Re: Duty cycle and Even Harmonics

>>>>>... I agree with you, generally speaking, but will like to have confirmation of it. I start having doubts on my understanding of this matter.

No doubts it is confusing, I suggest pulling out a basic Fourier Series Expansion math/engineering textbook and go through the derivation of calculating the coefficients from the expansion using sines/cosines as the basis function on various pulse duty cycles. In this context I need not do any Fourier Transforms or FFT's. The Fourier series expansion follows Parseval's Theorem for the discrete case.

What most texts and wikipedias do is assume a pulse, and do a series expansion of the harmonics using sine waves, thus the common saying that a square wave only has odd harmonics. That is very true when the duty cycle is 50% since the Fourier expansion assumes periodicity (repeats and why I quoted the term 'periodic' earlier). That can be seen on the fundamental frequency by looking at a sine wave and the positive sine lobe lines up with the positive pulse, and the negative sine lobe lines up with the negative pulse. The second harmonic would have a positive AND and negative sine lobe for the positive pulse, and similarly on the negative pulse thus they cancel out when added to the whole set of harmonics.

When you shift the duty cycle to say 33%, where the positive pulse is less in length than the negative pulse, the positive sine lobe is too big for the pulse and the negative sine lobe is too small for the negative pulse. The second harmonic would not cancel in this case and would be needed to synthesize the aggregate waveform.

Based on the derivation of the coefficients that are a function of the harmonic number and the duty cycle in the more general case of a pulse waveform, a 25% (75% the complement version) is the duty cycle looks like the maximum value for the energy in the second harmonic. What I failed to tell you earlier is for this case, the third harmonic is less than compared to the 50% duty cycle, but I did not want to confuse.

Assuming a harmonic analysis for one pulse, and another harmonic analysis for another pulse is valid as long as in the composite waveform superposition properties apply and are periodic, since these properties are linear in nature. Then again, nothing is perfect so harmonics leak in in some form or fashion.

Other basebanded waveforms, like what is used with a zapper, that one can compare to are pulse coded modulation or pulse width modulation signals that can have different duty cycles and even another periodicity of the information pattern, but that is another story that throws even more harmonics into play. The same properties apply to inherently lower duty cycle radar signals, but translated up in carrier frequency.

Hey if this helps you in any designs, how about a cut? Just kidding, hope it helps you and the zapper community in general.