## 2 Suggested Topology and Its Operation
### Output Voltage and Frequency Control
#### 2.3.2 Output Frequency Control  
The output frequency regulation in discrete steps can be achieved by operating the proposed topology in non-inverting and inverting states according to the requirements. Figures 9 and 10 depict the gating control schemes for output frequency regulation realized in buck and boost mode respectively.  
Figure 8: Output rms voltage control: (**a**) half-wave control; (**b**) full-wave control; (**c**) on-off cycle control; (**d**) PWM chopping.  
In Figure 9a-c, the frequency at the output is controlled to one-half, one-third, and one-fourth of the input respectively. The required outputs can be realized by forcing the developed circuit to operate in non-inverting and inverting states. These characteristics are ensured by setting the control signals \(c_{1}\) and \(c_{2}\) to logic high for the first and second half intervals of the output voltage. The instantaneous characteristics of the output voltage waveforms of Figure 9a-c can be mathematically realized in Equations (11) to (13) respectively.  
Figure 9: Output step down frequency control with respect to input frequency: (**a**) one-half; (**b**) one-third; (**c**) one-fourth.  
\[v_{o}(\omega t) = \left\{\begin{array}{cc}+V_{m}\sin(\omega t)&0\leq\omega t\leq\pi& \&\&3\pi\leq\omega t\leq 4\pi\\ -V_{m}\sin(\omega t)&\pi\leq\omega t\leq 3\pi\end{array}\right. \tag{11}\] \[v_{o}(\omega t) = \left\{\begin{array}{cc}+V_{m}\sin(\omega t)&0\leq\omega t\leq \pi&\&2\pi\leq\omega t\leq 4\pi\\ &5\pi\leq\omega t\leq 6\pi\end{array}\right.\] (12) \[v_{o}(\omega t) = \left\{\begin{array}{cc}+V_{m}\sin(\omega t)&\pi\leq\omega t\leq 2 \pi&\&\&4\pi\leq\omega t\leq 5\pi\\ &0\leq\omega t\leq\pi&\&2\pi\leq\omega t\leq 3\pi\end{array}\right.\] (13) \[v_{o}(\omega t) = \left\{\begin{array}{cc}+V_{m}\sin(\omega t)&0\leq\omega t\leq \pi&\&2\pi\leq\omega t\leq 3\pi\\ &5\pi\leq\omega t\leq 6\pi&\&7\pi\leq\omega t\leq 8\pi\\ &\pi\leq\omega t\leq 2\pi&\&\&3\pi\leq\omega t\leq 5\pi\\ &6\pi\leq\omega t\leq 7\pi\end{array}\right.\]  
The same control algorithm can also be employed for frequency step-up operation. Figure 9(a),b shows the gating schemes to have output frequency two and three times the input frequency. For example, for the two-time output frequency realization as shown in Figure 9(a), the output voltage in phase and out phase in every half cycle of the input voltage. While in the case of tree time output frequency, the voltage at the output is non-inverted and inverted two and one-time respectively in each half cycle of the input voltage.  
The control input \(c_{1}\) is logic high once we require non-inverted input at the output while to invert the input, the control input \(c_{2}\) has to be set to logic high. The nature of the instantaneous output voltage of Figure 9(a),b with the reference of the input voltage is mathematically depicted in Equations (14) and (15) respectively.  
\[v_{o}(\omega t) = \left\{\begin{array}{cc}+V_{m}\sin(\omega t)&0\leq\omega t\leq \frac{\pi}{2}&\&\&\frac{3\pi}{2}\leq\omega t\leq 2\pi\\ -V_{m}\sin(\omega t)&\frac{\pi}{2}\leq\omega t\leq\frac{3\pi}{2}\end{array}\right. \tag{14}\] \[v_{o}(\omega t) = \left\{\begin{array}{cc}+V_{m}\sin(\omega t)&0\leq\omega t\leq \frac{\pi}{3}&\&\frac{2\pi}{3}\leq\omega t\leq\frac{4\pi}{3}\\ &\frac{5\pi}{3}\leq\omega t\leq 2\pi\\ -V_{m}\sin(\omega t)&\frac{\pi}{3}\leq\omega t\leq\frac{2\pi}{3}&\&\frac{4 \pi}{3}\leq\omega t\leq\frac{5\pi}{3}\end{array}\right. \tag{15}\]  
Negative and positive values of the instantaneous output voltage illustrate inverting and non-inverting operations.