design and modelling of electromagnetic actuation in mems

The University of Southern Denmark

Design and Modelling of

Electromagnetic Actuation in

MEMS Switches

Author:

Romans Safonovs

Supervisors:

Jost Adam

Roana Melina de Oliveira Hansen

1st June 2017

Abstract

Magnetic actuation is a promising approach to operate MEMS switches due to larger

forces than electrostatic actuation, resulting in a possibility of a better performance.

The purpose of this work is to design and model such MEMS switch, which can be

fabricated at Fraunhofer ISIT. The switch should also be minimized for the power

needed to actuate, given the manufacturing constraints.

Modelling has been performed with FEM software package COMSOL Multiphysics.

The simulation results were also compared to the results from analytical solutions.

Two concepts were developed, which were fully minimized according to the fabrica-

tion constraints of Fraunhofer ISIT. The first one is a much more compact solution and

it requires 4.25 µW of power and 13.8 mA of current to actuate. The second concept,

however, is a bit more challenging to fabricate and requires more space, but is able to

operate with just 180 nW of power and 1.3 mA of current.

Also, this work covers some of the research done in magnetic MEMS switch industry,

theoretical background of magnetic actuation and the process of modelling such sys-

tems in COMSOL, combining both numerical methods and analytical ones, as well

as efficiency improvements of those simulations. In addition to that, it includes some

additional simulations that examine influence of deformation of a square-shaped coil

on magnetic force that it exerts and simulations that derive position and dimensions of

a rectangular-shaped coil under a magnetic film to generate the biggest possible forces.

ii

Preface

This work has been carried out as a part of Master of Science in Engineering - Mechat-

ronics programmme at the University of Southern Denmark, between September 2016

and June 2017.

I would like to thank my friends, parents and girlfriend for supporting me through-

out this process. Also, I would like to thank Fabian Lofink of Fraunhofer ISIT, for his

insight, experience and recommendations regarding fabrication of MEMS.

1st June 2017

Romans Safonovs

iii

Contents

List of Tables vi

List of Figures vii

1 Introduction 1

2 Theoretical Background 3

2.1 Magnetic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Magnetic Field of a Coil . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Cantilever Beam Deflection . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Electric Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.6 MEMS Fabrication Processes . . . . . . . . . . . . . . . . . . . . . . . 9

3 State of the Art 12

4 Concept of a Magnetic MEMS Switch 19

5 Modelling of a Magnetic Actuation 21

5.1 Coil and Magnetic Film Modelling . . . . . . . . . . . . . . . . . . . . 21

5.2 Cantilever Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Model Verification 26

6.1 Calculating the Magnetic Force . . . . . . . . . . . . . . . . . . . . . . 26

6.2 Calculating the Deflection . . . . . . . . . . . . . . . . . . . . . . . . . 28

iv

Contents

7 Modelling Efficiency 30

7.1 Exploiting the Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.2 Magnetic Force Calculation . . . . . . . . . . . . . . . . . . . . . . . . 31

7.3 Meshing Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8 Design and Optimization of a Magnetic MEMS Switch 34

8.1 Fabrication Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

8.2 Magnetic Switch Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 35

8.3 Optimization Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 37

8.4 Results and Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9 Conclusion and Future Work 44

Bibliography 45

A Calculations and Derivations 48

A.1 Magnetic Field for a Rectangular Electric Loop . . . . . . . . . . . . . 48

A.2 Derivation of a Magnetic Force . . . . . . . . . . . . . . . . . . . . . . 51

A.3 Calculations for a Cantilever Beam . . . . . . . . . . . . . . . . . . . . 52

B Additional Simulations 55

B.1 Cantilever Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 55

B.2 Relation Between a Magnetic Film Length and a Magnetic Force . . . . 56

B.3 Optimal Coil Position Under a Magnetic Film . . . . . . . . . . . . . . 57

B.4 Coil Shape Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

C MATLAB Scripts 62

C.1 Magnetic Force Calculation . . . . . . . . . . . . . . . . . . . . . . . . 62

C.2 Result Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

v

List of Tables

3.1 Performance comparison of magnetic MEMS switches and microactuators 14

8.1 Fabrication constraints of Fraunhofer ISIT . . . . . . . . . . . . . . . . 34

8.2 Parameter values solved for in the simulations . . . . . . . . . . . . . . 38

B.1 Dimensions of two models to be analyzed for stress . . . . . . . . . . . 55

vi

List of Figures

1.1 Operating principle of a magnetic MEMS switch proposed in this work 2

2.1 Magnetic field dB due to an electric current I. Source: [6] . . . . . . . 4

2.2 Magnetic field B at point P due to an electric current I in a straight wire 5

2.3 Cantilever beam under a distributed load w . . . . . . . . . . . . . . . 6

2.4 Standard micromachining process flowchart. Source: [12] . . . . . . . . 9

2.5 Photolithographic process. Source: [13] . . . . . . . . . . . . . . . . . . 10

2.6 Surface (a) and bulk (b) micromachining processes. Source: [15] . . . . 11

3.1 Switching principles for ohmic and capacitive devices. Source: [2] . . . 13

3.2 Concept scheme of a magnetic microactuator [19]. Source: [19] . . . . . 14

3.3 Concept scheme of a magnetic microactuator [20]. Source: [20] . . . . . 15

3.4 Concept scheme of a magnetic MEMS switch [22]. Source: [22] . . . . . 16

3.5 Concept scheme of a latching magnetic MEMS switch [23]. Source: [23] 17

3.6 Concept scheme of a latching magnetic MEMS switch [24]. Source: [24] 17

4.1 Cross-section of the proposed magnetic MEMS switch . . . . . . . . . . 19

4.2 Modelling domain of the proposed magnetic MEMS switch . . . . . . . 20

4.3 Sub-problem that solves the device for magnetic force . . . . . . . . . . 20

4.4 Sub-problem that solves the device for deflection . . . . . . . . . . . . . 20

5.1 Dimensions of the magnetic film and the coil . . . . . . . . . . . . . . . 21

5.2 3D model of the magnetic film and the coil . . . . . . . . . . . . . . . . 21

5.3 Modelling domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.4 Influence of having an infinite domain versus not having one . . . . . . 22

vii

List of Figures

5.5 Magnetization curve of cobalt . . . . . . . . . . . . . . . . . . . . . . . 23

5.6 Convergence study for coil and magnetic film modelling, n is a mesh size

parameter, the bigger it is, the smaller mesh becomes . . . . . . . . . . 24

5.7 Dimensions of the cantilever . . . . . . . . . . . . . . . . . . . . . . . . 24

5.8 Cantilever setup diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.9 Convergence study for cantilever modelling, n is a mesh size parameter,

the bigger it is, the smaller mesh becomes . . . . . . . . . . . . . . . . 25

6.1 Magnetic field in the center of a square electric loop due to 1 A of current 27

6.2 Force diagram of a cantilever beam under force Fmagnetic . . . . . . . . 28

7.1 Modelling domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.2 Convergence of COMSOL’s method of magnetic force calculation, n is

a mesh size parameter, the bigger it is, the smaller mesh becomes . . . 32

7.3 Convergence of COMSOL’s method of magnetic force calculation com-

pared to convergence of analytic-based solution, n is a mesh size para-

meter, the bigger it is, the smaller mesh becomes . . . . . . . . . . . . 33

7.4 Mesh of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8.1 Minimized dimensions of the device according to fabrication constraints 35

8.2 Top-view dimensions of Concept 1 . . . . . . . . . . . . . . . . . . . . . 36

8.3 Top-view dimensions of Concept 2 . . . . . . . . . . . . . . . . . . . . . 36

8.4 Simulation results of Concept 1 for different parameter configurations

when l = 1 and hfilm = 3 µm . . . . . . . . . . . . . . . . . . . . . . . . 40

8.5 Simulation results of Concept 2 for different parameter configurations

when l = 1 and hfilm = 3 µm . . . . . . . . . . . . . . . . . . . . . . . . 41

8.6 Expanded simulation results of Concept 2 for different parameter con-

figurations when l = 1 and hfilm = 3 µm . . . . . . . . . . . . . . . . . 41

8.7 Simulation results of Concept 1 for different parameter configurations . 42

8.8 Simulation results of Concept 2 for different parameter configurations . 43

viii

List of Figures

A.1 Magnetic field dB due to an electric current I. Source: [6] . . . . . . . 48

A.2 Magnetic field B at point P due to an electric current I in a straight wire 49

A.3 Rectangular electric loop, generating a magnetic field at point P . . . . 49

A.4 Force diagram of a cantilever beam under force Fmagnetic . . . . . . . . 53

B.1 Surface stresses of Concept 1 . . . . . . . . . . . . . . . . . . . . . . . . 56

B.2 Surface stresses of Concept 2 . . . . . . . . . . . . . . . . . . . . . . . . 56

B.3 Relation between length of a rectangular loop and force that it exerts . 57

B.4 Relation between coil dimensions and magnetic force that it exerts upon

constant magnetic film. The coil has 1 loop . . . . . . . . . . . . . . . 58


constant magnetic film. The coil has 2 loops . . . . . . . . . . . . . . . 59


constant magnetic film. The coil has 3 loops . . . . . . . . . . . . . . . 60

B.7 Deformed square loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

B.8 Relation between deformation and magnetic force . . . . . . . . . . . . 61

B.9 Comparison of methods of increasing magnetic forces . . . . . . . . . . 61

ix

1. Introduction

Every electric and electronic device takes use of switches: from flashlights and toasters

to mobile phones and power supplies. They allow or block current flow between two

conductors and are used to convert and manage energy and to route signals. And it

is logical that there is an ongoing research trying to find an ideal switch, which would

have zero on resistance, infinite off resistance, zero energy and time required to change

state, infinite lifetime, infinite power-handling capability and minimal size [1].

Of course, achieving such kind of characteristics is impossible due to laws of physics,

but it gives an idea, which parameters should be minimized and maximized and by

improving them, it would generally lead to a better power efficiency and smaller size

of any device that uses switches.

Efficiency is important, since it allows a system to accomplish the same task by

using less energy, which leads to many beneficial effects, such as lower operational

costs, lower pollution and others, depending on the device. For example, in case of

switching regulators, higher efficiency means less heat buildup, allowing for smaller and

lighter power supplies.

