PIEZOELECTRIC ENERGY HARVESTERS FOR WIRELESS SENSOR NETWORKS by Michael Renaud

KATHOLIEKE UNIVERSITEIT LEUVEN FACULTEIT INGENIEURSWETENSCHAPPEN DEPARTEMENT ELEKTROTECHNIEK – ESAT Kasteelpark Arenberg 10, B-3001 Leuven - Belgi¨ e

PIEZOELECTRIC ENERGY HARVESTERS FOR WIRELESS SENSOR NETWORKS Jury: Prof. Dr. Ann Haegemans, chair Prof. Dr. Robert Mertens, promoter Prof. Dr. Chris Van Hoof, promoter Prof. Dr. Eric van den Bulck Prof. Dr. Ronnie Belmans Prof. Dr. Robert Puers Dr. Paolo Fiorini Prof. Dr. Jo De Boeck

Dissertation submitted in partial fulfillment of the requirements for the degree of ”doctor in de ingenieurswetenschappen” by Michael Renaud

October 2009

in collaboration with

VZW

Interuniversitair Micro-Elektronica Centrum vzw Kapeldreef 75 B-3001 Leuven (Belgi¨e)

ÂŠKatholieke Universiteit Leuven - Faculteit Toegepaste Wetenschappen Arenbergkasteel, B-3001 Heverlee (BelgiÂ¨ e)

Alle rechten voorbehouden. Niets van deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt door middel van druk, fotocopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaande schriftelijke toestemming van de uitgever.

ISBN 978-94-6018-130-6

Wettelijke depotnummer: D/2009/7515/113

Acknowledgments Before the start of this text I would like to express my sincere gratitude to everybody who helped me during my research on this PhD Thesis. First of all, I wish to express my appreciation and thanks to my direct supervisor Dr. P. Fiorini for his valuable guidance, support and patience throughout my research. I would not have been able to conclude this Ph.D. without his help. I should also specifically thank him for his great efforts on reviewing my papers. I would then like to thank Prof. C. van Hoof and Prof. R. Mertens for offering me the opportunity to carry on this research and helping me with the tasks related to my Ph.D. I am grateful to the members of the reviewing committee (Prof. R. Belmans, Prof. E. van den Bulck, Prof. R. Puers and Prof. J. de Boeck) for their useful comments and corrections on the dissertation. I am also thankful to Prof. A. Haegemans for chairing the defense. I would like to thank the members of various teams I cooperated with. To start, many thanks go to Tom Sterken for sharing his valuable insights and for all the constructive discussions we had. I would like to thank Bert Dubois and Chikhi Abdelhafid for helping me getting accustomed to the cleanroom environment. Thanks to Vladimir Leonov and Ziyang Wang to create a pleasant team environment. Many thanks also go to Stanislaw Kalicinski and Vladimir Cherman for their help on the characterization of devices. I would also like to thank many colleagues from the Holst Centre in the Netherlands (R. Elfrink, M. Goedbloed, T. Kamel and R. van Schaijk to name a few) for the fruitful collaboration which is still ongoing. I am also appreciative of the work realized by the master students that I mentored during this Ph.D. research (L. de Vreede, V. Prins, H. Toreyin and A. Bayoumi). I am grateful to IMEC and the Katholieke Universiteit Leuven (K.U. Leuven). Both provided me the necessary means to conduct my Ph.D. research in an excellent environment and gave me the opportunity to present my results in high level international conferences. I would like to thank my friends Gregory, Raquel and Bert for the fun we had when we were all together. Also, thanks to my aquatic pets which helped

me relaxing when the pressure was too high. Finally, many thanks to my parents for their endless support. Michael Renaud Leuven, October 2009

To my grandmother

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Abstract Recent years have seen important developments in the field of wireless sensor networks. Such networks have already found applications in the field of health monitoring and smart environment. Electrochemical batteries are usually implemented for powering the sensor nodes. Depletion of batteries leads to a loss of functionality of the sensor node which can result in critical situations if not detected in due time. The costs of replacing batteries located in a remote or hostile environment is also high. Furthermore, electrochemical batteries contain dangerous chemicals possibly released in the environment because of the costs of recycling. Therefore, so-called energy harvesters gained interest in the last decade. Energy harvesters act by converting part of the energy available in the environment into useful electrical power. They are self replenished and do not need replacement. Common environmental energy sources are light, heat, vibrations, human motion or wind. This thesis is focused on harvesting energy from vibrations and human body motion. Devices adapted to each situation are designed, fabricated and characterized. Their active element is a piezoelectric bender and a detailed model of such a transducer is derived. The devices designed for harvesting energy from the human motion are based on the impact of a rigid body on piezoelectric cantilevers. An output power of 47 µW is obtained for a device of dimensions 3.5*2*2 cm3 weighting 60 g rotated over 180◦ each second. Also, a power of 600 µW is measured when the harvester is placed on the hand of a person and forcibly shaken at a frequency of approximately 7 Hz for an amplitude of 10 cm. A large amount of the volume occupied by the prototype of the harvester can be eliminated and an output power density of 10 µW/cm3 or 4 µW/g is estimated for an optimized device undergoing the aforementioned rotary motion. These figures are multiplied by a factor 12 when a 7 Hz frequency, 10 cm amplitude linear motion is considered. The devices designed for harvesting energy from vibrations in machinery environment are based on resonant piezoelectric beams. Both aluminum nitride (AlN) and lead zirconate titanate (PZT) are considered as piezoelectric materials. If parasitic dissipations can be maintained below a certain threshold, it is

demonstrated that MEMS harvesters based on thin film AlN or PZT compete with ceramic PZT based devices in terms of power generation. Micromachined piezoelectric harvesters are manufactured and characterized. An output power in the range of 50 ÂľW for an approximated volume of 0.3 cm3 is obtained with AlN and PZT MEMS harvesters, which is enough to power low consumption sensor nodes. As a conclusion, the results presented in this PhD thesis give an important contribution to the optimization of piezoelectric energy harvesters, both at the experimental and theoretical level.

Nederlandstalige samenvatting De laatste jaren hebben we een enorme ontwikkeling gezien van draagbare elektronica, zoals draadloze sensornetwerken die onder andere hun toepassing vinden in de gezondheidszorg en in slimme omgevingen. In het algemeen worden elektrochemische batterijen gebruikt als voeding voor de sensoren in zo’n netwerk. Deze hebben als nadeel hun hoge kostprijs en het feit dat ze gemaakt zijn uit schadelijke materialen die meestal tussen het restafval belanden wegens de hoge recyclagekost. Bovendien kan het vervangen van deze batterijen op afgelegen en gevaarlijke plaatsen ook een groot kostenplaatje met zich meebrengen. Daarom is er een groeiende interesse voor de zogenaamde energie collectoren. Energie collectoren kunnen energie die aanwezig is in de omgeving omzetten in bruikbaar elektrisch vermogen. Voorbeelden van omgevingsenergie zijn licht, warmte, mechanische trillingen, menselijke beweging en wind. Het werk in deze thesis focusseert op het verzamelen van energie komende van mechanische trillingen en menselijke bewegingsenergie. Verschillende oogstprincipes zijn nodig om een efficiente energieverzameling te verwezenlijken van deze twee energiebronnen; daarom werden twee types apparaten gemodelleerd, gefabriceerd en gekarakteriseerd. Beide types van collectoren zijn gebaseerd op het principe van piezo-elektrische transductie. Het apparaat ontwikkeld om energie te halen uit menselijke beweging is gebaseerd op het fenomeen van impact op een star lichaam van piezo-elektrische cantilevers. Een vermogen van 47 µW kan behaald worden met een apparaat met een dimensie van 3.5*2*2 cm3 en een gewicht van 60 g en dit geplaatst op de hand en elke seconde geroteerd over een hoek van 180◦ . Een groot deel van het volume van dit prototype kan gelimineerd worden en de uitgangsvermogendichtheid kan geschat worden op 10 µW/cm3 of 4 µW/g voor een geoptimaliseerd systeem dat dezelfde roterende beweging ondergaat. Deze waarden kunnen vermenigvuldigd worden met een factor 12 wanneer een lineaire beweging beschouwd wordt met een frequentie van 7 Hz en een amplitude van 10 cm. Het apparaat ontwikkeld om energie te halen uit trillingen is gebaseerd op een resonantiesysteem gemaakt uit een verpakte piezo-elektrische balk. In de

vii

veronderstelling van een enkele harmonische trilling als input, wordt aangetoond dat de beste prestaties behaald worden bij resonantie of anti-resonantie frequentie van de apparaten. Ook wordt aangetoond dat zowel hoog als laag gekoppelde systemen bruikbaar zijn als de parasitaire verliezen onder een welbepaalde drempelwaarde gehouden kunnen worden. MEMS piezo-elektrische collectoren worden gefabriceerd en gekarakteriseerd. Een uitgangsvermogen van rond de 50 ÂľW werd bereikt met een volume van ongeveer 0.3 cm3 met AlN en PZT MEMS collectoren, wat voldoende energie is om sensor knopen te voeden die een laag vermogen nodig hebben. Tot slot wordt dit werk als zeer nuttig gezien met nieuwe en belangrijke resultaten op het vlak van optimalisatie van piezo-elektrische energie collectoren, zowel op experimenteel als theoretisch vlak.

viii

Symbols and Abbreviations Symbols A0

Amplitude of the input acceleration (m.s−2 )

Real part of the quasi static clamped capacitance of the piezoelectric layer in a laminated piezoelectric beam (F)

Cp0

Quasi static clamped capacitance of the piezoelectric layer in a laminated piezoelectric beam (F)

Cpf

Clamped capacitance of an unsupported slab of piezoelectric material (F)

Cpc

Clamped capacitance of the piezoelectric layer in a laminated piezoelectric beam (F)

dij

Components of the piezoelectric charge constant tensor (C.N−1 )

Effective piezoelectric charge constant (C.N−1 )

Electrical displacement tensor (A.m−2 )

External viscous damping term for the motion of the beam (N.s.m−1 )

Normalized electrical damping term (-)

Dem

Normalized electromechanical damping term (-)

Normalized mechanical damping term (-)

External viscous damping term for the motion of the missile (N.s.m−1 )

Components of the electrical displacement tensor (A.m−2 )

Effective coefficient of restitution (-)

Effective piezoelectric constant (C.m−2 ) ix

Ef f

Harvester effectiveness (%)

Electrical field tensor (V.m−1 )

Coercive electrical field (V.m−1 )

Components of the electrical field tensor (V.m−1 )

Eel

Energetic term for the definition of k31 (J)

EIeq

Equivalent of the quotient of the area moment of inertia over the tensile compliance (multilayer beams) (N.m2 )

Concentrated force applied on the beam (N)

Gravity field (=9.81m.s−2 )

Figure of merit (=K2 Qm )

GAeq

Equivalent of the quotient of the area of the cross section over the shear compliance (multilayer beams) (N)

Thickness of the piezoelectric layer in the piezoelectric beam (m)

Thickness of the elastic layer in the piezoelectric beam (m)

Thickness of the body attached to the cantilever (m)

Area moment of inertia (m4 )

Shear force (N)

Real part of the quasi static lumped stiffness of the beam (N.m−1 )

Quasi static lumped stiffness of the beam (N.m−1 )

Lumped stiffness of the beam (N.m−1 )

Indentation stiffness (N.m−3/2 )

kij

Electromechanical coupling factor of the mode ij (-)

Effective transverse electromechanical coupling factor ()

k31

Transverse electromechanical coupling factor (-)

Generalized electromechanical coupling factor (-)

Length of the piezoelectric beam (m)

Length of the body attached to the cantilever (m)

Mass of the beam (kg)

Effective mass of the structure (kg)

Mass of the body attached to the cantilever (kg)

Mass of the missile (kg)

Bending moment due to mechanical efforts (N.m)

Bending moment due to an applied voltage (N.m)

Denominator of the Laplace transform of the voltage across a resistively shunted bender (-)

Distributed load applied on the beam (N.m)

Generated power (W)

P opt

Optimum generated power (W)

PR opt

Optimum generated power at resonance (W)

PA opt

Optimum generated power at anti-resonance (W)

Charges developped by the piezoelectric material (C)

Mechanical quality factor (-)

Electrical quality factor (-)

Roots in s of O (-)

Rayleigh quotient (-)

Radius of curvature of the beam (m)

Load resistor (Ω)

Radius of curvature of the missile (m)

Ropt

Optimum load resistor (Ω)

Rm opt

Optimum load resistor in multiple impacts situation (Ω)

Effective compliance for the beam (Pa−1 )

sij

Components of the compliance tensor (Pa−1 )

sE ij

Components of the compliance tensor under constant electrical field (Pa−1 )

sD ij

Components of the compliance tensor under constant electrical displacement (Pa−1 )

Effective compliance for the missile (Pa−1 )

Effective compliance for the elastic material (Pa−1 )

sE p

Effective compliance under constant electrical field for the piezoelectric material (Pa−1 )

sD p

Effective compliance under constant electrical displacement for the piezoelectric material (Pa−1 )

Strain tensor (-)

Components of the strain tensor (-)

Time (s)

Contact time (s)

Time interval between two successive impacts (s)

Stress tensor (Pa)

Components of the stress tensor (Pa)

f1 T

Average longitudinal stress in the piezoelectric layer (Pa)

Longitudinal component of the displacement field (m)

Relative velocity of impact (m.s−1 )

Velocity at the position of the center of mass of the body attached to the cantilever (m.s−1 )

Velocity at the position of the center of mass of the body attached to the cantilever just before an impact (m.s−1 )

Velocity at the position of the center of mass of the body attached to the cantilever just after an impact (m.s−1 )

Velocity at the position of the missile just before an impact (m.s−1 )

Velocity at the position of the center of the missile just after an impact (m.s−1 )

Potential difference across the electrodes of the piezoelectric beam (V)

Transverse component of the displacement field (m)

Position dependent component of the transverse displacement (m)

Time dependent component of the transverse displacement (m)

Width of the piezoelectric beam (m)

Experiment label (-)

xii

Input displacement (m)

Position of the neutral axis (m)

Amplitude of the input displacement (m)

Macroscopic piezoelectric constant (C)

Real part of the quasi static lumped piezoelectric transformation factor (N.V−1 )

Γ0

Quasi static lumped piezoelectric transformation factor (N.V−1 )

Γc

Lumped piezoelectric transformation factor (N.V−1 )

Deflection along the length of the body attached to the cantilever (m)

δi

Indentation (m)

δg

Deflection at the position of the center of mass of the body attached to the cantilever (m)

δM

Position of the missile (m)

δR opt

Deflection at the position of the center of mass of the body attached to the cantilever at resonance and for the optimum load (m)

δA opt

Deflection at the position of the center of mass of the body attached to the cantilever at anti-resonance and for the optimum load (m)

εij

Components of the permittivity tensor (F.m−1 )

εTij

Components of the permittivity tensor under constant stress (F.m−1 )

εSij

Components of the permittivity tensor under constant strain (F.m−1 )

εTp

Effective permittivity under constant stress (F.m−1 )

εSp

Effective permittivity under constant strain (F.m−1 )

Efficiency of the energy conversion for the impact harvester (%)

θy

Shear angle (rad)

Timoshenko’s correction factor for the shear compliance (-)

Characteristic wavelength of the bending wave (m)

Amplitude of the displacement of the mass (m)

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Electromechanical coupling correction (-)

Polarization direction (-)

Density (kg.m−3 )

ρe

Average density (kg.m−3 )

Load parameter (-)

Ψopt

Optimum load parameter (-)

ΨR opt

Optimum load parameter at resonance (-)

ΨA opt

Optimum load parameter at anti-resonance (-)

Angular frequency of the input vibration (rad.s−1 )

ω s0

Short circuit resonance angular frequency of the piezoelectric beam (rad.s−1 )

ω o0

Open circuit resonance angular frequency of the piezoelectric beam (rad.s−1 )

ωe

Cut off angular frequency of the RC circuit (rad.s−1 )

ωm

Average angular frequency of the motion undergone by an impacted bender (rad.s−1 )

ωs

Angular frequency frequency of a piezoelectric beam shunted by a resistor (rad.s−1 )

Ω

Normalized angular frequency of the input vibration (-)

Ωc

Contact angular frequency (rad.s−1 )

Ωs0

Normalized short circuit resonance angular frequency of the piezoelectric beam (-)

Ωo0

Normalized open circuit resonance angular frequency of the piezoelectric beam (-)

Abbreviations BCB

Benzocyclobutene

CMOS

Complementary Metal Oxide Semiconductor

EMC

Electromechanical coupling factor

GEMC

Generalized electromechanical coupling factor

Integrated circuit

LPCVD

Low pressure chemical vapor deposition

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MEMS

Microelectromechanical system

PSD

Position Sensitive Detector

PVDF

Polyvinylidene fluoride

PZT

Lead zirconate titanate

SEM

Scanning Electron Microscopy

SPE

Small piezoelectricity approximation

SOI

Silicon on insulator

SSHI

Synchronous switch harvesting on inductor

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Contents

Acknowledgments

Abstract

Nederlandstalige samenvatting

vii

Symbols and Abbreviations

Contents

xvii

1 Introduction 1.1

Energy harvesting: state of the art . . . . . . . . . . . . . . . .

1.1.1

Solar energy . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.2

Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.3

Vibrations and motion . . . . . . . . . . . . . . . . . . .

Piezoelectric inertial vibration energy harvesters . . . . . . . .

1.2.1

Resonant systems: machine environment . . . . . . . . .

1.2.2

Non resonant systems: human environment . . . . . . .

Scope and organization of the thesis . . . . . . . . . . . . . . .

2 Theory and lumped model of piezoelectric laminated beams

1.2

1.3

2.1

History, basic definitions and linear constitutive equations of piezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

2.1.1

History and basic definitions . . . . . . . . . . . . . . .

2.1.2

Constitutive equations of linear piezoelectricity . . . . .

2.1.3

Dissipative and non linear effects in piezoelectric materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Relevant piezoelectric materials and corresponding simplifications of the constitutive equations . . . . . . . . .

The constitutive equations of piezoelectric laminated beams . .

2.2.1

Elastic laminated beams . . . . . . . . . . . . . . . . . .

2.2.2

Piezoelectric laminated beams

2.1.4 2.2

2.3

Constitutive matrix and electrical network representation of piezoelectric beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.1

Geometry of the harvesters and applied mechanical efforts 48

2.3.2

Concepts of the constitutive matrix and electrical equivalent network . . . . . . . . . . . . . . . . . . . . . . . .

Constitutive matrix and equivalent electrical network of the piezoelectric harvesters . . . . . . . . . . . . . . . .

Generalized electromechanical coupling factor . . . . . .

2.3.3 2.3.4 2.4

. . . . . . . . . . . . . .

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Manufacturing and primary characterization of MEMS piezoelectric harvesters 61 3.1

3.2

Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1

General description of the manufactured devices . . . .

3.1.2

Process flow . . . . . . . . . . . . . . . . . . . . . . . . .

Characterization . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.1

3.3

General concept for the determination of the network parameters . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.2

Quasi static characterization methods . . . . . . . . . .

3.2.3

Steady-state characterization methods . . . . . . . . . .

3.2.4

Transient characterization methods . . . . . . . . . . . .

3.2.5

Typical values of the network parameters and estimation of the material properties . . . . . . . . . . . . . . . . .

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

4 Design and analysis of the human environment vibration energy harvester 83 4.1

Modeling of the impact based harvester . . . . . . . . . . . . .

4.1.1

Development of the model . . . . . . . . . . . . . . . . .

4.1.2

Theoretical optimization of the generated power: analytical perspectives . . . . . . . . . . . . . . . . . . . . . .

Theoretical optimization of the generated power: numerical perspectives . . . . . . . . . . . . . . . . . . . . . .

Experimental measurements . . . . . . . . . . . . . . . . . . . .

102

4.2.1

Coefficient of restitution . . . . . . . . . . . . . . . . . .

102

4.2.2

Comparison of the model predictions with experimental measurements . . . . . . . . . . . . . . . . . . . . . . . .

103

Characterization of a prototype of the human environment harvester . . . . . . . . . . . . . . . . . . . . . . .

104

4.1.3 4.2

4.2.3 4.3

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

5 Design and analysis of the machine environment vibration energy harvester 109 5.1

5.2

5.3

Theoretical analysis of the harvesterâ&#x20AC;&#x2122;s output power . . . . . .

110

5.1.1

Resistive load . . . . . . . . . . . . . . . . . . . . . . . .

110

5.1.2

Alternative loads . . . . . . . . . . . . . . . . . . . . . .

120

Experimental characterization of the harvesters . . . . . . . . .

122

5.2.1

Output power . . . . . . . . . . . . . . . . . . . . . . . .

122

5.2.2

Non linear effects . . . . . . . . . . . . . . . . . . . . . .

125

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

6 General conclusions and future work

129

List of Publications

135

Bibliography

137

xix

Chapter

Introduction Recent years have seen important developments in the field of wireless sensor networks. A first important example of such networks are the so called body area networks [1, 2]. It consists of an array of sensors located on the human body (Figure 1.1). The sensors are aimed primarily at measuring relevant information related to the health of a person, such as blood pressure or oxygen level, and at transmitting the corresponding data to medical authorities through existing communication networks. Three examples illustrate the potential of this approach: continually monitoring blood pressure for patients with hypertension can significantly increase medication compliance [3], realtime processing of electrocardiograph traces can be very effective at revealing early stages of heart disease [4], closed-loop control of insulin administration for diabetic patients would significantly reduce the risk of hypoglycemia [5]. A second important example of application for wireless sensor networks consists in smart environments and automated buildings. Arrays of sensors can for example be used to monitor human presence or environment characteristics and trigger appropriate responses (automatic light switching, intrusion detection, fire alarm). Power autonomy of the sensors nodes is essential for the success of wireless sensor networks and requires the development of low-power electronics and long life energy sources other than electrochemical batteries. Depletion of batteries leads to a loss of functionality of the sensor node which can result in critical situations if not detected in due time. The costs of replacing batteries located in a remote or hostile environment is also high. Furthermore, electrochemical batteries contain dangerous chemicals possibly released in the environment because of the costs of recycling. Therefore, clean, renewable and portable energy sources gained attention during the last ten years and the so called energy harvesters were investigated as an alternative to electrochemical batteries. Energy harvesters act by converting part of the energy available in 1

1.1 Energy harvesting: state of the art

Figure 1.1 - The technology vision for body area networks [6].

the environment into useful electrical power. They are self replenished and do not need replacement. Common environmental energy sources are light, heat, vibrations, human motion or wind. This thesis is focused on harvesting energy from vibrations and human body motion. This chapter is organized in the following way: in a first section, the state of the art in the field of energy harvesting is presented. The second subsection is focused on energy harvesting from mechanical vibrations or motion using a piezoelectric transduction principle. The different architectures that can be implemented are first discussed in details. Then, the basic ideas related to the devices studied in this thesis and the motivations for their development are presented. Finally, the plan of this thesis is given.

1.1

Energy harvesting: state of the art

This work is focused on small scale energy harvesters. By using the term ”small scale”, we refer to both small power ratings (<1mW) and to small sized devices (cm3 ). Recent advances in small scale energy harvesting applications are described in this section. Different sources of energy and the corresponding harvesting devices are presented in separate parts. The description is focused on the most common and promising sources of energy; fuel cells, microturbines and ambient electromagnetic waves (other than light) harvesting are not treated.

1. INTRODUCTION

Some references related to the latter subjects are the work of Kolanowski [7], of Oâ&#x20AC;&#x2122;Hayre [8] and of Li [9].

1.1.1

Solar energy

The principle of photovoltaic cells is known since the work of Becquerel in the 19th century. The first solar cells were manufactured by Fritts in 1883. The modern age of photovoltaic cells started with the work of Pearson, Chapin and Fuller in the 1950â&#x20AC;&#x2122;s. The efficiency of their devices was however not exceeding 5% and research efforts have been realized for increasing their efficiency while decreasing the costs of manufacturing. The efficiency of solar cells depends on optical and electrical properties of the semiconductor; the optimization of their performances required then an intensive material research and optimization. A balance between cost and efficiency is also an important issue for the widespread use of solar cells. In the 1970s, Berman designed significantly less expensive solar cells by using a poorer grade of silicon and packaging the cells with cheaper materials. GaAs appeared in the following years as a very good, but very expensive, candidate for increasing the efficiency. Photovoltaic energy still costs more than classical power sources; different roads towards lowering its costs are under investigation: research companies try to either grow the silicon into a shape that eliminates most of the slicing requirements or to deposit photovoltaic material onto an inexpensive but rigid support structure such as ceramic, glass or steel. Also, thin film technologies (related to the IC field) and mass production allows a lower cost of fabrication.

1.1.2

Heat

A review of recent thermoelectric generators is given by Hudak in [10]. The short description presented here is derived from his article. Thermoelectric power generators produce electrical power from temperature differences between two substrates. They have been successfully developed since the beginning of the 20th century for large scale power generation by using waste heat from industrial processes [11]. The use of turbines is one of the simplest methods to extract energy from heat but it is not the sole. Indeed, electrical power can be generated from temperature gradients exploiting the Seebeck effect: when two faces of plates made of different conductor materials (i.e. a thermocouple) are set into mechanical contact, an electrical potential difference is observed between faces if they have different temperatures. Small-sized scale thermoelectric generators based on this effect have only been investigated during the past decade. Thermocouples for power generation are normally made by the association of a n and p type doped elements. The most widely used thermoelectric materials for decades have been n and p type alloys of bismuth telluride. Wristwatches powered only by thermoelectric effects have already been commercialized [12]. The development of MEMS manufacturing tech-

1.1 Energy harvesting: state of the art

nologies led recently to the investigation of less performing while cheaper and easier to integrate materials such as silicon germanium [13]. An interesting application of thermopiles in the field of energy harvesting was recently proposed by Torfs and Leonov [14]: a commercial pulse oximeter is fully powered by the heat dissipated by the human body. The full system is able to transmit data to a neighboring computer. As a rough estimate, the achievable power density of thermoelectric generator in human environment is in the range of 20 µW/cm2 .

1.1.3

Vibrations and motion

Mechanical vibrations or motions are present almost everywhere and are therefore attractive sources of energy for producing electrical power. The dynamo of a bike is a well known example of motion energy harvesting. Self powered wristwatches based on a rotating heavy mass and on an electromagnetic principle were also developed during the 19th and 20th century. The conversion of mechanical to electrical energy is achieved through electromechanical transducers: the most common implemented transduction mechanisms are electromagnetic [15–19], electrostatic [20–23] and piezoelectric [24–31]. Alternative transduction methods such as magnetostriction [32, 33] were also proposed but remain marginal. The electromagnetic transduction is based on the classical Faraday’s law of induction which states that the motion of a coil in a magnetic field generates an electrical current in the coil. Capacitive transduction is obtained with a movable plate capacitor with one of the electrodes connected to a source of electrical charges. According to Gauss’ law, a change in the values of the capacitor results in a motion of charges so that a useful current can be produced. Finally, the piezoelectric mechanism depends on the crystalline nature of the material; piezoelectric materials develop electrical charges when deformed. Motion-driven generators fall into two distinct categories: those using direct application of an external force and those using inertial forces acting on a movable proof mass. A large amount of energy can easily be extracted from the former kind of harvesters and several applications focused on harvesting energy from human motion have already been commercialized. The best known example is the harvesting shoe [34, 35]. An illustration of an electromagnetic implementation of this idea is given in Figure 1.2. The power generated by these shoes exceeds a few watts, enough to supply energy to many kinds of applications. Several wind-up based charging systems were also recently proposed with GSM or flashlight loader amongst them. One of the most promising applications consists in the human powered laptop associated with the ”One laptop per child” project. The charging mechanism presented in Figure 1.2 is based on a hand crank system that has been recently abandoned due to its poor efficiency in favor of a more compact off-laptop design that uses a pull string to spin a small generator. As a last example of direct force energy harvesters, Keawboonchuay [26] developed piezoelectric generators that can be incorporated into a special kind of ammunitions. Energy is generated when the munitions impact the ground. Poulin gives some additional examples of direct

1. INTRODUCTION

Electromagnetic generator Figure 1.2 - Examples of direct force energy harvesters: hand crank GSM reloader, hand crank powered laptop, electromagnetic shoe.

force harvesters in [36]. Most of the direct force motion energy harvesters are however bulky and do not fulfill the needs of wireless sensor networks. On the opposite, harvesters based on inertial mechanisms can be made relatively small. As for the former type, inertial generators are most of the time based on electromagnetic, electrostatic or piezoelectric transduction. The classical design of inertial harvesters is based on a resonant scheme similar to the one implemented in accelerometers and the device should oscillate along one of its fundamental modes in order to deliver maximum power. In a first approximation, these systems can be represented by a dashpot mass spring system coupled with some kind of electrical damping corresponding to the energy harvesting process [37]. Such harvesters received a lot of attention over the last years. Most of these harvesters are manufactured by conventional technologies. However, MEMS fabrication was recently implemented with the advantage of low cost mass production [20, 24]. Reviewing the literature on resonant inertial energy harvesters is a colossal task due the abundance of articles. Fortunately, general reviews can be found [38–40]. This thesis is focused on piezoelectric energy based devices analyzed in details later. In the remainder of this section, a brief overview of the characteristics of environmental vibration sources and of the principles implemented for inertial harvesters are given. Ubiquitous vibration sources vary considerably in dominant frequency. The resonant inertial harvesters should be tuned to this frequency. Roundy [41] presents the capabilities for a number of vibration sources indicating that the

1.1 Energy harvesting: state of the art

amplitude and frequency varies from 12 m.s−2 at 200 Hz for a car engine compartment to 0.2 m.s−2 at 100 Hz for the floor in an office building with the majority of sources measured having a fundamental frequency in the range 601000 Hz. Vibrations present in most environments are not purely harmonic but rather broadband. In their basic form, the resonant harvesters described above produce a reasonable amount of electrical power only if they are excited at their fundamental frequency and they are then not adapted to extract efficiently the energy from broadband input vibrations. A wider frequency response can be obtained using higher order resonators and multiple mass spring systems. Designs of this type are proposed in [42]. In [43], Roundy proposed a methodology for tuning resonant inertial harvesters by varying the spring constant or the value of the proof mass. Challa [44] described a passively tunable device in which the resonant frequency of a piezoelectric cantilever is shifted by the application of magnetic forces. Arrays of piezoelectric cantilevers with different resonance frequencies to broaden the response spectrum were also proposed [45]. The resonant or broadband harvesters presented above are adapted to relatively high frequency vibrations occurring in an ”industrial” environment. In the case of the human body, it is not possible to speak of vibrations but rather of motion characterized by low frequencies and high amplitude. It is difficult to design devices resonating at such low frequencies. Also, oppositely to the industrial vibration situation, the amplitude of the external motion is larger than the displacement of the proof mass. Miao analyzes this problematic in [46] and proposes an alternative design based on electrostatic transduction depicted in Figure 1.3(a). In this system, the variable capacitor is made of a fixed and movable plate, the latter constituting also the proof mass of the system. The proof mass sticks to the fixed plate through electrostatic forces till the external acceleration results in an inertial force higher than the sticking one. At this moment, the mass is released and discharging occurs through contact pads when it has reached the other plate. A hybrid piezoelectric and electrostatic low frequency energy harvester is proposed in [47]. It is an electrostatic oscillator suspended by piezoelectric springs. Rotational rather than linear internal motion is also adapted for low frequencies of the input motion, with wristwatch generators or bike dynamo as the most notable examples. Analysis of the possible operating modes and power limits of rotating mass generators are presented in [48] and [49]. Alternative spring designs which allow reducing the operating frequency are also proposed by Hu [50]. Finally, several principles of conversion of low input frequencies were presented: Kulah [51] describes a device made of a large mass oscillating at low frequencies associated with high frequencies oscillating cantilevers Figure 1.3(b). The mass is ferromagnetic and small cubes of metal were attached to the high frequency cantilevers. When the large magnetic mass oscillates, it alternatively catches and release the cantilevers which undergo free vibrations when not attract by the mass. Transduction is realized through electromagnetic means. A ratchet-pawl type system following a similar approach but implementing piezoelectric transduc-

1. INTRODUCTION

(a)

(b) Figure 1.3 - Examples of inertial harvesters for low frequency vibration energy harvesting proposed by (a) Miao [46] and (b) Kulah [51].

tion was proposed in [52]. Also, as shown by several authors [53â&#x20AC;&#x201C;59], impact based harvesters are adapted to low frequency input motion. This principle is investigated into details in this thesis.

1.2

Piezoelectric inertial vibration energy harvesters

Different designs should be used to harvest energy from the vibrations present in industrial environment and from the motion of the human body. Resonant harvesters are adapted to the former case, while non resonant devices produce in general more power in the latter situation. Therefore, the present section is divided into two parts each dealing with the different principles. Note that most of the points that are rapidly presented in this section are investigated in details in further chapters.

1.2 Piezoelectric inertial vibration energy harvesters Piezoelectric capacitor

Input vibration

Proof mass

Elastic beam

(a) Effective mass

Piezoelectric element Piezoelectric element Energy V harvesting Energy circuit V harvesting circuit

Displacement limit +/‐ Λ

Parasitic damping Parasitic damping

Structural stiffness

(b) Figure 1.4 - (a) Schematic of the typical structure for vibrations energy harvesting, (b) lumped model of a resonant piezoelectric harvester.

