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Semi-empirical model of the losses in HTS tapes carrying AC currents in AC magnetic fields applied parallel to the tape face
Physica C 349 (2001) 225±234
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Semi-empirical model of the losses in HTS tapes carrying AC currents in AC magnetic ®elds applied parallel to the tape face N. Magnusson Electric Power Engineering, Royal Institute of Technology, Teknikringen 33, SE-100 44 Stockholm, Sweden Received 27 April 2000; received in revised form 10 July 2000; accepted 11 July 2000
Abstract The AC losses are an important parameter of high-temperature superconductors (HTSs) to be used in electric power applications. To obtain an optimised design of such applications, models of the AC losses are needed. This article describes a semi-empirical model of the AC losses in HTS tapes as a function of temperature, magnetic ®eld, transport current and frequency. The magnetic ®eld (applied in parallel to the tape face) and the transport current are in phase with each other. The model considers ¯ux-¯ow losses (based on the power law dependence) and hysteresis losses (based on the critical state model). Results obtained by the model are compared to experimental data obtained calorimetrically on a BSCCO/Ag tape in the temperature interval 41±94 K. Using the model, minima in losses per unit carried current and unit length are determined as a function of temperature and magnetic ®eld. The currents at which these minima occur represent optimal operating currents of the conductor with respect to the losses. The model can be used to determine the losses in the inner part of the winding of e.g. a transformer coil. In the outer part of such a winding, large AC losses occur due to the magnetic ®eld component perpendicular to the tape face. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 84.70+p; 85.25 K Keywords: High-temperature superconductors; AC losses; Power applications
1. Introduction The development of high-temperature superconductors (HTSs) is intensive at present. The performance level of these conductors approaches the level which is required for the usage in electric power applications [1]. At this stage, attention should be paid to questions concerning how to utilise the HTSs in such applications. In power
E-mail address: [email protected] (N. Magnusson).
applications (except for power cables), the conductor is wound in some kind of coil con®guration in which it is exposed to a magnetic ®eld considerably larger than the self-®eld of the conductor. For tape-shaped HTSs, the magnetic ®eld in the major part of the coil is oriented parallel the tape face. To optimise the design of an electric power component in a speci®c application based on HTSs, the cost of both the AC losses and the conductor have to be considered. Any HTS will then have an optimal working point in the temperature and current. To ®nd this optimum, a model of the losses, Pac , is needed. Such a model can be of the form:
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 0 ) 0 1 5 3 8 - 0
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Pac Pac T ; B; I; f ;
N. Magnusson / Physica C 349 (2001) 225±234
1
where T is the temperature; B B t, the magnetic ®eld; I I t, the transport current and f, the frequency. When comparing the losses of dierent HTSs or between HTSs and conventional conductors, the losses per unit carried current and unit length (W/ A m) is a relevant ®gure of merit. Also, when comparing the losses at dierent operating temperatures, the cooling penalty factor (the number of W needed in a cooling device to remove one W dissipated at the operating temperature of the HTS) has to be taken into account. A conventional, oil-®lled, transformer has typically a current density of 2±4 A/mm2 and losses of 40±80 mW/A m in its copper windings. At 77 K, the cooling penalty factor is about 10, and hence an HTS operating at this temperature with losses of 4±8 mW/A m has equally high losses as the copper conductor. To limit the losses to only 10% of the corresponding losses in copper, losses of the order of 0.4±0.8 mW/A m can be tolerated. At 45 K, the cooling penalty factor is about 20, and hence the tolerable losses are of the order 0.2±0.4 mW/A m. At low temperatures, the cooling penalty factor is much higher (about 240 at 4.2 K), and although there exist good conductors operating in the liquid helium range, the high cooling penalty factor makes them uneconomical for the use in AC electric power applications. The losses of multi-®lamentary BSCCO/Ag HTS tapes have several sources. Flux-¯ow losses arise in the ®laments in the presence of a transport current. The irreversible motion of vortices in the ®laments causes hysteresis losses. Eddy current losses and coupling losses are associated with locally induced currents (due to external ®elds or self-®elds) ¯owing entirely or partly in the silver or silver alloy matrix. Models of the hysteresis losses in superconductors are generally based on the critical state model [2]. An important parameter in this model is the critical current, Ic . In HTSs, the transition to resistive behaviour does not occur sharply. Instead, Ic is usually determined by an electric ®eld, E, criterion (1 lV/cm). For a slab geometry, the hysteresis losses were calculated within the critical
state model ÿ assuming a sharp transition to the resistive behaviour ÿ for the case with an AC magnetic ®eld applied parallel to the slab surface [2] and for the case with such a ®eld in combination with an AC transport current (Ipeak 6 Ic ) [3]. Under self-®eld conditions, the hysteresis losses due to an AC transport current (Ipeak 6 Ic ) were calculated for strips and elliptical wires [4]. These models have shown to give results in good agreement with the experimental data of the hysteresis losses obtained from tape-formed HTSs in the external magnetic ®elds [5], in self-®eld [6,7] and in combinations of magnetic ®elds and transport currents [8]. However, to utilise HTSs in power applications, they should operate in the vicinity of the critical current. In this high-current regime, the losses due to ¯ux-¯ow become signi®cant. These losses have therefore to be incorporated in an AC loss model to be used in the design of electrotechnical components. In this work, a semi-empirical model of the losses, due to ¯ux-¯ow and to hysteresis, in an HTS tape is presented. The eddy current losses are neglected since they have shown to be small compared to the total losses at power frequencies [9]. The hysteresis losses include the losses caused by the coupling of ®laments (coupling losses). The losses are modelled as function of temperature, AC magnetic ®eld applied parallel to the tape face, AC transport current and frequency. The magnetic ®eld and the transport current are in phase with each other. The model is compared to experimental AC loss results obtained calorimetrically on a BSCCO/Ag tape in the for applications interesting temperature interval 41±94 K. Also, considering the losses, model predictions of an optimal working point in current are given for the conductor for dierent magnetic ®elds and as function of temperature. 2. Experimental The AC losses were measured with a calorimetric system which allows measurement of the AC losses as function of temperature, magnetic ®eld, transport current and frequency [10,11]. The sample was a non-twisted, multi-®lamentary, Bi-
N. Magnusson / Physica C 349 (2001) 225±234
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T and B dependencies. The hysteresis losses are based on the equations by Carr for transport currents in combination with ®elds parallel to an in®nite slab [3]. These equations are also extended to include currents larger than the critical current. To account for the magnetic ®eld component perpendicular to the tape face, due to the self-®eld, the equation derived by Norris for a superconducting strip [4] is used. 3.1. Flux-¯ow losses
Fig. 1. Sample with heater and thermometer.
