On the interaction of an eddy current coil with a right-angled conductive wedge

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1 Iowa State University From the SelectedWorks of John R. Bowler 2010 On the interaction of an eddy current coil with a right-angled conductive wedge John R. Bowler, Iowa State University Theodoros Theodoulidis Available at:

2 1034 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 4, APRIL 2010 Interaction of an Eddy-Current Coil With a Right-Angled Conductive Wedge Theodoros Theodoulidis 1 and John R. Bowler 2 Department of Mechanical Engineering, University of Western Macedonia, Kozani, Greece Center for Nondestructive Evaluation, Iowa State University, Ames, IA USA 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. Index Terms Analytical modeling, conductive wedge, eddy currents, impedance calculation, nondestructive evaluation. I. INTRODUCTION AN important problem in electromagnetism is to determine the quasi-static field of an induction coil in the presence of an electrical conductor. In the case where the conductor is an infinite plate [1], an infinite tube [2] or a uniform sphere [3], exact expressions can be found for the field using analytical procedures starting with the separation of variables. Such results, once found are invaluable since they are part of a timeless, readily accessible resource for the engineering community to use. Analytical solutions are naturally limited in scope but they often have a role in solving part of a more complex problem. For example in the general area of measurement science where accurate numerical results are needed but also in cases where purely numerical models continue to be used [4], [5]. In eddy-current nondestructive evaluation, there is 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 block. The need arises because cracks are likely to form at corners where stress concentrations occur. To approach the problem of finding the electromagnetic field at a flawless edge, consider first a conductive quarter space. A quarter space is a special case of an infinite wedge that has surfaces meeting at a right angle. In radiation physics, the scattered field due to a plane wave incident on an infinite perfectly conducting wedge of arbitrary angle can be written as an exact expression in cylindrical coordinates involving Bessel functions of non-integer order [6]. Because the perfectly conducting wedge is impenetrable to radiation, there is no need to consider the internal field. Similarly, solutions exist for hard and soft (Neumann and Dirichlet) boundary conditions for wedge scattering of an acoustic wave [6]. Manuscript received July 29, 2009; revised September 13, 2009; accepted November 04, First published November 20, 2009; current version published March 19, Corresponding author: T. Theodoulidis ( theodoul@uowm.gr; theodoul@ieee.org). Digital Object Identifier /TMAG For a penetrable wedge, it is necessary to consider general forms for both the internal and external field since they are interdependent, and then try to match them according to continuity conditions at the interfaces. It turns out that the interface conditions cannot be satisfied easily. For example, if internal and external solutions are expressed as series containing Bessel functions, the matching across the boundary cannot be done term-by-term. In other words, the interface conditions cannot be satisfied by matching a term from the internal solution with a corresponding term of the same order from the external solution. This is because terms of the same order representing the radial dependence in the two series have different arguments. There are ways of dealing with the issue but they are not straightforward and have not so far produced an exact solution. Consequently, wave scattering at a penetrable wedge remains an open problem [7]. In the absence of an exact treatment, we sought a relatively simple, computationally efficient engineering solution for the quarter space problem. For applications to eddy-current testing we consider an induction coil excitation at a frequency where displacement current is negligible. Mindful of the fact that matching the internal and external solutions is troublesome in cylindrical coordinates, a cartesian coordinate system is used. The direction is normal to the horizontal surface of the conductor and the direction is normal to the vertical surface, Fig. 1. To simplify the task of matching solutions at the interfaces, artificial boundaries are introduced that limit the problem domain in the and directions. The truncated domain is divided into regions and the solution expressed in terms of elementary functions using Fourier series expansions in each region. It is possible to ensure continuity of the tangential magnetic field and the normal magnetic flux at the vertical interface (, ) on a term-by-term basis. By truncating the series expansions, we arrive at a point where there are three finite sets of expansion coefficients. These are found from the continuity conditions governing the magnetic field at the plane.in this case, term-by-term matching is not possible but by taking moments of the continuity equations we get a system of matrix /$ IEEE

