Design, Fabrication, and Dynamic Modeling of a Printed
Circuit Based MEMS Accelerometer
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
John E. Rogers
Certificate of Approval:
Ramesh Ramadoss, Co-Chair
Assistant Professor
Electrical and Computer Engineering
John Y. Hung, Co-Chair
Professor
Electrical and Computer Engineering
Bodgan M. Wilamowski
Professor
Electrical and Computer Engineering
George T. Flowers
Interim Dean
Graduate School
Design, Fabrication, and Dynamic Modeling of a Printed
Circuit Based MEMS Accelerometer
John E. Rogers
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
May 10, 2007
Design, Fabrication, and Dynamic Modeling of a Printed
Circuit Based MEMS Accelerometer
John E. Rogers
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
John Elvin Rogers was born in Columbus, Georgia in 1982. He grew up in Smiths,
Alabama where he graduated from Smiths Station High School in 2002. John then attended
Auburn University where he graduated in December 2005 with a Bachelor of Electrical
Engineering degree. He then entered graduate school at Auburn University in Spring 2006.
His research interests are in nonlinear systems and control, the design and fabrication of
Micro-Electro Mechanical Systems (MEMS), Silicon-on-Insulator (SOI) and Printed Circuit
Based (PCB) Fabrication Techniques, and monolithic integration of MEMS with electronics.
John?s interests in MEMS has brought him to study various modeling issues associated with
these devices that are still unstudied by engineers.
iv
Thesis Abstract
Design, Fabrication, and Dynamic Modeling of a Printed
Circuit Based MEMS Accelerometer
John E. Rogers
Master of Science, May 10, 2007
(B.E.E., Auburn University, 2005)
76 Typed Pages
Directed by John Y. Hung & Ramesh Ramadoss
A MEMS capacitive-type accelerometer fabricated using printed circuit processing tech-
niques is presented. A KaptonR? polymide film is used as the structural layer for fabricat-
ing the MEMS accelerometer. The accelerometer proof mass along with four suspension
beams are defined in the KaptonR? polyimide film. The proof mass is suspended above a
RT/DuroidR? (TeflonR?) substrate using a spacer. The deflection of the proof mass is de-
tected using a pair of capacitive sensing electrodes. The top electrode of the accelerometer
is defined on the top surface of the KaptonR? film. The bottom electrode is defined in the
metallization on the RT/DuroidR? substrate. The initial gap height between the bottom
electrode and the KaptonR? film is approximately 41.8 ?m. For an applied external acceler-
ation/deceleration (normal to the proof mass), the proof mass deflects towards or away from
the fixed bottom electrode due to inertial force. This deflection causes either a decrease
or increase in the air gap height thereby either increasing or decreasing the capacitance
between the top and the bottom electrodes.
v
An example PCB MEMS accelerometer with a square proof mass of membrane area
6.4 mm?6.4 mm is reported. The measured resonant frequency of 375 Hz and the Q-factor
in air is 1.5.
The ability to build MEMS accelerometers using low-cost printed circuit processing
techniques allows for integration of electronics, suitability for high-volume manufacturing,
and large surface area applications for low-g accelerometers. These are all key advantages
for using PCB MEMS accelerometers.
vi
Acknowledgments
I?d like to first and foremost dedicate this to the memory of my father, James Rogers.
He has always been a big inspiration in my life and has inspired me to be who I am today,
as he was an Electrical Engineer too. I?d also like to give a big thanks to my parents, Elesa
and Randy Curenton, for always having been there for me and helping me learn to live life
for what it is. I?d also like to thank my grandparents, Mary and John Rogers, for always
pushing me to do my best and strive for more in life. I?d also like to give thanks to my
grandparents, Elenor and Elvin Nix, and my aunt and uncle, Debbie and Eddie Nix. Last,
but not least I?d like to thank my fiancee, Brittany Michelle Camp. She has been there for
me through it all and inspires me to be everything I am.
A special thanks goes to Dr. John Y. Hung and Dr. Ramesh Ramadoss for their guid-
ance and wisdom. Without their kindness, knowledge, and having given me the opportunity
to work with them, none of this work would have been made possible. I?d also like to thank
my colleague, Phil M. Ozmun, for his help and insight in our research of the analysis and
fabrication of various MEMS devices. My final thanks goes to Ms. Madhurima Maddella
for her assistance and help with the fabrication of the accelerometer.
vii
Style manual or journal used Journal of Approximation Theory (together with the style
known as ?aums?).
Computer software used The document preparation package TEX (specifically LATEX)
together with the departmental style-file aums.sty, Microsoft PowerPoint for graphics.
viii
Table of Contents
List of Figures xi
1 PCB MEMS Accelerometer 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Materials, Configuration, & Analysis . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Spring Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Proof Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Natural Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4 Damping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.5 Quality Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1 Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.2 Spacer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.3 KaptonR? Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Detailed Fabrication Process . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Photolithography Masks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Fabrication Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Experimental Characterization 25
2.1 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Conclusions & Future Work 42
3.1 Il Buono . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Il Brutto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Il Cattivo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Bibliography 44
ix
A Rigid Body Model for MEMS Accelerometer 46
A.1 Electrical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.2 Mechanical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A.3 Mechanical Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . 49
B Rayleigh-Ritz Method for Determining Equivalent Mass of a Flexible
Structure 50
C Linear State Variable Analysis for PCB MEMS Accelerometer 53
C.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
C.2 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
C.3 State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
C.4 Equilibrium State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
C.5 Linearized State Variable Model . . . . . . . . . . . . . . . . . . . . . . . . 55
C.6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
C.7 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
C.8 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
C.9 Stabilizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
C.10 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
C.11 Linear State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
C.12 Observer Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
C.13 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
D MATLAB Code 63
x
List of Figures
1.1 Photograph of the fabricated PCB MEMS Accelerometer integrated with
capacitive readout chip. Courtesy AMSTC-Auburn University . . . . . . . . 2
1.2 Sequence of Layers for PCB MEMS Accelerometer . . . . . . . . . . . . . . 3
1.3 Deflection of the PCB MEMS Accelerometer . . . . . . . . . . . . . . . . . 4
1.4 Schematic of the PCB MEMS Accelerometer Top view (not to scale) . . . . 6
1.5 Schematic of the PCB MEMS Accelerometer Cross-sectional view (not to scale) 7
1.6 A 2 mil thick KaptonR? film used for fabrication of the PCB MEMS Ac-
celerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Kapton Mask used in Photolithography . . . . . . . . . . . . . . . . . . . . 19
1.8 CopperTop Mask used in Photolithography . . . . . . . . . . . . . . . . . . 20
1.9 Subini Mask used in Photolithography . . . . . . . . . . . . . . . . . . . . . 21
1.10 Subcop Mask used in Photolithography . . . . . . . . . . . . . . . . . . . . 22
2.1 A Photograph of an LDS model V408 electromechanical shaker (courtesy
Auburn University). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Full photograph of PCB MEMS Accelerometer. Courtesy AMSTC-Auburn
University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Laser Interferometric Measurement System . . . . . . . . . . . . . . . . . . 28
2.4 Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer
(First Test, Mechanical Response) . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer
(Second Test, Electrical Response) . . . . . . . . . . . . . . . . . . . . . . . 30
2.6 Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer
(Second Test, Mechanical Response) . . . . . . . . . . . . . . . . . . . . . . 31
xi
2.7 Magnitude Plot of the Experimental and Theoretical Results using Second-
Order Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.8 Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer
(Third Test, Electrical Response) . . . . . . . . . . . . . . . . . . . . . . . . 34
2.9 Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer
(Second Test, Mechanical Response) . . . . . . . . . . . . . . . . . . . . . . 37
2.10 Four pole, three zero, one delay transfer function (MATLAB SYSID) . . . . 38
2.11 Eight pole, six zero, three delay transfer function (MATLAB SYSID) . . . . 39
2.12 Eight pole, seven zero, one delay transfer function (MATLAB SYSID) . . . 40
2.13 Fourth order pole-zero placement transfer function . . . . . . . . . . . . . . 41
A.1 Cross-sectional view of PCB MEMS device . . . . . . . . . . . . . . . . . . 46
B.1 Shape Function for Suspension Beam . . . . . . . . . . . . . . . . . . . . . . 50
C.1 Normalized pull-down voltage curve . . . . . . . . . . . . . . . . . . . . . . 55
C.2 Block diagram for linear state feedback control . . . . . . . . . . . . . . . . 60
C.3 Output response for linear state feedback control . . . . . . . . . . . . . . . 61
C.4 Block diagram for estimated state feedback control . . . . . . . . . . . . . . 61
C.5 Output response for estimated state feedback control . . . . . . . . . . . . . 62
C.6 Estimation error for estimated state feedback control . . . . . . . . . . . . . 62
D.1 PCB MEMS Accelerometer Parameters MATLAB Code . . . . . . . . . . . 63
D.2 PCB MEMS Accelerometer Paramaters MATLAB Code (Cont.) . . . . . . 64
xii
Chapter 1
PCB MEMS Accelerometer
1.1 Background
Micro-Electro-Mechanical Systems (MEMS) is a technology that integrates many engi-
neering fields: electrical, mechanical, materials, computer science, and control systems [1].
