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Scaling law in voltage — current characteristics of hard superconductors
From scaling laws for pinning we have suggested scaling laws for voltage - current characteristics o f type II superconductor~ After linearization of Jc dependence on magnetic field B, temperature T and stress e, they lead to exponential dependence of the voltage E on these parameters In order to obtain genera/scaling laws, then the temperature, field and stress dependence of the denominator Jo in E = E o exp [(J-Jc)/Jo] should also be determined. The generalization of the instability criterion leads to the conclusion that the prestrain of the superconducting material can enhance or deteriorate the stability of the winding.
Scaling law in voltage- current characteristics of hard superconductors I. H R i s n i k
and
S. Tak~ics
Key words: superconductor, scaling law, voltage-current characteristics
Voltage-current characteristics of technical superconductors have been intensively studied in previous years, ~-n because the E-J relation is a principal characteristic of each electrotechnical material. It was pointed out 12-16 that in many technical superconductors at constant temperature T, magnetic induction B and strain e the E-J characteristic is an exponential function of the form
( J - Jco(T, B, e) ) E = E o exp
Jo (r, B, e)
and following la
K(e)'~B~(e) with
B~c2(e) =
Bc~2(em)( I
--
a[e[ u)
e
=
ec
Fp(t, b, e) "~f(b). A(t). K(e)
(2)
where b = B/B*e2(e, T), t = T/Too and Be~2(e, T) is the bulk value for the upper-critical field obtained from an extrapolation of the pinning force at practical current densities. =° + In superconductors with isotropic pi&ning forces, Fp does not depend on the orientation of B with regard to the superconductor structure and f(b) and A(0 can be e x p r e s s e d as ~7 "~" bP(1 -- b) q
A(t) "" B~(t) with "" 1 -
--
a[el"),
-- e m
where ec is the strain acting on the whole conductor and em is that value of e¢ at whichP for given t is maximum.
In technical superconductors NbTi, NbaSn, VaGa and Nb3Ge, Joo as function of T, B and e can ~e det_$rmi~ed by scaling laws for the volume pinning force Fp = J¢ x B as follows 17-22
B¢2(t)
Bc~m(l
The combination of the temperature and stress scaling laws is still an open question. The different approaches are not always consistent) 9-22 The best solution seems to be that of =
Scaling laws
f(b)
=
(3)
(I)
where Joo is the current density in the superconductor at electric field Eo and Jo is a material parameter.
(2c)
t2
(2a)
(2b)
T
2
(4)
with [Bc*2( e ) ] t / w to(e) = Too L
(5)
Jo(T, B, e) changes very slowly with T, B and e. Therefore, substituting expressions (2)-(5) into (1) and dividing it by Eo we obtain in the first approximation the scaling law for E--J characteristics: E _ (Jc(bo, to, eo) e = E--o - e x p [ J o ( b o , to, Co)
IJe(bo, "to, Co)
c,bte,
Jc(bo, to, Co)
]1
=
0011-2275/83/060314-03 $03.00 © 1983 Butterworth & Co (Publishers) Ltd. 314
CRYOGENICS . JUNE 1983
B
~'*,(To)
(1 - alelU) n = exp
B~2(eo~To)
\-YLJo/ J
rc(bo, to, eo)
Jo(bo, to, eo)
Jc(bol tot eo)
To
(1 -aleolU) n
\ Bc*2m(To)
1 - r--~o T2
1
~o (1 -alelU) a/w
= (1 - alel u)
2
12[ rco/J
1-
(6)
B B~c2(To) B~2(eolTo)
/
Bc~m(To)
We see that in this case E is an exponential function not only of J, but of t, b, and e, too ie of T, B and e. Such a dependence of E on J, B and T has been observed in is and t6 and on e inls. From (6) one can determine the intervals for t, b, and e in which (7) gives E values with an accuracy better than eg -+ 10%. Equation (6) or (7) can allow the generalisation of the stability criterion defined in 14
=(1 - a l e l u)
1
Be~2m(To)
1-(To(e)
with
B~c2(e, To)
q
[a(eJ)
To2 1 - --
7rR2 [ at
a(eJ) + aa(eJ) ] '-~-] >
+ ~
a~
2~rR aT
(9)
~o
Values of the parameters n, p, q, l, a and Bc2(em, 4.2 K) for different materials taken from ts and 19 are given in Table 1. The values of u ~ 1.7 and w ~ 3 are nearly material independent. The most convenient choice for To should be To = 4.2 K.
