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Analysis of AC loss properties of HTS coated-conductor with magnetic substrate under external magnetic field using FEM
Physica C 468 (2008) 1739–1742
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Analysis of AC loss properties of HTS coated-conductor with magnetic substrate under external magnetic field using FEM M. Umabuchi a, D. Miyagi a,*, N. Takahashi a, O. Tsukamoto b a b
Department of Electrical and Electronic Engineering, Okayama University, Tsushimanaka 3-1-1, Okayama 700-8530, Japan Faculty of Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan
a r t i c l e
i n f o
Article history: Available online 9 July 2008 PACS: 74.78.W 84.71.Mn Keywords: Finite element method Non-linear magnetic property HTS coated-conductor Magnetic substrate AC magnetization loss
a b s t r a c t It is very important to understand AC magnetization loss of coated-conductor with magnetic substrate under the external magnetic field which has an arbitrary angle to the wide face of coated-conductor for designing realistic apparatuses. In this paper, the current distribution and AC magnetization loss in the superconducting layer of the magnetic substrate conductor are analyzed using the finite element method taking account of both the non-linear E–J characteristics and the non-linear magnetic property of the substrate. The influence of magnetic substrate on AC magnetization losses under the external AC magnetic field which has an arbitrary angle to the wide face of coated-conductors is examined. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Recently, HTS coated-conductors with ferromagnetic Ni-alloy substrates are promising as low-cost conductors because the Ni-alloy can be well-texturized by a simple process. However, the magnetic substrate influences the current distribution and AC loss [1,2]. The influence of the magnetic substrate on AC magnetization loss of coated-conductor has been examined experimentally [3,4]. On the other hand, the report of the analysis of AC magnetization loss of the coated-conductor with magnetic substrate using numerical calculation is few [5]. It is very important to understand AC magnetization loss characteristic of coated-conductor with magnetic substrate under the external magnetic field which has an arbitrary angle to the wide face of coated-conductor for designing realistic apparatuses. In this paper, the current distribution and AC magnetization losses in the superconducting layer of the magnetic substrate conductor are analyzed using the finite element method taking account of both the non-linear E–J characteristics and the nonlinear magnetic property of the substrate [5]. AC magnetization losses of the HTS conductor with magnetic substrate, under the external AC magnetic field which has an arbitrary angle to the wide face of coated-conductors, are compared with those of non-mag-
* Corresponding author. Tel.: +81 86 251 8121; fax: +81 86 251 8258. E-mail address: [email protected] (D. Miyagi). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.05.195
netic substrate, and the influence of magnetic substrate on AC magnetization losses is examined. 2. Finite element model The magnetic field is analyzed using the edge-based hexahedral finite element method (A formulation, A: magnetic vector potential (defined by B = rot A, B: magnetic flux density)) which was developed by us [5]. The mesh with brick elements is used. The governing equation is a complicated non-linear one having two kinds of non-linear parameters: permeability l and conductivity r. The permeability l is changed by the flux density B, and the conductivity r follows the non-linear E–J characteristic. The Newton–Raphson (N–R) method was employed only for the non-linear iteration of permeability l in the substrate (the soft magnetic material). An under relaxation method was applied in considering the non-linearity of conductivity r in the superconducting layer. Fig. 1 shows the schematic illustration of numerical analysis model. To examine the influence of the magnetic property of a substrate on the magnetization loss, the losses of four kinds of coatedconductor models with different magnetic properties of the substrate are calculated. The external magnetic field which has an arbitrary angle is impressed to the wide face (x–z plane) of the HTS conductor. The frequency is 50 Hz. Table 1 shows the specifications of these models. It is assumed that the model is infinite in the z direction. Model A has the non-magnetic substrate. Models B
M. Umabuchi et al. / Physica C 468 (2008) 1739–1742
y
100
Superconducting layerr
x z
Substrate
Fig. 1. Schematic illustration of numerical analysis model. The external magnetic field with an arbitrary angle, h, is impressed to the wide face of the HTS conductor.
Table 1 Specifications of analysis Model
A
B
C
SC layer width thickness (mm mm) 10.0 0.0008 Substrate width thickness (mm mm) 10.0 0.1 SC layer: aspect ratio 12500 Critical current density Jc (A/m2) 1.5 1010 Critical current Ic (A) 120 n value 14 Frequency (Hz) 50 Conductivity of substrate (S/m) 0 1.11 106 Relative permeability of substrate lr 1 20 100
D
10-1 10-2
Brandt Bean model (Infinite plate)
-3
10
10-4 10-5 10-6 10-7
θ = 0 deg. θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg.
