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From: ehsjr <ehsjr@verizon.net>
Newsgroups: sci.electronics.design
Subject: Re: Omega
Date: Sun, 30 Jun 2024 16:35:34 -0400
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On 6/30/2024 3:44 AM, Cursitor Doom wrote:
> Gentlemen,
> 
> For more decades than I care to remember, I've been using formulae
> such as Xc= 1/2pifL, Xl=2pifC, Fo=1/2pisqrtLC and such like without
> even giving a thought as to how omega gets involved in so many aspects
> of RF.  BTW, that's a lower-case, small omega meaning
> 2*pi*the-frequency-of-interest rather than the large Omega which is
> already reserved for Ohms. How does it keep cropping up? What's so
> special about the constant 6.283 and from what is it derived?
> Just curious...

You've had a number of answers - but not really answering at
the "gut" level.  Why is 2 pi so important - how does omega
get involved in so many aspects of RF?

Every one of the formulas you mentioned has to do with frequency.
The unit of measurement for that is Hertz which is CYCLE(s) per
second.
A cycle's length is 360 degrees regardless of frequency.
A CIRCLE's length is 360 degrees regardless of frequency.
A circle's length is also 2*pi*r regardless of frequency.
Therefore a CYCLE's length (a.k.a wavelength a.k.a. omega)
is also 2*pi*r long.

So 2*pi is used in the conversion between the number of
degrees (time) and distance (length displacement) or
"How much happened ?" (length displacement)
"and how long did it take?" time (frequency).

That's what some call the "gut level" understanding aas
to why 2*pi appears so often.  If you use the math a lot
over time it becomes less mysterious - if that's the right
term. I guess you develop an intuitive understanding or
something like that.

Ed