Theory of eddy current inversion

Abstract

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.

Conductivity Imaging in Plates Using Current Injection Tomography

“…2 is referred to as the nonlinear term, and a'V x E is referred to as the linear term. The problem is nonlinear since both the conductivity and the electromagnetic field in regions of varying conductivity are simultaneously unknown [1]. For DC injected currents, the linear term is zero according to Faraday's Law.…”

“…The task of reconstructing an unknown distribution of electrical conductivity is widely recognized as a central theoretical problem in eddy-current nondestructive evaluation [1). Rather than using an eddy-current method, we address this problem using DC injection of current into conductive materials.…”

Determining The Step-Change Conductivity Profiles Within Layered Metal Structures Using Inductance Spectroscopy

“…Equations (1)(2)(3)(4)(5)(6)(7)(8) are an accurate description of the forward problem. However, to evaluate them in practice involves integration over a large zone and the computation process is slow.…”

“…Another approach is eddy current inspection, which infers the conductivity profile from impedance (or inductance) data [3][4][5][6][7][8][9]. The principle of these inverse methods [5][6][7][8][9] is to fit the inductance data (measured or simulated) to a complete analytical solution using simplex search. Therefore, these methods limit their reconstructed results to certain profiles.…”

Determining Conductivity and Thickness of Continuously Varying Layers on Metals Using Eddy Currents

“…Consequently, it is desirable to develop nondestructive methods for characterizing near-surface properties, such as the electrical conductivity and magnetic permeability. In this paper we present an eddy current method to determine the structure of continuously changing surface layers Recently, several groups [1][2][3][4][5][6][7] have studied the use of eddy-current testing to characterize samples produced by coating an otherwise uniform plate of metal with a single metal layer (e.g. cladding or a metallic paint).…”