3 THEODOULIDIS AND BOWLER: INTERACTION OF AN EDDY-CURRENT COIL WITH A RIGHT-ANGLED CONDUCTIVE WEDGE 1035 a boundary element method [5], [13], [14], we need to compute: i) the incident field; that is, the eddy-current density at the crack location but without the crack present and ii) an appropriate Green s function defining the electric field produced in the conductive wedge by an embedded electric dipole. Such computations, based on the methods described in the present paper, are currently under development. Fig. 1. A cylindrical coil above the edge of a right-angled conductor. equations for the expansion coefficients. We call the procedure the truncated region eigenfunction expansion (TREE) method. In an earlier paper, we gave an account of the edge problem in which a truncation was in the direction only [8]. Although the truncation in is not essential, the bi-truncation used here simplifies the analysis and the numerical procedure. Using symmetry considerations, the solution was extended to the case of a plate [9] and then a through thickness slot [10]. In the present study, we report further developments in a different direction including: 1) the reformulation of the problem by expressing the quasi-static electromagnetic field and coil impedance in the form of a double series to simplify calculations and improve the control of convergence; 2) an improved method for the rapid and accurate computation of complex eigenvalues; and 3) consideration of a wide range of air-cored coils. We give expressions for the magnetic field, eddy-current density and coil impedance variation with position in terms of the source coefficients that characterize the coil. These coefficients depend only on the isolated coil magnetic field and can be found by applying any one of a number of methods, for example by using the Biot-Savart law or via a numerical code. Typical coil configurations that can be treated analytically are the tilted cylindrical coil and the two coils in driver-pickup mode [11], including rectangular coils and other shapes. Preliminary work on coil orientation appeared in [12] where the case of a tilted cylindrical coil was examined. Also in previous work [8], the theory was compared with experimental measurements in order to validate the model, the agreement being to within about 2%. In this paper, we do not present further measurements but instead compare our results with those from a numerical solution of the problem computed using a commercial 3-D finite-element package. In this way we can check both the numerical calculations of the impedance and predictions of the induced eddy-current density. At the same time, one gains an appreciation of the comparatively short computation times required with our approach. The motivation behind the present work is not only to find a quasi-analytical and rapid solution of the quarter space eddycurrent problem but more significantly to lay the foundations for the efficient evaluation of edge crack signals. Following established theory for modeling ideal or very narrow cracks with II. ANALYSIS Consider Fig. 2, which shows a coil located above a right-angled, truncated conductive quarter-space having a conductivity and a permeability equal to that of free space. The coil is excited by a time harmonic current varying as the real part of. We seek expressions for the impedance change of the coil due to induced current and the electromagnetic field in the conductor. In the context of eddy-current testing, the former gives the observed signal and the latter potentially determines the probe-flaw interaction. In the new approach, the solution domain for the boundary value problem is truncated in both and directions as shown in Fig. 2. Thus, the solution domain extends from 0 to in the -direction and from 0 to in the -direction. The presence of four boundary surfaces at and means that we have a wide choice of possible combinations of boundary conditions. The choices include one which defines a perfect magnetic insulator, (subscript stands for normal component) and one which defines a perfect electric insulator, (subscript stands for tangential component). For the configuration examined here, where the edge at and the coil are located far from the boundaries, these choices have a negligible effect on numerical values of the field close to the edge and the coil. However, they determine the eigenfunctions and eigenvalues used to represent the solution. In this work we consider a magnetic insulator at, and an electric insulator at,. This choice leads to Fourier series solutions in both and that do not require zero order terms. The absence of such terms simplifies the solution somewhat and has a negligible effect on the edge field. The electromagnetic field problem is formulated here using a scalar decomposition for the magnetic flux density. Although the magnetic field is determined in full, we do not attempt to derive explicitly the conservative component of the electric field associated with surface charge because it is not needed for finding the coil impedance. For the nonconductive region the magnetic field is expressed as the gradient of a scalar potential, where satisfies the Laplace equation. The potential can be considered as the superposition of the isolated coil potential and a contribution originating from the eddy currents in the conductive region,. For the region between the lowest point of the coil and plane, the source potential may be written (1) where,, and. The source coefficients,, are prescribed by first solving the Poisson equation for the coil in a uniformly

