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Electromagnetic field analysis of rectangular high Tc superconductor with large aspect ratio
Physica C 412–414 (2004) 1050–1055 www.elsevier.com/locate/physc
Electromagnetic field analysis of rectangular high Tc superconductor with large aspect ratio N. Enomoto *, N. Amemiya Faculty of Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya, Yokohama 240-8501, Japan Received 29 October 2003; accepted 15 December 2003 Available online 28 May 2004
Abstract AC loss reduction can substantially influence the economic feasibility of superconducting power devices. For AC loss reduction, understanding the AC loss characteristics of high Tc superconductors is essentially required, and numerical electromagnetic field analysis of high Tc superconductors is very useful tool for this purpose because the electromagnetic phenomena inside the superconductors can be visualized. This work focuses on electromagnetic field analysis of rectangular high Tc superconductors with a large aspect ratio. A numerical code based on the finite element method was tuned to perform electromagnetic field analysis of high Tc superconductors with a large aspect ratio. The optimization of mesh systems is required for accurate and efficient numerical analysis. AC losses in high Tc superconductors with AC transport current and/or an AC external magnetic field are estimated through numerical analysis. The numerically calculated AC losses are compared with analytical values at several reference points to study the validity and error of the numerical analysis. 2004 Elsevier B.V. All rights reserved. PACS: 02.70.Dh; 84.70.p; 85.25.Kx Keywords: Total loss; AC loss; Finite element method; Numerical analysis; Thin film; YBCO
1. Introduction AC loss properties determine whether high Tc superconductors can be put to practical use. The reduction of AC loss is a key issue for the realization of AC applications with high Tc superconductors. Therefore, the quantitative evaluation of AC loss is important for the development of
*
Corresponding author. Tel.: +81-45-339-4125; fax: +81-45338-1157. E-mail address: [email protected] (N. Enomoto).
superconductors and their AC applications in electrical power apparatuses. Although experimental approaches have often been used, there are also many advantages in a numerical one [1–6]. Numerical analysis by the finite element method (FEM) that can almost approximate the form of an object is an effective means for the evaluation of AC loss properties because AC loss properties depend on the cross-sectional form of the superconductor. This helps us with an understanding of the electromagnetic phenomena inside superconductors and can evaluate details concerning AC loss properties of a superconductor.
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.12.079
N. Enomoto, N. Amemiya / Physica C 412–414 (2004) 1050–1055
The authors have been developing a series of numerical models for superconductors by FEM to simulate the electromagnetic phenomena in superconductors and to calculate their AC losses. Although several formulations for the eddy current problem were proposed with various numerical conversions, the authors’ numerical models are formulated with the current vector potential T and the magnetic scalar potential X giving its validity for the handled problem [1,7]. This work numerically calculates total AC losses in a superconductor with a rectangular cross-section having an aspect ratio of 500. The calculated total AC losses in a superconductor with or without carrying an AC transport current and/or exposed to an AC magnetic field are compared with each analytical value.
2. Theoretical method 2.1. T–X formulation The total magnetic field H is the sum of the external magnetic field H 0 and the magnetic field generated by the eddy current H s where H ¼ H 0 þ H s:
ð1Þ
H s is related to the eddy current density J as r H s ¼ J:
ð2Þ
The current vector potential T is defined by J r T:
ð3Þ
From Eqs. (1)–(3), the magnetic scalar potential X can be defined as H H 0 þ T rX:
ð4Þ
Then, Faraday’s law and magnetic Gauss’s law are formulated using T and X as 1 o r T ¼ l0 ðH 0 þ T rXÞ; r ð5Þ r ot
1051
30mm Vacuum region
0.02mm 10mm 30mm
Superconductor region imitating YBCO thin film
Fig. 1. Overview of analysis region for numerical calculation. The aspect ratio of the superconductor region is set to 500 to imitate YBCO thin film.
