- Email: [email protected]
Three-dimensional field measurements of levitated HTS and the modeling of the shielding current
Physica C 426–431 (2005) 731–738 www.elsevier.com/locate/physc
Three-dimensional field measurements of levitated HTS and the modeling of the shielding current E. Ito
a,c,* ,
Y. Komano a,c, K. Sawa a, Y. Iwasa b, N. Sakai c, I. Hirabayashi c, M. Murakami c,d
a
b
Department of System Design Engineering, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Japan Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA c Superconductivity Research Laboratory, 1-10-13, Shinonome, Koto-ku, Tokyo, Japan d Department of Materials Science and Engineering, Shibaura Institute of Technology, 3-9-14 Shibaura, Minato-ku, Tokyo, Japan Received 23 November 2004
Abstract We developed a shielding current distribution model in a bulk superconductor with the aim of estimating the levitation force. First, we directly measured the magnetic field in axial and radial directions above a bulk Y–Ba–Cu–O superconductor levitated by an electromagnet using a three-dimensional Hall sensor. We then calculated the field distribution by assuming two types of the shielding current distributions and compared the results with the measured values. The field distributions based on the model in that the shielding currents are flowing concentrically at the surface were not in good agreement with the measured data. We then assumed that the shielding currents are not uniform and the boundary has the form of nth hyperbola, which could reproduce the measured field distribution reasonably well. 2005 Published by Elsevier B.V. PACS: 74.72.Bk; 74.76.Bz Keywords: Levitation; Bulk superconductor; Y–Ba–Cu–O; Shielding current distribution
*
Corresponding author. Address: Department of System Design Engineering, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Japan. Tel.: +81 045 563 1141x43057; fax: +81 045 566 1720. E-mail address: [email protected] (E. Ito). 0921-4534/$ - see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.physc.2005.05.027
1. Introduction The magnetic levitation system with a HTS bulk and an electromagnet can be used for various engineering applications such as a flywheel energy
732
E. Ito et al. / Physica C 426–431 (2005) 731–738
storage system. For the design of the levitation devices used for practical applications, it is necessary to estimate the levitation force or the tolerable mass of the system for safety operation. The levitation force has been estimated using either the magnetization or the critical currents of the superconductors. Since the magnetization is dependent on the sample size, the force calculation based on the critical currents is more useful. When the superconductor is exposed to the external field, the currents are induced at the surface as to shield the external field. Here, the product of the induced currents (J) and the external field (B) gives the levitation force: F = J · B. According to the critical state model, the depth where the currents flow can be calculated from the critical current density, which is equal to the field gradient. To a first approximation, we assumed that the currents are flowing at the surface over the same distance from the edge for the entire disk, that is, the flowing layer is parallel to the sample edge. This is similar to the case of a coil current. Such an approximation provides fairly good results and the values are in good agreement with the measured values when the external field is small. However, the deviation from the measured value increases with increasing the external field. This is due to the fact that the field penetration is not
uniform. We thus assumed that the boundary of the field penetration is curved and has the form of the nth order hyperbola instead of a straight line. We measured the levitation force of Y–Ba– Cu–O superconductor disk and directly measured three-dimensional magnetic field distributions on the surface. The data were compared with the simulation results based on two models.
2. Experimental Fig. 1 shows a schematic illustration of the levitation force measurement system that consists of a balance and a load cell. The field was applied with an electromagnet composed of Bi2223 double pancake coils that operate below 30 K cooled by a cryo-cooler. The load cell is placed apart from the levitation coil by using a balance, so that we can measure the levitation force without receiving the influence of magnetic field generated by the Bi2223 magnet. The superconductor used for force measurements was a Y–Ba–Cu–O disk 47 mm in diameter and 15 mm in height, prepared by the top-seeded melt-growth process. We used a three-dimensional Hall probe sensor in order to measure the magnetic field distribution on the disk surfaces. The Hall sensor unit has three
Fig. 1. Experimental levitation system that is used for the measurements of the field distribution.