MEMS can be classified as either sensors or actuators, and in many applications are a
combination of the two. MEMS were first introduced in the early 1960s as discrete open-
loop pressure sensors [2]. Typical advantages of MEMS devices include small-size (in the
range of 300 nm to 300 ?m), light-weight, low-cost, high performance, and low-power con-
sumption [3]. MEMS have found applications in many industries including automotive,
biomedical, aerospace, and communications, among many others. MEMS have proven to
be a revolutionary technology in many application areas including accelerometers, gyro-
scopes, pressure sensors, displays, inkjet nozzles, and fluid pumps. Along with the advances
in MEMS technology has been a growing interest in the control of MEMS actuators to
achieve extended range of motion [4].
In recent years, there has been a considerable interest in development of meso-scale (on
the order of ? 10 mm) MEMS devices fabricated using printed circuit processing techniques
known as Printed Circuit Based MEMS or PCB MEMS [5]. In PCB MEMS technology,
organic polymer materials are used as substrates, sacrificial layers, and structural layers for
fabrication of MEMS devices. PCB MEMS enable monolithic integration of MEMS devices
and electronics using low-cost, conventional printed circuit techniques. The advantages of
1
Figure 1.1: Photograph of the fabricated PCB MEMS Accelerometer integrated with ca-
pacitive readout chip. Courtesy AMSTC-Auburn University
PCB MEMS components include low-cost, ease of integration with electronics, suitabil-
ity for high-volume manufacturing, and large surface area applications. Recently, PCB
MEMS devices such as flow sensors, tactile sensors [6], pressure sensors [7], salinity sensing
system [8], RF MEMS switches [5], and tunable antennas [9] have been demonstrated by
various research groups.
One type of PCB MEMS device is known as a PCB accelerometer, shown in Figure 1.1.
The accelerometer makes use of organic polymers such as KaptonR? film for the structural
layer, PolyflonTM bonding film for the spacer layer, and RT/DuroidR? (TeflonR? for the
substrate, as shown in Figure 1.2. The proof mass and its four suspension beams, are defined
in the KaptonR? polyimide film. A PCB accelerometer is a MEMS-based accelerometer
that moves in an out-of-plane motion or a motion orthogonal to the plate mass as shown in
2
Figure 1.3. An accelerometer can be used to measure acceleration of a moving object as well
as velocity and position by integrating electronics. The accelerometer may also be used as
a pressure sensor. The ability to build accelerometers using low-cost, conventional printed
circuit techniques allows for monolithic integration of MEMS with electronics. Materials,
configuration, analysis, fabrication, and experimental characterization of an example PCB
MEMS accelerometer are discussed in the following sections.
Figure 1.2: Sequence of Layers for PCB MEMS Accelerometer
1.2 Materials, Configuration, & Analysis
The materials, configuration, and analysis of the PCB MEMS accelerometer are dis-
cussed in the following subsections. The materials subsection discusses what materials are
3
used for the PCB MEMS accelerometer. The configuration subsection discusses what the
structure is comprised of and how it performs. The analysis subsection discusses the physics
behind the structure and how the parameters are derived.
Figure 1.3: Deflection of the PCB MEMS Accelerometer
1.2.1 Materials
The materials used in fabrication of the PCB MEMS accelerometer include KaptonR?,
PolyflonTM, and RT/DuroidR? (TeflonR?). The KaptonR? E polyimide (a polymer of imide
monomers) film is available from DuPont and has a permittivity, epsilon1r, of 3.1 at 1 kHz with a
3 ?m copper cladding. KaptonR? E is a premium performance polyimide film for use as a
dielectric substrate in flexible printed circuits and high density interconnects. KaptonR? E
is a preferred dielectric film for very fine circuitry due to its high modulus and a coeffi-
cient of thermal expansion equivalent to that of copper. KaptonR? E also has excellent
electrical characteristics and chemical etchability. The PolyflonTM bonding film is available
from Daikin Industries and is used as a spacer film. PolyflonTM has minimal deformation
under loads, is a good electrical insulator (dielectric breakdown strength), and has good
4
transparency. The RT/DuroidR? 6002 substrate is available from Rogers Corporation and
has a permittivity, epsilon1r = 2.94 at 10 GHz, a dissipation factor, tan ? = 0.0012 at 10 GHz with
a 1/4 oz (9 ?m) of copper cladding. RT/DuroidR? 6002 is a microwave material with low
loss for excellent high frequency performance, extremely low thermal coefficient of dielectric
constant, and excellent electrical and mechanical properties.
1.2.2 Configuration
The top and cross-sectional views of the PCB MEMS accelerometer are shown in
Fig. 1.4 and Fig. 1.5. The proof mass of the accelerometer is defined in the 2 mil (50.8 ?m)
thick KaptonR? polyimide film. The proof mass comprises of a square membrane supported
by four suspension beams. The proof mass is suspended above a 30 mil (762 ?m) thick
RT/DuroidR? substrate using a 2 mil (50.8 ?m) thick PolyflonTM bonding film (spacer). For
an applied external acceleration/deceleration (normal to the proof mass), the proof mass
deflects towards or away from the substrate due to inertial force. The deflection of the proof
mass is detected using a pair of capacitive sensing electrodes as shown in Figure 1.3. The
top sensing electrode of the accelerometer is defined in the 3 ?m thick (standard thickness
available from DuPont) copper metallization on the top surface of the KaptonR? film. The
bottom sensing electrode is defined in the 9 ?m thick (standard thickness available from
Rogers Corporation) copper metallization on the RT/DuroidR? substrate. Standard copper
metallization materials available from the manufacturers were used in this work. The top
and bottom sensing electrodes have different thicknesses due to the standard thicknesses
available from their respective manufacturers. The spacer film determines the nominal air
gap height between the KaptonR? film and the bottom sensing electrode. The nominal gap
5
Figure 1.4: Schematic of the PCB MEMS Accelerometer Top view (not to scale)
height is approximately 41.8 ?m (50.8 ?m spacer - 9 ?m bottom sensing electrode). The
deflection of the membrane causes either a decrease or increase in the air gap thereby either
increasing or decreasing the capacitance between the top and the bottom electrodes. The
sensing leads of width 100 ?m (chosen by design) are used for connecting the electrodes to
the capacitance read-out chip.
