Excerpt

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)

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- Zoltan Nagy (Author), 2006, Numerical Approaches To 3D Magnetic MEMS, Munich, GRIN Verlag, https://www.grin.com/document/183339

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