Difficulty: Medium
Correct Answer: X = R
Explanation:
Introduction / Context:Servomotors used in control systems should offer a reasonably linear torque–speed characteristic around the operating region to simplify design and ensure faithful tracking. Two-phase AC servomotors are designed with intentionally high rotor resistance to achieve this property.
Given Data / Assumptions:
Concept / Approach:The air-gap power and electromagnetic torque depend on rotor impedance R + jX_s (seen at slip frequency). A linear torque–speed region is favored when the reactive and resistive parts are comparable—avoiding excessive current phase angles and sensitivity to slip. The design rule-of-thumb is to make rotor reactance roughly equal to rotor resistance at the significant slip frequency, i.e., X ≈ R, which straightens the curve over a practical operating range.
Step-by-Step Solution:
Consider T ∝ (rotor current)^2 × (resistive component / synchronous speed).Rotor current I ∝ V / √(R^2 + X^2); torque phase sensitivity reduces when R and X are comparable.With X ≈ R the angle of rotor current is ~45°, yielding a smoother, less peaky torque variation vs slip.Verification / Alternative check:
Manufacturer data sheets for servomotors show linearized speed–torque when resistance is high and X is of similar magnitude, validating X = R as a practical criterion.Why Other Options Are Wrong:
X ≪ R: mostly resistive, low power factor changes and poor torque utilization.X ≫ R: highly reactive, torque becomes very sensitive and non-linear near low slips.R = 0 impossible for real rotor; X = 2R does not generally produce the best linearity.Common Pitfalls:
Assuming “large R only” ensures linearity; actually, balance with X is what shapes the characteristic.Final Answer:
X = R
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