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Park and Clarke Transformation: ABC to Alpha Beta Zero

Park and Clarke Transformation: ABC to Alpha Beta Zero


We are going to delve into Park and Clarke transformations. These transformations are fundamental for converting ABC (three-phase) quantities into DQ0 (Park transformation) or Alpha Beta Zero (Clarke transformation). These transformations are particularly useful in control systems, allowing easier manipulation and control of AC quantities by converting them into DC quantities.

Understanding Park and Clarke Transformations

Park Transformation (ABC to DQ0)

The Park transformation converts three-phase AC quantities (ABC) into DC quantities (DQ0). This transformation simplifies the control system by converting AC signals into a form that is easier to handle. By working with DC quantities, control parameters become simpler to manage.

Clarke Transformation (ABC to Alpha Beta Zero)

The Clarke transformation converts three-phase AC quantities into two-phase quantities (Alpha Beta Zero). This transformation is useful for various control applications, including motor control and inverter control, where two-phase quantities are easier to manage than three-phase ones.

Implementation in MATLAB Simulink

Generating Three-Phase Sine Wave

First, we need to generate a three-phase sine wave in MATLAB Simulink. This involves creating three sine wave blocks with appropriate phase shifts.

  1. Sine Wave Configuration:

  • Amplitude: 1

  • Frequency: 50 Hz (converted to radians per second as 2π2\pi2π)

  • Phase Angles:

  • Phase A: 0

  • Phase B: −120∘-120^\circ−120∘ or −120/360×2π-120/360 \times 2\pi−120/360×2π

  • Phase C: 120∘120^\circ120∘ or 120/360×2π120/360 \times 2\pi120/360×2π

Using Park and Clarke Transformation Blocks

MATLAB provides built-in blocks for these transformations. You can find them by searching ABC to DQ0 and ABC to Alpha Beta0.

  1. Park Transformation Block (ABC to DQ0):

  • Requires three-phase input.

  • Requires angular velocity (ωt\omega tωt), generated using a Phase Locked Loop (PLL) block.

  1. Clarke Transformation Block (ABC to Alpha Beta0):

  • Requires three-phase input.

  • Also uses the angular velocity from the PLL block.

Measuring Outputs

To observe the transformation, we measure the outputs using scopes. We will configure a scope to display three signals:

  • Input (ABC)

  • Output of Park Transformation (DQ0)

  • Output of Clarke Transformation (Alpha Beta0)

Inverse Transformations

We will also demonstrate the inverse transformations:

  • Inverse Park Transformation (DQ0 to ABC)

  • Inverse Clarke Transformation (Alpha Beta0 to ABC)

Setting Up the Simulation

  1. Connect the outputs of the Park and Clarke transformation blocks to their respective inverse transformation blocks.

  2. Use the same angular velocity (ωt\omega tωt) for the inverse transformations.

  3. Configure additional scopes to measure the outputs of the inverse transformations.

Simulation Results

Initial Simulation

After setting up the simulation, we observe the following:

  • Input (ABC): The original three-phase sine wave.

  • Output of Park Transformation (DQ0): DC quantities derived from the three-phase input.

  • Output of Clarke Transformation (Alpha Beta0): Two-phase quantities derived from the three-phase input.

Inverse Transformation Results

The outputs of the inverse transformations should match the original input (ABC), confirming the accuracy of the transformations.

Practical Applications

These transformations are essential in various applications:

  • Inverter Control: Simplifying the control logic by converting AC quantities to DC.

  • Motor Control: Easier manipulation and tuning of control parameters.


Park and Clarke transformations are powerful tools for simplifying control systems by converting three-phase AC quantities into more manageable forms. By leveraging MATLAB Simulink, we can efficiently implement and visualize these transformations.

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