Concentrated Winding Calculator: Layout, Winding Factor & MMF

The layout uses the star of slot EMFs, with common winding-factor and MMF routines from em.windings. Slot k (zero based) is at mechanical angle γₖ = 2πk/Q and electrical angle pγₖ, with p = P/2. Equal-turn coils span slots k and k+1. Every-tooth double layer assigns the start-side slot EMF to a phase belt and places the opposite return side in the next slot. Alternate-tooth single layer assigns the complete coil EMF to the nearest signed belt. Belt boundaries use a deterministic tie convention. Symmetric even-phase systems additionally split identical signed phasor groups between opposite phase axes, reversing the coil when changing to the opposite axis. Each half receives the same oriented phasor multiset; this admits the published symmetric six-phase 6N4P layout ACEdfb (Zheng et al., attached p. 4, Table 1). Caruso et al. (2018), attached pp. 2–7 and 13–14, describe the symmetry, star/WDT construction and coil connections; this tool uses star assignment, not their complete optimization procedure.

For phase A and mechanical harmonic ν,

The sum includes both active sides; σ is their current direction. At zero pitch factor both the pitch and winding factors vanish; the harmonic table prints distribution factor 0 as a plotting convention, not an independently defined ratio. The working order is ν=p. Fundamental phase phasors, normalized by their prescribed current axes, must coincide; coil counts must also match.

Balanced-current loading and MMF spectra use the library's sums over all phases, including cancellation of non-excited harmonics. The plotted MMF amplitude is k_loading,ν·p/ν, so the working value equals the working factor for a balanced multiphase winding. For one phase, the sum of the two counterrotating components is divided by two in the normalized spectrum. The spatial waveform is the cumulative slot current with its circumference mean removed, in coil-side × current units per turn. Static 0°/90° quadratures are cached and combined as F(γ,ωt)=F₀ cos(ωt)+F₉₀ sin(ωt). The bore contour is a normalized schematic of the working MMF, not air-gap flux density.

The coincident-star count is gcd(Q,p). The separate MMF periodicity is the GCD of nonzero balanced-current spatial orders, evaluated with the common winding-factor routine. Cogging order N_c=lcm(Q,P); Zhu and Howe (2000), attached pp. 3 and 6, give the slot/pole indicator C=QP/N_c and uniform continuous-skew attenuation

with β in mechanical radians and the zero-skew limit 1. Use ν=N_c for fundamental cogging, ν=p for working EMF. The first ideal cogging null is β=2π/N_c. C equals gcd(Q,P); a larger C generally worsens cogging only for otherwise comparable machines. Neither indicator predicts N·m.

Each physical reference coil Sₖ→Fₖ carries current out of the page in slot k and into the page in slot k+1 (slot Q+1 wraps to 1). Negative letters reverse the traversal. Parallel paths split every identical oriented coil phasor group evenly; the GCD of group multiplicities gives the admissible divisors. Series-chain order is increasing tooth number within each branch. Each symbol in the connection drawing represents the complete phase, with branch details in the schedule.

For symmetric even phase counts m≥4, let n be the number of tooth coils: Q for double layer, Q/2 for alternate-tooth single layer. Equal opposite-axis sets require n/m to be integral and [n/gcd(n,P)]/(m/2) to be integral. The generated coil counts and actual fundamental phase phasors are then checked explicitly. Other arrangements retain the library's star-balance rules. The factors q=Q/(Pm) always use the actual number of phases; some six-phase tables use a three-phase SPP normalization instead.

With currents iⱼ/I_peak=cos(ωt−φⱼ), Kirchhoff's current law requires a star return current i_N=Σiⱼ, positive from winding common point N toward the drive. Its peak ratio is |Σexp(−iφⱼ)|. Only when this phasor sum vanishes may N float for the prescribed currents. Otherwise the drawing connects N to a controlled drive return; independent phase bridges are an alternative. This check is over the entire electrical cycle, not only the current animation angle.

The star diagram draws compact group IDs G1, G2, … in order of first slot occurrence. The lookup table lists each group's raw electrical angle and every coincident slot. More than 24 distinct groups use the table without perimeter labels, so dense stars remain readable. No long slot list is drawn around the circle.

Uniformly spaced radial-flux slots; one-slot, equal-turn tooth coils; ideal linear iron, no flux barriers or skew-induced axial leakage. Sinusoidal currents have equal peak magnitudes and the listed phase angles. All wound teeth have the entered turn count. A floating star is assumed only when the phase-current phasor sum is zero. A nonzero sum requires the displayed controlled neutral return, or independent phase bridges; this common-mode current is part of the imposed-current model. For delta, additional circulating zero-sequence currents driven by back-EMF harmonics are outside this model.

Identical alternating magnets have half-wave symmetry. Equal oriented fundamental coil phasors then also match the odd rotor-EMF harmonics in the parallel branches. Manufacturing asymmetry, even EMF harmonics, unequal wire resistance and unequal leakage are excluded, so the parallel-path multiplicity is an ideal electrical option, not approval of a physical parallel connection.

The star-of-slots layout, MMF and skew integral are implemented from electromagnetic phasor sums and the cited method; the explicit S/F lead convention, phasor balance residual and conservative phasor-multiset branch allocation are derived here. No torque or loss correlation is fitted to the reference cases.

Supports 3–96 slots, even total pole counts 2–192, and 1–12 phases, with every-tooth double layer or alternate-tooth single layer. All coils span one slot. Invalid equal-occupancy, single-layer, phase-axis or zero-working-pitch combinations are blocked with input messages. Empty-slot optimization, unequal turns, distributed/variable-span coils and arbitrary user-edited layouts are outside this tool.

For even phase counts, equally spaced currents use 360°/m with opposite-axis coil sharing; the multiple-system option follows the library's prime-factor grouping. Two quadrature phases use 0°/90°. The six-phase multiple-system arrangement uses 0°,30°,120°,150°,240°,270° (A,C,E form one star; B,D,F form the other); symmetric six-phase currents use 0°,60°,120°,180°,240°,300°. Phase names A…L are used throughout. A single phase gives a pulsating, not a self-starting rotating field. A symmetric two-phase 0°/180° choice is rejected.

Overpitch one-slot coils, such as 12N14P, are intentional in the concentrated-winding family and are not rejected merely because one slot exceeds rotor pole pitch. The tool exposes the numeric coil schedule rather than the library's generic layout warning for this normal tooth-coil case.

The waveform and stator animation are unskewed; skew affects the scalar EMF result and attenuation chart only. Harmonics are plotted from mechanical order 1 through 2Q, not an infinite-spectrum THD. At high slot counts tooth text is omitted from the schematic; use the complete schedule. Continuous skew assumes identical axial slices and does not model stepped skew, eccentricity or end effects.

Cogging exists with no winding current and depends strongly on geometry. Changing star/delta or rewinding the same iron does not change the ideal slot/pole cogging indicator. A predicted skew null cancels the ideal periodic terms; imperfections can leave residual cogging. Phase-bus connections are given for star (all phase counts) and delta (three phases only). The drive must supply the listed phase currents and provide the computed sum current whenever it is nonzero. In that case N must connect to a suitable controlled drive return (for example an actively controlled neutral leg), or the windings must be fed by independent phase bridges. An unconnected floating star cannot realize those currents; N is a drive winding connection, not protective earth. Check physical coil polarity and open-circuit branch voltages before connecting a rewind.