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The tensor complex permeability of a multi-turn coil with elliptic cross-section is analytically expressed. In field analysis, a multi-turn coil can be modeled by the uniform material that has the present tensor complex permeability. It is shown that the frequency characteristic of the present tensor complex permeability is in good agreement with that evaluated by finite element method applied to a unit cell of the multi-turn coil region.

The authors introduce a new method to evaluate the complex permeability of a multi-turn rectangular coil. To obtain the complex permeability of a rectangular coil in a closed form, it is approximated as an elliptic coil. Because the rectangular coil has different complex permeabilities in the vertical and horizontal directions, the complex permeability have to be defined in a tensor form. It suffices to discretize the coil region into rather coarse finite elements without considering the skin depth in contrast to the conventional finite element method.

The proposed method is shown to give the impedance of multi-turn coils which is in good agreement with results obtained by the conventional finite element (FE) analysis. By extending the proposed approach, the authors can easily perform 3D FE analysis without difficulty in discretization of the coil region with fairly fine finite elements. Moreover, they found that the approximation of rectangular coils as the elliptic coils is valid for analysis of quasi-static fields using this homogenization method.

The novelty of this study is in the approximation of the rectangular coils with elliptic coils, and the complex permeability for them is formulated here in a closed form. The proposed formula includes that for the round coils. Using the present method, the authors analyze the rectangular coils without fine discretization.

Inductances of complex coils, in the presence of linear materials only, can be computed by discretizing coils into simpler elements, whose magnetic behavior is analytically expressible, and suitably combining elementary contributions. Reliable results require high numbers of elements. In such cases, advantages can be taken from Graphic Processor Unit (GPU) capabilities of dealing efficiently with high numbers of repeated simple computational tasks. The purpose of this paper is to set up a fast and prompt numerical procedure to cope with the above described task.

The coils are first decomposed into current segments, taking into account accuracy, relative position and shape of coils to determine the number of segments. An analytical formula is then used to compute elementary contributions using GPUs to speed up the process, and finally superposition is used to recover the result.

The main advantages of the proposed approach are first demonstrated using simple examples, with analytical solutions, to validate the method accuracy and promptness, then more complex cases are taken to demonstrate its generality.

The method is intrinsically limited by the linearity assumption, excluding the presence of magnetic materials. The adopted formulas require in addition that coils must lie in free space.

The proposed method can help in the design of complex coils or coils systems, where the performance depends on total magnetic energy or magnetic forces among coils.

The paper presents an original implementation in GPU-based computational environment of a procedure to compute inductances, based on the superposition of a high number of current segments. The procedure includes an original method to self-adaptively define number and position of current segments used in the coils discretization.