## 5 Single-Phase Direct Frequency Controller  
Single-phase direct AC-AC converters may also be employed to govern the output frequency in discrete steps. In these power converting topologies, the frequency at the output may be increased and decreased with respect to source frequency. Here the frequency step-up outputs are considered to explore the use of the pulse selective approach for their power quality concerns. Figure 7a,b depicts the outputs having frequencies two and three times the input frequency respectively. Figure 7: Variable frequency outputs with respect to input (**a**) double frequency and (**b**) triple frequency. The period of the required component of output in Figure 7a is equal to the period of the waveform as its average value during the output periods is zero. The harmonic coefficients of this output are calculated in Equations (31) and (32) by selecting the output pulse for the '\(\pi\)' period. \[a_{n}\ =\frac{1}{\pi}\left[\int\limits_{0}^{\pi/2}2V_{m}\cos(2n\omega t)\sin( \omega t)d(\omega t)-\int\limits_{\pi/2}^{\pi}2V_{m}\cos(2n\omega t)\sin(\omega t )d(\omega t)\right]=0 \tag{31}\]  
\[b_{n}\ \ =\frac{1}{\pi}\left[\int\limits_{0}^{\pi/2}2V_{m}\sin(2n\omega t)\sin( \omega t)d(\omega t)-\int\limits_{\pi/2}^{\pi}2V_{m}\sin(2n\omega t)\sin(\omega t )d(\omega t)\right] \tag{32}\]  
These coefficients can also be formulated by considering for '\(2\pi\)' period as in Equations (33) and (34). \[a_{n}\ =\frac{1}{2\pi}\left[ \begin{array}{c}\int\limits_{0}^{\pi/2}2V_{m} \cos(n\omega t)\sin(\omega t)d(\omega t)-\int\limits_{\pi/2}^{3\pi/2}2V_{m} \cos(n\omega t)\sin(\omega t)d(\omega t)\\ +\int\limits_{3\pi/2}^{2\pi}2V_{m}\cos(n\omega t)\sin(\omega t)d(\omega t) \end{array} \right]=0 \tag{33}\]  
\[b_{n}\ \ =\frac{1}{2\pi}\left[ \begin{array}{c}\int\limits_{0}^{\pi/2}2V_{m} \cos(n\omega t)\sin(\omega t)d(\omega t)-\int\limits_{\pi/2}^{3\pi/2}2V_{m} \cos(n\omega t)\sin(\omega t)d(\omega t)\\ +\int\limits_{2\pi}^{2\pi}2V_{m}\cos(n\omega t)\sin(\omega t)d(\omega t) \end{array} \right] \tag{34}\]  
\[=\left\{\frac{-2V_{m}}{\pi(n-1)}\cos(\frac{n\pi}{2})-\frac{-2V_{m}}{\pi(n+1)} \cos(\frac{n\pi}{2})\quad\text{for}\ \ n=\ 2,4,6,\ldots\]  
The output waveform of Figure 7b has a non-zero average value in the period of the required output voltage component as its output pulses are asymmetric along the \(y\)-axis. It means that the period of the required component is not the same as the period of the output pulses. Power quality analysis with the period of the output pulse leads to incorrect results. To tackle this issue, select the period of the waveform at which the average value of the waveform becomes zero and it is observed to be '\(2\pi\)'. The harmonic coefficients depending upon this period are evaluated in Equations (35) and (36). \[a_{n}\ =\frac{1}{2\pi}\left[ \begin{array}{c}\int\limits_{0}^{\pi/3}2V_{m} \cos(n\omega t)\sin(\omega t)d(\omega t)-\int\limits_{\pi/3}^{2\pi/3}2V_{m} \cos(n\omega t)\sin(\omega t)d(\omega t)\\ +\int\limits_{2\pi/3}^{4\pi/3}2V_{m}\cos(n\omega t)\sin(\omega t)d(\omega t)- \int\limits_{4\pi/3}^{5\pi/3}2V_{m}\cos(n\omega t)\sin(\omega t)d(\omega t) \\ +\int\limits_{5\pi/3}^{2\pi}2V_{m}\cos(n\omega t)\sin(\omega t)d(\omega t) \end{array} \right]=0 \tag{35}\]  
\[b_{n}\ \ =\frac{1}{2\pi}\left[ \begin{array}{c}\int\limits_{0}^{\pi/3}2V_{m} \sin(n\omega t)\sin(\omega t)d(\omega t)-\int\limits_{\pi/3}^{2\pi/3}2V_{m} \sin(n\omega t)\sin(\omega t)d(\omega t)\\ +\int\limits_{2\pi/3}^{4\pi/3}2V_{m}\sin(n\omega t)\sin(\omega t)d(\omega t)- \int\limits_{4\pi/3}^{5\pi/3}2V_{m}\sin(n\omega t)\sin(\omega t)d(\omega t) \\ +\int\limits_{5\pi/3}^{2\pi}2V_{m}\sin(n\omega t)\sin(\omega t)d(\omega t) \end{array} \right] \tag{36}\]  
\[=\left\{ \begin{array}{c}-\frac{2V_{m}}{\pi(n+1)}\left[\sin(\frac{n\pi}{3})+ \sqrt{3}\cos(\frac{n\pi}{3})\right]\\ +\frac{2V_{m}}{\pi(n-1)}\left[\sin(\frac{n\pi}{3})-\sqrt{3}\cos(\frac{n\pi}{3} )\right]\end{array}