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STRESSES IN COILS FOR STRONG PULSED MAGNETIC FIELDS
F. Herlach, G. de Vos, J. Witters
To cite this version:
F. Herlach, G. de Vos, J. Witters. STRESSES IN COILS FOR STRONG PULSED MAGNETIC FIELDS. Journal de Physique Colloques, 1984, 45 (C1), pp.C1-915-C1-921.
�10.1051/jphyscol:19841187�. �jpa-00223663�
STRESSES I N C O I L S FOR STRONG PULSED MAGNETIC F I E L D S F . Herlach, G . de Vos and 3. Witters
UniversitO Leuven, Belgium
Resum@
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Notre s o l u t i o n a n a l y t i q u e pour l e s tensions e l a s t i q u e s dans des m e s anisotropiques a @ t & etendue pour i n c l u r e l a deformation p l astique.Cette c o n d i t i o n peut se p r o d u i r e dans des bobines 2 champ p u l s e @ l e v @ . En augmentant l e champ magnetique, l a deformation p l a s t i q u e commence
a
l 1 i n t @ r i e u r e t s'etend
a
t r a v e r s l a bobine e n t i e r e dans une gamme e t r o i t e de quelques Teslas. Dans c e t t e c o n d i t i o n , l a bobine p e u t @ t r e c o n t r a i n t e par un renforcement e x t e r i e u r l e q u e l d o i t supporter l e s tensions a d d i t i o n n e l - les. Quelques bobines o n t 6 t 6 eprouv6es jusqu'au p o i n t de r u p t u r e e t l e s r e s u l t a t s sont compares avec 1 es c a l c u l s.
A b s t r a c t
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Our a n a l y t i c a l s o l u t i o n f o r t h e stresses i n a n i s o t r o p i c c o i l s i s extended i n t o t h e r e g i o n o f p l a s t i c deformation. This c o n d i t i o n may occur i n pulsed f i e l d c o i l s f o r v e r y h i g h f i e l d s . With r i s i n g magnetic f i e l d , t h e p l a s t i c deformation begins a t t h e i n n e r r a d i u s and spreads over t h e e n t i r e c o i l w i t h i n a narrow range o f o n l y a few Tesla. I n t h i s c o n d i t i o n , the c o i l can s t i l l be h e l d t o g e t h e r by an e x t e r n a l reinforcement which then must support a d d i t i o n a l s t r e s s . A number of c o i l s has been t e s t e d t o t h e breaking p o i n t and t h e r e s u l t s are compared t o t h e c a l c u l a t i o n s .I
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INTRODUCTIONI n a previous paper, we have given an a n a l y t i c a l s o l u t i o n f o r t h e r a d i a l and tangen- t i a l stresses i n magnetic f i e l d c o i l s w i t h a n i s o t r o p i c modulus o f e l a s t i c i t y / I / . When t h i s i s used t o discuss t h e performance o f c o i l s f o r very h i g h pulsed magnetic f i e l d s , i t i s found t h a t stresses i n these c o i l s o f t e n exceed t h e l i m i t i n g value where p l a s t i c deformation occurs. We have t h e r e f o r e extended t h i s s o l u t i o n t o i n - clude p a r t i a l and complete p l a s t i c deformation.
The d i f f e r e n t i a l equations
describe t h e e l a s t i c deformation o f a c y l i n d e r w i t h d i f f e r e n t modulus o f e l a s t i c i t y i n t h e r a d i a l and t a n g e n t i a l d i r e c t i o n s . The a x i a l s t r e s s oz i s neglected.
We r e a l i z e t h a t t h i s i s a crude approximation f o r q u a n t i t a t i v e c a l c u l a t i o n s b u t on the o t h e r hand t h e m a t e r i a l parameters a r e anyhow n o t w e l l known under t h e extreme c o n d i t i o n s t o which a pulsed f i e l d c o i l i s subjected. Our c a l c u l a t i o n s a r e intended t o p r o v i d e u s e f u l g u i d e l i n e s f o r the trends t h a t determine t h e design o f a successful c o i l
.
