Rotating field in AC motor

A sinusoidally distributed stator winding produces a sinusoidal air-gap magnetic field. One phase creates a standing field. Three windings separated by \(120^\circ\) in space and supplied by currents separated by \(120^\circ\) in time create a resultant field that rotates continuously around the air gap.

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Sinusoidal conductor distribution

Let \(n_a(\theta)\) be the phase-\(a\) conductor density in conductors per radian. An ideal sinusoidal distribution for a 2-pole motor is

n_a(\theta)=\hat{N}\sin\theta .

The positive half of the distribution contains \(N_t\) series turns:

N_t=\int_0^\pi \hat{N}\sin\theta\,d\theta =2\hat{N}, \qquad \hat{N}=\frac{N_t}{2}, \] \[ n_a(\theta)=\frac{N_t}{2}\sin\theta.

Sinusoidally distributed phase-a conductors in a twelve-slot stator — Phase-\(a\) distribution with 12 slots.

Sinusoidally distributed phase-a conductors in a twenty-four-slot stator — Phase-\(a\) distribution with 24 slots.

Sinusoidally distributed phase-a conductors in a forty-eight-slot stator — Phase-\(a\) distribution with 48 slots.

Detail of phase-a conductors in the twelve-slot winding — Zoomed-in phase-\(a\) winding with 12 slots.

Detail of phase-a conductors in the twenty-four-slot winding — Zoomed-in phase-\(a\) winding with 24 slots.

Detail of phase-a conductors in the forty-eight-slot winding — Zoomed-in phase-\(a\) winding with 48 slots.

Example: Consider winding with \(N_t=48\)

For 48 turns, \(\hat{N}=24\). Condensing the continuous distribution into \(30^\circ\) slot groups gives

\begin{aligned} N_{a,1} &=\int_0^{\pi/6}24\sin\theta\,d\theta \approx3.22\longrightarrow3,\\ N_{a,2} &=\int_{\pi/6}^{\pi/3}24\sin\theta\,d\theta \approx8.78\longrightarrow9,\\ N_{a,3} &=\int_{\pi/3}^{\pi/2}24\sin\theta\,d\theta =12. \end{aligned}

Symmetry completes the positive half as \(3,9,12,12,9,3\). The opposite half contains the same numbers with the conductor direction reversed. A dot denotes current out of the page; a cross denotes current into the page.

Three spatially displaced phase windings

Phase \(b\) is the phase-\(a\) winding rotated by \(120^\circ\), and phase \(c\) is rotated by \(240^\circ\):

\begin{aligned} n_a(\theta)&=\frac{N_t}{2}\sin\theta,\\ n_b(\theta)&=\frac{N_t}{2}\sin\!\left(\theta-\frac{2\pi}{3}\right),\\ n_c(\theta)&=\frac{N_t}{2}\sin\!\left(\theta-\frac{4\pi}{3}\right). \end{aligned}

Phase-a winding distributed around the stator — Phase-\(a\) winding.

Phase-b winding distributed around the stator — Phase-\(b\) winding.

Phase-c winding distributed around the stator — Phase-\(c\) winding.

Detail of the phase-a winding distribution — Detail of the phase-\(a\) winding.

Detail of the phase-b winding distribution — Detail of the phase-\(b\) winding.

Detail of the phase-c winding distribution — Detail of the phase-\(c\) winding.

Placing the three spatially displaced phase windings in the same stator gives the combined winding shown below.

The three distributed phase windings occupying the same stator — All three phase windings in the stator.

Detail of all three phase winding distributions in common stator slots — Detail of the combined three-phase winding.

Air-gap magnetic field

With very high iron permeability, the magnetic-field intensity in the iron is nearly zero and the magnetomotive-force drop appears across the two air-gap crossings of an Ampèrian loop:

\oint \mathbf H\cdot d\boldsymbol\ell=N_{\mathrm{enc}}i, \qquad 2l_gH_g(\theta)=N_{\mathrm{enc}}(\theta)i.

The loop encloses all \(N_t\) turns on the magnetic axis. Therefore

H_{g,\max}=\frac{N_t i}{2l_g}, \qquad H_a(\theta,t)=\frac{N_t i_a(t)}{2l_g}\cos\theta.

Reversing the chosen positive radial direction changes only the sign. The \(90^\circ\) spatial shift remains: conductor density is largest where the magnetic field crosses zero, and the magnetic field is largest on the phase magnetic axis.

Phase \(a\)

Increasing the slot count makes the stepped field closer to a sinusoid.

