IEEE TRANSACTIONS ON MAGNETICS, VOL. 24, NO. 2, MARCH 1988 933

ELECTRODYNAMICS OF SUPERCONDUCTORS WITH REAL TRANSITION CHARACTERISTIC

E.N.Aksenova, G.L.Dorofeev and E.Yu.Klimenko 1.V.Kurchatov Institute of Atomic Energy,Moscow 123 182,USSR

Processes of electromagnetic diffusion in mono- and multifila- mentary superconducting wires having an exponential transition cha- racteristic from a superconducting to normal state are studied experi- mentally and theoretically.

1. Problem Statement The behaviour of inhomogeneous superconductors (SC) in ac

external and intrinsic magnetic fields is of a considerable interest from the viewpoint of practical application of these material;. Since there is no general microscopical theory of inhomogeneous SC, elec- trodynamics has a phenomenological character. The accuracy of the theory of dissipative characteristics and stability depends on that of the description of a real transient characteristic (KTC) of SC, which is used as a constitutive relation completing a system of the Maxwell equations in constructing electrodynamics.

The real form of a SC wire transition to the normal state was studied almost beginning from the moment the technical supercon- ductivity became known [ 1,2]. The interest aroused by this problem is due to revealing the nature of the high current-carrying capacity (CC) and determination of the CC limit for technical SC. The criti- cal state model (CSM) [3], in the framework of which the solutions of basic electrodynamical problems were obtained [4-81, is a zero approximation of the RTC. At the same time, the CSM electrodyna- mics requires more exact definition in considering a broad range of the magnetic field variation rates and, in a number of cases, yields only a qualitative pattern of the current distribution, e.g. for an ac field being logitudinal to the composite [ 7,8].

Presently, there are two different mathematical descriptions of the RTC, which are used commonly: (1) in terms of the transcendental functions [2,9]

(1) ~ = j p e x p ( - ~ T -T + B(T) + - j ( B J )

TO Bo Jo

(2) in terms of the power functions [10,11] of the form

E = KIn (2)

On the basis of the calculations performed [ 121 the authors of the present work reckon that the smearing of the RTC, i.e. their differ- rence from the CSM step characteristic, is connected with the longi- tudinal or volume inhomogeneity of the wire SC properties, which results in an exponential current-voltage characteristic. Therefore, in studying the SC wire behaviour in ac quasi-stationary magnetic fields, use was made of SC exponential transient characteristic (1) as a cons- titutive reiation, which in the dimensionless form (where j is norma- lized to jc) is as follows:

j = 1 - t - B + 2621nE = j co + 26’1nE (3)

The factor 26 the smoothing of the SC transient characteristic, t = T/Tc, B is nor- malized to Bc.

The isothermality condition which is satisfied at least at the frequencies f < 1 ( Tk is connected with the wire thermal

conductivity; Th, with the heat removal conditions at the boundary) transforms (1) into (4)

= To/Tc = Bo/Bc =jo/jc numerically characterizes

rk +‘h

(4)

where jc, is the current density in the wire at the level of the electri- cal field E, and magnetic one B.

A general equation for the electrical field in dimensionless coor- dinates h3s the f o r -

rotZE = - ulEl - u ~ ~ E N - 26’(&- - [rot E I ) -

+ + J

E J (5)

ul, uj1 are the matrix conductivity along the filaments and that per- pendicular to them.

11. Monofilamentary Wire. Saturated Multifilamentary Wire. The nonlinearity of the second-order vector equation ( 5 ) is eli-

minated at a linear change of the external magnetic field =BI + Bt, B = const. In the case of a harmonically changing exter- nal magnetic field of a large amplitude Bm > B (B is the penetra- tion field) E = &(t) - c(r), where e(r) is the cogdingate function coinciding with the stationary distribution of E. The applicability re- gion and t+ accufacy of the solution obtained are specified by the condition E/E < B/Bo which for Be = Bmsinwt is violated in the

n = 0, 1 ... The analytical solutions obtained for the mono filamentary wire

in the longitudinal, transverse and rotating external fields of the lar- ge amplitude, B >Bp, and with the ac transport current, as well as those for the shKlding-current saturated multifilamentary wire in the longitudinal and transverse magnetic fields differ from the solutions of the CSM electrodynamics in corrections which arise due to the account taken of the logarithmical dependence of the current density on the local electrical field and are expressed via the RTC smoothing parameter 26’, namely: (1) the slope of the magnetic field penetra- tion front depends logarithmically on the external magnetic field rate of variation:

