- Email: [email protected]
or exposed to DC or AC external magnetic field
Physica C 310 Ž1998. 30–35
Finite element analysis of AC loss in non-twisted Bi-2223 tape carrying AC transport current andror exposed to DC or AC external magnetic field Naoyuki Amemiya a
a,)
, Kengo Miyamoto a , Shun-ichi Murasawa a , Hideki Mukai b, Kazuya Ohmatsu b
DiÕision of Electrical and Computer Engineering, Yokohama National UniÕersity, 79-5 Tokiwadai, Hodogaya, Yokohama, 240-8501, Japan b Sumitomo Electric Industries, 1-1-3 Shimaya, Konohana, Osaka, 554-0024, Japan
Abstract AC losses in Bi-2223 superconducting tapes carrying AC transport current andror exposed to DC or AC magnetic field are calculated with a numerical model based on the finite element method. Superconducting property is given by the E–J characteristic represented by a power law using equivalent conductivity. First, transport loss and magnetization loss are calculated numerically and compared with measured values. The calculated losses almost agree with the measured losses. Frequency dependencies of calculated and measured transport losses are compared with each other. Next, the influence of DC external magnetic field on the transport loss is studied. DC external magnetic field reduces n that is an exponent in the power law connecting resistivity and current density. The numerically calculated transport loss increases with increasing DC magnetic field. Finally, the total loss of superconducting tape carrying AC transport current in AC magnetic field is calculated. In the perpendicular magnetic field, the calculated total loss is lager than the sum of the transport loss and the magnetization loss, while they almost agree with each other in the parallel magnetic field. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Numerical analysis; Transport loss; Finite element method; Flux penetration
1. Introduction In practical electrical power apparatuses used at commercial frequencies, high Tc superconductors carry AC transport current in AC external magnetic field. Therefore, quantitative evaluation of their AC loss is one of the key issues in the research and
)
Corresponding author. Tel.: q81-45-339-4119; Fax: q81-45338-1157; E-mail: [email protected]
development of superconductors and their applications. Although experimental techniques for the transport loss measurement and the magnetization loss measurement are almost established w1,2x, no reliable and accurate technique to measure the total loss of superconductors carrying AC transport current in AC external magnetic field has been developed yet. The authors have been developing a series of numerical codes for the electromagnetic analysis of superconductors based on the finite element method.
0921-4534r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 4 2 8 - 6
N. Amemiya et al.r Physica C 310 (1998) 30–35
Their purpose is numerical estimations of AC losses substituting for AC loss measurements. The main objective of this paper is to describe the FEM analysis of AC losses in a non-twisted silversheathed Bi-2223 superconducting tape carrying AC transport current andror exposed to DC or AC magnetic field. Numerically calculated AC losses are compared with experimental results. In the numerical analysis, electric field Ž E . –current density Ž j . characteristics of superconductor are given with a power law. The magnetic field dependence of E–J characteristics is also taken into account. First, the transport loss is calculated, and is compared with the measured value. Its frequency dependency is also discussed. Next, the calculated and measured magnetization losses are compared with each other. Then, the influence of DC magnetic field on the transport loss is presented. Finally, the total loss in Bi-2223 tape carrying AC transport current in AC external magnetic field is calculated, and compared with the sum of the transport loss and magnetization loss.
2. Numerical method The numerical model is formulated with current vector potential T and magnetic scalar potential V by the finite element method. They are given as, J'==T ,
Ž 1. Ž 2.
H s H0 q T y =V
where H0 is the external magnetic field. Superconducting property is given by a power law, E s Ec
J
n
ž /
Ž 3.
Jc
where Jc is a critical current density and Ec s 1 = 10y4 Vrm. Then, an equivalent conductivity of superconductor ssc is derived as,
ssc s
J s E
Jcn Ec
Ž==T .
lyn
.
Ž 4.
Elements in the filamentary region are treated as a mixture of superconductor and silver matrix whose equivalent conductivity is given as,
s s lfr ssc q Ž 1 y lfr . sAg
Ž 5.
31
where l fr and sAg are the fraction of superconductor in the filamentary region and the conductivity of the silver matrix, respectively. Then, Faraday’s law and modified magnetic Gauss’s law are written as,
==
ž
1
s
E
/
= = T s ym
Et
Ž H0 q T y =V . ,
Ž 6.
E ==m
Et
Ž H0 q T y =V . s 0.
