IEEE TRANSACTIONS ON MAGNETICS, VOL. 25, NO. 2, MARCH 1989

LOSSES IN SUPERCONDUCTING WIRES WITH INHOMOGENEOUS AND

ANISOTROPIC MATRIX CONDUCTIVITY OVER CROSS SECTION

E.N. Aksenova

I.V. Kurchatov Institute of Atomic Energy, Moscow, 123182, USSR

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Methods of calculating energy losses for multifilamentary wires with anisotropic and inhomogeneous conductivity of a matrix over the cross section are developed, in particular, for wires with a cluster assembly.

1. Introduction The analysis of electrodynamical processes taking place in

a wire in a transverse ac magnetic field is, as a rule, of primary importance from the viewpoint of losses and stability of dynamical SMS, since in the majority of the SMS the wires operate Kecisely in the transverse ac magnetic fields induced by neighbouring kayers of the winding and considerably exceeding the self - field of the wire. The successes in understanding general regularities of the transverse magnetic field penetration into a multifilamentary SCW are connected with the model of anisotrpic continuum [ l ] , in the framework gf which the multifilamentary wire is considered as continuum with anisotropic conductivity along the filaments, SI1, and in the direction perpendicular to them, 3,. The averaging over dimensions (d+A), where d is the filament diameter, A is the thickness of the matrix normal metal, surrounding the SC filament, and the use of the critical state model ( CSM ) for the description of SC properties of the wire [ 2] made it possible to solve the system of Maxwell equations for a composite in a transverse magnetic field and to qualitatively describe the pooess of formation of a region saturated with a macroscopic current and dissipative characteristics, having divided them into three types quasihysteresis losses of penetration in the wire saturated layer, hysteresis losses in filaments of the internal nomaturated region and eddy - current losses in the matrix. The extension of the serviceability dynamic range requires the creation of wires with fine filaments of the diameter less than or of the order of 1 Mm with the aim of decreasing the hysteresis losses. The wire diameter is preserved at the level of a few tenths of mm and the coeffjcl?nt of wire filling with a supercoductor A , of the order of tens of per cent due to employment of multiple (cluster) assemblies, as a rule. Moreover, to decrease the eddy -current losses, it is required to increase the matrix resistivity. However, since the decrease of the matrix conductivity U results in decreasing thermal conductivity and heat removal conditions, one often makes use of highly resistive elements (barriers, spacers, etc.) [ 31 to increase the effective transverse conductivity of the matrix or, vice versa, copper channelsare introduced into the highly resistive matrix of the wire to ensure the heat removal of the power produced in the process of the wire operation [4 1 . The use of cluster assemblies with inhomogeneous distribution of filaments over the wire cross section and combined matrices makes significant corrections to the processes of diffusion and dissipation of energy in the transverse magnetic field [ 5,6,7 ] requires generalization of the model of hisotropic continuum for the cases of anisotropic and inhomogeneous matrix conductivity over the cross section of the composite.

2. Composite with Anisotropic Conductivity of Matrix

The case of matrix anisotropic conductivity uV f ut in the azimutal $ and radial $+ directions is most interesting from a practical viewpoint. An ideal case is represented by a matrix designed so that U' = uI + 0, U , = or >> up, i.e. the heat removal is ensured in the radial direction, and in the azimuthal one there is the minimization of the cooperative currents. For generality, let there be no SC filaments in wire core of the radius r l r having normal conductivity u4.

The distribution of the electric field components

should be found analogously[ 11 in the form of a sum of the solenoidal field V 6 = & + under the condition that the external field Be = B t ( cosv S, - sinq $). The equation for determining the potential containing SC filaments, has the form:

$= -firsin&, and the vortexless one

+ =

in the external part of the cross section,

+ div j l = o where f = 1 + (2nr / L)' , L is the twist pitch

In the CSM framework:

(here u3 is the effective longitudial conductivity along the filaments). The account taken of the exponential nature of the real current - volage characteristic (CVC) signifkantly compli - cates the process of finding analytical solutions and, even in the case of homogeneous conductivity over the cross section of a wire positioned in the ac transverse magnetic field, it requires the use of numerical methods. In taking into account the real CVC of SC filaments

