THE GENERALIZED THEOR~ OF PRACTICAL ELECTRIC MACmNES

by

JOHN C. DUNFIELD

A Thesis submitted to the Faculty of Graduate Studies and Research in partial fu1fi1ment of the requirements for the degree of Doctor of,Philosophy.

Department of Electrical Engineering McGill University Montreal, Quebec.

~ John C. Dunfield 1967

October 1966'

ABSTRACT

The inductance matrices often used in the study of electric

machines are, except for sorne special cases, inadequate to define

their behaviour in a11 but a very cut;sory manner because of space

harmonics of airgap flux density and mmf, arising from non-simple

machine geometryand winding configuration. Test data confirms the

usefulness of inclusion of these effects and illustrates sorne con­

straint8 imposed by the magnetically non-linear material surrounding

the coils. The two basic transformations of generalized machine

theory, the slip-ring to commutator primitive and the polyphase to

two-phase, are extended to include harmonics in a linear machine.

They give insight into the physical processes involved but are of

little value for numerical solution. Several means of direct numer~'

ical solution o~ systems of equations repreaenting electric machines

are applied with special attention to modified dq axis equations of

synchronous machines. Test data confirma the aurprisingly large

harmonic effects which can occur.

i

ACKNOWLEDGEMENTS

The author wishes to elcpress his deep appreciation to Dr.

T. H. Barton for his guidance and encouragement during the course of

the work. He is indebted to his colleagues for many long hours of

invaluable discus~ion, especially J. E. Buchan, B. A. Howarth and

ii

P. Sylvester, and to the technical staff of the Department of Electrical

Engineering for their assistance in the construction of equipment.

Tnanks are due to Miss C. Hennessy for her excellent ~yping.

To my wife, Elizabeth, and our three children Stephen, George and

Deborah l owe much appreciation for many long hours away from the

home.

The author wishes to express his gratitude to McGill University

and in particular to the Department of Electrical Engineering for the

establishment of an atmospherebeneficial to research and to the~atlonal

Reeearch Counci1 of Canada who provided financial assistance without

which the project would not have been feasible.

_ 1

\ 1

iii

CLAIM OF ORIqINALITY

.. To the best of the author's know1edge, the fo11owing

contributions are original:

(1) Derivation of the inductance matrix of e1ectric

machines with comp1exairgap geometry and winding distribution.*

(2) Investigation of the influence of mmf and flux

density harmonics in terms of a two phase slip-ring primitive to

two-phase commutator primitive e1ectric machine.**

(3) Investigation of mmf and flux density harmonics in

terms of an n phase to nlO phase slip-ring primitive electric machine.***

(4) Numerical solution of the machine equations iQ slip-ring

primitive form for a practical electric machine and consequent exper-

imental verification.****

(5) Derivation of approximate equations which illustrate

modifications required of the c1assical dq axis method of ana1ysis.****

(6) A method of continuous m~asurement of induçtance of an

e1ectric machine rather th an a point by point determination.

(7) Determination of the dq axis equiva1ent of a damper

winding by frequency response methods.

60 * I.E.E.E, P.A.S., paper by Barton and Dunfie1d •

** Paper to be presented at Winter 1967 I.E.E.E., P.A.S. Conference.

*** Paper to be presented at Winter 1967 I.E.E.E., P.A.S. Conference.

**** Paper submitted to I.E.E.

iv

TABLE OF CONTENTS

ADSTRACT ',i

ACKNOW ... EDGEMENTS il ' CLAIM' OF ORIGINALITY iii

TABLE OF CONTENTS iv LIST OF ILLUSTRATIONS vii LIST ,OF TABLES x

NOMENCLATURE xii CHAPTER 1 - 1 NTRODUC'l'I ON 1 CHAPTER 2 - INDUCTANCES OF A PRAC'l'ICAL SLIP-RING PRIMITIVE 8

PART l, AN ANALYTICAL STUDY 8 Introduction 8 Definition of Inductance 9 Radial Airgap Flux Density in an

Electric Machine Il The General Inductance Expression 14 Slip-Ring Primitives 16 Windings on the Salient Member 16 Windings on Opposite Sides of the Airg~p 18 Windings on the Cylindrical Member 18 Two Phase Windings 19 Three Phase Windings 19 Inductance Ratios in the Primitive Machines 20 Conclusions 24

PART 'II, AN EXPERIMENTAL STUDY 25 Introduction 25 The Test Machine 25 Inductance Measurements 26 ' Inductance Identification 28 The Field Inductance, 28 The Field Stator Hutual Inductances 30 The Stator Inductances 35 Correlation of the Stator Inductances 40 The Specifie Airgap permeance 41 The Airgap Equivalent of the Field Winding 43 Correlation of Theory and practice 46 Conclusions 48

v

(

PART III , THE DAMPER WINDING EQUIVALENT CIRCUIT 50 An Experiment 51 The Analysis 60

CHAPTER 3 - AXIS TRANSFORMATION FOR PRACTICAL PRIMITIVE MACHINES 62

l NTRODUCTI ON 62

PART '1, THE TWO PHASE TRANSFORMATI ON 64 Tne dand qAxis Equivalents of a Single

Winding 64 The MMF Equivalence of the Slip-Ring and Commutator Primitives 65 Voltage Equivalence 68 Resistance and Leakage Inductance 70 Equivalence of Impedances 71 Transformation to the Commutator Primitive 75 Torque Relationships 77 Torque of the Cbmmutator Primitive 78

Summary 82 Physical Analysis 84 Conclusions 87

PART '11, THE POLYPHASE, TO TWO AXIS TRANSFORMATION 88 The Two Axis Equivalent of a Polyphase

Winding 88 The General Connection Matrix 90 Voltage Transformation 91 Impedance Transformation92 Impedance Associated with the Airgap Flux 94 Resistance and Leakage Inductance 94 Torque th 95 Harmonies Higher than the N Neglected 95 The Ideal Winding 97 The Three Phase Winding 97 Harmonies Higher than the Third Neg1ected 98 The Squirrel Cage Winding 99 Conclusions 100

CHAPTER 4 - NUMERI CAL SOLUTI ON OF SYNCHRONOUS MACHI NE STEADY-STATE PERFORMANCE 101 Solution by 4 Point Runge-Kuttu Method 105 Transcendental Equations Solved by Genera1ized

Newton-Raphson Method 112 CHAPTER 5 - THE THREE' WIM STAR CONNECTED SYNCHRONOUS MACHI NE . 123

The ory 123

," "

,. )

,.PO., \ }

Comparison of Ca1cu1ation Methpds Experirnent Conclusions

CHAPTER 6 - THE FOUR WIRE STAR CONNECTED SYNC1:mONOUS ~CHINE

Theory Comparison of Calculation Methods Experiment Conclusions

CHAPTER 7 - CONCLUSIONS

BIBLIOGRAPHY

APPENDIX l - THE OPEN CIRCUIT CHARACTERISTIC

APPENDIX II - mO-PHASE MACHINE INDUCTANCE RATIOS

~PENDIX III - INFLUENCE OF SATURATION

APPENDIX IV - CONNECTION OF THREE-PHASE WINDING TO SIMUl.ATE A MACHINE WITH. 1200 SPREAD

APPENDIX V - INDUCTANCE IN TERMS OF RADIAL AIRGAP FLUX DEN SI TY AND MME'

/U'PENDIX VI - REQUIRED NUMBER OF TWO- PHASE MARMONI C CURRENTS

APPENDlX VII - MOMENT OF lNERTIA

APPENDIX VIII - MODIFIED dq AXIS PHASOR DIAGRAM FOR THREE WIRE STAR CONNECTED STATOR

vi

127 129 135

136

136 136 138 140

144

148

A-1

A-3

A-5

A-7

A-IO

A-12

A-14

. A-15

\ 1

Figure 2-1

Figure 2-2

Figure 2-3

Figure 2-4

Figure 2-5

Figure 2-6

Figure 2~7

Figure 2-8

Figure 2-9a

Figure 2-9b

Figure 2-10

LIST OF ILLUSTRATIONS

The Angular Notation

The Slip-Ring Primitive

Inductance Ratios as Function of Coil Span

M32/L2 for Various q

Inductance Measuring Circu:J.t

Field Self Inductance and Stator-Field Hutual Inductance vs. Field Current

Field-Stator Mutual Inductance vs. Stator

13

17

22

23

27

29

Current 31

Stator-Field Mutual Inductance vs. Position for If = 0.5 Amp. 32

Stator-Field Mutual Inductance for If = 0.5 Amp. 33

Field-Stator Mutual Inductance for Ia = 6.0 Amp. 34

Direct and Quadrature Axis Inductance vs. Stator Current 36

Figure 2-11 Stator Self Inductance and Stator Mutual Induct-ance vs. Position for la = 6.0 Amp. 37

Figure 2-l2a Stator Self Inductance vs. Position for la = 6.0 Amp. 38

Figure 2-12b Stator Mutual Inductance vs. Position for la = 6.0 Amp. 39

Figure 2-13 Specifie Airgap permeance vs. Position for Three Separate Positions of the Magnetizing Coil Relative to the Direct Axis and at a Magnetizing Current of 6.0 Amp. 42

Figure 2-14 Harmonie Components of Specifie Airgap permeance as a Function of Stator Current 44

Figure 2-15

Figure 2-16

Airgap Equivalent Field tolinding

Measureme,nt of Yin

45

52

vii

Figure 2 .. 17

Figure 2-18

Figure 2-19

Figure 2-20

Figure 2-21

Figure 3-1

Figure 3-2

Figure 3-3

Figure 3-4

Figure 3-5

Figure 3-6

Figure 4-1

Figure 4-2

Figure 5-1

Figure .5-2

Figure 5-3

Figure 5-4

Figure 6-1

Figure 6-2

Magnitude of Admittance of a Stator Winding

Phase of Admittance of a Stator Winding

Magnitude of Admittance of Rotor Winding

Phase of Admittance of Rotor Winding

A Single Passive Coupled Circuit

The d and q Axis Equivalent of a Single Winding

The Slip-Ring Primitive

Reciprocal Harmonie Inductance

Stator Current Waveforms for a Two Phase Wound Rotor Induction Motor Operator at a Slip of 0.5 from a Balanced '!'wo Phase Supply Having an RMS Phase Voltage of 120. AU Four Windings are Identical, 90° Spread, 180° Pitch

The N-ph.ase Winding and Its '!'wo-Axis Equivalent

vi,ii

The Three Phase Winding aud Its 'l'wo-Axis Equivalent

53

54

55

56

58

66

67

73

85

89

93

The Synchronous Machine with Damper Windings 102

Phasor Diagram of Kron's Ideal Machine 111

Three Wlre Star Measurements 130

Line Current and Mutual Voltage of Three Wire Star Synchronous Machine at a Load Angle of -25.6° 131

Computed and Measured Line Current, Third Harmonic Neutral Voltage, Torque and Sixth Harmonic of Field Current for Various Load Angles 133

Computed Sixth Harmonic of E1ectromagnetic Torque and Damper Winding Currents J.34

Line Current and A.C. Component of Field Current for a Four Wire Star Connection of Synchronous Machine at a Load Angle of 22° 139

Predicted and Experimental Values of Electro­magnetic Torque, Fundamental and Third Harmonic Line Current, and Sixth Harmonic Field Current for the Four Wire Star Connected Synchronous Machine at Various Load Angles 141

Figure 6-3

Figure Al-l

Figure A2·l

Figure A3-l

Figure A4-1

Figure A4-2

Figure A8-l

)

Predicted Sixth Harmonie Damper Winding Currents and Electromagnetic Torque at Various Load Angles for the Four Wire Star Connected Synchron­ous Machine

Synchronous Machine Open Circuit Characteristic

M22/L2 for Various Currents

M32/L2 for Various Currents

Measurement of Self Inductance of Approximately 1200 Spread Windings

Measurement of Mutual Inductance of Approx­imately 1200 Spread Windings

Modified dq Axis Phasor Diagram for Three Wire Star Connected St~tor

ix

143

A-2

A-4

A-6

A-8

A-9

A-16

( )

Table 2-1

Table 2-2

Table 2-3

Table 2-4

Table 2-5

Table 2-6

Table 2-7 '-

Table 2-8

Table 3-1

Table 3-2

Table 3-3

Table 5-1

Table 5-2

Table 5-3

Table 6-1

)

LIST OF TABLES

Winding Factoœof the Experimental Machine

Comparison of Predictedand Measured Inductance Ratios

Winding Factors of the Airgap Equivalent Field Winding

Correlation of Theory and Practice

Time Constants Determined by Frequency Response Analysis with a Stator Windihg Along the Direct Axis and Along the Quadrature Axis of the Field Winding

Self Inductance of Stator Winding and Field Winding by Electronic Fluxmeter and Frequency Response Analysis

Self Inductance and Resistance of Equivalent Damper Windings and Maximum Value of Stator

x

41

41

46

47

59

59

Damper Mutual Inductance 60

Direct Axis Damper l'linding Equivalent from Field Measuremente 61

Effect of the Magnetic Field Created by Balanced Stator Currents for a Slip ~f O.~ 86

Effect of the Magnetic Field of the f/2 Component of the Rotor Currents 86

Transformation Coefficients Used by Various Authors 98

Comparison of Runge-Kutta and Newton-Raphson Values for ~ = _10 0

128

Comparison of Computed and Measured 5th and 7th Stator Line Current ijarmonics 132

Harmonie Analysie of Additiona1 Terms 132

Comparison of Runge-Kutta and Newton-Raphson Solution for 5 = _10 0

137

Table 6-2

Table 6-3

Comparison of Computed and Averaged Measured 5th and 7th Stator Line Current Harmonies

Additiona1 Measurable Harmonies

xi

138

140

a, b, c, s

A

Ac

B (11.)

c

C

D

F (1')

G

i

f

h, k

l

L, M

L<1"

L

m, r, k

P

p

~ (11-), Q ('Il)

q

~. ", R/L )

1

xii

NOMENCLAWRE

suffix signifying windings of the cylindricnl member of the machine

connection matrix

cylindrical surface area of the airgap

radial airgap flux density

suffix signifying the commutator primitive

connection matrix

mean diameter of the cylindrical member of the airgap

airgap nnnf

torque matrix

winding current

field winding

winding factors

even series index

winding inductance

leakage inductance matrix

effective axial length of coils in slot along the air gap

odd series index - when used as a subscript denotes harmonic of air gap nnnf; as a superscript denotes harmonic of radial airgap flux density

d/dt operator

power

specific airgap permeance

slots per pole

resistance/inductance matrix

( sI.., s2, f

T

T

U

v

x

y

y

Z

0(,

0<.,(3 /'\

(:l "',-~t

fi

&

6

0 , ()

e I{

f tu

W

1t

suffix signifying windings of the salient member of the machine

electromagnetic torque

time constant

stored energy

voltage

xiii

general position along the airgap measured from the direct axis

pole pairs

admittance

impedance matrix

phase angle

displacement between a winding axis and the direct axis

constant of integration

angular separation of phases

hysteresis angle

load angle

displacement between two winding axes

coil span

position of direct axis relative to stator vhase

angle of datum rotor winding relative to the direct axis

flux linkage

angle between adjacent elots

total turns 1 pole

angular frequency of stator voltage

general position along airgap measured from the direct axis

l "

1.

CHAPTER l

The first electric machine of mod.ern form was a connnutator

machine constructed by Sturgeon and reported in 1838 in the Annals of

Electricityl. In 1886 the theory of magnetic circuits 'was applied to

2 the design of electric TIk~chines by J. and E. Ropkinson and Kapp. Demand

for alternating current became significant st the end of the nineteenth

century and, as a result, considerable effort was expended towards the

proper construction ~nd analysis of these electric machines. Despite

this effort by some of the best minds of this era many problems arose,

3 sorne of which remain with us today. For example, Oberretl states that

"the second induction motor constructed by Dolivo Dobrowolsky in 1889

suffered from poor running up".

During the first decade of the ~gentieth century significant

456 7 contributions were made by Carter , Steinmetz , Blondel , and Lyon in

8 terms of machine analysis and by Fortescue in terms of proper machine

utilization. These authors as weIl as others set the stage for three

9 significant papers presented in the ·1920's. Wieseman considered the

graphicsl determination of magnetic fields in electric machines. Doherty

and NicklelO extended Blondel's two axis method to a machine with nnnf

harmonics. 11 Park improved the equivalent circuit concept of an electric

machine.

By the 1930's much of the foundation of electric machine desigq

and analysis was established. Each class of machine was considered essent-

ially on separate grounds. However, the design of electric machines in-

variably required calculations based upon the magnetic circuit and the

2.

prediction of performance invariably utilized an equivalent circuit point

of view in which reactances were substituted for fluxes.

In 1930, Kroril2. published the first of many papers. He advo­

cated that the· basic similarit~es ben7een the vast majority of electric

machines should be emphasised rather th an the differences. He assigned

a geometric instead of circuit significance to electric machines, view-

ing reactances as tensor operators in a vector space with current axes.

Kron showed that the slip-ring primitive ac machine and the commutator

primitive dc machine were related by a suitable; transform of axes. After

a time lag of near1y a decade, the genius of Kron was recognized and many

authors, some using Kron's controversial notation, began to extend this

approach.Authors such as Kron, Adkins13 , Gibbà14 , Lynnl5 ,.Fitzgerald

d i 1 17 1 18 d iff19 Whi d d 20 Y 21 an ~ ngs ey ,Bew ey , Tang an Cosgr , te·an Woo son , u ,

and Messerle22 contributed mu ch in terms of the breadth of application

and the re-evaluation of it for pedagogic purposes. 23 Higgins gives an

excellent bibliography of contributions to generalized machine theory.

In parallel with the Kronian type analysis, the classical treat-.

ment of electric machines has also been extended. Most noteworthy of

24 25 26 27 . works in this are a are those by Alger ,Concordia ,. Langsdorf' ,and

Kuhlmann28 •. The two basic approaches to the ana1ysis of e1ectric machines

tend to complement and supplement each other.

By the end of the 1950's and beginning of the 1960's a trend

was established in which a re-examination of electric machine analysis

in relation to practical problems which hitherto had either been ignored

or were treated very superficially was begun. 29 Chalmers puts many of

the problems of the ana1ysis of electric machines in practica1 perspective.

-

c

3.

The areas of concern may be summarized as the problems of:

(a) non-linear iron

(b) commutation

(c) complex machine geometry

(cl) hysteresis and eddy currents

(e) winding configuration

(f) abnormal operation

Contributions to basic understanding and analysis in these sixareas will

be considered in turne

Analysis of the affect of saturation of ac electric machines in

30 31 terms of adjusted or saturated values of reactance ' originated in

32 the 1930's.Hamdi-Sepen extended the method for two axis considerations

by ascribing direct axis and quadrature axis saturation factors as weIl

as direct axis and quadrature axis saturation coupling factors. 33 Silvester

proved that the concept of inductance is valid in terms of the slip-ring

primitive model of an electric machine in the presence of a saturation

34 35 type non-linearity. Erdelyi et al ' demonstrated that the complete

magne tic state of an electric machine may be evaluated by numerical methods

if sùfficient information about the machine such as machine currents, geo-

metrical configuration of the airgap and suitable knowledge of the iron

characteristics is available. In the author's opinion, it is the last

two papers which outline the type of procedure necessary to calculate

accurately the influence of saturation in an electric machine. The draw-

back to this approach is the length of time required for solution even on

a modern high speed digital computer. Thus a compromise approach of an

extension of the method of Hamdi-Sepen might be more practical. It would,

)

4.

however, probably be necessary to map experimentally the state of the mach-

ine considered for aIl combinations of excitation that would be expected

to occur in operation.

36 Alger and Bewley summarize the classical treatment of commu-

tation - that of treating it as a separate problem di~tinct from other

aspects of machine performance. Jones and Barton37 showed that a linear

slip-ring machine of constrained geometry and winding distribution with

periodically altered rotor connections possessed the mathematical prop-

erties requi.red to enable its replacement by a commutator mach~ne. Thus

the transformation between Kron's two primitive machines could be con-

38 sidered as more th an simply a fortuitous application of Floquet's

theorem-regarding the transformation of differential equations with time-

varying coefficients to a simpler forme

Modern contributions to the solution of problems arising from

39 non-simple machine geometry include the papers of Angst and Oldenkamp , 40 41 42 43 Ginsberg et al , Oberretl , Ahamed and Erdelyi ,and Robinson . The

first two papers -deal with modifications to the two-axis theory to include

first order additional pole shape effects which result in third harmonic

voltage generation. The body of this thesis contains a consideration

whièh encompasses the approach of these papera but which is more general.

In fact, the third harmonic voltage generated by the machine analysed

arises from a .machine state neglected by these authors but related to a 41 phenomena recently emphasized by Oberretl in conjunction with induction

motor asynchronous crawling torques. Ahamed and Erdelyi use an approach 34 35 previously outlined ' • Robinson recognizes the .importance of airgap

geometry in terms of the interrelationship of radial airgap flux density

c·

5.

harmonics, nmlf harmonics, terminal voltages, winding currents and torque but

fails to link these quantities together in a forro suitable for the production

of numbers. This thesis contains an analysis of h~rmonicB which leads not

only to third harmonie voltage prediction but to other related influences

of harmonies upon terminal quantities such as voltage, current and torque.

Historically the work of Steinmetz is a classic in the consider-

ation of problems of magnetic hysteresis. 44 Recently Nagy used an elliptic

approJcimation and Slemon45 a trapeziodal approximation to a B-H loop to

evaluate hysteresis motor operation. However, the area is still very much

an open field and one of significance, as the inductance measurements of

Chapter 2 indicate.

Silvester46

and others listed in the references of this author

have considered in depth the problem of eddy currents. The analysis is, in

general, -constrained to very::simple situations although Silvester makes the

point that frequency domain techniques may be used to measure an equivalent

47 set of circuits to represent the eddf currents used by Bharali and Adkins

48 49 50 and Ewart et al • Others such as Kuyper and Walker have given approx-

imate methods of calculating the power 108s resulting from pole face eddy

currents of high-speed turbine driven generators.

51 Liwschitz - Garik has considered the mmf harmonics produced by

the stator and damper windings and the resulting synchronouB torques which

52 can prevent starting. Nasar has erroneously claimed to justify the assump-

tion of separate equivalent machines for each space harmonic in terms of

53 torque production. Buchanan has obtained an equivalent circuit for a

single-phase motorhaving space harmonics in its magne tic field. 54 Chalmers

has shown how A.C. machine windings may be arranged to reduce harmonic content.

(

r 1

.... ~

6.

51; V. P. Anempodistov et al J have considered harmon~cs in design of 750 MW

56 57 58 turbo generators. Koenig , Prescott and E1-Kharashi ,Jones ,Carter

et a159 and Barton and Dunfield60 measured flux linkages and obtained

machine inductances from these measurements which were of a form not in

agreement with that assumed by Kron and his disciples. The messurements . 61

were verified by the theoretical work of Robinson and B~rton and Dun-

60 field an~as will be seen, arise because of winding configuration and

airgap geometry. The implications of machine harmonics enabled the prim-

itive transformation matrix to be put on a strong physical basis as out-

1ined in Chapter 3.although the resultant form is of little value as far

as computations are conèerned.

Much emphasis has been placed on steady state behaviour in terms

of machine analysis because of transient Dlodel complexity and difficulties

in ana1ysie. Various transformations are availab1e to simp1ify the ana-

62 1Y8is. The works of Clarke , Kimbark and contributors to discuss1.on of

63 64 65 66 67 68 his paper·' ,Wagner and Evans , Lyon ,Rsiao ,Hwang ,and Rao

and Ra069 best summarize considerations regarding transformations of

variables during various types of fault conditions. Analysis of three-

phase to two-phase slip-ring transformation given in Chapter 3 establishes

a p~ysical reason for the transformation of Clarke and related transfor-

mations in terms of machine harmonics. 70

Wahl and Ki1~ore and Maginniss

and Schultz7l have con~idered transient performance of induction motors

uti1izing axis transform methods and assuming no interaction of electrica1

and mechanical differential equations. Recently S1ater et a172 and Smith

and Sriharan73 have calculated transient performance of induction machines

using a Runge-Kutta integration procedure. Chapters 4, 5 and 6 are con-

7.

cerned with the numerical solution Qf synchronous machine steady state

performance for cases where the simplifying assumptions of Kron are

not applicable. Sorne of the techniques are quite general and may be

applied to the prediction of transient performance of electric machines.

Expression of basic differential equations of electric machines 1s seen

to be preferable to an over-reliance upon tradit10nally accepted'models.

, .......

c

8.

INDUCTANCES OF A PRACTICAL SLIP-RING PRIMITIVE

PART 1,. AN ANALYTICAL STUDY

. INTRODUCTION

It is now more than thirty years since Kron unveiled his gener-

alized electric machine theory and, while interest in it was only slowly

aroused, it is now firmly established as an analytic tool. The original

theoretica1 treatment, a1though ful1y explored by Kron, was 1eft by him

14 15 in a somewhat undi.gestib1e state, but interpretation by Gibbs , Lynn ,

20 White &nd Woodson and many others have so c1arified the basic theoretical

issues that attention can now be fruitfu11y turned to the closer corre1~

ation of the structure of Kron's primitives with the non-ideal structure

of practical machines. In this context may be cited the work of Jones37 ,58,

59 61 Carter et al , and Robinson • The latter three papers are aIl concerned

with the inductances of actual machines and their variation with saturation

level and rotor position, a Most pertinent topic since the basic the ory

deals with a linear machine of~~ circumscribed airgap geometry and

winding distribution.

20 White and Woods on' gave a detailed theoretical treatment of

the harmonic effects of practical windings in a uniform airgap machine

but did not cover the crucial complication of saliency. 56 Jones was the

firet to show by measurement that the standard assumptions were far from

practical rea1ity for the case of cornmutator machines. This study had

important implications regarding the physical nature of the cornmutator

58 primitive which were discussed in a subsequent paper • A Most va1uab1e

c

9.

by-product of Jones' 'original work was the development of a verypowerful

experimental tool for the determination of inductance in a ferromagnetic

environment and in the presence of permanently closed eddy current path,

57 a method discovered independently by Prescott and El-Karashi • The

measurements of induct.ances of a salient pole alternator made by Carter

'59 et al also indicated grave discrepancies between th~ory and practice.

Their results were, however, difficult to interpret with precision since

the measurements of inductance were made by standard a-c techniques and

were therefore subject to the effects of permanently closed, coupled

circuits such as damper windings and solid iron and to waveform distort-

ion due to non-linear magnetization characteristic. These problems are 1

avoided by Jones' fluxmeter techniques. Frequency response techniques

may be used to de termine the effects of permanently closed, coupled

circuits.

61 Recently Robinson made a theoretical study of machine in-

duc tances based upon conductor and air gap permeance distributions and

was able to draw extensive conclusions which are backed by some exper~

imental evidence. The same theoretical conclusions are reached in this

section as Robinson's, but by a route whic~ becaus~ based on more trad-

itional concepts, is perhaps more easi1y grasped and yields results of

complete generality in a more compact forme The experimental investi-

gations of Parts II and III have a1so enabled conclusions to Qe drawn

which are not based on·idealized permeahce waves as are those of

Robinson.

DEFI NI TI ON OF INDUCTANCE

The circuits of electric machines are invariably embedded in

10.

a u~gnetically non-linear environment and any theoretical treatment

which is to be tested in practice must take cognizance of this facto

Probably the Most fruitful approach to the production of numerical

results ia to consider the d and q axis fluxes produced by aIl the cir~

13 cuits acting together as i8 done by Adkins • However, it is far easier

to perform analytical work by considering the contribution to the flux

of each separate circuit through its self inductance and its mutual

inductance, as attested to by the almast universal use of this approach

by proponents of the ge~eralized theoL~. It thenbecomes necessary to

consider the validity of the concept of, inductance in the non-linear

case. Not surprisingly the answer to this problem depends upon the mean-'

ing to be allocated to inductance. Three possible definitions are avail-

abl~ from the linear case, flux linkages per unit current, rate of change

of flux linkage with current and twice the ratio of electromaglletic stored

energy to the square of current. AlI these definitions can be applied to

particular non-linear situations with' profit, however"it ia theauthor's

experience that the'f.irst .. is mÇ)st generally suitable ,forelect:t:icmachine

,analysie and,this isthe definition adopted hare.

33 Silvester has, shawn that this preferred definition, together

with the requirement that inductance be a unique, continuous, finite

function of aIl circuit currents, leads to a .unique inductance matrix for

the non-linear case. The non-linear inductance matrix is extremely com-

plex,and is not as well'adapted to numerical computation as the d and q

axis flux approach but it daes establish the validity of the algebraic

operations normally e~plGyed in the development~f the generalized the ory

of ,machines.

CI 11.

Much of the complexity of Silvester's approach is avoided in the present section by considering only one coil to be excited at any one time. A technique for including an approximation to the saturation

effects associated with,this coil is presented in Part II.

RADIAL AIRGAP FLUX DENSITY IN AN ELECTRIC MAClUNE

The electric machine to be considered corresponds to Kron's

primitives in that it has a salient member to be designated by s, and

a cylindrical or non-salient member to be designated by c, the adjectives salient and cylindrical being preferred to stator and rotor since the

the ory is applicable no matter whether the saliency be on the rotor or

stator. The usua1 d.and q axes fixed to the salient member are tlsed as

reference. axes.

'fbe radial airgap flux density can be expressed as the product

of a specifie airgap permeance and an airgap mmf. The form of the airgap specifie permeance depends on the length of the airgap and the shape of the salient member but for practical machines, which are magnetically

symmetrica1 about bath the d.and q axes, it can be expressed in terms of the electrical angular disp1acement }t from the d axis by an even cosine

series; 1. e. ,

P (1t) = Po + p 2 cos 2 ~ + P 4 cos 4"}t + ... c»

p (']t) = ~ lJcos 1~ 1=0

2-1

where l is an even integer •.

10 Doherty and Nickel estab1ished that, for the relative1y short airgaps employed in practical machines, radial airgap flux density can be expressed as the product of an airgap mmf and a unique specifie air gap

74 permeance; a conclusion supported by the later work of Bet·dey if the

12.

machine is assumed to contain a 1inear iron flux path. The first order

effectaf saturation of the machine iron can be considered bY'assuming

that the specifie afrgap permeance can be a function of the airgap mmf

excitation level but not of the excitation, i.e., the airgap specific

permeance remains symmetrical about the d and q axes. A1though the

latter assumption has no theoretical justification, it has been found

to hold true in practice over a considerable excitation range, experi-

mental ev:l.dence of this being given in Part II.

A symmetrical winding in slots whi~h are not skewed having LJ~

total turns per pole and whose axis is inclinedat 0( electrical degrees

to the direct axis, as indicated in Figure 2-1 excited by·a current i,

creates a magnetomotive force at the airgap which May be expressed as

a function of the angle 1'-C<by an odd cosine series, 1. e.

cos (1l- .... p() - h3 cos 3 (1'-cC) + hS

œ

F (1'0) = L Fm cos m (~-o() mel

where m is an odd integer

3 -S-

cos 5 (~-ot) ••• }

~ is the winding factor of the mth harmonie.

2-2

2-3

AlI authorities. agree on the numerical value of the winding

factor but there is considerable variability regarding its signe Rere,

winding factor is defined.as the algebraic ratio of the mth ~ harmonie

produced by the actual winding to that produced by a full pitched con-

centrated winding having the same number of turns per pole. Thus if

the two mmf's are in phase, the sign of the winding factor is positive,

13.

CI AXIS

FIGURE 2-1 THE ANGULAR NOTATI ON

,.."

whereas if theyare in antiphase it is negative. The appropropriate

expression for a winding having a coil span of ~ , placed in q slots "

per pole, adjacent slots having an electrica1 angu1ar separation of t' ia

sin Dl; sin m 'frf' _ ____ 2 ___ ,..--...I!I.o" _

h CI

m q sin mU sin .!!!l-,- 2

14.

The airgap flux density is obtained as the product of equations

2-1 and 2-2 ClO 00

B (1t) c LI L P /) Fm cos ll, cos m (1!-0() 11l= JJ. =0 ~

or, by simple trigonometric expansions

B <~) a t: 1; !II./m f cos (<1+ m)'J - mot) +

cos «R - m)')I. + me() 1 R being even and m odd.

2-4

Thus the interaction of an mmf harmonic with a permeance harmonic generates

two flux harmonics, one of arder 1 + m and the other of order .A - m, a

givenair gap flux harmonic being the combination of components arising

from U1S?lY such interactions. Further, aince l is even and m odd, a11

flux harmonics are odd.

THE GENERAL INDUCTANCE EXPRESSION

The airgap flux created by theoC winding links th~ other

mach:l.ne windinga and thereby contributea ta their mutual induçtance.

Other contributions of win ding inductance come from various leak~ge

fluxes, but these can be taken into account by the Iater addition of

other terme.

The flux linkage "'e( of the flux characteri:z:ed by equation

2-4 with a second winding of ~~turns per pole whose axis is inclined

(i

)

15.

at f electrical degrees to the direct axis and whose winding factor to

the r th flux harm0nic is k : r

À == f!. foC .R=o m=l

where Ac = 1l'DL is the total cylindrical surface area of the airgap,

D being the mean diameter of the air gap and L the effective

axial length of windings.

Evaluation of the integrals and simplification of the resulting

expressions yields the mutual inductance L fOC of the two windings as:

co . CIO

L == 4 Ac Woc W@ ~ 2: PA cos J.;: r ~m k.Q +m ~ 7T 2. Jl =0 m=l l 'Tkil

cos (mo( - (.I.+m)P> _:m l~~m cos ( md + (~ -m) P)} 2-5

This expres~ion leads to resu~ts which, allowing for the

difference in notation, are identical with Robinson's. However, the

expression is cumbersome and can be made more useful and compact by

taking the summations of the two righthand terme separately. Thus, the

last term can be sunnned over a dummy variable r replacing the odd integer

m: GO

~ rel cos (ro(+ (l-r),)

The dummy variable r can be replaced by -m which, by noting

that h = h and that cos (-6) = cos 9 yields -m _C\\\

~ .hm kil. +nt cos (mo<.- (li +m)A) ~ m ~ t"

16.

'!his expression is, except·for the 1imits of summation, identi~

cal with the first term of equation 2-5. Interchanging the order of

summation yie1ds

4 Ac W~W~

7T2

"'00 +00 L.L. P.Q cos Jhr hm k..Q+m .1=0 m=-tO 2 m .Il. +m-

cos (mo(- (~+m)p) 2-6

A simi1ar expression can be obtained for ~@ which, sinceJand m range over a11 possible values, yie1ds a resu1t identical with the above

although the terms are in different order, thus eotablishing tbe necessary reciprocity of mutual inductance.

To avoid constant repetition in subsequent work the 8ign~is +co +00 taken to denote ;?:. L where m 1s an odd and..l is an even integer. ~. 0 m=-ex> SLIP-RING PRIMITIVES

It is worthwhi1e defining primitive machines ~o simp1ify an

inevitably complexnomenc1ature. This is done for two phase and 'three

phase slip-ring primitives by Figure 2-2. It 1s to be noted that the

angle of the datum roto~ winding relative to the direct axis is àenoted by 9.

Equation ~6 may now be applied to three specifie cases:

(a) when both windings are on the salient member;

(b) wh en the two windings are on opposite sides of the airgap;

(c) when both windings are on the cylindrical member.

WINDINGS ON THE SALIENT MEMBER

Here the angles 0( and ~ are fixed so that the inductances are, as expected, inde pendent of rotor position. LU is obtained by putting

both 0( and ~ equal to zero and km = hm; L22 b.y putting 0( and fJ equal to n/2 and hm = km; and L2l by putting 0( equal to zero and (J equal to ."./2

17.

(/) -x « -

i~ equation 2-6. Thus,

cos 17f hm h.{) +m "2 ID l+m

WINDINGS ON OPPOSITE SIDES OF THE AIRGAP

18.

2-7

2-8

2-9

l t 1s convenient, a1though not necessary, to take the 0( wind-

ing as on the cylindrical member; the angle P is then a fixed quantity

ando( a variable dependent on the rotor position. The inductance is,

as expected, represented by an odd harmonic series.

In applying the general equation to the primitive machine it

is only necessary to consider the mutuals L14 and ~24' windings 3 and

B, band c differing from 4 only by a fixed angular disp1acement. For

winding 1,' is zero; for winding 2, p is1r/2; and for winding 4,9(is e.

Thus,

L = 4 Ac W14J 4 L PJ cos .l1T hm k,q +m cos me 14 "Ir 2 T m -:y:;;-

2-10

L24 = 4 ~2fJ)4 ~ Pl sin mlT hm ki +m sin me '2 ID l+m

2-11

The values of k in equation 2-10 ar.~ those for winding land

in equation 2-11 for winding 2.

WINDINGS ON THE CYLINDRICAL MEMBER

In this case the angleso( and ~ are both dependent'on the salient

member position. They do, however, have a fixed angu1ar re1ationship.

6 = ~ - 0(. Thus it is convenient to make the substitution {J= 0( + ~

(

; ,_.

19.

in equation 2:..() and also to note that, since the windings are balanced,

CA = 6J~ and lt = h. l'hus, P n n

cos .. hr.hm h.Q +ID cosellO( +& (11 -tin») 2 Iii' ..Q +m

The inductance ia here represented by anerenharmonic series, again in accord with expectations •

. 'IWo Phase Windine! ;

The self inductance L44 is obtained by putting 0(= 9 and & = 0 and the mutual inductance L34 by putting 0( = e and S = 7r 12:

2 .. hr cos Jl L44 =71iz Ac iAJ 4 L P

A cos hm hJ +m e

T Iii .l+m 2

sin ~ Q 1 ~ 4 AcW 4 2:P.Jl sin m1T hm h.f+m ~34 ii2 T m "T-kil 2-14

It is of interest to consider the component sin (m.,.. 12)

2-12

hm/m h R +ml1l +m of the latter equation. Again replacing m by a dummy variable r yields sin (rn 12) hr/r h Jl+rl .f +r. As m. has aU possible odd values it is alw~ys possible ta choose a value of r = -(l+m), corresponding ta

any particular value of m, ta yield - sin (1 +ID) 7f 12 h .1 +ml ~ +m ~/m. Expanc;1ing the sine term and noting that .JI. is even yields - cos· ( l1112)

sin (m 71' 12) hm/m h R +ml .Q +m. ThuB if J. is amuI tiple of four this term will cancel the original term and the sumef the series is zero. There are therefore no harmonics of the mutual for these values of J i.e., the zero, fourth, eighth; etc.

Three Phase Windings

The self inductance of the 'a' winding is identical in form with L44 and the self inductances of windings·'b' and 'c' differ only by the

Il.

20.

appropriate phase. The mutual inductance Lab is obtained by putting

b '" 2 7r /3 and L by putting tf '" -2 Tf /3. ac

L '" 4 AcW a 2, P-t cos 111" hm h Il +m co~ JI. e + (~+m) 271") ab i'- tt- 2 ID Jl +m T

INDUCTANCE RATIOS IN THE PRIMITIVE MAClUNES

The simplification of the equations brought about by trans-

formati9n from the slip-ring to the eommutator primitive is an essential

part of generalized machine theory. This transformation 18 affected by

harmonie components of the various inductances but, as the series in

practice converge rapidly, it is acceptable to ignore the harmonies in

most cases. nle inductances of the eylindrical member present an exçept-

ion since the slip-ring to commutator primitive transformation requires

that the second harmonie eomponénts of the self and mutual inductances,

for bath the two phase and three phase primitives, be of equal magnitude.

This is far from the case except in highly idealized situations.

If L2 ia the amplitude of the second harmonie camponent of

self inductance and M22 and M32 are the correspo~ding amplitudes for the

mutual inductance of the ~o phase and three phase primitives, then

from equations 2-13,.2-14 and 2-15: cc

~ sin m1f hm hm+2 M m=-tD T m ïii+2' .22

'" ~ ~ ~2 2-16

L2 mc-co m nt+2 te

~ cos (m+l) 2n' hm hm;-2 M32 3 rn ~ '" m=-(lC)

L2 ~ 2-17

hm ~+2 m=-eo m m-1-2

2-

, .

Cl

21.

The ratio of the constant componcnt, M30 , of the three. phase

primitive mutual inpuctance to that of the self inductance is also of

some interest, although the value is modified somewhatin practice by

leakage components.

2-18

lt ia not possible to quote generally applicable values for

these ratios because of the independent influences of winding pitch

and distribution on the winding factor. However, much can be deter-

min~d from sorne special cases.

Firstly, for the ideel case of sinusoidally distributed

windings, ~ is zero for aIl values of m other than unity. The ratios

then becorne:

1.0

2-19

These are the values necessary for the simple transformation. Cons id-

erable divergence from these ideel values occurs in practice, as

indicated by Figure 2-3, which shows the ratios for equ~tions 2-16, 2-17

and 2-18 plotted as:functions of coil span for windings uniformly dis n

tributed over phase bel:ts of 60°,90° and 120°. Concentration of the

conductors in a finite number of slats results only in minor modification

of the values. Figure 2-4·indicates the variation of M32/L2 for several

values of slots per phase per pole as functions of coiI span.

(1) o

2.5

2.0

@

~132 6 () SPREAD l2

M22 ---fr-cC[ 1.5 lŒ:

---r---+----~1--~4 1 ~ . . 7 ::;;;;- • l l2 90 SPREAD

I?JJ U 1.0

@

M32 120 SPREAD z 41: 6 ~ M30 0 ;:) 0.5 /. -l 120 SPREAD Cl ~ - .. 0 -~ M3() 60

0SPIREAD

~ • 1 lO 1 . 1 1- 11ft 1 Ol 0.5 0.6 0.7 0.8 0.9 1.0

COlL SPA-N POLEPITCH

FIGURE 2-3 INDUCTANCE RATIOS AS FONCTION OF COlL SPAN

N N .

23.

r--------r--~--~rT~-+------~------r_----_+----~3

0)

ci

cr

ifJ ':) Q r1J 0:: 0

§Ç a:: ~

!lll~ i' ci

~ .

~------~------r-----_+------~------r_----_+----~~ Ô

~------~------~----~------~------~----------~8 o M

o c\J

o d

ou.v~ 3:>NV.l.::>nONI

w ,~ ~ 0: ü)

..J

8

"'" 1 N

; t,!) H ~

)

24.

In general ,t:here are wide divergences from the ideal values

expressed in equation 2-19. There are, however, some significant

exceptions; thus it will be noted that good agreement is obtained for

any 'tvinding spread when the coU pitch 1s 2/3 and for any coU pitch

when the winding spread is 120°. These are situations where the third

harmonie component of the winding nnn~ is zero. They emphasize the

important role played by this harmonic in the inductance expressions

and also suggest a reason for the Buccess of the idealized theory in

the prediction of the behaviour of practical three phase machines, the

third harmonic effect in such machines being zero except in those rare

cases where zero sequence current flows.

CONCLUSIONS

The expressions developed for the inductance of electric

machines indicate that large divergences from those for the idealized

primitive machines are likely in practice. This divergence is fully

supported by experimental evidence and has important theoretical con­

sequences aince the simple sliprring to ccmmutator primitive transf9r­

mations cannot then be performed.

25.

( PART II. AN EXPERIMENTAL STUDY

INTRODUCTION

Part l of this chapter discussed the theoretical implications

of winding mmf and airgap permeance harmonies for the self and mutual inductances of windings. This part describes the results of an exper~

imental investigation designed to check these conclusions and to

establish the inductances of some test machines. Part III of this

chapter is concerned with measurement of parameters of coupled coils

with no accessilble terminaIs.

While the data recorded here is the result of a very detailed investigation regarding a particular salient pole machine, it is in

accord with the behaviour of many other machines recorded by the

authors and other investigators in references 56, 57, 59 and 61.

THE TEST MACJ:nNE

The machine is a three phase, salient pp le alternator rated

at 220 V, 3 KW, 0.8 p.f. lagging.

The field winding is on the rotor and there are four salient

poles each having a 600 turn exciting wi,nding" a 2 turn search coU at -"'" each end of the main field winding and a damping cage in the pole face.

The 48 slot stator carries a balanced, three phase, double

layer windin~ of 5/6 pitch and 52 turns per pole and phase.

The stator ie also equipped with a complete set of full pitch search coils mounted immediately in the slot openings so that it will be possible to measure, as closely as possible, the radial component

of the airgap flux. The slot flux can be measured by connecting adjacent coils in series opposition.

26.

