## Abstract

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.

## 1. Introduction

An important problem in electromagnetism is the determination of the quasi-static field of an induction coil in the presence of a conductive test-piece. In case the test-piece has the form of an infinitely extending plate, exact analytical expressions can be found for the field and the coil impedance using the separation of variables method (Dodd & Deeds 1968). Such results are very useful in a number of engineering fields including eddy-current non-destructive evaluation (NDE). Nevertheless, there is also a need to compute the field of inductive probes near edges where two surfaces meet at an angle as, for example, at the edge of a conductive plate or a block. The need arises because cracks are likely to form at corners where stress concentrations occur. The eddy-current inspection of cracks near edges is a very important issue for a number of industrial applications, particularly for those involved in the aerospace domain. The development and propagation of fatigue cracks in the vicinity of aircraft fasteners owing to mechanical stresses may have detrimental effects on the structural integrity of the aircraft. The inspection is performed either with the fastener removed and the coil inside the borehole or with the fastener installed and the coil scanning the area from above.

A simplified version of the latter geometry is the subject of this work, i.e. a borehole in a conductive half-space with the coil moving freely above the planar surface. Simulation of the configuration is based on the solution of Maxwell's equations, and in general can be done with the finite-element method (FEM). However, the extensive use of FEM is not encouraged for engineering purposes, for a number of reasons, namely the usually small thickness of the plates, which results in an increased mesh size, and the fact that the problem needs to be solved for a large number of coil positions above the borehole. Also, the resulting computation time is prohibitive for the solution of the inverse problem when a large number of forward solutions is required. This is the reason for seeking a rapid analytical or semi-analytical method for calculating the electromagnetic field of a coil above a half-space with a borehole. In the absence of an exact treatment, we sought a relatively simple, computationally efficient engineering solution for this problem. Note that for applications to eddy-current testing, we consider an induction coil excitation at a frequency where displacement current is negligible.

Analytical solutions to the eddy-current induction problem in infinite layered media of either planar or cylindrical geometry using eigenfunction expansions have been around since the classic work of Dodd & Deeds (1968). Following that approach and making use of the addition theorem for Bessel functions, the eddy-current problem can be tackled in infinite but not symmetrical configurations like the one of an off-axis bobbin coil in an infinite eccentric tube (Skarlatos & Theodoulidis 2010).

In a number of recent contributions, modal solutions to eddy-current problems in configurations of finite extent have been achieved, thanks to a technique based on the artificial truncation of the (originally infinite) problem domain, an approach usually referred to in the literature as the truncated region eigenfunction expansion (TREE) (Theodoulidis & Kriezis 2006). In case of a parabolic problem, like eddy-current induction, this approximation can be applied without significantly affecting the accuracy of the solution. It should be noted that it is the same approach which is usually adopted in numerical techniques based on volume meshing, like FEM. The approximation error introduced by the domain truncation can be arbitrarily reduced by increasing the size of the considered domain, with the consequent increase, however, in the number of modes that have to be taken into account. Using the TREE approach, Theodoulidis & Bowler (2005) have addressed the induction problem of a coil over a right-angled conducting corner.

Concerning cylindrical structures, truncation can be applied either in the axial direction (Theodoulidis 2004; Bowler & Theodoulidis 2005) for the treatment of conducting tubes and rods of finite extent, or in the radial one, e.g. for the modelling of ferrite probes over planar media or the inspection of a cylindrical borehole using axis-symmetrical coils (Theodoulidis & Kriezis 2006). The problem of the borehole inspection using an off-axis cylindrical coil (either parallel or perpendicular to the hole axis) is tackled by Theodoulidis & Bowler (2008), where a second-order potential approach is applied in combination with axial truncation. Nevertheless, that solution is restricted to configurations where the coil is entirely contained inside the cylindrical surface defined by the tube wall, and thus it can be applied only when the inspection is performed from the interior of the hole.

The contribution of this paper is the semi-analytical treatment of the same geometry, but this time assuming that the probe is located above the half-space. In contrast to the configuration considered in Theodoulidis & Bowler (2008), matching of the internal (conductive space) and external (air) solutions proves to be troublesome for this particular coil location. This is the reason for adopting an engineering approach, where the matching procedure on the two interfaces is performed in two successive steps. Furthermore, the proposed approach leads to a transcendental equation (the equation for the evaluation of the problem eigenvalues), which is very efficient because it does not involve Bessel functions, as in the two-dimensional case with simple radial truncation (Theodoulidis & Kriezis 2006). The problem of an infinite cylindrical hole, yet parallel to the interface, has been studied in Skarlatos & Theodoulidis (2011). Together with the analysis presented herein and the one in Theodoulidis & Bowler (2008), they can be considered as complementary works offering a class of analytical solutions for different variations of the general problem of a hole inside an infinite half-space.

