## 4 Design Guidelines and Loss Analysis
### Buck-Boost Components Selection  
Brief guidelines to select the buck-boost circuit components are illustrated. Based on the average method analysis of the inductor in steady state, the value of the inductor can be obtained as  
\[L=V_{L}\frac{\Delta T}{\Delta I}=\frac{V_{on}DT}{\Delta I}=\frac{V_{on}\;D}{ \Delta I\;f} \tag{4}\]  
where \(V_{on}\) is the voltage across the inductor in the ON state, \(D\) the duty cycle which can be calculated by Equation (3), \(\Delta I\) the current ripple, and \(f\) the switching frequency. Defining a ripple ratio factor \(r\) with an optimum value of 0.4 achieves a trade-off between inductor size and output capacitor current stress [29].  
\[\Delta I=rI_{L}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Inverter Topology in [27]** & **Inverter Topology in [23]** & **The Proposed Inverter** \\ \hline Two switches in series carry the load current during the active states of the positive and negative power cycles. & Only one switch carries the load current during the active states of the positive and negative power cycles. & Only one switch carries the load current during the active states of the positive and negative power cycles. \\ \hline It does not charge the flying capacitor during the active state of the negative power cycle. & It does not charge the flying capacitor during active states. & It charges the flying capacitor during all operating states. \\ \hline No RB-IGBT required. & It required two switches with bipolar voltage-blocking capability, such as RB-IGBT. & No RB-IGBT required. \\ \hline No proposed way to control or reduce the intrusion current during the charging of the flying capacitor & The intrusion current during the charging of the charging of the flying capacitor & In the proposed inverter, the flying capacitor charging current can be controlled using the buck–boost inverter because the capacitor charges through the inductor. Based on the nature of the inductor, in that it tries to oppose the current change, the slope of the current can be controlled. \\ \begin{tabular}{c} flying capacitor \\ \end{tabular} & \
\begin{tabular}{c} cannot be fully controlled; it ranges \\ from 2 Iac, max to 5 Iac, max \\ \end{tabular} &
\begin{tabular}{c} where \(I_{o}\) is the output current of the buck–boost inverter, in this case \(I_{ac,max}\), \(D\) the duty cycle, and r the ripple ratio. Hence, the maximum current can be controlled by inductor value because \(r=\frac{V_{on}\;D}{\Delta I\;I_{L}f}\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparative table between the proposed inverter and similar inverters.  
where \(I_{L}\) is the average current of the buck-boost inductor and \(I_{o}\) is the output current. Substituting (5) into (4), the value of the inductor will be  
\[L=\frac{V_{on}\ D}{r\ I_{L}\ f} \tag{6}\]  
The buck-boost diode average current and switch RMS current are addressed in Equations (7) and (8), respectively, while the voltage blocking capability must be at least rated as _2 Vdc_ for both devices  
\[I_{D}=I_{o} \tag{7}\]  
\[I_{sw\_rms}=I_{L}\ \sqrt{D\bigg{(}1+\frac{r^{2}}{12}\bigg{)}} \tag{8}\]  
The flying capacitor (output capacitor of the buck-boost) is sized here based on the rated RMS current, because the voltage ripple requirement is not critical in this application; therefore  
\[I_{out\_rms}=I_{o}\ \sqrt{\frac{D+\bigg{(}\frac{r^{2}}{12}\bigg{)}}{1-D}} \tag{9}\]  
where \(I_{out\_rms}\) is the conventional buck-boost output capacitor RMS current. Since the flying capacitor is discharged only in the negative half cycle in the proposed inverter, the RMS current of the flying capacitor \(I_{C_{FC\_rms}}\) can be averaged to  
\[I_{C_{FC\_rms}}\approx\frac{I_{out\_rms}}{2} \tag{10}\]