o [ ( h &- 1 ) f 11 = o ( h + 2), oh, o ( h - 2) . Ordinarily the (h f 2 ) o frequencies will be ignored.

19-8. Damping and Synchronizing T0rques.7~~ The equation of small oscillations may be split into its voltage and torque components as in Eq. 1949. If the terminals are short-circuited, Ae, = 0, and if the inertia is ignored, J = 0, these equations may be solved to give the electric torque

Under oscillating conditions a t one frequency, ho, upon substituting p = jho, Eq. 19-51 is a complex number :

AT', = ( T s + jhoT,) A0 (19-52)

in which T D is called the damping torque and Ts the synchronizing torque. The machine will hunt when T D is negative.

In terms of ' T ~ and Ts the equation of small torque is

where the approximation holds for small values of TD/Ts. These equations are solved by successive trials, starting with an assumed

value for h and ending with a calculated value of h in Eq. 19-54. When the two values are substantially equal, the calculation is complete.

PROBLEMS

Ch. 191 SMALL OSCILLATIONS A N D HUNTING 22 1

19-2. Derive Eq. 19-3 by substituting (ifl + Aifl), etc., in Eq. 19-1, multiply out, drop product terms of higher order such as A@ Aiv, and subtract the original equation (19-1).

19-3. Starting with the equation Ae, = f aS Aip, show that the transformation of the motional impedance given in Eq. 19-8 is indeed fa,B' as given in Eq. 19-12.

194. Reconcile Eq. 19-31 with Eq. 19-12. 19-5. Verify the entries in Table 19-1 and the various frequencies described

in the paragraph following Eq. 19-50. 19-6. Show that Eq. 19-53 does imply Eq. 19-54.

19-1. Confirm all equations in the chapter and verify in detail each step in the derivations.

SYNCHRONOUS MACHINES7*'

df 20-2. The Steady-StateVoltages. Under steady-state conditions, the voltage on the field winding is d-c, and the voltages on the short- circuited amortisseur windings are, of course, zero. It is customary to refer the armature voltages to an injnite bus, assuming the field of the alternator to be ahead of the bus field

$q' by the torque angle 8. The angle 6 is positive U

for a generator and negative for a motor, and

20-1. The Transient Impedance (Stationary Axes). The analysis for a synchronous machine may start from either the first (quasi-holonomic) or second (holonomic) kind of generalized machine, but it is generally simpler to start with the first kind. The impedance for a generator with C.W. rotation, direct axis salient pole field (df), amortisseur windings on both axes (dk, qk), and stationary armature (da, qa) is found from Eq. 13-5 by changing all signs (to convert from motor to generator) and adding additional terms for the amortisseur windings, thus :

df dk qk da '?a

its magnitude is proportional to the load on Fig. 20-1. Machine on an

infinite bus. the machine.

f

dk

Z m n = qk

da

qa

Art. 20-31 SYNCHRONOUS MACHINES 223

An inJinite bus is defined as a d-c excited fundamental frequency generator having zero armature resistances and inductances, and no amortisseur windings. Putting

The matrix has been arranged differently from that of Eq. 13-5 in order to facilitate a future elimination of axes. If there is no amortisseur winding in an axis, the corresponding row and column are deleted. If a current is known to be zero (for example, an open-circuited winding), the corresponding

/d' column is deleted.

in Eq. 20-1, i t is seen that the only armature voltage is

L

f - L P ~ ~ f k p --I - " f k ~ -'kd - L k d ~

along the q'-axis of the infinite bus, and i t is constant. If this voltage lags the generator q-axis by an angle 6 , the voltage matrix

of the generator is

1 - - " f d ~ I-

- M f d p

-MfdpB

Thus the voltages along the d- and q-axes are constant, or d-c voltages, if 6 is constant.

If the machine is running below synchronous speed, so that pe = vw, where v is a fraction less than unity,

-rk, - L,P

lnhpe

- M ~ P

- M k d p

-MkdpB

and the voltage matrix becomes

- " k d ~

Under sudden short-circuit conditions, the cancellation of the armature voltages by the short circuit is equivalent to the application of unit functions (in the Heaviside sense), and the voltage matrix is

-M,P

df dk qk da !?a

20-3. Subsynchronous Speed. If a machine is connected to an infinite bus but is running below synchronous speed, its speed is p0 = vw, where v is a fraction (v < 1.0). The fundamental frequency currents applied to the armature develop an mmf rotating in the forward direction a t synchronous speed with respect to the stationary armature. But, since the field poles (and therefore the d-, q-axes) are rotating forward a t speed vw, the armature

I -r - L d ~ I L Q ~ B --

- L , ~ B I - r - Lap

E 0 0 - e sinsot e cos sot (204)


mmf is rotating forward a t speed (w - vo) = so , or slip frequency (s < 1.0), with respect to the d- , q-axes. Therefore the voltages induced in these axes by this armature mmf are a t slip frequency, and in all hp terms of the transient impedance matrix i t is necessary to substitute

and for the speed voltages, represented by the Lp6 terms, i t is necessary to substitute

Lpe = Lvo = v x

The a-c phasor quantities, a t slip frequency, corresponding to the voltage terms in Eq. 2 0 4 are

d - -e sin sot - j B = e cos sot

If the field is open-circuited during subsynchronous speed, the df column in Eq. 20-6 is deleted.

20-4. Synchronous Speed and Steady State. At synchronous speed v = 1.0, and in all speed voltage terms p6 = o. All armature currents in the d- and q-axes are a t zero frequency, so that p = 0 in the da and qa columns of Eq. 20-1. The field voltage, E, and current, I, are d-c, so that p = 0 in the df column. The amortisseur voltages are zero, since these are permanently short-circuited windings. Then from Eq. 20-1 the voltage equations for the amortisseur windings are

If the field is short-circuited, the equation of voltage is found by making the above substitutions in Eq. 20-1, which then becomes (putting X , = o M d k , etc.)

df dk qk da

from which i t is seen that idk - .ak = 0 - %


-jaX, 9'

vX ,

-r - jax , 1 :,a

That is, there are no currents or voltages in the amortisseur windings, and the corresponding rows and columns of Eq. 20-1 may be deleted. The equation of voltage is simply

da

(20-6)

-jsXfd

-jsxod

-r - jaXd

-vXd

df -r f - jaX, - jaXfk

dk - jaXfk -rkd - jaXkd

= qk

da -jaXfd -jaXad

The last term (-XfdI) is the open-circuit induced voltage, ef. The terms Xaiq and -Xdid are speed voltages (not reactance drops) generated by rotation of the conductors of one axis through the armature reaction field in the other axis.

The admittance matrix is

-rh - jaXm

v x ,

-jax,q

and the currents are

d f da

20-5. Elimination of Axes. The rotor axes (df, dlc, qk) of Eq. 20-1 may be eliminated by any of the methods of Chapter 11, or simply by solving the first three rows for the currents in these axes and substituting them in the last two rows. Regardless of the method, the reduced voltage and impedance matrices become

-- -- (20-12)


in which there are the operational inductances

When the resistances are neglected, the operational impedances are called the direct and quadrature subtransient reactances.

The terminal voltages, by Eq. 20-12, are

ed = G ( ~ ) ~ E - Lr + L d ( ~ ) ~ l i d f L g ( ~ ) ~ 8 i g (20-14) eg = G(pIp8E - Ld(p)p0id - [r + L9(p)plia

The torque exerted on the stationary armature is then the sum of the products of the p8 voltage terms and the corresponding currents ; thus

When t,here are no amortisseur windings, the inductance expressions reduce to

in which is defined the Jield time constant,

bf To = - rf

and the short-circuit inductance,

'J f

Under steady-state conditions, putting p = 0 and pe = w, Eqs. 20-13 give

Ld(0) = Ld, LQ(0) = L,, G(0) = M f d h

and Eq. 20-12 becomes

(20-19) e cos 6 - EXfd/rf -xd -r

Art. 20-61 S Y N C H R O N O U S MACHINES 227

and the torque is

Solving Eq. 20-19 for the currents by multiplying both sides by the inverse of the impedance matrix,

204. Equivalent Circuits of the Operational Impedances. If all mutual inductances in an axis are equal, and self-inductances are expressed

( a ) Direct axis

( b Quadrature axis

Fig. 20-2.

as the sum of leakage and mutual inductances, then Eqs. 20-13 may be expressed in the form

where Lad = Mfd = Mkd = Mfk are mutual inductances of the d-axis.

Laa = Mkg. L,, L f , Lk,, Lk, are leakage inductances of armature, field, and amortisseur.

These expressions are now seen to correspond to the equivalent circuits of Fig. 20-2.


20-7. Equivalent Circuit of the Synchronous Machine (Steady- If it is assumed, as in the preceding section, that all mutual

inductances in the d-axis are equal to Lad and all mutual inductances in the q-axis are equal to La,, the impedance matrix for a motor-reversing all signs in Eq. 20-6-becomes

This matrix can be converted to a symmetrical form by the following substitutionls of forward and backward current components (if, ib)

idf - 'df - '8 dj

i d k = ( j f k + jbk)/2 dk

jqk - - 1 -j(ifk - ibk)/2 0, c;, = - qk 2

id" = (ifa + jba)/2 do

iqa = +(;fa - ba 3 i )I- Q"

This is the rotating field, or 2-phase symmetrical component, transformation of synchronous machine theory. *

If the impedance (Eq. 20-23) is transformed by Eq. 20-24 as c ,ZmnC, . and a t the same time the voltage equations (rows of Zmtn,) are divided by s for t,he df, fk, and bk, by (s + v) for the fa, and by (s - v) for the ba rows,


df fk bk fa ba

idf I j f k j bk j j f a jba

in which ef = (e, + je,) and eb-= (e, - je,). The torque matrix follows from Eq. 20-19 upon transforming the terms having v as a coefficient; i t is

I

(20-28) The cross-flux matrix then is

(20-26)

Since the impedance matrix of Eq. 20-27 is symmetrical, i t may be represented by the equivalent circuit of Fig. 20-3a. This may be verified by tracing through the meshes in accordance with Kirchhoff's second law (after putting Xd = Xad + X,, Xq = Xuq + Xt, Xkd = Xad f Xkd, Xk, = Xaq + Xkq, and Xf = Xu, + xf).

The cross-flux (Eq. 20-29) is represented in this equivalent circuit by the voltages E; and Ei, as can be verified by tracing out on the diagram.

09#,. = OGmtn,in' =

fa 4

ba

(20-29) The torque, per phase, is

2jXadidf +j(Xod+Xw)ifk+j(Xod-Xw)ibk+ j (Xd+Xu) i fa+ j (Xd-Xu) ik

-2jX,idf -j(X,-X,)ifk-fiX,,+X,,)P-j(Xd -Xu)i fa- j (Xd+X,) iba - =B


( a ) Equivalent circuit for armature voltages

( b ) Equivalent circuit for field voltage

Fig. 20-3. Equivalent circuit of the salient pole synchronous machine at non-synchronous speed.

Art. 20-a SYNCHRONOUS MACHINES 23 1

In addition to the currents flowing in response to the applied voltages ef and e,, there are the currents flowing in response to the d-c excitation E. The amortisseur currents ifk and ibk are zero, and the other currents are constant under this condition. Therefore, putting 8 = 0 in Eq. 20-23, deleting the k-axes, and transforming by Eq. 20-24,

df fa ba

Eliminating the df-axis,

(20-34)

The equivalent circuit for the d-c excitation is shown in Fig. 20-3b. The torques (in synchronous watts) which would be measured by watt-

meters in the equivalent circuits with armature applied voltages (W, and W,), and with field voltages (W, and W,) are given in Pig. 20-3, and their sum is the total torque. When both field and armature are excited, the two sets of fluxes and currents produce slip-frequency torques, given by the watts and vars W, + jQ5, etc. The crest of the oscillating torque per phase is To.

In machines with smooth air gaps, X,, = X,,, the capacitance element of Fig. 20-3 becomes a short circuit.


At synchronous speed, s = 0, three of the branches in Fig. 20-3 become open circuits, and the two circuits may be combined as shown in Fig.. 2 0 4 . The presence of a negative resistance in this network does not permit a direct representation on a calculating board.

The elimination of the field axis in Eq. 20-31 and the elimination of the amortisseur windings lead to more simplified equivalent circuits, but the matter will not be pursued further here.

Fig. 20-4. Fig. 20-5. Two-phase machine.

