T-S characteristics and condition for maximum torque

Torque-slip characteristics

The torque-slip curve shows the variation of the torque of a 3-phase induction motor with slip, at constant supply 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)

• At standstill ($s = 1$): starting torque $T_{st}$, with high rotor current but a poor power factor.
• At small slip: $sX_2\ll R_2$, so $T\propto s$. The curve is nearly a straight line, and this is the stable operating region (full-load slip 2–5%). Load increases give a small fall in speed.
• At $s = s_m$: the torque reaches its maximum value $T_{max}$.
• At large slip: $T\propto 1/s$, so torque decreases as $s$ rises. This part is unstable for normal loads.
• At synchronous speed ($s = 0$): torque is zero.

Condition for maximum torque

Let $K = \dfrac{3E_2^2}{\omega_s}$. Then

$$T = K\,\frac{sR_2}{R_2^2 + s^2X_2^2}$$

For the maximum value, $\dfrac{dT}{ds} = 0$:

$$\frac{dT}{ds} = K\,\frac{(R_2^2 + s^2X_2^2)R_2 - sR_2(2sX_2^2)}{(R_2^2 + s^2X_2^2)^2} = 0$$ $$R_2^2 + s^2X_2^2 - 2s^2X_2^2 = 0 \;\Rightarrow\; R_2^2 = s^2X_2^2$$ $$s_m = \frac{R_2}{X_2}$$

So the torque is maximum when the rotor resistance equals the rotor reactance at slip $s_m$, that is $R_2 = s_mX_2$ (the rotor power factor is then $0.707$ lagging).

Substituting $s = s_m$:

$$T_{max} = K\,\frac{s_mR_2}{2R_2^2} = \frac{3E_2^2}{2\omega_sX_2}$$

$T_{max}$ is independent of the rotor resistance, is proportional to $V^2$ and inversely proportional to $X_2$.

Also: $\dfrac{T}{T_{max}} = \dfrac{2ss_m}{s^2 + s_m^2}$.