Forced Oscillations — Question 3

Question 3

Maximum charge response occurs below maximum current response

For fixed drive peak Vp, define $\omega_0^2=1/(LC)$ and $\alpha=R/(2L)$. The charge amplitude is $Q=(V_p/L)[(\omega_0^2-\omega^2)^2+4\alpha^2\omega^2]^{-1/2}$.

1. Find the nonzero frequency of maximum charge response and the condition for its existence.

2. Find the frequency of maximum current amplitude $I=\omega Q$.

Question 3 - Solution

Maximum charge response occurs below maximum current response

(a) Method and calculation

$$\frac d{d\omega}[(\omega_0^2-\omega^2)^2+4\alpha^2\omega^2]=4\omega(\omega^2-\omega_0^2+2\alpha^2).$$$$\boxed{\omega_{Q,\max}=\sqrt{\omega_0^2-2\alpha^2}\quad(\alpha<\omega_0/\sqrt2)}.$$

(b) Interpretation and further calculation

$$I=\frac{V_p}{\sqrt{R^2+(\omega L-1/(\omega C))^2}}.$$$$\boxed{\omega_{I,\max}=\omega_0,\quad I_{\max}=V_p/R}.$$

Physical check

The word resonance must specify the measured response; charge and current peaks generally differ.