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Levitation performance of high-Tc superconductor in sinusoidal guideway magnetic field
Physica C 468 (2008) 2345–2350
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Levitation performance of high-Tc superconductor in sinusoidal guideway magnetic field W. Liu *, J.S. Wang, H. Jing, M. Jiang, J. Zheng, S.Y. Wang Applied Superconductivity Laboratory, Southwest Jiaotong University, Chengdu 610031, PR China
a r t i c l e
i n f o
Article history: Received 22 February 2008 Received in revised form 17 July 2008 Accepted 26 August 2008 Available online 2 September 2008 PACS: 84.71.Ba Keywords: Levitation performance Permanent magnet guideway Sinusoidal magnetic field
a b s t r a c t The vertical component of the Halbach array’s magnetic field exhibits a sinusoid distribution because of the closed magnetic flux area between two neighbouring poles, so this field can be regarded as the sinusoidal magnetic field. This article mainly discusses the influence of the closed flux region on the levitation performance of the bulk high-temperature superconductor (HTS). Moreover, the levitation performance is compared between the closed and diverging region of magnetic flux. The experimental results can be analyzed by the magnetic circuit theory and the frozen-image model. The analysis indicates that the closed region of magnetic flux can influence the levitation performance of bulk HTS obviously and provide an extra useful guidance force. These conclusions are helpful to optimize the HTS Maglev system. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In many applications, the bulk high-Tc superconductor (HTS) is used with the permanent magnet guideway (PMG), such as flywheel, levitation transportation, and so on. The first man-loading HTS Maglev test vehicle [1,2] used the PMG which utilized iron to focus the magnetic flux, and this PMG type is very popular in the levitation transportation system [3,4]. Only half of the magnetic energy of this PMG is converged on the upper surface because iron lacks the magnetization orientation. In order to improve the levitation performance of bulk HTS, the Halbach array is introduced in the system. The main characteristic of Halbach array is to use the permanent magnet (PM) to converge the magnetic flux, and it can concentrate most of the magnetic energy on the upper surface [5,8]. Another notable feature of Halbach array is the alternate magnetization direction of the polar PM, which causes the sinusoidal wave shape of the vertical component of the magnetic field. The polar PM is a kind of PM which is used to concentrate the magnetic flux in the Halbach array. This is a great difference between the present PMG and Halbach array, so the magnetic field of Halbach array can be denoted as the sinusoidal magnetic field. Unlike the present PMG, the fluxes of Halbach array are closed between those two neighbouring poles. The closed flux area may change the levitation performance of bulk HTS, so it becomes an important factor * Corresponding author. Tel.: +86 28 87601794; fax: +86 28 87603310. E-mail address: [email protected] (W. Liu). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.08.011
to affect the HTS Maglev system. The comparison and discussion about levitation and guidance force of HTS bulk between the closed and diverging areas of magnetic flux will be included in this article. 2. Experiment and results The Halbach PMG and the origin of Cartesian coordinates are shown in Fig. 1a. Assumed that there is no flux gradient in the longitudinal direction, the magnetic field can be divided into two components: Bz is the vertical component and Bx is the horizontal component. As shown in Fig. 1b, the wave shape of Bz obviously exhibits a sinusoidal trace. Two single domain melt-textured YBCO bulks are used in the experiment, and one is 50 mm in diameter and 12 mm thick, corresponding values for the other is 30 mm and 18 mm, as shown in Fig. 1c [6,7]. All the experiments are made on the SCML-2 test system [8,9]. In the experiment, the center of YBCO bulk is kept coincident with the test point. For the present PMG, there is only one peak of Bz above the upper surface where the YBCO bulk can achieve the best levitation performance [10]. But for the Halbach PMG, there exist two peaks of Bz, and one peak of Bx, as shown in Fig. 1b. Because of the symmetric character of these two peaks of Bz, only the peak of Bz at x = 25 mm is tested. The levitation force was tested firstly. To avoid the influence produced by the trapped flux, the levitation force was tested under the zero field-cooling (ZFC) condition. The experimental result is shown in Fig. 2.
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Fig. 2. (a) is the experimental results of levitation force for the YBCO bulk with diameter 50 mm, and (b) is the result for the YBCO bulk with diameter 30 mm.
Fig. 1. The structural sketch of cross section for the Halbach PMG is shown in (a), and the wave shape of its magnetic field’s vertical and horizontal components is shown in (b); (c) is the YBCO bulk used in the experiment.