Previously, there were mainly two groups of switches: electromechanical relays

and solid-state relays. While electromechanical ones have better off-isolation, linearity

and low on resistance, solid-state switches are smaller, cheaper, faster and consume

less power [1]. But in recent years, micro-electro-mechanical systems (MEMS) have

become good candidates for combining good qualities of both above-mentioned switch

groups. There is a number of ways they can be operated: using electrostatic forces,

electrothermal expansion forces and magnetic forces. Electrostatically actuated MEMS

switches are easy to fabricate and hence, are the most common type of MEMS switches,

1

Introduction

but they are limited by their relatively high actuation voltage and small displacements.

Electrothermally actuated switches are the slowest type of switches, however they

produce the best contact forces , resulting in low contact resistance [1].

Magnetic actuation, however, is a promising approach to operate MEMS switches,

due to forces of higher magnitude and longer range than of electrostatic ones, even

though current designs are slower and consume a more power [2]. Operation mechanism

of such MEMS switches is illustrated below (Figure 1.1).

Figure 1.1: Operating principle of a magnetic MEMS switch proposed in this work

The goal of this work is to design and model a magnetic MEMS switch in such

a way that it is feasible in terms of fabrication capabilities of Fraunhofer Institute

of Silicon Technologies (Fraunhofer ISIT) and requires the least amount of power to

activate, possibly pushing the capabilities of such devices at microscale.

In the following chapters, first, I will cover concepts used throughout the project,

such as: magnetic force, magnetic field, cantilever beam deflection, electric power,

finite element method and microfabrication techniques. After that, I will explore state

of the art. Then, I will go though the process of modelling of a magnetic MEMS switch

(Figure 1.1) in COMSOL and compare the results to analytical solutions. And finally,

I will apply fabrication constraints of Fraunhofer ISIT and optimize all parameters of

the model for the smallest power dissipation. In the end, I will draw conclusions based

on the results and briefly sum up the work.

2

2. Theoretical Background

In this chapter I will briefly go through the most important concepts and formulas,

which were used or touched upon during this project, related to such fields as: magnetic

force, magnetic field, cantilever beam deflection, electric power, finite element method

and microfabrication techniques.

2.1. Magnetic Force

The goal of this work is to model a magnetically actuated MEMS switch, so it is

important to understand mechanisms that govern operation of a such device. Magnetic

switches usually contain two actuation components: an excitation coil and a magnetic

material. The coil generates magnetic field and pulls the magnetic material, which is

attached to a cantilever, closing the switch (Figure 1.1). A magnetic force acting on

an electric loop with magnetic moment m under an external magnetic field B can be

calculated as [3]:

F = ∇(m ·B). (2.1)

But Equation 2.1 only calculates the force acting only on a single loop. If there

is a magnetic material with volume V and magnetization per volume M , it can be

expressed as a collection of infinitely small loops, meaning that the total magnetic

force acting of a magnetic material in an external magnetic field is equal to [4]:

Ftotal =

∫∇(M ·B) dV. (2.2)

COMSOL calculates the force using the so-called Maxwell’s stress tensor S [5]:

3

Theoretical Background

Ftotal =1

µ0

∫∫dAS · n (2.3)

where n is a unit vector pointing out of a unit surface dA, and S is defined as:

S =1

µ0

B2x −B2/2 BxBy BxBz

ByBx B2y −B2/2 ByBz

BzBx BzBy B2z −B2/2

(2.4)

where B2 = B2x +B2

y +B2z .

2.2. Magnetic Field of a Coil

A magnetic force is obviously dependent on magnetic field, as discussed in chapter

above. And it is logical that first, a magnetic field generated by a coil must be calculated

first. The Biot-Savart’s law is an equation, describing the magnetic field generated by

an electric current and holds only in magnetostatic problems [6]:

dB =µ0

4π

Idl × ~x|~x|3

(2.5)

where dB, µ0, I, dl and ~x are elemental flux density, current, vacuum permeability

(4π × 10−7), element of length and position vector respectively (Figure 2.1).

Figure 2.1: Magnetic field dB due to an electric current I. Source: [6]

Using Equation 2.5 we can calculate magnetic fields due to a straight current wire

[4]:

4


B =µ0I

4πd(cosα1 + cosα2) (2.6)

where d, α1 and α2 are distance to the wire and angles that are shown on Figure 2.2

respectively. This equation can be expanded to calculate a magnetic field exerted by a

square loop simply by splitting the loop into 4 straight fragments, calculating magnetic

field for each one of them and adding the results together. See Appendix A.1 for a

detailed derivation of a magnetic field due to a rectangular electric loop.

Figure 2.2: Magnetic field B at point P due to an electric current I in a straight wire

COMSOL calculates the magnetic field by directly solving Gauss’s law for magnet-

ism and Ampere’s circuital law (static version, in this case):

∇ ·B = 0 (2.7)

∇×B = µ0J (2.8)

where J is current density.

2.3. Cantilever Beam Deflection

In addition to calculating the magnetic force, it is also necessary to calculate how much

the cantilever bends under it in order to model a magnetic actuation of a switch in

Figure 1.1. The Euler-Bernoulli beam theory is a simplification of the linear theory of

elasticity, which calculates deflection of beams due to lateral loads [7]:

5


d2

dx2

(EI

d2y

dx2

)= w(x) (2.9)

where E, I, y, x and w are the Young’s modulus, second moment of inertia, deflection,

position and load respectively (Figure 2.3).

Figure 2.3: Cantilever beam under a distributed load w

COMSOL calculates stresses using the Cauchy’s equilibrium equation [8]:

0 = ∇ · S + F (2.10)

where S is stress tensor and F is external force. And finally, a strain tensor T is

calculated using the Hooke’s law for linear elastic materials [8]:

S = cT (2.11)

where c is a fourth order tensor.

2.4. Electric Power

Power dissipation is an important figure of merit for any switch. It says how much

power is needed for it to actuate. Electric power P is defined as:

P = V I (2.12)

6


where V and I are voltage and current respectively. So, using the Ohm’s law (I = V/R),

power P , dissipated in a wire (or a coil) can be calculated as:

P = I2R (2.13)

where I and R are the current and resistance of the wire respectively. In case of

magnetic switches, power dissipation comes from a coil, carrying electric current needed

for to generate such magnetic field that the cantilever is attracted, closing the switch

(Figure 1.1). Resistance of the wire R can be calculated using the following formula:

R =ρL

A(2.14)

where ρ, L and A are wire material’s resistivity, length and cross-section respectively.

2.5. Finite Element Method

The laws of physics are usually expressed in terms of partial differential equations

(PDEs). However, in most cases it cannot be done analytically. Instead, there are

different numerical methods that calculate an approximation of the real solution of the

PDEs. The finite element method (FEM) is one of those [9].

FEM is a powerful tool for solving differential equations. This method can easily

deal with complex geometries and higher-order approximations of the solution [10]. It

revolves around an idea where a function u can be approximated by a function uh using

linear combinations of basis functions [9]:

u ≈ uh (2.15)

uh =∑i

ξiϕi (2.16)

where ξi and ϕi are the coefficients and basis functions respectively.

7


This process starts with rewriting the original PDE as its weak form by multiplying

both sides of it by a test function v, and integrating, thus getting the variational

formulation. The test function v must be square integrable and zero at Dirichlet

boundaries [8].

At this stage the equation exists on continuum, the global domain of the prob-

lem. Therefore, the collection of admissible functions and trial functions span infinite-

dimensional functional spaces.

Next step is to discretize the weak form by subdividing its space into smaller sub-

domains or elements, defining the finite element formulation. This step is equivalent to

projection of the weak form of PDEs onto a finite-dimensional subspace. The notations

uh and vh represent the finite-dimensional equivalent of u and v respectively [11]. Since

uh belongs to space of vh, it can be rewritten as the following linear combination:

uh =∑i

ξiϕi (2.17)

where ϕi and ξi are the basis functions and coefficients to be determined respectively.

By using those basis functions as load functions, the finite element formulation can be

rewritten as:

Aξ = b (2.18)

where A is the stiffness matrix and b is the load vector, both of which can be calculated.

By solving Equation 2.18, it is possible to find ξ, which allows the solution of Equation

2.17.

In essence, FEM represents the original domain of the problem as a collection of

elements. Then, for each element, substitutes the original PDE problem by a set

of simple equations that locally approximate the original equations. After that, it

applies boundary conditions for boundaries of each element and assembles the resulting

equations and boundary conditions into a global system of equations that models the

entire problem. And finally, solves the resulting system of equations [11].

8


2.6. MEMS Fabrication Processes

Microfabrication owes most of its techniques to the semiconductor industry, the same

tools can be used for MEMS fabrication (Figure 2.4). The core process includes cleaning

the substrate, deposing a thin film, applying mask, etching and repeating, if needed

[12].

Figure 2.4: Standard micromachining process flowchart. Source: [12]

Photolithography is the single most important process that enables MEMS to be

produced reliably with microscopic dimensions and in high volume. It is the process of

transfer of a pattern on to a material and is arguably the most important step in the

microfabrication process. The essentials of the photolithographic process are illustrated

in Figure 2.5 [13, 14].

The process begins by selecting a substrate material and geometry, typically a

single-crystal silicon wafer (Figure 2.5 (a)). Next, the substrate is coated by a pho-

9


Figure 2.5: Photolithographic process. Source: [13]

tosensitive polymer, photoresist (Figure 2.5 (b)). A mask, consisting of a transparent

supporting medium with patterned opaque regions, is used to cast a detailed shadow

onto the photoresist. The regions receiving an exposure of ultraviolet light are chemic-

ally altered (Figure 2.5 (c)). After exposure, the photoresist is immersed in a solution

that removes either the exposed regions or the unexposed regions (Figure 2.5 (d)).

After the wafer is dried, the photoresist can be used as a mask for a subsequent de-

position (Figure 2.5 (e)) or etch (Figure 2.5 (f)). Lastly, the photoresist is removed

(Figures 2.5 (g) and 2.5 (h)) [13, 14].

Typically, fabrication of MEMS takes advantage of one of two common MEMS

photolithography processes: bulk or surface micromachining [15].

Structures created from bulk micromachining are predominantly made within a

silicon substrate, which is used as the sacrificial layer (Figure 2.6 (a)) [13].

Surface micromachining is a process of fabricating three-dimensional structures on

10


the surface of the silicon substrate using multilayer depositions and patterning of these

thin films (Figure 2.6 (b)) [13].

Figure 2.6: Surface (a) and bulk (b) micromachining processes. Source: [15]

11

3. State of the Art

The past 25 years have seen MEMS transition from being a research curiosity to a

multibillion dollar commercial enterprise. At the same time, there have not been many

commercially successful MEMS, compared to the number of prototype devices created

as part of research. In large part this is because MEMS development is still in an

exploratory phase, where all ideas are considered worth examining [16].