For a c o i l w i t h constant c u r r e n t d e n s i t y , the s o l u t i o n f o r e l a s t i c deformation i s given by
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19841187
C1-916 JOURNAL
DE
PHYSIQUE1 0 r = CIX
+
c2x2+
c 3 x - p - I+
c+xq-o 4 = 2c1x
+
3c2x2-
pc3x-P-l+
q ~ 4 x q - 1-
(ax+
bx2)w i t h
(Ba
-
Bi)(Ba-
aBi)Pa
- Bi) 2a = 2 b = -
vo (a-1) 2
Po ( * - I )
and w i t h t h e f o l l o w i n g symbols and d e f i n i t i o n s : a1 i n n e r r a d i u s o f t h e s o l e n o i d
a2 o u t e r r a d i u s of t h e solenoid
a %/a1
r r a d i u s of a l a y e r i n t h e c o i l x r / a l
j c u r r e n t d e n s i t y
I 2af (a-1) B j t o t a l c u r r e n t
B magnetic i n d u c t i o n , Bi a t i n n e r radius, Ba a t o u t e r
o r r a d i a l s t r e s s (compressive = p o s i t i v e ) , uri a t i n n e r radius,
era
a t o u t e r o4 t a n g e n t i a l s t r e s s along a x i s o f w i r eG 1 i m i t i n g s t r e s s where p l a s t i c deformation occurs P
Er r a d i a l modulus o f e l a s t i c i t y E+ t a n g e n t i a l modulus o f e l a s t i c i t y
Er/E4
v P o ~ s s o n ' s r a t i o 6 ( a x i a l l e n g t h ) (2al)
- 1 - 1
S I u n i t s a r e used w i t h uo = 4a x 1 0 - ' ~ s A m
.
Stress i s given i n k i l o b a r o r gigapascal, where 1 GPa = 10 kbar.P l a s t i c deformation i s described by t h e von Mises equation
i n combination w i t h equ. (1) which gives the r e l a t i o n between t h e d i f f e r e n t forces a c t i n g on any p i e c e o f w i r e i n the c o i l . The combination of (1) and ( 8 ) cannot be solved a n a l y t i c a l l y because o f the n o n l i n e a r i t y .
We t h e r e f o r e have brought i t i n t o t h e form
where the a x i a l s t r e s s i s again neglected i n order t o be c o n s i s t e n t w i t h t h e approximation made f o r t h e s o l u t i o n w i t h e l a s t i c deformation. These equations can e a s i l y be i n t e g r a t e d by numerical methods, s t a r t i n g a t t h e i n n e r surface where t h e a x i a l s t r e s s can be s p e c i f i e d as i n i t i a l c o n d i t i o n ; i n most cases t h i s w i l l be zero.
I n a s o l i d c o i l w i t h r a d i a l transmission o f s t r e s s , p l a s t i c deformation w i l l always occur f i r s t a t t h e i n n e r r a d i u s . We have used a f o u r t h order Runge-Kutta i n t e g r a t i o n /2/ on a Hewlett Packard 9836 desktop computer. The t r a n s i t i o n p o i n t between t h e regions w i t h p l a s t i c and e l a s t i c deformation i s determined by a double r o o t f i n d e r method. F i r s t , t h e i n t e r s e c t i o n o f t h e two curves f o r t h e t a n g e n t i a l s t r e s s M i t h p l a s t i c and e l a s t i c deformation i s determined. Then, t h e c o e f f i c i e n t s C3 and C4
(equs. 3, 4 and 7 ) a r e m o d i f i e d t o match t h e r a d i a l stresses o f t h e two s o l u t i o n s a t t h e t r a n s i t i o n p o i n t . Now, the i n t e r s e c t i o n of t h e curves f o r t h e t a n g e n t i a l s t r e s s i s determined f o r t h e m o d i f i e d e l a s t i c s o l u t i o n and t h e procedure i s repeated u n t i l the value x t i s found where both t h e r a d i a l and t h e t a n g e n t i a l stresses a r e equal.
The procedure converges r a p i d l y and t h e t r a n s i t i o n i s smooth such t h a t t h e p o s i t i o n o f xt
IS
n o t c r i t i c a l . This i n d i c a t e s t h a t o u r s o l u t i o n w i l l be a reasonable approximation o f r e a l i t y where the t r a n s i t i o n between p l a s t i c and e l a s t i c behaviour i s n o t as sharp as we have assumed. An a l t e r n a t i v e c r i t e r i o n f o r t h e t r a n s i t i o n p o i n t between t h e two s o l u t i o n s i s g i v e n by t h e i n t e g r a l- 2 .
31T 3 6 T 38T LOT L1 T L1.L T
F i g . 1 Radial and t a n g e n t i a l stresses F i g . 2 Radial and t a n g e n t i a l stresses f o r a c o i l w i t h a = 3 , u 6 k b , f o r t h e same c o i l as i n f i g . 1, a t e = 1, f = I a t differen! a g n e t i c h i g h e r magnetic f i e l d s .
f i e l d s .