INDUCTANCE MEASUREMENTS

As discuBsed in Part l, the definition of inductance 1S taken to be flux linkages per unit current, a modification of the fluxmeter

56 57 technique of Jones and Prescott and EI-Kharashi being used to measure flux· linkage. This technique relies on the intcgration of induced emf resulting from a polarity reversaI of a winding current in a coupled

circuit and is unaffected by the presence of permanently coupled passive

circuits and by saturation type non-linearity. The low input impedance drawback of the conventional fluxmeter can be avoi.ded by performing the

necessary integration with an electronic integrator, a high impedance

instrument, as described inreference 75. However, as this reference IJlélY

not be readily available, a brief description of the method will be

given.

The equipment, as set up to measure self inductance, ia ~hown

in Figure 2-5. The Maxwell-Rayleigh bridge is balanced by adjustment

of the Latto arma Rl and R2' the drift rate of the integrator being ,

then ideally zero, in practice very small. The reversing awitch, which

ia motorizeQ for convenience, is then operateçl. At the end of the

current reversaI the bridge is aga in in balance but, during reversaI,

the emf induced by the changing flux in th~ inductanceLxproduces a

voltage which ia integrated by the detector. The final output of the

integrator is then a measure of the change in flux linkage and is

unaffected by other passive circuits coupled to Land by saturation, x these merely affecting the rate at which the output is established.

The measurement of mutual inductance is similar to that of

self inductance but ia simpler in that the bridge c.ircui t is not

necessary.

Oc Supply

1

-!!!!!!!!I Motori zed Reversing·

Switch 5

c

Electronic Integrator

Trigger Pulse

FIGURE 2-5

1:: Magnitude ofsteady-state current

_ R,RS RX- ·R·

2

LX=RC. R1+ Ri 0 V·~ -.1 . R2 - 2L

INDUCTANCE MEASURING CIRCUIT

N

" >

i\

28.

Added conveniences are the switch S which may be used to short­circuit the input to the integrator except during current reversaI and the oscilloscope trigger pulse. The timing of these various operations

ia governed by cam operated switches.

The winding inductances were also measured continuously as

functions of rotor position by plotting flux linkage at a constant

current, and rotor position on an XY recorder. The effect of damper

winding eddy currents on these results was eliminated by driving the

rotor at a very.low speed of appro~imately l rpm. Unlike results

obtained by current reversaI these show the effects of magne tic hyster­

esis, a most interesting feature which will be commented on later.

1 NDUC.TANCE 1 DENTI FI CATI Jlli

In g~neral the three phases of the machine are identified

by the letter s or specifically by the letters a, band c. The field

is denoted by f. Rence a subscripted inductance Mfa signifies the

mutual inductance between the ·field and the a phase.

The point at which the direct axis coincides with the axis

of the a phase is taken as the origin of rotor positon.

THE FIELD INDUCTANCE

The inductance varied with excitation current, i.e. with

saturation level, in the expected manner as shown in Figure 2-6. The

inductance at first rises with increasing current as the initial bend

of the magnetization characteriatic is traversed, then the inductance

decreases as the knee of the characteristic is reached and the machine

begins to saturate. Although the stator slots were not skewed, the

rotor inductance was found to be inde pendent of rotor positioq. The

% .. -1

J2 ·a-Iaoot- . .1

elt 800~ % ::e t;

::lE

.-

~

._~

lf

~

1\11 $ f ~

4t 400p--------~------~~----~---~--------t-

-----. ft "==",",,,.,M,~ ••••• I.,'''bL'. L"","".I"'" """,, .. L."..ii.'"O.L ..... "'_& ..... '.,b .. ".!JO. hl ...... d .......... L-,.;,...

FIGURE 2-6

0.5 1.0 1.5 2.0 FIELD C URRENT v A

FIELD SELF INDUCTANCE AND· STATOR-FIELD MUTUAL INDUCTANCE VS FIELD CURRENT

N \0 o

,/

'-

30.

absence of any tooth ripple effect ia attributed to the semi-closed

stator slots and weIl chosen pole arc.

THE FIEJ~D STATOR MU'l'UAL INDUCTANCES

These resulta are both current and position dependent. The

value of the mutual inductance between the stator a winding and the

field winding is shm~n in Figure 2-6. as a function of field current

and as a function of stator current in Figure 2-7 for zero rotor angle, i.e. when the axes of the two windings coincide. Since mutual induct-

ance must be reciprocal under similar magnetic conditions, these two

curves enable the ratio of field turns per pole to effective winding

turns per pole to he determined. The value thus found 'is 10.8, in

reasonable accord with the value of Il.55 computed from the design

data.

The position dependence of aIl three stator rotor mutuals for a rotor current of 0.5 ampere is sh~ in Figure 2-8. The three curves

are aIl similar and bear the expected phase relationship. llereafter,

, only one set of resul ts is illustrated. The waveforne of these rotor

stator mutual inductances have negligible harmonie content even at the

highest saturation, and are typical of aIl machines in the author's

experience.

Similar resulte to the -above sh~ :.~_n !!'igl\re 2-9 were taken

on a XY recorder with the rotor continuously, slowly turning, firat

forwards then backwarda.

A lag of about JO ia apparent in Figure 2-9b due to hyster-

esis in the rotor irone The corresponding lag in Figure 2-98 due to hysteres1s in the stator iron 1a too small to be seen due to the much

/ \..

t

Cl o O. v--

,

, o N

on ~

en 0-E m

"-....... C CU

C .... r-'-:.a U L. .a ~

31.

E-l

1 ~ ~ CIl

CIl :::-f;I;l

~ E-l 0 ::;, !@ H

1 ~

~ CIl

'0 S t:

400

.r:. E ~ (.)

~ --c c -

-800

F'IGURE 2-8

80 100 Position

. STATOR-FIELD MUTUAI. INDUCTANCE VS POSITION FOR I.e = 0_ S AMP_

w ~ . .

flJ­C6

~

80

Ga800

FIGURE 2-9a

180 -POSITION.

STATOR-FIELD MUTUAL INDUCTANCE FOR

If = 0.5 AMP.

" 360 DfEGRIE_E S

w w

:r: ~500

m et-

~

œ.500

FIGURE 2-9b

1

POSITION,

FIELD-STATOR MUTUAL INDUCTANCE FOR l = 6.0 AMP. a

360 1

DEGRE ES

1,,0.) ~

35.

higher magnetie quality of the stator laminations. It should be noted

tbat the hysteresis angle results from the combination of airgap and

non-excited member and is therefore much smaller than for the iron alone.

The open circuit characteristic and the field stator mutual

inductance are related to one another. This relationship is considered

in Appendix 1.

THE STATOR INDUCTANCES

The direct and quadrature axis self inductances of a stator

winding are sh'own, as funetions of winding eurrent, in Figure 2-10.

The effects of magnetic saturation, while not so pronouneed, are similar

to those of Figure 2-6.

The self inductance L of the red phase and its mutual induct­a

an~e M with the c phase are shawn as functions of rotor position in ca

Figure 2-11 "t\lhich app1ies for an a phase current of 6 amperes. Similar

waveforms are obtained at other values of current. The second harmonie

.nature of the variation and the appreciable higher harmonie content are

apparent.

Typical continuous XY plots of these inductances are shawn in

Figures 2-12a and 2-12b. These curves have the Bame general form as the

curves of. Figure 2-11 but with modifications due to hysteresis. The

influence of hystere~is is particularly marked in the cal3e of the

mutual inductance of Figure 2-12b which shows both horizont~l and vert-

iea1 displacernents.

lt is possible to make sorne quantitative correlations of the

influence of hysteresis on thesereS\;l:lts .... If the a phase is excited

and,the rotor is rotated forw~rd, rotor hysteresis causes the magnetic

8

X6 ::E

w

" z <t4 lm­(.)

:l Cl Z .."...

20

FIGURE 2-10

Ld

2 4 6 STATO"R 'CURREI~Tt A

DIRECT AND QUADRATURE AXIS INDUCTANCES VS STATOR CURRENT

8 10

W 0\

80

:x;

I&B o z40 ~ (J ::') c z --

FIGURE 2-11

80 120 POSITION, 1

~

200 DEGREES

STATOR S-.:.'LF INDUCTANCE AND STATO'R MUTUAL INDUCTANCE VS POSr.ION FOR 1.., = 6.0 AMPS.

W -...J

, .. ~

%40 :!

:20 ..si

FIGURE 2-12a

45 90. POSITION. MECHo

135 D EGRLElES

STATOR SELF INDUCTANCE VS POS! TI ON FOR l a = 6. 0 AMPS.

ISO

w 00 .

.. 0 zm o i= -(1),

2

39.

JI --I,------+-~~~~::::..--_L 8

2

• o an

8

40.

axis of the coil to be advanced, thus increasing the coupling with the

b phase and reducing it with the c phase. When the rotor is moved back­

wards the reverse situation occurs. Taking·a simplified view of hyster­

esis, if the shi ft in axis is~o electrical, then the angular separation

between the forward and reverse curves will be 2~ and the ratio of the

two mean mutual inductances is cos (120 + â ) /cos (120 - & ). Thus, the

vertical shift in inductance and the horizontal shift in phase can be

correlated. The angu1ar displacement of the curves of Figure 2-l2b

yields a value of 5 of 3.5°, which indicates a ratio of amplitude of

1.24. This compares weIl with the actual value of 1.2.

CORREl·ATlON OF THE STATOR INDUCTANCES

It has bean noted in Part l that the ratios of the harmonic

components of the stator mutual and self inductances are solely depend­

ent on the winding configuration and not at aIl on the airgap permeance.

It is therefore possible at this stage to make a preliminary correlation

between the ory and practice.

The winding factors for this machine are listed in Table 2-1.

Substitution of these values into the appropria te one of equations

2-15, 2-17 or 2-18 of Part l yields the predicted values of the harmonic

ratios. These are listed and compared with the measured values for

the zero, second and fourth arder harmonies in Table 2-2. It is to

be noted that the agreement between the two sets of values is good, the

worst case being represented by the fourth harmonie which is difficult

to determine experimentally because of its small amplitude. Harmonics

higher than the fourth are too small for reliable experimental deter­

mination.

{ 1

41.

Table 2-1. Winding Factors of the Experimental Machine

Harmonic, m 1 3 5 7 9 11

Winding Factor, h .925 .462 .053 .041 .191 .122 m

Table 2-2. Comparison of Predicted and Mea8ured Inductance Ratios

Inductance Ratio, Mutual/Self

Harmonic predicted Measured

0 .456* .472

2 1. 765 1.86

4 .613 .72

* The airgap value of the zero order component of self inductance was obtained from the measured value by subtracting the leakage inductance 4.35 mh.

'!'he good correlation'. between theory and practice obtained

here is gratifying, but it only touches on a small aspect of the

measurements. Accurate prediction of the absolute values of aIl the

inductances provides a much more stringent test of the theory. However,

before this test can be applied it is necessary to determine the air gap

permeance and the airgap equbralent of the field winding.

THE SPECIFIC AIRGAP PERME~

The speci.fic air gap permeance i8 defined as the ratio of the

radial airgap flux density to the airgap mmf and was determined by

exciting the machine from the stator, i.e. with windings of known mmf

and measuring the resultant mean tooth flux with the search coils. The

N

Ec ,~ ln. '-Q..

~ i 1.2 cu ~

M '0 ~

x Ces QJ

o c co QI

E t-

rf. De4

1 l~

/. /.

, ..... "

----0·-

Co i l Axi~ Along d Ax is Coi[ Axis Along q Axis

Coit Axis Midway Between

~

d . Ax is and q Ax is -t------+-----_+_

0,\ 1 1'} ,~ 0 O_e_._ ... ~

.-.... -.,.- ."

15 30 45 60 Position, mechanical degrees

FIGURE 2-1.3 SPECIFIC AIRGAP PERMEAN'CE VS POSITION FOR THREE SEPARATE POSITIONS OF THE MAGNETIZING COlL RELATIVE TO THE DIRECT AXIS AND AT A MAGNETIZING CURRENT OF 6.0 AMP.

75 90 .po ~

43.

waveform of the airgap flux density was calculated from these values.

This waveform was divided, point by point, by the known airgap mmf, to

give the waveform of the specifie airgap permeance. Measurements were

taken at several different excitation levels with the axis of the

exciting coil in various directions to investigate the uniqueness of

the specifie permeance series. Figure 2-13 shows typical results taken

at ~ current of 6 amperes and with the coil axis along the direct axis,

the quadrature axis, and midway between:these two axes. When weight

is given to the fact that flux measurements in the low flux density

zones close to 90° from the coil axis are subject to appreciable in­

accuracies, the results are seen to be remarkably consistent. It is

of interest to note the large dip in specifie permeance on the direct

axis. This results from two holes drilled into the poles along the

direct axis forbolts which hold the pole to the shaft.

The first three harmonie components of specifie air gap per­

meance are shown, as functions of direct axis stator coil current in

Figure 2-14. The permeance components show the expected effects of

saturation.

THE AIRGAP EQUIVALENT OF THE FIELD WINDING

The above procedure can be reversed to obtain the airgap

e'quivalent of the field winding, the flux density produced by a given

rotor excitation current being divided by the specifie airgap permeance.

This distribution of equivalent field turns per pole is shawn in Figure

2-15, the winding factors for this equivalent rotor winding being tab­

ulated in Table 2-3. The distribution of Figure 2-15 is approximately

of trapezoidal form having the expected amplitude of 600 turns over the

N fl8 Ec ~5 L..~ œJ:.l., ..cE Ql} nJ 3:

('f'J),

'0 r-

x cu

(1,6

~ QA tU cu E L­W

0..

Q,2

-1 1 1 1 - 1

(PO

--- :..-

.

tP:l ..

-- - - ~ .....;

-- ~ -

r--

-(P, 1 1 1

f 1 ~ J 1 ft ft R ~-------6 7 8 . 9 10

stator Current v amps FIGURE 2-14 HARMONIC COMPONENTS OF SPECIFIC AIRGAP PERMEANCE. AS A FUNCTION OF STATOR

CURRENT ---------_., .. _._._-_._--_._ ... -.--~-"._--

~ ~

-90

; Ji

-60 -30

FIGURE 2-15

mmf ·amp

600

'.

400 ~!~----+-

Experi mental

200 ~ -- TrapezoidaL Approxima tion

30 60 Position, eLectrîcal degrees

AIRGAP EQUIVALENT FIELD WiNDING 1

90 .po. \.Il

!

46.

pole arc. Table 2-3 includes the winding factors of the trapezoidal

form for the purpose of comparison with the actual equivalent rotor

winding. The author believes that, in the absence of a definitive

method of establishing the equivalent rotor winding, this approxi-

mation may be used with profit.

Table 2-3. Winding Factors of the Airgap Equivalent Field Winding

Harmonie Order m 1 3 5 7

~quivalent Field Winding iFactors-Actual hm .939 .521 .015 .200

~quivalent Field Winding Wactors-Trapezo~dal Approximation hm .955 .637 .191 -.136

This discrepancy between the mmf of the actual rotor winding

and the equivalent field winding at the airgap is explainable in terms

of field leakage fluxes. In the reglon between the pole tips, the

reluctanee o~ the airgap becomes appreciable compared with the reluct-

ance of the winding leakage path. Thus a lower level of airgap flux

density is recorded by the search coils which are distributed along

the stator side of the airgap. Clearly, this equivalent rotor winding

is that which must be used to calculate the mutual inductance between

field and stator, and the component of field self inductance not attrib-

utable to field winding leakage fluxes.

CORRELATI ON OF THEORY AND PRACTICE

The winding information of Table 2-1 and 2-3 and the permeance

data of Figure 2-14 enable predictions of the various inductances to be

made according to the equations of Part l. Comparison between predicted

( '\

47.

and measured inductance values appears in Table 2-4 which app1ies to

an excitation 1eve1 of 6 stator amperes or comparable field current.

Appendix II gives more co11aborating evidence for a two phase machine.

Table 2-4. Correlation of Theory and Practice

Harmonic Stator Self Stator Mutua1 Stator-Rotor Field Self Inductance Inductance Inductance Mutua1 Induct- Inductance Order

0

1

2

3

4

Mi 11ihenry Mi 11ihenry ance Mi11ihenry Henry

Expt.* Ca1c. Expt. Ca1c. Expt. Calc. Expt.

55.75 56.8 -26.3 -25.9 0 0 10.2

0 0 0 0 812 826 0

10.7 11.8 19.8 20.8 0 0 0

0 0 0 0 - -14.2 0

3.0 2.20 -2.2 -1.35 0 0 0

* The airgap value of the zero order component of self inductance was obtained from the measured value by sub­tracting the leakage inductance of 4.35 rob. This leak­age inductance was determined fram measurements of stator fundamenta1 voltage and current with the machine stator three wire star connected, the rotor dr'iven at synchron­ous speed and the stator search coi1s used ta detect a nu1l of fundamenta1 airgap flux. This test a1so indicated an effective field-stator turne ratio of 5.4 or in terms of the design turns after sccounting for a 3/2;t2 factor, a field-stator turns ratio of 11.46, a value in accord with those quoted ear1ier. A Potier test with a delta connected stator gave an incorrect value of leakage inductance of 13.1 rob. The error arises because of surprising1y large time harmonics of current that can f10w in a ba1anced synchronous machine. This i8 a matter which is considered in Chapter 6.

** Includes a value of 1eakage inductance of 0.95 h obtained by camparing flux 1inking the stator and field for field excitation.

Ca1c,~*

10.6

0

0

0

0

,

48.

The excellent agreement between the ory and practice is typical

of aIl reasonable excitation levels for this machine. The constraint of

saturating iron is considered in Appendix III.

CONCLUSIONS

The the ory of machine inductances based on the concepts of

equivalent air gap windings and mmf and permeance harmonies which is

developed in Part l is weIl substantiated by measurements. Although only

data for one machine is given here the the ory has been applied with equal

success to a typical d.c. machine and to the machine described by Carter

et a159 •

A consequence of this work is th?t wide divergences, from the

behaviour of the inductances of the idealized primitives are to be ex-

pected in practice. This divergence may be particularly attributed to

the interaction of the third harmonic component of mmf with the first

few specific air gap permeance harmonics. Appendix IV illustrates a

method of connection of the stator windings to simulate a machine with

stator windings of 1200 spread. In this configuration, both the stator

self and stator mutual inductance measurements showed negligible higher

harmonic content, and amplitudes and ratios in excellent accord with

predicted values. Thus the discrepancies between the ideal Kronian

inductance representations and the observed inductance behaviour results

essentially from the existence of a thtrd harmonic component of stator

mmf. Other mmf harmonics do play a part but this is small due to the

rapid convergence of the inductance series when one or more windings

on the cylindrical member are involved.

The divergence from the assunœd behaviour puts in doubt the

process of primitive transformation since the latter absolutely relies

(

49.

on the assumed behaviour. It is, therefore, necessary to investigate

the implications of these results in this area, a matter considered in

Chapter 3.

50.

PART III, THE DAMPER WINDING EQUIVALENT CIRCUI T

The measurement of stator and rotor winding inductances

presented in the previous section using an electronic fluxmeter is

not affected by the presence of passive coupled circuits. These cir-

cuits are present in many electric machines. They result from closed

eddy current paths and squirrel cage type windings. The question of

the location of these passive coupled circuits and, in an engineering

sense, the modelling of them will be considered in this chapter for

the machine tested in Part II of this chapter.

Data available from the manufacturer of the machine showed

that it had an amortisseur winding of six copper bars embedded in the

76 pole face of the laminated rotor. Kinitisky has modelled damper

bars of salient pole machines as two separate windings - one on the

direct axis of the machine and the other along the quadrature axis.

77 Sylvester has shawn that eddy currents in solid iron may

be modelled as an infini te series of coupled windings. The quality of

material and thickness of lamination of the stator and rotor material

are different, the stator of the machine being designed to carry the

main pulsating flux. It is therefore assumed that there is negligible

stator eddy currents but that the rotor may contain significant eddy

77 current path. A calculation based on the work of Sylvester for

the rotor material with airgap neglected suggests a break frequency

of the order of 1 cps for the first equivalent eddy current winding.

Nagy has illustrated that an elliptic approximation to the

51.

B-H curve of magnetic material leads to the concept of complex per­

meability and, therefore, of a frequency dependent hysteresis resist­

ance and reactance.

An experimental approach utilizing frequency techniques was

devised to determine the first approximation of the influence of the

damper bars and rotor eddy currents although the complicating affect

of hysteresis was neglected., in the analysis of the resul ts.

AN EXPERIMENT

The input admittance of a stator winding with its axis first

along the direct axis and then its axis along the quadrature axis of

the machine and of the rotor winding was determined with the aid of

78 the circuit of Figure 2-16. The power amplifier constructed by Birch

was a cyclo-converter capable of producing an output power of the order

of 5 kw from 0 to 50 cps. A low pass fil ter attenuated components of,

higher frequency than the source frequency. Phase angle was deter­

mined with the aid of a storage oscilloscope by adjusting the phase

shifter for a nUll in phase as indicated by a straight line Lissajous

pattern for Vin and lin and subtracting the readings indicated on a

calibrated phase shifter. Amplitude was determined by ratio of

amplitudes indicated by the oscilloscope.

Fig~res 2-17 and 2-18 show the normalized amplitude and the

phase plot of Yin for a stator axis along the direct axis and the

quadrature axis for about two and one-half decades. Figures 2-19 and

2-20 respectively show the normalized amplitude and the phase plot

of Yin for the rotor winding. These experimental results are curve

fitted with t.hree break points of equation 2-20 - that is one zero

52.

.E -\

, j

1

( )

fi~orm. 1 db. ,

-5 o DA Measurements

D QA Measurements -10

-15

FIGURE 2-17

F HZ. 10

V~GNITUDE OF ADMITTANCE OF A STATOR WINDING

il

r~

100

I.n W .

.. " N )

%

, )

o o

0 ........

-

>:Ji ~

e QJ t; ua CI CD :i

<C Q

0

~

54.

§g ~ z ~ ~ ~ E-i CIl

<Xl ~ 0

rz:I

~ E-I

~ f:l rz:I

~ p,.

-= • ~

00 r-I

1 N

~ ::1 en Cl

t! CD :E

ct 0 Q

0 0 0 V CD , • ,

· '0 "0

0 ==

,t

~',

aa. ' '

" ,55. '

t

~ z

m M

~ M

~ ~

~ ~ ~

' . m 1

o o -o o

o

56.

