Some really fun Bullard Plots showing some cool harmonics.

A Pulse Width Modulation plot showing the pulse width going from 99% to 1% in just a few steps.

An animation showing a notch distortion at the peak of a wave showing that Even (red) harmonics are dependent on assymetry. Notice that the Odd (blue) harmonics don't change at all during the entire animation. In fact, the Even harmonics only show how asymmetrical the distortion is, they don't prove anything else!

An animation that shows how a 4% distortion placed at various places from the peak to the zero crossing causes very different harmonics.

I changed the way I create Bullard plots, so now the main signal under scrutiny sits in the forground in black, with the 36 harmonics in the background. Applied to the above example, this really helps you see the signal, unlike the way it works in the older version above where you can barely see the signal.

And here I recreate the one above that. Again, Even harmonics only appear when the distortion is asymmetrical. That makes THD, which is the sum of all harmonics rather misleading.

In this example I show how a glitch that moves through a device transfer funciton changes the spectral signature depending on the angle where the distortion impacts the sinewave.

When only one glitch occurs in a sinewave it's seen as an impulse, and the spectral response is a flat spectrum all the way up, regardless of angle.

This Bullard plot shows how the angle where a glitch occurs is not really what causes the humps and notches in the spectrum according to the Bullard Harmonic Solution. The first part of this animation duplicates the above example where the angle makes no difference when there is only one glitch. However, when the distortion is placed in a device transfer funciton, it is automaticly hit twice, once on the way up and once on the way down. That means that the harmonics have to compromise to create the wave, as some harmonics can't help make the pair of glitches no matter what the starting phase is, because of the distance between the low spots depending on frequency. So, nature reduces their amplitudes to near zero (if not zero) to prevent them from acting counterproductive in the creation of the wave.

Proof of Bullard Laws of Harmonics #1 and #2. I boosted the amplitudes of the harmonics and modified the phases to change the shape of the wave. But since I didn't change the Harmonic Signature the angle where the wave changed didn't budge. Notice that when the wave goes from convex to concave, you can't see the phase change in the spectrum becuase we never show you the phases of the harmonics.

Proof of Bullard Laws of Harmonics #5 and Daver's Law. No matter what I do to this cosine wave with distortion, the phases (those two blue lines near the bottom) always give me either 0 degrees or 180 degrees.

And for those who think this is all made up crap, a comparison between a real spectrum taken from a clipped sine wave on an Applicos ATX7006 and the simulation in Excel. They match perfectly.

Proof of Bullard Laws of Harmonics #2. As I change the angle at which I clip the top of a sine wave off, the harmonic signature changes. Absolute proof of Law #2.

In a fun experiment, I move a distortion from the positive peak to the negative peak of a sine wave. How did I do it? I flipped the phases of the Even harmonics in stages. Then I flip them back again. I flipped them by (essentially) adding 180° to the phase value in the frequency domain, and I made sure the value never exceeded 360° by using the MOD operator. This proves that a spectrum retains the concept of time using the phases of the harmonics.

Another way of looking at the trick of moving a distortion from one side to the other by flipping the phases of the Even harmonics.