Piezoelectric Energy Harvesters for Wireless Sensor Networks

KATHOLIEKE UNIVERSITEIT LEUVENFACULTEIT INGENIEURSWETENSCHAPPENDEPARTEMENT ELEKTROTECHNIEK – ESATKasteelpark Arenberg 10, B-3001 Leuven - Belgie

PIEZOELECTRIC ENERGY HARVESTERSFOR

WIRELESS SENSOR NETWORKS

Jury:Prof. Dr. Ann Haegemans, chairProf. Dr. Robert Mertens, promoterProf. Dr. Chris Van Hoof, promoterProf. Dr. Eric van den BulckProf. Dr. Ronnie BelmansProf. Dr. Robert PuersDr. Paolo FioriniProf. Dr. Jo De Boeck

Dissertation submitted in partialfulfillment of the requirementsfor the degree of ”doctor in deingenieurswetenschappen”

by

Michael Renaud

October 2009

in collaboration with

VZW

Interuniversitair Micro-Elektronica Centrum vzwKapeldreef 75B-3001 Leuven (Belgie)

©Katholieke Universiteit Leuven - Faculteit Toegepaste WetenschappenArenbergkasteel, B-3001 Heverlee (Belgie)

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All rights reserved. No part of this publication may be reproduced in any form by print,photoprint, microfilm, or any other means without written permission from the publisher.

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 toeverybody who helped me during my research on this PhD Thesis.

First of all, I wish to express my appreciation and thanks to my direct super-visor Dr. P. Fiorini for his valuable guidance, support and patience throughoutmy research. I would not have been able to conclude this Ph.D. without hishelp. I should also specifically thank him for his great efforts on reviewing mypapers.

I would then like to thank Prof. C. van Hoof and Prof. R. Mertens foroffering me the opportunity to carry on this research and helping me withthe tasks related to my Ph.D. I am grateful to the members of the reviewingcommittee (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. Iam also thankful to Prof. A. Haegemans for chairing the defense.

I would like to thank the members of various teams I cooperated with. Tostart, many thanks go to Tom Sterken for sharing his valuable insights andfor all the constructive discussions we had. I would like to thank Bert Duboisand Chikhi Abdelhafid for helping me getting accustomed to the cleanroomenvironment. Thanks to Vladimir Leonov and Ziyang Wang to create a pleasantteam environment. Many thanks also go to Stanislaw Kalicinski and VladimirCherman for their help on the characterization of devices. I would also like tothank 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 fruitfulcollaboration which is still ongoing. I am also appreciative of the work realizedby the master students that I mentored during this Ph.D. research (L. deVreede, V. Prins, H. Toreyin and A. Bayoumi).

I am grateful to IMEC and the Katholieke Universiteit Leuven (K.U. Leu-ven). Both provided me the necessary means to conduct my Ph.D. research inan excellent environment and gave me the opportunity to present my resultsin high level international conferences.

I would like to thank my friends Gregory, Raquel and Bert for the fun wehad when we were all together. Also, thanks to my aquatic pets which helped

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me relaxing when the pressure was too high. Finally, many thanks to myparents for their endless support.

Michael RenaudLeuven, October 2009

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To my grandmother

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Abstract

Recent years have seen important developments in the field of wireless sensornetworks. Such networks have already found applications in the field of healthmonitoring and smart environment. Electrochemical batteries are usually im-plemented for powering the sensor nodes. Depletion of batteries leads to a lossof functionality of the sensor node which can result in critical situations if notdetected in due time. The costs of replacing batteries located in a remote orhostile environment is also high. Furthermore, electrochemical batteries con-tain dangerous chemicals possibly released in the environment because of thecosts of recycling. Therefore, so-called energy harvesters gained interest in thelast decade.

Energy harvesters act by converting part of the energy available in theenvironment into useful electrical power. They are self replenished and donot need replacement. Common environmental energy sources are light, heat,vibrations, human motion or wind. This thesis is focused on harvesting energyfrom vibrations and human body motion. Devices adapted to each situation aredesigned, fabricated and characterized. Their active element is a piezoelectricbender and a detailed model of such a transducer is derived.

The devices designed for harvesting energy from the human motion arebased on the impact of a rigid body on piezoelectric cantilevers. An outputpower of 47 µW is obtained for a device of dimensions 3.5*2*2 cm3 weighting60 g rotated over 180 each second. Also, a power of 600 µW is measuredwhen the harvester is placed on the hand of a person and forcibly shaken at afrequency of approximately 7 Hz for an amplitude of 10 cm. A large amountof the volume occupied by the prototype of the harvester can be eliminatedand an output power density of 10 µW/cm3 or 4 µW/g is estimated for anoptimized device undergoing the aforementioned rotary motion. These figuresare multiplied by a factor 12 when a 7 Hz frequency, 10 cm amplitude linearmotion is considered.

The devices designed for harvesting energy from vibrations in machinery en-vironment are based on resonant piezoelectric beams. Both aluminum nitride(AlN) and lead zirconate titanate (PZT) are considered as piezoelectric mate-rials. If parasitic dissipations can be maintained below a certain threshold, it is

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demonstrated that MEMS harvesters based on thin film AlN or PZT competewith ceramic PZT based devices in terms of power generation. Micromachinedpiezoelectric harvesters are manufactured and characterized. An output powerin the range of 50 µW for an approximated volume of 0.3 cm3 is obtained withAlN and PZT MEMS harvesters, which is enough to power low consumptionsensor nodes.

As a conclusion, the results presented in this PhD thesis give an importantcontribution to the optimization of piezoelectric energy harvesters, both at theexperimental and theoretical level.

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Nederlandstalige samenvatting

De laatste jaren hebben we een enorme ontwikkeling gezien van draagbareelektronica, zoals draadloze sensornetwerken die onder andere hun toepassingvinden in de gezondheidszorg en in slimme omgevingen. In het algemeen wor-den elektrochemische batterijen gebruikt als voeding voor de sensoren in zo’nnetwerk. Deze hebben als nadeel hun hoge kostprijs en het feit dat ze gemaaktzijn uit schadelijke materialen die meestal tussen het restafval belanden we-gens de hoge recyclagekost. Bovendien kan het vervangen van deze batterijenop afgelegen en gevaarlijke plaatsen ook een groot kostenplaatje met zich mee-brengen. Daarom is er een groeiende interesse voor de zogenaamde energiecollectoren.

