What is the Clarke Transformation in the Field-Oriented Control (FOC) Algorithm?

Electromechanical Motors, Actuators, Solenoids and Drivers
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In the Field-Oriented Control (FOC) algorithm, the Clarke Transformation is a core coordinate transformation step for simplifying three-phase variables and achieving decoupling. Its function is to convert physical quantities (such as current and voltage) in the three-phase stationary coordinate system (abc coordinate system) to the two-phase stationary coordinate system (αβ coordinate system), laying the foundation for the subsequent independent control of torque and magnetic flux.

Core Logical Framework of Sensorless Field-Oriented Control (FOC) for PMSM

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(Image source: Microchip)

1. Physical Background and Coordinate Definition of the Clarke Transformation

The Clarke Transformation is the first-stage preprocessing of the FOC algorithm. By mapping variables from the three-phase stationary coordinate system to the two-phase stationary coordinate system, it achieves dimensionality reduction and decoupling, providing a mathematical basis for the subsequent Park Transformation and the decoupling control of torque/magnetic flux. The figure below shows the simplification process from 3D to 2D.

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(Image source: Microchip)

  • Left Side: Three-Phase Stationary Coordinate System (abc Coordinate System)
    It corresponds to the three-phase windings (a, b, c phases) of the motor stator, which are symmetrically distributed at 120° intervals in space (the angle between the a, b, and c axes in the diagram is 120°). In actual control, the three-phase currents ia, ib, and ic are coupled in this coordinate system (satisfying the three-phase balance condition: ia + ib + ic = 0).

  • Right Side: Two-Phase Stationary Coordinate System (αβ Coordinate System)
    It is a virtual two-dimensional coordinate system. The α-axis coincides with the axis of the a-phase winding (remaining stationary), and the β-axis is perpendicular to the α-axis (leading the α-axis by 90°). Both axes remain stationary in space and are independent of the motor rotor.

2. Mathematical Principle and Transformation Formula

The core of the Clarke Transformation is to map three-phase variables to the two-phase coordinate system through mathematical formulas while ensuring power conservation before and after the transformation.

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  • Input: Any two phases of the three-phase current (e.g., ia, ib). Since ic = -ia - ib can be derived from the balance condition, no additional detection is required.
  • Output: The α-axis current iα (directly corresponding to the a-phase current) and the β-axis current iβ (the vertical component synthesized from the a-phase and b-phase currents).

3. Core Role in FOC

The role of the Clarke Transformation can be summarized in two points:

  • Dimensionality Reduction and Decoupling: It simplifies complex three-phase variables (3 coupled quantities) into two-dimensional variables (2 independent quantities), significantly reducing the complexity of the control algorithm. For example, there is no need to handle the dynamic changes of three-phase currents simultaneously; only iα and iβ need to be controlled.

  • Laying the Foundation for Subsequent Transformations: The iα and iβ obtained after the Clarke Transformation are variables in the stationary coordinate system. Later, through the Park Transformation (rotating coordinate system transformation), they can be further converted into id (magnetic flux component) and iq (torque component) that rotate synchronously with the rotor. Ultimately, this enables the independent control of torque and magnetic flux (similar to the armature and excitation control of DC motors).

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