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What is sharpness of resonance ? Derive equation of Q-factor |
| Answer» <html><body><p></p>Solution :Frequency of resonance is independent to resistance R but sharpness of resonance is inversely proportional to R. <br/> Sharpness of resonance `= ( omega_(0))/( <a href="https://interviewquestions.tuteehub.com/tag/2-283658" style="font-weight:bold;" target="_blank" title="Click to know more about 2">2</a> <a href="https://interviewquestions.tuteehub.com/tag/delta-947703" style="font-weight:bold;" target="_blank" title="Click to know more about DELTA">DELTA</a> omega)` <br/> `= ( omega_(0) 2 L )/( 2 xx R ) = ( omega_(0) L ) /( R )`but `( omega_(0) L )/( R ) ` is <a href="https://interviewquestions.tuteehub.com/tag/called-907796" style="font-weight:bold;" target="_blank" title="Click to know more about CALLED">CALLED</a> the quality factor Q of a circuit. <br/> `:.` Quality factor `Q = ( omega_(0) L )/( R )` <br/> `:. ( omega_(0))/( 2 Delta omega ) = Q = ( " Resonant frequency ")/( "<a href="https://interviewquestions.tuteehub.com/tag/bandwidth-14801" style="font-weight:bold;" target="_blank" title="Click to know more about BANDWIDTH">BANDWIDTH</a>") = ( omega_(0))/( omega_(2) - omega_(1))` <br/> `:. 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))`. <br/> So, larger the value of Q the smaller is value of `2 Delta omega` or bandwidth and <a href="https://interviewquestions.tuteehub.com/tag/sharper-7711972" style="font-weight:bold;" target="_blank" title="Click to know more about SHARPER">SHARPER</a> is the resonance. <br/> Putting`L = ( 1)/( omega^(2) C 0` <br/> `Q = ( omega_(0) xx 1 )/( R xx omega_(0)^(2) C )` <br/> `:. Q = ( 1)/( omega_(0) RC )`<br/> (1) For series LCR circuit, Q-factor, <br/> `Q = ( 1)/( R ) sqrt(( L )/( C )) = ( omega_(0) L )/( R ) = ( 1)/( omega_(0) CR )`<br/> (2) For parallel LCR circuit, <br/> `Q = ( V_(L ))/( V_(R )) ` or `( V _(C ))/( V_(R ))`<br/> 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. <br/> So, less sharp the resonance, less is selectivity of the current or vice versa.<br/> If quality is larger, means R is low or L is large the circuit is more sleective.</body></html> | |