How does it know the angles?

Published on July 29, 2019

Copyright © Dan P. Bullard

I had a dream. Not like Martin Luther King, the other kind, like a nightmare. At the time I didn't know it was a nightmare, but I was suspicious because Paul Botsford (Credence, Maxim) told me in my dream that "Everyone knows that the phases of the harmonics of a distorted sine wave are 0 degrees, 90, 180 or 270, it's nothing new!" It's not true of course, because I asked the experts at Electrical Engineering Stack Exchange. These "experts" didn't have an answer, just a bunch of excuses why they wouldn't bother looking into it, or why my motives where suspect. I admit right here, my motives were pure: I want to know if anyone else has figured this out before, because this phenomenon has been staring you guys in the face for as long as Matt Mahoney has been doing FFTs, and that's a long damn time! I mean, what the hell were you guys doing for the last 50, 60 or 70 years? Watching paint dry?

Along with the dream, I had another notion: Could this law be true even for single events, like noise spikes, or glitches? Well, I was disappointed; the answer is no and the reason is pretty simple. Let me show you.

Above you can see the Bullard plot of what would be an asymmetrical crossover distortion, except for one thing: It doesn't actually impact the wave in the two places it should. Let's look at a real distortion again, from an earlier article:

Notice how the zero crossing distortion impacts not just the dead center of the wave, but the first little bit of the wave at the start. Now for my "impulse" zero crossing wave.

Notice how the start of the wave is undisturbed? I did that by pasting the distorted zero crossing in the center of a perfect sine wave. Now look again at the harmonics of both of those Bullard plots. You can see how the upper one, the "real" example with a real transfer function failure, the harmoics have to compromise to make both of those notches, the one at the start of the wave and the one in the middle. The Even (red) harmonics get involved too, and you can see how it's a bit of a choreographed dance to make those harmonics all play together nicely to create both of those distortions. But in the lower one, the fake wave that could never happen in a real circuit (since a failure in the transfer function would be struck as the wave goes up as well as when it comes down from the positive peak) allows all the harmonics to work together to build that one disturbance in the wave, the one in the center of the wave. But notice that there is no compromising here, there is no need! There is one, and only one distortion that the harmonics have to make, and so it's treated the same way as an impulse is treated, the starting phase of each harmonic can be whatever it wants to to create that wave. The harmonics in the other wave have to work together, to compromise, and so they are phase restricted, as this table shows.

Notice that the phases of all the harmonics in the real wave obey Bullard Laws of Harmonics #5, all the phases are either 0 degrees, 90, 180 or 270, but in the fake, impossible wave, the harmonics are all over the place. That's because the distortion at the starting point of the wave wasn't there, as it would have been if there were a real transfer function problem. The wave doesn't "know" where 0 degrees is, or 90, or 180, or 270 for that matter. But it can tell when something is 180 degrees away, which is what happens with the real wave. With the fake wave, the made up wave (OK, both of them are made up, but one of them can appear in real life, the other can't) there is no reference point, and hence no need to compromise the harmonic amplitudes and phases to make two distortions, those harmonics have it easy, because they only have to create one distortion. The real world example has to pick the phases and amplitudes carefully to get the two distortions in the wave, and if this was a symmetrical distortion, as often happens because of matched transistors, then the Even harmonics would be excused, because in addition to helping make the distortions, they also keep the distortion from appearing on the other side of the wave, in this case, the negative side. Without the Even harmonics, those distortions would be smaller, but they would also be repeated on the negative side, since that's mostly what the Even harmonics do.

It's not really that hard, trust me! OK, if you hang out at Electrical Engineering Stack Exchange you probably won't get it, but if you keep an open mind and don't shut out good ideas just because your professor didn't know about it, you can have a shot at learning new things. Hey, it's so easy, a high school graduate can do it, and explain it!