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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:
)...240(7sin7
1)240(5sin
5
1)240(3sin
3
1)240sin(
2
)...120(7sin7
1)120(5sin
5
1)120(3sin
3
1)120sin(
2
...7sin7
1)90(5sin
5
13sin
3
1sin
2
0000
0000
0
CC
BB
AA
NiF
NiF
NiF
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.
)...(7sin7
1)5sin(
5
1)sin(
32 tttNIF
FFFF
total
CBAtotal
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.
)...240(2
sin2
)...120(2
sin2
...2
sin2
0
0
pNiF
pNiF
pNiF
CC
BB
AA
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)
)...
2(
2sin
32 t
p
pNIFA
The above formulation may seem cumbersome at
first. Defining mechanical angle θm, electrical angle θe,
mechanical speed ωm and electrical speed ωe; 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
)...
2(
2sin
32 t
p
pNIF
emA
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 fe is given a distinct name, "synchronous
speed". Synchronous speed in radians per second is
defined as:
p
fs
2
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
p
fs
60
