Superconducting cables are described by current sheets using the continuum model of W. Carr Jr. [1] and assuming an anisotropic conductivity. Two different situations are considered: a) finite length of cable in spatially independent magnetic field; b) infinitely long cable in a periodical magnetic field

Hartmann, R.A.

Klundert, L.J.M. van de

Rem, P.C.

English

van de Klundert, L.J.M.

NARCIS 

Numerical solutions of the current distribution in superconducting cables

University of Twente Research Information

1584 IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-23, NO. 2, MARCH 1987 NUMERICAL SOLUTIONS OF THE CURRENT DISTRIBUTION I N  SUPERCONDUCTING CABLES R . A .  Hartmann, P.C. Rem, L . J .M.  van de Klundert ,  Twente University  of  Technology Department of Applied Physics, P.O. Box 217, 7500 AE Enschede, The Netherlands. Abstract Superconduct ing  cables  are descr ibed by c u r r e n t   s h e e t s  using  the  continuum model of W .  Carr Jr. [ l ]  and  as- suming  an  a isotropic   conduct ivi ty .  Two d i f f e r e n t  s i t u a t i o n s  are considered: a )  f i n i t e  l e n g t h  o f  c a b l e  i n  s p a t i a l l y   i n d e p e n d e n t   m a g n e t i c   f i e l d ;   b )   i n f i n i t e l y  long  cab le  in  a per iodica l  magnet ic  f ie ld .  Introduction The a.c. loss of  long  superconducting  cables  has  been s t u d i e d  by G. Ries and S. TakAcs C2,3,41. T h e i r   c a l c u l a t i o n  is  based on a l o c a l i z a t i o n  of t h e  Biot  & Savart  law. Represent ing  the  current   densi ty  by a Taylor  expansion,  they  derived a set of p a r t i a l  d i f -  f e r e n t i a l   e q u a t i o n s .   I n   t h i s  way, f i n i t e   c a b l e s   i n  spa t ia l ly  independent  magnet ic  f ie ld  a re  t rea ted .  Thei r  r e s u l t s  were exper imenta l ly   ver i f ied  by K.  Kwasnitza and B. Bruzzone [SI. A good qual i ta t ive   agreement  was found  between  measurements  and the  theory  of G .  Ries and S. TakAcs. I n   t h i s   p a p e r  we w i l l  show a n  a l t e r -  nat ive  approach  using  Fourier  series fo r   ep resen t ing  bo th  f i e ld  and  cu r ren t  dens i ty .  Theory F i r s t  we will cons ider   the  case o f   an   i n f in i t e ly   l ong  superconduct ing cable  in  a per iodica l  magnet ic  f ie ld :  2Lsthe period of the f i e l d  a l o n g  t h e  a x i s  of the  cable .  This  problem  can be t r e a t e d   a n a l y t i c a l l y .  The cor res -  ponding   so lu t ions   for  j z  and j are: + L2 j z ( $ , z , t )  = ~ (-al ,,sin+ cos(-) - nnz L u R a n  0 where 2  2 P  P a = o ( L  +(2aR)  )/(2nRL ) L = t w i s t  l ength  R = Radius o = conduct iv i ty  P 1 ^ A  2L nL - ~ T ~ B  [ (--- + $)+io.rnSISK) nL- 5 lw- rnB 1 .  *A a = 2,n l--((-- + + + i w T  s s ] 1 2L nL 2 4 nL P n I K  T = nnRLapo/L s = I ( ” ” R ) - - I ( - )  L nnR I O L  n a R 1 L  S = K  (-) + - K  (-1 nnR L naR K O L  n n R 1  L with I , K  , I  and K modified  Besselfunctions.  