1.2.3 Analysis
To understand what?s happening dynamically with a MEMS device a knowledge of
some basic underlying principles is essential. The electrial and mechanical dynamics of
lumped element systems have been studied thoroughly by engineers. The various physical
6
Figure 1.5: Schematic of the PCB MEMS Accelerometer Cross-sectional view (not to scale)
parameters of the PCB MEMS accelerometer are shown in Fig. 1.4 and Fig. 1.5. The area
of the square KaptonR? membrane is d?d. The length and width of the suspension beams
are l and w, respectively. The reduction distance between the square area of the KaptonR?
membrane and the top electrode is r. In this work, the hypothesis is that the accelerometer
may be modeled as a second-order mass-spring-damper system (Appendix A)
m?x = ?c?x?kx+ epsilon1AV
2
2
1
(g ?x)2 (1.1)
where m is the effective mass, c is the damping constant, k is the spring constant, epsilon1 is the
permittivity of the surrounding gas, A is the accelerometer area, V is the applied voltage,
and g is the nominal gap distance.
1.3 System Parameters
The estimation of parameters in a system is essential in the analysis process. They
can be estimated by knowing information about the geometry, material properties, envi-
ronmental conditions, etc. If the second-oder model above holds and the parameters can
7
be estimated, a linear state variable control technique can be used to reject disturbrances
that would detriment the accelerometer (Appendix C). Some of the system parameters of
the example PCB MEMS accelerometer along with their corresponding values are shown in
Table 1.1. These parameter values (proof mass, spring consant, damping constant, natural
frequency) along with a few others will be derived below.
Table 1.1: Designed System Parameters for the Example PCB MEMS Accelerometer
Parameter Name Value Units
m effective mass 4.499?10?6 kg
k spring constant 20.32 N/m
c damping constant 18.21?10?2 N-sec/m
fo natural frequency 338 Hz
g gap distance 41.8 ?m
1.3.1 Spring Constant
The mechanical spring constant for a system is defined by the geometry and materials
properties of the spring. A system is usually designed around known spring types (i.e.
cantilever beams, fixed-fixed beams), load types (i.e. uniform loads, point loads), and
materials (i.e. silicon, KaptonR?) for ease of calculations. This allows for ease of estimation
when designing. It should be noted that the ability to estimate the spring constant allows
for the calculation of other unknown parameters (i.e. damping) in the experimental results.
The spring constant for the given accelerometer has four cantilever beams with distributed
point loads (Figure 1.4).
8
The spring constant of the bi-material KaptonR?-copper suspension beams of the ac-
celerometer is given by
k = 48EIl3 (1.2)
where EI is the equivalent flexural rigidity [10] given by
EI = (wcEct
2c)2 + (wkEkt2
k)
2 + 2wcwkEcEktctk(2t2c + 3tctk + 2t2
k))
12(wcEctc +wkEktk) (1.3)
where Ek and tk are the Young?s modulus and thickness of the KaptonR? film, Ec and tc are
the Young?s modulus and thickness of copper, wk and wc are the widths of the KaptonR?
and copper in the suspension beams, and l is the length of the suspension beams.
1.3.2 Proof Mass
The effective mass, or proof mass, is the total mass that will experience motion as a
result of an inertial force. On the accelerometer the plate mass will displace, but there is
also spring displacement. For any one spring, a fraction of the spring mass will displace from
one end of the pinned or fixed side. The question then arises of how much of the spring
will displace? This can be calculated using the Rayleigh-Ritz Method [11] for vibration
frequency. The Rayleigh-Ritz Method is based on the principle of energy conservation
and is used for calculating the vibration frequency of systems with distributed masses.
Experimentally, the distributed mass can be solved for if one knows the vibration frequency
and the spring constant. The Rayleigh-Ritz Method for the accelerometer is solved for in
9
Appendix B. The results from the Rayleigh-Ritz Method show the effective proof mass is
m = mp + 1335mb (1.4)
where mp is the mass of the KaptonR? membrane with the copper top electrode and mb is
the total mass of all four suspension beams (along with sense leads).
1.3.3 Natural Frequency
The vibration frequency, or natural frequency, is the frequency at which a device natu-
rally vibrates or oscillates. The accelerometer should be operated below this frequency (in
other words the accelerometer must be shaken below this frequency) in order to operate as
an accelerometer. The natural frequency of the accelerometer can be expressed
?o =
radicalbigg
k
m (1.5)
where k is the effective spring constant and m is the effective proof mass of the accelerometer.
1.3.4 Damping Constant
The accelerometer moves in an out-of-plane motion or a motion orthogonal to the plate
mass. Squeeze-film damping is the dominant damping mechanism in this configuration.
The squeeze-film damping refers to the energy dissipated in displacing the gas molecules
between the moving beam or plate and the substrate. The squeeze-film damping constant
10
for a square membrane [11] is
c = 0.42?d
4
g3 (1.6)
where ? is the viscosity of air (= 18.27 ?Pa.s at 20oC), d is the side length of the plate
mass (KaptonR? film), and g is the nominal gap height between the bottom electrode and
the KaptonR? film.
1.3.5 Quality Factor
The quality factor is a measure of how well a system dissipates energy. The quality
factor is defined as the energy stored over the power loss at resonance. Thus a higher quality
factor indicates a lower rate of energy dissipation at resonance. The quality factor, Q, of
the accelerometer is given by
Q = ?omc (1.7)
where ?o is the resonant frequency, m is the proof mass, and c is the damping constant of
the accelerometer.
In this work, an example PCB MEMS accelerometer with a square membrane of area
6.4 mm?6.4 mm is considered. The length l and width w of the suspension beams are
5.8 mm and 1 mm, respectively. The reduction distance between the square area of the
KaptonR? membrane and the top electrode is 270 ?m. The initial gap height g is approxi-
mately 41.8 ?m. These parameter values were chosen based on using equations (2.2)-(2.7)
to design an accelerometer with a low-frequency resonance for low-g applications. This
11
means that the spring to mass ratio given by Equation (2.5) should be small. The various
system parameters can be calculated and are shown in Table 1.1. In the following sections,
fabrication and experimental characterization are discussed.
1.4 Fabrication
The PCB accelerometer is fabricated using printed circuit techniques. The fabrication
of the accelerometer has three layers: a substrate layer, spacer layer, and polyimide layer
(Figure 1.5). They will be discussed in further detail below. The substrate layer is made of
RT/DuroidR?, a TeflonR? material. The spacer is made of a PolyflonTM bonding layer. The
polyimide layer is made of KaptonR? film.
