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.FINITE ELEMENT ANALYSIS OF A DEFECTIVE INDUCTION MOTOR.

A D i s s e r t a t i o n P re se n t ed t o

The G r ad u a te F a c u l t y o f

The C o l l e g e o f E n g i n e e r in g a nd T ec hn ol og y o f O h io U n i v e r s i t y

I n P a r t i a l F u l f i l m e n t

o f t h e R e qu ir e me n ts f o r t h e D eg re e

D o c t o r o f P h i l o s o p h y

Clarence Nwabunwanne Ob iozor ,-J u n e 1987

FINITE ELEMENT ANALYSIS O F A DEFECTIVE INDUCTION MOTOR

CLARENCE N . OB I OZOR

This Dissertation has been approved

f o r the Department of El ect ri cal and Computer Engineering

and the Col le qe of Engineering and Technology

Q R ~ )Associate Professe4/0f Ele ctr ica l EngiRWri ng

Dean, Col 1ege of Engineering and Techno1 ogy

CLARENCE MWABUNWANNE OBIOZOR

Al l Rights Reserved

OBIOZOR, CLARENCE N . June 1987. E l ec t r i c a l E n g i n ee r i n g

F in i t e E lement Analys is of a Defe ct iv e Ind uct ion Motor

( 130 P P ) .

Dir ec to r of D is s e r t a t i on : Dr. Nasser J a le e l i

Th is d i s se r t a t io n p rov ides a methodology f o r the computa tion

of f lu x d i s t r ib u t io n in defec t iv e induc t io n machines. Having ob ta ined

t h e f l u x d i s t r i b u t i o n f o r th e a p p li e d v o l t a g e , t h e s t a t o r c u rr e n t i n

each phase f o r any load can be ca lc ul at ed , and hence i t can bede te rmined i f c on t inua t ion of the ope ra t ion o f th e de fe c t iv e machine

u nd er t h e ap p l i ed l oad i s s a f e .

The method01ogy i s based o n the use of Maxwell ' s e qua t ion s t o

d e r i v e a u n i f i e d e q u a t io n . T h i s eq u a t i o n r e l a t e s t h e s p ace an d t i m e

d e r i v a t i v e s o f t h e m ag ne ti c v ec t o r p o t en t i a l (MVP) of each point

w i t h i n t h e m achin e t o t h e d en s i t y of t h e ap pl i ed c u r r e n t a t t h e

p o i n t . A pp ly in g t h e m ethod of f i n i t e e l em en t s t o t h i s eq u a t i o n a t

d i f f e r e n t s e c t i o n s of t h e machine l e a d s t o a gl o b a l e q u a t i o n . In

t h i s d e r i v a t i o n , s a t u r a t i o n a t any p o in t o f t h e m achine and a t

any i n s t a n t o f t im e i s f u l l y a cc ou nte d f o r .

The g loba l equa t ion i s a s e t of n on l in ear t ime domain d i f f e r e n t ia l

e q u a t i o n s . A s tep-by-s tep numer ica l method i s employed t o in te gr a t e

t h i s g lo b a l e q u a t i o n . T h i s p r o ce s s y i e l d s t h e v a l u e o f MVP fo r any

p oi n t of th e machine a t any in s ta n t of t ime. The computer program

deve loped in th i s work t o c a r r y o u t t h e ab ov e t a s k s i s v a l i d a t e d

by a p p l y i n g i t t o sim p1 e e l e c t r o m a g n e t i c s y s t e m s . I t i s t h en u sed

t o p r o d u c e M V P c o n t o u r s of an i n d u c t i o n ma ch in e f o r t h r e e d e f e c t s .

Approved -( S i g n a t u r e of G e c t o r )

ACKNOWLEDGEMENTS

I am g r e a t l y i n d e b t e d t o Dr. N a ss er J a l e el i f o r h i s p a t i e n c e ,

g u i d a n c e a n d e n c o u r a g e m e n t t h r o u g h o u t t h i s d i s s e r t a t i o n . I a1 so wish

t o e x p r e s s my g r a t i t u d e t o M rs. F a r id e h J a l e el i f o r h e r we1 1 w i s h e s .

T h i s d i s s e r t a t i o n i s d e d i c a t e d t o my w i f e May and my s on M ar ti n

f o r t h e i r s u p p o r t , u n d e r s t a n d i n g and m o t iv a t i on t h r o u g h ou t t h e c o u r s e

of t h i s wo rk .

F i n a l l y , my t h a n k s g o t o t h e m em bers of s t a f f , D e pa rt m en t o f

El e c t r i c a l and C omputer Eng in ee r ing , Oh io U n i ve rs i t y , who p rov ided

t h e f a c i l i i e s t o c a r r y o u t t h e work.

T A B L E O F CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . . . . .BSTRACT i v

. . . . . . . . . . . . . . . . . . . . . . . .CKNOWLEDGEMENTS v i

L I S T OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . x

. . . . . . . . . . . . . . . . . . . . . . . .I S T OF FIGURES x i

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 11.1 G e n e r a l . . . . . . . . . . . . . . . . . . . . . . . 11 . 2 C on t e n t and C o n t r i b u t i o n o f t h i s D i s s e r t a t i o n . . . . 5

2 FIE LD EQUATIONS AND F I N I T E ELEMENT APPROXIMATION . . . . . 7

2 .1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . 7

2 . 2 G e n e r a l MVP E q u a t i o n . . . . . . . . . . . . . . . . . 8

2 .3 U n i f i e d E q u a t i o n f o r t h e I n d u c t i o n M ac hi ne . . . . . . 0

2 . 3 . 1 S t a t o r S l o t s o f a n I n d u c t i o n M ac hi ne . . . . . 0

2 . 3 .2 A i r Gap a nd S t a t o r I r o n R e gi on s . . . . . . . . 3

. . . . . . . . . . . . . .. 3 . 3 S o l i d R o t o r R e gi o n 1 4

. . . . . . . . . . . . . . . . . . .. 4 E l e m e n t E q u a t i o n 1 7

. . . . . . . . .SOLUTION METHODOLOGY AND COMPUTER PROGRAM 24

. . . . . . . . . . . . . . . . . . . . .. 1 I n t r o d u c t i o n 24

. . . . . . . . . . . . .. 2 S o l u t i o n o f G l o ba l E q u a t i o n 24

3 .3 C on to ur s o f t h e M a g n e t i c V e c t o r P o t e n t i a l s . . . . . . 6

3 . 4 C o m p ut a t i o n o f W i n d in g C u r r e n t s . . . . . . . . . . . 28

3 . 5 A D e s c r i p t i o n o f t h e C om pu te r P ro gr am s . . . . . . . . 2

. . . . . . . .. 5 . 1 D e s c r e t i z a t i o n P r o g r a m . MESHGEN 3 2

3 . 5 . 2 M a i n P r o g r a m . F ETIM E . . . . . . . . . . . . . 3

4 VALIDATION O F THE COMPUTER PROGRAMS . . . . . . . . . . . 40

4 . 1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . 40

4 . 2 V a l ida t ion o f t he P rogra m f o r a L ine a r Ca se . . . . 4 1

4 . 3 P rogram Re su l t s f o r an I nd uc to r E nc lo se d by I ron

. . . . . . . . . . . . . . . . . . . . . . . .ore 44

4 . 4 A M a gn eti c C i r c u i t w i t h a n Air Gap . . . . . . . . . 4 9

4 . 5 S i m u l a t i o n o f a S o l id Ro to r I nduc t ion Moto r w i th

No D e fe ct . . . . . . . . . . . . . . . . . . . . . 56

5 SIMULATION RESULTS FOR A DEFECTIVE SOLID ROTOR

INDUCTION MOTOR . . . . . . . . . . . . . . . . . . . . . 65

5 . 1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . 65

5 . 2 D i s c ~ n n e c t i o nof One of th e Two Pa ra l l e l C oi ls in

Phase AB . . . . . . . . . . . . . . . . . . . . . . 66

. . . . . .. 3 S ho r t Ci rc u i t o f Some Turns of Phase AB 56

. . . . . . . . . . . .. 4 D isc on ne cti on of Two Phases 73

. . . . . . .CONCLUSION A N D SUGGESTIONS FOR FURTHER W O R K 83

6 . 1 Conc lus ion . . . . . . . . . . . . . . . . . . . . . 83

. . . . . . . . . . . .. 2 S ugge s t ions f o r F u r the r W or k 84

. . . . . . . . . . . . . . . . . . . . . . . . . . .E F E R EN CE S 85

APPENDIX

EXPANSION O F V x . J x A = J . . . . . . . . . . . . . . 94'J

. . . . . . . . . . . . . . . . . .. MATHEMATICAL FORMULAE 96

. . . . . . . . . . . . . . . . .. l V e c t o r I d e n t i t i e s 96

. . . . . . . . . . . . . . . . . .. 2 Green 's Theorem 96

v i i i

APPENDIX

B . 3 stoke' s Theorem . . . . . . . . . . . . . . . . . . 96

B . 4 Integration Formuale for a Triangl e . . . . . . . . 98

C DERIVATION OF MVP WITHIN A TRIANGLE . . . . . . . . . . . 100

. . . . . . . .DERIVATION O F EQUATION (2.47) FROM (2.43) 105

E DERIVATION OF A TIM E FUNCTION FOR THE FLUX IN A

MAGNETIC CIRCUIT . . . . . . . . . . . . . . . . . . . . 12

LIST O F TABLES

2.1 Constants r, 3 , a ndy f o r Di f f e r e n tR e g ions of a nInductionMachine. . . . . . . . . . . . . . . . . . . . 17

4.1 Comparison of the Solutions Obtained fo r theTemperature of Diffe re nt Nodes a t t = 1.2 Hours . . . . . 4 3

4 . 2 D a t a f o r t h e S o l i d R o t o r I n d u c t i o n M a c h i n e . . . . . . . 56

4 . 3 The Values of n fo r Each Phase . . . . . . . . . . . . . 609

5.1 The Values of n f o r Each Phase When Coil AB1 i s9

Disconnected . . . . . . . . . . . . . . . . . . . . . . 68

5.2 The Values of n f o r Each Phase When F ifty Percent of9

One o f the Two Parall el AB Coils i s Bridged Over . . . . 74

5 . 3 The Values o f n f o r Each Phase When Phases AB a n d B C9

are Disconnected . . . . . . . . . . . . . . . . . . . . 7 9

LIST OF FIGURES

F i g u r e P a g e

. . . . . . . . .. 1A

S e c t i o n o f t h e C o n s i d e re d S o l i d R o t o r 1 5

2 . 2 T w o - D i m e n s i o n a l R e g i o n Q. Bo u n d e d b y a Co n to u r 7 . . . . 1 8

2 . 3 A R e g i o n R. D iv id ed i n t o T r i a n ~ u l a r le m e n t s . . . . . . 2 1

3 . 1 C o n to u r P l o t s f o r a M a g n e t i c C i r c u i t . E ac h P a i r o fL i n e s D i f f e r by a S p e c i f i c V a l u e o f MVP . . . . . . . . . 27

. . . . . . . . . . . . . . . . . . .. 2 A S i n g l e T u r n C o i l 29

. . . . . . . . . . . . .. 3 A Fl ow ch ar t o f P rog ram MESHGEN 34

3 . 4 A S e c t i o n o f a n I n d u c t i o n M a ch i ne Sh o wi ng S e c t o r s an d. . . . . . . . . . . . . . . . . . . . .u a dr i 1 a t e r a l s 36

. . . . . . . . . . .. 5 A F l o w c h a r t o f b l a in P r o g r a m FETIME 3 7

4 . 1 T he F i n i t e E l e m en t Mesh f o r t h e E xa mp le C o n s i d e r e d i n. . . . . . . . . . . . . . . . . . . . . . .e c t i o n 4 .2 4 2

4 . 2 F i n i t e E l em e nt Mesh When t h e C o i l i s C e n t r a l l y P l a c e di n t h e C o r e . . . . . . . . . . . . . . . . . . . . . . . 45

4 . 3 MVP C o n t o u r s f o r t h e I n d u c t o r When t h e C o i l i s. . . . . . . . . . . .y m m e t r i c a l l y P l a c e d i n t h e C o re 16

. . .. 4 F i n i t e E l e m e n t M esh When t h e C o r e i n N ot S y m m e t r i c 47

4 . 5 MVP C o n t o u r s f o r t h e I n d u c t o r w i t h U n sy m m et r i c I r o nC o r e . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 . 6 F i n i t e E l em e nt Mesh f o r the M a g n e t i c C i r c u i t C o n s id e r ed. . . . . . . . . . . . . . . . . . . . .n S e c t i o n 4 . 4 50

4 . 7 C o n t o u r P l o t s f o r t h e M a gn e t i c C i r c u i t w i t h a n AirGap . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4 . 8 T he V a r i a t i o n o f t h e F lu x E s t a b l i s h e d i n the C o i l f o r. . . . . . . . . . .h e M a gn e t ic C i r c u i t i n F i g u r e 4 . 6 5 4

V a r i a t i o n o f F lu x D e n s i t i e s f o r Two E l e ~ e n t s n t h eI ron Core of the Magne t ic Ci rcu i t Shown in

. . . . . . . . . . . . . . . . . . . . . . .i gu r e 4 . 6 5 5

F i n i t e Element Mesh of th e Co nside red Sol id Rotor. . . . . . . . . . . . . . . . . . . . .nd uc tio n Motor 57

