## 3 Improved Modulation Strategy Analysis of Six-Switch Converter Based on Time Average Modulation
So, the average current \(i_{4}(t)\) can be written as:  
\[i_{4}(t)=i_{s}(t)(d_{4}+d_{6}-1)+(i_{s}(t)-i_{o}(t))(1-d_{6}) \tag{6}\]  
Substitute (5) and (6) into (4), average current \(i_{c}\) can be figured out as:  
\[i_{c}(t)=(d_{1}-d_{4})i_{s}(t)-(d_{6}-d_{3})i_{o}(t) \tag{7}\]  
Modulation reference for rectifier is defined as:  
\[v_{rA}(t)=m_{r}\sin(\omega t+\theta_{1})+(1-m_{r}) \tag{8}\]  
Modulation reference for inverter is defined as:  
\[v_{rX}(t)=m_{i}\sin(\omega t+\theta_{1}+\varphi)-(1-m_{i}) \tag{9}\]  
where \(m_{r}\) and \(m_{i}\) are modulation indices for the rectifier and inverter, respectively. Assuming unity input power factor operation condition, \(\varphi\) is phase angle difference between rectifier reference and inverter reference. Duty radios \(d_{1}\), \(d_{3}\), \(d_{4}\), \(d_{6}\) can be calculated through similar triangle theory:  
\[\frac{d_{1}T_{s}}{T_{s}}=\frac{1+m_{r}\sin(\omega t+\theta_{1})+(1-m_{r})}{2} \tag{10}\]  
\[d_{1}=\frac{1+m_{r}\sin(\omega t+\theta_{1})}{2}+d_{off1} \tag{11}\]Similarly,  
\[d_{4}=\frac{1-m_{r}\sin(\omega t+\theta_{1})}{2}+d_{off1} \tag{12}\]  
\[i_{c}(t)=i_{1}(t)-i_{4}(t) \tag{4}\]0  
\[i_{c}(t)=i_{1}(t)-i_{4}(t) \tag{4}\]1  
Parameters \(d_{off1}\) and \(d_{off2}\) are offset values utilized for moving up/down modulation reference of rectifier/inverter. Finally, substitute \(d_{1}\), \(d_{3}\), \(d_{4}\) and \(d_{6}\) into Equation (7):  
\[ \begin{array}{l}i_{c}(t)=m_{r}\sin(\omega t+\theta_{1})i_{s}(t)-m_{i}\sin( \omega t+\theta_{1}+\varphi)i_{o}(t)\\ =m_{r}\sin(\omega t+\theta_{1})I_{s}\sin\omega t\\ -m_{i}\sin(\omega t+\theta_{1}+\varphi)I_{o}\sin(\omega t+\theta_{1}+\varphi- \theta_{o})\\ =(\frac{1}{2}m_{r}I_{s}\cos\theta_{1}-\frac{1}{2}m_{i}I_{o}\cos\theta_{o})\\ +\frac{1}{2}m_{i}I_{o}\cos(2\omega t+2\theta_{1}+2\varphi-\theta_{o})\\ -\frac{1}{2}m_{r}I_{s}\cos(2\omega t+\theta_{1})\end{array}  \tag{15}\]  
where \(i_{s}(t)\) is expressed as \(I_{s}\) sin\(wt\), the phase angle of inversion modulation reference is \((wt+\theta_{1}+\varphi)\), so the output voltage \(v_{xy}\) is also \((wt+\theta_{1}+\varphi)\). Since the output current \(i_{o}\) lags behind the inversion output voltage \(v_{xy}\), the output current's phase angle is \((wt+\theta_{1}+\varphi-\theta_{o})\). \(\theta_{o}\) is the power factor angle of inverter load, thus \(i_{o}(t)\) can be expressed as \(I_{o}sin(wt+\theta_{1}+\varphi-\theta_{o})\). The angle relationship of proposed converter is shown below. The constant term in Equation (15) is identical to zero based on the assumptions of no switching or capacitor loss. \[ \begin{array}{l}i_{c}(t)=\frac{1}{2}m_{i}I_{o}\cos(2\omega t+2\theta_{1}+2 \varphi-\theta_{o})\\ -\frac{1}{2}m_{r}I_{s}\cos(2\omega t+\theta_{1})\end{array}  \tag{16}\]  
Amplitude of current \(i_{c}\) can be gotten as below through expansion of (16). \[I_{c}=\sqrt{\frac{1}{4}m_{i}^{2}I_{o}^{2}+\frac{1}{4}m_{r}^{2}I_{s}^{2}-\frac {1}{2}m_{i}m_{r}I_{o}I_{s}\cos(2\varphi-\theta_{o}+\theta_{1})} \tag{17}\]  
It is observed from (17) that when phase angle \(\varphi\) reaches \((\theta_{o}-\theta_{1})/2\), minimization of AC power variation can be achieved in DC-link voltage. Therefore, \(\varphi\) is equal to \((\theta_{o}-\theta_{1})/2\) in this paper and modulation indices for rectifier and inverter are supposed to be approximately the same \((m_{r}=m_{i}=m)\). \(I_{cmin}\) under traditional control can be written as:  
\[I_{c\min}=\frac{1}{2}m|I_{o}-I_{s}| \tag{18}\]  
\(I_{c}^{\prime}\) under improved control can be attained in the same way. When the switching states of \(ST_{1}\), \(ST_{2}\), \(ST_{3}\) are \([0  
Similarly, when the switching states of \(ST_{4}\), \(ST_{5}\), \(ST_{6}\) are \(\{011\}\), \(i_{4}{}^{\prime}(t)=0\); when the switching states are \(\{101\}\), \(i_{4}{}^{\prime}(t)=i_{s}(t)\) with the duration of \(((d_{1}+d_{3}+d_{4}+d_{6})/2-1)T_{s}\); when the switching states are \(\{110\}\), \(i_{4}{}^{\prime}(t)=i_{s}(t)-i_{o}(t)\) with the duration of \((1-d_{6})T_{s}\).