## 2 Materials and Methods
### Method I with Constant Frequency Varying Phase-Shift Modulation  
Method I is based on phase-shifted modulation with a fixed switching frequency. Generally, the two bridges are operated actively with a phase-shift between them with the typical steady-state waveforms shown in Figure 3. The two voltage sources \(v_{\text{p}}\) and \(v_{\text{s}}\) are HF ac voltages generated by the two full-bridge converters, respectively. It is assumed  
Figure 1: Variations of voltage, current, and power in a typical charging process.  
Figure 2: The circuit diagram of a DBRC converter.  
that those two voltages are square-wave voltages with 50% duty cycle, and they can be expressed as  
\[v_{p} = \sum_{n=1,3}^{\infty}\frac{4V_{i}}{n\pi}\sin(n\omega_{s}t)\] \[v_{s} = \sum_{n=1,3}^{\infty}\frac{4V_{o}}{n\pi}\sin(n\omega_{s}t-n\phi)\]  
where \(\omega_{s}=2\pi f_{s}\) is the switching angular frequency (\(f_{s}\) is the switching frequency), \(\phi\) is the phase-shift angle by which \(v_{p}\) leads \(v_{s}\). \(v_{T}\) is the resonant tank voltage, which is the difference between \(v_{p},\ n_{t}v_{s}\).  
Using FHA(fundamental harmonics approximation), the steady-state analysis of the resonant converter can be performed easily with consideration of the fundament harmonics of voltages/currents only. According to [25; 26], the active power of the converter can be expressed as  
\[P=\frac{8V_{i}n_{t}V_{o}\sin\phi}{\pi^{2}X_{t}} \tag{1}\]  
where \(X_{t}=\omega_{s}L_{s}-(\omega_{s}C_{s})^{-1}\) is the resonant tank impedance. Therefore, the output current and voltage can be derived respectively as  
\[I_{o} = \frac{8n_{t}V_{i}\sin\phi}{\pi^{2}X_{t}} \tag{2}\] \[V_{o} = \frac{8n_{t}V_{i}R_{L}\sin\phi}{\pi^{2}X_{t}} \tag{3}\]  
Figure 3: The typical steady-state waveforms of Method I.  
It is known that the output current \(I_{o}\) is the average value of the sinusoidal resonant current \(i_{r}\) after active rectification of the secondary bridge.  
\[I_{o}=\frac{1}{\pi}\int_{0}^{\pi}n_{i}i_{r}(\omega_{s}t)d\omega_{s}t=\frac{2}{ \pi}n_{i}I_{rp}\cos\beta \tag{4}\]  
where \(\beta\) is the phase angle by which \(v_{s}\) lags \(i_{r}\). Therefore, the peak resonant current is calculated as  
\[I_{rp}=\frac{\pi I_{o}}{2n_{t}\cos\beta} \tag{5}\]  
In addition, the peak resonant capacitor voltage is evaluated as  
\[V_{cp}=I_{rp}\cdot\frac{1}{\omega_{s}C_{s}} \tag{6}\]  
These two equations can be used to find voltage/current stresses of resonant tank components and switches.  
Seen from (2), the output current \(I_{o}\) is related to the input voltage \(V_{i}\), switching frequency \(\omega_{s}\) and the phase-shift \(\phi\) while it is independent of the equivalent load resistance \(R_{L}\). Regardless of variation of \(R_{L}\), \(I_{o}\) can be maintained constant as long as \(V_{i}\), \(f_{s}\), \(\phi\) are fixed. Meanwhile the output voltage naturally rises from \(V_{omin}\) to \(V_{omax}\) with the increasing equivalent load resistance \(R_{L}\) according to (3). In other words, the DBRC is able to act as a current source. Therefore, it can be concluded that the CC charging stage can be easily realized without active control measurement due to the inherent feature of the converter itself.  
To achieve CV operation, the variation of output resistance in (3) should be compensated by other variables. The controllable parameters include \(\omega_{s}\) and \(\phi\). For the sake of simplicity, the phase-shift \(\phi\) is used as the sole control parameter in this strategy. Apparently, the term \(\sin\phi\) should be reduced to compensate for the increase of \(R_{L}\) in the CV stage to regulate output voltage. On the other hand, the declination of \(\sin\phi\) will decrease the output current \(I_{o}\) as expected according to (2).