Despite the fact that only cylindrical air-cored coils with the coil axis normal to the upper surface of the half-space (pancake coils) are considered in this work, the solution is not restricted to this specific type of excitation. In fact, it can be applied to any AC excitation current distribution provided the corresponding expansion coefficients in free space can be properly calculated.

The motivation behind the present work is not only to find a quasi-analytical and rapid solution of the borehole in the half-space eddy-current problem but more significantly to lay the foundations for the efficient evaluation of signals emanating from cracks located at the edge of the borehole. To the best of the authors' knowledge, the state of the art concerning the calculation of crack signals in the proximity of the borehole/fastener edge consists either of the brute meshing of the whole region via FEM or of the application of the volume integral method (VIM), where both crack and hole are considered as ‘flaws’, and they are discretized simultaneously using Green's function of infinite planar multi-layered media (Knopp *et al.* 2004, 2006, 2009; Reboud *et al.* 2010). The perspective of this work faces the alternative of applying the boundary element method (BEM) on the crack surface only. Following established theory for modelling ideal or very narrow cracks with BEM (Bowler & Theodoulidis 2009), we need to compute (i) the incident field that is the eddy-current density at the crack location but without the crack, and (ii) an appropriate Green's function defining the electric field produced near the edge by an embedded electric dipole. The solution for the first requirement is described in the current paper. Work is underway for the development of a similar approach regarding the second requirement.

## 2. Description of the problem and governing equations

Let us consider a non-magnetic half-space of finite conductivity *σ* containing an infinite cylindrical hole of radius *ρ*_{a}. The hole axis is normal to the half-space interface, as shown in figure 1. We assume that both the interior of the borehole and the part of the space above the conductor are filled with air.

The eddy-current flow is induced in the half-space by a cylindrical coil excited by harmonic current *e*^{jωt}. We assume that the frequency is sufficiently low in order for the quasi-static approximation to be valid, the notion ‘sufficiently’ being determined with respect to the free-space wavelength. More precisely, the wavelength in the free space should be large compared with the dimensions of the coil and the hole, which is always the case in eddy-current testing applications.

The axis of the coil is normal to the interface. Extension to more complicated coil geometries and orientations is possible as long as the modal expansion coefficients for the excitation term related to the coil can be determined; such geometries will not be considered here.

The problem solution is based on the calculation of the magnetic field throughout the problem region. The magnetic flux density in air is expressed in terms of a scalar potential 2.1which satisfies the Laplace equation 2.2

In the conductive half-space, the magnetic field is expressed via the second-order potential
2.3where *W*_{a,b} satisfy the Helmholtz equation
2.4The magnetic flux density in the conducting regions is derived from the potential expressions via the relation
2.5where *k*^{2}=*jωμ*_{0}(*σ*−*jωε*_{0})≈*jωμ*_{0}*σ*, *ε*_{0} and *μ*_{0} being the permittivity and the permeability of the free space, respectively.

## 3. Formal solution

Since the coil is located in the air half-space above the conducting piece, the natural choice for the construction of the formal solution would be to distinguish three subregions: the air half-space above the conductor (wherein the coil moves), the hole and the conductor itself. This choice would dictate a radial truncation scheme and a subsequent matching of the modal solutions for the different subregions on the half-space horizontal interface. Yet, as it can be shown, the solution in the hole-conductor ensemble with this approach is formally equivalent to the solution of a dielectric waveguide. Hence, from established knowledge (Chew 1995), it is impossible to arrive to a non-trivial field solution for the general non-symmetrical case when the condition of vanishing normal current on the conductor horizontal interface is imposed (for zero coil offset instead, the configuration becomes axis-symmetric, normal currents disappear and the approach is indeed applicable).

An alternative choice would be the one followed in Theodoulidis & Bowler (2008), where the hole and the corresponding overhead cylindrical air column are treated as a unique subregion. This choice works well for arbitrary field configurations since the terms involving *W*_{a} and *W*_{b} can be treated independently, and the zero normal current condition on the conductor interface is imposed *a priori* during the construction of the formal solution; it delivers, however, constraints with respect to the coil radial location, which makes the treatment of the considered inspection situation very difficult. A possible way out here would be to split the coil into two parts (for the positions where the coil intersects the cylindrical surface defined by the hole) and treat them as separate sources. Yet, such an approach is cumbersome and lacks mathematical elegance.

