What's with the humps?

Published on May 3, 2019

Copyright © Dan P. Bullard

As I mentioned in a previous article I had seen this humping pattern several times before, but my friend (who held an MSEE degree) told me to ignore it because it was just "an artifact." But the pattern looked too mathematical for me to ignore, and seeing as no one else seemed to have a reasonable explanation for it, I suspected that the secret to understanding harmonic distortion laid in that pattern, and as it turns out, I was right.

In my books Distortion and Harmonics I walk you through the discovery processes so you too can understand how I came up with the Bullard Harmonic Solution. But there is another way to understand why we have these humps and notches, let me show you here.

Below I show a Bullard plot of a wave I distorted with a 4% chunk taken out of the wave at some seemingly random point in the waveform.

With the help of the dashed lines I placed there, you can see how every single harmonic visible seems to dip right at the missing chunks equally, the one on the front and the one on the back side of the sine wave. However, there are some harmonics missing. The most obvious one that is missing is the 9th harmonic right below the distortions. Looking at the spectrum, you can see that this forms a notch, a low spot in the harmonic amplitudes of the wave.

This low spot in the spectrum would have been considered an inexplicable anomaly by my buddy with an MSEE degree. But I know why it's not just an inexplicable anomaly, that's how I know that everyone else out there is wrong about harmonics. Here is the reason.

In this Bullard plot I gained up the very small 9th harmonic so we can see it. And why is the 9th so small we can't see it? Because its presence would not be particularly helpful to the creation of those distortions; it would have countered the missing chunks, totally ruining the signal. So the reason for the notch in the above spectrum is not some inexplicable anomaly, it's that the low amplitude harmonics just happen to be the wrong phase (no matter which starting phase they have) to be helpful in the creation of the distortion. And that notch happens where it does in the spectrum due to the angle where the sinusoid impacted the distortion.

Now, it's easy to imagine that some all-powerful being is hand-picking the harmonic amplitudes and initial phases to create these distoritons; call it the God of Harmonics. But really, it's just the laws of nature that make it all happen, Fourier told us about it long ago. I did nothing more than uncover the mysteries with the Bullard Laws of Harmonics. You are welcome.