## 4 Output Current Control
### Capacitor Current Feedback Design  
In Section 4.2, the influence of filter capacitor is neglected when the PI parameters are designed. However, the LC filter together with the load inductance forms the LCL filter, which leads to resonance. To analyze the stability of the ARSI, the influence of filter capacitor cannot be ignored. The filter capacitor current feedback is introduced to damp the resonance. Because of the delay caused by the computation, the system is no longer a minimum phase system. To design the inner capacitor current loop, a discrete model must be built instead of a continuous model. Figure 10 shows the model of the inverter, which is obtained from Figure 6. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline DC voltage \(V_{s}\) & 80 V \\ Switching frequency \(f_{s}\) & 200 kHz \\ Dead time \(t_{dead}\) & 0.2 μs \\ Load & 3.7 \(\Omega\), 4.87 mH \\ Resonant inductor \(L_{r}\) & 2.2 \(\mu\)H \\ Resonant capacitor \(C_{r}\) & 2.7 nF \\ Filter inductor \(L_{f}\) & 22 \(\mu\)H \\ Filter capacitor \(\hat{C}_{f}\) & 1 \(\mu\)F \\ Maximum output current \(l_{o,\max}\) & 8 A \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of the circuit. Figure 10: Model of ARSI in digital control. Figure 9: Bode plot of the ARSI with different PI parameters: (**a**) different \(K_{p}\); (**b**) different \(K_{i}\). The closed-loop and open-loop transfer functions can be obtained as follows. (The detailed proof is presented in Appendix A.) \[T(z)=\frac{i_{o}(z)}{i_{o}^{*}(z)}=\frac{G_{dc}(z)G_{c}(z)G_{P}(z)}{1+K_{cf}G_{ dc}(z)G_{T}(z)+G_{dc}(z)G_{c}(z)G_{P}(z)} \tag{23}\]  
\[G(z)=\frac{i_{o}(z)}{i_{o}^{*}(z)-i_{o}(z)}=\frac{G_{dc}(z)G_{c}(z)G_{P}(z)}{1+ K_{cf}G_{dc}(z)G_{T}(z)}, \tag{24}\]  
where  
\[G_{c}(z)=K_{p}+\frac{K_{i}}{1-z^{-1}}\]  
\[G_{dc}(z)=z^{-1}\]  
\[G_{T}(z)=K_{pvm}\sqrt{\frac{C_{f}}{L_{f}}}\frac{(z-1)\text{sin}\varphi}{z^{2}- 2z\text{cos}\varphi+1}\]  
\[G_{P}(z)=K_{pvm}\frac{-z(R_{o}F\text{cos}\varphi+L_{o}\omega_{res}F\text{sin} \varphi-R_{o}F)+R_{o}F-R_{o}F\text{cos}\varphi+L_{o}\omega_{res}F\text{sin} \varphi}{z^{2}-2z\text{cos}\varphi+1}\]  
\[F=\frac{L_{f}C_{f}}{L_{o}^{2}+L_{f}C_{f}R_{o}^{2}}\]  
Figure 10:0  
Figure 10:1  
In the z-domain, the root locus is used to analyze the stability of the system. The relevant expressions can be developed from Equation (23) as follows:  
Figure 10:2  
Equation (19) can be transformed as follows:  
Figure 10:3  
From Equation (26), the root locus related to \(K_{cf}\) can be plotted in Figure 10:4 When the capacitor current feedback is not used, namely \(K_{cf}\) = 0, the closed-loop poles are outside the unit circle, which demonstrates that the system is unstable. As \(K_{cf}\) increases, the system becomes stable. However, if \(K_{cf}\) is too large, the system becomes unstable again. Figure 12 shows the Bode plot with different \(K_{cf}\) which is obtained from Equation (24). In the magnitude-frequency characteristic, \(K_{cf}\) hardly influences the low-frequency characteristic. As \(K_{cf}\) increases, the open-loop gain decreases slightly around the crossover frequency \(\omega_{c}\). Besides, the resonance peak is damped. In the phase-frequency characteristic, the phase margin decreases with the increase of \(K_{cf}\). When \(K_{cf}\) is large enough, the phase-frequency curve crosses \(-180^{\circ}\) after it drops to \(-180^{\circ}\). This makes the system unstable according to the stability criterion. In order to stabilize the system, the active damping gain \(K_{cf}\) must meet the requirements as follows. (The detailed proof is presented in Appendix B.) Figure 10:5  
Figure 10:6 Root locus of variation in active damping gain \(K_{cf}\) using a dual-loop controller. Figure 10:7 Open-loop Bode plot with different active damping gain \(K_{cf}\). where  
Figure 10:8  
Figure 10:9  
Figure 9:0