## 2 Electrical Modeling of BTS in Free Space
### Electromagnetic Coupling Model  
In the BTS, the OAU and balise antenna are approximately rectangular. The specifications [24] constrain the electrical dimensions of the balise, with the antenna model measuring 390 mm in length and 200 mm in width. When the antenna perimeter \(l\) satisfies the relationship expressed in Equation (1) with the signal wavelength \(\lambda\), it can be assumed that the current distribution on both antennas is uniform, with equal amplitude and direction at all points. \[l<\frac{\lambda}{2\pi} \tag{1}\]  
Figure 2 illustrates a three-dimensional Cartesian coordinate system (\(xyz\)) established with the center of the balise designated as the origin \(O\). Rectangle \(ABCD\) depicts the balise,  
while rectangle \(EFGH\) represents the OAU. Point \(P\) indicates the electrical center of the OAU. The balise is installed in the middle of the track, the position is unchanged, and the OAU is moved. Rectangles \(E_{1}F_{1}G_{1}H_{1}\) and \(E_{2}F_{2}G_{2}H_{2}\) represent the relative positions of the OAU with respect to the balise at two distinct time points. The train speed velocity vector \(\overrightarrow{v}\) aligns with the positive direction of the \(Ox\) axis. The relative height between the OAU and the balise is \(h\). The magnetic flux density generated by the balise at any point \(P_{1}\) within the OAU can be expressed by Equation (2):  
\[B=\frac{\mu I}{4\pi}\oint_{C}\left(\frac{d\overrightarrow{l}\times \overrightarrow{r}}{r^{3}}\right) \tag{2}\]  
In the equation, \(I\) represents the current of the transponder antenna, \(C\) denotes the curve formed by rectangle \(ABCD\), \(Id\overrightarrow{l}\) represents the current element, and \(\overrightarrow{r}\) denotes the vector distance from any point \(P_{1}\) to the current element. Let the length of side \(AB\) of rectangle \(ABCD\) be \(2a\), the length of side \(CD\) be \(2b\), the length of side \(EF\) of rectangle \(EFGH\) be \(2c\), and the length of side \(GH\) be \(2d\). Point \(O_{1}\) is the projection of point \(P_{1}\) on plane \(ABCD\), and \(P_{1}O_{2}\) is perpendicular to line \(CD\). The magnetic flux density generated by the balise along the \(Z\)-axis at point \(P_{1}\) can be expressed by Equation (3):  
\[B_{z}=B_{z\text{AB}}+B_{z\text{BC}}+B_{z\text{CD}}+B_{z\text{DA}}=I\cdot N(a,b, l,u,x,y,h) \tag{3}\]  
\[N(a,b,h,\mu,x,y)=\frac{\mu}{4\pi}\left\{ \begin{array}{l}\frac{b+y}{(b+y)^{2 }+h^{2}}\cdot\left[\frac{a+x}{\sqrt{\left(a+x\right)^{2}+(b+y)^{2}+h^{2}}}+ \frac{a-x}{\sqrt{\left(a-x\right)^{2}+(b+y)^{2}+h^{2}}}\right]\\ +\frac{a+x}{\left(a+x\right)^{2}+h^{2}}\cdot\left[\frac{b+y}{\sqrt{\left(a+x \right)^{2}+(b+y)^{2}+h^{2}}}+\frac{b-y}{\sqrt{\left(a+x\right)^{2}+(b-y)^{2} +h^{2}}}\right]\\ +\frac{b-y}{\left(b-y\right)^{2}+h^{2}}\cdot\left[\frac{a+x}{\sqrt{\left(a+x \right)^{2}+(b-y)^{2}+h^{2}}}+\frac{a-x}{\sqrt{\left(a-x\right)^{2}+(b-y)^{2} +h^{2}}}\right]\\ +\frac{a-x}{\left(a-x\right)^{2}+h^{2}}\cdot\left[\frac{b+y}{\sqrt{\left(a-x \right)^{2}+(b+y)^{2}+h^{2}}}+\frac{b-y}{\sqrt{\left(a-x\right)^{2}+(b-y)^{2} +h^{2}}}\right]\end{array} \right\}\right\}\]  
Then, the magnetic flux received by the OAU \(\varphi\) can be expressed by Equation (4):  
\[\varphi=\int_{-c}^{c}\int_{-d}^{d}B_{z}dxdy=I\cdot\int_{-c}^{c}\int_{-d}^{d}N(a,b,h,\mu,x,y)dxdy \tag{4}\]  
Derive Equation (5) based on the relationship between the magnetic flux and mutual inductance. \[M(a,b,h,\mu,x,y)=\int_{-c}^{c}\int_{-d}^{d}N(a,b,h,\mu,x,y)dxdy \tag{5}\]  
Figure 2: Schematic diagram of electromagnetic coupling model between OAU and balise. When a train provides an appropriate telepowering signal, the voltage \(U\) of the received signal on the OAU can be approximated as the electromotive force induced at that position when the train is stationary [25].