A series combination of a resistor of 50 Ω, an inductor of (20 ÷ π) H and a capacitor of (5 ÷ π) μF is connected across a voltage source of 230 V, 50 Hz. The impedance of the circuit is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 50 Ω
In a series LCR circuit the impedance depends on the difference of the two reactances, so find both at 50 Hz first.
\( \displaystyle X_L = 2\pi f L = 2\pi \times 50\,\mathrm{Hz} \times \frac{20}{\pi}\,\mathrm{H} = 2000\,\Omega \)
\( \displaystyle X_C = \frac{1}{2\pi f C} = \frac{1}{2\pi \times 50\,\mathrm{Hz} \times \frac{5}{\pi}\times10^{-6}\,\mathrm{F}} = 2000\,\Omega \)
\( \displaystyle Z = \sqrt{R^{2} + (X_L - X_C)^{2}} \)
\( \displaystyle Z = \sqrt{(50\,\Omega)^{2} + 0} = 50\,\Omega \)
Since \( X_L = X_C \), the circuit is at resonance and the impedance equals the resistance. The source voltage of 230 V is not needed.