20-8. Two-Phase Alternator Referred to Moving Axes.8 The impedance and inductance matrices of a synchronous machine with rotating armature referred to stationary axes are

The transformation matrix to rotating holonomic axes fixed to the slip rings, and its derivative are

ds a b US cls a


By the general transformation formula for the impedance, and for 8, = 8,

ds a b q8

rd, + Ld,p M,(cos 8 p - sin 8 p8) 1 - M d ( s i n 8 p + c o s 8 p 8 ) 0 I

+ 2(Ld. - L.,) sin 8 cos 8 p 8

i Ma(sin 8 P + cos 13 P O )

The V,,,, matrix comprises all terms having p8 as a coefficient. Eq. 20-37 may be brought into simpler form by the substitutions

(COS Op - sin 8 p8)i = p(cos 8 i)

(sin 8 p + cos 8 p8)i = p(sin 8 i )

[sin 8 cos 8 p + (cos2 8 - sin2 8)p8]i = p(sin 8 cos 8 i)

(cos2 8 p - 2 sin 8 cos 8 p8)i = p(cos2 8 i)

(sin2 8 p + 2 sin 8 cos 8 p8)i = p(sin2 8 i)

Hebce, with the understanding that p refers to both the currents and 8 terms,

ds a b 'P ,

czs rds f L d s ~ -pMd sin 8 1 pMd cos 8 0

a pMd cos 8 r + p ( A + B cos 28) - p B sin 28 pM, sin 8 z,,,, = .

b -pMds in 8 - p B sin 28 + P ( A - B cos 20) P M , cos 8

'P 0 pM, sin 8 pM, cos 8 1 pas + L ~ P


The new inductance matrix L,,,, comprises all terms having p as a coefficient, the new resistance matrix comprises all r terms, and the new voltage equation becomes

The saliency of the poles gives rise to the second harmonic term of Eq. 20-38. For a machine with smooth air gap, L,, = LdT, B = 0, A = L drt

and all the double-frequency terms in Eq. 20-38 drop out. The torque matrix is found by transforming the p8 terms of Z,, (the G,,

matrix). It is ds a b 98

20-9. Single-Phase A l te rna t~r .~ The single-phase alternator may be regarded as a 2-phase machine with one phase opened. Hence, deleting the b column a,nd row of Eq. 20-38, the single-phase impedance matrix is seen to be

a,.,. = c:,c;,a,* = b -Md cos 0 - A - B cos 20

pM, sin 0 r,, + LWP

Under steady-state conditions when ids = I (d-c field current) and i a = ~ a ~ i m t (a-c current in the slip rings), it is necessary to substitute p = 0

in the ds and qs columns, and p = jw in the a and b columns. Then the terms jwL may be replaced by corresponding reactances jX if desired.

A - B cos 20

B sin 20

20-10. Three-Phase Synchronous Machine (Holonomic Axes).24 Either the inductance matrix may be set up directly from physical considerations pertaining to the holonomic frame, or else transformed from the quasi-holonomic frame. Since the latter procedure was employed in the case of the2-phase machine in the preceding section, the first procedure will be followed in the present case and then i t will be confirmed by the other method.

Fig. 20-6b shows an ideal 3-phase machine in which the a, b, c phases are 120" apart and phase a is centered a t an angle 8 ahead of the d-axis. Owing to the saliency of the poles, the rotor inductances will vary (sinusoidally, by assumption) from a maximum value L,, when the phase is centered on the pole to a minimum value L,, when the phase is centered on the interpolar axis (Fig. 20-6c). Thus the inductance passes through a complete cycle for every 180" rotation of the armature phase, and its variation

M , cos 0

- M , sin 0


is therefore second harmonic. Let Ll = (La, + LqT)/2 be the average and L, = (La, - LaT)/2 be the variation of the inductance. Then the rotor self-inductances take the form

La = Ll + L, cos 28

L, = Ll + L, cos 2(8 + 120") (2042)

L, = Ll + L, cos 2(8 + 240")

( a ) Quasi - holonomic (b) Holonomic ( c ) Variation of inductance

Fig. 20-6.

The mutual inductance between a phase and a stator (d or q) axis varies sinusoidally, causing flux linkages

4d = Ldsids + MdS[cos 8 iQ + cos (8 + 120") ib + cos (8 + 240") iC]

ias + +,,[sin 8 ia + sin (8 + 120") ib + sin (8 + 240")iC] (2043)

49 = LQS The mutual inductances between phases are determined by finding the

flux linkages of one phase due to a unit current in another. Using the notation for the flux linkage of a due to a unit current ib,

1q,, = +,(,) = 4 d ( b ) COS 6 + 4 d b ) sin 8 = MdT cos (8 + 120") cos 8 + MqT sin (8 + 120") sin 8

= -MI + M , cos 2(8 + 60")

Jloc = +a(c) = +d(e) cos 8 + 4qcc ) sin 8 = Ma, cos (8 + 240") cos 8 + M,, sin (8 + 240") sin 8 (2044)

= -M, + M, cos 2(e + 120")

= +b(c) = 4d(c, cos (8 + 120") + +g,c, sin (8 + 120") = iM,, cos (8 + 240") cos (8 + 120")

+ M,, sin (8 + 240") sin (8 + 120")

= -MI + M, cos 2(8 + 180")


The holonomic inductance matrix then is

Ll + a1 Mds cos0 1 4 cos 26

J f d , cos (0 + - M I + .I;' M2 cos 2::;)

Ll + L, cos 2(0 +

120")

~ f , , cos (6 + 1 240')

- M I + / A&, sin (6 + M , cos 26 1

I

Ll + I M,,, sin (0 + L , cos 2(0 + ! 240')

240') I DlQ, sin (6 + I M,. sin (6 + L,, I

120°) I 240') 1

Now let this be confirmed by a transformation from the quasi-holonomic frame. In that frame the three phases must be represented by three separate rotor windings, as shown in Fig. 20-6a. The inductances are all constants, and the transformation matrix follows upon comparison of Figs. 20-6a and 20-6b. Thus

ds d r l dr2 dr3 qr3 qr2 qr1 qs

I

-- I

I _ _ _-- --

Art. 20-1 I] SYNCHRONOUS MACHINES

ds a b

where

The holonomic inductance matrix checks (Eq. 2 0 4 6 ) exactly when computed by

L,.,, = C,".CEfLrnn (20-48)

Comparing the two methods, it is evident that the transformation from the quasi-holonomic frame required little thinking beyond establishing the transformation matrix and leads to the final result by a routine process. Moreover, the second harmonic terms came in automatically, But setting up the inductance matrix directly in the holonomic frame required considerable appeal to the physics of the situation and, in particular, careful determination of the stator and rotor mutual inductances and recognition of the second harmonic variation of the rotor self-inductances. In all machines whose analysis permits a quasi-holonomic reference frame, it is generally simpler to start from that frame.

In treatises on synchronous machines,14 it is customary to replace the self- and mutual inductances by "3-phase" values, or some other combinations, but no purpose is served here by making these substitutions.

?he holonomic torque matrix is

The holonomic impedance matrix and voltage equation are

z,,,, = (Rrnpn, + PL,,, ,) (20-49)

em, = R,,,,in' + p(Lrn,,.in') (20-50)

From here on, the analysis of a synchronous machine follows the pattern first formulated by R. H. Park and given in detail in Concordia's book.14

20-1 1. Interconnection of Synchronous Machines.'y8 When two or more synchronous machines are interconnected, a transformation matrix


may be set up from the diagram of connections representing the interconnections in terms of the slip ring currents, that is, referred to holonomic axes:

im = C'",jm' (20-51)

Both the old currents im (comprising all currents in the group of machines) and the new currents im' are with respect to moving axes. Each of them may be referred to stationary axes by appropriate transformations :

j m = c;,,jm" (20-52) i m 3 = cr'p (20-53)

Then jm" = c;"jm = c;"c;,jm' = c,rn"c~,cr';a = c,mP'ja (20-54)

,4s an example, consider the two simple synchronous machines, say a generator and a motor, of Fig. 20-7. The second machine is running a t an angle 6 = 0, - 0, ahead of the first machine. The transformation matrices of Eqs. 20-51, 20-52, and 20-53 are then (only the rotor axes need to be considered in these transformations, since the stator currents are independent in the two machines and do not change to new values)

machine axes, c;, =

stationary axes

\ mf \ m/!

cr , ] cos 0, sin 13, I I bl 1 - sin 0, cos B,

8 1 I I

( 1 , I cos 0, 1 sin 6, I

i'"'paxes,i = ;: t cos el sin 1 C,"' moving to --

stationary -sin 0, cos 0,

The transformation matrix of the interconnection in terms of stationary axes then is (by Eq. 20-54)

-cos 6 , sin 6 -7

Art. 20-1 I] SYNCHRONOUS MACHINES 239

It is to be noted that this final result could have been written directly from Fig. 20-7 by placing the (dr,, qr,) axes of the first machine on the sketch of the second machine an angle 6 ahead and reversing the currents.

The same matrix results for rotating field machines if the poles of the first machine are an angle 6 ahead of those of the second machine.

Fig. 20-7. Two synchronous machines.

The voltage and impedance matrices for two machines, after the field and amortisseur axes have been eliminated, follows from Eq. 20-12 :

d l

q~

edl - G,(p)pEl - ed2 cos 6 + G,(p)pE, cos 6 - eq2 sin 6 + G Z ( P ) P ~ ~ E Z sin 6

egl - G,(p)pO,E, + ed, sin 6 - G,(p)pE2 sin 6 - e,, cos 8 + Gz(P)P&Ez cos 6

(20-58) zap = c;"c;"z,," =

d l u 1


However, in Eq. 20-58 e,, cos 6 = e,, and e,, sin 6 = -e,, so that some terms are canceled.

If the machines are similar and running a t the same speed, so that Gl(p) =

G,(p), L,, = L,,, r, = r,, and pel = PO,, the equations simplify considerably.

20-12. Ignoring the Quadrature Axes.8 For balanced voltages and smooth air gap machines the armature voltages and currents in the q-axis are identical with those in the d-axis, but occur 90' later in time. The q-axis may then be eliminated by the polyphase matrix of Eq. 11-17:

The voltage, current, and impedance matrices motor then become

elf d a

for a simple synchronous

The same result may be obtained by simply noticing that the currents in the q-axes are the same as those in the corresponding d-axes but lagging by 90". Hence -j times a q-column is subtracted from a d-axis.

20-13. Hunting and oscillation^.^^^ The motional impedance for the simple generalized machine is given by Eq. 19-28'. The steady-state voltage a t synchronous speed is (from Eq. 20-3),

The law of transformation for the motional impedance from a quasi-holonomic reference frame is given by Eq. 19-23. But for the synchronous machine

Art. 20-131 S Y N C H R O N O U S MACHINES 24 1

with stationary reference axes the transformation matrix has constant elements Cz, = 6:,, so that Eq. 19-23 reduces to

During hunting, the angle 6 varies, and, since 8 = ot + 6,

The transient impedance matrix for a synchronous machine with rotating field and amortisseur windings is given in Eq. 20-1. The torque matrix comprises the terms in Eq. 20-1 having p8 as a coefficient:

Gmn =

d a --- (20-65)

qa -"df -"dk - L d

The bordering column and row of the motional impedance matrix (Eq. 19-28) then are, respectively,

The motional impedance (Eqs. 19-10 and 20-1) then is (using s for the he coordinate) .


while the changes in voltage and current are

If the rotor axes df, dk, and qk are eliminated, the equation of oscillations reduces to

-'. - L d ( ~ l ~ L a l ~ ) ~ O 1 ydp - 00s 4

- -Ld(p)p0 / -r. - & ( P I P i yhp + e sin d (20-70)

i g L d ( ~ ) - yd - i d L q ( ~ ) - ya Jp2

Aid° A84 i

.! A6

in which G(p), L,(p), and L,(p) are as defined in Eq. 20-13. For sustained hunting, substitute p = jhw in Eq. 20-70.

20-14. Interconnected Synchronous Machines Os~illating.'.~ If the rotors of both machines in Fig. 20-7 are oscillating, then AO, - AO, = Ad. With respect to stationary axes, the reference axes of the first machine may be considered fixed while the axes of the second machine oscillate by A6 about their mean position. Then AC:, is not zero. Alsopd = p(02 - 8,) = 0 for the average velocity, and therefore

and p Act, = 0. Since the steady-state currents along the stationary axes are constant, piy' = 0, and as there are no impressed voltages on the slip rings, ae,/aO = 0.

The transformation matrix is the same as in Eq. 20-56 with sl and s, rows and columns added:

Art. 20441 SYNCHRONOUS MACHINES 243

The motional impedance (Eq. 19-24) in this case reduces to

ac;. ac:, 9 a.B. = C:L';,9aB + C,".Sa, - iy' + -

ad* (20-72)

a d in which p refers only to Aia'.

