# A High-Conversion Ratio Multiphase Converter Realized with Generic Modular Cells
## 4 Extension for Higher Order Converters  
The cell of Figure 2 enables converter extension along multiple directions. The first extension addresses a higher conversion ratio. It is achieved by chaining several cells one after another. The number of cells is defined as \(n\) (Figure 5), and the chaining procedure follows the method outlined in Section 2, using the cell described in Figure 2, where all inductors are connected in parallel at the output, and the high-side voltage of the flying capacitor of a preceding cell connects to the high-side switch (\(S_{\text{xH}}\)), of the next one (Figure 1). To provide a discharge path for the last cell capacitor, one additional switch per converter is added \(S_{(n-1)-(n)}\) Figure 5.  
The expression for the conversion ratio, in a generalized form for a single module with \(n\) cells in series, \(1\times n\) topology, and assuming equal duty cycle is applied to all the cells, takes the form of (19):  
\[M_{n}=\frac{V_{\text{o}}}{V_{in}}=\frac{D}{n+1}, \tag{19}\]  
where \(n\) is the number of cells of one converter module, and \(D\) is the common duty cycle applied to all the cells. Average capacitor voltages per phase can be expressed as follows:  
\[V_{Ci}=\frac{n-(i-1)}{n+1}V_{in}, \tag{20}\]  
where \(i\) is the index of the cell number. For example, in a two-cell topology, \(n=2\)[20], the conversion ratio \(M\) is equal to \(M=D/3\) and capacitor voltages are \(V_{C1}=2V_{\text{in}}/3\) and \(V_{C2}=V_{\text{in}}/3\). For a three-phase case, \(n=3\), the ratio \(M\) equals to \(M=D/4\), and the average capacitor voltages are \(V_{C1}=3V_{\text{in}}/4\), \(V_{C2}=2V_{\text{in}}/4\), and \(V_{C3}=V_{\text{in}}/4\), as summarized in Section 3.1. The average inductors' currents for each phase (except phase \(n\)-1) can be represented as follows:  
\[I_{Li}=\frac{1}{n+1}I_{out}, \tag{21}\]  
Figure 5: Circuit diagram of a multiphase buck converters topology of an \(m\times n\) order. where the average inductor current for phase \(n\)-1 will be twice that of the other phases (assuming again equal duty cycle) due to a slightly different operation mode of the last two cells. For example, for a single module, \(m=1\), and two cells, \(n=2\), the average inductors currents are \(I_{L1}=I_{\text{out}}/3\) and \(I_{L2}=2I_{\text{out}}/3\), and for the case of three cells, \(n=3\), inductors currents at steady state are \(I_{L1}=I_{\text{out}}/4\), \(I_{L2}=I_{\text{out}}/2\), and \(I_{L3}=I_{\text{out}}/4\), as mentioned in Section 3.2. The second extension covers the power capability of the converter, i.e., output current. Higher power to the load is facilitated by paralleling modules. Paralleling is carried out on the module level, where the inputs and the outputs of all the modules are connected in parallel to the source and the load, respectively. In this manner, the outputs of the modules share the same capacitor. The number of modules in parallel are referred to here as \(m\) (Figure 5). The total output current in the case of the \(m\times n\) system can be calculated by summing up all output currents from all the modules, where each module output current consists of the sum of its inductors currents. The expression for multi module total output current is summarized in (22):  
\[I_{outT}=\sum_{\text{Module}=1}^{m}\left(\sum_{\text{Ph}=1}^{n}I_{L(Ph)}\right) \tag{22}\]  
A parallel connection of multiple modules can significantly lower the voltage and current component stresses, enabling a higher output power with better efficiency. The third expansion provides a degree of freedom where the per-phase duty ratio is independently set. This assists in balancing the currents along the cells, facilitates targeted current sharing control that could be unbalanced, for example, and enables a better thermal management. The current mismatch between the phases can be adjusted by tuning the duty cycle of each phase, as mentioned in Section 3.2.