DEVELOPMENT OF MEMS-BASED PIEZOELECTRIC ...ufdcimages.uflib.ufl.edu/UF/E0/01/56/46/00001/vs_a.pdf1 DEVELOPMENT OF MEMS-BASED PIEZOELECTRIC CANTILEVER ARRAYS FOR VIBRATIONAL ENERGY

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DEVELOPMENT OF MEMS-BASED PIEZOELECTRIC CANTILEVER ARRAYS FOR VIBRATIONAL ENERGY HARVESTING

By

ANURAG KASYAP V. S.

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

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Copyright 2007

by

Anurag Kasyap V.S.

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To my father

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ACKNOWLEDGMENTS

Financial support for the research project was provided by NASA.

First, I thank my advisor Dr. Louis N. Cattafesta for his guidance and support, which was

vital for completing my dissertation. I also thank my co-advisor Dr. Mark Sheplak for advising

and guiding me with various aspects of the project. I would also like to thank Dr. Toshi Nishida

for helping me understand the electrical engineering aspects of the project.

Drs. Khai Ngo and Bhavani Sankar deserve special thanks for finding time to help me out

with the project whenever I approached them. I thank all the members of the Interdisciplinary

Microsystems group, especially fellow students Steve Horowitz and Yawei Li for their help with

my research.

I also thank the University of Florida Department of Aerospace Engineering, Mechanics,

and Engineering Science for their financial support.

Finally, I want to thank my family and friends for their endless support, particularly my

parents whose affection and encouragement has been the driving force for my success as a

student and more importantly as a person.

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TABLE OF CONTENTS page

ACKNOWLEDGMENTS ...............................................................................................................4

LIST OF TABLES...........................................................................................................................8

LIST OF FIGURES .......................................................................................................................10

ABSTRACT...................................................................................................................................16

1 INTRODUCTION...................................................................................................................18

Energy Reclamation................................................................................................................18 Energy Resources and Harvesting Technologies ............................................................19 Self-Powered Sensors......................................................................................................21

Vibration to Electrical Energy Conversion.............................................................................23 Transduction Mechanisms...............................................................................................24

Electrodynamic transduction....................................................................................29 Electrostatic transduction .........................................................................................31 Piezoelectric transduction ........................................................................................33

Microelectromechanical Systems (MEMS)............................................................................40 Piezoelectric MEMS........................................................................................................45

Objectives of Present Work ....................................................................................................46 Organization of Dissertation...................................................................................................46

2 PIEZOELECTRIC CANTILEVER BEAM MODELING AND VALIDATION..................48

Piezoelectric Composite Beam...............................................................................................49 Analytical Static Model ..........................................................................................................54

Static Electromechanical Load in the Composite Beam .................................................55 Experimental Verification of the Lumped Element Model ....................................................70

3 MEMS PIEZOELECTRIC GENERATOR DESIGN.............................................................84

Power Transfer Analysis.........................................................................................................84 Nondimensional Analysis.......................................................................................................87

Scaling Theory...............................................................................................................110 Validation of Scaling Theory ........................................................................................116

Extension to MEMS .............................................................................................................122 Design of Test Structures ..............................................................................................122

Test devices ............................................................................................................127

4 DEVICE FABRICATION AND PACKAGING..................................................................130

Process Flow.........................................................................................................................130 Process Traveler ............................................................................................................147

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Packaging..............................................................................................................................148 Vacuum Package ...........................................................................................................148 Open Package ................................................................................................................150

5 EXPERIMENTAL SETUP...................................................................................................153

Ferroelectric Characterization Setup ....................................................................................153 Piezoelectric Characterization .......................................................................................155

Electrical Characterization....................................................................................................156 Blocked Electrical Capacitance, ebC and Dielectric Loss, eR ......................................156

Mechanical Characterization ................................................................................................159 Electromechanical Characterization .....................................................................................163 Open Circuit Voltage Characterization ................................................................................164 Voltage and Power Measurements .......................................................................................166

6 EXPERIMENTAL RESULTS AND DISCUSSION............................................................168

Ferroelectric Characterization ..............................................................................................168 Blocked Electrical Impedance Measurements......................................................................179 Lumped Element Parameter Extraction................................................................................185

Method 1........................................................................................................................188 Method 2........................................................................................................................197 Method 3........................................................................................................................201

Results and Discussion .........................................................................................................206 PZT-EH-09 ....................................................................................................................206 PZT-EH-07 ....................................................................................................................212

Summary and Discussion of Results ....................................................................................215

7 CONCLUSIONS AND FUTURE WORK...........................................................................224

Conclusions...........................................................................................................................224 Future Work..........................................................................................................................230

Second Generation Design Procedure ...........................................................................232 Electromechanical Conversion Metrics.........................................................................233

A EULER-BERNOULLI BEAM ANALYSIS: VARIOUS BOUNDARY CONDITIONS..237

Euler Bernoulli Beam ...........................................................................................................237 Cantilever Beam (Clamped-Free Condition).................................................................237 Clamped-Clamped Beam (Fixed-Fixed Condition) ......................................................239 Pin-Pin Beam (Simply Supported) ................................................................................243

B DISSIPATION MECHANISMS FOR A VIBRATING CANTILEVER BEAM................248

Introduction...........................................................................................................................248 Overall Mechanical Quality Factor ......................................................................................249 Dissipation Mechanisms.......................................................................................................250

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Airflow Damping...........................................................................................................251 Intrinsic region : .....................................................................................................252 Molecular region : ..................................................................................................252 Viscous region........................................................................................................253

Support Losses...............................................................................................................254 Surface Dissipation........................................................................................................254 Volume Loss..................................................................................................................254 Squeeze Damping Loss .................................................................................................255 Thermoelastic Dissipation .............................................................................................255

Analytical model ....................................................................................................257

C TRANSFORMATION OF COORDINATES FOR RELATIVE MOTION........................265

D ELECTRICAL IMPEDANCE FOR A PIEZOELECTRIC MATERIAL............................267

E CONJUGATE IMPEDANCE MATCH FOR MAXIMUM POWER TRANSFER.............270

F UNDESTANDING THE PHYSICS OF THE DEVICE.......................................................274

G FABRICATION LAYOUTS.................................................................................................274

LIST OF REFERENCES.............................................................................................................282

BIOGRAPHICAL SKETCH .......................................................................................................292

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LIST OF TABLES

Table page 1-1 Conjugate power variables for different energy domains..................................................26

1-2 Vibration based energy harvesters characterterized for power..........................................40

2-1 Material properties and dimensions for a homogenous aluminum beam. .........................61

2-2 Material properties and dimensions for a piezoelectric composite aluminum beam.........65

2-3 Material properties and dimensions for a homogenous aluminum beam. .........................70

2-4 Measured and calculated parameters for the homogenous beam.......................................71

2-5 Measured and calculated parameters for the homogenous beam with a proof mass. ........74

2-6 Material properties and dimensions for a piezoelectric composite aluminum beam.........75

2-7 Measured and calculated values for a PZT composite beam.............................................76

2-8 Measured and calculated parameters for a PZT composite beam with a proof mass. .......77

2-9 Comparison between experimental and theoretical values for power transfer. .................82

3-1 List of all device variables that are described in the electromechanical model.................88

3-2 Dimensional representation of all the device variables. ....................................................89

3-3 Primary variables used in the dimensional analysis. .........................................................90

3-4 List of independent ∏ groups. ...........................................................................................93

3-5 Final set of nondimensional groups involving response parameters. ..............................109

3-6 Material dimensions and properties of composite beam for FEM validation..................117

3-7 Static lumped element parameters from FEM and LEM to validate the scaling analysis.............................................................................................................................121

3-8 Properties and dimensions used for designing MEMS PZT devices...............................123

3-9 Material properties of piezoelectric composite beam. .....................................................128

3-10 Designed MEMS PZT structures. ....................................................................................129

4-1 Residual stress measurements for the PZT pattern process (source : ARL)....................133

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4-2 DRIE recipe conditions for top side etch.........................................................................136

4-3 DRIE recipe conditions for back side etch. .....................................................................142

4-4 Process traveler for the fabrication of micro PZT cantilever arrays................................147

5-1 Reported polarization results (ref: ARL) .........................................................................156

5-2 Data acquisiton parameters for mechanical characterization...........................................163



6-1 Comparison of ARL's reported hysteresis parameters with measured values. ................178

6-2 Dielectric parameters of all tested design geometries on the device wafer. ....................182

6-3 LEM parameters extracted using experimental data.......................................................185

6-4 LEM parameters extracted using Method 1.....................................................................197



6-7 LEM parameters extracted for PZT-EH-09-01................................................................207

6-8 Extracted LEM parameters for PZT-EH-09-03. ..............................................................212

6-9 LEM parameters extracted for PZT-EH-07-02................................................................212

6-10 Comparison between theory and experiments for PZT-EH-07. ......................................215

6-11 Comparison between theory and experiments for PZT-EH-09 devices. .........................216

6-12 Quality factors for PZT MEMS devices. .........................................................................219

A-1 LEM parameters and bending strain for various beams subjected to a point load. .........246

A-2 LEM parameters and bending strain for various beams subjected to uniform load. .......247

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LIST OF FIGURES

Figure page 1-1 Schematic of a typical vibration to electrical energy converter.........................................24

1-2 An electromagnetic vibration-powered generator (adapted from Glynne-Jones and White 2001). ......................................................................................................................30

1-3 Deformation of a piezoceramic material under the influence of an applied electric field. ...................................................................................................................................33

1-4 A nonlinear piezoelectric vibration powered generator (adapted from Umeda et al, 1997). .................................................................................................................................38

1-5 Schematic of the proposed cantilever configuration for energy reclamation. ...................44

2-1 Schematic of a piezoelectric composite beam subject to a base acceleration....................50

2-2 Overall equivalent circuit of composite beam. ..................................................................52

2-3 Schematic of the piezoelectric cantilever composite beam. ..............................................55

2-4 Free body diagram of the overall configuration. ...............................................................56

2-5 Free body diagram of the composite beam where the self weights are replaced with equivalent loads. ................................................................................................................57

2-6 Static model verified with the ideal solution for a homogenous beam solved for self weight.................................................................................................................................61

2-7 Static model verified with the ideal solution for a homogenous beam solved for tip load.....................................................................................................................................62

2-8 Deflection modeshape for a composite beam subjected to an input voltage. ....................66

2-9 Experimental setup for verifying the electro-mechanical lumped element model for meso-scale cantilever beams..............................................................................................72

2-10 Comparison between experiment and theory for tip deflection in a homogenous beam (no tip mass).......................................................................................................................73

2-11 Comparison between theory and experiments for the tip deflection in a homogenous beam with tip mass.............................................................................................................74

2-12 Frequency response of a piezoelectric composite beam (no tip mass) ..............................77

2-13 Frequency response for a piezoelectric composite beam (mp=0.476 gm). ........................78

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2-14 Output voltage for an input acceleration at the clamp. ......................................................80

2-15 Output voltage for varying resistive loads. ........................................................................81

2-16 Output power across varying resistive loads. ....................................................................82

3-1 Thévenin equivalent circuit for the energy reclamation system ........................................85

3-2 Schematic of the MEMS PZT device. ...............................................................................88

3-3 Meshed PZT composite cantilever beam for FEM validation. ........................................117

3-4 Short circuit natural frequency for a PZT composite beam.............................................118

3-5 Short circuit compliance for a PZT composite beam.......................................................119

3-6 Effective mechanical mass for a PZT composite beam. ..................................................120

3-7 Effective piezoelectric coefficient for a PZT composite beam........................................121

3-8 Schematic of a single PZT composite beam. ...................................................................123

4-1 Deposit 100 nm blanket SiO2 (PECVD) on SOI wafer...................................................131

4-2 Sputter deposit Ti/Pt (20 nm/200 nm) as bottom electrode..............................................131

4-3 Spin coat sol-gel PZT (125/52/48) over the wafer using a spin-bake-anneal process.....132

4-4 Deposit and pattern Pt for top electrode using liftoff. .....................................................132

4-5 Pattern opening for access to bottom electrode and wet etch PZT using PZT Etch mask. ................................................................................................................................133

4-6 Ion milling of PZT and bottom electrode using Ion Milling mask as pattern..................133

4-7 Deposit Au (300 nm) and pattern bond pads using Bond Pads mask and wet etching....134

4-8 Sidewall profiles on topside of a 4" Si test wafer. ...........................................................136

4-9 Wet etch exposed oxide with BOE and DRIE to BOX from top.....................................137

4-10 Sidewall profiles for backside etching using DRIE.........................................................138

4-11 Curved edges during backside DRIE...............................................................................139

4-12 Onset of silicon grass during a backside etch run............................................................140

4-13 Sidewall profiles for a backside etch on a test wafer.......................................................141

4-14 Pattern proof mass on the backside and DRIE to BOX. ..................................................143

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4-15 Schematic of final released device...................................................................................144

4-16 SEM pictures of a PZT-EH-07 released device. ..............................................................145

4-17 SEM pictures of a PZT-EH-09 released device. ..............................................................145

4-18 Sidewall profiles of released devices. ..............................................................................146

4-19 Schematic of the bottom of vacuum package for MEMS PZT devices...........................149

4-20 Schematic of glass top for vacuum package. ...................................................................149

4-21 An isometric view of the overall vacuum package..........................................................150

4-22 Schematic of open package for MEMS PZT devices. .....................................................151

4-23 Picture of the open package. ............................................................................................152

5-1 Schematic for ferroelectric characterization. ...................................................................154

5-2 Experimental setup for ferroelectric characterization......................................................155

5-3 Schematic for blocked electrical impedance measurement. ............................................158

5-4 Experimental setup for electrical impedance characterization. .......................................158

5-5 Experimental setup for mechanical and electromechanical characterization. .................160

5-6 Experimental setup for vibration and velocity measurements with LV. .........................161

5-7 Experimental setup for open circuit voltage measurements. ...........................................165

5-8 Experimental setup for open circuit voltage measurements. ...........................................165

5-9 Experimental setup for voltage and power measurements. .............................................166

6-1 A typical P-E hysteresis loop for a piezoelectric material (adapted from Cady 1964)....169

6-2 A typical ε-E curve for a piezoelectric material. .............................................................170

6-3 Polarization, capacitance and input voltage waveforms for PZT-EH-02-1-1..................171

6-4 Hysteresis plots for PZT-EH-02-1-1................................................................................172

6-5 Pr and Vc for different applied voltages for PZT-EH-02-1-1...........................................173

6-6 Normalized Ceb for PZT-EH-02-1-1 during the hysteresis test. ......................................174

6-7 Leakage current for PZT-EH-02-1-1 subjected to 10V DC.............................................175

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6-8 Poling of PZT-EH-02-1-1 at 5V for different times. .......................................................176

6-9 Poling of PZT-EH-02-1-1 at different temperatures........................................................178

6-10 Variation of Ceb and tanδ with dc bias and a constant sinusoid, 500 mV at 100 Hz. ......179

6-11 Variation of Ceb and tanδ with source amplitude at 100Hz. ............................................180

6-12 Ceb and εr for MEMS PZT devices on wafer before release for a) PZT-EH-01 (106 geometries) b) PZT-EH-02 (16 geometries) c) PZT-EH-03 (15 geometries) d) PZT-EH-04 (14 geometries) e) PZT-EH-05 (16 geometries) ..................................................183

6-13 Ceb and εr for MEMS PZT devices on wafer before release for a) PZT-EH-06 (150 geometries) b) PZT-EH-07 (12 geometries) c) PZT-EH-08 (22 geometries) d) PZT-EH-09 (108 geometries)...................................................................................................184

6-14 Flowchart for method 1 to extract the LEM parameters from the experimental data......190

6-15 Low frequency electromechanical response data compared with curve fit to extract dm. ....................................................................................................................................194

6-16 Comparison between experiment and LEM based curve fit around resonance for a) electromechanical response b) short-circuit mechanical response ..................................195

6-17 Low frequency curve fit compared with experiment to extract Ceb.................................195

6-18 Comparison between experiment and curve fit for low frequency open circuit voltage response to extract Mm. ....................................................................................................196

6-19 Experimental data and curve fits for open circuit voltage response compared around resonance..........................................................................................................................196

6-20 Flowchart for parameter extraction using Method 2........................................................199

6-21 Experimental data and curve fits for open circuit voltage response and free electrical impedance compared around resonance. .........................................................................201

6-22 Flowchart for LEM parameter extraction implementing Method 3.................................203

6-23 Comparison between experiment and LEM based curve fit for short circuit mechanical and electromechanical response around resonance. .....................................205

6-24 Experimental data and curve fits for open circuit voltage response compared around resonance..........................................................................................................................205

6-25 Comparison between model and experiments for PZT-EH-09-01. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance

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response D) Open circuit voltage response E) Normalized output voltage and power across resistive loads at resonance...................................................................................208

6-26 Comparison between model and experiments for PZT-EH-09-02. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive loads at resonance...................................................................................209





A-1 Schematic of a cantilever beam. ......................................................................................237

A-2 A schematic of clamped-clamped beam. .........................................................................240

A-3 Free body iagram of a clamped-clamped beam. ..............................................................240

A-4 Schematic of a pin-pin beam............................................................................................243

A-5 Free body diagram for a simply supported beam.............................................................243

B-1 A simple schematic of the cantilever beam. ....................................................................251

C-1 Vibrating cantilever beam in an accelerating frame of reference. ...................................265

D-1 Blocked electrical impedance in a parallel network representation.................................267

D-2 Blocked electrical impedance in a series network representation ...................................269

E-1 Thevenin equivalent representation connected to a external complex impedance. .........270

E-2 Thevenin equivalent representation connected to a resistive load...................................273

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F-1 Schematic of the composite beam energy harvester. .......................................................275

F-2 Free body representation of the device as a two degree of freedom system....................275

F-3 Electromechanical circuit representation of the energy harvester. ..................................276

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

DEVELOPMENT OF MEMS-BASED PIEZOELECTRIC CANTILEVER ARRAYS FOR VIBRATIONAL ENERGY HARVESTING

By

Anurag Kasyap V.S.

May 2007

Chair: Louis Cattafesta Cochair: Mark Sheplak Major: Aerospace Engineering

In this dissertation, the development of a first generation MEMS-based piezoelectric

energy harvester is presented that is designed to convert ambient vibrations into storable

electrical energy. The objective of this work was to model, design, fabricate and test MEMS-

based piezoelectric cantilever array structures to harvest power from source vibrations.

The proposed device consists of a piezoelectric composite cantilever beam

( 2Si SiO Ti Pt PZT Pt ) with a proof mass at one end. The proof mass essentially translates the

input base acceleration to an effective deflection at the tip relative to the clamp, thereby

generating a voltage in the piezoelectric layer (using 31d mode) due to the induced strain. An

analytical electromechanical lumped element model (LEM) was formulated to accurately predict

the behavior of the piezoelectric composite beam until the first resonance.

First, macro-scale PZT composite beams were built and tested to validate the LEM. In

addition, a detailed non-dimensional analysis was carried out to observe the overall device

performance with respect to various dimensions and properties. Various first generation test

structures were designed using a parametric search strategy subject to fixed vibration inputs and

constraints.

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The proposed test structures thus designed using the electromechanical LEM were

fabricated using standard sol gel PZT and conventional surface and bulk micro processing

techniques. The devices have been characterized with various frequency response measurements

and the lumped element parameters were extracted from experiments. Finally, they were tested

for energy harvesting by measuring the output voltage and power at resonance for varying

resistive loads.

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CHAPTER 1 INTRODUCTION

This dissertation discusses the modeling, design, fabrication, and characterization of an

array of micromachined piezoelectric power generators to harness vibration energy. The

reclaimed power is rectified and stored using a power processor (Taylor et al. 2004, Kymissis et

al. 1998) for subsequent use by, for example, sensors. The details of this concept are discussed

in subsequent sections. This chapter begins with an introduction to energy reclamation, various

available resources, and harvesting technologies. Then, a detailed description is presented

concerning energy reclamation from vibration and its uses in various fields such as self-powered

sensors, human-wearable electronics and vibration control. Finally, it concludes with motivation

for microelectromechanical systems (MEMS) and piezoelectricity as the tools for this research.

An in-depth literature survey is presented to familiarize the reader with the previous and current

work in these fields.

Energy Reclamation

Conservation of energy is a fundamental concept in physics along with the conservation of

mass and Newton’s laws. The law of conservation of energy states that energy can neither be

created nor destroyed but only converted from one form to another. A useful description of this

law in a thermodynamic system is the first law of thermodynamics. It states that the difference

between the total rate of inflow of energy into a system minus the total rate of outflow of energy

from the system (to the surroundings) equals the time rate of change of energy contained within

the system. Therefore, energy reclamation, by definition, relates to converting any form of

energy that is otherwise lost to the surroundings into some form of useful power.

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Energy Resources and Harvesting Technologies

There are two classes of available energy sources, renewable and non-renewable. Non-

renewable sources, as the name suggests, include all that have a limited supply such as oil, coal,

natural gas, etc. These sources take thousands or millions of years to form naturally and cannot

be replaced once consumed. They have constituted the major part of the United States (U.S.)

power supply for a long time. But, with increasing technology and society’s ever-growing

consumption of energy, these sources could soon be exhausted (National Energy Policy Report,

2001). Hence, it is an ecological and economical necessity to investigate alternate sources of

energy to meet societal demands. Consequently, research in the past few decades has focused on

using an alternate form, called renewable resources, to meet the demand, such as optical, solar,

tidal, etc.

Jan Krikke, in his editorial article in “Pervasive Computing” (2005) reviews the current

situation in energy harvesting technologies. Many companies in the US, Europe, and Japan are

steadily involved in this area as there exists a general fascination with energy scavenging from

ambient sources. Many energy harvesting concepts are already available such as a self-reliant

house (powered by solar energy that operates all appliances in the house) and a camel fridge,

which uses solar energy to operate a refrigerator used to store (below o8 C ) and transport

vaccines in African nations.

Previous studies have successfully shown that energy can be reclaimed from renewable

sources such as solar and tidal energy (Saraiva 1989). Solar cells are an existing technology that

is extensively used in self-powered watches, calculators, and rooftop modules for houses. Solar

energy has also been harvested on a smaller scale from an array of micro-fabricated photovoltaic

cells to produce an overall open circuit voltage of 150 V and a short circuit current of 2.8 Aμ

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(Lee et al. 1995). While solar energy has been widely explored and implemented, it becomes

difficult to generate power in dark areas.

Even though renewable sources can serve as a substitute for the usual power supply

resources, energy is still wasted in the form of heat, sound, light and vibrations that can be

further reclaimed, at least partially, for future use. For example, thermal energy was generated

from a 20.75 0.9 cm× bismuth-telluride thermoelectric junction to produce 23.5 Wμ for a

temperature difference of 20 K (Stark and Stordeur, 1999). Qu et al. in 2001 designed and

fabricated a thermoelectric generator, 316 20 0.05 mm× × , consisting of multiple micro Sb-Bi

thermocouples embedded in a 50 mμ epoxy film capable of producing 0.25 V from a

temperature difference of 30 K . Kiely et al. (1991; 1994) designed a low cost miniature

thermoelectric generator consisting of a silicon on sapphire and silicon on quartz substrate.

Another thermoelectric power generator based on silicon technology produced 1.5 Wμ with a

temperature difference of 10 C (Glosch et al. 1999). Of all the renewable sources, optical and

thermal energy have been the most popular and widely implemented, even in micro power

requirements. However, in applications where light and thermal energy are not readily available,

alternate sources need to be considered such as mechanical energy. In addition, an advantage for

mechanical energy conversion over thermal conversion is that, ideally, it does not require any

heat isolation. In addition, scaling thermal systems to microscale possesses fundamental

limitations such as thermal related noise due to thermal fluctuations, temperature based

adsorption, etc. (Devoe 2003).

In recent years, extensive research has been conducted on harvesting undesirable

vibrational energy. Although most efforts have been in the area of mesoscale energy harvesting,

the focus on microscale has gained importance lately. The energy thus claimed from vibrational

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sources can be stored and later used to power various devices. In the past, efforts in energy

reclamation from vibrations have largely focused on the available energy in human ambulation

(Starner 1996). Reclaiming energy from human ambulation has generated immense interest

primarily because of its ability to power artificial organs and human wearable electronic devices.

Growing interest in the area of human-wearable electronic devices creates a need for portable

power sources for these devices. Starner and Paradiso (2005) describes various sources from

humans for energy harvesting such as body heat, breath, blood-pressure, walking, etc. In

addition, heel strike, limb movement, and other gait-related activities are useful sources of strain

energy and can be used as alternate methods for powering artificial organs (Antaki et al. 1995).

This could replace conventional portable batteries that are currently restricted by energy

limitations, especially for prolonged usage. In addition, batteries are often bulky and possess a

limited shelf life and could be potentially hazardous due to chemicals. The development of

MEMS technology has led to a wide range of applications for micro actuators and sensors (see,

for example, Senturia 2000). It also has enabled implantation of these devices into various host

structures, such as medical implants and embedded sensors in buildings and bridges (Mehregany

and Bang 1995). In most of these applications, the devices need to be completely isolated from

the outside world. These remote devices, along with their accompanying circuitry, have their

own power supply that is usually powered by batteries. The strides achieved in battery

technology have not sufficiently matched the improvements in integrated circuit technology.

Therefore, developing a micro-scale self-contained power supply offers great potential for

applications in remote systems.

Self-Powered Sensors

The ever-reducing size of CMOS circuitry and correspondingly lower power consumption

have also provided immense opportunities to design and build micro power generators that can

22

be ideally integrated with CMOS. Simultaneous research is also being carried out to develop

new chip technology to lower the power requirement for electronic equipment (Krikke 2005).

The need for self-contained power generators has led to the development of “self-powered

systems” that is an important application for energy reclamation, and is currently gaining

widespread importance (Shenck and Paradiso 2001). Self-powered systems possess an inherent

mechanism to extract power from the ambient environment for their operation. The main

objective of self-powered systems is to utilize a generator that can convert energy from an

ambient source to electrical energy as long as sufficient energy is available in the ambient

source. Consequently, the primary features of self-powered systems include power generation,

energy extraction, and storage. Ideally, a self-powered device should possess high power density

for given size constraints. Attempts to build perpetual motion machines date back to as early as

the 13th century when the conservation laws had not yet been formulated. Glynne-Jones and

White (2001) provide a review on available energy resources for self-powered sensors such as

vibrations, optical, thermoelectric, etc.

Next, some of the relevant work carried out in the field of self-powered sensors is

examined. As mentioned earlier, heel strike is a resource for strain energy that can be

electromechanically transformed into electrical power. Consequently, shoe-mounted devices

have been developed and tested that convert strain energy induced during heel strike and store it

as electrical energy. Kymissis et al (1998) and Shenck and Paradiso (2001) designed two novel

piezoelectric devices to harness power that were embedded in a shoe. Furthermore, vibrations

when available are excellent potential sources for energy harvesting. Meso-scale energy

reclamation approaches include rotary generators (Lakic 1989), a moving coil electromagnetic

generator (Amirtharajah 1998), and a dielectric elastomer with compliant electrodes (Pelrine

23

2001). Single meso-scale piezoelectric cantilevers (Ottman et al. 2002) and stacks (Goldfarb and

Jones 1999) have been investigated for energy reclamation but were not operated in a stand-

alone, self-powered mode. Another source for power harvesting is mechanical energy from fluid

flow. Taylor et al. (2001) designed an energy harvesting eel that was approximately 1 m long

using a piezoelectric polymer to convert fluid flow and vortex-induced strain to generate power.

In addition, Allen and Smits (2001) investigated the feasibility of utilizing a piezoelectric

membrane in the wake of a bluff body to induce oscillations in the structure generating a

capacitance build-up that acts as a voltage source to power a battery in a remote location. Power

generation from ocean waves has also been investigated involving very large-scale piezoelectric

generators (Smalser 1997). As a result, there is a clear indication that energy reclamation from

strain energy is a promising field in terms of research and applications. The focus of this

dissertation is to study the possibility of using vibrational mechanical energy as a potential

source for energy reclamation on a micro scale. Ambient vibration sources, such as household

appliances, machinery equipment, and HVAC ducts typically occur at frequencies in the range of

100’s of Hz with an acceleration amplitude of 1-10 m/s2 (Roundy et al. 2003).

Vibration to Electrical Energy Conversion

Continuing the discussion on converting vibrational energy to electricity, this can be

achieved using a transduction mechanism that effectively converts energy from the mechanical

domain to the electrical domain. A simple schematic of a power generator based on vibration is

shown in Figure 1-1. The device consists of a spring-mass-damper system acting as a single

degree of freedom system with an input vibration that results in an effective displacement ( )z t .

24

Figure 1-1: Schematic of a typical vibration to electrical energy converter.

The following equation is used to represent the behavior of the above system that basically

converts the kinetic energy of a vibrating structure to electrical energy by virtue of the relative

motion between the base and the inertial mass.

Mz Rz Kz My+ + = − (1.1)

where z is the relative deflection, y is the input displacement, M is the inertial mass, K is the

spring constant and R is the effective damping in the system that accounts for mechanical and

electrical losses. The above model does not include nonlinear effects and is thus valid only

under the constraints of linear system theory. It also does not specify the electromechanical

transduction mechanism with which the kinetic energy is converted to electrical power. These

mechanisms are discussed in detail in the following sections.

Transduction Mechanisms

Vibrational energy reclamation can be achieved conceptually using different transduction

mechanisms. Any transduction mechanism relates to energy conversion from one form to

25

another. For example, it can involve coupling of two or more energy domains such as

electrostrictive coupling (Uchino et al 1980), electromagnetic (Hanagan 1997; Kato 1997) and

electromechanical coupling (Lee 1990). In his Ph.D. dissertation, Roundy (2003) calculates the

theoretical maximum and the practical maximum for the energy densities of various transduction

mechanisms, namely piezoelectric, electrostatic, and electromagnetic. The expressions were

obtained from the basic governing equations of each of the materials and calculated using

maximum yield stress for the piezoelectric, the electric field for capacitive, and the maximum

magnetic field for electromagnetic materials as the respective upper limits. In his summarized

results, he found that piezoelectric materials possess a practical maximum energy density of

317.7 mJ cm , which is almost four times that of the other transducers. The following

paragraphs provide some basic discussion on transducer theory and explain electromechanical

transduction mechanism in detail.

A typical transducer is represented using different energy domains associated with power

flow from one domain to another. Modeling the energy transfer between domains enables a

better representation of the transducer behavior. The net power flow between two elements

describing the device is represented as a product of two terms called the conjugate power

variables (Senturia 2000).

P e f= ⋅ , (1.2)

where e is effort and f is flow. Next, a generalized momentum can be defined by integrating

the effort over time and is represented as

( ) ( )0

0 .t

p e t dt p= +∫ (1.3)

26

Similarly, a generalized displacement is defined that is associated with the flow variable, given

by

( ) ( )0

0 .t

q f t dt q= +∫ (1.4)

Here, ( )0p and ( )0q are the initial momentum and displacement in the element respectively.

Consequently, the energy in the element is given by the product of flow and momentum or effort

and displacement as

.E q e p f= ⋅ = ⋅ (1.5)

The ratio between effort and flow results in the generalized complex impedance of the element.

.e f Z= (1.6)

Some examples of conjugate power variables for various energy domains (Senturia 2000) are

listed in Table 1-1.

Table 1-1: Conjugate power variables for different energy domains.

Angular velocityTorqueRotational Mechanical

CurrentVoltageElectrical

Flux rate mmfMagnetic

Entropy rateTemperatureThermal

Volumetric flowPressureIncompressible flow

VelocityForceTranslational Mechanical

FlowEffortEnergy domain

Angular velocityTorqueRotational Mechanical

CurrentVoltageElectrical

Flux rate mmfMagnetic

Entropy rateTemperatureThermal

Volumetric flowPressureIncompressible flow

VelocityForceTranslational Mechanical

FlowEffortEnergy domain

( )[ ],i I A

( )[ ]F N

( )[ ]V V

( )[ ]T K ( ) ( ) 1S J Ks −⎡ ⎤⎣ ⎦

( ) 3 1,q Q m s−⎡ ⎤⎣ ⎦

( )[ ]Vφ( )[ ]Aη

( ) 2P Nm−⎡ ⎤⎣ ⎦

( ) 1,u U ms−⎡ ⎤⎣ ⎦

( )[ ]Nmτ ( ) 1sω −⎡ ⎤⎣ ⎦

A transducer is broadly classified into energy conserving and non-energy conserving

transducers (Hunt 1982, Fischer 1955). They can be classified further on factors such as

linearity, reciprocity etc. Electromechanical transducers are classified based on force generation

27

due to the interaction between electric field and charge or magnetic field and current. For

electromechanical transduction, there are five major linear energy conserving transducers,

namely, electrodynamic, electrostatic, piezoelectric, magnetic, and magnetostrictive.

All linear conservative transducers are generally represented using simple two-port

network theory (Rossi 1988) expressed in impedance or admittance notation. Here, the

impedance form is explained to discuss the various transduction mechanisms. The governing

equations for an electromechanical transducer are

.eb em

me mo

Z TV IT ZF U⎡ ⎤⎡ ⎤ ⎡ ⎤

= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

(1.7)

The blocked electrical impedance is defined as,

0

,ebU

VZI =

= (1.8)

where 0U = indicates that the device is mechanically restricted or “blocked” from any motion.

Alternatively, the free-electrical impedance,

0

efF

VZI =

= (1.9)

is defined as the electrical impedance when the device is “free” or not subjected to any

mechanical load. The coupling terms are defined as open circuit electromechanical transduction

impedance and the blocked mechanical-electro transduction impedance, represented as

0

emI

VTU =

= (1.10)

and

0

,meU

FTI =

= (1.11)

28

respectively. The electromechanical transducer is defined to be reciprocal when the cross

diagonal coefficients in Eq. (1.7) are equal, me emT T= . moZ is defined as the open-circuit

mechanical impedance expressed as the ratio between mechanical force and resulting velocity for

zero current

0

.moI

FZU =

= (1.12)

Alternatively, the ratio between the force and velocity while preventing any voltage from

building up defines the short circuit mechanical impedance

0

.msV

FZU =

= (1.13)

Both forms of mechanical and electrical impedances expressed in Eqs. (1.8)-(1.9) and Eqs. (1.12)

-(1.13) are related to each other as

( )21ms moZ Z κ= − (1.14)

and

( )21 ,ef ebZ Z κ= − (1.15)

where 2κ is defined as the electromechanical coupling coefficient that relates the amount of

energy converted from electric domain to mechanical domain. The coupling coefficient

represents the ideal effectiveness of an electromechanical transducer is defined as

2 .em me

eb mo

T TZ Z

κ = (1.16)

Two-port network theory can also be represented with a corresponding set of coefficients

in the admittance form. For reciprocal transducers, em meT T= , which implies that the

29

electromechanical conversion from an applied voltage to velocity and applied force to resulting

current are equal.

Electromechanical transducers are commonly represented using equivalent circuits with

lumped elements and will be explained in detail in Section 2.1. Some of the widely used

electromechanical transduction mechanisms for energy harvesting involve electromagnetic

(specifically electrodynamic), electrostatic and piezoelectric phenomenon that are explained

next.

Electrodynamic transduction

Electrodynamic transduction occurs when energy conversion is produced by motion of a

current carrying electric conductor subject to a constant magnetic field. This phenomenon is

characterized by Laplace’s law (Beranek 1986, Tilmans 1996), which defines the force on the

electric conductor in terms of the current and the magnetic field through the relation

( ).magF L I B= × (1.17)

Here, magF is termed as ‘Lorentz force’, I is the current, B is the magnetic field and L is the

length of the conductor. Conversely, the motion of the conductor in the presence of a magnetic

field leads to a voltage generation across its terminals, given by Lenz’s law

( ).V L U B= × (1.18)

In Eq. (1.18), U is the velocity of the conductor and V is the generated voltage. Combining

these two laws in a two-port representation yields,

0

.0

V BL IF BL U⎡ ⎤ ⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(1.19)

30

Since ebZ and moZ for this system are identically zero a direct coupling between electrical and

mechanical domains exists. So, an electrodynamic transducer is linear, reciprocal, and direct.

Another mechanism called the electromagnetic transduction is proposed in Figure 1-2

(Glynne-Jones and White 2001, Glynne-Jones et al, 2004). This transduction in nonlinear, but

can be linearized about its mean state to be represented as a linear, reciprocal transducer. The

linearization is valid for small variations in current and magnetic field that are possible by

biasing the electrical conductor with an initial current (Tilmans 1997).

Figure 1-2: An electromagnetic vibration-powered generator (adapted from Glynne-Jones and White 2001).

El-Hami et al. (2001) designed an electromagnetic generator comprised of a magnetic core

mounted on the tip of a steel beam. When an input vibration is supplied to the structure, the

beam vibrates, thereby inducing current in the coil. They report an output power of 0.53 mW

for an input displacement magnitude of 25 mμ at 322 Hz . The overall volume of the device

was 30.24 cm . In 2000, Li et al. presented a micromachined generator that had a permanent

31

magnet mounted on a spring structure and generated 10 Wμ at 2 V DC for an input vibration

amplitude of 100 mμ at 64 Hz from a volume of 31 cm . Williams and Yates in 1996 designed

an electromagnetic generator ( )5 5 1 mm mm mm× × that had a predicted power output of 1 Wμ

at 70 Hz and 0.1 mW at 330 Hz for an input vibration amplitude of 50 mμ . Shearwood and

Yates in 1997 designed an electromagnetic generator based on a polyimide membrane 2 mm in

diameter that could generate 3 Wμ of RMS power at a resonant frequency of 4.4 kHz .

Rodriguez et al. (2005) presented their work on the design optimization of an

electromagnetic vibrational generator to scavenge Wμ ’s- mW ’s of power in the frequency range

between 10 Hz to 5 kHz . The design proposed in their work consists of a movable magnet

mounted on a resonant membrane that induces a current in a fixed planar coil.

Electrostatic transduction

Electrostatic transduction is the conversion of energy that is produced by varying the

mechanical stress to generate a potential difference between two electrodes. An example for this

transduction is a simple parallel plate capacitor.

If we assume that one plate is moving relative to the other (generally stationary), due to an

external load, the variation in gap generates a capacitance given by

( ) ( )eAC t

x tε

= (1.20)

where ε is the permittivity of the medium separating the plates, A is the area and ( )x t is the

distance between the plates that changes about an initial mean distance. The voltage generated

between the terminals due to this is

( ) ( )( )e

Q tE t

C t= (1.21)

32

where ( )Q t is the accumulated charge in the capacitor. From Eq. (1.21), we know that the field

has a nonlinear relation with charge and displacement, which implies that it is nonlinear with

current and velocity. In addition, the force generated also follows a nonlinear relation with the

flow variables. However, the coupled equations can be linearized for small variations about a

mean initial condition, generally achieved by applying a bias voltage to the plates (Rossi 1988,

Tilmans 1997) or by storing a permanent charge using an electret (Boland et al 2003). The final

linearized set of equations are expressed in the two-port form as

1

.1

oeo o

o

o m

Vj CV Ij x

VF Uj x j C

ω ω

ω ω

−⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥

=⎢ ⎥ ⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

(1.22)

Here, oE and ox are electric field and distance between the plates. mC is the mechanical

compliance that relates the force and velocity and eoC is the mean capacitance. Since the effort

variables are originally calculated using charge and distance, jω is the integration factor in the

frequency domain to convert them to current and velocity. Although the cross terms in the

matrix are same, diagonal terms do exist, which implies indirect coupling between the electrical

and mechanical domains for an electrostatic transducer. Hence, this system of equations

represents a linear, reciprocal and indirect transduction mechanism.

In electrostatic transduction, a relative deflection induces charge between the electrodes

that can be converted to power. For example, at the micro-scale, a MEMS variable capacitor has

been designed and fabricated to harvest vibrational energy with a chip area of 21.5 1.5 cm× and

a reported net power output of approximately 8 Wμ (Meninger et al. 2001).

