Journal of ELECTRICAL ENGINEERING, VOL 57. NO 8/S, 2006, 181-184
OPTIMIZATION OF ACTUATOR WITH PERMANENT MAGNET
FROM VIEWPOINT OF THEIR ARRANGEMENT AND TOTAL MASS
OF MAGNETIC CIRCUIT
Ivo Doležel*  Pavel Karban**  Bohuš Ulrych**
Actuators with permanent magnets represent modern and prospective devices with widely controllable parameters as the lift and
magnetic force acting on its movable parts. The paper represents an introductory study of such a device with more permanent magnets connected in series. Solution of its mathematical model is illustrated on a typical example whose results are discussed.
Keywords: electromagnetic actuator, permanent magnet, magnetic field, optimization.
1 INTRODUCTION
Actuators are electromechanical converters widely
used in various technical applications such as transport,
automated production systems, robotics, nuclear engineering and many others. Their principles are based on several
ways of generation of mechanical forces as a consequence
of various electromagnetic phenomena (details can be
found, for example, in [1], [2]).
The paper presents a study providing information necessary for an optimal design of a DC electromagnetic actuator with a system of permanent magnets, whose principal arrangement is in Fig. 1. The actuator consists of several main parts. Its yoke 3 is made of magnetically soft
material. Magnetic flux Φ is produced by a ring-shaped
permanent magnet 2 magnetized in the radial direction.
The working coil 1 carrying direct current I w is placed in
the air gap of the magnetic circuit. The coil is affected by
Lorentz force FL whose direction depends on the direction of the mentioned current and orientation of the permanent magnet.
Operation characteristics of the actuator strongly depend on both its structure and magnetic properties of the
used materials.
principal scheme of a typical actuator of this kind including its basic nominal dimensions is shown in Fig. 2.
Fig. 2. The considered arrangement of an actuator with three permanent
magnets in series, 1 cylindrical working coil, 2 main permanent magnet,
3 and 6 auxiliary ring (permanent magnet or soft ferromagnetic material), 4 internal shell of the yoke (ferromagnetic material), 5 external
shell of the yoke (ferromagnetic material), 7 nonmagnetic leading shell
of the coil, 8 air
The aim of the paper is to evaluate this type of actuator
from the viewpoints of its arrangement and magnetic properties of the used materials, the objective function being the
highest possible force effects. This evaluation is carried out
only by solution of a certain set of alternatives and consequent judgment of the results from the viewpoint of the objective function. So we do not deal with the optimization
process in the correct sense, but just with finding and validation of the backgrounds for the optimization algorithm.
2 MATHEMATICAL MODEL
Fig. 1. Principal scheme of a DC electromagnetic actuator with a permanent magnet, 1 cylindrical working coil, 2 permanent magnet, 3
cylindrical yoke
The further text – representing natural continuation of
paper [2] aimed at actuators with one permanent magnet –
is devoted to modified arrangements of similar actuators
with several permanent magnets connected in series. The
As the device is considered axisymmetric, its mathematical model is formulated in cylindrical coordinates r , t , z .
2.1 Definition area
The definition area consists of five subdomains Ω1 − Ω5 ,
as follows:

*Czech Technical University, Faculty of Electrical Engineering, Department of Electrical Power Engineering, Technick8 2, 166 27 Praha 6,
E-mail{[email protected]}
**University of West Bohemia, Faculty of Electrical Engineering, Department of Theory of Electrical Engineering, Univerzitní 26, 306 14 Plzeň,
E-mail{ulrych, [email protected]}
ISSN 1335-3632 © 2006 FEI STU
182
I. Doležel et al: OPTIMIZATION OF ACTUATOR WITH PERMANENT MAGNET FROM VIEWPOINT OF THEIR…
Ω1 –
the cross-section of coil 1 characterized by current
A ( r , z ) = r0 0 + t 0 At ( r , z ) + z 0 0
density J ext and magnetic permeability µ0 .
Ω2 –
J ext = r0 0 + t 0 J c + z 0 0,
(2)
the cross-section of the ring of permanent magnet 2
characterized by coercive force H c and remanence Br , or
by permeability µ = Br / H c and orientation of the magnet
with respect to coordinate system r , t , z .
Ω3 –
the cross-section of the magnetic yoke 4, 5 or also
3, 6 characterized by nonlinear dependence B ( H ) .
Ω4 –
the cross-section of the nonmagnetic leading shell 7
of coil 1 characterized by permeability µ0 .
Ω5 –
air of permeability µ0 .
This system of subdomains is bounded by a fictitious boundary Γ ∞ characterized by the Dirichlet condition for vector
potential in the form A = 0 .
The numerical values of the physical parameters of particular
subdomains are given in Tab. 1 and Figs. 2, 3.
nominal current I w = 3.927 A
current density J w = 5 A/mm2
coefficient of filling κ = 0.785
RECOMA 28
see [7],
V1
where V1 is the volume of coil 1. In our case FL = z 0 FL .
2.3 Boundary conditions
The boundary conditions provide unambiguousness of solution of system (1)–(3). They are – with respect to antisymmetry of magnetic field along the axis of the arrangement
and continuity of vector of magnetic flux density B (r , z )
along the fictitious boundary Γ ∞ – expressed in the form
rA t ( r , z ) = 0
(4)
3 COMPUTER MODEL AND ACHIEVED ACCURACY
OF SOLUTION
Table 1: Basic physical parameters of used materials
Part
Material
Parameter and its value
coil 1
Cu wire
d = 1 mm, N z = 750 *
permanent
magnets
2 or also
3 or 6
The Lorentz force acting on coil 1 carrying current of density J ext that is placed in magnetic field B (r , z ) is given by
relation
FL = ∫ ( J ext × B ) dV
(3)
H c = 720 kA/m, Br = 1070 mT
permeability µR = 1.05
H c = 868 kA/m, Br = 1190 mT
FeNdB, quality N35
permeability µ R = 1.091
see [11]
magnetic
Fe 99.9% [8]
magnetization curve
circuit 4, 5 VACOFLUX 50 [10]
see Fig. 3
or also 3, 6 steel “T“ [3]
leading
Kevlar [9]
permeability µr = 1
shell 7
*The number of turns corresponds to the basic arrangement in Fig. 2.
The mathematical model presented in the previous
paragraph was solved by the FEM-based professional
code QuickField [6] supplemented with several own procedures. Particular attention was paid to the geometrical
convergence of solution. In order to achieve the accuracy
of the Lorentz force FL at the level of three nonzero valid
digits it was necessary to use a mesh containing 80000–
150000 elements, in dependence on particular shift ξ (see
Fig. 2) of field coil 1. An illustration of typical results –
distribution of force lines in one of the considered alternatives of the actuator – is in Fig. 4.
2,5
B(T)
2,0
1,5
VACOFLUX 50
1,0
Fig. 4. Distribution of force lines ( J w = 5 A/mm 2, ξ = 0 ,
steel "T"
0,5
lc = 50 mm, three permanent magnets RECOMA 28, shell Fe 99.9%)
Fe 99,9 %
0,0
0
2000
4000
6000
4 ILLUSTRATIVE EXAMPLE
8000
H(A/m)
Fig. 3. Magnetization characteristic B ( H ) of the considered ferromagnetic
materials
in the coil J c = 3.925 A/mm2, when coefficient of filling is
2.2 Differential equations
The differential equation for magnetic field in the system
generally reads (see, for example, [3-5])
1

curl  curl A − H c  = J ext
µ

where for cylindrical coordinates (see also par. 4)
Considered is the actuator depicted in Fig. 2 in common with its nominal dimensions. The coil carries current
I w = 3.925 A of density J w = 5 A/mm2 and current density
(1)
κ = 0.785 ( J c ≈ J ext , see (2)). Other data can be found in
Tab. 1. In association with evaluation of the Lorentz force
FL (that should be as high as possible) we studied the
following influences:
replacement of a ferromagnetic part of magnetic circuit by permanent magnets,
183
Journal of ELECTRICAL ENGINEERING, VOL 57. NO 8/S, 2006
field current I w (or its density J w ) and
length lc of the field coil.
for ry,max = 75.6 – which represents an increase by 20% –
substantially suppressed.
