# Analysis of AC DC Four-Switch Boost-Buck Battery Charger Converter for EV Applications
## 4 Numerical Analysis  
Table 1 represents the converter parameters that will be used for the numerical analysis. For the boost mode, the output voltage is 400 V, and 300 V for the buck mode. In circuit design, reactive elements such as inductors and capacitors were carefully selected to achieve specific ripple criteria. For boost mode operation, inductance \(L_{1}\) can be calculated from  
\[L_{1}=\frac{D_{1}V_{g\_max}}{\Delta iL_{1}F_{s}}, \tag{8}\]  
where \(\Delta iL_{1}\) is the inductor and \(L_{1}\) is the current ripple. \(V_{g\_max}\) is the maximum input voltage.  
However, for the buck mode, the second inductor \(L_{2}\) can be calculated from  
\[L_{2}=\frac{D_{2}\big{(}V_{g\_max}-V_{o}\big{)}}{\Delta iL_{2}F_{s}}, \tag{9}\]  
where \(\Delta iL_{2}\) is the second inductor and \(L_{2}\) is the current ripple.  
The output capacitor \(C_{o}\) can be calculated from the following [17]:  
\[\Delta V_{o}=\frac{P_{O}}{\omega C_{o}V_{o\_rms}}, \tag{10}\]  
\[C_{o}=\frac{I_{o}}{\omega\Delta V_{o}}, \tag{11}\]  
where \(P_{O}\) is the output power, \(\omega\) is the line angular frequency, and \(\Delta V_{o}\) is the peak-to-peak output voltage ripple.  
Under small-ripple assumption, the intermediate capacitor voltage ripple must be similar to the output voltage ripple. With this ripple choice, the discontinuous capacitor voltage mode is avoided. Figure 5a,b display the Bode plots for the transfer function \(G_{igd_{lu}}(s)\) and \(G_{vd_{lu}}(s)\) in the boost mode. The pole-zero mappings of these transfer functions are shown in Figure 5c and Figure 5d, respectively. It is observed that transfer function \(G_{igd_{lu}}(s)\) is the minimum phase and has no RHPZ but the \(G_{vd_{lu}}(s)\) transfer function has RHPZ.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Input voltage peak \(V_{g}\) & 300 V \\ Input voltage frequency \(F_{line}\) & 50 Hz \\ Output voltage \(V_{o}\) & 200–400 V \\ Max output current \(I_{o}\) & 8 A \\ Switching frequency \(F_{s}\) & 200 kHz \\ Intermediate capacitor \(C\) & 8 uF \\ Output capacitor \(C_{o}\) & 800 μF \\ Inductance \(L_{1}\) & 0.5 mH \\ Second inductance \(L_{2}\) & 0.5 mH \\ \hline \hline \end{tabular}
\end{table}
Table 1: Four-switch boost-buck PFC converter parameters.  
Furthermore, these transfer functions exhibit high resonance undamped peaks occurring between 1 kHz and 3 kHz. These two resonance peaks below switching frequency appear in fourth-order two-inductor converter topologies under small-ripple assumption. These high resonance peaks can lead to system instability by causing oscillations and amplifying harmonic signals at their central frequencies. Managing the non-minimum phase system can be challenging, but one approach is to decrease the bandwidth and push the cut-off frequency away from the resonance undamped frequency to improve stability.  
Figure 6a,b depict the Bode plots for the transfer functions \(G_{igd_{lu}}(s)\) and \(G_{vd_{lu}}(s)\) in the buck mode. The pole-zero mappings of these transfer functions are shown in Figure 6c and Figure 6d, respectively. Similar to the boost mode case, the Bode plots of the buck mode transfer functions also exhibit a high undamped resonance frequency around 3 kHz. Moreover, in this case, both transfer function \(G_{igd_{lu}}(s)\) and \(G_{vd_{lu}}(s)\) demonstrate a minimum phase characteristic.  
The analysis of these transfer functions and their corresponding Bode plots provides insights into the frequency response characteristics of the converter in both the boost and buck modes. Therefore, to see the effects of these high undamped behaviors in the duty cycle to input current \(G_{igd_{lu}}(s)\) and duty cycle to output voltage \(G_{vd_{lu}}(s)\) transfer functions and the non-minimum phase characteristic of \(G_{vd_{lu}}(s)\), simulation for the time response behavior of the converter is conducted in the next section.