1.2.1

Resonant systems: machine environment

Inertial resonant systems based on piezoelectric transduction are typically made of an elastic structure attached to piezoelectric elements such as patches. As illustrated by Figure 1.4(a), the most common implementation for frequencies of input vibrations in the range of hundreds of Hertz consists of an elastic cantilever supporting one or several piezoelectric layers sandwiched between metallic electrodes. Such structures are referred to as unimorphs or bimorphs [60]. The piezoelectric capacitors are connected to a load circuit in which energy is stored or dissipated. A proof mass is attached to the tip of the beam. An external vibration is applied to the clamped end of the cantilever and results in an inertial force on the proof mass. Extensive reviews of such piezoelectric harvesters have been proposed in the literature by Sodano [61], Anton [62] and Cook-Chennault [63]. Overall, such a system can be approximated by a dashpot mass spring system coupled to some kind of damping resulting from the piezoelectric energy harvesting process. The corresponding mechanical model is given in Figure 1.4(b). The first investigations of such a structure for power generation were proposed by Glynne-Jones [64] and Elvin [65]. Kasyap [66] proposed a piezoelectric vibration energy harvester designed for producing energy from acoustic sources. From this moment, the race for developing both theoretical models and prototypes of resonant piezoelectric harvesters was launched. Sodano [67] was the first to demonstrate that this type of system was able to

1. INTRODUCTION

recharge commercial batteries. He also compared the efficiency of PZT (Lead Zirconate Titanium) and of piezoelectric fibers for the same application in [61]. He found that PZT was a better candidate for power generation. Piezoelectric materials suitable for energy harvesting were investigated in the following years. It was understood that the electromechanical coupling factor is a crucial parameter and should be made as high as possible. The electromechanical coupling factor represents the mechanical energy transformed into electrical energy or vice versa during a quasi static cycle defined in [68]. For a same material, different values of the electromechanical coupling factor are found depending on the direction of the deformation and of the developed electrical field. The electromechanical coupling factor is labeled as kij . The indices i and j refer respectively to the direction of the developed electrical field and the direction of the deformation (in the principal frame of reference). Ikeda [69] distinguished 13 different possible values of the electromechanical coupling factor but the most commonly applied modes of deformation are the 31 (direction of deformation perpendicular to the developed electrical field) and the 33 (direction of deformation parallel to the developed electrical field). Detailed discussions on the electromechanical coupling factor are developed in the next chapters. Ceramic PZT and its variants are still the most used materials because of their unequally attained electromechanical coupling factor and of the maturity of their manufacturing process. However, PZT ceramics are brittle and susceptible to crack propagation. Therefore, piezoelectric polymers received attention during the last decade. A common piezoelectric polymer is polyvinylidene fluoride (PVDF). PVDF is extremely flexible when compared to PZT. Lee [70, 71] developed electroded PVDF films. Mohammadi [72] developed a fiber-based piezoelectric (piezofiber) material consisting of PZT fibers of various diameters (15, 45, 120 and 250Âľm) that were aligned, laminated, and molded in an epoxy. Piezofiber power harvesting materials have also been investigated by Churchill [73] who tested a composite consisting of unidirectionally aligned PZT fibers of 250Âľm diameter embedded in a resin matrix. Because of their low electromechanical coupling factor, piezoelectric polymers and fibers however perform worse than PZT ceramics. They have applications requiring high flexibility such as the piezoelectric shoe described in Figure 1.2. Recently, the development of MEMS technologies inspired the development of materials obtained by thin film deposition processes. PZT received again most of the attention, but alternative materials such as AlN (Aluminum Nitride) [24, 74] were also proposed because they are easier to integrate into conventional IC manufacturing processes. The investigation of high coupling piezoelectric materials is not the only element required for developing high performance vibration energy harvesters. The design of the transducer itself and the understanding of its behavior are also crucial points. Therefore, a lot of research on the design of such devices was done over the last decade. The simplest structure for piezoelectric vibrations harvesters consists in a cantilever supporting one or several piezoelectric capacitors. The performances of these structures are optimized by maximizing

1.2 Piezoelectric inertial vibration energy harvesters

their generalized electromechanical coupling factor. The generalized electromechanical coupling factor represents an extension for composite structures of the electromechanical coupling factor concept. This parameter is discussed in details later. Also, the parasitic dissipations have intuitively to be reduced [75]. Cho [76, 77] has shown that the generalized electromechanical coupling factor was depending on the electrode coverage of the piezoelectric layer in the case of rectangular membranes. In case of a cantilever, the author of this thesis demonstrates that the optimum of the generalized electromechanical coupling factor is obtained by choosing a particular ratio between the thicknesses of the piezoelectric and elastic materials. It was shown by Karakaya [78] that the residual stress can also have an influence on the generalized electromechanical coupling factor. Other geometries than a simple cantilever were also studied. Mateu [79] and Roundy [43] analyzed the possibility of using a triangular cantilever rather than a rectangular one. They both found that a triangularly shaped piezoelectric beam delivered more power for the same input force. Ericka [80] has also investigated power generation from a circular membrane. Kim [81] developed a novel circular configuration for power harvesting called a piezoelectric ’cymbal’ in which two dome-shaped metal end-caps are bonded on either side of a piezoelectric circular plate. Using this configuration, the stress applied to the piezoelectric material when compressed is more evenly distributed than in a conventional structure. In this way, the efficiency of the power harvester increases. Rather than modifying the geometry of the typical rectangular cantilever, Mossi [82] proposed a configuration in which an initial stress is applied to the cantilever. In this case, the cantilever has an initial curvature from which the performances in terms of power generation depend. Baker [83] proposed a new configuration in which a piezoelectric beam is compressed and fixed at both ends with pin connections. This so-called ”bistable” system device generates power by switching from one stable mode to another. Experimental results show that the bistable beam has a broader band of response than the classical configuration. Cornwell [84] described the concept of attaching an auxiliary structure for being able to actively modify the resonance frequency of the harvester. Following the same principle, Roundy [43] further investigated the idea of tuning the resonant frequency of a piezoelectric device to match the frequency of ambient vibrations. It was however shown that an active tuning system would never result in a net increase of the delivered power. In order to eliminate this problem, Wu [85] proposed a passive tuning approach based on two piezoelectric cantilevers and a micro controller. One of the piezoelectric bender was used for tuning, while the other served as generating element. The performances of a piezoelectric energy harvester can be improved through development of high performances materials and on a dedicated electromechanical design. A last important point for optimizing such a system consists in the characteristics of the load circuit. Indeed, the basic load circuit typically implemented, mainly for verification of the physical model, consists in a simple resistor. It is however possible to improve the performances and to condi-

1. INTRODUCTION

tion effectively the generated power by using more elaborated load circuits. Ng [86, 87] developed a power harvesting circuit to extract energy from a cantilever beam piezoelectric harvester. The voltage generated by the piezoelectric material is first rectified by a diode and then stored into a buffer capacitor. A voltage monitoring circuit is connected to the buffer capacitor and releases energy from the capacitor in burst mode. Han [88] developed a power harvesting circuit made of two stages consisting of a rectifier followed by a DC-DC converter. When experimentally compared to the traditional diode-resistor rectifier, the proposed conditioning circuit extracted over 400% more power. Ottman [89] and Lesieutre [90] implemented a switching DC-DC step-down converter in the power harvesting circuit. It was shown that the advantages of this method depend on the frequencies of the input vibration. Ammar [91] developped an adaptive algorithm for controlling the duty cycle of a DC-DC buck converter. When comparing the charging time of a battery, it was demonstrated that the proposed circuit led to faster charging than the conventional rectifier. Lefeuvre [92] proposed a method in which the extraction of the electric charge from a piezoelectric device is synchronized with the system vibration. The circuit used for this approach is made of a rectifying diode bridge and of a flyback switching mode DC-DC converter. A control circuit senses the voltage across the diode rectifier and activates the flyback converter when that voltage reaches its maximum. The charges are at this moment allowed to flow in a battery. When the electric charges on the piezoelectric element become zero, the control circuit stops the converter and the corresponding energy transfer. Such a circuit gave experimentally an important increase of the power when compared to a simple load resistor. In further studies, Badel [93] and Guyomar [94] investigated another method of synchronizing the electric charge extraction with the vibrations of the system. This new non linear technique was labeled as â&#x20AC;?synchronous switch harvesting on inductorâ&#x20AC;? (SSHI). The SSHI circuit contains an electronic switch triggered on the maximum and minimum displacements of the piezoelectric device. The switching device associated in series with an inductor is placed in parallel or series with the piezoelectric capacitor before the rectifying diode bridge. The SSHI method was theoretically and experimentally compared to a standard circuit containing only a diode bridge rectifier and capacitor. It was shown that the SSHI circuit is capable of multiplying the efficiency by a factor 4 over the standard circuit in the case of low generalized electromechanical coupling factor. All structures described above are based on the 31 deformation mode. The coupling coefficient of most piezoelectric materials is higher in 33 than in 31 modes. Therefore, harvesters based on 33 coupling coefficients were developed. The simplest example consist in a stack of piezoelectric capacitors as illustrated by Figure 1.5(a). These structures are however very stiff and their resonance frequencies are well above those found in an industrial environment. They are more adapted to direct force harvesters than to inertial ones. Jeon [95], Zhou [96] and Dutoit [97] proposed an interesting alternative approach based on a cantilever supporting a piezoelectric capacitor with interdigitated electrodes as

1.2 Piezoelectric inertial vibration energy harvesters

Â Â

(a)

(b)

Figure 1.5 - Examples of structures making use of the 33 mode of deformation. (a) Piezoelectric stack, (b) piezoelectric cantilever with interdigitated electrodes [97].

given in Figure 1.5(b). This electrodes configuration allows using the 33 mode with a relatively compliant structure. To conclude this section, an overview of the performances of some of the existing piezoelectric vibration energy harvesters based on an inertial resonant principle and on a cantilever configuration assuming a simple resistor as load circuit is given in Table 1.1. The proposed table is adapted from the review described by Mitcheson [38]. Most harvesters presented in Table 1.1 are based on ceramic PZT but a few results of thin film materials are also included. PZT is again the material of choice in this case, but problematic from a point of view of process integration. Therefore, AlN was also recently investigated. The results proposed in Table 1.1 suggest that the performances of AlN in terms of energy harvesting are much below those of PZT. This is due to the fact that AlN devices, developed later, were not properly engineered, and it is not related to fundamental material properties. It was demonstrated in [24] that for a device operating at the same frequency, the output power delivered for PZT and AlN based harvesters is similar assuming that parasitic dissipations are not too large. However, the former material delivers more current while the latter results in higher voltages. Comparing the performances of the different devices described in Table 1.1 is not easy, as the output power depends strongly on the values of the proof mass, the frequency of the input vibration and the volume of the device so that it is difficult to perform a benchmarking of the existing systems. Therefore, Mitcheson [38] defined a metric that he designated as harvester effectiveness Eff. It consists in the ratio of the generated power over the maximum achievable power when the displacement of the proof mass me is equal to Î&#x203A; (Figure 1.4b).

1. INTRODUCTION

Table 1.1 - Comparison of the performances of existing inertial piezoelectric vibration energy harvesters. Generator Proof Input Input Manufacturing Raw Reference volume mass frequency acceleration technology and power Eff (cm3 ) (g) (Hz) (m.s−2 ) material (µW) (%) [64] 0.53 80.1 Ceramic PZT 1.5 [28] 1 8.5 120 2.3 Ceramic PZT 80 7.3 [28] 1 7.5 85 2.3 Ceramic PZT 207 14 [28] 1 8.2 60 2.3 Ceramic PZT 365 34 [29] 4.8 52.2 40 2.3 Ceramic PZT 1700 [30] 9 50 1 Ceramic PZT 180 [87] 0.2 0.96 100 72.6 Ceramic PZT 35.5 This work 0.8 3.4 75 1 Ceramic PZT 50 12 [31] 0.0006 0.0015 609 64 Thin film PZT 2.16 [24] 0.3 0.035 1798 23 Thin film PZT 40 2.7 This work 0.3 0.011 1383 5 Thin film PZT 3 16 [24] 0.3 0.035 320 0.2 Thin film AlN 0.05 20 This work 0.3 0.011 660 1 Thin film AlN 0.05 5 [98] 0.0002 1511 4 Thin film AlN 0.026

Eff is defined by (1.1) in which A0 and ω are respectively the amplitude and frequency of the input acceleration. The values of Eff are reported in Table 1.1 when the data given in the references were sufficient to compute it. The harvester effectiveness for the micromachined devices has the same order of magnitude than the macroscopic devices. For the devices presented in [24], it varies between 2.7% for the tested PZT element and 16% for the AlN one. The latter harvester efficiency is similar to the values obtained for macroscopic PZT based devices and it appears therefore that AlN is a promising material for micromachined energy harvesters. Eff = 2

Measured power me ωA0 Λ

(1.1)

In this section, an overview of the operating principle and designs related to resonant inertial piezoelectric vibration harvesters was presented. These systems are adapted for high frequency and low amplitude vibrations found mainly in industry. In the next section, inertial harvesters relevant for producing energy from low frequency and high amplitude motion characteristic of the human body are described.

1.2.2

Non resonant systems: human environment

Harvesters based on a resonant scheme are not adapted for the characteristics of human motion and devices based on other principles have to be developed for human powered sensor nodes. These systems are referred to as non resonant harvesters. They did not get much attention from researchers and only a few authors presented piezoelectric based inertial non resonant harvesters. Umeda [53] was the first to investigate non resonant piezoelectric harvesters

1.3 Scope and organization of the thesis

by analyzing the transfer of energy from a steel ball impacting a piezoelectric membrane. The piezoelectric transducer consisted of a 19 mm diameter, 0.25 mm thick piezoelectric ceramic bonded to a bronze disc 0.25 mm thick with a diameter of 27 mm. This work determined that the optimum efficiency, assuming a purely resistive load circuit, was about 10% for an optimum load of 10 kâ&#x201E;Ś. Most of the energy was returned to the ball which bounces off the transducer after the initial impact. Later research further explored the feasibility of storing the charges developed under impact on a capacitor or battery [53]. The output of the generator was connected to different values of load capacitors via a bridge rectifier. It was shown that the performances of the generator to charge the capacitor depended upon the value of the latter and on its initial voltage. The generator was also attached to nickel cadmium, nickel metal hydride and lithium ion batteries. The charging characteristics were found to be unaffected by the battery type. Umeda [55] proposed a commercial application of the principle by successfully implementing a self powered door alarm system illustrated in Figure 1.6(a). Atsushi [99] deposited a patent for rotary structures containing small steel balls attached to a central axis by straight metal wires. When the frame of the device is rotated, the balls impact on piezoelectric membranes located on the inner surface of the frame. Cavallier [58] and Takeuchi [59] studied experimentally an equivalent device but obtained a very low efficiency because of a badly designed piezoelectric converter. Anecdotic applications were also developed such as for example the piezoelectric ear ring presented in Figure 1.6(b) which was used to power electroluminescent diodes attached to the rings. Finally, the author of this thesis determined analytically the optimum parameters of the piezoelectric transducer for shock excitation in [56]: it was shown that the generalized electromechanical coupling factor and the mechanical quality factor were the most important parameters for power generation. To our knowledge, a single other approach to non resonant harvesters based on piezoelectric transduction has been described: Rastegar [52] investigated a device where a low frequency vibrating mass spring system transfer its energy to piezoelectric cantilevers through a ratchet type interaction: as illustrated in Figure 1.7, an energy transfer teeth attached to a slow moving mass allows to successively deflect and release piezoelectric benders. While the large mass is not in the neighborhood of the cantilevers, they are allowed to vibrate freely along their own resonance frequency. This device was however only analyzed in a theoretical way and no experimental measurements are available.

1.3

Scope and organization of the thesis

It was shown that inertial piezoelectric vibration harvesters are an interesting approach for powering sensor nodes in wireless networks. Electrical energy is provided through conversion of the ambient vibrations or motion and the need for the replacement of electrochemical batteries is eliminated. Many different

1. INTRODUCTION

(a)

(b)

Figure 1.6 - Examples of non resonant inertial harvesters based on piezoelectric transduction. (a) Self powered door alarm system [55], (b) piezoelectric ear rings.

Figure 1.7 - Non resonant harvester proposed by Rastegar [52].

1.3 Scope and organization of the thesis

approaches have been proposed for high frequency energy harvesting. However, few MEMS devices have been proposed and one of the major goals of this thesis is to manufacture and investigate such a harvester. The second goal of this thesis is propose a device adapted to harvest energy from the low frequency motion observed on the human body. Both harvesters developed are based on the piezoelectric transduction effect and more particularly on piezoelectric laminate bending structures. The principles of piezoelectricity are well understood and several models representing this type of transducers already exist. However, energy harvesters have mainly been investigated during the last decade and some confusions or misunderstanding are often found in the models which represent piezoelectric energy harvesters. Particularly, the approximations related to the classical model are generally not well taken into account. Therefore, the second chapter is dedicated to the description of the theory of piezoelectric laminates. The equations describing the behavior of piezoelectric benders are first derived; based on their solution, lumped models in the form of impedance matrices and equivalent electrical networks are proposed. In the third chapter, the fabrication by MEMS technologies of piezoelectric cantilevers designed for energy harvesting in a resonant configuration is presented. Such devices are manufactured using both thin film AlN and PZT piezoelectric layers. The basic characterization of the produced devices consists in determining experimentally the lumped parameters derived in Chapter 2 and is also proposed in Chapter 3. No experimental procedure for determining these parameters has been proposed in literature. Therefore, we developed a complete plan of experiments to this aim. This characterization can be realized by a combination of static, transient and steady state measurements and is performed on the fabricated MEMS piezoelectric benders but also on commercial ceramic PZT based structures. Finally, the material properties of the piezoelectric materials are extracted from the measured values of the parameters of the lumped model. Chapter 4 is dedicated to the modeling, fabrication and characterization of a low frequency inertial harvester designed for producing power from human motion. The developed device is based on the impact of a rigid body on piezoelectric benders. No published work proposing a detailed model of a piezoelectric impact harvester including the description of the impact mechanism and of the resulting behavior of the piezoelectric bender exists. Therefore, a complete model of such a vibration harvester is developed and validated by a series of measurements performed on a macroscopic prototype. Note that this harvester is realized by conventional precision machining and the implemented piezoelectric transducers are bought from commercial companies. An output power of 600 ÂľW is obtained (using a resistive load) for a device of dimensions 3.5*2*2 cm weighting 60 g placed on the hand of a person and shaken at a frequency of approximately 10 Hz with 10 cm amplitude. Also, a power of 47 ÂľW is measured when the harvester is rotated of 180â&#x2014;Ś each second.

1. INTRODUCTION

In Chapter 5, the model of inertial resonant piezoelectric harvesters adapted to high frequencies corresponding to a machine environment is developed and analyzed based on the theory of piezoelectric laminates presented in Chapter 2. Design parameters for optimizing the performances of the device are derived. Resonance and anti resonance behaviors are theoretically studied in details. Efficient load circuitries are also discussed. Then, the fabricated AlN and PZT MEMS resonant harvesters are characterized experimentally. It is found that the developed model results in a good approximation of the measured data. Raw output powers in the range of 40 ÂľW are measured, which is enough to power simple electronic applications. Finally, some elements related to the future investigations are given: particularly, non linear effects due to large input vibrations or parasitic dissipations related to the presence of a package around the harvesters are discussed and investigated experimentally.

1.3 Scope and organization of the thesis

Chapter

Theory and lumped model of piezoelectric laminated beams Piezoelectric laminated beams constitute the transduction element between mechanical and electrical energy in the two types of investigated energy harvesters. The model of such transducers is developed in this chapter. It is organized as follow: in the first section, a short literature review and history of piezoelectricity is proposed, followed by the derivation of its linear constitutive equations. Common non linear effects and intrinsic dissipations occurring in piezoelectric materials are also discussed. The section is concluded by the presentation of the piezoelectric materials used in this thesis and of the corresponding simplifications of the constitutive relations. The second section describes the derivation of the equations governing the dynamics of multi layered beam structures. Finally, in a third section, analytical solutions of the previous equations are developed for the particular situation of a cantilever loaded by a distributed mass. These solutions are arranged in the form of an impedance matrix and an electrical equivalent network. Some parts of the derivations described can be found in classical textbooks. However, the field of piezoelectric harvesters is a recent area of research and a publication proposing a detailed derivation of the piezoelectric multilayer beam equations in view of such applications does not yet exist.

2.1 History, basic definitions and linear constitutive equations of piezoelectricity

Force

O‐ Si+

Si+

‐

O‐

Si+

Si+ ‐

O‐ Si+

Voltage

Figure 2.1 - Illustration of the piezoelectric effect as understood by Kelvin.

2.1

2.1.1

History, basic definitions and linear constitutive equations of piezoelectricity History and basic definitions

Coulomb proposed in the 18th century a conjecture which states that electricity might be produced when a mechanical pressure is applied to a material. Hauy and Becquerel performed experiments in order to prove this conjecture. However, it was not possible in their measurements to make the difference between the electrical charges created by friction or contact electricity and those resulting from a possible electromechanical phenomenon. The Curies were, in 1880, the firsts to demonstrate a relation between the symmetries in crystalline materials and the electrical charges appearing at the surface of some crystals when mechanically stressed. Their experiments were performed on Rochelle salt, tourmaline and quartz amongst others. The effect discovered by the Curies is commonly referred to as the direct piezoelectric effect, i.e. electrical charges results from mechanical efforts. The so-called converse piezoelectric effect (mechanical stresses results from applied electrical field) was mathematically derived from thermodynamics by Lippmann in 1881. The same year, the Curie brothers confirmed experimentally the existence of the converse effect and started developing a few laboratory applications. In 1893, Lord Kelvin proposed an atomic model to explain the observed phenomena. The piezoelectric effect as understood by Kelvin is illustrated in Figure 2.1 considering a quartz unit cell. When no mechanical stress is applied to the crystal, the positive and negative electrical charges present on the different atoms share the same barycenter, so that no net electric displacement or field is observed at the surface of the cell. The barycenter of the positive and negative charges does not coincide when the cell of quartz is deformed, so that an electrical polarization is developed at the surface of the crystal. It was understood by Kelvin that piezoelectricity occurs only in non centrosymmetric crystals. The complete definition of the crystal classes in which piezoelectric effects occur was published by Voigt in 1910 (Lehrbuch der Kristallphysik), on the basis of the work done by Duhem, P¨ockels and Neumann. As

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 21

32 crystal symmetry groups

21 non centrosymmetric

11 centrosymmetric

Non piezoelectric

20 piezoelectric

1 non piezoelectric

10 pyroelectric

Ferroelectric

Non ferroelectric

Figure 2.2 - Piezoelectricity and crystal classes.

illustrated by Figure 2.2, he determined that from the 21 non centro-symmetric crystal classes, 20 of them lead to piezoelectric effects. It was also shown that 10 of the piezoelectric crystal classes exhibit pyroelectric effects, i.e. they possess a spontaneous electrical polarization which depends on the temperature. Above a certain temperature referred to as Curie temperature, a pyroelectric crystal undergoes a phase transition and does no longer exhibit spontaneous polarization. In the next decades, it was discovered that the spontaneous polarization can be reversed under the action of a strong applied electrical field in some of the pyroelectric crystals. By analogy with ferromagnetism, this effect has been designated as ferroelectricity. The work of Voigt became a standard reference for piezoelectricity and the tensor algebra and notations introduced are today still widely in use. The complexity of the mathematics required to design piezoelectric systems led to a delay in the development of practical applications. The first engineering application of the piezoelectric effect was made by Langevin in 1917. In the World War I context, his goal was to develop a system able to detect submarines and his work on piezoelectric transducers led to the development of SONAR (SOund Navigation And Ranging) and to further advances in the field of ultrasonics. During the 1920-40’s, most of the classic piezoelectric applications (microphones, accelerometers, ultrasonic transducers, bender element actuators, phonograph pick-ups, signal filters, etc.) were conceived and brought into practice. Some important names associated with this period are Cady, who developed the quartz oscillator and published reference books in the field [100– 103], Butterworth and Van Dyke, who proposed the first electrical network model of a piezoelectric resonator thus simplifying the design of piezoelectric transducers [104–106], Mason who extended the electromechanical representation of piezoelectric systems and developed alternative formulations of the

2.1 History, basic definitions and linear constitutive equations of piezoelectricity

piezoelectric theory [107–111]. Until the 40’s, piezoelectricity had only been observed in crystals. A technologic revolution occurred when it was discovered during World War II (in the U.S., Japan and the Soviet Union), that certain ceramic materials exhibited dielectric and piezoelectric constants having the same order of magnitude than those of common cut crystals. Piezoelectric ceramics are prepared by heat treatment of metallic oxide powders and have a polycrystalline structure. They require an electrical poling process in order to exhibit a piezoelectric effect. Indeed, as described in Figure 2.3(a), the material consists before poling of misaligned domains. When the material is deformed, electrical charges appears locally at the boundaries between the domains, but due to charge cancellation, no net piezoelectric effect is observed at the surface of the sample. The poling process is illustrated in Figure 2.3(b): metallic electrodes are attached at two opposite faces of the sample and a strong electrical field is applied on the electrodes so that the different domains partially align their polar axis along the electrical field. After removing the electrical field, the different domains keep a certain alignment and a noticeable piezoelectric effect is observed. Note that the poling process is only relevant for polycrystalline ferroelectric materials. A piezoelectric but not ferroelectric polycrystalline material (as for example quartz) with randomly oriented grains can not be poled; it can only exhibit macroscopic piezoelectric properties if the growth orientation of the different grains is well controlled. Independently of the required poling step, manufacturing technologies much cheaper than those existing for growing crystals were developed for piezoelectric ceramics in the 40’s and a renewed interest was observed at the level of scientific and engineering research. In the following years, the barium titanate and lead zirconate titanate (PZT) class of materials were developed. During that period, the relations between the electromechanical coupling and the perovskyte crystalline structure corresponding to PZT material were also understood. Doping of these materials with metallic impurities was also investigated and successfully implemented. Some of the most important applications related to this era are powerful sonar, piezo ignition systems, small and sensitive microphones, relays and signal filters. From 1965 till the beginning of the 80’s, most of the engineering successes were achieved in Japan. The field of applied piezoelectricity shifted from purely military and academic interests to a wide variety of everyday life applications as for example smoke and intrusion alarms or TV remote controls. The 80’s have then seen the opening of new commercial markets for piezoelectric applications in each part of the world. Applications were found in the field of automotive, actuation, aeronautics and more recently of energy harvesting. A new field of application for piezoelectric materials has been induced recently by the industry of MEMS. In terms of energy density, the piezoelectric effect is not affected by miniaturization, so that it constitutes a principle of choice for designing small scales actuators, sensors or vibration energy harvesters. Strong efforts have been done in order to integrate piezoelectric materials in the form of thick or thin films into silicon wafer based batch processes. Non ferroelectric

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 23

Electrodes

Random dipoles

Monocrystal with single polar axis

Polarization

Polycrystal with random polar axis (a)

Surviving polarity (b)

Figure 2.3 - (a) Comparison between mono and poly-crystalline material, (b) illustration of the poling process for polycrystalline material.

and piezoelectric materials such as for example aluminum nitride (AlN) or zinc oxide (ZnO) are fairly easy to integrate into a conventional IC process flow. On the opposite, strong difficulties are encountered for PZT type materials which require high temperature processing. Furthermore, contamination problems are encountered with PZT. In modern applications, piezoelectric transducers exist in a wide variety of shapes and are excited in a wide variety of vibrations modes. Some of these shapes, modes of vibration, corresponding range of frequency and applications are presented in Figure 2.4. Flexural vibrations piezoelectric transducers (cantilever, membranes) allow designing systems performing in a range of a few Hz till tens of kHz. Typical applications related to this mode of vibration are buzzer, cooler, motion actuators and vibration energy harvesters as it will be discussed in this thesis. Length, thickness and area vibrations of piezoelectric plates or discs are used for filters and resonators in the range of tens of kHz till a few MHz. Filters and resonators performing from a MHz till tens of MHz are obtained by designing thickness shear or thickness trapped vibration. Finally, in order to operate at higher frequencies (above GHz), surface acoustic waves are implemented. The two types of vibration energy harvesters that are analyzed in this thesis are based on flexural piezoelectric elements and more precisely on beam structures. In the remainder of this chapter, the basis of the model of the piezoelectric benders which will be implemented later to analyze and optimize the performances of the energy harvesters is developed.

2.1 History, basic definitions and linear constitutive equations of piezoelectricity

Vibration mode

Frequency (Hz)

Applications

1K 10K 100K 1M 10M 100M 1G

Flexural vibrations

Lengthwise vibrations

Piezoelectric buzzers

kHz filters

kHz resonators

MHz filters

MHz resonators

SAW filters

Area vibrations

Radius vibrations

Thickness shear vibrations Thickness trapped vibrations Surface acoustic wave

Figure 2.4 - Modes of vibrations and frequency range for modern piezoelectric applications.

2.1.2

Constitutive equations of linear piezoelectricity

The constitutive equations of linear piezoelectricity can be derived from classical mechanics and thermodynamics. The approach proposed in this subsection is based on the classical continuum approach. Relatively large transducers are dealt with (≥5 mm2 ) and there are no particular needs of describing the microscopic theory of piezoelectricity (such a theory was proposed by Born and Huang [112]). The different steps required to derive the classical constitutive equations of piezoelectricity are given. Details have been extensively given in the literature as for example by Mason [111] or Cady [103]. By definition, piezoelectric materials exhibit elastic, dielectric and coupled elastic-dielectric phenomena, so that it is necessary to discuss first the elastic and dielectric continuum. The classical elastic continuum is described by the well-known relations between stresses and strains in a solid. For a Cartesian element of volume, the force acting on each surface of the cube can be decomposed into three components directed along the different axis x, y and z of the Cartesian coordinates system. Weak equilibrium considerations (i.e. the body forces are neglected) impose that the forces applied on two opposite surfaces cancel out so that from the 18 stress components obtained from the decomposition of surface forces, only 6 are independent. Three of the stress components (T1 , T2 and T3 in engineering notation, the subscript 1, 2 and 3 referring to respectively the x, y and z axis for the convention used here) tend to change

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 25 the elementary volume without distorting it (tensile stresses). The three other components of the stress tensor tend to distort the elementary cube of material without changing its volume (shear stresses). The stresses can be considered as intensive thermodynamic variables. To each stress component corresponds an extensive variable usually defined as the elongation or distortion per unit length along a relevant axis, also known as strain Si . The relation between the stresses and strains in the absence of piezoelectric effect, also known as constitutive equation of elasticity, depends on the properties of the material considered. The constitutive relation is formally established using thermodynamics principles considering adiabatic or isothermal (differences between the material constants in adiabatic or isothermal situations are negligible for solids [110]) reversible transformations. For linear elasticity, the material properties tensor relating the stresses to the strains is denominated as the compliance tensor (rank 2, symmetric). Its components are labeled as sij with i=1..6 and j=1..6. The fundamental laws governing the physics of dielectrics are the quasistatic form of Maxwell’s equations. These equations involve 4 thermodynamic variables which are the electrical field, the electrical displacement, the magnetic field and the magnetic strength. In the classical theory of dielectrics, the magnetic field and its conjugated variable are neglected (because of the dimensions of the considered system and of the relatively low involved frequencies), so that the dielectric continuum is totally described by the electrical field E and displacement D. In case of linear dielectrics, the electrical field is related to the displacement by the so called permittivity tensor (rank 2, symmetric). Its components are labeled as εij with i=1..3 and j=1..3. Piezoelectric materials are both elastic and dielectric, so that two intensive (E and T) and two extensive (D and S) thermodynamic variables describe the state of the system in an adiabatic or isothermal situation. From thermodynamics, each variable can be expressed as an explicit function of two others in the same way as proposed for the dielectric and elastic continuum. Because of the number of variables, it is possible to derive different forms of the constitutive equations. Throughout this thesis the convention and notations proposed in the IEEE standards on piezoelectricity [68] are used. The common strain-charge form is given in (2.1) and (2.2) using the compressed matrix notation, in which sE ij represents a compliance term of the material under constant electrical field (this condition is indicated by the superscript E), dij is a charge constant relating the amount of dielectric displacement created by a given stress and εTij is the electrical permittivity of the medium under constant stress (this condition is indicated by the superscript T). Equations (2.1) and (2.2) represent respectively the converse and the direct piezoelectric effect. Note that Einstein’s summation convention is used in (2.1) and (2.2) as in the following parts of this chapter. Si = sE ij Tj + dij Ej

(2.1)

Di = dij Tj + εTij Ej

(2.2)

2.1 History, basic definitions and linear constitutive equations of piezoelectricity

The order 3 tensor containing the electromechanical coupling terms can be represented by a 6*3 matrix so that one can obtain the matrix expanded form of the converse and direct linear constitutive equations of piezoelectricity.      E E E E s11 sE sE T1 S1 12 13 s14 s15 s16 E E E E    S2   sE sE 23 s24 s25 s26   T2     12 s22 E E E E E   S3   sE s33 s34 s35 s36   T3      13 s23 E E E E    S4  =  sE s24 sE 34 s44 s45 s46   T4     14 E E E E E E  S5   s15 s25 s35 s45 s55 s56   T5  E E E T6 S6 sE sE sE 26 36 s46 s56 s66 16  d11 d21 d31  d12 d22 d32       d13 d23 d33  E1   + (2.3)  d14 d24 d34  E2   E3  d15 d25 d35  d16 d26 d36

 

  D1 d11  D2  =  d21 D3 d31

d12 d22 d32

εT11 +  εT12 εT13 

εT12 εT22 εT23

d13 d23 d33

d14 d24 d34

d15 d25 d35

  εT13 E1 εT23   E2  E3 εT33

 d16   d26    d36  

T1 T2 T3 T4 T5 T6

       

(2.4)

Some common alternative forms of the constitutive equations of piezoelectricity are given below. The superscripts S, T, D and E represent respectively a constant strain, stress, electric displacement and electrical field condition. The differences between the values of the material properties under these different conditions are not negligible as it was the case under adiabatic and isothermal situations. It is then important to use the superscripts describing these conditions. Explicit relations between some of the material constants are given later when dealing with the particular case of laminated piezoelectric beams. Ti = cE ij Sj − eij Ej Di = eij Sj + εSij Ej

(2.5)

Ti = cD ij Sj − hij Dj S Ei = −hij Tj + βij Dj

(2.6)

Si = sD ij Tj + gij Dj T Ei = −gij Tj + βij Dj

(2.7)

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 27

2.1.3

Dissipative and non linear effects in piezoelectric materials

Different kinds of dissipative and non linear effects occur in piezoelectric materials. Some are common to both ferroelectric and non ferroelectric piezoelectric materials, others are peculiar to ferroelectric materials. The losses have been neglected until here by considering a reversible, adiabatic or isothermal situation. Linear behaving dissipation mechanisms induce a phase shift between the input and the response of the system. This situation is easily described from a formal point of view. The dissipations and the resulting phase shift between ”input and output” are introduced in the system by setting complex valued components in the elastic, dielectric and electromechanical tensors. The use of complex components to represent losses at a phenomenological level can be justified from relaxation and domain wall motion phenomena described by microscopic theories [113–115]. A detailed analysis of the implications of this methodology on the modeling of piezoelectric materials is proposed by Mezheritsky [116]. A first type of non linear effect observed in both ferroelectric and non ferroelectric piezoelectric materials comes from the high order terms neglected in the derivation of the linear equations of piezoelectricity. Some of them can no longer be neglected when relatively large signal inputs are considered. In the particular case of ferroelectric materials, a fundamental reorganization of the microstructure occurs under high signals and it gives rise to hysteretic effects. As the poling process necessary to obtain piezoelectric ceramics is based on this effect, it is important to give a concise and simple explanation of the observed hysteretic behavior. A detailed analysis of hysteresis effects in piezoelectric materials is proposed by Damjanovic [117]. As introduced previously, ferroelectric materials are a special case of pyroelectric materials possessing a spontaneous polarization. In ferroelectrics, the spontaneous polarization possesses two thermodynamic states of equilibrium and its direction can be reversed through the application of a strong electrical field. Each grain in a ferroelectric polycrystalline material is made of several so-called Weiss domains consisting of a collection of crystalline cells with aligned spontaneous polarization (Figure 2.5). Before a poling process is carried on, the misalignment of the different Weiss domain in a crystal results in a low value of the macroscopic polarization, if any. This effect has even more influence on ceramics, as the influence of the misalignment of Weiss domains is reinforced by the different orientations of the grains. The influence of the poling process on the microstructure of a ferroelectric ceramic is illustrated by Figure 2.6. In a ”virgin” state, i.e. before any electrical field has been applied to the prepared material, the polarization in the different Weiss domains of each grain are misaligned, so that no net electrical charges are observed at the surface of the ceramics. This situation corresponds

2.1 History, basic definitions and linear constitutive equations of piezoelectricity

Figure 2.5 - Weiss domains in a grain.

to the point a in Figure 2.6. As the applied electrical field increases, the material behaves as a dielectric (linear or non linear) in the path (ab). More and more different grains and domains tend to align their polarization during this phase until saturation occurs at point b. The maximum number of aligned domains is obtained at this point and a further increase of the electrical field does not affect the surface charges. The influence of the irreversibility of the rearrangement process is observed when one starts decreasing the applied electrical field: the electrical displacement does no longer follow the path (ab). Because of the strong coupling that now exists between the microscopic polarizations and because of the thermodynamic stability of this state, the different domains oppose a strong resistance to any change in their orientations. Therefore, the material conserves a permanent (also known as remanent) polarization when the electrical field is set to zero (point c): the material is now polarized. Between the point b and d, the material again behaves as a normal dielectric. In order to reverse the orientation of the electrical dipoles, the magnitude of the applied electrical field should be decreased further till reaching the so called coercive field Ec corresponding to point d in Figure 2.6: the system reaches at this moment an unstable state in which the dipoles already reoriented imposes a cascade reversal of polarization on the remaining of the unmodified domains and the system reaches the state described by the point e. As in the previously discussed saturation state, a further decrease of the electrical field does not influence the net charges developed. A behavior equivalent to the one previously described occurs when the amplitude of the applied electrical field again increases and a new reversal of polarization is observed when the positive value of the coercive field is reached. Practically, several cycles are applied on prepared ceramics in order to increase the values of the remanent polarization and piezoelectric properties. Some of the characteristic dissipative and non linear effects occurring in piezoelectric and ferroelectric materials have been briefly discussed in this paragraph for sake of completeness. However, for the applications presented in this thesis, the applied and resulting fields are small with respect to the coercive

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 29

D d

‐Ec

E e

‐Dr

Figure 2.6 - Illustration of hysteresis effects observed in ferroelectric material.

field (Ec is in the range of tens of kV.cm−1 for PZT [118]), so that the non linear effects due to hysteresis and to high order terms in the constitutive equations can be neglected. In other words, it is considered in the following of this work that a relevant model of our devices can be developed under the assumption of the linear constitutive equations given in (2.3) and (2.4).