2223 HTS tape produced by the Nordic Superconducting Technologies. The cross-section area of the tape was 0:21 3:4 mm2 . The sample was wound in a single turn coil with a diameter of 0.30 m, and hence the sample length was 0.94 m. The magnetic ®eld was applied parallel to the tape face and perpendicular to the current (Fig. 1). The magnetic ®eld and the transport current were sinusoidal and in phase with each other. To determine the AC losses, the temperature rise of the sample caused by the AC losses was measured and compared to the temperature rise caused by a reference heater. DC current-to-electric ®eld, Idc ±Edc , measurements were performed electrically with a four probe technique in the same setup. The distance between the voltage taps was 0.81 m.
3. Semi-empirical model The model, which was partly introduced in Ref. [12], gives the AC losses as function of T, B, I and f, where B and I are simple periodic functions in phase with each other. The model takes ¯ux-¯ow losses (modelled frequency independent) and hysteresis losses (modelled proportional to the frequency) into account. The ¯ux-¯ow losses are modelled with an Idc ±Edc power law dependence including ®tting parameters determined from measured Idc ±Edc characteristics to account for the
The AC ¯ux-¯ow losses, Pff , are determined from the equation, Z tper 1 IEdc dt; 2 Pff tper 0 where I is the transport current; Edc Edc T ; B; I, the electric ®eld and tper , the period time. At high transport currents, part of the current is carried by the silver alloy matrix. Pff is then expressed as Z tper ÿ 1 3 Ifil Edc IAg Edc dt; Pff tper 0 where Ifil T ; B; I is the current carried by the HTS ®laments and IAg T ; B; I, the current carried by the silver alloy matrix. IAg is determined by Edc IAg RAg ;
4
where RAg RAg T is the resistance per unit length of the silver alloy matrix. Edc is determined from Idc ±Edc measurements at currents, where IAg is negligible compared to Ifil . Edc is modelled by n Ifil : 5 Edc a Ic dc Here a is 1 lV/cm determined by the chosen standard critical current criterion. The DC critical current Ic dc Ic dc T ; B (Fig. 2) is determined by ( Ic self ; 6 Ic dc min 1 I0 1B=B ; 0 where Ic self Ic self T is the measured self-®eld critical current ®tted to experimental data at
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N. Magnusson / Physica C 349 (2001) 225±234
Fig. 2. Magnetic ®eld dependence of the critical current. The symbols represent measured values. The lines are obtained by ®tting the data points to Eq. (6).
Fig. 4. Magnetic ®eld dependence of the exponent n in Eq. (5). The symbols represent measured values. The lines are obtained by ®tting the data points to Eq. (7).
obtained at dierent temperatures. The exponent n n T ; B is obtained from experimental data of Idc ±Edc curves at dierent temperatures and magnetic ®elds. The magnetic ®eld dependence of n is similar to the magnetic ®eld dependence of Ic [15]. Therefore, the magnetic ®eld dependence is modelled as n n0
Fig. 3. The measured self-®eld critical current, Ic self , as function of temperature. The line is a linear ®t to the measured values of Ic self .
dierent temperatures (Fig. 3). I0 I0 T and B0 B0 T are, at dierent temperatures, ®tted to experimental data for B > 20 mT; at lower applied magnetic ®elds the self-®eld has a substantial in¯uence on the measured critical current [13,14]. The temperature dependence of I0 and B0 are accounted for by ®tting a linear ®fth degree polynomial, respectively, to the values of I0 and B0
1 ; 1 B=Bn
7
where n0 n0 T and Bn Bn T are ®tted to the experimental data of n B at dierent temperatures (Fig. 4). The temperature dependence of n0 and Bn are accounted for by ®tting linear polynomials, respectively, to the values of n0 and Bn obtained at dierent temperatures. 3.2. Hysteresis losses 3.2.1. Parallel ®eld The hysteresis losses due to the combination of a transport current (Ipeak < Ic ) and a magnetic ®eld applied parallel to the tape face are calculated from the equations by Carr [3] for a superconducting slab employing the critical state model. The losses per cycle and unit volume are expressed as
N. Magnusson / Physica C 349 (2001) 225±234
i B2p h b i3 b ÿ i3 ; for i 6 b 6 1; 3l0 " 2 B2p 1 ÿ i 2b 3 i2 ÿ 4 1 ÿ i3 12i2 3l0 bÿi # 3 1 ÿ i ÿ 8i2 ; for i < 1 6 b; 2 b ÿ i i B2p h 3 3 i b i ÿ b ; for b 6 i 6 1; 3l0 8
Qkvol
where Bp Bp T is the penetration ®eld; Ic , the critical current; b Bpeak =Bp and i Ipeak =Ic . These equations apply to simple periodic functions for transport current (Ipeak < Ic ) and magnetic ®eld in phase with each other. For Ipeak > Ic , the motion of vortices is assumed to be reversible ÿ and thus not contribute to the hysteresis losses ÿ for the part of the cycle when I > Ic . The hysteresis losses can then be expressed as B2p b 8 ; for 1 6 i 6 b; Qkvol 3l0 i 2 3 h i Bp 1 i b3 i ÿ b3 ; 3l0 i
229
currents, the losses due to the perpendicular component of the self-®eld dominate the losses. In the model, these losses, Q? , (losses per cycle and unit length) are accounted for by the equation by Norris for a strip superconductor [4]: Ic2 l0 1 ÿ i ln 1 ÿ i 1 i ln 1 i ÿ i2 ; p for i < 1: 11
Q?
Assuming again that the motion of vortices is reversible for the part of the cycle when I > Ic , Eq. (11) can be extended to Q?