C i r c u i t D iag ram of t h e S t a to r Co il of a N on- D e fe c tive. . . . . . . . . . . . . . . . . . . .nduc t ion Machine 58

Rota t ing Magne t ic F ie1 d of an I n d u ct io n Nachi ne '+ii hNo D e f e c t . . . . . . . . . . . . . . . . . . . . . . . . 61

P h a s e C u r r e n t s of a N on- D e fe c tive I nduc t ion w a c h in e . . 4

D e f e c t i n P ha se AB , Where Coil A B , i s D i s c o n n e c t e d. . . 7

M agne tic Fi el d of an I nd uc ti on Machine When One of t h e. . . . .wo P ar a l le l C oi l s o f Phase AB i s D i sc o nn e ct ed 69

Waveform of th e C ur re nt in Each Phase of an In du cti onMachine When One of t h e Two P a r a l l e l C o il s of Ph ase AB

. . . . . . . . . . . . . . . . . . . . .s D i s co n n e ct e d 71

F i f t y P e r c e n t o f One o f th e Two P ar a ll el Coil s of. . . . . . . . . . . . . . . .hase AB i s Br idged O ver 7 2

Con tou r P lo t s f o r an I ndu c t ion Mac hine When F i f ty P e r c e n tof One of t h e Two P a ra l 1 e l C oi l s of Phase A B i s B r i d g e d

. . . . . . . . . . . . . . . . . . . . . . . . . .ver 75

V ar i a t i on of C ur re n t in Each Phase of an In du c t io nMachine When F i f t y Pe rc en t of One of t h e Two Par al l elCo i l s o f P ha se AB i s Bridged Over . . . . . . . . . . . . 77

. . . . . . . . . . . .i sconnec t ion of Phases A B and BC 78

Con tour P l o t s of an In du cti on Machine When Phases AB andBC a r e D i sc onne c te d . . . . . . . . . . . . . . . . . . . 80

V a r i a t i o n o f C u r r e n t i n P h a s e C A of an Induction MachineWhen Phases AB and B C a r e D i sc onne c te d . . . . . . . . . 82

Two-Dimensional Region 1, Bounded by Y Over Ldhich. . . . . . . . . . . . . . . . .reen ' s Theorem Appl i e s 97

Figure Page

8 . 2 A Triangul a r El ement Ze . . . . . . . . . . . . . . . . . 99

C . l A Triangular Element. 2e. Showing MVPs . . . . . . . . . 01

E . l A Magnetic Ci rc ui t with I t s Exc itatio n System . . . . . . 13

Chapter 1

INTRODUCTION

1.1 General

A single-phase equivalent ci rc ui t i s derived fo r a three-phase

induction machine, when nonl in ea ri t i e s a r e neg lec ted and the machine

i s assumed t o have symmetry in st ru c tu re . The parameters of t h i s

c i r c u i t ar e obtained by allowing simp1 ify ing assumptions in th e flu x

dist r ibu tion in the a i r gap. This cir cu it i s mainly used fo r the

conceptual understanding of th e machine behavio r. I t i s a ls o, when

approximate so lut ion i s allowed, used t o pred ict th e performance of

the machine a t various loads.

To obt ain a s e t of more acc ura te parameters f o r the machine,

a det ai le d deriva tion of f lux di st ri bu ti on and some account of

nonl ine ari t i e s a re required. These requi rements can be sa t i s f i e d

by applying Maxwell equations on d i f f eren t se ct io ns of the machine.

The resu lt i s a second order part ial dif fer ent ial equation, the

solution of which can provide much more information related to the

operatio n of the machine than those obta inab le from the equ ival ent

c i r c u i t .

I n the 1960's with the av ai la bi li ty of computers, the fi n i t e

element method emerged as a useful numerical method of so lv ing problems

in mathematical phys ics and engi nee ring . Since the f i r s t mention of

the " f i n i t e element" name [ I ] - [ Z ] , i t has been used with succes s in

the areas of str uc tur al mechanics, fl u id flow and heat conduction.

A n e x t e n s i o n o f t h e f i n i t e e le m e n t m ethod t o m a gn e ti c f i e l d p ro ble ms

h a s en j oy e d i n c r e a s i n g p o p u l a r i t y s i n c e S i l v e s t e r a nd C ha ri [3] used

i t t o a n a l yz e a t ra n s fo r m e r. I t i s a d es ig n to ol c u r r e n t l y used t o

s u p p ly g r e a t d e t a i l s r e g ar d i ng th e m a g n e t i c f i e 1 d d i s t r i b u t i o n f o r

n o n - d e f e c t i v e t h r e e - p h a s e mach ines which a r e f ed by a ba lanced th re e-

p h a s e v o l t a g e s o u r c e . I t i s u se d t o o p t i m i z e t h e d e s i g n p a r am e t e r s such

a s t h e n um ber a nd d im e n s io n s o f s l o t s i n e l e c t r i c a l m a ch in e s [ 4 ] -[ 5 ] ,

a nd f o r a c c u r a t e c o m pu ta ti on of p a r a m e te r s i n c l a s s i c a l c i r c u i t model s

[ 6 ] - [ 8 ] . In t h i s d i s s e r t a t i o n , a f i n i t e e l e m e nt -b a se d method i s d ev el op e d

and t a i l o r e d f o r t h e a n a l y s i s of a d e f e c ti v e i n d u c t i o n m ac hi ne .

An ind uc t io n machine may become de fec t iv e wh i le in s e r v i ce . The

d e f e c t may be du e t o t h e s h o r t c i r c u i t o f some t u r n s o f a c o i l , d i s -

c o n n e c t io n of a c o i l o r a p h as e , u n sy m m et ri ca l a i r g a p , b r ok en r o t o r

bar s in cage mo to r s , e t c . When th e s i z e of th e machine i s huge , t ak ing

i t o f f - l i n e i s v er y e x p e n si v e . I t ca n m ean, f o r e xa m pl e, t h a t a power

pl an t must be sh ut down. For some p la n ts , th e co st may amount to

several thousands of do1 1 a r s p e r h o u r .

I n o r d e r t o o p t i m a l l y co pe w it h t h i s s i t u a t i o n , t h e p l a n t m anagem ent

n e ed s t o know to w ha t l e v e l t h e d e f e c t i v e m a ch in e c an s t i l l b e u t i l i z e d

w i t h o u t f u r t h e r e s c a l a t i o n of t h e d e f e c t . The d e s i r e t o a ns w er s uc h

a q u e s t i o n h a s l e d pow er e n g i n e e r s t o s ee k me th od s t o a n a l y z e d e f e c t i v e

mach i n es .

Will iamson and Sm ith [9] used a g raph of t he ro to r of a s q u i r r e l

c a g e i n d u c t io n m o to r t o d e t e r m in e t h e n um ber o f u nknown c ~ r r e n t s n

th e m ac hin e f o r a g iv e n r o to r b a r and end r i n g f a u l t s . F or an i d e a l

st at or winding, rel ati ons hip between t hese cu rre nts and the applied

vo lta ge was made by use of coupl ing impedances between the r otor and

st at or . These impedances ar e derived from the speci fie d resi sta nce

and reactance of the machine.

A1 though s atu rat ion e ff ec t was neglected in the an al ys is , t he

authors suggested that correction coul d be made by using satirrated

values of re sis tan ces and reactances. Therefore, accuracy of anal ysi s

in such a case will be limit ed t o the accuracy of pre dic tin g th e

saturated parameters.

I n t h e i r anal ysi s of induction machines with s t a t o r winding

fau l t s , Will iamson and M rzoian [ l o ] devel oped a Fourie r ser ies -based

method which they used t o der iv e the coupl ing impedance between the

ro to r and st a to r . They employed these impedances to est ab l ish a

re1 at io ns hi p between the appl ied vol tage and the curr en t harmonics.

This ana lys is did not account fo r satur ati on of the iron core, as the

ro to r and s t a t o r were modelled by two concen tri c smooth cyl inde rs of

i nf i ni te l y permeabl e i ron.

The methods proposed by Williamson e t a l . f o r the an al ys is of

defec tive induc tion machines cannot be used f o r many def ec ts including

non-uniform a i r gaps. Even f o r th e defe cts f o r which the se methods

a re developed, the fl ux densi ty in t he machine cannot be repre sented.

Therefore, f o r some of t he defect s t ha t cause severe satur ati on in

some port ions of the machine, t h i s author be1 ieves tha t a more

sophi stic ated method such as th at presented in t hi s d is se rt at io n

i s needed t o val id at e t hese proposed methods.

A m ethod based on f i n i t e e le me nt i s t a i l o r e d i n t h i s d i s s e r t a t i o n

t o e na bl e an a c c u ra t e p r e d i c t i o n o f f l u x d e n s i t y d i s t r i b u t i o n i n t h e

m a ch in e w i t h a ny d i s c u s s e d t y p e o f d e f e c t . I t w i l l a cc ou nt f o r

s a t u r a t i o n e ve ry wh er e i n t h e m ac hin e. W i t h t h i s m eth od , r a d i a l f o r c e s

a nd t h e p o s s i b l e r e s u l t i n g v i b r a t i o n c a n b e de t er m in e d. From a

k no wle dg e of f l u x d e n s i t y d i s t r i b u t i o n o f a d e f e c t i v e m ac hi ne u n de r

s a t u r a t i o n , t h e new o p e r a t i n g p o i n t of t h e l o a d- m a ch i n e s y st e m c an

b e f o un d . W i th t h i s i n f o r m a t i o n , i t i s p o s s i b l e t o compute t h e s a f e

l o a d w h ic h t h e m a ch in e ca n s u p p o r t w i t h o u t f u r t h e r d amage.

F r e q ue n c y d om ai n m e th o ds may b e u s e d t o a n a l y z e a n o n - d e f e c t i v e

m a ch in e . H ow ev er , i n a d e f e c t i v e ma ch in e, some p o r t i o n s o f t h e

m a ch i ne may become h e a v i l y s a t u r a t e d a n d t h e c u r r e n t may s u b s t a n t i a l l y

d e v i a t e fr om s i n u s o i d a l . U nd er t h i s c i r c u ms t a nc e , t h e s i n u s o i d a l

v a r i a t i o n o f w a ve fo rm s assumed i n f r e q u e n c y do ma in a n a l y s i s i s n o

l o n g e r v a l i d . T h e r e f o r e , a t i m e do ma in f i n i t e - e l e m e n t b as ed m et ho d

i s d e ve lo pe d i n t h i s d i s s e r t a t i o n f o r t h e a n a l y s i s o f d e f e c t i v e

i n d u c t i o n m a c h i n e s .

One m a j o r d i f f e r e n c e e x i s t s b e tw ee n t h e a n a l y s i s o f a n on -

d e f e c t i v e a nd d e f e c t i v e m ac hi ne . I n t h e f o r m e r c a t e g o r y , a ba l a nc e d

t hr ee -p ha se c u r r e n t i s assumed t o f l o w i n t h e w i n d in g i n o r d e r t o

c om pu te t h e m a c h in e p a r a m e t e r s . B a se d o n t h i s a ss u m p t i o n , t h e f l u x

d i s t r i b u t i o n i s o b t a i ne d . I n t h e l a t t e r c a t e go r y o f p ro b le m , t h e

a pp l i e d v o l t a g e i s t h e o n l y known i n p u t q u a n t i t y . The c u r r e n t i n

t h e t h r e e ph ase s a r e d i f f e r e n t , unknown a nd t h e i r v a r i a t i o n s w i t h

t i m e h a v e t o b e d e t e r m in e d .

1 . 2 C o n t en t and C o n t r i b u t i o n of t h i s D i s s e r t a t i o n

Maxwell ' s e q u a t i o n s i n t h e t im e d om ain a r e u se d t o d e r i v e a

u n i f i e d e q u a t i o n r e l a t i n g t h e m ag n et ic f i e l d a nd c u r r e n t a t any p o i n t

i n t h e c r o s s s e c t i o n o f t h e i n d u c t i o n m a c h i n e . The only unknown

v a r i a b l e i n t h i s e q u a ti o n i s t h e m ag ne ti c v e c to r p o t e n t i a l ( M V P ) .

T h i s eq u a t io n r e l a t e s t h e f i r s t and s ec on d s pa ce d e r i v a t i v e and t h e

f i r s t tim e d e r i v a t iv e of t h e MVP t o t h e a p p l i e d c u r r e n t . A v e r s i o n

of such a uni f ied equa t ion has been de r ived by Odamura [11] f o r s o l v i n g

s a t u r a t e d t r a v e l 1 ing wave prob l ems. However, t a i l o r i n g a f i n i t e

e l e m en t- ba se d d yn am ic model f o r t h e a n a l y s i s of r o t a r y d e f e c t i v e

i n d u c t i o n m a ch in es i s c o n s i d e re d t o b e t h e c o n t r i b u t i o n of t h i s

d i s s e r t a t i o n .

In Chapte r 2 , a g l o b a l e q u a t io n i s d e r i v e d fro m t h e u n i f i e d

e q u a t i o n . To s o l v e t h i s e q u a t i o n a t any ti m e s t e p , an i t e r a t i v e m ethod

t o g e t h e r w it h t h e B-H c h a r a c e r t i s t i c of t h e c o r e a r e u se d t o com pute

f o r an y p o i n t . The n u m e ri c al t e c h n i q u e u se d f o r t h i s p u r po s e a n d

t h e i n t e g r a t i o n o f t h e g l o b a l e q u a t i o n i s p re s e n t e d i n C h a p te r 3 .

