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H y s t e r e s is l o s s o f a r o u n d s u p e r c o n d u c t o rc a r r y i n g a d . c . t r a n s p o r t c u r r e n t i n a na l t e r n a t i n g t r a n s v e r s e f i e l dK . K a n b a r aD e p a r t m e n t o f P h y s ic s , Co l le g e o f Hu m a n i t i e s a n d Sc i e n c e s , N i h o n Un i v e r s i t y, Sa k u r a j o u s u i3 - 2 5 - 4 0 , Se t a g a y a k u , T o k y o 1 5 6 , J a p a nReceived 23 February 1987The hys te res is l oss in a cy l i nd r i ca l ha rd s u p e r c o n d u c t o r is i n v e s t ig a t e d t h e o r e t i c a l l y a d o p t i n gthe c r i t i ca l s ta te mode l fo r the case tha t an a l te rna t ing f ie ld is a p p l ie d t ransverse to thec o n d u c t o r c a r r y i n g a d . c. t r a n s p o r t c u r re n t . T h e s h a p e o f t h e c r o s s - s e c t i o n o f t h e s h i e l d e dr e g i o n i n t h e s u p e r c o n d u c t o r i s a p p r o x i m a t e d t o a n e l l i p s e . T o s i m u l a t e t h e w i n d i n g o fs u p e r c o n d u c t i n g m a g n e t s , t h e m a g n e t i c i n t e r a c t i o n b e t we e n t h e s u p e r c o n d u c t o r c o m p o -nents i s taken in to accou n t . The to ta l hysteresis loss is analysed a s a s u m o f t h e m a g n e t i z a t io nloss and the loss due to the dynamic res i s tance . The ca lcu la ted l o s s e s f o r a w i d e r a n g e o fa m p l i t u d e o f t h e a l t e r n a t i n g f ie ld are g i v e n i n g r a p h s wh i c h c l e a r l y s h o w h o w t h e t r a n s p o r tc u r r e n t a n d t h e i n t e r a c t i o n b e t we e n t h e s u p e r c o n d u c t o r s a f fe c t t h e hysteresis .K e y w o r d s : s u p e r c o n d u c t o r s ; m a g n e t i c f i e ld s ; h y s t er e s is lo ss

W h e n a n o n - i d e a l t y p e 2 s u p e r c o n d u c t o r i s e x p o s e d t o a na l t e r n a t i n g m a g n e t i c f i e l d , a n e l e c t r o m a g n e t i c e n e r g yd i ss ip a t io n c a l l e d h y s t e re s i s l o ss o c c u r s i n t h e su p e r -c o n d u c t o r . T h i s l o s s d e p e n d s c l e a r ly o n t h e s h a p e o f th es u p e r c o n d u c t o r a n d t h e a m p l i t u d e o f t h e a l t e r n a t i n g fi e lda n d i t s d i r e c t i o n . I n s i n g l e c o r e s u p e r c o n d u c t i n g w i r e ,h y s t e r e si s l os s o n l y o c c u r r e d i n t h e w i r e i f t h e e d d yc u r r e n t s i n n o r m a l m e t a l f o r s ta b i l i z a t io n w e r e n e g le c t e d .I n a m u l t i f i l a m e n t a r y s u p e r c o n d u c t i n g w i r e , t h e e l e c t r o -m a g n e t i c e n e r g y d i s s i p a t i o n c o n s i s ts o f t h e h y s t e r e s i sl o s s e s a n d t h e l o s s e s g e n e r a t e d b y t h e c o u p l i n g c u r r e n t sb e t w e e n s u p e r c o n d u c t i n g f il a m en t s. H o w e v e r , w h e n t h ea l t e r n a t i n g f i e l d w a s v a r i e d s o s l o w l y t h a t t h e c o u p l i n gc u r r e n t s c o u l d b e n e g l e c t e d e a c h f i l a m e n t s e e m e d i n d e -p e n d e n t o f t h e o t h e r f i l a m e n t s e le c t ri c a ll y . H o w e v e r ,m u l t i f i l a m e n t a r y s u p e r c o n d u c t i n g w i r e s c a n n o t a l w a y sb e u s e d a s a b o v e a n d t h e s t i m a t i o n o f h y s t e r e s is l os s i si m p o r t a n t i n v a r i o u s o t h e r a p p l i c a t i o n s o f m u l t i fi l a -m e n t a r y s u p e r c o n d u c t i n g w i r es .H y s t e r e s i s l o s s i n t h e s u p e r c o n d u c t o r c a r r y i n g n ot r a n s p o r t c u r r e n t h a s a l r e a d y b e e n i n v e s t i g a t e d b y m a n ya u t h o r s. M o r g a n ' s t h e o r y 1 is p o p u l a r w i t h r e g a r d t o t h eh y s t e r e s is l o ss o f a n o n - i d e a l t y p e 2 s u p e r c o n d u c t o r o fr o u n d c r o s s - s e c t i o n c a r r y i n g n o t r a n s p o r t c u r r e n t .S e v e r a l t h e o r i e s h a v e a l s o b e e n p u b l i s h e d b y K a t o e t a l . 2 ,Ash k in 3 , W i l so n e t a l . 4 a n d Z e n k e v i t c h e t a l . 5 .K a t o e t a l f l h a v e a d o p t e d t h e c r i ti c a l s ta t e m o d e l s w i t ha c o n s t a n t c r i t i c a l c u r r e n t d e n s i t y a n d c o n s i d e r e d ac o n t o u r i n t e g r a l i n t h e s h i e l d e d r e g i o n . C o n s e q u e n t l y ,t h e y h a v e v e r i f ie d th a t a n a p p r o p r i a t e c i rc l e c a n a p p r o x i -m a t e t o t h e c r o s s - s e c t i o n o f th e s h i e l d e d r e g i o n a n do b t a i n e d a f o r m u l a f o r t h e h y s t e r e s i s l o s s b a s e d o n t h i sa p p r o x i m a t i o n . H o w e v e r , t h e y h a v e n o t c a l c u la t e d t h e

