Example

Question:

Suppose the frequency of the source in the previous example can be varied. (a) What is the frequency at which resonance occurs? (b) Calculate the impedance, the current, and the power dissipated at the resonant condition.

Solution:

(a) Resonance frequency: \[ \omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{25.48 \times 10^{-3} \times 796 \times 10^{-6}}} = 222.1~\text{rad/s} \] \[ \nu_r = \frac{\omega_0}{2\pi} = \frac{221.1}{2 \times 3.14} = 35.4~\text{Hz} \] (b) At resonance, impedance \(Z = R = 3\,\Omega\):
Rms current at resonance: \[ I = \frac{V}{Z} = \frac{283/\sqrt{2}}{3} = 66.7~\text{A} \] Power dissipated: \[ P = I^2 R = (66.7)^2 \times 3 = 13.35~\text{kW} \] Thus, power dissipated at resonance is more than in Example 7.8.