Considering Capacitive Effects in Foil Winding Homogenization

TL;DR

The paper addresses the limitation of standard foil winding homogenization, which neglects capacitive effects at high frequencies. It extends the mqs-based homogenization by incorporating a capacitive current term expressed through the voltage function, without introducing new degrees of freedom. The authors derive a coupled PDE system and its Ritz-Galerkin discretization, and demonstrate, via a Cartesian test case and a pot-inductor example, that capacitive effects cause resonances and a transition to displacement-current-dominated behavior at high frequencies. This capacitive foil winding model improves accuracy for high-frequency transformer and inductor design while incurring only a modest computational overhead.

Abstract

In conventional finite element simulations, foil windings with a thin foil and many turns require many mesh elements. This renders models quickly computationally infeasible. With the use of homogenization approaches, the finite element mesh does not need to resolve the small-scale structure of the foil winding domain. Present homogenization approaches take resistive and inductive effects into account. With an increase of the operation frequency of foil windings, however, capacitive effects between adjacent turns in the foil winding become relevant. This paper presents an extension to the standard foil winding model that covers the capacitive behavior of foil windings.

Figures (12)

• Figure 1: Schematic representation of a (tube type) foil winding with $N=5.0$ turns. The geometry with the two terminals highlighted in purple is shown in \ref{['fig:geometry_foil_winding']}. The cross section of the foil with the conductor in gray and the insulation material in white is shown in \ref{['fig:cross_section_foil']}.
• Figure 2: Schematic representation of the foil winding domain $\Omega_{\mathrm{fw}}$. The local coordinate system $(\alpha,\beta,\gamma)$ is shown at one instance of the cross section, highlighted in purple. An exemplary representation of $\Gamma(\alpha)$ for one specific $\alpha$ is highlighted in blue.
• Figure 3: Illustration of two turns, $k-1$ and $k$, of a foil winding in the local coordinate system, located at $\alpha_{k-1}$ and $\alpha_{k}$. For convenience, the insulation thickness is increased, and the height is reduced. Quantities and surfaces associated with conductive currents are highlighted in blue and associated with capacitive currents are highlighted in purple. The cross sections of the conducive domains are denoted with $\hat{S}_{k-1}$ and $\hat{S}_{k}$, respectively. Conductive currents are denoted with $I_{\mathrm{c}}$ and the capacitive current with $I_{\mathrm{cap}}$.
• Figure 4: Potential $\phi$ inside turn $k-1$ and $k$ of the foil winding. The potential drop is given by $\Phi(\alpha_{k})$. The magnitude of the displacement field $\vec{D}_{\mathrm{cap}}$ between the two turns is visualized by the length of the purple arrows, pointing from the higher to the lower potential of the adjacent turns.
• Figure 5: Seven quadratic B-splines, $B_i$, on the interval $[-1,1]$.