GATE 2022 EE Q.15: solution and answer

GATE 2022 · EEQ.15 · 1 mark · MCQ

The valid positive, negative and zero sequence impedances (in p.u.), respectively, for a 220 kV, fully transposed three-phase transmission line, from the given choices are

(A)
1.1, 0.15 and 0.08
(B)
0.15, 0.15 and 0.35
(C)
0.2, 0.2 and 0.2
(D)
0.1, 0.3 and 0.1

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Answer and solution

ANSWER

The valid set is $Z_{1} = 0.15$ , $Z_{2} = 0.15$ and $Z_{0} = 0.35$ p.u., which is option (B).

Official answer: BSource: Official answer key, GATE 2022 EE (IIT Kharagpur)

A fully transposed transmission line is a static, balanced, passive network. Its impedance to a set of currents does not depend on their phase order, so $Z_{1} = Z_{2}$ . This rules out (A), where $Z_{1} = 1.1$ and $Z_{2} = 0.15$ , and (D), where $Z_{1} = 0.1$ and $Z_{2} = 0.3$ .
Only (B) and (C) have $Z_{1} = Z_{2}$ , so the zero-sequence impedance decides between them.
Write $Z_{s}$ for the self impedance of one phase conductor and $Z_{m}$ for the mutual impedance between two phases. For a transposed line, $Z_{1} = Z_{2} = Z_{s} - Z_{m}$ and $Z_{0} = Z_{s} + 2 Z_{m}$ . Hence $Z_{0} - Z_{1} = 3 Z_{m}$ .
Zero-sequence currents are equal and in phase in all three conductors, so their fluxes add and the currents return through earth and the ground wires. The mutual impedance $Z_{m}$ is therefore positive, and $Z_{0} > Z_{1}$ . For an overhead line, typically $Z_{0} \approx 2$ to $3.5 Z_{1}$ .
Option (C) has $Z_{0} = Z_{1}$ , which needs $Z_{m} = 0$ . That is not possible for a real line, so (C) is invalid.
Option (B) has $\frac{Z _{0}}{Z _{1}} = \frac{0.35}{0.15} \approx 2.3$ , which lies in the typical range. Hence the answer is (B).

KEY IDEA

For a transposed line, $Z_{1} = Z_{2} = Z_{s} - Z_{m}$ and $Z_{0} = Z_{s} + 2 Z_{m}$ . Since $Z_{m} > 0$ , $Z_{0}$ is always larger than $Z_{1}$ , typically 2 to 3.5 times as large.

WATCH OUT FOR

Treating a line like a rotating machine, where $Z_{1} \neq = Z_{2}$ , or assuming all three sequence impedances can be equal as in option (C). Equal sequence impedances would mean no mutual coupling between conductors, which does not happen on a real overhead line.

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GATE 2022 EE Q.15: solution and answer

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