An Exact Analytical Approach to Mutual Inductance Calculation in Wireless Power Transfer Systems Lying on Earth Structures

A new exact analytical expression is derived for the mutual inductance of two co-axial flat coils positioned on a conducting planar medium. The expression comes from converting the Fourier–Bessel transform describing the magnetic coupling between two generic turns belonging to the coils into a finite integral and then expanding the resulting integrand according to Gegenbauer addition theorem. This allows for term-by-term analytical integration, which expresses the inductance as an infinite sum of products of two modified Bessel functions, depending on the radii of the turns and the wavenumbers in the air space and the material medium. It is seen how the new developed formulation exhibits significantly better accuracy than previous well-established analytical approaches, the latter being subject to restrictive hypotheses, such as the magnetoquasistatic approximation. Moreover, in light of its superior performance in terms of computation time, the new solution offers a promising middle ground by combining the accuracy of numerical methods with the efficiency of analytical formulas.