Forced Oscillations — Question 7

Question 7

Reciprocal detuning gives equal current but unequal charge amplitudes

A series RLC circuit is driven at either $\omega_a=\omega_0/2$ or $\omega_b=2\omega_0$ by the same peak voltage. Let $X_0=\omega_0L=1/(\omega_0C)$.

1. Compare the two current amplitudes and their phase lags relative to voltage.

2. Compare the charge amplitudes.

Question 7 - Solution

Reciprocal detuning gives equal current but unequal charge amplitudes

(a) Method and calculation

$$X_a=X_0(1/2-2)=-3X_0/2,\quad X_b=X_0(2-1/2)=3X_0/2.$$$$\boxed{I_a=I_b=\frac{V_p}{\sqrt{R^2+(3X_0/2)^2}},\quad\phi_b=-\phi_a}.$$

(b) Interpretation and further calculation

$$Q=I/\omega\Rightarrow Q_a/Q_b=\omega_b/\omega_a.$$$$\boxed{Q_a=4Q_b}.$$

Physical check

Frequency weighting distinguishes a charge response from a current response even when impedance magnitudes agree.