The kingdom of MEMS actuators mainly consists of four families: electrostatic,

piezoelectric, thermal and magnetic [16]. Each actuation principle has its own ad-

vantages and disadvantages. The choice and the optimization of an approach should

be made according to the requirements of a particular application [17]. Electrostat-

ical actuation is the most frequently applied principle combining versatility and simple

technology. It needs neither additional elements like coils or cores, nor special materials

like shape-memory-alloys or piezoelectric ceramics, consumes little power, easily integ-

rated and controlled [18]. On the contrary, the other types of actuators are more robust

and more capable of producing larger forces [17]. Although nearly every permutation

of activation principle and device has been tried, only a few have been leaving the re-

search laboratories [18]. Still, MEMS actuators are used in many fields, such as optics,

medicine, communication systems, automation, aerospace, medicine etc. [17]. Typical

MEMS devices are micropumps, microvalves, microrelays, microgrippers, micromirrors

etc. [18].

In scope of this project I am going to inspect MEMS actuators that are used as

switches. MEMS switches are substantially different from PIN semiconductor diodes

or FET switches, although the purpose of both types of devices is to vary the imped-

ance of an electrical path in a controlled fashion. While solid state devices employ

12

State of the Art

electric fields to vary the conductivity of a channel, effectively closing or opening a

conduction line, MEMS switches utilize mechanically moving parts to physically vary

the distance between two conductive elements of a signal line in order to make or break

an ohmic contact, in the case of ohmic switches, or to increase or decrease the enclosed

capacitance,in the case of capacitive switches (Figure 3.1) [2].

Figure 3.1: Switching principles for ohmic and capacitive devices. Source: [2]

MEMS relays generally offer a number of advantages over solid state devices such

as higher off-state isolation and low power consumption, depending on the actuation

scheme. On the other hand, MEMS switches suffer from a series of problems in terms

of reliability, particularly exacerbated by failure mechanisms such as self-actuation,

stiction, electromigration, microwelding, etc. MEMS devices additionally carry the

burden of needing an appropriate packaging solution that guarantees functionality and

reliability, which potentially increases manufacturing costs. All of these challenges are

currently subjects for process improvements and optimization [2].

As discussed previously, there are four main groups of MEMS actuators. Same can

be applied to MEMS switches. In this work I am going to focus on electromagnetic

MEMS switches.

Magnetic MEMS are based on the interaction between sources of electromagnetic

or magnetic forces such as coils or permanent magnets and microstructures fabricated

with magnetic materials. The strong interest in the application of such old and well-

established physics to microscale components lies in the advantages offered by magnetic

forces over conventional electrostatic components at smaller scales [2].

There is a number of magnetic MEMS switches reported in literature (Table 3.1).

13

State of the Art

Table 3.1: Performance comparison of magnetic MEMS switches and microactuators

Device Power/energy required Latching

Ahn and Allen [19] 832µW No

Judy and Muller [20] 1.6 W No

Wright et al. [21] 19 mW No

Taylor et al. [22] 33 mW No

Ruan et al. [23] 93µJ Yes

Cho et al. [24] 40.3 µJ Yes

Glickman et al. [1] 13 mW No

Kohl and Gray [25] 85µJ Yes

Lu et al. [26] 50 µJ Yes

One of the first ones dates back to 1993, when Ahn and Allen [19] suggested a mag-

netic microactuator, which would have a magnetic core with a conductor coil wrapped

around it (Figure 3.2). A current through the coil would generate a magnetic flux in

the core and the air gap, generating a magnetic force in the air gap, which would at-

tract the cantilever to the bottom contact. With a cantilever, which was 2.5 µm thick,

25µm wide and 780 µmlong, they achieved a deflection of 6µm by applying 800 mA to

the coil, which results in 832 µW power consumption.

Figure 3.2: Concept scheme of a magnetic microactuator [19]. Source: [19]

14

State of the Art

A few years later, in 1997, Judy and Muller [20] introduced a concept of local

magnetic forces generated by a coil (Figure 3.3). It requires 500 mA to rotate the plate

with magnetic material by 45, which is 450 µm wide and 450µm long, resulting with

1.6 W power dissipation.

Figure 3.3: Concept scheme of a magnetic microactuator [20]. Source: [20]

In the same year, Wright et al. [21] demonstrated an electromagnetic actuator,

designed to drive microrelays, that includes a permalloy cantilever beam and a planar

coil. For the cantilever which was 1000 µm long, it would require 24 mA to achieve a

deflection of 4µm, resulting with 19 mW power dissipation.

Later, in 1998 Taylor et al. [22] develop a device that has a single-layer coil that

actuates an upper movable magnetic plate (Figure 3.4). The device shows great per-

formance in terms of actuation force, thus resulting in a low contact resistance. The

minimum current for actuation was 180 mA, resulting in an actuation power of 33 mW

with contact resistance as low as 22.4 mΩ. The upper plate is 3.5 mm long and 1.95 mm

wide. This device could also carry 1.2 A of current through the relay contacts.

In 2001, Ruan et al. [23] designed a magnetically actuated microrelay with latching

functionality. A bistable configuration is achieved due to two magnetic effects. A

hinged magnetic cantilever is put on top of a planar coil, and the whole assembly is

then mounted on top of a permanent magnet (Figure 3.5).

Microcantilevers are highly anisotropic structures that strongly favor magnetization

15

State of the Art

Figure 3.4: Concept scheme of a magnetic MEMS switch [22]. Source: [22]

along their easy axis corresponding to their length. A magnetized cantilever experiences

a torque that aligns its magnetization to the field axis and orientation. The device

utilizes the embedded coil to force the cantilever in one of two possible magnetic states,

depending on the polarity of the applied voltage, thus magnetizing the cantilever along

its length in either direction. The bias field provided by the permanent magnet then

exerts a torque on the hinged cantilever, pulling either of its ends towards the bottom

substrate while the other end is pushed upwards. Once the cantilever is forced in a

magnetic state, and is consequently aligned to the bias field, the electrical current can

be turned off, as the magnetization is then induced solely by the bias field.

The bistable latching device offers great performance in terms of power consump-

tion, as only short current pulses are needed to excite the magnetic cantilever. To

change state, it requires 79 mA of current for 0.2 ms, which results in 93µJ energy

consumption. Low contact resistance values are also measured, around 50 mΩ.

The work reported in 2005 by Cho et al. [24] represents an attempt to integrate

electromagnetic and electrostatic actuation mechanisms in a single device, with the

16

State of the Art

Figure 3.5: Concept scheme of a latching magnetic MEMS switch [23]. Source: [23]

aim of providing latching functionality and ensuring low power consumption. The

device comprises an insulating movable membrane fabricated as a stack of nitride and

a microcoil (Figure 3.6). When the device is immersed in a uniform magnetic field,

it is possible to inject a current in the coil to generate a magnetic moment, which

experiences magnetic forces that align its orientation to the external field lines.

Figure 3.6: Concept scheme of a latching magnetic MEMS switch [24]. Source: [24]

When the suspended membrane is actuated, it approaches the plates and a voltage

can then be applied to exert a force between them. This can act as a latching mechan-

ism that maintains the on state by simply applying a voltage to charge the capacitor.

17

State of the Art

The membrane is 500µm wide, 700µm long and 1 µm high, in addition to coil

thickness of 3 µm. To change state the device requires 53 mA, which results in 40.3 µJ

of energy needed. The electrostatic hold voltage is as low as 3.7 V and the contact

resistance was determined to be 0.5 Ω.

Of course, there are many more other designs for magnetic MEMS switches [1, 25,

26], but the ones covered in this chapter represent most of the ideas that currently

exist.

18

4. Concept of a Magnetic MEMS Switch

This project has started with an idea, where a magnetic MEMS swtich would have a

conductive cantilever (polysilicon with Young’s modulus of 162.8 GPa) with a magnetic

film (cobalt with remanent magnetization of 1.75 T) attached to it, while there is planar,

rectangular coil (copper with resistivity of 1.68× 10−8 Ω m) beneath it, that creates a

magnetic field, attracting the cantilever and closing the switch (Figure 1.1). The cross-

section of the suggested device can be seen below (Figure 4.1).

Figure 4.1: Cross-section of the proposed magnetic MEMS switch

To simplify the problem, in this work I am only focusing on modelling the deflection

of the cantilever and the magnetic force due to the magnetic field exerted by the coil,

so in Figure 4.2 I show which region of the device is discussed in scope of this project.

Magnetic actuation in this case involves two branches of physics: classical mechan-

ics, which deals with calculation of cantilever deflection due to magnetic forces applied

to the magnetic film, and electromagnetism, which deals with calculation of magnetic

forces due to the magnetic field as well as the calculation of that magnetic field exerted

by the current in the coil.

So it was pretty logical for me to divide this model into two simpler ones, each

dealing only with one branch of physics. The one that calculates the magnetic force

19

Concept of a Magnetic MEMS Switch

Figure 4.2: Modelling domain of the proposed magnetic MEMS switch

exerted by the coil includes the coil itself and the magnetic film (Figure 4.3). This force

is then “fed” to the model that calculates the deflection, which in its turn includes the

cantilever itself and the magnetic film, to which the force is applied (Figure 4.4).

Figure 4.3: Sub-problem that solves the device for magnetic force

Figure 4.4: Sub-problem that solves the device for deflection

At this stage, the dimensions are not defined yet. The only set in stone thing is that

the cantilever beam should not be longer or wider than 70 µm, which is a constraint

that I set for the device, so that in total it is not bigger than ≈ 100 µm.

20

5. Modelling of a Magnetic Actuation

The modelling problem can be divided into two simpler subproblems: modelling of the

coil and magnetic forces and modelling of the cantilever’s deflection.

5.1. Coil and Magnetic Film Modelling

To solve a problem in COMSOL, usually, there are well-defined steps to do that. First,

I have created a 3D model according to Figure 4.3, although only with one loop of wire

for the sake of simplicity. The dimensions for now were chosen arbitrarily, but within

reason (Figures 5.1).

Figure 5.1: Dimensions of the magnetic film and the coil

Figure 5.2: 3D model of the magnetic film and the coil

21

Modelling of a Magnetic Actuation

Then the model (Figure 5.2) was put inside of an air domain, which in its turn is

put inside of an infinite domain to simulate infinite space around the model to “damp”

the magnetic field beyond the air domain (Figure 5.3).

