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Estimation of the shielding current distribution in levitated YBCO bulk superconductor
Physica C 392–396 (2003) 634–638 www.elsevier.com/locate/physc
Estimation of the shielding current distribution in levitated YBCO bulk superconductor T. Nishikawa a
a,b,*
, Y. Komano a,b, K. Sawa a, Y. Iwasa c, N. Yamachi b, M. Murakami b
Department of System Design Engineering, KEIO University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama, Japan b ISTEC, Superconductivity Research Laboratory, 1-16-25 Shibaura, Minato-ku, Tokyo, Japan c Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, 02139 Cambridge, MA, USA Received 13 November 2002; accepted 27 January 2003
Abstract The high temperature superconductor (HTS) has very high Jc value and can trap the very high magnetic field. The YBa2 Cu3 Ox (YBCO) bulk superconductor is one of these attractive HTS and expected for many applications. In our previous study, to estimate the shielding current distribution in levitated YBCO bulks, we directly measured the levitation force and values of magnetic field on the surface of levitated YBCO bulk superconductor by using load cell and Hall probes. In this paper, we proposed shielding current distribution based on the axisymmetric 3-dimensional Bean-model and calculated the levitation force and magnetic fields on the surface of bulk superconductor. These calculated results were compared with experimental data. 2003 Published by Elsevier B.V. PACS: 74.72.Bk; 74.76.Bz Keywords: Shielding current distribution; Levitation force; Bean model; YBCO bulk
1. Introduction Bulk superconductors have significant potential for various applications such as magnetic bearing and non-contact transport, since superconductors can stably levitate over permanent magnets or vice versa. However, so long as permanent magnets are used as the magnetic source, the levitation force
*
Corresponding author. Tel.: +81-45-563-1141x43057; fax: +81-45-566-1720. E-mail address: [email protected] (T. Nishikawa). 0921-4534/$ - see front matter 2003 Published by Elsevier B.V. doi:10.1016/S0921-4534(03)00995-X
and the levitation height cannot be controlled. Iwasa [1] proposed the so-called ‘‘electromaglev’’ system in which an electromagnetic coil was used as the magnetic source. The system is attractive for practical applications, since the levitation force and levitation height can be controlled by simply changing the coil current. We constructed a levitation system with a Bi2 Sr2 Ca2 Cu3 O10 superconducting coil as a magnetic source for achieving a large levitation force without heat generation [2]. In this levitation system, we have succeeded in controlling the electromagnetic force and the levitation height by
T. Nishikawa et al. / Physica C 392–396 (2003) 634–638
changing the coil current. We further developed the force measurement system in which we can directly measure the levitation force and the magnetic field on the surface of the levitated bulk superconductors [3]. From these measurements, we have confirmed that the flux pinning properties strongly influence the levitation forces. We also derived a theoretical model to simulate the levitation behavior for a levitated bulk superconductor. In this study, we measured the levitation force and the magnetic field, which was compared with the model that we derived.
2. Experimental 2.1. Apparatus and samples We used YBa2 Cu3 Ox bulk superconductors grown by the top-seeded melt-growth process. The samples were single grains of 46 mm diameter and 15 mm or 7 mm in thickness. The levitation coil was composed of a Bi2 Sr2 Ca2 Cu3 O10 double pancake coil that operates below 30 K and is cooled by a cryo-cooler. Seven Hall sensors were mounted on the bulk surface with the inter-sensor distance of 2.5 mm (as shown in Fig. 1). The experimental system consists of a balance, a load cell and Hall sensors used in the previous study [4] as shown in Fig. 2. 2.2. Procedures In the levitation force measurements, the electric currents were passed through a levitation coil that generated the field Bfc . Next, the superconductor was placed at the center of the coil and
Fig. 1. Configuration of plural hall sensors.
635
Fig. 2. Experimental apparatus.
cooled with liquid nitrogen in the presence of Bfc . A previous study [5] demonstrated the importance of field-cooling for stable levitation of large bulk superconductors. We then increased the coil current to levitate the YBCO sample and measured the levitation force. We repeated increasing and decreasing field processes for three times.
3. Results and discussion Fig. 3 shows the levitation forces during increasing and decreasing field processes for the sample 1 and sample 2. It is notable that both curves exhibit hysteresis, which is explained in terms of irreversible field penetration and trapping in the superconductor. The magnetic field change on the surface of the sample 1 and sample 2 for three cycles are shown in Figs. 4 and 5, respectively. The external field is also plotted as a dotted line in the figure. One can see the field strength
Fig. 3. Levitation force (sample 1 and sample 2).