ysis region consists of a superconductor region and a vacuum region as Fig. 1. The conductivity of the vacuum region is set to 1 S/m which is much smaller than the equivalent conductivity of superconductor. The transport current can be given by the boundary condition [1]. The boundary is placed far from the superconductor. The small but finite conductivity of the vacuum region 1 S/m mitigates the influence of the assumed uniform tangential component of T on the boundary. Even if the vacuum region has conductivity, little current flows there because the conductivity of the superconductor region is much higher. Eqs. (5) and (6) are discretized spatially by a mesh system and in a time domain by the backward-difference method to obtain the system matrix equations to calculate the temporal evolution of T and X. Both potentials are calculated at each node that comprises each element. The potentials in each element are approximated by the node basis function Nj ðx; yÞ as T¼
3 X
Nj ðx; yÞT j ;
ð7Þ
Nj ðx; yÞXj ;
ð8Þ
j¼1
r l0 ðH 0 þ T rXÞ ¼ 0;
ð6Þ
where r is the conductivity and l0 is the magnetic constant [1,7]. In this numerical model, the anal-
X¼
3 X j¼1
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N. Enomoto, N. Amemiya / Physica C 412–414 (2004) 1050–1055
where T j and Xj are the nodal value of T and X in each element. Since the linear triangle was adopted as the element type, Nj ðx; yÞ is defined at each node as follows: 1 aj þ b j x þ c j y ; Nj ðx; yÞ ¼ ð9Þ 2D where D is the area of the element and aj ; bj , and cj are determined as Nj ðx; yÞ ¼ 1 at the node and Nj ðx; yÞ ¼ 0 at the other two nodes that comprise the element [1].
Fig. 2. Enlargement of edge part of the superconductor region for finite element model.
2.2. E–J characteristics of the superconductor E–J characteristics of the superconductor are represented with n Jsc E ¼ E0 ; ð10Þ Jc where E is the electric field intensity, Jsc is the current density in superconductor, Jc is the critical current density, and E0 ¼ 1 104 V/m. By setting Eqs. (3) and (10) in Ohm’s law, the following equation is derived: r¼
jJ j Jcn ¼ jr T j1n : j E j E0
ð11Þ
Since Eq. (5) depends on r; r is determined iteratively at each time step by the Newton–Raphson method. AC loss Q (J/m3 /cycle) can be obtained from Eqs. (3) and (11), and the following equation: Z J J dt: ð12Þ Q¼ r cycle 2.3. Finite element model The two-dimensional finite element model is constructed as in Fig. 1. The superconductor carrying transport current is set in a vacuum and exposed to the parallel or perpendicular transverse magnetic field. The magnetic field and the transport current are in phase with each other. Fig. 2 shows the edge part of the superconductor region in the finite element model. Table 1 shows specifications of superconductor and conditions of analysis. The dependence of critical current density on the magnetic field is neglected in this analysis.
Table 1 Specifications of superconductor and conditions of analysis Critical current n value Transport current/critical current (It =Ic ) External magnetic field Frequency of current and magnetic field
100 A 20 0 or 0.7 0 or 50 mT 50 Hz
3. Numerical result and discussion Figs. 3 and 4 show the current density distribution in the superconductor exposed to a perpendicular magnetic field at the midpoint and peak of cyclic field, respectively, while Figs. 5 and 6 are for a parallel magnetic field. Their scales are expanded by 100 in the direction of the thickness only for legibility. The superconductor carries only the transport current in (a) (Figs. 3(a) and 5(a) are the same, Figs. 4(a) and 6(a) are the same.), only the magnetization current in (b), and both transport and magnetization currents in (c). The current density in (c) was visualized as being compounded with (a) and (b) roughly. Table 2 shows AC loss calculated from the temporal evolution of current using Eq. (12). AC loss is not the sum of the transport loss in superconductor exposed to no applied field and the magnetization loss in superconductor carrying no transport current, unlike the current distribution. For a superconductor with the critical current Ic carrying only the transport current with the amplitude It numerical AC loss was compared with the analytical value for a strip model given by Norris [8]. The formula is as follows:
N. Enomoto, N. Amemiya / Physica C 412–414 (2004) 1050–1055
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Fig. 3. Instantaneous current density distribution in superconductor at midpoint of cyclic field with (a) It =Ic ¼ 0:7, (b) perpendicular magnetic field of 50 mT, and (c) their simultaneous application. The scale of the thickness direction is expanded by 100.