10-8 10-9 10-10 10-3
10-2
10-1
Field amplitude B m (T) 100
Non-linear
1000 800
μr
Magnetization loss (J/m/cycle)
B = Bm sin ωt θ
600
Magnetization loss (J/m/cycle)
1740
10-1 10-2 -3
10
10-4 10-5 10-6 10-7
θ = 0 deg. θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg.
10-8 10-9 10-10 10-3
400
Brandt Bean model (Infinite plate)
10-2
10-1
Field amplitude B m (T) 200 0 0
0.02
0.04
0.06
0.08
0.1
B (T) Fig. 2. B–lr curve of the substrate. The B–lr curve is a magnetic property of the Nialloy substrate obtained at the room temperature [1].
and C have magnetic substrates of which the relative permeability lr is constant (=20, 100). Model D has the substrate with the nonlinear magnetic property. Fig. 2 shows the B–lr curve [1] used for magnetic property of the substrate of model D. The B–lr curve is the magnetic property of the Ni-alloy substrate determined from measurements using a single sheet tester at the room temperature. It is noted that the B–H curve at 77 K is different from that at the room temperature. Therefore, this study is focused on an influence of non-linear magnetic property such as magnetic saturation on AC magnetization loss. The mesh divided into about 50 layers with brick elements along the y direction in the superconducting layer is used. 3. Numerical analysis and discussion 3.1. Effect of relative permeability of substrate on AC magnetization loss of superconducting layer Fig. 3 shows the magnetization losses of three kinds of models with substrates having different relative permeability. In Fig. 3, the parameter h is the angle of the external AC magnetic field between the field direction and the wide face of a conductor. h = 0° and 90° denote that the external magnetic field is applied in parallel and perpendicular to the wide surface of the conductors respec-
Magnetization loss (J/m/cycle)
100 10-1 10-2
Brandt Bean model (Infinite plate)
-3
10
10-4 10-5 10-6 10-7
θ = 0 deg. θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg.
10-8 10-9 10-10 10-3
10-2
10-1
Field amplitude B m (T) Fig. 3. Magnetization losses of three kinds of models with substrates having different relative permeability (50 Hz). The parameter h is the angle of AC magnetic field vector. (a) Model A (non-magnetic) (b) Model B (lr = 20) (c) Model C (lr = 100).
tively. The magnetization losses per unit length per cycle (J/m/ cycle) are plotted against the amplitude Bm of the external AC magnetic field. The Brandt’s theoretical curve which is the analytical values of the magnetization loss of infinitely thin superconducting strip in a perpendicular transverse magnetic field is plotted [6]. Moreover, the magnetization loss of 0.8 lm thick infinite plate of superconductor in a parallel transverse magnetic field calculated based on the Bean model is also plotted. Fig. 4 shows the magnetic flux density at the center of the substrate under the external AC magnetic field. As shown in Fig. 3, the magnetization losses of the conductors with the magnetic substrate (models B and C) are smaller than the loss of the conductor with the non-magnetic substrate (model
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M. Umabuchi et al. / Physica C 468 (2008) 1739–1742
Flux density in the substrate (T)
2
center
θ = 11.33 deg. 1.5
( × 1010 A/m2) 2.00
(i)
θ = 0 deg.
Superconducting layer
edge
1.60
θ = 31.01 deg.
1.20 (a) Model A (non-magnetic)
θ = 90 deg.
0.80
1
0.40 (b) Model B (μr = 20)
0.5
z Substrate
(c) Model D (nonlinear) 0 -3 10
-2
10
10
-1
Field amplitude B m (T)
0
y
x
(× 10 010 A/m2) 1.00
(ii) center
Superconducting layer
edge
0.60
Flux density in the substrate (T)
5
θ = 0 deg. 4
0.20 (a) Model A (non-magnetic)
θ = 11.33 deg.
-0.20
θ = 31.01 deg. 3
-0.60 (b) Model B (μr = 20)
θ = 90 deg.
