On the distribution of a transport current inside a multifilamentary superconducting wire in a rapidly changing transverse magnetic field

For the current distribution inside a multifilamentary superconducting wire carring a dc transport current in a rapidly changing transverse magnetic field, inconsistencies with the existing models are shown by the following experimental evidence: when a transverse magnetic field is applied, the distribution of transport current is not unaltered but is forced to concentrate into the inner circular cross section region during a characteristic time constant cc called the coupling time constanL Secondly, the characteristic time constant for the transport current distribution inside the inner region to approach a uniform distribution is not Cc but a new time constant cl called the "uniforming time constant" though the variation in the distribution does not occur unless the external magnetic field changes with time. It is shown that the m o d e / o f the current distribution based on the above experimental evidence exhibits a remarkable difference from the existing models, especially for the wires containing very fine superconducting filaments.

On the distribution of a transport current inside a multifilamentary superconducting wire in a rapidly changing transverse magnetic field F. Sumiyoshi, K. Koga, H. Hori, F. Irie, T. Kawashima and K. Yamafuji Key words: cryogenics, superconducting wire, transport currant, magnetic field

Multifilamentary superconducting wires used as magnet windings are subject to various kinds of external conditions on transport currents and magnetic fields. While many studies ~-6 on the current distribution inside the wire under these conditions have been made, there still remain unsolved problems about the current distribution when a rapidly changing field is applied to the wire carrying a dc transport current. In this case an additional shielding current is induced and flows through normal metal matrices among filaments, ~ and two different models have been proposed 2-4 for the current distribution. Soubeyrand et al. 2 and Campbell 3 assumed that the shielding current is superposed on the dc transport current. Ogasawara et al.? on the other hand, assumed that the transport current concentrates and distributes uniformly in the inner region of the wire. Neither of these assumptions, however, have been confirmed theoretically or experimentally. The purpose of this paper is to elucidate the distribution of transport current inside a multifilamentary wire in a rapidly changing transverse field. For this purpose it may be instructive to mention first the results of our recent work s on the distribution of a transport current inside a multifilamentary wire in a slowly changing field. When external magnetic pulses are applied successively, it has been known ~ that the distribution of the transport current approaches uniformity. For slowly varying pulses, we derived5 the following partial differential equation for the transport current distributionj(r, 7") in the cylindrical coordinates: a/ _ l 1 ~ ( a j ) r (1) ~'IO a~' k~ r ar with 0011-2275/83/011619-06 CRYOGENICS . NOVEMBER 1983

rl°

(2)

k~ D

where D is the 'diffusion constant' given by af

2#0 X/c D =

af

/,toX/c

I/}el

for a single wire,

(3)

I/}el

for a wire wound closely into a solenoid

(4)

In the above expressions, r w is the radius of the filament bundle in the wire; k, is the first positive zero of the Bessel function Jr, ie, J.(k0 = 0; I/}el is the absolute value of the time derivative of the external flux density; af is the root of the cross section of a filament; ~ is the magnetic permeability of the vacuum; h is the volume fraction of filaments in a filament bundle; andjc is the critical current density of filaments. It is to be noted that any variation in the distribution of transport current does not occur during the time interval when the external field does not change with time, ie, Be = 0, and hence the time }'in (I) does not represent the real time but the reduced time which is defined by the sum of the changing time of the external field. Equation (1) indicates that an initially localized distribution changes into the uniform one by the time constant ri0 defined by (2). We shall call rx0 the 'uniforming time-constant' hereafter. When the solution of (1) is expressed in the form of a series of exp(-/~/rl0), the higher terms of l _> 2 usually contribute to the solution only by an amount of less than a few percent. It has been confirmed experimentally5 that the present equation can quantitatively describe the varia-

$03.00 © 1983 Butterworth 8- Co (Publishers) Ltd. 619

seen easily from (2) that the effective uniforming time- " constant in the rapid-field case is not qo itself but is % given by

fion in the distribution of transport current as will be shown in the next section. In the existing models for the rapid sweep case, the above-mentioned effects of variation in the transport-current distribution with the uniforming time-constant r m have not been taken into account. In this paper, therefore, we propose a new model of the current distribution, and the results of experiments carded out to confirm the present model are also mentioned.

