IEEE Transac t ions on Magnetics vel. MAC-13, NO. 5, September 1977 1125

FINITE ELEMENT ANALYSIS OF THE SKIN EFFECT I N CURRENT CARRYING CONDUCTORS

M.V.K. Chari and Z.J. Csendes General Electric Company

Schenectady, New York 12301

Abs t r ac t 4. The p e r m e a b i l i t y o f t h e i r o n p a r t s is s i n g l e v a l u e d

The c u r r e n t d e n s i t y d i s t r i b u t i o n i n a conductor of f i n i t e s i z e i s a f f e c t e d by the presence of eddy 5. The conduct iv i ty o f the media i s cons t an t . c u r r e n t s i n t h e c o n d u c t o r . T h i s phenomenon, gene ra l ly known as s k i n e f f e c t , c a u s e s ohmic l o s s e s i n t h e con- ductors and alters the i r magnet ic induct ion . This Der iva t ion of the Two-Dimensional Di f fus ion Equat ion

and t i m e i n v a r i a n t .

6 . Tempera ture e f fec ts a re neglec ted .

p a p e r p r e s e n t s a n a n a l y s i s o f s k i n e f f e c t phenomena u s i n g t h e t r i a n g u l a r f i n i t e e l e m e n t method. The re- s u l t s o b t a i n e d b y t h i s method a r e compared wi th t hose o f c l a s s i c a l o n e - d i m e n s i o n a l a n a l y s i s f o r ( i ) a con- d u c t o r i n f r e e s p a c e a n d ( i i ) a c o n d u c t o r i n t h e presence of an iron boundary.

In t roduc t ion

The presence of eddy c u r r e n t s i n c o n d u c t i n g m e d i a a f f e c t s t he magne t i c i nduc t ion and t he cu r ren t dens i ty d i s t r i - b u t i o n i n t h e r e g i o n . T h i s phenomenon can produce sig- n i f i c a n t l o s s e s i n t h e c o n d u c t o r s , and i s , t h e r e f o r e , o f i n t e re s t i n e l ec t r i ca l equ ipmen t des ign . Of t h e var ious methods o f ana lys i s in ex is tence , a good number are based on a one-dimensional analysis and only in a few cases of geometrical symmetry, a two-dimensional analysis has been a t tempted by c lass ical methods. For g e n e r a l a p p l i c a t i o n t o i r r e g u l a r c o n t o u r s and m a t e r i a l boundaries such as t h o s e t h a t o c c u r i n t y p i c a l e l e c t r i - cal equipment , a numerical method is necessary. Earlier work7 o n . t h e s u b j e c t r e l a t e s t o s k i n e f f e c t i n s l o t em- bedded conductors us ing e igenvalue ana lys i s . In th i s pape r an ana lys i s employ ing t r i angu la r f i n i t e e l emen t s i s p r e s e n t e d i l l u s t r a t i n g a novel technique of de t e r - m i n i n g t h e e d d y c u r r e n t d i s t r i b u t i o n i n c u r r e n t c a r r y - ing conductors .

Sec t ion 1 d e a l s w i t h t h e t h e o r e t i c a l a n a l y s i s o f t h e f i n i t e element method and i n S e c t i o n 2 , a comparison is made be tween t he r e su l t s ob ta ined by t h i s method and those o f t he c l a s s i ca l one -d imens iona l ana lys i s fo r i ) a conduc to r i n f r ee - space and i i ) a c o n d u c t o r i n t h e presence of an iron boundary.

Sec t ion 1

Basic Assumptions

The a n a l y s i s p r e s e n t e d i n t h i s p a p e r is based on the following assumptions:

The c u r r e n t i n t h e c o n d u c t o r is assumed t o b e d i r ec t ed a long t he conduc to r and is assumed t o v a r y s i n u s o i d a l l y i n time.

The magne t i c vec to r po ten t i a l has on ly a 2 d i r e c t e d component and does not vary i n magni- t ude a long t he l eng th o f t he conduc to r . It a l s o v a r i e s s i n u s o i d a l l y i n time.

Disp lacement cur ren ts a re neglec ted a t power f r e - quencies.