Figure 5.3: Modelling domain

It is necessary to apply magnetically insulated boundary conditions for the outer

boundaries to solve the model and if there was not the infinite domain, then I would be

forced to apply them to the air domain, which would in most cases affect the solution.

See Figure 5.4 for difference between having an infinite domain and not.

Figure 5.4: Influence of having an infinite domain versus not having one

The model was set up for solving the Maxwell’s equations for stationary problems

using the “Magnetic Fields” interface. The outer boundary of the infinite domain is

magnetically insulated, meaning that the magnetic field does not exit the modelling

domain, but, like it was discussed above, that is not a problem because of how the

22


infinite domain “damps” the field. The coil is prescribed with some current density,

following its shape in clockwise direction.

The magnetic film’s material is cobalt with remanent magnetization of 1.75 T (Fig-

ure 5.5), pointing in −z direction, so that when the coil has the current flowing in

clockwise direction, like defined above, the film would experience attractive forces.

Figure 5.5: Magnetization curve of cobalt

The model was meshed using normal COMSOL settings, with a bit denser mesh

in and around the magnetic film. Then it was solved using the “Stationary Solver” of

COMSOL and the force acting on the film was calculated to be −3.7 µN.

However, it was not clear if the solution is converged here, so it was necessary to do

the convergence study by solving the same model multiple times, each time reducing

mesh element size. The solution is converged, when reducing mesh size does not affect

the solution and it stabilizes, like it is shown in Figure 5.6. So the final result for this

model configuration is around −4.5 µN.

5.2. Cantilever Modelling

Solving the cantilever for displacement is a simpler task, as it does not require a medium

to be modelled in. Other than that, the procedure is similar to the one described in

the section above.

23


Figure 5.6: Convergence study for coil and magnetic film modelling, n is a mesh size

parameter, the bigger it is, the smaller mesh becomes

First, the dimensions for the cantilever beam must be chosen. In Chapter 5.1, I

have set the magnetic film to be 10 µm wide, 10µm long and 1µm high. I wanted to

model a very simple rectangular beam, so I chose it to be 10µm wide as well, 50 µm

long and 0.5 µm high with a magnetic film attached to the top of it (Figure 5.7).

Figure 5.7: Dimensions of the cantilever

Then I have used the “Solid Mechanics” interface of COMSOL, which is based

on solving the equations of motion to get results such as displacements, stresses and

strains. In that interface, I have applied such boundary conditions for the system as

evenly distributed force on the magnetic film and how the cantilever is fixed, so that

the system would look as in Figure 5.8.

24


Figure 5.8: Cantilever setup diagram

And as a final step, like in Chapter 5.1, I have performed a mesh convergence study,

continuously reducing the mesh size while solving the model, using the “Stationary

Solver”, as there are not any time-dependent variables. And when the solution is

converged, it becomes the final result. For example, in this case, if I apply total force

of 4.5 µN pointing in −z direction, the maximum deflection would be around 9.27µm

(Figure 5.9).

Figure 5.9: Convergence study for cantilever modelling, n is a mesh size parameter,

the bigger it is, the smaller mesh becomes

25

6. Model Verification

After I have performed the simulations in Chapers 5 and got converged results, I have

decided to compare those results with analytic solutions, using methods discussed in

Chapters 2.1, 2.2 and 2.3, for the sake of checking if the simulations’ results were

correct.

6.1. Calculating the Magnetic Force

In Chapter 2.1 I have covered two methods of calculating the magnetic force acting

upon a magnet. The second method, which takes use of Maxwell’s stress tensors

(F = 1µ0

∫∫S · n dA) is being used by COMSOL, while I will use

F =

∫∇(M ·B) dV (6.1)

to calculate the force and compare my results with the simulation’s results.

First, I needed to calculate the magnetic field exerted by a square electric loop. To

do that, I had to apply the Biot-Savart’s law (Chapter 2.2) for a straight wire:

B =I

4πd(cosα1 + cosα2) (6.2)

which holds for an infinitely thin wire, which is a reasonable approximation for most

cases. And by splitting the loop into 4 straight fragments, calculating magnetic field

due to each one of them and adding the results together, I came up with a magnetic

field as a function of position relative to the loop (Appendix A.1).

26

Model Verification

For example, if I have a loop that is 10µm wide and 10 µm long and carries a

current of 1 A (like in Chapter 5.1), the magnetic field in the center would be around

9× 104 A m−1 (Figure 6.1).

Figure 6.1: Magnetic field in the center of a square electric loop due to 1 A of current

As I am interested only in forces along z axis, Equation 6.1 takes the form of:

Fz = Mz

∫∂Bz

∂zdV. (6.3)

from which follows that to calculate the magnetic force F in z direction, I need to

take partial derivatives with respect to z of the z components of the magnetic field B,

integrate the results across the volume V and multiply it by magnetization Mz.

Calculation of the derivative and integration have been done numerically in MAT-

LAB, as the expression for the magnetic field became quite lengthy.

So, if I try to replicate the model from Chapter 5.1, assuming thin wires (resulting

with distance between the loop and the magnetic film to be 1.5 µm instead of 1 µm),

then the resulting magnetic force would be −4.58µN, compared to −4.5 µN, which was

calculated using COMSOL.

From this result, two things can be concluded: that the simulations in COMSOL

were performed correctly and that even though the results were close, the geometry of

27

Model Verification

the wires of the electric loop has its effect, since the results from COMSOL converge a

little bit away from the value that was analytically calculated (Figure 5.6).

6.2. Calculating the Deflection

To calculate a deflection of a cantilever, I am using the Euler-Bernoulli equation (Equa-

tion 2.9), assuming that the Young’s modulus and second moment of inertia are con-

stant across the beam’s length:

EId4y

dx4= w(x). (6.4)

To efficiently model a beam, a load diagram is needed. In Figure 6.2 I assume that

second moment of inertia and the Young’s modulus are constant here, meaning that

the cantilever will be solved without having the magnetic film attached to it.

Figure 6.2: Force diagram of a cantilever beam under force Fmagnetic

The magnetic force Fmagnetic is evenly distributed between x = 40 µm and x =

50µm. Its equivalent can be the magnetic force Fmagnetic applied at x = 45 µm. The

system is in an equilibrium, so the net force and moment are both zero. Using this,

the reaction force and moment can be easily found:

∑F = 0 = −Fmagnetic + Freaction (6.5)

28

Model Verification

Freaction = Fmagnetic (6.6)

∑M = 0 = rFmagnetic −Mreaction (6.7)

Mreaction = rFmagnetic (6.8)

where r = 45µm. The load function w(x), using the singularity functions, would then

take a form of:

w(x) = −Fmagnetic〈x− r〉−1 + Freaction〈x− 0〉−1 −Mreaction〈x− 0〉−2. (6.9)

According to Equation 6.4, w(x) has to be integrated 4 times to find the deflection

function y(x):

y(x) =1

EI

(− Fmagnetic

6〈x− r〉3 +

Freaction6〈x− 0〉3 − Mreaction

2〈x− 0〉2

). (6.10)

Substituting (6.6) and (6.8):

y(x) =1

EI

(− Fmagnetic

6〈x− r〉3 +

Fmagnetic6

〈x− 0〉3 − rFmagnetic2

〈x− 0〉2)

(6.11)

which becomes the final expression for deflection y(x).

The material used for the cantilever beam is a polycrystalline silicon (polysilicon),

which has the Young’s modulus of 162.8 GPa and second moment of inertia:

Ix =bh3

12=

(10× 10−6 )(0.5× 10−6 )3

12= 0.104 166× 10−24 m4 (6.12)

where b and h are width and height of the cantilever respectively.

So, if there is a magnetic force Fmagnetic = 4.5 µN, the maximum deflection would

be y = −9.40µm, which is pretty close to the simulated deflection of y = −9.27µm.

Hence, I can conclude that both, the simulation and the calculations were correct.

29

7. Modelling Efficiency

Another important topic in this project is how efficiently the problems are solved.

Finding a converged solution of a simulation can take quite a bit of time and computer

memory. Reducing those is crucial if I want to optimize the model, as I will have

to run those simulations a number of times, while changing its parameters (Chapter

reference). And if the model is not solved efficiently, it may take days for it to find

the solutions, provided that a computer does not run out of memory. So, there are

two things I came up with that can be used to improve this situation: symmetry and

analytic solution for the magnetic force.

7.1. Exploiting the Symmetry

The first thing I noticed is that if the coil and magnetic film are of rectangular shape

(Figure 5.3), it has two planes of symmetry, meaning that it is possible to model only

a quarter of that model (Figure 7.1) and still get the correct magnetic force just by

multiplying the result by 4.

The boundary condition for the surfaces, where the “cut” was made is Magnetic

Insulation, meaning that the magnetic field is forced to be parallel to the boundary,

while the currents is forced to be perpendicular to the boundary. In essence, it creates

a mirror symmetry plane for the magnetic field (n×A = 0, where n and A are a vector

perpendicular to a surface of a boundary and a magnetic vector potential respectively).

If I try to simulate the model from Chapter 5.1 using the symmetry, then, as a

result, I will get the same magnetic force of −4.5 µN, but 4 times faster (Figure 7.2).

30

Modelling Efficiency

Figure 7.1: Modelling domain

7.2. Magnetic Force Calculation

Another idea to improve the calculation speed for the model was to calculate the

magnetic force as in Chapter 6.1, but purely numerically in COMSOL and compare if

this solution converges faster than in Figure 7.2.

From Figure 7.3 it can be easily concluded that if I am using F =∫∇(M · B) dV

instead of F = 1µ0

∫∫S ·n dA (COMSOL’s method) to calculate the magnetic force, the

results are more stable and converge faster, reducing the simulation time substantially.

As a result, in the future, I will be using the analytic solution method.

7.3. Meshing Improvements

Meshing is a process that I have barely talked about in this work, even though it is

one of the most important things during FEM. Especially in this work, it is crucial to

set up the mesh properly and effectively.

For simple geometries, like for a cantilever, the default COMSOL settings work

just fine, producing pretty accurate results even without mesh refinement. However,

when modelling a coil and a magnetic film, the model becomes much more complex

31


Figure 7.2: Convergence of COMSOL’s method of magnetic force calculation, n is a

mesh size parameter, the bigger it is, the smaller mesh becomes

(Figure 7.4). It would include an air sphere, a bigger sphere of air with infinite domain

properties, a coil and a magnetic film. It results with a model of a relatively big size.

