Eddy-current loss in a fully-stabilized multifilamentary superconducting wire

Eddy-current loss in a fully-stabilized multifilamentary superconducting wire Q.F. Zhang, F. Sumiyoshi, H. Nagaishi, F. Irie*, K. Miyaharat and K. Yamafuji Department of Electronics, Kyushu University, Fukuoka 812, Japan *Research Institute of Superconducting Magnet, Kyushu University, Fukuoka 812, Japan tDepartment of Electrical Engineering, Kumamoto University, Kumamoto 860, Japan Received 28 April 1984 A theoretical expression is provided for the ac loss in a fully-stabilized multifilamentary superconducting wire with a thick coating of aluminium or copper. It is shown that for a wire with a large amount of stabilizer, the skin effect in the stabilizer seriously affects the ac loss not only at high frequencies but also at low frequencies. In fact, the loss due to the eddy current along the wire axis induced by the skin effect predominates over the coupling-current loss if the twist pitch of filaments is shorter than the effective perimeter of the whole wire even in the low frequency range. In the middle frequency range, as used for a high-speed pulse magnet, the above feature in the ac loss is retained even if the twist pitch is not so short. The present feature in the ac loss is quite different from the known one for ordinary multifilamentary wires without a large amount of stabilizer. The present theoretical results show quantitative agreements with the results of our experimental measurements. Keywords: superconductivity; eddy currents; superconductors

Multifilamentary superconducting wire with a large amount o f a l u m i n i u m as a stabilizer has been used for the windings o f a dc magnet for high energy physicsL Because of its high stability, this type of wire has also been used in other applications, accompanied by a recent advance in the technique for coating aluminium on the wire surfacC ,3. For example, Irie et aL" used aluminiumstabilized Nb3Sn wires for the windings of a high-field pulse magnet. In these applications including the case of dc magnets, aluminium-stabilized wires experience external magnetic fields changing with time. Some theoretical investigations have been carried out on the coupling-current loss in the wire with a large amount of stabilizer. Turck 5,6 estimated the loss in this type of wire with a circular or a rectangular cross section, and concluded that the effect of a thick stabilizer on the ac loss can be described by using an effective transverse conductivity related to the coupling-time constant. In his calculation of the ac loss, however, the stabilizer only plays a role by providing a bypass for the coupling current flowing in the cross section perpendicular to the wire axis. Such a method of estimation of the ac loss is permissible only when the skin effect of the stabilizer can be disregarded. The profile of the skin effect in the stabilizer is clearly different from the well-known one 7 in the normal-metal matrix inside the filamentary region of the wire, and hence the skin effect in a composite system with a large amount of stabilizer should be re-examined carefully. Since the skin effect in the stabilizer for an external transverse field tends to induce the eddy current along the wire axis in addition to the coupling current in

0011-2275/85/030129-10 $03.00 © 1985 ButterworthEr Co (Publishers) Ltd

the cross sectional plane perpendicular to the wire axis, the profile of the ac loss is expected to become quite different from those formerly described 5,6. The purpose of this paper is to discuss the ac loss in a fully-stabilized multifilamentary wire. First we calculate the ac loss by taking into account the skin effect in a thick coating surrounding the filamentary region in the wire, where we adopt C a r f s continuum model 7 to describe the electromagnetic properties of the filamentary region. Next, we make a comparison of the present theory with the experimental results on the ac loss in an aluminiumstabilized Nb3Sn wire in a small ac field. Finally, we discuss the external conditions that the additional eddycurrent loss induced by the skin effect predominates over the coupling-current loss.

A theoretical c u r r e n t loss

expression

for

an

eddy-

For the fully-stabilized multifilamentary wire, we choose a cylindrical wire composed of two regions, i.e. a filamentary region and a stabilizer region, as shown in F/g. 1. We also adopt Carr's continuum model ~ which has often been used to describe the electromagnetic behaviour of the filamentary region. As in the case of an ordinary multifilamentary wire without a thick stabilizer, the current sheath, called the saturated region, in which the induced shield currents flowing in superconducting filaments are saturated, also appears in the present case at the outer boundary of the filamentary region. We confine our calculation in this paper to the case where the

Cryogenics 1985 Vol 25 March

129

Eddy-current loss in a superconducting wire: Q.F. Zhang et al. where Om is the electric conductivity of a normal matrix in the filamentary region; h the volume fraction of superconductors in this region; OAl,OCu the electric conductivity of aluminium and copper, respectively, ~ the magnetic permeability of vacuum. It is noted that al and p~ can take a wide range of values according to the kind of practical multi filamentary wire concerned e.g. wire with mixed matrices, Nb3Sn wire with a superconducting barrier, and

X

r2 i=3

SO o n .

