Maximum staring torque of induction motor is obtained when

Question

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Option A: $R_2 = X_2$

Explanation & Concepts:
The maximum starting torque of a 3-phase induction motor occurs when the rotor resistance per phase ($R_2$) equals the standstill rotor reactance per phase ($X_2$). Under starting conditions, the slip $s = 1$, so substituting $s = 1$ into the general condition for maximum torque ($R_2 = sX_2$) yields $R_2 = X_2$.

Working Principle: Torque in an induction motor depends on the rotor current, the flux, and the rotor power factor angle. At starting ($s=1$), the maximum torque condition is achieved by maximizing the active power dissipated in the rotor. Differentiating the starting torque expression with respect to $R_2$ and setting it to zero gives $R_2 = X_2$, where the rotor impedance phase angle becomes $45^\circ$.

Key Formulas: $T_{st} = \frac{3}{2\pi N_s} \cdot \frac{V^2 R_2}{R_2^2 + X_2^2}$ — Starting torque equation of an induction motor; $s_{m} = \frac{R_2}{X_2}$ — Slip at which maximum torque occurs; $T_{max} = \frac{3}{2\pi N_s} \cdot \frac{V^2}{2 X_2}$ — Maximum (breakdown) torque expression (independent of $R_2$)

Key Points: At standstill or starting, the slip $s = 1$.; Maximum torque at starting is achieved when $R_2 = X_2$.; Maximum breakdown torque $T_{max}$ is independent of $R_2$, but the slip at which it occurs depends directly on $R_2$.; Slip ring (wound rotor) induction motors use external rotor resistance to satisfy $R_2 = X_2$ for high starting torque.

Advantages: Provides high starting torque for heavy-load operations.; Reduces starting current drawn by the motor when external rotor resistance is used.

Disadvantages: Adding permanent rotor resistance increases copper losses ($I^2R$) during normal running conditions.; Reduces motor efficiency during continuous operation if resistance is not cut out.

Applications: Slip ring induction motors used in cranes, hoists, and elevators.; Industrial drives requiring high starting torque under loaded conditions.

Additional Info: Option B ($R_2 = sX_2$) represents the general condition for maximum torque at any running slip $s$, not strictly at starting ($s=1$).. Option C ($R_2 = s/X_2$) and Option D ($sR_2 = X_2$) are mathematically incorrect formulations of the torque equation condition.

Answer: A is correct — Maximum starting torque occurs when the standstill rotor resistance per phase ($R_2$) equals the standstill rotor reactance per phase ($X_2$).

Exam Tip: Remember that while $R_2 = X_2$ maximizes torque at starting ($s=1$), the maximum available torque value $T_{max}$ itself is independent of $R_2$ and depends only on standstill reactance $X_2$ and supply voltage $V$.

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Option B: $R_2 = sX_2$

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Option C: $R_2 = \frac{s}{X_2}$

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Option D: $sR_2 = X_2$

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