Analytical Solutions to Eddy-Current Probe-Coil Problems

TLDR

Eddy currents are generated by a circular coil of rectangular cross‑section driven by a constant‑amplitude alternating current. The authors derive analytic solutions for two geometries—a coil above a layered semi‑infinite slab and a coil encircling a coated cylindrical rod—expressed as integrals of first‑order Bessel functions that yield the vector potential and related electromagnetic quantities. Analytic solutions for axially symmetric eddy‑current problems in these two canonical configurations were obtained, encompassing numerous special cases, and the calculated coil impedance for a layered plane conductor agrees excellently with experimental measurements.

Abstract

Solutions have been obtained for axially symmetric eddy-current problems in two configurations of wide applicability. In both cases, the eddy currents are assumed to be produced by a circular coil of rectangular cross section, driven by a constant amplitude alternating current. One solution is for a coil above a semi-infinite conducting slab with a plane surface, covered with a uniform layer of another conductor. This solution includes the special cases of a coil above a single infinite plane conductor or above a sheet of finite thickness, as well as the case of one metal clad on another. The other solution is for a coil surrounding an infinitely long circular conducting rod with a uniformly thick coating of another conductor. This includes the special cases of a coil around a conducting tube or rod, as well as one metal clad on a rod of another metal. The solutions are in the form of integrals of first-order Bessel functions giving the vector potential, from which the other electromagnetic quantities of interest can be obtained. The coil impedance has been calculated for the case of a coil above a two-conductor plane. The agreement between the calculated and experimental values is excellent.