## 3 DT Model of the Electromagnetic Coupling Process of the BTS
### Modeling and Analysis of Matching Circuit  
In order to ensure the transmission efficiency of telegram signals and reduce the signal distortion caused by reflection and group delay, the antennas of BTS are usually impedance matched. The equivalent circuit of the uplink signal transmission antennas using an L-type matching circuit is shown in Figure 5.  
Figure 4: Schematic diagram of mutual inductance model between balise antenna and onboard antenna. For the uplink transmitting antenna of the balise, \(Z_{1}\) is the input impedance, \(C_{1}\) and \(C_{2}\) are the series and parallel tuning capacitors, respectively, \(R_{1}\) represents the sum of the equivalent resistance of the transmitting antenna and the series loss resistance, and \(L_{1}\) represents the equivalent inductance of the transmitting antenna. \(i_{1}\) is the current flowing through the balise transmitting antenna. For the onboard receiving antenna, \(L_{2}\) represents the equivalent inductance of the onboard receiving antenna. \(i_{2}\) is the induced current on the onboard receiving antenna. \(R_{2}\) represents the sum of the equivalent resistance and the series loss resistance of the onboard receiving antenna. \(C_{3}\) and \(C_{4}\) are the series and parallel tuning capacitors, respectively. \(Z_{2}\) and \(Z_{3}\) are resistors for impedance matching, \(Z_{4}\) is the load impedance. \(L_{3}\) and \(C_{5}\) are used to filter the tele-powering signal. \(M\) is the mutual inductance between the balise transmitting antenna and the onboard receiving antenna. This article focuses on the analysis of the equivalent circuit of the onboard receiving antenna. The purpose of the parallel tuning capacitor \(C_{4}\) is to reduce the relatively large impedance connected in parallel on the right side so that the real part is equal to \(R_{2}\). The quality factor \(Q_{r}\) of the receiving antenna can be changed by adjusting the resistance of \(R_{2}\). The role of the series tuning capacitor \(C_{3}\) is to resonate with the antenna equivalent inductance \(L_{2}\) and the parallel capacitor \(C_{4}\), thereby canceling the reactance part of the circuit. Since the onboard receiving antenna usually uses a single-turn rectangular planar loop antenna, its equivalent capacitance can be ignored, and we can calculate the equivalent inductance \(L_{2}\)[24] as follows:  
\[L_{2}=\frac{\mu_{0}}{\pi}\left\{ \begin{array}{c}l_{\rm a}\cdot\ln\left[ \frac{2l_{\rm a}l_{\rm b}}{d_{\rm c}\cdot\left(l_{\rm a}+\sqrt{l_{\rm a}^{2}+ b_{\rm b}^{2}}\right)}\right]+l_{\rm b}\cdot\ln\left[\frac{2l_{\rm c}l_{\rm b}}{d_{ \rm c}\cdot\left(l_{\rm b}+\sqrt{l_{\rm a}^{2}+l_{\rm b}^{2}}\right)}\right]\\ -2\left[l_{\rm a}+l_{\rm b}-\sqrt{l_{\rm a}^{2}+l_{\rm b}^{2}}\right]+\frac{l_ {\rm a}+l_{\rm b}}{4}\end{array} \right\}, \tag{2}\]  
where \(l_{\rm a}\) and \(l_{\rm b}\) are the sizes of the long and short sides of the rectangular planar loop antenna respectively, and \(d_{\rm c}\) is the diameter of the equivalent cylindrical conductor with the same cross-sectional perimeter of the antenna conductor. \[d_{\rm c}=\frac{2(t+w)}{\pi}, \tag{3}\]  
where \(t\) is the thickness of the rectangular loop antenna conductor and \(w\) is the width of the conductor. The quality factor \(Q_{\rm r}\) of the matching circuit can be calculated as follows:  
\[Q_{\rm r}=\sqrt{\frac{Z_{\rm total}}{R_{2}}-1}, \tag{4}\]  
where \(Z_{\rm total}\) can be calculated according to the circuit structure:  
\[Z_{\rm total}=Z_{2}+\frac{Z_{3}\left(j\omega L_{3}+Z_{4}+\frac{1}{j\omega C_{ 5}}\right)}{j\omega L_{3}+Z_{4}+\frac{1}{j\omega C_{5}}+Z_{3}}. \tag{5}\]  
Figure 5: Equivalent circuit of uplink signal transmission antennas. Therefore, once the quality factor \(Q_{r}\) is defined, the total resistance \(R_{2}\) of the series circuit can be calculated. The reactance value \(X_{s}\) of the matching circuit in series with the antenna and the reactance value \(X_{p}\) of the matching circuit in parallel with the antenna can be expressed as:  
\[\left\{