E. Ito et al. / Physica C 426–431 (2005) 731–738
733
sured points shown in Fig. 2. If the z and r components (Bz and Br) of the fields generated from the Bi2223 coil are subtracted from these values, one can obtain the z and r components of the fields (Bzsc and Brsc ) produced by the shielding currents flowing in the superconductor: Bzsc ¼ Bzex Bz ;
Bulk
Bulk Measurement point
Fig. 2. Measurement points of z and r components of magnetic fields above a bulk Y–Ba–Cu–O disk.
probes that are installed as to be able to measure the magnetic fields in three rectangular axis directions. Using the Hall sensor unit, we measured the magnetic field both in radial and horizontal directions at various places above the bulk superconductor. The range for field measurements was 24 mm wide along a radial direction from the center to the edge and 10 mm high from the top surface with an interval of 2 mm. The measured points of magnetic fields are schematically shown in Fig. 2. The measurements of the levitation force and the magnetic field distribution on the sample surface were carried out according to the following procedure. First, the Y–Ba–Cu–O disk in the normal conducting state was placed at the center of the Bi2223 superconducting magnet. Then the field was generated by passing the coil currents to 65 A, and the sample was cooled by liquid nitrogen, which is the field-cooling process. The field was further increased by passing currents to 185 A followed by the measurements of the force and the magnetic field distribution around the sample surface.
Brsc ¼ Brex Br .
Fig. 4(a) and (b) shows respective z and r components (Bzsc and Brsc ) of the fields that are generated by the shielding currents induced in the superconductor. The top curves in both graphs correspond to the fields on the Y–Ba–Cu–O disk or the nearest measuring points. The second curves from the top are for the fields at the gap of 2 mm from the top surface. Likewise, the third curves for the gap of 4 mm and the other curves are for the gap of 6, 8, and 10. As the gap increases, the field strength decreases for both z and r directions. It is interesting to note that the z component has the peak at the center, while the r component has the peak near the sample edge. Such a field distribution resembles that generated from an open coil. If one can simulate the precise shielding current distribution based on the measured values of z and r components above the superconductor, it will be possible to obtain the levitation forces. First, we assumed that the current are flowing concentrically at the surface, where the magnetic field penetrates in parallel to the side edge of a Y–Ba–Cu–O disk. For this case, the Y–Ba–Cu–O cylindrical disk can be regarded as a coil with the outer diameter of 2a, the inner diameter of 2(a d), the height b, and the self-inductance L. Here, d is the penetration depth. The total current I which flows in the disk is given by I ¼ J c db;
ð1Þ
where Jc is the critical current density. The total current is also given by the following equation: I¼
/ Bz pa2 ¼ ; LðdÞ LðdÞ
ð2Þ
3. Results and discussion
where L is the self-inductance. Using Eqs. (1) and (2), one can obtain the following relation:
Fig. 3(a) and (b) shows respective z and r components (Bzex and Brex ) of magnetic fields at the mea-
dLðdÞ ¼
Bz pa2 . bJ c
ð3Þ
734
E. Ito et al. / Physica C 426–431 (2005) 731–738 0.35 0.3 0.25
Bz(T)
0.2
z=0mm z=2mm
0.15
z=4mm 0.1
z=6mm z=8mm
0.05 0
z=10mm 0
5
10
15
20
25
15
20
25
r(mm)
(a) 0
Br(T)
-0.02
-0.04
z=0mm z=2mm
-0.06
z=4mm z=6mm z=8mm
-0.08
z=10mm -0.1
0
5
10
r(mm)
(b)
Fig. 3. (a) Axial and (b) radial components of magnetic fields above the Y–Ba–Cu–O disk.