A general expression for H i in each of two regions inside the wire (i = 1 or 2) can be obtained by solving the following equation resulting from a combination of Equations (2) and (1)

~v~ - t~ yT~ = 0

(5)

with kl_

%1 '3 Fig. 1 A fully-stabilized multifilamentary wire. /=-1, the filamentary region; /=-2. the stabilizer region; and /=-3, the outer vacuum region. At the boundary between 1 and 2, there exists

(6)

= 0

(7)

= 0

(S)

If we use the cylindrical coordinates (r, O, z) and assume d/dz=-O as is usually adopted, the general solutions of Equations (5) and (7) are given by

Hit

Expressions for electromagnetic fields

1/2

In solving Equation (5), we must take into account the auxiliary equations derived directly from Equations (1) and (2) as

the saturated region with a very thin thickness

saturation region is not very thick. This restriction may be permitted from the practical point of view, because the applicable range of this restriction covers a wide externalfield condition for practical use.

1 + ] . 8t = [ 1 ~

= ~

{Atn[I2n(kir)-I2n+2(kf)]

n=0

(9a)

General expressions. A long time after an external tr_ansversemagnetic field with a sinusoidal wave form of He(t) = Hm exp q2rrft) was applied to the wire in the x direction, the electric and magnetic fields also vary with time, t, and with the frequency, f, as ~//exp (j2rr~) and Hi exp (/'2r0q), where the subscript i specifies each region shown in Fig. 1, i.e. the filamentary region (i = 1), the stabilizer region (i = 2), or the outer vacuum region (i = 3). Then the set of Maxwell equations for E i and H i are given by

Hie

V x E t = -/'2

Ha = 2 . ,

(1)

+ Gn [K~(kir)-K2n*u(kir)]}

=- ~

cos(2n + 1)0

{Ain[I2n(kf)+I2n+2(kir)]

.~1=0

(9b)

+ Cln [K2n(kir) +K2n+2(kir)]} sin(2n + 1)0

[BtnI2n+l(ktr ) + DlnK2n+l(kir)] sin(2n + 1)0

n=0

~ × Hi = o ~

(2)

(9c)

In the above equations, the electric conductivity o i and the magnetic permeability/zi take the following values

where In(kit) and Kn(kir) are the modified Bessel functions of the nth order; A i n, Bi n, Ci n. and D i n are constants to be determined, except for Ct, = Din = 0. In the above expressions for J~i, the symmetricities of the wire with respect to thex a n d y axes in Fig. 1 are already taken into account. Then the general expressions for the electric field E i = (Eir, Eio, Eiz) can directly be obtained by substituting Equation (9) into Equation (2) as

1-X

1 +X

(3a)

02 = OAI or OCu

(3b)

03 = 0

(3c)

1- X I + X /an <~ /al ~ /,to

(4a)

Eir = ki 2et

/a2 = ~to

(4b)

I1,3 = I,to

(4c)

130

Cryogenics 1985 Vol 25 March

{ Bi. [/2,,(k~O -/2.+~ (kit)] n..~O

- Din [K2n(ktr)-K2n+2(kir)] } cos(2n + 1)0

(10a)

Eddy-current loss in a superconducting wire: O.F. Zhang et al. Eio

(E2z-Elz)r=, 1 = 0

{Btn [I~n(kir) + I2n+2(kir)]

_ 20 i

n=0

(lOb)

- Din [K2n(kir) +K2n+2(kir)]} sin(2n + 1)0

Et z = _ 2k___j Z Oi

[AinI2n+l (kit) - CmK2n+l (ktr)]

n = o

Equation (13a) results from the two boundary conditions, //20 - H~0 = - J s : and H~z - H~z = Js0 at r = r~, and also the request that the superconducting shielding current-~s must flow in the direction along the twisted superconducting filaments

Jso = AJsz; A = 2nrl/Lp (10c)

(13e)

(14)

Lp is the twist pitch of the wire. Since the wire concerned is carrying no net transport current, we can assume that no superconducting current flows in the filamentary region. Then all the current in the filamentary region is flowing_.~nside the normal matrices with a current density of tr~E~, and is composed of the coupling current a.._~sa bypass current of superconducting shielding current, Js, and of the eddy current caused by the skin effect in the normal matrices. But the degradation of the magnetic field across a very thin saturated region due to the skin effect is negligible, and hence almost all the current flowing at the outer boundary of the filamentary region is the coupling current which cannot flow along the superconducting filaments, because the electric field should be zero in the superconducting filaments without carrying current. Thus, we reach an additional constraint given by where

sin(2n + i)0 For the vacuum region (i = 3), however, we must use the Laplace equation, V 2 H 3 = 0. in place of Equation (5). Then the general expressions for ~ and k:"-~ can be obtained by taking into consideration t_.hat ~ should approach the applied magnetic field H m as r goes to infinity. The resultant expressions are given by

H3r = ?~

[Hm6no

-

-

Canr-(~n+2)] cos(2n + 1)0

n= o

-/30

( 1 1 a)

~"

"~ --

[Hm6no +Canr -(2n+2)] sin(2n + 1)0 (1 lb)

n = o

( E l z -t- ,~kElO )r=r I = 0

(15)