rl = rio(1 - 1~)2

A main difference between the present model and that proposed by Soubeyrand et al. 2 and CampbelP lies in the fact that in the present model, the transport current which is initially uniformly distributed is forced to concentrate into inside the inner region III by development of saturated regions within the time scale Zc, and hence the shapes of I~ and Ii are not asymmetric but symmetric. If we consider the case where an external transverse field is increased monotonically, the distribution of the shielding current must be symmetric in the steady state, as pointed out by Ogasawara et al? based on the fact that the time variation of the flux enclosed by two neighbouring filaments is equal to zero in the steady state. Thus the appearance of asymmetric

A n e w model of current distribution In this section we shall propose a new model of the current distribution inside a multifilamentary wire carrying a dc transport current in a rapidly changing transverse magnetic field. A typical current distribution inside the unclad wire in an increasing field is shown schematically in Fig. l, where the areas denoted by Ir and Ii represent the saturated regions, the area II specifies the intermediate region, and the area III is the inner region. In the present model, two saturated regions with shielding currents of an average density of X/~, locate symmetrically with respect to the central axis in the direction of a transverse external field. Whereas we do not have any reliable expression for the shapes of these saturated regions at present their thicknesses are approximately given by flrw !sin 01 so far as relatively thin saturated regions are concerned, where 0 is the azimuthal angle measured from the direction of the external field, and fl is given by /~ = r~l/~ Itaoap,,,

(8)

&

~

{

i

:

'

.

l

r l roa

(5)

In this expression, Hpw = (2/rr))~Jc rw is the value of transverse external fieldnecessary for a flux front to reach the centre of the wire when the electromagnetic behaviour of the wire is virtually replaced by that of an equivalent single-core wire, and Vc is the coupling timeconstant given by'

re =

(1/8rr2)laoolL~

for a single wire

(6)

(1/4rr2)~0 ozL~

for a wire wound closely into a solenoid

(7)

coil

where o I is the transverse conductivity of the wire and Lp the twist pitch. The appearance of the symmetric saturated regions induced by the application of a rapidly varying external field makes the uniform distribution of transport current concentrate into inside the inner region, the process of which is characterized by the coupling time-constant Vc. During this process, the transport current inside the intermediate region II should decay rapidly following shrinkage of the inner region III, because the transport current cannot flow in the region II without voltage drop along the wire axis. As for the distribution of the transport current j(r, t) inside the inner regions III, it can be easily deduced that any further restriction other than those used in the derivation of (1) does not appear even in a rapidly changing external field. Thus we can assume that (1) is also applicable for describing the variation of the transport-current distribution inside the inner region even for a rapid-field case. Taking account that the radius of the inner region is (1 - - r ) rw, it can be

620

"7.

It

.,- __

7/-7/ZT/Z

-' I Fig. 1

Current distribution in the wire carrying a dc transport current in a

rapidly changing transverse magnetic-field. The transport current is concentrated into the inner region III during 0 < t ,~ re, and then this concentrated transport-current distributes uniformly over the inner region during r I. Shaded and dotted areas represent the regions where the transport and the shielding currents are flowing, respectively.

CRYOGENICS. NOVEMBER 1983

Model

x~ a

,I

I I I v I t

t

t

6

t~

t

~"

ID e

II

b

ti

t

Soubeyrand et ol2

i I Ogasoworo et al4

i

Present model

Fig. 2 Differences between the present model and the existing models for the case of r c << r1. The current distributions are illustrated for the various situations of Be(t )

saturated regions is only possible during the transient state before the steady state is attained. The elimination of this possibility has not been theoretically successful due to a difficulty in the application of irreversible thermodynamics to this problem. But the appearance of asymmetric saturated regions is a direct consequence of an intuitive assumption that the transport current distribution is kept unaltered even if the shielding currents are induced, and the unreality of this assumption can be confirmed experimentally as will be shown in the next section. The current pattern in an increasing field shown schematically in Fig. l is essentially the same as that adopted by Ogasawara et al? The present model differs from that proposed by Ogasawara et al? only in the distribution of transport current inside the inner region: (A) the characteristic time-constant subject to the variation in the transport current distribution inside the inner region is not vc but q. (B) The distribution of transport current inside the inner region is kept unaltered during the time interval in which B e = 0 is retained, and hence the area of the inner region is also kept unaltered during the above time interval. while the transport current distribution is always assumed to approach the uniform distribution within the characteristic time-constant rc in the Ogasawara model. Some typical examples showing the differences between these models are illustrated in Fig. 2 for the case of re < < q.

