## 4 Results and Discussion
Figure 8: Impedance variation in (%) at 10 kHz with (**a**) depth = 0.3 mm, (**b**) depth = 0.5 mm, and (**c**) depth = 1 mm. Figure 10: Impedance variation in (%) at 40 kHz with (**a**) depth = 0.3 mm, (**b**) depth = 0.5 mm, and (**c**) depth = 1 mm. Figure 9: Impedance variation in (%) at 20 kHz with (**a**) depth = 0.3 mm, (**b**) depth = 0.5 mm, and (**c**) depth = 1 mm. The method applied here to characterize the thin cracks relies on the fact that the eddy-current response depends on the probe excitation frequency. By considering the operating frequency, detection and the optimum frequency, the penetration depth of eddy currents was significant. Therefore, the eddy-current response would be sensitive to the crack surface in-depth direction. When \(\mu_{r}=1\) the impedance  
Figure 11: Impedance variation in (%) at 50 kHz with (**a**) depth = 0.3 mm, (**b**) depth = 0.5 mm, and (**c**) depth = 1 mm. Figure 12: Impedance variation in (%) at 100 kHz with (**a**) depth = 0.3 mm, (**b**) depth = 0.5 mm, and (**c**) depth = 1 mm. Figure 13: Impedance variation in (%) at 300 kHz with (**a**) depth = 0.3 mm, (**b**) depth = 0.5 mm, and (**c**) depth = 1 mm. signal is smooth for each frequency. Fine cracks do not appear, or are almost indistinguishable, as shown by the results of Figures 8-11. Therefore, the results obtained for constant relative magnetic permeability cannot provide right information regarding the presence or absence of thin cracks in the weld area. In contrast, with \(\mu_{r}\
eq 1\) this model is able to reproduce the experimental results, while giving a better prediction of cracks. The shape of the impedance signal is not the same for each depth and each frequency. The signal presents fluctuations at the peak, which corresponds to the change of environment (weld and crack). Thus, this is comprehensible as it corresponds to a presence of the thin cracks in the weld area. Figure 14 shows three shapes of shallow mountains that correspond to the three cracks referred as d0.3, d0.5 and d1.0. A small peak at 180 mm on the X axis indicates a small defect as shown in Figure 5:0 The output signal of the Nortec 500 equipment is entered into the DAQ and expressed as a 3D surface graph using LabVIEW as given in Figure 5:1  
Figures 12 and 13 correspond respectively to 100 and 300 kHz. At these frequencies, the impedance of the probe is maximum, and the current flow is stronger at the surface and decreases rapidly in-depth direction. Thus, the current response would be particularly sensitive to surface cracking. The same signal was obtained with the two relative magnetic permeabilities. An insignificant signal difference was observed with 100 kHz. The simulation results reproduce the trend and the shape of the experimental signals; however, a small deviation can be observed that is probably due to several parameters; the universe of the experimental is in 3D while the simulations are in 2D, the machining of the cracks, the lift-off and another factor come from the impedance measurement carried out. The impedance variation as a function of the probe displacement reflects the change in the distribution of physical properties over the part of the pipe being inspected. The proposed approach is validated by a comparison and shows a satisfactory concordance with the experimental ones in all cases for all the frequencies used, which proves:  
* Eddy currents are well adapted to the detection of thin surface cracks under the stress of the heat welding which affects the relative magnetic permeability locally. * The validity of the modeling and the analysis approach.