2.1.4

Relevant piezoelectric materials and corresponding simplifications of the constitutive equations

The general characteristics of the piezoelectric materials implemented in the devices developed for this thesis and the corresponding simplifications of the constitutive equations are presented in this section. Depending on the considered type of energy harvester, different families of piezoelectric materials are implemented for the manufacturing of prototypes. For the human environment energy harvester, solely commercial PZT obtained by conventional ceramics technologies are used. For the machine environment energy harvester, two different piezoelectric materials consisting of commercial thin film PZT and thin film AlN grown at IMEC, Belgium (http://www2.imec.be) and the Holst Centre, The Netherlands (http://www.holstcentre.com/) are used. The goal of this work is to develop prototypes of energy harvesters based on relevant phenomenological models, so that little time is devoted to the study and optimization of the manufacturing of the different materials. However, as the presented thesis is done in parallel with the development of a deposition process for thin film AlN dedicated to energy harvesting situation, some words and further analysis of the produced AlN are proposed in Chapter 3. On the other hand, all PZT based materials used for this work are obtained from commercial companies and no details about their processing are given. The materials belonging to the PZT family exhibit a perovskyte type crystal structure. They are solid solution of general formula PbZr1−x Tix O3 and

2.1 History, basic definitions and linear constitutive equations of piezoelectricity

the ratio of Ti4+ to Zr4+ ions determines the phase of the solution. The ratio is generally chosen close to 50% so that the so called morphotropic boundary is reached. It has been shown that the piezoelectric properties are maximal for this composition [119]. The underlying phenomena occurring during this phase transition are yet not well understood. It has even be suggested that a new phase exists at the morphotropic boundary [120]. Two PZT materials are implemented: thick ceramic PZT obtained from Piezo Inc, US (http://www.piezo.com) and thin film PZT deposited on silicon wafers from Inostek, Korea (http://inostek.com/). The former material is a ferroelectric ceramic. Before poling, it is isotropic because of the randomly aligned grains. The process of electrical poling influences the texture of the film and induces elements of symmetry similar to those of hexagonal polar crystal of the symmetry group 6mm [121]. The latter material is grown by the sol gel method and consists in a columnar polycrystalline arrangement. The growth of the material is controlled so that the polar axes of the grains are aligned in the same direction and the averaged anisotropy of this type of PZT is also equivalent to the one of crystals belonging to the 6mm group [122]. PZT thin films exhibits high piezoelectric constants but are not easily integrated into conventional CMOS process, which have to be considered if it is desired to integrate the conditioning electronics with the manufacturing of the developed MEMS harvesters. These difficulties result from the high required processing temperatures and from contamination problems. For this reason, other thin films materials, such as zinc oxide and aluminum nitride, were investigated in the recent years. AlN is a good dielectric which has been reported to grow for temperatures between 100 and 900â&#x2014;Ś C [123]). The suitability of its piezoelectric properties depends of the application considered: AlN is a poor candidate for actuators, but relevant for sensing or energy harvesting devices, as demonstrated along this thesis. Research on AlN deposition processes with a focus on energy harvesting is ongoing at the Holst Centre and the AlN based harvesters studied during this work are manufactured by this institute. A SEM photography of an example of deposited AlN is given in Figure 2.7. As for PZT thin films, the growth direction of the grains is perpendicular to the seed plane. Again, symmetry elements similar to the ones found in hexagonal crystals are observed [122]. The different piezoelectric materials used belong to the same symmetry group. Symmetry considerations allow reducing the number of independent components in the material properties tensor [124] and the constitutive equations of piezoelectricity can be simplified for the materials studied to (2.8) and (2.9). Elastic materials are also dealt with. They are used as support for the piezoelectric layers. It is considered that they are either isotropic (brass alloy, stainless steel) or transverse isotropic (silicon) so that their compliance matrices can be written in the same form as the one given in (2.8) (with s33 =s11 and s12 =s13 for the isotropic case).

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 31

AlN

Figure 2.7 - SEM photography of an AlN thin film layer.

       

S1 S2 S3 S4 S5 S6

sE 11   sE 12   E   s13 =   0     0 0  



   +   

sE 12 sE 11 sE 13 0 0 0

0 0 0 0 d15 0

sE 13 sE 13 sE 33 0 0 0

0 0 0 d15 0 0

0 0 0 sE 44 0 0 

d31 d31 d33 0 0 0

0 0 0 0 sE 55 0

0 0 0 0 0 sE 66

       

  D1 0  D2  =  0 D3 d31 εT11  0 + 0 

0 0 d31 0 εT11 0

0 0 d33

       

   E1    E2    E3 

(2.8)

 

T1 T2 T3 T4 T5 T6

0 d15 0

d15 0 0

  0 E1 0   E2  E3 εT33

 0   0   0  

T1 T2 T3 T4 T5 T6

       

(2.9)

2.2

2.2 The constitutive equations of piezoelectric laminated beams

The constitutive equations of piezoelectric laminated beams

The equations describing the problem at the mesoscopic scale have been derived. A macroscopic model is required for analyzing the performances of the piezoelectric transducers implemented in our energy harvesters. Representations of the macroscopic behavior of simple piezoelectric vibrators was initiated by the work of Mason [109â&#x20AC;&#x201C;111], Butterworth [104] and van Dyke [105, 106]. However, few relevant models were proposed for piezoelectric laminated beam until the work of Smits [125, 126], who initiated a renewal of interest for the theoretical analysis of such structures. In the following year, a large amount of publications adapting and slightly refining the results of Smits were proposed [127â&#x20AC;&#x201C;130]. Most of the derivations proposed in this section are based on the work of Smits, but new theoretical elements are also included. In the following, the equations describing the macroscopic behavior of elastic laminated beam are first derived. Then, the obtained equations are adapted for the case of laminated beam containing elastic and piezoelectric layer(s).

2.2.1

Elastic laminated beams

A brief outline of the procedure implemented in this section is described here. Hypothesis related to the geometry and to the mechanical loading conditions of the beam allowing a reduction of the number of relevant stress and strain components are first defined. Second, the assumed distribution of the stress and strains is presented. Then, the mesoscopic variable stress and strains are related to corresponding macroscopic variables (efforts and displacements) using the stress resultants method and kinematic considerations in the framework of a small displacement approximation. The equation governing the dynamical behavior of an elastic beam is derived by introducing the constitutive equations of elasticity in the obtained relations. Finally, the results are extended to inhomogeneous beams made of layers of different materials. A mechanical beam, as it is usually defined, is illustrated by Figure 2.8(a). It is first considered to be a 3D solid element with one dimension much smaller than the two others. In the following, the small dimension is referred to as the thickness hs of the structure belonging to the z axis of the chosen coordinates system. As hs is much smaller than the two other dimensions, the thickness shear and strains can be neglected: S3 =T3 =0. Second, a beam is characterized by the boundary conditions applied on the lateral faces parallel to the yz plane: both faces are constrained by a couple of compatible boundary conditions which can not be both free. At the opposite side, the two faces belonging to the zx plane are not constrained and free to deform, except in the close neighborhood of the yz constrained surfaces. The most common example of a compatible couple of boundary conditions corresponds to a clamped-free (cantilever) configuration.

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 33

Loading P(x) x

P(x)

z0 W

Free face

Constrained face (a)

(b)

Figure 2.8 - (a) Schematic representing the definition of a mechanical beam, (b) simplified 2D model.

An exhaustive list of all the possible couples is given in the literature by Roark [131]. The length l of the beam is now defined as the dimension in the direction perpendicular to the constrained faces and the width W as illustrated in Figure 2.8(a). It is considered in beam theory that the distributed mechanical load P is applied on the faces belonging to the xy plane. In the classical analysis of mechanical beams, it is also assumed that the cross sections along yz planes retains the same shape and are symmetric with respect to the xz plane at W/2. These cross sections are taken as rectangular. The applied load is uniform along the width of the structure and is directed solely along the z axis in the case of a force or around y in the case of an external torque. In common beam problems, the shear stresses/strains couples occurring in the planes different than the zx one are neglected (S4 =S6 =T4 =T6 =0). Finally, it is supposed that the width is small compared to the length of the beam, so that the faces in the xz plane are free to expand (except close to the boundary conditions) and the stress developed along y is negligible: T2 =0. This condition is referred to as plain stress. In a following section, a simple refinement is proposed to adapt the problem to plain strain condition. From all previous simplifications, the problem is independent of the position on the y axis. The beam can be described by the 2D model of Figure 2.8(b). Furthermore, the constitutive equations of elasticity can be reduced to (2.10). The reduction of the coordinate dependency of the different stress and strains is also given. Note that the lateral strain S2 =s12 T1 is not a variable of primary interest in the analysis of beams. S1 (x, z) s11 0 T1 (x, z) = (2.10) S5 (x, z) 0 s55 T5 (x, z) The modeling work is developed assuming a small displacement approximation: the spatial frame of reference (Eulerian formulation) is supposed to remain equivalent to the material frame of reference (Lagrangian formulation) in the deformed state. An important consideration resulting from this assumption consists in the fact that an effort applied in a direction perpendicular to top surface of the beam in the non deformed state is supposed to remain per-

2.2 The constitutive equations of piezoelectric laminated beams

x z0

P(x)

J(x) P(x)dx

P(x)dx

J(x+dx) M(x)

T5(x)

T5(x+dx)

T1(x)

T1(x+dx)

Figure 2.9 - Assumed behavior of the shear and tensile stresses.

pendicular to this surface in the deformed state. The 2D beam described by Figure 2.8(b) can conceptually be represented as a superposition of straight longitudinal fibers deformed into continuous curves under the application of a load. The assumption of continuous displacements and of a curved shape suggests that, depending on their position along the thickness of the beam, the local length of some fibers is increased while the length of some others is diminished, i.e. the sign of the strain depends of the position along z. By analyzing the curvature of the collection of fibers, it is possible to show first that the maximum and minimum longitudinal tensile strains S1 occur in the most outer fibers of the beam, and second that a specific fiber referred to as neutral axis undergoes zero longitudinal strain. The neutral axis location is denominated as z0 . In homogeneous beams, the neutral axis is the central fiber of the structure. The theory of Timoshenko [132–134] is implemented for the next derivations, i.e. it is assumed that the variation of the longitudinal and shear strains S1 and S5 show respectively a linear and constant behavior along the thickness of the beam, with S1 being zero on the neutral axis. As illustrated by Figure 2.9, the stresses T1 and T5 can be represented in the same way because of the considered simplified linear constitutive equations. To determine the equations describing the behavior of the beam, equilibrium considerations applied on a differential cross sectional element of volume Whs dx of the beam have first to be elaborated. An applied mechanical load induces stresses in the beam. A common method of establishing relations between the stresses and the load consists in introducing the so called stress resultants: the distribution of the stresses in the beam is statically equivalent to forces and moments applied on the different cross sections. Because of the restrictions imposed on the geometry and loading of the beam, the only non zero stress

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 35 resultants are a force directed along the z axis and a moment inducing a rotation around y. These elements are referred to as shear force Jz and bending moment My respectively. The bending moment is the static equivalent of the distribution of the longitudinal stress T1 , while the shear force is the one of the distribution of the shear strain T5 . The bending moment and shear force can be written as (2.11) and (2.12). The origin of the z axis is taken at the lowest fiber of the beam in the given expressions. A list of explicit expressions of My and Jz for different types of loading P and boundary conditions are given by Roark [131] in the small displacement approximation. In the remainder of this thesis, the classical sign convention defined by Timoshenko [133] for the shear force and bending moment in a beam is used: if the fibers located above the neutral axis undergo compression, My is defined as positive; if the shear forces induce a negative vertical motion of the cross section, Jz is defined as positive. Zhs My (x) = ±W

zT1 (x, z) dz

(2.11)

T5 (x, z) dz

(2.12)

Zhs Jz (x) = ±W 0

A relation between the stresses and the relevant intensive macroscopic variables is obtained. The macroscopic variables corresponding to the tensile and shear strain S1 and S5 are classically defined as the displacement and distortion fields of a segment belonging to the elastic continuum. The kinematics of a portion of a beam according to Timoshenko theory is illustrated by Figure 2.10. In this theory, it is assumed that a segment (belonging to the yz plane) of material remains a segment after deformation. Practically, the initial segment is rather transformed into a curve by the deformation process, but Timoshenko’s theory has been proven to give a good approximation of the behavior of beams. The relations between displacements and strains are obtained by a geometrical analysis of the transformation undergone by the previously discussed segment. Lets define the segment AB with the points A and B with respective coordinates (x,z0 ) and (x,z). The segment AB is perpendicular to the neutral axis and the point A belongs to the neutral axis. After deformation, the points A and B are respectively displaced to A’(x+u(x,z0 ),z+w(x,z0 )) and B’(x+ u(x,z),z+w(x,z)), in which u and w are the displacement fields respectively along the x and z axis. Due to the nature of the structure, the lateral displacement is assumed negligible. The strain S3 is neglected in the proposed model, so that the length of A’B’ is equal to the one of AB and w(x,z)=w(x,z0 )=w(x). In the hypothesis of small deformations, θ0 can be approximated by dw/dx-θy . Positive angles correspond to counter clockwise rotations. A geometrical analysis of Figure 2.10 combined with the previous assumptions leads to the so called kinematic equation of the beam. dw (x) (2.13) u (x, z) ≈ u (x, z0 ) − (z − z0 ) dx

2.2 The constitutive equations of piezoelectric laminated beams

u(x,z)

x w(x,z)

B’ Portion of a beam before and after deformation

u(x,z0)

A w(x,z0)

Neutral axis before and after deformation

A’ θy

θ0

Figure 2.10 - Displacement fields in a portion of a beam according to Timoshenko’s theory.

The longitudinal strain S1 is defined as the elongation per unit length in the x direction and can be written as S1 =du/dx+(z-z0 )dθy /dx. Combining this relation with (2.13) leads to S1 (x, z) = − (z − z0 )

d2 w (x) dθy (x) − dx2 dx

(2.14)

Positive longitudinal strains correspond to elongated fibers. The strain S5 corresponds to the rotation field θy shown in Figure 2.10 with S5 =θy . w is the main displacement of interest in the analysis of beams and not attention is given to the longitudinal one u. Both relations between the stresses and the bending moment/shear force and between the strains and displacement fields are obtained. It is now possible to link the displacement to the mechanical efforts by introducing the constitutive equations of elasticity relating stress and strain in the previous results. Combining (2.10), (2.11), (2.12) and (2.14): My (x) =

d2 w (x) dθy (x) − dx2 dx

W s11

Zhs z (z − z0 )dz

(2.15)

Jz (x) = −

W hs θy (x) s55

(2.16)

In (2.15), the term Zhs z (z − z0 )dz 0

(2.17)

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 37 is referred to as the area moment of inertia Ii of a beam cross section. For homogeneous rectangular beams, Ii =Whs 3 /12. In order to obtain a theoretical model predicting accurately the behavior of a beam, Timoshenko has shown that the compliance term s55 involved in the definition of the shear force given in (2.16) has to be weighted by a factor κ depending on the cross section of the beam. This correction is required to compensate the error induced on the calculations by the approximation of constant shear strain along a cross section. A commonly accepted value for a rectangular cross section is κ=5/6 [133]. The final equations describing the dynamic behavior of the beam are obtained by combining (2.15)and (2.16) with dynamical equilibrium considerations on the elements of Figure 2.9 for the vertical and rotary motions. The equations can be rearranged in (2.18) and (2.19), in which ρ represents the density (assumed uniform) of the material and A is the cross section area. An external viscous type damping is present through an additional term Da dw/dt, while the internal damping is represented through complex values of the compliances involved in the equation. The right-hand side of (2.18) contains all elements related to an eventual distributed load applied on the surface of the beam. In the case of concentrated load, all vanish. The terms related to the shear correction are easily identifiable through the presence of the compliance term s55 . The larger the value of this compliance, the smaller the influence of the shear effects. The coupled time-space derivative term and the one in dx4 w/dt4 are related to the rotary inertia of the cross sections of the beam. If a quasi static situation without distributed load is considered, one easily obtains the well known static beam equation Ii /s11 dx4 w/dt4 =0. According to (2.19), if shear is considered in the static and load free situation, it becomes constant over the length of the beam. In the considered case, it can be shown for example by FEM simulations that the value of the shear is constant over most of the length of the beam, but varies abruptly near the constrained boundaries. However, the differences in the prediction of the general behavior due to this effect are negligible. Ii κs55 ∂ 2 P (x, t) κs55 ∂ 2 P (x, t) + ρI i s11 A ∂x2 A ∂t2 4 4 Ii ∂ w (x, t) κs55 ∂ w (x, t) = − ρIi 1 + 4 s11 ∂x s11 ∂t2 ∂x2 2 ∂w (x, t) ∂ w (x, t) + Da +ρA 2 ∂t ∂t P (x, t) −

1 ∂θy (x, t) ∂ 2 w (x, t) + P (x, t) = ρ κs55 ∂x ∂t2

(2.18)

(2.19)

The piezoelectric bending structures considered in this thesis consist of beams made of a superposition of piezoelectric and elastic layers. Before introducing the piezoelectric effect in the problem, it is judicious to discuss the

2.2 The constitutive equations of piezoelectric laminated beams

P(x) s11,2 s55,2

s11,1 s55,1

Figure 2.11 - Assumed behavior of the shear and tensile stresses in a multilayered beam.

behavior of beams consisting of several linear elastic layers. The analysis is limited to a simple theory in which the phenomena occurring in the interface are neglected and it is assumed that the different layers do not slip towards each other so that the shear and longitudinal strains are continuous along the thickness of the beam. The thicknesses of the interfaces are considered negligible. In this simple theory, the shear and longitudinal stresses are discontinuous at the interfaces, because of the different compliances of the materials. Furthermore, the stress gradient of T1 and the value of T5 (constant in Timoshenko theory) are not equal in the different layers. This situation is illustrated by Figure 2.11. The relevant compliances terms are labeled s1,11 and s1,55 for the first layer, s2,11 and s2,55 for the second. The determination of the position z0 of the neutral axis is not as trivial as in the case of the monolayer. z0 can be determined by imposing the conditions that the longitudinal force resulting from the distribution of the stress T1 over a cross section is zero and that T1 (z0 )=0 from the definition of the neutral axis. Simple closed form expressions of z0 are proposed by Weinberg [129]. An explicit form of this parameter is given later for the particular structures studied. In the framework of the proposed theory, the single layer beam problem is adapted to multilayer by reconsidering the expression of the bending moment and shear force given in (2.15)and (2.16). These two equations can be rewritten for the structure depicted in Figure 2.11 as 2 d w (x) dθy (x) My (x) = W − dx2 dx   h 1 Z Zh2 1 1  z (z − z0 )dz + z (z − z0 )dz  (2.20) s1,11 s2,11 0

Jz (x) = −W θy (x)

h2 h1 + κ1 s1,55 κ2 s2,55

(2.21)

The example given above considers a two layers beam. In order to establish a model relevant for an arbitrary number n of elastic layers, the definitions given in (2.22) and (2.23) are used (h0 =0). EIeq and GAeq represents equivalent for multilayer beam of respectively the ratio of the moment of inertia over the

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 39 longitudinal compliance and the quotient of the cross section area over the effective shear compliance.

EIeq

Zhi n X 1 =W z (z − z0 )dz s i=1 i,11

(2.22)

hi−1

GAeq = W

n X i=1

hi κi si,55

(2.23)

Following these definitions, the equations describing the dynamical equilibrium of a homogeneous elastic beam given in (2.18) and (2.19) can finally be rewritten for multilayered structures as Ii ∂ 2 P (x, t) EIeq ∂ 2 P (x, t) + ρe 2 GAeq ∂x GAeq ∂t2 ∂ 4 w (x, t) EIeq ∂ 4 w (x, t) =EIeq − ρ e I + A i ∂x4 GAeq ∂t2 ∂x2 2 4 ∂ w (x, t) ∂w (x, t) A ∂ w (x, t) + ρeA + Da +e ρ2 Ii 4 2 GAeq ∂t ∂t ∂t P (x, t) −

(2.24)

GAeq ∂θy (x, t) + P (x, t) = 0 (2.25) A ∂x in which ρe represents the average density along the thickness of the laminate. Note that the expressions A and Ii representing the cross section area and the area moment of inertia now have to be written considering the total thickness of the structure. The equations describing the behavior of an elastic beam in terms of relevant macroscopic variables are obtained by combining the constitutive equations of elasticity with equilibrium and kinematics consideration. In the next subsection, an equivalent approach for deriving the representative equations of the dynamics of a piezoelectric laminated beam is followed.

2.2.2

Piezoelectric laminated beams

In the previous subsection, the constitutive equations of elasticity are simplified and an equation describing the macroscopic behavior of a multilayer elastic beam is derived. Equivalent manipulations are performed in order to derive the constitutive equations of beams made of a superposition of elastic and piezoelectric rectangular layers. In this section, the simplifications of the constitutive equations of piezoelectricity related to the particular types of piezoelectric laminates studied are presented. Then, the transverse electromechanical coupling factor is discussed before presenting the polarization

layer

Elastic layer z

Metallic

2.2 The constitutive equations of electrodes piezoelectric laminated beams

Piezoelectric layer

Piezoelectric layer Elastic layer

Elastic layer z

Metallic electrodes

(a)

(b)

Figure 2.12 - (a) Common configuration of a piezoelectric Piezoelectric laminated beam, (b) interdigitated electrodes configuration. layer Elastic layer z

Metallic

y and electrodes arrangement scheme related to piezoelectric unimorphs and bielectrodes morphs. Finally, the equations describing the macroscopic electrodynamics of piezoelectric laminates are derived.

In this work, it is considered that the piezoelectric layer(s) is configured as a classical capacitor, i.e. sandwiched between a pair of electrodes. This allows imposing electrical boundaries conditions by fixing the voltage or the charge on the electrodes. The electrodes are assumed to be located on the xy plane (Figure 2.12(a)) and the piezoelectric material is poled along the z axis (ferroelectrics) or grown in such a way that the polar axis of most of the grains is oriented along the z axis (non ferroelectrics). If fringing effects occurring on the lateral sides of the element are neglected, the components of the electrical field and displacement vanished along y and x, so that the constitutive equation of dielectrics is reduced to the simple form D3 =εT33 E3 or D3 =εS33 E3 . Although it is not discussed further, it is interesting to present an alternative and relatively unconsidered electrode configuration, shown in Figure 2.12(b). This design requires depositing electrodes on a single surface of the device, which can be advantageous in the case of micro fabrication. It also involves piezoelectric coefficients different than those appearing in our applications. However, the performances in terms of energy harvesting obtained from it do not compete for the moment with the ones resulting from the classical type of electrodes configuration [97]. The constitutive equations of piezoelectricity for the piezoelectric laminated beam of Figure 2.12(a) can be obtained by combining the simplified equations of dielectrics and of elasticity. Assuming all the elements of symmetry that were introduced in the analysis of the elastic beam, the electrical field can be written as E3 (x,z) and the dielectric displacement as D3 (x). The constitutive equations of piezoelectricity given in (2.8) and (2.9) can be simplified to the form given in (2.26) and (2.27). It can be seen that a single piezoelectric charge constant is involved in the problem; the piezoelectric material is excited in the

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 41 so called d31 mode. E S1 (x, z) s11 = S5 (x, z) 0

0 sE 55

T1 (x, z) T5 (x, z)

d31 0

D3 (x) = d31 T1 (x, z) + εT33 E3 (x, z)

E3 (x, z)

(2.26) (2.27)

Before continuing the derivation of the equations of laminated piezoelectric beams, the concept of the electromechanical coupling factor is now introduced. As described above, several forms of the constitutive equations of piezoelectricity do exist. The relations between the materials constants involved in the different forms are too complex to be of practical use in the non simplified expressions, but they can now be established in the framework of the simplified equations (2.26) and (2.27). Not all the relations between the constants are expressed but only those which are of primary interest for the understanding of the problem. Combining the different forms of the constitutive equations, one can obtain the relation between compliance and permittivity under constant stress (also referred to as ”free” material properties) and constant strain (also referred to as ”clamped” material properties). d231 E sD = s 1 − (2.28) 11 11 T sE 11 ε33 d231 S T (2.29) ε33 = ε33 1 − E T s11 ε33 The term k31 2 =d31 2 /(εT33 sE 11 ) in (2.28) and (2.29) is referred to as transverse electromechanical coupling factor. Its physical meaning can be understood by considering a simple and unsupported axial piezoelectric transducer configured for d31 mode operation undergoing a tension or compression directed along its axis 1. An ideal quasi-static thermodynamic cycle, as defined in the IEEE standards on piezoelectricity [68], is applied to the transducer (Figure 2.13). In the first part of the cycle, the piezoelectric element is short-circuited and a compressive or tensile stress T1 is applied along the axis 1 so that a strain S1 is developed along the same axis, and a mechanical energy EM is stored in the transducer (slope equal to sE 11 ). In a second phase, the piezoelectric transducer is open-circuited and free to return to a zero stress configuration (slope equal to sD 11 ). During this phase, a quantity Em of mechanical energy is ”consumed’ and a quantity Eel of electrical energy is developed in the structure. Finally, the cycle is completed by shunting the electrodes of the piezoelectric element to a perfect load in which Eel is integrally dissipated. Obviously EM =Em +Eel . One can easily compute the fraction Eel /EM found to be equal to the piezoelectric material transverse coupling factor k31 2 . The analysis is focused on specific cases of piezoelectric laminates. Both families of structures considered are illustrated in Figure 2.14. In both situations, the thickness of the electrodes is assumed negligible. The first structure is referred to as a unimorph [60] and consists in an elastic layer supporting a

2.2 The constitutive equations of piezoelectric laminated beams

Opened

S1 Eel

Shorted

Shunted

Eel = k312 Eel + Em

Eel k312 = Em 1 − k312

T1 Figure 2.13 - Quasi-static thermodynamic cycle illustrating the definition of k31 2 and k31 2 /(1-k31 2 ).

single piezoelectric capacitor. One of the electrodes is grounded. Due to manufacturing convenience it is generally the one located at the interface between the elastic and piezoelectric layers. The other(s) are connected to a prescribed charge or voltage. The piezoelectric laminate presented in Figure 2.14(b) is known as bimorph and is manufactured by attaching a piezoelectric capacitor on each opposite sides of the elastic layer. It is common to connect the capacitors in order to limit inputs and outputs of the system and to improve the performances. The bimorph is symmetric with respect to the x axis and the same piezoelectric materials are used for the two capacitors. Depending on their position in the beam and on the direction of their poling, the piezoelectric layers in the bimorph can develop either positive or negative electrical charges. In this case, appropriate schemes of polarization orientation and capacitors connection have to be implemented so that the charges developed in the different piezoelectric layers do not cancel. The two basic possible schemes are presented for the bimorph in Figure 2.14(b). Series operation imposes opposite directions of poling Π (also known as X poled) in the two piezoelectric layers so that the developed electrical fields have the same directions. Parallel operation requires the same direction of poling Π (also known as Y poled). The sign of the longitudinal strain in a given piezoelectric layer can vary depending on the boundary conditions and on the type of load. In this case poling and connection scheme must be reconsidered. For example, considering a clamped-clamped unimorph loaded by a concentrated force applied at the center of the structure, some of the upper fibers of the beam undergo tensile strain while others undergo compressive strain. It can be shown that the total strain along the x axis cancels, so that no net voltage or charges are developed if the piezoelectric layer consists of a single capacitor and has constant polarization. The solution consists in either dividing the layer in several capacitors

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 43

hp hs

(a)

hp hs hp

(b)

Figure 2.14 - (a) Schematic of a piezoelectric unimorph, (b) schematic of a piezoelectric bimorph.

or implementing a non constant polarization scheme. In this thesis, cantilever beams are dealt with and they require no special adjustment of the polarization or electrodes configuration. The constitutive equations of piezoelectricity are now rewritten in terms of the voltage developed across the arrangement of piezoelectric capacitors. The following derivations are based on the case of the unimorph of Figure 2.14(a). The derivations are similar for a bimorph. Therefore, only their results are discussed. The potential difference between the electrodes is defined as dV/dz=-E3 . Integrating the expression of the electrical field extracted from (2.27), (2.30) is obtained, in which Te1 represents the average longitudinal stress along the thickness of the piezoelectric layer. Because T1 is considered to vary linearly, Te1 (x)=T1 (x,hs +hp /2). V =−

hp D3 (x) hp d31 f + T T1 (x) εT33 ε33

(2.30)

The simplified constitutive equations of piezoelectricity can be rewritten in terms of the potential difference as S1 (x, z) = sE 11 T1 (x, z) − d31

V d2 f1 (x) − T31 T1 (x, z) − T hp ε33

f1 (x) − εT33 V D3 (x) = d31 T hp

(2.31)

(2.32)

In the following derivation, the term in d31 2 /(εT33 ) in (2.31) is neglected. This simplification constitutes the so-called small piezoelectricity approximation and is valid for material with low electromechanical coupling coefficient [135]. Simple refinements based on literature results are proposed later. The relevant expressions of the mesoscopic constitutive equations in the case of piezoelectric laminated beams are obtained. The next step consists in deriving relations describing the macroscopic behavior of the unimorph. In the

2.2 The constitutive equations of piezoelectric laminated beams

T1 =

hp hs

d31 V E h s11 p

Figure 2.15 - Longitudinal stresses occurring in a piezoelectric unimorph under the sole action of an applied voltage.

previous subsection, the macroscopic variables corresponding to the stress and strain in the case of laminate elastic beams are introduced. For the considered piezoelectric laminates, the macroscopic mechanical variables do not differ. The dynamical equilibrium equations given in (2.24) and (2.25) still yields. However, the definitions of the bending moment and shear force have to be redefined by considering the piezoelectric effect. The expression of the bending moment due to the piezoelectric effect can be found by analyzing the stresses occurring in a cross section of a unimorph undergoing solely an applied voltage (Figure 2.15). According to (2.31), the longitudinal stresses in the piezoelectric layer resulting from an applied voltage is d31 /(sE 11 hp )V. These stresses exist solely in the piezoelectric layer and result in a bending moment applied on the cross section of the beam. This moment exists because of the opposition that the elastic layer exhibits against longitudinal deformations. For the materials considered, an applied voltage does not induce shear stress: the beam undergoes pure bending. The bending moment Mv developed under the action of an applied voltage is found by integrating the product of the distance from the neutral axis of a fiber belonging to the piezoelectric layer and the stress corresponding to this fiber. Its expression is Mv = −αV (2.33) with

d31 α=W E s11

hp hs + − z0 2

(2.34)

According to the conventions used in this thesis, a positive voltage corresponds to a downward curvature. The dynamic equation of the beam including the piezoelectric bending moment represents somewhat a macroscopic equivalent of the mesoscopic converse equation and an equivalent for the direct equation should be elaborated. The macroscopic variable corresponding to the electrical field has already been defined. It is the potential difference V. The dielectric displacement can be linked