CRYOGENICS. N O V E M B E R 1 9 8 3

Experimental confirmation of the present model Experiment I. Since the characteristic timeconstant which is subject to the variation in the transport current distribution is the most important concept introduced by the present model, it may be instructive to illustrate a typical example of our observed data 5 for the case of a slowly changing external field. It has been pointed out by Ogasawara et al. 6 that the transport current distribution is forced to approach a uniform distribution by an application of a series of successive magnetic pulses with a slow sweep rate. If we start from a localized transport current distribution shown schematically in Fig. 3b, the current distribution gradually approaches the uniform distribution shown schematically in Fig. 2a as the number, n, of the applied pulses is increased. The variation in the current distribution affects the amount of flux A~(n) passing through the wire from the right hand side to the left hand side during the period of the nth pulse, and (1) yields5 A¢(n) = AO(,~) + ~

4(/)exp(-n/nt)

(9)

l=l

where z~(~,) = 2reBmh

(10)

621

I

where y is determined by the condition for the conservation of the total transport current I t as

,

'1

O

1.3

!1 - 0) 2 - it ]1/2

v =

(15)

t-~

//xL

Then the value of A~(n) is given by (9), except that q~(1) given by (12) is replaced by x/,~ - 1 - x/~ Jo (k~) ~(/) = --4 re B m

"

o

ks [Jo (kt)] 2

(I 6)

b [(1 -/'t) 7Jr (kiT) + (1 -/3) + (1-/3)J, (kl(1 - #))]

<> (3 0

I

I

I

I

5

I0

15

20

<

>

2,5

/7

Fig. 3

a - The dependence of AlP on n for the initial current distribution, b. The solid line represents theoretical values given from (9)-(11) and (12), where ]t " 0 . 6 8 and B m = 0.2 T. A sample wire (No 2) and an arrangement of the experiments I-III in the present paper are the same as those in our previous paper5

nt =

( r d 2 a t ) (kl//cO:

(11)

As can be seen from Fig. 4a, the observed results show a quantitative agreement with the theoretical values obtained from (9) and (16). Since the current distribution in the Soubeyrand model shown in Fig. 2c is a direct consequence of their intuitive assumption that the distribution of transport current is kept unaltered regardless of the presence or absence of shielding currents, the present experiment shows clearly the unreality of their assumption. Whereas the current distribution by the Ogasawara model happens to be the same as that by the Soubeyrand model in the situation shown in Fig. 2c, the difference between the Ogasawara model and the present model results from the facts (A) and (B) mentioned in the previous section, and the present experiment shows a failure of the Ogasawara model at least for this case.

and ~(/) = 4 r f B m

(1

-

x / , , - 1 - ~ / ~ J o q , t)

kdJo(ki)]:

(12)

it) 112 Jt (kt(l - ]'t) 112)

In the above expressions, rf is the radius of a filament, B m ----I Be IAt is the height of each pulse, kt is the l-th positive zero of the Bessel function Ji, and jr is the average transport current density normalized by X/c defined as

it

=

h/~rr~ X/c

(13)

It can be seen in Fig. 3a that the observed value of A¢(n) approaches A¢(**) which is the value of A ¢ for the uniform distribution of the transport current shown schematically in Fig. 2a, and the quantitative agreement between the observed and theoretical values seems to be satisfactory. The present example, together with other observed datad guarantees a quantitative reliability of the expression of rl0 given by (2). Experiment II. The situation shown in Fig. 2c is the one most suitable to experimentally confirm the present model. If a series of successive magnetic pulses with a low sweep rate is applied starting from the current distribution shown in Fig. 2c, the value of A~(n) should remain equal to Ate(**) independent of n, so far as any of the existing models can describe the real current distribution. On the other hand, according to the present model the starting current distribution is approximately given by

i(r,O) {!forO
~Jc

622

for "r < r/rw ~< 1 -/3 for 1 -/3 < r/r w < 1

(14)