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T. Nishikawa et al. / Physica C 392–396 (2003) 634–638
Fig. 6. Schematic illustration of the shielding current distribution.
directions can be written as Eqs. (2) and (3), respectively. Fig. 4. Magnetic field on the surface of sample 1.
dr ¼
Bz Bzfc ; kl0 Jc
ð2Þ
dz ¼
Br Brfc ; kl0 Jc
ð3Þ
where k is the Nagaoka coefficient and Brfc and Bzfc are r and z direction components of Bfc , respectively. 3.2. Magnetic field induced by shielding current We estimated the magnetic field induced by shielding current using Eq. (4): Bsc ¼ Bex Bz ;
Fig. 5. Magnetic field on the surface of sample 2.
gradually decreases from the edge toward the sample center. The field strength inside the Hall sensor of the position 4 was almost equal to the position 4, so that we did not plot the data. 3.1. Flux penetration In the process of increasing the external field, the shielding current is induced as to cancel the variation of the field in a levitated bulk superconductor (Fig. 6). Taking account of symmetry, governing equation rot B ¼ l0 J c can be rewritten as Eq. (1). l0 Jc# ¼
dBz dBr ; dr dz
ð1Þ
where Br and Bz are r and z axis components of the magnetic field, Jc the critical current density. Based on this, the penetration depth for radial and z
ð4Þ
where the Bsc is the magnetic field by shielding current, Bex and Bz are the external field and the measured field, respectively. We also estimated the magnetic field using the Biot–SavartÕs law (shown in Eq. (5) in that the penetration depth of each layer was estimated using Eqs. (2) and (3)). # " l0 I 1 a2 r 2 z 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K þ Bsc ¼ E ; 2 2p 2 ða rÞ z2 ða þ rÞ þ z2 ð5Þ where K and E are the complete elliptic integrals of the first and the second kinds, respectively. And I is the shielding current flowing through each layer and a is the radius of the each current loop. We then compared the calculated values using Eq. (5) with the experimental data. Figs. 7 and 8 show the results of comparison for each sample.
T. Nishikawa et al. / Physica C 392–396 (2003) 634–638
Fig. 7. Comparison of the field by shielding current (sample 1).
637
Fig. 9. Comparison of the levitation force (both samples).
where T is the thickness of sample, R is the radius of the sample. Here we used a Jc value of 1.5 · 108 A/m2 for each sample, which was selected based on the best fitting. Fig. 9 shows the calculated levitation forces. One can see that simulated levitation forces are in agreement with experimental data, showing that our assumption that the currents are flowing as multiple loops is applicable to the present levitation phenomenon. 4. Summary Fig. 8. Comparison of the field by shielding current (sample 2).
The calculated values were in good agreement with the measured data for both samples. 3.3. Levitation force The levitation force of a bulk superconductor was calculated for both samples. For estimation, we assumed that Jc is constant in the bulk superconductor and further used the Jc value as a fitting parameter. Here the shielding current was assumed to flow as multiple thin current loops and the Lorentz force generated from each current loop was integrated. Thus the total levitation force is given by Eq. (6). Z T Z R F ¼ Jc Br 2pr dr dt 0
þ
Z
Rdr T
Z 0
Acknowledgements This work was supported by the New Energy and Industrial Technology Development Organization (NEDO) as Collaborative Research and Develop of Fundamental Technologies for Superconductivity Applications. References
R
Jc Br 2pr dr dt; T dz
The levitation force of bulk superconductors exhibited a hysteresis loop during increasing and decreasing field processes. We performed numerical simulations of the magnetic field and the levitation forces, which were in good agreement with experimental data when Jc values were used as a fitting parameter.
ð6Þ
[1] Y. Iwasa, paper presented at ISSÕ96, Sapporo, Japan, 1996.
638
T. Nishikawa et al. / Physica C 392–396 (2003) 634–638
[2] K. Sawa, H. Horiuchi, K. Nishi, Y. Iwasa, H. Tsuda, H. Lee, K. Nagashima, T. Miyamoto, M. Murakami, H. Fujimoto, paper presented at 4th International Symposium on Magnetic Suspension, Gifu, Japan, 1997. [3] Y. Tachi, N. Uemura, K. Sawa, Y. Iwasa, K. Nagashima, T. Otani, T. Miyamoto, M. Tomita, M. Murakami, Supercond. Sci. Technol. 13 (2000) 850.
[4] T. Nishikawa, K. Sawa, N. Yamachi, M.Murakami, paper presented at The 14th Symposium on Electromagnetics and Dynamics, Okayama, Japan, 2002. [5] K. Nishi, Y. Tachi, K. Sawa, Y. Iwasa, K. Nagashima, T. Miyamoto, M. Tomita, M. Murakami, paper presented at ISSÕ98, Fukuoka, Japan, 1998.
