9th World Congress on Structural and Multidisciplinary Optimization
June 13 - 17, 2011, Shizuoka, Japan
Shape Optimization of Pole Shoes in Interior Permanent Magnet Synchronous Machines
Jae Seok Choi1*, Kazuhiro Izui1, Shinji Nishiwaki1, Atsushi Kawamoto2 and Tsuyoshi Nomura2
1
Kyoto University, Kyoto,Japan, 1* jschoi.ok@gmail.com
2
Toyota Central R&D Laboratories Inc., Aichi, Japan
1. Abstract
This study presents a method for shape optimization of rotor pole shoes in interior permanent magnet (IPM)
machines, to effectively produce a sinusoidal distribution of the air-gap flux density. A sinusoidal field in the
air-gap offers several advantages, such as reduction of torque ripple when using sinusoidal phase currents, and
reduction in cogging torque and higher harmonic components of back electromotive force (EMF). In this study, the
optimization process is achieved based on a phase field model using the reaction-diffusion equation. Approaches
using the reaction-diffusion equation are a kind of shape optimization method, and thus have an interface-tracking
property similar to level set-based optimization methods where the Hamilton-Jacobi equation is used. Due to this
interface-tracking property, the design sensitivities are applied only at the diffuse interfacial layer. In
highly-nonlinear magnetic field problems such as IPM machines, regions with meaningful design sensitivity
values sometimes lie outside the interfacial layer, and as a result, it may become impossible to update the phase
field. Therefore, in our approach, the design sensitivities are diffused in the radial direction, using a diffusion
equation, so that they can be expanded at the phase transition region. The evolution of the phase field in time is
based on the implicit finite element method, and the augmented Lagrangian method is used to deal with the volume
constraint.
2. Keywords: Interior permanent magnet machines, Shape optimization, Phase field model
3. Introduction
With the development of powerful permanent magnet (PM) materials, such as those employing neodymium, PM
motors have been extensively used in a wide variety of applications. PM motors are usually classified into two
groups: 1) brushless DC PM motors, and 2) PM synchronous motors. Brushless DC PM motors are driven by
current with rectangular waveforms, hence, trapezoidal back electromotive force (EMF) and a strong pulsating
torques are produced. On the other hand, PM synchronous motors are driven by sinusoidal currents so the back
EMF is also sinusoidal. Ideally, PM synchronous motors should have a sinusoidal air-gap flux density distribution
[1]. The flux density distribution in the air-gap is affected by the magnet flux as well as the winding distribution,
and desired flux distributions can be achieved by appropriately shaping the magnets or rotor poles [2-4].
PM synchronous motors that incorporate a sinusoidal air-gap flux distribution have a number of advantages
[2]. By making the phase EMF more sinusoidal, torque ripple can be reduced. Also, it is possible to achieve a low
cogging torque. Moreover, such motors have smaller eddy current losses in the stator teeth, compared with
brushless DC PM motors, and thus offer improved machine efficiency.
Until recently, most research pertaining to the optimization of motors has been based on size optimization,
but gradient-based optimization methods using design sensitivities are increasingly popular now. Lee et al. [5]
designed switched reluctance motors based on the density method, which is a kind of topology optimization
method. Kwack et al. [6] optimized the stator shoes of interior permanent magnet (IPM) motors using a level
set-based design method. Labbé et al. [7] maximized the torque-to-weight ratio of PM synchronous motors based
on the density method.
Herein, we present a shape optimization method for rotor pole shoes in IPM motors, to obtain a sinusoidal
flux density distribution in the air-gap between the rotor and the stator. The shape optimization is based on the
phase field model using a reaction-diffusion equation. This optimization method has an interfacial-tracking
property, due to the use of double well potential, but in contrast to level set-based methods using the
Hamilton-Jacobi equation, ours does not require a reinitialization scheme [8,9].
4. Optimization algorithm
4.1. Topology optimization using a reaction-diffusion equation
Reaction-diffusion equations are mathematical models that can describe many complicated natural phenomena
that occur due to the propagation of interfaces between two or more substances. These equations express two
processes, namely, a ‘diffusion’ process describing how substances spread in space, reducing their concentration,
and a ‘reaction’ process that describes how substances react with each other locally. The reaction term may be
1
modified to accommodate specific problems such as population dynamics [10], combustion [11] and chemical
reactions [12].
Recently, reaction-diffusion equations have been used in topology optimization and shape optimization
methods [8,9,13]. Optimization methods based on a reaction-diffusion equation tend to have simpler and more
intuitive implementations compared with level set-based optimization methods. Choi et al. [9] proposed a
topology optimization method using a reaction-diffusion equation in which design sensitivities are simply
employed in the reaction term, as in the following equation:

