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Applicability of the normal distribution for calculating voltage-current characteristics of superconductors
On the basis of the longitudinally &homogeneous multifilament superconductor model the normal distribution function for the critical current values along a conductor is considered. A conclusion has been made that it is correct to apply the normal distribution for calculating voltage-current characteristics (VCC) within a wide range of inhomogeneities.
Applicability of the normal distribution for calculating v o l t a g e - c u r r e n t characteristics of superconductors E.Yu. Klimenko and A.E. Trenin Keywords: superconductivity, critical current values, inhomogenaity
Recently it has been found that the real form of superconductor transition to the normal state due to temperature, magnetic field or current increase, is essential for calculating the electric field limit value in a superconductor L2, conductor stability parameters 3 or current redistribution parameters in a multifilament conductorL One can account for the form of this transition, satisfactorily described by the exponential function, by the longitudinal or volume inhomogeneities of a superconductor employing the normal (Gaussian) distribution for its description. This method is used to describe the local critical current distribution along the length of an inhomogeneous conductor2 and that of critical temperatures of elements in an equivalent circuit reflecting the superconductor volume inhomogeneityL It is evident that the normal distribution contains nonphysical values of the parameters, since the current and temperature critical values cannot be less than zero and greater than some limit value. In order to check in which cases it is correct to use the normal distribution. We compared the latter with the results of computer simulation of real distributions limited by the zero and maximum distribution parameter values. For this comparison a model of the longitudinally inhomogeneous multifilament superconductor a was used. Let such a conductor contain I filaments in a matrix of a normal metal, and let a unit of the conductor length be divided into N sections. Let any filament be either sound or broken (defective) at each of the sections. By distributing the defects between the sections at random it is possible to simulate the critical current longitudinal inhomogeneity in the conductor, if we assume the sound wire critical current to be equal to unity and that of the defective one to zero, at each of the sections. In so doing, the maximum critical current turns out to equal I and the minimum is zero. Assuming a total number of defects, it is possible to investigate the dependence of the critical current distribution on the density of these defects.
0011-2275/85/010027-02 Cryogenics. January 1985
Calculations have been carried out using the computer for I = 50 and N = 300. For this purpose the cross section in which the defect was placed, was chosen at random by generating a uniformly distributed random value for a f x e d number of defects. After that the normalized discrete distribution function F (I c, C) was formed, which determined the share of sections with the given critical current value if the total number of defects for a unit of the conductor length was 1.5 × 104 C, Fig. I presents the results of such a calculation for three values of C. Each distribution has been obtained by averaging lO attempts. The vertical lines denote the functions F and the dashed line indicates the normal distributions
F ,o,o °
1
(C) 0xp-
20=(0°)=]]
- oo < / ~ < , ~
'°° I C=93.3%
C=40%
C=3.3%
fi
_8 o i 4o I
~
dl/ /k 0
20
3o I¢
40
45
\
50
Fig. 1 The critical current distribution function of individual cross sections for defect concentration C (vertical lines). The Gaussian approximation is shown by dashed lines
$03.00 © 1985 Butterworth Et Co (Publishers) Ltd 27
8
C=95.3%
80% 66.6%53.3%
40% 26B% 13.3%66%33*/.
0.5 0.4 0.3 0.2 0.1 0
m
12
i
I0
__
%8 6
;'S~ ! t ~ i
4
~iiect
C~c°ni:~l°n:it.!!
2
50
40 "50 20 I0 C
Fig. 2 The dependence of the normal distribution parameters 8, o2, M on the defect concentration
numerically equal to the value, which is excessive over the critical one. The shape of the characteristics obtained is in good compliance with the type of the multifilament conductor real VC(Y. It follows from the computer experiment which we performed that the normal distribution application for calculating the VCC of inhomogeneous superconductors is correct over a broad region of inhomogeneities, while the nonphysical parameter values do not affect the type of transition characteristics due to the low probability of their appearance.
with the parameters M and tr selected as follows so M(C'9 =
Ic=O so o2
=
The authors are from the I.V. Kurchatov Institute of Atomic Energy, Moscow, USSR. Paper ~eceived May 1984.
