Magnetic levitation for hard superconductors

Magnetic levitation for hard superconductors Alexander A. Kordyuka) Institute of Metal Physics, 252680 Kiev 142, Ukraine ~Received 12 June 1997; accepted for publication 23 September 1997! An approach for calculating the interaction between a hard superconductor and a permanent magnet in the field-cooled case is proposed. The exact solutions were obtained for the point magnetic dipole over a flat ideally hard superconductor. We have shown that such an approach is adaptable to a wide practical range of melt-textured high-temperature superconductors’ systems with magnetic levitation. In this case, the energy losses can be calculated from the alternating magnetic field distribution on the superconducting sample surface. © 1998 American Institute of Physics. @S0021-8979~98!01501-1# The study of systems with levitation has provoked particular interest before1 and after2–4 the discovering of high temperature superconductors ~HTS! and especially today when melt-textured HTS technology is actively developed.5,6 Earlier7 we described the elastic properties of the point magnetic dipole over a granular HTS sample. We showed that in such a system the granular HTS at 77 K may be considered as a set of small isolated superconducting grains in calculating elastic properties7 and energy losses.8 We obtained the information about granular structure and intragrain magnetic flux motion from the investigation of the resonance frequencies7 and damping coefficients9 for different modes of the permanent magnet ~PM! forced oscillations. The melt-textured large grain HTS samples that are actively studied now are very different from granular ones in levitation properties. First, they have very strong pinning resulting in the absence of the effect of the PM rise above HTS sample at its cooling. Second, the small isolated grains approximation does not work for large grains. In this article, the absolutely hard superconductor approach is used. The sense of this approach is to use the surface shielding currents to calculate the magnetic field distribution outside the superconductor and to obtain from it the elastic properties of the PM-HTS system. The magnetic field inside such an ideal superconductor B~r! does not change with PM displacements. The feasibility of this approximation is determined by the condition d!L, where d is the field penetration depth and L is the character system dimension ~first the distance between PM and HTS!. With such an approximation, this problem has an exact analytical solution for the case of a magnetic dipole over a flat superconductor in the field cooled ~FC! case. To describe the FC behavior of the PM, the advanced mirror image method was applied. The method is illustrated by Fig. 1. Its distinction from the usual one, which is applied to the type-I superconductors, is in the using of the frozen PM image that creates the same magnetic field distribution outside the HTS as the frozen magnetic flux does. From the uniqueness theorem, the magnetic field distribution in an area with no induced currents is uniquely determined by the normal field component on its boundary. In other words, the distribution of this component determines the PM-HTS intera! Electronic mail: kord@imp.kiev.ua 610 J. Appl. Phys. 83 (1), 1 January 1998 action, and in turn is determined by the PM initial position ~FC position!. For the PM with initial position r0 5(x 0 ,y 0 ,z 0 ) and magnetic moment m0 ~see Fig. 1! that generates the magnetic field H(r2r0 , m0), the normal magnetic field component on HTS surface r5(x,y,0) is equal to the same component of its reverse image with r* 0 5(x 0 ,y 0 , 2z 0 ) and m* 0 ~the operation * maps any vector symmetrically about r surface! H z ~ r2r0 , m0 ! 5H z ~ r2r* 0 ,2 m* 0 !. ~1! It is required that H z ( r) should be unchanged at any PM displacements d r5r1 2r0 ~m0 → m1 , Fig. 1! from initial position. To do this, the presence of another image with m* 1 is required. This image moves with PM to r* position H @ z( r 1 * )50 . Thus, the interaction be, 2r1 , m1 )1H z ( r2r* m # 1 1 tween the PM and shielding current can be described by the interaction of the PM with net field of two images Him~ r! 5H~ r2r* 0 ,2 m* 1 , m* 0 ! 1H~ r2r* 1 !, ~2! and for the field outside and inside the superconductor we can write: B~ r! 5H~ r2r0 , m0 ! 1Him~ r! , B~ r! 5H~ r2r0 , m0 ! ~ for z.0 ! ~ for z,0 ! . ~3! ~4! FIG. 1. The advanced mirror image method illustration. 