GATE 2020 EE Q.49: solution and answer

GATE 2020 · EEQ.49 · 2 marks · NAT

A cylindrical rotor synchronous generator has steady state synchronous reactance of 0.7 pu and subtransient reactance of 0.2 pu. It is operating at $(1 + j 0)$ pu terminal voltage with an internal emf of $(1 + j 0.7)$ pu. Following a three-phase solid short circuit fault at the terminal of the generator, the magnitude of the subtransient internal emf (rounded off to 2 decimal places) is ______ pu.

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Answer and solution

ANSWER

The magnitude of the subtransient internal emf is 1.02 pu.

Official answer: 1.01 to 1.03Source: Official answer key, GATE 2020 EE (IIT Delhi)

Because rotor flux linkages cannot change instantaneously at the instant of a fault, the emf behind the subtransient reactance must be evaluated using the prefault load current — first found from the steady-state circuit.
Prefault steady-state relation: $E = V + j X_{d} I$ , with $X_{d} = 0.7 pu$ , $V = 1∠ 0^{\circ} pu$ , $E = 1 + j 0.7 pu$ .
Solve for the prefault current: $I = \frac{E - V}{j X _{d}} = \frac{( 1 + j 0.7 ) - ( 1 + j 0 )}{j 0.7} = \frac{j 0.7}{j 0.7} = 1∠ 0^{\circ} pu$
The subtransient internal emf is the emf behind the subtransient reactance, evaluated at this same prefault current: $E^{''} = V + j X_{d}^{''} I$ , with $X_{d}^{''} = 0.2 pu$ .
Compute: $E^{''} = (1 + j 0) + j 0.2 (1) = 1 + j 0.2 pu$
Magnitude: $∣ E^{''} ∣ = 1^{2} + 0. 2^{2} = 1.04 \approx 1.02 pu$

KEY IDEA

Flux linkages (and hence the emf behind the subtransient reactance) cannot change instantaneously at the instant of fault, so $E^{''}$ is computed using the prefault load current together with the subtransient reactance, evaluated at the prefault terminal voltage.

WATCH OUT FOR

Using the subtransient reactance to recompute a "prefault" current, or trying to use the given internal emf $E = 1 + j 0.7$ directly as $E^{''}$ , instead of first finding the actual prefault line current from the steady-state relation $E = V + j X_{d} I$ and then applying that same current with $X_{d}^{''}$ .

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GATE 2020 EE Q.49: solution and answer

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