33

Piezoelectric transduction

Piezoelectricity, by definition, is a property of certain materials to physically deform in the

presence of an electric field or, conversely, to produce an electric charge when mechanically

deformed. Piezoelectricity occurs due to the spontaneous separation of charge within the crystal

lattice (Cady 1964). This phenomenon, referred to as spontaneous polarization, is caused by a

displacement of the electron clouds relative to their individual atoms, as well as a displacement

of the positive ions relative to the negative ions within the crystal structure, resulting in an

electric dipole. There are a wide variety of materials that exhibit this phenomenon, including

natural quartz crystals and even human bone. During electrical polarization, the material

becomes permanently elongated in the direction of the poling field (polar axis) and

correspondingly reduced in the transverse direction. Applying a voltage in the direction of the

poling voltage produces further elongation along the axis and a corresponding contraction in the

transverse direction subject to its Poisson’s ratio. This effect is depicted in Figure 1-3, which

shows a piezoelectric material under the influence of an electric field; P is the poling direction

and V is the externally applied voltage.

ExpansionContraction

P P PV V

V=0

Figure 1-3: Deformation of a piezoceramic material under the influence of an applied electric field.

Piezopolymers and piezoceramic materials are typically used as transducers for

piezoelectric energy harvesting applications. Piezoelectric materials possess a unique property

34

that makes them a viable option for electromechanical transducers. Applying an external electric

field across the piezoelectric material induces a mechanical strain in the material, thereby

enabling them to function as actuators. Conversely, when the piezoelectric material is

mechanically deformed, the resulting strain produces a voltage that allows them to operate as a

sensor. This strain/electric field characteristic of a piezoelectric material is termed as the

“piezoelectric effect.” Materials with good piezoelectric properties possess high coupling

between the mechanical and electrical domains. This effect can be generated using

piezopolymers, such as polyvinyledene fluoride (PVDF), or piezoceramics, such as lead

zirconium titanate (PZT), Zinc Oxide (ZnO), Aluminum Nitride (AlN) and Barium Titanate.

For any linear piezoceramic material (IEEE Standard on Piezoelectricity, 1987), the

constitutive governing equations can be expressed as

Tk kj j ik iS d Eε σ= + (1.23)

and

.i iq q ij jD d Eσ γ= + (1.24)

In the above equations, kε is the mechanical strain, jσ is the stress, iD is the electric

displacement, iE is the electric field applied to the ceramic, kjS is the proportionality constant

between the stress and strain (and is the reciprocal of the elastic modulus of any material), ijγ is

defined as the dielectric permittivity at constant stress, and ikd is the piezoelectric coefficient.

The material constants S , ,d and γ are defined as shown below for a piezoceramic due to its

crystal structure (IEEE Standard on Piezoelectricity, 1987)

35

11 12 13

12 11 13

13 13 33

44

44

66

0 0 00 0 00 0 0

,0 0 0 0 00 0 0 0 00 0 0 0 0

S S SS S SS S S

SS

SS

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(1.25)

15

15

31 31 33

0 0 0 0 00 0 0 0 0 ,

0 0 0

dd d

d d d

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

(1.26)

and

11

11

33

0 00 00 0

γγ γ

γ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

(1.27)

For a typical piezoceramic patch, the electric field is often applied vertically across the

ends of the piezoceramic in the 3-direction, while the stress acts in the 1-direction for the

composite beam. Therefore, we extract index 1k = from Eq. (1.23) and 3i = from Eq. (1.24),

since 1 0,σ ≠ 2 3 0σ σ≅ ≅ , 3 0,E ≠ and 1 2 0E E≅ ≅ . Substituting the matrices for the constants

and expanding the constitutive equations for the one-dimensional case results in

1 11 1 31 3S d Eε σ= + (1.28)

and

3 31 1 33 3.D d Eσ γ= + (1.29)

Rewriting the above equations to express strain in terms of deflection x , stress in terms of

the force applied F , electric field in terms of an applied voltage V , and the electric

displacement in terms of charge q induced in the piezoceramic simplifies them to

ms mx C F d V= ⋅ + ⋅ (1.30)

and

36

,m efq d F C V= ⋅ + ⋅ (1.31)

where 0

msV

xCF =

= is the short circuit compliance, 0

efF

qCV =

= is the free electrical capacitance,

and 0

mF

xdV =

= is an effective piezoelectric constant. Equations (1.30) and (1.31) will be used to

model the composite cantilever beam in this dissertation. Equations (1.30) and (1.31) when

expressed in frequency domain provide the two-port network equations in admittance matrix

form as

.mms

efm

j dj CU Fj Cj dI Vωωωω

⎡ ⎤⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦ (1.32)

Piezoelectric materials, especially PZT, exhibit good strain sensitivity and possess an

elastic modulus ( )e.g., 60 GPa that is comparable to many structural materials. This property is

essential for effective strain transfer between the layers, which occurs when there is a good

impedance match between the piezoceramic and the shim material. However, PZT is a brittle

material and cannot withstand large strains without fracturing unlike PVDF, which is very

flexible and easy to handle and shape (Starner 1996). PVDF can sustain higher strains and

exhibits higher stability over long periods of time. However, the disadvantage of using PVDF

instead of PZT is the fact that it has a very low electrical permittivity and, therefore, a much

lower coupling factor. Due to this, the electrical response of the device, such as output voltage,

power, and overall efficiency are significantly lower. Also, the working frequency range, which

can be defined as the difference between the open and short circuit resonance for the device is

greatly decreased due to poor electromechanical coupling.

A very common application of piezoceramics is that of a bending motor composed of a

layer of piezoceramic bonded to a host material. The piezoelectric material is assumed to be

37

firmly attached to the cantilever beam to ensure continuity in strain across the interface (Crawley

and deLuis 1987). Thus, when a voltage is applied to the piezoceramic, an induced moment is

concentrated at the ends of the piezoceramic patch. The maximum induced strain is given by the

expression

31p fieldd Eε = ⋅ (1.33)

where 31d is the piezoelectric constant, fieldE is the externally applied electric field, and pε is the

strain induced in the piezoceramic. The curvature of a bending motor is due to the expansion of

one layer and the contraction of the other. This phenomenon occurs due to an induced moment

(Crawley and De Luis 1987) when voltage is applied to the piezoceramic.

Umeda et al. (1996, 1997) performed theoretical and experimental characterizations of a

piezoelectric generator based on impact energy reclamation. In their studies, an oscillating

output voltage resulting from an input mechanical impact was rectified and stored in a capacitor.

With an initial voltage of over 5 V a maximum efficiency of 35 % was achieved with a

prototype generator. The working principle employed in their design is based on a steel ball that

freely falls toward the center of a circular membrane consisting of bronze and piezoceramic that

vibrates on impact resulting in an alternating current in the ceramic. A schematic representing

their structure is redrawn in Figure 1-4 for reference.

Ramsay and Clark (2001) performed a detailed design study on piezoelectric energy

harvesting for bio-MEMS applications. Their design employed a simple geometry for

harnessing energy from blood flow in the body. The proposed structure consisted of a square

PZT-5A plate that is connected to the blood pressure on one side and a chamber with constant

pressure on the other. Preliminary results reported an output power of 2.3 Wμ from a

( )1 1 9 cm cm mμ× × plate. It was also reported in their work that the device has a mechanical

38

advantage in converting applied pressure to working stress for piezoelectric conversion, when it

functions in the 31-mode than in the 33-mode.

Figure 1-4: A nonlinear piezoelectric vibration powered generator (adapted from Umeda et al, 1997).

Glynne-Jones et al. (2001) and White et al. (2001) designed a thick film piezoelectric

composite beam structure that generated 3 Wμ of power at 90 Hz from ambient vibrations. An

another paper by the same authors measured 2 μW at 80 Hz for a maximum amplitude of 0.9 mm

across an optimal resistive load of 333 kΩ. Their device consisted of a macro-scale piezoelectric

composite beam that was tapered along its length to ensure constant stress distribution at any

point on its length. In 2004, James et al. investigated two applications for two self-powered

sensors, namely a liquid crystal display and an infra-red link to transmit the data output. The

required energy for the prototypes was derived from a 0.17 g – 0.23 g vibrating source at 102 Hz.

In another application of piezoelectric energy harvesting, Hausler and Stein (1984)

proposed a device that basically consisted of a roll of PVDF material that can be attached

between body ribs. They were designed in such a way that regular breathing induced a strain in

39

the material thereby producing power. It was tested on a dog by surgically implanting the

device, thereby generating micro-watts of power from the breathing.

Roundy and Wright in 2004 designed a piezoelectric vibration generator consisting of a

cantilever bimorph bender with a proof mass at its end. Their design was aimed at generating

enough energy from a 1 cm3 to power a 1.9 GHz radio transmitter from the same vibration

source. Their design was predicted to produce 375 μW from a vibration source of 2.5 m/s2 at 120

Hz. The lumped element model (LEM) introduced in their work was unconventional and used

stress as the effort variable unlike force which is the standard effort function for LEM

representation. Correspondingly, strain rate was used as the flow variable in the representation.

Sood et al. (2004) developed a piezoelectric micro power generator (PMPG) that is based

on a piezoelectric layer deposited and patterned on a membrane consisting of SiO2 and SiNx,

followed by a ZrO2 diffusion barrier. The two electrodes for the PZT layer are formed using an

inter-digitated top electrode (IDT) with Pt/Ti that makes use of the d33 mode (described later in

this chapter) to extract power. The premise governing their device was that the d33 coefficient is

much higher than 31d of a piezoelectric material. This potentially results in a higher voltage, but

the power density and input acceleration levels are not available directly for comparison with

other available d31 configurations. The maximum measured power using a direct charging circuit

consisting of a full-bridge rectifier and a capacitor occurred at 5 MΩ of load resistance. The

corresponding output voltage and power were 2.4 DCV and 1.01 Wμ respectively (Jeon et al.

2005).

Another application for a self-powered piezoelectric device is a Strain Amplitude

Minimisation Patch (STAMP) damper that uses piezoelectric elements as sensor, actuator and

power source. Konak and Powlesland (2001) presented their analysis on this device that

40

combined the vibration control aspect of a piezoelectric element along with its energy generation

characteristic producing a self-powered vibration damper.

Table 1-2 compiles all the reported energy harvesters discussed in this chapter that

generated power from vibration sources using different transduction mechanisms. The columns

list the authors, the vibration source (which was mostly resonant in nature), the size of the

device, and the overall power harvested.

Table 1-2: Vibration based energy harvesters characterterized for power. Ambient source Size or Mass Power Sood et al. 10 @ 13.9 g kHz 170 260 m mμ μ× 1.01 Wμ Shearwood et al. 500 @ 4.4 nm kHz 2.5 2.5 700 mm mm mμ× × 0.3 Wμ Chandrakasan et al. 500 @ 2.5 nm kHz 500 mg 8 Wμ Li et al. 100 @ 64 m Hzμ 31 cm 10 Wμ Roundy et al. 0.25 @ 120 g Hz 28 3.6 8.1 mm mm mm× × 375 Wμ White et al. 0.9 @ 80 mm Hz 2.2 Wμ Marzencki et al. 0.5 @ 204 g Hz 2 2 0.5 mm mm mm× × 38 nW El Hami et al. 25 @ 322 m Hzμ 30.24 cm 0.53 mW Ching et al. 200 @ 60 -110 m Hzμ∼ 31 cm∼ 200 830 Wμ−

Stark et al. 20T KΔ = 267 mm 20 Wμ

Next, a brief introduction to the application of piezoelectric materials in microsystems is

presented followed by the proposed PZT based micro energy harvester.

Microelectromechanical Systems (MEMS)

Some of the earliest ideas about MEMS were initiated by Richard Feynman in his popular

speech “There is plenty of room at the bottom” delivered in 1960 (Feynman 1992) followed by

“Infinitesimal machinery” (Feynman 1993). In the early 1960’s, silicon gained a lot of attention

as a material for microsystems due to its excellent properties that suit both electrical and

mechanical applications (Peterson 1982). Micromachining is based on fabrication techniques

that are used in silicon integrated chips but adds numerous other fabrication techniques as well.

41

This ability to batch fabricate numerous such devices in each step is a potentially significant

advantage of microfabrication in MEMS.

Another major advantage of MEMS is that their small size enables suitability for micro

applications that were not possible prior to the advent of MEMS. However, there are some

significant considerations, such as packaging for structural robustness, operation in harsh

environment, and power requirements that may limit their feasibility in certain applications

(Angell et al. 1983). Recently, smart structures that incorporate MEMS devices were

investigated for their importance and use in aerodynamic structures, spacecraft, and vehicles for

structural health monitoring (Schoess 1995).

The structural configuration adopted for the device described in this dissertation is that of a

piezoelectric composite cantilever beam with an integrated proof mass that functions along the

lines of conventional accelerometers. Significant research has been invested in understanding a

cantilever beam arrangement for energy harvesting (Kim et al. 2004).

The performance of a piezoelectric cantilever bimorph in the flexural mode has also been

analyzed for scavenging ambient vibration energy (Jiang et al. 2005). Their analysis calculates

the output voltage, power, and the device efficiency of the composite beam with a concentrated

tip mass subjected to a harmonic clamp motion. The analytical dynamic model implemented in

their work can be used to design the device appropriately to tune the frequency and increase the

power. However, their work is purely theoretical and does not provide any experimental data for

validation. In addition, model assumes the end mass as a concentrated point load and does not

account for its finite stiffness. This dissertation also aims to first develop an analytical model that

can be used as a design tool for specific energy harvesting applications. Furthermore, the validity

of the model is investigated for various canonical structures both at mesoscale and MEMS. It

42

uses a different modeling technique called lumped element modeling that, subject to the

assumption that it is valid until the first bending mode, is analytically simpler. This technique is

applicable when the device is small compared to the characteristic length scale of the distributed

physical system.

A cantilever configuration is chosen for our energy harvester because it provides the

maximum average strain when subjected to a specific load (Appendix A). In addition, a

cantilever beam has a lower natural frequency compared to beams with other boundary

conditions (Roundy et al 2003). An explanation of these reasons along with a proof is provided

in Appendix A, where beams subject to different boundary conditions and loads are analyzed to

estimate their average strain and natural frequencies. Therefore, it provides an opportunity to

model a slightly different configuration with variable piezoelectric dimensions from the shim

layer. In addition, the proof mass, which is generally large (especially for MEMS structures), is

modeled to account for its mass and its stiffness providing a complete accurate model. In

addition, the analytical model developed can be utilized as a tool to design cantilever based PZT

energy harvesters for specific applications. The lumped element modeling technique is

investigated in more detail in 2.1.

A simple schematic of the proposed configuration is shown in Figure 1-5. The structure

basically consists of a cantilever beam with a proof mass and a thin film of piezoelectric material

deposited on the beam. When the device is subjected to base vibrations, the inertial mass

vibrates relative to the base causing bending in the beam. The strain thus resulting from this

relative motion is converted to an effective output voltage by virtue of the piezoceramic

transducing element. The piezoceramic layer converts the mechanical strain induced due to the

vibrations into voltage due to the piezoelectric effect. However, even though the design of the

43

device is similar to an accelerometer, it is implemented and operated as a resonant sensor. In

other words, the device needs to be “tuned” to the input vibration frequency so that it operates

near its resonance frequency to generate maximum power, unlike a conventional accelerometer

that operates across a wide bandwidth far removed from its resonance. Therefore, the goal of our

design is to maximize the performance of the accelerometer device at its resonance. To provide

a brief insight in this area, several investigations have demonstrated the feasibility of fabricating

silicon accelerometers. The basic structure usually consisted of a silicon cantilever with a proof

mass made of silicon or is gold plated to increase the sensitivity (Seidel and Csepregi 1984).

Different transduction mechanisms for accelerometers such as piezoresistors, piezoelectric films,

and electrostatic coupling have been studied in detail, and the advantages and disadvantages of

these transducers have been already been published in the literature (Polla 1995).

Piezoelectric accelerometers are of interest to us due to their low power dissipation and

high electromechanical coupling (Polla 1995). However, the major drawback of this design is

the difficulty in processing and integration with electrical circuitry. Piezoresistive sensors have

much higher dissipation and noise floor even though the processing is relatively straightforward

and CMOS compatible. Capacitive accelerometers are favorable in many aspects such as noise,

power of dissipation and ease of processing, but are sensitive to dimensional tolerance (Polla

1995; Polla et al 1996). In MEMS, it is difficult to achieve small and accurate dimensions and a

considerable uncertainty exists in the material properties and final dimensions of the device.

44

Cantilever Beam

Proof Massao

Piezoelectric Element

V

Mm

Cms

y x

Rm

Figure 1-5: Schematic of the proposed cantilever configuration for energy reclamation.

Initially accelerometers were fabricated using conventional bulk micromachining

techniques. This approach has a clear advantage in the fact that large proof masses can be etched

out of a silicon substrate. However, disadvantages with this approach arise during front to back

alignment and passivation for integrated circuitry (DeVoe and Pisano 2001). Additionally, this

process consumes a larger die area for bulk etching which is undesirable for batch fabrication.

On the other hand, surface micromachining uses standard VLSI techniques and therefore does

not pose the above problems. DeVoe and Pisano presented their work on the design, fabrication,

and characterization of surface micromachined piezoelectric accelerometers (PiXLs) that

consisted of thin film Zinc oxide (ZnO) as the piezoelectric material. In addition, they describe

some guidelines for robust design based on device sensitivity and resonant frequency. Using a

cantilever without a proof mass whose resonant frequency was 3.3 kHz , their results reported a

sensitivity of 0.95 fC g . Addition of a proof mass significantly improved the sensitivity to

13.3 fC g and 44.7 fC g , but decreased the corresponding resonant frequencies to 2.23 kHz

45

and 1.02 kHz respectively. The cantilever accelerometer was modeled using classical Euler-

Bernoulli beam theory similar to the method adopted in our design. However, the model

described in their work assumed that the thickness of the piezoelectric layer is negligible

compared to the thickness of the beam. Additionally, it assumes that the ZnO is deposited across

the length of the structure and that the elastic moduli of the two materials are comparable in

magnitude. This assumption holds true when the active layer is ZnO and the beam is made of

silicon. However, in our design where PZT is the piezoelectric layer, the elastic moduli of the

two materials are significantly different, and therefore a detailed static electromechanical model

is derived for our cantilever composite beam.

Piezoelectric MEMS

Silicon is an excellent material for MEMS due to its good mechanical properties such as

elastic modulus and density (Peterson 1982). As a result, most of the devices that are fabricated

in MEMS consist of a silicon substrate. Initially, thick film piezoelectric layers were imprinted

on micromachined silicon substrates to form the desired structure (Allen et al 1989).

Thicknesses in the range of 100 mμ can be achieved with this process leading to much higher

actuation forces compared to conventional thin film piezoelectric micro actuators (Barth el al

1988, Terry 1988). Zinc oxide was often used as the piezoelectric material for most applications

until Lead Zirconate Titanate (Pb(Zrx, Ti1-x)O3) gained acceptance. x is the percentage

composition of Zr in PZT. It was observed that when x lies between 0.52 and 0.55, the material

exhibited high dielectric constants and electromechanical coupling (Wang et al, 1999). PZT has

been extensively studied and used lately due to its excellent electromechanical coupling and

piezoelectric properties. Piezoelectric thin films in micro systems are used in a wide variety of

applications such as micro actuators (Lee et al. 1998; Zurn et al. 2001), micro mirrors (Cheng et

46

al. 2001), micropumps (Nguyen et al 2002), microphones (Lee et al. 1996), micro accelerometers

(DeVoe and Pisano 2001), fiber bulk wave acoustic resonators (Nguyen et al. 1998) etc.

Objectives of Present Work

The following chapters describe in detail the lumped element modeling technique used to

represent the composite beam and discuss its use in designing an optimal energy harvesting

device to harness maximum power from a vibrating device. The electromechanical lumped

element model thus developed is validated using meso-scale experiments. Furthermore, a

scaling theory is developed to observe the device behavior as it is reduced in size to a MEMS

scale, which is verified using finite element analysis. In addition, the fabrication process adopted

to build the devices and their characterization will be presented.

The main contributions for this dissertation are as follows:

• A complete static analytical model of a cantilever composite beam validated using FEM and experiments on candidate devices.

• Electromechanical lumped element model of a piezoelectric energy harvester, intended to provide a design optimization tool for complete circuit simulation with power processors.

• A first generation fabrication of a MEMS PZT cantilever array is realized for vibrational energy harvesting.

• Design, fabrication and testing of a stand alone MEMS device to demonstrate energy reclamation.

Organization of Dissertation

The dissertation is organized as follows. Chapter 2 describes in detail the static

electromechanical model of the composite beam structure. In particular, lumped element

modeling is used to obtain the various electromechanical parameters that represent the system.

Chapter 3 discusses the detailed non-dimensional analysis and the design formulation for the

device. Chapter 4 discusses the fabrication process adopted to build arrays of the MEMS

piezoelectric generators. Chapter 5 describes the experimental setup and characterization

47

procedure for testing energy reclamation devices. Chapter 6 describes the experimental results.

Chapter 7 concludes the dissertation with a summary and discussion of future work.

48

CHAPTER 2 PIEZOELECTRIC CANTILEVER BEAM MODELING AND VALIDATION

The objective of this dissertation is to model and design MEMS piezoelectric cantilever

composite beams with an integrated proof mass to reclaim energy from base vibrations.

Consequently, these structures will be optimally designed to extract maximum power from the

vibrations, subject to some design constraints. The ultimate goal is to eventually use an array of

such structures to obtain sufficient power to operate self-powered sensors. This chapter describes

the static electromechanical modeling of the composite beam using conventional Euler-Bernoulli

beam theory. The shortcomings of this approach are that it does not model nonlinear effects due

to large deflections and neglects rotary inertia effects. In addition, fabrication-induced inplane

residual stresses are neglected in the model. These stresses exist in MEMS structures due to

thermal stresses and other sources that mainly occur during layer depositions and other high

temperature treatment. For the purpose of this first generation effort, we assume that Euler-

Bernoulli theory is adequate to model, design and characterize the device. However, future

models may include the above effects to implement a more complete model.

Some of the earlier work in this area was concentrated on modeling and testing a canonical

cantilever mesoscale composite beam without any proof mass that was excited at its tip with a

load (Kasyap 2002). For a known force input, the amount of power generated at the ends of the

PZT was used in a flyback converter circuit to reclaim power (Kasyap et al. 2002). However, the

previous configuration cannot be directly used in real applications because of the nature of its

loading condition. In all practical applications, the energy reclamation device should be directly

49

attached to the vibrating surface. This alters the complete setup as the device is loaded at the

clamp due to the vibrations, instead of the tip.

In the new configuration shown in Figure 2-1, the test structure modeled consists of a

piezoelectric composite cantilever beam with a proof mass attached to its tip. The composite

beam, when directly attached to a vibrating surface, places the whole structure in an accelerating

frame of reference. The proof mass essentially converts the input base acceleration into an

effective inertial force at the tip that deflects the beam, thereby inducing mechanical strain in the

piezoceramic (Yazdi et al. 1998). This strain produces a voltage in the piezoceramic that is

converted into usable power with the help of an energy reclamation circuit. The motion of the

beam depends on the size of the proof mass. If the proof mass is relatively small compared to

the effective mass of the beam, it reduces to a cantilever beam subject to an acceleration at the

clamp instead of its tip. Alternatively, if the proof mass is very large compared to the effective

mass of the beam, it results in large deflections in the beam and consequently, large strains at the

clamp. This configuration will be favorable for energy reclamation because a piezoelectric

patch, when attached to the beam, converts the induced strain into electrical charge. However, if

the proof mass is comparable to the actual effective mass of the beam, the motion of the beam

resembles that of a rigid body and, therefore, might not induce any strain in the beam. These

issues are clarified via the model described below.

Piezoelectric Composite Beam

In this analysis, the test structure consists of a piezoelectric (PZT) composite cantilever

beam with a proof mass attached to it as shown in Figure 2-1.

50

Figure 2-1: Schematic of a piezoelectric composite beam subject to a base acceleration.

In Figure 2-1, oa is the input acceleration, extf is the excitation frequency of the base

vibration and V is the resulting voltage from the piezoceramic. The composite beam is modeled

using the lumped element modeling technique described in Hunt (1982) and Rossi (1988). This

approach is valid in general when the characteristic wavelength of the bending waves is very

large compared to the geometric length scale and, in the case of a cantilever composite beam, is

valid up to at least the fundamental bending resonance frequency (Merhaut, 1979). This

approach simplifies the partial differential equations governing the system to coupled ordinary

differential equations.

In addition, the lumped element modeling technique is useful in analyzing and designing

coupled energy domain transducer systems. In this approach, we use equivalent circuit elements

to effectively represent the coupled electromechanical behavior of the device. These circuit

analogies enable efficient modeling of the interaction between different energy domains in a

system. Furthermore, the tools developed for circuit analysis can be utilized for representing and

solving a coupled system with different energy domains.

51

A piezoelectric composite beam represents an electromechanical system that can be

separated primarily into two energy domains consisting of electrical and mechanical parts.

These two energy domains interact in the equivalent circuit via a transformer as shown in Figure

2-2. The circuit is obtained by lumping the distributed energy stored and dissipated in the

system into simple circuit elements. In this electromechanical circuit, force and voltage are the

generalized effort variables, while velocity and current are the generalized flow variables

(Senturia 2000). An impedance analogy is used to represent the circuit, in which case all

elements that share a common effort are connected in parallel, and the elements that share a

common flow are connected in series. When the composite beam is subject to a mechanical

load, the strain induced in the piezoelectric material generates a voltage, which represents the

conversion from the mechanical to the electrical domain. Conversely, the composite beam can

be driven with an ac voltage that causes it to vibrate due to the piezoelectric effect. This

represents a conversion from the electrical to the mechanical domain.

Figure 2-2 represents the entire equivalent circuit consisting of mechanical and electrical

lumped elements representing the composite beam. All elements are labeled and defined in the

figure. In the notation shown in Figure 2-2, the first subscript denotes the domain ( m for

mechanical and e for electric), while the second subscript denotes the condition ( s for short

circuit and b for blocked). Using the described notation, for example, msC is defined as short-

circuit mechanical compliance, and ebC is the blocked electrical capacitance of the piezoceramic.

F is the effective force applied to the structure that is obtained by the product of input

acceleration and effective mass lumped at the tip, U is the relative tip velocity with respect to

the base, V is the voltage, and I is the current generated at the ends of the piezoceramic. All

52

the parameters are obtained by lumping the energy at the tip using the relative motion of the tip

with respect to the clamp/base.

Figure 2-2: Overall equivalent circuit of composite beam.

The beam is represented as a spring-mass-damper system by lumping the energy (kinetic

and potential) in the beam to an equivalent mass and compliance. The mechanical mass and

compliance of the structure can be equated to an equivalent electrical inductance and

capacitance. Similarly, mechanical damping is analogous to electrical resistance. However,

mechanical damping cannot be easily estimated from first principles although it is a critical

parameter for resonant behavior in structures. The same holds true for electrical losses in the

device, modeled using eR .

In principle, the fundamental operation of any power generator is effectively dependent on

the nature of the mechanism by which the energy is extracted. Most microgenerators reported to

date can be classified into velocity-damped resonant generators (VDRGs) or Coulomb-damped

resonant generators (CDRGs) as described in Mitcheson et al. (2004). VDRG represents the

damping effect as a function of the velocity characterized by a viscous force, while CDRG

53

represents the same effect using a coulomb frictional force. Analytical expressions for the

dissipated power for these two cases are derived in Mitcheson et al. (2004) that provide an

estimate for the available power. It should be noted that the aforementioned damping

mechanisms are resonant in nature and that while VDRG is widely used and linear, CDRG is a

nonlinear representation, although a closed-form solution is available (Den Hartog 1931; Levitan

1960). An alternate class, namely, Coulomb-force parameteric generator (CFPG) is also

suggested in their work that operates in a non-resonant manner.

For the purpose of our analysis, a VDRG implementation is adopted that represents the

damping phenomenon using a viscous effect with an effective damping coefficient. Damping

coefficients are typically estimated from experimental modal analysis and include effects such as

viscous dissipation, boundary condition non-ideality, thermoelastic dissipation, etc. (Srikar and

Senturia 2002). A detailed analysis of various damping mechanisms is discussed and

corresponding empirical relations are presented in Appendix B. The mechanical damping in the

system is obtained from the damping ratio using the expression

2 ,mm

ms

MRC

ζ= (2.1)

where mM is the effective mechanical mass of the composite beam (discussed in Section 2-2),

and ζ is the mechanical damping ratio. However, the dielectric loss of the piezoelectric material

can be estimated using an empirical expression provided in Jonscher (1999)

( )

1 ,tan 2e

n eb

Rf Cδ π

= (2.2)

where nf is the natural frequency of the system. tanδ is the loss tangent defined as the ratio of

resistive and reactive parts of the impedance. The theory behind dielectric loss in piezoelectric

54

materials is described in Mayergoyz and Bertotti (2005) and Jonscher (1999). In this case, the

electrical damping is assumed to be in parallel with the capacitor. A discussion about this and an

alternate representation for electrical impedance is provided in Appendix D.

All the other parameters in the circuit in Figure 2-2 are obtained analytically. The main

purpose for modeling the device as a beam is to obtain the lumped parameters such as msC , mM ,

and φ that characterize the circuit. The following sections describe in detail the process of

lumped parameter extraction.

Analytical Static Model

The composite beam clamped at one end is analyzed from first principles using linear

Euler-Bernoulli beam theory described in, for example, Beer and Johnston (1992). Therefore,

shear and rotary inertia effects are neglected. Another assumption is that plane sections remain

planar and no geometric nonlinearity exists in the structure. The following section presents the

static analytical model for the composite beam to calculate the lumped element parameters

represented in Figure 2-2. In this analysis, the composite beam is solved for its static equilibrium

to obtain its transverse deflection for all the static loads acting on it, which permits calculation of

the lumped element parameters.

When a base acceleration is applied to the structure, the Euler-Bernoulli governing

equations used in our analysis remain valid. However, a Galilean transformation of coordinates

is carried out, described in detail in Appendix C, to transform to a local coordinate system that

treats relative motion of the beam with respect to the clamp. In addition, the effect of the base

mass is also analyzed in Appendix F.

55

Static Electromechanical Load in the Composite Beam

The static electromechanical model is used to calculate the effective short circuit

compliance, mass and piezoelectric coefficient of the composite beam. A simple schematic of

the composite beam configuration for this case is shown in Figure 2-3. The cantilever composite

beam is analyzed for all the static mechanical loads acting on it from first principles using the

bending beam equation.

Figure 2-3: Schematic of the piezoelectric cantilever composite beam.

In the figure, , and p sL L l are the lengths of piezoceramic, beam (i.e., shim) and the proof mass,

respectively. The applied voltage will induce a mechanical strain and, hence, a bending moment

at the ends of the piezoceramic as described in Cattafesta et al. (2000). The induced moment

oM , in the composite beam due to an applied voltage to the piezoceramic is given by the

expression

( )31 21 2 .2o p app p pM E d V b c t= − − (2.3)

56

Here, pE is the elastic modulus of piezoceramic, 31d is the piezoelectric coefficient, appV is the

applied voltage to the piezoceramic, 2c is the location of the neutral axis from the bottom of

piezoceramic given by the expression

2

22 2 ,

pss s p p

s s p p

ttE t t Ec

E t E t

⎛ ⎞+ +⎜ ⎟⎝ ⎠=

+ (2.4)

where sE and st represent the elastic modulus and the thickness of the shim. Similarly, pb and

pt are the width and thickness of piezoceramic, respectively. The free body diagram for the

above configuration, shown in Figure 2-4, essentially replaces the mass of the composite beam as

an equivalent uniform load due to its weight.

Figure 2-4: Free body diagram of the overall configuration.

where 1 2 3, and q q q are the equivalent linear load densities ( )N m in the composite, shim, and

proof mass sections, respectively.

Since the configuration represented in Figure 2-4 is assumed to be a linear system, it can

be simplified and solved analytically for the deflection using Euler-Bernoulli beam theory.

57

Figure 2-5 represents the simplified free body diagram for the composite beam. In Figure 2-5,

the clamp is replaced with a reaction bending moment ( )rM and reaction force ( )R .

Figure 2-5: Free body diagram of the composite beam where the self weights are replaced with equivalent loads.

As indicated in Figure 2-5, the composite beam is uniformly loaded in a piecewise fashion

over its total length, ( )sL l+ . Consequently, each of the uniform loads shown in Figure 2-4 can

be replaced with an effective static load density defined as

,mq whgρ= (2.5)

( )1 ,s s s p p pq t b t b gρ ρ= + (2.6)

and

2 .s s sq t b gρ= (2.7)

Here, and s sb t are the width and thickness of the shim, and w h are the width and thickness of

the proof mass, and p, and s mρ ρ ρ are the densities of piezoceramic, shim, and proof mass,

respectively. Assuming static equilibrium for the beam in the Figure 2-5, we can obtain

expressions for the reaction force and bending moment at the clamp as

( )1 2p s pR q L q L L ql= + − + (2.8)

58

and

2 2 2

1 2 .2 2 2

p s pr s

L L L lM q q ql L⎡ ⎤− ⎛ ⎞= − + + +⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦ (2.9)

Let us now divide the composite beam in Figure 2-5 into 3 sections. Section 1 consists of

the composite ( )0 px L≤ ≤ , the second section consists of the shim ( )p sL x L≤ ≤ , and the third

section is the proof mass ( )s sL x L l≤ ≤ + . The Euler-Bernoulli equation for the beam is then

solved using free body diagrams to obtain the bending moment and shear force in each of the

sections (Beer and Johnston 1992). Furthermore, the bending moment can be integrated using

the Euler-Bernoulli equation to obtain the mode shape. The governing equations for the sections

are

( ) ( )2 21

12 ; 0 ,2r o pc

w x xEI M Rx M q x Lx

∂= + − − ≤ ≤

∂ (2.10)

( ) ( ) ( )22

21 22 ; ,

2 2pp

r p s p ss

x LLw xEI M Rx q L x q L x L

x−∂ ⎛ ⎞

= + − − − ≤ ≤⎜ ⎟∂ ⎝ ⎠ (2.11)

( ) ( ) ( ) ( )

2

223

1 22

2 ;

2 2 2

s ps p

p sr p s s sm

L LL L x

Lw x x LEI M Rx q L x q q L x L l

x

⎛ + ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟∂ −⎛ ⎞ ⎝ ⎠⎝ ⎠= + − − − − ≤ ≤ +⎜ ⎟∂ ⎝ ⎠

(2.12)

Here, ( ) ( ) ( ), and c s m

EI EI EI are the flexural rigidity in each of the three sections. Further,

( ) ( ) ( )1 2 3, and w x w x w x are the transverse deflections in each of the sections at a distance x

from the clamp. The two clamped boundary conditions and four matching conditions, shown in

Eq. (2.13), are obtained from the clamped boundary condition and by matching the deflection

and slope at each of the interfaces between the sections

59

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

11

0

1 2

1 2

2 3

2 3

0 0,

,

,

,

.

p p

s s

x

p p

x L x L

s s

x L x L

w xw

x

w L w L

w x w xx x

w L w L

w x w xx x

=

= =

= =

∂= =

∂

=

∂ ∂=

∂ ∂

=

∂ ∂=

∂ ∂

(2.13)

The Euler-Bernoulli equations shown in Eqs. (2.10)-(2.12) can now be solved to obtain a

piecewise continuous deflection mode shape for the beam, that is represented as

( ) ( )( ){ }( ) ( )

2 2 22

1 2341 21

1

2 2 2,

24 6 2

p s ps o

p s p

c c c

L L L lq q ql L M xq L q L L ql xq xw x

EI EI EI

⎧ ⎫⎛ ⎞−⎪ ⎪⎛ ⎞+ + + +⎜ ⎟⎨ ⎬⎜ ⎟⎜ ⎟+ − + ⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭= − + − (2.14)

( ) ( ){ }

( ) ( )( )( ) ( )

22

2342 1 32 2 4

2

2 2,

24 6 2

ss

s

s s s s s

L lq ql L xq L ql x C C xq x C Cw x

EI EI EI EI EI

⎧ ⎫⎛ ⎞⎪ ⎪⎛ ⎞+ +⎨ ⎬⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ +⎪ ⎪⎝ ⎠⎩ ⎭= − + − + + (2.15)

and

( ) ( )( )( )

( )( )

( )( ) ( )

23 245 7 6 8

3 ,24 6 4

s s

m m m m m

q L l x q L l x C C x C Cqxw xEI EI EI EI EI

+ + + += − + − + + (2.16)

where the integration constants 1 2 3 4 5 6 7, , , , , ,C C C C C C C and 8C are given by the following

expressions

60

( ) ( )( )

( )( ) ( )( )

( )( )

( )

21 2 1 2

22

2 1 2 2

3

24

32

5 1

42

6 2

3

7 3

3 1 ,6

3 2 8 6 1 ,24

1 2 2 ,2

1 ,2 3 4 2

,6

,24

16 2

pp s s p

pp s p s

ps p o

po sp

s

s

s

LC L q q q L L L

LC L q q q L L L

LC ql L L l M

LM LlC L ql

q LC C

q LC C

qLC C q

α α

α α α

α α

α α

γ

γ

γ

⎡ ⎤= − + − −⎣ ⎦

⎡ ⎤= − − + − −⎣ ⎦

⎡ ⎤= − − + −⎣ ⎦

⎡ ⎤⎛ ⎞= + − − −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦⎡ ⎤

= −⎢ ⎥⎣ ⎦⎡ ⎤

= +⎢ ⎥⎣ ⎦

= + + ( )( )

( )( )4 28 4

1 ,

1 1 4 6 .24

s s

s s s

lL L l

C C qL qlL L l

γ

γ γ

+ −

⎡ ⎤= − + − +⎣ ⎦ (2.17)

In the above equation, α is defined as the ratio between the flexural rigidity moduli in sections 1

and 2, ( ) ( )s cEI EI⎡ ⎤⎣ ⎦ , and γ is the ratio between the flexural rigidity modulus in sections 2 and

3, ( ) ( )m sEI EI⎡ ⎤⎣ ⎦ .

To check the validity of this general result, the deflection modeshape thus obtained is

verified by comparing with a few simple special cases. The first ideal case used to verify the

static electromechanical solution is that of a homogenous cantilever beam subjected to its self-

weight. The resulting deflection for this ideal case is given as (Beer and Johnston 1992)

( ) ( )2 2 2

.4 24 6

sw s ssw

s

q x L L xxw xEI

⎛ ⎞= − + −⎜ ⎟

⎝ ⎠ (2.18)

where, swq is the uniform load acting on the homogenous beam as a result of its own weight.

The static short-circuit solution is verified by setting the input voltage and the proof mass size to

zero. The piezoelectric patch is thus absent in this solution and represents a homogenous beam

61

that deflects due to its self-weight. The two solutions are plotted for a test case consisting of an

Aluminum beam (Al 6061). The properties and dimensions used in the simulations are listed in

Table 2-1. As indicated in Figure 2-6, the static solutions are identical.

Table 2-1: Material properties and dimensions for a homogenous aluminum beam. Elastic modulus ( )sE 73 GPa

Density ( )sρ 2718 3kg m

Length of the beam ( )sL 127 mm

Width of the beam ( )sb 6.35 mm

Thickness of the beam ( )st 1.02 mm

Figure 2-6: Static model verified with the ideal solution for a homogenous beam solved for self-weight.