Another co n t r ib u t io n i s th e development of a computer program

which u t i l i z es th e methodology of C hapte r s 2 and 3 to compute the

f l u x de ns i t y eve rywhere in the machine . The program i s deve loped in

th e t ime domain so th a t th e cu r r en t waveform in eve ry winding can be

p r e d i c t e d u n de r s a t u r a t e d c o n d i t i o n s . U sing t h e a pp l i e d v ol t a g e ,

t h e p ro gram p r o du c es a s o l u t i o n o v e r a s p e c i f i e d t im e i n t e r v a l .

I t p ro du ce s c o n t o u r p l o t s a t each t im e s t e p t o d e s c r i b e t h e f l u x

d e n s i t y d i s t r i b u t i o n o v e r a s o l i d r o t o r i n d u c t i o n m a ch in e w it h a

stat iona ry roto r . This is a f i r s t ste p towards the derivation of

the torque-speed c ha ra ct er is ti c fo r defecti ve machines. For th is

derivation the f lux density dist r ibu tion must be calculated a t

di ff er en t roto r speeds. From the torque-speed cha ra ct er is ti c,

th e new operati ng point of th e machine can be determined. Based on

th is and th e computed cu rr en ts , the plant engineer can decide whether

he should remove the def ec ti ve machine from se rv ic e immediately, o r

l e t i t remain u nt il the next scheduled maintenance.

The need f o r th e development of f i n i t e element-based computer

programs fo r magnetic f i e l d an al ys is has become inc rea sin gl y recognized

in recent years among e l ec t r i ca l power enqineer s. Thi s need has

brought about some production grade commercially available programs

over the past two years [12] . These programs may be grea t ly enhanced

if the methodology developed in this dissertation for the analysis of

de fe cti ve induction machines i s included.

FIELD EQUATIONS AND F IN IT E ELEMENT APPROXIMATION

2 . 1 I n t r o d u c t i o n

The o b j e c t i v e of t h i s d i s s e r t a t i o n i s t h e de ve lo pm en t o f a

m e t h o d o lo g y by w h ic h a d e f e c t i v e i n d u c t i o n m ac h in e may b e a n a ly z e d .

The m et ho d s o ug h t i n t h i s w or k i s b as ed on t h e d e t e r m i n a t i o n o f t h e

m ag ne ti c f i e l d d i s t r i b u t i o n i n t h e m ach in e .

I n t h i s c ha p t e r , t h e t h e o r y o f e l e c t r o m a g n e t i c f i e 1 ds i s e mp lo ye d

t o e s t a b l i s h a r e l a t i o n s h i p b et we en t h e m a gn e t i c q u a n t i t i e s and t h e

a p p l i e d c u r r e n t f o r e v e r y r e g i o n of t h e ma ch in e. An e q u a t i o n w i t h

o n l y one u nk no wn q u a n t i t y , t h e MVP , i s d e r i v e d f ro m t h e e s t a b l i s h e d

r e l a t i o n s h i p s . C o ns i de r a t i o n o f t h i s e qu a t i o n a t d i f f e r e n t s e c t io n s

of a s o l i d r o t o r i n d u c t i o n m ot o r l ea ds t o a u n i f i e d e q ua t io n f o r t h e

mach i ne .

I n t h i s w ork , t h e a x i a l l e n g t h o f t h e m ac hin e i s assumed t o be

much l o n g e r t h a n t h e m a c h i n e' s d i a m e t e r . The c o n t r i b u t i o n o f t h e

m a g ne t ic f i e l d due t o e nd c on n e ct o rs w hi ch c o nn e ct t h e a x i a l c o n d u c t o r s

i s n e g l e c t e d . W i th t h e s e a s s um p t io n s , t h e v a r i a t i o n of t h e m a g n e t ic

f i e l d i n t h e a x i a l d i r e c t i o n c an b e n eg le c t ed , a nd h en ce t h e a n a l y s i s

may b e c o n f i n e d t o t w o d i m e n s io n s .

I n t h e n e x t s e c t io n , M ax we ll ' s e q u a t i o n s a r e e mpl o ye d t o e s t a b l i s h

a r e l a t i o n s h i p be tw ee n t h e MVP and t h e d e n s i t y o f t h e a p p l i e d c u r r e n t .

I n S e c t i on 2.3 t h i s e qu at io n i s a p p l i e d t o t h e s t a t o r s l o t ,

a i r gap, s t a t o r i r o n and s o l i d r o t o r i r o n o f an i n d u c t i o n mach ine .

From t he se , a u n i f i e d e q u a t io n i s o b ta i ne d . The l a s t s e c t i o n o f t h i s

c h a p t e r d ea l s w i t h t h e d e r i v a t i o n o f e le m en t e q ua t io n s i n wh ic h

G a l e r k i n ' s m e th od [33] i s a p p l ie d t o t h e u n i f i e d e qu a t i on . The e r r o r

b et we en t h e a p p ro x im a t e s o l u t i o n f o r t h e MVP and t h e t r u e s o l u t i o n

i s m i n i mi ze d . T h i s p ro c es s y i e l d s a s e t o f a l g e b r a i c e q u at io n s.

2 . 2 G e n e r a l MVP E q u a t i o n

An e q u a t i o n wh ic h r e l a t e s t h e MVP t o t h e a p pl i e d c u r r e n t d e n s i ty

a t e v e ry p o i n t i n t h e m ac hin e i s d e r i v e d f ro m M a x w e l l ' s e q u a t i o ns .

C o n s i d e r i n g t h e f r e q u e n c y o f p ow er s ys te m s, t h e d i s p l a c e m e n t c u r r e n t

i s n eg le c t ed i n t h i s d e r i v a t i o n .

F o r 1ow f r e q u e n c y a p p l i c a t i o n s , M a x w e ll ' s e q u a t i o n s may be

w r i t t e n i n p o i n t f o rm and i n t i m e dom ain as [34 ] :

where :

x , y a n d ; r e u n i t v e c t o r s a l on g t h e t h r ee - d i m e ns i on a l o r th o g on a l

c o o r d i n a t e a xe s.

E i s t h e e l e c t r i c f i e l d i n t e n s i t y a t t h e p o i n t .

B i s t h e ma gn et ic f l u x d e n s i t y a t t h e p o i n t .

fi i s t h e m a g n e t i c f i e l d i n t e n s i t y a t t h e p o i n t .

J i s t h e t o t a l c u r r e n t d e n s i t y a t t h e p o i n t .

A1 1 t h e f i e l d q u a n t i t i e s , E , B , a n d J a r e v e c t o r s a nd f u n c t i o n s o f

space a nd t i m e. I n a d d i t i o n t o e q u a t i o n s ( 2 . 1 ) t h ro u gh ( 2 . 4 ) ,

t h e r e a r e c o n s t i t u t i v e r e l a t i o n s b etw e e n t h e f i e l d q u a n t i t i e s a nd

t h e m a t e r i a l p r o p e r t y o f t h e m edium . T he se a r e :

w h e re :

u i s t h e p e r m e a b i l i t y o f t h e m a t e r i a l , a nd c ha ng es w i t h B .

o i s t h e c o n d u c t i v i t y .

The MVP , A , may b e u s e d t o t r a n s f o r m e q u a t i o n s ( 2 . 1 ) t h r o u q h ( 2 . 7 )

t o a s i n g l e e q u a t i o n w i t h o n l y o ne unknown q u a n t i t y . The MVP i s

d e f i n e d t o b e a v e c t o r s a t i s f y i n g [35] :

U si ng f r o m e q u a t i o n ( 2 . 6 ) i n ( 2 . 8 ) g i v e s :

E q ua t io n (2 . 9 ) i s us ed i n ( 2 .2 ) a nd t h e r e s u l t i s a g e n er a l MVP

e q u a t i o n :

2 . 3 U n i f i e d E q u a ti o n f o r t h e I n d u c t i o n M ach in e

In t h i s s e c t i o n , t h e abo ve eq u a ti o n i s s i m p l i f i e d . I t i s a pp li e d

t o e v er y r e g io n o f a s o l i d r o t o r i n d u c t i o n mo to r t o y i e l d a u n i f i e d

e q u a t i o n f o r t h e m a ch in e.

T he f o l l o w i n g a s s u m p t i o n s a r e made i n o r d e r t o s i m p l i f y e q u a t i o n

( 2 . 1 0 ) :

a ) T he a x i a l l e n g t h o f t h e m ac h in e i s much l o n g e r t h a n t h e

d i am e t er and t h e a o p l i e d c u r r e n t d e n s i t y i s a x i a l l y d i r e c t e d . The M V P

i s a l s o assum ed t o be a x i a l l y d i r e c t e d .

b ) C o n t r i b u t i o n o f t h e m ag n e ti c f i e l d d ue t o end c o n d u c t o r s

i s n eg l e c t e d .

c ) F e r r o m a g n e t i c a n d c o n d u c t i n g r e g i o n s a r e i s o t r o p i c i n x and y

d i r e c t i o n s .U nder a s su m p ti on a ) t h e c u r r e n t d e n s i t y i s w r i t t e n a s :

5 = J Z ( 2 . 1 1 )

S i m i l a r l y t h e NVP i s w r i t t e n a s :

U nd er t h e a bo ve a s s u m p t i o n s , e q u a t i o n ( 2 . 1 0 ) i s s im p1 i f i e d i n

Appendix A . The r e s u l t f ro m ( A . 6 ) i s :

2 . 3 . 1 S t a t o r S l o t s o f a n I n d u c t i o n M achine

The t o t a l c u r r e n t d e n s i t y a t any p o i n t w i t hi n t h e s t a t o r

s l o t s i s r e so l ve d i n t o t h e ap pl i e d and i n du ce d c u r r e n t d e n s i t i e s .

T h es e co m po ne nt s a r e em plo ye d i n ( 2 . 1 3 ) t o y i e l d t h e MVP e q u a t i o n

f o r p o i n t s i n t h i s r e g io n of t h e i n d u c t i o n m ot or .

Use of ( 2 . 8 ) in ( 2 . 1 ) g i v e s :

S i n c e t h e c u rl of t h e g r a d i e n t o f any s c a l a r f u n c t i o n i s z e r o ( s e e

B . 2 ) ,

where.I'

i s a s c a l a r p o t e n t i a l .Mu1

t i p l y i n g e q u a t io n ( 2 . 1 5 ) byc:

g i v e s :

Resol vin g s7A' i n t o tw o c o m p o n en t s, t h e f o l l ow ing e q u a t i o n i s

o b t a i n e d :

where :

J0 i s t h e a p p l i e d c u r r e n t d e n s i t y .

.I i s a s c a l a r p o t e n t i a l .

Then, us ing J f o r oE from ( 2 . 7 ) , ( 2 . 1 7 ) g i v e s :

Taking t h e d i v e r g e n c e of t h e a bo ve e q u a t i o n g i v e s :

From equa t ion ( 2 . 4 ) , the term on the 1 e f t ha nd s i d e an d t h e s e c o n d

term on t h e r i g h t han d s i d e of e q u a t io n ( 2 . 1 9 ) a r e b oth z e r o . T h e re -

f o r e , t h e e q u at io n i s r ed uc ed t o :

A theo rem o f v ec to r a na ly s i s due to Helmhol t z [2 7] , [ 5 0 ] , s t a t e s

t h a t a v e c t o r i s s p e c i f i e d when i t s d i v e r g e nc e and c u r l h av e been

s p e c i f i e d . The curl of A i s s p e c i f i e d by ( 2 . 8 ) , and ac c o rd i ng t o

H elm hol t z ' s t h e o re m , t h e d iv e r g e n c e o f A h as t o be s p e c i f i e d . Choice

of t h e d iv e r g e n c e of A i s a r b i t r a r y [4 7] , [5 0 ]. I t i s u s u a l l y m ade

s o a s t o a c h i e v e some s i m p l i f i c a t i o n i n t h e r e l a t i o n b e tw ee n t h e

m a g ne ti c v e c t o r p o t e n t i a l and t h e c u r r e n t d e n s i t y . A c h o i c e o f :

t o g e t h e r w i t h ( 2 . 8 ) f u l l y s p e c i f i e s A . C o n s i d e r i n q e q u a t i o n ( 2 . 2 0 ) ,

t h i s c h o i c e r e q u i r e s .I o s a t i s f y t h e L a p l a c e ' s e q u a t i o n :

A s o l u t i o n which s a t i s f i e s ( 2 . 2 2 ) and s i m p l i f i e s (2.17) i s :

W ith t h i s c h o i c e , e q u a t i o n ( 2 . 1 7 ) b ec om es :

Use of t h e sam e s o l u t i o n i n ( 2 . 1 8 ) g i v e s :

A x ia l c om p on en ts of t h e a b ov e e q u a t i o n a r e u se d f o r t h e r i g h t han d

s i d e of ( 2 . 1 3 ) . S i n c e J = IJ f o r s t a t o r s l o t s , t he a p p li ca b le0

e q u a ti o n f o r e ve ry p o i n t w i t h in t h i s r e g i o n i s :

2 .3 .2 Ai r Gap and S ta to r I ron Reqions

As c o n d u ct io n and t h e a p p l i e d c u r r e n t i n t h e a i r a r e z e r o ,

t h e r i g h t han d s i d e o f ( 2 . 1 3 ) may be s e t t o z e r o t o g i v e t h e f i e l d

e q u a ti o n f o r any p o i n t i n t h e a i r ga p a s :

S i m i l a r l y , c o nd u ct io n i n t h e s t a t o r c o r e i s n e g l i g i b l e due t o t h e

l a m i n a t e d s t r u c t u r e o f t h e c o r e . T he e q u a t i o n w hich i s a p p l i c a b l e

f o r t h i s r e g i o n i s o b ta i n e d fro m ( 2 . 1 3 ) by s e t t i n g J t o z e r o ,

hence :

2 .3 . 3 S o l i d R o t o r R e gi o n

The c u r r e n t d e n s i t y a t a p o i n t P o f t h e s o l i d r o t o r

shown i n F i g . 2 . 1 may b e w r i t t e n a s [ 36 ] :

where :

i s t h e v e l o c i t y o f t h e p o i n t .

