Field computations of inductive sensors with various shapes for semi-analytical ECT simulation Christophe REBOUD a,1, Theodoros THEODOULIDIS b, a CEA, LIST, Département Imagerie Simulation pour le Contrôle, Centre de Saclay, F-91191 Gif-sur-Yvette cedex, France b MEANDER Group, Department of Mechanical Engineering, University of Western Macedonia, Bakola & Sialvera, 50100 Kozani, Greece Abstract. A semi-analytical approach is proposed for the rapid calculation of electromagnetic fields induced in a planar stratified media by inductive sensors with complex shapes. Theoretical aspects of the method and new shapes modelled are presented, then comparisons of results obtained in classical cases with reference data are discussed. Finally, an application of this development to the simulation of eddy current testing with a semi-analytical model is detailed. Keywords. Eddy current testing, semi-analytical modelling, TREE method, field computations, complex sensors, planar stratified media 1. Introduction Modelling of eddy current testing (ECT) techniques is now commonly used in industry for various purposes, like design of new probes, interpretation of experimental data or support to the evaluation of inspection procedures. CEA LIST and the Meander group of University of Western Macedonia University have collaborated for some years in order to develop semi-analytical ECT simulation tools. These tools are proposed to industrial partners into the CIVA software developed at CEA LIST. Among the different classes of modelling methods, semi-analytical integral formulations realize a compromise between the accuracy and speed of purely analytical methods and the generality and versatility of purely numerical ones. In general, such methods, like the Volume Integral Method (VIM) [1], [2] or the Boundary Element Method (BEM) [3], [4] are organised around the solution of an integral equation governing the interaction between the excitation field and the local inhomogeneity that constitutes the flaw. The response of the receiving part of the ECT probe is then calculated using the well-known reciprocity principle [5] or integral operators [6]. The efficiency of these methods usually depends on two factors. The first one is the use of analytical expressions of the interaction operator that can be derived when the piece inspected presents canonical geometrical properties. 1 Corresponding Author: Christophe REBOUD, CEA, LIST, Centre de Saclay, F-91191 Gif-sur-Yvette cedex, France; E-mail: christophe.reboud@cea.fr
Figure 1. Typical configuration considered for the calculation of electromagnetic fields emitted in layer i by a sensor located in layer 1. The second one is the fast computation of excitation fields induced by the probe inside the piece. This paper presents a general approach for the computation of electromagnetic fields emitted by complex inductive sensors inside a planar piece. The medium considered here is conductive, stratified and possibly magnetic. It is assumed to be sufficiently large in transverse directions, so that edge effects can be neglected. After a brief presentation in the first section of the fields expressions derived using the method developed at University of Western Macedonia, called Truncated Region Eigenfunctions Expansion (TREE) [7], validations of the results obtained are discussed in the second section. Examples of use with simulation tools based on VIM and developed at CEA LIST in complex cases are then detailed in the third section. Finally, perspectives opened by this work in terms of new ECT configurations addressed and further applications are introduced. 2. Calculations with the TREE method of electromagnetic fields induced in planar multi-layered structures 2.1. Principle of the TREE method Let us consider a planar stratified medium made of N layers with respective electric conductivities and magnetic permeabilities (σ i, µ i ) 1 i N. By convention, layers number 1 and number N correspond to air above and below the piece, respectively, and the sensor is driven at the angular frequency ω by a volumetric current density J(r) located in layer 1, as shown in Figure 1. The principle of the TREE method, illustrated in Figure 2, consists in solving Maxwell equations using the separation of variables into a finite box of dimensions h x and h y along directions x and y, respectively. This box is chosen large enough in transverse directions x and y so that the solution calculated may be set to zero at its boundaries (and outside of the box). For the sake of clarity, the coil size in Figure 2 has been exaggerated. The introduction of a finite box leads to expressions of the fields as series instead of the integrals corresponding to the infinite case. From the computational point of view, when choosing finite numbers of terms N i and N j in the x and y directions, the electric field E l emitted in layer l > 1 can be accurately approximated by the expression of equation (1),
