The performance of an induction motor in relation to the rotor resistance is explained by analyzing the equivalent circuit. The complete development and analysis requires quite a few textbook pages with numerous equations and diagrams. Here are the essentials related to this problem from Fitzgerald, Kingsley, Umans, Electric Machinery 4th ed.:
The equivalent circuit is simplified to eliminate the magnetizing branch that would appear between a and b in the diagram below.
The torque equation is developed using the principle that electrical power supplied to the circuit is equal to mechanical power delivered from the circuit plus losses in the circuit. The mechanical power developed in the rotor is the power developed in the variable resistor that represents the power conversion, R2(1-s)/s. The circuit represents one phase of a three phase, wye connected motor.
The torque vs. speed curve can be drawn from the above equation. Note that q1 is the number of phases and omega s is the synchronous speed in radians per second.
The slip at maximum torque and the maximum torque can also be calculated:
From the above equations, it can be seen that the slip at maximum torque is directly proportional to rotor resistance R2, but the value of the maximum torque is independent of R2. When R2 is increased by adding external rotor resistance the value of maximum torque is constant, but the speed at which it occurs can be controlled using the value of rotor resistance as shown below: