Mutual Impedance of Thin-Wire Circular Loops in Near-Surface Applications

This paper presents an exact analytical approach to the calculation of the mutual impedance between two coplanar loop antennas located in close proximity to a conducting medium. Evaluation of the integral representation for the impedance is carried out by replacing the square of the first-order Bessel function of the integrand with its power series expansion. Then, Bessel's differential equation is used to convert the powers of the radial wave number, resulting from the expansion, into linear differential operators with respect to the spacing between the loops. This makes it possible to perform term-by-term integration and obtain a series representation for the impedance. The proposed formula is tested by comparison with the outcomes from the small-loop approximation for the impedance and those provided by the finite-difference time-domain simulations. What emerges is that the new solution is far more accurate than the small-loop approximation, and that, at the same time, it is significantly less time consuming than purely numerical techniques.