Experiment III. The same kind of experiment can also be carried out starting from the situation shown in Fig. 2e. According to the present model, the starting current distribution is approximately given by

/(r, O) = ({~t f o r O ~ < r / r w ~ l - 1 3 Xic

(17)

[ 0 for 1 -/3 ~ r/rw <~ 1

w h e r e jr = . / t / ( l - - fl)2. Then the value of AO(n) is

again given by (9), where ~(/) should be replaced by

(~q) = - 4 rf BmX/Tr - 1 - X/n Jo (kt) kt[Jo(kl)] 2 It Jl (k/(1 -/3)'(18) As can be seen from Fig. 4b, the observed results again show a quantitative agreement with the theoretical values based on the present model. As can be seen from Figs 2c and 2e, the difference in AO(2) between Figs 4a and 4b reflects the localized and the uniform distribution of transport current inside the inner region. One may recognize that the amount of difference in Ate(2) clearly exceeds experimental errors, and more detailed quantitative confirmation may not be necessary because a quantitative reliability of the expression for r I has already been guaranteed by the experiment I. Discussion

In this section we shall discuss conditions under which the present model applies and some problems related to the uniforming time-constant. The applicability of the present model. The present model is applicable only when the following conditions are satisfied:

CRYOGENICS. NOVEMBER 1983

I.I

Three, 27rrw < < L p ( # ~o). The coupling time constant is assumed to be longer than the time constant of the skin effect, defined as the inverse of the frequency at which the skin depth of the normal metal matrix is equal to the wire radius. This condition is satisfied with usual wires, except for non-twisted wires which are not available for practical use. While the difference between the present model and the existing ones becomes remarkable for a limiting case of ri0 > > rc, the present model holds true regardless of this condition. It is also to be noted that an explicit expression for fl given by (5) for relatively thin saturated regions are used only to make a quantitative comparison between the present theory and the observed data. The present model is also valid for thicker saturated regions if we interpret fl as simply the m a x i m u m thickness of saturated regions. Values of the uniforming time-constant Here we should like to emphasize that the uniforming timeconstant is usually longer than the coupling timeconstant for the multifilamentary wires used practically. In Table 1 examples of the calculated values of the uniforming time-constant at 6 T are shown for various field conditions and various configurations of NbTi multifilamentary wires, where the critical current density of filaments is assumed to be Jc "~ 1.3 × 109 A m -2. Taking into account that the coupling time constant rc is of the order of 1 ms at most, we c a n say that the case of r c < < rio is the ordinary one in practical cases. This condition of rc < < rlo is satisfied more easily for the wires with a filament diameter in the order of submicrons, s For the case of Nb3Sn multifilamentary wires, this condition is also satisfied in the pulse field with the sweep rate of about 10 T s-l. 9 When we discuss ac losses and instabilities of such a wire with fine filaments, we must take account that the effect of the uniforming time constant on the current distribution lasts for a long time, so that the current distribution in such a wire tends to be localized in any external conditions.

-

1.0

E 0.9

<3 0.8

0.7

R

0.6

>

>a 0

I

I

5

I0

I

I

15

20

25

/7

I.I

1.0

E

0.9

0.8

Current distribution in the case of simultaneous sweep of magnetic field and current Here we shall briefly

0.7

0.6

>

I

I

5

I0

I

I

15

20

25

n

Fig. 4 The d e p e n d e n c e s of AO on n after applying the external fieldpulse with high sweep-rate of 0.8 T s-1 for the various initial currentdistributions; a - Fig. 2c, b - Fig. 2e. The solid lines represent the theoretical values given by using (16) and (18) respectively, w h e r e Jt = 0.88, B m == 0.2 T,/~ = 0.15, r I = 0.9 s and r e = 0.08 s. The broken line represents the theoretical values corresponding to both the Soubeyrand and the Ogasewara models

One, fl < 1 - - j t 1/2. The averaged current densisty in the inner region must be smaller than the critical one. When fl > 1 --.it ]/2, a profile of the shieldingcurrent distribution is different from the present model. Two, I Be I > > / x 0 Hpw. The self-field around the wire, which is at most/to Hpw, must be negligibly small compared with the strength of external field. If this condition is not satisfied, the fundamental equation (1) is not available for describing the change in the transport current.