The v e l o c i t y o f P m a y b e r e s o l v e d i n t o x and y componen ts as :

U = - r dr S i n ? = - y ,x r

( 2 . 3 0 )

where :

o i s t h e a n g u la r v e l o c i t y o f t h e r o t o rr

Use o f e q u a t io n (2 .2 4 ) w i t h t h e f a c t t h a t J o = 0 n t h e s o l i d r o t o r

g i v e s :

AE = - - 2 ( 2 . 3 2 )

S u b s t i t u t i n g ( 2 . 32 ) f o r 5 a n d ( 2 . 8 ) f o r B i n ( 2 .2 9 ) g i v e s :

E x p a n d i n g t h e s eco nd te r m on t h e r i g h t ha nd s i d e o f t h i s e q u a t i o n

a nd e q u a t i n g t h e a x i a l c om po ne nt s g i v e s :

Figure 2 . 1 A section of t h e considered solid rotor.

E q ua ti on s ( 2 . 3 0 ) and ( 2 . 3 1 ) a r e u s e d , r e s p e c t i v e l y , f o r U x an d U inYt h e a bo ve e q u a t io n t o y i e l d :

The a b ov e e q u a t i o n i s u s ed f o r J in ( 2 . 1 3 ) . The r e s u l t i s a r e la t i o n -

s h i p which a p p l i e s t o e v e r y p o i n t i n t h e s ol i d r o t o r a s :

For t h e pu rp os e of f i n i t e e le m en t d e s c r e t i z a t i o n , i t i s c on v e ni en t t o

w r i t e o ne e q u a ti o n f ro m w hich e q u a ti o n s ( 2 . 2 6 ) , ( 2 . 2 7 ) , ( 2 . 2 8 )

and (2 .3 6 ) may be de r iv ed . Th i s equa t io n which i s named the un i f ie d

e q u at io n i s w r i t t e n a s :

T h e c o n s t a n t s 7 , '3 and y a r e g i ve n i n T a b le 2 . 1 . I n t h e ne x t s e c t i o n ,

G a l e r k i n ' s m ethod i s a pp l i e d t o t h e u n i f i e d e q u a t io n i n o r d e r t o

d e r i v e t h e e l e m e n t e q u a t i o n .

T a b l e 2 . 1

CONSTANTS r, 3 A N D - F O R DIFFERENT REGIONS

3F AN INDUCTION MACHINE

I/ s t a t o r S l o t

/ ~ i rap

/ S t a t o r I r on

! S ol i d R o t o r

2 .4 E lemen t Eau a t ion

The u n i f i e d e q u a t io n d e r i v e d i n t h e p r ev i o u s s e c t i o n i s c o n v e r te d

i n t o a s e t of a l g e b r a i c eq u a t i o n s u s in q G a l e r k i n ' s m eth od . T h i s i s

a m eans of o b t a i n i n g an a p pr o x im a t e s o l u t i o n t o a p a r t i a l d i f f e r e n t i a l

e q u a t i o n . I t d o es s o by r e q u i r i n g t h e e r r o r b etw ee n t h e a p p r o x im a t e

s o l u t i o n and t h e t r u e on e b e o r th o g o na l t o t h e i n t e r p o l a t i n g f u n c t i o n s

u s ed in t h e a p p r o x i m a t i o n . L e t A b e a n a p p r o x i m a t e s o l u t i o n t o ( 2 . 3 7 )

f o r a p o i n t w i t h in t h e r eg io n :, bounded by a co n t ou r a s in

F i g u r e 2 . 2. I f A i s s u b s t i t u t e d i n t o ( 2 . 3 7 ) , i t w i l l n o t, i n g e n e r a l ,

s a t i s f y t h e e q u a t i o n , a nd t h e f o l lo w i n g i s o b t ai n e d :

where R i s t h e r e s i d u al o r e r r o r . The s m a l l e r t h e R , t he more

a c c u r a t e l y A r e p r e s e n t s t h e M V P a t t h e c o r re sp o nd in g ? o i n t i n f. The

F i g u r e 2 .2 T w o - d i m e n s i o n a l r e g i o n 3, bounded by a c o n t o u r :.

g e n e r a l m e t ho d o f w e i g ht e d r e s i d u a l [38] may be u se d t o o b t a i n i o r

d i f f e r e n t p o i n t s w i t h i n t h e s o l u t i o n d omain . T h i s m et ho d r e q u i r e s

t h a t :

where :

W i s a w e ig h t i n g f u n c t i o n t o b e s p e c i f i e d s u b se q ue n tl y .

Hence,

T h i s i n t e g r a l w i t h t h e r e s o l u t i o n i n t ro d u ce d i n E q ua t i on ( B . 3 ) becomes:

w here :

n i s t h e o u tw a rd no rm al t o ? i n F i g u r e 2 .2 .

I n t h e s t u d i e s t o b e p r es en te d i n t h i s d i s s e r t a t i o n , t h e c o n t o u r 1i s s o ch ose n t h a t t h e c o n t r i b u t i o n o f t h e m a gn e ti c f i e 1 d due t o

c u r r e n t s o ur ce i n 2 i s n eg l i g i b l e b eyo nd T. T h i s , t o g e t h e r w i t h

( 2 . 8 ) , i n d i c a t e s t h a t :

I n a f i r s t -o rde r f i n i t e element approximation of ( 2 . J l ) , the

region : s divided i nto a number of tr ia nq ul ar elements, :,, as in

Figure 2.3. Over each element, A i s assumed t o vary 1 inear ly a n d i t s

values a t any poin t i s shown by ie . The weighting function \!4 f o r t h i s

element i s si mi la rl y shown by W e [ 39 ] . Furthermore, , nd : re assumed

constan t over each tr ia ng le . Yith these assumptions and (2 . 42 ), ( 2 . 4 1 )

nay be written after rearrangement as:

' dxdy=y >Y/

subscript "e" refers to any triangle within the domain.

M i s the to ta l number of t ri ang le s within the domain.

J i s the average current density in t r ia ng le "e" .Oe

u e Y Ad e ' - f e are the values of ;, S and v , respectively, in

t r i ang l e "en .

I n Appendix C , the assumption that A e var ies 1 inearly within each

tr ia ng le i s used t o derive the following expression:

F i g u r e 2 . 3 A r e g i o n 2 , d i v id e d i n t o t r i a n g u l a r e l e m e n t s

where:

A A . and f f k a r e t h e v a l u e s o f ie t no des i , j an d k .3

N i, N . and N a r e i n t e r p o l a t i o n ( o r sh ap e) f u n c t i o n s g i ve n by

N = - - ; ( a t b x + c y ) , f o r m = i , j and km 2 - m m m

where:

2 i s t h e a r e a o f t h e e l e m e n t , g i ve n by ( B . 5 ) .

a b a nd c m f o r e ac h n ode a r e s ome c o n s t a n t s d e f i n e d i n ( C . 1 2 )m y rn

t o ( C . 2 0 ) .

G a l e r k i n ' s neth hod i s m ade c o m p l e t e by s e l e c t i n s t h e w e i g h ti n g

f u n c t i o n We t o be N i , N . and N k one a t a t i n e [ 3 9] , [ a l l , a n d i n t e g r a t i n g3

t h e t er m s of e q u a t i o n ( 2 . 4 3 ) f o r e le m e n t " e " . These se l e c t i o n s may be

u se d t o d e f i n e :

In Appendix D , ( 2 . 4 4 ) and e v e r y row of ( 2 . 4 6 ) a r e s u b s t i t u t e d i n t o

( 2 . 4 3 ) and t h e i n t e g r a l s a r e e v al u a te d t o y i e l d f o r e l em e nt " e " :

where :

[ ge l i s a 3x3 m a t r i x , ( s e e 0 . 1 4 ) .

[ P e l i s a 3 x3 m at r i x, ( s e e 0 . 3 3 ) .

[Eel i s a 3x 1 colum n v e c t o r , ( s e e 0 . 1 7 ) .

A p p l i c a t i o n o f ( 2 . 4 7 ) t o e ac h o f t h e M e l e m e n t s i n :, a n d s u b s t i t u t i o n

o f t h e s e c o n t r i b u t i o n s i n t o ( 2 . 4 3 ) , y i e l d s t h e fo l l ow i n g g l o b al m a t r i x

e q u a t i o n :

where :

[ P G ] i s an nxn g l o b a l m a t r i x .

[ aG ] i s an nun g l o b a l m a t r i x .

[IG]nd [A] a r e n x l gl o b al v e c t o r s .-n i s t h e t o t a l num ber o f n o d es i n 1 and on ? .

Chapter 3

SOLUTION METHODOLOGY A N D COMPUTER PROGRAM

3.1 Introduction

The obje ctiv e of th is di ss er ta ti on i s to develop a method fo r

the ana lys is of a defe ctiv e induction machine. A global differential

equation was derived f o r the machine in Chapter 2. In this chapter ,

a method t o s olve th e equation numerically i s provided. Two computer

programs developed t o hand1 e a1 1 the computations involved and generate

the required resul t s are described.

I n Section 3.2, a step-by-step method used to solve the global

equation i s present ed. Having obtained the sol ut io n, Section 3 . 3

describes th ei r use in plott ing MVP contou rs. In Section 3 . 4 the

procedure used t o compute the c urre nts in th e s ta t or windings i s

presented. In the l a s t sec ti on , two flow cha rts, one f o r each computer

program developed in t h i s di ss er ta ti on , are presented. The functio n

of each block in the ch art i s described .

3.2 Solut ion of the Global Equation

The global equation was derived in Chapter 2 and in th is se ction,

a method used t o solve t h i s equation i s presented. The equat ion i s

reproduced here for convenience as:

T he ab o ve e q u a t i o n i s s o l v e d by c e n t r a l d i f f e r e n c e me th od [4i].

T h is i n v o l v e s e v a l u at i o n of t h e v e c t o r p o t e n t i a l and i t s d e r i v a t i v e ,

a s w ell a s t h e r i g h t ha nd s i d e of ( 3 . 1 ) a t t h e m i d p oi n t of t h e t im e

i n t e r v a l . The resul t i s :

Hence:

[ P G I n t l i n ( 3 . 3 ) i s a f u n c t i o n o f an d h e nc e d ep en ds on t h e v a l u e

of [&Intl . T h e r e fo r e , t h e fo l lo w i ng i t e r a t i v e o r o c e s s i s u se d t o

compute [!Intl a t each s t e p :

- [A],

K K t1( C Q G I / ~ t[ P G l n + l / 2 )A] +l

= ( LQ , 3 / i t - [ P G l n ) W n+ I n [ k l n ) / 2 ( 3 . 4 )

S o l u t i o n ( 3 . 4 ) i s t ak e n t o ha ve c on ve rg ed a t t h e k - t h i t e r a t i o n , i f :

wher e s i s a t o l e r a n c e . The n e x t s e c t i o n d e s c r i b e s how t h e s o l u t i o n

i s u s e d t o p l o t MVP c o n t o u r s .

3 . 3 C o n to u rs of t h e M a g ne t i c V e c t o r P o t e n t i a l s

S o l u t i o n of t h e g l o b a l e q u a t i o n g i v e s t h e v a l u e s o f t h e NVP f o r

a l l no de s o v e r t h e c r os s s e c t i o n o f t h e m ac hin e. The l o c u s o f t h e

p o i n t s w h i c h h av e a s p e c i f i c v a l u e o f MVP c an b e d e t e r m i n e d b y

i n t e r p o l a t i o n . P l o t t i n g t h e s e p o i n t s w h i ch ha ve t h e same MVP g i v e s

a c o n t ou r f o r t h a t v a l ue . C on to ur p l o t s p r o v id e a v e r y u s e f u l t o o l

t o u nd e rs ta nd t h e s t a t e of m ag n et ic c i r c u i t s r e g a rd i n g f l u x d e n s i t y

d i s t r i b u t i o n a nd s a t u r a t i o n , e s pe c ia l l y when the MVP v a l u e a s s o c i a t e d

w i t h a c o n to u r l i n e d i f f e r s f r o n t h e tw o a d j a c en t ones by a s p e c i f i c

v a l u e .

A r e l a t i o n s h i p b et we en t h e f l u x d e n s i t y an d c o n t o u r 1 n e s o f

MVP may b e s e en when t h e f o r m e r i s e x p r e s s e d i n t e r m s o f t h e MVP.

