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COMPEL
28,2
272
Received October 2007
Revised August 2008
Accepted August 2008
Characterization and optimization
of a permanent magnet
synchronous machine
Peter Sergeant, Guillaume Crevecoeur, Luc Dupré and
Alex Van den Bossche
Department of Electrical Energy, Systems and Automation,
Ghent University, Gent, Belgium
Abstract
Purpose – The first purpose of this paper is to identify – by an inverse problem – the unknown
material characteristics in a permanent magnet synchronous machine in order to obtain a numerical
model that is a realistic representation of the machine. The second purpose is to optimize the machine
geometrically – using the accurate numerical model – for a maximal torque to losses ratio. Using the
optimized geometry, a new machine can be manufactured that is more efficient than the original.
Design/methodology/approach – A 2D finite element model of the machine is built, using a
nonlinear material characteristic that contains three parameters. The parameters are identified by an
inverse problem, starting from torque measurements. The validation is based on local
BH-measurements on the stator iron.
Findings – Geometrical parameters of the motor are optimized at small load (low-stator currents) and
at full load (high-stator currents). If the optimization is carried out for a small load, the stator teeth are
chosen wider in order to reduce iron loss. An optimization at full load results in a larger copper section
so that the copper loss is reduced.
Research limitations/implications – The identification of the material parameters is influenced
by the tolerance on the air gap – shown by a sensitivity analysis in the paper – and by 3D effects,
which are not taken into account in the 2D model.
Practical implications – The identification of the material parameters guarantees that the
numerical model describes the real material properties in the machine, which may be different from the
properties given by the manufacturer because of mechanical stress and material degradation.
Originality/value – The optimization is more accurate because the material properties, used in the
numerical model, are determined by the solution of an inverse problem that uses measurements on the
machine.
Keywords Finite element analysis, Electric motors, Magnetic devices, Electromagnetism
Paper type Research paper
COMPEL: The International Journal
for Computation and Mathematics in
Electrical and Electronic Engineering
Vol. 28 No. 2, 2009
pp. 272-285
q Emerald Group Publishing Limited
0332-1649
DOI 10.1108/03321640910929218
1. Introduction
Nowadays, compact permanent magnet synchronous motors (PMSM) are used for
applications that require a high-power density, such as electric vehicles and
compressors. Because the volume and weight are very critical, the magnetic material in
these machines is exploited in a highly saturated state. In recent literature, the majority
This work was supported by the FWO projects G.0322.04 and G.0082.06, by the GOA project
BOF 07/GOA/006 and the IAP project P6/21 funded by the Belgian Government. The first author
is a postdoctoral researcher for the “Fund of Scientific Research Flanders” (FWO).
of the numerical models of such a motor takes into account the nonlinearity of the
magnetic material. In Henneberger et al. (1997), saturation effects were studied in a
PMSM. In Stumberger and Hribernik (2000), the characteristics of a PMSM are
predicted by using a standard 2D finite element model (FEM). In Lee et al. (2006), the
motor is described using the voltage equation wherein the inductances of d- and q-axis
are also determined by a nonlinear FEM. Another nonlinear FEM was developed in de
Belie et al. (2006) for the study of the cross-saturation and stability of a surface PMSM.
Vivier et al. (2005) couples the electromagnetic model with mechanic and acoustic
models in order to study noise.
Most of the above-mentioned models use nonlinear material characteristics that are
obtained from the manufacturer of the steel sheets, or from measurements of B-H
characteristics performed in an Epstein frame or in a single sheet tester. However,
these characteristics change inside the material due to mechanical stresses on the
lamination and due to material degradation resulting from the cutting or punching of
the yoke (Crevecoeur et al., 2007). Therefore, it is more accurate to determine the
material properties of the magnetic material when it is already inside the machine.
In this paper, the numerical model (Section 2) is used to characterize the material
behaviour from an inverse problem (Section 3). The material behaviour is also
identified by local measurements of magnetic field and magnetic induction – measured
directly on the stator yoke – in Section 4, in order to validate the results found by the
inverse problem. The paper concludes by two optimizations of geometrical parameters
in Section 5.
