## 2 System Design and Modeling  
According to the introduction in the previous section, the theoretical model for the air layer, metallic foreign object layer, and coal layer is established based on the actual site conditions. Figure 3a shows the side view of the truncated region eddy current field model of the single-turn excitation coil and feeding metallic foreign objects. First, a cylindrical coordinate system or \(\varphi x\) is established, dividing the field into four regions: I, II, 1, and 2. These regions represent the area where the single-turn coil is far from the detection target along the \(x\)-axis, the area where the single-turn coil is close to the detection target along the \(x\)-axis, the metallic foreign object region, and the feeding coal layer region, respectively. Regions I and II can be combined into region 0. As shown in the diagram, the radius of the single-turn coil is \(r_{0}\), the distance between the coil and the metallic foreign object layer is \(x_{0}\), the radius of the truncated region is h, and the thickness of the metallic foreign object layer is d.  
Next, the magnetic vector potential \(\overset{\rightarrow}{A}\) is taken as the object of solution. Since the cylindrical coil with a rectangular cross-section is axially symmetric about the \(x\)-axis, the magnetic vector potential only has an azimuthal component \(A_{\varphi}(r,x)\). Based on the Laplace operators for scalars and vectors in cylindrical coordinates [25; 26], and using [PERSON]'s  
Figure 3: Side view of the truncated region of (**a**) the single-turn coil, and (**b**) the rectangular cross-section coaxial excitation and receiving coils with Hall sensors. function [27], the expression for the azimuthal component of the magnetic vector potential can be derived from [PERSON]'s equations:  
\[\frac{\partial A_{\varphi}}{\partial r}+\frac{\partial^{2}A_{\varphi}}{\partial x ^{2}}-\frac{A_{\varphi}}{r^{2}}-\mu\sigma\frac{\partial A_{\varphi}}{\partial t }+\mu I\delta(r-r_{0})\delta(x-x_{0})=0 \tag{1}\]  
where \(\sigma\) is the electrical conductivity. \(\mu=\mu_{0}\mu_{r}\), \(\mu\) is the magnetic permeability, with \(\mu_{0}\) being the permeability of free space and \(\mu_{r}\) being the relative magnetic permeability, and \(I\) is the amplitude of the excitation current. Then, taking \(t\) as the independent variable, the Fourier transform of the azimuthal component of the magnetic vector potential is performed to convert the time domain to the frequency domain, denoted as:  
\[\widetilde{A}=\int_{-\infty}^{+\infty}A_{\varphi}(t)\mathrm{e}^{-j\omega t} \mathrm{d}t \tag{2}\]  
Taking \(r\) as the independent variable, the Hankel transform is performed on \(\widetilde{A}(r,x)\) over the interval [0, h], denoted as:  
\[\widetilde{A}_{\mathrm{H}}(\lambda_{i},x)=\int_{0}^{h}\widetilde{A}(r,x)r \mathrm{J}_{1}(\lambda_{i}r)\mathrm{d}r \tag{3}\]  
where \(\lambda_{i}\) is the \(i\)-th positive root of \(\mathrm{J}_{1}(\lambda_{i}h)=0\). Using integration by parts and the general solution of the first-order Bessel function, the expression becomes:  
\[-\lambda_{i}^{2}\widetilde{A}_{\mathrm{H}}(\lambda_{i},x)+\frac{\partial^{2} \widetilde{A}_{\mathrm{H}}(\lambda_{i},x)}{\partial x^{2}}-s^{2}\widetilde{A} _{\mathrm{H}}(\lambda_{i},x)+\mu\widetilde{I}r_{0}\mathrm{J}_{1}(\lambda_{i} r_{0})\delta(x-x_{0})=0 \tag{4}\]  
where in region 0, \(s^{2}=0\), and in regions 1 and 2, \(s^{2}=j\omega\mu\sigma\). According to the general solution of ordinary differential equations, and satisfying the conditions that \(C(\lambda_{i})\) approaches 0 as \(x\rightarrow+\infty\) and \(\mathrm{D}(\lambda_{i})\) approaches 0 as \(x\rightarrow-\infty\), we obtain:  
\[\widetilde{A}_{\mathrm{H}}(\lambda_{i},x)=\sum_{i=1}^{\infty}\left[C(\lambda _{i})e^{\gamma_{k}x}+D(\lambda_{i})e^{-\gamma_{k}x}\right] \tag{5}\]  
where \(\gamma_{k}=\sqrt{\lambda_{i}^{2}+j\omega\mu_{0}\mu_{r}\alpha_{k}}\) (\(k=0\), 1, 2).