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In this paper we present a modification of an existing analytical model for a long surface crack on a conductive plate. This is actually a thin skin model and the challenge for its successful implementation is the accurate calculation of the magnetic field on the defect-free surface on a conductor. We can now combine analytical expressions with numerical results to calculate the coil impedance changes, due to the defect existence. Using the numerical results by FEM for the coil’s magnetic field into the final impedance change analytical expression we can simulate eddy current probes with complex shape that are difficult to be described analytically.

Eddy current based approaches have been investigated for a wide range of inspection applications. Dodd-Deeds model and the truncated region eigenfunction expansion (TREE) method are widely applied in various occasions, mostly for the cases that the sample is relatively larger than the radius of the sensor coil. The TREE method converts the integral expressions to the summation of many terms in the truncated region. In a recent work, the impedance of the co-axial air-cored sensor due to a plate of finite radius was calculated by the modified Dodd-Deeds analytical approach proposed by authors. In this paper, combining the modified analytical solution and the TREE method, a new finite region eigenfunction expansion (FREE) method is proposed. This method involves modifying its initial summation point from the first zero of the Bessel function to a value related to the radius of the plate, therefore makes it suitable for plate with finite dimensions. Experiments and simulations have been carried out and compared for the verification of the proposed method. Further, the planar size measurements of the metallic circular plate can be achieved by utilising the measured peak frequency feature.

Non-destructive tests were conducted on welded fatigue specimens prepared using the same steel material and welding method as the one used in the manufacture and repair procedures of a KRUPP SchRs 600 bucket wheel excavator to reveal any defects present. The chemical composition, the mechanical properties, tendency to cracks and the microstructure of the bucket wheel material were determined using appropriate tests. The initiation of cracks and their subsequent growth during fatigue testing of the welded specimens was studied using ultrasound testing (UT) and a metallographic examination in order to investigate the causes of failure during service and predict fatigue life of the bucket wheel welded parts. It was found that the welding method used produces welds with numerous discontinuities that can only be detected using ultrasound techniques.

The magnetostatic vector fields in terms of harmonic scalar potentials scattered by near–surface air inclusion of arbitrary shape, embedded in a conductive ferromagnetic medium, are investigated. The hollow inclusion is illuminated by a current–carrying coil, which serves as the primary field. The do-main of interest is separated into homogeneous subdomains under the assumption of a suitable truncation of the region of magnetostatic activity at a long distance from the incident source. Therein, the field is considered negligible and consequently a perfect magnetic boundary condition is implied. On the other hand, the introduced methodology addresses the full coupling between the two interfaces, i.e. the plane that distinguishes the half–space ferromagnetic material from the open air and the arbitrary surface among the inclusion and the ferromagnetic region. To this end, continuity conditions are applied in a rigorous way, while the expected behavior of the fields, either as ascending or as descending are taken into account. The scattering problem is solved by means of a modal approach, where potentials associated with the half–space are expanded via cylindrical harmonic eigenfunctions, while those related with the inclusion’s arbitrary geometry admit a generalized–type formalism, being the key to our method. However, since the transmission conditions involve potentials with different eigenexpansions, we are obliged to rewrite cylindrical to generalized functions and vice versa, obtaining handy relationships in terms of easy–to–handle integrals, where orthogonality then would be feasible. Once done, the calculation of the exact solutions leads to infinite linear algebraic systems, whose solution is achieved trivially through standard cut–off techniques. Thus, we obtain the implicated fields in a general analytical and compact fashion, independent of the inclusion’s geometry. In order to demonstrate the efficiency of the analytical model approach, we assume the degenerate case of a spherical inclusion, whereas the air–cored coil simulation via a numerical procedure validates our generalized method. The calculation is very fast, rendering it suitable for use with parametric inversion algorithms.

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.

The non-linear induction problem in ferromagnetic media is solved using the fixed-point iteration method, where the linearized problem at each iteration is treated by means of a modal approach. The proposed approach does not require meshing of the solution domain, which results in fast computations comparing to conventional mesh-based numerical techniques. Both harmonic and pulse excitations are considered via Fourier and Laplace transform, respectively. An efficient method for the fast computation of the inverse Laplace transform of the magnetic polarization signals is also devised based on the generalized pencil-of-function (GPOF) method. Although being restricted to one dimensional configurations, the present work provide the tools for the treatment of two and three dimensional problems, whose study is under way.

