A.c. loss measurements in high-tc superconductors

Applied SuperconductivityVol. 3, No. 6, pp. 339-349, 1995

0964~1807(95)00084-4

Pergamorl

Copyright 0 1996 EiskvierScience Ltd Printed in Great Britain. All riphts reserved 09%1087/95%9.50+ 0.00

A.C. LOSS MEASUREMENTS IN HIGH-T, SUPERCONDUCTORSt E. BeCHIN,’

J. BOCK,2 G. DUPERRAY,’

D. LEGAT’ and p. F. HERRMANN’

‘Alcatel Alsthom Recherche, Route de Nozay, 91460 Matcoussis, France; ‘Hoechst AG, Chemicals Division, D50354 Hihth, Germany (Received 30 May 1995; in revised form 16 September 1995)

Abstract-Low a.c. loss is a key issue for electrical engineering applications of high-T, superconductors at industrial frequencies. Different material options are characterized by three different a.c.. loss measurement techniques which approach as much as possible the application conditions of the conductors. Melt-textured-growth YBa2Cus07--x bulk samples, suitable for magnetic bearings, are characterized in an external a.c. field by magnetization measurements. Melt-casted Bi$r&aaCu30s_-x samples and powder-in-tube Bi&CazCusOtc_, tapes are suitable for current leads or power cables. In these applications the conductors are exposed to a.c. transport currents. The associated losses in self field are measured by a very sensitive electrical measurement technique. Finally, a calorimetric method is necessary when larger conductors (for instance Bi&CazCu30s_, tubes) are tested under transport currents I,, generating a transverse magnetic field Ha I,, as encountered .in magnet windings for SMES, transformers or generators. The results show that the a.c. losses are sufficiently low for self field applications at industrial frequencies and a comparison of the different high T, superconductors is given. The results show further that the a.c. losses are essentially hysteretic and can be modeled using the Bean model.

NOMENCLATURE thickness of the sample (m) external magnetic field (A/m) d.c. magnetic field (A/m) penetration field (A/m) amplitude of the a.c. magnetic field (A/m) critical current (A) d.c. measured critical current (1 pV/cm, 77 K) (A) heater current (A) transport current (A) amplitude of the a.c. transport current (A) critical current density (A/m’) d.c. measured critical current density (1 pV/cm, 77 K) (A/m’) critical current density deduced from magnetic measurement (A/m’) root mean square current density (A/m’) transport current density (A/m’) amplitude of the a.c. transport current density (A/m’) distance between voltage tapes (m) magnetization (A/m) a.c. losses s(W) resistive losses (W) pressures on both sides of a manometer (Pa) losses per cycle and per volume unit (J/m3) sample section (m’) part of the voltage of the sample in phase with 1, (V) heater voltage (V) sample volume (m’) outer or hmer diameter (m) frequency (Hz) Abbreviations Y-123 YBa&U3O:r _-* Bi-2212 Bi2Sr$ZaCu20e_, Bi-2223 Bi2Sr2Ca2Cus0ic--x

7 A part of this work is supported by BRITE (EURAM). IPW3:6-c

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340 SMES HTSC PIT MTG VSM

superconducting magnet energy storage high-T, superconductor powder in tube melt textured growth vibrating sample magnometer

1.

INTRODUCTION

In the presence of an alternating transport current and/or an alternating external magnetic field, the superconducting materials present energy dissipation. It is necessary to quantify and to minimize these losses in view of a.c. applications which can he divided into two classes: the applications where the wire sees low or no magnetic field [ 1, 21 such as current leads, fault current limiters or power cables, and those where the wire sees a large magnetic field [2] such as magnetic bearings or windings for superconducting magnet energy storage (SMES), transformers or generators. Different methods can be used to characterize the a.c. losses of high-T, superconductors (HTSC), depending on the samples (shape, material, etc.) and on their applications. Magnetization measurements are used for the characterization of superconductors subjected to an external alternating magnetic field [3-51. This method is useful to characterize melt textured YBa2Cu307_-x (Y-123) [6] which is seen as the most promising superconductor for bearings. The a.c. losses of conductors used for self field applications can be measured using the electric method [7-91. In this paper, we present the results obtained on melt-casted BizSr2CaCuzOs_, (Bi-2212) conductors [lo], used, for example, as current leads. Results obtained on powder-intube silver-sheathed Bi&-&!azCuaOic_, (Bi-2223/Ag) multi-filamentary tapes [ 1 l] are also reported. One of the target self field applications of these tapes is power cables. For applications with transport current in the presence of a significant magnetic field we used the calorimetric method. This method, initially developed for low temperature superconductors [ 121, has been tested by measuring the losses of Bi-22 12 tubes submitted to an alternating current in phase with an external a.c. magnetic field.

