DC-Biased Homogenized Harmonic Balance Finite Element Method

TL;DR

The paper extends the homogenized harmonic balance finite element method (HomHBFEM) to handle dc-biased excitations in laminated ferromagnetic cores, addressing saturation effects via a refined, saturation-aware homogenization. A look-up table derived from 1-D FE simulations provides frequency- and saturation-dependent parameters that feed into a modified reluctivity tensor, enabling accurate, time-efficient simulations without resolving laminate-scale skin depths. Verification against fine transient simulations across 50 Hz–10 kHz and multiple saturation scenarios shows good accuracy (often <10% losses) and substantial DoF/time reductions at moderate saturation, with degradation at higher saturation levels. The work promises practical, fast nonlinear eddy-current analyses for DC-biased devices like PETRA IV fast corrector magnets, while highlighting the need for further improvements in highly saturated regimes.

Abstract

The homogenized harmonic balance finite element (FE) method enables efficient nonlinear eddy-current simulations of 3-D devices with lamination stacks by combining the harmonic balance method with a frequency-domain-based homogenization technique. This approach avoids expensive time stepping of the eddy-current field problem and allows the use of a relatively coarse FE mesh that does not resolve the individual laminates. In this paper, we extend the method to handle excitation signals with a dc bias. To achieve this, we adapt the original homogenization technique to better account for ferromagnetic saturation. The resulting formula for the homogenized reluctivity is evaluated using a look-up table computed from a 1-D FE simulation of a lamination and containing the average magnetic flux density in the lamination and the corresponding skin depth. We compare the results of the proposed method to those from a fine-mesh transient reference simulation. The tests cover different levels of ferromagnetic saturation and frequencies between 50 Hz and 10 kHz. For moderate ferromagnetic saturation, the method gives a good approximation of the eddy-current losses and the magnetic energy, with relative errors below 10%, while reducing the required number of degrees of freedom at 10 kHz by 1.5 orders of magnitude. This results in a reduction in simulation time from 2 days on a contemporary server to 90 minutes on a standard workstation.

Figures (14)

• Figure 1: Electron beam in a particle accelerator (left) with FC magnets (center) and their laminations (right).
• Figure 2:
• Figure 3: Flow chart of the HomHBFEM. The index $n$ indicates the harmonic order and superscript $i$ the iteration number. Blue boxes are implemented in Python, red boxes in GetDP.
• Figure 4: First harmonic of the magnetic flux density in a lamination with $d = 1mm$ at $f = 1kHz$. The numerical simulation uses a Brauer curve for the $B$--$H$ curve, the analytical solution assumes a power law fitted to the Brauer curve.
• Figure 5: 3-D model of the laminated inductor.