Analysis of the Magnetomotive Force of a
Three-Phase Winding with Concentrated Coils
and Different Symmetry Features
Dieter Gerling
University of Federal Defense Munich, Neubiberg, 85579, Germany
Email: Dieter.Gerling@unibw.de
Abstract- Permanent magnet machines with windings concentrated
around each tooth gain more and more importance. In this paper,
the Magnetomotive Force of a three-phase winding with
concentrated coils will be analysed. As an example, a machine with
a two-coil zone width with opposing flux is regarded. It will be
shown that there are two different winding alternatives with four
different rotor pole numbers that can generate a constant torque.
All these alternative realizations are characterized as
fractional-slot windings. This analysis may serve as a basis for the
machine design concerning fundamental torque, as well as for
acoustic noise calculation because a major part of the harmonic
content of the air-gap field is identified.
I.
WINDING TOPOLOGY
The following figure 1 illustrates the winding topology under
investigation. As every tooth contains a winding, this is usually
referred to as two-layer winding (against a single-layer winding,
when just every other tooth carries a coil).
The investigations described in this paper will be conducted
regarding the following assumptions:
All coils will have the same number of turns.
All teeth have equal width and they are equally spaced.
The three-phase ( m = 3 ) currents are symmetric and
⎛
⎝
2π ⎞
⎛
⎝
4π ⎞
i B = ˆI sin ⎜ ωt −
i C = ˆI sin ⎜ ωt −
-
INTRODUCTION
Especially for Permanent Magnet (PM) motors the winding
configuration with concentrated coils around each tooth gets
more and more into the focus. Most often, such motors are
designed with a single-coil zone width [1-2], only sometimes
more than one coil per zone are regarded [3-4]. The special
interest in such motors mainly comes from the fact that simple
and therefore very cost-effective production possibilities arise
using such a winding topology. In addition, the end winding is
very small resulting in less volume, weight, costs, and losses. A
typical application of a PM motor with concentrated coils is
described in [5]. On the other hand, there are some technical
drawbacks against a sophisticated distributed winding,
especially the high harmonic content in the Magnetomotive
Force (MMF). Therefore, the MMF-distribution will be
analysed in this paper for the example of a three-phase winding
with concentrated coils and two-coil zone width with opposing
flux.
II.
purely harmonic:
i A = ˆI sin ( ωt )
-
-
⎟.
(1)
3 ⎠
⎟
3 ⎠
As the winding harmonics shall be investigated, the
slotting effect will be neglected. Therefore, the winding
distribution can be approximated by an electric loading
on a smooth stator surface, concentrating the current of a
single slot into the mid-point of the slot opening.
The winding distribution shown in figure 1 for each
phase is defined being “positive” for the following
analysis, i.e. the complete winding distribution shown in
figure 1 will be denoted with “+A +B +C”.
As an example, a stator with Z = 24 teeth will be
considered.
phase A
phase B
phase C
Fig. 1. Analysed winding topology.
III.
TORQUE GENERATION
The torque of electrical machines can be calculated from the
electric loading distribution (or MMF-distribution) and the flux
density distribution. For PM-machines, the rotor flux density
distribution and the stator MMF-distribution are essential.
Generally, both distributions are non-sinusoidal and they
contain a fundamental wave and an infinite number of harmonic
waves.
Torque is generated, if the ordinal number of a MMF-wave
and the ordinal number of a flux density wave coincide.
We will see in the following, that (under the exemplary
assumptions made above) there can be a symmetry on a quarter
circumference of the motor or on the half circumference. The
number of rotor pole pairs on the regarded minimum symmetry
must coincide with the ordinal number of the main MMF-wave
(relative to this symmetry) to generate a time-independent
torque.
IV.
A.
