Maximum torque condition and T-s curves for different voltages

The developed torque of a 3-phase induction motor at slip $s$ is

$$T = \frac{K\, s\, E_2^2 R_2}{R_2^2 + (sX_2)^2}, \qquad K = \frac{3}{2\pi n_s}\ (n_s \text{ in rps})$$

where $E_2$, $X_2$ are rotor emf and reactance per phase at standstill and $R_2$ is rotor resistance per phase.

Condition for maximum torque

For a given motor and supply, $E_2$, $R_2$ and $X_2$ are constant; only $s$ varies. Divide numerator and denominator by $s$:

$$T = \frac{K E_2^2 R_2}{\dfrac{R_2^2}{s} + s X_2^2}$$

$T$ is maximum when the denominator $y = \dfrac{R_2^2}{s} + sX_2^2$ is minimum:

$$\begin{aligned} \frac{dy}{ds} &= -\frac{R_2^2}{s^2} + X_2^2 = 0 \\ s^2 X_2^2 &= R_2^2 \\ s_m &= \frac{R_2}{X_2} \end{aligned}$$

Since $\dfrac{d^2y}{ds^2} = \dfrac{2R_2^2}{s^3} > 0$, $y$ is minimum and $T$ is maximum.

Condition: torque is maximum when $R_2 = s X_2$, i.e. rotor resistance equals rotor reactance under running conditions.

Maximum torque:

$$\begin{aligned} T_{max} &= \frac{K (R_2/X_2) E_2^2 R_2}{R_2^2 + R_2^2} = \frac{K E_2^2}{2 X_2} \end{aligned}$$

Conclusions:

1. $T_{max}$ is independent of $R_2$; $R_2$ only changes the slip at which it occurs.
2. $T_{max} \propto 1/X_2$.
3. $T_{max} \propto E_2^2 \propto V^2$.
4. For maximum starting torque, $R_2 = X_2$ ($s_m = 1$).

(With stator impedance included: $s_m = R_2'/\sqrt{R_1^2 + (X_1 + X_2')^2}$.)

T–s characteristics for different supply voltages

Since $E_2 \propto V$, at any slip $T \propto V^2$, while $s_m = R_2/X_2$ does not depend on $V$.

 T            V1 > V2 > V3
 |         .--.        V1
 |       .'    '.
 |      /  .--.  '.    V2
 |     / .'    '. '.
 |    / /  .--.  '. '.  V3
 |   / / .'    '. '. '.
 |  / / /        '.'. '.
 |_/_/_/_____:_____'-'-'--- s
 0           sm             1
          (same for all V)

Example: a 10% fall in voltage reduces $T_{max}$ and $T_{st}$ to $0.9^2 = 0.81$ of their values, i.e. by 19%. This is why induction motors are sensitive to supply voltage dips, and why reduced-voltage starters (star–delta, auto-transformer) reduce starting torque.