# Eddy current testing

The eddy current method is one of the numerous non-destructive testing methods that are widely used in scientific research and industry for controlling the quality of materials and products without damaging or impairing the test objects [a2], [a7]. This method is based on the law of electromagnetic induction discovered in 1831 by the English physicist M. Faraday. Many important experiments, which contributed to the understanding of the principles of electromagnetism, preceded Faraday's discovery and were made by A.M. Ampère, D.-F.-J. Arago, J.-B. Biot, F. Savart, and H. Öersted. According to the principle of electromagnetic induction, an electric current is induced in a closed conducting circuit located in an alternating magnetic field. It is found experimentally that currents are induced not only in a conducting contour, but also in a nearby continuous metallic medium if the magnetic flux changes. These currents are usually called eddy currents, or Foucault currents, after the French scientist L. Foucault who discovered this phenomenon.

A typical setup for eddy current testing is as follows. An excitation coil carrying an alternating current is placed near an electrically conducting medium to be tested. The current flow through the coil generates a varying magnetic field (called the primary field). The primary field induces varying eddy currents in the electrically conducting medium. The eddy currents, in turn, produce a varying magnetic field (called the secondary field). In the case of non-magnetic conducting media, the secondary field opposes the primary field. The effects of the secondary field can be read from the variation of the output signal of the excitation coil or from the output signal of a second coil (called a detector coil) situated nearby.

The output signal of the detector coil depends on several parameters, such as the magnitude and frequency of the alternating current, the electrical conductivity and magnetic permeability of the medium, as well as the relative position of the coil with respect to the medium. It also reflects the presence of inhomogeneities (called flaws) in the medium. Eddy currents change the amplitude and phase of the coil impedance. The output signal is conveniently represented by an impedance diagram, which is a plot in the $XY$-plane, of the variations of the amplitude and the phase of the coil impedance, $Z=X+jY$, $j=\sqrt{-1}$, which can be resolved into its real and imaginary components, $X$ and $Y=\omega L$, called the resistive and the reactive component, respectively; $L$ is the coil inductance and $\omega$ is the current frequency.

The eddy current method possesses many advantages which explain its popularity. No mechanical contact is required between the eddy current probe and the test material. The eddy current penetration depth, that is, the inspection depth, can be changed by modifying the frequency of the excitation current. The method is highly sensitive to small defects and can be used, in particular, for dimensional and conductivity measurements. Other advantages are its low cost and automatized measurements for a wide class of applications. However, the eddy current method has its limitations, the most important one being that it can be applied only to conducting materials.

The most general problem of flaw detection by eddy current methods can be formulated as the problem of determining the size, form and other properties of a flaw from the change in the output signal of a moving eddy current probe. This is an inverse problem, which is often ill-posed. Solutions of inverse problems for determining the conductivity and thickness of layers in a multi-layer medium can be found, for example, in [a8], [a9]. The inverse problem cannot be successfully solved without some knowledge of the response of an eddy current probe to a flaw, that is, without comparison with the solution of the corresponding forward problem.

Extensive theoretical investigation of eddy current methods was initiated in the 1950s. The study was motivated by numerous applications, since the determination of flaws in a conducting medium is an important quality control problem in industry. The few cases where the change in impedance can be evaluated in closed form, correspond to a symmetrically positioned probe with respect to the conducting medium, as in the following examples [a3], [a4], [a11], [a12]:

a) a probe situated above, and parallel to, a multi-layer medium;

b) a probe situated above a multi-layer sphere with the coil axis going through the centre of the sphere; and

c) a co-axially situated probe inside (or outside) a multi-layer tube. In practice, however, one is interested in asymmetric problems having a vector potential in an arbitrary direction. In the presence of a flaw in the conducting medium, symmetry breaks down and no analytical solution is known. Numerical methods, such as volume integrals, finite elements and wavelets are powerful tools for solving eddy current testing problems [a6]. Many commercial codes are also available. Although numerical methods can handle problems with complex geometry, in some cases their use is limited. One of the important practical problems is the detection of cracks in a conducting medium. A typical crack has very small opening in comparison with its height and length. In this case, fine meshes may be needed for accurate modeling of the response of the eddy current probe. In some instances, eddy current problems can be solved by approximate analytical methods (for instance, perturbation methods [a1], asymptotic methods based on the geometrical theory of diffraction [a5], etc.). In many cases such approximate results match well with accurate numerical predictions.

A widely used perturbation method in eddy current testing [a1] is as follows. Since a flaw is usually considered as an anomaly in a uniform conducting medium, one can assume that the properties of the flaw (that is, the electric conductivity and/or magnetic permeability) differ only slightly from those of the surrounding medium. Then it is natural to introduce small parameters $\epsilon=1-\sigma_1/\sigma_2$ and $\delta=1-\mu_1/\mu_2$, where $\sigma_1$, $\sigma_2$, and $\mu_1$, $\mu_2$ are the conductivities and magnetic permeabilities of the flaw and the surrounding medium, respectively. In this case, it follows from the Maxwell equations, which are used to compute the change in impedance of an eddy current probe due to a flaw, that one has a regularly perturbed problem, so that the solution can be expanded in a perturbation series and the terms of the series can be computed step by step. The zeroth-order solution corresponds to the flawless case, whose analytical solutions are well known.

An alternative approach is to use the Lorentz reciprocity theorem [a7]. It can be shown under rather general assumptions that the change in impedance of the eddy current probe can be expressed in terms of an integral over the region of the flaw, where the integrand is proportional to the dot product of the electric fields in the flawed sample and the reference sample. However, this formula contains the unknown field in the flawed sample. Several approximations are used to overcome this difficulty. In the Born approximation, the unknown field is replaced by the field in a reference sample in order to obtain a formula which gives the same result as the first-order approximation in the perturbation approach described above. Recently, the so-called layer approximation [a10] has been used to approximate the field in the region of the flaw by introducing in the conducting medium an additional layer whose height and conductivity are equal to those of the flaw. Using the known analytical solution for the case of a multi-layer medium, one can compute the change in impedance in this case. The layer approximation has been favourably compared [a10] with experimental data and numerical modeling by the volume integral method. In practice, one need only know some integral characteristics of the solution which, in some cases, can be expressed in terms of computationally suitable formulas. These solutions indicate, on the one hand, that perturbation methods can be successfully used to solve complicated asymmetrical three-dimensional electrodynamics problems and, on the other hand, that they are first-order approximations to the solutions of the original problems. In several cases, numerical methods and perturbation methods complement each other. The use of mathematical models together with the exact form of the solution facilitates the study of the influence of several of the parameters of the test problem on the characteristics of the output signal and on the testing process. Such a task may be costly and difficult, if not impossible, to achieve experimentally.

#### References

 [a1] M.Ya. Antimirov, A.A. Kolyshkin, R. Vaillancourt, "Mathematical models in non-destructive eddy current testing" , Publ. CRM Montréal (1997) [a2] J. Blitz, "Electrical and magnetic methods of nondestructive testing" , Adam Hilger (1991) [a3] V.V. Dyakin, V.A. Sandovsky, "Theory and computations of superposed transducers" , Nauka (1981) (In Russian) [a4] V.G. Gerasimov, "Electromagnetic control of one-layer and multi-layer products" , Energiya (1972) (In Russian) [a5] N. Harfield, J.R. Bowler, "A geometrical theory for eddy-current non-destructive evaluation" Proc. R. Soc. London Ser. A , 453 : 2 (1997) pp. 1121–1344 [a6] N. Ida, "Numerical modelling for electromagnetic non-destructive evaluation" , Chapman&Hall (1995) [a7] "Electromagnetic methods of nondestructive testing" W. Lord (ed.) , Nondestructive Testing Monogr. and Tracts , 3 , Gordon&Breach (1985) [a8] J.C. Moulder, E. Uzal, J.H. Rose, "Thickness and conductivity of layers from eddy current measurements" Review of Scientific Instruments , 63 : 6 (1992) pp. 3455–3465 [a9] S.M. Nair, J.H. Rose, "Reconstruction of three-dimensional conductivity variations from eddy current (electromagnetic induction) data" Inverse Problems , 6 : 6 (1990) pp. 1007–1030 [a10] R. Satveli, J.C. Moulder, B. Wang, J.H. Rose, "Impedance of a coil near an imperfectly layered metal structure: The layer approximation" J. Appl. Physics , 79 : 6 (1996) pp. 2811–2821 [a11] R.L. Stoll, "The analysis of eddy currents" , Clarendon Press (1974) [a12] J.A. Tegopoulos, E.E. Kriezis, "Eddy currents in linear conducting media" , Studies Electr. Electronic Engin. , 16 , Elsevier (1985)
How to Cite This Entry:
Eddy current testing. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Eddy_current_testing&oldid=32526
This article was adapted from an original article by A.A. KolishkinR. Vaillancourt (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article