The usefulness of the paper is that a geometrical optimization of the machine is
carried out after the detailed characterization of the machine. The characterization is
necessary in order to obtain a valid optimization with reliable results. It deals with the
development of a numerical model that produces approximately the same values of
torque and losses as the experimentally obtained values in the same working conditions
of the machine. Such a numerical model makes it possible to carry out accurate
simulations of the effect of a changed geometry to the machine performance and
efficiency. By using this accurate model, the result of a geometrical optimization is
reliable. The identification of the material parameters for a PMSM can be generalized to
any type of electromagnetic device (electrical machines, transformers, inductors, etc.).
2. Finite element model
2.1 Description
Figure 1 shows the geometry of the modelled machine with four poles. Details can be
found in Table I. The model is a 2D FEM with the vector potential A as unknown:
1
dA
¼ J s:
7£A þs
ð1Þ
7£
dt
m
Here, s is the conductivity, m(B) is the permeability and Js is the forced current density
in the three phase stator windings. The magnetic induction B equals f £ A. In the
magnets, the equation is written in terms of the magnetization M of the magnets:
1
7 £ A 2 7 £ M ¼ 0:
ð2Þ
7£
m0
The eddy currents in the magnet sðdA=dtÞ are not taken into account.
Magnet
synchronous
machine
273
COMPEL
28,2
50
Stator iron
40
d
Rotor iron
30
Magnet
20
y [mm]
274
10
q
0
Mesh interface
−10
−20
−30
Figure 1.
Cross-section of the PM
machine showing the
interface between stator
and rotor mesh
−40
−50
Stator
Rotor
Table I.
Specifications of the
PMSM
Air gap
−60
– 40
–20
0
x [mm]
20
Outer diameter
Inner diameter
Lamination thickness
Stack width
Number of slots
Resistance per phase
Number of turns per phase
Outer diameter
Yoke diameter
Shaft diameter
Minimal magnet thickness
Maximal magnet thickness
Number of poles
Diameter mesh interface
Air gap
40
60
90 mm
53 mm
0.5 mm
25 mm
15
0.193 V
9
51 mm
44 mm
6 mm
2 mm
3 mm
4
52 mm
1.0 mm
During rotation, the mesh of the stator remains fixed to the stator and the mesh of the
rotor remains fixed to the rotor. This means that the mesh of the rotor rotates relative
to the coordinate system of the stator, corresponding with the transformation:
"
# "
#
#"
xrotor
cosðvtÞ 2sinðvtÞ xstator
¼
;
ð3Þ
yrotor
ystator
sinðvtÞ cosðvtÞ
with v being the angular frequency. In the middle of the air gap – as shown in Figure 1
and Table I – an interface is created where the fixed stator mesh and the rotating rotor
mesh are related by a suitable boundary condition: continuity of the vector potential in
the stator coordinate system. This technique is called the arbitrary
Lagrangian-Eulerian method, which is well-explained in Braess and Wriggers (2000),
where it is applied to fluids. It has the advantage that during rotation of the machine, it
is not necessary to remesh the whole domain at every time step.
2.2 Validation based on electromotive force
At constant speed and no-load, the three electromotive forces (emf’s) generated by the
machine were measured and compared with the emf’s found by a FE time domain
simulation. The shape of the waveforms and the phase shift between the fundamental
components of the phases allows a validation of the winding scheme: the measured and
simulated waveforms and phase shifts should be almost equal if the scheme of the
windings is modelled correctly. The amplitude of the waveforms depends not only on
the layout of the windings, but also on the magnetization of the magnets and the
characteristic of the magnetic material. The magnetization of the magnet was chosen
equal to the typical magnetization of NdFeB magnets (1.15 T). The material
characteristic is determined in paragraph 4.
Figure 2 shows the measured and simulated emf in the three phases.
