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An analytical model of an axisymmetric eddy current ferrite-cored probe above a multi-layered conducting half-space has been developed using a procedure in which the domain of the problem is truncated radially. This means that solutions can be expressed in the form of generalized Fourier-Bessel series. The expansion coefficients are found by matching the field across the interfaces between the subregions of the problem. Initially, the magnetic vector potential of a simple circular current filament is expanded in a series form. The solution is then modified to accommodate an infinitely long coaxial ferrite core, and the principle of superposition is invoked to derive a coil field from the filament field in the presence of the core. Next, we consider a semi-infinite core and then one of finite length. Finally, the effects of a multi-layered conductor are included. Numerical predictions of probe impedance have been compared with experimental data showing excellent agreement. [ABSTRACT FROM AUTHOR]

The electromagnetic field of a cylindrical eddy current probe coil near the open end of a borehole in a conductor has been calculated analytically accounting for edge effects. Calculations of the coil impedance as a function of position and excitation frequency have been made allowing theoretical results to be compared with experimental measurements. Comparisons have been carried out for special cases in which a cylindrical coil has its axis either perpendicular or parallel to the axis of the hole. In the approach used, the field is expressed in terms of transverse electric and transverse magnetic potentials defined with respect to the axis of the hole. The domain of the problem is truncated in the axial direction in order to express the solution in the form of eigenfunction expansions. The truncation modifies the original unbounded domain problem, but the additional boundaries can be made as remote from the coil as desired so that they have a negligible effect on numerical estimates of the coil field. The truncated region approach has proved to be accurate and computationally efficient but more significantly, it allows new solutions to be found for problems that are otherwise analytically intractable. The results model eddy current inspections of boreholes including edge effects at the opening of the hole. [ABSTRACT FROM AUTHOR]

Presents a study which described the theory of eddy current inversion. Overview of the forward problem; Reconstruction of an arbitrary conductivity distribution; Finite number of flaw parameters.

Presents a study that described an inversion scheme of eddy-current interaction in an ideal crack metal. Analysis of the impedance variation; Details of the local edge solution; Evaluation of the impedance gradient.

We have derived analytical expressions for the quasi-static time-harmonic electromagnetic field of a circular coil used to induce eddy currents in a tube or pipe, the axis of the coil being perpendicular to that of the conductor. The results are used to determine coil impedance variations with frequency and position due to induced current. The analysis is based on a derivation of the transverse electric field of a circular current filament in free space expressed as a single layer potential. Integration of the filament field yields the coil field and its impedance in the presence of a tube. We have used the theory and a numerical approximation in fitting the frequency dependence of experimental coil impedance and thereby determined accurate estimates of the electrical conductivity of an inconel tube and the coil lift-off. Finding these parameters is a preliminary step in evaluating coil interactions with flaws. [ABSTRACT FROM AUTHOR]

We have derived an analytical solution for the transient potential drop due to a step function excitation of a four-point probe on a conducting plate. Similar expressions have already been developed based on a previous analysis for a conducting half-space. The purpose of this article, however, is to extend the theory to measurements on conductors of arbitrary thickness and thereby broaden the practical applicability of the technique. The results are useful for non-destructive measurements of the conductivity, permeability and wall thickness of metals. Further applications of the technique include monitoring material loss due to corrosion and measurement of factors that affect the electromagnetic properties of materials such as mechanical stress. [ABSTRACT FROM AUTHOR]

The impedance and electromagnetic field of a circularly cylindrical coil at an arbitrary position and orientation inside a cylindrical conductor, such as a borehole or tube, has been evaluated using analytical expressions. Previously, such expressions were available for two specific cases: one where the axis of the coil is parallel to the axis of the conductor and the other where the axes of the coil and conductor are perpendicular. In this paper, we consider the general case accounting for an arbitrary coil position and orientation. The analytical solution is made possible by successive transformations of the expressions for the coil field between spherical and cylindrical coordinate systems. The extended model is valuable for simulating eddy current inspections of boreholes and tubes with internal probes accounting for lateral movement and tilt. [ABSTRACT FROM AUTHOR]

This Letter presents an analytical expression for the capacitance of a curved patch capacitor whose electrodes conform to the curvature of a long, homogeneous, cylindrical dielectric rod. The capacitor is composed of two infinitely long curved electrodes, symmetrically placed about a diameter of the cylinder cross-section. The resulting capacitance per unit length depends on both the dielectric properties of the material under test and the capacitor configuration. A practical capacitance measurement is also presented, with appropriately guarded finite electrodes. Very good agreement between measured and theoretically predicted capacitances were observed, to within 2.4 percent. The analytical result presented in this Letter can be applied for extremely rapid evaluation of rod permittivity from measured capacitance. [ABSTRACT FROM AUTHOR]

