The aim of this work is to present and discuss an
original way to analyze a synchronous permanent magnet
micromotor (SPMM) for design purposes. The analysis of the
magnetic field distribution in the motor, whose geometry would
require a 3D modeling, is instead carried out with the aid of two
2D finiteelement (FE) simulations: one axisymmetric and one
on the crosssection or transverse plane. To validate the
suitability of the proposed method for torque computation, a
3Dfield solution is also presented. Computed results of
distribution of magnetic induction, as well as the torque
developed by the motor in both 2D and 3D simulations, have
shown good agreement with measurements in a prototype
machine.
3604
IEEE TRANSACTIONS ON MAGNETICS, VOL
34,
NO.
5,
SEPTEMBER
1998
Finite Element Analysis
of a
Syn ermanent Magnet Micromotor throug ransverse Planar Simulations
Adrian C. Agiiero, Fernando A. Actis C1MM:Centro de Investigacion de Materiales
y
Metrologia, AV. VClez Sarsfield
1561
C.C. Central
884,
5000
Cordoba, Argentina Viviane Cristine Silva, JosC Roberto Cardoso, Silvio
I.
Nabeta LMAG/PEA Escola Polittcnica da Universidade de
SLo
Paulo, AV. Prof. Lucian0 Gualberto,
T
3,
n0158,05508900
Sb
aulo SP, Brazil
bstract
The aim of this work
is
to present and discuss an srcinal way to analyze a synchronous permanent magnet micromotor SPMM) for design purposes. The analysis
of
the magnetic field distribution in the motor, whose geometry would require
a
3D modeling, is instead carried out with the aid of two
D
finiteelement
FE)
simulations: one axisymmetric and one on the crosssection or transverse plane.
To
validate the suitability of the proposed method for torque computation, a 3Dfield solution is also presented. Computed results
of
distribution of magnetic induction, as well as the torque developed by the motor in both
2D
and
3D
simulations, have shown good agreement with measurements in a prototype machine.
Index
terms
Claw pole, synchronous permanent magnet micromotor, finite element analysis
of
synchronous machines.
I.
INTRODUCTION The SPMM is
a
lowcost device employed as a time reference, which provides rotating movement with low torque. Its main feature is its constant speed for
a
given supply frequency, independent
of
the load until it is in an overload state. In this condition, the motor loses synchronization with consequential speed reduction and oscillations. Typical applications include chart recorders, programming devices, valve drives, etc. Figure 1 illustrates the winding arrangement, stator claw poles and permanent magnet multipole ring rotor, which are magnetized along the radial direction
[
121. Performance characteristics such as developed torque can be obtained with the aid of 2D and 3D FE analyses by assuming
a
magnetostatic behavior, since in both simulations one is interested in the steady state operation of the synchronous motor. Although 2D representations
of
SPMMs provide less accurate results when compared to 3D modeling, they allow
faster
and
owercost simulations
[3].
II.2D FE MODELS The steady state operation of a SPMM has been simulated as a 2D magnetostatic phenomenon using a 2D FE package
(FLUX2D) [4].
The domain studied consists
of
a 16pole
Manuscript received November
3, 1997.
A.
AgUero,
fax.
+5451699459;
email:
aguero@com.uncor.edu;
J.
R.
The authors
would like
to
acknowledge
the
support
of
Tamyr S.A. Cardoso, fax:
+5511814 2092;
email:
cirdoso@pea.usp.br.
prototype synchronous motor with a permanentmagnet ring rotor (refer to Appendix for the SPMM dimensions and data). Figure 2 presents a schematic layout of the motor geometry in
two
views: axisymmetric and cross section, displayed by the geometric modeler of the
FE
package.
Fig.
1
Topology of
the
prototype
SPMM
1
Stator
housing,
2
coil,
3
claw
poles,
4
shortcircuiting ring (copper),
5
permanentmagnet
rotor
The prototype machine can be seen in Fig. 2 and has the following constructive features:

the claw poles are made from electrical steel with a 1 mm thickness; the coil has 11000 turns and is made of copper wire with a
0.06
mm diameter; the magnet has a remanent induction of 0.2 T. The first simulation assumes axial symmetry (Fig. 3a) and enables
the
determination
of
the normal component
of the
flux density distribution,
Bn
on the portion of the claw surface facing the permanent magnet. After solving the axisymmetric problem, the calculation of
Bn
was carried out along a path spanning from the top to the bottom of the permanent magnet (see dashed line
in
Fig. 3a). The plot of
Bn
along this path can be seen in Fig.
4.
This curve will be used in the next step
of
the methodology: the 2D planar simulation. To calculate the torque, a second 2D simulation assuming planar symmetry was carried out using the geometry shown in Fig. 3b, which represents a cross section of the prototype
00189464/98 10.00
998
IEEE
3605
machine, taken in the midheight
of
the magnets. Along the
external
boundary (dashed line
of
Fig. 3a) a non homogeneous Dirichlet boundary condition was imposed, i.e. a fixed magnetic vector potential (the state variable), which is numerically equal to the magnetic flux per meter crossing the cylindrical surface represented by the vertical dashed line
of
Fig. 3a.
Fig.
2.
The
SPMM
prototype.
This magnetic flux can be determined with the aid
of
the
Bn
curve
of
Fig.
4,
plotted along the vertical dashed line in the axisymmetric simulation. The magnetic
flux
was calculated by taking the “mean value” of this curve, i.e. the value
of
Bn
which gives a rectangular area equal to the area delimited by the curve in Fig.
4.
