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Effect of Slotting in Pm Electric MachinesGu Qishan a & Gao Hongzhan aa Department of Electrical Engineering, Harbin Institute of Technology, Harbin, People'sRepublic of ChinaPublished online: 22 Jan 2007.

To cite this article: Gu Qishan & Gao Hongzhan (1985): Effect of Slotting in Pm Electric Machines, Electric Machines & PowerSystems, 10:4, 273-284

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EFFECT OF SLOTTING IN PM ELECTRIC MACHINES

GU QISHAN and GAO HONGZHAN

Deparrmenr of El.%rrical Engineering Harbin lnsrirure of Technology Harbin, People's Republic of China

ABSTRACT

The e f f e c t o f s l n t t i n q i n pe rmanen t magnet e l e c t r i c mach ines d i f f e r s f rom t h a t i n wound f i e l d ones. An a n a l y t i c a l l y e x t e n d e d f i e l d model f o r s l o t t e d permanent magnet mach ines i s p r e s e n t e d i n t h i s p a p e r . Ry u s i n g t h e model , t h e C a r t e r ' s c o e f f i c i e n t f o r p e r - manent magnet machines can b e e a s i l y deduced f r o m t h e C a r t e r ' s c l a s s i c a l f o r m u l a . The c l o s e d - f o r m s a l u t i o n t o t h e p r o b l e m a t hand i s e x a c t and much s i m p l e r t h a n t h a t o f p r e v i o u s l y a p p r o x i m a t e me- t h o d s .

INTROOUCTION

W i t h t h e r a p i d deve lopment o f new permanen t magnets( such a s r a r e e a r t h and f e r r i t e magnets 1, t h o a p p l i c a t i o n s o f permanent magnet (PN) e x c i t a t i o n i n e l e c t r i c machines a r e now i n c r e a s i n g l y g r o w i n g . Many new s e r i e s o f f r a c t i o n a l - h o r s e p o w e r PM m o t o r s a r e w e l l e s t a b l i s h e d . The power l i m i t bet lueen PM and wound f i e l d ma- c h i n e s i s r a p i d l y mouinq t o w a r d s much h i g h e r l e v e l due t o t h e use o f r a r e e a r t h permanent magnets .

The a i r qap f i e l d o f s l o t t e d PM mach ines i s one o f t h e b a s i c f i e l d phenomena i n PM machine s t u d i e s . The e f f e c t o f s l o t t i n g on t h e a i r gap f i e l d i n PW mach ines , i n w h i c h magnets f a c e d i r e c t l y wit,h t h e a i r gap, d i f f e r s f rom t h a t i n wound f i e l d ones. Computa- t i o n o f t h e m a q n e t i c f i e l d f o r s l o t t e d PM mach ines w i t h f i n i t e s l o t d e p t h has been made by u s i n g t h e n u m e r i c a l ( 1 ) o r c l a s s i c a l l y a n a l y - t i c a l ( 2 ) methods. B u t . b o t h methods have t o s o l v e a l a r q e s e t o f a l g e b r a i c e q u a t i o n s and hence a r e a l s o cumbersome f o r d e s i g n p l l r - ooses .

F o r t h e sake o f p o i n t i n g o u t c e r t a i n r e l a t i o n s h i p s and d i f f e r - ences between a s l o t t e d PM machine and i t . s wound f i e l d c o u n t e r p a r t , an a n a l y t i c a l l y e x t e n d e d f i e l d model f o r s l o t t e d PM machines i s p r e s e n t e d i n t h i s p a p e r . 8 ~ u s i n g t h e model , t h e C a r t e r ' s c o e f f i c i e n t f o r PM machines can be e a s i l y deduced f r o m t h e C a r t e r ' s c l a s s i c a l

Elsctric Mashins and Power Systems. 10:273-284. 1985 Copyright B 1985 by Hemisphere Publishing Carporation

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274 G. QISHAN AND G. HONGZHAN

f o r m u l a . The c l o s e d - f o r m s o l u t i o n t o t h e p r o h l e m a t hand i s e x a c t and much s i m p l e r t h a n t h a t o f n r e v i o u s l y a p p r o x i m a t e methods.

