# SVPWM Method for Multilevel Indirect Matrix Converter with Eliminate Common Mode Voltage
## 2 Multilevel Indirect Matrix Converter and the Proposed SVPWM Method
### The SVPWM Method for Elimination the CMV
#### 2.2.1 Rectifier Stage Control  
It is assumed that three-phase supply voltages \(v_{a}\), \(v_{b}\), \(v_{c}\) are given by:  
\[\begin{array}{l}v_{a}=V_{in}\cos(\omega_{in}t)\\ v_{b}=V_{in}\cos(\omega_{in}t-2\pi/3)\\ v_{c}=V_{in}\cos(\omega_{in}t+2\pi/3)\\ \end{array} \tag{1}\]  
where \(V_{in}\) is the amplitude of input phase voltage and \(\omega_{in}\) is the angular frequency of the input side.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{5}{c}{**Switching State**} & \multicolumn{3}{c}{**DC-Link Voltage**} \\ \hline \(S_{ap1}\) & \(S_{bp1}\) & \(S_{cp1}\) & \(S_{an1}\) & \(S_{bn1}\) & \(S_{cn1}\) & \(V_{PO}\) & \(V_{ON}\) \\ \(S_{ap2}\) & \(S_{bp2}\) & \(S_{qr2}\) & \(S_{an2}\) & \(S_{bn2}\) & \(S_{cn2}\) & & \\ \hline
1 & 0 & 0 & 0 & 1 & 0 & \(v_{ab}\) & \(v_{ab}\) \\
1 & 0 & 0 & 0 & 0 & 1 & \(v_{ac}\) & \(v_{ac}\) \\
0 & 1 & 0 & 0 & 0 & 1 & \(v_{bc}\) & \(v_{bc}\) \\
0 & 1 & 0 & 1 & 0 & 0 & \(v_{ba}\) & \(v_{ba}\) \\
0 & 0 & 1 & 1 & 0 & 0 & \(v_{ca}\) & \(v_{ca}\) \\
0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The switching states at rectifier stage and the DC-link voltage.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Inverter**} & \multicolumn{3}{c}{**Switching State of Power Switches**} & \multirow{2}{*}{**Output-Phase Voltage \(V_{XO}\)**} \\ \cline{2-2} \cline{4-5}  & & \(X\) = (\(A\), \(B\), \(C\))** & & \\ \cline{2-2} \cline{4-5}  & \(S_{X1}\) & \(S_{X2}\) & \(S_{X3}\) & \(S_{X4}\) & \\ \hline
1 & 1 & 1 & 0 & 0 & \(V_{PO}\) \\
0 & 0 & 1 & 1 & 0 & 0 \\ \(-1\) & 0 & 0 & 1 & 1 & \(-V_{ON}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The state of inverter stage and the output voltage according to the switching state.  
The current space vectors of the rectifier are illustrated in Figure 4. There are six active vectors with fixed direction and three zero vectors. Each active vector represents the connection of the input phase voltage to the positive, neutral or negative point of DC-link bus. In each sector, the reference current space vector is synthesized by two neighbor active vectors.  
We assume that the reference current vector is located in sector 1 as shown in Figure 4. The duty cycles \(d_{\gamma}\), \(d_{\delta}\) of two active vectors \(I_{ab}\), \(I_{ac}\) are given below:  
\[d_{\gamma}=m_{i}\sin(\pi/6-\alpha_{i});d_{\delta}=m_{i}\sin(\pi/6+\alpha_{i}) \tag{2}\]  
where \(m_{i}\) is the modulation index in the rectifier stage and \(\alpha_{i}\) is the phase angular of the reference input current.  
Due to the elimination of zero vectors in the modulation of the rectifier stage to obtain the highest of DC-link voltage, the duty cycles of two active vectors is modified as following:  
\[d_{x}=\frac{d_{\gamma}}{d_{\delta}+d_{\gamma}};d_{y}=\frac{d_{\delta}}{d_{ \delta}+d_{\gamma}} \tag{3}\]  
The local average value of DC-link voltage in one sampling period which is generated by the upper and lower rectifier can be written as:  
\[V_{PO}=V_{ON}=d_{x}v_{ab}+d_{y}v_{ac}=\frac{3V_{in}^{2}}{2v_{a}} \tag{4}\]  
Therefore, the average DC-link voltage is  
\[V_{DC}=V_{PO}+V_{ON}=3V_{in}^{2}/v_{a} \tag{5}\]  
Similarly, Table 3 summarizes the average of DC-link voltage, duty cycles of active vectors according to the input current sector.  
Figure 4: The space vector diagram of the rectifier stage.