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Modeling of hysteretic behavior of the levitation force between superconductor and permanent magnet
Accepted Manuscript Modeling of hysteretic behavior of the levitation force between superconductor and permanent magnet Xing-da Wu, Ke-Xi Xu, Yue Cao, Shun-bo Hu, Peng-xiang Zuo, Guan-dong Li PII: DOI: Reference:
S0921-4534(12)00411-X http://dx.doi.org/10.1016/j.physc.2012.12.006 PHYSC 1252231
To appear in:
Physica C
Received Date: Revised Date: Accepted Date:
20 April 2012 9 November 2012 16 December 2012
Please cite this article as: X-d. Wu, K-X. Xu, Y. Cao, S-b. Hu, P-x. Zuo, G-d. Li, Modeling of hysteretic behavior of the levitation force between superconductor and permanent magnet, Physica C (2012), doi: http://dx.doi.org/ 10.1016/j.physc.2012.12.006
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Corresponding Author: Xing-da Wu Postal Address: School of Information Engineering, Guangdong Medical College, No. 2, Eastern Wenming Road, Zhanjiang, Guangdong Province, China Postal Code: 524023 Telephone Number: 86-0759-2388572 E-mail Address of the Corresponding Author: [email protected]
1
Modeling of hysteretic behavior of the levitation force between superconductor and permanent magnet
Xing-da Wu a, Ke-Xi Xu b,*, Yue Cao b, Shun-bo Hu b, Peng-xiang Zuo b and Guan-dong Li b
a
School of Information Engineering, Guangdong Medical College, Zhanjiang 524023, People’s
Republic of China b
Department of Physics, Shanghai University, Shanghai 200444, People’s Republic of China
*E-mail: [email protected]
Abstract
The hysteretic behavior of the levitation force between a permanent magnet and a melt-textured-growth YBCO bulk has been investigated under both zero-field cooling (ZFC) and field cooling (FC) processes. It is found that both in ZFC and FC measurements, the hysteresis loop for the first descent/ascent cycle of magnet is relatively larger than that for the second or third cycle, and the hysteresis loops for Cycle 2-4 have the same area. These results can be qualitatively understood in terms of the critical state model. To describe these experimental results, we develop an updated frozen-image model, which is obtained by modifying the change rules of the vertical movement image in the advanced frozen-image model proposed by Yang et al.. Comparing with the advanced frozen-image model proposed by Yang et al., our model can not only give the
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hysteretic characteristic in the first descent-ascent cycle of magnet, but also show the hysteresis loops with the same area for the second and subsequent cycles.
PACS: 74.25.Ha; 74.81.Bd
Keywords: Hysteretic behavior; Levitation force; Vertical movement; Frozen-image model
1. Introduction
It has been well known that a permanent magnet (PM) can be stably levitated above a bulk high temperature superconductor (HTS) operating at liquid nitrogen temperature. This fascinating magnetic levitation is attractive for various industrial applications such as noncontact superconducting bearings [1], flywheel energy storage systems [2, 3] and motors[4]. In order to better design the magnetic levitation system between the PM and the HTS, thoroughly understanding the characteristics of the levitation force is necessary, as a result, many experimental measurements as well as theoretical calculations have been performed[5-14]. To describe the field cooling behavior of the levitation force, Kordyuk[5] proposed a frozen-image model by using the frozen PM image that created the same magnetic field distribution outside the HTS as the frozen magnetic flux did. The relationships between levitation force and vertical displacement in the different initial cooling heights can be obtained by the frozen-image model. However, as pointed out by Hull and Cansiz [8], this model can not predict hysteresis in the levitation force, since it does not account for the change of the magnetic flux inside HTS when the
3
PM moves. Actually, force-displacement hysteresis is the most common feature of the magnetic levitation, and has been found in many experimental studies [6-8]. To give the hysteretic characteristic in the levitation force, Yang and Zheng proposed an advanced frozen-image model (we call it “Yang’s model” in this paper), in which a magnetic dipole named vertical movement image was added [12]. Subsequently Zhang et al. [13] modified the location of the vertical movement image to describe the relationship between the maximum levitation forces and the initial cooling heights; and their new model can give the result that the maximum levitation forces tend to saturation with the cooling heights increasing, which has been observed in Yang et al.’s levitation force measurements [15]. Recently this frozen-image model is further improved by J. Zhou et al. to describe the influence of lateral moving speed on vertical force in a HTS levitation system [14]. More and more experimental phenomena can be qualitatively described by Yang’s model based on the above-mentioned improvement. However, according to Yang’s model, the maximum levitation forces in different descent/ascent cycles of PM do not equal to each other. This result is not in accord with the experimental fact that the maximum levitation force in the first descent/ascent cycle is equal to that in the second one [16]. Therefore, Yang’s model needs to be further improved. In this article, we first present the experimental results of the hysteretic behavior of the levitation force between a permanent magnet and a melt-textured-growth YBCO bulk for both zero-field cooling (ZFC) and field cooling (FC) cases, and then propose an updated frozen-image model by modifying the change rules of the vertical movement image in Yang’s model. Our new model can not only give the hysteretic characteristic in the first descent/ascent cycle of magnet, but also show the hysteresis loops with the same area for Cycle 2-4, which is
4
agreement with the experimental results.
2. Experimental procedure
For levitation force measurements, a hexagon-shaped YBCO bulk, with a face diagonal length of 34 mm and a thickness of 12 mm, was prepared by the top-seeded melt-growth process. Fabrication details of YBCO bulk samples have been reported elsewhere [17-18]. The temperature dependence of the resistivity for the sample is shown in Fig. 1(a), it can be seen that the critical temperature (Tc) is 89.5 K, with a relatively narrow temperature transition width. The trapped magnetic field exhibits only a single peak as shown in Fig. 1(b), indicating that there is no weak link in the sample. The levitation forces were measured at liquid nitrogen temperature by a device designed by the General Research Institute Metals (Beijing). The YBCO bulk was fixed inside an open-topped Styrofoam container, and a cylindrical NdFeB PM was placed above YBCO bulk. The YBCO bulk and the PM were axisymmetrically fixed in their coaxial line in vertical direction. The PM was 30 mm in both diameter and thickness, and with surface field of 0.5 T. In the process of PM moving toward or away from YBCO bulk, the distance between the bottom surface of PM and the top surface of YBCO bulk was measured by a displacement sensor, and the repulsive force or attractive force between them is measured by the pressure sensor. Both ZFC and FC experiments were carried out in our study. In the ZFC measurements, YBCO bulk was cooled by a distance of 50 mm from the PM (where its magnetic field is negligible), and then the PM was approximated to the YBCO bulk by a speed of about 1 mm/s.