The guidance force was tested secondly, under the field-cooling (FC) condition. Because of different flux capturing capacity, the FC height for the 50 mm YBCO bulk was 15 mm, and 10 mm for the 30 mm YBCO bulk. To avoid the influence caused by the diamagnetism of HTS bulk in the vertical motion, the guidance forces were measured directly at FC heights. The experimental result is shown in Fig. 3. From the experimental results, it is obvious that the levitation force of YBCO bulk at the peak of Bz is larger than that at the peak of Bx, especially for the 30 mm YBCO bulk. Oppositely, the guidance force of YBCO bulk at the peak of Bx is larger than that at the peak of Bz, no matter the diameter of HTS bulk is big or small.
Another notable phenomenon is that the guidance force of YBCO bulk appears remarkable asymmetry. As Fig. 3a shows, when the 30 mm-diameter YBCO bulk moving horizontally at the test position x = 25 mm, the maximal guidance forces at the left and right terminal point are nearly the same; when the test position approaches the peak of Bx, the asymmetry is more visible, like at the test position x = 15 mm, and the maximal guidance force at the right terminal point is nearly 1.5 multiples as much as that of the left terminal point; but the asymmetry disappears under the condition at the peak of Bx. The asymmetry is also observed for the YBCO bulk with bigger diameter, as shown in Fig. 3b. To avoid the edge of YBCO bulk beyond the central axial line of PMG in the FC process, the test positions are chosen as x = 25 and 30 mm, and the asymmetry appears when the YBCO bulk moves at x = 25 mm. 3. Analysis and discussion As Fig. 2 shows, the YBCO bulk can generate the larger levitation force at the peak of Bz. The main reason can be explained by the magnetic circuit theory, because the levitation force originates from the compression of magnetic fluxes caused by the diamagnetism of bulk HTS, and the magnetic fluxes is contained in the magnetic circuits. As Fig. 4a shows, the Halbach PMG includes three main magnetic circuits: M1, M2 and M3, and the peak positions of Bz and Bx
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Fig. 3. The experimental result of the guidance force: (a) is the result of the YBCO bulk with diameter 30 mm, and FC height 10 mm; (b) is the result of the YBCO bulk with diameter 50 mm, and FC height 15 mm. The insert of (b) is the result of the YBCO bulk with diameter 50 mm and FC height 30 mm.
are represented by A and B respectively. When the YBCO bulk moves vertically at position A, it compresses the magnetic circuits M1 and M2; and when it moves vertically at position B, it compresses the magnetic circuit M2 only. The YBCO bulk at position A can utilize more magnetic energy and produce larger levitation force. Fig. 4b shows the distribution of the magnetic flux for the Halbach PMG, which is calculated by the FEM software [6] The absolute values of magnetic density |B| along the lines at positions 1,2,3 from z = 0–30 mm are shown in Fig. 4c. It is clear that |B| of those positions are nearly the same, which means that the circle numbers of the magnetic fluxes are nearly the same for M1, M2 and M3. The levitation force of the YBCO bulk at position A should be twice as much as that of position B, without the consideration of leakage magnetic flux. This phenomenon is visible only for the YBCO bulk with smaller diameter, as shown in the inset of Fig. 2. The main reason can be explained by the demagnetizing effects [11]. A demagnetizing region caused by the diverging fluxes will generate in the bulk HTS when the bulk descending at position A, as shown in Fig. 5. This demagnetizing region will not be apparent under the condition at position B, because of the closed fluxes there. As Fig. 4b shows, for the 50 mm YBCO bulk, Bz at right and left sides of the YBCO bulk will change directions during the test pro-
Fig. 4. (a) is the schematic illustration of magnetic circuit theory; (b) is the distribution of the magnetic flux for the Halbach PMG; (c) is the absolute value of the magnetic density along the lines at positions 1, 2, 3. The contour of the YBCO bulk at position A is also shown in (b).
cess, and the induced supercurrent which flows in the demagnetizing region will be opposite to that of the bottom layer, as shown in Fig. 5a. But for the 30 mm YBCO bulk, the direction of Bz at these two sides of YBCO bulk will not reverse, so the induced supercurrent will have the same direction for the upper and bottom layers, as shown in Fig. 5b. More detail can refer to reference [12]. Referring to the basic calculating equation of levitation force: *
F lev ¼
Z
*
*
J B x dv: V
ð1Þ
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W. Liu et al. / Physica C 468 (2008) 2345–2350
Fig. 5. The structural sketch of the demagnetizing effect; (a) is for the 50 mm YBCO bulk and (b) is for the 30 mm bulk.
