DFRT explanation of no starting torque

According to double field revolving theory, the pulsating field of a single-phase winding is equivalent to two equal fields, each of half the maximum value, rotating in opposite directions at synchronous speed. At standstill these two fields produce equal and opposite torques, so the net starting torque is zero and the motor is not self-starting.

Resolution of the pulsating field

The flux of a single-phase winding at space angle $\theta$ is

$$\begin{aligned} \phi(\theta, t) &= \phi_m \cos\omega t \cos\theta \\ &= \frac{\phi_m}{2}\cos(\theta - \omega t) + \frac{\phi_m}{2}\cos(\theta + \omega t) \end{aligned}$$

• First term: field of amplitude $\phi_m/2$ rotating forward (anticlockwise) at $\omega$.
• Second term: equal field rotating backward (clockwise) at $\omega$.

     t = 0         t = T/8        t = T/4
   phi_f phi_b   phi_f   phi_b    phi_f  phi_b
     ^  ^          \     /         <--  -->
     |  |           \   /
   resultant      resultant      resultant
   = phi_m        = 0.707phi_m   = 0
   (vertical)     (vertical)     (sum cancels)

The two vectors always add along the winding axis, so the resultant only pulsates between $+\phi_m$ and $-\phi_m$.

Slips with respect to the two fields

If the rotor runs at $N$ in the forward direction:

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

Torques

Each field acts like the field of a three-phase motor and produces a torque–slip curve:

• Forward torque $T_f$ (positive) by the forward field
• Backward torque $T_b$ (negative) by the backward field
• Net torque $T = T_f - T_b$

 T |         Tf
   |        /\
   |   ____/  \     Tnet (solid)
   |  /    \   \
 --+-/------+---\------> N
 -Ns \   /  0    +Ns
   |  \_/  Tb
   |
 s:  2      1     0

Why not self-starting

• At standstill $N = 0$: $s_f = s_b = 1$.
• Both fields cut the rotor at the same speed, induce equal rotor emfs and currents, and produce equal and opposite torques: $T_f = T_b$.
• Net starting torque $T_{st} = T_f - T_b = 0$. The rotor just vibrates and hums.

Behaviour once started

If the rotor is turned forward by some means, $s_f < 1$ and $s_b > 1$. The forward torque increases and the backward torque falls (the backward rotor current is at nearly $2f$ and largely reactive), so $T_f > T_b$ and the motor accelerates to near synchronous speed. It can run equally well in either direction, depending on the initial push.

In the equivalent-circuit form, at $s = 1$, $Z_F = Z_B$ and $P_{gF} = P_{gB}$, giving $T = (P_{gF} - P_{gB})/\omega_s = 0$.

Remedy

Make the motor temporarily two-phase: add an auxiliary winding displaced $90^\circ$ in space carrying a current displaced in time (split-phase, capacitor-start, PSC, CSCR) or use shading coils. This produces a true rotating field and hence starting torque.