Ž 7.
These equations are discretized to obtain the system matrix equations to calculate the distribution of T and V . The details of the numerical method are shown in Ref. w3x.
3. Experimental method Transport loss of the superconducting tape is measured by a standard voltage tap method w1x. The voltage taps are attached to the edge of the tape, and the distance between the lead wires and the edge is 20 mm. Magnetization loss of the superconducting tape is measured by a modified saddle type pick up coil. Magnetic field up to 120 mT is produced by a racetrack coil without iron core wound with copper wire. The coil is cooled in liquid nitrogen with sample superconducting tapes. The length of the sample, the magnet bore, and the saddle coil are 210 mm, 140 mm, and 50 mm, respectively. Both in the transport loss measurement and magnetization loss measurement, a lock-in amplifier technique w1x and direct waveform integration technique w2x are used simultaneously to cross check the measured values. E–J curve derived from the transport voltage–current curve is fitted between E s 10y5 –10y4 Vrm to Eq. Ž3. to determine Jc and n. Four sample pieces are tested in the experiments. Samples No. 1, and No. 2 are the samples for the transport loss and magnetization loss measurements, respectively. Their Jc and n are measured at the same time as the loss measurements with the racetrack coil without iron core. Jc and n of samples No. 3 and No. 4 are measured in a broader range of the magnetic field that is parallel and perpendicular to the tape wide face, respectively, that are produced by another coil with iron core. These data are used in
N. Amemiya et al.r Physica C 310 (1998) 30–35
32
4.2. Calculated and measured transport losses
Fig. 1. Magnetic field dependency of critical current density and n value.
numerical calculations of AC losses in superconducting tapes carrying AC transport current in external magnetic field ŽFig. 1..
4. Numerical and experimental results 4.1. Specifications of sample superconducting tapes and their Jc and n Specifications of a silver sheathed Bi-2223 superconducting tape tested in this study are listed in Table 1. Temperature is 77 K. Jc and n of sample No. 1 at self magnetic field condition are 2.03 = 10 8 Arm2 and 19.0, respectively. Jc and n of samples No. 2, No. 3, and No. 4 are plotted against the magnitude of the magnetic field. These values are used in the following numerical calculations.
Transport loss is calculated and measured with sample No. 1. In Fig. 2, the numerically calculated and measured transport losses are plotted against the transport current divided by the critical current, ItrIc , when frequency, f, is 60 Hz. The analytical value for an elliptical superconductor based on the Bean’s critical state model w4x is also plotted in this figure. The numerical and measured values almost agree with each other. In Fig. 3, the calculated and measured transport losses are plotted against frequency, when ItrIc s 0.5. The transport loss per cycle decreases with increasing frequency. Such frequency dependency cannot be explained by the Bean’s critical state model. Since the equivalent conductivity of superconductors is finite due to their small n, the current density at the peripheral increases with increasing frequency, so as a skin effect, and this decreases in the loss per cycle. 4.3. Calculated and measured magnetization losses in parallel magnetic field Magnetization loss is calculated and measured with sample No. 2 in the AC magnetic field parallel to the tape wide face at f s 60 Hz. Numerical calculations are made with Jc and n measured at the DC magnetic field that agrees with the root mean square
Table 1 Specifications of superconducting tape Size Type of superconductor Number of filament Matrix Žconductivity. Cross-section of superconductor Fraction of superconductor in filamentary region Critical current at 0 T, 77 K
3.5 mm=0.25 mm Bi-2223 61 Silver Ž3.33=10 8 Srm. 0.22=10y6 m2 0.415 ; 2=10 8 Arm2 Fig. 2. Calculated and measured transport losses vs. It r Ic .
N. Amemiya et al.r Physica C 310 (1998) 30–35
33
agrees well with the measured losses, especially in a fully penetrated high magnetic field region. There, the Bean’s critical state model applied to the superconducting slab overestimates the magnetization loss. 4.4. Calculated transport loss in tape carrying AC transport current in DC magnetic field The transport loss in the superconducting tape carrying AC transport current in the parallel and perpendicular DC magnetic fields is calculated with samples No. 3 and No. 4, respectively. In Fig. 5, the transport losses per unit length of the samples nor-
Fig. 3. Frequency dependence of calculated and measured transport losses.
of the applied AC magnetic field. In Fig. 4, the numerically calculated and measured magnetization losses are plotted against the amplitude of the external magnetic field, Bm , together with the analytical value given by the Bean’s critical state model for a slab of superconductor. In the analytical model, the thickness of the slab is assumed to be 0.195 mm that equals the estimated thickness of the filamentary region, and its critical current density is assumed to equal the average critical current density in the filamentary region. The numerically calculated loss
Fig. 4. Calculated and measured magnetization losses vs. Bm where external magnetic field is parallel to tape wide face.