2. MATERIALS

AND METHODS

The textured Y- 123 samples were fabricated at Alcatel Alsthom Recherche by the melt-textured growth (MTG) method [6]. In large samples, transport currents up to 1160 A have been measured in a field of 1 T. In small samples, the d.c. measured critical current density JFc (1 pV/cm, 77 K) was found to be up to 3 x 10 A/m2 under self field conditions and 2.3 x lo* A/m2 in a field of 2 T [ 131. For the magnetization measurements, the dimensions of the samples were * 10 mm long with 1 mm2 section. The Bi-2212 tubes were fabricated by Hoechst using the melt-casting method [lo]. Typical dimensions of the tubes which have been characterized are 35/28 to 70/58 mm outer/inner diameter, and length 100 to 200 mm. The d.c. critical current density reaches values up to lo7 A/m2 in self field. Some of the tubes have been cut in bars with “I” or “U” shapes of %20 mm2 cross-section for electrical measurements at different frequencies (Fig. 1). The d.c. critical current 1,“” of these bars is approximately 100 A. This current level is within the range of our variable frequency current supply. The Bi-2223/Ag multi-filamentary tapes were fabricated by Alcatel Alsthom Recherche, using the powder-in-tube (PIT) method [l 11. The results presented in this work were obtained on 24-filament silver-sheathed tapes with rectangular shaped cross-section, with filaments of 10-25 pm x 200-500 urn cross-section. The d.c. critical current density in the superconducting filaments reaches lo* A/m2 in self field (1 pV/cm). For the electric measurements, they were cut into samples of length ~45 mm. An overview of the different samples and of the different methods for a.c. loss characterization is given in Table 1. The magnetization measurements allow the electromagnetic characterization of small samples submitted to an external magnetic field. The magnetic response of the superconductor is related to

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in high-T, superconductors

/

/

341

/

Fig. 1. Fabrication of “U” and “I” shaped samples from Bi-2212 tubes for electrical a.c. loss measurements.

screening currents induced in the sample, generating a magnetization A4. These measurements present a high sensitivity and have the advantage that no contacts for current supply or voltage sensing are needed. This greatly simplifies the sample preparation and explains why this method was frequently used after high-T, superconductors were discovered. The a.c. losses per cycle and per unit volume are given by the hysteretic loop area: Q = p,, $M(H) dH. For moderate frequencies v, the hysteretic losses are proportional to v. The a.c. losses P&v) were obtained by multiplying by v the measured losses per cycle. For a sample of thickness 2e completely penetrated, the “magnetic” critical current density can be determined by J,, o( M/e [ 14, 151. Unfortunately, J,,,, deduced from the magnetic measurements, does not (in general) correspond to Table 1. Conductors, materials and a.c. measurements methods. (P,=: a.c. losses, HI external magnetic field, HP: penetration field, HDC: continuous rield, Ho: amplitude of the alternating field, Ia: amplitude of the alternating transport current I, which is related to the overall current density Jt, IcDC: d.c. critical current, v: frequency, V: sample volume) Measurements

Samples

Magnetization (H//)

MTG Y-123

O~&~DC);

%l uw at 50 Hz

j$

Ho 9 HP

Q P.,(T); Ho >>sHP

Sensitivity

for YmlO mm3

M Bi-2212 with “I” shape

Electric (I,) aPa&);

10I P

Bi-22 12 with “U” shape 0.1 uv ~O.l,...,lO~W forI,=l,...,lOOA

++

@Pa&); 105 &uc ++ Bi-2223/Ag multi-filamentary v Electric (&,v = 50 Hz)

