For the circuit of the figure below, let C = 17.5 nF, L = 21 mH, and R = 88.5 .
ID: 2030023 • Letter: F
Question
For the circuit of the figure below, let C = 17.5 nF, L = 21 mH, and R = 88.5 .
(a) Calculate the oscillation frequency of the circuit once the capacitor has been charged and the switch has been connected to point a.
kHz
(b) How long will it take for the amplitude of the oscillation to decay to 10.0% of its original value?
ms
(c) What value of R would result in a critically damped circuit?
Explanation / Answer
a)
Angular frequency of damped oscillation is
w' = sqrt(1/LC - R^2/4L^2)
w' = sqrt[1/(21 x 10^-3 x 17.5 x 10^-9) - 88.5^2/(4 x 21 x 21 x 10^-6)]
w' = sqrt(2721088435 - 4440051)
w' = 5.2121 x 10^4 rad/s
Therefore oscillation frequency for the circuit is
f = wi/2pi
f = 5.2121 x 10^4 rad/s / 2pi
f = 8.295 KHz
b)
amplitude of the damped oscillation is
A = Aoe^(-R/2L)t
ln(A/Ao) = -(R/2L)t
t = [-2L*ln(A/Ao)] / R
Amplitude of the the damped oscillation decreases by the 10%
A/Ao = 0.10
t = (-2 x 21 x 10^-3 x ln(0.10)) / 88.5
t = 1.09 mA
c)
R = sqrt(4L/C)
R = sqrt[(4 x 21 x 10^-3) / 17.5 x 10^-9)]
R = 2190.89 ohms