Fig. 5. Influence of DC external magnetic field on transport loss; Ža. in parallel DC magnetic field and Žb. in perpendicular DC magnetic field.
34
N. Amemiya et al.r Physica C 310 (1998) 30–35
malized by the square of their critical current are plotted against ItrIc for various DC external magnetic field where f s 50 Hz. The transport loss of elliptical superconductor based on the Bean’s critical state model is also plotted w4x. The normalized transport loss increases with increasing external magnetic field. The application of the perpendicular magnetic field of Bm s 200 mT results in the 60% increase of the normalized transport loss, when ItrIc s 0.5. The DC external magnetic field decreases the critical current density and the n of superconductor, but it does not change the shape of the current distribution in the tape. The decrease in the critical current density does not change the loss normalized by the square of Ic . The decrease in n should decrease the normalized transport loss. The calculated current profiles show that the maximum current density decreases with increasing applied magnetic field, that is decreasing n. This should cause the increase in the normalized transport loss. 4.5. Calculated total loss in superconducting tape carrying AC transport current in AC magnetic field The total loss in the superconducting tape carrying AC transport current in the parallel or perpendicular AC magnetic field is calculated with samples No. 3 and No. 4, respectively. In this analysis, the equivalent conductivities are changing temporally with temporally changing external magnetic field following to the magnetic field dependencies of Jc and n shown in Fig. 1. In Fig. 6Ža. and Žb., the total losses in the parallel and perpendicular magnetic fields are plotted against Bm , respectively, where f s 50 Hz. The sum of the numerically calculated transport loss and magnetization loss is also shown for ItrIc s 0.5. In the parallel magnetic field, this sum almost agrees with the total loss calculated numerically. However, in the perpendicular magnetic field, this sum underestimates the loss in a small Bm region. For example, when ItrIc s 0.5 and Bm s 5 mT, the calculated total loss and the simple sum are 229 Jrm3rcycle and 130 Jrm3rcycle, respectively. In the high magnetic field region, the external magnetic field dominates the total loss, and the transport current does not substantially influence the total loss. The threshold over which the total loss is dominated by the external magnetic field should be determined by the tape size. For the 3.5 mm = 0.25 mm tape, the
Fig. 6. Total AC loss in superconducting tape carrying AC transport current in AC external magnetic field; Ža. in parallel magnetic field and Žb. in perpendicular magnetic field.
thresholds in the parallel and perpendicular magnetic fields are 100 mT and 50 mT, respectively.
5. Conclusion Transport loss, magnetization loss, and the total loss in Bi-2223 tape carrying AC transport current in AC magnetic field were calculated numerically, and some of them were compared with experimental results. The calculated and measured transport losses almost agree with each other. The calculated and measured magnetization losses agree with each other
N. Amemiya et al.r Physica C 310 (1998) 30–35
well. The application of the DC magnetic field, especially the perpendicular magnetic field, decreases in n and increases in the transport loss. In the parallel magnetic field, the total loss in Bi-2223 tape carrying AC transport current in AC magnetic field almost agrees with the sum of the transport loss and the magnetization loss. In the perpendicular magnetic field, the total loss is much larger than this sum. However, in a high field region, the total loss is dominated by the external magnetic field.
35
References w1x M. Ciszek, B.A. Glowacki, A.M. Campbell, S.P. Ashworth, W.Y. Liang, IEEE Trans. Appl. Superconduct. 7 Ž1997. 314. w2x S. Fukui, Y. Kitoh, T. Numata, O. Tsukamoto, J. Fujikami, K. Hayashi, Adv. Cryog. Eng. 44 Ž1998. to be published. w3x N. Amemiya, S. Murasawa, N. Banno, K. Miyamoto, in these Proceedings ŽTopical ICMC ’98 AC Loss and Stability, 10–13 May 1998, Enschede, The Netherlands. Physica C 310 Ž1998. 16. w4x W.T. Norris, J. Phys. D ŽApplied Physics. 3 Ž1970. 489.
