An<i>H</i>-formulation-based three-dimensional hysteresis loss modelling tool in a simulation including time varying applied field and transport current: the fundamental problem and its solution

Abstract

When analytic solutions are not available, finite-element-based tools can be used to simulate hysteresis losses in superconductors with various shapes. A widely used tool for the corresponding magnetoquasistatic problem is based on the H-formulation, where H is the magnetic field intensity, eddy current model. In this paper, we study this type of tool in a three-dimensional simulation problem. We consider a case where we simultaneously apply both a time-varying external magnetic field and a transport current to a twisted wire. We show how the modelling decisions (air has high finite resistivity and applied field determines the boundary condition) affect the current density distribution along the wire. According to the results, the wire carries the imposed net current only on the boundary of the modelling domain, but not inside it. The current diffuses to the air and back to the boundary. To fix this problem, we present another formulation where air is treated as a region with 0 conductivity. Correspondingly, we express H in the air with a scalar potential and a cohomology basis function which considers the net current condition. As shown in this paper, this formulation does not fail in these so-called AC-AC (time varying transport current and applied magnetic field) simulations.