The use of arrayed eddy current (EC) sensors in Non Destructive Testing (NDT) provides high speed inspection and better space resolution by miniaturization of their coils. The arrayed sensors can make a measurement of large surfaces without a scan, as illustrated in the Figure 1, which results in a gain in time and measurement noise reduction; on the other hand, compared to conventional EC sensors, arrayed EC sensors provide more information about the defect characteristics.

There are several configurations of arrayed eddy current sensors [1–3]; when their coils are fed separately, the effect of the adjacent coils is negligible; the modeling approach is then the same as for a single coil sensors. In this work, we consider an arrayed sensor in which the coils are connected in series and fed simultaneously by a current source as shown in Figure 2. The advantages of such configuration are:

- The synchronization of the supply and the measurement is not required for the electronic component.

The investigation is done by the measurement of the impedance variation of each coil. The purpose is to determine a crack shape and size using the measurements provided by such a sensor in a real time investigation.

The inversion method proposed is based on the iterative solving of the direct problem; it is thus important to have a fast tool to solve the latter. The use of the 3D finite element method would be very expensive in terms of memory space and CPU time. On the other hand, the analytic models lack the flexibility to handle complex geometries. In this work, we use the ideal crack model [4–6], generalized to arrayed eddy current sensors [7]. In the ideal crack model, the effect of the crack is represented by a current dipole layer on its surface, evaluated by an integral equation involving the electric dyadic Green’s functions and the normal incident electric field on the crack surface. The impedance variation of each coil is evaluated using the reciprocity principle. The evaluation of the dyadic Green’s function matrix is made independently of the iterative procedure of inversion; this makes the inversion to be very fast. On the other hand, a fast calculation of the incident field, produced by all the coils of the arrayed sensor, on the crack surface is achieved by making a translation and a superposition of the 2D axisymmetric finite element results obtained for one coil [7].

Figure 3 describes the modeled system. It is constituted of a (3 × 4) matrix of identical coils situated above a conductive plate characterized by a conductivity σ and the free space permeability μ₀. The plate contains an ideal crack of a surface S with an arbitrary shape described in Figure 3. The arrayed sensor coils are fed in series by a current source with a time harmonic variation i s ( t ) = 2 I s e j ω t. Table 1 gives the numerical values of the fixed parameters of the system [9].

The direct problem is based on the generalization of the ideal crack model to an arrayed eddy current sensor [7], which we recall briefly in this section. Firstly, the electric field induced by a single coil in the unflawed piece is calculated using the 2D axisymmetric finite element method. The electromagnetic problem formulation is given by (1), involving the magnetic vector potential A and the current source density Js. The total electric field E^T induced by all the coils constituting the arrayed sensor is then obtained by (2), making a spatial translation and a superposition of the results obtained for the single coil [8]. In (2), Aφ is the magnetic vector potential solution of (1) for one coil, nc is the number of coils constituting the arrayed sensor, “ox_k, oy_k” are the center coordinates of the coil k, “x,y,z” are the Cartesian coordinates of the computing point, r_k is the distance between the computing point and the axis of the coil k, and sign(I_sk) indicates the direction of the current in the coil k.

Once the total normal incident field E N T on the surface S of the ideal crack is determined, we calculate an equivalent current dipole p normal to this surface S by using the following integral equation [9]: (3) E n T ( r 0 ) − lim r → r 0 { j ω μ 0 ∫ S G nn ( r , r ′ ) p ( r ′ ) d s } = 0 ; r 0 ∈ S   ,where: (4) G nn ( r , r ′ ) = n ^ . G ¯ ( r , r ′ ) . n ^ .

In (4), n̑ is the vector normal to the surface S, and Ḡ (r, r′) is the electric dyadic Green function satisfying Equation (5) and subjected to the same continuity conditions as the electric field. In (5), Ī = x̂x̂ + ŷŷ + ẑẑ is the unit tensor and k² = jωμ₀σ: (5) ∇ × ∇ × G ¯ ( r , r ′ ) − k 2 G ¯ ( r , r ′ ) = I ¯ δ ( r − r ′ )

The integral equation (3) is solved using the moment method. The crack surface is subdivided into (N = n_L × n_d) rectangular elements of equal surfaces S_e; the dipole density is considered constant in each element. We obtain the following matrix equation: (6) { E } = [ G ] { P } .

The vectors E and P are of dimension (N); containing respectively the values of E n ( k ) and p for the N elements of the crack grid. The matrix G is of dimension (N × N); its elements are calculated as follows: (7) g ( i , j ) = ∫ s j G nn ( r i , r j ) d S ,       ( i = 1 , .. N , j = 1 , .. N ) .

Using the reciprocity principle, the impedance variation of a coil k of the arrayed sensor is given by the following equation: (8) Δ Z ( k ) = − 1 I s 2 ∫ s E n ( k ) p   d s

In (6) E n ( k ) and p are scalars representing, respectively, the part of the normal electric field induced by the coil k on the surface S, and the normal current dipole solution of (3). The discrete form of (8) is given by: (9) Δ Z ( k ) = − S e I s 2 ∑ i = 1 N E ( i ) P ( i ) .

We have presented a fast crack profile reconstitution procedure using arrayed eddy current sensor data. The use of the reciprocity and superposition principles allows a fast resolution of the direct problem. In the inverse problem, which is based on an iterative solving of the direct one, we adopted a coarse approximation of the crack profile which is represented by only a few discrete values; this makes the inversion procedure very fast when using genetic algorithms. This method can give a real time inspection when implemented in an embedded NDT hardware.