Furthermore, the piecewise solution can be verified for a homogenous beam subjected to a

tip load. From conventional theory, for this ideal case, the deflection modeshape in a beam due

to point static tip load is given as (Thomson, 1993)

62

( ) ( )2 3

,2 6

tiptipload s

s

q x xw x LEI

⎛ ⎞= −⎜ ⎟

⎝ ⎠ (2.19)

where tipq l is the equivalent tip load. In the piecewise static solution, the piezoelectric patch is

absent and the input voltage is set to zero to generate a similar configuration as before but with a

proof mass that contributes an effective tip load. Similar to the previous case, Eq. (2.19) was

calculated for the test beam listed in Table 2-1, and the deflections are plotted in Figure 2-7.

Again, the solutions match.

Figure 2-7: Static model verified with the ideal solution for a homogenous beam solved for tip load.

Now, the complete static mechanical model representing the PZT composite beam has

been verified for various test cases. As described in Section 0, the purpose of obtaining a

complete electromechanical model is to calculate the lumped element parameters in the circuit

63

shown in Figure 2-2. The advantage of this solution model is that all the parameters are

analytical and their scaling dependence on the dimensions can also be obtained which will be

further useful in optimizing the structure for maximum power output.

The static deflection mode shape can now be used to estimate an equivalent effective mass

and compliance that can replace the composite beam as a simple single degree of freedom

system. For this configuration, we emphasize that only the mechanical loads are considered, and

the piezoceramic is electrically shorted. This configuration effectively eliminates the electrical

side from the lumped element circuit represented in Figure 2-2, leading to short-circuit electrical

condition for the piezoceramic. The potential energy associated with distributed strain energy in

the composite beam is given by the expression (Thomson 1993)

( ) ( ) ( ) 22

20

,2

sL l E x I x d w xPE dx

dx

+ ⎛ ⎞= ⎜ ⎟

⎝ ⎠∫ (2.20)

where ( )E x and ( )I x are the local elastic modulus and moment of inertia of the section. The

above integral equation is determined in each of the three sections and summed to obtain the

total potential energy in the beam,

( ) ( ) ( ) ( ) ( ) ( )

2 2 22 2 21 2 3

2 2 20

.2 2 2

p s s

p s

L L L lc s m

L L

EI EI EId w x d w x d w xPE dx dx dx

dx dx dx

+⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠∫ ∫ ∫ (2.21)

Similarly, the total kinetic energy in the composite beam is given by the integral

expression

( ) ( )( )2

0

,2

sL lL x

KE w x dxρ+

= ∫ (2.22)

64

where Lρ is mass length of the section and ( )w x is the velocity in the section. We calculate

the kinetic energies in the individual sections and add them to obtain the total kinetic energy in

the composite beam

( ) ( ) ( )2 2 21 2 3

02 2 2

p s s

p s

L L L lLc Ls Lm

L L

KE w x dx w x dx w x dxρ ρ ρ +

= + +∫ ∫ ∫ . (2.23)

Lumping the overall potential strain energy at the tip yields an effective short circuit

mechanical compliance for the composite beam

( )2

.2

Ftipms

wC

PE= (2.24)

Using the same analogy, an effective mass for the composite beam from its deflection shape is

obtained by lumping the kinetic energy of the beam at its tip

( )2

2 .F

m

tip

KEMw

= (2.25)

where, ( )3Ftip sw w L l= + is the resulting tip deflection of the beam due to its self-weight

calculated from Eq. (2.16). The natural frequency of the composite beam is calculated from the

effective mass and compliance using the expression

1 1 .2n

ms m

fC Mπ

= (2.26)

Next, the electromechanical behavior of the general solution, Eq. (2.14) - (2.17), is

validated for the case when the piezoceramic composite beam is subjected to an applied voltage.

In oder to validate the solution, the proof mass is neglected for this special case. This

configuration corresponds to the cantilever piezoelectric actuator described in Kasyap (2002).

The actuator deflection is determined for a test specimen comprised of a piezoceramic patch

65

attached to an aluminum shim. The dimensions and properties for this beam are listed in Table

2-2.

Table 2-2: Material properties and dimensions for a piezoelectric composite aluminum beam. Length of the beam ( )sL 101.60 mm

Width of the beam ( )sb 6.35 mm

Thickness of the beam ( )st 1.02 mm

Elastic modulus of PZT ( )pE 62 GPa

Density of PZT ( )pρ 2500 3kg m

Length of the PZT patch ( )pL 25.40 mm

Width of PZT ( )pb 6.35 mm

Thickness of PZT ( )pt 0.51 mm

Piezoelectric coefficient ( )31d -274 X 10-12 m V

Relative permittivity ( )rε 3400

Rewriting the static solution from Eq. (2.14) - (2.16) for the case when all mechanical

loads are neglected in the composite beam yields

( ) ( )2

1 2o

c

M xw xEI

⎛ ⎞= − ⎜ ⎟

⎝ ⎠ (2.27)

and

( ) ( ) ( )2 3 .2

o p p

c

M L Lw x w x x

EI⎛ ⎞

= = − −⎜ ⎟⎝ ⎠

(2.28)

Figure 2-8 compares the deflection using the two methods mentioned above. As indicated

in the figure, the modeshapes match exactly, which indicates that the electromechanical static

model accurately represents the structure in the absence of a proof mass.

66

Figure 2-8: Deflection modeshape for a composite beam subjected to an input voltage.

Next, we can use the modeshape for the piezoelectric actuator with a proof mass to

calculate the effective piezoelectric coefficient, which is defined as the tip deflection resulting

from an applied unit voltage. Since we need to obtain the electromechanical coupling between

the input voltage and the resulting deflection, the static deflection of the composite beam due to

all the mechanical loads is subtracted from the overall deflection. However, we assume the

system to be linear, and the solutions can be superimposed. Consequently, the effect of the

voltage on the deflection can be decoupled from the overall equations. Therefore, the resulting

tip deflection due to an input voltage is given as

( ) 2

Vtip o p pm s

app c

w M L Ld L

V EI⎛ ⎞

= = − −⎜ ⎟⎝ ⎠ ,

(2.29)

67

where, Vtipw is the tip deflection due to the applied voltage. After obtaining mM , msC , and md ,

the rest of the parameters in the circuit, such as , , and m eb eR C Rφ , can be easily obtained as

shown below, since they are simple analytical expressions related to these elements. In the

electromechanical circuit shown in Figure 2-2, φ is defined as the turns ratio for the

transformation between the electrical and mechanical domains and is given by the expression

.m

ms

dC

φ = − (2.30)

Next, as described earlier ebC is given by

2

1 ,meb ef

ef ms

dC CC C

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠ (2.31)

where efC is the free capacitance of the piezoceramic

.pef

p

AC

tε=

. (2.32)

Here, ε is the dielectric permittivity in the piezoceramic and pA is the surface area of the

piezoceramic. The resistive elements in the circuit are calculated using Eqs. (2.1) and (2.2).

Therefore, all but two of the lumped element parameters in the circuit have been analytically

obtained from the static electromechanical model. Only the mechanical damping and electrical

loss are estimated using the empirical relations provided in Eqs. (2.1) and (2.2). It should be

noted here that a viscous damping model assumed in this model, represented with an effective

damping ratio does not capture all loss mechanisms. A more detailed in-depth study of various

damping mechanisms is provided in Appendix B. The empirical relations and the estimated

68

values for the tested MEMS devices are also presented in Table 6-12 along with their

experimentally extracted damping.

Now, we can represent the composite beam in the lumped element circuit and simulate it

for various loading conditions. In the overall configuration, the input to the system is an

effective acceleration applied at the clamp, which is replaced with an equivalent inertial force in

the circuit (Yazdi et al. 1998). This effective force in the single degree of freedom system is

defined as the product of the effective mass and the acceleration of the center of mass of the

system. In this analysis, it has been assumed that the input acceleration is equal to the

acceleration of the center of mass. As will be demonstrated, this assumption has proven to be

fairly accurate in predicting the dynamic response until the first resonance with experimental

results. Therefore, the equivalent force is given as

.m o mF M a M g= + (2.33)

In the above equation, the first term corresponds to the dynamic input force due to applied

acceleration at the clamp. The second term is the static load on the beam acting due to gravity

which indicates the static deflection of the beam. This term is, however, not used for dynamic

simulations to predict the output voltage and current in the equivalent circuit. Therefore, the

input dynamic mechanical power is given as

,inP FU= (2.34)

where U is the relative instantaneous velocity of the tip with respect to the base. In Figure 2-2,

the device is connected to an external load circuit to reclaim power in a real application. Solving

for the input velocity in the circuit from U F overall impedance= , we obtain

( )( )2

1 .m o eb Le

eb L ms e L

M a Z RU

Z R Z Z Rφ+

=+ +

(2.35)

69

In the above expression, LR is the external load which is assumed to be purely resistive in

our analysis (Taylor 2004), ebZ is the blocked electrical impedance, and emsZ is the short circuit

mechanical impedance represented in the electrical domain that are given by the following

expressions

1

e

ebeb

eeb

Rj CZ

Rj C

ω

ω

=+

(2.36)

and

2

1 1 ,ems m m

ms

Z j M Rj C

ωφ ω

⎛ ⎞= + +⎜ ⎟

⎝ ⎠ (2.37)

where ω is the frequency of excitation in [ ]rad s . Therefore, the input power supplied to the

composite beam is obtained from Eq. (2.34) as

2

1 .m oin

eeb Lms

eb L

M aP Z R ZZ R

φ⎛ ⎞

= ⎜ ⎟⎝ ⎠ +

+

(2.38)

The input power to these structures when calculated using Eq. (2.38) based on the input

base acceleration gives an understanding about the amount of mechanical energy available for

conversion to the electrical domain. The conversion of mechanical power to electrical energy is

related to the coupling factor defined as

2

2 .m

ef ms

dC C

κ = (2.39)

The coupling factor determines the amount of electrical energy available in the

piezoceramic that can be reclaimed (Ikeda 1990).

70

All the analytical expressions for the electromechanical circuit elements have been derived

and presented for a composite beam to complete the circuit in Figure 2-2. Since all the lumped

parameters excluding the damping ratio in the circuit are obtained analytically and are dependent

on the material dimensions and properties, a detailed scaling analysis is carried out in the next

chapter to provide a motivation for designing MEMS devices. Furthermore, a simple design

strategy is presented to model and design these devices for characterizing energy reclamation

from vibrations.

Experimental Verification of the Lumped Element Model

This section summarizes meso-scale experiments to validate the lumped element model

and the corresponding electromechanical circuit for power generation. First, results are

presented for experiments carried out to verify the electromechanical lumped element model for

its mechanical and electrical behavior. Finally, overall power transfer estimates are obtained

experimentally and compared with the theoretical predictions to validate the model.

Initially, experiments were conducted with a clamped aluminum beam that was mounted

on a vibrating surface (LDS dynamic shaker model V408) to verify the dynamic and static

lumped element model. The material dimensions of the test specimen used are listed in Table 2-

3. The properties are listed in Table 2-1

Table 2-3: Material properties and dimensions for a homogenous aluminum beam. Length of the beam 127 mm

Width of the beam 6.35 mm

Thickness of the beam 1.02 mm

Length of the proof mass 3.17 mm

Width of proof mass 6.35 mm

Thickness of proof mass 8.64 mm

71

All the lumped element parameters (Figure 2-2) were obtained experimentally to validate

the model as follows. Static tests were carried out by loading the tip with known masses (that

were measured using an OHAUS mass balance with resolution of 0.1 mg± ) and the tip

deflection was measured using a Micro-Epsilon laser displacement sensor (OPTONCDT series

2000). An average compliance of the composite beam was obtained by calculating the ratio

between the resulting tip deflection and the static load at the tip for all the masses. The difference

between the estimated and calculated theoretical value using the properties and dimensions is

listed in Table 2-4. A simple impact test was carried out to obtain a damped impulse response to

estimate the natural frequency of the specimen. The natural frequency thus obtained using the

logarithmic decrement method (Craig 1981) was 50.5 Hz. From the measured natural frequency

and the effective compliance, the effective mass was calculated to be 0.523 gm.

Table 2-4: Measured and calculated parameters for the homogenous beam. CALCULATED MEASURED UNCERTAINTY

Effective mass of the beam, MM

0.540 gm 0.523 gm 3.1 %

Effective compliance of the beam, CMS

0.018 m/N 0.019 m/N 5.5 %

Natural Frequency, Fn 50.9 Hz 50.5 Hz 0.8 %

The structure was then mounted on a vibration shaker as shown in Figure 2-9 that was used

to excite the composite beam over a frequency range.

72

Figure 2-9: Experimental setup for verifying the electro-mechanical lumped element model for meso-scale cantilever beams.

The input acceleration to the structure was measured using an impedance head (Bruel &

Kjaer type 8001). The resulting tip deflection was measured using the displacement sensor. To

check mass loading effects in the impedance head, the input acceleration measured with the

impedance head was initially compared with the results obtained from a displacement sensor

measurement at the same point. It was observed that the results matched very well over the

frequency range. The measured resonant frequency and the compliance were then used to adjust

the mass of the model to match the predicted natural frequency. Figure 2-10 shows a plot of the

frequency response function between the tip deflection (measured with the displacement sensor)

and input acceleration. The magnitude, phase and coherence are indicated in the plot along with

a comparison with the lumped element model predictions. The results were found to match well

until beyond the first resonance. The parameter plotted in the figure is the transfer function

73

between the input acceleration and resulting tip deflection. The observed resonant frequency was

50.5 Hz. The frequency response using the LEM is calculated using the expression

( ) 11

m otip

m mms

M awjj M R

j C

ωωω

ω

=+ +

(2.40)

The damping ratio was estimated to be 0.005 by matching the response peaks at the resonant

frequency.

0 20 40 60 80 10010

-4

10-2

100

mag

(m/s

/m/s

2 )

exptLEM

0 20 40 60 80 100-200

0

200

phas

e(de

g)

0 20 40 60 80 1000

0.5

1

cohe

renc

e

frequency (Hz)

Figure 2-10: Comparison between experiment and theory for tip deflection in a homogenous beam (no tip mass).

As indicated in the above plots, the response was accurately predicted using the lumped

element model. Similar experiments were carried out for a homogenous beam with a known

proof mass attached at its tip. The addition of the tip mass to the system leads to an inertial tip

load during the vibration. This load will act as shear force along with the static load of the tip

74

mass due to gravity. The measured tip mass was 0.476 0.1 g mg± . A similar vibration

experiment was carried out for this device, and the tip deflection was measured as a function of

frequency for an input acceleration to the clamp. Figure 2-11 shows a comparison of the

frequency response function between theory and experiment. As seen in the figure, the plots

match well and the resonant frequency was measured to be approximately 37 Hz. The reduction

in the resonant frequency is due to the addition of the tip mass.

0 20 40 60 80 10010

-4

10-2

100

mag

(m/s

/m/s

2 )

exptLEM

0 20 40 60 80 100-200

0

200

phas

e(de

g)

0 20 40 60 80 1000

0.5

1

cohe

renc

e

frequency (Hz)

Figure 2-11: Comparison between theory and experiments for the tip deflection in a homogenous beam with tip mass.

Table 2-5: Measured and calculated parameters for the homogenous beam with a proof mass. CALCULATED MEASURED UNCERTAINTY


0.948 gm 0.974 gm 2.7 %


0.020 m/N 0.019 m/N 5 %


75

The estimated and calculated lumped element parameters for the homogenous beam with a

proof mass are listed in Table 2-5 along with the estimated uncertainty in the values.

After verifying the lumped element model for the two cases mentioned, they were

extended to a piezoelectric composite beam. The dimensions of the piezoelectric composite

beam are listed in Table 2-6. The same tip mass was used for these experiments.

Table 2-6: Material properties and dimensions for a piezoelectric composite aluminum beam. Length of the beam 103.38 mm

Width of the beam 6.35 mm

Thickness of the beam 0.51 mm

Elastic modulus of PZT 66 GPa

Density of PZT 7800 kg/m3

Length of the PZT patch 25.40 mm

Width of PZT 6.35 mm

Thickness of PZT 0.51 mm

Piezoelectric coefficient -190 X 10-12 m/V

Relative permittivity 1800

Length of the proof mass 3.17 mm

Width of proof mass 6.35 mm

Thickness of proof mass 8.64 mm

Before performing the vibration experiments, the static loading test and the impact test

were conducted as before to measure the mechanical compliance and the natural frequency.

These values are compared with the theoretically calculated parameters using the dimensions and

properties in Table 2-7. The relative uncertainties are observed to be higher than the

homogenous beam and can be attributed to the bond layer and the uncertainties in the PZT

dimensions (Mathew 2001). A detailed uncertainty analysis along the lines of what was

76

described in Kasyap (2002) can be carried out to obtain better estimates. However, for the

purpose of this validation, we use the values measured.

Table 2-7: Measured and calculated values for a PZT composite beam. CALCULATED MEASURED UNCERTAINTY


0.183 gm 0.176 gm 3.9 %


0.038 m/N 0.041 m/N 7.8 %


Effective piezoelectric coefficient, deff

-1.28e-6 m/V -1.18e-6 m/V 7.8 %

Blocked electrical capacitance

5.01 nF (Cef = 5.06 nF)

4.86 nF (Cef = 4.88 nF)

3.1 %

A vibration experiment was then carried out using the composite beam by mounting it on

the vibrating shaker. The clamped base was harmonically excited, and the tip deflection was

measured using a displacement sensor. Figure 2-12 plots the frequency response and a

comparison with the LEM. The response was observed to match well through the first

resonance. The mass of the composite beam was calculated using the compliance and measured

natural frequency. Some of the reasons for higher discrepancy in the composite beam are

attributed to the fact that the LEM does not include the epoxy bond layer that was used to attach

the PZT with shim. In addition, it does not incorporate the small gap that is provided between the

clamp and PZT to prevent any potential shorting during vibration. However, this will not occur

in the MEMS device as the clamp will form a part of the substrate itself. This will be understood

better in Figures 4.16 and 4.17.

77

0 20 40 60 80 10010

-4

10-2

100

mag

(m/m

/s2 )

exptLEM

0 20 40 60 80 100-200

0

200

phas

e(de

g)

0 20 40 60 80 1000

0.5

1

cohe

renc

e

frequency (Hz)

Figure 2-12: Frequency response of a piezoelectric composite beam (no tip mass)

A similar experiment was conducted with the composite beam that has a proof mass

attached to its tip. The measured and calculated parameters for the specimen are listed along

with the uncertainties in Table 2-8.

Table 2-8: Measured and calculated parameters for a PZT composite beam with a proof mass. CALCULATED MEASURED UNCERTAINTY


0.598 gm 0.623 gm 4.1 %


0.043 m/N 0.041 m/N 4.6 %


Effective piezoelectric coefficient, deff

-1.30e-6 m/V -1.18e-6 m/V 7.8 %

Blocked electrical capacitance

5.01 nF (Co = 5.06 nF)

4.86 nF (Co = 4.88 nF)

3.1 %

78

Figure 2-13 shows a plot of the frequency response function between the tip deflection and

input acceleration and, compares with the theoretical predictions using static LEM. As is evident

from the plot, the response matches well with the predictions and the resonant frequency was

found to be at 30 Hz.

0 20 40 60 80 10010

-4

10-2

100

mag

(m/m

/s2 )

exptLEM

0 20 40 60 80 100-200

0

200

phas

e(de

g)

0 20 40 60 80 1000

0.5

1

cohe

renc

e

frequency (Hz)

Figure 2-13: Frequency response for a piezoelectric composite beam (mp=0.476 gm).

Based on the above observations, it can be concluded that the lumped mechanical model

that was developed is sufficiently accurate in predicting the dynamic behavior of the meso-scale

composite beam.

To further verify and validate the electromechanical LEM, the same PZT aluminum

composite beam (without the proof mass) was characterized for both its mechanical and

electrical response. The dimensions and properties of the composite beam are listed in Table 2-6

and therefore not reproduced again.

79

All the lumped element parameters obtained previously in Table 2-7 are used for

subsequent validation. The damping ratio for the system was adjusted to match the peaks at

resonance in the response obtained both experimentally and the lumped element model (Figure

2-12). The resulting damping ratio was estimated to be 0.015 and this value was subsequently

used in the analysis. The increase in damping ratio from a homogeneous beam to the composite

beam is attributed to the added losses in the piezoelectric material and the epoxy layer.

To measure the effective piezoelectric coefficient, an ac voltage was applied to the PZT

and the resulting response at the tip was measured using the laser displacement sensor. Ideally,

the deflection needs to be measured at dc, but since it is difficult to perform this experimentally,

the response was measured at very low frequencies (~ 20 Hz) where the response is flat. This

value was used as the effective piezoelectric coefficient (dm) for subsequent calculations.

The free electrical impedance of the composite beam was measured using a vector

impedance meter, and an effective free capacitance was obtained as a result. However, the value

for dielectric loss was not measured experimentally and an empirical relation was used (Eq. 2.5)

to estimate its value. Therefore, all the lumped elements that can be estimated experimentally

were thus obtained and these values were used in the lumped element model to generate the

overall response and predict its output characteristics. The resulting values are shown in Table

2-7 and were compared with the theoretical values. Reasonable agreement (better than 8%) was

obtained between the measured and calculated values.

To validate the electrical behavior of the composite beam, another experiment was

conducted wherein the resulting voltage across the PZT was measured as a function of the input

acceleration at the clamp. The frequency response function thus obtained is shown in Figure 2-

80

14. As indicated in the figure, the model matched well with the measured response indicating the

validity of the complete lumped element model.

Figure 2-14: Output voltage for an input acceleration at the clamp.

Finally, after verifying the lumped element model for the frequency response, a sinusoidal

acceleration signal was input at resonance, and the resulting output voltage was measured across

a range of resistive loads to measure the output power. The results for the measured RMS

voltage are shown in the following figure. The plot indicates the voltage generated for unit

acceleration input (1 m/s2 RMS) as a function of different resistive loads varying from 10 KΩ to

1 MΩ. As indicated in the figure, the output voltage increases and saturates to a constant value

called the open circuit voltage as the load increases. In the shown plot, the output voltage is

normalized with the input acceleration to compare with the experimental values.

81

Figure 2-15: Output voltage for varying resistive loads.

Figure 2-16 shows the output power ( )2L LV R generated at the PZT calculated from the

measured voltage across the resistive loads for the same input conditions as in Figure 2-15. As

indicated in the plot, the power reaches a maximum value at an optimal resistance which occurs

when it is equal to the input impedance of the composite beam. It was observed that the optimal

load thus estimated is approximately 404 kΩ which is close to the theoretical value, 450 kΩ.

These measured and calculated values for the voltage and power are listed in the following Table

2-9.

82

Table 2-9: Comparison between experimental and theoretical values for power transfer. Estimated Measured

Optimal load 450 kΩ 404 kΩ

Voltage (per unit acceleration) 0.91 V/m/s2 0.85 V/m/s2

Corresponding output power 1.83 µW/m2/s4 1.78 µW/m2/s4

Figure 2-16: Output power across varying resistive loads.

As seen in the above plots, the lumped element model results agree within 10%. This

discrepancy is either within experimental uncertainty or acceptably small for design purposes. It

should be noted here that the mechanical and electrical damping are not accurately characterized

or known. Using these results and conclusions, the LEM is extended next to the micro-scale by

scaling down all the dimensions of the structure proportionally. The input acceleration is also

scaled proportionally to operate the composite beam in the linear region so that the model can be

used to predict its behavior. The next chapter describes in detail the scaling analysis for the

83

composite beam indicating the dependence of all the lumped element parameters with the

dimensions as they scale down. The next chapter also describes the motivation for fabricating

these structures using MEMS and the inherent advantages in their performance.

84

CHAPTER 3 MEMS PIEZOELECTRIC GENERATOR DESIGN

In this chapter, a detailed dimensional analysis is presented for the piezoelectric composite

beam. Then, a scaling theory is developed based on the dimensional analysis to determine the

response of the structure when it is scaled down in size. The objective behind developing a

dimensional analysis and scaling theory is to provide a tool that enables better understanding of

the device behavior as a function of dimensions and properties. In addition, it can be used as a

tool to optimally design a first generation device aimed at specific applications. Next, a design

strategy is formulated for the composite beam based on a given set of input parameters. In

addition, each of the proposed designs is optimized using a parametric search procedure

described in this chapter, subject to design and fabrication constraints, but without any

conventional optimization techniques.

Power Transfer Analysis

Recall that all of the lumped parameters calculated in the previous chapter, with the

exception of the empirical damping coefficient, are analytical functions of the material properties

and device dimensions. The equivalent circuit model for the composite beam can now be

attached to an external circuit to harness power. The external circuit has an electrical impedance

associated with it, which determines the amount of power that can be reclaimed from the

composite beam. For the sake of our simulations, we assume that the external circuit is purely

resistive (Taylor et al. 2004; Horowitz et al. 2002) and is represented as shown in Figure 3-1.

Since most energy harvesters seek to reclaim and store energy (e.g., via a battery) that is later

dissipated, a resistive load works best for analyses. In addition, most energy reclamation circuits

85

present a purely resistive load to the generator. Figure 3-1 is the overall equivalent

electromechanical circuit drawn as its Thévenin equivalent. From elementary circuit analysis, it

can be proven that maximum power transfer occurs when the complex load impedance LZ is the

complex conjugate of the Thévenin impedance *THZ (derived in Appendix D). In the present

case, in which the external circuit presents a purely resistive impedance, the optimal load

resistance LR equals the magnitude of the Thévenin impedance in order to maximize power

transfer (Appendix D). The Thévenin voltage is defined as the open circuit voltage, and the

Thévenin impedance as the short circuit impedance (Irwin 1996) across the output and is

calculated from the original representation shown in Figure 2-2.

Figure 3-1: Thévenin equivalent circuit for the energy reclamation system

In the circuit, ThV is the equivalent Thévenin voltage, which is

2 2 2

1

,

11

e

eb

eeb

The

eb m m

me

eb

Rj C F

Rj C

V Rj C j M R

j CRj C

ωφ

ω

ω ωφ ωφ φ

ω

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟+⎜ ⎟⎝ ⎠=

+ + ++

(3.1)

86

and the Thévenin impedance, ThZ , is

2 2 2

2 2 2

11

.

11

e

eb m m

me

ebTh

e

eb m m

me

eb

Rj C j M R

j CRj C

Z Rj C j M R

j CRj C

ω ωφ ωφ φ

ω

ω ωφ ωφ φ

ω

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟ + +⎜ ⎟⎜ ⎟⎝ ⎠+⎜ ⎟⎝ ⎠=

+ + ++

(3.2)

All the parameters in the above equations are defined in the equivalent circuit in Figure 2-2.

Consistent with the above discussion, we assume that the output load is optimal and is therefore

equal to the Thévenin impedance as given by the expression (Appendix E)

.L ThR Z= (3.3)

The current across the load can be obtained from Ohm’s law as

.ThL

Th L

VIZ R

=+

(3.4)

The rms power across the load is defined as the product of the load and the square of the rms

current, given by the expression

2_

1 ,2L rms L LP I R= (3.5)

where L LI R is the voltage across the load. In addition, the input rms mechanical power to the

device was calculated using the expression

1 1 .2 2in

FP U F Uφφ

= = (3.6)

From Eq. (3.5) and Eq. (3.6), we can calculate the overall electromechanical efficiency of the

power transfer across the resistive load

87

2

.L LL

in

I RPP F U

η = = (3.7)

Substituting the lumped element expressions for these parameters in terms of material

properties and device dimensions provides the desired scaling dependence of power and

efficiency but results in expressions that, because of their algebraic complexity, do not provide

any significant physical insight. Instead, dimensional analysis is used below for the scaling

analysis to optimally design an energy reclamation device and corresponding external circuit that

can harness maximum energy from the piezoelectric composite beam.

Nondimensional Analysis

A list of the all the variables in the electromechanical model are listed below in Table 3-1

that describe the dynamic behavior of a piezoelectric composite beam. First, a set of primary

variables are selected that incorporate the basic dimensions such as length, time, etc. Next, all the

other variables used to describe the composite beam are expressed as nondimensional groups.

These groups are later used to nondimensionalize the response functions such as modeshape,

LEM parameters, etc. providing the dependent Π groups.

A schematic of the device with all dimensions and properties is shown below in Figure 3-

2. The dimensions and properties have already been discussed in Section 2.2.1.

88

Figure 3-2: Schematic of the MEMS PZT device.

Table 3-1: List of all device variables that are described in the electromechanical model. Variable Description

,s sE ρ Material properties of shim

31, , ,p p rE dρ ε Material properties of piezoelectric layer

,m mE ρ Material properties of proof mass , ,s s sL b t Geometric dimensions of shim , ,p p pL b t Geometric dimensions of piezoelectric layer

, ,l w h Geometric dimensions of proof mass tan ,δ ζ Dielectric loss tangent and mechanical damping coefficient

,oa f Vibration parameters, acceleration and frequency

Due to the fabrication process that was designed for the devices, the following conditions

hold true, namely,

s pb b b= = (3.8)

The width of the piezoelectric layer and shim are assumed to be same to simplify the analysis. In

addition, the shim and proof mass are assumed to be made from the same material. Therefore,

and s m s mE E ρ ρ= = (3.9)

89

For the scaling analysis carried out here, we make the following assumption that simplifies

the derivation: namely, pt and h are fixed in the analysis due to fabrication constraints that

restrict the thickness of the piezoelectric layer and the proof mass. The thickness of the proof

mass is formed from the substrate and therefore is equal to the wafer thickness. The thickness of

the PZT layer was restricted by ARL process capability, which was 1 mμ at the time. Listing the

remaining variables, we obtain the following tabulated parameters with their dimensions as

indicated in Table 3-2.

Table 3-2: Dimensional representation of all the device variables. Variable Dimensional units

sE 1 2ML T− −⎡ ⎤⎣ ⎦ sρ 3ML−⎡ ⎤⎣ ⎦ pE 1 2ML T− −⎡ ⎤⎣ ⎦ pρ 3ML−⎡ ⎤⎣ ⎦

31d 1 1 2M L T Q− −⎡ ⎤⎣ ⎦ ε 1 3 2 2M L T Q− −⎡ ⎤⎣ ⎦

sL [ ]L b [ ]L

st [ ]L

pL [ ]L l [ ]L w [ ]L

oa 2LT −⎡ ⎤⎣ ⎦ ef 1T −⎡ ⎤⎣ ⎦

For the dimensional analysis, the following independent primary variables were defined.

These parameters were chosen to include the primary dimensions of length, time, mass, and

charge. All the other parameters will be expressed using these primary variables.

90

Table 3-3: Primary variables used in the dimensional analysis. Independent variable

Dimension

sL [ ]L corresponds to "length" dimension

sρ 3ML−⎡ ⎤⎣ ⎦ includes the "mass" dimension

sE 1 2ML T− −⎡ ⎤⎣ ⎦ includes the "time" dimension

31d 1 1 2M L T Q− −⎡ ⎤⎣ ⎦ corresponds to the "charge" dimension

The remaining variables are now nondimensionalized using the 4 repeating variables to

obtain independent dimensionless “Π ” groups as listed below. These Π groups will be used to

nondimensionalize the piecewise deflection solution obtained in Eq. 2.9-2.11. Furthermore, the

analysis will be extended to nondimensionalize the LEM parameters in the equivalent circuit

model to finally investigate the device behavior for various topologies.

1p

s

EE

= Π (3.10)

2p

s

ρρ

= Π (3.11)

We know that pef

p

AC

tε = which implies that ε is dimensionally represented as F

m⎡ ⎤⎢ ⎥⎣ ⎦

.

Therefore,

1 3 2 2M L T Qε − −⎡ ⎤= ⎣ ⎦ (3.12)

and the corresponding nondimensional Π group is

3231sE d

ε= Π (3.13)

91

Equation (3.13) is a measure of the coupling between the electrical and mechanical domain

and in the ideal case, can be reduced to Eq. 2.33. Furthermore, the device dimensions are scaled

as

4s

bL

= Π (3.14)

5s

s

tL

= Π (3.15)

6p

s

LL

= Π (3.16)

7p

s

tL

= Λ (3.17)

From Eq. (3.15) and Eq. (3.17), we obtain

77

5

p p s

s s s

t t Lt L t

Λ= = = Π

Π (3.18)

8s

lL

= Π (3.19)

9s

wL

= Λ (3.20)

From Eq. (3.14) and Eq. (3.20),

99

4

s

s

Lw wb L b

Λ= = = Π

Π (3.21)

10s

hL

= Λ (3.22)

From Eq. (3.15) and Eq. (3.22),

92

1010

5

s

s s s

Lh ht L t

Λ= = = Π

Π (3.23)

Similarly, the external vibration parameters, such as the acceleration and excitation

frequency, are nondimensionalized as

11o

s

s s

aE

L ρ

= Π⎛ ⎞⎜ ⎟⎝ ⎠

(3.24)

121e

s

ss

fE

L ρ

= Π⎛ ⎞⎜ ⎟⎝ ⎠

(3.25)

All the above derived independent primary Π groups are listed below in Table 3-4. Next,

the location of the neutral axis in the composite section ( )0 px L< < measured from the bottom

can be expressed as (Chapter 2)

( )2 2

2

2.

2s s s s p p p

s s p p

E t E t t E tc

E t E t+ +

=+

(3.26)

Dividing the above expression by 2s sE L and nondimensionalizing with respect to shim

thickness yields

( ) ( )

2 2

21 5 7 1 5 7

2, , , ,

2

p p ps s

s s s s s

p ps s

s s s

t E tt tL L L E Lc f f

E tL tL E L

⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠= = Λ Λ Λ = Π Π Π⎛ ⎞

+⎜ ⎟⎝ ⎠

(3.27)

Similarly, 1c is denoted as the position of the neutral axis from the top

1 2s pc t t c= + − (3.28)

and in nondimensional form becomes

93

( ) ( )1 21 5 7 1 5 71 , , , ,p

s s s

tc c f fL L L

= + − = Λ Λ Λ = Π Π Π (3.29)

Table 3-4: List of independent ∏ groups. Π group Dependent variables

1Π p

s

EE

2Π p

s

ρρ

3Π 231sE d

ε

4Π s

bL

5Π s

s

tL

6Π p

s

LL

7Π p

s

tt

8Π s

lL

9Π wb

10Π s

ht

11Π o

s

s s

aE

L ρ⎛ ⎞⎜ ⎟⎝ ⎠

12Π 1

e

s

ss

fE

L ρ⎛ ⎞⎜ ⎟⎝ ⎠

The bending moments of inertia in each section, 1 2, , and s s p mI I I I are nondimensionalized

as

( ){ }331 1 23s p

bI c c t= + − (3.30)

94

( )

3 3

1 2

11 4 5 74 , , ,

3

p

s s s ss

s

tc cbL L L LI f

L

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪+ −⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭= = Π Π Π Π (3.31)

( ){ }332 23p p

bI c c t= − − (3.32)

( )

3 3

2 2

1 4 5 74 , , ,3

p

s s s sp

s

tc cbL L L LI

fL

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪− −⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭= = Π Π Π Π (3.33)

3

2 12s

sbtI = (3.34)

( )

3

24 54 ,

12

s

s ss

s

tbL LI f

L

⎛ ⎞⎜ ⎟⎝ ⎠= = Π Π (3.35)

3

12mwhI = (3.36)

( )

3

4 5 9 104 , , ,12

s sm

s

w hL LI f

L

⎛ ⎞⎜ ⎟⎝ ⎠= = Π Π Π Π (3.37)

The linear mass densities of each section are now nondimensionalized as

slen s st bρ ρ= (3.38)

( )4 52 ,slen s

s s s s

t b fL L L

ρρ

= = Π Π (3.39)

( )clen s s p pt t bρ ρ ρ= + (3.40)

( )2 4 5 72 , , ,slen p ps

s s s s s s

tt b fL L L L

ρ ρρ ρ

⎛ ⎞= + = Π Π Π Π⎜ ⎟⎝ ⎠

(3.41)

mlen mwhρ ρ= (3.42)

95

( )4 5 9 102 , , ,slen m

s s s s s

w h fL L L

ρ ρρ ρ

= = Π Π Π Π (3.43)

Next, the rigidity moduli in each of the beam sections are nondimensionalized as

( ) 2s ssEI E I= (3.44)

( ) ( )2

4 54 4 ,s s

s s s

EI I fE L L

= = Π Π (3.45)

( ) 1s s p pcEI E I E I= + (3.46)

( ) ( )1

1 4 5 74 4 4 , , ,p pc s

s s s s s

EI E II fE L L E L

= + = Π Π Π Π (3.47)

( ) m mmEI E I= (3.48)

( ) ( )4 5 9 104 4 , , ,m m m

s s s s

EI E I fE L E L

= = Π Π Π Π (3.49)

Let us now define another nondimensional parameter for acceleration due to gravity as

s

s s

gE

L ρ

= Ψ⎛ ⎞⎜ ⎟⎝ ⎠

(3.50)

Nondimensionalizing the uniform line load on the beam due to its weight, we obtain

ss lenq q g= (3.51)

( )4 52 ,slens

ss s s ss s

q g fEE L LL

ρρ

ρ

⎛ ⎞⎛ ⎞⎜ ⎟

= = Π Π Ψ⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟

⎝ ⎠

(3.52)

cc lenq q g= (3.53)

( )2 4 5 72 , , ,clenc

ss s s ss s

q g fEE L LL

ρρ

ρ

⎛ ⎞⎛ ⎞⎜ ⎟

= = Π Π Π Π Ψ⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟

⎝ ⎠

(3.54)

96

mm lenq q g= (3.55)

( )4 5 9 102 , , ,mlenm

ss s s ss s

q g fEE L LL

ρρ

ρ

⎛ ⎞⎛ ⎞⎜ ⎟

= = Π Π Π Π Ψ⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟

⎝ ⎠

(3.56)

In the static solution for the deflection of the composite beam described in Chapter 2, we

defined two constants

( )( )

( )

( ) ( )2

1 4 5 7

2

, , ,s

s s s

cc

s s

EIEI E LC f

EIEIE L

= = = Π Π Π Π (3.57)

and

( )( )

( )

( ) ( )2

4 5 9 10

2

, , ,m

m s s

ss

s s

EIEI E LD f

EIEIE L

= = = Π Π Π Π (3.58)

The Euler-Bernoulli equations that were solved earlier to obtain a piecewise continuous

deflection modeshape for the beam (in Chapter 2), are rewritten here as

( )( )

( )

2 224 3 2

1

24 6 2 2 2 2 2p ps

c c p s s p m c s m s o

c

L LLx x l xq q L q L L q l q q q l L M

w xEI

⎡ ⎤⎛ ⎞ ⎛ ⎞⎡ ⎤− + + − + − + − + + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦=

(3.59)

( ) ( ){ }

( ) ( )( )( ) ( )

22

341 3 2 4

2

2 2,

24 6 2

ss m s

s s ms

s s s s s

L lq q l L xq L q l x C C xq x C Cw x

EI EI EI EI EI

⎧ ⎫⎛ ⎞⎪ ⎪⎛ ⎞+ +⎨ ⎬⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ +⎪ ⎪⎝ ⎠⎩ ⎭= − + − + +

(3.60)

and

( ) ( )( )( )

( )( )

( )( ) ( )

23 245 7 6 8

3 .24 6 4

m s m sm

m m m m m

q L l x q L l x C C xq x C Cw xEI EI EI EI EI

+ + + += − + − + + (3.61)

97

where the integration constants 1 2 3 4 5 6 7, , , , , ,C C C C C C C and 8C are given by the following

expressions

( ) ( )( )

( )( ) ( )( )

( )( )

( )

21

22

2

3

24

3

5 1

4

6 2

3

7 3

3 1 ,6

3 2 8 6 1 ,24

1 2 2 ,2

1 ,2 3 4 2

,6

,24

6

pp s c s s s p

pp c s s s p s

pm s p o

po sp m

s s

s s

m s

LC L q q q L L L

LC L q q q L L L

LC q l L L l M

LM LlC L q l

q LC C

q LC C

q LC C

α α

α α α

α α

α α

γ

γ

γ

⎡ ⎤= − + − −⎣ ⎦

⎡ ⎤= − − + − −⎣ ⎦

⎡ ⎤= − − + −⎣ ⎦

⎡ ⎤⎛ ⎞= + − − −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦⎡ ⎤

= −⎢ ⎥⎣ ⎦⎡ ⎤

= +⎢ ⎥⎣ ⎦

= + + ( )( )

( )( )4 28 4

1 1 ,2

1 1 4 6 .24

m s s

m s m s s

q lL L l

C C q L q lL L l

γ

γ γ

+ −

⎡ ⎤= − + − +⎣ ⎦ (3.62)

Removing the effect of moment due to applied voltage to the piezoelectric layer and

nondimensionalizing the deflections with length yields,

( ) ( ) ( ) ( )1 2 31 2 4 5 6 7 8 9 10, , , , , , , , ,

s s s

w w wf

L L Lψ ψ ψ

ψ= = = Π Π Π Π Π Π Π Π Π Ξ (3.63)

Substituting for ( )3Ftip sw w L l= + and nondimensionalizing, we obtain

( )1 2 4 5 6 7 8 9 10, , , , , , , ,Ftip

s

wf

L= Π Π Π Π Π Π Π Π Π Ξ (3.64)

Integrating the deflection across the length of the beam results in the total potential and

kinetic energies, represented as

98

( ) ( ) ( ) ( ) ( ) ( )

2 2 22 2 21 2 3

2 2 202 2 2

p s s

p s

L L L lc s m

L L

EI EI EId w x d w x d w xPE dx dx dx

dx dx dx

+⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠∫ ∫ ∫ (3.65)

and

( ) ( ) ( )2 2 21 2 3

02 2 2

p s s

c s m

p s

L L L llen len len

L L

KE w x dx w x dx w x dxρ ρ ρ +

= + +∫ ∫ ∫ (3.66)

Nondimensionalizing the energies yields

( ) 21 2 4 5 6 7 8 9 103 5 , , , , , , , ,

s s s s

PE KE fE L Lρ

= = Π Π Π Π Π Π Π Π Π Ξ (3.67)

From the energies, the short circuit compliance and mass are extracted and

nondimensionalized as

( )1 2 4 5 6 7 8 9 103 , , , , , , , ,mms s s

s s

MC E L fLρ

= = Π Π Π Π Π Π Π Π Π (3.68)

The natural frequency shown in Eq. 2.24 is nondimensionalized as

( )1 2 4 5 6 7 8 9 103

3

1 1 , , , , , , , ,21

n

s s msms s s

s s s s ss

f fL ME C E L

E L LLρπ

ρ ρ

= = Π Π Π Π Π Π Π Π Π⎛ ⎞⎜ ⎟⎝ ⎠

(3.69)

Furthermore, the angular frequency , nω ( )2 nfπ= is a function of the same Π groups and

follows Eq. (3.69). The Rayleigh mechanical damping in the system using its empirical relation

listed in Eq. 2.1 is nondimensionalized as

( ) ( )1 2 4 5 6 7 8 9 1032

2 , , , , , , , ,1

m m n

s ss s s s

ss

R M fLL E E

L

ωζρρ

ρ

= = Π Π Π Π Π Π Π Π Π⎛ ⎞⎜ ⎟⎝ ⎠

(3.70)

It should be noted here that the mechanical damping model in the system does not

accurately represent all loss mechanisms. Some of the general damping losses were investigated

99

and presented in Appendix B. Although the LEM in this dissertation still assumes a Raleigh

viscous damping effect with an equivalent damping ratio, other damping losses are also studied

here for their scaling behavior. The loss due to air flow in the viscous region due to device

vibration is given by an empirical relation derived in Appendix B as

12

2 2

12 .6 1

s sn s s

aeq

eq s

Ek b tQ

RR L

ρ

πμδ

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞+⎜ ⎟

⎝ ⎠

(3.71)

All the variables are defined in Appendix B. For any structure operating at fixed conditions,

and nk μ remain constant. So, the nondimensional form simplifies to

( )225 4

12 31

s s sna

eq

eq s

ss

L EkQR

R LL

L

ρμπ

δ

⎛ ⎞ Π Π= ⎜ ⎟⎜ ⎟ ⎛ ⎞⎝ ⎠ ⎜ ⎟

+⎜ ⎟⎜ ⎟⎝ ⎠

(3.72)

The losses at the support that arise due to the transmitted energy through the clamp during

flexural vibration is empirically given as

3

0.23 ,sc

s

LQt

⎛ ⎞= ⎜ ⎟

⎝ ⎠ (3.73)

and in nondimensional form is

3

5

10.23cQ⎛ ⎞

= ⎜ ⎟Π⎝ ⎠ (3.74)

The surface losses are given as

( )2 3

s ss

s s

bt EQb t Eδ

=+ Δ

(3.75)

100

where sEΔ is the difference between the adiabatic and isothermal Young’s modulus of the

material. It is also known as the dissipation modulus of the surface layer whose thickness is

given by δ . The nondimensional form of Eq. (3.75) is

( )

4 5

4 52 3s

ss

s

EQEL

δΠ Π

=ΔΠ +Π

(3.76)

Similarly, the volume losses are

sv

s

EQE

=Δ

(3.77)

For scaling purposes, we assume that the dissipation modulus scales proportional to the elastic

modulus of the material. In this analysis, the squeeze film damping is neglected as the vibrations

occur in free space without influence of walls around the device. Finally the empirical form for

thermoelastic loss in a vibrating structure is given as

2

180 ,st

s p s n s

EkQC t f Eρ

=Δ

(3.78)

where k is thermal conductivity and pC , the specific heat capacity at constant pressure for the

material. This expression in Eq. (3.78) is a simpler form of the actual expression derived in Eq.