,"

)

)

57.

and two poles.

K ( 1 + Ta s) 2-20

The fit is adequate for aIl but the higher frequencies. At

frequencies of the order of 50 cps the amplitude response is in agree-

ment, but the phase fit begins to deviate from the experimental results.

This is a manifestation of the influence of additional coupled circuits.

The circuit of Figure 2-21 depicts a coil with one passive

coupled circuit. The input admittance of this coii is of the form of

equation 2-21.

where

Gl

Tl

T2

y .. = 1n

=

=

=

T3 =

2-21

l/R 1

LI/RI

L2/R2 2

M /RIR2

Comparison of equations 2-20 and 2-21 leads to four equations for the

four unknowns Gl , Tl' T2, and T3 •

Gl = K

T2 = Ta

Tl + T2 = Tb + Tc

Tl T2 - T3 = Tb Tc

The results of this analysis are given in Table 2-5.

2-22

2-23

2-24

2-25

58.

E-f

§ p::

~ N ~ N !:il ...:1 0:

....J g u f ~ H CIl :! ~ \ !:il

t3 z !ri <Xl 0:::

\

.-1 N

1 N

~ ~

)

59.

Table 2-5*. Time Constants Determined by Frequency Response Analysis with ~ Stator Winding Along the Direct Axis and Along

the Quadrature Axis and of the Field Winding

WINDING EXClTED Tl T2 T3

msec msec msec

Stator l'Ti th Direct Axis 47.7 27.5 .807 winding axis Quadrature along the Axis 36.6 11.8 .139

Field Winding 249.7 29.5 3.61

* The value of Gl was,in accord with the dc winding resistance pO'f' .68 mhos for the stator winding and .0263 mhos for the rotor winding.

The values of Tl contribute no information regarding the

equivalent damper winding, but enable data correlation with the

fluxmeter measurements of the second section of this chapter. A

comparison of values is given in Table 2-6. The agreement is reason-

ably good especially if the simplifying assumptions and unavoidable

small variation of saturation level for the frequency response measure-

ments 'are considered.

Table 2-6. Self Inductance of Stator Winding and Field Wlnding by Electronic Fluxmeter and Frequency Response Analysis

: , , l"

i

Ld Lq

mh mh

:Fluxmeter * 73.8 53.4

Frequency Response 70.2 53.8

* At 6 stator amperes and comparable field excitation.

Lf

h

10.2

9.18

(

, - -

60.

ANALYSIS

It is convenient to be able to assign actual numbers to the

equivalent damper winding parameters. The leakage inductance has been

measured as 4.35 mh for the stator winding and .95 henry for the rotor

winding. With reference to measurements of the stator winding M(&) is then known from equation 2- 26.

= - 4.35 mh 2-26

The values of the direct axis and quadrature axis damper winding para-

meters are calculated from equation 2-26 and the numbers of table 2-6

and are presented in table 2-7.

Table 2-7. Self Inductance and Resistance of Equivalent Damper Windings and Maximum Value of Stator

Damper Mutual Inductance

Axis Aligmœn t M L2 R2 mh mh ..tl.

Direct Axis Damper 65.8 100.3 3.65.

Quadrature Axis Damper 49.5 141.2 11.97

The direct axis damper win ding referred to the field is calculated

using data from the last row of table 2-5 and equation 2-27. These

results may be correlated with measurements at the stator terminaIs by

M ~ Lf - .95 h 2-27 referring the field data to the stator with the field-stator turns

ratio of Il.7. This data is presented in table 2-8. Comparison with

the direct axis damper winding parameters of table 2-7 shows reassuring

)

agreement.

Table 2-8. Direct Axis Damper Winding Equivalent from Field Measurements

M L2 R2

mh mh ,.S'l-

Measurements 8250 14600 496

Referred to Stator 60.2 106.5 3.62

61.

For the purpose of evaluating synchronous machine performance,

Chapters 5 and 6, the damper winding data referred to the stator will

be used with the mutual inductance between the direct axis damper

winding and the field of value (65.8)x(11.7) mh or 770 mh.

,,--, \. }

,-' ')

62.

CHAPTER 3

AXIS TRANSFORMATIONS FOR PRACTICAL PRIMITIVE MAClITNES

l NTRODUCTI ON

The generalized the ory of electric machines may be considered

as the extension of normal static circuit the ory to the case of circuits

in relative motion with its practical realization characterized by the

transformation from a moving reference frame attached to the rotor of

the machine under study, to the quasi-stationary reference frame of its

co~nutator equivalent. For the majority of situations occurring in

practice this results in conversion of the differential equations des-

cribing the system from the periodic time varying coefficient to the

constant coefficient type, an enormous mathematical simplification.

The transformation normally encountered is that of the two-

phase slip-ring primitive to the two-phase commutator primitive as

described by Kron in his early works. There appears to be little

incentive on both physical and mathematical grounds to pursue other

79 . Cil. transformations such as that of Stigant to dia'g0tytze matrices. A

rigorously circumscribed machine geometry is assumed with w:l.ndings that

produce only fundamental components of mmf and with airgap permeance

comprising only the zero and second harmonies. These constraints are

not usually directly apparent but appear in the guise of equal ampli-

tudes of the second harmonie components of the two phase winding self

58 59 61 and mutual inductances. Jones , Carter et al ,Robinson and Barton

60 and Dunfield in a paper based on Chapter 2 have aIl demonstrated that

the inductances of practical machines show wide divergences from the

( )

'.

63.

above simple types of variation due to the neglected harmonies of mmf

and permeance.

Although the above restrictions appear severe, the the ory

developed on this basis is in agreement with the classical theories

andthere has therefore been little incentive to investigate this

aspect of the problem, attention being more profitably directed to

extension of the generalized the ory in breadth rather than depth. 20 White and Woodson appear to be the only investigators who have con-

sidered this aspect of the slip-ring to commutator primitive transfor-

mation but their work is restricted to uniform airgap machines and is

not developed to astate suited to the numerical solution of problems.

Omission of the salient pole situation and restriction to

purely theoretical studies lende a deceptive simplicity to the. topic.

Consideration of mmf harmonics,even without the added complexity of

perme~nce harmonies, results in an extremely complex situation in

which the efficiency of the dq axis transformation is lost. It is the

intention in the next section to develop the general transformation $nd

to illustrate the problems which arise in its application.

( 1

\,

,/" ,

64.

PARTI - THE mo PHASE TRANSFORMATION

THE d AND q AXiS EQUIVALENTS OF A SINGLE WINDING

The equivalent airgap mmf of a winding whose axis is inelined

at an angle 0< to the direct axis, may be expressed by an odd cosine

series in the angle x - 0(, x being the angular dis placement of a point

in the airgap from the direct axis • ... co

F(x) = ~ l w ~ i sin m 7f cos m(x - IX ) mc-co 1T m '2

where m is an odd integer

w is the number of "'l1inding turns per pole

hm is the winding factor for the mth harmonie

i is the winding current.

3-1

l twill be noted that the harmonie series eovers the range - Go to + OC>

rather than the more usual range of 0 to 00 • This simplifies work

at a later stage when the produet of harmonie series is taken. The mth

harmonie of equation 3-1 may be expanded into direct and quadrature

axis eomponents.

2 hm sin m i F c- w Tf cos mo( cos mx dm 7T m '2 3-2a

2 ~ i sin m OC m(x - '!) Fqm ct ;r w ID

cos 2

3-2b

l t is evident that currents of i cos m 0( and i sin m Zf sin m 0< passed

th respeetively through identical d,and q axis m harmonie windings hav-

ing w hm sin m ~ 1 (11 m/2) turns per harmonie pole will reproduee

th exaetly the m harmonie mmf.

Sinee both the winding turns per pole and the winding currents

are dependent on the order of the harmonie, a pair of sueh windings, is

() "

)

)

65.

required for every mmf harmonie, a situation represented diagrammati-

cally in Figure 3-1. Figure 3-1 shows the actual winding, Figure 3-lb shows it decomposed into its harmonie equivalents, aIl carrying the

same current and ther.efore series connected, and Figure 3-lc shows the

d and q axis equivalent, aIl the windings carrying different currents

and therefore separate.

THE MMF EQUIVALENCE OF THE SLIP-RING AND COMMUTATOR PRIMITIVES

The practical slip-ring primitive is, like the ideal one,

the simplest complete two phase machine. It differs from the idesl

primitive in having mmf harmonics higher than the second" and is

depicted in Figure 3-2 so that the windings nmy be identified. The

windings a and b constitute a ba1anced two phase pair on the cylind-

rical member whi1e the windings sI and s2 on the sa1ient member may

be different. Extension of the arguments of the previous section shows that currents of

i dm = ia cosme( - ib sin m 1" sin m 0(

and

i qm ~ ia sin m ~ sin mO< + ib cos MO<

passed through the mth harmonic d and q axis equiva1ent windings will

reproduce the mth harmonic mmf of the machine.

cos me( - sin m"" /2 sin MD<

3-3 sin m Tf /2 cos MO(' sin m'a(

an equation which maybe inverted to yield

r'- \.

}

(\ /

66.

/

~. ) '. "

92 ~

FIGURE 3-2

67.

t Il AXIS·

.-t Si

. \- 0(4)1

.. \a

THE SLIP-RING PRIMI TIVE

'"

)

68.

i: ~ -- cos m 0< sin m 7f /2

sin m 0<

ib - sin m 7T /2 cos m 0(

sin mc:>(

The connection matrix for the mth harmonic Cm is therefore

cos m 0( sin m 71 /2 sin m 0<

3-5 -sin m 7f /2 cos m D(

sin m 0(

It shou1d be noted that ~is orthogonal so that

CI 3-6

The actua1 current matrix i is re1ated to the mth harmonic

current matt'ix of the commutator equivalent, i cm' by i == Cmicm• While

i is invariant it is useful, as a reminder that the physica1 basis of

the transformation is the mth harmonic of mmf, to write i m for i. Thus

3-7

and 3-8

where

VOLTAGE EQUIVALENCE

The above procedure endows. the commutator primitive with an

airgap mmf, and hence flux, identica1 with that of the slip-ring

machine. It is a1so necessary that the dynamic behaviour of the two

rotors be identica1, a condition requiring the reproduction of the

e1ectric intensity in the airgap. This can be accomp1ished direct1y,

in a manner similar to that just emp10yed for the mmf equiva1ence, but

Kron's method, using the invariance of total instantaneous power, is

more convenient. To this end it is noted that the orthogona1ity of

(

\.

69.

sine functions ensures that instantaneous power is on1y produced by

the interaction of mmf and flux waves of the same numerical order.

Waves of different numeriea1 order interaet to produce forces which

vary round the rotor periphery but the resultant of these forces is

at aIl times zero.

With the above concept in mind the voltages applied to the

slip-ring machine, represented by a matrix V, are subdivided into com­

ponents due to winding resistance, Ve , due to leakage reaetance, Vd'",

and due to flux harmonies Vr , r being an odd integer denoting the

order Qf the harmonie. Thus +00

V = Vr + V<f + ~ vr 3-9 re-et)

~ th Interaction of the m mmf harmonie and the .1 permeanee harmonic (l being

an even integer) produees flux harmonies of order ~ + m andJ - m. Thus

each mmf harmonie, in combinat ion with the appropriate permeanee har-

th r monie, eontributes to the r flux harmonie. The voltage V may there-

fore be analyzed further into eomponents V~ •

CIO

Vr = L 3-10

m=-œ

r The voltage matrix Vm, i.e. the voltages indueed in the windings by

th th the component of the r flux harmonie produced by the m mmf harmonic,

is the basic voltage eomponent of the machine. Its equivalent in the

eommutator primitive will be identified by the use of a subscript c

r as Vcm •

In the original machine, sinee aIl the harmonic component

th windings are in series, the power assoeiated with the r mmf harmonic

70.

ili ili and that portion of the r flux harmonie contributed by the m mmf r

harmonie is Vmt i while the same quantity in the commutator prim­r

r itive is Vcmt i cr - The requirement of power invariance therefore

yields

fromwhich, since i r = i = Cr'-icr (equation 3-7)

Vc~ = Crt V~

RESISTANCE AND LEAKAGE INDUCTANCE

3-11

The rate ,of dissipation of energy as heat by the actual wind-

ing and the energy stored in the lealçage fields are

=

...

Conservation of energy requires that Rck' the resistance of

the commutator primitive and L~ck' the leakage inductance of the

commutator primitive, be related respeetively to Rand L <f by

't= Pp ek ... Pp

t= U< ek ... U4$"

where kâenotes the order of harmonie

Since i ... Ck i ek

and P p ek ... i ekt Rek i ck

U4" ek = 1/2 i ekt ~ek i ek '

the relationships between Rand Rek and L« and Lçck are

R =2: Ckt Rek Ck k

Ld =~ Ckt Lcr ek Ck

71.

Although it is mathematically possible to assigna form to

the matrices R land L.- 1 which will conform with the above equations, cc. u Ct

the author , in the absence of physical criteria for making the sel-

ection, has chosen to leave R and ~ untransformed, L.e.

+ V<r :::: (R + LeS") i 3-12

EQUIVALENCE OF IMPEDANCES

The voltage component V: i6 related to the current matrix,

r i m by the impedance matrix zm. This defines the coupling, by way of

the appropriate permeance harmonic, between the mth harmonic mmf wave

th and the r harmonic flux wave.

:::: 3-13

Substitution of expressions 3-7 and 3-11 yields

Zc~ :::: C~ ~ Cm 3-14

r The inductances of Zm are determined fram a generalization of the

derivation of Chapter 2, a matter discussed in Appendix V. For a pair fr

of windings « and p La( m is

'r v L.Jm == 0 l'l.... W.GI cos (rA + me() 3-15 ~ r+m "'m "'r r

the total mutual inductance between the two wir.dings being obtained by

sunnnillg both of the odd integers m and r independently over the range

- IJO to + (O. l t should be noted in this context that

')f ~ (r+m) = ~ r+né w'" (-m) = l'lc( m

and w f (-r) :: wp r

A point which, while not relevant to the present discussion,

becomes of importance in the later work on torque, should be noted here.

C'r ()(r Reciprocity does not apply between L~mand L~ m but rather between

~, ')

'-,

~ r o(m Lo< m' and Lts r That this anomaly is' apparent rather than real is

shawn by the following argument. The voltage Vr of equation 3-13 m

arises because the mth mmf harmonie reacts with the,ith permeance

harmoni~~ being equal to r - m, to produce a component of the rth

flux harmonic. th The latter reacts with the r harmonic winding to

72.

produce the voltage ~. Thfu process is represented diagrammatically

by the upper row of Figure 3-3. The lower row of this figure shows

the reciprocal relationship. Thus the postulation of turns ~ rin the

upper row implies that a cur.rent ip will produce an mmfF~ m in the

lower' row. Since double range series, are used, the permeance P of r+m,

the upper row implies an identical permeance p-(r+m) of the lower row

and hence the production of the fluXqPm. Additionally, when equation

3-13 is summed over aIl values of m and r so as to yield the complete

machine behaviour, the reciprocity requirement is fulfilled.

Equation 3-15 is used to de termine the harmonie inductance

matrices of the slip-ring primitive of Figure 3-2 by introduction of

the correct angles 0( and ~ and the correct number of turns. Thus to

establish the self inductance of winding a, the angle@ is put equal

toO( and the number of turns w pris put equal to Wo( r. To obtain

the mutual inductance between winding b and a, the angle, is put

equal to~ + ~ and again wp r is equal to w~r. To obtain the mutual

inductance between windings b and sI, the angle ~ is put equal to

0( + ~ and the angle oc. equal to zero, etc.

The resulting inductance matrix, L~, for couplings associated , ~ ~

with the component of the r flulc harmonic produced by the m mmf

harmonic is given in equation 3;16.

winding cuwrent

harmonie winding

~-<"~

.mmf . permeance flux harmonie winding harmonie· harmonie harmonie VI inding 'volfa gel

Ir . ig 'iF' wm 'IiP Fm .,. I=r+m 11'. 6· ~ Vlfr ...

.. d".

r ~Vm

. mb

" " " " ""torque producing . ~'" '" interactions

/ " /' " , ln

:aD-- Fr .. I=m+r ... .+ . fjlllm Y3m ~ w,

FIGURE 3-3 RECIPROCAL HARMONIC INDUCTANCE

"m rr

"

-..J W .

81

s2

r L = m

2f r+m

a

b

74.

sI s2 a b 3-16

Ws1m ws1:r wam ws 1r -sin m7T 12

1 cos me:< wam ws1r

sin m 0(

-sin m1T 12 -sinr1f/2 -sin m 7(12 sin r1T 12

ws2m ws2r wam ws 2r Walll ws2r

sin m 0( cos me;(

ws 1m war -sin m1r 12 wam war -sin mTll2

cos ra< ws2m wai:' cos (m+r)o( w W am ar

sin r 0( sin (m+r)o(

-sin r1112 -sin mN 12 -sin r 'If 12 -sin mn/2 sin r 'ff/2

-sin r7T12

ws 1m war ws2m war wam war wam war

sin r D( cos rc( sin (m+r)D( cos (m+r)K

Equation 3-16 can be checked by considering the c1assical

slip-ring primitive whose windings on1y produce fundamental components

of mmf and whose airgap permeance is described by only the zero.and

second order harmonics. Flux harmonics of first and third order are

produced but as the windings can respond on1y to fundamental compon-

ents,.on1y the cases of m = ± 1 and r = ± 1 need be considered. The

four situations then represented by equation 3-16 yie1d the complete

inductance matrix as

1 1 \.

)

75.

- ('if 0 + ~ 2)

wal wsII

sino<

( go - lf2)

wal ws2l

cos~

2 -wal (f 2

L=2 2 ( II 0 + l( 2) wal (~ 0 - ~ 2) wal « 0 wal +

wsll cos 0< ws2l sin 0( 2

';J 2 wal cos 20< sin 20<

2 Ko wal - nf 0 + ~ 2) ( li 0 + ~ 2)

2 ~2 -wal

walwsll wal ws2l sin 20<. -W;l 't 2

sinD< cose( cos 20<

a result entirely in accord with the impedance of Kron's primitive.

TRANSFORMATION TO THE COMMUTATOR PRIMITIVE

The inductance matrix of equation 3-16 transforms according

to equation 3-14 to yield the corresponding inductance matrix for the

commutator equivalent. From equation 3-5 it is seen that the approp­

riate values of Cm and C~ are

C = m

1

1

cos m 0( sin m1l /2 sin ml)(

-sin m 7f /2 cos mO< sin mo(

3-17

3-18

(

( .

\

76.

;

1:

1 r

C t = cos r f>( -sin r 7T /2 3-19

sin rD(

sin r1T /2 cos r 0(

sin rK

In accomp1ishing the transformation it is essentia1 to remember the

location of the differentia1 operator d/dt = P implicit in equation

r r 3-14 •. TI1US th~ required matrix is C tpLm Cm" Its eva1uation, wliich

is at the best tedious, is most readi1y accomplished by expansion

fo11owed by partitioning in the way made familiar by Kron's ear1y work.

TItus,

3-20

yielding

Ws 1m wslrP wam wslrP 3-21

-sin m71 -sin m7T sin r 7T 2'" 2'" 2"

sin r 7( W W

2" am s2r

ws2m ws2rP p

ws 1m warP -sin m 71 /2 wam warP -sin m 71/2

ws2m war wam war r 0< r~

-sin r7T /2 -sin mll' -sin r1( /2 -sin m 'Ir sin r 7T' 2'" '- -2 2

sin r 71 "'2

ws1m w ws2m war wam war wam warP ar

rO<-Il

P rO<

and for Kron's slip-ring primitive

Z =2 c

2 wsl l

(~o +~ 2):P

('g 0 +~ 2)

wsll walP

- (d 0 + è(2)

wsll wal • 0(

( do - t 2)

2 ws21 p

( 'if 0 - 'lf 2)

ws21 wal 0

1><

( ~o -'le 2)

wa21 wal p

TORQUE RELATIONSHIPS

77.

Of 0 + g 2)wsll

walP

Cd 0 - '(f 2)

ws 2l walP

( 2f 0 + (1 2) ( il 0 - t 2)

2 2 walP W .'al

~

- ( If 0 +l$ 2) ( 60 - ~ 2)

. wal 2 0

wal 2 p eX

AlI standard worka on generalized machine theory show that

the output torque of a rotating machine is

T = 1/2Y it ~ i ~.

3-23

the matrix dL/doC being the torque matrix, G. Implicit in the torque

equation is, the requirement that the inductance matrix be symmetric.

This condition i~ not fulfilled by the commutator primitive but, by

• defining its torque matri~ as the coefficient ofe( in the impedance

matrix, its torque can be expressedas

3-24

This problem is raised here because, as already noted, the

r basic inductance matrix l;n is.asymmetric and cannot be employed in

3-2.

(

78.

ëquation3-23. The physieal basis of this can be seen from Figure 3-3

in whieh, from a torque production viewpoint, the' upper and lower-rows,

th hitherto independent, are cros~ linked by the interaction of the m

th harmonic_mmf and flux and the r harmonic mmf and flux. Thus it is

no longer possible, either on mathematieal or physical grounds, to

keep the two rows of Figure 3-3 separate and torque must-be expressed as

1/2 (y it d(~ + L~) .i) do(

Sinee two torque components are ineluded in this expression, it is

evident that the total torque, T, is

T - 1/2 ~ 'f m=-Q) r=-oo

3-25

3-26

Since, in conjunction with L!, i represents the mthand it the r th mmf

harmonie, equation 3-25 can be more clearly-written

T~ + ~ = -3-27

1!ORQUE :OF'THE COMMtl'rATOR PRIMITIVE.; -

The required transformation to the commutator primitive is

now obvious

d~ T~ + ~ ~ 1/2 Y f J.ert c\ dO<

Expanding

3-28

)

r Cr dLm

t dO<

C = Ifr+m m

-sin m7r/2

W W . ar sIm

r

and

m cmdLr .. C ID

t"dO[ r

't m+r

-sin r 11'/2

war ws2m

r

-sin r11' /2

wam wsIr

m

79.

