dynamic simulation and design of rf-mems switches made of steel

DYNAMIC SIMULATION AND DESIGN OF

RF-MEMS SWITCHES MADE OF STEEL

A PROJECT REPORT

SUBMITTED IN PARTIAL FULFILLMENT

REQUIREMENTS FOR THE DEGREE OF

Master of EngineeringMaster of EngineeringMaster of EngineeringMaster of Engineering

IN

FACULTY OF ENGINEERING

BY

GAURAV NAIR

Department of Mechanical Engineering

Indian Institute of Science

BANGALORE - 560012

June, 2012

ii

Acknowledgements

Many people deserve my heart-felt thanks for their help during my graduate work. First

and foremost, I would like to thank my advisor, Prof. G. K. Anathasuresh, for his

enduring support and encouragement. I have appreciated his ability to guide me toward

appropriate solutions to the problems I have had to face. I also would like to thank him

for letting me use his codes for solving coupled-electromechanical problems. The switch

I was working on was designed by Subajit Banerjee and was fabricated in collaboration

with the University of Michigan, Ann Arbor, by Fatih Mert Ozkeskin. I would like to

thank them both for their prompt reply whenever I had any doubts. I would also like to

thank Harish Varma for his help in debugging Matlab codes, Puneet Singh for his help in

generating solid models in Solidworks. I would also like to thank Sudhanshu Shekhar for

his support in modeling of coupled electrostatic problems in CoventorWare. I would also

like to thank Rakesh Pathak for the help with formatting of the final report. The members

of M2D2, M2 and CONE lab, who gave valuable inputs and suggestions during group

meetings. I would also like to thank Dr. Adarsh V. K. for his help with post processing of

output files of ANSYS.

Last but not the least; I would also like to thank the Indian Institute of Science for

equipping me with required skills, knowledge, and facilities for pursuing my research.

iii

Abstract

The focus of this research is on analysis, and re-design of contact type Radio Frequency

Micro-Electro-Mechanical Systems (RF-MEMS) switch that has high power-handling

(250 W) capability. This work is a continuation of Banerjee’s ME project work (2011)

that improved the switch through shape optimization by minimizing the pull-in voltage

without compromising the recoil force. This work began by analyzing the discrepancy

between the experimental and simulated pull-in voltages of Banerjee’s optimized switch.

Analysis that allowed the possibility of tilting of the switch during assembly supported the

experimental values of pull-in voltage. This insight pointed to the need to have caution

during the assembly of the switch. The subsequent work was focused on improving the

performance of the switch. Increasing the contact gap and inclusion of holes on the

switch are suggested to decrease the contact resistance and switching time respectively,

without altering the optimal shape of the switch. A novel design concept is proposed by

including a contact bump that increases the stiffness after the pull-in of the switch by

reducing the effective length of the beam of the switch.

The analysis that supported the aforementioned design improvements of the switch

includes the estimation of performance characteristics such as pull-in voltage, switching

time, recoil force, and contact parameters. The Recoil force is evaluated using finite

element (FE) beam model as well as 3D FE modeling. The contact parameters are

evaluated using 3D FE modeling. Pull-in voltage is evaluated using a combination of the

FE model (both 3D and 1D), relaxation scheme, and the bisection algorithm. The

switching time is obtained by solving the Euler-Bernoulli beam equation with the help of

normal mode summation method for which the damping parameters are evaluated using

the modal projection method.

The outcome of the work is a new set of improved designs for the switch that could be

prototyped using the same process as that used for Banerjee’s switch in 2011.

iv

Contents 1. Introduction .................................................................................................................................. 1

1.1. Background and Motivation ............................................................................................ 1

1.2. Scope of the work ............................................................................................................ 2

1.3. Organization of the Report ............................................................................................... 3

2. Literature Review......................................................................................................................... 4

2.1. RF-MEMS switches and reliability.................................................................................. 4

2.2. Dynamics and squeeze film damping .............................................................................. 4

2.3. Contact resistance ............................................................................................................ 7

2.4. Modeling of contact when considering beam dynamics .................................................. 8

2.5. High recoil force .............................................................................................................. 8

3. Description of the problem and formulation .............................................................................. 10

3.1. Description of the problem ............................................................................................ 10

3.2. 1D lumped model dynamics .......................................................................................... 12

3.3. Beam model with damping ............................................................................................ 14

3.4. Recoil force .................................................................................................................... 17

3.5. Contact resistance ............................................................................................................... 19

4. Results and discussion ............................................................................................................... 20

4.1. Lumped model ............................................................................................................... 20

4.2. Beam model ................................................................................................................... 20

4.3. Discussion ...................................................................................................................... 22

4.3.1. Pull-in voltage discrepancy .................................................................................... 22

4.3.2. Improvement in contact resistance ......................................................................... 24

4.3.3. Improvement of the Pull-in time ............................................................................ 25

4.3.4. Recoil force ............................................................................................................ 26

5. Recoil force enhancement with nonlinear stiffness ................................................................... 27

5.1. Contact bumps ............................................................................................................... 27

5.2. Redesign of switches ...................................................................................................... 30

6. Summary and Conclusions ........................................................................................................ 33

Appendix ........................................................................................................................................ 34

A1. Modal projection method in Ansys and input files ........................................................ 34

A2. Matlab codes .................................................................................................................. 39

A3. Manuscript in preparation for possible submission to a journal .................................... 49

Bibliography .................................................................................................................................. 58

v

List of figures

Fig.2.1. Fixed-fixed type switch used in [25] displaying the nonlinear spring damper

foundation which is active only under compression. 8

Fig.3.1. Scanning Electron Microscopes (SEM) micrographs for Design 2 in

Table 1. (a) Cantilever is shown upside down with recessed regions. (b)

Assembled structure with 6 µm gap between the cantilever and the

ground electrode shown in the inset. 10

Fig.3.2. Dimensions of re-designed shapes which were fabricated. 11

Fig.3.3. Lumped approximation of a MEMS switch as a parallel-plate capacitor. 12

Fig.3. 4. Electrostatic force and the elastic restoring force for different actuation

voltages. It can be seen that the electrostatic force curve intersects the elastic

force curve at only one point (at 00.33z g= ). 13

Fig.3.5. Time response of the switch described in Fig. 3.3 for different values of ζ . 13

Fig.3. 6. Simplified model of the switch for beam FE analysis. 14

Fig.3. 7. Contributions of different modes during actuation. (a) Mode 1. (b) Mode 2. (c)

Mode 3. (d) Mode 4. (e) Mode 5. 16

Fig.3. 8. Pull-in time vs. voltage for the cantilever switch. The increase of actuation

voltage by 1.2 V reduces the pull-in time substantially and further increase in

actuation voltage has no significant effect on the pull-in time. 17

Fig.3. 9 (a) Linearly tapering switch with dimensions similar to [6]. (b) Pull-in voltage

and recoil force with taper angle of the beam with gaps and both as 8 µm. 18

Fig.3. 10. Cantilever beam coming into with the contact pad. Inset shows the hills of the

contacting surfaces coming in contact which reduces the actual area of contact. 19

Fig. 4.1. Time response of the six designs. The sharp rise in the displacement indicates

that pull-in has occurred and the corresponding time is taken as the pull-in

time. 21

Fig.4.2. (a)The beam tilts about the edge of the resting surface (axis shown) by 0.14o.

(b) Exaggerated 2D sectional view depicting the beam tilting by θ . 23

Fig.4.3. Pull-in voltage for different configurations obtained from experiments and

simulation by varying the actuation gap. 23

Fig.4.4. Effect on pull-in voltage with some amount of initial tilt given to the beam

which could occur during the assembly process. 24

vi

Fig.5.1. Cantilever with solid guide which exhibits nonlinear stiffness. 27

Fig.5.2 Electrostatic and elastic force acting on the beam for various values of

displacements applied to the tip of the beam. The applied voltage is 108.5 V

which is the pull-in voltage for the beam with zero slope. 28

Fig.5.3. Minimum tip displacement required for different slopes of the beam when the

pull-in voltage is applied. A cubic curve fit for the scattered data occurring due

to the approximate evaluation of the pull-in voltage due to the bisection

method. 28

Fig.5.4. The increase in recoil force due to the inclusion of contact bump. At 0o slope of

the beam the recoil force is more than double that of the configuration without

the bump. 29

Fig.5.5. The side view of the beam with the region where the thickness is to be reduced.

All other dimensions are the same as in [8]. 30

Fig.5.6. Switch with contact bump attached to the beam placed at 10 µm distance from

the actuation electrode. 32

Fig.A2.1 Pressure distribution for the beam mentioned in Section 3. For the first mode. 34

vii

List of Tables

Table 3.1. Data for the six optimized re-designed switches. 11

Table 3.2. Results of experiments of six optimized and redesigned switches. 11

Table 4.1. Lumped parameters and performance characteristics of the six designs. 19

Table 4.2. Results of simulation of six optimized and re-designed switches. 20

Table 4.3. Pull-in voltage when actuation gap is 3 µm and 0.0660 tilt is present. 21

Table 4.4. Contact force and resistance for contact gap of 6 µm. 23

Table 4.5. Q-factor for the first mode. 24

Table 4.6. Pull-in time for the with and without holes. 24

Table 5.1 Pull-in voltage for existing switch and the switch with reduced

thickness shown in Fig. 5.5. 29

Table 5.2. Recoil force for the existing switch, the modified switch with reduced

thickness and the modified switch with contact bump in mN. 29

Table 5.3. Recoil force comparison for existing switch with the increased contact

gap and the modified switch. 29

1

Chapter 1

Introduction

1.1. Background and Motivation

Micromachined radio-frequency (RF) switches, known as RF MEMS

(microelectromechanical systems) switches, have received much attention from academia

and industry ever since the MEMS field came into research focus [1,2]. The importance

of RF MEMS switches is underscored by the fact that they can be co-located with

digitally controllable circuit elements with a small footprint on the chip. Among different

types of actuation used, electrostatic actuation is the most common because of its low

power consumption. While linear behavior, low power consumption, low insertion loss,

high isolation, and low manufacturing cost are benefits of electrostatic micromechanical

switches, there is room for improvement in power handling capacity, switching times,

reliability, and actuation voltages as compared to purely electronic switches [3,4].

Switches made of silicon have power handling capacity of tens of W and beyond this

they tend to fail by stiction or adhesion. Also environmental factors require silicon

switches to be packaged, which proves to be expensive. Keeping these reasons in mind,

metals are used as the structural element in micro-switches [5]. In particular, steel was

assembled on a printed circuit board (PCB) [6] and the switch was shown to have up to

250 W of dissipative power capacity and occupied a small footprint of 6 mm2. Here, the

actuation voltage under cold-switching condition was over 300 V, actuation voltage

required for the functioning (hot-switching conditions) of this switch was 130 V, and the

actuation time was in ms range.

Increasing the actuation (i.e., switching) voltage reduces the switching time. But it

also increases the contact force that may cause increased adhesion and wear [7].

Therefore, Pt-Rh was used on the contact pads in [4] and [6]. Although it helped, the

problem remains that Pt-Rh is expensive and makes fabrication and assembly difficult.

Banerjee et al. [8] addressed the adhesion problem with a mechanical approach. That is,

they increased the recoil or spring-back force of the switch so that the switch can spring

back upon turning off the actuation voltage by breaking any unwanted fused connections

formed due to stiction or adhesion. Thus, it not only reduced the actuation voltage but

also alleviated problems associated with adhesion and wear.

RF-MEMS switches are aimed to replace the p-i-n diodes and field-effect

transistor (FET) diode switches which have the limitations of high insertion loss and low

isolation but have the advantage of very low switching times that is of the order of µs.

The switches mentioned in [6] have switching times in the ms range. Damping in the

micro-scale has significant effect on the response of a switch under actuation. The

millimeter order of dimensions and the micro-scale gaps between the switch and the

substrate account for very large damping forces compared to the electrostatic forces.

2

Improving the Q-factor leads to reduced damping force and hence decrease in the

switching times.

1.2. Scope of the work

This work considers the following four aspects that further improve the switch designs

reported by Ozkeskin [9] and Banerjee [10].

i. This work begins with the analysis of six switches prototyped at the University of

Michigan, Ann Arbor, USA by Ozkeskin [9] for which the designs were provided

by Banerjee [10] by means of optimization. The gap used in their switches is in

the range of a few microns but the planar dimensions of the switch are in the

millimeter range. Thus, slight error in the planar dimensions or manual assembly

can lead very big change in the gap between the actuation electrode and the

switch. The performance characteristics of the switch are highly dependent on the

gap as the electrostatic force is inversely proportional to the square of the gap [9].

The experimentally reported pull-in voltages are about one-third the values of the

simulated pull-in voltages reported in [9] and [10]. Since there is considerable

difference between the simulation and experimental results, there is a need to

investigate the reason behind this discrepancy. In this work, we investigate

various possibilities that could lead to the decrease in pull-in voltage after

assembly of the switch.

ii. Contact resistance plays an important role in ensuring the reliability of an RF-

MEMS switch. Most switches fail when the contact resistance increases beyond 5

Ω. In practice, in order to have realiable performance in repeated operation,

contact resistance of less than 1 Ω is preferred [16]. No study had been conducted

on the contact resistance that occurs during the closing of Ozkeskin’s [9] switch.

In this work, evaluation of contact resistance and redesign of the switch based on

the contact resistance are pursued. Contact parameters, i.e., the contact force and

contact area, are evaluated by means of 3D FEA and Holm’s model [17] is used

for evaluating the contact resistance.

iii. Fast switching time is a desired characteristic of a good RF-MEMS switch. Pull-

in times were measured during the experiment but this was not simulated by

Banerjee [10]. This work considers the dynamic modeling of the switch to

estimate the pull-in time. This requires us to consider the squeezed-film effect due

the air in the narrow gaps of the switch [11]. As coupled 3D simulation of

squeezed-film effects with the deformation of the electrostatically actuated body

of the switch is time-consuming, normal mode summation method [13] in

conjunction with modal projection method [14] are used in this work to develop a

quick method to capture the dynamic response of the switch. The method is also

3

extended to handle the presence of holes in the beam of the switch, which reduces

the squeezed-film effect without disturbing the optimal width profile of the beam.

iv. By noting that the recoil force, which is important for preventing the adhesion

problem, has positive correlation with the stiffness of the switch, Banerjee [10]

had used optimization of the width profile of the beam of the switch to increase

the recoil force. In this work, a new approach is proposed to increase the recoil

force by increasing the stiffness of the beam after the pull-in by incorporating a

contact bump that decreases the effective length of the beam. This approach

substantially increases the recoil force while keeping the actuation voltage low.

1.3. Organization of the Report

Literature on RF-MEMS switches and their reliability, dynamic modeling of MEMS

switches with squeeze film effect and methods to evaluate contact resistance are briefly

presented in Chapter 2. Chapter 3 contains the description of the problem and various

models and methods used for evaluation of the performance characteristics, i.e., the pull-

in voltage, contact resistance, pull-in time, and the recoil force. Chapter 4 contains the

results of the analysis that give the performance characteristics of the switch.

Discrepancy between the results obtained by simulation and experiment are discussed

here. Also discussed are some improvements related to contact parameters and switching

times. Described in Chapter 5 is a new method to improve the recoil force. This chapter

also contains the details of the redesigned switch and its comparison with the existing

designs. Concluding remarks are in Chapter 6. Appendices contain the input files of

ANSYS software, MATLAB codes, and a manuscript in preparation for possible

submission to a journal.