According to Eq. (1), the demagnetizing regions of Fig. 5a will contribute to reduce the total levitation force, but the upper layer of Fig. 5b will not. So compared with the 30 mm YBCO bulk, the difference of levitation force between position A and position B for the 50 mm YBCO bulk is not obvious. To explain the experimental results of the guidance force, the frozen-image method [13–15] is more suitable because it distinguishes the trapped flux part of the bulk HTS from the diamagnetism part. In the frozen-image model, after the FC process, when the bulk YBCO descends vertically, it suffers the repulsive force from its diamagnetic mirror image; when the bulk moves laterally, it suffers the attractive force from its frozen mirror image. In this article, the frozen-image model only includes the frozen dipoles which represent the trapped flux of the YBCO bulk because the measurements of the guidance force are without the vertical displacement. As Fig. 6a shows, when the YBCO bulk cools at position A, it captures magnetic fluxes with one direction. But when the YBCO bulk cools at position B, it captures the magnetic fluxes which change their directions, so it includes two parts with opposite poles. In the Fig. 6b and c, the initial position of YBCO bulk is represented by the dark gray circle. The light gray circle P1 represents the imaging dipole which forms in the PM area after the FC process. There exists another magnetic dipole P2, with the same magnetic energy but opposite pole to P1.
Fig. 6. The schematic illustration of the frozen-image method; (a) is the FC process for bulk YBCO; (b) is the condition under which the bulk YBCO moves at position A; (c) is the condition under which the bulk YBCO moves at position B.
As Fig. 6b shows, when the YBCO bulk moves to the position 1, it only suffers the pulling force which comes from P1; but when the YBCO bulk moves to the position 2, near or beyond the central axial line of the PMG, it suffers the pulling force coming from P1 and the pushing force coming from P2. As Fig. 6c shows, there exist three tensile forces when the YBCO bulk moves from position B. When the YBCO bulk moves towards right, its left part suffers the pulling force from P1 and its right part suffers the pulling force from P2, but there also exists a pushing force between its left part and P2. Those three tensile forces also exist when the motion direction is left, so the guidance force is symmetric at position B. Because the capture fluxes are assembled at the central area of YBCO bulk, the test height will influence the symmetry of guidance force at position A. When the test height is lower, the linking part of magnetic flux between YBCO bulk and PMG is limited in a narrow space, so when the bulk moves towards right, the influence of
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Fig. 7. The process and result about the magnetic field scanning at position A and B; The left part of (a) represents the capture flux of the YBCO bulk for position A and the right part of it represents that for position B; (b) is the 2D map across the center of the YBCO bulk.
P2 is weak in the test range and the asymmetry is not obvious. When the test height is higher, the linking part of magnetic flux is bigger and the influence of P2 is stronger, so the asymmetry is more obvious, as shown in the inset of Fig. 3b. It should be noticed that under some conditions, the guidance force of YBCO bulk at the peak of Bx is nearly 1.5 multiples as much as that at the peak of Bz. So it is necessary to analyze the magnetic tensile force among these imaging dipoles. The magnetic strength of the YBCO dipole, P1 and P2 are represented as mh, m1 and m2 respectively. F1m and F2m present the vector forces between the YBCO dipole and P1, P2, where r is the position vector:
F 1m ¼
F 2m ¼
m1 mh ðr h r1 Þ jr 1 r h j3 m2 mh ðr h r2 Þ jr 2 r h j3
;
ð2Þ
:
ð3Þ
In those equations, m1 and m2 should equal each other because of the translational symmetric characteristic. The magnetic tensile forces are determined by different mh at position A and B. The trapped magnetic field of the 50 mm YBCO bulk is scanned by the hall probe using the SCML-1 test system [10], and the results are shown in Fig. 7a and b. The scanning course is shown in Fig. 7c.
