Wednesday, March 6, 2013

Back to basics: Urms (part II)

So, let's revisit our half rectified sine wave example.

DC component of the half rectified sine wave

This is given by

URMS of the half rectified sine wave

As seen in part I, this is

RMS value of the alternating component

Now, the next question is: how "strong" is the total alternating component in the half rectified sine wave? 

In part I we established that URMS is splitted into two parts: a continuous component and the sum of sines. The latter, the sum, is the total alternating component of the signal. The formula is

where C=UDC. Substituting, we arrive at the RMS value for the alternating part

Tying this with Fourier

This gives the RMS value for the total alternating component, but what if we are interested in the RMS value of a specific harmonic? 

The Fourier Transform will be our tool: it transforms a time-domain signal to a frequency-domain one, and the Fourier Series gives us the amplitude of each harmonic.

For a half rectified sine wave of period T, the amplitudes are given by:

Here is a table of the first 5 harmonics amplitudes relative to the peak amplitude of the original signal.

Half rectified sine wave - first 5 harmonics Blogger Table caption
Harmonic Amplitude

1 0.5
2 0.21221
4 0.042441
6 0.018189
8 0.010105

The following graph shows the harmonic amplitudes.

Remembering that the RMS value for a sine signal is the peak intensity divided by the square root of two, we have
Half rectified sine wave - first 5 harmonics, RMS and cumulative RMS Blogger Table caption
Harmonic RMS Amplitude Cumulative RMS
1 0.35355 0.35355
2 0.15005 0.38408
4 0.030011 0.38525
6 0.012862 0.38546
8 0.0071454 0.38553

After just a few harmonics, we are almost at the calculated total value already.