Torque-slip curve: starting, maximum and operating region

The torque–slip (speed) characteristic of a 3-phase induction motor is the graph of developed torque against slip (or speed) at constant supply voltage and frequency. It follows from

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

 T
 |              B  Tmax
 |            .---.
 |          .'     '.
 |    Tfl  /         '.
 |   ...../ A          '._
 |       /|               '-.  C
 |      / |                  '-- Tst
 |     /  |    unstable          |
 |    /   | <-- region -->       |
 |   /    |                      |
 |  /     |                      |
 |_/______|______________________|__ s
 0  sfl   sm                     1
 N=Ns                           N=0
 |<-normal->|
   operating

Starting torque (point C, $s = 1$, $N = 0$)

$$T_{st} = \frac{K E_2^2 R_2}{R_2^2 + X_2^2}$$

At standstill the rotor reactance $X_2$ is large compared with $R_2$, so rotor power factor is low and starting torque is moderate (about 1.5–2 times full-load torque for cage motors), even though starting current is 5–7 times full-load. Increasing $R_2$ (slip-ring motors) raises $T_{st}$, up to $T_{max}$ when $R_2 = X_2$.

Maximum torque (point B)

Putting $dT/ds = 0$ gives $R_2 = sX_2$:

$$s_m = \frac{R_2}{X_2}, \qquad T_{max} = \frac{K E_2^2}{2X_2}$$

$T_{max}$ is the pull-out (breakdown) torque, usually 2–3 times full-load torque. It is independent of $R_2$ and proportional to $V^2$. If load torque exceeds $T_{max}$, the motor stalls.

Regions of the curve

1. Normal operating (stable) region, $0 < s < s_m$: $sX_2 \ll R_2$, so $T \approx (KE_2^2/R_2)s$, i.e. $T \propto s$ — almost a straight line. The full-load point A lies here, at slip 2–5%. If load increases, speed falls a little and torque rises to match: stable, nearly constant speed.
2. Unstable region, $s_m < s < 1$: $sX_2 \gg R_2$, so $T \approx KE_2^2R_2/(sX_2^2)$, i.e. $T \propto 1/s$ — a rectangular hyperbola. A rise in load lowers speed, which lowers torque further, so the motor slows to a stop. The motor only passes through this region while accelerating.
3. Beyond the motoring range: $s < 0$ ($N > N_s$) gives generating action; $s > 1$ (rotor driven against the field) gives braking (plugging).