Discussion
The speed of a three-phase induction motor is controlled by varying both voltage and frequency. What is the slip for maximum torque?
\(R_r' = 0.38\ \Omega\) = rotor per phase resistance reflected on stator side
\(R_s = 0.66\ \Omega\) = stator per phase resistance
\(\beta = 1\) = frequency ratio
\(X_s = 1.14\ \Omega\) = stator per phase reactance
\(X_r' = 1.71\ \Omega\) = rotor per phase reactance reflected on stator side
Answer: A
The slip at which the induction motor develops its maximum torque is
\[ s_m = \frac{R_r'}{\sqrt{R_s^2 + (X_s + X_r')^2}} \]
With the frequency ratio \(\beta = 1\) (rated frequency) the reactances have their normal values.
\[ \begin{aligned} s_m \\ &= \frac{0.38}{\sqrt{0.66^2 + (1.14 + 1.71)^2}} \\ &= \frac{0.38}{\sqrt{0.4356 + 8.1225}} \\ &= \frac{0.38}{2.925} = 0.1299 \end{aligned} \]
The answer is 0.1299, option A. The values 0.562 and 0.661 would come from leaving out the stator reactance, and 0.235 has no basis in the data.
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