Finite element analysis of AC loss in non-twisted Bi-2223 tape carrying AC transport current andror exposed to DC or AC external magnetic field Naoyuki Amemiya a
a,)
, Kengo Miyamoto a , Shun-ichi Murasawa a , Hideki Mukai b, Kazuya Ohmatsu b
DiÕision of Electrical and Computer Engineering, Yokohama National UniÕersity, 79-5 Tokiwadai, Hodogaya, Yokohama, 240-8501, Japan b Sumitomo Electric Industries, 1-1-3 Shimaya, Konohana, Osaka, 554-0024, Japan
Abstract AC losses in Bi-2223 superconducting tapes carrying AC transport current andror exposed to DC or AC magnetic field are calculated with a numerical model based on the finite element method. Superconducting property is given by the E–J characteristic represented by a power law using equivalent conductivity. First, transport loss and magnetization loss are calculated numerically and compared with measured values. The calculated losses almost agree with the measured losses. Frequency dependencies of calculated and measured transport losses are compared with each other. Next, the influence of DC external magnetic field on the transport loss is studied. DC external magnetic field reduces n that is an exponent in the power law connecting resistivity and current density. The numerically calculated transport loss increases with increasing DC magnetic field. Finally, the total loss of superconducting tape carrying AC transport current in AC magnetic field is calculated. In the perpendicular magnetic field, the calculated total loss is lager than the sum of the transport loss and the magnetization loss, while they almost agree with each other in the parallel magnetic field. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Numerical analysis; Transport loss; Finite element method; Flux penetration
1. Introduction In practical electrical power apparatuses used at commercial frequencies, high Tc superconductors carry AC transport current in AC external magnetic field. Therefore, quantitative evaluation of their AC loss is one of the key issues in the research and
)
Corresponding author. Tel.: q81-45-339-4119; Fax: q81-45338-1157; E-mail: [email protected]
development of superconductors and their applications. Although experimental techniques for the transport loss measurement and the magnetization loss measurement are almost established w1,2x, no reliable and accurate technique to measure the total loss of superconductors carrying AC transport current in AC external magnetic field has been developed yet. The authors have been developing a series of numerical codes for the electromagnetic analysis of superconductors based on the finite element method.
0921-4534r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 4 2 8 - 6
N. Amemiya et al.r Physica C 310 (1998) 30–35
Their purpose is numerical estimations of AC losses substituting for AC loss measurements. The main objective of this paper is to describe the FEM analysis of AC losses in a non-twisted silversheathed Bi-2223 superconducting tape carrying AC transport current andror exposed to DC or AC magnetic field. Numerically calculated AC losses are compared with experimental results. In the numerical analysis, electric field Ž E . –current density Ž j . characteristics of superconductor are given with a power law. The magnetic field dependence of E–J characteristics is also taken into account. First, the transport loss is calculated, and is compared with the measured value. Its frequency dependency is also discussed. Next, the calculated and measured magnetization losses are compared with each other. Then, the influence of DC magnetic field on the transport loss is presented. Finally, the total loss in Bi-2223 tape carrying AC transport current in AC external magnetic field is calculated, and compared with the sum of the transport loss and magnetization loss.
2. Numerical method The numerical model is formulated with current vector potential T and magnetic scalar potential V by the finite element method. They are given as, J'==T ,
Ž 1. Ž 2.
H s H0 q T y =V
where H0 is the external magnetic field. Superconducting property is given by a power law, E s Ec
J
n
ž /
Ž 3.
Jc
where Jc is a critical current density and Ec s 1 = 10y4 Vrm. Then, an equivalent conductivity of superconductor ssc is derived as,
ssc s
J s E
Jcn Ec
Ž==T .
lyn
.
Ž 4.
Elements in the filamentary region are treated as a mixture of superconductor and silver matrix whose equivalent conductivity is given as,
s s lfr ssc q Ž 1 y lfr . sAg
Ž 5.
31
where l fr and sAg are the fraction of superconductor in the filamentary region and the conductivity of the silver matrix, respectively. Then, Faraday’s law and modified magnetic Gauss’s law are written as,
==
ž
1
s
E
/
= = T s ym
Et
Ž H0 q T y =V . ,
Ž 6.
E ==m
Et
Ž H0 q T y =V . s 0.
Ž 7.
