The constant value of 0.637 in the formula for calculating the average value of a sinusoidal alternating current waveform (I_avg = I_peak × 0.637) is derived from the mathematical properties of a sine wave.
In an ideal sine wave, the peak value represents the maximum amplitude or magnitude of the waveform. The average value, on the other hand, is obtained by taking the arithmetic mean of all the instantaneous values over a complete cycle.
For a perfect sine wave, the instantaneous value of the waveform varies sinusoidally between its positive and negative peaks. The mathematical expression for a sinusoidal waveform is given by:
I(t) = I_peak × sin(ωt)
Where: I(t) is the instantaneous current at time t, I_peak is the peak value of the current, ω is the angular frequency of the waveform.
To find the average value of this waveform, we need to calculate the mean value of the absolute magnitude of the waveform over a complete cycle. Since the sine function oscillates symmetrically around the zero axis, the average value can be derived by integrating the absolute value of the waveform over a complete cycle and dividing it by the period.
The integral of the absolute value of a sine function over a complete cycle is equal to 2 divided by π (2/π). Dividing this by the period T (which is the reciprocal of the frequency, f) gives us:
I_avg = (2/π) / T = (2/π) / (1/f) = 2f/π
However, in practice, the formula for calculating the average value of an alternating current waveform is often simplified by using an approximation factor. This factor takes into account the area under a half-cycle of the sine wave and converts it to the equivalent area of a square wave.
The ratio of the area under a half-cycle of a sine wave to the area of a rectangle with the same peak value is approximately 0.637. Hence, this factor is commonly used to estimate the average value of a sine wave without having to perform the full mathematical integration.
Therefore, the constant value of 0.637 in the sine wave average formula is an approximation that provides a quick and convenient method to estimate the average value of a sinusoidal alternating current waveform.