My two cents intertwined below ...
At 01:38 PM 4/21/99 -0400, Scott Douglas wrote:
>Hi All,
>
>Recently an interesting discussion came up about harmonics. A general
>disagreement followed. We hope you all can offer some insight and perhaps
>help us settle the question.
>
>The question is numbering of harmonics.
>
>One side says that given a fundamental frequency of 200 MHz, the first
>harmonic is 400 MHz, the second harmonic is 600 MHz and the third harmonic
>is 800 MHz.
>
>The other side says that given a fundamental frequency of 200 MHz, the
>first harmonic is 200 MHz (or same as fundamental), the second harmonic is
>400 MHz and the third harmonic is 600 MHz.
----------
I've always referred to the integer number of the
harmonic as the number that's multipied to the
fundemental. The first harmonic in my scheme of
things would indeed by the fundemental. I have
never dealt with anyone who would tell me that the
third harmonic of 200 MHz would be 800 MHz.
And indeed, when the fourier expansion is done,
this is in fact the case. i.e.
Square Wave = a1*sin(wt) + a3*sin(3wt) + a5*sin(5wt) + ...
ANY "Fields and Waves"/Electromagnetics book worth
it's salt on fourier expansions will cover this.
----------
>The other part of the discussion revolved around even and odd harmonics.
>
>One side says that even harmonics are lower amplitude than the odd
>harmonics, the other side says odd harmonics are lower amplitude than even
>harmonics.
>
>All discussions assumed non-sinusoidal sources, generally our sources are
>square- or modified-square waves.
>
>Can someone shed some light on harmonic numbering and if possible, point
>to a reference material that specifies this?
----------
The even or odd being lower than the other
is certainly an odd observation. <grin>
In general <weasel word alert>, as the coefficients
decrease in an exponential manner (as long as the
factors include even and odd components), I don't
understand this observation.
If what's being observed is the sum of all wave energy
from the odds being higher than the sum of all wave
energy from the evens, then that's quite possibly true
since the odd sums will usually include the fundemental
which is highest anyway. But again, I don't understand
the observaton.
For instance, an absolutely perfect sawtooth wave
with peak values at +V and -V produces odd and even
harmonics. Such as
y-axis
|
/ +V| /
/ | /
/ | /
----------------+------------- x-axis
/ | /
/ | /
/ -V| /
The fourier expansion of this is
f(t) = + (2V/pi)*{sin(wt) - 1/2sin(2wt)
+ 1/3sin(3wt) - 1/4sin(4wt) + ...}
Odds add in amplitude to be
(2V/pi)*(1 + 1/3 + 1/5 + ...)
Evens add in amplitude to be
(2V/pi)*(1/2 + 1/4 + ...)
The odds have it hands down since they
start with 1 and evens start with only 1/2.
Also, important to note is that an absolutely
perefect square wave produces nothing but odd
harmonics. Deviate the slightest from an
absolutely perfect square wave and the
expansion starts producing even harmonics of
amplitude related to how imperfect the
square wave ends up being.
Not sure if I answered your question.
----------
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