Magnetic Radial Force Density of the PM Machine
with 12-teeth/10-poles Winding Topology
Gurakuq Dajaku
Dieter Gerling
FEAAM GmbH
D-85577 Neubiberg, Germany
Tel: +49 89 6004 4120, Fax: +49 89 6004 3718
E-mail: Gurakuq.Dajaku@unibw.de
Homepage: http://www.unibw.de/EAA
Institute for Electrical Drives
University of Federal Defense Munich
D-85577 Neubiberg, Germany
Tel: +49 89 6004 3708, Fax: +49 89 6004 3718
E-mail: Dieter.Gerling@unibw.de
Homepage: http://www.unibw.de/EAA
Abstract- This work presents an accurate and detailed analysis of
the magnetic radial force density of a PM machine with 12teeth/10-poles winding topology. The air-gap radial force density
distribution of the studied PM machine, as function of angular
position and corresponding space harmonics (modes) is analysed
using a combination of finite elements (FE) and analytical
methods. It is shown that, due to the presence of a large number
of low and high order space MMF harmonics, low frequency
modes of vibration are excited in this type of machines.
I.
INTRODUCTION
Permanent magnet synchronous machines (PMSM) with
non-overlapping concentrated windings have become a
competitive alternative to PMSMs with distributed windings
for certain applications. The concentrated winding machines
have potentially more compact designs compared to the
conventional machine designs with distributed windings, due
to shorter and less complex end-windings. A PM machine with
12-teeth and 10-poles is illustrated in the following figure 1. Its
stator winding differs from that of conventional PM machines
in that the coils which belong to each phase are concentrated
and wound on adjacent teeth, as illustrated in figure 1, so that
the phase windings do not overlap.
Using fractional slot windings different combinations of
numbers of poles and numbers of teeth are possible. However,
the magnetic field of these windings has more space
harmonics, including sub-harmonics. These unwanted
harmonics leads to undesirable effects, such as localised core
saturation, noise and vibration and eddy current loss in the
magnets, which are the main disadvantages of these winding
types. The radial force density distribution on the stator
surface, which results from the air-gap magnetic field under
no-load (open-circuit) and on-load conditions, is the main
cause of electromagnetically induced noise and vibration.
Therefore, the aim of this work was to analyse the radial
magnetic forces of a PM machine with concentrated
12-teeth/10-poles winding topology. Both, the finite elements
(FE) and the analytical methods are used during the following
analysis.
978-1-4244-4252-2/09/$25.00 ©2009 IEEE
Fig. 1. PM machine with concentrated 12-teeth / 10-poles winding
II. MMF DISTRIBUTION OF THE PM MACHINE WITH 12-TEETH
10-POLES WINDING TOPOLOGY
In machines with fractional slot windings, the windings are
not sinusoidally distributed and the resulting air-gap flux
density distribution may be far from being sinusoidal.
Analysing the MMF (magnetomotive force) and its harmonics
is of interest for the electric machines. One part of the air-gap
harmonics is a result of the MMF winding harmonics.
The following figure 2 show the stator and rotor MMF
waveform and corresponding space harmonics of a PM
machine with 12-teeth / 10-poles winding topology. It is
shown that the 1st, 5th, 7th, 17th and 19th are the dominant space
harmonics for this winding type. For the 10-pole machine,
however, only the 5th stator space harmonic interacts with the
field of the permanent magnets to produce continuous torque.
The other MMF space harmonics, in particular the 1st, 7th, 17th,
etc., which have relatively large magnitudes, may cause
undesirable effects, such as noise and vibration, rotor losses
etc..
