## 6 Validation of the Generated Harmonics
### Validation through Simulation and Analytical Results  
The Simulink-based environment is used to validate the computed harmonics coefficients for various output voltage control schemes. Figure 8 shows the output voltage waveforms realized with phase-angle, on-off cycle, and \(PWN\) control. The output voltage waveform (see Figure 8a) is simulated with a firing-delay of \(\pi/2\). With this control, the input voltage at the output is chopped from '0' to '\(\pi/2\)', which only generates a DC component and even and odd harmonics as depicted in Table 2. The DC component is negative as the area under the negative half cycle is greater than its positive half cycle area. The output voltage waveform (see Figure 8b) is simulated with a firing-delay of \(\pi/2\). With this control, the input voltage at the output is chopped from \({}^{\prime}0^{\prime}\) to \({}^{\prime}\pi/2^{\prime}\) and from \({}^{\prime}\pi^{\prime}\) to \({}^{\prime}3\pi/2\). This control generates low-frequency harmonics as the switching frequency is the same as that of the input source. These generated harmonics are verified from the simulated and mathematical results of Table 3 with a firing delay control of \(\pi/2\). Here it can be observed that the maximum value of the voltage fundamental component, third, fifth, seventh, ninth, and eleventh harmonics are approximately 90 V, 48 V, 16 V, 16 V, 10 V, and 10 V, respectively with a 150 V peak input voltage. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Harmonic Order (\(\mathbf{u}\))**} & \multirow{2}{*}{**Harmonic Frequency (Hz)**} & \multicolumn{2}{c}{**Mathematical Results**} & \multicolumn{2}{c}{**Simulink Results**} \\ \cline{3-6}  & & **Magnitude (V)** & **Phase-Angle (Deg)** & **Magnitude (V)** & **Phase-Angle (Deg)** \\ \hline
0 & 0 & \(-23.87\) & 0 & \(-23.90\) & 0 \\
1 & 50 & 115 & 348 & 114.20 & \(-12\) \\
2 & 100 & 35.58 & 154 & 35.42 & 154 \\
3 & 150 & 23.87 & 90 & 23.71 & 90 \\
4 & 200 & 13.12 & 14 & 13.09 & 13.76 \\
5 & 250 & 7.95 & 270 & 7.93 & \(-90\) \\
6 & 300 & 8.29 & 170 & 8.26 & 170 \\
7 & 400 & 7.95 & 90 & 7.91 & 90 \\
8 & 450 & 6.11 & 7.11 & 6.08 & 7 \\
9 & 500 & 4.77 & 270 & 4.70 & \(-90\) \\
10 & 550 & 4.84 & 174 & 4.83 & 175 \\
11 & 600 & 4.77 & 90 & 4.74 & 90 \\
12 & 650 & 4.0 & 4.7 & 4.0 & 4.6 \\
13 & 700 & 3.41 & 270 & 3.39 & \(-90\) \\
14 & 750 & 3.43 & 175 & 3.42 & 176 \\
15 & 800 & 3.41 & 90 & 3.38 & 90 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Harmonic coefficients of output unipolar voltage control with a phase-angle of \(\pi/2\). Figure 8: Simulation waveforms of the output voltage: (\(\mathbf{a}\)) phase-angle unipolar control; (\(\mathbf{b}\)) phase-angle bipolar control; (\(\mathbf{c}\)) on-off cycle control; (\(\mathbf{d}\)) PWM control. As already remarked, the on-off cycle control is another voltage control technique used to govern the load voltage's RMS value. This control scheme is also employed at a low switching frequency that results in the generation of low-frequency harmonics. The output voltage waveform of Figure 8c is obtained with an equal (two) number of 'on' and 'off' cycles. The generated harmonics of this output voltage waveform are tabulated in Table 4. These harmonics are also verified from the mathematically-computed results. The generated harmonics with phase-angle or on-off cycle control cannot be moved at high frequency as their switching frequencies cannot be increased. This problem is tackled with PWM control. Here, the amplitude of the output voltage is controlled through the PWM control, and the frequency of the harmonics is governed through the switching frequency. Figure 8d shows the output voltage waveform obtained with PWM control having a 50% pulse width and switching frequency six times (300 Hz) as that of the input (50 Hz). This control shifts the harmonics at switching frequency and their integer multiple, as can be viewed from the results of the harmonics of Table 5. It can be observed from Table 5 that harmonics are shifted to 300 and 900 Hz.