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When a minimal gap of 0.2 mm was reached the moving direction was inverted, and it was returned immediately with the same speed. This descent-ascent cycle was repeated four times. For FC refrigeration process the YBCO bulk was cooled in an initial distance of 0.2 mm from the PM, then the PM was elevated. When a maximal gap of 50 mm was reached, the PM was brought back to the initial distance above YBCO bulk. This ascent-descent cycle was also repeated four times.
3. Experimental results and discussion
Fig. 2 shows levitation force-distance curves of YBCO bulk measured for four continuous descent/ascent cycles of PM. YBCO sample cooled by ZFC (in Fig.2(a)) and FC methods (in Fig.2(b)), respectively. Fig. 2 shows three features of the experimental results. Firstly, the levitation force – distance curve always shows a small hysteresis loop during each descent/ascent cycle of the PM. Secondly, the hysteresis loop for the first descent/ascent cycle of the PM is larger than that for the second, third, and fourth cycles; and the levitation force – distance curves for the second, third and fourth cycles almost overlap each other. Thirdly, in ZFC case, four levitation force – distance curves, corresponding to the first, second, third and fourth ascent of the PM respectively, overlap each other; and in FC case, four levitation force – distance curves for the first, second, third and fourth descent of the PM overlap each other. These results can be qualitatively understood in terms of the Bean critical state model, which considers the critical current density Jc as constant [19]. In the ideal Bean model, superconducting persistent currents induced by the varying external field, flow in the outer ring of the sample. The radial thickness of this ring depends only on Jc and the magnitude of external field. Fig. 3 shows
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distributions of magnetic field and induced current in a rectangle HTS sample subjected to the external magnetic field in ZFC measurements. When the PM locates at the highest position h0 away from YBCO sample, the magnetic field that the sample feels can be neglected. No magnetic flux will be trapped in YBCO sample after cooling the sample to liquid nitrogen temperature. As the PM descends from the highest position h0, magnet field in the vicinity of superconductor is increased, a surface layer of superconducting shielding current is formed, and some magnetic flux lines penetrate the superconductor. Fig.3(a) shows the induced current distribution when the PM is at a certain height h1 between the highest and lowest positions in the first descent. The induced current and the magnetic field distributed in HTS are in accordance with Maxwell’s equation. When the PM descends to the lowest position h2, the largest induced current is obtained (as shown in Fig.3(b)), resulting in the largest levitation force. Then the PM ascends from the lowest position, the trapped flux is difficult to exit due to the flux-pinning effect, so the levitation force during the ascent is always smaller than the one during the descent. A reverse induced current at the outer rim of superconductor is generated during the ascending process of the PM. Fig. 3(c) and 3(d) show the induced current distribution when the PM is at the height h1 and h0 of the first ascent. Fig. 3(e) and 3(f) show the induced current distribution when the PM is at the height h1 and h2 of the second descent. Due to the trapped magnetic field, the levitation force during the second descent is always smaller than the one during the first descent, except that when the PM locates at the lowest position. As can be seen in Fig. 3(b) and 3(f), the induced current distribution is identical, indicating the same maximum levitation force can be obtained when the PM descend to the lowest position h2 in the first and second descent. It is easy to find that the distribution changes in magnetic field and induced current during the second ascending process are the same as the one in
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the first ascent. In further descent-ascent cycles, the current changes affect only the same region as in the second cycle, reversing the current flow at the approach of the PM and again at its retreat. Therefore, the levitation force-distance curves for the second, third, and fourth descent-ascent cycles overlap each other. The experimental results for the FC measurement can be also discussed qualitatively based on the Bean model. When the PM is moved away from the HTS after field cooling, an attractive force (negative) due to the trapped field is created. The distributions of the trapped magnetic field and superconducting current in the HTS when the PM ascends to the highest position h0 are shown in Fig.4(a) (solid line). The dashed lines in Fig.4(a) represent the trapped field distribution when the PM ascends to a certain height h1 between the highest and lowest positions. We assume that this superconducting current flows clockwise. Then, when the PM descends, a new layer of shielding current with a reverse direction (i.e. anticlockwise) appears at the outer rim of the HTS. Fig.4(b) and Fig.4(c) show the distributions of magnetic field and current in the HTS when the PM descents to the height h1 and the lowest position h2, respectively. It can be found that some magnetic flux penetrate the HTS during the descending process, and there are two layers of current in the HTS, leading to repulsive force (positive) and attractive force respectively. As the PM approaches the surface of the HTS, the anticlockwise current becomes stronger and stronger, and the clockwise current changes in an opposite way, resulting in a gradual increase in the total levitation force. The largest levitation force is obtained when the PM descends to the lowest position. Then the PM ascends for the second time, the third layer of current flowing clockwise appears at the outer rim of the HTS. Fig.4(d) shows the induced current distribution when the PM ascends to the height h1. Since more magnetic flux is trapped for any given height, the levitation
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force during the second ascending process is always smaller than the one during the descending process. Similarly, since less magnetic flux is trapped for any given height, the attractive force during the second ascent is always smaller than that during the first one. When the PM ascends to the highest position h0, the anticlockwise current vanishes, and the distributions of the magnetic field and current in the HTS are the same as of Fig. 4(a) (solid line). Then the second descent process begins, it is easy to find that the distribution changes in the magnetic field and current during the second descent process are the same as that in the first descent. In further ascent-descent cycles, the current changes are the same as that in the second cycle. Therefore, a larger hysteresis loop of levitation force-distance curve is observed in the first ascent-descent cycle of PM, and the levitation force-distance curves for the second, third, and fourth ascent-descent cycles overlap each other.
