## 2 System Description  
### Power Stage Circuit  
The studied converter is shown in Figure 1. It is based on the conventional single DC-DC buck converter but it is enriched with additional cells. The voltage stress on the switches for the conventional two-level single-cell buck converter is the total input voltage \(v_{g}\). The number of cells determines the number of voltage levels that the converter can handle. Therefore, for the 3-cell 4-level buck converter, the voltage stress on the switches is \(v_{g}/3\). The size of the inductance can be reduced by \(3^{2}=9\) while getting the same inductor current ripple as a conventional buck converter operating with the same switching frequency. If the inductance value is maintained the same as the one used in a conventional buck converter, the current ripple will be reduced by \(3^{2}=9\) in the 3-cell buck converter operating with the same switching frequency. Therefore, compared to a two-level single-cell buck converter for the same current ripple, the multi-cell buck converters have better power density because of using a smaller inductor. Figure 2 shows an illustrative inductor current waveforms for the conventional buck converter with a switching period \(T\) and inductance \(L\) and two different inductor choice for the 3-cell buck converter. Each pair of switches \(\mathrm{S}_{i}\) and \(\overline{\mathrm{S}}_{i}\) (\(i=1,2,3\)) indicates a cell. Switches \(\mathrm{S}_{i}\) and diodes \(\overline{\mathrm{S}}_{i}\) are activated in a complementary manner. When \(\mathrm{S}_{i}\) is on, \(\overline{\mathrm{S}}_{i}\) is off and viceversa. The voltages in the cells must be balanced to reduce the stress in these devices [2; 3]. The presence of switching devices in power electronic converters produce a cyclic change of structure when defined conditions of time and variables are fulfilled. The different switch \(\mathrm{S}_{i}\) positions define the different circuit configurations \(\mathcal{C}_{u_{1}u_{2}u_{3}}\) among which the converter toggles depending on the values of \(u_{1}\), \(u_{2}\) and \(u_{3}\). The triple subscript notation that was adopted in this study correspond to the state of the different switches that is in turns related to the values the command signals \(u_{i}\). Namely, \(u_{i}\) defines the state of the switches \(\mathrm{S}_{i}\) for \(i=1,2,3\). If \(u_{i}=1\), \(\mathrm{S}_{i}\) is closed (on) and if \(u_{i}=0\), \(\mathrm{S}_{i}\) is open (off). For instance the state (1,1,1) for the switches \(\mathrm{S}_{1}\), \(\mathrm{S}_{2}\) and \(\mathrm{S}_{3}\) defines configuration \(\mathcal{C}_{111}\) for which all the switches \(\mathrm{S}_{i}\) (\(i=1,2,3\)) are closed and the state \((0,0,0)\) for the same switches defines configuration \(\mathcal{C}_{000}\) for which all the switches are open. For optimization of the state variable ripples, the driving signals are phase shifted \(2\pi/3\)[2]. The switching from one configuration to the following forms a fixed sequence as according to the operation modes. Each configuration \(\mathcal{C}_{u_{1}u_{2}u_{3}}\) is described by an affine time invariant differential equations in the form of \(\dot{x}=\mathbf{A}_{u_{1}u_{2}u_{3}}\mathbf{x}+\mathbf{B}_{u_{1}u_{2}u_{3}}\). \(\mathbf{A}_{u_{1}u_{2}u_{3}}\) and \(\mathbf{B}_{u_{1}u_{2}u_{3}}\) are the system matrices and vectors during each configuration \(\mathcal{C}_{u_{1}u_{2}u_{3}}\) for all the possible combinations of the values of \(u_{i}\) for \(i=1,2,3\) for a specific mode of operation. \(\mathbf{x}\) is the vector of state variables \(i_{L}\), \(v_{1}\) and \(v_{2}\). The mathematical model can be obtained by applying standard Kirchhoff's voltage and  
Figure 1: Topology of flying capacitor three-cell buck converter. Figure 2: Inductor current waveforms for the conventional buck converter with a switching period \(T\) and inductance \(L\) and two different inductor choice for the 3-cell buck converter operating at the same switching period.