C1-918 JOURNAL DE PHYSIQUE
F i g . 3 Radial and t a n g e n t i a l stresses f o r the same c o i l as i n f i g . 1, a t a f i e l d of 40 T and w i t h an e x t e r n a l l y a p p l i e d s t r e s s o f 1 kb.
which o n l y depends on t h e magnetic f i e l d , t h e e x t e r n a l l y a p p l i e d stresses, al and a.
This can be used f o r a cross-check.
As an example, r e s u l t s f o r a t y p i c a l small pulsed f i e l d c o i l a r e shown i n f i g s . 1 and 2. I t i s e v i d e n t how t h e t r a n s i t i o n p o i n t moves r a p i d l y from t h e i n n e r t o t h e o u t e r r a d i u s as t h e magnetic f i e l d i s increased over a range o f o n l y a few.tesla.
When t h e e n t i r e c o i l i s undergoing p l a s t i c deformation, t h e s o l u t i o n o f ( 8 ) and (1) gives t h e r a d i a l s t r e s s a t t h e o u t e r r a d i u s t h a t must be supported by an e x t e r n a l r e i n f o r c i n g c y l i n d e r t o prevent the c o i l from exploding ( f i g . 2). F i g . 3 gives an example f o r the e f f e c t o f an o u t e r reinforcement (simulated by t h e a p p l i c a t i o n o f a given r a d i a l s t r e s s ) on t h e t r a n s i t i o n p o i n t which i s s h i f t e d back towards t h e i n n e r radius.
The numerical c a l c u l a t i o n s demonstrate t h a t i n many cases of p r a c t i c a l i n t e r e s t t h e t a n g e n t i a l s t r e s s i s almost constant and equal t o the l i m i t i n g s t r e s s f o r p l a s t i c deformation. Therefore, the magnetic f i e l d where t h e e n t i r e c o i l j u s t becomes p l a s t i c a l l y deformed can be estimated from the equation
which has been d e r i v e d under t h i s assumption. T h i s assumption always overestimates t h e l i m i t i n g f i e l d .
I n f i g . 2 we show t h e e f f e c t o f i n c r e a s i n g the magnetic f i e l d much beyond t h i s l i m i t . Through t h e r e l a t i o n w i t h the r a d i a l s t r e s s (equ. 8 ) t h e r e i s an i n c r e a s i n g d e v i a t i o n o f the t a n g e n t i a l s t r e s s from t h e e l a s t i c - p l a s t i c l i m i t . F i n a l l y , t h e t a n g e n t i a l s t r e s s becomes p o s i t i v e and approaches t h e value o f t h e r a d i a l s t r e s s , i n d i c a t i n g t h a t t h e m a t e r i a l begins t o behave l i k e a l i q u i d . I n our s o l u t i o n , t h e f i n a l t r a n s i t i o n i n t o t h i s s t a t e occurs q u i t e suddenly. T h i s d i s c u s s i o n i s o f course somewhat academic because i n p r a c t i c e a c o i l w i l l f a i l l o n g before t h i s c o n d i t i o n i s approached.
I 1 1
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COIL DESIGN AND EXPERIMENTAL RESULTSTheedesign o f our p r e s e n t l y used wire-wound c o i l s i s shown i n f i g . 4.
Copper w i r e w i t h r e c t a n g u l a r cross s e c t i o n i s used t o o b t a i n a good f i l l i n g f a c t o r and a good transmission o f s t r e s s w i t h o u t s l i p p i n g .
The c o i l s are wound "wet" i . e . t h e epoxy impregnation i s a p p l i e d w i t h a brush d u r i n g t h e winding procedure. T h i s a l l o w s t h e use o f a f i l l e d epoxy ( S t y c a s t 2850 FT) w i t h improved heat c o n d u c t i v i t y and a c o e f f i c i e n t o f expansion matched t o t h a t o f copper.
