Harmonic signatures II

In a previous article, I explained that in many cases, harmonics come from the spectral content of the feature removed from a sinusoid. Some types of distortion, such as clipping remove parts of the wave and the spectral content of the distorted sinusoid exactly matches the spectral content of the feature removed. Here I demonstrate it again using a new tool that I created for interested users of Excel. Because Excel is nearly universal, I thought it was best prove my findings on harmonics using Excel to make it easier for people around the world to try to replicate my findings. There are lots of other tools, Matlab, ATE proprietary tools, etc but they are not nearly as available to everyone, if you have Excel you have no excuse to not try some of these experiments. Mind you, this tool is a very simplified version of what I used to write my dissertation, Distortion, The Cause Of Harmonics And The Lie Of THD. To create the experiments I used to write the book I created a huge Excel program that contained a 10,000 point VLookup table (to replicate the transfer function of the device in question) and a huge number of ancillary tools to help me understand what is going on. It was so complex that it took at least 30 minutes to do one experiment and get through the FFT on 2048 points, so researching the book was a massive undertaking, in some cases taking days to get a single result. When one of my readers emailed me with a subject line "Your Distortion Book: A++" and requested the Excel code, I had to turn him down because it would have been too hard to explain to him how to use the bloody thing. But I take customer comments to heart, so I finally found a way to simplify it and now show you how it can prove my points.

In the above waveform you can see that the very top of the wave has been clipped off, and below you can see the spectrum. When the very peak of a waveform is clipped, the number of humps is very small. If you clip further down on the sinusoid you get more humps. And as I point out in the book, the number of humps you get if you clip at exactly 90 degrees is zero, that is you get a totally flat spectrum. This is what led me to distill the Bullard Harmonic Solution that predicts the spectral signature of a distorted wave. It's all about the angle, nothing else controls the number of humps, and I am the only person to ever have deduced this fact. Now for the spectrum, complete with color coding (blue for odd harmonics, red for even harmonics):

Four humps, that's due to the angle where I clipped the top of the wave. Now to prove my point, here is the "clipping," as if I had taken a pair of scissors and cut off the peak, laid the clipping aside and performed an FFT on it alone.

The peak is almost invisible, but there it is, just the very tip-top of the sinusoid. Now, what does the spectrum look like?

It's an exact replica of the spectrum above, with one difference, there is almost no energy in the fundamental bin! That's because there is no sinusoid, just the clipping. So clipping a sinusoid removes a chunk of the wave, and the harmonic content of the spectrum matches the harmonic content of that specific feature that was clipped off! How could the two spectra be the same? Because the phases are not identical! It's pretty simple, but we tend to forget about the phase when looking at the spectrum. If you paste that almost invisible clipping back onto the distorted sinusoid, you don't get double the harmonics, you get no harmonics because they are out of phase canceling each other out. Never forget, every spectrum contains two parts, and the phase plays a part too.

Doing this experiment was easy, the supporting web page explains how to do it, so now you have no excuse. You have a computer, you have Excel, you can validate my conclusions (or not if you can) and prove me right or prove me wrong. But I doubt you can prove me wrong, even if you have a Master's degree ;-)