T h i s i s g i v e n i n (A .2 ) a s :

F o r e xa mp le , t h e a bo ve e q u a t i o n s u gg e s ts t h a t a t P, Q a n d R i n

'* - 0. F o r t h e s ei g u r e 3.1, t h e B i s p e r p e nd i c u l a r t o t h e x - a x i s , + -7Y

3Ap o i n t s , B = -;in d t h e f o l l o w i n g c o n c l u s i o n s c an b e made:

F i g u r e 3 . 1 C o n t o u r p l o t s f o r a m a g n e t i c c i r c u i t . Each pa i r of

1 n e s d i f f e r by a s p e c i f i c v a l ue o f W I P .

where Bp, B and B R a r e t h e f l u x d e n s i t i e s a t p o i n ts ? , 0 and R ,4

r e s p e c t i v e l y .

( b ) The m ate ria l medium a t Q i s m a g n e t ic a l l y m ore s a t u r a t e d

t ha n t h a t a t R .

Hence, t he con tou r 1 i nes imp1 i c i t l y r e p r e s e n t t h e f l ux d e n si t y

d i s t r i b u t i o n . C l o se r l i n e s s ug g es t h i g he r f l u x d e n s i t i e s i n t h e

r e g i o n . Contour p l o t s of MVPs a r e employed in C hap ter s 4 and 5 t o

d i s p la y t h e d i s t r i b u t i o n of f l u x d e n s i t i e s a nd t h e i r v a r i a t i o n s i n

t ime .

3 . 4 Computation o f Winding Currents

A m eth od t o i n t e g r a t e t h e g l o b a l e q u a t i o n i n t i m e a nd o b t a i n

a s o l u t i o n f o r t h e M V P was p resen ted in Sec t io n 3 . 2 . The so lu t ion

i s u se d i n t h i s s e c t i o n t o com pu te d i f f e r e n t t y p es of c u r r e n t i n a

s in g1 e t u r n c o i l .

A s i n g l e t u r n c o i l of t h e s t a t o r c a rr y in g a c u r r e n t , i , i s shown

i n F i g u r e 3 . 2 . The v o l t a g e e qu a ti o n f o r t h e c o i l i s :

where:

v ' i s t h e i n s t a n t a n e o u s v a l u e o f t h e v o l t a g e a c r o s s t h e c o n si d er e d

r ' i s t h e r e s i s t a n c e of t h e turn.

$ i s t h e f l u x w hich l i n k s t h e turn.

P f Side ionductar

Endconnect0 r

Figure 3 . 2 A single turn c o i l .

S o l u t i o n o f t h e g l o b a l e q u a t i o n g i v e s t h e YVPs a t e v e r y no de i n t h e

s o l u t i o n d omain . I n o r d e r t o comp ute t h e c u r r e n t i n e ve r y w i n d in g

f r o m t h e r e s u l t s , i t i s n e ce ss ar y t o e x pr es s t h e f l u x 2 i n ( 3 . 7 ) i n

t e rm s of t h e s e MVPs. T h i s i s a c h i e v e d b y i n t e g r a t i n g b o t h s i d e s of

( 2 . 8 ) o v e r t h e s u r f a c e S o f t h e s i n g l e t u r n c o i l i n F i g u r e 3 .2 :

A p p l y i n g (B.4) o t h e a bo ve e q u a t i o n g i v e s :

w he re t h e c o n t o u r i n t e g r a l on t h e r i g h t ha nd s i d e o f ( 3 .9 ) i s t a k e n

a r o u n d t h e c u r v e C , b o u n d i n g t h e s u r f a c e S .

The i n t e g r a l f o r t h e en d c o nn e c t o r a t one end o f t h e t u r n c a n ce l s

t h a t a t t h e o t h e r end. A ss um pt io n ( a ) o f S e c t i o n 2 .3 i s u se d t o

e v a l u a t e t h e i n t e g r a l a l o n g t h e r e m a in i n q tw o s i d e c o n d u ct o r s of t h e

t u r n . H en ce , ( 3 . 9 ) r o u n d t h e c o n t o u r C i n F i g u r e 3 . 2 g i v e s :

where:

i i s t h e a x i a l l e n g t h o f t h e t u r n .

A1 and A 2 a r e t h e MVPs a t t h e t w o s i d e s o f t h e c o i l .

E q u a t i o n ( 3 . 1 0 ) i s us ed i n ( 3 . 7 ) t o g i v e t h e v o l t a g e a c ro s s t h e s i n g l e

t u r n c o i l a s :

The above equation i s w r i t t e n f o r e ver y s ing1 e tu r n c o i l c onnec te d

i n s e r i e s i n a pha se and the to t a l w inding c u r r e n t i i s ob ta ine d a s :

where:

i 0 i s t he a ppl i e d c u r r e n t i n th e w ind ing .

v i s t he in s t a n ta ne ous v o l t a ge a c r os s th e tu r ns c onne cted in

s e r i e s .

r i s t h e t o t a l r e s i s t a n c e of t h e t u r n s c on n ec te d in s e r i e s .

ind ic a t e s a sum mation f o r a l l s in g le tu r n c o i l s c onnec te de

i n s e r i e s t o make one phase.

A j , A k a r e th e MVPs a t the two si d e s of any of the co il s connected

i n s e r i e s .

The second term of th e ri g h t hand si d e of ( 3 . 1 2 ) i s t he induc e d

c u r r e n t . I n s i m u l at i o n s t u d i e s , t h r e e c u r r e n t s , namely t h e t o t a l

c u r r e n t i , t h e a pp l i ed c u r r e n t i 0 and t h e i n du ce d c u r r e n t a r e p l o t t e d

to show t h e i r va r i a t io ns wi th t ime . The procedures desc r ibed in th is

and previous sect ions are implemented in the second computer program

d e s c r i b e d i n t h e n e x t s e c t i o n .

3 . 5 A D es cr i p t i on of th e Computer Programs

Two computer programs developed in t h i s work and used t o ob ta in

th e MVPs w i th in an i n d u c t io n m a c hine a r e d e s c r ib e d i n t h i s s e c t i o n .

The f i r s t one i s a prog ram w hich d i v i d e s t h e c r o s s s e c t i o n of t h e

i n d u c t i o n m ac hin e i n t o t r i a n g l e s w it h s p e c i f i c p r o p e r t i e s which a r e

g i v e n s u b s e q u e n t l y . Th is program i s employed i n Ch apter 4 t o

d e s c r e t i z e t h e c r o s s s e c t i o n of a n i n d u c t io n m a ch in e u se d i n s im u la -

t i o n s t u d i e s . O th e r p ro gr am s a l s o d e v elo pe d an d u s ed t o d e s c r e t i z e

re c t an gu la r domains fo r some examples in Chap te r 4 a r e n o t d e s c r i b e d .

T he s ec on d pro gra m d e s c r ib e d i n t h i s s e c t i o n u t i l i z e s t h e m eth od

o f S e c t i o n 3 . 2 t o i n t e g r a t e t h e q l o b al e q u a t i o n i n t i m e . T he o u t p u t

o f t he p rog ram i s mainly th e MVP a t d i f f e r e n t nodes f o r d i f f e r e n t

t im e s t e p s . T he pro gra m a1 s o em plo ys t h e d e r iv a t i o n s p r e s e n t e d i n

S e c t i o n 3 . 4 t o c om pute t h e c u r r e n t s i n t h e s t a t o r w i n di n gs .

3 . 5 . 1 D es cr e t iz a t i on P rog ram, MESHGEN

The mesh generation program, MESHGEN d e s c r i b e d h e r e i s

devel oped with th e fo l 1 o w in g p r o p e r t i e s :

( a ) The e l e me nts g e n e r a t e d a r e a c u t e a n g le d t r i a n g l e s , b e c au s e

e l e m e n t s c l o s e t o an e q u i l a t e r a l s h a pe p ro d uc e m ore a c c u r a t e r e s u l t s

( b ) I f t h e r e g i o n ha s c ur ve d b o u n d a r i e s , t h e s i d e s of t h e

e l e m e n t s a lo n g t h e b o un da ry a r e assum ed to a p p r o x im a te ly r e p r e s e n t

t h e c u r v a t u r e .

( c ) The boundar ies between any two regions of d i f f e r e n t m a t e r i al

p r o p e r t i e s a r e made t o c o i n c i d e w i th t h e s i d e s of t h e e l e m e n t s .

A flowchart of program MESHGEN i s shown in Figure 3 . 3 . A complete

s e t of data describ ing th e domain t o be des cre ti zed i s read. They

are: radius of the rot or, sl ot opening, s lo t angles, sl o t depth,

st at o r inner and outer ra d ii , number of s l ot s and material prop erti es

of d i f f e r en t subdomains of the region. The domain i s divided int o

sectors as in Figure 3 . 4 . Each sect or i s divided in to quadr ila ter als .

These quadrilaterals are subdivided into acute angled triangles

according to ( a ) above. For example, the program will subdivide

quadri la teral Q , Figure 3 . 4 , in the form shown by case A , a n d not as

that in case B . Element numbers, node numbers a n d material properties

are assigned and saved in a n array. If a l l quadri la terals within

one sector are considered, the program checks whether all the sectors

have been considered. The above process i s repeated unt il every

se ct or has been considered a f t e r which execution termina tes. The

next s ec ti on descr ibe s a flowchart of the second computer program

used to int egr ate th e global equation.

3 . 5 . 2 Main Program, FETIME

A flowchar t of the main program, FETIME, i s shown in

Figure 3 . 5 . I t reads element da ta generated f o r th e domain by program

M E S H G E N . Based on the data, the program sets u p pointers for diagonal

elements of the matr ices in the global equat ion. This process

exp loi ts th e sparse nature of th e global matrix by assigning s tora ge

only for non-zero elements above the diagonal. The program estimates

the required storage. If the memory i s not s u f f i c i en t , t he program

I1 Read ciomain d a t a : r o t o r r a d i u s , s t a t o ro u t e r and i n n e r r a d i i , s l o t o p en in g, s l o td e p t h , n um ber o f s l o t s , an d m a t e r i a lp r o p e r t i e s .I

1D i vi de d or a in i n t o s e c t o r s . I

I D iv id e a s e c t o r i n t o a u a d r i l a t e r a l s .

D i vi de a q u a d r i l a t e r a l i n t o a c u t e a n g le dt r i a n q u l a r e l e m en ts .i

umber each e leme nt and nodes of t h e e l en e n t .

C o n s i d e r a n o t h e rq u a d r i l a t e r a l

F i g u r e 3 . 3 A fl o w c h a rt of proqrarnPlESPGEM.

Ass ign m a te r i a l p r ope r tyto e ac h e l e m e n t .

Save element number, nodenumbers a n d ma t e r i a1p r o p e rt y i n an a r r a y .

C ons ide ra n o t h e rs e c t o r

i I Aw

Save element numbers,node numbers and materialproper ty on a d i s c

F igure 3 . 3 ( c onc lude d)

F i g u r e 3 . 4 A s e c t i o n o f a n i n d u c t i o n m ac h i ne s ho wi n g j e c t o r s a nd

q u a d r i l a t e r a l s . Q u a d r i l a t e r a l Q i s s u b - d i v i d e d i n t o

t r i a n g l e s i n t h e f o r r show n b y c a se A a b o v e . E a s e S

i s a v o i d e d b y t h e p r o q r a n .

F i g u r e 3 . 5 (concluded)

t e r m i n a t e s a u t o m a t i c a l 1 y . O t h e r w i s e , t h e p r og r am commences t o i n t e g r a t e

t h e g l o b a l e q u a t i o n .

KThe NVPs, A a r e u se d t o u p da t e t h e g l o b a l m a t r i x e q u a t i o n f o r

t h e k - t h i t e r a t i o n and n - t h t i m e s te p . T h i s e q ua t io n i s s e t u p

a cc o r d i ng t o ( 3 . 4 ) a f t e r t h e m a t r i x [ Q G ] , [ PG ] a n d t h e co l um n v e c t o r

[k] r e a ss em bl ed f r o m ( 0 . 1 4 ) , ( 0 . 3 3 ) a nd ( 0 . 1 7 ) . S o l u t i o n o f

( 3 . 4 ) i s a c co m p li sh e d u s i n g l o w e r a nd u p pe r ( L U ) d e c o m p o s i t i o n o f

t h e l e f t h an d s i d e o f t h e e q u a t i o n a nd s ub se qu en t b ac kw ar d s u b s t i t u -

t i o n . S o l u t i o n a t each i t e r a t i o n i s c he ck ed f o r c o nv er ge nc e a c c o rd i n g

t o ( 3 .5 ) . I f c on ve rg en ce i s a ch ie ve d, t h e n e x t t i m e s t e p i s c o n s i d er e d

and t h e above p r oc es s i s r e p e at e d u n t i l t h e s p e c i f i e d t i m e i n t e r v a l

i s e x ha u st e d.

A t t h e e nd o f t h e t i m e i n t e r v a l , t h e p r og ra m c om pu te s t h e

w i n d i n g c u r r e n t a s d e s c r i b e d i n S e c t i o n 3 . 4. I t u s e s e q u a t i o n ( 3 . 1 2 )

t o c om pute t h e t o t a l c u r r e n t w h i l e u s i n g t h e second q u a n t i t y o n t h e

r i g h t h an d s i d e t o com pute t h e i n du c ed c u r r e n t . The a p p l i e d c u r r e n t

i s com puted f r o m (3 . 1 3 ) . Each o f t h e t h r e e c u r r e n t s i s p l o t t e d a g a i n s t

t i m e u s i n g a s e p a ra t e p l o t t i n g p ro gra m.

