T-N characteristics; effect of rotor resistance and voltage

Torque-speed characteristics

The torque of an induction motor is

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

 T
 |            Tmax
 |          _.-*-._
 |   Tst  /         \
 |    *--'            \
 |   /                  \
 +--+------+----------------+--> N
  N=0   N(Tmax)             Ns

• At $N=0$ the torque is the starting torque, which is moderate.
• The torque rises as the motor accelerates until the maximum (breakdown) torque occurs at $s_m = R_2/X_2$.
• From there to $N_s$ the torque falls nearly linearly to zero. Rated operation is on this falling part at 2 to 5 % slip, so speed changes very little with load.
• At $N_s$ the torque is zero.
• The part left of $T_{max}$ is unstable; the part right of it is stable.

Effect of rotor resistance

• $T_{max} = \frac{3E_2^2}{2\omega_s X_2}$ does not depend on $R_2$.
• The slip at maximum torque $s_m = R_2/X_2$ increases with $R_2$, so the peak shifts to lower speed.
• Starting torque rises with $R_2$ and is maximum when $R_2 = X_2$.
• Full-load speed falls and rotor copper loss rises (lower efficiency).
• Used in wound-rotor motors for high-torque starting and speed control.

 T   R2 low   R2 med   R2 = X2
 |   peak at high N     peak at N = 0

Effect of applied voltage

$E_2 \propto V_1$, so $T \propto V_1^2$ at any speed.

• Both $T_{st}$ and $T_{max}$ vary as $V^2$ (a 10 % voltage drop gives about 19 % less torque).
• $s_m$ does not change, so the peak stays at the same speed.
• At a fixed load torque, a lower voltage gives a higher slip, a higher rotor current and more heating.
• Reduced-voltage starting (star-delta, autotransformer) reduces starting current but also starting torque.