v a l u e o f t h e l o s s b a s e d o n t h i s r e f i n e d a p p r o x i m a t i o nb e c a u s e t h i s f o r m u l a c o n t a i n s i n t e g r a t i o n s t o o c o m -p l i c a t e d to e v a lu a t e .A s h k i n 3 d e t e r m i n e s t h e s h a p e o f t h e s h i e ld i n g c u r r e n tr e g i o n b a s e d o n n u m e r i c a l c a l c u l a ti o n b y c o m p u t e r . T h i si s b a se d o n th e p r in c ip l e t h a t n o c h a n g e o f f i e ld in t h es h i e ld e d r e g i o n o c c u r s c o r r e s p o n d i n g t o c h a n g e s o f th ee x te rn a l f i e ld . Ash k in e s t im a te s t h e h y s t e re s i s l o ss b a se do n t h i s c o m p u t e r g e n e r a t e d s h a p e .W i l s o n e t a l . 4 h a v e i n t r o d u c e d a f o r m u l a o f h ys t e r e si sl o s s c o r r e s p o n d i n g t o t h e a r e a e n c l o s e d b y t h e m a g n e t i z -a t i o n l o o p b a s e d o n t h e i r a p p r o x i m a t i o n . T h e y a s s u m et h a t t h e c r i t i ca l c u r r e n t d e n s i t y i s c o n s t a n t a n d t h e c r o s s -se c t io n i s t h e sa m e a s i n F i g u r e 1 . H o w e v e r , t h e r e l a t i o nb e t w e e n t h e c h a n g e i n e x t e r n a l f i e ld a n d t h e w i d t h o f th es h e i l d i n g c u r r e n t r e g i o n i s n o t a p p r o p r i a t e , s i n c e i ta s s u m e d t h a t t h e m a x i m u m a b s o l u t e v a l u e o f t h e s h ie l d -in g c u r re n t d e n s i ty i s e q u a l t o t h e c r i t i c a l c u r re n t d e n s i ty .I n t h e t h e o r y d e s c r i b e d i n t h i s P a p e r , t h e v a l u e o f t h esh i e ld in g c u r re n t d e n s i ty i s twic e t h a t o f th e c r i t i c a lc u r r e n t d e n s i t y , b e c a u s e t h e c u r r e n t t u r n s o v e r t o s h i el d .Z e n k e v i t c h e t a l . 5 a s s u m e t h a t t h e m a g n e t i z a t i o n d u e t oth e sh i e ld in g c u r re n t sa t i s f i e s a se m i -e m p i r i c a l fo rm u lai n c l u d i n g t h r e e c o n s t a n t s , w h i c h a r e d e t e r m i n e d f r o m t h em a x i m u m v a l u e o f t h e m a g n e t i z a t i o n a n d t h e v a l ue s o f th em a g n e t i z a t i o n a n d t h e h y s te r e s i s l o s s c o r r e s p o n d i n g t ot h e c a s e w h e r e t h e s h i e l d i n g c u r r e n t r e g i o n i s m i n i m u m .T h e i r t h e o r y a g r e e s w i t h t h e i r e x p e r i m e n t e x c e l l e n t l y ;h o w e v e r , i t i s di f fi c u lt t o e x p a n d t h e i r t h e o r y i n t o t h e c a s et h a t t h e s u p e r c o n d u c t o r i s c a r r y i n g t r a n s p o r t c u r r e n t , a st h e s e m i - e m p i r i c a l f o r m u l a t a k e s n o a c c o u n t o f t h is .I n s u p e r c o n d u c t i n g m a g n e t s , t h e s u p e r c o n d u c t i n g w i r ec a r r i e s a c u r re n t wh ic h e x c i t e s t h e m a g n e t i c f i e ld . Th e re -

