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.
“The space vector is not a mathematical trick. It is the machine’s own memory of what it is.”
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 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. Difference between machine types is merely a matter
1. The Inadequacy of the Single-Phase Gaze
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.
$$\vec{x}_s = \frac{2}{3} \left( x_a + a x_b + a^2 x_c \right)$$ It is the machine’s own memory of what it is
$$T_e = \frac{3}{2} p \cdot \text{Im} { \vec{\psi}_s \cdot \vec{i}_s^* } = \frac{3}{2} p (\vec{\psi}_s \times \vec{i}_s)$$
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.
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 Inadequacy of the Single-Phase Gaze The three-phase
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:
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.