Bi-2212 tubes

@P&l,); r, ); 10:5 Ipc ++

n

Calorimetric (H 1. +ZJ

Bi-2212 tubes

@P&H, I,); 10* $=

II

8)Jl’l)tH

ml pv *l,..., 5 mW forl,=l,..., 5 kA

EC: 100 mW

E. BBGHM et al.

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the critical current density J,DCdetermined by d.c. transport measurement in self field. This is why such measurements should be used only for material development and should be avoided for conductor characterization in applications where a macroscopic transport current is required. In this case, the electrical and calorimetric methods described below are more adequate. Two experimental set-ups have been used for the magnetization measurements. One (set-up 1, Table 1) consists of a classical measurement of the magnetization loop with two pick up coils [ 161. The sample is exposed to an alternating external magnetic field, superposed on a continuous magnetic field. This method was used for the measurement at 77 K of the d.c. field (Hoc) dependency of the a.c. losses Pa, due to an alternating 2.14 Hz magnetic field of amplitude Ha (Fig. 2). The variations of the a.c. field correspond to the “minor” curves while the increase or the decrease of the dc. field correspond to the “major” curve. These experimental conditions are close to the ones found at higher frequencies in magnetic bearings and flywheels. Flux lines of an external field generated by a permanent magnet are pinned. in the superconductor and inhomogeneity of the field in the azimuthal direction or vibrations can induce magnetic field variations. The bearing is submitted to a mechanical load which results in a certain equilibrium distance between the permanent magnet and the superconductor. This distance z corresponds to an external magnetic field Hoc(z). An increase of the mechanical load induces a new equilibrium distance and a new Hnc (“major” curve). Variations of amplitude Ho of the a.c. magnetic field (“minor” curves) induce losses in the superconductor which can become a technical problem (decay of rotational speed, increase of temperature, etc.). We can distinguish two regimes of depinning of flux lines in the superconductor which are related to a critical value of the field gradient: In the first one (1 in Fig. 2), the critical field gradient is exceeded only in a limited volume of the sample beginning at the surface. Using the Bean model [17, 181, the a.c. losses per volume unit, for a simplified geometry of an infinite plate of thickness 2e, can be described by: P vH3 acO<0*

v

eJ,

In the second domain (2 in Fig. 2), the critical field gradient is exceeded everywhere in the sample. Here, the Bean model [ 17, 181 gives for the a.c. losses per volume unit, for a simplified geometry of an infinite plate of thickness 2e:

pa,cx vH&,,. v

V = (4.9 x 3.01 x 0.56) mm3

-0.3

0

0.3

0.6

0.9

1.2

IO-H (T) Fig. 2. 2.14 Hz magnetic measurements at 77 K using pick up coils: experimental curves for different HD, (“minors”) and reconstituted curve (“major”) [ 131. MTG Y-123 sample in magnetic field perpendicular to the slabs. 1: tbe critical field gradient is exceeded only in a limited volume beginning at the sample’s surface. 2: tbe critical field gradient is exceeded in all the sample.

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There is a critical amplitude of magnetic field variation of the order of M where the flux lines are depinned in the whole sample, The second experimental setup (set-up 2 in Table 1) uses a vibrating sample magnetometer (VSM). Here, the external magnetic field is constant and the signal in the pick-up coils is induced by the sample motion. The VSM system is equipped with a continuous flow cryostat which allows the temperature dependence of the a.c. losses to be measured. Our measurements were made with external fields larger than the penetration field Hp. The electric measurement is used for the characterization at different frequencies (set-up 3 in Table 1) of the self field a.c. losses related to the transport current (4 in Table 1). This method needs contacts for the current leads, with low contact resistance Rt, and contacts for the voltage taps. Our voltage measurements are realised with phase sensitive lock-in detection. The a.c. losses are related to the real part (in phase with the current) of the voltage signal while the out of phase signal is related to the inductance of the measurement circuit. The second is always present, which requires a high accuracy for the separation of the two contributions. Our measurement set-up is shown in Fig. 3: a signal generator is used to control the current amplifier (current and frequency values). The maximum current value is 130 A,, with frequency from d.c. to 1 kllz. A low inductance shunt is used to measure the current in the circuit. We measure the transport current It with a second lock-in amplifier. This lock-in is used to check that the phase shift remains negligible with regard to the reference signal given by the generator. Phase errors were observed to be less than 0.2 degrees. The first lock-in amplifier is used to measure the voltage taps of the sample. The a.c. losses are given by Pat = UIt, where U is the-in-phase part of are performed at different frequencies. the voltage of the sample. The measurements Measurements were also performed using this method but with a 50 Hz/220 kVA power supply instead of the signal generator and the current amplifier (5 in Table 1). The maximum current value in this case is 8000 A,, which allows the characterization of large samples. For these measurements the reference signal is given by the shunt. This electric method is very sensitive: the voltage sensitivity is % 0.1 uV for the first measurements with v < 500 Hz and x 1 pV for the 50 Hz measurements. However, this method is not suitable for the measurement of large conductors with a coil geometry or for the measurement of conductors submitted to an external a.c. magnetic field. In this case eddy current losses induced in surrounding metallic materials become non-negligible and would disturb the measurement. Here, the use of a calorimetric method is required.

