Bullard Plot of distorted wave highlighting the 14th harmonic

Why Law #5 is so important

Published on March 13, 2020

Copyright © Dan P. Bullard

Having a newfound appreciation of LinkedIn and their preservation of #freespeech, I decided to write yet another helpful article that might dilute my books, but who cares? LinkedIn has been good to me, compared to Quora, StackExchange and AllAboutCircus.

In this article I will explain why Bullard Laws of Harmonics #5 is so important, and why we would not see those mysterious humping patterns when we look at the spectrum of a distorted wave if it weren't for Law #5. As we all know by now, Bullard Laws of Harmonics #2 says:

The Harmonic Signature is the result of the angle where the sinusoid impacts the distortion as predicted by the Bullard Harmonic Solution.

The Harmonic Signature refers to the humping pattern we see when we look at the spectrum of a distorted wave, like this:

The harmonic amplitudes rise and fall at various frequencies based on the what the angle the stimulating sine wave was at when it hit the distortion. So in the above animation where I am changing the point in the transfer function where I clip from 62.8° to 66.8° on the incoming sine wave, the spectrum has these high points and these low points, and they move as the angle changes, absolute proof of Law #2. But then the question becomes why does this happen? The harmonics have two axes of freedom, amplitude and phase, so why can't they both be varied to create the distorted wave? The reason is because both of those axes are not 100% variable. The phase angle of the harmonics is constrained by Law #5.

The harmonics of a distorted sine wave will always start at 0, 90, 180 or 270 degrees relative to the fundamental, no exceptions.

I'm not going to get into why that is the case now, my book More On Harmonics For Morons explains it pretty well, but I'll show you why the constraints of Law #5 make Law #2 true.

At the top of this article is a Bullard plot that I created by distorting 400 points in the transfer function which holds the value of the incoming sine wave at 0.28 volts on a 1V peak sine wave. That actually results in just 26 * 2 or 52 points in a 2048 point sine wave getting stuck at 0.28 volts, 26 points on the way up and 26 points on the way down. Realize that the transfer function is 10001 points, a lot bigger than the 2048 point sine wave, so 400 points in the transfer function does not necessarily equate to 400 points (or 800 points) of the sine wave being disturbed. Anyway, what you can see is that the distortion lines up pretty well with the 14th harmonic, which appears to be almost flat in this Bullard plot. In fact, this is so important, let me show it here too.

I noticed this looking at one of my animations and I thought it would be an excellent way to make this point. I enhanced the thickness of the 14th harmonic (red, my color code for Even harmonics) so you could see what I'm talking about. That would be one of those nulls I was talking about, a low point in the harmonic signature, and it follows the angle where the incoming sine wave impacts the distortion in the transfer function. Notice there are several "nulls," one at the 9th (blue for Odd) harmonic, the 19th, 27th, 32nd and so one all the way through the spectrum. They do follow a pattern, predicted by the Bullard Harmonic Solution (which I never give away in these articles, you have to have some reason to buy my books).

So the question is, why is the 14th harmonic so small compared to the other harmonics? OK, let's artificially boost the amplitude of the 14th harmonic without letting it impact the distorted wave. Just magnify it by a factor of 10 so you can see why it's so small.

Here it is magnified by 10 and you can see why the "God of Harmonics" didn't make it bigger. At the first distortion on the far left you can see that it's going down while the distortion is trying to recover the normal sinusoid shape, and moving toward the right, you can see that it's going up when the distorted wave is going flat. That is not helpful in creating this wave, and remember, that is the point of harmonics: Distorted waves do not make harmonics, harmonics make distorted waves. The wave is distorted by the distortion in the transfer function, but as Fourier pointed out over a hundred years ago, harmonics make waves, waves do not make harmonics. This is a crucial point, never forget it. However the distorted wave looks, it looks that way because of the harmonics making it look that way.

So we might be able to have more amplitude in the 14th harmonic, but we would have to change the phase. Now remember, there are only 4 phases allowed, and the 14th harmonic is already using one of the allowed phases, specifically 270°. So let's say we change the phase without letting it impact the wave. In other words, cheat and change the phase of the 14th harmonic but don't let that impact the wave that we have, just to see if changing the phase would be helpful in creating the wave or not. So let's flip the phase to 90°.

Here you can see that the 14th harmonic starts at the positive peak, whereas last time it started at the negative peak. Now you can see that this phase, 90°, wouldn't be that helpful either, at the far left the 14th harmonic would be defeating the impact of all the other harmonics trying to hold the level of the distorted wave flat. If the 14th harmonic were this big and in this phase, it would be pushing the sine wave to continue rising, and moving over to the distortion on the right, it would be going against the other harmonics that are trying to make the flat spot as the sine wave descends, which is counter productive to what our distortion is trying to do to the incoming sine wave.

OK, only two more phases left, first let's look at 180°.

This is a real problem for the distorted wave. You can see that on the left, the 14th harmonic would be pushing the distorted wave up while the other harmonics are trying to keep it flat at 0.28 volts, but look to the right! Over there it's doing exactly the opposite! That is not good! We can't allow a harmonic to do two different things to a wave, imagine the chaos that would ensue! That would be nuts. So now you will understand why using the only remaining phase, 0° won't work either.

So looking at the left side, you can see how the 14th harmonic being at this phase might help that first notch in the wave, it would have the opposite effect when the wave is coming down and gets flattened at 0.28 volts by the other harmonics. This just won't work!

So, because Bullard Laws of Harmonics #5 is The Law, there are only four values that can be chosen by the God of Harmonics, 0°, 90°, 180° and 270°°. And in this case none of them work very well, so the best solution is to make the 14th harmonic small. Maybe not infinitely small, that does happen sometimes, just not in this case. When the harmonic is especially useless in creating the distortion, the amplitude goes to zero, and in my book I show you a case like this. But I figured that this was a good place to show you what I talk about in the book, because I want you to have some knowledge about harmonics. You sure as hell aren't going to get anything useful out of the other sites I mentioned above, unless you see my name in the byline.