Fig. 5. Instantaneous current density distribution in superconductor at midpoint of cyclic field with: (a) It =Ic ¼ 0:7, (b) parallel magnetic field of 50 mT, and (c) their simultaneous application. The scale of the thickness direction is expanded by 100.
Fig. 4. Instantaneous current density distribution in superconductor at peak of cyclic field with: (a) It =Ic ¼ 0:7, (b) perpendicular magnetic field of 50 mT, and (c) their simultaneous application. The scale of the thickness direction is expanded by 100.
Fig. 6. Instantaneous current density distribution in superconductor at peak of cyclic field with: (a) It =Ic ¼ 0:7, (b) parallel magnetic field of 50 mT, and (c) their simultaneous application. The scale of the thickness direction is expanded by 100.
Q¼
Ic2 l0 ð1 iÞ ln ð1 iÞ þ ð1 þ iÞ ln ð1 þ iÞ i2 ; Sp ð13Þ
where i ¼ It =Ic and S is a cross-sectional area of the superconductor. AC loss in a perpendicular magnetic field was compared with the analytical value for a strip model given by Brandt and
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N. Enomoto, N. Amemiya / Physica C 412–414 (2004) 1050–1055
Table 2 Comparison between numerical and analytical results It =Ic
Perpendicular magnetic field (mT)
Parallel magnetic field (mT)
Analytical total loss (J/m3 /cycle)
Numerical total loss (J/m3 /cycle)
0.7 0 0 0.7 0.7
0 50 0 50 0
0 0 50 0 50
1.02 · 103 2.22 · 105 4.57 · 102 – 1.57 · 103
9.34 · 102 2.40 · 105 4.58 · 102 2.99 · 105 1.27 · 103
Indenbom [9]. The formula for the peak field H is as follows: 4l0 a2 Jc dH H g Q¼ ; ð14Þ S Hc where gðxÞ ¼ ð2=xÞ lnðcosh xÞ tanh x;
ð15Þ
Hc ¼ Jc d=p, d is the thickness of the superconductor, and a is the half width of the superconductor. AC loss in a parallel magnetic field was compared with the analytical value given by Bean’s critical state model for a slab of the superconductor [10]. The formula is as follows: 21 Q ¼ dl0 Jc H 1 ; ð16Þ 3b where b ¼ 2H =ðJc dÞ. AC loss in the superconductor carrying a transport current in a perpendicular magnetic field cannot be compared with any analytical value because there is no analytical model for suitable approximation about this condition. AC loss in the superconductor carrying a transport current in a parallel magnetic field was compared with the analytical value for an infinitely large slab of the superconductor given by Carr [11]. The formula is as follows: Q¼
2B2p 3l0 4i
vð3 þ i2 Þ 2ð1 i3 Þ þ 6i2 2
ð1 iÞ3 ðv iÞ
2
ð1 iÞ2 ðv iÞ
! for v > 1;
ð17Þ
where, Bp ¼ l0 Jc d=2 is the penetration field of a slab with thickness of d and v ¼ l0 H =Bp However, this analytical value is given for the infinite slab, and does not take into account the loss due to a
perpendicular component of magnetic field generated by transport current in a superconductor tape with a finite width. At high transport current and low magnetic field, the perpendicular component of the self-field dominates the total AC loss in the handled superconductor with the rectangular cross section having a high aspect ratio. Here, it is assumed that the loss due to the perpendicular component of self-field is given by Norris’s strip model Eq. (13) [12]. Then, we make the sum of AC loss in Eqs. (13) and (17) as the analytical value in this condition. Numerical and analytical AC losses reasonably agree with each other, but there are some numerical errors. The finite element model, spatially discretized, is thought to have substantially influenced the numerical result. A vector potential distribution cannot be approximated correctly by the finite element model. The base function for the potential used differs from the actual potential base and the elements are too distorted. Therefore, the finite element model is the cause for the error in the numerical result.
