## 3 Vertical Cracks with Small Eddy Current Penetration Depth
Figure 4a shows the temperature after different pulse lengths. As the diffusion length \(d_{th}\) increases with time, the heated region around the crack also becomes broader and has an extension of \(2d_{th}\) on both sides of the crack. At the crack position, the phase is higher (see Figure 4b) than that at the sound surface and the minimum occurs around \(d_{th}/2\), which means that, with increasing pulse length, the width of the detectable phase signal of the crack increases. The deeper the crack, the more it represents a barrier for the eddy current and for heat diffusion. It can be seen from Equation (4) that the temperature at the crack position increases with depth \(d\)  
Figure 4: Temperature (**a**) and phase difference (**b**) around a crack (\(d=1\) mm) as shown in previous figures, calculated for different pulse lengths using the simu.2 model. Figure 3: Temperature (**a**) and phase difference (**b**) around a crack calculated with the three models for the same case as in Figure 1 and in Figure 2.  
Figure 5 compares the temperature and phase distribution around cracks with different depths. These simulations were carried out with all three models, and results of simu.2 are shown in the figure (\(f\) = 200 kHz, \(\delta\) = 0.034 mm). Comparing the results of the three model types shows a very good agreement. How the temperature and the phase at the crack position depend on the crack depth is depicted in Figure 6a,b. In Figure 6c,d, the abscissa is \(\alpha\), which is the ratio of the crack depth to the thermal diffusion length. This means, e.g., that \(d\) = 1 mm and \(t_{pulse}\) = 0.1 s results in \(d_{th}\) = 2.16 mm and \(\alpha\) = \(d\)/\(d_{th}\) = 0.46; if \(t_{pulse}\) = 1 s, then \(d_{th}\) = 6.8 mm, and we have \(\alpha\) = \(d\)/\(d_{th}\) = 0.147. As the temperature increases monotonously with the crack depth, it would theoretically be suitable for determining the crack depth from measurements. On the other hand, inhomogeneous heating or variations in the emissivity, e.g., small scratches or ground surfaces, have a strong influence on the measured temperature, and the defect depth determination from the temperature increase is not reliable. In contrast, in the phase image, most of these factors become negligible, and the crack depth can be much more reliably determined. According to Figure 6d, the phase contrast increases approximately up to a crack depth of \(d_{th}/3\) with a phase contrast of 27\({}^{\circ}\) at this depth. For deeper cracks, the phase contrast is saturated. This means that the optimal pulse duration is dependent on the crack depth, which should be resolved in the experiment. For example, for steel with \(t_{pulse}\) = 0.1 s, one has a very good resolution of up to \(d_{th}/3\) = 2.16/3 mm = 0.72 mm and with \(t_{pulse}\) =1 s up to \(d_{th}/3\) = 6.8/3 mm = 2.27 mm. If the pulse length is short, then shallow cracks are well resolved, but the same signal occurs for deeper cracks. If the pulse length is long, then deeper cracks can be resolved, but on the other hand, the signal and the resolution become low for shallow cracks. Figure 5: Temperature (**a**) and phase difference (**b**) around cracks with different depths after a 0.1 s pulse duration. Figure 6: \(Cont.\)