4 1036 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 4, APRIL 2010 Fig. 2. Side and top view of a normal cylindrical coil above the right-angled conductor in a truncated solution domain. nonconductive truncated domain [12], [15]. Everywhere above the plane (2) The magnetic flux density below, is represented in terms of the second order vector potential as [16] where (3) and denotes the directed unit vector. The transverse electric and transverse magnetic potentials, and respectively, satisfy the Laplace equation for a nonconductive source free region and the Helmholtz equation for a homogeneous conductive region, where.by satisfying the insulator boundary conditions at the truncation boundaries, one finds that (4) Next we consider the eigenvalue problem referred to earlier. From the requirements that the normal component of the magnetic flux density and the tangential magnetic fields are continuous at, we get two relationships. From the roots of the first, the eigenvalues and hence the are found. The complex eigenvalues do not depend on the variable, which means that their numerical computation needs to be carried out only once. A numerical scheme for their evaluation is given in Section IV. The second relationship, (8) gives the coefficients. The expansion coefficients,, and,havea linear relationship with the source coefficients,. In finding these relationships, it is helpful to refer to the following expressions for the magnetic flux density. For the components are written (7) (9) (5) (10) where and. The values of are evaluated from an eigenvalues equation described below. Note that in (5), provision is made for satisfying continuity conditions at the vertical face of the conductor (at ) by having the same dependence in the two parts of. For the TM potential, we have and for (11) (6) where and. The potential is not sought for the region, since it is not needed for an evaluation of the magnetic flux density. (12)

5 THEODOULIDIS AND BOWLER: INTERACTION OF AN EDDY-CURRENT COIL WITH A RIGHT-ANGLED CONDUCTIVE WEDGE 1037 where we recall, represents the source coefficients characterizing the isolated coil and represents the contribution of the eddy-current density induced in the right-angled conductor. Note that in the case of a conductive half-space (20) (13) (14) From the continuity of,at, we derive a matrix equation (15) where the bold symbols and are column vectors containing the expansion coefficients for a particular value of. For computational purposes these are limited to components. The have been formed into an diagonal matrix.in deriving (15), we have equated expressions for above and below the plane. Then taken moments by multiplying the resulting equation by, and integrating with respect to and over the truncated plane. These integrals are either eliminated using the orthogonality properties of the sine function or used to form the matrix, the elements of which are given in Appendix A. Similarly, one uses the continuity of and to get and (16) (17) In (15) (17),,, and are square matrices. The definitions of these and the formal solution of the matrix system are given in the Appendix. III. IMPEDANCE CHANGE AND ELECTROMAGNETIC FIELD The general expression for the impedance change caused by the presence of the conductive edge can be derived by using a reciprocity relation and written in the following form [9]: The total coil impedance is the superposition of the impedance change due to the wedge and the impedance of the isolated coil (coil in free space). The latter is calculated from expressions in [15]. The magnetic field in all regions as well as the eddy-current density in the conductor can be calculated from the expressions that relate and to. Here, we provide the expression for the induced eddy-current components in the conductive quarter space. Thus, for and (21) (22) (23) The magnetic field is calculated in the various problem areas using (9) (14). IV. THE SOURCE COEFFICIENT FOR A TILTED CYLINDRICAL COIL In addition to the case where the coil axis is normal to the upper surface of the conductor, as studied in [8], it is of interest to compute other coil configurations as shown in Fig. 3, where the coil type is designated with reference to the alignment of its axis with respect to the coordinate system. Derivations of coil potentials for a uniform truncated domain are described in [15]. For the Z-coil, whose axis is normal to the upper surface of the conductor, the source coefficients in the truncated domain are given by (24) (18) Substituting from (1), (2) and using Parseval s theorem for Fourier series gives (19) where with denoting the Bessel function of order 1. Here are the coordinates of the coil center and is the excitation current density with denoting the number of wire turns and the coil height. The lift-off, i.e., the distance of the lowest point of the Z-coil to the upper conductor surface, is related to the height of the coil center.

