When something oscillates peaks at its maximum amplitude when applied with energy. Examples of resonance are a couple on a squeaky bed, a ringing bell, or a car with no shocks on a bumpy road.

A resonant frequency can be many things. One application of resonant frequencies can be the amplification of sound through a tube of the correct length. Yet it can also be much more physical

Consider a swing. There is a frequency of pushing a person on the swing that will cause them to rise higher, and there are frequencies that will end up accomplishing very little. Any of us who have pushed a swing know that you must push to add momentum, not subtract. The fall of the Tacoma Narrows Bridge is an interesting example of this.

See Resonance if you are not familiar with it

The Resonant frequency of a system is defined in two ways which though similar are not equivalent

One way is to define it as the frequency at which the system must be driven in order for the oscillations to have the maximum amplitude .

Another way to define it is as the frequency at which the system must be driven in order for there to maximum energy transfer between the driving oscillator and the system

In the case of an undamped system, the resonant frequency is just the natural frequency of the system. However the moment you introduce damping this is not so anymore(I still have to figure out a satisfactory physical reason behind this).

If you have an undamped system described by the differential equation:
d^2(x)/dt^2 + g*d(x)/dt + w0^2*x = 0
then the natural frequency when it is undergoing free oscillations is:
sqrt(w0^2 - g^2/4)
However the maximum amplitude is obtained at
sqrt(w0^2 - g^2/2)
and the maximum energy transfer at w0.
In electronics, the frequency that a series circuit is resonant is where XL = XC.

This frequency can be determined using the formula:


                   1
 f = ----------------------------------
      2 * pi * (squareroot of (L * C))


     where:
        f = frequency
       pi = 3.14159...
        L = inductance in henries
        C = capacitance in farads

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.

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