Extracto
Diploma Thesis
Numerical Approaches
To
3D Magnetic MEMS
Zolt´
an Nagy
Institute of Robotics and Intelligent Systems
Swiss Federal Institute of Technology Zurich (ETH)
200607
i
Anyone who has never made a mistake
has never tried anything new.
Albert Einstein (1879-1955)
Preface
ii
Preface
I would like to express my gratitude to the persons who motivated, encouraged,
and supported me throughout all my studies and especially during the work on
this diploma thesis. In particular, I am indebted to
Prof. Nelson, for giving me the opportunity to carry out this work at his lab-
oratory.
Olga¸
c Ergeneman and Dr. Jake Abbott, my supervisors who always had
an open door for me and my problems.
The IRIS research group, who gave me the feeling of being part of them,
rather than just another student not washing his coffee cup.
Simon, Matthias and Gian-Marco, who shared the labspace and their coffee
brakes with me.
The Tonipolsterei Football Team, for giving me the optimal balance to study
and work.
My parents, who supported me during my studies.
Sylwia, for her love and patience, especially during the final stage of this work.
All the other people, whom I forgot to mention.
Abstract
iii
Abstract
The present work investigates the potential of the finite element method (FEM) in
the design process of magnetic Micro-Electro-Mechanical-Systems (MEMS). The
magnetic forces and torques acting on a magnetic body are of great importance
in wireless actuating principles. Good models are required to allow for precise
and predictable motion of the magnetic body. However, analytical results are
only available for simple geometries and experiments are often time consuming
and may have a certain number of uncertain parameters that may influence the
results. Numerical methods, and in particular the finite element method, offer
the possibility to study a magnetic body with known material properties in a well
defined environment.
Consequently, in this work, a method is proposed to calculate the net body
torque on arbitrarily shaped bodies in a homogeneous magnetic field using the
commercial finite element software Ansys . In addition, a procedure to de-
termine the demagnetization factors of these bodies is given. The code is first
validated by the known analytical results for an ellipsoid. As an application,
the demagnetization factors, as well as the net magnetic torque on brick shaped
bodies and the IRIS Microrobot are calculated. A method is proposed to predict
the torque acting on the Microrobot analytically. However, experimental results
are necessary to confirm this method.
Furthermore, Ansys is used to model magneto-structural coupling that is, the
motion and deformation of a magnetic body due to an external magnetic field.
Two devices are presented (as case studies rather than as actual design concepts),
the magnetic resonator and the magnetic scratch drive actuator (MSDA). A quasi-
analytical model for the static deflection of the magnetic resonator is given and
good agreement with the finite element model is obtained. The MSDA is modeled
to show the potential of Ansys in modeling MEMS devices, as additional to the
coupling effects, contact elements and spring elements are introduced. Again,
experimental results are required.
Zusammenfassung
iv
Zusammenfassung
Die vorliegende Arbeit untersucht das Potenzial der Methode der Finiten Ele-
mente (FEM) in der Entwurfsphase von magnetischen Mikro-Elektro-Mechanischen-
Systemen (MEMS). F¨
ur drahtlose Antriebstechniken sind magnetische Kr¨
afte und
Momente, die auf einen magnetischen K¨
orper einwirken von grosser Bedeutung.
Gute Modelle sind notwending um eine pr¨
azise und vorhersagbare Bewegung des
magnetischen K¨
orpers zu erm¨
oglichen. Analytische Resultate liegen jedoch nur
f¨
ur einfache Geometrien vor, und Experimente sind oft zeitintensiv und haben
eine gewisse Anzahl von unsicheren Parametern, die das Resultat beeinflussen
k¨
onnten. Numerische Methoden, insbesondere die Methode der Finiten Elemente,
erlauben es, einen magnetischen K¨
orper mit bekannten Materialeigenschaften in
einer wohl definierten Umgebung zu untersuchen.