Ic2 l0 2 ln 2 ÿ 1 p
for i P 1:
12
To obtain the losses per unit length, P? , Eqs. (11) and (12) are multiplied by the frequency, P? fQ? :
13
Eqs. (8) and (9) can, for a superconducting slab of in®nite size, be derived from the critical state model by an integration of the local current density and the local electric ®eld over the slab width and one cycle [16]. To obtain the losses per unit length, Pk , the Eqs. (8) and (9) are multiplied by the frequency and an eective area CA,
3.2.3. Fitting parameters The model of the hysteresis losses contains three ®tting parameters: Bp , C and Ic . Bp is ideally determined by 1=2l0 Jc w, where Jc is the critical current density and w, the width of the slab in the Bean model. However, to get a reasonable ®t to the experimental data for a multi-®lamentary tape, Bp need to be determined empirically from measured AC losses due to magnetic ®elds. CA(C 6 1) depends on e.g. the ®ll-factor and the ®lament architecture. In the model, Bp and C are determined by ®tting Eq. (10) to the experimental data at zero current. The temperature dependence of Bp is accounted for by a linear dependence with Ic self T [18]:
Pk fCAQkvol ;
Bp I c
for b 6 i; i P 1:
9
10
where C is a dimensionless ®tting parameter and A, the cross-section area of the tape. 3.2.2. Perpendicular ®eld The self-®eld of the tape (generated by the transport current) contains a component perpendicular to the tape face. The AC losses due to a perpendicular ®eld is much larger than the AC losses due to a parallel ®eld of equal magnitude [17]. At low magnetic ®elds and high transport
self :
14
For the tape in this investigation, with Ic Fig. 3, Bp is determined by Bp 0:28 10ÿ3 Ic
self ;
self
from 15
where the constant has the dimension V s/A m2 . C is generally independent of the temperature [18] and for the investigated tape, C was found to be 0.59. The model critical current, Ic , is generally a ®tting parameter which is strongly related to the
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DC critical current. To reduce the amount of needed AC loss measurements, Ic can be based on a DC critical current. It can then be discussed as to which DC critical current to be used. In the presence of an applied magnetic ®eld, the critical current decreases. In the AC case, the question is, should Ic be based on the DC critical current at peak ®eld, zero ®eld, self-®eld or some other ®eld? Ic in¯uences the model results most when Bpeak =Bp is small compared to Ipeak =Ic . At these relatively low ®elds, the decrease in critical current with applied ®eld is moderate. Ic is therefore determined by Ic self . 3.3. Total losses The total losses, Ptot , are expressed as a sum of the ¯ux-¯ow losses, the hysteresis losses due to the parallel magnetic ®eld and the hysteresis losses due to the perpendicular component of the self-®eld: Ptot Pff Pk P? :
16
Fig. 5 shows the magnetic ®eld dependence of the dierent loss components at T 77 K, Ipeak =Ic self 1.0 and f 50 Hz. At low ®elds the losses are dominated by P? . As the applied ®eld is increased, Pk becomes more important, and it dominates the losses between 26 and 100 mT (peak). At higher ®elds, Pff becomes dominant as the DC critical current decreases.
Fig. 5. Total losses (Ð) and the dierent loss components as function of magnetic ®eld. The diamonds represent measured values (Ppara Pk ; Pperp P? ). The transport current value corresponds to Ipeak =Ic self 1:0. Here, the losses P? are constant with the magnetic ®eld. This behaviour is due to the assumption of constant Ic with ®eld. In practice P? varies with B as Ic varies in Eqs. (11) and (12).
and 1.3) in Fig. 7. In Fig. 8, the AC losses are plotted versus the current for three dierent magnetic ®elds (Bpeak 0, 54 and 110 mT). The losses increase rapidly for currents larger than Ic dc B. Fig. 9 displays the temperature dependence of the losses at four dierent currents (Ipeak 0, 40, 60 and 80 A). As the temperature increases, Ic dc de-
4. Results 4.1. Comparison between model results and experimental data In Figs. 6±9 results obtained by the model are compared to experimental data. Fig. 6 shows the expected linear frequency dependence of the losses. For I 0, the loss curve has its intersection with the P-axis in the origin, while for Ipeak =Ic self 1.1, the intersection with the P-axis corresponds to the value of the losses present also in DC (Pff ). The magnetic ®eld dependence of the losses is shown for four dierent currents (Ipeak =Ic self 0, 0.7, 1.0
Fig. 6. Frequency dependence of the losses. The symbols represent measured values and the lines are given by the model.
N. Magnusson / Physica C 349 (2001) 225±234
Fig. 7. Magnetic ®eld dependence of the losses. The symbols represent measured values and the lines are given by the model.
Fig. 8. Transport current dependence of the losses. The symbols represent measured values and the lines are given by the model.
creases and for currents larger than Ic dc the losses increases rapidly. The losses for I 0 decrease, due to decreasing Bp (Ic dc ), with increasing temperature (Eq. (8) with b > 1). 4.2. Minimising the losses To ®nd an optimal working point with respect to AC losses for a conductor, the AC losses per unit carried current and unit length (W/A m) are
231
Fig. 9. Temperature dependence of the losses. The symbols represent measured values and the lines are given by the model.
Fig. 10. Model predictions of the losses per A m in the tape as function of current.
discussed. Fig. 10 shows these losses as function of current for four dierent magnetic ®elds at 77 K. The predictions are given for magnetic ®elds up to 200 mT although the ®tting parameters were obtained from measurements with ®elds up to only about 100 mT. The minima of the losses occur at dierent currents for dierent magnetic ®elds (Fig. 11). At low magnetic ®elds, these minima occur in a region where the hysteresis losses are dominant and the current yielding the minima increase with
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Fig. 11. The currents (peak values) at which the losses per A m (in Fig. 10) have a minimum as function of magnetic ®eld.
Fig. 12. Model predictions of the power needed in a cooling device to remove the losses in the HTS tape as function of current for a tape exposed to a magnetic ®eld of 100 mT (peak).
magnetic ®eld. For higher magnetic ®elds, the ¯ux¯ow losses become dominant due to the decrease in the critical current. The currents at which the minima occur are then limited by the critical current and decrease with magnetic ®eld. To compare the AC losses at dierent temperatures, the losses are multiplied by the cooling penalty factor to obtain the power needed at the ambient temperature to remove the power dissipated at the operating temperature. The cooling penalty factor, Fcp , can be approximated from the Carnot factor, Fcarnot Tamb ÿ T =T (Tamb is the ambient temperature), and an assumed eciency of the cooling device g 30%, Fcp
1 300 ÿ T ; 0:3 T
17
with Tamb 300 K. Fig. 12 shows the losses required in a cooling device as function of current for dierent temperatures when the tape is exposed to an applied ®eld of 100 mT. For a given applied magnetic ®eld, a tape has a minimum in the required cooling power for every temperature. In Fig. 13, these minima are given as function of temperature for four dierent magnetic ®elds. The currents at which the minima occur vary with temperature and are shown in Fig. 14.
Fig. 13. Model predictions of the minima in the power needed in a cooling device to remove the looses in the HTS tape as function of temperature. The currents at which these minima occur depend on the temperature (Fig. 14).
5. Discussion The semi-empirical model presented in this article can be a valuable tool in the design of electrotechnical devices based on HTSs; especially for the part of e.g. a transformer or a reactor winding exposed to an axial magnetic ®eld. To ®nd
N. Magnusson / Physica C 349 (2001) 225±234
Fig. 14. The currents (peak values) corresponding to the minima in the cooling losses plotted in Fig. 13.
