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With induction motors, we know that the speed depends on the number of poles and also the supply frequency. Now, the slipring motor is also an induction motor and should follow the same principles, but it doesn't and that confuses me.

Consider a slipring motor with resistor banks that are cut out of the rotor circuit in stages. At starting, the torque is much higher and gradually decreases as the resistance is lowered. But at the same time, the speed increases as the resistors are cut out of the rotor circuit.

How does this happen and why does it happen? Also, does this mean that the resistance of the rotor circuit affects the speed of the motor?

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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.

enter image description here

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.

enter image description here

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:

enter image description here

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:

enter image description here

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With induction motors, we know that the speed depends on the number of poles and also the supply frequency.

Don't forget slip. Slip is also a major factor.

Adjusting the resistance affects how much slip is present between the motor speed and synchronous speed, amongst other factors.

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