Energie collectoren kunnen energie die aanwezig is in de omgeving omzettenin bruikbaar elektrisch vermogen. Voorbeelden van omgevingsenergie zijn licht,warmte, mechanische trillingen, menselijke beweging en wind. Het werk indeze thesis focusseert op het verzamelen van energie komende van mechanis-che trillingen en menselijke bewegingsenergie. Verschillende oogstprincipes zijnnodig om een efficiente energieverzameling te verwezenlijken van deze twee en-ergiebronnen; daarom werden twee types apparaten gemodelleerd, gefabriceerden gekarakteriseerd. Beide types van collectoren zijn gebaseerd op het principevan piezo-elektrische transductie.

Het apparaat ontwikkeld om energie te halen uit menselijke beweging isgebaseerd op het fenomeen van impact op een star lichaam van piezo-elektrischecantilevers. Een vermogen van 47 µW kan behaald worden met een apparaatmet een dimensie van 3.5*2*2 cm3 en een gewicht van 60 g en dit geplaatstop de hand en elke seconde geroteerd over een hoek van 180. Een groot deelvan het volume van dit prototype kan gelimineerd worden en de uitgangsver-mogendichtheid kan geschat worden op 10 µW/cm3 of 4 µW/g voor een geop-timaliseerd systeem dat dezelfde roterende beweging ondergaat. Deze waardenkunnen vermenigvuldigd worden met een factor 12 wanneer een lineaire be-weging beschouwd wordt met een frequentie van 7 Hz en een amplitude van 10cm.

Het apparaat ontwikkeld om energie te halen uit trillingen is gebaseerd opeen resonantiesysteem gemaakt uit een verpakte piezo-elektrische balk. In de

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veronderstelling van een enkele harmonische trilling als input, wordt aange-toond dat de beste prestaties behaald worden bij resonantie of anti-resonantiefrequentie van de apparaten. Ook wordt aangetoond dat zowel hoog als laaggekoppelde systemen bruikbaar zijn als de parasitaire verliezen onder een wel-bepaalde drempelwaarde gehouden kunnen worden. MEMS piezo-elektrischecollectoren worden gefabriceerd en gekarakteriseerd. Een uitgangsvermogenvan rond de 50 µW werd bereikt met een volume van ongeveer 0.3 cm3 metAlN en PZT MEMS collectoren, wat voldoende energie is om sensor knopen tevoeden die een laag vermogen nodig hebben.

Tot slot wordt dit werk als zeer nuttig gezien met nieuwe en belangrijke re-sultaten op het vlak van optimalisatie van piezo-elektrische energie collectoren,zowel op experimenteel als theoretisch vlak.

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Symbols and Abbreviations

Symbols

A0 Amplitude of the input acceleration (m.s−2)

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

Cp0 Quasi static clamped capacitance of the piezoelectriclayer in a laminated piezoelectric beam (F)

Cpf Clamped capacitance of an unsupported slab of piezo-electric material (F)

Cpc Clamped capacitance of the piezoelectric layer in a lam-inated piezoelectric beam (F)

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

dp Effective piezoelectric charge constant (C.N−1)

D Electrical displacement tensor (A.m−2)

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

De Normalized electrical damping term (-)

Dem Normalized electromechanical damping term (-)

Dm Normalized mechanical damping term (-)

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

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

e Effective coefficient of restitution (-)

ep Effective piezoelectric constant (C.m−2)

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Eff Harvester effectiveness (%)

E Electrical field tensor (V.m−1)

Ec Coercive electrical field (V.m−1)

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

Eel Energetic term for the definition of k31 (J)

Em Energetic term for the definition of k31 (J)

EM Energetic term for the definition of k31 (J)

EIeq Equivalent of the quotient of the area moment of inertiaover the tensile compliance (multilayer beams) (N.m2)

F Concentrated force applied on the beam (N)

g Gravity field (=9.81m.s−2)

G Figure of merit (=K2Qm)

GAeq Equivalent of the quotient of the area of the cross sectionover the shear compliance (multilayer beams) (N)

hp Thickness of the piezoelectric layer in the piezoelectricbeam (m)

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

H Thickness of the body attached to the cantilever (m)

Ii Area moment of inertia (m4)

Jz Shear force (N)

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

k0 Quasi static lumped stiffness of the beam (N.m−1)

kc Lumped stiffness of the beam (N.m−1)

ki Indentation stiffness (N.m−3/2)

kij Electromechanical coupling factor of the mode ij (-)

kp Effective transverse electromechanical coupling factor (-)

k31 Transverse electromechanical coupling factor (-)

K Generalized electromechanical coupling factor (-)

l Length of the piezoelectric beam (m)

L Length of the body attached to the cantilever (m)

x

mb Mass of the beam (kg)

me Effective mass of the structure (kg)

mt Mass of the body attached to the cantilever (kg)

M Mass of the missile (kg)

My Bending moment due to mechanical efforts (N.m)

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

O Denominator of the Laplace transform of the voltageacross a resistively shunted bender (-)

P Distributed load applied on the beam (N.m)

P d Generated power (W)

P opt Optimum generated power (W)

PRopt Optimum generated power at resonance (W)

PAopt Optimum generated power at anti-resonance (W)

Q Charges developped by the piezoelectric material (C)

Qm Mechanical quality factor (-)

Qe Electrical quality factor (-)

ri Roots in s of O (-)

R Rayleigh quotient (-)

Rb Radius of curvature of the beam (m)

RL Load resistor (Ω)

RM Radius of curvature of the missile (m)

Ropt Optimum load resistor (Ω)

Rmopt Optimum load resistor in multiple impacts situation (Ω)

sb Effective compliance for the beam (Pa−1)

sij Components of the compliance tensor (Pa−1)

sEij Components of the compliance tensor under constantelectrical field (Pa−1)

sDij Components of the compliance tensor under constantelectrical displacement (Pa−1)

sM Effective compliance for the missile (Pa−1)

ss Effective compliance for the elastic material (Pa−1)

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sEp Effective compliance under constant electrical field forthe piezoelectric material (Pa−1)

sDp Effective compliance under constant electrical displace-ment for the piezoelectric material (Pa−1)

S Strain tensor (-)

Si Components of the strain tensor (-)

t Time (s)

tc Contact time (s)

ta Time interval between two successive impacts (s)

T Stress tensor (Pa)

Ti Components of the stress tensor (Pa)

T1 Average longitudinal stress in the piezoelectric layer(Pa)

u Longitudinal component of the displacement field (m)

U0 Relative velocity of impact (m.s−1)

