GATE 2017 EC – Question 44

Networks, Signals and Systems · Sinusoidal Steady State Analysis · 2 marks · Numerical answer

The figure shows an RLC circuit excited by the sinusoidal voltage $100\cos(3t)$ Volts, where $t$ is in seconds. The ratio $\dfrac{\text{amplitude of } V_2}{\text{amplitude of } V_1}$ is ________.

A source $100\cos 3t$ in series with a $4\ \Omega$ resistor and a $1$ H inductor, across which $V_1$ is measured. They are followed by a $5\ \Omega$ resistor in parallel with a $\frac{1}{36}$ F capacitor, across which $V_2$ is measured.

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Correct answer: 0.91 to 0.93

Explanation

At $\omega = 3$, the inductor has impedance $j3$ and the capacitor has $\frac{1}{j3/36} = -j12$. The series part is $4 + j3$ with magnitude 5. The parallel part is $\frac{5(-j12)}{5 - j12}$, whose magnitude is $\frac{60}{13} = 4.615$. Both voltages are across parts carrying the same current, so the ratio of amplitudes is $\frac{4.615}{5} = 0.923$.