400
FL (N)
the shape of magnetization characteristic of ferromagnetics used,
parameters H c , Br of the permanent magnets,
oversaturation of shell 5 that is a function of its external radius ry,max ,
350
RECOMA 28
300
FeNdB
250
200
150
The influence of the number of permanent magnets in
series is apparent from Fig. 5. The force FL obviously
grows with this number. A weak nonlinearity is caused by
local oversaturation of external yoke 5.
100
50
0
0
10
20
300
perm. magnet 2
2, 3
2, 3, 6
FL (N)
250
200
30
40
50
Fig. 6. Static characteristics of the actuator – influence of parameters of
permanent magnets ( J w = 5 A/mm2 , lc = 50 mm, ry,max = 63 mm, three
permanent magnets, shell Fe 99.9%)
150
This results in a growth of FL , as follows from Figs. 8
and 9. But this growth is not too high, something below
20%.
100
50
0
0
10
20
30
40
50
Fig. 5. Static characteristics of the actuator – influence of the number of
permanent magnets ( J w = 5 A/mm2 , lc = 50 mm, ry,max = 63 mm, three
permanent magnets RECOMA 28, shell Fe 99.9%)
The influence of curves B ( H ) of ferromagnetic material used for parts 4 and 5 (Fig. 2) follows from Tab. 2.
Although this influence is not negligible, its significance
is not decisive. The best material is Fe 99.9% [8].
Table 2: Influence of ferromagnetic materials used for the yoke on the
Lorentz force acting on the coil ( J w = 5 A/mm2 , lc = 50 mm,
ry,max = 63 mm, three permanent magnets)
236
permanent magnet RECOMA 28
45
Fe 99.9% [8]
236
permanent magnet RECOMA 28
45
VACOFLUX 50 [10]
236
permanent magnet RECOMA 28
45
steel “T” [3]
(*) Characteristics B ( H ) see Fig. 3
4500
4000
3500
3000
2500
2000
1500
1000
500
0
ry,max = 63 mm
75,6 mm
0
P1
25
50
75
100
(mm)
125
P2
Fig. 7. Distribution of relative permeability along line P1P2 , see Fig. 2
( J w = 5 A/mm2 , ξ = 0 , lc = 50 mm, three permanent magnets FeNdB,
shell Fe 99.9%)
FL
(N)
288.9
247.6
237.6
380
375
370
FL (N)
materials of permanent
magnet and magnetic circuit (*)
µr (-)
ξ (mm)
position
(see Fig. 2)
ξ (mm)
365
360
355
350
The influence of parameters H c , Br of the selected
permanent magnets (RECOMA 28 or FeNdB) is obvious
from Fig. 6 showing the corresponding static characteristics. It is clear that magnets FeNdB have higher value of
coercive force H c (see Tab. 1), produce stronger magnetic field and, consequently, higher values of FL . The
influence of eventual oversaturation of the magnetic circuit is obvious from Figs. 7, 8 and 9. Fig. 7 shows the dependence of oversaturation (low values of µr ) on the radius ry,max of the external shell 5 (Fig. 2). While for
ry,max = 63 mm the over saturation is high (in accordance
with the map of force lines in Fig. 4), the same effect is
63
65
67
69
71
73
75
77
ry,max (mm)
Fig. 8. Dependence of the Lorentz force on external radius ry,max of
outer shell 5 , see Fig. 2 ( J w = 5 A/mm 2, ξ = 0 , lc = 50 mm, three permanent magnets FeNdB, shell Fe 99.9%)
The influence of field current I w (or corresponding field
current density J w ) can be seen in Fig. 10. Depicted are
the static characteristics for both nominal current
I w = 3.927 A and current by 20% higher I w = 4.71 A. Increase of force FL is now grater than 20%.
The influence of length lc of the field coil is presented
184
I. Doležel et al: OPTIMIZATION OF ACTUATOR WITH PERMANENT MAGNET FROM VIEWPOINT OF THEIR…
in Fig. 11 that contains two static characteristics for the
nominal length lc = 50 mm (see Fig. 2) and length greater
by 20% lc = 60 mm.
400
ry,max = 63 mm
FL (N)
350
300
250
75,6 mm
Acknowledgments
200
150
Financial support of the Grant Agency of the Czech Republic (project No. 102/04/0095) and Ministry of Education
of the Czech Republic (Research Project MSM 4977751310)
is gratefully acknowledged.