-sin r'1l'/2 3-29

Wam ws lr

m

-sin m 7T/2

wam 'tV's2r

m

-sin r 11' /2

W w am ar

(m+r)

-sin m 11/2

W W . am ar

(m+r)

3-30

-sin m 7T /2

W w . ar sIm

m

-sin r 71' /2

war ws 2m

r

-sin m1T/2 -sin mll /2

wam ws 2r war wam

m (m+r)

-sin r7(/2

war wam

(m+r)

,.'-

80.

Adopting Kron's definition of the torque matrix of the commutator

" primitive as the coefficient of ()( , reference to equation 3- 21 shows

m ili that'Gcr ' the torque matrix associated with the r flux harmonie

th and Gc~ , that'associated with the mflux harmonie are

r Gcm =

')J mf-r

G TIl ... cr

)1 m+r

and

-r sin m'lr '2

r sin mlT /2

w w w w s2m ar am ar

-r' sin"r rr /2, -r sin r'R'/2

w w ,sIm ,ar wam war

-m sin r1T12 -m sin rTT 12 -

ws2r wa!Il w w ar am

-m sin m1T12 -m sin m7( 12

ws1r wam war wam

Comparison of equations 3-29, 3-30, 3-31 and 3-32 shows that

r dLr r Ct m C III Gm+G

"""do< m cr cmt

'm di.m =Gr+G m

Ct r Cr do( cm crt

Making these substitutions in equation 3-26 yields

'3-31

3-32

81.

'l'r + Tmr = 1/2 Y f i (G m r )

Ïll crt cr + G cmt

+ i (G r + G m ) cmt cm crt

Since each of the four components of this expression is a

scalar, the whole expression can be condensed to

y (iert Gc~ iem + i emt Ge~ i er J 3-33

For the conventional primitive on1yvalues of m and r of ± 1

need be considered and from equation 3-28·

T 1/2 i 5 2 1 2 ~l 2 -1 = Y c1t 1 Gcl + Gc_l + GC_l

where it has been noted that iC(_l) = icI-

(-1) 1 1 Since G ( 1) = Gland G 1 ' c - c 'c-

I 1 -1 ] T= 2 Y i c1t ( Gc1 + Gc1 icI

From equation 3-31

-ws 21 wal

-wsll wa1

w w s21 al

-wsl1 wa1

2 . -w al

-w 2 al

w 2 al

2 -wa1

82.

Hence

T = 2 Y i clt ( ({ 0 ~ '12) (11 0 - ({ 2)

ws2l wal w 2 al

- ( ~ 0 + )/2) - (U 0 + 11 2)

2 wsll wal wal

a result in accord with Kron's.

SUMMARY

The transformation from a two phase slip-ring machine with

mmf and permeance harmonics, to an equivalent commutator primitive has

been established and it has been shawn to reduce to Kron's classic case

when the appropriate restrictions areapplied. However, establishment

of the transformation does not guarantee its utility and in fact the

general transformation, in complete cOld;rast to Kr on 's restricted form,

is more complexthan the original problem.

The original four winding slip-ring primitive is described

electrically by four linear simultaneous first order differential

equations with periodic time varying coefficients and, while analytic

solutions to such equations are not in general kn~7n, solutions to

specific situations are readily obtained by numerical techniques.

The d and q axis equivalent is described by a leak~ge impedance

equation and 4n simultaneous first or der equationswith constant

coefficients, n being the order of the highest significant mmf harmonic.

)

)

83.

The leakage impedance equation is entirely in terme of the

original variables. as·· noted in equation 3-12 •

. V e + V < = (R + Le( p) i

From equation 3-13

v r c = ~

m=-n

which appears to yield8n equations for the 2n values of r. However,

. symmetry can be employed to reduce these to 4n.

Elation at the enormous simplification implicit in the trans-

formation from time dependent to constant coefficients is rapidly

dissipated when it is realized that the driving voltages are unknown.

Thus the voltage transformation equation 3-11

r Cr Vr Vcm = t m

requires knowledge of ". Unfortunately this is only available, from

equation 3-13, after the problem is solved since, although 2:. V~ is

known, the individual components are note While it is theoretically

possible to obtain an analytic solution to this dilemma, in reality the

problem of relating harmonic voltage components back' to the known term-

inal voltages and forward to the unknawn winding currents is incredibly

complexe The successive approximation type of solution offers an

alternative to direct analytic solution but due to the large number of

unknowns encountered in even a simple problem of this type the optimi-

zation process will be slow.

It might be thought that the transformation would be useful

for the non-salient machine since there is then no interaction between

mmf and flux harmonics of different nUUlerical order. However, as the

84.

complex waveforms of Figure 3-4 indicate, even this modest hope is

unlikely to be realized. These waveforms show the stator currents for

a balanced two phase two pole induction motor with rotor windings

short circuited and stator windings connected to a balanced two phase

120 volt, 60 cps source and were obtained by numerical solution of the

problemin slip-ring primitive forme AlI four windings were sssumed

identical each comprising full pitch coils uniformly spread over 90°. ,

The winding resistance is 2 ohms,self inductance 0.21 henry and leak-

age inductance 0.01 henry.

The waveforms are steady state solutions obtained by a Runge-

Kutta integration procedure for a slip of 0.5; more complexwaveforms

are obtained when the speed of rotation is not thus simply related ta

the syUchronous speed. The appreciable harmonic content of the wave-

forma is immediately apparent and is perhaps not surprising, what is

surprising is the lack of balance in the fundamental component of the

stator currents. Measurements on a two phase machine have given qual-

itative confirma tien of these appreciable departures from sinusoidal

waveformae

A qualitative physical analysis can help clarify these results

. and illustrates the complexity of the problem •

. PHYSICAL ANALYSIS

Consider balanced two phase currents of frequency f to be

flowing in the stator windings. These create two pole, six pole and

10 pole fields rotating with speeds noted in Table 3-1. Consider now

the f/2 component of the rotor emf. This will cause balanced rotor

currents to flow Which produce fundamental, third and fifth harmonic

(

\.

10

-10

85.

CURRENT A.

FIGURE 3-4 STATOR CURRENT WAVEFORMS FOR A TWO PHASE WOUND ROTOR INDUCTION MOTOR OPERATING AT A SLIP OF 0.5 FROM A BALANCED TWO PHASE SUPPLY HAVING AN RMS PHASE VOLTAGE OF 120. ALL FOUR WINDINGS ARE IDENTICAL, 900 SPREAD, 1800 PITCH.

",<"

fields as enumerated in Table 3-2.

Table 3-1. Effect of the Magnetic Field Created by Balanced Stator Currents for a Slip of 0.5

;

Sta~or Speed Speed Induced Rotor Voltage nnnf Relative Relative Harmonie to to Frequency Sequence

Stator Rotor

1 f f/2 f/2 Pos. 3 -f/3 -5f/6 5f/2 PoS. 5 f/5 -3f/10 3f/2 Neg.

Table 3-2. Effect of the Magnetic Field of the f/2 Component of the Rotor Currents

Rotor Speed Speed Induced Stator Voltage nnnf Relative Relative Harmonie to to Frequency Sequence

Rotor Stator

l f/2 f f POSe 3 -f/6 f/3 f Neg. 5 f/lO 3f/5 3f POSe

86.

The fundamental component of the rotor mmf induces positive

sequence stator emfs and so interacts normally with the original

stator currents. The third harmonie component induces negative

sequence voltages of the original frequency and hence disturbs the

balance of the original currents. The fifth harmonie component in-

duces positive sequence voltages of three times the original freq-

uency and is responsible for the third harmonie component of stator

current. It should also be noted that this simple harmonie relation-

ship only exists at certain specifie speeds. At other speeds sub-

(' \.

87.

harmonies modulate both the stator currents and the rotor currents -

both sets of currents being balanced.

CONCLUSIONS

The dq axis transformations for a slip-ring machine having

mmf and permeance harmonies has been derived and has been shawn to re-

duce to Kron's classic case when the appropriate restrictions are

applied.

The general transformation throws light on flux and mmf inter-

actions and on torque production in such complex structures but fails

as an analytic tool since the transformed problem is more complex than

the original.

The question naturallyarises as to why Kron's technique

worka in practice. This is a fortuitous outcome of the almo,st universal

use of three phase machines in which third harmonie interactions are

eliminated by suitable three phase winding connections. The next part

of this Chapter is concerned with n phase to two phase transformations,

a matter of interest with regard to both three phase systems and

machines with squirrel cage rotors.

"

)

88.

PART II, THE POLYPHASE TO l'WO AXIS TR.ANSFORMATION

The three phase to two phase transformation will be con-

sidered. Some. insight into the physical nature of the 0( pC and

related transformations is gained. Although, with the exception of

squirrel cages, three phase windings are now almost universal, certain

aspects of the analysis are clarifiedif the generalized N,to two phase 1

transformation is first de:t'ived. This process is therefore adopted.

THE NO AXIS EQUIVALEN'r OF A POLYPHASE WINDING

The general polyphase winding comprises N similar phases

uniformly distributed around the airgap 6f the machine with an electrical

angular separation, , , of 2~/N. For identification the phases will

be numbered from l to N with the axis of the first phase constituting

the datum direction as indicated in Figure 3-5a.

To simulate the magnetic effect of these windings, currents th iam,and ibm are passed through orthogonal m harmonie windings of

Figure 3-5b. G7neralizing equation 3-3 and a8signing ~ ~ ~ sin m2?T / ~ turns to the a and b axis windings gives

N

lam = L ~ cos m(n-l) S' in n=l

N

ibm = L ~sinm7f/2sinm(n-I)S in n=l

3-34

where Km is a turns ratio of value that will be assigned when orthogon-

ality of transformation i8 discussed.

In matrix notation

i CI

2m 3-35

89.

)

:1 z :) .... -

11"\ 1

"'l----.. ct ~ C

M ...-.-

~ a -~

~

t~ ... ~

~ )h

90.

where " ....

th ~ ia a 2 x N matrix whose n column is Km cos m(n-l)$"

I<m sin m7T12 sin m(n-l)&

Since the matriJe A ie non-invertable, the orthogonal currents may be expressed in terms of the N phase currents but in general not

visa-versa. This· .is an expression of the fact that a given mmf can be

produced by many different current combinationa in the N phase winding

but by only a unique set of pairs in the two phase system. The number

of orthogonal currents required for complete modelling is not unlimited

being N if N is odd and N/2 if N is even as shawn in Appendix VI.

THE GENERAL CONNECTI ON MATRIX

Restricting consideration to systems having an odd number of

phases, the matrix relating the actual currents, in' and the N currents 'iab, of the two phase equivalent is A2N compounded of the individual

matrices

iab <= A2N in 3-36

where A2N ~ f Al. A3' •••• AN-2' ANI J

91.

1 and AN is the first row of AN in accordance with equation 5 of

Appendix VI.

The square matrix A2N may he inverted so as to express the

actual currents in terms of the equivalent system currents. The trans-

formation matrix so ohtained will be called a connection matrix, CN2 •

=

The turns ratio factor ~ is chosen to assure that A2N is

orthogonal as

m .;,. N

m = N

Thus

where

CN2 (n, m) =./2iN cos m(n-l)6

CN2 (n, m+l) = /2ïN sin mI'" sin m(n-l) 6

m being lmy odd integer between land N-2 inclusive and

CN2 (n, N) = ~

3-38

VOLTAGE TRANSFORMATION th The m mmf harmonie by interaction with an appropria te per-

th meance harmonie, produces a component of the r flux harmonie. This

flux can interact on1y with the r th harmonicwindings and induces a

voltage V~ in them, che subscript m signifying the order of the mmf wave

and the. superscript r the order of the flux wave.

Th mbi i f h 1 Vr d the r th e co nat on 0 t e vo tage component man

)

92.

r th harmonie winding current i r yields the component Pm of the r harmonie

power pro Physical identity between the actual machine and its ortho-

gonal winding equivalent demands that this power component be invarient,

i.e.

::

which, by application of equation 3-35 yields

Vh~ = i\t V2~ 3-39

The N phase voltages are therefore expressible in terms of the two phase

voltages, a fact which causes no surprise.when it is remembered that the

harmonie flux wave has only two degreea of freedom, being conlpletely

defined by its amplitude and phase.

r th The terminal voltage, V , of the r harmonie winding is the

sum of the components

Vr =2": V~ = Art ~ V r

n m m 2m

V. r = Art V r 3-40 n 2

The terminal voltage of a group of series conneeted harmonie

windings is obtained by·a sununation which takes.aeeount of the alterations

of winding polarity in the orthogonal equivalent. An expression incorp-

orating this feature is unwieldy and it is more eonvenient to.assign

signs to the harmonie voltages by inspection of a connection diagram

such.as Figure 3-6.

IMPEDANCE TRANSFORMATION

In discussing impedance transformations it is convenient to

separate the winding inductances due to harmonie eomponents of the airgap

"' ---

fDATUM

!DATUM

1

'. 1

al ,~ 1

R .. 03

1

ft

12 + 1

Ld' a

Lq

1 . if t~t~1 brR LA"

~~ ...

.15 17

r ' --5 Il 11 \CI

w . b

FIGURE 3-6 THE THREE PHASE WINDING AND ITS TWO-AXIS EQUIVALENT

94.

, ,. flux from the resistances and leakage inductances.

IMPEDANCE ASSOCIATED WITH THE AIRGAP FLUX

The ratio, number of turns per harmonic component of the

actual winding to that for the two phase equivalent is fixed by the ortho-

gonality requirement of equation 3-37. The airgap components of the two

phase equivalent windings are readily derived from the actual values by

using the phase appropriate to the winding axis and by increasing the

amplitude by the product of the turns ratios of the pair of windings

concerned.

RESISTANCE AND LEAKAGE INDUCTANCE

Resistance and leakage inductance are properties of a winding

as a whole and cannot be divided between the various harmonic components.

These properties can, however, be allocated to the N series groups of

the two phase equivalent.

Considering the case of resistance, invariance of resistive

power loss requires that

Applying equation 3-38

Rab = CN2t RN CN2

and since ~ = Rll and CN2 is orthogonal

Rab = RN 3-41

In a similar manner but using the criteria of invariance of magne tic

stored energy it can be shown that

3-42

The two phase equivalent system of Figure 3-6 is therefore

f"

.... )

95.

completed by the addition of the phase resistance and leakage inductance

in series with each of the coil groups.

TORQUE

The torque, T!, is produced by the interaction of the r th mmf

th th wave with the r flux component due to the m mmf wave must be invar-

iant under transformation. Thus the torque matrix G~ of the origin.al

r system and G2m of its two phase equivalent are related by

3-43

Since the transformation matrices are not time dependant, the

torque matrix is in each case the differential coefficient of the induct-

ance matrix with respect to rotor angle.

HARMONICS HIGHER THAN THE Nth NEGLECTED

It has been established that the electromagnetic behaviour of

a balanced N phase winding requires the flow of N distinct currents in

N distinct windings disposed in a pair of orthogonal axes. Because of

this complexity, problemsolving is not in general simplified by trans-

formation, the most direct solution being obtained by the application

of numerical techniques to the original equations. However, when it

is recognized that the harmonic content of real windings diminishes

rapidly with increase in harmonic order, a very considerable simplifi-

cation can be obtained.

Thus if the rate of harmonic attenuation is such that aIl

th harmonic components higher than the N can be neglected, the connect-

ion matrix CN2 can be employed for aIl transformations instead of the

transformation matrix ~.

96.

The "oltage matrix, Vab ' of the two phase equivalent is the

sum of the leakage impedance component, Va~' and the harmonie components,

LVrb r a

cr ~ r V b -1- L-V b a r a

In this case, ~ V2~ has the particularly simple configuration

~vr -= r· ab

V 1 a

V 1 b.

V3

a

· · · V N

a

The actual voltages, VN

' are related to those of the two

phase equivalent by

which can be written

VN ~ CN2 Va: + 1 Ait 1 A3t 1······· 1 ANt 1 V l

a

V 1 b

· · ·

3-45

V N a 3-46

Since 1 AIt 1 A3t 1······ 1 ~t 1::: CN2 equation 3-46 by comparison with

3-44 and 3-45 may be written

97.

th Renee for situations in which harmonies higher than the N can be

neglected.

==

iN c CN2 iab 3-47

Zab = CN2t ZN CN2 and the standard ana1ytic techniques can be emp1oyed.

THE IDEAL WINDING

Aparticu1ar1y simple situation resu1ts when the windings are

idea1 in that they produce no harmonies. Then Vn is zero when n ~ 1.

20 This is a more genera1 case than that considered by White and Woodson

who app1ied the additiona1 restriction that airgap permeance harmonies

higher than the second were neg1igib1e.

THE THREE PHASE WI ND! NG

The harmonie winding interconnections of the two phase equi-

valent of three phase winding are shows in Figure 3-6.

The transformation matrix is

1 cos m 21T cos m47T 3 -3-

sin !!l!!: sin m217 sin m"" sin m47T 2 ~ 2"" -3-

and m rf. 3

m::: 3

The connection martix re1ating the actua1 currents and the

three currents of the two phase equiva1ent are

')

98 •

r~ . . -

h - 1/.[6 1//2 1

rr 3-49

- 1/[6 - l/JT .J... .. J3

Essentially this is the 'orthogonal transformation employed

by a number of authors62 , 63, 67, 68, 69, 80 Numerical coefficients

differ in magnitude and sign as indicated in Table 3-3 for several

authors. In support of the present proposaI may be cited its develop-

ment from consideration of winding harmonies and its orthogona1ity.

Table 3-3. Transformation Coefficients Used by Various Authors

Multiply Co1umns of CN2 by

Authors Col. No. 1 Col. No. 2 Col. No.

Hwang 1 1 1 Lewis 1 1 1/./3 Clarke /JIn. -~.n .f3 Kimbark .r3lfi - 3/./2 fil 2 Boyajian ../3112 -Ji lin

HARMONICS HIGHER THAN THE THIRD NEGLECTED

3

The threephase winding whose harmonies higher than the third

may be neg1ected 'is of ·the particularly simple class. The winding has

the particu1ar1y simple two phase equivalent shawn in Figure 3-9, the

laws of the transformation being given by equation 3-47 with the conn-

ection matrix of equation 3-48. They have the great tnerit of separating

the fundamenta1 and third harmonie effects. In the situation normal1y

considered, when the third harmonie, in addition to aIl others, is

neglected the two phase equivalent becomes particularly simple aince

the a3 winding comprises solely resistance and leakage reactance and

is completely decoupled from the other windings. One is therefore

99.

faced with the standard two phase problem plus an additional independent

circuit consisting of the constant leakage impedance R + L<rF. In the star connected system with isolated neutral ia3 is

zero and the neutral voltage is Va3/;-3: The two phase equations are

therefore particularly easily solved by the standard methods and the

neutral voltage is then readily obtained.

In the mesh connected system it is Va3 which is zero and the

equations are more difficult to solve because of the coupling between

third and fundamental mmf harmonies .via the permeance harmonies. The

mesh current in this case is J3 ia3' The neglect of harmonies of

order higher than the third is somewhat extreme and application of this

method will not be puraued. Rather, an alternate approach that ia out­

lined in Chapter 4 will be used for the production of numbers.

THE SQUIRREL CAGE WINDING

The squirrel cage winding is a particular case of multiphase

winding perruanently short circuited. The number of phases i8 equal to

the number of bars per pole pair and is commonly large. Thus the approx­

imation of neglecting' ,a11 harmonica higher than the N - 2 is therefore

particularly weIl, justified and the transformation of equation 3-47 may

be applied. From the voltage transformation it ia apparent that the

short circuited nature of the original winding ensurea that each of

the two phase harmonie equivalent windings is short circuited so that

~-

\

100.

the squirre1 cage i8 equiva1ent to a set of idea1, ba1anced, short

circuited two phase winding8, one for each significant harmonic. The

complete winding, therefore, responds simp1y to comp1icated airgap

flux waveforms with none of the comp1exities entai1ed by the multiple

armature reaction effects of a wound two or three phase winding. The

induction motor equiva1ent circuit in which the harmonic effects appear

as a string cf series connected circuits fo11ows direct1y.

CONCLUSIONS

nle electromagnetic behaviour of a ba1anced N phase circuit

cannot in general be simu1ated by a balanced two phase winding but

rather required N orthogonal windings (N/2 if N i8 even) carrying N

inde pendent currents. It is impossible to derive the voltages of these

latter windings from the voltages app1ied to the actua1 windings un1ess

the winding current8 are known. Since in the majority of practica1 pro-

blems, voltages are known and currents are unknown, the transformation

is of 1itt1e merit.

However, if the rate of attenuation of harmonic amplitude

th permits neglect of harmonics higher than the N ,the above restriction

no longer holds, the orthogonal equivalent voltages being derived from

the actual phase voltages. The merit of the orthogonal equivalent,

separation of air gap space harmonic effects, then has full play and per-

mits, for example, ready computation of the neutral voltage of a star

connected system or of the circulating current of a mesh connected

system. The same consideration e8tablishes the validity of the convent-

ional harmonic equivalent circuit of a squirrel cage induction motor,

but shows that such a simple representation would, in general, be

incorrect for a machine with wound secondary phases.

('

'"

)

CHAPTER 4

NUMERICAL SOLUTION OF SYNCHRONOUS MACHINE STEADY-STATE PERFORMANCE

101.

The form of the inductance coefficients of the synchronous

machine were not those of Kron's ideal machine as reported in Chapter

2. The attempt to extend Kron's.algorithm of a transformation to the

commutator primitive resulted in a form more comp1ex than the original.

Thus, in the absence of a suitab1e a1gorithm, there was no recourse but

to solve the equations of performance of the synchronous machine as

they appear in slip-ring' primitive forme

The configuration of the synchronous machine is defined by

Figure 4-1, where a, band c represent the three stator.windings, f

the field windingand 1 and 2 respectively the direct axis and quarl-

rature axis damper windings. The equations re1ating the voltages and

currents of the machine are given by equation 4-1.

AlI the quantities of equation 4-1 were measurab1e and ,

therefore significant except Maf which was undectab1y sma11. The

Msf' term is included for generality of analysis as Ginsberg'et'a140

. 39 and Angst and 01denkamp have observed significant affects resu1ting

from higher stator-field mutual harmonics. It is assumed that the

stator supply is of the infinite bus type and that the field is

supplied from a voltage source with output impedance added to that of

102.

DA 1 ~

1

f \

FIGURE· 4-1 THE SYNCHRONOUS MACHINE WITH DAMPER WINDINGS

103.