4

Chapter 2

Literature Review

2.1. RF-MEMS switches and reliability

RF-MEMS switches achieve a short or open circuit by mechanical motion of a structural

element. This mechanical motion is achieved by means of electrostatic, piezoelectric,

magnetostatic force or deformation due to thermal effects. Among these, electrostatic

type switches have shown reliable operations for 0.1-100 GHz applications. However,

RF-MEMS suffer from a few limitations. These include slow switching time, low power

handling capability, high packaging cost and relatively low reliability. As the input power

to the switch increases the reliability of the switch comes down. This is because in high

power applications the failure occurs due to stiction or adhesion. A lot of research has

gone into finding solutions to these problems. Some of them include change in material

properties and inclusion of heat sinks. Pt-Rh contact element along with a heat sink made

up of four micro-rods connected to the contact element via a ball[6] was able to address

the problem but the complexity involved was high. Purely mechanical means for

overcoming the problem of stiction and adhesion was explored by Banerjee et al. [8] by

posing and solving an optimization problem. They considered the recoil force, force with

which the switch will come back once the actuation voltage is turned off was taken as the

objective function. A volume constraint was imposed on the amount of material to be

used so that the switch remains within 6 mm2 foot-print area. It was found that linearly

tapering profile was optimum and six such linear profiles with varying taper angle were

fabricated and tested at the University of Michigan, Ann Arbor, USA. Details of the

variational formulation and optimization are given in [10]. The structural element used

there was steel that again scores over silicon switches. This is because silicon switches,

unlike steel switches, require hermetic packaging in inert atmospheres, which increases

the cost.

2.2. Dynamics and squeeze film damping

Banerjee et al. [8] in their design did not take into account the dynamic behavior of the

switch. Experimental observation revealed that the pull-in times for these switches were

below 8 ms which is a significant improvement over the switch in [6] which had a pull-in

time of 16 ms. Dynamics of parallel-plate and torsional switches are studied in detail in

[19] where it is proved that dynamic pull-in voltage is lower than the static pull-in

voltage. For the undamped case dynamic pull-in voltage for the parallel-plate is 92% of

the value of static pull-in voltage and is 91% of the value of static pull-in voltage for the

torsional switch. The analysis of undamped system is of little use as at the microscale

where the effect of squeeze film damping is significant. Squeeze film damping occurs

due to the change in pressure distribution on the surface of switch as the fluid film

5

between the switch and the substrate is compressed. The pressure distribution is governed

by the Reynold’s equation [11], given by

t

p

Phdt

dh

hy

p

x

p

∂

∂+=

∂

∂+

∂

∂

0

2

0

3

0

2

2

2

2 1212 µµ (2.1)

where p is the excess pressure in the film, µ the coefficient of viscosity, h the

instantaneous thickness of the film, 0h the initial thickness of the film and 0P the

reference pressure. When the mean free path of the fluid, λ , is comparable to the

thickness of the film the viscosity of the fluid becomes a function of Knudsen number,

Kn , given by 0/ hλ . An empirical relation for effective viscosity was given by Veijola et

al. [20]

159.1658.91 Kn

eff+

=µ

µ (2.2)

Starr exploited the similarity between the Eqn 2.1 and the transient heat equation

[21]. Finite element formulation of the transient heat problem is well developed [22] and

available in many commercial softwares like COMSOL Multiphysics (www.comsol.com)

and ANSYS (www.ansys.com). The transient heat equation is given by

t

TCQ

y

T

x

Tk p

∂

∂+−=

∂

∂+

∂

∂ρ

2

2

2

2

(2.3)

where k is the thermal conductivity, Q the heat source density, ρ the density and pC

the specific heat and T the temperature variable.

In order to evaluate the damping force acting on the structure we have to evaluate

the pressure at every instant as the beam deforms. This leads to a coupled problem where

the state variables wvu ,, and p are all being functions of spatial coordinates, zyx ,, and

time, t . Too many variables lead to very large computation times. The usage of beam

elements and normal mode summation method [13] reduce the number of variables

considerably. The dynamic response for a beam is given by

ed F

t

wc

t

wA

x

wEI =

∂

∂+

∂

∂+

∂

∂2

2

4

4

ρ (2.3a)

where, eF is the electrostatic force on the beam, EI the flexural rigidity, dρ the mass per

unit length of the beam, c the damping coefficient. Now discretization by finite element

formulation leads to the equation of the form

eMw + Cw + kw = Fɺɺ ɺ (2.3b)

6

where M,C and K are the mass, damping and stiffness matrices respectively. The

displacement vector, w , can be written as a linear combination of eigen-vectors, iφ and

generalized coordinates, iη where Ntoi 1= . N is the number of modes to be taken into

consideration. Let p be the matrix containing all eigen-vectors of the system and η be

the vector of the generalized coordinates. Then the displacement vector w can be written

as pη . Using the orthogonality property of the eigen-vectors [13], Eqn. (3b) on

premultiplicaion by Tp is reduced to a set of linear equations with iη as the variables

given by

T

eMη+ Cη+ Kη= p Fɶɶ ɶɺɺ ɺ (2.3c)

where Mɶ and Kɶ are diagonal due to the property of orthogonality. Cɶ can also be

diagonalised if the damping matrix, C , is expressed as a linear combination of mass and

stiffness matrices i.e. α β= +C M K [13]. This idea of applying a proportional damping is

also known as Rayleigh damping. If the damping ratio is known for the modes of for the

first and last modes of interest, the matrix C can be generated by finding the coefficients

βα & . For a one degree of freedom spring, mass and dashpot system with mass m ,

damping coefficient c given by km βα + and stiffness k and forcing term F the

equation of motion is given by

m

Fx

m

kxkm

mx =+++ ɺɺɺ )(

1βα (2.4)

Comparison of the coefficient of xɺ of Eqn. (2.4) with the equation of a second order

system given by

mFxxx nnn /2 2 =++ ωωζ ɺɺɺ (2.5)

gives the equation of damping ratio as

+= n

n

n βωω

αζ

2

1 (2.6)

If the range of interest is the first N modes then two linear equations containing α and

β the variables can be formed by writing Eqn. (2.6) for the first and the th

N mode.

Solution of the two linear equations gives

2

1

2

111

)(2

ωω

ωζωζωωα

−

−=

N

NNN (2.7)

7

2

1

2

11 )(2

ωω

ωζωζβ

−

−=

N

NN (2.8)

Thus, if the natural frequency and the damping ratio of the first and last mode under

consideration are known the dynamic response of the beam can be obtained. In [23] it is

mentioned that at least five modes are required to capture accurately the response of a

beam in motion. The electrostatic force can be easily evaluated by making use of parallel-

plate approximation but damping force remains a concern. Hung and Senturia, in [24],

have simulated the beam undergoing deformation due to electrostatic force with a fluid

between the beam and the electrode but the solution cannot be generalized to any switch.

In [23,25], constant damping is applied for all modes to simplify analysis. A very quick

and easy method is suggested by Mehner et al. [14] where the modal projection method is

used to extract the damping ratio associated with a particular mode. The DMPEXT

command macro of the commercial software ANSYS readily extracts the damping ratio

for different structures with different boundary conditions. Inclusion of perforations in

the structural element reduces the effect of damping considerably. In [15] the area ratio

and number of holes are varied and a parametric study is conducted to evaluate the

damping ratio.

2.3. Contact resistance

While the switching time has no effect on the reliability of the switches, the contact

parameters are very important for ensuring reliable operation of the switch. A lower

contact area leads to high contact resistance which produces a lot heat given by tRI c

2 ,

where I is the current, cR the contact resistance, and t the time. In fact, this is the very

principle of spot welding which is used to attach two sheets of metal together by passing

a high current through a very small area on the sheets of metal [26]. Thus, there is a very

high chance of the switch getting stuck to the transmission line if the contact resistance is

high. It has been reported in [16] that switches usually fail when the contact resistance

exceeds 5 Ω. But, for design purposes contact resistances below 1 Ω are desired. Bromley

and Nelson, in [27] have experimentally verified that the apparent area of contact has no

effect on the contact resistance and it is the contact force that controls the contact

resistance. This is due the surface roughness of either surfaces coming in contact. Only

the hills of either contact surfaces come in contact leading to much lower actual contact

area and a much high contact resistance. Holm in [17], has given an expression for the

contact resistance if area of contact is considered to be of circular shape with radius a

aRc

2

ρ= (2.9a)

For dissimilar materials coming into contact an expression for the contact resistance is

given by [28]

aRc

4

21 ρρ +=

Thus, once the contact force is known Eqn. (2.9b) can be u

resistance.

2.4. Modeling of contact when considering beam dynamics

A very popular method to model contact is to model the surface against which the beam

will contact as a foundation of

beam touches the surface and

spring is commonly simulated using

Sigmoid function [30] and

Vyasarayani et al., in [25],

is the vertical displacement of the beam and

Heaviside function acting on the gap function will give a non

1dw > . When the beam impacts the surface some losses occur due to the impact and

give rise to deformation dependent damping

the contact function defined in [25

1),( txgkF n

cc

+=

where impµ is the coefficient of deformation dependent damping. The value of

subtracted from the right hand side of Eqn. (2.3a) to give the governing equation of beam

undergoing electrostatic actuation and contacting with a surface.

Fig.2.1. Fixed-fixed type beam

active only under compression

2.5. High recoil force

A lot of research has been conducted on the failure of MEMS switche

adhesion especially with respect to the material

power application switches

8

Thus, once the contact force is known Eqn. (2.9b) can be used to evaluate the contact

Modeling of contact when considering beam dynamics


a foundation of nonlinear springs which become active only whe

beam touches the surface and only react on compression [22], [25] and [29

spring is commonly simulated using step functions such as Heaviside funct

] and simulated stiffness has high value of spring constant,

, in [25], first defined a gap function given by ),( txg =

is the vertical displacement of the beam and 1d is the gap as shown in Fig. 2.1. Thus,

viside function acting on the gap function will give a non-zero value only when

. When the beam impacts the surface some losses occur due to the impact and

give rise to deformation dependent damping [31] which is proportional to

t function defined in [25] takes the form

)),(( txgHt

gimp

∂

∂+ µ

is the coefficient of deformation dependent damping. The value of

ht hand side of Eqn. (2.3a) to give the governing equation of beam

undergoing electrostatic actuation and contacting with a surface.

beam used in [25] displaying the nonlinear spring damper foundation which is

compression.

High recoil force

A lot of research has been conducted on the failure of MEMS switches due to stiction and

adhesion especially with respect to the material [32]. Pt-Rh is used in a number of hig

power application switches [6]. Pt-Rh being a very expensive material, Banerjee et

(2.9b)

sed to evaluate the contact


nonlinear springs which become active only when the

only react on compression [22], [25] and [29]. This type of

Heaviside function or

high value of spring constant, ck .

1dw −= where w

is the gap as shown in Fig. 2.1. Thus, the

zero value only when

. When the beam impacts the surface some losses occur due to the impact and

which is proportional to t

g

∂

∂. Finally,

(2.10)

is the coefficient of deformation dependent damping. The value of cF is

ht hand side of Eqn. (2.3a) to give the governing equation of beam

used in [25] displaying the nonlinear spring damper foundation which is

s due to stiction and

used in a number of high

Banerjee et al. [8]

9

used a purely mechanical approach for improving reliability and thereby avoiding Pt-Rh.

Even if the switch adheres to the transmission line the optimum design ensures that any

bonds formed between the transmission line and the switch will be broken by the recoil

or the spring back force, which is maximized by design.

Chapter 3

Description of the problem and formulation

3.1. Description of the problem

An optimized profile with high recoil force was designed

Ozkeskin [9] at the University of Michigan, Ann Arbor, USA. Even though the previo

design by Ozkeskin [6] did not require any sort of packaging , this design was superior to

the previous design as it did not contain the complexity

require any contact element like Pt

maximizing the recoil force

profile, and the other was a curve

work and the two solutions obt

The cantilever beam of the switch was photochemically etched from 50

SS304 foils (Kemac Technology Inc., CA). The recesses, as shown in Fig. 3.1(a), of 6

µm and 4 µm were machined using micro

Perforations of 300 µm diameter were located for the alignment and the attachment on

the PCB (Fig. 3.1(a)). A 600 µm

Circuits Inc., CO). Metal interconnect traces of 70 µm

and the contact pad. In such PCBs, 4

base, and 0.25-µm thick outer gold layer provided an electrical contact. Through vias were

located on the PCB for the subsequent attachment of the cantilever.

Fig.3.1. Scanning Electron Microscopes (SEM) micrographs for Design 2 in Table 1. (a) Cantilever is

shown upside down with recessed regions. (b) Assembled structure with 6 µm gap betwe

and the ground electrode shown in the inset.

For the assembly, alignment posts (1000 µm

machined from gold wire using µEDM, and

on the PCB (Fig. 10a). Conical

cantilever was assembled over the posts (Fig. 3.1(

epoxy (Creative Materials). The flatness of the cantilever was maintained during the

10

Description of the problem and formulation

Description of the problem

An optimized profile with high recoil force was designed Banerjee [10] and fabricated by

University of Michigan, Ann Arbor, USA. Even though the previo

] did not require any sort of packaging , this design was superior to

the previous design as it did not contain the complexity of the heat sink and did not

require any contact element like Pt-Rh which is expensive. The optimization problem of

maximizing the recoil force, in [10] led to two solutions. One was linearly tapering

other was a curve that was also almost linear. The details of optimization

work and the two solutions obtained are in [10].

The cantilever beam of the switch was photochemically etched from 50


4 µm were machined using micro-electrodischarge machining (


A 600 µm-thick Rogers 4003 was used as the substrate (Advanced

). Metal interconnect traces of 70 µm - thick Cu provided bias electrodes

and the contact pad. In such PCBs, 4-µm thick Ni was used as an adhesion layer on the Cu

µm thick outer gold layer provided an electrical contact. Through vias were

located on the PCB for the subsequent attachment of the cantilever.

Scanning Electron Microscopes (SEM) micrographs for Design 2 in Table 1. (a) Cantilever is

shown upside down with recessed regions. (b) Assembled structure with 6 µm gap betwe

and the ground electrode shown in the inset.

For the assembly, alignment posts (1000 µm-height; 300 µm

machined from gold wire using µEDM, and they were tightly fitted into the isolated

on the PCB (Fig. 10a). Conical wire tips facilitated the insertion of the cantilever. The

ssembled over the posts (Fig. 3.1(b)) and fixed by applying conductive


and fabricated by

University of Michigan, Ann Arbor, USA. Even though the previous

] did not require any sort of packaging , this design was superior to

of the heat sink and did not

expensive. The optimization problem of

led to two solutions. One was linearly tapering

linear. The details of optimization

The cantilever beam of the switch was photochemically etched from 50 µm-thick


electrodischarge machining (µEDM).


thick Rogers 4003 was used as the substrate (Advanced

thick Cu provided bias electrodes

µm thick Ni was used as an adhesion layer on the Cu

µm thick outer gold layer provided an electrical contact. Through vias were

Scanning Electron Microscopes (SEM) micrographs for Design 2 in Table 1. (a) Cantilever is

shown upside down with recessed regions. (b) Assembled structure with 6 µm gap between the cantilever

height; 300 µm-diameter), were

tightly fitted into the isolated-vias

wire tips facilitated the insertion of the cantilever. The

and fixed by applying conductive


11

assembly process with the help of a high resolution laser displacement sensor (Keyence

LK-G32).

TABLE 3.1

DATA FOR THE SIX OPTIMIZED AND RE-DESIGNED SWITCHES

Design No. Slope (°) 1W (in µm) 2W (in µm)

1 -13.7 2231 572

2 -6.1 1769 1042

3 -0.9 1463 1354

4 1.7 1308 1510

5 6.9 1000 1823

6 14.5 538 2293

1l = 500, 2l = 500, 3l = 1430, 4l = 130, 5l = 600, 6l = 200, and L = 3400 (all values are in µm).

Six such designs were fabricated with the shape shown in Fig. 3.2. Dimensions are

provided in Table 3.1. Figure 3.1 depicts Design 2. Ozkeskin conducted experiments on

these six designs to obtain the performance characteristics. Results of the experiment are

included in Table 3.2.

