Electrical Machines And Drives A Space Vector Theory Approach Monographs In Electrical And Electronic Engineering Review

Let a three-phase system (voltages, currents, flux linkages) be represented by a single complex time-varying vector in a stationary two-dimensional plane (the $\alpha\beta$-plane). For a set of phase quantities $x_a, x_b, x_c$ satisfying $x_a + x_b + x_c = 0$, the space vector is defined as:

$$\vec{v}_s = R_s \vec{i}_s + \frac{d\vec{\psi}_s}{dt} + j \omega_k \vec{\psi}_s$$ Let a three-phase system (voltages, currents, flux linkages)

The three-phase machine is one entity. Its state is a rotating complex number. Unbalance, harmonics, and switching states (inverters) become geometric loci, not case-by-case trigonometric expansions. The factor $2/3$ ensures that the magnitude of

“The space vector is not a mathematical trick. It is the machine’s own memory of what it is.” synchronous). The torque expression unifies as:

where $a = e^{j2\pi/3}$. The factor $2/3$ ensures that the magnitude of $\vec{x}_s$ equals the peak amplitude of a balanced sinusoidal phase quantity.

where $\omega_k$ is the speed of the chosen reference frame (stationary, rotor, synchronous). The torque expression unifies as: