Skewing Effect on the PM Flux-Linkage High Harmonics of the PM Machines with
Delta Winding
DAJAKU Gurakuq
Skewing Effect on the PM Flux-Linkage High Harmonics of the PM
Machines with Delta Winding
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 deals with the analysis of the high-harmonic contents on the PM flux-linkage and induced
back-emf of a PM machine at no-load and with delta connected winding topology. The skewing effect
on these unwanted high-harmonics is investigated for a continuously skewed and a discrete skewed
rotor topology. The FE results show that the PM flux-linkage of the studied machine contains high
harmonics of odd order. When the phase windings are connected in delta the PM flux-linkage and
resulting induced back-emf contains high harmonics of order 3rd and 9th. Further, the analysis
performed here show that using skewed rotor topology these high harmonics clearly can be reduced.
Keywords
Permanent Magnet Motor, Skewing Effect, Delta Connection Winding, Circulating Current, Finite
Element Calculation.
1 Introduction
It is well known that the air-gap flux density of permanent magnet (PM) machines is not pure
sinusoidal; it consists of time and space harmonics, slotting effect harmonics, etc. Therefore the
flux-linkage and the resulting induced voltage in each phase becomes distorted, and the waveform is
no longer sinusoidal. The distortion is mainly due to an undesired air-gap high harmonics whose
frequency is n-times that of the fundamental frequency. With a wye connection, the distorting
line-to-neutral harmonics do not appear between the lines because they effectively cancel each other.
Unfortunately, when a delta connection is used, the harmonic voltages do not cancel, but are add up.
Because the delta is closed on itself, they produce a high-harmonics circulating current, which
increases the I 2 ⋅ R losses. Further, high-harmonics circulating current have an negative effect on
others performances of electric machines, such as rotor losses, torque ripple, magnetic noises etc..
Different methodology for reducing these harmonics by selecting the appropriate magnet shape and
dimensions has been introduced in many literatures, e.g. in [1] and [2]. It is shown in [2] that in order
to keep the circulating currents at a tolerable level a magnet covering of two thirds of pole must be
used.
However, in this work the skewing effect on the high-harmonic contents of the flux-linkage and
induced voltage of a PM machines at no-load and with delta connected winding is investigated. A
continuously skewed and a discrete skewed rotor topology is taken into consideration. The analysis of
the flux-linkage high-harmonics is performed using analytical and finite element (FE) methods.
EPE 2009 - Barcelona
ISBN: 9789075815009
P.1
Skewing Effect on the PM Flux-Linkage High Harmonics of the PM Machines with
Delta Winding
DAJAKU Gurakuq
2. No-load PM flux-linkage
In a PM machine there is a flux due to the magnets which links all windings in turn, and gives the flux
linkage ψ m in each phase, even when there is no current flowing. The flux-linkage in the stator
windings due to the magnets of a PM machine considering the high-harmonics can be described as, eg.
phase-u:
∞
ψ Um (θ ) = ∑ kψˆ PM ⋅ sin ( kθ )
(1)
k
where, with θ is denoted the rotor position in electrical degree, and kψˆ PM is the peak value of k-th
harmonic of the PM flux-linkage.
For the electric machines with delta winding topology the phase windings are connected in series, and
the total flux linking the stator windings is the sum of flux-linkage of each phase. Thus,
ψ Δm (θ ) = ψ m (θ ) + ψ m (θ ) + ψ m (θ )
U
V
(2)
W
Substituting equ. (1) into (2) and after some trigonometric transformation, the total PM flux linking
the stator windings for the delta winding topology is:
∞
ψ Δm (θ ) = 3 ⋅ ∑ hψˆ PM ⋅ sin ( h ⋅θ ) ;
g = 1,3,5,...
(3)
h =3 g
The above equ. (3) shows that the harmonics h = 3 g exist in the PM flux linking the stator windings if
the phase windings of the PM machine are connected in delta.
3. Skewing effect on the harmonic contents of PM flux-linkage
While skewing is effective in reducing the harmonic content in the flux-linkage and back-emf
waveforms, as well as in reducing the torque ripple, skewing either stator teeth or rotor magnets results
in a 3-D magnetic field distribution in the active region of a machine. Figure 1 shows three cross
sections of a PM machine with skewed stator. The stator core is assumed having a skewing angle
α skew . As is shown skewing changes the relative position between the stator and rotor along the axial
length of the machine. Hence, the parameters of the machine are not constant along the motor axis.
The magnetic field distribution also varies.
Figure 1: Cross sectional views of a PM machine at different axial-z location.