Based on this relation, combining with the Bio– Savart law, one can calculate the z and r components of the field at the height of z and the distance from the center r: Brsc ða; r; zÞ ¼
l0 I z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p 2 r ða þ rÞ þ z2 ( ) a2 þ r2 þ z2 KðkÞ þ EðkÞ ; ð4Þ 2 ðz rÞ þ z2
lI 1 Bzsc ða; r; zÞ ¼ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p 2 r ða þ rÞ þ z2 ( ) a2 r 2 z 2 KðkÞ þ EðkÞ ; ðz rÞ2 þ z2
ð5Þ
where K and E are the complete elliptic integrals of the first and the second kinds, respectively. For the calculation, we need to know the Jc value of the sample. Based on the sand-pile model [1], we estimated Jc from the trapped-magnetic field by the following equation: Jc ¼
z 1 oBtrap ðrÞ ; l0 or
ð6Þ
where r is radial distance from the center of the sample and Bztrap ðrÞ is the z component of the trapped-magnetic field, which is saturated. The present sample exhibited a peak trapped field of 0.8 T at 77 K. It was then drawn from Eq. (6) that Jc of the present sample is 0.95 · 108 A/m2 at 77 K.
E. Ito et al. / Physica C 426–431 (2005) 731–738
735
0.25 z=0mm z=2mm
0.2
z=4mm z=6mm
Bscz (T)
0.15
z=8mm z=10mm
0.1 0.05 0
-0.05 0
5
10
(a)
15
20
25
15
20
25
r (mm) 0.2 z=0mm z=2mm
0.15
z=4mm
Bscr (T)
z=6mm z=8mm
0.1
z=10mm
0.05
0 0
5
10
(b)
r (mm)
Fig. 4. (a) Axial and (b) radial components of magnetic fields produced by shielding currents flowing in the Y–Ba–Cu–O disk.
Next, we estimated the fields generated from the superconductor and the results are plotted in Fig. 5(a) and (b), in which the measured data are also plotted for comparison. One can see that calculated data points agree reasonably well with the empirical data, although some deviation is observed. Such a discrepancy is ascribable to oversimplified hypothesis of current distribution in that currents are supposed to flow over the same distance from the sample edge in the entire disk. Hence, we assumed that the boundary of the current flowing region is not straight and approximated by the nth order hyperbola. Here, the boundary can be expressed by Eqs. (7) and (8):
1 1 h0 ; a rn a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dn þ dn ð4=h0 þ dn Þ a¼ . 2
z ¼ dðrÞ ¼
ð7Þ
ð8Þ
Fig. 6 shows schematic illustration of this model along with a simple current flowing model that we employed above. The current flowing region or the flux-penetrated area is shaded in this figure. For the present simulation, the exponent n was not fixed initially but considered as a fitting parameter, which was determined as to best fit the empirical data.
736
E. Ito et al. / Physica C 426–431 (2005) 731–738 0.25
z=0mm z=2mm z=4mm
0.2
z=6mm z=8mm
Bscz (T)
0.15
z=10mm
0.1 0.05 0 -0.05 5
0
10
a
15
20
25
15
20
25
r (mm) 0.2 z=0mm z=2mm
0.15
z=4mm
Bscr (T)
z=6mm z=8mm
0.1
z=10mm
0.05
0 0
5
10
b
r (mm)
Fig. 5. Calculated (a) axial and (b) radial components of magnetic fields produced by shielding currents flowing in the Y–Ba–Cu–O disk by assuming that the current flowing region is parallel to the disk edge.
Fig. 6. Schematic illustrations of the current flowing regions.
Fig. 7(a) and (b) shows the z and r components of the fields generated from the supposed current distribution with n = 0.495. The empirical data points are also plotted in the figure. One can see that the simulated field values are in good agree-
ment with the measured data, which contrasts with the results shown in Fig. 4. These results show that the present model taking account of non-uniform current distribution is more suitable for the description of the current flow inside the superconductive disk. Based on the current distribution obtained here, we calculated the levitation force, which is the product of the shielding current and the external magnetic field, and is obtained by the following equation: F ðtÞ ¼ 2pbJ
Z
a
ad
rBrex ðr; tÞ dr;
ð9Þ
E. Ito et al. / Physica C 426–431 (2005) 731–738
737 z=0mm
0.25
z=2mm z=4mm
0.2
z=6mm z=8mm
Bscz (T)
0.15
z=10mm
0.1 0.05 0
0
5
a
10
15
20
15
20
r (mm) 0.2
z=0mm z=2mm z=4mm
Bscr (T)
0.15
z=6mm z=8mm z=10mm
0.1
0.05
0 0
5
b
10
25
r (mm)
Fig. 7. Calculated (a) axial and (b) radial components of magnetic fields produced by shielding currents flowing in the Y–Ba–Cu–O disk by assuming that the boundary of the current flowing region has the form of nth hyperbola.
where Brex ðr; tÞ is the z component of the external magnetic field. The levitation force of 16.3 N was obtained from these data, which agrees well with the empirical value of 16.4 N.