On the bas i s o f the above assumpt ions , Maxwell's equa- t i o n s i n t h e c u r r e n t c a r r y i n g r e g i o n become

v x H = J = J + J s (1)

v x E = - a w t (2) Where Je is t h e eddy cu r ren t dens i ty and Js is t h e con- d u c t i o n c u r r e n t d e n s i t y

The c o n s t i t u t i v e r e l a t i o n s are

J = UE ( 4 ) where E is t h e e l e c t r i c f i e l d and pruo t h e perme- a b i l i t y . D e f i n i n g a v e c t o r A s u c h t h a t

B = V x A (5)

t he fo l lowing r e l a t ionsh ips may be obtained from equat ions (2) a n d ( 5 ) . a f t e r t h e n e c e s s a r y s u b s t i t u t i o n s

v x E = - a / a t ( v x A) ( 6 )

E = - a A / a t + VQ (7) o r

Wri t ing VQ i n terms of Js, t h e r e is

Since a s i n u s o i d a l v a r i a t i o n o f A has been assumed, equat ion ( 7 ) reduces to the form

E =-juA + Js /u (9 ) Subs t i tu t ion of equa t ions ( 3 ) , ( 4 ) , (5) and ( 6 ) i n t o equat ion (1) y i e l d s t h e f o l l o w i n g e x p r e s s i o n

l /prpo (V x V x A) = - juwA + Js (10) Using the wel l known v e c t o r i d e n t i t y

( V x V x A) = V(V.A) - V2A

t h e r e r e s u l t s by s u b s t i t u t i n g f o r V - A = 0,

l /prpo V ~ A = j u d - J, (11) Equation (11) is the required two-dimensional dif fusion e q u a t i o n s a t i s f y i n g t h e s p e c i f i e d b o u n d a r y c o n d i t i o n s f o r l i n e a r c o n d i t i o n s .

V a r i a t i o n a l Method a n d t h e F i n i t e Element Formulation

The two-dimensional diffusion equation can be expressed i n v a r i a t i o n a l t e r m s by t h e e n e r g y f u n c t i o n a l ( r e f . 4 ) F = / . f S V / 2 I VA 1' ds + //, k2/2 A2 d s - /fS. J s A d s where v = l/pruo and k2 = jwo ( 1 2 )

The min imiza t ion o f t he above func t iona l y i e lds t he s o l u t i o n t o t h e f i e l d p r o b l e m s a t i s f y i n g n a t u r a l boun- da ry cond i t ions , p rov ided t ha t i t s Euler equa t ion is i d e n t i c a l t o t h e o r i g i n a l d i f f u s i o n e q u a t i o n ( v i d e r e f . 1). The EuLer equat ion of a two-dimensional energy f u n c t i o n a l is g iven by ( re f . 2 )

a / a X (aF/aAx) + a/ay (aF/aAy) - aF/aA = 0 (13)

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where A, and % a r e p a r t i a l d e r i v a t i v e s of A a long t he x and y a x e s . S u b s t i t u t i n g f o r F i n equa t ion (11) from equa t ion (lo), one ob ta ins a l a x (v a ~ / a ~ ) + a / a y (v aA/ay) - K ~ A + . J ~ = o (14) Equation ( 1 4 ) is i d e n t i c a l t o t h e d i f f u s i o n e q u a t i o n (ll), s i n c e v = l / p rp0 - i s c o n s t a n t i n a f i r s t o r d e r f i n i t e e l e m e n t so tha t min imiza t ion of the energy f u n c t i o n a l i n e q u a t i o n ( 1 2 ) y i e l d s t h e r e q u i r e d solu- t i o n t o t h e problem.

As is customary i n f i n i t e e lement ana lys i s , the two d imens iona l reg ion of in te res t is d iv ided i n to tri- ang les ensu r ing t ha t t he t r i ang le edges co inc ide w i th t h e m a t e r i a l i n t e r f a c e s and boundaries . The r e l u c t i - v i t y a n d c o n d u c t i v i t y a r e assumed t o b e c o n s t a n t i n e a c h t r i a n g u l a r r e g i o n . I f t h e p o t e n t i a l f u n c t i o n i n e a c h t r i a n g l e is d e f i n e d t o b e a f i r s t o r d e r p i e c e - wise polynomial, then i t s v a l u e a t any p o i n t i n t h e r eg ion of t h e t r i a n g l e w i l l b e a l i n e a r i n t e r p o l a t e of i t s ve r t ex va lues . Thus

A(x,y) = 1 / 2 A E (ai + b i x + Ciy)Ai where i , j , m a r e t h e v e r t i c e s o f a t r i a n g l e and a ,b and c are geomet r i ca l cons t an t s de f ined by t he r e l a t ions

m

i (15)

a . = x .

b i = Y j - Ym c i = 31 - xj (16)

S u b s t i t u t i n g f o r A from equation (15) i n e q u a t i o n (12) and s e t t i n g i t s d e r i v a t i v e w i t h r e s p e c t t o each nodal p o t e n t i a l A i , A . and equal to zero , the fo l lowing expres s ion is d e r l v e d a f t e r c o n s i d e r a b l e a l g e b r a i c manipulation.