F ( , u)
(1.1)
  2 
in T :   (0, T ) ,
t


(1.2)
0
in T :   (0, T ) ,
nˆ
where   R N (N=2) is a bounded domain with boundary  ,  is the density field determining the material
properties and u is the system response. The right-hand side of Eq. (1.1) is composed of a diffusion term and a
reaction term, F /  , which is the gradient of the augmented Lagrangian F . The diffusion coefficient 
determines the thickness of the diffuse interfacial layer. As this value is reduced, the layer thickness is
decreased.
In general, an optimization problem for a structural topology design is written as
minimize F  , u( ) 

subject to G ( ) 
  dx  V
req

 0,
(2)
0    1,
where F ( , u) and G( ) represent the objective function and the volume constraint, respectively.
The augmented Lagrangian F for the above case is defined as
r
F ( , u,  )  F ( , u)  G( )  G( ) 2 .
(3)
2
Here,  is the Lagrange multiplier and r is a penalty parameter. The augmented Lagrangian is minimized using
the reaction-diffusion equation. The gradient of the Lagrangian with respect to the density field is given by
F ( , u,  )
F ( , u) G( )
  rG( ),


(4)



where  is a parameter normalizing the design sensitivity F /  , and is defined as

 dx

F / 
.
(5)
L2 (  )
4.2. Phase field-based shape optimization
For the shape optimization, the gradient of the Lagrangian F /  is simply modified as
F ( , u,  )
F ( , u)
G( )
  rG( ) ,
 30
f ( )  f ( ) 



where f is a double well potential and is defined as
(6)
(7)
f ( )   2 (1   ) 2 .
The double well potential makes the values of the sensitivities close to global minima very small, hence the
interface-tracking property of the phase field model.
4.3. Diffusion of the design sensitivities
Design methods that use the phase field model are a kind of shape optimization, similar to level set-based design
methods. In the proposed method, the updating of the design variables is possible only in the diffuse interfacial
layer. In this study, the design domain is a rotor pole, in which a permanent magnet is embedded. A
neodymium-based permanent magnet material is assumed so that the portions close to barriers are saturated. For
the core design in highly nonlinear problems such as IPM motor designs, regions having meaningful sensitivity
values sometimes lie outside the interfacial layer. Therefore, we blur the design sensitivities in the radial direction,
using the following diffusion equation:
2

(8)
   c  .
t
Here, c is a diffusivity tensor for diffusing the sensitivity in the radial direction. The tensor for the x-directional
diffusion is defined as
0 
1
(9)
c0  
,
0 0.001
and the diffusion in a specific direction is achievable using the following diffusivity tensor
1 c0,11  c0, 22  (c0,11  c0,22 ) cos 2
c( )  
(c0,11  c0, 22 ) sin 
2


x
1 
,


cos
 2
2
c0,11  c0, 22  (c0,11  c0,22 ) cos 2 
 x y
(c0,11  c0, 22 ) sin 2

.