In order to address the above issues, the original problem is tackled in two successive steps. First, the hole is ignored and the simplified two-layer problem is solved. Next, the hole is re-introduced, and the constructed solution for the air region is artificially extended inside the hole volume. In this way, the field continuity across the horizontal interface is ensured, although a discontinuity is introduced across the hole walls. Considering the latter as an equivalent source which produces a secondary field, the axial-truncation approach of Theodoulidis & Bowler (2008) can now be applied without further concern. The described decomposition is schematically depicted in figure 2.

Let us first consider the two-layer problem, i.e. the plate without the hole. The potential solution in the *z*≥*c* region (cf. figure 2) is expressed as the sum of two terms *ϕ*_{s}+*ϕ*_{d}, where *ϕ*_{s} is the potential solution for the coil in the free space and *ϕ*_{d} stands for the ‘reflection’ term owing to half-space. Assuming that the computational domain is truncated at the *ρ*=*ρ*_{L} limit using a perfect magnetic conductor (PMC) condition (yet considered unbounded in *z*-direction),^{1} the potential expressions can be written
3.1and
3.2It should be noticed here that (3.1) holds only for the region of the free space below the coil (i.e. *c*≤*z*≤*z*_{0}−*l*/2, where *l* is the coil thickness). For the regions above and at the same height as the coil, the expressions for the *z* dependence should be modified to guarantee a convergent solution (Theodoulidis & Kriezis 2006). The transmitted field inside the conductor (*z*≤*c*) is expressed via the second-order potential
3.3with . The *W*_{b} term is zero for the two-layer problem (Theodoulidis & Kriezis 2006). The *κ*_{n} eigenvalues are determined by imposing zero tangential field on the *ρ*=*ρ*_{L} boundary, which yields
3.4

The introduction of the hole will produce a perturbation to the above solution, which will be compensated by the additional potentials *ϕ*_{0,1} and . Assuming a perfect electric conductor (PEC) condition on the *z*=0 surface and a PMC on the *z*=*h* limit (the computational domain being considered unconstrained along the radial direction this time), the modal expansions for the these potentials are given by the expressions
3.5
3.6
3.7
and
3.8with and . Notice that, choosing different conditions at the two truncation limits automatically suppresses the zero-order modes (i.e. the TEM ones), so no special action is needed for their proper account.

The PEC condition on the *z*=0 plane is already incorporated in the above expansions. The PMC condition on the *z*=*h* plane yields as before
3.9which implies: . The *r*_{n} values are determined by demanding a zero normal component of the eddy-current flow on the *z*=*c* boundary. The latter is derived from the second-order potential via the relation:
3.10which implies
hence . The remaining eigenvalues *p*_{n} and *q*_{n} will be determined by imposing the field continuity across the conductor interface *z*=*c*. Their calculation will be examined in §3*a*.

It could be claimed at this point that the above choice of the truncation conditions is mathematically incoherent in the sense that whereas the expansions (3.1)–(3.3) assume an infinite domain along the *z*-direction, the same direction is truncated for the construction of the expansions (3.5)–(3.8) (and vice versa for the *ρ* direction). This is a justified objection in terms of mathematical rigorousness. It should be recalled though that the domain truncation approach is merely an engineering approximation. Acknowledging this, once it is recognized that the field is negligible near the truncation limits, either in *z*- or *ρ*-direction, we can add (or remove) any condition at will, which is convenient for the mathematical treatment of the problem, without significantly altering the solution.

The total solution in *ρ*≥*ρ*_{a}, *z*≥*c* can thus be written as the sum of the above terms, namely
3.11Similarly in the *ρ*≤*ρ*_{a} region, we have
3.12Inside the conductor, the total potential can be written as
3.13It should be pointed that despite the fact that an infinite two-layer medium has been assumed for the derivation of the *ϕ*_{d} and *W*_{a} potential expressions, these are not taken into account in (3.12), which provides the solution in the *ρ*≤*ρ*_{a} region. This is, however, a valid choice since the field in each subregion can be freely expressed as any superposition of partial solutions, provided that the latter span a complete function basis, and assure the continuity of the fields across all the inter-region boundaries of the solution domain.

