Medium
Suppose the frequency of the source in the previous example can be varied.
$(a)$ What is the frequency of the source at which resonance occurs?
$(b)$ Calculate the impedance,the current,and the power dissipated at the resonant condition.
$(a)$ What is the frequency of the source at which resonance occurs?
$(b)$ Calculate the impedance,the current,and the power dissipated at the resonant condition.
(N/A) The frequency at which resonance occurs is given by the formula $\omega_{0} = \frac{1}{\sqrt{LC}}$.
Substituting the values $L = 25.48 \times 10^{-3} \, H$ and $C = 796 \times 10^{-6} \, F$:
$\omega_{0} = \frac{1}{\sqrt{25.48 \times 10^{-3} \times 796 \times 10^{-6}}} = 222.1 \, rad/s$.
The resonant frequency in Hertz is $v_{r} = \frac{\omega_{0}}{2\pi} = \frac{222.1}{2 \times 3.14} \approx 35.4 \, Hz$.
$(b)$ At resonance,the inductive reactance equals the capacitive reactance $(X_{L} = X_{C})$,so the impedance $Z$ is equal to the resistance $R$:
$Z = R = 3 \, \Omega$.
The rms current at resonance is $I = \frac{V_{rms}}{Z} = \frac{V_{peak} / \sqrt{2}}{R} = \frac{283 / 1.414}{3} \approx 66.7 \, A$.
The power dissipated at resonance is $P = I^{2}R = (66.7)^{2} \times 3 \approx 13.35 \, kW$.
Substituting the values $L = 25.48 \times 10^{-3} \, H$ and $C = 796 \times 10^{-6} \, F$:
$\omega_{0} = \frac{1}{\sqrt{25.48 \times 10^{-3} \times 796 \times 10^{-6}}} = 222.1 \, rad/s$.
The resonant frequency in Hertz is $v_{r} = \frac{\omega_{0}}{2\pi} = \frac{222.1}{2 \times 3.14} \approx 35.4 \, Hz$.
$(b)$ At resonance,the inductive reactance equals the capacitive reactance $(X_{L} = X_{C})$,so the impedance $Z$ is equal to the resistance $R$:
$Z = R = 3 \, \Omega$.
The rms current at resonance is $I = \frac{V_{rms}}{Z} = \frac{V_{peak} / \sqrt{2}}{R} = \frac{283 / 1.414}{3} \approx 66.7 \, A$.
The power dissipated at resonance is $P = I^{2}R = (66.7)^{2} \times 3 \approx 13.35 \, kW$.
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