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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
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FINITE ELEMENT ANALYSIS O F A DEFECTIVE INDUCTION MOTOR
B Y
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
a
a
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CLARENCE MWABUNWANNE OBIOZOR
Al l Rights Reserved
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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
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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 )
a
a
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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.
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T A B L E O F CONTENTS
Page
. . . . . . . . . . . . . . . . . . . . . . . . . . . .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
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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
A1
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
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De
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
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LIST O F TABLES
Page
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
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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
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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
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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
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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.
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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
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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.
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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 .
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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
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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.
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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 ,
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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 :
,. A
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.
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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 :
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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 .
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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 :
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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 :
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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 :
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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 :
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Figure 2 . 1 A section of t h e considered solid rotor.
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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 .
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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
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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 :.
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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 :
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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/
where
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:
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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
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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
3 k1
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 :
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[ 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 ? .
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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:
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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 :
1 -
- [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 :
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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:
3 X
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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 .
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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
turn.
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.
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P f Side ionductar
Endconnect0 r
.
A,
Figure 3 . 2 A single turn c o i l .
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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 :
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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 .
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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
1411.
( 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 .
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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
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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
I1 1
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 .
1
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.
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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
1 Yes
Save element numbers,node numbers and materialproper ty on a d i s c
F igure 3 . 3 ( c onc lude d)
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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 .
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F i g u r e 3 . 5 (concluded)
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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
-n
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.
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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 :
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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 .
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- 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 &='- - - - - - - - - - -
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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
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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 .
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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 .
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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 .
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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:
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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 .
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51
I
/
I1
I'
/-
i= t +ZA t i I0
I,
\ ?I\
\,--
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
F i g
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Figure 4 . 7 ( c o n c l u d e d ) .
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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.
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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.
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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 .
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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
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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.
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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 .
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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
where
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
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6
t= o t = t O + L t
L
-1
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
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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
down.
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 .
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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.
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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.
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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.
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I C o i l BC2
y--/ S*
I I C
1 1II
IwIo i l BC 1
C
C
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.
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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
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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.
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F i g u r e 5 . 2 ( c o n c l u d e d ) .
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Legend
0 BlU s u 1 m r
a 4 suelotr ;
i
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 .
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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.
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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
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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.
24 60
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Figure 5 . 5 ( c o n c l u d e d ) .
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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 . .
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Coi 1
disconnec
I-' :disconnected
Coil BC1
C
0
F igu r e 5 . 7 Diconnection of phases AB and 8 C .
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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
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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
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F i g u r e 5.8 ( c o n c l u d e d )
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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 .
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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,
. and
( 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 .
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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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REFERENCES
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IEEE Trans. , Vol . PAS-104, No. 7 , J u l y 1 9 8 5 , p p. 1 7 9 7- 1 80 3 .
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M i n n i c h , S . H . , C h a r i , N . V . K . a n d B e r k e r y , J . F . , " O p e r a t i o n a l
I n d u c t a n c e s o f T u r b i n e - G e n e r a t o r by t h e F i n i t e El e m en t M et ho d, "
IEEE Pa pe r 8 2 , WM 227-9 .
W i l l ia m s o n , S . a n d S m i t h , A . C. , " S t e a d y - S t a t e A n a l y s i s of T h r e e-
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W i l l i a m s o n , S . a n d M i r z o i a n , K . , " A n a l y s i s o f C a ge I n d u c t i o n
Y o t o r s w i t h S t a t o r W in di ng F a u l t s , " IEEE P a p e r 8 4 , SM 6 7 4- 8 .
Odamura, M . a n d I t o , M . , "Up-W ind F i n i t e E l e m en t S o l u t i o n o f
S a t u r a t e d T r a v e l 1 n g M a g n e t i c F i e l d P r o b l e m s , " N a t i o n a l C o n v e n t i o n ,
I n s t i t u t e of E l e c t r i c a l E n g i n e e r s i n J a p a n , No. 1 0 , 1 97 9,
pp . 819 -825 .
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C h a r i , M . V . K . , " F i n i t e E l em e n t A n a l y s i s o f N o n l i n e a r M a g ne t ic
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197 1, pp . 2362-2372.
B r a u e r , J o h n R . , " S a t u r a t e d M ag ne ti c E ne rg y F u nc t i o na l f o r F i n i t e
E le m en t A n a l y s i s o f El e c t r i c a l M a c h in e s, " IEEE P ap e r C75, 1 51- 6.