CRYOGENICS.NOVEMBER 1983

describe the effects of the uniforming time-constant on the simultaneous sweep case. For this case, Ogasawara et aL to considered that the concentration of the transport current into the inner region may also occur and that the concentrated transport current may o distribute uniformly when B e rf > > Bs rw,11 where Bs is the time variation of the self field at the wire surface. According to the present result however, the reduced time of r I may be necessary to make the concentrated current distribute uniformly even for the case of simultaneous sweep of magnetic field and current.

Table 1. E x a m p l e s of the values of the u n i f o r m i n g time-constant, s Be,

T s -1

2 rw, m m

0.5

1

2rf,/xm 1

10

100

0.5

5

50

1

40

4

0.4

20

2

0.2

10

4

0.4

0.04

2

0.2

0.02

100

0.4 0.04 0.004

0.2 0.02 0.002

623

Conclusions In summary, profiles of the present model of the current distribution are given as follows. The transport current is concentrated into the inner region immediately following the extension of the saturated region induced by the shielding current. This mechanism is characterized by the coupling timeconstant %. The concentrated transport current is distributed uniformly in the inner region after the reduced time of the uniforming time-constant rl, where the reduced time is defined by the sum of the time intervals during which the external magnetic field is changing. When the change in the external field stops, the distribution of the transport current is kept unaltered while the shielding current decays. The dynamic resistance loss for each field p u l s d may be seriously affected by the variation in the transport current distribution, which will be discussed in detail in a following paper, including the case of simultaneous sweeps of external field and transport current.

Authors FS, KK, HH, FI and KY are from the Department of Electronics, Kyushu University,

624

Fukuoka 812, and T K is from the Department of Electronics, Fukuoka Institute of Technology, Fukuoka 812-11, Japan. Paper received 20 June 1983.

References 1 Cart, W.J. JrJAppl Phys 45 (1974) 929 2 Soubeyrand, J.P., Turek, B. IEEE Trans Magn MAG-15 (1979) 248 3 Campbell, A.M. Cryogenics 21 (1981) 107 4 Ogasawara, T., Takahashi, Y., Kanbara, IC, Kubota, Y., Yasohama, K., Yasukoehi, K. Cryogenics 20 (1980) 216 5 Sumiyoshi,F., Hori, H., Irie, F., Kawashima,T. Cryogenics 23 (1983) 373 6 Ogasawara, T., Yasukochi, IC, Sayama, S. Cryogenics 16 (1976) 89 7 Sumiyoshi,F., Irie, F., Yoshida, K. JAppl Phys 51 (1980) 3807 8 Ogasawara, T., Kubota, Y., Makiura, T., Akaehi, T., Hisanari, T., Oda, Y., Yasukochi, K. IEEE Trans Magn (to be published) 9 Irie, F., Yamafuji, K., Takeo, M., Sumiyoshi, F., Miyake, Y., Noguehi, K., Kazawa, Y., Saito, R. Proc ICMC (1982) 477 10 Ogasawara, T., Itoh, M., Kubota, Y., Kanbara, K., Takahashi, T., Yasohama, K., Yasukochi, K. [EEE Trans Magn MAG-17 (1981) 967 11 This condition is rewritten by using Vjo as rto/t I << (4/v/Trk2~).Ic/l m, O.151jl,,. 0.151¢/I,~ is the maximum transport current and t I is the rise time from 0 to Im. The applicable condition of the Ogasawara model lbr this case should be corrected as r[ -----rio (1-fl)z << re

CRYOGENICS. NOVEMBER 1983