Be =

t = Bo/(2wBm) vicinity of the moments of time t o = H 1 +nL

E( P o )

E(P 1 B = Be - j c o ( ~ o - ~ ) - 262[(po-p)lnE(po) +In -1

Here jco= jco(ro), P =

(2) the losses in the ac external magnetic field depend logarithmical- ly on the rate of its variation, so that in changing the rate from B, tu B, a relative increment in the losses is

I 0.05 0.5 5

E; T / 8

Fig.1 Losses W versus logarithm of rate of changing external magnetic field B with different amplitudes for mono (1-3) and niultifilamentary (4,5) wires.

These effects were observed experimentally for monofilamentary with an increased CC and saturated multifila- mentary conductors (fig.1). The loss measu- rement was made by using calorimetry with 0.1-mW sensitivity. It is described in detail in [13]. In the case of the self field generated by the ac transport current, the rate dependence of losses is weakly appea- red, since the current- carrying layer decreases with increasing I but the local current density grows ; (3) upon cessation of the magnetic field vari- ation the diffusion of the magnetic flow to the SC continues at a damping rate. Such a ~-

magnetic relaxation of the SC niobium cylinder was observed by Kim with collaborators back in 1962 [ 14;;

0018-9464/88/0300-0933$01.0001988 IEEE

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Table 1. Analytical Expressions for Electrical E, Magnetic B Fields and Heat Release Power P in SC Wires with Exponential Current-Voltage Characteristic

I ,

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(4) the RTC smearing weakly affects the coordination dependence of electromagnetic fields and currents and, practically, becomes no- ticeable only in the vicinity of the zero points oT’ the current and field components. Therefore, in most cases a modified characteris- tic (4’) taking into account the RTC smoothing only via the rate dependence

can be used for thick layers embraced by the shielding currents. A more detailed description of the peculiarities of the SC wire satura

j = jcl +jo In (B/Bl) (4’)

. ted state, connected with the RTC smoothing, is given in [ 151.

111. Incomplete Filling of Cross Section with Shielding Currents. Nonsaturated Case

It is much more difficult to solve the problem with an incomp- lete penetration of shielding currents, which is realized: (1) at small amplitudes of the external magnetic field, Bm<Bp, in the monofilamentary wire; (2) in the process of increasing the longitudinal magnetic field impo- sed on the twisted multifilamentary wire, B < B*; (3) at low rates of changing the magnetic field B<&rO/u1)(2n/L)2, being transverse to the twisted composite, with the amplitude Bm>(pOjcd/2), ;.e. a field of the filament penetration. Only the se- cond out of these three problems can be solved analytically taking into account the characteristic (4’).

of a numerical solution of eq. ( 5 ) show that the increase of the RTC smoothng results in growing nonlinearity of the penetrating field front, depth of its penetration and phase shift of E and B. The fre- quency increase is accompanied by the decreasing shielding layer and losses per cycle, which is proportional to the characteristic smearing. Thus, the losses in the SC with the smeared characteristic can differ noticeably both in the direction of increasing values at low frequen- cies and in that of decreasing ones at higher frequencies from losses in the SC with an ideal CSM characteristic (the dashed in fig.2).

For low amplitudes of the external ac magnetic field the results

5 o 50

7 ;&

The consideration of the composite saturation process in the longitudi- nal ac magnetic field is simplified considerably without taking into acco- unt the matrix currents. Then the electromagnetic field distribution with the Elp, E,, B , B, compo- nents and’the current fol- lows the system of equa- tions (7) given below. The conductor cross section is divided at first into two regions: the outer one r1 < K r o with the shielding current jse=jc +joln(E(ro)/ /El) , in compfiance with the characteristic (4’) and the inner one r Q rl carry- ing the compensation cur-

Fig.2. Losses per cycle of field change rent jsm=4rrBe/(Lpohf3~2). versus degree of RTC smearing jo/jc and frequency.

