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GATE 2022 EC – Question 38

Networks, Signals and Systems · Sinusoidal Steady State Analysis · 2 marks · Multiple choice

For the circuit shown, the locus of the impedance $Z(j\omega)$ is plotted as $\omega$ increases from zero to infinity. The values of $R_1$ and $R_2$ are:

Diagram for GATE 2022 EC question 38
  1. $R_1=2\ \text{k}\Omega,\ R_2=3\ \text{k}\Omega$
  2. $R_1=5\ \text{k}\Omega,\ R_2=2\ \text{k}\Omega$
  3. $R_1=5\ \text{k}\Omega,\ R_2=2.5\ \text{k}\Omega$
  4. $R_1=2\ \text{k}\Omega,\ R_2=5\ \text{k}\Omega$

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Correct answer: (A) $R_1=2\ \text{k}\Omega,\ R_2=3\ \text{k}\Omega$

Explanation

$Z=R_1+\frac{R_2}{1+j\omega R_2C}$. At $\omega=0$ the capacitor is open and $Z=R_1+R_2=5\ \text{k}\Omega$; at $\omega\to\infty$ it is a short and $Z=R_1=2\ \text{k}\Omega$. So $R_1=2\ \text{k}\Omega$ and $R_2=3\ \text{k}\Omega$. The locus is a semicircle of diameter $R_2=3\ \text{k}\Omega$, whose radius $1.5\ \text{k}\Omega$ matches the peak in the plot.