Magnet
synchronous
machine
275
2.3 Losses in the bearings, the magnets and the yoke
From the input power at no-load, the several loss terms can be identified. Figure 3
shows the measured electromagnetic power at no-load and the several simulated loss
contributions. The constant term in the figure represents the losses of the bearings and
hysteresis losses. Furthermore, there is a loss term caused by Joule losses in the stator
conductors – these losses are very low at no-load – and a linear term representing the
eddy current losses in the stator yoke. The eddy current loss in the stator yoke is
calculated based on the equation from Bertotti (1998) for the classical loss:
P cl 1 2 2 2
¼ p sd B f ;
6
f
ð4Þ
where d is the thickness of the lamination, f the frequency and B the amplitude of the
induction in case of sinusoidal induction waveforms. In Figure 3, the Joule losses in the
stator are shown by integrating equation (4) over the whole stator, using the typical
4
Emf [V]
2
0
–2
–4
u
v
w
0
0.02
0.04
0.06
Time [s]
0.08
0.1
(b)
(a)
Note: Both in the simulation (a) and in the measurement (b), the time scale is 10 ms/div and the voltage
scale is 1 V/div
Figure 2.
(a) Simulated and
(b) measured induced emf
of the machine at no-load
and 625 rpm
COMPEL
28,2
0.18
0.16
0.14
0.12
P/f [J]
276
Other
0.1
0.08
Eddy currents stator yoke
0.06
Figure 3.
Measured input power at
no-load (circular markers)
as a function of the
frequency f and separation
into several contributions
Joule loss coils
0.04
Hysteresis & bearings
0.02
0
0
20
40
60
Frequency f [Hz]
80
100
conductivity of the material. After adding these contributions, a difference is seen with
the measurements, denoted in the figure by “Other”. This term increases approximately
linearly with f and includes the eddy current losses in the rotor yoke and the magnets
that are not taken into account explicitly in the FEM. Especially, the magnets may in
this configuration have a major contribution as their width equals a quarter of the rotor
circumference (Figure 1). Finally, the loss in the power stage of the brushless DC drive
(Dorf, 2005) is also a contribution to the total loss that is not modelled in the
numerical model. Thanks to the low currents at no-load, the losses in this converter
are almost negligible. In order to improve the correspondence between the model and
the experimental machine, the conductivity was fitted so that equation (4) equals
the part of the measured contribution that increases linearly. Therefore, s should not be
interpreted as a conductivity any more, but as a parameter that accounts for all loss
terms that depend on f 2 except the Joule losses in the stator windings.
3. Identification of material parameters by local magnetic measurements
We identified the “real” material characteristics of the PMSM through the use of local
magnetic measurements. In this way, we are able to compare the measurements of the
real magnetic material with the “substitute” material characteristics obtained by the
inverse problem in the next section. In order to do that, we removed the rotor and
performed measurements on the stator. An excitation winding was wound around the
stator yoke as shown in Figure 4 and magnetic measurements were carried out. It is
difficult to measure the magnetic induction through the application of a B-coil around
the stator: the surface enclosed by the winding is unknown due to the presence of the
coils on the stator. Therefore, we use a local flux measurement technique. The
conventional way of performing local magnetic flux measurements on soft magnetic
materials is the use of search coils, see, e.g. Enokizono and Fujiyama (1998). This
method however is destructive, since holes need to be drilled. The needle probe method
is a non destructive method where needles are applied on the surface of the magnetic
Local B-needles
Excitation
winding
Magnet
synchronous
machine
277
Stator
Figure 4.
The stator of the machine
with added excitation
winding, showing also the
needles for the
B-measurement
material (Loisos and Moses, 2001). The potential difference between the needles Vn is
dependent on the magnetic material properties and can be calculated using Faraday’s
law (Crevecoeur et al., 2008):
dFsn
;
ð5Þ
Vn ¼ 2
dt
with Fsn being the flux through the surface Sn. The surface Sn is half the surface that is
considered when applying a search coil at the positions of the needles. The magnetic
field in the stator was deduced using Ampre’s law: H ¼ nI =l with n ¼ 100 being the
number of turns of the enforced coil, I being the enforced current and l < 250 mm.
We applied two needles with a distance of 4.5 mm between each other on the surface
of the stator. These needles for the B-measurement can be seen in Figure 4.
