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The Fringing Effect in PM Electric MachinesGu Qishan a & Gao Hongzhan aa Harbin Institute of Technology, Harbin, People's Republic of ChinaPublished online: 24 Oct 2007.

To cite this article: Gu Qishan & Gao Hongzhan (1986) The Fringing Effect in PM Electric Machines, Electric Machines & PowerSystems, 11:2, 159-169, DOI: 10.1080/07313568608909170

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THE FRINGING EFFECT IN PM ELECTRIC MACHINES

G U QISHAN and GAO HONGZHAN Harbin Institute of Technology Harbin, People's Republic of China

ABSTRACT

To p r e d i c t t h e f r i n g i n g f l u x d i e t r i b u t i o n a l o n g t h e a x i a l l e n g t h of a permanent magnet e l e c t r i c machine, a segmented perma- n e n t magnet model, i n which magnets f a c e a s l o t t e d and i n f i n i t e l y permeable boundary, i s p r e s e n t e d i n t h i s paper. By us ing t h e model and t h e method of F o u r i e r expans ion , t h e a n a l y t i c a l s o l u t i o n t o t h e f r i n g i n g f i e l d problem can be e a s i l y determined. With t h e a i d of t h e a n a l y t i c a l r e s u l t s , t h e e f f e c t i v e armature l e n g t h f o r per - manent magnet e l e c t r i c machines due t o t h e a x i a l e x t e n s i o n of t h e magnet may be obta ined.

INTRODUCTION

A t p r e s e n t , permanent magnet e l e c t r i c machines a r e most wide- l y used i n i n d u s t r i a l , m i l i t a r y and domest ic a p p l i c a t i o n s . I n ord- e r t o p r e d i c t t h e i r performance, t h e knowledge of t h e b a s i c f i e l d phenomena i n t h e air gap o f PM e l e c t r i c machines i s necessary .

The f r i n g i n g e f f e c t i s one of t h e b a s i c f i e l d phenomena i n PM e l e c t r i c machines. Owing t o t h e e x i s t e n c e of t h e magnet wi th low p e r m e a b i l i t y i n PM machines, t h e i r f r i n g i n g f i e l d d i s t r i b u t i o n i n t h e v i c i n i t y of t h e a i r gap d i f f e r s from t h a t i n wound f i e l d ones. I n o r d e r t o e s t i m a t e t h e e f f e c t of a x i a l e x t e n s i o n of t h e mamet . i t i s a l s o n e c e s s a r y t o compute t h e f r i n g i n g f i e l d d i s t r i b u t i o n . Although t h e f r i n g i n g f i e l d problem i n wound f i e ld ,mach ines was w e l l so lved by u s i n a t h e conformal t r ans fo rma t ion s i x t y y e a r s ago, t h e f r i n g i n g e ? f e c t i n PM ones has h a r d l y e v e r been- i n v e s t i - ga ted . The on ly in fo rma t ion on t h e e f f e c t o f a x i a l e x t e n s l y of t h e magnet i n PM machines is t h e e m p i r i c a l e s t i m a t i o n g iven i n . I n o r d e r t o analyze t h e e f f e c t of magnet and a i r gap geometry on t h e f r i n g i n g f i e l d d i s t r i b u t i o n , t h e a n a l y t i c a l method i s much more economic and e f f i c i e n t as compared w i t h t h e d i s c r e t i z e d f i e l d model based numer ica l methods.

The aim of t h e paper i s t o s e a r c h o u t an i d e a l i s e d permanent magnet f i e l d model s u i t a b l e f o r a n a l y t i c a l hand l ing of t h e f r i n g - i n g f i e l d problem. A segmented PM model i s p resen ted i n t h i s pa- per . By us ing t h e model and t h e method o f F o u r i e r expans ion , t h e a n a l y t i c a l s o l u t i o n t o t h e f r i n g i n g f i e l d problem can be e a s i l y

Electric Machiner and Power Sysrern% 11 :159-169. 1986 Copyright 0 I 986 by Hemisphere Publishing Corporation

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G. QISHAN AND G. HONGZHAN

determined.

FIELD MODEL

Fig.1 shows the longitudinal sec t ion of a PM e l e c t r i c machine, i n which the axial length o f the magnet i s general ly longer than t h a t of the armature core. To solve the problem ana ly t i ca l ly , a segmented PM model, i n which the magnets face a s l o t t e d and i n f i - n i t e l y permeable boundary. i s proposed, as shown i n Fig.2. The

y L m 1 1 back i ron -

2 magnet

1 3 core

Fig.1 The longi tudina l sec t ion of a PM e l e c t r i c machine

Fig.2 The segmented permanent magnet model

f i e l d model i s very much a l i k e t o t h a t one used t o analyze the a i r gap f i g l d of a s l o t t e d PM e l e c t r i c machine along the armature per- iphery . The only difference between them i e t h a t the magnet of i n f i n i t e length has been replaced by a s e r i e s of segmented magnets extending t o i n f i n i t y along the machine axis . Let 8 ' be the s p l i t width between segmented magnets; s,hg--- the fictitious s l o t width and depth; L ,L --- the a x i a l magnet and core lengths. Then the period of thef magnet system may be expressed as

It i s c l e a r t h a t the ac tua l magnet eystem may be represented by one of r e p e t i t i v e elements of the magnet system, provided t h a t the following condition holds, 1.e.

where g i s the a i r gap length; hm i s the magnet height.

