FOC Mathematical Transforms

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Responsibilities

Is responsible for: - Converting three-phase stator currents to a two-phase stationary frame (Clarke transform) - Converting between the stationary αβ frame and the rotor-synchronous dq frame (Park transform) - Providing a composite Clarke → Park (and inverse) operation that evaluates trigonometric functions exactly once per call - Converting a two-component voltage vector in the αβ frame to three-phase PWM duty cycles (Space Vector Modulation) - Supplying fast sine and cosine approximations via a pre-computed lookup table for use in the FOC hot path

Is NOT responsible for: - Reading or writing hardware peripherals (ADC, PWM, encoder) - Managing PID controllers or setpoints - Determining rotor position — electrical angle must be supplied by the caller - Any control strategy decisions (torque, speed, or position setpoints) - Scaling voltage vectors to physical units — inputs and outputs are normalised fractions relative to the DC bus

Component Details

Clarke Transform — Three-Phase to Stationary αβ Frame

The Clarke transform is a geometric projection that reduces a balanced three-phase system to an equivalent two-axis representation. It requires no knowledge of rotor position.

Forward transform (currents 3-phase → αβ):

The amplitude-invariant Clarke transform uses all three measured phases, so it stays correct when the currents are not perfectly balanced:

Iα = (2/3)·(Ia - (Ib + Ic)/2)
Iβ = (Ib - Ic) / √3

When the system is balanced (Ia + Ib + Ic = 0) this reduces to the familiar two-sensor form Iα = Ia, Iβ = (Ia + 2·Ib)/√3. The implementation uses the three-phase form unconditionally.

All three phase currents are measured. AdcPhaseCurrentMeasurement converts three ADC samples and delivers them as a PhaseCurrents triple, so no phase is reconstructed from the balance constraint.

Inverse transform (voltages αβ → 3-phase):

Reconstructs the three-phase voltage references from Vα and Vβ, using the same geometric relationships in reverse. The output of the inverse Clarke is fed into the Space Vector Modulator.

Invariants: - No balance assumption is required on the input; a nonzero common-mode current appears only in the zero-sequence component, which the two-axis output discards. - The transform is purely linear and involves no state — every call is independent.

Park Transform — Stationary αβ to Rotor-Synchronous dq Frame

The Park transform rotates the stationary αβ frame to align with the rotor magnetic field. The result is a coordinate frame that is stationary relative to the rotor, so DC steady-state values represent constant torque and flux components.

Forward transform (αβ → dq):

Id =  Iα·cos(θe) + Iβ·sin(θe)
Iq = −Iα·sin(θe) + Iβ·cos(θe)

The d-axis (direct) current component is aligned with the rotor flux. For a surface-permanent-magnet synchronous motor (SPMSM), the d-axis setpoint is held at zero to maximise torque per ampere. The q-axis (quadrature) current is proportional to electromagnetic torque.

Inverse transform (dq → αβ):

Vα = Vd·cos(θe) − Vq·sin(θe)
Vβ = Vd·sin(θe) + Vq·cos(θe)

The inverse Park transform is applied after the PID controllers produce voltage demands Vd and Vq, converting them back to the stationary frame for SVM processing.

Electrical angle:

The electrical angle θe is obtained by multiplying the mechanical rotor angle θm by the motor’s pole-pair count P:

θe = θm · P

The caller is responsible for supplying the correct θe.

Invariants: - The same θe (and therefore the same cos/sin pair) must be used for both the forward Park in the current-sensing path and the inverse Park in the voltage-output path within a single control cycle. - The transform is stateless — no memory of previous cycles.

Composite ClarkePark — Chained Transform with Single Trigonometric Evaluation

The ClarkePark composite combines the forward Clarke and forward Park transforms (or their inverses) into a single operation. The motivation is efficiency: cos(θe) and sin(θe) are computed exactly once and reused for both the Park rotation in the forward direction and the inverse Park rotation.

The forward composite takes (Ia, Ib, θe) and produces (Id, Iq) directly.
The inverse composite takes (Vd, Vq, θe), performs inverse Park to (Vα, Vβ), then inverse Clarke to (Va, Vb, Vc), producing all three phase voltages in one call.

This is the form used by all inner-control-loop implementations in this project.

Space Vector Modulation — αβ Voltage Vector to PWM Duty Cycles

Space Vector Modulation (SVM) maps a two-component voltage reference vector in the αβ plane to three symmetrical PWM duty cycles. The implementation is sector-free: it reaches the same duty cycles as classical sector-and-dwell-time SVM by injecting a common-mode offset, which is algebraically equivalent and has no branch on sector index.

Inverse Clarke:

The reference is first expanded to three phase voltages, vA = Vα, vB = -Vα/2 + (√3/2)·Vβ, vC = -Vα/2 - (√3/2)·Vβ.

Common-mode (min-max) injection:

The offset vCommon = -(max(vA,vB,vC) + min(vA,vB,vC)) / 2 is added to all three phases. Because it is identical on every phase it cancels in every line-to-line voltage, so it changes no motor current while centring the waveform. This is what recovers the linear range that classical SVM obtains from symmetric null-vector splitting: a factor 2/√3 ≈ 1.155 more than sinusoidal PWM reaches on the same bus. In the per-unit convention used here — inputs relative to V_dc/√3 — that extended range is the unit circle, which is why the modulation limiter clamps on |V| ≤ 1 rather than on 2/√3.

Duty cycle output:

Each offset phase voltage is scaled by 1/√3, biased by 0.5 to centre it in the PWM period, and clamped: d = clamp(v/√3 + 0.5, 0, 1). A reference of unit magnitude therefore spans exactly [0, 1].

Constraints: - Input Vα, Vβ are per-unit relative to V_dc/√3 (dimensionless, inside the unit circle for linear operation). - Output duty cycles are in the range [0.0, 1.0]; values outside this range are clamped, which is how over-modulation is handled. - Being branch-free in the sector, the output is continuous across every sector boundary by construction. - SVM does not accept angles directly — it operates only on the (Vα, Vβ) pair.

Fast Trigonometry — Lookup-Table Sine and Cosine

Direct hardware transcendental functions (sin, cos) introduce variable latency and are unsuitable for a 20 kHz ISR. A 512-entry pre-computed lookup table (LUT) covering one full period [0, 2π) provides both sine and cosine with bounded, cycle-predictable evaluation time.

LUT characteristics: - 512 entries, each a 32-bit floating-point value - Stored in ROM (read-only, constexpr) at a 16-byte memory alignment boundary - Total ROM footprint: 512 × 4 bytes = 2 048 bytes (2 KB) - Covers exactly one full period

Range handling: the angle is not normalised to [0, 2π) before indexing. It is scaled by 512/2π, truncated toward negative infinity, and the resulting index is masked with size - 1, so the wrap is a bitwise AND on the index rather than a modulo on the angle. This is why there is no fmod in the hot path. The consequence to know is the domain limit it leaves: the intermediate is an int32_t, so the result is meaningless for angles beyond roughly 2.6 × 10⁷ radians. Electrical angle reaches the transforms already wrapped, so this bound is not approached in the control loop.

Cosine reuses the sine table — there is no separate cosine table — but it is not an index offset. It adds π/2 to the angle and performs the whole lookup again, so a cosine costs a full sine plus one addition, and evaluating both for the same angle does the range reduction twice.

Interpolation: Linear interpolation between adjacent LUT entries provides accuracy sufficient for FOC applications; the quantisation error is below the noise floor of 12-bit ADC current sensing.

Interfaces

Provided

Required

Data Model

Block Diagram

Constraints & Limitations