( 1 1 c)

H3z=0 and

Ear = kA 02

Danr -(2n+2) cos(2n + 1)0

(12a)

Danr-(2n+2) sin(2n + 1)0

(12b)

n=0

Eao = kA O2

n=0

Eaz = - k~ -

Hmr6no - 2n+ l r-(2n+l)

-

02

(12c)

Note that the above constraint is not generally satisfied inside the filamentary region where the eddy current due to the skin effect cannot be disregarded compared with a dispersed coupling current. At the outer boundary of the stabilizer region, r = r2, the boundary conditions are given by

(1t3o-H2o)r=r,

= 0

(16a)

(H3z-H2z)r=.~

= 0

(16b)

(E3o-E20)r=r2 = 0

(16c)

(E3z-E2z)r=r,

(16d)

= 0

sin(2n + 1)0 where 8. 0 is the Kronecker delta.

Boundary conditions and additional constraints. T h e constants included in the general expressions forE/and//,. given by Equations (9) - (12) can be determined with the aid of the boundary conditions and some additional ones. Before describing these conditions, we must mention an approximation adopted here for the saturated region existing at the boundary between the two regions of i = 1 and 2. To describe the shielding current in the saturated region with a very thin thickness ds, we consider here the sheath current density per unit length in this region,-~ ~(O, JaO,Js:), instead of the current density per unit area, js = Js/ds. Under the approximation taking the limit ofdJrl --" 0, the shape of the outer boundary of the filamentary region (i --- 1) can be regarded as a cylinder wi_~ththe radius % Then the boundary conditions onHi andEi a t r = ra are given by

~(~o -H,o),=,, = - ( ~ = -H~=),=,,

(E:o-E,o)r=,,

= 0

Now we have eight conditions given by Equations (13), (15) and (16) for eight sets of coefficients, A~..B~n, A2n, B2n, C2., D2n, C3. and D3.. Substituting Equations (9) - (12) into these conditions and eliminating C3n and D3n, we get the following equation for the remaining six sets of coefficients PnC, = H ,

(17a)

with

CTn = (A in, B In, A 2n , B2n , C2n , D2n),

(17b)

HTn = (0, O, O, Hm6no, 0, 0),

(17c)

where the expression for the 6 × 6 matrix F,, is shown in Table 1, and the superscript T specifies the transposed matrix. From Equation (17c), we at once get Co = P01Ho

(13a)

and

(13b)

Cn = 0

forn = 0,

(18a)

for n 1> 1.

(18b)

Cryogenics 1 9 8 5 Vol 25 March

131

Eddy-current loss in a superconducting wire: Q.F. Zhang et al. Table 1

Elements of a matrix, I n = (I'~nqn) 1

2

3

4

5

6

1

-A[12n(klrl) +12n+2(k~r~ )]

/2n*1(klr1)

A[12n(k2rl) +12n+2(k2r~)]

-/2.+1(k2r1)

A[K2n(k2rO +K2n+2(k2r ~)]

--~2~,+1(k2rl)

2

0

12n(k~r~) +12n.2(k ~rt)

0

-(z[12n(k2r~) +12..2(k2r ~)]

0

e[K2n(k2rl) +K2n.2(k2r 1)]

qn P.

3

/2n+1{k~r~)

0

--0~/2n+1(k2rl)

0

tZK2n+l (k2rl)

0

4

0

0

12n(k2r2)

0

K2n(k2r2)

0

5

0

0

0

/2n+1(k2r2)

0

K2n+I (k2r2)

6

4/2n*l(klr1)

A[12n(k lrl) +12.+2(k~rO

0

0

0

0

Expressions for each component of Co are given in Appendix 1.

E2r

_ 1 k2 2 o~

/~2o + 2

k~r ~ l

cos0

(23a)

Low-frequency ac loss

In this section we shall derive the expressions for the ac loss in the low-frequency region, other than the intrinsic hysteresis loss in the superconducting filaments inside the filamentary region. If we define the frequencies characterizing the skin effect in the filamentary region and the stabilizer region by fez = (2n/.q azr~) -~

(19a)

/ca = [2n/a~o: (r: - rt)~ ] -~

(19b)

1 /q (B"2o - 2 D2o I sin0 =

}

E2z = - 2k2o~

k2,42or - k2r ]sinO

(23b)

(23c)

with ~10

then the low-frequency region can be defined bYf
o-;

-

-- -

-

2 3 klk2rl

o =

D2 o

(24a)

D2 o

(24b)

2oa

/c2r 2 2

1

I k2(r2 - r,)l < 1.

In the present situation, the expressions for the electric and magnetic fields obtained in the previous subsection are reduced to the following simple ones

A2 o = H a ~2o =

Hlr = Alo cos0

(20a)

Hlo = --~1 o sin0

(20b)

1 klB1 orsinO

-

(25b)

~s I D2o C~ o = ~1 k]r]Hm _ 2k2r

(25c)

2Ak2rl D2o

1 kl /~oCOS0 2 ol

B20

(20c)

=

Elr

2 ~2

~2r2

(25a)

A2s ( kl)_2(rl]' k,k2r] ot + -~2 \-r'22]

(21a)

/-/,.

=

+ (l+al]s o2 ] (25d)

k, Blo sin8 Elo = - 21 ol

(21b)

E,= ffi - k ~ ,~t orsin0

(21c)

Ol

and ffi k~ol/klO 2 = (#2011#102)112

s = 1 + (rl ]r2)2 H2r =

o - 2 k~r2 j cos0

H2o = H~z =

132

e o/

(22a)

o + 2 k~r2 /sinO

(22b)

k2B2 or + k2r J sin0

(22c)

Cryogenics 1985 Vol 25 March

(26a) (26b)

To calculate the ac loss per cycle per unit volume of the whole wire, we also have to retain the eddy-current loss resulting from the skin effect, because the eddy-current loss in the stabilizer region is amplified by the volume ratio of (~ -- r~)/~ compared with the coupling-current loss in the filamentary region. To distinguish the eddycurrent loss from the coupling-current loss, we shall divide the ac loss into two parts as

Eddy-current loss in a superconducting wire: Q.F. Zhang et al.

Wi = Wic + WiE f o r i ---- 1 and 2

(27)

with

°i 2nr~f

WiC

f

[ r~ r~ in ( r ~ ) ] ~ gae = 1 - 4 r~ +r"--'~t+ 4 r4-r----~

fio (IEtr 12 + IEw I~)rdr,

dO

rfi

o

(28a) Wi E --

01

f

2rrr~f

f rti IEia 13rdr,

(29)

WIC -P WIE

with

(36a)

/~ - ~ ~ +#1 (36b)

+ 4 r4-r~

(28b)

~-

=

~ [ r~ ta2e = [ 1 - 4 r ~ + r ~

iO

dO J2~r 0

where rii and rio are the inner and the outer radius of the ith region, respectively. Substituting Equations (21) and (24) into (28) and noting gli 0 and rio ----- rl, we get the ac loss in the filamentary region, W~, as Wt

and

~-1

~'¥-~I/

J ~

Since normal metals with high electric conductivities are usually chosen as stabilizers, the couplingcurrent loss, W2c, in the stabilizer region is much larger than Wlc in the filamentary region for a fully-stabilized wire with r~ ~, r~. Thus the ac loss in the stabilizer region predominates over the ac loss in the filamentary region, irrespective of the ratio of W2E/W2c. To compare the eddy-current loss, W2E, with the coupling-current loss, W20 in the stabilizer region, it is convenient to give an approximate expression for W2E as

Wic = 2~r{r~ ] = o, ~r21 oae

f]ffc~ laxeH2m 1 +(f/fie,)2

1 f¢~ Wlc WIE - 8 fc2

(30a) f2~r r ~ - r ~ r] (30b)

=

(37a)

21r r~-zr] f la2H2ma t f e l "¢f'~fc2,fec3 r~ fec3 (37b)

4~'/0~ o~L~

(31a)

4~./U~eo~eL ~

(31b)

~qe - - 2#: /.tx #a * #2

(32a)

r~ - r ~ o ~ = o~ + o~ r~ +r~

(32b)

fe=

fel

W~E %

where

fc~

f ~ e /.t2H2m a t f fe 3 /~e

and

Equation (30a) represents the coupling-current loss in the normal-metal matrices with the effective conductivity, cr~e,and the effective permeability, Pie, and is the same as derived by Turck s. As expected, the eddy-current loss, WIE, is negligibly small compared with the couplingcurrent loss, W~c, owing to the ratio of the coupling frequency,fcl, to the skin frequency, fc2, in the filamentary region. The ac loss in the stabilizer region, W2, can be obtained in a similar way as

W, =

W2¢ + W~E

(33)

with

current loss in a hollow cylinder made of a normal metal with the effective permeability, #2¢, and the effective skin frequency, fcC3.Since the external field can penetrate into the filamentary region almost freely atf'gf e, the deviation of #l from P2 modifies the permeability of the hollow cylindrical region,/h, to the effective one, #2e Note that ~t2e reduces to #2 if#~ = #2. At the frequencies off~.~f,Cfc2,f~, however, the, external field can scarcely penetrate into the filamentary region, and hence the effective permeability, ~e, reduces to #2e which can be obtained by putting #l = 0 in the expression for ~ze given by Equation (36). High-frequency ac loss

At frequencies much higher than the skin frequency of the stabilizer, i.e. f ~,fcC3, the external field can scarcely penetrate into the filamentary region. Thus, the total loss is almost the same as the high-frequency limit of the eddycurrent loss in a rod of radius r2 made of normal metal with permeability Pa, and is given by [ f \-1/2 /~H2m W -'2" 2rt ~-~--~)

(38)

where for3 is the skin frequency of the rod with the radius r=

W2c- r]-r~

d+,';

