Thermal Stability of Superconductors

I INTRODUCTION When a superconduct ing magnet i s quoted t o be " s t a b l e " , t h i s a lmost c e r t a i n l y means t h a t no-matter-what mechanical , thermal o r e l e c t r i c a l p e r t u r b a t i o n induces a t r a n s i t i o n f rom the superconduct ing s t a t e t o t h e normal s t a t e , t he sur round ing medium i s capable o f e x t r a c t i n g t h i s sudden q u a n t i t y o f energy t o g e t h e r w i t h t h e subsequent energy generated by Jou le e f f e c t , t h e magnet then be ing brought back t o i t s o r i g i n a l s t a t e . However, t he d i v e r s i t y o f cryomagnet ic systems makes i t d i f f i c u l t t o develop a genera l model o f t he s t a b i l i t y o f t h e o p e r a t i n g mode. A f t e r a s h o r t o u t l i n e o f t he c l a s s i c a l s t a b i l i t y c r i t e r i a , some spec ia l systems f o r which these c r i t e r i a a re n o t a p p l i c a b l e a re presented. A few impor tan t p o i n t s a r e examined, some o f which a re genera l , such as t he MPZ concept and c r i t i c a l energy, and some o f which a re more s p e c i f i c t o c e r t a i n c o i 1 and conductor c o n f i g u r a t i o n s c u r r e n t l y employed e.g. " t r a n s i e n t " s t a b i l i t y and s t a b l e normal zones. I n t h e l i g h t o f a l l t h e so lved problems, f u r t h e r work which can be madeissuggested. I 1 OPERATIONAL CONDITIONS SOUGHT. LIMITS OF APPLICATION OF GENERALLY ACCEPTED STABILITY CRITERIA 1 . S t a b i l i t y o f an e q u i l i b r i u m p o s i t i o n o f a system ................................................ It i s u s e f u l t o cons ide r t he problem o f t h e thermal s t a b i l i t y o f a superconductor i n t h e general framework o f t h e s t a b i l i t y o f a system so as t o be a b l e t o de f i ne an a p p r o p r i a t e language ( 1 , 2 ) . The s t a t e o f a superconductor i s charac ter i z e d by i t s temperature d i s t r i b u t i o n T. A s t a t e o f e q u i l i b r i u m To i s a s t a t i o n a r y s o l u t i o n o f t he heat equa t i on w i t h boundary c o n d i t i o n s . A r e a l system i s exposed t o p e r t u r b a t i o n s , and t h e ques t i on o f t h e s t a b i l i t y o f t h e e q u i l i b r i u m s t a t e thus a r i s e s : i f i t i s assumed t h a t t he system i s i n an i n i t i a l s t a t e Ti , s u f f i c i e n t l y c l o s e t o To, o r t h a t i t i s sub jec ted t o t h e a c t i o n o f a smal l p e r t u r b a t i o n , t h e problem i s t o know whether t h e system w i l l r e t u r n t o To. If t h i s i s t he case, t h e To s t a t e i s ( a s y m p t o t i c a l l y ) s t a b l e o r a t t r a c t i v e . An e v i d e n t a t t r a c t o r i s t h e un i f o rm temperature d i s t r i b u t i o n T = T , the ba th temperature. 2 . ~ttractlye_reglon-of-~-?ta!!1e-?tkte It i s p o s s i b l e t o have a s t a b l e system which does n o t behave c o r r e c t l y when p e r t u r b a t i o n s exceed a c e r t a i n magnitude. The system may t h e r e f o r e be uns tab le i n p r a c t i c e . The ques t i on o f t h e s i z e o f t h e r e g i o n o f a t t r a c t i o n o r t he r e g i o n o f s t a b i l i t y o f t he a t t r a c t o r thus a r i s e s . I f the superconduct ing s t a t e i s t he o n l y a t t r a c t o r , g l o b a l