The impedance of an eddy-current probe changes when the current it induces in an electrical conductor is perturbed by a flaw such as a crack. In predicting the probe signals, it is expedient to introduce idealizations about the nature of the flaw. Eddy-current interaction is considered with an ideal crack having a negligible opening and acting as a impenetrable barrier to electric current. The barrier gives rise to a discontinuity in the electromagnetic field that has been calculated by finding an equivalent electrical source distribution that produces the same effect. The choice of source is between a current dipole layer or a magnetic dipole layer; either will give the required jump in the electric field at the crack. Here a current dipole layer is used. The strength of the equivalent source distribution has been found by solving a boundary integral equation with a singular kernel. From the solution, the probe impedance due to the crack has been evaluated. Although analytical solutions are possible for special cases, numerical approximations are needed for cracks of arbitrary shape. Following a moment method scheme, numerical predictions have been made for both rectangular and semielliptical ideal cracks. These predictions have been compared with experiments performed on narrow slots used to simulate ideal cracks. Good agreement has been found between the calculations and the measurements.

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The time-harmonic electromagnetic field in an electrically conductive right-angled wedge due to an inductive excitation by circular coil in air has been calculated. Using a formulation in Cartesian coordinates, the problem domain is truncated in a dimension whose axis is normal to a wedge face, and an approximate series solution found using elementary functions satisfying Maxwell's equations in the quasi-static limit. The coil impedance variation with position and frequency is calculated and compared with measurements made on a coil near the edge of a large aluminium block which approximates the effect of a conductive quarter-space. The comparison between theory and experiment shows very close agreement.

The inverse eddy current problem can be described as the task of reconstructing an unknown distribution of electrical conductivity from eddy-current probe impedance measurements recorded as a function of probe position, excitation frequency, or both. In eddy current nondestructive evaluation, this is widely recognized as a central theoretical problem whose solution is likely to have a significant impact on the characterization of flaws in conducting materials. Because the inverse problem is nonlinear, we propose using an iterative least-squares algorithm for recovering the conductivity. In this algorithm, the conductivity distribution sought minimizes the mean-square difference between the predicted and measured impedance values. The gradient of the impedance plays a fundamental role since it tells us how to update the conductivity in such a way as to guarantee a reduction in the mean-square difference. The impedance gradient is obtained in analytic form using function-space methods. The resulting expression is independent of the type of discretization ultimately chosen to approximate the flaw, and thus has greater generality than an approach in which discretization is performed first. The gradient is derived from the solution to two forward problems: an ordinary and an "adjoint" problem. In contrast, a finite difference computation of the gradient requires the solution of multiple forward problems, one for each unknown parameter used in modeling the flaw. Two general types of inverse problems are considered: the reconstruction of a conductivity distribution, and the reconstruction of the shape of an inclusion or crack whose conductivity is known or assumed to be zero. A layered conductor with unknown layer conductivities is treated as an example of the first type of inversion problem. An ellipsoidal crack is presented as an example of the second type of inversion problem.

Eddy current induced in a metal by a coil carrying an alternating current may be perturbed by the presence of any macroscopic defects in the material, such as cracks, surface indentations, or inclusions. In eddy-current nondestructive evaluation, defects are commonly sensed by a change of the coil impedance resulting from perturbations in the electromagnetic field. This paper describes theoretical predictions of eddy-current probe responses for surface cracks with finite opening. The theory expresses the electromagnetic field scattered by a three-dimensional flaw as a volume integral with a dyadic kernel. Probe signals are found by first solving an integral equation for the field at the flaw. The field equation is approximated by a discrete form using the moment method and a numerical solution found using conjugate gradients. The change in probe impedance due to a flaw is calculated from the flaw field. Predictions of the theory are compared with experimental impedances due to eddy-current interaction with a rectangular surface breaking slot. Good agreement is found between predictions and the measurements.