What is the condition for maximum torque in relation to slip (S) in a three phase induction motor?

Concept:

The expression for Torque produced in the induction motor is

\(T = \frac{{3 \times 60}}{{2\pi {N_s}}} \times \frac{{sE_2^2{R_2}}}{{R_2^2 + {{\left( {s{X_2}} \right)}^2}}}\)

Condition to get maximum torque:

\(\begin{array}{l} \frac{{dT}}{{ds}} = 0\\ \frac{d}{{ds}}\left( {\frac{{3 \times 60}}{{2\pi {N_s}}} \times \frac{{sE_2^2{R_2}}}{{R_2^2 + {{\left( {s{X_2}} \right)}^2}}}} \right) = 0\\ \frac{d}{{ds}}\left( {\frac{{s{R_2}}}{{R_2^2 + {s^2}X_2^2}}} \right) = 0\\ \frac{{\left( {R_2^2 + {s^2}X_2^2} \right){R_2} - s{R_2}\left( {0 + 2sX_2^2} \right)}}{{{{\left( {R_2^2 + {s^2}X_2^2} \right)}^2}}} = 0\\ R_2^2 - {s^2}X_2^2 = 0\\ {R_2} = s{X_2} \end{array}\)

The slip at maximum torque is, s = R₂ / X₂

Important Points

The torque slip characteristic curve is divided into three regions:

• Low slip region
• Medium slip region
• High slip region

Low slip region:

• At synchronous speed, slip = 0, therefore the torque is zero.
• When there is a light load, the speed is very near to synchronous speed.
• The slip is very low and (sX20)2 is negligible in comparison with R2. Therefore
• \(T = \frac{{{K_1}s}}{{{R_2}}}\)

i.e. T ∝ S

Medium slip region:

• As the slip increases, the speed of the motor decreases with the increase in load.
• The term (sX20)2 becomes large.
• The term R22 may be neglected in comparison with the term (sx20)2 and the torque equation becomes
• \(T = \frac{{{K_3}{R_2}}}{{sX_{20}^2}}\)

i.e. \(T \propto \frac{1}{s}\) 

Hence for heavy loads, torque is inversely proportional to slip.

High slip region:

• Beyond the maximum torque point, the value of torque starts decreasing. As a result, the motor slows down and stops.
• At this stage, the overload protection must immediately disconnect the motor from the supply to prevent damage due to the overheating of the motor.
   

Note: The motor operates for the values of the slip between s = 0 and s = sm, where sm in the value of slip corresponding to maximum torque.