0 0 1 1 - 2 2 7 5 / 8 7 / 1 1 0 6 2 1 - 1 0 $ 0 3. 00(~'~ 1987 Butte rwo rth & Co (P ubl ishers) L td C r y o g e n ic s 1 9 8 7 V o l 2 7 N o v e m b e r 621

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Hysteresis loss of a rou nd su perco nduc tor . K. Kanbara

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Figure I A p p r o x i m a t e d d is t r i b u t io n o f t h e s h i e l d i n g c u r re n ts . T h es h a d e d a r e a i n d i c a t e s t h e s h i e l d e d r e g i o n w i t h a n e l l i p t i c a l s h a p e

fo re , i t is i mp or t a n t t o es t i m at e t he l o ss o f t he super -c o n d u c t o r c a r r y i n g t h e t r a n s p o r t c u r r e n t . P u b l i s h e dt heo r i es o f t he hys t e r es i s lo s s fo r a sup erco ndu c t o r w i t h ar o u n d c r o s s - s e c t i o n c a r r y i n g a t r a n s p o r t c u r r e n t a r ef ew 7 '8 . As f a r as we kn ow, t he rou nd c ro ss - sec t i on ha susua l l y been r ep l aced b y an equ i va l en t s l ab c ro ss - sec t ioni n t h i s case 9 -11 . W i t h t h i s p rocedu re , so me a r t i f ic i a l i ti esoccu r , as t he equ i va l en t s l ab c ro ss - sec t i on changes w i t ht h e s t r e n g t h o f th e m a g n e t i c i n t e r a c t io n b e t w e e n s u p e r -c o n d u c t o r s i n t h e m a g n e t w i n d i n g .F o r a s u p e r c o n d u c t o r c a r r y i n g n o t r a n s p o r t c u r r e n t ,hys t e r es i s l o s s i s equa l t o t he a r ea enc l o sed by i t sm a g n e t i z a t i o n l o o p a s w e l l a s t h e f e r r o m a g n e t i c m a t e r i a ls .F o r a s u p e r c o n d u c t o r c a r r y i n g a t r a n s p o r t c u r r e n t , t h ehys t e r es i s l o s s i s no t ob t a i ned f rom on l y t he magnet i z -a t i o n l o o p a s t h e d y n a m i c r e s is t a n c e l o ss l 2 e x p e r i en c e d b yt he sou rce o f t he t r ans po r t cu r r en t occu r s . Thus , i n t hec a s e o f w i r es c a r r y i n g t r a n s p o r t c u r r e n t , t h e m e t h o d b a s e don t he c r i t i ca l s t a t e model s hou l d be se l ec t ed because i f t hed i s t r ib u t i o n o f c u r r e n t i n a s u p e r c o n d u c t o r i s k n o w n , t h ec r it ic a l s ta t e m o d e l l e a d s u s t o t h e v a l u e o f t h e d y n a m i cresi s tance.

In t he fo l l owi ng , t he hys t e r es i s l o s s is es t i ma t ed ba sedon t he c r i t i ca l s t a t e model fo r t he case t ha t a super -con duc t o r o f rou nd c ro ss - sec t i on is ca r ry i ng a d .c .t r ansp o r t cu r r en t i n an a l t e rna t i ng f ie l d wh i ch i s tr ans -v e r s e a n d s m a l l b y c o m p a r i s o n t o t h e b i a s f i e l d . F o r as i ng le co re w i r e, t he i n f luence o f t he mag net i c m om en t s o fo t her w i r es i n t he w i nd i ng i s cons i dered . F o r a mu l t i f i l a -m e n t a r y w i r e , t h i s t r e a t m e n t c a n b e a p p l i e d t o t h em a g n e t i c i n t e r a c t io n b e t w e e n f i la m e n t s d u e t o t h e f i e ld o ft h e m a g n e t i c m o m e n t s o f f i la m e n t s .

