Analytical modeling for transient probe response in pulsed eddy current testing Mengbao Fan, Pingjie Huang à , Bo Ye, Dibo Hou, Guangxin Zhang, Zekui Zhou Department of Control Science & Engineering, State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, China article info Article history: Received 19 August 2008 Received in revised form 5 January 2009 Accepted 15 January 2009 Available online 29 January 2009 Keywords: Multilayered conductive structures Pulsed eddy current testing Transient probe response Analytical modeling Gibbs phenomenon abstract An improved analytical model by the Fourier method for transient eddy current response is presented. In this work, an alternative approach is considered to solve the harmonic eddy current problem by the reflection and transmission theory of electromagnetic waves, thus a more concise closed-form expression is expected to be obtained. To reduce the inherent Gibbs phenomenon, a harmonic order- dependent decreasing factor is employed to weight the Fourier series (FS) representation. It is shown that the developed model is promising to be used as a fast and accurate analytical solver for the transient probe response and is helpful to gain a deep insight into pulsed eddy current (PEC) testing. & 2009 Elsevier Ltd. All rights reserved. 1. Introduction Real-time inspection and evaluation of multilayered conduc- tive structures are extremely important for accident reduction and product quality assurance, especially for safety-critical fields. Among most common nondestructive testing (NDT) techniques, eddy current testing (ECT), based on Faraday’s Induction Law, has gained wide acceptance because of its good accuracy, high speed and low cost. Conventionally, ECT is usually carried out by driving the probe coil with a single- or multi-frequency sinusoidal signal. Recently, PEC testing, which uses pulsed excitation, has been paid much attention in research and development work in NDT. In contrast to traditional ECT, PEC testing has some advantages in terms of frequency content, inspection time and analysis of probe response [1]. Since the probe response contains information of the specimen under test, it is the signal of interest and importance. Analytical modeling for probe response is generally accepted as a very effective way to understand the underlying physical process of ECT [2]. For axisymmetric configuration, the theoretical study for PEC response has been done for more than 3 decades. All the work concerned falls into two categories. One is to perform the inversion of Laplace transform (LT). The other is to extend the solution of harmonic response with help of Fourier transform (FT) or FS. For the former case, transient eddy current density in conductive half-space is formulated both in 2-D rectangular coordinates and in axisymmetric cylindrical coordinates at early stage [3,4]. Subsequently, Bowler [5,6] and de Haan et al [7] find the closed-form expression for PEC response resulting from a conductive half-space and single plate, respectively. Later a novel semi-analytical approach is employed to implement the inverse LT based on a systematic way of bracketing and computing the poles of the LT expression [8]. However, for transient signal due to layered specimen, its expression in complex frequency domain may be too complicated and perhaps the corresponding exact inverse LT is not available now and still to be investigated. So, for prediction of PEC signal caused by multilayered conductive structures, it is preferred to choose FT- or FS-based modeling techniques, because the analytical model of harmonic eddy current field caused by multilayered specimen has been estab- lished successfully [9,10]. Fourier superposition-based modeling technique has been extensively used in calculation of PEC signal [11–13]. For this case, the formulation of harmonic probe response is the key point. Although the closed-form solutions to both probe impedance and normal magnetic field have been given [9,10,13], it is worth noting that they are solved either by Cheng’s matrix method [9] or recursive method [10], usually described in an infinite integral. It suffers from determining infinite integration range. Moreover, using the matrix method to compute the probe response requires considerable matrix operations, which is the possible reason for round-off errors and overflows. The impedance model by the recursive method has not been proved strictly and completely [14], and the eddy current distribution in conductive plates is not available now. ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ndteint NDT&E International 0963-8695/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2009.01.005 à Corresponding author. Tel: +86 057187952241; fax: +86 057187951219. E-mail address: huangpingjie@zju.edu.cn (P. Huang). NDT&E International 42 (2009) 376–383