DFRT and making single phase motor self-starting

Double field revolving theory

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.

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

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

 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 vertical components cancel and the horizontal components add up, giving the pulsating flux $\phi_m\cos\omega t$.

Rotor behaviour

Each field induces rotor currents and produces its own torque. For 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 forward and backward torques become unequal. The backward field induces rotor currents of frequency $(2-s)f$; at this high frequency the rotor reactance is large, so these currents are small and lag heavily, and the backward torque falls. So $T_f > T_b$ in the direction of rotation, and the motor keeps accelerating towards $N_s$.
• 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 therefore 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 the reverse for negative speeds).

Making a single-phase induction motor self-starting

Since a single-phase winding gives no starting torque, a rotating field must be created at starting. This is done by adding an auxiliary (starting) winding displaced $90^\circ$ electrically from the main winding and making its current differ in phase from the main winding current. The methods are:

1. Resistance split-phase: the auxiliary winding has higher resistance and lower reactance than the main winding. The currents differ in phase by about $30^\circ$, giving a weak rotating field and a moderate starting torque. A centrifugal switch cuts the auxiliary winding out at about 75 % speed.
2. Capacitor-start: a capacitor in series with the auxiliary winding gives a phase difference near $90^\circ$ and a high starting torque (3 to 4.5 times full load).
3. Capacitor-start, capacitor-run: a capacitor stays in the circuit while running (a smaller one than at starting), giving good torque and power factor.
4. Permanent split capacitor: the same capacitor is used for starting and running. Starting torque is low, but running is smooth and quiet.
5. Shaded pole: copper rings around part of each pole delay the flux in the shaded part, giving a small rotating-field effect. Used for very small motors (fans, toys).

 Main winding ------+
                    +---- supply
 Aux. winding --[C]-+-(centrifugal switch)