## 2 Harmonic Analysis of DC-Link Capacitor Current for One VSI
 \tag{19}\]When \(m+n=2h+1\) (odd), the \(A_{mn}\) coefficient is zero according to Equation (11). Considering the restrictions on m and n, when m is an even integer and \(n=6h\pm 1\), there is a harmonic content due to the trigonometric identities. Similarly, if m is an odd integer and \(n=6h\pm 2\), there is also a harmonic content. A further simplified version of Equation (19) can be written as  
\[ \begin{array}{ll}i_{nw\_4 th}=\frac{3}{2}A_{mn}I_{0}cos(m(w_{c}t+\theta_{c})+( n+1)w_{0}t-\O)&\{m=2h_{1},n=6h_{2}-1\\ \frac{3}{2}A_{mn}I_{0}cos(m(w_{c}t+\theta_{c})+(n-1)w_{0}t+\O)&\{m=2h_{1},n=6h_{ 2}+1\\ \frac{3}{2}A_{mn}I_{0}cos(m(w_{c}t+\theta_{c})+(n+1)w_{0}t-\O)&\{m=2h_{1}+1,=6h_ {2}-4\\ \frac{3}{2}A_{mn}I_{0}cos(m(w_{c}t+\theta_{c})+(n-1)w_{0}t+\O)&\{m=2h_{1}+1,=6h_ {2}+4\\ \end{array}  \tag{20}\]  
Since the first and third terms of the inverter input current are zero, and the second term of the inverter input current is the average current supplied by the battery, the total harmonic content of the dc-link capacitor for a three-phase VSI is equal to the fourth term of the inverter input current  
\[i_{cap}=i_{inv\_4 th}\;\;for\;a\;three\;-\;phase\;VSI \tag{21}\]  
As shown in Figure 2, the main harmonic component of the dc-link capacitor current is the second carrier harmonic, and it is maximum in the middle range of the modulation index and at a power factor close to unity. The second carrier harmonic component of the dc-link capacitor current for a VSI can be simplified using Equation (20), with its restrictions (m, n) and trigonometric identities, as follows:  
\[i_{cap\_2 nd}^{*}=\frac{3}{2}A_{nm}I_{0}cos(m(w_{c}t+\theta_{c})+(n+1)w_{0}t- \O)+\frac{3}{2}A_{nm}I_{0}cos(m(w_{c}t+\theta_{c})+(n-1)w_{0}t+\O) \tag{22}\]  
\[i_{cap\_2 nd}^{*}=3A_{2 nd}^{*}I_{0}^{*}cos(2w_{c}t+2\theta_{c1})cos\O_{1} \tag{23}\]  
where \(i_{cap\_2 nd}^{*}\) is the second carrier harmonic components for the first VSI. \(A_{2 nd}^{*}\) is the second carrier harmonic component coefficient of the dc-link capacitor current (\(m=2h_{1}\) and \(n=6h_{2}\pm 1\)) for the first VSI. \(I_{0}^{*}\) is the output phase current, \(\theta_{c1}\) is the initial phase angle of the carrier harmonic component, and \(\O_{1}\) is the power factor angle of the first VSI. Figure 2: The harmonic components of the dc-link capacitor current for various modulation indices: **(a)** pf = 1, **(b)** pf = 0.966, **(c)** pf = 0.866, and **(d)** pf = 0.707. The harmonic components of the dc-link capacitor current for different modulation indices with a power factor of 0.966 are given in Table 2. The magnitude of the harmonics are normalized according to the phase currents. As shown, the most significant harmonic component is the second carrier harmonic, followed by the fourth, sixth, eighth, and tenth carrier harmonics, similar to the results in Figure 2.  
Reduction in DC-Link Capacitor Current by Phase Shifting of Carrier Wave for Dual PMSM Driver System  
In the dual VSIs, two PMSMs are powered by two VSIs, both of which use the same dc-link voltage and dc-link capacitor, as shown in Figure 1. Hence, the current of the dc-link capacitor is equal to the sum of the harmonic component of each of the VSIs. Since the most significant harmonic component is the second carrier harmonic, if the magnitude of the second carrier harmonic is decreased, the dc-link capacitor current can be reduced. The second carrier harmonic of the dc-link capacitor current for the dual VSIs dual PMSMs system can be represented, using Equation (23), as follows:  
\[i_{cap\_2 nd}=i_{cap\_2 nd}^{*}+i_{cap\_2 nd}^{**} \tag{24}\]  
\[i_{cap\_2 nd}=3A_{2 nd}^{*}I_{0}^{*}cos(2w_{c}t+2\theta_{c1})cos\Theta_{1}+3A_{2 nd}^{**}I_{0}^{*}cos(2w_{c}t+2\theta_{c2})cos\Theta_{2} \tag{25}\]