And if I wanted to do a convergence study by refining mesh, the computation time

could grow out of control very fast. So it was necessary to determine which regions

were required to be refined in that study, and which not.

For example, the infinite domain layer should just have around 5 elements in its

thickness to work as intended, while a magnetic film would have to have a very dense

mesh, since a gradient of a magnetic field must be taken inside of it and having a coarse

mesh there would severely affect the results. An electric coil, in its turn, does not

require a dense mesh, since not much is calculated inside of it, because it is prescribed

with current density already, so it basically acts as a boundary condition. And lastly,

an air domain also does not require a dense mesh, since a magnetic field approximations

are very well done and the geometries are not complex at all.

Of course, convergence studies must be performed either way, but deciding which

mesh regions should be dense right away in advance, definitely helps with speeding up

the simulations, as it eliminates the condition when the program refines mesh across

the whole geometry, while only one piece of it actually needs it.

32


Figure 7.3: Convergence of COMSOL’s method of magnetic force calculation compared

to convergence of analytic-based solution, n is a mesh size parameter, the bigger it is,

the smaller mesh becomes

Figure 7.4: Mesh of the model

33

8. Design and Optimization of a Magnetic

MEMS Switch

In this chapter I will describe how I performed the design and optimization of the

switch, starting with defining fabrication constraints, then specifying dimensions for

the two proposed concepts, running the simulations to find the best set of parameters

and showing the results.

8.1. Fabrication Constraints

One of the goals of this project is to design a magnetic switch, which is viable in terms of

fabrication in Fraunhofer ISIT. To determine the possibilities of their technologies, me

and Roana arranged a meeting with Fabian Lofink, who is responsible for fabrication

of microcantilevers at Fraunhofer ISIT. With his insight, we were able to determine

minimum and maximum dimensions for the switch that can be manufactured, as well

as some other properties (Table B.1).

Table 8.1: Fabrication constraints of Fraunhofer ISIT

Minimum (µm) Maximum (µm)

Horizontal dimension 1.5 unlimited

Vertical dimension 0.5 3

Distance between coil turns 1.5 unlimited

Resolution 0.5 0.5

Distance between contacts 0.5 unlimited

34

Design and Optimization of a Magnetic MEMS Switch

If I try to apply those constraints to the schematics of a proposed switch design

(Figure 4.1), then additional constraints arise: minimum gap between the cantilever

and the contact becomes 0.5 µm and that minimum distance between the coil and the

cantilever becomes 1 µm, since the contact is minimum 0.5 µm above the coil.

8.2. Magnetic Switch Concepts

Using the information from Chapter 8.1, I was able to determine most of the dimensions

for a minimized magnetic switch, which would have to consume the least amount of

power. The sketch for the overall switch setup has already been developed in Chapter

4 (Figure 4.1).

Figure 8.1: Minimized dimensions of the device according to fabrication constraints

The maximum length and width for a cantilever were decided to be 70 µm, since I

have wanted the whole device to be less than 100 µm wide and 100 µm long. Hence,

the cantilever length becomes 70µm and its height — 0.5 µm, as it is most beneficial

to have it as long as possible, since that will allow for easier deflection, resulting in

smaller power consumption. The gap between the cantilever is also 0.5 µm, hence the

distance between a coil and a magnetic film is 1.5 µm. This distance is also required

to be minimized, since the closer the magnetic film is to the coil, the stronger forces

35


would be generated. Going for the smallest possible gap is also essential, as the smaller

it is, the less force is required to close the contact. The side dimensions can be seen

on Figure 8.1. But when it comes to deciding how wide the cantilever should be, two

different concepts arise. Concept 1 states that its width should be as small as possible

— 1.5 µm (Figure 8.2). Concept 2 states that the total width of the cantilever should

be 70 µm, but with having two arms, 1.5 µm each (Figure 8.3).

Figure 8.2: Top-view dimensions of Concept 1

Figure 8.3: Top-view dimensions of Concept 2

Concept 1 is easier to fabricate and more compact, but Concept 2 will allow for

easier deflection and hence smaller power consumption. The coil’s inner width in each

concept should be calculated according to Appendix B.3:

ain =

afilm − d(l − 1)− wl, if l = 1, 3

afilm − d(l − 2)− w(l − 1), if l = 2

(8.1)

where afilm, d, l, w are width of the magnetic film, distance between coil loops, number

of coil loops and width of the wires in the coil. The distance between coil loops d is

1.5 µm, as dictated by the fabrication constraints, providing that the best performance

36


is required. Note that minimum inner width of the coil ain cannot be less than d, so

if it becomes less than d, when being calculated according to (8.1), then it is set to be

equal to d — 1.5 µm. Similar equation has been derived for the inner length of the coil:

bin =

bfilm − d(l − 1)− wl, if l = 1, 3

bfilm − d(l − 2)− w(l − 1), if l = 2

(8.2)

where bfilm is the magnetic film’s length. And like with ain, bin cannot be smaller than

d, so if it is, after calculating it with (8.2), it becomes equal to d, which in this case

would be 1.5 µm.

Still, there are some parameters left undefined. For example, it is unknown what

is more beneficial, to have a larger cross-section of the wires in the coil to get lower

resistance, or to have them smaller, to achieve a more “concentrated” force from them.

The optimal number of loops is also unknown, as well as the optimal height (hfilm) and

length of the magnetic film, as increasing those would result in a more stiff cantilever.

The width of the magnetic film (and the cantilever, as they are equal), in its turn,

depends on the concept (1.5 µm for Concept 1 and 70µm for Concept 2).

8.3. Optimization Methodology

To determine which configuration of parameters gives the best results, I have conducted

a series of simulations in COMSOL.

The simulations were performed both for Concept 1 and Concept 2, while solving

for all combinations of parameters (from their minimum values to their maxumum),

that were left undefined (Table 8.2). The methodology of modelling this problem is

the same as described in Chapter 5 - dividing the system into 2 subsystems, but this

time using more effective approaches as described in Chapter 7.

In a model that solved an electric coil for a magnetic force, which it exerts, para-

meter bfilm was swept through only 2 values, while a model that solved a cantilever

37


Table 8.2: Parameter values solved for in the simulations

Minimum Maximum Number of steps

Wire’s width (w) 1.5 µm 7µm 12

Wire’s height (h) 0.5 µm 3µm 6

Magnetic film’s length (bfilm) 25.5 µm 70 µm 2 or 90

Magnetic film’s height (hfilm) 1 µm 3 µm 3

Number of loops in the coil (l) 1 3 3

for its deflection, has swept this parameter through 90 values. The reason for that is

because when solving the coil, the magnetic force is linearly dependent on the magnetic

film length, assuming that the coil’s inner length changes as well, according to (8.2).

This relation has been shown in Appendix B.2.

Another thing that is worth covering is the minimal value of bfilm, which is 25.5 µm,

which is vastly different from 1.5 µm, like the constraints allow. This was done in order

to satisfy equation (8.2), so that bin ≥ d for all parameters. And like the results will

show later, the optimal bfilm is bigger than 25.5 µm anyway.

For each concept, a mesh convergence study was performed only once for the fol-

lowing parameter values: w = 1.5 µm, h = 0.5 µm, bfilm = 70 µm, hfilm = 1 µm, l = 1,

which would give the thinnest and most prolonged model. And the mesh element sizes,

which were determined in this study that lead to a converged result, were used for sim-

ulating other parameter configurations, since I have assumed that the required mesh

element size will not be smaller in other simulations.

Like I have already mentioned, simulation of each configuration of a magnetic switch

is split into two models: one that solves a cantilever beam for the deflection due to

some force, and one that solves a coil for the magnetic force that it exerts upon a

magnetic film due to some current. Force and current in those respective simulations

are arbitrary, in this case: 10 nN and 1 A. The goal of those simulations is to find a

relation between how much current is applied and how much magnetic force is generated

38


as well as between how much force is applied to a cantilever and how big deflection

it causes. But those relations are linear (Equations (2.5) and (2.9)), so it allows me

to extrapolate those results to see how much force is needed to achieve a maximum

deflection of 0.5 µm, and knowing the required force, it also gives me the information

of how much current is needed to achieve this force.

For example, if a force of 10 nN gives a deflection of 1 µm and a current of 1 A

gives a force of 1 µN, then the required force to achieve 0.5 µm deflection is 5 nN, and

consequently, the required current becomes 0.005 A.

Using this principle, I have calculated the required current to actuate a switch

(reach a deflection of 0.5 µm), which also gives the information of required power to

do that. And by comparing the required power for each configuration of parameters, I

was able to determine which one is the best in terms of power consumption. MATLAB

script that processes the results from simulations the way it was described above, can

be found in Appendix C.2.

8.4. Results and Findings

The results from simulations have been compiled in a number of plots, for Concept

1 (Figure 8.7) and Concept 2 (Figure 8.8). Although they may not provide accurate

data, they clearly show, which configuration of parameters gives the best performance

in terms of power consumption. Of course, power here is a function of all parameters,

however on those figures, its dependency on bfilm is not shown. In essence, each data

point there should also be a function of bfilm, but instead, the heat maps here already

show the global minimum of those functions with a corresponding bfilm value with it.

From Figure 8.7 it can be seen that the best results for Concept 1 are when l =

1 and hfilm = 3 µm. After a closer inspection of that plot (Figure 8.4), the least

amount of power is consumed when w = 5.5 µm and h = 3 µm. This configuration

of parameters results with 4.25 µW power consumption when bfilm = 30 µm and the

39


current is 13.8 mA, which also is a global minimum of the system, since it may only

be smaller when h > 3 µm or when hfilm > 3 µm, but that is not possible because

of fabrication constraints at Fraunhofer ISIT. But in general, increasing hfilm and h

benefits the system.

Results of simulations for Concept 2 are presented on Figure 8.8. A plot with

the results when l = 1 and hfilm = 3 µm shows the best performance in terms of

power consumption, however upon closer inspection of that plot, it can be seen that a

better configuration may be found by increasing w. After running the simulations for

Concept 2 again, with l = 1 and hfilm = 3 µm, but now sweeping w from 4 µm to 11 µm

instead (Figure 8.6), the global minimum of power has been found when w = 10 µm

and h = 3 µm, which is 180 nW when bfilm = 25.5 µm and the current is 1.3 mA.