4. Modeling
4.1 Analysis based on Yang’s model According to the frozen-image model, the PM is considered as a point magnetic dipole m1, which is above the HTS top surface, in addition, there are two images below the HTS top surface: one is the diamagnetic mirror image m2 created by screening currents in the HTS, and the other is the frozen image m3 created by trapped flux in the HTS at certain initial cooling process. The diamagnetic mirror image m2 moves when the PM moves so that its lateral position equals that of the PM and its vertical height below the HTS surface equals the height of the PM above the surface. Once formed, the frozen image m3 does not change and move and has a fixed position
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below the HTS surface equals the height of the PM above the surface at the initial cooling position [8]. The frozen-image model does not account for flux flow, and can not predict hysteresis in the levitation force. To describe the hysteretic characteristic in the levitation force, Yang and Zheng modified the frozen-image model by adding a magnetic dipole m4 named vertical movement image, which indicates that the magnetic flux changes in the HTS as the PM moves vertically [12]. The schematic diagram of this advanced frozen-image model and detailed calculation can be found in Ref. [12]. When the PM moves vertically, the levitation force between the PM and superconductor can be described as
Fz ( z ) = A[
2 2 2 m2 − m3 − m4 ] 4 4 (2 z ) ( z + h) ( z + z0 ) 4
(1)
where A = (3μ 0 m1 ) / 4π , z0 is the minimum vertical distance between the magnet and the HTS and h is the initial field cooling height. The magnitude of the magnetic moment of the frozen image is equal to that of the diamagnetic image (m3 = m2) [8, 12]. The variable z is the distance between the PM and HTS. The quantity of the vectors m1, m2 and m3 are fixed for a given PM/HTS system, so the levitation force is determined only by the distance z and the vertical movement image m4. To simplify the problem, a linear rule of m4 is given in Yang’s model. For ZFC case, when the PM descends from the initial cooling position z = h to the lowest position z = z0, the magnetization of superconductor is diamagnetic, the rule of m4 is given as
m4 = −a1 (h − z )
(2)
where a1 is a positive proportional constant. The negative maximum of m4 is
− a1 (h − z0 ) when
the PM arrives at the lowest position. When the PM ascends from the lowest position to the initial 10
cooling height, the magnetization of the superconductor is paramagnetic, m4 is given as
m4 = −a1 (h − z0 ) + a2 ( z − z0 )
(3)
where a2 is a positive proportional constant. The difference between
a1 and a2 externalizes the
hysteretic characteristic in the levitation force. According to Ref. [12], when the PM descends in the second time, the rule of m4 is given as
m4 = −a1 (h − z0 ) + a2 (h − z0 ) − a1 (h − z ) The final value of m4 is
(4)
− a1 (h − z0 ) + (a2 − a1 )(h − z0 ) when the PM arrives at the lowest
position; obviously it is not equal to the one when PM arrives at the lowest position in the first descent, meaning that the maximum levitation force in the first descent is different from the one in the second descent. When the PM ascends in the second time, the rule of m4 is described by
m4 = (a2 − 2a1 )(h − z0 ) + a2 ( z − z0 ) .
(5)
Then the PM descends in the third time, the rule of m4 is given as
m4 = 2(a2 − a1 )(h − z0 ) − a1 (h − z )
(6)
When the PM arrives at the lowest position, the final value is − a1 ( h − z0 ) + 2( a2
− a1 )(h − z0 ) ,
which is also different from the two previous values. For FC case, when the PM ascends from the initial cooling position z = z0 (h = z0) to the highest position z = zm, the magnetic field which the superconductor feels becomes weak and the magnetization of the superconductor is paramagnetic, so the sign of m4 is positive. The rule of m4 is given as
m4 = b1 ( z − z0 )
(7)
where b1 is a positive proportional constant. When the PM arrives at the maximum height zm, the positive maximum of m4 is b1 ( zm − z0 ) . When the PM descends from the maximum height zm to
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the lowest position z0, the magnetization of HTS is diamagnetic, m4 is given as
m4 = b1 ( zm − z0 ) − b2 ( zm − z )
(8)
where b2 is a positive proportional constant. The difference between b1 and b2 externalizes the hysteretic effect in the levitation force. When the PM arrives at the lowest position, the value of m4 is (b1 − b2 )( zm − z0 ) . According to Ref. [12], when the PM ascends in the second time, the rule of m4 is given as
m4 = b1 ( zm − z0 ) − b2 ( zm − z0 ) + b1 ( z − z0 )
(9)
The final value of m4 is (2b1 − b2 )( zm − z0 ) when the PM arrives at the highest position. Then the PM descends in the second time, the rule of m4 is described by
m4 = (2b1 − b2 )( zm − z0 ) − b2 ( zm − z )
(10)
The final value of m4 is 2(b1 − b2 )( zm − z0 ) when the PM arrives at the lowest position; obviously it is not equal to the one when PM arrives at the lowest position in the first descent, meaning that the maximum levitation force in the first descent is different from the one in the second descent. When the PM ascends in the third time, the rule of m4 is described by
m4 = 2(b1 − b2 )( zm − z0 ) + b1 ( z − z0 )
(11)
Then the PM descends in the third time, the rule of m4 is given as
m4 = (3b1 − 2b2 )( zm − z0 ) − b2 ( zm − z )
(12)
When the PM arrives at the lowest position, the final value is 3(b1 − b2 )( zm − z0 ) , which is also different from the two previous values. Therefore, in order to correctly describe the hysteretic characteristic in the second and subsequent descent/ascent cycles of PM, the change rule of m4 in both ZFC and FC cases needs to be modified.
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4.2 Updated frozen-image model When the PM descends from the initial cooling position z = h to the lowest position z = z0, the rule of m4 can be given as
m4 = −a1 (h − z )0.8 The negative maximum of m4 is
(13)
− a1 (h − z0 )0.8 when the PM arrives at the lowest position z0.
When the PM ascends from the lowest position to the initial cooling height, the magnetization of the superconductor is paramagnetic, m4 is given as
m4 = −a1 (h − z0 )0.8 + a2 ( z − z0 )0.8
(14)
Then the PM descends in the second time, the magnetic field that the superconductor feels becomes strong again, so the value of m4 decreases. In order to simplify the problem, the decreasing rule of m4 in the second descent is regarded as the same as the increasing rule in the first ascent. The rule of m4 is given as
m4 = −a1 (h − z0 ) 0.8 + a2 (h − z0 ) 0.8 − a2 (h − z ) 0.8
(15)
When the PM arrives at the lowest position in the second descent, the final value of m4 is − a1 ( h − z0 ) 0.8 , which is equal to the one in the first descent. In the second ascent, the rule of m4 can be described by Eq. (14). In the subsequent descent-ascent cycles, the rule of m4 for the descending process can be described by Eq. (15), and the rule for the ascending process can be described by Eq. (14). If Eqs. (13), (14) and (15) are substituted into Eq. (1), respectively, three levitation force distance curves can be obtained as shown in Fig. 5: the top curve gives the dependence of levitation force on the distance z for the first descent of PM, the bottom curve gives the
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dependence of levitation force on the distance for the ascending process in every descent-ascent cycle of PM, and the middle one gives the dependence of levitation force on the distance for the descending process in the second and subsequent cycles of PM. In order to compare with the results reported in Ref. [12], in the calculations above the parameters are selected as: m1 = m2 = m3 = 5.0×10-3 A m2, a1 = 0.1 A m, a2 = 5.0 A m, and the maximum and minimum height of PM are 30 and 0.5 mm, respectively. As can be seen from Fig. 5, the levitation force - distance curves obtained by our update model agree with the ones obtained in the experimental curves shown in Fig. 2(a) qualitatively. For FC case, when the PM ascends from the initial cooling position z = z0 (h = z0) to the highest position z = zm, the rule of m4 is given as
m4 = b1 ( z − z0 )0.3
(16)
The positive maximum of m4 is b1 ( zm − z0 )
0 .3
when the PM arrives at the maximum height zm.
