## 2 Harmonic Analysis of DC-Link Capacitor Current for One VSI
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **M = 0.2** & **M = 0.4** & **M = 0.6** & **M = 0.8** & **M = 1** & **M = 1.154** \\ \hline \(\rm{f_{c}\pm 6w_{0}}\) & 0 & 0 & 0.008 & 0 & 0 & 0 \\ \hline \(\rm{f_{c}\pm 3w_{0}}\) & 0 & 0.008 & 0.017 & 0.029 & 0.042 & 0.052 \\ \hline \(\rm{f_{c}}\) & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \(2\rm{f_{c}\pm 6w_{0}}\) & 0 & 0.006 & 0.019 & 0.038 & 0.056 & 0.065 \\ \hline \(2\rm{f_{c}\pm 3w_{0}}\) & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \(2\rm{f_{c}}\) & 0.2872 & 0.5013 & 0.5885 & 0.5282 & 0.3383 & 0.1352 \\ \hline \(3\rm{f_{c}\pm 3w_{0}}\) & 0.006 & 0.019 & 0.028 & 0.025 & 0 & 0.016 \\ \hline \(4\rm{f_{c}\pm 6w_{0}}\) & 0 & 0.02 & 0.033 & 0.018 & 0.028 & 0.06 \\ \hline \(4\rm{f_{c}\pm 3w_{0}}\) & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \(4\rm{f_{c}\pm 3w_{0}}\) & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \(4\rm{f_{c}}\) & 0.2506 & 0.2641 & 0.035 & 0.2082 & 0.2313 & 0.082 \\ \hline \(5\rm{f_{c}\pm 3w_{0}}\) & 0.008 & 0.018 & 0 & 0.023 & 0.028 & 0 \\ \hline \(6\rm{f_{c}\pm 6w_{0}}\) & 0.006 & 0.022 & 0 & 0.0439 & 0.0138 & 0.04 \\ \hline \(6\rm{f_{c}\pm 3w_{0}}\) & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \(6\rm{f_{c}}\) & 0.1962 & 0.0234 & 0.17 & 0.0153 & 0.162 & 0.069 \\ \hline \(7\rm{f_{c}\pm 3w_{0}}\) & 0.009 & 0 & 0.020 & 0.009 & 0.02 & 0.012 \\ \hline \(8\rm{f_{c}\pm 6w_{0}}\) & 0.009 & 0.01 & 0.0329 & 0.008 & 0.03 & 0.03 \\ \hline \(8\rm{f_{c}\pm 3w_{0}}\) & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \(8\rm{f_{c}}\) & 0.1321 & 0.104 & 0.012 & 0.109 & 0.096 & 0.06 \\ \hline \(9\rm{f_{c}\pm 3w_{0}}\) & 0.009 & 0.008 & 0.010 & 0.017 & 0.008 & 0.012 \\ \hline \(10\rm{f_{c}\pm 6w_{0}}\) & 0.0112 & 0.011 & 0.008 & 0.0247 & 0.034 & 0.024 \\ \hline \(10\rm{f_{c}\pm 3w_{0}}\) & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \(10\rm{f_{c}}\) & 0.0677 & 0.093 & 0.097 & 0.0768 & 0.039 & 0.052 \\ \hline \end{tabular}
\end{table} 
Table 2: Harmonic component of the dc-link capacitor current with pf = 0.966 for different modulation indices. where \(i^{**}_{cap\_2 nd}\) is the second carrier harmonic components for the second VSI. \(A^{**}_{2 nd}\) is the coefficient of the second carrier harmonic component of the dc-link capacitor for the second VSI, respectively; \(I^{**}_{0}\) is the output phase currents, \(\theta_{c2}\) is the initial phase angle of the carrier harmonic component, and \(\mathcal{O}_{2}\) is the power factor angle of the second VSI. In order to reduce the total dc-link capacitor current, the second carrier harmonics of VSIs must cancel each other out. If a phase difference is introduced between the cosine terms of the second carrier harmonics in Equation (25), these terms can cancel each other out. Note that for sinusoidal signals, maximum cancellation occurs when there is a \(\pi\) radian difference between the cosine terms. But because the frequency of the second carrier harmonics is twice that of the fundamental, there should be a \(\pi/2\) phase difference between the carrier waves, as shown in Figure 3. Since the switching frequency of the VSIs is constant, there must be a \(\pi/2\) phase difference between the initial phases of the carrier harmonics for maximum cancellation of the second carrier harmonics, which can be described by the following equations:  
\[cos(2w_{c}t+2\theta_{c1})=-cos(2w_{c}t+2\theta_{c2}) \tag{26}\]  
\[2\theta_{c2}-2\theta_{c1}=\pm\pi,\,\pm 3\pi,\,\pm 5\pi\dots\dots. \tag{27}\]  
\[\Delta\theta_{c}=\theta_{c2}-\theta_{c1}=\pm\frac{\pi}{2},\,\pm\frac{3\pi}{2},\,\pm\frac{5\pi}{2}\dots\dots. \tag{28}\]  
The harmonic spectrum of the dc-link capacitor current for the dual VSI dual PMSM system without phase difference is shown in Figure 4a. The harmonic magnitude is normalized according to the phase current of the VSIs. While the second carrier harmonic content is doubled in the dual VSI dual PMSM system with no phase difference between the carrier waves, the second carrier harmonic content is diminished in the dual VSI dual PMSM system with a \(\pi/2\) radian phase difference between the carrier waves, as shown in Figure 4b.