In the following paragraphs, we derive the relations for the calculation of the expansion coefficients by imposing the field continuity across the two interfaces of the problem.

### (a) Continuity at the *z*=*c* interface

First, we consider the continuity of the tangential magnetic field and the normal magnetic flux density components across the half-space interface (*z*=*c*) for the two layer problem. The continuity relations read
3.14for *H*_{ρ},
3.15for *H*_{φ}, and
3.16for *B*_{z}.

Substituting the modal expansions (3.1)–(3.3) into the potential expressions, and taking into account the orthogonality of the *e*^{jmφ} functions, as well as of the Bessel functions, namely
3.17where *δ*_{ℓ,n} is Kronecker's delta, we obtain from the first two conditions
3.18whereas the last one yields,
3.19and consequently the expansion coefficients and are expressed in terms of the excitation coefficients via the relations
3.20and
3.21

Concerning the *ϕ*_{1}, derived terms, given that the *ρ* dependence is the same for both potentials, the continuity of the fields across the *z*=*c* interface is assured if we set
3.22The proportionality coefficient *a*_{n} is determined by imposing the continuity of the *H*_{ρ},*H*_{φ} and *B*_{z} components for each mode separately (i.e. by replacing *ϕ*_{s}+*ϕ*_{d} and *W*_{a} in (3.14)–(3.16) by *ϕ*_{1} and , respectively), which after elimination of the expansion coefficients and yields
3.23The eigenvalues of the expansions are the roots of the complex transcendental equation
3.24which can be solved numerically (see §6).

### (b) Continuity at the (*ρ*=*ρ*_{a}) interface

The continuity of the normal magnetic flux density and the tangential magnetic field across the hole-conductor interface leads to
3.25for *B*_{ρ},
3.26for *H*_{φ}, and
3.27for *H*_{z}.

Substituting the modal expressions (3.5)–(3.8) in the boundary condition relations and taking again the orthogonality of the *e*^{jmφ} functions into account, we obtain
3.28
3.29
and
3.30where *u*(.) is the Heaviside step function. The terms *σ*_{m}(*z*),*J*_{ϕ}(*z*) and *J*_{z}(*z*) can be viewed as equivalent magnetic charge and electric current sources produced by the discontinuity of the *ϕ*_{s},*ϕ*_{d} and *W*_{a} terms at the considered interface.^{2} Their concrete expressions are obtained by subtracting the corresponding field terms:
3.31
3.32
and
3.33

In order to derive the matrix equation which will determine the unknown coefficients, we weight the continuity equations (3.28), (3.29) with and (3.30) with , taking into account the orthogonality of the trigonometric functions 3.34and we obtain 3.35 3.36 and 3.37with the mode-coupling matrices 3.38 3.39 and 3.40and the source terms 3.41 3.42 and 3.43where 3.44and are the corresponding moment integrals for the terms. Note that all integrals appearing in (3.38)–(3.40) have closed-form expressions given in Theodoulidis & Bowler (2005), and that the integral in (3.44) leads to a simple analytical expression as well.

In matrix notation, we can now write 3.45 3.46 and 3.47with and For the formation of the above matrix equations, it has been tacitly assumed that the number of involved modes is finite. In other words, the infinite series have to be truncated in order to be able to treat them numerically.

It should be explicitly noticed at this point, that the above-presented analysis is valid only for the case that the coil is moving above the half-space interface, i.e. *z*_{0}−*l*/2≥*c*. If the coil enters in the borehole, then the correct solution approach is the one given in Theodoulidis & Bowler (2008) (under the condition of course that the coil location is constrained by the cylindrical surface defined by the hole).

## 4. Calculation of the excitation term

The expansion coefficients depend only on the probe geometry and position, as already mentioned. For the case of a pancake coil considered here, these can be easily obtained from the potential expression for the coil in the free space, which in the coil local coordinate system (*ρ*^{′},*z*^{′}) can be written (Theodoulidis & Kriezis 2006, p. 96)
4.1

In fact, Theodoulidis & Kriezis (2006) give the expression for the magnetic vector potential *A*. The above relation for the scalar one is obtained by simply interchanging the J_{1} and J_{0} functions (the two potentials differ by a differentiation). Applying the addition theorem of the Bessel functions now, we can change the coordinate system to the one of figure 2, so we obtain for :
4.2where *ρ*_{1} and *ρ*_{2} stand for the inner and outer radii of the coil, respectively, *l* is its height and *N* its number of turns. The coil is assumed to have an off-axis shift equal to *d*, whereas *z*_{0} denotes the vertical position of its centre (cf. figure 2), resulting to a lift-off of *z*_{0}−*l*/2. The excitation current is *I*_{0}.

