Induction motor working and torque equation

A 3-phase induction motor works on electromagnetic induction: the stator's rotating field induces current in the rotor, and the interaction produces torque.

Working

1. A balanced 3-phase supply to the 3-phase stator winding produces a rotating magnetic field of constant magnitude moving at $N_s = 120f/P$.
2. The field cuts the rotor conductors and induces emf in them.
3. The rotor circuit is closed, so rotor current flows.
4. Current-carrying conductors in the field experience force; by Lenz's law the rotor turns in the direction of the field to reduce relative motion.
5. The rotor settles at speed $N < N_s$; slip $s = (N_s - N)/N_s$. If $N = N_s$, no emf, no torque.

Torque equation

At slip $s$: rotor emf $= sE_2$, rotor reactance $= sX_2$.

$$I_2 = \frac{sE_2}{\sqrt{R_2^2 + (sX_2)^2}}, \qquad \cos\phi_2 = \frac{R_2}{\sqrt{R_2^2 + (sX_2)^2}}$$

Torque $T \propto \phi I_2\cos\phi_2$ and $E_2 \propto \phi$:

$$\begin{aligned} T &= k\,\phi\,I_2\cos\phi_2 \\ &= k' E_2 \cdot \frac{sE_2}{\sqrt{R_2^2 + (sX_2)^2}}\cdot\frac{R_2}{\sqrt{R_2^2 + (sX_2)^2}} \\ &= \frac{K\, s\, E_2^2 R_2}{R_2^2 + (sX_2)^2} \end{aligned}$$

Using air-gap power, $T = \dfrac{P_{ag}}{\omega_s} = \dfrac{3I_2^2R_2/s}{2\pi N_s/60}$, so $K = \dfrac{3}{2\pi N_s/60}$.

Starting torque ($s = 1$): $T_{st} = \dfrac{K E_2^2 R_2}{R_2^2 + X_2^2}$.

Maximum torque: $s_m = R_2/X_2$, $T_{max} = \dfrac{KE_2^2}{2X_2}$.

Relation between speed and torque

Put $s = (N_s - N)/N_s$:

• Near synchronous speed (small $s$): $sX_2 \ll R_2$, so $T \approx \dfrac{K E_2^2}{R_2}\, s = \dfrac{K E_2^2}{R_2}\cdot\dfrac{N_s - N}{N_s}$. Torque rises linearly as speed falls — stable region.
• Low speeds (large $s$): $T \approx \dfrac{K E_2^2 R_2}{s X_2^2}$, i.e. $T \propto 1/s$. Torque falls as speed falls — unstable region.

 N
 Ns|'-._
   |    '-.  normal running
   |       '.
 Nm|--------: Tmax (pull-out)
   |       .'
   |     .'   unstable
   |   .'
  0|_.'___________ T
     Tst

So the motor runs at almost constant speed from no load to full load, like a DC shunt motor.