What is the frequency at which the impulse response to a RLC circuit oscillates?

[iVenky]

What is the frequency at which the impulse response of a RLC circuit oscillates?

Let the resonant frequency of the RLC circuit be ωo and the exponential damping coefficient be 'α' (alpha) then the natural resonant frequency is given by ωd.

Now my question is if we have under damped system i.e. α < ωo then the impulse response would oscillate for some time before it settles. At what frequency will this oscillate? I believe that it should be ωd basically because the transient response of an under damped system is given by

v(t) = A e^-αt (B cos(ωd t) + C sin(ωd t) )

But I tried using Spice and I checked the value of the freuqency and it is equal to ωo. Is that because ω0 is approximately equal to ωd ?

So at what frequency will the system oscillate?

Thanks in advance.

 

[BradtheRad]

The formula normally used is:

f = the reciprocal of 2 Pi √( LC )

Ohmic resistance does not affect frequency. It is cancelled in the process of merging the time constant formulae for inductance (L/R) and capacitance (RC).

Since your formula has voltage at the left, then the frequency of oscillation is based on the volt level on the capacitor (in the midst of charging/discharging from one moment to the next), and having it match the emf put up by the coil (in the midst of admitting current from one moment to the next).

- - - Updated - - -

Here's a couple of links that discuss LC tank calculations:

http://www.physicsforums.com/showthread.php?t=482987

http://www.physicsforums.com/showthread.php?t=556245

 

[rsashwinkumar]

@BradtheRad : I think, what iVenky told is right... wd is approx equal to undamped frequency, and thats the reason the SPICE shows up the result as he mentioned. As wd = wn*sqrt(1-zeta^2), and as for underdamped, zeta<1, it approxes to wn. And it is the only possible explanation i guess.

- - - Updated - - -

@BradtheRad : I think, what iVenky told is right... wd is approx equal to undamped frequency, and thats the reason the SPICE shows up the result as he mentioned. As wd = wn*sqrt(1-zeta^2), and as for underdamped, zeta<1, it approxes to wn. And it is the only possible explanation i guess.

 

[LvW]

Re: What is the frequency at which the impulse response of a RLC circuit oscillates?

But I tried using Spice and I checked the value of the freuqency and it is equal to ωo. Is that because ω0 is approximately equal to ωd ?
So at what frequency will the system oscillate?

For the classical and idealized RLC resonant circuit (series/parallel: All losses concentrated in a single series/parallel damping resistor Rs resp. Rp) the natural frequency (step or impulse response) wd is always somewhat smaller than the resonant frequency wo=1/sqrt(LC) caused by an additional loss term under the sqrt. However, for a circuit with small losses (Rs small resp. Rp large) both frequencies are nearly identical. In this case it is not easy to the the difference as a result of SPICE simulations.
In oscillators, the loss term is compensated by regenerative feedback. Thus, the circuit oscillates at wo.