Dementsprechend wird in dieser Arbeit eine Methode vorgeschagen, um das
Gesamtmoment auf einen beliebigen K¨
orper, in einem homogenen Magnetfeld
mit Hilfe der kommerziellen FEM Software Ansys zu bestimmen. Zustzlich wird
gezeigt, wie die Demagnetisierungsfaktoren von diesen K¨
orpern berechnet werden
k¨
onnen. Der Code wird zun¨
achst mit den bekannten analytischen Resultaten f¨
ur
ein Ellipsoid validiert. Als Anwendung werden sowohl die Demagnetisierungs-
faktoren, als auch das Gesamtmoment auf Quaderf¨
ormige K¨
orper und den IRIS
Microrobot bestimmt. Eine Methode, um das Moment auf den Microrobot ana-
lytisch vorherzusagen wird vorgestellt. Es sind jedoch experimentelle Resultate
erforderlich um die Methode zu best¨
atigen.
Des Weiteren wird in Ansys magneto-mechanische Kopplung, d.h. die Be-
wegung und Verformung eines magnetischen K¨
orpers in einem ¨
ausseren Mag-
netfeld modelliert. Zwei devices werden vorgestellt, der magnetische Resonator
und der magnetische scratch drive actuator (MSDA). Das vorgeschlagene quasi-
analytische Modell f¨
ur die statische Verformung des magnetischen Resonators
stimmt gut mit dem Finite Elemente Modell ¨
uberein. Der MSDA wird modelliert
um das Potenzial von Ansys in der Modellierung von MEMS devices zu zeigen,
da zus¨
aztlich zu den Kopplungseffekten noch Kontakt-, und Federelemente ver-
wendet werden. Auch hier sind Experimente erforderlich um die Resultate zu
best¨
atigen.
Contents
v
Contents
Abstract
iii
Zusammenfassung
iv
List of Tables
vii
List of Figures
vii
1 Introduction
1
1.1
Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Structure of the Report . . . . . . . . . . . . . . . . . . . . . . . .
2
Nomenclature
1
2 Theoretical Considerations
3
2.1
Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Magnetic Force and Torque
. . . . . . . . . . . . . . . . . . . . .
7
2.3
Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3 Finite Element Model
11
3.1
Introduction to the Finite Element Method . . . . . . . . . . . . .
11
3.2
ANSYS and the Maxwell Stress Tensor . . . . . . . . . . . . . . .
11
3.3
The Meshing
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.4
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.5
Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.6
Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.7
Scripting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4 Demagnetization Factors and Magnetic Torque
22
4.1
Demagnetization Factors . . . . . . . . . . . . . . . . . . . . . . .
22
4.2
Magnetic Torque . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.3
Saturation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Contents
vi
5 Coupled Magneto-Structural Analysis
37
5.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
5.2
Implementation of Magneto-Structural Coupling . . . . . . . . . .
38
5.3
Magnetic Resonator . . . . . . . . . . . . . . . . . . . . . . . . . .
42
5.4
Magnetic Scratch Drive Actuator . . . . . . . . . . . . . . . . . .
50
6 Summary and Outlook
56
References
58
A Mathematical Expressions
60
B ANSYS Script to determine the Magnetic Torque on the IRIS
Microrobot
62
C Modeling BH curves using the Langevin Function
72
List of Tables
vii
List of Tables
1
The KEYOPT(1) setting for the Solid5 element
. . . . . . . . . .
39
2
Schematics of the indirect coupling . . . . . . . . . . . . . . . . .
40
3
Dimensions and parameters of the magnetic resonator . . . . . . .
42
4
Dimensions and parameters of the modeled MSDA
. . . . . . . .
53
List of Figures
1
The considered magnetic and air domains. . . . . . . . . . . . . .
6
2
Force and torque calculation using the Maxwell stress tensor . . .
9
3
Analytical model of the torque on a prolate ellipsoid in a constant
external magnetic field . . . . . . . . . . . . . . . . . . . . . . . .
10
4
Mesh of the prolate ellipsoid that is used to validate the finite
element code. (Only 1/8 of the model is shown) . . . . . . . . . .
13
5
Example of a BH curve defined in Ansys
. . . . . . . . . . . . .
17
6
The Solid5 and Solid96 Finite Elements . . . . . . . . . . . . .
19
7
The Solid98 and Plane13 Finite Elements . . . . . . . . . . . .
