Re: Omega

On Sun, 30 Jun 2024 16:35:34 -0400, ehsjr wrote:

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.

There's something about electronics and its associated mathematics that
I've always found challenging to be honest. Many years ago when I studied
medicine and then law I didn't have to do any real work at all to speak
of. I just absorbed those subjects easily without even trying. I was often
baffled as to why so many students struggled with the topics in those
subjects and why they clearly had to work so hard for so long to keep up
with the class. I'd skip lectures and assignments with gay abandon but
still get top marks. Not so with the hard sciences! Now I can finally
empathise with my contemporaries back in those days who had to really make
a serious effort to grasp the principles and who just about scraped
through. They would never go on to excel in those fields just as I have
never and will never excel at electronics. Still, no one can be
outstanding at everything in today's world and the days of the true
polymaths are long gone.