## 2 Electrical Modeling of BTS in Free Space
### Circuit Modeling Analysis
\[Z_{2}^{\prime}=\frac{\left(j\omega L_{2}+\frac{j\omega L_{4}}{1-\omega^{2}C_{ 4}L_{4}}+\frac{1}{j\omega C_{5}}+R_{3}\right)\left(j\omega C_{6}+j\omega C_{7} \right)+1}{\left(j\omega C_{7}+j\omega C_{8}\right)\left(1-\omega^{2}C_{6}L_{ 2}+\frac{\omega^{2}C_{6}L_{4}}{\omega^{2}C_{4}L_{4}-1}+\frac{C_{6}}{C_{5}}+R_{ 3}j\omega C_{6}\right)-\omega^{2}C_{7}C_{8}\left(j\omega L_{2}+\frac{j\omega L _{4}}{1-\omega^{2}C_{4}L_{4}}+\frac{1}{j\omega C_{5}}+R_{3}\right)} \tag{9}\]  
According to KCL, the relationship between currents \(I\) and \(I_{1}\) can be expressed as  
\[I=\left(1+Z_{1}j\omega C_{1}+\frac{j\omega^{3}M^{2}C_{1}}{Z_{22}}\right)\,I_{1} \tag{10}\]  
Combining Equations (7), (8) and (10), the equivalent circuit voltage equation for the OAU and the balise in Figure 3 can be expressed as:  
\[\left\{ \begin{array}{l}\left(R_{1}+R_{1}Z_{1}j\omega C_{1}+\frac{j\omega^{3 }M^{2}R_{1}C_{1}}{Z_{22}}+Z_{1}\right)I_{1}-j\omega MI_{2}=U_{1}\\ -j\omega MI_{1}+\frac{\left(Z_{2}^{\prime}+j\omega L_{5}\right)I_{2}}{1+\left( j\omega C_{9}+\frac{1}{k_{4}}\right)\left(Z_{2}^{\prime}+j\omega L_{5}\right)}=0 \end{array} \right. \tag{11}\]  
The parameters of the inductors and capacitors in the circuit are designed based on the operating resonant frequency of the BTS. Given that the antenna exhibits inductive behavior at the operating frequency, the inductance values of antennas \(L_{1}\) and \(L_{2}\) should be considered during impedance matching. When the antenna frequency is 4.234 MHz, the sum of the imaginary part at the transmitting port \(j\omega R_{1}Z_{1}C_{1}+j\omega^{3}M^{2}R_{1}C_{1}Z_{22}^{-1}+Z_{1}\) is approximated as 0, and the real part is approximated as \(R_{2}\). For the receiving port, the sum of the imaginary part \(\left(Z_{2}^{\prime}+j\omega L_{5}\right)\left[1+\left(j\omega C_{9}+R_{4}^{-1 }\right)\left(Z_{2}^{\prime}+j\omega L_{5}\right)\right]^{-1}\) is approximated as 0, and the real part is approximated as \(R_{3}+R_{4}\). Therefore, the relationship between voltages \(V_{1}\), \(V_{2}\), and impedance \(Z_{in}\) can be expressed as:  
\[\left\{ \begin{array}{l}V_{1}=I_{1}R_{2}-j\omega MI_{3}\\ V_{2}=-I_{3}R_{3}+j\omega MI_{1}\end{array} \right. \tag{12}\]  
\[Z_{in}=R_{2}-j\omega M\frac{I_{3}}{I_{1}} \tag{13}\]  
When the balise transmitting power is \(P\), the current \(I_{1}\) in the balise and the current \(I_{3}\) in the load of the OAU satisfy the relationship expressed in Equation (14). \[\left\{ \begin{array}{l}P=\frac{U_{1}^{2}}{R_{1}}\\ I_{1}=\frac{U_{1}}{R_{1}+Z_{in}}\\ I_{3}=\frac{V_{2}}{k_{4}}\end{array} \right. \tag{14}\]Equations (12) and (14) can be respectively substituted into Equation (13) to derive the relationship between mutual inductance \(M\) and load voltage \(V_{2}\), so:  
\[j\omega M=\frac{V_{2}+I_{2}R_{3}}{I_{1}}=\frac{V_{2}(R_{1}+Z_{in})\Big{(}1+\frac {R_{1}}{R_{4}}\Big{)}}{\sqrt{R_{1}P}} \tag{15}\]  
Substituting Equation (5) into Equation (15) enables the derivation of the relationship expression for the voltage \(V_{2}\) across the load. \[V_{2}=\frac{j\omega\sqrt{R_{1}P}f_{-c}^{c}\int_{-d}^{d}N(a,b,h,\mu,x,y)dxdy}{( R_{1}+Z_{in})\Big{(}1+\frac{R_{3}}{R_{4}}\Big{)}} \tag{16}\]  
From Equation (16), it can be observed that after proper circuit matching, the voltage of the load is influenced by the dimensions of the OAU and the balise, their relative spatial positions, and the input power of the balise. However, it is independent of the reactive components in the matching circuit. The load at the receiving port of the OAU needs to effectively utilize the available power transmitted from the balise. In order to describe the transmission characteristics of the BTS, the insertion loss parameter is used. The calculation formula is as follows:  
\[S_{21}(dB)=20\ddagger g_{10}\bigg{|}\frac{\omega\int_{-c}^{c}\int_{-d}^{d}N(a,b,h,\mu,x,y)dxdy}{R_{1}}\bigg{|}+20\ddagger g_{10}(\alpha) \tag{17}\]  
where \(\alpha=\Big{(}1+Z_{in}R_{1}^{-1}\Big{)}\Big{(}1+R_{3}R_{4}^{-1}\Big{)}\).