Pac Pac T ; B; I; f of an HTS tape in a for power applications relevant region of T, B, I and f, the measurements required as input to the model are limited to: (1) DC I ÿ E measurements as function of T and B and (2) AC loss measurements as function of magnetic ®eld at one temperature and one frequency without any transport current present. The DC I±E measurement are (despite of the rather large number of data points needed) relatively easy to perform. Also, avoiding the need of measurements of the AC losses in combinations of magnetic ®elds and transport currents, the experimental eorts are reduced considerably by the use of the model compared to an experimental determination of the losses. In the model, losses due to the parallel ®eld (from the externally applied ®eld and from the self®eld) and the losses due to the perpendicular component of the self-®eld are considered separately. The losses due to the parallel ®eld are calculated from the situation in an in®nite slab and the losses due to the perpendicular component from the situation in a strip in self-®eld. The approach is similar to the one used by Ashworth and Suenaga, who (for parallel applied ®elds) separated the losses due to the externally applied magnetic ®eld (calculated from an in®nite slab in externally applied ®eld only) and the losses due to
233
the self-®eld (calculated from an ellipse in self®eld) [19]. Both these approaches seem to give results in agreement with experimental data. The model apply to a single HTS tape exposed to the self-®eld of the tape and an external magnetic ®eld applied parallel to the tape face. In the coil winding of an electric power device, the situation is somewhat dierent. The perpendicular component of the self-®eld in one turn is reduced by the ®eld from the other turns. This reduction may be important for the losses at low applied magnetic ®elds and a compensation of the loss component P? should be done in that region. Also, at the coil ends the external magnetic ®eld contains a component perpendicular to the tape face. To calculate the losses for this region of the winding, a model considering also the ®eld angle is needed. In Section 4, minima in the losses are shown as function of current and temperature for dierent magnetic ®elds. For high temperatures and low magnetic ®elds, these minima occur at relatively low currents. In the design of an electric power device also the cost of the wire should be considered. The optimal operating current will then, especially in the low ®eld region, increase. Furthermore, the low critical current at high temperatures results in an optimal operating temperature of the investigated conductor possibly in the temperature range 45±80 K for magnetic ®elds of the order of 100 mT. 6. Conclusions The semi-empirical model presented describes the AC losses as function of temperature, magnetic ®eld applied in parallel to the tape face, transport current and frequency. In the model the hysteresis losses were based on the critical state model and the ¯ux-¯ow losses on a power law dependence. Fitting parameters were obtained from DC I±E measurements at dierent temperatures and magnetic ®elds and from AC loss measurements as function of magnetic ®eld without transport current at one temperature and one frequency. The results obtained with the model showed good agreement with experimental data obtained calorimetrically on a Bi-2223 HTS tape in the
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N. Magnusson / Physica C 349 (2001) 225±234
intervals T 41±94 K, Bpeak < 125 mT, Ipeak =Ic dc < 1:2 and f < 125 Hz. With the model, an optimal operating current with respect to the losses can be predicted for dierent operating temperatures. At 77 K, 125 mT (peak) and 50 Hz, the minimum in losses was 2 mW/A m for the investigated tape. This level of losses is about a factor of ®ve higher than what can be tolerated in an application. However, this tape was not primarily designed for AC use. It can be expected that a conductor optimised for AC use has good chances to meet the loss criterion. Acknowledgements This work was supported by the Swedish National Energy Administration. The author would like to acknowledge the Nordic Superconductor Technologies for providing the HTS tape and Prof. Sven H ornfeldt, ABB Corporate Research, for valuable discussions regarding both the experimental work and the modelling. References [1] A.P. Malozemo, W. Carter, S. Fleshler, L. Fritzemeier, Q. Li, L. Masur, P. Miles, D. Parker, R. Parella, E. Podtburg, G.N. Riley, M. Rupich, J. Scudiere, W. Zhang, IEEE. Trans. Appl. Supercond. 9 (1999) 2469. [2] C.P. Bean, Rev. Mod. Phys. 36 (1964) 31. [3] W.J. Carr, IEEE Trans. Magn. 15 (1979) 240.
[4] W.T. Norris, J. Phys. D 3 (1970) 489. [5] J. Orehotsky, K.M. Reilly, M. Suenaga, T. Hikata, M. Ueyama, K. Sato, Appl. Phys. Lett. 60 (1992) 252. [6] S. Fleshler, L.T. Cronis, G.E. Conway, A.P. Malozemo, T. Pe, J. McDonald, J.R. Clem, G. Vellago, P. Metra, Appl. Phys. Lett. 67 (1995) 3189. [7] A.E. Mahdi, T. Huges, C. Beduz, Y. Yang, R.L. Stoll, J.K. Sykulski, P. Haldar, R.S. Sokolowski, A. Power, IEEE. Trans. Appl. Supercond. 7 (1997) 1658. [8] J.J. Rabbers, D.C. van der Laan, B. Ten Haken, H.H.J. Ten Kate, IEEE. Trans. Appl. Supercond. 9 (1999) 1185. [9] S. Stavrev, B. Dutoit, Physica C 310 (1998) 86. [10] N. Magnusson, S. H ornfeldt, Rev. Sci. Instrum. 69 (1998) 3320. [11] N. Magnusson, N. Sch onborg, A. Wolfbrandt, S. H ornfeldt, Improved experimental set-up for AC loss measurements on HTSs carrying transport currents in applied magnetic ®elds at variable temperatures, ICMC2000 11±15 June 2000, Rio de Janeiro, Brazil, Physica C (2001). [12] N. Magnusson, S. H ornfeldt, IEEE. Trans. Appl. Supercond. 9 (1999) 785. [13] S. Sprea®co, L. Gheradi, S. Fleshler, D. Tatelbaum, J. Leone, D. Yu, G. Snitchler, IEEE. Trans. Appl. Supercond. 9 (1999) 2159. [14] N. Magnusson, G. Mugala, N. Sch onborg, S. H ornfeldt, J. Low Temp. Phys. 117 (1999) 1519. [15] B. Dutoit, M. Sj ostr om, S. Stavrev, IEEE. Trans. Appl. Supercond. 9 (1999) 809. [16] N. Magnusson, AC Losses in High-temperature Superconducting Tapes: Calorimetric Measurement System and Semi-empirical Modelling, PhD Thesis, Royal Institute of Technology, 2000. [17] M.P. Oomen, J. Rieger, M. Leghissa, H.H.J. Ten Kate, Appl. Phys. Lett. 70 (1997) 3038. [18] N. Magnusson, N. Sch onborg, S. H ornfeldt, Inst. Phys. Conf. Ser. 167 (2000) 891. [19] S.P. Ashworth, M. Suenaga, Physica C 329 (2000) 149.
