## 2 Numerical Analysis of Generated Harmonics
The next step change of the output frequency is one-third, where the period of the output voltage is three times that of the input voltage. There are three positive and three negative half cycles of the output voltage that are arranged in the three cycles of the input voltage. This sort of output voltage is realized by inverting the second and first half cycles of the first and third cycles of the input voltage as can be viewed from Figure 5b. The magnitude of the non-inverted output pulses is set to half of the input, as shown in Figure 5a. This arrangement helps to suppress the fundamental frequency harmonic from the output voltage (as investigated in [21]). But this adjustment creates power quality disturbances in the input current. The harmonic indexes of the output voltage as computed in [21] of Figure 5a are highlighted in the following set of equations. \[V_{o}=\frac{V_{m}}{2}\sqrt{1+k^{2}} \tag{2}\]1  
\[V_{o}=\frac{V_{m}}{2}\sqrt{1+k^{2}} \tag{2}\]2  
The values of the harmonic factor for the value of \(k=0.5\) and \(k=1\) are 55% and 68%, respectively, as evaluated in [21]. The wave shape characteristics of the input current for the value of \(k=1\) are nearly sinusoidal, but this is non-sinusoidal for the value of \(k=0.5\). That is explored with the help of the next set of numerically computed equations. \[V_{o}=\frac{V_{m}}{2}\sqrt{1+k^{2}} \tag{2}\]3  
\[V_{o}=\frac{V_{m}}{2}\sqrt{1+k^{2}} \tag{2}\]4  
\[V_{o}=\frac{V_{m}}{2}\sqrt{1+k^{2}} \tag{2}\]5 Output voltage for one-third frequency operation with respect to input for voltage (**a**) regulated form and (**b**) non-regulated form. The first part is the required fundamental component, and the second part is undesirable harmonic components. It is observed from Equations (15) and (17) that for the required value of \(k=0.5\), once, we try to improve the harmonic factor of the output voltage by eliminating the harmonic at the fundamental frequency, then at the same time, the value of the required component of the input current decreases and undesired harmonic components increase. The values of harmonic and power factor for \(k=0.5\) are computed as 36% and 0.94, respectively. The values of the harmonic and power factor are zero and unity for the value of \(k=1\). Here it can be observed that improvement of the output voltage quality degrades the quality concerns of the input current. This analysis can also be extended to the one-fourth output frequency requirement. Here, four output half cycles are arranged into positive and negative values during the four input cycles of the input voltage. Positive half cycles of the output voltage are arranged by non-inverting and inverting the positive and negative half cycles of the first two cycles of the input voltage. This pattern is reversed for the generation of negative half cycles of the output voltage as here input positive and negative half-cycles are inverted and non-inverted at the output. The power quality index of the output voltage is improved by adjusting the amplitudes of some half cycles, as can be seen from Figure 6a. The output rms voltage and its harmonic components are numerically calculated as  
\[V_{o}=\frac{V_{m}}{2}\sqrt{1+k^{2}} \tag{2}\]6  
\[V_{o}=\frac{V_{m}}{2}\sqrt{1+k^{2}} \tag{2}\]7  
It is seen that there is the elimination of harmonic components in the output voltage at source frequency. The magnitude of the output voltage required component and harmonic components depend on the value of \({}^{\prime}\!K\). The harmonic factor of the output voltage for a value of \(k=0.5\) is calculated at nearly 52%. The sinusoidal behavior of the input current deviates from its required level once the harmonic factor of the output voltage is improved. Its total rms value and harmonic series are evaluated as  
\[V_{o}=\frac{V_{m}}{2}\sqrt{1+k^{2}} \tag{2}\]8  
\[V_{o}=\frac{V_{m}}{2}\sqrt{1+k^{2}} \tag{2}\]9  
With the help of these two equations, the harmonic and power factor for \(k=0.5\) are evaluated as 33% and 0.94, respectively.