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Implement *dq* to *αβ* transformation

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

The Inverse Park Transform block computes the inverse Park transformation
of the orthogonal direct and quadrature axes components in the rotating *dq*
reference frame. You can configure the block to align either the *d*- or
*q*-axis with the *α*-axis at time *t* =
0.

The block accepts the following inputs:

*d-q*axes components in the rotating reference frame.Sine and cosine values of the corresponding angles of transformation.

It outputs the two-phase orthogonal components in the stationary *αβ*
reference frame.

The figures show a rotating *dq* reference frame and the
*α-β* axes components in an *αβ* 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 inverse Park transformation.

When the

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

When the

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

where:

$${f}_{d}$$ and ${f}_{q}$ are the direct and quadrature axis orthogonal components in the rotating

*dq*reference frame.${f}_{\alpha}$ and ${f}_{\beta}$ are the two-phase orthogonal components in the stationary

*αβ*reference frame.

Park Transform | Discrete PI Controller with anti-windup and reset | DQ Limiter | ACIM Feed Forward Control | Space Vector Generator | Sine-Cosine Lookup | PMSM Feed Forward Control