Analytical Method for Predicting the Air-Gap
Flux Density of Dual-Rotor PermanentMagnet (DRPM) Machine
W.Xie, G.Dajaku*, D.Gerling
Institute for Electrical Drives and Actuators, University of Federal Defense, Munich
Werner-Heisenberg-Weg 39
85577 Neubiberg, Germany
Tel: (+49)-89-60044126, fax: (+49)-89-60043718
e-mail: Xie.Wei@unibw.de
harmonics and slotting effect were analyzed, the leakage
flux is represented by the leakage coefficient.
This paper presents the principle and process of building
equivalent magnetic circuit. Based on the geometry and
path of magnetic field, DRPM machine works like two
conventional machines in series [1], [6]. In section three,
finite-element method (FEM) also verifies the
independence of the separate flux path of inner part and
outer part. Through the separate principle, the equivalent
magnetic circuits can be analyzed separately. Reference [3]
has built equivalent magnetic circuit which takes into
account leakage flux based on single surface-mounted
permanent-magnet (SMPM) machines. Reference [5] has
analyzed the equivalent magnetic circuit taking account
slotting effect and harmonic of interior-type permanentmagnet (IPM) machines. However, for the equivalent
magnetic circuit of DRPM machine, there is nearly no
research on the equivalent magnetic circuit. This paper
introduces the complete equivalent magnetic circuit of
DRPM machine. Compared with the accuracy, the easy
implementation is also important, thus the analytical
method has ignored the saturation of iron. And finally, the
analytical model was verified by comparison with FEM
results.
Abstract-- In this paper an analytical method for designing
DRPM machine is presented. As well known, DRPM machine
has high efficiency, high torque density and can be used as
Electric Vehicle Drive and so on, thus developing a simple and
accurate analytical method is necessary. The aim of this paper
is to calculate the air-gap flux density taking into account the
harmonics of the winding current, the winding type and the
influence of radial and parallel magnetized magnets. A
magnetic equivalent circuit of DRPM machine is developed,
and both inner and outer magnet circuits are analyzed
separately. Compared with finite-element method (FEM), the
analytical method gets an accurate result.
Keywords – DRPM Machine, Analytical Method, Air-Gap
Flux Density, Slotting Effect
I.
INTRODUCTION
T
HIS paper introduces an analytical method of DualRotor Permanent-Magnet Machines (DRPM) design.
DRPM machines can substantially improve the
efficiency due to the greatly shortened end windings and
sizably boost the torque density by doubling the working
portion of the air gap as well as optimizing the machine
aspect ratio [1]. Fig.1 shows the geometry of DRPM
machine. Due to the advantages, many applications can be
envisaged in the future. Several research work on designing
and optimizing DRPM machines have already been done
[1]-[3]. However, there is no accurate and simple analytical
method to design or calculate the DRPM performance
before going to the step of finite element method (FEM)
until now, so this paper introduces a mathematical method
to predict the air-gap flux density of DRPM machine. Airgap flux density distribution has an important position for
the machine design process. In order to improve the
accuracy of analytical method, many earlier works have
taken into account the leakage flux, the influence of
different magnetization, the slotting effect, the saturation in
stator and rotor, the winding harmonics and so on [3]-[5].
But considering about the accuracy, the influence of the
TABLE I
REQUIREMENT AND ASSUMPTIONS
Parameters
Number of pole pair
Diameter/Length
Outer rotor outside radius
Outer rotor inside radius
Outer rotor magnet inside radius
Outside radius of stator
Inside radius of stator
Inner rotor magnet outside radius
Inner rotor magnet inside radius
Inside radius of inner iron
W.Xie is with the Institue of Electical Drives, University of Federal
Defense Munich, Germany (e-mail: Xie.Wei@unibw.de).
G.Dajaku is Senior Scientist with FEAAM GmbH, D-85577
Neubiberg, Germany (e-mail: Gurakup.Dajaku@unibw.de).
D.Gerling is Head of the Institute of Electrical Drives, University of
Federal
Defense
Munich,
Germany
(e-mail:
Dieter.Gerling@unibw.de).
978-1-4673-0141-1/12/$26.00 ©2012 IEEE
2758
value
2
1.3
87.5
78.9
71.9
71.2
30.2
29.6
22.1
10.6
units
mm
mm
mm
mm
mm
mm
mm
mm
Fig.3. Equivalent magnetic circuit of DRPM machine.
Fig.1. Cross-section of the dual-rotor PM machine
II.
ANALYTICAL METHOD
In this section the analytical method to predict the airgap flux density of DRPM is introduced. Since the
topology has two “separate” air gaps and two different flux
paths, thus the design equations can be separated into two
portions: one is for the inner rotor, inner permanent magnet,
and the stator; the other one is for the outer rotor, outer
permanent magnet, and the stator [1]. As shown in Fig.2,
the green lines represent the main flux path. Two flux loops
use different paths: loop 1 represents the flux path of outer
part; loop 2 represents the flux path of inner part; and the
black circuits present the leakage flux.
