Evaluation of the Inductive Coupling between Coplanar Concentric Coils in the Presence of the Ground

An analytical approach is presented that allows deriving an exact series-form representation for the flux linkage between two physically large concentric circular coils located on a lossy soil. The expression comes from a three-step analytical procedure. First, the integral expression for the flux linkage is converted into a double integral consisting of a finite and a semi-infinite integral. Next, the semi-infinite integral is recognized to be a well-known tabulated Sommerfeld integral, which may be analytically evaluated straightforwardly. Finally, applying Lommel's expansion allows rewriting the remaining finite integral as a sum of elementary integrals amenable to analytical evaluation. As a result, the flux linkage between the two coils is given as a sum of spherical Hankel functions of the wavenumber in the air and in the ground, multiplied by a coefficient depending on the geometrical dimensions of the coils. The accuracy and robustness of the proposed formulation is tested by comparing its outcomes with those generated by numerical integration of the complete integral representation for the flux linkage and with the results provided by previous analytical approaches to the same problem. It is found that the use of the derived expression for the inductance makes it possible to obtain significant time savings as compared to numerical quadrature schemes.