Fig.3.2. Dimensions of re-designed shapes which were fabricated.

TABLE 3.2

RESULTS OF EXPERIMENTS OF SIX OPTIMIZED AND RE-DESIGNED SWITCHES

Design No Pull-in voltage (V) Pull-in time (ms)

1 46.25 4-5

2 40.95 4-6

3 35.75 5-6

4 34.1 5-7

5 32.8 5-8

6 28.15 6-8

3.2. 1D lumped model dynamics

A computationally efficient way to model a system for dynamic analysis is to obtain the

lumped parameters for the system and solve it as a

a spring [33]. By lumping the mechanical elements the system can be ap

a single degree-of-freedom system whose governing equation is given by

2

0

2( )

AVmz kz

g z

ε+ =

−ɺɺ

Figure 3.3 shows the schematic of the lumped system where

mass, A the overlap area between the switch and the actuation electrode,

voltage and g the gap. Neglecting the

the system shown in Fig. 3.3. Figure 3.4 shows the electrostatic force

restoring force elF for various values of

value piV only one equilibrium position is found (intersection of the curves

where as when the voltage is set to

the left and unstable on the right. Thus it can be said that

applied voltage beyond which

term and solving the equation

which also is the point of intersection betwe

curve for piV as the actuation voltage. On substitution of

term as zero, we can find the closed form expression for

A

kgVpi

0

3

0

27

8

ε=

Fig.3.3. Lumped approximation of a MEMS switch as a

The solution of the lumped system given by Eqn. (3.1) gives an estimate of the dynamic

pull-in voltage. The system is solved

method for different actuation voltages. The time response of the switch for different

actuation voltages is given in Fig. 3.5. It can be seen that pull

approximately 92% of the valu

12

1D lumped model dynamics


the system and solve it as a parallel-plate capacitor constrained by

. By lumping the mechanical elements the system can be ap

freedom system whose governing equation is given by 2

22( )

AV

g z

Figure 3.3 shows the schematic of the lumped system where k is the stiffness,

the overlap area between the switch and the actuation electrode,

the gap. Neglecting the zɺɺ term in Eqn. (3.1) we get the static equation of

the system shown in Fig. 3.3. Figure 3.4 shows the electrostatic force

for various values of z . It can be seen that when voltage is set to a

only one equilibrium position is found (intersection of the curves

where as when the voltage is set to piV8.0 two equilibrium positions are found, stable on

the left and unstable on the right. Thus it can be said that piV is the critical value of the

applied voltage beyond which the stable solution ceases to exist. By neglecting the

term and solving the equation 0/ =dzdV and solving z we get /0gz =

which also is the point of intersection between the electrostatic force and elastic force

as the actuation voltage. On substitution of 3/0gz = into Eqn. (3.1) with

term as zero, we can find the closed form expression for piV given by

Lumped approximation of a MEMS switch as a parallel-plate


in voltage. The system is solved in the time domain using Runge


actuation voltages is given in Fig. 3.5. It can be seen that pull-in of the switch occurs at

approximately 92% of the value of piV . The damping which has a significant effect at


capacitor constrained by

. By lumping the mechanical elements the system can be approximated into

freedom system whose governing equation is given by

(3.1)

is the stiffness, m the

the overlap area between the switch and the actuation electrode, V the actuation

term in Eqn. (3.1) we get the static equation of

the system shown in Fig. 3.3. Figure 3.4 shows the electrostatic force eF and elastic

. It can be seen that when voltage is set to a

only one equilibrium position is found (intersection of the curves eF and elF )

two equilibrium positions are found, stable on

is the critical value of the

the stable solution ceases to exist. By neglecting the zɺɺ

3/ as the solution

en the electrostatic force and elastic force

into Eqn. (3.1) with zɺɺ

(3.2)

capacitor.


in the time domain using Runge-Kutta 4th

order


in of the switch occurs at

. The damping which has a significant effect at

13

the micron level is not considered here but if the value of the damping ratio ζ is known

the damping coefficient c can be easily obtained as kmζ2 . The evaluation of ζ for

different modes is discussed in the next section.

Fig.3.4. Electrostatic force and the elastic restoring force for different actuation voltages. It can

be seen that the electrostatic force curve intersects the elastic force curve at only one point (at

00 33.z g= ).

Fig.3.5. Time response of the switch described in Fig. 3.3 for different values of piVV / .

3.3. Beam model with

By referring to Fig. 3.6 that

Bernoulli beam equation that governs its static deformation as

(

2 2

2 22

d d w wEI d x d l

dx dx b

= + ≤ ≤ +

ɶ

where ( )w x is the transverse displacement of the beam,

modulus with E indicating the Young’s modulus and

moment of inertia, 0ε the permittivity of free space,

beneath the beam, d the distance from the fixed end of the cantilever beam to the start of

the actuation electrode,

actuation electrode, and L

the transverse electrostatic force per unit length of th

Parallel-plate approximation with fringing field appro

the electrostatic force acting on the beam.

Fig.3.6. Simplified model of the switch for beam FE analysis.

Equation (3.3) cannot be solved analytically because o

Finite element modeling is used to discretize Eq. (3.3) and then solved iteratively starting

with the initial zero deformation. It is a standard method to solve this equation and to find

the pull-in voltage at which the beam wou

solution ceases to exist. There is a small difference here because the beam is shown to be

slightly thicker at the free end with contact gap equal to

can be closed before the portion of the beam above the actuating electrode actually

touches and shorts with the electrode.

To validate the results of the beam analysis, 3D finite element modeling was used

to solve the coupled problems of electrostatics and elastos

software (www.cowentorware.com

14

with damping

that depicts a simplified version of the switch, we write Euler

Bernoulli beam equation that governs its static deformation as

( )

2

0

2

0 for

1 0.65 for2

0 for

a

x d

bd d w wEI d x d l

dx dx bg w

d l x L

ε φ

<

= + ≤ ≤ +

−

+ < ≤

is the transverse displacement of the beam, (/ 1E E= −ɶ

indicating the Young’s modulus and ν the Poisson’s ratio,

the permittivity of free space, b the width of the beam,

the distance from the fixed end of the cantilever beam to the start of

the actuation electrode, ϕ the applied electric potential between the beam and the

L the total length of the beam. The term on the right hand side is

the transverse electrostatic force per unit length of the beam that deflects the beam [3

plate approximation with fringing field approximation [34] is used for modeling

the electrostatic force acting on the beam.

. Simplified model of the switch for beam FE analysis.

quation (3.3) cannot be solved analytically because of the nonlinear force term.



in voltage at which the beam would snap down catastrophically because a stable


slightly thicker at the free end with contact gap equal to c ag g< . Hence, the contact gap

osed before the portion of the beam above the actuating electrode actually



to solve the coupled problems of electrostatics and elastostatics in the CoventorWare

www.cowentorware.com). In addition to the pull-in (and hence actuation)

depicts a simplified version of the switch, we write Euler-

(3.3)

)2

/ 1 ν= − the biaxial

the Poisson’s ratio, I the area

the width of the beam, ag the gap

the distance from the fixed end of the cantilever beam to the start of

pplied electric potential between the beam and the

the total length of the beam. The term on the right hand side is

e beam that deflects the beam [33].

] is used for modeling

. Simplified model of the switch for beam FE analysis.

f the nonlinear force term.



ld snap down catastrophically because a stable


. Hence, the contact gap

osed before the portion of the beam above the actuating electrode actually


tatics in the CoventorWare

in (and hence actuation)

15

voltage, we also need the contact and recoil forces and switching time. Contact force is

the force felt by the contacting portion of the beam with the actuation voltage present.

The recoil force, as noted earlier, is defined as the force with which the beam would

spring back up after the actuation voltage is set to zero. The switching time is obtained by

performing the dynamic analysis.

The normal mode summation was discussed in the previous chapter. Time

response of the beam is evaluated using the ODE45 solver of Matlab

(www.mathworks.com) by normal mode summation. Only five modes are used to model

the system [23]. To estimate the damping, we use the modal projection method [14] to

evaluate the damping coefficient, ς . The steps used in damping parameter extraction in

ANSYS by modal projection method are described in Appendix A1. The structural

element is modeled in ANSYS as a cantilever with length 2900 µm, 1400 µm width and

50 µm thickness (similar to design 3 mentioned in Section 3.1) using SOLID45 elements

with properties of SS304. The fluid film thickness is assumed to be 6 µm and the fluid is

modeled using FLUID136 elements with properties of air. The beam is assumed to be

open at three sides that are not fixed and closed at the side that is fixed. Care has to be

taken so that we only extract the modes arising due to bending of the beam. The modal

analysis of the switch by keeping the left end of the beam fixed and the top and bottom

sides with roller supports reveals that the first five bending modes are 1,2,4,7 and 11. We

obtain the damping coefficient for the first mode, 1ς , as 3.204 and the fifth mode, 5ς , as

0.0014. This value of 1ς corresponds to a Q-factor value of 0.16 at resonance which is a

very conservative value for this design. This value of Q-factor can be easily improved to

a higher value by introduction of holes in the beam that reduces the effect of squeeze film

damping. The contour plots of pressure distribution on the beam surface are included in

Appendix A1.

Proportional damping is applied in the form of Rayleigh damping. The damping

matrix, C , is evaluated as α β+M K , where Mand K are mass and stiffness matrices

and α and β are constants and obtained by fixing the damping ratio, ς , equal to 3.204

for the first and 0.0014 for the fifth [22]. The electrostatic force is applied as considering

each element of the beam as a parallel-plate. The dynamic response is given by Eqn.

(2.3b) which can be converted into Eqn. (2.3c). Equation (2.3c) is solved using ODE45

solver of Matlab. The pull-in time is taken as the time taken for the switch to first touch

the contact pad. As the input voltage is increased, it is observed that the switch first

closes when the voltage is 110.8 V. This is taken as the dynamic pull-in voltage. Fig. 3.7

shows the contribution of each mode to the dynamic response of the switch. It can be

seen that mode 1 has the most significant contribution. Increasing the voltage beyond this

value results in decrease of the pull-in time as shown in Fig. 3.8. It can also be seen that

increasing the actuation voltage beyond 112 V does not have any significant effect on the

pull-in time.

16

Fig.3.7. Contributions of different modes during actuation. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5.

17

Modeling contact is unnecessary for the estimation of pull-in time. However, in

order to simulate lift-off, the beam must initially be in contact with substrate. For this we

make use of the contact model suggested by Vyasarayani et al. [25] (described in the

previous chapter). Lift-off is simulated by first making the beam pull-in by applying a

voltage beyond the pull-in voltage and letting it settle down to an equilibrium condition.

Then using this solution as the initial condition for displacements and setting the

actuation voltage to zero volts we can simulate the lift-off phenomenon.

3.4. Recoil force

The Recoil force is evaluated by assuming that the entire region of the beam intended for

contact is in contact with the transmission line (cantilever stopper of Fig. 2.1). This can

be achieved by prescribing the displacement at the nodes of the beam elements intended

for contact. The sum of the reaction force at these nodes gives the force with which the

switch will spring back i.e. the force that will be experienced by any bonds (if formed)

between the switch and the transmission line.

Fig.3.8. Pull-in time vs. voltage for the cantilever switch. The increase of actuation voltage by

1.2 V reduces the pull-in time substantially and further increase in actuation voltage has no

significant effect on the pull-in time.

As discussed in the previous chapter, linear width profile is optimum for

maximizing the recoil force. Now, in order to reduce pull-in voltage, we take the linear

profile shown in Fig. 3.9(a) and vary its slope in re-designing the switch. The slope is

taken as L

WW

2

12 − where L is the length of the tapering region, 2800 µm in Fig. 3.9(a).

Here, the contact element is of fixed size (600 µm×2000 µm). The thickness was taken as

50 µm and the gap was taken as 8 µm throughout. The w

based on the slope while some minimum and maximum values are obeyed. The actuation

electrode was kept at a distance of 1150 µm from the fixed end. The electrode length was

taken to be 1350 µm in all cases. The contact occurs

µm at the end of the beam.

39 values for the taper angle (slope) of the beam were taken and each case was

simulated using beam FE modeling for pull

(www.comsol.com) for computing the recoil force with 3D model. The entire length of

the contact pad is assumed to be in contact with the transmission line while evaluating the

recoil force. Figure 3.9b shows the computed values for the pull

force. The recoil force is non

profile has negative or positive slope.

Fig.3.9(a) Linearly tapering switch with dimensions similar to [

recoil force with taper angle of the beam with gaps

18

50 µm and the gap was taken as 8 µm throughout. The widths at either end get decided



taken to be 1350 µm in all cases. The contact occurs at the contact element of length 600



simulated using beam FE modeling for pull-in voltage and recoil force and COMSOL

) for computing the recoil force with 3D model. The entire length of


recoil force. Figure 3.9b shows the computed values for the pull-in voltage

force. The recoil force is non-monotonic because beams bend differently when the width

has negative or positive slope.

(a) Linearly tapering switch with dimensions similar to [8]. (b) Pull

per angle of the beam with gaps cg and ag both as 8 µm.

idths at either end get decided



at the contact element of length 600


in voltage and recoil force and COMSOL

) for computing the recoil force with 3D model. The entire length of


in voltage and recoil

monotonic because beams bend differently when the width

]. (b) Pull-in voltage and

(b)

(a)

19

3.5. Contact resistance

After the switch pulls in, the hills of either surface come in contact as shown in Fig. 3.10.

The actual area that bears all the load, denoted by bA , is small compared to the apparent

area of contact. The contact resistance is dependent on bA . In [17], the expression for the

load bearing area is given as

Cb

FA

nH= (3.4a)

where, CF is the contact force, H is the hardness of the material and n is an empirical

index which usually lies between 0 and 1 (commonly 0.1 to 0.3 when contact pressures

are not too small). Thus, it can be said that the apparent area of contact has no effect on

the contact resistance. In fact, in [27], Bromley and Nelson have experimentally showed

that decrease of apparent contact area by many orders of magnitude only had a marginal

effect on the contact resistance. To estimate the load bearing area we make use of a

conservative value of 1n = and the hardness of SS304 is taken from [14] as 2800 MPa.

Again in [12], an expression for the contact resistance is obtained by assuming a circular

contact area. Now, if ρ is the resistivity of the material and the load bearing area is

considered to be a circle with radius, a , the contact resistance can be approximated by

2CR

a

ρ= (3.4b)

This expression is valid when the contacting materials are of the same type. In case of

different materials the contact resistance is given by [28]

1 2

4CR

a

ρ ρ+= (3.4c)

where 1ρ and 2ρ are the resistivities of the metals coming in contact. In our case, the

metals coming under contact are SS304 and Gold (see Section 3.1). Taking the values of

resistivities as 7.2 x 10-7

Ω-m and 2.33 x 10-8

Ω-m [36] for SS304 and gold respectively,

the contact force required for a contact resistance of 1 Ω evaluated from Eqn. (3.4a) and

(3.4c) is 300 µN.

Fig.3.10. Cantilever beam coming into with the contact pad. Inset shows the hills of the

contacting surfaces coming in contact which reduces the actual area of contact.

20

Chapter 4

Results and discussion

4.1. Lumped model

Equation (3.1) gives the governing equation for the dynamics of a lumped system without

damping. With damping Eqn. (3.1) changes to

2

0

2

0

)(2 zg

AVkzzczm

−=++

εɺɺɺ (4.1)

The lumped parameters are evaluated by performing modal analysis on the six designs.

Table 4.1 shows the values of the lumped parameters for the six designs. The damping

ratio obtained for the first mode in Section 3.3 was 3.204 which can be used to evaluate

the damping coefficient c given by kmζ2 for each design. Using these values for the

lumped parameters m , c and k ,Eqn. (4.1) can be solved using Runge-Kutta 4th

order

method [37]. The time response for the six designs is shown in Fig. 4.1.

