The three-phase voltages, currents, and fluxes of AC motors can be analyzed in terms of complex space vectors. With regard to the currents, the space vector can be defined as follows:

• Assuming that i_a, i_b, and i_c are the instantaneous currents in the stator phases, then the complex stator current vector is defined in Equation 22.
  Equation 22. ${\overline{i}}_s=i_a+\alpha i_b+{\alpha }^2{i}_c$

  where

  • $\alpha =e^{j\frac{2}{3}\pi }$ and ${\alpha }^2=e^{j\frac{4}{3}\pi }$ represent the spatial operators

Figure 3-6 shows the stator current complex space vector.

Figure 3-6 Stator Current Space Vector and Component in (a, b, c) Frame

In Figure 3-6, (a, b, and c) are the three-phase system axes. This current space vector depicts the three-phase sinusoidal system. This still needs to be transformed into a two-time invariant coordinate system. This transformation can be split into two steps:

• Equation 23. $(a, b)\Rightarrow (\alpha ,\beta )$
  (Clarke transformation) which outputs a 2-coordinate time-variant system
• Equation 24. $\left(\alpha ,\beta \right)\Rightarrow (d, q)$
  (Park transformation) which outputs a 2-coordinate time-invariant system