AC losses in single-core superconducting composite conductors

Kes/Kenas a f u n c t i o n

of t e m p e r a t u r e

According to BCS the energy gap at absolute zero is given by

2A(0) = o~B Tc where a = 3.53, but experimentally 6 the value ofot is found to vary between about 3.1 and 4.2. Mtihlschlegel 2 has calculated z~(T)/A(0) as a function of t = TIT c for the BCS value ot = 3.53. We have used his results to calculate Kes/Ken as a function of t for this value of or, as well as for the values 3.1 and 4.2. The results can easily be extended to other values of ~. The procedure is as follows: let the values of TIT c be t/" and let the corresponding values of A(T)/A(0), be x/. Let the values of Table 1 be Yi = A(T)/kB T and (Kes/Ken)i. Then A(T) = yikB T = x/~(O) = x1~/2)k B T c and

(3)

7"cp

where ~ is the digamma function, Tc and Tcp the critical temperatures of the alloy and pure host, and X a parameter that for the AG case is equal to × = O.14(Tcp/Tc) (ni/ncr), where n i is the magnetic impurity concentration and her the concentration needed to depress Tc to 0 K. Equation 3 holds also for other cases 7 with different expressions for ×. We give in Table 3 the parameter × and the AG ratio (ni/ncr) as a function of (Tc/Tcp) for (Tc/Tcp) from 1.00 to 0.04 in steps of 0.02. We are grateful to Prof Peter Lindent'eld for many suggestions and helpful discussions. References

y, (;) (:s) If for x/and t/we use Mtihischlegel's results, the values of Yi and hence (Kes/Ken)i can be calculated for any a. The results for ~ = 3.1, 3.53, and 4.2 are shown in Table 2 for t from 0.16 to 1.0 insteps of 0.02. T h e depression o f the critical t e m p e r a t u r e w i t h magnetic i m p u r i t i e s

Abrikosov and Gor'kov 3 calculated the depression of Tc for a superconductor containing magnetic impurities. Their result can be written as

1 Bardeen,J., Riekayzen, G., Tewordt, L. PhysRev 113 (1959) 982 2 Miihlschlegel,B. ZPhysik 155 (1959) 313 3 Abrikosov,A. A., Gofkov, L.P. Zh Eksp TeorFiz 39 (1960) 1781 ; (English translation) Soviet PhysicsJETP 12 (1961) 1243 4 Rhodes, P. Proc Roy Soc (London) A204 (1950) 396 5 Bardeen,J., Cooper, L. N., Sehrieffer, J. R. Phys Rev 108 (1957) 1175 6 See Meservey,R., Schwartz, B. B. Superconductivity,Vol 1 (ed Parks, R. D.) (Marcel Dekker, New York, 1969) 141 7 (a) Parks, R. D. Superconductivity,Proceedingsof the Advance Summer Study Institute, McGill University, Vol II (ed Wallace, P. 11.) (Gordon and Breach, New York, 1969) 625 (b) Miiller-Hartman, E., Zittartz, J. Phys Rev Lett 26 (1971) 429 8 Ramos, E. D., Sanehez, D. H. 'Tabulation of the BRT Function for the Thermal Conductivity Ratio and the AG Function for the Depression of Tc', Rutgers University (available on request from the authors)

AC losses in single-core superconducting composite conductors K. Shiiki, K. A i h a r a , M. K u d o , and F. Irie

The problem of ac losses in superconducting wires is one which has received the attention of a large number of cryogenic engineers. Superconductors have been used for large powerful dc magnets because of their ability to carry large dc currents without energy loss. In ac fields, however, superconductors give rise to considerable losses. In recent years great interest has been shown in many ac applications such as superconducting ac power transmission lines. A number of investigations on the loss in superconducting wires have been carried out, 1, 2 but a loss mechanism has not yet been proposed for composite conductors. In this paper we study the loss occurring in single-core composite conductors from the frequency dependence of the loss. E x p e r i m e n t a l details

Characteristics of samples are listed in Table 1. Samples were copper-clad wires and bare superconducting wires of KS, KA, and MK are with the Central Research Laboratory, Hitachi Limited, Kokubunji, Tokyo, Japan. F| is with the Department of Electronics, Kyushu University, Fukuoka, Japan. Received 1 March 1974.