ORIGINAL FIELD NOOEL

F i g . 1 shows t h e d e w e l o p i n g s c h e m a t i c a f a s l o t t e d Pf l mach ine . The f o l l o w i n g a s s u m p t i o n s a r e made t o s i m p l i f y t h e a n a l y s i s :

1 The e f f e c t o f c u r v a t u r e i s n e g l e c t e d . 2 . V a r i a t i o n s i n t h e a x i a l d i r e c t i o n a r e i g n o r e d . 3 . A l l l a y e r s e x t e n d t o i n f i n i t y i n t h e i x - d i r e c t i o n . h . A l l s l o t s e x t e n d t o i n f i n i t y i n t h e - y - d i r e c t i o n . 5 . A 1 1 t e e t h and t h e b a c k i r o n a r e c o n s i d e r e d t o be i n f i n i t e l y p e r -

meable . i . e . A = m . 6. A n i s o t r o o i c oe rmanen t maonets a r e u s e d a n d u n i f o r m l v m a o n e t i z e d - - . t o s a t u r a t i o n i n t h e p r e f e r r e d d i r e c t i n n . d ? i c n c o i n c i d e s ~ i t n t h e y - O l r e c t i o n i n F i q . 1 . Tne c o n s t i t u t i o n e q u a t i o n o f t h e maonets aav be e x p r e s s e d a s

B =/UmHy + A N = Hy t M ) ( 1 )

where N t h e m a g n e t i z a t i o n v e c t o r o f t h e magnet , I N ( = 8 /A; t h e p e r m e a b i l i t y o f t h e magnet.' w h i c h i s assume8 t o be&.

I

F i g . 1 D e v e l o p i n g s c h e m a t i c o f a s l o t t e d PN e l e c t r i c mach ine

Recause o f t h e l o w p e r m e a b i l i t y o f t h e magnet, t h e i n t e r f a c e between t h e magnet and t h e a i r gap i s n o l o n g e r c o n s i o e r e d as an e q u i p o t e n t i a l s u r f a c e . As a consequence, r h e p h y s i c a l mode l o f t h e p r o b l e m s h o u l d c o n s i s t o f t h r e e l a y e r s : t h e pe rmanen t magnet l a y e r , t h e a i r gap l a y e r and t h e t o o t h l a y e r .

The o b j e c t i v e o f t h e p a p e r i s t o f i n d o u t an a l t e r n a t i v e f i e l d model , by w h i c h t h e a i r gap f i e l d s o l u t i o n can b e e a s i l y o b t a i n e d . F o r s i m p l i c i t y , . w e f i r s t c o n s i d e r a t w o - l a y e r mode l a s t h e o r i g i n a l f i e l d model . I t c o n s i s t s o f a PPl l a y e r and an a i r gap l a y e r s p r e a d - i n g o v e r a s l o t p i t c h , shown by t h e d o t t e d l i n e i n f i g . 1 . The vo- lume c u r r e n t d e n s i r y v a n i s h e s w i t h i n t h e u n i f o r m l y m a g n e t i z e d mag- n e t . Thus, t h e f o l l o w i o t g L a p l a c e e q u a t i o n i s now v a l i d i n b o t h PN and a i r gap - r e g i o n s .

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SLOTTED PERMANENT MAGNET MACHINES

F i g . 2 O r i g i n a l f i e l d Model

The b o u n d a r y c o n d i t i o n s o f t h e o r i g i n a l f i e l d model, shown i n F i g . 2, a r e as f o l l o w s :

Y = hm . @ (x .hm) = 0 Y =-Q . @ ( x , - 9 ) = f ( x )

,'A/2, 39- = 0 - a x

where f ( x ) i s an unknown h a r m o n i c f u n c t i o n o f t h e s c a l a r m a g n e t i c p o t e n t i a l due t o s l o t t i n g .