B.22. Expanding for the natural frequency and simplifying leads to

2

380 ,s sst

s p s s

ELkQC t E

ρρ

=Δ

(3.79)

The nondimensional form of the quality factor can be expressed as

25

180 ,st

p s s s

EkQC E tρ

=Π Δ

(3.80)

101

For the mechanical characterization, the frequency response between the input acceleration and

the resulting deflection is already shown in Eq. 2.37. Normalizing the impedance terms,

1m m

ms

j M Rj C

ωω

⎛ ⎞≅ ≅⎜ ⎟

⎝ ⎠ in the function yields

( ) ( ) ( ) ( )1 2 4 5 6 7 8 9 102 2 2

1, , , , , , , ,m m ms

s s s s s s s s s

j M R j C fL E L E L Eω ω

ρ ρ ρ≅ ≅ = Π Π Π Π Π Π Π Π Π (3.81)

Substituting this back into the original equation, we obtain the nondimensional form of the

response as

( )1 2 4 5 6 7 8 9 10 11, , , , , , , , .tip

s

wf

L= Π Π Π Π Π Π Π Π Π Π (3.82)

Now, for an applied voltage, neglecting all mechanical loads, the moment resulting at the

PZT due to the voltage is nondimensionalized as

3

231

3131

1 22

p pin s so

ss s s s

E tV E LcbM d LE L L L dd

⎛ ⎞= − −⎜ ⎟

⎝ ⎠ (3.83)

( )1 4 5 7 133 , , ,o

s s

M fE L

= Π Π Π Π Π (3.84)

The effective piezoelectric coefficient of the composite is a function of 31d and is


( ) 2

Vtip o p pm s

app appc

w M L Ld L

V EI V⎛ ⎞

= = − −⎜ ⎟⎝ ⎠

(3.85)

102

( )( )

3

1 4 5 6 731

4

31

1 , , , ,2

o p

pm s s s

sappc

ss s

M LLd E L L f

d LVEILE L

d

⎛ ⎞⎜ ⎟

= − − = Π Π Π Π Π⎜ ⎟⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠

(3.86)

Extending the analysis to the electromechanical parameters, the transduction factor is


3131

m

ms s

ms ms s s

dd d d E LC C E L

φ = − = − (3.87)

( )1 2 4 5 6 7 8 9 1031

, , , , , , , ,s s

fd E L

φ= Π Π Π Π Π Π Π Π Π (3.88)

The free and blocked electrical capacitances are similarly expressed in terms of

nondimensional groups as

2

23131

p

p s s sef s s

pp

s

L bL b d E L LC d E Ltt

L

εε

= = (3.89)

( )1 2 3 4 5 6 7 8 9 10231

, , , , , , , , ,ef

s s

Cf

d E L= Π Π Π Π Π Π Π Π Π Π (3.90)

and

2

2 2231312

31

m

efmeb ef s s

ms s s ms s s

dCd dC C d E L

C d E L C E L

⎛ ⎞⎜ ⎟

= − = −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

(3.91)

( )1 2 3 4 5 6 7 8 9 10231

, , , , , , , , ,eb

s s

C fd E L

= Π Π Π Π Π Π Π Π Π Π (3.92)

The empirical form of the dielectric loss in the piezoelectric layer is represented as

103

( )2

2 3131

1 1 1tan tan

1

eebeb s

sss ss

ss

R CC Ed Ed E LEL

ωωδω δ ρ

ρ

= =

⎛ ⎞⎜ ⎟⎝ ⎠

(3.93)

( ) ( )231 1 2 3 4 5 6 7 8 9 10, , , , , , , , ,s

e ss

ER d E fω ρ = Π Π Π Π Π Π Π Π Π Π (3.94)

Finally, the electromechanical coupling coefficient is nondimensionalized as

( )

2

2 22 31

1 2 3 4 5 6 7 8 9 10

231

, , , , , , , , ,m

m

efef msms s s

s s

dd d fCC C C E L

d E L

κ = = = Π Π Π Π Π Π Π Π Π Π (3.95)

The overall short circuit mechanical impedance expressed in the electrical domain, given

in Chapter 2, is rewritten here as

2

1 1 .ems m m

ms

Z j M Rj C

ωφ ω

⎛ ⎞= + +⎜ ⎟

⎝ ⎠ (3.96)

( ) ( ) ( ) ( )

2

2 2 2 2 231

31

11 .s s se m m ms

mss s s s s s s s s s s

s s

L E j M R j CZd E L L E L E L E

d E L

ρ ω ωρ ρ ρφ

⎛ ⎞⎜ ⎟= + +⎜ ⎟⎛ ⎞ ⎜ ⎟⎝ ⎠⎜ ⎟

⎝ ⎠

(3.97)

Nondimensionalizing the impedance results in

( )1 2 4 5 6 7 8 9 10

231

, , , , , , , ,1

ems

s

s s

Z f

E d Eρ

= Π Π Π Π Π Π Π Π Π⎛ ⎞⎜ ⎟⎝ ⎠

(3.98)

Similarly, the blocked electrical impedance is also nondimensionalised yielding

,1

e

ebeb

eeb

Rj CZ

Rj C

ω

ω

=+

(3.99)

104

( )

1tane

eb

RCδ ω

≅ (3.100)

( ) ( )2 2

31 31 1 2 3 4 5 6 7 8 9 101 , , , , , , , , ,

tans s

e s ss seb

E ER d E d E fCρ ρδ ω

≅ = Π Π Π Π Π Π Π Π Π Π (3.101)

and,

( )1 2 3 4 5 6 7 8 9 10

231

, , , , , , , , ,1

eb

s

s s

Z f

E d Eρ

= Π Π Π Π Π Π Π Π Π Π⎛ ⎞⎜ ⎟⎝ ⎠

(3.102)

In the resulting overall Thévenin equivalent circuit (shown in Figure 3-1), the Thévenin

impedance is obtained by combining the two impedances in parallel as

2 231 31

231

2 231 31

1 11 .

1 1

ems eb

s se

s s s sms eb sth ee

ms ebms eb s s

s s

s s s s

Z Z

E d E E d EZ ZZZ ZZ Z E d E

E d E E d E

ρ ρρ

ρ ρ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠= =

+ +⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(3.103)

Therefore,

( )1 2 3 4 5 6 7 8 9 10

231

, , , , , , , , , .1

th

s

s s

Z f

E d Eρ

= Π Π Π Π Π Π Π Π Π Π⎛ ⎞⎜ ⎟⎝ ⎠

(3.104)

Similarly, the Thévenin voltage is the open circuit voltage across the piezoelectric layer

given as

105

2 331

31

31

2 231 31

1

.

1 1

eb

s m

s seb m o s s o sth ee

sms ebms eb

s s s ss s

s s s s

Z

ME d EZ M a L a LV EZ ZZ Z d

E L d L

E d E E d E

ρρφφ

ρρ ρ

⎛ ⎞⎜ ⎟⎝ ⎠= =

+ +⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(3.105)

So, the dimensionless form is

( )1 2 3 4 5 6 7 8 9 10 11

31

, , , , , , , , , .th

s

V fLd

= Π Π Π Π Π Π Π Π Π Π Π (3.106)

As evident in the above equation, the voltage is directly proportional to the input

acceleration, 11Π . The optimal load for maximum power transfer occurs when the output

resistance equals the absolute value of the Thévenin impedance.

231

231

11

s thL th

s s s

s s

ZR ZE d E

E d E

ρρ

= =⎛ ⎞⎜ ⎟⎝ ⎠

(3.107)

and

( )1 2 3 4 5 6 7 8 9 10

231

, , , , , , , , , .1

L

s

s s

R f

E d Eρ

= Π Π Π Π Π Π Π Π Π Π⎛ ⎞⎜ ⎟⎝ ⎠

(3.108)

The current across the load resistor is given as

3131

2 231 31

1 1

th

s

th sL s s

th Lth L s

s s

s s s s

VL

V d EI L E dZ RZ R

E d E E d E

ρρ ρ

= =+ +

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(3.109)

106

and

( )1 2 3 4 5 6 7 8 9 10 11

31

, , , , , , , , , .L

ss s

s

I fEL E dρ

= Π Π Π Π Π Π Π Π Π Π Π⎛ ⎞⎜ ⎟⎝ ⎠

(3.110)

The output voltage across the load resistance is given as

31

31 231

1sL L

L L Ls s

s ss s s

LI RV I RdEL E d

E d Eρ

ρ

= =⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(3.111)

and

( )1 2 3 4 5 6 7 8 9 10 11

31

, , , , , , , , , .L

s

V fLd

= Π Π Π Π Π Π Π Π Π Π Π (3.112)

The output rms power across the load is obtained as the product of the load current and

voltage, represented as

2

3131

12

sL LL L L s s

s sss s

s s

EV IP V I E LL EL E ddρ

ρ

= =⎛ ⎞⎜ ⎟⎝ ⎠

(3.113)

( ) 21 2 3 4 5 6 7 8 9 10 11

2

, , , , , , , , , .L

ss s

s

P fEE Lρ

= Π Π Π Π Π Π Π Π Π Π Π (3.114)

The tip velocity can be obtained by solving for the current in the Thévenin equivalent as

( ) 2 .eb L m o

e eeb ms L ms eb

Z R M aUZ Z R Z Z φ

+=

+ + (3.115)

Expressing the velocity in dimenionless terms simplifies Eq. (3.115) to

107

( )1 2 3 4 5 6 7 8 9 10 11, , , , , , , , , .s

s

U fEρ

= Π Π Π Π Π Π Π Π Π Π Π (3.116)

Therefore, the input rms mechanical power is the product of the force and the resulting tip

velocity given as

23

1 12 2

m o sin s s

ss s ss

s s s

M a EUP FU E LEL EL

ρ ρρ ρ

= = (3.117)

and the nondimensional form for the input power is simplified to

( ) 21 2 3 4 5 6 7 8 9 10 11

2

, , , , , , , , , .in

ss s

s

P fEE Lρ

= Π Π Π Π Π Π Π Π Π Π Π (3.118)

All the response parameters derived so far are generalized expressions and will simplify

near resonance when the reactive portion of the mechanical impedance approaches zero. Hence,

at resonance, only the real part of the impedance remains, which is directly related to the

dissipative mechanisms in the device. Since the LEM uses very basic simplified empirical

relations for the mechanical damping and electrical loss, the expressions do not provide any

useful physical insight. Therefore, they have not been simplified and/or presented here.

The overall electromechanical efficiency expressed as the ratio between the output and

input power is observed to be independent of the base acceleration (i.e., not a function of 11Π ).

This holds true subject to the constraints in the model that implies a linear relation between the

input acceleration and the tip deflection, which is valid for small deflections. The efficiency is

given as

( )1 2 3 4 5 6 7 8 9 10, , , , , , , , , .L

in

P fP

η = = Π Π Π Π Π Π Π Π Π Π (3.119)

108

Another quantity that is generally used to parameterize energy harvesters is the power

density, usually represented as amount of power generated per unit volume. In some literature,

power density is expressed with respect to mass and/or volume. The exact usage depends on

specific application that poses a restriction on volume or mass of the energy harvester. The

power densities with respect to volume and mass are expressed in Eqs. (3.120) - (3.123)

2

3

,v

L

ss s

s s sLD

s s

s

PEE L

E EPP volumevolume LL

ρρ

= = (3.120)

and in nondimensional form, is represented as

( ) 21 2 3 4 5 6 7 8 9 10 11, , , , , , , , , .vD

s s

s s

Pf

E EL ρ

= Π Π Π Π Π Π Π Π Π Π Π (3.121)

When expressed with respect to mass, the power density is

2

3

,m

L

ss s

s s sLD

s s s

s s

PEE L

E EPP massmass LL

ρρ ρ

ρ

= = (3.122)

And, in nondimensional form, is represented as

( ) 21 2 3 4 5 6 7 8 9 10 11, , , , , , , , , .mD

s s

s s s

Pf

E EL ρ ρ

= Π Π Π Π Π Π Π Π Π Π Π (3.123)

109

Table 3-5: Final set of nondimensional groups involving response parameters.

Dependent Π groups Functional dependence

ms s sC E L , 3m

s s

MLρ

, 1

n

s

ss

fE

L ρ⎛ ⎞⎜ ⎟⎝ ⎠

( )1 2 4 5 6 7 8 9 10, , , , , , , ,f Π Π Π Π Π Π Π Π Π

31

mdd

( )1 4 5 6 7, , , ,f Π Π Π Π Π

31 s sd E Lφ ( )1 2 4 5 6 7 8 9 10, , , , , , , ,f Π Π Π Π Π Π Π Π Π

231

ef

s s

Cd E L

, 231

eb

s s

Cd E L

, ( ) 231

se s

s

ER d Eω ρ ( )1 2 3 4 5 6 7 8 9 10, , , , , , , , ,f Π Π Π Π Π Π Π Π Π Π

2κ ,

231

1eb

s

s s

Z

E d Eρ⎛ ⎞

⎜ ⎟⎝ ⎠

,

231

1L

s

s s

R

E d Eρ⎛ ⎞

⎜ ⎟⎝ ⎠

( )1 2 3 4 5 6 7 8 9 10, , , , , , , , ,f Π Π Π Π Π Π Π Π Π Π

231

1

ems

s

s s

Z

E d Eρ⎛ ⎞

⎜ ⎟⎝ ⎠

( )1 2 4 5 6 7 8 9 10, , , , , , , ,f Π Π Π Π Π Π Π Π Π

31

L

ss s

s

IEL E dρ

⎛ ⎞⎜ ⎟⎝ ⎠

,

31

L

s

VLd

, s

s

UEρ

,

31

th

s

VLd

( )1 2 3 4 5 6 7 8 9 10 11, , , , , , , , ,f Π Π Π Π Π Π Π Π Π Π Π

2

L

ss s

s

PEE Lρ

, 2

in

ss s

s

PEE Lρ

( ) 21 2 3 4 5 6 7 8 9 10 11, , , , , , , , ,f Π Π Π Π Π Π Π Π Π Π Π

vD

s s

s s

P

E EL ρ

, mD

s s

s s s

P

E EL ρ ρ

( ) 21 2 3 4 5 6 7 8 9 10 11, , , , , , , , ,f Π Π Π Π Π Π Π Π Π Π Π

η ( )1 2 3 4 5 6 7 8 9 10, , , , , , , , ,f Π Π Π Π Π Π Π Π Π Π

110

All the above derived Π variables are summarized in Table 3-5 that list the response functions

in terms of the original independent Π groups.

Scaling Theory

Let us now use the above dimensional analysis to develop a scaling theory for reducing the

size of the structure from macro-scale to micro-scale. This analysis will enable us to investigate

the performance metrics of the energy harvester as a function of miniaturization. For this

analysis to hold true, it is assumed that the linear Euler-Bernoulli equation is valid for micro-

scale devices. This is an important assumption because the external source acceleration remains

constant for an application and the micro device is subjected to potentially large vibrations. This

results in a large deflection and the device can easily begin to operate in nonlinear region.

We define a variable scale factor called " "s that covers the meso-scale ( ), 1typically s =

range to the micro-scale ( ), 0.001typically s = . This would, for example, cover the typical

dimensions of the structure from a meter down to a millimeter. All the other length scale

dimensions, such as the width and thickness of the beam and the size of the piezoceramic, are

assumed to be similarly scaled in proportion to the scale factor ( )s . For the purpose of this

exercise, the material properties are assumed to be constant over the scale. It should be noted

here that the materials used for mesoscale piezoelectric devices are generally Aluminum,

Stainless steel, Brass etc while the MEMS devices are mostly made of Silicon. In addition, the

properties for bulk piezoelectric material are well known and are generally higher than thin film

PZT.

31, , , , , = constants s p pE E dρ ρ ε (3.124)

and

, , , , , , , .s s p pL b t L t l w h s∝ (3.125)

111

Using this assumption, the nondimensional groups (Table 3-4) 1 2 3, and Π Π Π are constant

as they represent the material properties. In addition, 4 5 6 7 8 9 10, , , , , and Π Π Π Π Π Π Π remain

constant as both the numerator and denominator scale proportionally. If the input acceleration is

assumed to be constant, which is true because the input base vibrations are independent of the

harvesting device scale, and the material is fixed, 11Π and 12Π are proportional to s . Based on

these trends, the dependence of all the nondimensional parameters described earlier can be

evaluated as a function of the scale factor. The effective short-circuit mechanical compliance

expressed in Eq. (3.68) remains constant across the scale as all the dependant Π groups are

constant.

1constant, which implies .ms s s msC E L Cs

= ∝ (3.126)

Similarly, the nondimensional effective mass varies as

( )333 constant, which implies .m

m ss s

M M L sLρ

= ∝ ∝ (3.127)

The mechanical damping in the system can be expressed as

( )

2 2

2constant, implies .m

m s

s s s

R R L sL E ρ

= ∝ ∝ (3.128)

Again, it is noted that the mechanical damping model is not perfectly valid, but is calculated

nevertheless to complete the overall scaling analysis. In reality, overall damping is dependent on

various mechanisms described in Eq. (3.71) to Eq. (3.80) that have to be studied for their scaling

behavior. The air flow loss quality factor is scaled as

( )2

25 constant

2 3a m n

s s

Q k P kE ρ

= Π = (3.129)

112

which will remain constant for no change in pressure and length to thickness ratio. This holds

true for a specific material operating at the same conditions such as temperature, molar mass, etc.

Next, the support dissipation is given as

3

5

10.23 constantcQ⎛ ⎞

= =⎜ ⎟Π⎝ ⎠ (3.130)

The support losses remain constant as long as the length to thickness ratio is maintained.

However, it is inversely proportional to the thickness and could become significant if the

thickness and length are of same order. The surface dissipation mechanism is shown in Eq.

(3.131) and is related to the thickness of the thin film layer on the substrate. If all the dimensions

are proportionally scaled, they bear no effect on surface dissipation.

( )

4 5

4 52 3s

ss

s

EQEL

δΠ Π

=ΔΠ +Π

(3.131)

Next, the volume loss is studied, which is only dependent on material properties, specifically its

elastic modulus and therefore is independent of scaling here.

,sv

s

EQE

=Δ

(3.132)

Finally, thermoelastic dissipation shown in Eq. (3.133) has a significant on the size of the device.

It is related to the material properties such as modulus and density. In addition, it depends on the

length and thickness of the device. But when all dimensions are scaled proportionally, it

increases with a decrease in thickness. Therefore, the thermoelastic quality factor increases at

reduced scale. This observation is particularly important here as the MEMS PZT energy

harvesters are shown in Chapter 6 to posses a much higher Q as opposed to the meso-scale

113

devices. This can be attributed to the fact that thermoelastic damping becomes significant at

micro-scale and dominates most other loss mechanisms. Hence, a higher Q was measured.

25

180 ,st

p s s s

EkQC E tρ

=Π Δ

(3.133)

Next, the natural frequency of the device is represented as

1 1constant, implies 1

nn

ss

ss

f fL sE

L ρ

= ∝ ∝⎛ ⎞⎜ ⎟⎝ ⎠

(3.134)

From Eq. (3.69), the angular frequency nω is proportional to s . The normalized

frequency response between the input acceleration and the resulting deflection, represented in

Eq. (3.82)is 11tip

s

ws

L∝Π ∝ . Therefore,

2tip sw L s s∝ ∝ (3.135)

Therefore, the tip deflection scales down as the square of the dimensions, thereby increasing the

operating range for the device in its linear region. The linear behavior limit is usually expressed

as a function of the ratio between the tip deflection and length for a cantilever beam (Section

3.3.1). Now, the nondimensional effective piezoelectric coefficient of the composite (shown in

Eq. (3.86)) is a function of 31d and scales as

31

constant, implies constant.mm

d dd

= = (3.136)

Similarly, the transduction factor that relates the electrical domain and the mechanical

domain is nondimensionalized in Eq. (3.88) and scales as

114

31

constant, implies .ss s

L sd E L

φ φ= ∝ ∝ (3.137)

The free and blocked electrical capacitances are similarly scaled with s as

2 231 31

; constant, implies ;ef ebef eb s

s s s s

C C C C L sd E L d E L

= ∝ ∝ (3.138)

According to this scaling theory here, the dielectric permittivity and piezoelectric coefficients are

assumed to remain constant. These properties are highly dependent on the fabrication process

and are therefore subject to large variations at the micro-scale. The nondimensional dielectric

loss in the piezoelectric layer, shown in Eq. (3.94) is

( ) 231 constant, implies constant.s

e s es

ER d E Rω ρ = = (3.139)

In addition, the electromechanical coupling coefficient 2κ also remains constant versus

scale. Furthermore, the power transfer parameters can also be reduced to expressions that show

their dependence on the scale factor. A similar analysis is carried out to reduce these parameters

to obtain a simple relation with the scale factor. The short-circuit mechanical impedance,

blocked electrical impedance and the Thévenin equivalent impedance remain constant with s as

2 2 231 31 31

, , constant.1 1 1

ems eb th

s s s

s s s s s s

Z Z Z

E d E E d E E d Eρ ρ ρ

=⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(3.140)

Similarly, the nondimensional Thévenin voltage and the output voltage across the load are

represented as

211

31 31

, , implies, , .th Lth L s

s s

V V s V V sL sL Ld d

∝Π ∝ ∝ ∝ (3.141)

115

The optimal load for maximum power transfer, which is the magnitude of the Thévenin

impedance also remains constant with s . Again, the optimal load is a direct measure of the

overall damping in the system and therefore follows the damping assumptions made in the

model. For a better estimate, each of the losses needs to be included into the model and their

scaling behavior studied. The current across the load resistor given by the ratio of output voltage

and load scales as

211

31

, implies, .LL s

ss s

s

I s I sL sEL E dρ

∝Π ∝ ∝ ∝⎛ ⎞⎜ ⎟⎝ ⎠

(3.142)

The output rms power across the load and input rms mechanical power scale with respect to s as

2 2 2 2 411

2 2

, , implies, , .inLL in s

s ss s s s

s s

PP s P P s L sE EE L E Lρ ρ

∝Π ∝ ∝ ∝ (3.143)

The overall electromechanical efficiency remains constant with the scale factor and is

independent if all the device dimensions are scaled proportionally. The power densities with

respect to volume and mass are expressed in Eqs. (3.120) - (3.123) scale as

2 211, , which implies ,v m

v m

D DD D

s s s s

s s s s s

P Ps P P s

E E E EL Lρ ρ ρ

∝Π ∝ ∝ (3.144)

To summarize, a comprehensive nondimensional tool was developed using the analytical

LEM that was investigated for its scaling behavior for miniaturization. This tool will enable

variation of various individual properties and dimensions to study the device response for

specific conditions and applications.

116

Validation of Scaling Theory

The developed scaling theory needs to be validated, and the predicted results and response

of various parameters need to be corroborated. To verify the scaling theory experimentally would

mean designing test specimens over an appreciable dimensional range followed by detailed

characterization to extract the lumped element parameters, which is difficult and time

consuming. In addition, obtaining custom made commercial PZT patches over the required

dimensional scale is fairly expensive. Instead, an FEM validation is easier to carry out by

changing the device dimensions and performing simulations.

A simple validation was carried out by simulating a sample PZT-Aluminum composite

beam using ABAQUS to calculate the static lumped element parameters such as natural

frequency, short-circuit mechanical compliance and mass and the effective piezoelectric

coefficient (Table 3-7). A PZT cantilever composite beam clamped at one end with a proof mass

at the other end was assumed with arbitrary dimensions. The material dimensions and properties

of the base test specimen simulated are listed in Table 3-6. The composite beam was modeled

using 3-D solid elements that consisted of 20-node quadratic hexahedran elements for shim and

proof mass and 20-node quadratic hexahedran piezoelectric elements for the PZT layer. The final

meshed structure is shown in Figure 3-3.

The computed values compared favorably with the analytical values. The sample was

then reduced proportionally in all it dimensions to micro-scale, and the dependence of these

parameters was observed and compared with the predicted trend.

117

Table 3-6: Material dimensions and properties of composite beam for FEM validation. Material Dimensions (s=1)

Length of beam 1 m

Width of beam 0.2 m

Thickness of beam 0.1 m

Length of PZT 0.5 m

Width of PZT 0.2 m

Thickness of PZT 0.05 m

Length of proof mass 0.2 m

Width of proof mass 0.05 m

Thickness of proof mass 0.05 m

Material Properties

Elastic modulus of shim 73.263 GPa

Density of shim 2718.33 kg/m3

Elastic modulus of PZT 66 GPa

Density of PZT 7800 kg/m3

Piezoelectric coefficient -190e-12 m/V

Dielectric constant 1800

Figure 3-3: Meshed PZT composite cantilever beam for FEM validation.

118

Simulations were performed for the short-circuit case to obtain the eigenvalue solution for

the beam. From the modeshape, the deflections along the center line were extracted and

integrated to estimate the kinetic and potential energies. Lumping these energies at the tip

yielded the short circuit mechanical compliance and mass. The eigenvalue analysis directly gives

the natural frequency of the beam. Another simulation was carried out to obtain the static

deflection due to an applied voltage to the PZT. The tip deflection corresponding to unit applied

voltage is the effective piezoelectric coefficient ( )md of the composite beam. Assuming that

1s = represents the base structure, the dimensions of the beam were proportionately scaled down

1000 times in regular intervals and simulations were performed to estimate the lumped element

parameters as described before.

Figure 3-4: Short circuit natural frequency for a PZT composite beam.

Figure 3-4 shows the short circuit mechanical natural frequencies obtained directly from

the eigenvalue solution using FEM. The scale varied from the base dimensions down to the

micro-scale, for example, the length of the beam varied from 1 m to 1 mm. As predicted from the

119

scaling theory, the natural frequencies increased with decreasing dimensions given by 1nf s∝ ,

verified by the results in Figure 3-4.

The difference between the natural frequencies from FEM and those estimated from theory

can be attributed to the fact that the model employs a static modeshape while the FEM is based

on the frequency dependent modeshape.

Since 3-D stress elements were used, the deflection across the width of the beam is not

constant and therefore, the deflections along the center line are extracted from the eigen

modeshape. The 2-D deflection shape thus obtained is integrated to obtain the stored potential

energy, which is lumped at the tip to estimate an effective compliance of the beam. This was

repeated for all the test cases and the results are presented in Figure 3-5. As indicated in the

figure, the short circuit compliance increases proportionally with decreasing s , given by

1msC

s∝ as predicted with the scaling theory.

Figure 3-5: Short circuit compliance for a PZT composite beam.

120

In Figure 3-5, the compliance predicted from FEM is lower than the model because the

FEM predicts higher natural frequencies, shown in Figure 3-4. Higher natural frequency implies

lower compliance. Next, the mechanical mass is calculated from the natural frequency and

compliance using

( )2

12

mn ms

Mf Cπ

= (3.145)

The results obtained for all the test cases are plotted in Figure 3-6. As indicated in the

plots, the mass scales down as 3mM s∝ , since all the three dimensions vary proportionally

decreasing the overall volume cubically. However, the masses from both methods are nearly

equal, primarily because the differences in natural frequency and compliance compensate

inversely to minimize the difference in mass.

Figure 3-6: Effective mechanical mass for a PZT composite beam.

Finally, from the static voltage simulation where the deflection was calculated for a unit

applied voltage to the PZT, the tip deflection was directly extracted and compared with the

model. The results are shown in Table 3-7. The effective piezoelectric coefficient remained

121

constant over the scaling range validating the trend predicted from the model (Figure 3-7). In this

case, both models being static, the values match very well within 2% .

Figure 3-7: Effective piezoelectric coefficient for a PZT composite beam.

The results represented by the above plots are summarized in Table 3-7. The columns indicate

the LEM parameters for various scales represented with s . The static LEM parameters thus

obtained matched with the predicted trend and therefore serve as a good validation for our design

methodology.

Table 3-7: Static lumped element parameters from FEM and LEM to validate the scaling analysis.

Fn (Hz) Cms (m/N) Mm (kg) dm (m/V)

s FEM Theory FEM Theory FEM Theory FEM Theory

1 72 67 2.17e-7 2.42e-7 22.20 23.07 -1.60e-8 -1.57e-8

0.5 145 126 4.30e-7 5.73e-7 2.80 2.76 -1.60e-8 -1.57e-8

0.1 724 616 2.17e-6 3.09e-6 0.022 0.022 -1.60e-8 -1.57e-8

0.05 1452 1231 4.29e-6 6.20e-6 2.80e-3 2.70e-3 -1.60e-8 -1.57e-8

0.01 7260 6154 2.13e-5 3.10e-5 2.26e-5 2.16e-5 -1.60e-8 -1.57e-8

0.005 14521 12310 4.29e-5 6.20e-5 2.80e-6 2.69e-6 -1.60e-8 -1.57e-8

0.001 72604 61540 2.14e-4 3.10e-4 2.24e-8 2.20e-8 -1.60e-8 -1.57e-8

122

Now that we have verified the scaling analysis using ABAQUS, we can design the test

specimens and make reasonably accurate predictions for its response characteristics, subject to

the validity of the model assumptions and boundary conditions.

Extension to MEMS

After completing the scaling analysis for the lumped element and power transfer

parameters, we now explore this concept on a micro-scale. Some of the features observed in

MEMS structures that were suggested by the scaling theory described earlier are as follows.

First, some parameters that control the power harvesting aspects, such as device efficiency and

coupling factor, remained constant even though the device was reduced in size. This holds true

in the ideal case and is not valid if the piezoelectric properties vary at micro-scale. However,

parameters such as overall power delivered to a load and even power density decrease due to

scaling and miniaturization. On the other hand, an advantage of microfabrication is the feasibility

to batch-fabricate many devices simultaneously after standardizing the process. Furthermore,

fabricating arrays of such devices enables parallel or series connection of the electrodes to

enhance the output voltage or current to meet the micro power processor requirements.

Design of Test Structures

In order to design the test specimens at the micro-scale, the target natural frequencies

used in this design analysis are 100 Hz , 1 kHz and 10 kHz . These frequencies were selected to

broadly cover the widely occurring vibration frequencies (Chapter 1). The dimensions of the

composite beam are designed to match this frequency. The properties of the materials are fixed.

The composite beam is comprised of a PZT thin film deposited using sol gel process on bulk

silicon shim. The material properties used in this analysis were obtained from existing literature

(Horowitz 2005) and ARL and are listed in Table 3-8.

123

Table 3-8: Properties and dimensions used for designing MEMS PZT devices. Elastic modulus of Silicon 169 GPa Density of Silicon 2330 kg/m3 Elastic modulus of PZT 60 GPa Density of PZT 7500 kg/m3 Piezoelectric coefficient -45 X 10-12 m/N Relative permittivity 900 Damping ratio 0.01 tanδ 0.02

The objective of this design formulation is to build test structures to match the target

natural frequencies. In addition, they are designed for specific acceleration

levels ( )0.1 , 1 , 10g g g that are indicative of ambient acceleration levels in many applications.

The acceleration levels act as the criterion to enforce the linear elastic limit. The LEM is valid

for small deflections and will fail to predict the behavior if the resonant deflection for a specific

acceleration is beyond the elastic limit. Furthermore, after estimating the amount of power

available for reclamation in these beams, the dimensions of each are optimized to obtain

maximum power. The schematic (elevation and plan) of a typical cantilever composite beam

that is designed is shown in Figure 3-8.

Figure 3-8: Schematic of a single PZT composite beam.

124

The design constraints that are considered for the design formulation are listed below. • Our goal is to fit as many such cantilever structures as possible in

( )1 1 500 cm cm mμ× × , which is the overall dye size. Therefore, we divide the

dye further into smaller cells, typically of the size, ( )3 3 mm mm× or

( )5 5 mm mm× squares according to the size of the composite beam that consist of

multiple cantilevers connected to a single proof mass.

• Due to the constraint on the dye size, the height of the proof mass is fixed as

500 mμ which is the thickness of the wafer. For simplicity, the width of the

piezoelectric layer and the shim are considered equal in the design formulation. In

addition, the thickness of the piezoelectric layer is fixed at 0.5 mμ due to fabrication

constraints.

• The resulting natural frequency of the structure should match the external vibration

frequency for maximum power generation.

• Clearance needs to be provided to the structure on the bottom and side wall for it to

vibrate. For simplicity, we assume that the clearance at the bottom needs to be less

than the tip deflection at resonance. In addition, the mechanical damping

predominantly controls the amplitude of the tip deflection at resonance. We have

assumed a damping ratio of 0.01 in our design analysis which held true for most

meso-scale cantilever structures verified experimentally. However, as discussed

earlier in this chapter, the damping phenomena are different at microscale and could

vary, depending on how the loss mechanisms scale with device size.

• The main constraint of the analysis is the linear elastic limit. In order to stay below

2 % of the linear elastic limit, the tip deflection of a cantilever beam must be less

125

than 0.4 times the actual length (Appendix D). Therefore to incorporate this in our

design, the dynamic response from the lumped element model is used to predict the

tip deflection at resonance. This tip deflection, which reaches a maximum at

resonance, is ensured to stay within the linearly elastic range.

• In addition, the stress fracture limit for Silicon was considered as a constraint. The

yield strength for Silicon is 7 GPa (Petersen 1982) and the stress limit for the

devices was designed to be less than 10 % of the yield stress. Since no yield strength

data was readily available for thin film PZT, only Silicon stress state was analyzed to

meet the constraint.

• Finally, the design has to meet the energy reclamation circuit requirements. The

minimum overall open circuit voltage should be 2 V and the output current should

be at least 20 Aμ . The blocked electrical capacitance should be less than

4eb

m

CRπω

≤ (Shengwen 2005).

A simple optimization technique was adopted to design the test structures. However, for

the second generation devices designed in the future, a detailed and thorough optimization needs

to be carried out. The technique adopted here is rudimentary and is based on a parametric

analysis, by varying the dimensions and observing the overall response subject to constraints.

The process that was adopted is listed in the following steps.

• The overall available area is 1 1 cm cm× . As described earlier, this total area is divided

into various cells. The first step involves deciding the area of each cell. From the scaling

analysis, we observed that as the dimensions become smaller, the natural frequency

increases. So for the lowest natural frequency, the whole dye is used as a single cell. For

126

1 kHz cantilevers, the cell size was typically assumed to be 3 5 mm mm× that leads to

6 cells or3 3 mm mm× , which leads to 9 such cells in the overall dye.

