Question 3.40: An inductive coil has a resistance of 10 Ω and inductance of......
An inductive coil has a resistance of 10 Ω and inductance of 100 mH. This coil is connected in parallel with a capacitor of 20 μF. A variable frequency power at 100 V is applied across this parallel circuit. Calculate the frequency at which the circuit will reasonate. Also calculate the Q-factor, dynamic impedance of the circuit, and resonant current.
Step-by-Step
Resonant frequency, f_0=\frac{1}{2 \pi} \sqrt{\frac{1}{L C}-\frac{\mathrm{R}^2}{\mathrm{~L}^2}}
Substituting the given values,
\begin{aligned} & \begin{aligned} \mathrm{f}_0 & =\frac{1}{2 \pi} \sqrt{\frac{1}{100 \times 10^{-3} \times 20 \times 10^{-6}}-\frac{(10)^2}{\left(100 \times 10^{-3}\right)^2}} \\ & =112.48 \mathrm{~Hz} \end{aligned} \\ & \text { Q-factor }=\frac{2 \pi \mathrm{f}_0 \mathrm{~L}}{\mathrm{R}}=\frac{6.28 \times 112.48 \times 100 \times 10^{-3}}{10}=7.06 \end{aligned}Dynamic impedance, \mathrm{Z}_0=\frac{\mathrm{L}}{\mathrm{CR}}=\frac{100 \times 10^{-3}}{20 \times 10^{-6} \times 10}=500 \Omega
Current at resonance, \mathrm{I}_0=\frac{\mathrm{V}}{\mathrm{Z}_0}=\frac{100}{500}=0.2 \mathrm{~A}
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