4. Conclusions The levitation force of a bulk superconductor can be precisely calculated if one knows the shielding current and the external field distributions, in that the force is the product of the current and the field. However, the current distribution is not
uniform inside a superconductive disk, so that the calculation based on the uniform current distribution model fails to obtain correct levitation forces. We first measured the field distribution above the levitated Y–Ba–Cu–O disk and then estimated the fields generated from the superconductive disk. A simple model in that the shielding current is assumed to be uniform could not reproduce the field distribution. Then we assumed that the boundary of the current flowing region or the field penetration has the form of nth hyperbola and applied the model to the calculation of the field distribution. With this method, we could
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E. Ito et al. / Physica C 426–431 (2005) 731–738
reproduce the field distribution above the superconductor reasonably well, showing that such a model is appropriate for obtaining true current distributions in the superconductive disk.
References [1] K. Nagashima, T. Higuchi, J. Sok, S.I. Yoo, H. Fujimoto, M. Murakami, Cryogenics 37 (1997) 577.
Further readings [1] Y. Iwasa, H. Lee, K. Sawa, M. Murakami, Adv. Supercond. 9 (1997) 1379. [2] Y. Iwasa, H. Lee, Cryogenics 37 (1997) 807. [3] Y. Yoshida, M. Uesaka, K. Miya, Int. J. Appl. Electoromagn. Mater. 5 (1994) 83. [4] T. Tamegai, Y. Iye, I. Oguro, K. Kishio, Physica C 213 (1993) 33. [5] Y. Komano, K. Sawa, Y. Iwasa, N. Yamachi, M. Murakami, I. Hirabayashi, Physica C 412 (2004) 729.
Three-dimensional field measurements of levitated HTS and the modeling of the shielding current E. Ito
a,c,* ,
Y. Komano a,c, K. Sawa a, Y. Iwasa b, N. Sakai c, I. Hirabayashi c, M. Murakami c,d
a
b
Department of System Design Engineering, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Japan Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA c Superconductivity Research Laboratory, 1-10-13, Shinonome, Koto-ku, Tokyo, Japan d Department of Materials Science and Engineering, Shibaura Institute of Technology, 3-9-14 Shibaura, Minato-ku, Tokyo, Japan Received 23 November 2004
Abstract We developed a shielding current distribution model in a bulk superconductor with the aim of estimating the levitation force. First, we directly measured the magnetic field in axial and radial directions above a bulk Y–Ba–Cu–O superconductor levitated by an electromagnet using a three-dimensional Hall sensor. We then calculated the field distribution by assuming two types of the shielding current distributions and compared the results with the measured values. The field distributions based on the model in that the shielding currents are flowing concentrically at the surface were not in good agreement with the measured data. We then assumed that the shielding currents are not uniform and the boundary has the form of nth hyperbola, which could reproduce the measured field distribution reasonably well. 2005 Published by Elsevier B.V. PACS: 74.72.Bk; 74.76.Bz Keywords: Levitation; Bulk superconductor; Y–Ba–Cu–O; Shielding current distribution
*
Corresponding author. Address: Department of System Design Engineering, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Japan. Tel.: +81 045 563 1141x43057; fax: +81 045 566 1720. E-mail address: [email protected] (E. Ito). 0921-4534/$ - see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.physc.2005.05.027
1. Introduction The magnetic levitation system with a HTS bulk and an electromagnet can be used for various engineering applications such as a flywheel energy
732
E. Ito et al. / Physica C 426–431 (2005) 731–738
storage system. For the design of the levitation devices used for practical applications, it is necessary to estimate the levitation force or the tolerable mass of the system for safety operation. The levitation force has been estimated using either the magnetization or the critical currents of the superconductors. Since the magnetization is dependent on the sample size, the force calculation based on the critical currents is more useful. When the superconductor is exposed to the external field, the currents are induced at the surface as to shield the external field. Here, the product of the induced currents (J) and the external field (B) gives the levitation force: F = J · B. According to the critical state model, the depth where the currents flow can be calculated from the critical current density, which is equal to the field gradient. To a first approximation, we assumed that the currents are flowing at the surface over the same distance from the edge for the entire disk, that is, the flowing layer is parallel to the sample edge. This is similar to the case of a coil current. Such an approximation provides fairly good results and the values are in good agreement with the measured values when the external field is small. However, the deviation from the measured value increases with increasing the external field. This is due to the fact that the field penetration is not
uniform. We thus assumed that the boundary of the field penetration is curved and has the form of the nth order hyperbola instead of a straight line. We measured the levitation force of Y–Ba– Cu–O superconductor disk and directly measured three-dimensional magnetic field distributions on the surface. The data were compared with the simulation results based on two models.
