## 4 The Phase-Shifted Carriers Modulation  
In order to define the size of the battery/SC stages inductors (L1, L2), the ripple currents of these inductors at the boundaries of the operating voltage range for both battery and SC need to be considered. By considering the converter output voltage of Vo, which is defined by the application, the operating voltage range of the battery can be considered as wide as from 0.75 Vo at the lowest SoC up to 1.5 Vo at the highest SoC. Accordingly, the operating voltage range of the PV array considering these conditions can be selected to be from 0.5 Vo up to 0.75 Vo. Accordingly, four operation boundary points of the battery stage inductor (L2) can be obtained, A, B, C, and D. At points A and B, the PV voltage is at its minimum; however, at point A the battery is at the highest SoC and at point B the battery is at lowest SoC. On the other hand, at points C and D, the PV voltage is at its maximum; however, at point C, the battery is at the highest SoC, and at point D, the battery is at the lowest SoC. The corresponding duty cycles for the battery and PV switches and the current ripple (\(\Delta\Pi\)) of the inductor L2 at each of these boundary points at a switching frequency (\(\mathrm{f_{s}}\)) are listed in Table 2, and the relevant waveforms are shown in Figure 10a. \begin{table}
\begin{tabular}{c c c c} \hline \(\mathbf{V_{PV}}\) & \(\mathbf{V_{batt}}\) & \(\mathbf{1.5\ Vo}\) & \(\mathbf{0.75\ Vo}\) \\ \hline \multirow{4}{*}{0.5 Vo} & \(\mathrm{d3}\approx 0.33\) & & \(\mathrm{d3}\approx 0.67\) \\  & \(\mathrm{d5}=0.5\) & & \(\mathrm{d5}=0.5\) \\  & \(\mathrm{\Delta I}|_{\theta=-0}=\frac{0.5\mathrm{V_{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{ \mathrm{  \mathrm{   }}}}}}{}{}}{}}{}}{}}}}}}}\) & \(\mathrm{\Delta I}|_{\theta=0}=\frac{0.375\mathrm{V_{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{   \mathrm{  }}}}}}{}{}}}{}}}{{}}}{}}{}}}}}}\) & \(\mathrm{\Delta I}|_{\theta=\frac{5\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{ \mathrm{   }}}}}}}{}{}}}}{{}}}{{}}}{{}}}{}}}}}}}\) & \(\mathrm{\Delta I}|_{\theta=\frac{5\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{       }}}}}}}{}{}{}}{}}{}}{{}}}{{}}}{}}}}}}}\) \\ \hline \multirow{4}{*}{0.75 Vo} & \(\mathrm{d3}=0.5\) & & \(\mathrm{d3}=1\) \\  & \(\mathrm{d5}=0.25\) & & \(\mathrm{d5}=0.25\) \\ \cline{1-1}  & \(\mathrm{\Delta I}|_{\theta=0}=\frac{0.5\mathrm{V_{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{ \mathrm{ \mathrm{     }}}}}}{}}{}{}}{}{}}{}{}{}{}{}{}{}{}{}{}{{}}{{}}{{}}{{}}{{}}{{}{{}}{{} {}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}{}{}{}{}{}{}{{}}{{}}{{} {}}{{}}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{In order to minimize the inductor L2 current ripple, a phase-shifted carrier modulation (PSCM) technique was employed by adopting a phase-shift angle of (\(\theta^{\text{o}}\)) between the PWM carriers of the switches S5 and S3 to minimize the voltage across the inductor and consequently its current ripple. The angle should be varying based on the value of the duty cycles to avoid the deadlock situation that would prevent power flow to/from the battery, which occurs when d\({}_{3}\) = (1 - d\({}_{5}\)) and \(\theta^{\text{o}}\) = 360 d\({}_{5}\).