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 45 to the surface electrical charges Q accumulated on the electrodes. Q is found by integrating the dielectric displacement given by the direct equation (2.32) over the surface of the electrodes. To perform the integration, the expression of Te1 (x) derived from the SPE form of (2.31) is first introduced in (2.32). The obtained equation depends on the strain S1 replaced by its expression in terms of the vertical displacement field w as given in the kinematic equation (2.13). Integration of the latter leads to the macroscopic direct equation of piezoelectric beams: dw (l) dw (0) − − (θy (l) − θy (0)) − Cpf V (2.35) Q = −α dx dx In this equation, Cpf =εS33 Wl/hp is the clamped (motion restrained) capacitance of an unsupported slab of the piezoelectric layer. It is shown later that the theoretical expression of the clamped capacitance of the same piezoelectric layer coupled to an elastic one is different from Cpf . The fundamental equations describing the macroscopic behavior of a piezoelectric laminated beam are derived. These equations allow a correct representation of the behavior in a large range of frequencies and for a large variety of geometries respecting the beam definition given previously. However, analytical solutions for these equations are not easily found. The purpose is to develop relatively simple closed form formulas allowing the understanding of the influence of the geometry and of the material properties on the performances of the energy harvesters. Shifting right now to numerical analysis would not fulfill this requirement. Therefore, the analysis is focused on a simplified form of the fundamental equations of the piezoelectric beam. The devices are either excited by a harmonic signal in the neighborhood of their fundamental resonance frequency (industrial environment harvesters) or undergo an impact which, as shown in Chapter 4, excites mainly the fundamental vibration mode of the beam (human environment harvesters). Shear and rotary inertia effects generally occur at high frequencies, well above the fundamental mode [136], and they can be reasonably neglected in the analysis. Also, it is assumed that the parasitic damping could be represented by the sole use of complex material properties (complex EIeq for internal dissipations and complex ρe for external ones). The simplified fundamental equations that are investigated in the following considering time dependant variables are P (x, t) = EIeq

∂ 2 w (x, t) ∂ 4 w (x, t) + ρ e A ∂x4 ∂t2

Q (t) = −α

dw (l, t) dw (0, t) − dx dx

θy (x, t) = 0

(2.36)

− Cpf V (t)

(2.37)

(2.38)

2.2 The constitutive equations of piezoelectric laminated beams

Before solving the equations derived for representative cases, simple methods for relaxing the SPE and for shifting the problem from a plain stress to a plain strain approximation are proposed. Also, expressions of EIeq and z0 are given for unimorph and bimorph. The approximations used in the derivation of the constitutive equations to be relaxed in this subsection are plain stress and small piezoelectricity assumptions. Plain stress simplification is used when deriving (2.10). This assumption is valid for long and slender beam. However, in some cases, short and wide beam are implemented. The problem can be represented in this case by a more meaningful 2D assumption consisting in neglecting the lateral strain S2 rather than the lateral stress T2 . This assumption is referred to as plain strain. Tadmor [137] has shown that the results reached for plain stress are easily adapted to plain strain by introducing effective values of the material properties. The effective longitudinal compliances are indicated for the elastic and piezoelectric material respectively as ss and sE p . The effective permittivity under constant stress and the effective charge constant of the piezoelectric layer are labeled respectively as εTp and dp . Their expressions in plane stress and plane strain are  E  ss = ss,11 sE p = sp,11 Plane stress  T dp = d31 εp = εT33   s2s,12   s = s − s s,11  ss,22  

E sE p = sp,11 −

   d232  T T   εp = ε33 − sE p,22

sE dp = d31 − d32 p,12 sE p,22

2 sE p,12 sE p,22

(2.39) Plane strain

The effective transverse electromechanical coupling factor has to be computed with the effective values of the material properties and is labeled as kp =dp /(sp εTp )1/2 . When deriving the macroscopic form of the constitutive equations, the small piezoelectricity approximation is used by omitting the electrical field induced by mechanical deformation in the converse equation (2.31). This simplification can have repercussions in modeling modern piezoelectric materials which exhibit large electromechanical coupling effect. As shown by Tadmor [137], the SPE can be relaxed by defining an effective moment of inertia for the piezoelectric layer: the value of Ii has simply to be replaced by Ii (1+ξ) in which ξ=(kp )2 /(1-kp 2 ). This consideration has an incidence on the expression of EIeq and z0 involved in the constitutive parameters of the model. Combining the results of Tadmor with those of Weinberg [129], one can obtain the closed form expressions of EIeq , GAeq and z0 for the piezoelectric structures depicted in Figure 2.14. They are given in (2.40) for the unimorph and in (2.41) for the symmetric bimorph. The reference of the z axis is taken at the lowest fiber of

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 47 the beam. z0 =

2 2 sE h p hs + ss hp + 2ss h p s E 2 sp hs + ss hp

  EIeq = W

sE p



h4s + (1 + ξ) (ss ) h4p +

2 2 6h h sE p ss hp hs (4 + ξ) hp + 4hs + p s E E 12sp ss sp hs + ss hp

(2.40)



z0 = hp + h2s EIeq

sE h3 + 2ss hp (4 + ξ) h2p + 3h2s + 6hp hs =W p s 12sE p ss

(2.41)

A comparison of the values of EIeq obtained from the basic and refined theory is proposed in Figure 2.16. For the material properties, values representative of strong coupling structures that will be investigated in later chapters are used (PZT-5A and generic brass alloy). The total thickness and the width of the beam are set to a fixed value so that a comparison can easily be achieved. It is clearly seen that for both unimorph and bimorph the plain stress and plain strain approximations results in important differences. The former simplification results in lower values of the predicted stiffness and of the fundamental resonance frequency. In accordance with common sense, the SPE has a noticeable influence for structures in which the bulk of the beam consists of piezoelectric material. Relaxing the SPE induces higher predicted values of the stiffness because of the additional stress term in (2.31). According to the conclusions presented in this paragraph, one should carefully analyze the geometry and boundary conditions of a given problem and determine according to it the correct expressions of the effective material properties. In the next section, the simplified fundamental equations of the piezoelectric beam are solved for the particular case of a mass loaded cantilever which is representative of the devices manufactured during this thesis. The obtained solutions are arranged in the form of an impedance matrix, which is particularly useful for phenomenological analysis. Also, an alternative representation of the impedance matrix based on electrical equivalent networks is presented.

2.3

Constitutive matrix and electrical network representation of piezoelectric beams

In this section, closed form solutions of the equations derived previously are elaborated for the particular structures implemented. The section is organized as follow: the geometry of the piezoelectric laminates and the type of mechanical efforts relevant for our energy harvesting situations are described.

EIeq

2.3 Constitutive matrix and electrical network representation of piezoelectric 48 beams

0.2

0.4

0.6

hp/(hp+hs)

(a)

0.8

1.0

0.2

0.4

0.6

0.8

1.0

2hp/(2hp+hs)

(b)

Figure 2.16 - Values of EIeq computed for (a) unimorph, (b) symmetric bimorph. Solid and dotted lines correspond respectively to the plain stress and plain strain cases considering the SPE, dashed and dotted dashed lines correspond to the plain stress and plain strain cases when the SPE is relaxed. The scale of the ordinates axis is linear and arbitrary in both graphics. The material properties are ss,11 = ss,22 =10 pPa−1 , ss,12 =3 pPa−1 , sE p,11 = −1 , sE =5.7 pPa−1 , d =d =175 pC/N, εT /ε =1700. sE =16.4 pPa 31 32 0 p,22 p,12 33

The general method of resolution of the equations developed in the previous sections and the representation of the solutions in the form of an impedance matrix or equivalent electrical network are then presented. Finally, the concept of generalized electromechanical coupling factor is introduced.

2.3.1

Geometry of the harvesters and applied mechanical efforts

The energy harvester consists, in the most general way, of a piezoelectric cantilever beam with a mass mt attached at its tip (Figure 2.17). A unimorph is considered in the derivation of the model, but results of the analysis for bimorphs are also presented later. The attached mass mt has a large thickness H compared to the one of the beam, so that it is assumed that no strains occur along the length L of the mass. Therefore, the piezoelectric layer(s) covers only the length l of the laminated beam. The deflection along the top surface of the beam and of the mass are denoted respectively as w and δ. The deflection w of the beam is defined by a curve, while the deflection along the attached mass follows a straight line. According to the kinematic relations introduced previously and considering a small beam tip angle, the different deflections are related by δ(x)=w(l)+(x-l)dw(l)/dx. Finally, mb represents in Figure 2.17 the total mass of the piezoelectric beam (mb =e ρWl(hp +hs )). The mechanical efforts that are applied on the piezoelectric structures are now defined. They are different in the case of the machine and human environment harvester. In the former situation, the clamped end of the cantilever undergoes a har-

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 49 z

hp hs Clamped boundary

l x

w(x)

Piezoelectric layer Elastic layer

L δ(x)

mt H

Figure 2.17 - Schematic of the piezoelectric laminate implemented in the energy harvesters.

monic motion Z(t)=Z0 sin(ωt)which results in apparent z directed forces per unit volume in the beam and in the tip mass when the computations are realized in the frame of reference attached to the bender. Steady state is considered in the analysis of this harvester. The inertial forces acting on the beam can be represented by a uniform pressure P applied all along the length of the beam (Figure 2.18(a)). In the following, it is supposed that the mass mb of the beam is small compared to the one of the tip mass mt , so that the discussed distributed pressure is ignored in the analysis of the machine environment harvester and P=0. In this case, the left hand side of (2.36) vanishes. The force distributed over the volume of the mass can be represented by a resultant F acting at the centre of gravity G. In the case of the human environment harvester, the mechanical efforts consist in the result of the impact of a moving object on the beam. It is shown later that the collision can be represented within some approximations as a concentrated pulse type force F=U(t). It is assumed that this force is applied at middle of the mass (or at the tip of the beam if no additional mass is attached). For both inertial and impact situations, the mass-cantilever system can be represented by a simple cantilever of the same length, while the tip of this equivalent beam undergoes a vertical force F and a torque FL/2, assuming symmetry of the mass and small displacements (Figure 2.18(b)). It is judicious to define now characteristic dimensions of the beams analyzed. The particular types of piezoelectric cantilever configurations that are investigated can be described by three different sets of characteristic dimensions and values of the beam and attached masses: in the case of the machine environment harvester, MEMS fabricated short and wide cantilevers supporting a long attached mass are investigated. PZT and AlN materials are used for the piezoelectric layer. For this type of harvester, long and slender PZT based commercial beams supporting a short attached mass are also considered. In the human environment harvesters, the last type of described cantilever without attached mass is used. PZT is the sole piezoelectric material implemented in

P=−

mb d 2 Z l dt 2

F=U(t)

2.3 Constitutive matrix andL/2 electrical network representation of piezoelectric x 50 beams x P=−

mb d 2 Z l dt 2

F=U(t)

2 dL/2 δG P = −M dt 2

F L/2

F = −M

d δG dt 2

(a)

(b)

F L/2

F symbols refer to Figure 2.18 - (a) Mechanical efforts applied on the energy harvesters (blue the machine environment harvester, red symbols to the human motion energy harvester), (b) simplified representation of the applied efforts for both cases. Table 2.1 - Characteristics of the three types of piezoelectric cantilever implemented in the vibration energy harvesters. A

Application field

Machine environment

Human environment

Manufacturing Material Representative dimensions

MEMS AlN, PZT W=7 mm, l=2 mm, hp=1 μm, hs=100 μm L=7 mm, T=650 μm mt =100 mg mb =5 mg

Machine environment Human environment Conventional PZT W=5 mm, l=4 cm, hp=600 μm, hs=200 μm, L=5mm, T=1 cm mt =3 g mb =500 mg

Case label Illustration

Representative beam and tip mass

Conventional PZT W=5 mm, l=4 cm, hp=600 μm, hs=200 μm L=5 mm, T=1 cm mt =0 g mb =500 mg

the design of the latter harvester. There are then three particular situations for the piezoelectric cantilevers implemented in this work. They are labeled as case A, B and C. Representative characteristics of the 3 different situations are summarized in Table 2.1.

2.3.2

Concepts of the constitutive matrix and electrical equivalent network

In this section, the general principles of the constitutive matrix and equivalent network representation are discussed first. Then, explicit expressions of the constitutive matrix components are elaborated. Finally, the generalized

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 51 electromechanical coupling factor of the piezoelectric beams is introduced. It is important to define what is meant by constitutive matrix of the piezoelectric beam. The idea consists in relating local values of the intensive macroscopic variables describing the system to the local values of the corresponding extensive variables by simple linear relationships, which can be expressed in the form of a symmetric matrix. It can be considered that a so called lumped model is obtained in this way. This method was initially developed by Smits [125][126] and is adapted here for the geometry and efforts described in Figure 2.18. In this situation, the extensive variables are limited to the force F and the voltage V, while the intensive variables are the displacements w or δ and the charges Q. Q and V have already been ”lumped” when deriving the macroscopic fundamental equations of piezoelectric beams. The location of the applied force has already been defined (Figure 2.18(b)) and it is only required to define a point of observation for the deflection. It is chosen in the following as the position δ g of the center of mass of the attached body (or tip of the beam when L=0). Rather than using a true impedance matrix representation as Smits, the model is developed in the form of a heterogeneous admittance matrix described in (2.42). This particular representation allow developing a convenient form of the equivalent electrical network approach of the problem. In (2.42), kc represents the lumped mechanical stiffness of the beam, Cpc the clamped capacitance of the piezoelectric layer (this time coupled to an elastic layer, Cpc 6=Cpf ) and Γc a macroscopic piezoelectric conversion factor, macroscopic ”equivalent” of e31 . F kc Γc δg = (2.42) Q Γc −Cpc V The impedance matrix of (2.42) can be represented in the form of an electrical equivalent circuit. This type of representation has some general advantages: it first allows representing the problem in terms of lumped parameters belonging to a single engineering domain. Network representation provides a single domain tool which requires only a basic understanding of the physical phenomena occurring in the other domain. These models are furthermore very general, as they can be applied to many kinds of electromechanical transducers, as shown by Tilmans in [138][139]. Second and more important, the network representation can be easily integrated into modern simulation software which allow building models where the transducer is coupled to complex mechanical or electrical structures. This type of model is particularly useful in the field of energy harvesting, as the transducer has to be connected to power management electronics. Electrical network representations of piezoelectric transducers have been initiated by the work of Butterworth and van Dyke [104][105], where a model representing the behavior of the structure near to resonance was proposed. Mason was the first to introduce a representation taking into account wave propagation phenomena [108]. Redwood [140] and Krimholtz [141] proposed refined representations of Mason’s model for thickness mode piezoelectric transducers

2.3 Constitutive matrix and electrical network representation of piezoelectric 52 beams

dδg/dt

1/kc

Γc : 1

dQ/dt Cpc

Figure 2.19 - Electrical network representation of a piezoelectric beam.

using transmission lines analogies. The equivalent network representation of piezoelectric bending structure was exhaustively analyzed by Ballato in [142] and the circuit used in the analysis is inspired from his work. Making use of Kirschoff’s laws for transforming the constitutive matrix and adding the dynamic component due to the attached mass mt , the dynamical equilibrium of the piezoelectric beam can be represented in the form of the equivalent two ports electrical network shown in Figure 2.19. The left side of the circuit corresponds to mechanical parameters, the right to electrical. The mechanical stiffness kc is represented in the electrical domain by a capacitor 1/kc and the attached mass by an inductance of value mt . The electromechanical conversion related to the piezoelectric effect is represented by a perfect transformer of ratio Γc . In the particular situation of vibration and motion energy harvesters, the mechanical ports of the network are connected to a source of mechanical energy, steady state vibrations or impulse, while the electrical ports are connected to an electrical load in which energy is stored or dissipated. Complex values of the different components allow representing the parasitic losses occurring in the system. Explicit expressions of the different parameters involved in the constitutive matrix and in the equivalent electrical network should now be determined. This process goes through the solutions of the beam and charge equations (2.36) and (2.37) described in the next section.

2.3.3

Constitutive matrix and equivalent electrical network of the piezoelectric harvesters

General methods for solving the beam equation are given for example in [136]. The charge equation is solved without difficulties once a solution of the beam equation is available. The dynamical behavior corresponding to the piezoelectric benders implemented in the two types of investigated energy harvesters F are different in essence. In the case of the machine environment harvester, the device operates according to a steady state principle, while for the human environment harvester, the repeated impacts of the moving body on the bender results in free-vibrations and a transient type behavior. In both cases, it can be

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 53 assumed that the displacement is separable in space and time. For the steady state situation, the displacement can be written as w(x,t)=wx (ω,x)wt (ω,t), in which ω is the angular frequency of the input motion. The time dependant part of the solution can be written in complex form as wt (ω,t)=Cexp(j(ωt+ψ)) in which j is the complex unity, ψ a phase angle and C a constant related to the amplitude and frequency of the input. The solution for multi harmonic vibrations is found through Fourier decomposition of the input and superposition principle. For the transient case, results of modal analysis [136] suggest that w(x,t) can be expressed as w(x,t)= Σ wxi (ω i ,x)wti (ω i ,t) in which ω i is the ith eigenfrequency, wxi (ω i ,x) is the related modal shape of the beam and wti (ω i ,t) is the related time component. It is possible to write wti (ω i ,t)=C exp((ξ i +jω i )t) in which ξ i is a dissipative factor. It is shown in Chapter 4 that one can assume in the discussed situation that the impact on the beam excites solely the fundamental mode at ω 0 (this frequency depends on the electrical boundary applied on the electrodes in the case of a piezoelectric laminated beam). The solutions of the steady state and transient problems have then an equivalent form and can be elaborated by a single type of analysis described in the following. The derivation of the impedance matrix does not require an immediate investigation of the time dependant part of the displacement. This analysis is performed later in the chapters dealing with the optimization of the different harvesters. For both transient and steady-state cases, the time components in the beam equation can be eliminated yielding (2.43) with ω=ω 0 for the transient situation. λ represents the characteristic wavelength of the bending wave in the beam. 1 d4 wx (ω, x) − 4 wx (ω, x) = 0 (2.43) dx4 λ s 1 4 EIeq (2.44) λ= √ ρeA ω General solutions of (2.43) can be written as wx (ω, x) = C1 sinh

x λ

+ C2 sin

x λ

+ C3 cos

x λ

+ C4 cosh

x λ

(2.45)

in which the Ci are real constants. These constants depend however on the electrical and mechanical boundary conditions of the system. Four boundary conditions are required for obtaining expressions of the Ci coefficients. Two different sets of boundary conditions are required for determining the expressions of the constitutive matrix parameters. When considering that the electrodes of the piezoelectric bender are short circuited, i.e. V=0, the matrix constitutive equation is reduced to F kc = δg (2.46) Q Γc

2.3 Constitutive matrix and electrical network representation of piezoelectric 54 beams In this situation, the relevant mechanical boundary conditions are defined as follow (in the frame of reference attached to the bender): at the clamped end, no displacement is possible so that wx (ω,0)=0. There are also no deflection angle at the clamped end so that dwx (ω,0)/dx should also be null. At the free end of the equivalent cantilever of Figure 2.18b, a shear force F and a moment FL/2 result from the force applied on the attached mass. The boundary conditions for this situation are w (0) = 0 dw (0) =0 dx F L d2 w (l) =− dx2 EIeq 2 d3 w (l) F = dx3 EIeq

(2.47)

In order to simplify the expressions of the equations developed in the remainder of this chapter, the following definitions are used. Ac = cos λl Ach = cosh λl As−ch = sin λl cosh λl As = sin λl (2.48) Ash = sinh λl Ac−ch = cos λl cosh λl Ac−sh = cos λl sinh λl As−sh = sin λl sinh λl It is now possible to determine the expressions of the coefficients Ci of by solving the system of equations defined by the boundary conditions combined with (2.45): λ2 λ (Ac + Ach ) + L2 (As − Ash ) F C1 = −C4 = 2EIeq (1 + Ac−ch ) λ2 −λ (As + Ash ) + L2 (Ac + Ach ) F C2 = −C3 = 2EIeq (1 + Ac−ch )

(2.45) (2.48)

(2.49) (2.50)

As expressed by (2.46), F=kc δ g in the considered situation. As δ g =wx (ω,l)+ L/2 dwx (ω,l)/dx, it is now possible to determine the expression of kc : kc =

4EIeq (1 + Ac−ch ) λ ((L2 + 4λ2 ) As−ch + (L2 − 4λ2 ) Ac−sh + 4λLAs−sh )

(2.51)

From (2.46), Q=Γc δ g for a short circuited bender. By inserting the expression of wx obtained above in the charge equation (2.37) and by performing some additional algebraic manipulations, one can derive Γc =

2α (L (As−ch + Ac−sh ) + 2λAs−sh ) (L2 + 4λ2 ) As−ch + (L2 − 4λ2 ) Ac−sh + 4λLAs−sh

(2.52)

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 55 Two components of the constitutive matrix have been obtained by considering the behavior of a short circuited piezoelectric bender. The constitutive matrix is symmetric and a single parameter is missing, being the clamped capacitance of the piezoelectric layer. It is obtained by analyzing the behavior of a beam undergoing an applied voltage without applied mechanical force (F=0). In this case, the matrix constitutive equation is reduced to the form given in 2.53. δg −Γc /kc = V (2.53) Q −Cpc − Γ2c /kc The relevant mechanical boundary conditions are defined as follow: no displacement neither rotation are allowed at the clamped end, i.e. wx (ω,0)=0 and dwx (ω,0)/dx=0. As described by (2.33), a voltage applied across the electrodes of the piezoelectric beam results in a bending moment Mv =-αV along the beam, while no shear force exists. For this situation, the boundary conditions are w (0) = 0 dw (0) = 0 dx αV d2 w (l) = − dx2 EIeq d3 w (l) =0 dx3

(2.54)

By the same method used for the short circuited bender, the coefficients Ci are first derived: αλ2 (Ash − As ) C1 = −C4 = V (2.55) 2EIeq (1 + Ac−ch ) C2 = −C3 = −

αλ2 (Ac + Ach ) V 2EIeq (1 + Ac−ch )

(2.56)

The value of the dynamic clamped capacitance Cpc is obtained by solving the charge equation (2.37): Cpc = Cpf −

EIeq

((L2

4λ2 ) A

4α2 λ3 (1 + Ac−ch ) (2.57) 2 2 s−ch + (L − 4λ ) Ac−sh + 4λLAs−sh )

It was verified that the expression of Γc found with the boundary conditions corresponding to the clamped bender was the same than the one obtained with short circuit condition. The three different components of the constitutive matrix are now available. The computations are done assuming mechanical wave propagation phenomena and the so called dynamic constitutive matrix of the piezoelectric bender has been obtained. When the length of the attached mass is null, the components of the derived matrix correspond to those found by Smith [126]. The obtained formulas are tedious and developed for the purpose of studying the response of the harvester excited around high order modes of oscillation. However, in

2.3 Constitutive matrix and electrical network representation of piezoelectric 56 beams the present work, the devices are excited around their resonance frequency and a quasi-static approximation (mechanical wave propagations neglected, i.e. λ→∞) is justified. It was verified by FEM analysis that the quasi static formulas give reasonable estimates of the constitutive parameters for the three types of bender described in Table 2.1. The quasi static constitutive parameters are denoted as k0 , Γ0 and Cp0 . In order to introduce the effects of parasitic damping in the quasi static form of the model of Figure 2.19, complex values of the parameters are used in the analysis proposed in the next chapters. The approach is limited to complex values of k0 and Cp0 which can be rewritten as k0 =k(1+j/Qm ) and Cp0 =Cp (1+j/Qe ) in which Qm and Qe are the quality factors related to respectively the mechanical and dielectric dissipations (Qm >0, Qe <0). The real part k of the quasi static stiffness, the quasi static transformation factor Γ and the real part Cp of the quasi static capacitance are k=

Γ=

3EIeq 1 + 23 Ll + 3α 1 +

2l 1 +

Cp = Cpf −

3L 2 l

4EIeq

L l

3 L2 4 l2

(2.58)

(2.59)

3 L2 4 l2

α2 l 1 + 32 Ll +

3 L2 4 l2

(2.60)

The equivalent network describing the dynamic equilibrium of the piezoelectric bender has to be slightly rearranged when the quasi static approximation is considered. Indeed, in the dynamical form of the circuit, the mass of the beam is taken into account by the component kc . The quasi static version of this parameter is not related to the mass of the beam mb . This problem can be solved by replacing the mass mt of the attached body by an effective mass me in the circuit of Figure 2.19. An expression of me can be obtained by first elaborating a formula for the mechanical angular resonance frequency ω 0 of the beam. It can be understood from the equivalent network that a piezoelectric beam disposes of two characteristics frequency: one corresponding to a short and the other to an open circuit arrangement. In order to avoid confusion between the two values, the short and open circuit resonance are labeled as ω s0 and ω o0 respectively. It will be shown in later chapters that the relation between the two values can be approximately written as ω o0 = ω s0 (1+Γ0 2 /(k0 Cp0 ))1/2 . From the circuit of Figure 2.19 (with mt replaced by me ), it is possible to write ω s0 =((k0 /me )1/2 , so that the determination of ω s0 allow finding an expression for the effective mass of the beam. In this case, ω s0 is not easily found from the eigenvalues problem because of the attached mass resulting in a highly non linear characteristic equation. Rather, the Rayleigh quotient method is used [136]. It gives an upper bound to the fundamental frequency. The Rayleigh quotient R can be defined as R=Wp /Wk =(ω s0 )2 , in which Wp and Wk represent respectively the maximum potential and kinetic energy present in the system. For the situation described

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 57 by Figure 2.18(b) and in short circuit configuration, (ω s0 )2 can be written as Zl

EIeq

2 (ω0s )

d2 w (x) dx2

= mb l

Zl 0

l+L Z mt 2 2 w (x) dx + δ (x) dx L

(2.61)

The quasi static form of the expression of wx (ω,x) found when computing the parameters kc and Γc of the impedance matrix have to be used in (2.61) in order to determine the closed form expression of (ω s0 )2 given below. 3 L 3 L2 3EIeq 1 + + 2 l 4 l2 2  (ω0s ) =  (2.62) 2 33 m l3 1 + 91 L + 21 L   140 b 66 l 44 l2     2 3 4 L + 21 L + 3 L +mt l3 1 + 3 L + 63 2 3 4 16 8 4 l l l l The correctness of (2.62) was investigated by performing 3D finite elements simulation (FEM) on a simple singled layer elastic beam. We observed that the proposed closed form expression yields very small error for the geometrical situations studied in this work. A noticeable case where (2.62) does not predict correctly the resonance frequency is related to structures for which a short and extremely thick mass is attached to the cantilever (”L shaped”). By computing k0 /(ω s0 )2 , the expression of the effective mass given below is obtained. L 21 L2 1 + 91 66 l + 44 l2 33 mb me = 2 2 140 1 + 32 Ll + 43 Ll2 L2 21 L3 3 L4 + + 1 + 3 Ll + 63 2 3 4 16 l 8 l 4 l (2.63) +mt 3L 3 L2 2 1 + 2 l + 4 l2 It can be observed that when no mass is attached to the beam (mt =0, L=0), the familiar formula me =33/140mb is obtained. At the opposite, when the attached mass is very large compared to the one of the beam but is concentrated (L=0), we find me =mt . In this part, the constitutive parameters are derived solely for piezoelectric unimorphs (Figure 2.14(a)). The proposed results can easily be adapted for the symmetric bimorphs of Figure 2.14(b) by reconsidering the expression of EIeq , A and Ii taking into account the corrected dimensions of the cantilever. Also, Γ0 and Cp0 have to be replaced by respectively 2Γ0 and 2Cp0 for the

2.3 Constitutive matrix and electrical network representation of piezoelectric 58 beams bimorph parallel configuration. Cp0 has to be replaced by Cp0 /2 for the bimorph series configuration. The bimorph parallel or series arrangement has a strong influence on the performances of the bimorph in terms of actuation or sensing applications. On the opposite, it is shown later that using of a series or parallel bimorph configuration does not have important consequences in energy harvesting situations.

2.3.4

Generalized electromechanical coupling factor

The longitudinal electromechanical coupling factor k31 2 and the related effective coefficient k31 2 /(1-k31 2 ) have been introduced previously. They define the amount of mechanical energy transformed into electrical energy for a piezoelectric axial transducer during a quasi static thermodynamic cycle. Equivalent variables exist for multilayered piezoelectric beams. The generalized electromechanical coupling factor K which is the equivalent of k31 2 /(1-k31 2 ) [143]. Because of physical considerations, the generalized electromechanical coupling factor of a bimorph or unimorph can only be smaller than the coefficient k31 2 /(1-k31 2 ) characteristic of its piezoelectric layer. In these structures, the piezoelectric layer(s) is supported by a purely elastic one, so that only a fraction of the developed mechanical strain produces electrical charges. Also, the structure undergoes bending deformations and not purely tensile or compressive ones. K can be expressed in terms of the equivalent network parameters as K2 =

Î&#x201C;2 kCp

(2.64)

It is shown in the chapters dealing with the optimization of the harvesters that the generalized electromechanical coupling factor is an important parameter related to the output power of the devices. It has been demonstrated by the author of this thesis in [56] that the generalized electromechanical coupling factor depends solely on the thicknesses ratio, on the compliances ratio ss /sE p and on the transverse electromechanical coupling factor k31 2 . Furthermore, K2 is almost directly proportional to k31 2 . The generalized electromechanical coupling factor allows representing the performances of the beam in terms of energy harvesting for any piezoelectric material (transverse isotropic) and independently of the width and length of the cantilever. For each compliances ratio, a particular value of the relative thickness leads to a maximum of K2 . In the case of the bimorph, a series or parallel arrangement of the piezoelectric layers does not have any influence on the value of the generalized electromechanical coupling factor and then on the energy harvesting capabilities of the device. The transverse electromechanical coupling factor k31 is in the range of 0.4 for modern PZT ceramics and the maximum values of K that can theoretically be obtained are approximately 0.2 for unimorphs and 0.35 for bimorphs.

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATED BEAMS 59

2.4

Conclusion

In this chapter, the history and the basic concepts of the piezoelectric effect are presented. Then, the linear constitutive equations of piezoelectricity are derived, based on the classical continuum description. Non linear phenomena and simplifications of the constitutive equations are also described. Piezoelectric cantilevers constitute the electromechanical transduction element in the vibration energy harvesters studied during this thesis. Therefore, the constitutive equations of piezoelectricity are combined with Timoshenkoâ&#x20AC;&#x2122;s beam theory in order to elaborate the fundamental equations of piezoelectric multilayered benders. This complete derivation had not been proposed in the literature. Both dynamic and quasi static solutions of the latter equations are derived and arranged in the form of a constitutive matrix and of an equivalent electrical network. A complete representation of a cantilever loaded by a distributed mass which is the basis of many harvesters currently investigated is developed. Finally, the generalized electromechanical coupling factor, a crucial parameter for the performances of the piezoelectric beams in terms of energy harvesting, is defined and a clear interpretation of the difference between the material and structure coupling factor is proposed. The derived equivalent electrical circuit constitutes the base of the model used to analyze the characteristics and performances of the energy harvesters. Experimental methods for measuring the values of parameters involved in the electrical circuit model are described in Chapter 3. The optimization of the human environment energy harvester is presented in Chapter 4 and the one of the machine environment harvester in Chapter 5.

2.4 Conclusion

Chapter

Manufacturing and primary characterization of MEMS piezoelectric harvesters The fabrication by MEMS technologies and the primary characterization of piezoelectric harvesters are presented. As described in Table 2.1, different technologies were involved in the manufacturing of the piezoelectric benders implemented in the harvesters. However, the devices obtained by conventional processing (cases B and C) are bought from commercial companies and their fabrication is not presented. The manufacturing process for the MEMS devices is described in a first part of this chapter. In a second part, static, steady state and transient methods of characterization for piezoelectric benders are presented. The characterization methods presented in this chapter aim at determining the values of the parameters involved in the impedance matrix and the corresponding electrical network model presented in Chapter 2. The output power characteristics of the harvesters are not discussed in this part, but in the subsequent ones.

3.1

Manufacturing

MEMS processing appears to be an interesting fabrication technique because of the possible mass production and the corresponding cost reduction. It has however to be understood that the output power of the proposed harvesters strongly depends of their mass and then of their volume. The thickness of the devices being limited by the one of the silicon wafer, a high mass and output power corresponds to a large occupied surface per device. The author of this 61

3.1 Manufacturing

Â (a)

(b)

Figure 3.1 - (a) Top view picture of unpackaged harvesters, (b) picture of glass packaged harvesters.

thesis is only marginally involved in the device fabrication. His contribution is limited to the mask design and to the development of some process step. After this initial work, the investigations are concentrated on the design and characterization of the harvesters. The main part of the fabrication work on PZT devices has been done by our colleague Andreas Schmitz in the framework of a post doctoral research. His results have been published in literature [144]. The AlN based harvesters are manufactured by the WATS division of IMEC at the Holst Centre, The Netherlands (http://www.holstcentre.com). A description of the work performed by the Holst Centre can be found in [78] and [24]. Because the manufacturing process of the devices investigated during this thesis is already described in the literature, this section is limited to the presentation of the most important elements in the fabrication of the harvesters. A general description of the devices is proposed in a first subsection which is followed by a presentation of the different process steps.

3.1.1

General description of the manufactured devices

The design of the presented piezoelectric devices is similar to the classical design of bulk micromachined accelerometers and consists of a mass connected to a thin beam and a vibrating package. The process includes a zero level package which is obtained by protecting the wafer holding the piezoelectric generator by a top and bottom wafer attached through benzocyclobutene (BCB) bonding. When the package moves under the action of an external vibration, the piezoelectric beam is deflected and electrical charges are produced. These charges are allowed to flow in a load circuit connected to the contact pads. In this way, electrical energy is harvested from vibrations. Pictures of the fabricated harvesters are shown in Figure 3.1. Several designs of piezoelectric devices differing in geometry and electrical connections are manufactured. The devices are equipped with masses of dif-

3. MANUFACTURING AND PRIMARY CHARACTERIZATION OF MEMS PIEZOELECTRIC HARVESTERS

ferent lateral dimensions. Here, the thickness of the mass is limited to the one of the device wafer. Alternative flows requiring the manual assembly of an external heavier mass were proposed in the literature [95]. The introduction of such a step has to be evaluated in terms of fabrication cost and generated power. Harvesters resonating at various frequencies are realized by modifying the length or the thickness of the bending beam. The targeted values of the resonance frequencies are in the range of 100 Hz to 2 kHz. The devices were designed either as a single piezoelectric capacitor or as a series connection of four piezoelectric capacitors. The latter devices allow a higher output voltage at the price of a lower output current and of higher electrical output impedance. An overview of the geometrical parameters of the different fabricated harvesters is given in Table 3.1. Table 3.1 - Dimensions of the fabricated devices.

