Magneto-thermally Coupled Field Simulation of Homogenized Foil Winding Models

TL;DR

The paper tackles the challenge of simulating foil windings that exhibit strong electromagnetic–thermal coupling while avoiding the computational burden of resolving every thin foil turn. It introduces a homogenized magneto-thermal model using an $A$-$\phi$ formulation for the MQS sub-problem and a transient heat equation for the thermal sub-problem, with temperature-dependent conductivity and Joule losses providing the coupling term $q_v$. A weakly coupled iterative scheme allows different time-step sizes for the magnetic and thermal solvers, and the homogenization reduces the material description to diagonal tensors aligned with local coordinates. The approach is validated against Comsol and demonstrated on a pot-type transformer, where a hot spot near the yoke air-gap is predicted, illustrating the method’s practical utility for design and safety assessments in foil-winding devices.

Abstract

Foil windings have, due to their layered structure, different properties than conventional wire windings, which make them advantageous for high frequency applications. Both electromagnetic and thermal analyses are relevant for foil windings. These two physical areas are coupled through Joule losses and temperature dependent material properties. For an efficient simulation of foil windings, homogenization techniques are used to avoid resolving the single turns. Therefore, this paper comprises a coupled magneto-thermal simulation that uses a homogenization method in the electromagnetic and thermal part. A weak coupling with different time step sizes for both parts is presented. The method is verified on a simple geometry and showcased for a pot transformer that uses a foil and a wire winding.

Figures (8)

• Figure 1: Schematic representation of a foil winding. \ref{['fig:FW Schematic']} shows the foil winding domain $\Omega_\mathrm{fw}$ with the local coordinate system $(\alpha,\beta,\gamma)$. Its coordinates are directed perpendicular to the turns, in winding direction and in the direction towards the tips of the turns, respectively. The constant cross section is highlighted in purple and the surface $\Sigma(\alpha)$ (see \ref{['eq:def_foil_cut']}) is illustrated for a fixed $\alpha$ in blue. The cross section of one turn is highlighted in teal and shown in detail in \ref{['fig:FW Cross Section']}, with the conducting material in gray and the insulation material in white. (Both figures are adapted from Bundschuh2024ac).
• Figure 2: Weak coupling scheme. It holds $k\Delta t_\mathrm{mg}=\Delta t_\mathrm{th}$ with $k\in\mathbb{N}$.
• Figure 3: Geometry of the verification model with the dimensions in the table. The foil winding (purple) is centered within a box filled with air. On the boundary $\Gamma$, an electric bc and an isothermal bc is applied to the magnetic and thermal problem, respectively. (Adapted from Bundschuh2024ac).
• Figure 4: Relative error between the resolved thermal internal energy $U_\mathrm{res}$ and the homogenized thermal inner energy $U_\mathrm{hom}$ at the time instance $t_\text{end}$ over the number of mesh elements $N_{\mathrm{e}}$.
• Figure 5: Geometry and dimensions of the pot transformer. The inner foil winding and the outer wire winding (both in light gray) are surrounded by the yoke (in dark gray). The yoke has an air gap in the center limb and is filled with air (white).