Approximate Closed-Form Formula for Calculating Ohmic Resistance in Coils of Parallel Round Wires With Unequal Pitches

An approximate closed-form formula for calculating the ohmic resistance of a circular multiloop coil with unequal pitches is presented. Skin effect and proximity effect are included in the formula. The proximity effect is expressed as a proximity factor obtained using transverse magnetic fields applied to a wire from the rest of the wires. For verification, the optimum dimension for minimum resistance of wires with an equal pitch is compared with the previous results, and both results agree. The formula is applied to calculate the ohmic resistance of helical and spiral coils and is verified by a 2-D finite-element-method simulation. Both calculation and simulation results are consistent as well. As a practical application, a spiral coil with unequal pitches is designed for uniform mutual inductance, and it is optimized for the lowest resistance using the formula. The measured ohmic resistance of the designed coil also agrees with the calculated and simulated results. The results show that the formula can be well applied to designing circular multiloop coils with minimum ohmic loss in wireless-power-transfer systems.