When the PM descends from the maximum height zm to the lowest position z0, the magnetization of HTS is diamagnetic, m4 is given as
m4 = b1 ( zm − z0 )0.3 − b2 ( zm − z )0.3
(17)
Then the PM ascends in the second time, the magnetic field that the superconductor feels becomes weak again, so the value of m4 increases. In order to simplify the problem, the increasing rule of m4 in the second ascent is regarded as the same as the decreasing rule in the first descent. The rule of m4 is given as
m4 = (b1 − b2 )( zm − z0 )0.3 + b2 ( z − z0 )0.3 The final value of m4 is b1 ( zm − z0 )
0 .3
(18)
when the PM arrives at the highest position. Then the PM
descends in the second time, the rule of m4 is described by
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m4 = b1 ( zm − z0 )0.3 − b2 ( zm − z )0.3
(19)
The change rule of m4 during the second descent is the same as that during the first descent, suggesting that the levitation force-distance curves for the first and second descent of PM will overlap each other. In the third ascent, the rule of m4 can be described by Eq. (18). In the subsequent descent-ascent cycles, the rule of m4 for the descending process can be described by Eq. (19), and the rule for the ascending process can be described by Eq. (18). If Eqs. (16), (17) and (18) are substituted into Eq. (1) respectively, and the calculation parameters are selected as: m1 = m2 = m3 = 5.0×10-3 A m2, b1 = 5.0×10-2 A m, b2 = 1.0 A m, h = z0 = 0.5 mm, zm = 30 mm, three levitation force- curves can be obtained as shown in Fig. 6: the bottom curve gives the dependence of levitation force on the distance z for the first ascent of PM, the top curve gives the dependence of levitation force on the distance for the descending process in every ascent-descent cycle of PM, and the middle curve gives the dependence of levitation force on the distance for the ascending process in the second and subsequent cycles of PM. As can be seen from Fig. 6, the levitation force –distance curves obtained by our update model agree with the experimental results shown in Fig. 2(b) qualitatively.
5. Conclusion
The hysteretic behavior of the levitation force during multiple descent/ascent cycles of the permanent magnet has been investigated under both ZFC and FC processes. It is found that the hysteresis loop for the first descent/ascent cycle of magnet is larger than that for the second and subsequent cycles, and the levitation force - distance curves for the second, third, and fourth
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cycles almost overlap each other. It is proved that Yang’s model can not predict the hysteretic characteristic in the levitation force during multiple descent/ascent cycles of the PM. An updated frozen-image model is developed by modifying the change rules of the vertical movement image in Yang’s model. Our model is able to describe the experimental results in both ZFC and FC cases correctly, and at the same time retains the simplicity of the frozen-image model.
Acknowledgments
This work is supported by “Doctoral Initiating Project of Guangdong Medical College (B2011008)”, and partially supported by the “Shanghai Leading Academic Discipline Project, No. S30105” and the “Chinese National Programs for High Technology Research and Development, No. 2009AA03Z205”.
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Fig. 1 The superconductivity of the YBCO sample: (a) the temperature dependence of resistance, (b) trapped magnetic field at 77K Fig. 2 The levitation force-distance curves of YBCO bulk measured for four continuous descent/ascent cycles of PM. YBCO bulk cooled by ZFC (a) and FC (b) methods, respectively. The arrows mean the direction of PM moving relative to YBCO sample. The numbers 1, 2, 3 and 4 beside the arrows indicate the cycle number of PM. Fig. 3 The dependence of magnetic field and induced current in Bean’s model for ZFC measurement, with PM locating at different positions away from HTS: (a) a certain height h1 between the highest and lowest positions in the first descent, (b) the lowest position h2 of the first descent (the first ascent), (c) the height h1 of the first ascent, (d) the highest position h0 of the first ascent (the second descent), (e) the height h1 of the second descent, (f) the lowest position h2 of the second descent Fig. 4 The dependence of magnetic field and induced current in Bean’s model for FC measurement, with PM locating at different positions away from HTS: (a) the highest position h0 (solid line) and a certain height h1 between the highest and lowest positions (dashed line) of the first ascent (the first descent), (b) the height h1 of the first descent, (c) the lowest position h2 of the first descent (the second ascent), (d) the height h1 of the second ascent Fig. 5 Levitation force vs distance between superconductor and PM in ZFC case from our updated model. Fig. 6 Levitation force vs distance between superconductor and PM in FC case from our updated model.
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B
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Reference: PHYSC 1252231 Editorial reference: PHYSC_PHYSC-D-12-00127 To be published in: Physica C: Superconductivity and its applications Title: Modeling of hysteretic behavior of the levitation force between superconductor and permanent magnet
Highlights: 1 Experimental results on hysteretic behavior of the levitaion force are presented. 2 Hysteresis loop for the first descent/ascent cycle of magnet is largest. 3 Hysteresis loop for the second and subsequent cycles almost overlap each other. 4 Yang’s frozen-image model can not describe this characteristic of levitation force. 5 An updated frozen-image model is developed to describe these experimental results.
