Re: Duty cycle and Even Harmonics

AHA!!! The missing word - PHASE

Not shown in most online Fourier waveform apps and not captured by most FFT algorithms is the fact that there are *four* (4) components to a Fourier series expansion:

Odd sine harmonics
Even sine harmonics
Odd cosine harmonics
Even cosine harmonics

Note that for this discussion, there is no DC offset to any fundamental or harmonic. By definition, a sine wave is symmetrical about zero.

A sine harmonic is defined as having zero phase angle with respect to the fundamental. That is, at time t-0, the leading edge zero crossing of the harmonic in question is right on top of the leading edge zero crossing of the fundamental. For cosine harmonics, the harmonic is at its positive peak when the fundamental is at the zero crossing.

One way to view the difference between symmetrical and non-symmetrical square waves is the addition of cosine harmonics. I've seen exactly one website that does this correctly, but of course I can't find it now.

PZ is right about the sampling aperture of most FFT engines. It is a universally overlooked source of error. I once played with a LeCroy for a week. Spoiled me for life.

Rule of thumb: rapid rise and fall times require odd harmonics. Slopes require even harmonics. In one of these posts there is a statement about the difference between sawtooth and triangle waves. It is wrong.

Side note to PZ - saw the UZ photo on your site. Where are the "stabilized wave" components? Also, we may need to have a discussion about the *apparent* offset of the UZ waveform under load...

ak