1 Jym - %yj

J .

aF/aAi = 1/4A2 ~ J C V [L] [A] + C N i N j A i - ZANiJ,) ds (17) where [L] =[ ( b i b i + c i c i ) , ( b i b j + c i c j ) ,

(bib, + c~c,) I (18) Nk = (ak + bkx f Cky) 1-41 = [Ai, A j , Am] (19 1

The f u n c t i o n a l is d i f f e r e n t i a t e d w i t h r e s p e c t t o A. and A,,, i n a similar manner to the above . J

I f the above process is repeated for each and every t r i a n g l e i n t h e two d imens iona l reg ion , an equat ion similar to equa t ion ( 1 7 ) w i l l be obtained which w i l l inc lude Poissonian , Laplac ian and d i f fus ion reg ions . Formal ly the resu l t ing equat ion can be expressed as

v [SI [A] + k2 [TI [A] = J [A/31 (20) where [SI and [TI can be evaluated f rom the geometr i - c a l c o e f f i c i e n t s of t h e i n d i v i d u a l t r i a n g l e s .

Boundary Conditions

The above var ia t iona l formula t ion conta ins homogen- eous Neumann boundary condi t ions unless another boundary condition i s exp l i c i t l y spec i f i ed . The re - f o r e , i t is only necessary to spec i fy f ixed po ten- t i a l boundar ies ( f lux- l ine boundar ies ) on the domain A f u r t h e r n e c e s s a r y c o n d i t i o n t h a t must be s a t i s f i e d is t h a t t h e i n t e g r a l o f t h e e d d y - c u r r e n t d e n s i t y ove r t he r eg ion must equal zero. This can be ex- pressed as

t i a l s o b t a i n e d by so lv ing equat ion ( Z O ) , on ly t he r e f e r - e n c e l e v e l and no t t he so lu t ion t o t he p rob lem w i l l b e changed. Fu r the r , t he impos i t i on of t h i s boundary con- d i t i o n s a t i s f i e s e q u a t i o n ( 2 1 ) a u t o m a t i c a l l y . The eddy- c u r r e n t v e c t o r p o t e n t i a l i s then given by

A i A i - JA*ds//ds (24) Magnetic Induction and Reluctivity

S ince B = V x A and A has only a component a long t he Z-axis , the value of / B I o b t a i n e d f o r t h e two-dimen- s iona l p roblem is

S u b s t i t i t u t i n g f o r A from equation (15) in equation (25) , i t is s e e n t h a t B is independent of x and y , and hence i s c o n s t a n t i n e a c h t r i a n g l e . The va lue o f B is , t h e r e f o r e , o b t a i n e d a s

IBI = 1 / 2 4 Y J b i A i + bjAj + b m h ) 2 + ( c iAi + c.A. + c,%)~ (26)

S i n c e t h e r e l u c t i v i t y is a func t ion of / B 1 . i t is a l s o J J

i ndependen t o f t he coo rd ina te sys t em fo r t h i s f i r s t o rde r f i n i t e element problem.

Sect ion 2

Applicat ions

To a s c e r t a i n t h e e f f i c a c y o f t h e method f o r s o l v i n g two- d imens iona l d i f fus ion problems, the fo l lowing two s imple c a s e s a r e c o n s i d e r e d f o r which c losed form so lu t ions a r e r e a d i l y a v a i l a b l e as a bas i s for compar ison .

i. A t h i n f l a t c o n d u c t o r i n f r e e s p a c e .

ii. A conduc to r i n t he p re sence of an iron boundary.

In bo th of the above cases , the magnet ic induct ion in the one-dimensional c losed form analysis is assumed t o have only an x component a s shown i n F i g . 1.

Z J

Fig . 1 .a S t r ip Conductor in Free Space .

0 = {Je - ds = pjE*ds = - jwpIA-ds (21) However, s ince the above equat ion cannot be exact ly s a t i s f i e d , a r e s i d u a l R w i l l r e s u l t s u c h t h a t

R = JJe* ds = - jWpJA * ds (22)

Fig. 1.b Conductor i n t he p re sence o f an I ron Boundary.

The closed form expressions f o r t h e c u r r e n t d e n s i t y d i s t r i b u t i o n f o r t h e a f o r e s a i d two cases were devel- .oped a n d t h e r e s u l t s a r e compared wi th t hose o f t he f i n i t e e l e m e n t a n a l y s i s i n F i g s . 2, 3 and 4 .