In eddy current testing, a flaw in a metal is detected when it gives rise to a change in the electrical impedance of the probe that induces current in the material. Theoretical models and computer codes have been developed to predict the probe signals as an aid to improving inspections and the interpretation of measurements. Model calculations can be efficient for a restricted class of problems in which the conductor geometry is simple, such as an infinite plate or tube. The computational cost is usually low in such cases because dedicated Green's kernels are available, allowing numerical approximations of integral equations to be found using only a few unknowns to represent the field in the flaw region. In this study, the aim has been to perform eddy current calculations on corner cracks efficiently using an approximate Green's function for a conductive quarter-space, thereby extending the class of problems that benefit from the use of a dedicated kernel. The properties of the kernel mean that numerical solutions based on boundary or volume elements can be found for an edge crack by rendering as a discrete approximation only the field at the surface of the flaw or the field within it respectively. Volume element calculations have been carried out to determine the field at a corner crack and from it the probe response. Comparisons of the calculated probe impedance due to edge notches show good agreement with experimental measurements. [ABSTRACT FROM AUTHOR]

Cracks in conductors are detected through changes in the impedance of a coil that induces current in the material. In order to gain insight into the physics of the inspection, we have developed a theoretical and computational model that predicts the signals due to cracks in circular cylindrical holes using a boundary element calculation. In formulating the problem, the electromagnetic field is decomposed into transverse electric and transverse magnetic scalar modes. The effect of a planar crack in an electromagnetic field is represented by an electric current dipole layer orientated normal to the crack surface. The dipole density is determined by the integral equation whose dyadic kernel ensures that the tangential electric and magnetic fields are continuous at the surface of the hole. Instead of solving this equation, a numerical approximation is found in the form of a discrete system of linear algebraic equations formed using either boundary and volume elements depending respectively, on whether the crack opening is negligible or not. Because the kernel embodies the interface conditions at the surface of the hole, a discrete approximation of the field is only necessary in the flaw domain which means that relatively few unknowns are needed. The probe impedance variation has been computed for both ideal cracks, defined as having negligible opening but impenetrable to current, and open cracks/slots. Open crack model predictions of coil impedance variations with position relative to a semi-elliptical axial crack are in good agreement with measurements. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

Electromagnetic fields due to excitation coils in boreholes containing cracks have been calculated using a boundary element method formulated in terms of a scalar decomposition of the field. By using a Green's kernel that ensures continuity of the tangential electric and magnetic field at the cylindrical surface, it is only necessary to render the flaw region in a discrete form. The effect of an ideal crack, one with zero opening but acting as a perfect barrier to eddy currents, is here represented by a current dipole layer orientated normal to the crack surface. An integral equation determines the source density on the crack and a numerical solution is found using boundary elements. The coil impedance change due to the crack is then computed using an expression based on a reciprocity principle. [ABSTRACT FROM AUTHOR]

We have determined the quasi-static time-domain dyadic Green's function for an electric source in a conductive plate for use in electric field integral equations. Starting with a frequency-domain representation, we constructed the dyadic kernel from electric and magnetic scalar potentials defined with respect to a preferred direction normal to plate. The final time-domain expression has three parts: a free-space term, multiple image terms, and partial reflection terms. The free-space fundamental solution is expressed in terms of a three-dimensional Gaussian bell curve satisfying the diffusion equation. Similarly, the image terms are expressed in the same form with coordinates shifted so that each Gaussian curve centers on an image point. In order to carry out the inverse Laplace transform from the frequency domain to the time domain, we expand the partial reflection functions as asymptotic series before transforming them analytically. The resulting expression for dyadic Green's kernel can be evaluated efficiently and with well-controlled accuracy. [ABSTRACT FROM AUTHOR]

We describe the calculation of eddy currents in a two-layer conducting rod of finite length excited by a coaxial circular coil carrying an alternating current. The calculation uses the truncated region eigenfunction expansion (TREE) method. By truncating the solution region to a finite length in the axial direction, we can express the magnetic vector potential as a series expansion of orthogonal eigen-functions instead of as a Fourier integral. The restricted domain can be arbitrarily large to yield results that are as close to the infinite domain results as desired. Integral form solutions for an infinite rod are well known and relatively simple. For a finite length cylindrical conductor, however, additional boundary conditions must be satisfied at the ends. We do this by comparing series expansions term by term to match the solutions across the end of the cylinder. We derive closed-form expressions for the electromagnetic field in the presence of a finite two-layer rod. A special case of the solution is that for a conductive tube. We illustrate the end effect by calculating the coil impedance variation with respect to the axial location of the coil. The results are in very good agreement with those obtained by using a two-dimensional finite-element code. [ABSTRACT FROM AUTHOR]

Presents information on a study which developed a boundary integral vector potential formulation to evaluate eddy-current interactions with three-dimensional finite cracks in conductors. Vector potential integral formulation; Formal solution to Laplace's equation; Numerical implementation; Results and discussion on the study; Conclusion.

In this article, the multilevel adaptive cross approximation (MLACA) algorithm is presented to accelerate the boundary element method (BEM) for eddy current nondestructive evaluation (NDE) 3D problems involving arbitrary shapes. The Stratton-Chu formula, which does not have the low frequency breakdown issue, has been selected for modeling. The equivalent electric and magnetic surface currents are expanded with Rao-Wilton-Glisson (RWG) vector basis functions while the normal component of the magnetic field is expanded with pulse basis functions. The MLACA compresses the rank deficient matrices with the ACA and the butterfly algorithm. We improve the efficiency of MLACA by truncating the integral kernels after a certain distance and applying the multi-stage (level) algorithm adaptively based on the criteria for different operators to further decrease the memory and CPU time requirements while keeping almost the same accuracy comparing with the traditional MLACA. The proposed method is especially helpful to deal with the large solution domain issue of the BEM for eddy current problems. Numerical predictions are compared with the analytical, the semi-analytical predictions and the experimental results for 3D eddy current NDE problems of practical interest to demonstrate the robustness and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, the nested kernel degeneration (NKD) based integral equation fast solver is proposed to compress the dense system matrix generated by boundary element method (BEM) for calculating eddy current signals in 3D nondestructive evaluation (NDE) problems. The near and diagonal block interactions are computed and stored by full matrices. However, for the far block interactions, not like in the single level kernel degeneration-based method that all the interaction matrices are compressed by the interpolations, the nested approximation framework makes the expression of the higher-level matrices by the ones at leaf levels and the transfer matrices between neighboring levels. Different integral kernels are degenerated and nested to ensure the performance of NKD based method. The robustness and efficiency of the proposed solver are validated and verified by making numerical predictions and comparing them with the ones achieved by analytical method, numerical methods and experiments. Our numerical experiments show that the NKD solver is more efficient than the butterfly solver (multilevel adaptive cross approximation algorithm (MLACA)) for electrically small eddy current NDE problem. With the NKD based method, O (N) computational complexity is achieved for multiscale low frequency eddy current simulations without sacrificing accuracy, where N is the number of unknowns. So far, to our best knowledge, the proposed method with linear complexity is the most efficient integral equation-based BEM forward solver to predict the eddy current signals from cracks and flaws. [ABSTRACT FROM AUTHOR]

An arc-electrode capacitive sensor has been developed for quantitative characterization of permittivity of cylindrical dielectric rods. The material property of the cylindrical test piece can be inversely determined from the sensor output capacitance based on a theoretical model. For the modeling process, the electrostatic Green's function due to a point source exterior to a dielectric rod is derived. The sensor output capacitance is numerically calculated using the method of moments (MoM), in which the integral equation is set up based on the electrostatic Green's function. Numerical calculations on sensor configuration optimization are performed. Calculations also demonstrate the quantitative relationship between the sensor output capacitance and the test-piece dielectric and structural properties. Capacitance measurements on different dielectric rods with different sensor configurations have been performed to verify the validity of the numerical model. Very good agreement (to within 3%) between theoretical calculations and measurement results is observed. [ABSTRACT FROM PUBLISHER]

In this article, the kernel independent H -matrix method is applied as the mathematical framework. The dense matrix generated by the selected integral equation, without low frequency breakdown problem for solving 3D arbitrary shaped eddy current nondestructive evaluation (NDE) forward problems, is compressed by the truncated and degenerated (TD) kernel function method. It results in a significant reduction in computational burden with nearly linear complexity. The kernel functions (Green's function) are degenerated by Lagrange polynomials which leads to two advantages: firstly, the double integrals are separated in two single integrals, and secondly, they are represented in a factorized form with good accuracy control. Meanwhile, due to the nature of the kernel function with exponential decay in the lossy medium, it is truncated to improve the overall performance. The TD kernel function-based boundary element method (BEM) fits well for solving eddy current NDE problems over the adaptive cross approximation based BEM, especially for detections of flaws or slots in planar objects. Several numerical predictions calculated by the proposed method are compared with those achieved by others such as the experiment, analytical and semi-analytical methods from benchmarks. The robustness and efficiency are demonstrated with excellent agreements and reduced complexity. [ABSTRACT FROM AUTHOR]