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From the measured values, the peak value of Bz at position A is about 0.27 T and that at position B are 0.13 T and 0.13 T. We use the peak value of Bz to simply present the strength of mh. According to Eqs. (2) and (3), when the YBCO bulk moves horizontally to position 1, the pulling force between it and the two imaging dipoles for position A and B equals each other. But there exists an extra pushing force when the YBCO bulk moves at position B, as shown in Fig. 6c. The pushing force is about half of the total pulling force and can enhance the guidance force, so the guidance force at position B is nearly 1.5 multiples as much as that at position A. It should emphasize that this pushing force does not exist for the current PMG type which is similar to the left movement from position A. When the test height is higher, the pushing force between bulk and the imaging dipoles is also obvious at position A, as shown in the inset of Fig. 3b. The pushing force also appears for the 30 mmdiameter YBCO bulk at the FC position x = 15 mm, as shown in Fig. 3a. From the above discussion, the different levitation performance between the peak of Bx and Bz can be analyzed qualitatively by the magnetic circuit theory and the frozen-image model. 4. Conclusion From the experimental result and discussion, it is obvious that the levitation performance of the YBCO bulk above the Halbach PMG is different from the peak of Bz and Bx. According to the theory of magnetic circuit, the YBCO bulk can obtain the best levitation force at the peak of Bz because it can make use of the most of mag-
netic fluxes and energy. By the analysis of frozen-image method, it is clear that the YBCO bulk can obtain the largest guidance force at the peak of Bx because of the extra magnetic pushing force. The extra magnetic pushing force does not exist for the current PMG type. To acquire larger guidance or levitation force above the Halbach PMG, the HTS bulks should be concentrated at the peak of Bx or Bz respectively. Acknowledgement This work is supported by the National High Technology Research and Development Program of China (2007AA03Z210), and the National Natural Science Foundation in China (50677057). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
J.S. Wang et al., Physica C 378–381 (2002) 809. Z.Y. Ren, J.S. Wang, et al., Physica C 378–381 (2002) 873–876. L. Schultz et al., IEEE Trans. Appl. Supercond. 15 (2) (2005) 2301. M. Okano et al., J Phys.: Conf. Ser. 43 (2006) 999. J.R. Hull, IEEE Trans. Appl. Supercond. 9 (2) (1999) 1229. P. Schatzle et al., Mater. Sci. Eng. B. 53 (1998) 95. L. Shlyk et al., Physica C 390 (2003) 325. H. Jing, J.S. Wang, et al., Physica C 463 (2007) 426. S.Y. Wang et al., IEEE Trans. Appl. Supercond. 17 (2) (2007) 2067. J.S. Wang et al., IEEE Trans. Appl. Supercond. 11 (1) (2001) 1801. A. Sanchez, C. Navau, IEEE Trans. Appl. Supercond. 9 (2) (1995) 2195. N. Del Valle et al., Appl. Phys. Lett. 90 (2007) 042503. A.A. Kordyuk, J. Appl. Phys. 83 (1) (1998) 610. J.R. Hull, A. Cansiz, J. Appl. Phys. 86 (11) (1999) 6396. J.R. Hull, Supercond. Sci. Technol. 13 (2000) R1.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Levitation performance of high-Tc superconductor in sinusoidal guideway magnetic field W. Liu *, J.S. Wang, H. Jing, M. Jiang, J. Zheng, S.Y. Wang Applied Superconductivity Laboratory, Southwest Jiaotong University, Chengdu 610031, PR China
a r t i c l e
i n f o
Article history: Received 22 February 2008 Received in revised form 17 July 2008 Accepted 26 August 2008 Available online 2 September 2008 PACS: 84.71.Ba Keywords: Levitation performance Permanent magnet guideway Sinusoidal magnetic field
a b s t r a c t The vertical component of the Halbach array’s magnetic field exhibits a sinusoid distribution because of the closed magnetic flux area between two neighbouring poles, so this field can be regarded as the sinusoidal magnetic field. This article mainly discusses the influence of the closed flux region on the levitation performance of the bulk high-temperature superconductor (HTS). Moreover, the levitation performance is compared between the closed and diverging region of magnetic flux. The experimental results can be analyzed by the magnetic circuit theory and the frozen-image model. The analysis indicates that the closed region of magnetic flux can influence the levitation performance of bulk HTS obviously and provide an extra useful guidance force. These conclusions are helpful to optimize the HTS Maglev system. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In many applications, the bulk high-Tc superconductor (HTS) is used with the permanent magnet guideway (PMG), such as flywheel, levitation transportation, and so on. The first man-loading HTS Maglev test vehicle [1,2] used the PMG which utilized iron to focus the magnetic flux, and this PMG type is very popular in the levitation transportation system [3,4]. Only half of the magnetic energy of this PMG is converged on the upper surface because iron lacks the magnetization orientation. In order to improve the levitation performance of bulk HTS, the Halbach array is introduced in the system. The main characteristic of Halbach array is to use the permanent magnet (PM) to converge the magnetic flux, and it can concentrate most of the magnetic energy on the upper surface [5,8]. Another notable feature of Halbach array is the alternate magnetization direction of the polar PM, which causes the sinusoidal wave shape of the vertical component of the magnetic field. The polar PM is a kind of PM which is used to concentrate the magnetic flux in the Halbach array. This is a great difference between the present PMG and Halbach array, so the magnetic field of Halbach array can be denoted as the sinusoidal magnetic field. Unlike the present PMG, the fluxes of Halbach array are closed between those two neighbouring poles. The closed flux area may change the levitation performance of bulk HTS, so it becomes an important factor * Corresponding author. Tel.: +86 28 87601794; fax: +86 28 87603310. E-mail address: [email protected] (W. Liu). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.08.011
to affect the HTS Maglev system. The comparison and discussion about levitation and guidance force of HTS bulk between the closed and diverging areas of magnetic flux will be included in this article. 2. Experiment and results The Halbach PMG and the origin of Cartesian coordinates are shown in Fig. 1a. Assumed that there is no flux gradient in the longitudinal direction, the magnetic field can be divided into two components: Bz is the vertical component and Bx is the horizontal component. As shown in Fig. 1b, the wave shape of Bz obviously exhibits a sinusoidal trace. Two single domain melt-textured YBCO bulks are used in the experiment, and one is 50 mm in diameter and 12 mm thick, corresponding values for the other is 30 mm and 18 mm, as shown in Fig. 1c [6,7]. All the experiments are made on the SCML-2 test system [8,9]. In the experiment, the center of YBCO bulk is kept coincident with the test point. For the present PMG, there is only one peak of Bz above the upper surface where the YBCO bulk can achieve the best levitation performance [10]. But for the Halbach PMG, there exist two peaks of Bz, and one peak of Bx, as shown in Fig. 1b. Because of the symmetric character of these two peaks of Bz, only the peak of Bz at x = 25 mm is tested. The levitation force was tested firstly. To avoid the influence produced by the trapped flux, the levitation force was tested under the zero field-cooling (ZFC) condition. The experimental result is shown in Fig. 2.
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Fig. 2. (a) is the experimental results of levitation force for the YBCO bulk with diameter 50 mm, and (b) is the result for the YBCO bulk with diameter 30 mm.
Fig. 1. The structural sketch of cross section for the Halbach PMG is shown in (a), and the wave shape of its magnetic field’s vertical and horizontal components is shown in (b); (c) is the YBCO bulk used in the experiment.
The guidance force was tested secondly, under the field-cooling (FC) condition. Because of different flux capturing capacity, the FC height for the 50 mm YBCO bulk was 15 mm, and 10 mm for the 30 mm YBCO bulk. To avoid the influence caused by the diamagnetism of HTS bulk in the vertical motion, the guidance forces were measured directly at FC heights. The experimental result is shown in Fig. 3. From the experimental results, it is obvious that the levitation force of YBCO bulk at the peak of Bz is larger than that at the peak of Bx, especially for the 30 mm YBCO bulk. Oppositely, the guidance force of YBCO bulk at the peak of Bx is larger than that at the peak of Bz, no matter the diameter of HTS bulk is big or small.