W2E =

It can be seen easily that the above expression for W2E at

f , g f ~ is the same as the low-frequency limit of the eddy-

o2 Wlc

o,

2~r r~ -r-------~21 f ('~e]#~e) +(f/f~')2l~2H2m r] f h I +~/YZ)2

(34a)

(34b)

f r 3 = (27r/~2o2r~)-a

(39)

Genera/expression for ac loss With the aid of Equations (i 0), (17), (i 8), (27), (28) and (A1) - (A6), we can get the general expressions for the ac loss as

where 4

1% = ,~moo2(r~ + r~)

(35)

wit =

u, ( ~I 12 Im(S1 [Bxo 12)

(40a)

Cryogenics1985 Vo125 March 133

Eddy-current loss in a superconducting wire: Q.F. Zhang

W,E

=

2~r/a~ ~

~r Uo 'w~¢ = ~-

Im(S~ IA~ o 12)

Im(Pt IB2 o I~

+

(40b)

P2 ID2 o 12+ P~B2 oD2 o) (4la)

e t al.

the values of eddy-current loss in the hollow cylinder of aluminium subject to the transverse ac magnetic field is also referred to for the sake of comparison of losses with each other. As mentioned in the previous section, we can see inFig. 2 that the hollow-cylinder loss is nearly equal to W2c both in the low-frequency range off,~fce~ and in the high frequency range o f f ~, fc~, while in the middle frequency range offc~ ,~f,~fce3 this kind of loss becomes very large compared with W2E except for the wire with r 2 ~ r 1.

--

(41b) where (42a)

S~ = k ~ r ~ I ~ t('Iox ~ +12 ~ ~) S~ = k~rtlo

~

tI11 t

(42b)

Pt = k ~ r 2 I t : ~ ( I o ~ + I 2 2 2 ) -

k2r~Ix 2 ~(Io2 ~ +I22 ~)

(43a) k2r2Ki 2 ~(Ko : ~ +/(222)

(43b)

k~ r,Kt 2 ~(Ko 2 ~ + K2 2 1)

Comparison with experiments To confirm the theoretical results given in the previous section, we carried out measurements of ac losses in an aluminium-stabilized NbsSn wire. In this section, the theoretical results are compared not only with the present experimental data for a short sample but also with the data for the coil which had been reported in the previous papeP. Experiment 1 The sample wire used in this experiment is the same as the wire wound into a high-field pulse-magnet developed at the Research Institute of Superconducting Magnet in Kyushu University4, The cross section and a specification 101

~=

k2r2I~ 2 2(Ke 22 +K22 2)

~ ( K o : +

a

=

(43c)

kzrz(Ioz: +I~2~)K~2~

10-'

kzr2Io 2 2I~ ~ 2 - kzr~Io ~ t ~I'~t

Q2 = k2rjK6 2 ~K~ ~ ~ - k : r x K o ~ t K~ 2 Qa

=

k2r2Io2 2Kl : 2 -k2rlI02

IK1 2 1

(44a)

== O ..J

io-2

(44b)

Low frequency region

i0-4

/

Middle frequency = .~"recjiorl

/

I0'

1

iO-Z

I ~1

iO-I

100 f, Hz

fre~lue cy = ~.. re(~%

\

{a

IO-o_ 4 A10-3

(44c)

In the above expressions, the abbreviated notation o f f 0 , 2 represents Io(k,r2) for example; Im represents the imaginary parts; and the upper bar specifies the complex conjugate. Some examples of the numerically obtained results are shown in Fig. 2 for the fully-stabilized wire with the matrix of CuNi and the stabilizer of aluminium, where a m ( = 0rcuN0 ~-- 7.1 × 106 g~-' m-'; a2(= aAD= 3 × 101°0 -1 m-t; p, = / ~ ; X = 0.6 and r, -----0.25 mm. From Fig. 2, we can see the overview of loss-frequency characteristics, where the loss in the stabilizer predominates over the loss in the filamentary region and frequency-characteristic curves of the former loss have two peak frequencies, i.e.fee, andfc~. Since the lower peak frequency,fi e, depends upon the twist pitch and the higher one, fee3, depends upon the outer radius of the wire, the feature of loss becomes complex accordingly, as the twist pitch approaches the outer radius. The increment of the loss in the fully-stabilized wire, which has already been pointed out experimentally by some investigators, is mainly caused by the frequency characteristics of W2c with the lower peak frequency determined by the conductivity of the stabilizer. In Fig. 2,

Cryogenics 1985 Vol 25 March

/

t0-3

+ k 2 r 2 I t 2 ~Ko ~ 2 - k2r~I~ 2 ~Ko ~

134

---~/

10o

x +K:~)

k2 r l ( / 0 2 1 "P I2 2 1)/~1 2 !

