Damped Resonance Frequency OF series LCR circuit

[I_am_learning]

the solution for current I, for series LCR circuit is
I = (E/Z)sin(wt+\phi)
Where Z = \sqrt{R^2 + (X_{L}-X_{C})^{2}}
So for Resonance (i.e. maximum Current Amplitude) of LCR Circuit the necessary condition seems to be
X_{L}=X_{C}
Which gives \omega=1/\sqrt{LC}

But some text-books and wikipaedia have given that the damped resonace frequency is

where

How is this relation Derived ?

 

[jmb]

Not sure about your solution. What driving voltage are you assuming?

Anyway the undamped resonant frequency is the frequency the undriven system would oscillate with when there is no damping (R=0). When the system is underdamped and undriven then its oscillations consist of a sinusoidal component multiplied by an exponentially decaying envelope. The frequency of the sinusoidal terms in this expression is called the damped resonant frequency. It is found just by solving the harmonic motion equations with a damping term and then identifying the sinusoidal component.

See http://en.wikipedia.org/wiki/Damping" for how to do this for a mass-spring-damper oscillator. You should be able to get the result for an LCR circuit just by replacing the terms used there with the specific coefficients of the damped harmonic oscillator differential equation that arise in LCR circuits (e.g. L corresponds to m, R to c, and C corresponds to 1/k).

 

[nasu]

The first frequency is the frequency of the undamped oscillator. It is also the frequency of the resonance for a driven damped oscillator.

The second one is the frequency of a "free" (not driven) damped oscillator.

 

[Bob S]

See swept frequency plot of voltage-driven and current-driven series resonant circuit. Also see
http://en.wikipedia.org/wiki/RLC_circuit