## Park Transformation

The Park transformation used in Simscape™ Electrical™ Specialized Power Systems models and functions corresponds to the definition provided in [1].

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 following reverse transformation is used:

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

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

The transformations are 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.

### References

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