C u r r e n t d i s t r ib u t i o nApproximat ion by an e l l ipseIn t he a l t e rna t i ng m agne t i c f ie l d , t he sh i e l d i ng cu r r en t i si n d u c e d o n t h e s u r fa c e o f t h e s u p e r c o n d u c t o r a n d i ts

n e i g h b o u r h o o d t o s h ie l d th e i n n e r r e g i o n f ro m t h e c h a n g eo f t he ex t e rna l f i e l d . Th i s r eg i on i s ca l l ed t he sh i e l d i ngcu r r en t r eg i on and t he i nner r eg i on i s ca l l ed t he sh i e l dedreg i on .I f t he am p l i t ude o f the a l t e rna t i ng f i e ld is very smal l , t hesh i e l d i ng cu r r en t r eg i on i s very t h i n and t he sh i e l dedr e g i o n is a l m o s t t h e w h o l e o f t h e s u p e r c o n d u c t o r . T h e r e -fo re , f o r t he round superconduc t o r , t he c ro ss - sec t i on o ft he sh i e l ded r eg i on i s a l so round . I f t he am p l i t ude i s veryl a rge , so t ha t t he sh i e l d i ng cu r r en t i s i nduced i n t he who l eo f t h e s u p e r c o n d u c t o r , t h e s h i e l d e d r e g i on v a n i s h e s o n a nax i s o f the r oun d c ro ss - sec t ion . O f cou r se , t h i s ax is i spara l l e l t o t he a l t e rna t i ng f ie ld . Then , j u s t befo re i tvan i shes , t he c ro ss - sec t i on o f t he sh i e l ded r eg i on can beregarded as a l i ne segmen t w i t h i t s l eng t h equa l t o t hed i a m e t e r .Acco rd i ng l y , i t i s r ecogn i zed t h a t a pp ro p r i a t e e l li p sesare ab l e t o r ep resen t t he above l i mi t s . Therefo re , i t i sr e a s o n a b l e t o a p p r o x i m a t e t h e c r o s s - s e c ti o n o f a n ysh i e l ded r eg i on t o an e l l i p se as Figure 1. A non-d i m ens i on a l f ac t o r , q , i s i n t rodu ced f ro m t he w i d t h , W, o ft he sh i e l d i ng cu r r en t r eg i on . In t he fo l l owi ng , t h i s f ac t o r i su s e d t o s p e ci fy th e s t a t e o f t h e s u p e r c o n d u c t o r

1= R _- -W) 2 1w h e r e R i s t h e r a d i u s o f t h e s u p e r c o n d u c t o r . T h e d e g r e e o ff i tness o f t h i s app ro x i m at i on i s descr i bed i n t he A ppend i x .

Mag net iza t ion p rocess and cu r ren t d is t r ibu t ionIn t he case t ha t t he su perc ond uc t o r i n a l a rge b i as fi e l d i sexposed t o t he a l t e rna t i ng f i e l d wh i ch i s smal l by com-par i son , we can neg l ec t t he depen denc e o f t he c r i ti ca lc u r r e n t d e n s i t y o n t h e s t r e n g t h o f th e m a g n e t i c f i e ld a n dm a y a c c e p t t h e a p p r o x i m a t i o n t h a t i t is a l w a y s c o n s ta n t .On t he app rox i m at i on o f t he sh i e l ded reg i on by e l li p ses ,t h e c u r r e n t d i s t r i b u t i o n s c o r r e s p o n d i n g t o e a c h s t a t e o fm a g n e t i z a t i o n a r e e x p l a i n e d b e l ow .I f t h e s u p e r c o n d u c t o r i s c a r r y i n g n o t r a n s p o r t c u r r e n t ,t h e d i s t r ib u t i o n s a r e a s s h o w n i n Figures 2 and 3 . The s i gnsi n t he f i gu res o f the cu r r en t d i s t r i bu t i o n i nd i ca t e t he s i gns

