To Mike Sawyer: I sent a private reply to your private email but it was
rejected by your server so I am sending it here.
Mike,
Yes you are wrong in your thinking, the approximate formulae that many hams
believe to be exact are approximate at all frequencies.
For example, you probably believe that the formula for the resonant
frequency of a tuned circuit is 1/(2*pi*SQRT(LC)) because you were taught
that.
Well that is true for a series LCR circuit but it is NOT true for a parallel
tuned circuit.
It does give approximately the right answer for a HIGH Q parallel tuned
circuit but the lower the Q the more erroneous the result will be.
If you take the case of a parallel circuit having a series L and R in one
branch and a lossless C in the other branch the formula for the resonant
frequency is:
(SQRT((1/LC)-(R*R/L*L)))/(2*pi)
This differs from the original formula due to the R*R/L*L factor which
modifies the resonant frequency.
In the special case where R=0 the formula becomes identical to the original
formula.
In the case of a high Q circuit, R is negligible and so the R*R/L*L term is
often dropped yielding the approximate formula that many hams are familiar
with.
It's difficult to write mathematical formulae in an email so I will not
attempt the formula for the more general case of a series L and R in one
branch and a series C and R in the other branch. Suffice it to say that that
case also approximates to the original formula above when the resistance is
negligible, in other words a high Q circuit.
Here is a web page that will allow you to plug in values for the parallel
resonant circuit above:
http://www.cvs1.uklinux.net/cgi-bin/calculators/tuned_circuit.cgi
You will have to experiment a little bit, when you get down to some workable
values that produce a low Q (say around 3) try changing R but keeping L and
C unchanged and you will see the resonant frequency vary. Right there you
can see that Xl and Xc are not numerically equal at resonance.
73 Jim, AF6O
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