Another notable phenomenon is that the guidance force of YBCO bulk appears remarkable asymmetry. As Fig. 3a shows, when the 30 mm-diameter YBCO bulk moving horizontally at the test position x = 25 mm, the maximal guidance forces at the left and right terminal point are nearly the same; when the test position approaches the peak of Bx, the asymmetry is more visible, like at the test position x = 15 mm, and the maximal guidance force at the right terminal point is nearly 1.5 multiples as much as that of the left terminal point; but the asymmetry disappears under the condition at the peak of Bx. The asymmetry is also observed for the YBCO bulk with bigger diameter, as shown in Fig. 3b. To avoid the edge of YBCO bulk beyond the central axial line of PMG in the FC process, the test positions are chosen as x = 25 and 30 mm, and the asymmetry appears when the YBCO bulk moves at x = 25 mm. 3. Analysis and discussion As Fig. 2 shows, the YBCO bulk can generate the larger levitation force at the peak of Bz. The main reason can be explained by the magnetic circuit theory, because the levitation force originates from the compression of magnetic fluxes caused by the diamagnetism of bulk HTS, and the magnetic fluxes is contained in the magnetic circuits. As Fig. 4a shows, the Halbach PMG includes three main magnetic circuits: M1, M2 and M3, and the peak positions of Bz and Bx
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Fig. 3. The experimental result of the guidance force: (a) is the result of the YBCO bulk with diameter 30 mm, and FC height 10 mm; (b) is the result of the YBCO bulk with diameter 50 mm, and FC height 15 mm. The insert of (b) is the result of the YBCO bulk with diameter 50 mm and FC height 30 mm.
are represented by A and B respectively. When the YBCO bulk moves vertically at position A, it compresses the magnetic circuits M1 and M2; and when it moves vertically at position B, it compresses the magnetic circuit M2 only. The YBCO bulk at position A can utilize more magnetic energy and produce larger levitation force. Fig. 4b shows the distribution of the magnetic flux for the Halbach PMG, which is calculated by the FEM software [6] The absolute values of magnetic density |B| along the lines at positions 1,2,3 from z = 0–30 mm are shown in Fig. 4c. It is clear that |B| of those positions are nearly the same, which means that the circle numbers of the magnetic fluxes are nearly the same for M1, M2 and M3. The levitation force of the YBCO bulk at position A should be twice as much as that of position B, without the consideration of leakage magnetic flux. This phenomenon is visible only for the YBCO bulk with smaller diameter, as shown in the inset of Fig. 2. The main reason can be explained by the demagnetizing effects [11]. A demagnetizing region caused by the diverging fluxes will generate in the bulk HTS when the bulk descending at position A, as shown in Fig. 5. This demagnetizing region will not be apparent under the condition at position B, because of the closed fluxes there. As Fig. 4b shows, for the 50 mm YBCO bulk, Bz at right and left sides of the YBCO bulk will change directions during the test pro-
Fig. 4. (a) is the schematic illustration of magnetic circuit theory; (b) is the distribution of the magnetic flux for the Halbach PMG; (c) is the absolute value of the magnetic density along the lines at positions 1, 2, 3. The contour of the YBCO bulk at position A is also shown in (b).
cess, and the induced supercurrent which flows in the demagnetizing region will be opposite to that of the bottom layer, as shown in Fig. 5a. But for the 30 mm YBCO bulk, the direction of Bz at these two sides of YBCO bulk will not reverse, so the induced supercurrent will have the same direction for the upper and bottom layers, as shown in Fig. 5b. More detail can refer to reference [12]. Referring to the basic calculating equation of levitation force: *
F lev ¼
Z
*
*
J B x dv: V
ð1Þ
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Fig. 5. The structural sketch of the demagnetizing effect; (a) is for the 50 mm YBCO bulk and (b) is for the 30 mm bulk.
According to Eq. (1), the demagnetizing regions of Fig. 5a will contribute to reduce the total levitation force, but the upper layer of Fig. 5b will not. So compared with the 30 mm YBCO bulk, the difference of levitation force between position A and position B for the 50 mm YBCO bulk is not obvious. To explain the experimental results of the guidance force, the frozen-image method [13–15] is more suitable because it distinguishes the trapped flux part of the bulk HTS from the diamagnetism part. In the frozen-image model, after the FC process, when the bulk YBCO descends vertically, it suffers the repulsive force from its diamagnetic mirror image; when the bulk moves laterally, it suffers the attractive force from its frozen mirror image. In this article, the frozen-image model only includes the frozen dipoles which represent the trapped flux of the YBCO bulk because the measurements of the guidance force are without the vertical displacement. As Fig. 6a shows, when the YBCO bulk cools at position A, it captures magnetic fluxes with one direction. But when the YBCO bulk cools at position B, it captures the magnetic fluxes which change their directions, so it includes two parts with opposite poles. In the Fig. 6b and c, the initial position of YBCO bulk is represented by the dark gray circle. The light gray circle P1 represents the imaging dipole which forms in the PM area after the FC process. There exists another magnetic dipole P2, with the same magnetic energy but opposite pole to P1.