Figure 8.4: Simulation results of Concept 1 for different parameter configurations when

l = 1 and hfilm = 3 µm

40


Figure 8.5: Simulation results of Concept 2 for different parameter configurations when

l = 1 and hfilm = 3 µm

Figure 8.6: Expanded simulation results of Concept 2 for different parameter config-

urations when l = 1 and hfilm = 3 µm

41


Fig

ure

8.7:

Sim

ula

tion

resu

lts

ofC

once

pt

1fo

rdiff

eren

tpar

amet

erco

nfigu

rati

ons

42


Fig

ure

8.8:

Sim

ula

tion

resu

lts

ofC

once

pt

2fo

rdiff

eren

tpar

amet

erco

nfigu

rati

ons

43

9. Conclusion and Future Work

As a result of this work I have successfully designed and modelled two concepts of

a magnetically actuated MEMS switch, which can be fabricated at Fraunhofer ISIT.

Their designs have been minimized for required actuation power. Concept 1 requires

4.25 µW of power and 13.8 mA of current to actuate, while Concept 2 requires just

180 nW of power and 1.3 mA of current. Even though the difference in power between

these two concepts is substantial, it is not quite clear which one is better, since Concept

1, in its turn, is about 46 times narrower than the other one, so that would be more of

a question of application and additional constraints.

Actuation power, current and dimensions are not the only parameters that describe

a magnetic MEMS switch. Some other important ones are on and off resistances,

switching speed, lifetime, contact force, maximum frequency. So, in the future, more

simulations have to be performed. For example, a frequency analysis of the structure

has to be done to calculate resonant frequencies. In addition to that, it also must be

calculated how much current can the switch carry. This is needed to fully characterize

the system, so that it can be fairly compared to others that are currently on the market.

This work has also covered some of the research performed in this direction, theor-

etical background of magnetic actuation and the process of modelling such systems in

COMSOL, combining both numerical methods and analytical ones, as well as efficiency

improvements of those simulations. Also, it includes some additional simulations that

examine influence of deformation of a square-shaped coil on force that it exerts and

show that deforming the wire does increase the force and, in fact, is more power efficient

than if this force is increased by adding more current.

44

Bibliography

[1] M. Glickman, P. Tseng, J. Harisson, T. Niblock, I. B. Goldberg and J. W. Judy,

“High-performance lateral-actuating magnetic mems switch”, Journal of Micro-

electromechanical Systems, vol. 20, pp. 842–851, 4 2011.

[2] G. Schiavone, M. P. Y. Desmulliez and A. J. Walton, “Integrated magnetic mems

relays: Status of the technology”, Micromachines, vol. 5, pp. 622–653, 3 2014.

[3] L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media. Pergamon

Press, 1960.

[4] B. Wagner and W. Benecke, “Microfabricated actuator with moving permanent

magnet”, in IEEE Micro Electro Mechanical Systems, 1991, pp. 27–32.

[5] COMSOL, Ac/dc module user’s guide.

[6] J. D. Jackson, Classical Electrodynamics. John Wiley & Sons, Inc., 1962.

[7] J. M. Gere, Mechanics of Materials. Thompson Learning, Inc., 2004.

[8] M. G. Larson and F. Bengzon, The Finite Element Method: Theory, Implement-

ation, and Practice. Springer, 2010.

[9] COMSOL. (n.d.). Detailed explanation of the finite element method (fem), [On-

line]. Available: https://www.comsol.dk/multiphysics/finite-element-

method.

[10] H. P. Langtangen. (2013). Introduction to finite element methods, [Online]. Avail-

able: http://hplgit.github.io/INF5620/doc/pub/sphinx-fem/._main_

fem000.html.

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http://se.mathworks.com/help/pde/ug/basics-of-the-finite-element-

method.html.

[12] M. J. Jackson, Micro and Nanomanufacturing. Springer, 2007, p. 4.

[13] J. W. Judy, “Microelectromechanical systems (mems): Fabrication, design and

applications”, Smart Materials and Structures, vol. 10, no. 6, pp. 1115–1134,

2001.

[14] K. F. Lei, “Chapter 1: Materials and fabrication techniques for nano- and micro-

fluidic devices”, in Microfluidics in Detection Science: Lab-on-a-chip Technolo-

gies, The Royal Society of Chemistry, 2015, pp. 1–28.

[15] P. S. Waggoner and H. G. Craighead, “Micro- and nanomechanical sensors for

environmental, chemical, and biological detection”, Lab Chip, vol. 7, pp. 1238–

1255, 10 2007.

[16] D. J. Bell, T. J. Lu, N. A. Fleck and S. M. Spearing, “Mems actuators and

sensors: Observations on their performance and selection for purpose”, Journal

of Micromechanics and Microengineering, vol. 15, S153–S164, 7 2005.

[17] H. Fujita, “Microactuators and micromachines”, Proceedings of the IEEE, vol. 86,

pp. 1721–1732, 8 1998.

[18] E. Obermeier and E. Thielicke, “Microactuators and their technologies”, Mechat-

ronics, vol. 10, pp. 431–455, 4–5 2000.

[19] C. H. Ahn and M. G. Allen, “A fully integrated surface micromachined magnetic

microactuator with a multilevel meander magnetic core”, Journal of Microelec-

tromechanical Systems, vol. 2, pp. 15–22, 1 1993.

[20] J. W. Judy and R. S. Muller, “Magnetically actuated, addressable microstruc-

tures”, Journal of Microelectromechanical Systems, vol. 6, pp. 249–256, 3 1997.

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Bibliography

[21] J. A. Wright, Y.-C. Tai and S.-C. Chang, “A large-force, fully-integrated mems

magnetic actuator”, in Proceedings of International Solid State Sensors and Ac-

tuators Conference (Transducers ’97), vol. 2, 1997, pp. 793–796.

[22] W. P. Taylor, O. Brand and M. G. Allen, “Fully integrated magnetically actu-

ated micromachined relays”, Journal of Microelectromechanical Systems, vol. 7,

pp. 181–191, 2 1998.

[23] M. Ruan, J. Shen and C. B. Wheeler, “Latching micromagnetic relays”, Journal

of Microelectromechanical Systems, vol. 10, pp. 511–517, 4 2001.

[24] I.-J. Cho, T. Song, S.-H. Baek and E. Yoon, “A low-voltage and low-power rf

mems series and shunt switches actuated by combination of electromagnetic and

electrostatic forces”, IEEE Transactions on Microwave Theory and Techniques,

vol. 53, pp. 2450–2457, 7 2005.

[25] P. A. Kohl and G. D. J. Gray, “Magnetically bistable actuator: Part 1. Ultra-low

switching energy and modeling”, Sensors and Actuators A: Physical, vol. 119,

pp. 489–501, 2 2005.

[26] S. Lu, T.-H. Lin, S. Paul and H. Lu, “A study on the performance and reliability

of magnetostatic actuated rf mems switches”, Microelectronics Reliability, vol. 49,

pp. 59–65, 1 2009.

[27] K. J. Hemker and W. N. J. Sharpe, “Microscale characterization of mechanical

properties”, Annual Review of Materials Research, vol. 37, pp. 93–126, 2007.

47

A. Calculations and Derivations

A.1. Magnetic Field for a Rectangular Electric Loop

The Biot-Savart’s law is an equation, describing the magnetic field generated by an

electric current and holds only in magnetostatic problems [6]

dB =µ0

4π

Idl × ~x|~x|3

(A.1)

where dB, µ0, I, dl and ~x are elemental flux density, current, vacuum permeability

(4π × 10−7), element of length and position vector respectively (Figure A.1).

Figure A.1: Magnetic field dB due to an electric current I. Source: [6]

Using (A.1) we can calculate magnetic fields due to a straight current wire [4]

B =µ0I

4πd(cosα1 + cosα2) (A.2)

where d, α1 and α2 are distance to the wire and angles that are shown on Figure A.2

respectively. This equation can be expanded to calculate a magnetic field exerted by a

square loop simply by splitting the loop into 4 straight fragments, calculating magnetic

field for each one of them and adding the results together.

48

Calculations and Derivations

Figure A.2: Magnetic field B at point P due to an electric current I in a straight wire

On Figure A.3, a diagram of a rectangular loop is displayed. Magnetic field at point

P due to wire 1 (Figure A.3) is calculated using Equation A.2:

Figure A.3: Rectangular electric loop, generating a magnetic field at point P

B1 =µ0I

4π

1√b21 + z2

(a1√

b21 + z2 + a21+

a2√b21 + z2 + a22

). (A.3)

Its x, y and z components are then:

49


B1x = 0, B1y = −B1z√

b21 + z2, B1z = B1

b1√b21 + z2

. (A.4)

Magnetic field at point P due to wires 2, 3 and 4 are calculated analogously:

B2 =µ0I

4π

1√a22 + z2

(b1√

a22 + z2 + b21+

b2√a22 + z2 + b22

)(A.5)

B2x = B2z√

a22 + z2, B2y = 0, B2z = B2

a2√a22 + z2

(A.6)

B3 =µ0I

4π

1√b22 + z2

(a1√

b22 + z2 + a21+

a2√b22 + z2 + a22

)(A.7)

B3x = 0, B3y = B3z√

b22 + z2, B3z = B3

b2√b22 + z2

(A.8)

B4 =µ0I

4π

1√a21 + z2

(b1√

a21 + z2 + b21+

b2√a21 + z2 + b22

)(A.9)

B4x = −B4z√

a21 + z2, B4y = 0, B4z = B4

a1√a21 + z2

. (A.10)

To get the total magnetic field at point P , all magnetic fields due to wires 1, 2, 3

and 4 must be added together:

Btotal = B1 +B2 +B3 +B4. (A.11)

Same thing can be done if only specific components of the magnetic field are re-

quired:

Btotalx = B1x +B2x +B3x +B4x (A.12)

Btotaly = B1y +B2y +B3y +B4y (A.13)

Btotalz = B1z +B2z +B3z +B4z . (A.14)

50


A.2. Derivation of a Magnetic Force

A magnetic force exerted by a magnetic field on a single moving charge is defined by:

dF = qv ×B (A.15)

where q is charge, v its velocity and B is external magnetic field. It can be rewritten

as:

dF = I(dl ×B) (A.16)

where I is electric current flowing through a wire segment dl. The total force on the

whole wire would be:

F = I

∫(dl ×B) (A.17)

which is called the Laplace force. This expression can be written as an integral over a

surface bounded by the wire:

F = I

∮(dl ×B). (A.18)

Using Stokes’ theorem, dl is replaced by the operator dµ×∇, obtaining∮dl×B =∫

(dµ×∇)×B. Now:

(dµ×∇)×B = −dµ(∇ ·B) +∇(dµ ·B) =

= −dµ(∇ ·B) + dµ× (∇×B) + (dµ · ∇)B.(A.19)

∇ ·B = 0, according to Gauss’s law, and ∇×B = 0 in space outside the currents,

according to Ampere’s law. Thus:

F = I

∫(dµ · ∇)B. (A.20)

Magnetic moment of the loop is:

51


m = I

∫dµ (A.21)

with dµ = 12r×dl, where r is the position vector. The product dµ is equal in magnitude

to the triangular area formed by the vectors r and dl.