T h e . c o h t i n u i t y c o n d i t i o n s a t t h e i n t e r f a c e EF w i t h i n t h e model a r e a s f o l l o w s :

Then, E q . ( l ) can be r e w r i t t e n a s

U s i n g t h e boundary and c o n t i n u i t y c o n d i t i o n s , t h e f i e l d s o l u - t i o n i n b o t h r e g i o n s may be s o l v e d and e x p r e s s e d i n te rms o f t h e unknown o o t e n t i a l f u n c t i o n f ( x ) a t s l o t t e d s u r f a c e CO (see Aooendix A ). F o r t h e

m X

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G. OISHAN AND G. HONGZHAN

F o r t h e a i r gap r e g i o n ,

EXTENDED FIELD MODEL

I f we e x t e n d t h e d e f i n i n g r e g i o n o f t h e a i r gap p o t e n t i a l f u n c - t i o n + i n t h e o r i g i n a l f i e l d model t o ' - t h e PM l a y e r , t h e n t h e mag- n e t i c g c a l a r p o t e n t i a l a t t h e boundary A8 ( a t y = h m ) becomes

0

T h i s l e a d s t o a new f i e l d model, w h i c h c o n t a i n s o n l y one s i n - g l e - l a y e r a i r gap r e g i o n i n s t e a d o f t h e ~ r i g i n a l d o u b l e - l a y e r one, as shown i n F i g . 3. The new model i s s e t up by e x t e n d i n g t h e a n a l y - t i c f u n c t i o n o f t h e a i r gap p o t e n t i a l t o t h e Pfl r e g i o n i n . t h e o r i g i - n a l model , so i t i s c a l l e d t h e a n a l y t i c a l l y e x t e n d e d f i e l d mode l , i n l uh ich a new p o t e n t i a l f u n c t i o n qe may be d e f i n e d .

F i g . 3 E x t e n d e d f i e l d model

S t a r t i n g f r o m t h e e x t e n d e d f i e l d model i n F i g . 3 , t o g e t h e r w i t h i t s new boundary c o n d i t i o n s :

i t may be p r o v e d t h a t ( see Appendix B ) b o t h a i r gap p o t e n t i a l f u n c - t i o n s a r e i d e n t i c a l i n a c t u a l a i r gap r e g i o n , namely

O D )

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SLOTTED PERMANENT MAGNET MACHINES

Fig. 4 Carter's coefficient for PN machines k =f(-,----- S , crn x h + g m

s m Fig. 5 Carter's coefficient k versus - and - - c m h 9

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278 G. OISHAN AND G. HONGZHAN

As f a r a s t h e a i r gap f i e l d d i s t r i b u t i o n i s concerned , f r o m E q . ( l O ) i t i s c l e a r t h a t , t h e e x t e n d e d f i e l d model i s e q u i v a l e n t t o t h e o r i g i n a l one. Thus. Eq. ( lO)may b e c a l l e d t h e e q u i v a l e n c e c o n d i t i o n f o r b o t h f i e l d mode ls . I n f a c t , t h e e x t e n d e d f i e l d model can b e s e t up by c o n v e r t i n g t h e d i s t r i b u t e d PM s o u r c e w i t h i n t h e o r i g i n a l model i n t o an i d e a l i z e d and lumped one l y i n g on t h e bound- a r y AB i n F i g . 3.