We analogously amplified the signal with a gain of 9,180 and performed an analog
integration of the potential difference. The measurements were carried out with a
frequency of 1 Hz (quasi-static). Figure 5 shows the single-valued (peak H to peak B values)
Magnetic induction B (T)
1.5
1.25
1
0.75
0.5
0.25
0
0
0.5
1
1.5
Magnetic field H (A/m)
2
× 104
Figure 5.
Locally measured
magnetic material
characteristics at 1 Hz
COMPEL
28,2
BH-curve recovered using the proposed method. We remark that there exist errors on the
measurements: there is an error on the surface Sn and an error by applying Ampere’s law.
By twisting the wires of the needles, we minimized the influence of the flux coupled
with the air surface between the needles (surface Sn includes also the air surface between
the needles).
278
4. Identification of the material parameters of the yoke by solving an
inverse problem with torque measurements as input
In the simple case that the material in the machine is assumed to be linear, the “average
machine permeability” can be determined from a measurement of current and torque at
standstill.
In the realistic case of nonlinear material, the permeability changes from point to
point, depending on the magnetic field. The identification of the material characteristic
is much more complicated and has to be carried out by solving an inverse problem. The
input data are a set of measurements at standstill: the torque is measured for P
amplitudes of the stator current and for N mechanical angles of the rotor, shown in
Figure 6 by markers. The parameters to optimize are q ¼ {H 0 ; B0 ; n}, which describe
the constitutive law of the permeability:
"
n21 #21
B0
B
mðBÞ ¼
1þ
:
B0
H0
ð6Þ
We fitted the H0, B0 and n parameters to the measurements (Figure 5). The parameters
ðmÞ
in equation (6) have the following values: H ðmÞ
0 ¼ 1; 882 A=m, B0 ¼ 1:02 T and
ðmÞ
n ¼ 6:63.
The inverse problem requires the minimization of the several objectives in order to
find the optimal values q * of the parameter vector q in the parameter space Vq:
N X
P
n
o
X
F 2i; j ;
q* ¼ H *0 ; B*0 ; n * ¼ arg min
q[Vq
i
j
Simulated
Experiment
0.7 8A
Figure 6.
Measured and simulated
torque at standstill as a
function of the mechanical
rotor angle and the input
current obtained by
applying a DC voltage
between two phases
Torque [Nm]
0.6 7A
6A
0.5
5A
0.4
4A
0.3 3A
0.2 2A
0.1 1A
0
0
Note: Machine in Y
10
20
30
Mechanical rotor angle [degrees]
40
ð7Þ
q* ¼ arg min
q[Vq
N X
P h
X
i
i2
meas
T fem
ðH
;
B
;
n
Þ
2
T
;
0
0
i; j
i; j
ð8Þ
j
If N rotor angles and P current amplitudes are considered. The parameter space Vq is
limited to a realistic range of all three parameters in equation (6): 10 , H 0 , 700,
1:0 , B0 , 2:0 and 1 , n , 10. The minimization is a nonlinear least squares
problem. We employed an algorithm based on the interior-reflective Newton method
(Coleman and Li, 1996), which uses gradients. The gradients are calculated in a
numerical way by the introduction of a small variation in each of the parameters and
by starting from the converged solution. The step size of the variation should be a
compromise: on the one hand, it should be large enough to overcome inaccuracies in the
numerical model (that discretizes the domain in a mesh of triangular cells); on the other
hand, it should be small to have an accurate estimation of the gradient in the
considered point. For the three parameters H0, B0 and n, the steps for the variation are
2 A/m, 0.05 T and 0.2, respectively. The calculation of the gradients is very fast as only
a few iterations of the linear solver are needed instead of about 20 for the initial
calculation.
For the above optimization, we used an alternative cost function Fij. If we try to
with the measured torques T meas
align the calculated torques T fem
i; j
i; j , possible
measurement errors are not eliminated when using a limited number of N angles.
