## 4 Results and Discussion  
The specific objective of this work is to evaluate and improve the ability to detect narrow cracks in carbon steel weld joints with the eddy current testing technique. So, it is critical to investigate, identify, and control the formation and growth of such cracks to reduce the threat of unexpected failure. For this purpose, very accurate solutions and measurement data are required. It is obvious that numerical analysis plays a crucial role in the solution of the ECT probe. This is why the finite element method is used. The radius of the tube of the carbon steel specimen under investigation is considerably bigger than the size of the probes and the surface area is relatively larger than the probes, so the wall of the tube can be considered as a plate, and the problem is simplified and can be modeled as two-dimensional geometry [25; 43; 44]. On the other hand, the configuration shown in Figure 6a (3D configuration) is streamlined, as shown in Figure 6c (2D configuration), which is representative of the specimen, the single coil, and the narrow crack. The corresponding schematic structure of the device is depicted in Figure 7, in which the various parts of the device are listed directly. In Figure 8a, the entire geometry is then covered by a finite element mesh, consisting of triangular elements obtained by a mesh generator based on the MATLAB Partial Differential  
Figure 6: Same section views with (**a**) 3D configuration, (**b**) 3D configuration with long defect, and (**c**) amended 3D configuration to a 2D configuration. Figure 7: Solving domain and boundary conditions. Equation Toolbox. This toolbox enables the generation of a mesh using the Delaunay triangulation algorithm. The selection of an appropriate mesh density for the reliability analysis is important, especially around the crack region, to achieve reliable results. Due to the symmetrical distribution of these properties, the distribution represented covers a path of 20 mm from the center length of the inspected part to the last inspected point, as illustrated in Figure 9.  
Considering the need for detecting the deeper crack to achieve larger skin depth and to control the surface of inspected pipe, a compromise is made by considering the operating frequency, detection, and optimum frequency, with a lift-off of 0.1 mm. The distance and scan pitch simulated were identical to the experimental parameters. A fixed scan distance of 40 mm was chosen so that the cracks are located at the half-center of the weld zone along the scan trajectory. The impedance signal amplitudes obtained for all the used frequencies are normalized and plotted versus the probe displacement. For each crack depth, the corresponding impedance signal amplitude is normalized by the impedance signal amplitude of the 1 mm crack depth. The normalized impedance was computed by \(Delta\)\(Z(\%)=(Z-Z_{0})/(Z-Z_{1.0})_{max}\), where \(Z_{0}\), \(Z\), and \(Z_{1.0}\) are the impedances of the raw material without crack, the impedance from the HAZ and the weld with the crack, and the impedance  
Figure 8: (**a**) Finite element mesh; (**b**) Excitation probe configurations. Figure 9: Physical properties distributions: (**a**) Electrical conductivity distribution; (**b**) Relative permeability distribution. Figure 14: Impedance variation in (%) at 20 kHz with (**a**) depth = 0.3 mm, (**b**) depth = 0.5 mm, and (**c**) depth = 1 mm. Figure 12: Impedance variation in (%) at 40 kHz with (**a**) depth = 0.3 mm, (**b**) depth = 0.5 mm, and (**c**) depth = 1 mm. Figure 13: Impedance variation in (%) at 10 kHz with (**a**) depth = 0.3 mm, (**b**) depth = 0.5 mm, and (**c**) depth = 1 mm. As shown in Figures 10-12, the output signals from the ECT measurement experiments are noised. However, the narrow cracks being visible and the noise appearing in the signal would not be considered sufficiently continuous to be called cracks. In Figure 12a, there are noise signals that would be confused with cracks.