AC windings and the rotating magnetic field

Induction and synchronous machines, which together convert most of the world's electrical energy, share the same stator: a laminated steel cylinder with slots holding a three-phase winding. When balanced three-phase currents flow in that winding, they produce a magnetic field of constant strength that rotates smoothly around the air gap, without any moving parts. This rotating field is the foundation for Lessons 7 to 11. This lesson explains how AC windings are arranged in slots, why the field rotates and at what speed, how the distribution and pitch of the coils shape the field and the induced voltage, why harmonics arise and how winding design suppresses them, and what happens when the supply is unbalanced or a phase is lost.

Poles, electrical degrees and synchronous speed

A machine with \(p\) poles has \(p/2\) pairs of north and south poles around the air gap. One pole pair corresponds to one full cycle of the field pattern, so it is convenient to measure positions in electrical degrees:

In a 4-pole machine, one mechanical revolution is 720 electrical degrees. A field that advances 360 electrical degrees each cycle of the supply turns \(2/p\) of a revolution per cycle, so its speed, the synchronous speed, is

Distributed windings

If each phase had a single coil, the air-gap flux density produced by it would be a rectangular wave, rich in harmonics. Practical AC windings instead spread each phase over several slots under each pole. The number of slots per pole per phase is

where \(S\) is the number of slots and \(m\) the number of phases. A 36-slot, 4-pole, three-phase stator has \(q = 3\). Adjacent slots are separated by the slot angle \(\gamma = 180° \times p/S\) electrical, here 20°. The MMF of the coils in adjacent slots is shifted by \(\gamma\), so the step-shaped MMF of each phase approximates a sine wave much better than a single coil would. The cost is that the EMFs of the coils in a group are not quite in phase, so their phasor sum is slightly less than their arithmetic sum. The ratio is the distribution factor:

For \(q = 3\) and \(\gamma = 20°\), \(k_d = \sin 30° / (3 \sin 10°) = 0.960\).

Short pitching

A full-pitch coil spans exactly one pole pitch, 180 electrical degrees, so its two sides sit under opposite poles and their EMFs add directly. Most windings are short-pitched (chorded): each coil spans less than a pole pitch, typically 5/6 or 7/9 of it. The EMFs of the two coil sides then add with a small phase difference, giving a pitch factor

where \(\rho\) is the coil span as a fraction of the pole pitch. Short pitching reduces the fundamental only slightly but suppresses selected harmonics strongly, and it shortens the end windings, saving copper and reducing leakage. The winding factor combines both effects, \(k_w = k_p k_d\), and the EMF per phase is

where \(N\) is the number of series turns per phase and \(\Phi\) the flux per pole, as in the synchronous machine of Lesson 9.

import numpy as np

# Pitch and distribution factors of a three-phase winding: 4 poles, 36 slots (q = 3 slots per pole per phase)
slots, poles, phases = 36, 4, 3
q = slots // (poles * phases)
gamma = np.pi * poles / slots                 # slot angle in electrical radians (20 degrees)

def k_d(h):
    return np.sin(h * q * gamma / 2) / (q * np.sin(h * gamma / 2))

def k_p(h, pitch):                            # pitch as a fraction of full pitch
    return np.sin(h * pitch * np.pi / 2)

print(f"slot angle {np.degrees(gamma):.0f} electrical degrees, q = {q}")
print("pitch     h=1     h=3     h=5     h=7    (winding factor k_w = k_p k_d)")
for slots_pitch in (9, 8, 7):
    pitch = slots_pitch / 9
    row = "  ".join(f"{abs(k_p(h, pitch) * k_d(h)):.3f}" for h in (1, 3, 5, 7))
    print(f"{slots_pitch}/9    {row}")
# EMF per phase: E = 4.44 f N k_w Phi
f, N, Phi = 60, 96, 0.032
for slots_pitch in (9, 8, 7):
    kw = k_p(1, slots_pitch / 9) * k_d(1)
    print(f"pitch {slots_pitch}/9: fundamental EMF = 4.44 x {f} x {N} x {kw:.3f} x {Phi} = {4.44 * f * N * kw * Phi:.0f} V per phase")

