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!
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