6 1038 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 4, APRIL 2010 Fig. 3. The three types of coil configurations according to the direction of their axes. For the case of the X-coil, whose axis is in the -direction and therefore parallel to the upper surface of the conductor, the source coefficients are given by Fig. 4. Distribution of complex eigenvalues and search strategy. (25) where with denoting the modified Bessel function of order 1. In terms of the lift-off, the height of the X-coil center is given by. For the case of the Y-coil, whose axis is in the -direction and therefore also parallel to the upper surface of the conductor, the source coefficients are given by intermediate values of one gets a hybridization of the two sets in which one set is dominant depending on whether is large or small. For we get the merged set shown in Fig. 4. A robust method for locating initial estimate of the is one based on a procedure given by Delves and Lyness [17]. The key step uses Cauchy s theorem to test whether a pole or several poles of a given function lies in a prescribed region of the complex plane. By finding the poles of a function, one implicity locates the zeros of its reciprocal. Thus, if is a closed contour in the complex plane which does not pass through a zero of one can show using Cauchy s theorem that (26) Again, in terms of the lift-off, the height of the X-coil center is given by. Note also the similarity between the expressions of the source coefficients for all three coil configurations. V. COMPUTATION OF COMPLEX EIGENVALUES In order to ensure that numerical results are accurate, a reliable method for computing complex eigenvalues satisfying (7) is needed. Once initial estimates have been found, an iterative method based on the Newton-Raphson algorithm can quickly hone in on the precise values. The need for a reliable procedure, therefore, reduces to one of finding a set of starting values that lead to a unique and complete set of final values in a predefined range. One can get an indication of where to find the roots by considering two limiting cases depending on the value of ; the horizontal width of the conductive region. For the conductor vanishes and (7) is satisfied if which is the same condition that is applied to find the, therefore in this case. In the other limit, and. The limiting cases generate two sets of eigenvalues, in one set the values are real and positive and the other set starts close to on the complex plane and tends toward and thus the values of the other set on the positive real axis, with increasing index. At (27) where is an analytic function inside and on the contour enclosing region and the summation is over the enclosed poles. In addition, are the zeros of in raised to the th power. The path integral, evaluated for gives the number of zeros of enclosed by contour : (28) A numerical integration inevitably gives rise to errors but one only need compute with sufficient accuracy to identify unambiguously the integer. It is best to work with at most a few poles in each region, say 3 or fewer, and if exceeds the chosen limit, one can simple reduce the size of the region to capture fewer poles. One strategy that may be adopted is to choose a region of integration small enough that there is only one pole in it. Then it is easy to locate it accurately. If there are several poles in and the number, is known from, we need calculate only to locate them by constructing the th order polynomial [18] (29)

7 THEODOULIDIS AND BOWLER: INTERACTION OF AN EDDY-CURRENT COIL WITH A RIGHT-ANGLED CONDUCTIVE WEDGE 1039 Fig. 5. Real and imaginary part of the normalized impedance of a Z-coil as it is moved above a conductive quarter-space with x 0 c denoting the distance of the coil center from the edge. using Newton s recursive formula (30) For example, if we have (31) with a root, whereas if we have (32) with roots,, and if we have (33) with roots, and. One can find the roots of higher order polynomials via a number of available techniques but they tend to decrease in efficiency and accuracy as increases. Note that the polynomial does not approximate the function, it simply has the same roots as the function in. Hence, we have replaced the problem of finding the zeros of a given analytic function in region with the problem of finding the zeros of a given polynomial in the same region. The above method allows us to find several roots simultaneously, thus avoiding the need to isolate zeros which may be difficult when they are close together or tightly grouped in the complex plane. In our case the analytic function is given by (7) and is rewritten as (34) Noting that all eigenvalues are located in the area between the dashed lines and the real axis in Fig. 4, we employ the above scheme and search the area by dividing it in subareas with a trapezoidal or rectangular shape as appropriate. In each subarea we calculate the complex roots by computing the repeated integrals of (27) along its bounding path. The procedure continues up to a certain maximum or when the maximum set Fig. 6. Amplitude contours of eddy-current density for three coil positions at the y = h =2 plane. Numbers in each graph show the maximum amplitude. number of eigenvalues has been reached. The path integrals need not be computed with great accuracy since they are used to compute initial estimates of the zeros prior to refinement using a Newton-Raphson routine.