VF ~ ~1 ~a ~b ~c if

0 ~1 Zn ZIa Zlb Zlc il

0 Z22 Z2a Z2b Z2c i 2

Va ~a ZIa Z2a Zaa Zab Zac ia 4-1

Vb ~b Zlb Z2b Zab ~b ~c ib

Vc ~c Zlc Z2c Zac ~c Zcc ic

where

VF = VF

Va = fi V cos W t

Vb = /2 V cos W t - 120

Vc = J2 V cos W t + 120

~ = RF + ~P

~1 ... ~ii'1P

Z:Fa = Msf P cos 9 + Maf' P cos 3 9

~Ib = Maf P cos (9 - 120) + Maf ,

p cos 3 9

~c 1::1 Maf P cos (9 + 120) + Msf' P cos 3 9

Zn ... RI + LIP

ZIa ... Ms1 P cos 9

Zlb = Ms1 P cos (9 - 120)

Zlc = Ms1 P cos (9 + 120)

Z22 = R2 + L2p

Z2a = - Ms2 P sin Q

Z2b = - Ms2 p sin (9 - 120)

Z2c = - Ms2 p sin (9 + 120)

Zaa = Ra + (L4'" + LO) P + L2P cos 2 9 + L4P cos 4 Q

Zab = - Mop + M2P cos 2(9 - 60) - M4P cos 4 (9 - 60)

= - cos 4 (9 + 60)

104.

Zbb ,= Ra + (L'I( + LO)p + L2P cos 2(9 + 60)+L4P cos 4(9 + 60)

= - Mop + ~P cos 2 Q - Mt.P cos 4 9

The first derivative of e is a constant related to the syn-

chronous speed of the machine.

where

Thus

o e ::1 y 6.)syn

o

e = w

y = no. pole pairs

W syn = synchronous speed

W = circulal!' frequ~cy of applied stator voltages

= wt + ~

~ = constant of integration

4-2

4-3

The angle ~ is related to S,the load angle of the machine,if

5 is defined in terms of the applied stator voltage rather than this

voltage after resistance and leakage reactance drops. Specifically

5 = 3 7f /2 - ~ 4-4

where & is a positive quantity for motor operation and a negative

quantity for generator operation.

The problem of prediction of steady state machine performance

is complicated by the fact that the voltages are known rather than the

currents. Equation 4-1 may be written as equation 4-5 where it 1s to

be remembered thatX is a 6 x 6 array of terms of differential equations

\- .•

105.

with periodic time varying coefficients, v is a 6 x l matrix containing

the knowns which are, in general, periodic functions of time and i a 6 x l

matrix representing the unknown currents.

v = Xi 4-5

An approximate solution of the very simple differential

68 equations with variable coefficients of Hwang discouraged the author

from attempting to solve equation 4-5 without resort to numerical

methods. Utilization of the excellent computer facilities (IBM 7044)

at McGi11 University enabled equation 4-5 to be solved numerically. A

brief discussion of the metb.ods used follows, a consideration of the

constraints imposed on equation 4-5 by various stator winding connect-

ions is left to the following chapters.

SOLUTION BY 4 POINT RUNGE-KUTTA METHOD

83 The four point Runge-Kutta method was chosen because of

several factors

(a) ease of starting

(b) reasonable efficiency

(c) ease of programming

It should be noted that step size choice had to be conservative since

84 the error estimate capability of this method, as outlined by Warten ,

are not as good as that of predictor-corrector methods.

Since a linear machine has been sssumed, the operator p

does not act on the inductance coefficients and X of equation 4-5 may

be expanded to yield

" X = (A) p + (R + e G) 4-6

\ , ~ ,_ ....... -'--.~.

, ~ ~1 +Msf cos 3 9

-4- Msf cos 9

Mpl LI Ms1 cos 9

L2 -Ms2 sin 9

,

A= Msf cos 9 Ms1 cos 9 -Ms2 sin 9 (Ler + LO)

+Msf cos 9 +L2 cos 2 9

+L4 cos 4 9

Msf cos(Q-120) Ms1 cos(9-120) -Ms2 sin(9-120)

-Mo , +M2 cos 2(9-60)

+~f cos 3 9 -M4 cos 4(9-60)

Msf cos(9+120) -Mo , Ms1 cos(9+120) -Ms2 sin(Q+120) +M2 cos 2(9+60) ~-sf cos 3 9

-M4 cos 4( 9 ",,0) _._ .. _ .... _--~- - -~-- ----

, +Msf cos 3 Q

+ Msf cos(9-120)

Ms1 cos(9-120)

-Ms2 sin(9-120)

-Mo +M2 cos 2(9-60)

-M4 cos 4(9-60)

(L d' + LO)

+L2 cos 2(9+60)

+L4 cos 4(9+60) . -

-Mo +M2 cos 2 9

-M4 cos 4 9 -

," "

4--7

+Msf cos 3.9

+ Msf cos (9+120)

Msf cos (9+120)

-Ms2 sin(9+120)

-Mo 1

+M2 cos 2 (9+60)1

-M4 cos 4(9+60)

-Mo +M2 cos 2 9

-Mf,. cos 4 9·

(L4' + LO)

+L2 cos 2(9-60)

+L4 cos 4(9-60) --- -

.... o 0\ .

107.

RF

RI

R2

R = Ra 4-8

Ra

Ra

, .. )

CI

r--.. /"-,

,~.

, -3Ms f sin 3 9

- Msf sin 9

- Ms1 sin Q

- Ms2 cos 9

G - Mgf sin 9 -Ms1 sin 9 -Ms2 cos 9 -2L2 sin 2 9 1

-3Msf sin 3 9 -4L4 sin 4 9

- Mgf sin(9b 120) -2M2 sin 2(9-60) , -Mg1 sin(9-120) -Mg2 cos (9-120)

-3Msf sin 3 9 +4M4 sin 4(Q.60)

- Msf sin(9+120) -2M2 sin '2(9+60) , -Ms1 sin(9+120) -Ms2 cos (9+120)

-3Mgf sin 3 9 +4~ sin 4 (Q+60) - --- -

, -3Ms f sin 3 9

- Msf sin(9-120)

- Ms1 sin(~-120)

- Ms 2 cos (9~ 120)

-2M2 sin 2(9-60)

+4M4 sin 4(9-60)

- 2L2 sin 2 (9+60)

-4L4 sin 4(9+60)

-2M2 sin 2 9

+4Mq. sin 4 Q - - -

4-9

, -3Msf sin 3 9

- Mgf sin{9+120)

- Ms1 sin(9+120)

- Ms 2 cos (-Q+120.)

-2M2 sin 2(9+60)

+4M4

sin 4(9+60)

-2M2 sin 2 9

-4~ sin 4 9

~2L2 sin 2(9~60)

-4L4 sin 4(9~~~) -- - - - ---

t-' o 00 .

) -- '

,,-)

109.

It is to be noted that G i8 the torque matrix fromwhich

the torque may be determine4.

y T = 2 it G i 4-10

Substituting 4-6 into 4-5 yields D

V= A (pi) + (R + 8 G) i

or V = A (pi) + Bi 4-11 o

where B = R + e G.

Rearranging equation 4-11 gives

(pi) = (V - Bi) 4-12

whieh is in a form readily recognizable as being amenable to stepping

out a solution.

Solution of equation 4-12 proeeeds by assuming a starting time,

a time inerement and a set of initial eurrents. Sinee we are interested

only in the steady state solution at a partieular load angle, there is

no eoupling between the eleetrieal differential equations and the meeh-

anieal differential equation whieh might be eonsidered to be of the

form of equation 4-13. Although the results of the analysis indieate

that:a 6tl1 time harmonie of eleetromagnetie torque is produeed, one

need not fear that this torque will be translated into asignifieant

6th harmonie perturbation of G because of the low pass nature of equation

4-13 and the faet that the inertia of thi.s machine set ia not negligible

as reported in Appendix VII.

•• • T = J e + K. 9 + TL 4-13

J = inertia

K = viacous friction of this màehine set ..

TL = load torque

'-..

(

' ..

110.

At any particular time,. to, the entries in the matrices A,

B and V are known and thus A-l. (an ill-conditioned determinant re-

quires that physical reality be violated since real windings have

non-zeroleakage inductance) may be determined numerical.ly so that

after the time increment 6 t, the currents may be determined as

= i6t + i to with the Runge-Kutta method. Three points

are of interest in this contexte The first comment relates to the

time increment. If A t is chosen to be very coarse, solution in-

stability can occur. A reasonable trade-off between solution time,

accuracy and round-off error leads to a choice of â t suchthat about

8 points per cycle of highest significant frequency are required.

The second comment relates to the initial choice of machine

currents. A realistic choice of initial currentsleads to a minimal

artificial transient computing interval. The direct axis inductance

and the quadrature axis inductance can be determined from equation 2-6

for Kron' s ideal machine simply by setting hm = 0 for 1 m,.,..1 and

P.Il = 0 for ~> 2. The well known phasor diagram 17, 27 of figure 4- 2

then results since the three phase slip-ring to the two-phase commutator

primitive transformation is applicable with stator resistance and leak-

age reactance neglected of Kron's ideal machine may be used to evaluate

ini.tial guesses of the starting values currents as outlined in equations

4-14 to 4-26.* Better starting values may be determined by using the

equations of the next section.

= 4-14

= 4·-15

* Stator win ding resistanceand leakage inductance neglected.

111.

)

FIGURE 4-2 PHASOR illAGRAM OF KRON' S l DEAL MACH! NE

112.

=j ID2 2

4-17 la + IQ

oc: = S - tan- 1 ln/IQ 4-18

ia = fila cos W t -0( 4-19

ib = /2la cos W t - 0( - 120 4-20

ic = fila cos W t -0< + 120 4-21

Xsf = tA) Msf 4-22

Xd = 3/2 (La + Lb)L\,. 4-23

Xq = 3/2 (La ,. Lb)W 4-24

La = +4 A W 2 Po (h1)2 4-25 H2 c a

Lb +2

Ac Wa2 P2 (h1)2 4-26 = ~ 7r

The third point is that a steady state solution is indicated

when the currents at time to and to + T are in agreement, T being the

period of the forcing function.

Other methods of numerical solution of equations of electric

machines such a~ with predictor-corrector techniques may be more

efficient. 81 Of special interest are the techniques ,now coming to

fore in conjunction with control problems formulated in state space

notation. A restriction to steady state operation allm~s an improve-

ment in computing efficiency of many orders of magnitude. This approach

is considered in the next section.

TP~SCENDENTAL EQUATIONS SOLVED BY A GENERALIZEn NEWTON-RAPHSON METHOn

Ana1ysis of results of the solution of equation 4-12 for

both motor and generator action and for several load'ang1es revea1ed

the time harmonies present in the steady state solution were rapidly

attenuated. Since on1y the steady state solution was desired, another

method of solution was sought with a trade-off between solution time

and accuracy being made.

An approach simi1ar in sorne respects to one out1ined by

White and Woodson20

was fo11owed. The form of the stator currents

113.

and rotor currents were assumed known. Analysis of Runge-Kutta resu1ts

revea1ed that the significant current harmonies were the fundamenta1,

third, fifth and seventh stator current harmonies and the sixth harmonie

of both field and damper winding currents.

i a

1

Fz cos

(61A) t - 0( F6)

cos (6cJ t - 0( 16)

cos (6W t

- 0( 26)

cos cos(.?4AJ t (6.1 t - 0(3) -ot 1)

co~ t cos{3 li) t -0<1 - 0< 3) -120)

cos(iA.I t cos(iw t -0( 1 - 3)

+1.20)

cos(>w t cosqoJ t - 0( 5) - 0( 7)

cos(>W t co(7w t -0(5 -0(7

+120) -120)

cosC)14J t CO(7,'.AJ t -olS - fJ(.7

-120) +120)

IFO

I F6 4-2

116

1 26

Il

13

15

17

114.

Substitution of equation 4-27 into equa.tion 4-1 yielded the

following information:

(a) =

(b) the three stator rows yield identieal equations

(e) the right hand side of the resultant equation

eontained terms equalin.frequeney to the applied

voltage of that partieular row, termsequal in

frequeney to the assumed eurrents, and terms at

higher frequeneies than the assumed frequéneies.

A set of equalities were established by negleeting the latter;

in aIl there were seven equations, four equations from the stator funda-

mental, third, fifth, and seventh harmonies and three equations of the

6th harmonie respeetively in the field, lst damper winding and 2nd

damper winding. These seven equations eaeh eontain sine and eosine

terms. Eaeh equation yielded two equations sinee both the sine and

eosine terms on the left hand side and on the right hand side must be

equal. These fourteen equations are given by equation 4-28 in terms

of the fourteen unknowns - Il' 0(1' 13' 0(3' 15, 0(5' 17, 0< 7' I F6, 0<' F6,

(

·115.

Wno ,fs/nat,- ~6léS~ ""~cQ

J~OS; X4CoS~ otj",)(C1;

"~~21~ ec,S9.P'''~ <éSo(4~fi 1;

V~~ ,.fâtS"0(, -1-

~,s.SiN~, XCI! • X'q Sl/)O(; V?: ~~Xë" sli7~ o"'QP' ~St;'2~ ~/Q#P~3

"'" ..lB ,

3.r66(éS~ ~ "',.;"0(.1 ~~6° I-..1Xc CI

,. W~· -KsI' ~

'€11 -~-3Xc° -.1~ • t'oS~~ ("4S'fP,I~ Cc.s;r~ ~S.3 f.oS 4.6'''~ C:CSeJ(/J 6 Zs

, 3~~$i~ RatJs~ )fsl" Jfl~. ",.3X66 ' SKc -

Y2. -Qtj -..?"fo 13~4""~~ 5'1.b2/~ 'iQ~"~ f/Q"I~ :J'/n~{I ..1'/;"~,.,~ 3' '1 ~

17'

0 -5~e" 5 %t6 " R.7lhot'S--. -S~· .. -/7.)(S'/. -%X.s"" ~~. (OSll,#~ a,s~.~ j"J4~ ~S.?f',t~ ~,,4.,t~ t:/)sl'''' ~6' -'Ï;,p~' z:

~

0 -4"Xt!c S'X""" ,feoStl/S $)(~ • ~·Xsl' ~ • ..\SI" 4i2 Ks2 '

S/~l'l-q, .:S'Ù)2~-~ ~";~",'.i.J~ $-"'Jtd'~ "'''~,4'''f s/I)f'~~ c::'Ôf4'''qz .,~ ~

T -/, 0 -7X'~· -7~· R SI/) q"". -~K~. "~~/' %~2'

tos9,&'.~ ~~-~ -7~~ é'~-~ ~1'-ttl6 S"/JI-~~ - ~ 1-

-7Xc - -7~ .. ~~.s~oJ -~~~ -,% ,xs/" -?Z Xii .0 SI;'~"'A .:t'Â~~ 74fSiA~ Sh/';tY- ~~,4-~ (bs,P'-~ 3 ~ 4-28 1

-~Xsf· !? X'.rl'~ A}41~ XJF .. ..,.,9%911' () "lq.1I-~ jiQ/-~ S/~/-IDt, +. • .5/nq;, :r'1'~~

Il

,fj:a,S~ 0 -.1 X.rl'. -9Xs/. -?Ks-I' .. -KtF • CD>J/~ot: CtJS!-ers W~~"7 ~~~~ ~S"pV, -. '';

0 -~Xs/· 9Xr/, ~r · ~/ ~.r'Y" ~4~-q'$ SI4~"'0t7 ~/)K~

~ 4C ~ SÙI{)(It.

0 ".9~/. -9 XSI9 -.kiF" ~S;~" ~.s-,6'-~ ~4S'I"'~ Cd$t:r',:;: ~~~ ({

0 -9...\j-z • 9XsZ 0 Rzt.qJq'~

"" 61S,D' .. «s âSI'~~ ~SI';'lY~

0 9XSz· 9Xs2 • Rz $"i14'~ $i/),II-~ :J//},o~T -)(écoSlYu

116.

with xa = le) (LO + Mo) + L4( W

xaa = W(LO - 2Mo) + L([ W

xb = W(M2 + L2/ 2)

xbb = W(M2 - L2) /2

Xc = W(Mt. + L4) /2

~6 = 6 Lp W

Equation 4-28 reverts to standard form if Kron's idea1 machine

i8 assumed. Resistance and 1eakage reactance will be neg1ected to simp1ify

the equations and for the idea1 case

Mo = Lo/2

, xsf = xaa = xbb = Xc = xcc = 0

Thus xa = 3/2 W Lo

xb = 3/2 W L2

"

117.

4-29 since

and

Under these constraints equation 4-28 co11apses to equation

13 = 15 = 17 = I F6 = 116 = 1 26 = 0

I FO = if

X sf IF V + fi sin' = xa la sin 0( 1 - xb la sin 2 fi' + c( 1

Substituting ~ = 3 'Tt /2 - S equations 4-29 may be rewritten as

xsf IF sin ~ _- 1 ~ 1 2 ~ ..J o xa a cos~ 1 - xb a cos 0 - ~ 1 fi

Returning to Figure 4-2, reso1ving a10ng and in quadrature to V yie1ds

V = xsf IF 3 fi cost; +'2 W {La - Lb) la sin~ cos 6-0(

- 1 W (La + Lb) la sin ~ - 0{ cos ~ 2

~ W la (La - Lb) cos.6 cos ~ - 0<. = xsf IF sin' fi

Expanding and rearrang!ng

... xsf IF sin' 0 = =n

- 1 lA) la (La + Lb) sin S - 0( sinS 2

3 3 + 2' W La la cos IX - 2' W Lb la cos 2S

4-29

4-31

-0(

xsf IF cos, ~ W la La

3 2' - 0( V = 12 + sin 0( - 2' W Lb la sin

4-32

Since Xa = 3/2 WLa and xb = 3/2 W Lb, equations 4-30 and 4-32 are

equal.

(, , !

118.

A closed form solution of equation 4-28 is not possible since

transcendental relationships are'involved, but a soluti.on may be obtained

using numerical methods85 such as the Newton-Raphson technique86 •

Subtracting the left hand side of 4-28 from both sides yields

a set of equations of the form

Yi = Fi (xi) i = 1, , 14 4-33

where Yi is zero for aIl i when the correct set of XilS are found.

Equation 4-33 may be expanded about initial values xOl ' ••• ,

x014 and taking only the first terms of the eJcpansion yields

o = FI ( ) + (xl - xOl) ~ FI xOl' ••• , x014 --() xl

, x014

+ ...

o = F14 (xOl' ••• , x014) + ••• + (x14 - x014) ~ F14 j ~ x14 x01 ' ••. , xOl~

4-34

which may be rearranged as

= ~ xj 4-35

with i = 1, ••• , 14 j = 1, ••• , 14

(

\.

(

119.

~ Fi The matrix -:;:-­

CI Xj may be inverted numerically to yield -1

lhus equation 4-35 is rewritten as

~ xj = -1

~'i and ~ Fil '(, Xj can be evaluated by starting from sorne

initial Xoj' Thus ~Xj is determined using 4-36. This in turn yields

a new set of starting values = Xoj + 6 x j and the process is

repeated until 2: À j F j =, where f is a small number j=1, ••• ,14

close to zero. The À j are weighting factors chosen on the basis of

equation sensitivity - for an initial study the y may be set to unity.

4-36

This method yields results which converge much more quickly

th an those of a Runge-Kutta solution. Typically a run, of 10 minutes

might be required for a poor'choice of initial conditions for the Runge-

Kutta program to evaluate the i'sand T and a particular load angle.

The Newton-Raphson method, while requiring considerably more

set-up time to determine the equations to be solved, yields results,

i.e. the i's of the form given hithertofore and T, for the same starting

conditions and load angle in an execute time of the order of 70 milli-

seconds.

Once the current magnitudes and phase angles have been

determined, they may be substituted into torque relationships which

result from a closed form evaluation of equation 4-10. A constant

component of torque, equation 4-37, a sixth harmonic component of

torque, equation 4-38, are produced. The expressions are given for a

4 pole machine.

Teons =

Tl =

T2 =

Tl + T2

-3,[2 MsfIFO Il sin (, +()( 1) - 6Lb 1 12

sin(2p + 20< 1)

12 I1I3Lbb sin(2~ + 0(3 - 0( 1)+ 12 I3I SLbb sin(2' +0( S -0( 3)

-12.ISI 7Lb sin(2p+O<7 - o(S)

-3 Msf I F6 (1 S sin<fl + o(F6 - D( S)+ 17 sin(fJ +~ - I)(F6»

-3 Ms1 1 16 (IS sin(p+O<16 -O(S)+ 17 sintl +0(7 -0(16»

-3 Ms2 1 26 (IS eoscp + 0)6 - C(S)+ 17 eos(t' +0(7 - 0«26»

- 24 I1I3Le sin (4 ~ + 0<1 + 0(3)

- 24 III SL cc sin (4 ~ + oC S - 0(1)

-24 I3I7Le sin(4~ + 0(7 - 0(3)

120.

4-37

T6th = 12 1113 (Lbb sin(~t + 2f -0(1 - 0(3)- 2Le sin(6CcJt + 4' +0(1 - ~3»

+12 1317 (Lbb sin (6Wt + 2 P + 0(3 - 0(7)+ 2Le sin (6fA)t - 4 ~ - D<3 - 0(7» 2

-12 Il Lee sin(6wt + 4~ - 20(1) 2

+12IS Lee sin(~t - 20(S + 4(:1>

+12 I1I7Lb sin(6lAJt - 2p-O<l -0(7)

-6 ~f I FO (I S sin(661t +P - c< S)- 17 sin (6tüt - ~ - 0( 7»

-6 Msf IF6I1 sin(p +CX(1)eos(6Wt -O('F6)

-6 Ms1I16Il sin(p +o(l)eos(6sJt -0(16)

-6 Ms2I26I1 eos(p +o(l)eos (6wt - 0(26) 4-38

)

121.

The 6th harmonie of torque can rare1y 1ead to stabi1ity pro-

b1ems because of the low break frequencies that genera11ycharacterize

the mechanica1 system. In other words this, for a 60 cps system, 360

cps variation of e1ectromagnetic torque is rare1y trans1ated into shaft

butput"torque.