Device label Dev1 Dev2 Dev3 Dev4 Dev5 Dev6 Dev7 Dev8 Dev9 Dev10 Dev11

3.1.2

Number of capacitors 1 4 1 4 1 4 1 4 1 4 1

Lateral dimensions of the mass (µm*µm) 3000*3000 3000*3000 3000*3000 3000*3000 5000*5000 5000*5000 5000*5000 5000*5000 7000*7000 7000*7000 7000*7000

Width of the beam (µm) 3000 3000 3000 3000 5000 5000 5000 5000 7000 7000 7000

Length of the beam (µm) 2670 2670 3365 3365 2235 2235 2832 2832 2526 2526 1996

Process flow

The first steps of the proposed process differ for PZT and AlN based harvesters because of the different chemical nature, reactivity and compatibilities of the two materials. The manufacturing methods for AlN and PZT based harvesters merge into a single one after the patterning of the piezoelectric capacitors. In the following, the process of the AlN and PZT devices is described separately till this process step. Then, the last and common steps of the fabrication are presented. The silicon wafers used for the AlN based harvesters have a (100) surface orientation and are coated with 500 nm thermal silicon oxide and 100 nm LPCVD silicon nitride. In the first step of the process, a 100 nm platinum bottom electrode with a titanium adhesion layer is deposited by sputtering and patterned by a lift-off technique [145] in order to define the bottom electrodes of the piezoelectric capacitors. In the second step, a layer of AlN (200-1000 nm) is sputtered on the wafer according to the parameters defined in [146].

3.1 Manufacturing

The AlN coating is then shaped by a wet etching process based on an OPD262 solution. Because the AlN layer is thicker than the electrodes, it is etched in such a way that the contact pad of the top electrode is held on top of the piezoelectric material. In this way, the probability of step coverage related failure is reduced. In the next step of the process, a 100 nm conductive layer of aluminum is sputtered on the wafer and etched according to classical wet techniques respecting the integrity of Pt and AlN, so that the top electrode of the device is patterned without damaging the bottom electrodes and the piezoelectric material. Finally, the top nitride, oxide and silicon are patterned and dry etched to predefine the shape of the cantilever beam. The depth of the trench defines the thickness of the beam and it is between 20 and 100 µm for our devices. The steps described in this paragraph are illustrated in Figure 3.2. For the PZT based harvesters, silicon wafers coated with 500 nm thermal oxide, 100nm nitride and a 1 µm PZT layer (sandwiched between metallic electrodes) are bought from Inostek, Korea (http://inostek.com/). Details of the deposition process are given on the company’s website. Both electrodes on Inostek’s wafers are made of platinum. For these devices, the first step of the process consists in patterning and etching the top Pt layer by reactive ion etching in order to define the top electrode and first bond contact of the device. The PZT layer is then etched by the successive action of a buffered HF/HCl mixture and concentrated HF [147]. The thickness of the resist used in the lithography process and the etch time must be appropriate for this step, as hydrochloric acid etch organic compounds. In the next step, the bottom platinum layer is patterned and etched by the same technique as the top electrode to form the bottom one. Finally, as in the AlN process flow, the shape of the cantilever beam is predefined by patterning and dry etching the top nitride, oxide and silicon. The steps described in this paragraph are illustrated in Figure 3.3. The release of the PZT and AlN based devices are then done by the same method. The back silicon nitride and oxide are first patterned and etched to define the limit of the KOH wet etching used in the next step for shaping the beam and the attached mass. Si3 N4 is KOH resistant and can be used as a hard mask for such a chemical etching. The top side of the wafer supporting the piezoelectric capacitor is protected from KOH attack by being maintained in a special vacuum holder. The hard mask of nitride should be carefully designed in order to prevent the convex corners of the mass to be attacked: for a (100) oriented silicon wafer, an anisotropic KOH attack etches only the (100) and (110) planes, while the (111) planes are essentially untouched. As a consequence convex corners of the substrate are completely undercut (etching perpendicular to the (110) planes). As the etching velocities in the <100> and <110> directions are similar, large parts of convex corners are etched away and the volume of the mass can be reduced drastically. In order to avoid this problem, simple corner compensation structures have to be implemented [148][149]. The KOH etch is stopped a few micrometers before reaching the top trenches in order to avoid damaging the piezoelectric layer and of the electrodes. The final release is done by dry etching of the top silicon for the PZT devices

3. MANUFACTURING AND PRIMARY CHARACTERIZATION OF MEMS PIEZOELECTRIC HARVESTERS Process step

Side view

Top view

Silicon wafer coated with oxide and nitride

Deposition and patterning of the bottom electrode

Deposition of the piezoelectric layer

Si3N4 SiO2 Pt

AlN

Deposition and patterning of the top electrode

Predefinition of the cantilever beam

Figure 3.2 - Description of the first steps in the manufacturing of the AlN based harvesters. by wet etching based on tetramethylammonium hydroxide (TMAH) for the and AlN harvesters. The beam and the mass are also slightly etched during this last step. The process steps relative to beam shaping and release are pictured in Figure 3.4.

The last part of the manufacturing process consists in the packaging of the released devices. This is done by adhesive bonding of KOH preshaped wafers to the device wafer, as illustrated in Figure 3.5. BCB (BenzoCycloButene), a photopatternable polymer, is used as adhesive layer. The harvesters are ready to be connected to an electrical load via the contact pads accessible through the opening in the top capping wafer. Note that, for demonstration purposes, glass package wafers were used in the devices of Figure 3.1(b). The description of the manufacturing process of MEMS piezoelectric vibration energy harvesters is proposed in this section. The process is based on bulk micromachining of the silicon wafer. Sputtered AlN and PZT grown by solgel methods are used as piezoelectric materials. The devices are packaged by

3.1 Manufacturing

Process step

Side view

Top view

Silicon wafer coated with oxide, nitride and a Pt/PZT/Pt layer

Patterning of the bottom electrode

Si Si3N4 SiO2 Pt PZT

Patterning of the piezoelectric layer

Patterning of the bottom electrode

Predefinition of the cantilever beam

Figure 3.3 - Description of the first steps in the manufacturing of the PZT based harvesters.

Process step

Side view

Back view

Patterning and etching of backside nitride and oxide

Si SiO2

KOH etching of back cavities

Si3N4

Release of the cantilevers

Figure 3.4 - Release of the cantilevers.

3. MANUFACTURING AND PRIMARY CHARACTERIZATION OF MEMS PIEZOELECTRIC HARVESTERS

Process step

Side view

Back view

Silicon wafer coated with oxyde and nitride

Patterning and etching of back oxyde and nitride

KOH etching of back cavities

BCB

Patterning and etching of top oxyde and nitride

SiO2 Si3N4

Top view

KOH etching of top cavities

Packaging of the device wafer

Figure 3.5 - Preparation of the capping wafers and packaging of the harvester.

3.2 Characterization

being encapsulated between capping wafers. The developed process has been proven to be reliable and to result in functional devices. Some flaws are however present and are being eliminated: the precision of the KOH attack used to shape the mass and the bending beam is limited by the fact that this step is time controlled. In order to avoid this problem, SOI (Silicon On Insulator) wafers will be used in the future as the support of the piezoelectric capacitors. In this way, the KOH etch will be slowed down when reaching the oxide layer and the thickness of the beam will be more precisely controlled. Also, it is shown in the next chapters that the proposed packaging approach introduces a large amount of mechanical dissipations, due to squeeze film damping in the enclosure. To eliminate this problem, future devices will be packaged under vacuum. In the next section, experimental methods for measuring values of the equivalent network parameters of the fabricated devices are described and implemented on different samples.

3.2

Characterization

Abundant literature on the characterization of length or thickness piezoelectric vibrators is available [68, 121, 150]. This is however not the case for piezoelectric bending structures. Recently, some analyses related to this problem were proposed in [151–153]. However, the methods described by these authors were developed for the purpose of characterizing actuators and they do not allow obtaining all the information required. Therefore, specific methods of characterization fulfilling the requirements of this analysis are developed. Thanks to these methods, it is possible to determine experimental values of the parameters of the equivalent network model of the harvesters (Chapter 2). In this section, the general concept of the proposed characterization is first described. Then, static, steady-state and transient methods are presented in three separate subsections. Finally, a summary of the characteristics values of the measured parameters are presented for the three types of studied piezoelectric benders (MEMS PZT and AlN devices and commercial ceramic PZT beams).

3.2.1

General concept for the determination of the network parameters

The methods of characterization proposed are based on a simple approach: in a first step, the particular form of the equivalent network model is ignored and it is assumed that the behavior of the system is represented by a ”black box” linking the two mechanical (force F and displacement δ g ) and the two electrical variables (voltage V and charge Q) of the system. Mathematically speaking,

3. MANUFACTURING AND PRIMARY CHARACTERIZATION OF MEMS PIEZOELECTRIC HARVESTERS

Table 3.2 - The different possible combinations of measurements.

Constrained Measured Case label

V=0 Q=0 F=0 δg=0 (Short circuit) (Open circuit) (Free bender) (Clamped bender) F δg F δg F δg F δg V Q V Q V Q V Q F δg F Q Q δg F V V Q V δg δg Q V F X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12

the black box model corresponds to a set of two equations (assumed linear) for four unknowns. For obtaining a determined system, it is then necessary to constrain two of the four variables. Due to the electromechanical nature of the system, it is in fact necessary to constrain one mechanical and one electrical variable. For characterization purposes, one of the constrained variables is fixed (usually to 0) while the other is varied and the free variables are measured. For example, a force can be applied on a short circuited bender (F varied, V=0) and the force-deflection and the force-charge relations can be measured. In total, 16 different relations can be obtained. Some of them are however reciprocal and 12 are found to be independent. The 12 possible characterization experiments are presented in Table 3.2. Note that for the steady state and transient methods, Q is replaced by the injected current I=dQ/dt more easily measured in these situations. It is shown in the following that some of the experiments of Table 3.2 are difficult to implement in practice. Furthermore, it is not necessary to perform all of them in order to obtain the full set of parameters (however, multiplying the number of performed experiments might increase the accuracy of the values obtained). The different methods of characterization are discussed in the next subsections. Results of the realized measurements are presented in details for a commercial bender and MEMS PZT and AlN devices. The tested commercial bender is a bimorph bought from Piezo Systems Inc. (http://www.piezo.com). Its dimensions are given later in this chapter. A compact steel mass of 3 g is glued to its tip and the assumption of concentrated attached mass can reasonably be applied for this structure. The tested MEMS benders are Dev1 sample using PZT and AlN. While the lateral dimensions of the characterized MEMS benders are the same, the thickness of the silicon beam are different (approximately 80µm and 45µm for the PZT and AlN device respectively).

3.2.2

Quasi static characterization methods

The theoretical expressions corresponding to the measurements described in Table 3.2 are given for a static situation in Table 3.3. All experiments involving the measurement of the voltage or charges are complex to perform because of charges leaking phenomena: the charges developed in the piezoelectric layer tend to be dissipated through the electrodes of the structure or through the read out electronics. The decay of the charges follows an exponential behavior

3.2 Characterization

Table 3.3 - Theoretical relations for the different experiments in the static case.

Case label Theoretical relation

F ΓF Q= δg = k k

X3 Q = Γδ

X4 δ=

(

k 1+ K 2

)

(

K 2F

Γ 1+ K 2

)

Γδ g Cp

Case label Theoretical relation

(

)

Q = Cc 1 + K 2 V

ΓV k

δ=

(

K Q

Γ 1+ K

)

X10

X11

Q = C pV

F = ΓV

X12 F=

ΓQ Cp

governed by a time constant estimated to be 10 ms for the commercial piezoelectric bender. For the MEMS devices, the low amplitude of the developed charge or current in quasi static mode makes these measurements even more difficult. In these conditions, extremely rapid and precise force and displacement actuators and dedicated read out electronics (such as the one used by Dubois [154]) are necessary to perform the experiments X2, X3, X5 and X6. However, rapid actuators would induce transient phenomena and the expressions of Table 3.3 might no longer be valid. Similar limitations are encountered for experiment X4: indeed, the stiffening of the structure due to the generalized electromechanical coupling represented by K2 is annihilated by the leakage of charges. For the experiments X7 and X10, dedicated read out electronics are required. The goal is to provide easy to implement techniques of characterization and no efforts are devoted to the development of such elements. Finally, a charge generator was not available, so that the practical static methods of characterization for the piezoelectric benders are limited to experiments X1 and X8 (X11 does not bring additional information). It can be understood that the easily achieved static measurements only allow obtaining the values of the stiffness k and of the transformation factor Γ. The experiments X1 and X8 are realized with a setup consisting of a capacitive contact force sensor (1 mN resolution) attached to a micromotor (16 nm resolution)(Figure 3.6). Contact with the sample is done with a metallic needle attached to the force sensor. The controller of the micromotor monitors the displacement applied to the structure while the sensor measures the reaction force of the sample. The chosen observation point is located near the tip of the beam for the commercial element. For the MEMS bender, the displacement is applied at the tip of the beam (the limits of the beam correspond to the limits of the piezoelectric capacitor easily found by observation through a microscope). The deflection at the center of the mass δ g is then estimated from the theoretical relation established in Chapter 2. The results of X1 and X8 are proposed respectively in Figure 3.7 and Figure 3.8. k=749N/m and Γ=4.6*10−4 N/V is obtained for the ceramic PZT

3. MANUFACTURING AND PRIMARY CHARACTERIZATION OF MEMS PIEZOELECTRIC HARVESTERS

Micromotor

Force sensor Needle

Sample Figure 3.6 - Illustration of the experimental setup used for performing the quasi static experiments X1 and X8.

150 Displacement (μm)

Displacement (μm)

500 400 300 200 100

100

200 Force (mN)

300

400

(a)

15 20 Force(mN)

(b)

Figure 3.7 - Results of X1 in static mode for (a) the PZT commercial bender, (b) the AlN MEMS bender.

Displacement (μm)

6 5 4 3 2 1

10 15 Voltage (V)

(a)

30 Voltage (V)

(b)

Figure 3.8 - Results of X8 in static mode for (a) the PZT commercial bender, (b) the AlN MEMS bender.

3.2 Characterization

bender, while k=190N/m and Γ=2.2*10−5 N/V is found for the thin film AlN bender. Γ is directly related to the performances of the device in actuation mode; because of the low thickness of the piezoelectric material compared to the one of the substrate in the MEMS devices, this type of piezoelectric bender is much less adapted to actuation than the commercial ones. However, the situation is different for energy harvesting as demonstrated in the next chapters. It has been shown in this section that static methods of characterization are in practice limited for piezoelectric benders, principally due to the phenomena of charge leakage. The only parameters of the network model easily measured are the stiffness k and the conversion factor Γ. In spite of the limitations of the static methods of characterization, the static experiment X1 is extremely important as it allows determining the stiffness k, which can not be directly obtained from the steady state and transient methods.

3.2.3

Steady-state characterization methods

The concept of the proposed steady state methods of characterization is similar to the static methods: one electrical and one mechanical variable are constrained and the others measured. The measurements are realized by using sinusoidal signals (voltage or mechanical vibrations) with a frequency near to the fundamental resonance frequency. The obtained relations are then fitted with the theoretical expressions obtained by analyzing the network model. Fitting is realized here by a characteristic point approach, i.e. the theoretical and experimental relations are only fitted at frequencies corresponding to some remarkable points (maxima and minima) and not over a continuous range of frequency (except for the measurements which are not frequency dependant). Due to the steady state nature of the problem, one should determine for each specific experiment the frequency dependence of the amplitude and phase of the measured variable. Amongst others, it is also possible to use a real and imaginary part representation rather than an amplitude-phase one. The choice of the most relevant representation depends on the considered experiment. Indeed, depending on the situation, the simplicity of the expressions corresponding to the remarkable points of the measured curves depends on the chosen representation. Also, for some of the experiments of Table 3.2, it is advantageous to normalize or multiply the measurements to the frequency of oscillation in order to compute the parameters of the network model. This information is presented when necessary in the tables. The number of remarkable points depends on the considered measurement. For the experiments X3, X6, X10, X11 and X12 of Table 3.2, no frequency dependence is observed and theoretical expressions very similar to the ones obtained for the static case are found. They are given in Table 3.4 taking into account the dielectric and mechanical parasitic dissipations by the intermediary of the electrical and mechanical quality factors Qe and Qm . These experiments

3. MANUFACTURING AND PRIMARY CHARACTERIZATION OF MEMS PIEZOELECTRIC HARVESTERS

are interesting as they allow almost direct measurements of Γ, Qe and Qm . However, several practical limitations are again encountered: the experiments X11 and X12 require a dynamic force sensor with a high resolution (<100 µN for precise measurements) and which does not perturb the dynamics of the tested sample. For the MEMS piezoelectric benders, which have a small effective mass (<100 mg), such high end sensors might exist but are absolutely not standard measurement equipment and these experiments are skipped. They are furthermore not crucial, as Γ and Qe can easily be determined by other means. X3 can be done using as input a controlled mechanical vibration source (a loudspeaker is a cheap example) connected to the tip of the bender while the other end of the test sample is attached to a mechanical ground. X6 can be done using the same experimental arrangement. However, the devices are small and relatively fragile and the handling required for performing experiments X3 and X6 often led to the failure of the devices, so that they are avoided. Finally, the experiment X10 is the sole measurement which is easily done for both types of piezoelectric benders. It can be performed by using an impedance analyzer (a HP4294 was used in our case) and, for example, plain silicon wafers as top and bottom clamps. X10 allows obtaining the value of the clamped capacitance and the electrical quality factor. The results of X10 are given for the MEMS AlN and commercial bender in Figure 3.9. It can be seen that the value of the measured parameter is approximately constant over frequency, even if a slight shift is observed for the real part. For the ceramic PZT bender Cp =2.6 nF and Qe =-30 are determined. For the MEMS structure, Cp =640 pF and Qe =-110 are obtained. A part of the dielectric losses can be attributed to the electrical connections between the tested devices and the impedance analyzer but not all. Indeed, it can be seen that the dielectric quality factor of the MEMS harvesters is higher than the one of the commercial samples while the same electrical connections were used. It is then clear that dielectric dissipations intrinsic to the harvesters exist. It can be supposed that these losses are larger for the commercial structures because they were made of a larger volume of piezoelectric material, so that the probability of defect in the material is increased. Until now, only the experiments leading to non frequency dependant relations have been described. In the case of the experiments X1, X2, X4, X5, X7, X8 and X9, the theoretical relations between the applied and measured variables follow a different behavior, typical of resonant systems, described by mathematical functions disposing of maxima and minima. For X4, X5 and X9, the theoretical relations are extremely tedious when both dielectric and mechanical dissipations are considered and we disregard them as the precise extraction of the parameters is made difficult by the complexity of the relations. Therefore, the analysis is limited to X1, X7 and X8. For these three experiments, the theoretical relations corresponding to the maximum of the amplitude and to the maximum and minimum of the real part are derived. For these three remarkable points, the theoretical expressions of the elements listed below are summarized in Table 3.5 and Table 3.6.

3.2 Characterization

Table 3.4 - Theoretical relations for the experiments exhibiting no frequency dependence in steady state mode.

X3 Measured value: I

X6 Measured value: V*ω ΓQe

Γv

Amplitude

C p 1 + Qe

Phase

ΓQe 2

Γv

Imaginary part

(

C p 1 + Qe 2

ΓQe

(

C p 1 + Qe 2

X11 Measured value: F

Phase

)

ΓQe

ArcTan (1/ Qe ) −

Cp Qe

C pV

ArcTan ( Qe ) ΓQe

(

ΓV

Imaginary part

)

C p 1 + Qe 2

Real part

X12 Measured value: F*ω

ΓV

Amplitude

C p 1 + Qe 2

ArcTan (1/ Qe )

Real part

X10 Measured value: I/ω

C p 1 + Qe 2 ΓQe 2

(

C p 1 + Qe 2

)

-9

-10

6 4

4 2 -11

I /ω (A.s)

I / ω (A.s)

2 -10

6 4 2

4 2 -12

-11

6 4

-13

460

480 ω(rad/s)

(a)

500

4000

4200 ω (rad/s)

4400

(b)

Figure 3.9 - Results of X10 in steady state mode for (a) the PZT commercial bender, (b) the AlN MEMS bender. V=0.1 V in both cases. The solid line corresponds to the real part, the dashed line to the imaginary part.

3. MANUFACTURING AND PRIMARY CHARACTERIZATION OF MEMS PIEZOELECTRIC HARVESTERS

• At the maximum of the amplitude: – Frequency – Value of the amplitude – Corresponding value of the phase • At the maximum of the real part: – Frequency – Value of the maximum real part – Corresponding value of the imaginary part • At the maximum and minimum of the imaginary part: – Frequency – Value of the maximum and minimum of the imaginary part – Corresponding value of the real part The experiments X1 require a dynamic force as input. In order to ”simulate” it, a known mechanical vibration is applied to the clamped end of the device. By means of inertia, an apparent force is developed along the length of the beam and of the attached mass. However, as explained in Chapter 2, the inertial force on the beam itself can be neglected for the tested samples and it is relevant to only consider the resulting force on the attached mass. The applied force can be expressed as F=me A0 in which me and A0 are the values of respectively the effective mass and the amplitude of the acceleration of the input vibration. For experiment X1, the deflection of the beam was monitored using an optical setup consisting of a laser beam and of a position sensitive detector (PSD). The results of X1 are given in Figure 3.10. The values of the resonance frequencies ω s0 are found to be equal to 4153rad.−1 for the MEMS bender and to 469rad.s−1 for the commercial one. It is now possible to determine the effective mass of the structure reminding that ω s0 =(k/me )1/2 . For the AlN MEMS device, k was previously found equal to 190 N/m, so that we can derive me = 11 mg. The effective mass is in this case equivalent to the large mass attached at the tip of the cantilever. For the PZT commercial element, we obtained me =3.4 g. According to the equations derived in Chapter 2, me = 33/140mb +mt when a concentrated attached mass mt is assumed. The latter was equal to 3 g for the tested bender, so that we can estimate mb to 1.7 g, which corresponds roughly to the values obtained from the dimensions and material properties of the piezoelectric beam. It is also possible to determine the mechanical quality factor from the maximum of the amplitude: Qm =640 is found for the MEMS harvester and Qm =48 for the commercial structure. The frequencies of maximum and minimum of the imaginary part corroborated these values.

3.2 Characterization

Table 3.5 - Theoretical relations for experiments X1 and X2 in steady state mode.

X2 X1 Measured value: I/ω Measured value: δg Maximum of the amplitude

ω0s

Frequency

ω0s

Value

( )

Corresponding phase

Qm Γ

2 me ω0s

( )

me ω0s

0 Maximum real part

Frequency

ω0s Qm

Value

( )

me ω0s

Corresponding value of the imaginary part

ω0s 1−

( )

2 2me ω0s

Value of the real part

( )

2 2me ω0s

0 Imaginary part Minimum Maximum

1 Qm

Value

( )

me ω0s

0 Maximum

Frequency

ω0s Qm Γ

ω0s 1+

−

1 Qm

( )

2 2me ω0s

ω0s 1−

( )

2 2me ω0s

1 Qm

Qm Γ

( )

2 2me ω0s

Qm Γ

Minimum

( )

2 2me ω0s

ω0s 1+

−

1 Qm

Qm Γ

( )

2me ω0s

Qm Γ

( )

2me ω0s

-4

3.5x10

1.2

0.8

-40

0.4

-80 440

460 480 ω (rad/s)

(a)

3.0 40

2.5 2.0

1.5

-40

1.0 -80

0.5

500

4120

Phase (°)

1.6

Amplitude of δg (m)

Phase (°)

Amplitude of δg (m)

2.0x10

4140 4160 ω (rad/s)

4180

(b)

Figure 3.10 - Results of X1 in steady state mode for (a) the PZT commercial bender (A0 =1 m.s−2 ), (b) the AlN MEMS bender(A0 =9.8 m.s−2 ). The solid line corresponds to the amplitude of the deflection, the dashed line to its phase.

3. MANUFACTURING AND PRIMARY CHARACTERIZATION OF MEMS PIEZOELECTRIC HARVESTERS

Table 3.6 - Theoretical relations for experiments X7 and X8 in steady state mode. X7 X8 Measured value: I/ω Measured value: δg Maximum of the amplitude

ω0s

Frequency

Qm Γ

( )

Value

me ω0s

Corresponding phase

0 Maximum real part

Frequency

ω0s

ω0s Qm Γ

⎛ 1 ⎞ −C p ⎜ − K 2Qm ⎟V ⎝ Qe ⎠

Value Corresponding value of the imaginary part

Frequency

Value of the real part

Cp V

Imaginary part Maximum Minimum

Imaginary part Minimum Minimum

ω0s 1−

1 Qm

⎛ K 2Qm ⎞ C p ⎜1 + ⎟ V ⎜ 2 ⎟⎠ ⎝

Value

( )

me ω0s

ω0s 1 +

1 Qm

⎛ K 2Qm ⎞ C p ⎜1 − ⎟ V ⎜ 2 ⎟⎠ ⎝

⎛ 1 K 2Qm ⎞ ⎛ 1 K 2Qm ⎞ −C p ⎜ − − ⎟⎟ V −C p ⎜⎜ ⎟ V ⎜Q 2 ⎠ 2 ⎟⎠ ⎝ e ⎝ Qe

ω0s 1−

1 Qm

Qm Γ

( )

2me ω0s

Qm Γ

( )

2 2me ω0s

ω0s 1+ -

1 Qm

Qm Γ

( )

2me ω0s

Qm Γ

( )

2me ω0s

X7 is easily performed with a classical impedance analyzer as for the clamped electrical impedance measurement corresponding to experiment X10. The obtained measurements are illustrated in Figure 3.11. The real-imaginary parts representation allows determining Qm =670 and Qm =58 for respectively the MEMS and commercial bender. Also, from the values of the clamped capacitance and of the electrical quality factor derived previously, one can determine K2 =2.9*10−3 for the MEMS structure and K2 =1.2*10−1 for the commercial one. Assuming the values of k and Cp found in the previous experiments, Γ=1.9*10−5 N/V for the MEMS harvester and Γ=4.8*10−4 N/V for the commercial element. These values are in agreement with those measured with static experiments. The last performed steady state measurement, X8, is done with the same PSD setup used for X1. The obtained results are depicted in Figure 3.12. The extracted values of the different parameters were Qm =660, Γ=2.5*10−5 N/V for the MEMS sample and Qm =64, Γ=5.5*10−4 N/V for the commercial one. The value of the mechanical quality factor is similar to those found from X1, X2 and X7. For Γ, the obtained results corroborate those found from the static

3.2 Characterization

-12

-9

100x10

1.5x10

80 I /ω (A.s)

I /ω (A.s)

1.0 0.5 0.0

60 40 20

-0.5 440

450

460 470 ω (rad/s)

480

0 4120

490

(a)

4140 4160 ω (rad/s)

4180

(b)

Figure 3.11 - Results of X7 in steady state mode for (a) the PZT commercial bender (V=0.1 V), (b) the AlN MEMS bender(V=0.1 V). The solid line corresponds to the real part, the dashed line to the imaginary. -4

-4

4.0x10

2.0

1.6

1.2 0.8

-40

0.4

-80

ω (rad/s)

(a)

3.0

2.0

-40

1.0

-80 0.0 4120

440 450 460 470 480 490

80 Phase (°)

Amplitude of δg (m)

80 Phase (°)

Amplitude of δg (m)

2.4x10

4140

4160

4180

ω (rad/s)

(b)

Figure 3.12 - Results of X8 in steady state mode for (a) the PZT commercial bender (V=5 V), (b) the AlN MEMS bender(V=5 V). The solid line corresponds to the amplitude of the deflection, the dashed line to its phase.

implementation of X8 and from the steady state implementation of X7. It has been shown in this section that the steady state methods of characterization allow determining most of the parameters of the equivalent network representation of the piezoelectric benders. The different experiments lead to some variation of the measured parameters, because of the deviations from the model and because of experimental errors. However, the spread of the measurements was reasonable and their averages can be considered as a representative value. The steady state characterization approach proposed in this section is very simple and can be refined by, for example, considering more characteristic points and also their derivatives. The obtained values might be processed by an algorithm performing statistical analysis. At the moment, such an algorithm has been written solely for the clamped and free impedance analysis. In the future, it will be combined with other routines analyzing the results of the other measurements.

3. MANUFACTURING AND PRIMARY CHARACTERIZATION OF MEMS PIEZOELECTRIC HARVESTERS

3.2.4

Transient characterization methods

For piezoelectric benders, transient methods of characterization are limited due to the complexity of the analytical treatment of the transient response of the devices. As shown in Chapter 4, it is not possible to obtain a simple closed form of the transient behaviour in a general case. It is not relevant to monitor the open circuit voltage or short circuit current resulting from a shock, as the possible presence of dielectric losses makes the behavior of these two variables relatively complex. For the same reason, it is difficult to realize a precise analysis of the waveform of the deflection resulting from an initial voltage or current. Therefore, the sole parameters that can be easily obtained from transient techniques are the fundamental resonance frequency and the mechanical quality factor. These parameters can be obtained by applying an impulse force or by setting initial velocity or displacement conditions to the short-circuited structure and by monitoring the time dependence of the deflection after the impulse: the so called free oscillation behavior of the deflection is measured. This measurement is done with the PSD setup described previously. The mechanical behavior of a piezoelectric bender in short circuit conditions is equivalent to the one of a purely elastic beam. In this case, the free oscillations follow a pseudo sinusoidal behavior described by s

δg (t) = e−ω0 /2Qm t s " A cos ω0s

1 t 1− 4Q2m

! + B sin ω0s

!# 1 t 1− 4Q2m

(3.1)

in which A and B are real constants. Classical methods such as the logarithmic decrement [151] or a simple fit with an exponential function are used to determine the ratio ω s0 /2Qm . From this ratio and from the free oscillation frequency ω s0 (1-1/4Qm )1/2 the values of ω s0 and Qm are obtained. ω s0 =4150 rad.s−1 and Qm =710 for the MEMS device, while ω s0 =468 rad.s−1 and Qm =62 for the ceramic PZT bender. In the next section, typical values of the network parameters for the different fabricated MEMS devices and commercial samples are summarized and compared. Also, the material properties of the elastic and piezoelectric layers implemented in the piezoelectric benders are estimated from the performed measurements.

3.2.5

Typical values of the network parameters and estimation of the material properties

The different characterization experiments done on the AlN based Dev1 sample have been repeated for a large number of samples (PZT and AlN based MEMS harvesters and ceramic PZT commercial devices). The obtained results were relatively homogeneous and corresponded to the expected values for most of

3.2 Characterization

the samples. In order to perform a comparison of the values of the parameters, Table 3.7 shows measured values of the equivalent network parameters for the AlN Dev1 sample and the commercial bimorph characterized in the previous sections. Also, the parameters for a PZT Dev1 sample are given in the same table. It is difficult to draw a direct conclusion from the values of the parameters, as the dimensions of the three devices are quite different. However, it can easily be said from the values of Γ that the PZT devices, both thin film and ceramic, are more adapted to force actuation application than the AlN devices. The displacement actuation performances are controlled by the ratio Γ/k similar for the three devices, but with a small advantage for the ceramic PZT bender. The sensing performances are governed by the ratio Γ/Cp and the AlN MEMS device performs in this case better than the PZT MEMS structure, but not as well as the commercial bender. The mechanical dissipations are mainly due to air damping and increase with the size of the devices. It is then observed that the mechanical quality factor of the commercial device is much smaller than for the MEMS. The MEMS devices tested in this chapter are unpackaged. The packaged devices have a lower mechanical quality factor, similar to the one of the commercial structures. It is planned to package the MEMS devices under vacuum in order to avoid this problem. The values of the electrical quality factor are in the same range, but slightly smaller for the PZT devices. Also, it can be seen that the value of K2 is much larger for the commercial device than for the MEMS. This result is mainly due to the low thickness of the piezoelectric material compared to the one of the substrate in the MEMS devices. Indeed, as shown in Chapter 2, the generalized electromechanical coupling factor depends solely on the thicknesses and compliances ratio and on the piezoelectric constant k31 . K2 is an important parameter for power generation and one could argue that the MEMS devices should have been designed for optimum K2 . However, technical problems limit the thickness of the deposited piezoelectric layer and the beam should not be too thin in order to avoid fracture phenomena. Furthermore, it is shown in Chapter 5 that, for the machine environment harvester, it is not necessary to optimize the generalized electromechanical coupling factor if parasitic dissipations can be maintained below a certain threshold. Combining the measured values of the different parameters with their theoretical expressions in terms of dimensions and material properties developed in Chapter 2 ((2.58), (2.59), (2.60), (2.40)and (2.41)), the effective compliance of the substrate ss , the effective permittivity εTp and the effective piezoelectric constant ep =dp /sE p of the piezoelectric material can be estimated. In order to choose the correct equations, it is important to remind that the MEMS devices are unimorphs while the ceramic PZT beam are symmetric bimorphs. Also, the thickness of the piezoelectric material was neglected for the MEMS devices and ξ=0 in (2.40) is assumed for both thin film and ceramic structures. For the AlN MEMS harvesters, the values obtained are quite similar to those

3. MANUFACTURING AND PRIMARY CHARACTERIZATION OF MEMS PIEZOELECTRIC HARVESTERS

found in the literature [78]. For the PZT MEMS harvesters, the compliance of the silicon differs slightly from the one found from the analysis of the AlN device, probably because the etching of the beam is imperfect and results in thickness unhomogeneities. Inostek (http://inostek.com/), the supplier of the PZT covered wafers, do not provide the material properties of their PZT so that it was not possible to compare the obtained values with existing data. For the PZT commercial structure, the thickness of the piezoelectric material is not negligible as it is the case for the MEMS ones. Therefore, the values of the piezoelectric material compliance given on the manufacturer website (sp =17 pPa−1 ) are used to estimate ss . The measured value of the permittivity and of ep corresponds approximately to the manufacturer’s data. Unfortunately, it is not possible to estimate the values of the electromechanical coupling factor k31 in the case of the MEMS harvesters as the compliance of the piezoelectric material is unknown. For the PZT commercial structure, k31 =0.33. Table 3.7 - Values of the network parameters and estimated material properties for the different types of tested samples. The standard deviation of the measurements is indicated when possible.

