A closed-form formula for 2-D ohmic losses calculation in SMPS transformer foils

A new formula aimed at calculating ohmic losses in switch-mode power supply (SMPS) transformers is presented. It is based on intensive two-dimensional (2-D) finite element method (FEM) simulations, the results of which have been summarized in a closed-form formula following a "semi-empirical" approach. The main benefit of this new formula, specifically intended for industrial designers, is to combine the precision of 2-D models with the ease-of-use and speed of calculation of one-dimensional (1-D) models, on the whole frequency range. It accurately covers cases where the classical Dowell's formula significantly underestimated the losses, specifically those with significant edge effect in foil windings. Experimental validation and discussion of accuracy is provided. At the moment, the formula is only valid for one layer of foil located between a zero and a maximum of the magnetomotive force but a similar approach could be applied with success to other types of windings. Furthermore, the analytical expression proposed in the article, based on Maxwell equations, can be used as a stand-alone tool to model the real behavior of any type of winding. More accurate understanding of the 2-D field is also possible thanks to the direct link established between the losses and the geometrical data of the winding.