A Two-Dimensional Winding Model for Planar Inductors Balancing Accuracy and Computational Speed

This paper presents a winding model for predicting winding losses in planar inductors. In transformer windings, the magnetic field can often be simplified to a one-dimensional problem, as the net current flowing into the core window is nearly zero. However, this simplification is not applicable to inductors, where two-dimensional (2D) magnetic fields are always present. The 2D field arises from two primary sources: the non-zero total current flowing into the core window and the fringing field around air gaps. Consequently, a 2D modeling approach is essential for accurate inductor analysis. Existing methods are often computationally intensive or simplified for a specific situation; this work aims to provide a method that balances accuracy and computational speed. This paper proposes an easily applicable 2D numerical method, based on Green's function and Gaussian quadrature, to solve the quasi-static magnetic field problems in planar windings. Various quadrature methods are compared, leading to the selection of Lobatto quadrature for this application. The proposed 2D numerical method is then incorporated into a winding model to construct the impedance matrix for planar windings. This comprehensive model can calculate losses resulting from non-uniform 2D magnetic fields on foil windings and accounts for current sharing among parallel conductors. The method is validated through comparisons with 2D finite element method (FEM) simulations and measurements performed on a planar inductor with parallel windings.