+ 2 n aj, 2 n . 2 n azo 1 divjk = A-- + us- [- Brcoslp + -- L a ~ '

Lf"* a V Lf where us is the matrix effective conductivity along the

filaments, being, approximate, equal to some mean one between the u1 and u2 conductivities to an accuracy of the filling coefficient of the external part of the cross section with SC filaments, us =[(I - A)/( l + A].[(ul + 0 , ) / 2 1.

form brings about corrections depending on the magnetic field, current density and dissipated power determined in the framework of the CSM, and, in the first approximation, it can be performed by means of a simple replacement of the

fSM = jc value in the resultant expressions for the losses

As shown in ref. [ 81, the account of the CVC real form

E B with j, = jo[ln+ - - 1, where 7, jo, Bo = const.

BO

In connection with all mentioned above, the analysis of the electric-field distribution physical pattern will be performed in the CSM framework with subsequent account of the j&b) dependence in calculating the dissipated power. The electric field potential @ i in the internal (free of SC filaments) region has the form: -6 i = As I cow and should be joined at the point r l with determined from eq.(l) in compliance with conventional conditions of joining, which, for the potentials ai, ae, have the form (2):

in the external region:

0, = ( -SrL/2n + U)coslp U is a function of the radius r and is found from the equation:

1 2 n uu L f ? L f

U](u" + fUI) - u g + (-y - 3 - 5 1 = B-( 2 n r U, - U2 )

001 8-9464/89/0300-2 101 $01 .oO 0 1989 IEEE

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tw general solution of which is U = Al I,(kr) + Az K, (h),

where k = 2 ~ [ u , / ( f u ~ ) ] ~ / ~ / L , U = d m

If kr = 2 n r [ ~ ~ / ( f o ~ ) ] ” ~ / L >> 1, and it is precisely this case

that arouses interest, the solution

satisfies eq.(3) to an accuracy of terms of a higher order of

smallness than 6 Lr (1 - uz /ul )/(2nk2 rz ), while the I, and K, approximation (through exponents) has the accuracy of (kr)-3 ” , hence, it is quite tolerable. The coefficients Al , Az, A3 are determined from the conditions of the joining (2) and bounhy condition (4)

(4)

The approximate expressions for the electric field components are as follows

E, = - S r sinq

. L U - U 1 E , ~ = B-(I + L4 -)sinq

2n 01 krl

At that, the saturation pattern will be different depending on the u1 and u4 & m , a e Ell has local maxima at r l , ro . If u1 > u4 i.e. the wire has a high resistivity core, the process of saturation resembles the process of saturation of SC hollow cylinder. In the opposite case if the core is filled with the high conductivity (u4 > u l ) alloy, then during the process of saturation there appear skgle polar mm. of saturation near rl and ro, their screeng currents being closed through the core of the wire.

As seen from the expressions obtained for electrical fields (5), the radial u1 and azimuthal uz conductivities in the first approximation at small f3 (i.e. when kro >> 1) make proportional contributions t o the eddy - current losses, so that:

At that the saturation layer thickness l / k should be determined taking into account j,(B) from the real CVC.

losses in the region of the wire cross section, containing the SC filaments, where Ell + 0, i.e. each filament carries the current in the positive and negative directions.

Besides, one should take into account the hysteresis

3. AC Losses in Conductors with

Cluster Assembly

It is evident that in SC wires with the duster assembly, even in the case of an identical material of the matrix in clusters and between them, the effective transverse conductivities outside and inside the clusters will differ owing to the difference in the filling coefficients, which should, according to the results of

calculating electric fields (5) a t the inhomogeneous transverse conductivity over the wire cross section, result in local maxima of the electric field at the cluster boundaries, i.e. t o the process of parallel saturation of the wire as a whole and clusters in the transverse ac magnetic field. It is apparant that this effect will get more pronouced in the presence of resistive barriers between the clusters.‘he scheme of calculation of losses in wires with the cluster assemblies is based on subdivision of a complex composite system into homogeneous subsystems and application of the model of anisotropic continuum to them. Thus, the wire is treated as an ensemble of the clusters and a cluster, as an ensemble of filaments. At that, one should take into account that the processes of current overflow and saturation of elements take place in parallel i.e. each saturated filament exchanges current of its own cluster and of the wire as a whole.,T.he exchange process proceeds in compliance with the effective conductivities in the clusters, u1 , and in the wire, oz. In increasing the rate of the external transverse saturation process goes through the phases shown in Fig. 1 (a - d), if the conductivity u1 noticeably exceeds uz .