CRYOGENICS . JUNE 1974

Nb-62.5Ti-2.5Zr superconducting ternary alloy. The composites consisted of about 250/am diameter superconducting wire with a 34-126/am thick copper layer. The bare superconducting wires were prepared after removal of Table 1 Wire diameter,

SC diameter,

Cu thickhess,

Cu resistivity,

/am

/am

/am

~cm

25C

319

250

34

25S

250

250

(bare)

50C

356

254

51

50S

254

254

(bare)

80C

407

252

78

80S

252

252

(bare)

-

125C

505

253

126

1.05 x 10 "s

125S

253

253

(bare)

-

Samples

1.28 x 10 .8 1.21 x 10 .8

1.24 x 10 "8

343

the copper layer by concentrated HNO3. Notations S and C for the sample name denote (bare) superconducting wire and composite conductor. The resistivity of the copper layer at about 12 K was about 1.2 x 10"s I2 cm. The ac losses were measured by a boil-off method at 4.2 K in the frequency range 2 0 - 5 0 0 Hz for sinusoidal currents up to 100 Arms. Wires were wound bifilarly on a bakelite bobbin and were placed in a nylon chamber immersed in liquid helium. They were connected to copper terminal plates, which also acted as a heat sink, outside the nylon chamber. When currents were applied to the wire, ac magnetic fields generated around the wire caused losses and boiled off the liquid helium in the nylon chamber. The mass of the evaporated helium gas conveyed by a nylon pipe was measured by a mass flowmeter.

1 = IOOA o •

5OC SOS

"E D

C

cr I ._o.-

= ._o-

Results and discussions

Results of samples No 50S and No 50C are shown in Fig.1. Similar results were obtained for other samples. Fig.1 shows the linear relations between the loss per cycle Qt/f (W cm "1 Hz "1) and the frequencyf(Hz). This leads to the experimental relation

Q t = a f + b.f 2

I" 0

"

T I00

= 2O0

=-

1 300 f , Hz

I 4O0

I

500

Fig.1 Measured loss per cycle Ot/f versus frequency f for the samples No 50C and No 50S. The result shows the linear relation between them; Ot = a f + b f2

(1)

where a and b are functions of the magnitude of ac current, and are obtained by the method of least squares. The values of the f'trst term in the composite conductors, which are independent of the copper thickness, are shown in Fig.2 as a function of the peak magnetic field Hm (Oe) at the surface of the core superconductor. Assuming that the currents only pass through the superconductors, we obtain

+ 25c 0

50c 80c [] 125c

_d

td: =N

-r

~E U

Hm

x/2 I -

5r

/

(2)

where I is the current (Arms) and r the radius of superconducting filament (era). The loss per cycle Qt[fin the bare superconducting wires is independent of the frequency f ( b = 0) and is proportional to Hm 3's as in previous work, 1 shown in Figs 1 and 3. The results shown in Fig.2 agree with those shown in Fig.3 within the limits of experimental error. This means that the first term of (1) shows the hysteresis loss in the superconductor. The copper cladding does not change the hysteresis loss in the superconductor in this case. The second term may be regarded as an effect of copper cladding. Fig.4 shows the square frequency dependence component of the loss per unit copper volume qe (W cm "3 Hz "2) as a function of the peak magnetic field H m (Oe). The solid lines show the calculated values from the following equation which represents the eddy current loss in the normal metal

qe

= KH'Zmt2 P

(3)

where t is the thickness of the copper layer (cm) and p the copper resistivity (I2 cm). The constant K is 6.5 x 10"16, which is determined to agree with the experimental value.

344

I

I

500

2000 Hm, O¢

Fig.2 Linear frequency dependence component of loss in the composites

The stun effect in the copper layer is almost negligible in our experimental case. To the zeroth order approximation, the square frequency dependence component can be interpretated in terms of the eddy current loss in the copper, as shown in Fig.4. Consequently, the ac losses in the singlecore superconducting composite conductor can be represented as the sum of the hysteresis loss in the superconductor and the eddy current loss in file copper layer. The deviation of the experimental results from the calculated values of (3) for the eddy current loss, however, seems to be significant: the deviation becomes noticeable as the thickness of the copper layer gets thinner, though the experimental error is very large in this region. The Hm dependence of the experirnental curves seems to have the

CRYOGENICS

. JUNE

1974

io" -P 2 5 s

/

0 50S A 8OS [] 125S

"r

/t=

IC5"

a/

E (J

Id

j/

,/

500

"1"

3=

I

I

2000

H,O¢ I000

Fig.3 Measured loss in the bare superconducting wire, which agrees the result shown in Fig.2 within experimental error

form/Pro, where a is much greater than 2. One of the authors (Irie) suggests that the superconductor distorts the magnetic field in the copper layer and that the eddy current loss in the composite conductor is modified from that in the free space. The calculation on the basis of a theory shows ~ (t~ > 2) dependence of the eddy current loss) The authors wish to thank Drs K. Kuroda and H. Kimura for their discussions and encouragements. We are also indebted to Dr G. Kamoshita for his support of this work.

Hint 0 ¢ Fig.4 Square frequency dependence component of loss in the composites. The solid lines show the calculated values of the eddy current loss in the copper layer

References

1 Rhodes, R. G., Rogers, R. C., Seebold, P~ T. A. Cryogenics 4 (1964) 206 2 Pech,T., Duflot, J. E, Fournet, G. Phys Lett 16 (1965) 201 3 Irie, F. (Forthcoming)

Dielectric loss of liquid helium R. LI. Nelson

The loss tangent of a capacitor A with liquid helium dielectric and stainless steel electrodes was investigated by the author 1 : a sample of polytetrafluoroethylene (PTFE), 'Fluon', was found to have a value of approximately +2.5/arad relative to the capacitor, at 60 Hz, 4.2 K and low electric stress. The absolute loss tangent of 'Fluon' is about 1 to 2 microradians at the same frequency and temperature.2, 3 Therefore, it seems reasonable to assume that the loss tangent of capacitor A, and hence the helium dielectric, was < 1/arad at low electric stress. At values greater than 4 to 5 MV m "1 (rms), however, the loss of the capacitor appeared to increase. 1 In order to investigate the variation of the loss tangent of liquid helium with electric stress further, and to obtain a reliable loss reference for measurements at higher voltages, a low-stressed, co-axial cylindrical capacitor, B, was built for use with a liquid helium dielectric. (A PTFE sample, reference, at 4.2 Kwas considered. It was thought, how-

The author is with the Central Electricity Research Laboratories, Kelvin Avenue, Leatherhead, Surrey, UK. Received 1 March 1974.

C R Y O G E N I C S . JUNE 1974

ever, that deterioration of the vacuum-deposited contact electrodes 1 with repeated thermal cycling could result in inaccuracies. 2,4) The inner cylinder served as the highvoltage (hv) electrode. The outer cylinder was in three sections; the middle section was the measuring electrode and the two end sections were earthed, guard electrodes. An electrode separation of ~2 mm was obtained using two annular spacers. To prevent errors due to leakage currents the spacers were positioned between the guard and hv cylinders. The electrode material was stainless steel with a finely machined surface. The difference between the loss tangents of the capacitor B and capacitor A was measured at 4.2 K and 60 Hz, using the method described in a previous paper. 1 The electrode separation of capacitor A was *180/am and both capacitances were %130 pF. The measured difference was 'x,O.0/arad for voltages up to 700 volts (rms). The detector resolution was about 0.3/arad. The loss of capacitor A increased (relative to B) for higher voltages; that is when the stress exceeded 4 MV m "1 in A. This increase was a property of the stressed helium dielectric as confirmed by the following experiment with another capacitor: the loss

345