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Semi-empirical model of the losses in HTS tapes carrying AC currents in AC magnetic ®elds applied parallel to the tape face N. Magnusson Electric Power Engineering, Royal Institute of Technology, Teknikringen 33, SE-100 44 Stockholm, Sweden Received 27 April 2000; received in revised form 10 July 2000; accepted 11 July 2000
Abstract The AC losses are an important parameter of high-temperature superconductors (HTSs) to be used in electric power applications. To obtain an optimised design of such applications, models of the AC losses are needed. This article describes a semi-empirical model of the AC losses in HTS tapes as a function of temperature, magnetic ®eld, transport current and frequency. The magnetic ®eld (applied in parallel to the tape face) and the transport current are in phase with each other. The model considers ¯ux-¯ow losses (based on the power law dependence) and hysteresis losses (based on the critical state model). Results obtained by the model are compared to experimental data obtained calorimetrically on a BSCCO/Ag tape in the temperature interval 41±94 K. Using the model, minima in losses per unit carried current and unit length are determined as a function of temperature and magnetic ®eld. The currents at which these minima occur represent optimal operating currents of the conductor with respect to the losses. The model can be used to determine the losses in the inner part of the winding of e.g. a transformer coil. In the outer part of such a winding, large AC losses occur due to the magnetic ®eld component perpendicular to the tape face. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 84.70+p; 85.25 K Keywords: High-temperature superconductors; AC losses; Power applications
1. Introduction The development of high-temperature superconductors (HTSs) is intensive at present. The performance level of these conductors approaches the level which is required for the usage in electric power applications [1]. At this stage, attention should be paid to questions concerning how to utilise the HTSs in such applications. In power
E-mail address: [email protected] (N. Magnusson).
applications (except for power cables), the conductor is wound in some kind of coil con®guration in which it is exposed to a magnetic ®eld considerably larger than the self-®eld of the conductor. For tape-shaped HTSs, the magnetic ®eld in the major part of the coil is oriented parallel the tape face. To optimise the design of an electric power component in a speci®c application based on HTSs, the cost of both the AC losses and the conductor have to be considered. Any HTS will then have an optimal working point in the temperature and current. To ®nd this optimum, a model of the losses, Pac , is needed. Such a model can be of the form:
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 0 ) 0 1 5 3 8 - 0
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Pac Pac T ; B; I; f ;
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1
where T is the temperature; B B t, the magnetic ®eld; I I t, the transport current and f, the frequency. When comparing the losses of dierent HTSs or between HTSs and conventional conductors, the losses per unit carried current and unit length (W/ A m) is a relevant ®gure of merit. Also, when comparing the losses at dierent operating temperatures, the cooling penalty factor (the number of W needed in a cooling device to remove one W dissipated at the operating temperature of the HTS) has to be taken into account. A conventional, oil-®lled, transformer has typically a current density of 2±4 A/mm2 and losses of 40±80 mW/A m in its copper windings. At 77 K, the cooling penalty factor is about 10, and hence an HTS operating at this temperature with losses of 4±8 mW/A m has equally high losses as the copper conductor. To limit the losses to only 10% of the corresponding losses in copper, losses of the order of 0.4±0.8 mW/A m can be tolerated. At 45 K, the cooling penalty factor is about 20, and hence the tolerable losses are of the order 0.2±0.4 mW/A m. At low temperatures, the cooling penalty factor is much higher (about 240 at 4.2 K), and although there exist good conductors operating in the liquid helium range, the high cooling penalty factor makes them uneconomical for the use in AC electric power applications. The losses of multi-®lamentary BSCCO/Ag HTS tapes have several sources. Flux-¯ow losses arise in the ®laments in the presence of a transport current. The irreversible motion of vortices in the ®laments causes hysteresis losses. Eddy current losses and coupling losses are associated with locally induced currents (due to external ®elds or self-®elds) ¯owing entirely or partly in the silver or silver alloy matrix. Models of the hysteresis losses in superconductors are generally based on the critical state model [2]. An important parameter in this model is the critical current, Ic . In HTSs, the transition to resistive behaviour does not occur sharply. Instead, Ic is usually determined by an electric ®eld, E, criterion (1 lV/cm). For a slab geometry, the hysteresis losses were calculated within the critical
state model ÿ assuming a sharp transition to the resistive behaviour ÿ for the case with an AC magnetic ®eld applied parallel to the slab surface [2] and for the case with such a ®eld in combination with an AC transport current (Ipeak 6 Ic ) [3]. Under self-®eld conditions, the hysteresis losses due to an AC transport current (Ipeak 6 Ic ) were calculated for strips and elliptical wires [4]. These models have shown to give results in good agreement with the experimental data of the hysteresis losses obtained from tape-formed HTSs in the external magnetic ®elds [5], in self-®eld [6,7] and in combinations of magnetic ®elds and transport currents [8]. However, to utilise HTSs in power applications, they should operate in the vicinity of the critical current. In this high-current regime, the losses due to ¯ux-¯ow become signi®cant. These losses have therefore to be incorporated in an AC loss model to be used in the design of electrotechnical components. In this work, a semi-empirical model of the losses, due to ¯ux-¯ow and to hysteresis, in an HTS tape is presented. The eddy current losses are neglected since they have shown to be small compared to the total losses at power frequencies [9]. The hysteresis losses include the losses caused by the coupling of ®laments (coupling losses). The losses are modelled as function of temperature, AC magnetic ®eld applied parallel to the tape face, AC transport current and frequency. The magnetic ®eld and the transport current are in phase with each other. The model is compared to experimental AC loss results obtained calorimetrically on a BSCCO/Ag tape in the for applications interesting temperature interval 41±94 K. Also, considering the losses, model predictions of an optimal working point in current are given for the conductor for dierent magnetic ®elds and as function of temperature. 2. Experimental The AC losses were measured with a calorimetric system which allows measurement of the AC losses as function of temperature, magnetic ®eld, transport current and frequency [10,11]. The sample was a non-twisted, multi-®lamentary, Bi-
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227
T and B dependencies. The hysteresis losses are based on the equations by Carr for transport currents in combination with ®elds parallel to an in®nite slab [3]. These equations are also extended to include currents larger than the critical current. To account for the magnetic ®eld component perpendicular to the tape face, due to the self-®eld, the equation derived by Norris for a superconducting strip [4] is used. 3.1. Flux-¯ow losses
Fig. 1. Sample with heater and thermometer.