The transient response from a transmitter–receiver coil system inside a multi-layer cylindrical conductive configuration is obtained. The particular set-up applies to well logging as well as to eddy current tube testing. In this work, a number of improvements are presented to existing models for an efficient calculation of the induced voltage. These include: domain truncation, novel treatment of arbitrary number of layers in order to avoid computational overflows and efficient time response calculation. The latter is based on a combination of Laplace inversion techniques for short- and long-time transient responses. This article is part of the theme issue ‘Advanced electromagnetic non-destructive evaluation and smart monitoring’.

The non-linear induction problem in an infinite ferromagnetic pate is studied theoretically by means of the truncated region eigenfunction expansion (TREE) for the 2D case. The non-linear formulation is linearised using a fixed-point iterative scheme, and the solution of the resulting linear problem is constructed in the Fourier domain following the TREE formalism. The calculation is carried out for the steady-state response under harmonic excitation and the harmonic distortion is derived from the obtained spectrum. This article is meant to be the theoretical part of a study, which will be complemented by the corresponding experimental work in a future communication.

We report on further measurements and a numerical solution for a proposed benchmark problem involving an eddy-current nondestructive evaluation configuration. This is a simplified version of the inspection of fastener holes in aircraft structures, comprising combinations of plates with through holes and a crack. Previously, only line (B-scan) data were provided for the coil impedance, while in this paper, we provide precision surface (C-scan) data as well as details on the evaluation of the critical parameters of the problem. Furthermore, the experimental data are cross validated with numerical ones using a model that combines an integral formulation and the FEM approach.

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.

Carson’s integral is used in power systems analysis for evaluating the earth-return impedance of overhead conductors above homogeneous earth. Contrary to the widespread belief that it does not evaluate to a closed-form expression, a scrutiny of Carson’s historical (1926) paper reveals that such an expression exists and was actually presented therein. Instead of using the original expression that involves the evaluation of just one Struve and one Bessel function, modern approaches wander between simplified but crude approximations of the integral and numerical evaluations by adaptive quadrature methods. In this paper we present the facts and show that neither of these approaches are necessary, since we can readily and rapidly evaluate the integral (to computer machine accuracy) by using the closed-form expression.

Expressions are presented for the inductance of a curved circular spiral coil in free space and the change in impedance when the coil is wrapped around the surface of a magnetic, conductive cylinder. Inductance and impedance measurements of a thin conformable coil on a cylindrical Al-alloy rod were used to test the validity of these expressions. The theoretical predictions are in very good agreement with the experimental measurements over the frequency range 1 kHz–10 MHz. The findings are compared with previous work using rectangular spiral coils and it was found that the change in normalized impedance at high frequencies for a circular coil was almost identical to that obtained for a similarly-sized square coil. The implications for use in eddy-current nondestructive testing are discussed.

Currently, there are three TEAM Workshop benchmark problems involving eddy current nondestructive evaluation configurations, Nos. 8, 15, and 27. Regarding the first two, they have ceased to constitute challenging problems since they can be routinely solved with existing commercial software. In this paper, we present precision impedance measurements in a configuration involving a system of plates with through holes and a crack and propose them for benchmarking numerical codes. The benchmark represents a simplified version of eddy current inspection of fastener holes in aircraft structures, provides a lot of experimental data, and overall constitutes a more challenging problem.

International audience; Simulation of flaw detection by eddy-current testing requires the calculation of the eddy-current signal due to the presence of the flaw. In this paper, the surface integral method (SIM) is applied for the calculation of the crack signal in work-pieces composed of a conducting plate with a borehole. The adopted approach makes use of a Green’s operator that takes into account the effect of the borehole, thus the integral formalism is restricted to the crack surface only. The truncated region eigenfunction expansion (TREE) method is followed for the construction of the Green’s operator. Comparisons of the computed crack signal with numerical simulations carried out using a hybrid VIM-SIM approach and experimental results show excellent agreement.

Conference of 42nd Annual Review of Progress in Quantitative Nondestructive Evaluation, QNDE 2015, Incorporating the 6th European-American Workshop on Reliability of NDE ; Conference Date: 26 July 2015 Through 31 July 2015; Conference Code:119404; International audience; In this paper, a numerical solution of WFNDEC 2015 eddy current benchmark is presented. In particular, the Surface Integral Equation (SIE) method has been employed for numerically solving the benchmark problem. The SIE method represent an effective and efficient alternative to standard numerical solver like Finite Element Method (FEM) when electromagnetic fields need to be calculated in problems involving homogeneous media. The formulation of SIE method allows to properly solve the electromagnetic problem by meshing the surface of the media instead to the complete media volume as done in FEM. The surface meshing enables to describe the problem with a smaller number of unknowns with respect to FEM. This property is directly translated in an obvious gain in terms of CPU time efficiency.