100
50
0
0
10
20
ξ (mm)
30
40
50
REFERENCES
Fig. 9. Dependence of the static characteristic on external radius
ry,max of outer shell 5, Fig. 2 ( J w = 5 A/mm2, ξ = 0 , lc = 50 mm,
FL (N)
three permanent magnets FeNdB, shell Fe 99.9%)
450
400
350
300
250
200
150
100
50
0
Jw = 5 A/mm2
6 A/mm2
Fig. 10. Dependence of the static characteristic on field current I w of
density J w ( lc = 50 mm, ry,max = 63 mm, three permanent magnets
FeNdB, shell Fe 99.9%)
FL (N)
case of long shifts while their dimensions remain still acceptable. These forces may be influenced mainly by the
total magnetomotive force of the field coil (par. 4.5 and
4.6) and by quality (the coercive force) of the permanent
magnets (par. 4.3). Next research in the domain will be
aimed at possibilities of using combined arrangement of
permanent magnets (in serial-parallel connection).
450
400
350
300
250
200
150
100
50
0
lc = 50 mm
60 mm
0
10
20
ξ (mm)
30
40
50
Fig. 11. Dependence of the static characteristic on length lc of the field
coil ( J w = 5 A/mm2 , ry,max = 63 mm, three permanent magnets FeNdB,
shell Fe 99.9%)
Even here, the increase of force FL is substantial and
exceeds 20%. On the other hand, this increase is not as
significant as in case of higher current (as was shown in
the previous paragraph).
5 CONCLUSION
The results show that electromagnetic actuators with
permanent magnets provide relatively high forces even in
[1] JANOCHA, H.: Aktuatoren: Grundlagen und Anwendungen, SpringerVerlag, Berlin, 1992 (in German).
[2] MAYER, D. –– ULRYCH, B.: Analysis of an Electromagnetic Actuator
with Permanent Magnet, AEEE 5 No. 2 (2006), 301–305.
[3] MAYER, D. –– POLÁK, J.: Methods of Solution of Electric and Magnetic Fields, SNTL/ALFA, Prague 1983 (in Czech).
[4] HAŇKA, L.: Theory of Electromagnetic Field, SNTL/ALFA, Prague
1975 (in Czech).
[5] OPERA 2D User Guide, Chapter 5, Vector Fields Ltd., Oxford, UK.
[6] www.quickfield.com.
[7] Firm materials, RECOMA, DEUTSCHE CARBONE AG, Frankfurt,
Germany.
[8] HASSDENTEUFEL, J. –– KVĚT, K. et al: Materials for Electrical
Engineering, SNTL, Prague 1967 (in Czech).
[9] www.azom.com.
[10] www.vakuumschmelze.com
[11] www.elidis.com.
Received 5 December 2006
Ivo Doležel (Prof., Ing., CSc.) born in 1949, obtained his Eng. degree from the Faculty of Electrical Engineering (Czech Technical
University in Prague) in 1973 and after 28 years in the Institute of
Electrical Engineering of the Academy of Sciences of the Czech
Republic he returned back to the Czech Technical University. His
interests are aimed mainly at electromagnetic fields and coupled
problems in heavy current and power applications. He is an author
or co-author of one monograph, about 250 papers and several large
program packages.
Pavel Karban (Ing.) born in 1979, finished his studies at the
Faculty of Electrical Engineering, University of West Bohemia
in Pilsen, in 2002. In 2003 he started doctoral studies in the Department of Theory of Electrical Engineering (supervisor Prof.
Doležel). His topic is computational electromagnetics, particularly integral models for solution of low-frequency magnetic
fields. He is an author and co-author of more than 30 papers and
several large program packages.
Bohuš Ulrych (Doc., Ing., CSc.) born in 1937, works in the
Department of Theory of Electrical Engineering at FEL UWB in
Pilsen. His professional interests are aimed at modern numerical
methods of solution of electromagnetic and coupled problems.
Author and co-author of more than 200 papers and several
textbooks. Co-investigator of several grant projects (GA CR and
MSMT). An author of a lot of user's SW for calculation of
electromagnetic fields and coupled problems.