## 2 System Description
### RF AC-DC Converter with Input Impedance Matching
\[i_{in,\,\Pi}\left(t\right) =i_{in,\,I}\left(T_{I}\right)\cdot\cos\omega_{\Pi}\left(t-T_{I} \right)-\frac{V_{con}}{\omega_{\Pi}L_{1}}\sin\omega_{\Pi}\left(t-T_{I}\right)\] \[+\frac{V_{in}\cos\omega_{in}T_{I}}{\left(\omega_{in}^{2}-\omega_{ \Pi}^{2}\right)L_{1}}\left[\omega_{in}\cos\omega_{\Pi}\left(t-T_{I}\right)- \omega_{in}\cos\omega_{in}\left(t-T_{I}\right)\right] \tag{28}\] \[+\frac{V_{in}\sin\omega_{in}T_{I}}{\left(\omega_{in}^{2}-\omega_{ \Pi}^{2}\right)L_{1}}\left[\omega_{in}\sin\omega_{in}\left(t-T_{I}\right)- \omega_{\Pi}\sin\omega_{\Pi}\left(t-T_{I}\right)\right]\]  
where \(\omega_{\Pi}\) is the resonant angular frequency in Mode II and can be calculated by  
Figure 8:0  
Because the smoothing capacitance \(C_{5}\) is large, \(\omega_{\Pi}\) can be ignored. Then (28) is simplified to  
Figure 8:1  
In addition, the capacitor voltages \(v_{cur},v_{cv},v_{cp}\) and \(v_{cq}\) all remain unchanged in this mode. Mode II ends when the polarity of the input voltage changes from positive to negative. * Operation Mode III  
In this mode, the input current flows in two paths: \(C_{2}\to C_{1}\to L_{1}\) and \(C_{4}\to C_{3}\to L_{1}\), which are the opposite directions to those in Mode I. The input current in Mode III is expressed by  
Figure 8:2  
where \(T\) is the period of the input voltage. The capacitors \(C_{2}\), \(C_{3}\) and \(C_{1}\), \(C_{4}\) are gradually discharged and charged, respectively, by the input current. Once the capacitor voltages \(v_{cc}\), \(v_{cp}\) decrease to zero and \(v_{cur},v_{cq}\) increase to \(V_{con}\), the next operation mode starts. The duration of Mode III is the same to that of Mode I. * Operation Mode IV  
As the capacitor voltages \(v_{cv}\) and \(v_{cp}\) decrease to zero, the diodes \(D_{2}\) and \(D_{3}\) turn on. The path of the input current changes to \(D_{2}\to C_{5}\to D_{3}\to L_{1}\). The input current in Mode IV is expressed by  
Figure 8:3  
Mode IV ends when the polarity of the input voltage changes from negative to positive. Then the Mode I of the next working period starts. According to (25) and (30), the amplitude and phase of the input current are determined by the resonant parameters: the capacitance \(C\), inductance \(L_{1}\), the relation between the input voltage amplitude \(V_{in}\) and rectifier output voltage \(V_{con}\). Therefore, the intended input impedance of the AC-DC converter, which is the same as the output impedance of the matched acoustic-electric channel, can be obtained by designing these parameters properly. The design procedure contains eight steps as follows [26]. * Step 1: Derivation of the input voltage amplitude \(V_{in}\)  
Because the input impedance of the AC-DC converter is to be matched to the output impedance of the matched acoustic-electric channel, which is 50 \(\Omega\), the input current of the AC-DC converter will be in-phase with the input voltage. The amplitude of the input voltage, \(V_{in}\), can be calculated by Figure 8:4  
where \(Z_{in,\ AC-DC}=(50+j0)\)\(\Omega\) is the intended input impedance of the AC-DC converter, and \(P_{in}\) is the input power of the AC-DC converter. * Step 2: Decision of the resonant-type rectifier output voltage \(V_{con}\) When deciding the output voltage of the resonant-type rectifier, \(V_{con}\), two requirements should be met. One is that \(V_{con}\) must be within the rated input voltage range of the DC-DC converter. The other is that \(V_{con}\) must be higher than \(V_{in}\), because the resonant-type rectifier is a kind of step-up converter [25]. * Step 3: Calculation of the voltage ratio \(\alpha_{V}\) The voltage ratio \(\alpha_{V}\) is defined as Figure 8:5 (34) As mentioned above, the relation between \(V_{in}\) and \(V_{con}\) is one of the resonant parameters. \(\alpha_{V}\) would be remained constant through the operation of the AC-DC converter.