slot angle 20 electrical degrees, q = 3
pitch     h=1     h=3     h=5     h=7    (winding factor k_w = k_p k_d)
9/9    0.960  0.667  0.218  0.177
8/9    0.945  0.577  0.140  0.061
7/9    0.902  0.333  0.038  0.136
pitch 9/9: fundamental EMF = 4.44 x 60 x 96 x 0.960 x 0.032 = 785 V per phase
pitch 8/9: fundamental EMF = 4.44 x 60 x 96 x 0.945 x 0.032 = 774 V per phase
pitch 7/9: fundamental EMF = 4.44 x 60 x 96 x 0.902 x 0.032 = 738 V per phase

For harmonic \(h\), the factors become \(k_{p,h} = \sin(h\rho\pi/2)\) and \(k_{d,h} = \sin(hq\gamma/2)/(q\sin(h\gamma/2))\). A 7/9 pitch cuts the 5th harmonic winding factor from 0.218 to 0.038 while reducing the fundamental by only 6%; an 8/9 pitch is a compromise that suppresses the 7th. Designers choose the pitch to suppress the harmonics that matter most: the 5th and 7th in motors (they cause torque dips and noise) and, in generators, the harmonics that would distort the voltage waveform. Third harmonics and their multiples cancel in the line-to-line voltages of a wye-connected winding, because they are in phase in all three phases, so they matter less.

Why the field rotates

Each phase winding, carrying a sinusoidal current, produces an MMF wave that is sinusoidal in space and fixed in position, but whose amplitude pulsates with the current: phase a produces \(F_a = F_m \cos\omega t \cos\theta\). Such a pulsating wave can be written as the sum of two waves of half the amplitude rotating in opposite directions:

Phases b and c are displaced by 120° in space and their currents by 120° in time. When the three phases are added, the forward-rotating components are all in step and add to \(\tfrac{3}{2} F_m\), while the backward-rotating components are at 0°, 120° and 240° to each other and cancel:

The result is an MMF wave of constant amplitude, one and a half times the peak of one phase, rotating at \(\omega\) electrical radians per second, which is synchronous speed. Swapping any two supply leads reverses the phase sequence and so reverses the direction of rotation, which is how motor direction is changed.

import numpy as np

# Three windings spaced 120 electrical degrees, carrying balanced three-phase currents:
# the total MMF is a wave of constant amplitude 1.5 F_max that rotates at synchronous speed.
theta = np.linspace(0, 2 * np.pi, 3601)        # position around the air gap (electrical radians)
def total_mmf(t, ia_scale=1.0, ib_scale=1.0, ic_scale=1.0):
    w = 2 * np.pi * 60
    F = (ia_scale * np.cos(w * t) * np.cos(theta)
         + ib_scale * np.cos(w * t - 2 * np.pi / 3) * np.cos(theta - 2 * np.pi / 3)
         + ic_scale * np.cos(w * t + 2 * np.pi / 3) * np.cos(theta + 2 * np.pi / 3))
    k = np.argmax(F)
    return F[k], np.degrees(theta[k])

print("balanced currents:")
for ms in (0, 1, 2, 3, 4):
    t = ms / 1000
    peak, where = total_mmf(t)
    print(f"  t = {ms} ms: peak MMF {peak:.3f} (per unit of one phase) at {where:6.1f} electrical degrees")
print(f"  the peak moves {360 * 60 / 1000:.1f} electrical degrees per ms: 360 electrical degrees per cycle")
print("phase c open (single phasing): currents in a and b only, equal and opposite")
for ms in (0, 2, 4, 6, 8):
    t = ms / 1000
    w = 2 * np.pi * 60
    F = np.cos(w * t) * (np.cos(theta) - np.cos(theta - 2 * np.pi / 3))
    k = np.argmax(F)
    print(f"  t = {ms} ms: peak MMF {F[k]:.3f} at {np.degrees(theta[k]):6.1f} degrees")

