A circuit's resonant
frequency isn't very difficult to determine. For LC circuits, the omega value is just 1/sqrt(L * C). The resonant frequency occurs when the output voltage is at its maximum possible value. The magnitude of the
system function, |H|, must be at its maximum value for this to happen.
When you have the value of the system function in terms of omega, you can find the value for omega which will make the magnitude the largest. Usually this can be done by inspection, but for complicated functions it is necessary to take the derivative and solve for the maximum in the manner taught in basic calculus: set the derivative equal to zero and solve. The values you get for omega will be minima or maxima.
So why exactly is the resonant frequency of an LC circuit equal to 1/sqrt(L * C)? Let's take a look at a standard LC circuit:
L
_ _ _
/ \ / \ / \
----------\_/-\_/-\_/-----------------O
| |
| | +
/+\ |
/ \ |
| V |Vi: ----- Vo
\ / voltage C
\-/ source -----
| | -
| |
---------------------------------------O
|
----- Ground
---
-
Figuring out the system function for this circuit is easy. The impedance of the
inductor is just L * S, and the
capacitor's impedance is 1 / (S * C). Using a
voltage divider formula, the system function becomes:
1
-----
S * C 1
H(S)= ---------------- = ---------------
1 1 + LCS2
----- + (L * S)
S * C
And we know that S = j * omega, where j is the square root of negative one, so we get:
1
H(j(omega)) = -----------------
1 - LC(omega2)
Since there is no imaginary part to this function, it is its own magnitude. So, we have to look at this and determine which value of omega will produce the largest H. The closer that the LC(omega2) term gets to 1, the larger H will be. If that term does hit 1, we will in theory have a system function magnitude of infinity. In order to obtain this, omega must be equal to 1 / sqrt (L * C). Omega is equal to (2 * pi * frequency), so the resonant frequency can easily be derived from that relation once you solve for omega. Most circuits you will find will usually adhere to some common pattern of resonant frequency. But whenever there is doubt, this technique can be applied to any circuit.