Forced Oscillations — Question 10

Question 10

An undamped off-resonant switch-on retains beats instead of settling

A lossless LC loop starts from q=i=0 under $V_p\cos\omega t$, with $\omega\ne\omega_0$.

1. Find the exact charge including the natural-frequency term.

2. Factor the result to expose the slow modulation when the frequencies are close.

Question 10 - Solution

An undamped off-resonant switch-on retains beats instead of settling

(a) Method and calculation

$$q_p=\frac{V_p\cos\omega t}{L(\omega_0^2-\omega^2)},\quad q_h=-\frac{V_p\cos\omega_0t}{L(\omega_0^2-\omega^2)}.$$$$\boxed{q=\frac{V_p(\cos\omega t-\cos\omega_0t)}{L(\omega_0^2-\omega^2)}}.$$

(b) Interpretation and further calculation

$$\cos a-\cos b=-2\sin[(a+b)/2]\sin[(a-b)/2].$$$$\boxed{q=\frac{2V_p\sin[(\omega_0+\omega)t/2]\sin[(\omega_0-\omega)t/2]}{L(\omega_0^2-\omega^2)}}.$$

Physical check

Without resistance the natural component does not disappear, so a unique attracting steady state is absent.