## 3 Improved Modulation Strategy Analysis of Six-Switch Converter Based on Time Average Modulation  
It is assumed the inversion and rectification have the same power factor. Under the traditional six-switch converter control strategy, one switching cycle includes eight working states, followed by FS, RS, FS (BS in state B), IS, FS, IS, FS (BS in state B), RS. Two RS (or IS) are too close, and there is only one FS between them, so that the DC side capacitor is continuously charged (or discharged). A large capacitor is required, which greatly limits practical applications of the six-switch converter. The key point of the proposed improved modulation method is to exchange rectification state and inversion state in the second half of the switching cycle by logic control, so that the order of the eight states in a cycle becomes FS, RS, FS (BS in state B), IS, FS, RS, FS (BS in state B), IS. Therefore, the energy on the capacitor can quickly accumulate and release in a complete switching cycle, achieving the effect of 'fast charging and discharging'. Figure 6 shows a larger version of the DC side capacitor voltage \(v_{d}\) under two modulation strategies. Each linear section in the curve corresponds to FS, RS, IS, and BS, respectively. According to the input/output power balance theory and supposing \(V_{S}\approx V_{AB}\approx V_{XY}\), the Equation (2) can be obtained by vector relations. Where \(X_{1}\) is the input impedance  
Figure 5: Schematic diagram of the switching state: (**a**) state A; (**b**) state B.  
Figure 6: Partially enlarged view of \(v_{d}\) in switching period. and \(X_{2}\) is the output impedance. \(\theta_{1}\) represents the phase angle of rectification modulation reference and \(\theta_{0}\) is the power factor angle of inverter load. When the radio of \(X_{1}/X_{2}\) is about 0.2, the unit input power factor can be ensured. \[ \begin{array}{l}V_{AB}I_{S}\cos\theta_{1}=V_{XY}I_{0}\cos\theta_{0}\\ \Rightarrow\theta_{1}=-\sin^{-1}\left(\frac{X_{1}}{X_{2}}\cos\theta_{0}\right) \end{array}  \tag{2}\]  
The voltage fluctuation of the DC side capacitor is caused by the fluctuating current flowing through the capacitor, as shown in Equation (3). When the amplitude of \(i_{c}\) becomes smaller, the capacitor which used to keep \(v_{d}\) relatively stable also decreases. In order to verify the effectiveness of the improved six-switch converter modulation strategy, the \(i_{c}\) under the traditional and improved modulation strategies are calculated and compared respectively. \[v_{d}(t)=\frac{1}{C_{d}}\int_{0}^{t}i_{c}(t)dt+V_{d} \tag{3}\]  
The equation of Kirchhoff's Current Law at point P of six-switch converter in Figure 1 is:  
\[i_{c}(t)=i_{1}(t)-i_{4}(t) \tag{4}\]  
In (4), \(i_{1}(t)\) and \(i_{4}(t)\) are currents flowing through switch \(ST_{1}\) and \(ST_{4}\) based on the predefined reference direction. The average value of current \(i_{1}(t)\) can be calculated according to the switching state of the six-switch converter. When the switching states of \(ST_{1}\), \(ST_{2}\), \(ST_{3}\) are \(\{011\}\), \(i_{1}(t)=0\); when the switching states becomes \(\{101\}\), \(i_{1}(t)=i_{s}(t)\) with the duration of \((d_{1}+d_{3}-1)T_{5}\); when the switching states are \(\{110\}\), \(i_{1}(t)=i_{s}(t)-i_{o}(t)\) with the duration of \((1-d_{3})T_{5}\). Therefore, the average current \(i_{1}(t)\) can be written as:  
\[i_{1}(t)=i_{s}(t)(d_{1}+d_{3}-1)+(i_{s}(t)-i_{o}(t))(1-d_{3}) \tag{5}\]  
Similarly, when the switching states of \(ST_{4}\), \(ST_{5}\), \(ST_{6}\) are \(\{011\}\), \(i_{4}(t)=0\); when the switching states are \(\{101\}\), \(i_{4}(t)=i_{s}(t)\) with the duration of \((d_{4}+d_{6}-1)T_{5}\); when the switching states are \(\{110\}\), \(i_{4}(t)=i_{s}(t)-i_{o}(t)\) with the duration of \((1-d_{6})T_{5}\).