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GATE 2023 EE – Question 54

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

The circuit shown in the figure is initially in the steady state with the switch $K$ in open condition and $\bar K$ in closed condition. The switch $K$ is closed and $\bar K$ is opened simultaneously at the instant $t=t_1$, where $t_1>0$. The minimum value of $t_1$ in milliseconds, such that there is no transient in the voltage across the 100 μF capacitor, is ___________ (Round off to 2 decimal places).

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Correct answer: 1.56 to 1.58

Explanation

Before switching, the capacitor sits across the parallel 10 Ω and the current source, with the battery loop shorted by $\bar K$. The parallel impedance is $10\parallel(-j10)=5-j5=7.07\angle-45^\circ\ \Omega$, so $v_C(t)=7.07\sin(1000t-45^\circ)$ V. After switching, $K$ shorts the source and the 10 Ω, so the capacitor charges through the 10 Ω from the 5 V battery to a final voltage of 5 V. There is no transient only if $v_C(t_1)$ already equals 5 V: $7.07\sin(1000t_1-45^\circ)=5$, so $1000t_1-45^\circ=45^\circ$, i.e. $1000t_1=\pi/2$ and $t_1=1.57$ ms.