Field and current driven versions of Brandt method for calculating transport ac loss of superconducting cylinder and strip

Abstract

As an elegant and fast numerical tool for solving time-dependent electromagnetic field problems in hard superconductors, Brandt’s method has played an important role in understading the magnetic behavior of superconducting strips, discs, bars and cylinders in various aspect ratios. However, the application of this convenient method was mainly in magnetization processes. Traditionally, the solution of current transport problem needs to introduce a driving electric field $E_a$, which requires a low efficiency iterative process and $E_a$ itself was not clearly explained. In this work, three integral algorithms based on the Brandt’s method are developed to deal with current transport problems, which directly adopt the applied current as a boundary condition. Namely the current (I)-driven version and two current-field-driven versions A and B. Moreover, the arbitrary applied magnetic field can also be included in the I-driven version. The derivation with all necessary formulas for the methods are given in this work. As an example, the new methods, as well as the traditional method are used for calculating transport ac loss Q of a superconducting cylinder or strip obeying a power-law relation of $E\propto J^n$ as a function of a given $I\left(t\right)$. Derived from the Ampère law and the differential rather than the integral expression of the Faraday law, the current-driven version can be used for more accurate and much quicker computation. Being an intermediate quantity, $E_a\left(t\right)$ in the two current-field-driven versions is accurately calculated under the given $I\left(t\right)$, but version B is much quicker than A. Problems relating to $E_a\left(t\right)$ and Q stabilization process are discussed.