The motional impedance for each synchronous machine is the same as that for the generalized machine of Eq. 19-28. If the last column of Eq. 19-29 is multiplied b y p and correspondingly A0 is substituted for Aia = Ape, and if then the stator axes are eliminated, there finally results for the two machines before interconnection

d r 1 qrl

VdlP /

The first term of Eq. 20-72 is

c:,qpcg. =

dr 1

Lblpel

6 )p + (L;,, sin8 6 + L:,, cos8 6)p& ydlp

+ (L;,, - L;,,) sin 6 cos 6 p02 + (L;,, - Li,,) sin 6 cos 6 p

The transformatl'on matrix (Eq. 20-71) is a function of 6 = 8, - el, and therefore


Then, in the second term of Eq. 20-72, ~1 82

- This result becomes evident without performing the matrix multiplications if it is remembered that C$(aC$/aO) is the rotation matrix of Eq. 13-24. Multiplying Eq. 20-76 by (C:,2YaB), the second term of Eq. 20-72 becomes

s 1 82

i q T 2 [ L k sin 8 P O , - cos 6(r,, + L & r z ~ ) l -iq*[L&,, sin 6 p 8 , - cos 6(r,, + L;,zp)] + ide[L;,2 cos 6 P O , + sin 6(rr2 + L~, ,p ) l -id*[L& cos 6 p 8 , + sin 6(r,, + Li,,p)] -- -

iqT2[L;,2 ~ 0 s 6 P ~ Z + sin 8(r,, + L&,ap)l - iqT2[L~, , cos 6 p 6 , + sin 6(r,, + L ; , , ~ ) ] -idr2[L;,2 sin 6 P O , - cos 4r , , + L;,,p)l +ide[L,,, sin 6 p 8 , - cos 6(r,, + L,,,p)]

-- ion VTZ ( i Lip , - yd2) + idrZ(idr2L;,r + yip,) -iq*(iq*L' dre - Y ~ Z ) - id*(idr2L~12 + % a )

(20-77)

The last term of Eq. 20-72 contains the pre-oscillating steady-state voltage,

drl qrl s l dr2 qr2 s2

e, = ~ 8 , i S ' = aB B earl %I

O ed,2 e m 0 (20-78)

Multiplying Eq. 20-78 by Eq. 20-75, the last term of Eq. 20-72 becomes

drl -ear, sin 6 + e,,, cos 6 e,,, sin 6 - e,,, cos 6 I --- (20-79)

qrl -edT, cos 6 - eQ,, sin 6 e,, cos 6 + e,., sin 6

The motional impedance for the two machines is then the sum of the equations

TZ.,, = (20-74) + (20-77) + (20-79) (20-80)

PROBLEMS

20-1. Confirm all equations in the chapter and verify in detail each step in the derivations.

20-2. If the machine in Fig. 20-1 is running above synchronous speed and is oscillating at an angular velocity hw about its mean speed, what is the voltage matrix corresponding to Eq. 20-3?

20-3. The equivalent circuits of Fig. 20-2 define the subtransient impedances of a synchronous machine. What are the equivalent circuits for the transient impedances of Eq. 20-169

Ch. 201 SYNCHRONOUS MACHINES 245

20-4. The transformation matrix (Eq. 20-24), defines the old currents in terms of the new currents (if, ib). By reading the columns, does this matrix also define the (if, ib) currents in terms of the (id, i q ) currents?

20-5. From Eq. 20-29 identify E; and Ei and from Eq. 20-34 identify E; and El in Fig. 20-3.

20-6. Prove that To in Fig. 20-3 gives the crest of the oscillating torque. 20-7. Show that Eq. 20-38 follows directly from Eqs. 20-35 and 20-36 if

Z,,,, = C ~ , Z m n C ~ , and the p in Z,, refers to both the currents and 6 terms. 20-8. If Ldr = Lqr, then B = 0 in Eq. 20-40 and the torque terms due to

salient poles vanish. is cuss this reluctance torque. 20-9. Check Eq. 20-46 by Eq. 20-47. 20-10. Verify Eq. 20-56 by following the suggestion given directly below

that equation. Show the appropriate diagrams of connection. 20-1 1. Simplify Eqs. 20-57, 20-58, and 20-59 for the case where the two

machines are similar and running at the same speed. 20-12. Obtain Eq. 20-61 without employing the polyphase matriu, but

justify the procedure. 20-13. Is Eq. 20-64 true if ot is not constant? 20-14. Write out the complete motional impedance matrix of Eq. 20-80.

INDUCTION MACHINES

21-1. Stationary Axes77* The equations of performance for both single and polyphase induction machines are most easily derived by starting with the quasi-holonomic generalized machine. Induction machines have smooth air gaps, and their impedance, torque, and voltage matrices referred to stationary axes are

ds dr ar as

I

If the rotor is short-circuited (as in a squirrel-cage induction motor), or closed through external impedances (as in a wound rotor induction motor with external resistors), the applied rotor voltages vanish, e,, = e,, = 0.

If the motor is operating with a-c applied voltages and a t slip s = (1 - v), the steady-state matrices are found by substituting p = jm and p8 = vw. Then Lp = jwL = jX and Mp8 = vwM = v X m and

Art. 21-21 INDUCTION MACHINES 247

21-2. Two-Phase Symmetrical Comp0nents.~*8 Let 2-phase symmetrical (revolving field) components (Fig. 21-1) be introduced by the transformation matrix

Fig. 21-1. Two-phase Ps symmetrical components.

Y

The steady-state matrices become (Z,.,, = CiTE,Z,,) :

1s l r 2s 2r

5%

7 + jX,(l + v )

1s

1 z,,,. = -

2s

2r

(21-4)

l r - j x m - j x , warn,,, = -

2r

Let the l r row of Z,.,, and em, be divided by (1 - v) = s and the 2r row

m-

7 , + j x ,

1 - v )

jXm

T + jX, ( l - v )

r , + j X ,

I j x m ( l + V )


Expressing the self-reactances as sums of leakage and mutual reactances ( X , = x, + X , and X , = x, + X,), doubling the impedances, and halving the currents, these equations correspond to the equivalent circuit of Fig. 21-2.

The torque, in synchronous watts, is

and is given as the sum of the wattmeter readings for the rotor resistance elements.

4 ixs i x , -

Fig. 21-2. Equivalent current of the induction motor.

The rotor axes are short-circuited and therefore may be eliminated by the short-circuit matrix or otherwise. The impedance matrix (Eq. 21-5) then reduces to

in which 2, and 2, are, respectively, the positive and negative sequence impedances.

21-3. Balanced Polyphase Voltages. When the applied stator voltages are balanced (equal magnitudes and the q-axis voltage lagging the d-axis voltage by 90' in time), the phenomena occurring in the q-axis are exactly the same as those which occurred in the d-axis a quarter cycle earlier. Then the polyphase matrix given by Eq. 11-17 may be employed and the equations

Art. 21-41 I N D U C T I O N MACHINES 249

of performance greatly simplified. Multiplying the impedance matrix by the polyphase matrix,

In the steady state, substituting p = j o , p0 = vo, and (1 - v) = s,

For an applied voltage e,, = d the currents and torques are

This motor is a special case of the double squirrel-cage induction motor discussed later. I ts equivalent circuit is the same as that in Fig. 21-7 with the r, and x, elements deleted. I t may also be regarded as a special case of the split-phase motor of Fig. 21-10, where Z = 0, n = 1, and its equivalent circuit is that of Fig. 21-11d.

21-4. Three-Phase Induction Motor.8 The stator phase windings a, b, and c of a 3-phase induction motor are spaced 120' apart, and each phase

M A C H I N E ANALYSIS [Ch. 21

GJ 9 qsl qs2 eqr ern. ( a ) Moving axes ( b ) Stationary axes

Fig. 21-3. Three-phase induction motor.

is assumed to belong to its own winding layer (Fig. 21-3), so that the transformation matrix between stationary and moving axes is

eos 30' 1 ! I The impedance matrix, referred to stationary axes, is (putting Z = r + Lp)

Art. 21-41 I N D U C T I O N MACHINES

The impedance matrix referred to moving axes is

where

A = ( - M p + & ~ p 8 ) / 2 B = ( - M p - A ~ p 8 ) / 2

If the rotor axes (dr, qr) are eliminated,

a

Z,"," = e

(

in which

Transforming to symmetrical sequence axes (0, 1, 2) b y

252

there results

M A C H I N E ANALYSIS

0 1 2

[Ch. 21

(21-16)

in which

The steady-state matrix is given by substituting p = jw and pe =

(1 - s)w, in which s is the slip. If a t the same time 3-phase constants are used by substituting

L, = jL, + JL,, M , = $L, - $Lo, QW = M 2

the steady-state matrix takes the form

which is identical with Eq. 21-7, except for the zero sequence component impedance.

The torque tensor comprises the terms involving s in Eq. 21-17 and is

and the torque, in synchronous watts per phase, is * ," ,"

T = Re (wGm,,,,-i i n ) (21-19)

Art. 21-51 I N D U C T I O N MACHINES 253

The impedance matrix (Eq. 21-17) could also have been found by adding to the impedance matrix of Eq. 21-2 extra zero sequence rows and columns,

0s TO, + jXo8 (21-20)

Or To? + XW

and then transforming by the sequence matrix,

21-5. Single-Phase Induction M o t ~ r . ~ ~ ~ The connection diagram of this motor, referred to stationary axes, is shown in Fig. 2 1 4 . I ts transient

The voltage matrix is 0s 1s 2s

impedance and torque matrices are the same as for the generalized machine with only three axes:

Zmn = dr (21-23)

H e d s - je,.) (21-22) eo

For a smooth air gap, L, = L,, only the - M term of G,, contributes to the

H e d , + je,J

toraue. The steady-state impedance follows upon substituting p = jo and pe =

vw; or is the same as that in Eq. 21-2 with the qs row and column deleted. The applied voltage is e,,.


If 2-phase symmetrical component's are introduced, then for a smooth air gap machine, as in Fig. 21-1, the impedance matrix becomes (after dividing the l r row by s and the 2r row by 2s)

and e,, = e,, = ed /2

It is simpler to apply 2-phase components to the rotor only, leaving the stator axis ds unchanged.

r, i x , i x ,

Fig. 21-4. Fig. 21-5. Equivalent circuit of the single-phase induction motor.

Hence

Putting X, = X,, + x , and X, = X , + x,, where x, and x , are leakage impedances, the equivalent circuit is as shown in Fig. 21-5.

The torque is * *

A n 2 1 4 1 I N D U C T I O N MACHINES

The motional impedance Tap of the single- phase induction motor is the same as that of the generalized machine (Eq. 19-29), after putting L d , = L , , = L , , L d r = L , , = L r , M d = M * = M , deleting the qs column and row, and putting iQs = 0.

or1

21-6. Double Squirrel-Cage Induction Motor.8 The connection diagram of this motor, referred to stationary axes, is shown in Pig. 21-6. I ts impedance matrix is the same as that of the Fig. 21-6. generalized machine with one stator and two rotor layers of windings :

Under steady-state conditions, where p = jw and pe = vw, the q-axes may be eliminated by the polyphase matrix, or simply by substituting iq =

-jid and combining columns with the same id :

+ j m i jxmz 1 --

Z ,,,, = drl jsX,, r, + jsX, jsX,, (21-29) ---

I

I The torque matrix is


If the rotor rows of Eq. 21-29 are divided through by the slip s, and if

in which xr is the leakage reactance of both rotor windings and xl and x2 are the individual leakage reactances of the two rotor windings, the equivalent circuit is then seen to be that of Fig. 21-7.

Fig. 21-7. Equivalent circuit of the double squirrel-cage induction motor.