where q is density of heat transferred from the conductor surface to the helium. The first two terms on the left hand side at increasing magnet current are positive while the third term related to the strain change can be positive or negative. It is negative if
a~
> 0 and eo < 0
(10a)
ae ~7- < 0 and eo > 0
(10b)
al
Discussion
or
In spite o f the very different form of the temperature, field and stress dependence of Jc, in (6) they have some similar features, a9'2°'22 There exist considerable large intervals where the dependence on t, b, e is well expressed with the linear ones. In addition, in the most practical cases the changes of t, b, e are small, therefore Jo can be well approximated by the first two terms of Taylors series around t o, bo, eo with Jo being nearly constant. Then (1) can be expressed as E = E o (to, 6o, eo, Jo)
Me
Mc
expt
Me
~ - ( b - b o ) - ~ - 0(6-%)}
J-Jc(to, bo, eo, Ec)- ~ - o ( t - t o ) Jo(to, bo, Co)
J = Eo (to, bo, %) exp joo
Jeo joo
At too
Ab boo
A6 } eoo
This means that prestrain o f the superconducting material - due to different thermal expansion coefficients o f the superconductor composite components and/or o f the coil construction materials as well as that executed during the coil winding - together with the strain due to electrodynamic and other forces during the magnet operation can enhance or deteriorate the winding stability.
Authors The authors are from Electrotechnical Institute, Electrophysical Research Centre, Slovak Academy o f Sciences, 842 39 Bratislava, Czechoslovakia. Paper received February 1983.
References (7)
1
Kim, Y.B., Hempstead, C.F., Strnad, A.R. Phya Review
2 3
Anderson,P.W.PhysRev Lett 9 (1962) 309 Webb,W.W.JApplPhys42 (1971) 107 Kim, Y.B., Hempstead, C.F., Slrnad, A.R. Phys Rev 131
(1963) 528 where
Joo
Joo = Jo (tot bol eo),
4
5 6
boo = ~
goo
,
&o eoo- (af~_)o
CRYOGENICS. JUNE 1983
(8)
7 8 9
(1963) 2486 Beasley, M.R., Labusch, R., Webb, W.W. Phys Rev 181 (1969) 682 Kim, Y.B., Hempstead, C.F., Strnad, A.IL Rev Mod Phys 36 (1964) 43 Kim,Y.B.ProclCECI, Tokyo, Kyoto (1967) Wade, J.M.A. PhilMag 20 (1969) 168 Matsunaga, S., Yano, S., Ushio, S., Ushio, K., Nakaya, S. Proc ICECI Tokyo, Kyoto (1967) 178
315
Table 1.
