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OPTIMUM PARTITION OF POWER TO MAXIMIZE FIELD OF MAGNET SYSTEMS
R. Weggel
To cite this version:
R. Weggel. OPTIMUM PARTITION OF POWER TO MAXIMIZE FIELD OF MAGNET SYSTEMS.
Journal de Physique Colloques, 1984, 45 (C1), pp.C1-881-C1-884. �10.1051/jphyscol:19841179�. �jpa-
00223654�
OPTIMUM PARTITION OF POWER TO MAXIMIZE FIELD OF MAGNET SYSTEMS*
R . J . Weggel
Francis B i t t e r National Magnet Laboratory, Massachusetts I n s t i t u t e of Technology, Cambridge, MA 0 2 1 3 9 , U.S.A.
R6sum6
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Pour o b t e n i r l e champ magndtique maximum p r o d u i t par un systOme de b o b i n a g e s , i l f a u t r d p a r t i r 1 ' 6 n e r g i e d i g o n i b l e de s o r t e que t o u t e s l e s bobines a i e n t l a meme e f f i c a c i t e d i f f g r e n t i e l l e . S i l a r d s i s t a n c e de chaque bobine e s t independante de l ' e n e r g i e , t o u t e s l e s bobines p r o d u i r o n t l e mike champ p a r u n i t e d16nergie. Mais s i une bobine c h a u f f e , e l l e ne p o r t e r a que (1-x)' f o i s l e courant Q t a b l i auparavant, x Q t a n t l'augmentation f r a c t i o n n a i r e de r e ' s i s t a n c e , Les systemes oit l e conducteur d o i t Q t r e de'plac6 3 cause du renforcement n d c e s s a i r e pour r e s i s t e r aux h a u t e s f o r c e s appliqu6es r e q u i e r e n t une r e ' p a r t i t i o n d ' g n e r g i e encore p l u s marqu6e.Abstract
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To maximize t h e magnetic f i e l d from a system of c o i l s , one should p a r t i t i o n t h e a v a i l a b l e power s o t h a t a l l c o i l s have t h e same incremental e f f i c i e n c y . I f t h e r e s i s t a n c e of every c o i l i s independent of power, t h e n a l l c o i l s should g e n e r a t e t h e same f i e l d per u n i t power. But i f a c o i l h e a t s up, it should c a r r y only (I-x)' a s much c u r r e n t a s b e f o r e , where x i s t h e f r a c t i o n a l i n c r e a s e i n r e s i s t a n c e . Systems i n which conductor must be d i s p l a c e d by reinforcement, i n o r d e r t o w i t h s t a n d high s t r e s s e s , c a l l f o r an even more marked r e a l l o c a t i o n of power.Scope
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Suppose t h a t one wishes t o a l l o c a t e power among a number of c o i l s of f i x e d dimensions, i n o r d e r t o maximize t h e t o t a l f i e l d generated a t some s p e c i f i e d p o i n t . To do so, each power consumption Pn should match i t s r e s p e c t i v e f i e l d c o n t r i b u t i o n Bn a t t h e given p o i n t , s o t h a t every incremental e f f i c i e n c y , En = dBn/dPn,
w i l le q u a l every o t h e r . Such a system t h e n w i l l have no ''weak l i n k s , " no c o i l s of below- average e f f i c i e n c y from which t o t r a n s f e r power t o c o i l s i n which t h a t same power would g e n e r a t e more f i e l d .
I n t h e f o l l o w i n g paragraphs we d e r i v e formulas f o r f o u r p r o g r e s s i v e l y more s o p h i s t i - c a t e d models f o r incremental e f f i c i e n c y . The f i r s t two c a s e s apply t o c o i l s which a r e not s t r e s s l i m i t e d ; t h e l a s t two c o n s i d e r c o i l s i n which conductor s t r e n g t h i s i n c r e a s e d i n p r o p o r t i o n t o t h e s t r e s s which t h e c o i l i s t o experience. The f i r s t c a s e r e q u i r e s t h a t t h e c o i l r e s i s t a n c e be e s s e n t i a l l y independent of t h e l e v e l of e n e r g i z a t i o n ; t h e second c a s e allows t h e r e s i s t a n c e t o i n c r e a s e as an a r b i t r a r y f u n c t i o n of t h e power consumed, provided t h a t one knows t h e r a t e of i n c r e a s e near t h e intended o p e r a t i n g p o i n t . The t h i r d c a e p e r m i t s t h e c o i l r e s i s t a n c e t o depend n o t only on power but a l s o on s t r e s s , s o long a s t h e s t r e s s can be considered p r o p o r t i o n a l t o t h e c u r r e n t i n t h e c o i l ; according t o one technique f o r e s t i m a t i n g s t r e s s , t h i s r e q u i r e s t h a t t h e average f i e l d s e e n by t h e c o i l be independent of p e r t u r b a t i o n s from t h e optimum p a r t i t i o n of power. The f i n a l c a s e acknowledges t h a t t h e ambient f i e l d seen by any c o i l s u r e l y w i l l be i n f l u e n c e d by t h e c u r r e n t i n t h a t c o i l . It i g n o r e s t h e e f f e c t of a l l o c a t i n g power among t h e remaining c o i l s , but n o n e t h e l e s s should give a very good g u i d e l i n e t o t h e optimum a l l o c a t i o n of power i n any system, a t l e a s t s o l o n g a s c u r r e n t does not spontaneously r e d i s t r i b u t e i t s e l f under t h e i n £ luence of temperature g r a d i e n t s o r magnetoresistance.