• The number of structures in each cell has to be decided. For convenience, it was assumed

that there were 3 cantilever beams attached to a single proof mass at their tips.

Consequently, this number ( )n varied anywhere from 1-4 for the designs to generate

enough power from the device.

• We can choose the width of the proof mass, ( )w after providing enough clearance at the

edges. Similarly, we can also choose the overall length of the

structure ( ) sL l clamp+ + . The size of the clamp in all cases was assumed to be

anywhere between 0.5 mm and 1 mm , depending on the overall length of the composite

beam.

• Once the number of beams ( )n and w have been established, the width of the beam can

be calculated ( ) s pb and b .

• Using the dimensions of accelerometers listed in DeVoe and Pisano (2001) and Vizvary

(2001), some typical ratios for s sL b (5, 10) and s sL t (50, 100, 200) were obtained to

act as base dimensions. For some cases, the dimensions of existing meso-scale aluminum

beams (Table 2-6, Table 2-7) were reduced to micro scale to estimate these typical ratios.

Using these typical ratios and width of the beam, basic length and thickness was

calculated.

• To obtain the required natural frequency, the lengths and thicknesses of the beam and

proof mass were varied by trial and error. Some knowledge about how these dimensions

affect the mass and compliance of the beam (from the scaling analysis) was useful in

127

arriving at the final dimensions. For example, if we wish to decrease the natural

frequency, we need to either increase the compliance or the mass. We can increase the

length of the beam to increase its compliance, which would restrict the length of the proof

mass as the overall length is fixed. In addition, increasing the thickness of the proof mass

will also increase the mass which will in turn increase the natural frequency.

Figure 3-9 shows the top view of a typical cell that has a 3-element array of such

composite beam structures. In the figure, n is the number of individual cantilevers present in

one cell structure, which is generally 3 for most of the designs.

Figure 3-9: Layout for a single cell comprising of an array of composite beams. Test devices

Using the above design methodology, various test structures were designed. The

following table lists all the properties of the test specimens designed to achieve the target natural

frequencies.

128

Table 3-9: Material properties of piezoelectric composite beam. Elastic modulus of Silicon ( )sE 169 GPa

Density of Silicon ( )sρ 2330 3kg m

Elastic modulus of PZT ( )pE 60 GPa

Density of PZT ( )pρ 7500 3kg m

Piezoelectric coefficient ( )31d -100 X 10-12 m V

Relative permittivity ( )rε 1000

Table 3-10 shows the dimensions of the test structures designed to achieve the target natural

frequencies for a specific input accelerations. The predicted natural frequency is obtained from

the calculated compliance and mass using the dimensions and properties selected. It was

difficult to design an optimal structure for 0.1 g , due to the restrictions in area and natural

frequency. However, the same structures that were designed for 1 g and 10 g can be used for

0.1 g , but would generate less power compared to the other designs. In the table, the first row

lists the targeted frequency and the corresponding input acceleration. Since, the design and

optimization procedure adopted here is slightly crude, the resulting designs possess different

natural frequency. While performing the experiments, the input vibrations applied to the

structure will be altered to match the frequencies. All the other values listed in the table are self-

explanatory. The last row indicates the number of cells ( )n in 1 1 cm cm× and number of

cantilevers per cell ( )m . The nomenclature used to denote the 9 different designs are indicated in

the first row of Table 3-10.

129

Table 3-10: Designed MEMS PZT structures. PZT-EH- 1 2 3 4 5 6 7 8 9

Ls (mm) 1 4 6 6 6 0.6 1 7 0.5

bs (mm) 0.2 1 2 2 1 0.3 1 1 0.5

ts (um) 12 12 12 12 12 12 12 12 12

Lp (mm) 1 3 6 6 4 0.4 1 5 0.5

bp (mm) 0.2 1 2 2 1 0.3 1 1 0.5

tp (um) 1 1 1 1 1 1 1 1 1

l (mm) 1 4 3 2 0.5 0.2 2.5 0.5 1.8

w (mm) 0.8 3 3 3 3 0.8 4 2 0.8

h (um) 500 500 500 500 500 500 500 500 500

# of cantilevers/chip

3 3 3 3 3 3 1 4 3

Fn (Hz) 354 25 30 40 68 3055 127 66 501

normalized P_load (nW/g) per cantilever

3 540 480 250 49 5.75E-05 88 35 3

130

CHAPTER 4 DEVICE FABRICATION AND PACKAGING

This chapter begins with a detailed description of the fabrication process of micro

piezoelectric structures and the associated microfabrication challenges. A final process flow

consisting of 6 Photosciences 5090CRSL masks and the recipes for device release are presented.

The device is based on a Silicon-on-Insulator (SOI) wafer consisting of Si/SiO2/Ti/Pt/PZT/Pt as

the layered structure. The fabrication involved uses standard sol gel PZT and conventional

surface micromachining techniques. The thin films were deposited using sputtering or

evaporation techniques, and sol gel PZT is deposited using a 3-step process involving spin, bake

and anneal steps. The final step involved patterning the beams from the top and proof mass from

the backside and deep reactive ion etching to release the devices. Furthermore, a discussion of

packaging used in the experimental characterization of the test devices is provided followed by

the mounting mechanism employed for the devices in the package.

Process Flow

The proposed test structures designed using the electromechanical lumped element model

were fabricated using conventional surface and bulk micro processing techniques (Madou 1998).

The first three major steps in the fabrication, which involved the PZT layer deposition and

patterning, were carried out at the Army Research laboratory (ARL) through MEMS Exchange.

ARL has a standard PZT process (Piekarski et al. 1999) available, from which the process flow

was devised and the corresponding mask-sets were generated. Their process involves three

masks for depositing and patterning the PZT layer along with its electrodes. The fabrication was

carried out on a 4′′ Silicon On Insulator (SOI) wafer that had a 0.4 mμ buried oxide layer

131

separating the overlayer from the bulk silicon substrate. The active overlayer thickness was

12 mμ of Silicon.

First, the base SOI wafer was RCA cleaned to remove surface particles and impurities.

Silicon dioxide ( )20.1 , m SiOμ was deposited on the wafer using the Plasma Enhanced

Chemical Vapor Deposition (PECVD) technique as shown in Figure 4-1. The 2SiO layer is

necessary to prevent possible lead diffusion into the silicon overlayer during the PZT anneal

step. Any lead diffusion will immensely affect the performance of the PZT and could result in

the loss of its piezoelectric property.

SiO2 (100 nm)Si (12 um) SiO2(BOx)

Figure 4-1: Deposit 100 nm blanket SiO2 (PECVD) on SOI wafer.

Next, Titanium ( ), 20 Ti nm and Platinum ( ), 200 Pt nm are sputter deposited on the

whole wafer to form the bottom electrode, as shown in Figure 4-2. A Titanium seed layer is

provided to ensure good adhesion between Pt and the substrate.


Ti/Pt (20 nm/200 nm)

Figure 4-2: Sputter deposit Ti/Pt (20 nm/200 nm) as bottom electrode.

Next, sol gel PZT ( )Lead, / Zirconium, / Titanium,Pb Zr Ti of composition ( )125 / 52 / 48

is deposited using a 3-step process involving spin, bake and anneal steps as shown in Figure 4-3.

The sol gel is initially spun at a certain rotational speed (typically at 3000 rpm for 30 s ) in a

spinner, after which the wafer is pyrolized on a hot plate at 350 C for 50 60 s− . For higher

132

thicknesses of PZT, multiple spin and bake steps are carried out before the final anneal step is

done at a much higher temperature (typically o700 C for 1 min in the presence of 2O ) in a rapid

thermal anneal (RTA) furnace. These multiple spin and bake steps are adopted to maintain

consistent PZT properties even at higher thickness.

PZT(1 um)SiO2 (100 nm)Si (12 um) SiO2(BOx)


Figure 4-3: Spin coat sol-gel PZT (125/52/48) over the wafer using a spin-bake-anneal process.

Next, the top electrode for the PZT ( ), 200 Pt nm is patterned using a lift-off photolithography

technique with the Top Electrode mask as shown in Figure 4-4.

Pt (200 nm)PZT(1 um)



Top_electrode mask

Figure 4-4: Deposit and pattern Pt for top electrode using liftoff.

In the next step, openings are patterned on the device areas to provide access to the

bottom electrode. The exposed PZT in those areas is wet etched revealing the bottom Pt

underneath as shown in Figure 4-5. A PZT Etch mask was used to pattern the access holes on

the electrode. The PZT in open areas is etched using a combination of hydrofluoric acid/

hydrochloric acid/de-ionized water that had an etch rate of 23 nm s . The residues left behind in

the previous etch were removed using dilute nitric acid and hydrogen peroxide.

133




PZT_mask

Figure 4-5: Pattern opening for access to bottom electrode and wet etch PZT using PZT Etch mask.

The final step in the PZT process at ARL involved patterning the electrode area using an

Ion Milling mask. The exposed PZT and bottom /Pt Ti are physically etched using Ion milling

process resulting in the PZT features along with top and bottom electrodes with provisions for

bond pads. The profile for this step is shown in Figure 4-6




Ion_Milling mask

Figure 4-6: Ion milling of PZT and bottom electrode using Ion Milling mask as pattern.

After the PZT process was completed, the wafers were returned for device release steps,

which were carried out at UF. The residual stresses in each layer patterned as provided by ARL

are listed in Table 4-1.

Table 4-1: Residual stress measurements for the PZT pattern process (source : ARL).

Film As Deposited Stress

PECVD Oxide (0.1 μm film) -45.7 MPa Annealed

Bottom Ti/Pt (20/200 nm film) 558.1 MPa Annealed

134

PZT (1000 nm film) 109.7 MPa

Oxide/Ti/Pt/PZT Stack 175 MPa*

Top Pt (200 nm film) -15 – 0 MPa*

* as provided for a general process. The rest were measured specifically for this process. Before performing the release, Gold ( )Au was deposited on the wafer using e-beam

deposition technique. Alternatively, gold can also be deposited using the sputtering technique,

but an e-beam was chosen for our process due to its availability. Then, patterns were transferred

to the wafer to protect the bond pads as shown in Figure 4-7. The remaining exposed Au areas

were removed using Transene Gold ( Au ) Etchant (Potassium Iodide solution) protecting the

bond pads that were patterned using the Bond Pads mask.




Bond_Pads mask

Figure 4-7: Deposit Au (300 nm) and pattern bond pads using Bond Pads mask and wet etching.

In the next step, the beams are released from the top. A positive photoresist (PR), AZ 1512

was first spun at 4000 rpm for 40 s to approximately deposit 1.5 mμ of PR. This was followed

by soft baking the wafer for 60 s on a hotplate at o95 C . The Beam Etch mask is used to

pattern the release trenches for the beams with Karl Suss MA-6 Mask Aligner, following which,

the wafer was hard baked for 60 min at o90 C . Subsequently, the pad oxide deposited in the

first step is wet etched using BOE ( )2: 6 :1HF H O = . Removal of this oxide is critical as any

residual oxide in the exposed open areas will prevent silicon etching during topside release.

Consequently, 12 mμ of overlayer is removed using Deep Reactive Ion Etching (DRIE) with the

135

STS-MESC Multiplex ICP etcher. DRIE is a highly anisotropic etching process that involves

number of etch passivation cycles to achieve the vertically directional profile (also called

Bosch process). During the etch cycle, highly reactive 6SF gas is used along with 2O to perform

a nearly isotropic etch of Si . In the deposition cycle, 4 8C F is used as a passivation layer to

protect the etched area from further etching. Then, the process switches to an etch cycle where

the energetic plasma ions are collimated and bombarded to remove the passivation layer from the

bottom of the previously etched trench. This is repeated for a prescribed number of cycles to

achieve a specific etch depth. The process parameters can be chosen such that aspect ratios of

greater than 20 :1 can be obtained. Even side walls with fairly negative or positive slope angles

can be achieved depending on the application. Initially, a standard DRIE recipe recommended by

STS was used to observe the etch profiles on 4′′ test wafers. The process parameters were varied

to eventually obtain a standard recipe for our layout. Some issues encountered during this

iterative process are shown here. In Figure 4-8, an SEM picture of a sidewall profile for PZT-

EH-06 is shown that was taken at an angle of o40 C . It also shows the corresponding profile for

PZT-EH-09 on the right. The scalloped lines on the sidewall can be seen, which indicate the 15

etch passivation cycles.

136

PZT-EH-09PZT-EH-06 @ 40 deg

Figure 4-8: Sidewall profiles on topside of a 4" Si test wafer.

In our process, the etching from the top stops at the buried oxide layer (BOX) as shown in

Figure 4-9. The etch selectivity in DRIE with Si and 2SiO is approximately 1:100 and

therefore, the BOX layer acts as an etch stop. The process conditions used for top side release are

listed in Table 4-2.

Table 4-2: DRIE recipe conditions for top side etch. Gas Flow Rates 4 8 6 285 , 130 , 13 C F sccm SF sccm O sccm= = =

Platen power 12 W

Pressure ( ) 83.6%manual control

Etch : Passivation 12 : 7

Number of cycles 13

DRIE was carried out for 12 cycles before the buried oxide layer was revealed. An extra

etch passivation cycle was performed to over etch and remove any residues.

137

Photoresist(5 um)Pt (200 nm)PZT(1 um)



Beam_Etch mask

Figure 4-9: Wet etch exposed oxide with BOE and DRIE to BOX from top.

After the beam release step, the remaining PR was removed using Acetone and

subsequently rinsed in methanol and de-ionized (DI) water. First, AZ 1529 was spun on the

patterned wafer 4000 rpm for 40 s to approximately deposit 2.9 mμ∼ which protects the

topside. The wafer was soft baked on a hotplate for 60 s at o95 C . Next, AZ 9260 positive

photoresist was spun on the backside of patterned wafer at 2000 rpm for 40 s to approximately

deposit 9.8 mμ∼ of PR. After soft baking for 30 min at o90 C , the proof masses were

patterned on back side using EVG-620 mask aligner with the Proof Mass mask, followed by a

hard bake for 60 min at o90 C . The EVG-620 mask aligner enables back to front alignment to

ensure that the beams and the proof masses were perfectly aligned with each other. The patterned

wafer was then etched from the backside using DRIE technique.

First, as carried out previously, this process was performed on 4 SOI′′ test wafers to

optimize the process and recipe conditions. Some of the phenomenon observed during many

iterations carried out to optimize the process are explained below with pictures.

For the backside release, the patterned wafer is typically mounted on a carrier wafer,

( Si or pyrex ) and loaded in the DRIE machine. To prevent any thermal loading on the wafer

that could potentially produce uneven etching, AIT cool grease 7016 was used to attach the

138

patterned wafer to the carrier wafer. The carrier wafer is required for long and/or through etches

to provide mechanical integrity during clamping of process wafer.

Figure 4-10: Sidewall profiles for backside etching using DRIE.

In Figure 4-10, on the left, shown etched area corresponds to PZT-EH-06 on a test wafer.

Two regions are zoomed in and shown on the right to indicate the sidewall profiles for the proof

mass. As is evident from the figure, vertical striations are observed. In this test, the process wafer

was attached to a carrier wafer using AiT Cool-Grease 7016 that was applied only along the

edges of the wafer preventing any cool grease from contaminating the features on the topside.

139

Figure 4-11: Curved edges during backside DRIE.

Figure 4-11 shows another etch area corresponding to a larger device that clearly indicates

curved edges near the corners of the proof mass. Initial efforts with this test run, shown in Figure

4-10 and Figure 4-11 produced highly non-uniform etching across the wafer with the features in

the center of the wafer etching much slower than the areas further away. The process conditions

for this run remained the same as listed in Table 4-2. It was hypothesized at this point that

thermal effects on the wafer were causing the highly non-uniform etching. In the next test run, a

blank 4 bulk Si′′ wafer was patterned with the Proof Mass mask. Since the Si wafer was not

patterned on the topside, cool grease was applied evenly over the entire wafer. This time,

however, the cool grease was spread evenly at an elevated temperature ( )o50 C∼ to ensure

uniform contact across the wafer and provide good thermal dissipation. The process conditions

140

remained the same as Table 4-2 except that the etch passivation ratio was changed to 13 : 7

with an overrun of 1: 0.5 . The overrun allows compensation for the ramp in flow rate from zero

to specified rate for the gases during etch and passivation cycles.

Figure 4-12: Onset of silicon grass during a backside etch run.

SEM pictures were obtained again by cleaving the process wafer at different locations to

observe the etch results. Figure 4-12 shows an etch area that indicated the onset of Si grass. A

small area from the etch bottom is magnified on the right that clearly shows 10 mμ∼ of grass. A

further magnified image is provided to show the scallops in the grass from etch passivation

cycles. In addition, it also indicates increasing width of grass with pointed tops.

141

Figure 4-13: Sidewall profiles for a backside etch on a test wafer.

In Figure 4-13, the sidewall profiles of two etch areas are provided that show a significant

negative profile with curved corners and concave bottom. The non-uniform etching across the

wafer persisted even in this test run, with measured etch lags as large as ( )60 80 mμ−∼ for 350

etch passivation cycles. In the layout, the etch areas in the center are several orders higher than

those further away. The largest etch feature was 270 mm∼ and the smallest was 20.5 mm∼ that

corresponded to dimensions of 10 mm and 200 mμ respectively. In addition, the exposed Si

area was calculated to be approximately 38.8 % of the overall wafer area.

Based on our layout design and discussions with STS, it was understood that to evenly etch

through 500 mμ of the wafer with the current pattern would be difficult. A more detailed

process optimization needed to be performed to carefully investigate the effects of etching by

varying each of the recipe parameters individually. In view of the time constraint for fabrication,

it was decided to dice the patterned wafer into individual dyes and etch them separately to

release the devices. The dicing process involves attaching the process wafer to a carrier wafer

using hot melt glue and sawing. The diced wafer pieces can be removed from the carrier wafer

by heating and melting the glue that binds the two. An added advantage with prior dicing is that

if the devices are diced after release, any damage to the suspended cantilevers during the

mechanical sawing process will lower the overall yield of the structures.

142

Another test run was carried out on a 4 bulk Si′′ wafer that was patterned using the

Proof Mass mask. The wafer was later diced into small dyes consisting of a few devices of each

design. These dyes were mounted on a carrier wafer using cool grease and separately etched

using DRIE. The non-uniformity in etching was minimal with etch lags of less than 10 cycles.

For this process, the recipe parameters remained the same as Table 4-2, except that the

etch passivation ratio was 13 : 7 without any overrun. The final process conditions are listed in

Table 4-3.

Table 4-3: DRIE recipe conditions for back side etch. Gas Flow 4 8 6 285 , 130 , 13 C F sccm SF sccm O sccm= = =

Platen 12 W

Pressure ( ) 83.6%manual control

Etch : 12 : 7

Number of 350∼

After ensuring that the above methodology of release worked for all device sizes, the same

process was employed on the original PZT process wafer that had been released from the top.

The wafer was first patterned on the backside using the photolithography process described

previously with Proof Mass mask. It was diced into individual dyes that consisted of 1 or 2

devices using the dicing saw. The dyes were mounted on a pyrex carrier wafer using cool grease

to ensure good thermal dissipation. DRIE was then performed until the BOX layer was revealed

to release the structures from the backside as shown in Figure 4-14. The number of cycles to

release varied from 350 to 400 depending on the designs.

143

Photoresist(10 um)

Photoresist(5 um)Pt (200 nm)PZT(1 um)



Proof_Mass mask

Figure 4-14: Pattern proof mass on the backside and DRIE to BOX.

DRIE was followed by immersing carrier wafer and the chip in Acetone for a few hours to

separate the chip and strip the photoresist. The chip was then attached to a 2 Si′′ wafer using AZ

9260 photoresist to protect the PZT on the topside. The wafer was then etched in 6 :1 BOE to

remove the 0.4 mμ BOX layer. The PR binding the chip to the carrier wafer was stripped using

acetone, followed by chip rinse in DI water and dry to obtain the final PZT devices. A schematic

of the devices when released is shown in Figure 4-15. In the figure, 3 cantilevers are shown

attached to a single proof mass. The number of cantilevers per proof mass typically varies

between 1 4− depending on the designs listed in Table 3-10. It should be noted that extra care

was taken to ensure that BOE never came in contact with the PZT, as initially, some released

devices appear shorted and HF is known to etch PZT.

144




Figure 4-15: Schematic of final released device.

Shown below in Figure 4-16 and Figure 4-17 are SEM pictures of two designs that were

released using DRIE. Several devices of these designs were released, packaged and characterized

for mechanical, electrical and electromechanical response. The experimental setup and results

are presented in later chapters (Chapter 5 and Chapter 6). Figure 4-16 shows two views of PZT-

EH-07 that consists of a PZT composite beam, 1 1 mm mm× attached to a single proof mass,

2.5 4 mm mm× . Figure 4-17 shows PZT-EH-09 comprising of three beams each

0.5 0.5 mm mm× attached to one proof mass, 1.8 2.4 mm mm× .

145

Figure 4-16: SEM pictures of a PZT-EH-07 released device.

Figure 4-17: SEM pictures of a PZT-EH-09 released device.

Although the devices released completely, the recipe used for DRIE is still not optimized

and issues such as negative profile, slightly concave bottom and excessive undercut were not

addressed in detail. From discussions with STS, it was found that a negative profile cannot be

avoided with large etch areas due to reentry of gases during the etch step. In addition, it was

suggested that increasing the platen power by 2 3 W− will ensure a more directional and flatter

etch bottoms. Furthermore, reducing the pressure during etch step to 40 30 mT− and 22 mT

during the passivation step may help the etch profiles. However, since the modified DRIE recipe

seemed to release the devices albeit imperfect sidewalls (see Figure 4-18) current process was

continued for release of subsequent devices. The suggestions for increased power and lowered

pressure could be explored in future to optimize the process to obtain straighter walls.

146

Furthermore, the layout and device locations on the wafer need to be rearranged to obtain a more

symmetric pattern to ensure uniform etching.

Figure 4-18 shows SEM pictures of sidewall for PZT-EH-07. As is evident in the figure,

the sidewalls exhibit vertical striations that have been magnified on the right. The scallops

indicate etch passivation cycles that are approximately 1.1 1.2 mμ−∼ each. In addition, a

significant undercut is observed as shown in the figure below. Furthermore, insufficient

passivation occurs in some areas of the sidewall that is shown on the right bottom of Figure 4-18.

Figure 4-18: Sidewall profiles of released devices.

147

Process Traveler

A complete process flow that was finally used to fabricate the MEMS PZT devices is

summarized in Table 4-4.

Table 4-4: Process traveler for the fabrication of micro PZT cantilever arrays

1 Start with SOI wafer with 4000 A° oxide (12 mμ of silicon overlayer)

2 Deposit 100 nm of SiO2 using PECVD

3 Spin deposit bottom electrode. Deposit 20 200 nm nm of Ti/Pt using sputtering technique.

4 Spin coat PZT (125/52/48) solution at 3000 RPM for 30 s . Pyrolize at 350 C° for 1 min. Repeat spin/pyrolize 3 times to achieve 1 mμ thick PZT. Furnace anneal at 700 C° for 1 min (O2).

5 Spin photoresist on front surface and pattern top electrode using the same thickness for the top electrode as the bottom electrode. Deposit 200 nm Pt using sputtering technique. Strip resist to liftoff Pt using Acetone

6 Etch PZT in 3:1:1 ammonium biflouride/hydrochloric acid/DI water using etch mask. Etch PZT residues left behind by previous etch with dilute nitric acid/hydrogen peroxide etchant

7 Spin photoresist on topside and pattern to etch the bottom electrode using Ion milling.

8 Deposit Au (300 nm) and pattern bond pads using Bond Pads mask and wet etching.

9 Spin photoresist on topside and pattern to etch the beams. Plasma etch (DRIE) to BOX from the top to release the beams

10 Deposit thick photoresist on the bottom along with protecting frontside. Pattern bottom for etching the proof mass. Plasma etch (DRIE) from backside that stops on BOX layer

11 Etch oxide (BOE) on backside to remove oxide. Finally, strip photoresist on top and bottom using Acetone to release the device

148

Packaging

The individual chips after release, were then packaged for experimental characterization.

Two packages were designed for this purpose, namely, a vacuum package and an open package.

The chips were mounted in a Lucite package, described next, and the bond pads were connected

to copper leads that allow for external electrical access during experimental measurements.

Vacuum Package

A two-part vacuum package was devised to minimize any fluid loading on the device that

will result in a higher mechanical damping at resonance. Figure 4-19 shows a schematic of the

bottom piece of the vacuum package that basically consisted of a 2 2 1′′ ′′ ′′× × Lucite base and a

1 1 0.5′′ ′′ ′′× × center cavity with 5 mm wide support to mount the chips. The base has a threaded

hole at the bottom to facilitate rigid mounting to a vibrating surface (for eg., Bruel and Kjaer

mini shaker 4810) and a 1 4′′ vacuum port. The vacuum package is significantly larger than the

device itself, primarily to provide considerable contact between the Lucite base and the top cover

during pump down. A rubber gasket is also provided between the base and cover to hold the

vacuum for a long time. Copper posts go through the Lucite base and extend on the other side to

serve as external leads.

149

1 in

1 in

5 mm

2 in

2 in

2 in

2 inA A

B B

Bottom ViewTop View

Sectional view at BB Sectional view at AA

1 mm

10-32 threaded hole(0.25 in)

Pressure tap for vacuum

1 in

0.5 in

LUCITE BOTTOM

Figure 4-19: Schematic of the bottom of vacuum package for MEMS PZT devices.

Figure 4-20 shows the top part of the package comprising of a glass/plexiglass cover to

essentially act as a transparent surface for laser based measurements during characterization.

2 in

0.5 in

GLASS TOPTop View

Bottom View

Figure 4-20: Schematic of glass top for vacuum package.

A 3-D schematic of the completely assembled vacuum package is shown in Figure 4-21.

The devices are mounted on the center support in the cavity using a two-part silver epoxy

150

(Epotek H2OE). The silver epoxy was cured in an oven at o90 C for 90 min to provide a rigid

support to the device. Although curing time can be reduced at higher temperatures, the above

temperature was chosen to stay below the glass transition temperature for Lucite ( )o100 C . The

electrode bond pads of the chips were connected to the copper posts with gold wires using a

wire-bonder. In areas where wire-bonding was difficult, silver epoxy was used to create the

contacts. The package as shown in the Figure 4-21 has a threaded hole in the bottom to be

mounted on a vibrating shaker for characterization.

Glass

Lucite bottom

External leads (Cu)

Device

Wire bonds (Au)

Figure 4-21: An isometric view of the overall vacuum package.

Open Package

The open package shown in Figure 4-22 is similar to the vacuum package without the

transparent top part and the vacuum port. Since this package will not be pumped down, a smaller

package was devised comprising of a 1 1 1′′ ′′ ′′× × Lucite base and a 0.5 0.5 0.5′′ ′′ ′′× × center cavity

151

with 3 mm wide support to mount the chips. The package also has a threaded hole in the bottom

to be mounted on a vibrating shaker for characterization.

0.5 in

0.5 in

3 mm

1 in

1 in

1 in

1 inA A

B B

Bottom ViewTop View

Sectional view at BB Sectional view at AA

1 mm

10-32 threaded hole(0.25 in)

1 in

0.5 in

LUCITE BOTTOM

Figure 4-22: Schematic of open package for MEMS PZT devices.

An optical photograph of the open package is provided in Figure 4-23. In the figure, 2

devices of PZT-EH-09 are mounted on the package which is attached to vibrating surface (a

shaker) on the bottom. Gold wires are used to provide contact with the bond pads on the

individual devices. The wires were connected to the copper posts with Silver epoxy using the

curing process described earlier.

152

Cu postsMEMS PZT device

B & K shaker External copper leads

Lucite base

Figure 4-23: Picture of the open package.

Following fabrication and packaging of the MEMS PZT cantilever devices, experimental

setups were designed to completely characterize them for lumped element parameter extraction.

Furthermore, the devices were characterized for their frequency response and power generation.

The next chapter describes these experimental setups and the data acquisition conditions

employed for the experiments.

153

CHAPTER 5 EXPERIMENTAL SETUP

Prior to nondimensionalizing the lumped element model and developing the scaling theory,

the model was validated experimentally on candidate aluminum-PZT composite beams at a

macro-scale as described in Chapter 2. This chapter continues from there and describes the

experimental setup and procedures used to characterize the MEMS PZT cantilever generators.

First, the devices were tested for their dielectric and piezoelectric properties. Next, various

characterization experiments were designed to extract all of the lumped element parameters to

complete the equivalent circuit. The values extracted are ultimately compared (Section 6.5) with

the predictions from the model, providing clear insight and direction for further analysis and

validation. Detailed descriptions of the mechanical, electrical, and electromechanical

characterization setups are provided in this chapter along with the data acquisition methods

employed. Finally, methods to excite the devices at resonance are described, in which the output

voltage and power with respect to input acceleration were measured across varying resistive

loads.

Ferroelectric Characterization Setup

Before releasing devices from the processed wafer that was returned from the Army

Research Laboratory (ARL), it was characterized initially for the dielectric and ferroelectric

properties of the PZT layer. The wafer was mounted in a probe station and the electrical

measurements were taken using a Radiant Technologies Precision LC Ferroelectric Tester. Two

probes were used to access the top and bottom electrodes of the PZT layer on the wafer. A

bipolar triangular voltage waveform is applied to the PZT layer for a specified time, typically

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10 ms ( )100 Hz , and the current during that period is measured. The triangular waveform was

generated by cycling increments of voltage determined by the number of points for one whole

time period. Then, the current is integrated across the area (i.e., using a value input by the user)

to obtain the accumulated charge in the area to give an effective polarization. The voltage signal

was divided into either 101 or 201 points to cover the entire time period and the polarization for

each value was obtained and plotted. This process was carried out for peak voltages ranging

from 1 60 V V− . Experiments were not extended for higher voltages beyond the saturation

voltage of the PZT thin film. The polarization saturated at approximately 60 V for all tested

PZT layers. A simple schematic of the setup used is shown in Figure 5-1 and the results of these

experiments are presented in the next chapter. A photograph of the actual setup used is shown

here in Figure 5-2. The area and the thickness of the dielectric were specified to calculate the

remnant polarization and coercive fields.

current

voltage

Radiant Ferroelectric Tester Base

PZT wafer

voltage

current

probes

microscope

Figure 5-1: Schematic for ferroelectric characterization.

155

Figure 5-2: Experimental setup for ferroelectric characterization.

Piezoelectric Characterization

The piezoelectric material on the processed wafer is not poled. Hence, we need to polarize

the PZT to the recommended voltage for a recommended duration (at a specified temperature if

applicable). The literature (Kholkin et al. 1998, Roeder et al. 1998) suggests a polarizing

temperature of slightly below the Curie temperature and using a strong electric field. Although

ARL recommends applying 5-10 V for 5-10 minutes to polarize the PZT at room temperature, it

reports that the PZT is usually partially poled during fabrication and may not need polarization

for most of their applications. Therefore, using the Radiant (Precision LS) Ferroelectric

Materials Analyzer, hysteresis tests were conducted to obtain the hysteresis for the PZT layer

and estimate its remnant polarization and coercive fields. The hysteresis curve is plotted with

polarization as a function of the applied voltage on the x axis− . The applied voltage profile in

this experiment is a triangular waveform described earlier that cycles from a negative peak to a

positive value in increments over time. The remnant polarization is the intercept on the y axis− ,

and the coercive field is obtained from the intercept on the x-axis where the polarization changes

156

orientation. These values were then compared with the values provided by ARL for the

fabricated wafers that are listed in Table 5-1. If the remnant polarization is considerably larger,

the devices need not be polarized. These experiments that capture the hysteresis behavior of

PZT were repeated on known geometries on the wafer after poling at different voltages for

different times and elevated temperatures to observe their effect. The hysteresis behavior along

with results for the experiments conducted to extract the relevant parameters is further explained

in detail in chapter 6.

Table 5-1: Reported polarization results (ref: ARL) Thickness of PZT 1.016 10%mμ ±

Remnant Polarization (Pr) 22.5 μC/cm2

Coercive Field (Vc) 80 kV/cm

Electrical Characterization

Next, electrical impedance measurements were carried out to estimate the electrical

lumped element parameters in the circuit. The first two parameters that are measured are the

blocked electrical capacitance, ebC , and the dielectric loss, eR , in the PZT layer. Furthermore,

the released devices after packaging are characterized for their free electrical impedance ( )efZ .

The packaged devices are generally connected in series or parallel if they contain an array of

PZT cantilevers. So, the overall free electrical impedance for the connection is measured.

Blocked Electrical Capacitance, ebC and Dielectric Loss, eR

Blocked electrical impedance measurements were obtained from the process wafer before

device release. These measurements give an estimate of the blocked capacitance and dielectric

157

loss of the device geometries. An effective relative permittivity value for the piezoelectric

material can be extracted from the measured capacitance using the dimensions of the devices.

ebC , defined in Eq. 2.25, is the capacitance in the piezoelectric layer when the device is

blocked (no resulting motion). So, measuring the capacitance before releasing the structures

from the wafer directly yields ebC . A HP 4294A Precision Vector Impedance Analyzer was used

to measure the blocked electrical impedance. This response is curve-fitted using an assumed

circuit topology (e.g., consisting of a capacitor and resistor in parallel). An effective value for

ebC and eR is thus extracted from the response.

In the probe station (shown in Figure 5-3), the fixture compensation was initially

performed by calibrating for short and open circuits to eliminate any residual impedance of the

cables and probes. Then the two probes were connected to the top and bottom electrode and the

blocked electrical impedance was measured using the Impedance Analyzer. A known input

voltage, ( )500 ebmV for Z was applied to the PZT, and the corresponding current was

simultaneously measured across a frequency range. First, an experiment was conducted to

measure the electrical impedance as a function of the applied voltage varying from 10 1 mV V− ,

and it was observed that the capacitance varies with the input amplitude. Therefore, an

intermediate value of 500 mV was chosen to characterize all measurements. A schematic of the

experimental setup is shown in Figure 5-3.

158

Figure 5-3: Schematic for blocked electrical impedance measurement.

A photograph of the experimental setup used for the impedance measurements is shown in

Figure 5-4. The setup basically consists of a probe station on which the process wafer is mounted

and probes are used to provide contacts to the electrodes.

Figure 5-4: Experimental setup for electrical impedance characterization.

The start frequency for the impedance analyzer is fixed by the instrument at 40 Hz . The

impedance was thus measured across a frequency range of 40 1000 Hz Hz− . The real and

159

imaginary parts of the impedance were extracted to estimate an effective dielectric loss factor,

tanδ , which is given as

( )( )

Retan

Im 1e

eb

Z RRZ X j C

δω

= = ≡ (5.1)

In the LEM, the dielectric loss in the PZT layer is calculated using an empirical relation

given as ( )

1tan 2e

eb

RfCδ π

= , where tanδ was assumed to be 0.04 (from existing literature) for

our predictions. The actual values for tanδ will be extracted for each of the devices from

experimental measurements. From the blocked impedance experiments, an effective value for

tanδ can be estimated using Eq. (5.1) and compared with the assumed tanδ . The new value

will be used henceforth to estimate other lumped element parameters in the circuit.

It should also be noted that the top and bottom electrode extend into the clamp (shown in

the process flow in Figure 4-16). Therefore, the measured capacitance includes the PZT material

on the clamp that acts like a parasitic capacitance for the electromechanical characterization.

Mechanical Characterization

Overall, six total devices that consisted of two devices of type PZT-EH-07 and four

devices of PZT-EH-09 were characterized. PZT-EH-09-1 and PZT-EH-09-2 belong to Design-9

that consists of 3 cantilever beams attached to a single proof mass. In these two devices, all the

PZT electrodes are connected in parallel for characterization. PZT-EH-09-3 and PZT-EH-09-4

are also of the same design type, but only one PZT electrode is used as external electrical

connection. This will compare the performance of the device for one PZT versus the PZT layers

connected in parallel. Furthermore, PZT-EH-07-1 and PZT-EH-07-2 are of design type 7. The

device dimensions for the two designs are listed for reference in Table 3-10. A detailed

160

schematic of the experimental setup used to characterize the PZT energy harvesters is shown in

Figure 5-5.

Figure 5-5: Experimental setup for mechanical and electromechanical characterization.

In the setup shown in a picture in Figure 5-6, the packaged device is mounted on the

Bruel & Kjaer mini shaker (type 4810). The whole setup is mounted underneath the microscope

using a lab jack that provides the flexibility to move the shaker both horizontally and vertically.

This arrangement is necessary to get the device in focus and also to traverse to different spots on

the device.

161

Figure 5-6: Experimental setup for vibration and velocity measurements with LV.

The MEMS PZT device was rigidly mounted on the mini shaker using a 1" 10-32 threaded

screw with a locking nut. The schematic of the package and the mounted devices were explained

in Section 4.2. The whole setup is mounted underneath an Olympus BX60 optical microscope

with a 5x objective lens to perform velocity measurements. A Polytec scanning laser

vibrometer (LV) as shown in Figure 5-6 is connected to the microscope to obtain velocity

measurements. The laser vibrometer consists of a microscope adapter (Model OFV 074) that

enables scanning very small areas with the laser. The laser spot can be as small as 10 mμ with a

100x magnification lens. The laser also consists of a fiber interferometer (Model OFV 511), a

scanner controller (Model MSV-Z-040) and a vibrometer controller (Model OFV 3001s) that

effectively enables velocity measurements across a scanning range. A signal is generated using

the waveform generator inside the LV that is used to drive the shaker through a B&K power

amplifier (type 2718). The resulting acceleration to the device is measured by focusing the laser

on the clamp of the MEMS device that is rigidly mounted to the package. This gives an estimate

of the input acceleration to the device mimicking an input base vibration. Sample records are

obtained for a periodic chirp input and its corresponding Fourier Transform (FFT) was

162

calculated. A uniform window function was applied for the data with 1600 FFT lines and

averaged 50-100 times to minimize any random errors during data acquisition. The data

acquisition parameters are summarized in Table 5-2. Next, the device was horizontally traversed

and the laser was focused on the tip of the cantilever beam and the corresponding tip velocity

was measured for the same input conditions. A relative tip motion was calculated by subtracting

out the clamp motion. Consequently, a frequency response was obtained between the input

acceleration and resulting tip deflection.

The resonant frequency is extracted from the peak in the frequency response spectrum. A

nominal mechanical damping, mR , in the system is estimated by matching the frequency

response peaks at resonance between the model and experiment. Since damping affects only the

resonance, we can obtain the value directly by altering the value of ζ so that the resonance

peaks match.

For all PZT-EH-09 devices, the frequency response was measured over 0 1 kHz− that was

divided with 1600 FFT lines resulting in a frequency resolution of 0.625 Hz . The number of

averages for the frequency response of PZT-EH-09-1 and PZT-EH-09-2, was 50, while 100

averages were measured for PZT-EH-09-3 and PZT-EH-09-4. In the case of PZT-EH-07 devices,

the frequency response was obtained for 0 0.5 kHz− using 800 FFT lines with a frequency

resolution of 0.625 Hz . 100 averages were obtained for all measurements with PZT-EH-07-2

and PZT-EH-07-3. Since a periodic chirp signal was used as an input, a uniform window

function was used for all tested devices

163

Table 5-2: Data acquisiton parameters for mechanical characterization. Waveform Perodic chirp

Frequency Range 0-1 kHz

Number of FFT lines 1600

Window Uniform

Resolution 0.625 Hz

Number of Averages 50 or 100

During the experiment, the laser sensitivity and the measurement range were adjusted to

ensure good signal quality and resolution. Measurements were carried out with the PZT

electrodes shorted to avoid any electromechanical effect. A corresponding open circuit

measurement was also taken, but the coupling between the mechanical and electrical domains is

minimal resulting in minimal changes in the response or resonance.

