## Wednesday, March 6, 2013

### Back to basics: Urms (part II)

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.