The Series RLC Circuit — Question 3

Question 3

Resonance can produce large internal voltages with a modest source

A series circuit has R=5.00 ohms, L=.200 H, C=20.0 microfarads, and source peak 10.0 V. It is driven at resonance. Define quality factor $\mathcal Q=\omega_0L/R$.

1. Find resonance frequency, peak current, and both reactive voltage peaks.

2. Find total instantaneous stored field energy in steady state.

Question 3 - Solution

Resonance can produce large internal voltages with a modest source

(a) Method and calculation

$$\omega_0=1/\sqrt{LC}=500\,s^{-1},\quad I_p=V_p/R=2.00\,A.$$$$\boxed{\mathcal Q=20.0,\quad V_{L,p}=V_{C,p}=\mathcal QV_p=200\,V}.$$

(b) Interpretation and further calculation

$$U=\tfrac12LI_p^2\cos^2\omega_0t+\frac{I_p^2}{2C\omega_0^2}\sin^2\omega_0t.$$$$\boxed{U=\tfrac12LI_p^2=.400\,J\quad\text{at every phase}}.$$

Physical check

Equal large reactive voltages cancel in the source sum; their individual magnitudes need not be below the source voltage.