The s i t u a t i o n  n = 0, corresponding to  a homogeneous mag- n e t i c   f i e l d , c a n  be t r e a t e d   s e p a r a t e l y .  T h i s  r e s u l t s  i n  the fo l lowing   expres s ions   fo r  j z , j +  and the  l o s s  per  cycle ,  Po : 0 0 1  1 j +  = O ... ~ B ” L  R Po = 1, li_ 2 L 1+p aL w/4n l2 P O P  and  fo r  P t h e  loss  per  cycle:  The last expression  corresponds  exact ly  w i t h  t h e  one obtained by G. Ries and S. TakAcs for t h e   s t a t i o n a r y  s i t u a t i o n .  Manuscript received September 30, 1986. Figure 1 shows t h a t   r e s u l t   ( 4 )   c o r r e s p o n d s   n i c e l y  with t h a t  of S. Takdcs  except  for the s ingu la r i ty   nea r  2L/L = 1, which  does  not  ccur i n   t h e   a n a l y t i c a l  P approach. 0018-9464/87/0300-1584$01.0001987 IEEE 1585 current   f lows  based on t h e   t y p i c a l   l e n g t h  Lp t h e  o r ig ina l   r ec t angu la r  one still  exists due t o   t h e  end e f f e c t s  of t h e   f i n i t e   c a b l e .   F i g u r e  2e is the  analy-  t i c a l   e q u i v a l e n c e  of 26 as time  goes t o   i n f i n i t y .  For t h i s  limit the   cu r ren t   d i s t r ibu t ion   can  be descr ibed a n a l y t i c a l l y  by: The second problem, a f i n i t e   c a b l e  i n  a magnetic f ie ld ,  can  be solved by r e p r e s e n t i n g  t h e  c u r r e n t  d e n s i t y  a n d  t h e  a p p l i e d  f i e l d  i n  a Fourier  series. (10)  B 2 ( t )  -A s i n g   s i n ( L )  nnz Unlike the treatment  of S. TakBcs, the  Biot  & Savart  law is used on the whole c y l i n d e r  t o  o b t a i n  a Fourier  s e r i e s  f o r  the induced   f i e ld  Br. The main advantage  of t h i s  approach is that  a s e t  of more coax ia l   cy l inde r s  may be t rea . ted.  To obta in  a Four i e r  coe f f i c i en t  of the expansion of t h e  induced  f ie ld ,  a four-dimensional  integral  has t o  be solved.  It  is p o s s i b l e   t o   r e d u c e   t h i s   t o  a two- dimensional   integral .  One ge ts   he   fo l lowing   mat r ix  expression:  I ( 1 1  1 which y i e l d s  B:,nf as a func t ion  of j n  as  desc r ibed  by t h e  Bio t  & Savar t  law. Solving (VXE), = -fir i n t o  (1 2 )  -L j z  = 2 aB * A  ((cos(-)sin(-)-Lsin(-)cos(-))cosg L 2nL 2nz z 2 r L  2nz + P P L L P LP L =& sin(-)) L 2nL s i n ( r )  2nz + s i n ( F ) c o s ( p ) c o s g )   ( 1 7 )  P P P P and the wel l  known result of the  loss P: f i r s t  given by C. Ries and S. Takics. Figure 3 shows the  momentary loss,P,as a func t ion  of time i n  a  homogeneous magnet ic   f ie ld .  F ina l ly   f i gu re  4 shows the  frequency  dependence  of  the loss per cycle P f o r  d i f f e r e n t  c a b l e  l e n g t h  2L. Conclusions a system of o r d i n a r y   l i n e a r   d i f f e r e n t i a l   e q u a t i o n s  is derived:  Calcu la t ions  on the cu r ren t   d i s t r ibu t ion .   