Parameter values for different materials n
p
q
Bcm (4,2 K), T
/
a(e, < O)
a(eo > O)
Nb3Sn
1
0.5-1,2
2
22
2,5-2.6
900
1250
N b-Mf/Cu-Sn-Ga
1.2
0,36
t .13
24.8
V3Ga
1.4
0.4-0.5
1
21.2
3.5
450
650
Nb3Ge
1,6
0,5-0.7
2-2.24
24
2.5
NbTi
4
0.6-1
0,7-1,2
10.5
2-2,5
0
23
10 11 12 t3 14 15
Voelket, F. Particle A ccel I (1970) 205 Gilbert, W.S. 1971 Particle Accel Conf. Chicago, Illinois (1971) 629 Alexejev~kij, N.E., Hl~nik, I , Dubrovin,A.V. JETP 54 (1968) 84 Hli~nik,L, Pol~lk, M., Krempasky, L. Phys Stat Sol a 16 K (1973) 153 Pol~k, M., ladsnik, I., Ktempasky, L. Cryogenics 13 (1973) 702 Do~ofejev, G.L., Imenilov, A.B., Klimenko, EJu. Preprint IAE-2987, I.V. Kurchatov Atomic Energy Institute, Moscow (1978)
17
Dorofejev, G.L., Imenilov, A.B., Klimenko, EJu. Cryogenics 20 (t980) 307 Kmmer, EJ.ApplPhys44 (1973) 1360
18
Clark,A.F., EIdn, LW. IEEE Trans on Mag MAG-13 (1977) 38
19
Eldn, J.W. IEEETransonMagMAG-17(t981) 758
20
Eldn, J.W. Cryogenics 20 (1980) 611
16
21
CampbeU,A.M.,Evetts, J.E.Adv Phys 21(1972) 199
22
Luhman, T., Welch, D.O. Filamentary A 15 Superconductors, Plenum Publ Corp New York (1980) 171
| ~ilD'qm" " ientl¥, the Tectr°LnireC.a o,,,~ninq conCUI~ j ~,~ test, meaSU !i.
Jointlym _. ensiveshovnng ..= ~ost comp'eh~'~..~ts and s e r w ~ equ=~, .na~_o~iants p ~ ~-`~t h i s all-~
_
,u
,
,~reat e x h
"e1%u~ ~
e UI<-.
to bring Y ° ~ . r a d i c a . ' ~ . y imp,'-'-t e c h n o l o g y u ' ~ . ' c t and proceSS.
ition deah g
.
P
temperature
~ - - , " ~ r 81%(~ ~
.-r"\_
~(~
Londons pre ,. tune 14-17, 1983. --O1%t~Ol ~ Y ~ ' - ; - ~ n c e ~"
_...= Cordelia"
O1% ~
•
_~
~ -el%Soil a l % ~ .
ternP.
control PL_~nd ~ecUO~ t~+ 19ractice" eve1%t o~ 11%st~
316
i
Is
uou.r ,2
close,
~9.~6.5.30 da~l¥, ~ ' ' ' - " 4 00Pro' _ ~ international~,~ntact" T~4de,-~ ~ ~Ivmouth Boao, %.,v*~ " t d , ~ 1 * '~ "E~uhibmgnsL "_" Te3.08Zg4671 TavistOck, uevou. -~ "
CRYOGENICS . JUNE 1983
Scaling law in voltage- current characteristics of hard superconductors I. H R i s n i k
and
S. Tak~ics
Key words: superconductor, scaling law, voltage-current characteristics
Voltage-current characteristics of technical superconductors have been intensively studied in previous years, ~-n because the E-J relation is a principal characteristic of each electrotechnical material. It was pointed out 12-16 that in many technical superconductors at constant temperature T, magnetic induction B and strain e the E-J characteristic is an exponential function of the form
( J - Jco(T, B, e) ) E = E o exp
Jo (r, B, e)
and following la
K(e)'~B~(e) with
B~c2(e) =
Bc~2(em)( I
--
a[e[ u)
e
=
ec
Fp(t, b, e) "~f(b). A(t). K(e)
(2)
where b = B/B*e2(e, T), t = T/Too and Be~2(e, T) is the bulk value for the upper-critical field obtained from an extrapolation of the pinning force at practical current densities. =° + In superconductors with isotropic pi&ning forces, Fp does not depend on the orientation of B with regard to the superconductor structure and f(b) and A(0 can be e x p r e s s e d as ~7 "~" bP(1 -- b) q
A(t) "" B~(t) with "" 1 -
--
a[el"),
-- e m
where ec is the strain acting on the whole conductor and em is that value of e¢ at whichP for given t is maximum.
In technical superconductors NbTi, NbaSn, VaGa and Nb3Ge, Joo as function of T, B and e can ~e det_$rmi~ed by scaling laws for the volume pinning force Fp = J¢ x B as follows 17-22
B¢2(t)
Bc~m(l
The combination of the temperature and stress scaling laws is still an open question. The different approaches are not always consistent) 9-22 The best solution seems to be that of =
Scaling laws
f(b)
=
(3)
(I)
where Joo is the current density in the superconductor at electric field Eo and Jo is a material parameter.
