Published on January 12, 2019

Copyright © **Dan P. Bullard**

I can hear it now. "Your experiments with Excel prove nothing, you need to do experiments on real devices." Sure buddy, because we all know that electronics is completely unrelated to math, and I'd like you to be the first person to prove Fourier wrong. But one reason I can't do real life experiments on actual devices is that I was fired by Applicos for pursuing my Harmonics Hobby, which they thought was a dead end. I mean, nobody in history had been able to solve it. The applications engineers at Applicos, bless their hearts were completely silent when I asked them for help in solving the mystery of harmonics, and so, because I wouldn't back down in my battle with Bruce Tibbetts, Master of Science in Electrical Engineering from Teradyne, I was sent packing. I mean, if nobody in the history of science had been able to solve this problem, who could expect a US Navy vet and high school graduate to solve it? I clearly didn't know what I was talking about.

And yet I did solve it, despite not having access to an Applicos ATX 7006, which got me into this battle in the first place. On it, I did my standard experiment that I had started doing back in 1992 (one year before the Audio Precision book came out). On any new mixed signal tester I had to use, I would test Match Mode (a digital feature) and I would test the FFT by clipping a sine wave on one side, then the other, then symmetrically on both sides. When I did it on the Applicos ATX 7006 I got the same result I had gotten on a plethora of other testers. Watch the movie if you want to see the experiment in action. But since I got the ax, I no longer had a tester to experiment with, so all my experiments had to be done in Excel. I considered MatLab or something obscure like that. I even tried to get Credence's AWT up and running, but to no avail. Excel was the better choice anyway because it's so universal, and in fact, far more accurate and flexible. I mean, good luck trying to color code your Odd and Even harmonics on another platform. Excel does it all easily and it was free, and as my buddy Steve at Chip Test Solutions says, "If it's free, it's for me!"

So the question was, is it legit? I based two books on it, so it had better be, but I thought I should do an experiment, duplicating one of the tests I did on the Applicos ATX 7006. In the video I used the tester's DC offset feature to clip the top of the waveform off at the input to an AD7671 Analog to Digital Converter. If you watch the video it's a simple thing, but replicating it in Excel can be tricky because I don't know exactly which voltage the AD7671 cut the peak off at, so I had to do a successive approximation to get the two spectra exactly right. The reason is because of Bullard Laws of Harmonics Law #2; *The Harmonic Signature is the result of the angle where the sinusoid impacts the distortion as predicted by the Bullard Harmonic Solution.* That makes the angle the critical factor in duplicating the spectrum, and I don't know the angle that I clipped at, because I don't have an ATX 7006. But with a few iterations I was able to nail it at 62.6464513 degrees. Here is the Applicos plot of the clipped waveform:

And now the Excel simulation.

The two waves look pretty similar, but the proof is in the spectrum, because if the clipping doesn't happen at exactly the same place on the wave, the spectrum will be different as predicted by the Bullard Harmonic Solution. First the Applicos spectrum:

And now for my (superior) color coded Excel version, produced by doing an FFT on the wave above.

You can see that **I totally nailed it**, it's an exact replica of the Applicos spectrum, and all I had to do was to run a few iterations to make sure my angle was right. The angle is what controls the look of the spectrum, and believe it or not, I am the first person in the history of the world to deduce this fact.

Now, can I prove that it's the angle that makes those two spectra look the same? Yep, pretty easily. How's about instead of clipping at 62.6464513 degrees, I clip at 62.4476596 degrees, just 0.2 degrees difference? Will it matter to the *harmonic signature*? You be the judge.

Now you might say that the two look identical, and you would be wrong. Let's compare. First, a closeup of the Applicos FFT comparing the difference between the 9th harmonic and the 10th harmonic. Since they are not color coded, you'll have to take my word for it.

They are almost exactly the same amplitude. Now for my first example at 62.6464513 degrees, again, comparing the 9th and 10th harmonic.

Again, they are at almost exactly the same amplitude. Now for the last example, clipped at 62.4476596 degrees, just 0.2 degrees off.

The 10th harmonic is higher than the 9th harmonic by a significant amount, 2.5dB exactly. Just 0.2 degrees made this happen, and that is simply a slight change in the DC offset for any instrument, in fact in voltage terms it was just 1.5mV given a 2Vpp sinusoid. Why was I fired? Because Bruce told Applicos that a DC offset could not impact harmonics, and apparently they believed him, because he had an MSEE degree and I didn't. We know for a fact that this change in harmonic signature is not due to the slightly increased **area** of the distortion, because in this article I proved that **area** controls the amplitude of the harmonics as a group, not individual harmonics. The only way to change the amplitudes of the **individual harmonics** is to change the *angle* in accordance with the Bullard Harmonic Solution, and given a fixed amplitude sinusoid, that would require changing the DC offset, the one thing that Bruce Tibbetts claimed would not impact harmonic amplitudes. He was wrong, I was right. I got fired and he kept his job.

Kind of makes you wonder why people pay so much money for a crappy college degree when the answer to the only unsolved riddle in electronics was solved by a guy like me, just a Navy vet who worked on RADAR, SOSUS, HF radio and Crypto. (I could tell you more but then I would have to kill you.) Let's face it, I am a genius, and the rest of you, well, you are welcome to bask in my glory from afar.