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~ W AFigure 2 R e p r e s e n t a ti o n o f th e m a g n e t i z a t i o n p r o c es s a nd c u r re n td i s t r i b u t i o n ( /T = 0 , A H A < A H p )622 Cryogenics 987Vol 27 November

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. . . . W p = D / 2F i g u r e 3 Rep resentat ion f the mag netizat ion processand cur rentdis tr ibut ion ( I T = 0 , z~HA > A H p ) , wh ere D is the d iameter o f thesuperconduc tor

o f th e z c o m p o n e n t o f th e c u r r e n t i n d u c e d b y t h ea l t e r n a t i n g f i e l d , a n d t h e d o t t e d a r e a s a r e t h e r e g i o n ss h i e ld e d f r o m t h e c h a n g e o f t h e f ie l d t h r o u g h o u t t h em a g n e t i z a t i o n p r oc e s s.I f th e a m p l i t u d e , A H A , o f t h e a l t e r n a t i n g f i el d is n o tl a r g e e n o u g h t o s a t u r a t e t h e m a g n e t i z a t i o n , t h a t i s , i f t h ea m p l i t u d e A H A i s s m a l l e r t h a n t h e a m p l i t u d e A H p , t h em a g n e t i z a t i o n a n d t h e c u r r e n t d i s t r i b u t i o n a f t e r a s e c o n dr u n a r e a s s h o w n i n Figure 2 . H e r e , WA i s t h e w id th o f t h es h i e l d i n g c u r r e n t r e g i o n w h i c h i s t h e m o s t d e v e l o p e dt h r o u g h t h is p r o c e s s . T h e n u m b e r s o n t h e f i g u r e o f t h ec u r r e n t d i s t r i b u t i o n i n d i c a t e t h e m a g n e t i z a t i o n s t a t e .I n t h e c a s e t h a t A H A i s l a r g e r t h a n A H p , t h e m a g n e t i z -a t i o n a n d t h e c u r r e n t d i s t r i b u t i o n a f t e r a s e c o n d r u n a r e a ss h o w n i n Figure 3 . I f n o t r a n s p o r t c u r r e n t f l o w s , t h es h i e l d in g c u r r e n t r e g i o n c a n d e v e l o p i n t o t h e w h o l e o f t h es u p e r c o n d u c t o r , s o t h a t i t s w i d t h g r o w s u p t h e l e n g t h o ft h e r ad iu s .I f t h e s u p e r c o n d u c t o r i s c a r r y i n g a d . c. t r a n s p o r tc u r r e n t , i t f o l lo w s o n e o f t h e m a g n e t i z a t i o n p r o c e s s e ss h o w n i n Figures 4 a n d 5 . T h e c a s e o f o n ly u n s a t u r a t e dm a g n e t i z a t i o n o c c u r r i n g is s h o w n i n Figure 4 w h e r e t h eh a t c h e d p a r t s a r e t h e r e g i o n s s h i e l d e d t h r o u g h o u t t h ep ro ce ss ( as in Figure 2). T h e c u r r e n t d i s t r i b u t i o n s i n t h e s er e g io n s r e m a i n c o n s t a n t t h r o u g h o u t t h e p r oc e s s a n d t h et r a n s p o r t c u r r e n t f l ow s i n t h is r e g i o n . O f c o u r s e , t h em a g n e t i c f i e l d i n d u c e d b y t h e t r a n s p o r t c u r r e n t a l s or e m a i n s t h r o u g h o u t t h i s p r o c e s s . T h u s , WA i s t h e w id th o ft h e s h i e l d i n g c u r r e n t r e g i o n w h i c h i s t h e m o s t d e v e l o p e dt h r o u g h o u t t h i s p r o c e s s ( a s i n Figure 2).

I r a d . c . t r an sp o r t c u r r e n t f l o ws , t h e l a rg es t w id th , Wp , o ft h e s h i e l d i n g c u r r e n t r e g i o n m u s t b e s h o r t e r t h a n t h er a d i u s , R , o f t h e s u p e r c o n d u c t o r . S i n c e t h e s u p e r c o n d u c -t o r is c a r r y i n g t h e t r a n s p o r t c u r r e n t c o n s t a n t l y r e g a r d l e s so f w h e t h e r t h e m a g n e t i z a t i o n i s s a t u r a t e d o r n o t , i t isu n d e r s t o o d e a s i l y t h a tW~ = R(1 - i ) (1)a n di = IT /Ic (2 )