Estimation of the shielding current distribution in levitated YBCO bulk superconductor T. Nishikawa a
a,b,*
, Y. Komano a,b, K. Sawa a, Y. Iwasa c, N. Yamachi b, M. Murakami b
Department of System Design Engineering, KEIO University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama, Japan b ISTEC, Superconductivity Research Laboratory, 1-16-25 Shibaura, Minato-ku, Tokyo, Japan c Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, 02139 Cambridge, MA, USA Received 13 November 2002; accepted 27 January 2003
Abstract The high temperature superconductor (HTS) has very high Jc value and can trap the very high magnetic field. The YBa2 Cu3 Ox (YBCO) bulk superconductor is one of these attractive HTS and expected for many applications. In our previous study, to estimate the shielding current distribution in levitated YBCO bulks, we directly measured the levitation force and values of magnetic field on the surface of levitated YBCO bulk superconductor by using load cell and Hall probes. In this paper, we proposed shielding current distribution based on the axisymmetric 3-dimensional Bean-model and calculated the levitation force and magnetic fields on the surface of bulk superconductor. These calculated results were compared with experimental data. 2003 Published by Elsevier B.V. PACS: 74.72.Bk; 74.76.Bz Keywords: Shielding current distribution; Levitation force; Bean model; YBCO bulk
1. Introduction Bulk superconductors have significant potential for various applications such as magnetic bearing and non-contact transport, since superconductors can stably levitate over permanent magnets or vice versa. However, so long as permanent magnets are used as the magnetic source, the levitation force
*
Corresponding author. Tel.: +81-45-563-1141x43057; fax: +81-45-566-1720. E-mail address: [email protected] (T. Nishikawa). 0921-4534/$ - see front matter 2003 Published by Elsevier B.V. doi:10.1016/S0921-4534(03)00995-X
and the levitation height cannot be controlled. Iwasa [1] proposed the so-called ‘‘electromaglev’’ system in which an electromagnetic coil was used as the magnetic source. The system is attractive for practical applications, since the levitation force and levitation height can be controlled by simply changing the coil current. We constructed a levitation system with a Bi2 Sr2 Ca2 Cu3 O10 superconducting coil as a magnetic source for achieving a large levitation force without heat generation [2]. In this levitation system, we have succeeded in controlling the electromagnetic force and the levitation height by
T. Nishikawa et al. / Physica C 392–396 (2003) 634–638
changing the coil current. We further developed the force measurement system in which we can directly measure the levitation force and the magnetic field on the surface of the levitated bulk superconductors [3]. From these measurements, we have confirmed that the flux pinning properties strongly influence the levitation forces. We also derived a theoretical model to simulate the levitation behavior for a levitated bulk superconductor. In this study, we measured the levitation force and the magnetic field, which was compared with the model that we derived.
2. Experimental 2.1. Apparatus and samples We used YBa2 Cu3 Ox bulk superconductors grown by the top-seeded melt-growth process. The samples were single grains of 46 mm diameter and 15 mm or 7 mm in thickness. The levitation coil was composed of a Bi2 Sr2 Ca2 Cu3 O10 double pancake coil that operates below 30 K and is cooled by a cryo-cooler. Seven Hall sensors were mounted on the bulk surface with the inter-sensor distance of 2.5 mm (as shown in Fig. 1). The experimental system consists of a balance, a load cell and Hall sensors used in the previous study [4] as shown in Fig. 2. 2.2. Procedures In the levitation force measurements, the electric currents were passed through a levitation coil that generated the field Bfc . Next, the superconductor was placed at the center of the coil and
Fig. 1. Configuration of plural hall sensors.
635
Fig. 2. Experimental apparatus.
cooled with liquid nitrogen in the presence of Bfc . A previous study [5] demonstrated the importance of field-cooling for stable levitation of large bulk superconductors. We then increased the coil current to levitate the YBCO sample and measured the levitation force. We repeated increasing and decreasing field processes for three times.
3. Results and discussion Fig. 3 shows the levitation forces during increasing and decreasing field processes for the sample 1 and sample 2. It is notable that both curves exhibit hysteresis, which is explained in terms of irreversible field penetration and trapping in the superconductor. The magnetic field change on the surface of the sample 1 and sample 2 for three cycles are shown in Figs. 4 and 5, respectively. The external field is also plotted as a dotted line in the figure. One can see the field strength
Fig. 3. Levitation force (sample 1 and sample 2).
