Solved GATE Paper

GATE 2008 Electric Circuits Questions and Solutions

Instructor: Prof. Mithun Mondal Institution: BITS Pilani Subject: Electric Circuits
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Question 01

Question 1

The Thevenin's equivalent impedance \(Z_{th}\) between the nodes P and Q in the following circuit is

GATE 2008 Electric Circuits Q1 circuit diagram
Circuit for GATE 2008 Electric Circuits Q1
  1. \(1\,\Omega\)
  2. \(1+s+\frac{1}{s}\)
  3. \(2+s+\frac{1}{s}\)
  4. \(\frac{s^{2}+s+1}{s^{2}+2s+1}\)

Solution

  • Method: To find \(Z_{th}\), deactivate the independent sources (10V source is shorted, 1A source is open). The impedances in the s-domain are \(Z_L = sL = s(1) = s\) and \(Z_C = 1/(sC) = 1/(s(1)) = 1/s\).
  • Equivalent Impedance: The circuit consists of two parallel branches: Branch 1 is a series R-L \((1\,\Omega + s)\) and Branch 2 is a series R-C \((1\,\Omega + 1/s)\).
  • Calculation: The equivalent impedance is calculated as the parallel combination:
    Equation
    \[Z_{th} = (1+s) || \left(1+\frac{1}{s}\right) = \frac{(1+s)\left(1+\frac{1}{s}\right)}{(1+s)+\left(1+\frac{1}{s}\right)}\]
    Equation
    \[Z_{th} = \frac{(1+s)\left(\frac{s+1}{s}\right)}{2+s+\frac{1}{s}} = \frac{\frac{(s+1)^2}{s}}{\frac{2s+s^2+1}{s}}\]
    Equation
    \[Z_{th} = \frac{(s+1)^2}{s^2+2s+1} = \frac{(s+1)^2}{(s+1)^2} = 1\,\Omega\]
A
Final Answer
Correct answer: (1) \(1\,\Omega\).
Question 02

Question 2

The condition for maximum power transfer to a load impedance \(Z_L = R_L + jX_L\) from a source with internal impedance \(Z_S = R_S + jX_S\) is:

  1. \(R_L = R_S\) and \(X_L = X_S\)
  2. \(R_L = R_S\) and \(X_L = -X_S\)
  3. \(R_L = |Z_S|\) and \(X_L = 0\)
  4. \(Z_L = R_S\)

Solution

  • For a complex source impedance, maximum power is transferred when the load impedance is the conjugate of the source impedance, i.e., \(Z_L = Z_S^*\).
B
Final Answer
Correct answer: (2) \(R_L = R_S\) and \(X_L = -X_S\).
Question 03

Question 3

A two-port network is reciprocal if its admittance parameters satisfy which of the following conditions?

  1. \(y_{11} = y_{22}\)
  2. \(y_{12} = -y_{21}\)
  3. \(y_{12} = y_{21}\)
  4. \(y_{11}y_{22} - y_{12}y_{21} = 1\)

Solution

  • A two-port network is considered reciprocal if the ratio of the output current (Port 2) to the input voltage (Port 1) equals the ratio of the input current (Port 1) to the output voltage (Port 2) under appropriate open/short circuit conditions. In terms of the Y-parameters (Admittance parameters), this condition is \(y_{12} = y_{21}\).
  • The condition \(y_{11} = y_{22}\) indicates a symmetric network.
C
Final Answer
Correct answer: (3) \(y_{12} = y_{21}\).
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GATE Electric Circuits