# dq0_to_abc Transformation

Perform Park transformation from dq0 reference frame to abc reference frame

## Library

powerlib_extras/Measurements, powerlib_extras/Discrete Measurements

 Note:   The Transformations section of the Control and Measurements library contains the dq0 to abc to block. This is an improved version of the dq0_to_abc Transformation block. The new block features a mechanism that eliminates duplicate continuous and discrete versions of the same block by basing the block configuration on the simulation mode. If your legacy models contain the dq0_to_abc Transformation block, they will continue to work. However, for best performance, use the dq0 to abc block in your new models.

## Description

The dq0_to_abc Transformation block performs the reverse of the so-called Park transformation, which is commonly used in three-phase electric machine models. It transforms three quantities (direct axis, quadratic axis, and zero-sequence components) expressed in a two-axis reference frame back to phase quantities. The following transformation is used:

$\begin{array}{c}{V}_{a}={V}_{d}\mathrm{sin}\left(\omega t\right)+{V}_{q}\mathrm{cos}\left(\omega t\right)+{V}_{0}\\ {V}_{b}={V}_{d}\mathrm{sin}\left(\omega t-2\pi /3\right)+{V}_{q}\mathrm{cos}\left(\omega t-2\pi /3\right)+{V}_{0}\\ {V}_{c}={V}_{d}\mathrm{sin}\left(\omega t+2\pi /3\right)+{V}_{q}\mathrm{cos}\left(\omega t+2\pi /3\right)+{V}_{0},\end{array}$

where ω = rotation speed (rad/s) of the rotating frame.

The transformation is the same for the case of a three-phase current; you simply replace the Va, Vb, Vc, Vd, Vq, and V0 variables with the Ia, Ib, Ic, Id, Iq, and I0 variables.

The dq0_to_abc Transformation block is used in the model of the Synchronous Machine block where the stator quantities are referred to the rotor. The Park transformation then eliminates time-varying inductances by referring the stator and rotor quantities to a fixed or rotating reference frame. The Id and Iq currents represent the two DC currents flowing in the two equivalent rotor windings (d winding on the same axis as the field winding, and q winding in quadratic) producing the same flux as the stator Ia, Ib, and Ic currents.

## Inputs and Outputs

`dq0`

Connect to the first input a vectorized signal containing the sequence components [d q 0] to be converted.

`sin_cos`

Connect to the second input a vectorized signal containing the [sin(ωt) cos(ωt)] values, where ω is the rotation speed of the reference frame.

`abc`

The output is a vectorized signal containing the three-phase sinusoidal quantities [phase A phase B phase C], in the same units as the `dq0` input signal.

## Example

See the example of the abc_to_dq0 Transformation block for an example using the dq0 to abc block (the improved version of the dq0_to_abc Transformation block).

## References

[1] Krause, P. C. Analysis of Electric Machinery. New York: McGraw-Hill, 1994, p.135.