Re: Duty cycle and Even Harmonics

Hi, dsss27
Hi, Parazapper

I’m a zapper’s designer doing as much as I can to make zappers as effective as possible (electrically speaking, for sure, for sure!) and duty cycle is an important aspect of this efficiency, in way to produce the highest possible number of different harmonics from a single fundamental frequency.
Calculation of FFT to find how much (and which ones) harmonics I can made out of a single frequency are far away from my mind, since computers permits to do this fast and easy.
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This discussion is VERY important to me.
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Parazapper, your way to see/understand asymmetrical square waves is simple, appealing, and makes some sense to my understanding of “how calculate quickly harmonics” but goes against what I have been learning/using until now.

Lets tale your example : 1kHz (1000uS period), or 1.333kHz (750uS period) 50/50 will produce only odd harmonics. I agree.
1.333kHz 66.6/33.3 will have 500us “on”, 250uS “off”.
As per your calculation, this wave form will produce :
500uS => 1kHz, 3kHz, 5kHz, 7kHz, 11kHz, 13kHz, … Odd harmonics only of 2x 500uS=1kHz
250uS => 2kHz, 6kHz, 10kHz, 14kHz, … Odd harmonics only of 2x 250uS=2kHz
Mixing them, in order : 1kHz, 2 kHz, 3 kHz, 5 kHz, 6 kHz, 7 kHz, 10 kHz, 11 kHz, 13 kHz, 14 kHz, …
All this from a periodic frequency of 1.333kHz

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When using FFT calculations, I found a quick way to verify results.
50/50 is a 1/2 fraction, giving energy on the “1” and missing the “2” meaning odd only.
66/33 is a 1/3 fraction, giving energy on the 1, and 2 and missing energy on the 3.
This will give me harmonics on 1, 2, x, 4, 5, x, 7, 8, x,10 …
75/25 is a 1/4 fraction, giving energy on the 1, 2, 3 and missing energy on the 4
This will give me harmonics on the 1, 2, 3, x, 5, 6, 7, x, 9, …
Note : 75/25 and 25/75 seems to give same results, but 75/25 gives more DC current on the output, a plus on a zapper.

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I has been using an industrial oscilloscope, able to show harmonics in the 60Hz line, a real nuisance on industrial control. The oscilloscope is a Tektronix THS720P, a very serious trademark giving trusted information.
Feeding it with a frequency generator having different offsets and duty cycles gives results following “my” numbers on 50/50, 2/3, 3/4, … … 9/10

Simulation programs using theoretical wave forms (precision is not a problem) gives same results.

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To anyone having a deep understanding of this topic :
If I’m wrong, please, let me know, where and why, because it will affect the way I design my zappers, and build my publicity, a publicity I want as accurate as possible.
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Dsss27, I agree with you, generally speaking, but will like to have confirmation of it.
I start having doubts on my understanding of this matter.

=> Also, the impedance of your skin/body interface will soften the rising edge and that will leak (in digital filter theory) some even harmonics energy since it is not a pure rectangle.

This point, easily visible on the output of a zapper under human load, has lead to the use of High Pass Filters on the output, to boost the “weak / leaked harmonics – The so called “super stabilized wave form” circuits.