Torque-slip characteristic and linear operating region

The T–s characteristic shows how the torque developed by a 3-phase induction motor varies with slip (speed) at constant voltage and frequency. It follows from

$$T = \frac{K\, s\, E_2^2 R_2}{R_2^2 + (sX_2)^2}$$

 T
 |          Tmax
 |         .--.
 |       .'    '.
 |      /  B     '.
 |     /           '-._
 |    / A              '-- Tst (C)
 |   /     |
 |  /stable|  unstable
 | /       |
 |/________|______________ s
 0        sm              1
 (Ns)                    (N=0)

Regions of the curve

1. Low-slip region ($0 < s < s_m$): $sX_2$ is small compared with $R_2$, so

$$T \approx \frac{K E_2^2}{R_2}\, s \quad\Rightarrow\quad T \propto s$$

The curve is almost a straight line through the origin (region OA–B). 2. Maximum torque (B): at $s_m = R_2/X_2$, $T_{max} = KE_2^2/(2X_2)$ — pull-out torque, usually 2–3 times full-load torque. 3. High-slip region ($s_m < s \le 1$): $sX_2 \gg R_2$, so

$$T \approx \frac{K E_2^2 R_2}{s X_2^2} \quad\Rightarrow\quad T \propto \frac{1}{s}$$

The curve is a rectangular hyperbola, ending at starting torque $T_{st}$ at $s = 1$ (C). 4. For $s < 0$ the machine generates; for $s > 1$ it brakes (plugging).

Why the motor operates only in the linear portion

1. Stability: in the linear region, if load torque increases, the speed falls slightly, slip rises and the motor torque rises to meet the new load; a new steady point is reached. So $dT/ds > 0$ gives stable operation. In the region beyond $s_m$, an increase in load reduces speed, slip increases and the motor torque falls; the speed keeps falling until the motor stalls. Operation there is unstable.
2. Near-constant speed: in the linear region slip is only 2–5%, so speed changes very little from no load to full load, which most drives need.
3. Efficiency: rotor copper loss $= s \times P_{ag}$. At low slip this loss is small; at high slip most air-gap power is wasted as heat in the rotor.
4. Current and heating: rotor current rises sharply with slip; at high slip the current is near starting value (5–7 × FL), which would overheat the machine.
5. Power factor: rotor power factor $R_2/\sqrt{R_2^2 + (sX_2)^2}$ is high at low slip and poor at high slip.

Hence the motor is designed so that full-load torque lies on the linear part, well below $T_{max}$, leaving a margin for temporary overloads.