Calculation of earth-return impedance for underground conductors is based on the efficient evaluation of Pollaczek's integral. So far, the established practice involved either approximate expressions or numerical integration algorithms. Apart from extensive comments on the latter, this paper presents for the first time three exact solutions in the form of converging series. One of the series guarantees a specific relative accuracy for any parameter range with a specific number of terms and can be regarded as the definite solution that will hopefully settle the issue of computing Pollaczek's integral both rapidly and accurately. The exact solution provides the means to assess the validity of the impedance formula.

Calculations are carried out to evaluate pulsed eddy-current interaction with a crack in a planar conductor. The field is excited by a coil carrying current pulses and crack signals are detected by measuring changes in the magnetic field above the conductor. The model is based on the Fourier superposition concept, where the current pulses are decomposed into distinct frequencies. An existing boundary element method model for eddy current inspection of narrow cracks is used for each frequency and the final transient crack signal is synthesized by the individual frequency responses. The model is verified by a 3D finite element commercial package with a time-stepping technique. Computation times for the presented approach are orders of magnitudes shorter and allow rapid parametric studies of the transient signals for identification of principal component characteristics as well as for inversion purposes.

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.

The eddy-current induction problem in a conducting half-space with a vertical borehole is studied and a solution based on modal techniques is presented. Owing to the particular characteristics of the specific geometry, the straightforward application of the truncated region eigenfunction expansion method (a well-established tool for the analytical solution of such problems) presents significant difficulties. Therefore, a modified truncation approach has been adopted that treats the problem in two steps, each one based on the superposition principle. The approach proves to be very efficient and numerically robust. Results for the coil impedance are in excellent agreement with those obtained with the finite-element method. From the engineering point of view, the geometry simulates the eddy-current inspection of fastener holes in aircraft structures when coil scanning is performed from the surface. The presented solution constitutes an important step towards the development of a complete model for fastener hole crack inspections.

The analytical solution for the eddy-current induction in a conducting half-space containing an infinitely long cylindrical hole parallel to the interface is derived by means of the second-order vector potential and the domain truncation technique. The eddy currents are induced by a cylindrical coil which can be placed either in the nonconductive region above the half-space or inside the hole or in both locations simultaneously. Analytical expressions for the coil impedance are derived in both cases. The problem is treated as a coupled scattering problem and the derived analytical expressions permit a clear physical interpretation of the different terms, thus enhancing the understanding of underlying physics. The validity of the solution is demonstrated by comparisons to results obtained with the finite element method.

PurposeThe paper aims to develop expressions for calculating the mutual impedance between isolated conductors buried in homogeneous earth. The conductors have finite length and arbitrary position.Design/methodology/approachThe conductors are represented by the use of elementary electric dipoles. Well‐known existing expressions are employed for the electric field of these dipoles. The induced voltages are evaluated and the final expressions for the mutual impedance are derived. The resulting expressions involve infinite double integrals, evaluated by using adaptive quadratures that are, however, time consuming. Therefore, an alternative approach is followed involving Sommerfeld integrals (SI) for representing the electric field of a dipole and a recently devised method for computing the SI, in the spatial domain, by using calculation of discrete complex images.FindingsFinal expressions for parallel and perpendicular conductors were derived and numerical results for several values of frequency, conductors' length and horizontal distance between them, were produced. Comparison to results produced with the well‐known Pollaczek formula showed excellent agreement.Research limitations/implicationsFor future research, it is possible to use the developed expressions for earthing systems study, where the grounding grid is discretized and the moment method is invoked.Originality/valueCurrently, the formulas used for calculating mutual impedance are valid for parallel conductors of infinite length. With the present work, accurate expressions are given for finite length and arbitrary horizontal positioned conductors. In addition, the use of SI and the discrete complex image method results in a rapid and efficient tool for massive impedance calculations.

The complex impedance of an air-cored coil in a conductive tube with eccentric inner and outer cylindrical surfaces is calculated. The analytic expressions for the induced fields and the impedance variation due to the eddy-current flow inside the tube wall are derived using a second-order potential approach. The addition theorem of Bessel functions is employed to perform the transition between the local coordinate systems that conform to the boundaries of the structure. Although the model can be used for any coil shape and orientation, we focus our study on the configuration of a bobbin coil with axis parallel to the axes of the tube surfaces, but not necessarily coinciding with either of them. The results of the presented analysis are verified by a finite-element-method (FEM) solution.