1.4.1 Substrate
The substrate chosen is a 30 mils (762 ?m) thick RT/DuroidR? with 9 ?m thick cop-
per metallization. The bottom electrode for capacitive sensing is defined in the copper
metallization on the RT/DuroidR? substrate.
1.4.2 Spacer
The spacer layer provides the required spacing between the substrate and the KaptonR?
polyimide layer. Hence, the thickness of the spacer film determines the up-position gap
height. A 2 mil (50.8 ?m) thick PolyflonTM bonding film is used as the spacer layer. The
bonding film was cut to create openings for the movable membrane with suspension beams.
A milling machine could be employed for designs with small characteristic dimensions.
12
1.4.3 KaptonR? Film
A 2 mil (50.8 ?m) thick KaptonR? film with 3 ?m thick copper metallization is used as
the structural layer for the accelerometer. The KaptonR? film contains copper metallization
on both sides as shown in Figure 1.6. A 150-250?A thick nichrome seed layer is present
between the KaptonR? film and the copper layer. The film is essentially a sandwich of
copper-nichrome-kapton-nichrome-copper. The two major steps in the fabrication of the
KaptonR? film are discussed in detail below.
Figure 1.6: A 2 mil thick KaptonR? film used for fabrication of the PCB MEMS Accelerom-
eter
Plasma Etching of KaptonR? Film
Fabrication starts with rinsing of the KaptonR? film using Acetone then Methanol
followed by cleaning using diluted sulfuric acid to remove any oxide growth on the copper
surfaces. The bottom side copper is used to define the mask for DRIE processing. A piece
of Dynaflex wafer grip film is attached to a silicon wafer by heating it on a hotplate to
110oC. The Dynaflex wafer grip film is used as an adhesion layer to mount the KaptonR?
film to the wafer. When the wafer is sufficiently hot (110oC), the KaptonR? is carefully
attached by hand to the wafer in a way as to smooth out all air pockets. The next step is
13
to spin-coat photoresist (PR) onto the KaptonR? film and soft bake. The KaptonR? film is
then exposed and developed using the KaptonR? layer mask.
The copper is then etched off using an industry standard copper etchant such as CE-200.
The photoresist is then removed and the nichrome is fully etched off from the KaptonR? layer
pattern. The wafer is then ready for the Deep Reactive Ion Etch (DRIE) process. The wafer
is put into the DRIE where the etcher will etch through the KaptonR? and nichrome/copper
on the back side. The equipment used for the DRIE process is an STS AOE (Advanced
Oxide Etcher). The configuration of the gases used for this etch in Auburn University?s
AMSTC facility is 8 sccm carbon tetrafluoride (CF4) and 35 sccm oxygen with 500 Watts of
radio frequency (RF) power. Using these processing parameters it takes about 70 minutes
to etch a 2 mil thick KaptonR? layer. The bottom copper and nichrome are etched fully
using chemical etchants. Then, the wafer grip is dissolved using Amyl Acetate and the
KaptonR? film is removed by hand from the wafer.
Top Electrode Patterning
At this stage, the KaptonR? film contains copper and nichrome layers on only one side.
The KaptonR? side is then attached to the dicing tape. The film is then attached to a silicon
wafer and photoresist is spin-coated. The top sensing electrode mask is used to define the
top electrode in the copper layer on the KaptonR?. Finally, the nichrome is removed.
Thermal Compression Bonding
Thermo-compression bonding is performed using a Carver Press consisting of two
platens. The platens are heated using heaters, whose temperature is sensed by a ther-
mocouple. The fixture consists of two steel plates with alignment holes on them. The
14
substrate forms the bottom most layer, the spacer is the middle layer and the KaptonR?
film forms the top most layer in this structure. The different layers are aligned by aligning
the holes created on the three layers during the fabrication process described earlier. This
unit is now placed between the press platens. The bonding is performed at a pressure of
80 psi (a load of 200 lbs.) and a temperature of 130oC. Both pressure and temperature are
maintained for 5 min during bonding. Before pressure is released, the assembly is cooled
down to the room temperature in a process known as annealing.
1.5 Detailed Fabrication Process
Table 1.2: PCB MEMS Accelerometer
Process Method Time
Kapton Film Cleaning
1. Cleaning Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
2. Oxide Removal 250mL of H2O + 2 drops of H2SO4
DI Rinse 30 Sec
N2 Dry
3. Cleaning Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
Stick Kapton Film on Wafer
1. 4? Wafergrip Application Apply Wafergrip on wafer
2. Soft Bake Hot plate - 110oC 30 Sec
3. Stick Film on Wafer Smooth Kapton Film on wafer
15
Define Kapton Layer
1. Spin Photoresist S1813 Photoresist 30 Sec
2500 RPM
500 r/s ramp
2. Soft Bake Hot plate - 110oC 60 Sec
3. UV Exposure Mask Aligner - expose 30 Sec
Kapton Layer mask
4. Develop CD-30 developer 45 Sec
DI Rinse 30 Sec
N2 Dry
5. Hard Bake Hot plate - 110oC 60 Sec
6. Wet Etch CE-200 etchant 8 Sec
DI Rinse 30 Sec
N2 Dry
7. Remove Photoresist Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
8. Wet Etch NaOH + KMNO4 mixture 50 Sec
DI Rinse 30 Sec
N2 Dry
9. Cleaning Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
Deep Reactive Ion Etch (DRIE)
1. DRIE 4:1 CF4:O2 proces 75 Min
2. Cleaning Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
3. Remove/Clean Kapton Film Remove Kapton Film from Wafer
Clean using Amyl Acetate
4. Cleaning Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
16
Define CopperTop Layer
1. Spin Photoresist S1813 Photoresist 30 Sec
2500 RPM
500 r/s ramp
2. Soft Bake Hot plate - 110oC 60 Sec
3. UV Exposure Mask Aligner - expose 1:30 Min
CopperTop mask
4. Develop CD-30 developer 45 Sec
DI Rinse 30 Sec
N2 Dry
5. Hard Bake Hot plate - 110oC 60 Sec
6. Wet Etch CE-200 etchant 8 Sec
DI Rinse 30 Sec
N2 Dry
7. Remove Photoresist Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
8. Wet Etch NaOH + KMNO4 mixture 50 Sec
DI Rinse 30 Sec
N2 Dry
9. Cleaning Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
Define Subini Layer
1. Cleaning Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
2. Oxide Removal 250mL of H2O + 2 drops of H2SO4
DI Rinse 30 Sec
N2 Dry
3. Cleaning Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
4. Tape RT/Duroid Apply blue tape to one side of RT/Duroid
5. Spin Photoresist S1813 Photoresist 30 Sec
2500 RPM
500 r/s ramp
6. Soft Bake Hot plate - 110oC 60 Sec
17
7. UV Exposure Mask Aligner - expose 30 Sec
Subini mask
8. Develop CD-30 developer 45 Sec
DI Rinse 30 Sec
N2 Dry
9. Hard Bake Hot plate - 110oC 60 Sec
10. Wet Etch CE-200 etchant 8 Sec
DI Rinse 30 Sec
N2 Dry
11. Remove Photoresist Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
Define Subcop Layer
1. Cleaning Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
2. Spin Photoresist S1813 Photoresist 30 Sec
2500 RPM
500 r/s ramp
3. Soft Bake Hot plate - 110oC 60 Sec
4. UV Exposure Mask Aligner - expose 30 Sec
Subcop mask
5. Develop CD-30 developer 45 Sec
DI Rinse 30 Sec
N2 Dry
6. Hard Bake Hot plate - 110oC 60 Sec
7. Wet Etch CE-200 etchant 8 Sec
DI Rinse 30 Sec
N2 Dry
8. Remove Photoresist Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
9. Wet Etch NaOH + KMNO4 mixture 50 Sec
DI Rinse 30 Sec
N2 Dry
10. Cleaning Acetone then Methanol
DI Rinse 30 Sec
N2 Dry
18
1.6 Photolithography Masks
Figure 1.7: Kapton Mask used in Photolithography
19
Figure 1.8: CopperTop Mask used in Photolithography
20
Figure 1.9: Subini Mask used in Photolithography
21
Figure 1.10: Subcop Mask used in Photolithography
22
1.7 Fabrication Issues
Fabrication of a PCB MEMS accelerometer raises many issues. Some of the fabrication
issues can be attributed to the fabrication equipment. For example, fine temperature control
and pressure control are hard to achieve. If a process calls for a specific temperature and
pressure to be maintained over a time period, then undesirable results could be formed and
cause further defects in other fabrication steps. There can also be issues with the chemicals
used. For example, if a certain chemical is used as an etchant and has been used for several
cycles, then the etchant will not etch at the same rate and if the time is kept constant the
result will be an unfinished surface. This issue occured in all of the chemical processes due
to recycling of the chemicals and was delt with by overetching until it was visibly clear
that the process was done. The environmental conditions such as temperature, pressure,
moisture can also effect the chemicals, but are usually controlled in a microfabrication clean
room.
