Asymmetrical zero crossing distortion Bullard plot

Is Law #5 for real?

Published on July 27, 2019

Copyright © Dan P. Bullard

When I make groundbreaking discoveries, my mind races to find an exception. In this article where I introduce Law #5, I used a peak clipping distortion to make my case. In my last article I used a proxy for a peak clipping, a glitch distortion right at the same point as one of the peak clippings. And I explained the effect of the phases based on symmetry around the peak at 90 degrees, which makes plenty of sense. But, what about zero crossing distortion, which is about as far from the 90 degree point of a wave as you can get? OMG, did I screw up? Could Law #5 be wrong? Well, the only way to find out is do some experiments.

So I created an asymmetrical zero crossing distortion of 500 samples in the transfer function which translated into 33 samples of zeros at the zero crossing on the positive side of the sine wave. That's just the way it works, my transfer function is 10,001 samples long and my waveform is only 2048 samples long, which is compatible with the Excel FFT requirements. I found through experiments that the transfer function, which is continuous in the real world, must be deeper than the stimulating sine wave, and 10,001 samples seemed like a good compromise between 2049 and infinity.

So I made this wave (above), and then ran my FFT, and the results came back with numbers I had hoped for.

There ya go, the answers meet Bullard Laws of Harmonics #5 perfectly.

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

Even with this crazy wave, the law holds true. But why? Let's look at the Bullard plot from above again. Note that I gained up the harmonics by a factor of 200.

Since the distortion is right near near the 180 degree point in the wave it's still related to 90 degrees. Plus, while you could generate that distortion with just Odd harmonics, you can't prevent the distortion from showing up on the negative side without bringing in the Even harmonics, and that requires symmetry around 180 degrees, which, again, is a multiple of 90 degrees. See why my law holds true? OK you don't believe, so let me try to make an even crazier distortion, with 500 samples zeroed out on the positive side of the transfer function and 781 samples (just a random number, determined by how long I dragged the select box around the transfer function column in Excel) zeroed out on the negative side of the transfer function. That results in this Bullard plot:

Now, because it's very difficult to eyeball the phases of the harmonics, let's look at the phases of this wave along with the first crazy wave.

Is that amazing or what? You could look at all the angles for both of these waves, there is not one number in those columns that is anything other than one of the four numbers defined in Law #5. 0 degrees, 90 degrees, 180 degrees or 270 degrees. It has to be this way, period! That is the definition of a law, it doesn't have exceptions! And it amazes me that I discovered this law in the space of 8 hours looking for something entirely different! But that's the way it is, I'm not Charles Darwin, I'm not going to milk my research for 30 years before I publish. I know how fast someone can swoop in and steal your ideas, your inventions, your discoveries. No way am I going to let that happen again, and the good news is, you benefit by being able to learn something totally new and unknown to the mind of man for the price of a few minutes of reading. You are welcome.