## 4 Conventional SVM for Nine-Phase Voltage Source Inverter  
This section reviews the conventional SVM scheme for the nine-phase VSI given in [23] to address the CMV problems adequately. In this scheme, eight active vectors and the two zero vectors (0, 511) are utilized in each sector to obtain the reference vector \(\left(v_{\alpha\beta}^{*}\right)\). The eight active vectors are selected from \(\alpha-\beta\) subspace to eliminate the components of the x-y subspaces [24]. Then, the selected vectors should be organized to minimize the switching stresses and obtain symmetrical SVM. Figures 2(a) and 3(a) show the selected vectors in sector 1, the corresponding switching sequence, and the CMV waveform, respectively. The eight vectors' duty cycles can be determined by  
\[ \begin{bmatrix}v_{\alpha}^{*}\\ v_{\beta}^{*}\\ v_{\beta}^{*}\\ v_{\alpha\gamma}^{*}\\ v_{\beta\gamma}^{*}\\ v_{\gamma\delta}^{*}\\ v_{\beta\gamma}^{*}\\ v_{\gamma\delta}^{*}\\ v_{\beta\gamma}^{*}\\ v_{\beta\delta}^{*}\\ \end{bmatrix} = \begin{bmatrix}v_{1}\alpha&v_{2}\alpha&v_{3}\alpha&v_{4} \alpha&v_{5}\alpha&v_{6}\alpha&v_{7}\alpha&v_{8}\alpha\\ v_{1}\beta&v_{2}\beta&v_{3}\beta&v_{4}\beta&v_{5}\beta&v_{6}\beta&v_{7}\beta&v_ {8}\beta\\ v_{1}x1&v_{2}x1&v_{3}x1&v_{4}x1&v_{5}x1&v_{6}x1&v_{7}x1&v_{8}x1\\ v_{1}y1&v_{2}y1&v_{3}y1&v_{4}y1&v_{5}y1&v_{6}y1&v_{7}y1&v_{8}y1\\ v_{1}x2&v_{2}x2&v_{3}x2&v_{4}x2&v_{5}x2&v_{6}x2&v_{7}x2&v_{8}x2\\ v_{1}y2&v_{2}y2&v_{3}y2&v_{4}y2&v_{5}y2&v_{6}y2&v_{7}y2&v_{8}y2\\ v_{1}x3&v_{2}x3&v_{3}x3&v_{4}x3&v_{5}x3&v_{6}x3&v_{7}x3&v_{8}x3\\ v_{1}y3&v_{2}y3&v_{3}y3&v_{4}y3&v_{5}y3&v_{6}y3&v_{7}y3&v_{8}y3\\ \end{bmatrix}\begin{bmatrix}d_{\alpha 1}\\ d_{\alpha 2}\\ d_{\alpha 3}\\ d_{\alpha 4}\\ d_{\beta 1}\\ d_{\beta 2}\\ d_{\beta 3}\\ d_{\beta 4}\\ \end{bmatrix}  \tag{6}\]  
where the subscripts \(\alpha 1\to\alpha 4\), and \(\beta 1\to\beta 4\) refer to the eight active vectors (\(\alpha 1\equiv 451\), \(\alpha 2\equiv 385\), \(\alpha 3\equiv 487\), \(\alpha 4\equiv 256\), \(\beta 1\equiv 449\), \(\beta 2\equiv 483\), \(\beta 3\equiv 384\), \(\beta 4\equiv 503\)), and the subscripts \(1\alpha\to 8\alpha\), \(1\beta\to 8\beta\), \(1x1\to 8x1\), \(1y1\to 8y1\), \(1x2\to\)\(8x2\), \(1y2\to 8y2\), \(1x3\to 8x3\), \(1y3\to 8y3\) refer to the components of the vectors in different subspaces. Figure 3: Selected space-vectors of sector 1 in \(\alpha-\beta\) subspace corresponding to the traditional and proposed SVM technique. Note that, the asterisks (*) denote the reference. The reference voltage vectors for the \(x-y\) planes of (6) are set to zero to nullify the harmonic components. Hence, the duty cycles of the active vectors for sinusoidal output voltages can be determined by solving the matrix given in (6) for \(V_{x1y1}^{*}=V_{x2y2}^{*}=V_{x3y3}^{*}=0\). Solving this matrix yields  
\[ \begin{cases}d_{a1}=0.6737v_{a}^{*}-1.6972v_{\beta}^{*}\\ d_{a2}=0.5920v_{a}^{*}-1.4913v_{\beta}^{*}\\ d_{a3}=0.4388v_{a}^{*}-1.1055v_{\beta}^{*}\\ d_{a4}=0.2348v_{\alpha}^{*}-0.5916v_{\beta}^{*}\\ d_{\beta 1}=1.8043v_{\beta}^{*}\\ d_{\beta 2}=1.5891v_{\beta}^{*}\\ d_{\beta 3}=1.1783v_{\beta}^{*}\\ d_{\beta 4}=0.6260v_{\beta}^{*}\end{cases}  \tag{7}\]  
If the summation of the duty cycles corresponding to \(\alpha\)- and \(\beta\)-axis is assumed to be \(d_{\alpha}\) and \(d_{\beta}\), respectively, hence  
\[ \begin{cases}d_{a1}+d_{a2}+d_{a3}+d_{a4}=d_{\alpha}\\ d_{\beta 1}+d_{\beta 2}+d_{\beta 3}+d_{\beta 4}=d_{\beta}.\end{cases}  \tag{8}\]  
Accordingly  
\[ \begin{cases}d_{a1}/d_{\alpha}=d_{\beta 1}/d_{\beta}=0.1205\\ d_{a2}/d_{\alpha}=d_{\beta 2}/d_{\beta}=0.2268\\ d_{a3}/d_{\alpha}=d_{\beta 3}/d_{\beta}=0.3055\\ d_{a4}/d_{\alpha}=d_{\beta 4}/d_{\beta}=0.3473\end{cases}  \tag{9}\]  
\[\left\{ \begin{array}{c}d_{\alpha}=V_{o}^{*}\sin(\pi/9-\theta)/(V_{l}\sin( \pi/9))\\ d_{\beta}=V_{o}^{*}sin(\theta)/(V_{l}\sin(\pi/9))\end{array} \right. \tag{10}\]  
where \(V_{l}\) is the largest active vector magnitude, which equals \(0.64V_{dc}\)[22], \(\theta\) indicates the reference vector position, and \(V_{o}^{*}\) is the reference vector length.