Electromagnetic Modeling of Superconductors With Commercial Software: Possibilities With Two Vector Potential-Based Formulations

In recent years, the <inline-formula><tex-math notation="LaTeX">$H$</tex-math></inline-formula> formulation of Maxwell's equations has become the de facto standard for simulating the time-dependent electromagnetic behavior of superconducting applications with commercial software. However, there are cases where other formulations are desirable, for example for modeling superconducting turns in electrical machines or situations where the superconductor is better described by the critical state than by a power-law resistivity. In order to accurately and efficiently handle these situations, here we consider two approaches based on the magnetic vector potential: the <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula>-<inline-formula><tex-math notation="LaTeX">$A$</tex-math></inline-formula> formulation of Maxwell's equations (with power-law resistivity) and the implementation of a quasi critical state model (QCSM) with a steep <inline-formula><tex-math notation="LaTeX">$E$</tex-math></inline-formula>-<inline-formula><tex-math notation="LaTeX">$J$</tex-math></inline-formula> relationship limited at <inline-formula><tex-math notation="LaTeX">$J_{\rm c}$</tex-math></inline-formula>. In this article, we extend the <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula>-<inline-formula><tex-math notation="LaTeX">$A$</tex-math></inline-formula> formulation to thick conductors so that large coils with different coupling scenarios between the turns can be considered. We also discuss the QCSM in terms of its ability to calculate ac losses: in particular, we investigate the dependence of the calculated ac losses on the frequency of the ac excitation and the possibility of using quick one-step (instead of full-cycle) simulations to calculate the ac losses.