and  loss  i n  i n f i n i t e l y l o n g  cables w i t h  a n  a r b i t r a r y  a p p l i e d  f i e l d  c a n  appl ied  f i e l d  i n  a Fourier  series. For i n f i n i t e l y  long ( 1 4 )  be done by represent ing  the cur ren t   dens i ty   and   the  We have solved t h i s  system f o r  a pa r t i cu la r   ca se  treated totally analyticallv. c a b l e s  w i t h ' p e r i o d i c a l  a p p l i e d  f i e l d  the  problem can be n 2L/L = 26 f o r  two d i f f e r e n t  a p p l i e d  f i e l d s .  . P  3 Results Figure  2a..  .2d shows t h e   c u r r e n t   d i s t r i b u t i o n  of a f in i t e   supe rconduc t ing   cab le   w i th   l eng th  2L i n  a homogeneous magnet ic  f ie ld .  B ( t )  = B t A ^ A  D i r e c t l y   a f t e r   a p p l y i n g   t h e   m a g n e t i c   f i e l d  on the  v i r g i n   c a b l e ,   t h e   c u r r e n t   d i s t r i b u t i o n  is t h a t  of  ar ec t angu la r  shape ,  i n  o rde r  t o  sc reen  t h e  appl ied  f i e ld  t o   t h e  maximum. After  a while, the  twist of t he   cab le  becomes more v i s i b l e   i n   t h   c u r r e n t   d i s t r i b u t i o n  because  of a decrease of necess i ty   for   sc reening   the  f i e l d .  After  some time it is even   poss ib le   tha t   before   the  c u r r e n t  flow reaches the end of the cable  it r e t u r n s  t o  its s t a r t i n g   p o i n t .   F i n a l l y   a l l  modes have  been a c t i -  vated, resu l t ing   main ly   in   cur ren t   f lows   wi th   typ ica l  l ength  Lp t h e  t w i s t  l ength  of the  cable .   Besides  these (15) References [11 Carr Jr.,  W.J., J. Applied Phys.  45 [2]  Ries, G. ,  Takdcs, S. ,  I E E E  Trans. C31 TakBcs, S., Cryogenics 22 (1982) 661 [4l Takics, S., Cryogenics 24 (1984) 237 C5l Kwasnitza, K . ,  Bruzzone, B . ,  Proc. (1  981 ) 2281 . (1985)  489. ( 1  974 1 929. Magn. MAC-1 7 MT-9, Zurich 1586 5 I 3 2 1 0 0 5 10 15 20 25 2*L/LP Figure 1 .  The loss  P i n  a s inusoida l  ac  f ie ld  dependent  on  2L/Lp, the  wavelength of t h e  f i e l d  2L and t h e   t w i s t l e n g t h  Lp,  w i t h   r e s p e c t   t o   t h e  loss  i n  a homogeneous a c  f i e l d .  . ~~~~ Figure  2a. The c u r r e n t   d i s t r i b u t i o n  of a f i n i t e   l e n g t h  of cab le  (2L)  in  a .  homogeneous ( l inear   wi th  t ime  increas ing)   f ie ld   for  severa l   t ime-  poin ts .  (L /2rR = -; t he   f i gu res  made  by s c a l i n g  R a h i f i c i a l l y )  t = 0 . 5 ~ .  n Figure 2b. t = 3. Figure  2c. t= 57. Figure 2d. t = 1 0 ~ .  1587 1.20 1 .oo 0.90 eo 0.60 L e. 40 0.20 0.00 0 S 10 1s 20 25 TIME Figure 3. The  momentary loss P as a fUnCtiOn of time. P depends on a homogeneous(with time increasing)  magnet ic  f ie ld ,  (T = 2 J~:c .  ) . 3 .oo 2.so .- I 2 2.00 8 Y %O 1.50 Q 1 .oo 0.30 0.00 0 1 2 J ' 4  S FREQUENCY Figure 4 .  The l o s s  P pe r   cyc le   i n   an   ac   f i e ld  as a func t ion  of the frequenby w f o r   d i f f e r e n t  cable  lengths  ( the value of 2L/Lp is given a t  the corresponding curves) .  

661 [4l Takics,

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Numerical solutions of the current distribution in superconducting cables

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