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Fig. 2: PM machine with 12-teeth/10-poles winding topology; Stator and Rotor MMF distribution and corresponding space harmonics
III. ELECTROMAGNETIC SOURCE OF NOISE
The radial force density distribution on the stator surface,
which results from the air-gap magnetic field under no-load
(open-circuit) and on-load conditions, is the main cause of
electromagnetically induced noise and vibration, and can be
evaluated analytically by Maxwell’s stress method [1]. Thus,
f rad (θ s , t ) =
1
⎡ Br2 (θ s , t ) − Bθ2 (θ s , t ) ⎤⎦
2µ0 ⎣
to produce the effective torque. Therefore, taking into account
the rotor and stator space MMF harmonics the air-gap flux
density due to magnets and stator currents can be presented as:
∞
∗
⎛
⎞
p
Bm (θ s , t ) = ∑ η Bˆ m ⋅ cos ⎜η ∗ psθ s − η ∗ s ω t − ϕη ∗ ⎟
p
η =1
R
⎝
⎠
pR
∗
⋅η,
η =
η = 1, 3, 5,...
ps
(1)
where, f rad is the radial component of force density ( N/m2 ),
Br and Bθ are the radial and tangential components of the
magnetic flux density in the air-gap, µ 0 is the permeability of
free space, θ s is the angular position at the stator bore, and t is
the time.
Different aspects of magnetic noise have been subjects of
many papers ( [1 to 4]), and it is well established that a major
part of this noise is due to the space harmonics in the magnetic
flux.
∞
Bi (θ s , t ) = ∑ ν Bˆi ⋅ cos (ν psθ s − ω t − ϕν )
(2)
(3)
ν =1
where, ps is the number of stator pole-pairs and pR = 5 is the
true number of rotor pole-pairs, η is the rotor space harmonic
order, ϕη is the phase angle of rotor MMF harmonics, ω = pω R
is the stator current angular frequency, ν is the stator space
harmonic order and ϕν is the phase angle of stator MMF
harmonics.
III.1 AIR-GAP FLUX DENSITY COMPONENTS
It is important to underline here that for the studied PM
machine the MMF stator and rotor working harmonics are
different; the 5th harmonic of the stator interacts with the 1st
harmonic (fundamental) of the rotor to produce the effective
torque). This type of the machine can be considered so as the
both stator and rotor have the same number of pole-pairs,
th
pr = ps = 1 , and the 5 MMF harmonic of the stator (5xps=5)
III.2 RADIAL MAGNETIC FORCE DENSITY COMPONENTS
For the sake of simplicity the harmonic content and
frequencies of the radial force density will be examined in
following by considering the contribution due to the radial flux
density components alone. Analogous equations like for the
radial components hold true for the tangential components.
Thus from the equ. (1) we get,
interacts with the 5th MMF harmonic of the rotor (5x pr =5)
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f rad (θ s , t ) =
2
1
⎡ Bm,r (θ s , t ) + Bi,r (θ s , t ) ⎤⎦
2µ0 ⎣
(4)
Using
equs.
η ∗ = ( p R / ps ) ⋅ η = 5 ⋅ η
number of radial force waves:
•
IV. AMPLITUDE, FREQUENCY AND ORDER OF MAGNETIC
FORCE DENSITY
(2) and (3) into (4), and for
leads to three groups of the infinite
2
⎡⎣ Bm,r (θ s , t ) ⎤⎦
of
harmonics j , k ,5 ⋅ η = 5, 15, 25,...
The product
the rotor MMF
According to the equs. (5) to (8) the following Table-1
summarises all the resultant time-varying radial force
components, in terms of their magnitude, frequency, and
spatial order.
TABLE-1: Magnitude, frequency, and spatial order of time-varying
components of radial force density
1 ∞ ∞ jˆ kˆ
frad ,η∗ (θs , t ) =
∑∑ Bm,r ⋅ Bm,r ⋅
4µ0 j =5 k =5
⎤
ps
⎪⎧ ⎡
⎨cos ⎢( j − k ) psθs − ( j − k ) ωt − (ϕ j − ϕk ) ⎥
p
R
⎦
⎩⎪ ⎣
(5)
⎡
⎤ ⎪⎫
p
+ cos ⎢( j + k ) psθs − ( j + k ) s ωt − (ϕ j + ϕk )⎥ ⎬
pR
⎣
⎦ ⎪⎭
•
The product 2 ⋅ Bm,r (θ s , t ) ⋅ Bi,r (θ s , t ) of the stator ν and
rotor η ∗ harmonics
frad (θs , t ) =
1 ∞ ∞ ν ˆ η∗ ˆ
∑∑ Bi,r ⋅ Bm,r ⋅
2µ0 ν =1 η∗ =5
⎧⎪ ⎡
⎤
⎛
∗
∗ ps ⎞
⎨cos ⎢(ν − η ) psθs − ⎜1 − η
⎟ωt − ϕν − ϕη∗ ⎥
pR ⎠
⎥⎦
⎝
⎪⎩ ⎢⎣
(
)
⎡
⎛
p ⎞
+ cos ⎢(ν + η∗ ) psθs − ⎜1 + η∗ s ⎟ωt − ϕν + ϕη∗
pR ⎠
⎢⎣
⎝
(
•
(6)
⎤⎫
)⎥⎥⎬⎪⎪
⎦⎭
2
The product ⎡⎣ Bi,r (θ s , t )⎤⎦ of the stator MMF harmonics
j, k ,ν = 1, 3, − 5, 7,...