0021-8979/98/83(1)/610/3/$15.00 © 1998 American Institute of Physics TABLE I. The values of the coefficients k and g are calculated from ~7! and ~8!. m direction k2 g 0 245/128 245/128 3/4 245/16 105/16 2165/64 9/32 3/32 3/8 0 0 245/32 275/256 215/256 105/32 225/64 215/64 2165/64 Mode s k0 x5y 3/8 z x y z k1 mi z mi x FIG. 2. The levitation force acting on the magnetic dipole with for the different FC positions z 0 vs its distance to the superconducting surface z 1 from Eq. ~6!. mi z proximation can be determined from the critical state model ~the thickness of the layer carrying the critical current J c must be well less than PM-HTS distance! and for the above configuration z 0 @d5 The above relations are true for any shape of PM where m indicates the direction of volume magnetization. For the case of the point magnetic dipole, the analytical solution can be obtained. The force acting on the magnetic dipole F F~ r1 ! 5 ~ m1 •¹ ! Him5 2 ~ m1 •¹ ! ¹ 1 S 3 u r2r* 0u 3 u r2r* 1u DG W5k ~5! r5r1 ~6! The F z (z 1 ) dependencies for different z 0 are presented in Fig. 2. The elastic properties of PM-HTS system ~oscillation frequencies v and nonlinearities g! can be obtained from the expansion of F( d r) on d 5 d s/z 0 ~for any mode s5x,y,z): 2 3 4 F s ~ d ! 52m m2 z 24 0 ~ k 0 d 1k 1 d 1k 2 d ! 1O ~ d ! , v 5 v 0 @ 11 g ~ A/z 0 ! 2 # , S D 3k 2 5 k 1 g5 2 8k 0 12 k 0 v 05 m A k0 mz 50 2 ~7! , ~8! , where m is PM mass and A is the PM amplitude. The values of the coefficients k and g are presented in the Table I. The other useful side of this approach is the opportunity to obtain the surface distribution of alternating field h( r) and consequently—energy losses. From ~3!, this field only has a component parallel to the HTS surface h( r)5h r ( r) and is determined by the PM and its moving image ~Fig. 1! h ~ r,s ! 52A S ] H r~ r ! ]s D . ~9! z5z 0 For the melt-textured HTS where the energy losses have predominantly hysteretic nature,5 the feasibility of such apJ. Appl. Phys., Vol. 83, No. 1, 1 January 1998 c m3 3 A , J c z 10 0 ~11! where k'0.83. From here, the reversed Q factor of the PMHTS system8 is can be calculated symbolically for any PM displacement and m direction. For example, for r1 5(x 0 ,y 0 ,z 1 ) and mi z F z ~ z 1 ! 56 m2 @~ 2z 1 ! 24 2 ~ z 1 1z 0 ! 24 # . ~10! where c is speed of light. As this takes place, and as the dimension of the HTS sample is much more than d, the energy loss per square w( r)5(c/24p 2 )h 3 ( r)/J c . Then from ~9! after integrating the energy loss per PM oscillation period ~for mi z case and z mode for example!, @ 2 m* 0 !# 1 ~ r2r* @ m* 1 !# 1 1 ~ r2r* c h ~ r,s ! , 4p Jc Q 21 5 k c m A. p k 0 J c z 50 ~12! The condition ~10! is much stronger than is necessary to validate the use of the described approach for melt-textured HTS. Even for J c ;104 A/cm2 and for h;100 Oe, the penetration depth d;0.1 mm. The experimental values of resonance frequencies that we obtained for PM over the single domain HTS sample system ~m50.021 g, m51.2 G cm3, z 0 52.5 mm! are in complete agreement with theoretical ones.10 If the melt-textured sample has more than one domain, the resonance frequencies and levitation forces are appreciably reduced. Thus the elastic properties of the PMHTS system can be used to obtain the information about ‘‘granularity’’ of such HTS samples but to determine from this the number of domains per sample the additional investigations are needed. Therewith, the energy losses depend only slightly on such granularity and critical current density in the thin layer d under HTS surface can be determined from ~12! with a high accuracy. 1 A. F. Hebard, Rev. Sci. Instrum. 44, 425 ~1973!. F. C. Moon, M. M. Yanoviak, and R. Ware, Appl. Phys. Lett. 52, 1534 ~1988!. 3 S. A. Basinger, J. R. Hull, and T. M. Mulcahy, Appl. Phys. Lett. 57, 2942 ~1990!. 4 R. Grosser, J. Jäger, J. Betz, and W. Schoepe, Appl. Phys. Lett. 67, 2400 ~1995!. 5 Z. J. Yang, J. R. Hull, T. M. Mulcahy, and T. D. Rossing, J. Appl. Phys. 78, 2097 ~1995!. 2 Alexander A. Kordyuk 611 6 T. Sugiura and H. Fujimori, IEEE Trans. Magn. 32, 1066 ~1996!. A. A. Kordyuk and V. V. Nemoshkalenko, Appl. Phys. Lett. 68, 126 ~1996!. 8 A. A. Kordyuk and V. V. Nemoshkalenko, J. Supercond. 9, 77 ~1996!. 7 612 J. Appl. Phys., Vol. 83, No. 1, 1 January 1998 9 V. V. Nemoshkalenko and A. A. Kordyuk, Low Temp. Phys. 21, 791 ~1995!. 10 A. A. Kordyuk and V. V. Nemoshkalenko, Phys. Metal. Adv. Technol. ~to be published!. Alexander A. Kordyuk