Anomalous fractional diffusion equation for magnetic losses in a ferromagnetic lamination

During a full magnetization cycle and under a collinearity situation, the magnetic losses in a ferromagnetic are observable by plotting the average magnetic flux density or magnetization as a function of the tangent magnetic excitation. This highly frequency-dependent magnetic signature is called hysteresis cycle, and its area is equal to the energy consumed during the magnetization cycle. The physical mechanisms behind this energy conversion are complex as they interfere and take place at different geometrical scales. Microscopic Eddy currents due to domain wall variations play an important role, as well as the macroscopic Eddy currents due to the excitation field time variations and ruled by the magnetic field diffusion equation. From the literature on this topic, researchers have been proposing simulation models to reproduce and understand those complex observations. Even if all these losses contributions are physically interconnected, most of the simulation models available in the literature are based on the magnetic losses separation principle where each contribution is considered separately. Physically, the Weiss domains distribution and movements distort the diffusion process which becomes anomalous. In this article, the standard magnetic field diffusion equation is modified to take into account such anomaly. The first-order time derivation of the magnetic induction diffusion is replaced by a fractional-order time derivation. This change offers flexibility in the simulation scheme as the fractional order can be considered as an additional degree of freedom. By adjusting precisely this order, very accurate simulation results can be obtained on a very broad frequency bandwidth for the prediction of the iron losses in a ferromagnetic material.