Scattering of the copper layer is another issue in fabrication. Scattering means the
copper layer begins to crack and break under the high temperatures present in the DRIE
process. This happens due to the large difference in thermal coefficients of expansion be-
tween copper and KaptonR?. This issue wasn?t considered before fabrication, thus the
fabrication steps were altered to use the backside of the KaptonR? film for the copper layer.
To avoid this issue for top and backside copper metallizations aluminum must be deposited
before DRIE and removed after DRIE.
Residual stress is another possible issue in printed circuit techniques due to the com-
bination of KaptonR? with nichrome and copper. Residual stress occurs when a thin film
is deposited on a substrate and has in-plane stress. Plane stress is caused by mismatches
23
in thermal expansion of the film and sustrate, which can lead to deformation of the device.
This happened when the various layers were pressure bonded in the thermo-compression
bonding stage. The gap increased and warped due to these mismatches. These effects could
have been supressed by having copper layers on both sides of the KaptonR? to even out the
mismatches.
Another issue with KaptonR? is the fact that it is a thin polyimide film. This means
that every process that requires the film to be moved by hand carries with it the potential
to bend the film and make permanent dents thus creating further warpage of the film. This
has the potential to happen in the chemical etching stages where the KaptonR? film must be
moved back and forth with wafer tweezers in order for the etchant to be effective. There is
also the potential for warping in the drying of the KaptonR? film. Every time the KaptonR?
film is cleaned or etched it must go through deionized water then dried with nitrogen (N2)
gas. The N2 gas has a high pressure rate and when spread across the KaptonR? film has
the potential to create dents. These fabrication issues can and do effect the modeling of the
device as they will change the various parameters (i.e. mass, spring constant).
24
Chapter 2
Experimental Characterization
2.1 Experimental Set-up
A photograph of the fabricated MEMS accelerometer is shown in Fig. 2.2. The MEMS
accelerometer was characterized using an LDS model V408 electromechanical shaker shown
in Figure 2.1. For testing purposes, the accelerometer substrate was mounted onto a plex-
iglass fixture as shown in Figure 2.2 with a mounting screw attached at the center of the
bottom surface. The plexiglass is attached to the threaded hole in the shaker head. The
shaker vibrates the MEMS accelerometer at a chosen amplitude over a specified frequency
range. In response to the applied external acceleration/deceleration, the proof mass vibrates
in a direction normal to the substrate.
The mechanical displacement of the proof mass was measured by reflecting a laser
beam off of the proof mass using a laser interferometric measurement system as shown
in Figure 2.3. The experiment yields y(f), the motion of the membrane as a function
of frequency. The TeflonR? substrate was used as a reference frame and a second laser
interferometric measurement system was used to measure the motion of the reference frame
as a function of frequency, x(f), by reflecting a laser beam off of the reference frame. The
signals x(f) and y(f) were recorded simultaneously using a signal analyzer. The signal
analyzer provides the transmissibility spectrum of the MEMS accelerometer by computing
the transfer function T(f) = y(f)/x(f). The transmissibility spectrum is defined as the
ratio of the magnitudes of the displacement of the membrane (output) and the reference
frame (input) over a range of frequencies otherwise known as the transfer function.
25
Figure 2.1: A Photograph of an LDS model V408 electromechanical shaker (courtesy
Auburn University).
2.2 Experimental Results
The measured magnitude of the transfer function as a function of frequency for the PCB
MEMS accelerometer is shown in Figure 2.4. From this plot, the resonant frequency and the
quality factor of the MEMS accelerometer were found to be 375 Hz and 1.5, respectively.
The measured resonant frequency is reasonably close to that of the calculated value of
338 Hz (refer Table 1.1). The measured Q is higher than that of the estimated value due to
a larger air gap height caused by the thermal expansion of various layers in the accelerometer
during fabrication. Using Equation (2.6), the effective air gap height can be estimated to
be 125 ?m.
26
Figure 2.2: Full photograph of PCB MEMS Accelerometer. Courtesy AMSTC-Auburn
University
The first experimental results for the PCB MEMS accelerometer were taken using the
set-up described above using two lasers to measure the displacements of both the membrane
and the reference frame. The mechanical results for this set-up are shown in Figure 2.4.
The second experimental results were taken of the electrical characteristics as well as more
mechanical characteristics. The electrical characteristics were taken by having the input
signal be the reference frame and the output signal be the electrical output from the capac-
itance to voltage or C-V chip. The C-V chip detects a capacitance from the accelerometer
and converts it to a voltage in by a linear amount (1 V/pF) and the output is that amount
added to a bias voltage of approximately 2.25 V. The change in deflection of the membrane
27
Figure 2.3: Laser Interferometric Measurement System
creates a change in capacitance which results in a voltage change. The signal analyzer then
creates a transfer function of the output voltage over the reference frame displacement.
C = epsilon1Ag (2.1)
?C
?g = ?
epsilon1A
g2 ?g (2.2)
Vout = Vbias +C( 1V1pF ) (2.3)
For low frequency there?s a small change in displacement, ?g, which results in a small change
in capacitance. The bias voltage, Vbias, falls out in the transfer function and only the voltage
28
Figure 2.4: Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer
(First Test, Mechanical Response)
due to capacitance change is considered. The electrical results for this set-up are shown in
Figure 2.5. The results show a linear increase in voltage of 20dB/decade for low frequency.
The plot then rolls off due to the mechanical response.