balanced currents:
  t = 0 ms: peak MMF 1.500 (per unit of one phase) at    0.0 electrical degrees
  t = 1 ms: peak MMF 1.500 (per unit of one phase) at   21.6 electrical degrees
  t = 2 ms: peak MMF 1.500 (per unit of one phase) at   43.2 electrical degrees
  t = 3 ms: peak MMF 1.500 (per unit of one phase) at   64.8 electrical degrees
  t = 4 ms: peak MMF 1.500 (per unit of one phase) at   86.4 electrical degrees
  the peak moves 21.6 electrical degrees per ms: 360 electrical degrees per cycle
phase c open (single phasing): currents in a and b only, equal and opposite
  t = 0 ms: peak MMF 1.732 at  330.0 degrees
  t = 2 ms: peak MMF 1.263 at  330.0 degrees
  t = 4 ms: peak MMF 0.109 at  330.0 degrees
  t = 6 ms: peak MMF 1.104 at  150.0 degrees
  t = 8 ms: peak MMF 1.718 at  150.0 degrees

Unbalance and single phasing

When the supply voltages are unbalanced, the currents contain a negative-sequence component, which produces a field rotating backward at synchronous speed. For a motor running forward, this backward field induces rotor currents at nearly twice supply frequency and produces a braking torque. Because the motor's impedance to negative sequence is low (similar to its locked-rotor impedance), a small voltage unbalance causes a large current unbalance: typically, each 1% of voltage unbalance causes 6 to 10% current unbalance, and the extra rotor and stator heating rises roughly with twice the square of the percent unbalance. NEMA MG 1 recommends derating motors when voltage unbalance exceeds 1% and not operating them above 5%.

Single phasing, the loss of one supply phase from a blown fuse or broken conductor, is the extreme case. As the code above shows, the remaining two phases carry equal and opposite currents and produce a pulsating field that does not rotate: it is the sum of equal forward and backward fields. A motor that is already running keeps turning, because the forward field dominates at low slip, but draws about 1.7 to 2 times its normal current in the remaining phases and overheats; a stopped motor cannot start and draws locked-rotor current. Overload relays sized correctly will trip, and electronic motor protection relays detect phase loss and unbalance directly (Lesson 12).

Space harmonics, slotting and their effects

Even with distributed, short-pitched windings, the MMF contains small harmonics of orders \(h = 6k \pm 1\) (5, 7, 11, 13, …) in a three-phase winding. The 5th harmonic field rotates backward at one fifth of synchronous speed and the 7th forward at one seventh. In induction motors these fields produce parasitic torques that can create dips in the torque-speed curve near one seventh of synchronous speed, where a lightly loaded motor may "crawl". Slot openings modulate the air-gap permeance and produce slot harmonics, which cause magnetic noise and, if the numbers of stator and rotor slots are poorly chosen, cogging, in which the motor locks at low speed. Designers avoid these problems by choosing slot combinations carefully and by skewing the rotor bars by about one slot pitch, which averages out the slot harmonics at a small cost in fundamental torque.

Winding construction and connections

Small and medium machines use random-wound coils of round enamelled wire, inserted into slots lined with insulating paper. Large machines and those above about 1 kV use form-wound coils of rectangular insulated conductors, pre-shaped and taped with mica-based insulation that resists partial discharge. Double-layer windings, with two coil sides per slot, allow any pitch and are standard for form-wound machines. Coil groups of each phase are connected in series or in parallel paths to suit the rated voltage, and the phases are connected in wye or delta. Many motors bring out six or nine leads so they can be connected for two voltages (for example, 230 V in delta or parallel wye and 460 V in series wye) or started wye and run in delta (Lesson 7).