S i l v e s t e r , P . , C a ba y an , H .S. a n d B row ne, B . T . , " E f f i c i e n t T e c h n i qu e s
f o r F i n i t e El e m en t A n a l y s i s o f E l e c t r i c M a c h i n es , " IEEE T r a n s . ,
Vo l. PAS-92, No. 4 , 19 73, pp. 1274-1 280.
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1 7 . C s e n d e s , Z . J . a n d C h a r i , M . V . K . , " G e n er a l F i n i t e E le m en t
A n a l y s i s of R o t a ti n g E l e c t r i c M a ch in e s, " I n t e r n a t i o n a l C o n fe r en c e
on Nu m er ic a l M e t ho ds i n E l e c t r i c a l a n d M a g n e t i c F i e1 d Probl ems,
S an M a r g h e r i t a L i g u r e ( I t a l y ) , J u n e 1 -4 , 1 9 76 , pp . 1 99 -2 09 .
18. W i l l i a m s o n , S . a n d R a l p h , J.W ., " F i n i t e E le m en t A n a l y s i s f o r
N o n l i n e a r M a g n e t i c F i e l d P r o bl e m s w i t h C om plex C u r r e n t S o u r c e s , "
IEE P r oc . , Vo l . 129 , P t . A , No. 6, 1982, pp. 391-395.
1 9 . W i l l ia m s o n S . a n d R a l ph , J. W ., " F i n i t e E l e me n t A n a l y s i s f o r a n
I n d u c t i o n M o to r Fed fr o m a C o n s t a n t - V o l t a g e S o u r c e , " IEE P r o c . ,
Vol . 130 , P t . 8 , No. 1, J a n u a r y 1 9 8 3 , pp . 18-24 .
2 0 . I t o , M . , F u j i m o t o , N . , T a k a h a s h i , N . a n d M i y a t a , T . , " A n a l y t i c a l
Model f o r M a g n e ti c F i e l d A n a l y s i s o f I n d u c t i o n M ot or P e r f o r -
man ce," IEEE T ra n s . , Vol. FAS-100, No. 11, November 1981,
pp . 4580-4590.
21 . Hanna l l a , A . Y . an d McDonal d , D . C . , " A N odal M eth od f o r t h e S o l u t i o n
o f T r a n s i e n t F i e l d P r ob l ems i n E l e c t r i c a l M a c h i n e s ," IEEE T r a n s .
on Magne t i cs , Vo l . MAG-11 , No. 5 , Septe mber 1975 , pp . 1544-1546.
2 2 . H a n n a l l a , A . Y . and McDonald, D . C . , " N um e ri ca l A n a l y s i s o f T r a n s i e n t
F i e l d P r ob le m s i n E l e c t r i c a l M a c h i n es , " IEE P r o c . , V o l. 1 2 3 ,
No. 9 , September 1976 , pp . 893-898 .
2 3 . C h a r i , M . V . K . , Minn ich , S .H. , Tandon , S .C . , Cse ndes , Z . J . and
B e r k e r y , J . , "L oa d C h a r a c t e r i s t i c s o f S y n ch ro no u s G e n e r a t o r s by
t h e F i n i t e E l em e n t M e t h od , " I EEE T r a n s . , Vol . PAS-100, No. 1,
J a n u a r y 1 9 8 1 , p p . 1 - 1 3 .
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Gruch, John C . , J r . a n d Zyval o s k i , G e o r g e, " T r a n s i e n t Two -
D i m e ns i on a l H e a t C o n d u c t io n P r o b l em s S o l v e d by t h e F i n i t e E l em e n t
M e th o d" , I n t e r n a t i o n a l J o u r n a l f o r Numeri ca1 Methods in Enq inee r ing ,
Vol . 8 , 1974 , pp . 481-494 .
M a t s c h , L e a n d e r W . , E l e c t r o m a q n e t i c a n d E l e c t r o m e c h a n i c a l M a c h i n e s ,
2 nd e d i t i o n ( C ro w el 1 , H a r p e r a n d Row, 1 9 7 1 ) .
Sl emon, G . R . a n d S t r a u g h e n , A . , El e c t r i c a l M a c h i ne s ( A d d is o n -
Wesl ey Pub1 i sh in g Company, 1980).
0' el l y , D. , " T h e o r y a n d P e r f o r m a n c e of S o l i d - R o t o r I n d u c t i o n a nd
H y s t e r e s i s Mach ines , " IEE P ro c . , Vol . 123 , No. 5 , May 1976 ,
pp. 428-429.