with the filaments

magnetic field variation, the wire

Figure 1. scheme ofschidding of extemal magnetic field by n i u l w s(: wire with cl* a?s?mtiy.

To take into account the effect of redistribution of the coopemtive currents and the dependence of the current density j, on the rate k together with the concepts about the partial schielding of internal regions of the multifilamentary wire by external ones and separation of losses into hysteresis, quasihysteresis and ccopzdive ones, it is required to employ numerical methods. Approximately, at an arbitrary rate B < Bc2 ( the rate of magnetic field variation, a t which the clusters are saturated) the overall losses are described by the expression

2R ro-R j 2R r - r r - R j, W=S[-[1-(-)2]& + - [ l - ( L ) Z ] ( L ) ~ +

d ro IC d rl ro Jc

+(”- r -R )’ (-)’ rl-r 1 + 4ohBm, X1(-)’ rl-r (“-p r -R (g)’ L ( I + 91 ) + ro r1 r l ro

I

r . B 3 1 4B,,,i 2 2

*. = - -[- + 2(Ali + Azi) - - ( A i i +AZ 2i) - 2e-T/4Ti x

Ali = 2 - exp(-T / 4 ri) ;

A2i = [ 2 - exp(+/4ri)] exp(</2ri) - 2 ;

rl is the cluster radius, S is the hysteree losses in filaments, T is the period of magnetic field variation. jc is the mean current critical density calculated according to the CVC of the wire [9,10], vi is a coefficient connected with the energy accumulated in the circuits formed by SC filaments and coupling through the matrix. The index i = 1 corresponds to a coupling in the cluster in the circuit with the inductance L, and characteristic t i e T , = ul L1, and i = 2 corresponds to the eddy - current in the wire as a whole with the circuit inductance Lz and time T~ = uz . For the case of T % 71 ( low rates of the magnetic field variation) q= 5~~/ (2T) ; in the opposite case, r i S T, vi = 4ri/T. [(rl - I)/ r l ] and [([ro - R)/ro] take into account the decrease of volumes in which the proposed scheme of calculation divides the losses into cooperative and hysteresis ones in filaments, while in the saturated region of the cross section, which the geometric factors

[ 1 - cy)'] and [ 1 - p- 7 correspond to, the losses are I O

calculated as quasihysteresis ones with the effectively lower current

increasing the rate of the magnetic field variation higher than &, the losses in the whole wire could be calculated so as if it were a solid superconductor with the effective density of the critical current ( Fig. I d )

Significant conditions for the rightfulness of the proposed scheme are as follows : (1) the presence of inhomogeneous conductivity in the transverse cross section of the wire, which allows us to combine filaments into groups ; (2) the uniform character of filling the wire with clusters and clusters with filaments

The approximate character of the given model consists in the neglect of real shape of the saturation region [lOj, which is corsidaedtobe circular. It is evident that the larges! errors appear in this case in the vicinity of the saturation rates BC2, Bc3. The calculations performed by us differ from the experimental results for different wires in the 6 - hC2 region by no more than 10 % and qualitatively convey the shape of the rate dependence of losses in wires with different designs.

on losses is calculated according to the described scheme and

One of the most important conclusions of the present work is the necessity of taking into account the possibility of cluster saturation at B < 8,. which may arise due to high A , o and the inhomogenesity of fine SC filaments which can conside- rably increase the quasihysteresis losses as compared with calcula- tions in the framework of the model of anisotropic continuum with homogeneoqs distribution of filaments.