Phase-a air-gap field approaching a sinusoidal distribution

Phase \(b\)

Phase \(c\)

A purely mathematical constant-current snapshot

Suppose, purely mathematically, that the same constant current is applied to all three phases. This is not feasible in a three-wire system because the instantaneous currents must satisfy \(i_a+i_b+i_c=0\). The corresponding field distribution due to each phase would be

\begin{aligned} H_a(\theta)&=H_m\cos\theta,\\ H_b(\theta)&=H_m\cos\!\left(\theta-\frac{2\pi}{3}\right),\\ H_c(\theta)&=H_m\cos\!\left(\theta-\frac{4\pi}{3}\right). \end{aligned}

The three distributions are separated by \(120^\circ\) in space and their point-by-point sum is zero.

Phase-a, phase-b, and phase-c spatial magnetic-field distributions separated by one hundred twenty degrees

A single phase produces a standing field

Apply a sinusoidal current to phase \(a\):

i_a(t)=I_m\cos\omega t.

With \(H_m=N_tI_m/(2l_g)\), the physical air-gap field is

H_a(\theta,t)=H_m\cos\omega t\cos\theta.

Its spatial axis remains fixed while its magnitude rises, collapses to zero, reverses, and rises again. It is a standing field. The product-to-sum identity decomposes it into two traveling components:

\cos A\cos B =\frac12\left[\cos(A-B)+\cos(A+B)\right], \] \[ H_a(\theta,t) =\frac{H_m}{2} \left[ \cos(\theta-\omega t)+\cos(\theta+\omega t) \right].

In the tube, two physical traveling waves sum to produce a standing wave. Here the physical quantity is the standing magnetic field, and it is mathematically decomposed into equal forward- and reverse-traveling components. Because their magnitudes are equal, neither direction is preferred.

Animated phase-a winding as the sinusoidal current changes — Spatially and temporally sinusoidally distributed air-gap field of the phase-\(a\) winding as its current varies.

Animated phase-a standing magnetic field around the air gap — Standing air-gap field distribution in space.

Animated phase-a sinusoidal current — Sinusoidal phase-\(a\) current distribution in time. The marker tracks the current magnitude at the instant for which the air-gap field is visualized.

Animated stator view of the forward and reverse components of the phase-a field — Spatial decomposition of the phase-\(a\) standing air-gap field into equal forward- and reverse-traveling fields in the stator.

Animated forward and reverse traveling components of the phase-a standing field — Forward- and reverse-traveling air-gap field distributions in space. Their sum is the phase-\(a\) standing field.

Animated phase-a current corresponding to the decomposed field — Sinusoidal phase-\(a\) current distribution in time. The marker tracks the current magnitude at the instant for which the decomposed air-gap fields are visualized.

Tesla’s rotating-field insight

The single-phase field was the obstacle: it pulsed, but it would not travel. Nikola Tesla wrestled with the problem until the work affected his health. While recovering in Budapest, he walked through a park with a friend at sunset and recited a passage from Goethe’s Faust:

The glow retreats, done is the day of toil
It yonder hastes, new fields of life exploring
Ah, that no wing can lift me from the soil
Upon its track to follow, follow soaring!
— Goethe, Faust

Tesla felt much the same. He believed he had done everything needed to make the motor rotate, yet the laws of physics seemed to have placed a cage around it, preventing his rotor from soaring. Then, as he later recalled, the idea arrived in a flash. Seen through the imagery of the poem, the setting sun was not simply extinguished at the horizon; it continued onward. The magnetic field could do the same. Instead of collapsing and rising again on one fixed axis, it could set on one axis while rising on another, always opening a path ahead of the rotor. A second winding placed \(90^\circ\) away, carrying a current shifted by \(90^\circ\) in time, makes its field rise as the first field falls. The field peak moves continuously from one axis to the next, and the rotor can chase it like a bird following the setting sun. In that poetic sense, Tesla had given wings to the rotor. He drew the idea in the sand for his friend and showed how the field could reverse. This became Tesla’s two-phase motor concept; Galileo Ferraris developed the rotating-field principle independently.

Balanced three-phase currents create a rotating field

For a positive phase sequence,

\begin{aligned} i_a(t)&=I_m\cos\omega t,\\ i_b(t)&=I_m\cos\!\left(\omega t-\frac{2\pi}{3}\right),\\ i_c(t)&=I_m\cos\!\left(\omega t-\frac{4\pi}{3}\right). \end{aligned}

The winding axes and current waveforms have the same \(120^\circ\) displacement. Each phase still produces a standing field, and each standing field can be separated into two counter-traveling components.

Phase \(a\)

Animated phase-a current — Phase-\(a\) current.

Phase \(b\)

Phase \(b\) is displaced by \(120^\circ\) in both space and time relative to phase \(a\). Its winding and sinusoidal current therefore produce the following standing field:

H_b(\theta,t) =H_m\cos\!\left(\omega t-\frac{2\pi}{3}\right) \cos\!\left(\theta-\frac{2\pi}{3}\right).

Animated phase-b winding and standing air-gap field — Phase-\(b\) winding and its standing air-gap field.