frad ,ν (θ s , t ) =
1 ∞ ∞ jˆ kˆ
∑∑ Bi,r ⋅ Bi,r ⋅
4µ0 j =1 k =1
⎧
⎨cos ⎡⎣( j − k ) psθ s − (ϕ j − ϕk )⎤⎦
⎩
V. RADIAL FORCE DENSITY OF THE STUDIED PM MACHINE
UNDER LOAD
(7)
⎫
+ cos ⎡⎣( j + k ) psθ s − 2ωt − (ϕ j + ϕk )⎤⎦ ⎬
⎭
In accordance with equs. (4) to (7), the magnetic forces per
unit area can be presented in following general form:
∞
frad (θ s , t ) = ∑ m Fˆrad ⋅ cos ( m ⋅ θ s − ωmt − ϕm )
The following figure 3 shows the cross section of the
studied PM machine. The stator of this machine type consists
of 24 slots in which a concentrated winding topology presented
in the second section is located, and the rotor consists of 3x20
rectangular permanent magnets insetted in the rotor core in
V-form.
(8)
m=0
where m Fˆrad is the amplitude of the radial magnetic force (morder), ω m is angular frequency and m = 0, 1, 2, 3,... are
corresponding order of radial magnetic forces.
Fig. 3: Cross section and the no-load flux density of the studied PM machine
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Using 2D ANSYS FE program the field distribution inside
the studied PM machine is obtained. The FE simulations are
performed under iˆ = 460 A , δ = 20° load operation condition.
The following figure 4 shows the distribution of radial and
tangential air-gap flux density components and its
corresponding space harmonics. Afterward, the radial forces
are calculated according to the equ. (4). Figure 5 shows the
resulting radial force density distribution as function of θ s and
its corresponding space harmonics. As is shown, the radial
force density under load condition contains all even space
harmonics of order 0, 2, 4, 6, 8,… as a result of the interaction
between the odd space harmonics in the permanent magnet
field and the armature reaction field distribution. The strongest
radial force harmonics are 0, 2, 10, and 12.
separately. To take into account the effect of the armature
reaction flux density on the flux density due to magnets and
vice versa when these flux density components are determined
separately, the fixed permeability method (FPM) [5, 6] is used
during magnetic field analysis with ANSYS.
VI. CONTRIBUTION OF THE FLUX DENSITY COMPONENTS
The following figure 6 shows the distribution of the
tangential and radial flux density components due to magnets
and its corresponding space harmonics when the PM machine
operate under the load condition. From the waveforms it can be
seen that the permanent magnet creates a significant radial
component of flux density. There are changes in the radial
components around the regions directly below stator slots.
Comparing the space harmonics in figure 6 it is shown that 5th ,
7th, 15th, 17th, 19th, 25th, etc. are the dominant harmonic
components of the radial flux density. It is important to
underline here that the air-gap PM flux density consists of the
rotor MMF harmonics, slotting effect harmonics, and the
harmonics due to the magnetic saturation. The rotor MMF
harmonics of the studied machine are: 5th, 15th, 25th,…. which
agrees with the rotor space harmonic of the PM flux density
given in the equ. (5). However, the rest of harmonics in the
figure 6 are the spatial harmonics due to slotting effect, e.g.
3rd, 7th, 17th, etc..
