Forced Oscillations — Question 4

Question 4

An ideal undamped resonant drive produces secular growth

An ideal LC loop with zero resistance is driven from rest by $V_p\cos\omega_0t$, where $\omega_0=1/\sqrt{LC}$. Initially q=i=0.

1. Find q(t) using a trial solution proportional to $t\sin\omega_0t$.

2. Find current and the leading large-time stored energy.

Question 4 - Solution

An ideal undamped resonant drive produces secular growth

(a) Method and calculation

$$(d^2/dt^2+\omega_0^2)(t\sin\omega_0t)=2\omega_0\cos\omega_0t.$$$$\boxed{q=\frac{V_p}{2L\omega_0}t\sin\omega_0t}.$$

(b) Interpretation and further calculation

$$\boxed{i=\frac{V_p}{2L\omega_0}[\sin\omega_0t+\omega_0t\cos\omega_0t]}.$$$$\boxed{U(t)=\frac{V_p^2}{8L}t^2+O(t)\quad(t\to\infty)}.$$

Physical check

No finite steady resonant amplitude exists when the model has no loss.