The magic of an AC motor isn't in any single component, but in how stationary parts work together to create something that moves. The key is combining two different kinds of shifts: a physical shift in space and a timing shift in the electrical supply.

Imagine three separate coils of wire, arranged in a circle inside the motor's housing. They aren't right next to each other. Instead, they are physically displaced, set 120 degrees apart. This is the spatial phase shift. Each coil is fixed in place.

Now, we power these coils with a three-phase AC supply. The currents flowing into each coil are identical in magnitude and frequency, but they peak at different times. Each current waveform is offset from the next by one-third of a cycle, or 120 degrees. This is the temporal phase shift.

When current flows through a coil, it generates a magnetic flux. Because the current is AC, this flux is a pulsating vector. It grows to a maximum, shrinks to zero, reverses direction, and repeats. Our goal is to see what happens when we add these three pulsating, spatially separated magnetic fluxes together.

Each phase produces a magnetic flux that varies sinusoidally with time. We can represent these fluxes as vectors. The direction of each vector is fixed along the axis of its coil, but its magnitude changes. Let's write them down, where $Φ_m$ is the maximum flux from any single phase, and $ω$ is the angular frequency of the AC supply.

The total, or resultant, flux $Φ_r$ is the vector sum of these three individual fluxes. To find this sum, we resolve each vector into its horizontal (x) and vertical (y) components and add them up. The horizontal component of a vector is its magnitude times the cosine of its angle, and the vertical component is its magnitude times the sine of its angle.

Let's calculate the total horizontal component, $Φ_x$.

Using the trigonometric identity $\cos(X) + \cos(Y) = 2 \cos(\frac{X+Y}{2}) \cos(\frac{X-Y}{2})$, the terms $\cos(\omega t - 120^\circ) + \cos(\omega t - 240^\circ)$ simplify to $-\cos(\omega t)$.

Substituting this back in gives us:

Now for the total vertical component, $Φ_y$.

Using the identity $\cos(X) - \cos(Y) = -2 \sin(\frac{X+Y}{2}) \sin(\frac{X-Y}{2})$, the vertical component simplifies dramatically.

We now have the components of our resultant flux vector. To find its total magnitude, $Φ_r$, we use the Pythagorean theorem: $Φ_r^2 = Φ_x^2 + Φ_y^2$.

The angle of this resultant vector, $θ$, is given by $\arctan(Φ_y / Φ_x)$, which simplifies to $\arctan(-\tan(\omega t)) = -\omega t$. The negative sign indicates a clockwise rotation at a constant angular velocity $ω$. Stationary coils and time-shifted currents have produced a magnetic field of constant magnitude, rotating smoothly in space.

What if we only had two phases? In a two-phase system, the coils are spatially separated by 90 degrees, and the currents are temporally shifted by 90 degrees.

If you run through the same vector summation, you'll find the horizontal component is $Φ_x = Φ_m \cos(\omega t)$ and the vertical component is $Φ_y = Φ_m \sin(\omega t)$.

The magnitude of the resultant flux is:

This mathematical superposition is the principle behind how we turn electrical energy into smooth, continuous rotational motion.