I am trying to understand Park's Transform also known as the dq0 transform. Although there are people who argue that they are not the same but in principle.

Anyway, Park's transform:

\(

v_{d} = \frac{2}{3}\cdot (i_{a} \cdot cos(\theta) + i_{b}\cdot cos(\theta-120) + i_{c}\cdot cos(\theta+120))\\

v_{q} = -\frac{2}{3}\cdot (i_{a} \cdot sin(\theta) + i_{b}\cdot sin(\theta-120) + i_{c}\cdot sin(\theta+120))\\

v_{0} = \frac{1}{3}\cdot (i_{a} + i_{b} + i_{c})\\

\)

I understand where the cosine and sines come from but not where the factors \(\frac{2}{3}\\) and \(\frac{1}{3}\\) come from.

Is there someone who knows?

The reason for my question is that I can not find the original explanation paper which is by Park out of 1928.