4 . 2 V a l i d a t i o n o f t h e P ro gram f o r a L i n e a r C ase

In t h i s s e c t i o n , t h e c lo s e d form s o l u t i o n f o r a 1 i n e a r p a r t i a l

d i f f e r e n t i a l e q ua ti on i s us ed t o v a l i d a t e a p or t i o n o f t h e p r o g r a m .

The c o n s i d e r e d p a r t i a l d i f f e r e n t i a l e q u a t i o n i s shown below a n d d e s c r i b e s

t h e t e m p e r a tu r e d i s t r i b u t i o n T , i n a r e c t a n g u l a r d om ain :

S u b j e c t t o t h e b o u n d a r y c o n d i t i o n s :

a n d i n i t i a l t e m p e r a t u r e :

where :

K x an d K a r e t he rm a l c o n d u c t i v i t i e s t a k en t o be 1 . 2 5 .Y

L, and L a r e t h e l e n g t h an d w i dt h of t h e s o l u t i o n do main b o thYt a ke n t o b e 3 m .

The s o l u t i o n o f ( 4 . 1 ) t h ro u g h ( 4 . 3 ) o v e r t h e r e g i o n shown i n

F i g ur e 4 . 1 i s d e s i re d . A n a l y t ic a l s o l u t i o n t o t h e problem i s 14 21 :

where :

F i g u r e 4 . 1 T he f i n i t e e l em e nt mesh f o r t h e e xa mp le c o n s i d e r e d i n

t h i s s e c t i o n . I t c o n s i s t s o f 20 0 t r i a n g l e s w i t h 12 1

n od es . D i me n si o ns a r e g i v e n i n m e t e r s .

- h C - - h - - I

c o m m h m ~ m a m m a m m n o r - m i n cc c m ~ w m a m w a m m mu , N N hh = ca 0 e - m o b e m m m m m m u a m m -- CC , 1 h

d o d o o ~ o ~ o ~ o Q d ~ o ~ ~o 4 3 w

1- - - - - - - - - - -0 0 h m m a L O N a m m m m o -3CU -?a h n = to c m ~ m ~ h a N - h a 3 - Q W C U N CUN c oo c m m w w m m 0 0 0 0 c c c o c . o a ~ m = C c. . . . . . . . . . . . . . . . . . .0 0 0 0 0 0 0 0 - 4 - 3 3 - a 0 o c c c c o cJ- - - - - - - - - - -- - - - - - - - - - -

0 0 N m OCU N IO 0 - C Q U U N U 2 U C U 0 0 3 Cc o a m h w O m d m m a o m m a : S G m m = c0 0 q q m m N - a m a - 3 e m e -. . . . . . . . . . ? yqc: C 3. .0 0 0 0 0 0 -- - - - 3 - 3 e - o = C C c o

- - - - - - - - - - -- - - - - - - - - - A

o c o m w m 0 - a m m m ~ m -3 - m n n m c cc c e m N - - m m n n~ m m COI - - m m o ca 0 m m 0 0 T O W W hh w a a m 3 0 m m S Z. . . . . . . . . . . . . . . . . . . . . .0 0 0 0 4- 4- - - - h e - -+ 4 - a 0 c c- - - - - - - - - - -- - - - - - - - - - -

c o m o m m o w m m O N m m o a m ' n m c D C

0 0 -10 h w m w O N N - mew C C ~ a 9 0 = C0 0 LDm 0 0 a a P s h =a 2 m h n a 3 C ) m m C =. . . . . . . . . . . . . . . . . . . . . .0 0 0 0 - + 3 - - - - c 3 - 4 3 - - 0 c a c- - - - - - - - - - -

- - - - - - - - - - . -C C LDO 0 3 0 - Q ' N m m N '.Om C U a -o c n m - - a m m m m o m m - 3 r u - 'S2 G Zo c m m c c s rm Q W hh 9'.0 c m o c m u , C =. . . . . . . . . . . . . . . . . . . . . .0 0 0 0 3 4 - 4 - 4 - e - 3 4 - - - o t =,7- - - - - - - - - - -- - - - A n . - - . - - - . --

3 3-7 N C D C U - a - = =8 8 % % Z Z 2 g S Z = G - c c. r 4 4 G i T c z0 0 a - 3 QOq - - 'f- r i c T c^ . N e : : C ' -- = z. . . . . . . . . . . . . . . . . . . . .0 0 0 0 3 0 3 -

- -- 4 3, c- S C c = = S

- - - - - - - - - - -C ~ h ~ L D ~ - 3 ~ m ~ O ~ a ~ W ~?zh%=z0 0 N N N N W W He h ' . O iW - 2 % F N OIU = =c o m m ~ w m m t o o c o o -.- . o m m c 2.7

d d 6 6 6 6 6 6 - 2 & 6 = & & &- - - - - - - - - - -m e - - - - - - - - . -

c o d m h m m m m n u,o o n cd i? hn m n 5,-C C hh @JN O m m m '.OG c rm a m m ru h h - _

o c - - m m -.J m s o m m m err m n - - c s

d d d d d d d d 46 j& & d j 66 &='- - - - - - - - - - -

Table 4.1 shows the values of T a t t = 1 . 2 h r f o r d i f f e r e n t nodes

shown in Figu re 4. 1. The number enclo sed in pare nt he sis a r e obta ined

from e qua t ions ( 4 .4 ) and ( 4 . 5 ) , w h i l e t hose w i thou t a r e p roduce d by

th e program. The res ul t s compare fa vo ra bl y. The maximum dev ia ti on

from t h e a n a l y t i c a l r e s u l t i s 3.4 6 p e r c e n t .

4 .3 P rogram Re sul t s f o r an Ind uc to r Enc losed by I ron Core

I n t h i s s e c t io n , t he c ompu te r progra m i s u sed t o produce contours

f o r two d i f f e r e n t e n c lo s u r e s c o n s id e r e d f o r a 9 6 - t u r n c o i l . The t o t a l

r e s i s t a n c e o f t he c o i l i s 0 .862 ohms. The f i n i t e e le me n t mesh f o r

these two cases are shown in Figures 4.2 and 4.4. T he c on tou r p lo t s

of t h e MVPs f o r t h e s e c o n f i g u r a t i o n s a r e g iv e n i n F i g u r es 4 . 3 and

4 . 5 , r e s p e c t i v e l y .

T he c on tou r p lo t s i n F igu r e 4 .3 a r e p roduce d f o r an a pp l i e d

c o i l v o l t a g e of v = 50 "7S in (,t + 9), with 3 = 7 / 2 . T he se ind ic a t e

a r ea sonab l e pa t t e r n o f f l u x d i s t r i b u t io n f o r t he symme tr ic al 1y p1 aced

c o i l .When v = 10 Y?? S in ( w t + 3 ) , with E=7.;/12, i s a p pl ie d t o t h e

ind uc t or shown in F igu re 4 . 4 , th e conto urs a r e changed t o t h a t shown

in F igure 4 .5 . This new pa t t e r n seems t o have a r easo nable dev ia t io n

from th a t i n F igu r e 4 .3 due t o t he change in t he po s i t i o n o f t h e c o i l .

The compute r p rogram has been appl ied in th is sec t ion to two

s imp le ma gne t ic c i r c u i t s f o r w hich e l e c t r i c a l e ng ine e r s ha ve a good

f e e l i n g o f t h e f l u x d e n s i t y d i s t r i b u t i o n s . R e s u l t s p ro du ce d by th e

F i g u r e 4 .2 F i n i t e e le m en t mesh when t h e c o i l i s c e n t r a l l y p l a c e d

i n t h e c o r e . Mesh c o n s i s t s o f 9 50 el em e nt s w i t h 520

n od es . C o i l l o c a t i o n i s shaded. D im e ns io ns a r e g i v e n

i n c e n t im e te rs .

F i g u r e 4.4 F i n i t e e le m e n t mesh when t h e c o r e i s n o t s y m m e tr ic . I t

h a s 1 14 0 e l em e n ts w i t h 6 20 n od es . C o i l l o c a t i o n i s

shaded. D im e ns io ns a r e g i v e n i n c e n t i m e t e r s .

F i g u re 4 . 5 MVP c o n to u rs f o r t h e i n d u c t o r w i th u n s y m e t r i c i r o n c o r e .- 5 1 1

Contour 1 i n e s i n 10 IJb/m. i? t = - - .

24 60to = 0 . 3 s .

p ro g ra m h a ve n o t be en o u t o f e x p e c t a t i o n s , t h e r e b y s u p p o r t i n q t h e

v a l i d i t y o f t h e p ro gra m t o t h e e x t e n t e xa min ed . I n t h e n e x t s e c ti on ,

t h e p ro gra m i s a p p l i e d t o a m a gn e ti c c i r c u i t w i t h a n a i r gap. R e s u l t s

g e n e r a t e d b y t h e p ro g ra m a r e co mpa red w i t h a n a l y t i c a l r e s u l t s .

4 . 4 A M a gn e ti c C i r c u i t w i t h an A i r Gap

A f i n i t e e le me nt mesh f o r t h e m ag n et ic c i r c u i t w i t h a n a i r gap

c o n s id e r e d i n t h i s s e c t i o n i s shown i n F i g u r e 4 .6 . The c o i l em plo ye d

i s t h e same o ne u se d i n t h e p r e v i o u s s e c t i o n .

F i g u r e 4 . 7 g i v e s c o n t o u r p l o t s o f t h e MVPs f o r h a l f a c y c l e when

v = 4 0 "7 S i n ( w t + 8) i s a pp l i e d t o t h e c o i l . The c u r v a t u r e o f t h e

c o n t o u r 1 n e s a r ou nd t h e a i r gap i n d i c a t e s t h e f r i n g i n g e f f e c t .

Append i x E g i v e s a n a pp ro x i ma te c l o se d f o rm s o l u t i o n f o r t h e

m a gn e ti c c i r c u i t when t h e a p p l i e d v o l t a g e i s :

v = \Im S i n + i)

T h e f l u x 3 i n t h e c o i l f rom ( E . 1 3 ) i s :

Vm(o Cos O - 3 S i n 3 ) - 3 tN ( s 2 + 3 2 )

"m+ N(WZtp2)

CB S i n ( w t + 6) - 9 Cos (w t + 9 ) ;

where:

Figure4 . 6

F in i t e e le m en t mesh f o r t he m a gne ti c c i r c u i t c onside r edin t h i s s e c t i o n . I t has 760 elements and 420 nodes.

Coil i s shaded . Dimensions a r e g iven in cent im ete r s .

i= t +ZA t i I0

r'/ -,

,,/ * I

t = t Q + 4 p t i t = t 0 + 5 1 t I

u r e 4 . 7 C o n t o u r p l o t s f o r t h e m a g n e t i c c i r c u i t w i t h a n a i r g a o .1 1

Contour 1 i n e s i n Ub/m. I t = - - . to = 0 s .24 60

Figure 4 . 7 ( c o n c l u d e d ) .

m0 i s t he f l ux a t t = 0 .

R i s t h e r e s i s t an ce of t h e c o i l .

N i s the number of tu rns in the co i l .

Ac i s the area of the ferromagnetic core.

ec i s the magnetic path leng th through th e core.

g i s the l eng th of the a i r gap.

p i s the permeabili ty of the core.

is the permeabili ty of free space.

A i s t h e e f f ec t i v e a r ea o f t h e a i r gap.g

Figure 4 .8 shows the variation of flux in the core of the magnetic

circuit shown in Figure 4.6 when v = 40 fl s in ( w t t 8) i s ap pl ied

t o t h e c o i l . One of t he cu rves shown i s obt ain ed by th e use of

equation ( 4 . 7 ) . The othe r one i s produced by the f i n i t e element-

based computer program developed in this work. The closeness of

these curves support the soundness of the computer program. The

small dif fe ren ce s seen in the peaks a re due to the approximations

allowed in Appendix E t o der ive ( 4 . 7 ) .

I n Figure 4 .9 , f lux dens i t ies are p lot ted agains t t ime for two

elements in the iron core. These p l o t s , which a re produced by th e

program show initial transients as suggested by ( 4 . 7 ) . This may

ind icat e t ha t the program i s sui tab1 e f or t ra ns ien t as well as s teady

s t a t e s im u l at io n s . I n

the next sect ion, the program is applied t oa non-defective induction machine.

F i g u r e4 .8

The v a r i a t i o n o f t h ef l u x

e s t a b l i s h e d i n t h e c o i l f o r

t h e m ag ne t i c c i r c u i t i n F i g u r e 4 . 6 . T h e a p o r o x i m a t e

p l o t i s g e n e r a t e d f ro m e q u a t i o n ( 4 . 7 ) . The o t h e r o l o t

i s p r o d uc e d b y t h e p r o gr a m.

F i g u r e 4 . 9 V a r i a t i o n s of f l u x d e n s i t i e s f o r two el em e n ts i n t h e

i r o n c o r e of t h e m a g n e t ic c i r c u i t show n i n F i g u r e 4 . 6 .

4 . 3 S i m u l a t i o n o f a S ol i d R o t o r I n d u c t i o n W ot or w i t h >lo D e f e c t

I n t h i s s e c t i o n , a n on - de f ec t iv e s o l i d r o t o r i n d u c t i o n mo to r i s

s i m u l a t e d . D a ta f o r t h e m a ch i ne t a k e n f r om 0 ' K e l l y [ 4 5 ] i s g i v e n

i n T a b l e 4 . 2 . F i g u r e 4 . 1 0 shows a c r o s s s e c t i o n o f t h e ma c hi ne

d i s c r e t i z e d i n t o f i n i t e e l em e nt s . A c i r c u i t d i a g ra m of t h e s t a t o r

w i n d in g s i s shown i n F i g u r e 4 . 1 1 .

