Double field revolving theory

Double field revolving theory states that a pulsating (alternating) magnetic field of maximum value $\phi_m$ can be replaced by two fields, each of constant magnitude $\phi_m/2$, rotating in opposite directions at synchronous speed $N_s = 120f/P$.

Pulsating field resolved

The single-phase stator winding produces a flux that only alternates along its axis:

$$\phi = \phi_m \cos\omega t$$

This is equal to the sum of two rotating phasors $\phi_f$ (forward) and $\phi_b$ (backward), each $\phi_m/2$, turning at $\omega$ in opposite directions:

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

 wt = 0      wt = 90 deg     wt = 180 deg
  ^ ^          <-- -->          v v
  | |  phi_f  phi_b               | |
  sum = phi_m  sum = 0       sum = -phi_m
 (both along  (cancel)       (both reversed)
  the axis)
 phi_f turns anticlockwise, phi_b clockwise

At every instant the vertical components add and the horizontal components cancel, so the resultant is the pulsating field.

Torque at standstill - not self-starting

• At standstill each rotating field induces equal rotor currents (slip = 1 for both), so the forward torque $T_f$ and backward torque $T_b$ are equal and opposite.
• Net starting torque $T = T_f - T_b = 0$. Hence a single-phase induction motor is not self-starting.

Torque when running

If the rotor turns at speed $N$ in the forward direction:

$$s_f = \frac{N_s - N}{N_s} = s, \qquad s_b = \frac{N_s + N}{N_s} = 2 - s$$

Using the induction motor torque expression for each field:

$$T_f = \frac{K s R_2}{R_2^2 + (sX_2)^2}, \qquad T_b = -\frac{K (2-s) R_2}{R_2^2 + \left((2-s)X_2\right)^2}$$ $$T = T_f + T_b$$

• Near normal speed $s$ is small, $2 - s \approx 2$: the backward rotor current sees a large frequency $(2-s)f$ and high reactance, so $T_b$ is small and $T_f \gg |T_b|$. The motor keeps running in whichever direction it was started.

        T
        |            _
        |         .-' \   forward run
        |      .-'     |  (Tf > Tb)
 -Ns ---+----.'--------+---> speed
     |  |  .' 0        Ns
      \_|.'
 reverse run      net T = 0 at N = 0
 (mirror image)

The resultant torque-speed curve passes through zero at standstill and is symmetrical: positive in the forward direction, negative in the reverse direction.

Equivalent circuit (based on DFRT)

The rotor is split into two halves, one for each field: forward half with $0.5R_2'/s$ and $0.5X_2'$, backward half with $0.5R_2'/(2-s)$ and $0.5X_2'$, each with $0.5X_m$ in parallel, in series with the stator $R_1 + jX_1$.

 o--R1--X1--+--0.5X2'--0.5R2'/s ------+
 V          |  (0.5Xm parallel)  forward
            +--0.5X2'--0.5R2'/(2-s)---+
            |  (0.5Xm parallel)  backward
 o----------+-------------------------+

Hence, an auxiliary means (split phase, capacitor, shaded pole) is needed only to start the motor; once running, the forward field dominates.