This paper presents an analytical model of a copper-tube-based resonator coil used for wireless power transfer (WPT) applications. Some well-known techniques for calculating skin-effect-caused resistance, mutual inductance, self-inductance and self-capacitance are discussed in terms of their applicability for resonator modeling. Additionally, a new approach for proximity effect modeling is proposed, that results in a computationally efficient expression for proximity factor. Finally, all these elements are combined to derive a simple optimization function. In order to verify the derivations, the analytical outcomes are compared to FEM and experimental results for a planar spirally-wound, six-turn tubular coil.
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