T a b l e 4 . 2

DATA FOR THE SOLID ROTOR INDUCTION MACHINE

S t a t o r R o t o r

P h a s e s

S l o t s

C o n d u c t o r s

R a t e d C u r r e n t

C o r e L e n g t h , mA i r G a p, rn

R a d i u s , m

S l o t W id th , rnS l o t D e p t h , rnToo th Wid th , rn

ID i a m e t e r , m 1 0 . 0 5 3 5

A x i a l L e n g t h , m 1 0 .09533

IC o n d u c t i v i t y , S ie m en si m 4 . 6 9 6 8 x l o 6

Figure 4 . 1 0 Fin i t e e l ement mesh of the cons ide red so l id ro to r

i n d u c t i o n m o to r. I t c o n s i s t s of 1 4 2 1 elements with 7 5 1

nodes.

Coil BC2

Iw-Coil BC1

F igur e 4 .1 1 C ir c u i t d i a g r a m o f t h e s t a t o r c o i l s of a n o n - d e f e c t i v e

induc t ion machine .

The co n d u c to r s o f ea ch p ha se a r e d i s t r i b u t e d i n s l o t s t o g i v e a

s i n u s o i d a l v a r i a t i o n o f t h e m a g ne ti c f i e l d f o r a b al an ce d t h r e e -

p has e c u r r e n t i n t h e s t a t o r w in d in g . I n o r d e r to a c h i e ve t h i s , t h e

fo l lo win g e q u a t i o n s u g g e s t e d by S l emon a n d S t ra u g h e n [4 4 ] i s u se d

t o c om pu te t h e f r a c t i o n o f t h e n um ber o f c o n d u c t o r s i n t h e q - t h s l o t

o v e r t h e t o t a l num ber o f t u r n s o f o ne o f t h e t wo c o i l s i n p h a s e AB:

Cos 3 d 5

S i s t h e t o t a l number o f s l o t s .

n i s t h e f r a c t i o n o f t h e n um ber of c o n d u c t o r s i n t h e q - t h s l o t9

b e lo n g in g t o c o i l A B 1 o r A B 2 o v e r t h e t o t a l num ber o f t u r n s

i n t h e sam e c o i 1 .

To co mp ute t h e num ber o f c o n d u c t o r s o f c ~ i l s C , , BC2, CA I an d C A 2 i n*

t h e q - t h s l o t , 3 i n ( 4 . 9 ) i s r e p l ac e d by ( ? - 2 71 3) f o r t h e f i r s t two

c o i l s a n d ( 9 + 2 ~ 1 3 ) o r th e se co nd two c o i l s . A p p l i ca t io n of ( 4 . 9 )

t o a 2 4 - s l o t n o n - d e f e c t i v e i n d u c t i o n m ac hi ne g i v e s t h e n f o r d i f f e r e n t9

p h a s e s a s sh own in T a b le 4 .3 .

T he m a c h in e d e s c r i b e d a b o v e , w i t h t h e n g iv en i n T a b l e 4 . 3 ,9

i s s u b j e c t e d t o a b a la n ce d t h r e e - p h a s e v o l t a g e o f 60 Hz and 63.5 V

l i n e t o l i n e . W hile t h e r o t o r i s k ep t s t a t i o n a r y , t h e M'IP c o n t o u r

p l o t s p ro du ce d f o r h a1 f a c y c l e o f t h e a p p l i e d v o l t a g e a r e show n i n

F i g u r e 4 . 1 2 . As e x p e c t e d , t h e c o n t o u r s p r od u c ed by t h e pr og ra m a r e

t= o t = t O + L t

Figure 4 . 1 2 Rotating m agnetic f i e l d of an ind uct ion machine with1 1

no d ef ec t . Contour 1 i es in Ub/m. A t =* -

4 60 "t s = 3 s .

t = t o + 2 C t t = t 0 + 3 L t

t = t o + 5 A t t = t o + 6 C t

r o t a t i n g a t t h e a n g ul a r v e l o c i t y o f 1 2 0 7 r a d i a n s p e r s e c o n d . T h es e

c o n t o u r s a r e t a k en a f t e r t h r e e s e c on d s when t h e t r a n s i e n t s h ave d i e d

V a r i a t i o n s o f t h e s t a t o r c u r r e n t s w i th t i m e a r e shown i nF i g u r e 4 . 1 3 f o r e ac h o f t h e t h r e e p h a s e s . T h es e c u r r e n t s ha ve e q u al

a m p l i t u d e s a s e xp e c t ed f o r t h i s c a s e w he re no d e f e c t h as o c c u r r e d .

I n t h e n e x t c h a p t e r , t h i s m a ch in e i s s i m u l a t e d f o r v a r i o u s s t a t o r

w i n di n g d e f e c t s . R esul t i ng p h as e c u r r e n t s a n d MVP c o n t o u r o l o t s a r e

c o m p a r e d w i t h t h o s e g i v e n i n t h i s s e c t i o n .

F i g u r e 4.13 Phase c u r r e n t s o f a n o n - d e f e c t i v e i n d u c t i o n m a ch in e.

Chapter 5

SIMULATION RESULTS FOR A DEFECT IVE SOL ID ROTOR INDUCTION MOTOR

5 . 1 Introduction

To develop a method f o r the ana ly si s of de fe ct iv e induction

motors, a unifi ed equation relat ing the M V P a t every point of the

machine t o th e applied cur ren t was derived in Chapter 2 . A computer

program u t i l izi ng the numerical methods de scr ibed in Chapter 3 i s

developed. The program was used to compute th e fl ux d is t r ibut ion

f o r di ff er en t systems in Chapter 4. These res ul ts support th at the

program works properly. The program i s used in th i s chapte r to

compute the waveform of th e cu rre nt in every phase f o r a sol id ro to r

induction motor with di ff er en t st at or winding de fe ct s. Contour pl ot s

a t many time step s ar e plot ted f o r each d efe ct.

The initial MVP value f o r al l nodes i s taken to be zero. This

produces a tra ns ien t in the values computed f o r M'JPs. The st ar t in g

time of the cycle over which MV P contours are plo tte d i s chosen t o

be th re e seconds so t h a t al l tr an si en ts have damped out.

I n Section 5 . 2 , one of t he two pa ra ll el c o i l s of Phase AB shown

in Figure 4 . 1 1 i s considered disconnected. In Section 5.3, f i f t y

percent of one of the two parallel coils in phase AB i s considered

bridged over. The def ect simulated in Section 5.4 i s a disconnection

of two of the th ree phase windings.

5.2 Disconnect ion of One of the Two Paral le l Coil s in Phase AB

The machine considered in t h i s sec tio n i s the same as th at

described in Section 4.5. Figure 5.1 shows the stator circuit of

th e machine when the def ec t i s th e disconnect ion of one of th e two

para1 1el coils in phase A B .

This i s modelled by se tt in g n f o r the disconnected coil equal9

to zero as given in Table 5.1. The remaining n ' s and i t s counter-9

par ts f o r th e oth er phases ar e computed from ( 4. 9 ) .

A voltage of 63.5 V rms, 60 Hz i s appl ied to t h i s defect ive

machine. St ar ti ng from 3 s , contour pl ot s of MVPs are shown in

Figure 5.2.

I n th is simulation, the rotor i s assumed to be sta tio nar y.

However, u i s changed according t o th e value of B a t any po int and

a t every time step . I n contrast with those shown in Figure 4.12 f o r

a non-defective machine, the contour plots here show that the rotating

magnetic fie1 d i s non-symmetric. The va ri at io n of the cur re nt s in

each phase i s shown in Figure 5.3. This shows unequal ampli tudes fo r

the to ta l cur ren t in each phase. As expected, t hi s i s in con tra st

with the results shown in Figure 4.13 when the machine has no de fect .

5.3 Short Circuit of Some Turns of Phase AB

A de fe ct in t he s t a to r winding which occurs when a por tio n of

the winding of one coi l i s bridged over i s simulated in th i s s ect ion .

For th i s example, f i f t y percent of coil AB1 i s assumed to be bridged

over. This condition i s il lu st ra te d in Figure 5.4.

I C o i l BC2

y--/ S*

IwIo i l BC 1

F i g u r e 5 . 1 D e f e c t i n phase AB, w h e r e c o i l AB1 i s d i sc o nn e c t e d.

T a b l e 5 . 1

T H E V A L U E S O F n F O R E A CH P H A S E W HE N C O I L A B 1 I S D I S C O N N E C T E Dq

S L O T P H A S E A B P H A S E B C 1 P H A S E C A

Figure 5.2 Magnetic field of an induction machine when one of the

two parallel coils in phase AB is disconnected. Contour1 1

lines in lov4 Wb/rn. Lt = * s to = 3s.

F i g u r e 5 . 2 ( c o n c l u d e d ) .

Legend

0 BlU s u 1 m r

a 4 suelotr ;

Legend

IMWCCD C U R R t R T

10141 CU l I D l

a 4mnn N R ~ ~ U

Fi g u re 5 . 3 Wavefom of t he cu rr e n t in each phase of an ind uc tio n

machine when one of th e two pa ra l l e l c o i l s of phase AB

i s d i s c o n n e c t e d .

I n order t o model th is de fe ct , the n ' s in s l o t s 1 t o 1 2 of4phase AB shown in Table 4 . 3 are reduced by f i f t y per cent . The new

n ' s are given in Table 5.2 . Contour pl ots of the MVPs fo r an9

applied l i ne voltage of 63.5 V rms, 60 Hz are given in Figure 5 . 5 .

The resulting currents in each phase are shown in Figure 5.6.

5.4 Disconnection of Two Phases

The de fec t considered in t hi s s ect ion i s the disconnection of two

of the thr ee phase windings. Figure 5.7 shows the st a t o r c i r c u i t when

phases A6 and BC are disconnected.

The def ec t i s modelled by s e tt i n g the n ' s of phases A6 and BC fo r4every s l o t to zero as in Table 5.3. Application of 63.5 V rms,

60 Hz three-phase voltage r es ul ts in the contour plo ts shown in

Figure 5 .8 . These pl ots show, as exp ected, t ha t the magnetic fi el ds

have symmetry about the s ta ti o na ry axi s of t he phase CA coil. Although

the amp1 it ud e of th e magnetic f ie ld ar e changing in time, th e f i e l d

patt ern does not ro ta te in space. Figure 5.9 shows the va ri at io n

of current in phase C A , with time.

T a b l e 5 .2

THE VALUES OF n q FOR EACH PHASE WHEN FIFTY PERCENT

OF ONE PARALLEL COIL OF PHASE AB I S E R I D GE D OVER

PHASE AB PHASE BC PHASE CA

t=to+4At t=t0+5AtL J

Figure 5 . 5 Contour p lo ts fo r an induc tion machine when f i f t y percent

of one of t he two pa ra ll el c o i l s of phase A B is bridged1 1

over. Contour li ne s ar e in Ub/m. \ t = - * - .to = 3s.

Figure 5 . 5 ( c o n c l u d e d ) .

Fi g u re 5 .6 Var ia t ion o f cu r ren t in each phase o f an i n d u c t i o n

mach ine when f i f t y perc en t o f on e o f the two para l l e l

c o i l s of p h ase AB i s b r id g ed o v e r . .

disconnec

I-' :disconnected

Coil BC1

F igu r e 5 . 7 Diconnection of phases AB and 8 C .

T a b l e 5 . 3

T H E V A L U E S O F n q F O R E A C H P H A S E WHEN P H A S E S A B A N D B C

A R E D I S C O N N E C T E D

SLOT 1I P H A S E A B P H A S E B C 1 P H A S E C A

Figure 5.3 Contour alots of an induction machine when phases AB and

BC are disconnected. Contour 1 ines are in 'bIb/rn.

1 1\ t = - * - s. to = 3s.- 24 60

F i g u r e 5.8 ( c o n c l u d e d )

i e g enc ,

I U@U:ED CIISnEV

3 TOTAL C 2 P O C N T

O APQL ED C - R SM '

F i g u r e 5.9 V a r i a t i o n o f c u r r e n t i n p hase CA o f a n i n d u c t i o n m a ch in e

when phases AB and BC a r e d i s c o n n e c t e d .

Chapter 6

CONCLUSION A N D SUGGESTIONS FOR FURTHER W O R K

6 . 1 Conclus ion

A m e th od ol og y f o r t h e a n a l y s i s of a d e f e c t i v e i n d u c t i o n m oto r

h as been p r e s e n t e d . M a x we ll 's e q u a t i o n s a r e a p p l i e d t o d i f f e r e n t

s e c t i o n s of t h e m a ch in e. T he se e q u a t i o n s a r e m a n i p u la t e d t o a r r i v e

a t a un i f ied t ime domain equat ion in which the M V P i s t h e o n l y

unknown v a r i a b l e . T he c ro s s s ec t i o n of t h e m ach in e i s d iv id ed i n to

t r i a n g u l a r e l e m e n t s . U sing G al e r k i n ' s f i n i t e e l em ent m eth od , t h e

u n i f i e d e q u a t i o n i s t ra n s f or m e d i n t o a g l o b a l t im e dom ain d i f f e r -

e n t i a l e q u a t i o n . A s t ep - b y -s t ep i n t e g r a t i o n a lg o ri th m i s u t i l i z e d

t o y i e l d an i t e r a t i v e n u m er ic al p r oc e du r e f o r s o l v in g t h e g lo b a l

eq ua t io n. Based on th e above methodology, a computer program i s

devel oped.

