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 plays 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.

Toroidal massive ferromagnetic cores are used in a wide range of electromagnetic applications, such as current sensors, inductances, static converters and filters. Growing interest exists in the industrial field with regards simulation tools to reduce experimental campaigns and improve product knowledge and performance. Accurate simulation results require a consideration of precise electromagnetic laws, such as the exact non-linear magnetic behavior of toroidal magnetic cores. Under the influence of an external surface magnetic field that was created by a surrounding coil, the local magnetic state through a ferromagnetic core cross-section was ruled by a combination of magnetic domain kinetics and external magnetic field diffusion. Conventional methods to simulate magnetic behavior are based on a separation of magnetic contributions, where microscopic Eddy currents from domain wall motions and macroscopic currents from external magnetic field variations are considered separately. This separation is artificial, because both loss mechanisms occur simultaneously and interact. In this study, an alternative solution was proposed through the resolution of a two-dimensional anomalous fractional magnetic field diffusion. The fractional order constitutes an additional degree of freedom in the simulation scheme, which can be identified by comparison with the experimental results. By adjusting this order, accurate local and global simulation results can be obtained on a broad frequency bandwidth and allow for the precise prediction of the dynamic magnetic behavior of a toroidal massive magnetic core.

An alternative sensing solution is described to measure local magnetic hysteresis cycles through a laminated magnetic core. Due to the reduced space gap separating two successive laminations, it is impossible to interpose the usual oversize magnetic sensors (wound coil, Halleffect sensor). In this study, the space issue has been solved by printing the needle probe method for the magnetic state monitoring and by using a micrometric Giant Magneto Resistance (GMR) for the magnetic excitation measurement. An instrumented magnetic lamination including the non-invasive monitoring solution has been built and moved successively to every lamination position of the whole laminated ferromagnetic core. A precise cartography of the hysteresis losses has been reconstructed from all these local measurements and the average values compared to the classic measurement methods obtained with a wound coil. The relative agreement between the experimental results observed opened doors to large improvement in the estimation of magnetic losses and in the design of magnetic circuits.