An LCR series circuit has inductor L = 15 mH, Capacitor \( C = 22\,\mu\mathrm{F} \) and resistor \( R = 200\,\Omega \). Select the correct value for impedance of the circuit at the source frequency of 1 kHz.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 218\,\Omega \)
In a series LCR circuit the inductive and capacitive reactances are opposite in phase, so the impedance is
\( \displaystyle Z = \sqrt{R^{2} + (X_L - X_C)^{2}} \)
\( L = 15\,\mathrm{mH} = 1.5\times10^{-2}\,\mathrm{H} \)
\( C = 22\,\mu\mathrm{F} = 2.2\times10^{-5}\,\mathrm{F} \)
\( \nu = 1\,\mathrm{kHz} = 10^{3}\,\mathrm{Hz} \)
\( \displaystyle X_L = 2\pi\nu L = 2\pi \times 10^{3} \times 1.5\times10^{-2} = 94.2\,\Omega \)
\( \displaystyle X_C = \frac{1}{2\pi\nu C} = \frac{1}{2\pi \times 10^{3} \times 2.2\times10^{-5}} = 7.2\,\Omega \)
\( X_L - X_C = 87.0\,\Omega \)
\( \displaystyle Z = \sqrt{200^{2} + 87.0^{2}} = \sqrt{40000 + 7569} \approx 218\,\Omega \)