# 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
#### 2.2.2 Inverter Stage Control
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Space Vectors** & **Switching States** & \(V_{\mathbf{a}}\) & \(V_{\mathbf{\beta}}\) & **Type of Vector** &
\begin{tabular}{c} **Vector** \\ **Magnitude** \\ \end{tabular} \\ \hline \(\overset{\rightarrow}{V_{0}}\) & (-1,-1,-1) & 0 & 0 & \\ \(\overset{\rightarrow}{V_{0}}\) & (0,0,0) & 0 & 0 & Zero vector & 0 \\  & (1,1,1) & 0 & 0 & \\ \(\overset{\rightarrow}{V_{1}}\) & (1,0,0) & \(V_{DC}/3\) & 0 & \\ \(\overset{\rightarrow}{V_{1}}\) & (0,-1,-1) & \(V_{DC}/3\) & 0 & \\ \(\overset{\rightarrow}{V_{2}}\) & (1,1,0) & \(V_{DC}/6\) & \(\sqrt{3}V_{DC}/6\) & \\ \(\overset{\rightarrow}{V_{2}}\) & (0,0,-1) & \(V_{DC}/6\) & \(\sqrt{3}V_{DC}/6\) & \\ \(\overset{\rightarrow}{V_{3}}\) & (0,1,0) & \(-V_{DC}/6\) & \(\sqrt{3}V_{DC}/6\) & \\ \(\overset{\rightarrow}{V_{3}}\) & (-1,0,-1) & \(-V_{DC}/6\) & \(\sqrt{3}V_{DC}/6\) & Small vector & \(\frac{1}{3}\)V\({}_{\text{DC}}\) \\ \(\overset{\rightarrow}{V_{4}}\) & (0,1,1) & \(-V_{DC}/3\) & 0 & \\ \(\overset{\rightarrow}{V_{4}}\) & (-1,0,0) & \(-V_{DC}/3\) & 0 & \\ \(\overset{\rightarrow}{V_{5}}\) & (0,0,1) & \(-V_{DC}/6\) & \(-\sqrt{3}V_{DC}/6\) & \\ \(\overset{\rightarrow}{V_{5}}\) & (-1,-1,0) & \(-V_{DC}/6\) & \(-\sqrt{3}V_{DC}/6\) & \\ \(\overset{\rightarrow}{V_{6}}\) & (1,0,1) & \(V_{DC}/6\) & \(-\sqrt{3}V_{DC}/6\) & \\ \(\overset{\rightarrow}{V_{6}}\) & (0,-1,0) & \(V_{DC}/6\) & \(-\sqrt{3}V_{DC}/6\) & \\ \(\overset{\rightarrow}{V_{7}}\) & (1,0,-1) & \(V_{DC}/2\) & \(\sqrt{3}V_{DC}/6\) & \\ \(\overset{\rightarrow}{V_{8}}\) & (0,1,-1) & 0 & \(\sqrt{3}V_{DC}/3\) & \\ \(\overset{\rightarrow}{V_{9}}\) & (-1,1,0) & \(-V_{DC}/2\) & \(\sqrt{3}V_{DC}/6\) & Medium Vector & \(\frac{\sqrt{3}}{3}\)V\({}_{\text{DC}}\) \\ \(\overset{\rightarrow}{V_{10}}\) & (-1,0,1) & \(-V_{DC}/2\) & \(-\sqrt{3}V_{DC}/6\) & \\ \(\overset{\rightarrow}{V_{11}}\) & (0,-1,1) & 0 & \(-\sqrt{3}V_{DC}/3\) & \\ \(\overset{\rightarrow}{V_{12}}\) & (1,–1,0) & \(V_{DC}/2\) & \(-\sqrt{3}V_{DC}/6\) & \\ \(\overset{\rightarrow}{V_{13}}\) & (1,–1,-1) & \(2V_{DC}/3\) & 0 & \\ \(\overset{\rightarrow}{V_{14}}\) & (1,1,-1) & \(V_{DC}/3\) & \(\sqrt{3}V_{DC}/3\) & \\ \(\overset{\rightarrow}{V_{15}}\) & (–1,1,-1) & \(-V_{DC}/3\) & \(\sqrt{3}V_{DC}/3\) & Large vector & \(\frac{2}{3}\)V\({}_{\text{DC}}\) \\ \(\overset{\rightarrow}{V_{16}}\) & (–1,1,1) & \(-2V_{DC}/3\) & 0 & \\ \(\overset{\rightarrow}{V_{17}}\) & (–1,–1,1) & \(-V_{DC}/3\) & \(-\sqrt{3}V_{DC}/3\) & \\ \(\overset{\rightarrow}{V_{18}}\) & (1,–1,1) & \(V_{DC}/3\) & \(-\sqrt{3}V_{DC}/3\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Space vectors, switching states and type of vectors. Figure 5: The space vector diagram of the 3-level T-Type NPC inverter stage. In the three-phase load, the CMV can be defined as a voltage between motor neutral point and ground point of power supply. The CMV can be described as follows:  
\[ \begin{array}{l}v_{A}-v_{CM}=Ri_{A}+L(di_{A}/dt)\\ v_{B}-v_{CM}=Ri_{B}+L(di_{B}/dt)\\ v_{C}-v_{CM}=Ri_{C}+L(di_{C}/dt)\end{array}  \tag{7}\]  
where \(v_{CM}\) is the common mode voltage; \(i_{A}\), \(i_{B}\), \(i_{C}\) are three-phase output currents; \(R\)\(L\) are resistor and inductor of the load. Under the assumption that the sum of output currents is equal to zero, \(i_{A}+i_{B}+i_{C}=0\). The CMV can be obtained as follows:  
\[v_{CM}=\frac{v_{A}+v_{B}+v_{C}}{3} \tag{8}\]  
From (7), Table 5 shows the CMV according to the state of the inverter stage. From Table 5, it can be easily seen that the magnitude of CMV has seven values: 0, \(\pm\)V\({}_{\text{DC}}\)/6, \(\pm\)V\({}_{\text{DC}}\)/3, \(\pm\)V\({}_{\text{DC}}\)/2. In these 27 vectors, one zero and six medium vectors produce the ZCMV. The implementation of ZCMV modulation using an SVPWM approach is shown in Figure 6. It is assumed that the reference output voltage vector is in sector 1.