FOC: three phase currents into two DC values

Why change coordinates

The three phase currents are sine waves that keep changing with the rotation speed, so it is hard to make a PI controller follow them directly. Seen from a frame that rotates with the rotor, these currents become steady DC values, and DC is easy to control with PI.

Order of the transforms

1. Clarke transform: ia, ib, ic → iα, iβ (stationary two-axis frame)
2. Park transform: rotate by the rotor electrical angle θ → id, iq (rotating two-axis frame)
3. PI control: id → target 0 (surface-magnet motor), iq → target torque current
4. Inverse Park transform: vd, vq → vα, vβ
5. SVPWM: vα, vβ → duty of the three phases

iq = -iα·sinθ + iβ·cosθ
id =  iα·cosθ + iβ·sinθ

What the d and q axes mean

• q-axis current: produces torque (torque = 1.5 · p · λ · iq)
• d-axis current: along the flux. Held at 0 it gives maximum torque for the same current; made negative it allows field-weakening operation at high speed

What you need

• An accurate electrical angle θ: an encoder, interpolated Hall sensors, or a sensorless observer
• A fast current loop: the whole calculation above every PWM period (10 to 20 kHz)
• Accurate current sensing and timing

Compared with six-step it has less torque ripple, is quieter and more efficient, which is why electric vehicles, industrial servos and high-end drone ESCs use FOC.

In one sentence

Seen from a frame that rotates with the rotor, the three-phase AC currents look like two DC values. DC is easy to handle with PI controllers, so FOC changes coordinates, controls there, and transforms back.

What happens in one period

Order	Stage	Input → output
1	Current measurement	Shunt resistors and ADC → ia, ib (ic = −ia − ib)
2	Clarke transform	ia, ib → iα, iβ (stationary two axes)
3	Park transform	iα, iβ, angle θ → id, iq (rotating two axes)
4	Two PI controllers	id error, iq error → vd, vq
5	Inverse Park transform	vd, vq, θ → vα, vβ
6	SVPWM	vα, vβ → duty of the three phases

These six stages repeat every PWM period (usually 10 to 20 kHz).

The d and q axes

• q-axis current (iq): at right angles to the magnet flux. It produces torque. Torque is proportional to iq.
• d-axis current (id): in the same direction as the magnets. It does not contribute to torque, so it is normally controlled to 0. Making it negative to hold down the back-EMF at high speed is field-weakening control.

3D labRun what you have read so far→

Transform code

// Clarke: three-phase → stationary two-axis (assumes ia + ib + ic = 0)
float i_alpha = ia;
float i_beta  = (ia + 2.0f * ib) * 0.57735027f;      // 1/√3

// Park: stationary two-axis → rotating two-axis
float s = sinf(theta), c = cosf(theta);
float id =  i_alpha * c + i_beta * s;
float iq = -i_alpha * s + i_beta * c;

// Inverse Park: rotating two-axis → stationary two-axis
float v_alpha = vd * c - vq * s;
float v_beta  = vd * s + vq * c;

theta is the electrical angle. It is the mechanical angle times the number of pole pairs, and you have to find, once, the electrical-angle offset at the point where the encoder reads 0 and add it.

What happens when the angle is wrong

With an angle error δ, torque falls by a factor of cos δ.

Angle error	Remaining torque	Symptom
0°	100%	Normal
30°	87%	More current, more heat
60°	50%	Noticeably weak
90°	0%	Current flows but nothing turns
Over 90°	Negative	Turns backwards or runs away

If the torque is low for the current you are putting in, or the motor runs unusually hot, suspect the angle offset first. You can see this relationship for yourself by moving the angle-error slider in the site's FOC lab.

Compared with six-step

	Six-step	FOC
Current shape	Steps	Sine wave
Torque ripple	Large	Small
Sensors needed	Hall sensors	Current sensors + precise angle
Computation	Little	Trigonometry and two PI controllers
Use	Drones, power tools	Electric vehicles, robot joints, appliances

Problems that come up in practice

• Current sampling instant: measuring at a switching instant picks up noise. Trigger the ADC at the centre of the PWM period.
• PI output limit: limit the combined magnitude of vd and vq so it does not exceed the available voltage, and stop integrating when saturated.
• Current sensor offset: measure the ADC value at zero current at start-up and subtract it.
• Wrong pole-pair count: the electrical angle is wrong, so the motor shakes at low speed and loses force at high speed.