Method
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A l l t h e d e r i v a t i o n s i n v o l v e t h e method of Lagrange m u l t i p l i e r s , We wish t o-
maximize t h e t o t a l f i e l d Bt from a system of N c o i l s . I f , a s p o s t u l a t e d , one f o r b i d s* ~ u ~ ~ o r t e d by t h e National Science Foundation
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19841179
Cl-882 JOURNAL DE PHYSIQUE
spontaneous r e d i s t r i b u t i o n of c u r r e n t , t h e n t h e f i e l d c o n t r i b u t i o n Bn from each c o i l n w i l l be p r o p o r t i o n a l t o i t s c u r r e n t : I n : Bn = bn In, where bn i s t h e c o n s t a n t of p r o p o r t i o n a l i t y . S i m i l a r l y , t h e t o t a l power consumption Pt i s t h e sum of N i n d i v i d - u a l power consumptions Pn, each p r o p o r t i o n a l t o 12: Pn = Rn I:, where Rn, t h e r e s i s t a n c e of c o i l n, may be considered c o n s t a n t (Case I ) , o r t o depend on t h e c u r r e n t i n c o i l n, e i t h e r because of h e a t i n g only (Case 2), or because of t h e com- bined e f f e c t s of h e a t i n g and r e s i s t a n c e i n c r e a s e s a s s o c i a t e d with r e q u i r e d i n c r e a s e s i n conductor s t r e n g t h (Cases 3 and 4 ) .
Define a f u n c t i o n F = Pt
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2 y Bt, where y i s t h e unknown (but c o n s t a n t ) Lagrange m u l t i p l i e r ; d i f f e r e n t i a t e F w i t h r e s p e c t t o t h e unknown c u r r e n t s In; and s e t t o zero:N
I n a r r i v i n g a t t h e right-hand e q u a t i o n , we have r e s t r i c t e d R, t o depend only on In, s o t h a t we can focus on one c o i l a t a time, e l i m i n a t i n g t h e summation and dropping a l l s u b s c r i p t s .
Case 1
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I n Base 1 we assume t h a t R i s independnt of I; hence dR/dI = 0, and:-
2 R I = 2 y b; o r , e q u i v a l e n t l y , I = y b / R. (2) One's choice of y determines t h e t o t a l power consumption of t h e system. This i s t h e s o l u t i o n f o r t h e c a s e of r e s i s t a n c e independent of c u r r e n t .
Case 2
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I n Case 2, i n which each c o i l ' s r e s i s t a n c e depends upon t h e power which i t-
consumes, l i n e a r i z e t h e dependence by means of t h e f i r s t two terms i n t h e Taylor s e r i e s :where Ro i s t h e zero-power i n t e r c e p t ("base r e s i s t a n c e " ) and R i s t h e s l o p e , around t h e expected o p e r a t i n g p o i n t , of t h e tangent t o t h e R-vs-P curve. P Ro t y p i c a l l y w i l l be s l i g h t l y g r e a t e r t h a n R(O), t h e c o i l r e s i s t a n c e a t z e r o power: t h e r e s i s t a n c e of a c o i l t e n d s t o r i s e e v e r more g r a d u a l l y w i t h i n c r e a s i n g power, as h e a t - t r a n s f e r c o e f f i c i e n t s improve w i t h i n c r e a s i n g bulk water temperature. For t h e same reason,
w i l l t e n d t o be l e s s t h a n t h e s l o p e of t h e s e c a n t l i n e (R
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R(0)) / P.) i n t o (1) and d i f f e r e n t i a t i n g :
The s m a l l e r t h e q u a n t i t y R ~ I ~ , t h e more r a p i d w i l l be convergence i n s o l v i n g e q u a t i o n (4) i t e r a t i v e l y .