The mechanical characteristics were taken in the same set-up as the first experimental
results. There are three plots shown in Figure 2.6 that represent three measurements taken
over the membrane at various positions for statistical purposes. These plots were taken by
the signal analyzer over a long period of time to insure a sufficiently rich set of data for
statistical analysis. The results were taken on a different day and match up with the results
29
Figure 2.5: Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer
(Second Test, Electrical Response)
taken in the first experiment and thus show the experimental results are replicable. Note
however that these results are not ensured for replicated fabrication due to the fabrication
issues described in Section 1.5
The results show a resonant frequency around 375 Hz. The results also show a quality
factor or Q factor of 1 to 1.5. These results vary from those seen in Table 1.1. By closer
inspection of Figure 1.1 it can be seen that the amount of copper left on the beams is
much more than that shown in Figure 1.4. This will change the spring constant as well as
the effective mass. By using the WYKO Profiler in Auburn University?s CAVE lab it was
30
Figure 2.6: Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer
(Second Test, Mechanical Response)
observed that the plate was warped and the actual gap distance could not be calculated.
A distributed gap then had to be calculated to represent the overall warpage and was
calculated to be 125 ?m. This gap is more than three times that of the original gap.
This mismatch can be explained by the thermal compression stage of fabrication. PCB
devices will experience an expansion in the gap distance due to thermal compression due
to mismatches in thermal coefficients of copper and KaptonR? [12]. These changes due to
fabrication will change the system parameters as shown in Table 2.1.
31
Table 2.1: Actual System Parameters for the Example PCB MEMS Accelerometer
Parameter Name Value Units
m effective mass 4.499?10?6 kg
k spring constant 24.93 N/m
c damping constant 6.8?10?3 N-sec/m
fo natural frequency 375 Hz
g gap distance 125 ?m
There also is seen in the frequency results a second resonant frequency around 1100 Hz
in some of the results. In the ideal case the accelerometer would move only in an out-of-
plane motion. This resonant frequency may correspond to a torsional motion. Torsional
motion has the potential to excite other higher frequency harmonics. There also is a dip
before the resonant frequency that may be associated with under-damping. The accelerom-
eter doesn?t seem to have a second-order model as described in Section 1.2.3. As to be
described in the following section, the accelerometer seems to fit better with a higher-order
model. This is most likely due to the accelerometer having more of a flexible structure
than a hypothesized rigid structure. Rigid structures can more accurately be described
with second-order models, but flexible structures have more of a wave-like nature to them.
Flexible structures can thus better be described with distributed parameters rather than
lumped-element parameters. The torsional motion notion helps supplement this argument.
The quick roll-off at frequencies above the second resonant frequency also seems to support
a higher-order model. The accelerometer with the given actual parameters in Table 2.1
should have a second-order response seen in Figure 2.7, but it is obvious that the system
has a higher-order response. These issues will be further discussed in the next section.
32
Figure 2.7: Magnitude Plot of the Experimental and Theoretical Results using Second-
Order Model
A third test was done of the PCB MEMS accelerometer. The test set-up used a refer-
ence accelerometer as an input and the output voltage from the PCB MEMS accelerometer
as the output. The electromechanical shaker was excited at random frequencies in the range
of 10-1600 Hz. The test results are shown in Figure 2.8. The results show a linear ratio be-
tween the output voltage and the input acceleration for a given frequency. At the resonant
frequency (375 Hz), for an input acceleration of 1 g an output of approximately 4 mV is
given. The accelerometer should be operated in the low-frequency range (30-400 Hz) where
33
the curve is mostly flat in order for the accelerometer to give consistant results over the
range.
Figure 2.8: Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer
(Third Test, Electrical Response)
34
2.3 System Identification
The experimental data is examined in a control systems analysis technique known as
system identification. MATLAB?s System Identification Toolbox was used to develop some
higher-order models to fit the experimental data in order to get an insight into the relative
order of the system. The System Identification Toolbox is brought up in a GUI window of
MATLAB?s command window using the ?Ident? command. The GUI program allows the
user to import time domain or frequency domain experimental data. The experimental
data used was one of the mechanical response plots from the second test (Figure 2.9). The
System Identification Toolbox has three methods for estimating models: parametric estima-
tion, process model estimation, and nonparametric estimation. The parametric estimation
method was chosen due to its ability to allow the user to select the order of the polynomial
for the transfer function. The user then has the ability to select the number of poles, zeros,
and time delays.
In Figure 2.10 a transfer function with four poles, three zeros, and one delay was
estimated to fit the experimental data curve. The curve seems to look like a second-order
response, but having a resonant peak at the second resonant point. In Figure 2.11 a transfer
function with eight poles, six zeros, and three delays is shown. The curve fits better with
the experimental data and even seems to show the first resonant peak, but doesn?t show
the dip before the first resonant peak. In Figure 2.12 a transfer function with eight poles,
seven zeros, and one delay is shown. The curve fits really well with the experimental data
showing the first and second resonant points, and the dip before the first resonant point.
Although MATLAB?s System Identification Toolbox can be used to develop higher-
order models it is important to have a controls background to interpret the data. All three
35
of the generated transfer functions by MATLAB were unstable in discrete time showing zeros
and poles outside the unit circle. It is uncertain how MATLAB does numerically analysis of
the experimental data and many of the coefficients are extremely large which could lead to
the stability issues. A controls background can allow one to estimate the system order by
pole-zero placement using classical control system techniques [13]. A fourth order system
was estimated by placing two pairs of complex poles at the resonant points and a pair of
complex zeros at the dip before the first resonant point. The placed pole-zero system is
shown in Figure 2.13. The system matches the experimental data pretty well for a quick
estimate. The conclusion can be made by observing these models that the system is in fact
higher-order. A second-order model gives a pretty good estimate, but a higher-order model
gives a much better estimate of the actual system.
36
Figure 2.9: Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer
(Second Test, Mechanical Response)
37
Figure 2.10: Four pole, three zero, one delay transfer function (MATLAB SYSID)
38
Figure 2.11: Eight pole, six zero, three delay transfer function (MATLAB SYSID)
39
Figure 2.12: Eight pole, seven zero, one delay transfer function (MATLAB SYSID)
40
Figure 2.13: Fourth order pole-zero placement transfer function
41
Chapter 3
Conclusions & Future Work
3.1 Il Buono
A MEMS-based accelerometer was fabricated using printed circuit processing tech-
niques. The design, fabrication, and mechanical characterization of the PCB MEMS ac-
celerometer were discussed. The resonant frequency of an example accelerometer with a
square membrane of area 6.4 mm?6.4 mm was measured to be 375 Hz. The attractive
feature of PCB MEMS is that it enables monolithic integration of MEMS devices with
electronics using conventional printed circuit techniques. The advantages of the proposed
PCB MEMS technology include low-cost, ease of integration with electronics, suitability for
high-volume manufacturing, and large area applications.
3.2 Il Brutto
The experimental results of the fabricated MEMS accelerometer show that some of the
actual parameters didn?t match well with the designed parameters. This is largely due to
the two issues: 1) excess copper on the plate that increased the stiffness of the springs, and
2) the large gap increase due to thermal expansion of copper and KaptonR?. More devices
should be fabricated using other fabrication techniques to try to eliminate this large gap
change. The fabrication of PCB MEMS devices is still at the experimental stage and a
single technique hasn?t been developed for various devices thus many tests must be done to
refine the process for any one device.
42
3.3 Il Cattivo
The experimental results of the fabricated MEMS accelerometer show a higher-order
model than the predicted second-order model. This is most likely due to the fact that the
structure is flexible. Fabrication of smaller square plates should be done to see if/when the
structure will become rigid. Every rigid structure has a point where it becomes flexible due
to the large surface area to thickness ratio and material properties. It was not hypothesized
that the MEMS accelerometer would be flexible.