QI

Eq. (37a)

N

,

IOI

"ca

L\~,

10z

IO3

10z

103

•

\

IO4

b

ioo io 4

u~

8

10-z

..J

_

/jJJ/J

x=,~.\ \

iO-s 10-4 10-5

10"4

I0 "3

IO-z

I0"

I0 ° f, Hz

101

IO4

Fig. 2 Examples o f the numerically obtained f r e q u e n c y characteristics of ac losses, w h e r e a I = am(1 + k)/(1 - ,~); O'm(=O'CuNi ) = 7.1 x 106 ~-1 m-l; a 2 ( : aA,) : 3 X 101° ~-1 m-~; R 1 = 2 5 0 /~m; X = 0.6. (a) R 2 = 1.5 R T, L = 5 0 mm; (b) T2 : 2.5 R 1. , Loss ( W ' ) f o r Lp : 6 0 mm; loss (W") for Lp = 5 mm

Eddy-current loss in a superconducting wire: Q.F. Zhang et al. Cu N b~ Sn

o ~ **o

,4- -1

Bronze

AL\

®

i

1"¢7 4_J

r E E to

Cu

~

Nb

Nb3Sn

I I

Bronze

I

rL ..

/L.J

(~) '~-1 I

Amp

I

Fig. 3 A cross section of the aluminium-stabilized Nb3Sn multifilamentary wire

Liq. He

a

are shown in Fig. 3 and TableZ respectively. This Nb3Sn multifilamentary wire with a thick aluminium coating was manufactured by the Hitachi Electric Co Ltd by means of the 'extrusion with front tension' method 8, resulting in a good metallic bonding between aluminium and the outer surface of the filamentary region, ie, copper. A straight sample wire, 160 m m in length, was specially prepared for this experiment since the application of bending strain changes the conductivity of the aluminium coat as well as the critical current density of Nb3Sn filaments in the wire. The sample wire was mounted inside a race-track magnet and then the magnetic field was applied to the wire in the direction shown in

Fig. 3. For measuring ac losses in such a rigid superconductor, we developed a new system composed of the superconducting race-track magnet, a superconducting transformer and a superconducting heat switch for the persistent current mode, which are illustrated in Fig. 4. This compact system has a wide measurable frequency

He (t) Table 2

Specification of AI stabilized superconductor 3

Cross section of conductor

7.5 mm x 2.5 mm - 0 . 5 mmR

Cu-bronze-Nb3Sn core

6.0 mm x 0.9 mm - 0 . 3 mmR

Nb3Sn filament

4 . 3 / x m diameter x 7 3 8 1 3

(Nb3Sn+Nb)/bronze ratio

2.0

CLt/(N b3Sn+Nb+bronze) ratio

0.67

AV(Cu+Nb3Sn+Nb+bronze ) 2.5 ratio Twist pitch, Lp

50 mm

/ cat 12 T

108OA

1=-

b Fig. 4 System for measuring ac losses of a large conductor without flexibility. 1, A superconducting magnet of a race-track type; 2, a heat switch; 3, a superconducting transformer; 4, a sample conductor; 5, pick-up coils; 6, a dump resistor; 7, A / D converters; 8, a computer. (a) Schematics of the measuring system; (b) sequence of a generation of the small ac magnetic field superposed on the dc field

Cryogenics 1 9 8 5 V o 1 2 5 March

135

Eddy-current loss in a superconducting wire: Q.F. Zhang et al. ioo ~.ViZT)

/~.X

~,.~

¢"

\/ %

.,

. . . . .

I0 -I

I

I ~Wie {2Ti l~t I2T) x

• Hd¢=0l

.

I

-

/

\

,

,o-= , - / / / /

x

I'lilill

,

I0 -I

~

l ll,"~,l

I

I0 0

•

0.1

o

o.25

,~ ,

.... I

2.0

, , , ..... I

I01

102

line in Fig. 5 represents a set of the curves of loss-frequency characteristics which is shifted both to the down side by a factor of I/1.72 and to the left hand side by a factor of 1/1.38. As has been pointed by some investigators 6,9J°, this correction factor seems to be reasonable considering the demagnetization effect because we estimated here the loss of the wire with a rectangular cross section by using the present theory for the wire with a circular ~ross section. We shall not discuss the meaning of these factor in more detail because the purpose of the present paper is only to elucidate the loss-frequency characteristics. As can be seen in Fig. 5, a good agreement between theory and experiment is given with respect to the frequency characteristics.

