Shortcut Methods
JEE Numerical:
1. An LCR circuit has L=1H, C=100μF, and R=10Ω. Calculate the resonant frequency.
$$f_r=\frac{1}{2π\sqrt{LC}}=\frac{1}{2π(1)(100×10^{-6})}$$
$$f_r=159.2Hz $$
2. A 100mH inductor, a 10μF capacitor, and a 10Ω resistor are connected in series to a 100V, 50Hz AC source.
$$Z=\sqrt{R^2+(X_L-X_C)^2}$$
$$Z=\sqrt{10^2+(2π50×0.1-1/2π50×100×10^{6})^2}$$
$$Z=\sqrt{100+2500-10}=44.7Ω$$
$$I=\frac{E}{Z}=\frac{100}{44.7}=2.24A$$
CBSE Numerical:
1. A 50Hz alternating current source of 100V is connected to a series LCR circuit containing R = 100Ω, L = 0.2H, and C = 100μF. Determine the impedance of the circuit.
$$Z=\sqrt{R^2+(X_L-X_C)^2}$$
$$Z=\sqrt{100^2+(2π50×0.2-1/2π50×100×10^{6})^2}$$
$$Z=\sqrt{10000+100-10}=99.9Ω$$
2. Calculate the resonant frequency of a series LCR circuit having L=10mH, C=400μF, and R=20Ω.
$$f_r=\frac{1}{2π\sqrt{LC}}=\frac{1}{2π(10×10^{-3})(400×10^{-6})}$$
$$f_r=79.6Hz $$