o f t h e g l a s s f i b r e sheet covering t h e outermost l a y e r o f the c o i l , i n order t o o b t a i n t h e b e s t p o s s i b l e transmission o f r a d i a l s t r e s s , and f o r keeping t h e i n n e r r a d i u s o f t h e reinforcement as small as possible. A c r i t i c a l p o i n t i s a t t h e spot where t h e w i r e comes o u t o f the c o i l a t t h e i n n e r l a y e r ; a t t h i s place i t i s subjected t o t h e f u l l magnetic f o r c e b u t n o t supported by t h e o u t e r l a y e r s . As much as possible, t h i s w i r e i s l e a d along t h e magnetic f i e l d l i n e s , and a d d i t i o n a l reinforcement i s provided by wrapping i t w i t h g l a s s f i b r e t h r e a d and s t e e l w i r e . Another c r i t i c a l p o i n t are t h e contacts. We found AMP W-crimp
@
connections b o t h p r a c t i c a l and s u f f i c i e n t l y strong. The crimp connection i s made i n t h e c l o s e v i c i n i t y o f t h e c o i l and i s embedded i n c a s t epoxy w i t h a reinforcement by loose g l a s s f i b r e s . To t h e e y e l e t o f t h e crimp connector, a s t r o n g brass s t r i p i s b o l t e d which extends o u t of t h e epoxy and which i s d i r e c t l y connected t o t h e cables l e a d i n g t o the c a p a c i t o r bank.f i g . 4 Design o f a standard c o i l as i t i s now i n use a t our l a b o r a t o r y w i t h a 70 kJ, 3.5 kV c a p a c i t o r bank. Copper w i r e 2 x 1 mm, s t e e l band ( y i e l d
s t r e n g t h 12.7 kb) 2 x 0.5 mm, 250 t u r n s , 16.4 mm i n n e r diameter, 52.5 mm o u t e r diameter, 56 mm l e n g t h , inductance 0.8 mH, r e s i s t a n c e 78 ma a t 77 K and 228 ma a t 300 K, peak f i e l d 42 T a t 3 kV charging voltage, 9.1 ms pulse d u r a t i o n (ha1 f p e r i o d )
.
F i g . 5 The i r r e v e r s i b l e change o f the inductance d u r i n g t h e b r e a k - i n o f two c o i l s .
o : standard c o i l as described i n f i g . 4 x : copper w i r e 1.5 x 2.6 mm, 163 turns,
16.7 mm i n n e r diameter, 43.5 mm o u t e r diameter, 56 mm l e n g t h , i n - ductance 0.244
mH,
r e s i s t a n c e 32 mn a t 77 K and 99 ma a t 300 K, peak f i e l d 40 T a t 2.5 kV charging voltage, 7.9 ms pulse d u r a t i o n*
: c o i l explodedC1-920 JOURNAL DE PHYSIQUE
Our c a l c u l a t i o n s as w e l l as p r a c t i c a l experience have shown t h a t t h e most i m p o r t a n t s i n g l e f a c t o r f o r o b t a i n i n g t h e h i g h e s t p o s s i b l e f i e l d i s t h e mechanical s t r e n g t h o f the w i r e i t s e l f . For c o i l s t h a t generate magnetic f i e l d s w i t h a l o n g p u l s e d u r a t i o n (from several m i l l i s e c o n d s t o seconds), t h e e l e c t r i c a l performance depends c r i t i c a l l y on t h e r e s i s t i v i t y /3/. Therefore, a compromise must be made between good e l e c t r i c a l c o n d u c t i v i t y and h i g h mechanical strength; f o r t h e known m a t e r i a l s these two
p r o p e r t i e s f o l l o w opposite tendencies. I n r e c e n t years, a number o f s p e c i a l copper a l l o y s have become a v a i l a b l e t h a t combine increased mechanical s t r e n g t h w i t h reaso- nab1 e c o n d u c t i v i t y
,
examples a r e copper w i t h a1 uminium oxide ( ~ l idcop)@ o r copper- zirconium. The l a t t e r a l l o y has been used a t the I n s t i t u t e f o r S o l i d S t a t e Physics, Tokyo /4/. Another approach t o t h i s problem i s t h e use o f composites w i t h pure copper f o r c o n d u c t i v i t y and a d i f f e r e n t m a t e r i a l f o r mechanical s t r e n g t h . I n t h e USSR, c o i l s have been made w i t h superconducting w i r e as i t i s manufactured f o r making s t a b i l i z e d superconducting c o i 1 s /5/. I n t h i s a p p l i c a t i o n , t h e superconducting core i s o n l y used f o r i t s mechanical strength; i t i s n o t p r a c t i c a l t o cool a l a r g e pulsed f i e l d c o i l by l i q u i d helium. A t our l a b o r a t o r y we have developed a technique o f simultaneously winding a s t e e l band o f t h e same w i d t h on t o p o f t h e copper w i r e w i t h r e c t a n g u l a r cross s e c t i o n . This i s a good combination b u t t h e r e i s an i n s u l a t i o n problem as t h e s t e e l w i r e i s n o t f u r n i s h e d w i t h an i n s u l a t i n g coating. The s t e e l w i r e i s e l e c t r i c a l l y connected i n