Further developments in a Boundary Element Method model for eddy current inspection of ideal or narrow cracks are presented. First, the required Green’s function is rapidly computed in the spatial domain using the concept of discrete complex images. The approach is based on the analytical evaluation of the Sommerfeld type Integrals through the use of the Generalized Pencil of Function method. Computation times are recorded in order to show the remarkable reduction in the moment matrix fill time as well as in the total computational time. Second, the model is extended to arbitrary crack depth profiles. Theoretical results are compared to published benchmark measurements of the coil impedance change due to a semi-elliptic and an epicyclic crack and excellent agreement is achieved.

Eddy current testing is one of the most widely used methods in non-destructive testing for the inspection of conductive materials. Numerical modelling of eddy current testing has emerged as an important approach alongside experimental studies. This paper investigates an application of numerical modelling and experimental study as a means of the quantitative non-destructive evaluation (QNDE) of defects in conductive samples. There are two methods of measuring eddy current response, more commonly by measuring the change in impedance of the eddy current probe coil, or as used in this work, by measuring the change in magnetic field directly using magnetic field sensors such as superconducting quantum interference devices, giant magneto resistance, or as in this case Hall sensors. Specifically, measurements made using an eddy current probe containing an excitation coil and a Hall sensor, experimentally obtained using an X–Y scanner table, are compared with a numerical (finite element method) model. The discrepancies between the experimental tests and the numerical models have been analysed and explained, which is an important factor in engineering applications of QNDE.

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.

An existing boundary element method model for eddy current inspection of ideal or narrow cracks is revised. Using some lately devised concepts on efficiently and rapidly simulating canonical eddy current geometries, we show how this model can be made easier to implement and faster to calculate. The approach is largely based on a novel method for rapidly calculating the Green's function and reducing the matrix fill-time, and also on a similar method for rapidly computing the incident electric field. As a result, the model has been made an order of magnitude faster than the existing one without sacrificing accuracy. We pay attention to numerical analysis details and analyze issues that so far have not been clarified. The validity of our approach is also verified by the experiment. Although we have tested the model against published data, we have also produced our own precision measurements for surface and through-the-thickness slots in plates with the coils performing scans along and across the slots at various frequencies.

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.

Eddy-current inspection for nondestructive evaluation has traditionally been investigated in terms of coil impedance signals via theoretical and experimental methods. However, advanced eddy-current techniques use solid-state sensors such as Hall devices, giant magnetoresistive sensors, anisotropic magnetoresistive sensors, and superconducting quantum interference devices for magnetic field measurement to achieve better sensitivity and high temporal and spatial resolution in material evaluation and characterization. Here, we review the Dodd and Deeds integral model and use the truncated region eigenfunction expansion (TREE) method for computation of the magnetic field. This results in series expressions instead of integral ones. Thus, the computation is both simplified and speeded up so that it becomes convenient for solving one-dimensional eddy-current inverse problems. We compare the theoretical results from the analytical model with the results from a numerical simulation based on the finite-element method in terms of accuracy and computation time.

International audience; A fast model for the eddy-current signal calculation of thin cracks at the vicinity of a conducting tube edge is presented. The model is based on the surface integral equation formalism using a dedicated Green’s function. The approach followed for the construction of the Green’s kernel and the primary field is the Truncated Region Eigenfunction Expansion (TREE) method.

International audience; A semianalytical solution to the eddy-current induction problem in a conducting, nonmagnetic plate near a cylindrical through-holeis derived by means of the second-order vector potential and the domain truncation technique. The construction of a solution, which simultaneously satisfies the field continuity across the cylindrical surface of the hole and the planar interfaces of the plate, proves to be a very challenging problem, and thus, a special superposition of partial subsolutions is proposed. The presented solution is part of a more general development aiming at the calculation of the probe interaction with material flaws located in the vicinity of the hole by means of the integral equation formalism. These tasks will be the subject of a future paper. The comparison of the semianalytical results against both finite-element simulations as well as experimental data demonstrates excellent agreement.

A fast integral equation model for the eddy-current signal calculation of thin cracks breaking the wall of a borehole in a conducting plate is presented. The model is based on the surface integral equation formalism using a dedicated Green’s function, which takes into account the effect of the borehole. The approach followed for the construction of the Green’s kernel and the primary field is the Truncated Region Eigenfunction Expansion (TREE) method.