The T2 component of constant torque constitutes a 10ss torque,

whereas the Tl component is the idea1 torque. It is of some value to

compare the expected torque from equations 4-37 and 4-38 with that for

torque derived for Kron's idea1 machine. For this situation

T2 = T6th = 0 and since ~ = 3 7T 12 - & and ~ = 3/2 L2' Tcons becomes

Tcons = .3-(2 MsfIFOI 1 cos(o( 1 - S) + 9L2112 sin(20< 1 - 2 ~) 4-39

Consider the phasor diagram of Figure 2. The power input is

p = 3 VI cos 0(

or in phasor form, were * denotes the comp1ex conjugate .... .,100

p = 3· Re V • 1*

For :a :y pole machine with supply frequency w the synchronous

speed is wlY and thus the torque is

T = pl w/Y

3Y ~ ~ T = ~ Re V • 1*

Substituting for V and l from

T 3Y Ref xsf1f = W fi

T 3Y Re { xsfif = w fi

the phasor diagram of Figure

- xdld + j xqIQ (IGa + jI D) ..

- xèd + j XqIQ (I~ - JIn> J

4-2 yie1ds

J

4-40

122.

Rearranging

T = 3'Y f ()~if - Ld 1 sin(~ - o(») 1 cos($ - c()

+ Lq 12 sin([; - 0{) cos(b - Dt) }

which yie1ds equation 4-41 for a 4 pole machine, which is seen to be

identica1 with equation 4-39 since 0(1 = 0( , I FO = if and L2 = Lb

T = 3~ Maf if 1 cos(6 - IX) + 9Lb 12 sin(20( - 2~) 4-41

'.

c:

123.

CHAPTER 5

THE THREE-WIRE STAR CONNECTED SYNCHRONOUS MACHINE

THEORY

The type of connection of the stator windings imposes a

constraint on the order of harmonies which may be expected to exist

in a synchronous machine in steady state. Since for the three-wire

star connection the sum of the stator currents is zero, equation 5-1,

the exist~nce of trip1en time harmonies of stator current is prec1uded.

5-1

Thus equation 4-1 must be transformed so that a solution

takes cognizance of this facto It may be transformed by a CT Z C type

transformation where C is given by equation 5-2 and the new impedance

matrix is given hy equation 5-3.

if 1 if

il 1 il

i 2 1 i 2

ia 1 ia

ib 1 ib 1...---

ic - 1 - 1 5-2

,"

124.

VF ~ ~1 ~a-~c ZFb-~c

0 ~1 Z11 Zla- Z1c Zlb- Zlc ..

0 Z22 Z2a- Z2c Z2b- Z2c

~a-~c Zla- Z1c ZZa- Z2c Za + Zc . Zc+Zab va-vc ==

- 2Zac -Zac-Zbc

~b-~c Zlb- Zlc Z2b- Z2c Zc+Zab Zb + Zc vb-vc

- Zbc- Zac - 2Zbc

Equation 5-3 may be .' arranged fora Runge-Kutta type

solution fo11owing the procedure. out1ined in Chapter 4. The equation

necessary for the Newton-Raphson type solution may be determined and

is simi1ar to that of equation 4-28 of Chapter 4 as depicted by

equation 5-4.

5-3

)

... !

IXsf'IFo ct

JE cosfJ

V+Xsf1Po

o~

0

()

