Double field revolving theory

Double field revolving theory (DFRT)

A single-phase winding carrying alternating current produces a pulsating (alternating) flux along a fixed axis: $\phi = \phi_m\cos\omega t$. According to the double field revolving theory, this pulsating flux is equivalent to two rotating fluxes, each of constant magnitude $\phi_m/2$, rotating at synchronous speed $N_s = \frac{120f}{P}$ in opposite directions.

$$\phi_m\cos\omega t = \frac{\phi_m}{2}e^{j\omega t} + \frac{\phi_m}{2}e^{-j\omega t}$$

• $\phi_f$: the forward-rotating component (in the direction of the rotor).
• $\phi_b$: the backward-rotating component (in the opposite direction).

 phi_f = phi_m/2 -->  (anticlockwise)
          \   /
   phi_b   \ /  resultant = phi_m cos(wt)
 = phi_m/2  X   along fixed axis (pulsating)
 (clockwise)

At every instant the components perpendicular to the winding axis cancel, and the components along the axis add, giving the pulsating flux $\phi_m\cos\omega t$.

Rotor behaviour

Each field induces rotor currents and produces its own torque. For a rotor speed $N_r$ in the forward direction:

• Slip with respect to the forward field: $s_f = s = \frac{N_s - N_r}{N_s}$
• Slip with respect to the backward field: $s_b = 2 - s$

The net torque is $T = T_f - T_b$.

• At standstill ($s = 1$): $s_f = s_b = 1$, so $T_f = T_b$ and the net torque is zero. The motor is not self-starting.
• If the rotor is started in either direction, the two torques become unequal. The backward field acts on the rotor at a high slip ($2 - s$), so the rotor current it induces has a high frequency, is limited mainly by reactance and has a low power factor; the backward torque is therefore small. So $T_f > T_b$ in the direction of rotation, and the motor keeps accelerating to near $N_s$.
• At $N_s$ ($s=0$): $T_f = 0$, but $T_b$ is not zero (slip 2), so the net torque is slightly negative; the motor runs at a speed a little below $N_s$.

  T
  |   Tf              net T = Tf - Tb
  |   .--.            _.-'-._
  |  /    \          /       \
  +-/------\--------/---------\---> N
  -Ns  Tb   0      (start)     Ns

The net torque curve passes through zero at standstill and is positive at positive speeds (and negative at negative speeds).