Use of ( 3 ) l e a d s t o t h e more compact e x p r e s s i o n :
where r i s t h e r e s i s t a n c e r a t i o R/Ro = ( 1
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Rp I ~ ) -1.
Note t h a t t h e optimum c u r r e n t p r e d i c t e d f o r a c o i l which h e a t s up i s s m a l l e r by a f a c t o r r m 2 , compared t o a c o i l of c o n s t a n t r e s i s t i v i t y .Case 3
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I f a t i t s intended o p e r a t i n g c u r r e n t any c o i l must w i t h s t a n d a s t r e s s g r e a t e r t h a n t h a t which can be supported s a f e l y by t h e most conductive p r a c t i c a l conductor ( i . e . , c o p p e r ) , t h e n i t s c o e f f i c i e n t Ro no longer can be considered independent of t h e s t r e s s which t h e c o i l i s t o experience; t h e c o i l should be designed w i t h a conductor which i s s t r o n g enough t o w i t h s t a n d t h e d e s i g n s t r e s s w i t h a f a c t o r of s a f e t y which e q u a l s t h a t of every o t h e r s t r e s s - l i m i t e d c o i l i n t h e system. Again l i n e a r i z e t h e dependence: Ro = Roo+
Rs S, where S i s t h e s t r e s s i n t h e c o i l , and Roo and Rs a r e , r e s p e c t i v e l y , t h e z e r o - s t r e s s i n t e r c e p t and s l o p e , a t t h e expected o p e r a t i n g p o i n t , of t h e curve of base r e s i s t a n c e v e r s u s s t r e s s . I n Case 3 assume t h a t t h e s t r e s s i n a c o i l i s p r o p o r t i o n a l t o i t s c u r r e n t , S = s I, where s i s t h e c o n s t a n t of p r o p o r t i o n a l i t y , s o t h a t : Ro = Roo+
Ri I, where Ri = s Rs i s t h e r a t e a t which t h e r e q u i r e d base r e s i s t a n c e i n c r e a s e s w i t h i n c r e a s i n g c u r r e n t .Comparison w i t h (5) r e v e a l s t h a t t h i s e q u a t i o n f o r s t r e s s - l i m i t e d c o i l s p r e d i c t s an optimum c u r r e n t which i s l e s s t h a n t h a t f o r copper c o i l s by a f a c t o r R / (R
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% R i I ) . A s i m i l a r e q u a t i o n a r i s e s i f , i n s t e a d of a l i n e a r i n c r e a s e i n r e s i s t a n c e , one assumes a l i n e a r d e c r e a s e i n conductance, t h e e c i p r o c a l of r e s i s t a n c e : Co = Coo-
Cs S =Coo
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Ci I, g i v i n g R = (Coo-
Ci I )-F ,
where Coo i s t h e z e r o - s t r e s s i n t e r c e p t , and C, and Cia r e s p e c t i v e f y a r e t h e r a t e s of d e c r e a s e of conductance w i t h i n c r e a s i n g s t r e s s and c u r r e n t . P a r a l l e l i n g t h e s t e p s l e a d i n g t o ( 6 ) :A l t e r n a t i v e l y , one could have d e r i v e d (7) from ( 6 ) , simply by r e p l a c i n g Ri i n (6) by dRo/dI = d / d I (Coo
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Ci I)".Case 4
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The f i n a l c a s e assumes a l i n e a r i n c r e a s e i n r e s i s t a n c e o r decrease i n conductance, but recognizes t h a t t h e s t r e s s i n a c o i l depends not only on i t s c u r r e n t b u t a l s o on t h e ambient f i e l d t o which i t i s exposed. This ambient f i e l d t e n d s t o depend r a t h e r s t r o n g l y on t h e c u r r e n t i n t h e g i v e n c o i l , but l e s s s o on t h e a l l o c a - t i o n of power among t h e o t h e r c o i l s i n t h e system. I f we i g n o r e changes i n ambient f i e l d not a t t r i b u t a b l e t o t h e c o i l i t s e l f , t h e n s = so+
si I, o r S = ( s o + s i I ) I, and hence:where R 1 = so Rs measures t h e r a t e of i n c r e a s e i n base r e s i s t a n c e with ambient f i e l d h e l d c o n s t a n t , and R2 = si Rs measures t h e a d d i t i o n a l r e s i s t a n c e a s s o c i a t e d w i t h t h e
i n c r e a s e i n ambient f i e l d . Again f o l l o w i n g t h e s t e p s r e s u l t i n g i n ( 6 ) :
Note t h a t t h e optimum c u r r e n t p r e d i c t e d f o r s t e s s - l i m i t e d c o i l s i n f l u e n c e d by t h e i r
5
own s e l f f i e l d i s s m a l l e r , because of t h e R2 I term, t h a n t h a t f o r c o i l s not s o i n £ luenced.
