Inverse transforms

Inverse Park and inverse Clarke transforms produce three phase-voltage commands.

  • signal
  • Gradara equations
  • 6 ports
BlockInverse transformsKindinverseParkLibraryControl

Description

Converts d/q voltage commands into three phase-voltage commands: an inverse Park rotation by theta followed by the amplitude-invariant inverse Clarke transform.

It is the exact inverse of Park transform followed by Clarke transform for a set without zero sequence, so the output amplitude equals √(vd² + vq²).

Example

PMSM · Field-oriented control 1500 rpm speed command with cascaded d/q current control. A 48 V averaged inverter drives a surface PMSM and a mechanical load that increases at 0.45 s. Ideal current and rotor-position feedback; continuous controllers; no PWM ripple or sensor noise.

Speed commandrpm → rad/sSpeed errorKp · speedKi · speedIntegralPI sumCurrent limitq-axis errorq-axis PIid* = 0 Ad-axis errord-axis PIInverse Park / ClarkeAveraged inverterPMSMMechanical loadClarkePark
In Gradara, select a Inverse transforms block and press F1, then choose Open example. It opens as a new model in My models, ready to run.

Also in this example:SumSubtractConstantLimited integratorCurrent PIClarke transformPark transformThree-phase inverterPMSMMechanical loadStepGainSaturation

Ports

Inputs 3

  • dvdsignal · V

    d-axis voltage command, in volts.

  • qvqsignal · V

    q-axis voltage command, in volts.

  • θethetasignal · rad

    Electrical rotor angle θe, in radians.

Outputs 3

  • avasignal · V

    Phase a voltage command, in volts.

  • bvbsignal · V

    Phase b voltage command, in volts.

  • cvcsignal · V

    Phase c voltage command, in volts.

Equations

Modelica
alpha = vd · cos(theta) − vq · sin(theta)
beta = vd · sin(theta) + vq · cos(theta)
va = alpha
vb = −alpha/2 + (√3/2) · beta
vc = −alpha/2 − (√3/2) · beta

Implementation

The block’s Modelica model
Real alpha;
Real beta;
alpha = vd*cos(theta)-vq*sin(theta);
beta = vd*sin(theta)+vq*cos(theta);
va = alpha;
vb = -alpha/2 + sqrt(3)*beta/2;
vc = -alpha/2 - sqrt(3)*beta/2;

Assumptions and limitations

  • The outputs always sum to zero: no common-mode injection or space-vector modulation.
  • No voltage limit or overmodulation handling; limit vd and vq upstream or in the inverter.

Tips

  • Use the same theta as the Park transform on the measurement side.

Used in

These larger examples use it too. Open them from Examples in the app.

  • PMSM · Field-oriented control

See also

Select a block in Gradara and press F1 to open its page offline.