## 2 Electrical Modeling of BTS in Free Space
### Circuit Modeling Analysis  
To optimize the transmission of uplink FSK signals and reduce the potential for signal distortion that can occur due to reflections, it is standard practice to implement impedance matching between the OAU and the balise. This impedance matching approach is crucial for maintaining the signal integrity and ensuring reliable communication. The equivalent circuit diagram representing the antenna configuration for uplink FSK signal transmission is shown in Figure 3. This circuit utilizes a \(\pi\)-type matching network, which is a common and effective method for achieving the desired impedance match [26]. The \(\pi\)-type matching network consists of passive components, allowing for the flexible adjustment of impedance characteristics to optimize signal transfer between the OAU and balise components of the system. In the uplink FSK signal chain of the BTS, \(U_{1}\) represents the equivalent voltage source at the transmitting port, \(I\) denotes the total current in the balise antenna, \(V_{1}\) signifies the voltage across the transmission channel, and \(V_{2}\) indicates the voltage across the load \(R_{4}\). \(Z_{int}\) stands for the input impedance of the transmitting port, while \(Z_{1}\) represents the transmitting antenna impedance. \(R_{1}\) is the equivalent resistance of the transmitting circuit. \(C_{1}\)-\(C_{3}\) and \(L_{3}\) are the capacitors and inductor, respectively, used for tuning and matching at the transmitting port. \(L_{1}\) represents the equivalent inductance of the balise antenna, \(R_{2}\) denotes the sum of the balise antenna's equivalent resistance and the series loss resistance, and \(I_{1}\) is the current flowing through the balise antenna. For the OAU port, \(Z_{2}\) represents the impedance of the receiving antenna, \(L_{2}\) represents the equivalent inductance of the receiving antenna, and \(I_{2}\) is the induced current in the antenna. \(R_{3}\) is the sum of the antenna's equivalent resistance and the series loss resistance. \(C_{4}\)-\(C_{6}\) and \(L_{4}\) are the capacitors and inductor for tuning and matching at the receiving port. \(C_{7}\)-\(C_{9}\) and \(L_{5}\) form a composite \(\pi\)-type impedance matching network, which can filter the telepowering signals of the downlink. \(C_{8}\) and \(C_{9}\) are variable capacitors to adapt to mismatches introduced during transmission under various environmental conditions. \(I_{4}\) is the current flowing through the load impedance \(R_{4}\). \(Z_{11}\) and \(Z_{22}\) represent the total impedance of the balise transmitting port and the OAU receiving port. \(M\) is the mutual inductance between the OAU and the balise. Figure 3: Equivalent circuit of uplink FSK signal. In Figure 3, \(Z_{1}\) represents the series branch impedance of the balise port, with \(Z_{11}\) and \(Z_{1}\) expressed by Equation (7). \(Z_{2}\) denotes the parallel load impedance of the OAU receiving port, with \(Z_{22}\) and \(Z_{2}\) expressed by Equation (8). \[\left\{ \begin{array}{l}Z_{11}=R_{1}+\frac{Z_{1}}{1+Z_{1}j\omega C_{1}}\\ Z_{1}=\frac{j\omega L_{3}}{1-\omega^{2}C_{3}I_{3}}+\frac{1}{j\omega C_{2}}+j \omega L_{1}+R_{2}\\ \end{array} \right. \tag{7}\]  
\[\left\{ \begin{array}{l}Z_{22}=\frac{Z_{2}^{\prime}+j\omega L_{5}}{1+\left(j \omega C_{9}+\frac{1}{k_{4}}\right)\left(Z_{2}^{\prime}+j\omega L_{5}\right)} \\ Z_{2}=\frac{Z_{2}^{\prime}+j\omega L_{5}}{1+j\omega C_{5}\left(Z_{2}^{\prime}+ j\omega L_{5}\right)}\end{array} \right. \tag{8}\]  
In Equation (8), \(Z_{2}^{\prime}\) is derived from the circuit structure and is expressed by Equation (9).