4. Conclusions AC losses in a superconductor with a rectangular cross section having a high aspect ratio were calculated numerically and compared with analytical values. Although there are some numerical errors, numerical and analytical AC losses mostly agree with each other. Using FEM, AC losses can be given even if there is difficulty calculating AC losses analytically or measuring them experimentally. The numerical models based on FEM are useful tools to investigate electromagnetic
N. Enomoto, N. Amemiya / Physica C 412–414 (2004) 1050–1055
phenomena inside superconductors and to estimate their AC losses.
References [1] N. Amemiya, S. Murasawa, N. Banno, K. Miyamoto, Physica C 310 (1998) 16. [2] N. Amemiya, Y. Ohta, Physica C 357–360 (2001) 1134. [3] J. Paasi, M. Lahtinen, IEEE Trans. Appl. Supercond. 7 (1997) 322.
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[4] N. Amemiya, K. Miyamoto, N. Banno, O. Tsukamoto, IEEE Trans. Appl. Supercond. 7 (1997) 2110. [5] S. Fukui, M. Ikeda, T. Sano, H. Sango, M. Yamaguchi, T. Takao, IEEE Trans. Appl. Supercond. 11 (2001) 2212. [6] N. Nibbio, IEEE Trans. Appl. Supercond. 11 (2001) 2627. [7] C.J. Carpenter, Proc. IEE (1977) 1026. [8] T. Norris, J. Phys. D 3 (1970) 489. [9] M.N. Wilson, Superconducting Magnets, Oxford University Press, New York, 1983, p. 162–174. [10] E.H. Brandt, M. Indenbom, Phys. Rev. B 48 (1993) 12893. [11] W.J. Carr, IEEE Trans. Magn. 15 (1979) 240. [12] N. Magnusson, Physica C 349 (2001) 225.
Electromagnetic field analysis of rectangular high Tc superconductor with large aspect ratio N. Enomoto *, N. Amemiya Faculty of Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya, Yokohama 240-8501, Japan Received 29 October 2003; accepted 15 December 2003 Available online 28 May 2004
Abstract AC loss reduction can substantially influence the economic feasibility of superconducting power devices. For AC loss reduction, understanding the AC loss characteristics of high Tc superconductors is essentially required, and numerical electromagnetic field analysis of high Tc superconductors is very useful tool for this purpose because the electromagnetic phenomena inside the superconductors can be visualized. This work focuses on electromagnetic field analysis of rectangular high Tc superconductors with a large aspect ratio. A numerical code based on the finite element method was tuned to perform electromagnetic field analysis of high Tc superconductors with a large aspect ratio. The optimization of mesh systems is required for accurate and efficient numerical analysis. AC losses in high Tc superconductors with AC transport current and/or an AC external magnetic field are estimated through numerical analysis. The numerically calculated AC losses are compared with analytical values at several reference points to study the validity and error of the numerical analysis. 2004 Elsevier B.V. All rights reserved. PACS: 02.70.Dh; 84.70.p; 85.25.Kx Keywords: Total loss; AC loss; Finite element method; Numerical analysis; Thin film; YBCO
1. Introduction AC loss properties determine whether high Tc superconductors can be put to practical use. The reduction of AC loss is a key issue for the realization of AC applications with high Tc superconductors. Therefore, the quantitative evaluation of AC loss is important for the development of
*
Corresponding author. Tel.: +81-45-339-4125; fax: +81-45338-1157. E-mail address: [email protected] (N. Enomoto).
superconductors and their AC applications in electrical power apparatuses. Although experimental approaches have often been used, there are also many advantages in a numerical one [1–6]. Numerical analysis by the finite element method (FEM) that can almost approximate the form of an object is an effective means for the evaluation of AC loss properties because AC loss properties depend on the cross-sectional form of the superconductor. This helps us with an understanding of the electromagnetic phenomena inside superconductors and can evaluate details concerning AC loss properties of a superconductor.
