How does the duty cycle of the input signal affect the range of quality factors?
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Figure 7: Periodic rectangular pulse trains were input to a system with a 3 dB cutoff frequency of 750 kHz, and quality factors* of the output waveforms were computed. The two curves for the 80% case correspond to quality factors computed using the ideal high and low state widths. The lines connecting the data points are guides for the eye.
* Curves are labeled with the word "High" or "Low" to indicate that quality factor values were calculated using the high-state or low-state, respectively, widths of the ideal rectangular pulses.
Quality Factor Measurements
The blue curve in Figure 7 plots the quality factors calculated for the output pulses when the input waveforms had 50% duty cycles and repetition rates between 25 kHz and 750 kHz. The quality factor computed for the 83.3 kHz rep rate case was 0.87, which compares well with the maximum possible quality factor of one. The high quality factor computed for the 83.3 kHz case supports the use of the 9X rule, which refers to the minimum recommended ratio between the 3 dB cutoff frequency and the rep rate of the input rectangular pulse train.
The 0.58 quality factor computed for the 250 kHz, 50% duty cycle case supports the qualitative observation that despite some pulse distortion in the output signal, the pulse profile was more rectangular than sinusoidal. This was the case even though only two signal components from the input signal had frequencies within the 750 kHz system bandwidth. When the rep rate was 750 kHz, the output waveform appeared sinusoidal and the quality factor was zero.
Quality factors for the 20% duty cycle case are plotted in red. The 0.68 quality factor for an 83.3 kHz rep rate corresponded to an identifiably rectangular pulse profile. However, there was more pulse distortion than for the 83.3 kHz rep rate case with the 50% duty cycle, in which the pulse had a quality factor of 0.87. This illustrates that reducing the duty cycle, while maintaining the rep rate results in increased output pulse distortion. To obtain a 0.87 quality factor, the rep rate of the 20% duty cycle waveform would have to be lowered to approximately 33 kHz.
When the rep rate of the 20% duty cycle waveform was increased to approximately 250 kHz, the quality factor became zero. Quality factors corresponding to higher repetition rates were negative. Physically, this means that the rise time was longer than input pulse width. Under these conditions, it can be challenging to accurately measure rise time.
The green (80% High) curve demonstrates that when the duty cycle of the input signal is >50%, the minimum quality factor is positive. Since the scale does not extend to zero, it is difficult to compare these quality factors with other duty cycle cases. In addition, the quality factor rates the shape of the pulse, but not the low-state segments of the output waveform, which are more significantly distorted. To address these concerns, the quality factor can instead be computed using the low-state widths. These alternate 80% data were plotted in gold and resembled the 20% duty cycle data, as expected.
The quality factors plotted in Figure 7 concisely show the influence of pulse width, repetition rate, and input duty cycle on the distortion of the output pulse. When other parameters are held constant, output signal distortion increases when:
- Rep rate is increased.
- Duty cycle is decreased.
- System bandwidth is decreased.
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