The coefficients

density A , js for the cluster and A I s for the w k as a whole.-In

hja.

The effect of the construction and material of the matrix

represented in Fig. 2 for the wire with D = 0.85 mm and L= 5mm

in the clusters

1 I 1 I ,

0 1 0 2 0 . 3 0 4 0 5 B;T/8

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Figure 2. Dependence of energy losses W on transverse magnetic field variation rate B for wires with different construction.

~~

Curve N u d m O f Numberof 01; ua ; No filaments in cluster clusters 1 / (0hm.m) 1 / (Ohmem)

1 2970 1 3.4 + 109 3.4 .io9

2 55 54 8.3 . io9 0.9.109 3 2970 1 0.9.109 0.9 .io9

4 2970 1 1.1 . io9 1.1 . io9

2.2.109 1.1 . io9 5 55 54

4. Conclusion

The performed analysis of diffusion of the magnetic flux and dissipative characteristics for the multifilamentary wire in the transverse ac magnetic field has shown that: (1) in the case of the matrix anisotropic conductivity the processes of current coupling go in parallel in the opposite directions z,, making contribution to losses, which is proportional to U , ; (2) the presence of the matrix conductivity inhomogeneity causes perculiarities in the saturation process, namely, the longitudial-to-the filaments electric field has local maxima at the outer radius ro and at the inner one r,; (3) this circumstance explains the process of parallel saturation of groups of filaments and the wire as a whole, taking place in the wire with cluster assembly; (4) at that the rise of hysteresis losses ratio to the filament diameter with decreasing the latter for d - lpm can be accounted for by the cluster saturation and increase of filament inhomogeneity in their thinning without recourse to the proximity effect.

The author expresses her gratitude to Dr. E.Yu.Klimenko for the problem statement and valuable remarks.

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References --

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W.J.Carr, Jr.,"AC loss in a twisted filamentary supercoducting wire", J. AppL Phys., vol. 45, MO. 2, pp. 929 - 938, 1974. C.P.Bean, "Magnetisation of hard superconductors", Phys. Rev. Lett., voL 8, No. 6, pp. 250 - 253, 1962. R.A.Popley, D.J.Sambrook, C.K.Walters and M.N.Wilson, "A new superconducting composite with low hysteresis loss", in Proc. of ASC 1972, IEEE Publ. No. 72CH06855-TABSC, pp. 516 - 517. E.N.Aksenova, et al., '' Superconducting wire", Bulletin of Inventions and Discoveries ( in Russian), vo1.46. pp. 1987. B.Turck, "Effect of the respective positions of filament bundles and stabilizing copper on coupling losses in superconductor composites", Cryogenics, vol. 22, No. 7, pp. 446-468, 1982. Yu.P.Agapov, et al., "Study of energy losses in conductors with inhomogeneous distribution of superconduting filaments", Preprint IAE - 2913, Moscow, 1979,25 p. S.A.Egorov, "Eddy currents in multifilamentary superconductors with squared shape of transverse cross section", Preprint SRIEFA n - - 0540, Leningrad, 1982,lO p. E.N.Aksenova, G.L.Dorofeev and E.Yu.Klimenko, "Diffusion of magnetic flux in superconducting wires with exponential currentvoltage characteristic", in Proc. of I1 All-Union Conf. on Appl. Supercondutivity, Leningrad, USSR, 1984, vol. 2, pp. 105 - 112. Yu.P.Agapov, et al., "On connection between magnetization of multifilamentary superconducting wire and its currentvoltage characteristic" Dokl. Akad. Nauk SSSR, vol. 254, No. 4, pp. 862 - 864,1980. E.N.Aksenova, et al., "Electrophysical and structural propertties of multifilamentary superconducting niobium-titanium wire'', ICFA Worcshop on SC Magnets and Cryogenics, May 13,1986, pp. 28 - 30. D.Ciazynsky and B.Turck, "Theoretical and experimental study of the saturation of a superconducting composite under fast changing magnetic field", Cryogenics, vol. 24, No. 10, pp.5 07 pp. 507 - 514; 1984.