TABLE 4.1

LUMPED PARAMETERS AND PERFORMANCE CHARACTERISTICS OF THE SIX DESIGNS

Design

No

K (N/m) m (1e-7 Kg) A (µm2) Static pull-in

voltage (V)

From Eqn.

(3.2)

Pull-in time

without

damping

(ms)

Pull-in time

with

damping

(ms)

1 2733.22 2.086 1987042 99.71 0.038 4.40

2 2384.24 2.972 2002200 92.78 0.048 5.42

3 2146.26 3.581 2012553 87.79 0.056 6.04

4 2021.25 3.883 2017580 85.10 0.060 6.43

5 1758.07 4.531 2027711 79.17 0.070 7.18

6 1348.95 5.535 2042869 69.09 0.088 8.44

It can be seen that the values of pull-in times are in the millisecond range and are

in agreement with pull-in times measured during experiments. Thus, lumping gives us a

reasonable model to predict the performance characteristics of the switch.

4.2. Beam model

The performance characteristics obtained from the simulation are given in Table 4.2. It

can be seen that the pull-in voltage gradually comes down as the slope of the beam

increases. The decrease can be attributed to the decrease in stiffness of the beam. The

dynamic pull-in voltage also observes the same trend and is 13-16% lower than the static

pull-in voltage. The decrease in dynamic pull-in voltage as compared to the static pull-in

voltage decreases as the slope is increased. This can be explained by the fact that the net

vibrating mass increases as the slope of the beam increases from negative to positive

value.

21

TABLE 4.2

RESULTS OF SIMULATION OF SIX OPTIMIZED AND RE-DESIGNED SWITCHES

Design

No

Pull-in voltage

(V)

3D model

Pull-in

voltage (V)

1D model

Dynamic pull-in

voltage (V) 1D

model

Pull-in

time (ms)

Lift-off

time (ms)

1 120.94 125.31 105.05 1.79 0.95

2 110.63 113.42 95.98 2.26 1.15

3 103.44 105.76 90.25 2.61 1.30

4 99.69 101.94 87.07 2.79 1.35

5 92.19 94.28 80.87 3.19 1.50

6 81.25 82.38 71.13 3.90 1.75

TABLE 4.3

CONTACT PARAMETERS EVALUATED AT 5V ABOVE THE STATIC PULL-IN VOLTAGE

Design No Actuation voltage (V) Contact force (µN) Contact resistance (Ω)

1 125.94 121.84 1.58

2 115.63 120.26 1.59

3 108.44 119.22 1.60

4 104.69 110.34 1.66

5 97.19 123.81 1.57

6 86.25 111.55 1.65

Fig. 4.1. Time response of the six designs. The sharp rise in the displacement indicates that pull-in has

occurred and the corresponding time is taken as the pull-in time.

There is a significant difference between the pull-in voltage obtained through

simulation and the pull-in voltage measured by experiments (compare Table 4.2 and

Table 3.2). We discuss a few possible reasons for this discrepancy in the next section.

22

RF-MEMS switches are usually operated a few volts above the pull-in voltage.

So, we operate the switches at 5 V above the pull-in voltage to obtain the contact

parameters and evaluate the resistance using equation (3.5a) & (3.5b). The values of

contact force, area and resistances obtained are shown in Table 4.3. It can be seen that the

contact resistance is higher than 1 Ω but below 5 Ω for all the six designs. A simple re-

design is suggested in the following section which addresses this problem.

The pull-in and lift-off times obtained are in the range of a few milliseconds

which itself is an improvement over the switch in [6]. But this switch has mm range

planar dimensions and micron range gaps, giving rise to significant damping forces.

Presence of holes in the structural element of the switch can significantly reduce the

effect of damping on the structure. The effect of holes is discussed in the next section by

making a parametric study of a couple of parameters.

4.3. Discussion

4.3.1. Pull-in voltage discrepancy

It can be seen that there is a considerable difference between the pull-in voltage evaluated

by simulation and experiment i.e. the values of pull-in voltage obtained from experiments

is lower than that obtained by simulation. As mentioned in Section 3.1, conical wire tips

are used to facilitate assembly. This conical shape of the alignments posts gives rise to

clearances that can in turn make the beam tilt about the edge of its resting surface

possible as shown in Fig. 4.2. Since conductive epoxy is used for assembly, this tilt is

likely to remain after curing.

TABLE 4.3

PULL-IN VOLTAGE WHEN ACTUATION GAP IS 3µm and 0.066o TILT IS PRESENT

Config No 3 µm gap 0.066 o

tilt

1 44.69 47.41

2 40.62 41.18

3 37.81 37.33

4 36.56 35.45

5 33.75 31.72

6 29.64 26.03

An initial study suggested that the pull-in voltages of the simulation agree within

8% of the experimental values of pull-in voltage when the gap between the actuation

electrode and the beam, ag , is reduced to 3 µm from 6 µm. The results of simulation with

reduced gap is shown in Table 4.3 and the comparison with experimental results and

simulation with 6 µm gap is given in Fig. 4.3(a) and (b). It can be geometrically verified

that a tilt of approximately 0.14o is sufficient to bring the mid-point of the part of the

switch above the actuation electrode to 3 µm. This makes the average gap between the

switch and the actuation electrode as 3 µm. To facilitate this rotation the required taper on

the alignment posts is just 0.14

techniques used.

Fig. 4.2. (a)The beam tilts about the edge of the resting surface (axis shown) by 0.14

sectional view depicting the beam tilting by

Fig. 4.3. Pull-in voltage for different configurations obtained from experiments and simulation by varying

the actuation gap.

Going further with the assumption that the cantilever tilts during assembly, we try

to vary the tilt of the beam from 0 to 0.14

shown in Fig. 4.4 for design 1

voltage of 47.41 V, which is within 2.5% of the value of pull

experiment. This tilt of 0.066

23

lignment posts is just 0.14o, which is within the tolerances of the fabrication

The beam tilts about the edge of the resting surface (axis shown) by 0.14o.

beam tilting by θ .

in voltage for different configurations obtained from experiments and simulation by varying


t of the beam from 0 to 0.14o. This results in a decrease of pull

design 1. It can be seen that a tilt value of 0.066o

voltage of 47.41 V, which is within 2.5% of the value of pull-in voltage obtained

experiment. This tilt of 0.066o

when applied to all six designs results in pull

(a)

(b)

, which is within the tolerances of the fabrication

. (b) Exaggerated 2D

in voltage for different configurations obtained from experiments and simulation by varying


. This results in a decrease of pull-in voltage as

results in pull-in

in voltage obtained by

results in pull-in voltages

(Table 4.3) which are within 3 to 12% of the pull

experiment. The large dimensions (mm range) of the switch make the pull

the switches extremely sensitive to the value of the tilt that might occur during assembly.

Fig. 4.4. Effect on pull-in voltage with some amount of initial tilt given to the beam which could occur

during the assembly process.

4.3.2. Improvement in contact

The contact resistance obtained is

desired value of 1 Ω was not achieved. Improved contact resistance calls for enhanced

contact force. Force analysis at static equilibrium of a switch under contact suggests that

higher the electrostatic force act

Thus, to increase the electrostatic force on the switch at equilibrium without altering the

geometry of the switch or increasing the operating voltage, we suggest an increase in

contact gap from 4 µm to 6

contact gap is increased to 6

CONTACT FORCE AND RES

Config No. Actuation voltage

1

2

3

4

5

6

24

) which are within 3 to 12% of the pull-in voltage values obtained by

experiment. The large dimensions (mm range) of the switch make the pull


in voltage with some amount of initial tilt given to the beam which could occur

Improvement in contact resistance

The contact resistance obtained is less than 5 Ω for all the six configurations but the

was not achieved. Improved contact resistance calls for enhanced


higher the electrostatic force acting on the beam, the higher will be the contact force.



o 6 µm. Table 4.4 shows the increase in contact force when the

contact gap is increased to 6 µm which leads to contact resistance values of below 1

TABLE 4.4

ONTACT FORCE AND RESISTANCE FOR CONTACT GAP OF 6 µm

Actuation voltage

(Volts)

Contact force (µN) Contact resistance

125.94 678.28

115.63 711.11

108.44 733.05

104.69 708.83

97.19 711.94

86.25 661.73

in voltage values obtained by

experiment. The large dimensions (mm range) of the switch make the pull-in voltage of


in voltage with some amount of initial tilt given to the beam which could occur

for all the six configurations but the

was not achieved. Improved contact resistance calls for enhanced


higher will be the contact force.



shows the increase in contact force when the

m which leads to contact resistance values of below 1 Ω.

Contact resistance

(Ω)

0.67

0.65

0.64

0.65

0.65

0.68

25

4.3.3. Improvement of the Pull-in time

The p-i-n diode and FET switches have switching times in the range of µs. In [18] it is

reported that many MEMS switches have switching times around 2-40 µs. Table 3.2

reveals that the six designs of the switches have pull-in times of a few ms. Thus, there is

need to improve the switching times. Increase in the actuation voltage is a very easy way

to bring down the pull-in time but beyond a few volts the increase of actuation voltage

has almost negligible effect on pull-in time. Moreover we are looking for a switch with

lower actuation voltages thus increasing the actuation voltage is not a good idea. Altering

the geometry of the switch will lead to non-optimal shape of the structural element thus

affecting the recoil force and the pull-in voltage. The Q-factor of the switch has a

significant effect on the switching time. As mentioned in Section 3.3, the Q-factor

obtained for the switches was 0.16 which is a very low value. It is mentioned in [2] that

the switching time decreases considerably for a change in quality factor from 0.2 to 2, but

beyond 2 the effect of increase in Q-factor on pull-in time becomes less predominant. A

very high value of Q-factor increases the settling time for the beam and thus a Q-factor of

1 is recommended. Thus, we aim to improve the Q-factor of the switch in order decrease

the switching time.

It is a common practice to include holes in MEMS structures for the purpose of

releasing of oxides (etch holes) during fabrication and also to decrease the effect of

damping. Eung-Sam Kim et al. in [15] have made a study of effect of holes in a structure

moving like a parallel-plate. The parameters they used for the study are the area ratio

(i.e., percentage area of holes to the area of the plate) and the number of holes. It was

found that increasing the number of holes was more effective than increasing the area

ratio of the holes. Including holes in the area above the actuation electrode will reduce the

electrostatic force acting on the switch we only include the holes in the area above the

contact pad. This would decrease the area available for contact but we know that the area

of contact is dependent only on the contact force and not the apparent area of contact

[27].

The air passing through the holes is modeled using FLUID 138 elements in

ANSYS which are usually used to model fluid flow behavior through short channels.

Another benefit of these elements is that they can be used in conjunction with FLUID

136 elements which we used to model the thin-film. As in [15] we make use of the same

parameters for our study. The area above the contact pad (600 µm ×1400 µm) is taken as

the base area. We vary the number of holes as 1, 4, 9 and 16 and take area ratios as 0.05,

0.1 and 0.25. Usually it is very difficult to get circular holes by etching but we make use

of circular holes to ease the process of node selection for applying the boundary

conditions and new element generation. If we use circular holes, then by selecting a local

coordinate system with origin at the center of the hole and coordinate system of polar

type, we can get the details of all the nodes on the circumference and it is easier to

operate

26

TABLE 4.5

Q-FACTOR FOR THE FIRST MODE

NO OF HOLES/AREA

RATIO

0.05 0.1 0.25

1 0.26 0.30 0.39

4 0.32 0.36 0.44

9 0.41 0.44 0.52

16 0.46 0.51 0.55

TABLE 4.6

PULL-IN TIME FOR THE SWITCH WITH AND WITHOUT HOLES

CONFIG NO WITHOUT HOLES WITH 16 HOLES

1 1.79 0.48

2 2.26 0.61

3 2.61 0.71

4 2.79 0.76

5 3.19 0.87

6 3.90 1.07

Table 4.5 shows that we are able to bring the Q-factor to a value of 0.55 when we

employ 16 holes with an area ratio of 0.25. This value of Q-factor corresponds to a

damping ratio value of 0.9 for the first bending mode. Applying Rayleigh damping with

1ζ and 5ζ as 0.9 and 0.017 respectively we obtain the pull-in times for all the six

configurations (see Table 4.6). Almost one order of magnitude reduction is visible by the

incorporation of holes into the structural element. As mentioned earlier in this section the

use of circular holes is to ease the process of node selection. Now if we replace all

circular holes with squares such that the sides of the squares are tangents to the circle we

get square holes. There will be a slight increase in the area ratio and region of the holes

will not overlap. Thus, giving a smaller value of damping ratio and hence pull-in time.

4.3.4. Recoil force

The geometry and the material of the structural element has a significant effect on the

recoil force. The recoil for the six designs is evaluated as mentioned in Section 3.4. It is

found that the recoil force ranges between 7 and 9 mN. As reported by Ozkeskin [9],

these values of recoil force are sufficient to overcome the problems of stiction and

adhesion. Since the designs are obtained by the method of optimization, any change in

the geometry of the switch will reduce the recoil force and thus, increase the chances of

failure by stiction or adhesion. In the next chapter we introduce a new design in which,

no changes are made to the structural element and still we get an increase in the recoil

force.

27

Chapter 5

Recoil force enhancement with nonlinear stiffness

5.1. Contact bumps

The recoil force of a structure has a positive correlation with the stiffness of the structure

undergoing deformation. Keeping this in mind we try to increase the stiffness of the

structure. But, a higher stiffness leads to a higher pull-in voltage. In order to avoid the

increase in pull-in voltage we reduce the effective length of the cantilever after pull-in

has occurred. Thus, increasing the stiffness and thereby the recoil force. This idea of

nonlinear stiffness is also mentioned in [12] where a cantilever which, when deflected,

lies against a solid guide and thus shortening its effective length and becoming stiffer.

Figure 5.1 shows the cantilever with the solid guide. Since it is difficult to obtain shapes

of solid guide shown in Fig. 5.1, we replace it with a contact bump placed at a suitable

distance from the fixed end.

Fig. 5.1. Cantilever with solid guide which exhibits nonlinear stiffness.

We make use of the system used in [8] where the beam length is taken as 3400

µm, thickness 50 µm, actuation gap 8 µm and contact gap 8 µm. In order to design the

contact bump we displace the point on the beam which is above the end of the actuation

electrode (denoted by tip in this Section) by a value δ . We evaluate the electrostatic

force, eF , occurring on the beam due to this deflection and also the elastic force, elF ,

which tries to oppose the electrostatic force. Figure 5.2 shows the values of eF , elF and

ele FF − for a voltage of 108.5 V which is the pull-in voltage for zero slope switch [8].

The pull-in condition is identified by the following conditions:

1. piele FF δδ => , i.e. electrostatic force is greater than the elastic force.

2. 0)(

>−

= pid

FFd eleδδ

δ.

28

Fig. 5.2. Electrostatic and elastic force acting on the beam for various values of displacements applied to

the tip of the beam. The applied voltage is 108.5 V which is the pull-in voltage for the beam with zero

slope.

Fig. 5.3. Minimum tip displacement required for different slopes of the beam when the pull-in voltage is

applied. A cubic curve fit for the scattered data occurring due to the approximate evaluation of the pull-in

voltage due to the bisection method.

It can be found out from Fig 5.2 that the preceding conditions are satisfied for

2.88δ ≥ i.e. 2.88pi mδ µ= . Figure 5.3 shows the values of api g/δ for varying slopes of

the beam when the pull-in voltage for the configuration is applied. We fit a cubic curve

29

to the resulting data. The scatter in the results is due to the fact that pull-in voltage is

evaluated using the bisection method and thus is not exact but only within a tolerance that

we have specified. It can be seen that for a slope of -28o the minimum tip displacement

required is ag44.0 . For all other configurations the minimum tip displacement required

for pull-in conditions to be satisfied is below ag44.0 . We make use of a value of 0.66a

g

tip displacement which is above ag44.0 and thus involves a factor of safety.

