Understanding the Peak Factor of Sinusoidal Waveforms

The question asks for the peak factor of a sinusoidal waveform. Let's break down what this means and how to calculate it.

What is Peak Factor?

The peak factor, also known as the crest factor, is a dimensionless measure used in electrical engineering to describe the characteristics of an alternating current (AC) waveform. It is defined as the ratio of the peak (or maximum) value of the waveform to its RMS (Root Mean Square) value.

Mathematically, the peak factor (PF) is expressed as:

$$ \text{Peak Factor} = \frac{\text{Peak Value}}{\text{RMS Value}} $$

Calculating Peak Factor for a Sinusoidal Waveform

Consider a sinusoidal voltage waveform given by the equation:

$$ v(t) = V_{peak} \sin(\omega t) $$

Where:

• $ V_{peak} $ is the maximum instantaneous value (peak value) of the voltage.
• $ \omega $ is the angular frequency.
• $ t $ is the time.

For any sinusoidal waveform, the RMS value is related to the peak value by the following formula:

$$ V_{RMS} = \frac{V_{peak}}{\sqrt{2}} $$

Now, we can substitute these values into the peak factor formula:

$$ \text{Peak Factor} = \frac{V_{peak}}{V_{RMS}} = \frac{V_{peak}}{\frac{V_{peak}}{\sqrt{2}}} $$

Simplifying the expression, we get:

$$ \text{Peak Factor} = \sqrt{2} $$

Numerical Value and Options Analysis

The square root of 2 ($ \sqrt{2} $) is approximately equal to 1.414.

Let's look at the options provided:

• 1.414: This is the approximate value of $ \sqrt{2} $, which is the calculated peak factor for a sinusoidal waveform.
• 3.14: This is the approximate value of $ \pi $.
• 1.57: This is the approximate value of $ \frac{\pi}{2} $.
• 1.11: This is the approximate value of the form factor (or amplitude factor) for a sinusoidal waveform, which is the ratio of RMS value to the average value ($ \frac{V_{RMS}}{V_{avg}} $).

Therefore, the correct peak factor for a sinusoidal waveform is $ \sqrt{2} $, which is approximately 1.414.

Conclusion

The peak factor quantifies how much the peak of a waveform exceeds its RMS value. For the standard sinusoidal waveform, this ratio is consistently $ \sqrt{2} $, approximately 1.414.