Clarke Transform
How it works
The transform multiplies the phase vector [a, b, c] by a constant 2×3 matrix (or 3×3 when the zero-sequence is included). In the amplitude-invariant form: α = (2/3)·(a − b/2 − c/2), β = (2/3)·(√3/2)·(b − c), and the zero-sequence γ = (1/3)·(a + b + c). For a balanced system where a + b + c = 0 the zero-sequence vanishes and the 2×2 variant suffices. The result is two quantities (α, β) in a stationary reference frame that still vary sinusoidally in time but are mutually orthogonal. The inverse Clarke transform reconstructs the phase quantities from (α, β, γ).
Problem solved
Directly controlling three mutually dependent, sinusoidally varying phase quantities (a, b, c) is complex. The Clarke transform reduces this set to two independent orthogonal coordinates (α, β), removing the redundancy of the third phase (in a balanced system a+b+c = 0) and setting up the subsequent Park transform.
Components
A constant 2×3 (or 3×3) matrix with a 2/3 factor (amplitude-invariant form) or √(2/3) factor (power-invariant form).
Stationary-frame component coincident with the phase-a axis.
Stationary-frame component orthogonal to the α axis.
Equals (1/3)(a+b+c); zero in a balanced three-phase system.
Evolution
Blaschke's work on field-oriented control made the Clarke and Park transforms a standard stage in AC drive control.
Hyperparameters (configurable axes)
Choice between the amplitude-invariant form (2/3 factor) and the power-invariant form (√(2/3) factor).
2×2 variant (α, β only) versus 3×3 variant (α, β, 0).