Torque-slip characteristic and effect of rotor resistance

The torque–slip characteristic of a 3-phase induction motor is the curve of developed torque against slip (or speed) at constant supply voltage and frequency.

Torque equation

$$T = \frac{K\, s\, E_2^2\, R_2}{R_2^2 + (s X_2)^2}$$

where $E_2$ and $X_2$ are rotor emf and reactance per phase at standstill, $R_2$ is rotor resistance, $s$ is slip and $K = 3/(2\pi n_s)$ with $n_s$ in rps.

Shape of the curve

 T
 |          Tmax
 |         .--.
 |       .'    '.
 |      /        '.
 |     /           '-.
 |    /               '-. Tst
 |   /  stable  |  unstable
 |  /           |
 | /            |
 |/_____________|____________ s
 0             sm           1
(N=Ns)                    (N=0)

1. Low-slip region ($s \ll s_m$, normal running): $sX_2 \ll R_2$, so $T \approx K s E_2^2/R_2$, i.e. $T \propto s$. The curve is nearly a straight line through the origin. This is the stable operating region; speed falls only slightly as load rises.
2. Maximum torque: occurs at $s_m = R_2/X_2$, with $T_{max} = K E_2^2/(2X_2)$. It is also called pull-out or breakdown torque.
3. High-slip region ($s > s_m$): $sX_2 \gg R_2$, so $T \approx K E_2^2 R_2/(s X_2^2)$, i.e. $T \propto 1/s$. The curve is a rectangular hyperbola. Operation here is unstable: if load increases, speed falls, torque falls further, and the motor stalls.
4. At $s = 1$ the torque is the starting torque $T_{st} = K E_2^2 R_2/(R_2^2 + X_2^2)$.

Effect of rotor circuit resistance

When $R_2$ is increased, the whole curve keeps the same peak height but the peak moves to the right (towards $s = 1$).

Using $T_{st}/T_{max} = 2a/(1 + a^2)$ with $a = R_2/X_2$, for $X_2 = 1\ \Omega$:

• Maximum torque is unchanged: $T_{max} = KE_2^2/(2X_2)$ does not contain $R_2$.
• Slip at maximum torque increases: $s_m = R_2/X_2$, so higher $R_2$ shifts the peak towards $s = 1$ (lower speed).
• Starting torque increases up to $R_2 = X_2$, when $s_m = 1$ and $T_{st} = T_{max}$. Beyond that, starting torque falls again.
• Starting current decreases and starting power factor improves.
• Full-load slip increases, so running efficiency falls (rotor copper loss $= s P_{ag}$) and speed regulation becomes poorer.

This is why slip-ring motors use external rotor resistance only during starting (or for speed control) and short it out in normal running. Double-cage and deep-bar motors get the same effect automatically.