GATE 2022 EE Q.64: solution and answer

GATE 2022 · EEQ.64 · 2 marks · NAT

A star-connected 3-phase, 400 V, 50 kVA, 50 Hz synchronous motor has a synchronous reactance of 1 ohm per phase with negligible armature resistance. The shaft load on the motor is 10 kW while the power factor is 0.8 leading. The loss in the motor is 2 kW. The magnitude of the per phase excitation emf of the motor, in volts, is ________. (round off to nearest integer).

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Answer and solution

ANSWER

The magnitude of the per-phase excitation emf is 245 V.

Official answer: 240 to 248Source: Official answer key, GATE 2022 EE (IIT Kharagpur)

Phase (star) voltage: $V_{p h} = \frac{400}{3} = 230.94 V$ , taken as reference, $230.94∠ 0^{\circ} V$ .
Total electrical input power drawn by the motor = shaft output + losses = $10 + 2 = 12 kW$ (three-phase total).
Line current from $P = 3 V_{L} I_{L} cos ϕ$ (for a star connection, line current = phase/armature current): $I_{L} = \frac{12000}{3 \times 400 \times 0.8} = 21.651 A$
For a motor, a leading power factor means the armature current leads the applied voltage: $cos^{- 1} (0.8) = 36.8 7^{\circ}$ , so $I = 21.651∠36.8 7^{\circ} = 17.320 + j 12.990 A$
With negligible armature resistance, the motor's voltage equation is $V = E + j X_{s} I$ , so $E = V - j X_{s} I$
Substituting: $E = 230.94 - j (1) (17.320 + j 12.990) = 230.94 - (- 12.990 + j 17.320) = 243.93 - j 17.32 V$
Magnitude: $∣ E ∣ = 243.9 3^{2} + 17.3 2^{2} = 59500.5 + 299.98 \approx 244.54 V$
Rounded to the nearest integer: $E \approx 245 V$ .

KEY IDEA

For a synchronous motor with negligible armature resistance, the phasor equation is V = E + jXsI (current taken as flowing into the motor), so E = V - jXsI. Input electrical power equals shaft output plus internal losses, and for a leading power factor the current phasor leads the reference voltage.

WATCH OUT FOR

Forgetting to add the 2 kW loss to the 10 kW shaft output before computing the line current (using 10 kW instead of 12 kW input power), or using the wrong sign convention for a leading power factor (taking the current phase angle as -36.87° instead of +36.87°, since for a motor a leading power factor means the current phasor leads the voltage phasor).

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GATE 2022 EE Q.64: solution and answer

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