25
S0921-4534(12)00411-X http://dx.doi.org/10.1016/j.physc.2012.12.006 PHYSC 1252231
To appear in:
Physica C
Received Date: Revised Date: Accepted Date:
20 April 2012 9 November 2012 16 December 2012
Please cite this article as: X-d. Wu, K-X. Xu, Y. Cao, S-b. Hu, P-x. Zuo, G-d. Li, Modeling of hysteretic behavior of the levitation force between superconductor and permanent magnet, Physica C (2012), doi: http://dx.doi.org/ 10.1016/j.physc.2012.12.006
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Corresponding Author: Xing-da Wu Postal Address: School of Information Engineering, Guangdong Medical College, No. 2, Eastern Wenming Road, Zhanjiang, Guangdong Province, China Postal Code: 524023 Telephone Number: 86-0759-2388572 E-mail Address of the Corresponding Author: [email protected]
1
Modeling of hysteretic behavior of the levitation force between superconductor and permanent magnet
Xing-da Wu a, Ke-Xi Xu b,*, Yue Cao b, Shun-bo Hu b, Peng-xiang Zuo b and Guan-dong Li b
a
School of Information Engineering, Guangdong Medical College, Zhanjiang 524023, People’s
Republic of China b
Department of Physics, Shanghai University, Shanghai 200444, People’s Republic of China
*E-mail: [email protected]
Abstract
The hysteretic behavior of the levitation force between a permanent magnet and a melt-textured-growth YBCO bulk has been investigated under both zero-field cooling (ZFC) and field cooling (FC) processes. It is found that both in ZFC and FC measurements, the hysteresis loop for the first descent/ascent cycle of magnet is relatively larger than that for the second or third cycle, and the hysteresis loops for Cycle 2-4 have the same area. These results can be qualitatively understood in terms of the critical state model. To describe these experimental results, we develop an updated frozen-image model, which is obtained by modifying the change rules of the vertical movement image in the advanced frozen-image model proposed by Yang et al.. Comparing with the advanced frozen-image model proposed by Yang et al., our model can not only give the
2
hysteretic characteristic in the first descent-ascent cycle of magnet, but also show the hysteresis loops with the same area for the second and subsequent cycles.
PACS: 74.25.Ha; 74.81.Bd
Keywords: Hysteretic behavior; Levitation force; Vertical movement; Frozen-image model
1. Introduction
It has been well known that a permanent magnet (PM) can be stably levitated above a bulk high temperature superconductor (HTS) operating at liquid nitrogen temperature. This fascinating magnetic levitation is attractive for various industrial applications such as noncontact superconducting bearings [1], flywheel energy storage systems [2, 3] and motors[4]. In order to better design the magnetic levitation system between the PM and the HTS, thoroughly understanding the characteristics of the levitation force is necessary, as a result, many experimental measurements as well as theoretical calculations have been performed[5-14]. To describe the field cooling behavior of the levitation force, Kordyuk[5] proposed a frozen-image model by using the frozen PM image that created the same magnetic field distribution outside the HTS as the frozen magnetic flux did. The relationships between levitation force and vertical displacement in the different initial cooling heights can be obtained by the frozen-image model. However, as pointed out by Hull and Cansiz [8], this model can not predict hysteresis in the levitation force, since it does not account for the change of the magnetic flux inside HTS when the
3
PM moves. Actually, force-displacement hysteresis is the most common feature of the magnetic levitation, and has been found in many experimental studies [6-8]. To give the hysteretic characteristic in the levitation force, Yang and Zheng proposed an advanced frozen-image model (we call it “Yang’s model” in this paper), in which a magnetic dipole named vertical movement image was added [12]. Subsequently Zhang et al. [13] modified the location of the vertical movement image to describe the relationship between the maximum levitation forces and the initial cooling heights; and their new model can give the result that the maximum levitation forces tend to saturation with the cooling heights increasing, which has been observed in Yang et al.’s levitation force measurements [15]. Recently this frozen-image model is further improved by J. Zhou et al. to describe the influence of lateral moving speed on vertical force in a HTS levitation system [14]. More and more experimental phenomena can be qualitatively described by Yang’s model based on the above-mentioned improvement. However, according to Yang’s model, the maximum levitation forces in different descent/ascent cycles of PM do not equal to each other. This result is not in accord with the experimental fact that the maximum levitation force in the first descent/ascent cycle is equal to that in the second one [16]. Therefore, Yang’s model needs to be further improved. In this article, we first present the experimental results of the hysteretic behavior of the levitation force between a permanent magnet and a melt-textured-growth YBCO bulk for both zero-field cooling (ZFC) and field cooling (FC) cases, and then propose an updated frozen-image model by modifying the change rules of the vertical movement image in Yang’s model. Our new model can not only give the hysteretic characteristic in the first descent/ascent cycle of magnet, but also show the hysteresis loops with the same area for Cycle 2-4, which is
4
agreement with the experimental results.
2. Experimental procedure
For levitation force measurements, a hexagon-shaped YBCO bulk, with a face diagonal length of 34 mm and a thickness of 12 mm, was prepared by the top-seeded melt-growth process. Fabrication details of YBCO bulk samples have been reported elsewhere [17-18]. The temperature dependence of the resistivity for the sample is shown in Fig. 1(a), it can be seen that the critical temperature (Tc) is 89.5 K, with a relatively narrow temperature transition width. The trapped magnetic field exhibits only a single peak as shown in Fig. 1(b), indicating that there is no weak link in the sample. The levitation forces were measured at liquid nitrogen temperature by a device designed by the General Research Institute Metals (Beijing). The YBCO bulk was fixed inside an open-topped Styrofoam container, and a cylindrical NdFeB PM was placed above YBCO bulk. The YBCO bulk and the PM were axisymmetrically fixed in their coaxial line in vertical direction. The PM was 30 mm in both diameter and thickness, and with surface field of 0.5 T. In the process of PM moving toward or away from YBCO bulk, the distance between the bottom surface of PM and the top surface of YBCO bulk was measured by a displacement sensor, and the repulsive force or attractive force between them is measured by the pressure sensor. Both ZFC and FC experiments were carried out in our study. In the ZFC measurements, YBCO bulk was cooled by a distance of 50 mm from the PM (where its magnetic field is negligible), and then the PM was approximated to the YBCO bulk by a speed of about 1 mm/s.
5
When a minimal gap of 0.2 mm was reached the moving direction was inverted, and it was returned immediately with the same speed. This descent-ascent cycle was repeated four times. For FC refrigeration process the YBCO bulk was cooled in an initial distance of 0.2 mm from the PM, then the PM was elevated. When a maximal gap of 50 mm was reached, the PM was brought back to the initial distance above YBCO bulk. This ascent-descent cycle was also repeated four times.