A minimum distance of 5 micro inches (127 µm) should be maintained between

any two components to be placed on the PCB. This comes as a limitation of the PCB

being used. The contact bump is placed at a distance of 1020 µm, i.e., as close as possible

to the start of the actuation electrode (starts at a distances of 1150 µm) so as to have

maximum decrease in the effective length of the beam and the gap between the beam and

the bump, cbg , is taken as the displacement that would occur if the point on the beam at

the end of the actuation electrode (the tip) is made to displace by 0.66a

g . Thus, the

contact bump comes in contact with the substrate once the part of the beam at the end of

actuation electrode deforms by 0.66a

g . As in [8], we apply the Dirichlet boundary

condition to the entire region intended for contact. The point above contact bump is

displaced by cbg . The Recoil force obtained for different slopes of the beam is shown in

Fig. 5.4. It can be seen that the Recoil force increases substantially due to the inclusion of

the bump.

Fig. 5.4. The increase in recoil force due to the inclusion of contact bump. At 0

o slope of the beam the

recoil force is more than double that of the configuration without the bump.

30

It can be easily verified from Fig. 5.4 that for a beam slope of 0o we get an

increase from 11.2 mN to 22 mN which is a 96% increase. Now, this improvement only

enhances the recoil force and the pull-in voltage remains the same. Thus, incorporation of

the contact bump in our existing six configurations will only increase the recoil force and

the pull-in voltage will remain the same. Thus, we reduce the thickness of the beam in the

region shown in Fig. 5.5.

5.2. Redesign of switches

The reduction in pull-in voltage due to the reduced thickness of the switch is shown in

Table 5.1. But this reduction of pull-in voltage comes with a decrease in recoil force that

can be overcome with the addition of the contact bump as shown in Fig. 5.5. The contact

bump is placed at distance of 370 µm, which again is the farthest possible from the fixed

end thus giving the maximum reduction in the effective length. The gap between the

beam and the contact bump, cbg , is taken as the displacement that would occur at

distance of 370 µm along the beam when the tip is displaced by 0.66a

g . As discussed in

Section 4.3.2, the contact gap, c

g , is kept as 6 µm instead of 4 µm in order keep the

contact resistance below 1 Ω.

TABLE 5.1

PULL-IN VOLTAGES FOR EXISTING SWITCH AND THE SWITCH WITH REDUCED THICKNESS IN THE REGION

SHOWN IN FIG. 5.5

CONFIG NO. EXISTING SWITCH [8] REDUCED THICKNESS PERCENTAGE

DECREASE

1 125.31 110.13 13.78

2 113.42 99.34 14.17

3 105.76 92.38 14.49

4 101.94 88.90 14.66

5 94.28 81.93 15.08

6 82.38 71.08 15.90

Fig. 5.5. The side view of the beam with the region where the thickness is to be reduced. All other

dimensions are the same as in [8].

Thickness to be reduced from 50 µm to 44 µm

cg ag

Contact bump

Tip cbg

130 µm

31

TABLE 5.2

RECOIL FORCE FOR EXISTING SWITCH, THE MODIFIED SWITCH WITH REDUCED THICKNESS AND MODIFIED

SWITCH WITH CONTACT BUMP IN mN

CONFIG NO. EXISTING

SWITCH

MODIFIED

SWITCH

(REDUCED

THICKNESS)

MODIFIED

SWITCH WITH

CONTACT BUMP

PERCENTAGE

INCREASE

1 8.51 7.40 14.29 67.92

2 8.55 7.42 14.64 71.23

3 8.47 7.34 14.72 73.79

4 8.40 7.27 14.71 75.12

5 8.18 7.08 14.59 78.36

6 7.67 6.63 14.16 84.62

Table 5.2 shows the recoil force for the existing switch, the modified switch with

reduced thickness, and the modified switch with the reduced thickness and the contact

bump. As expected, due to the reduction in stiffness, the recoil force drops when the

thickness is reduced. It can be seen from Table 5.1 and 5.2 that for configuration 6, there

is a 16% decrease in pull-in voltage and an 84% increase in recoil force when we reduce

thickness and include the contact bump. The thickness can further be reduced from 44 µm

to reduce the pull-in voltage and still have similar recoil force but the manufacturing

feasibility needs to be checked. In fact the thickness can be reduced to as low as 40 µm

and still have a 56.45% (7.67 to 12 mN) increase in recoil force and 25.5% (82.38 V to

61.38V) decrease in pull-in voltage for configuration 6. Further decrease in thickness

needs to be checked for manufacturing feasibility.

TABLE 5.3

RECOIL FORCE COMPARISON FOR EXISTING SWITCH WITH INCREASED CONTACT GAP AND THE MODIFIED

SWITCH

CONFIG NO EXISTING SWITCH MODIFIED SWITCH PERCENTAGE INCREASE

1 12.76 14.29 11.99

2 12.82 14.64 14.20

3 12.70 14.72 15.91

4 12.59 14.71 16.84

5 12.28 14.59 18.91

6 11.51 14.16 23.02

It can be argued that the increase in recoil force is not solely due to the inclusion

of contact bump but also due to the increase in contact gap, c

g , to 6 µm from 4 µm

which facilitates more deformation and thus more recoil force. So, we evaluate the recoil

force for the existing six configurations again with 6c

g mµ= . We can see that there is an

increase 12 – 23% (see Table 5.3) from configuration 1 to 6. Thus, we can claim that the

inclusion of the contact bump gives substantial increase in the recoil force.

The minimum limit of 127 µm distance between the components on the PCB

prevents further decrease in effective length of the cantilever. This limitation can be

overcome by making the contact bump a part of the beam rather than being a separate

32

component attached to the PCB. Figure 5.6 shows the idea of having the contact bump as

a part of the beam.

Fig. 5.6. Switch with contact bump attached to the beam placed at 10 µm distance from the actuation

electrode.

The benefits of having the contact bump on the beam itself are multiple. First, as

mentioned, is the increase in recoil force due to further reduction in the effective length.

Second is the fact the required gap between the bump and the PCB will be more as

farther the location of the bump more will be the deformation before the beam pulls-in.

cg ag

Contact bump Tip cbg

PCB

33

Chapter 6

Summary and Conclusions

Stiction and adhesion are two problems that limit the performance of a micromachined

switch. Using special materials that reduce adhesion and modifying the contacting

surface to eliminate stiction are usually followed. In this work, we provided a new

mechanical design-based approach to solve the problem by incorporating a contact bump

that reduces the effective length of the switch after pull-in has occurred. The new

approach improves the recoil force, the force with which the switch goes out of contact

upon turning off the activation voltage. Increase in stiffness due to the contact bump gave

us the option of reducing the stiffness of the switch by reducing the thickness of the beam

which in turn reduced the pull-in voltage. Contact resistances were evaluated to be of the

value of a few Ω, which is not desired for reliable operation of the switch as the number

of switching cycles increases. A simple redesign of increasing the contact gap was

suggested which brings the contact resistance below 1 Ω. Normal mode summation

method along with the modal projection method was used to obtain the time response of

the switches. The pull-in time of the switch was found to be a few ms, which is a

significant improvement over the previous switches reported by Ozkeskin [6]. The effect

of inclusion of holes to reduce the switching time was discussed. It was found that

increasing the number of holes as well as the area ratio reduced the squeezed-film effect.

Also discussed is that even a minor error in manual assembly may cause substantial

difference in the performance of the switch. This was a result of analyzing the reason for

discrepancy between the simulated and experimental values of the pull-in voltage of the

previous switch. Tilting that occurs during assembly is argued as the reason for

discrepancy and this led a guideline for more accurate assembly.

34

Appendix

A1. Modal projection method in Ansys and input files

Modal projection techniques provide an efficient method for computing damping

parameters for flexible bodies. The Modal Projection Technique is the process of

calculating the squeeze stiffness and damping coefficients of the fluid using the

eigenvectors of the structure. In the modal projection method, the velocity profiles are

determined from the mode-frequency response of the structure.

The basic steps in performing the analysis are as follows:

1. Build a structural and thin-film fluid model and mesh.

2. Perform a modal analysis on the structure.

3. Extract the desired mode eigenvectors.

4. Select the desired modes for damping parameter calculations.

5. Perform a harmonic analysis on the thin-film elements.

6. Compute the modal squeeze stiffness and damping parameters.

7. Compute modal damping ratio and squeeze stiffness coefficient.

8. Display the results.

Steps 4-7 have been automated using the DMPEXT command macro. Step 8 is available

through the MDPLOT command macro.

Fig. A2.1 Pressure distribution for the beam mentioned in Section 3. For the first mode.

Input file for Beam without holes

/batch,list

35

/PREP7

/title, Damping Ratio calculations for a Beam

/com uMKS units

ET, 1,136,1 ! 4-node option, High Knudsen Number

ET,2,45 ! Structural element

s_l=1450 ! Half Plate length (um)

s_w=700 ! Plate width

s_t=50 ! Plate thickness

d_el=6 ! Gap

pamb=.1013 ! ambient pressure (MPa)

visc=1.83e-11 ! viscosity kg/(um)(s)

pref=.1013 ! Reference pressure (MPa)

mfp=64e-3 ! mean free path (um)

Knud=mfp/d_el ! Knudsen number

mp,visc,1,visc ! Dynamic viscosity gap

mp,ex,2,193e3 ! SS304

mp,dens,2,8000e-18

mp,nuxy,2,.29

r,1,d_el,,,pamb ! Real constants - gap

rmore,pref,mfp

! Build the model

rectng,-s_l,s_l,-s_w,s_w ! Plate domain

TYPE, 1

MAT, 1

smrtsize,4

AMESH, all ! Mesh plate domain

esize,,1

type,2

mat,2

real,2

vext,all,,,,,s_t ! Extrude structural domain

nsel,s,loc,x,s_l

nsel,a,loc,y,s_w

nsel,a,loc,y,-s_w

nsel,r,loc,z,-1e-9,1e-9

d,all,pres ! Fix pressure at outer plate boundary

nsel,all

esel,s,type,,1

nsle,s,1

! nsel,u,cp,,1,5

cm,FLUN,node

allsel

nsel,s,loc,x,-s_l

d,all,ux

d,all,uy

d,all,uz

allsel

nsel,s,loc,y,-s_w

nsel,a,loc,y,s_w

d,all,uy

allsel

/solu

antype,modal ! Modal analysis

modopt,lanb,7 ! Extract lowest seven eigenmodes

eqslv,sparse

mxpand,7 ! Expand lowest two eigenmodes

36

solve

fini

/post1

RMFLVEC ! Extract eigenvectors

fini

/solu

DMPEXT,1,1,,'EIG' ! Extract damping ratios for 1st mode

Finish

Input file for Beam with holes

/batch,list

/PREP7

/title, Damping Ratio calculations for a Perforated Beam

/com uMKS units

ET, 1,136,1 ! 4-node option, High Knudsen Number

ET,2,45 ! Structural element

ET, 3,138,1 ! Circular hole option, Hugh Knudsen Number

s_l=1450 ! Half Plate length (um)

s_w=700 ! Plate width

s_t=50 ! Plate thickness

nholes = 4 !Number of holes

Ar=.25 !Area ratio

Ah = Ar*1400*600/nholes !Area of each hole

pi=acos(-1)

c_r= sqrt(Ah/pi) ! Hole radius

d_el=6 ! Gap

pamb=.1013 ! ambient pressure (MPa)

visc=1.83e-11 ! viscosity kg/(um)(s)

pref=.1013 ! Reference pressure (MPa)

mfp=64e-3 ! mean free path (um)

Knud=mfp/d_el ! Knudsen number

mp,visc,1,visc ! Dynamic viscosity gap

mp,visc,3,visc ! Dynamic viscosity holes

mp,ex,2,193e3 ! Stainless steel SS304

mp,dens,2,8000e-18

mp,nuxy,2,.29

r,1,d_el,,,pamb ! Real constants - gap

rmore,pref,mfp

r,3,c_r,,,pamb ! Real constants - hole

rmore,pref,mfp

! Build the model

rectng,-s_l,s_l,-s_w,s_w ! Plate domain

*do,i,1,sqrt(nholes)

*do,j,1,sqrt(nholes)

cyl4, 640+150+(i-1)*300,-700+(j-1)*700+350,c_r

*enddo

*enddo

ASBA, 1, all

TYPE, 1

MAT, 1

smrtsize,2

AMESH, all ! Mesh plate domain

! Begin Hole generation

*do,i,1,2

*do,j,1,2

37

nsel,all

*GET, numb, node, , num, max ! Create nodes for link elements

N, numb+1,640+150+(i-1)*300,-700+(j-1)*700+350

N, numb+2,640+150+(i-1)*300,-700+(j-1)*700+350, s_t

TYPE,3

MAT, 3

REAL,3

NSEL, all

E, numb+1, numb+2 ! Define 2-D link element

ESEL, s, type,,1

NSLE,s,1

local,11,1,640+150+(i-1)*300,-700+(j-1)*700+350

csys,11

NSEL,r, loc, x, c_r ! Select all nodes on the hole circumference

NSEL,a, node, ,numb+1

*GET, next, node, , num, min

CP, (2*i-2)+j, pres, numb+1, next

nsel,u,node, ,numb+1

nsel,u,node, ,next

CP, (2*i-2)+j, pres,all !Coupled DOF set for constant pressure

csys,0

*enddo

*enddo

! End hole generation

esize,,1

type,2

mat,2

real,2

vext,all,,,,,s_t ! Extrude structural domain

nsel,s,loc,x,s_l

nsel,a,loc,y,s_w

nsel,a,loc,y,-s_w

nsel,r,loc,z,-1e-9,1e-9

d,all,pres ! Fix pressure at outer plate boundary

nsel,all

esel,s,type,,3

nsle,s,1

nsel,r,loc,z,s_t

d,all,pres,0 ! P=0 at top of plate

dlist,all

esel,s,type,,1

nsle,s,1

nsel,u,cp,,1,4

cm,FLUN,node

allsel

nsel,s,loc,x,-s_l

d,all,ux

d,all,uy

d,all,uz

allsel

nsel,s,loc,y,-s_w

nsel,a,loc,y,s_w

d,all,uy

allsel

fini

/solu

38

antype,modal ! Modal analysis

modopt,lanb,7 ! Extract lowest seven eigenmodes

eqslv,sparse

mxpand,2 ! Expand lowest seven eigenmodes

solve

fini

/post1

RMFLVEC ! Extract eigenvectors

fini

/solu

DMPEXT,1,1,,'EIG' ! Extract damping ratio for 1st mode

Finish

39

A2. Matlab codes

P1. Finds out the pull-in voltage for the six configurations using bisection method.

clf % Clear graphics window

clear all % Clear all variables

clc % Clear command window

hold off % No hold on the graphics window

% This script needs the following scripts to run

% matcut.m, veccut.m, thickness.dat

% veccut.m

% Enter data

length_taper=2900;

length_in = 3400; % in micron units

load thickness.dat

Y_in = 193E3/(1-.29^2); % Young's modulus of 155 GPa

% Initial gap in microns

g0 = 6; % Actuation gap

Cg0 = 4;% Contact gap

%----------------------------------------------------------------

----------

%---------------------Configuration details----------------------

----------

W1=[2231.34 1769.29 1462.58 1308.43 1000.53 538.48]; % Width at

fixed end

W2=[572.14 1041.69 1353.78 1510.39 1823.33 2292.88]; % Width at

free end

theta=(W2./2-W1./2);

theta=(1/length_in)*theta*180/pi; % Taper angle

V_PI=zeros(1,size(theta,2));