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~ W p : p ( 1 - i )F igu re 5 Representat ion f the magnet izat ion processand cur rentdistribu tion (/T =#0, H A > AH p)w h e r e I x, I c a n d i a r e t h e t r a n s p o r t c u r r e n t , t h e c r it i c a lc u r r e n t , a n d t h e n o r m a l i z e d t r a n s p o r t c u r r e n t , r e s p e c t -i v e l y . I n t h i s c a s e , t h e f i e l d i n d u c e d b y t h e t r a n s p o r tc u r r e n t a l s o r e m a i n s t h r o u g h o u t t h i s p r o c e s s .Sh ie ldi n g f i e l d a n d ma g n e t ic m o me n tS i n c e t h e c u r r e n t d i s t r i b u t i o n s a r e d e t e r m i n e d a s a b o v e ,b a s e d o n B i o t S a v a r t ' s l a w , it is p o s s i b l e t o e s ti m a t e t h ef i e l d s i n d u c e d b y t h e s h i e l d i n g c u r r e n t . T h e s h i e l d i n gc u r r e n t m i g h t c o n s i s t o f it s z c o m p o n e n t o n l y , t h e r e f o r et h e s h i e l d in g f ie l d H s o a t t h e c e n t r e o f t h e s u p e r c o n d u c t o ri s sp ec i f i ed as an i n t eg ra lH s = 2 n d x d yJ(x , y ) x2 + y2 (3 )w h e r e J(x , y ) i s t h e s h i e l d i n g c u r r e n t d e n s i t y a n d t h es u r f a c e i n t e g r a l i s c a r r i e d o u t o v e r t h e w h o l e c r o s s -s e c t i on . I t s h o u l d b e n o t e d t h a t t h e v a l u e o f th e s h i e l d i n gc u r r e n t d e n s i t y i s 0 o r + 2 J c .

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H y s t e r e s is lo s s o f a r o u n d s u p e r c o n d u c t o r : K . K a n b a r aU s i n g t h e a p p r o x i m a t i o n t o a n e l l i p se fo r t h e s h ie l d e dr eg ion , th i s in tegr a l can be do ne ana ly t ica l ly : (1 ) in thecase of the inc r eas ing ex te r na l f i e ld

f ; f R(= x2m 2)~s o - 2 J c d x Y d y,J( a- ~(1 --x2/R2) ~ X2 + y2

- ~ o J c R 1 - - t a n - 1 q (4 )~ qand ( 2) in the case of the decr eas ing ex te r na l f i e ld

4 ( 1 __lq)H s o = - t~oJc R 1 tan (5)I n a d d i t i o n , t h e m a g n e t i c m o m e n t , M , o f th e s u p e r -c o n d u c t o r p e r u n i t l e n g t h i n d u c e d b y t h e s h i e l d i n gc u r r e n t a t a n y t i m e c a n b e e s t i m a t e d , p r o v i d e d t h a t t h em a g n e t i c m o m e n t i s z e r o w h e n t h e a l t e r n a t i n g f i e l d i sm i n i m u m

f f s d x d y J ( x , y ) y (6)T h e a b o v e i n t e g r a l a l s o c a n b e i n t e g r a t e d a n a l y t i c a l l yu s i n g t h i s a p p r o x i m a t i o n : (1 ) i n t h e c a s e o f t h e i n c r e a s i n gexte r na l f i e ld

f R IR(1 - x 2 / R 2 ) M = - 4 # o J c d x y d y- R , ) ( g - R ' ) ( 1 - x 2 / g 2 ) ~= - a # o J c R ~ ( 2 R - i f )= _ ~ l ~ o J c R a q ( 1 + q2) - i ( 7 )

