Quality factor in RLC circuit

The quality factor Q is defined as Q = wo/(dw)where wo is the resonance frequency, and dw (delta w) is the bandwidth, which is defined as the difference between the two frequencies at which the power dissipated at the resistor R falls to half of its maximum value. At max power, the supplied voltage Vo and current I are in phase. When the phase angle between Vo and current I is +45 degrees (Vo leads I) or -45 degrees (Vo lags I), the power dissipated at R is half of the maximum power. When Vo leads I by 45 degrees, we have, tan(45) = (wL – 1/wC)/Ri.e. wL - 1/wC = R or (LC)w^2 - (RC)w – 1 = 0solve for w gives w = (R/2L) + square-root[(R/2L)^2 + 1/(LC)]For small values of R (light damping), w = (R/2L) + square-root[1/LC]But square-root[1/LC] = wo, the resonance frequencyHence, w = wo + (R/2L) When Vo lags I by 45 degrees, we have, tan(-45) = (wL – 1/wC)/Ri.e. 1/wC – wL = R or (LC)w^2 + (RC)w – 1 = 0solve for w gives w = -(R/2L) + square-root[(R/2L)^2 + 1/(LC)]Again, for small values of R (light damping), w = -(R/2L) + square-root[1/LC]hence, w = wo - (R/2L) The bandwidth dw = [wo + (R/2L)] – [wo - (R/2L)] = R/Lhence, Q = wo/dw = (wo)L/R


2011-05-05 22:56:26 補充：
Re: your supplemnetary question.
Yoy may discard the assumption, as you could still arrive at the result without it. If R is large, the resonance peak would be much broadened.