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% READ INPUT from files

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

nodenos = 290*2+1; % Number of nodes

node(:,1) = [1:1:nodenos]';

node(:,2) = [0:length_taper/(nodenos-1):length_taper]';

node(:,3) = zeros(nodenos,1);

elem(:,1) = [1:1:nodenos-1]';

elem(:,2) = [1:1:nodenos-1]';

elem(:,3) = [2:1:nodenos]';

elem(:,4) = Y_in*ones(nodenos-1,1);

% Read in displacement boundary condition data from the file

disp.dat

load dispbc.dat

load thickness.dat

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% PRE-PROCESSING

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

NNODE = size(node,1); % Number of nodes

40

% Nodal coordinates

nx = node(:,2);

ny = node(:,3);

NELEM = size(elem,1); % Number of elements

ncon = elem(:,[2 3]); % Nodal connectivity

E = elem(:,4);

t = thickness;

for i =1 : NELEM

if nx(ncon(i,2)) <= 500

t(i) = 44;

elseif nx(ncon(i,2)) <= 1950

t(i) = 44;


t(i) = 50;


t(i)= 46;

else

t(i) = 50;

end

end

% Applying Boundary conditions

Nfix = size(dispbc,1);

j = 0;

for i = 1:Nfix,

j = j + 1;

dispID(j) = (dispbc(i,2)-1)*3+dispbc(i,3);

end

[dispID sortIndex] = sort(dispID);

dispVal=[0 0 0];

% Compute the lengths of the elements

for ie=1:NELEM,

eye = ncon(ie,1);

jay = ncon(ie,2);

L(ie) = sqrt ( (nx(jay) - nx(eye))^2 + (ny(jay) - ny(eye))^2

);

end

% Initialization

max_iter = 50;

V=60;

for configNo=1:6

configNo

w1=W1(configNo);

w2=W2(configNo);

for i=1:NELEM

b1=w1+(w2-w1)/length_in*length_in/340*(50+i-1);

b2=w1+(w2-w1)/length_in*length_in/340*(50+i);

width(i,1)=(b1+b2)/2;

end

A = width.*t; %cross-section area of the beam

Inertia= width.*t.^3/12; %Area moment of inertia of the beam

% Arrange force information into a force vector, F

F = zeros(3*NNODE,1); % Initialization

41

U = zeros(3*NNODE,1);

U_prev = U;

flag=10;

V_inc=20;

while (flag~=0)

if flag==1

V=V+V_inc;

elseif flag==2

V=V-V_inc;

end

delta_U = 100.0; % Initialized to a large value

fV = 8.854e-12 * V^2/2*1E6; % Force per unit area per

fV2 = 8.854e-12 * 0^2/2*1E6;

for relax_iteration = 1:max_iter,

F = zeros(3*NNODE,1); % Initialization to zero.

if abs(delta_U) < 1e-6,

flag =1;

break;

end

% Computation of electrostatic forces on nodes

for i = 101:390

gi = g0 - U( 3*(i-1) + 2);

gj = g0 - U( 3*i + 2 );

gave = (gi+gj)/2;

eforce = fV * width(i) * L(i) / gave^2;

eforce = eforce * (1 + 0.65*gave/width(i));

F( 3*(i-1) + 2 ) = F( 3*(i-1) + 2 ) + eforce/2;

F( 3*i + 2 ) = F( 3*i + 2 ) + eforce/2;

end

for i = 417:540

gi = Cg0 - U( 3*(i-1) + 2);

gj = Cg0 - U( 3*i + 2 );

gave = (gi+gj)/2;

eforce = fV2 * width(i) * L(i) / gave^2;

eforce = eforce * (1 + 0.65*gave/width(i));

F( 3*(i-1) + 2 ) = F( 3*(i-1) + 2 ) + eforce/2;

F( 3*i + 2 ) = F( 3*i + 2 ) + eforce/2;

end

%

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% SOLUTION

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Call fembeam.m to solve for the following.

% Deflections: U

[U,R] =

fembeam(A,L,E,nx,ny,ncon,NELEM,NNODE,F,dispID,dispVal,Inertia

);

delta_U = max( abs(U(2:3:3*NNODE)) ) - max(…

abs(U_prev(2:3:3*NNODE)) );

U_prev = U;

end

42

% Bisection method

if relax_iteration==max_iter

V_inc=V_inc/2;

flag=2;

if V_inc<=1e-4

flag=0;

end

end

end

V_PI(configNo)=V;

end

plot(theta,V_PI);

P2. Evaluates the pull-in time for the six configurations clear all

hold off

clc

W1=[2231 1769 1463 1308 1000 538];

W2=[572 1042 1354 1510 1823 2293];

VDC_all=[125.31;113.42;105.76;101.94;94.28;82.38]; % Static pull-

in voltages

for configNo=1:6

VDC=VDC_all(configNo,:);

pullin_time=zeros(1,size(VDC,2));

for volt_index=1:size(VDC,2)

global GK GM GK1 GC evec1 L B ic voltage v_time FM Flag

freq H...

vdc vac omega gap e_force_elems e_force_contact_elems

force_bump_elem ncnt...

kc NInd nu Contactgap NNODE NELEM

sigma_r=-0;% residual stress

NELEM = 290*2; % Number of elements

w1=W1(configNo);

w2=W2(configNo);

B=zeros(NELEM,1);

length_tot=3400;

length=length_tot-500;

% Width profile of switch

for i=1:NELEM

b1=w1+(w2-w1)/length_tot*((length_tot-

length)/(length/NELEM)…

+i-1)*length/NELEM;

b2=w1+(w2-w1)/length_tot*((length_tot-…

length)/(length/NELEM)+i)*length/NELEM;

B(i)=(b1+b2)/2;

end

nu=.8;% Deformation dependent damping

NInd=3;

kc=4.7e11; %Contact stiffness

load thickness.dat;

H = thickness; % Thicknesses of beam elements in microns

43

E = 193e3/(1-.29^2); % Units: microN/micron^2

rho = 8000e-18; % Units: microKg/micron^3

vdc =VDC(volt_index);

vac = 0.0;

omega = 1e6*2*pi;

gap = 6;

Contactgap=4;

bumpNode=51;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

NNODE = NELEM + 1;

nx = [0:length/NELEM:length];

ny = zeros(size(nx));

% Nodal connectivity

for ielem=1:NELEM,

ncnt(ielem,1) = ielem;

ncnt(ielem,2) = ielem+1;

end

% Thickness of the beam

for i =1 : NELEM

if nx(ncnt(i,2)) <= 500

H(i) = 50;

elseif nx(ncnt(i,2)) <= 1950

H(i) = 44;


H(i) = 50;


H(i)= 46;

else

H(i) = 50;

end

end

e_force_elems =[]; % Elements on which electrostatic

force is applied

e_force_contact_elems=[];% Elements on which contact

force is applied

for i = 1:NELEM

if H(i) == 44

e_force_elems = [e_force_elems i];

elseif H(i) == 46

e_force_contact_elems =[e_force_contact_elems i];

end

end

force_bump_elem=bumpNode-1;

% Displacement boundary condition specification

dispID = [1 2 ];

dispVal = [0 0];

% Lengths of elements

for ie=1:NELEM,

eye = ncnt(ie,1);

jay = ncnt(ie,2);

44

L(ie) = sqrt ( (nx(jay) - nx(eye))^2 + (ny(jay) -

ny(eye))^2 );

end

% Call the fem m-file

% K = stiffness matrix

% M = inertia (mass) matrix

[K,

M]=fem2(B,H,L,E,rho,nx,ny,ncnt,NELEM,NNODE,dispVal,dispID,0);

% Compute mode shapes and natural frquencies

% evec = eigenvector matrix

% eval = diagonal matrix containing the eigenvalues

D = inv(K)*M;

[evec eval] = eig(D);

% Take only first few modes for simulation

FM = 5; % Number of modes considered

% Normalize mode shapes so that the maximum transverse

% displacement is unity.

sz = size(evec,2);

evec1=zeros(sz,FM);

for i=1:FM

evec1(:,i)= evec(:,i) / max(abs(evec(1:2:sz,i)));

end

% Generalized inertia matrix

GM = evec1'*M*evec1;

% Generalized stiffness matrix

GK = evec1'*K*evec1;

GK1=evec'*K*evec;

freq = sqrt(diag(GK)./diag(GM));

% Damping matrix by Rayleigh damping

dc1 = 3.204;

dc2 = .14e-2;

w1 = freq(1);

w2 = freq(FM);

alpha = 2*w1*w2*(dc1*w2-w1*dc2)/(w2^2-w1^2);

beta = 2*(dc2*w2-dc1*w1)/(w2^2-w1^2);

C=alpha*M+beta*K;

GC=evec1'*C*evec1;

T0 = 0.0;

Tfinal = Ncycles * 2*pi/freq(1);

Tfinal = 10e-3;

Tspan = [T0:Tfinal/100: Tfinal];

options = odeset('RelTol',1e-4,'AbsTol',[1e-4 1e-4 1e-

5]);

Y0 = zeros(1,2*FM);

ic = 0; % Index used to keep track of voltage as a

function of time

tic;

[T, Y] = ode45('stfew',Tspan, Y0);

toc;

% Take FM modes

idvec = 1:2:2*FM;

x_soln = Y(:,idvec);

45

x_soln = evec1(:,1:FM)*x_soln';

figure(8)

tip=2*(ncnt(e_force_contact_elems(size(e_force_contact_el

ems,2)),…

2))+1;

plot(T, x_soln(tip,:),'Linewidth',2.5);

xlabel('Time (sec)');

ylabel('Displacement of end of Contact part (\mum)')

title('Time response')

x_dot = evec1(:,1:FM)*Y(:,2:2:2*FM)';

figure(5)

[XEnd TEnd]=size(x_soln);

for i=1:TEnd

plot(-x_soln(1:2:XEnd,i));

axis([0 NELEM+1 -gap 2]);

F(i)=getframe;

end

end

grid on

pullin_time(volt_index)=T(size(T,1));

end

figure(configNo)

plot(VDC,[pullin_time]*1e3,':','Linewidth',2);hold on;

plot(VDC,pullin_time*1e3,'.r');

xlabel('Voltage (Volts)','fontsize',12);

ylabel('Pull-in time (milli-seconds)','fontsize',12);

title('Pull-in time variation with voltage','fontsize',12);

grid on

P3. Lumped system dynamics

global V_in e A g_0 k b m R;

K=[2733.22 2384.24 2146.26 2021.25 1758.07 1348.95] ; %Lumped

stiffness

M=[2.086234 2.971885 3.520828 3.882725 4.530887 5.534692]*1e-7;%

Lumped mass

Ar=[1987042 2002200 2012553 2017580 2027711 2042869]*1e-12;% Area

of… actuation

styl =['-:-:-:']

for configNo = 1:6;

k =K(configNo);

m = M(configNo);

b=2*3.204*sqrt(k*m); % Damping coefficent obtained from

damping ratio = 3.204

A =Ar(configNo);

g_0=6e-6; % Actuation gap

e=8.854e-12;

V_pullin=sqrt(8*k*g_0^3/27/e/A) % Static pull-in voltage

obtained from… formula

[~,a]=size(V);

46

for i=1:a

V_in=V(i);

[T,Y]=ode45(@dynamicpullin,[0 25e-3],[0 0]);

plot(T*1e3,[Y(:,1)]*1e6,[styl(configNo)

'k'],'Linewidth',3);

hold on

end

grid on

end

xlabel('Time (seconds)','fontsize',12)

ylabel('z (\mum)','fontsize',12)

function dy=dynamicpullin(t,y)

global V_in e A g_0 k b m R;

dy=zeros(2,1);

dy(1)=y(2);

dy(2)=(V_in^2*e*A/2/(g_0-y(1))^2-k*y(1)-b*y(2))/m;

P4. Recoil force evaluation

clear all % Clear all variables

clc % Clear command window

hold off % No hold on the graphics window

%----------------------------------------------------------------

----------


----------

length_taper=2900;

disp =[];

length_in = 3400; % in micron units

W1=[2231.34 1769.29 1462.58 1308.43 1000.53 538.48];

W2=[572.14 1041.69 1353.78 1510.39 1823.33 2292.88];

theta=(W2./2-W1./2);

theta=(1/length_in)*theta*180/pi;

Y_in = 193E3/(1-.29^2); % Young's modulus of 155 GPa

% Initial gap in microns

g0 = 6;

gc = 4;

load thickness.dat


RecoilForce=zeros(1,size(theta,2));

RecoilForceWithBump=zeros(1,size(theta,2));

nodenos = 291; % Number of nodes (do not change this!)

node(:,1) = [1:1:nodenos]';

node(:,2) = [0:length_in/(nodenos-1):length_in]';

node(:,3) = zeros(nodenos,1);

elem(:,1) = [1:1:nodenos-1]';

elem(:,2) = [1:1:nodenos-1]';

elem(:,3) = [2:1:nodenos]';

elem(:,4) = Y_in*ones(nodenos-1,1);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

47

% PRE-PROCESSING

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Identify the number of nodes, X and Y Coordinates of the nodes

NNODE = size(node,1);

nx = node(:,2);

ny = node(:,3);

% Identify the number of elements and form an element

connectivity array,

% the cross-section and Young's modulus arrays.

NELEM = size(elem,1);

ncon = elem(:,[2 3]);

E = elem(:,4);

t = thickness;

% Compute the lengths of the elements

for ie=1:NELEM,

eye = ncon(ie,1);

jay = ncon(ie,2);

L(ie) = sqrt ( (nx(jay) - nx(eye))^2 + (ny(jay) - ny(eye))^2

);

end

for i =1 : NELEM

if nx(ncon(i,2)) <= 500

t(i) = 40;


t(i) = 40;


t(i) = 50;


t(i)= 46;

else

t(i) = 50;

end

end

for configNo=1:size(theta,2)


j = 0;

for i = 1:Nfix,

j = j + 1;

dispID(j) = (dispbc(i,2)-1)*3+dispbc(i,3);

end

fixNode=291-20-1-60;

for i=1:61

j=j+1;

dispID(j)= (fixNode-1)*3+2;

fixNode=fixNode+1;

end

[dispID sortIndex] = sort(dispID);

dispVal=[0 0 0 g0*ones(1,61)];


j = 0;

for i = 1:Nfix,

j = j + 1;

48

dispID1(j) = (dispbc(i,2)-1)*3+dispbc(i,3);

end

dispID1(j+1)=3*(195-1)+ 2;

dispVal1=[0 0 0 g0*.66];


j = 0;

for i = 1:Nfix,

j = j + 1;

dispID2(j) = (dispbc(i,2)-1)*3+dispbc(i,3);

end

dispVal2=[0 0 0];

w1 = W1(configNo);

w2 = W2(configNo);

for i=1:NELEM

b1=w1+(w2-w1)/340*(50+i-1);

b2=w1+(w2-w1)/340*(50+i);

width(i,1)=(b1+b2)/2;

end

A = width.*t; %cross-section area of the beam

Inertia= width.*t.^3/12; %Area moment of inertia of the beam

F = zeros(3*NNODE,1); % Initialization

[U,R] =

fembeam(A,L,E,nx,ny,ncon,NELEM,NNODE,F,dispID,dispVal,Inertia);

RecoilForce(configNo)=sum(R(4:size(R,2)));

[U,R] =

fembeam(A,L,E,nx,ny,ncon,NELEM,NNODE,F,dispID1,dispVal1,Inertia);

dispID2(4)=(3*(38-1)+2);

dispVal2(4)=U(3*(38-1)+2);

disp = [disp U(3*(38-1)+2)];

fixNode=291-60-20-1;

j=size(dispID2,2);

for i=1:61

j=j+1;

dispID2(j)= (fixNode-1)*3+2;

fixNode=fixNode+1;

end

dispVal2=[dispVal2 g0*ones(1,61)];

[U,R] =

fembeam(A,L,E,nx,ny,ncon,NELEM,NNODE,F,dispID2,dispVal2,Inertia);

RecoilForceWithBump(configNo)=sum(R(5:size(R,2)));

clear dispID dispID1 dispID2 dispVal dispVal1 dispVal2

end

plot(1:6,RecoilForce*1e-3,'s-k',1:6,RecoilForceWithBump*1e-3,'o-

k','Linewidth',2.5);

legend('Without Bump','With Bump','Location','Northwest');

xlabel('Taper angle (Degrees)');ylabel('Recoil force (mN)')

grid on

49

Abstract— The focus of this paper is on analysis, optimization,

re-design, fabrication, and testing of an existing contact type

Radio Frequency Micro-Electro-Mechanical Systems (RF-

MEMS) switch that has high power-handling (250W) capability.