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FRINGING EFFECT IN PM ELECTRIC MACHINES 161

This impl ies t h a t an a r t i f i c i a l boundary A A ' f a r enough from t h e end s u r f a c e of t h e magnet must be s p e c i f i e d t o approximate t h e ac- t u a l f i e l d d i s t r i b u t i o n , as shown i n Fig.2.

For symmetry, we can cons ide r only a h a l f of one r e p e t i t i v e element sp read ing over one s l o t p i t c h A, as shown i n Fig.?. The fo l lowing assumptions a r e made t o s impl i fy t h e a n a l y s i s : 1 . V a r i a t i o n s a long t h e armature pe r iphery a r e ignored. This l e a d s

t o a two-dimensional f i e l d a n a l y s i s . 2. Armature core and the back i r o n a r e considered t o be i n f i n i t e l y

permeable, i.e.&,= w.

3. Anisot ropic permanent magnets(such ae f e r r i t e s , SmCo e t c . ) a r e used and uniformly magnetized t o s a t u r a t i o n i n t h e p3efe r red d i r e c t i o n , which co inc ides wi th t h e y - d i r e c t i o n i n Fig.3. The c o n s t i t u t i o n equa t ion of t h e magnets may be expressed as follows:

By =/UmHy + / U O M =porn Y + M) ( 3 )

where /Urn = / u ~ , i . e . t h e pe rmeab i l i ty of t h e magnet is assumed

t o b e y o ; M t h e magnet iza t ion v e c t o r of t h e magnet, I M ( = Br$uo.

4. s / 2 , s ' / 2 >> (a, + g).

5. (A- s ) /Z >> (hm + g ) , then B (W2.0) = Brhm/(hm + g ) . Y 6. he>>(hm + g ) , o r i n t h e l i m i t i n g case hs = .o.

Fig.3 One h a l f of a r e p e t i t i v e element

For uniformly magnetized magnets, t h e volume c u r r e n t d e n s i t y vanishes. Consider ing t h e assumption 1 and 2 , t h e problem at hand l e a d s t o s o l v i n g t h e fo l lowing Laplace 'e equat ion

According t o t h e assumption 3, i . e . & , = 3' we may cons ide r

t h a t t h e r e g i o n 3 is wholly f u l l of magnet mate i a l , b u t t h e f i c t i - t i o u s p o r t i o n of magnet i s unmagnetized. It can be seen from Pig.2,

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162 G. QISHAN AND G. HONGZHAN

the s p a t i a l d i s t r i b u t i o n of t h e source f u n c t i o n Mix) i s a homopolar one. The magnet izat ion v e c t o r M(x) i n Fig.3 may be expressed i n terms of t h e Four ie r expansion

MAGNETIC SCALAR POTENTIALS

The magnetic s c a l a r p o t e n t i a l s i n d i f f e r e n t r eg ions may be expressed as follows:

< > > For s l o t r e g i o n n l : / x i = s / 2 , hs = y = 0 -

Vl(x.y) = [Alish(i$y) + B l i c h ( i ~ y ) ] cos( iTx) i=1

+ *lo9 + B ~ o (6) > > For air gap r e g i o n f 1 2 : = = -g

For magnet r e g i o n f 1 3 : -g 2 y 2 -($ + g) -

For s i m p l i f y i n g t h e a n a l y s i s , we def ine an a n a l y t i c a l l y exten- ded func t ion V (x ,y ) wi th the pe r iod 2s f o r t h e r e i o n R and i t s neighbouring o&siextending from x = i s / 2 t o x = ~ s y and l e t

Vle(x'y) = V1(x,y) ( 9 )

< 7 For extended s l o t r e g i o n f l : 1x1 = 8, hs 2 y = 0 From Eqs.(6) and (91, we gk%

4 The boundary c o n d i t i o n now is used: Vle(x.hs) = 0. 1x1 = 8. Then

a sh[iZ(y-h )] ,

V ( x e y ) = Z A ----!----2-cos(~?x) + ~ ~ ~ ( y - h ~ ) (12) l e i=1 l i ch ( i2hs ) s

Considering Eq. (10) and v l e ( t s / 2 ,Y) = 0 , y i e l d s

A 1 , 2 i = 0 , A I 0 = 0.

