## 3 DT Model of the Electromagnetic Coupling Process of the BTS
### Modeling and Analysis of Matching Circuit
\begin{array}{l}X_{s}=Q_{r}R_{2}\\ X_{p}=\frac{Z_{\text{total}}}{Q_{r}}\end{array} \right., \tag{6}\]  
According to the resonant frequency of the designed circuit, the parameters of the corresponding inductor and capacitor components can be calculated. \[\left\{ \begin{array}{l}L_{s}=\frac{X_{s}}{\omega}=\frac{Q_{r}R_{2}}{2\pi f_ {0}}\\ C_{p}=\frac{1}{\omega X_{p}}=\frac{Q_{r}}{2\pi f_{0}Z_{\text{total}}}\end{array}  \right., \tag{7}\]  
In Equation (7), \(L_{s}\) represents the inductance in the series circuit, and \(C_{p}\) represents the capacitance in parallel with the output impedance (\(C_{p}=C_{4}\)). Because the antenna is inductive, its inductance \(L_{2}\) is not equal to the calculated value of the above impedance-matched series inductance \(L_{s}\). Therefore, the inductance of the antenna itself can be used for impedance matching. For the remaining inductance \(\Delta L=L_{2}-L_{S}\), when \(\Delta L\) is greater than 0, the imaginary part is matched to 0 through series capacitance, so \(C_{3}\) can be calculated as follows:  
\[C_{3}=\frac{1}{\omega^{2}(L_{2}-L_{s})}=\frac{1}{2\pi f_{0}(2\pi f_{0}L_{2}-Q _{r}R_{2})}. \tag{8}\]  
The double-mesh equation of the entire equivalent circuit can be expressed as:  
\[\left\{ \begin{array}{l}\left(R_{1}+j\omega L_{1}+\frac{1}{j\omega C_{1}}+ \frac{Z_{0}}{j\omega C_{2}Z_{01}+1}\right)i_{1}-j\omega Mi_{2}=U_{0}\\ j\omega Mi_{1}-\left(R_{2}+j\omega L_{2}+\frac{1}{j\omega C_{3}}+\frac{Z_{ \text{total}}}{j\omega C_{4}Z_{\text{total}}+1}\right)i_{2}=0\end{array} \right.. \tag{9}\]  
When the onboard receiving antenna is tuned at the center frequency of 4.234 MHz, the sum of the imaginary parts of \(j\omega L_{2}+\frac{1}{j\omega C_{3}}+\frac{Z_{\text{total}}}{j\omega C_{4}Z_{ \text{total}}+1}\) should be 0, and the real part of \(\frac{Z_{\text{total}}}{j\omega C_{4}Z_{\text{total}}+1}\) is equal to \(R_{2}\), so:  
\[i_{2}=\frac{j\omega Mi_{1}}{2R_{2}}. \tag{10}\]  
When \(\Delta L\) is less than 0, series inductance can be used to increase the inductance of the series circuit, so as to offset the capacitive reactance of the parallel circuit to achieve a matching state. At this time, the resonant current \(i_{2}\)' of the onboard receiving antenna at the center frequency can be calculated by the same way:  
\[i_{2}\)' = \(i_{2}=\frac{j\omega Mi_{1}}{2R_{2}}, \tag{11}\]  
Therefore, the port voltage of shunt capacitor \(C_{4}\) is:  
\[U_{Z_{\text{total}}}=\frac{Z_{2}-j\omega C_{4}Z_{\text{total}}^{2}}{2R_{2}(1+ \omega^{2}C_{4}{}^{2}Z_{\text{total}}{}^{2})}\times j\omega Mi_{1}. \tag{12}\]  
Combining (1), (5), and (12), the port voltage of the load impedance \(Z_{4}\) after matching and filtering can be obtained:  
\[U_{Z_{4}}=\frac{2A_{g}f_{0}u_{0}i(Q_{r}+j)}{\cdot\frac{\sin(ax_{3})}{\alpha} \cdot\frac{\sin(y_{0})}{\beta}}\cdot\frac{\sin(x_{1})}{\beta}\cdot\frac{\sin( y_{1})}{\beta}dadb. \tag{13}\]where \(A_{\rm z}=\frac{Z_{3}Z_{\rm d}}{(Z_{2}Z_{3}+Z_{2}Z_{4}+Z_{3}Z_{4})}\). According to (13), it can be seen that the voltage of the uplink signal received by the onboard receiving antenna after good impedance matching is related to the quality factor of the antenna and has nothing to do with the reactance components in the matching circuit. The voltage is also affected by the size of the balise transmitting antenna and the onboard receiving antenna, the relative spatial position between the antennas, and the current flowing through the transmitting antenna. The received uplink signal voltage is proportional to the current intensity on the balise transmitting antenna.