Fig. 3. Schamatic view of the measurement set up for electrical self field a.c. loss measurements.

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E. BBGHIN et al. Calibrated

i I

1

Fig. 4. Schematic view of the calorimetric measurement set up for a.c. loss measurements with transport current Z, and transversal magnetic field H 0: I,.

The calorimetric method (set-up 6 in Table l), widely employed for the a.c. losses measurements of the low-Z’, superconductors, is based on the measurement of the gas evaporation due to the heat transfer to the cryogenic liquid. The classical measurement in helium uses a flow meter [ 191. Nevertheless, the enthalpy of evaporation at the boiling point of nitrogen is much larger than the enthalpy of evaporation of helium, which results in a low evaporation rate for nitrogen. Therefore, the gas flow is small and a method with an enhanced sensitivity for the measurement at 77 K of the pressure was developed (Fig. 4). This method, based on a high sensitivity differential manometer and a needle valve, is a comparative measurement: first, we measure the pressure difference when the HTSC sample is submitted to an alternating transport current and a variable magnetic field due to an external magnetic field coil. Then the current Z,, supplied to a heater situated in the liquid nitrogen bath, is raised until the same evaporation rate is reached. The equivalent a.c. losses are then given by Pa,= P,= U,Z, where U, is the heater voltage determinated using a four point measurement and P, are the resistive losses of the heater. The needle valve allows different magnitude of the N2 pressure p2 (p2 - p1 = 0.01 to 10 Torr in our measurement). It was checked that the differential pressure and the temperature of the gas on both sides of the manometer have no incidence on the results. Nevertheless, the height of liquid nitrogen above the sample modifies the boiling point ( x 0.5 K/m) and it is necessary to carry out the measurements with low variations of the liquid nitrogen level. Due to heat gradients which remain in the liquid nitrogen, the boiling cannot be reproduced at low dissipation rates. The limit for a reproductive measurement is situated near 0.1 W which is sufficient for large tubes or coils. 3. RESULTS

AND

DISCUSSION

To facilitate the comparison of the results, the a.c. losses are normalized with respect to the sample volume Yand the external magnetic field amplitude Ho multiplied by u. for magnetic

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345

measurements. For the electrical measurements the a.c. losses are normalized with respect to the sample volume and the root mean square current density J,,,,,. This allows the comparison of results obtained on different materials or with different methods. 3.1. MagneticJie.ld dependence

of a.c. losses in A4TG Y-123

The results ob!.ained by the magnetization measurements of MTG Y- 123 samples, using pick-up coils (set-up 1 in Table 1) are shown in Fig. 5 where the 50 Hz normalized inductive losses at 77 K vs the continuous magnetic field value are reported. The magnetic field modulation was superior to the penetration field for these measurements. In the case presented in Fig. 5, the external magnetic fields (continuous and alternating) were parallel to the texturation planes of the MTG Y-123 sample. The plot shows that, in these conditions, the ac. losses can be described with the Bean model [ 17, 181: 2

P

o< veJ,

for Ho > Hp.

VCLOHO

Using the Kim formula [20] for the magnetic field dependence of the critical current density, the above relation becomes: P ac= VpoHo

a’(T) Bo + POHDC

'

where the parameter a’(7) is related to vortex pinning of the flux line lattice. 3.2. Temperature dependence

of a.c. losses in MTG Y-123

Figure 6 shows the measured temperature dependence of the 50 Hz losses of a MTG Y-123 submitted to an external magnetic field with Ho greater than the penetration field Hp. This plot was obtained by the magnetic measurement using the VSM (set-up 2 in Table 1). The external magnetic field, parallel to the slabs of the textured sample of thickness 2e, was u&Jo = 1.8 T. Our results are in good agreement with the empirical law found in Ref. 3, which were obtained in similar conditions: 2

P

=

ye-flT,

VFOHO

where #I is the ternperature dependence and y the a.c. losses level. We found y M 7.0 lo7 W/m3 T and B m 0.05 K- ‘. Yunhui Xu et al. [3] found y * 1.2 lo7 W/m3 T and /Ix 0.05 K- ’ for a GdBaCuO sample, y % 1.O lo7 W/m3 T and /I = 0.1 K- * for a sintered YBaCuO. These /I values confirm that the sintered YBaCuO is more sensitive to temperature variations than the MTG

0

1

2

3

6

7

8

Fig. 5. Continuous magnetic field dependent a.c. losses at 77 K and 50 Hz deduced from 2.14 Hz magnetic measurement. MTG Y-123 sample in magnetic field parallel to the slabs.

