ELSEVIER 8 August 1996 Physica C 268 (1996) 46-52 PHYSICA Vertical force, magnetic stiffness and damping for levitating type-II superconductors Alvaro Sanchez *, Caries Navau Grup d'Electromagnetisme, Departament de F[sica, UniversitatAutOnoma Barcelona, 08193 Bellaterra (Barcelona), Spain Received 8 February 1996; revised manuscript received 19 June 1996 Abstract The analytical expressions for the vertical force experienced by a superconducting cylinder situated above a cylindrical permanent magnet are derived within the framework of the critical-state model. The force, with an appropiate normalization, depends on a single parameter, the penetration field of the superconductor, which is proportional to its critical-current density. The results are applied to calculate the vertical magnetic stiffness and damping, from which the conditions for achieving a stable equilibrium of the system and the characteristics of the hysteretic losses associated to small displacements of the superconductor are discussed. Keywords: Levitation; Applications of high-Tc superconductors;Magnetization Magnetic levitation of superconductors is one of the most promising fields with regard to their technologi- cal applications at short term [ 1 ]. Big efforts are be- ing done in various directions, such as improving the material properties of the superconductors for meet- ing the requirements for each particular application, or measuring the levitation forces in carefully designed experiments in order to gain information about the basic parameters that will allow the best device per- formance. These measurements have helped to under- stand the general features of the levitation, first stud- ied by Brandt [2], while at the same time have raised some questions which remain unclear up to now. Our work intends to clarify some of these questions, with a double aim: to provide a systematic framework for modeling superconducting levitation, and to derive the analytical expressions describing the vertical magnetic * Corresponding author. Fax: + 34 3 581 1350; e-mail: alvar@elema.uab.es. force and two of the most important parameters, mag- netic stiffness and damping, under some simple but useful approximations. We will consider a type-II superconducting cylinder (SC) which is displaced over a cylindrical permanent magnet (PM) along their common axis, in the z di- rection. We will use cylindrical coordinates (r, ~b, z ), with the origin located at the center of the upper side of the PM. We assume that the PM has uniform mag- netization .A,4 in the z direction. We will be only in- terested in the region close to the axis, where it can be demonstrated that the components of the H field created by the PM are such that Hz >> Hr, and H z is roughly independent of r. Then, the field in this re- gion can be approximated by the value of the field on the axis, which has only z-component M( z+b Hz = -~- v/a 2 + (z + b) 2 (1) 0921-4534/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved P11S0921-4534(96)00400-5