# A High-Conversion Ratio Multiphase Converter Realized with Generic Modular Cells
## 3 Steady State Analysis
### CCM-DCM Boundary  
For a new converter operation in the continuous conduction mode (CCM), the inductor current needs to be always positive. The boundary between CCM and discontinuous conduction mode (DCM) is considered when the inductor current falls to zero at the end of the switching cycle. Setting the \(I_{L\text{ (min)}}\) to zero [48; 49; 50], results in an expression for the average inductor current for the CCM-DCM boundary:  
\[I_{L(av)}=\frac{1}{T}\bigg{(}\frac{1}{2}I_{\text{max}}D_{on}T+\frac{1}{2}I_{ \text{max}}D_{off}T\bigg{)}=\frac{1}{2}I_{\text{max}}\Big{(}D_{on}+D_{off} \Big{)}. \tag{14}\]  
Assuming that the duty cycles are equal for all converter phases, the average current of inductors \(L_{1}\), \(L_{2}\), \(L_{3}\), can be expressed as \(I_{L1}=V_{\circ}/4R_{\circ}\), \(I_{L2}=V_{\circ}/2R_{\circ}\), and \(I_{L3}=V_{\circ}/4R_{\circ}\), respectively, according to (11). At steady state, the \(\Delta I_{L}\) in the charge stage is equal to the discharged stage. For inductor \(L_{1}\), the expression for \(I_{L1(\text{max})}\) can be summarized as follows:  
\[I_{L1(\text{max})}=\frac{(V_{in}-V_{C1})-V_{0}}{L_{1}}D_{on}T, \tag{15}\]  
The boundary condition is calculated for each inductor from the charge or discharge equations using the expressions in (2), (3), and (4). Substituting (15) into (14), using Equation (2), and after some manipulations, the boundary condition can be summarized as follows:  
\[\begin{array}{c}\frac{V_{\circ}}{4R_{\circ}}=\frac{(V_{in}-V_{\circ})-V_{0}} {2L_{1}}D_{lon}T\Big{(}D_{lon}+D_{1 off}\Big{)}=\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\frac{ V_{0}}{2L_{1}}D_{1 off}T\Big{(}D_{lon}+D_{1 off}\Big{)}\end{array}. \tag{16}\]  
Note that at the boundary between the CCM to DCM modes, the sum of the duty ratios can still be written as \(D_{\text{1}on}+D_{1 off}=1\); hence, Equation (16) can be also expressed as follows:  
\[L_{\text{1}min}=\frac{2R_{\circ}(1-D_{1 on})}{f_{S}}, \tag{17}\]  
where \(L_{\text{1}(\text{min})}\) is the minimum inductance required for CCM mode operation. The boundaries for \(L_{2}\) and \(L_{3}\) can be calculated using Equations (3), (4), (11), and (14), in the similar manner as described for \(L_{1}\). The expressions for \(L_{2}\) and \(L_{3}\) are summarized below:  
\[\begin{array}{c}L_{\text{2}min}=\frac{R_{\circ}(1-D_{2m})}{f_{S}},\\ L_{\text{3}min}=\frac{2R_{\circ}(1-D_{3m})}{f_{S}}.\end{array} \tag{18}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & & & & **Stress** \\ \hline
**Switch** & **I** & **II** & **III** & **IV** & **V** & **VI** & & **\(V_{\text{max}}\) [V]** \\ \hline \(S_{\text{1L}}\) & \(V_{in}\)-\(V_{\text{C1}}\) & 0 & 0 & 0 & 0 & 0 & 12 \\ \hline \(S_{\text{1H}}\) & 0 & \(V_{in}\)-\(V_{\text{C1}}\) & \(V_{in}\)-\(V_{\text{C1}}\) & \(V_{in}\)-\(V_{\text{C1}}\) & \(V_{in}\)-\(V_{\text{C1}}\) & 12 \\ \hline \(S_{\text{2L}}\) & 0 & 0 & \(V_{\text{C3}}\) & 0 & 0 & 0 & 12 \\ \hline \(S_{\text{2H}}\) & \(V_{in}\)-\(V_{\text{C2}}\) & \(V_{\text{C1}}\)-\(V_{\text{C2}}\) & 0 & \(V_{\text{C1}}\)-\(V_{\text{C2}}\) & \(V_{\text{C1}}\)-\(V_{\text{C2}}\) & 24 \\ \hline \(S_{\text{2-3}}\) & \(V_{\text{C3}}\) & \(V_{\text{C3}}\) & 0 & \(V_{\text{C3}}\) & \(V_{\text{C2}}\) & \(V_{\text{C3}}\) & 24 \\ \hline \(S_{\text{3L}}\) & 0 & 0 & 0 & 0 & \(V_{\text{C2}}\)-\(V_{\text{C3}}\) & 0 & 12 \\ \hline \(S_{\text{3H}}\) & \(V_{\text{C2}}\)-\(V_{\text{C3}}\) & \(V_{\text{C2}}\)-\(V_{\text{C3}}\) & \(V_{\text{C2}}\) & \(V_{\text{C2}}\)-\(V_{\text{C3}}\) & 0 & \(V_{\text{C2}}\)-\(V_{\text{C3}}\) & 24 \\ \hline \hline \end{tabular} The blue shade indicates the highest stress per switch.  
\end{table}
Table 1: Switches voltages of each state in the 3 ph. converter.