21
8
Validation of the procedure to determine the demagnetization factors 23
9
Demagnetization factor n
x
of brick shaped bodies compared to
their equivalent ellipsoids . . . . . . . . . . . . . . . . . . . . . . .
24
10
Validation of the calculation of the magnetic torque . . . . . . . .
26
11
Magnetic torque on brick shaped structures
. . . . . . . . . . . .
27
12
Finite Element Model of the Microrobot . . . . . . . . . . . . . .
28
13
Torque per volume on the Microrobot predicted by FEM and com-
pared to analytical results . . . . . . . . . . . . . . . . . . . . . .
29
14
Fit of the relationships between the demagnetization factors and
the size of the equivalent ellipsoid and
E . . . . . . . . . . . . . . 30
15
Mapping between the Microrobot, its equivalent ellipsoid and the
ellipsoid E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
16
Different torque results for brick with b/a = 0.8 and c/a = 0.3420.
33
17
The BH curve allows to study saturation effects on the torque
behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
List of Figures
viii
18
Experimental results for the torque on the microrobot compared
to the analytical value for an ellipsoid . . . . . . . . . . . . . . . .
35
19
Model of the magnetic resonator . . . . . . . . . . . . . . . . . . .
43
20
Mesh of the the magnetic resonator . . . . . . . . . . . . . . . . .
43
21
Static deflection of the magnetic cantilever calculated by FEM and
compared to the analytical model . . . . . . . . . . . . . . . . . .
47
22
The first two mode shapes of the magnetic resonator. . . . . . . .
49
23
Bode plot showing the frequency response of the magnetic resonator 49
24
The electrostatic scratch drive actuator . . . . . . . . . . . . . . .
50
25
Working principle of the ESDA. See text for explanations.
. . . .
51
26
Model of the Magnetic Scratch Drive Actuator . . . . . . . . . . .
52
27
Mesh of the Magnetic Scratch Drive Actuator . . . . . . . . . . .
54
28
Deflection of the MSDA plate. . . . . . . . . . . . . . . . . . . . .
55
29
Total reaction force at the bushingsubstrate interface . . . . . . .
55
1
1
Introduction
The integration of magnetic materials onto flexible silicon supports has been
demonstrated several years ago [1]. Moreover, these structures have been suc-
cessfully actuated by external magnetic fields. Thus, magnetic MEMS (Micro-
Electro-Mechanical-Systems) offer the potential for wireless sensing and actuation
applications.
However, due to the planar fabrication technology (lithography), only two
dimensional devices with a given thickness can be produced. 3D MEMS with
arbitrary shapes in all three directions are difficult to fabricate. One approach
to overcome this limitation is to assemble three dimensional bodies from planar
structures.
The knowledge of the forces and torques acting on a magnetic body that
is placed in a magnetic field is necessary in order to predict its motion and
consequently to allow for successful control strategies. For example, the Institute
of Robotics and Intelligent Systems (IRIS) at ETH Z¨
urich has proposed a 3D
MEMS Microrobot assembled from electroplated magnetic parts that is actuated
by external magnetic fields [2]. As one of the applications of the Microrobot is
supposed to be eye surgery, a precise and predictable motion is required.
As will be shown later, the calculation of forces and torques is straightforward
as long as the magnetization of the body is known. However, the magnetization
depends on the magnetic field inside the magnetic body, which in turn is a func-
tion of the external field and the shape of the body. Hence, analytical solutions
are only available for simple geometries and magnetic field configurations, e.g.
the torque on an ellipsoid in a constant external magnetic field can be predicted.
1.1
Scope of the Thesis
Currently, several projects at IRIS aim at deepening the understanding of the
magnetic properties of microassembled 3D magnetic bodies. Amongst others,
the dependency of the torque on the shape of the mentioned Microrobot is ex-
perimentally investigated in a semester project [3].
1.2
Structure of the Report
2
The present work will contribute to this by proposing a finite element method
(FEM) to predict the torques acting on the Microrobot. Since no experimental
results are available yet, the code is validated by the analytical results for an el-
lipsoid. Similarly, a method for the determination of the demagnetization factors
of symmetrical bodies is proposed.
