Rotating magnetic field, slip and standstill torque equation

Rotating magnetic field, synchronous speed and slip

• Rotating magnetic field (RMF): when a balanced 3-phase supply is given to a 3-phase winding whose phases are displaced by 120° in space, the three pulsating fluxes (displaced 120° in time) combine into a resultant flux of constant magnitude $1.5\,\phi_m$ that rotates around the air gap at constant speed.
• Synchronous speed: the speed of the RMF, $N_s = \dfrac{120 f}{P}$ rpm. For 50 Hz, 4 poles: 1500 rpm.
• Slip: the rotor runs at $N < N_s$. The relative speed as a fraction of $N_s$ is the slip:

$$s = \frac{N_s - N}{N_s}$$

At standstill $s = 1$; at synchronous speed $s = 0$; full-load slip is 2–5%. Rotor frequency $f_2 = sf$.

Derivation of standstill torque

Torque is proportional to the flux, the rotor current and the rotor power factor:

$$T \propto \phi\, I_2 \cos\phi_2$$

At standstill ($s = 1$), per phase: rotor emf $E_2$, resistance $R_2$, reactance $X_2 = 2\pi f L_2$.

      R2        X2
  +--[===]----[~~~]--+
  |                  |
 (~) E2      I2 -->  |
  |                  |
  +------------------+

Rotor impedance and current:

$$Z_2 = \sqrt{R_2^2 + X_2^2}, \qquad I_2 = \frac{E_2}{\sqrt{R_2^2 + X_2^2}}$$

Rotor power factor:

$$\cos\phi_2 = \frac{R_2}{\sqrt{R_2^2 + X_2^2}}$$

The rotor emf is produced by the stator flux, so $E_2 \propto \phi$, i.e. $\phi = E_2/k_1$. Substituting:

$$\begin{aligned} T_s &= k\,\phi\, I_2\cos\phi_2 \\ &= k\cdot\frac{E_2}{k_1}\cdot\frac{E_2}{\sqrt{R_2^2 + X_2^2}}\cdot\frac{R_2}{\sqrt{R_2^2 + X_2^2}} \\ &= K_1\,\frac{E_2^2 R_2}{R_2^2 + X_2^2} \end{aligned}$$

where $K_1 = k/k_1$. From the power method, $T = P_{ag}/\omega_s$ with $P_{ag} = 3I_2^2R_2$ at $s = 1$, so $K_1 = \dfrac{3}{2\pi N_s/60}$.

Notes:

1. $T_s \propto E_2^2 \propto V^2$: starting torque is very sensitive to supply voltage.
2. Starting torque is maximum when $\dfrac{d T_s}{dR_2} = 0$, i.e. $R_2 = X_2$ — achieved by adding rotor resistance in slip-ring motors.