f Hz Fig. 5 Measured ac losses of the aluminium-stabilized Nb3Sn wire for the small ac magnetic field in a bias field. Each curve is a theoretical one

range and a relatively large sample space. Using this system, we measured the loss-frequency dependences for the case of a small ac field with the amplitude of 0.07 T superimposed to a dc bias field ofHac = 0 - 2 T. The measured values ofac losses are shown inFig. 5. These data normalised by p~H2m were checked to be independent o f H m w h e n H m = 001 - 0.07 T, which means that the observed losses are mainly the eddy-current loss and the coupling-current loss induced in the normal metal matrix or the aluminium coat, and the hysteresis loss is negligibly small. We can find that each lossfrequency characteristic curve has a similar feature to the one for the ordinary coupling-current loss, ie, a Debyetype curve shifts to the right hand side with the increase of the bias field. To explain these data theoretically, we calculated the theoretical values of losses by assuming that the theoretical ones for the sample wire with a rectangular cross section can be estimated by the present theory for the cylindrical wire. In Fig 5, we show the theoretical results obtained numerically by substituting the values of conductivities of o"m = O'Cu = (rl(l -- h')/(l -t- ~k') and tr~ = irA] at various bias fields shown in Table 3, the permeabilities for a small-field amplitude case of /xI =/.q(1 - h')/(1 + h') and #2 = / ~ , the effective volume fraction ofh' = 0.80, the effective radii oft, = 1.31 x l0 -3 m and r2 = 2.44 x l0 -3 m, and the twist pitch ofLp = 5 x l0 -2 m into Equations (40) - (44). In this calculation, we adopted the effective volume fraction, h', which is defined as the ratio of the region within Nb3Sn and bronze enclosed by the Nb barrier to the all filamentary region, instead of the ordinary volume fraction of superconductors, h. The solid

Table 3

Values of conductivities at various bias fields [10 a f l -1 m-l]

Experiment 2 We have already carried out another experiment for the high-field pulse magnet wound by the same aluminium stabilized Nb3Sn multifilamentary wires as those used in the present experiment described in the previous subsection ~. The performance of this magnet is generation of a pulse field of a trapezoidal or a triangular waveform with its ramp rate of 0.5 T/0.1 s in a bias field of 12 T. Substituting crm = 2.2 × 109 N -I m-l; ~r2= 1.4 x 10l° f~-l m-l; and t t l / h = / ~ , into Equations (31b) and (35), we have fc~ = 0.15 Hz and fc3 = 29 Hz. Since the operation mode of this magnet approximately corresponds to the frequency of 5 Hz, we can see that the winding in this magnet suffers an external magnetic field in the middle frequency range. To confirm the obtained theoretical results especially for the middle frequency range, we rearranged the data given in Reference 4, and shown in Fig. 6. The loss values in Fig. 6 are normalized by /~ , where represents the averaged value of the square of the local-field amplitude at the winding in the pulse magnet (Taking account of the analogy of the wave form of pulse fields to the sinusoidal one, we determined here the amplitude as a half of the amplitude of pulse fields at local points in the magnet windings.) For the sake of an accurate calorimetric loss measurement, the measuring condition of the pulse magnet was chosen to generate trapezoidal magnetic fields with a height of 1 T in a bias field of 3.45 T regardless of the m a x i m u m performance. However, these data are sufficient for the present purpose. In Fig. 6, the theoretical results are also shown for a comparison with the experiments, where this theoref, Hz

107' IO°I

I0°

I

,o

ttm(= ~rr~)

tt2(= (;rAm)a

0 0.1 0.25 2 4 12

17 16 15 8.1 5.4 2.2

47 41 34 20 17 14

~'hese values of o 2 are estimated by using the values at 12 and the magnetic field dependence of the 9 9 , 9 9 9 % aluminium after applying 1% strain 5 times 3

136

Cryogenics 1985 Vol 25 March

io-tl 10-i

I0z ~

o w~

Bias field (T)

I0' ,

I

i..'i""i i i l i l l I0 °

toi , i -

i\l

/

liili,l~ I01 I/~, s-I

,

i

, i illi,I 102

I

Fig. 6 Measured ac losses of the high-field pulse magnet wound by the aluminium-stabilized Nb3Sn wie when the magnet generates the trapezoidal magnetic field shown in this figure 4. Each curve is a theoretical one for Hdc = 4 T, where 1 / 2 t o = f is assumed

Eddy-current loss in a superconducting wire: Q.F. Zhang et al. tical curve is shifted to the down side by a factor of 1/2.38 and to the right hand side by a factor of 1.60. The former factor can be accepted by taking account of the demagnetization effect, but the latter factor can hardly be interpreted at the present stage. It may originate from some deviation of the magnetic field at the winding in the pulse magnet. As can be seen in Fig. 6, however, a good agreement between theory and experiment was obtained for the frequency characistics, which results in a confirmation of the quantitative availability of the present theoretical expressions of the ac loss to practical wires.