Especially, the torque ripple that is shown in Figure 6 seems not to be reproduced by
the model: probably, the torque ripple is not caused by slot effects (which are modelled
correctly in the FEM), but by a small excentricity in the rotor. These measurement
errors result in an inaccurate solution of the inverse problem. In order to eliminate this
problem, we used the following alternative cost function:
h
i
fit meas
;
ð9Þ
F i;j ¼ T fem
i; j ðH 0 ; B0 ; nÞ 2 T i; j
meas
the torque obtained by fitting the measurement curve to the following
with T fit
i; j
function:
meas
ðar ; am;i ; bm;i ; cm;i Þ ¼ am;i ðar 2 bm;i Þcm;i ;
T fit
i; j
ð10Þ
with fitting parameters gm;i ¼ {am;i ; bm;i ; cm;i } and rotor angle ar. Indeed, it is
important to capture the main tendency of the torque as a function of ar for a certain
current. Figure 7 shows the relative change of the fitting parameters gm,i for different
enforced currents.
FEM-based optimization is possible due to the fact that the calculated torque
profiles are sensitive to the used three material parameters:
fem
fem
›T fem
i; j ðH 0 ; B0 ; nÞ ›T i; j ðH 0 ; B0 ; nÞ ›T i; j ðH 0 ; B0 ; nÞ
;
;
;
›H 0
›B0
›n
meas
are different from zero. By aligning the measurements T fit
(with
i; j
low-measurement error), and the calculated torque profiles (sensitive to magnetic
material parameters), we are able to recover the magnetic material parameters.
Magnet
synchronous
machine
279
COMPEL
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280
In the optimization, we used N ¼ 4 angles, which were chosen in such a way that
the dependence of the torque as a function of the rotor angle is included. P ¼ 4 (I ¼ 2,
4,6, 8 A) current amplitudes were chosen. The reason to choose only P ¼ 4 currents and
N ¼ 4 angles instead of P ¼ 8 and N ¼ 10 in Figure 6, is to obtain a reduction in
computational time. Indeed, the calculation of the cost value requires the evaluation of
NP FEMs – see double summation in equation (8). As the minimization algorithm
needs many evaluations of this cost value, it is useful to reduce N and P, on condition
that the remaining angles and currents are equally distributed in the available range of
angles and currents. In order to have global optimization, we employed the local
optimization algorithm, the nonlinear least squares interior-reflective Newton method
with gradients, and started the algorithm at three different starting points.
An important parameter, which has to be accounted for when recovering the
material parameters, is the remanent induction Brem of the magnets in the numerical
simulations. This parameter has a large influence on the torque simulations. We
illustrate this by calculating the following cost:
PN PP
i
Cost ¼
kF i;j k
;
NP
j
ð11Þ
for different remanent inductions of the magnets, using the measured material
ðmÞ
ðmÞ
characteristics H ðmÞ
. Table II shows the cost values for different
0 , B0 , and n
remanent inductions. The cost is minimal for Brem ¼ 1.25 T. When recovering the
material characteristics of the PMSM, using Brem ¼ 1.15 T, unrealistic values (2.4 T at
10 kA/m) for the material characteristics were obtained: H *0 ¼ 482, B*0 ¼ 1:61 T and
n * ¼ 7:3. This means that the forward model is not accurately modelling the
Relative change
1
Figure 7.
Relative change of the
fitting parameters gm for
different enforced
currents, relative to the
values at 8 A
Table II.
Cost of measured to
calculated torque profile
for different remanent
inductions with air gap
1 mm
cm
0.8
bm
0.6
am
0.4
0.2
0
1
2
3
4
5
6
7
8
Current (A)
Note: The numerical values of gm for 8 A are: am(8A) = 0.0823, bm(8A) = 0.6573
and cm(8A) = 0.5798
Brem(T)
1.15
1.20
1.25
Cost (11)
0.0446
0.0340
0.0241
Note: The measured material characteristics are used as parameters in the forward calculations
actual machine. This was also observed in the convergence history where the change of
cost from the initial value to the final iteration, was only approximately 1.2 per cent.
However, when using Brem ¼ 1.25 T, the values shown in Table III are obtained for an
air gap length of 1 mm. The convergence history is shown in Figure 8 and a large
difference is observed between the cost of the initial value and the cost of the final
iteration. This means that the magnetic material parameters have a large effect on the
cost and thus make it possible to recover the material properties within the
measurement error range (which was minimized by using the alternative cost function
equation (9)) of the torque profiles T fit meas.
