Steady State Stability of drive

When the motor torque equals the load torque, the motor-load system is in equilibrium. As long as the speed is that of stable equilibrium, the drive will function in a steady state at this speed. The steady-state speed-torque curves of the motor and load have been developed into a concept called steady state stability of drive, which eliminates the need to solve differential equations that are only applicable to the drive’s transient operation. In the majority of drives, the motor’s electrical time constant is insignificant in comparison to its mechanical time constant. Since the motor can be assumed to be in electrical equilibrium during transient operation, steady-state speed-torque curves also apply to transient operation.

Let’s take the Steady State Stability of Drive at Equilibrium Point A in Fig. 2.9(a) as an example. When the operation returns to the equilibrium point after a slight deviation from it caused by an issue with the motor or load, that situation is referred to as stability. Let Δωm decrease in speed as a result of the disturbance. Since the motor’s torque is greater than the load’s torque at the new speed, the motor will accelerate and resume operation in the A mode. Similar to this, an increase in Δωm’s speed brought on by a disturbance will cause the load’s torque to exceed the motor’s, causing a deceleration and the return of operation to point A. As a result, at point A, the drive is steady-state stable.

Now let’s look at equilibrium point B, which occurs when the same motor drives a different load. When speed drops, the load torque exceeds the motor torque, drive slows down, and the operating point shifts away from B. When operating at B, increasing speed will cause the motor torque to be greater than the torque of the load, which will cause the operating point to move away from B. B is an unstable point of equilibrium as a result. The Steady State Stability of Drive of Points C and D shown in Figs. 2.9(c) and (d) may be similarly examined by readers.

The discussion above suggests that an equilibrium point will be stable. When a speed increase causes the load torque to exceed the motor torque,. Or when the following condition is met at the equilibrium point:

Inequality (2.24) can be derived by an alternative approach. Let a small perturbation in speed, Δω_m, results in ΔT and ΔT_l perturbations in T and T_l respectively. Then from Eq. (2.2)

Subtracting (2.2) from (2.25) and rearranging terms gives

The speed torque curves of the motor and load can be assumed to be straight lines for small perturbations. Thus

where (dT/dω_m) and (dT_l/dω_m) are respectively slopes of the steady-state speed-torque curves of motor. And load at operating point under consideration. Substituting Eqs. (2.27) and (2.28) into (2.26) and rearranging the terms yields

This linear differential equation is first order. The solution to Equation (2.29) will be if the initial deviation in speed at time zero is (Δωm)0. As t approaches infinity, an operating point will become stable as Δωm approaches zero. The exponent in Equation (2.30) must be negative for this to occur. The inequality of Eq. (2.24) results from this.