Measured parameters

Dimensions

Estimated material properties

3.3

k0 (N.m−1 ) me (mg) Γ (µN.V−1 ) Cp0 (nF) Qm (-) Qe (-) K2 (-) ω s0 (rad.s−1 ) W (mm) l (mm) L (mm) hs (µm) hp (µm) ss (pPa−1 ) ep (C.m−2 ) εTp /ε0 (-)

AlN MEMS 190 11.2±0.4 22±3 0.64±0.01 656±15 -110±13 0.0036±0.0006 4120±74 3 2.37 3 45 0.8 6.6 0.9±0.1 8.5±0.2

PZT MEMS 830 11.4±0.2 170±24 71±1.6 1200±23 -63±9 0.00057±0.00003 8691±89 3 2.37 3 80 1 8.5 4.1±0.5 1130±25

Ceramic PZT 749 3390±100 496±47 2.58±0.03 57±8 -30±6 0.114±0.008 470±7 3.2 31.8 7 340 270 14.7 11.9±1.1 1850±21

Conclusion

In this chapter, the manufacturing by MEMS technologies of piezoelectric harvesters and the experimental determination of their corresponding equivalent network parameters is presented. A robust process flow is developed. It results in functional devices but still needs some improvements at the level of etch homogeneity and packaging. Particularly, the proposed ambient atmosphere packaging approach results in low mechanical quality factors and will

3.3 Conclusion

be improved in the future by implementing a vacuum packaging method. Static, transient and steady state experimental methods for determining the equivalent network parameters are then described and implemented on MEMS PZT and AlN based harvesters and on a commercial ceramic PZT structures. No complete procedure for the characterization of the network parameters of piezoelectric beams existed before this work and methods are developed to this aim. From a theoretical point of view, a large amount of experiments can be realized. However, from a practical point of view, some of them are difficult to be done and only experiments that can be realized with commonly available laboratory equipments are presented. It is shown that the static methods allow measuring the stiffness k and the transformation factor Î&#x201C;. All the parameters but k can be determined from steady state measurement, which are for most of them easily implemented. Finally, simple transient methods allow only extracting the resonance frequency Ď&#x2030; s0 and the quality factor Qm . Characteristic values of the network parameters are presented for the three types of structures studied. The material properties of the piezoelectric materials are estimated from these measurements. It is shown that PZT is more adapted to actuation situation than AlN, but, at the opposite, that the latter material has better performances in terms of sensing applications. It is now important to analyze the capabilities of the piezoelectric benders when energy harvesting is considered. This is done in the next two chapters, first for the human environment device and then for the machine environment.

Chapter

Design and analysis of the human environment vibration energy harvester In this chapter, the design and the characterization of the impact type harvester described in the introductory chapter of this thesis is presented. For sake of clarity, the general principles of the presented device are repeated. Classical resonant systems are not easily adapted to the human environment because of the low frequency of the ”vibrations” occurring in this case. Therefore, the energy harvester presented is based on a different principle which allows obtaining a reasonable amount of generated electrical power for low frequency excitation. A conceptual representation of the human environment generator is depicted in Figure 4.1. It consists basically of a frame containing a channel which guides a free sliding mass (referred to as the ”missile” in the following) and of two piezoelectric benders located at the extremity of the frame. The benders are cantilevers in our case. When the frame is shaken, the mass occasionally impinges on the piezoelectric structures and a part of the kinetic energy accumulated by the moving object is transformed into electrical energy through the impact on the piezoelectric structures. The output voltages of the benders are processed by an electronic block which can also contain the powered application. This type of harvester has been already discussed by a few authors: Umeda [53–55] pioneered the analysis of the energy generated by the impact of a steel ball on a piezoelectric membrane. Keawboonchuay studied high power impact piezoelectric generator that can be incorporated into ammunitions [26]. The author of this thesis presented some analysis related to the impact harvester for human applications in [56, 57, 155]. Cavallier [58] and Takeuchi [59] described an experimental analysis of an equivalent device.

4.1 Modeling of the impact based harvester Free moving object

Guiding channel

Processing electronics & powered application

Piezoelectric benders Figure 4.1 - Conceptual representation of the human environment vibration energy harvester.

From our knowledge, no other authors have proposed results on the subject. Furthermore, none of the existing publications proposes a detailed model of the device including the description of the impact mechanism and of the resulting behavior of the piezoelectric bender. Therefore, a simple but complete model of such a harvester is developed in a first part of this chapter. The model is based on the equivalent network representation of the piezoelectric benders described in Chapter 2. The experimental validation of the model and the characterization of a prototype are presented in a second part.

4.1

Modeling of the impact based harvester

In order to develop the equivalent network model of the piezoelectric benders, it was introduced in chapter 2 that, in the framework of some approximations, the behavior of an elastic beam after having been impacted by a moving object can be represented by fundamental free oscillations. The details of these approximations and relevant derived information are described in a first part of this subsection. The equations describing the dynamics of the full system are also derived. In a second part, particular types of load circuitry are introduced in the problem and the output power of the harvester is theoretically optimized for some simple input vibrations types and load circuitry situations.

4.1.1

Development of the model

Modeling the impact between two solid objects is a complex subject which can not be easily treated without considering specific applications. Low velocity impacts are relevant in this thesis. They do not result in extreme reconfigurations of the considered system and they can be classified along the four categories described by Stronge [156]. The simplest representation of impact phenomena is achieved by the so called stereo-mechanical model: in this case,

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER

the dynamics of two perfectly rigid and smooth spherical particles is predicted by applying the momentum conservation principle established by Newton. For bodies with a finite stiffness, one should consider the local deformations that occur in the neighborhood of the impact surface, assumed to remain small. The Hertz theory is used to describe this situation. A third category consists in the transverse impact of a rigid body on a flexible element. This case is the most relevant in the analysis of the harvester. A fourth category of impacts described by Stronge consists in axial impact on flexible bodies and is not of interest here. The proposed analysis is limited to one dimensional collision on a flexible beam. It is assumed that the velocities of the objects and the efforts resulting from the collision are directed perpendicularly to the large faces the beam. The contact surfaces are supposed to be perfectly smooth so that the friction mechanisms are neglected. The different phases of the impact phenomena are described in Figure 4.2: first, in (a), the moving object approaches the beam with a velocity directed along the normal to surface of the bender. In this application, this velocity depends on the particular excitation applied to the frame of the device of Figure 4.1. The impact process itself is described by Figure 4.2(b) and consists of two separate phases: in the first, the two objects tend to interpenetrate each other and a local compression referred to as indentation Î´ i is observed in the area surrounding the contact surfaces. The contact area becomes larger as the magnitude of the indentation increases. The compression phase ends when the amplitude of the restoring elastic force F(t) is large enough to induce a local expansion of the two bodies tending to repulse each other at this moment. During the contact time, a radial wave due to the indentation starts propagating away from the impact location. It is assumed in this work that the contact area and the magnitude of the indentation are small so that the corresponding radial wave does not have an important influence on the macroscopic behavior of the two bodies and it is ignored. Another wave consisting in a vertical displacement is initiated in the beam during the impact process. It is referred to as bending wave and results from the exchange of momentum between the two objects. The impact of the missile on the beam might excite a large number of oscillation modes. However, for the configuration studied in this chapter, it is shown later that most of the energy transferred to the beam during the impact is confined to the fundamental mode of vibration, so that only the fundamental bending wave is considered. After the expansion phase (Figure 4.2(c)), the two objects separate with velocities directed along opposite direction. The missile also oscillates along its own eigenmodes after the separation. As the moving object is assumed to be very stiff, these oscillations do not have influence on the general dynamics. In the beam, the bending wave propagates further away from the impacted vertical segment. The behavior of the bender prior to a second impact consists in free oscillations. It is assumed that the time necessary for a â&#x20AC;?standingâ&#x20AC;? wave to be established is negligible compared to the fundamental period of the beam so that the displacement induced at the impact location is

4.1 Modeling of the impact based harvester Compression

Expansion F(t)

F(t) ‐F(t)

‐F(t)

Indentation wave

Bending wave (a) (b) (c)

Figure 4.2 - The different phases of the impact of a stiff body on a flexible beam.

considered in the analysis to propagate instantaneously to the other points of the beam. This last simplification might not be correct in case of an extremely flexible or long bender. Also, the contact time tc is considered to be so short that the deflection undergone by the beam during the contact is neglected. A closed form expression of the contact force F(t) can be found using Hertz model [157]. In this approximated theory, it is assumed that the dynamics of two curved contact surfaces during an elastic impact can be represented by a single degree of freedom model consisting of two lumped masses connected by a non linear spring ki creating a contact force equal to ki δ i 3/2 . In a first approximation, the lumped mass associated with the impacted segment of the beam corresponds to its effective mass me . For the impacting missile, it corresponds to its full mass M. The expression of the indentation stiffness ki obtained by this method is given by (4.1), in which Rb and RM represent the curvature of respectively the beam and missile impact surfaces at the initial contact (in the limit case of a flat impact surface on the beam, Rb →∞), while sb,13 , sb,33 , sM,13 and sM,33 correspond to the relevant components of the compliance tensors of the two objects (the subscript b refers to the bender, M to the missile), according to the direction convention given in Chapter 2. q 4 ki = 3

1−

sb,13 sb,33

−1 Rb−1 + RM

−1

2 !

sb,33 +

1−

sM,13 sM,33

2 !

(4.1)

sM,33

An approximated closed form expression of the contact force during the compressionexpansion phase can be obtained by the method of Lee [158]. Assuming that the impulse can be approximated during the contact period by half a sinusoid of angular frequency Ωc , the contact time can be written as tc =π/tc , i.e. F(t)=Fc sin(Ωc t) for ti <t< ti +tc in which Fc is the maximum amplitude of the impact force and ti is the time at which the objects collide. The expressions of

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER

Fc and Ωc found by Lee’s method are 

3/5 2 0.5M U 0  Fc = 1.437ki  M ki 1 + m e 2 !1/5 ki2 U0 M Ωc = 1+ me 1.306M 2

(4.2)

(4.3)

U0 represents the relative velocity of the two bodies just before impact (U0 = vg -vM ). In order to assess the relevance of the different approximations related to the description of the impact, numerical values of the different characteristic parameters are computed by using the material properties and geometries of the problem. Representative dimensions of the piezoelectric cantilevers implemented in the human environment harvester are given in Table 2.1. For the numerical application, the radius of curvature of the impacting missile at the contact surface is RM =5 mm while Rb tends to infinity (flat surface). M=4 g and me =0.4 g corresponding to the case C of Table 2.1. U0 =1 m.s−1 and the fundamental resonance frequency of the beam ω s0 =1700 rad.s−1 . For the indentation related compliances, the relevant values of steel for the moving mass and of PZT for the beam are used (sb,33 =18.8 pPa−1 , sb,13 = -7.22 pPa−1 , sM,33 =5 pPa−1 ,sM,13 = -1.5 pPa−1 ). In this case, it is found that ki =4.6*109 N/m3/2 , Ωc =1.64*105 rad.s−1 and Fc =60 N. The contact time tc and the maximum indentation are equal to 20 µs and 5.5 µm respectively. The assumptions of negligible duration of the contact and of negligible indentation magnitude are then justified. The numerical application is carried out considering the geometries of a C type bender, but the same conclusions can be reached with type B. The first assumption used in Chapter 2 when developing the equivalent network of the piezoelectric bender is proven; i.e. the duration of the contact is much smaller than the fundamental period of the beam. The impact force of a moving object on the beam can then be represented by a pulse type load. It is also assumed in the derivations of Chapter 2 that the impact process excites solely the fundamental mode of vibration of the beam. The validity of this assumption is verified by performing 2D plane stress FEM simulations of the beam C of Table 2.1. A triangular mesh and quadratic elements are used. The analysis is purely mechanical and piezoelectric effects are not considered. A transient solver is used. Elastic behavior and small deformations are assumed. The first boundary condition of the model corresponds to the clamped end of the cantilever and no displacements or rotations are allowed on this segment. The second boundary condition consists of the pulse force F(t), as expressed in (4.2), applied on the free end point. Because of the 2D plane stress approximation, it is implicitly assumed that this force is applied on the full width of the cantilever, rather than being concentrated on a single point. This consideration does not change the conclusions related to the single mode approximation.

4.1 Modeling of the impact based harvester

Position of the tip of the beam (m)

-3

5x10

Time(s)

Figure 4.3 - Position of the tip of the beam C obtained from FEM (solid line) and by the lumped model (dashed line) in the case of a pulse force applied at the tip.

In Figure 4.3, the tip deflection as predicted by the FEM model is compared to the one found by a single degree of freedom representation (”mechanical” part of the equivalent circuit developed in Chapter 2). It is observed that the high order frequency components of the response have a small effect on the first periods of oscillation of the beam, but they do not have any influence afterwards. The single mode approximation can then be implemented without yielding noticeable errors in the general dynamics of the system. The results of the analysis are only presented for a C type beam, but they are also true for a B type one. As the contact time is so short, it is also assumed that the impact process and the corresponding pulse results in a quasi instantaneous redefinition of the velocity of the impacted segment in the beam and of the bulk of the moving object. In this model, the dynamic behavior of the system is then discontinuous at the moment of the collision. The velocity of the objects before (vg 0 0 and vM ) and after (vg and vM ) impact can be obtained by considering that the variation of momentum occurring during the compression-expansion phase should equate the time integral of the pulse F(t):

me vg − vg = M vM − vM

tZ i +tc

F (t) dt

(4.4)

The momentum conservation principle is sufficient to determine an expression of the velocities before and after impact thanks to the expression of F(t) (4.2). However, this expression is only valid for perfectly elastic impact in which no energy dissipations occur during the compression-expansion phase. In practice, complex dissipation mechanisms result from the collision. The elastic energy can for example be transformed into heat by viscoelastic or internal friction phenomena or induce local plastic deformations. In this case, the value of the

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER

time integral of the impulse decreases. The study of these loss mechanisms is a complex subject which is beyond the scope of this thesis. However, it is possible to establish a simple phenomenological representation of the dissipations by defining the energetic coefficient of restitution e: for each body, it corresponds to the square root of the ratio of the work Wc done by F(t) on the contact surfaces during the compression over the work We done by the contact surfaces in the expansion phase [156]. It is possible to write eb =(We,b /Wc,b )1/2 for the beam and eM =(We,M /Wc,M )1/2 for the missile. eb and eM correspond to the energy dissipated in each body during the compression-expansion. From these definitions, it is also possible to define an effective coefficient of restitution ee relating the energy present in the system before (Wc,b +Wc,M ) and after (We,b +We,M ) impact as ee2 =

e2b Wc,b + e2M Wc,M Wc,b + Wc,M

(4.5)

This representation is particularly useful here as the duration of the impact process is very small. ee depends on the effective masses, on the curvatures and areas of the contact surfaces, on the material properties and on the relative collision velocity (e e decreases if U0 increases [159, 160]). No efforts are devoted for finding an explicit expression of the coefficient of restitution. Instead, the values of ee are determined experimentally. Considering the fact that Wc,b +Wc,M is equal to the initial quantity of energy before impact, the energy Wd dissipated during the impact is Wd =

1 me M 1 − ee2 U02 2 me + M

(4.6)

From the energy conservation principle and from the first equality of (4.4), it is possible to express the velocities of the bodies after impact in terms of their velocities before impact as 0

vg =

vM =

me vg + M vM − eeM (vg − vM ) M + me

(4.7)

me vg + M vM + eeme (vg − vM ) M + me

(4.8)

The results of the developed model are very similar to those coming from the simple approach of Newton for two impacting spheres [161]. All the elements necessary to establish the simplified system of equations describing the dynamics of the human environment harvester are now available. If a simple representation of the losses which occur during the sliding of the free mass is assumed, the dynamics of the missile are described in the referential of the device’s frame by the simple ordinary differential equation Md2 δ M /dt2 +Dv dδ M /dt=Fext in which Dv is a viscous damping factor, δ M represents the position of the mass respectively to one of the piezoelectric benders

4.1 Modeling of the impact based harvester •

δ g1

•

δM

1/k0

•

Γ : 1

δM=δg1 ?

•

Fext

δg2

e δM=δg2 ?

Figure 4.4 - Equivalent electrical network representation of the human environment vibration energy harvester.

(δ M is assumed one dimensional in our model) and Fext is the apparent force resulting from the motion of the frame (the model is developed in the referential attached to the device’s frame). The dynamics of the bender are represented by the quasi static solutions of the beam equation developed in Chapter 2 (it is assumed that the low frequency motion undergone by the frame do not affect noticeably the behavior of the beams). The behavior of the full system is obtained by coupling the two previous equations with the developed impact representation which consists in redefining the bodies velocities at the moment of collision. By similarities with the considerations expressed in Chapter 2, it is possible to develop the electrical network model of the impact energy harvester depicted in Figure 4.4 in which the indices 1 and 2 correspond to the two benders. The electrical equivalent of the piezoelectric beams has already been discussed previously. Dielectric and piezoelectric dissipations are neglected in this case so that Γ and Cp are real valued, while the stiffness k0 =k(1+j/Qm ) has a complex component representing the mechanical parasitic losses. In case of the missile, its dynamics are represented by the series association of an inductance M and a resistor Dv . The impact coupling between missile and benders is represented through sensing/actuating type elements (indicated by the encircled symbol ee) which determine if the moving mass enters into contact with one of the benders. In this case, the velocities of the missile and of the impacted bender are reinitialized according to (4.7) and (4.8). The base of the model that used for the analysis of the human motion energy harvester has been developed in this section. In the next, specific types of load circuitry and of input forces are introduced so that the power delivered to the load can be analyzed and optimized from a theoretical point of view. The remainder of the modeling part is divided in two sections: in a first one, efforts are devoted in order to establish analytical elements which allow obtaining

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER

1/k0

Γ: 1

•

Cp0

δ g (0) = vg '

Figure 4.5 - Simplified situation considered for the analytical analysis of the harvester.

useful insights on the behavior of the device. However, an analytical approach of the problem is limited to some very simple situations in which repeated impacts are not considered. Therefore, numerical analyses based on the circuit of Figure 4.4 are carried out for more complex and representative cases in a second part.

4.1.2

Theoretical optimization of the generated power: analytical perspectives

It is impossible to develop an analytical model representing the complete behavior of the human environment harvesters for a simple reason: the equations of motion are transcendental. Indeed, sinusoidal components constitute the motion of the beam while the motion of the missile contains (at least) one 0 linear component vM after an impact. No closed form solutions are known for transcendental relations, so that it is impossible to determine an analytical expression of the collision times (except for the first one). The simplified situation considered in this part is illustrated in electrical network form by Figure 4.5. It consists of a single impact approximation in which the missile hits one of the benders and does not interact with it afterwards. In this case, the piezoelectric beam undergoes after impact unperturbed 0 free oscillations resulting from the induced velocity vg (the bender is assumed at rest before collision). Also, a purely resistive load RL is assumed in this analytical approach. It can be understood that the bender behaves after an 0 input impulse as if an initial quantity of mechanical energy E0 =1/2me (vg )2 is injected. In an ideal case, the average output power of the system per impulse can then be defined as Pd =E0 /ta , in which ta represents the time interval between two consecutive impacts or shocks. The bender should be designed in such a way that its output power is the closest possible to this theoretical limit. F The optimization process goes through the solution of the free oscillation problem described above. The detailed computations of the proposed analysis are given in [56] and only the most important steps of the derivation are presented. Applying Kirchhoff’s laws to the circuit of Figure 4.5 and a Laplace transformation to the

4.1 Modeling of the impact based harvester

obtained equations leads to the expressions of the deflection δ g and voltage V given below. L (δg ) = 

me vg0 (1 + Cp RL s)

 k + ( k s + kCp RL + Γ2 RL )s Q ω  m 0    k 2 3 + me + Cp RL s + me Cp RL s Qm ω0s

L (V ) = 

(4.9)

me vg0 ΓRL s

 (4.10) k + ( k s + kCp RL + Γ2 RL )s+ Q ω m 0     k 2 3 s + me Cp RL s me + Cp RL s Qm ω0 s represents the Laplace variable and the L function indicates the Laplace transform of the corresponding variable. All the parameters present in (4.9) and (4.10) have been defined in Chapter 2. The work is now focused on the expression of the voltage across the load, which is more relevant for the purpose of studying the power dissipated into the load. In order to reduce the number of variables involved in the denominator of (4.10), the dimensionless parameters defined in Table 4.1 can be used and the Laplace transform of the voltage can be rewritten as s RL Γ s vg0 ω0 h i L (V ) = s  (4.11)  s + 2 ω0 1 + Ψ(1 + K ) + Q1 s  2 m ω0 3    + Ψ ss (1 + QΨ ) ss ω0 ω0 m Ψ represents the ratio of the fundamental resonance frequency ω s0 to the cut-off angular frequency ω e of the electrical RC circuit, Qm is the mechanical quality factor of the system and K is the GEMC of the piezoelectric unimorph (defined in Chapter 2). In order to approach the optimum output power per impulse defined in the previous section, it should be insured that most of the initial amount of energy present in the bender after an impact is transferred to the load and not dissipated by parasitic mechanisms. Because the parasitic dissipations and the energy harvesting process compete in time for transforming the initial amount of mechanical energy, it can be assumed from the definition of the mechanical quality factor and of the GEMC that optimum harvesting performance is achieved when K and Qm are maximized. Secondly, an electrical load appropriately matched to the system should be defined, so that the electrical damping is optimum. It is possible to obtain an analytical expression of the power dissipated into the load by applying an inverse Laplace transform to (4.11) in order to obtain the expression in the time-domain of the voltage V. The average power Pd dissipated into the load resistor can then be written as Z ta 1 2 V (t) dt (4.12) Pd = RL ta 0

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER

Table 4.1 - Dimensionless parameters used in the representation of the voltage across the load.

Short circuit mechanical angular frequency

ω0s =

k me

Cut-off angular frequency of the RC circuit

ωe =

1 RL C p

Ratio of the mechanical and electrical angular frequency

Ψ=

Generalized electromechanical coupling factor (GEMC)

K2 =

ω0s ωe Γ2 kCp

in which ta represents the time interval between two consecutive impacts or shocks. The value of the optimum load resistor is found by determining the maximum of (4.12) in terms of RL . The exposed procedure is relatively complex from the mathematical point of view and it is preferred to follow a different approach, based on reasonable approximations and giving better insights in the physics of the problem. It has been stated previously that the damping effects (parasitic and energy harvesting process) compete in time, so that the rate at which energy is extracted from the harvesting process (or in other words, the instantaneous harvested power) has to be maximized, independently of the amplitude of the parasitic dissipations. Furthermore, it can be seen from the denominator of (4.11) that the parasitic damping and the energy harvesting process act independently, as the coefficients of s/ω s0 and of s3 ω s0 3 do not contains cross terms of Qm and Ψ. Then, the optimization of the system can be performed by analyzing the parasitic damping free behavior of the system and by finding the parameters which lead to the shortest possible time required for dissipating integrally the initial quantity of energy E0 . From (4.12), the shortest possible time to transform the initial amount of energy corresponds to the shortest possible settling time for the voltage. The parasitic free expression of the voltage is obtained by assuming Qm →∞ in (4.11): s 0 RL Γ s vg ω0 L (V ) = s ω0 1 + [Ψ(1 + K 2 )] s + s2 + Ψ s3 ω0s ω0s 2 ω0s 3

(4.13)

In order to proceed, it is necessary to determine the time domain form of the voltage V(t). To this aim, the nature of the roots of the denominator of (4.13), which will be subsequently referred to as O(s), are analyzed. In the case of second-order systems, O(s) would be a quadratic polynomial in s and the influence of the nature of its roots on the transient characteristics of a system is well-known: a couple of complex conjugates roots means that the system has a

4.1 Modeling of the impact based harvester

Table 4.2 - The different possible forms of the time domain expression of the voltage. Sign of the discriminant of O(s)

Nature of the roots

Form of the time-domain expression

1 real r1 2 complex r2, r2*

C1e r1t + e Re ( r2 ) t ⎡⎣C2 cos ( Im( r2 )t ) + C3 sin ( Im(r2 )t ) ⎤⎦

3 single real r1, r2, r3

C1e r1t + C2 e r2t + C3e r3t

1 double real r1 1 single real r2

(C1t + C2 )e r1t + C3e r2t

1 triple real r1

(C1t 2 + C2 t + C3 )e r1t

pseudo-oscillatory behavior, a double real root that it is critically damped and two real and different roots that the system is overdamped. In free oscillations cases, the critically damped state is the one for which the system comes the most rapidly at rest, without oscillating. In the following it is shown that it is also theoretically possible to reach the equivalent of a critically damped behavior in the investigated system, which is a third order one. The different forms of the time domain expression of (4.13) are given in Table 4.2 considering the possible values of the discriminant of O(s). In this table, the symbols Re and Im indicate respectively the real and imaginary part of the corresponding variable. The coefficients Ci are real numbers. It has been shown in the literature that the shortest settling-time of thirdorder systems is obtained when the system parameters are such that the roots of O(s) are near the triple real root [162]. This state corresponds to the last line of Table 4.2. The values of K and Ψ corresponding to a triple real root r1 for O(s) can be obtained by comparing the coefficients of O(s) with those of the√polynomial (Cs-r1 )3 , in which C is a real constant. They are K2 =8 and Ψ=(3 3)−1 . As stated previously and as it is shown more clearly in the next section, such high values of the generalized electromechanical coupling factor can not be obtained, so that this behavior, although theoretically possible, can not be reached in a practical application. Overdamped type behaviours (2nd and 3rd lines of Table 4.2) requires higher values of K than for the critically damped one and it is then also practically impossible to reach it. Therefore, the time domain expression V(t) of (4.13) has necessarily in practical applications the form given in the first line of Table 4.2, equivalent of the pseudo-oscillatory behaviour of second order systems. It consists in the sum of an exponentially decaying term and of a pseudo-oscillatory one. The roots of the characteristic equation of the system consist in a real one r1 , and in two complexes conjugated ones r2 and r2 ∗ . The real parts of the different roots can only be negative, as

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER

they represent the inverse of the time constants of a system which has no physical possibilities to diverge. V(t) can then be written as V (t) = C1 er1 t + eRe(r2 )t {C2 Cos [Im(r2 )t] + C3 Sin [Im(r2 )t]}

(4.14)

Now that the form of the time domain expression of the voltage across the load is known, efforts should be done to find an explicit expression of the different parameters involved in it, in order to define their influence on the settling time of the structure. A method proposed in [161] is adapted to the problem. This method allows obtaining analytical expressions of the value of the load resistor leading to the minimum settling time of (4.14) and of the different parameters involved for this electrical loading condition. This method is based on the assumption that the minimum settling time is obtained when one or both of the time constant involved in (4.14) (1/Re (r2 ) and 1/Re (r1 )) is minimized. This hypothesis has been checked numerically and is valid for values of K2 smaller than unity, which is representative of all practical applications. By this method, it is found that, in the given conditions, the expression of the optimum load Ropt is 1 Ropt = (4.15) K2 s ω0 Cp 1 + 2 The corresponding angular frequency ω s of the shunted piezoelectric bender is s K2 K2 (1 − ) (4.16) ωs = ω0s 1 + 2 8 It is finally possible to derive the explicit expression of V(t):  s exp (−ω 0 t) 2 s  Γvg0  + exp − K ω0 t  4 V (t) = − 2Cp ω0s   2 K cos(ωs t) + 1 − 2 sin(ωs t)

     

(4.17)

The correctness of the analytical expression given in (4.15), (4.16) and (4.17) has been checked by performing numerical simulations with the software Mathematica. V(t) is first computed by performing a numerical inverse Laplace transform on (4.13) considering various values of the parameters Ψ, RL , K2 , ω s0 and Γ and it is verified that (4.13) leads to a correct approximation of the voltage when parasitic dissipations are ignored. It is observed that the expression of the shunted resonance frequency is correct within a few %. The energy dissipated into the resistive load (Pd *ta ) is then computed using (4.12) for ta →∞ (the expression of the voltage taking into account parasitic dissipations was used in this case). It is first observed that, even in the presence of parasitic dissipations, the value of the optimum load resistor Ropt obtained from numerical simulations is in close agreement with the one given by (4.15) if K2 is smaller than 1. Finally, it is seen that, independently of the values of

4.1 Modeling of the impact based harvester

the other parameters, the energy harvested increases monotonically with K2 and Qm . Several observations can then be done on (4.17). It is first seen that the time constant of the exponential term is just equal to the inverse of the short circuit and parasitic free natural frequency of the structure ω s0 . Second, the time constant of the pseudo-oscillatory term depends on both the generalized electromechanical coupling factor and natural frequency. Finally, the three coefficients related to the exponential term, to the exponentially decaying sinus term and to the exponentially decaying cosines one are in practice almost equal. In this case, minimizing the settling time of the structure means minimizing both the time constants of the exponential term and of the pseudo oscillatory ones. This can be done by maximizing the generalized electromechanical coupling factor and the natural frequency ω s0 . However, this conclusion is not absolute for ω s0 , because the model is valid when the duration of the input impulse is much shorter than the oscillation period of the bender. With high natural pulsation of the structure, it might be practically difficult to find applications in which our model is relevant. Also, the addition of an extra mass at the tip of the bender (see Table 2.1) leads to a reduction of the natural frequency, but also to an increase of the initial energy stored in the structure, so that a compromise has to be found. It is shown in this part that the performances of the piezoelectric benders in terms of impact energy harvesting are optimized by maximizing the generalized electromechanical coupling factor, by limiting the parasitic losses and by appropriately tuning the value of the load. This analysis is carried out in the framework of an over simplified situation. However, the conclusions that are reached gives useful insights for the more representative situations investigated by numerical means in the next section.

4.1.3

Theoretical optimization of the generated power: numerical perspectives

The goal of the presented numerical analysis consists in determining general rules for optimizing the output power of the harvester. The behavior of the device is very complex (it might even become chaotic [163]) and involves an important number of independent variables. It is not relevant to analyze individually the influence of each of the constituent on the behavior. In the previous section, it is shown that the performances of the system are optimized when the generalized electromechanical coupling factor and the quality factor of the piezoelectric bender are maximum. This conclusion has been obtained in the framework of a simplified situation but it appears intuitively that it should still yield for multiple impact case. Therefore, the parameters defining the generalized electromechanical coupling factor and the quality factor are set in the following to fixed values representative of the benders that are implemented in the experiments. k, Γ, Cp and Qm are set to respectively 1000 N/m, 0.5

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER

mN/V, 2.6 nF and 50 (and then K2 =0.12). The dissipations which oppose the motion of the missile (Dv ) are not taken into account in the present computations as it is clear that they have a negative influence on the output power of the system and they have to be minimized. The assumed input motion of the frame in the numerical analysis consists of quasi instantaneous 180◦ rotations occurring every second. Between these rotations, the frame is supposed to be oriented in such a way that the sliding channel is aligned with the gravity field g, so that the missile undergoes a gravitational force Fext =-M*g which makes it moving from one of the benders to the other (the action of gravity on the beams is neglected). The considered input motion is representative of a movement that can easily be achieved when a device is placed on a human wrist. Other situations are studied in the experimental part of this chapter. For the missile travelling distance Λ, it is found by varying the values of this parameter in the numerical simulations, that the performances of the harvester in terms of conversion efficiency decrease slightly as Λ increases. This effect can be explained in the following way: if the travel distance Λ becomes too large, the missile might not have reached its rest position on the second bender when the second rotation occurs. In this case, a part of the potential energy of the mass has not been transferred to the piezoelectric bender and the efficiency of the energy conversion is reduced. It does not necessarily mean a lower output power but in any case a lower output power per unit volume. However, for values of Λ ranging from 10 mm till 50 mm, the differences in the prediction of the model in terms of electromechanical conversion efficiency are noticeable but no exceeding a few %, so that the analysis is limited to the constant value Λ=15 mm. From all the previously expressed considerations, the variables that are left for a parametric analysis of the harvester consists of the masses M and me , the coefficient of restitution ee and the characteristics of the load circuitry. It is determined numerically that when the natural period of oscillation of the beam is much smaller than the period of the rotations of the frame, the influence of the masses is coupled and can be studied through the reduced parameter M/me . Indeed, for a fixed coefficient of restitution and electrical load, one obtains the same performances for constant M/me independently of the particular values chosen for the different masses. Concerning the electrical load, a simple type of circuit consisting in a pure resistor is considered. The situation is complicated by the ”two sided” nature of the system. If the piezoelectric benders are shunted by the same load circuit, they have to be connected in a parallel or series arrangement. In this case, when one of the benders is impacted, a part of the electrical energy it develops flows towards the second bender rather than towards the load. This effect reduces inevitably the efficiency of the conversion. This problem might easily be eliminated by designing for example electrical switches which close and open in a relevant way. For matter of simplicity, it is considered that each bender is connected to its own load. Furthermore, the missile is assumed at rest on one of the bender before each rotation. It is possible to estimate the power generated by the harvester by studying solely

4.1 Modeling of the impact based harvester

5 4

3 2

-2

1 1

-4x10 0.05

Voltage (V)

Beam and missile displacement (mm)

0.10

0.15 Time (s)

(a)

0.20

0.05

0.25

0.10

0.15

0.20

0.25

Time (s)

(b)

Figure 4.6 - (a) Illustration of the motion of the beam and of the missile after a rotation of the frame, (b) corresponding output voltage on the piezoelectric capacitor (open circuited).

the behavior of the system between two successive rotations. As the period of the considered rotations is 1 s, the average output power corresponds to the energy dissipated or stored into the load during that time. Before presenting the results related to the output power, some observations on the general behavior of the system are proposed. The motion of the missile and bender after a rotation of the frame is illustrated in Figure 4.6(a). It can be seen in the inset of the figure that each ”genuine” impact is followed by a multitude of short time related collisions until the missile is ejected from the neighborhood of the bender. At this moment, the beam oscillates freely till a second genuine impact occurs. Depending on the values of the different parameters, the missile might apparently ”stick” to the bender during a short duration. In a perfectly inelastic case (e e=0), the missile and the beam merge into a single oscillator just after impact. They separate when the acceleration of the beam becomes null. For other values of the coefficient of restitution, the sticking parts of the behavior are difficult to predict in a general way and depend on all different characteristic parameters. However, when the effective mass of the bender is negligible compared to the missile, the behavior is very similar to the inelastic one independently of the value of the coefficient of restitution. The voltage developed on the electrical ports (open circuited) of the piezoelectric beam is reported in Figure 4.6(b). Each genuine impact results in a sharp peak of the voltage followed by decaying sinusoidal oscillations. The secondary collisions tend to distort the initial peak and they can have a strong influence on this part of the waveform of the voltage. For sake of clarity, the simplified circuit representation of the harvester corresponding to the situation that is investigated in this part is given in Figure 4.7. Multiple impacts are here taken in consideration (not being the case in the previous section). The parametric analysis that is performed consists in determining the influence of M/me and RL on the efficiency of the energy conversion. The initial energy E0 present in the system at the beginning of a cycle

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER •

δ g (0) = 0

δ g ( 0) = 0 δ M ( 0) = Λ

•

δ M (0) = 0

•

δM

•

δg

1/k0

Γ : 1 V

δM=δg ?

