## 2 Proposed Cycloconverter
### Harmonic Analysis of Proposed Cycloconverter  
In this subsection, the details of the harmonics spectrum of the proposed topology have been illustrated. The output voltage of a conventional cycloconverter can be written in terms of Fourier series as below:  
\[V_{0}(\omega t)=a_{0}+\sum_{n=1}^{\infty}a_{n}cos(\frac{n\omega t}{m})+b_{n}sin( \frac{n\omega t}{m}) \tag{1}\]  
where, \(m\) is the conversion ratio and n is the harmonic order. Here,  
\[ \begin{array}{c}a_{0}=\frac{1}{6\pi}\int_{0}^{\pi}V_{0}d(wt)\\ =\frac{1}{6\pi}\left[\int_{0}^{\pi}(V_{m})sinwtd(wt)+\int_{\pi}^{2\pi}(V_{m}) sinwtd(wt)+\int_{2\pi}^{3\pi}(V_{m})sinwtd(wt)+\right.\\ \left.\int_{3\pi}^{4\pi}(-V_{m})sinwtd(wt)+\int_{4\pi}^{5\pi}(-V_{m})sinwtd(wt) +\int_{5\pi}^{6\pi}(-V_{m})sinwtd(wt)\right]=0\end{array}  \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Different** & **T1** & **T2** & **T3** & **T4** & **T5** & **Output** \\ \hline Mode 1 & 1 & 0 & 0 & 1 & 0 & \(V_{m}/2\) \\ Mode 2 & 0 & 1 & 1 & 0 & 0 & \(V_{m}\) \\ Mode 3 & 1 & 0 & 0 & 0 & 1 & \(V_{m}\) \\ Mode 4 & 0 & 0 & 1 & 1 & 0 & \(V_{m}/2\) \\ Mode 5 & 0 & 0 & 1 & 1 & 0 & \(-V_{m}/2\) \\ Mode 6 & 1 & 0 & 0 & 0 & 1 & \(-V_{m}\) \\ Mode 7 & 0 & 1 & 1 & 0 & 0 & \(-V_{m}\) \\ Mode 8 & 1 & 0 & 0 & 1 & 0 & \(-V_{m}/2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Switching states of (1/3) and (1/4) frequency conversion of Input Frequency of the proposed model. Figure 5: Cycloconverter detailed operation of the eight modes for (1/3) and (1/4) frequency conversion of input frequency. \[ \begin{array}{c}a_{n}=\frac{1}{3\pi}\int_{0}^{6\pi}V_{0}cos(\frac{n\omega t}{m} )d(wt)\\ =\frac{1}{3\pi}[\int_{0}^{\pi}(V_{m})sinwt\ cos(\frac{n\omega t}{m})d(wt)\\ +\int_{2\pi}^{2\pi}(V_{m})sinwt\ cos(\frac{n\omega t}{m})d(wt)\\ +\int_{2\pi}^{3\pi}(V_{m})sinwt\ cos(\frac{n\omega t}{m})d(wt)\\ +\int_{3\pi}^{4\pi}(-V_{m})sinwt\ cos(\frac{n\omega t}{m})d(wt)\\ +\int_{4\pi}^{5\pi}(-V_{m})sinwt\ cos(\frac{n\omega t}{m})d(wt)\\ +\int_{5\pi}^{6\pi}(-V_{m})sinwt\ cos(\frac{n\omega t}{m})d(wt)]\\ =0\end{array}  \tag{3}\]  
and,  
\[ \begin{array}{c}b_{n}=\frac{1}{3\pi}\int_{0}^{6\pi}V_{0}sin(\frac{n\omega t} {m})d(wt)\\ =\frac{1}{3\pi}[\int_{0}^{\pi}(V_{m})sinwt\ sin\ (\frac{n\omega t}{m})d(wt)+\int_{ \pi}^{2\pi}(V_{m})sin(\frac{n\omega t}{m})d(wt)\\ +\int_{2\pi}^{3\pi}(V_{m})sinwt\ sin(\frac{n\omega t}{m})d(wt)\\ +\int_{3\pi}^{4\pi}(-V_{m})sinwt\ sin(\frac{n\omega t}{m})d(wt)\\ +\int_{4\pi}^{5\pi}(-V_{m})sinwt\ sin(\frac{n\omega t}{m})d(wt)\\ +\int_{5\pi}^{6\pi}(-V_{m})sinwt\ sin(\frac{n\omega t}{m})d(wt)]\end{array}  \tag{4}\]  
after simplification,  
\[V_{0}(\omega t)=\frac{V_{m}}{3}sin(\omega t)+\sum_{ \begin{array}{c}n=1,5,7 \ldots\\ n\
eq 3\end{array} }^{\infty}\frac{24V_{m}sin(\frac{n\omega t}{3})}{\pi(m^{2}-n^{2 })}sin(\frac{n\omega t}{m}) \tag{5}\]  
The first term of Equation (5) represents the third harmonics components, and the remaining term represents the other harmonics component without third harmonics value. The fundamental component of the conventional cycloconverter can be written as:  
\[V_{01}(\omega t)=\frac{24V_{m}sin(\frac{\omega t}{3})}{\pi(m^{2}-1)}sin(\frac {\omega t}{3}) \tag{6}\]  
In the case of the proposed one-third cycloconverter, the output has six half cycles, but their values are same as a conventional cycloconverter. The output voltage of the second and fifth half cycle is \(V_{m}\) and the output of the remaining half cycles is \(\frac{V_{m}}{2}\). Therefore, the output of the proposed cycloconverter can be written as below in terms of Fourier series expansion. \[V_{0}(\omega t)=\sum_{ \begin{array}{c}n=1,5,7\ldots\\ n\
eq 3\end{array} }^{\infty}\frac{18V_{m}sin(\frac{n\pi}{3})}{\pi(m^{2}-n^{2})} sin(\frac{n\omega t}{3}) \tag{7}\]  
The fundamental frequency of output voltage can be obtained by using \(n=1\).