Combining a fractional diffusion equation and a fractional viscosity-based magneto dynamic model to simulate the ferromagnetic hysteresis losses

Magnetic losses in a laminated ferromagnetic core have been studied for years. However, magnetization mechanisms are complex, and the ideal model is still lacking. Classic resolution in the time domain combines a 1D magnetic diffusion equation with a viscosity-based magneto dynamic material law (first-order differential equation). This simultaneous resolution has already been solved by matrix inversion: the diffusion equation temporal term is replaced by its differential equation expression. It leads to a fast solution but overestimates the excess losses linked to the dynamic of the magnetic domains wall motions. Improved material laws using power operators have alternatively been tested. However, it is impossible to regroup the magnetic field terms on the same side of the final equation, and the resolution can only go through complex iterative methods (fixed point, Newton-Raphson). In this study, we propose to combine a fractional diffusion equation and a fractional viscosity-based magneto dynamic differential equation. Matrix resolution is possible, such as an accurate simulation of the dynamic behavior by adjusting the fractional order. The space term of the diffusion equation being solved by space discretization, the combined resolution leads to local information (excitation and induction fields). The number of dynamic parameters is limited but large enough for excellent simulation results.