636
T. Nishikawa et al. / Physica C 392–396 (2003) 634–638
Fig. 6. Schematic illustration of the shielding current distribution.
directions can be written as Eqs. (2) and (3), respectively. Fig. 4. Magnetic field on the surface of sample 1.
dr ¼
Bz Bzfc ; kl0 Jc
ð2Þ
dz ¼
Br Brfc ; kl0 Jc
ð3Þ
where k is the Nagaoka coefficient and Brfc and Bzfc are r and z direction components of Bfc , respectively. 3.2. Magnetic field induced by shielding current We estimated the magnetic field induced by shielding current using Eq. (4): Bsc ¼ Bex Bz ;
Fig. 5. Magnetic field on the surface of sample 2.
gradually decreases from the edge toward the sample center. The field strength inside the Hall sensor of the position 4 was almost equal to the position 4, so that we did not plot the data. 3.1. Flux penetration In the process of increasing the external field, the shielding current is induced as to cancel the variation of the field in a levitated bulk superconductor (Fig. 6). Taking account of symmetry, governing equation rot B ¼ l0 J c can be rewritten as Eq. (1). l0 Jc# ¼
dBz dBr ; dr dz
ð1Þ
where Br and Bz are r and z axis components of the magnetic field, Jc the critical current density. Based on this, the penetration depth for radial and z
ð4Þ
where the Bsc is the magnetic field by shielding current, Bex and Bz are the external field and the measured field, respectively. We also estimated the magnetic field using the Biot–SavartÕs law (shown in Eq. (5) in that the penetration depth of each layer was estimated using Eqs. (2) and (3)). # " l0 I 1 a2 r 2 z 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K þ Bsc ¼ E ; 2 2p 2 ða rÞ z2 ða þ rÞ þ z2 ð5Þ where K and E are the complete elliptic integrals of the first and the second kinds, respectively. And I is the shielding current flowing through each layer and a is the radius of the each current loop. We then compared the calculated values using Eq. (5) with the experimental data. Figs. 7 and 8 show the results of comparison for each sample.
T. Nishikawa et al. / Physica C 392–396 (2003) 634–638
Fig. 7. Comparison of the field by shielding current (sample 1).
637
Fig. 9. Comparison of the levitation force (both samples).
where T is the thickness of sample, R is the radius of the sample. Here we used a Jc value of 1.5 · 108 A/m2 for each sample, which was selected based on the best fitting. Fig. 9 shows the calculated levitation forces. One can see that simulated levitation forces are in agreement with experimental data, showing that our assumption that the currents are flowing as multiple loops is applicable to the present levitation phenomenon. 4. Summary Fig. 8. Comparison of the field by shielding current (sample 2).
The calculated values were in good agreement with the measured data for both samples. 3.3. Levitation force The levitation force of a bulk superconductor was calculated for both samples. For estimation, we assumed that Jc is constant in the bulk superconductor and further used the Jc value as a fitting parameter. Here the shielding current was assumed to flow as multiple thin current loops and the Lorentz force generated from each current loop was integrated. Thus the total levitation force is given by Eq. (6). Z T Z R F ¼ Jc Br 2pr dr dt 0
þ
Z
Rdr T
Z 0
Acknowledgements This work was supported by the New Energy and Industrial Technology Development Organization (NEDO) as Collaborative Research and Develop of Fundamental Technologies for Superconductivity Applications. References
R
Jc Br 2pr dr dt; T dz
The levitation force of bulk superconductors exhibited a hysteresis loop during increasing and decreasing field processes. We performed numerical simulations of the magnetic field and the levitation forces, which were in good agreement with experimental data when Jc values were used as a fitting parameter.
ð6Þ
[1] Y. Iwasa, paper presented at ISSÕ96, Sapporo, Japan, 1996.
638
T. Nishikawa et al. / Physica C 392–396 (2003) 634–638
[2] K. Sawa, H. Horiuchi, K. Nishi, Y. Iwasa, H. Tsuda, H. Lee, K. Nagashima, T. Miyamoto, M. Murakami, H. Fujimoto, paper presented at 4th International Symposium on Magnetic Suspension, Gifu, Japan, 1997. [3] Y. Tachi, N. Uemura, K. Sawa, Y. Iwasa, K. Nagashima, T. Otani, T. Miyamoto, M. Tomita, M. Murakami, Supercond. Sci. Technol. 13 (2000) 850.
[4] T. Nishikawa, K. Sawa, N. Yamachi, M.Murakami, paper presented at The 14th Symposium on Electromagnetics and Dynamics, Okayama, Japan, 2002. [5] K. Nishi, Y. Tachi, K. Sawa, Y. Iwasa, K. Nagashima, T. Miyamoto, M. Tomita, M. Murakami, paper presented at ISSÕ98, Fukuoka, Japan, 1998.