s t a b i l i t y occurs ; f o r a l l pe r tu rba t i ons , the conductor w i l l r e t u r n spontaneously t o t he superconduct ing s t a t e . I f t h e r e i s another a t t r a c t o r however, s t a b i l i t y i s l i m i t e d ; i t i s thus u s e f u l to know what p e r t u r b a t i o n s c o n s t i t u t e t h e boundary o f t he a t t r a c t i v e reg ion o f t b e superconduct ing s t a t e ;.such p e r t u r b a t i o n s a r e des ignated " c r i t i c a l p e r t u r b a t i o n s . When the p e r t u r b a t i o n 1 ~ e s i n s l d e t h e Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19841103 C1-504 JOURNAL DE PHYSIQUE a t t r a c t i v e reg ion o f t h e superconduct ing s t a t e , t he conductor i s s t a b l e ; t h e normal zone i s absorbed. Otherwise, t h e temperature d i s t r i b u t i o n tends towards t h e o t h e r a t t r a c t o r ; the conductor i s uns tab le ; the normal zone propagates u n t i l t h e second s t a b l e p r o f i l e i s es tab l i shed . I n o r d e r t o e x a c t l y i d e n t i f y the s t a b i l i t y r e g i o n and t o p r e d i c t t h e t r a n s i e n t response o f t he conductor, i t i s necessary t o use numer ica l models.However, approx imate a n a l y t i c a l s o l u t i o n s a r e u s e f u l t o acqu i re an unsders tand ing o f t h e conduc to r ' behav ior , t o app rec ia te t h e e f f e c t o f va r i ous p h y s i c a l parameters and geometr ies on s t a b i l i t y and t o compare d i f f e r e n t conductors. 3 . Staymt~-~rctf TI" Several q u a n t i t a t i v e express ions f o r t h e s t a b i l i t y o f a superconductor coo led b y a he l ium b a t h have been fo rmu la ted f rom t h e b a s i c i deas developed b y Z.J.J. S t e k l y and J.L. Zar ( 3 ) and f rom the equal areas theorem o f B.J. Maddock and a l . ( 4 ) . The l a t t e r au tho rs p resen t a model e n a b l i n g a s t a b i l i t y c r i t e r i o n , which takes conduct ion a long t h e conductor I n t o account, t o be d e f i n e d g r a p h i c a l l y . It i s a p p l i c a b l e when t h e power t r a n s f e r r e d t o t h e o u t s i d e and d i s s i p a t e d i n t he conductor depends e x p l i c i t l y o n l y on the temperature o f t h e conductor . Furthermore, the conductor i s t r e a t e d as a one-dimensional i n f i n i t e l y l o n g w i re . The c r i t e r i o n d e f i n e s t h e c o l d end recove ry c u r r e n t Ir and assures g l o b a l conductor s t a b i l i t y f o r a l l c u r r e n t s l e s s then Ir. For c u r r e n t s i n excess o f Ir, the conductor can, i n f a c t , r e t u r n t o t he superconduct ing s t a t e p rov ided t h a t t he p e r t u r b a t i o n has a f i n i t e ampl i tude. Conductor s t a b i l i t y i s l i m i t e d . M. N. Wi lson and Y . Iwasa extended the equal areas theorem t o t he normal f i n i t e zones case ( 5 ) . For a l l c u r r e n t s i n excess o f Ir, a s t a t i o n a r y temperature p r o f i l e e x i s t s f o r which t h e Jou le power i s i n e q u i l i b r i u m w i t h t h e power e x t r a c t e d by conduct ion and by t h e e x t e r n a l environment. Th i s s t a t i o n a r y s o l u t i o n i s des ignated "minimum p r o p o g a t i n g zone" (MPZ) ( 6 ) . As w i l l be seen l a t e r , t h e uns tab le s t a t i o n a r y s t a t e MPZ g i v e s i n f o r m a t i o n about the