Steady state stability

dt

dJ

dt

dJ

dt

JdTT m

mmLM

)(

For drives with constant inertia

0dt

dJ

dt

dJTT m

LM

The drive is said to be in equilibrium if the torque developed by the motor is exactly equal to the load torque.

If TM = TL drive will operate in steady state at this speed provided it is the speed of stable equilibrium.

τe<< τm during transient operation drive is in electrical equilibrium so steady state torque speed curve is applicable.

If the drive comes out of the state of equilibrium due to some disturbance, it comes back to steady state for stable equilibrium

but for unstable equilibrium the speed of the drive increases uncontrollably or decreases to zero.

When the drive coming out of the state of equilibrium preserves it steady state at different speed (lying in small range), it is said to be in neutral range.

The stability of the motor load combination is defined as the capacity of the system which enables it to develop forces of such a nature as to restore equilibrium after any small departure.

Equilibrium state of the drive mainly disturbs because of the following two types of disturbances:

1.Changes from the state of equilibrium takes place slowly and the effect of either the inertia or the inductance is insignificant – Steady state stability.

2.Sudden and fast changes from the equilibrium state so effect of both inertia and inductance can not be neglected- Dynamic or transient stability

Criteria for steady state stability

Let the equilibrium of the torques and speed is  

                        and the small deviations are                           

                                                                                       

After the displacement from the equilibrium state the torque equation becomes,

Considering the small deviation, changes can be expressed as a linear function of change in speed,

From the torque equation, where all quantities are expressed

in terms of their deviations from the equilibrium,                                                

Solution is,

Where,        

is the initial value of the deviation in speed.

For the stable system the exponent must be negative, so speed increment will disappear with time.

The exponent will always be negative if,                      

Criteria for the steady state stability is

for a decrease in the speed, the motor torque must exceeds the load torque

and

for increase in speed the motor torque must be less than the load torque.

Load torque       results in a stable equilibrium point,

and the load torque       results in an unstable situation.

Criteria for the steady state stability is for a decrease in the speed, the motor torque must exceeds the load torque and for increase in speed the motor torque must be less than the load torque.

To check the stability at an operating point of the motor, if an increase in speed brings greater increase in load torque than the motor torque, the speed will tend to decrease and return to its original value, so operating point will be a stable point else operating point will be an unstable point.

                                                                     

                                                                                                                                     

Various Speed and Torque Curves of Motor and Load

•Cases (d), (e) and (f) represents unstable operation of drive.

•Case (g) represents indeterminate condition.

•Cases (a), (b) and (c) represents stable operation of drive.

Fundamentals of Electric DrivesGopal K Dubey , Narosa 2nd editionPg 1- 10 Ch-1 , Introduction.

Parts of electric drives, advantages, choice, status of ac-dc drives.

Pg 11- 20 , 23- 27 , Ch-2 Dynamics of Electric Drives.

Torque eqn., Eq values of drive parameters, components of load torque, classification, steady state stability, load equalization.

Pg 60-64 dc motor

67-75 starting, braking

87-92 speed control

97 – 107 and 111 – 118 Controlled rectifiers