8 1040 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 4, APRIL 2010 Fig. 7. Distribution of the eddy-current density amplitude on the upper surface of the right-angled conductor as the coil moves above and away from the edge. The three figures correspond to the three coil positions shown in Fig. 6. VI. RESULTS Results presented here were computed using Mathematica and MATLAB for the Z-coil, Fig. 3. The coil is driven by a current ma and its parameters were as follows. The inner radius, mm, outer radius, mm, lift-off mm, height mm, and is the number of wire turns. The coil free-space self inductance was found to be mh, [15]. The conductor is assumed to have a conductivity MS/m, equal to that of an aluminium alloy. It is useful to establish guidelines for deciding the extent of the truncated domain defined by and and the number of terms in the and summations, and, respectively. Reference to the case of the half-space conductor was very helpful in this respect. Theoretical results from the double series expressions were compared with results from the integral expressions [15] for an unbounded region. It was observed that for and we obtained an agreement of the order of 1% for a frequency of 1 khz and with. Usually only 40 terms in the series expansions are enough and the square matrices are then The TREE approach clearly makes the truncated quarter-space calculation computationally efficient. For example, the calculations of the coil impedance for 40 coil positions regularly spaced with respect to the edge takes only a few seconds using a typical Pentium IV personal computer. The criterion with ensures that the coil is always relatively far from the boundaries at,,, and thus its impedance is not significantly affected by them. The theoretical results for the coil impedance were compared to results from the 3D-FEM package Comsol for a frequency of 1 khz and gave good agreement. Fig. 5 shows the normalized impedance of the coil as it is moved above the edge in the direction. The normalizing factor is the isolated coil reactance. For the finite-element calculations we used the AC/DC, Quasi-Statics, Electromagnetic, Electric and Induction Currents, Time-harmonic analysis, ungauged AV module with and automatic fine meshing for each coil position. This resulted in about 20,000 elements and about

9 THEODOULIDIS AND BOWLER: INTERACTION OF AN EDDY-CURRENT COIL WITH A RIGHT-ANGLED CONDUCTIVE WEDGE 1041 degrees of freedom. The computation time for each coil position was about 35 s with a quad-core Pentium IV personal computer. It is expected that as the frequency increases the accuracy of the finite-element calculation decreases because of the need for a finer grid to maintain accuracy. In the TREE method however, results remain consistently accurate over a range of frequencies. In comparing the TREE predictions of the eddy-current density with those found using finite-element calculations at 1 khz the results agreed to with 1%. In calculating the eddy-current density using TREE we need many more terms in the -direction, of the order of, whereas in the -direction still gives good results. The behavior of the current density in the conductive wedge is illustrated in Fig. 6. The diagram shows contour plots of the amplitude of the component of eddy-current density on the plane inside the wedge. This plane passes vertically through the center of the coil where, due to symmetry, there are no and eddy-current components. It is evident from these results how the eddy-current distribution changes as the coil moves away from the edge. Fig. 7 shows the distribution of the eddy-current amplitude on the upper surface of the wedge for three coil positions showing how the eddy-current pattern changes and the amplitude decreases as the coil moves away from the edge. VII. CONCLUSION An existing model of the field induced by a coil above a right-angled conductor has been enhanced by improving both numerical implementation and scope. We are now able to formally express the edge effect for any coil and we have provided explicit expressions for three cylindrical coil orientations. The model is still open for further developments including: i) the use of closed form expressions for source coefficients for noncylindrical coils; ii) its extension to magnetic media; and iii) the extension to driver pickup probes. Furthermore, impedance boundary conditions may be used to get the coil response at very high frequencies. Regarding the edge crack problem, we have so far accomplished one of the two steps in the solution, as described in the Introduction. Work is currently underway to obtain the Green s function expression for an electric dipole embedded in the wedge. APPENDIX A The elements of the matrices appearing in the system are defined as follows: (37) (38) (39) (40) Note that all of the above matrices have a common characteristic: they are independent of the variable and therefore they need to be formed just once. For the solution of the system (15) (17), if we write (15) in the form (41) and replace in (16) and (17), after some manipulation the matrix system can be written as where for each The solution is (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (35) (36) (52) Note that for each, there are two matrix inversions to be performed, and. The second matrix is illconditioned and is necessary to use a numerical method utilizing Singular Value Decomposition for its inversion. The routines pinv and PseudoInverse were used in MATLAB and Mathematica, respectively.

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