(2c)
t2
(2a)
(2b)
T
2
(4)
with [Bc*2( e ) ] t / w to(e) = Too L
(5)
Jo(T, B, e) changes very slowly with T, B and e. Therefore, substituting expressions (2)-(5) into (1) and dividing it by Eo we obtain in the first approximation the scaling law for E--J characteristics: E _ (Jc(bo, to, eo) e = E--o - e x p [ J o ( b o , to, Co)
IJe(bo, "to, Co)
c,bte,
Jc(bo, to, Co)
]1
=
0011-2275/83/060314-03 $03.00 © 1983 Butterworth & Co (Publishers) Ltd. 314
CRYOGENICS . JUNE 1983
B
~'*,(To)
(1 - alelU) n = exp
B~2(eo~To)
\-YLJo/ J
rc(bo, to, eo)
Jo(bo, to, eo)
Jc(bol tot eo)
To
(1 -aleolU) n
\ Bc*2m(To)
1 - r--~o T2
1
~o (1 -alelU) a/w
= (1 - alel u)
2
12[ rco/J
1-
(6)
B B~c2(To) B~2(eolTo)
/
Bc~m(To)
We see that in this case E is an exponential function not only of J, but of t, b, and e, too ie of T, B and e. Such a dependence of E on J, B and T has been observed in is and t6 and on e inls. From (6) one can determine the intervals for t, b, and e in which (7) gives E values with an accuracy better than eg -+ 10%. Equation (6) or (7) can allow the generalisation of the stability criterion defined in 14
=(1 - a l e l u)
1
Be~2m(To)
1-(To(e)
with
B~c2(e, To)
q
[a(eJ)
To2 1 - --
7rR2 [ at
a(eJ) + aa(eJ) ] '-~-] >
+ ~
a~
2~rR aT
(9)
~o
Values of the parameters n, p, q, l, a and Bc2(em, 4.2 K) for different materials taken from ts and 19 are given in Table 1. The values of u ~ 1.7 and w ~ 3 are nearly material independent. The most convenient choice for To should be To = 4.2 K.
where q is density of heat transferred from the conductor surface to the helium. The first two terms on the left hand side at increasing magnet current are positive while the third term related to the strain change can be positive or negative. It is negative if
a~
> 0 and eo < 0
(10a)
ae ~7- < 0 and eo > 0
(10b)
al
Discussion
or
In spite o f the very different form of the temperature, field and stress dependence of Jc, in (6) they have some similar features, a9'2°'22 There exist considerable large intervals where the dependence on t, b, e is well expressed with the linear ones. In addition, in the most practical cases the changes of t, b, e are small, therefore Jo can be well approximated by the first two terms of Taylors series around t o, bo, eo with Jo being nearly constant. Then (1) can be expressed as E = E o (to, 6o, eo, Jo)
Me
Mc
expt
Me
~ - ( b - b o ) - ~ - 0(6-%)}
J-Jc(to, bo, eo, Ec)- ~ - o ( t - t o ) Jo(to, bo, Co)
J = Eo (to, bo, %) exp joo
Jeo joo
At too
Ab boo
A6 } eoo
This means that prestrain o f the superconducting material - due to different thermal expansion coefficients o f the superconductor composite components and/or o f the coil construction materials as well as that executed during the coil winding - together with the strain due to electrodynamic and other forces during the magnet operation can enhance or deteriorate the winding stability.
Authors The authors are from Electrotechnical Institute, Electrophysical Research Centre, Slovak Academy o f Sciences, 842 39 Bratislava, Czechoslovakia. Paper received February 1983.
