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GATE 2017 EE – Question 39

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 · Multiple choice

Two passive two-port networks are connected in cascade as shown in figure. A voltage source is connected at port 1.

[Figure: Two-port network 1 and two-port network 2 in cascade. Port 1 has voltage $V_1$ and current $I_1$, port 2 has voltage $V_2$ and current $I_2$, and port 3 has voltage $V_3$ and current $I_3$.]

Given

$V_1 = A_1V_2 + B_1I_2$

$I_1 = C_1V_2 + D_1I_2$

$V_2 = A_2V_3 + B_2I_3$

$I_2 = C_2V_3 + D_2I_3$

$A_1, B_1, C_1, D_1, A_2, B_2, C_2$, and $D_2$ are the generalized circuit constants. If the Thevenin equivalent circuit at port 3 consists of a voltage source $V_T$ and an impedance $Z_T$, connected in series, then

Diagram for GATE 2017 EE question 39
  1. $V_T = \frac{V_1}{A_1 A_2}$, $Z_T = \frac{A_1B_2 + B_1D_2}{A_1A_2 + B_1C_2}$
  2. $V_T = \frac{V_1}{A_1A_2 + B_1C_2}$, $Z_T = \frac{A_1B_2 + B_1D_2}{A_1A_2}$
  3. $V_T = \frac{V_1}{A_1 + A_2}$, $Z_T = \frac{A_1B_2 + B_1D_2}{A_1 + A_2}$
  4. $V_T = \frac{V_1}{A_1A_2 + B_1C_2}$, $Z_T = \frac{A_1B_2 + B_1D_2}{A_1A_2 + B_1C_2}$

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Correct answer: (D) $V_T = \frac{V_1}{A_1A_2 + B_1C_2}$, $Z_T = \frac{A_1B_2 + B_1D_2}{A_1A_2 + B_1C_2}$

Explanation

Substituting the second pair of equations into the first gives $V_1 = (A_1A_2 + B_1C_2)V_3 + (A_1B_2 + B_1D_2)I_3$. With port 3 open ($I_3 = 0$), $V_T = \frac{V_1}{A_1A_2 + B_1C_2}$. The Thevenin impedance is the ratio of the coefficients, $Z_T = \frac{A_1B_2 + B_1D_2}{A_1A_2 + B_1C_2}$.