t h e s a t u r a t e d v a l u e , M p , i s s im p l y e v a l u a t e d a s b e l o wMp = - 8 /~oJcRS( 1 - i 2 ) ( 8)and ( 2) in the case of decr eas ing ex te r n a l f i e ldM = M m i . + ~ # o J c R f f ' ( 2 R - i f) (9)w h e r e M i ni i s t h e m i n i m u m o f t h e m a g n e t i z a t i o n , w h i c h i sd i a m a g n e t i c a n d W i s w i d t h o f t h e r e g i o n s h i e l d e d f r o mt h e d e c r e a s e o f t h e e x t e r n a l f ie ld . I f th e m a g n e t i z a t i o n i sn o t s a t u r a t e d , M i n i . i s e v a l u a t e d f r o m F i g u r e 4M m i n = 8 ~ o J c R a q A ( 1 + q2 ) 1 (10)w her e qA i s de f ined asqA -- (R - ~VA) 1 (11)I f t h e m a g n e t i z a t i o n i s s a t u r a t e d , M m ~n i s o b t a i n e d e a s i lyMm i , = Mp ( 12)Magnet ic in teract ion betweensuperconductorsS i n c e e a c h s u p e r c o n d u c t o r i n t h e w i n d i n g h a s m a g n e t i cm o m e n t s i n d u c e d b y s h i e l d in g c u r r e n t s, i t is s u b j e c t t o t h ef i e l d p r o d u c e d b y o t h e r s u p e r c o n d u c t o r s . Z e n k e v i t c h e tal . s f ir s t c o n s i d e r e d t h i s e f f ec t o n t h e e s t i m a t i o n o f t h et r ansver se a l te r n a t ing f ie ld los s f or s ing le cor e w i r es. Theyh a v e s u g g e s t e d t h a t t h e r e a r e r e l a t i o n s b e t w e e n t h eampl i tude , A H , of the ex te r na l f i e ld w hich a f f ec t s thew i n d i n g , a n d t h e a m p l i t u d e , A H A , w h i c h a f f e c ts th es u p e r c o n d u c t o r i n t h e w i n d i n g , c o n s i d e r e d a s b e l o wA H A - A H = G A A M ( ~ R 2 ) - 1 (13)

A M = M ( H A . . . ) - M(HA.mi.) (14)1 2 cos 2 qJ . , . - 1

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~ 'm . = Y " ( 1 6)rm nr . . = (X 2 + y2) (17 )X , ,, = m A X (18)Y . = n A Y (19)w h e r e G k~ i s t h e g e o m e t r i c a l fa c t o r i n t r o d u c e d b y t h e m , Z 'i m p l i e s t h e s u m o v e r t h e w h o l e o f t h e w i n d i n g w i t h o u t t h es u p e r c o n d u c t o r c o n s i d e r e d i t s el f, a n d A X a n d A Y a r el a t ti c e p a r a m e t e r s o f th e w i n d i n g s . N o t e t h a t G A i nE q u a t i o n ( 13 ) i s th e a r i t h m e t i c a l a v e r a g e o f t h e f a c t o r G k~.I n t h e a b o v e m e t h o d , i t i s a s s u m e d t h a t e a c h s u p e r -c o n d u c t o r i n t h e t w o - d i m e n s i o n a l l a t t i c e , a s i n F i g u r e 6 ,i n t e r a c t s w i t h th e o t h e r s u p e r c o n d u c t o r s w h o s e m a g n e t i cm o m e n t s a r e t h e s am e . T h is m e t h o d c a n b e a p p l i e d t o a n yw inding s t r uc tur e in pr inc ip le . H ow ever , s ince i t i sa s s u m e d t h a t e a c h m a g n e t i c m o m e n t h a s t h e s a m em a g n i t u d e a n d d i r e c t i o n , s o m e r e s t r i c t i o n s o r i g i n a t i n gf r o m t h e a b o v e a s s u m p t i o n m i g h t o c c u r o n t h e e s t im a t i o no f t h e d i s t r i b u t i o n o f t h e l o ss .I n t h e t r e a t m e n t u s e d b y Z e n k e v i t c h e t a l . , t h e s t r e n g t ho f t h i s i n t e r a c t i o n i s d e t e r m i n e d b y t h e G a factor .H o w e v e r , t h e i r e s t i m a t i o n o f t h i s f a c t o r i s i n c o n v e n i e n tp r a c t ic a l l y , si n c e t h is i s r e p r e s e n te d a s a s u m m a t i o n o f t h et e r m s o f th e i n t e r a c t i o n o v e r t h e t w o - d i m e n s i o n a l l a t t ic ei n c l u d e d b y m a n y p o i n t s . T h u s , it is c o n v e n i e n t t o e x p a n dt h e s u m t o a n i n t e g r a l a n d a p p r o x i m a t e t h e a v e r a g e O f Gk~by th e f ac to r G w hich i s the v a lue o f Goo a t the cen t r e o f thew i n d i n g . T h e r e f o r e , G c a n b e s p e ci fi e d a s a n i n t e g r a t i o n a ss h o w n i n F i g u r e 7

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