Avery Fineman wrote:

In article , Peter John Lawton
writes:

I am pondering why the energy
available for higher harmonics is less than for the fundamental and also
how your program works out this energy.

My program was developed while at RCA Corporation, specifically in
the time period of winter 1973-1974 using the core of three ideal
waveforms: rectangular, rising triangle, falling triangle. They relate
to a singular waveform using a time-delay formula multiplier so that
the rising triangle butts up to (in time) to the start of the rectangular
waveform and the falling triangle starts at the end of the rectangular
waveform. Entry is rise-time (the rising triangle), fall-time (the falling
triangle), and 50% amplitude pulse width which is the rectangular
waveform length and the length of the rising and falling triangles
adjusted for their inputted times. [draw it out to see it better]
Each basic waveform generates its own Fourier coefficient set. All
sets are simply added algebraically. Mathematically okay to do that.

A quick form of proof of that is to use a simple frequency-to-time
transform that works at each specified point in time along the
repetition period of the waveform. The original was a time-to-
frequency transform, mathematically different than the opposite.
If a reconstruction of the frequency-to-time results in the original
entry specifications, then it is called accurate enough. I didn't
derive the reconstruction transform since it was already in a book.
Neither did I derive any of the basic ideal waveforms which were
already in the ITT Blue Bible. The delay multiplier used to set
rise, fall, and 50% width was another book value, simplified to
faster calculation simplicity because the original was a math
problem thing with more terms than needed.

Thanks, that's clear.

Its like pushing the baby on the swing in the park, you only need to
give it the occasional push or pull in the right direction. A 5f
resonator gets has to go for 2.5 cycles in between refuelling from a
square-wave (1:1) of frequency f.

Use any analogue you want. I don't agree with the above, but
feel free and I not going further on that...

OK. I'll just say that on reflection I realise that it's not an analogy
- it's a bona-fide case of extracting a harmonic from a repetitive pulse
waveform.

What do you mean by intuition here?
My intuition suggests to me that as the rise and fall times get shorter,
the energy available for the harmonics approaches that for the
fundamental. In other words, as a square wave approaches perfection it

For any ideal rectangular shape, the harmonic energies have a
(SinX / X) locus. That's explained in textbooks also. Harmonics
of a repetitive waveform Fourier transform will NEVER have more
energy than the fundamental. That's also basic book stuff.

If the rise and/or fall times are finite, the harmonics will drop their
energy levels compared to the zero rise and fall time ideals. As the
rise and fall times get longer and longer the harmonic energy gets
less and less. By the time one gets to a sinusoid waveshape,
there are NO harmonics in any Fourier transform, its all
fundamental frequency (1 / repetition-period).

I'm talking intuition not Fourier.
BTW your earlier comment on shortage of energy at higher harmonics may
be exacerbated by the lower Q of LC resonators at higher frequencies.

Recess.
OK

Peter
Len Anderson
retired (from regular hours) electronic engineer person