2-D Analytical Copper Loss Model for PCB and Copper Foil Magnetics with Arbitrary Air Gaps

The modeling and optimization of copper losses are the basis of magnetic designs. By solving 2-D diffusion and Laplace equations, this article proposes a 2-D analytical copper loss model for printed circuit board inductors and transformers with arbitrary air gaps. The copper loss resistance is decoupled into skin, proximity, and fringing resistances. The fringing resistance is caused by the inhomogeneity of magnetic induction expressed as the Fourier series. Based on the proposed model, some conclusions are derived theoretically. For example, the fringing resistance of transformers is independent of the arrangements of primary and secondary windings, and fringing resistance increases with winding width, which could be a theoretical reason why matrix transformers should be used in some applications. The proposed model is verified by simulation with a 6.0% average error and by experiment with a 7.7% average error.