Eddy-Current Loss Model for Soft Magnetic Composite Materials Considering Particle Size Distribution
Joonas Vesa, Leo Hyvärinen, Paavo Rasilo
Abstract
Dynamic magnetization curves of soft magnetic materials are often written in terms of magnetic flux density <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX"> $b$ </tex-math> </inline-formula> and magnetic field strength <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX"> $h$ </tex-math> </inline-formula> as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX"> $h = \mathcal {H}(b) + c {\text {d} b}/{\text {d} t}$ </tex-math> </inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX"> $\mathcal {H}$ </tex-math> </inline-formula> is a static magnetization model and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX"> $c$ </tex-math> </inline-formula> is a real number describing eddy-current effects. In this article, an analytical derivation for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX"> $c$ </tex-math> </inline-formula> is presented for soft magnetic composite materials. The parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX"> $c$ </tex-math> </inline-formula> will depend explicitly on the conductivity of the material particles as well as the geometry of the particles, described by mean particle volume, variance of the particle volumes, volume fraction of the material and insulation thicknesses. No experimental or empirical parameters appear in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX"> $c$ </tex-math> </inline-formula> in our treatment.