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.12.079
N. Enomoto, N. Amemiya / Physica C 412–414 (2004) 1050–1055
The authors have been developing a series of numerical models for superconductors by FEM to simulate the electromagnetic phenomena in superconductors and to calculate their AC losses. Although several formulations for the eddy current problem were proposed with various numerical conversions, the authors’ numerical models are formulated with the current vector potential T and the magnetic scalar potential X giving its validity for the handled problem [1,7]. This work numerically calculates total AC losses in a superconductor with a rectangular cross-section having an aspect ratio of 500. The calculated total AC losses in a superconductor with or without carrying an AC transport current and/or exposed to an AC magnetic field are compared with each analytical value.
2. Theoretical method 2.1. T–X formulation The total magnetic field H is the sum of the external magnetic field H 0 and the magnetic field generated by the eddy current H s where H ¼ H 0 þ H s:
ð1Þ
H s is related to the eddy current density J as r H s ¼ J:
ð2Þ
The current vector potential T is defined by J r T:
ð3Þ
From Eqs. (1)–(3), the magnetic scalar potential X can be defined as H H 0 þ T rX:
ð4Þ
Then, Faraday’s law and magnetic Gauss’s law are formulated using T and X as 1 o r T ¼ l0 ðH 0 þ T rXÞ; r ð5Þ r ot
1051
30mm Vacuum region
0.02mm 10mm 30mm
Superconductor region imitating YBCO thin film
Fig. 1. Overview of analysis region for numerical calculation. The aspect ratio of the superconductor region is set to 500 to imitate YBCO thin film.
ysis region consists of a superconductor region and a vacuum region as Fig. 1. The conductivity of the vacuum region is set to 1 S/m which is much smaller than the equivalent conductivity of superconductor. The transport current can be given by the boundary condition [1]. The boundary is placed far from the superconductor. The small but finite conductivity of the vacuum region 1 S/m mitigates the influence of the assumed uniform tangential component of T on the boundary. Even if the vacuum region has conductivity, little current flows there because the conductivity of the superconductor region is much higher. Eqs. (5) and (6) are discretized spatially by a mesh system and in a time domain by the backward-difference method to obtain the system matrix equations to calculate the temporal evolution of T and X. Both potentials are calculated at each node that comprises each element. The potentials in each element are approximated by the node basis function Nj ðx; yÞ as T¼
3 X
Nj ðx; yÞT j ;
ð7Þ
Nj ðx; yÞXj ;
ð8Þ
j¼1
r l0 ðH 0 þ T rXÞ ¼ 0;
ð6Þ
where r is the conductivity and l0 is the magnetic constant [1,7]. In this numerical model, the anal-
X¼
3 X j¼1
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N. Enomoto, N. Amemiya / Physica C 412–414 (2004) 1050–1055
where T j and Xj are the nodal value of T and X in each element. Since the linear triangle was adopted as the element type, Nj ðx; yÞ is defined at each node as follows: 1 aj þ b j x þ c j y ; Nj ðx; yÞ ¼ ð9Þ 2D where D is the area of the element and aj ; bj , and cj are determined as Nj ðx; yÞ ¼ 1 at the node and Nj ðx; yÞ ¼ 0 at the other two nodes that comprise the element [1].
Fig. 2. Enlargement of edge part of the superconductor region for finite element model.
