# Analysis of Non-Minimum Phase System for AC/DC Battery
## 3 Converter Analysis
### Simulation Results  
The minimum phase condition in (10) is satisfied for the boost mode, however the converter can also operate in buck mode, which needs to be validated. In circuit analysis of boost mode, the intermediate capacitor \(C\) is considered as an output capacitor in this mode, so the condition of stability has no serious effect in the PF. On the other hand, in buck mode, this stability condition will deteriorate the PF, as the intermediate capacitor \(C\) is an input capacitor in this mode. To prove that, the converter is simulated in PSIM using the parameters presented in Table 2. Figure 7 shows the converter current waveform \(i_{L1}(t)\) and its normalized Fast Fourier Transform (FFT) plot showing the fundamental frequency  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Input voltage peak \(V_{g}\) & 300 V \\ \hline Input voltage frequency \(F_{line}\) & 50 Hz \\ \hline Output voltage \(V_{o}\) & 200 V–400 V \\ \hline Output current \(I_{o}\) & 4–8 A \\ \hline Switching frequency \(F_{s}\) & 200 kHz \\ \hline Intermediate capacitor \(C\) & 8 μF \\ \hline Output capacitor \(C_{o}\) & 800 μF \\ \hline Input inductance \(L_{1}\) & 0.5 mH \\ \hline Second inductance \(L_{2}\) & 0.5 mH \\ \hline \hline \end{tabular}
\end{table}
Table 2: Converter parameters.  
(100 Hz). Figure 7a shows a high switching frequency overlapped to the rectified grid signal. This sub-harmonic oscillation of the switching frequency appears in the FFT between 2 kHz and 3 kHz, the same range where the system has the RHP zero and the high resonance peak. This frequency appears in Figure 3a, where the high peak resonance is exactly between the same frequency range 2 kHz to 3 kHz wherein the system is non-minimum phase.  
By applying the minimum phase condition in (10) for the boost mode, the system will be minimum phase and closed-loop will be stable with no RHP zero. The condition is fulfilled when the output capacitor \(C_{o}\) is equal to 8 uF and intermediate capacitor C is equal to 800 uF. Figure 8 shows the inductor current waveform \(i_{L1}(t)\) and its normalized FFT, wherein the current waveform in plot and its FFT show the absence of the previous switching frequency sub-harmonics. This validates the proposal for eliminating the RHP zero in  
Figure 5: Transfer function \(Gi_{L1}d_{-}u(s)\) with minimum phase condition for boost mode: (**a**) Bode plot. (**b**) Pole-zero map.  
Figure 6: Pole and zero map of numerator \(Gi_{L1}d_{-}u(s)\) at different output power.  
Figure 7: Inductor current \(i_{L1}(t)\) in boost mode without minimum phase condition and its normalized FFT: (**a**) Inductor current waveform \(i_{L1}(t)\). (**b**) Normalized FFT of Inductor current waveform \(i_{L1}(t)\).  
boost mode and the correct dynamic behavior of the converter as a PFC. Nevertheless, the feasibility of the proposal condition must be studied in the buck mode. Figure 9a shows the converter operating in buck mode with parameters not fulfilling minimum phase condition in (10). It proves that the buck mode is working, and the current shape is sinusoidal and following the reference with high PF. Figure 9b shows the inductor current waveform when the minimum phase condition in (10) is utilized. The input current waveform \(i_{L1}(t)\) doesn't follow the current reference, and has a distortion factor that leads to lower PF. As observed in the simulation results, modifying the converter parameters to ensure minimum phase characteristic in boost mode of operation makes the converter not feasible as a PFC when it operates in buck mode. Therefore, another strategy must be studied to ensure minimum phase characteristic and feasible operation as PFC in both operating modes.