Fig.4. Equivalent magnetic circuit without leakage flux.
According to Fig.2, the inner portion can be assumed as
a complete single rotor surface mounted PM (SMPM)
machine. And according to reference [3], [4], the equivalent
magnetic circuit (EMC) was build, as shown in Fig.3. In
order to simplify the model, the leakage flux can be
represented by the leakage coefficient. Thus the circuit can
be simplified to Fig.4. Based on the basic theory, Equation
(1) is used, and it is simple, easy and widely used in many
literatures for the analysis of electric machines. The air-gap
flux density is produced by the magneto motive force
(MMF) and the air-gap permeance. Here B ( x, t ) is the flux
density, Λ ( x ) is permeance and Θ ( x , t ) is the magneto
motive force (MMF). In this paper, the improved analytical
method taking into account the permeance based on EMC
and MMF based on stator winding and magnet was
analyzed [4]. And it is denoted that for the inner portion,
the air-gap flux density depends on the equation (2).
Fig.2. Flux path of DRPM.
B ( x, t ) = 2 Λ ( x ) ⋅ Θ ( x, t )
(1)
Bg1 (φs1 , t ) = Λ1 (φs1 ) ⋅ [Θs _1 (φs1 , t ) + Θr _1 (φs1 , t )]
(2)
Where
φs1 is the displacement along the stator circumference,
Λ 1 (φ s1 ) is the function of inner flux path permeance,
Θs _1 (φs1 , t ) is inner MMF due to inner stator winding,
Θr _1 (φs1 , t ) is inner MMF based on inner rotor magnet.
2759
In order to enable the analytical solution and simplify the
analytical model, the following assumptions are made.
• End effects are ignored.
• The
permanent
magnets
have
linear
demagnetization characteristic.
• Stator and rotor back-iron has infinite permeability,
the saturation effects of the back iron are
neglected.
• The airspace between the magnets and iron teeth
has the same permeability as the magnets.
• The eddy current effect is neglected.
A.
Permeance
The flux permeance depends on the path of flux. As the
rotor rotates, the flux permeance of some type of machine
will change cyclically, for example inset PM machine. But
in this paper for DRPM, because of the rotor magnet is on
the surface of the rotor iron, and the permeance of magnet
is nearly equal to air, the permeance can be assumed as a
constant. Fig.5 shows the flux path of inner part. R0 is the
magnetic resistance of inner air gap, R1 is the magnetic
resistance of inner stator teeth, R2 is the magnetic
resistance of inner magnet, R3 is the magnetic resistance of
inner rotor yoke, R4 is the magnetic resistance of inner
stator yoke. The magnetic resistance can be computed from
the relationship of length of the magnetic path, the
permeability and the cross-sectional area. Equation (3) is
used to calculate the average value of inner flux permeance
but ignores the cross-sectional area, since the destination is
just to determine the distribution of the air-gap flux density
[2]. For the outer EMC, the component of the outer circuit
is nearly the same with inner one, thus in this paper, only
the inner part of DRPM machine is analyzed.
lgap1 is length of inner air gap,
H s1_ teeth is length of inner stator teeth,
lr1_ Yoke is length of flux path in inner rotor yoke,
ls1_ Yoke is length of flux path in stator yoke.
From the geometry data Table, the result of inner
permeance has been analyzed. According to the
mathematical model, the permeance is a constant with the
different rotor position.
As well known the permeance of inner flux path in Fig.5
does not consider the stator slotting effect, but in fact, the
stator slots not only influence the flux amplitude but also
influence the distribution of the permeance. As shown in
Fig.6, it can be seen that the detail of flux path of circle 1
(C1), and R s 0 represents the magnetic resistance of flux
path in slot. Because of the topology, both right and left
slots have the flux path (line 1) in Fig.6. As a result of
slotting effect the total permeance will decrease (which
means the magnetic resistance will increase) and the
permeance in the slot position will be much smaller than
average as shown in Fig.7 [9] . A new analytical method
was introduced as shown in equation (4), which is used to
calculate the value of permeance produced by only slotting
effect [4]. It considers the harmonic of slot permeance.
Fig.7 shows the result of inner permeance taking into
account slotting effect. It can be seen that in slot position,
the permeance has a clear downward trend which means the
magnetic reluctance amplitude becomes higher.
Fig.6. Inner flux path taking into account the slotting effect.
∞
Λ s 0 = Λ s 0,0 + ∑ Λ s 0, k ⋅ cos( k ⋅ N s ⋅ P ⋅ φs1 )
k =1
Fig.5. Flux path and EMC of inner part.
