GATE 2023 EE Q.22: solution and answer

GATE 2023 · EEQ.22 · 1 mark · MCQ

The four stator conductors (A, A′, B and B′) of a rotating machine are carrying DC currents of the same value, the directions of which are shown in the figure (i). The rotor coils $a$ - $a^{'}$ and $b$ - $b^{'}$ are formed by connecting the back ends of conductors ' $a$ ' and ' $a^{''}$ ' and ' $b$ ' and ' $b^{''}$ ', respectively, as shown in figure (ii). The e.m.f. induced in coil $a$ - $a^{'}$ and coil $b$ - $b^{'}$ are denoted by $E_{a - a^{'}}$ and $E_{b - b^{'}}$ , respectively. If the rotor is rotated at uniform angular speed $ω$ rad/s in the clockwise direction then which of the following correctly describes the $E_{a - a^{'}}$ and $E_{b - b^{'}}$ ?

GATE 2023 EE question 22: Two sub-figures. — Figure from the GATE 2023 EE paper. It is part of the question.

(A)
$E_{a - a^{'}}$ and $E_{b - b^{'}}$ have finite magnitudes and are in the same phase
(B)
$E_{a - a^{'}}$ and $E_{b - b^{'}}$ have finite magnitudes with $E_{b - b^{'}}$ leading $E_{a - a^{'}}$
(C)
$E_{a - a^{'}}$ and $E_{b - b^{'}}$ have finite magnitudes with $E_{a - a^{'}}$ leading $E_{b - b^{'}}$
(D)
$E_{a - a^{'}} = E_{b - b^{'}} = 0$

Answer and solution

ANSWER

Both E_a-a' and E_b-b' are identically zero (Option D), because the diametrically opposite stator conductors carrying current in the same direction produce a flux pattern with 180° periodicity, so each rotor coil's two sides always see the same radial field and their series-connected e.m.f.s cancel exactly.

Official answer: DSource: Official answer key, GATE 2023 EE (IIT Kanpur)

Model each stator conductor as an infinite line current, and let $B_{r} (θ)$ be the net radial flux density these four currents produce at the rotor surface, with $θ$ measured clockwise from the top.
Reading the figure clockwise from the top: A (current into the page) at $θ = 0^{\circ}$ , A $^{'}$ (out of the page) at $θ = 9 0^{\circ}$ , B (into the page) at $θ = 18 0^{\circ}$ , B $^{'}$ (out of the page) at $θ = 27 0^{\circ}$ .
A and B are $18 0^{\circ}$ apart and carry current in the SAME direction; A $^{'}$ and B $^{'}$ are also $18 0^{\circ}$ apart and carry current in the SAME direction. This is unlike the usual 2-pole dc-machine layout (where diametrically opposite conductors carry OPPOSITE currents, giving $B_{r} (θ + 18 0^{\circ}) = - B_{r} (θ)$ ): here instead $B_{r} (θ + 18 0^{\circ}) = B_{r} (θ)$ for every $θ$ — the field pattern repeats every $18 0^{\circ}$ rather than reversing sign.
As the rotor turns at $ω$ rad/s, the motional e.m.f. induced in an axial rotor conductor at angular position $θ$ is $e (θ) = - ω R l B_{r} (θ)$ , where $R$ is the rotor radius and $l$ the axial length (the $v \times B$ sign convention is the same at every $θ$ ; only the local $B_{r}$ value changes).
Coil $a - a^{'}$ has its two sides diametrically opposite, at $θ$ and $θ + 18 0^{\circ}$ . Like any normal series-connected full-pitch armature coil, its net e.m.f. is $E_{a - a^{'}} = e (θ) - e (θ + 18 0^{\circ})$ — a convention that makes the two sides of a coil in an ordinary 2-pole field ADD, precisely because $B_{r}$ there reverses sign $18 0^{\circ}$ away.
Using $B_{r} (θ + 18 0^{\circ}) = B_{r} (θ)$ from step 3: $E_{a - a^{'}} = - ω R l [B_{r} (θ) - B_{r} (θ)] = 0$ at every instant and for every rotor position $θ$ — not merely zero on average.
The same argument applies to coil $b - b^{'}$ , whose sides sit at $θ + 9 0^{\circ}$ and $θ + 27 0^{\circ}$ — also exactly $18 0^{\circ}$ apart — giving $E_{b - b^{'}} = 0$ identically as well.
Therefore $E_{a - a^{'}} = E_{b - b^{'}} = 0$ for all $ω$ and all rotor positions, independent of time — option (D).

KEY IDEA

A full-pitch rotor coil's net e.m.f. is the algebraic difference of the motional e.m.f.s at its two diametrically-opposite sides. That difference is finite only if the stator-produced radial flux density reverses sign 180° away (the ordinary 2-pole case); here, because A & B (0°,180°) carry the same-direction current and A′ & B′ (90°,270°) carry the same-direction current, the flux density instead repeats identically every 180°, so the difference — and hence the coil e.m.f. — is zero at every instant for both coils.

WATCH OUT FOR

Assuming any arrangement with two conductors "into the page" and two "out of the page" 90° apart must be a standard 2-pole field (where diametrically opposite points see opposite-sign flux), and therefore concluding the coil EMFs add up to a finite value — missing that here it is conductors 180° apart (A & B, A' & B') that share the SAME current direction, giving a field pattern that repeats every 180° instead of reversing sign.

From the GATE 2023 EE paper. Every question of this paper · Download the full paper · All 8 DC machines questions, with notes

GATE 2023 EE Q.22: solution and answer

Answer and solution

More DC machines questions