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Obtain an equation for sharpness of resonance in an L-C-R series AC circuit and what is quality factor Q ? And explain bandwidth. |
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Answer» Solution :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 ofthe sharpness of resonance. The smaller the `Delta omega` the sharper or narrower is the resonance. |
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