Analytical Methods and Formulas for Modeling High Temperature Superconductors

The electrodynamics of type-II superconductors is considered in the quasi-static limit when the concept of the so-called critical state is well applicable to the superconductors. We review analytical techniques and formulas used to describe the critical states in the superconductors of the various shapes characteristic of high-temperature superconducting materials. In particular, it is explained how 3-D critical-state problems for thin planar superconductors can be reduced to more simple 2-D problems. Within this approach, the critical states in a thin infinitely long superconducting strip are analytically described not only in the simplest case of a constant critical current density jc but also in the case of anisotropic flux-line pinning and for simple model dependence of jc on the magnetic-field magnitude. We also present the analytical solutions of the critical-state problem for superconducting strips with ferromagnetic substrates and for various arrays of superconducting strips with a transport current or in an external magnetic field. In particular, one of the discussed configurations of the strips can model superconducting power transmission cables. We also describe some analytical solutions of the electrodynamic equations for type-II superconductors with power-law voltage-current characteristics.