2.2. E–J characteristics of the superconductor E–J characteristics of the superconductor are represented with n Jsc E ¼ E0 ; ð10Þ Jc where E is the electric field intensity, Jsc is the current density in superconductor, Jc is the critical current density, and E0 ¼ 1 104 V/m. By setting Eqs. (3) and (10) in Ohm’s law, the following equation is derived: r¼
jJ j Jcn ¼ jr T j1n : j E j E0
ð11Þ
Since Eq. (5) depends on r; r is determined iteratively at each time step by the Newton–Raphson method. AC loss Q (J/m3 /cycle) can be obtained from Eqs. (3) and (11), and the following equation: Z J J dt: ð12Þ Q¼ r cycle 2.3. Finite element model The two-dimensional finite element model is constructed as in Fig. 1. The superconductor carrying transport current is set in a vacuum and exposed to the parallel or perpendicular transverse magnetic field. The magnetic field and the transport current are in phase with each other. Fig. 2 shows the edge part of the superconductor region in the finite element model. Table 1 shows specifications of superconductor and conditions of analysis. The dependence of critical current density on the magnetic field is neglected in this analysis.
Table 1 Specifications of superconductor and conditions of analysis Critical current n value Transport current/critical current (It =Ic ) External magnetic field Frequency of current and magnetic field
100 A 20 0 or 0.7 0 or 50 mT 50 Hz
3. Numerical result and discussion Figs. 3 and 4 show the current density distribution in the superconductor exposed to a perpendicular magnetic field at the midpoint and peak of cyclic field, respectively, while Figs. 5 and 6 are for a parallel magnetic field. Their scales are expanded by 100 in the direction of the thickness only for legibility. The superconductor carries only the transport current in (a) (Figs. 3(a) and 5(a) are the same, Figs. 4(a) and 6(a) are the same.), only the magnetization current in (b), and both transport and magnetization currents in (c). The current density in (c) was visualized as being compounded with (a) and (b) roughly. Table 2 shows AC loss calculated from the temporal evolution of current using Eq. (12). AC loss is not the sum of the transport loss in superconductor exposed to no applied field and the magnetization loss in superconductor carrying no transport current, unlike the current distribution. For a superconductor with the critical current Ic carrying only the transport current with the amplitude It numerical AC loss was compared with the analytical value for a strip model given by Norris [8]. The formula is as follows:
N. Enomoto, N. Amemiya / Physica C 412–414 (2004) 1050–1055
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Fig. 3. Instantaneous current density distribution in superconductor at midpoint of cyclic field with (a) It =Ic ¼ 0:7, (b) perpendicular magnetic field of 50 mT, and (c) their simultaneous application. The scale of the thickness direction is expanded by 100.
Fig. 5. Instantaneous current density distribution in superconductor at midpoint of cyclic field with: (a) It =Ic ¼ 0:7, (b) parallel magnetic field of 50 mT, and (c) their simultaneous application. The scale of the thickness direction is expanded by 100.
Fig. 4. Instantaneous current density distribution in superconductor at peak of cyclic field with: (a) It =Ic ¼ 0:7, (b) perpendicular magnetic field of 50 mT, and (c) their simultaneous application. The scale of the thickness direction is expanded by 100.
Fig. 6. Instantaneous current density distribution in superconductor at peak of cyclic field with: (a) It =Ic ¼ 0:7, (b) parallel magnetic field of 50 mT, and (c) their simultaneous application. The scale of the thickness direction is expanded by 100.