Electromechanical Characterization

The effective piezoelectric coefficient, defined as the static tip deflection for an applied

voltage to the PZT, is obtained from the electromechanical characterization. However, it is

difficult to apply a static voltage and measure a static deflection because the voltage dissipates

across the PZT in the form of dielectric loss. Alternatively, the effective piezoelectric

coefficient, md , can be experimentally determined from the low frequency ( )~ 10 50 Hz−

asymptotic dc response of the device when excited using an ac voltage.

For the electromechanical characterization, the same setup shown in Figure 5-5 was used.

Instead of having the PZT electrodes shorted, they act as the input to apply a voltage that results

in a tip velocity measuring using the laser vibrometer. An input periodic chirp signal across

prescribed frequency range was used to electrically excite the structure and the frequency

response between the input voltage and the resulting tip deflection was extracted. The data

acquisition parameters for this experiment are listed in Table 5-3.

164

Table 5-3: Data acquisiton parameters for mechanical characterization. Waveform Periodic chirp



Window Uniform

Resolution 0.625 Hz


The low frequency response of the experiment that measures the tip deflection per unit

voltage directly provides an effective piezoelectric coefficient ( )md .

Open Circuit Voltage Characterization

All of the LEM parameters are extracted using a least squares fit to the experimentally

obtained frequency response with the LEM equations described in Chapter 2. The details about

this extraction are summarized in Chapter 6. Each of the frequency response obtained from the

mechanical, electrical, and electromechanical characterization provides the corresponding LEM

parameters in each energy domain.

Next, the open circuit voltage is measured across a prescribed frequency range for a

known input acceleration. A periodic chirp signal is generated using the SRS spectrum analyzer

that is applied to the device mechanically through the LDS dynamic shaker (V408) driven using

an LDS power amplifier (PA100E-CE) as shown in Figure 5-7. A photograph of the

experimental setup is provided in Figure 5-8.

165

acceleration

OPTICAL TABLE

Package with PZT Device

voltage

Power Amplifier

SHAKER

Spectrum Analyzer Imp. Head

Figure 5-7: Experimental setup for open circuit voltage measurements.

Figure 5-8: Experimental setup for open circuit voltage measurements.

The input acceleration is measured using a Bruel & Kjaer impedance head (model 8001)

that simultaneously measures force and acceleration. The resulting voltage across the PZT is

measured using the spectrum analyzer. The frequency response is calculated between the input

acceleration and output open circuit voltage. This extracted response provides an estimate of

166

how much voltage per unit acceleration can be generated across the range. The data acquisition

parameters used for this experiment are listed in Table 5-4.

Table 5-4: Data acquisiton parameters for mechanical characterization. Waveform Periodic chirp



Window Uniform

Resolution 0.625 Hz


Voltage and Power Measurements

Finally, a sinusoidal signal was generated using the waveform generator at the measured

resonance frequency of each device to excite it mechanically through the shaker. The

experimental setup, similar to the one described before for the vibration measurements is shown

in Figure 5-9.

Figure 5-9: Experimental setup for voltage and power measurements.

The input acceleration at this frequency was measured at the clamp using the vibrometer,

similar to the process described in 0. A low vibration level on the order of 0.1 g is used to

167

ensure safe operation of the device at resonance. The resulting output voltage at the PZT was

measured across various resistive loads ( )1 750 k kΩ− Ω to observe the voltage and power

characteristics. Plots of rms voltage and power normalized to the input acceleration as a function

of the load resistance are generated. The output voltage increases with increasing output

resistance and finally approaches the open circuit voltage. On the other hand, the output rms

power increases and reaches a maximum for an optimal output load and then decreases with

increasing resistance.

168

CHAPTER 6 EXPERIMENTAL RESULTS AND DISCUSSION

A detailed experimental characterization of the MEMS PZT power generators is presented

in this chapter. The overall goal of the experimental procedure is to extract all possible LEM

parameters in the electromechanical circuit (Figure 2.2) by performing various experiments and

using the extracted values to predict the overall voltage and power output of the device and

compare with experimental results.

Ferroelectric Characterization

Prior to releasing the devices from the process wafer, polarization tests (Section 5.2) were

performed to measure the piezoelectric properties. First, polarization hysteresis was

characterized by applying a static voltage to the piezoelectric layer and measuring the current in

the film. The current in the PZT is integrated with respect to time to obtain the stored charge.

Polarization, which is defined as charge stored per unit area, is usually expressed in units of

2C cmμ . An ideal hysteresis loop for a piezoelectric material is shown in Figure 6-1 which

represents a typical polarization-electric field (P-E) plot. Hysteresis occurs when the material

exhibits multi-valued behavior of any property (Megaw 1957). In Figure 6-1, for every value of

electric field there exists two distinct values for the polarization depending on the direction of the

hysteresis cycle. Various important parameters in the hysteresis loop (Cady 1964), denoted by

labeled points in the figure, are described below.

169

Pr

-Pr

Pm

-Ec

Ec

Ps

Pola

rizat

ion

(P, µ

C/c

m2 )

Electric Field (E, V/m)

Figure 6-1: A typical P-E hysteresis loop for a piezoelectric material (adapted from Cady 1964).

Points a and d are maximum polarization points in positive and negative directions

represented by mP± . Another parameter sP , defined as saturation polarization is obtained as the

intersection of the tangent to the hysteresis loop at point a and the polarization axis. Points b

and e are called retentivity points represented by rP and rP− . These points are defined as the

residual polarization in the material when the applied electric field is zero. Finally, points c and

f are defined as coercivity points represented by cE and cE− . These points are defined as the

electric coercive fields required to reverse the direction of polarization in the piezoelectric

material.

Next, a dielectric characterization is performed to calculate the capacitance and the relative

permittivity of the piezoelectric material. Shown in Figure 6-2 is a typical εr-E plot that

170

represents the behavior of relative permittivity as a function of the electric field. In the figure, cE

and cE− define the coercive fields when the relative permittivity of the material approaches a

minimum. Furthermore, rε is the residual relative permittivity when the material is not

subjected to any field (Cady 1964).

Figure 6-2: A typical ε-E curve for a piezoelectric material.

The results for a sample PZT geometry (PZT-EH-02-1-1) measured on the process wafer

are presented here, and the hysteresis parameters are compared to the values specified by ARL.

Figure 6-3 shows the polarization and capacitance plot corresponding to the input triangular

voltage waveform. As seen in the plots, when the polarization crosses zero (corresponding to the

coercive voltage) capacitance of PZT reaches its maximum.

171

Figure 6-3: Polarization, capacitance and input voltage waveforms for PZT-EH-02-1-1.

Hysteresis tests were performed for waveforms with different maximum voltages and the

results are presented in the above figure. In Figure 6-4, the polarization in the material is plotted

as a function of the applied voltage. As expected the PZT layer exhibits highly nonlinear

hysteresis behavior similar to the one shown in Figure 6-1.

172

Figure 6-4: Hysteresis plots for PZT-EH-02-1-1.

Tests were conducted for different voltages as shown in the above figure. It can be seen

that the polarization saturates with increasing voltage. The points where the curve intersects the

horizontal and vertical axes are extracted for each voltage and are plotted in Figure 6-5.

Hysteresis parameters such as , and m r cP P V± ± defined earlier are shown in the figure below.

All parameters are observed to saturate as the voltage increases. The units for r and mP P± are

2C cmμ while cV± is expressed as V in Figure 6-5. These results are plotted as a function of

increasing input voltage along x axis− .

173

Figure 6-5: Pr and Vc for different applied voltages for PZT-EH-02-1-1.

The measured remnant polarization for PZT-EH-02-1-1 is 226.34 C cmμ . The coercive

electric field, cE is obtained from the coercive voltage as c pV t , where pt is the thickness of

PZT. Therefore, the measured coercive field is obtained from the saturated 6.6 cV V= , which

corresponds to an electric field of 63.34 kV cm for 1.04 mμ thick PZT layer. The maximum

polarization observed was 253.25 C cmμ for a maximum applied voltage of 60 V . From the

measured charge, the capacitance of the PZT layer is calculated for all the cases. The normalized

capacitance expressed as capacitance per unit surface area of the PZT layer is plotted as a

function of applied voltage in Figure 6-6. The curve exhibits a typical nonlinear behavior similar

to Figure 6-2, where the static blocked capacitance is extracted at zero voltage. The shape of the

174

curve is generally symmetrical about the 0V = axis and the maximum capacitance occurs at the

coercive field.

Figure 6-6: Normalized Ceb for PZT-EH-02-1-1 during the hysteresis test.

Finally, a leakage test was conducted on PZT-EH-02-1-1 where 10 V dc was applied for

1000 ms and the resulting current was measured. Figure 6-7 shows a plot between the measured

current and time. The steady state current is 81.18 10 A−× that corresponds to a dielectric

resistance of 88.41 10 × Ω for the applied voltage. It should be noted that the dielectric resistance

tends to vary inversely with frequency and the value presented here is measured at dc. The

resistivity for the PZT feature is obtained from p

p

AR t as 112.69 10 cm× Ω− , where pA is the

surface area of the PZT and R is the measured dielectric resistance.

175

Figure 6-7: Leakage current for PZT-EH-02-1-1 subjected to 10V DC.

Next, poling tests were carried out to observe its effect on the piezoelectric properties.

Initially, PZT-EH-02-1-1 was poled by subjecting the PZT feature to 5 dcV at room temperature

for 5 min, 10 min, and 15 min respectively. After each poling step, hysteresis tests were

conducted using the ferroelectric tester at different voltages. The remnant polarization, coercive

voltages and maximum polarization parameters were recorded and are plotted in Figure 6-8. No

significant change/improvement in the polarization was observed while poling at room

temperature and the remnant polarization of the material remained unchanged. The results were

similar for tests that were carried out at 10 dcV and 15 dcV for the above listed time periods.

176

Figure 6-8: Poling of PZT-EH-02-1-1 at 5V for different times.

Another poling test was carried out to observe the effect of temperature on piezoelectric

properties. The wafer was heated to a specific temperature, and PZT-EH-02-1-1 feature was

poled with a constant 5 dcV while the wafer was allowed to cool to room temperature. The

experiment was performed at room temperature, o50 C , and o100 C . Previous work in thin film

PZT have performed poling at elevated temperatures upto o150 C (Roeder et al. 1998, Kholkin

et al. 1998). To achieve maximum polarization, the PZT should be poled at a temperature

slightly below the curie point. Since the curie temperature for our thin films is not know nor

supplied from ARL, polarization tests were only conducted at o100 C . Future work can extend

poling to higher temperatures to observe any notable improvement in remnant polarization.

Moreover, ARL recommends room temperature poling for 10 15 min− with 3 5− times the

177

coercive voltage. In addition, there will be some degradation of the piezoelectric coefficients

with time after poling. This phenomenon known as aging occurs primarily since thin film

materials are not single domain materials and some degree of self alignment with the

ferroelectric domains remains inherent, even after poling. For reliability, it is necessary to keep

track of the time allowed between the end of the poling procedure and the beginning of the

measurement. If this time is not kept consistent, each device may have aged to a different degree

prior to the measurement and in between measurements (Polcawich and Troiler-McKinstry

2000). More information can be found in literature on poling and its effect on piezoelectric

properties. However, for this dissertation, those different poling conditions have not been

explored in detail. After each poling step mentioned above, hysteresis tests were conducted for

the feature with the ferroelectric tester. The extracted hysteresis parameters are shown in Figure

6-9. Clearly, the remnant polarization remains within 10 % of the unpoled value, and it is

therefore concluded that the PZT is sufficiently poled during the fabrication process as reported

by ARL.

178

Figure 6-9: Poling of PZT-EH-02-1-1 at different temperatures.

Table 6-1 compares the measured values piezoelectric hysteresis parameters with those

reported by ARL for the PZT process. Since the values are similar and not much difference was

observed with post-fabrication poling, all subsequent devices were tested directly after release.

Table 6-1: Comparison of ARL's reported hysteresis parameters with measured values. Hysteresis parameters Measured at

ARL Measured at UF

Remnant polarization rP 222.5 C cmμ

226.34 C cmμ

Coercive field cE 80 kV cm 63.34 kV cm

179

Blocked Electrical Impedance Measurements

Since it was established from the hysteresis that the capacitance is a function of the applied

voltage, a similar test was carried out here to observe the variation of ebC and tanδ (Eq. 5.1)

with parameters such as source amplitude and dc bias. First, a constant sinusoidal voltage of

500 mV at 100 Hz was applied to the PZT-EH-1-3-02 feature, and the dc bias was varied from

0 to 40 dcV . The electrical impedance was simultaneously measured using HP4924a Vector

Impedance Analyzer, and the blocked capacitance and dielectric loss were extracted and are

plotted in Figure 6-10. A point averaging scheme with 10 samples at each frequency was used to

measure the data. Both ebC and tanδ were observed to monotonically decrease with bias.

Figure 6-10: variation of Ceb and tanδ with dc bias and a constant sinusoid, 500 mV at 100 Hz.

180

A least squares fit was performed to obtain an empirical relation between ebC and tanδ

with dc bias as shown in Figure 6-10. The expression for ebC was observed to fit best with

( )48.3 1092.21 10 eb dcC V F−− ×−= × and tanδ was fitted using ( ) 0.01421.24 10 dcV −−× , where dcV is the

dc bias applied to the PZT.

Another test was performed at zero bias where a sinusoidal voltage at 100 Hz was

applied to the PZT with increasing source amplitude. Figure 6-11 shows a plot of ebC and tanδ

along with their curve fits, generated using least squares method.

Figure 6-11: Variation of Ceb and tanδ with source amplitude at 100Hz.

Here, a quadratic polynomial was used to fit the data and the empirical expressions are

( )29 22.16 10 0.04 0.02 1 eb s sC V V F−= × + + and ( )22 2tan 0.84 10 0.72 0.29 1s sV Vδ −= × + + , where

181

sV is the applied source amplitude. It should be noted that the coefficients of the fitting functions

in both cases represent the asymptotic values of ebC and tanδ . However the overall functional

form is complicated and needs to be explored further to understand the general dependence. This

exercise was carried out only on one device geometry just to demonstrate the highly nonlinear

dependence of the electrical and piezoelectric properties (hysteresis tests) on the dc bias,

frequency of operation and voltage amplitudes. More information on the nonlinear dependence

of piezoelectric properties can be found in Damjanovic (1997a; 1997b), Takahashi et al. (1998),

Masys et al. (2003) etc.

For the blocked impedance measurements, an intermediate value for source amplitude,

500 mV , was chosen and applied across the electrodes of the piezo layer, and the resulting

current is simultaneously measured using the impedance analyzer. The electrical impedance was

measured across a frequency range of 40 1000 Hz− that was divided into 201 points. A point

averaging scheme is employed for the measurements that cycles through 10 averages at every

point to measure the final impedance. From the measured complex electrical impedance, the real

and imaginary parts were curve-fitted using the least squares technique to extract an effective

blocked capacitance ( )ebC and dielectric loss tangent ( )tanδ . The fitting function used in the

technique consists of a parallel combination of a capacitor and a resistor per the lumped element

model shown in Figure 2.1. Using the thickness of the PZT layer 1.02 0.1 mμ± and the surface

areas of the design geometries, the effective dielectric permittivities are calculated. The results

obtained from these experiments are shown in Figure 6-12 and Figure 6-13. A summary of the

estimated parameters are listed in Table 6-2. In Table 6-2, the areas are calculated from the

theoretical dimensions in the design structures.

182

Table 6-2: Dielectric parameters of all tested design geometries on the device wafer.

Device Geometry ( ) ebC nF ( )2 pA mm rε tanδ

PZT-EH-01 2.07 0.03± 0.26 931 15± ( ) 21.29 0.17 10−± × PZT-EH-02 27.15 0.43± 3.34 955 15± ( ) 21.69 0.15 10−± × PZT-EH-03 98.19 2.05± 12.05 957 20± ( ) 21.79 0.24 10−± × PZT-EH-04 102.44 1.43± 12.39 972 14± ( ) 22.30 2.14 10−± × PZT-EH-05 35.72 0.42± 4.43 948 12± ( ) 21.55 0.19 10−± × PZT-EH-06 1.60 0.02± 0.20 936 11± ( ) 21.27 0.17 10−± × PZT-EH-07 12.08 0.20± 1.47 962 16± ( ) 21.53 0.28 10−± × PZT-EH-08 43.26 0.60± 5.31 957 13± ( ) 21.59 0.15 10−± × PZT-EH-09 2.59 0.04± 0.32 951 15± ( ) 21.43 0.22 10−± ×

Therefore, from the blocked electrical impedance measurements performed on the process

wafer, the overall dielectric permittivity was estimated as 943 18± and the dielectric loss tangent

was measured to be ( ) 21.41 0.46 10−± × . These values will be used to estimate the blocked

capacitance ( )ebC and dielectric resistance ( )eR to complete the LEM representation for

individual tested devices. Another observation that can be drawn from these results is that the

permittivity and dielectric loss tangent both tend to increase with the PZT area. However, this

conclusion is just based on these specific results and can only be substantiated with more

experiments and further investigation.

183

Figure 6-12: Ceb and εr for MEMS PZT devices on wafer before release for a) PZT-EH-01 (106 geometries) b) PZT-EH-02 (16 geometries) c) PZT-EH-03 (15 geometries) d) PZT-EH-04 (14 geometries) e) PZT-EH-05 (16 geometries)

184

Figure 6-13: Ceb and εr for MEMS PZT devices on wafer before release for a) PZT-EH-06 (150 geometries) b) PZT-EH-07 (12 geometries) c) PZT-EH-08 (22 geometries) d) PZT-EH-09 (108 geometries)

Next, the PZT MEMS devices that were released and packaged were characterized to

extract all the lumped element parameters described in Section 2.1. Various frequency response

measurements were carried out for each device and least square error curve fits were generated

using the functional forms of the response described in 2.3. The LEM parameters were estimated

from the functional fits and compared with theory. Finally, output voltage and power

measurements were obtained for each device operating at resonance, subjected to fixed source

acceleration. These experimental results were compared with those predicted using the extracted

LEM parameters.

185

In the next section, various algorithms for lumped element parameter extraction are

investigated in detail. The methods are first presented and discussed after which each of these

methods was tested on a single device (PZT-EH-09-02). The method that produced the best

results was then adopted and applied for all other devices.

Lumped Element Parameter Extraction

In this section, a detailed description of the procedure adopted for the lumped element

parameter extraction is provided. First, a list of all the parameters required to complete the

electromechanical circuit is presented followed by a list of various experiments performed to

characterize the behavior of the MEMS PZT device. Next, three different data extraction

algorithms are presented along with the corresponding results and discussion. One of the

presented methodologies is selected as the common extraction procedure and is applied for all

experimental data for the devices. The results thus estimated for the LEM parameters for each of

the MEMS devices are presented with their uncertainties.

The lumped element parameters that need to be estimated for completing the

electromechanical circuit (Figure 2.1) are again listed below in Table 6-3.

Table 6-3: LEM parameters extracted using experimental data. mM Effective mechanical mass

msC Short circuit mechanical compliance

mR Effective mechanical damping

md Effective piezoelectric coefficient

ebC Blocked electrical capacitance

eR Dielectric loss

nF Natural frequency

The device responses obtained in the characterization experiments described in Chapter 5

are used to extract these parameters. The equations from the LEM that will be used to represent

the behavior of the device are described next. These equations (simplified and applied to

186

different frequency regions) are used to fit the measured experimental response with an effective

set of LEM parameters.

The electromechanical characterization experiment is conducted to measure the

frequency response between the input voltage to the PZT and resulting tip displacement. From

the LEM described in Section 2.2, the frequency response can be represented using Eq. (6.1),

where tipw and V are the relative tip displacement and input voltage, respectively.

20

.

1 2

tip m

F

n n

w dV

jω ωζω ω

=

=⎛ ⎞

− +⎜ ⎟⎝ ⎠

(6.1)

From the short circuit mechanical characterization where the relative tip displacement

tipw was measured as a function of the input base displacement basew , the resulting frequency

response is inherently dimensionless and is shown in Eq. (6.2).

22

20

.

1 2

tip n

base V

n n

ww

j

ωω

ω ωζω ω

=

=⎛ ⎞

− +⎜ ⎟⎝ ⎠

(6.2)

The free electrical impedance of the PZT device was experimentally measured and the

response obtain from the lumped element model is

2

2 22

2

1 1 2tan

.2 11 1 2

tan tan tan

eb n n

ef

m

n n n n ms eb

jC

Zdj

C C

ω ωζω δ ω ω

ω ζ ω ω ωζω δ ω δ ω ω δ

⎡ ⎤⎛ ⎞⎢ ⎥− +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦=

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥− − + − + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎣ ⎦

(6.3)

187

The final experiment conducted to extract the LEM parameters is the frequency response

between the open circuit voltage across the PZT and the input base acceleration. The model

when solved for this acceleration response follows Eq. (6.4)

2 22

2

tan .2 11 1 2

tan tan tan

m m

oc eb

o m

n n n n ms eb

M dV Ca dj

C C

δ

ω ζ ω ω ωζω δ ω δ ω ω δ

=⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥− − + − + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎣ ⎦

(6.4)

These equations and their asymptotic limits will be used in the following section. In

particular, they are applied to the measured response in specific frequency regions such as “low

frequency region”, near “resonance” etc. This approach simplifies the expressions facilitating

easy parameter extraction via the method of least squares. Next, three potential parameter

extraction algorithms are presented with a discussion on the extraction procedure for each. In

addition, flowcharts are provided for each to understand the overall process better.

The general extraction algorithm to estimate the LEM parameters is listed below in the

order they need to be obtained

1. Extract md (Eq. (6.5)) from the low frequency response of the

electromechanical characterization results.

2. Extract efC from the low frequency response of the free electrical impedance

measurements. It should be noted that since the capacitance depends on the

source amplitude, these measurements should ideally be obtained at similar

amplitude levels.

3. Extract the short circuit natural frequency ( )scf and open circuit natural

frequency ( )ocf of the device.

188

4. Extract 2κ from the extracted frequencies using the relation, 2 1 sc

oc

ff

κ = − .

5. Calculate msC from md , efC and 2κ using the relation, 2

2 m

ef ms

dC C

κ = .

6. Calculate mM from msC and nf using Eq. (6.12).

7. Extract ζ and tanδ by fitting the responses around resonance.

In addition, ebC can be extracted from the process wafer before releasing the individual

devices. Furthermore, the extraction steps assume that 31d and ε are constant with frequency

which is generally not true. It should be noted here that for damping estimates, experiments can

be conducted to separate the individual mechanisms. For example, conducting the short circuit

characterization will yield a purely mechanical damping factor, while the open circuit condition

included electrical losses also. If these experiments are carried out under vacuum, air flow

dissipation can be estimated and neglected in the model.

However, since the coupling coefficient of these devices is very small, the short circuit

resonance and open circuit resonance could not be separated. Hence, alternate techniques were

devised and implemented. Three different algorithms were designed to extract the parameters

that are described in detail in the following sections.

Method 1

This method aims to obtain the LEM parameters using the algorithm listed below in the

order they are extracted.

1. Extract md (Eq. (6.5)) from the low frequency response of the

electromechanical characterization results.

189

2. Extract ebC (Eq. (6.8)) from the low frequency response of the free electrical

impedance measurements.

3. Extract mM (Eq. (6.10)) from the low frequency response of open circuit

voltage measurements.

4. Extract an effective nf and ζ by simultaneously fitting both short circuit

mechanical (Eq. (6.2)) and electromechanical (Eq. (6.11)) responses near

resonance.

5. Calculate msC from mM and nf using Eq. (6.12).

6. Extract an effective tanδ by fitting the open circuit voltage response (Eq.

(6.13)) around resonance. All other previously estimated parameters will be

included in the model.

190

Experimental response spectra

1ef

eb

ZCω

=

oc m m

o eb

V M da C

=

( )21

2ms

n m

Cf Mπ

=

oc

o

m m

eb

VaF M dC

=

mdebC

mM

,nf ζ

msC

tanδ

0f → 0f →

0f →

rf f→

rf f→

, , , , , , tanm eb m n msd C M f Cζ δ

0

tipm

F

wd

V=

=

021

1 2

tip

F

m

n n

wV

djω ωζ

ω ω

= =⎛ ⎞

− +⎜ ⎟⎝ ⎠

22

20 1 2

tip n

base V

n n

ww

j

ωω

ω ωζω ω

=

=⎛ ⎞

− +⎜ ⎟⎝ ⎠

Figure 6-14: Flowchart for method 1 to extract the LEM parameters from the experimental data.

191

A flowchart that shows the extraction procedure pictorially is shown in Figure 6-14.

Next, the details of all fitted functions in different frequency regions in each step are described.

From Eq. (6.1) the response at very low frequencies (approaching dc) reduces to

0

.mF

z dV =

= (6.5)

The data at low frequencies is thus obtained using least squares via linear regression at

low frequencies, which directly provides md , the effective piezoelectric coefficient. The

uncertainty in the estimate is also obtained from the regression model for the fit. It should be

noted that the experimental data is very noisy at low frequency with low coherence. For

example, the coherence measured for PZT-EH-02 was approximately 0.5 0.6− for frequencies

close to 100 Hz . The poor coherence reflects in the uncertainty estimates for the extracted

parameters.

Next, the free electrical impedance, efZ is simplified for low frequencies to

2

1tan .

11 1tan

ebef

m

ms eb

CZdj

C C

ω δ

δ

=⎡ ⎤

+ +⎢ ⎥⎣ ⎦

(6.6)

Here, 2m

ms eb

dC C

is defined as the coupling factor ( )2κ which is very small and therefore we

assume 2

1 1m

ms eb

dC C

+ ≅ . This holds true especially for these devices as the coupling between

electrical and mechanical domains is unfortunately poor, and no significant shift is observed

between short circuit and open circuit response. This observation was confirmed when

192

measurements were conducted for short-circuit and open-circuit resonant frequencies and no

shift was noted. From Eq. (6.6), the magnitude of the impedance is further simplified as

2

1tan

11tan

ebef

CZ ω δ

δ

=+

(6.7)

The reported tanδ for PZT in literature is generally 2 % 4 %− which was verified

during the blocked electrical impedance measurements on MEMS PZT devices (Chapter 6).

Hence, it is assumed that 2 2

1 11tan tanδ δ

+ ≈ simplifying the electrical impedance at low

frequency to

1ef

eb

ZCω

= (6.8)

Consequently, fitting 1

efZω with a straight line on a Bode plot provides an effective ebC .

The uncertainty in the estimate is directly obtained from the linear regression model.

Next, the open circuit voltage response at low frequencies is simplified to

2

tan11 1

tan

m m

oc eb

o m

ms eb

M dV Ca dj

C C

δ

δ

=⎡ ⎤

+ +⎢ ⎥⎣ ⎦

(6.9)

Using the same assumptions employed in the response for efZ , the response follows

oc m m

o eb

V M da C

= (6.10)

193

Since both md and ebC have already been experimentally extracted, fitting the data at low

frequency directly yields the effective mass mM of the device.

The short circuit mechanical response is inherently dimensionless shown earlier in Eq.

(6.2). In addition, dividing Eq. (6.1) by md renders it dimensionless as

021

1 2

tip

F

m

n n

wV

djω ωζ

ω ω

= =⎛ ⎞

− +⎜ ⎟⎝ ⎠

(6.11)

Both functions, Eqs. (6.2) and (6.11), are simultaneously fitted near the short-circuit

resonance to extract an effective damping ratio ζ and natural frequency nf . Note that if

2n

nfω

π⎛ ⎞=⎜ ⎟⎝ ⎠

is the extracted resonant frequency, it does not, by definition match the measured

resonance rf . They are related as ( )21 2r nf f ζ= − . rf corresponds to the frequency where the

magnitude of the data reaches its maximum. The fits are generated by minimizing the root mean

square (rms) error of the data. Here, ζ was estimated to be very small for all the devices.

Therefore, it is assumed that the estimate for the natural frequency is exact. Consequently, only

the uncertainties in ζ are estimated by adjusting its value such that the peaks match at

resonance. The upper and lower bounds on damping ratio, namely, uζ and lζ , are calculated

from the functions Eqs. (6.2) and (6.11) such that the peaks at resonance match in both the data

and fit.

From the previously estimated nf and mM , an effective compliance msC is calculated

using

194

( )2

12

msn m

Cf Mπ

= (6.12)

Finally, the open circuit voltage response at resonance is fitted using the non-

dimensionalized form of Eq. (6.4) as shown in Eq. (6.13) to extract an effective tanδ by

minimizing the rms error. The frequency range of interest in this cases depended on the number

of points. It was ensured that atleast 10 points were chosen for the fit.

2 22

2

1tan

2 11 1 2tan tan tan

oc

o

m mm

ebn n n n ms eb

Va

M d djC C C

δω ζ ω ω ωζω δ ω δ ω ω δ

=⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥− − + − + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎣ ⎦

(6.13)

The fits obtained using the above mentioned steps are shown below along with the actual

measured data. The results presented here in Figure 6-15-Figure 6-19 are for just 1 device, PZT-

EH-02.

Figure 6-15: Low frequency electromechanical response data compared with curve fit to extract dm.

195

Figure 6-16: Comparison between experiment and LEM based curve fit around resonance for a) electromechanical response b) short-circuit mechanical response

Figure 6-17: Low frequency curve fit compared with experiment to extract Ceb.

196

Figure 6-18: Comparison between experiment and curve fit for low frequency open circuit voltage response to extract Mm.

Figure 6-19: Experimental data and curve fits for open circuit voltage response compared around resonance.

The final extracted parameters for PZT-EH-02 are listed in the following Table 6-4 using

the above described method.

197

Table 6-4: LEM parameters extracted using Method 1. LEM parameter Estimated value Uncertainty

md 73.24 10 m V−× 3 %

nf 502.97Hz ζ 43.20 10−× 41.15 10−×

ebC 97.51 10 F−× 1 %<

mM 62.34 10 kg−× 8 %

msC 24.28 10 m N−× 7 % tanδ 0.72 0.25

The above method although appears to fit data well, predicts a much higher tanδ than

expected. This essentially contradicts the initial assumption that tanδ is small. Therefore,

another formulation was adopted to extract the parameters and is presented next.

Method 2

This method was proposed and implemented primarily to get a better estimate of tanδ

by simultaneously fitting more functions around resonance by minimizing the error. The

algorithm is listed below describing the steps involved.

1. Extract md (Eq. (6.5)) from the low frequency response of electromechanical

characterization results.

2. Extract ebC (Eq. (6.8)) from the low frequency response of free electrical




4. Extract an effective nf and ζ by simultaneously fitting both short circuit

mechanical (Eq. (6.2)) and electromechanical responses (Eq. (6.11)) around

resonance.


198

6. Extract an effective tanδ by fitting both open circuit voltage response and

free electrical impedance (Eqs. (6.14)-(6.15)) around resonance. All other

previously estimated parameters will be included in the model. It was

assumed that fitting both functions simultaneously will produce a better

estimate for tanδ .

199


0

tipm

F

wd

V=

= 1ef

eb

ZCω

=

oc m m

o eb

V M da C

=

( )21

2ms

n m

Cf Mπ

=

021

1 2

tip

F

m

n n

wV

djω ωζ

ω ω

= =⎛ ⎞

− +⎜ ⎟⎝ ⎠

22

20 1 2

tip n

base V

n n

ww

j

ωω

ω ωζω ω

=

=⎛ ⎞

− +⎜ ⎟⎝ ⎠

1 2

1 2

eb ef

n n

C ZF

j

ω

ω ωζω ω

=⎡ ⎤⎛ ⎞⎢ ⎥− +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

2

oc

o

m m

eb

VaF M dC

=

mdebC

mM

,nf ζ

msC

tanδ

0f → 0f →

0f →

rf f→

rf f→


Figure 6-20: Flowchart for parameter extraction using Method 2.

200

Steps 1-5 remain same as the ones described in Method 1 and therefore are not repeated.

The main difference between the two methods is in Step 6, where both electrical impedance and

open circuit voltage are fitted simultaneously fitted (using Eqs. (6.4) and (6.3)) that results in

dimensionless expressions to extract an effective tanδ minimizing the overall rms error of the

fit. The fitted non-dimensional functions are listed in Eqs. (6.14) and (6.15).

1 2 22

2

1tan

2 11 1 2tan tan tan

oc

o

m mm

ebn n n n ms eb

VaF M d djC C C

δω ζ ω ω ωζω δ ω δ ω ω δ

= =⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥− − + − + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎣ ⎦

(6.14)

2 2 2 22

2

1

tan

2 11 2 1 1 2

tan tan tan

eb ef

m

n n n n n n ms eb

C ZF

dj j

C C

ω δ

ω ω ω ζ ω ω ωζ ζ

ω ω ω δ ω δ ω ω δ

= =

− + − − + − + +⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎛ ⎞ ⎤⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎝ ⎠ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(6.15)

The fits obtained using this algorithm, remain similar as Method 1, except the fit obtained

in Step 6 that fits both normalized electrical impedance and open circuit voltage. The results

obtained for PZT-EH-02 using this step are shown in Figure 6-21. As seen in the figure, 2F does

not fit as well as 1F . This could be due to the fact that the absolute values of the magnitudes are

lower for 2F . In addition since the frequency resolution for the electrical impedance is poor

( )4.77 Hz∼ , it is possible that the exact resonance is not captured and hence the low value for

the response.

201

Figure 6-21: Experimental data and curve fits for open circuit voltage response and free electrical impedance compared around resonance.

The final extracted parameters for PZT-EH-02 obtained with method 2 are listed in the

following Table 6-5.


md 73.24 10 m V−× 3 %

nf 502.97Hz ζ 43.20 10−× 41.150 10−×

ebC 97.51 10 F−× 1 %<

mM 62.34 10 kg−× 8 %

msC 24.28 10 m N−× 7 % tanδ 0.71 0.25

Although Method 2 produces a better fit than Method 1 at resonance and is more

accurate, it still over-predicts tanδ with no significant improvement from the previous estimate.

Therefore, another approach is proposed and presented next to obtain a better estimate for tanδ

while maintaining the goodness of the fits.

Method 3

This method was proposed and implemented to get a better estimate of tanδ by

simultaneously fitting more functions around resonance by minimizing the error. The algorithm

202

is listed below describing the steps involved. A flowchart for the steps listed below is provided in

Figure 6-22.

1. Extract md (Eq. (6.5)) from the low frequency response of electromechanical

characterization results.

2. Extract ebC (Eq. (6.8)) from the low frequency response of free electrical




4. Extract an effective nf , ζ and tanδ by fitting all four experimental

responses (Eqs. (6.16)-(6.19)), namely, short circuit mechanical response,

electromechanical response, free electrical impedance and open circuit

voltage response around resonance. Fitting all functions simultaneously

should provide the best possible estimate for tanδ .


203


0

tipm

F

wd

V=

= 1ef

eb

ZCω

=

oc m m

o eb

V M da C

=

( )21

2msn m

Cf Mπ

=

mdebC

mM

, , tannf ζ δ

msC

0f → 0f →

0f →

rf f→

rf f→


3 ef ebF Z Cω=02

tip

F

m

wV

Fd

==10

tip

base V

wF

w=

= 4

oc

o

m m

eb

VaF M dC

=

Figure 6-22: Flowchart for LEM parameter extraction implementing Method 3.

Since Steps 1-3 essentially remain the same even in this method, they are not repeated. In

step 4, all four measured responses are non-dimensionalized and simultaneously fitted to the four

204

fitting functions listed below in Eqs. (6.16)-(6.19). The fit are generated such that the overall rms

error is minimizing resulting in the extraction of nf , ζ and tanδ .

22

1 20

,

1 2

tip n

base V

n n

wF

wj

ωω

ω ωζω ω

=

=⎛ ⎞

− +⎜ ⎟⎝ ⎠

(6.16)

02 2

1 ,

1 2

tip

F

m

n n

wV

Fd

jω ωζω ω

== =⎛ ⎞

− +⎜ ⎟⎝ ⎠

(6.17)

22 2

31 2

1 1 2tan n n

ef ebF Z Cd d

ω ωζδ ω ω

ω

⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟− +⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎣ ⎦= =+

(6.18)

and

41 2

1tan ,

oc

o

m m

eb

VaF M d d dC

δ= =+

(6.19)

where 1d and 2d are used to simplify the above expressions and are represented as

22

1

222

2 2

21tan

1 1 2tan tan

n n

m

n n ms eb

d

ddC C

ω ζ ωω δ ω

ω ωζδ ω ω δ

⎡ ⎤⎛ ⎞⎢ ⎥= − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞= − + +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

(6.20)

In 2d , msC is expressed in terms of mM and nf . Again, nf is assumed to be an accurate

estimate and uncertainties are obtained only for ζ and tanδ using the same process described

205

in Method 1. After extracting mM and nf , the compliance, msC is obtained using Eq.(6.12). The

fitted plots around resonance for all functions are shown below for PZT-EH-02.

Figure 6-23: Comparison between experiment and LEM based curve fit for short circuit mechanical and electromechanical response around resonance.

Figure 6-24: Experimental data and curve fits for open circuit voltage response compared around resonance.

The final extracted parameters for PZT-EH-02 obtained with method 3 are listed in the

following Table 6-6.

206


md 73.24 10 m V−× 3 %

nf 503.01 Hz ζ 43.79 10−× 40.98 10−×

ebC 97.51 10 F−× 1 %<

mM 62.34 10 kg−× 8 %

msC 24.28 10 m N−× 8 % tanδ 0.41 0.12

It was decided to adopt method 3 to extract LEM parameters for all tested devices as it

gave the best fit with reasonable parameter estimates. Using Method 3, fits were generated for all

experimental characterization curves and are presented in the next section.

Results and Discussion

The experimental response curves generated for each device are simultaneously compared

with the model predictions in the figures. In addition, the output voltage and power measured at

resonance across different resistive loads are compared with the model. The experimental setups

and data acquisition parameters for each characterization were described previously in Chapter 5.

PZT-EH-09

Using Method 3, the LEM parameters that describe the device behavior were first

extracted and are listed in Table 6-7 for PZT-EH-09-01 and PZT-EH-09-02.

207

Table 6-7: LEM parameters extracted for PZT-EH-09-01. PZT-EH-09-01 PZT-EH-09-02 LEM parameter Estimated value Uncertainty Estimated value Uncertainty

md 72.91 10 m V−× 4 % 73.24 10 m V−× 3 %

nf 481.04 Hz 503.01 Hz ζ 42.81 10−× 41.24 10−× 43.79 10−× 40.98 10−×

ebC 97.56 10 F−× 1 %< 97.51 10 F−× 1 %<

mM 72.90 10 kg−× 20 % 62.34 10 kg−× 8 %

msC 0.38m N 20 % 24.28 10 m N−× 8 % tanδ 0.47 0.24 0.41 0.12 These two devices belong to the same design and were characterized identically with all

three PZT layers connected in parallel and hence should ideally perform similar. The plots that

show the comparison between measured and model predictions are shown in Figure 6-25 and

Figure 6-26 for the two devices.

208

Figure 6-25: Comparison between model and experiments for PZT-EH-09-01. A) Short circuit

mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive loads at resonance

209

Figure 6-26: Comparison between model and experiments for PZT-EH-09-02. A) Short circuit mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive loads at resonance

210

Next, the different characterization plots compared with fitted models for PZT-EH-09-03

and PZT-EH-09-04 are shown in Figure 6-27 and Figure 6-28.


mechanical response B) Electromechanical response C) Free electrical impedance response D) Open circuit voltage response E) Normalized output voltage and power across resistive loads at resonance.

211



212

The extracted LEM parameters for these devices are shown in Table 6-8.