CARTER'S COEFFICIENT FOR PM MACHINES

4s above ment ioned , t h e e x t e n d e d f i e l d model i s much s i m p l e r t h a n t h e o r i g i n a l one. F u r t h e r m o r e , i t i s t h e v e r y model , w h i c h was w e l l s o l u e d by C a r t e r ( 3 ) a t t h e b e g i n n i n g o f t h e c e n t u r y . As a con- sequence. by u s i n g t h e model , t h e C a r t e r ' s c o e f f i c i e n t f o r PM ma- c h i n e s can be e a s i l y d e r i v e d f r o m t h e C a r t e r ' s c l a s s i c a l f o r m u l a , p r o v i d e d t h a t t h e a i r gap o f t h e model i s s u b s t i t u t e d f o r ' t h e a c t u a l one. The e x t e n d e d a i r gap o f t h e model i s

g e x t = hm + 9 ( 1 1

The C a r t e r ' s c o e f f i c i e n t f o r PM machines i s

where Bg max

t h e a i r gap f l u x d e n s i t y o f a smooth a r m a t u r e ;

B t h e a v e r a g e a i r qap f l u x d e n s i t y o f a s l o t t e d one; Q

. % t h e s l o t - w i d t h r e d u c t i o n f a c t o r .

S S

A f a m i l y o f c u r v e s kcm' f ( ~ ' f i - ~ 6 - ) i s o b t a i n e d f r o m E q . ( l Z ) and ( 1 3 ) . a s shown i n F i g . 4. I t i s n o t e d t h a t , k depends upon t h e sum o f hmandg. B u t , k c m i s i n d e p e n d e n t o f t h e F a t l o hm/g w i t h t h e sum b e i n unchanged. The weak ly dependence o f k on t h e r a t i o h m / Q i n ? 2 ) may be due t o t h e t r u n c a t i o n e r r o r i n c m t h e a p p r o x i m a t e s o l u t i o n .

F i g . 5 shows t h e d i f f e r e n c e i n C a r t e r ' s c o e f f i c i e n t between t h e PM and t h e wound f i e l d mach ines . T a k i n g t h e r a t i o h m / g as a p a r a m e t e r and w i t h t h e a c t u a l a i r gap g r e m a i n i n g c o n s t a n t , t h e e f f e c t o f t h e magnet h e i g h t h on k i n F i g . 5 i s a p p a r e n t and con- s i d e r a b l e as h / Q i n c r e a s e s . m ~ h e n C R m / g = ~ , k c m r e d u c e s t o k - - - t h e

C C a r t e r ' s c o e f f T c i e n t i n wound f i e l d machines.

COMPARISON OF FIELD SOLUTIONS

To v e r i f y t h e a n a l y t i c r e s u l t s o b t a i n e d b y t h e e x t e n d e d f i e l d model , a compar i son o f a i r gap f i e l d p u l s a t i o n s due t o s l o t t i n g w i t h t h a t o b t a i n e d b y t h e f i n i t e d i f f e r e n c e method h a s been made. F i g . 6 shows t h e n o r m a l i s e d s c a l a r p o t e n t i a l and t h e r a d i a l a i r gap f l u x d e n s i t y d i s t r i b u t i o n a l o n g t h e i n t e r f a c e EF. The a n a l y t i c f i e l d s o l u t i o n i n t h e e x t e n d e d model i s computed by u s i n g t h e C a r t e r ' s c o n f o r m a l t r a n s f o r m a t i o n r e s u l t s . The n u m e r i c a l s o l u t i o n i s o b t a i n - ed f r o m a f i n e r g r i d , s o i t seems t o be a c c e o t a b l e F o r compar i son . The a n a l y t i c r e s u l t s i n t h e e x t e n d e d F i e l d model a r e i n good a g r e e - ment w i t h t h e n u m e r i c a l ones i n t h e o r i g i n a l f i e l d model as shown

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i n F i g . 6 .

1 .

F i g . 6 The n o r m a l i s e d s c a l a r p o t e n t i a l and t h e r a d i a l

a i r gap f l u x d e n s i t y d i s t r i b u t i o n a l o n g t h e

i n t e r f a c e EF. s 0.6, h / g = 20, h = 0 . 0 4 m , hm=O.Olm m

0 - - - - a n a l y t i c method a - - - - n u m e r i c a l method, i n d u c t i o n

- - - - n u m e r i c a l method, p o t e n t i a l .