Cp RL

‐Mg

Figure 4.7 - Circuit representation of the gravity drop of the missile on a piezoelectric bender shunted by a resistive load.

can be approximated by the potential energy of the missile Mgλ with g=9.8 m.s−2 and λ=15 mm. The energy EL dissipated in the load resistor during a period of the motion is obtained by integrating V(t)2 /RL . The efficiency η is defined as EL /E0 and the average output power Pd is equal to E0 /ta (ta =1 s). An analytical expression of the optimum load resistor has previously been found for a single impact approximation in (4.15) and it is first checked if this formula is valid in multiple impact situations. To this aim, the efficiency η is computed for a large variety of values of the coefficient of restitution and of the masses ratio, while considering a constant range of values for the load resistor. The chosen range was centered on the value resulting from the expression of Ropt given in (4.15). A few of the obtained results are presented in Figure 4.8. The abscissa of the presented graphics corresponds to the load resistor normalized to the optimum value (4.15). It can be seen in Figure 4.8 that the coefficient of restitution does not have a strong influence on the value of the optimum load, but on the other hand, that M/me plays an important role in this matter. When this parameter increases, the optimum load resistor in multiple impact situations Rm opt shifts towards higher values. This effect can be explained in the following way: it is assumed in the analytical derivations of the previous section that the missile is impacting the beam a single time and that the objects do not have any interactions afterwards. In this case, the beam oscillates solely along its fundamental frequency and the transfer of energy to the load resistor is optimized for the value of Ropt (4.15). In the multiple impacts situation, the mechanical behavior of the bender can be divided into three characteristic periods. In the first, the missile is not in the vicinity of the beam and the situation corresponds to the one just described. In the second, the beam is in the neighborhood of the beam and several short time related collisions occur (inset of Figure 4.6(a)). For the third type ones, the missile apparently sticks to the bender and the system oscillates at a frequency

100

4.1 Modeling of the impact based harvester

Efficiency (%)

20 10 0

0.1

4 6

1 RL/Ropt

(a)

4 6

0.1

4 6

1 RL/Ropt

4 6

(b) 40 Efficiency (%)

Efficiency (%)

40 30 20 10 0

0.1

4 6

1 RL/Ropt

4 6

30 20 10 0

0.1

4 6

1 RL/Ropt

4 6

(c) (d) Figure 4.8 - Efficiency of the energy conversion versus RL /Ropt , (a) ee=0.7, (b) ee=0.5, (c) ee=0.2, (d) ee=0. In each graph, the black solid line corresponds to M/m=0.2, the black dotted line to M/m=1, the black dashed line to M/m=2, the black dashed dotted line to M/m=5, the grey solid line to M/m=10 and the grey dotted line to M/m=20.

ω s (1+M/me )1/2 . The occurrence of these periods depends on most of the different parameters of the model. However, periods of the second type are observed in situations involving high coefficient of restitution and M/me ≈1, while the third type occur for high M/me and low coefficient of restitution. The third type characteristic periods of the motion introduce low frequency components into the frequency spectrum of the bender displacement. The expression of the optimum load resistor given in (4.15) is found considering only the periods of the first type. In order to adapt it to the multiple impacts situation and to the corresponding additional low frequency components, one can assume a representative or ”average” frequency of the motion. This average frequency ω m is inevitably smaller than the shunted resonance frequency ω s of the free bender when M/me 1. As Ropt is inversely proportional to the frequency of the oscillation undergone by the bender, the optimum load in multiple impacts cases Rm opt =Ropt ω s /ω m shifts towards higher values. This effect can clearly be observed in Figure 4.8 for large M/me independently of the value of the coefficient of restitution.

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER

101

Position (m)

40 30

Time (s)

0.1

1 M/me (a)

Position (m)

Efficiency at optimum load (%)

(b)

Figure 4.9 - (a) Efficiency of the energy conversion at the optimum load resistor, solid line: ee=0.7, dotted line: ee=0.5, dashed line: ee=0.2, (b) position of the beam and of the missile for points A and B.

From the obtained results, it is possible to estimate the maximum efficiency of the system to be approximately 40-50%. It is however not straightforward to establish a clear influence of the coefficient of restitution and of the masses ratio from the curves of Figure 4.8. It is observed that for a fixed value of ee, the efficiency does not vary monotonically with M/me and does not follow an equivalent behavior for different values of ee. Also, it can be seen that higher values of the efficiency can be obtained with low values of the coefficient of restitution, which is somewhat surprising. In order to clarify this point, the efficiency at the optimum load resistor versus the masses ratio and for different coefficients of restitution is plotted in Figure 4.9(a). In the low M/me range, several local maxima and minima are observed when large coefficients of restitution are considered. The presence of these peaks is explained by the complex dynamics of the system and by the repeated exchanges of energy between beam and missile. The dynamic situations corresponding to the points A and B in Figure 4.9(a) are illustrated in Figure 4.9(b). It can be seen that for the maxima of point A, the missile is ejected after the first impact in such a way that his position is just slightly above the one of the beam when it has reached its maximum amplitude. In this way, the quantity of energy transferred to the beam during the first impact remains in the bender and can be dissipated during the free oscillations. At the opposite, for the point B, the configuration is such that, after the first collision, the beam hits the missile when it has reached its maximum velocity and it transfer back a large part of its initial energy to the moving object. The quantity of energy left in the beam for harvesting is then much smaller for the point B than for the point A. Equivalent reasoning can be developed for the other maxima and minima. These peaks do not appear for low

102

4.2 Experimental measurements

or large mass of the missile, as in this case the complex multi impact behavior does not exist and is reduced to a â&#x20AC;?stickingâ&#x20AC;? situation. It is also observed from Figure 4.9(a) that when M/me becomes very large, the efficiency for the different values of the coefficient of restitution merges. Finally, for middle range values of the masses ratio, it is advantageous (counter intuitively) to design a system with a low coefficient of restitution in order to optimize the efficiency. It is shown in this part that the efficiency of the human environment energy harvester is optimized by maximizing the generalized electromechanical coupling factor, limiting the parasitic losses, choosing an appropriate load and implementing an appropriate masses ratio. In the best conditions and neglecting parasitic losses, an optimum efficiency in the range of 40-50% is predicted. In the next section, the conclusions obtained from the model are verified experimentally and a prototype of the impact harvester is tested on the human body.

4.2

Experimental measurements

Most of the parameters of the network model of the impact harvester can be found through the characterization methods proposed in chapter 3. However, the coefficient of restitution is a parameter requiring specific measurements. Therefore, a simple method of measuring ee is proposed for the materials and structures implemented in this thesis. Then, the predictions of the model are compared to experimental measurements in the simple rotary motion described previously. Finally, a prototype of the impact harvester is mounted on the human body and its performances are measured.

4.2.1

Coefficient of restitution

The value of the coefficient of restitution ee is obtained using a simple method in which the velocity of a steel missile was measured before and after impact by means of laser detection. The impacted piezoelectric beam is clamped so that there is no need to measure the velocity of the bender. The principle of the method is illustrated by Figure 4.10(a). The missile is dropped from an arbitrary height (the chosen values is 15 mm for most of the experiments) on the surface of the clamped piezoelectric bender. A laser beam coupled to a light detector allows detecting the successive times t1 and t2 at which the moving object crosses and exits the beam, so that from the basic law of motion, its velocity v2 at the time t2 can be written as (2gLM +v1 2 )1/2 with v1 =LM /(t2 t1 )-g(t2 -t1 )/2. The impact time t3 is determined thanks to the surface wave created at the moment of the collision and which results in a voltage developed across the open circuit piezoelectric laminate. The distance between the laser beam and the piezoelectric bender LL is chosen so that it is just a bit larger than LM . In this way, the dissipations due to air damping can be neglected in

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER

103

PSD

Laser

LL t2

Voltage(V)

Clamped piezoelectric beam

t3 Impact

-10 (a)

t2 t3 t4

10 Time(s)

-3

30x10

(b)

Figure 4.10 - (a) Illustration of the method for determining the coefficient of restitution, (b) measurement plot for the coefficient of restitution. The black and red curves correspond respectively to the voltage across the light detector and across the electrodes of the piezoelectric beam.

this portion of the motion of the missile and its velocity just before impact v3 can reasonably be approximated by (2g(LL -Lm )+v2 2 )1/2 . In the same way, the 0 velocity just after impact v3 can be estimated from the time interval between 0 the impact and t4 so that v3 =(LL -LM )/(t4 -t3 )+g(t4 -t3 )/2. If one considers now the expression involving the coefficient of restitution given in (4.8), it is 0 seen that in case of the impact of a missile on a clamped beam (vg =vg =0 0 and me M), it is possible to write ee= -v3 /v3 , i.e. the energetic coefficient of restitution is equivalent to the kinematic one [156]. A typical measurement plot obtained from this method is given in Figure 4.10(b). The voltage across the light detector becomes null when the missile passes in front of the beam. As indicated on the graphic, the different times related to the instants illustrated in Figure 4.10(a) are easily obtained from the described measurement. In this experiment, the angle of drop of the missile is not perfectly controlled so that small variations are observed on the measured values of ee. The standard deviation of the measurements was approximately equal to 5%, so that it is reasonable to consider the average of the distribution as a representative value for the next comparisons. The average value of ee was found to be equal to 0.55.

4.2.2

Comparison of the model predictions with experimental measurements

The prediction of the model concerning the voltage developed across the electrodes of the piezoelectric beam and the power dissipated in the load resistor are compared with experimental measurements using the arrangement described in

104

4.2 Experimental measurements

Figure 4.11(a). The equivalent network parameters of the piezoelectric beam used in these experiments were k=749 N/m, me =400 mg, M=4 g, Qm =59, Γ=0.5 mN/V, Cp =2.6 nF (they are those of the ceramic PZT bender characterized in Chapter 3, without attached mass) and ee=0.55 is assumed. In a first series of experiments, the efficiency of the energy conversion versus the value of the load resistor is measured for a drop distances of 15 mm. The results are reported in Figure 4.11(b). It is observed that the theoretical and experimental optimum values of the load coincide well. Rm opt =400kΩ≈2/(ω 0 Cp ), corresponding approximately to the value found in Figure 4.8(b) for ee=0.5 and M/me =10), but the predicted efficiency is approximately 50% higher than the measured one. This result is explained because of the losses occurring during the motion of the missile that are neglected in the numerical computations. They play an important role on the general behavior of the device. These parasitic losses are however difficult to estimate and to represent in a proper way, as they are mainly due to the imperfectly controlled direction of impact and of bounce. Indeed, it can be easily understood from Figure 4.11(a) that, in the measurement setup used, the missile after a bounce might not move in a direction perfectly perpendicular to the beam and it can hit the sides of the guiding channel. Friction phenomena resulting from this effect decrease dramatically the efficiency of the system. This assumption is verified by performing a second series of experiments in which the theoretical and experimental time dependence of the voltage across the piezoelectric beam and of the energy dissipated in the optimum load are compared (Figure 4.12). The model gives a very good estimation of the voltage and efficiency for the first impact. However, the second impact occurs after a shorter time than the predicted one and the corresponding efficiency step is smaller than expected. This effect becomes more pronounced for the following collisions. It appears that at each bounce, an important part of the kinetic energy of the missile is dissipated because of the parasitic phenomena described above. In spite of the presence of this parasitic damping mechanism, the conclusions related to the optimum bimorph’s parameters remains valid. The experiments performed in this section are meant to establish a comparison between the predictions of the model and the measurements in a simple experimental situation. In the next part, a similar characterization is performed on a prototype of the impact harvester.

4.2.3

Characterization of a prototype of the human environment harvester

A conceptual representation and a picture of the manufactured prototype of the impact energy harvester are given in Figure 4.13. The housing of the device is made of Teflon while aluminum is used for the clamps and the closing caps. The missile is made of steel (M=4 g) and has an oblong shape, so that it occupies approximately half of the length of the guiding channel equal to 30

Conversion efficiency (%)

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER

Vertical stage

Missile

Load and measurement electronics

Release pad

Guiding channel Piezoelectric bender

30 20 10 0 4

Laser

105

PSD

6 8

10 Load resistor (Ω)

(a)

(b)

Conversion efficiency (%)

Figure 4.11 - (a) Description of the experimental setup, (b) theoretical (solid line) and experimental (markers) efficiency of the energy conversion.

60 Voltage(V)

40 20 0 -20 -40 -60 0.0

0.1

0.2 0.3 Time(s)

(a)

0.4

0.5

30 20 10 0 0.0

0.2

0.4 0.6 Time(s)

0.8

(b)

Figure 4.12 - Voltage (a) and efficiency (b) waveforms for a gravity drop. For all the different graphics, the black lines correspond to the theoretical expectations while the red ones correspond to experimental measurements.

mm. A part of the housing of the prototype might be eliminated or replaced by the powered application and conditioning electronics, so that the output power per unit volume or mass can be improved. The piezoelectric cantilevers implemented in the prototype are not optimized for energy harvesting because of a lack of materials. The characteristics corresponding to the network model were k=6000 N/m, me =600 mg, Qm =40, Γ=1 mN/V and Cp =9.6 nF. The corresponding value of the generalized electromechanical coupling factor is 1.7*10−2 . As shown in the following, this relatively low value of K2 reduces the efficiency of the energy conversion. For the coefficient of restitution, the same value of 0.55 that was considered in the previous experiments (the same piezoelectric materials are implemented). Reliability problems are observed in the testing of the prototype because of the large deflection and dynamical stress resulting from the impact. In order to limit this problem, small magnets are attached on each cantilever and on the closing cap in such a way that they repulse

106

4.2 Experimental measurements

(a)

(b)

Figure 4.13 - (a) Conceptual representation of the harvester prototype, (b) photography of the actual prototype.

each other. This simple method allows implementing a non linear spring which smoothes the motion of the cantilever when it approaches the closing cap. In normal mode of operation, the magnets remain relatively far from each other and they do not have a strong influence on the general dynamics and efficiency of the system. The harvester prototype was first tested according to the simple rotary motion described previously. In this case and for the proposed configuration, the kinetic energy of the mass when hitting one of the cantilevers is Mgλ=593 µW. The theoretical and experimental values of the conversion efficiency are plotted in Figure 4.14. Because of the low value of the generalized electromechanical coupling factored, the maximum theoretical value of the efficiency is reduced to 13% instead of 40% in Figure 4.14(a). The discrepancy between the prediction of the model and the measurements is slightly lower than for the situation studied in the previous subsection and the maximum experimental value of the efficiency is found to be around 8%. As before, this discrepancy is due to the friction of the missile when sliding in the guiding channel and to the angle of bounce of the moving object after impact. A single rotation of the device is performed in these measurements but one can estimate that during the repeated rotary motion at 1 Hz, the total power dissipated in the resistive loads would be approximately 95 µW. The device is also tested when it is attached to the hand of a person and forcibly shaken. The frequency of the applied motion is estimated to 7 Hz while the amplitude is approximately 10 cm. This extreme situation is obtained by simulating a strong ”scratching” motion. Such vibrations might also be observed in some sportive situations (off road motor biking) and profes-

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENT VIBRATION ENERGY HARVESTER

107

600

Output power (μW)

Conversion efficiency (%)

10 8 6 4 2 3

6 8

10 Load resistor (Ω)

(a)

6 8

500 400 300 3

6 8

10 Load resistor (Ω)

6 8

(b)

Figure 4.14 - (a) Efficiency of the energy conversion when the prototype is rotated over 180◦ each second, (b) generated power when the device is forcibly shaken.

sional activities (jackhammer). Also, the developed device might be useful for particular industrial applications such as weaving machines. The output power of the device under these conditions and versus the load resistor is presented in Figure 4.14(b). A maximum of 600 µW was measured. In such erratic conditions of excitation, the behavior of the power does not follow a smooth curve. However, it is observed that for resistor ranging from 3 kΩ till 60 kΩ, values above 400 µW are obtained. This amount of power is sufficient to supply energy to low consumption applications.

4.3

Conclusion

In this chapter, the design and characterization of a prototype of harvester able to produce energy from the motion of human limbs is presented. The proposed harvester is based on the impact of a moving object on piezoelectric bending structures. An output power of 600 µW is obtained for a device of dimensions 3.5*2*2 cm3 weighting 60 g placed on the hand of a person and shaken at a frequency of approximately 7 Hz for a 10cm amplitude. Also, a power of 47 µW is measured when the harvester is rotated over 180◦ each second. A large amount of the volume occupied by the prototype of the harvester can be eliminated and one can estimate an output power density of 10 µW/cm3 or 4 µW/g for an optimized device undergoing the previously described rotary motion. These figures can be multiplied by a factor 12 when a 7 Hz frequency, 10 cm amplitude linear motion is considered. It is shown that in order to realize an efficient impact energy harvester, one should apply the following design guidelines: • In order to maximize the efficiency of the energy conversion, the generalized electromechanical coupling factor and the mechanical quality factor of the piezoelectric transducer have to be made as high as possible by

108

4.3 Conclusion choosing appropriate materials and dimensions. Also, the parasitic dissipations in the motion of the missile have intuitively to be minimized. â&#x20AC;˘ The resonance frequency of the bender should be high enough so that the amount of energy transferred by the moving object during an impact can be dissipated prior to a second impact. â&#x20AC;˘ The missile should be made as stiff as possible in order to avoid storing energy into vibrations of the missile rather than in vibrations of the piezoelectric transducer. â&#x20AC;˘ In order to limit reliability problems due to high stresses and deflections resulting from high amplitude excitations, a damping system has to be implemented in the neighborhood of the piezoelectric elements (repulsing magnets are used here).

Chapter

Design and analysis of the machine environment vibration energy harvester The output power characteristics of the harvesters designed for generating electrical power from high frequency and low amplitude vibrations (machine environment) are presented. As explained in Chapter 1, the developed harvesters are resonant devices which have to be excited in the neighborhood of their fundamental resonance frequency for optimizing the production of electrical power. In this thesis, the investigations are limited to single harmonic mechanical excitations. In real life applications, broadband spectrum of input vibrations should be considered. In the framework of a first tentative of optimization, it is however assumed that the spectrum of most ubiquitous vibrations contains a dominant frequency which can be roughly approximated by a single frequency. The basic principles of energy extraction from harmonic vibrations by piezoelectric elements have been developed in the seventies for purposes different than producing useful electrical power: piezoelectric elements were primarily used for attenuating parasitic and undesired vibrations in mechanical machinery. The goals of the vibration damping and energy harvesting fields are different by essence but most of the principles developed for the former domain are applicable in the latter, particularly when the design of an effective load circuitry is considered. The energy harvesting field received little attention till the beginning of the second millennium. The amount of publications related to the subject however increased exponentially in the following years. Roundy [28, 164] was the first to propose an analytical approach to the optimization of the generated power. The expressions that he developed are often taken as a reference for 109

110

5.1 Theoretical analysis of the harvesterâ&#x20AC;&#x2122;s output power

elaborating simple estimation but it is shown in this chapter that they are only valid in particular and restrictive conditions. Approaches equivalent to the one of Roundy and refinements of the obtained results were proposed by numerous authors such as Richards [75], DuToit [97], Mitcheson [165, 166] or the author of this thesis [24]. Advanced but purely numerical methods were also recently proposed by Liao in [167]. The approach proposed here is based on the analysis of the equivalent network circuit developed in Chapter 1. It is organized as follow: in a first part, a theoretical analysis of the expected performances of the harvesters is presented. The analysis is concentrated on purely resistive load circuits. Alternative linear and non linear signal conditioning methods are only briefly discussed. In a second part, the output power of the devices is measured experimentally. Both the MEMS fabricated and commercial benders based harvesters (Chapter 3) are characterized.

5.1

Theoretical analysis of the harvesterâ&#x20AC;&#x2122;s output power

The elements proposed here are based on the steady state behavior of the piezoelectric benders and classical methods of harmonic analysis are implemented. Rather than utilizing the Laplace transform which is particularly adapted for transient situations (see Chapter 4), the complex transform is used. The quasistatic version of the equivalent circuit proposed in Chapter 2 is taken as a basis for performing the proposed analysis and is repeated in Figure 5.1 for sake of clarity. The force F acting on the mechanical side of the device has been replaced by the inertial force me sin(Ď&#x2030;t) due to the input vibration applied to the clamped end of the piezoelectric cantilevers. ZL represents the electrical load circuitry connected to the harvester and in which energy is stored or dissipated. When dealing with loads made of linear components, ZL also correspond to the impedance of the load circuit. Dielectric and parasitic dissipations are taken into account and k0 =k(1+jQm ) and Cp0 =Cp (1+jQe ). In the following, the analysis of the power dissipated by the harvesters through a perfect resistor is first presented in details. In a second part, alternative power conditioning methods are briefly discussed.

5.1.1

Resistive load

The resistive load is the simplest to be imagined and in this case ZL =RL . When the bender oscillates under the action of the external vibration, the charges developed in the piezoelectric layer are allowed to flow through RL . The corresponding energy dissipated per cycle is equivalent to the output power of the harvester. Roundy [41] was the first to analyze this problem for the scope of energy harvesting, even if the same principle was understood since long in

5. DESIGN AND ANALYSIS OF THE MACHINE ENVIRONMENT VIBRATION ENERGY HARVESTER me

1/k0

Γ: 1 Cp0

−me A0 sin (ωt )

111

Figure 5.1 - Equivalent circuit model of a piezoelectric bending structure excited by a sinusoidal vibration.

the field of vibration absorption [143]. Assuming steady state behavior, the average power Pd dissipated in the resistor can be written as ω Pd = 2π

2π/ω Z

V (t) |V (ω)| = RL 2RL

(5.1)

in which the upper score indicated the complex transform of the corresponding variable. It is then necessary to determine first an expression of the voltage drop across the load resistor before obtaining the one of the output power. Applying Kirschoff’s laws in the frequency domain to the circuit of Figure 5.1 leads to me A0 = me ω 2 + k0 δt + ΓV (5.2) V = jωΓδt − jωCp0 V RL

(5.3)

This system can be solved for the complex voltage and deflection. me A0 RL Γs  j j j k 1+ Q + Cp 1 + Q k 1 + Q RL + Γ2 RL s m m   e j +me s2 + me Cp 1 + Q RL s3

V =

(5.4)

j me A0 1 + Cp 1 + Q RL s e  (5.5) δg =  j j j k 1+ Q + Cp 1 + Q k 1 + Q RL + Γ2 RL s m m   e j +me s2 + me Cp 1 + Q RL s3 e In order to simplify the expressions, the non dimensional parameters introduced F 4.1 are also used in this chapter. According to the definitions of in Table Table 4.1, Ψ represents the ratio of the mechanical resonance frequency ω s0 to the cut-off angular frequency ω e of the electrical RC circuit corresponding to the load resistor coupled with the piezoelectric capacitor and K is the generalized electromechanical coupling factor of the piezoelectric unimorph. The symbol

112

5.1 Theoretical analysis of the harvester’s output power

Ω =ω/ω s0 is also used in the following. The amplitude of the voltage and of the deflection can now be written as |V | =

A0 Γ 2

Cp (ω0s ) s ∗

|δg | =

with

Ω2 Ψ2 2

(Ω2 − 1) + Ω2 Ψ2 (Ω2 − 1 − K 2 ) + Dm + De + Dem

(5.6)

A0 2

(ω0s ) v u 2 2 2 u Ω Ψ 1 + Q 2ΩΨ e u 1− + u Qe Q2e t ∗ 2 2 (Ω2 − 1) + Ω2 Ψ2 (Ω2 − 1 − K 2 ) + Dm + De + Dem 1 ΩΨ 2 Dm + 2ΩΨ K + Qm 2Qm 2 2ΩΨ Ω2 − 1 ΩΨ 1− De = − Qe 2Qe 2ΩΨ 1 1 Dem = − − ΩΨ − K2 Qe Qm Qm 2Qe Qm 1 = Qm

(5.7)

(5.8)

(5.9) (5.10)

The terms defined in (5.8), (5.9) and (5.10) are related to the parasitic dissipations. Dm is solely linked to the parasitic mechanical dissipations, while De depends only on the dielectric losses. Dem is a coupled term and is different from zero only if both mechanical and dielectric dissipations are considered. It can also be seen that at the short circuit resonance (Ω=1), the pure dielectric term De vanishes while the coupled term Dem does not. All the elements necessary for computing the output power Pd of the harvester have been established. By combining (5.1) and (5.6), one can elaborate the expression of Pd given below. Pd =

Ω2 K 2 Ψ me A20 2 s 2ω0 (Ω2 − 1) + Ω2 Ψ2 (Ω2 − 1 − K 2 )2 + Dm + De + Dem

(5.11)

In the absence of parasitic losses, two ”resonance” frequencies expressed by the terms ω 2 -1 and ω 2 -1-K2 are clearly identifiable. In short-circuit configuration, RL =0 and Ψ is then also null in (5.11), so that the term Ψ2 Ω2 (ω 2 -1-K2 ) vanishes from the denominator. (ω 2 -1) corresponds to the normalized shortcircuit fundamental normalized frequency Ωs0 =1. In a situation close to an open-circuit one, Ψ is very large and Ψ2 Ω2 (ω 2 -1-K2 ) dominates the denominator of (5.11). It corresponds to the normalized open-circuit fundamental

5. DESIGN AND ANALYSIS OF THE MACHINE ENVIRONMENT VIBRATION ENERGY HARVESTER

113

frequency Ωo0 =(1+K2 )( 1/2)=1 of the bender. Ωs0 and Ωo0 can also be approximately defined as the frequencies at which the impedance seen from the electrical ports of Figure 5.1 are respectively minimum and maximum. The shift between the short and open circuit characteristic frequencies is pronounced if the generalized electromechanical coupling factor is large. Without parasitic dissipations, a non physical result is obtained from (5.11) when the bender is short circuited and excited at Ωs0 or when the bender is open circuited and excited at Ωo0 , i.e. Pd →∞. This non-physical result is explained by the fact that the deflection of the bender is also mathematically infinite in these cases, so that in a practical situation, even if the parasitic dissipations are extremely small, the output power is limited by physical constraints such as for example the yield limit of the structure or the limited space available in a package. It has however been shown in Chapter 3 that parasitic dissipations exist in the devices and have to be incorporated in the model. It is interesting to get some insights on the behavior of the different losses terms Dm , De and Dem as they have a direct negative influence on the amplitude of the output power. In Figure 5.2, the amplitude of these terms is plotted versus the normalized frequency Ω considering the values of K2 , Qm and Qe measured in Chapter 3 for the MEMS AlN sample and for the commercial PZT one (Table 3.7). For any value of Ψ (and of the load resistor), it can be observed that out of resonance, the dielectric term De dominates the losses for both devices, even if the difference is more pronounced for the MEMS devices. However, near the short circuit resonance, the values of De decrease sharply till becoming null at Ω=Ωs0 =1. The effect of the dielectric dissipations is reduced as the the impedance of the piezoelectric vibrator reaches a minimum at the resonance frequency. In this case, the mechanical losses term Dm dominates. In any situation, the electromechanical variable Dem is at least 10 times smaller than Dm and can be neglected. It would also be tempting to neglect De as the devices should intuitively be excited around their resonance to generate a large output. However, it is shown later that the frequency corresponding to the maximum output power does not correspond exactly to Ωs0 and vary in the interval [Ωs0 , Ωo0 ]. In some theoretical cases, two frequencies of maximum power can even be observed so that neglecting De might lead to noticeable errors. Assuming the simplification proposed in the previous paragraph, it is now possible to determine the frequency(ies) and the characteristics of the load leading to the optimum output power. The mathematical manipulations are easier by determining first the optimum value of the load resistor (or of Ψ). It is obtained by computing the derivative of (5.11) with respect to Ψ, equating the corresponding expression to 0 and solving the obtained equation with respect to Ψ. The determined expression of the optimum of Ψ is given in (5.12). If Ω=1 and if a small generalized electromechanical coupling factor is assumed, the familiar expression Ropt =(ω s0 Cp )−1 is obtained. v u 2 1 2 1u Q2m + (Ω − 1) t (5.12) Ψopt = (Ω2 −1)2 1 Ω 2 2 2 Q2 + (Ω − 1 − K ) + Q2 m

114

5.1 Theoretical analysis of the harvester’s output power Amplitude of the damping terms

Amplitude of the damping terms

-2

-4

-6

-8

0.6

0.8 (a)

1.0

1.2

1.4

-1

-3

-5

-7

0.6

0.8

1.0

1.2

1.4

(b)

Figure 5.2 - Amplitude of the damping terms Dm (dashed line), De (solid line) and Dem (dotted line) versus the normalized frequency Ω. The blue lines refer to Ψ=0.1, the black lines to Ψ=1 and the red lines to Ψ=10. (a) corresponds to a Dev1 MEMS AlN bender, while (b) corresponds to a ceramic PZT structure.

The output power corresponding to Ψ=Ψopt is obtained by combining (5.11) and (5.12). Numerical analysis of this formula shows two distinct behaviors depending on the values of the quality factor and of the generalized electromechanical coupling factor. They are illustrated in Figure 5.3 and Figure 5.4: for structures with high Qm and high K2 (K2 >0.01 for Qm =10000, K2 >0.05 for Qm =1000, K2 >0.1 for Qm =100, K2 >0.5 for Qm =10), two maximums of the output power are observed. As noted by DuToit [97], the presence of two power peaks (which also correspond approximately to displacement peaks) is related to the fundamental resonant and anti resonant characteristics of piezoelectric materials and transducers. The first maximum corresponds approximately to the minimum impedance (seen from the electrical side) frequency (Ω≈Ωs0 ) and the optimum load related to this situation has also a minimum value because of power transfer matching considerations. At the opposite, the optimum load corresponding to the second maximum (Ω≈Ωo0 ) has a relatively large impedance. Therefore, the dielectric losses have a stronger influence on the anti resonance power peak than on the resonance one. It can be seen that for |Qe |>100, the dielectric dissipations do not have an influence on the characteristics, but that for |Qe |<100, they tend to diminish the amplitude of the anti resonance power peak. At extremely low values of the electrical quality factor, the latter peak can even vanish. A second type of behavior is observed for low Qm or low K2 structures: a single maxima of power is observed and the anti resonance peak does not exist (Figure 5.4(a)). This can be explained either because the amplitude of the resonance power peak is strongly reduced by the mechanical dissipations (low Qm ) or because the frequency shift between resonance and anti resonance is so small that the power peaks merge together (low K2 ). Also, the values of Ψopt (Figure 5.4(b)) follows an almost perfect hyperbole Ψopt =Ω−1 (1+K2 )−1/2 , except close to Ω=1 where a down scaled version of the behavior observed in Figure 5.3(b) can be observed. From the results obtained

5. DESIGN AND ANALYSIS OF THE MACHINE ENVIRONMENT VIBRATION ENERGY HARVESTER

115

Optimum load parameter Ψ

Optimum output power

10 9 8 7 6

0.8

0.9

Ω0 1.1 1.0

Ω0 1.3 1.2

1.4

(a)

0.1

0.8

0.9

Ω0 1.1 1.0

Ω0 1.3 1.2

1.4

(b)

Figure 5.3 - Behaviour of the output power at optimum load (a) and of the optimum load (b) for high Qm -high K2 structures. The solid lines refers to Qe →∞, the dashed lines to Qe =-100, the dotted lines to Qe =-10.

Optimum load parameter Ψ

Optimum output power

0.1

0.01

0.001

0.8

0.9

Ω0 1.0 (a)

1.1

1.2

1.3

1.4

0.8

0.9

s Ω 1.0 0

1.1

1.2

1.3

1.4

(b)

Figure 5.4 - Behavior of the output power at optimum load (a) and of the optimum load (b) for low Qm or low K2 . The curves are not presented for different values of Qe , as in this case, it does not have any influence on the behavior.

in Chapter 3, it is possible to determine the type of behavior that should correspond to the different types of piezoelectric bender described in Table 2.1. From the values of the generalized electromechanical coupling factor and quality factor given in Table 3.7, it can be concluded that the MEMS devices (AlN or PZT based) should follow the second type of behavior described above. The situation is less clear for commercial ceramic PZT benders. The values of K2 and Qm are just high enough to observe the anti resonance peak. However, dielectric dissipations are also relatively important for the type B bender and it is shown that the available samples do not allow a clear visualization of two distinct peaks. The expression obtained for the output power at optimum load is relatively tedious and some simplifications are introduced. The remainder of the analysis is focused on the resonance and anti resonance power peaks and it is assumed that the former is obtained at Ωs0 and the latter at Ωo0 . As the impedance of

116

5.1 Theoretical analysis of the harvester’s output power

the piezoelectric vibrator is minimum at Ωs0 , the dielectric dissipations have a negligible influence on the resonance power peak for both high Qm -high K2 and low Qm or low K2 situations, so that Qe →∞ is assumed for the optimum s power PR opt at Ω0 . The dielectric dissipations can however not be neglected when considering the anti resonance power peak PA opt which corresponds to a maximum of the impedance of the piezoelectric vibrator (only relevant for high A Qm -high K2 structures). The expressions of PR opt and Popt considering these simplifications and those of the corresponding optimum load parameter are given below. 1 me A20 Qm R r (5.13) Popt = 4ω0s 1 1+ 1+ 4 2 K Qm ΨR opt = p

A Popt

1 + K 4 Q2m p −K 2 Qm Qe 1 + K 2 me A20 Qm = 2 4 2 4ω0s −K p Qm Qe + K Qm + (1 + K 4 Q2m ) (Q2e + K 4 Q2m ) v u 1 + K 4 Q2m u A Ψopt = u t K 4 Q2m 1+ 1 + K2 2 Qe

(5.14)

(5.15)

(5.16)

The expression found for PR opt corresponds to the one obtained by Roundy [28] except for the fact that, in the given reference, the material coupling factor k31 2 is erroneously used instead of the generalized electromechanical coupling factor K2 . This confusion has a strong impact on the estimations of the device performances. Indeed, it is shown in Chapter 2 that the generalized electromechanical coupling factor is only a fraction of k31 2 , because the piezoelectric material is coupled with elastic material in a piezoelectric composite structure and because it undergoes bending and not purely tensile deformations. Both expressions of the optimum power depend primarily on the effective mass of the structure and on the frequency and amplitude of the input vibrations. This dependence is expressed by the term me A0 2 /(4ω s0 ). This term does not represent an upper limit for the power as the additional multiplicative terms present A in the expressions of PR opt and Popt are not necessarily lower than 1 (for the resonance, the limit is approximately Qm ). However, me , A0 and ω s0 are the first parameters to be taken into consideration when designing the harvesters. For analyzing the influence of the other parameters on the performances of the device, numerical evaluations of (5.13) till (5.16) are performed. In Figure 5.5, the graphics describing the values of the optimum load parameter Ψopt are presented. The curves corresponding to the resonance behavior are represented by red solid lines. The curves corresponding to the anti resonance behavior are represented by solid (Qe →∞), dashed (Qe =-100) and dotted (Qe =-10) black lines. According to (5.14), for a given Qm , the optimum

5. DESIGN AND ANALYSIS OF THE MACHINE ENVIRONMENT VIBRATION ENERGY HARVESTER

117

load parameter decreases as the generalized electromechanical coupling factor increases in the resonance case. This is due to the fact that the minimum of the impedance of the piezoelectric vibrator observed at the resonance frequency decreases when the generalized electromechanical coupling factor is increased. 2 ΨR opt is very close to 1 for small values of K while the optimum resonance power is obtained for very low load impedances when K2 is large. At the opposite of the characteristics observed at resonance, the optimum anti resonance load pa2 rameter ΨA opt increases continously with K when the dielectric dissipations are 2 neglected. An increase of K results in larger absolute values of the impedance of the system at the anti resonance, so that the optimum load resistor has also larger values. However, when dielectric dissipations are included, the optimum load parameter increases till reaching a maximum and decreases afterwards. This effect can be explained as follow: the presence of dielectric dissipations is represented in our model by a complex valued capacitor Cp0 =Cp (1+jQe ) in the network model of Figure 5.1. As a possible alternative schematization, one could represent Cp0 by the parallel combination of a real valued capacitor Cp and of a real valued resistor Rel =-Qe /(ω o0 Cp ). As K2 becomes larger, the impedance of the piezoelectric vibrator at anti resonance increases while Rel decreases (because ω o0 =ω s0 (1+K2 )1/2 ). When the impedance of the piezoelectric element becomes too large, the load resistor RL only ”‘see”’ Rel as input impedance and the optimum load is at this moment equal to Rel . A 2 s In Figure 5.6, the values of PR opt and Popt normalized to me A0 Qm /(4ω 0 ) are presented. In terms of output power at resonance, it is always worth to design a low mechanical loss structure. Independently of the values of the generalized electromechanical coupling factor, the output power of the device increases monotonically with Qm . When mechanical dissipations are large, it is important to make the generalized electromechanical coupling factor as high as possible so that the process of energy extraction can compete with the parasitic dissipations. On the other hand, in low parasitic losses situation, the output power is constant over a large range of the generalized electromechanical coupling factor and it is only necessary in this case to design of the harvester so as to exceed the lower boundary limit of K2 . It was shown in Chapter 2 that the value of K2 depends as well on the relative dimensions and on the material properties, particularly on the piezoelectric constant d31 . As the output power does not depend on the generalized electromechanical coupling factor above a certain range, it is then possible to adjust the other parameters (particularly to match the frequency of the input vibrations) while not perturbing the performances. This observation is extremely important, as it suggest that low K2 structure (MEMS devices) can compete with high coupling structure (ceramic PZT bender) in terms of power generation when similar mass and resonance frequency are considered.

