The Impedance of a Loop Near a Conducting Half-Space

The change in complex impedance between an ideal one-turn circular coil located above and oarallel to a conducting half-space with respect to a similar isolated coil has been calculated. From this result a series expansion of the integrand allows the solution to be approximated by terms expressed as complete elliptic integrals. Results have been calculated for the chanoe in imoedance as a function of the lift-off distance and the conductivity of the half-space for a coil of representative.radius. INTRODUCTiON The eddy current method of nondestructive evaluation entails the induction of eddy currents in a conductive test object by a time-varying field produced by a suitable distribution of impressed curents (vi a an excitation ur primary coi 1), and the detection of the resultant field, usually by an inductive search coil which may be either a separate secondary coil or the primary coil itself. (See Fig. 1.) The method is ordinarily used at frequencies sufficiently low to neglect effects due to displacement current; hence a theoretical analysis entails calculating either a transfer impedance for a primary coil and secondary coil in the presence of the test object, or the calculation of the self impedance of a primary coil in the presence of the test object. In practice one often needs only the change in impedance produced by the test object or by chan9es in the nominal properties of the test object (e.g. changes in its geometry or position with respect to the test coil or coils, or distributed or localized changes in the resistivity of the test object). The most general case, allowing arbitrary configurations of primary and secondary coils and arbitrary test objects can be handled only by numerical methods. Certain idealized arrangements can be treated analytically either exactly or in useful approximation. In virtually all cases of practical interest, the analysis eventually reduces to the evaluation of certain integrals which cannot be expressed in closed form in terms of standard transcendental functions. In this paper we discuss the case of a one-turn circular coil located above and parallel to the surface of a homogeneous conductive half-space. From the standard boundary value problem approach we obtain the general expression for the change in coil impedance, ~z. produced by the half space; ~Z is given in terms of an integral over a separation parameter. A series expansion of one term in the integrand permits the integral to be expressed as a series of terms each of which is expressible in terms of complete elliptic coaxial cylindrical test objects. Such brute force numerical procedures are valuable for design purposes, but have the disadvantage of somewhat concealing the essentially simple manner in which the final result depends upon the parameters of the problem. The approach taken here, while less universal than the purely numerical approach; results in relatively simple, thou9h approximate and restricted, formulas for ~Z in terms of the basic parameters of the problem. For illustrative and comparative purposes, some selected numerical examples are also given. integrals. The leading terms of this series approxiFig. 1 Geometrical confiquration of loop near a mate ~Z asymptotically for sufficiently small values conductor. ·· of skin depth of the halfspace. The problem addressed here has previously been treated by Cheng [1] who evaluated ~Z by numerical methods for various choices of the relevant parameters. Similarly, Dodd and Deeds [2] have devised a digital computer program capable of handling circular test coils in the presence of layered planar and THEORETICAL ANALYSIS The basic geometry of the problem is shown in Fiq. 1 and consists of a loop radius r 0 oriented parallel to and at a distance~ above homoaeneous halfspace of conductivity cr. Beginning with the basic eauation for the vector potential