# Torque-Slip Characteristic Stability Analysis for Induction Motor

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!

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.

• Hi Charles! Thank you for answering! I was looking at this forum and saw your brilliant answers at other questions. But I am still doubtful, I understand that the motor torque=load torque and hence no additional torque is necessary to overcome the load, but my question is whether that point A is stable or not? And the question also asked the about the condition that should be satisfied for the stability of the operating point A? Thank you for your kind answer. – Prietess Jun 23 '17 at 12:53
• And also, is my understanding that I wrote above is correct? (This part: "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". ) Thanks Charles. – Prietess Jun 23 '17 at 13:00
• It is correct to say that increasing torque causes to operating point to move to a higher slip, lower speed operating point. However, that is really not why point A is a stable operating point. Point A is a stable operating point because, without a change in motor or load characteristics, there is no force that would tend to move the operating point. Also, there is no tendency for the operating point to change with very little change in characteristics as would be the case when the load curve is at the peak of the motor curve as illustrated by Tony Stewart. – Charles Cowie Jun 23 '17 at 14:00
• Perhaps I can add an illustration to my answer a little later. – Charles Cowie Jun 23 '17 at 14:09
• It has been added. – Charles Cowie Jun 23 '17 at 15:46

The reasoning doesn't require the notion of slip. Consider two cases:

• Motor torque higher than load countertorque → the speed increases, which leads to a lower motor torque.
• Motor torque lower than load countertorque → the speed decreases, which leads to a higher motor torque.

What you see at a stable operating point is not only the equilibrium of motor torque and load countertorque, but also the conclusion a motor torque which is too high for a given load countertorque will lower itself by the M/n characteristic of the motor. And vice-versa.

This is true for all motor-load combinations, not only those which operate on slip as an AC asynchronous motor does.

If the constant load torque is below the zero-speed torque (that the motor can produce) then point A is achieved when the motor is switched on. If the load torque is higher than the zero-speed torque then there is a problem and the motor would never run at the required speed.

However, if the higher load torque were applied once the motor was up and running then the system would be stable because it could back-down (a bit) on the graph without going past the point of no return (maximum torque).

(rev C)

We know from linear and rotational acceleration that F=ma. Thus a constant speed is achieved (stable) when the forces are equal at a point on the load line.

Where it is unstable is below the Tmax. This is a boundary conditions where Tload < Tmax and where equal determines RPM with no acceleration force and constant velocity. The other is starting up if the load is between Tmin and Tmax, it wont start up, so startup caps and coils are used to boost the motor current until a speed where the transfer switch has about the same torque as Tmax

Otherwise the motor would never start up and the load torque must be less than the starting torque to start up.

Now the slip effects cause this low torque at 0 RPM and this loss of torque declines with rising speed ( ie torque rises) such that when it generates BEMF which ends up limiting the voltage drop internally and thus current drops with available torque.

Past this peak, Tmax Torque is now dominated by BEMF as the slip effects are now minimal but depends on the magnetic design. ## analogy

Just like a voltage regulator with resistance. Voltage drops with more load current V=Voc-IR or like a spring in the linear stable part of the curve that deflects down x with force in the same direction, F for x=F/k for some spring constant, k.

Thus we see there two factors that reduce torque in AC induction motors.

1) Slip frequency which demands a boost circuit for startup and

2) back EMF , which generates a voltage opposite to the applied voltage and for Induction Motors it becomes 0 torque at some integer ratio of the line frequency and number of poles.

This optimal peak ratio of RPM / max RPM depends on the motor design and there are standard torque curves that differ from your example.

The maximum power is at a higher RPM than Tmax since Power = speed * Torque when using MKS units.

• Hi Tony! Thank you for answering! But I am still confused, you explained about the physical meaning of slip or BEMF, which is helpful. But my question is whether point A is stable?the question also asked the about the condition that should be satisfied for the stability of the point A? And also, is my understanding that I wrote above is correct? (This part: "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"） – Prietess Jun 23 '17 at 12:57
• When available torque = demand load the RPM shifts to a stable point always when the curve is negative. Like any negative feedback system, It is stable. when slope reverses, it becomes positive feedback and stalls. – Tony Stewart Sunnyskyguy EE75 Jun 23 '17 at 13:02
• Oh ok Thank you. THen how about the part asking 'state the condition that should be satisfied for the stability of the operating point A'? Based on your explanation, it should be always stable right?　Then why the question still ask some kind of boundary condition to satisfy the stability at that point? – Prietess Jun 23 '17 at 13:10
• constant load torque. < Max torque avail. in motor spec. gives constant RPM. dynamic load modulates speed. Then if load exceeds max i.e. > avail torque , inertia and duration of load define resulting RPM change, otherwise stall. So boundary depends on load/max torque, duration and rotational inertia. – Tony Stewart Sunnyskyguy EE75 Jun 23 '17 at 13:12