-1.00
y (c) Model D (nonlinear)
2
center -2
10
10
Superconducting layer
x
(× 1010 A/m2) 2.00 2.00
(iii)
1
0 -3 10
Substrate z
edge
1.20 1.60
-1
Field amplitude B m (T)
0.40 1.20
(a) Model A (non-magnetic)
-0.40 0.80
Fig. 4. Maximum flux density at the center of the substrate under the external AC magnetic field. (a) Model B (lr = 20) (b) Model (C) (lr = 100).
-1.20 0.40
(b) Model B (μr = 20)
y (c) Model D (nonlinear)
Substrate z
x
Fig. 5. Current distributions in the cross-section of superconducting layer of models A, B and D at the instant when the amplitude of the perpendicular or parallel external AC magnetic field is the maximum. 1/2 of the superconducting layer is illustrated. The thickness is enlarged to 400 times the actual one for an easier readability of the picture. (i) h = 90° (Bm = 0.01 T), (ii) h = 0° (Bm = 0.002 T) and (iii) h = 0° (Bm = 0.1 T).
100
Magnetization loss (J/m/cycle)
A) in the case of h = 90° and Bm < Bp = 12 mT, where Bp is the full penetration magnetic field for magnetic field perpendicular to wide face of a conductor [6]. However, the magnetization loss characteristics of models A, B and C are close to the Brandt curve and almost the same. In the case of h = 31.01° and Bm > Bp, the magnetization losses of the conductor with the magnetic substrate (models B and C) are close to the loss of the conductor with the non-magnetic substrate (model A). Therefore, the magnetization loss is hardly affected by the magnetism of the substrate at h > 30° and Bm > Bp. Fig. 3b and c show that the magnetization losses of models B (lr = 20) and C (lr = 100) are remarkably increased due to the magnetism of the substrate when the angle of external AC magnetic field to the wide face of a coated-conductor becomes small. The reason is as follows: the component of the flux perpendicular to the wide face of a conductor increases in the superconducting layer when lr is high and h becomes small because the magnetic flux is concentrated in the substrate as shown in Fig. 4. Fig. 5 shows the current distributions at the cross-section of superconducting layer of models A, B and D at the instant when the amplitude of the external AC magnetic field is the maximum. 1/2 of the superconducting layer in width is illustrated. The thickness is enlarged to 400 times the actual one for an easier readability of the picture. As shown in Fig. 5i, the shielding current of the model B under the perpendicular external AC magnetic field is smaller than that of the model A due to the influence of the magnetic substrate. But, these current distributions are almost the same. On the other hand, in the case of h = 0° the current distribution of model B shows the significant concentration of the current at the edge of conductor at Bm = 2 mT as shown in Fig. 5ii and iii.
-2.00 0
10-1 10-2
Brandt Bean model (Infinite plate)
-3
10
10-4 10-5 10-6 10-7
θ = 0 deg. θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg.
10-8 10-9 10-10 10-3
10-2
10-1
Field amplitude B m (T) Fig. 6. Magnetization losses of model D with substrate having non-linear magnetic property (50 Hz). The parameter h is the angle of AC magnetic field vector.
The current distributions of model A and model B are different at all. The current concentration causes the increase in the magneti-
M. Umabuchi et al. / Physica C 468 (2008) 1739–1742
a
Flux density in the substrate (T)
1742
0.3
θ = 0 deg. 0.25 0.2
θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg.
0.15 0.1 0.05 0 -3 10
-2
10
10
-1
tization loss characteristics of model D are similar to that of model C in the region of Bm < 2 mT and h < 31.01° as shown in Fig. 6. However, in the region of Bm > 2 mT, the magnetization loss characteristics are close to that of model A with non-magnetic substrate as the external AC magnetic field increases. In 2 mT < Bm < 20 mT and h = 31.01°, and Bm > 10 mT and h = 11.33°, the magnetization losses of model D are smaller than that of model A. The reason is as follows: in the region of Bm < 2 mT, the relative permeability at the center of the substrate is above 50, however, in the region of Bm > 2 mT, the relative permeability at the center of the substrate is decreased as the flux density in the magnetic substrate approaches the saturation flux density as shown in Fig. 7b. As shown in Fig. 5ii, the current distribution of model D is similar to that of model B at Bm = 2 mT. However, the current distribution of model D is close to that of model A at Bm = 100 mT as shown in Fig. 5iii.