WINDING ALTERNATIVES
General Remarks
The following four alternatives are possible, if the winding
distribution described in chapters 1 and 2 is used:
Case 1: +A +B +C –A –B –C
(minimum symmetry: 12 slots,
half machine circumference)
Case 2: +A –B +C –A +B –C
(minimum symmetry: 12 slots,
half machine circumference)
Case 3: +A +B +C +A +B +C
(minimum symmetry: 6 slots,
quarter machine circumference)
Case 4: +A –B +C +A –B +C
(minimum symmetry: 6 slots,
quarter machine circumference)
The change of two phases just changes the direction of the
MMF-waves (and therefore the rotational direction of the rotor),
but not the general MMF-distribution. As this rotational
direction is not of interest here, this will not be regarded in the
following. The four alternative cases mentioned above will be
investigated separately in the following sections.
B. Case 1: +A +B +C –A –B –C
The following figure 2 shows the MMF-distribution for the
time steps ωt = 0 and ωt = π / 2 ; on the horizontal axis the
slots are numbered, on the vertical axis the relative MMF-value
Θ x is given (p.u. value).
Fig. 3 shows the fourier components of these
MMF-distributions. The calculation of the fourier components
is done by using the saltus function method described in [6]. On
the horizontal axis are given the fundamental and harmonic
wave numbers, on the vertical axis is shown the amplitude Θi
of each wave.
It becomes obvious from these figures that the amplitudes are
not constant in time, so that it is not possible to generate a
time-independent torque. This winding distribution is not useful.
Fig. 2. MMF-distribution Θ x for case 1 and ωt = 0 (left) and ωt = π / 2 (right).
Fig. 3. Fourier components of the MMF-distribution Θ i for case 1 and ωt = 0 (left) and ωt = π / 2 (right).
C.
Case 2: +A –B +C –A +B –C
The MMF-distribution and their fourier components for case
2 are shown in the following figures 4 and 5.
It can be shown in general, that for this case the amplitudes of
the fundamental wave and the harmonic waves are constant (the
points in time shown in these figures serve as a hint). Therefore,
with this winding distribution a constant torque can be
generated.
Fig. 4. MMF-distribution Θ x for case 2 and ωt = 0 (left) and ωt = π / 2 (right).
Fig. 5. Fourier components of the MMF-distribution Θ i for case 2 and ωt = 0 (left) and ωt = π / 2 (right).
Fig. 5 demonstrates that there are two dominant fourier
components: ordinal number 5 and ordinal number 7. The
respective amplitudes are: A 5 = 0.713 and A 7 = 0.509 ,
resulting in a ratio of A 5 / A 7 = 1.4 . Nevertheless, the
amplitude of the electric loading (which is essential for the
torque generation) is the same for both harmonics, as the
MMF-distribution is calculated from integrating the electric
loading distribution over one pole pitch. To get an overview,
which harmonic should be used for torque generation, the
following figure of the fourier components of the electric
loading is very useful (as just the p.u. data are of interest, each
MMF-harmonic is multiplied with the respective number of
pole-pairs):
Fig. 6. Fourier components of the electric loading distribution A i for case 2.
In general this winding distribution can generate a
time-independent torque with a 20-pole rotor (ordinal number 5
means 5 complete sine-waves or 10 half waves per half
circumference, resulting in a 20-pole configuration) or a 28-pole
rotor. Of course, even the other harmonics can generate a
time-independent torque with the appropriate rotor pole number,
but as the required frequency is much higher this does not make
sense because of higher losses (eddy current losses, hysteresis
losses, and switching losses).
D. Case 3: +A +B +C +A +B +C
The MMF-distribution and their fourier components for case
3 are shown in the following figures 7 and 8:
Fig. 7. MMF-distribution Θ x for case 3 and ωt = 0 (left) and ωt = π / 2 (right).
Fig. 8. Fourier components of the MMF-distribution Θ i for case 3 and ωt = 0 (left) and ωt = π / 2 (right).