The torque is

21-7. Moving Axes.8 Let referred matrix

Fig. 21-8.

the to

2-phase induction motor (Fig. 21-8) be moving axes by the transformation

Art. 21-71 I N D U C T I O N MACHINES

Then the impedance Z,, of (21-1) becomes

M p sin 8 a l M p c o s 8 ~ r + ~ , 0 1 1 (21-34)

b - M p sin 0 r + L,p M p cos 8

Pa 0 M p sin 8 M p cos 0 r, + L,p

ds a b '2s

Under balanced conditions, when i b = -kia and i'JS = +ids, the matrix reduces to

7. + L8p M p cos 8 - M p sin 8

in which j = 2/---I refers to fundamental frequency in the stator and k =

0

ds a

6 1 to slip frequency in the rotor. The torque matrix comprises the p8 terms of Eq. 21-34, after differentiat-

ing the cos 8 and sin 8 terms ; for example, Mp cos 8 = (- M sin 8 p0 + M cos 8 p). Then

ds a b 98

r, + Lap ; MPE" -

M ~ E - ' ~ r + L,p

ds

Z,.," = a

However, in a smooth air gap machine the a and b columns do not contribute to the torque and may be ignored. Thus, under balanced conditions, all that remains is

-1 (21-37) a,,:,. = a -jM(cos 8 - j sin 8 ) = a ME-'^

r~ + Lap

Mp(cos 8 - j sin 8 )

The steady-state impedance follows from Eq. 21-35 upon writing id* = Id8Pt and ia = I a & O t , putting the dWt and cksWt inside the impedance

Mp(cos 8 + k sin 8 )

7 + LTP


mat,rix, and operating on the products with p. The impedance becomes

Also

21-8. Double-fed Induction M o t ~ r . ~ If the stator of an induction motor is supplied with voltages a t fundamental frequency, o , and the rotor with

OS voltages a t slip frequency, so , the rotor will revolve a t a speed p8 = (1 - s)o.

\pBIZu The impedance and torque matrices referred to stationary axes are given in Eq. 21-1, and the inductance matrix, L,., is simply the coefficients of p in Eq. 21-1. The air gap flux revolves a t synchronous speed, o , and the

X voltage matrix therefore is (if 6 is \bs the angle between stator and rotor

Fig. 21-9. fluxes)

ds dr qr P

em = -e, sin wt -e, sin (wt + 6) e, cos (at + 6) e, cos wt

Select new axes (a , 6) revolving a t synchronous speed and let 8, = o t be the instantaneous displacement of these axes. The transformation matrix then is (Fig. 21-9)

as ar br bs !

as cos e I I , -sin 0,

dr 1 cos 0, -sin 8, c;, = --I- (21-41)

qr 1 sin 8, cos 9,

I qs sin el i I ' cos 0,

I I

The new voltage matrix becomes

Art. 21-91 INDUCTION MACHINES 259

The law of transformation of the impedance to rotating axes gives act. z,,,, = c;,c,",z,, + C;,Lmn - pel = ,301

Under steady-state conditions, the impressed voltages are constant in Eq. 2142, since 6 is constant, and, putting p = 0, p8, = o , and p(81 - 8) =

the impedance becomes us ar br bs

I !

as rs 1 1 - -x, pp -- ----

ar z ,,,, = (21-45)

bs X ,

21-9. The Capacitor and Split-Phase Motor.8 The capacitor motor has a two-phase stator, one winding of which is supplied with single-phase a-c voltage, and the other winding, having n times as many turns, has an impedance Z (usually a capacitor) in series. The rotor has a closed winding

;'Is! & or squirrel cage. Although the second stator phase is usually closed through its impedance, for the sake of generality it will be considered here as having an applied voltage. The diagram of connections is

iqs+

shown in Fig. 21-10. The impedance, torque, and voltage matrices of the generalized machine are given in Eq. 21-2, but this needs to be modified by multiplying the qs row and column by n to account for the difference in turns between the

I3 Fig. 21-10.

qs- and ds-axes (which is all that a turn-ratio transformation matrix would accomplish). Then in addition the series impedance Z must be added to the stator impedance in the qs-axis.


If the rotor columns of Z,,,, are eliminated, the impedance matrix of ... .. Eq. 21-49 reduces to

Is 2s

zm+,. =

in which the sequence impedances are

The equivalent circuits corresponding to Eqs. 2 1 4 9 and to Eq. 21-50 are shown in Fig. 21-11, together with two special cases. The E,, and E,, of Eq. 21-50 are easily verified on the equivalent circuit.

21-10. Shaded-Pole Motor.8 The shaded-pole motor of Fig. 21-12 has a single-phase stator winding, ds2, and a short-circuited stator winding, a, considered to be in a different layer. The squirrel-cage rotor is represented by a pair of short-circuited brushes:

I sin a I I

Fig. 21-12. I I The Gansient impedance matrix of this machine, which is the same as that of the generalized machine, is

Art. 21-1 I ] INDUCTION MACHINES 263

Under steady-state conditions, substituting p = jo and p0 = vw, the impedance is

rs + jXds cos2 a I

jX,, cos u jXmd cos a jX,, sin a + jX,, sinZ u I

I I I I jX,, sin a

v x m d l 1 - flXmd cos a v X d ' + jxgr I

The torque tensor is

as2 ds l

21-1 1. Motional Impedance of the Polyphase Induction M o t ~ r . ~ The motional impedance matrix of the genemlized machine, referred to stationary axes (d, q), was given in Eq. 19-28. For the polyphase induction motor with smooth rotor, L,, = La, = L,, M, = M, = M, L,, = La, = L,. Let moving axes (a, b ) , rotating a t synchronous speed,pe, = o, be introduced, as in Fig. 21-9 and Eq. 21-41 :

dr 1 cos 0, -sin 0,

CE, = qr 1 1 sin el 1 cos el , cos ' qs sin el

8 1


The new motional impedance then is, putting 8, - 8 = 6,

ac;, 3?a.s. = c , " , ~ , , C ' , B ~ + CK, Lmn - pel (19-31)

a h as ar br bs s

When the applied voltages or torques are of hunting frequency, hw, the currents and velocities are also of hunting frequency. Substituting p = jhw, pel = w , p 6 = p(8 , - 8 ) = sw and multiplying the last row and column by w , the motional impedance becomes

bs X , Xm T , + jhX. 0

s ibrXm -ia'Xm jhw3J

The currents in the bordering row and column are those corresponding to the steady-state conditions prevailing before hunting. The transient impedance matrix was given in Eq. 21-43, and the steady-state impedance in Eq. 2 1 4 5 , from which the prehunting steady-state currents may be found as

in' = (Zmtn,)- lemf (21-59) and substituted in the bordering row and column.

When fundamental frequency voltages are applied to the stator and slip frequency voltages to the rotor, then both the stator and rotor air gap fields revoke a t synchronous speed, but the stator field leads the rotor field by an angle

6 = 8 , - 8 , - e in which 8, is the position of the stator flux, (8 + 8,) the position of the rotor flux, and 0 the displacement of the rotor. The applied voltages then are

Art. 21-12] I N D U C T I O N MACHINES 265

The rate of change, if 6 is varying, is

I n accordance with Eq. 19-3, the term appears on the left (or voltage) side of the equation of oscillations. However, it may be transferred to the right-hand side,

as ar br bs s

aea I

- = -e cos 6 ; -e sin 6 ' 0 0

ae Ae, = 3?,,Ai@ - -' A6 as

as

and placed in the s column of the motional impedance matrix. Thus matrices (21-58) and (21-59) are augmented in the s column by -e cos 6 in the a, row and by -e sin 6 in the b, row.

21-12. Indhction Motor with Oscillating Load.8 Let the steady- state veloci$y, v w , and torque, T , be constant and the applied voltages and currents be of fundamental frequency, w . If the superimposed oscillatory torque, AT, is of hunting frequency, hw, the corresponding velocity changes Aiu will also be of this frequency, but the current changes Aim will have the two product frequencies ( h f 1)w. That is, the frequencies involved between

(21-61)

the variables are as shown in Table 21-1.

TABLE 21-1

I 1 Steady State I Oscillating 1

Mechanical vw, T Constant 1 A N , A? Hunting (hw) 1 c t i c 1 e, i Fundamental ( w ) Ae, Ai F'roduct ( h * 1)-

The motional impedance of the unbalanced induction motor is the same as that of the generalized machine (Eq. 19-28).

Assuming the current changes to have frequencies ( h f l ) w , and the average speed p8 = vw, the impedance matrix (Eq. 19-28), excluding the bordering row and column, upon substituting p = j (h f 1)w becomes

The voltage changes have frequencies ( h f 1)w.


In the bordering row and column the steady-state currents are complex numbers of fundamental frequency, w. If the s coordinate is taken as ApO/w = A(vw)/w = Av and if the torque is measured in synchronous watts, it is necessary to multiply the bordering row and the bordering column of Eq. 19-28 by w. Then for current changes of frequencies ( h + l ) w and ( h - l ) w the products of these changes by the steady-state fundamental frequency currents yield torque frequencies

( h + l ) w w = hco and ( h + 2)co

( h - l)o f w = hw and ( h - 2)w

of which the ( h f 2)w frequencies will be ignored. But the product of two quantities of different frequencies,

A cos (wlt + a ) = Re ,4si("lt+") = Re Aei"lt

B cos (w,t + B) = Re ~ $ ( " a ~ + ~ ' = Re Bsi"zt

yields two frequencies (w , f w,) :

AB AB cos (w,t + a ) cos (w,t + B ) = - cos [(o, + co,)t + a + B ]

2

Hence the product having the sum of the two frequencies is found by multiplying the two complex numbers together, while the product having the difference of the'two frequencies is found by multiplying one complex number by the conjugate of the other. Accordingly, the bordering column of Eq. 19-28 is taken as

ds dr P ' PS

I column: s 1 0 iarXqp + iaaXmq - i d d - i d d 1 0 I (21-64)

I I I I I for (h + l ) w frequency of Ae,

Art. 21-12] I N D U C T I O N M A C H I N E S

column: s

Likewise, the bordering row of Eq. 19-28 is taken as

for (h - 1 ) o frequency of AT

for ( h - l ) w frequency of Ae,

row: 1 i d - X ) - ig8Xmq I idr(XdI - Xq,) + idaXmd I -CrXmq l(21-67)

for (h + 1)o frequency of AT

(21-65) *

* I * * i

If, now, the motional impedance equation is expressed in the form

n v

A e a ' = 2 a8 A i ~ = m ~ l = ~ ~ u Ae, (21-68)

the current changes hin for each set of frequencies may be solved for separately in terms of the voltage changes and then substituted in the torque change :

Ain = (Zmn)-'(he, - Zm,Aiv) (21-69)

0 0

Ae, = Z,,Ain + Jo3Aiv (21-70)

iqrXq, + iaSXma , -idsXmd - idrXdr

PROBLEMS


21-2. Obtain Eq. 21-17 by using Eqs. 21-20 and 21-21. 21-3. Derive the motional impedance matrix for the single-phase induction

motor. 214 . Derive the motional impedance matrix for the double squirrel-cage

induction motor. 21-5. Derive the motional impedance matrix for the capacitor and split-phase

motor. 21-6. Derive the motional impedance matrix for the shaded-pole motor. 21-7. From its impedance matrix develop the steady-state equivalent circuit

for the double-fed induction motor. 21-8. From its impedance matrix develop the steady-state equivalent circuit

of the polyphase induction motor. 21-9. Solve Eqs. 21-69 and 21-70 formally for the change in velocity.

COMMUTATOR MACHINES

There is a great variety of commutating machines of both the a-c and d-c type. Since such machines have stationary brushes, their performance equations are most easily found by starting with the quasi-holonomic generalized machine (the primitive machine of the f i s t kind). The analysis for a number of such machines will be given in this chapter, but primarily from the point of view of using tensor methods for establishing the machine equations and not with any intention to explore detailed performance characteristics.

22-1. The Repulsion M o t ~ r . ~ * ~ The repulsion motor (Pig. 22-1) has an a-c field on the d-axis, a smooth air gap, and a short-

l d circuited pair of brushes on the commutator shifted an

ids! angle u from the d-axis. The corresponding primitive machine is that of Fig. 1 3 3 , having an impedance

I (Eq. 1 3 3 ) with the qs column and row deleted: thus

Fig. 22-1. - M P ~ -&PO T, + L,P

Its transformation tensor is d s a

Art. 22-11 COMMUTATOR MACHINES 269

The impedance of the motor then is d s a

The torque tensor comprises the coefficients of p0 in Eq. 22-3, or it can be verified by direct transformation of a,, out of Eq. 22-1 :

d s Z ,,,, =C:,C:,,Z,, =

a

ds dr

a,,,, = c:,q.amn = c : . ~ , - The admittance tensor is the inverse of Eq. 22-3, or

268

7, 4- L ~ P - M p cos u ym'n' = ( k , ) - l d8 81 m1 (22-5)

a - M ( p cos a - p0 sin a ) r. + Lbp

where

D = (L,L, - M2 cos2 u)p2 + (r,L, + r,L, + M2 sin u cos up8)p + r,rr (22-6)

The only applied voltage is e,, = e, since e, = 0 for the short-circuited brush axis, and therefore

id, - (rr + L,p)e

(22-3) ?a + Lap

M ( p cos a - p0 sin a )

- M cos u pe + M sin a pee ia D

M p cos u

T, + L,p

and for constant speed, p0, the explicit solutions are given by the rules of operational calculus (or Laplace transforms).

