## 7 Common-Mode Voltage Analysis  
As far as this paper is concerned with the CMV magnitude, the CMV waveform and its instantaneous function, \(v_{cm}\) is considered in every section of the sample time for the conventional and proposed schemes. For example, the CMV waveforms of Figure 4 for the first sector are considered. It can be seen that \(v_{cm}\) shows a staircase waveform. The mean square value of the CMV for one sample time, \(v_{cm-MS}^{2}\) is determined as follows [13]  
\[v_{cm-MS}^{2}=\frac{1}{T_{s}}\int_{t_{0}}^{t_{0}+T_{s}}v_{cm}^{2}dt. \tag{15}\]  
where \(T_{s}\) is the sampling time. Based on (13) and the CMV waveforms of Figure 4, the mean square CMV, \(v_{cm-MS}^{2}\) can be written as  
\[v_{cm-MS}^{2}=D_{PWM}(V_{dc}/18)^{2}. \tag{16}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Sector**} & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** & **15** & **16** & **17** & **18** \\ \hline \multirow{9}{*}{**Sectors**} & a & N & P & P & P & P & P & P & P & N & N & P & P & P & P & P & P & P & P & N \\  & b & P & N & N & P & P & P & P & P & P & P & N & N & P & P & P & P & P & P \\  & c & P & P & P & N & N & P & P & P & P & P & P & P & P & N & P & P & P & P & P \\  & d & P & P & P & P & P & N & N & P & P & P & P & P & P & P & P & N & N & P & P & P \\  & e & P & P & P & P & P & P & P & N & N & P & P & P & P & P & P & P & N & N \\  & f & N & N & P & P & P & P & P & P & P & N & N & P & P & P & P & P & P & P \\  & g & P & P & N & N & P & P & P & P & P & P & P & N & N & P & P & P & P & P \\  & h & P & P & P & P & P & N & N & P & P & P & P & P & P & P & N & N & P & P \\ \hline \multirow{9}{*}{**Sectors**} & a & P & P & P & P & P & P & P & P & P & P & N & N & N & N & N & N & N & N \\  & b & N & N & P & P & P & P & P & P & P & P & P & N & N & N & N & N & N & N \\  & c & N & N & N & P & P & P & P & P & P & P & P & P & N & N & N & N & N & N \\  & d & N & N & N & N & N & P & P & P & P & P & P & P & P & N & N & N & N & N \\  & e & N & N & N & N & N & N & N & N & P & P & P & P & P & P & P & P & N & N \\  & f & P & N & N & N & N & N & N & N & N & N & N & P & P & P & P & P & P & P \\  & g & P & P & P & N & N & N & N & N & N & N & N & N & P & P & P & P & P & P \\  & h & P & P & P & P & P & P & N & N & N & N & N & N & N & N & P & P & P & P \\  & i & P & P & P & P & P & P & P & N & N & N & N & N & N & N & N & N & P & P & P \\ \hline \hline \end{tabular} (P) letter denotes \(v_{tri}\) and the (N) letter denotes the opposite carrier of \(v_{tri}\).  
\end{table} 
Table 6: Selection of Carrier Waves for The CBPWM of the Analyzed PWM Schemes. where  
\[D_{\text{SVM}}=\left\{9^{2}(d_{0}+d_{511})+7^{2}(d_{256}+d_{503})+5^{2}(d_{384}+d_ {487})+3^{2}(d_{385}+d_{483})+d_{449}+d_{451}\right. \tag{17}\]  
\[D_{\text{AZS}}=\left\{5^{2}(d_{264}+d_{247})+7^{2}(d_{256}+d_{503})+5^{2}(d_{384 }+d_{487})+3^{2}(d_{385}+d_{483})+d_{449}+d_{451}\right. \tag{18}\]  
\[D_{\text{SV-10L}}=1 \tag{19}\]  
Based on (14) to (17), Figure 6a shows the dependency of RMS-CMV with the modulation index and the angle \(\theta\) in the first sector for the analyzed PWM schemes. As can be seen, the AZS scheme gives a lower CMV than the conventional SVM scheme, whereas the proposed SVM-10L scheme gives the minimum CMV with a constant magnitude of any value of \(M\). Consequently, the total RMS-CMV for the fundamental period, \(V_{cmf-RMS}\) is determined from  
\[V_{cm}(M)=\sqrt{\frac{9}{\pi}\int_{0}^{\pi/9}v_{cm-MS}^{2}(M,\ \theta)d\theta}. \tag{20}\]  
Solving (20) for the analyzed PWM schemes yields  
\[\frac{V_{cm}^{2}}{V_{dc}^{2}}=\left\{\  
Figure 7: Simulation results for a nine-phase motor based on the conventional SVM and proposed schemes.