Field amplitude B m (T) 4. Conclusions
b Relative permeability
1000
We studied the magnetization loss characteristics of HTS coated-conductor with magnetic substrate under the external AC magnetic field which has an arbitrary angle to the wide face of coated conductors. The current distributions and the magnetization losses are analyzed using the finite element method taking account of the non-linear E–J characteristic and the non-linear magnetic property of the substrate. The obtained results can be summarized as follows:
100
10
1 -3 10
θ = 0 deg. θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg. -2
10
10
-1
Field amplitude B m (T) Fig. 7. Maximum flux density and the relative permeability at the center of the substrate under the external AC magnetic field. (a) Maximum flux density at the center of the substrate and (b) relative permeability at the center of the substrate.
zation loss and shows the existence of the flux perpendicular to the wide face of a conductor. 3.2. Effect of non-linear magnetic property of substrate on AC magnetization loss of superconducting layer The magnetization losses of model D with the substrate having non-linear magnetic property are plotted in Fig. 6. Fig. 7 shows the magnetic flux density and the relative permeability at the center of the substrate under the external AC magnetic field. As shown in Figs. 3b,c and 6, the magnetization loss characteristics of model D is close to that of models B, C in the case of h = 90°. This is because that the current distribution of model D is similar to that of model B as shown in Fig. 5i. On the other hand, the magne-
(a) The effect of the magnetism of the substrate on the magnetization losses is remarkable as the angle of external AC magnetic field vector to the wide face of a coated-conductor becomes small. (b) The magnetization loss is hardly affected by the magnetism of the substrate when the angle of external AC magnetic field to the wide face of a coated-conductor is above about 30°. (c) The magnetization loss characteristic of the superconducting layer with a magnetic substrate is close to that with a nonmagnetic substrate as the external AC magnetic field increases and the flux density in the magnetic substrate approaches the saturation flux density.
References [1] O. Tsukamoto, Z. Li, H. Suzuki, M. Liu, K. Yamagishi, M. Ciszek, T. Takao, K. Ohmatsu, IEEE Trans. Appl. Supercond. 17 (2007) 3144. [2] D. Miyagi, Y. Amadutsumi, N. Takahashi, O. Tsukamoto, IEEE Trans. Appl. Supercond. 17 (2007) 3167. [3] M. Suenaga, Q. Li, Appl. Phys. Lett. 88 (2006) 262501. [4] O. Tsukamoto, M. Liu, S. Odaka, D. Miyagi, K. Ohmatsu, Physica C 463–465 (2007) 766. [5] D. Miyagi, M. Umabuchi, N. Takahashi, O. Tsukamoto, IEEE Trans. Appl. Supercond. 18 (2008) 1374. [6] E.H. Brandt, M. Indenbom, Phys. Rev. B (1993) 12893.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Analysis of AC loss properties of HTS coated-conductor with magnetic substrate under external magnetic field using FEM M. Umabuchi a, D. Miyagi a,*, N. Takahashi a, O. Tsukamoto b a b
Department of Electrical and Electronic Engineering, Okayama University, Tsushimanaka 3-1-1, Okayama 700-8530, Japan Faculty of Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan
a r t i c l e
i n f o
Article history: Available online 9 July 2008 PACS: 74.78.W 84.71.Mn Keywords: Finite element method Non-linear magnetic property HTS coated-conductor Magnetic substrate AC magnetization loss
a b s t r a c t It is very important to understand AC magnetization loss of coated-conductor with magnetic substrate under the external magnetic field which has an arbitrary angle to the wide face of coated-conductor for designing realistic apparatuses. In this paper, the current distribution and AC magnetization loss in the superconducting layer of the magnetic substrate conductor are analyzed using the finite element method taking account of both the non-linear E–J characteristics and the non-linear magnetic property of the substrate. The influence of magnetic substrate on AC magnetization losses under the external AC magnetic field which has an arbitrary angle to the wide face of coated-conductors is examined. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Recently, HTS coated-conductors with ferromagnetic Ni-alloy substrates are promising as low-cost conductors because the Ni-alloy can be well-texturized by a simple process. However, the magnetic substrate influences the current distribution and AC loss [1,2]. The influence of the magnetic substrate on AC magnetization loss of coated-conductor has been examined experimentally [3,4]. On the other hand, the report of the analysis of AC magnetization loss of the coated-conductor with magnetic substrate using numerical calculation is few [5]. It is very important to understand AC magnetization loss characteristic of coated-conductor with magnetic substrate under the external magnetic field which has an arbitrary angle to the wide face of coated-conductor for designing realistic apparatuses. In this paper, the current distribution and AC magnetization losses in the superconducting layer of the magnetic substrate conductor are analyzed using the finite element method taking account of both the non-linear E–J characteristics and the nonlinear magnetic property of the substrate [5]. AC magnetization losses of the HTS conductor with magnetic substrate, under the external AC magnetic field which has an arbitrary angle to the wide face of coated-conductors, are compared with those of non-mag-
