Damped Oscillations in an RLC Circuit — Question 3

Question 3

Critical damping reaches zero charge without crossing it after release from rest

A series circuit has $R=2\sqrt{L/C}$ and is released from $q=Q_0>0$, $i=0$. Let $\omega_0=1/\sqrt{LC}$.

1. Find q(t) from the repeated-root solution.

2. Find the time and magnitude of maximum current.

Question 3 - Solution

Critical damping reaches zero charge without crossing it after release from rest

(a) Method and calculation

$$q=(A+Bt)e^{-\omega_0t},\quad A=Q_0,\quad B=\omega_0Q_0.$$$$\boxed{q=Q_0(1+\omega_0t)e^{-\omega_0t},\quad i=-Q_0\omega_0^2t e^{-\omega_0t}}.$$

(b) Interpretation and further calculation

$$\frac d{dt}|i|=Q_0\omega_0^2e^{-\omega_0t}(1-\omega_0t)=0.$$$$\boxed{t_{|i|\max}=1/\omega_0,\quad |i|_{\max}=Q_0\omega_0/e}.$$

Physical check

For these initial conditions q stays positive and decreases monotonically.