Damped Oscillations in an RLC Circuit — Question 2

Question 2

Successive charge extrema identify damping and the undamped frequency

Positive charge maxima in an underdamped RLC loop are .0100 s apart and decrease by a factor of two each time. L=.200 H.

1. Infer alpha, R, and the damped angular frequency.

2. Find the capacitance, retaining the damping correction.

Question 2 - Solution

Successive charge extrema identify damping and the undamped frequency

(a) Method and calculation

$$e^{-\alpha T_d}=1/2\Rightarrow\alpha=\ln2/.0100=69.3147\,s^{-1}.$$$$\boxed{R=2L\alpha=27.7259\,\Omega,\quad\omega_d=2\pi/.0100=628.319\,s^{-1}}.$$

(b) Interpretation and further calculation

$$\omega_0^2=\omega_d^2+\alpha^2,\quad C=[L(\omega_d^2+\alpha^2)]^{-1}.$$$$\boxed{C=12.5129\,\mu F}.$$

Physical check

The extrema are separated by the damped period, not the undamped period.