* Corresponding author. Tel.: +81 86 251 8121; fax: +81 86 251 8258. E-mail address: [email protected] (D. Miyagi). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.05.195
netic substrate, and the influence of magnetic substrate on AC magnetization losses is examined. 2. Finite element model The magnetic field is analyzed using the edge-based hexahedral finite element method (A formulation, A: magnetic vector potential (defined by B = rot A, B: magnetic flux density)) which was developed by us [5]. The mesh with brick elements is used. The governing equation is a complicated non-linear one having two kinds of non-linear parameters: permeability l and conductivity r. The permeability l is changed by the flux density B, and the conductivity r follows the non-linear E–J characteristic. The Newton–Raphson (N–R) method was employed only for the non-linear iteration of permeability l in the substrate (the soft magnetic material). An under relaxation method was applied in considering the non-linearity of conductivity r in the superconducting layer. Fig. 1 shows the schematic illustration of numerical analysis model. To examine the influence of the magnetic property of a substrate on the magnetization loss, the losses of four kinds of coatedconductor models with different magnetic properties of the substrate are calculated. The external magnetic field which has an arbitrary angle is impressed to the wide face (x–z plane) of the HTS conductor. The frequency is 50 Hz. Table 1 shows the specifications of these models. It is assumed that the model is infinite in the z direction. Model A has the non-magnetic substrate. Models B
M. Umabuchi et al. / Physica C 468 (2008) 1739–1742
y
100
Superconducting layerr
x z
Substrate
Fig. 1. Schematic illustration of numerical analysis model. The external magnetic field with an arbitrary angle, h, is impressed to the wide face of the HTS conductor.
Table 1 Specifications of analysis Model
A
B
C
SC layer width thickness (mm mm) 10.0 0.0008 Substrate width thickness (mm mm) 10.0 0.1 SC layer: aspect ratio 12500 Critical current density Jc (A/m2) 1.5 1010 Critical current Ic (A) 120 n value 14 Frequency (Hz) 50 Conductivity of substrate (S/m) 0 1.11 106 Relative permeability of substrate lr 1 20 100
D
10-1 10-2
Brandt Bean model (Infinite plate)
-3
10
10-4 10-5 10-6 10-7
θ = 0 deg. θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg.
10-8 10-9 10-10 10-3
10-2
10-1
Field amplitude B m (T) 100
Non-linear
1000 800
μr
Magnetization loss (J/m/cycle)
B = Bm sin ωt θ
600
Magnetization loss (J/m/cycle)
1740
10-1 10-2 -3
10
10-4 10-5 10-6 10-7
θ = 0 deg. θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg.
10-8 10-9 10-10 10-3
400
Brandt Bean model (Infinite plate)
10-2
10-1
Field amplitude B m (T) 200 0 0
0.02
0.04
0.06
0.08
0.1
B (T) Fig. 2. B–lr curve of the substrate. The B–lr curve is a magnetic property of the Nialloy substrate obtained at the room temperature [1].
and C have magnetic substrates of which the relative permeability lr is constant (=20, 100). Model D has the substrate with the nonlinear magnetic property. Fig. 2 shows the B–lr curve [1] used for magnetic property of the substrate of model D. The B–lr curve is the magnetic property of the Ni-alloy substrate determined from measurements using a single sheet tester at the room temperature. It is noted that the B–H curve at 77 K is different from that at the room temperature. Therefore, this study is focused on an influence of non-linear magnetic property such as magnetic saturation on AC magnetization loss. The mesh divided into about 50 layers with brick elements along the y direction in the superconducting layer is used. 3. Numerical analysis and discussion 3.1. Effect of relative permeability of substrate on AC magnetization loss of superconducting layer Fig. 3 shows the magnetization losses of three kinds of models with substrates having different relative permeability. In Fig. 3, the parameter h is the angle of the external AC magnetic field between the field direction and the wide face of a conductor. h = 0° and 90° denote that the external magnetic field is applied in parallel and perpendicular to the wide surface of the conductors respec-
Magnetization loss (J/m/cycle)
100 10-1 10-2
Brandt Bean model (Infinite plate)
-3
10
10-4 10-5 10-6 10-7
θ = 0 deg. θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg.
