Formulas Regarding Radio Waves

Date: 11/17/2004 at 21:52:42
From: Fred Cain
Subject: High frequency radio antennas

Why does part of the equation to find the resonant frequency have the 
notation of 2 times pi times the square root of the L times C, the 
answer of which is divided into 1?  There are several formulas that 
have to do with transmission of radio waves that have this 2pi in 
them.  I have passed the advanced ham examination in Canada but have 
yet to find any explanation of why this 2pi is there and what it 
signifies.  The Q factor also has 2pi times f times L which is the 
inductance.   

There is nothing confusing about this other than I would like to know 
rather than just knowing it by rote.  An explanation would round this
out a great deal.

Date: 11/23/2004 at 12:36:33
From: Doctor Douglas
Subject: Re: High frequency radio antennas

Hi Fred.

In this context, it basically comes down to the fact that there are
2*pi radians in a full circle.  For the trigonometry to work out, the 
natural unit to measure the angles is the radian, as it makes many
formulas simpler to work in radians rather than degrees:

   sin(x) = x - x^3/3! + x^5/5! - ...      for x in radians

You can of course convert the angles to degrees (pi radians = 180 
degrees), but then factors of pi and 180 will appear in trigonometry
formulas that operate in degrees.  You can also convert the angle 
to "revolutions" or "cycles", and then factors of 2 and pi can appear.  
This is what happens in your formula 1/sqrt(L*C):

   w = 1/sqrt(L*C),     

where w is the angular resonant frequency and has units of radians
per second.  Of course we work in the real world and would probably
rather count cycles than radians, so we convert to real cycles
(as in AM or FM cycles):

   f = w/(2*pi) = 1/[2*pi*sqrt(L*C)].

If we are being careful, we say that angular frequency w is measured
in rad/sec and that f is measured in cycles/sec (cps).  If we are 
being sloppy, we simply say that w and f are both measured in Hz.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/24/2004 at 21:07:01
From: Fred Cain
Subject: Thank you (High frequency radio antennas)

Doctor Douglas,

Thank you for taking the time to answer my question.  It really is
quite elegant how it all works!