## 3 Results
### Frequency Dependence of the Angular Scan Gradient  
As we have shown in the previous subsection and will show here, the phase of the rf signal in some cases does not contain enough information to make object composition identification. However, combining measurements of the angular dependence and frequency dependence of the rf signal amplitude allows estimation of the lift-off value and subsequently material composition identification. Recording the full angular variation of the inductive signal is time-consuming and might be impractical in some scenarios. Instead, because of the linear character of the signal's angular dependence, we use the absolute value of the gradient of this dependence that could be evaluated, in principle, based on only two measurements. Figure 8 shows the gradient of the angular amplitude dependence as a function of the operating frequency recorded for copper, aluminium, brass, and steel plates recorded at a lift-off of 3 mm. It is worth pointing out that each point on this plot represents an object-orientation independent measurement of the signal amplitude at a given frequency. For frequencies higher than 17.5 kHz all data sets show a linear dependence on frequency (linear fits indicated with dotted lines). This linear dependence represents the scenario when the object response has a purely surface character, i.e., the skin depth is smaller than the object thickness, and the increase of the signal comes from the frequency dependence of Faraday's induction. This dependence is generic across all materials, which is reflected in the parallel slopes above 17.5 kHz. For the frequencies below 17.5 kHz for the sets representing copper, brass and aluminium there is a departure from linear dependence. Figure 6: Plots of the amplitude (**a**) and phase (**b**) of measured rf signals generated by scans over the 40 \(\times\) 40 mm\({}^{2}\), 2 mm thick ferrite plate for lift-off’s of 1.5 mm, 3 mm, 4 mm, 6 mm, 8 mm, 10 mm, 13 mm, 16 mm, and 25 mm. The images were recorded at an operating frequency of 29 kHz. Figure 7: Amplitude (**a**) and phase (**b**) rf signal extracted from Figure 7 as a function of lift-off. The measurement was performed at operating frequency of 29 kHz. The red diamonds, green triangles and blue circles represent the data recorded over the vortex radii with coordinates (41, 50) mm to (41, 70) mm, (50, 41) mm to (70, 41) mm, and (41, 40) mm to (41, 20) mm, respectively. This is caused by the change of character of eddy currents generation, i.e., transition from purely surface to volume generation. In other words, a decrease in efficiency of Faraday induction with decreasing frequency is compensated by the increase of the object volume that contributes to the secondary field generation. The value of the frequency where this change is observed is defined by the ratio between skin depth and object thickness. Steel shows slightly different characteristics than the others. It has the lowest conductivity, which accounts for the reduced gradient (due to smaller secondary field amplitude), but also the smallest skin depth due to its permeability. This results in the extension of the linear character of the gradient's frequency dependence at low frequencies. It is important to stress, that we observe a non-zero gradient at low frequencies where the skin depth is bigger than the thickness of the object, i.e., at 4 kHz, because there are still eddy currents circulating on the object's surface. As discussed in ref. [12] the actual value of the gradient is defined by the ratio between surface and volume generated components of the secondary field. The distinction between volume and surface characteristics of an object response provides a better understanding of the phase measurement in the context of composition identification.