GATE 2022 EE Q.31: solution and answer

GATE 2022 · EEQ.31 · 1 mark · NAT

In the circuit shown below, the switch $S$ is closed at $t = 0$ . The magnitude of the steady state voltage, in volts, across the $6 Ω$ resistor is ________. (round off to two decimal places).

GATE 2022 EE question 31: Three branches are connected in parallel between a left vertical rail and a right vertical rail. — Figure from the GATE 2022 EE paper. It is part of the question.

Answer and solution

ANSWER

The steady-state voltage across the $6 Ω$ resistor is $5.00 V$ .

Official answer: 4.95 to 5.05Source: Official answer key, GATE 2022 EE (IIT Kharagpur)

Topology. Three branches join the same two nodes, the left rail and the right rail. They are the source branch (the $10 V$ battery, switch $S$ and $2 Ω$ in series), the $R C$ branch ( $1 μ F$ in series with $10 Ω$ ) and the resistor pair ( $6 Ω ∥ 3 Ω$ ).
DC steady state. Once the switch has closed and the transient has died out, $v_{C}$ is constant, so $i_{C} = C \frac{d v _{C}}{d t} = 0$ . The capacitor behaves as an open circuit and the $10 Ω$ branch carries no current. Neither element affects the steady-state answer.
Equivalent load. $R_{e q} = 6 ∥ 3 = \frac{6 \times 3}{6 + 3} = 2 Ω$
Series loop. The source, the $2 Ω$ resistor and $R_{e q}$ now form a single loop: $I = \frac{10}{2 + 2} = 2.5 A$
Voltage across the pair. $V_{6Ω} = I R_{e q} = 2.5 \times 2 = 5 V$ Voltage division gives the same result: $10 \times \frac{2}{2 + 2} = 5 V$ .
Parallel elements share the same voltage, so the steady-state voltage across the $6 Ω$ resistor has magnitude $5.00 V$ .

KEY IDEA

In DC steady state a capacitor carries no current, so it acts as an open circuit. Any resistor in series with it (here the $10 Ω$ ) carries no current either, and the $R C$ branch drops out. The circuit is then just the source, $2 Ω$ and $6 ∥ 3 = 2 Ω$ in one loop, and the voltage splits equally, giving $5 V$ .

WATCH OUT FOR

Counting the $10 Ω$ resistor in the steady-state network. Putting it in parallel with $6 ∥ 3$ gives $R_{e q} = 1.667 Ω$ and about $4.55 V$ . A second mistake is to find the current in the $6 Ω$ resistor ( $5/6 \approx 0.83 A$ ) and report that instead of the voltage. A third is to use the initial ( $t = 0^{+}$ ) condition, where the uncharged capacitor acts as a short.

From the GATE 2022 EE paper. Every question of this paper · Download the full paper · All 20 Transients questions, with notes

GATE 2022 EE Q.31: solution and answer

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