T-S characteristics and effect of rotor resistance

The torque-slip characteristic shows how the torque developed by a 3-phase induction motor varies with slip (from $s = 1$ at standstill to $s = 0$ at synchronous speed), at rated voltage and frequency.

Torque of a 3-phase induction motor at slip $s$ (per-phase equivalent):

$$T = \frac{3}{\omega_s}\cdot\frac{sE_2^2R_2}{R_2^2 + (sX_2)^2}\ \text{N·m}$$

where $E_2$ and $X_2$ are the rotor emf and reactance per phase at standstill and $\omega_s$ is the synchronous speed in rad/s.

 Torque T
   |          Tmax
   |         .--*--.
   |  Tst   /   |   \          Operating
   |    *--'    |    \          region
   |            |     \___ ___ (stable)
   |            |     
   +------------+-----------+----> slip s
   1 (standstill) sm       0 (sync)

Regions of the curve

1. Starting point ($s = 1$): the torque is the starting torque $T_{st}$. Here the rotor reactance $sX_2$ is large, so the rotor power factor is low and the torque is less than the maximum.
2. Low-slip (operating) region ($0 < s < s_m$): the rotor reactance is small, so $sX_2\ll R_2$ and $T\approx \dfrac{3sE_2^2}{\omega_sR_2}\propto s$. The curve is almost a straight line. The motor works stably here: if the load rises, the speed falls slightly, slip increases and torque increases to meet it. The full-load slip is only about 2–5%.
3. Maximum torque point ($s = s_m$): the pull-out or breakdown torque $T_{max}$ occurs at $s_m = R_2/X_2$ and has the value

$$T_{max} = \frac{3E_2^2}{2\omega_sX_2}$$

4. High-slip region ($s_m < s < 1$): torque decreases as slip increases because $sX_2$ dominates ($T\propto 1/s$). This region is unstable for normal loads; if the load torque exceeds $T_{max}$, the motor stalls.

Effect of rotor resistance

$T_{max}$ does not depend on $R_2$, but the slip at which it occurs does: $s_m = R_2/X_2$.

 T
 |      R2 low    R2 med    R2 high
 |      .-*-.     .-*-.      .-*-.
 |     /     \   /     \    /     \
 |    /       \ /       \  /       \
 +---+--------+---------+--+--------+--> s

• Increasing $R_2$ (wound-rotor motor with external resistance) moves the peak toward higher slip (toward $s = 1$). When $R_2 = X_2$, $s_m = 1$ and maximum torque occurs at starting.
• The starting torque increases (up to $T_{max}$ when $R_2 = X_2$; more resistance beyond that reduces it again) and the starting current falls (better rotor power factor).
• The slope in the operating region becomes less steep, so the speed regulation worsens (more slip for the same torque), and the running efficiency falls due to $I_2^2R_2$ loss.
• Hence external rotor resistance is used for starting and speed control in slip-ring motors, and then cut out for running.