John R. Bowler

Theory of eddy current inversion

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 express...

Eddy‐current interaction with an ideal crack. I. The forward problem

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 ...

Eddy‐current probe impedance due to a volumetric flaw

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 wit...

Pulsed eddy-current response to a conducting half-space

Eddy-current nondestructive evaluation commonly carried out using single frequency time harmonic excitations, but a pulsed excitation offers a simple and effective alternative. The pulse signals have been calculated for a probe coil whose current rises and falls exponentially, approximating a square wave when the exponential time constant is small. Predictions of the induced electromotive force (EMF) across a coil above a half-space conductor and of the magnetic field on the coil axis have been compared with experiments. The comparison shows excellent agreement between theory and experiment.

A theoretical and computational model of eddy-current probes incorporating volume integral and conjugate gradient methods

A general three-dimensional computational model of ferrite-core eddy-current probes has been developed for research and design studies in nondestructive evaluation. The model is based on a volume integral approach for finding the magnetization of the ferrite core excited by an AC-current-carrying coil in the presence of a conducting workpiece. Using the moment method, the integral equation is approximated by a matrix equation and solved using conjugate gradient techniques. Illustrative results are presented showing the impedance characteristics and field distributions for practical eddy-current probe configurations. >

Eddy current calculations using half‐space Green’s functions

A simple scalar representation is used to describe the electromagnetic field in the quasi‐static limit for an arbitrary time‐harmonic source current above an imperfectly conducting half‐space. Solutions are given in terms of half‐space scalar and dyadic Green’s functions. The general results are then used to derive analytical expressions for the fields arising from circular filaments and extended sources whose axes of symmetry are parallel to the surface of the conductor. These tangent coil solutions have applications in the theory of inductive sensors, particularly for eddy current nondestructive testing.

Interaction of an Eddy-Current Coil With a Right-Angled Conductive Wedge

A fundamental problem in eddy-current nondestructive evaluation is one of finding the quasi-static electromagnetic field of a cylindrical coil in the vicinity of the edge of a metal block. Although the field can be calculated numerically, an effective analytical approach can potentially provide a better understanding of the edge fields and form the basis of a procedure for solving a whole class of related edge problems including edge structures that contain corner cracks. One can represent the metal block as a conductive quarter space in an unbounded region. However, it has been found that the analysis is more straightforward if the problem domain is truncated in two dimensions. With the domain boundaries far from both the coil and the corner, the truncation has a negligible effect on the solution near the edge but the field calculation becomes much easier. A double Fourier series representation of the field is used, as in the case of a rectangular waveguide problem. The field in the conductor is then matched at the interfaces with that in air to determine the expansion coefficients that are used to represent the field in different parts of the domain. In this way we have derived expressions for the magnetic field, the induced eddy-current density and the coil impedance at arbitrary position and orientation of the coil.

Eddy-current interaction of a long coil with a slot in a conductive plate

We describe a truncated-domain method for calculating eddy currents in a plate with a long flaw. The plate is modeled as a conductive half-space and the flaw is a long slot with a rectangular cross section. A long two-dimensional (2-D) coil carrying an alternating current is aligned parallel to the slot. The coil impedance variation with frequency is determined for an arbitrary coil location. The electromagnetic field due to a long coil above a conductive half-space can be expressed as integrals of trigonometric functions. For a half-space with a long slot, however, additional boundary conditions must be satisfied at the slot walls. The truncated-domain method makes this possible by recasting the problem in a finite domain; as a result, the Fourier integral is replaced by a series. The domain can be made arbitrarily large, thereby yielding results that are numerically as close to the infinite domain solution as desired. We have used the truncated domain approach to study both eddy-current flaw interactions and edge effects in the limiting case of a very wide and deep slot. We confirmed the theoretical predictions by comparing them with results of a 2-D finite element calculation and of experiments.

Eddy current coil interaction with a right-angled conductive wedge

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 Maxwells 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.

Evaluation of probe impedance due to thin-skin eddy-current interaction with surface cracks

Crack detection using eddy-current nondestructive testing is often carried out at frequencies such that the skin depth of the induced current is much smaller than the crack dimensions. The induced current then flows in a thin skin at the conductor surface and at the faces of a surface crack. In the case of a crack that acts as an impenetrable barrier to electric current, the electromagnetic field at the crack surface can be represented, at an arbitrary frequency, in terms of a potential which satisfies a two-dimensional Laplace equation. The boundary conditions required in the solution of the Laplace equation have not yet been determined for the general case, but we have derived approximate boundary conditions which are applicable in the thin-skin regime. The conditions derived are valid for cracks in materials of arbitrary permeability. From the harmonic solutions of the Laplace equation, the impedance change of the excitation coil due to the defect has been calculated for cracks in aluminum and ferromagnetic steel. Comparisons between predictions and experimental measurements on rectangular slots show good agreement, thus corroborating the theory and the numerical calculations.