## 4 Theoretical Simulation and Experimental Verification
### Single-Probe Testing Experiment  
In order to better observe the interaction between the rectangular cross-section coaxial excitation, receiving coil, Hall sensor, and the moving metal foreign object, a single-probe testing experimental platform is built, as shown in Figure 6. The signal generator, with a power amplifier, outputs a high-level square wave signal with a 12 V amplitude and a 50% duty cycle. The square wave signal is applied to the excitation coil of the single probe, and the output signals from the receiving coil and Hall sensor are sent through a signal conditioning module to the data acquisition card. The signal conditioning module is a second-order low-pass filter circuit, characterized by fast attenuation and strong filtering effects. The data acquisition card used is Smacq USB-4650 (Smacq Technologies Co., Ltd., Beijing, China), a high-speed acquisition card that transmits data via USB. It has 8 synchronized analog input channels with a sampling rate of 500 kSa/s, a resolution of 16 bits, and an input voltage amplitude of \(\pm\)10 V. It is equipped with a DAQUS 2.0 driver and can be programmed using LabVIEW 2021 or MATLAB 2023b software. The PC and data acquisition card are connected via USB. The system block diagram is shown below. Figure 4: Typical PEC signals with non-ferromagnetic metals; (**a**) receiving coil voltage signals; (**b**) magnetic induction signals of Hall sensor. Figure 5: Typical PEC signals with ferromagnetic metals; (**a**) receiving coil voltage signals; (**b**) magnetic induction signals of Hall sensor. The PC is connected to the controller to manipulate the three-dimensional moving platform. The three-dimensional moving platform is responsible for the horizontal and vertical positioning of the single probe and the uniform movement of the metal foreign object. The single-probe coil is a cylindrical hollow coil precisely wound with copper wire, and the Hall sensor is placed at the center of the single probe, as shown in Figure 7a. The Hall sensor selected is a proportional linear Hall effect magnetic sensor with a high-speed detection bandwidth of 20 kHz, a magnetic sensitivity of 51.5 mV/mT, and a range of \(\pm\)42 mT. In the experiment, the sensitivity of the coil and Hall sensor is affected by temperature [31]. The resistance of the copper hollow coil is proportional to temperature. As the temperature of the Hall sensor rises from \(-40\)\({}^{\circ}\)C to 120 \({}^{\circ}\)C, its magnetic sensitivity increases from 46 mV/mT to 56 mV/mT. Therefore, data collection should only be performed after the temperature of the single-probe coil has stabilized. The parameters of the coil used in the experiment are shown in Table 1.  
The sample dimensions are determined based on the maximum discharge particle size of the SANDVIK dual-roll screening crusher CR610/14-35, which is 300 mm. The length \begin{table}
\begin{tabular}{c c c} \hline \hline
**Coil Parameters** & **Receiving Coil** & **Excitation Coil** \\ \hline Inner radius (mm) & 2.6 & 7.0 \\ Outer radius (mm) & 5.0 & 12.0 \\ Length (mm) & 3.2 & 7.1 \\ Number of turns & 100 & 745 \\ Liftoff (mm) (same as Hall sensor) & 5 & 5 \\ Inductance value (mH) & 0.06 & 9.1 \\ \(\tau_{0}\) (\(\mu\)s) & - & 250 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Single-probe parameters. Figure 6: Single-probe testing experiment; (**a**) experimental platform; (**b**) block diagram of the system. Figure 7: Detailed view of single-probe and samples; (**a**) bottom view of the single-probe; (**b**) seven test samples for experiment. and width are 300 mm \(\times\) 300 mm, with thicknesses chosen as 3 mm, 5 mm, 7 mm, 10 mm, and 12 mm. The materials of the samples are selected from common metal foreign objects found in coal mine crushers, such as alloy steel Mn13, 42 CrMo, copper alloy beryllium bronze, aluminum alloys 4A01, 5A05, 6061, and 7075, as shown in Figure 7b. The relative magnetic permeability of the ferromagnetic alloy steel 42 CrMo is taken as 60, while the relative magnetic permeability of other metals is 1.