How to calculate the winding factor of a given winding distribution

There is only one calculation. Draw every conductor of a phase as a vector in the electrical-angle plane, add the vectors, and divide by how many there are. Everything else — distribution factor, pitch factor, harmonic winding factors — falls out of that one sum. This article works the method end to end on two very different machines: a 12-slot / 8-pole tooth-coil winding and a 48-slot / 8-pole single-layer winding.

What a winding distribution actually is

A winding distribution is the complete answer to three questions, asked for every slot in the stator:

1. Which phase does the conductor in this slot belong to — A, B or C?
2. Which direction does its current flow — into the page (a go side, written A+) or out of it (a return side, A−)?
3. How many conductor sides does the slot hold — one (single-layer) or two (two-layer)?

Coils are then formed by pairing a go side with a return side. The number of slots between the two sides is the coil span y. But note the ordering: the distribution — the phase and polarity of every conductor — is the primary object. Coil span is downstream of it. This ordering matters a great deal for the single-layer example later.

Mechanical angle, electrical angle, and the slot angle

Slots are cut at equal mechanical intervals around the bore. With Q slots, the mechanical pitch between adjacent slots is:

But the rotor's flux wave does not care about geometry — it cares about poles. A rotor with P poles (p = P/2 pole pairs) completes p full magnetic cycles in one mechanical revolution. So one mechanical degree is worth p electrical degrees, and the slot angle — the electrical phase shift between the EMFs of two adjacent slots — is:

Slot n therefore sits at electrical angle θ_n = (n − 1) × α, taken modulo 360°. Two slots that are physically far apart can share the same electrical angle; two slots that are neighbours can be almost in antiphase. The electrical angle is the only thing the winding factor sees.

Two derived quantities follow immediately and are used throughout:

The vector rule

As the rotor turns, the fundamental flux wave sweeps past the stator and induces in the conductor of slot n an EMF of the form Ê·sin(ωt − θ_n). Every conductor produces the same amplitude; only the phase differs. That is exactly a phasor, and phasors add as vectors.

Assign to each conductor i of the phase a unit vector:

The sign c_i = −1 is a physical current reversal, and for the fundamental it is equivalent to rotating the vector by 180°. It is convenient to fold the sign into the angle and speak of an effective angle θ_i^eff = θ_i + 180° for return sides. All the tables below list that effective angle.

Now sum, and normalise by the number of conductors:

The denominator is the EMF you would get if every conductor of the phase were somehow at the same electrical angle — the theoretical maximum. The numerator is what the winding actually delivers. Their ratio is the winding factor, and it is bounded by 0 ≤ k_w ≤ 1 by construction. No memorised formula is required, and the rule works for integer-slot, fractional-slot, single-layer, two-layer, chorded, concentric and unbalanced windings alike.

Where k_d and k_p come from

The familiar distribution factor and pitch factor are not independent physics. They are what you get when you evaluate the same sum in two stages instead of one.

Stage 1 — add within each coil. Group the conductors into coils. A coil is one go side and one return side, so its vector is the sum of two unit vectors. If the coil spans β electrical degrees, elementary trigonometry gives its magnitude as 2·sin(β/2). Divide by 2, the arithmetic sum of the two sides, and you have the pitch factor:

Stage 2 — add the coil vectors. The coils of one phase are generally not all at the same angle. Their vector sum divided by their arithmetic sum is the distribution factor:

Multiply the two stages back together and you recover the original conductor sum exactly:

This is a factorisation, not a decomposition into separate effects. As the 48-slot example will show, the same physical winding can be grouped into coils in two different ways that give completely different k_d and k_p values — and identical k_w. The vector sum is the invariant; the split is bookkeeping.

Harmonics come free

For the ν^th harmonic of the air-gap field, the flux wave has ν times as many cycles per revolution, so every slot's phase shift scales by ν. The conductor polarity c_i is a wiring direction and does not scale. Hence:

Example 1 — 12 slots, 8 poles, two-layer (q = 0.5)

This is the archetypal fractional-slot concentrated winding (FSCW): every tooth carries its own coil, coil span y = 1 slot, two coil sides per slot. It is the winding of choice for hub motors, drone motors and many servo motors, because the end-turns are short and the coils can be machine-wound onto segmented teeth.

Slot electrical angles

With α = 120°, the twelve slots collapse onto just three distinct electrical directions:

Three directions, 120° apart, is precisely a three-phase set. That is what makes 12s/8p work: the slot star has exactly three spokes, and each phase takes one spoke. Since GCD(12, 8) = 4, the machine is four identical repetitions of a 3-slot / 2-pole base unit.

Phase allocation

Assign coils cyclically — coil n is wound around tooth n with its go side in slot n and its return side in slot n+1:

Phase A therefore owns four coils: 1→2, 4→5, 7→8 and 10→11 — eight conductor sides in total.

