Clarke / Park Transform Simulator
The Clarke/Park Transform Simulator provides an interactive 4-panel visualization of the coordinate transformations used in Field-Oriented Control (FOC). Watch 3-phase currents convert to Clarke αβ fixed-frame coordinates, then to Park dq rotating-frame coordinates where AC becomes DC — the fundamental insight enabling precise motor torque control. Free, no signup required.
① 3-Phase Currents (Ia, Ib, Ic): 3-phase vectors A, B, C at 120° intervals. Each current varies sinusoidally. Combining all three produces a rotating resultant vector (yellow).
② Clarke: Clarke: 3-phase → 2-phase orthogonal (α,β). Resultant decomposed into α (horizontal) and β (vertical). Vector tip traces a circle = AC.
③ Park: Park: apply rotor angle θ to α,β → d,q. d-axis = magnet direction, q-axis = torque direction. Vector appears stationary = DC!
④ Time Waveforms: Top: Iα,Iβ (AC sine waves). Bottom: Id,Iq (constant = DC!). Park transform magic — AC becomes DC.
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What are Clarke and Park Transforms?
The Clarke and Park transforms are mathematical coordinate transformations fundamental to Field-Oriented Control (FOC) of AC motors. The Clarke transform converts 3-phase currents (Ia, Ib, Ic) into a 2-phase orthogonal system (Iα, Iβ), simplifying three 120°-spaced variables into two perpendicular components. The Park transform then rotates the αβ frame by the rotor angle θ to produce d,q coordinates that are fixed to the rotor. In the dq frame, sinusoidal AC currents appear as constant DC values — Id along the magnet flux axis and Iq along the torque-producing axis. This DC-like representation enables simple PI controllers to regulate motor torque and flux independently, which is the fundamental insight that makes FOC the dominant motor control algorithm for BLDC and PMSM drives.
How to Use the Clarke/Park Simulator
- Drag the electrical angle (θ) slider to rotate the rotor and observe all four panels update simultaneously
- Watch Panel ① to see how 3-phase current vectors combine into a single rotating resultant vector
- Observe Panel ② (Clarke) where the resultant is decomposed into orthogonal α and β components — still rotating (AC)
- See Panel ③ (Park) where applying rotor angle θ makes the vector appear stationary — d,q values become constant (DC!)
- Check Panel ④ waveforms: top shows sinusoidal Iα,Iβ while bottom shows flat Id,Iq lines — proving Park transforms AC to DC
- Enable auto-rotate to watch the continuous transformation, or use Freeze θ to see what happens when Park transform uses wrong angle
Frequently Asked Questions
Why does Park transform convert AC to DC?
The Park transform rotates the coordinate frame at the same speed as the rotor's electrical frequency. Since the current vector also rotates at this frequency, it appears stationary in the rotating dq frame — just like a passenger on a merry-go-round appears stationary to another passenger. This converts sinusoidal signals into constant values that are much easier to control with PI regulators.
What is the difference between d-axis and q-axis?
The d-axis (direct) is aligned with the rotor's permanent magnet flux direction. The q-axis (quadrature) is 90° electrical ahead of the d-axis. In FOC, the q-axis current (Iq) directly controls torque production, while the d-axis current (Id) controls the flux. For maximum efficiency in PMSM drives, Id is typically set to 0 (Id=0 control), directing all current to torque production.
What happens when the Freeze θ button is pressed?
Freezing θ simulates using an incorrect rotor angle — as if the encoder failed. The Park transform no longer rotates at the correct speed, so the dq values are no longer constant DC. Instead, they oscillate as AC signals. This demonstrates why accurate rotor position sensing is critical for FOC.
What is the Clarke transform formula?
The Clarke transform converts 3-phase to 2-phase: Iα = Ia, Iβ = (Ia + 2·Ib) / √3. The Park transform then applies: Id = Iα·cos(θ) + Iβ·sin(θ), Iq = -Iα·sin(θ) + Iβ·cos(θ), where θ is the electrical rotor angle.