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

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

v′

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

vM Velocity at the position of the missile just before animpact (m.s−1)

v′

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

V Potential difference across the electrodes of the piezo-electric beam (V)

w Transverse component of the displacement field (m)

wx Position dependent component of the transverse dis-placement (m)

wt Time dependent component of the transverse displace-ment (m)

W Width of the piezoelectric beam (m)

Xi Experiment label (-)

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Z Input displacement (m)

z0 Position of the neutral axis (m)

Z0 Amplitude of the input displacement (m)

α Macroscopic piezoelectric constant (C)

Γ Real part of the quasi static lumped piezoelectric trans-formation factor (N.V−1)

Γ0 Quasi static lumped piezoelectric transformation factor(N.V−1)

Γc Lumped piezoelectric transformation factor (N.V−1)

δ Deflection along the length of the body attached to thecantilever (m)

δi Indentation (m)

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

δM Position of the missile (m)

δRopt Deflection at the position of the center of mass of thebody attached to the cantilever at resonance and for theoptimum load (m)

δAopt Deflection at the position of the center of mass of thebody attached to the cantilever at anti-resonance andfor the optimum load (m)

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

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

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

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

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

η Efficiency of the energy conversion for the impact har-vester (%)

θy Shear angle (rad)

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

λ Characteristic wavelength of the bending wave (m)

Λ Amplitude of the displacement of the mass (m)

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ξ Electromechanical coupling correction (-)

Π Polarization direction (-)

ρ Density (kg.m−3)

ρ Average density (kg.m−3)

Ψ Load parameter (-)

Ψopt Optimum load parameter (-)

ΨRopt Optimum load parameter at resonance (-)

ΨAopt Optimum load parameter at anti-resonance (-)

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

ωs0 Short circuit resonance angular frequency of the piezo-electric beam (rad.s−1)

ωo0 Open circuit resonance angular frequency of the piezo-electric beam (rad.s−1)

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

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

ωs Angular frequency frequency of a piezoelectric beamshunted 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 ofthe piezoelectric beam (-)

Ωo0 Normalized open circuit resonance angular frequency ofthe piezoelectric beam (-)

Abbreviations

BCB Benzocyclobutene

CMOS Complementary Metal Oxide Semiconductor

EMC Electromechanical coupling factor

GEMC Generalized electromechanical coupling factor

IC 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 i

Abstract v

Nederlandstalige samenvatting vii

Symbols and Abbreviations ix

Contents xvii

1 Introduction 1

1.1 Energy harvesting: state of the art . . . . . . . . . . . . . . . . 2

1.1.1 Solar energy . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Vibrations and motion . . . . . . . . . . . . . . . . . . . 4

1.2 Piezoelectric inertial vibration energy harvesters . . . . . . . . 7

1.2.1 Resonant systems: machine environment . . . . . . . . . 8

1.2.2 Non resonant systems: human environment . . . . . . . 13

1.3 Scope and organization of the thesis . . . . . . . . . . . . . . . 14

2 Theory and lumped model of piezoelectric laminated beams 19

2.1 History, basic definitions and linear constitutive equations ofpiezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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2.1.1 History and basic definitions . . . . . . . . . . . . . . . 20

2.1.2 Constitutive equations of linear piezoelectricity . . . . . 24

2.1.3 Dissipative and non linear effects in piezoelectric materi-als . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.4 Relevant piezoelectric materials and corresponding sim-plifications of the constitutive equations . . . . . . . . . 29

2.2 The constitutive equations of piezoelectric laminated beams . . 32

2.2.1 Elastic laminated beams . . . . . . . . . . . . . . . . . . 32

2.2.2 Piezoelectric laminated beams . . . . . . . . . . . . . . 39

2.3 Constitutive matrix and electrical network representation of piezo-electric beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3.1 Geometry of the harvesters and applied mechanical efforts 48

2.3.2 Concepts of the constitutive matrix and electrical equiv-alent network . . . . . . . . . . . . . . . . . . . . . . . . 50

2.3.3 Constitutive matrix and equivalent electrical network ofthe piezoelectric harvesters . . . . . . . . . . . . . . . . 52

2.3.4 Generalized electromechanical coupling factor . . . . . . 58

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Manufacturing and primary characterization of MEMS piezo-electric harvesters 61

3.1 Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1.1 General description of the manufactured devices . . . . 62

3.1.2 Process flow . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.1 General concept for the determination of the networkparameters . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.2 Quasi static characterization methods . . . . . . . . . . 69

3.2.3 Steady-state characterization methods . . . . . . . . . . 72

3.2.4 Transient characterization methods . . . . . . . . . . . . 79

3.2.5 Typical values of the network parameters and estimationof the material properties . . . . . . . . . . . . . . . . . 79

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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4 Design and analysis of the human environment vibration en-ergy harvester 83

4.1 Modeling of the impact based harvester . . . . . . . . . . . . . 84

4.1.1 Development of the model . . . . . . . . . . . . . . . . . 84

4.1.2 Theoretical optimization of the generated power: analyt-ical perspectives . . . . . . . . . . . . . . . . . . . . . . 91

4.1.3 Theoretical optimization of the generated power: numer-ical perspectives . . . . . . . . . . . . . . . . . . . . . . 96

4.2 Experimental measurements . . . . . . . . . . . . . . . . . . . . 102

4.2.1 Coefficient of restitution . . . . . . . . . . . . . . . . . . 102

4.2.2 Comparison of the model predictions with experimentalmeasurements . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2.3 Characterization of a prototype of the human environ-ment harvester . . . . . . . . . . . . . . . . . . . . . . . 104

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5 Design and analysis of the machine environment vibration en-ergy harvester 109

5.1 Theoretical analysis of the harvester’s output power . . . . . . 110

5.1.1 Resistive load . . . . . . . . . . . . . . . . . . . . . . . . 110

5.1.2 Alternative loads . . . . . . . . . . . . . . . . . . . . . . 120

5.2 Experimental characterization of the harvesters . . . . . . . . . 122

5.2.1 Output power . . . . . . . . . . . . . . . . . . . . . . . . 122

5.2.2 Non linear effects . . . . . . . . . . . . . . . . . . . . . . 125

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6 General conclusions and future work 129

List of Publications 135

Bibliography 137

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Chapter1Introduction

Recent years have seen important developments in the field of wireless sen-sor networks. A first important example of such networks are the so calledbody area networks [1, 2]. It consists of an array of sensors located on thehuman body (Figure 1.1). The sensors are aimed primarily at measuring rel-evant information related to the health of a person, such as blood pressure oroxygen level, and at transmitting the corresponding data to medical authori-ties through existing communication networks. Three examples illustrate thepotential of this approach: continually monitoring blood pressure for patientswith hypertension can significantly increase medication compliance [3], real-time processing of electrocardiograph traces can be very effective at revealingearly stages of heart disease [4], closed-loop control of insulin administrationfor diabetic patients would significantly reduce the risk of hypoglycemia [5]. Asecond important example of application for wireless sensor networks consistsin smart environments and automated buildings. Arrays of sensors can for ex-ample be used to monitor human presence or environment characteristics andtrigger appropriate responses (automatic light switching, intrusion detection,fire alarm).

