Semianalytical and Analytical Formulas for the Classical Loss in Granular Materials With Rectangular and Elliptical Grain Shapes

Granular soft magnetic materials, such as ferrites or soft magnetic composites, are widely spread in modern electrical engineering applications. An important loss contribution is the classical one. It is known that in these materials, two kinds of classical loss must be distinguished: eddy currents flowing at the scale of the whole sample (therefore called macroscopic), and current lines inside the grains (called microscopic). For the macroscopic eddy currents computation, the sample cross sections are often square or rectangular. For eddy currents prediction inside the grains, two cases can be distinguished: 1) the case of low density materials, for which circular or elliptical grain shapes are often realistic and 2) the case of high density materials. In this last class of granular materials, it is more difficult to identify a precise grain shape, because the deformations occurring during the compaction process often give to the grains a random shape. For carrying out the eddy current computation, rectangular shapes are often considered, because they allow a complete filling of the available space. To our knowledge, analytical eddy current formulas only exist for cylindrical or spherical regions. For other shapes, such as squares, rectangles, or ellipses, the finite element method must be used to compute the eddy current distribution, which can be time consuming. This paper overcomes this difficulty by proposing respectively a semianalytical formula for classical loss in rectangular geometries, and an analytical formula for elliptical cases.