For the anti resonance, it has been stated previously that the corresponding power peak was only observed for high Qm -high K2 characteristics. Therefore, the curves corresponding to the anti resonance variables merge together with the resonance curves when the generalized electromechanical coupling factor

118

5.1 Theoretical analysis of the harvester’s output power

Ψopt

10 10

-1

10 -1

-2

-3

4 6

-2

4 6

-1

4 6

-3

4 6

-2

4 6

-1

4 6

(a) (b) 2

Ψopt

10 10

-1

-2

-3

4 6

-2

4 6

-1

4 6

-3

4 6

-2

4 6

-1

4 6

(c) (d) Figure 5.5 - Optimum load parameter Ψopt. (a) corresponds to Qm =10, (b) to Qm =100, (c) to Qm =500 and (d) to Qm =1000. The black solid, dashed and dotted lines represent the anti resonance characteristics for respectively Qe →∞, Qe =-100 and Qe =-10. The red lines represent the resonance characteristics.

is small. For high values of K2 , the behaviors are however totally different. Independently of the amplitude of Qm , PA opt increases monotonically with the generalized electromechanical coupling factor when the dielectric dissipations are neglected and there is no saturation of the normalized power. When dielectric dissipations are included, a maximum of PA opt is found. For the same reasons that were expressed when discussing the optimum load parameter, above a certain value of K2 , the largest amount of the electrical energy produced by the piezoelectric element is shared between the dielectric loss resistor Rel and the load resistor RL rather than being shared between the capacitor Cp and RL . It is also interesting to investigate the output power per unit displacement of the system. Indeed, the power density of the harvester has to be maximized for miniature electronic applications. The graphics of the power density normalized to me A0 ω s0 and assuming the expressions of the optimum load that have been proposed previously are given in Figure 5.7. At resonance, the power density increases with K2 till saturating for high values of K2 as it was the case for the power. The situation is different for the anti resonance: the power density possesses an inflexion point but increases monotonically with the gener-

5. DESIGN AND ANALYSIS OF THE MACHINE ENVIRONMENT VIBRATION ENERGY HARVESTER

119

Normalized optimum power

-1

-2

-3

4 6

-2

4 6

-1

4 6

-1

-2

-3

4 6

-2

4 6

-1

4 6

Normalized optimum power

(a) (b)

-1

-2

-3

4 6

-2

4 6

-1

4 6

-1

-2

-3

4 6

-2

4 6

-1

4 6

(c) (d) Figure 5.6 - Output power at optimum load normalized to me A0 2 Qm /(4ω s0 ). (a) corresponds to Qm =10, (b) to Qm =100, (c) to Qm =500 and (d) to Qm =1000. The black solid, dashed and dotted lines represent the anti resonance characteristics for respectively Qe →∞, Qe =-100 and Qe =-10. The red lines represent the resonance characteristics.

alized electromechanical coupling factor. Also, it can be seen that the dielectric dissipations have a negligible influence on the behavior of this parameter. It is now important to determine what the best option for power generation is between resonance and anti resonance operating frequency. The anti resonance peak does not exist for low values of the generalized electromechanical coupling factor so that this discussion does not make sense in this situation. For large enough K2 , it can be seen from Figure 5.6 that it is always advantageous to work at anti resonance when the dielectric dissipations are ignored. However, it was shown in Figure 5.3a that the dielectric losses had a strong inR fluence on PA opt but did not have noticeable effects on Popt , so that the previous conclusion does not hold when Qe is not taken as infinite. Indeed, in high dielectric dissipation and low mechanical losses situations, PR opt has larger values A than Popt . The optimum load parameter Ψopt for the anti resonance is always much larger than for the resonance. In the former case, extremely large load impedance is required for optimum power generation so that the current delivered to the load element is very small. At the opposite, low shunt impedance is needed at resonance so that the generated voltage is relatively small in this

5.1 Theoretical analysis of the harvester’s output power 0

-1

Normalized optimum power density

120

-1

-2

-3

4 6

-2

4 6

-1

4 6

-2

-3

4 6

-2

4 6

-1

4 6

6 5 4 3 2

-1

-3

4 6

-2

4 6

-1

4 6

Normalized optimum power density

(a) (b) 0

6 5 4 3 2

-1

-3

4 6

-2

4 6

-1

4 6

(c) (d) Figure 5.7 - Power par unit displacement at optimum load normalized to me A0 ω s0 . (a) correspond to Qm =10, (b) to Qm =100, (c) to Qm =500 and (d) to Qm =1000. The black solid, dashed and dotted lines represent the anti resonance characteristics for respectively Qe →∞, Qe =-100 and Qe =-10. The red lines represent the resonance characteristics.

case. The relative amplitude of the deflection in the two discussed situations depends on the value of Qe . Finally, in terms of power density and for high K2 structures, it is always largely advantageous to operate at anti resonance.

5.1.2

Alternative loads

The purely resistive load analyzed in the previous section is the simplest that can be imagined and is useful for demonstrating experimentally the relevance of the developed circuit model. However, the impedance of the piezoelectric harvester can not match fully the impedance of the load circuit as the former contains imaginary components so that the transfer of energy is not optimized. One can then imagine that a ”complex” load consisting in a parallel or series association of linear components such as inductors, capacitors or resistors should lead to better performances. The imaginary part of the impedance of the piezoelectric harvester seen from the electrical side is positive so that the imaginary part of the impedance of the load circuit should be negative in order

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to reach the matching condition. Then, it is only possible to use inductors in a series or parallel association with a dissipative load resistor. Hagood [143] has shown in the field of vibration absorption that it was indeed possible to largely increase the performances of the harvester with the use of an impedance matching circuit. However, the values required for the inductors, for both series and parallel scheme, are so large (>1000H for our MEMS devices) that it is practically impossible to implement it. For real applications, the situation is then limited to the purely resistive load case described above when linear conditioning electronics is assumed. In the situation described above, the delivered power is pure AC and the portable electronics applications that can be powered with it are limited. For this reason, efforts have been done in the literature for proposing more effective interfaces based on non linear conditioning. Lefeuvre [92] proposed an interesting analysis of 4 different conditioning circuitries. The first one consists in a standard rectifying approach and includes a diode rectifier bridge and a filter capacitor. The piezoelectric element is open circuited when the rectifier bridge is blocking, i.e. when the absolute value of the voltage across the electrodes of the piezoelectric element is lower than the rectified DC voltage VDC . In steady state operation, the average current through the filter capacitor Cr is null, so that the absolute value of the electric charge outgoing from the piezoelectric element during a period T is equal to the average current flowing through the load RL . The maximum power that can be obtained from this method is equal to me A0 2 K2 /(2Ď&#x20AC;Ď&#x2030; s0 ). The delivered power is then in any cases smaller than the one obtained with a pure resistive load, but consists in a DC voltage which is more adapted to practical applications. The second approach has been denominated as synchronous charge extraction principle and consists in removing periodically the electric charge accumulated on the piezoelectric capacitor Cp , and then to transfer the corresponding amount of electrical energy to the load or to the energy storage element. The piezoelectric element is let most of the time in open circuit configuration and the charge extraction phases occur when the electronic switch S is closed: the electrical energy stored in the piezoelectric capacitor is then transferred into the inductor L. When the electric charges in the piezoelectric bender vanish, the switch is re-opened and the energy stored into the inductor L is transferred to the smoothing capacitor Cr through the diode D. The inductor L is chosen to get duration of the charge extraction phase much shorter than the vibration period. The output power corresponding to this interface is equal to more than 5 times the one obtained by the basic rectification circuit. Based on the work of Guyomar [94], Lefeuvre also studied the so called SSHI (Synchronized Switch Harvesting on Inductor) interface which can be implemented in a series or parallel arrangement. The Series-SSHI interface leads to low matching load impedances, while the Parallel-SSHI interface leads to higher matching load impedances. The SSHI interface is composed of a non-linear processing circuit connected with the piezoelectric electrodes and the input of the rectifier bridge. The non-linear processing circuit is composed of an inductor L in series with an electronic switch S. The electronic switch

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5.2 Experimental characterization of the harvesters

is briefly turned on when the mechanical displacement reaches a maximum or a minimum. At these triggering times, an oscillating electrical circuit L-Cp is established. The period of these oscillations is chosen much smaller than the mechanical vibration period. The switch is turned off after half an electrical period, resulting in a quasi-instantaneous inversion of the voltage V. Theoretical and experimental investigations have shown that this method allows in some cases to multiply the delivered power by a factor 15 and is then a very promising conditioning circuit. Shu [168] proposed a refined analysis of this method and demonstrated that the SSHI interface is particularly relevant for low coupling structures. The MEMS harvesters that are studied during this thesis belong to this category of devices and the SSHI interface appear particularly adapted for these harvesters, even if its practical implementation is not straightforward. Some efforts have also been done in order to realize conditioning electronics based on IC manufacturing technologies. These approaches are particularly useful in the case of the MEMS harvesters, as the processing circuit and the harvesters could be fabricated in the same clean room environment (in the future, monolithic integration may also become a relevant option). A few examples of the results published in the literature are the work of Dallago, who proposed a high efficiency integrated AC-DC converter [169], based on CMOS technology and described in [170]. Han [88] proposed an interface based on charge pumping realized through switches, MOSFETS and capacitors. Ottman [171] described an integrated step down converter which allows increasing the rate of charging of a battery by three times compared to a standard interface. Finally, Dâ&#x20AC;&#x2122;Hulst proposed in [172] a buck-boost topology, working in discontinuous conduction mode. The circuit performs an AC-DC conversion as well as presenting the correct electrical impedance to maximize the power output of the harvester. This design has been implemented in a 80 V CMOS technology and proved to have a conversion efficiency of more than 60%. Alternative load to a purely resistive one have been described in this subsection. It was shown that is possible in this way to increase or adapt in a better way the power produced by the harvester to a real life application. The experimental validation of the theoretical predictions is however limited in this thesis to the case of a purely resistive load and of the model described in the previous subsection. This experimental analysis is presented in the next section.

5.2

5.2.1

Experimental characterization of the harvesters Output power

It is demonstrated in the previous section that the power delivered by the vibration harvesters to a resistive load is optimized when the frequency of the

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input vibration matches the short circuit resonance frequency in the case of the presented harvesters, so that the experimental analysis is limited to this particular frequency of excitation. In the following, the theoretical predictions concerning the voltage and the output power of the devices are first compared with experimental measurements. In a second subsection, some non linear effects that are not included into the model, particularly the effect of the package on the MEMS devices, are discussed. The output voltage and corresponding power is measured at resonance for the three tested devices while varying the value of the resistive load (and then of Ψ ). The results are reported for the AlN MEMS, for the PZT MEMS and for the commercial PZT harvester respectively in Figure 5.8, Figure 5.9 and Figure 5.10. As the power is computed according to (5.1), the deviation between theory and experiments is inevitably larger for the power than for the voltage. The fit between the model and the measurements is very good in the case of the AlN devices, both for the amplitude of the expected signals and for the value of the optimum load parameter Ψopt . The amplitude of the input acceleration is equal to 1 m.s−2 and a maximum power of approximately 200 nW is measured for a load resistor of 110 kΩ (Ψopt =0.28). The fit is less convincing for the two PZT based devices and a relatively important deviation from the model is observed in terms of amplitude. It can be seen from Figure 5.9 and Figure 5.10 that the values of the optimum load resistor coincides approximately with the expected ones, but that the amplitudes of the output voltage and of the power are lower than those predicted by the model. This deviation has also been reported by most of the authors who have investigated PZT based harvesters such as for example Gao [173] and Kasyap [66]. There might be multiple possible reasons for the observed deviation but the fact that it occurs only for both thick and thin PZT suggests that it is linked to the material itself. At the opposite of AlN, PZT is ferroelectric and one can imagine that important phenomena occurring at the microstructural level are not taken into account in our analysis. Also, linear constitutive equations of piezoelectricity are assumed. It is shown in the next subsection that non linear effects exist for large amplitudes of the input acceleration. Finally, it is also possible that some dissipation mechanisms neglected in our approach (such as for example the piezoelectric dissipations described by Mezheritsky [116]) do exist and do have a non negligible influence on the output power and voltages. Despite the deviation between the theoretical predictions and the experimental measurements observed for the PZT based devices, the developed model still results in a reasonable estimation of the performances of the harvesters, assuming small amplitude of the input vibrations. The level of the input accelerations used in the presented experiments is quite low and resulted in output power below the µW for the AlN MEMS harvesters. For higher amplitude of the exciting vibrations, output power in the range of 50 µW are obtained for both PZT and AlN MEMS devices, which is enough to power low consumption sensor nodes [24].

124

5.2 Experimental characterization of the harvesters -1

-7

2.0x10

3.0

Power (W)

Voltage (V)

4.0x10

2.0 1.0

1.6 1.2 0.8 0.4

-2

6 8

-1

4 6 8

-2

4 6 8

-1

4 6 8

Ψ Ψ (a) (b)

Figure 5.8 - Output voltage (a) and power (b) at resonance for the Dev1 AlN MEMS har vester with an input acceleration A0 =1 m.s−2 . The solid lines correspond to the theoretical predictions while the markers indicate the experimental measurements. -1

-6

3.0x10

1.2

Power (W)

Voltage (V)

1.6x10

0.8 0.4

2.0 1.0 0.0

6 8

-1

6 8

5 6

-1

4 5 6

Ψ Ψ (a) (b)

Figure 5.9 - Output voltage (a) and power (b) at resonance for the Dev1 PZT MEMS har vester with an input acceleration A0 =5 m.s−2 . The solid lines correspond to the theoretical predictions while the markers indicate the experimental measurements. -5

5.0x10 Power (W)

Voltage (V)

6 4 2

4.0 3.0 2.0 1.0

6 8

-1

6 8

-2

6 8

-1

10 Ψ Ψ (a) (b) 10

6 8

Figure 5.10 - Output voltage (a) and power (b) at resonance for the PZT commercial har vester with an input acceleration A0 =1 m.s−2 . The solid lines correspond to the theoretical predictions while the markers indicate the experimental measurements.

5. DESIGN AND ANALYSIS OF THE MACHINE ENVIRONMENT VIBRATION ENERGY HARVESTER 1

Voltage (V)

10 Voltage (V)

125

-1

-2

-1

10 10 -2 A0 (m.s )

-1

-2

4 6

10 10 -2 A0 (m.s )

(a) (b)

4 6

Figure 5.11 - Amplitude of the open circuit voltage at resonance vs. input acceleration for a) the commercial PZT bender, b) the AlN MEMS bender. The solid lines correspond to a linear approximation and the markers represent the experimental measurements.

5.2.2

Non linear effects

As stated previously, large amplitude of the input vibration might results in a non linear behavior. This effect was investigated by measuring the amplitude of the open circuit voltage of the harvesters at resonance for different values of the input acceleration. The results of this experiment are reported in Figure 5.11(a) for the ceramic PZT device and in Figure 5.11(b) for the AlN MEMS harvester. In the proposed graphics, the solid line represents the theoretical linear variation of the voltage versus the input acceleration. Experiments and theory fit well for low levels of the input but a discrepancy is observed at large values of A0 : the open circuit voltage is lower than expected from the linear model. The deviation from the model is higher for the PZT commercial device than for the AlN MEMS one. Furthermore, the non linear deviation occurs for lower amplitudes of the input. This phenomena results from the dimensions of the commercial bender. This piezoelectric bender was made of a long beam whose behavior deviates quite rapidly from the small deformations Euler Bernoulli beam theory on which the developed model is based. It was introduced in Chapter 3 that the MEMS fabricated piezoelectric harvesters are meant to be packaged in order to limit the potential threats from the environment and to include a constraint to limit the maximum deflection of the beam. The package consists in a top and bottom wafer containing a cavity. The air contained in the cavity is not allowed to flow smoothly around the piezoelectric beam in the case of a packaged device and additional squeeze damping exists in the system. In order to estimate the influence of the latter damping, the output power at optimum load of the same harvester is measured in the three different following conditions (the characterized device is an AlN MEMS device having dimensions different from the sample investigated previously): • Unpackaged device under atmospheric pressure

126

5.3 Conclusion 8x10

-6

Power (W)

6 5 4 3 2 1 0 0.98

0.99

1.00

1.01

1.02

Figure 5.12 - Output power of the same AlN MEMS harvester for the optimum resonance resistive load versus the normalized frequency of the input vibration and for different environmental conditions. The solid line corresponds to a packaged device under vacuum conditions, the dashed line to an unpackaged device under atmospheric conditions, the dotted line to a packaged device under atmospheric conditions.

• Packaged device under atmospheric pressure. • Packaged device under vacuum (≈1mbar). The results of this experiment are given in Figure 5.12. By comparing the measurements corresponding to the packaged and unpackaged device under atmospheric pressure, it can be seen that the additional damping due to the package reduces the output power of more than 50% and in the same way, the quality factor of the device is reduced to half its original value. The vacuum experiment was realized by placing a packaged device into a vacuum chamber, but it is planned in the future to realize vacuum packaged devices. However, the manufacturing process of the harvesters is not advanced to this level at this moment, but it is believed that the experiments done in the vacuum chamber are representative of the performances that will be obtained with vacuum packaged devices. The output power of the devices should be multiplied by a factor 4 in these conditions. The parasitic dissipations are reduced in vacuum condition and the output power for a same input acceleration is intuitively larger than for the two others situations. Also, it can be seen that the frequency of maximum power shifts towards lower values when the amplitude of the parasitic dissipations increases (the reference Ω=1 was taken for the vacuum experiment). This result is in accordance with the fact that parasitic mechanical dissipations tend to decrease the value of the resonance frequency.

5.3

Conclusion

In this chapter, the behavior of the output power of the piezoelectric vibration energy harvesters is analyzed, based on the equivalent network model developed

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in the previous chapters. When assuming a single harmonic input vibration, it is shown that the best performances are obtained at the resonance or anti resonance frequency of the devices. However, the anti resonance performances depend strongly on the generalized electromechanical coupling factor and on the dielectric dissipations. It is not possible to study experimentally the anti resonance behavior for the available samples, either because of too low generalized electromechanical coupling factor in the case of the MEMS devices or because of too large dielectric dissipations in the case of the ceramic PZT piezoelectric bender. Therefore, the analysis is focused on the resonance characteristics. The output power delivered to a purely resistive load at resonance depends primarily on the effective mass of the bender, the frequency and the amplitude of the input vibrations, but also on the generalized electromechanical coupling and the mechanical quality factor, which have not necessarily to be maximized, but only to be set above a certain value. This conclusion is important as it means that low coupling structures (such as AlN or PZT MEMS harvesters) can achieve the same performances than high coupling structures (such as ceramic PZT harvesters). Efficient load circuitries are also discussed. In some cases, specially designed shunt circuit, such as for example the SSHI interface, allows strongly improving the performances of the harvesters. MEMS devices based on thin film PZT and AlN materials and commercial piezoelectric bender made of ceramic PZT are experimentally characterized. It is demonstrated that the developed model gives a good estimation of the measured data, but that some discrepancies are observed, particularly at high level of input acceleration for which non linear effects are observed. Finally, it is shown that the packaging of MEMS AlN harvesters creates additional parasitic dissipation mechanisms which reduce drastically the output power of the devices. Therefore, it is planned to package the devices under vacuum in the future. Preliminary experiments suggest that the output power can be increased by a factor 4 in this case. At the present moment, output power in the range of 50 ÂľW are obtained with AlN MEMS harvesters, which is enough to power low consumption sensor nodes.

128

5.3 Conclusion

Chapter

General conclusions and future work In Chapter 1, it is demonstrated that energy harvesters are a promising renewable source of electrical energy for portable electronics applications such as wireless sensor networks. They act by converting the energy available in the environment into useful electrical power. Common environmental energy sources are light, heat, vibrations, human motion or wind. The work realized during this thesis is focused on harvesting energy from vibrations and human body motion. The energy produced by these sources can be transformed following a â&#x20AC;?directly applied forceâ&#x20AC;? or an inertial scheme. The latter type of devices is investigated, as they offer more freedom in terms of possible applications. Piezoelectric transduction is the principle chosen for the electromechanical energy conversion, because of its relative simplicity of implementation compared to electromagnetic or electrostatic transduction. It is also determined that different designs of the harvesters are required for extracting efficiently energy from vibrations and from the motion of the human body. The former type of input is made of relatively high frequency and low amplitude components, while low frequency and high amplitude motion is characteristic of the latter. Resonant devices are adapted to vibration energy harvesting. They are based on a classical mass/spring/dashpot system which must be excited in the neighborhood of its resonance frequency to deliver maximum power. Such devices are manufactured by MEMS technologies and characterized. In the case of body motion energy harvesting, a non resonant device based on the impact of a rigid body on piezoelectric beams is investigated. A prototype is fabricated and characterized. Chapter 2 is focused on the modeling of piezoelectric beams. The constitutive equations of piezoelectricity are combined with Timoshenkoâ&#x20AC;&#x2122;s beam theory

129

130 in order to elaborate the fundamental equations of piezoelectric multilayered benders. From our knowledge, this complete derivation had not been proposed in the literature. Both dynamic and quasi static solutions of the latter equations are derived and arranged in the form of a constitutive matrix and of an equivalent electrical network. The electrical network representation is particularly useful in the field of energy harvesting, as the devices has to be coupled with complex conditioning electronics. A complete representation of a piezoelectric cantilever loaded by a distributed mass, which is the basis of many harvesters currently investigated, is developed. Finally, the generalized electromechanical coupling factor defining the amount of mechanical energy transformed into electrical energy (or vice-versa) by a piezoelectric bender during a cycle is defined and a clear interpretation of the difference between the material and structure electromechanical coupling factor is proposed. Confusion between these two factors is often found in the literature. It can have great repercussions on the estimations of the model. It was also shown that the generalized electromechanical coupling factor depends only on the material properties of the support and piezoelectric materials and on the thickness ratio of the two materials, but not on the lateral dimensions of the structure. In Chapter 3, the manufacturing by MEMS technologies of piezoelectric harvesters and the experimental determination of their corresponding equivalent network parameters is presented. A robust process flow is developed. It results in functional devices but they still need some improvements at the level of etch homogeneity and packaging. Particularly, the proposed ambient atmosphere packaging approach results in low mechanical quality factors and will be improved in the future by implementing a vacuum packaging method. Static, transient and steady state experimental methods for determining the equivalent network parameters are then described and implemented on MEMS PZT and AlN based harvesters and on commercial ceramic PZT structures. No complete procedure for the characterization of the network parameters of piezoelectric beams was existing before this work and methods are developed to this aim. From a theoretical point of view, a large amount of experiments can be realized. However, from a practical point of view, some of them are difficult to be carried out and experiments that can be realized with commonly available laboratory equipments are presented. It is shown that all the parameters but the stiffness of the beam can be determined from steady state measurements, which are for most of them easily implemented. The determination of the stiffness of the beam requires a quasi-static experiment. Characteristic values of the network parameters are then presented for the three types of structures (MEMS thin film AlN unimorph, MEMS thin film PZT unimorph and commercial thick film PZT bimorph) studied during this thesis. The material properties of the piezoelectric materials are estimated from these measurements. The e31 piezoelectric constant is found to be 0.9 C.mâ&#x2C6;&#x2019;2 , 4.1 C.mâ&#x2C6;&#x2019;2 and 11.9 C.mâ&#x2C6;&#x2019;2 for respectively the thin film AlN, the thin film PZT and the thick film PZT. It is shown that PZT, either thick or thin, is more adapted to actuation situation than AlN, but, at the opposite, that the latter material has better performances in terms

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of sensing applications. The generalized electromechanical coupling factor is equal to 4.10−3 for the MEMS AlN harvester, 6.10−5 for the PZT MEMS device and 1.10−1 for the commercial bender. These values do however not contain enough information to perform a direct comparison in terms of energy harvesting. This is done in the next chapters. The human environment harvester, based on the impact of a rigid body on piezoelectric beams, is investigated in Chapter 4. A complete model of the impact of a mass on a piezoelectric bender based on the equivalent network representation previously obtained is established for the first time. Analytical and numerical analysis of the developed model was realized. The following conclusions are obtained: • In order to maximize the efficiency of the energy conversion, the generalized electromechanical coupling factor and the mechanical quality factor of the piezoelectric transducer have to be made as high as possible by choosing appropriate materials and dimensions. Also, the parasitic dissipations in the motion of the missile have intuitively to be minimized. • The influence of the masses ratio and of the impact coefficient of restitution is very complex. However, when the mass of the impacting body is small compared to the one of the beam, better performances are obtained by implementing an elastic impact. At the opposite, the performances are counter intuitively optimized for inelastic impact when the mass of the impacting body is large compared to the one of the beam. • The resonance frequency of the bender should be high enough so that the amount of energy transferred by the moving object during an impact can be dissipated prior to a second impact. • In order to limit reliability problems due to high stresses and deflections resulting from high amplitude excitations, a damping system has to be implemented in the neighborhood of the piezoelectric elements (repulsing magnets are used in our case). As the conditions of large generalized electromechanical coupling factor are required for optimizing the performances, commercial ceramic piezoelectric benders are implemented in the prototype of the human motion energy harvester. Because of their low generalized electromechanical coupling factor, the fabricated MEMS devices can clearly not give good results in this situation. An output power of 600 µW is obtained for a device of dimensions 3.5*2*2 cm3 weighting 60 g placed on the hand of a person and shaken at a frequency of approximately 7 Hz for a 10 cm amplitude. Also, a power of 47 µW is measured when the harvester is rotated of 180◦ each second. A large amount of the volume occupied by the prototype of the harvester can be eliminated and one can estimate an output power density of 10 µW/cm3 or 4 µW/g for an optimized device undergoing the previously described rotary motion. These

132 figures can be multiplied by a factor 12 when a 7 Hz frequency, 10 cm amplitude linear motion is considered. These results are the highest published for inertial human motion energy harvesters. In Chapter 5, the behavior of the output power of the piezoelectric vibration energy harvesters (machine environment) is first theoretically investigated. For the first time, dielectric dissipations are included in the model. No publications taking into account this mode of parasitic mechanism has been proposed. When assuming a single harmonic input vibration, it is shown that the best performances are obtained at the resonance or anti-resonance frequency of the devices. In terms of power per unit displacement, it is demonstrated that it was always advantageous to work at anti-resonance, at the price of a high impedance optimum load (low current). However, the anti-resonance performances depend strongly on the generalized electromechanical coupling factor and on the dielectric dissipations. It is not possible to study experimentally the anti resonance behavior for the available samples, either because of too low generalized electromechanical coupling factor in the case of the MEMS devices or because of too large dielectric dissipations in the case of the commercial PZT piezoelectric bender. Therefore, the analysis is focused on the resonance characteristics. The output power delivered to a purely resistive load at resonance is not affected by dielectric dissipations and depends primarily on the effective mass of the bender and on the frequency and amplitude of the input vibrations, but also on the generalized electromechanical coupling factor and on the mechanical quality factor. At the opposite of what is observed in the case of the human environment harvester, it is shown that the generalized electromechanical coupling factor and the mechanical quality factor do not have necessarily to be maximized, but only to be set above a certain value. This conclusion is important as it means that thin film AlN or PZT MEMS devices can achieve performances similar to those of ceramic PZT structures if the parasitic mechanical dissipations can be reduced below a certain threshold. In the remainder of Chapter 5, the behavior of the output power at resonance of MEMS devices based on thin film PZT or AlN materials and commercial piezoelectric bender made of ceramic PZT are experimentally characterized. It is demonstrated that the developed model gives a good estimation of the measured data. Some discrepancies are however found, particularly at high level of input acceleration for which non linear effects were experimentally observed: the generated voltage predicted by the linear model is clearly larger than the measured one in this situation. Finally, it is shown that the packaging of MEMS AlN harvesters creates additional parasitic dissipation mechanisms which reduce drastically the output power of the devices. At the present moment, output power in the range of 50 ÂľW are obtained for an approximated volume of 0.3 cm3 with AlN and PZT MEMS harvesters, which is enough to power low consumption sensor nodes. These results are amongst the highest published data related to the output power of MEMS piezoelectric energy harvesters. The research on the developed MEMS piezoelectric harvesters continues at

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Imec and at the Holst Centre. Particularly, vacuum packaged devices have been produced. As mentioned in the last chapter, vacuum packaging allows increasing the produced power by a factor 4 by limiting the parasitic mechanical dissipations. New problems are however encountered: due to the large displacement of the mass, it impacts in most of the situations on the inner side of the package. The effect of this phenomenon on the reliability of the devices is currently investigated. A general study of the fracture and fatigue behavior is also being carried on. Fracture is a critical issue for the manufactured devices. They are fragile, particularly when subjected to undesired mechanical shocks. Fatigue effects need also to be studied, as the energy harvesters are meant to remain functional for periods exceeding several years. From a literature review, it is determined that fatigue in micromachined piezoelectric laminated beams is the most susceptible to occur in the interfaces between the different materials. Geometries different than a rectangular beam such as for example tapered cantilevers or membranes are also investigated for the purpose of increasing the power generated by the devices. To the same aim, structures based on 33 piezoelectric coefficients rather than on the 31 coefficients are investigated. Also, efforts are done for developing viable conditioning electronics for rectifying the AC power delivered by the piezoelectric vibration harvesters. Finally, new designs of devices for harvesting efficiently the energy from broadband spectrum mechanical vibrations are studied.

134

List of Publications Journal Publications 1. M. Renaud, P. Fiorini and C. van Hoof, “Optimization of a piezoelectric unimorph for shock and impact energy harvesting”, Smart Mater. Struct., vol. 16, pp. 1125–1135, 2007. 2. M. Renaud, K. Karakaya, T. Sterken, P. Fiorini, C. van Hoof and R. Puers, “Fabrication, modeling and characterization of MEMS piezoelectric vibration harvesters”, Sens. and Actuat. A, vol. 145-146, pp. 380–386, 2008. 3. M. Renaud, P. Fiorini, R. van Schaijk and C. van Hoof, “Harvesting energy from the motion of human limbs: the design and analysis of an impact-based piezoelectric generator”, Smart Mater. Struct., vol. 18, pp. 943–961, 2009. 4. K. Karakaya, M. Renaud, M. Goedbloed and R. van Schaijk, “The effect of the built-in stress level of AlN layers on the properties of piezoelectric vibration energy harvesters”, J. Micromech. Microeng., vol. 18, 2008.

Conference Publications 1. M. Renaud, T. Sterken, P. Fiorini, R. Puers, K. Baert and C. Van Hoof, “Scavenging energy from human body: design of a piezoelectric transducer”, in Proc. of the Conf. on Solid State Sens., Act. and Microsys., Transducers’05, 2005, pp. 784–787. 2. M. Renaud, T. Sterken, A. Schmitz, P. Fiorini, R. Puers and C. Van Hoof, “Piezoelectric Harvesters and MEMS Technology: Fabrication, Modeling and Measurements”, in Proc. of the Conf. on Solid State Sens., Act. and Microsys., Transducers’07, 2007, pp. 891–894. 3. M. Renaud, P. Fiorini, R. van Schaijk and C. Van Hoof, “An Impact Based Piezoelectric Harvester Adapted to Low Frequency Environmen135

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List of Publications tal vibration”, in Proc. of the Conf. on Solid State Sens., Act. and Microsys., Transducers’09, 2009, pp. 2094-2097.

4. A. Schmitz, T. Sterken, M. Renaud, P. Fiorini, R. Puers, and C. van Hoof, “Piezoelectric Scavengers in MEMS Technology: Fabrication and Simulation”, in Proc. of Power MEMS, pp. 61–64, 2005. 5. N. El Ghouti, R. Vullers, A. Schmitz and M. Renaud, “Piezoelectric energy scavengers for powering wireless autonomous transducer systems”, in Proc. Smart Systems Integ., 2007. 6. S. Matova, R. Elfrink, R. van Schaijk and M. Renaud, “Modeling and validation of piezoelectric AlN harvesters”, in Eurosensors XXII, 2008. 7. R. Elfrink, M. Renaud, T. Kamel, C. de Nooijer, M. Jambunathan, M. Goedbloed, D. Hohlfeld, S. Matova and R. van Schaijk, “Vacuum packaged MEMS piezoelectric vibration energy harvesters”, accepted for presentation at Power MEMS 2009. 8. R. Elfrink, V. Pop, D. Hohlfeld, T. Kamel, S. Matova, C. de Nooijer, M. Jambunathan, M. Goedbloed, L. Caballero, M. Renaud, J. Penders and R. van Schaijk, “First autonomous wireless sensor node powered by a vacuum-packaged piezoelectric MEMS energy harvester”, accepted for presentation at IEDM 2009.

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