(10)
5. Numerical examples
For the design model, we consider a two-dimensional IPM motor with 3-phase winding, 8 magnetic poles and
24 stator slots. Fig. 1 shows a 1/8 model of the motor. The outer diameters of the rotor and the stator are 80mm and
120mm, respectively. The remanent flux density of the permanent magnet material used in this study is 1.1T. The
entire analysis domain and design domain are discretized using 34,008 and 21,856 triangular three-node elements,
respectively. The magnetic vector potential and the density field are computed using finite element analysis. For
the time discretization of the density field, we use the implicit scheme, which is accurate for order 1 in time and
order 2 in space.
(a)
(b)
Figure 1: Schematic diagram of the design model (air-gap: 0.6mm, stack length: 65mm); (a) definition of the
design domain, and (b) flux line plot at a rotor angle of 180°E
5.1. Formulation of the optimization problem
The rotor and stator core are assumed to be soft iron. For the optimization of the rotor pole shoe, the magnetic
reluctivity is interpolated as follows [14]:
(11)
 ( , B 2 )   0   ( B 2 )  0  .


Here,  0 and  are the reluctivity of air and soft iron, respectively.
To obtain the optimal rotor pole shape that will produce a sinusoidal flux density in the air-gap, the
optimization problem is formulated as
minimize F ( , Az )  
 air  gap

B
rad

2
 B* d
subject to G ( )    dx  Vreq  0,
(12)

0    1,
a ( Az , Az )  l ( Az )
for Az V , Az V .
Here, Brad is the radial directional flux density norm in the air-gap. The target flux density B * is defined as
B  2 Brms sin  E , Brms 
*

 air  gap

2
Brad
d
d
,
(13)
 air  gap
where  E represents the angle in electrical measure ( 0   E  180 ). In this study, the rotor position is
3
represented based on the electrical angle.
5.2. Optimization results
Figs. 2(b) and (c) show the original design sensitivity and the diffused sensitivity, respectively. The design
sensitivity is computed based on the adjoint variable method. As shown in Fig. 2(b), high intensity sensitivity
originally appears only in the solid phase, which inhibits the phase transition. To resolve this problem, the
sensitivity is radially blurred using the diffusion equation. As a result, the initial sensitivity is expanded into the
diffuse interfacial layer, even though the sensitivity amplitude is reduced, as shown in Fig. 2(c).
(a)
(b)
(c)
Figure 2: Radially diffused sensitivity ( Vreq  0.8 , iteration number of 30); (a) density field distribution, (b)
original sensitivity, and (c) diffused sensitivity
Fig. 3 shows the optimization results for two volume constraint values ( Vreq  0.75, 0.8 ). Two optimized
configurations, respectively shown in Figs. 3(a) and (b), are nearly the same. The air gap has a minimum size at the
d-axis and shrinks gradually toward both ends of the magnet. The flux density distribution in the air gap is close to
that of a sine wave, as shown in Figs. 3(c) and (d), although there are small higher harmonic components.
(a)
(b)
(c)
(d)
Figure 3: Optimization results; (a) optimized configuration for Vreq  0.75 , (b) optimized configuration for
Vreq  0.8 , (c) optimized air-gap flux density ( Vreq  0.8 ), and (d) flux line of the optimized model ( Vreq  0.8 )
5.3. Performances of optimized models
As expected, the cogging torque, torque ripple and higher harmonics of back EMF are decreased in the optimal
configurations. Fig. 4 shows the performances of the optimized IPM motor models. The cogging torques are
remarkably reduced, by a ratio of from 1:13 to 1:35. Torque ripple values are decreased by 59.3-71.6% and average
torque values are slightly increased. To calculate the back EMF, we assume a constant speed of 500rpm. As shown
in Figs. 4(c) and (d), first order harmonics are increased while higher harmonics are increasingly suppressed.
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(a)
(b)
(c)
(d)
Figure 4: Performance measures of optimized models; (a) cogging torque profiles, (b) torque ripple profiles, (c)
back EMF, and (d) number of harmonics
6. Conclusions
In this paper, we presented a shape optimization method for rotor pole shoes using a phase field model to produce
a sinusoidal flux density in the air gap. To obtain meaningful sensitivity values in the diffuse interfacial layer, the
original design sensitivity was radially blurred using the diffusion equation. Cogging torque and torque ripple were
remarkably reduced in the optimized models, and higher harmonics of the back EMF were also decreased.
7. Acknowledgements
The authors are deeply grateful for the support received from JSPS Scientific Research (C), No. 19560142, and
JSPS Fellows.
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