The instantaneous torque is T = Q,,,,i""i'" = - M sin a id%" (22-8)

Under steady-state conditions, with an a-c voltage e applied, substituting p = jo and p8 = v o in the previous equations gives the corresponding a-c values. Then Eqs. 22-7 become

ids - (rr + jXJe D

(-jXm cos cr + vX, sin u)e ia = (22-9)

D D = (-XJr + X z cos2 u + r,r,) + j(rJs + rJ, + vX; sin a cos u)

and the synchronous torque by Eq. 1 3 4 5 is

eX,(v sin u + j cos a) = Re [ (-X, sin a)

D* D e2Xk sin u(X, cos u - r,v sin a)

- - (r,r, + X i cos u - X,X,) + (rJS + rJT + Xzv sin u

(22-10)


22-2. Repulsion Motor with Two Sets of Brushes8 Fig. 22-2 shows a repulsion motor with two sets of brushes connected in series. The transformation matrix is

C$, = dr (22-1 1)

sin cc + sin j3

The steady-state impedance then is

in which

r8 + jx, 2jXm cos y cos 6

a 2Xm cos y ( j cos 6 - v sin 6) 4(r, + jX,) cosa y =

d = - - a + ' - mean brush axis 2

2y = t!? - a = brush separation

If the brush separation is zero, y = 0, a = 8, and 6 = E, and Eq. 22-12 reduces to Eq. 22-3 if e, and i" are divided by 2 to take into consideration the fact that the rotor circuit is entered twice in Fig. 22-2 and only once in w:- 00 1

22-3. Compound-wound D-C Motor.8 The compound-wound d-c motor with interpole winding is shown in Fig. 22-3, and its transformation matrix is

ds2 f

Art. 22-31 COMMUTATOR MACHINES

'P if-

Fig. 22-2. Fig. 22-3.

I ts transient impedance f&lows upon transforming from the generalized machine with stationary axes (Eq. 13-51) as

I ts torque matrix comprises the terms in Eq. 22-14 having p6 as a coefficient,

Under steady-state conditions, p = 0, the impedance matrix reduces to

, The impedance for a series motor is given by canceling the ds2 row and column.

The impedance for a shunt motor is given by putting n,, = 0. The impedance for a machine without an interpole winding is given by

putting n, = 0. The torque equation is


From a combination of Eqs. 22-14 and 22-17, the acceleration matrix may be set up, using s for the p0 coordinate and transferring the p0 terms to that column; thus

If i t is assumed that the currents in the s row and column are known and remain constant during acceleration, this equation is readily solved. If the actual variation of the current is to be taken into account, Eq. 22-18 can be solved step by step.

The motional impedance of the d-c machine, by Eq. 19-10, is

For oscillations a t a single frequency, hw, upon substituting p = jhw and p8 = w , and multiplying the bordering row and column by w, the motional impedance matrix becomes

Art. 22-41 C O M M U T A T O R MACHINES 273

224. Schrage M o t ~ r . ~ The connection diagram of the 4-phase Schrage motor is shown in Fig. 2 2 4 . This motor actually has an inner rotor connected to slip rings, but the slip rings have been replaced by stationary brushes on which slip frequency voltages are assumed to be impressed. The outer rotor has a commutator on which bear two pairs o l brushes shifted angles a and ,fl from the d-axis and two pairs of brushes shifted angles a and ,fl from the

Fig. 22-4.

q-axis. These brushes are connected to the stator coils as shown. Assuming that each stator coil has n turns for each rotor winding turn, and projecting all currents on the d- and q-axes, respectively, the transformation matrix is seen to be

Under steady-state a-c conditions the voltages induced in all stationary axes by the C.C.W. revolving field due to the inner rotor currents will be of slip frequency, and the voltages and currents in the quadrature axes will lead those in the direct axes by 90". The polyphase matrix which eliminates the

274 MACHINE ANALYSIS [Ch. 22 Art. 22-51 COMMUTATOR MACHINES

quadrature axes then is The torque matrix is

7 f q ; l ;, f j4nM sin y .5-j6

f 1 am,,,, = (22-26) P$ = -- (22-22) dr2 2jnM'

dr2

Applying this to Eq. 22-21,

retained.

El in which only those terms which actually contribute to the torque have been

k n' f dr2

ds

drl

dr2 P::Ci, =

qr2

qr1

Ps

2n

+ E - ~ B

j(&ja + E-@)

j2n

1

j

(22-27) @i@ Fig. 22-5.

22-5. The Squirrel-Cage Repulsion Motor.8 This single-phase a-c motor has a squirrel-cage rotor in addition to the commutator winding of the simple repulsion motor (Fig. 22-1). Its circuit diagram is shown in Fig. 22-5, and its transformation matrix is

(22-23) ds a dr2 qr2

The impedance matrix for a generalized machine with two rotor layers and CE, = dr2

a smooth air gap is qr2

ds dr 1 dr2 P.2 qr 1 qs

dr2 ds

ds

dr2

z,, =

~2

qrl

1

--

-- sin a

E

1

1

dr 1

--

qr 1

The impedance matrix (Eq. 22-24), with the qs column and row deleted, is used as the starting point for this motor. In that matrix, putting p = jco and p0 = vw for steady-state conditions, and transforming by Eq. 22-27, there results

ds a dr2 qr.2

r , + L,P

MP 1 r.1 + 4 1 ~

P 1 M.P

cos a --

I M.P M?PO I L.,pO MpB

r,, + L,P 1 L,PO 1 M , ~ O 1 wo

Xm,( j sin u + v cos a )

v&2

rr2 + 5%

(22-24) The transient impedance matrix is then a

* z ,.,. = 6;,)2;?z,,c;c".P;: Z,,,, =

dr2 The steady-state matrix follows upon substituting p = jsw, p0 = vw,

s + v = 1, and putting (a - /3)/2 = 6, y = 90 - (a + 8)/2; it is qr2

--

I -- ---Me -M& 1 -Lr2p8 r., + Lap 1 M,p 1 M3p

M P ~ - L ~ ~ P ~ / -1wee / M,P rr1 + 4 , p / ~p I

M p

2jXm, sin ys-j8 (22-28) Z,,,,, = (22-25)

dr2 2njXml + ZjX,, sin y d B rr2 + j x12 The only applied voltage is e,, = e, the other axes being short-circuited.

M'P 1 0

X,,( j cos a - v sin a )

rrz + jXr2

- -

- vx,2

Xm( j cos a - v sin a )

j X k

- v x ;

0 9;

T,I + jXvl

Xm,(j cos a + v sin a )

X,,( j sin a - v cos a )

r , + j X , 1 MP 0 0

0

j X , cos a M'P T8 + &P ds

0


The torque matrix (of which only the first column contributes to the torque) is

COG,,,, = dr2 X,, sin u

The motional impedance of the squirrel-cage repulsion motor is

r, + Lap M cos u p

M,(cos u p + sin u p0)

iaM sin u id8M sin a + i'=*M1

iq*M, cos u M,(cos u p M,(sin u p -ideM, sin a - sin a p e ) + cos u pe ) -id8M sin a

Fig. 22-6.

0 JP

(22-29a)

22-6. Fynn-Weichsel Motor.8 This motor has a double rotor, a commutator with one pair of brushes feeding a stator coil in the d-axis, and a short-circuited stator coil in the q-axis. It is a simpler machine than the Schrage motor, but it has similar characteristics. I ts circuit diagram is given in Fig. 224 , and its transformation matrix in (22-30) below. I ts analysis proceeds along the same lines as for the Schrage motor, starting with the impedance (Eq. 22-24),

Art. 22-71 COMMUTATOR MACHINES

but the polyphase matrix (Eq. 22-22) is not applicable.

22-7. Frequency C o n ~ e r t e r . ~ A rotary converter armature inside a smooth stator with no windings may be driven mechanically, and i t will convert the frequency applied to its slip rings to some other frequency a t its brushes. Its connection diagram is shown in Fig. 22-7, and its transformation

- for the generalized machine with windings on

matrix is d a b q

(22-31)

matrices the rotor

d c:, =

q

only and a smooth air gap are Fig. 22-7.

The transient impedance and inductance

becomes a c , z,.,, = c:z,,q, + c:,L,, - ~ e = ae

0

1

1

0

d d q

zmn = dGl Lmn = d L O (22-32)

( r + L p ) sin 8 (r + L p ) cos 8 - L c o s O ~ O + L s i n O p O

P - L P ~

cOse I --sine -- sin 8

Referred to moving reference axes (by Eq. 16-17), the impedance matrix

p . 0

cos 8

L


where p operates on terms to its right as well as on the column currents, except for the velocity factor p0.

Under steady-state a-c conditions with fundamental frequency applied to the slip rings, the b- and q-axes may be eliminated by the polyphase matrix :

d a b

(22-34) a

in which j refers to a 90' phase difference a t fundamental frequency and k refers to a 90° phase difference a t slip frequency. The impedance matrix then is

-

(22-35)

To find the steady-state matrix put p€J = (1 - s)o and 0 = (1 - s)ot +

d a

6; then in the d column (where the current is of slip frequency) put id = IdeksWt, and in the a column (where the current is of fundamental frequency) put ia = ladut. Then, when all exponentials are placed inside the matrix

d Z,."" =

a

and operated on by p, there results (after putting k = j and canceling the exponential time functions in the currents and voltages)

r + Lp + kLp8 I ( r + Lp)(cos 8 - j sin 8 ) + L(sin 8 + j cos 8)pe

( r + Lp)(cos 8 + k sin 8)-/ 7 + LP

22-8. Synchronous C~nverter .~ The synchronous converter diagram of connections is shown in Fig. 22-8, and its transformation matrix is given by

Art. 22-91 COMMUTATOR MACHINES 279

I ts impedance may be referred to moving axes, as in the case of the frequency converter, by the formula

22-9. Scherbius Advan~er.~ This machine has a commutator rotor and a smooth stator without windings. I t s connection diagram is shown in

Fig. 22-8. Fig. 22-9.

Fig. 22-9, and its impedance and torque matrices, referred to stationary axes, are

dr 99.

Under balanced polyphase conditions the currents and voltages in the q-axis lag those in the d-axis by 90" in time, and therefore, putting iQr = -jidr and then combining columns, there results

dr

or in the steady state, putting p = j o and p0 = vw,

Z,,,, = dr T + j X ( 1 - v ) 0 (22-41)

1 The torque is zero, as would be expected for a smooth air gap machine with nu stator windings, and i t must therefore be driven by an auxiliary adjustable speed drive. When

v = 1 (synchronous speed), the advancer acts as a resistance v < 1 (below synchronous speed), the advancer acts as a reactance v > 1 (above synchronous speed), the advancer acts as a capacitance

I It may therefore be used to adjust power factor.

I


22-10. Leblanc Advan~er.~ This polyphase machine has a commutator rotor, a smooth air gap, and polyphase stator windings. Its connection diagram is shown in Fig. 22-10, and its transformation matrix is

a b

ds n 0

dr cos a -sin a

qr sin a cos u

P 0 n

Applying Eq. 2242 to the impedance and torque matrices of the simpler generalized machine (Eq. 13-3), and substituting p = jw and p0 = vw for steady-state conditions, there results

a b

nVa + jXJ + (rv + jXr) vxv + nvxm cos

+nxm(2j cos u - v sin u ) I nV, + jXA + (rr + jXT) +nXm(2j cos u - v sin a )

-

a b

a -nM sin a L, + nM cos a B,,,,, = (22-44)

b - L , - n M c o s u -nM sin cx >

Under balanced conditions, where ib = -jia, upon combining columns,

Obviously, the phase may be advanced or retarded by changing the brush shift a, by changing the slip s, or by both.

22-1 1. Shunt Polyphase Commutator Motor.8 The connection diagram for this motor is given in Pig. 22-1 1. The transformation matrix is

ds 1

dr cos u -sin a

P7 sin u cos u

q.3 1

Art. 22-1 11 COMMUTATOR MACHINES

Fig. 22-10. Fig. 22-1 1.

