T-S characteristics, max torque, rotor resistance effect

Torque-slip characteristics

The torque of a three-phase induction motor is

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

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

 T
 |    R2 small     R2 larger
 |      Tmax ______ Tmax
 |       .-*-.   .-*-.
 |      /     \ /     \
 |  Tst1       Tst2 (higher)
 +--+---------+--------------> s
   s=1      sm1 sm2       0

The curve rises from $T_{st}$ at $s=1$ to a maximum and then falls to zero at $s = 0$. It is nearly linear near $s=0$ (the stable region), and the motor runs at low slip there.

Condition for maximum torque

Differentiate $T$ with respect to $s$ and equate it to zero:

$$\frac{dT}{ds} = 0 \;\Rightarrow\; R_2^2 - s^2X_2^2 = 0 \;\Rightarrow\; s_m = \frac{R_2}{X_2}$$

So the torque is maximum when the rotor resistance equals the rotor reactance at that slip ($R_2 = s_mX_2$). Substituting:

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

Effect of rotor resistance

• $T_{max}$ is independent of $R_2$; only its position changes.
• $s_m \propto R_2$: with higher $R_2$ the maximum torque occurs at a higher slip (lower speed).
• Starting torque rises as $R_2$ increases (up to $R_2 = X_2$, where $T_{st} = T_{max}$).
• The full-load slip increases, so the speed falls; the rotor copper loss rises and the efficiency falls.

This is why slip-ring motors use external rotor resistance for starting and speed control.