Computationally-Efficient, Generalized Expressions for the Proximity-Effect in Multi-Layer, Multi-Turn Tubular Coils for Wireless Power Transfer Systems

Wireless power transfer (WPT) based on magnetic induction is used in numerous applications where physical contact between the power source and the load is not desired. For efficient power transfer, the resonant coils should have a low equivalent series resistance at the resonant frequency and have a high packing factor while being simple to manufacture. Coils made from hollow copper tubes might be an acceptable alternative to Litz wire designs due to low skin-effect resistance, easy manufacturing, and simplicity in implementing active cooling; however, the lack of an analytical model for complex coil designs poses a difficulty in systematically assessing its benefits and limitations. This paper presents a new method, based on the Fourier series, for evaluating proximity-effect losses in a multi-turn, multi-layer tubular coil. The model evaluates the proximity factor G_p as a function of coil and tube parameters, which can be incorporated into the design and optimization procedures. The derivations are supported by simulations that compare analytic and finite element models (FEM) of current density distribution in the coil. The model is further validated via experimental measurements of the resulting equivalent series resistance for two prototype coils.