Fig. 4: Air-gap flux density of the studied PM machine under load
Fig. 6: Air-gap PM flux-density under load
Fig. 5: Radial force density under load
The above figures 4 and 5 show the total flux density in the
middle of the air-gap of the studied machine and resulting
radial force density under load operation. However, to show
the contribution of each component of the flux density (PM
field and the current field) on the radial forces in following
analysis these components and their harmonics are investigated
Further the following figure 7 shows the distribution of the
tangential and radial flux density components due to stator
currents and its corresponding space harmonics when the PM
machine operates under the load condition. It is shown that the
armature reaction field contains harmonics of order 1st, 5th, 7th,
13th, 15th, etc.. For the 10-pole machine, however, only the 5th
stator space harmonic MMF interacts with the field of the
permanent magnets to produce continuous torque. The other
MMF space harmonics, in particular the 1st, 7th, 17th, etc.,
which have relatively large magnitudes, may cause undesirable
effects, such as localised core saturation, noise and vibration
2081
and eddy current loss in the magnets, which are the main
disadvantages of this winding type. In analogy to the PM
air-gap flux density, also the current flux-density is effected
from the slotting effect, rotor permeance effect and the
saturation condition in the machine.
fields. It is shown that most of the results obtained from the FE
analysis agree with analytical formulation resumed in the
Table-1.
magnet field
a)
armature field
Fig. 7: Air-gap current flux-density under load
When the air-gap flux density is determined, the magnetic
radial force density can be obtained according to the
expressions (4) to (8) given in the section 3.2 of this paper.
Figure 8 shows the components of the magnetic radial force
densities and corresponding harmonic order (modes) of the
studied machine under load condition. The simulation results
show that all force density components have the same
harmonic order (odd order).
b)
Interaction between magnet
and armature field
c)
Fig. 8: Radial force density components under load
Fig. 9: Radial force density of the studied machine: magnitude, frequency and
spatial order (modes)
Further, the following figures 9-a) to 9-c) show the results
for the frequencies and harmonic order of the radial force
density components obtained as result of the magnet field,
armature field and the interaction of the magnet and armature
The obtained results show that the radial force density
distribution of the studied machine contains a very strong 2nd
order space harmonic which results mostly from the interaction
of the 5th working harmonic field with the 7th harmonic.
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Therefore, since this type of machines contains this “two-pole”
force density harmonic it may be more susceptible to resonant
vibrations within its operating speed range. The rotor speed
that may induce the resonant vibration in such a type of
machine will be much lower than that for inducing resonances
in conventional AC machines with p>2 [4].
VII. CONCLUSION
This work deals with an accurate and detailed analysis of the
magnetic radial force density of a PM machine with 12-teeth
10-poles winding topology. A combination of analytical and
FE methods is used during analysis of radial forces. The
air-gap flux density components are determined using 2D
ANSYS program. Afterwards, the radial force density
component are calculated using analytical expressions. Further,
using FFT analysis the harmonic orders of the air-gap flux
density and resulting radial forces are obtained. It is shown
that, due to the presence of a large number of low and high
order space MMF harmonics, low frequency modes of
vibration are excited in this type of machines.
REFERENCES
[1] Gieras, J. F.; Wang, C.; Lai, J. C.: “Noise of Polyphase Electrical Motors”,
Taylor & Francis Group, 2006. ISBN 0-8247-2381-3.
[2] Ellison A. J., Yang S. J.: “Effects of Rotor Eccentricity on Acoustic Noise
from Induction Machines”. Proc. IEE, Vol. 118, No.1, 1971.
[3] Wallace A.: “Current Harmonics and Acoustic Noise in AC AdjustableSpeed Drives”. IEEE Trans. Ind. App., Vol., No 2, 1990.
[4] Wang, J.; Xia, Zh. P.; Howe, D.; Long, S. A..: “Vibration Characteristics
of Modular Permanent Magnet Brushless AC Machines”. IEEE IAS
Annual Meeting, 2006, Tampa, Florida, USA.
[5] Kwak, S. J.; Kim, K. J.; Jung, H. K.: “The characteristics of the magnetic
saturation in the interior permanent magnet synchronous motor”,
International Conference on Electrical Machines, 2004 Cracow-Poland
(ICEM 2004).
[6] Bianchi, N.; Bolognani, S.: “Magnetic models of saturated interior
permanent magnet motors based on finite element analysis”, The IEEE
IAS Annual Meeting, 1998.
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