May 28, 1987 - LIL,. Figure 2. The dependence of i, on L/L, at 4nEc/Ro5, = 0.04. Substituting equation (30) ... The dependence of i, on E,/RoB, for the untwisted.
J. Phys. D: Appl. Phys. 20 (1988) 826-830. Printed in the UK
R G Mints and A L Rakhmanov Institute of High Temperatures, 127412, Moscow, USSR Received 28 May 1987
Abstract. The superconducting state stability in twisted multifilamentary wire is investigated. The case of varying transport current and transverse magnetic field is considered. It is shown that the current-carrying capacity I, of multifilamentary superconducting composites increases with the decrease of the twist pitch L and I, attains its maximum value at L less than some critical value L,.
1. Introduction
Thecurrent-carryingcapacity of superconductors is limited by the superconducting state stability relative to perturbations of a different physical nature in many cases of practical interest. If the intensity of these perturbations is not too high, then the maximum transport current I , is determined by the superconducting state stability with respect to small disturbances (Mints and Rakhmanov 1981,1984). The problemhasalready beenconsidered by the present authors for the case when the twisting of thesuperconducting wire is neglected(Mints and Rakhmanov 1982,1984). It has been shown that to explain the high values of I , Zs (where I, is the critical current) one has to take into account the behaviour of the current-voltage characteristics of a hard superconductor in the region of low values of the electric field E . At low E this I-V characteristic may be presented in the form (Polaket a1 1973)
-
j
=
is + il W E / & )
(1)
where j is the current density, j s = j ( E o ) is the critical current density and the ratioj l / j s is usually of the order of (1-3) x lo-'. Then, the differential conductivity of the superconductor is a ( E ) = j l / E . Assuming that jl = 107Am-* and E = V m-lone finds that a ( E ) = 10'' Q-' m-', which is muchhigher than the normal matrixconductivity.Sincethesuperconductingstate stability increases with the increase of longitudinal conductivity (Mints and Rakhmanov 1981), then the value of I , depends on the electricfield E induced by external sources (Mints and Rakhmanov 1982, 1984). In this paper the maximum transport currentI , for a twisted multifilamentary superconducting wire placed in a transverse magnetic field B,(t) is found. In 8 2 the 0022-37271881050826
+ 05 $02.50 @ 1988 IOP Publishing Ltd
stability criterion for a twisted wire is obtained. In 8 3 theelectric field distributioninatwistedcomposite with a transport currentis discussed and Q 4 investigates the dependence of I on the twist pitch L . Finally, in 8 5 the equation for ,Z is found for the case of a wire with a sufficiently small twist pitch L .
2. Stability criterion
In order to find the stability criterion in the case of a twisted composite superconductor the standard method of stability investigation may be used (Mints and Rakhmanov 1984). Let us consider a twisted multifilamentary wire of field B,(t). It is radius R. in atransversemagnetic known (Carr 1983) that in thissituationthecurrent jll and a density j has both a longitudinal component transverse component j,. Taking into account the current in the normal matrix one can write
il = is + il M E I E o ) + allEli
il
= a,E,
(2)
where Ell and E , are the longitudinal (along the filaments) and transverse components of the field E and GI!,U , are the longitudinal and transverse conductivities of the composite. In practical situations one may suppose that cqEl1, a,E_ 6 j l 4 is. We consider here only extended disturbances (such as b,) and therefore can neglect heat conduction along the composite in the following derivation of the stability criterion. The consideredinstabilitycauses temperatureandelectromagnetic field perturbationsto of these increase in acorrelatedmanner.Eachone
Current-carrying capacity of superconducting composites
processes is characterised by itsrespectivediffusion coefficient,namely thethermal diffusioncoefficient D,= K / V and the magnetic diffusion coefficient D,= (voa)", where v is the heat capacity and K is the heat conductivity of thecomposite.Let us introducethe parameter t:
(3) Forsuperconductingcomposites T + 1. Accordingly, fast heating of composites occurs under conditions of frozen-inmagnetic flux. Then,one canassume that in the initial stage of the instability development the perturbation of the current density 6j is equal to zero, i.e.
3. Electric field distribution
Theelectricproperties of twistedmultifilamentary composites have been studied extensively (Carr 1983). It has been shown that in the case of the wire placed in a varying transverse magnetic field the cross section of theconductor maybedivided intotwo regions:an external or 'saturated' region in which Ell is non-vanishing and an interior one in which Ell = 0. To find the analytical solution for Ell in the 'saturated' region in a general case is impossible. The purpose of the present calculations is to find Ell in some limiting cases being, however, of practical interest. Let us suppose that the value of g, is not too high:
ai
6j=-6EIl +--E, +-6T=O (4) dEIl aE, aT where 6E and 6 T are infinitesimally small perturbations of the field E and temperature T. Equation (4) allows us to find the relation between 6E and ST. Using this relation and equations (1) and (2) to obtain the Joule heating term GTa(Ej)/aT, we find the heat equation in a linear approximation as
where a is the effective field variation, t o= ,uoL20,/ 8n2is the characteristic time of the resistive current decay in a twisted wire (Carr 1983), B, = 2j~~,$~/.7dis the magnetic flux penetration field and B, is the amplitude of the Ba(t)variation in the considered process. (10) one can write Maxwell's Under conditions equation in the form (Carr 1983) curlB
The heat conductivity of the composite is usually high and W = 2 W a O /
