## 4 Theoretical Simulation and Experimental Verification
### Single-Probe Testing Experiment
The electrical conductivity of the samples is measured using the Autosigma3000 from General Electric Company (Boston, MA, USA), and the conductivity parameters are shown in Table 2.  
Figure 8 above shows the output signal curves of the ferromagnetic alloy steel 42 CrMo at five different thicknesses, with the probe lift height set to 5 mm. Figure 9 extracts the relationship between the key characteristic quantities and the thicknesses from Figure 8. The results indicate that as the sample thickness increases, the peak value gradually decreases, while the peak time gradually increases. The differential output signals U_ref_(_t_) and _Bref_(_t_) from the receiving coil and Hall sensor exhibit the same characteristics. The two results corroborate each other, allowing for more accurate classification and identification of ferromagnetic metals. The simulation used a CPU E-2276M (Intel Corporation, Santa Clara, CA, USA) and MATLAB 2023b, and the simulated data closely matched the measured results. Figures 10 and 11 above shows the relationship between key characteristics of pulsed eddy current differential signals and the conductivity and thickness of non-ferromagnetic metals. The results differ from those of ferromagnetic metals. It can be seen that, for non-ferromagnetic metals of the same thickness, as the conductivity increases, both the peak value and peak time also increase. When the conductivity of non-ferromagnetic metals is the same, increasing the thickness leads to higher peak values and longer peak times. However, the effect of conductivity on these two features is more significant than the effect of thickness, whether considering the peak value or the time to peak. This allows for more effective extraction of the conductivity and thickness characteristics of non-ferromagnetic metals. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material Name** & **Mn13** & **42 CrMo** & **Beryllium Bronze** & **4A01** & **5A05** & **6061** & **7075** \\ \hline Conductivity (MS/m) & 3.69 & 6.49 & 12.76 & 18.02 & 20.12 & 23.26 & 25.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Conductivity of common metal foreign objects in coal mines. Figure 8: PEC differential signals of alloy steel 42 CrMo with different thicknesses; (**a**) receiving coil differential voltage signals; (**b**) magnetic induction differential signals of Hall sensor. Figure 11: Three-dimensional surface plots between key characteristics of pulsed eddy current differential magnetic inductance signals and the conductivity and thickness of non-ferromagnetic metals; (**a**) peak magnetic inductance; (**b**) time to peak. Figure 10: Three-dimensional surface plots between key characteristics of pulsed eddy current differential voltage signals and the conductivity and thickness of non-ferromagnetic metals; (**a**) peak voltage; (**b**) time to peak. Figure 9: Relationship between key characteristic quantities of PEC differential signals and the thicknesses of alloy steel 42 CrMo; (**a**) peak voltage and its corresponding time to peak; (**b**) peak magnetic inductance and its corresponding time to peak.