E. BBGHINet al.

346

F “E g ^o 3 L ‘I d

108 _III1,IIII,IIII,IIII,IIII,IIII,IIII,IIII- . T=77K _ la v=50Hz la . .. l l* Pac/(V~HO) = 7 107 e-O.05 T ‘.a. l* -8. l. 4 .N 107 - H \ l* j T l* 00 .*. -B 2e l-. . WHO= 1.8T l. . . Jcm(77K. OT) - 2 108 A/m2 -* 9. V=(I.21 xO.37x4.82)mm3

: -

\. loa ““““‘1”“1”“1”““““‘~“~““~’ 0 10 20 30

40

50

60

70

80

T Kl Fig. 6. Temperature dependent a.c. losses at 50 Hz deduced from VSM magnetic MTG Y-123 sample in magnetic field parallel to the slabs.

measurement.

material. This result is in general related to J, limitation due to a high density of weak links in sintered samples. For the y value, our result is higher than those of Ref. 3. The explanation can be given by the Bean model [17]: the a.c. losses divided by the external magnetic field (i.e. y) are proportional to eJ, when H> HP [ 181. These results are in good agreement with the fact that the critical current density obtained in MTG YBaCuO is at least one magnitude higher than in sintered YBaCuO and GdBaCuO. 3.3. Frequency dependence of a.c. losses in Bi-2212 and Bi-2223/Ag The frequency dependence of 77 K a.c. losses in Bi-2212 and Bi-2223/Ag samples was studied using the electric measurement (set-up 3 in Table 1). The hysteretic losses are known to be proportional to the frequency while the eddy current losses are proportional to the square of the frequency v. Figure 7 shows that the a.c. losses of a Bi-22 12 “I” sample have a linear frequency dependence: P, m av + b with b % 0 if Zt << Z, and b > 0 if Zt z Z,. All measured HTSC samples show this behaviour, demonstrating that the a.c. losses are essentially hysteretic when the current is small compared to the critical current. When the current increases above Z, a flux flow dissipation component is added. These results are in good agreement with those of Ref. 7. 8 ‘,,,,,,,’

,,,,,,

,,,,

T=77K

,,,,,,,,,d Irms = 13OA /’

V = (6.0 x 4.7 x 40) mm3 I,m = 188 A

0

50

100

-0

150

200

250

300

V WI Fig. 7. Frequency dependent a.c. losses measured at 77 K in a Bi-2212 “I” shape submitted to an alternative transport current with the electric method.

A.C. loss measurements ,

10-3 t_

I -

I

I

in high-T, superconductors

,,,,,

347

,

T=77K v=50Hz

/’

r_

-

4 E

105

5 105

106

5106

Jms [A/m21 Fig. 8. Transport current density dependent a.c. losses at 77 K and 50 Hz for different samples measured with the electric method (Bi-2212 tube) or deduced by linear interpolation from electric measurement at 40Hz and 60 Hz (Bi-2212 “U” shape, Bi-2212 “I” shape and Bi-2223/Ag multifilamentary). Also figuring the curve featuring the behaviour: P,,/VJ,, a J&,.

3.4. Tmnsport current dependence

of a.c. losses in Bi-2212 and Bi-2223/Ag

In Fig. 8 we plot the normalized losses at 50 Hz vs the current density, for different samples. These results were obtained by linear interpolation between 40 and 60 Hz electric measurements (Bi-2212 with “I”’ or “U” shape and BL2223/Ag multi-filamentary, set-up 4 in Table 1) or by direct 50 Hz measurements (&,t 70 mm Bi-2212 tube, set-up 5 in Table 1). Table 2 shows the section and the critical current density of the measured samples. As we have seen before, using the Bean model [ 17, 181 the losses divided by the volume, for a simplified geometry of an infinite plate of thickness 2e submitted to an external field of amplitude HO, can be described by: P vH3 acpJ_!?

v

4

when H,, << Hp.

