Quasi-Static Explicit Expression for the Flux Linkage Between Noncoplanar Coils in Wireless Power Transfer Systems Above a Lossy Ground

This work deals with the proposal to derive an explicit analytical expression for the mutual inductance between two coaxial circular coils placed above a conducting medium. The integral representation for the inductance is evaluated starting from expanding the product of the exponential factor and one of the Bessel functions of the integrand into a power series of the integration variable, which coincides with the radial wavenumber. Then, Bessel's differential equation is used to turn the generic power of the integration variable into a linear differential operator with respect to the radius of the source coil. This allows to convert the semiinfinite integral describing the inductance into an infinite sum of simpler integrals whose analytical evaluation is straightforward. Thus, after carrying out term-by-term integration, the mutual inductance is expressed in series form as a combination of spherical Hankel functions and products of cylindrical Bessel functions. The outcomes from the obtained formula for the inductance are compared with the data provided by both numerical integration of the original field integral and the previously published quasi-static solution to the same problem, valid for the special case that the two coils lie at the air-ground interface under the assumption of small receiving coil. The plotted curves reveal that, accuracy being equal, the proposed solution is less time consuming than numerical quadrature schemes, and that it provides accurate results even in the situations where numerical integration fails.