Fractional operators for the magnetic dynamic behavior of ferromagnetic specimens: An overview

Abstract

This paper reviews the use of the fractional derivative operators for the dynamic magnetization of ferromagnetic specimens. Magnetic behaviors in ferromagnetic specimens are strongly nonlinear and frequency dependent. Magnetism has an atomic origin but the magnetic behavior as observed at the human scale is highly affected by phenomena occurring at larger scales. Under the influence of an external magnetic field, the homogeneity of a ferromagnetic sample magnetization is linked to the excitation dynamics. Models and simulations in this domain are strongly needed, as they provide theoretical explanations and allow us to anticipate complex phenomena, difficult to observe in a practical way. On the one hand, such multi-scale dynamical behaviors can hardly be taken into account with the usual mathematical operators. On the other hand, correct simulation results on large frequency bandwidths can be obtained using fractional derivative operators. The use of fractional derivatives can be envisaged through different approaches: Lump models based on time fractional differential equations is one option, and fractional anomalous diffusion equations is another. In this manuscript, these two methods are detailed and compared. Theoretical results are compared to experimental ones, and conclusions and perspectives are drawn such as possible improvements.

Interlaminar faults in a GOFeSi laminated magnetic core: measurements and simulations

“…This technique combines the resolution of the magnetic diffusion equation with a fractional order differential equation as the material law. Remarkable simulation results were achieved across a broad frequency range, exceeding 10 kHz [4], using minimal parameters (only two for the frequency-dependent contribution). This article details the successful extension of this simulation method to a stack of laminations incorporating ILFs and predicts their impact on the resulting magnetic core performance.…”

Modeling Dynamic Hysteresis Curves in Amorphous Magnetic Ribbons

“…The consideration of both loss terms related to eddy currents flowing in bulk (macroscale) and around moving domain walls (the so-called excess or abnormal losses) as a single quantity in which the dependence on excitation frequency is given as a fractional allows one to avoid a number of interpretational problems, particularly as far as the quasi-static regime is considered [19,26,27]. Fractional power laws related to anomalous diffusion in ferromagnets have recently been the subject of considerable research, and [28][29][30] can be mentioned as representative examples.…”

Skin Effect and Losses in Soft Magnetic Sheets: From Low Inductions to Magnetic Saturation

“…Most of them are either based on suitably modified Steinmetz laws or manipulations of the loss separation formulas. It has been suggested, for example, to change the power law of the classical loss component to extend the frequency of the loss separation formula [11] or to express the Maxwell's diffusion equation by adopting a fractional induction derivative [12]. Coefficients for the loss components variable with peak induction and frequency have also been proposed [13] [14].…”