## 1 Introduction
[16] calculated the transient response of the induced coil above an infinite conductive plate in the time domain by combining a semi-analytical modal expansion method with Euler integration scheme. [PERSON] et al. [17] proposed a new method called the Sturm-Liouville Global Function (SLGF) for calculating complex eigenvalues in eddy current problems, which can obtain results in a very short time. [PERSON] et al. [18] proposed a new Finite Region Eigenfunction Expansion (FREE) method, which can achieve planar dimension measurement of metallic plates by utilizing peak frequency features. [PERSON] et al. [19] used the lift-off point of intersection (LOI) in the pulsed eddy current response signal of ferromagnetic test pieces to detect their conductivity, magnetic permeability, and thickness. [PERSON] et al. [20] obtained the log-log method (LLM) by taking the logarithm of power fitting equation, where the linearity and slope can be used to detect metal plates with different conductivities. [PERSON] et al. [21] proposed a high conductivity resolution alloy classification method based on single-frequency eddy current (EC) sensing technology. [PERSON] et al. [22] proposed a Pulse Eddy Current Testing (PECT) method that combines multidimensional features and classification algorithms to identify bearing steel, rebar steel types, and heating defects. [PERSON] et al. [23] innovatively combined Eddy Current Testing (ECT) with deep learning, utilizing the nonlinear fitting ability of neural networks to distinguish between five types of metal materials. [PERSON] et al. [24] introduced the conductivity point of intersection (CPI) feature for noncoaxial T-R sensors in PEC, where the magnitude of the CPI is primarily related to the sensor parameters and sample thickness, and the thickness of the sample and the lift-off of sensor were precisely inverted using the improved Newton-Raphson method. The mining truck directly dumps the raw coal material into the ore receiving hopper. A large amount of raw coal material is transported into the crusher through the plate feeder. Due to the vibration of the plate feeder and large density of raw coal material, metallic foreign objects tend to sink to the bottom of the material during transportation. Therefore, in this paper, a rectangular-section coil and Hall sensor array are arranged beneath the outlet of the plate feeder to conduct quantitative detection and analysis of feeding metallic foreign objects, as shown in Figure 1. This study is focused on the SANDVIK dual-roll screening crusher CR610/14-35 (Sandvik Rock Processing Solutions, Stockholm, Sweden), with removable eagle-beak-shaped crusher teeth and a maximum discharge particle size of 300 mm, as shown in Figure 2.  
To address the complex issues of material type and size of feeding metallic foreign objects, and to improve the robustness of the detection system, a reflective double-coil  
Figure 1: Structure diagram of the open-pit coal mine crushing station (1—Mining Truck, 2—Ore Receiving Hopper, 3—Plate Feeder, 4—Protective Steel Structure, 5—Electrical Control Room, 6—Detection Probes Array, 7—Dual-roll Screening Crusher, and 8—Belt Conveyor). Figure 2: Structure diagram of the dual-roll screening crusher (1—Wear Plates for Front and Side Walls, 2—Crusher Tooth Rolls, 3—Drive Motor, 4—Hydraulic Coupling, 5—Reducer, and 6—Coupling). and Hall sensor array was designed to quantitatively classify and identify the characteristics of metallic foreign objects. A theoretical analysis model was established, internal design parameters were optimized, an engineering prototype was fabricated, and on-site experimental verification was conducted. The rest of this paper is organized as follows: Section 2 presents the frequency domain analytical expressions for the induced voltage and magnetic induction intensity of the metallic foreign object truncated pulsed eddy current field. Section 3 calculates the time-domain expressions for the induced voltage and magnetic induction intensity of the reflective field under actual square wave current signals.