Damped Oscillations in an RLC Circuit — Question 10

Question 10

Critical damping does not forbid a zero crossing for arbitrary initial current

A critically damped circuit has $q(0)=Q_0>0$ and $i(0)=-2\omega_0Q_0$, where $\omega_0=1/\sqrt{LC}$.

1. Find the charge and its first zero crossing.

2. Find the later charge minimum and show that the response still is not periodic.

Question 10 - Solution

Critical damping does not forbid a zero crossing for arbitrary initial current

(a) Method and calculation

$$q=(A+Bt)e^{-\omega_0t},\quad A=Q_0,\quad B=i_0+\omega_0Q_0=-\omega_0Q_0.$$$$\boxed{q=Q_0(1-\omega_0t)e^{-\omega_0t},\quad t_{q=0}=1/\omega_0}.$$

(b) Interpretation and further calculation

$$i=Q_0\omega_0(\omega_0t-2)e^{-\omega_0t}=0\Rightarrow t=2/\omega_0.$$$$\boxed{q_{\min}=-Q_0e^{-2};\quad q\to0^-\ \text{without further crossings}}.$$

Physical check

Nonoscillatory means no repeated oscillation; it does not rule out a crossing caused by an initial impulse.