## 2 Theoretical Investigation of Monitoring Models
### Analysis of the Spatial Magnetic Field Distribution of Permanent Magnets  
Consider a rectangular permanent magnet with dimensions a \(\times\) b \(\times\) c positioned in a three-dimensional Cartesian coordinate system, where the magnet is magnetized unidirectionally along the \(z\)-axis to a state of uniform saturation. At this point, the magnetization vector \(M\) within the magnet can be approximated as a constant vector. Based on [PERSON]'s molecular circulation theory, the magnetic field at any point in the external space is generated by the collective contributions of all of the loop molecular currents within the permanent magnet. After uniform magnetization, the molecular current effects inside the magnet neutralize one another and, thus, do not contribute to the magnetic field. From a macroscopic perspective, it can be approximated that only the surface molecular currents are responsible for generating the external magnetic field. Furthermore, as the magnetization along the z-direction reaches full saturation, the molecular current effects on the upper and lower surfaces of the rectangular permanent magnet also cancel one another. Consequently, the magnetic field at any point in the external space is primarily generated by the closed current loop ABCDA on the side surface of the magnet, as illustrated in Figure 2.  
Figure 1: Sensor structure and monitoring schematic. Figure 2: Molecular circulation model of the rectangular permanent magnet. Assume that the current intensity in the loop is \(I\) and the current density is \(J\). The magnetic field source point of the permanent magnet is located at (0, 0, 0), and the coordinates of any point in the external space of the magnet are P (x, y, z). According to [PERSON]'s circuital law and the Biot-Savart law, the magnitude of the magnetic field at a given point in space produced by the current loop A\({}^{\prime}\)B\({}^{\prime}\)C\({}^{\prime}\)D\({}^{\prime}\)A\({}^{\prime}\) is as follows:  
\[B_{x(A^{\prime}\text{B}^{\prime}\text{C}^{\prime}\text{D}^{\prime}A^{\prime})}= B_{x(A^{\prime}\text{B}^{\prime})}+B_{x(\text{B}^{\prime}\text{C}^{\prime})}+B_{x( \text{C}^{\prime}\text{D}^{\prime})}+B_{x(\text{D}^{\prime}A^{\prime})} \tag{1}\]  
Using segment A\({}^{\prime}\)B\({}^{\prime}\) as an example, consider a micro-segment of the current element \(I\)d\(I\) (\(I=J\)d\(z_{0}\), \(|\)d\(I|=\) dy); the magnetic field it generates at any spatial point is as follows:  
\[\text{d}B_{x(A^{\prime}\text{B}^{\prime})}=\frac{\mu_{0}}{4\pi}\frac{\text{d} I\times\mathbf{e_{r}}}{r^{2}}=\frac{\mu_{0}\times J\text{d}z_{0}}{4\pi}\frac{ \text{d}I\times\mathbf{e_{r}}}{r^{2}} \tag{2}\]  
where \(\mu_{0}\) is the vacuum permeability, with a value of 4\(\pi\)\(\times\)\(10^{-7}\) (H/m); \(I\)d\(I\) denotes the length vector of the current element; \(\mathbf{e_{r}}\) represents the unit vector in the direction of the radius vector \(\mathbf{r}\); and \(r\) denotes the distance from point P to the current element. Based on the monitoring principle outlined in Section 2.1, the probe detects the magnetic induction density in the z-direction, denoted as \(B_{x}\). Consequently, the analytical expression of the z-direction component is the focus of the analysis. For segment AB, we have:  
\[B_{x(\text{AB})}=\frac{\mu_{0}J}{4\pi}\int_{0}^{\text{h}}\int_{0}^{b}\frac{a- x}{\left[\left(x-a\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2} \right]^{3/2}}\text{d}y\text{d}z_{0} \tag{3}\]  
where:  
\[K=\frac{\mu_{0}J}{4\pi} \tag{4}\]  
\[\psi_{i}(\psi_{1},\psi_{2},\psi_{3})=\frac{\psi_{i}}{\left(\psi_{1}{}^{2}+\psi _{2}{}^{2}+\psi_{3}{}^{2}\right)^{3/2}}(i=1,2,3) \tag{5}\]  
Then,  
\[B_{x(\text{AB})}=-K\int_{0}^{\text{h}}\int_{0}^{b}\psi_{1}(x-a,y-y_{0},z-z_{0 })\text{d}y_{0}\text{d}z_{0} \tag{6}\]  
Similarly, the magnetic field intensities excited by segments BC, CD, and DA can be calculated.