## 4 Experimental Results for Vertical Cracks in Ferro-Magnetic Steel  
For the experiments, a 10 kW induction generator with an excitation frequency of 200 kHz was used. The induction coil is a so-called Helmholtz coil consisting of two circular windings placed from each other at a distance (see Figure 7). This coil geometry has the advantage that the magnetic field in the region between the windings is approximately homogenous [10], and the induced eddy current flows roughly parallel to the coil circles. The workpiece to be inspected is placed in this middle part. We have manufactured several coils with different sizes and an appropriately sized coil is selected for each sample. An additional advantage of this coil geometry is that the infrared camera has an undisturbed view of the sample without occlusion by the coil. The generator works as a resonance circuit; the external capacitance and the inductivity of the coil determines the resonance frequency, which is then the working frequency of the generator. It is important to note that the simulations and the measurements were set up in a way such that the induced eddy current flows perpendicular to the crack direction. Additional investigations were carried out on how the signal is influenced if this angle is less than 90\({}^{\circ}\)[10, 23]. A possible setup was also presented using two Helmholtz coils with different excitation frequencies, whereby the detection becomes independent of the crack orientation. The infrared camera has a cooled InSb detector, which is sensitive in the wavelength range 1.5-5 \(\upmu\)m and whose frame rate is 383 images/s with a frame resolution of 320 \(\times\) 256 pixels. The generator and the camera are controlled by PLC (programmable logic controller) to guarantee a reproducible and synchronized procedure. Figure 6: Temperature (**a**) and phase difference (**c**) depending on crack depth; relative temperature contrast (**b**) and phase difference (**d**) depending on the relative crack depth \(\alpha\), calculated with Equations (4)–(7) for ferro-magnetic steel material parameters. Figure 7: Helmholtz coil with a wire, for which the results are shown in Figure 8.  
Figure 8 shows the results for a steel wire with a diameter of 10.8 mm. The crack is a very long defect, mainly along the whole wire with a constant depth. The wire was cut at one position and a microscopic cross section was made (see Figure 8a), showing a vertical crack with a depth of 0.35 mm. For the measurement, \(t_{pulse}\) = 0.1 s was selected. The phase image is depicted in Figure 8b. The crack has a constant depth along the wire, as the phase contrast is very homogeneous. Figure 8c shows three profiles through this phase image, the distance between the positions of the profiles is about 23 mm, proving the homogeneity of the crack. In this diagram, the calculated phase profile for a depth of 0.35 mm is also plotted, showing very good agreement with the measured ones. In the experiment, the IR camera image has a resolution of 0.22 mm/pixel, so the phase peak at the crack position is not a perfect fit. It is important to note that the induction generator cannot immediately switch the heating pulse on, but measurements show that, during the first approximately 40 ms, the induction current increases linearly. This behavior was also taken into consideration in the simulation model. Additionally, the wire with a diameter of 10.8 mm is not a \"semi-infinite body\" as considered in the models. Investigations of inductive heating for wires regarding the pulse length and the diameter have been published previously [24]. These results showed that for 0.1 s, the wire with a diameter of 10.8 mm behaves as a semi-infinite body. In the simulation models, the crack is assumed to be a long crack, which can be modeled in 2D cross sections. The question arises how long a crack has to be so that the models remain valid. If a crack is short, then hot spots occur at its ends as the eddy current at the surface flows around the crack ends.