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

The space vector theory, first crystallized by Kovacs and Racz in the 1950s and later refined by Depenbrock, Leonhard, and Vas, offers not merely an alternative method but the canonical language for electromechanical energy conversion in polyphase systems.

$$T_e = \frac{3}{2} p \cdot \text{Im} { \vec{\psi}_s \cdot \vec{i}_s^* } = \frac{3}{2} p (\vec{\psi}_s \times \vec{i}_s)$$

“The space vector is not a mathematical trick. It is the machine’s own memory of what it is.” The space vector theory, first crystallized by Kovacs

Difference between machine types is merely a matter of flux generation: $\vec{\psi}_s = L_s \vec{i}_s$ (IM), $\vec{\psi}_s = L_s \vec{i} s + \vec{\psi} {PM}$ (PMSM), or $\vec{\psi}_s = L_s \vec{i}_s + L_m \vec{i}_r'$ (DFIM). The drive —the control algorithm—does not need to know the difference beyond the flux linkage map.

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: The drive —the control algorithm—does not need to

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

This monograph does not seek to replace the classic texts of Fitzgerald, Leonhard, or Novotny & Lipo. Rather, it aims to re-center the student and practitioner onto the structural invariant : the rotating space vector is the real physical quantity; the three phase windings are merely its projection sensors. From this vantage point, electrical drives become a branch of applied vector calculus, not a catalog of special cases. The torque expression unifies as: This monograph does

$$\vec{x}_s = \frac{2}{3} \left( x_a + a x_b + a^2 x_c \right)$$

When coupled to a voltage-source inverter, the space vector approach reveals the finite set of discrete stator voltage vectors ($V_0$ to $V_7$). The machine’s response—current trajectory, torque ripple, flux drift—is simply the integral of: