ON NON-UNIFORM DISTRIBUTIONS OF TRANSPORT CURRENT IN MULTIFILAMENTARY SUPERCONDUCTING WIRES

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ON NON-UNIFORM DISTRIBUTIONS OF

TRANSPORT CURRENT IN MULTIFILAMENTARY SUPERCONDUCTING WIRES

F. Irie, F. Sumiyoshi, Q. Zhang, K. Yamafuji, T. Kawashima

To cite this version:

F. Irie, F. Sumiyoshi, Q. Zhang, K. Yamafuji, T. Kawashima. ON NON-UNIFORM DISTRIBUTIONS

OF TRANSPORT CURRENT IN MULTIFILAMENTARY SUPERCONDUCTING WIRES. Journal

de Physique Colloques, 1984, 45 (C1), pp.C1-547-C1-550. �10.1051/jphyscol:19841111�. �jpa-00223580�

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ON NON-UNIFORM D I S T R I B U T I O N S OF TRANSPORT CURRENT I N MULTIFILAMENTARY SUPERCONDUCTING WIRES

F. Irie, F. Sumiyoshi, Q.F. Zhang, K. Yamafuji and T. ~ a w a s h i m a * Kyushu U n i v e r s i t y , Dept. o f Electronics Eng., 10-1 Hakozaki 6-Chome, Higashi-ku, Fukuoka-shi 812, Japan

* ~ n s t i t u t e o f TechnoZogy, Fukuoka, Japan

~6sum6

-

Nous avons htudi; l e s v a r i a t i o n s de l a d i s t r i b u t i o n du courant dans l e s supraconducteurs f i l a m e n t s mu1 t i p l e s , dues ?I 1 'a c t i o n de champs magn6tiques e x t 6 r i e u r s v a r i a b l e s . Le changement d'une d i s t r i b u t i o n non=

uniforme ;3 une d i s t r i b u t i o n uniforme s'accompagne de p e r t e s q u i deviennent notables lorsque l e courant de t r a n s p o r t e s t proche du courant c r i t i q u e . A b s t r a c t

-

We s t u d i e d the change o f c u r r e n t d i s t r i b u t i o n s i n mu1 t i f i lamentary superconducting w i r e s induced by the changing e x t e r n a l magnetic f i e l d . The change o f d i s t r i b u t i o n from non-uniform i n t o u n i f o r m one i s accompanied by losses which become remarkable f o r the t r a n s p o r t c u r r e n t near t h e c r i t i c a l c u r r e n t .

Current d i s t r i b u t i o n s i n a mu1 ti fi l a m e n t a r y superconducting wi r e have been s t u d i e d e x t e n s i v e l y i n connection w i t h t h e i r ac losses and i n s t a b i l i t i e s /1,2,3,4/. A non=

u n i f o r m d i s t r i b u t i o n o f t r a n s p o r t c u r r e n t undesirable f o r s t a b l e magnets can be made u n i f o r m by a p p l y i n g some transverse f i e l d - p u l s e s w i t h a low ramp r a t e /1,2/.

On

t h e o t h e r hand, t h e changing e x t e r n a l magnetic f e i l d w i t h a h i g h ramp r a t e makes t h e t r a n s p o r t - c u r r e n t d i s t r i b u t i o n concentrate i n t o t h e i n n e r r e g i o n o f the w i r e wi t h i n t h e c o u p l i n g time-constant /3,4/. The non-uniform c u r r e n t d i s t r i b u t i o n due t o the concentration approaches t o a u n i f o r m one i n t h e i n n e r r a g i o n by a p p l y i n g some f i e l d changes. Those changes o f c u r r e n t d i s t r i b u t i o n were described q u a n t i t a t i v e l y by u s i n g the "uniforming time-constant" i n t r o d u c e d by us r e c e n t l y /4/. The purpose o f t h i s paper i s t o e l u c i d a t e t h e l o s s brought w i t h those changes o f t h e c u r r e n t d i s - t r i b u t i o n . I n t h i s paper, a d i s t i n c t i o n between a low and a h i g h ramp r a t e s i s made by u s i n g t h e s a t u r a t i o n c o n d i t i o n a t which the c u r r e n t i n some f i l a m e n t s i n the outermost l a y e r begins t o be s a t u r a t e d by t h e induced s h i e l d i n g c u r r e n t .

I

-

THE UNIFORMING TIME-CONSTANT AND CURRENT DISTRIBUTIONS

According t o our previous paper / 4 / , when the changing e x t e r n a l t r a n s v e r s e - f i e l d Be i s a p p l i e d t o t h e w i r e w i t h a non-uniform t r a n s p o r t - c u r r e n t d i s t r i b u t i o n , the d i s - t r i b u t i o n i s made u n i f o r m w i t h i n t h e u n i f o r m i n g time-constant -r,, given by

where rw i s t h e r a d i u s o f the f i l a m e n t bundle i n the w i r e , kl = 3.83 the f i r s t p o s i - t i v e zero o f the Bessel f u n c t i o n i . e . , J l ( k l ) = 0 ,

( e e l

the absolute value o f t h e ramp r a t e o f Be, a f 2 = lT rf t h e cross s e c t i o n a l area o f a

filament,.^^

the magnetic p e r m e a b i l i t y o f vacuum, A the volume f r a c t i o n o f f i l a m e n t s i n a f i l a - ment bundle, jc th e c r i t i c a l c u r r e n t d e n s i t y o f f i l a m e n t s , and

5

the c o l l e c t i v e i n z t e r a c t i o n f a c t o r o f windings

( 5

= 1 2.2). This change o f c u r r e n t d i s t r i b u t i o n s j ( r , t ) can be described by the f o l l o w i n g p a r t i a l d i f f e r e n t i a l equation i n the c y l i n d r i c a l coordinates :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19841111

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Cl-548 JOURNAL DE PHYSIQUE

I t i s t o be noted t h a t t h e time

%

i n Eq.(2) does n o t r e p r e s e n t the r e a l time b u t the reduced time which i s d e f i n e d by the sum o f t h e changing time o f t h e e x t e r n a l f i e 1 d : s i n c e the d i s t r i b u t i o n o f the t r a n s p o r t c u r r e n t i s h o l d when

be

= 0.

I n t h e case o f f i e l d pulses w i t h a l o w ramp r a t e , a non-uniform c u r r e n t d i s t r i b u - t i o n i s changed w i t h a pulse number n. A s o l u t i o n o f Eq. ( 2 ) w i t h the i n i t i a l d i s - t r i b u t i o n o f j = 0 f o r 0.5 r 5 rSCt(O) and j = X j c f o r rsat(0)

<

^r^{5 rw}under the condi- t i o n o f ~ [ r ;

-

r g a t ( 0 ) ] ~ ~ c = IS given by

( 1 - j t ) ' 1 2 J l ( k i 11-3t)1/2 )

j = j t - 2 n

Xjc_

k i [Jo (kill J P -

1

^;nl = r I o / 2 t l (3) rw

where jt i s the t r ~ n s p o r t c u r r e n t normalized by t h e c r i t i c a l c u r r e n t o f t h e w i r e I c = A j C ~ r i , i . e . , j t = It-I,, J o the Bessel f u n c t i o n s o f t h e f i r s t kind, t 1 the r i s e o r t h e f a l l time o f t h e t r a p e z o i d a l e x t e r n a l f i e l d pulse.

I n t h e case o f f i e l d pulses w i t h a h i g h ramp r a t e , on the o t h e r hand, t h e area of the s a t u r a t e d r e g i o n i n the w i r e i s extended by the induced s h i e l d i n g c u r r e n t . The p r o f i l e o f the c u r r e n t d i s t r i b u t i o n s i n t h i s case i s d i f f e r e n t from t h a t f o r t h e case o f a low ramp r a t e . When an e x t e r n a l f i e l d i s applied, t h e d i s t r i b u t i o n o f the t r a n s p o r t c u r r e n t i s n o t kept u n a l t e r e d b u t i s forced t o concentrate i n t o the i n n e r r e g i o n o f c i r c u l a r cross s e c t i o n d u r i n g the c o u p l i n g time-constant T,. The concen- t r a t e d c u r r e n t which i s i n i t a l l y l o c a l i z e d near the boundary between the i n n e r and the o u t e r regions approaches t o a u n i f o r m one i n the i n n e r r e g i o n a f t e r the reduced time e u a l t o t h e e f f e c t i v e u n i f o r m i n g time-constant o f r I = T (1

-

^B),, where

i

E ⁷

rcl

e / / p o ~ j c r w . When the change o f the e x t e r n a l f i e l d I0stops, t h e d i s t r i b u - t i o n o f t h e t r a n s p o r t c u r r e n t i s k e p t unaltered, w h i l e t h e s h i e l d i n g c u r r e n t decays.

I 1

-

LOSS ESTIMATION FOR THE W I R E CARRYING DC TRANSPORT CURRENTS

I n t h i s section, we w i l l c a l c u l a t e losses i n t h e case t h a t the w i r e c a r r y i n g a dc t r a n s p o r t c u r r e n t i s exposed t o the successive e x t e r n a l f i e l d - p u l s e s . Now, we con- s i d e r a s i n g l e l a y e r e d solenoidal c o i l w i t h a l o n g a x i a l l e n g t h enough t o n e g l e c t the edge e f f e c t , as shown i n Fig.1. When t h e n - t h t r a p e z o i d a l f i e l d pulse w i t h a l o n g f l a t - t o p time i s a p p l i e d t o t h e wire, t h e l o s s d e n s i t y p e r one pulse W(n) given by u s i n g a s l a b approximation as W(n) = ( l / 2 r )

I

(P, - P l ) d t

-

can be separated i n t o the dynamic r e s i s t a n c e l o s s WD(n)

/ I /

and the m a g n e t l z a t ~ o n l o s s WM c o n s i s t i n g o f t h e i n t r i n s i c h y s t e r e s i s l o s s Wh and t h e c o u p l i n g c u r r e n t l o s s Wc as W(n) = Wn(n) +WM w i t h

I n the above equations. P and E a r e the Poynting v e c t o r and the e l e c t r i c f i e l d a t each surface of 1 o r 2, r e s p e c t i v e l y , WSt(n) t h e s t o r e d energy d e n s i t y p e r one pulse given by

1

a

B

(6) HI =

AjcrWFt

the s e l f - f i e l d a t t h e w i r e s u r f a c e due t o t h e t r a n s p o r t c u r r e n t It, B

t h e magnetic f l u x d e n s i t y i n the w i r e , 6 the d i s t a n c e between an e l e c t r i c center l i n e and a geometric one i n the f i l a m e n t a t t h e center o f t h e wire.

The case o f a low ramp r a t e : The dynamic r e s i s t a n c e l o s s i n t h i s case i s given from Eqs.(3) and ( 4 ) by

1 - j 2 J l ( k l ( 1

-

^Jt)'/') ( A - 1- A J ~ ( ~ I ) ) n

w D ( n ) = w D ( m ) { l + 2

-

^exp(-

K) 1.

jt k ~ C J 0 ( k 1 ) I 2

(1

-

~ ( n ) ( 7 )

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most 14% f o r various values o f jt. As can be seen i n Eq.(7), t h e value o f t h e dynamic r e s i s t a n c e l o s s depends upon t h e p u l s e number n, and approaches t o t h e con- s t a n t value o f W(-) w i t h t h e decay constant o f nl. On the o t h e r hand, t h e magneti- z a t i o n l o s s i n t h i s case c o n s i s t s o f t h e i n t r i n s i c h y s t e r e s i s l o s s i n f i l a m e n t s Wh, and the c o u p l i n g c u r r e n t l o s s WC. According t o t h e e x i s t i n g theory /3/, the magne- t i z a t i o n l o s s independent o f n 1s given by WM = Wh + Wc w i t h

Wh =

X

jcrf B m ( l -

St2

) f o r B,, >> 2 j c r f (1

-

jt ) and

W, = L ~ i v ( 1 - v ) ; v = L

u

⁰ ^t¹

The case o f a h i g h ramp r a t e : The t h e o r e t i c a l v a l u e o f t h e dynamic r e s i s t a n c e l o s s i n t h i s case QD n can be obtained i n the same manner as the s l o w l y changing f i e l d case. Taking a k c k n t t h a t the r a d i u s of the i n n e r r e g i o n i s ( 1

-

3)rw, we o b t a i n t h e t h e o r e t i c a l expressions as,

i D ( n ) = ( 1

- a ) ?

wD(n) (1 0)

where wD(n)' represents the reduced l o s s expression given by s u b s t i t u t i n g the e f f e c - t i v e u n i f o r m i n g time-constant T i n s t e a d o f T i n t o Eq. ( 7 ) . The dynamic r e s i s t a n c e l o s s i n t h i s case has t h e depenaence o f t h e Iopu1se number as i s f o r t h e s l o w l y changing f i e l d case. The magnetization l o s s i n t h i s case WM, consistin_g o f the c o u p l i n g c u r r e n t l o s s

oc,

t h e h y s t e r e s i s l o s s i n t h e s a t u r a t e d r e g i o n Ws, and the i n t r i n s i c h y ~ t e r e s i s l o s s i n the i n n e r and t h e i n t e r m e d i a t e regions Wh, i s given by

G M = Q

c +Ws+Wh w i t h

-11'4

e,

⁺

tis

⁼

-

V - ( I - e ) ] ( I - ~ ) ^B

,

u

⁰ ⁽¹¹

and

ill

~ W h ( 1

- ⁹⁾ .

111

-

COMPARISON WITH EXPERIMENTS

I n order t o c o n f i r m t h e t h e o r e t i c a l expressions o f losses obtained i n Sec.11, we c a r r i e d o u t l o s s measurements o f a t w i s t e d NbTi m u l i i f i l a m e n t a r y w i r e w i t h a Cu ma- t r i x . Taking account t h a t the s a t u r a t i o n c o n d i t i o n o f t h e w i r e i s about 0.18T/sec, we measured losses f o r two kinds o f successive f i e l d pulses; i ) The case o f pulses w i t h a low ramp r a t e o f 0.08T/sec, i i ) the case o f pulses w i t h a h i g h ramp r a t e o f 0.6 T/sec (

6

= 0.11 )

.

The method o f t h e measurement i s t h e same as t h a t i n t h e p r e - vious paper /2/.

For the i n i t i a l c u r r e n t d i s t r i b u t i o n f u l l y l o c a l i z e d near t h e surface, observed losses i n t h e case i a r e shown i n Fig.2. For t h e i n i t i a l c u r r e n t d i s t r i b u t i o n u n i - form i n the whole w i r e , observed losses i n t h e case ii are a l s o shown i n Fig.2. We can seen i n t h i s f i g u r e , as was p r e d i c t e d , t h a t t h e dynamic r e s i s t a n c e losses change w i t h the p u l s e number n and the magnetization loss, on the o t h e r hand, does n o t . These data are i n good agreement w i t h t h e t h e o r e t i c a l values w i t h respect t o the dependence on

n.

I V

-

DISCUSSION

I n t h i s s e c t i o n , we s h a l l discuss t h e e f f e c t s o f t h e non-uniform c u r r e n t d i s t r i b u - t i o n on t h e l o s s f o r p r a c t i c a l cases. We u s u a l l y adopt t h e w i r e design t h a t the c o u p l i n g c u r r e n t l o s s i s n o t so l a r g e compared w i t h t h e i n t r i n s i c h y s t e r e s i s and the dynamic r e s i s t a n c e losses. For t h e discussion o f t h e e f f e c t , t h e r e f o r e , i t i s

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C1-550 JOURNAL DE PHYSIQUE

s u f f i c i e n t t o make a comparison between t h e i n t r i n s i c h y s t e r e s i s l o s s and t h e dynam- i c r e s i s t a n c e loss. Using Eqs. ( 4 ) , ( 6 ) and ( 8 ) given f o r the s l o w l y changing case, t h e contour map o f these losses i s shown i n Fig.3 w i t h r e s p e c t t o jt and B,. We can see i n Fig.3 t h a t t h e p u l s e number dependence o f the dynamic r e s i s t a n c e l o s s i s r e - markable f o r t h e w i r e w i t h a l a r g e rIO such as the w i r e w i t h a small r a d i u s o f f i l a - ments o r a l a r g e r a d i u s o f wires.

References

/1/ Ogasawara, T., Yasukochi, K., Takahashi, Y., Yasohama, K. and Kubota, Y . IEEE Trans. Magn. MAG-15 (1979) 236

/2/ Sumiyoshi, F., H o r i , H., I r i e , F. and Kawashima, T. Cryogenics

23

(1983) 373 /3/ Ogasawara, T., Takahashi, Y., Kanbara, K., Kubota, Y., Yasohama, K. and

Yasukochi, K. Cryogenics

20

(1980) 216

/4/ Sumiyoshi, F., Koga, K., H o r i , H., I r i e , F., Kawashima, T. and Yamafuji, K. ( t o be pub1 i s h e d i n Cryogenics).

I I I x 1 0 3

A W M ( = W ~ + W , )

2 1 . 5 -

U

Z

-

_3;

-

Fig.1 Slab approximation o f s i n g l e

1 ayered s o l e n o i d a l c o i 1

,

^a

where B1=Be and B2=B,+2~,HI. 3

Fig.2 The dynamic r e s i s t a n c e l o s s and t h e magnetization l o s s of the w i r e (sample No.2 /2/)

w i t h 2rf=6.8pm, 2rw=410pm,

-

¹

A=0.63 and I,=220A a t 2.5 T.

The losses i n t h e low ramp r a t e case, WD and WM, were

measured f o r Bm=0.16T, and o I I I - 0

t h e losses i n t h e h i g h ramp o 5

l o n 1 5

r a t e case, W and WM, were f o r Bm=0.06 T. g o l i d l i n e s r e - present t h e t h e o r e t i c a l values f o r t h e case o f jt=0.68.

j t j t

Fig.3 Contour maps of losses a t 6 T

,

~ ( n ) [ J l c y c l e m33, f o r t h e w i r e w i t h

2rf=lpmun,

2rw=500pm, X=0.6, and j c = l .3x109 ~ / m ~ . (a) - n=2, ( b )

-