Forced Oscillations — Question 2

Question 2

The steady-state phase moves from elastic response to inertial response

A series RLC circuit is driven by $V_p\cos\omega t$. Its charge amplitude is $Q=V_p/\sqrt{(1/C-L\omega^2)^2+(R\omega)^2}$, and charge lags the drive by delta.

1. Find low-frequency and high-frequency limits of Q and delta.

2. Find charge and current at the natural frequency $\omega_0=1/\sqrt{LC}$.

Question 2 - Solution

The steady-state phase moves from elastic response to inertial response

(a) Method and calculation

$$\omega\to0:\quad\boxed{Q\to CV_p,\quad\delta\to0}.$$$$\omega\to\infty:\quad\boxed{Q\sim V_p/(L\omega^2),\quad\delta\to\pi}.$$

(b) Interpretation and further calculation

$$\omega=\omega_0:\quad q=\frac{V_p}{R\omega_0}\sin\omega_0t.$$$$\boxed{i=\dot q=(V_p/R)\cos\omega_0t}.$$

Physical check

At the natural frequency, current is in phase with the drive while charge lags by a quarter-cycle.