Power autonomy of the sensors nodes is essential for the success of wire-less sensor networks and requires the development of low-power electronicsand long life energy sources other than electrochemical batteries. Depletion ofbatteries leads to a loss of functionality of the sensor node which can resultin critical situations if not detected in due time. The costs of replacing bat-teries located in a remote or hostile environment is also high. Furthermore,electrochemical batteries contain dangerous chemicals possibly released in theenvironment because of the costs of recycling. Therefore, clean, renewable andportable energy sources gained attention during the last ten years and the socalled energy harvesters were investigated as an alternative to electrochemicalbatteries. Energy harvesters act by converting part of the energy available in

1

2 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 donot need replacement. Common environmental energy sources are light, heat,vibrations, human motion or wind. This thesis is focused on harvesting energyfrom vibrations and human body motion.

This chapter is organized in the following way: in a first section, the stateof the art in the field of energy harvesting is presented. The second subsectionis focused on energy harvesting from mechanical vibrations or motion usinga piezoelectric transduction principle. The different architectures that can beimplemented are first discussed in details. Then, the basic ideas related to thedevices studied in this thesis and the motivations for their development arepresented. 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 sizeddevices (cm3). Recent advances in small scale energy harvesting applicationsare described in this section. Different sources of energy and the correspondingharvesting devices are presented in separate parts. The description is focused onthe most common and promising sources of energy; fuel cells, microturbines andambient electromagnetic waves (other than light) harvesting are not treated.

1. INTRODUCTION 3

Some references related to the latter subjects are the work of Kolanowski [7],of O’Hayre [8] and of Li [9].

1.1.1 Solar energy

The principle of photovoltaic cells is known since the work of Becquerel in the19th century. The first solar cells were manufactured by Fritts in 1883. Themodern age of photovoltaic cells started with the work of Pearson, Chapin andFuller in the 1950’s. The efficiency of their devices was however not exceeding5% and research efforts have been realized for increasing their efficiency whiledecreasing the costs of manufacturing. The efficiency of solar cells depends onoptical and electrical properties of the semiconductor; the optimization of theirperformances required then an intensive material research and optimization. Abalance between cost and efficiency is also an important issue for the widespreaduse of solar cells. In the 1970s, Berman designed significantly less expensivesolar cells by using a poorer grade of silicon and packaging the cells with cheapermaterials. GaAs appeared in the following years as a very good, but veryexpensive, candidate for increasing the efficiency. Photovoltaic energy stillcosts more than classical power sources; different roads towards lowering itscosts are under investigation: research companies try to either grow the siliconinto a shape that eliminates most of the slicing requirements or to depositphotovoltaic material onto an inexpensive but rigid support structure such asceramic, glass or steel. Also, thin film technologies (related to the IC field) andmass production allows a lower cost of fabrication.

1.1.2 Heat

A review of recent thermoelectric generators is given by Hudak in [10]. Theshort description presented here is derived from his article. Thermoelectricpower generators produce electrical power from temperature differences be-tween two substrates. They have been successfully developed since the be-ginning of the 20th century for large scale power generation by using wasteheat from industrial processes [11]. The use of turbines is one of the simplestmethods to extract energy from heat but it is not the sole. Indeed, electri-cal power can be generated from temperature gradients exploiting the Seebeckeffect: when two faces of plates made of different conductor materials (i.e. athermocouple) are set into mechanical contact, an electrical potential differenceis observed between faces if they have different temperatures. Small-sized scalethermoelectric generators based on this effect have only been investigated dur-ing the past decade. Thermocouples for power generation are normally madeby the association of a n and p type doped elements. The most widely usedthermoelectric materials for decades have been n and p type alloys of bismuthtelluride. Wristwatches powered only by thermoelectric effects have alreadybeen commercialized [12]. The development of MEMS manufacturing tech-


nologies led recently to the investigation of less performing while cheaper andeasier to integrate materials such as silicon germanium [13]. An interesting ap-plication of thermopiles in the field of energy harvesting was recently proposedby Torfs and Leonov [14]: a commercial pulse oximeter is fully powered by theheat dissipated by the human body. The full system is able to transmit data toa neighboring computer. As a rough estimate, the achievable power density ofthermoelectric 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 there-fore attractive sources of energy for producing electrical power. The dynamo ofa bike is a well known example of motion energy harvesting. Self powered wrist-watches based on a rotating heavy mass and on an electromagnetic principlewere also developed during the 19th and 20th century. The conversion of me-chanical 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 transduc-tion methods such as magnetostriction [32, 33] were also proposed but remainmarginal. The electromagnetic transduction is based on the classical Faraday’slaw of induction which states that the motion of a coil in a magnetic fieldgenerates an electrical current in the coil. Capacitive transduction is obtainedwith a movable plate capacitor with one of the electrodes connected to a sourceof electrical charges. According to Gauss’ law, a change in the values of the ca-pacitor results in a motion of charges so that a useful current can be produced.Finally, the piezoelectric mechanism depends on the crystalline nature of thematerial; piezoelectric materials develop electrical charges when deformed.

Motion-driven generators fall into two distinct categories: those using di-rect application of an external force and those using inertial forces acting ona movable proof mass. A large amount of energy can easily be extracted fromthe former kind of harvesters and several applications focused on harvesting en-ergy from human motion have already been commercialized. The best knownexample is the harvesting shoe [34, 35]. An illustration of an electromagneticimplementation of this idea is given in Figure 1.2. The power generated bythese shoes exceeds a few watts, enough to supply energy to many kinds ofapplications. Several wind-up based charging systems were also recently pro-posed with GSM or flashlight loader amongst them. One of the most promisingapplications consists in the human powered laptop associated with the ”Onelaptop per child” project. The charging mechanism presented in Figure 1.2is based on a hand crank system that has been recently abandoned due toits poor efficiency in favor of a more compact off-laptop design that uses apull string to spin a small generator. As a last example of direct force energyharvesters, Keawboonchuay [26] developed piezoelectric generators that can beincorporated into a special kind of ammunitions. Energy is generated when themunitions impact the ground. Poulin gives some additional examples of direct

1. INTRODUCTION 5

Electromagnetic generator

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

force harvesters in [36].

