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:

$\begin{matrix} V_{a} = V_{d} \sin (ω t) + V_{q} \cos (ω t) + V_{0} \\ V_{b} = V_{d} \sin (ω t - 2 π / 3) + V_{q} \cos (ω t - 2 π / 3) + V_{0} \\ V_{c} = V_{d} \sin (ω t + 2 π / 3) + V_{q} \cos (ω t + 2 π / 3) + V_{0}, \end{matrix}$

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

The following reverse transformation is used:

$\begin{matrix} V_{d} = \frac{2}{3} (V_{a} \sin (ω t) + V_{b} \sin (ω t - 2 π / 3) + V_{c} \sin (ω t + 2 π / 3)) \\ V_{q} = \frac{2}{3} (V_{a} \cos (ω t) + V_{b} \cos (ω t - 2 π / 3) + V_{c} \cos (ω t + 2 π / 3)) \\ V_{0} = \frac{1}{3} (V_{a} + V_{b} + V_{c}), \end{matrix}$

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 V_a, V_b, V_c, V_d, V_q, and V₀ variables with the I_a, I_b, I_c, I_d, I_q, and I₀ variables.

References

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