T h i s p ro gram which i s v a l i d a t ed i n C h ap t e r 4 has th e un ique

c a p a b i l i t i e s t o compute t h e f o l lo w i n g , f o r d i f f e r e n t s t a t o r d e f e c t s :

( a ) M V P a t ev e ry n od e of t h e m achine ,

( b ) f l u x d e n s i t i e s t h r ou g ho u t t h e c r o s s s e c t i o n of t h e m ac hin e,

( c ) d i f f e r e n t t y p e s of c u r r e n t s i n t h e t h r e e p ha s es of t h e m ac hin e

I n t h e s e c o m p u t at i on s , t h e r o t o r i s assum ed t o be s t a t i o n a r y .

The value of 9 f o r e v er y p o i n t i s c om pu te d a s a f u n c t i o n of B a t t h e

p o i n t .

I n Chapter 5 , the program i s appl ied t o a specific induction

motor with various s ta to r de fects. For each case, MVP contours

which imp1 i c i t l y show the fl ux density di st ri bu ti on s a re p lot ted

a t differe nt t ime steps. Also, the varia tio n of curr ent fo r each

phase i s plo tted.

The torque-speed c ha ra c te ri s t ic of the defec tive machine can be

derived when the program developed in t h i s work i s extended t o a1 low

the rot atio n of the ro to r. With the k n ow1 edge of t h i s and the mechanial

load supported by the machine, i t i s poss ible t o predict whether the

defe ctive machine can s afely continue i t s operation or not. Hence,

formulat ion of the methodology presented in t h i s work and the

computer program based on i t provides ele ctri cal engineers with a

powerful t o o l for a detailed analysis of a defective induction machine.

6 . 2 Suggestions fo r Further Work

The computer program developed in t h i s work may be enhanced for

the following capabil i i e s :

( a ) To allow the consideration of d if fe re nt types of ro to rs ,

with various kinds of defects.

( b ) To di re c tl y compute the torque fo r any speed of the ro to r.

( c ) To compute vibration forces on the machine.

Also, the man/machine in terface of the program may be improved.

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A P P E ND I C E S

AP P END IX A

1EXPANSION OF V x - 7 Y A = J

The as sumpt ions out1 ined in S ec t io n 2 .3 to ge th e r wi th

e qu a t i ons ( 2 . 11 ) and ( 2 . 1 2 ) a r e u se d i n t h i s A ppendix t o expa nd

e q u a t i o n ( 2 . 1 0 ) . U sing e qua t i on ( 2 . 12 ) t he c u r l o f A i s given by:

T h e r e f o r e :

Hence:

-, 1 :A 1 Ai n e q u a n t i t i e s - - an d - - na ve no v a r i a t i on a l ong t he z d i r e c t i o n .

L X - - Y

T h e r e f o r e , t h e f i r s t t1,vo t e rms o n t h e r i g h t hand s i d e of e a u a t i o n ( 4 . 4 )

a r e z e r o , h e nc e:

1El i m i na t i ng T :( - : A an d 2 be tween equa t ions ( 2 . lo), (2 .11) and

( A . 5 ) and can cel1 ing z f rom bo th s i de s of t h e r e s u l t i n 9 e qua t i on

g i v e s :

MATHEMATICAL FORMULAE

B . l V e c t o r I d e n t i t i e s [35]

' i * C x ' i - 0

f o r any v e c t o r f u n c t i o n 7 and s c a l a r f u n c t i o n V .

B.2 G re en ' s Theorem [ 39 ]

where:

g , U an d V a r e s c a l a r f u n c t i o n s i n a tw o -d im e ns io na l r e g i o n ,

2 , bounded by a c on tou r r .

n i s a u n i t o ut w ard n orm al t o T as i n F i g u re B . 1 .

B.3 Stoke ' s Theorem [ 50 ]

where:

f i s a v e c t o r , S i s an a r b i t r a r y s u r f a c e bounded by t h e

co n t o u r c dT i s a lo ng c .

F i g u r e B . l Two-dimensional region > bounded by ? over which Gre en ' s

theorem a p p l i e s .

8.4 I n t e g r a t i o n Fo rmu lae f o r a T r i a n g l e [Sl]

F o r a t r i a n g u l a r e l e me n t ne , w i t h v e r t i c e s a t ( x . . ) , ( x . y . )1 1 J ' J

a n d ( x , y ) , F i g u r e 5 . 2 , o f a r e a I, h e f o l l ow in a a p p l y :k k

where :

N i , N . and N k a r e sh a p e f u n c t i o n s d e f i n e d i n e q u a t i o n s ( C . 9 )J

t h r o u g h ( C . 11).

Ji,( N ~ ) " ' ( N ~ ) ~ ( N ~ ) ~x dy = rn ! n! p ! 20

e (m+n+p+2) !

where :

r n , n and p a r e p o s i t i v e i n t e g e r s a nd ! d e n o t e s f a c t o r i a l .

Figure 8.2 A triangular element y e .

s h o w i n gMVP s .

i g u r ec . 1 A

t r i a n g u l a r e l e me n t ,. I , ,

E q u a t i o n s ( C . 2 ) t h r o u g h ( C .4 ) a r e s o l v e d s imui t a n e o u s l y f o r t h e

c o n s t a n t s a , b and c. The r e s u l t i s :

where:

1A = 7 - y i x j + XjYk - YjXk + XkYi - ykxi) ( C m

S u b s t i t u t i n g a, b and c f r o m e q u a t i o n ( C. 5) i n t o (C . 1 ) g i v e s :

1Be = [ ( x ~ Y ~y j x k ) + ( Y ' - YI(lx ( x k - x j ) ~ I H i

1+ [ ( Y ~ ~ ~ - Y ~ ~ ~ )( y k - y i ) ~ +

1+ [ ( x i y j - y i x j ) + ( y . - y J ) x + ( x - x ~ ) ~ ] ~ ~

E qua t ion ( C .7 ) may be w r i t t e n a s :

where:

N i, N . and N k a r e i n t e r p o l a t i n g ( o r s ha pe ) f u n c t i o n s , g i v en by:3

1+ b i x t c i y ) , i n Q e

0 e l s e w h e r e

J ' el sewhere

0 el sewhere

( C . 1 0)

( C . 1 2 )

( C . 1 3)

( C . 1 4 )

( C . 1 5 )

( C . ! 6 )

(C. 1 7 )

(C.18)i j 1 jk = X . Y - y - x

(C. 0)

E q u a t i o n ( C .8) o g e t h e r w i t h e q u a t i o n s ( C .9 ) t h r o u gh ( C . 20) p r o v i d e a

d e f i n i t i o n f o r t h e a pp ro x im a te s o l u t i o n , 8,.

E v e r y r o w o f ( D . 3 ) A i s u s e d f o r W e a n d ( C . 9 ) - ( C . 1 1 ) f o r Yi , NN k

'ide r_.w j'

t o g i v e- e

s :1̂x 3Y

U s e of ( D . 4 ) - ( D . 7 ) a n d (B.5) f o r t h e f i r s t i n t e g r a l on t h e l e f t hand

s i d e o f ( D . l ) w r i t t e n w i t h each r o w of ( 0 . 3 ) a s W e g i v e s :

where:

Mith th e use of ( 0 . 2 ) f o r de and every row of ( 0 . 3 ) f o r Ue, t he sec ond

i n t e g r a l on t h e l e f t hand s i d e o f ( D . l ) may be w ri t t en a s :

The f i r s t row of t h e m a t r i x i n ( D . 1 1 ) i s e v a l u a t e d u s i n g ( 8 . 8 ) a s :

( D . 1 2 )

The r em a in in g e l e m e n t s of ( D . 1 1 ) a r e s i m i l a r l y e v a l u a t e d t o y i e l d :

( D . 1 3 )

kine r e :

r o e ] =-2 1 1 ( D . 1 4 )

Every row of ( 0 . 3 ) i s used f o r Lie i n t h e t h i r d t er m on t h e ? e f t hand

s i d e o f ( D . 1 ) t o g i v e :

( D . 1 5 )

The f i r s t row o f ( D . 1 5 ) i s e v a l u a te d by s e t t i n g rn = 1 , and n = p = 0

i n ( 6 . 8 ) :

( D . 16 )

S i m i l a r l y , t h e l a s t two rows of ( D . 1 5 ) a r e e v a l u a t e d t o g i v e t h e

column vector [ F ] a s :-e

( D . 1 7 )

Using t he t h r e e va lue s g ive n in ( 0 . 3 ) f o r Ye, t he l a s t te rm on th e

1 e f t hand s i d e of ( D . 1 ) may b e w i r t t e n a s :

(C.18)

where:

(D. 9)

(D. 0)

Using every row of (0.3) o r Q e , (8.6) o r x and ( 0 . 2 ) f o r A , , (0.19)

becomes:

T h e r e f o r e :

(D. 2)

The f i r s t di ago n a l e l em en t of [MI may be evaluated as shown below:

i1 ( N f x i + N N . x + P N x ) - dx dy

i ~ j k k 2 ~"

E v a l u a t i o n of o t h e r e l em e nt s o'f [ M I w i t h a s i m i l a r a pp ro ac h g i v e s :

where :

x a = ~ x . + x . + x1 J k

S i m i l a r l y ( 8 . 7 ) i s u se d f o r y, ( 0 . 3 ) f o r w e a n d ( D .2 ) f o r A e i n

( D. 2 0) t o e v a l u a t e [ D l as :

(D . 24)

( D . 25)

( D . 26)

(D. 27)

( D . 28)

where :

APPENDIX E

DERIVATION OF A TIME FUNCTION FOR THE FLUX

IN A MAGNETIC CIRCUIT

Fi g u re E . l shows a magnet ic c i r c u i t with a co i l on one l imb and

an a i r g ap on t h e o t h e r . The co i l i s ex c i t ed by a s i n u s o i d a l v o l t ag e

s o u r c e , V m S in ( .Lt i ) , w here i s t h e an g u l a r f r eq u en cy and 2 i s

t h e p h as e an g l e .

A ssu m ptio ns i n t h i s d e r i v a t i o n a r e :

( a ) L eakage f l u x e s a r e n e g l i g i b l e s o t h a t a l l t h e f l u x $ a r e

c o n fi n ed t o t h e c o r e and l i n k a l l t h e t u r n s o f t h e c o i l .

( b ) Permeabil i t y of t h e co re i s i n d ep en d en t of t h e 1 eve1 of

t h e f l u x d e n s i t y .

( c ) F r i n g i n g e f f e c t a ro u nd t h e a i r g ap can b e accom m od ated by

apply ing an a i r gap c o r r e c t i o n f a c t o r s u gg e st ed by Matsch [43] .

For the coi l shown in Figure E . l , K i r c h h o f f ' s v o l t a g e e q u a t i o n

may be w r i t t en f o r t h e t e rm i n a l co n d i t i o n a s :

( E . 1)

where:

i i s th e co i l c u r r e n t .

N i s the number of t u rn s .

R i s t h e r e s i s t a n c e of t h e c o i l .

The total magnetomotive force (mmf) in the magnet ic c i r c u i t may be

w ri t t e n a s :

Figure E . 1 A magnetic ci rc u it with i t s exc itati on system. The

depth o f the core is b .

where:

Hi s t h e m agne t ic f i e l d i n t e n s i t y i n t h e a i r gap .g

g i s t h e 1e ng th o f t h e a i r gap.

H c i s t h e m ag n e t i c f i e l d i n t e n s i t y i n t h e c or e .

L C i s t h e a ve ra ge f l u x p a t h 1e n g th t h r o u g h t h e c o r e .

U nd er a s su m pt io n ( a ) , t h e l e v e l of t h e f l u x e s t a b l i s h e d i n t h e c o r e

i s t h e same as t h a t i n t h e a i r gap. E q u a t i o n ( E . 2 ) may b e w r i t t e n

where:

9 s t h e p erm ea bi l i t y o f t h e c o re .

a i s t h e w id t h o f t h e c or e.

b i s t h e d e pt h o f t h e c or e .

A i s t h e e f f e c t i v e a re a o f t h e a i r gap g iv e n by [43 ] :g

T h e r e f o r e :

S u b s t i t u t i n g i r o m e q u a t i o n ( E. 6) i n t o e q u a t i o n ( E . l ) g i v e s :

Theref o r e :

L e t :

then equat ion ( E . 8) becomes:

d6-t t 64 = "m S in (,t + 9 ) ( E . 1 0 )

Equat ion ( E . l O ) i s a f i r s t o rd e r d i f f e r e n t i a l e q ua ti on whose s o l u t io n

" - ijta = [2 S in ( i t + 7) - C O S ( h t + -)I + Ae

where:

A i s a c o n s t a n t d e t er m in e d s u b s e q u e n t l y .

I f a t t = 0 , 5 = a,, then (E . l l ) g i v e s A a s :

( b Cos 9 - 3 S in $ )A = " + N ( , 2 + $ 2 ) " ( E . 1 2 )

Using (E .12) fo r A i n ( E . 1 1 ) , t h e c o m p l e t e s o l u t i o n f o r 3 i s o b ta in e d

+ " m ' 3 S i n ( ,t + 5 ) - - c o s ( , t + ? ) ;

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