An Analytical Correction to Dowell's Equation for Inductor and Transformer Winding Losses Using Cylindrical Coordinates

Analytical expressions that have been derived to calculate ac resistance and losses of magnetic devices at high frequencies have primarily used a one-dimensional approximation, and have used Cartesian coordinates or approximations that are equivalent to Cartesian coordinates. The result is Dowell's equation. This paper examines the effect of winding curvature on the losses relative to Dowell's equation by solving the problem completely in cylindrical coordinates. The error in Dowell's equation relative to a full cylindrical solution is found to be as great as 33% for some designs. A mathematical model is derived for the relative error between the two solutions, which can be used to assess the accuracy of Dowell's equation for a particular design, as well as a correction factor for Dowell's equation. The corrected Dowell's equation using the model is validated using finite-element analysis simulations as well as previously published experimental data. This paper is accompanied by Python codes to generate all data and figures used in the paper, a Python object that implements the derived mathematical model, and Lua scripts to run the finite-element simulations using FEMM.