Interactive 3D visualization demonstrating the fundamental mathematical transformation in power systems analysis
This interactive 3D visualization shows the Clarke transformation in action. The black axes represent the three-phase quantities (a, b, c), while the blue axes show the transformed coordinates (α, β, 0). The red line traces the trajectory of the three-phase voltage vector in 3D space. You can adjust the frequency and amplitude using the sliders and pause/resume the rotation to better examine the transformation.
The Clarke transformation (also known as the alpha-beta transformation) is a mathematical technique used in electrical engineering to simplify the analysis of three-phase systems. It transforms the three-phase quantities (a, b, c) into two orthogonal components (α, β) plus a zero-sequence component (0).
The transformation is represented by the following matrix equation:
[α, β, 0]T = C × [a, b, c]T
Where the Clarke transformation matrix C is:
C = ⅔ × [ 1, -½, -½ ; 0, √3/2, -√3/2 ; ½, ½, ½ ]
This mathematical transformation offers several advantages:
In balanced three-phase systems, the zero-sequence component is zero, meaning all the information is contained in the α-β plane.
The Clarke transformation is fundamental in field-oriented control (FOC) of AC motors. It transforms three-phase stator currents into a stationary reference frame before the Park transformation rotates them to a synchronous reference frame.
In power systems, the Clarke transformation helps in analyzing unbalanced systems, fault detection, and power quality assessment by separating symmetrical components.
Used in control algorithms for inverters, rectifiers, and STATCOM devices to simplify the control structure and improve dynamic performance in power conversion systems.