2. Experimental Fig. 1 shows a schematic illustration of the levitation force measurement system that consists of a balance and a load cell. The field was applied with an electromagnet composed of Bi2223 double pancake coils that operate below 30 K cooled by a cryo-cooler. The load cell is placed apart from the levitation coil by using a balance, so that we can measure the levitation force without receiving the influence of magnetic field generated by the Bi2223 magnet. The superconductor used for force measurements was a Y–Ba–Cu–O disk 47 mm in diameter and 15 mm in height, prepared by the top-seeded melt-growth process. We used a three-dimensional Hall probe sensor in order to measure the magnetic field distribution on the disk surfaces. The Hall sensor unit has three
Fig. 1. Experimental levitation system that is used for the measurements of the field distribution.
E. Ito et al. / Physica C 426–431 (2005) 731–738
733
sured points shown in Fig. 2. If the z and r components (Bz and Br) of the fields generated from the Bi2223 coil are subtracted from these values, one can obtain the z and r components of the fields (Bzsc and Brsc ) produced by the shielding currents flowing in the superconductor: Bzsc ¼ Bzex Bz ;
Bulk
Bulk Measurement point
Fig. 2. Measurement points of z and r components of magnetic fields above a bulk Y–Ba–Cu–O disk.
probes that are installed as to be able to measure the magnetic fields in three rectangular axis directions. Using the Hall sensor unit, we measured the magnetic field both in radial and horizontal directions at various places above the bulk superconductor. The range for field measurements was 24 mm wide along a radial direction from the center to the edge and 10 mm high from the top surface with an interval of 2 mm. The measured points of magnetic fields are schematically shown in Fig. 2. The measurements of the levitation force and the magnetic field distribution on the sample surface were carried out according to the following procedure. First, the Y–Ba–Cu–O disk in the normal conducting state was placed at the center of the Bi2223 superconducting magnet. Then the field was generated by passing the coil currents to 65 A, and the sample was cooled by liquid nitrogen, which is the field-cooling process. The field was further increased by passing currents to 185 A followed by the measurements of the force and the magnetic field distribution around the sample surface.
Brsc ¼ Brex Br .
