Damped Oscillations in an RLC Circuit — Question 9

Question 9

A damped frequency specification is stricter than merely requiring oscillation

A series RLC oscillator must have damped angular frequency at least .980 times its ideal natural frequency. L and C are fixed.

1. Derive the largest allowed resistance.

2. Compare this limit with the critical resistance.

Question 9 - Solution

A damped frequency specification is stricter than merely requiring oscillation

(a) Method and calculation

$$\omega_d/\omega_0=\sqrt{1-\zeta^2}\ge.980.$$$$\boxed{\zeta\le\sqrt{1-.980^2}=.198997}.$$

(b) Interpretation and further calculation

$$R=2\zeta\sqrt{L/C},\quad R_{\mathrm{crit}}=2\sqrt{L/C}.$$$$\boxed{R_{\max}=.198997R_{\mathrm{crit}}=.397995\sqrt{L/C}}.$$

Physical check

An oscillator may be underdamped yet have a frequency substantially below its ideal LC value.