Lars Sjöberg

Taha El Hajji, Lars Sjöberg

This paper presents a comprehensive 2D analytical model of a toroidal magnetic ring with a square cross-section, subjected to sinusoidal excitation. By applying Maxwell's equations in local Cartesian coordinates and utilizing a complex permeability framework, the exact analytical expressions for the internal magnetic field, flux, complex impedance, and losses are derived. The model rigorously separates eddy current losses, hysteresis losses, and winding losses, explicitly accounting for the skin effect and complex permeability within the conductive core using separation of variables and hyperbolic functions. Furthermore, parameter for apparent permeability is expressed to map the core behavior onto simplified linear material models. The derivations establish a mathematical foundation highly suitable for standardized material characterizations, such as Brockhaus and Iwatsu ring measurements, by avoiding the heavy computational cost of 2D and 3D Finite Element Analysis.

Taha El Hajji, Aleksandr Nadkin, Stefan Skoog et al.

Variable flux memory motors, which employ Low Coercive Force (LCF) magnets, achieve extended high-efficiency operation through controllable magnetization states. To address the need for a unified approach to defining and comparing the magnetization state (MS) across material and motor levels, this paper proposes four MS definitions: two based on intrinsic material properties-magnetic flux density B and magnetic polarization J-and two based on motor-level quantities-fundamental flux linkage and back-EMF components. These definitions are evaluated across the id, iq operating plane using finite element analysis on an interior PMSM with a hybrid magnet configuration (LCF and HCF: High Coercive Force) and a defined circuit setup. The results clarify the relationship between material-level behavior and measurable motor quantities. The proposed framework provides guidance for selecting appropriate MS metrics depending on the application objective, whether for material analysis, control implementation, or condition monitoring in variable flux machines.