POTENTIAL DROPS I N PN PlACHINES

Because t h e i n t e r f a c e between t h e magnet and t h e a i r gap i s n o t a n e q u i p o t e n t i a l s u r f a c e , so t h e p o t e n t i a l d r o p a c r o s s t h e a i r gap v a r i e s a l o n g t h e w h o l e s l o t p i t c h , a s shown i n F i g . 7c . The f i e l d model o f PNmachines must be s i m p l i f i e d f u r t h e r i n m a g n e t i c c i r c u i t c a l c u l a t i o n s . We f i r s t c o n s i d e r a m a g n e t i c c i r c u i t e l e m e n t a l o n g t h e t o o t h c e n t e r l i n e BFD, w h i c h c o n s i s t s o f a magnet and an a i r gap e l e m e n t o f w i d t h d x , a s shown i n F i g . 7 a . L e t t h e maximum f l u x d e n s i t y o f t h e magnet be Bm max = 9 , w h i c h r e m a i n s con- s t a n t a l o n g t h e l i n e 8FD. F u r t h e r m o r e , m a x l e t t h e mean f l u x den- s i t y o f t h e magnet be Em= B . Thus, t h e a i r gap p o t e n t i a l d r o p a t x = ' h / 2 may b e w r i t t e n a s

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G. OISHAN AND G. HONGZHAN

max

. x

max

F i g . 7 The a i r gap f l u x d e n s i t y and p o t e n t i a l d r o p

v e r s u s x

The i n t e r n a l p o t e n t i a l d r o p w i t h i n t h e magnet e l e m e n t a t x = r h / 2 i s

1 F .= - B = l 0 k h

m rnax hm jo m crn m ( 1 5 )

y o

T h e r e f o r e , t h e e q u i v a l e n t a i r gap and magnet h e i g h t may b e d e f i n e d r e s p e c t i v e l y a s f o l l o w s :

ge = kcmg ( 1 6 )

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The t o t a l MMF consumed. i . e . t h e i d e a l i s e d n o - l o a d MMF o f t h e magnet , i s

I t i s n o t e d t h a t , F r e m a i n s unchanged a l o n g t h e who le s l o t p i t c h and Fc = FJrhm/L+,. herefo fore, F and Fm. a t t h e s p e c i f i c p o - s i t i o n s x=+X/2 may be a c c e p t e d as t h e 9 p o t e n t l h l d r o p s i n m a g n e t i c c i r c u i t c a l c u l a t i o n s f o r PM mach ines . The e q u i v a l e n t c i r c u i t i s shown i n F i g . Ba.

a b

F i g . 0 The e q u i v a l e n t m a g n e t i c c i r c u i t

F o r c o n v e n i e n c e , e s o e c i a l l y i n PM g r a p h i c a l methods, E q . ( l B ) may be r e w r i t t e n a s

g . i s c a l l e d t h e c a l c u l a t e d a i r gap, c o n t a i n i n g t h e i n c r e m e n t o f t h e e q u i v a l e n t magnet h e i g h t due t o " o n - u n i f o r m f i e l d d i s t r i b u t i o n i n t h e magnet . The e q u i v a l e n t c i r c u i t fromEq<19) i s shown i n F i g . 8 b . I t i s n o t e d t h a t , b o t h F ' and g . a r e t h e c a l c u l a t e d v a l u e s , a n d n o t t h e p h y s i c a l q u a n t i t i e s . ' I t i e l c l e a r t h a t , t h e S h a l a b y ' s e f f e c t i v e a i r gap i s t h e sane a s g . i n t h i s p a p e r . I t seems t h a t d e f i n i n g t w o d i f f e r e n t C a r t e r ' s c o e f f i c i e n t s i n ( 1 ) w o u l d be u n n e c e s s a r y . .