The second 2D simulation yielded the flux density distribution, which can be seen in Fig.
5
through contours of flux lines. The torque vs. load angle characteristic has also been determined in one pole pitch with the aid
of
this simulation. The result will be presented later on in this work. 111.3D
FE
MODEL The complicated shape of the claw poles usually needs a threedimensional analysis. However, results from a 3DFE modeling
of
the same prototype motor have presented
a
very close agreement with the proposed methodology, namely the
two
2D simulations. The 3D simulation presented in the sequence was carried out using a 3D finite element package, FLUX3D
[5]
Fig.
6
shows the 3D
FE
mesh. Again, the phenomenon was assumed as a magnetostatic one, since the interest is the calculation of synchronous torque in steady state condition. Therefore, only one pole pitch needs to be modeled. Figure
8
illustrates the distribution of the flux density
vectors
in a cut plane, parallel to the
x y
plane
of
global co ordinate system, which cuts the zaxis at z
=
15 mm (shown in Fig.
7).
This case is equivalent to the second
2D
FE model (cross section). Figure
9b
shows the
fl~k
ensity vectors in another cut plane, which is normal to plane
xOy
(shown in Fig. Sa).
This
is equivalent to the axisymfinetric
2D
simulation.
I
\
Shaft
‘Path where
B
is firstly calculated
boundary condition
was
impoted, whose value was determined in the previous axisymmetric simulatiop.
1
2
3
4
5
6
7
h 102mm)
(path parallel to magnet
height)
Fig
4
Normal
flux
density plotted along the bold dashed line shown
in
Fig
2
3606
Fig.
5.
Flux lines resulting from the second (planar)
2D
simulation.
Flg
6.
Showing one pole pitch
of
the prototype machine with the
3D
FE
mesh Fig
8
Distribution
of
the flux density vectors in the cut plane shown in
Fig
7
Fig. 7.
A
cut plane (parallel toxOy plane of global coordinate system) in the
3D
motor geometry.
z
0
a=
.50
...
I
..

.............
.........
>... *I::::: ::
1 :
............
....
.............
.......
...............
..........
..............
...............
..............
...............
Fig
9
a)
A
radial cut plane (normal toxOy plane
of
global coordinate system) in the
3D
motor geometry;
b)
distribution of the flux density vectors in this cut plane
3607
111
RESULTS
IV.
C~ONCLUSIONS
A
curve of the reduction in torque when varying the air gap in the range 0.2 0.8 mm is plotted in Fig.
10,
which exhibits a comparison between the values issued from the second 2D simulation and from experimental measurements. The 3D simulation for an airgap of
0.8
mm yielded a torque of
0.01277
kg.cm. The error is less than
8
in comparison with
2D
simulation and the tests in the prototype machine. Table
I
shows the mesh data and CPU times. The great difference
of
total CPU time between the 2D and 3D simulations for nearly the same number of nodes can be noticed. This is because the 3D model leads to 3 times as many unknowns as the 2D model for the same number of nodes (3 components
of
magnetic vector potential in 3D versus 1 component in 2D).
0 06
Oo5A
Simulation
a
Test
001
4
0 2
0 3
0.4
0 5
0 6
0 7
0
8
Airgap
[mm]
Fig.
10.
Torque variation with airgap length: 2D FE analysis and experimental results. TABLE I MESH DATA
ND
CPU TIME
FOR
HE
2D
AND
3D FE SIMULATIONS 2D (planar) 3D Number
of
nodes 10.097 10.498 Number
of
elements
53
089
55
198
Number of nonlinear iterations
5
10
Total CPU time
a
15 140
a
Computer used: IBM PC compatible Pentium
166
MHz
64
MB RAM
problem, namely the of synchronous torque was carried out with the d of a
2D
FE computer package and validated by
3D
FE pqckage. The accuracy of the results issued from the 2D dpproach has been verified by measurements in a protobpe and proved to be sufficient, thereby avoiding the need bf the costly 3D FE analysis.
~PPENDIX
SPMM DI~ENSIONAL
ATA
Rated power
2.75
W Current
15
mA
Based speed
375
rpm Rated voltage 220
v
Frequency
S
HZ
Number of phases
1
STATOR Outer diameter 43 mm Winding type ]multilayer Number of coils
1
Phase resistance 7200
R
Self inductance 4.2
H
ROTOR Magnet outer diameter
2
1.5
mm
Magnet inner diameter
16.8
mm Magnet height
8.6
mm Magnet material Sintered
u
ferrite Airgap length
10.20.8
mm Number of poles
11
ACKNOWLEDGEMENT
The authors wish to expkess their most sincere gratitude to
R.
Ottolini for constructidn of the prototype utilized
in
this investigation and R. Moyabo, who performed the test.
I
MFERENCES
[l]
J Vogel,
“Grundlagen der elektrischen Antriebstechnik mit Berechnungsbeispielen”,
Dr.
Alfred Huthig Verlag, Heidelberg, pp.
I
A.
Viorel, Csapo,
E
M’ inescu, L. Szabo, “Claw pole brushless D
C.
motor for a variable speed drive system”, Intelligent Motion,
[3]
S
RH
Hook,
CompuierAided Analysis and Design
of
Electromagnetic Devices, Isevier, New York, 1989 [4] FLUX2D CAD package or 2D Electromagnetic Field Computation, CEDRAT [5] FLUX3D CAD package
lfor
3D Electromagnetic Field Computation, CEDRAT.
356358,
1985. [2] Nurnberg,
DD
127131,
19
r
,
l