Taking into account the skewing effect the machine parameters can be presented as a 3-D function of
the rotor position θ and axial length z of the machine. Therefore, the flux-linkage per length due to
magnets of a PM machine with skewed stator or rotor topology can be described using the following
equ. (4). Thus, for the k-harmonic we have (in following the phase-u is considered)
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ISBN: 9789075815009
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Skewing Effect on the PM Flux-Linkage High Harmonics of the PM Machines with
Delta Winding
⎡ ⎛
α skew ⎞ ⎤
⎣ ⎝
laxial
2D
ψ Um (θ , z ) = kψˆ PM
⋅ sin ⎢ k ⎜θ +
k
DAJAKU Gurakuq
(4)
z ⎟⎥
⎠⎦
The subscribt 2D presents the machine parameters obtained with the 2-D analysis (parameters/per
length). In the following analysis the PM flux-linkage of a skewed PM machine is derived under the
following assumptions:
•
The origin of the general reference frame is taken to be at the midpoint of the axial length of
the machine,
•
The axial variation of the saturation due to skewing is neglected, and
•
Integrating the expression for the flux-linkage per length (equ. (4)) over the active length of
the machine, − l a x ia l ≤ z ≤ l a x ia l , gives the resulting PM flux-linkage.
2
2
Thus, the k-harmonic of PM flux-linkage of a machine with skewing is:
PM
ψ Usk,m (θ ) = kξ skew
⋅ kψˆ PM ⋅ sin (θ )
(5)
k
2D
PM
where, kψˆ PM = kψˆ PM
⋅ laxial and with kξ skew
is denoted the skewing factor due to permanent magnets for
the k-th harmonic,
⎛ α ⎞
PM
= si ⎜ k ⋅ skew ⎟
ξ skew
2 ⎠
⎝
(6)
k
The above equ. (5) shows that using skewing stator or rotor topology the harmonic components of the
PM
. The following figure 2 show skewing factors for
PM flux linkage are reduced by skewing factor kξ skew
st
rd
th
the 1 (fundamental), 3 and 9 harmonic components of the PM flux-linkage.
1
k=1
0.8
0.6
PM
ξ skew
k
k=3
0.4
0.2
k=9
0
-0.2
-0.4
0
50
100
150
α skew
200
250
300
350
[degree]
Figure 2: PM skewing factors vs. skewing angle.
Referring to the figure 2 it is shown that 9th harmonic in the PM flux linkage can be eliminated using a
skewing angle α skew = 40° , α skew = 80° , α skew = 120° etc., however the 3rd harmonic can be eliminated
using a skewing angle α skew = 120° , α skew = 240° etc.. Figure 2 shows that the skewing effect reduce
also the 1st (fundamental) harmonic of the PM flux-linkage that results on the reduction of the
effective torque of the machine.
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ISBN: 9789075815009
P.3
Skewing Effect on the PM Flux-Linkage High Harmonics of the PM Machines with
Delta Winding
DAJAKU Gurakuq
4. FE results
In following analysis the FE methods are used to determine the flux-linkage of a PM machine with
interior magnets in the rotor. The following figure 3 shows the geometry and the flux-lines distribution
of the studied machine. The stator of this machine type consists of 24 slots in which a concentrated
winding topology is located, and the rotor consists of 3x20 rectangular permanent magnets insetted in
the rotor core in V-form.
a)
b)
Figure 3: Studied PM machine; a) Cross-section view, b) Flux-lines distribution at no-load.
Using FE results for the PM flux linkage and according to the equ. (2), the total PM flux-linkage of the
studied PM machine with delta connected winding topology is shown in following figure 4. It is
shown from the figure 4-a that the dominant harmonics in the delta flux-linkage are the 3rd and 9th
harmonic which agrees with the analytical formulation described in (3). Further, using the relation
e m = dψ m / dt the induced back-emf on the stator phase can be calculated. Figure 4-b shows the
no-load induced back-emf and corresponding harmonic contents of the studied PM machine at 6000
rpm and for the case when the phase windings are connected in delta. At high speed these high
harmonic voltage induce high harmonic circular currents in the stator windings which damage the
performances of the machine and also make problems during control of the machine.
a)
b)
Figure 4: PM machine with delta winding; a). ψ Δm (θ ) ; b). eΔm (θ ) at n=6000 rpm
In following analysis the skewing effect on the 3rd and 9th high-harmonics of the delta PM flux-linkage
and induced back-emf is investigated. An continuously skewed and a discrete skewed rotor topology is
considered during the following analysis.