3. Experimental results and discussion
Fig. 2 shows levitation force-distance curves of YBCO bulk measured for four continuous descent/ascent cycles of PM. YBCO sample cooled by ZFC (in Fig.2(a)) and FC methods (in Fig.2(b)), respectively. Fig. 2 shows three features of the experimental results. Firstly, the levitation force – distance curve always shows a small hysteresis loop during each descent/ascent cycle of the PM. Secondly, the hysteresis loop for the first descent/ascent cycle of the PM is larger than that for the second, third, and fourth cycles; and the levitation force – distance curves for the second, third and fourth cycles almost overlap each other. Thirdly, in ZFC case, four levitation force – distance curves, corresponding to the first, second, third and fourth ascent of the PM respectively, overlap each other; and in FC case, four levitation force – distance curves for the first, second, third and fourth descent of the PM overlap each other. These results can be qualitatively understood in terms of the Bean critical state model, which considers the critical current density Jc as constant [19]. In the ideal Bean model, superconducting persistent currents induced by the varying external field, flow in the outer ring of the sample. The radial thickness of this ring depends only on Jc and the magnitude of external field. Fig. 3 shows
6
distributions of magnetic field and induced current in a rectangle HTS sample subjected to the external magnetic field in ZFC measurements. When the PM locates at the highest position h0 away from YBCO sample, the magnetic field that the sample feels can be neglected. No magnetic flux will be trapped in YBCO sample after cooling the sample to liquid nitrogen temperature. As the PM descends from the highest position h0, magnet field in the vicinity of superconductor is increased, a surface layer of superconducting shielding current is formed, and some magnetic flux lines penetrate the superconductor. Fig.3(a) shows the induced current distribution when the PM is at a certain height h1 between the highest and lowest positions in the first descent. The induced current and the magnetic field distributed in HTS are in accordance with Maxwell’s equation. When the PM descends to the lowest position h2, the largest induced current is obtained (as shown in Fig.3(b)), resulting in the largest levitation force. Then the PM ascends from the lowest position, the trapped flux is difficult to exit due to the flux-pinning effect, so the levitation force during the ascent is always smaller than the one during the descent. A reverse induced current at the outer rim of superconductor is generated during the ascending process of the PM. Fig. 3(c) and 3(d) show the induced current distribution when the PM is at the height h1 and h0 of the first ascent. Fig. 3(e) and 3(f) show the induced current distribution when the PM is at the height h1 and h2 of the second descent. Due to the trapped magnetic field, the levitation force during the second descent is always smaller than the one during the first descent, except that when the PM locates at the lowest position. As can be seen in Fig. 3(b) and 3(f), the induced current distribution is identical, indicating the same maximum levitation force can be obtained when the PM descend to the lowest position h2 in the first and second descent. It is easy to find that the distribution changes in magnetic field and induced current during the second ascending process are the same as the one in
7
the first ascent. In further descent-ascent cycles, the current changes affect only the same region as in the second cycle, reversing the current flow at the approach of the PM and again at its retreat. Therefore, the levitation force-distance curves for the second, third, and fourth descent-ascent cycles overlap each other. The experimental results for the FC measurement can be also discussed qualitatively based on the Bean model. When the PM is moved away from the HTS after field cooling, an attractive force (negative) due to the trapped field is created. The distributions of the trapped magnetic field and superconducting current in the HTS when the PM ascends to the highest position h0 are shown in Fig.4(a) (solid line). The dashed lines in Fig.4(a) represent the trapped field distribution when the PM ascends to a certain height h1 between the highest and lowest positions. We assume that this superconducting current flows clockwise. Then, when the PM descends, a new layer of shielding current with a reverse direction (i.e. anticlockwise) appears at the outer rim of the HTS. Fig.4(b) and Fig.4(c) show the distributions of magnetic field and current in the HTS when the PM descents to the height h1 and the lowest position h2, respectively. It can be found that some magnetic flux penetrate the HTS during the descending process, and there are two layers of current in the HTS, leading to repulsive force (positive) and attractive force respectively. As the PM approaches the surface of the HTS, the anticlockwise current becomes stronger and stronger, and the clockwise current changes in an opposite way, resulting in a gradual increase in the total levitation force. The largest levitation force is obtained when the PM descends to the lowest position. Then the PM ascends for the second time, the third layer of current flowing clockwise appears at the outer rim of the HTS. Fig.4(d) shows the induced current distribution when the PM ascends to the height h1. Since more magnetic flux is trapped for any given height, the levitation
8
force during the second ascending process is always smaller than the one during the descending process. Similarly, since less magnetic flux is trapped for any given height, the attractive force during the second ascent is always smaller than that during the first one. When the PM ascends to the highest position h0, the anticlockwise current vanishes, and the distributions of the magnetic field and current in the HTS are the same as of Fig. 4(a) (solid line). Then the second descent process begins, it is easy to find that the distribution changes in the magnetic field and current during the second descent process are the same as that in the first descent. In further ascent-descent cycles, the current changes are the same as that in the second cycle. Therefore, a larger hysteresis loop of levitation force-distance curve is observed in the first ascent-descent cycle of PM, and the levitation force-distance curves for the second, third, and fourth ascent-descent cycles overlap each other.
