Motor control Clarke（ α/β） Derivation of equal amplitude transformation

Motor control Clarke Derivation of equal amplitude transformation

Clarke The transformation is a three-phase plane coordinate system 0ABC To two-phase plane rectangular coordinate system 0 α β Transformation . Generally, we know that the coordinate system of three-phase current is the mutual difference of three-phase 120° The coordinate system of ：

In general, Kirchhoff's law :ia+ib+ic =0, We convert the three-phase plane coordinate system to the two-phase plane direct coordinate system as follows ：

From the picture ：

Further calculation ：

from ia+ib+ic =0, Make one step calculation to get ：

Further calculation shows that ：

Because what we do is equal amplitude transformation , So there must be Iα=Ia, So the equation has to do the following , Multiply the left side of the equation by one (2/3):

So in the formula （2/3） It is called transformation coefficient . Eventually The transformation formula is deduced as ：