Justify synchronous motor is not self starting

A 3-phase synchronous motor with its field excited cannot start by itself because the average torque on the stationary rotor is zero.

Justification

1. When 3-phase supply is switched on, the stator produces a magnetic field that immediately rotates at synchronous speed, $N_s = 120f/P$ (e.g. 1500 rpm for a 4-pole, 50 Hz machine).
2. The rotor is excited by DC, so it has fixed N and S poles, and it is at rest with large inertia.
3. Instant 1: suppose stator poles are placed so that stator N is just ahead of rotor S. Unlike poles attract, so the rotor gets a torque in, say, the clockwise direction.
4. Half a cycle later (0.01 s at 50 Hz): the stator field has moved by one pole pitch; now a stator S pole faces the rotor S pole. Like poles repel and the torque is anticlockwise.
5. The torque thus reverses every half cycle. The heavy rotor cannot pick up speed in 0.01 s to follow the field, so it only vibrates.
6. The average torque over one cycle is zero, and the motor does not start.

 Instant 1                 Instant 2 (T/2 later)
  stator: N      S          stator: S      N
          |      |                  |      |
  rotor : S      N          rotor : S      N
  attract -> clockwise       repel -> anticlockwise
         average torque = 0

Condition for running

Continuous torque in one direction is possible only when the rotor poles move at the same speed as the stator field, so that they stay locked to opposite stator poles. Therefore the rotor must first be brought close to $N_s$ by another means.

How it is started

• Damper winding (induction motor starting), then DC field is switched on.
• Pony motor on the same shaft.
• Slip-ring induction motor method.
• Variable frequency supply (VFD) raising frequency from zero.