Finally, the coupling of the magnetic and the mechanical field is examined to
provide a method for the optimization of magnetic MEMS devices. Two devices
are investigated, a magnetic resonator and the magnetic scratch drive actuator.
These studies are only feasibility studies to show the capabilities of the finite
element code and there is no intention to give design propositions.
1.2
Structure of the Report
Section 2 presents the theoretical background of this work. The governing equa-
tions, that is, Maxwell's equations in the magnetostatic case, as well as the mag-
netic force and torque equations are introduced and applied to the case of a
uniformly magnetized ellipsoid.
The basics of the finite element method are discussed in Section 3. The usage
of the finite element software Ansys [4] for magnetostatics, as well as specific
topics such as the meshing, the boundary conditions, and material parameter
settings are explained.
The results of the finite element simulations on the calculation of the de-
magnetization factors and the global body torque are presented in Section 4. In
addition, the torque on the Microrobot is predicted and compared to the analyt-
ical torque expression of an ellipsoid.
Coupled Field Analysis is introduced in Section 5. After presenting the basics
on the coupling of different physical fields in the finite element code, two devices,
the magnetic resonator, and the magnetic scratch drive actuator are studied.
2 Theoretical Considerations
3
2
Theoretical Considerations
After introducing Maxwell's equations to describe electric and magnetic fields, the
constitutive relationships for magnetic materials are given. Next, boundary and
continuity equations are discussed. Furthermore, two equivalent expressions for
the net magnetic force and torque acting on a soft magnetic body are presented.
Finally, the analytical expression for the net magnetic torque on an ellipsoid is
derived.
2.1
Magnetostatics
2.1.1
Maxwell's Equations
The behavior of electromagnetic fields as well as their interactions with matter
are described by Maxwell's equations, which in the differential form are given by
1
Gauss's Law
·D =
(2.1)
Gauss's Law for Magnetics
·B = 0
(2.2)
Faraday's Law of Induction
×E = -
dB
dt
(2.3)
Amp`
ere's Law
×H = J +
dD
dt
(2.4)
where H and E are the magnetic and electric field respectively, D and B are the
electric
2
and magnetic flux density, and and J are the free electric charge and
free current density.
We will consider the special case with no electrical charges ( = 0), no electric
fields (E = 0), no currents (J = 0) and static fields (
d(·)
dt
= 0). Then Maxwell's
equations reduce to
·B = 0
(2.5)
×H = 0
(2.6)
1
See Appendix A for an overview of the used mathematical symbols
2
D is also referred to as the electric displacement
2.1
Magnetostatics
4
2.1.2
Magnetic Materials
In a magnetic material (as well as in vacuum) B and H are related by the con-
stitutive law
B = H
(2.7)
where is the magnetic permeability tensor. This relationship can be anisotropic
and nonlinear ( = (H)). For air (treated as vacuum) and a linear soft magnetic
material (2.7) reduces to [5]
B
air
=
0
· H
air
(2.8)
B
m
=
0
r
· H
m
(2.9)
where
0
is the free space permeability (scalar) and
r
is the relative permeability
tensor that reduces to a scalar in the isotropic case. The subscript m refers to a
soft magnetic material. When dealing with magnetic materials in magnetic fields
the quantities B
air
and H
air
are often referred to be the external (or applied)
fields, that is the field without the magnetic material, whereas B
m
and H
m
are
considered to be the internal fields.
A magnetic material in an applied magnetic field H
air
responds to the field
by producing a magnetic field: its magnetization M. Depending on their demag-
netization properties, magnetic materials can be classified into two groups:
Hard Magnetic materials need high external fields to reduce the magnetiza-
tion and are thus difficult to demagnetize. This fact is also described by
designating hard magnetic materials as having a high coercivity.
Soft Magnetic materials on the other hand show a low coercivity and are con-
sequently easily demagnetized.
In the following, we will only concentrate on soft magnetic materials. The mag-
netization M of a soft magnetic material is defined by
M = B
m
/
0
- H
m
= (
r
- 1)H
m
:= H
m
(2.10)
where
1 is the unity tensor, the second equality is obtained using (2.9), and is
the magnetic susceptibility tensor defined by :=
r
- 1 that relates the mag-
2.1
Magnetostatics
5
netization of a material to its internal field H
m
.