Fig. 4(a) and (b) shows respective z and r components (Bzsc and Brsc ) of the fields that are generated by the shielding currents induced in the superconductor. The top curves in both graphs correspond to the fields on the Y–Ba–Cu–O disk or the nearest measuring points. The second curves from the top are for the fields at the gap of 2 mm from the top surface. Likewise, the third curves for the gap of 4 mm and the other curves are for the gap of 6, 8, and 10. As the gap increases, the field strength decreases for both z and r directions. It is interesting to note that the z component has the peak at the center, while the r component has the peak near the sample edge. Such a field distribution resembles that generated from an open coil. If one can simulate the precise shielding current distribution based on the measured values of z and r components above the superconductor, it will be possible to obtain the levitation forces. First, we assumed that the current are flowing concentrically at the surface, where the magnetic field penetrates in parallel to the side edge of a Y–Ba–Cu–O disk. For this case, the Y–Ba–Cu–O cylindrical disk can be regarded as a coil with the outer diameter of 2a, the inner diameter of 2(a d), the height b, and the self-inductance L. Here, d is the penetration depth. The total current I which flows in the disk is given by I ¼ J c db;
ð1Þ
where Jc is the critical current density. The total current is also given by the following equation: I¼
/ Bz pa2 ¼ ; LðdÞ LðdÞ
ð2Þ
3. Results and discussion
where L is the self-inductance. Using Eqs. (1) and (2), one can obtain the following relation:
Fig. 3(a) and (b) shows respective z and r components (Bzex and Brex ) of magnetic fields at the mea-
dLðdÞ ¼
Bz pa2 . bJ c
ð3Þ
734
E. Ito et al. / Physica C 426–431 (2005) 731–738 0.35 0.3 0.25
Bz(T)
0.2
z=0mm z=2mm
0.15
z=4mm 0.1
z=6mm z=8mm
0.05 0
z=10mm 0
5
10
15
20
25
15
20
25
r(mm)
(a) 0
Br(T)
-0.02
-0.04
z=0mm z=2mm
-0.06
z=4mm z=6mm z=8mm
-0.08
z=10mm -0.1
0
5
10
r(mm)
(b)
Fig. 3. (a) Axial and (b) radial components of magnetic fields above the Y–Ba–Cu–O disk.
Based on this relation, combining with the Bio– Savart law, one can calculate the z and r components of the field at the height of z and the distance from the center r: Brsc ða; r; zÞ ¼
l0 I z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p 2 r ða þ rÞ þ z2 ( ) a2 þ r2 þ z2 KðkÞ þ EðkÞ ; ð4Þ 2 ðz rÞ þ z2
lI 1 Bzsc ða; r; zÞ ¼ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p 2 r ða þ rÞ þ z2 ( ) a2 r 2 z 2 KðkÞ þ EðkÞ ; ðz rÞ2 þ z2
ð5Þ
where K and E are the complete elliptic integrals of the first and the second kinds, respectively. For the calculation, we need to know the Jc value of the sample. Based on the sand-pile model [1], we estimated Jc from the trapped-magnetic field by the following equation: Jc ¼
z 1 oBtrap ðrÞ ; l0 or
ð6Þ
where r is radial distance from the center of the sample and Bztrap ðrÞ is the z component of the trapped-magnetic field, which is saturated. The present sample exhibited a peak trapped field of 0.8 T at 77 K. It was then drawn from Eq. (6) that Jc of the present sample is 0.95 · 108 A/m2 at 77 K.
E. Ito et al. / Physica C 426–431 (2005) 731–738
735
0.25 z=0mm z=2mm
0.2
z=4mm z=6mm
Bscz (T)
0.15
z=8mm z=10mm
0.1 0.05 0
-0.05 0
5
10
(a)
15
20
25
15
20
25
r (mm) 0.2 z=0mm z=2mm
0.15
z=4mm
Bscr (T)
z=6mm z=8mm
0.1
z=10mm
0.05
0 0
5
10
(b)
r (mm)
Fig. 4. (a) Axial and (b) radial components of magnetic fields produced by shielding currents flowing in the Y–Ba–Cu–O disk.
Next, we estimated the fields generated from the superconductor and the results are plotted in Fig. 5(a) and (b), in which the measured data are also plotted for comparison. One can see that calculated data points agree reasonably well with the empirical data, although some deviation is observed. Such a discrepancy is ascribable to oversimplified hypothesis of current distribution in that currents are supposed to flow over the same distance from the sample edge in the entire disk. Hence, we assumed that the boundary of the current flowing region is not straight and approximated by the nth order hyperbola. Here, the boundary can be expressed by Eqs. (7) and (8):
1 1 h0 ; a rn a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dn þ dn ð4=h0 þ dn Þ a¼ . 2
z ¼ dðrÞ ¼
ð7Þ
ð8Þ
Fig. 6 shows schematic illustration of this model along with a simple current flowing model that we employed above. The current flowing region or the flux-penetrated area is shaded in this figure. For the present simulation, the exponent n was not fixed initially but considered as a fitting parameter, which was determined as to best fit the empirical data.
