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GATE 2024 EE – Question 40

Electrical Machines · Synchronous machines: cylindrical and salient pole machines, performance and characteristics, regulation and parallel operation of generators, starting of synchronous motors · 2 marks · Multiple choice

A 3-phase, 11 kV, 10 MVA synchronous generator is connected to an inductive load of power factor $(\sqrt3/2)$ via a lossless line with a per-phase inductive reactance of 5 Ω. The per-phase synchronous reactance of the generator is 30 Ω with negligible armature resistance. If the generator is producing the rated current at the rated voltage, then the power factor at the terminal of the generator is

  1. 0.63 lagging.
  2. 0.87 lagging.
  3. 0.63 leading.
  4. 0.87 leading.

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Correct answer: (A) 0.63 lagging.

Explanation

Rated phase voltage $V_t=11000/\sqrt3=6351$ V and rated current $I=\dfrac{10\times10^6}{\sqrt3\times11000}=524.9$ A. Take $I$ as reference and let the load voltage be $V_L\angle30^\circ$ (lagging load, pf 0.866). Then $V_t=V_L\angle30^\circ+j5I=0.866V_L+j(0.5V_L+2624)$. Setting $|V_t|=6351$: $V_L^2+2624V_L-3.345\times10^7=0$, so $V_L=4618$ V. Then $V_t=3999+j4933$, whose angle relative to $I$ is $51^\circ$, so the terminal power factor is $\cos51^\circ=0.63$ lagging. (The synchronous reactance does not affect this.)