What is sharpness of resonance ? Derive equation of Q-factor

Explanation

Frequency of resonance is independent to resistance R but sharpness of resonance is inversely proportional to R.
Sharpness of resonance `= ( omega_(0))/( 2 Delta omega)`
`= ( omega_(0) 2 L )/( 2 xx R ) = ( omega_(0) L ) /( R )` but `( omega_(0) L )/( R ) ` is called the quality factor Q of a circuit.
`:.` Quality factor `Q = ( omega_(0) L )/( R )`
`:. ( omega_(0))/( 2 Delta omega ) = Q = ( " Resonant frequency ")/( "Bandwidth") = ( omega_(0))/( omega_(2) - omega_(1))`
`:. 2 Delta omega = ( omega_(0))/( Q)` where `omega_(2)` and `omega_(1)` are the frequency at the current amplitude of `(I_("rms") )_("max")` with a value of `(1)/( sqrt(2))`.
So, larger the value of Q the smaller is value of `2 Delta omega` or bandwidth and sharper is the resonance.
Putting `L = ( 1)/( omega^(2) C 0`
`Q = ( omega_(0) xx 1 )/( R xx omega_(0)^(2) C )`
`:. Q = ( 1)/( omega_(0) RC )`
(1) For series LCR circuit, Q-factor,
`Q = ( 1)/( R ) sqrt(( L )/( C )) = ( omega_(0) L )/( R ) = ( 1)/( omega_(0) CR )`
(2) For parallel LCR circuit,
`Q = ( V_(L ))/( V_(R )) ` or `( V _(C ))/( V_(R ))`
If the resonance is less sharp not only is the maximum current less the circuit is close to resonance for a larger range `Delta omega` of frequencies and the tuning of the circuit will not be good.
So, less sharp the resonance, less is selectivity of the current or vice versa.
If quality is larger, means R is low or L is large the circuit is more sleective.