Most of the direct force motion energy harvesters are however bulky and donot fulfill the needs of wireless sensor networks. On the opposite, harvestersbased on inertial mechanisms can be made relatively small. As for the formertype, inertial generators are most of the time based on electromagnetic, electro-static or piezoelectric transduction. The classical design of inertial harvestersis based on a resonant scheme similar to the one implemented in accelerometersand the device should oscillate along one of its fundamental modes in order todeliver maximum power. In a first approximation, these systems can be repre-sented by a dashpot mass spring system coupled with some kind of electricaldamping corresponding to the energy harvesting process [37]. Such harvestersreceived a lot of attention over the last years. Most of these harvesters aremanufactured by conventional technologies. However, MEMS fabrication wasrecently implemented with the advantage of low cost mass production [20, 24].Reviewing the literature on resonant inertial energy harvesters is a colossaltask due the abundance of articles. Fortunately, general reviews can be found[38–40]. This thesis is focused on piezoelectric energy based devices analyzedin details later. In the remainder of this section, a brief overview of the charac-teristics of environmental vibration sources and of the principles implementedfor inertial harvesters are given.

Ubiquitous vibration sources vary considerably in dominant frequency. Theresonant inertial harvesters should be tuned to this frequency. Roundy [41]presents the capabilities for a number of vibration sources indicating that the


amplitude and frequency varies from 12 m.s−2 at 200 Hz for a car engine com-partment to 0.2 m.s−2 at 100 Hz for the floor in an office building with themajority of sources measured having a fundamental frequency in the range 60-1000 Hz. Vibrations present in most environments are not purely harmonic butrather broadband. In their basic form, the resonant harvesters described aboveproduce a reasonable amount of electrical power only if they are excited at theirfundamental frequency and they are then not adapted to extract efficiently theenergy from broadband input vibrations. A wider frequency response can beobtained using higher order resonators and multiple mass spring systems. De-signs of this type are proposed in [42]. In [43], Roundy proposed a methodologyfor tuning resonant inertial harvesters by varying the spring constant or thevalue of the proof mass. Challa [44] described a passively tunable device inwhich the resonant frequency of a piezoelectric cantilever is shifted by the ap-plication of magnetic forces. Arrays of piezoelectric cantilevers with differentresonance frequencies to broaden the response spectrum were also proposed[45].

The resonant or broadband harvesters presented above are adapted to rel-atively high frequency vibrations occurring in an ”industrial” environment. Inthe case of the human body, it is not possible to speak of vibrations but ratherof motion characterized by low frequencies and high amplitude. It is difficultto design devices resonating at such low frequencies. Also, oppositely to theindustrial vibration situation, the amplitude of the external motion is largerthan the displacement of the proof mass. Miao analyzes this problematic in[46] and proposes an alternative design based on electrostatic transduction de-picted in Figure 1.3(a). In this system, the variable capacitor is made of a fixedand 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 ex-ternal acceleration results in an inertial force higher than the sticking one. Atthis moment, the mass is released and discharging occurs through contact padswhen it has reached the other plate. A hybrid piezoelectric and electrostaticlow frequency energy harvester is proposed in [47]. It is an electrostatic oscil-lator suspended by piezoelectric springs. Rotational rather than linear internalmotion is also adapted for low frequencies of the input motion, with wrist-watch generators or bike dynamo as the most notable examples. Analysis ofthe possible operating modes and power limits of rotating mass generators arepresented in [48] and [49]. Alternative spring designs which allow reducing theoperating frequency are also proposed by Hu [50]. Finally, several principlesof conversion of low input frequencies were presented: Kulah [51] describesa device made of a large mass oscillating at low frequencies associated withhigh frequencies oscillating cantilevers Figure 1.3(b). The mass is ferromag-netic and small cubes of metal were attached to the high frequency cantilevers.When the large magnetic mass oscillates, it alternatively catches and releasethe cantilevers which undergo free vibrations when not attract by the mass.Transduction is realized through electromagnetic means. A ratchet-pawl typesystem following a similar approach but implementing piezoelectric transduc-

1. INTRODUCTION 7

(a)

(b)

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

tion was proposed in [52]. Also, as shown by several authors [53–59], impactbased harvesters are adapted to low frequency input motion. This principle isinvestigated into details in this thesis.

1.2 Piezoelectric inertial vibration energy har-vesters

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

8 1.2 Piezoelectric inertial vibration energy harvesters

Proof mass Piezoelectric capacitor

Elastic beam Input vibration

Energy harvesting circuit

V Parasitic damping

Piezoelectric element

(a)

Energy harvesting circuit

V

Effective mass

Displacement

limit +/‐ Λ

Parasitic damping

Structural stiffness

Piezoelectric element

(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 madeof an elastic structure attached to piezoelectric elements such as patches. Asillustrated by Figure 1.4(a), the most common implementation for frequenciesof input vibrations in the range of hundreds of Hertz consists of an elasticcantilever supporting one or several piezoelectric layers sandwiched betweenmetallic electrodes. Such structures are referred to as unimorphs or bimorphs[60]. The piezoelectric capacitors are connected to a load circuit in which en-ergy 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 resultsin an inertial force on the proof mass. Extensive reviews of such piezoelectricharvesters have been proposed in the literature by Sodano [61], Anton [62] andCook-Chennault [63]. Overall, such a system can be approximated by a dash-pot mass spring system coupled to some kind of damping resulting from thepiezoelectric energy harvesting process. The corresponding mechanical modelis given in Figure 1.4(b). The first investigations of such a structure for powergeneration were proposed by Glynne-Jones [64] and Elvin [65]. Kasyap [66] pro-posed a piezoelectric vibration energy harvester designed for producing energyfrom acoustic sources. From this moment, the race for developing both theoret-ical 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 9

recharge commercial batteries. He also compared the efficiency of PZT (LeadZirconate Titanium) and of piezoelectric fibers for the same application in [61].He found that PZT was a better candidate for power generation. Piezoelec-tric materials suitable for energy harvesting were investigated in the followingyears. It was understood that the electromechanical coupling factor is a crucialparameter and should be made as high as possible. The electromechanical cou-pling factor represents the mechanical energy transformed into electrical energyor vice versa during a quasi static cycle defined in [68]. For a same material,different values of the electromechanical coupling factor are found dependingon the direction of the deformation and of the developed electrical field. Theelectromechanical coupling factor is labeled as kij . The indices i and j referrespectively to the direction of the developed electrical field and the direction ofthe deformation (in the principal frame of reference). Ikeda [69] distinguished13 different possible values of the electromechanical coupling factor but themost commonly applied modes of deformation are the 31 (direction of defor-mation perpendicular to the developed electrical field) and the 33 (direction ofdeformation parallel to the developed electrical field). Detailed discussions onthe electromechanical coupling factor are developed in the next chapters.