Under balanced polyphase conditions, the voltages and currents in the b- and q-axes are lagging those in the a- and d-axes, respectively, by 90' in time, and therefore, substituting ib = -jia and iQa = -jid* and combining columns with the same currents, the transformation matrix becomes

cos a + j sin a = efu

Applying Eq. 2248 to the impedance and torque matrices of the simpler generalized machine (Eq. 13-3), and substituting p = jw and p8 = v o for the steady state, there results

r, + 5% X m ( j cos u - sin a ) (22-49)

r, + jsXr

ds a

- M ( s i n a + j c o s a ) -jL, (22-50)

// If the ds-axis is connected in series with the a-axis, and the qs-axis is connected in series with the b-axis, this machine becomes identical with the Leblanc advancer.

PROBLEMS

22-1. Confirm all equations in t h e chapter and veri fy in detail every step in t h e derivations.


22-2. Carry out the steady-state analysis for the repulsion motor with two sets of brushes (Fig. 22-2) in the same way as was done for the simple repulsion motor (Fig. 22-1).

22-3. Carry out the analysis for a shunt-wound d-c motor, and check the results at each stage of the development as a special case of the analysis for the compound-wound motor of Fig. 22-3.

224. Carry out the same analysis for a series-wound d-c motor, and check the results at each stage of the analysis as a special case of the analysis for the compound-wound motor of Fig. 22-3.

22-5. Derive Eq. 22-23 without using the polyphase matrix in Eq. 22-22. 22-6. Why does only the Grst column of Eq. 22-29 contribute to the torque? 22-7. Show that the Scherbius advancer is a special case of the Lebhnc

advancer. 22-8. Develop the impedance and torque matrices for the Fynn-Weichsel

motor of Fig. 22-6. 22-9. Will the L, term of Eq. 22-50 contribute to the torque? 22-10. At what angle of brush shift will the torque be a maximum for the

shunt polyphase commutator motor?

INTERCONNECTED MACHINES

Machines of all types may be electrically interconnected, or operated in combination, for such purposes as energy conversion, speed control, changing phase or frequency, and power factor correction. The interconnection between machines may also involve static networks as part of a combined system.

If the transient impedance, inductance, and torque tensors of several individual machines and networks are represented by lZmn, ,Z,,, . . . , ,L,,, BLmn, . . . , and ,arnn, ,Gmn, . . . , respectively, and are interconnected in such fashion that the transformation matrix of the connections is q., then the transient impedance and torque tensors of the entire group are

23-1. D-C Machine~.~3 Two compound d-c machines connected in series are shown in Fig. 23-1, and the corresponding transformation matrix is

d h d ' ~ 8'

d82

j

S

ds'2

f'

sf

The transient, acceleration, and motional impedance matrices for a single compound d-c machine are given in Eqs. 22-14, 22-18, and 22-19, respectively. Putting

= r, i- nzrdSl + nirgs

L = La, + n,2Lds1 + n&, + 2ngM, 283

1

IT-- ---

1

I l 1 ---

I 1 ---

---


for the first machine and using primes for the corresponding quantities of the second machine, the impedances of the two interconnected machines may be combined in a single matrix and transformed by matrix (23-3) to give the impedance for the pair of machines. (In this simple connection the transformation matrix multiplication may be dispensed with and columns

Fig. 23-1.

combined after putting if = if' = i h ) . The motional impedance transforms directly as 3a,p, = C ~ 2 2 ' ~ ~ C j ~ since the transformation matrix contains only constants. The transient, acceleration, and motional impedance matrices for the interconnected machines then are

M;(id' + R' + J'p n;ih)

Art. 23-21

d

INTERCONNECTED MACHINES

M' id' a( M;ih + 2n;ih)

R' + J'p U I n matrices (23-5) and (23-6) the steady-state currents id, id', and ih must be determined from the solution of Eq. 2 3 4 . The applied voltages e,,,, e,&,, and eh = e , + e,. are all - --- constant.

P L

23-2. Kapp Vibrat~r . ' .~ The Kapp vibrator consists of a polyphase slip ring induction motor with

------ each of its rotor phases connected to a d-c motor, as shown in Fig. 23-2 for one phase only. The rotor of the d-c motor oscillates a t slip frequency, but it does not rotate. The Fig. 23-2. rotor of the induction motor rotates at a uniform speed, but i t does not oscillate. The connection matrix is

286 MACHINE ANALYSIS [Ch. 23 Art. 23-31 INTERCONNECTED MACHINES 287

The motional impedance of the d-c machine, as a special case of Eq. 22-19, that is, without series or commutating field, is

Since the induction motor does not oscillate, its motional impedance is the same as its transient impedance (Eq. 21-35),

ds2 a

Combining Eqs. 23-8 and 23-9 and transforming by Eq. 23-7, there results

The pre-oscillating current of the d-c motor, appearing in the bordering row and column, is idS1 = edsl/rdsl and is constant. The current ids2 in the stator of the induction motor is of fundamental frequency. The current if is of slip frequency. Since the d-c motor does not rotate, the average velocity is zero, pOl = 0. Therefore

Jf2p(Ekez Aif) = M ~ ( ~ " ~ - s ' " ~ AIfEkSut) = kmJf2 AIfskut . M,p(E-iez Aids2) = ~ ~ ~ ( ~ - j ( l - s ) ~ t A ~ d s t E jut ) = jsmMZ AIds2&jsot

Then, multiplying the bordering row and column by w, the steady-state motional impedance is seen to be

dsl f (792 s

If the s-axis is eliminated (the mechanical torque is zero), the motional impedance reduces to

(23-12)

The oscillating d-c rotor introduces a capacitance in the rotor impedance proportional to the square of field flux idslMdl linked with the rotor, and' inversely proportional to the slip s and to the inertia J1.

23-3. Phase Advancer.73 Fig. 23-3 shows (for one phase only) a polyphasefrequency converter fed from the slip rings of a polyphase induction motor. The connection matrix is dr2

dsl a1 a2

I Fig. 23-3.

The impedance of the induction motor is given by Eq. 21-35, and that of the frequency converter by Eq. 22-35. The impedance of the group is

dsl a1 a2

The current in axes a1 and dr2 is of slip frequency, sw. Hence, putting O2 = O1 + 6, ial = Ialdsot, ia2 = Ia2dwt, and idsl = Idsldot, there results for


the steady-state impedance of the group

23-4. Cascade Induction Motors.738 Two induction motors on the same shaft, interconnected so that the stator of the second motor is fed from the wound rotor of the first motor, comprise a cascade set. The two motors

Fig. 23-4.

may have a different number of pairs of poles, Pl and P,, respectively. The connection diagram is shown in Fig. 2 3 4 , and the connection matrix is

dsl a dr2

The transient impedance of the first machine lZm, is given by Eq. 21-35, and that of the second machine ,Zm, by Eq. 21-8. Then, for the group,

Art. 23-51 INTERCONNECTED MACHINES 289

The currents in axes a1 and dr2 are of slip frequency, s = 1 - Plv, where v is the ratio of the actual rpm to the 2-pole synchronous speed (3600 for 60 cycles). Since the two machines are on the same shaft, pOl/Pl = pO,/P, = vw. Substituting idsl = IdS1dwt and ia = the steady-state impedance becomes

Z,,,, =

jXm2(1 - P1v)

dr2 jX,,(l - Plv - P,v) r,, + jX& - P1v - P B )

(23-18) The transient and steady-state torque matrices of the group are

23-5. Cascade Induction and Commutator motor^.^^^ Fig. 23-5 shows a polyphase series commutating motor fed from the slip rings of a

L 1

Fig. 23-5.

polyphase induction motor and on the same shaft. is

d s l a

The connection matrix

(23-20)


The transient impedance ,Zmn of the induction motor is given by Eq. 21-35, and the impedance ,Zmn of the commutator motor is

. f

The transient impedance of the group is z,,,, = C;,(,Zmn + ,Zmn)C;, =

dsl a

Assuming a different number of pairs of poles PI end P, for the two motors, putting s = 1 - Plv, p8,/Pl = pO,/P, = v o = (1 - s)w, and substituting ia = Iaajsmt in the a-axis, the steady-state impedance becomes

The transient and steady-state torque matrices of the group are

23-6. Power Selsyns.7.8 Power selsyns comprise two polyphase induction motors running a t the same speeds and with a constant angle 6 = 8, - 8, between the positions of their rotors. The connection diagram (for one phase only, assuming balanced conditions) is shown in Fig. 23-6, and the transformation matrix of the interconnection is

Art. 23-61 INTERCONNECTED MACHINES 29 1

The sum of the individual impedances of the two machines, referred to their own axes, from Eq. 21-35, is

dsl

Fig. 23-6.

The impedance of the interconnected group (by Eq. 23-1) is

Transforming from rotating rotor axis a to stationary axis d by dl d d2

there results


Since p(~-jei) = ~ - j ~ ( ~ - jp8)i and 8, - 8, = 6, this matrix becomes

Under steady-state conditions all currents are of fundamental frequency o, and, upon putting p = jw and p - jp8 = jo - j (1 - s)o = jso, there results

If the d2 row is multiplied by ~- j%nd the d2 column by d8, and if the d row is divided by s, Eq. 23-32 assumes the diagonal form :

The torque matrices corresponding to Eqs. 23-27 and 23-33 are

Art. 23-71 INTERCONNECTED MACHINES 293

The equivalent circuit corresponding to (23-33) is shown in Fig. 23-7. The torques a3.e

The motional impedance formula for oscillating induction motors, with stationary axes, must include the terms for the time variation of the sinusoidal pre-oscillating currents, as derived in Eq. 19-24; that is

act, .,, ac;. .,. ac:, f~",,,. 2 c,~c,B.~Z",@ + C:*SaP - + C:*Lap- P' f 1 ax, ax,' ab

The Zap for two generalized machines with stationary axes is given in Eq. 20-73. The transformation matrix for two oscillating induction motors

is the same as that given in Eq. 20-71 for two oscillating synchronous machines. All except the penultimate term of Eq. 23-37 were calculated in the case of the two oscillating synchronous machines in Eq. 20-80. This

I term gives !

- L;?, cos 6 (piqTa) L;,, cos 6 (pigr2)

+ Lb,, sin 6 (pidr2) - L:12 sin 6 (pidr2) (23-38)

Lir2 sin 6 (piqra) - L;,, sin 6 (pigr2)

+ Li,, cos 6 (pidr') - Lb,, cos 6 (pidr2)

Hence the.motiona1 impedance for two oscillating power selsyns is the sum of the four matrices :

23-7. Differential Selsyn~.~98 Consider three polyphase induction motors interconnected as shown in Fig. 23-8. Machines No. 2 and No. 3 are

k driven a t different arbitrary speeds vz and v,, and machine No. 1 follows a t a


constant angle 6 such that 8, = 8, + 8, - 6. Under steady-state conditions, fundamental frequency is applied to axes dsl and ds3, and slip frequency

Fig. 23-8.

currents slw flow in axis a, where sl = 1 - v, - v3. The connection matrix for the group is

dsl a ds2 ds3

The combined impedance of machines No. 1 and No. 2 has already been giveh in Eq. 23-28; for all three machines it is

dsl a ds2 ds3 a3

1 ~ , p ~ j ~ l 0 0 1

Art. 23-71 INTERCONNECTED MACHINES

Then, transforming by Eq. 23-40,

dsl a ds2 ds3

Under steady-state conditions, putting id" = Idfl&jot, ia = laejslot, j d d = 1d82Ejs~~t jdS3 = 1d83&wt

2 , 8, = (1 - sl)mt - 6, 8, = (1 - s1)ot - (1 - s,)ot, and 8, = (1 - s3)wt, the matrix above reduces to

The torque matrices for the group are

dsl ds2 ds3

(2345)

I

a

COG,,,, =

ds2

I - ~ x , ~ E ' ~

0

jX,,

0

0

-jXrn3


23-8. Arnplidyne Voltage Regulator.' This regulator comprises an amplidyne, a stabilizing transformer, and an exciter, interconnected as

Fig. 23-9.

shown in Fig.,23-9. Its connection matrix is

Neglecting the induced voltage (Lp and Mp terms) in both the stator and rotor windings of the amplidyne, as well as the (r, + Lg) term for the


exciter, the impedance matrices for the three devices are

1 2 3 d Q

By C$(,Zrnn + ,Zmn + ,Zrn,)q,, the impedance of the interconnected

The applied voltages are f 2

!

The voltage equation may be rewritten by employing the identity


in which Kkj is the inverse of R,,. Then

[Ch. 23

in which Ti, = KkiLmk = time constant matrix

= KkiGm, = amplification matrix

Eq. 23-51 substitutes resistance drops for currents, and this form of the voltage equation is equivalent to multiplying each current in each axis by the resistance in the corresponding diagonal term of Eq. 2348, and then compensating by dividing the column of Eq. 2 3 4 8 by the diagonal resistances. Then Eq. 2 3 4 8 becomes

Upon elimination of the 3-, q-, c-, and p-axes (there are no voltages impressed on these axes), there results


in which p, is the over-all amplification factor and

Eq. 23-53, a cubic (p3) in the time operator, gives the change in exciter voltage Aef corresponding to a change hea impressed on the control field.

