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

$$\frac{d\vec{\psi}_s}{dt} = \vec{v}_s - R_s \vec{i}_s$$

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. For a set of phase quantities $x_a, x_b,

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: No sinewave mythology required

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. No sinewave mythology required.

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

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

The art of modern drive control (field-oriented control, direct torque control, model predictive control) reduces to selecting, in real time, the inverter switching state that minimizes a cost function of the flux and torque errors. No sinewave mythology required.