Although the found solution yields the best correspondence between the torques
calculated by FEM and the measured torques, the material characteristic may not be
completely correct because of geometrical tolerances, especially on the air gap. A
simulation was carried out with small variations of the air gap in order to illustrate the
variation of the torque for the same material characteristic. Figure 9 shows the
simulated torque profiles for different air gaps (0.75, 1, 1.25 mm) and same material
characteristic (q ¼ {482, 1.61, 7.3}) at Brem as a function of the rotor angle. Table III
shows the optimal magnetic parameters for all considered air gaps, i.e. the parameters
that result in the best match with the measurements. In Figure 10, the recovered
BH-curves are shown and compared with the locally measured material properties.
We remark a large difference in the H0-values and the H ðmÞ
0 . The H0-value has a large
influence on the height of the permeability for low-magnetic fields. The difference
between the values are due to measurement errors in the torque profile and
Air gap (m)
H *0
B*0
n*
0.75
1
1.25
823
1057
868
0.974
1.069
1.210
10.74
9.53
11.92
7
Magnet
synchronous
machine
281
Table III.
Optimal magnetic
parameters for different
air gaps for Brem ¼ 1.25 T
× 10–3
6
Cost
5
4
3
2
1
1
2
3
4
5
6
7
8
9 10 11 12
Number of iterations
Notes: Here, q(k) ={H0 ,B0 , m(k)} are the parameters in the k-th
iteration. Brem = 1.25 T and air gap length is 1 mm in the forward
calculations
Figure 8.
Convergence history of the
gradient nonlinear least
squares interior-reflective
Newton method with
P P
cos t ¼ Ni Pj
ðkÞ
ðkÞ
ðkÞ
½T fem
i;j ðH 0 ; B0 ; n Þ
fit meas 2
2T i;j
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282
the approximation of the model to “reality”. We remark that the air gap has a large
influence on the saturation induction (mainly determined by the B0-parameters).
5. Geometrical optimization of the PMSM
Now that all properties of the motor are known, a geometrical optimization is carried
out for maximal ratio torque to losses. Notice that an optimization of the machine
makes sense only because the numerical model used for optimization represents the
machine well, thanks to the preliminary identification of the material parameters. The
use of the optimization is to determine an optimized geometry of the machine in order
to have a better performance. After the optimization, a new machine can be
manufactured using the optimized geometry. Evidently, in order to be valid, the
magnetic material and the manufacturing processes for the new machine should be the
same as for the initial machine. We want to improve the following geometrical
parameters: the thickness of the magnets w, the width of the stator teeth dt and the
angle f of the stator current I q þ jI d ¼ I ejf . We denote the geometrical parameters as
0.8
Torque [Nm]
Figure 9.
Simulated torque at
standstill for the optimized
material parameters
(optimized for 1 mm air
gap) as a function of the
air gap that is known only
approximately
measurements
0.75 mm
0.7 8A
1 mm
0.6
0.5
1.25 mm
6A
0.4
5
10
15
20
Mechanical rotor angle [degrees]
1.5 air gap = 1.25 mm
25
air gap = 1 mm
Magnetic induction B (T)
air gap = 0.75 mm
Figure 10.
Recovered single-valued
BH-characteristics for
several air gap length
using Brem ¼ 1.25 T
1
0.5
Local measurements
0
0
0.5
1
1.5
Magnetic field H (A/m)
2
2.5
× 104
p ¼ {w; d t ; f} with parameter space Vp, which is given by all valid combinations of
the parameters: 1.0 , w , 5.0 mm, 3.0 , dt , 20.0 mm and 2 908 , f , 908. Next
to the three optimized parameters, there are other quantities changing that are
dependent on these parameters. The slots with the stator windings become smaller as
the width of the stator teeth increases, because the circumference and the number of
slots are fixed. The cylindrical yoke of the rotor becomes smaller if the thickness of the
magnets increases. The d- and q-components of the currents change with the phase
angle of the current. The objective is to obtain a high ratio of torque to losses:
p * ¼ arg max
p[Vp
Tð pÞ
;
3RI 2 þ P cl ð pÞ
Magnet
synchronous
machine
283
ð12Þ
with R the resistance of the stator windings (Table I) and Pcl given by equation (4).
