Approaches for Tree-Co-Tree Gauged $T$ –$\varphi$-Formulated Eddy-Current Problem in Superconductor Hysteresis Loss Simulations

Abstract

Hysteresis losses of superconductors can be computed based on the well-known eddy-current model. When using this model, the only difference to the conventional electromagnetic modeling is the strong nonlinear relation between the electric field and the current density. In addition, due to the highly complex structures of superconducting wires, the computation is still usually done in two dimensions. Furthermore, this represents a big part of practical cases. In this paper, we scrutinize the modeling decisions that can be done when using the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> -φ formulation, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> and φ are the vector potential for the current density and the magnetic scalar potential, respectively. We particularly consider variations of the formulation for determining net currents in superconductor and external magnetic field. We consider conductors in parallel, in series, or completely unrelated. Additionally, in an external field, we study coupled and completely uncoupled filaments. We compare the computational efficiency of different type tree-co-tree decompositions and demonstrate the use of a thick cut. The results show that a well-chosen formulation can be used to solve more versatile problems than typically needed for conventional eddy-current computations.