One of the most fundamental aspects of rotating AC machinery is the rotating magnetic field.  In a 3 phase machine, there are 3 coils.  Each coil is connected to a phase, and therefore the magnetic field from each coil is different in time.  let us consider 3 coils, all with a temporal phase shift (a phase shift in time, brought about by the phase shift inherent in 3 phase power from phase to phase) and all in the same plane, as in the drawing below:

            All of the magnetic fields will be on the same axis.  Since there is no angle introduced by the coils, the only difference in magnetic fields will be due to the temporal phase shift introduced by the three phase power.  If we imagine that all of the magnetic fields produced by the coils lie in the X-Y plane, or more specifically along the Y axis since each coil will be in the X-Z plane, centered at the origin, and there is no space between the coils (not physically possible, but it makes the math easier.)  The magnetic field due to one coil can be written as:

            The net magnetic field will be the result of the sum of each magnetic field.  This can be expressed as:

The Cosine functions can be broken apart using the trig identity:

            Cos(a-b)=Cos(a)*Cos(b)+Sin(a)*Sin(b)

Thus the equation becomes:

Evaluating the Sines and Cosines gives us:

After evaluating the function, it is clear to see that the net resulting field will be:

Thus the resulting magnetic field due to 3, 3-phase coils all placed in the same plane will be zero.  However, if the 3 coils are rotated 120 degrees spatially from each other, as in the picture below:

Then a new net magnetic field can be calculated.  The resulting magnetic field from each coil can be represented as below:

Grouping the X and Y components together we get:

Using the same trig identity as before, we can further decompose these functions to become:

Evaluating the Sines and Cosines:

After simplification:

If you evaluate this function at different values of ‘t’ you will find that the vector will rotate Counterclockwise as depicted in the parametric plot below, evaluated from 0 to 3/2*pi-0.5:

            This rotating magnetic field is the cornerstone of rotating machinery.