Improved performance characteristics in terms of higher recoil

force and lower actuation voltage are obtained while ensuring

moderate contact force and small switching times. In order to

fulfill the high power-handling capabilities, the switch is

assembled on the Printed Circuit Board (PCB) and is made of

stainless steel. The Recoil force is evaluated using finite element

(FE) beam model as well as 3D FE modeling while the other

performance characteristics such as pull-in voltage, pull-in time,

and contact force are evaluated using a combination of the FE

model, relaxation scheme, and bisection method. Structural

optimization of the switch, in the framework of calculus of

variations, gave a nearly linear tapering profile of the beam,

which would maximize the recoil force. It was noted that there is

significant decrease in pull-in voltage because of the optimum

profile without any compromise in recoil force. Six switches with

optimized width profile were fabricated and tested for pull-in

voltage and pull-in time, the simulated and measured

characteristics are compared, and reasons for discrepancies were

analyzed pointing to improvements in modeling and fabrication.

Index Terms— EDM, optimization, photochemical etching,

pull-in voltage, recoil force, RF-switch, and stainless steel

I. INTRODUCTION

icromachined radio-frequency (RF) switches, known as

RF MEMS (microelectromechanical systems) switches,

have received much attention from academia and industry ever

since the MEMS field came into research focus [1] & [2]. The

importance of RF MEMS switches is underscored by the fact

that they can be co-located with digitally controllable circuit

elements with a small footprint on the chip. Among different

types of actuation used, electrostatic actuation is the most

common because of its low power consumption. While linear

behavior, low power consumption, low insertion loss, high

isolation, and low manufacturing cost are benefits of

electrostatic micromechanical switches, there is room for

improvement in power handling capacity, switching times,

reliability, and actuation voltages as compared to purely

electronic switches [3] & [4]. Switches made of silicon have

power handling capacity of tens of W and beyond this they

tend to fail by stiction or adhesion. Also environmental factors

require silicon switches to be packaged, which proves to be

expensive. Keeping these reasons in mind, metals are used as

the structural element in micro-switches [5]. Steel was

assembled on a printed circuit board (PCB) [6] and the switch

was shown to have up to 250 W of dissipative power capacity

and occupied a small footprint of 6 mm2. Here, the actuation

voltage under cold-switching condition was over 300 V,

actuation voltage required for the functioning (hot-switching

conditions) of this switch was 130 V, and the actuation time

was in ms range.

Increasing actuation (i.e., switching) voltage reduces the

switching time. But it also increases the contact force that may

cause increased adhesion and wear [7]. Therefore, Pt-Rh was

used on the contact pads in [4] and [6]. Although it helped, the

problem remains that Pt-Rh is expensive and makes fabrication

and assembly difficult. In this paper, we address the adhesion

problem with a mechanical approach. That is, we increase the

recoil or spring-back force of the switch so that the switch can

spring back upon turning off the actuation voltage by breaking

any unwanted fused connections formed due to stiction or

adhesion. Thus, we not only reduce the actuation voltage but

also alleviate problems associated with adhesion and wear.

II. PROBLEM STATEMENT AND THE SCOPE OF WORK

The model of the RF-switch considered is depicted in Fig. 1.

This switch, also called a relay switch, closes the RF

transmission line upon actuation. The actuating electrode at the

bottom is located away from the contact pad so that the beam

of the switch does not contact the actuation electrode. Figure 2

shows a solid model of an RF switch [6] in which four micro

rods serve as fins to dissipate heat while handling high power.

The performance of the switch is improved but the fabrication

becomes complex.

In order to re-design the switch we consider modifying the

mechanical element of the switch without increasing the

complexity or cost of fabrication. For re-designing the

mechanical element, we use the width profile of the beam as

the design variable and the recoil force as the objective

function. A constraint on the volume is applied so that the size

of the switch remains small and within 6 mm2 area. The

designed switch should be amenable for micro electro

discharge machining (µEDM) and PCB assembly as was the

case with switched reported in [4],[6] & [8].

An Optimized Steel Micro-switch for

Low Switching Voltage and High Recoil Force

Subhajit Banerjee, Gaurav Nair, Fatih M. Ozkeskin, Yogesh Gianchandani, and

G. K. Ananthasuresh

M

A3. Manuscript in preparation for possible submission to a journal

50

Fig. 1. Schematic of a resistive switch assembled onto a printed circuit board

(PCB). The contact pad closes the RF transmission lines. The actuation and

contact gaps are exaggerated to show the arrangement of various elements

clearly

Fig. 2. Solid model of the steel switch with RF transmission lines and

electrodes on a PCB [6].

III. ANALYSIS

A. Pull-in voltage

By referring to Fig. 3 that depicts a simplified version of the

switch, we write Euler-Bernoulli beam equation that governs

its static deformation as

( )

22 2

0

2 2 2

0 for

1 0.65 for2

0 for

a

x d

bd d w wEI d x d l

bdx dx g w

d l x L

ε φ

<

= + ≤ ≤ +

− + < ≤

ɶ (1)

where ( )w x is the transverse displacement of the beam,

( )2

/ 1E E ν= −ɶ the biaxial modulus with E indicating the

Young’s modulus and ν the Poisson’s ratio, I the area

moment of inertia, 0

ε the permittivity of free space, b the

width of the beam, a

g the gap beneath the beam, d the

distance from the fixed end of the cantilever beam to the start

of the actuation electrode, ϕ the applied electric potential

between the beam and the actuation electrode, and L the total

length of the beam. The term on the right hand side is the

transverse electrostatic force per unit length of the beam that

deflects the beam [9]. Parallel-plate approximation with

fringing field approximation [10] is used for modeling the

electrostatic force acting on the beam.

Equation (1) cannot be solved analytically because of the

nonlinear force term. Finite element modeling is used to

discretize Eq. (1) and then solved iteratively starting with the

initial zero deformation. It is a standard method to solve this

equation to find the pull-in voltage at which the beam would

snap down catastrophically because a stable solution ceases to

exist. There is a small difference here because the beam is

shown to be slightly thicker at the free end with contact gap

equal toc a

g g< . Hence, the contact gap can be closed before

the portion of the beam above the actuating electrode actually


Fig. 3. Simplified model of the switch for beam FE analysis.

The pull-in voltage calculated by solving Eq. (1) is shown in

Fig. 4 as a function of a

g and for different values ofc

g . It can

be seen that varying c

g makes little difference because the

overhanging part of the beam beyond actuating electrode

simply moves down with zero curvature until contact happens.

To validate the results of the beam analysis, 3D finite element

modeling was used to solve the coupled problems of

electrostatics and elastostatics in the CoventorWare software

(www.cowentorware.com). Three instances of this are shown

in Fig. 4 to confirm that beam modeling is reasonable. We use

beam modeling here because it helps in design and

optimization as shown in the next section. In addition to

estimating the pull-in (and hence actuation) voltage, we also

need contact and recoil forces. Contact force is the force felt

by the contacting portion of the beam with the actuation

voltage present. The recoil force, as noted earlier, is defined as

the force with which the beam would spring back up after the

actuation voltage is set to zero. This is shown in Figs. 5a-b.

These two forces can be estimated using 3D FE modeling as

well as beam modeling; of course, the latter would make

simplifying assumptions to get an analytical formula to be used

in design optimization.

Fig. 4. Comparison of pull-in voltage for different gaps in beam FE code and

Coventorware’s CoSolve.

B. Contact resistance

The reliability of contact type RF-MEMS switches depends

on the contact force and contact resistance at the interface of

materials. The contact resistance is a result of the surface

roughness at the interface. Very low value of surface area of

contact implies very high value of contact resistance. In [11], it

is mentioned that RF-MEMS switches tend to fail when

x

d

l

L

φ

51

contact resistance is about 5-10 Ω. In general, contact

resistance below 1 Ω is required for reliable operation of the

switch.

Fig. 5. Simplified models for computing (a) the contact force, and (b) recoil

force. Contact force is the reaction force experienced by the beam when it is

pulled down with an actuation voltage. Recoil force is the force with which

the beam springs back up upon turning off the actuation voltage.

After the switch pulls in, the hills of either surface come in

contact as shown in Fig. 6. The actual area that bears all the

load, denoted by b

A , is small compared to the apparent area of

contact. The contact resistance is dependent on b

A . In [12],

the expression for the load bearing area is given as

C

b

FA

nH= (2a)

where, C

F is the contact force, H is the hardness of the

material and n is an empirical index which usually lies

between 0 and 1 (commonly 0.1 to 0.3 when contact pressures

are not too small). Thus, it can be said that the apparent area of

contact has no effect on the contact resistance. In fact, in [13],

Bromley and Nelson have experimentally showed that

decrease of apparent contact area by many orders of magnitude

only had a marginal effect on the contact resistance. To

estimate the load bearing area we make use of a conservative

value of 1n = and the hardness of SS304 is taken from [14] as

2800 MPa. Again in [12], an expression for the contact

resistance is obtained by assuming a circular contact area.

Now, if ρ is the resistivity of the material and the load bearing

area is considered to be a circle with radius, a , the contact

resistance can be approximated by

2C

Ra

ρ= (2b)

Fig. 6. Cantilever beam coming into with the contact pad. Inset shows the

hills of the contacting surfaces coming in contact which reduces the actual

area of contact.

This expression is valid when the contacting materials are of

the same type. In case of different materials the contact

resistance is given by [15]

1 2

4C

Ra

ρ ρ+= (2c)

where 1

ρ and 2

ρ are the resistivities of the metals coming in

contact. In our case, the metals coming under contact are

SS304 and Gold (see Section V). Taking the values of

resistivities as 7.2 x 10-7

Ω-m and 2.33 x 10-8

Ω-m for SS304

and Gold respectively, the contact force required for a contact

resistance of 1 Ω evaluated from Eqn. (2a) and (2c) is 300 µN.

C. Pull-in time

Fast switching is a desirable feature for RF-MEMS switches.

For power-handling requirements of 20-50 mW the switching

time for most of the RF-MEMS switches in existence is around

2-40 µs [16]. But here, we aim for switching times in the range

of few ms. It is mentioned in [2] that the switching time

decreases considerably for a change in quality factor, Q , from

0.2 to 2, but beyond 2 the effect of increase in Q-factor on

pull-in time becomes less predominant.

The beam is modeled using beam elements [17] and solved

using the modal summation method. It is mentioned in [18]

that to capture the dynamics of a beam in flight at least five

modes are required but if contact occurs we need to take at

least ten modes to ensure accurate results [19].

To estimate the damping, we use the modal projection

method [20] to evaluate the damping coefficient, ς . The

structural element is modeled in ANSYS (www.ansys.com) as

a cantilever with length 2900 µm,1400 µm width and 50 µm

thickness (similar to configuration 3 mentioned in section V)

using SOLID45 elements with properties of SS304. The fluid

film thickness is assumed to be 6 µm and the fluid is modeled

using FLUID136 elements with properties of air. The beam is

assumed to be open at three sides that are not fixed and closed

at the side that is fixed. We obtain the damping coefficient for

the first mode, 1

ς , as 3.204 and the fifth mode, 5

ς , as 0.0014.

This value of 1

ς corresponds to a Q-factor value of 0.16 at

resonance which is a very conservative value for this design.

This value of Q-factor can be easily improved to a higher

value by introduction of holes in the beam that reduces the

effect of squeeze film damping. Proportional damping is

applied in the form of Rayleigh damping. The damping matrix,

C , is evaluated as M Kα β+ , where M and K are mass and

stiffness matrices and α and β are constants and obtained by

fixing the damping ratio, ς , equal to 3.201 for the first and

0.0014 for the fifth [21]. The electrostatic force applied is the

same as in Eqn 1. The dynamic response is given by

eMw Cw Kw F+ + =ɺɺ ɺ (3)

where, e

F is the electrostatic force on the beam given by the

right hand side of Eqn (1). Equation (3) is solved using

ODE45 solver of Matlab (www.mathworks.com). As the input

voltage is increased, it is observed that the switch first closes

when the voltage is 110.8 V. This is taken as the dynamic pull-

52

in voltage. Increasing the voltage beyond this value results in

decrease of the pull-in time as shown in Fig. 7.

Fig. 7. Pull-in time vs. voltage for the cantilever switch.

IV. OPTIMIZATION AND RE-DESIGN

A. Optimum profile for the width of the beam

As discussed in Section 1, maximizing the recoil force

alleviates the problems associated with stiction and adhesion

in switches. While it is possible to pose this as a general

structural optimization problem to maximize the recoil force

and minimizing the actuation voltage, here we follow a simpler

approach. We vary only the width profile of the beam as it

does not warrant in changes in prototyping used to make the

switch shown in Fig. 2 by Ozkeskin and Gianchandani [4] &

[8]. We consider only the elastic behavior rather than coupled

behavior involving mechanical and electrostatic domains. With

these simplifications, we pose the optimization problem in the

framework of calculus of variations.

( )2

0

recoil

1

0

2 max

0

3 2

max

1Minimize

Subject to

: 0

: 0

Data: /12, , , / (1 ) , , , ( ), ( )

L

b

L

l u

L

l u

L xdx

b

F E

M mdx

b E

t b dx V

t t E E L V M x m x

α

α

α ν

−

=∆

Λ ∆ − ≤

Λ − ≤

= ∆ = −

∫

∫

∫

ɶ

ɶ

ɶ

(5)

where the expression for was obtained by computing the

reaction force for an applied displacement of the tip using

Clayperon’s energy theorem [22], i.e., half the work done by

the external force is equal to the strain energy of an elastic

body at static equilibrium.

( )

( )

22

recoil

0

recoil recoil 2

0

1

2 2

L

L

F L xdx

b EF F

E L xdx

b

α

α

−

∆∆ = ⇒ =

−

∫

∫

ɶ

ɶ (6)

The width ( )b x is the optimization variable. 1

Λ is the estimate

for the Lagrange multiplier corresponding to the deflection

constraint. Here, l

M is the bending moment in the beam for the

electrostatic load for the applied potential, φ . u

m is the

bending moment due to an unit load applied at the tip. The

expression for the deflection of the tip indicated in the first

inequality constraint can be obtained by using the unit dummy

load theorem from mechanics [22]. The second inequality

constraint restricts the volume of the material used. This is to

prevent the problem from becoming unbounded and to achieve

economy of material and keeping the size of the switch small.

maxV is the maximum amount of material allowed.

2Λ is the

estimate of the Lagrange multiplier associated with the second

constraint. The rest of the symbols were defined earlier or in

Eq. (5).

By writing the Lagrangian for the constrained minimization

problem posed in Eq. (5) and then writing the Euler-Lagrange

necessary conditions, including the complementarity

conditions, for this problem, we get

( )2

1 2 max

0 0 0

1L L L

l uL x M m

L dx dx t b dx VbE b Eα α

−= + Λ ∆ − + Λ −

∆ ∫ ∫ ∫ɶ ɶ

(7)

( )2

1 22 20l u

L x M mt

E b E bα α

−− + Λ + Λ =

∆ɶ ɶ (8a)

1 1

0

0, 0

L

l uM m

dxAEα

Λ ∆ − = Λ ≥

∫ (8b)

2 max 2

0

0, 0

L

t b dx V

Λ − = Λ ≥ ∫ (8c)

Four possibilities exist now depending on at least one, both, or

none of the constraints are active.

