GATE EE 2020 Set 1 — Question 58

Given: windings A, B, C (20 turns each) and X (2 turns) share one magnetic circuit. From the figure, each of A, B and C is connected directly across its own phase source ($230\angle 0^\circ$, $230\angle -120^\circ$, $230\angle 120^\circ$ V), the sources sharing a common return.1. What each winding imposes on the core
With an unsaturated core and ideal (resistanceless) windings, a winding connected across a source fixes the flux it links: $v_k = \pm N\,\dfrac{d\phi}{dt}$, the sign set by its winding sense. All three windings link the same core flux $\phi$, so each would require$$N\frac{d\phi}{dt} = \pm V_A, \qquad N\frac{d\phi}{dt} = \pm V_B, \qquad N\frac{d\phi}{dt} = \pm V_C$$2. The conditions contradict each other
$V_A$, $V_B$ and $V_C$ are equal in magnitude but $120^\circ$ apart, so $\pm V_A$, $\pm V_B$ and $\pm V_C$ can never all be the same phasor, whatever senses are assigned. No single core flux satisfies all three windings at once; with ideal sources and windings the circulating currents would be unbounded, so the no-load condition the question asks about does not exist.3. Consequence for winding X
The voltage across X is $V_X = N_X\,\dfrac{d\phi}{dt}$, which needs a definite core flux. Since the data do not define one, no unique value of $V_X$ follows (different simplifying assumptions, such as adding the three source voltages with the drawn senses, give different numbers).Answer: The data are inconsistent, so no unique RMS voltage across X can be determined; every response is accepted.