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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.

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. To capture the coupled capacitive-inductive effects, the Darwin model is invoked in the frequency domain, which only neglects the radiation effects of the full Maxwell system. In this work, we are interested in the comparison of the different solvers, both iterative and direct, for ungauged and gauged finite-element (FE) systems for Darwin model. In addition, to handle the huge FE resulting matrix due to the industrial application, a novel formulation is proposed.

Dans les dernières années, la modélisation des composants magnétiques et électriques suscite beaucoup d’intérêt dans la recherche scientifique. Un modèle magnétodynamique est suffisant pour décrire le comportement des machines électriques dans les basses fréquences, mais, avec l’augmentation des fréquences en électronique de puissance, les machines sont soumises à des tensions hautes fréquences, cela nécessite une modélisation des isolants surtout en raison du vieillissement auquel ils seront exposés. L’objectif de ces travaux est de calculer le champ électrique dans les milieux non conducteurs, en particulier, les effets capacitifs. En effet, vu que les modèles classiques tels que la magnétostatique et la magnéto-quasistatique ne prennent pas en compte la modélisation de ces effets, il est indispensable de mettre en œuvre des formulations en potentiels adaptées dans le code_Carmel pour calculer simultanément les champs électriques et magnétiques, telle que le modèle de Darwin. Ce modèle est capable de capturer les effets capacitifs-inductifs couplés à des fréquences intermédiaires, en particulier, autour de la fréquence de résonance. En revanche, l’électrostatique et l’électro-quasistatique sont parmi les modèles connus qui sont capables à modéliser les effets capacitifs. En outre, un problème de structure mince de Darwin est résolu en utilisant la méthode des éléments coques, appliquée à la fois aux éléments de nœuds et d’arêtes, afin de modéliser des circuits imprimés à des fréquences élevées. Ces éléments sont dérivés de la dégénérescence des éléments prismatiques de Whitney et elles peuvent être appliqués facilement à une formulation pour résoudre une coque mince. Différentes applications industrielles ont été présentées afin de valider les résultats de simulation obtenus par le modèle de Darwin en les comparant aux résultats de mesures.