Winding factor and MMF

Layout using the star of slot EMFs: slot k has electrical angle k·2πp/Q; the positive belt of phase j is centered at its current phase φ_j, the negative belt at φ_j + π, the belts splitting the circle evenly. Symmetric arrangement: φ_j = 360°·j/m; for an even m the negative belt of a phase is the positive belt of its opposite phase, so belts are 360°/m wide. Asymmetric arrangement: m/P systems of P phases (P the largest prime factor of m, a two-phase system being 0°, 90°) shifted by 180°/m, belts 180°/m wide. Double layer winding: an upper-layer side in slot k and its return side in the lower layer of slot k + y. Concentrated single layer: coils on the odd teeth, with phases assigned by coil EMF. With the currents at electrical angle ωt phase j carries cos(ωt − φ_j); the MMF is the sum of the slot currents along the bore, and its harmonic of order ν a wave that travels at 1/ν of the electrical speed, forwards or backwards: the working one (ν = p) with the rotor, the others against it or ahead of it, which is where their losses and noise come from. The winding factor of order ν is the current-loading harmonic of an m-phase system with balanced sinusoidal currents, normalized by the number of sides per phase; MMF of harmonic ν equals kw·p/ν. Flux barriers: a reluctance network of the stator. The bore is cut into pieces at slots and barriers, each with an air-gap reluctance inversely proportional to its width; the yoke behind a slot has no reluctance, behind a barrier r times that of the narrowest piece. Mesh fluxes around slots and barriers give the MMF across the air gap of each piece, and its harmonics the equivalent winding factor, which can exceed 1. A barrier given by its size: r = (w_b / h_b)·(πD / Q / 2) / g_eff, the reluctance of the air it puts across the yoke over that of the air gap under half a slot pitch, with no fringing.

Balanced sinusoidal currents; uniformly spaced slots; infinitesimally narrow slot openings; iron of infinite permeability (no saturation); flux barriers act only in the yoke, as a reluctance r.

Integer Q/(m·t) and equal numbers of coil sides in all phases (balance); single layer windings require an even number of sides per phase (even Q/m); a symmetric two-phase system is a single-phase one (use the asymmetric arrangement); a distributed single layer winding with an even number of phases needs the asymmetric arrangement; double layer coil span 1…Q−1; barriers behind every other tooth need an even number of slots.