# Why is the torque vs slip speed characteristic of an induction motor costant with respect to the frequency of the voltages applied?

I'm currently studying the induction motor and I have some troubles to understand which causes this thing.

Let $$w_2 = w_1 - w_r$$ be the slip velocity and $$s = w_2 / w_1$$ be the slip, where w_1 is the speed of the stator magnetic field and w_r the speed of the rotor electromagnetic variables. Let also the amplitude of the three voltages applied to the motor be constant. Why is the characteristic T vs w_2 constant with respect to the frequency of the applied voltages?

In the following the characteristic which I'm referring to.

I'm more interested in a qualitatively explanation than a proof that involves only calculus but both are welcome.

• I'm trying to understand the relevance of the diagram you attached. If that diagram isn't quite the one then why show it. Show what you really mean and don't be ambiguous. – Andy aka May 29 '18 at 15:34
• @Andyaka, for a fixed frequency the only difference in the two diagrams is the scale. However the Torque vs Slip characteristic change if we change the frequency of the voltages applied to the motor, while the Torque vs Slip characteristic doesn't. If you think that I should delete the picture I'll do, but I didn't find any image on the internet that represent the characteristic that I mentioned. – gvgramazio May 29 '18 at 15:57
• @Andyaka, I changed the image. Now is exactly the curve that I'm referring in the question. – gvgramazio May 29 '18 at 16:14
• Ehhh? How come in that diagram W_2 is larger in magnitude across the x-axis than W_1? Where is the frequency in the graph? What does "Break" mean. If you want this answering I would spend time finding a diagram or link to some page that explains what you are trying to understand? Where is T vs W_2 constant in that diagram? – Andy aka May 29 '18 at 16:26
• @Andyaka, w_2 is the x-axis, w_2 is greater than w_1 if w_r is less than 0. It's the same of saying slip greater that 1. – gvgramazio May 29 '18 at 16:33

It is torque vs. slip speed hat is constant, not torque vs per unit or percent slip. Slip speed is stator field speed minus rotor speed. Torque vs. slip speed is only constant if the applied voltage is proportional to the applied frequency. Maximum torque is approximately proportional to the square of the voltage/frequency ratio. If that ratio is too high, the stator iron will be magnetically saturated. If the ratio is too low, the torque may not be sufficient tor drive the load. Torque is proportional to slip speed because slip speed determines the rotor current and thus the rotor magnetic field.

For additional information, look at answers to: Condition for max torque

• So if torque vs slip is constant if if I keep voltage/frequency ratio constant then I can say that torque vs w_2 is constant if I change the frequency without changing the voltage? – gvgramazio May 29 '18 at 16:50
• No. Keeping the V/f ratio constant means changing the voltage whenever the frequency is changed. If W2 is stator field speed minus rotor speed, W2 is slip. – Charles Cowie May 29 '18 at 16:53
• I revised my answer to clarify slip speed vs. per-unit slip. – Charles Cowie May 29 '18 at 17:25
• This happens because keep v/f ratio constant means to almost keep the flux constant (at least not at low frequency)? – gvgramazio May 29 '18 at 17:31
• Yes. Constant V/f keeps the flux almost constant. Controlling the V/f more carefully can do a better job of keeping the flux constant. – Charles Cowie May 29 '18 at 17:42

AFAIK, Speed is proportional to f* of applied current but independent of voltage which affects available max torque ...

The current is proportional to torque but both inverse with speed (BEMF) except near stall where it's torque drops to zero and needs a Motor Start cap.

Thus* Slip ratio is independent of frequency of voltage.

• When you say speed, are you referring to the speed of the rotor w_m = w_r / p where p is the number of polar couples for each phase? When you say slip ratio, do you mean w_2 or w_2 / w_1? – gvgramazio May 29 '18 at 16:03