GATE 2024 EE Q.54: solution and answer

GATE 2024 · EEQ.54 · 2 marks · NAT

A 3-phase star connected slip ring induction motor has the following parameters referred to the stator: $R_{s} = 3 Ω$ , $X_{s} = 2 Ω$ , $X_{r}^{'} = 2 Ω$ , $R_{r}^{'} = 2.5 Ω$ . The per phase stator to rotor effective turns ratio is 3:1. The rotor winding is also star connected. The magnetizing reactance and core loss of the motor can be neglected. To have maximum torque at starting, the value of the extra resistance in ohms (referred to the rotor side) to be connected in series with each phase of the rotor winding is ________ (rounded off to 2 decimal places).

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Answer and solution

ANSWER

The extra resistance to be added in series with each phase of the rotor winding (referred to the rotor side) is 0.28 Ω.

Official answer: 0.26 to 0.29Source: Official answer key, GATE 2024 EE (IISc Bengaluru)

Per-phase equivalent circuit (stator resistance/reactance and stator-referred rotor resistance/reactance, magnetizing branch neglected): $R_{s}$ , $X_{s}$ in series with $\frac{R _{r}^{'}}{s}$ , $X_{r}^{'}$ .
Condition for maximum torque as the rotor resistance is varied: $\frac{R _{r}^{'}}{s} = R_{s}^{2} + (X_{s} + X_{r}^{'})^{2}$ .
At starting, $s = 1$ , so the total stator-referred rotor resistance required is $R_{r, req}^{'} = R_{s}^{2} + (X_{s} + X_{r}^{'})^{2} = 3^{2} + (2 + 2)^{2} = 9 + 16 = 5 Ω$ .
Extra stator-referred rotor resistance needed: $Δ R^{'} = 5 - R_{r}^{'} = 5 - 2.5 = 2.5 Ω$ .
Convert to the actual rotor-side value using the turns ratio $a = 3$ (stator:rotor), since a rotor quantity referred to the stator satisfies $R^{'} = a^{2} R_{actual}$ : $Δ R_{actual} = \frac{Δ R ^{'}}{a ^{2}} = \frac{2.5}{9} = 0.2778 Ω$ .
Rounded to 2 decimal places: $Δ R_{actual} = 0.28 Ω$ .

KEY IDEA

For a slip-ring induction motor with the magnetizing branch neglected, the per-phase equivalent circuit is a series combination of $R_{s}$ , $X_{s}$ , $R_{r}^{'} / s$ and $X_{r}^{'}$ (all referred to the stator). Torque is maximum (as a function of variable rotor resistance) when $\frac{R _{r}^{'}}{s} = R_{s}^{2} + (X_{s} + X_{r}^{'})^{2}$ . At starting, $s = 1$ , so this directly gives the total stator-referred rotor resistance needed for maximum starting torque.

WATCH OUT FOR

Forgetting to convert the required EXTRA resistance back from the stator-referred frame to the actual rotor-side value using $R_{actual} = R^{'} / a^{2}$ (with $a = 3$ , the stator:rotor turns ratio) — the question explicitly asks for the resistance "referred to the rotor side," not the stator-referred value.

From the GATE 2024 EE paper. Every question of this paper · Download the full paper · All 17 Induction machines questions, with notes

GATE 2024 EE Q.54: solution and answer

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