Ceramic PZT and its variants are still the most used materials because oftheir unequally attained electromechanical coupling factor and of the maturityof their manufacturing process. However, PZT ceramics are brittle and suscep-tible to crack propagation. Therefore, piezoelectric polymers received attentionduring the last decade. A common piezoelectric polymer is polyvinylidene fluo-ride (PVDF). PVDF is extremely flexible when compared to PZT. Lee [70, 71]developed electroded PVDF films. Mohammadi [72] developed a fiber-basedpiezoelectric (piezofiber) material consisting of PZT fibers of various diameters(15, 45, 120 and 250µm) 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 of250µm diameter embedded in a resin matrix. Because of their low electrome-chanical coupling factor, piezoelectric polymers and fibers however performworse than PZT ceramics. They have applications requiring high flexibilitysuch as the piezoelectric shoe described in Figure 1.2. Recently, the develop-ment of MEMS technologies inspired the development of materials obtained bythin film deposition processes. PZT received again most of the attention, butalternative materials such as AlN (Aluminum Nitride) [24, 74] were also pro-posed because they are easier to integrate into conventional IC manufacturingprocesses.

The investigation of high coupling piezoelectric materials is not the onlyelement required for developing high performance vibration energy harvesters.The design of the transducer itself and the understanding of its behavior arealso crucial points. Therefore, a lot of research on the design of such deviceswas done over the last decade. The simplest structure for piezoelectric vibra-tions harvesters consists in a cantilever supporting one or several piezoelectriccapacitors. The performances of these structures are optimized by maximizing


their generalized electromechanical coupling factor. The generalized electrome-chanical coupling factor represents an extension for composite structures of theelectromechanical coupling factor concept. This parameter is discussed in de-tails later. Also, the parasitic dissipations have intuitively to be reduced [75].Cho [76, 77] has shown that the generalized electromechanical coupling factorwas depending on the electrode coverage of the piezoelectric layer in the caseof rectangular membranes. In case of a cantilever, the author of this thesisdemonstrates that the optimum of the generalized electromechanical couplingfactor is obtained by choosing a particular ratio between the thicknesses of thepiezoelectric and elastic materials. It was shown by Karakaya [78] that theresidual stress can also have an influence on the generalized electromechanicalcoupling factor.

Other geometries than a simple cantilever were also studied. Mateu [79]and Roundy [43] analyzed the possibility of using a triangular cantilever ratherthan a rectangular one. They both found that a triangularly shaped piezo-electric beam delivered more power for the same input force. Ericka [80] hasalso investigated power generation from a circular membrane. Kim [81] devel-oped a novel circular configuration for power harvesting called a piezoelectric’cymbal’ in which two dome-shaped metal end-caps are bonded on either sideof a piezoelectric circular plate. Using this configuration, the stress appliedto the piezoelectric material when compressed is more evenly distributed thanin a conventional structure. In this way, the efficiency of the power harvesterincreases. Rather than modifying the geometry of the typical rectangular can-tilever, Mossi [82] proposed a configuration in which an initial stress is appliedto the cantilever. In this case, the cantilever has an initial curvature from whichthe performances in terms of power generation depend. Baker [83] proposeda new configuration in which a piezoelectric beam is compressed and fixed atboth ends with pin connections. This so-called ”bistable” system device gen-erates power by switching from one stable mode to another. Experimentalresults show that the bistable beam has a broader band of response than theclassical configuration. Cornwell [84] described the concept of attaching anauxiliary structure for being able to actively modify the resonance frequency ofthe harvester. Following the same principle, Roundy [43] further investigatedthe idea of tuning the resonant frequency of a piezoelectric device to match thefrequency of ambient vibrations. It was however shown that an active tuningsystem would never result in a net increase of the delivered power. In order toeliminate this problem, Wu [85] proposed a passive tuning approach based ontwo piezoelectric cantilevers and a micro controller. One of the piezoelectricbender was used for tuning, while the other served as generating element.

The performances of a piezoelectric energy harvester can be improved throughdevelopment of high performances materials and on a dedicated electromechan-ical design. A last important point for optimizing such a system consists inthe characteristics of the load circuit. Indeed, the basic load circuit typicallyimplemented, mainly for verification of the physical model, consists in a simpleresistor. It is however possible to improve the performances and to condi-

1. INTRODUCTION 11

tion effectively the generated power by using more elaborated load circuits. Ng[86, 87] developed a power harvesting circuit to extract energy from a cantileverbeam piezoelectric harvester. The voltage generated by the piezoelectric mate-rial is first rectified by a diode and then stored into a buffer capacitor. A volt-age monitoring circuit is connected to the buffer capacitor and releases energyfrom the capacitor in burst mode. Han [88] developed a power harvesting cir-cuit made of two stages consisting of a rectifier followed by a DC-DC converter.When experimentally compared to the traditional diode-resistor rectifier, theproposed conditioning circuit extracted over 400% more power. Ottman [89]and Lesieutre [90] implemented a switching DC-DC step-down converter in thepower harvesting circuit. It was shown that the advantages of this methoddepend on the frequencies of the input vibration. Ammar [91] developped anadaptive algorithm for controlling the duty cycle of a DC-DC buck converter.When comparing the charging time of a battery, it was demonstrated that theproposed circuit led to faster charging than the conventional rectifier. Lefeuvre[92] proposed a method in which the extraction of the electric charge from apiezoelectric device is synchronized with the system vibration. The circuit usedfor this approach is made of a rectifying diode bridge and of a flyback switchingmode DC-DC converter. A control circuit senses the voltage across the dioderectifier and activates the flyback converter when that voltage reaches its max-imum. The charges are at this moment allowed to flow in a battery. When theelectric charges on the piezoelectric element become zero, the control circuitstops the converter and the corresponding energy transfer. Such a circuit gaveexperimentally an important increase of the power when compared to a sim-ple load resistor. In further studies, Badel [93] and Guyomar [94] investigatedanother method of synchronizing the electric charge extraction with the vibra-tions of the system. This new non linear technique was labeled as ”synchronousswitch harvesting on inductor” (SSHI). The SSHI circuit contains an electronicswitch triggered on the maximum and minimum displacements of the piezoelec-tric device. The switching device associated in series with an inductor is placedin parallel or series with the piezoelectric capacitor before the rectifying diodebridge. The SSHI method was theoretically and experimentally compared to astandard circuit containing only a diode bridge rectifier and capacitor. It wasshown that the SSHI circuit is capable of multiplying the efficiency by a factor4 over the standard circuit in the case of low generalized electromechanicalcoupling factor.

