Question

In an induction motor, maximum torque occurs when:

• A. $R_2 = X_2$
• B. $R_2 = s X_2$
• C. $\frac{R_2}{s} = X_2$
• D. $\frac{R_2}{X_2} = s_m$

Hint:

Remember the slip for maximum torque: \(s_m = R_2 / X_2\).
At starting (where \(s = 1\)), the condition for maximum starting torque is \(R_2 = X_2\).
Adding external resistance to the rotor circuit of a slip-ring induction motor increases the slip \(s_m\) at which maximum torque occurs, which is useful for high starting torque applications.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the condition under which maximum torque is produced in a three-phase induction motor, in terms of rotor resistance (\(R_2\)), standstill rotor reactance (\(X_2\)), and slip (\(s\)).

Step 2: Key Formula or Approach:
The torque equation of a three-phase induction motor is:
\[ T = \frac{3}{\omega_s} \cdot \frac{V^2 \cdot (R_2 / s)}{(R_1 + R_2 / s)^2 + (X_1 + X_2)^2} \] Using the simplified rotor-only model, the torque is proportional to:
\[ T \propto \frac{s E_2^2 R_2}{R_2^2 + (s X_2)^2} \] To find the slip at which maximum torque occurs (\(s_m\)), we differentiate torque with respect to slip \(s\) and set it to zero, which yields:
\[ R_2 = s_m X_2 \]

Step 3: Detailed Explanation:

• The condition for maximum torque in an induction motor is that the rotor resistance per phase is equal to the rotor reactance per phase under running conditions.

• Under running conditions at slip \(s_m\), the rotor reactance becomes \(s_m X_2\), where \(X_2\) is the standstill rotor reactance.

• Therefore, the condition is:
\[ R_2 = s_m X_2 \]
• Rearranging this equation to solve for the slip at maximum torque (\(s_m\)):
\[ s_m = \frac{R_2}{X_2} \]
• This can also be written in the form:
\[ \frac{R_2}{X_2} = s_m \]
• Comparing this with the options, option (D) is the correct representation of this condition.

Step 4: Final Answer:
Maximum torque occurs when \(\frac{R_2}{X_2} = s_m\).