2223 HTS tape produced by the Nordic Superconducting Technologies. The cross-section area of the tape was 0:21 3:4 mm2 . The sample was wound in a single turn coil with a diameter of 0.30 m, and hence the sample length was 0.94 m. The magnetic ®eld was applied parallel to the tape face and perpendicular to the current (Fig. 1). The magnetic ®eld and the transport current were sinusoidal and in phase with each other. To determine the AC losses, the temperature rise of the sample caused by the AC losses was measured and compared to the temperature rise caused by a reference heater. DC current-to-electric ®eld, Idc ±Edc , measurements were performed electrically with a four probe technique in the same setup. The distance between the voltage taps was 0.81 m.
3. Semi-empirical model The model, which was partly introduced in Ref. [12], gives the AC losses as function of T, B, I and f, where B and I are simple periodic functions in phase with each other. The model takes ¯ux-¯ow losses (modelled frequency independent) and hysteresis losses (modelled proportional to the frequency) into account. The ¯ux-¯ow losses are modelled with an Idc ±Edc power law dependence including ®tting parameters determined from measured Idc ±Edc characteristics to account for the
The AC ¯ux-¯ow losses, Pff , are determined from the equation, Z tper 1 IEdc dt; 2 Pff tper 0 where I is the transport current; Edc Edc T ; B; I, the electric ®eld and tper , the period time. At high transport currents, part of the current is carried by the silver alloy matrix. Pff is then expressed as Z tper ÿ 1 3 Ifil Edc IAg Edc dt; Pff tper 0 where Ifil T ; B; I is the current carried by the HTS ®laments and IAg T ; B; I, the current carried by the silver alloy matrix. IAg is determined by Edc IAg RAg ;
4
where RAg RAg T is the resistance per unit length of the silver alloy matrix. Edc is determined from Idc ±Edc measurements at currents, where IAg is negligible compared to Ifil . Edc is modelled by n Ifil : 5 Edc a Ic dc Here a is 1 lV/cm determined by the chosen standard critical current criterion. The DC critical current Ic dc Ic dc T ; B (Fig. 2) is determined by ( Ic self ; 6 Ic dc min 1 I0 1B=B ; 0 where Ic self Ic self T is the measured self-®eld critical current ®tted to experimental data at
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Fig. 2. Magnetic ®eld dependence of the critical current. The symbols represent measured values. The lines are obtained by ®tting the data points to Eq. (6).
Fig. 4. Magnetic ®eld dependence of the exponent n in Eq. (5). The symbols represent measured values. The lines are obtained by ®tting the data points to Eq. (7).
obtained at dierent temperatures. The exponent n n T ; B is obtained from experimental data of Idc ±Edc curves at dierent temperatures and magnetic ®elds. The magnetic ®eld dependence of n is similar to the magnetic ®eld dependence of Ic [15]. Therefore, the magnetic ®eld dependence is modelled as n n0
Fig. 3. The measured self-®eld critical current, Ic self , as function of temperature. The line is a linear ®t to the measured values of Ic self .
dierent temperatures (Fig. 3). I0 I0 T and B0 B0 T are, at dierent temperatures, ®tted to experimental data for B > 20 mT; at lower applied magnetic ®elds the self-®eld has a substantial in¯uence on the measured critical current [13,14]. The temperature dependence of I0 and B0 are accounted for by ®tting a linear ®fth degree polynomial, respectively, to the values of I0 and B0
1 ; 1 B=Bn
7
where n0 n0 T and Bn Bn T are ®tted to the experimental data of n B at dierent temperatures (Fig. 4). The temperature dependence of n0 and Bn are accounted for by ®tting linear polynomials, respectively, to the values of n0 and Bn obtained at dierent temperatures. 3.2. Hysteresis losses 3.2.1. Parallel ®eld The hysteresis losses due to the combination of a transport current (Ipeak < Ic ) and a magnetic ®eld applied parallel to the tape face are calculated from the equations by Carr [3] for a superconducting slab employing the critical state model. The losses per cycle and unit volume are expressed as
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i B2p h b i3 b ÿ i3 ; for i 6 b 6 1; 3l0 " 2 B2p 1 ÿ i 2b 3 i2 ÿ 4 1 ÿ i3 12i2 3l0 bÿi # 3 1 ÿ i ÿ 8i2 ; for i < 1 6 b; 2 b ÿ i i B2p h 3 3 i b i ÿ b ; for b 6 i 6 1; 3l0 8
Qkvol
where Bp Bp T is the penetration ®eld; Ic , the critical current; b Bpeak =Bp and i Ipeak =Ic . These equations apply to simple periodic functions for transport current (Ipeak < Ic ) and magnetic ®eld in phase with each other. For Ipeak > Ic , the motion of vortices is assumed to be reversible ÿ and thus not contribute to the hysteresis losses ÿ for the part of the cycle when I > Ic . The hysteresis losses can then be expressed as B2p b 8 ; for 1 6 i 6 b; Qkvol 3l0 i 2 3 h i Bp 1 i b3 i ÿ b3 ; 3l0 i
229
currents, the losses due to the perpendicular component of the self-®eld dominate the losses. In the model, these losses, Q? , (losses per cycle and unit length) are accounted for by the equation by Norris for a strip superconductor [4]: Ic2 l0 1 ÿ i ln 1 ÿ i 1 i ln 1 i ÿ i2 ; p for i < 1: 11
Q?
Assuming again that the motion of vortices is reversible for the part of the cycle when I > Ic , Eq. (11) can be extended to Q?
