Draw and explain torque-slip characteristics

The torque-slip characteristic shows how the electromagnetic torque of a three-phase induction motor varies with slip, from $s=1$ (standstill) to $s=0$ (synchronous speed).

$$T = \frac{3}{\omega_s}\cdot\frac{s E_2^2 R_2}{R_2^2 + (sX_2)^2}$$

 T
 |          Tmax
 |        _.-*-._
 |  Tst _-'      `-._
 |   *-'              `-.   <- rated
 |  /                      `*  load T_fl
 +--+--------+-------------+--> s
  s=1       s_m           s=0
  (unstable)  (stable region)

(slip decreases to the right, so speed increases to the right)

Regions of the curve

1. Starting torque ($s = 1$):

$$T_{st} = \frac{3}{\omega_s}\cdot\frac{E_2^2 R_2}{R_2^2 + X_2^2}$$

It is the torque at standstill and must exceed the load torque for the motor to start. 2. Maximum torque ($s = s_m = R_2/X_2$):

$$T_{max} = \frac{3E_2^2}{2\omega_s X_2}$$

It is independent of $R_2$ and is also called pull-out or breakdown torque, typically 2 to 3 times full-load torque. 3. Normal operating region ($0 < s < s_m$): here $sX_2 \ll R_2$, so $T \approx \frac{3E_2^2}{\omega_s R_2}\,s$, which is a straight line. The torque is proportional to slip. A small drop in speed gives a large rise in torque, so the speed stays nearly constant (shunt-like behaviour). This region is stable and the motor runs here with slip of about 1 to 5 %. 4. Unstable region ($s_m < s < 1$): the torque falls as slip increases (reactance dominates, $T \propto 1/s$). If the load torque is more than the electromagnetic torque, the motor slows further and stalls.

For stability, the load torque line must cut the motor curve in the normal region.