## 5 Design Example  
The circuit parameters are listed in Table 2. The specifications are given as the crossover frequency \(f_{c}\approx 10\) kHz and the phase margin \(\varphi_{m}>45^{\circ}\). (1) Determination of the initial resonant current \(I_{r}\):  
According to Equation (1), the minimum initial resonant current is given as \(I_{r\_\min}>2.16\) A. The minimum initial resonant current and the initial resonant current for AZVS are selected as \(I_{r\_\min}=2.5\) A and \(I_{r\_A}=5\) A, respectively. Therefore, when \(i_{lf,l}<-2.5\) A, \(S_{\tau 1}\) is not operated, achieving NZVS of \(S_{1}\) and \(S_{4}\) according to Table 1. Otherwise, \(S_{\tau 1}\) would be turned on properly to realize AZVS of \(S_{1}\) and \(S_{4}\). Regarding \(S_{2}\) and \(S_{3}\), when \(i_{lf,u}<2.5\) A, \(S_{\tau 2}\) must be turned on to create AZVS conditions. When \(i_{lf,u}>2.5\) A, \(S_{2}\) and \(S_{3}\) can achieve NZVS. (2) Determination of the upper limit and lower limit of the DPWM (\(V_{upperlimit}\) and \(V_{lowerlimit}\))  
The frequency of the DPWM clock is 120 MHz. Thus, the carriers are specified as \(V_{c\max}=300\) and \(V_{c\min}=0\) to achieve 200 kHz switching frequency. To simplify the calculation, the filter inductor current ripple is neglected. Therefore, the maximum required auxiliary current can be obtained from Equation (5) as \(I_{Lrm\_max}=I_{r\_A}+I_{o\_\max}=13\) A. The maximum charging time \(t_{dt\_\max}\) can be calculated from Equation (6) as \(t_{dt\_\max}=357.5\) ns. The upper limit can be calculated from Equation (12) as \(V_{upperlimit}\approx 266\) by using the improved loading scheme. Thus, the lower limit \(V_{lowerlimit}\) and maximum duty ratio \(D_{\max}\) are 34 and 0.867, respectively. Regarding the conventional loading scheme, the upper limit, lower limit, and maximum duty ratio are 257, 43, and 0.857, respectively. Table 3 shows a comparison of the parameters between the conventional and improved loading scheme. The range of the duty ratio is improved by utilizing the improved loading scheme. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Conventional Loading Scheme** & **Improved Loading Scheme** \\ \hline Maximum Duty Ratio \(D_{\max}\) & 0.857 & 0.867 \\ Minimum Duty Ratio \(D_{\min}\) & 0.143 & 0.133 \\ Upper Limit \(V_{upperlimit}\) & 257 & 266 \\ Lower Limit \(V_{lowerlimit}\) & 43 & 34 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the parameters between the conventional and improved loading scheme. 3. Determination of the PI regulator parameters and active damping gain \(K_{\zeta\!f}\)  
The active damping gain \(K_{\zeta\!f}\) will influence the stability of the system and the phase margin will decrease due to the increase of \(K_{\zeta\!f}\) according to the analysis in Section 4.3. However, the influence of the filter capacitor is neglected to simplify the design of the PI regulator in Section 4.2. Therefore, in order to meet the requirement \(\varphi_{m}>45^{\circ}\), the phase margin is specified as \(60^{\circ}\) due to the capacitor current feedback. Thus, the parameters of the PI regulator can be calculated from Equation (21). The results are that \(K_{p}=3.6522\) and \(K_{i}=70{,}999\). To ensure the system's stability, the capacitor current feedback gain \(K_{\zeta\!f}\) must meet the requirement of \(0.0158<K_{\zeta\!f}<0.119\) according to Equation (27). The feedback gain is selected as \(K_{\zeta\!f}=0.05\). Based on the above parameters, the Bode plots of the single loop and double loop can be obtained from Figure 13. The crossover frequency of the single loop is \(-62{,}900\) rad/s and the phase margin is \(60^{\circ}\). This is in agreement with the design specifications. However, the LC filter leads to resonance, which makes the phase-frequency curve cross \(-180^{\circ}\) after it drops to \(-180^{\circ}\). According to the stability criterion, the single-loop system is unstable. Regarding the double-loop control, the crossover frequency is \(-61{,}000\) rad/s and the phase margin is \(46{.}4^{\circ}\).