Discussion As already shown in the previous sections, the main loss in the fully-stabilized wire is the ac loss in the stabilizer region. Therefore, it is instructive to compare two components of the ac loss in this region, i.e. W2c and W2E. As can be seen in Equation (28), W2c is the couplingcurrent loss due to the bypass current of the shielding current induced in the filamentary region, while W2E is the eddy-current loss induced by the skin effect in the stabilizer region. In the low frequency region o f f < f ~ l
[2m" e 1 2

.L

_ r~ +r~

! '

2r,

(45)

References I

2 3 4 5 6 7 8

9 10

Yamada,R., Kishimoto, T., Mori, S., Noguchi, N., Yoshizaki, R., Kawakami, H., Kondo, K., Hirabayashl, H., Morimoto, K., Wake, M., Yamamoto, A., Aihara, IC, Asano, H., Kazawa, Y., Kimura, H., Miyake, Y., Ogata, H., Saito, R. and Suzuki, S. Proc IC1".'C9 (1982) 221 Royet,J.M., Scudiere, J.D. and Schwall, R.E. IEEE TransMagn (1983) MAG-19 761 lrie, F., Yamafuji, K., Takeo, M., Sumiyoshi, F., Miyake, Y., Noguchi, K., Kazawa, Y. and Saito, R. Proc ICMC (1982) 477 Irie, F., Yamafuji, K., Takeo, M. and Sumiyoshi, F. IEEE Trans Magn (1983) MAG-19 672 Turck, B. J Appl Phys (1979) 50 5397 Turck, B., Lefevre, F., Polak, M. and Krempasky, L. C~'ogenics (1982) 22 441 Cart, W., J. Jr. J Appl Phys (1974) 45 929 Mori, S., Yoshizaki, R., Kawakami, H., Kondo, K., Hirabayashi, H., Morimoto, K., Wake, M., Yamamoto,A., AiharL K., Kazawa,Y., Kimura, H., Miyake, Y., Ogala, H., Saito, R., Suzuki, S., Kephart, R. and Yamada, R. Adv Cryog Eng (1982) 27 151 Sumiyoshi, F., lrie, F. and Yoshida, K. J Appl Phys (1980) 51 3807 Soubeyrand, J.P. and Turck, B. IEEE Trans Magn (1979) MAGI5 248

Appendix The elements of Co in Equation (18) are given by using D20 as follows /122(K021 "l'l'K221)'l'Kl 22(/02 1 "1"/221) 1 D2 o Alo = ~- Aa I1111122

When the amplitude of the external field is large enough to satisfy the relation of P-i = #2 = ~ , the above ratio is written as W2E= ( 27rret 2

(46)

Even at low frequencies off I is fulfilled. (The solid line in Fig 2b corresponds to the case of2rrre/Lp = !.14.) As can be seen from Equation (45), this condition is widely satisfied in the practical fully-stabilized wires, while it is scarcely satisfied for an ordinary wire without stabilizer because r e reduces to r t and the condition of Lp < 2rrrl is hardly achieved from the metallurgical limitation for twisting. In the middle-frequency range offer < f < fff3, Equations (30a), (34a) and (37b) lead to

(AI)

Blo

I122(K021 + K 2 2 1 ) + K I 2 2 ( I 0 2 1 +/'221) =--tX

/122(/011 +1211)

(A2) h2o =

/122K121Hm 1122(/I2 iK022 +1o22K121)

z~kK022 [/122(K021 + K 2 2 1 ) +K122(/021

+¼

+/22 I)] D20

/122 (/12 iK022 +/022K121)

(m) K122

B20 = - I t 2 2 D2° W2[

-

/Z2e~2:27rre~2

(f)

2

D2 o

(A4)

(47'

112 aI122Hm C2o

W e can see that W2E/W2¢ > I can be satisfiedmore easily due to the existence of the factor off/f~ > I. This is because WzE is proportional to frequency in this middle frequency range, while W2¢ is inversely proportional to frequency. In the higher frequency range off>f c, the totalloss is almost the same as the eddy-current loss, W2E, in the stabilizer region, as expected.

-¼

Acknowledgements

with

The authors are thankful to Hitachi Co Ltd and Hitachi Cable Co Ltd for providing the sample wire. The present study is supported in part by the Grant in Aid for Scientific Research of the Education Ministry of Japan.

=

1122([o22K121 +112 iKo22)

Z~d022 [/122(K021 +K221)-KI22(]021 +I221)]D20

I122(Io22K121 +I121Ko22)

(A5)

D2 o = -- 4~"-1g~/111/122(]011 + ] 2 1 1 ) [ I 1 2 1 ( K 0 2 i + K 2 2 1 ) + K~2~(I02, +I22 *)]Hm

Cryogenics 1 9 8 5 V o 1 2 5 March

137

Eddy-current loss in a superconducting wire: Q.F. Zhang et al. /3 = - 1

detFo 0~

+ g, 22(lo2, +I22 ,)]

= 4II11(I022K121 +I121K022){(I011 +1211)

X {0/(/011 +1211)(I022K121 +1121K022)

x (I121K122-I122K121)

+ 1, i 1 [lo2 2 ( K o 2 , + K 2 2 1 ) - K o 2 2(1o2 t +12 2 1 ) ] } -

~1 ~ 1[I122(Ko21 +/G21) + K, 22(Io21 +I22~)]1

-

/

- A2(Iol 1 +1:11) [I122(Ko21 +K:21)

138

Cryogenics 1985 Vol 25 March

(A6) In the above equations, the notations of/o,2 and K2lo represent Io(klr2) and K2(klro), respectively.