The stability of the regulator may be determined by applying Routh's criterion, coefficients (11-32), or simply by solving for the roots of Eq. 23-53.

PROBLEMS

23-1. Cordinn all equations in the chapter and verify in detail every step in the derivations.

23-2. Develop the motional impedance matrix for the cascade induction motors of Fig. 23-4.

23-3. Develop the motional impedance matrix for the cascade induction and commutator motors of Fig. 23-5.

23-4. Develop the motional impedance matrix for the differential selsyns of Fig. 23-8.

23-5. Set up the transformation matrix and the transient impedance matrix for a synchronous generator connected to a polyphase induction motor.

23-6. Set up the transformation matrix and the transient impedance matrix for a synchronous generator connected to a series-wound a-c commutating motor.

23-7. Set up the transformation matrix and the transient impedance matrix for a shunt-wound exciter and a synchronous generator.

BIBLIOGRAPHY

1. MCCONNELL, A. J. Applications of the Absolute Differential Calculus. Blackie & Sons, Toronto, 1931. This book is unexcelled as an aid in learning the elements of tensor analysis and many of its applications. However, since the author assigns the development of many of the principles to the numerous problems throughout the text, i t is absolutely necessary for the reader actually to do the problems if he is to achieve real familiarity with the material-and this is a long job.

2. SOKOLNIKOFF, I. S. Tensor Analysis. John Wiley & Sons, Inc., New York, 1951. The same kind of book as McConnell's. Well adapted for a one-semester course.

3. EDDINGTON, A. S. The Mathemtical Theory of Relativity. Cambridge University Press, London, 1924. The first 75 pages of this book are devoted to tensor analysis. Written by one of the great minds of our times, it is a lucid and clear exposition of the subject.

4. LEVI-CIVITA, T. The Absolute Differential Calculus. Blackie & Sons, Toronto, 1927. Written by the principal student and associate of Ricci, the father of tensor analysis, this book is invaluable in presenting the pioneer point of view.

5. STIGANT, S. A. The Elements of Determinants, Matrices and Tensors for Engineers, McDonald & Co., Ltd., London, 1959. This is an introductory textbook written by an engineer for engineers, and can be perused with profit before studying the application of the methods to circuits, networks, and machines.

6. KRON, G. Tensor Analysis of Networks. John Wiley & Sons, Inc., New York, 1939. A comprehensive and original source in the application of matrices and tensor algebra to electrical networks. It does not make use of the differential or integral aspects of tensor analysis. As is true in all of Kron's work, the book employs a special brand of matrix algebra which has its own form and which borrows tensor rules. The writer favors this brand, as it is more versatile and flexible than conventional matrix algebra. The book is long and somewhat repetitious, but it is a classic.

7. &ON, G. A Short Course in Tensor Analysis for Electrical Engineers. John Wiley & Sons, Inc., New York, 1942. This little book of 250 pages is about half devoted to circuits and networks, and half to rotating machines. While it is essentially based on matrices and tensor algebra, some differential calculus of tensors is introduced. It contains many original ideas. This book has been reprinted (1959) by Dover Publications, Inc., as Tensors for Circuits and includes a complete bibliography of Kron's publications.

8. KRON, G. Application of Tensors to the Analysis of Rotating Electrical Machinery. General Electric Review, 1942. This book comprises a series of articles published by Kron in the Review from 1935 to 1938. It is mostly concerned with applications to rotating machines and covers just about every known machine in both the steady and transient state. Because it was a pioneer effort in which the author was developing his original ideas and concepts as he progressed, it is a difficult text to use for classroom work or from which to gain a knowledge of the field, but the material is all there for the reader who is willing to go after it.

9. &ON, G. Equivalent Circuits of Electric Machinery. John Wiley & Sons, Inc., New York, 1951. While this is not a book on tensor analysis per se, the author uses the basic philosophy of the tensor concept to evolve the most complete and comprehensive development of equivalent circuits for electrical machines yet presented. It is an excellent example of the power and generality of the tensor approach.

10. GIBBS, W. J. Tensors in Electrical Machine Theory. Chapman & Hall, Ltd., London, 1952. This British book is based closely on the publications of Kron, and covers essentially the same ground as Ref. 7 above. However, it includes a few chapters on the pure mathematics of tensors which are not present in Kron's books.

30 1

302 BIBLIOGRAPHY

11. KELLER, E. G. Mathematics of Modern Engineering. John Wiley & Sons, Inc., New,York, 1942. Chapter I1 of this book contains a readable and unified summary of Kron's treatment of the tensor theory of circuits and machines, together with some generalizations of the author's.

12. PEN-TUNG SAH. Dyadic Circuit Analysis. International Textbook Co., Scranton. Pa., 1939. This book is concerned with the vector and dyadic point of view in electrical machine analysis. Closely allied in concept and philosophy with the tensor point of view, it would undoubtedly have made engineering history had not Kron's more powerful approach quickly superseded it. Pen-Tung Sah must be credited with having been the first to introduce many of the concepts which are now the background of the "tensor point of view" in electric circuit and machine analysis.

13. LE CORBEILLER, P. Matrix Analysis of Electric Networks. Harvard University Press, Cambridge; Mass., 1950. This little monograph makes pleasant reading, but 'is not sufficient for a real understanding of the subject.

14. CONCORDIA, CHARLES. Synchronous Machines. John Wiley & Sons, Inc., New York, 1951. A text on the advanced synchronous machine theory developed by R. E. Doherty, C. A. Nickle, and R. H. Park, and which in effect is a generalized machine approach although neither tensors nor matrices are employed. The author is a recognized authority who has contributed many papers on synchronous machine theory over the past quarter century.

15. LYON, WALDO V. Transient Analysis of Alternating Current Machines. John Wiley & Sons, Inc., New York, 1954. A text on advanced machine theory based on "instantaneous symmetrical components" which covers the subject of machines a t an advanced level.

16. WHITE, D. C., and WOODSON, H. H. Electromechanical Energy Conversion. John Wiley & Sons, Inc., New York, 1959. This book on machine analysis uses matrices and the concepts of the generalized machine, but it avoids the use of tensor analysis. As a consequence the treatment is limited and restricted, the notation is complicated, and the many interesting physical interpretations brought to light by a tensor approach are lost.

17. TAKEUCHI, T. J. Matrix Theory of Electrical Machinery. The Ohm-Sha, Ltd., 1958. This book by a Japanese author (in English) is an advanced treatment of all types of electrical machines, including those with unbalanced windings, by matrix methods. Instantaneous symmetrical components and "commutation" matrices are used for each machine. The treatments of time harmonics and hunting are part~cularly interesting. X

18. Ku, Y. H. Electric Energy Conversion. The Ronald Press Co., New York, 1959. A text on advanced machine theory by an author who has contributed much to such theory over the past thirty years. I n spite of the title, the treatment is based - essentially on circuit analysis rather than energy considerations. Matrices are ? employed here and there.

19. ADKINS, B. The General Theory of Electrical Machines. John Wiley & Sons, Inc., New York, 1951. This book by a British author introduces the generalized machine concept, and uses matrices. I t is an excellent text for an undergraduate course, but hardly advanced enough for graduate work.

20. BEWLEY, L. V. Flux Linkages and Electromagnetic Induction. The Macmillan CO., New York, 1952. ++

21. BEWLEY, L. V. Alternating Current Machinery. The Macmillan Co., New York, 1949.

22. BEWLEY, L. V. Traveling Waves on TransmissionSysten,~. John Wiley & Sons, Inc., New York, 1951.

PAPERS AND ARTICLES

23. KRON, G. "Tensor Analysis of Rotating Machinery," 1932. Printed privately in Rumania. This 29-page monograph (in English) is of historical significance since it presents for the first time Kron's basic concepts of a generalized machine matrix (which he then called "the iron tensor"), and the connection transformation

BIBLIOGRAPHY 303

tensor (which he then called "the copper tensor"), together with applications t o a number of machines, and the formulation of one form of "the equation of motion." I n spite of the title, i t is based entirely on vectors, dyadics, and matrices, and contains neither tensors nor tensor analysis.

24. &ON, G. "Non-Riemannian Dynamics of Rotating Electrical Machinery," Jou-1 of Mathematics and Physics, 13 (May, 1934): 103-194. This is a highly sophisticated treatment of the subject, making full use of the advanced concepts of tensor analysis and differential geometry. It should not be studied without a prior knowledge of the mathematical aspects of tensor analysis, such as given in Ref. 1. But for the initiated it will be found to be the most satisfying and consistent treatment of machine theory in Kron's voluminous output.

25. KRON, G. "Tensor Analysis in Electrical Engineering," Bulletin Scientijque of the Association Belge des Ingknieurs Electriciens, March, 1936. Upon the award of the Fondation George Montefiore prize to Kron.

26. KRON, G. "Quasi-Holonomic Dynamical Systems," Physics, 7 (April, 1936). 27. KRON, G. "Invariant Form of the Maxwell-Lorentz Field Equations for Accelerated

Systems," Applied Physics, 9: (March, 1938). 28. KRON, G. "Equivalent Circuit of the Capacitor Motor," G.E. Review, 44 (September,

1941). 29. KRON, G. "Equivalent Circuit of the Salient Pole Synchronous Machine," G.E.

Review, 44 (December, 1941). 30. KRON, G., CRARY, S. B., and CONCORDIA, C. "The Doubly Fed Machine," A.I.E.E.

Trans. 61 (1942): 286-289. 31. KRON, G. L'Equi~alent Circuits of the Primitive Rotating Machine with Asym-

metrical Stator and Rotor," A.I.E.E. Trans. 66 (1947): 17-23. 32. KRON, G. "Tensorial Analysis and Equivalent Circuit of a Variable-Ratio Frequency

Changer," A.I.E.E. Trans. 66 (1947): 1503-1506. 33. &ON, G. "Steady-State Equivalent Circuits of Synchronous and Induction

Machines," A.I.E.E. Trans. 67 (1948): 175-181. 34. KRON, G. "Stationary Networks and Transmission Lines Along Uniformly Rotating

Reference Frames," A.I.E.E. Trans. 68 (1949): 690-696. 35. KRON, G. "Equivalent Circuit of the Shaded-Pole Motor," A.I.E.E. Trans. 69

(1950): 735-741. 36. KRON, G. 'LC1assifi~ation of the Reference Frames of a Synchronous Machine,"

A.I.E.E. Trans. 69 (1950): 720-727. 37. KRON, G. "Equivalent Circuits for the Hunting of Electrical Machinery," A.I.E.E.

Trans. 61 (1942): 290-296. 38. &ON, G. "Equivalent Circuits for Oscillating Systems and the Riemann-Christoffel

Curvature Tensor," A.I.E.E. Trans. 62 (1943): 25-31. 39. KRON, G., BODINE, R. B., and CONCORDIA, C. L'Self-Excited Oscillation of Capacitor-

Compensated Long-Distance Transmission Systems," A.I.E.E. Trans. 62 (1943): 41-44.

40. KRON, G. L'Steady-State and Hunting Equivalent Circuits of Long-Distance Transmission Systems," G.E. Review 56 (1943): 337-342.

41. KRON, G., and CONCORDIA, C. "Damping and Synchronizing Torques of Power Systems," A.I.E.E. Trans. 64 (1945): 366-371.

42. KRON, G. ",4 New. Theory of Hunting," A.I.E.E. Trans. 71 (1952): 859-866. 43. KRON, G. "Tensor Analysis of Integrated Transmission Systems," A.I.E.E. Trans.

60 (1951): 1239-1246; 61 (1952): 505-512, 814-821; 62 (1953): 827-838. 44. KRON, G. 'LDiakoptics-The Piecrjwise Solution of Large-Scale Systems," Electric

Journal (London, 1957-58, a serial). These articles describe the theory and applications of Kron's method of "tearing." This consists of analyzing a large and complex system by breaking i t down into manageable parts, solving each part in terms of the variables a t its terminals, and then matching the terminal quantities of the parts to find the solution for the whole.