a t t r a c t i v e r e g i o n of t he s t a b l e s t a t i o n a r y superconduct ing s t a t e . The hypothes is o f a conductor , which i s s e m i i n f i n i t e i n t he d i r e c t i p o f i t s a x i s , l eads t o two c o n d i t i o n s be ing imposed a t t h e i n f i n i t y l i m i t : T (x-, ) = T and aT/ a x = 0. These two c o n d i t i o n s e l i m i n a t e t h e s t a t i o n a r y s o l u t i o n f o r which a lmost t h e e n t i r e conductor 1s i n t h e normal s t a t e . I t i s , i n f a c t , i n t u i t i v e l y c l e a r t h a t i f T 5 T i s s t a b l e and MPZ unstab le , then another s t a b l e s o l u t i o n e x i s t s , o the rw ise t k e conductor whould be g l o b a l l y s t a b l e i n a l l cases : a l l so lut i o n s c o u l d o n l y converge t o t h e s t a b l e superconduct ing s t a t e . Th i s o t h q r s t a b l e s o l u t i o n can be c a l c u l a t e d by imposing a f i n i t e boundary and t h e T(x = L ) = Tb c o n d i t i o n . I f t h e conductor e x h i b i t s t ime dependent heat exchange c h a r a c t e r i s t i c s w i t h t he o u t s i d e environment, t he p r e v i o u s l y desc r i bed c r i t e r i a a r e n o t a p p l i c a b l e . For example, f o r i n t e r n a l l y coo led cables, t h e s t a b i l ~ t v concept i s d i f f e r e n t t o t h a t a p p l i c a b l e t o c o i l s coo led b y he l i um channels a c t i n g as a c o l d source. A s t a t i o n a r y s t a t e does no t , i n genera l , e x i s t f o r heat exchange. The k i n e t i c s o f t h e hea t exchange between t h e conductor and c o o l a n t and t h e e n t h a l p y a v a i l a b l e i n t h e l i m i t e d volume o f he l ium ad jacen t t o t h e conductor a r e o f p r ime importance f o r s t a b i 1 i t y (see paragraph V ) . Another c o n f i g u r a t i o n f o r which t h e hypothes is o f t he equal areas c r i t e r i o n a r e n o t v e r i f i e d i s t h e one i n which heat t r a n s f e r c h a r a c t e r i s t i c s a r e e x p l i c i t l y dependent on the x coo rd ina te . Th i s case i s encountered, i n p a r t i c u l a r , w i t h magnets coo led by channels i n s i d e t h e w ind ing. The i n s u l a t i n g spacers d e f i n i n g t h e channel geometry r e s u l t i n a heterogeneous coo lan t d i s t r i b u t i o n a long t h e conductor . Exper imental and t h e o r e t i c a l s t u d i e s i n d i c a t e t h e ex i s tence o f a r a t h e r unusual behav ior o f t he superconductor (see paragraph I V ) . F i n a l l y , a t t e n t i o n must be drawn t o t h e case o f compact magnets, coo led by conduct ion f rom t u r n t o t u rn , which i s n o t s t u d i e d i n d e t a i l here. The minimum energy o f t h e l o c a l p e r t u r b a t i o n necessary t o quench an impregnated superconduct ing c o i l i s t h e o r e t i c a l l y determined f rom the three-d imens iona l minimum p ropaga t i ng zone concept ( 7 ) . A heat conduct ion model i s developed by c o n s i d e r i n g t h e c o i l t o be a cont inuous a n i s o t r o p i c medium w i t h l o n g i t u d i n a l and t ransve rse thermal conduct i v i t i e s . I t should be no ted t h a t t h e model i s o n l y v a l i d i f t h e c a l c u l a t e d MPZ s i z e i s g r e a t e r than t h a t o f t h e conductor . 111 ATTRACTIVE REGION OF THE SUPERCONDUCTING STATE. ENERGY OF CRITICAL