Notice that only the angle of the stator current is optimized; the amplitude is chosen
a priori. Indeed, when optimizing a machine, it should be known if the machine will be
used mainly at small loads or mainly at full load. The results of the optimization are
different. We choose the amplitude of the current once equal to 2 A and once equal to
16 A. Table IV shows the optimal parameters obtained after the optimization. This
table shows that for high current, the thickness of the permanent magnets is increased
to produce more flux. For low current, the iron losses are dominant and can be reduced
by choosing thinner magnets. Furthermore, the width of the stator teeth is reduced for
high current to limit the losses in the copper: if the teeth are smaller, the cross-section
for the copper conductors increases because the total volume of the stator is constant.
6. Conclusions
From an inverse problem that uses suitable measurements as inputs, the material
properties of a permanent magnet synchronous machine are identified. To eliminate
inaccuracies in the measurements, the measurement inputs for the inverse problem are
replaced by a more smooth analytical function fitted through the measurement data.
The material characteristic obtained by the inverse problem is compared with the
characteristic obtained by local magnetic measurements on the stator yoke. With the
obtained material parameters, the 2D FEM predicts torques similar to the measured
ones for all considered rotor angles and current amplitudes. For rather weak fields, the
obtained characteristic deviates somewhat from the locally measured
BH-characteristic, but for strong fields, the correspondence between both
characteristics is good. With this model, an optimization of the geometry is carried
out to obtain maximal torque for minimal losses. It is observed that in case that the
machine is optimized for high-stator current, smaller stator teeth and thicker magnets
are chosen than in case the machine is optimized for low-stator current.
Geometrical parameters
Thickness of permanent magnets w (mm)
Width of stator teeth dt (mm)
Angle f of the stator current (8)
2A
16 A
2.8
7.4
2 25.24
3
6.5
2 27.48
Table IV.
Optimal geometrical
parameters for different
enforced currents
COMPEL
28,2
284
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About the authors
Peter Sergeant was born in 1978. In 2001, he graduated in Electrical and Mechanical Engineering
at the Ghent University, Belgium. In 2006, he received the degree of Doctor in Engineering
Sciences from the same university. He joined the Department of Electrical Energy, Systems and
Automation, Ghent University in 2001 as a Research Assistant. From 2006, he is a postdoctoral
researcher for the Fund of Scientific Research Flanders (FWO). His main research interests are
numerical methods in combination with optimization techniques to design nonlinear
electromagnetic systems, in particular actuators and magnetic shields. Peter Sergeant is the
corresponding author and can be contacted at: peter.sergeant@ugent.be
Guillaume Crevecoeur was born in 1981. He received the Physical Engineering degree from
Ghent University, Ghent, Belgium, in 2004. He joined the Department of Electrical Energy,
Systems and Automation, Ghent University, in 2004 as a doctoral student of the Special Research
Fund (BOF). His main research interests are numerical methods in electromagnetics and the
solution of inverse problems in electromagnetics for magnetic material characterization, source
localization and geometrical optimization.
Luc Dupré was born in 1966, graduated in Electrical and Mechanical Engineering in 1989 and
received the degree of Doctor in Applied Sciences in 1995, both from the Ghent University,
Belgium. He joined the Department of Electrical Energy, Systems and Automation, Ghent
University in 1989 as a Research Assistant. From 1996 until 2003, he has been a postdoctoral
researcher for the FWO. Since 2003, he is a Research Professor at the Ghent University. His
research interests mainly concern numerical methods for electromagnetics, especially in
electrical machines, modeling, and characterisation of magnetic materials.
Alex Van den Bossche received the MSc and the PhD degrees in Electromechanical
Engineering from Ghent University Belgium, in 1980 and 1990, respectively. He has worked
there at the Electrical Energy Laboratory. Since 1993, he is a Full Professor at the same
university in the same field. His research is in the field of electrical drives, power electronics on
various converter types and passive components and magnetic materials. He is also interested in
renewable energy conversion. He is co-author of the book Inductors and Transformers for Power
Electronics.
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