Table 6-8: Extracted LEM parameters for PZT-EH-09-03. LEM parameter Estimated value Uncertainty Estimated value Uncertainty

md 71.13 10 m V−× 5 % 87.60 10 m V−× 2 %

nf 477.99 Hz 473.76 Hz ζ 43.29 10−× 41.22 10−× 413.33 10−× 41.87 10−×

ebC 92.52 10 F−× 1 %< 92.36 10 F−× 3 %

mM 61.65 10 kg−× 14 % 61.92 10 kg−× 25 %

msC 26.72 10 m N−× 14 % 25.89 10 m N−× 25 % tanδ 21.48 10−× 20.62 10−× 21.50 10−× 20.57 10−×

PZT-EH-07

Next, similar results were obtained for PZT-EH-07-02 and PZT-EH-07-03 and are shown

in Figure 6-29 and Figure 6-30. The extracted LEM parameters for both devices are listed below

in Table 6-9.

Table 6-9: LEM parameters extracted for PZT-EH-07-02. LEM parameter Estimated value Uncertainty Estimated value Uncertainty

md 61.25 10 m V−× 8 % 61.66 10 m V−× 2 %

nf 126.60 Hz 128.85 Hz ζ 411.63 10−× 44.19 10−× 45.44 10−× 40.22 10−×

ebC 912.19 10 F−× 1 %< 912.41 10 F−× 1 %<

mM 64.68 10 kg−× 11 % 64.72 10 kg−× 17 %

msC 233.76 10 m N−× 11 % 232.30 10 m N−× 17 % tanδ 21.94 10−× 20.78 10−× 20.99 10−× 20.25 10−×

213



214



215

Summary and Discussion of Results

All the extracted lumped element parameters using detailed experimental characterization

are summarized and discussed here. Table 6-10: lists the various lumped elements extracted for

PZT-EH-07-2 and PZT-EH-07-3 and compares them with those calculated from the theoretical

model described in Chapter 2. The short circuit mechanical parameters such as mM , msC and nF

match very well with theory, thereby validating the beam model for current designs.

Furthermore, the electrical parameters, ebC and tanδ also match well with the model as

indicated in Table 6-10. However, the effective piezoelectric coefficient md is over predicted by

the model, and the experimentally measured values are considerably smaller. The results are

discussed in more detail later in this section.

Table 6-10: Comparison between theory and experiments for PZT-EH-07.

* assumed values in the model

Table 6-11 summarizes the lumped element parameters extracted for individual PZT-EH-

09 devices. It should be noted that the modeling procedure describe in 2.1 involves a single

cantilever beam structures with a proof mass at one end. However, PZT-EH-09 consists of 3 PZT

cantilever beams attached to a single proof mass. In Table 3-10, which lists the predicted lumped

PZT-EH-07- Theory 02 03

( ) mM mg 4.71 4.68 4.72

( ) msC m N 0.3371 0.3376 0.3230

ζ 0.01* 0.0012 0.0005

( ) nf Hz 126.3 126.6 128.8

( ) md m Vμ -1.91 -1.25 -1.66

( ) N Vφ μ 5.66 3.70 5.14 2κ 9.19 X 10-4 3.79 X 10-4 6.87 X 10-4

( ) ebC nF 11.84 12.19 12.41

tanδ 0.02* 0.0194 0.0099

216

element parameters for all designs, the values are indicated for single cantilever configuration.

So, if the design has multiple beams connected to a single proof mass, the proof mass is divided

equally between the number of cantilevers along its width. This essentially enforces the

condition that the beams are mechanically connected in parallel, an assumption that holds true

for small deflections. Therefore, the overall mass of the device is approximately three times the

mass of each cantilever configuration. Consequently, the masses of PZT-EH-09 devices

measured match well with the model prediction if the cantilever mass is multiplied by three due

to multiple beams.

Similar reasoning implies that the compliance of the single cantilever beam is thrice the

overall compliance of the device which has three beams. In such a case, the short circuit

compliance values measured for the devices match with the predicted compliance. However, the

natural frequencies remain the same and are close to theory.

Table 6-11: Comparison between theory and experiments for PZT-EH-09 devices.

* assumed values in the model

PZT-EH-09-01 and PZT-EH-09-02 had all the PZT beams connected in parallel which

would effectively add the individual capacitances. The extracted capacitances for these devices

PZT-EH-09- Theory 01 02 03 04

( ) mM mg 1.92 0.29 2.34 1.65 1.92

( ) msC m N 0.053 0.380 0.043 0.067 0.059

ζ 0.01* 0.0003 0.0004 0.0003 0.0013

( ) nf Hz 501.5 481.0 503.0 477.9 473.7

( ) md m Vμ -0.65 -0.29 -0.32 -0.11 -0.08

( ) N Vφ μ 4.15 0.76 7.47 0.72 1.36 2κ 1.4 X 10-3 2.93 X 10-5 3.18 X 10-4 1.24 X 10-4 0.46 X 10-4

( ) ebC nF 7.71 7.56 7.51 2.52 2.36

tanδ 0.02* 0.47 0.41 0.0148 0.015

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match well with the theoretical model. Alternatively, PZT-EH-09-03 and PZT-EH-09-04 had just

one PZT beam connected for external access and therefore, the capacitances are approximately

one-third the total capacitance.

From Table 6-10 and Table 6-11, it can be noted that although the mechanical LEM

parameters are reasonably close to the model, the mechanical damping measured is significantly

lower than assumed. In the LEM, an assumed value 0.01ζ = was used based on the measured

damping for meso-scale devices. However, the MEMS devices produce a much higher quality

factor ( )Q as confirmed by the low 'sζ (Candler et al, 2003). The quality factor and the

damping ratio are related as 1 2Q ζ= . So, using the various dissipation mechanisms studied in

Appendix B, the corresponding damping was obtained for the two distinct MEMS devices,

namely, PZT-EH-07 and PZT-EH-09. These mechanisms are investigated in some detail to

obtain empirical relations. It should be noted here that there are possibly many more

uninvestigated mechanisms that contribute to the overall mechanical damping in the device.

Only the most common ones such as air damping, thermoelastic damping, surface losses etc have

been included here. In addition, many different models for these individual losses exist in

literature. However, simple and generalized expressions were reported in this dissertation to

basically understand how these loss mechanisms scale to micro devices and decipher the

importance of each of these mechanisms in overall damping. This will help design and fabricate

these devices accordingly depending on the Q requirements of the device. In some cases, it may

be necessary to opt for a lower Q to expand the bandwidth to be able to operate over a wider

frequency range. Higher Q has an inherent disadvantage in resonant energy harvesting because a

minor change in either the source vibration frequency or the device resonance will result in a

large drop in power generation.

218

The expressions used for these loss mechanisms are greatly simplified and in general, are

applicable to cantilever beams with rectangular cross-sections. The PZT MEMS devices

compose of a beam section in addition to a proof mass at one end that complicates the total

effective dimensions. For this analysis, the total length of the device was used whereas the width

and thickness correspond to the dimensions of the silicon layer in the beam section. Hence, the

width and the thickness of the proof mass are neglected, which may result in inaccurate damping

estimates. The dissipation due to air flow in the viscous region (up to atmospheric pressure) is

derived by solving the flow past a string of spheres (Appendix B). Usually, the cantilever beam

is modeled using a string of spheres with an equivalent radius that is obtained by fitting

experimental data. Here, the radius of the sphere is based on the simplest approximation, the half

width of the beam. The support loss is smaller compared to the other losses primarily due to the

high length to thickness ratio. Again, the thickness of the beam is used here and not that of the

proof mass. It should also be noted that the surface and volume losses are considerable. The

surface loss primarily occurs due to thin films on a substrate, for e.g., PZT. However since all the

thermal properties such as conductivity and linear expansion coefficient are unknown for PZT,

the thickness of the beam was used which results in a fairly smaller Q. Therefore, the surface

dissipation is not particularly useful in understanding the physics of the damping in the device.

Furthermore, these relations are purely empirical and may not be applicable in that form to these

PZT MEMS devices. Hence, another Q was calculated using only three mechanisms that could

be important for the test structures, namely, dissipation due to air flow, support and thermoelastic

mechanism. These values for the two devices are also presented in the last rows of Table 6-12

using effQ and the corresponding damping ratio effζ .

219

The quality factors for different loss mechanisms are calculated using the empirical

relations in Appendix B for the two devices and presented in Table 6-12. In addition, the

equations used specifically to calculate the individual quality factors are listed in the table.

Table 6-12: Quality factors for PZT MEMS devices. Quality factor PZT-EH-07 PZT-EH-09 Equation # Air flow dissipation ( )airQ 536 31.04 10× B.13

Support dissipation ( )supQ 65.71 10× 61.62 10× B.15

Surface dissipation ( )surfQ 31.02 10× 31.01 10× B.16

Volume dissipation ( )volQ 36.12 10× 36.12 10× B.17

Thermoelastic dissipation ( )temQ 45.62 10× 43.42 10× B.22

Effective quality factor ( )totQ 330 466 B.5

Effective quality factor ( )effQ * 531 31.01 10× B.5

Effective damping ratio ( )effζ * 49.41 10−× 44.96 10−× B.1 * includes only three damping mechanisms

The newly calculated damping ratio is of the same order of magnitude as the extracted

value from experiments. Further experiments and a more detailed investigation of the dissipation

mechanisms is required to better understand and accurately predict the resonant behavior of these

devices. This exercise also confirms that the assumed value for ζ does not apply for these

devices. From the calculated quality factors, it appears that the air flow damping is the dominant

mechanism followed by thermoelastic damping. The air damping can be considerably minimized

by performing the characterization under vacuum in which case, the overall material damping

will approach the thermoelastic limit as described in Candler et al. (2003).

Next, the effective piezoelectric coefficients do not match well with theory. The effective

piezoelectric coefficient ( )md is extracted using the low frequency electromechanical response.

A very small amplitude voltage signal ( )0.1 V≤ was used to electrically excite the beam and

measure its response from dc to beyond resonance. Smaller voltage inputs were applied to

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prevent excessive deflections at resonance and, hence, risk damaging the devices. As a result, the

deflections at low frequency are significantly lower and the measured coherence was poor. For

better estimates of md , it is advisable to apply single frequency voltage inputs to the PZT and

measure deflection. Higher voltages can then be applied to obtain clean measurable deflections

with better coherence and therefore, more accurate values for md . Future work will concentrate

on more detailed characterization at low frequencies. In addition, there exists some PZT on the

clamp with electrodes that is not accounted for in the model as evident in Figure 4.16 and Figure

4.17. This additional parasitic capacitance was calculated to be significant, 32% for PZT-EH-07

and 23% for PZT-EH-09. These values are based on the added area of the electrode and PZT

whose dimensions were obtained from the design layouts. It is possible that the added PZT area

affects the device behavior in general, even for voltage and power generation. The parasitic

capacitance is smaller for the larger devices ( )10 %< and will eliminate its affect on the

deflection. This may still not explain the difference between the measured data and model.

Characterization of many devices of the same type is required to understand the

electromechanical behavior and obtain better repeatability of measurements between devices.

The extracted coupling factor, 2κ is dependent on msC , md and efC . The excessive

uncertainties in md and msC and the additional parasitic capacitances may have resulted in the

large differences. Factors such as residual stresses, non-uniformity in piezoelectric properties on

the wafer and further poling of the PZT may play an important role in finally determining the

actual piezoelectric parameters. Furthermore, the uncertainties associated with the fabrication

process for each batch of devices coupled with the dimensional tolerances of the structures need

to be considered for the differences in measurements.

221

The output power results are also included in the plots compared with the predicted

values using the fitted model. Although the model fitted the various individual frequency

responses well, the output power at resonance does not match well with model. Although some

explanation can be provided for these differences, they may not still compensate for the large

discrepancy. First, the extracted damping ratio and the resonant frequency using the model were

obtained from the least squares fits. These extracted frequencies do not match the resonant

frequencies where the data was obtained. The model predictions were based on the extracted

LEM parameters. Additionally due to very high 'Q s observed, the frequency resolution may

have to be improved considerably to excite the device at its true resonance to measure better

results. All these observations were drawn after the experiments were conducted and therefore

the option of redoing the measurements was not explored. This will have to be considered in

future work to obtain better match between model and experiments.

PZT-EH-09-03 and PZT-EH-09-04 appeared to significantly underachieve in its

piezoelectric and dielectric properties that led to significantly lower output voltages and power

compared to similar candidate devices. These results are evident in all the frequency response

plots presented before in Figure 6-25 - Figure 6-30. The wirebonding process that employs an

ultrasonic impact technique was used to provide electrical contacts for all devices. During this

process, one of the beams in PZT-EH-09-04 was damaged near its clamp that may have led to

some deterioration in its overall performance.

It was observed that method 3 (and other methods too) predict tanδ as expected (<0.02)

for 4 out of 6 devices. Only PZT-EH-09-01 and PZT-EH-09-02 give unreasonably high values

(0.41, 0.47) for tanδ . These devices had the three PZT layers connected in parallel, while the

other 4 devices that matched theory more closely were tested only with one PZT layer connected.

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This difference in electrical connection between the devices needs to be explored further to

understand if the parallel excitation adds impedance to change the response.

Another variation of method 3 was implemented to observe if it improved the parameter

extraction for tanδ . After the first set of parameters were estimated using method 3, a new ebC

was calculated using the low frequency fit using the extracted tanδ . This step resulted in a

lower ebC that was then used to estimate the rest of the parameters using an iterative method.

The idea of this procedure was to determine if an iterative procedure would obtained an

improved value for tanδ . However, this new approach resulted in the same value of tanδ as

before. The reason behind this is because mM is extracted from the low frequency voltage

response, and a lower ebC results in a lower mM , while md remains constant. This maintains the

ratio constant (Eq. (6.10)), and so the change in the value for ebC is compensated here and is not

propagated further in the curvefits. However, if ebC is extracted simultaneously with other

parameters, the problem becomes a complete parametric curve fit. The disadvantage with this

approach is that the low frequency behavior of the device that can be reduced to simple

analytical expressions is not leveraged to extract parameters individually.

To summarize, various devices of designs PZT-EH-07 and PZT-EH-09 were completely

characterized to obtain the LEM parameters and subsequently measure the output voltage and

power generated by each device operating at resonance. The next chapter essentially discusses

the conclusions of the present work involving the overall modeling, fabrication, and

characterization of MEMS PZT micro power generators. A discussion on the prediction of the

overall behavior with the LEM is provided, which explores some of the limitations in the model

in terms of predicting the behavior of micro devices, specifically at resonance. In addition, some

223

key aspects for improvement in design, modeling and fabrication are presented along with

directions and scope for future work.

224

CHAPTER 7 CONCLUSIONS AND FUTURE WORK

This chapter presents all the conclusions drawn from the research conducted in this

dissertation. Furthermore, suggestions and directions for future work are provided following the

concluding remarks.

Conclusions

A MEMS-based piezoelectric energy harvesting concept is developed and implemented in

this dissertation. First, a detailed literature review was conducted to study various energy

harvesting mechanisms and the current state of technology in the field. A device consisting of a

cantilever piezoelectric composite beam with a proof mass was chosen as the candidate energy

harvester design for this work. A detailed analytical model was developed using the lumped

element modeling technique to represent the structure in the form of an equivalent circuit. The

piezoelectric composite beam was modeled using Euler-Bernoulli beam theory that provided an

analytical piecewise solution for the deflection modeshape. The assumptions governing the

analytical solution are that the deflections are small compared to the beam length. In addition,

shear and rotary inertia affects are neglected. Furthermore, the model does not incorporate any

residual stress affects that result in an initial curvature of the device. The resulting modeshape

was then utilized by lumping the resulting potential and kinetic energies to obtain the mechanical

lumped element parameters. An effective transduction factor and piezoelectric coefficient for the

device was also obtained that relates the conversion of energy from the mechanical to electrical

domain and vice versa.

225

After the lumped element circuit parameters were obtained analytically, the LEM was

extended to investigate the dynamic behavior of the device and its overall response for voltage

and power output. The developed model was verified using meso-scale composite beams

comprised of PSI PZT-5A patches bonded to an Al-6061 beam. . The experimental data matched

favorably with the model predictions. Furthermore, the model was validated using FEM for static

LEM parameters.

Next, a dimensional analysis was carried out for the analytical LEM where all the circuit

parameters and response functions were expressed in terms of nondimensional Π groups. This

was done because all the response parameters are analytical but unwieldy functions of geometric

dimensions and material properties. Therefore, a scaling analysis was performed using the

nondimensional LEM to determine micro-scale effects on overall device performance. In the

scaling theory, the damping ratio was assumed to be constant as the device is scaled. However,

experiments and subsequent literature search confirmed that this assumption is invalid especially

at MEMS scale involving different materials. Most meso-scale devices use commercially

available bulk PZT ceramics attached to shims made of Aluminum, Brass, Stainless steel etc.

However, in MEMS, most devices are based on Silicon and sol gel PZT which has inferior

piezoelectric properties such as lower 31d and ε .

The developed LEM forms the basis for efficiently designing such piezoelectric composite

beam energy harvesters to maximize power output. In the dissertation, a parametric search

optimizations strategy was adopted to realize various designs intended to harvest energy from

specific harmonic vibrations with prescribed base frequency and acceleration levels.

Following the design stage, these MEMS devices were fabricated using conventional

surface and bulk microfabrication techniques. The fabrication stage involved developing a

226

process flow and mask sets that were used to realize these devices. The PZT deposition and

patterning was performed at ARL using a sol-gel process. The device wafers were initially

characterized for their piezoelectric and dielectric properties. Tests were also conducted to

polarize the PZT layer for different temperatures and voltages. However, not much improvement

in remnant polarization was observed from the original values reported by ARL.

Two different packages, namely open and vacuum, were implemented for the devices, after

fabrication and release from the process wafer. However, only the open packages were tested in

the experiments described in Chapter 6. The vacuum packages were primarily designed to

investigate the effect of air damping. Since the quality factors measured as such in the open

packages were very large, the vacuum packages were not tested. However, further experiments

are required to obtain quantitative estimates of the effects of air damping and mass loading.

A detailed experimental characterization was carried out on several packaged devices to

determine their frequency response for various operating conditions. A lumped element

parameter extraction was carried out to obtain the full set of parameters that complete the

equivalent circuit. Various extraction algorithms were investigated on a test device and the

method that provide reasonable values for these parameters was adopted. These experimentally

estimated parameters were compared with those obtained using the theoretical analytical model

using the base dimensions and properties. While the mechanical parameters matched favorably,

especially for the devices that were tested with a single PZT beam configuration, the results did

not match well when multiple PZT layers were connected in parallel.

To summarize the overall characterization results, the normalized voltage and power

generated for specific input accelerations for all the tested devices are shown in the following

227

table. The output voltage and power correspond to the optimal load as indicated in Table 7-1. As

expected, the normalized output power was found to be higher for devices with higher Q .

Table 7-1: Normalized voltage and power outputs for all the tested MEMS PZT energy harvesters.

Input accn. (m/s2)

Frequency (Hz)

Optimal load (kΩ)

Voltage (mV/m/s2)

Power (nW/(m/s2)2)

PZT-EH-07- 02 0.28 127 100 65 42 03 0.20 129 100 313 980 PZT-EH-09- 01 1.51 481 50 41 34 02 0.77 503 50 53 57 03 0.28 478 100 44 19 04 0.86 474 200 5 0.14

As confirmed with the experimental results on these microfabricated devices, the quality

factor is higher than anticipated during the design stage. Therefore, the measured deflections for

each of these devices were used to calculate the acceleration limits for onset of non-linearity,

fracture and endurance limits. The measured normalized tip deflections of each of the devices

with input base acceleration are listed in Table 7-2. First, the moment at the clamp where the

strain is a maximum is calculated for the specific device using the static analytical model.

Assuming that the strain is proportional to the tip deflection, it is scaled linearly with the

measured tip deflection at resonance, after which the corresponding bending stress at the clamp

is estimated. These stresses are also listed in the table. The fracture stress for silicon used in the

design procedure (Section 3.3.1) was 10 % of its yield strength, 7 GPa . This assumption is

valid as Baghdan and Sharpe (2002) measured the initial fracture strength of silicon for cyclic

tensile testing as 1.1 GPa while the endurance limit for 810 cycles was limited at 0.75 GPa .

Using the initial fracture stress limit and endurance limit, the acceleration limits are calculated

and listed in the following table. In addition, the limits for the onset of nonlinearity ( 2 % )

228

deviation are also included. As expected, the linear limit is reached before the endurance limit of

the devices.

Table 7-2: Acceleration limits for MEMS PZT energy harvesters based on different stress states. Acceleration limit based on stress due to

( )2m s PZT-EH

Normalized tip deflection ( )2m m s

Normalized bending stress ( )2MPa m s Endurance

limit ( )0.75 GPa

Fracture limit

( )1.1 GPa

Linear model limit

( )0.4tip sw L= 09-01 41.627 10−× 221 3.4 5.0 1.2 09-02 41.326 10−× 180 4.2 6.1 1.5 09-03 41.543 10−× 209 3.6 5.3 1.3 09-04 53.71 10−× 50 15.0 22 5.4 07-02 42.95 10−× 143 5.2 7.7 1.4 07-03 48.00 10−× 389 1.9 2.8 0.5

Finally, the overall power densities of the fabricated and tested MEMS devices are listed in

Table 7-3. The power density can be improved with some optimization in future generation

devices. However, the poor electromechanical coupling restricts the overall power generation

compared to meso-scale devices. The power densities of some of the reported energy harvesters

in literature and commercially available energy harvesters are also listed in the table for

comparison. It should be noted here that the power density values are mere ratios of power

generated per unit volume of the device. Since no universal metric is available that efficiently

compares the performance of energy harvesters, the acceleration values and operating

frequencies are also listed. As expected, higher acceleration produces larger power densities

since power scales as the square of base acceleration. In addition, frequency is also important

because, typically, lower frequency acceleration produces higher deflection as opposed to higher

frequency acceleration with the same magnitude.

229

Table 7-3: Power densities of reported and commercially available energy harvesters. Acceleration

(g) Frequency

(Hz) Power Density

(μW/cm3) Type Source

10.9 13900 37000 a Piezoelectric Jeon et al. 2005 0.25 120 70 ” Roundy et al 2003 0.25 120 375 ” Roundy et al 2004

0.24 a 80.1 16.8 ” Glynne-Jones et al 2001 0.24 120 1.1 ” Mide Inc 12 120 500 ” Mide Inc 0.1 57 16 a ” Microstrain Inc 0.1 21 122 Electromagnetic Ferro Solutions Inc 0.1 120 49 ” Perpetuum Co. 0.1 60 98 ” Perpetuum Co. 10.4 322 2208 ” El Hami et al 0.28 127 0.3 Piezoelectric PZT-EH-07-02 0.20 129 3.2 ” PZT-EH-07-03 1.51 481 17 ” PZT-EH-09-01 0.77 503 7.6 ” PZT-EH-09-02 0.28 478 0.3 ” PZT-EH-09-03 0.86 474 0.02 ” PZT-EH-09-04

a estimated (Beeby et al, 2006)

One of the main contributions of this dissertation is the development of a complete static

analytical model of a piezoelectric cantilever composite beam that was validated using FEM and

experiments on candidate devices. The equivalent electromechanical model for a piezoelectric

energy harvester provides a design tool for specific applications to maximize power transfer and

enable complete circuit simulation with power processors. Another major contribution of this

research is the realization of a first generation fabrication of a MEMS PZT cantilever array for

vibrational energy harvesting. In addition, the complete design, fabrication and testing of a stand-

alone device is demonstrated for energy reclamation. Furthermore, the novel aspect of this

research is the ability to connect multiple PZT layers either in series or parallel for power

enhancement.

230

Future Work

This section provides some suggestions for future work and for potential improvements in

piezoelectric cantilever beam energy harvesting, specifically in the areas of device modeling,

fabrication process, packaging and characterization.

Although the developed model worked well for predicting device response at the meso-

scale, it failed to match well for the MEMS devices. There are considerable fundamental

differences between the devices operating at these different scales. The fabrication process

involved for MEMS devices is completely different and invalidates some of the assumptions

incorporated in the model. One main factor is the inclusion of residual stresses in the device that

occur during fabrication. The residual stress gives raise to an initial curved shape of the device

and potentially places the device in the non-linear regime. In such a case, the model cannot be

directly applied. Hence, as a first step, residual stress should be considered in the model and a

modified analytical solution for the composite beam should be obtained. The resulting

modeshape may provide a better estimate for the LEM parameters. Furthermore, a plate theory

may have to be carried out as some of these devices perform better with higher PZT area in

which case, the length and width of the beam will approach a plate shape.

It is also assumed in the model that the width of the PZT and substrate are the same, while

a significant clearance ( )5 10 mμ−∼ is provided along the width in MEMS devices during

fabrication to prevent any misalignment and potential shorting of electrodes. Furthermore, the

model incorporates a two layer model neglecting the effect of the electrodes, whereas many

layers are present in MEMS fabrication that may have to be considered in the model. In addition,

the electrodes and PZT layer extend on to the clamp of the device for bond pads. This will lead

231

to a parasitic capacitance that could alter the electrical performance and hence, the predictability

of the model.

Ideally, more detailed experiments need to be conducted for the parallel PZT

configurations to investigate the relatively higher observed values for tanδ compared to the

single PZT beams. More experiments need to be conducted for each of the designs to verify the

repeatability of their performance and the extracted LEM parameters.

It was observed in the experiments that the electromechanical response consistently

measured a lower Q than a pure short-circuit mechanical response. However, the LEM fails to

capture this difference and provides only a mechanical ζ . It is proposed that there are additional

electrical losses that need to be modeled.

One of the packaged devices was damaged during operation. The package is generally

mounted on a vibrating shaker that, when turned on, provided an initial impulse. This impulse

broke the cantilever structure near its clamp. For later experiments, the packaged was mounted

after the shaker was setup and operational. However, to maintain the robustness of the device

under such operating conditions, mechanical stops need to be provided either during fabrication

or in packaging. This will prevent from excessive vibration of the beam when subject to sudden

impacts etc. Mechanical stops in the form of an outer frame may be provided to prevent vibration

due to higher order modes such as twisting motion. This may require that the device be in a

vacuum package which will benefit in eliminating air flow damping. If the gap between the

mechanical stop and the device is small, squeeze-film damping may become significant and

needs to be considered for overall damping in the system.

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Second Generation Design Procedure

For the second generation PZT device, a suggested design procedure that may be adopted

is listed below. This procedure is similar to the one discussed earlier in Section 3.3.1. A more

robust design strategy would involve a global deterministic or uncertainty based optimization

procedure (Gurav et al. 2004) that was not investigated in this dissertation. First, the overall

available area of the dye is denoted as A ( )for eg., 1 1 cm cm× . This total area is divided into

various cells comprised of individual chips.

1. The first step involves deciding the area of each cell.

a. Smaller cells for higher frequencies

b. Larger cells for lower frequencies

2. Assume the number of cantilevers per proof mass ( )for eg., 1..5n =

3. Assume a size for the clamp, for example, between 0.5 mm and 1 mm , depending on the

overall length of the composite beam.

4. After providing enough clearance around the edges,

a. choose the width of the proof mass, ( )w

b. choose the overall length of the structure, ( ) sL l clamp+ +

5. After establishing n and w , the width of the beam can be calculated ( ) s pwb and b n= .

6. Use nominal dimension ratios s sL b (5, 10) and s sL t (50, 100, 200) to act as base

dimensions for parametric search strategy.

7. To obtain the required natural frequency, very the lengths and thicknesses of the beam

and proof mass by trial and error.

233

a. Prior knowledge from scaling theory will be useful to easily obtain required

dimensions

8. For each of the designs, perform parametric analysis to maximize the output power of the

device. An alternate and more robust approach is to perform a constrained global

optimization with output power as the cost function.

a. For each of the designs, it has to be ensured that the device failure limits such as

endurance and fracture limits are not exceeded.

b. In addition, the individual damping mechanisms should be estimated for each of

the designs. This will help design the device for higher Q (lower bandwidth) or

higher bandwidth (lower Q ), depending on the application.

Electromechanical Conversion Metrics

To better understand the output power and electromechanical efficiency metrics of the PZT

energy harvesters, response expressions were derived at resonance to observe their behavior and

dependence on various device parameters. From the electromechanical LEM, the expressions for

output power and efficiency were obtained analytically as functions of the circuit parameters.

Since the analytical expressions are complicated in their general form, the following assumptions

are made to simplify the expressions and to present some physical insight. First, the frequency of

operation is assumed to be at its short circuit natural frequency ( )nf . It should be noted here that

the resonant frequency of the device varies between its short circuit and open circuit limit that is

determined by its effective coupling factor. Furthermore, the resonant frequency depends on the

external load that is connected to the device. However, for a poorly coupled device where the

difference between the short circuit and open circuit natural frequencies is negligible, for

example, in MEMS devices, the natural frequency assumption is reasonable.

234

Another assumption adopted in this analysis was that the damping due to dielectric loss ( )eR

is negligible and is therefore not considered. Since the LEM employs a parallel combination of

capacitor and resistor to model the PZT, the analytical expressions become complicated to

derive. In addition, the dielectric loss is higher compared to the mechanical damping in general

and is therefore neglected in most piezoelectric device models. Furthermore, at resonance the

mechanical mass and compliance cancel each other leaving only the damping in the system.

Consequently, the mechanical ( )ζ that is assumed in the model can be used to represent the

overall damping in the device including the electrical losses.

Using the above assumptions and using the Thevenin equivalent model discussed in Section

3.1, the optimal load for maximum power transfer occurs when the Thevenin impedance is

matched. Here, for a purely resistive load, it is equal to the magnitude of the impedance resulting

in the expression

( )24 2 2

1 2 .4 1

optn ef

RC

ζω κ ζ κ

=+ −

(7.1)

It is evident from Eq. (7.1) that the load is dominated by the capacitive behavior of the

piezoelectric material, but is dependent on the coupling and damping in the device. The output

power across the load is obtained as the ratio between the Thevenin voltage and the overall

impedance. These formulas were presented earlier in Section 3.1.

( )

2 2

22 4 2 2.

4 4 1

m oL

n

M aP κ

ζω κ κ ζ κ=

⎡ ⎤+ + −⎢ ⎥⎣ ⎦

(7.2)

The output power depends on the input acceleration, the overall mass of the device, device

coupling and damping in the system. However, the acceleration is generally fixed for a specific

235

application. The relative motion is dominated by the proof mass, which is amplified at

resonance. So, increasing the mass is restricted by limits of device failure due to stress at the

clamp during vibration. The output power ideally reaches its maximum when the effective device

coupling approaches unity and the damping in the system becomes zero. Since that is practically

impossible, the goal in the design should be to minimize the damping and maximize the device

coupling independently. Alternatively, the input power to the beam is calculated from the

product of the effective force and relative tip velocity, given as

( )

( ) ( )

24 2 22

122 24 2 2 2 4 2 2

8 1.

2 24 1 4 1

m oin

n

M aPκ ζ κ

ζωκ ζ κ κ κ ζ κ

+ −=

⎡ ⎤⎛ ⎞+ − + + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(7.3)

The device efficiency that represents the power conversion from the beam to the resistive load is

obtained from the ratio between Eqs. (7.2) and (7.3) and is expressed as

( )

( ) ( )

12

24 2 22

2 24 2 2 2 4 2 2

4 1.

2 8 1 4 1

κ ζ κκηκ ζ κ κ κ ζ κ

⎡ ⎤⎢ ⎥+ −⎢ ⎥=

⎡ ⎤⎢ ⎥+ − + + −⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

(7.4)

From Eq. (7.4), it can be noted that the device efficiency is only dependent on the effective

coupling factor and effective damping in the device. The device coupling factor is a function of

the material coupling coefficient and device geometry and properties. The functional dependence

of the device coupling with its dimensions and properties is not derived here due to the analytical

complexity of the expressions. In future design strategies, Eqs. (7.2) and (7.4) need to be

investigated and maximized for a given set of materials subject to the input conditions and

geometrical constraints.

236

Other potential future work involves further investigation of arrays of devices connected in

series/parallel to methodically study the power enhancement. An alternate application of this

device is as an accelerometer that can be studied using PZT-EH-06 that was designed to operate

beyond 1000 Hz. This will provide a reasonable operating range for the accelerometer and, if it

provides good performance, the design tool can be used to fabricate one for a larger range and

higher sensitivity.

237

APPENDIX A EULER-BERNOULLI BEAM ANALYSIS WITH VARIOUS BOUNDARY CONDITIONS

Euler Bernoulli Beam

The governing equation for a beam derived from Euler-Bernoulli theory is represented as

( )4

4 0w x

EIx

∂=

∂ (A.1)

Let us assume a general solution to be of the form

( ) 2 30 1 2 3w x a a x a x a x= + + + (A.2)

that describes the transverse deflection of a beam. 0 1 2, ,a a a and 3a are integration coefficients of

the modeshape. In this analytical exercise, beams with various boundary conditions are solved

for their exact deflection shapes, natural frequencies and maximum bending strain in the

following sections.

Cantilever Beam (Clamped-Free Condition)

Figure A-1: Schematic of a cantilever beam.

For a cantilever represented in Figure A-1, the following boundary conditions apply

238

( )( )

( )

( )

0

2

2

3

3

0 0

0

0

x

x L

x L

w

w xx

w xEI

x

w xEI P

x

=

=

=

=

∂=

∂

∂=

∂

∂=

∂

(A.3)

They enforce the deflection and slope at the clamp to zero. In addition, the moment at the tip is

zero and the vertical shear force represented by the 3rd derivative is the external load, P.

Applying the first 2 of the boundary conditions results in 0 1 0a a= = . When the next two

boundary conditions are applied, the rest of the coefficients can be calculated as

2 3; 2 6PL Pa aEI EI

= − = (A.4)

Substituting back for these coefficients results in the deflection modeshape as

( ) ( )2 3 2 32 6 6PL P Pw x x x x L xEI EI EI

= − + = − − (A.5)

From the above expression, the tip deflection is obtained by substituting ,x L= as

3

.3tipPLwEI

= − (A.6)

The effective mass and compliance using the deflection are obtained by calculating the

stored potential and kinetic energies in the beam and lumping them at the tip (described in detail

in Chapter 2). Using this method, the expressions for the compliance and mass of the beam are

3

3msLCEI

= (A.7)

and

239

33140m LM Lρ= (A.8)

It should be noted here that the effective mass is different from what is generally presented

in literature mainly because the model here is based on a static modeshape, while most

predictions of natural frequencies are based on the eigen modeshape. A static modeshape is used

here to be consistent with the actual modeling technique employed to represent the

electromechanical behavior of the piezoelectric composite beam (Chapter 2). From the mass and

compliance, the natural frequency is represented as

2

1 1 1 1 1402 2 11n

m ms L

EIfM C Lπ π ρ

= = (A.9)

The bending strain can be calculated from the deflection modeshape and is given as

( ) ( )2

2 00

xx

w x Pz z x Lx EI

ε=

=

∂= − = − −

∂ (A.10)

As indicated in the expression, the strain varies linearly along the length of the beam. In

addition, the bending strain is linearly increasing from zero at the neutral axis to the top and

bottom surface where it reaches the maximum (z is the vertical co-ordinate measured from the

neutral axis). Hence, the maximum stress and strain occurs at the clamp on the top or bottom

surface, when the bending moment is the highest and its value is

max 2PL tEI

ε = (A.11)

Clamped-Clamped Beam (Fixed-Fixed Condition)

Let us now analyze a beam with a different set of boundary conditions, clamped at both

ends as shown in Figure A-2.

240

Figure A-2: A schematic of clamped-clamped beam.

Drawing the free body diagram for the clamped-clamped beam and performing a force balance

results in (see Figure A-3)

Figure A-3: Free body iagram of a clamped-clamped beam.

In this case, let us assume the following boundary conditions that govern the beam for

0 .2Lx< <

241

( )( )

( )

( )

0

2

3

3

2

0 0

0

0

2

x

Lx

Lx

w

w xx

w xx

w x PEIx

=

=

=

=

∂=

∂

∂=

∂

∂=

∂

(A.12)

The above conditions mean that the beam is clamped at one end which enforces zero deflection

and slope. At ,2Lx = the slope is zero and the shear force is .2

P Imposing the first 2 of the

boundary conditions yields 0 1 0a a= = . The next conditions when applied result in the

coefficients as

2 3; 16 12

PL Pa aEI EI

= − = (A.13)


( ) ( )2 3 2 3 416 12 48

PL P Pw x x x x L xEI EI EI

= − + = − − (A.14)

The above deflection is valid for 0 2Lx< < and will be symmetrical about the center, 2

Lx = .

From the above expression, the maximum deflection occurs at the center is obtained by

substituting ,2Lx = as

3

.192cen

PLwEI

= − (A.15)

As described earlier, the effective mass and compliance using the deflection are obtained by

calculating the stored potential and kinetic energies in the beam and lumping them to the tip. It

should be noted here that the deflection shape represents only the half beam and therefore, the

242

energies are first obtained by integrating the deflection shape till ,2Lx = doubled to obtain the

total potential and kinetic energy in the beam. The modified equations for the energy are shown

in Eq.

( )

( )( )

222

20

22

0

22

2 .2

L

L

L

d w xEIPE dxdx

KE w x dxρ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

=

∫

∫

(A.16)

Lumping the energies at the tip results in the compliance and mass as

3

192msLC

EI= (A.17)

and

1335m LM Lρ= (A.18)

From the mass and compliance the natural frequency is represented as

2

1 1 10582 13n

L

EIfLπ ρ

= (A.19)


( ) ( )2

2 0, 20

48 x L

x

w x Pz z x Lx EI

ε=

=

∂= − = − −

∂ (A.20)

As indicated in the expression, the strain varies linearly along the length of the beam with its

maximum values occurring at the clamp and the center. In addition, the bending strain

approaches zero at ,4Lx = before it changes from compression to tension and vice-versa. The

maximum bending strain at the clamp or the center is

243

max 8 2PL tEI

ε = (A.21)

Pin-Pin Beam (Simply Supported)

Finally, let us analyze another set of boundary conditions for a beam, namely, pin-pin as shown

in Figure A-4.

Figure A-4:: Schematic of a pin-pin beam.

Drawing the free body diagram for the pin-pin beam and performing a equilibrium match results

in Figure A-5.

Figure A-5: Free body diagram for a simply supported beam.

In this case, lets assume the following boundary conditions that govern the beam for 0 2Lx< <

as

244

( )( )

( )

( )

2

20

2

3

3

2

0 0

0

0

2

x

Lx

Lx

w

w xEI

x

w xx

w x PEIx

=

=

=

=

∂=

∂

∂=

∂

∂=

∂

(A.22)

The above conditions mean that the deflection and moment for the beam at one end is zero. At

,2Lx = the slope is zero and the shear force is .2

P The main difference between a pin-pin and

a clamped-clamped condition is that one imposes a zero moment and the other imposes a zero

slope at the end. Imposing the first 2 of the boundary conditions yields 0 2 0a a= = . The next

conditions when applied result in the coefficients as

2

1 3; 16 12PL Pa a

EI EI= − = (A.23)


( ) ( )2

3 2 23 416 12 48PL P Pw x x x x L x

EI EI EI= − + = − − (A.24)

The above deflection is valid for 0 2

Lx< < and will be symmetrical about the center, .2Lx =

From the above expression, the maximum deflection occurs at the center is obtained by

substituting ,2Lx = as

3

.48cenPLw

EI= − (A.25)

245

As described earlier, the effective mass and compliance using the deflection are obtained by

calculating the stored potential and kinetic energies in the beam and lumping them to the tip,

which results in msC and mM as

3

48msLCEI

= (A.26)

and

1735m LM Lρ= (A.27)

From the mass and compliance the natural frequency is represented as

2

1 1 10542 17n

L

EIfLπ ρ

= (A.28)


( )2

2 22

2 LxLx

w x Pz z xx EI

ε=

=

∂= − = −

∂ (A.29)

As indicated in the expression, the strain varies linearly along the length of the beam starting

from zero at the clamp which was enforced using the boundary conditions. Hence, the maximum

stress and strain occurs at the center on the top or bottom surface, when the bending moment is

the highest and its value is

max 4 2PL tEI

ε = − (A.30)

The important parameters for all the various beams for an applied load, P are listed in Table A-1.

