I am having confusion for this stability question of induction motor.
The real question is here 
I searched and searched online, keep reading and reading, but still not enough to answer my doubt.
My understanding (that I learnt from here http://electricalbaba.com/understanding-induction-motor-stability/#comment-2645
) on why point A is stable is like this: "If I increase the load torque, as the load torque increases, the new operating point has higher slip compared with that of point A which in turn means that speed of Induction Motor has decreased, and hence operating point A is stable".
Is this reasoning correct? But how about the condition that should be satisfied for the stability of the operating point A?
Hope some people can kindly help me.
Thank you!
Best Answer
At the operating speed indicated by the point marked, the torque capability of the motor is equal to the torque required to turn the load. Thus the motor has no tendency to provide the additional torque that would be necessary to accelerate the load. Also there is no tendency for the load torque to overcome the torque provided by the motor and decelerate.
Any difference between the torque required to turn the motor at a given speed and the torque capability of the motor at that speed is applied to the inertia of the load as accelerating or decelerating torque. Any change to in the load or motor characteristic curve will move the operating point to the new intersection of the two curves.
Here is an illustration:
Note also that the motor curve can be expected to change. Normal (or abnormal) increases and decreases in the supply voltage will increase cause the available torque to increase and decrease. Motor warmup upon starting and ambient temperature changes also have a small effect on the motor's torque curve. That means that operation at the peak of the motor curve is not a stable operating point because the slightest decrease in motor torque or increase in load torque can cause the motor to stall. Of course operating continuously at that point is normally a severe overload condition.
Summary
The essential condition that defines a stable operating point is that it is the point where the torque requirement curve crosses the motor capability curve in a region where the motor torque capability is declining as speed increases. There are conditions that are more complicated than the diagram shows, but they are beyond the scope of the question. As the situation is presented, when the motor is switched on, it simply accelerates to the stable operating point. At all speeds lower than that point, the motor curve is above the load curve and accelerating torque is applied. At all speeds above that point, the motor curve is below the load curve and the deficit in motor torque allows the load torque to decelerate the load.