The time-harmonic induced current in a conductive plate has been calculated for an excitation by a normal coil near the plate edge. Using series expansions in elementary functions, the quasi-static solution for the plate is matched to the solution for the region outside the plate to ensure continuity of the tangential magnetic field and the normal magnetic flux density at the air–conductor interface. The continuity of the field is achieved by mode matching in a truncated domain. Predictions of the coil impedance as a function of position derived from this theory have been compared with experiment showing good agreement.

The electromagnetic field and impedance of a cylindrical eddy-current probe coil are calculated analytically for arbitrary coil orientation above a conductive half-space. The remarkably simple closed-form expressions are provided as a function of coil tilt angle. The effect of tilt on the impedance change produced by a long crack is also investigated by combining the analytical model with an existing thin-skin theory for surface crack inspections. Results for both cracked and uncracked conductors are expected to be useful for evaluation of movement-generated noise in eddy-current inspections.

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 eigenfunctions 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.

The well-established Dodd and Deeds models, that are widely used in eddy current nondestructive evaluation (NDE) and have served the industry for three decades, are being reformulated. Integral expressions for the electromagnetic field and the impedance of the eddy current coil are replaced by series expansions. As a result, computation time is considerably reduced, convergence is better controlled and computer implementation is greatly simplified. The analysis is performed for a coil over a conducting half-space and a coil inside a borehole with a conducting wall.

The magnetic and torque performance of a capacitor-run single-phase induction motor operated at constant torque under the optimal efficiency control, as presented in Part I, is analyzed. The magnetic field analysis demonstrates that magnetic saturation is considerably decreased with optimal efficiency control. Although triac-based optimal voltage controller introduces voltage and current harmonics, torque pulsations and acoustic noise are reduced due to magnetic flux weakening. The reduction of torque pulsations and acoustic noise is considerable in the low torque region where significant energy savings are also achieved. Selected calculated and experimental results are presented to validate the theoretical considerations and the resulting improvements.

This work investigates the problem of efficiency optimization in capacitor-run single-phase induction motors. The double-revolving-field concept is employed in the theoretical analysis and a relation between the main and auxiliary stator currents is derived that accomplishes optimal efficiency under constant torque operation. A triac-based drive with an optimal efficiency voltage controller is proposed. The controller is easily implemented since an experimental procedure is used for adjusting its parameters. Moreover, the proposed control scheme satisfies all of the prerequisites of simplicity, reliability, and cost-effectiveness that are imposed by the utilization of a single-phase motor. Several experimental results are presented to validate the effectiveness of the proposed efficiency optimization control method.

Eddy-current induction by an air-cored coil located inside a long cylindrical bolthole in an isotropic conductive medium is examined using a second-order vector potential formalism. Closed-form expressions are presented for (i) the induced electromagnetic fields and currents, and (ii) the change in coil impedance, ?Z, due to eddy-current induction in the surrounding conductor. In these expressions, the coil characteristics enter via a source function, Ds. Two alternative methods for computing Ds are described: in one, Ds is computed directly, and in the other, Ds is computed using a knowledge of the source magnetic flux density at the surface of the hole. The strengths and weaknesses of the two approaches are discussed. The non-axisymmetric case of a cylindrical coil with its axis perpendicular to the hole axis is examined in detail because of its significance in eddy-current non-destructive evaluation using bolthole probes. An explicit closed-form expression for ?Z in terms of the coil and bolthole parameters is derived for this important geometry. The validity of the theoretical calculations was tested by comparing the theoretical results with experimental measurements of ?Z as a function of frequency for a series of air-cored coils inside 10.0 and 6.4?mm diameter boltholes in Al alloy specimens. The theoretical and experimental results are in excellent agreement.

A model of an axisymmetric probe coil with a cylindrical ferrite core in the presence of a layered conducting half space is developed. The boundary value problem is formulated in terms of the magnetic vector potential, which is expanded in a series of appropriate eigenfunctions. The corresponding eigenvalues are the real roots of a function with the radius and the magnetic permeability of the core as parameters. The unknown coefficients of the series are computed by solving a matrix system, which is formed by applying the usual interface conditions. Expressions are derived for the induced eddy current density and the impedance of the coil. Numerical results for the latter are in good agreement with experimental data.

We present a development on an existing model for the electromagnetic interaction of a coil with a conductive cylinder. We can now calculate analytically both the electromagnetic field and the impedance of arbitrarily positioned probe coils. This is possible by using an appropriate expansion of cylindrical eigenfunctions in terms of eigenfunctions of the Cartesian coordinate system. The extended model is useful in simulating eddy current inspections of cylindrical structures under conditions of probe tilt as well as inspections with special coil orientations like the rotating probe coil.