Space vector representation of three phase signals in stationary and rotating frames

This demonstration illustrates the use of a complex space vector to represent a three-phase signal . It also shows the transformation of the 3-phase signal 'ABC' into an equivalent 2-phase system 'alpha_beta'. The only restriction to the 3-phase signal is that the zero-sequence component is zero i.e. fA+fB+fC=0.
The complex space vector in the stationary frame is defined as
Fs = 2/3 (fA + fB*exp(j2*pi/3) + fC*exp(-2j*pi/3)
whose cartesian components are
fa = Re (Fs)
fb = Im (Fs)
When expressed in a rotating frame at frequency wk, the space vector becomes
Fk = Fs*exp(-jwk*t)
whose cartesian components are
fd = Re(Fk)
fq = Im(Fk)
The inverse transformation from complex space vector back to the 3-phase signal is also demonstrated.
Finally, the real transformations of the ABC components to the components ab (in the stationary frame) and then dq (in the rotating frame) are shown.