An eddy current model of pot-cored coil for testing multilayer conductors with a hole

BULLETIN OF OF THE THE POLISH POLISH ACADEMY ACADEMY OF OF SCIENCES SCIENCES BULLETIN TECHNICAL SCIENCES, Vol. 68, No. 6, 2020 TECHNICAL SCIENCES, Vol. 68, No. 6, 2020 DOI: 10.1515/bpasts-2020-XXZZ 10.24425/bpasts.2020.135388 DOI: ELECTRONICS, TELECOMMUNICATION AND OPTOELECTRONICS An eddy current model ofcoil pot-cored coil multilayer An eddy current model of pot-cored for testing for testing conductors multilayer conductors with a holewith a hole G. TYTKO * G. TYTKO∗ Institute of Electronics, The Silesian University of Technology, ul. Akademicka 16, 44-100 Gliwice, Poland Institute of Electronics, The Silesian University of Technology, Akademicka 16, Gliwice, Poland Abstract. Pot-cored coils are commonly used as probes in eddy current testing. In this paper, an analytical model of such a coil placed over a three-layer plate with a hole been presented. The proposed modelling magnetic andofnon-magnetic conductive Abstract. Pot-cored coils are has commonly used as probes in eddysolution current enables testing. the In this paper, of anboth analytical model such a coil placed over plates that contain different types of hole, i.e. a through, a surface, an inner or a subsurface hole. The problem was solved by using the truncated a three-layer plate with a hole has been presented. The proposed solution enables the modelling of both magnetic and non-magnetic conductive regionthat eigenfunction expansion The analysis wasancarried outa in a cylindrical system which solution domain plates contain different types(TREE) of hole,method. i.e. a through, a surface, inner or subsurface hole.coordinate The problem was in solved bythe using the truncated was radially limited. With the employment of theThe filamentary the expressions for the magnetic vector potential, and subsequently for the region eigenfunction expansion (TREE) method. analysis coil, was carried out in a cylindrical coordinate system in which the solution domain impedance the cylindrical were obtained. final formulas presented form and potential, then implemented in Matlab. was radiallyoflimited. With thecoil employment of the The filamentary coil, thewere expressions forinthea closed magnetic vector and subsequently forThe the resistance and reactance values were compared results obtained the experiment and using finite methodininMatlab. the Comsol impedance of the cylindrical coil were obtained.with Thethe final formulas wereinpresented in a closed formtheand thenelement implemented The Multiphysics each of the compared cases, good agreement wasobtained obtained. resistance andpackage. reactanceInvalues were with the results in the experiment and using the finite element method in the Comsol Multiphysics package. In each of the cases, good agreement was obtained. Key words: non-destructive testing, eddy currents, coil impedance, ferrite core, truncated region eigenfunction expansion. Key words: non-destructive testing, eddy currents, coil impedance, ferrite core, truncated region, eigenfunction expansion. 1. Introduction Bringing a coil fed with alternating current closer to conductive material results in the induction of eddy currents. The presence of defects in the tested object disturbs the flow of eddy currents, which brings about changes in the impedance of the coil. The correct interpretation of these changes allows detecting flaws and estimating their geometric dimensions. What has proved to be extremely useful for this purpose are mathematical models. The employment of them makes it possible to carry out both the analysis of the obtained coil impedance components and a simulation of the measurement process. According to the author, analytical models are particularly effective, since they can be implemented in any programming language and used for calculations made directly in the measuring instrument during the test. In a series of articles, non-destructive and semi-destructive diagnostics of concrete structures [1–3], steel elements [4, 5] and composites [6] were discussed in detail. In non-destructive testing, the inspection of materials is usually carried out with the employment of probes containing coils. The probe in the form of filamentary coil [7–10], rectangular coil [11, 12], aircored coil [13–18], I-cored coil [19–22] and iron core coil [23] were studied. In eddy current testing, pot core probes are very frequently utilized [24, 25]. The closure of the magnetic flux inside the core makes the flux take direction towards the tested surface, which in turn causes an increase in its intensity in the near vicinity of the workpiece. The analytical model of the coil ∗ e-mail: *e-mail: Manuscript submitted 20XX-XX-XX, accepted publication Manuscript submitted 2020-05-26, revisedinitially 2020-08-11, initiallyfor accepted for publication 2020-09-11, in December 20XX-XX-XX, published published in ZZZZZZZZ 2020. 2020 Bull. Pol. Pol. Ac.: Ac.:Tech. Tech.68(6) 68(6) 2020 Bull. 2020 with such a core, placed over the conductive half-space was developed in [26]. However, the presented solution is insufficient because it facilitates making calculations only for objects of infinite thickness and containing an easily detectable surface hole. A comprehensive mathematical model intended for practical use in eddy current defectoscopy should allow us to model multilayer materials of finite geometric dimensions and containing a hole that may be situated anywhere in the workpiece. The novelty of this paper is an analytical solution that meets the aforementioned requirements. The developed mathematical model is designed for calculating the impedance components of an E-cored coil located above a three-layer plate with an inner hole (Fig. 1). A proper selection of workpiece parameters facilitates making calculations for both a magnetic and a non-magnetic plate containing an inner, a through, a surface or a subsurface hole. Fig. 1. E-cored coil located above a conductive plate with an inner hole 2. Methods The cross-section of a pot-cored coil of relative magnetic permeability µ f has been shown in Fig. 2. The coil was placed 1311 1 G. Tytko Tytko G. {J1 (pT r)} 0 ≤ r ≤ a1 {R1 (pT r)} a1 ≤ r ≤ a 2 {R1 (pT r)} c1 ≤ r ≤ c 2 A4 (r, z) = {R1 (pT r)} p−1 (e−pz C4 − epz B4 ) T {R 1 (p r)} at a distance l1 from the surface of a three-layer plate of magnetic permeability of µ6 , µ7 , µ8 and electrical conductivity σ6 , σ7 , σ8 . A hole with radius g and depth l3 − l2 was made in the middle layer of the plate. The solution domain was radially limited up to the value of parameter b. On the outer boundary r = b, the component of the magnetic vector potential Aϕ (b, z) = 0 satisfies the Dirichlet boundary condition. While using the cylindrical coordinate system, the problem was divided into 10 regions (towards component z) and 5 subregions (I–V). For the purpose of the analysis, a filamentary coil (r2 −r1 → 0, h2 −h1 → 0) was employed, whose all turns coiled in a circle of radius r0 were located at the distance (...truncated)