In addition, the magnetization can be used to relate the internal field H
m
to the
external field H
air
by [6]
H
m
= H
air
- N · M
(2.11)
where N is demagnetization tensor
3
that is diagonal if the coordinate frame is
aligned with the axes of symmetry (if existent) of the magnetic body:
N =
n
x
0
0
0
n
y
0
0
0
n
z
(2.12)
where n
i
is the demagnetization factor in the i-th direction.
By inserting (2.11) into (2.10) and assuming a constant and isotropic suscep-
tibility ( = 1, R) we find
M = H
m
= (H
air
- N · M).
(2.13)
Solving for M we can relate the magnetization to external field H
air
by
M = (
1 + N)
-1
H
air
:=
a
H
air
(2.14)
where
a
is the apparent susceptibility tensor defined by
a
=
1+n
x
0
0
0
1+n
y
0
0
0
1+n
z
(2.15)
Note, that all tensors are formulated with respect to the coordinate frame of the
magnetic body.
2.1.3
Boundary and Continuity Conditions
The transition of the magnetic field from one material to another is governed by
boundary and continuity conditions. We assume no electrical current flow and
3
The magnetic field N
· M is also referred to as the demagnetization field.
2.1
Magnetostatics
6
m
air
air,m
0
0
r
air
m
Figure 1: The considered magnetic and air domains.
the case of an isotropic, soft magnetic material with permeability
r
and magne-
tization M in a magnetic field H
air
(see Figure 1).
Mathematically, we consider the magnetic domain
m
and the air domain
air
.
The total domain =
air
m
is bounded by the union of their outer boundaries
=
air
m
. Finally,
air,m
is the boundary between
air
and
m
, that is
the boundary of the soft magnetic material that is in contact with air.
On the boundary conditions are defined as [7]
B
· n = 0
(2.16)
H × n = 0
(2.17)
where n is the normal unit vector. Thus, the normal component of B, as well as
the tangential component of H vanishes. Note, that in the finite element model,
we will prescribe these boundary conditions in order to achieve the desired field
on
air
(see section 3.4).
The continuity conditions on
air,m
require that
B
m
· n
m
+ B
air
· n
air
= 0
(2.18)
H
m
× n
m
+ H
air
× n
air
= 0
(2.19)
2.2
Magnetic Force and Torque
7
where n
air
and n
m
are opposed normal unit vectors, that is
n
air
=
-n
m
.
(2.20)
Thus, conservation of the normal component of the magnetic induction B and
the tangential component of the magnetic field H is demanded.
Writing B
i
and H
i
(i = air,m) in their normal and tangential components
B
i
= (B
i,n
, B
i,t
)
(2.21)
H
i
= (H
i,n
, H
i,t
)
(2.22)
we can rewrite (2.18) and (2.19) as
B
air,n
= B
m,n
(2.23)
H
air,t
= H
m,t
(2.24)
We now can express the normal and tangential component of the magnetization
on the boundary
air,m
as [5]
M
n
=
B
m,n
0
- H
m,n
=
B
air,n
0
- H
m,n
= H
air,n
- H
m,n
(2.25)
M
t
=
B
m,t
0
- H
m,t
=
B
m,t
0
- H
air,t
=
1
0
(B
m,t
- B
air,t
)
(2.26)
where in each line the second equality is obtained by applying the continuity
equations (2.23) and (2.24) respectively, and the last equality is given by the
constitutive law 2.8.
2.2
Magnetic Force and Torque
In general the net body force F and torque on a soft magnetic body in a magnetic
field H
air
=
1
0
B
air
can be found as [8]
F =
V
(M · )B
air
dV
(2.27)
=
V
[r × (M · )B
air
+ M × B
air
] dV
(2.28)
Final del extracto de 81 páginas
- Citar trabajo
- Zoltan Nagy (Autor), 2006, Numerical Approaches To 3D Magnetic MEMS, Múnich, GRIN Verlag, https://www.grin.com/document/183339
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