Similar analysis was carried out for a uniform loading condition on these beams. However, only

the results are summarized in Table A-2.

246

Table A-1: LEM parameters and bending strain for various beams subjected to a point load. Clamped-Free Pin-Pin Clamped-Clamped

( )w x

( )2 36

P x L xEI

− − ( )2 23 448

P x L xEI

− − ( )2 3 448

P x L xEI

− −

maxw 3

0.333 PLEI

− 3

0.021 PLEI

− 3

0.005 PLEI

−

msC

3

0.333 LEI

3

0.021 LEI

3

0.005 LEI

mM 0.236 LLρ 0.486 LLρ 0.371 LLρ

nf 2

10.568L

EIL ρ

2

11.582L

EIL ρ

2

13.619L

EIL ρ

( )xε

( )Pz x LEI

− − 2Pz xEI

− ( )48

Pz x LEI

− −

maxε 2

PL tEI

0.252

PL tEI

− 0.1252

PL tEI

As evident in the tables, a cantilever beam exhibits much higher compliance than beams bounded

on both ends. In addition, they possess a significantly lower natural frequency compared to the

other beams. Furthermore, a cantilever produces a higher bending strain for similar loading

condition making it an excellent choice for piezoelectric energy harvesting applications that

require higher strain to convert to voltage and lower frequencies to match ambient vibrations.

247

Table A-2: LEM parameters and bending strain for various beams subjected to uniform load. Clamped-Free Pin-Pin Clamped-Clamped

( )w x

( )2

2 24 624qx Lx x L

EI− − − ( )3 2 32

24qx L Lx xEI

− − + ( )2

2

24qx L x

EI− −

maxw 4

0.125 LqEI

− 4

0.013 LqEI

− 4

0.003 LqEI

−

msC

3

0.313 LEI

3

0.020 LEI

3

0.005 LEI

mM 0.257 LLρ 0.504 LLρ 0.406 LLρ

nf 2

10.562L

EIL ρ

2

11.572L

EIL ρ

2

13.573L

EIL ρ

( )xε

( )2

2qz x LEI

− − ( )2

qz x L xEI

− − ( )2 26 612

qz L Lx xEI

− +

maxε 2

0.52

L tqEI

− 2

0.1252

L tqEI

− 2

0.0832

L tqEI

248

APPENDIX B DISSIPATION MECHANISMS FOR A VIBRATING CANTILEVER BEAM

Materials can be broadly be classified into three categories based on how they respond to

external load. The first kind, “elastic” materials will completely recover the stored energy after

the load is removed and follow the Hooke’s law that relates stress and strain linearly with its

elastic modulus. The second kind called, “viscous” materials will lose the stored energy fully

once the load is removed. Here, the stress is related to the strain rate in terms of viscosity.

Finally, we have “viscoelastic” materials that lead to partial recovery of the stored energy and the

remaining is lost in the form of heat. To represent this behavior, the elastic modulus of the

material is expressed as a complex quantity, where the imaginary part corresponds to the lost

energy. This loss of energy is generally expressed as mechanical damping.

Introduction

Mechanical damping relates to the conversion of mechanical energy into heat. An

oscillating structure contains a combination of potential and kinetic energy that dissipates a part

of it during each cycle of motion. In this project, arrays of micromachined piezoelectric

cantilevers are being fabricated to harness electrical energy from source vibrations using a

mechano-electrical transduction mechanism by virtue of a piezoelectric layer. Since these

devices are resonant structures, extracting maximum energy at its resonance is critical for its

performance. However, at resonance, the performance of the device is greatly influenced by its

inherent mechanical damping related to the quality factor, Q. The output voltage and

consequently power in the device is directly related to the quality factor. Therefore, in the design

of MEMS devices, dissipation mechanisms may have detrimental effects on the quality factor.

249

Some of the main dissipation mechanisms that describe this loss of energy are surface and

volume losses, support losses, loss due to air flow (fluid damping), thermoelastic dissipation etc.

One of the major dissipation phenomena to consider in such micro-systems is the thermoelastic

damping which greatly influences the behavior of the vibrating micro-structures. So, a detailed

thermoelastic dissipation mechanism in cantilever beams has been studied. In addition, the

effects of supports and air damping have been analyzed and empirical relations for the individual

quality factors are presented. As part of future work, the model will enable a more efficient

design of resonant micro-structures

The overall mechanical damping is generally represented using any of the following

parameters, namely, effective quality factor ( )Q , damping ratio ( )ζ , loss factor ( )η , phase

angle between cyclic strain and stress ( )tanθ etc. All these parameters are inter-related using

the expression

3

1 1 1 22 tan

n

dB

f WQf W

πζ η θ

= = = = =Δ Δ

(B.1)

Here, nf and 3dBfΔ are the natural frequency and band-width corresponding to half power points

in the frequency spectrum.

Overall Mechanical Quality Factor

For a vibrating cantilever beam, the loss mechanisms can be broadly classified into two

types, namely, external and internal losses. The external losses include loss due to fluid damping

(air), radiation of the bending wave at the support (clamp). Losses such as thermoelastic loss and

surface loss due to material defects comprise the internal losses (Yang et al., 2002). The overall

quality factor is defined as

250

1

1 1n

i iQ Q=

= ∑ (B.2)

where Q is the total quality factor and iQ correspond to the quality factor of each loss

mechanism. For the analysis here, let us define the quality factor as

2 WQW

π=Δ

(B.3)

where, W is the stored energy in the beam and WΔ is the dissipated energy. If we include all

the dissipation mechanisms listed earlier, Eq. (B.3) can be written as

/

2TED support air fluid surface other

WQW W W W W

π=Δ + Δ + Δ + Δ + Δ

(B.4)

where, each WΔ represents an individual damping effect. Separating the above expression into

individual effects yields

/

/

12 2 2 2 2

1 1 1 1 1 1

support air fluid surface otherTED

TED support air fluid surface other

W W W WWQ W W W W W

Q Q Q Q Q Q

π π π π πΔ Δ Δ ΔΔ

= + + + +

= + + + + (B.5)

Here, the overall quality factor has the effect of all the individual quality factors connected in a

parallel impedance form. Therefore, the smallest quality factor of all dictates the overall

mechanical damping in the device.

Dissipation Mechanisms

Although there are various mechanisms by which energy is dissipated in a vibrating

structure (beam), some of the major phenomena are discussed here. In addition, a detailed

derivation is obtained for thermoelastic dissipation in a cantilever as it contributes a major

portion to the overall mechanical damping.

251

Airflow Damping

A simple schematic of the cantilever beam that is analyzed here is shown in Figure B-1. In

the figure, the dimensions are indicated and , ,E Iρ correspond to the elastic modulus, density

and moment of inertia of the homogenous beam.

Figure B-1: A simple schematic of the cantilever beam.

The drag force on a vibrating cantilever beam such as the one shown above due to fluid loading

is given as (Blom et. al, 1992)

( )1 2 1 2 1 2 where , are real constantsup j u uβ β β β β βω= + = − (B.6)

Consequently, the governing Euler-Bernoulli beam bending equation is modified into

( ) ( ) ( )4 21 2

4 2 0w x w x w x

EI A Lx L t tβ βρ ω

∂ ∂ ∂⎛ ⎞+ + + =⎜ ⎟⎝ ⎠∂ ∂ ∂

(B.7)

Here, ω is the angular frequency of vibration, A is the area of cross-section and ( )w x is the

resulting transverse deflection at x . The stored and dissipated energy in terms of the deflection

shape are given as

252

( )22

0

12

L

W A w x dxρ ω= ∫ (B.8)

and

( )21

0 0

.T L

W pudt w x dxL

β πωΔ = =∫ ∫ (B.9)

respectively. The quality factor from the ratio of the energies as represented in Eq. (B.3) is given

as

1

ALQ ρ ωβ

= (B.10)

In order to obtain 1β , three major regions are analyzed to cover the fluid pressure from vacuum

to atmospheric pressure and beyond.

Intrinsic region :

Very low pressure exists in this region which implies negligible damping due to air flow.

Molecular region :

Here, damping is caused by collision of air molecules and 1β is expressed as

132 where 9m a m

MLk bP kRT

βπ

= = (B.11)

where M is the molecular mass of the gas, T is the temperature and R is the universal gas

constant. Now, Q can be obtained as

22

12n

m a

k h EQk P L

ρ⎛ ⎞= ⎜ ⎟⎝ ⎠

(B.12)

In Eq.(B.12), aP is the pressure and nk is the constant for thn resonant mode.

253

Viscous region

This region includes pressure upto the atmospheric pressure and further thereof. The

damping here is solved using fluid mechanics principles for an incompressible viscous medium

(μ constant). An approximate model for this is implemented in Blom et al. (1992) that consists

of a string of spheres vibrating independently with infinite separation. This region is fairly

complex and can be further divided into sub-regions depending on operation pressure to

accurately model the damping. However, for the purpose of this investigation a more general and

simplified empirical relation is used (Blom et al. 1992) given by

12

2 2

12 .6 1

n

eqeq

Ek bhQ

RR L

ρ

πμδ

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞+⎜ ⎟

⎝ ⎠

(B.13)

Here, μ is the dynamic viscosity of the medium and δ is the width of the boundary layer

expressed as

1

22 ,a

μδρ ω

⎛ ⎞= ⎜ ⎟⎝ ⎠

(B.14)

where a aM PRT

ρ = is the density of the medium. In general, eqR is defined as the equivalent

radius of the sphere that is obtained from empirical measurements. However, in the simplest,

case it is assumed as the half width of a beam. Therefore, in the case of a cantilever beam,

.2eqbR = From Eq. (B.13), it can be inferred that for 1eqR δ , Q is independent of the

pressure, while for 1eqR δ , it is proportional to 1 aP . More information on airflow

damping and related work is summarized in Lin and Wang (2005).

254

Support Losses

In this mechanism, the vibration energy of a resonator is dissipated by transmission

through its support. It therefore exerts both shear and moment on the clamp during flexural

vibration. It is assumed that the elastic wave entering the support is propagated away and no

energy is returned, which ensures that the resonant modes are not affected. This assumption

holds true if the thickness of the beam is much smaller than the wavelength of the elastic wave,

which is already enforced in the model that used the lumped element approach. The support loss

also includes friction between connected surfaces such as bolted joints, clamps etc. The quality

factor due to support loss is given simply as (Hosaka et al. 1995)

3

0.23 .LQh

⎛ ⎞= ⎜ ⎟⎝ ⎠

(B.15)

This loss becomes insignificant for 100.Lh

Surface Dissipation

The dissipation due to surface effects becomes significant especially when the cantilever

thickness scales down leading to a much higher surface-to-volume ratio. This dissipation is

mostly caused due to induced surface stresses on the cantilever. The loss is enhanced when the

device has a thin surface layer (of thickness, sδ ) with high dissipative properties. The quality

factor for this mechanism is defined as

( )2 3s ds

bh EQb h Eδ

=+

(B.16)

where b is the width and dsE is the dissipation Young’s modulus of the surface layer.

Volume Loss

The empirical relation for volume loss is given in Yasamura et al. (2000) as

255

,ds

EQE

= (B.17)

where E is the elastic modulus and dsE is the dissipative modulus of the material.

Squeeze Damping Loss

The squeeze loss that usually occurs when device operates in a thin layer of fluid is

modeled using Reynolds equation in Hosaka et al., (1995). It employs a narrow bearing

approximation with no pressure gradient in the longitudinal direction. In addition, if we assume

that the vibration is much smaller than the gap, the squeeze loss is given as

3

2a o ng hQ

bρ ωμ

= (B.18)

where 0g is size of the gap and μ is the coefficient of viscosity.

Thermoelastic Dissipation

During flexural vibration of a structure, specifically a beam, tension and compression on

the top and bottom of the neutral axis results in a temperature gradient leading to thermoelastic

dissipation. Zener (1937) approximated this loss with a single relaxation peak and a

characteristic time proportional to the thermal diffusion time as shown below

( )

2

21 where and

1Ep

b kQ C

ωτ τ χχ ρωτ

= Δ ∝ =+

(B.19)

Here, 2

ad adE

pad

E E E E E TE CE E

αρ

− −Δ = = = , where adE is the unrelaxed adiabatic elastic modulus, E

is the isothermal elastic modulus and pC is the specific heat at constant pressure. For small

vibrations, assuming that adE E≈ and p vC C≈ results in the thermoelastic quality factor as

256

( )

2

21

1p

E TQ C

α ωτρ ωτ

=+

(B.20)

Eq. (B.20) is rewritten as ( ) ( )

2

2 21 1

1p

E TQ C

α τωρ τ ω

=+

to observe two limiting conditions, namely,

1. If 1ω τ , the structure remains in equilibrium resulting in no energy

dissipation.

2. If 1ω τ , the relaxation time is negligible which leads to minimal

dissipation.

Appreciable dissipation occurs only when ω and 1τ are of the same order in magnitude.

For example, a thin beam under flexure (with a rectangular cross-section) undergoes 98%∼ of

thermal relaxation through the first resonant mode whose relaxation time is given as

2

2

bτπ χ

= (B.21)

The quality factor due to thermoelastic dissipation can be represented in the form of an

empirical relation as (Lifshitz and Roukes 2000)

2

2 3

16 6 sinh sin

cosh coso

p

QE T

Cα ξ ξρ ξ ξ ξ ξ

=⎛ ⎞+

−⎜ ⎟+⎝ ⎠

(B.22)

where nf is the resonant frequency, E is the isothermal Young’s modulus, Eδ is the difference

between the adiabatic and the isothermal young’s modulii and χ is the thermal diffusivity

defined as pcχ κ ρ= . Here, κ is the thermal conductivity and pC is the specific heat at

257

constant pressure. ξ is defined as 2

ob ωξχ

= , where oω is the isothermal value of

eigenfrequency.

As evident from the above empirical relations, the loss due to the airflow can be minimized

by using the device in vacuum. In addition, the support losses can also be reduced by using

beams with high length to thickness ratios. The internal losses such as the thermoelastic

dissipation can be difficult to avoid for a specific beam and therefore need to be studied in detail

to model analytically. So, a detailed 2-D coupled beam bending and heat conduction solution is

provided for a homogenous beam to accurately estimate the thermoelastic damping.

Analytical model

For the purpose of this exercise, a thermoelastic model is obtained for a homogeneous

cantilever beam similar to the one described in Guo and Rogerson (2003). The model is derived

based on the following assumptions,

1. Linear Euler-Bernoulli beam theory that uses a small deflection model.

2. Pure bending with no rotary inertia and shear deformation.

3. The top and bottom surfaces are thermally insulated, 2

0.z h

dTdz =±

=

The cantilever beam that is analyzed here is shown in figure 1 with the dimensions

indicated.

( ), ,wu z w w x tx

∂= − =

∂ (B.23)

correspond to the displacement in axial (x) direction and transverse (z) direction.

( )2

2 , ,xwEz T x z t

xσ β∂

= − −∂

(B.24)

258

where E is the elastic modulus. β is the thermal modulus defined as [ ]Cβ α= , where [ ]C is

the compliance matrix and α is the coefficient of linear thermal expansion. In addition,

( )2

2

h

xh

M x b zdzσ−

= ∫ (B.25)

is defined as the bending moment in the beam. Now substituting the expression for the axial

stress in Eq. (B.25) yields the overall moment in the beam as

( ) ( )2 2

22

, ,h

h

wM x b Ez T x z t zdzx

β−

⎛ ⎞∂= − −⎜ ⎟∂⎝ ⎠∫ (B.26)

We know that the moment of inertia is given as

2 3

2

2 12

h

h

bhI bz dz−

= =∫ (B.27)

This simplifies Eq. (B.26) to

( ) ( )22

22

, ,h

h

wM x EI b T x z t zdzx

β−

∂= − −

∂ ∫ (B.28)

We denote ( )2

2

, ,h

h

K T x z t zdz−

= ∫ . Hereafter ( ), ,T x z t will be denoted as T for simplicity in the

derivation.

The equation of motion for a cantilever beam modeled using Euler-Bernoulli beam theory

is

2 4 2

2 4 2Lw w KEI b

t x xρ β∂ ∂ ∂

+ +∂ ∂ ∂

(B.29)

where Lρ is the linear mass density. The heat conduction equation including the thermoelastic

effect is denoted as (Fung, 1965)

259

2 2 2

2 2 2 0v oT T T wk k C T z

x z t t xρ β

⎛ ⎞∂ ∂ ∂ ∂ ∂+ − + =⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

(B.30)

where [ ]k W mK is the thermal conductivity, ρ is the material density and [ ]vC J kgK is the

specific heat at constant volume. Multiplying Eq. (B.30) by β and integrating along the

thickness of the beam will result in the following conduction equation

2 2 2 22 2 2

22 2 2

2 2 2 2

0h h h h

v oh h h h

wk Tzdz k Tzdz C Tzdz T z dzx z t t x

ρ β− − − −

⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎪ ⎪ ⎪ ⎪ ⎪ ⎪+ − + =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ ⎩ ⎭∫ ∫ ∫ ∫ (B.31)

substituting for K in the above equation simplifies it to

2 22 2

2 22 2

0h h

ov

h h

T IK T T K wk k z dz Cx z z t b t x

βρ− −

⎧ ⎫ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪+ − − + =⎨ ⎬ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎪ ⎪⎩ ⎭∫ (B.32)

The second term in the expression is expanded using the product rule. Since the top and bottom

surface of the beam is assumed to be thermally insulated, it follows that 2

0z h

Tz =±

∂=

∂, which

further simplifies Eq. (B.32) to

2 2

22 22

0h ovh

T IK K wk k T Cx t b t x

βρ−

⎛ ⎞∂ ∂ ∂ ∂− − + =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

(B.33)

Assuming a cubic polynomial dependence of temperature along the thickness of the beam, it can

be represented as

3 21 2 3 4T c z c z c z c= + + + (B.34)

From Eq. (B.34), 1c can be expressed as

( )2 2

4 3 21 2 3 4

2 2

,h h

h h

K Tzdz c z c z c z c z dz− −

= = + + +∫ ∫ (B.35)

260

which will simplify to

5 3

1 3 .80 12h hK c c= + (B.36)

If we assume that the thickness of the beam is very small, we can rewrite Eq. (B.36) as

5 3 3

21 3 1 3280 12 4

h

h

h h hK c c T c c h−

= + = = + (B.37)

This assumption leads to 2

2

212h

h

hK T−

= for small thickness, when h is very small compared to

the length of the beam. Now, the two governing equations can be rewritten by substituting for I

as

2 3 4 2

2 4 2 012

w bh w Kbh E bt x x

ρ β∂ ∂ ∂+ + =

∂ ∂ ∂ (B.38)

and

32 2

2 2 2

12 012

ov

T hK K K wk k Cx h t t x

βρ⎛ ⎞∂ ∂ ∂ ∂

− − + =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ (B.39)

Let us now introduce the following non-dimensional parameters to further transform Eqs. (B.38)

and (B.39).

2, , , , ,T Kx w tv Ex w v T KL h h E L Eβ βτ ρ= = = = = = (B.40)

Here, τ is the non-dimensional time and v is the non-dimensional velocity. All the parameters

represented as , , and x w T K are the corresponding non-dimensional terms.

Equation (B.38) will now be represented in their non-dimensional form as

2 3 4 2

2 2 3 4 22

012

bhL w bh w KE bEh L x x

v

ρτ∂ ∂ ∂

+ + =∂ ∂ ∂

(B.41)

261

which can be rewritten as

2 4 2

12 4 2 0w w KAx xτ

∂ ∂ ∂+ + =

∂ ∂ ∂ (B.42)

and

2 2

2 3 42 2 0K K wA K A Axξ τ τ

⎛ ⎞∂ ∂ ∂ ∂− − + =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

(B.43)

where the coefficients are defined by 2

1 212hA

L= ,

2

2 210 LAh

= , 2

3 vLA c vhk

ρ= and

23

4 212oA T h vEL kβ

= . The boundary conditions for the cantilever can be represented as

( ) ( )( ) ( )( ) ( )

0, 0, 0

1, 1, 0

0, 1, 0

w w

w w

K K

τ τ

τ τ

τ τ

′= =

′′ ′′′= =

= =

(B.44)

Let us assume a harmonic solution for the deflection and temperature as

( )( )

j

j

w w e

K K e

τ

τ

ξ

ξ

Ω

Ω

=

= (B.45)

where Ω is the non-dimensional frequency. Substituting the above harmonic solution into the

two governing equations, we obtain

( ) ( ) ( )21 0w A w Kξ ξ ξ′′′′ ′′−Ω + + = (B.46)

and

( ) ( ) ( ) ( )2 3 4 0K A K jA K j A wξ ξ ξ ξ′′ ′′− − Ω + Ω = (B.47)

Eliminating ( )K ξ′′ from equations (B.46) and (B.47), we obtain

( ) ( ) ( ) ( ) ( )21 4 2 3w A w j A w A jA Kξ ξ ξ ξ′′′′ ′′Ω − + Ω = + Ω (B.48)

262

Differentiating ( )K ξ twice with respect to ξ and substituting back in Eq. (B.46) wll yield the

overall coupled differential equation as

( ) ( ) ( ) ( ) ( )( )

21 42

12 3

0w A w j A w

w A wA jA

ξ ξ ξξ ξ

′′ ′′′′′′ ′′′′Ω − + Ω′′′′−Ω + + =

+ Ω (B.49)

The above equation can be simply expressed in the form

( ) ( ) ( ) ( )1 2 3 4 0a w a w a w a wξ ξ ξ ξ′′′′′′ ′′′′ ′′+ + + = (B.50)

where

11

2 3

42 1

2 32

32 3

24

AaA jA

j Aa AA jA

aA jA

a

= −+ Ω

Ω= +

+ Ω

Ω=

+ Ω

= −Ω

(B.51)

If we assume the solution to Eq. (B.50) is of the form

( ) ( ) ( )sinh coshw B Cξ λξ λξ= + (B.52)

the characteristic equation for Eq. (B.50) reduces to

6 4 21 2 3 4 0a a a aλ λ λ+ + + = (B.53)

Since 1 2 3 4, , and a a a a are complex constants, the roots of the above equation can be assumed to

be 1,2,3λ± and represented as

( ) ( ) ( )3

1sinh coshi i i i

iw B Cξ λξ λξ

=

= +∑ (B.54)

263

The above solution applies only the distinct roots and when multiple roots occur, the solution is

of the form

( ) ( ) ( )3

1sinh coshi i i i i i

iK B d C dξ λξ λξ

=

= +∑ (B.55)

where

( )( )

2 44 1

2 3 1, 2,3i i

i

i A Ad iA i A

λ λΩ + Ω −= =+ Ω (B.56)

Substituting Eqs. (B.54) and (B.55)

( ) ( )

( ) ( )

( ) ( )

3

1

3

1

32 2

1

33 3

1

3

1

3

1

0

0

sinh cosh 0

cosh sinh 0

0

sinh cosh 0

ii

i ii

i i i i i ii

i i i i i ii

i ii

i i i i i ii

C

B

B C

B C

C d

B d C d

λ

λ λ λ λ

λ λ λ λ

λ λ

=

=

=

=

=

=

=

=

+ =

+ =

=

+ =

∑

∑

∑

∑

∑

∑

(B.57)

Rewriting the above equations in the form of a matrix will yield

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( )

1 2 3

1 1 2 2 3 3

2 2 2 2 2 21 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3

3 3 3 3 3 31 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3

1 1 2 2 3 3

1 1 1

00

sinh sinh sinh cosh cosh cosh 0

cosh cosh cosh sinh sinh sinh 00

sinh

C C CB B B

B B B C C C

B B B C C CC d C d C dB d B

λ λ λ

λ λ λ λ λ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ λ λ λ λ λ

λ

+ + =

+ + =

+ + + + + =

+ + + + + =

+ + =

+ ( ) ( ) ( ) ( ) ( )2 2 2 3 3 3 1 1 1 2 2 2 3 3 3sinh sinh cosh cosh cosh 0d B d C d C d C dλ λ λ λ λ+ + + + = (B.58)

which simplifies to

264

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

1 2 32 2 2 2 2 2

1 1 2 2 3 3 1 1 2 2 3 3

1 2 33 3 3 3 3 3

1 1 2 2 3 3 1 1 2 2 3 3

1 1 2 2 3 3 1 1 2 2 3 3

0 0 0 1 1 10 0 0

sinh sinh sinh cosh cosh cosh0 0 0

cosh cosh cosh sinh sinh sinhsinh sinh sinh cosh cosh cosh

d d d

d d d d d d

λ λ λλ λ λ λ λ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ λ λ λ λ λλ λ λ λ λ λ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

[ ]

1

2

3

1

2

3

0

BBBCCC

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(B.59)

For a non-trivial solution, the determinant of the 6X6 matrix should be zero. The above set

of equations can be solved numerically by assuming a guess for Ω and substituting in the

determinant to obtain its value. An iteration procedure will be carried out to minimize the error

between the assumed value and calculated value. From the calculated value of Ω , which will be

a complex number, the real part corresponds to the natural frequency and the imaginary part

corresponds to the thermoelastic dissipation.

A detailed thermoelastic model of a homogeneous cantilever beam in bending was

obtained analytically using coupled set of equations. The solution was non-dimensionalized and

it was observed that the non-dimensional frequency is scale dependent when thermoelastic

effects are considered. The solution was completely formulated from first principles. However,

the numerical solution technique was not implemented to estimate the thermoelastic dissipation.

265

APPENDIX C TRANSFORMATION OF COORDINATES FOR RELATIVE MOTION

Let us consider a vibrating cantilever beam that is excited at its clamp. The resulting

configuration lies in an accelerating frame of reference with respect to the ground. A simple

schematic of a cantilever beam described above is shown in the following Figure C-1.

Cantilever Beam

Proof Massx

y

z

X

Y

Z

Global Co-ordinates Local Co-ordinates

0 sinclampd tω

Figure C-1: Vibrating cantilever beam in an accelerating frame of reference.

The governing dynamic equation describing the cantilever beam is expressed as

( ) ( ) ( )4 2

4 2

, ,, ,

W X t W X tEI A p X Y Z

X tρ

∂ ∂+ =

∂ ∂ (C.1)

Here, E is the elastic modulus of the material, I is the moment of inertia, ρ is the material

density and A is the cross-sectional area. ( ),W X t is the transverse dynamic deflection with

respect to the global coordinates. ( ), ,p X Y Z is the static load on the beam due to its self-weight,

where , and X Y Z are the global coordinates. The cantilever beam is subjected to an input

sinusoidal displacement at the clamp that causes it to vibrate. Let us assign local coordinates

, and x y z that describe the motion of the beam relative to its clamp position as shown in the

figure. Therefore, the relative vertical deflection with respect to the clamp can be expressed as

266

( ) ( ) ( ), , clampW X t w x t d t= + (C.2)

where ( ),w x t is the resulting deflection in the beam and ( )clampd t is the applied dynamic

displacement to the clamp. Substituting Eq. (C.2) in Eq. (C.1) results in the following expression

shown below.

( ) ( ) ( ) ( )

( )4 2

4 2

, ,, ,clamp clampw x t d t w x t d t

EI A p X Y ZX t

ρ⎡ ⎤ ⎡ ⎤∂ + ∂ +⎣ ⎦ ⎣ ⎦+ =

∂ ∂ (C.3)

Eq. (C.3) can be expanded as

( ) ( ) ( ) ( )24 2

4 2 2

, ,, ,clampd tw x t w x t

EI A A p X Y Zx t t

ρ ρ∂∂ ∂

+ + =∂ ∂ ∂

(C.4)

For a harmonic excitation at the clamp, ( ) 0 sinclamp clampd t d tω= , where 0clampd is the amplitude

and ω is the angular frequency, Eq. (C.4) can be rewritten as

( ) ( ) ( )4 2

204 2

, ,, ,clamp

w x t w x tEI A A d p X Y Z

x tρ ρ ω

∂ ∂+ − =

∂ ∂ (C.5)

( ) ( ) ( )4 2

204 2

, ,, , clamp

w x t w x tEI A p X Y Z A d

x tρ ρ ω

∂ ∂+ = +

∂ ∂ (C.6)

As evident in the above equation, the governing equations remain the same with the input

harmonic displacement acting like an effective inertial force on the right side.

267

APPENDIX D ELECTRICAL IMPEDANCE FOR A PIEZOELECTRIC MATERIAL

Let us consider the following circuit representation shown in Figure D-1 that models the

piezoelectric part of the device with an effective electrical impedance.

CebV

I Re

Figure D-1: Blocked electrical impedance in a parallel network representation.

The blocked electrical impedance of a piezoelectric material is represented as a parallel network

in the above representation. In the shown format, the dielectric loss represented as eR can be

empirically given using the relation

1 1tane

eb

RCδ ω

= (D.1)

where ebC is defined as the blocked electrical capacitance and ω is the frequency of oscillation

given as 2 fπ . tanδ is defined as the loss tangent for the impedance. In the shown format, the

overall impedance of the circuit is given as

268

1

e

ebeb

eeb

Rj CZ

Rj C

ω

ω

=+

(D.2)

which can simplified as

1

eeb

eb e

RZj C Rω

=+

(D.3)

( )( )2

11e eb e

ebeb e

R j C RZ

C Rω

ω

−=

+ (D.4)

Substituting for eR , the above equation simplifies to

2

1 1 1 11tan tan

1 11tan

ebeb eb

eb

ebeb

j CC C

Z

CC

ωδ ω δ ω

ωδ ω

⎛ ⎞−⎜ ⎟

⎝ ⎠=⎛ ⎞

+ ⎜ ⎟⎝ ⎠

(D.5)

( )

( )2

1 tan

1 taneb

eb

jCZ

δω

δ

−=

+ (D.6)

For small values of loss tangent, ( )21 tan 1δ+ ≈ , which further simplifies the impedance to

tan tan 1eb

eb eb eb eb

jZC C C j Cδ δ

ω ω ω ω= − = + (D.7)

The above expression looks like a series combination of a real quantity that can be represented as

an equivalent resistance and a complex quantity resulting from the capacitance. Hence, the

electrical impedance of a piezoelectric material can be alternatively represented as a series

combination as shown below in Figure D-2.

269

Ceb

V

I

Re’

Figure D-2: Blocked electrical impedance in a series network representation

Where, the overall impedance of the circuit is given as

' 1e

eb

Rj Cω

+ (D.8)

where ' tane

eb

RCδ

ω= .

Even though the series representation is mathematically valid, it may not accurately represent the

physical behavior of a piezoelectric material for the following reason. When a DC voltage is

applied to piezoelectric material, charge accumulates as the material is completely blocked for

any physical motion. Therefore, this charge dissipates due to the dielectric loss as a function of

time. Using a series representation breaks the circuit for DC as the impedance due to the

capacitor approaches infinity. However, the parallel network facilitates the physical phenomenon

of dissipation across the resistor, eR even though the capacitor branch becomes an open circuit in

Figure D-1.

270

APPENDIX E CONJUGATE IMPEDANCE MATCH FOR MAXIMUM POWER TRANSFER

Let us consider the following circuit representation shown in electrical impedance

thevenin equivalent form (Figure E-1).

V

I

ZL

Zth

Figure E-1: Thevenin equivalent representation connected to a external complex impedance.

The above plot shows a simple circuit representation of the electromechanical lumped element

model circuit simplified to its thevenin equivalent circuit. The thevenin voltage, thV and the

thevenin impedance, thZ are shown in the figure. LZ represents an arbitrary load at the output

across which, the output voltage is measured to reclaim power. The goal of this analysis is to get

an expression for LZ to maximize the power output.

First, let us assume that the thevenin impedance, thZ is a complex value represented as

r ith th thZ Z jZ= + (E.1)

Similarly, the thevenin voltage can be represented as

r ith th thV V jV= + (E.2)

271

If we assume the output load to be complex as well given by

LZ R jX= + (E.3)

the voltage across the load is given as

r i

r i

th ththL L

th L th th

V jVVV Z R jXZ Z Z jZ R jX

+= = +

+ + + + (E.4)

which simplifies to

( ) ( )

( ) ( )r i r i

r i

th th th thL

th th

V R V X j V X V RV

Z R j Z X

− + +=

+ + + (E.5)

( ) ( )

( ) ( )( ) ( )22

r i r i

r i

r i

th th th thL th th

th th

V R V X j V X V RV Z R j Z X

Z R Z X

− + += + − +

+ + + (E.6)

Collecting the real and imaginary parts results in

( ) ( ) ( )( ) ( )( ) ( )( ){ }( ) ( )22

r i r r i i r i r r i i

r i

th th th th th th th th th th th th

L

th th

V R V X Z R V X V R Z X j V X V R Z R V R V X Z XV

Z R Z X

− + + + + + + + − − +=

+ + +

(E.7)

The current across the load is given as

( )

r i r i

r i r i

th th th ththL

th L th th th th

V jV V jVVIZ Z Z jZ R jX Z R j Z X

+ += = =

+ + + + + + + (E.8)

which simplifies to

( ) ( ) ( ) ( ){ }

( ) ( )22r r i i i r r i

r i

th th th th th th th th

L

th th

V Z R V Z X j V Z R V Z XI

Z R Z X

+ + + + + − +=

+ + + (E.9)

The power measured across the load is the product between the voltage and complex conjugate

of the current given as

L L LP V I= (E.10)

272

where

( ) ( ) ( ) ( ){ }

( ) ( )22r r i i i r r i

r i

th th th th th th th th

L

th th

V Z R V Z X j V Z R V Z XI

Z R Z X

+ + + − + − +=

+ + + (E.11)

The real power is obtained by taking the real part of the complex product between the current

and voltage. Further simplification results in the following expression for the load power

( )( )

( ) ( )

2 2

22Re r i

r

r i

th thL L

th th

V V RP P

Z R Z X

+= =

+ + + (E.12)

For maximum power transfer, 0rLP

X∂

=∂

and 0rLP

R∂

=∂

.

( )

( ) ( )( )

2 2

22 2r i

r i

r i

th thL th

th th

V V RP Z X

X Z R Z X

+∂= − +

∂ + + + (E.13)

Equating the above result to zero yields the optimal value of X as

ithX Z= − (E.14)

( )( )

( )2 2

3r i

r r

r

th thL th

th

V VP Z R

R Z R

+∂= −

∂ + (E.15)

which results in the value of R as

rthR Z= (E.16)

Therefore, the optimal external load for maximum power transfer is

r iL th thZ Z jZ= − (E.17)

273

However, if we assume that the external load is purely resistive (shown in Figure E-2)

which is the nature of some standard circuits such as pulse width moderator circuit (PWM) as

discussed in Taylor 2004, then the real power assumes the expression

( )( )

( ) ( )

2 2

22Re r i

r

r i

th thL L

th th

V V RP P

Z R Z

+= =

+ + (E.18)

Vth

Ith

RL

Zth

Figure E-2: Thevenin equivalent representation connected to a resistive load.

For maximum power transfer, 0rLP

R∂

=∂

.

( )( ) ( )

2 2 22 2

22r i

r r i

r i

th thL th th

th th

Z Z RP V V

R Z R Z

+ −∂= +

∂ + + (E.19)

Equating the above result to zero yields the optimal value of R as

2 2r ith thR Z Z= + (E.20)

Therefore, the optimal external load for maximum power transfer is

2 2r iL th thZ R Z Z= = + (E.21)

274

APPENDIX F UNDERSTANDING THE PHYSICS OF THE DEVICE

The dynamic behavior of the energy harvester configuration comprised of a composite

cantilever beam with a proof mass at one and the clamp subjected a fixed acceleration is

discussed here. A schematic of the device is shown in Figure F-1.

Cantilever Beam

Proof Massao

PZT

Basefn

Figure F-1: Schematic of the composite beam energy harvester.

In the actual configuration shown above, the input acceleration is applied to the base that

translates into a relative motion of the proof mass with respect to the base. Consequently, this

device becomes a two degree of freedom system and can be represented as shown in Figure F-2.

Mp

Cms

y xRm

Mb

F

Figure F-2: Free body representation of the device as a two degree of freedom system.

Here, bM is the mass of the base, and pM is the proof mass. The two governing equations

describing the system are shown in Eqs. (F.1) and (F.2)

( ) ( )1b m

ms

M y F R y x y xC

= − − − − (F.1)

and

( ) ( )1p m

ms

M x R y x y xC

= − + − (F.2)

275

The device can then be represented using an equivalent circuit notation (similar to Figure 2-2) as

shown in Figure F-3. In the figure, the input force F applied to the base produces a base

velocity, Y , and a velocity of the proof mass, X , as shown in the circuit. Bending in the beam

occurs due to relative motion between the base and proof mass whose velocity is expressed using

( )Z Y X= − . As evident from the circuit behavior, when the base mass is large, there is no base

velocity, ( )0Y = . Similarly, if the impedance of the proof mass is large it does not move,

( )0X = . Furthermore, at low frequencies approaching dc, the device exhibits rigid body motion

as there is no relative motion between the proof mass and base, ( )0Z = .

FMp

Cms RmMb

X ZY

Figure F-3: Electromechanical circuit representation of the energy harvester.

Replacing ( ) ,z y x= − the equations of motion representing the two degree of freedom (Eqs.

(F.1) and (F.2)) become

1b m

ms

M y F R z zC

= − − (F.3)

and

1 .p mms

M x R z zC

= + (F.4)

276

Here, Eq. (F.3) describes the motion of the base and Eq. (F.4) models the proof mass.

Substituting for x with y z− , Eq. (F.4) simplifies to

( ) 1 .p mms

M y z R z zC

− = + (F.5)

Eq. (F.5) can be represented using a single degree of freedom system with relative motion

between the tip and base, z as the variable. The new governing equation becomes

1 .p p mms

M y M z R z zC

= + + (F.6)

Here, in pF M y= is the effective input inertial force on the composite beam and the proof mass,

as shown in Figure 2-2. In practice, the beam also has an effective mass. For the static

analytical approach employed in this thesis (Section 2.1), this mass is lumped with the proof

mass (i.e., they are added).

Both the two and single degree of freedom representations model the behavior of the

energy harvesting device. The only limitation with the simpler single degree of freedom system

employed in the thesis is that it does not explicitly include the effects of the base. In essence, it

is assumed that the device (including the base mass) does not load the vibrating device to which

it is attached. In this sense, it is like an accelerometer.

277

APPENDIX G FABRICATION LAYOUTS

Figure G-1: Top Electrode Etch Mask – Lift-off mask for patterning top electrodes

278

Figure G-2: PZT Etch Mask – Wet etching of PZT to reveal the bottom electrode pads

279

Figure G-3: Ion Milling Mask – Ion milling to pattern the PZT, top and bottom electrode

features.

280

Figure G-4: Beam Etch Mask – Front side DRIE to pattern the beams.

281

Figure G-5: Proof Mass Mask – Backside DRIE mask for patterning the proof mass.

282

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BIOGRAPHICAL SKETCH

Anurag Kasyap V.S. was born on September 17th 1978, in Chittoor, Andhra Pradesh, India.

He graduated from Space Central School, Sriharikota, India, in 1995. He obtained his

undergraduate degree in B.Tech (naval architecture) from Indian Institute of Technology,

Chennai, Tamil Nadu, India. He entered the University of Florida in the Fall of 1999 with a

graduate research assistantship and received his MS in Aerospace Engineering in 2002. He is

currently completing his doctoral degree at the University of Florida in the area of Piezoelectric

based energy harvesting and microelectromechanical systems (MEMS).

Documents

DEVELOPMENT OF MEMS-BASED PIEZOELECTRIC ...ufdcimages.uflib.ufl.edu/UF/E0/01/56/46/00001/vs_a.pdf1 DEVELOPMENT OF MEMS-BASED PIEZOELECTRIC CANTILEVER ARRAYS FOR VIBRATIONAL ENERGY