Ic2 l0 2 ln 2 ÿ 1 p
for i P 1:
12
To obtain the losses per unit length, P? , Eqs. (11) and (12) are multiplied by the frequency, P? fQ? :
13
Eqs. (8) and (9) can, for a superconducting slab of in®nite size, be derived from the critical state model by an integration of the local current density and the local electric ®eld over the slab width and one cycle [16]. To obtain the losses per unit length, Pk , the Eqs. (8) and (9) are multiplied by the frequency and an eective area CA,
3.2.3. Fitting parameters The model of the hysteresis losses contains three ®tting parameters: Bp , C and Ic . Bp is ideally determined by 1=2l0 Jc w, where Jc is the critical current density and w, the width of the slab in the Bean model. However, to get a reasonable ®t to the experimental data for a multi-®lamentary tape, Bp need to be determined empirically from measured AC losses due to magnetic ®elds. CA(C 6 1) depends on e.g. the ®ll-factor and the ®lament architecture. In the model, Bp and C are determined by ®tting Eq. (10) to the experimental data at zero current. The temperature dependence of Bp is accounted for by a linear dependence with Ic self T [18]:
Pk fCAQkvol ;
Bp I c
for b 6 i; i P 1:
9
10
where C is a dimensionless ®tting parameter and A, the cross-section area of the tape. 3.2.2. Perpendicular ®eld The self-®eld of the tape (generated by the transport current) contains a component perpendicular to the tape face. The AC losses due to a perpendicular ®eld is much larger than the AC losses due to a parallel ®eld of equal magnitude [17]. At low magnetic ®elds and high transport
self :
14
For the tape in this investigation, with Ic Fig. 3, Bp is determined by Bp 0:28 10ÿ3 Ic
self ;
self
from 15
where the constant has the dimension V s/A m2 . C is generally independent of the temperature [18] and for the investigated tape, C was found to be 0.59. The model critical current, Ic , is generally a ®tting parameter which is strongly related to the
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DC critical current. To reduce the amount of needed AC loss measurements, Ic can be based on a DC critical current. It can then be discussed as to which DC critical current to be used. In the presence of an applied magnetic ®eld, the critical current decreases. In the AC case, the question is, should Ic be based on the DC critical current at peak ®eld, zero ®eld, self-®eld or some other ®eld? Ic in¯uences the model results most when Bpeak =Bp is small compared to Ipeak =Ic . At these relatively low ®elds, the decrease in critical current with applied ®eld is moderate. Ic is therefore determined by Ic self . 3.3. Total losses The total losses, Ptot , are expressed as a sum of the ¯ux-¯ow losses, the hysteresis losses due to the parallel magnetic ®eld and the hysteresis losses due to the perpendicular component of the self-®eld: Ptot Pff Pk P? :
16
Fig. 5 shows the magnetic ®eld dependence of the dierent loss components at T 77 K, Ipeak =Ic self 1.0 and f 50 Hz. At low ®elds the losses are dominated by P? . As the applied ®eld is increased, Pk becomes more important, and it dominates the losses between 26 and 100 mT (peak). At higher ®elds, Pff becomes dominant as the DC critical current decreases.
Fig. 5. Total losses (Ð) and the dierent loss components as function of magnetic ®eld. The diamonds represent measured values (Ppara Pk ; Pperp P? ). The transport current value corresponds to Ipeak =Ic self 1:0. Here, the losses P? are constant with the magnetic ®eld. This behaviour is due to the assumption of constant Ic with ®eld. In practice P? varies with B as Ic varies in Eqs. (11) and (12).
and 1.3) in Fig. 7. In Fig. 8, the AC losses are plotted versus the current for three dierent magnetic ®elds (Bpeak 0, 54 and 110 mT). The losses increase rapidly for currents larger than Ic dc B. Fig. 9 displays the temperature dependence of the losses at four dierent currents (Ipeak 0, 40, 60 and 80 A). As the temperature increases, Ic dc de-
4. Results 4.1. Comparison between model results and experimental data In Figs. 6±9 results obtained by the model are compared to experimental data. Fig. 6 shows the expected linear frequency dependence of the losses. For I 0, the loss curve has its intersection with the P-axis in the origin, while for Ipeak =Ic self 1.1, the intersection with the P-axis corresponds to the value of the losses present also in DC (Pff ). The magnetic ®eld dependence of the losses is shown for four dierent currents (Ipeak =Ic self 0, 0.7, 1.0
Fig. 6. Frequency dependence of the losses. The symbols represent measured values and the lines are given by the model.
N. Magnusson / Physica C 349 (2001) 225±234
Fig. 7. Magnetic ®eld dependence of the losses. The symbols represent measured values and the lines are given by the model.
Fig. 8. Transport current dependence of the losses. The symbols represent measured values and the lines are given by the model.
creases and for currents larger than Ic dc the losses increases rapidly. The losses for I 0 decrease, due to decreasing Bp (Ic dc ), with increasing temperature (Eq. (8) with b > 1). 4.2. Minimising the losses To ®nd an optimal working point with respect to AC losses for a conductor, the AC losses per unit carried current and unit length (W/A m) are
231
Fig. 9. Temperature dependence of the losses. The symbols represent measured values and the lines are given by the model.
Fig. 10. Model predictions of the losses per A m in the tape as function of current.
discussed. Fig. 10 shows these losses as function of current for four dierent magnetic ®elds at 77 K. The predictions are given for magnetic ®elds up to 200 mT although the ®tting parameters were obtained from measurements with ®elds up to only about 100 mT. The minima of the losses occur at dierent currents for dierent magnetic ®elds (Fig. 11). At low magnetic ®elds, these minima occur in a region where the hysteresis losses are dominant and the current yielding the minima increase with
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Fig. 11. The currents (peak values) at which the losses per A m (in Fig. 10) have a minimum as function of magnetic ®eld.
Fig. 12. Model predictions of the power needed in a cooling device to remove the losses in the HTS tape as function of current for a tape exposed to a magnetic ®eld of 100 mT (peak).
magnetic ®eld. For higher magnetic ®elds, the ¯ux¯ow losses become dominant due to the decrease in the critical current. The currents at which the minima occur are then limited by the critical current and decrease with magnetic ®eld. To compare the AC losses at dierent temperatures, the losses are multiplied by the cooling penalty factor to obtain the power needed at the ambient temperature to remove the power dissipated at the operating temperature. The cooling penalty factor, Fcp , can be approximated from the Carnot factor, Fcarnot Tamb ÿ T =T (Tamb is the ambient temperature), and an assumed eciency of the cooling device g 30%, Fcp
1 300 ÿ T ; 0:3 T
17
with Tamb 300 K. Fig. 12 shows the losses required in a cooling device as function of current for dierent temperatures when the tape is exposed to an applied ®eld of 100 mT. For a given applied magnetic ®eld, a tape has a minimum in the required cooling power for every temperature. In Fig. 13, these minima are given as function of temperature for four dierent magnetic ®elds. The currents at which the minima occur vary with temperature and are shown in Fig. 14.
Fig. 13. Model predictions of the minima in the power needed in a cooling device to remove the looses in the HTS tape as function of temperature. The currents at which these minima occur depend on the temperature (Fig. 14).
5. Discussion The semi-empirical model presented in this article can be a valuable tool in the design of electrotechnical devices based on HTSs; especially for the part of e.g. a transformer or a reactor winding exposed to an axial magnetic ®eld. To ®nd
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Fig. 14. The currents (peak values) corresponding to the minima in the cooling losses plotted in Fig. 13.
