Half Wave vs Full Wave spectrum

# Bullard Laws of Harmonics #1

Published on April 11, 2020

My first LinkedIn article I ever posted used this example. But since then I have developed more skills in working with Excel, so let me post this one more time just to prove to you that Bullard Laws of Harmonics #1 is absolutely true, with no wiggle room.

### Harmonic amplitudes are proportional to the area of the distortion.

First I have a half wave rectified wave.

Now its spectrum:

The fundamental is the only Odd harmonic. Now, for the full wave rectified wave:

This wave has exactly twice the area of the half wave rectified wave, wouldn't you agree? Now for the spectrum:

No Odd harmonics at all. The energy in the fundamental has gone away entirely. And in its place is pure Even harmonics. This is obviously not true, it can't be because there must be a fundamental, but now the fundamental is the second harmonic of the original fundamental. If that was 60Hz (in the US) then the new fundamental is 120Hz and now the harmonics alternate Odd and Even. But we're not going to bother with that now, we are going to prove Law #1 absolutely, like this:

Here are the two spectra on the same plot. The half wave spectrum has the singular Odd harmonic and the lighter red Even harmonics. The full wave spectrum we know has no Odd harmonic, and the Even harmonics are now a darker shade of red. And since I used a punctuated line for the spectral lines, you can see there is a difference between the two. Take a wild guess what the difference is: Yep, it is exactly 6.02dB. Each bin, from DC to the highest harmonic shown (and even the ones not shown, the aliased ones) are precisely 6.02dB higher for the full wave spectrum than the half wave spectrum.

So, here we prove absolutely that Bullard Laws of Harmonics #1 is true, no doubt about it. If you double the area of a distortion, the amplitudes of the harmonics double. There is no better proof than this.