**Induction machine Dynamics:**

Induction machine Dynamics – In general, the mechanical time-constant for any machine is much larger than the electrical time constant. Therefore, the dynamic analysis can be simplified by neglecting the electrical transient without any loss of accuracy of results.

The circuit model of Induction machine Dynamics of Fig. 9.8(b) holds for constant slip but would also apply for slowly varying slip as is generally the case in motor starting. Figure 9.64 shows the typical torque-slip (speed) characteristic of a motor and also the load torque as a function of slip (speed). Each point on (T _{L}-s) characteristic represents the torque (frictional) demanded by the load and motor when running at steady speed. The motor would start only if T> T_{L} and would reach a steady operating speed of co_{o} which corresponds to T = T_{L}, i.e. the intersection point P of the two torque-speed characteristics. It can be checked by the perturbation method that P is a stable operating point for the load-speed characteristic shown. If for any reason the speed becomes more than co_{n}, (T- T_{L}) < 0 the machine-load combination decelerates and returns to the operating point. The reverse happens if the speed decreases below

During the accelerating period

where J = combined inertia of motor and load. Now

Therefore, Eq. (9.87) modifies to

Since, the term 1/(T — T_{L}) is nonlinear, the integration in Eq. (9.89) must be carried out graphically (or numerically) as shown in Fig. 9.65 for the case when s_{1}= 1 and s_{2}

= s_{o}. Since 1/(T— T_{L}) becomes 0. at s_{o}, the practical integration is carried out only up to 90 or 95% of s_{o} depending upon the desired accuracy.

Figure 9.66 shows how slip (speed) varies with time during the acceleration period reaching the steady value of s_{o}(ω_{0}) in time t_{A}, the accelerating time. Because of the nonlinearity of (T — T_{L}) as function of slip, the slip (speed)-time curve of Fig. 9.66 is not exponential.

**Starting on No-Load ****(T _{L}**

**= 0)**

In this particular case it is assumed that the machine and load friction torque T_{L} = 0.

Assuming stator losses to be negligible (i.e. R_{1} = 0), the motor torque as obtained from Eq. (9.22) is

Also from Eq. (9.24)

at a slip of (Eq. (9.23))

From Eqs (9.90) and (9.91),

Substituting Eq. (9.92) in (9.93)

Since T_{L} is assumed to be zero, the motor torque itself is the accelerating torque,

The time t_{A} to go from slip s_{1} to s_{2} is obtained upon integration of Eq. (9.95) as

The acceleration time for the machine to reach steady speed from starting can be computed from Eq. (9.96) with s_{1} = 1 and s_{2} = s, i.e.

**Optimum s**_{max,T} for Minimum Acceleration Time

_{max,T}for Minimum Acceleration Time

To find the optimum value of S_{max,T} for the machine to have minimum acceleration time to reach s_{2} from s_{1}, Eq. (9.96) must be differentiated with respect to S_{max,T}and equated to zero. This gives

For minimum acceleration time for the machine to reach any slip s from start, the optimum value of S_{max,T} is given by Eq. (9.98) with s_{1} = 1 and s_{2} = s. Then

Further, to enable us to compute the optimum value of the rotor resistance to accelerate the machine to slip s_{2} from s_{1}, Eq. (9.98) ifs substituted in Eq. (9.92) giving