Pac Pac T ; B; I; f of an HTS tape in a for power applications relevant region of T, B, I and f, the measurements required as input to the model are limited to: (1) DC I ÿ E measurements as function of T and B and (2) AC loss measurements as function of magnetic ®eld at one temperature and one frequency without any transport current present. The DC I±E measurement are (despite of the rather large number of data points needed) relatively easy to perform. Also, avoiding the need of measurements of the AC losses in combinations of magnetic ®elds and transport currents, the experimental eorts are reduced considerably by the use of the model compared to an experimental determination of the losses. In the model, losses due to the parallel ®eld (from the externally applied ®eld and from the self®eld) and the losses due to the perpendicular component of the self-®eld are considered separately. The losses due to the parallel ®eld are calculated from the situation in an in®nite slab and the losses due to the perpendicular component from the situation in a strip in self-®eld. The approach is similar to the one used by Ashworth and Suenaga, who (for parallel applied ®elds) separated the losses due to the externally applied magnetic ®eld (calculated from an in®nite slab in externally applied ®eld only) and the losses due to
233
the self-®eld (calculated from an ellipse in self®eld) [19]. Both these approaches seem to give results in agreement with experimental data. The model apply to a single HTS tape exposed to the self-®eld of the tape and an external magnetic ®eld applied parallel to the tape face. In the coil winding of an electric power device, the situation is somewhat dierent. The perpendicular component of the self-®eld in one turn is reduced by the ®eld from the other turns. This reduction may be important for the losses at low applied magnetic ®elds and a compensation of the loss component P? should be done in that region. Also, at the coil ends the external magnetic ®eld contains a component perpendicular to the tape face. To calculate the losses for this region of the winding, a model considering also the ®eld angle is needed. In Section 4, minima in the losses are shown as function of current and temperature for dierent magnetic ®elds. For high temperatures and low magnetic ®elds, these minima occur at relatively low currents. In the design of an electric power device also the cost of the wire should be considered. The optimal operating current will then, especially in the low ®eld region, increase. Furthermore, the low critical current at high temperatures results in an optimal operating temperature of the investigated conductor possibly in the temperature range 45±80 K for magnetic ®elds of the order of 100 mT. 6. Conclusions The semi-empirical model presented describes the AC losses as function of temperature, magnetic ®eld applied in parallel to the tape face, transport current and frequency. In the model the hysteresis losses were based on the critical state model and the ¯ux-¯ow losses on a power law dependence. Fitting parameters were obtained from DC I±E measurements at dierent temperatures and magnetic ®elds and from AC loss measurements as function of magnetic ®eld without transport current at one temperature and one frequency. The results obtained with the model showed good agreement with experimental data obtained calorimetrically on a Bi-2223 HTS tape in the
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intervals T 41±94 K, Bpeak < 125 mT, Ipeak =Ic dc < 1:2 and f < 125 Hz. With the model, an optimal operating current with respect to the losses can be predicted for dierent operating temperatures. At 77 K, 125 mT (peak) and 50 Hz, the minimum in losses was 2 mW/A m for the investigated tape. This level of losses is about a factor of ®ve higher than what can be tolerated in an application. However, this tape was not primarily designed for AC use. It can be expected that a conductor optimised for AC use has good chances to meet the loss criterion. Acknowledgements This work was supported by the Swedish National Energy Administration. The author would like to acknowledge the Nordic Superconductor Technologies for providing the HTS tape and Prof. Sven H ornfeldt, ABB Corporate Research, for valuable discussions regarding both the experimental work and the modelling. References [1] A.P. Malozemo, W. Carter, S. Fleshler, L. Fritzemeier, Q. Li, L. Masur, P. Miles, D. Parker, R. Parella, E. Podtburg, G.N. Riley, M. Rupich, J. Scudiere, W. Zhang, IEEE. Trans. Appl. Supercond. 9 (1999) 2469. [2] C.P. Bean, Rev. Mod. Phys. 36 (1964) 31. [3] W.J. Carr, IEEE Trans. Magn. 15 (1979) 240.
[4] W.T. Norris, J. Phys. D 3 (1970) 489. [5] J. Orehotsky, K.M. Reilly, M. Suenaga, T. Hikata, M. Ueyama, K. Sato, Appl. Phys. Lett. 60 (1992) 252. [6] S. Fleshler, L.T. Cronis, G.E. Conway, A.P. Malozemo, T. Pe, J. McDonald, J.R. Clem, G. Vellago, P. Metra, Appl. Phys. Lett. 67 (1995) 3189. [7] A.E. Mahdi, T. Huges, C. Beduz, Y. Yang, R.L. Stoll, J.K. Sykulski, P. Haldar, R.S. Sokolowski, A. Power, IEEE. Trans. Appl. Supercond. 7 (1997) 1658. [8] J.J. Rabbers, D.C. van der Laan, B. Ten Haken, H.H.J. Ten Kate, IEEE. Trans. Appl. Supercond. 9 (1999) 1185. [9] S. Stavrev, B. Dutoit, Physica C 310 (1998) 86. [10] N. Magnusson, S. H ornfeldt, Rev. Sci. Instrum. 69 (1998) 3320. [11] N. Magnusson, N. Sch onborg, A. Wolfbrandt, S. H ornfeldt, Improved experimental set-up for AC loss measurements on HTSs carrying transport currents in applied magnetic ®elds at variable temperatures, ICMC2000 11±15 June 2000, Rio de Janeiro, Brazil, Physica C (2001). [12] N. Magnusson, S. H ornfeldt, IEEE. Trans. Appl. Supercond. 9 (1999) 785. [13] S. Sprea®co, L. Gheradi, S. Fleshler, D. Tatelbaum, J. Leone, D. Yu, G. Snitchler, IEEE. Trans. Appl. Supercond. 9 (1999) 2159. [14] N. Magnusson, G. Mugala, N. Sch onborg, S. H ornfeldt, J. Low Temp. Phys. 117 (1999) 1519. [15] B. Dutoit, M. Sj ostr om, S. Stavrev, IEEE. Trans. Appl. Supercond. 9 (1999) 809. [16] N. Magnusson, AC Losses in High-temperature Superconducting Tapes: Calorimetric Measurement System and Semi-empirical Modelling, PhD Thesis, Royal Institute of Technology, 2000. [17] M.P. Oomen, J. Rieger, M. Leghissa, H.H.J. Ten Kate, Appl. Phys. Lett. 70 (1997) 3038. [18] N. Magnusson, N. Sch onborg, S. H ornfeldt, Inst. Phys. Conf. Ser. 167 (2000) 891. [19] S.P. Ashworth, M. Suenaga, Physica C 329 (2000) 149.