The vector table

Cross-check against the classical factors

Group the eight conductors into their four coils. Each coil spans β = y·α = 1 × 120° = 120°, so each coil vector has magnitude 2·sin(60°) = 1.7321 and lies at −30°. All four coil vectors are identical:

The fractional-slot form of the distribution factor agrees. Writing q = z/n in lowest terms (here q = 1/2, so z = 1, n = 2), the standard result k_d1 = sin(30°) / (z · sin(30°/z)) gives sin(30°)/sin(30°) = 1.000.

Harmonic content

Re-running the sum with θ → νθ exposes the well-known weakness of q = 0.5 windings:

Every odd non-triplen harmonic has the same winding factor as the fundamental. With only three distinct slot directions there is simply nothing left to cancel with. This is why FSCW machines carry large 5th and 7th MMF harmonics and why rotor magnet eddy-current loss is a first-order design concern in this topology — a cost the designer accepts in exchange for very short end-turns and simple manufacture.

Example 2 — 48 slots, 8 poles, single-layer (q = 2)

Now the opposite end of the design space: a classical distributed winding of the kind found in industrial induction motors and traction PMSMs.

Why there is only one single-layer winding here

In a two-layer winding you have a free parameter: the bottom layer can be shifted relative to the top by any number of slots, which is exactly what chording is. A single-layer winding has no second layer. Each slot holds one coil side, and that side's phase and polarity are fixed the moment you lay down the standard 60° phase belts. There is no shift to make.

What remains free is only how you pair the fixed go sides with the fixed return sides into coils. That choice changes the shape of the end-windings — and nothing else. The set of vectors being summed is identical either way, so k_w is identical either way. For Q = 48, P = 8 the two standard realisations are:

• Full-pitch (lap / "chain") realisation: every coil spans 6 slots.
• Concentric realisation: the two coils of each group are nested, spanning 7 and 5 slots. This is the usual choice in practice because concentric coils of different span can be inserted into the slots in one pass.

Both are the same winding. Presenting "48s/8p single-layer with 5/6 chording" as an option would be wrong — there is nothing to chord.

Phase allocation

With q = 2, each 60° phase belt occupies 2 consecutive slots, and the belt sequence over one electrical period (12 slots) is A+, C−, B+, A−, C+, B−:

This 12-slot block repeats four times, so phase A occupies:

The vector table

Because the pattern repeats every 12 slots (= 360° electrical), slots 1, 13, 25 and 37 all sit at 0°; slots 2, 14, 26 and 38 all sit at 30°. The return sides land on the same two directions once their sign is folded in: slot 7 is at 180°, and 180° + 180° = 0°.

All sixteen conductors reduce to eight vectors at 0° and eight at 30° — the two-slot phase belt is the only source of angular spread in the whole winding.

The same k_w, two different k_d / k_p splits

Here is the point that a memorised-formula approach obscures. Group the identical set of sixteen conductors into coils the two legitimate ways, and the classical factors come out completely differently — while their product does not move:

In the full-pitch grouping each coil is perfectly pitched (k_p = 1) but the two coils of a group sit 30° apart, so the whole penalty appears as distribution. In the concentric grouping the two coils are individually mis-pitched — one long by 30°, one short by 30°, and sin(105°) = sin(75°) = 0.9659 — but they end up perfectly aligned at 15°, so the whole penalty appears as pitch. The physics never changed. Only the accounting did.

Cross-check against the classical formula

Harmonic content

The 5th and 7th are attenuated by a factor of 3.7 relative to the fundamental — the payoff of distributing the phase over two slots. The 11th and 13th, however, are slot harmonics: at ν = 11 and 13 the harmonic slot angle να is congruent to ±α modulo 360°, so the vectors re-align and k_wν returns to 0.9659. Slot harmonics cannot be suppressed by distribution at all; skewing or a two-layer chorded design is the usual remedy.

Verification

Every number in this article was produced by direct vector summation in Python and then checked three ways: against the closed-form k_d = sin(qα/2)/(q·sin(α/2)) and k_p = sin(β/2) formulas, against the fractional-slot form of k_d for q = 1/2, and against the star-of-slots engine behind the Winding Analysis tool. All three routes return the same values:

No discrepancies were found. The vector sums are exact: √48/8 = √3/2 = 0.866025 for the 12-slot machine, and cos(15°) = 0.965926 for the 48-slot machine.

The method in five lines

1. Compute α = (P/2) × (360°/Q) and give slot n the angle (n−1)α mod 360°.
2. Write down the phase and polarity of every conductor side — this is the winding distribution.
3. Turn each conductor into a unit vector at angle θ_n, negated if it is a return side.
4. Add the vectors for one phase; divide the resultant magnitude by the conductor count.
5. For harmonic ν, repeat step 3 with νθ_n and keep the signs unchanged.

The distribution and pitch factors are worth knowing because they are how the literature talks, and because they tell you where a penalty comes from. But they are consequences of the vector sum, not inputs to it. When a winding is unusual — fractional q, concentric coils, unequal turns per coil, an unbalanced repair winding — the formulas run out of road and the vector sum does not.