Moreover, in self field we have HO oc eJO, where Jc is the current density amplitude. This leads to a behaviour which is indicated in Fig. 8: Pat

-~J&, VJiTlU

when J,,

< J,.

This model applies to the losses of the melt-cast Bi-2212 conductors. The high current measurements at 50 Hz are in excellent agreement with the low current measurements at variable frequency. For the silver-sheathed Bi-2223 conductors the agreement with the J,&, behaviour is less obvious. This might be related to the fact that the transport current density is too close to J,. Indeed, in self field and without twisting of the conductor, the multi-filamentarization and the Table 2. Section S, distance between voltage taas L, d.c. critical current and d.c. critical current density in self field (1 pV/cm) ICpcand J, of samples measured by the electric method Samples Bi-2212 “II” Bi-2212 “I” Bi-2223 I’ITt Bi-2212 tubes

Ip” (A)

J,” (10” A/m*)

s (mm*)

L (mm)

59.4-60.0 188-244 17.6 and 11.9 4500

4.4-5.7 6.7 4.1 and 5.2 3.8

10.5-13.5 28.2-36.6 4.29 and 2.30 1180

30-50 40-45 19 and 15 40

t Multi-filamentary: Overall section of the composite wire. The superconductor filling factor of the wires varies between 15 and 20%. $ Tube: I&= 57.8 mm; #,,=69.6 mm.

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348

IWO 5

10

b?l

15

20

25

30

35

500

600

700

l&C = 540 A with WHDC = 27 mT 0ext=35mm;Oint=28mm L=Mmm

0

100

200

300

400

IO IAl Fig. 9. Transport current amplitude and external magnetic field amplitude dependent a.c. losses at 77 K and 50 Hz measured with the calorimetric method.

presence of silver should not influence the a.c. losses. These a.c. losses, purely hysteretic, should be the same as for a bulk sample, when one takes the filling factor into account. For all the samples, the losses are always less than 10m3 W/Am at 77 K. Taking into account that the refrigerating system absorbs x 10 W, (electrical Watts) for the cooling of 1 W at 77 K, these losses are less than 1Oe2 W/Am at 300 K. For the copper at 300 K with Jt = 5 A/mm2 the resistive losses are M 8.5 x 10m2 W/Am. This shows that there is an economic advantage for a.c. applications of these materials at liquid nitrogen temperature. Finally, we measured the a.c. losses of an entire Bi-2212 tube, using the calorimetric method. The tube was submitted to an alternating 50 Hz transport current Zt of amplitude IO and an external magnetic field, of amplitude Ho, perpendicular to the tube and proportional to 4 : ,u,,H, = kl, (with k= 5 x IO-* T/A) (set-up 6 in Table 1). The normalized a.c. losses curve vs current amplitude or external magnetic field amplitude values (Fig. 9) has the same behaviour as the d.c. curve, showing the presence of flux flow for Zt above Z,. Critical currents up to 3000 A have been measured in self field in these Bi-2212 tubes of cross-section S= 346 mm2 (4ext ti 35 mm). In these samples the losses were too low to measure with currents below Z,. For the sample presented in Fig. 9 the d.c. critical current, measured in the presence of an external d.c. magnetic field ~&DC = 27 mT, is Z,DC= 540 A. This calorimetric method can be used to measure a.c. losses > 0.1 W of large samples (large tubes, coils, etc.) submitted to transport current and external magnetic field in liquid nitrogen. 4.

CONCLUSIONS

We have developed different a.c. loss measurement methods which allow the characterization of high-T, superconductors, in self field or in the presence of a magnetic field, depending on the target application. The a.c. losses in MTG Y-123 submitted to an external magnetic field are shown to decrease with increasing temperature exponentially, while the magnetic field dependence is close to the Kim formula. The results show that the a.c. losses below J, are essentially hysteretic and that they are proportional to the frequency. The losses can be well described using the critical state model: they are proportional to Z$ (or J$ when H0 < HP (or Jo < J,) and proportional to HO (or JO) when HO > HP (or Jo > J,). The losses of the measured conductors are shown to be low enough for various self field applications. Acknowledgements-The authors would like to thank Prof. P. Monod for the use of his VSM which allowed us to study the temperature dependence of the magnetization a.c. losses.

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