To All:
This thread has become quite cumbersome. With the fear of decreasing
clarity, I must jump I with my two cents worth.
1. Harmonic number: The Fourier Series of a periodic function looks
like
f(t) = a0/2 + SUM [an* cos(2*Pi*fn*t) + bn* sin(2*Pi*fn*t)], where
the SUM is done from n=1 to infinity (sorry I can't do subscripts in e-mail
text)
Therefore, there is an average value, a0, which is the DC value of
the periodic waveform, and the sinusoidal terms have frequencies that are
multiple of f1, which is the reciprocal of the period, T, of this periodic
function. The convention is that f1 is called the fundamental frequency, f2
is the second harmonic, f3 is the third harmonic, etc. There are no
frequencies below the fundamental, hence there is no non-zero frequency
associated with a0 (DC term). (If you identify frequencies below the
fundamental, then you have incorrectly identified the fundamental.)
Often, the amplitude of the harmonic decreases as the harmonic
number (frequency) increases, but there is no guarantee of this. This is
strictly a function of the waveform itself.
But, notice that each frequency has both a sine and cosine term.
Hence each harmonic frequency has a complex structure because it has these
two components, each of which is shifted in time by 180 degrees.
Note: There is NO f0 term in a Fourier Series. This is a
nomenclature used in communications theory and has been incorrectly inserted
into this discussion (Sorry, Ed).
2. Symmetry: If a periodic function is even, i.e., f(t) = f(-t), then
the bn coefficients are zero and you are left with only the cosine terms in
the series. If the periodic function is odd, i.e., f(t) = -f(-t), then the
an coefficients are zero and you only have the sine terms left in the
series. In either case, you still have a term for each harmonic frequency.
The strength of one harmonic vs. another is strictly dependant on the
original waveform, itself.
If, however, the periodic waveform has half wave symmetry, i.e., f(t) = -
f(t + T/2), where T = period of the waveform, or 1/f1, then the Fourier
Series will have NO EVEN harmonics, you will have only the fundamental, 3rd,
5th, etc. This is the case of the idealized square wave periodic waveform.
If the square wave is not ideal and there is some deviation in rise and fall
times, or in 50% duty cycle, then even harmonics (2nd, 4th, etc.) will
appear - and quickly.
Digital designers like to think that they are dealing with square wave,
hence they only have to deal with odd harmonics. In practice, especially as
device speeds increase, there are almost always asymmetries that give you
even harmonics to accompany the odd ones.
I hope this clarifies matters for someone. It makes me feel better.
Jim
Dr. Jim Knighten e-mail: [email protected]
<mailto:[email protected]>
Senior Consulting Engineer
NCR
17095 Via del Campo
San Diego, CA 92127 http://www.ncr.com <http://www.ncr.com>
Tel: 619-485-2537
Fax: 619-485-3788
-----Original Message-----
From: [email protected] [SMTP:[email protected]]
Sent: Monday, April 26, 1999 9:36 AM
Cc: [email protected]
Subject: Re[2]: Harmonics
Ed et al,
It seems to me that this thread is growing into a mountain and as a
whole is
providing a lot of individuals with the opportunity of allowing
themselves to
get more confused on this, oops excuse me, these subjects. Let's
try to
separate these subjects and maybe put some order to this seeming
small cell of
chaos and put this thread to rest.
HARMONICS:
The fundamental frequency is the first harmonic of a particular
fundamental
frequency (i.e., 1xfundamental frequency). Harmonics then usually
start with
the 2nd harmonic (2xfundamental) of the fundamental and go up from
there
(3rd=3x, 4th=4x, etc.). This is assuming that sub-harmonics and the
fundamental
waveshape characteristics are not significant issues. This is how
it was taught
in school and also how I've applied it since. It has worked well.
As for DC, we all know that its frequency component is zero(0). So,
assigning a
frequency designation to it is ludicrous in the practical sense.
F0:
As for f0, this scheme was never intentioned to be an indicator of
any
frequency, particularly a frequency fundamental. Many communication
systems
text books, for example, try to make it simple to illustrate and
describe
relationships between 2 or more frequencies and by doing so utilize
f0, f1, fn
accordingly. OK, sometimes in these texts f0 sometimes equates to a
fundamental
frequency but this is purely coincidence.
So, we can plainly see that some people are mixing these two
methods together
and getting something slightly slippery. Metaphorically speaking,
don't each
of us have to sift through the chaff to get to the wheat? And after
all, is
this f0 subject just a simple matter of semantics anyway.
Just some humble opinions of mine that I thought might serve some
useful
purpose pertinent to this thread.