I f one assumes i n c r e a s i n g s t r e s s t o imply d e c r e a s i n g conductance, t h e n Co = Coo
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C1 I-
C2 I ~ , which l e a d s t o :One can e x p r e s s (10) i n terms of c o e f f i c i e n t s r e l a t i n g cur e n t and c o n d u c t i v i t y ,
5
i n s t e a d of conductance: Co = (Co / co) (coo
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c1 I-
c 2 I ), where c o i s t h e "base c o n d u c t i v i t y " a t t h e o p e r a t i n g p o i n t , coo i s i t s z e r o - s t r e s s i n t e r c e p t , and cl and c2 a r e t h e f i r s t and second c o e f f i c i e n t s i n i t s power s e r i e s expansion w i t h r e s p e c t t o c u r r e n t . ThenCl = Co c l / c o and C2 = Co c 2 / c o y s o t h a t e q u a t i o n (10) becomes:Motivation
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Our i n t e r e s t i n t h e a l l o c a t i o n of power t o maximize f i e l d i n a magnet i n c l u d i n g s t r e s s - l i m i t e d c o i l s i s motivated p r i m a r i l y by t h e system sketched i n Fig.1. This i s a high-performance r a d i a l l y - c o o l e d i n s e r t t o a hybrid system whose o u t e r element i s a superconducting c o i l which g e n e r a t e s a s u b s t a n t i a l background f i e l d i n which t h e i n s e r t operates. A t t h e F r a n c i s B i t t e r National Magnet Laboratory, t h e
superconductor p r e s e n t l y g e n e r a t e s 7.0 t o 7.5 t e s l a s ; t h e i n s e r t i t s e l f g e n e r a t e s up t o 23 T a t 9 M?. An even more ambitious systerd which we have designed and a r e con- s t r u c t i n g f o r t h e U n i v e r s i t y of Nijmegen i s t o i n c o r p o r a t e a superconducting c o i l g e n e r a t i n g up t o 11 T ; t h e i n s e r t i s t o g e n e r a t e a t l e a s t 21 T a t 9 IW, f o r a t o t a l c e n t r a l f i e l d of 32 T , i n a 33 mm 0 room temperature bore.
Each i n s e r t i s cooled by r a d i a l grooves only 0.25 mm deep e t c h e d i n t o a broad f a c e of a l t e r n a t e p l a t e s . Because each passage removes s o l i t t l e conductor, one can
Cl-884 JOURNAL DE PHYSIQUE
a f f o r d t o employ a g r e a t many of them, t h e r e b y providing an abundance of c o o l i n g s u r f a c e . The magnet t h e r e f o r e i s not l i m i t e d s o much by h e a t f l u x as i s a t y p i c a l a x i a l l y - c o o l e d magnet. E f f i c i e n c y i s allowed t o be t h e governing c o n s i d e r a t i o n . Examples
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We have d i v i d e d t h e two-coil system shown i n Fig. 1 i n t o seven zones (and t h e i r m i r r o r images), t h e i n n e r c o i l i n t o f o u r , and t h e o u t e r c o i l i n t o t h r e e , zones of e q u a l length. Using t h e formulas j u s t descibed, we have p a r t i t i o n e d 9 I44 of power s o t h a t a l l c o i l s have t h e same incremental e f f i c i e n c y , a s descibed by Eqn. (11) i f t h e - c o i l i s s t r e s s l i m i t e d , and Eqn. (5) i f i t i s not. F i g u r e s (2a) and (2b) p l o t t h e r e s u l t i n g "secant e f f i c i e n c i e s " (Bn/ Pn) of each zone, a s a f u n c t i o n of t h e background f i e l d Ba g e n e r a t e d by a surrounding c o i l . The spread i n e f f i c i e n c i e s between t h e seven zones g i v e s an i n d i c a t i o n of t h e pronounced e f f e c t s of h e a t i n g and s t r e s s i n a f f e c t i n g t h e incremental e f f i c i e n c y of any high-performance c o i l . The new formulas should g r e a t l y f a c i l i t a t e t h e d e s i g n of Gaume c o i l s of e x c e p t i o n a l e f f i c i e n c y .ORIGIN
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AXISF i g u r e 1: Cross S e c t i o n of Two-Coil, Seven-Zone Radially-Cooled I n s e r t Magnet
F i g u r e s 2a & b: Secant E f f i c i e n c y Bn / Pn of Each Zone, Above, vs. Ambient F i e l d Ba
