Damped Oscillations in an RLC Circuit — Question 6

Question 6

Damping delays the first zero of charge beyond a quarter damped period

An underdamped circuit is released from q=Q0 and i=0, with $q=Q_0e^{-\alpha t}[\cos\omega_dt+(\alpha/\omega_d)\sin\omega_dt]$.

1. Find the first positive time at which q=0.

2. Compare it with the first current zero after release.

Question 6 - Solution

Damping delays the first zero of charge beyond a quarter damped period

(a) Method and calculation

$$\tan\omega_dt=-\omega_d/\alpha,\quad\pi/2<\omega_dt<\pi.$$$$\boxed{t_{q=0}=\frac{\pi/2+\tan^{-1}(\alpha/\omega_d)}{\omega_d}}.$$

(b) Interpretation and further calculation

$$i\propto-e^{-\alpha t}\sin\omega_dt\Rightarrow\boxed{t_{i=0}=\pi/\omega_d}.$$$$\boxed{T_d/4<t_{q=0}<T_d/2}.$$

Physical check

For zero damping the charge zero returns to T/4; finite damping changes the release phase.