10-8 10-9 10-10 10-3
10-2
10-1
Field amplitude B m (T) Fig. 3. Magnetization losses of three kinds of models with substrates having different relative permeability (50 Hz). The parameter h is the angle of AC magnetic field vector. (a) Model A (non-magnetic) (b) Model B (lr = 20) (c) Model C (lr = 100).
tively. The magnetization losses per unit length per cycle (J/m/ cycle) are plotted against the amplitude Bm of the external AC magnetic field. The Brandt’s theoretical curve which is the analytical values of the magnetization loss of infinitely thin superconducting strip in a perpendicular transverse magnetic field is plotted [6]. Moreover, the magnetization loss of 0.8 lm thick infinite plate of superconductor in a parallel transverse magnetic field calculated based on the Bean model is also plotted. Fig. 4 shows the magnetic flux density at the center of the substrate under the external AC magnetic field. As shown in Fig. 3, the magnetization losses of the conductors with the magnetic substrate (models B and C) are smaller than the loss of the conductor with the non-magnetic substrate (model
1741
M. Umabuchi et al. / Physica C 468 (2008) 1739–1742
Flux density in the substrate (T)
2
center
θ = 11.33 deg. 1.5
( × 1010 A/m2) 2.00
(i)
θ = 0 deg.
Superconducting layer
edge
1.60
θ = 31.01 deg.
1.20 (a) Model A (non-magnetic)
θ = 90 deg.
0.80
1
0.40 (b) Model B (μr = 20)
0.5
z Substrate
(c) Model D (nonlinear) 0 -3 10
-2
10
10
-1
Field amplitude B m (T)
0
y
x
(× 10 010 A/m2) 1.00
(ii) center
Superconducting layer
edge
0.60
Flux density in the substrate (T)
5
θ = 0 deg. 4
0.20 (a) Model A (non-magnetic)
θ = 11.33 deg.
-0.20
θ = 31.01 deg. 3
-0.60 (b) Model B (μr = 20)
θ = 90 deg.
-1.00
y (c) Model D (nonlinear)
2
center -2
10
10
Superconducting layer
x
(× 1010 A/m2) 2.00 2.00
(iii)
1
0 -3 10
Substrate z
edge
1.20 1.60
-1
Field amplitude B m (T)
0.40 1.20
(a) Model A (non-magnetic)
-0.40 0.80
Fig. 4. Maximum flux density at the center of the substrate under the external AC magnetic field. (a) Model B (lr = 20) (b) Model (C) (lr = 100).
-1.20 0.40
(b) Model B (μr = 20)
y (c) Model D (nonlinear)
Substrate z
x
Fig. 5. Current distributions in the cross-section of superconducting layer of models A, B and D at the instant when the amplitude of the perpendicular or parallel external AC magnetic field is the maximum. 1/2 of the superconducting layer is illustrated. The thickness is enlarged to 400 times the actual one for an easier readability of the picture. (i) h = 90° (Bm = 0.01 T), (ii) h = 0° (Bm = 0.002 T) and (iii) h = 0° (Bm = 0.1 T).
100
Magnetization loss (J/m/cycle)
A) in the case of h = 90° and Bm < Bp = 12 mT, where Bp is the full penetration magnetic field for magnetic field perpendicular to wide face of a conductor [6]. However, the magnetization loss characteristics of models A, B and C are close to the Brandt curve and almost the same. In the case of h = 31.01° and Bm > Bp, the magnetization losses of the conductor with the magnetic substrate (models B and C) are close to the loss of the conductor with the non-magnetic substrate (model A). Therefore, the magnetization loss is hardly affected by the magnetism of the substrate at h > 30° and Bm > Bp. Fig. 3b and c show that the magnetization losses of models B (lr = 20) and C (lr = 100) are remarkably increased due to the magnetism of the substrate when the angle of external AC magnetic field to the wide face of a coated-conductor becomes small. The reason is as follows: the component of the flux perpendicular to the wide face of a conductor increases in the superconducting layer when lr is high and h becomes small because the magnetic flux is concentrated in the substrate as shown in Fig. 4. Fig. 5 shows the current distributions at the cross-section of superconducting layer of models A, B and D at the instant when the amplitude of the external AC magnetic field is the maximum. 1/2 of the superconducting layer in width is illustrated. The thickness is enlarged to 400 times the actual one for an easier readability of the picture. As shown in Fig. 5i, the shielding current of the model B under the perpendicular external AC magnetic field is smaller than that of the model A due to the influence of the magnetic substrate. But, these current distributions are almost the same. On the other hand, in the case of h = 0° the current distribution of model B shows the significant concentration of the current at the edge of conductor at Bm = 2 mT as shown in Fig. 5ii and iii.
