## 2 System Description
### Simultaneous Conjugate Impedance Matching  
To minimize unwanted electrical reflections at both ports of the acoustic-electric channel, electrical impedance matching networks are applied. The matching networks simultaneously conjugate match the impedance of the ports to 50 \(\Omega\), which is a widely used characteristic impedance value in RF systems. The theory of simultaneous conjugate matching was firstly presented by [PERSON] [23] and firstly adopted in a UTMWPT system by [PERSON] et al. [17] The matching networks were designed according to the scattering matrix (S-parameters) of the acoustic-electric channel. For ease of analysis, a simplified model of the impedance matched acoustic-electric channel is shown in Figure 3. Note that the RF PA together with the CW signal source is simplified to a voltage source with an output impedance, \(Z_{S}\), the RF AC-DC converter and load circuit are simplified to a resistor, \(Z_{L}\). In addition, \(Z_{S}\) and \(Z_{L}\) both equal 50 \(\Omega\). Figure 1: Block diagram of the ultrasonic through-metal-wall power transfer system. The scattering matrix of the acoustic-electric channel, \(\mathbf{S}\), was measured using an AV36580A vector network analyzer (CETC41, Qingdao, Shandong, China). The TX transducer and RX transducer were connected to Port 1 and Port 2 of the network analyzer, respectively. The transducer power gain of the acoustic-electric channel, \(G_{T}\), is defined as the ratio of the power delivered to its load to the power available from its source. \(G_{T}\) can be calculated as  
Figure 2: Impedance and quality factor measurement results of the transducers (frequency range: 0.5–1.5 MHz): (**a**) Impedance modulus (yellow) and quality factor (cyan) of the TX (transmitter) transducer; (**b**) Impedance modulus (yellow) and quality factor (cyan) of the RX (receiver) transducer. \[G_{T}=\left|S_{21}\right|^{2} \tag{1}\]  
where \(S_{21}\) is one of the four elements of \(S\). The power transfer efficiency of the channel, \(\eta_{ch}\), is defined as a percentage and can be calculated directly from \(G_{T}\) as  
\[\eta_{ch}=G_{T}\times 100\% \tag{2}\]  
The impedance matrix of the channel, \(\mathbf{Z}\), can be derived from \(S\) as [24]  
\[Z_{11}=Z_{0}\frac{\left(1+S_{11}\right)\left(1-S_{22}\right)+S_{12}S_{21}}{ \left(1-S_{11}\right)\left(1-S_{22}\right)-S_{12}S_{21}} \tag{3}\]  
\[Z_{12}=Z_{0}\frac{2S_{12}}{\left(1-S_{11}\right)\left(1-S_{22}\right)-S_{12}S_ {21}} \tag{4}\]  
\[Z_{21}=Z_{0}\frac{2S_{21}}{\left(1-S_{11}\right)\left(1-S_{22}\right)-S_{12}S _{21}} \tag{5}\]  
\[Z_{22}=Z_{0}\frac{\left(1-S_{11}\right)\left(1+S_{22}\right)+S_{12}S_{21}}{ \left(1-S_{11}\right)\left(1-S_{22}\right)-S_{12}S_{21}} \tag{6}\]  
where \(Z_{0}\) is the reference impedance of the network analyzer and it was set to 50 \(\Omega\). Then the simultaneously matched source and load impedances, \(Z_{mS}\) and \(Z_{mL}\), can be calculated, respectively, by  
\[Z_{mS}=\frac{\alpha_{1}\pm\sqrt{\Delta}}{2\text{Re}\left\{Z_{22}\right\}} \tag{7}\]  
Figure 1:0  
where  
Figure 1:1  
Figure 1:2  
Figure 1:3  
where the \(\text{Re}\left\{Z\right\}\) and \(\text{Im}\left\{Z\right\}\) notations represent the real and imaginary parts of a complex impedance \(Z\). Care must be taken to select the appropriate signs of (7) and (8) to ensure that both \(Z_{mS}\) and \(Z_{mL}\) are positive. The impedances seen looking into Ports 1 and 2 under the conjugate matching condition,  
Figure 1:4 Simplified model of the impedance matched acoustic-electric channel. \(Z_{m1}\) and \(Z_{m2}\), would simply be the complex conjugate of the matched source and load impedances, respectively, i.e.,  
Figure 1:5  
Figure 1:6  
The scattering matrix of the simultaneously conjugate matched channel, \(S_{m}\), can be calculated by  
Figure 1:7  
where  
\[\mathbf{Z}_{m}=\left( \begin{array}{cc}Z_{mS}&0\\ 0&Z_{mL}\end{array} \right) \tag{15}\]  
\[\mathbf{F}=\left(