Winding distribution factor: why spreading conductors costs EMF

Distribution factor k_d measures the penalty for placing a phase's conductors in several slots rather than one. The conductors generate slightly out-of-phase EMFs that add as phasors — their vector sum is always less than their arithmetic sum, and k_d is the ratio between the two.

The phase belt: how slots are assigned to a phase

In a three-phase AC machine, the stator slots are divided among three phases — A, B, and C. Under each rotor pole, each phase occupies a phase belt: a group of consecutive slots whose conductors are all connected in series for that phase.

The number of slots in each phase belt is called slots per pole per phase, q:

When q = 1, every slot of a phase belt contains conductors from only one phase, and they are all at the same electrical angle — a concentrated winding. When q > 1, the phase belt spans q consecutive slots, each at a slightly different electrical angle — a distributed winding. Almost all large AC machines use distributed windings.

Electrical angle per slot and the phase belt spread

Every slot carries an EMF phasor whose phase angle is set by the slot's electrical position. Adjacent slots differ by the electrical angle per slot α:

The q slots of a phase belt are consecutive, so their EMF phasors span a total arc of (q − 1) × α in electrical degrees. The first slot of the belt is the reference at 0°; the last slot is at (q − 1) × α. Each slot's phasor is offset from its neighbour by exactly α.

Why phasor addition gives less than arithmetic addition

If both conductors of phase A's belt were in the same slot, their EMFs would be perfectly in phase and their series sum would equal 2 × E₁ — the arithmetic total. But they are not in the same slot; one is at 0° and the other at 30°. Adding two phasors of equal magnitude E₁ separated by 30°:

The 3.4% reduction is the price of distribution. In exchange, the winding benefits from lower MMF harmonics, shorter end-turns per conductor, and a more sinusoidal air-gap flux — advantages that far outweigh the small EMF penalty in most machine designs.

The distribution factor formula

The general result for q equally spaced phasors, each separated by α, follows from the geometry of a regular polygon inscribed in a circle. The vector sum of q unit phasors with step angle α equals:

The arithmetic sum of q unit phasors is simply q. Dividing gives the distribution factor:

Note that when q = 1, sin(α/2) / (1 · sin(α/2)) = 1 exactly — a concentrated winding has no distribution penalty. As q increases, k_d falls, but slowly: doubling q from 1 to 2 only reduces k_d from 1.000 to around 0.966 for typical slot/pole ratios.

Worked example: three machines, same pole count

Consider three 4-pole, 3-phase machines with different slot counts. All share P=4, m=3, so α = 2 × (360°/Q). Increasing Q increases q and distributes the winding more finely.

Notice that q·α/2 always equals 30° for these 4-pole, 3-phase machines regardless of slot count — this is because the phase belt always spans exactly 60° electrical (one sixth of the electrical cycle) when m=3. The numerator sin(30°) = 0.5 is constant; it is the denominator that grows as q increases and α decreases.

The distribution factor flattens quickly. Going from q=2 to q=∞ only costs an additional 1.1% in EMF. This is why practical machines rarely need more than q=3 or q=4 to achieve a near-ideal sinusoidal MMF distribution.

Distribution factor for harmonics

The distribution factor applies independently to every spatial harmonic of the air-gap MMF. For the ν^th harmonic, the electrical angle between adjacent slots is ν·α rather than α, so:

Higher harmonics have larger effective slot angles, which means larger angular spread across the phase belt, which means stronger phasor cancellation. In practice, k_d is much smaller for the 5th and 7th harmonics than for the fundamental:

The 5th harmonic distribution factor is 0.217 compared to 0.960 for the fundamental — a suppression ratio of about 4.4:1 from distribution alone, before any pitch factor contribution. This strong harmonic attenuation is the primary reason distributed windings dominate in machines where torque quality matters.

Distribution factor as a measure of winding quality

The distribution factor can be understood geometrically as the ratio of the chord to the arithmetic sum of the phasor lengths. Each phasor has the same length E₁. If all conductors were in one slot (q=1), all phasors would point in the same direction and their series sum would equal q×E₁ — the arithmetic sum. When conductors are spread across q slots, the phasors fan out slightly and their vector resultant — the chord from the start of the first phasor to the tip of the last — is always shorter than the arithmetic sum. k_d is the ratio between the two.

The diagram below shows why k_d stays close to 1 even for q=3: in a real machine with α=20°, the phasors barely deviate from vertical. The chord is only 5 units shorter than 3×42=126 — a 4% difference, not a dramatic one.

A higher k_d means more efficient use of copper — each conductor contributes more to the useful phase EMF. A lower k_d means more conductor material is needed to achieve the same voltage. For this reason, winding designers do not chase very high q values: the MMF harmonic benefit plateaus quickly while the manufacturing cost and slot count keep rising.

Relationship to winding factor

The distribution factor combines with the pitch factor k_p to give the overall winding factor k_w:

Both factors are less than or equal to 1, and both reduce the useful EMF relative to a hypothetical ideal winding. In a full-pitch concentrated winding (q=1, y=y_fp), both equal 1 and k_w=1. Every real distributed and/or chorded winding has k_w<1, and the winding factor directly scales the back-EMF constant, the torque constant, and the air-gap flux linkage.