3   Tuned Circuit Formula Part 2

Hopefully you will remember from my last piece the easy way to calculate frequency by taking
25,330, dividing this by L(µH) multiplied by C (pF) and then finding the square root of this to
give the answer in MHz.

But where did the original formula   _{f  =}           1           .           come from ? 
                                                             2 pi root LC

Well, the reactance of a C at a given frequency is given by   X_{C  =}         1      .
                                                                                                2 pi f C

The reactance of an L at the same frequency is given by         X_L  =  2 pi f L

What’s reactance?    AC resistance! (as against R for DC resistance)

At resonance, X_C and X_L  are equal, so we can write…   2 pi f L  ₌          1       .
                                                                                                  2 pi f C

Now if we multiply both sides by 2 pi f C we get…….     4 pi²  f²  L C =  1

To find  f²  we must get it on its own so we must divide both sides by 4 pi²  LC

This gives …..    f²   =            1       .
                                    4 pi²  L C

To get f  we must find the square root of the whole thing.

Then….. f  =            1          .            The real difficulty is that L must be in Henrys & C in Farads!
                     2 pi  root L C                    

Why 2 pi you may ask? ….Well, pi is  90 degrees of a circle or a quarter of a wavelength.

So 2pi must be half a circle or 180 degrees of an alternating current or half a wavelength!

As with all these formulae, ‘root’ means square root. This means a number which , when multiplied
by itself gives the number we started with. e.g. the square root of 4 is 2, because

 2 x 2 = 4. The square root of 64 is 8 because 8 x 8 = 64, 100 is the root of 10,000.

For more complex numbers you can use your calculator to do the hard work for you. Those of you
taking the next full licence exam (9^th Oct) should have calculators which have a square root button,
making life really easy.

Note, in L there will be a little bit of resistance, but that won’t affect the frequency.

Happier now ? No?   Well….. f²(MHz)   =     25,330  .   is a lot easier to use, eh?
                                                           LµH CpF

That’s it for this time. Mike G3JKX