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Implement *αβ* to *dq* transformation

**Library:**Motor Control Blockset / Controls / Math Transforms

The Park Transform block computes the Park transformation of two-phase
orthogonal components in a stationary *αβ* reference frame.

The block accepts the following inputs:

*α-β*axes components in the stationary reference frame.Sine and cosine values of the corresponding angles of transformation.

It outputs orthogonal direct and quadrature axis components in the rotating
*dq* reference frame. You can configure the block to align either the
*d*- or the *q*-axis with the *α*-axis at
time *t* = 0.

The figures show the *α-β* axes components in an *αβ*
reference frame and a rotating *dq* reference frame for when:

The

*d*-axis aligns with the*α*-axis.The

*q*-axis aligns with the*α*-axis.In both cases, the angle

*θ = ωt*, where:*θ*is the angle between the*α*- and*d*-axes for the*d*-axis alignment or the angle between the*α*- and*q*-axes for the*q*-axis alignment. It indicates the angular position of the rotating*dq*reference frame with respect to the*α*-axis.*ω*is the rotational speed of the*d-q*reference frame.*t*is the time, in seconds, from the initial alignment.

The figures show the time-response of the individual components of the
*αβ* and *dq* reference frames when:

The

*d*-axis aligns with the*α*-axis.The

*q*-axis aligns with the*α*-axis.

The following equations describe how the block implements Park transformation.

When the

*d*-axis aligns with the*α*-axis.$$\left[\begin{array}{c}{f}_{d}\\ {f}_{q}\end{array}\right]=\text{}\left[\begin{array}{cc}\mathrm{cos}\theta & \mathrm{sin}\theta \\ -\mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}{f}_{\alpha}\\ {f}_{\beta}\end{array}\right]$$

When the

*q*-axis aligns with the*α*-axis.$$\left[\begin{array}{c}{f}_{d}\\ {f}_{q}\end{array}\right]=\text{}\left[\begin{array}{cc}\mathrm{sin}\theta & -\mathrm{cos}\theta \\ \mathrm{cos}\theta & \mathrm{sin}\theta \end{array}\right]\left[\begin{array}{c}{f}_{\alpha}\\ {f}_{\beta}\end{array}\right]$$

where:

${f}_{\alpha}$ and ${f}_{\beta}$ are the two-phase orthogonal components in the stationary

*αβ*reference frame.$${f}_{d}$$ and ${f}_{q}$ are the direct and quadrature axis orthogonal components in the rotating

*dq*reference frame.

Inverse Park Transform | Clarke Transform | Sine-Cosine Lookup | Discrete PI Controller with anti-windup and reset | ACIM Feed Forward Control | ACIM Torque Estimator | PMSM Feed Forward Control | PMSM Torque Estimator