MAGNETIC INSTABILITIES IN MULTIFILAMENT SUPERCONDUCTING COMPOSITES IN A FAST TIME VARYING MAGNETIC FIELD

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HAL Id: jpa-00223581

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MAGNETIC INSTABILITIES IN MULTIFILAMENT SUPERCONDUCTING COMPOSITES IN A FAST

TIME VARYING MAGNETIC FIELD

D. Ciazynski

To cite this version:

D. Ciazynski. MAGNETIC INSTABILITIES IN MULTIFILAMENT SUPERCONDUCTING COM-

POSITES IN A FAST TIME VARYING MAGNETIC FIELD. Journal de Physique Colloques, 1984,

45 (C1), pp.C1-551-C1-554. �10.1051/jphyscol:19841112�. �jpa-00223581�

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JOURNAL

DE

PHYSIQUE

Colloque

C1,

suppldment au no I, Tome 45, janvier 1984 page C1-551

MAGNETIC INSTABILITIES IN MULTIFILAMENT SUPERCONDUCTING COMPOSITES IN A FAST T I M E VARYING MAGNETIC FIELD

D. C i a z y n s k i

CE;A/Sacla,u, UPh/PE-SZ'IPE, 91191 Gif-sur-Yvette Cedex, France

R6sumC - L ' a n a l y s e du comportement d 'un c o m p o s i t e s u p r a c o n d u c t e u r m u l t i f i l a - m e n t a i r e , soumis 5 un champ m a g n 6 t i q u e v a r i a b l e , permet d e dCf i n i r simplement un c r i t e r e d 1 i n s t a b i l i t 6 pour un c o m p o s i t e s e t r o u v a n t d a n s d e s c o n d i t i o n s a d i a b a t i q u e s . Les r 6 s u l t a t s donn6s par c e c r i t s r e s o n t e n bon a c c o r d a v e c c e u x o b t e n u s l o r s d e deux e x p 6 r i e n c e s d i f f C r e n t e s r 6 a l i s C e s s u r d e s compo- s i t e s d e KbTi.

A b s t r a c t - From t h e s t u d y o f t h e b e h a v i o u r o f a s u p e r c o n d u c t i n g m u l t i f i l a m e n t c o m p o s i t e i n a t i m e - v a r y i n g m a g n e t i c f i e l d , we have d e r i v e d a s i m p l e i n s t a b i - l i t y c r i t e r i a f o r a c o m p o s i t e under a d i a b a t i c c o n d i t i o n s . The r e s u l t s g i v e n by t h i s c r i t e r i a a r e i n good agreement w i t h t h o s e o b t a i n e d d u r i n g two d i f f e - r e n t e x p e r i m e n t s o n MbTi c o m p o s i t e s .

I

-

^THEORY

1) Behaviour of a s u p e r c o n d u c t i n g m u l t i f i l a m e n t c o m p o s i t e s u b j e c t t o a t i m e - v a r y i n g t r a n s v e r s e m a g n e t i c f i e l d

For t h e s a k e of s i m p l i f i c a t i o n we w i l l c o n s i d e r f i r s t a s l a b model which i s a good a p p r o x i m a t i o n f o r a t i g h t l a y e r winding ( g e n e r a l l y used f o r m a g n e t i z a t i o n measurements) / 1 / . An e x t e r n a l t r a n s v e r s e t i m e - v a r y i n g m a g n e t i c f i e l d Be i s a p p l i e d t o t h e c o m p o s i t e .

F i e l d and c u r r e n t d e n s i t y p r o f i l e s i n s i d e t h e c o m p o s i t e a r e p r e s e n t e d f i g u r e 1 . B i i s t h e f i e l d i n t h e c e n t e r of t h e c o m p o s i t e , Bf i s t h e r a d i u s o f t h e f i l a m e n t a r y r e g i o n , q i s t h e p r o p o r t i o n o f s u p e r c o n d u c t o r i n t h i s r e g i o n , and

6

is t h e t h i c k n e s s of t h e l a y e r o f s a t u r a t e d f i l a m e n t s ( c u r r e n t d e n s i t y : Q J c )

.

Using t h e s e n o t a t i o n s and Bd = I B e - ~ i l , we c a n w r i t e :

where B s i s t h e s u r f a c e f i e l d i n t h e m i d d l e p l a n e o f t h e c o m p o s i t e , y = 6/Rf and Sp = u o Q J c

Rf

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

F o r a s i n g l e round w i r e o r a n i s o t r o p i c winding t h e problem i s more complex b e c a u s e t h e f i e l d and c u r r e n t d e n s i t y p r o f i l e s a r e n o t e x a c t l y h o w n . I n a f i r s t approxima- t i o n , i f t h e p e n e t r a t i o n i s low

(y

$ 0.2), we c a n c o n s i d e r t h e s u r f a c e f i e l d i n t h e m i d d l e p l a n e t o b e : Bs

=

Be + Bd and i t f o l l o w s : IBs

-

B i l = 2 Bd = yBp s i n c e i n

t h i s c a s e :

y

= 2 Bd/Bp.

Then t h e f i e l d p e n e t r a t i o n i s low ( y 4 0.3) t h e d i f f e r e n t i a l e q u a t i o n g o v e r n i n g B i c a n be w r i t t e n / 2 / :

B i

+

T~ B i = Be (2)

,

w i t h

ii

= d B i / d t

,

T c i s t h e t i m e c o n s t a n t of t h e c o u p l i n g c u r r e n t s . I f t h e o u t e r m a t r i x s h e l l o f t h e c o m p o s i t e i s n e g l i g i b l e (

Kf

i.R ) , we c a n w r i t e i n a g e n e r a l way :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19841112

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C 1-552 JOURNAL

DE

PHYSIQUE

Tc =

$(A)* 2 ^,

^{where P}^{i s}t h e t w i s t p i t c h o f t h e c o m p o s i t e and Q~ i t s t r a n s v e r s e r e s i s t i v i t y . E = 1 f o r a s l a b ,

e

= 2 f o r a round w i r e / I / .

2) G e n e r a l i n s t a b i l i t y c r i t e r i a u n d e r a d i a b a t i c c o n d i t i o n s

Using t h e same kind of c a l c u l a t i o n s a s f o r s i n g l e c o r e s u p e r c o n d u c t o r s / 3 / , we c a n f i n d a v e r y s i m p l e i n s t a b i l i t y c r i t e r i a f o r t h e c o m p o s i t e under a d i a b a t i c c o n d i t i o n s . I n t h i s c a s e , however, t h e c r i t e r i a c a n n o t b e r e l a t e d t o t h e a m p l i t u d e Be ( b e c a u s e B i

#

0 ) b u t t o t h e r e l a t i v e f i e l d p e n e t r a t i o n t h i c k n e s s y , o r t o Bd = l ~ e - B i l . For c o m p o s i t e s w i t h low r e s i s t i v i t y m a t r i x ( e . g . c o p p e r ) , t h e t h e r m a l d i f f u s i v i t y Dth i s much l a r g e r t h a n t h e m a g n e t i c d i f f u s i v i t y I),,, and we assume a n i s o t h e r m a l b e h a v i o u r o f t h e whole c o m p o s i t e . For h i g h r e s i s t i v i t y m a t r i x (e.g. CuNi) : D t h < < D,,,, we assume a n a d i a b a t i c b e h a v i o u r o f t h e s a t u r a t e d r e g i o n o n l y .

I n t h e s e c o n d i t i o n s t h e i n s t a b i l i t y c r i t e r i a may b e w r i t t e n a s : y = y i , w i t h :

where Bf j =

J3

po C T; is t h e u s u a l f l u x jump f i e l d / 3 / , w i t h : C : a v e r a g e s p e c i f i c h e a t o f t h e c o m p o s i t e

\ T ~ =- J ~ / ( D J ~ / Y T )

.

A s f o r b o t h s l a b and round w i r e models t h e r e l a t i o n :

I B S -

^{~ i l}⁼^Y^Bp i s f u l f i l l e d , t h e i n s t a b i l i t y c r i t e r i a r e l a t e d t o Y i s v a l i d f o r b o t h c a s e s . For a g e n e r a l c r i t e r i a o n Bd, o n e must u s e t h e r e l a t i o n : Bd = y B p / c .

I1 - EXPERIMENTS

1) High r e s i s t i v i t y m a t r i x c o m p o s i t e

I n o r d e r t o u n d e r s t a n d t h e quench b e h a v i o u r o f a f a s t s u p e r c o n d u c t i n g s w i t c h , e x p e r i - m e n t s o n a CuNi m a t r i x NbTi m u l t i f i l a m e n t c o m p o s i t e were c a r r i e d o u t . The main cha-

r a c t e r i s t i c s o f t h e t e s t e d w i r e a r e summarized i n T a b l e I. The quench b e h a v i o u r o f a s i n g l e w i r e c a r r y i n g a d . c . c u r r e n t Io, and s u b j e c t t o a h i g h f r e q u e n c y s i n e t r a n s - v e r s e f i e l d : Be = Bm s i n w t was i n v e s t i g a t e d . The r e s u l t s show t h a t a quench o f t h e w i r e can be induced d u r i n g t h e f i r s t r i s e of t h e e x t e r n a l f i e l d . The quench o c c u r s a t a g i v e n l e v e l o f t h e m a g n e t i c f i e l d Bq which d e p e n d s o n l y on t h e t r a n s p o r t c u r r e n t

( f o r f & 5 . 5 kHz). The e x p e r i m e n t a l c u r v e Bq(Io) i s p r e s e n t e d f i g u r e 2.

For : Be = Bm s i n a t , e q u a t i o n ( 2 ) l e a d s t o :

y ( t ) = E

(2)

s i n $ [ c o s ( u t

-$) -

^{C O S $} e - t / ~ c ] w i t h t g

+

⁼^wTc

.

The t e s t e d w i r e h a s a t i m e c o n s t a n t T, = 185

u s

which g i v e s : WT, > 6 f o r

f = w/2n > 5.5 kHz. I n t h e e x p e r i m e n t a l c o n d i t i o n s we c a n so c o n s i d e r wTc >> 1 ( o r

$ =

n / 2 ) and y = E Be/Bp o r Bd = Be ( B i a 0 ) . So f o r f a s t t i m e - v a r y i n g m a g n e t i c f i e l d s ( W T ~ >> 1 ) t h e c o m p o s i t e behave a s a s i n g l e c o r e s u p e r c o n d u c t o r whose c r i t i c a l c u r r e n t d e n s i t y i s q J c . When t h e c o m p o s i t e c a r r i e s a d . c . c u r r e n t t h e i n s t a b i l i t y c r i t e r i a depends on t h e r e p a r t i t i o n o f t h i s c u r r e n t i n s i d e t h e c o m p o s i t e . For a s i n g l e w i r e , w i t h P >> IT Rf, we c a n assume I. t o be c a r r i e d o n l y by t h e o u t e r f i l a m e n t s b e c a u s e o f t h e s e l f - f i e l d e f f e c t / 4 / . For a w i r e i n a c o i l , t h e e x t e r n a l f i e l d e f f e c t t e n d s t o make t h e c u r r e n t d i s t r i b u t i o n more u n i f o r m i n s i d e t h e c o m p o s i t e 151. I n o u r t e s t c o n d i t i o n s we c a n w r i t e t h e i n s t a b i l i t y c r i t e r i a a s : 2 Bd + Bso = Y i Bp ( 4 1 , where Bso=p,,Io/2nRf i s t h e s u r f a c e f i e l d d u e t o 10. For t h e t e s t e d w i r e t h e i n s t a b i l i t y c r i t e r i a c a n b e f i n a l l y w r i t t e n a s : Be = Bq ( I o ) where : g q L ( I o ) = ( B f j - B s o ) / 2 ( 5 ) and Bfj = J3 1.1,

c

T; = 0.20 T a t 4.2 K. T h e o r e t i c a l c u r v e ( 5 ) a s w e l l a s r e s u l t s o b t a i n e d w i t h more s o p h i s t i c a t e d c a l c u l a t i o n s i n c l u d i n g r e s o l u t i o n o f e q u a t i o n ( 2 ) and a p p r o x i m a t e d f i e l d p r o f i l e s i n s i d e t h e c o m p o s i t e a r e p l o t t e d f i g u r e 2 . The a g r e e -

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ment w i t h t h e e x p e r i m e n t a l r e s u l t s i s v e r y good f o r I.

>

10 A . For lower v a l u e s o f I, t h e d i f f e r e n c e between t h e t h e o r e t i c a l and e x p e r i m e n t a l c u r v e s may be ex- p l a i n e d by t h e f a c t t h a t t h e f l u x jump d o e s n o t i n d u c e t h e quench o f t h e w i r e a s i t i s s e e n f o r s i n g l e c o r e s u p e r c o n d u c t o r s / 6 / , o r by a bad a p p r o x i m a t i o n o f t h e f i e l d p r o f i l e s i n s i d e t h e c o m p o s i t e a s t h e e x t e r n a l f i e l d p e n e t r a t e s more d e e p e r i n s i d e t h e c o m p o s i t e .

2) Low r e s i s t i v i t y m a t r i x c o m p o s i t e

During m a g n e t i z a t i o n measurements f l u x jumps and quenches were o b t a i n e d o n a c o p p e r m a t r i x NbTi m u l t i f i l a m e n t c o m p o s i t e . The main c h a r a c t e r i s t i c s o f t h e w i r e a r e summa- r i z e d i n T a b l e I. A t i g h t l a y e r winding was impregnated and p l a c e d i n a f a s t exponen- t i a l l y d e c r e a s i n g f i e l d : Be = Bm e x p ( - t / ~ ) . Example o f m a g n e t i z a t i o n c u r v e s w i t h and w i t h o u t f l u x jumps a r e p r e s e n t e d f i g u r e 3 . For t h i s f i e l d v a r i a t i o n l a w , we c a n c a l - c u l a t e t h e maximum v a l u e o f Y w i t h e q u a t i o n ( 2 ) : ym =

-

Rm exp

(2

Ln u) w i t h u = -r/rc.

B P 1 -u

As example, f o r u = 1, ym = 0.27 (Bm/Bp) and ym = 0.77 (Bm/Bp) f o r u =0.10

.

F o r t h e t e s t e d w i r e we have Y i = ( ~ f ~ / ~ ~ ) ~ / 3 w i t h Bfj = 0.18 T and Bp = 3.1 T which l e a d s t o Y i ==. For t h e t e s t c o n d i t i o n s : C = 1 and : B d i = y i Bp = 0.47 T.

E x p e r i m e n t a l r e s u l t s show t h a t no f l u x jumps c a n b e o b t a i n e d f o r Bm = 0.37 T even w i t h T = 1.2 ms (u = 0.06) ; which i s i n a g r e e m e n t w i t h t h e v a l u e of Bdi g i v e n b e f o r e . C a l c u l a t i o n s o f y ( e q u a t i o n ( 2 ) ) f o r d i f f e r e n t t e s t s show t h a t f l u x jumps o c c u r o n l y when Ym > 0.13 which i s a g a i n i n good a g r e e m e n t w i t h t h e v a l u e of Y i c a l c u l a t e d before.