- + i rot E = - B

where f = 1 + (2nr/L)’.

(7)

When the external magnetic field Be reaches the value

where js; = j+*ln(E(r2) /El) , there arises a region saturated with the compensation current in the vicinity of the wire axis with the radius r, . In increasing external field, the boundaries of the regions, satura- ted with the shielding current, rl , and compensation current, r2 , co- me closer (rl moves to the centre and r%, from the centre of the wi- re), while the intermediate region r2<r <rl is filled with the nonsatu- rated compensation current jsm. At the value of Be=B*=

= /.iohjs;L[l +l (h )’I3/’/(4n) the intermediate region disappers

since r 2 = r l = r 0 / a and the process of the wire saturation is

The distribution of the electromagnetic field components in the

C J .

2 L

completed.

process of saturation is presented in table 2 and fig.3. The account taken of the matrix zero conductivity does not change the picture substantially but results in a weak absolute shielding of the external field.

In the case of the composite arranged in the transverse ac mag- netic field a numerical solution of eq. (8)

(8) taking into account the RTC in the form of eq. (4), yields for the matrix isotropic conductivity a dumbbell-like form of the saturation boundary being similar to that of [ 6 ] .

In the case the wire core, r<rl, i s filled with material containing no SC filaments with the conductivity u1 being other than the con- ductivity u of the rest of the cross section, the electrical field paral- le1 to the filaments (fig.4) has local maxima in the points ro and rl

This circumstance can acco. unt for the process of paral- lel saturation of groups of filaments and the wire as a whole taking place in the wire with a cluster assembly.

Another important pecu- !iarity of the multifilamen- tary wire behaviour in the transverse magnetic field is the fact [ 161 that the hyste- resis losses appearing in length-inhomogeneous fila- ments of the nonsaturated region are proportional to the average critical density of the current, jc, determi- ned by the current-voltage characteristic. The a v e q e current critical density jc can noticeably (by tens of per cent) :xceed the current density j(B)corresponding to the electrical fjeld across the filament, E=Bd/4, and determined from the wire transport properties. On the contrary,the saturation lay- er thickness is determined by the transport properties, namely, is ipversely propor- tional to j(B). An dmormal- lY high level of losses in Wi- res with ultrafine filaments in the transverse magnetic field can be connected, in

Fig.3. Distribution of electrical (a) and magnetic (b) field components ~ e r multifdamentary composite cross section in its saturation in longitudinal magnetic field.

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Fig.4. Distribution of longitudinal electrical field and shielding currents in wire with core not filled with SC filaments and conductivity u1 in transverse ac magnetic field.

of the RTC.

mogeneity is not high.

IV. Summary

1. Corrections to the losses arising in monofia- mentary and saturated multifiamentary wires de- pend logarithmically on the rate of changing the external fie!d 4W/W(Bl )=

rimental observations of these dependences testify to the adequacy of the description of the local RTC of inhomogeneous

superconductorsby the ex- ponential form.

2. The saturation pro- cess of the twisted multi- filamentary wire in the longitudinal ac magnetic field passes the phase of formation of three regions in the cross section: a shielding-current saturated region, and compensation- current saturated and un- saturated ones.

3. In installations where the SC is subjected to a pulsed RF magnetic field with an amplitude Bm < B losses decrease with increasing smoothing

=j o /’ J~~ . ~ ( B z IB1). Expe-

P’ .

4. In the transverse ac magnetic field the composite saturation process (1) with the matrix conductivity inhomogeneous over the cross section may start simultaneously from the centre and periphery of the wire cross section, (2) with the cluster assembly it takes pla- ce at a parallel saturation of the cluster and the wire as a whole.

5. The calculation of the hysteresis losses in the unsaturated region filaments requires to take into account the average critical density of the current, determined using the RTC.

References

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VOL 2, pp. 105-112.

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Table 2 . Distribution of Electrical and Magnetic Field over Twisted Wire Cross Section in Its Saturation in Longitudinal AC Magnetic Field

r2 (Y r < rl