GATE 2022 EE Q.65: solution and answer

GATE 2022 · EEQ.65 · 2 marks · NAT

A 3-phase, 415 V, 4-pole, 50 Hz induction motor draws 5 times the rated current at rated voltage at starting. It is required to bring down the starting current from the supply to 2 times of the rated current using a 3-phase autotransformer. If the magnetizing impedance of the induction motor and no load current of the autotransformer is neglected, then the transformation ratio of the autotransformer is given by ________. (round off to two decimal places).

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Answer and solution

ANSWER

The transformation ratio of the autotransformer is x ≈ 0.63.

Official answer: 0.61 to 0.65Source: Official answer key, GATE 2022 EE (IIT Kharagpur)

Step 1 — Direct-on-line starting current: At rated voltage $V$ , the induction motor draws $I_{s t (D O L)} = 5 I_{r a t e d}$ at starting (since the magnetizing branch is neglected, starting current is limited only by fixed motor impedance).
Step 2 — Motor current with the autotransformer: With an autotransformer of transformation ratio $x$ , the motor terminal voltage is reduced to $x V$ . Since motor starting current is proportional to the applied voltage, the motor (winding) current becomes $I_{m o t or} = x \cdot I_{s t (D O L)} = 5 x I_{r a t e d}$
Step 3 — Supply (line) current: The autotransformer also steps down the current from the motor winding to the supply line by the same ratio $x$ (no-load current of the autotransformer neglected): $I_{l in e} = x \cdot I_{m o t or} = 5 x^{2} I_{r a t e d}$
Step 4 — Apply the given requirement: The supply current must be reduced to $2 I_{r a t e d}$ : $5 x^{2} I_{r a t e d} = 2 I_{r a t e d} \Rightarrow x^{2} = \frac{2}{5} = 0.4$
Step 5 — Solve for $x$ : $x = 0.4 \approx 0.6325$
Step 6 — Round to two decimal places: $x \approx 0.63$

KEY IDEA

An induction motor at starting behaves like a fixed impedance (magnetizing branch neglected), so its starting current is directly proportional to the terminal voltage applied to it. An ideal autotransformer of transformation ratio x reduces the motor's terminal voltage to xV and also reduces the line (supply) current by the same factor x relative to the motor's own current — so the current is stepped down twice: once by voltage reduction and once by the autotransformer's current ratio.

WATCH OUT FOR

Applying the ratio x only once (treating supply current = x × DOL current) instead of x² × DOL current, which gives x = 0.4 instead of the correct x ≈ 0.63 — forgetting that the autotransformer transforms current a second time between motor winding and supply line.

From the GATE 2022 EE paper. Every question of this paper · Download the full paper · All 17 Induction machines questions, with notes

GATE 2022 EE Q.65: solution and answer

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