## 2 The Topology and Modulation Principle of Six-Switch Converter  
The topology of the six-switch converter is shown in Figure 1. Each bridge arm is composed of three switches in series, and there are two sets of input and output. Compared with the traditional single-phase back-to-back AC-DC-AC converter, the proposed converter reduces two switches. In the proposed six-switch converter, the switch in the middle of each bridge arm is shared by the rectifier and inverter, and the voltage \(v_{XN}\) is no greater than \(v_{AN}\) at any time. As shown in Table 1, each bridge arm of the six-switch converter has three switching states. In order to meet the switching constraint requirements of the six-switch converter, the modified carrier-based PWM method is applied, and the modulation reference of the upper rectifier and the modulation reference of the lower inverter are compared with a common triangular carrier, respectively, as shown in Figure 2.  
The instantaneous value of the upper reference must be no less than the lower reference. The PWM signal of switch \(ST_{1}\) (\(ST_{3}\)) is generated by comparing the upper (lower) reference and the triangular carrier, and the PWM signal of switch \(ST_{2}\) is generated by logic operation. \[ST_{1}= \begin{cases}\begin{aligned} & ST_{2}=(ST_{1})XOR(ST_{3})\\ & 0,\quad v_{rA}\geq v_{cw}\\ \end{aligned},ST_{3}=\begin{cases}1,& v_{rX}\leq v_{cw}\\ 0,& v_{rY}>v_{cw}\\ \end{cases}\end{cases}  \tag{1}\]where \(v_{cuv}\) is the common triangular carrier, \(v_{rA}\) and \(v_{rX}\) are the upper and the lower modulation references, respectively. Figure 3 shows the modulation references of \(ST_{1}\), \(ST_{3}\), \(ST_{4}\), \(ST_{6}\) in a six-switch converter. It can be observed that due to the relative position changes of the four references in one line frequency cycle, the switching signals of the six switches also change. Therefore, the relative position in a line frequency cycle can be divided into four states (two for the first half cycle and two for the second half cycle) for analysis. Take the first half line frequency cycle of the modulation reference as an example to analyze the working states of the circuit. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Switching Status** & \(ST_{1}\) & \(ST_{2}\) & \(ST_{3}\) & \(v_{AN}\) & \(v_{\mathbf{\chi}N}\) \\ \hline
1 & 1 & 1 & 0 & \(v_{d}\) & \(v_{d}\) \\
2 & 0 & 1 & 1 & 0 & 0 \\
3 & 1 & 0 & 1 & \(v_{d}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Switching status of the six-switch converter. Figure 1: The six-switch converter topology. Figure 2: Generation of the single bridge arm PWM signal. There are four working states of the six-switch converter, which are introduced as follows. The rectification state (abbreviated as RS) is shown in Figure 4a, the inversion state (abbreviated as IS) is shown in Figure 4b, the freewheeling state (abbreviated as FS) is shown in Figure 4c, and the both-way state (abbreviated as BS) is shown in Figure 4d. When the relative positions of four modulation references are shown as state A in Figure 3, the six-switch converter has the first three working states in one switching cycle. When the relative positions of four modulation references are shown as state B in Figure 3, the six-switch converter has all four working states in one switching cycle. From the aforementioned six-switch converter modulation method, a schematic diagram of six switching states in one switching cycle can be achieved, as shown in Figure 5. The state A in Figure 5a corresponds to the state A in Figure 3, which means the relative positions of the modulation references are consistent. State B in Figure 5b corresponds to state B in Figure 3.  
Figure 4: Four working states of the six-switch converter: (a) rectification state; (b) inversion state; (c) freewheeling state; (d) both-way state. Figure 3: Reference modulated waves of \(ST_{1}\), \(ST_{3}\), \(ST_{4}\), \(ST_{6}\).