Fig. 6. The schematic illustration of the frozen-image method; (a) is the FC process for bulk YBCO; (b) is the condition under which the bulk YBCO moves at position A; (c) is the condition under which the bulk YBCO moves at position B.
As Fig. 6b shows, when the YBCO bulk moves to the position 1, it only suffers the pulling force which comes from P1; but when the YBCO bulk moves to the position 2, near or beyond the central axial line of the PMG, it suffers the pulling force coming from P1 and the pushing force coming from P2. As Fig. 6c shows, there exist three tensile forces when the YBCO bulk moves from position B. When the YBCO bulk moves towards right, its left part suffers the pulling force from P1 and its right part suffers the pulling force from P2, but there also exists a pushing force between its left part and P2. Those three tensile forces also exist when the motion direction is left, so the guidance force is symmetric at position B. Because the capture fluxes are assembled at the central area of YBCO bulk, the test height will influence the symmetry of guidance force at position A. When the test height is lower, the linking part of magnetic flux between YBCO bulk and PMG is limited in a narrow space, so when the bulk moves towards right, the influence of
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Fig. 7. The process and result about the magnetic field scanning at position A and B; The left part of (a) represents the capture flux of the YBCO bulk for position A and the right part of it represents that for position B; (b) is the 2D map across the center of the YBCO bulk.
P2 is weak in the test range and the asymmetry is not obvious. When the test height is higher, the linking part of magnetic flux is bigger and the influence of P2 is stronger, so the asymmetry is more obvious, as shown in the inset of Fig. 3b. It should be noticed that under some conditions, the guidance force of YBCO bulk at the peak of Bx is nearly 1.5 multiples as much as that at the peak of Bz. So it is necessary to analyze the magnetic tensile force among these imaging dipoles. The magnetic strength of the YBCO dipole, P1 and P2 are represented as mh, m1 and m2 respectively. F1m and F2m present the vector forces between the YBCO dipole and P1, P2, where r is the position vector:
F 1m ¼
F 2m ¼
m1 mh ðr h r1 Þ jr 1 r h j3 m2 mh ðr h r2 Þ jr 2 r h j3
;
ð2Þ
:
ð3Þ
In those equations, m1 and m2 should equal each other because of the translational symmetric characteristic. The magnetic tensile forces are determined by different mh at position A and B. The trapped magnetic field of the 50 mm YBCO bulk is scanned by the hall probe using the SCML-1 test system [10], and the results are shown in Fig. 7a and b. The scanning course is shown in Fig. 7c.
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From the measured values, the peak value of Bz at position A is about 0.27 T and that at position B are 0.13 T and 0.13 T. We use the peak value of Bz to simply present the strength of mh. According to Eqs. (2) and (3), when the YBCO bulk moves horizontally to position 1, the pulling force between it and the two imaging dipoles for position A and B equals each other. But there exists an extra pushing force when the YBCO bulk moves at position B, as shown in Fig. 6c. The pushing force is about half of the total pulling force and can enhance the guidance force, so the guidance force at position B is nearly 1.5 multiples as much as that at position A. It should emphasize that this pushing force does not exist for the current PMG type which is similar to the left movement from position A. When the test height is higher, the pushing force between bulk and the imaging dipoles is also obvious at position A, as shown in the inset of Fig. 3b. The pushing force also appears for the 30 mmdiameter YBCO bulk at the FC position x = 15 mm, as shown in Fig. 3a. From the above discussion, the different levitation performance between the peak of Bx and Bz can be analyzed qualitatively by the magnetic circuit theory and the frozen-image model. 4. Conclusion From the experimental result and discussion, it is obvious that the levitation performance of the YBCO bulk above the Halbach PMG is different from the peak of Bz and Bx. According to the theory of magnetic circuit, the YBCO bulk can obtain the best levitation force at the peak of Bz because it can make use of the most of mag-
netic fluxes and energy. By the analysis of frozen-image method, it is clear that the YBCO bulk can obtain the largest guidance force at the peak of Bx because of the extra magnetic pushing force. The extra magnetic pushing force does not exist for the current PMG type. To acquire larger guidance or levitation force above the Halbach PMG, the HTS bulks should be concentrated at the peak of Bx or Bz respectively. Acknowledgement This work is supported by the National High Technology Research and Development Program of China (2007AA03Z210), and the National Natural Science Foundation in China (50677057). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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