1 20, 0Λ > Λ = (9a)

1 20, 0Λ = Λ = (9b)

1 20, 0Λ = Λ > (9c)

1 20, 0Λ > Λ > (9d)

Let us first treat the two cases where the volume constraint

(i.e., the second constraint) is inactive. In the first case (Eq.

(9a)), 1

Λ , which is a constant, turns out to be a function of x

by virtue of Eq. (8a). Hence, it is not valid. The second case

(Eq. (9b)) is also invalid because it implies that ( )b x should

be infinite. Hence, we conclude that the volume constraint

must be active making the third and fourth cases valid. As per

Eq. (8a), the third case (Eq. (9b)) gives

( )

2

L xb

t Eα

−=

Λ ∆ɶ (10)

where 2

Λ can be computed using Eq. 5(c).

53

( )

( )

max

0 2

20

2

max max2

L

L

L xt dx V

t E

L xt dx

t E tL

V t E V

α

α

α

− = Λ ∆

− ∆ ⇒ Λ = =

∆

∫

∫

ɶ

ɶ

ɶ

(11)

By substituting for 2Λ from Eq. (11) into Eq. (10) gives

( )( )max

2

2

2L x Vb L x

tLt Eα

−= = −

Λ ∆ɶ (12)

Note that it shows that optimum width profile of the beam is

linear. The fourth case (Eq. (9d)) with Eq. (8a) gives

( )2

1

2

1l u

L xb M m

t Eα

− = − Λ ∆Λ ɶ

(13)

The constants 1

Λ and 2

Λ can be evaluated using the two

active constraints. Although it looks formidable and highly

nonlinear, it was found that this is also close to linear as shown

in Figs. 7a-b for some chosen data. While it is possible that it

may be substantially far from a linear width profile for some

other data, we choose to take the linear profile as the optimum

because: (i) manufacturing is simple and robust against

inaccuracies and (ii) keeping the actuation voltage low and

contact force moderate can be better done with the linear width

profile. The linear profile gives one parameter, i.e., the slope,

to vary to meet other performance criteria as discussed next.

B. Re-design of the switch

As discussed in the preceding sub-section, linear width

profile is optimum for maximizing the recoil force. Now, in

order to meet other criteria (reduce pull-in voltage without

compromising on recoil force), we take the linear profile

shown in Fig. 9 and vary its slope in re-designing the switch.

Here, the contact element is of fixed size. The thickness was

taken as 50 µm and the gap was taken as 8 µm throughout. The

widths at either end get decided based on the slope while some

minimum and maximum values are obeyed. The actuation

electrode was kept at a distance of 1150 µm from the fixed

end. The electrode length was taken to be 1350 µm in all

cases. The contact occurs at the contact element of length 600


39 values for the taper angle of the beam were taken and

each case was simulated using beam FE modeling for pull-in

voltage and COMSOL (www.comsol.com) for computing the

recoil force. The entire length of the contact pad is assumed to

be in contact with the substrate while evaluating the recoil

force. Figure 9 shows the computed values for the pull-in

voltage and recoil force. The recoil force is non-monotonic

because beams bend differently when the width profile has

negative or positive slope.

Fig. 8. Optimum width profiles (a) as per Eq. (12), (b) as per Eq. (13).

Fig. 9. Optimum width profiles (a) as per Eq. (10), (b) as per Eq. (11).

Fig. 10. Pull-in voltage and recoil force with taper angle of the beam with

gaps c

g and a

g both as 8 µm.

Figure 10 is instructive because it shows how pull-in (and

hence actuation voltage) can be decreased substantially

without compromising much on the recoil force. As shown in

Fig. 9, as we move from the current design (i.e., the design

reported in [6] and shown in Fig. 2) of negative slope of about

-20° to positive slope of about 8°, we get substantial (almost

45%) reduction in pull-in voltage whereas the recoil force

stays the same. Another re-design was decreasing the

actuation gap to 6 µm and the contact gap to 4 µm as it was

felt that this change can be handled in machining the steel foil

and in assembly. The final configuration is shown in Fig. 9.

The details for six design variants are shown in Table 1. The

pull-in voltages of these configurations were estimated using

the CoSolve module of CoventorWare. Details of how these

54

designs were arrived at are not presented here due to paucity

of space and interested readers may refer to [23]. All six

designs were fabricated and tested.

Highlights of fabrication and testing of the six design

variants presented in the preceding section are briefly noted in

the next section. Details can be found in [3].

V. FABRICATION AND TESTING

A. Fabrication

The cantilever beam of the switch was photochemically

etched from 50µm-thick SS304 foils (Kemac Technology Inc.,

CA). The recesses of 6 µm and 4 µm were machined using

micro-electrodischarge machining (µEDM). Perforations of

300 µm diameter were located for the alignment and the

attachment on the PCB (Fig. 10a). A 600 µm-thick Rogers

4003 was used as the substrate (Advanced Circuits Inc., CO).

Metal interconnect traces of 70 µm - thick Cu provided bias

electrodes and the contact pad. In such PCBs, 4-µm thick Ni

was used as an adhesion layer on the Cu base, and 0.25-µm thick

outer gold layer provided an electrical contact. Through vias

were located on the PCB for the subsequent attachment of the

cantilever.

Fig. 11. Dimensions of re-designed shapes which were fabricated.

TABLE 1

DATA FOR THE SIX OPTIMIZED AND RE-DESIGNED SWITCHES

Design No. Slope (°) 1

W (in µm) 2

W (in µm)

1 -13.7 2231 572

2 -6.1 1769 1042

3 -0.9 1463 1354

4 1.7 1308 1510

5 6.9 1000 1823

6 14.5 538 2293

1l = 500,

2l = 500,

3l = 1430,

4l = 130,

5l = 600,

6l = 200, and L = 3400

(all values are in µm).

For the assembly, alignment posts (1000 µm-height; 300 µm-

diameter), were machined from gold wire using µEDM, and

tightly fitted into the isolated-vias on the PCB (Fig. 10a).

Conical wire tips facilitated the insertion of the cantilever. The

cantilever was assembled over the posts (Fig. 10b) and fixed by

applying conductive epoxy (Creative Materials). The flatness of

the cantilever was maintained during the assembly process with

the help of a high resolution laser displacement sensor (Keyence

LK-G32).

B. Testing

The test circuit (see Figs. 11a-b) used the variable gate

actuation voltage (VG) and constant drain voltage (VD) as inputs.

Output voltage VOUT was monitored to detect the pull-in

condition. VG was increased in steps of 50 mV from 30 V to 45

V. When the test structure was in the off-state, VOUT was equal

to zero (denoted as VOUT–OFF). When a pull-in occurred, VOUT

was approximately VD/2 = 1 V due to the relationship given in

Fig. 11b (denoted as VOUT–ON). RON denotes the on-state

resistance and includes the contact resistance and the

parasitics. The test was run in air ambient atmosphere of air.

Pull-in occurred at 40.95 V. Further increase in actuation

voltage caused VOUT to remain constant at approximately 1 V

since RON was negligibly low compared to fixed resistors.

Fig. 12. Scanning Electron Microscopes (SEM) micrographs for Design 2 in

Table 1. (a) Cantilever is shown upside down with recessed regions. (b)

Assembled structure with 6 µm gap between the cantilever and the ground

electrode shown in the inset.

The six designs presented in the previous subsection were

also assembled tested. The results are shown in Fig. 12 and

Table 2. The notable thing is that the actuation voltage is

brought down to less than 82 V from the original value that

was more than 100 V.

55

Fig. 13. (a) Circuitry for testing, using variable gate actuation voltage (VG),

and constant drain voltage VD as the input. RON was the on-state resistance

that contact resistance and all the parasitic resistances. (b) Actuation concept

showing VOUT–OFF and VOUT–ON conditions.

C. Results

The performance characteristics obtained from the

simulation are given in table 2. It can be seen that the pull-in

voltage continuously comes down as the slope of the beam

increases. The decrease can be attributed to the decrease in

stiffness of the beam. The dynamic pull-in voltage also

observes the same trend and is 13-16% lower than the static

pull-in voltage. The decrease in dynamic pull-in voltage as

compared to the static pull-in voltage decreases as the slope is

increased. This can be explained by the fact that the net

vibrating mass increases as the slope of the beam increases

from negative to positive value.

TABLE 2

RESULTS OF SIMULATION OF SIX OPTIMIZED AND RE-DESIGNED SWITCHES

Config

No

Pull-in voltage

(V)

3D model

Pull-in

voltage (V)

1D model

Dynamic pull-

in voltage (V)

1D model

Pull-in

time (ms)

1 120.94 125.31 105.05 2.21

2 110.63 113.42 95.98 2.80

3 103.44 105.76 90.25 3.24

4 99.69 101.94 87.07 3.46

5 92.19 94.28 80.87 3.95

6 81.25 82.38 71.13 4.83

TABLE 3

CONTACT PARAMETERS EVALUATED AT 5V ABOVE THE STATIC PULL-IN

VOLTAGE

Config

No

Actuation voltage

(V)

Contact force

(µN)

Contact resistance

(Ω)

1 125.94 121.84 1.58

2 115.63 120.26 1.59

3 108.44 119.22 1.60

4 104.69 110.34 1.66

5 97.19 123.81 1.57

6 86.25 111.55 1.65

TABLE 4

RESULTS OF EXPERIMENTS OF SIX OPTIMIZED AND RE-DESIGNED SWITCHES

Config No Pull-in voltage (V) Pull-in time (ms)

1 46.25 4-5

2 40.95 4-6

3 35.75 5-6

4 34.1 5-7

5 32.8 5-8

6 28.15 6-8

RF-MEMS switches are usually operated a few volts above

the pull-in voltage. So, we operate the switches at 5 V above

the pull-in voltage to obtain the contact parameters and

evaluate the resistance using equation (2a) & (2b). The values

of contact force, area and resistances obtained are shown in

Table 3. It can be seen that the contact resistance is higher than

1 Ω but below 5 Ω for all the six configurations. A simple re-

design is suggested in the following subsection which

addresses this problem.

D. Discussion

It can be seen that there is a considerable difference between

the pull-in voltage evaluated by simulation and experiment i.e.

the values of pull-in voltage obtained from experiments is

lower than that obtained by simulation. As mentioned in

section V, conical wire tips are used to facilitate assembly.

This conical shape of the alignments posts gives rise to

clearances that can in turn make the beam tilt about the edge of

its resting surface possible as shown in Fig. 14. Since

conductive epoxy is used for assembly, this tilt may be

maintained after curing.

TABLE 5

PULL-IN VOLTAGE WHEN ACTUATION GAP IS 3µm and 0.066o TILT IS PRESENT

Config No 3 µm gap 0.066 o

tilt

1 44.69 47.41

2 40.62 41.18

3 37.81 37.33

4 36.56 35.45

5 33.75 31.72

6 29.64 26.03

An initial study suggested that the pull-in voltages of the

simulation agree within 8% of the experimental values of pull-

in voltage when the gap between the actuation electrode and

the beam, a

g , is reduced to 3 µm from 6 µm. The results of

simulation with reduced gap is shown in table 5 and the

comparison with experimental results and simulation with 6

µm gap is given in Fig. 13. It can be geometrically verified

that a tilt of approximately 0.14o is sufficient to bring the mid-

point of the part of the beam above the actuation electrode to 3

µm. Thus, making the average gap between the beam and the

56

actuation electrode as 3 µm. To facilitate this rotation the

required taper on the alignment posts is just 0.14o, which is

within the tolerances of the fabrication techniques used.

Fig. 14. Pull-in voltage for different configurations obtained from

experiments and simulation by varying the actuation gap.

Fig. 15(a). The beam tilts about the edge of the resting surface (axis shown)

by 0.14o. (b&c) Zoomed view of the top of the beam touching conical

alignment post.

Going further with the assumption that the cantilever tilts

during assembly, we try to vary the tilt of the beam from 0 to

0.14o. This results in a decrease of pull-in voltage as shown in

Fig. 16 for configuration 1. It can be seen that a tilt value of

0.066o results in pull-in voltage of 47.41 V, which is within

2.5% of the value of pull-in voltage obtained by experiment.

This tilt of 0.066o

when applied to all the six configurations

results in pull-in voltages (Table 5) which are within 3 to 12%

of the pull-in voltage values obtained by experiment. The large

dimensions (mm range) of the switch make the pull-in voltage

of the switches extremely sensitive to the value of the tilt that

might occur during assembly.

TABLE 6

CONTACT FORCE AND RESISTANCE FOR CONTACT GAP OF 6 µm

Config No. Actuation

voltage

(Volts)

Contact force

(µN)

Contact resistance

(Ω)

1 125.94 678.28 0.67

2 115.63 711.11 0.65

3 108.44 733.05 0.64

4 104.69 708.83 0.65

5 97.19 711.94 0.65

6 86.25 661.73 0.68

The contact resistance obtained is below 5 Ω for all the six

configurations but the desired value of 1 Ω was not achieved.

Improved contact resistance calls for enhanced contact force.

Force analysis at static equilibrium of a switch under contact

suggests that higher the electrostatic force acting on the beam,

higher will be the contact force. Thus, to increase the

electrostatic force on the switch at equilibrium without altering

the geometry of the switch or increasing the operating voltage,

we suggest an increase in contact gap from 4 µm to 6 µm.

Table 6 shows the increase in contact force when the contact

gap is increased to 6 µm which leads to contact resistance

values of below 1 Ω.

Fig. 16. Effect on pull-in voltage with some amount of initial tilt given to the

beam which could occur during the assembly process.

VI. CLOSURE

Stiction and adhesion are two problems that limit the

performance of a micromachined switch. Using special

materials that reduce adhesion and modifying the contacting

surface to eliminate stiction are usually followed. In this work,

we provided a mechanical approach to solve the problem by

designing the width profile of the beam of an electrostatically

actuated steel switch. The width profile of the beam was

optimized using structural optimization approach by posing a

calculus of variations problem to maximize the recoil force. It

was found that linearly tapering width is optimum and is

practical in view of meeting other performance criteria. By

varying the slope of the beam profile, it was shown that the

actuation voltage can be reduced substantially without

changing the recoil force much. The pull-in time of the switch

was found to be a few ms. Six design variants are finalized.

These had actuation voltage between 82 and 121 V. This is

considerably lower than the pull-in voltage in cold-switching

conditions for the switch in [6], which is more than 300 V and

will be reduced only to 150 V even with a reduced actuation

gap of 6 µm. All six designs were fabricated using

photochemical etching and micro-EDM and assembled onto a

PCB. Also discussed in the paper is that even a minor error in

manual assembly may cause substantial difference in the

57

performance of the switch and a simple re-design to improve

the contact resistance.

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58

Bibliography

[1] Goldsmith, C. L., Lin, T. H., Powers, B., Wu, W. R., and Norvell, B.,

“Micromechanical membrane switches for microwave applications,” IEEE MTT-S

Intl. Microwave Theory and Techniques Symp. Proc., pp.91–94, 1995.

[2] Rebeiz, G. M., RF MEMS: Theory, Design and Technology, Wiley, Hoboken, NJ,

2003.

[3] Ozkeskin, F. M., and Gianchandani, Y. B., “A Hybrid Technology for Pt-Rh and

SS316L High Power Micro-relays,” Proc. Solid-State Sensors, Actuators, and

Microsystems Workshop, Hilton Head Island, South Carolina, June 6-10, 2010.

[4] Ozkeskin, F. M., and Gianchandani, Y. B., “Double-cantilever Micro-relay with

Integrated Heat Sink for High Power Applications,” Proceedings of Power MEMS

2010, held in Leuven, Belgium, Nov. 30-Dec. 3, 2010, pp. 159-162.

[5] Zavracky, P. M., Majumder, S., and McGruer, N. E., “Micromechanical Switches

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