Animated phase-b standing magnetic-field distribution — Standing magnetic-field distribution of phase \(b\).

Animated phase-b sinusoidal current — Sinusoidal phase-\(b\) current.

Applying the product-to-sum identity separates the phase-\(b\) standing field into forward- and reverse-traveling components:

H_b =\frac{H_m}{2} \left[ \cos(\theta-\omega t) +\cos\!\left(\theta+\omega t-\frac{4\pi}{3}\right) \right].

Animated phase-b forward and reverse field components in the stator — Phase-\(b\) traveling field components in the stator.

Animated forward and reverse traveling components of the phase-b field — Forward and reverse phase-\(b\) components.

Animated phase-b current corresponding to its decomposed field — Phase-\(b\) current for the decomposed field.

Phase \(c\)

Phase \(c\) is displaced by \(240^\circ\) in both space and time relative to phase \(a\). Its winding and sinusoidal current produce the corresponding standing field:

H_c(\theta,t) =H_m\cos\!\left(\omega t-\frac{4\pi}{3}\right) \cos\!\left(\theta-\frac{4\pi}{3}\right).

Animated phase-c winding and standing air-gap field — Phase-\(c\) winding and its standing air-gap field.

Animated phase-c standing magnetic-field distribution — Standing magnetic-field distribution of phase \(c\).

Animated phase-c sinusoidal current — Sinusoidal phase-\(c\) current.

The same product-to-sum identity separates the phase-\(c\) standing field into its forward- and reverse-traveling components:

H_c =\frac{H_m}{2} \left[ \cos(\theta-\omega t) +\cos\!\left(\theta+\omega t-\frac{8\pi}{3}\right) \right].

Animated phase-c forward and reverse field components in the stator — Phase-\(c\) traveling field components in the stator.

Animated forward and reverse traveling components of the phase-c field — Forward and reverse phase-\(c\) components.

Animated phase-c current corresponding to its decomposed field — Phase-\(c\) current for the decomposed field.

All three phases together

Compare the three phases at the same instant. One traveling component is aligned in all three phases; the opposite component remains separated by \(120^\circ\).

Animated phase-a traveling field components — Phase-\(a\) traveling field components.

Animated phase-b traveling field components — Phase-\(b\) traveling field components.

Animated phase-c traveling field components — Phase-\(c\) traveling field components.

The machine views show the two counter-traveling components produced by each phase. The corresponding field plots can now be compared directly at the same instant.

Animated phase-a forward and reverse components — Forward and reverse components of phase \(a\).

Animated phase-b forward and reverse components — Forward and reverse components of phase \(b\).

Animated phase-c forward and reverse components — Forward and reverse components of phase \(c\).

Writing the phase-\(a\) field in the same form,

\begin{aligned} H_a&=\frac{H_m}{2} \left[\cos(\theta-\omega t)+\cos(\theta+\omega t)\right],\\[2mm] H_b&=\frac{H_m}{2} \left[\cos(\theta-\omega t)+ \cos\!\left(\theta+\omega t-\frac{4\pi}{3}\right)\right],\\[2mm] H_c&=\frac{H_m}{2} \left[\cos(\theta-\omega t)+ \cos\!\left(\theta+\omega t-\frac{8\pi}{3}\right)\right]. \end{aligned}

The three \(\cos(\theta-\omega t)\) terms are identical and add. The other three terms are separated by \(120^\circ\) and cancel:

\cos x +\cos\!\left(x-\frac{4\pi}{3}\right) +\cos\!\left(x-\frac{8\pi}{3}\right)=0, \] \[ H_{abc}(\theta,t)=H_a+H_b+H_c =\frac{3H_m}{2}\cos(\theta-\omega t) =\frac{3N_tI_m}{4l_g}\cos(\theta-\omega t).

One phase hands the field peak to the next in a continuous relay. The individual phase fields remain standing fields, but their sum is a single traveling field of constant amplitude.

Animated phase fields before the resultant is added — The three phase fields before adding the resultant.

Animated three phase fields and their rotating resultant — The three phase fields and their rotating resultant.

The rotating resultant can now be viewed together with the three-phase winding and the balanced phase currents.

Animated three-phase winding and rotating resultant magnetic field — Three-phase winding and rotating resultant field.

Animated balanced three-phase current waveforms — Balanced three-phase currents.

Animated individual phase fields and their rotating resultant — Individual phase fields and their rotating resultant.

Direction of rotation from phase order

Swapping any two phase connections reverses the time sequence while leaving the winding axes unchanged. The opposite traveling component then survives:

H_{acb}(\theta,t) =\frac{3N_tI_m}{4l_g}\cos(\theta+\omega t).

One direction	Opposite direction
\(abc,\ bca,\ cab\)	\(acb,\ cba,\ bac\)