Figure 2. Truncated region considered by the TREE method. Its dimensions h x and h y are chosen large enough so that the solution calculated may be set to zero at the boundaries. N i N j E l (x, y, z) = iωµ l h (s) (u i, v j ) R v j sin(u i x) cos(v j y) ij e γijz u i cos(u i x) sin(v j y), (1) i=1 j=1 0 with u i = iπ/h x, v j = jπ/h y, κ ij = u 2 i + v2 j, and γ ij = κ ij + iωµσ. The magnetic field H l emitted in the same layer can be derived with a similar expression. In equation (1), the term R ij stands for contributions due to reflections and transmissions at the interfaces of the medium and is calculated recursively [2]. More importantly, the source term h (s) (u i, v j ) corresponds to the contribution due to the sensor, with its shape, position and orientation. As sensors considered in this problem are not affected by the vicinity of the piece, i.e they do not contain any ferrite core or parts, this source term is the 2D Fourier transform of H 0z, the component normal to the interface of the magnetic field H 0 emitted in free space by the sensor. In other words, any sensor shape and orientation can be addressed by this approach, provided that the component H 0z can be computed in free space at the location of the interface between the piece and the layer containing the sensor. 2.2. Modeling ECT sensors with complex shapes A classical way of calculating the H 0z component in free space at position r due to a volumetric current density J(r ) in the region V consists in using the Biot-Savart law, recalled in equation (2). H 0 (r) = 1 ˆ 4π r V J(r ) (r r ) (r r ) 3 dr (2) However, when considering sensors with complex shapes and orientations, this particular calculation has to be carried out numerically. In order to avoid this, the magnetic field is approximated by the superposition of fields emitted by a set of current blocks [7]. The elementary shape selected in this work is the trapezoid, for which analytical expres-
Figure 3. a) Construction of sensors with complex shapes as a successions of elementary trapezoids. b) New shapes of sensors modelled with this approach. sions of H 0 are known [8]. The construction of some complex shapes of sensors with this technique is illustrated in Figure 3. After the calculation of the magnetic field in free space at the location of the first interface, the TREE method is applied to get the electric and magnetic fields in all layers of the piece under test. A very good performance in computation time is achieved for all new shapes modelled: D coils with or without rounded corners, rectangular coils with rounded corners, racetracks, meanders and spirals. The accuracy of these calculations is dependent on a small number of numerical parameters: numbers of modes N i and N j, dimensions h x and h y of the finite box and the numbers of trapezoids used for the approximation of rounded parts of the sensors. In practice, suitable values of these parameters can be set automatically for all parametric geometries considered in order to reach a high level of accuracy. 3. Numerical validations of the approach Among the numerical parameters cited previously, the one to which the results are the most sensitive is the number of trapezoids used to describe a rounded geometry, as other straight parts of the sensors can be described exactly with a succession of trapezoids. The accuracy of the method proposed has been tested in the standard case of a non tilted cylindrical coil with 328 turns located above a conductive plate. This particular case with a canonical coil shape has been chosen because analytical solutions [9] can be used as reference for comparisons. The coil inner and outer radii are 1.5 mm and 2 mm, respectively, and its height is 1 mm. The plate, which conductivity is 1 MS.m 1, is separated from the coil by a distance of 1 mm. Computations of the electric field in the plate have
Figure 4. Validation of electric fields computations carried out for a cylindrical coil at several frequencies. Results are compared with well-known analytical solutions proposed in this case by Dodd and Deeds [9]. been carried out at the frequencies of 100 Hz, 1 khz, 10 khz, 100 khz and 1 MHz by approximating the coil with 16 trapezoids, as shown in Figure 3 a). Quantitative comparisons, presented in Figure 4, versus reference results show a maximal discrepancy of 1 percent in amplitude for all frequencies. Hence, a quite accurate approximation has been obtained using trapezoidal blocks for the calculation. 4. Applications to semi-analytical modelling of non destructive testing techniques This new technique, proposed for the calculations of incident fields due to sensors with complex shapes in planar pieces, provides very fast and accurate results that may be used inside NDT simulation tools developed at CEA LIST. An application to the ECT simulation of a volumetric flaw with the Volume Integral Method (VIM) is presented in the next section. 4.1. Simulation of ECT signals due to volumetric flaws with the Volume Integral Method CEA has developed for many years semi-analytical tools dedicated to ECT simulation based on VIM. The configuration of interest in this application is the inspection of a flawed plate, with electrical conductivity σ 0 and magnetic permeability µ 0 = 4 π.10 7, containing a parallelepipedal notch that can be seen as a inhomogeneity σ(r ) in the region Ω of the plate. The probe is made of one cylindrical coil emitting at the angular frequency ω and two receiving D coils. The receivers are facing each other and function in differential mode. This type of sensor is commonly used at high frequencies in Aeronautics for the inspection of engine parts [10]. As illustrated in Figure 5, two cases are successively considered: first, the probe is positioned above the plate with a nominal orientation, and then a tilt angle of 20 is taken into account. The simulation of both cases with VIM is organised around the resolution of the state equation (3) derived from Maxwell equations and ruling interactions occurring in the plate.
