Perform Park transformation from three-phase (abc) reference frame to dq0 reference frame

powerlib_extras/Measurements, powerlib_extras/Discrete Measurements

The Transformations section of the Control and Measurements library contains the abc to dq0 block. This is an improved version of the abc_to_dq0 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 abc_to_dq0 Transformation block, they continue to work. However, for best performance, use the abc to dq0 block in your new models.

The abc_to_dq0 Transformation block computes the direct axis, quadratic axis, and zero sequence quantities in a two-axis rotating reference frame for a three-phase sinusoidal signal. The following transformation is used:

$$\begin{array}{c}{V}_{d}=\frac{2}{3}\left({V}_{a}\mathrm{sin}(\omega t)+{V}_{b}\mathrm{sin}(\omega t-2\pi /3)+{V}_{c}\mathrm{sin}(\omega t+2\pi /3)\right)\\ {V}_{q}=\frac{2}{3}\left({V}_{a}\mathrm{cos}(\omega t)+{V}_{b}\mathrm{cos}(\omega t-2\pi /3)+{V}_{c}\mathrm{cos}(\omega t+2\pi /3)\right)\\ {V}_{0}=\frac{1}{3}\left({V}_{a}+{V}_{b}+{V}_{c}\right),\end{array}$$

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 *V _{a}*,

This transformation is commonly used in three-phase electric
machine models, where it is known as a Park transformation [1]. It allows you to
eliminate time-varying inductances by referring the stator and rotor
quantities to a fixed or rotating reference frame. In the case of
a synchronous machine, the stator quantities are referred to the rotor.
I_{d} and I_{q} represent
the two DC currents flowing in the two equivalent rotor windings (d
winding directly on the same axis as the field winding, and q winding
on the quadratic axis), producing the same flux as the stator I_{a},
I_{b}, and I_{c} currents.

You can use this block in a control system to measure the positive-sequence
component *V*_{1} of a set of
three-phase voltages or currents. The *V _{d}* and

You can use the Math Function block and the Trigonometric Function
block to obtain the modulus and angle of *V*_{1}:

$$\begin{array}{c}\left|{V}_{1}\right|=\sqrt{{V}_{q}^{2}+{V}_{d}^{2}}\\ \angle {V}_{1}=\mathrm{atan}2\left({V}_{q}/{V}_{d}\right).\end{array}$$

This measurement system does not introduce any delay, but, unlike the Fourier analysis done in the Sequence Analyzer block, it is sensitive to harmonics and imbalances.

`abc`

Connect to the first input the vectorized sinusoidal phase signal to be converted [phase A phase B phase C].

`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.

`dq0`

The output is a vectorized signal containing the three sequence components [d q o], in the same units as the

`abc`

input signal.

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

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