0

0

0

0

0

0

0

0

125.

~~~~--~-----r----~----~---'I--l'(Sil1o(,- .. v " X4 CoSo(i /\(.'c

~Cas2M 'OS~1';fB I~~~~--~-----r----T-----r---~--

1 D0So(,-J \/ IX~ Sll)~" ,,~ ..

)(~"2~ Sli?tX$#lj; ~ -5..rcc" 'R SIÏ10â ," -5 Xb" .• Si2 ><sr. -% Xs-I" .5J.z XSi! " .z:, (f)S~ .. or,~~a,sc;(.s CoS2p.Jol,. Cosc:<~) li$ottl~P S/n~/f ~

-5)«<· RCbSoé;-I 5Xh" SXz }6-f· -V2~1 ~ ><52-f.rIÎJ9-~ ~Xq SlÎ1'D(;~i'J21-1D<1 Sin~r/J S'Ii, ~t?(J (Os ~ffJ { I--~~~~-+----~----+-----r----;~

"7Yb ~$I;' 0(7" "'?2 XsI, -~XsI- ~ ~.2' 1. CoS 2f-~7)(q~S09 ~f-~ ti:JSfJ-ott, .s1Î1,s-~ i't

-7.X6· ,f4JSc(rI'~~ ><SI- ~ XSI- ... ~ )($2' .z S;1J2p-~ 7~S'in\) 'infl-Aft $/Of-fX/6 CdS',B-", ~ .. .9 X'Sf'. 9 Xs{' ~ lÔS~ JV,:' Slnj1"'p(s s/~fl~ ~SliJ~ SI/) 0(/6

-9Xsf· -~XsI- I?c:.v;;~ -..%IF 0

CtJS;6'-~ tLJS(l-l-P6 ,~t4Js~ ,CtJS 0(/ t

-:-.9 X~, 9 XS/.· XtJ: • ~ Co~,,~ $/;'1-:- o<s S',/)#~oIr Si;" Dl}!', A;" ~Î'JDc/6

.. !J Xs1 • -9 XsI" - x/P - ,el S/n~, CtJsI-D(s CostS-I* t2ts D(}" X(Co.sAJ,

- 9 Xs2 • 9 Xsz • 4JSI-ocS CDs/~o(7

9 XS2." 9 Xs:z· S;n,d-~ Snflw?

5-4

It is to be noted that if the machine were to have inductance

coefficients arising only from the ~ and 4'2 permeance terms, the

current time harmonics greater than the fundamental would aIl be zero

and the standard phasor diagram derived in terms of the twoa~is

commutator primitive would be valid as long as Ld and Lq are defined in

terms of the.measured values as

= 5-5

The lack of symmetry of the inductance coefficients from Kron's

ideal form causes the generation of ident~ triplen voltages in each

phase of the stator windings. These voltages do not appear when voltage

is measured between the line terminaIs a, b or c but appear when voltage

is measured between line terminal and stator neutral or between stator

neutral and source neutral. These 'voltages can be troublesome because

they can give rise to a winding voltage gradient higher than that

expected on the basis of the applied stator voltage and thereby reduce

the stator insulation safety factor. A calculation procedure for the

triplen harmonic voltages will be outlined.

In terms of the Runge-Kutta technique, the difference between

any stator voltage on the left hand side of equation 4-1 and the product

of the resulting currents and impedance matrix of the same row yield

the triplen harmonic voltage variation between source neutral and machine

neutral. The same procedure may be used to evaluate the voltage for ,

the case of solution by Newton-Raphson method to yield Vnn ' n denoting ,

source neutral and n denoting the stator neutral.

)

127.

Vnn ' = 3V3 ' J2 cos 3lA1t + 3V3" J2 sin 3t.)t

V' 3 = (-Xbb sin 2 p - 0(1 + Xc sin 4~ +0( 1) Il

+ Xbb sin (2 p +0(5)15 - Xc sin(4 p +1)(7)17

X ' +....ê.L I FO sin 3 Il 3./2

V " 3 = (-Xbb cos(2 ~ -oC 1)+ Xc cos(4~ + oll») Il

-Xbb cose2 ~ +015)15 + Xc cos(4f9 +0(7)17

X ' +.....ê.L I FO cos 3 ~ 3 fi

COMPARISON OF CALCULATION METHODS

Additional constraints imposed by the practical machine make

determination of the j.nfluence of harmonics with an accuracy of the

order of one percent impossible without the addition of more facts

which complicate the computation procedure. Thus it is considered un-

desirable to use a high accuracy numerical method such as a Runge-Kutta

integration procedure to compute the compiete steady state performance

of a synchronous machine because of the computer'time cost penalty.

5-5

5-6

5-7

The equations solved by the Newton-Raphson method give more insight into

significant harmonic interactions and yield computed results in one-five

thousandths or better of the time. The influence of truncation error

of the number of harmonics assumed for the Newton-Raphson method is low

for this machine as the comparison of results of the two methods in

Table 5-1 indicates. The calculated results" presented for a parti-

cular load angle of -10 degrees,are typical of aIl load angles. The

equations leading to a Newton-Raphson solution are used for comparison

of measured data.

Table 5-1. Comparison of Runge-Kutta and Newton-Raphson Values for S =_10°

Amp1itude*

Parame ter Harmonie Runge- Newton-Order Kutta Raphson

128.

phase-Degrees

Runge- Newton-Kutta Raphson

Stator Current- 1 .926 .925 . 153.1 153.2

Amps 5 .033 .033 109.6 109.7

7 .015 .015 269.0 268.8

11 .001 226.9

Field Current- 6 .005 .005 9.4 '9.4

Amps 12 - -Torque - 0 -1.596 -1.597

N-M 6** .033 .033 108.8 . 109.2

da Damper 6 .009 .009 8.2 8.4

Current-Amps

qa Damper 6 .007 .010 277 .8 277.9

Current-Amps ,

Stator Neutra1 3 7.313 7.311 236.1 235.0

Voltage 9 .067 144.0

15 .002 270.6

* Values sma11er than ~001 ar.e' 'neg1é.etéd ,for 'Runge-Kutta solution

** Peak values

Table 5-1. Comparison of Runge-Kutta and . Newton-Raphson Values for S =. _10 0

Amplitude*

Parame ter Harmonie Runge- Newton-Order Kutta Raphson

128.

Phase-Degrees

Runge- Newton-Kutta Raphson

Stator Current- 1 .926 .925 . 153.1 153.2

Amps 5 .033 .033 109.6 109.7

7 .015 .015 269.0 268.8

11 .001 226.9

Field Current- 6 .005 .005 9.4 '9.4

Amps 12 - -Torque - 0 -1.596 -1.597

N-M 6** .033 .033 108.8 109.2

da Damper 6 .009 .009 8.2 8.4

Current-Amps

qa Damper 6 .007 .010 277.8 277.9

Current-Amps

Stator Neutra1 3 7.313 7.311 236.1 235.0

Voltage 9 .067 144.0

15 .002 270.6

* Values sma11er than ~OOlat.e' 'ueglé.etéd .for 'Runge-Kutta solution

** Peak values

)

129.

EXpERlMENT

The stator of the synchronous machine was connected three

wire star and various measurement apparatus depicted in Figure 5-1

utilized to measure synchronous performance. Briefly, load angle was

measured with a meter built along the lines of a phase angle meter.

This approach was judged to be simpler than some of the devices reported

i h li f i 1 d 1 87, 88, 89 A 2 1 h n t e terature or measur ng oa ang e • -po e tac 0-

meter was used to supply a reference signal to the load angle meter. It

was necessary ta divide the frequency of the applied voltage signal input

to the load angle by two, in order to compare signaIs at the same frequency.

Torque was detected with a torque meter similar in design ta that reported

90 by Barton and Ionides • The constant component of electromagnetic torque

was calculated fram shaft torque and no load torque. Photographs of line

current, field current, and stator neutral - source neutral voltage were

taken with the synchronous machine operating both as a motor and as a

generator. The accuracy of the measurements is considered to be ± 5%.

A stator line voltage of 208 volts and an average field current

of .5 amperes were used rather than the rated values to minimizethe

influence of saturation. Moderate saturation of the machine does not

significantly alter the validity of the model of the synchronous machine.

Measured and computed line current and neutral voltage for a

load angle of -25.6 degrees is depicted in Figure 5-2. The good

agreement between computed and measured amplitude and phase is typical

of comparisons at other load angles.

However, significant additional higher order effects than those

predicted can occur. 17 These are present as a result of slot harmonies ,

·60 Hz. Mains

Synchronous Machine Torque

~ -; J---- - Jl Jl Tube

Scapa'

.~n#

IL •

~

':"

Oc Machine

1 Power . Flo\'J

...",._ .....

Lo ad Angle Meter FIGURE 5-1 THREE WIRE STAR MEASUREMENTS

':

Ac Tachometer

1-' W o

· « 1-z 1IJ a: Il: ::,) o bJ

·2

z .... 2 -...J ::> I.IJ (!)

~. :.J o > <l

20

0:-20 1-::> 1LI z

" , 1 -'

FIGURE 5~2

180

LI NE CURRENT AND NEUTRAL VOLTAGE OF THREE WIRE STAR SYNCHRONOUS MACHINE AT A LOAD ANGLE OF - 25.6 0

360

..... W t-'

, .. ,_.-

~­

l ,

132.

e.g. the 21st harmonie of neutra1 voltage.

The experimenta1 reeu1ts were ana1yzed into their Fourier

series eomponents. The eomputed and measured ratios of 15/11 and

17/11 are given in Table 5-2. They are sma11 and essentia11y independ­

ent of load angle. The agreement between experimenta1 and eomputed

values of fundamenta1 of stator 1ine eurrent, third harmonie neutra1

voltage, constant eomponent of e1eetromagnetie torque and 6th harmonie

of field eurrent, for various load angles if Figure 5-3. Figure 5-4

shows the ea1eu1ated 6th harmonie of damper winding eurrents and e1eetro-

magnetie torque as a funetion of load angle. Table 5-3 depiets the

average resu1ts of a harmonie ana1ysis of measurab1e additiona1 harmonie

terms.

Table 5-2. Comparison of Computed and Measured 5th and 7th Stator Line Current Harmonies

% 15/11 % 17/11

Measured 3.8 2.7

Computed 3.6 1.6

Table 5-3. Harmonie Ana1ysis of Additiona1 Terms

Stator Current Neutra1 Voltage Field Current.

lU/Il 113/1 1 V9/V3 1 2*/16 14*/16

% 1.6 2.2 17 .5 80 21

/ <IL\

,--'

r 1

-'

• <[

-!:-':> MEASUREMENTS . . .. ~'«D ~, aU)() Z~~ rJ.:>_O

mOIa) '~TO •

. 11J) 2.L <D o •

A

Torque Neutral 3d Harmonie Voltage Fundammtal line Current 6 th Harmonie Field Current ..

Olq- m ..... 1 -' ',.. '1 '= ~ vt\ltO ' .... •

-80 -40 40 (5

o

FIGURE 5-3 c!oMPUTED AND MEASURED LlNE CURRENT, TlITRD HARMONIC NEUTRAL VOLTAGE, TORQUE Al.'ID SIXTH BARMONI C OF FIELD CURRENT FOR V ARr OUS LOAD ANGLES

, 80

1-' W W

~ ..

. c

• ~co 1 d!i

.!'I co --

.~ t ID . 0

•

8S .. N T 0 1 /_" / ~

•

"-'

~ w

, , :-: Il fi';::"

-80 -40

FIGURE 5-4

4- (l

COMPUTED SIXTH HARMONIC OF ELECTROMAGNETIC TORQUE AND DAMPER. WINffiNG CURRENTS

sa & @

135.

* These harmonies do not arise from calculation round-off

errors. Kinitisky89 shows that they may originate from

CONCLUSIONS

a partial turn to turn fault of a stator winding. Since

the stator currents appeared to be balanced, they are

assumed to originate from the stator bore being not quite

concentric with the rotor.

In summary, the harmonie effects are quite small with this

type of connection of the stator winding when it is operating in the

steady state and the use of the clasaical dq axis method of prediction

of performance is, justified provided the proper inductances are

assigned to the direct axis and quadrature axis armature reaction.

Appendix VIII gives a non-interacting modification to the classical

dq axis phasor diagram to de termine a first approximation to the third

harmonie neutral voltage.

, ..

'.

136.

CHAPTER 6

THE FOUR-WIRE STAR CONNECTED SYNCHRONOUS MACHINE

THEORY

The performance equations given in Chapter 4apply directly to

the four-wire star connected stator synchronous machine. This connect­

ion allows third harmonie stator line current to flow which, as inspect­

ion of equation 4-28 reveals, provides a strong link between fundamental

and fifth and seventh harmonies of stator current. The analysis of this

chapter applies equally weIl to a delta connected stator. The only

difference is that the third harmonies are not present in the line

currents since they circulate in the delta.

COMPARISON OF CALCULATION METHODS

The solution of equations by the Newton-Raphson method is to

be preferred to the more accurate but computer time consumingRunge­

Kutta solution. The truncation error is not significant as illustrated

by a comparison of typical results in Table 6-1. The Newton-Raphson

method ia used for the computation of steady state aynchronous machine

performance of Figures 6-2 and 6-3.

'\ ,

(~i

Table 6~1. Oompa~iaon of RUDgeMKutta and Newton"'Raphson Solution for (;;:; ... 1011 '. __ .At ., .• _ _ _. .' . _ . _ '. • . _ .

". - . . ..

Amplitudew

Para1l'.9ter Harmonie Runge ... Newton ... Order Kutta Raphson

Stator Current 1 1.047 1.047

Amps 3 1.096 1.086

5 .229 .224

7 .0367 .0158

9 .117

Il .022

13 .005

15 .012

Field Current 6 .024 .022

Amps 12 .001

Torque 0 ... 1.867 ';'1.861

N·M 6 .517** .467**

12 .054

da Dampel' 6 .0353 .038

Current Amps 12 .003

qa Damper 6 .105 .126

Current Amps 12 .011

137.

Phaae",Degreea

Runge ... Newton ... Kutta Raphson

154.7 154.6

314.0 314.0

110,5 110.7

89.8 98.6

246,7

4409

22.9

181.0

10.8 10.2

305.8

72.3 91.8

11.4

9.6 9.0

305.2

278.6 279.1

216.0

* Values sma11er than .001 are neg1eeted for Runge-~utta solution

** Peak amplitude

)

,-,

138.

EXPERIMENT

The apparatus of Figure 5-1 of Chapter 5 was used except the

source and stator neutrals were interconnected. Waveforms of line

current, neutral current, and field current were analyzed at a variety

of load angles with both motor and generator synchronous machine action

under derated conditions of 120 phase volts and an average field current

of .5 amperes. A sample set of data is shown for a load angle of 22

degrees iri Figure 6-1. The so11d lines are from a photograph and the

points are computed from& Runge-Kutta solution. 1t will be noted that

both amplitude and phase are in good agreement and that the third

harmonic of line current is c'omparable in magnitude with the funda-

mental 1ine current.

The fifth and seventh harmonics of 1ine currentare much 1arger

than those of the three-wire star connected synchronous machine. 15/11

and 1 7/11 are essentia11y independent of load angle and a comparison

of calculated and measured data 1s given in Table 6-2. The data is

computed with the Runge-Kutta method since higher stator harmonies, the

9th , llth, etc. are not neg1igible,

Table 6-2. Comparison of Computed and Averaged Measured 5th and 7th Stator Line Current Harmonics

% 15/11 % 17/11

Measured 16.3 4.4

Computed 21.9 3.5

0

<[

IF-Z lLLI~ a: ttt: :l (.)

lLfJ Z ."......

.,.J, ~ •

• « ~

..... ..,....

..... Z LLI Z 0 0..0 :lEU') 0 1

(,)

• (,) . <[

:'

1 C

FIGURE 6-1

180 360 r

wt

LINE CURRENT AND A.C. COMPONENT OF FIELD CURRENT FOR A FOUR WIRE STAR CONNECTlON OF SYNCHRONOUS MACmNE AT A LOAD ANGLE OF 22°

1-' W \0 .

140.

Figure 6-2 illustrates the reasonable agreement obtained

between experimental and calculated (Newton-Raphson method) results

for stator fundamental and third harmonic line current, electro-

magnetic torque, and 6th harmonic field current. The large harmonic

effects result in considerable losses and very inefficient operation.

Figure 6-3 depicts the supposed damper winding currents and 6th

harmonic of electromagnetic torque.

Additional harmonics of stator line current and field

current were measurable and are present in Table 6-3.

Table 6-3. Additional Measurable Harmonics

Stator Cllrrent Field Current

C,) 19/11 . Ill/Il 113/11 12/161: 14*/I6F

9% 2.2% 6% 12% 11%

* Decreasing with increasing IF6 ' values quoted for ~ CI 20.

CONCLUSIONS

Large harmonics which cannot be predicted in terms of a classical

two axis approach can result with either a delta or four-wire star conn-

ected synchronous machine. The most predominant effects are the third

harmonic of line current, the slowly converging additional harmonic terms,

and the resulting high losses. Many large salient pole synchronouB

machines have stator windings which are very nearly sin.usoidally dis-

tributed and thtis produce few harmonics. However, for large installations

losses of a fraction of a percent can be quite costly and every effort

to minimize them should be undertaken.

• :i z 1-

tP . <It

IfS 1 ft) -CD --., CD 1«

1.

-80 -40

o N -­•

• MEASIJRE~iENTS c CD ..... --

CD o

CI Torqu~

o Fundamental Lina Curfent Cl 3rd Harmonie Line CUflfsnt A 6th Harmonie Field Current

Ji .

40 b

80

t-' ~ t-'

FIGURE 6-2 PREDICTED AND EXPERIMENTAL VALUES OF ELECTROMAGNETIC TORQUE, FUNDAMENTAL AND3rd HARMQNIC LlNE CURRENT, AND SIXTH HARMONIC FIELD CURRENT FOR THE FOUR mRE STAR COl'."'NECTEDSYNCHRONOUS MACHINE AT VARIOUS LOAD ANGLES

~1

-80 FIGURE 6-3

,. "'

• • ..c 1 fi) :E

z ..!fI • t-G. --

'" GD N

'" , ... < ( ~ -

40 6 @ 80 PREDICTED SIXTH HARMONIC DAMPER. WINDING CURRENTS AND ELECTROMAGNETIC TORQUE AT VARIOUS LQ\ D ANGLES J!OR THE FOUR WIRE STAR CONNECTED SYNCHRONOUS MACHI NE

.... ~ N

For large synchronouB machines in which the assumption of

sinuBoidal distr1buted windings is not applicable, a delta or four­

wire star connection shoWdnot be used for the stator but rather a

three·wire star connection is to be preferred. If grounding of the

stator winding is required, a high neutral impedance iB in order to

minimize the third harmonie current.

143.

CHAPTER 7

CONCLUSIONS

144.

The modelling and analysis of electric machines is in the dawn

of a new era. The availability of advanced measuring equipment:and

techniques makes possible the determination and separation of signifi­

cant effects upon electric machine analysis. The accessibility of high

speed, large storage digital computing machines must surely lead ta

methods of analysis of electric machines which are not overly dependent

upon pedagogic techniques. Thus the classical method of analysis of

elèctric machines with simple equivalent circuits is, in general, only

seen ta be of use for a cursory analysis of steady state behaviour. On

the other hand, the framework of generalized machine the ory is consid­

ered ta be of sufficient scope that it may be modified as required ta put

it in a forro suitable for the calculation of bath the steady state and

the dynamical behaviour of electric machines. ~xpressed in this form

many of the techniques of astate space mode of analysis of control

systems are applicable.

Analysis of space harmonies of radial air gap flux density and

mmf produced by practical electrical machines are seen'to introduce

significant deviations from the form of the models of ac electric mach­

ines which are generally assumed. High power electric machines are often

constructed to minimize these harmonies by the use of appropriate winding

distributions and suitable airgap geometry forro, but even a small per­

centage loss in these machines can be costly. These deviations are mani­

fest by inductances of machine windings which are not of a forro assumed

145.

by Kron. Thus parameters of several machines were measured by an

electronic fluxmeter, a procedure which, when used in conjunction with

a frequency response te chn.i que , makes possible the modelling of electric

machines which have damper windings and indicates the realm of validity

of the linearity assumption. This assumption may readily be investi-

gated by mapping the state of the electric machine in current space by

using the electronic fluxmeter method, and is an area which should be

pursued with an eye to the modification of the two axis the ory of

32 synchronous machines in a manner similar to that proposed by Hamdi-Sepen.

The damper winding equivalent circuits model the first order

influence of both damper bars and eddy currents. The measurement of

inductance at constant current suggests that the influence of hyster-

esis is not negligible and should be modelled. This was not attempted

because of the difficulty of separating the hysteresis and the damper

bar effects with frequency response technique but is a matter worthy

of more consideration.

The analysie of the influence of harmonies is presented in two

parts. The first part is concerned with transformations linking primi-

tive machines in the presence of harmonies. The transformation of the

two-phase slip ring primitive to its two-phase commutator primitive

when space harmonies are not negligible is considered. It is seen that

although the equations which result shed sorne light on voltage and

current interactions and the· production of torque, the second primitive

machine does not represent either a machine realizable in terms of

practical commutator devices or a useful algorithme The special case

of a three-phase to two-phase slip ring transformation is considered.

146.

A three by three connection matrix is established by neglecting harmonics

higher than the third. Analysis of n-phase to two-phase slip ring trans-

formations shows why the harmonic equivalent circuit representation of

a squirrel cage induction motor is val id.

The second part relates to the affects of space harmonics on

the steady state behaviour of currents and torque. It is seen that

significant time harmonics of w:i.nding current and los ses may result.

These affects were observed in a wound rotor induction motor and in a

synchronous machine. Detailed data which confirms the utility of the

modelling procedure was given for a synchronous machine. It was seen

that the manner in which significant time harmonics manifest them-

selves was dependent upon the way the three-phase winding was connected;

the three wire star connection producing minimal harmonic cross-coupling

whereas the four wire star and three wire delta connection allowed con-

siderable harmonic interaction. The classical dq axis equivalent cir­

cuit was seen to be a good approximation and to allow the prediction of

winding currents and torque for a three wire star connected synchronous

machine aince the third harmonic winding factor was effectiv~ly zero. The

model of the synchronous machine was analyzed by two numerical method

approaches. The Runge-Kutta integration procedure was seen to give a

solution with no significant intrinsic error and to be appliable for

the determination of dynamics behaviour under fault conditions, but

required more time to produce numbers than the second method utilized.

This was the solution of transcendaI relationships resulting from the

assumed time response of the synchronous machine currents by a Newton-

Raphson technique. The applicability of this method is constra:l.ned to

147.

e1ectric machines in steady state or with e1ectrical time constants of

much shorter duration than mechanica1 time constants.

In sUtmllary, the modelling and ana1ysis of electri'c machines

has been extended both by the use of improved test methods and by

methods of solution suited to rapid evaluation of machine performance.

Real multi-winding machines were analyzed which, 9lthough'requiring

sorne compromises in,mode1 definition and accuracy, al10wed the determin-

ation of sorne practica1 effects of electric machines which hitherto had

not been considered in sufficient scope and depth. Regarding genera1ized

machine the ory , it is be1ieved that it is a more powerfu1 too1 in terms

of computation of machine performance than has been recognized. Undue

attention shou1d not be directed to the arrangement of electric machine

performance equations for the production of easily viewed mode1s, but

rather upon both their determination by information obtained either

directly or indirectly from tests of practica1 e1ectric machines and

ana1ysis by numerical methods.

BIELt bGltAPHY

1. Researches in E1ectro-Dynamics, Experimental and Theoretica1,

William Sturgeon. Anna1s of E1ectricity, Vol. II, Jan.

1838, No. 7.

148.

2. The E1ectrician, 1885-6; Proc. Inst. of Civil Engineers, 1885-6;

Jour. Soc. Te1egr. Engineers, Vol. XV, 1886, p. 524; phil.

Trans. 1886 pt. l, p. 331. Cited by Thompson, op. cita

p. 13, footnote.

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( \.

149.

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49. Pole Face Loss in Solid-Rotor Turbine Generators, W. W. Kuyper.

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June 1957, VoL 76, Part III, pp •. 275- 2810

52. Electromagnetic Theory of Electrical Machines, S. A. Nasar.

Proc. I.E.E., Vol. III, No. 6, June 1964, pp. 1123-1131.

53. An Equivalent Circuit for a Single-Phase MO,tor Having Space

Harmoni~s in its Magnetic Field, L. W. Buchanan. Trans.

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54. A.C. Machine Windings with Reduced Harmonic Content, B. J.

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55. Prob1ems in Design and Deve10pment of 750 MW Turbo Generator,

V. P. Anempodistov, E. G. Kasharshu and J. D. Urusov.

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56. Transient Response of Direct Current Dynamos, H. E. Koenig.

Trans. A.I.E.E., VoL 69, 1950, pp. 139-145.

57. A Method of Measuring Self Inductance Applicable to Large

Machines, J. C. Prescott and A. K. E1-~arashi. Proc.

LE.E., pt. A, Vol. 106, pp. 169,,173, 1959.

?8. An Analysis of Commutation for the Unified Machine Theory, C. V.

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59. The Inductance Coefficients of a Sa1ient Pole Alternator in

Relation to the Two-Axis The ory , G. W. Carter, W.!.. Leach,

J. Sudworth~ Proc. LE.E.,. pt. A, Vol. 108, pp. 263-270,

1961.

60. Inductances of a Practicd Slip-Ring Primitive l - An Ana1ytical

Study, II - An Experimental Study, T. H. Barton, J. C.

Dunfie1d. Trans. I.E.E.E., Vol. PAS-85, No. 2, Feb. 1966,

153.

pp. 140-151.

61. Inductance Coefficients of Rotating Machine Expressed in.Terms of

Winding Space Harmonies, R. B. Robinson. Proc.I.E.E.,

Vol. III, No. 4, pp. 769-774.

62. Simu1taneous Fau1ts on Three-Phase Systems, E. Clarke. A.I.E.E.

Trans., Vol. 50, 1931, pp. 919-941.

63. lwo-Phase Co-ordinates of a Three-Phase Circuit, E. W. Kimbark,

Trans., A.I.E.E., Vol. 58, 1939, pp. 894-910.

64. Power System Stabi1ity, Vol. III, E. W. Kimbark. John Wi1ey &

Sons, New York, 1956.

65. Symmetrica1 Components, Wagner and Evans. McGraw-Hi11 Book

Company, New York, 1933.

66. Applications of the Method of Symmetrica1 Components, W. V.

Lyon. MCGraw-Hi11 Book Company, New York, 1937.

67. Fau1t Ana1ysis by Modified Alpha, Beta and Zero Components,

T. Hsiao. Trans. A.I.E.E., Vol. 81, Part ~II, 1962,

pp. 136-146.

68. Unba1anced Operations in AC Machines, H. H. Hwang. I.E.E.E.,

Vol. PAS-84, No. 11, Nov. 1965, pp. 1054-1066.

69. Study of Symmetrica1 and Related Components Through the Theory

of Linear Vector Spaces, N. Dharma Rao and H. N. Ramachandra

Rao. Proe. I.E.E., Vol. 115, No. 6, June: 1966, pp. 1057-1063.

70. Transient Starting Torques in Induction Motors, A. M. Wahl and

L. A. Ki1gore. Trans., A.I.E.E., Vol. 60, 1941, pp. 1200-1209.

71. Transient Performance of Induction Motors, F. J. Maginniss and

N. R. Schultz. Trans. A.I.E.E., Vol. 63, 1944, pp. 641-646.

72. Digital Computation of Induction~Motor Transient Torque Patterns,

R. D. Slator, W. S. Wood, F. P. Flynn, R. Simpson. Proc.

I.E.E., Vol. 113, No. 5, ~y 1966, pp. 819-822.

,'.--

73. Transient Performance of the Induction Motor, 1. R. Smith and

S. Sriharan. Proc. LE.E., Vol. 113, No. 7, Ju1y 1966,

pp. 1173-1181.

154.

74. Two Dimensional Fields in E1ectrica1 Engineering, L. V. Bew1ey.

The MacMillan Co., New York, N.Y., 1952. Now pub1ished by

Dover Publications, N.Y., 1963.

75. Dynamic Circuit The ory , An'Experimenta1 Approach, T. H. Barton.

A.I.E.E. Conference Paper No. 62 - 1226.

76. Mutual Inductances of Synchronous Machines with Damper Windings,

V. A. Kinitsky. LE.E.E. Trans., PAS, Vol. 83, Oct. 1964,

pp. 997-1001.

77. Eddy-Current Modes in Linear Solid-Iron Bars, P. Silvester. Proc.

I.E.E., Vol. 112, No. 8, August 1965, pp. 1589-1594.

78. A High-powered Servo-Ana1yser, R. S. Birtch. Master's Thesis,

McGi11 University, Dec. 1965.

79. Matrix and Tensor Ana1ysis in E1ectrica1 Network Theory, S. A.

Stigant. MacDonald, London, 19640

80. A Basic Ana1ysis of Synchronous Machines, W. A. Lewis. Trans.

A.I.E.E., Vol. 77, pp. 436-456, August 1958.

81. Modern Control Theory, J. Tou. McGraw-Hi11, New York, 1964.

82. Mathematica1 Methode for Digital Computers, A. Ra1ston and Ho

Wi1f. Wi1ey, N.Y., N.Y., 1960.

83. Automatic Step-Size Control for Runge-Kutta Integration, R. M

Warten. I.B.M. Journal, October 1963, pp. 340-341.

84., Optimum Seeking Methods, D. J.'. Wilde. Prentice-Hal1, Inc.,

Eng1ewood C1iffs, N.J., 1964, Chapters 1 and 6.

85. A First Course in Numerica1 Ana1ysis, A. Ra1ston. McGraw-Hi11

Book Co., N. Y., N. Y. ,. 1965, Chapter 8.

86. Generator Load-Ang1e Measuring Equipment for Marchwood, N. S.

Annis. Brit. Conun.and E1ect., Vol. 6, No. 12, Dec. 1959,

\

pp. 350-353.

87. The Measurement of Transient Torque and Load Angle in Mode1'

Synchronous Machines, R. No Sudan, V. N. Manohar and

B. Adkins. Proc. I.E.E., Vol. 107A, 1960, pp. 51-60.

88. Deviee for Measuring the Angle Between the Voltage and the (EM]'

of a Synchronous Machine. M. Z. Gurgenidze and J. Ro

Swryngin. E1ektrichestro, No. 7, 1958, pp. 65-67.

155.

89. Ca1cu1ation of Interna1 Fau1t Currents in Synchronous Machines,

V. A. Kinitsky. Trans. I.E.E.E., May 1965, Vol. PAS~84,

No. 5, pp. 381-388.

90. A Precision Torquemeter Based on Magnetic Stress Aniosotropy,

T. H. Barton and R.J. lonides. Trans. I.E.E.E., Vol 0

PAS-85, No. 2, Feb. 1966, pp. '152-159.

(

"

r -

A-l

APPENDIX l

THE OPEN CIRCUIT CHARACTERISTIC

The stator open circuit phase voltage is equal to the rate of

change of flux linkage with. the field winding.

As seen in equation 2-8 the mutual inductance Msf between rotor and

stator consists of a harmonic series involving odd terms only.

00

Msf = ~ Mk COS K1t k=l

Combining land 2 yields es.

00

= 2: k=l

K ~ SINK)t

where ~ denotes the frequency of the stator voltage. The rms stator o

voltage is represented in equation 4.

E = + 2 2 1/2

9M + 2SMS + •••. ) 3

Since the harmonic content of both the measured and calculated mutuel

inductance between stator and rotor was .less than 1%, the expression

for the stator voltage may be simplified to that of equation 5.

E =

1

2

3

4

5

A comparison of predicted open circuit characteristic with measured points

is given in Figure Al-l. Small differences, Qf the order of ± 4%, are

explicable in terms of experimental error.

..-,., ,

320

l&Jl (!)

~ -'a4Q o > fJJ m 4 %160 Q.

80

o CALCULATED FROM STATOR FIELD MUTUAL INDUCTANCE 1

--MEASUREMENTS

0.4 OG8 1.2 L,ô

1 f A FIGURE Al-l SYNCHRONOUS MACHINE OPEN CIRaJIT CHARACTERISTIC

>. r->

2.0

A-3

APPENDIX II

ruO-PHASE MACIUNE INDUCTANCE RATIOS

The armature winding of a 2 hp., 110 V, l7.8A direct cu~rent

motor was reconnected to simulate a salient-pole machine with a two­

phase winding on the non-salient member. The windings so formed were

taken to be full-pitched and have a phase belt of ninety degrees.

Correlation of test and calculated data from equation 2-16

of the ratio of the second harmonic armature self inductance and armature

mutual inductance provided additional evidence of the validity of the

analysis. At the same time, the large errors which can arise as a result

of the neglect of saturation in the analysis became apparent. Figure

A2-l depicts the variation of inductance ration with excitation level

for this machine. A discussion of the influence of saturation upon

the experimental results ia delayed until Appendix III.

1\ A-4

V ~ •

...Il ;: , C Z

~ QA& ~

U ..... ..... «it! Cl antl

1 0 lai L

If )(

w

f!! 1 ~

Bd œ • Œ 1 :)

0 • en

::. 0 - , G:

== Œ: 0

'" -

La..

:I~ .. 2-1

l-

A :: N C\I fi)

"'" 0 V c. . • • •

o

- - - o

APPENDIX III

INFLUENCE OF SATURATION

In the calculation of inductance it is the permeance series

which is affected by saturation. A more generalized form of this

series th an 2-1 is

00

P = ~ P1I. cos Jl(1 - 17;. ) 1=0

A-5

where P 1. and Ill. are functions of both the magnitude of the resultant

MMF and its inclination, 0( , to the direct axis. Such a series, while

greatly complicating the problem, in practice adds little to the single-

coil excitation situation considered. As an example of this experimental

evidence based on equation 2-17 and depicted in Figure AJ-l shows that the

simplified permeance series adequately describes the experimental machine

for a stator-winding test current up to twice the rated rms value.

A consideration of saturation influence for multi-coil excit-

ation would greatly complicate the matter. Su ch a mapping was not

attempted. It is expected that if an approximate method of inclusion

of saturation of the machine under operating conditions is desired, con-

sideration of saturation in terms of a two axis approach might be more

fruitful, especially where the influence ûf t1me harmûnic8 1a small such

as in a three wire star conneèted stator as suggested by the work of

Hamdi-Sepen32•

M32./LZ. FOR VARIOUS CURRENTS

2.8

2.0 - " .

- - - ~ - - .... D_ ..

1.2

~<=-==- PREDICTED

EXPERIMENTAL

.4

:r 0\

4 a 12 AMPERES

FIGURE A3-1

__ . ___ • ________ ~. __ . __________ ,~ _____ . ...;.._____ '-_ ....... ....:..~ -'-_________ , ___ . _______ ....... ___ ~_. ________ --_0 ________ _

(

"

"

APPEl'~~X IV

CONNECTION OF THREE-PHASE WINDING TO SIMULATE A MACHINE WITH 1200 SPREAD

A-7

Each of the three stator windings of the synchronous machine

have, approximately, a 60 degree spread. Figures A4-l and A4-2 depict

connection of these windings to measure respectively the self and

mutual stator inductances of approximately a 120 degree spread winding.

It will be noted that the use of a high input impedance electronic

integrator permits an alternative to the Maxwell Rayleigh bridge method

of eliminating winding resistance voltage drops. A small non-inductive

resistance, comparable in magnitude with the winding resistance and

carrying the test current, is inserted so that the integrator ground is

common to both the winding under test and this resistance. With refer-

ence to Figures A4-1 and A4-2, variation of the integrator gain (R2)

permits nulling of the, influence of the test winding resistance drop.

Measurements of stator self and stator mutual inductance

showed negligible higher harmonic content, as weIl as amplitudes and

ratios in excellent accord with predicted values. Thus it may be con-

cluded that the discrepancies between the idesl Kronian ind.uctance

representations and the observed inductance behaviour essentially result

from the extstance of a third-harmonic component of stator mmf.

,-" ,--'

R S

RI

FIGURE A4-1

-•

" '\

RED

YEllOW

MEASUREMENT OF SELF INDUCTANCE OF APPROXIMATELY 1200 SPREAD WINDINGS

R2 c

~~

>. co

,

/-", 1

R S

R,

-•

FIGURE A4- 2

,r

R2

YELlOW .,

BLUE •

MEASUREMENT OF MUTUAL INDUCTANCE OF APPROXIMATELY 1200 SPREAD WINDINGS

c

:r \0

c

A-lO

APPENDIX V

INDUCTANCE IN TERMS OF RADIAL AIRGAP FlUX DENSITY AND MME'

The permeance series, 2-1, may be written as a double sided

series 00

~ cos 11l Q~) = L- QJ!

1=-00

where QI = P.i /2 1.:/: 0

QI = P~ .P= 0

The flux density is

00

L m=-oo

QR2

Fm f cos + cos

1. =-00

The sum of the second term is equal to that of the first term,

the order of the series merely being reversed. Thus

00

BE)!.) = L m=-oo <A. - m)"," +mo(

1.=-00

The component of the r th harmonic flux wave created by the mth

harmonic of the mmf wave is obtained by putting A = r+m in 1. r

Bm ("1-) = Qr+m Fm cos r1- + m 0(

Rence evaluating the winding inductance in a manner outlined

in Chapter 2 gives

1

A-11

~r Le( m = ~ ~c Wc<. W~ Qr+m sin m"lT /2 sin r1T /2 :m ;-

cos (rp + mo()

Simi1ar1y

0( r 4 L Ac fPIV WA Q sinm 71 sin r1T h / 1l'/ t9 m = -:;;:z ... f' r+m 2 2 r r A'tll m

cos rq{ + mf

It is evident that~: ri: ~~ i.e. reciprocity of components

does not ho1d unti1 the summation over a11 rand mare taken. However, o(m tir Lfr = Lo(m·

and

so that

The expre8sion for inductance is simp1ified by putting

Wc( m = WC)( ~ sin m1t /2 m

W pr = W~ ~r sin r tr/2

v 4Ac Q o r+m =.......---. r+m

7T 2

~: = ~ r+m ~ m W p r cos r~ + m 0(

2

3

A-12

APPENDIX VI

REQUIRED NUMBER OF TWO-PHASE HARMONIC CURRENTS

The trigonometric functions of the transformation matrix A

are such that when N i8 even

Am+2N = Am and when N i8 odd

where

Am+2N = U Am

U =QCJ LW

provided the turns ratio Km+2N i8 chosen equa1 to Km. Thus by appropriate series aiding or opposing connection of the

harmonie equiva1ent windings, N pairs of currents, i a1 to i a (1+2N) and

i b1 to ib(1+2N)' suffice for the ulophase equiva1ent.

Re1ationships exist within these groups of currents such that

when N is even

and when N is odd

~+j =

provided KN+j is equa1 to KN_j •

When N is odd the special case of j = 0 yie1ds N

iaN = KN S- i ~n

These latter two resu1ts reduce to N the number of independent

currents required for the two phase equiva1ent of a polyphase system

1

2

3

4

5

, ..

A-13

having an odd number of phases.

When N is even, pairs of windings share a common axis so that

their combined magne tic effect can be reproduced by either one carrying

the algebraic sum of the two currents. Thus the system has only N/2

degrees of freedom. The number of two phase currents is correspondingly

reduced by the fact that when N/2 is even

Axn+N :: Am

and when N/2 i8 odd

i\n.rN = U Am

6

7

A-14

APPEND1JC VII

MOMENT OF . l NERTIA

The moment of inertia of the test set was determined by two

methods; a retardation test and a frequency response test. The appli-

cation of frequency response techniques to the determination of inertia

is not common and this procedure will be outlined.

For small variations of speed and armature current about the

operating point of a dc machine, equation l re1ating armature current

and shaft speed is applicable.

T = C l(s) = (Js + K) w(s)

J = inertia Kgm - m2

l

Where

K = viscous friction n-m per radian per sec.

C = torque constant n-m per amp

l = armature current amp

Rearranging equation 1 yields equation 2 where T is the mechanica1 time

constant J/K

!..{& K (1 + Ts) w(s) = C 2

If ST »1 equation 2 may be written as

~ J w(s) = C s

Equation 3 is valid for frequencies greater than .1 cps for the test

machine and a value of inertia of .433 Kgm - m2 was determined. This

value is considered more accurate than that of ~468 Kgm - m2 determined

from a retardation test.

3

(-

APPENDIX VIII

MomFIED dq AXIS PHASOR DIAGRAMFOR THREE WIRE STAR CONNECTED STATOR

Assume zero stator winding resistance, zero P4 and higher

A-15

permeance terms, and sinusoidal stator-field mutual inductance. ~le

phasor diagram becomes essentially that of Figu~e 4-2,and torque may

be calcu1ated from 4-40. The neutral voltage is, from equation 5-5

V 1 = Re ( 3 -{2 l X e j (3' wt - (26 +0(1) nn l' bb"

Define" Il:

Vnn ' a voltage at the fundamental frequency related to Vnnl

by equation 2. 1 1

Vnn =

'1 Thus Vnn ia, from equation 1 and 2 of a form suitab1e for representat-

ion on the modified dq axis phasor diagram of

= J-::o . e j (wt -. ) 3V 2 Il ~b

where

1

2

3

,ft

v

FIGURE ~8-1

1 d

A-16

v' 1 nn

MOMFIED dq AXIS PHASOR DIAGRAM FOR THREE WIRE STAR CONNECTED STATOR