All structures described above are based on the 31 deformation mode. Thecoupling coefficient of most piezoelectric materials is higher in 33 than in 31modes. Therefore, harvesters based on 33 coupling coefficients were developed.The simplest example consist in a stack of piezoelectric capacitors as illustratedby Figure 1.5(a). These structures are however very stiff and their resonancefrequencies are well above those found in an industrial environment. They aremore adapted to direct force harvesters than to inertial ones. Jeon [95], Zhou[96] and Dutoit [97] proposed an interesting alternative approach based on acantilever supporting a piezoelectric capacitor with interdigitated electrodes as


(a)

(b)

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

given in Figure 1.5(b). This electrodes configuration allows using the 33 modewith a relatively compliant structure.

To conclude this section, an overview of the performances of some of theexisting piezoelectric vibration energy harvesters based on an inertial resonantprinciple and on a cantilever configuration assuming a simple resistor as loadcircuit is given in Table 1.1. The proposed table is adapted from the reviewdescribed by Mitcheson [38]. Most harvesters presented in Table 1.1 are basedon ceramic PZT but a few results of thin film materials are also included. PZTis again the material of choice in this case, but problematic from a point ofview of process integration. Therefore, AlN was also recently investigated. Theresults proposed in Table 1.1 suggest that the performances of AlN in termsof energy harvesting are much below those of PZT. This is due to the factthat AlN devices, developed later, were not properly engineered, and it is notrelated to fundamental material properties. It was demonstrated in [24] thatfor a device operating at the same frequency, the output power delivered forPZT and AlN based harvesters is similar assuming that parasitic dissipationsare not too large. However, the former material delivers more current whilethe latter results in higher voltages.

Comparing the performances of the different devices described in Table 1.1is not easy, as the output power depends strongly on the values of the proofmass, the frequency of the input vibration and the volume of the device so thatit is difficult to perform a benchmarking of the existing systems. Therefore,Mitcheson [38] defined a metric that he designated as harvester effectivenessEff. It consists in the ratio of the generated power over the maximum achievablepower when the displacement of the proof mass me is equal to Λ (Figure 1.4b).

1. INTRODUCTION 13

Table 1.1 - Comparison of the performances of existing inertial piezoelectric vibration energyharvesters.

Generator Proof Input Input Manufacturing RawReference 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 andfrequency of the input acceleration. The values of Eff are reported in Ta-ble 1.1 when the data given in the references were sufficient to compute it.The harvester effectiveness for the micromachined devices has the same orderof 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 macroscopicPZT based devices and it appears therefore that AlN is a promising materialfor micromachined energy harvesters.

Eff = 2Measured power

meωA0Λ(1.1)

In this section, an overview of the operating principle and designs related to res-onant inertial piezoelectric vibration harvesters was presented. These systemsare adapted for high frequency and low amplitude vibrations found mainly inindustry. In the next section, inertial harvesters relevant for producing energyfrom low frequency and high amplitude motion characteristic of the humanbody are described.

1.2.2 Non resonant systems: human environment

Harvesters based on a resonant scheme are not adapted for the characteristicsof human motion and devices based on other principles have to be developedfor human powered sensor nodes. These systems are referred to as non resonantharvesters. They did not get much attention from researchers and only a fewauthors presented piezoelectric based inertial non resonant harvesters.

Umeda [53] was the first to investigate non resonant piezoelectric harvesters

14 1.3 Scope and organization of the thesis

by analyzing the transfer of energy from a steel ball impacting a piezoelectricmembrane. The piezoelectric transducer consisted of a 19 mm diameter, 0.25mm thick piezoelectric ceramic bonded to a bronze disc 0.25 mm thick with adiameter of 27 mm. This work determined that the optimum efficiency, assum-ing a purely resistive load circuit, was about 10% for an optimum load of 10kΩ. Most of the energy was returned to the ball which bounces off the trans-ducer after the initial impact. Later research further explored the feasibility ofstoring the charges developed under impact on a capacitor or battery [53]. Theoutput of the generator was connected to different values of load capacitorsvia a bridge rectifier. It was shown that the performances of the generator tocharge the capacitor depended upon the value of the latter and on its initialvoltage. The generator was also attached to nickel cadmium, nickel metal hy-dride and lithium ion batteries. The charging characteristics were found to beunaffected by the battery type. Umeda [55] proposed a commercial applicationof the principle by successfully implementing a self powered door alarm systemillustrated in Figure 1.6(a). Atsushi [99] deposited a patent for rotary struc-tures containing small steel balls attached to a central axis by straight metalwires. When the frame of the device is rotated, the balls impact on piezoelec-tric membranes located on the inner surface of the frame. Cavallier [58] andTakeuchi [59] studied experimentally an equivalent device but obtained a verylow efficiency because of a badly designed piezoelectric converter. Anecdoticapplications were also developed such as for example the piezoelectric ear ringpresented in Figure 1.6(b) which was used to power electroluminescent diodesattached to the rings. Finally, the author of this thesis determined analyticallythe optimum parameters of the piezoelectric transducer for shock excitation in[56]: it was shown that the generalized electromechanical coupling factor andthe mechanical quality factor were the most important parameters for powergeneration.

To our knowledge, a single other approach to non resonant harvesters basedon piezoelectric transduction has been described: Rastegar [52] investigated adevice where a low frequency vibrating mass spring system transfer its energyto piezoelectric cantilevers through a ratchet type interaction: as illustrated inFigure 1.7, an energy transfer teeth attached to a slow moving mass allows tosuccessively deflect and release piezoelectric benders. While the large mass isnot in the neighborhood of the cantilevers, they are allowed to vibrate freelyalong their own resonance frequency. This device was however only analyzedin 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 interestingapproach for powering sensor nodes in wireless networks. Electrical energy isprovided through conversion of the ambient vibrations or motion and the needfor the replacement of electrochemical batteries is eliminated. Many different

1. INTRODUCTION 15

(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].


approaches have been proposed for high frequency energy harvesting. However,few MEMS devices have been proposed and one of the major goals of this thesisis to manufacture and investigate such a harvester. The second goal of thisthesis is propose a device adapted to harvest energy from the low frequencymotion observed on the human body.

