Brushless motors · interactive
How three currents turn a rotor
A brushless motor is a ring of stator teeth, one coil per tooth, inside a ring of permanent magnets. The drive feeds three phase currents into the coils. This model shows what those currents do: it lights every tooth with the current in its coil, extracts the resulting field, and lets the magnets chase it.
How to read the drawing
| Symbol | Element | Meaning | |
|---|---|---|---|
| 1 | Coil | Colour: phase A, B, C. Fill intensity ∝ |i|. ⊗ current into the page, ⊙ out of it. | |
| 2 | Tooth tip | Pole of the tooth at this instant: N (red) or S (blue). Opacity ∝ |Fk|, the tooth's magnetomotive force. | |
| 3 | Magnets | Rotor, 2p magnets, N and S alternating. | |
| 4 | Rotor axis | Direction of the rotor's S-pole axis, θr. | |
| 5 | Stator field | Nearest peak of the fundamental of the stator magnetomotive force. Length ∝ amplitude. | |
| 6 | Torque angle | δ, from (4) to (5), in electrical degrees. Torque ∝ sin δ; maximum at 90°. | |
| 7 | Gap strip | Air gap unrolled, 0 to 360° mechanical. Steps: Fk per tooth. Curve: its fundamental. Band: magnets. |
Rotating field, commutation and load
Currents are commutated from the measured rotor angle. Change the motor, the drive, the timing, the load. In hold target speed mode a speed loop sets the current for you; watch it change as the load changes.
—
torque angle
—° electrical
current
—per unit
torque
—per unit
torque ripple
—% of mean
electrical speed
—cycles/s
rotor speed
—rev/s
Things to try
- 12 teeth, 14 magnets, sinusoidal. The pole pattern laps the stator; the rotor turns 1/7 of a turn per lap. Both speeds are in the readouts.
- 3 coils, 2 poles. One lap of the pattern is one lap of the rotor, and the 90° torque angle is 90° on the drawing. The 12-tooth motor is the same physics folded seven times.
- Six-step. Square-wave currents, six pattern positions per cycle, torque angle swinging 60° to 120°, ripple about 14%. That is the hum of a cheap drive.
- Timing advance. Torque falls as cos(advance): at 60° you get half the torque for the same current and heat. A misaligned Hall sensor does this.
- Push the load. The rotor slows; the torque angle stays at 90° because the controller re-aims from the measured angle; speed settles where torque = load + friction. Load above the current amplitude stalls the motor.
- Hold target speed. Switch Control to "hold target speed". The current slider is now driven by a PI speed loop. Raise the load and watch the current rise to hold speed; push the load and watch it recover. Ask for more speed than the current limit allows (1 per unit) and the slider pins at the limit while the speed falls short.
What is computed
- Tooth k carries magnetomotive force Fk = (winding sign) × (phase current). Phases are assigned by the star-of-slots rule for the chosen tooth and magnet counts.
- The stator field is the fundamental of the Fk staircase around the gap, the spatial harmonic with p cycles per turn, p = magnet pairs. It is extracted by a discrete Fourier transform over the teeth every frame. Higher harmonics make no average torque.
- Torque = (fundamental amplitude) × sin(torque angle), per unit: 1 = the torque of a sinusoidal current set of amplitude 1.
- Rotor: J dω/dt = torque − load − Bω, electrical units, J and B chosen for legibility.
- Controller, sinusoidal: reads the rotor angle, adds 90° plus the timing advance, sets the three currents that put the fundamental's peak there. Six-step: picks the nearest of the six two-phase patterns.
- Speed loop: current = Kp·(ωtarget − ω) + Ki·∫(ωtarget − ω) dt, clamped to the drive's current limit, with anti-windup.
Not in the model
- No steel: no flux concentration, no saturation, no cogging. Newton-metres per ampere need a field simulation of the real cross-section.
- Magnet field assumed sinusoidal. A trapezoidal motor reverses which drive ripples.
- Ideal controller: exact angle, instant currents, no inductance, no sampling delay.