Question

A parallel resonant circuit has a capacitor 100pF in one branch and the impedance of 100 micro Henry plus a a resistance of 10 ohms in another parallel branch, calculate:

(a) resonant frequency 

(b) impendance of the circuit at the resonance and the line current at resonance 

(c) Q-quality factor of the circuitry

Answer 1

$\omega_o=\frac{1}{\sqrt{LC}}\\ \omega_o=\frac{1}{\sqrt{100\times{10^{-6}}\times{100\times{10^{-12}}}}}\\ \omega_o=10^7rads^{-1}\\ \omega_o=2\pi{f_o}\\ f_o=\frac{10^{7}}{2\pi}=1.591MHz$

b)

Impedance Z is

$Z=R\|2\pi{f_o}L\|\frac{1}{2\pi{f_o}C}\\ X_L=2\pi{f_o}L=10^7\times{100\times{10^{-6}}}=1000\Omega\\ X_C=X_L=1000\Omega\\ X_C\|X_L=500\Omega\\ Z=R\|500=\frac{500\times{10}}{500+10}=9.8\Omega\\$

c)$Q=\omega_oRC\\ Q=10^7\times{10}\times{100\times{10^{-12}}}=0.01\\$

Current cannot be calculated since there's no energy source given in the question.