A CONTRIBUTION TO THE UNDERSTANDING OF A.C. LOSSES IN MULTIFILAMENTARY WIRES

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A CONTRIBUTION TO THE UNDERSTANDING OF A.C. LOSSES IN MULTIFILAMENTARY WIRES

P. Rem, F. van Beckum, D. Dijkstra, L.J.M. van de Klundert

To cite this version:

P. Rem, F. van Beckum, D. Dijkstra, L.J.M. van de Klundert. A CONTRIBUTION TO THE UNDER-

STANDING OF A.C. LOSSES IN MULTIFILAMENTARY WIRES. Journal de Physique Colloques,

1984, 45 (C1), pp.C1-471-C1-474. �10.1051/jphyscol:1984196�. �jpa-00223752�

JOURNAL DE PHYSIQUE

Colloque C1, suppl6ment au n o 1, Tome 45, janvier 1984 page C1-471

A CONTRIBUTION TO THE UNDERSTANDING OF AmCe LOSSES IN MULTIFILAMENTARY WIRES

P.C. Rem, F.P.H. van ~eckum*, D. ~ijkstra* and L.J.M. van de Klundert -meate University of TechnoZogy, Department of AppZied Physics, Z ~ e p a r t m e n t of Applied Mathematics, P.O.B.

217,

7500 AE Enschede, The Netherlands

Resum6 - Nous presentons un modkle numsrique, base sur les dquations de Maxwell, pour calculer la dissipation d'un fil multifilamentaire, en champs magnetiques transverses.

Abstract - We present a numerical model, based on Maxwell's equations,for calculating transverse field losses in multifilamentary wiras.

Introduction

In the past few years, the problem of finding the electromagnetic field in a multi- filamentary wire, when placed in a changing perpendicular magnetic field, has been extensively studied. Although various authors [1,2,3:l have contributed approximate solutions to the problem, no one, to our best knowledge, has shown a way to solve it rigorously.

In this paper, we describe a method, which has yielded results in simple cases, but may also be applied to more general problems.

The method

In general the method is iterative in that the magnetic field components are assumed to be known functions of space and time. We first solve E0 from the simple equations:

( 1 )

a +

^:E = -r

i ; $ = o :

^:E = 0 , L

( 2 )

a $ ~ ; = r 8 z - g h 0 ;

$ = : : E ~ = o ,

L

(3)

E O = - ~ E O

0

2nr

z

which are essentially Maxwell's equations and the condition that E0 be perpendicular to the direction of the filaments. To find the real electric field

E ,

we divide the cross-section of the wire into three regions, as shown in figure 1 and write E as

( 4 ) E

= E O + V Q ~

;

i

= 0,1,2,

where superscript i distinguishes between the regions in figure 1.

figure 1. the qeometry.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984196

C1-472 JOURNAL DE _PHYSIQUE

I t i s e a s y t o s e e t h a t Q0 = 0 and Q1 depends o n l y on r s i n c e b o t h r e g i o n s a r e non- s a t u r a t e d and s o

( 5 ) V Q ' * ' - ~ =

o

; p = ( 0 , 2 l r r , ~ ) . P

C o n s e r v a t i o n o f c u r r e n t i n r e g i o n 1 r e l a t e s 0 1 ( r ) t o t h e s h a p e of t h e boundary r = r o ( $ ) between 1 and 2:

O n t h e boundary r = r O ( $ ) we have

w h i l e on t h e o u t e r e d g e , r = R:

I n r e g i o n 2 , t h e c o n d i t i o n of z e r o d i v e r g e n c e o f t o t a l c u r r e n t d e n s i t y , (normal and s u p e r c o n d u c t i n g ) ,

(9) d i v j = - d i v j

n s '

l e a d s t o a L a p l a c e problem f o r

I f , i n a d d i t i o n , we knew t h e component En o f E , normal t o r = r o ( $ ) , we would have a n o t h e r d i f f e r e n t i a l e q u a t i o n f o r Q1 and ro ( $ 1 .

