Spectral Element Method Modeling of Eddy Current Losses in High-Frequency Transformers

Abstract

This paper concerns the modeling of eddy current losses in conductive materials in the vicinity of a high-frequency transformer; more specifically, in two-dimensional problems where a high ratio between the object dimensions and the skin-depth exists. The analysis is performed using the Spectral Element Method (SEM), where high order Legendre–Gauss–Lobatto polynomials are applied to increase the accuracy of the results with respect to the Finite Element Method (FEM). A convergence analysis is performed on a two-dimensional benchmark system, for both the SEM and FEM. The benchmark system consists of a high-frequency transformer confined by a conductive cylinder and is free of complex geometrical shapes. Two different objectives are investigated. First, the discretizations at which the relative error with respect to a reference solution is minimized are compared. Second, the discretizations at which the trade-off between computational effort and accuracy is optimized are compared. The results indicated that by applying the SEM to the two-dimensional benchmark system, a higher accuracy per degree of freedom and significantly lower computation time are obtained with respect to the FEM. Therefore, the SEM is proven to be particularly useful for this type of problem.

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“…p-refinement. This feature is particularly useful for solving high-frequency eddy current problems where a high ratio between the object dimensions and the skin-depth exists [19]. Furthermore, the SEM applies Gaussian quadratures in order to perform numerical integration, whereas divergence operators are evaluated using Lagrangian basis functions.…”

“…Furthermore, the SEM applies Gaussian quadratures in order to perform numerical integration, whereas divergence operators are evaluated using Lagrangian basis functions. Additional details on the formulation and implementation of the method can be found in [14,15,19,31].…”

“…In [10], a single spectral element was used to model the losses induced in a highspeed rotating cylinder. In [19], it was demonstrated that by applying the SEM for the magnetic modeling of a high-frequency eddy current problem, with respect to the FEM, a higher accuracy per degree of freedom and lower computation time were obtained. In [11], a coupled FEM-SEM approach was applied to an induction machine, where the SEM was used for the non-linear modeling of eddy current losses in the rotor, and the FEM was used for the remaining machine parts.…”

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“…The SEM was originally applied to problems occurring in fluid dynamics and meteorology [28]. Over the past decades, the application of the method has spread to a wide variety of linear and non-linear engineering and mathematical problems [29], e.g., wave propagation, structural analysis, astrophysics, financial engineering, and more recently electromagnetic field modeling [27,30], including eddy current problems [16,20,31]. The formulation of the SEM for 2D Cartesian magnetostatic and thermal problems is presented in [27], of which the former has been applied to a linear synchronous actuator in [32].…”

“…Another example is observed in [33], where a coupled SEM-FEM approach is applied to investigate magnetoconvection for transformer cooling. In the context of high-frequency axisymmetric WPT systems, in [31], the SEM has been applied for the modeling of the losses that are induced in an electrically conductive and permeable cylinder, which surrounds the domain. The results have demonstrated that, compared to the FEM, the SEM obtains a higher accuracy per degree of freedom (DoF) and lower computation time.…”