Damped Oscillations in an RLC Circuit — Question 7

Question 7

A required number of visible cycles imposes a resistance limit

An underdamped oscillator must retain at least one-half its initial charge-envelope amplitude after N damped cycles. Define $\zeta=R/(2\sqrt{L/C})$.

1. Derive an exact upper bound on zeta.

2. Evaluate the bound for N=10 and express the corresponding R limit.

Question 7 - Solution

A required number of visible cycles imposes a resistance limit

(a) Method and calculation

$$e^{-\alpha NT_d}\ge1/2,\quad\alpha T_d=\frac{2\pi\zeta}{\sqrt{1-\zeta^2}}.$$$$\boxed{\zeta\le\frac{\ln2}{\sqrt{(2\pi N)^2+(\ln2)^2}}}.$$

(b) Interpretation and further calculation

$$N=10\Rightarrow\zeta_{\max}=.0110311.$$$$\boxed{R\le2(.0110311)\sqrt{L/C}}.$$

Physical check

The exact bound uses the damped period; weak-damping approximations are unnecessary here.