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GATE 2016 EE – Question 62

Electric Circuits · Transient response of DC and AC networks, sinusoidal steady-state analysis, resonance, two port networks, balanced three phase circuits, star-delta transformation, complex power and power factor in AC circuits · 2 marks · Numerical answer

In the circuit shown below, the supply voltage is $10\sin(1000t)$ volts. The peak value of the steady state current through the 1 $\Omega$ resistor, in amperes, is ________.

A $10\sin(1000t)$ source. A 4 Ω resistor and a 250 μF capacitor in parallel with a 4 mH inductor lead to a node. Another 2 μF capacitor in parallel with a 500 mH inductor connects the top node to the right-hand rail. A 1 Ω resistor and a 5 Ω resistor complete the network.

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Correct answer: 1

Explanation

At $\omega = 1000$ rad/s, the 250 μF capacitor has $X_C = \frac{1}{1000 \times 250 \times 10^{-6}} = 4\ \Omega$ and the 4 mH inductor has $X_L = 4\ \Omega$, so this parallel pair is in resonance and acts as an open circuit. Likewise the 2 μF capacitor ($X_C = 500\ \Omega$) and the 500 mH inductor ($X_L = 500\ \Omega$) are in parallel resonance and also act as an open circuit. The only path left for current is the series chain $4 + 1 + 5 = 10\ \Omega$. The peak current is $\frac{10}{10} = 1$ A, and it flows through the 1 Ω resistor.