3.4 Future Work
The new hypothesis now accepts that the structure is flexible and the new question
is to what size will the device become rigid. The ability to have a rigid structure will
allow for lumped-element modeling of the MEMS accelerometer. The need to conduct tests
on multiple devices of the same geometry, but different sizes will help to provide better
statistical data on the performance of PCB MEMS accelerometers and how they behave.
Lastly, someone should look into applying the science of flexible structures to the modeling
of PCB MEMS. This would give further insight into better predicting the behavior of the
system.
43
Bibliography
[1] B. Borovic, F. L. Lewis, W. McCulley, A. Q. Liu, E. S. Kolesar, and D. O. Popa,
?Control issues for microlectromechanical systems,? IEEE Control Systems Magazine,
vol. 26, pp. 18?21, April 2006.
[2] E. Bryzek, A. Abbott, D. C. Flannery, and J. Maitan, ?Control issues for mems,? IEEE
Conf. Decision and Control, vol. 3, pp. 3039?3047, 2003.
[3] C. T.-C. Nguyen, ?Frequency-selective mems for miniaturized low-power communi-
cation devices,? IEEE Transactions on Microwave Theory and Techniques, vol. 47,
pp. 1486?1503, August 1999.
[4] J. Rogers, P. Ozmun, J. Hung, and R. Dean, ?Bi-directional gap closing mems actuator
using timing and control techniques,? Proceedings of the 32nd Annual Conference of
the IEEE Industrial Electronics Society (IECON?06), pp. 3149?3154, November 2006.
[5] R. Ramadoss, S. L. S, Y. Lee, V. Bright, and K. Gupta, ?Rf mems capacitive switches
fabricated using printed circuit processing techniques,? IEEE/ASME Journal of Mi-
croelectromechanical Systems, vol. 15, pp. 1595?1604, December 2006.
[6] X. Wang, J. Engel, and C. Liu, ?Liquid crystal polymer (lcp) for mems: processes and
applications,? J. Micromech. Microeng., vol. 13, pp. 628?633, 2003.
[7] J. N. Palasagaram and R. Ramadoss, ?Mems capacitive pressure sensor fabricated using
printed circuit processing techniques,? IEEE Sensors Journal, vol. 6, pp. 1374?1375,
December 2006.
[8] D. Fries, G. Steimle, S. Natarajan, S. Ivanov, H. Broadbent, and T. Weller, ?Maskless
lithography pcb/laminate mems for a salinity sensing system,? Proc. Int. Microelectron.
Packag. Soc. (IMAPS) Workshop on Packag. MEMS and Related Micro Integr./Nano
Syst., 2002.
[9] R. Jackson and R. Ramadoss, ?A mems-based electrostatically tunable circular mi-
crostrip patch antenna,? Journal of Micromechanics and Microengineering, vol. 17,
pp. 1?8, January 2007.
[10] J. Soderkvist, ?Similarities between piezoelectric thermal and other internal means of
exciting vibrations,? Journal of Micromechanics and Microengineering, vol. 3, pp. 24?
31, 1983.
[11] M.-H. Bao, Micro Mechanical Transducers: Pressure Sensors, Accelerometers and Gy-
roscopes, vol. 8. New York: Elsevier Science, 2000.
44
[12] R. Jackson, ?Mems based tunable microstrip patch antenna fabricated using printed
circuit processing techniques,? Master?s thesis, Auburn University, August 2006.
[13] R. C. Dorf and R. H. Bishop, Modern Control Systems. Prentice Hall, 10 ed., 2004.
45
Appendix A
Rigid Body Model for MEMS Accelerometer
substrate
plate
c
x
r
x
o
A
k
Figure A.1: Cross-sectional view of PCB MEMS device
Shown above in Figure A.1 is a spring-mass-damper rigid body model for a typical
MEMS accelerometer. The system has two plates, a top plate and the substrate. The plate
area is defined by A. The spring constant for the system is k, the damping constant for the
system is c, and the nominal gap distance between the two plates is xo. When an external
force is applied to the substrate, a displacement r is seen and results in a displacement of the
top plate given by x. The electrical and mechanical dynamics for the spring-mass-damper
are described as follows:
A.1 Electrical Dynamics
The capacitance for a parallel plate capacitor is given by
C = epsilon1Ax
o ?x
(A.1)
46
Through the charge-voltage relationship
q = Cv = epsilon1Avx
o ?x
(A.2)
The work is defined by
W = qv2 = epsilon1Av
2
2
parenleftbigg 1
xo ?x
parenrightbigg
(A.3)
The electrostatic force is the change in work
Fe = ?W?x = epsilon1A2
parenleftbigg v
xo ?x
parenrightbigg2
(A.4)
where epsilon1 is the permittivity of the surrounding gas, A is the plate area, v is the applied
voltage, and xo is the nominal gap distance.
A.2 Mechanical Dynamics
The inertia force, also known as Newton?s second law
Finertia = m?x (A.5)
The damping force
Fdamper = ?c(?x? ?r) (A.6)
47
The spring force, also known as Hooke?s Law
Fspring = ?k(x?r) (A.7)
The mechanical force is the sum of the inertia, damper, and spring forces
Fm = Finertia +Fdamper +Fspring (A.8)
The differential equation for the mechanical force is thus
Fm = m?x+c?x+kx (A.9)
where m is the proof mass, ?x is the acceleration of the plate mass, c is the damping con-
stant, ?x is the velocity of the plate mass, ?r is the velocity of the substrate, k is the spring
constant, x is the displacement of the plate mass, and r is the displacement of the substrate.
Equating the mechanical and electrical dynamic force equations yields a stable state thus
producing an electromechanical model for the MEMS accelerometer
m?x = ?c?x?kx+ epsilon1A2
parenleftbigg v
xo ?x
parenrightbigg2
(A.10)
48
A.3 Mechanical Transfer Function
m?x+c(?x? ?r) +k(x?r) = 0 (A.11)
ms2X(s) +csX(s)?csR(s) +kX(s)?kR(s) = 0 (A.12)
(ms2 +cs+k)X(s) = (cs+k)R(s) (A.13)
X(s)
R(s) =
(cs+k)
ms2 +cs+k (A.14)
one zero at
s = ?kc (A.15)
two poles at
s = ?c?
?c2 ?4mk
2m (A.16)
49
Appendix B
Rayleigh-Ritz Method for Determining Equivalent Mass of a Flexible
Structure
Figure B.1: Shape Function for Suspension Beam
Shown above in Figure B.1 is the shape function for a suspension beam. The length of
the beam is given by L. The maximum displacement due to a force is given by xp.
The shape function for a suspension beam is given by
xb(y) = xp
bracketleftBig
3(yL)2 ?2(yL)3
bracketrightBig
(B.1)
The maximum potential energy
Epmax = 12kx2p (B.2)
The maximum kinetic energy
Ekmax = 12
bracketleftbigg
v2pmp +
integraldisplay
v2bdmb
bracketrightbigg
(B.3)
50
The change in mass can be re-expressed dmb = mbdyL
Ekmax = 12v2pmp + 12
integraldisplay
v2bmbdyL (B.4)
The velocity can be re-expressed v = ?x
Ekmax = 12x2p?2nmp + 12Lmb
integraldisplay
(?nxb)2dy (B.5)
Substituting for xb
Ekmax = 12x2p?2nmp + 12Lmb
integraldisplay
(?nxp
bracketleftBig
3(yL)2 ?2(yL)3
bracketrightBig
)2dy (B.6)
Re-writing the kinetic energy equation
Ekmax = 12x2p?2n
bracketleftbigg
mp + 1335mb
bracketrightbigg
(B.7)
By the principle of energy conservation
Epmax = Ekmax (B.8)
Equating the kinetic and potential energy
1
2kx
2
p =
1
2x
2
p?