Assuming that B is almost uniform, I can take it out of the integral in (A.20), and

after substituting (A.21) into (A.20), I get:

F = (m · ∇)B. (A.22)

Since m is constant:

F = ∇(m ·B). (A.23)

But in (A.23) I calculate the force acting only on a single loop and I need to find

force acting on a magnet with volume V and magnetization M . If I assume that magnet

is a collection of infinitely small current loops, then using (A.23), it can be said that

the magnetic force would be:

Ftotal =

∫∇(M ·B) dV. (A.24)

A.3. Calculations for a Cantilever Beam

The Euler-Bernoulli beam theory is a simplification of the linear theory of elasticity,

which calculates deflection of beams due to lateral loads [7]

d2

dx2

(EI

d2y

dx2

)= w(x) (A.25)

where E, I, y, x and w are the Young’s modulus, second moment of inertia, deflection,

position and load respectively.

According to Figure A.4, load w is expressed as:

52


Figure A.4: Force diagram of a cantilever beam under force Fmagnetic

w(x) = −Fmagnetic〈x− r〉−1 + Freaction〈x− 0〉−1 −Mreaction〈x− 0〉−2. (A.26)

where r = 45 µm. Calculating shear force V , bending moment M and deflection y(x):

− V (x) =

∫ x

−∞w(x) dx = −Fmagnetic〈x− r〉0+

Freaction〈x− 0〉0 +Mreaction〈x− 0〉−1 (A.27)

M(x) = −∫ x

−∞V (x) dx = −Fmagnetic〈x− r〉1+

Freaction〈x− 0〉1 +Mreaction〈x− 0〉0 (A.28)

EIdy(x)

dx=

∫ x

−∞M(x) dx = −Fmagnetic

2〈x− r〉2+

Freaction2〈x− 0〉2 +Mreaction〈x− 0〉1 + a (A.29)

EIy(x) =

∫∫ x

−∞M(x) dx2 = −Fmagnetic

6〈x− r〉3+

Freaction6〈x− 0〉3 +

Mreaction

2〈x− 0〉2 + ax+ b. (A.30)

53


Applying boundary conditions y(0) = 0 and dy(0)dx

= 0 leads to:

a = 0, b = 0. (A.31)

From equilibrium follows that:

∑F = 0 = −Fmagnetic + Freaction (A.32)

Freaction = Fmagnetic (A.33)

∑M = 0 = rFmagnetic −Mreaction (A.34)

Mreaction = rFmagnetic. (A.35)

The final expression for the deflection then is:

y(x) =1

EI

(−Fmagnetic

6〈x− r〉3 +

Fmagnetic6

〈x− 0〉3 +rFmagnetic

2〈x− 0〉2

). (A.36)

If the material of the cantilever is a polysilicon, its Young’s modulus E is 162.8 GPa

[27]. The cantilever’s cross-section is a rectangle, then its second moment of inertia is:

Ix =bh3

12=

(10× 10−6 )(0.5× 10−6 )3

12= 0.104 166× 10−24 m4 (A.37)

where b and h are the cantilever’s width and height. Assuming that the magnetic force

Fmagnetic is 4.5 µN, maximum deflection is:

ymax = −9.40 µm. (A.38)

54

B. Additional Simulations

B.1. Cantilever Stress Analysis

It is logical that when designing a cantilever, a stress analysis must be performed to

examine if the prescribed deflection causes a fracture or a permanent deflection. These

simulations were performed in COMSOL for both concepts, which were derived in

Chapter (reference) (Table).

Table B.1: Dimensions of two models to be analyzed for stress

Concept 1 Concept 2

Cantilever’s length 70 µm 70µm

Cantilever’s width 1.5 µm 70µm

Cantilever’s height 0.5 µm 0.5 µm

Magnetic film’s length 30 µm 25.5 µm

Magnetic film’s height 3 µm 3 µm

The material for the cantilever is polycrystalline silicon, which has an elastic limit

of up to 1.2 GPa [27], so the maximum stress must be below that number. After

simulations of Concept 1 (Figure B.1) and Concept 2 (Figure B.2), the maximum

stresses are 15.3 MPa and 14.5 MPa respectively, showing that the structure will not

be permanently deformed during operation. However, how that deflection affects the

magnetic film is unknown, since there is no data on stress-strain behavior for cobalt.

55

Additional Simulations

Figure B.1: Surface stresses of Concept 1

Figure B.2: Surface stresses of Concept 2

B.2. Relation Between a Magnetic Film Length and a Mag-

netic Force

There is a linear relation between magnetic film’s length and magnetic force that is

generated, assuming that the coil’s inner length is changing with the magnetic film’s

length linearly, for example, according to Equation 8.2. On Figure B.3 this linear

relation can be easily seen. It has been modelled with MATLAB script on basis of

Appendix C.1. The setup of this system is of little interest, this simulation just shows

the dynamics of the system while changing one of its parameters. The magnetic film’s

56


length has been swept from 25.5 µm to 70µm with 0.5 µm step size. This relation has

allowed

Figure B.3: Relation between length of a rectangular loop and force that it exerts

B.3. Optimal Coil Position Under a Magnetic Film

To decrease the number of parameters, for which the system should be modelled, it is

necessary to determine if some of them can be expressed in terms of others. In this

case, there must be a relation between the magnetic film’s position and size and coil’s

position and size, which gives the best performance. In case when the coil is infinitely

thin, it is best when the coil replicates the shape of the magnetic film, because there will

no be any counteracting vertical magnetic forces acting on the film. But in this case,

the wires in the coil have a non-zero width which also comes with distance between

loops in the coil, so it was necessary to determine the new rule of coil positioning for

a more complex situation.

Three simulations were performed in COMSOL. In the first simulation (Figure B.4),

the coil had 1 loop, the magnetic film was 40µm wide and 40µm long, while its inner

length and width have been swept from 30µm to 40 µm. Width of the wire is 5µm.

Both, the coil and the magnetic film are centered. The biggest force is generated when

the coil’s inner length and width are between 35 µm and 36µm.

57


Figure B.4: Relation between coil dimensions and magnetic force that it exerts upon

constant magnetic film. The coil has 1 loop

In the second simulation (Figure B.5), the setup is the same, except that the number

of loops in the coil is 2 and distance between loops is 1.5 µm. Coil’s inner length and

width have been swept from 19.5 µm to 37.5 µm. The biggest force is generated when

the coil’s inner length and width are between 34.5 µm and 35.5 µm.

In the third simulation (Figure B.6), the number of loops in the coil is 3. Coil’s

inner length and width have been swept from 7µm to 37µm. The biggest force is

generated when the coil’s inner length is between 22 µm and 23µm.

Ideally, it is necessary to find the best dimensions for the coil for each parameter

configuration, but to simplify the problem, two equations can be derived, which calcu-

late a very good approximation of the best dimensions of the coil for the corresponding

magnetic film:

ain =

afilm − d(l − 1)− wl, if l = 1, 3

afilm − d(l − 2)− w(l − 1), if l = 2

(B.1)

bin =

bfilm − d(l − 1)− wl, if l = 1, 3

bfilm − d(l − 2)− w(l − 1), if l = 2

(B.2)

58



constant magnetic film. The coil has 2 loops

where afilm, bfilm, d, l, w, ain, bin are width and length of the magnetic film, distance

between coil loops, number of coil loops, width of the wires in the coil, coil’s inner

width and length respectively. For example, if the magnetic film is 40µm wide and

long, the width of the wires in the coil is 5 µm, there are 2 loops in the coil and distance

between loops is 1.5 µm, then the coil’s inner width and length would be 35µm, which

is a pretty good approximation, taking into account that these formulas are meant

many different parameter configurations. They can also be valid for all odd or even

loop numbers respectively, but it has not been checked.

B.4. Coil Shape Study

In this work I have only focused on designing and modelling rectangular-shaped coils,

but I have also conducted a small study to discover how deforming a rectangular coil

may influence the magnetic force that it exerts upon a magnetic film.

Figure B.7 presents an initially rectangular coil that has been deformed. A number

of simulations were performed, sweeping through offset parameter from 1 µm to 4µm

59



constant magnetic film. The coil has 3 loops

(Figure B.8). It must be noted that the magnetic film’s shape follows the coil’s shape.

As it can be seen, the generated force increases the more the shape is deformed, keeping

total area constant.

Figure B.7: Deformed square loop

However, increasing offset also increases the power dissipation in the wire, so it

was necessary to see if deformation is more efficient than just increasing the current in

the wire. On Figure B.9 two graphs can be seen: one shows the results of deforming

60


Figure B.8: Relation between deformation and magnetic force

the wire, another one shows the results of increasing current of the wire before it was

deformed. In both cases the power dissipation are equal, which gives that deforming

the wire in that way is more efficient, since it gives a larger force.

Figure B.9: Comparison of methods of increasing magnetic forces

Even though tweaking coil shapes may result in a better performance, like it was just

shown, this concept has not been further investigated in this work, since it dramatically

complicates modelling, characterization , optimization and fabrication of the system

and I my opinion should be a research on its own.