T e s t s w i t h a d . c . t r a n s p o r t c u r r e n t I, = 500 A showed t h a t a f l u x jump c a u s e d t h e quench of t h e c o n d u c t o r . Although t h e s e l f s u r f a c e f i e l d Bso = 0.18 T was n o t n e g l i - g i b l e i n t h i s c a s e , t h e t r a n s p o r t c u r r e n t had l i t t l e e f f e c t o n t h e i n s t a b i l i t y c r i t e - r i a ( t h e c o n d i t i o n was ym > 0 . 1 2 ) . T h i s p r o p e r t y i s c e r t a i n l y d u e t o t h e b e t t e r r e p a r t i t i o n o f t h e d . c . c u r r e n t i n s i d e t h e c o m p o s i t e b e c a u s e o f t h e e x t e r n a l f i e l d o f t h e winding / 6 / .

111 - CONCLUSION

A s i m p l e i n s t a b i l i t y c r i t e r i a h a s been developped t o p r e d i c t t h e b e h a v i o u r of s u p e r - c o n d u c t i n g m u l t i f i l a m e n t c o m p o s i t e s i n a t i m e - v a r y i n g t r a n s v e r s a l m a g n e t i c f i e l d u n d e r a d i a b a t i c c o n d i t i o n s . I n t h e g e n e r a l c a s e , t h i s c r i t e r i a i s r e l a t e d t o t h e t h i c k n e s s of t h e l a y e r o f s a t u r a t e d f i l a m e n t s i n s i d e t h e c o m p o s i t e . For v e r y h i g h r a t e s bf f i e l d v a r i a t i o n , t h e c r i t e r i a c a n be r e l a t e d d i r e c t l y t o t h e a m p l i t u d e o f v a r i a t i o n of t h e e x t e r n a l f i e l d a s f o r s i n g l e c o r e s u p e r c o n d u c t o r s . T h e o r e t i c a l p r e d i c t i o n s a r e i n good a g r e e m e n t w i t h e x p e r i m e n t a l r e s u l t s .

REFERENCES

/ 1 / SOUBEYRAND J . P . , TURCK B., IEEE Trans. o n Mag. M A G - 2 (1979) 248.

/ 2 / RIES G . , IEEE T r a n s . o n Flag. MAG-2 (1977) 524.

/ 3 / WILSON M . N . e t a l . , J . P h y s i c s

D-2

(1960) 1517.

/ 4 / DUCHATEAU J . L . e t a l . , C r y o g e n i c s

16

(1976) 97.

/ 5 / FOURNET G . and BOYER L . , P r o c . 6 t h I n t . Cryogenic Eng. Conf. (1976) 451.

/ 6 / SAKAMOTO N. e t a l . , P r o c . 9 t h I n t . Cryogenic Eng. Conf. (1982) 535.

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JOURNAL

DE

PHYSIQUE

F i g . 1 : F i e l d and c u r r e n t d e n s i t y

-

p r o f i l e s f o r a s l a b .

T a b l e I: Main c h a r a c t e r i s t i c s o f t h e t e s t e d c o m p o s i t e s .

S2 NbTi Cu 1.32 1045 2 5 12.8 19.3 3.1 Composite

Superconductor Matrix

Conductor 0 (mm) Filament number Filament 0 (jlm) m i s t pitch P (m) r, (ms)

Bp (T)

Fig.Z:

Quench f i e l d on sample S1

d El

S I NbTi CuNi 0.25

6 1 20 25 0.185 0.75

- 1 . 1 1 . 4) ^Am: O S 7 L T , t = 2.4 n s

+

me.rureman\.s

.

-

- - -

( o r n u l c ( 5 )

8 -

^a.cvc.tc C ( L J C " I L ~ ~ ~ I ( ,

.+

F i g . 3 : M a g n e t i z a t i o n c u r v e s and f l u x jumps on sample S2.

I , , .

.

0 10 5 0

f.(Rf

-