## 7 Vertical Cracks with Large Eddy Current Penetration Depth
The deeper the crack, the more the eddy currents deviate into the material, so the phase difference increases as crack depth increases (Figure 22b). Figure 23a shows the relative temperature contrast for different penetration depths that were simulated for a crack with a depth of 1 mm. Three curves were calculated for ferro-magnetic steel (dash-dot lines) and the other three were for non-magnetic steel (solid lines). To be able to compare different materials, the abscissa is normalized to  
\[t_{dh}=\left(\frac{d}{2}\right)^{2}\cdot\frac{1}{\kappa} \tag{10}\]  
at which time the heat flows into a distance of \(d\) due to diffusion. In Figure 23a, it can be observed that  
* as penetration depth increases, the relative contrast becomes negative as the temperature at the crack position becomes lower than that at the sound surface;
* with increasing time, due to heat diffusion, the relative contrast decreases in absolute value. Figure 21: Temperature at the crack position and at the sound surface distribution for 0.1 s (**a**) and 1 s (**b**) heating pulses for the same scenario shown in Figures 18 and 20. Figure 18:0 (**a**) Phase distribution for different heating pulses for the same crack (\(d\) = 1 mm, \(\delta\) = 1.35 mm); (**b**) phase for different crack depths after a 0.1 s heating pulse. In Figure 23, the analytical calculation results of Equation (5) are also plotted with a dashed line relating to a penetration depth of \(\delta\) = 0 mm. Assuming a detection limit of +0.1 and \(-\)0.1 for the relative temperature contrast, a defect with a 1 mm depth can be detected with a positive \(C_{R}\) if the penetration depth is small compared to crack depth. Another possibility for detection is a negative \(C_{R}\) if the penetration depth is comparable with crack depth. However, in this second case, time duration of the possible observation is limited due to the equalization process of heat diffusion. In Figure 23b, phase differences are depicted for the same cases, and the temperature contrast is shown in Figure 23a. Generally, a negative temperature contrast results in a negative phase difference, and a positive \(C_{R}\) causes a positive phase difference. However, as the phase calculation considers the whole heating and cool-down duration, the detection of a crack from the phase is possible for a longer pulse duration than that from the temperature. For example, when \(\delta\) = 1 mm, a crack with a depth of 1 mm in non-magnetic steel in the temperature contrast can be detected up to \(t\) = 5 \(t_{dth}\)\(\sim\)0.31 s, and in the phase image up to \(t\) = 15 \(t_{dth}\)\(\sim\)0.93 s, assuming a detection limit of 10\({}^{\circ}\). As has been shown, the deeper the crack, the larger the effect it has on the eddy current distribution; consequently, it causes a larger phase difference in absolute value. Figure 24a compares the phase values for two penetration depths if the crack is 1 mm and 0.5 mm deep. \(\delta\) = 0.1 mm (dash-dot line) was calculated for ferro-magnetic steel, and \(\delta\) = 0.5 mm (solid line) for non-magnetic steel. It was observed that the phase values of the smaller crack were closer to zero, so are more difficult to detect. We defined two ratios \(\alpha\) (Equation (6)) and \(\beta\) (Equation (8)) that mainly determine the phase distribution. Comparing two cracks with a depth of \(d\) and \(d/2\), all distances, including the penetration depth and the diffusion length, have to be reduced to half to obtain the same phase distribution. To reduce the penetration depth to half, a fourfold higher excitation frequency is required; to reduce the diffusion length to half requires a reduction in the pulse duration to one-fourth. Thus, the phase distribution around both cracks becomes the same; however, for the smaller one, the distances are proportionally half of the larger one. This proportionality allows a normalized plotting of the phase contrast (Figure 24b).