61

C. MATLAB Scripts

C.1. Magnetic Force Calculation

1 c l o s e a l l ;

2 c l e a r a l l ;

3 c l c ;

4

5 I = −1; % cur rent

6 Mz = −1.75; % magnet izat ion o f the f i l m

7

8 l e n g t h l = 10e−6; % width o f the loop

9 width l = 10e−6; % length o f the loop

10

11 l e n g t h f = 10e−6; % length o f the f i l m

12 width f = 10e−6; % width o f the f i l m

13 h e i g h t f = 1e−6; % he ight o f the f i l m

14

15 d = 1 .5 e−6; % d i s t anc e between the c o i l and the f i l m

16

17 s t ep s = 101 ;

18

19 Bz = ze ro s ( steps , s teps , s t ep s ) ; % magnetic f i e l d , z component

20 Bx = ze ro s ( steps , s teps , s t ep s ) ; % magnetic f i e l d , x component

21 By = ze ro s ( steps , s teps , s t ep s ) ; % magnetic f i e l d , y component

62

MATLAB Scripts

22 normB = ze ro s ( steps , 1 ) ;

23

24 zz = 1 ;

25 f o r z = d : h e i g h t f /( steps −1) : d+h e i g h t f

26 xx = 1 ;

27 f o r x = l e n g t h l /2− l e n g t h f /2 : l e n g t h f /( steps −1) : l e n g t h l

/2+ l e n g t h f /2

28 yy = 1 ;

29 f o r y = width l /2−width f /2 : w idth f /( steps −1) : w id th l

/2+width f /2

30

31 a1 = l e n g t h l−x ;

32 a2 = x ;

33

34 b1 = y ;

35 b2 = width l−y ;

36

37 r1 = s q r t ( a1ˆ2+b1ˆ2+z ˆ2) ;

38 r2 = s q r t ( a1ˆ2+b2ˆ2+z ˆ2) ;

39 r3 = s q r t ( a2ˆ2+b1ˆ2+z ˆ2) ;

40 r4 = s q r t ( a2ˆ2+b2ˆ2+z ˆ2) ;

41

42 Bz( xx , yy , zz ) = I /(4∗ pi ) ∗( a1 /( a1ˆ2+z ˆ2) ∗( b1/ r1+b2/

r2 )+a2 /( a2ˆ2+z ˆ2) ∗( b1/ r3+b2/ r4 )+b1 /( b1ˆ2+z ˆ2) ∗(

a1/ r1+a2/ r3 )+b2 /( b2ˆ2+z ˆ2) ∗( a1/ r2+a2/ r4 ) ) ;

43 Bx( xx , yy , zz ) = I /(4∗ pi )∗z ∗ (1/( a1ˆ2+z ˆ2) ∗( b1/ r1+b2/

r2 )−1/(a2ˆ2+z ˆ2) ∗( b1/ r3+b2/ r4 ) ) ;

44 By( xx , yy , zz ) = I /(4∗ pi )∗z ∗ (1/( b1ˆ2+z ˆ2) ∗( a1/ r1+a2/

63

MATLAB Scripts

r2 )−1/(b2ˆ2+z ˆ2) ∗( a1/ r3+a2/ r4 ) ) ;

45

46 yy = yy+1;

47 end

48 xx = xx+1;

49 end

50 normB( zz ) = s q r t (Bz (51 ,51 , zz )ˆ2+By(51 ,51 , zz )ˆ2+Bx(51 ,51 , zz

) ˆ2) ; % magnetic f i e l d , norm

51 zz = zz +1;

52 end

53

54 gradM = ze ro s ( steps , s teps , s t ep s ) ; % grad i en t o f the magnetic

f i e l d , z component

55

56 f o r xx = 1 : s i z e (Bz , 1 )

57 f o r yy = 1 : s i z e (Bz , 2 )

58 f o r zz = 1 : s i z e (Bz , 3 )

59 temp ( zz ) = Bz( xx , yy , zz ) ;

60 end

61 gradM( xx , yy , : ) = grad i en t ( temp ) /( h e i g h t f / s t ep s ) ;

62 end

63 end

64

65 f o r c e = mean(mean(mean(gradM) ) ) ∗(10 e−6∗10e−6∗1e−6)∗Mz; %

magnetic f o r c e , z component

C.2. Result Processing

1 c l e a r a l l ;

64

MATLAB Scripts

2 c l o s e a l l ;

3 c l c ;

4

5 addpath ( ’ /home/romans/Documents/ Thes i s / Minimizat ion

S imulat ions ’ ) ;

6 addpath ( ’ /home/romans/Documents/ Thes i s / Cant i l eve r ’ ) ;

7

8 c = 1 ;

9 optimal model = ze ro s (9 , 6 ) ;

10

11 f o r i d l = 1 :3

12 f o r id h = 1 :3

13

14 i d t = ’ 1 ’ ;

15

16 %% read ing data from c a n t i l e v e r s imu la t i on

17

18 f i leName = fopen ( s t r c a t ( ’ c a n t i l e v e r t y p e ’ , i d t , ’ ’ , i n t 2 s t r (

id h ) , ’ . txt ’ ) , ’ r ’ ) ;

19 formatSpec = ’%f ’ ;

20 f i l e = f s c a n f ( f i leName , formatSpec ) ;

21 disp lacement = ze ro s ( l ength ( f i l e ) /2 ,2) ; % data from the

c a n t i l e v e r s imu la t i on

22

23 k = 1 ;

24 f o r i = 1 : l ength ( d i sp lacement )

25 disp lacement ( i , 1 ) = f i l e ( k ) ; k = k+1; % length o f the

magnetic f i l m

65

MATLAB Scripts

26 disp lacement ( i , 2 ) = f i l e ( k ) ; k = k+1; % maximum d e f l e c t i o n

due to 1e−8 N

27 end

28

29 f o r c e r e q u i r e d = ze ro s ( l ength ( f i l e ) /2 ,2) ; % f o r c e r equ i r ed to

ach ieve 0 .5 e−6 m d e f l e c t i o n

30 f o r c e r e q u i r e d ( 1 : l ength ( f o r c e r e q u i r e d ) ,1 ) = disp lacement ( 1 :

l ength ( d i sp lacement ) ,1 ) ;

31

32 f o r i = 1 : l ength ( d i sp lacement )

33 f o r c e r e q u i r e d ( i , 2 ) = (−1e−8)∗(−0.5) / disp lacement ( i , 2 )

∗(−1) ;

34 end

35

36 %% read ing data from c o i l s imu la t i on

37

38 f i leName = fopen ( s t r c a t ( ’ c o i l t y p e ’ , i d t , ’ ’ , i n t 2 s t r ( id h ) , ’

f o r c e l o o p ’ , i n t 2 s t r ( i d l ) , ’ . tx t ’ ) , ’ r ’ ) ;



41 c o i l d a t a = ze ro s ( l ength ( f i l e ) /4 ,5) ; % data from the c o i l and

magnetic f i l m s imu la t i on

42 k = 1 ;

43

44 f o r i = 1 : s i z e ( c o i l d a t a , 1 )

45 c o i l d a t a ( i , 1 ) = f i l e ( k ) ∗2 ; k = k+1; % length o f the

magnetic f i l m

46 c o i l d a t a ( i , 2 ) = f i l e ( k ) ; k = k+1; % width o f the wire

66

MATLAB Scripts

47 c o i l d a t a ( i , 3 ) = f i l e ( k ) ; k = k+1; % he ight o f the wire

48 c o i l d a t a ( i , 4 ) = f i l e ( k ) ; k = k+1; % magnetic f o r c e due to

1 A o f cur rent

49 end

50

51 f i leName = fopen ( s t r c a t ( ’ c o i l t y p e ’ , i d t , ’ ’ , i n t 2 s t r ( id h ) , ’

power loop ’ , i n t 2 s t r ( i d l ) , ’ . tx t ’ ) , ’ r ’ ) ;



54 k = 0 ;

55

56 f o r i = 1 : s i z e ( c o i l d a t a , 1 )

57 k = k+4;

58 c o i l d a t a ( i , 5 ) = f i l e ( k ) ; % power due to 1 A o f cur rent

59 end

60

61 %% i n t e r p o l a t i n g

62

63 v a r i a t i o n s = 72 ;

64

65 int temp = ze ro s ( s i z e ( c o i l d a t a , 1 ) / v a r i a t i o n s , 3 ) ;

66 i n t a r r a y = ze ro s ( s i z e ( displacement , 1 ) , v a r i a t i o n s , 2 ) ;

67

68 f o r l = 1 : v a r i a t i o n s

69 k = l ;

70 f o r i = 1 : s i z e ( c o i l d a t a , 1 ) / v a r i a t i o n s

71 int temp ( i , 1 ) = c o i l d a t a (k , 1 ) ;


67

MATLAB Scripts


74 k = k+v a r i a t i o n s ;

75 end

76 i n t a r r a y ( : , l , 1 ) = in t e rp1 ( int temp ( : , 1 ) , int temp ( : , 2 ) ,

d i sp lacement ( : , 1 ) ) ;

77 i n t a r r a y ( : , l , 2 ) = in t e rp1 ( int temp ( : , 1 ) , int temp ( : , 3 ) ,

d i sp lacement ( : , 1 ) ) ;

78 end

79

80 %% gain c a l c u l a t i o n

81

82 gain = ze ro s ( s i z e ( f o r c e r e q u i r e d , 1 ) , v a r i a t i o n s ) ;

83

84 f o r i = 1 : v a r i a t i o n s

85 gain ( : , i ) = f o r c e r e q u i r e d ( : , 2 ) . / i n t a r r a y ( : , i , 1 ) ;

86 end

87

88 %% requ i r ed power f o r 0 .5 e−6 m d e f l e c t i o n

89

90 power = i n t a r r a y ( : , : , 2 ) . ∗ ( ga in .∗ gain ) ;

91

92 %% requ i r ed f o r c e f o r 0 . 5 e−6 m d e f l e c t i o n

93

94 f o r c e = i n t a r r a y ( : , : , 1 ) . ∗ ( ga in ) ;

95

96 %% f i n d i n g optimal model

97

98 optimum = ze ro s ( v a r i a t i o n s , 6 , 9 ) ; % c o l l e c t i o n o f

68

MATLAB Scripts

c o n f i g u r a t i o n s , opt imized f o r power

99

100 f o r i = 1 : v a r i a t i o n s

101 f o r l = 1 : s i z e ( f o r c e r e q u i r e d , 1 )

102 i f ( ( power ( l , i ) < optimum( i , 5 , c ) ) | | ( optimum( i , 5 , c )

== 0) )

103 optimum( i , 1 , c ) = f o r c e r e q u i r e d ( l , 1 ) ; % length o f

the magnetic f i l m

104 optimum( i , 2 , c ) = c o i l d a t a ( i , 2 ) ; % width o f the

wire

105 optimum( i , 3 , c ) = c o i l d a t a ( i , 3 ) ; % he ight o f the

wire

106 optimum( i , 4 , c ) = f o r c e ( l , i ) ; % magnetic f o r c e

107 optimum( i , 5 , c ) = power ( l , i ) ; % power

108 optimum( i , 6 , c ) = 1∗ gain ( l , i ) ; % cur rent

109 end

110 end

111 end

112

113 c = c+1;

114

115 end

116 end

69