These equations are discretized to obtain the system matrix equations to calculate the distribution of T and V . The details of the numerical method are shown in Ref. w3x.
3. Experimental method Transport loss of the superconducting tape is measured by a standard voltage tap method w1x. The voltage taps are attached to the edge of the tape, and the distance between the lead wires and the edge is 20 mm. Magnetization loss of the superconducting tape is measured by a modified saddle type pick up coil. Magnetic field up to 120 mT is produced by a racetrack coil without iron core wound with copper wire. The coil is cooled in liquid nitrogen with sample superconducting tapes. The length of the sample, the magnet bore, and the saddle coil are 210 mm, 140 mm, and 50 mm, respectively. Both in the transport loss measurement and magnetization loss measurement, a lock-in amplifier technique w1x and direct waveform integration technique w2x are used simultaneously to cross check the measured values. E–J curve derived from the transport voltage–current curve is fitted between E s 10y5 –10y4 Vrm to Eq. Ž3. to determine Jc and n. Four sample pieces are tested in the experiments. Samples No. 1, and No. 2 are the samples for the transport loss and magnetization loss measurements, respectively. Their Jc and n are measured at the same time as the loss measurements with the racetrack coil without iron core. Jc and n of samples No. 3 and No. 4 are measured in a broader range of the magnetic field that is parallel and perpendicular to the tape wide face, respectively, that are produced by another coil with iron core. These data are used in
N. Amemiya et al.r Physica C 310 (1998) 30–35
32
4.2. Calculated and measured transport losses
Fig. 1. Magnetic field dependency of critical current density and n value.
numerical calculations of AC losses in superconducting tapes carrying AC transport current in external magnetic field ŽFig. 1..
4. Numerical and experimental results 4.1. Specifications of sample superconducting tapes and their Jc and n Specifications of a silver sheathed Bi-2223 superconducting tape tested in this study are listed in Table 1. Temperature is 77 K. Jc and n of sample No. 1 at self magnetic field condition are 2.03 = 10 8 Arm2 and 19.0, respectively. Jc and n of samples No. 2, No. 3, and No. 4 are plotted against the magnitude of the magnetic field. These values are used in the following numerical calculations.
Transport loss is calculated and measured with sample No. 1. In Fig. 2, the numerically calculated and measured transport losses are plotted against the transport current divided by the critical current, ItrIc , when frequency, f, is 60 Hz. The analytical value for an elliptical superconductor based on the Bean’s critical state model w4x is also plotted in this figure. The numerical and measured values almost agree with each other. In Fig. 3, the calculated and measured transport losses are plotted against frequency, when ItrIc s 0.5. The transport loss per cycle decreases with increasing frequency. Such frequency dependency cannot be explained by the Bean’s critical state model. Since the equivalent conductivity of superconductors is finite due to their small n, the current density at the peripheral increases with increasing frequency, so as a skin effect, and this decreases in the loss per cycle. 4.3. Calculated and measured magnetization losses in parallel magnetic field Magnetization loss is calculated and measured with sample No. 2 in the AC magnetic field parallel to the tape wide face at f s 60 Hz. Numerical calculations are made with Jc and n measured at the DC magnetic field that agrees with the root mean square
Table 1 Specifications of superconducting tape Size Type of superconductor Number of filament Matrix Žconductivity. Cross-section of superconductor Fraction of superconductor in filamentary region Critical current at 0 T, 77 K
3.5 mm=0.25 mm Bi-2223 61 Silver Ž3.33=10 8 Srm. 0.22=10y6 m2 0.415 ; 2=10 8 Arm2 Fig. 2. Calculated and measured transport losses vs. It r Ic .
N. Amemiya et al.r Physica C 310 (1998) 30–35
33
agrees well with the measured losses, especially in a fully penetrated high magnetic field region. There, the Bean’s critical state model applied to the superconducting slab overestimates the magnetization loss. 4.4. Calculated transport loss in tape carrying AC transport current in DC magnetic field The transport loss in the superconducting tape carrying AC transport current in the parallel and perpendicular DC magnetic fields is calculated with samples No. 3 and No. 4, respectively. In Fig. 5, the transport losses per unit length of the samples nor-
Fig. 3. Frequency dependence of calculated and measured transport losses.
of the applied AC magnetic field. In Fig. 4, the numerically calculated and measured magnetization losses are plotted against the amplitude of the external magnetic field, Bm , together with the analytical value given by the Bean’s critical state model for a slab of superconductor. In the analytical model, the thickness of the slab is assumed to be 0.195 mm that equals the estimated thickness of the filamentary region, and its critical current density is assumed to equal the average critical current density in the filamentary region. The numerically calculated loss
Fig. 4. Calculated and measured magnetization losses vs. Bm where external magnetic field is parallel to tape wide face.
