Winding layout calculator

Winding scheme

The scheme comes from the star of slot EMFs with 60° phase belts, as in the original calculator. The electrical angle between neighbouring slots is $\alpha = 180° \cdot 2p / Q$. Walking along the teeth (tooth-coil windings, $q = Q/(3 \cdot 2p) < 1$) or the slots ($q \ge 1$), the accumulated angle picks the belt: A (−30…30°), b (30…90°), C (90…150°), a (150…210°), B (210…270°) and c (270…330°). Upper case is a forward coil side, lower case a reversed one. A single-layer tooth-coil winding leaves every other tooth unwound (–).

The scheme is then normalised like the original: trailing A/a coils move to the front, it starts with A, phase B comes before C, and a distributed winding does not split a phase belt at its ends. In a distributed double layer the return layer is the slot pattern shifted by the shortening $s$. Each coil runs from slot $k$ to slot $k + y$, with $y = Q/2p - s$. A single layer with even $q$ uses concentric coils.

Winding factors

The balanced phase currents are $i_A = \cos\omega t$, $i_B = \cos(\omega t + 120°)$ and $i_C = \cos(\omega t - 120°)$. From them follow the slot currents $c_x$; a slot with two coil sides carries their mean. The winding factor of order $\nu$ (pole pairs) is

evaluated at $\omega t = 0$. The winding counts as balanced when the spectrum is the same at $\omega t = 108°$ (the original compares two instants) and every phase has the same numbers of forward and reversed coil sides.

The optional corrections are the ones of the original:

• slot opening: $k_{o,\nu} = \sin(\nu b_0/D)/(\nu b_0/D)$
• skew of $s_k$ slot pitches: $k_{sk,\nu} = \sin(\pi s_k \nu/Q)/(\pi s_k \nu/Q)$

Cogging steps per revolution: $\mathrm{LCM}(Q, 2p)$.

Coil connections

These are the rules of the original. The number of symmetry groups is $sym = \gcd(Q, 2p)$ for two layers, or $\gcd(\gcd(Q, 2p), Q/6)$ for one layer. Each group holds $(Q/3)/sym \cdot (layers/2)$ coils per phase. The allowed connections follow from that:

• one group: series only
• one coil per group: any connection
• otherwise odd $p$: series/antiparallel; even $p$: series/parallel

The branch count must divide $sym$, and a branch holds whole consecutive groups. A parallel arrangement is offered only when the groups hold equivalent coils (positions, spans, consistent polarity, non-zero EMF) and the oriented branch EMFs are equal. Groups of opposite polarity are traversed E → S. Coil labels carry their sign: lower case means connected reversed.

MMF and field

The MMF is the staircase of cumulative slot currents with its mean removed. Its harmonic of order $n$ is $F_n(\theta) = (A_n \sin n\theta - B_n \cos n\theta)/n$, where $A_n = \frac{1}{\pi}\sum c_k \cos n\theta_k$ and $B_n = \frac{1}{\pi}\sum c_k \sin n\theta_k$. Its crest lies at $n\theta = \operatorname{atan2}(B_n, A_n) + 90°$.

The speed of a harmonic relative to the working field is the drift of its crest as $\omega t$ advances, $(\Delta(n\theta)/n) / (\Delta(p\theta)/p)$. It equals $\pm p/n$, where the sign gives the direction, and 0 when the harmonic is not excited.

Animated views (ωt)

• Star of slot EMFs: one phasor per tooth coil (at its tooth) or per slot (layer 2 of a distributed winding), at the electrical angle of its position. The dash-dot lines are the magnetic axes of the phases: the crest of the working wave when only that phase carries current.

The coloured arrows lie along these axes, with a length equal to the instantaneous phase current. Their vector sum, the black arrow, is the field vector. It has constant length and turns with ωt, and its direction is the crest of the working wave.

• Circular view: the arrows mark the crests of the working wave. The rotor with 2p magnets turns with them, its north poles 90° (el.) behind the crests.
• MMF chart: the staircase, the working harmonic and a harmonic of your choice. With ν = 0 that is the one with the largest amplitude $k_{w,\nu}/\nu$ apart from the working one.

• Three-phase windings only, as in the original.
• Winding factors treat the conductors as concentrated at the slot centre. The slot opening and the skew enter only through the sin x/x factors above.
• All coils have the same number of turns. The original's turns-per-coil editor for its factor animation is left out; the chart over one current period replaces the animation for equal turns.
• The original's manual scheme entry is left out, because this builder has no text inputs. Schemes come from the slot and pole numbers only.
• Coil models of distributed windings need an integer pole pitch. Distributed windings with a fractional q (for example 36 slots / 8 poles) give the scheme and the winding factors only, as in the original.
• The rotor in the circular view is a sketch, not a field calculation. Its poles are placed 90° (el.) behind the stator field, which is the position for a current in the q axis (maximum torque per ampere).
• These are additions of this port: the MMF chart and its harmonics, the star of slot EMFs with the current vectors, the field arrows, the rotor, the outer-stator circular view and the connection schematic. The original instead draws the wiring of every coil end and offers inner/outer stator views and image export.
• The algorithms were transferred from the original JavaScript and are not derived here. The reference cases check the transfer against the original calculator, its README and classical distribution and pitch factors.

• The number of slots must be divisible by 3, the pole number must be even, and the two must differ.
• One layer is offered only for an even slot number. Coil shortening applies only to distributed double-layer windings and must stay below the pole pitch.
• For combinations without a balanced three-phase winding, the scheme and factors are still shown, with a warning.