## What is Resonant Frequency?

Any system that has at least a pair of complex conjugate poles has a natural frequency of oscillation. If the frequency of the system driving force coincides with the natural frequency of oscillation, the system resonates or resonant frequency and the system gives the maximum response.

Resonance is the phenomenon at which natural frequency coincides with the driving force and gives the maximum response. The resonant frequency is the frequency at which resonance happens. Resonance occurs in electrical systems when the system contains at least one inductor and one capacitor.

In this system, the phenomenon of cancellation of reactance when inductor and capacitor are in series or cancellation of susceptance when they are in parallel is termed as resonance. The circuit under resonance is purely resistive in nature and is termed as ‘resonant circuit’ or ‘tuned circuit’.

**Resonant Frequency**

A resonant frequency is the natural vibrating frequency of an object and denoted as ‘f’ with a subscript zero (f_{0}). When an object is in equilibrium with acting forces and could keep vibrating for a long time under perfect conditions, this phenomenon is resonance.

In our daily life example of a resonant frequency is a pendulum. If we pull back the pendulum and leave, it will swing out and return at its resonant frequency. Objects combine to form a system, this system can have more than one resonance frequency. The resonant frequency is termed as the resonance frequency.

The phenomena of resonant frequency used in the series circuit when the inductive reactance (X_{L}) is equal to the capacitive reactance (X_{C}). If the value of supply frequency is changed, we can observe that the value inductive reactance (X_{L}) and capacitive reactance (X_{C}) is also changed.

Inductive reactance (X_{L}) and capacitive reactance (X_{C}) are inversely proportional to each other. When we increase the frequency, the value of X_{L} increases, whereas the value of X_{C} decreases. When we decrease the frequency, the value of X_{L} decreases whereas the value of X_{C} increases.

At series resonance, when XL = XC. The mathematical equation of resonant frequency is:

**X _{L} = 2\(\pi\)fL; X_{C} = 1/2\(\pi\)fC**

**XL = XC**

**2π**** f _{0}L = 1/ 2\(\pi\)**

**f**

_{0}C ; f_{0}=1/2\(\pi\) \sqrt{LC}Where f_{0} is the resonant frequency, L is the inductance, C is the capacitance

### How to Calculate Resonant Frequency of an Object?

An object exposed to its resonant frequency can vibrate in sympathy with the sound. The wavefronts pushing on the object will arrive at just the right time to push the object with greater and greater amplitude in each cycle.

To get the clear idea of this concept one of the best examples is pushing a friend on a swing. If you push swing randomly, swing will not move very well but if you push the swing at a specific time, the swing will get higher and higher.

Another example to find the resonant frequencies is to place the object next to a speaker and place a microphone attached to an oscilloscope next to the object. Then play the sound in e the speaker at a given volume, and then without changing the volume slowly change the frequency.

Now observe the oscilloscope, you will observe that at certain frequencies the amplitude of the wave, which is proportional to the volume of the sound that the microphone is able to pick.

The frequency that is caught by the microphone will be greater than at surrounding frequencies. These are the resonant frequencies and are detectable as the sound energy absorbed by the object is re-emitted more efficiently at these frequencies.

**Solved Question for You**

Q: Compute the resonant frequency of a circuit whose inductance is 25mH and capacitance is 5\mu F?

**Ans: **Known values are,

L = 25mH = 25 x 10^{-3 }H

C = 5\mu F = 5 x 10^{-6} F

Formula for resonant frequency is,

f_{0}= **1/2\(\pi\) \sqrt{LC}**1/2π√L

f_{0}=1/2 ͯ 3.14√ (25 ͯ 10^{-3 }ͯ 5 ͯ 10^{-6})

= 450.384Hz