-2.00 0
10-1 10-2
Brandt Bean model (Infinite plate)
-3
10
10-4 10-5 10-6 10-7
θ = 0 deg. θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg.
10-8 10-9 10-10 10-3
10-2
10-1
Field amplitude B m (T) Fig. 6. Magnetization losses of model D with substrate having non-linear magnetic property (50 Hz). The parameter h is the angle of AC magnetic field vector.
The current distributions of model A and model B are different at all. The current concentration causes the increase in the magneti-
M. Umabuchi et al. / Physica C 468 (2008) 1739–1742
a
Flux density in the substrate (T)
1742
0.3
θ = 0 deg. 0.25 0.2
θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg.
0.15 0.1 0.05 0 -3 10
-2
10
10
-1
tization loss characteristics of model D are similar to that of model C in the region of Bm < 2 mT and h < 31.01° as shown in Fig. 6. However, in the region of Bm > 2 mT, the magnetization loss characteristics are close to that of model A with non-magnetic substrate as the external AC magnetic field increases. In 2 mT < Bm < 20 mT and h = 31.01°, and Bm > 10 mT and h = 11.33°, the magnetization losses of model D are smaller than that of model A. The reason is as follows: in the region of Bm < 2 mT, the relative permeability at the center of the substrate is above 50, however, in the region of Bm > 2 mT, the relative permeability at the center of the substrate is decreased as the flux density in the magnetic substrate approaches the saturation flux density as shown in Fig. 7b. As shown in Fig. 5ii, the current distribution of model D is similar to that of model B at Bm = 2 mT. However, the current distribution of model D is close to that of model A at Bm = 100 mT as shown in Fig. 5iii.
Field amplitude B m (T) 4. Conclusions
b Relative permeability
1000
We studied the magnetization loss characteristics of HTS coated-conductor with magnetic substrate under the external AC magnetic field which has an arbitrary angle to the wide face of coated conductors. The current distributions and the magnetization losses are analyzed using the finite element method taking account of the non-linear E–J characteristic and the non-linear magnetic property of the substrate. The obtained results can be summarized as follows:
100
10
1 -3 10
θ = 0 deg. θ = 11.33 deg. θ = 31.01 deg. θ = 90 deg. -2
10
10
-1
Field amplitude B m (T) Fig. 7. Maximum flux density and the relative permeability at the center of the substrate under the external AC magnetic field. (a) Maximum flux density at the center of the substrate and (b) relative permeability at the center of the substrate.
zation loss and shows the existence of the flux perpendicular to the wide face of a conductor. 3.2. Effect of non-linear magnetic property of substrate on AC magnetization loss of superconducting layer The magnetization losses of model D with the substrate having non-linear magnetic property are plotted in Fig. 6. Fig. 7 shows the magnetic flux density and the relative permeability at the center of the substrate under the external AC magnetic field. As shown in Figs. 3b,c and 6, the magnetization loss characteristics of model D is close to that of models B, C in the case of h = 90°. This is because that the current distribution of model D is similar to that of model B as shown in Fig. 5i. On the other hand, the magne-
(a) The effect of the magnetism of the substrate on the magnetization losses is remarkable as the angle of external AC magnetic field vector to the wide face of a coated-conductor becomes small. (b) The magnetization loss is hardly affected by the magnetism of the substrate when the angle of external AC magnetic field to the wide face of a coated-conductor is above about 30°. (c) The magnetization loss characteristic of the superconducting layer with a magnetic substrate is close to that with a nonmagnetic substrate as the external AC magnetic field increases and the flux density in the magnetic substrate approaches the saturation flux density.
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