Published on August 4, 2019

Copyright © **Dan P. Bullard**

In my very first LinkedIn article I compared the harmonics of a half wave rectified wave and a full wave rectified wave. Sixty articles later, I am returning to that topic, because I have since invented the Bullard plot which allows me to show you what is going on there to make each of the waves. Harmonics make the wave, waves do not make harmonics. Don't doubt me on that. Take away some of the harmonics and the wave changes.

The above image is a Bullard plot of a half wave rectified wave showing the target wave in black in the foreground, then the fundamental in blue on top, with no gain, and the harmonics that make up this wave, gained up by a factor of 10 in this case. In my Excel tool for creating Bullard plots, I have a cell where you can vary the gain applied to the harmonics because they can be very small, but in this case, they are not too small compared to the fundamental. Realize that the wave in black is 17 times larger than the fundamental wave, and that is just for clarity, so you can see what we are examining. Some of my earlier Bullard plots didn't look like this and it's hard to see the detail we are looking for in the target wave, so I settled on this look. I think it's better.

Notice how the second harmonic (second wave down from the top in red), which is quite large, seems to reinforce the positive half cycle of the target wave, but that the 4th harmonic (in red also because it's an Even harmonic) seems to counter it somewhat. Lower down, the 6th harmonic tends to counter the counter at the center of the peak of the target wave, and so on all the way down through the Even harmonics. The Odd harmonics are silent, this and the full wave rectified wave are the only two waves that have no Odd harmonics. In fact, in the real world, you would see Odd harmonics because of the diode drops in the rectifier, since they remove part of the positive (or negative, as the case may be) peaks, injecting distortion into the otherwise pristine half cycle waves. But here in MathLand we can ignore those anomolies and concentrate on what is really going on. While these Even harmonic waves seem to counter each other, they do two things: They don't interfere with the positive peak of the incoming wave, but they totally eliminate the negative half cycle of the incoming wave. Now, here is a full wave rectified wave in a Bullard plot:

What differences do you see between this wave and the one above? Well, in this one, all the Even harmonics are twice as large. The 2nd harmonic is in fact so large that it goes off the chart so to speak, so it's quite a bit larger than the 2nd harmonic in the half wave rectified Bullard plot. And as we know from my very first article mentioned above, all the Even harmonics here are exactly twice as big as the ones in the half wave example. You can eyeball it, but accept it, these harmonics are twice as big.

But, there is another difference that you may have missed. Take a look at the top blue line, the fundamental in this plot. It's dead flat isn't it? So, what were the Even harmonics doing in the half wave example vs what they are doing now? In the half wave example, the harmonics are helping the positive half cycle, because, remember, the fundamental is down by 50%, 6dB, since we removed half of the wave. Take a look at this graphic, where I show the original half wave rectified wave, TD Wave, in yellow along with the sum of the first 40 harmonics and the fundamental, Summed Wave, in blue. Then below that in green and red (not related to Odd or Even) are the sum of the Harmonics Only, in green, and the Fundamental only in red.

So the fundamental (in red) is half the amplitude of what it was because we removed half of the wave, meaning half of the area, half of the energy, again, Bullard Laws of Harmonics #1. But, the harmonics created do two things: They boost the the positive peak of the wave back to full amplitude, and they counter the negative peak of the remains of the fundamental, which completely flattens the wave, making it a flat DC level. The little bits of humpiness in the blue line are because I only summed together the first 40 harmonics, so the later harmonics, if they were included, would have eliminated those small but not insignificant anomalies.

But now, in the full wave rectified example above, the harmonics are twice as big, so they are replacing the positive half cycle that is now **missing**, because the fundamental isn't just half amplitude, it's flat as a pancake, because the first half cycle and the second half cycle are in opposition. That makes the fundamental totally disappear and become a totally flat line. And they now have nothing to counter on the negative half cycle of the fundamental, since there isn't one, so they are making an equally large positive half cycle to match the first half cycle. Look again, the Even harmonics in both half cycles are identical, and with no fundamental to muck things up, they make two identical half cycles, which happened because we doubled the amplitudes of all the Even harmonics and eliminated the fundamental.

The other way to look at it, as I did in the first article is to think about what happened to the energy in the fundamental bin. It was stolen and used to double all the Even harmonic amplitudes, causing a full amplitude full wave rectified wave which no longer has to fight the negative side of the fundamental, so you get two humps, not one.

Just fascinating stuff. I love exploring harmonics and I hope you do too.