The integral *χ*(*x*_{1},*x*_{2}) stems from the integration over the coil cross-section and has a closed-form solution in terms of Struve functions **H**_{n} (Abramowitz & Stegun 1972), namely
4.3

## 5. Impedance variation

The variation of the coil impedance as a function of the probe position is of great importance in NDE applications, since it is the physical parameter of interest when classical inductor probes are used. Its theoretical calculation is thus the main objective of every eddy-current testing analysis.

The coil impedance can be written
5.1where *Z*_{0} is the free-space coil impedance and Δ*Z* stands for the impedance variation owing to the presence of the conductor. The second term can be calculated by applying the reciprocity theorem (Auld *et al.* 1981):
5.2where *E*_{s},*B*_{s} are the electric and magnetic field in the absence of the conductor, and *E*_{ec},*B*_{ec} the corresponding fields in the presence of the conductor. The integration surface encloses the latter, and ** n** is the unit outward normal to

*S*. We choose the integration surface to coincide with the conductor surface at

*z*=

*c*. The remaining part of the surface is dictated by the truncation boundaries. Since the magnetic field is assumed to be zero on these boundaries, the only contribution to the closed integral that survives is the one of the conductor upper interface. Writing the magnetic field in air in terms of a scalar potential as in (2.1) for both terms and after some standard manipulations (Theodoulidis & Kriezis 2006) the above integral becomes 5.3where

*ϕ*

_{s}is the potential for the solution in air given by (3.1), whereas

*ϕ*

_{ec}stands for the solution in the presence of the conductor, for which we can write 5.4Substituting in (5.3), we obtain 5.5Substituting (3.1), (3.2) and (3.5), (3.6) and using the orthogonality of the

*e*

^{jmφ}functions, the first integral yields after some manipulations 5.6where it has been taken into account that and the symmetry of the Bessel functions J

_{−m}(.)=(−1)

^{m}J

_{m}(.).

In the same fashion, we obtain for the second integral 5.7and for the third one 5.8

The integrals involving Bessel functions in (5.7) and (5.8) have closed-form expressions (Abramowitz & Stegun 1972), namely 5.9for the first one, and the analogous expression for the second (with opposite sign for the first term).

The scalar potential in air is defined up to a constant term, the value of which is arbitrary and has no impact to the field solution. Yet care should be taken when dealing with the reciprocity theorem, since not only the gradient of the potential terms but also the potential values themselves are employed. Since, in addition, the potential in each subdomain is defined separately, ambiguities can arise, which will alter the value of the impedance calculated by the reciprocity theorem. To remove this ambiguity, one has to re-gauge the potentials in order to impose the same potential value on the interface points (which are common to both domains). To do so, we choose a reference point on the common interface of the subdomains, and we add a constant gauge term *V*_{g} in the potential expression for *ρ*≤*ρ*_{a} to set its value at the given point to be equal to the one calculated by the corresponding expression for *ρ*≥*ρ*_{a}. Let us choose (*ρ*_{a},*c*) as reference point. The continuity of the potential yields thus for the gauge term
5.10and the corresponding correction term to the impedance calculation will be (after substituting in (5.5) and carrying out the respective integration)
5.11Notice that only the *m*=0 mode contributes to (5.11), since *V* _{g} is constant. The choice of the reference point is arbitrary (except from the fact that it must lie on the *ρ*=*ρ*_{a} interface), something which is verified by the invariance of the impedance results for different point locations.

## 6. Numerical treatment and results

The numerical treatment of (3.45)–(3.47) requires the truncation of the infinite series to obtain matrix equations of finite dimension. The series truncation in modal solutions is a standard procedure thoroughly discussed in the cited literature, and therefore we shall not enter into many details here. The number of radial/axial modes taken into account for the following calculations was the same for all *κ*,*v*,*u*,*q* and *p* modes (yet not necessary), and was equal to 200 for all the cases studied. Attention, however, must be drawn to the number of *r* and *s* eigenvalues *N*_{r} since the respective modes span the *z*-interval, wherein the conductor lies, and not the hole domain. Thus, taking *N*_{r}∼*N*_{u} can lead the system close to singular. Keeping, however, the ratio *N*_{r}/*N*_{u} similar to the respective ratio of the geometrical dimensions of the conductor versus the truncated domain, i.e. *c*/*h*, the system condition number remains sufficiently low.