This alternative is the second possibility to generate
time-independent harmonic waves of the MMF-distribution (the
points in time shown above serve as a hint). Therefore, a
constant torque can be produced. Like in case 2 the fundamental
component is not the highest one. The same holds true for the
fundamental components of the electric loadings (please refer to
figures 6 and 9).
The highest MMF-amplitude is reached fort the ordinal
number 2 (i.e. 2 pole pairs for a quarter circumference).
Consequently, a 16-pole rotor has to be applied to use this
harmonic wave for torque production.
The fourier components of the electric loading of this
alternative are shown in the following figure 9 (demonstrating
that even a 32-pole rotor may be applied):
Fig. 9. Fourier components of the electric loading distribution A i for case 3.
Again, even the higher harmonics can be used for torque
production, but because of the required higher frequency this is
not useful.
As the vertical axis of figure 6 and figure 9 have the same
values one can deduce that the torque of case 2 (figure 6) is
higher. Although the frequency is higher in case 2 because of the
higher number of pole-pairs (20 or 28 against 16) most probably
it is reasonable to make use of the higher electric loading
harmonic to realize the maximum torque. In addition, the
fundamental wave (which is not used for torque production in
both cases) is much smaller in case 2.
E. Case 4: +A –B +C +A –B +C
The MMF-distribution and their fourier components for case
4 are shown in the following figures 10 and 11:
Fig. 10. MMF-distribution Θ x for case 4 and ωt = 0 (left) and ωt = π / 2 (right).
Fig. 11. Fourier components of the MMF-distribution Θ i for case 4 and ωt = 0 (left) and ωt = π / 2 (right).
Even this alternative shows time-dependent harmonic waves
of the MMF-distribution, resulting in time-dependent torque
generation. Again, this winding distribution is not useful.
F.
Summary
It could be shown that there are two winding alternatives
resulting in a time-independent MMF-distribution. In general,
both winding alternatives can be used for generating a constant
torque. Case 2 (+A –B +C –A +B –C) calls for a 20-poles rotor
or a 28-poles rotor, case 3 (+A +B +C +A +B +C) calls for a
16-poles rotor or a 32-poles rotor. As the stator is realized with a
3-phase ( m = 3 ), 24-slot ( Z = 24 ) design, we get the following
winding characteristics:
Case 2, 20-poles rotor ( 2 ⋅ p = 20 ):
q=
Z
24
= 0.4
2 ⋅ p ⋅ m 20 ⋅ 3
Case 2, 28-poles rotor ( 2 ⋅ p = 28 ):
q=
Z
=
24
≈ 0.29
2 ⋅ p ⋅ m 28 ⋅ 3
Case 3, 16-poles rotor ( 2 ⋅ p = 16 ):
q=
Z
2⋅p⋅m
=
=
24
16 ⋅ 3
= 0.5
Case 3, 32-poles rotor ( 2 ⋅ p = 32 ):
q=
Z
=
24
= 0.25
2 ⋅ p ⋅ m 32 ⋅ 3
All alternatives result in a fractional-slot winding. In case 2
and case 3 even a time-independent torque generation with an
integral slot winding is possible, but then the fundamental wave
with quite a low amplitude of the electric loading has to be used,
resulting in quite a low torque.
V.
CONCLUSION
In this paper, the Magnetomotive Force of a three-phase
winding with concentrated coils and two-coil zone width with
opposing flux has been analysed. It could be shown that there
are two winding alternatives resulting in a constant torque
generation. Having a machine with 24 stator slots, the first
winding alternative works with a 20-poles or 28-poles rotor, the
second winding alternative works with a 16-poles or 32-poles
rotor. All alternatives are featuring a fractional-slot winding.
Analysing the electric loading harmonics leads to the
conclusion that case 2 is advantageous concerning torque
production. To judge between the two alternatives of case 2
requires a detailed analysis of the respective electromagnetic
circuits (e.g. using FEM-calculations). Such an analysis shows
that in general the rotor with 20 poles gives the maximum
torque.
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