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In a series LCR circuit connected to an a.c. source of variable frequency and voltageV = V_(m) sin omega t draw a plot showing the variation of current (I) with angular frequency (w) for two different values of resistance R_(1) and R_(2) (R_(1) gt R_(2)).Write the condition under which the phenomenon of resonance occurs. For which value of the resistance out of the two curves, a sharper resonance is produced ? Define Q-factor of the circuit and give its significance. |
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Answer» Solution :A plot SHOWING variation of current (I) with angular frequency (`omega`) for two DIFFERENT values of resistances`R_(1)` and `R_(2)(R_(1) gt R_(2))` in a series LCR a.c. circuit is shown in the Fig. 7.33. Condition for resonance to occur is `X_(L) = X_(C)` `rArr` Angular frequency `omega_(0) =1/sqrt(LC)`.A sharper resonance is provided for smaller values of resistance. Thus, curve for resistance `R_(2)`is sharper than the curve for resistance `R_(1)` Q-factor of a resonant circuit is a measure of the "sharpness of resonance" and is defined as the RATIO of resonant angular frequency `omega_(0)`to the band width (`2 Deltaomega` ) of the circuit, where band width is the difference in angular frequencies`(omega_(0) + Deltaomega)` and `(omega_(0) -Deltaomega)` at which power is half the MAXIMUM power or current is `1/sqrt(2)` times the maximum current value at resonance. Mathematically, `Q = omega_(0)/(2.Deltaomega) = (omega_(0)L)/(R) = 1/(omega_(0)CR) = 1/R sqrt(LC)` 
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