Analytical Calculation of the Coupling Factor for Single and Multi-layered Circular, Square, and Hexagonal Wireless Power Transfer Coils

This paper proposes a method to compute the coupling factor of single and multi-layered coils having a square, hexagonal, or circular geometry, placed center to center to each other, such that there is no lateral or angular misalignment, and only vertical misalignments are of interest. Existing methods involve using 2D / 3D electromagnetic solvers, elliptical integrals, Bessel function, magnetic vector potential, or a combination of these methods. All these methods increase computational complexity and are not in a compact form. The calculation of the coupling factor requires the computation of mutual inductance and the self-inductance of the coils. The self-inductance of single-layered coils is calculated using Modified Wheeler's eq. and the self-inductance for multi-layered coils is calculated using a combination of Modified Wheeler's eq. (for the inductance of each layer) and elliptical integrals (for mutual inductance between layers). The mutual inductance of single and multi-layered coils is derived based on the series expansion of Neumann's eq. with an addition of a factor to reduce the maximum error from 37% to 6.64%. A side length transformation eq. is derived and proposed for equivalent coil transformation from a hexagonal coil to a circular coil or vice versa. The method proposed here requires only geometric quantities of the coils for the computation of mutual inductance. Simulation and experimental results showed excellent agreement with the proposed model of separation distances greater than the inner radius of the transmitter (Tx) or receiver (Rx) coil (whichever has a smaller inner radius). For simulation results, an error of up to 5% was seen for single-layered coils and an error of up to 6.64% was seen for multi-layered coils. For experimental results, the error was under 5% for double-layered air coils and 8.1% for single-layered coils with a ferrite core at separation distances greater than the inner radius of the coils. Hence, the coupling factor can be computed accurately with the proposed model at separation distances greater than the inner radius of Tx or Rx coil.