CURRENT DENSITY I A M P S I O m l 1140 1160 I180 1200 1220 1240 1260 r1645r1O41 I I I I I I I

Hence * res idua l = jA*ds//ds (23)

i 1 - FINITE ELEMENTMETHOD

5 m m DEEP

_-- CLASSICAL ONEDIMENSIONAL ANALYSIS

If Aresidual is now subtracted f rom each of the poten- Fig. 2.a Curren t Dens i ty Dis t r ibu t ion In A

Conductor In Free Space.

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1050 1100 1150 1200 1250 1300 1350 1400 ~ 1 6 . 4 5 ~ 1 0 ~ 1 CURRENT DENSITY l A M P S / O m l

I I I I I I 1

I - FINITE ELEMENTMETHOO --- CLASSICAL ONE DIMENSIONAL ANALYSIS

6.35 mm DEEP li I

Fig. 2 .b Current Densi ty Distr ibut ion In A Conductor In Free Space.

1000 1100 I200 1300 1400 1500 x16.45X1041 CURRENT DENSITY l A M P S / o m l

0.4 - c

0.3-

0.2 - \ \ - FINITE ELEMENT METHOD

CLASSICAL ONE DIMENSIONAL ANALYSIS

0

Fig. 2 .c Current Densi ty Distr ibut ion In A Conductor In Free Space.

C U R R E N T O E N S I T Y ( A M P S / O m )

d = 5mrn

- F I N I T E E L E M E N T M E T H O D --- C L A S S I C A L O N E D I M E N T I O N A L A N A L Y S I S

Fig. 3 .a Current Densi ty Distr ibut ion In A Conductor In The Presence Of An I ron Boundary.

Conclusions

The r e s u l t s o f t h e f i n i t e e l e m e n t a n a l y s i s compare we l l

C U R R E N T D E N S I T Y ( A M P S / o m ] 800 1000 1200 1400 1600 1800 XI6.45XlO? I I I I 1

4

d = lOmm

FINITE ELEMENT METHOD - - - C L A S S I C A L O N E D I M E N S I O N A L A N A L Y S I S

?

4.0

w

c)

Fig. 3 .b Current Densi ty Distr ibut ion In A Conductor In The Presence Of An I ron Boundary.

- FINITE ELEMENT METHOD ___ CLASSICAL ONE DIMENSIONAL ANALYSIS

Fig. 4 Var ia t ion Of A.C. Res i s t ance With a d

l a r g e l y depend on t h e number of elements chosen. A l so , s ince t he method g ives an approximat ion to the t r u e s o l u t i o n i n t h e l e a s t s q u a r e s e n s e , t h e l a r g e r t h e number of e lements , the c loser the boundary c o n d i t i o n s a r e s a t i s f i e d .

References

1.

2 .

3.

4 .

5.

w i th t hose of the c lass ica l one-d imens iona l ana lys i s for small conductor depths in bo th appl ica t ions . For h igher 6 - values of conductor depth, the one-dimensional solu- t i o n is n o t s t r i c t l y a p p l i c a b l e , a n d t h i s may account f o r t h e d i s p a r i t y .

Although i t is d i f f i c u l t t o e s t a b l i s h s t r i c t e r r o r bounds f o r t h e f i n i t e e l e m e n t method, excepting perhaps f o r homogeneous c a s e s , t h e d i s c r e t i z a t i o n e r r o r w i l l

7.

M.V.K. Chari , "Fini te Element Analysis of Non- l inear Magnet ic F ie lds in E lec t r ic Machines ," Ph.D. Disser ta t ion, McGil l Universi ty , Montreal , Canada, 19 70

F.B. Hildebrand, Methods of Applied Mathematics, Englewood C l i f f s , N . J . : P r e n t i c e Hall, 1965

O.C. Zienkiewicz, The F i n i t e Element Method i n Engineering Science, Mc-Graw Hill,London, 1971

P. Si lvester , "High-order polynomial t r iangular f i n i t e e l e m e n t s f o r p o t e n t i a l p r o b l e m s , " I n t . J. Engrg. Science, Vol. 7, pp. 849-861, 1969

M.V.K. Chari , "Fini te-Element Solut ion of the Eddy-current Problem i n Magnet ic Structures ," IEEE Trans. Vol. PAS-93, Number 1, 1973 J. Lammeraner and M. S t a f l , Eddy Currents , I l i f f e Books Ltd. , London, 1966

A. Konrad, J .L . Coulomb, J . C . Sabonnadiere and P .P . S i lves t e r , "F in i t e Element Analysis of S t e a d y S t a t e S k i n E f f e c t i n S l o t Embedded Conductor," A 76-189-1, IEEE Winter Meeting, Jan. 1976.