Figure 5. Typical sensor configuration used for the ECT inspection of aircraft engine parts. A probe composed by one cylindrical emitting coil and two D coils receiving in differential mode is scanning a flaw located in a plate. ˆ E p (r) = E t (r) + i ω µ 0 G(r, r ) [σ 0 σ(r )] E t (r ) dr (3) Ω This equation links the total electric field E t in Ω to the primary electric field E p emitted when no flaw is present. The integral operator G is the electric-electric Green dyad corresponding to the plate and verifying boundary conditions at infinity in the transverse directions and at the interfaces along the vertical direction. In order to determine the unknown field E t, the primary field E p has to be calculated first. after the resolution of equation (3) with a numerical procedure like the Method of Moments [11], the actual ECT signal, in this case the variation of tension V measured by the receiving D coils, is computed through the application of the reciprocity theorem, with the expression given in equation (4). In this equation, I stands for the current driving the emitting coil, and E D1, E D2 are fictitious electric fields that would be emitted by the receivers in the region Ω, would they be driven by a unitary current [5]. V = 1 I ˆ Ω [σ 0 σ(r)] E t (r) [E D1 (r) E D2 (r)] dr (4) In the non tilted case, the primary field E p has been calculated with an analytical formula, and fields E D1, E D2 are calculated with the TREE method. However, when the probe is tilted, all primary electric fields are calculated with the TREE method. Simulation results obtained for both configurations considered are shown in Figure 6. As expected, an important loss of amplitude is observed when the probe is tilted. Complete cartographies corresponding to 4000 positions of the probe were computed in less than 2 minutes, which is quite efficient compared to other numerical methods of simulation. Such rapidity is particularly appreciated when computing statistical quantities like Probability of Detection curves [12], for instance. Two remarks on this modelling approach can be made. First, the contribution of the probe is described by primary fields only and the calculation of fields due to arbitrarily oriented trapezoidal elements allows the simulation with VIM of configurations involving air cored probes with any position and orientation. The modelling of perturbations like random disorientation of the probe is also an important requirement when performing statistical studies in simulation. Secondly, the problem considered here can be fully described with the electric field because neither the piece or the flaw are magnetic. Oth-
Figure 6. Simulation results obtained in both configurations illustrated in Figure 5. Top: the cartography and 1D extraction obtained when the probe is not tilted. Bottom: results obtained when the probe is tilted by 20. As expected, an important loss of amplitude is observed in the latter case. erwise, in the general case, an additional integral equation involving the magnetic field H is required to solve the problem. 5. Conclusions A general semi-analytical method has been developed at University of Western Macedonia for the calculation of electromagnetic fields emitted by ECT air cored sensors into planar stratified pieces. Comparison with analytical solutions in standard cases have shown the high accuracy of this computation method, based on the approximation of the coil geometry with a succession of trapezoidal elements. Moreover, its implementation inside simulation tools developed at CEA LIST has led to the parametric modelling of several classical shapes of ECT sensors: D coils, racetracks, meanders and spirals. Simulation of complex cases with VIM can be carried out rapidly and perturbations of the signals due to changes of probe orientation are taken into account by the model.
This work opens up several perspectives. First, similar fields computations will be introduced in the cylindrical geometry, in order to address other families of industrial applications, like tubing inspection in the nuclear industry. Besides, meanders and spiral coils can be used in addition to static magnetic fields calculations in order to model the excitation produced by electromagnetic acoustic transducers (EMAT). The corresponding ultrasonic fields and flaw responses could then be simulated by semi-analytical models developed at CEA LIST. Finally, these fields computation will be used as part of a new semi-analytical model dedicated to the simulation of thin cracks in planar stratified media, which is also developed collaboratively by CEA LIST and the University of Western Macedonia. Acknowledgements This work was supported by the CIVAMONT project, aiming at developing scientific collaborations around the NDT simulation platform CIVA developed at CEA LIST. References [1] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: IEEE Press Series on Electromagnetic Waves (1995). [2] W. C. Chew and S.-Y. Chen, Response of a point Source embedded in a Layered Medium, IEEE Antennas and Wireless Propagation Letters, 2 (2003), 254-258. [3] J. R. Bowler, Inversion of open cracks using eddy-current probe impedance, Review of Progress in Quantitative Nondestructive Evaluation, 19A (2000), 529-533. [4] T. Theodoulidis, Developments in efficiently modelling eddy current testing of narrow cracks, NDT-E International, 43 (2010), 591-598. [5] G.D. Monteath, Applications of the Electromagnetic Reciprocity Principle, Pergamon Press (1973). [6] D. Prémel and G. Pichenot, Computation of the magnetic field due to a defect embedded in planar stratified media: application to AC field measurement techniques, Electromagnetic Non-Destructive Evaluation (XIV), Studies in Applied Electromagnetics and Mechanics, IOS Press (2010), to be published. [7] T. Theodoulidis, E. Kriezis, Eddy Current Canonical Problems (with applications to nondestructive evaluation), TechScience Press (2006). [8] S. Babic and C.Akyel, An improvement in the calculation of the magnetic field for an arbitrary geometry coil with rectangular cross section, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 18 (2005), 493-504. [9] C. Dodd, W. Deeds, Analytical Solutions to Eddy-Current Probe-Coil Problems, Journal of Applied Physics, 39 (6) (1968), 2829-2838. [10] S. Udpa, P. Moore, Editors, Nondestructive Testing Handbook, Third Edition, vol. 5: Electromagnetic Testing, ASNT (2007). [11] R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan; Florida: Krieger Publishing (1983). [12] C. Reboud, G. Pichenot, S. Paillard, and F. Jenson, Simulation and POD studies of riveted structures inspected using Eddy Current techniques, in Electromagnetic Non-Destructive Evaluation (XIII), Studies in Applied Electromagnetics and Mechanics, J. Knopp, M. Blodgett, B. Wincheski, N. Bowler Eds. Amsterdam: IOS Press (2009), 129-136.