2
n
bracketleftbigg
mp + 1335mb
bracketrightbigg
(B.9)
51
The resonant frequency as a function of effective spring constant and proof mass
?2n =
bracketleftBigg
k
mp + 1335mb
bracketrightBigg
(B.10)
The effective proof mass is thus
m = mp + 1335mb (B.11)
where mp is the plate mass and mb is the beam mass.
The plate and beam masses are given by
mp = ?Vp (B.12)
mb = ?Vb (B.13)
52
Appendix C
Linear State Variable Analysis for PCB MEMS Accelerometer
C.1 Motivation
Electrostatic Micro-Electro Mechanical Systems, or MEMS, are an interesting problem
in control systems in that they tend to have second-order nonlinear system dynamics that
resemble those of mass-spring-damper systems with a capacitive-like structure. If a structure
is rigid the following could be used to design a feedback control system. The objective is to
design a control system that stabilizes the output y = x, the actuator position.
C.2 Dynamic Model
Dynamics of the PCB MEMS Accelerometer are described by the nonlinear ordinary
differential equation
m?x = ?c?x?kx+ epsilon1oAV
2
2
1
(xo ?x)2 (C.1)
Electrostatics of the PCB MEMS Accelerometer are described by the nonlinear equation
kx = epsilon1oAV
2
2
1
(xo ?x)2 (C.2)
Let m = 4.375?10?6 kg, c = 6.7?10?3 N-sec/m, k = 24.791 N/m, A = 34.34 mm2, and
xo = 125 ?m.
53
C.3 State Variables
Let the state be
z =
?
?? x
?x
?
??
The plant input is u = V, and there is one output. Rewriting dynamic equation C.1 in
state variable form
?z = f(z,u) (C.3)
y = h(z) (C.4)
?z1 = z2
?z2 = ? cmz2 ? kmz1 + epsilon1oAV
2
2m
1
(xo ?z1)2
C.4 Equilibrium State
The equilibrium state ze, ue can be determined by observing the position versus applied
voltage graph (Figure C.1) for the electrostatic equation C.2.
The chosen equilibrium point satisfies equation C.2 ze = [10.5?10?6 0], ue = [150].
54
Figure C.1: Normalized pull-down voltage curve
C.5 Linearized State Variable Model
Find the linearized state variable model of the form
??z = A?z +B?u (C.5)
?y = C?z (C.6)
where ?z = z ?ze and ?u = u?ue.
Recall that there is one output variable, y = x.
A = ?zf|ze,ue =
?
?? 0 1
? km + epsilon1oAV 2m 1(xo?z1)3 ? cm
?
??
ze,ue
55
A =
?
?? 0 1
?4.6?106 ?1532
?
??
B = ?uf|ze,ue =
?
?? 0
epsilon1oAV
m
1
(xo?z1)2
?
??
ze,ue
B =
?
?? 0
0.80
?
??
C = ?zh|ze,ue =
bracketleftbigg
1 0
bracketrightbigg
ze,ue
=
bracketleftbigg
1 0
bracketrightbigg
C.6 Stability
Analyze stability of the linear state variable model (C.5), (C.6).
? ?z1 = z2
? ?z2 = ?5.6?106z1 + 119.3?1532z2 + 0.80u
Assume a positive-definite Lyapunov function
V(z) = z21 + 2z1z2 +z22
d
dtV(z) = (?zV)?z
?zV =
bracketleftbigg
2z1 + 2z2 2z1 + 2z2
bracketrightbigg
56
The function ?V(z) is negative-definite and thus solutions for ?z are asymptotically stable.
C.7 Controllability
rank
bracketleftbigg
B AB
bracketrightbigg
= 2 = dim(z)
The rank of (A, B) is equal to the dimension of z, thus the system is controllable.
C.8 Observability
rank
?
?? C
CA
?
?? = 2 = dim(z)
The rank of (A, C) is equal to the dimension of z, thus the system is observable.
C.9 Stabilizability
|sI ?A| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
s ?1
4.6?106 s+ 1532
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
= s2 + 1532s+ 4.6?106 = 0
s = ?766?2003i
The system is naturally stable, thus it is stabilizable.
57
C.10 Detectability
The system is naturally stable, thus it is stabilizable and detectable, also controllable
and observable.
C.11 Linear State Feedback
A linear state feedback ?u = ?K?z is to be designed to place eigenvalues of (A?BK)
at ?800?j800.
A?BK = AC =
?
?? 0 1
?4.6?106 ?0.80K1 ?1532?0.80K2
?
??
|sI ?(A?BK)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
s ?1
4.6?106 + 0.80K1 s+ 1532 + 0.80K2
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
= s2 + (1532 + 0.80K2)s+ (4.6?106 + 0.80K1) = 0
desire s = ?800?j800
(s+ 800)2 + (800)2 = s2 + 1600s+ 128?104 = 0
K1 = ?4.15?106,K2 = 85
K =
bracketleftbigg
K1 K2
bracketrightbigg
=
bracketleftbigg
?4.15?106 85
bracketrightbigg
58
C.12 Observer Gain
An observer gain L is to be designed so that the estimation error dynamics are char-
acterized by the eigenvalues ?900?j900.
A?LC = AO =
?
?? ?L1 1
?4.6?106 ?L2 ?1532
?
??
|sI?(A?LC)| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
s+L1 ?1
4.6?106 +L2 s+ 1532
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
= s2+(1532+L1)s+(4.6?106+1532L1+L2) = 0
desire s = ?900?j900
(s+ 900)2 + (900)2 = s2 + 1800s+ 162?104 = 0
L1 = 268,L2 = ?3.39?106
L =
?
?? L1
L2
?
?? =
?
?? 268
?3.39?106
?
??
C.13 Simulation
Simulate the closed loop behavior of the nonlinear system (C.3) under the linear state
feedback control designed in Section C.5. Let the initial condition be z(0) = [0.5?10?6 0]T.
Plot the time response of the GCA position x.
Simulate the closed loop behavior of the nonlinear system (C.3) under the estimated state
feedback control ?u = ?K??z. Use the observer designed in Section C.6. Let the initial
59
Figure C.2: Block diagram for linear state feedback control
condition be z(0) = [0.5 ? 10?6 0]T, and the initial value of the estimated state be ??z =
[0 0]T. Plot the time response of the GCA position x. Plot the estimation error x? ?x.
Since the damping in the problem is so small relative to the mass, increasing the damping
constant, c, would allow for better control. This is not usually an easy thing to do, but
by isolating the device within the right gas and pressure would do just this. The system is
naturally a very stiff system and controlling the damping will thus allow for better control
of K and L.
60
Figure C.3: Output response for linear state feedback control
Figure C.4: Block diagram for estimated state feedback control
61
Figure C.5: Output response for estimated state feedback control
Figure C.6: Estimation error for estimated state feedback control
62
Appendix D
MATLAB Code
Figure D.1: PCB MEMS Accelerometer Parameters MATLAB Code
63
Figure D.2: PCB MEMS Accelerometer Paramaters MATLAB Code (Cont.)
64