Dashboard › Deep Learning › Electrical Machines › Synchronous machines › The rotating magnetic field

The rotating magnetic field

Three pulsating MMFs 120° apart in space and time add up to one constant-magnitude rotating wave.

Freshman ~9 min

Step 1 — Three sinusoidal currents 120° apart in time

Animation speed 0.55×

|F_res| 0.00 ∠F_res 0° n_s 3000 rpm

Reference notes

Use Next → on the narrator above to step through six configurations: from three time-shifted currents, to spatial coils, to the resultant rotating MMF, and finally to synchronous speed.

What three currents 120° apart actually do

A balanced 3-phase supply produces three sinusoidal currents shifted by 120° in time:

i_a(t) = I_m·cos(ω t) i_b(t) = I_m·cos(ω t − 120°) i_c(t) = I_m·cos(ω t + 120°)

where ω = 2π·f is the angular frequency (314 rad/s for a 50 Hz supply). At any instant, the sum of the three currents is zero — they're "balanced".

The stator: three coils 120° apart in space

A 3-phase machine's stator carries three identical windings, called phases a, b, and c, displaced from each other by 120° around the air gap. Each winding, when its phase current is flowing, produces a magneto-motive force (MMF) directed along that winding's axis. The instantaneous MMF magnitude is proportional to that phase's current:

F_a(θ, t) = F_m·cos(ω t)·cos(θ)
F_b(θ, t) = F_m·cos(ω t − 120°)·cos(θ − 120°)
F_c(θ, t) = F_m·cos(ω t + 120°)·cos(θ + 120°)

Each individual phase produces a pulsating MMF — same axis in space, but its amplitude breathes up and down in time at frequency f.

Adding the three MMFs: the rotating field

Sum the three pulsating MMFs term by term and use the product-to-sum identity. The static (non-rotating) terms cancel, and what's left is one beautiful result:

F_resultant(θ, t) = (3/2)·F_m·cos(ω t − θ)

This is a constant-amplitude wave rotating at angular velocity ω. Its peak sits at the spatial angle θ = ω t at each instant — so it sweeps once around the stator per cycle of the supply.

Peak magnitude: 1.5 × the per-phase peak F_m.
Direction of rotation: from phase a → b → c (counter-clockwise in standard convention).
To reverse: swap any two of the three supply leads. The phase sequence becomes a → c → b, and the field rotates the other way.

Synchronous speed

For a machine with P magnetic poles wound into the stator, the field completes one full mechanical revolution every P/2 electrical cycles. So the mechanical rotational speed of the field is:

n_s = 120·f / P (rpm)

At 50 Hz: a 2-pole machine gives 3000 rpm, a 4-pole gives 1500 rpm, a 6-pole gives 1000 rpm, an 8-pole gives 750 rpm. Synchronous motors lock to exactly this speed; induction motors run slightly slower (the slip), and DC machines don't need this concept at all.

Why this matters

Synchronous machines have a DC-excited rotor that locks onto this rotating field — that's how they generate constant-frequency AC and lock to grid frequency.
Induction machines have a short-circuited rotor that the rotating field sweeps past, inducing rotor currents whose own field develops torque (Lenz). The rotor accelerates in the direction of the rotating field but can never catch it.
Why three phases? Two phases also produce a rotating field, but the magnitude pulsates. Three is the smallest balanced set that gives a constant-magnitude rotation — the reason 3-phase became the world standard for power transmission.

What you should take away. The rotating magnetic field is the single most important concept in polyphase machine analysis. Once you can see three pulsating sinusoidal MMFs adding to one steady rotating wave, every later lesson on synchronous and induction machines becomes "what does the rotor see?" instead of opaque algebra.

Keyboard shortcuts

→ next step · ← previous step
R replay narration · M mute / unmute · F fullscreen