Q¼
Ic2 l0 ð1 iÞ ln ð1 iÞ þ ð1 þ iÞ ln ð1 þ iÞ i2 ; Sp ð13Þ
where i ¼ It =Ic and S is a cross-sectional area of the superconductor. AC loss in a perpendicular magnetic field was compared with the analytical value for a strip model given by Brandt and
1054
N. Enomoto, N. Amemiya / Physica C 412–414 (2004) 1050–1055
Table 2 Comparison between numerical and analytical results It =Ic
Perpendicular magnetic field (mT)
Parallel magnetic field (mT)
Analytical total loss (J/m3 /cycle)
Numerical total loss (J/m3 /cycle)
0.7 0 0 0.7 0.7
0 50 0 50 0
0 0 50 0 50
1.02 · 103 2.22 · 105 4.57 · 102 – 1.57 · 103
9.34 · 102 2.40 · 105 4.58 · 102 2.99 · 105 1.27 · 103
Indenbom [9]. The formula for the peak field H is as follows: 4l0 a2 Jc dH H g Q¼ ; ð14Þ S Hc where gðxÞ ¼ ð2=xÞ lnðcosh xÞ tanh x;
ð15Þ
Hc ¼ Jc d=p, d is the thickness of the superconductor, and a is the half width of the superconductor. AC loss in a parallel magnetic field was compared with the analytical value given by Bean’s critical state model for a slab of the superconductor [10]. The formula is as follows: 21 Q ¼ dl0 Jc H 1 ; ð16Þ 3b where b ¼ 2H =ðJc dÞ. AC loss in the superconductor carrying a transport current in a perpendicular magnetic field cannot be compared with any analytical value because there is no analytical model for suitable approximation about this condition. AC loss in the superconductor carrying a transport current in a parallel magnetic field was compared with the analytical value for an infinitely large slab of the superconductor given by Carr [11]. The formula is as follows: Q¼
2B2p 3l0 4i
vð3 þ i2 Þ 2ð1 i3 Þ þ 6i2 2
ð1 iÞ3 ðv iÞ
2
ð1 iÞ2 ðv iÞ
! for v > 1;
ð17Þ
where, Bp ¼ l0 Jc d=2 is the penetration field of a slab with thickness of d and v ¼ l0 H =Bp However, this analytical value is given for the infinite slab, and does not take into account the loss due to a
perpendicular component of magnetic field generated by transport current in a superconductor tape with a finite width. At high transport current and low magnetic field, the perpendicular component of the self-field dominates the total AC loss in the handled superconductor with the rectangular cross section having a high aspect ratio. Here, it is assumed that the loss due to the perpendicular component of self-field is given by Norris’s strip model Eq. (13) [12]. Then, we make the sum of AC loss in Eqs. (13) and (17) as the analytical value in this condition. Numerical and analytical AC losses reasonably agree with each other, but there are some numerical errors. The finite element model, spatially discretized, is thought to have substantially influenced the numerical result. A vector potential distribution cannot be approximated correctly by the finite element model. The base function for the potential used differs from the actual potential base and the elements are too distorted. Therefore, the finite element model is the cause for the error in the numerical result.
4. Conclusions AC losses in a superconductor with a rectangular cross section having a high aspect ratio were calculated numerically and compared with analytical values. Although there are some numerical errors, numerical and analytical AC losses mostly agree with each other. Using FEM, AC losses can be given even if there is difficulty calculating AC losses analytically or measuring them experimentally. The numerical models based on FEM are useful tools to investigate electromagnetic
N. Enomoto, N. Amemiya / Physica C 412–414 (2004) 1050–1055
phenomena inside superconductors and to estimate their AC losses.
References [1] N. Amemiya, S. Murasawa, N. Banno, K. Miyamoto, Physica C 310 (1998) 16. [2] N. Amemiya, Y. Ohta, Physica C 357–360 (2001) 1134. [3] J. Paasi, M. Lahtinen, IEEE Trans. Appl. Supercond. 7 (1997) 322.
1055
[4] N. Amemiya, K. Miyamoto, N. Banno, O. Tsukamoto, IEEE Trans. Appl. Supercond. 7 (1997) 2110. [5] S. Fukui, M. Ikeda, T. Sano, H. Sango, M. Yamaguchi, T. Takao, IEEE Trans. Appl. Supercond. 11 (2001) 2212. [6] N. Nibbio, IEEE Trans. Appl. Supercond. 11 (2001) 2627. [7] C.J. Carpenter, Proc. IEE (1977) 1026. [8] T. Norris, J. Phys. D 3 (1970) 489. [9] M.N. Wilson, Superconducting Magnets, Oxford University Press, New York, 1983, p. 162–174. [10] E.H. Brandt, M. Indenbom, Phys. Rev. B 48 (1993) 12893. [11] W.J. Carr, IEEE Trans. Magn. 15 (1979) 240. [12] N. Magnusson, Physica C 349 (2001) 225.