Contents lists available at ScienceDirect Physica C: Superconductivity and its applications journal homepage: www.elsevier.com/locate/physc Modeling of a superconducting maglev in the Meissner state Alejandra Casimiro a , Fidel Gamboa-Perera a , Maria Jose Hernandez b , Tayde Mariana Rocha c , Victor Sosa ⁎ ,a a Cinvestav-IPN Unidad Mérida, Departamento de Física Aplicada, A.P. 73 Cordemex, Mérida, Yuc. C.P. 97310, México b Facultad de Ciencias Físico Matemáticas, BUAP, 18 Sur y Ave. San Claudio, San Manuel, Puebla, Pue. C.P. 75575, México c Escuela Superior de Ingeniería Química e Industrias Extractivas del IPN, Av. Instituto Politécnico Nacional s/n, Nuevo Industrial Vallejo, Cd. México, CDMX C.P. 07738, México ARTICLEINFO Keywords: Superconducting levitation Analytical Force calculation, ABSTRACT We built a small maglev by forming a closed track with permanent magnets and studied the motion of a su- perconductor levitating above the track. Dynamics of the superconducting specimen in both vertical and hor- izontal directions is studied theoretically by means of a model that describes the interaction between a super- conducting sample in the Meissner state and an applied field. It was assumed that it is possible to take into account the demagnetizing field by using a single average parameter. The calculations agree nicely with the measured variables. 1. Introduction Superconducting levitation is a fascinating phenomenon that offers the potential development of fast transportation trains. Werfel et. al. [1] and Ma et. al. [2] published nice reviews on this and others possible applications of this issue. There are two possible types of levitation, which depend on the cooling process: the first one is the levitation in the Meissner state, derived from the zero-field cooling (ZFC). In this process, the superconductor is cooled down far from the magnetic field of the maglev track. In the case of high temperature superconductors, i.e, those whose critical temperature is above 77 K, the cooling process consists typically in submerging the specimen in a liquid nitrogen bath. Once the sample is in the superconducting state, it is brought and left levitating above the magnetic track. The second type of levitation is the levitation in the mixed state, where the sample in the normal state is put above the magnets of the track using a thin, non magnetic se- parator. Then, the sample is cooled down and makes the transition to the superconducting phase (field cooling, FC). Because of the trapping of the field, when the separator is retrieved the superconductor remains levitated with a high lateral stability. Superconducting levitation in the mixed state has been widely studied [3]. Two magnetic field values deserve a special attention. One is the first critical field H c1 , which is the value of the local field H in at which the magnetic lines of the applied field start to penetrate the specimen, i.e., it behaves as a type I super- conductor for H in < H c1 . Different measurements of H c1 have been re- ported for a single crystal of YBCO at a temperature of 77 K and a uniform magnetic field parallel to the crystallographic direction c. Some results are: μ 0 H c1 = 7.5 mT [4], 5.6 mT [5], 13.8 mT [6], 6.2 mT [7], 8.0 mT [8]. In a polycrystal, it has been reported μ 0 H c1 = 4.0 mT [9]. The second special magnetic field is the so called penetration field H p . This is an important parameter used in the successful Bean’s model [10] to describe the critical state of superconductors. Basically, H p is the applied magnetic field (assumed uniform) at which the superconductor is fully penetrated. For a finite cylinder of radius R and thickness L,H p is given by [3]: = + + H JL ln RL R L ( /2) [2 / (1 4 / ) ] p c 2 2 1/2 . Here, J c is the critical current density of the superconductor. Important efforts have been made to build and characterize proto- type vehicles based on superconducting levitation [11,12]. By the other hand, small-scale maglevs built in a laboratory are usually used for demonstration purposes [13]. However, they have the advantage that measurements can be made to give useful information about the dy- namics of the system. Modeling the forces in a maglev system is not an easy task. To do this, it is necessary to know the response of the sample to an external magnetic field H a . In particular, in the Meissner state the magnetic induction is excluded from the sample: = + = + + = B μ H M μ H H M ( ) ( ) 0 in sc a d sc 0 0 . Here, H in is the internal magnetic field in the superconductor, M sc its volume magnetization and H d the demagnetizing field. From this condition, we conclude that the following equation is satisfied: = M H H sc a d (1) https://doi.org/10.1016/j.physc.2019.1353543 Received 19 June 2019; Received in revised form 13 September 2019; Accepted 7 October 2019 ⁎ Corresponding author. E-mail address: victor.sosa@cinvestav.mx (V. Sosa). Physica C: Superconductivity and its applications 567 (2019) 1353543 Available online 11 October 2019 0921-4534/ © 2019 Elsevier B.V. All rights reserved. T