Fig. 5. Influence of DC external magnetic field on transport loss; Ža. in parallel DC magnetic field and Žb. in perpendicular DC magnetic field.
34
N. Amemiya et al.r Physica C 310 (1998) 30–35
malized by the square of their critical current are plotted against ItrIc for various DC external magnetic field where f s 50 Hz. The transport loss of elliptical superconductor based on the Bean’s critical state model is also plotted w4x. The normalized transport loss increases with increasing external magnetic field. The application of the perpendicular magnetic field of Bm s 200 mT results in the 60% increase of the normalized transport loss, when ItrIc s 0.5. The DC external magnetic field decreases the critical current density and the n of superconductor, but it does not change the shape of the current distribution in the tape. The decrease in the critical current density does not change the loss normalized by the square of Ic . The decrease in n should decrease the normalized transport loss. The calculated current profiles show that the maximum current density decreases with increasing applied magnetic field, that is decreasing n. This should cause the increase in the normalized transport loss. 4.5. Calculated total loss in superconducting tape carrying AC transport current in AC magnetic field The total loss in the superconducting tape carrying AC transport current in the parallel or perpendicular AC magnetic field is calculated with samples No. 3 and No. 4, respectively. In this analysis, the equivalent conductivities are changing temporally with temporally changing external magnetic field following to the magnetic field dependencies of Jc and n shown in Fig. 1. In Fig. 6Ža. and Žb., the total losses in the parallel and perpendicular magnetic fields are plotted against Bm , respectively, where f s 50 Hz. The sum of the numerically calculated transport loss and magnetization loss is also shown for ItrIc s 0.5. In the parallel magnetic field, this sum almost agrees with the total loss calculated numerically. However, in the perpendicular magnetic field, this sum underestimates the loss in a small Bm region. For example, when ItrIc s 0.5 and Bm s 5 mT, the calculated total loss and the simple sum are 229 Jrm3rcycle and 130 Jrm3rcycle, respectively. In the high magnetic field region, the external magnetic field dominates the total loss, and the transport current does not substantially influence the total loss. The threshold over which the total loss is dominated by the external magnetic field should be determined by the tape size. For the 3.5 mm = 0.25 mm tape, the
Fig. 6. Total AC loss in superconducting tape carrying AC transport current in AC external magnetic field; Ža. in parallel magnetic field and Žb. in perpendicular magnetic field.
thresholds in the parallel and perpendicular magnetic fields are 100 mT and 50 mT, respectively.
5. Conclusion Transport loss, magnetization loss, and the total loss in Bi-2223 tape carrying AC transport current in AC magnetic field were calculated numerically, and some of them were compared with experimental results. The calculated and measured transport losses almost agree with each other. The calculated and measured magnetization losses agree with each other
N. Amemiya et al.r Physica C 310 (1998) 30–35
well. The application of the DC magnetic field, especially the perpendicular magnetic field, decreases in n and increases in the transport loss. In the parallel magnetic field, the total loss in Bi-2223 tape carrying AC transport current in AC magnetic field almost agrees with the sum of the transport loss and the magnetization loss. In the perpendicular magnetic field, the total loss is much larger than this sum. However, in a high field region, the total loss is dominated by the external magnetic field.
35
References w1x M. Ciszek, B.A. Glowacki, A.M. Campbell, S.P. Ashworth, W.Y. Liang, IEEE Trans. Appl. Superconduct. 7 Ž1997. 314. w2x S. Fukui, Y. Kitoh, T. Numata, O. Tsukamoto, J. Fujikami, K. Hayashi, Adv. Cryog. Eng. 44 Ž1998. to be published. w3x N. Amemiya, S. Murasawa, N. Banno, K. Miyamoto, in these Proceedings ŽTopical ICMC ’98 AC Loss and Stability, 10–13 May 1998, Enschede, The Netherlands. Physica C 310 Ž1998. 16. w4x W.T. Norris, J. Phys. D ŽApplied Physics. 3 Ž1970. 489.