The s o l u t i o n o f t h i s s e t o f e q u a t i o n s i s found by t h e f o l l o w i n g i t e r a t i o n scheme:

( a ) i n i t i a l g u e s s f o r En

( b ) s o l v e f o r r O ( $ ) , Q1 ( r ) u s i n g ( 6 ) and ( 1 0 ) ( c ) s o l v e t h e L a p l a c e problem

2 .

(4 1s known on r O ( $ ) by ( 7 ) ; Q 2 = 0 i f $ = n/2 by symmetry;

ara2

= -EO on r = R )

( d l compute

vQ2 I

from t h e c a l c u l a t e d 0 - v a l u e s i n r e q i o n 2 r=ro ( $

( e ) S i n c e V Q i s c o n t i n u o u s on t h e boundary by ( 7 ) we c a n r e c a l c u l a t e E from

( f ) Check on convergence, ( r e t u r n t o b)

( g ) p r i n t s o l u t i o n and c a l c u l a t e d v a l u e s o f t h e power and t h e induced c e n t r a l m a g n e t i c f i e l d .

At s t e p ( f ) En i s compared w i t h p r e v i o u s l y o b t a i n e d v a l u e s f o r En, and t h e i t e r a t i o n i s s t o p p e d when a convergence c r i t e r i o n i s s a t i s f i e d .

The r e s u l t s

The r e s u l t s shown a r e c a l c u l a t e d f o r t h e case t h a t t h e magnetic f i e l d i s uniform and a i s i s o t r o p i c and independent of r. I n t h i s c a s e t h e problem i s depending only on t h e two parameters

i *

and s o a r e t h e s c a l e d v a l u e s of t h e induced c e n t r a l magnetic f i e l d B and t h e power p*:

I f t h e e x t e r n a l f i e l d i s dominating, t h e approximation of t h e magnetic f i e l d by a uniform one with magnitude

i e x t

i s convenient. However i f B g e t s l a r g e r with r e s p e c t t o B

,

f l u c t u a t i o n s of e.g.

j c ( B ) become i n c r e a s i n g l y important and t h e problem should be i t e r a t e d f o r t h e t r u e magnetic f i e l d .

The f i g u r e s

For s e v e r a l v a l u e s of t h e parameters a and 6 f i g . 2-4 show t h e shape of t h e s a t u r a t e d region. For l a r g e v a l u e s o f B t h e s e d i f f e r markedly from e l l i p s e s .

F i g . 6 , 7 show t h a t t h e reduced v a l u e s of P* and ~ i * do not depend h e a v i l y on t h e parameters. Fig. 5 shows t h e f u n c t i o n Y , where

Y

i s d e f i n e d by E = EZez

+

V Y . Note t h a t a r Y = 0 a t t h e boundary. E a r l i e r s o l u t i o n s , o b t a i n e d by o t h e r a u t h o r s , f a i l e d t o s a t i s f y t h i s boundary c o n d i t i o n .

The a u t h o r s a r e indebted t o W. van H a l t e r e n f o r programming t h e numerical s o l u t i o n of L a p l a c e ' s problem.

References

[ I ] CAMPBELL, A.M., Cryogenics, Feb. (1981)107.

[ 2 1 KANBAFW, K., KUMBOTA, Y., OGASAWARA, T., PrOC. ICEC, (1981)715.

C31 MURPHY, J . H . , WALKER, M.S., Adv. Cryogenics Eng.

24

(19781406.

JOURNAL DE PHYSIQUE

P i g . 2: S a t u r a t e d r e g i o n when B = 1. F i g . 3 : S a t u r a t e d r e g i o n when

3

= 4.

~ J o i s i n d i c a t e d i n t h e f i g u r e . aJg i s i n d i c a t e d i n t h e f i g u r e .

F i ? . 4 : Saturated r e g i o n when 8 = 25. F i g . 5 : @ ( x , y ) ;

aJfl is i n d i c a t e d i n t h e f i g u r e .

---

9 ~ f R 1 S f l P i : l l F . in1 - .I

*

L I.

F i g . 6: P a s a f u n c t i o n of a and B. Fir;. 7: B s s a f u n c t i o n of a a n d 6 .