H r1_ teeth l gap1 H s1_ teeth
lr1_ Yoke ls1_ Yoke
1
)+
= 2⋅(
+
+
+
Λ10
μ0
μ0
μ0 * μ r
μ0 ⋅ μ r μ0 ⋅ μ r
(3)
Where
H r1_ teeth is thickness of inner magnet,
2760
(4)
-5
7.4
x 10
400
300
MMF based on inner stator winding [At]
Inner flux path permeance [H]
7.2
7
6.8
6.6
6.4
6.2
6
5.8
200
100
0
-100
-200
-300
0
1
2
3
4
5
Angle of Inner Rotor position [rad.elc.deg]
6
-400
7
Fig.7. Distribution of inner flux permeance taking into account the slotting
0
1
2
3
4
5
Angle of Inner stator position [rad.elc.deg]
6
7
Fig.9. MMF distribution of inner distributed winding topology with q=1.
effect.
C.
B.
Magnetomotive Force of Winding
Magnetomotive Force of Magnet
For any types of PM machines, knowledge of magnetic
field distribution is a prerequisite for predicting
performance parameters of PM machines, for example
torque, back-EMF, losses and so on [7]. For analytical
method, the distribution of different magnetization will
influence the waveform and accuracy of the calculation.
Fig.10 introduces radial and parallel magnetization
distributions. Fig.10-(a) represents parallel magnetization
distributions and waveform of radial magnetization is
shown in Fig.10-(b). Compared with radial magnetization,
parallel magnetization is much easier to be obtained [8]. In
this paper, parallel magnetization distribution is used.
After calculating the permeance, before predicting the
flux density, the MMF based on stator winding should also
be considered. Depending on the winding type, there will
be different MMF characteristics [4]. As well known, this
distribution of winding for DRPM machine can decrease
the end winding, and then increase the efficiency [1]. Now
there is a general expression for the MMF distribution
presented in equation (5) and (6) [4]. Here one layer
distributed winding topology and simulation result were
analyzed, as shown in Fig.8 and Fig.9.
Θ pm
2π
Θ pm
Fig.8. Distribution of stator wingding [6].
ξ w,ν = ξ p ,ν ⋅ ξ d ,ν
ν ⋅π
sin(
)
y1 ⋅ν ⋅ π
2m
)⋅
= sin(
τ p ⋅ 2 q ⋅ sin(ν ⋅ π )
2mq
Θ s _1 (φs1 , t ) = ∑
ν
m 2 w ⋅ ξ w,ν
⋅ ⋅
⋅ i ⋅ cos( wt −ν pφs1 + δ )
2 π
ν
2π
β mag
(5)
Fig.10. Parallel and radial magnetization and waveform [7].
As it is mentioned above, the distribution of rotor MMF
not only depends on the magnet topology but also on the
magnetization direction. For inner portion, the inner rotor is
nearly the same as the rotor of surface mounted PM
(SMPM) machine. After choosing the magnet type as
shown in Fig.10-(a), a general expression for the rotor
∧
MMF of magnet is presented in equation (7). ξ and ΘPM 1
denote the space harmonics and corresponding amplitudes.
The maximum of the MMF is based on the coercive field
intensity and thickness of the magnet, here βmag is magnet
pole width (pole arc).
(6)
Where:
ξ p,ν
is pitch factor,
y1 is pitch,
ν is harmonic times,
τ p is pole pitch,
ξ d ,ν is distributed factor,
ξw,ν
is winding factor.
∧
Θ r _1 (φR1 , t ) = ∑ Θ PM 1 ⋅ cos(ξ ⋅ p ⋅ φr1 )
ξ
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(7)
flux density due to winding
flux density due to magnet
outer air-gap flux density
0.8
0.6
Outer air-gap flux density [T]
∧
Θ PM 1
⎧
β mag
β mag ⎫
β mag ⋅ ΘPM 1 ⎪⎪ sin((ξ + 1) 2 ) sin((ξ − 1) 2 ) ⎪⎪
=
+
⎨
⎬
β
π
⎪ (ξ + 1) β mag
(ξ − 1) mag ⎪
⎪⎭
2
2
⎩⎪
(8)
Based on the geometry data, Fig.11 shows the
distribution of MMF based on inner magnet.
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
1
2
3
4
5
Angle of Outer rotor position [rad.elc.deg]
6
7
Fig.13. Distribution of outer air-gap flux density.
III. SIMULATION RESULT
In the following, the Finite Element Method (FEM) is
employed to verify the effectiveness of the analytical
model, redefine the geometry factors, find the suitable
performance, and verify if the machine can survive under
the stator saturation and the limitation of current density.
Thus the studied DRPM machine is analyzed using 2DMaxwell finite element software.
