In recent years, the $H$ 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 $T$-$A$ formulation of Maxwell's equations (with power-law resistivity) and the implementation of a quasi critical state model (QCSM) with a steep $E$-$J$ relationship limited at $J_{\rm c}$. In this article, we extend the $T$-$A$ 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.

中文翻译：

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使用商业软件对超导体进行电磁建模：两种基于矢量电位的公式的可能性

近年来， $H$麦克斯韦方程组的公式已成为使用商业软件模拟超导应用的时间相关电磁行为的事实标准。然而，在某些情况下，其他公式是可取的，例如用于对电机中的超导匝进行建模，或者用临界状态比用幂律电阻率更好地描述超导体的情况。为了准确有效地处理这些情况，这里我们考虑两种基于磁矢量势的方法：$T$——$A$ 麦克斯韦方程组（具有幂律电阻率）的公式化和具有陡峭曲线的准临界状态模型 (QCSM) 的实现 $E$——$J$ 关系限于 $J_{\rm c}$. 在本文中，我们扩展了$T$——$A$粗导体的公式化，以便可以考虑在匝间具有不同耦合情况的大线圈。我们还讨论了 QCSM 计算交流损耗的能力：特别是，我们研究了计算的交流损耗对交流激励频率的依赖性以及使用快速一步（而不是全周期）的可能性用于计算交流损耗的模拟。