The rotating magnetic field: how three-phase currents make a motor turn

One winding makes a pulsating field

Start with a single coil on the stator carrying an alternating current i = I_m × sin(ωt). The coil produces a magnetic field along a fixed axis, and the strength of that field follows the current: it grows, falls to zero, reverses and grows again, 100 times per second on a 50 Hz supply.

This is a pulsating field. It does not rotate; it just breathes in and out along one line. Mathematically it can be split into two equal fields rotating in opposite directions, one forward and one backward. They cancel each other's net rotation, so a rotor at rest feels no push in either direction. That is why a purely single-phase motor has no starting torque.

Three windings, 120° apart

Now place three identical windings, A, B and C, around the stator so that their axes are 120° apart in space. Feed them with three currents of equal amplitude that are shifted by 120° in time:

i_A = I_m × sin(ωt) i_B = I_m × sin(ωt − 120°) i_C = I_m × sin(ωt − 240°)

Each winding makes its own pulsating field along its own axis. The resulting field at any instant is the vector sum of the three. A neat result of the trigonometry is that the sum has a constant magnitude of 1.5 times the peak of one phase field, and its direction rotates around the stator at the supply's angular frequency.

You can check this at a few instants. At ωt = 0, phase A is zero, B is −0.866 I_m and C is +0.866 I_m, so the field points at right angles to the A axis. At ωt = 90°, A is at its peak, B and C are each −0.5 I_m, and the field points along the A axis. The field has moved by 90 electrical degrees, while its length stayed the same.

Synchronous speed: 120 × f / p

During one electrical cycle, the field turns through one pair of poles. A machine with p poles has p / 2 pole pairs, so the field makes 2 / p of a revolution per cycle. With f cycles per second and 60 seconds per minute, the speed in revolutions per minute is

N_s = 60 × f / (p / 2) = 120 × f / p

This is the synchronous speed.

• 2 poles at 50 Hz: 3000 rpm
• 4 poles at 50 Hz: 1500 rpm
• 6 poles at 50 Hz: 1000 rpm
• 8 poles at 50 Hz: 750 rpm
• 4 poles at 60 Hz: 1800 rpm

So the speed of the field depends only on frequency and pole count, never on load or voltage. To change speed you change the frequency, which is exactly what a variable frequency drive does, or you change the number of poles in the winding.

Reversing the direction

The field rotates in the order in which the phase currents reach their peaks, called the phase sequence. For the sequence A–B–C the field turns one way; for A–C–B it turns the other way.

So to reverse a three-phase motor you only need to swap any two of the three supply leads. Nothing changes in the motor itself. The rotor, whether it is a squirrel-cage rotor chasing the field or a magnet locked to it, follows the field and turns the other way.

The same applies to a three-phase generator: connecting it to a grid with the wrong phase sequence is a classic commissioning mistake, which is why phase-sequence meters exist.

How the rotor follows the field

A rotating field is only half the story; the rotor must respond to it. How it does so separates the main motor families.

• Induction motor: the field sweeps across the rotor bars, induces currents in them, and the interaction of those currents with the field produces torque. The rotor must run a little slower than the field for this to work, a difference called slip.
• Synchronous motor and PMSM: the rotor has its own magnetic poles (permanent magnets or a DC-fed winding). They lock onto the rotating field and turn at exactly N_s, with a load angle that grows as the load increases.
• Brushless DC motor: the inverter moves the stator field in discrete steps rather than smoothly, always keeping it ahead of the magnet rotor.
• Stepper motor: the stator field jumps from one pole to the next, and the rotor moves one step each time.
• Reluctance motors: the rotor has no magnets or windings, but it tries to align its low-reluctance path with the field.

The slip guide covers the induction case in detail.

Single-phase motors: faking a rotating field

Household supplies are often single phase, so single-phase induction motors need a trick to get a rotating field at start-up. A second, auxiliary winding is placed 90° from the main winding and fed through a capacitor, or with different resistance, so that its current is shifted in time. Two windings 90° apart with currents about 90° apart give an approximately rotating field, enough to start the rotor.

Once the motor is up to speed, a start winding may be disconnected by a centrifugal switch, and the rotor's own motion keeps the main winding's pulsating field working as a rotating one. A shaded-pole motor does a cruder version of the same thing with a copper ring around part of each pole. Because the field is never perfectly circular, single-phase motors have torque pulsation at twice the line frequency, and they are less efficient than three-phase motors of the same size.

Try it in the Curvo Studio simulators

Open the three-phase induction motor simulator. Change the Supply frequency slider and watch the Synchronous speed readout follow 120 × f / p. Change the Pole count in the nameplate (2 to 12 poles) and see how the same frequency gives a different speed. Switch the Phase sequence between ABC and ACB and notice that the rotor direction reverses. The scene legend marks phases U, V and W, and you can turn on the Magnetic field layer in the cutaway view to see the field pattern.

Then open the single-phase induction motor simulator and look at the Start winding and Torque pulsation readouts to see how a single-phase machine differs. Finally, the permanent magnet synchronous motor simulator shows a rotor that locks to the field; compare its Synchronous speed and Load angle readouts with the induction motor's Slip.

Try it in the simulator