Due to the advancement in the development of semiconductors used in the power converters, the printed circuit boards (PCBs) require an in-depth study of their electromagnetic behavior. To characterize the behavior of the PCBs, the Darwin model is employed, which can take into account all the coupled effects, namely resistive, inductive, and capacitive effects, at the intermediate frequencies. Nevertheless, the study of particular structures having a geometric dimension smaller than the others can create meshing difficulties. The modeling of thin structures by the finite element method requires the optimization of the mesh. To circumvent this issue, the shell elements for both node and edge elements are applied in this work. Finally, to validate the proposed approaches, two PCBs with different geometries are studied in both time and frequency domains, where the measurements for a single PCB are provided to compare with the numerical results.

The Darwin model, which simultaneously incorporates resistive, capacitive, and inductive effects but neglects the radiation one, has recently attracted more and more attention in the research area. For our industrial application needs, the finite-element (FE) system to solve derived from the Darwin model generally has a large size, which is beyond the capabilities of the direct solvers due to the memory limitation. In this work, a specially designed formulation amenable to an iterative solver is proposed for industrial applications. Moreover, a detailed comparison of different cases is carried out on two different examples.

The modeling of the capacitive phenomena, including the inductive effects becomes critical, especially in the case of a power converter with high switching frequencies, supplying an electrical device. At a low frequency, the electro-quasistatic (EQS) model is widely used to study the coupled resistive-capacitive effects, while the magneto-quasistatic (MQS) model is used to describe the coupled resistive-inductive effects. When the frequency increases, the Darwin model is preferred, which is able to capture the coupled resistive-capacitive-inductive effects by neglecting the radiation effects. In this work, we are interested in specifying the limits of these models, by investigating the influence of the frequency on the electromagnetic field distributions and the impedance of electromagnetic devices. Two different examples are carried out. For the first one, to validate the Darwin model, the measurement results are provided for comparison with the simulation results, which shows a good agreement. For the second one, the simulation results from three different models are compared, for both the local field distributions and the global impedances. It is shown that the EQS model can be used as an indicator to know at which frequency the Darwin model should be applied.

In order to design efficient power converter operating at high switching frequencies, the electromagnetic behavior of the printed circuit board (PCB) must be well known. Then, numerical approaches can be used to study the voltage and current waveforms as well as the current density distribution associated with a PCB. Considering a half-bridge converter connected to a passive electrical load, we propose a co-simulation based on the Cauer ladder network approach providing an equivalent circuit of the PCB combined with models of GaN transistors. The results from the co-simulation are compared with measurements at 1 MHz with an output power of 200 W.