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FRINGING EFFECT I N PM ELECTRIC MACHINES

The boundary c o n d i t i o n i s used f o r magnet region: V (x,-$-~) = 0. So Eq.(8) can be w r i t t e n a s 3

The c o n 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 y = -g now a r e used:

FURTHER DETERMINATION OF FOURIER COEFFICIENTS

I n o r d e r t o determine f u r t h e r t h e F o u r i e r c o e f f i c i e n t s i n Eqs. (13) and (18) . t h e c o n t i n u i t y c o n d i t i o n s at t h e i n t e r f a c e between R1 a n d R 2 now a r e used: D

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164 G . QISHAN A N D G. HONGZHAN

It i s noted t h a t the r i g h t hand of Eq.(19) o r (21) must be ex- panded i n t o the Fourier s e r i e s with the period X i n determining the Fourier coe f f i c i en t s f o r V2(x,y), whereas the r i g h t hand of Eq.(22) muat be expanded i n t o t h a t one with the period 2s i n determining the corresponding coef f ic ien ts f o r Vle(x,y).

After considerable mathematical manipulation, these Fourier coe f f i c i en t s can be wr i t ten a s

where A i O = $ AZ0;

t h [(2i-1 )$hs] = 1 f o r i n f i n i t e s l o t depth.

Pu t t ing i = l,'..,m f o r A and j = 1 ;..,n f o r A . we get a s e t of m+n+l algebraic equatiAAs with m+n+l unknowns?jFor the f i e l d problem with i n f i n i t e s l o t depth, Eqs.(23)-(25) may be rewr i t ten a s

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FRINGING EFFECT IN PM ELECTRIC MACHINES 165

Therefore , t h e air gap p o t e n t i a l f u n c t i o n may be approximated by t h e f i n i t e terms of F o u r i e r expansion.

EFFECTIVE ARMATURE LENGTH

The a i r gap f l u x d e n s i t y a t y = 0 is

2 r 2-r = - L + , [ ~ A ~ ~ 1 xcos(i--x) + A,jo a ' i=1 h (30)

The e f f e c t i v e armature l e n g t h is

The increment of armature l e n g t h due t o t h e a x i a l magnet ex- t e n s i o n i s

Some dimensions of t h e f i e l d model i n Fig.3, such as h +g. x and 8 , may be considered t o be c o n s t a n t and a r e given accor8 ing t o t h e assumptions 4 and 5 p r i o r t o t h e f i e l d computation. A s a re - sult, A l m ( o r i n terms of a - s t . ~ - s W and h / g ( o r i n terme of j V g ) a r e t h e on ly two independent v a r i a b l e s i n %qs.(26)-(28). Givin d i f f e r e n t va lues of a1 and $/g, t h e a l g e b r a i c equa t ions 26 - 28 can be e a s i l y solved by Gauss method. The s o l u t i o n t o Eqs.[26]-128]

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G. QISHAN AND G. HONGZHAN

Fig.4 The normal ised increment of a rmature

l e n g t h A l a e = f ( A l m , h m / g )

g i v e s a f ami ly o f c u r v e s a l = f ( A 1 , h /g ) , as shorn i n Fig.4. The computing t ime r e q u i r e d a e f o r dete%niWing 48 a i r gap f i e l d so lu - t i o n s ( p u t t i n g m=n=15 d u r i n g t h e f i e l d computat ion ) and t h e cor- r e spond ing v a l u e s o f a l a e on a 280 microcomputer i s about two hours.

The e f f e c t i v e armature l e n g t h is

Lae = La + Alae(hm + g )

From Fig.4 i t i s c l e a r t h a t :

1. The normal ised increment of a rmature l e n g t h A l becomes nega- t i v e when a 1 = 0. Th i s i m p l i e s t h a t L i s ali3%ys l e s s t h a n La i n PM machings when Lm = L . This i s dag t o t h e f a c t t h a t t h e i n t e r f a c e between t h e ma& and t h e a i r gap i s no l o n g e r an e q u i p o t e n t i a l s u r f a c e . On t h e c o n t r a r y , L is always g r e a t e r t h a n La i n wound f i e l d machines when t h e @&le l e n g t h L = La.

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FRINGING EFFECT I N PM ELECTRIC MACHINES 167

2. The e f f e c t o f a l on a 1 is predominant and t h e e f f e c t i v e n e s s of t h e a x i a l m a d e t ext@&sion i s q u i t e obvious i n PM machines. Furthermore, a knee p o i n t e x i s t s a t ~l * 1.4-1.6 i n t h e family of curves. It i s c l e a r t h a t A 1 must b&! chosen below t h e knee p o i n t f o r t h e c o s t - e f f e c t i v e d e of magnet m a t e r i a l .

