Magnetic Field Conforming Formulations for Foil Windings

TL;DR

This work extends foil-winding homogenization to magnetic field conforming formulations for both frequency- and time-domain FE analyses. It introduces full-$h$ and $h$-$\phi$ magnetic field conforming formulations, with the latter leveraging a magnetic scalar potential in non-conducting regions and a single global cut to enforce current conservation, complemented by an alternative $t$-$\omega$ discretization that uses isotropic resistivity. The proposed methods reproduce results from established $a$-$v$ FW and resolved models while dramatically reducing the number of degrees of freedom, achieving up to about $75\%$ DoF reduction in 3-D problems, and they are validated on 2-D axisymmetric and 3-D benchmark problems, as well as a nonlinear transient HTS coil scenario. This approach enables efficient, accurate modeling of large-scale foil-winding inductors and HTS coils using open-source FE tools, with demonstrated robustness across linear and nonlinear regimes.

Abstract

We extend the foil winding homogenization method to magnetic field conforming formulations. We first propose a full magnetic field foil winding formulation by analogy with magnetic flux density conforming formulations. We then introduce the magnetic scalar potential in non-conducting regions to improve the efficiency of the model. This leads to a significant reduction in the number of degrees of freedom, particularly in 3-D applications. The proposed models are verified on two frequency-domain benchmark problems: a 2-D axisymmetric problem and a 3-D problem. They reproduce results obtained with magnetic flux density conforming formulations and with resolved conductor models that explicitly discretize all turns. Moreover, the models are applied in the transient simulation of a high-temperature superconducting coil. In all investigated configurations, the proposed models provide reliable results while considerably reducing the size of the numerical problem to be solved.

Figures (10)

• Figure 1: FW homogenization: $N_{\text{c}}$ resolved conductors (left) replaced by a single homogenized bulk of thickness $L_{\alpha}$ (right).
• Figure 2: Illustration of entities involved in the discretization of the $t$-$\omega$ FW formulation, visualized on a structured hexahedral mesh of the homogenized bulk $\Omega_{ \text{c}}$. Left: cross-section of the homogenized bulk with the global cut basis function $\bm{c}_{\text{f}}$ and its support. Right: single layer $i$ of elements in the radial $\hat{\boldsymbol{\alpha}}$-direction with the layer cut-like basis function $\bm{c}_{\text{l},i}$ and its support. Nodal DoFs are in orange, remaining edge DoFs are in blue.
• Figure 3: Geometry (not to scale) of the 2-D axisymmetric verification problem: 20-turn foil winding inductor around a magnetic core ($\mu_{\text{r}} = 10$), both placed inside air. Units: mm.
• Figure 4: Voltage per turn in the 20-foil winding computed with the $h$-$\phi$ resolved, $h$-$\phi$ FW, full-$h$ FW and $a$-$v$ FW models. The FW model approximates the voltage with global polynomials of order $p=N_{\text{b}}-1 \in \{0,1,2,3\}$. The same mesh is used for all models.
• Figure 5: Current density distribution along the center of different (virtual) turns of the FW, computed with the various models. FW models consider a third-order global polynomial for $\Phi(\alpha)$. The same mesh is used for all models.