Explain resonance for an L-C-R series circuit and write its uses. In what kind of circuit will it possible ?

Explanation

For an L-C-R circuit driven with voltage of amplitude `V_(m)` and frequency `omega` the curren amplitude is given by,
`I_(m) = ( V_(m))/( Z) = ( V_(m))/( sqrt( R^(2) + (X_(C ) - X_(L))^(2)))`
where `X_(C ) = ( 1)/( omega C )` and `X_(L) = omega L `.
`I_(m) = ( V_(m))/( sqrt(R^(2)+ (( 1)/( omega C) - omega L )^(2)))` ....(1)
Hence, if `omega` is varied so current amplitude is varied at a particular frequency `omega _(0) , X_(C ) = X_(L)` and the impedance is minimum. `Z = sqrt( R^(2) + O^(2))= R ` . The value of amplitude of current is maximum. This is called resonance.
`:. ` If `X_(C ) = X_(L)`, then from equation (1),
`I_(m) = ( V_(m))/( R )` at resonant frequency amplitude is maximum and
`X_(C ) = X_(L)` or `(1)/( omega_(0) C ) \= omega_(0) L `
`:. omega _(0)^(2) =( 1)/( LC ) ` or `omega_(0) = ( 1)/( sqrt( LC ))`
Hence, at resonance frequency amplitude of current becomes maximum and it is` I_(m) = ( V_(m)) /( R )`.

Resonant circuits have a variety of applications for example in the tuning mechanism of a raido or a TV set . The antenna of a radio accepts signals from many broadcasting stations. The siglas picked up in the antenna acts as a source in the tuning circuit of the radio, so the circuit can be driven at many frequencies.
But , to hear one particular radio station, we tune the radio.
In tuning, we vary the capacitance of a capacitor in the tuning circuit such that the resonant frequency of the circuit becomes nearly equal to the frequency of the radio signal received. Hence, resonance occurs and amplitude of current becomes maximum.
The resonance phenomenon is exhibited by a circuit only if both L and C are present in the circuit. Only then do the voltages across L and C cancel each other and the current amplitude is `I_(m) = ( V_(m))/( R )` the total source voltage appearing across R `:`
This means that we cannot have resonance in a RL and RC circuit but resonance L and C both should be in circuit.