3. The e f f e c t of h /g on a1 i s far l e s s than t h a t o f A l . The weakly dependenEe of A 1 ?on t h e r a t i o $/g impl ies t h h the e f f e c t of air gap lengtfl i s no t s o s e n s i i v e i n PM machines wi th t h e sum ($ + g) be ing unchanged.

4. It is i n t e r e s t i n g t h a t AL i s independent of t h e a c t u a l arma- t u r e l e n g t h L . But, the % o r t e r t h e armature l e n g t h , t h e b e t t e r t h e ef fect ivefSeas of t h e axial magnet e x t e n s i o n , a s shown i n Table 1.

Fig.5 shows a p o r t i o n of the family of curves i n Fig.4 wi th magnifying s c a l e .

Fig.5 The normalised increment of armature

l e n g t h A l a e = f ( Alm.hm/g )

The maximum a x i a l magnet ex tens ion commonly used i n PM ma- ch ines i s about A L = 6-8 mm. Table 1 shows two des ign examples f o r PM machines, 811 dimensions of which a r e taken from2 and given i n mi l l ime te r s .

TABLE 1 Two des ign examples f o r PM machines

La A Lm hm+g g n l m hm/g A lac A

1. 21.9 7.3 6.36 0.76 1.15 7.4 0.80 5.1 27

2. 32.4 7.6 6.09 0.76 1.24 7.0 0.84 5.1 37.5

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168 G. OISHAN AND G. HONGZHAN

COMPARISON OF FIELD SOLUTIONS

Fig.6 shows a comparison of f i e l d s o l u t i o n s obta ined by d i f - f e r e n t computation methods. Curve 1 i n Fig.6 i s computed by t h e method of Four ie r expansion, whereas curve 2 by t h e f i n i t e d i f f e r - ence method. They a r e i n good agreement wi th each o t h e r . I n addi- t i o n , Fig.6 a l s o shows a comparison of f r i n g i n g flux d i s t r i b u t i o n s a long t h e a x i a l l e n g t h f o r both PM and wound f i e l d machine8 when L = L 0 La. Curve 7 denotes the y-component of t h e a i r gap induc- d o n b o n g the a i r gap c e n t e r l i n e y=-g/2 f o r wound f i e l d machines. whereas curve 1 r e p r e s e n t s t h a t one f o r PM machines wi th hpl/g = 4. The y-components of t h e a i r gap induc t ion B* a r e expresse i n nor-

gY

Fig.6 The f r i n g i n g f l u x distribution a long the a i r gap c e n t e r l i n e y = -g/2. 1.2--- f o r PM machines.

L = L m a ' hm/g = 4 , s / 2 = 7(%+g), (h-e)/2 = 6($+g); I--- computed by t h e method of Four ie r expansion, 2--- computed by t h e FDM. 3--- f o r wound f i e l d

machines, Lp = La.

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FRINGING EFFECT I N PM ELECTRIC MACHINES 169

malised form. From Fig.6 it i s c l e a r t h a t , because of the e x i s t e n - ce of the magnet wi th low permeab i l i ty , t h e i n d u c t i o n wi th in t h e a i r Ear, i n PM machines decays far e a r l i e r than t h a t i n wound f i e l d ones; ;hereas t h e f r i n g i n g f l u x o u t s i d e the a i r gap i n PM machines decays more r a p i d l y than t h a t i n wound f i e l d ones.

CONCLUSIONS

The segmented permanent magnet model and t h e a n a l y t i c a l method p resen ted i n t h i s paper f o r s o l v i n g t h e f r i n g i n g f i e l d problem i n PM machines i s economic and e f f i c i e n t as compared w i t h t h e numeri- c a l methods based on d i s c r e t i z e d f i e l d model. The accuracy and con- vergence a r e f u l l y acceptable . The a n a l y t i c a l r e s u l t s and t h e cur- ves A l = f ( ~ l . h /g) a r e of g r e a t g e n e r a l i t y and may be used t o predic*ethe e f f 8 c t i v e armature l e n g t h i n PM machines.

REFERENCES

Frey, "Appl ica t ions of Conformal Transformation t o P r a c t i c a l Problems i n E l e c t r i c Machines", A . d. E l . Ing. , Bd.4, Spr inger ,

B e r l i n , 1925. J. R. I r e l a n d . "Ceramic Permanent Magnet Motors", 1968. Gu Qishan and Gao Hongzhan, "The e f f e c t of S l o t t i n g i n PM Elec-

t r i c Machines". E l e c t r i c Machines and Power Systems, vol .10,

N0.4. 1985.

Manuscript received in final form, Augusf 13, 1985 Request reprints from Gu Oishan

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