Obtain an equation for sharpness of resonance in an L-C-R series AC circuit and what is quality factor Q ? And explain bandwidth.

Explanation

The amplitude of the current in the series L-C-R circuit is,
`I_(m) = ( V_(m))/( sqrt( R^(2) + (( 1)/( omega C ) - omega L)^(2))) = ( V_(m))/( sqrt( R^(2) + ( X_(C ) - X_(L))^(2)))`
when `omega = omega_(0) = ( 1)/( sqrt( LC ))` amplitude of curren is maximum
This maximum amplitude is `I_(m) = ( V_(m))/( R ) `.
For values of `omega `other than `omega_(0)` the amplitude of the current is less than the maximum value.
A value of `omega ` for which the current amplitude is `( 1)/( sqrt(2))` times its maximum value. At this value the power dissipated by the circuit becomes half.
`:.` If `I= ( I_(m))/( sqrt(2))` power becomes half and two values of `omega ` is obtained according to as given figure.

Variation of `I_(m)` with `omega ` for two cases (i) R = `100 Omega ` (ii) R = 200` Omega, L = 100 mH`.
There are two such values of `omega` say, `omega_(1) ` and `omega_(2)` one greater and the other smaller than `omega_(0)` and symmetrical about `omega_(0)`.
`:. omega_(1) = omega_(0) + Delta omega`
and `omega_(2) = omega_(0) - Delta omega`
The difference `omega_(1) - omega_(2) = 2 Delta omega` is called the bandwidth of the circuit.
The quantity `(omega_(0))/( 2 Delta omega )` is regarded as a measure of the sharpness of resonance.
The smaller the `Delta omega` the sharper or narrower is the resonance.