References (7)
1
Kim, Y.B., Hempstead, C.F., Strnad, A.R. Phya Review
2 3
Anderson,P.W.PhysRev Lett 9 (1962) 309 Webb,W.W.JApplPhys42 (1971) 107 Kim, Y.B., Hempstead, C.F., Slrnad, A.R. Phys Rev 131
(1963) 528 where
Joo
Joo = Jo (tot bol eo),
4
5 6
boo = ~
goo
,
&o eoo- (af~_)o
CRYOGENICS. JUNE 1983
(8)
7 8 9
(1963) 2486 Beasley, M.R., Labusch, R., Webb, W.W. Phys Rev 181 (1969) 682 Kim, Y.B., Hempstead, C.F., Strnad, A.IL Rev Mod Phys 36 (1964) 43 Kim,Y.B.ProclCECI, Tokyo, Kyoto (1967) Wade, J.M.A. PhilMag 20 (1969) 168 Matsunaga, S., Yano, S., Ushio, S., Ushio, K., Nakaya, S. Proc ICECI Tokyo, Kyoto (1967) 178
315
Table 1.
Parameter values for different materials n
p
q
Bcm (4,2 K), T
/
a(e, < O)
a(eo > O)
Nb3Sn
1
0.5-1,2
2
22
2,5-2.6
900
1250
N b-Mf/Cu-Sn-Ga
1.2
0,36
t .13
24.8
V3Ga
1.4
0.4-0.5
1
21.2
3.5
450
650
Nb3Ge
1,6
0,5-0.7
2-2.24
24
2.5
NbTi
4
0.6-1
0,7-1,2
10.5
2-2,5
0
23
10 11 12 t3 14 15
Voelket, F. Particle A ccel I (1970) 205 Gilbert, W.S. 1971 Particle Accel Conf. Chicago, Illinois (1971) 629 Alexejev~kij, N.E., Hl~nik, I , Dubrovin,A.V. JETP 54 (1968) 84 Hli~nik,L, Pol~lk, M., Krempasky, L. Phys Stat Sol a 16 K (1973) 153 Pol~k, M., ladsnik, I., Ktempasky, L. Cryogenics 13 (1973) 702 Do~ofejev, G.L., Imenilov, A.B., Klimenko, EJu. Preprint IAE-2987, I.V. Kurchatov Atomic Energy Institute, Moscow (1978)
17
Dorofejev, G.L., Imenilov, A.B., Klimenko, EJu. Cryogenics 20 (t980) 307 Kmmer, EJ.ApplPhys44 (1973) 1360
18
Clark,A.F., EIdn, LW. IEEE Trans on Mag MAG-13 (1977) 38
19
Eldn, J.W. IEEETransonMagMAG-17(t981) 758
20
Eldn, J.W. Cryogenics 20 (1980) 611
16
21
CampbeU,A.M.,Evetts, J.E.Adv Phys 21(1972) 199
22
Luhman, T., Welch, D.O. Filamentary A 15 Superconductors, Plenum Publ Corp New York (1980) 171
| ~ilD'qm" " ientl¥, the Tectr°LnireC.a o,,,~ninq conCUI~ j ~,~ test, meaSU !i.
Jointlym _. ensiveshovnng ..= ~ost comp'eh~'~..~ts and s e r w ~ equ=~, .na~_o~iants p ~ ~-`~t h i s all-~
_
,u
,
,~reat e x h
"e1%u~ ~
e UI<-.
to bring Y ° ~ . r a d i c a . ' ~ . y imp,'-'-t e c h n o l o g y u ' ~ . ' c t and proceSS.
ition deah g
.
P
temperature
~ - - , " ~ r 81%(~ ~
.-r"\_
~(~
Londons pre ,. tune 14-17, 1983. --O1%t~Ol ~ Y ~ ' - ; - ~ n c e ~"
_...= Cordelia"
O1% ~
•
_~
~ -el%Soil a l % ~ .
ternP.
control PL_~nd ~ecUO~ t~+ 19ractice" eve1%t o~ 11%st~
316
i
Is
uou.r ,2
close,
~9.~6.5.30 da~l¥, ~ ' ' ' - " 4 00Pro' _ ~ international~,~ntact" T~4de,-~ ~ ~Ivmouth Boao, %.,v*~ " t d , ~ 1 * '~ "E~uhibmgnsL "_" Te3.08Zg4671 TavistOck, uevou. -~ "
CRYOGENICS . JUNE 1983