p a r a l l e l t o t h e copper w i r e f o r a v o i d i n g s t r o n g p o t e n t i a l d i f f e r e n c e s between adjacent s t e e l and copper w i r e s . The problem then a r i s e s a t t h e end p l a t e s . During t h e f i e l d pulse, t h e c o i l s expand r a d i a l l y b u t i n t h e a x i a l d i r e c t i o n they c o n t r a c t . As a consequence, t h e c o i l may p u l l away a x i a l l y from t h e epoxy end plugs, l e a v i n g a gap i n which a surface discharge may e a s i l y develop. We have solved t h i s problem by winding t h e c o i l between n y l o n end p l a t e s w i t h a s p i r a l groove s t r u c t u r e which i s thus rep1 ic a t e d i n t h e epoxy. A f t e r removal o f t h e nylon p l a t e s , the c o i l i s p o t t e d i n epoxy which adheres much b e t t e r t o t h i s groove s t r u c t u r e . I n a d d i t i o n , we p r o v i d e a gap f i l l e d w i t h t h i n t e f l o n sheet a t a small d i s t a n c e from t h e s i d e w a l l s o f the winding a t a p o i n t where a gap may open up w i t h o u t danger o f flashover.The c o i l s are precooled by l i q u i d n i t r o g e n and d u r i n g t h e f i e l d p u l s e t h e y heat up t o above room temperature. We have conducted a number o f t e s t s on the response o f d i f f e r e n t i n s u l a t i n g m a t e r i a l s t o t h i s harsh treatment : Arodyn T i
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( e s t e r i m i d e ) , Kapton f i l m and d i f f e r e n t combinations w i t h g l a s s f i bre-epoxy. I n t h i s s e r i e s o f experiments where t h e c o i l s were t e s t e d t o t h e breaking p o i n t , no s i g n i f i c a n t d i f f e r e n c e between t h e t e s t e d m a t e r i a1 s c o u l d be detected.Each new c o i l undergoes a break-in procedure c o n s i s t i n g i n a sequence o f pulses w i t h i n c r e a s i n g v o l t a g e o f t h e c a p a c i t o r bank. A f t e r each pulse, t h e r e s i s t a n c e and inductance o f the c o i l a r e measured. While t h e r e s i s t a n c e mainly serves t o m o n i t o r t h e temperature o f t h e c o i l , t h e inductance gives a s e n s i t i v e i n d i c a t i o n o f any permanent change i n t h e shape of t h e c o i l . If a change i n t h e inductance i s observed, another p u l s e i s given a t t h e same v o l t a g e u n t i l t h e inductance i s s t a b l e a t the new value. Inductance data from such b r e a k - i n runs a r e given i n f i g . 5. The behaviour follows indeed t h e trends i n d i c a t e d by the c a l c u l a t i o n s . The observed onset o f p l a s t i c deformation corresponds t o a l i m i t i n g s t r e s s o f 2 kb f o r t h e copper c o i l w i t h a = 2.6 and t o 5 kb f o r t h e c o i l w i t h i n t e r n a l s t e e l reinforcement and a = 3.2. The observed peak f i e l d i n d i c a t e s t h e e f f e c t i v e n e s s o f work hardening and o f t h e e x t e r n a l reinforcement. It i s a m a t t e r o f personal judgment t o stop t h e break-in procedure a t a p o i n t where t h e c o i l i s w e l l c o n d i t i o n e d b u t n o t y e t damaged t o t h e p o i n t t h a t may l a t e r r e s u l t i n a sudden f a i l u r e o f the i n s u l a t i o n .
We had p o i n t e d o u t e a r l i e r t h a t an o u t e r reinforcement f o r an e l a s t i c c o i l would have t o be prestressed i n o r d e r t o be r e a l l y e f f e c t i v e . This i s n o t necessary when t h e c o i l undergoes some p l a s t i c deformation. I n t h i s case t h e c o i l i t s e l f remains i n a prestressed c o n d i t i o n a f t e r t h e a p p l i e d magnetic s t r e s s i s removed. T h i s accounts f o r t h e s u p e r i o r performance o f pulsed f i e l d c o i l s t h a t have undergone p l a s t i c de- f o r m a t i o n d u r i n g a c o n d i t i o n i n g process. I f p r o p e r l y made and t r e a t e d , such c o i l s can g i v e r e l i a b l e performance f o r many f i e l d pulses t h a t are o n l y s l i g h t l y below t h e peak f i e l d used i n t h e c o n d i t i o n i n g process.
We should l i k e t o t h a n k M r . R. Vens f o r h i s c o n t r i b u t i o n s t o t h e experimental work, and Mr. W. Wetekam from F & G, Arolsen (Germany) f o r p r o v i d i n g samples of copper w i r e w i t h d i f f e r e n t i n s u l a t i n g m a t e r i a l s .
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