Loop Impedance of Single-Turn Circular Coils Lying on Conducting Media

In the previous work, a rigorous mathematical approach was proposed to calculate the mutual impedance of two loop antennas positioned on a conducting planar medium. Although very efficient, the approach has been shown to fail when applied to the computation of the self-impedance of a single loop alone. The present work deals with the proposal to overcome the limitation of the previous approach, and to derive an exact explicit expression for the loop impedance of a single-turn circular coil on a lossy planar medium. The well-known integral representation for the impedance is turned into a double integral consisting of semi-infinite and finite integrals. Then, the semi-infinite integral is expanded into a sum of spherical Hankel functions of the loop radius, whose coefficients depend on the integration variable of the finite integral. This makes it possible to analytically evaluate the remaining sum of simpler finite integrals, and express the voltage as a series of spherical Hankel functions of the loop radius. Numerical examples show the advantages of the derived formula in terms of both accuracy and time cost.