The ratio *c*/*h* itself can be freely chosen. Its exact value, and that for the truncation limits *ρ*_{L} and *h*, seems only to be constrained by numerical issues. In this work, we have chosen the value 1/2 for the *c*/*h* ratio which provides adequate precision. The truncation limits *ρ*_{L} and *h* may also be fixed to values about 10 times the maximum of the hole radius and the coil external radius. It should be underlined here that the fact that the *c*/*h* is independent of the coil shift is another advantage of the adopted formulation, which fixes the matching surface on a direction normal to the coil displacement.

Concerning the number of angular modes *N*_{m}, it depends on the value of the coil radial shift. For the given coil dimensions and the hole radius, a number of 15 modes assures sufficient precision for displacements up to three times the coil external radius.

The solution to the transcendental equation (3.24) is performed using the technique proposed by Delves & Lyness (1967) and Lyness (1967) and is based on Cauchy's theorem. A thorough discussion of the procedure can be found in Theodoulidis & Bowler (2010).

The results of the semi-analytical solution for the impedance variation have been compared with the ones derived from the three-dimensional FEM package Comsol 3.5a for the typical inspection scenario depicted in figure 1. The half-space conductivity is *σ*=18.72 MS m^{−1} and the borehole radius is 5 mm. Figure 3 shows the comparison of the results for the coil impedance as a function of the probe shift from the hole axis. Two different coils are considered: the first one has inner and outer radii 2 mm and 4 mm, respectively, 1 mm height and is wound with 200 wire turns; the respective dimensions for the second one are 3 mm and 6 mm for the inner and outer radii, 1 mm for the thickness and the number of turns are 300. The coil lift-off in both cases remains constant and has the value of 0.2 mm. The frequency is taken to be 10 kHz. The comparison with FEM results shows an excellent agreement.

In order to get an image of the eddy-current flow patterns, these are visualized for different coil positions in figure 4. Here, the coil dimensions are the same with the ones of the first coil in figure 3 and the same for its lift-off. The considered frequency is 1 kHz. The current flow is calculated by taking the curl of the (3.3) and (3.7)–(3.8) and substituting the expansion coefficients determined from the solution of (3.45)–(3.47). The current patterns are visualized on the half-space interface (*z*=*c*) for the different coil shifts.

The computational time for all presented simulations was a few seconds using a standard PC workstation with dual processor at 3.16 GHz. It is recalled that the computational time is practically independent of the number of scanning positions. Actually, the addition of any new scanning points leaves the system matrix intact; it is translated to an additional inversion of the same system with a new excitation (right-hand vector). Hence, using LU variants for the system inversion, appending additional coil positions does not essentially burden the solver. For comparison, the computational time for the FEM solutions was *ca* 1 min per coil position, thus the semi-analytical solution is much faster.

## 7. Conclusion

The semi-analytical treatment of the eddy-current interaction of a coil above a conductive half-space with a vertical borehole is studied and solved. Owing to the geometry of the problem, direct application of the mode matching approach seems to be confronted by a number of mathematical complications. Nevertheless, these difficulties can be successfully overcome by reformulating the problem and treating it in two steps.

The proposed formulation is very efficient, since the derived transcendental equation for the computation of the problem eigenvalues is common for all angular modes, and therefore needs to be solved only once. Furthermore, it does not involve computations of Bessel functions, which would lessen the numerical performance.

This is a significant canonical geometry with important applications like inspection of fasteners in aircraft structures. Calculation of the induced fields and coil impedance has been done rapidly and efficiently for the first time in semi-analytical manner.

Work is underway for extending the presented formulation in order to address more interesting from the practical point of view geometries, like the one of a conducting plate of finite thickness, as well as to consider more complicated probe configurations than the pancake coil.

## Acknowledgements

The present work has been conducted in the context of the CIVAMONT collaboration (http://www-civa.cea.fr).

## Footnotes

↵1 In fact, this truncation is not necessary for the solution of the two-layer problem itself but it is indeed conducted in order to obtain a finite number of modes for the successive part of the analysis.

↵2 The equivalent current source is defined in this way in order to conform with the usual convention

=*J*×Δ*n*, Δ*H*being the discontinuity of the magnetic field and*H*the unit normal vector.*n*

- Received November 21, 2011.
- Accepted January 30, 2012.

- This journal is © 2012 The Royal Society