The tensor magnetic phase theory for mesoscopic volume structures of soft magnetic materials – Quasi-static and dynamic vector polarization, apparent permeability and losses – Experimental identifications of GO steel at low induction levels

Abstract In this contribution, we propose to cope with the problem of soft magnetic materials heterogeneity and non-uniformity in terms of domains structure. This non-uniformity expresses itself with space variations of domains and walls geometry and characteristic properties from the bulk towards the surface. We investigate the possibility of describing and predicting these changes from a mesoscopic point of view. We begin with an introduction of typical subdivisions and define a tensor state variable [Λ2] to represent the diversity of magnetic structures with domains and walls. We then explain the material structuring thanks to an energy balance between the mesoscopic magnetic exchange, magneto-crystalline anisotropy, self-magnetostriction anisotropy, stress induced anisotropy and the dipolar demagnetizing energy. We write every contribution as a function of [V2] = [Λ2]−1. After minimizing the total energy, we derive a formulation compatible with classical numerical methods. [Λ2] is deduced thanks to a partial differential equation and surface boundary conditions. When a time varying field is applied to the material, damping effects occur either in the mass or at the surface. Eddy currents induced within domains lead to consider a volume dissipation energy. The surface magnetic field is also dampened by both the static hysteresis mainly due to defects and the dynamic hysteresis which stems from eddy currents around magnetic walls, added to the an-hysteretic field. The surface magnetic field, magnetic structure, and thus the polarization being known on the external surface, time variations of the volume magnetic structure can be calculated within the mass. Using the static or dynamic magnetic field coupling at the surface, the magnetic polarization can be rebuilt in the mass to calculate the apparent magnetic permeability. Finally, finding the geometry and frequency dependent vector magnetic behavior and iron losses becomes possible. The tensor magnetic phase theory is able to account for the sensitivity of the magnetic structure to the geometry, the macroscopic anisotropy partly influenced by the metallography, the residual or induced stress, some surface effects such as the texture, the rugosity or even any scribing patterns, at a mesoscopic scale. Two test cases for GO and NGO electrical steels are presented. Sensitivity analysis on the test case with GO steel are discussed. Results are then compared to static and dynamic measurements of GO SiFe sheet samples. This paper contributes to the investigations carried out on the geometry dependent magnetic behavior of soft magnetic materials.