4. Modeling
4.1 Analysis based on Yang’s model According to the frozen-image model, the PM is considered as a point magnetic dipole m1, which is above the HTS top surface, in addition, there are two images below the HTS top surface: one is the diamagnetic mirror image m2 created by screening currents in the HTS, and the other is the frozen image m3 created by trapped flux in the HTS at certain initial cooling process. The diamagnetic mirror image m2 moves when the PM moves so that its lateral position equals that of the PM and its vertical height below the HTS surface equals the height of the PM above the surface. Once formed, the frozen image m3 does not change and move and has a fixed position
9
below the HTS surface equals the height of the PM above the surface at the initial cooling position [8]. The frozen-image model does not account for flux flow, and can not predict hysteresis in the levitation force. To describe the hysteretic characteristic in the levitation force, Yang and Zheng modified the frozen-image model by adding a magnetic dipole m4 named vertical movement image, which indicates that the magnetic flux changes in the HTS as the PM moves vertically [12]. The schematic diagram of this advanced frozen-image model and detailed calculation can be found in Ref. [12]. When the PM moves vertically, the levitation force between the PM and superconductor can be described as
Fz ( z ) = A[
2 2 2 m2 − m3 − m4 ] 4 4 (2 z ) ( z + h) ( z + z0 ) 4
(1)
where A = (3μ 0 m1 ) / 4π , z0 is the minimum vertical distance between the magnet and the HTS and h is the initial field cooling height. The magnitude of the magnetic moment of the frozen image is equal to that of the diamagnetic image (m3 = m2) [8, 12]. The variable z is the distance between the PM and HTS. The quantity of the vectors m1, m2 and m3 are fixed for a given PM/HTS system, so the levitation force is determined only by the distance z and the vertical movement image m4. To simplify the problem, a linear rule of m4 is given in Yang’s model. For ZFC case, when the PM descends from the initial cooling position z = h to the lowest position z = z0, the magnetization of superconductor is diamagnetic, the rule of m4 is given as
m4 = −a1 (h − z )
(2)
where a1 is a positive proportional constant. The negative maximum of m4 is
− a1 (h − z0 ) when
the PM arrives at the lowest position. When the PM ascends from the lowest position to the initial 10
cooling height, the magnetization of the superconductor is paramagnetic, m4 is given as
m4 = −a1 (h − z0 ) + a2 ( z − z0 )
(3)
where a2 is a positive proportional constant. The difference between
a1 and a2 externalizes the
hysteretic characteristic in the levitation force. According to Ref. [12], when the PM descends in the second time, the rule of m4 is given as
m4 = −a1 (h − z0 ) + a2 (h − z0 ) − a1 (h − z ) The final value of m4 is
(4)
− a1 (h − z0 ) + (a2 − a1 )(h − z0 ) when the PM arrives at the lowest
position; obviously it is not equal to the one when PM arrives at the lowest position in the first descent, meaning that the maximum levitation force in the first descent is different from the one in the second descent. When the PM ascends in the second time, the rule of m4 is described by
m4 = (a2 − 2a1 )(h − z0 ) + a2 ( z − z0 ) .
(5)
Then the PM descends in the third time, the rule of m4 is given as
m4 = 2(a2 − a1 )(h − z0 ) − a1 (h − z )
(6)
When the PM arrives at the lowest position, the final value is − a1 ( h − z0 ) + 2( a2
− a1 )(h − z0 ) ,
which is also different from the two previous values. For FC case, when the PM ascends from the initial cooling position z = z0 (h = z0) to the highest position z = zm, the magnetic field which the superconductor feels becomes weak and the magnetization of the superconductor is paramagnetic, so the sign of m4 is positive. The rule of m4 is given as
m4 = b1 ( z − z0 )
(7)
where b1 is a positive proportional constant. When the PM arrives at the maximum height zm, the positive maximum of m4 is b1 ( zm − z0 ) . When the PM descends from the maximum height zm to
11
the lowest position z0, the magnetization of HTS is diamagnetic, m4 is given as
m4 = b1 ( zm − z0 ) − b2 ( zm − z )
(8)
where b2 is a positive proportional constant. The difference between b1 and b2 externalizes the hysteretic effect in the levitation force. When the PM arrives at the lowest position, the value of m4 is (b1 − b2 )( zm − z0 ) . According to Ref. [12], when the PM ascends in the second time, the rule of m4 is given as
m4 = b1 ( zm − z0 ) − b2 ( zm − z0 ) + b1 ( z − z0 )
(9)
The final value of m4 is (2b1 − b2 )( zm − z0 ) when the PM arrives at the highest position. Then the PM descends in the second time, the rule of m4 is described by
m4 = (2b1 − b2 )( zm − z0 ) − b2 ( zm − z )
(10)
The final value of m4 is 2(b1 − b2 )( zm − z0 ) when the PM arrives at the lowest position; obviously it is not equal to the one when PM arrives at the lowest position in the first descent, meaning that the maximum levitation force in the first descent is different from the one in the second descent. When the PM ascends in the third time, the rule of m4 is described by
m4 = 2(b1 − b2 )( zm − z0 ) + b1 ( z − z0 )
(11)
Then the PM descends in the third time, the rule of m4 is given as
m4 = (3b1 − 2b2 )( zm − z0 ) − b2 ( zm − z )
(12)
When the PM arrives at the lowest position, the final value is 3(b1 − b2 )( zm − z0 ) , which is also different from the two previous values. Therefore, in order to correctly describe the hysteretic characteristic in the second and subsequent descent/ascent cycles of PM, the change rule of m4 in both ZFC and FC cases needs to be modified.
12
4.2 Updated frozen-image model When the PM descends from the initial cooling position z = h to the lowest position z = z0, the rule of m4 can be given as
m4 = −a1 (h − z )0.8 The negative maximum of m4 is
(13)
− a1 (h − z0 )0.8 when the PM arrives at the lowest position z0.
When the PM ascends from the lowest position to the initial cooling height, the magnetization of the superconductor is paramagnetic, m4 is given as
m4 = −a1 (h − z0 )0.8 + a2 ( z − z0 )0.8
(14)
Then the PM descends in the second time, the magnetic field that the superconductor feels becomes strong again, so the value of m4 decreases. In order to simplify the problem, the decreasing rule of m4 in the second descent is regarded as the same as the increasing rule in the first ascent. The rule of m4 is given as
m4 = −a1 (h − z0 ) 0.8 + a2 (h − z0 ) 0.8 − a2 (h − z ) 0.8
(15)
When the PM arrives at the lowest position in the second descent, the final value of m4 is − a1 ( h − z0 ) 0.8 , which is equal to the one in the first descent. In the second ascent, the rule of m4 can be described by Eq. (14). In the subsequent descent-ascent cycles, the rule of m4 for the descending process can be described by Eq. (15), and the rule for the ascending process can be described by Eq. (14). If Eqs. (13), (14) and (15) are substituted into Eq. (1), respectively, three levitation force distance curves can be obtained as shown in Fig. 5: the top curve gives the dependence of levitation force on the distance z for the first descent of PM, the bottom curve gives the
13
dependence of levitation force on the distance for the ascending process in every descent-ascent cycle of PM, and the middle one gives the dependence of levitation force on the distance for the descending process in the second and subsequent cycles of PM. In order to compare with the results reported in Ref. [12], in the calculations above the parameters are selected as: m1 = m2 = m3 = 5.0×10-3 A m2, a1 = 0.1 A m, a2 = 5.0 A m, and the maximum and minimum height of PM are 30 and 0.5 mm, respectively. As can be seen from Fig. 5, the levitation force - distance curves obtained by our update model agree with the ones obtained in the experimental curves shown in Fig. 2(a) qualitatively. For FC case, when the PM ascends from the initial cooling position z = z0 (h = z0) to the highest position z = zm, the rule of m4 is given as
m4 = b1 ( z − z0 )0.3
(16)
The positive maximum of m4 is b1 ( zm − z0 )
0 .3
when the PM arrives at the maximum height zm.
