The 3-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, i_c are the instantaneous currents in the stator phases, then the complex stator current vector is defined in Equation 17.

where

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

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

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

where

• a, b, and c are the three-phase system axes

This current space vector depicts the three-phase sinusoidal system which still needs to be transformed into a two time invariant co-ordinate system. This transformation can be split into two steps:

• $(\mathrm{a}, \mathrm{b})\Rightarrow (\alpha ,\beta )$ (Clarke transformation) which outputs a 2-coordinate time-variant system
• $\left(\alpha ,\beta \right)\Rightarrow (\mathrm{d},\mathrm{ }\mathrm{q})$ (Park transformation) which outputs a 2-coordinate time-invariant system