45. LAUDER, A. H. "Salient Pole Motors Out of Synchronism," A.I.E.E. Trans. 55 (1936): 636-649.

46. BEWLEY, L. V. "Resolution of Surges into Multivelocity Components," A.I.E.E. Trans. 54 (1935): 1199-1203.

304 BIBLIOGRAPHY

47. BEWLEY, L. V. "Tensor Algebra in Transformer Circuits," A.I.E.E. Trans. 56 (1937).

48. BEWLEY, L. V. "Traveling Waves on Power Systems," Bulletin of the American Illathematical Society, Vol. 48, No. 8, August, 1942: 527-538.

49. SHEN, D. W. C. LLOperational Impedance Matrices of n-Phase Partially Symmetrical Machines," Australian J o u m l of Scientific Research, 4 (1951): 544-559.

50. Ku, Y. H. "Transient Analysis of Rotating Machines and Stationary Networks by Means of Rotating Reference Frames," A.I.E.E. Trans. 70 (1951): 943-954.

51. Yu. Y. "The Impedance Tensor of the General Machine," A.I.E.E. Paper 56-122 (May, 1956).

52. Ku, Y. H. "Tensor Analysis of Unbalanced Three-Phase Circuits," A.I.E.E. Paper 52-126 (February, 1952).

53. CONCORDIA, C. "The Use of Tensors in Mechanical Engineering Problems," G.E. Review (July, 1936).

I N D E X

Absolute derivatives; see Derivatives, intrinsic

Admittance, 4, 18, 83 Affine connection, 129, 187 Amplidyne voltage regulator, 296 Apparent coils, 93 Axes

direct, 8, 144 elimmation of, 225 moving, 158, 256 quadrature, 8,144

Branches, 9 Brush shift, 159

Canonical equations, 5 Capacitor motor, 259 Cascade induction motor, 288, 289 Christoffel matrix, 187 Christoffel symbol, 120

holonomic, 171 quasi-holonomic, 183, 186 transformation of, 127

Classification components of oscillation, 211, 214 concepts, 198 machines, 162, 165 networks, 13, 102

Commutating machines, 268 compound-wound d-c, 270 frequency converter, 277 Fynn-Weichel motor, 276 Leblanc advancer, 280 repulsion motor, 268 repulsion motor with two sets of

brushes, 270 Scherbius advancer, 279 Schrage motor, 273 shunt polyphase commutator motor,

280 squirrel-cage repulsion, 275 synchronous converter, 278

Complete (orthogonal) networks, 12, 92, 102

adjacent-terminals, 93 all junction-pair, 104 all-mesh, 95 analysis, 94 apparent coils, 93

comparison, 104 conversion, 100 types of terminals, 93 variables, 92

Complex numbers elimination of, 133 product frequencies, 135, 266

Compound-wound d-c machine, 270 Conjugate, 48 Connection diagram, 7 Constraints, 41

equation of, 6, 31 junction-pair networks, 86

Coupling coefficients, 207 Covariant and contravariant, 110 Curl, 125 Curvature tensor, 125, 218

D-c machines, 270, 283 Decrement factors, 205 Derivatives

covariant, 120, 121 higher, 123 intrinsic (absolute), 121, 216 time, 134

Design constants mixed, 204

Differential selsyns, 293 Displacement of stator coil, 160 Divergence, 124 Double-fed induction motor, 258 Double frequencies, 135, 266 Double squirrel-cage motor, 255 Duals, 15 Dyadics, 6, 32

Energy relationships, 148

Field tensor, 185 Field time constant, 226 Flux

coil, 143 cross, 155 linkage, 145, 155

Forms, 30 differential, 30 invariancy, 31, 110 linear, 30 quadratic, 30

306 INDEX INDEX

Frequency converter, 277 Fynn-Weichel motor, 276

Generalized (primitive) machines basic concepts, 141 first kind (quasi-holonomic), 149, 200,

215 second kind (holonomic), 170, 200

Generalized per-unit concepts, 203 Gradient, 124 Group

of transformations, 50 property, 34, 156 theory, 107

Holonomic, 127, 167; see also Reference frames

equations, 171, 172, 178, 189, 192 Hunting ; see Oscillations Hurwitz's criterion, 137

Idemfactor (Kronecker delta, unit matrix), 19

Impedance, 4, 18 branch, 61 driving point, 38 leakage. 60 matrix

holonomic, 201 quasi-holonomic, 152, 154, 163, 222

mixed design constants, 206 motional, 211, 215, 241, 244, 263, 273,

284, 286, 293 synchronous machine, 222 transfer, 38 transformation of, 31

Indices, 105 covariant and contravariant, 31, 110 dummy, 20, 106 free, 106 raising and lowering, 118, 203 subscripts, 4, 105 superscripts, 4, 105 upper and lower. 31, 110

Induction machines, 246 balanced polyphase voltages, 248 capacitor motor, 259 double-fed, 258 double squirrel-cage, 255 equivalent circuit, 248, 254, 256, 261 motional impedance, 263 moving axes, 256 oscillating load, 265 shaded pole, 262 single-phase motor, 253 split phase, 259 stationary axes, 246 three-phase motor, 249

two-phase symmetrical components, 247, 251, 260

Interconnected machines amplidyne voltage regulator, 296 cascade induction, 288 cascade induction and commutator

motor, 289 d-C, 283 differential selsyns, 293 Kapp vibrator, 285 phase advancer, 287 power selsyns, 290 synchronous, 237, 242

Interconnection coils, 160 networks, 88 subnetworks, 35

Invariant form, 31, 110 power, 31, 84

Invariancy, 31, 84, 107, 110 Inverse, 22

Junction pairs, 9, 12 change of variables, 88 constraints, 86 definition, 12 interconnection, 88 networks, 83, 104 selection, 84

Junctions, 9

Kapp vibrator, 285 Kirchhoff's laws, 6

equivalence of, 44 Kronecker delta, 19

Lagrangian method, 167 equation of motion, 170

Laplacian, 124 Lauder's rule, 131 Leakage coefficient, 205 Leblanc advancer, 280

Magnetizing current, neglect of, 8, 42, 60 Matrices, 17

addition, 19 compound, 25 conjugate, 48 diagonal, 23 differentiation, 25 inverse, 22, 24 multiplication, 20 n-way, 17 polyphase, 134 product, 26 rotation, 156, 180

skew-symmetric, 24 solution of equations, 27 subtraction, 19 symmetric, 24 transpose, 24, 26 unit, 19

Maxwell-Lorentz equation, 183 Maxwell's equation, 178 Mesh network, 9, 12 Motion, equation of, 141, 171

generalized per unit, 203 holonomic, 171, 179, 192 Lagrangian, 170 matrix, 173, 178, 189 Maxwell-Lorentz, 185 non-holonomic, 174, 186, 194, 203 quasi-holonomic, 175, 185, 193

Motional impedance ; see Impedance

Networks, 3, 9, 12 classification, 102 complete, 12, 92 conversion, 100 junction-pair, 9, 12, 83, 102 mesh, 9, 12, 101, 102 open, 10 orthogonal, 12, 92 primitive, 13, 44, 53, 83, 102 sub-, 9, 35 symmetrical components, 53 topological considerations, 9 types, 12

Non-holonomic, 127, 167; see also Refer- ence frames

equations, 173 objects, 129

Notation, 3 matrix, 17 tensor, 4, 105

Orthogonal (complete) networks, 12, 92, 102

Oscillating load, induction motor, 265 Oscillations, small

equation of, 209 classification of components, 211, 214 invariant form, 216 transformation, 212

generalized machine, 215 motional impedance, 211, 215 sinusoidal disturbances, 219 stationary non-oscillating axes, 210

Phase advancer, 287 Polyphase matrix, 134 ' Postulates, generalization, 11 Power

equation of, 162

invariancy of, 31, 84 selsyns, 290

Primitive machines; see Generalized machines networks, 13, 44, 53, 83, 102

Quasi-holonomic, 149, 167; see also Ref- erence frames

equations, 151, 175, 176, 180, 183, 193, 204

Quasi-holonomic machine, 149 equation of voltage, 151 impedance matrix, 151, 152, 163

components, 154 modification, 153

induction matrix, 154, 164 resistance matrix, 154 torque matrix, 154, 155, 164, 188 transformation, 157

brush shift, 159 displaced coil, 160 interconnected coils, 160 moving axes, 158 slip rings, 159 successive, 161 turn-ratio, 158

two stator and two rotor winding layers, 150

voltage and current components, 150 winding layers, 150

Quotient law, 113

Raising and lowering indices, 118, 203 Reference frames (systems, axes)

explicit, 189 holonomic, 127, 168, 171, 172, 178, 189,

190, 192, 234 moving, 156, 158 non-holonomic, 127, 168, 173 quasi-holonomic, 149, 168, 175, 181, 183,

184, 193 rectilinear, 115

Repulsion motor, 268, 270, 275 Riemann-Christoffel curvature tensor,

126, 218 Routh's criterion, 136

Scherbius advancer, 279 Schrage motor, 273 Sets, 30, 31, 105 Shaded pole motor, 262 Short-circuit inductance, 205, 226 Short-circuit matrix, 131 Short-circuit theorem, 130 Shunt polyphase commutator motor, 280 Single-phase induction motor, 253 Sinusoidal disturbances, 219

308 INDEX INDEX

Slip rings, 159 Split phase induction motor, 259 Squirrel-cage repulsion motor, 273 Stability criteria, 135

Hunvitz's, 137 Routh's, 136

Steady-state calculations, 161 cascade induction motors, 289 differential selsyns, 295 induction machines, 246. 247, 249, 252,

254, 255, 257, 259, 260, 263 Kapp vibrator, 286 Leblanc advancer, 280 phase advancer, 288, 289 polyphase commutator motor, 281 power selsyns, 292 repulsion motors, 269, 270, 275 Schrage motor, 274 synchronous machine, 222, 224, 228

Subnetworks, 9, 35 Subscripts, 4 Successive transformation, 109, 161 Summation convention, 5, 106 Superscripts, 4 Symmetrical components, 8, 48, 49

basic networks, 53 transformation, 4 two-phase, 247

Synchronous machines, 222 converter, 278 elimination of axes, 225, 240 equivalent circuits, 227. 228, 230 forward and backward components,

228 hunting, 240 impedance matrix

forward and backward components, 229

motional, 241, 243, 244 moving axes, 233 transient, stationary axes, 222

interconnection, 237.242 operational impedance, 227 oscillation, 240, 242 quadrature axes ignored, 240 reactances, direct and quadrature. 226 single-phase, 234 steady-state voltage, 222 subsynchronous speed, 223 synchronous speed, 224 three-phase, 234 time constants, 226 torque, 229 two-phase, moving axes, 232

Tensor addition, 110

analysis, 105 associated, 118 composition, 111 conjugate, 48 contraction, 111 contravariant. 110 covariant, 110 covariant derivative, 121 curvature, 125, 218 definition. 109 differentiation, 115, 120 field, 185 fields, 119 group property, 109 higher derivatives, 123 indices, 105, 118 intrinsic derivative, 121 invariance, 107, 110 inverse, 112 magnitude, 117 metric, 116 mixed, 110 multiplication, 111 quotient law, 113 rotation, 156, 180 summation convention, 5, 106 torsion, 175, 180. 184, 188 transformation, 3, 108, 109, 113 weighted, 114

Tensor analysis, 105 Terminals, 93 Three-phase induction motor. 249 Topology, 9 Torque

damping, 220 equation of, 141, 146 holonomic. 173, 178, 190 193 matrix, 154, 162 non-holonomic, 175, 196 quasi-holonomic, 176, 181, 184!

194 reaction, 155 synchronizing. 220

Torsion tensor, 175, 180, 184, 188 Transformation, 3, 8, 30, 108

Christoffel symbol, 127 coordinates, 7, 30 current, 31 formula, 49 functional. 113 impedance, 31 linear, 108 moving axes, 158 primitive machine, 157, 200 successive, 109, 161 tensor, 9 turn-ratio, 158 voltage, 31

Transformers, multiwinding, 60 basic definitions, 60 coupling windings, 65 delta-quadruple-zigzag, 70 extended data, 74 forked auto-, 67 group of, 68 impedance, 60. 61 inscribed delta, 74 load ratio control circuit, 66 parallel windings, 64 series windings, 62 stub delta, 74 voltage equation, 61 wyedelta-zigzag, 69

Traveling waves, multivelocity, 75 differential equations, 75 impedances, 78

transition points, 77 Turn-ratio transformation, 158

Vectors angle between, 117 magnitude, 117

Voltage components. 196 equation of, 141, 142, 145

holonomic, 172, 178, 189, 193 non-holonomic, 174, 187, 195 quasi-holonomic, 151, 161, 180, 183,

193, 204 generated, 146 induced, 146

Winding layers, 150

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