Electromagnetic Characterization of Magnetic Ring: Case of Circular Cross-Section Shape

Abstract

This paper introduces a comprehensive two-dimensional analytical model of a toroidal magnetic ring with circular cross-section under sinusoidal excitation. Applying Maxwell's equations in local polar coordinates within a complex permeability, the model derives analytical expressions for the internal magnetic field, magnetic flux, complex impedance, and total losses. It rigorously separates the contributions of eddy current losses, hysteresis losses, and winding losses, while explicitly incorporating the skin effect in the conductive core via Bessel functions. An expression for the apparent permeability is also provided, enabling the nonlinear core behavior to be mapped onto simplified linear material models. The resulting analytical model offers a computationally efficient and accurate foundation for standardized magnetic material characterization, such as Brockhaus and Iwatsu ring measurements, as a powerful alternative to 2D and 3D finite element analysis.