736
E. Ito et al. / Physica C 426–431 (2005) 731–738 0.25
z=0mm z=2mm z=4mm
0.2
z=6mm z=8mm
Bscz (T)
0.15
z=10mm
0.1 0.05 0 -0.05 5
0
10
a
15
20
25
15
20
25
r (mm) 0.2 z=0mm z=2mm
0.15
z=4mm
Bscr (T)
z=6mm z=8mm
0.1
z=10mm
0.05
0 0
5
10
b
r (mm)
Fig. 5. Calculated (a) axial and (b) radial components of magnetic fields produced by shielding currents flowing in the Y–Ba–Cu–O disk by assuming that the current flowing region is parallel to the disk edge.
Fig. 6. Schematic illustrations of the current flowing regions.
Fig. 7(a) and (b) shows the z and r components of the fields generated from the supposed current distribution with n = 0.495. The empirical data points are also plotted in the figure. One can see that the simulated field values are in good agree-
ment with the measured data, which contrasts with the results shown in Fig. 4. These results show that the present model taking account of non-uniform current distribution is more suitable for the description of the current flow inside the superconductive disk. Based on the current distribution obtained here, we calculated the levitation force, which is the product of the shielding current and the external magnetic field, and is obtained by the following equation: F ðtÞ ¼ 2pbJ
Z
a
ad
rBrex ðr; tÞ dr;
ð9Þ
E. Ito et al. / Physica C 426–431 (2005) 731–738
737 z=0mm
0.25
z=2mm z=4mm
0.2
z=6mm z=8mm
Bscz (T)
0.15
z=10mm
0.1 0.05 0
0
5
a
10
15
20
15
20
r (mm) 0.2
z=0mm z=2mm z=4mm
Bscr (T)
0.15
z=6mm z=8mm z=10mm
0.1
0.05
0 0
5
b
10
25
r (mm)
Fig. 7. Calculated (a) axial and (b) radial components of magnetic fields produced by shielding currents flowing in the Y–Ba–Cu–O disk by assuming that the boundary of the current flowing region has the form of nth hyperbola.
where Brex ðr; tÞ is the z component of the external magnetic field. The levitation force of 16.3 N was obtained from these data, which agrees well with the empirical value of 16.4 N.
4. Conclusions The levitation force of a bulk superconductor can be precisely calculated if one knows the shielding current and the external field distributions, in that the force is the product of the current and the field. However, the current distribution is not
uniform inside a superconductive disk, so that the calculation based on the uniform current distribution model fails to obtain correct levitation forces. We first measured the field distribution above the levitated Y–Ba–Cu–O disk and then estimated the fields generated from the superconductive disk. A simple model in that the shielding current is assumed to be uniform could not reproduce the field distribution. Then we assumed that the boundary of the current flowing region or the field penetration has the form of nth hyperbola and applied the model to the calculation of the field distribution. With this method, we could
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reproduce the field distribution above the superconductor reasonably well, showing that such a model is appropriate for obtaining true current distributions in the superconductive disk.
References [1] K. Nagashima, T. Higuchi, J. Sok, S.I. Yoo, H. Fujimoto, M. Murakami, Cryogenics 37 (1997) 577.
Further readings [1] Y. Iwasa, H. Lee, K. Sawa, M. Murakami, Adv. Supercond. 9 (1997) 1379. [2] Y. Iwasa, H. Lee, Cryogenics 37 (1997) 807. [3] Y. Yoshida, M. Uesaka, K. Miya, Int. J. Appl. Electoromagn. Mater. 5 (1994) 83. [4] T. Tamegai, Y. Iye, I. Oguro, K. Kishio, Physica C 213 (1993) 33. [5] Y. Komano, K. Sawa, Y. Iwasa, N. Yamachi, M. Murakami, I. Hirabayashi, Physica C 412 (2004) 729.