When the PM descends from the maximum height zm to the lowest position z0, the magnetization of HTS is diamagnetic, m4 is given as
m4 = b1 ( zm − z0 )0.3 − b2 ( zm − z )0.3
(17)
Then the PM ascends in the second time, the magnetic field that the superconductor feels becomes weak again, so the value of m4 increases. In order to simplify the problem, the increasing rule of m4 in the second ascent is regarded as the same as the decreasing rule in the first descent. The rule of m4 is given as
m4 = (b1 − b2 )( zm − z0 )0.3 + b2 ( z − z0 )0.3 The final value of m4 is b1 ( zm − z0 )
0 .3
(18)
when the PM arrives at the highest position. Then the PM
descends in the second time, the rule of m4 is described by
14
m4 = b1 ( zm − z0 )0.3 − b2 ( zm − z )0.3
(19)
The change rule of m4 during the second descent is the same as that during the first descent, suggesting that the levitation force-distance curves for the first and second descent of PM will overlap each other. In the third ascent, the rule of m4 can be described by Eq. (18). In the subsequent descent-ascent cycles, the rule of m4 for the descending process can be described by Eq. (19), and the rule for the ascending process can be described by Eq. (18). If Eqs. (16), (17) and (18) are substituted into Eq. (1) respectively, and the calculation parameters are selected as: m1 = m2 = m3 = 5.0×10-3 A m2, b1 = 5.0×10-2 A m, b2 = 1.0 A m, h = z0 = 0.5 mm, zm = 30 mm, three levitation force- curves can be obtained as shown in Fig. 6: the bottom curve gives the dependence of levitation force on the distance z for the first ascent of PM, the top curve gives the dependence of levitation force on the distance for the descending process in every ascent-descent cycle of PM, and the middle curve gives the dependence of levitation force on the distance for the ascending process in the second and subsequent cycles of PM. As can be seen from Fig. 6, the levitation force –distance curves obtained by our update model agree with the experimental results shown in Fig. 2(b) qualitatively.
5. Conclusion
The hysteretic behavior of the levitation force during multiple descent/ascent cycles of the permanent magnet has been investigated under both ZFC and FC processes. It is found that the hysteresis loop for the first descent/ascent cycle of magnet is larger than that for the second and subsequent cycles, and the levitation force - distance curves for the second, third, and fourth
15
cycles almost overlap each other. It is proved that Yang’s model can not predict the hysteretic characteristic in the levitation force during multiple descent/ascent cycles of the PM. An updated frozen-image model is developed by modifying the change rules of the vertical movement image in Yang’s model. Our model is able to describe the experimental results in both ZFC and FC cases correctly, and at the same time retains the simplicity of the frozen-image model.
Acknowledgments
This work is supported by “Doctoral Initiating Project of Guangdong Medical College (B2011008)”, and partially supported by the “Shanghai Leading Academic Discipline Project, No. S30105” and the “Chinese National Programs for High Technology Research and Development, No. 2009AA03Z205”.
References
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Fig. 1 The superconductivity of the YBCO sample: (a) the temperature dependence of resistance, (b) trapped magnetic field at 77K Fig. 2 The levitation force-distance curves of YBCO bulk measured for four continuous descent/ascent cycles of PM. YBCO bulk cooled by ZFC (a) and FC (b) methods, respectively. The arrows mean the direction of PM moving relative to YBCO sample. The numbers 1, 2, 3 and 4 beside the arrows indicate the cycle number of PM. Fig. 3 The dependence of magnetic field and induced current in Bean’s model for ZFC measurement, with PM locating at different positions away from HTS: (a) a certain height h1 between the highest and lowest positions in the first descent, (b) the lowest position h2 of the first descent (the first ascent), (c) the height h1 of the first ascent, (d) the highest position h0 of the first ascent (the second descent), (e) the height h1 of the second descent, (f) the lowest position h2 of the second descent Fig. 4 The dependence of magnetic field and induced current in Bean’s model for FC measurement, with PM locating at different positions away from HTS: (a) the highest position h0 (solid line) and a certain height h1 between the highest and lowest positions (dashed line) of the first ascent (the first descent), (b) the height h1 of the first descent, (c) the lowest position h2 of the first descent (the second ascent), (d) the height h1 of the second ascent Fig. 5 Levitation force vs distance between superconductor and PM in ZFC case from our updated model. Fig. 6 Levitation force vs distance between superconductor and PM in FC case from our updated model.
18
0.7 0.6
30
0.5
Bz(T)
Resistance (mΩ)
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20
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10
0.2 0.1
0 0.0 0
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80
82
84
86
88
90
92
94
96
98
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10
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X(mm
Temperature (K)
(a)
)
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20
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m) Y(m
(b) Fig. 1
19
100 80 70 60
Levitation Force (N)
Levitation Force (N)
40
Cycle 1 Cycle 2 Cycle 3 Cycle 4
90
50 40
1
30 20 10 0 1,2,3,4
2,3,4
Cycle 1 Cycle 2 Cycle 3 Cycle 4
30 20 10
1,2,3,4
0 -10
2,3,4
-20 -30
1
-40
-10 0
5
10
15
20
25
30
35
40
45
0
50
5
10
15
20
25
30
35
40
45
50
Distance (mm)
Distance (mm)
(a)
(b) Fig. 2
20
B
B
B
Jc
Jc
Jc
(a)
B
(b)
(c)
B
Jc
(d)
B
Jc
Jc
(e)
(f) Fig. 3
21
B
B
B
Jc
(a)
Jc
(b)
B
Jc
(c)
Jc
(d)
Fig. 4
22
Fig. 5
23
Fig. 6
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Reference: PHYSC 1252231 Editorial reference: PHYSC_PHYSC-D-12-00127 To be published in: Physica C: Superconductivity and its applications Title: Modeling of hysteretic behavior of the levitation force between superconductor and permanent magnet
Highlights: 1 Experimental results on hysteretic behavior of the levitaion force are presented. 2 Hysteresis loop for the first descent/ascent cycle of magnet is largest. 3 Hysteresis loop for the second and subsequent cycles almost overlap each other. 4 Yang’s frozen-image model can not describe this characteristic of levitation force. 5 An updated frozen-image model is developed to describe these experimental results.
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