6.4 The MMF of Three- Phase In the diagram above there are three coils, arranged around the stator of a machine such that the angle between each of the phases is 120°. Assuming that the steel in the rotor and stator is infinitely permeable, the MMF produced in the air gap between the two sides of a coil will be constant. Each coil will produce a square wave MMF function, phase shifted by 120°.The MMF functions for each phase are:

The MMF functions will vary with both time and space, too. Although the above MMF functions may seem quite long, the MMF simplify significantly when summed to find the total MMF.

It can be seen from the above equation that: 1. The three pulsating MMF functions combine to create a rotating MMF function, with constant magnitude fundamental frequency component 2. The magnitude of the rotating MMF is 1.5 times the magnitude of the pulsating MMF components

3. All multiples of the third harmonics are eliminated 4. The magnitude of a higher space harmonic is inversely proportional to the harmonic number 5. Harmonic numbers 6n+1 where n is an integer rotate in the positive direction 6. Harmonic numbers 6n-1 where n is an integer rotate in the negative direction

Simplifications For the rest of the course we will neglect higher space harmonics and assume that a simple three coil arrangement is capable of producing a sinusoidal air gap MMF. This assumption is aided by the fact that most real machines are constructed with distributed windings which have been designed to minims space harmonic components.

Rotation Speed The rotating magnetic field in the earlier example can be thought of as two rotating magnetic poles, a north pole and a south pole. As the supply current waveform moves through 180 degrees, the 2-pole field moves through 180 degrees, and the locations of the north and south poles is reversed. When the current waveform has moved through 360 degrees, the 2-pole field has moved through 360 degrees.

There is no reason to limit the number of poles in a machine to two. If the number of coils is increased, the coils can be arranged so that the winding pattern occurs more than once around the circumference of the air gap. The original functions describing MMF variation with angular position.

In the above equation, p is the number of poles in the machine. The p/2 term indicates that the fundamental MMF repeats p/2 times around the circumference of the machine. Assuming sinusoidal supply with a balanced three-phase set, the analysis for the two pole field can be repeated to find a new functions describing MMF in terms of space (theta) and time (t)

The above formulation may seem cumbersome at first. Defining mechanical angle θ m, electrical angle θ e, mechanical speed ω m and electrical speed ω e ; it is possible to re-write the above equations in two forms. First, we must note that up to now, θ has been used for mechanical angles around the circumference of the machine and ω has been used to describe electrical supply frequency. Re-writing the above equation with the new symbols to differentiate between electrical and mechanical terms the fundamental MMF becomes

Finally, there can be a number of different mechanical speeds under consideration in an electrical machine. The mechanical speed of rotation of a magnetic field due to the fundamental electrical current, frequency f e is given a distinct name, "synchronous speed". Synchronous speed in radians per second is defined as:

It is common to use units of revolutions per minute, (rpm) to describe rotational speed. Speed in rpm is described using the symbol n and is related to radians per second using