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What is sharpness of resonance ? Derive equation of Q-factor |
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Answer» Solution :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. |
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