ze~0 (lc - M(C))2 F(Ie, C) =
Employing the well-known criterion X2,6 it has been found at the 5% level commonly used for such estimations that the hypothesis about the normal distribution-applicability is valid with 95% probability within a wide range of concentration values: 2.67% < C < 97.33%. Outside the limits the distributions are appreciably asymmetrical. In Fig. 2 the dependences of the distribution parameters M, ¢ and 8 = ~r/M on the defect concentration are given. Fig. 3 represents the calculated conductor VCC obtained on the assumption 2 that at each section the current exceeding the number of sound filaments flows over matrix and creates at this section a voltage being
28
Authors
References
1 Polak,M., Hlasnik,I., Krempassky,L Cryogenics 13 (1973) 702 2 Dorofejev,G.L, Imenitov,A.B.,Klimenko,E.Yu.Cryogenics 20 (1980)3O7 3 Klimenko, E.Yu., Martovetsky, N.N., Novikov, S.I. On stabilityof superconductorswith spreadtransitionto normal state.Doklady Akademii Nauk SSSR 261 (1981)1350(in Russian) 4 Dorefejev,G.L, Klimenko,E.Yu.Effectof multiwire superconductortransverseresistanceon its voltage-current characteristic.Doklady Akademii Nauk SSR (1979)248 (1979)97 (in Russian) 5 Klimenko,E.Yu.,Trenin,A.E.Cryogenics (to be published) 6 Bronstein,I.N., Semendyaev,K.A.Handbookon mathematics for engineersand studentsof highertechnicalschools. Nauka, Moscow(1981)(in Russian)
Cryogenics. January 1985
Applicability of the normal distribution for calculating v o l t a g e - c u r r e n t characteristics of superconductors E.Yu. Klimenko and A.E. Trenin Keywords: superconductivity, critical current values, inhomogenaity
Recently it has been found that the real form of superconductor transition to the normal state due to temperature, magnetic field or current increase, is essential for calculating the electric field limit value in a superconductor L2, conductor stability parameters 3 or current redistribution parameters in a multifilament conductorL One can account for the form of this transition, satisfactorily described by the exponential function, by the longitudinal or volume inhomogeneities of a superconductor employing the normal (Gaussian) distribution for its description. This method is used to describe the local critical current distribution along the length of an inhomogeneous conductor2 and that of critical temperatures of elements in an equivalent circuit reflecting the superconductor volume inhomogeneityL It is evident that the normal distribution contains nonphysical values of the parameters, since the current and temperature critical values cannot be less than zero and greater than some limit value. In order to check in which cases it is correct to use the normal distribution. We compared the latter with the results of computer simulation of real distributions limited by the zero and maximum distribution parameter values. For this comparison a model of the longitudinally inhomogeneous multifilament superconductor a was used. Let such a conductor contain I filaments in a matrix of a normal metal, and let a unit of the conductor length be divided into N sections. Let any filament be either sound or broken (defective) at each of the sections. By distributing the defects between the sections at random it is possible to simulate the critical current longitudinal inhomogeneity in the conductor, if we assume the sound wire critical current to be equal to unity and that of the defective one to zero, at each of the sections. In so doing, the maximum critical current turns out to equal I and the minimum is zero. Assuming a total number of defects, it is possible to investigate the dependence of the critical current distribution on the density of these defects.
0011-2275/85/010027-02 Cryogenics. January 1985
Calculations have been carried out using the computer for I = 50 and N = 300. For this purpose the cross section in which the defect was placed, was chosen at random by generating a uniformly distributed random value for a f x e d number of defects. After that the normalized discrete distribution function F (I c, C) was formed, which determined the share of sections with the given critical current value if the total number of defects for a unit of the conductor length was 1.5 × 104 C, Fig. I presents the results of such a calculation for three values of C. Each distribution has been obtained by averaging lO attempts. The vertical lines denote the functions F and the dashed line indicates the normal distributions
F ,o,o °
1
(C) 0xp-
20=(0°)=]]
- oo < / ~ < , ~
'°° I C=93.3%
C=40%
C=3.3%
fi
_8 o i 4o I
~
dl/ /k 0
20
3o I¢
40
45
\
50
Fig. 1 The critical current distribution function of individual cross sections for defect concentration C (vertical lines). The Gaussian approximation is shown by dashed lines
$03.00 © 1985 Butterworth Et Co (Publishers) Ltd 27
8
C=95.3%
80% 66.6%53.3%
40% 26B% 13.3%66%33*/.
0.5 0.4 0.3 0.2 0.1 0
m
12
i
I0
__
%8 6
;'S~ ! t ~ i
4
~iiect
C~c°ni:~l°n:it.!!
2
50
40 "50 20 I0 C
Fig. 2 The dependence of the normal distribution parameters 8, o2, M on the defect concentration
numerically equal to the value, which is excessive over the critical one. The shape of the characteristics obtained is in good compliance with the type of the multifilament conductor real VC(Y. It follows from the computer experiment which we performed that the normal distribution application for calculating the VCC of inhomogeneous superconductors is correct over a broad region of inhomogeneities, while the nonphysical parameter values do not affect the type of transition characteristics due to the low probability of their appearance.
with the parameters M and tr selected as follows so M(C'9 =
Ic=O so o2
=
The authors are from the I.V. Kurchatov Institute of Atomic Energy, Moscow, USSR. Paper ~eceived May 1984.
ze~0 (lc - M(C))2 F(Ie, C) =
Employing the well-known criterion X2,6 it has been found at the 5% level commonly used for such estimations that the hypothesis about the normal distribution-applicability is valid with 95% probability within a wide range of concentration values: 2.67% < C < 97.33%. Outside the limits the distributions are appreciably asymmetrical. In Fig. 2 the dependences of the distribution parameters M, ¢ and 8 = ~r/M on the defect concentration are given. Fig. 3 represents the calculated conductor VCC obtained on the assumption 2 that at each section the current exceeding the number of sound filaments flows over matrix and creates at this section a voltage being
28
Authors
References
1 Polak,M., Hlasnik,I., Krempassky,L Cryogenics 13 (1973) 702 2 Dorofejev,G.L, Imenitov,A.B.,Klimenko,E.Yu.Cryogenics 20 (1980)3O7 3 Klimenko, E.Yu., Martovetsky, N.N., Novikov, S.I. On stabilityof superconductorswith spreadtransitionto normal state.Doklady Akademii Nauk SSSR 261 (1981)1350(in Russian) 4 Dorefejev,G.L, Klimenko,E.Yu.Effectof multiwire superconductortransverseresistanceon its voltage-current characteristic.Doklady Akademii Nauk SSR (1979)248 (1979)97 (in Russian) 5 Klimenko,E.Yu.,Trenin,A.E.Cryogenics (to be published) 6 Bronstein,I.N., Semendyaev,K.A.Handbookon mathematics for engineersand studentsof highertechnicalschools. Nauka, Moscow(1981)(in Russian)
Cryogenics. January 1985