Fig.14-(a) shows the flux lines of DRPM machine on
2D-Maxwell. Outer flux loops are constituted with stator
yoke, stator teeth, outer air-gap, outer magnet and outer
rotor; inner flux loops contain stator yoke, stator teeth,
inner air-gap, inner magnet, and inner rotor. The flux loops
denoted the effectiveness of the assumption of Fig.2. And
Fig.14-(b) represents the flux vector for DRPM machine.
Fig.11. Distribution of MMF based on inner magnet.
According to equation (2), both of Θ r _1 (φR1 , t ) and
Θ s _1 (φs1 , t ) affect the air-gap flux density. In order to
increase the accuracy, not only the slotting effect, but also
the harmonic and magnetization type was considered.
Fig.12 and Fig.13 show the calculation result of inner part
air-gap flux distribution and outer part air-gap flux
distribution.
flux density due to winding
flux density due to magnet
Outer flux loop
inner air-gap flux density
Inner flux loop
0.5
0.4
Outer air-gap flux density [T]
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
Outer flux vector
-0.4
-0.5
0
1
2
3
4
5
Angle of Outer rotor position [rad.elc.deg]
6
Inner flux vector
7
Fig.12. Distribution of inner air-gap flux density.
Fig.14. Distribution of flux of DRPM machine; a). flux lines, b). flux
vector.
2762
With the same exciting current and magnetization of
magnet, Fig.15 shows the distribution of flux density of
DRPM machine. Compared with the simulation of FEM,
the implementation of analytical method is much easier.
The obtained result shows the agreement between
analytical and FEM method which also takes into account
the slotting effect, magnetization and harmonics.
[9]
machines," IECON 2011 - 37th Annual Conference on IEEE
Industrial Electronics Society, vol., no., pp.1776-1782, 7-10 Nov.
2011.
Andriamalala, R.N.; Razik, H.; Baghli, L.; Sargos, F.-M.; ,
"Eccentricity Fault Diagnosis of a Dual-Stator Winding Induction
Machine Drive Considering the Slotting Effects," Industrial
Electronics, IEEE Transactions on , vol.55, no.12, pp.4238-4251,
Dec. 2008.
Wei Xie was born in China, 1982. He graduated from the Northwestern
Polytechnical University in Xi’an, China in 2010 and is now at the
Institutes of Electrical Drives, University of Federal Defense, Munich,
Germany as a PhD student. His technical interest includes electromagnetic
field analysis and switched reluctance machines.
Gurakuq Dajaku; Dr.-Ing. Gurakuq Dajaku is with FEAAM GmbH,
Werner-Heisenberg-Weg 39, D-85577 Neubiberg, Germany, phone: +49
896004-4120, fax: -3718, e-mail: Gurakuq.Dajaku@unibw.de.
Born in 1974 (Skenderaj, Kosova), got his diploma degree in Electrical
Engineering from the University of Pristina, Kosova, in 1997 and his Ph.D.
degree from the University of Federal Defense Munich in 2006. Since2007
he is Senior Scientist with FEAAM GmbH, an engineering company in the
field of electric drives. From 2008 he is a Lecturer at the University of
Federal Defense Munich. His research interests include the design,
modelling, and control of permanent-magnet machines for automotive
application.
Dr. Dajaku received the Rheinmetall Foundation Award 2006 and the ITIS
(Institute for Technical Intelligent Systems) Research Award 2006.
Fig.15. Air-gap flux density due to one pole outer magnet of studied
DRPM machine.
IV. CONCLUSION
Dieter Gerling; Prof. Dr.-Ing. Dieter Gerling is head of the Institute of
Electrical Drives at the University of Federal Defense Munich, WernerHeisenberg-Weg 39, D-85579 Neubiberg, Germany (phone: +49 89 60043708; fax: -3718; email: Dieter.Gerling@unibw.de).
Born in 1961, Prof. Gerling got his diploma and Ph.D. degrees in
Electrical Engineering from the Technical University of Aachen, Germany
in 1986 and 1992, respectively. From 1986 to 1999 he was with Philips
Research Laboratories in Aachen, Germany as Research Scientist and later
as Senior Scientist. In 1999 Dr. Gerling joined Robert Bosch GmbH in
Bühl, Germany as Director. Since 2001 he is Full Professor at the
University
of
Federal
Defense
Munich,
Germany
(http://www.unibw.de/EAA/).
Air-gap flux density in PM machines is an important
parameter for predicting the performance of PM machines.
In this paper, for the distribution of air-gap flux density of
DRPM machine, the new analytical method takes into
account the harmonics of permeance, stator winding,
magnet and the slotting effect, and improves the accuracy.
The inner part and outer part are analyzed separately, and
the air-gap MMF components are also considered
separately for the stator winding and magnets. This paper
has denoted that this method is simple and effective for
DRPM machine design. The analytical method has been
verified by FEM calculations.
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