CONCLUSIONS

An a n a l y t i c a l l y e x t e n d e d f i e l d model f o r s l o t t e d PM mach ines i s p r e s e n t e d i n t h i s p a p e r . By u s i n t h e mode l , t h e C a r t e r ' s coe - f f i c i e n t f ~ r PM mach ines can be e a s i y y d e r i v e d f r o m t h e C a r t e r ' s c l a s s i c a l f o r m u l a w i t h o u t a c t u a l l y s o l v i n g t h e c o m p l i c a t e d bound- a r y - v a l u e o r o b l e m d e s c r i b e d bv t h e o r i o i n a l f i e l d mode l . The c l o s e d - f o r m f i e l d s o l u t i o n i n d i r e c t i y obtain:! by - t h e method i s e x a c t and much s i m p l e r t h a n t h a t o f p r e v i o u s l y a p p r o x i m a t e methods. T h i s p a p e r may be c o n s i d e r e d as a f u r t h e r g e n e r a l i z a t i o n o f t 5 e e a r l i e r work done bv F. W . C a r t e r .

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REFERENCES

G. QISHAN AND G. HONGZHAN

H . May a n d M. Sha laby , " F e l d k u r v e n und E i n f l u s s d e r N u t u n g i n p e r m a n e n t e r r e g t e n Synchronmaschinen" . A r c h i v fur E l e k t r o t e c h n i k , "01. 59, p . 243. 1977.

C. Kramer. "The e f f e c t o f s l o t t i n g i n p e r m a n e n t i c s y n c h r o n o u s mach ines" . P r o c . o f t h e I C E m . PM/l , p. 287. 1980.

F.W. C a r t e r . " A i r - G a p I n d u c t i o n " . E l e c t r i c Wor ld , "01. 38. p.884. 1901 .

APPENDIX A DERIVATION OF THE MAGNETIC SCALAR POTENTIAL FOR Pm

MACHINES

A p p l y i n g t h e boundary c o n d i t i o n s = 0 a t x = ' v 2 , t h e s i n e t e r m u a n i s h e s i n p o t e n t i a l exp ress ion . ' f he s o l u t i o n Eo E q . ( 2 ) becomes

F o r PM r e g i o n : @ ( x . h m ) = 0, a t y= hm.

S u b s t i t u t i n g i t i n t o Eq. (21 ) , we g e t

F o r a i r gap reg ion :C$(x . -g )= f ( x ) , a t y=-g. S u b s t i t u t i n g i t i n t o Eq. (21 ), we g e t

ZTI 2 n - '~i s i n h ( i - - g ) t x ) c o s ( i - - x ) d x

h h

-A209 + 820

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SLOTTED PERMANENT MAGNET MACHINES 283

From Eqs. ( 2 3 ) and ( 2 6 ) . t h e m a g n e t i c s c a l a r p o t e n t i a l i n Pm r e g i o n can b e w r i t t e n a s

m

(6)

From Eqs. ( 2 5 ) and ( 2 6 ) . t h e m a g n e t i c s c a l a r p o t e n t i a l i n a i r gap

APPENDIX B MAGNETIC SCALAR POTENTIAL I N EXTENDED FIELD NODEL

A p p l y i n g t h e new boundary c o n d i t i o n s Eq. ( 9 ) . we g e t

S u b s t i t u t i n g Eq. ( 2 8 ) i n t o Eq. ( 2 1 ) . t h e m a g n e t i c s c a l a r p o t e n t i a l i n t h e e x t e n d e d f i e l d model can b e w r i t t e n a s

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G. OISHAN AND G. HONGZHAN

C o m p a r i n g E q . ( 2 9 ) w i t h E q . (7), i t i s c l e a r t h a t b o t h p o t e n - t i a l e x p r e s s i o n s a r e o f t h e s a m e m a t h e m e t i c a l f o r m u l a t i o n . S o i t may b e c o n c l u d e d t h a t t h e a i r g a p p o t e n t i a l f u n c t i o n s F o r b o t h f i e l d m o d e l s a r e i d e n t i c a l i n a c t u a l a i r g a p r e g i o n , n a m e l y

Manuscript received in final fonn, December 10, 1984 Request reprints from Gu Oishan

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