GATE 2024 EC Q.60: solution and answer

GATE 2024 · ECQ.60 · 2 marks · NAT

For the two port network shown below, the value of the $Y_{21}$ parameter (in Siemens) is ______.

GATE 2024 EC question 60: A T-type two-port network. — Figure from the GATE 2024 EC paper. It is part of the question.

Answer and solution

ANSWER

$Y_{21} = 1.5 S$

Official answer: 1.5Source: Official answer key, GATE 2024 EC (IISc Bengaluru)

Definition: $Y_{21} = \frac{I _{2}}{V _{1}}_{V_{2} = 0}$ , so short-circuit port 2. Take the bottom wire (1'–2') as the reference node.
Terminal 2 is wired directly to the middle node N, so shorting port 2 forces $V_{N} = V_{2} = 0$ . The dependent source does not vanish, because it is controlled by $V_{1}$ , which is not zero.
Ohm's law in the 2 Ω branch (current $I_{1}$ flowing from terminal 1 towards N): $I_{1} = \frac{V _{1} - 0}{2} = 0.5 V_{1}$ .
Dependent source polarity: its '+' terminal is at the bottom (the reference, 0 V) and its '-' terminal is at the top, so the node between the 4 Ω resistor and the source sits at $0 - 8 V_{1} = - 8 V_{1}$ .
Ohm's law in the 4 Ω branch, taking current flowing down from N: $I_{4Ω} = \frac{0 - ( - 8 V _{1} )}{4} = 2 V_{1}$ .
KCL at node N (both port currents enter N, $I_{4Ω}$ leaves it): $I_{1} + I_{2} = I_{4Ω}$ , so $I_{2} = 2 V_{1} - 0.5 V_{1} = 1.5 V_{1}$ .
Therefore $Y_{21} = \frac{I _{2}}{V _{1}}_{V_{2} = 0} = 1.5 S$

KEY IDEA

For $Y_{21}$ , short port 2 ( $V_{2} = 0$ ). Here terminal 2 is wired straight to the middle node, so that node is at 0 V. The two branch currents then follow from Ohm's law. The $8 V_{1}$ source stays active because $V_{1} \neq = 0$ . KCL at the middle node, with both port currents taken as entering, gives $I_{2}$ .

WATCH OUT FOR

Misreading the polarity of the dependent source. Its '-' terminal is at the top, so the node above it is at $- 8 V_{1}$ . Taking '+' at the top puts that node at $+ 8 V_{1}$ , which gives $I_{4Ω} = - 2 V_{1}$ and the wrong answer $Y_{21} = - 2.5 S$ . The other common slips are taking $I_{2}$ as leaving terminal 2, which gives $- 1.5$ , and switching off the $V_{1}$ -controlled source when port 2 is shorted, which gives $- 0.5$ .

From the GATE 2024 EC paper. Every question of this paper · Download the full paper · All 11 Two-port questions, with notes

GATE 2024 EC Q.60: solution and answer

Answer and solution

More Two-port questions