## 2 Harmonic Analysis of DC-Link Capacitor Current for One VSI
The \(A_{mn}\) and \(B_{mn}\) coefficients should be evaluated for specific values of \"\(m\)\" and \"\(n\)\" for each PWM period. The double Fourier series coefficients of the switching function are represented as follows [22]:  
\[A_{mn}+jB_{mn}=\frac{1}{2\pi^{2}}\sum_{i=1}^{6}\int_{y_{n}(i)}^{y_{e}(i)}\int_{ x_{down}}^{x_{up}}S(t)*e^{j(mx+ny)}dxdy \tag{7}\]  
where the inner and outer integral limits are defined in Table 1. Since the SVPWM technique is composed of six parts for one fundamental cycle of the reference waves, the outer and inner integral terms are divided into six segments. The magnitude of the harmonic coefficients in Equation (6) can be calculated using Equation (7) and are summarized below [22]:  
\[A_{00}+jB_{00}=1 \tag{8}\]  
\[A_{01}+jB_{01}=\frac{M}{2} \tag{9}\]  
\[A_{0n}+jB_{0n}=M\frac{3\sqrt{3}}{2\pi(n^{2}-1)}sin\Big{(}n\frac{\pi}{6}\Big{)} sin\Big{(}n\frac{\pi}{2}\Big{)}\;\;for\;n=3,\;9,\;15\;\ldots. \tag{10}\]  
\[A_{mn}+jB_{mn}=\frac{4}{m\pi^{2}}\left[\begin{array}{c}\frac{\pi}{6}sin\big{(} (m+n)\frac{\pi}{2}\big{)}\Big{(}J_{n}\big{(}m\frac{3\pi}{4}M\big{)}+2 cos\frac{ \pi}{6}J_{n}\Big{(}m\frac{\sqrt{3}}{4}M\big{)}\Big{)}\\ +\frac{1}{n}sin\frac{\pi}{2}cos\frac{\pi}{2}sim\frac{\pi}{6}\Big{(}J_{0}\big{(} m\frac{3\pi}{4}M\big{)}-J_{0}\Big{(}m\frac{\sqrt{3}}{4}M\big{)}\Big{)}\Big{|}_{n \
eq 0}\\ +\sum\limits_{ \begin{subarray}{c}k=1\\ (k\
eq n)\end{subarray} }^{\infty}\left[ \begin{array}{c}\frac{1}{n+k}sin\big{(} (m+k)\frac{\pi}{2}\big{)}cos\big{(}(n+k)\frac{\pi}{2}\big{)}sin\big{(}(n+k) \frac{\pi}{6}\big{)}\\ \times\Big{\{}J_{k}\big{(}m\frac{3\pi}{4}M\big{)}+2 cos\big{(}(2n+3k)\frac{\pi} {6}\big{)}J_{k}\big{(}m\frac{\sqrt{3}}{4}M\big{)}\Big{\}}\Big{]}\end{array}  \right] \tag{11}\]  
where \(M\) is the modulation index; \(J_{(n)}(\xi)\) and \(J_{(0)}(\xi)\) are the Bessel functions of the first order \"\(n\)\" and \"\(k\)\", respectively. Coefficients of the imaginary part (\(B_{mn}\)) are zero for all equations. Equation (6) can be simplified and represented as in the following equations for each phase using Equations (8)-(11). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**i** & **y\({}_{\mathbf{s}}\)(i)** & **y\({}_{\mathbf{e}}\)(i)** & **X\({}_{\mathbf{down}}\)** & **X\({}_{\mathbf{up}}\)** \\ \hline
1 & \(\frac{2\pi}{3}\) & \(\pi\) & \(-\frac{\pi}{2}\Big{[}1+\frac{\sqrt{3}}{2}M\text{cos}\big{(}y+\frac{\pi}{6} \big{)}\Big{]}\) & \(\frac{\pi}{2}\Big{[}1+\frac{\sqrt{3}}{2}M\text{cos}\big{(}y+\frac{\pi}{6}\big{)} \Big{]}\) \\ \hline
2 & \(\frac{\pi}{3}\) & \(\frac{2\pi}{3}\) & \(-\frac{\pi}{2}\big{[}1+\frac{3}{2}M\text{cos}\,y\big{]}\) & \(\frac{\pi}{2}\big{[}1+\frac{3}{2}M\text{cos}\,y\big{]}\) \\ \hline
3 & 0 & \(\frac{\pi}{3}\) & \(-\frac{\pi}{2}\Big{[}1+\frac{\sqrt{3}}{2}M\text{cos}\big{(}y-\frac{\pi}{6} \big{)}\Big{]}\) & \(\frac{\pi}{2}\Big{[}1+\frac{\sqrt{3}}{2}M\text{cos}\big{(}y-\frac{\pi}{6}\big{)} \Big{]}\) \\ \hline
4 & \(-\frac{\pi}{3}\) & 0 & \(-\frac{\pi}{2}\Big{[}1+\frac{\sqrt{3}}{2}M\text{cos}\big{(}y+\frac{\pi}{6} \big{)}\Big{]}\) & \(\frac{\pi}{2}\Big{[}1+\frac{\sqrt{3}}{2}M\text{cos}\big{(}y+\frac{\pi}{6} \big{)}\Big{]}\) \\
5 & \(-\frac{2\pi}{3}\) & \(-\frac{\pi}{3}\) & \(-\frac{\pi}{2}\big{[}1+\frac{3}{2}M\text{cos}\,y\big{]}\) & \(\frac{\pi}{2}\big{[}1+\frac{3}{2}M\text{cos}\,y\big{]}\) \\ \hline
6 & \(-\pi\) & \(-\frac{2\pi}{3}\) & \(-\frac{\pi}{2}\Big{[}1+\frac{\sqrt{3}}{2}M\text{cos}\big{(}y-\frac{\pi}{6} \big{)}\Big{]}\) & \(\frac{\pi}{2}\Big{[}1+\frac{\sqrt{3}}{2}M\text{cos}\big{(}y-\frac{\pi}{6} \big{)}\Big{]}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Inner and outer limits of double Fourier series form [22]. \[S_{A}(t)=\frac{A_{00}}{2}+A_{01}\text{cos}(w_{0}t)+\sum\limits_{n=2}^{n=\infty}A_{0n }\text{cos}(nw_{0}t)+\sum\limits_{m=1}^{m=\infty}\sum\limits_{n=-\infty}^{n= \infty}A_{mn}\text{cos}(m(w_{c}t+\theta_{c})+nw_{0}t) \tag{12}\]  
\[