Best regards,
Ron Pickard
[email protected]
______________________________ Reply Separator
_________________________________
Subject: Re: Harmonics
Author: <[email protected]> at INTERNET
Date: 4/23/99 12:51 PM
Jeff:
If the Fundamental is the First Harmonic, and should be written as
F1 (and subse
ent harmonics F2 and F3 being double and triple the Fundamental
frequency), then
0, or the Zeroth Harmonic (being zero times the F1) is always zero
Hz (aka DC).
That sounds mathematically consistent.
However, I have seen, countless times, the use of F0 to represent
the fundamenta
frequency of a signal. For instance, a frequency step from F0 to F0
+ 1 MHz. Or
asin a sweep from 20% below F0 to 20% above F0.
Are we just looking at two different sides of the same elephant?
Ed
------------------------
From: Jeff Chambers <[email protected]>
Subject: Re: Harmonics
Date: Fri, 23 Apr 1999 19:41:52 +0100
To: Gary McInturff <[email protected]>,
"'[email protected]'"
<[email protected]>, Robert Macy <[email protected]>, Scott
Douglas
<[email protected] >
Cc: [email protected]
> I don't think anybody's said this yet. Fo is used to signify the
dc
> component of the waveform. If a square wave switches from 0 to 5V,
with a
> 50% duty cycle, Fo = 2.5V. F1 is the fundamental (1GHz in your
example), F2
> the second harmonic etc. The term 'first harmonic' is really a bit
> confusing, and 'the fundamental' is better.
>
> Jeff Chambers
>
> -------------------------------------------------------------
> Dr Jeff Chambers
> Westbay Technology Ltd
> Suppliers of EMC Design Software
> Tel: +44 1229 869 108
> Fax: +44 1229 869 108
> http://www.emcnet.com/westbay
> [email protected]
>
> Main St
> Baycliff
> Ulverston
> Cumbria LA12 9RN
> England
> -------------------------------------------------------------
> -----Original Message-----
> From: Gary McInturff <[email protected]>
> To: '[email protected]' <[email protected]>; Robert Macy
> <[email protected]>; Scott Douglas <[email protected]>
> Cc: [email protected] <[email protected]>
> Date: 23 April 1999 18:07
> Subject: RE: Harmonics
>
>
> >Jeeez, if we follow the convention of the harmonic being the
being written
> >as FX (and the fundamental F0) where the subscript is some
integer which
> >represents an harmonic and we include 0 as an integer, which is
an integer
> >by definition, then the fundamental or F0 is Fundamental times 0
or O
> >Hertz, for all frequencies. The first harmonic must then be what
we
> >traditionally call the fundamental. The first harmonic, F1 then
is the
> >Fundamental times 1 and both the fundamental and the first
harmonic are the
> >same.
> > By way of example. I choose 1 GHz (because it gives me heartburn
in
> >my equipment)
> > The fundamental F0 = 1 Ghz times 0 = 0 Ghz.
> > The first harmonic F1 = 1 Ghz times 1 = 1 Ghz
> > Life sort of gets back to normal as we hit the second
fundamental,
> >but again that depends on which side of the argument which
started all of
> >this, you believe.
> > The second harmonic F2 = 1 Ghz times 2 = 2 Ghz. Ad infinitum
> >and ad nausium.
> >
> > Take care (but don't write back I will be off playing golf in
> >California for the next week yea wooooo!)
> > Gary
> >
>
>
>
---------------End of Original Message-----------------
--------------------------
Ed Price
[email protected]
Electromagnetic Compatibility Lab
Cubic Defense Systems
San Diego, CA. USA
619-505-2780
Date: 04/23/1999
Time: 12:51:20
Military & Avionics EMC Services Our Specialty
Also Environmental / Metrology / Reliability
--------------------------
---------
This message is coming from the emc-pstc discussion list.
To cancel your subscription, send mail to [email protected]
with the single line: "unsubscribe emc-pstc" (without the
quotes). For help, send mail to [email protected],
[email protected], [email protected], or
[email protected] (the list administrators).
---------
This message is coming from the emc-pstc discussion list.
To cancel your subscription, send mail to [email protected]
with the single line: "unsubscribe emc-pstc" (without the
quotes). For help, send mail to [email protected],
[email protected], [email protected], or
[email protected] (the list administrators).
---------
This message is coming from the emc-pstc discussion list.
To cancel your subscription, send mail to [email protected]
with the single line: "unsubscribe emc-pstc" (without the
quotes). For help, send mail to [email protected],
[email protected], [email protected], or
[email protected] (the list administrators).
