effect of slotting in pm electric machines

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This article was downloaded by: [Fondren Library, Rice University ] On: 03 May 2013, At: 08:17 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Electric Machines & Power Systems Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/uemp19 Effect of Slotting in Pm Electric Machines Gu Qishan a & Gao Hongzhan a a Department of Electrical Engineering, Harbin Institute of Technology, Harbin, People's Republic of China Published online: 22 Jan 2007. To cite this article: Gu Qishan & Gao Hongzhan (1985): Effect of Slotting in Pm Electric Machines, Electric Machines & Power Systems, 10:4, 273-284 To link to this article: http://dx.doi.org/10.1080/07313568508909127 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

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Page 1: Effect of Slotting in Pm Electric Machines

This article was downloaded by: [Fondren Library, Rice University ]On: 03 May 2013, At: 08:17Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Electric Machines & Power SystemsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/uemp19

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

To link to this article: http://dx.doi.org/10.1080/07313568508909127

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form toanyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses shouldbe independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims,proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.

Page 2: Effect of Slotting in Pm Electric Machines

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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SLOTTED PERMANENT MAGNET MACHINES

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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SLOTTED PERMANENT MAGNET MACHINES

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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282

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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