## 2 System Description
### RF AC-DC Converter with Input Impedance Matching  
The RF AC-DC converter with input impedance matching efficiently converts the RF output of the RX matching network to a regulated DC voltage. This is the main superiority of our UTMWPT system over previous ones. The RF AC-DC converter used in the system is constructed from a resonant-type rectifier and a DC-DC converter as shown in Figure 8. The resonant-type rectifier was originally proposed by [PERSON] et al. [25] as a high DC voltage generator. [PERSON] et al. improved it and developed an AC-DC converter from it for a magnetic resonance coupling WPT system [26; 27]. We simplified the topology proposed by [PERSON] et al., and redesigned it for our UTMWPT system. The resonant-type rectifier increases the conduction angle of the input current and the DC-DC converter stabilizes the operating point of the resonant-type rectifier [25; 26; 27]. These two features together achieve the input impedance matching of the AC-DC converter to the acoustic-electric channel. A resonant inductor \(L_{1}\) and four resonant capacitors \(C_{1-4}\) are added to the resonant-type rectifier as compared with conventional CF-DFBRs. As shown in Figure 9, one working period of the AC-DC converter can be divided into four modes according to the conduction states of the diodes. Note that the DC-DC converter is drawn as an ideal voltage source with voltage \(V_{con}\) in Figure 9 because the DC-DC converter with large capacitance \(C_{5}\) can be assumed as an ideal voltage source [26]. The operation modes of the AC-DC converter can be mathematically analyzed [25; 26; 27]. The input current starts to flow while the input voltage increases from zero. At the beginning of this mode, the initial current through the resonant inductor \(L_{1}\) is assumed as zero, the resonant capacitors \(C_{1}\) and \(C_{4}\) have been charged during the Mode III in the previous working period. The voltages on \(C_{1}\) and \(C_{4}\) prevent the diodes \(D_{1}\) and \(D_{4}\) from turning on. Therefore, the input current flows through two paths: \(L_{1}\to C_{1}\to C_{2}\) and \(L_{1}\to C_{3}\to C_{4}\). According to circuit theory, the input current \(i_{in,1}\) is derived as  
Figure 8: Schematic of the RF AC-DC converter used in the system. Figure 7: Matching network topology and component values: (a) TX matching network; (b) RX matching network. \[i_{in,\,\rm I}\left(t\right)=\frac{\omega_{in}\omega_{1}^{2}CV_{in}}{\left(\omega_{ in}^{2}-\omega_{1}^{2}\right)}\left(\cos\omega_{1}t-\cos\omega_{in}t\right)+ \omega_{1}CV_{con}\sin\omega_{1}t \tag{25}\]  
where \(\omega_{in}\) is the angular frequency of the input voltage \(v_{in}\), \(V_{in}\) is the amplitude of \(v_{in}\), \(C=C_{1}=C_{2}=C_{3}=C_{4}\) is the resonant capacitance, and \(\omega_{1}\) is the resonant angular frequency in Mode I, which can be calculated by  
\[\omega_{1}=\frac{1}{\sqrt{L_{1}C}} \tag{26}\]  
The resonant capacitors \(C_{1}\), \(C_{4}\) and \(C_{2}\), \(C_{3}\) are gradually discharged and charged, respectively, by the input current in Mode I. Once the capacitor voltages \(v_{cu}\), \(v_{cq}\) decrease to zero and \(v_{cv}\), \(v_{cp}\) increase to the voltage of voltage source, \(V_{con}\), the next operation mode starts. The duration of Mode I, \(T_{\rm I}\), is equal to the discharging time of the capacitors \(C_{1}\), \(C_{4}\), and can be derived with numerical analytical approach from  
\[\frac{V_{con}}{2}\left(1+\cos\omega_{1}T_{\rm I}\right)+\frac{\omega_{1}V_{in }}{2\left(\omega_{in}^{2}-\omega_{1}^{2}\right)}\left(\omega_{1}\sin\omega_{ in}T_{1}-\omega_{in}\sin\omega_{1}T_{1}\right)=0 \tag{27}\]  
\(\bullet\) Operation Mode II  
As the capacitor voltages \(v_{cu}\) and \(v_{cq}\) decrease to zero, the diodes \(D_{1}\) and \(D_{4}\) turn on. The path of the input current changes to \(L_{1}\to D_{1}\to C_{5}\to D_{4}\). The input current charges the smoothing capacitor \(C_{5}\) and can be calculated by  
Figure 9: Operation modes of the AC-DC converter: (**a**) Mode I; (**b**) Mode II; (**c**) Mode III; (**d**) Mode IV.