Published on October 19, 2019
Copyright © Dan P. Bullard
I thought I had captured all the laws of harmonics so far, but as usual I was fooling around, trying an experiment and discovered yet another law. I had hints of it before, but couldn't reconcile what I was seeing, so I just ignored it. But now it is obvious, and I can't ignore it anymore.
I was trying to reconcile the above spectrum, where I clipped 4% off the top of the transfer function, and 3% off the bottom. Since the angles where the clipping happens are different, you can't expect those beautiful pictures we have come to know and love, the nicely humping patterns. In fact, this is what most real-world distorted spectra look like, because without a very high accuracy instrument like the Applicos ATX7006, you cannot be sure that you are clipping exactly where you think you are clipping. But thanks to Applicos, I found that the answer to harmonics is pretty simple. As I have said here on LinkedIn over and over again, the humping pattern comes from the angle where the distortion impacts the stimulating sine wave. The closer to 90 degrees, the fewer humps, and if you can clip at exactly 90 degrees, you get no humps, which is how I came to derive the Bullard Harmonic Solution. And when trying to prove that Even harmonics cancel out, I kept noticing that my harmonic phase angles were always nice even numbers: 0 degrees, 90, 180 or 270. Never 45 degrees, or 77.325 degrees, but always one of 4 different values, 0, 90, 180 or 270 degrees. Never have I seen any other number, and now I know why.
I created the waveform that created the above spectrum, here it is in time domain.
Again, 400 samples taken off the top of the transfer function, far fewer than that taken off the actual sine wave, since my transfer function is 10,001 samples and my waveform is only 2048 samples, and 300 samples taken off the bottom of the transfer function. That causes the clipping you see here, and if I do an FFT I get the ugly spectrum above at the top of this article. The spectrum is ugly because the Bullard Harmonic Solution predicts that the humping of the spectrum is due to the angle of the start of the distortion at the peaks (it's more complicated anywhere else, read the book). But for two angles, it gets very complicated, which is what I was trying to investigate when I made the discovery about the phases. No matter what crazy thing I do to the transfer function, the harmonics always start at either 0 degrees, 90 degrees, 180 degrees or 270 degrees. For a few hours I was confused, but then it hit me. The crucial factor in the Bullard Harmonic Solution is mentioned above, the fact that all harmonics are exactly the same amplitude if, and only if the distortion happens at 90 degrees (or 270 degrees). And since, as this article points out, any sine wave applied to a distorted transfer function has to hit the distortion twice, on it's way up, and on its way down (or on its way down and on its way back up if it happens in the lower half of the transfer function) it will be perfectly symmetrical around the peak of the fundamental. So if the sine wave hits the distortion at say, 66.91437 degrees, as happens in the wave above, we can be 100% certain that it will hit the distortion again exactly 23.0856 degrees after the peak at an angle of 113.0856 degrees. If the distortion happens at 250.0505 degrees as the lower half of the wave does above, you can bet your last dollar that the wave will hit that distortion again at 290 degrees and you will win, guaranteed. And because of that fact, here is what we get if we plot the amplitudes and phases of the first 17 harmonics for all three types of waves, the upper peak clipped, the lower peak clipped and both peaks clipped:
Look it over, you will not find a single value other than 0, 90, 180 and 270, not one. That's because all the harmonics must create their effects at an exact distance from 90 degrees or 270 degrees, as you can see here in a Bullard plot of the above wave:
Remember that this Bullard plot is a time domain representation of the spectrum you see at the top of this article, but only the first 36 harmonics. Look all you want, you will not find any of the harmonics starting at any point other than 0 degrees, 90, 180 or 270.
So, the Bullard Laws of Harmonics needs to be modified. The old Law 5 has go away to make room for this new law, which is:
So now, the Bullard Laws of Harmonics looks like this:
Sorry Law 5, you had to go away because the new law #5 only applies to distorted waves. It's still valid, but my laws are really meant mostly for distorted waves.
So, that makes 6 laws of physics I have discovered. When do I get my Nobel prize? My PhD from Stanford? My million dollar a year teaching job?