S i l v e s t e r , P . P . a n d F e r r a r i , R . L . , F i n i t e E le me nts f o r E l e c t r i c a l
E n g i n e e r s ( Ca mb ri d q e U n i v e r s i t y P r e s s , 1 9 8 3 ) .
C h a r i , M . V . K . a n d S i l v e s t e r , P . ( e d s . ) , F i n i t e Ele me nts i n
E l e c t r i c a l a n d M a q n e t ic F i e 1 d P r o b l ems ( Wi l e y , 1 9 8 0 ) .
K; a u s - J u r g e n , B a t h e , F i n i t e E le m en t P r o c e d ur e s i n E n g i n e e r i n g
A n a l y s i s ( P r e n t i c e Ha1 1 , 1 9 8 2 ) .
B r a n d l , P . , R e i c h e r t , K . and Vogt , W . Baden , "S im ula t io n of Turbo-
G e n e r a t o r s on S t e a d y - S t a t e L o a d , " Brown B o v e r i R e v ie w , 1 9 7 5 , 9 ,
pp. 444-449.
P l ons ey , R . a nd C o ll i n , R . E. , P r i n c i p l e s a nd A p p l i c a t i o n s o f
El ec tr om ag n et ic Fi el ds (McGraw-Hi 11 Book Company, 196 1) .Pl on us , Ma rt i n A. , Appl ie d El ec t r om ag ne t i cs ( !:&raw-Hi1 1 Book
Company, 1978) .
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5 2. H e u s e r , J . , " F i n i t e E le m e nt M ethod f o r T he rm al A n a l y s i s , " YASA
Pa pe r TN D-7274, November 1973 .
5 3 . B r a u e r , J o h n R . , " S im p l e E q u a t io n s f o r t h e M a g n e t i z a t i o n a nd
R e l u c t i v i t y C u rv e s o f S t e e l ," IEEE Trans . on Maqnet ics , Vol .MAG-11, No. 1, J a n u a r y 1 9 7 5 , p . 8 1 .
5 4 . F o g g i a , A . , S a b o n n ad i e r e , J . C . a n d S i l v e s t e r , P . , " F i n i t e E l e m e n t
S o l u t i o n of S a t u r a t e d T r a v e l 1 n g M a g n e t i c F i e l d P r ob le m s ," IEEE
T r a n s . , V o l . PAS-94, No. 3 , May / June 1975 , up . 866 -871 .
55 . Hue bner , Kenneth H . , T he F i n i t e E l em e nt M ethod f o r E n g i n e e r s
( J o h n W i le y a n d S o n s , 1 9 8 2 ) .
A l g e r , P . A . , I n d u c t i o n M a c h i n e s , T h e i r B e h a v i o r a n d U s es (New
Y or k: G or do n a n d B r e a ch S c i e n c e Pub1 i s h e r s , 1 9 6 5 ) .
5 7 . I t o , M o ta ya , F u j i m o t o , N . , Okuda, H . , T a k a h a s h i , T . a n d W a t a h i k i ,
S . , "E f f e c t o f Broken Bars on Unba lanced Magne t i c Pu l l and To rque
of I n d u c t i o n M o t o r s ," E l e c t r i c a l E n g i n e e ri n g i n J a p a n , Vol . 100 ,
1980, pp . 42-49 .
58 . Char i , M . V . K . a nd S i l v e s t e r , P . , " A n a l y s i s o f T urb o-A 1 t e r n a t o r
M a g n e t i c F i e l d s by F i n i t e E l e m e n t s , " IEEE T r a n s . , Vol . PAS-90,
No. 2, Ma rch/A pril 197 1, pp. 454-4 64.
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A P P E ND I C E S
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AP P END IX A
1EXPANSION OF V x - 7 Y A = J
U
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:
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-, 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
'c!
( 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 :
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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 .
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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 .
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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 .
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Figure 8.2 A triangular element y e .
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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 , ,
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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 :
or:
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 ~ ) ~ ] ~ ~
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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 )
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(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,.
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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
and-
s :1̂x 3Y
2x 2a
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:
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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 )
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kine r e :
7
: A
1- e-
;I
r o e ] =-2 1 1 ( D . 1 4 )
1Ii
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 :
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(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:
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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 :
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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 :
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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 .
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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
as :
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 :
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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
i s :
" - 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
a s :
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+ " m ' 3 S i n ( ,t + 5 ) - - c o s ( , t + ? ) ;