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ISBN: 9789075815009
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Skewing Effect on the PM Flux-Linkage High Harmonics of the PM Machines with
Delta Winding
DAJAKU Gurakuq
4.1 PM machine with continuously skewed rotor topology
In following the rotor of the studied PM machine is taken to be continuously skewed. The
analysis is based on the analytical method described in the previous sections and 2-D FE
results for the PM flux-linkage. Figure 5 shows the skewing effect on the delta PM flux-linkage of
the studied machine with delta connected winding topology. The obtained results show that the
skewing effect reduce the harmonic content in the flux-linkage. At α skew = 120° the delta PM flux
linking the stator windings is reduced to zero. This is true because the both PM skewing factors for the
3rd and 9th harmonic are zero at α skew = 120° (please refer to the figure 2).
ψ Δm (θ ,α skew )
[Vs ]
α skew = 120°
α skew [0° ÷ 160°]
θ [0° ÷ 360°]
Figure 5: Skewing effect on the delta PM flux-linkage, ψ Δm (θ ,α skew ) .
Further, the following figure 6 show the skewing effect on the delta flux-linkage ψ Δm (θ ) and the
induced back-emf eΔm (θ ) of the studied PM machine where different rotor skewing angles are taken
into consideration. The obtained results shows hat for α skew = 120° and continuously skewed rotor
topology the 3rd and 9th harmonics completely can be reduced.
b)
a)
Figure 6: Skewing effect on the PM machine with delta winding; a). ψ Δm (θ ) ; b). eΔm (θ ) at n=6000 rpm.
EPE 2009 - Barcelona
ISBN: 9789075815009
P.5
Skewing Effect on the PM Flux-Linkage High Harmonics of the PM Machines with
Delta Winding
DAJAKU Gurakuq
4.2 PM machine with discrete skewed rotor topology
It is important to underline here that the production of the continuously skewed rotor topology is
expensive, hence the skewed rotor mostly are realized using discrete skewing rotor. In following a
discrete skewing rotor topology with 60° skewing angle is studied. The rotor of the studied PM
machine is assumed to consists of five axially equal slices (five-step skewed rotor). For α skew = 60° ,
the distribution of the delta flux-linkage along the axial-length of the studied machine with five rotor
parts skewed axially is illustrated in the following figure 7. However, figure 8 compares the delta
flux-linkage and induced back–emf of the studied PM machine with (five-step skewed rotor) and
without skewed rotor. The obtained results shows that also the discrete skewing rotor topology has an
considerable effect on the unwanted flux-linkage harmonics.
ψ Δm (θ , z )
[Vs ]
−
laxial
l
≤ z ≤ axial
2
2
θ [0° ÷ 360°]
Figure 7: Five step skewed rotor; Delta PM flux-linkage distribution along axial length.
a)
b)
Figure 8: Five step skewed rotor topology; a). ψ Δm (θ ) ; b). eΔm (θ ) at n=6000 rpm.
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ISBN: 9789075815009
P.6
Skewing Effect on the PM Flux-Linkage High Harmonics of the PM Machines with
Delta Winding
DAJAKU Gurakuq
Conclusion
This work deals with the analysis of the high-harmonic contents on the PM flux-linkage and induced
back-emf of a PM machine at no-load and with delta connected winding topology. The skewing effect
on these unwanted high-harmonics is investigated for a continuously skewed and a discrete skewed
rotor topology.
The FE results show that the PM flux-linkage of the studied machine contains high harmonics of odd
order. When the phase windings are connected in delta the PM flux-linkage and resulting induced
back-emf contains high harmonics of order 3rd and 9th. At high speed these high harmonic voltage
induce high harmonic circular currents in the stator windings which damage the performances of the
machine and also make problems during control of the machine. Further, the analysis performed here
show that using rotor skewing topology the harmonic contents in the PM flux-linkage and induced
back-emf clearly can be reduced. It is shown that for α skew = 120° and continuously skewed rotor
topology the 3rd and 9th harmonics completely can be reduced. However, using a discrete rotor
topology these harmonics effectively can be reduced.
References
[1] J. De La Ree, N. Boules, "Magnet Shaping to Reduce Induced Voltage Harmonics in PM Machines With
Surface Mounted Magnets", IEEE Proc. PES 1989 Summer Meeting, Long Beach, CA, July 9 - 14, 1989.
[2] E. Nipp, "Examinations of Torque Interruptions and Circulating Currents in PM Synchronous Motors with
Switched Stator Windings," Proceedings of the International Conference on Electrical Machines,ICEM'98,
Istanbul, Turkey, pp. 273-278, September 2-4, 1998.
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ISBN: 9789075815009
P.7