Both harvesters developed are based on the piezoelectric transduction effectand more particularly on piezoelectric laminate bending structures. The princi-ples of piezoelectricity are well understood and several models representing thistype of transducers already exist. However, energy harvesters have mainly beeninvestigated during the last decade and some confusions or misunderstandingare often found in the models which represent piezoelectric energy harvesters.Particularly, the approximations related to the classical model are generallynot well taken into account. Therefore, the second chapter is dedicated to thedescription of the theory of piezoelectric laminates. The equations describingthe behavior of piezoelectric benders are first derived; based on their solution,lumped models in the form of impedance matrices and equivalent electricalnetworks are proposed.

In the third chapter, the fabrication by MEMS technologies of piezoelec-tric cantilevers designed for energy harvesting in a resonant configuration ispresented. Such devices are manufactured using both thin film AlN and PZTpiezoelectric layers. The basic characterization of the produced devices consistsin determining experimentally the lumped parameters derived in Chapter 2 andis also proposed in Chapter 3. No experimental procedure for determining theseparameters has been proposed in literature. Therefore, we developed a com-plete plan of experiments to this aim. This characterization can be realized bya combination of static, transient and steady state measurements and is per-formed on the fabricated MEMS piezoelectric benders but also on commercialceramic PZT based structures. Finally, the material properties of the piezo-electric materials are extracted from the measured values of the parameters ofthe lumped model.

Chapter 4 is dedicated to the modeling, fabrication and characterizationof a low frequency inertial harvester designed for producing power from hu-man motion. The developed device is based on the impact of a rigid bodyon piezoelectric benders. No published work proposing a detailed model of apiezoelectric impact harvester including the description of the impact mecha-nism and of the resulting behavior of the piezoelectric bender exists. Therefore,a complete model of such a vibration harvester is developed and validated by aseries of measurements performed on a macroscopic prototype. Note that thisharvester is realized by conventional precision machining and the implementedpiezoelectric transducers are bought from commercial companies. An outputpower of 600 µW is obtained (using a resistive load) for a device of dimensions3.5*2*2 cm weighting 60 g placed on the hand of a person and shaken at afrequency of approximately 10 Hz with 10 cm amplitude. Also, a power of 47µW is measured when the harvester is rotated of 180 each second.

1. INTRODUCTION 17

In Chapter 5, the model of inertial resonant piezoelectric harvesters adaptedto high frequencies corresponding to a machine environment is developed andanalyzed 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 PZTMEMS resonant harvesters are characterized experimentally. It is found thatthe developed model results in a good approximation of the measured data.Raw output powers in the range of 40 µW are measured, which is enough topower simple electronic applications. Finally, some elements related to thefuture investigations are given: particularly, non linear effects due to largeinput vibrations or parasitic dissipations related to the presence of a packagearound the harvesters are discussed and investigated experimentally.

Chapter2Theory and lumped model ofpiezoelectric laminated beams

Piezoelectric laminated beams constitute the transduction element betweenmechanical and electrical energy in the two types of investigated energy har-vesters. The model of such transducers is developed in this chapter. It isorganized as follow: in the first section, a short literature review and historyof piezoelectricity is proposed, followed by the derivation of its linear constitu-tive equations. Common non linear effects and intrinsic dissipations occurringin piezoelectric materials are also discussed. The section is concluded by thepresentation of the piezoelectric materials used in this thesis and of the cor-responding simplifications of the constitutive relations. The second sectiondescribes the derivation of the equations governing the dynamics of multi lay-ered beam structures. Finally, in a third section, analytical solutions of theprevious equations are developed for the particular situation of a cantileverloaded by a distributed mass. These solutions are arranged in the form ofan impedance matrix and an electrical equivalent network. Some parts of thederivations described can be found in classical textbooks. However, the field ofpiezoelectric harvesters is a recent area of research and a publication proposinga detailed derivation of the piezoelectric multilayer beam equations in view ofsuch applications does not yet exist.

19

202.1 History, basic definitions and linear constitutive equations of

piezoelectricity

Si+

Si+

Si+

O‐ O‐

O‐ Si+

Si+

Si+

O‐ O‐

O‐

Force

Voltage

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

2.1 History, basic definitions and linear consti-tutive equations of piezoelectricity

2.1.1 History and basic definitions

Coulomb proposed in the 18th century a conjecture which states that electric-ity 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 be-tween the electrical charges created by friction or contact electricity and thoseresulting from a possible electromechanical phenomenon. The Curies were,in 1880, the firsts to demonstrate a relation between the symmetries in crys-talline materials and the electrical charges appearing at the surface of somecrystals when mechanically stressed. Their experiments were performed onRochelle salt, tourmaline and quartz amongst others. The effect discovered bythe Curies is commonly referred to as the direct piezoelectric effect, i.e. electri-cal charges results from mechanical efforts. The so-called converse piezoelectriceffect (mechanical stresses results from applied electrical field) was mathemat-ically derived from thermodynamics by Lippmann in 1881. The same year, theCurie brothers confirmed experimentally the existence of the converse effectand started developing a few laboratory applications. In 1893, Lord Kelvinproposed an atomic model to explain the observed phenomena. The piezoelec-tric effect as understood by Kelvin is illustrated in Figure 2.1 considering aquartz unit cell. When no mechanical stress is applied to the crystal, the pos-itive and negative electrical charges present on the different atoms share thesame barycenter, so that no net electric displacement or field is observed at thesurface of the cell. The barycenter of the positive and negative charges does notcoincide when the cell of quartz is deformed, so that an electrical polarizationis developed at the surface of the crystal.

It was understood by Kelvin that piezoelectricity occurs only in non centro-symmetric crystals. The complete definition of the crystal classes in whichpiezoelectric effects occur was published by Voigt in 1910 (Lehrbuch der Kristall-physik), on the basis of the work done by Duhem, Pockels and Neumann. As

2. THEORY AND LUMPED MODEL OF PIEZOELECTRIC LAMINATEDBEAMS 21

32 crystal symmetry groups

11 centrosymmetric

Non piezoelectric

21 non centrosymmetric

1 non piezoelectric 20 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-symmetriccrystal classes, 20 of them lead to piezoelectric effects. It was also shown that10 of the piezoelectric crystal classes exhibit pyroelectric effects, i.e. they pos-sess a spontaneous electrical polarization which depends on the temperature.Above a certain temperature referred to as Curie temperature, a pyroelectriccrystal undergoes a phase transition and does no longer exhibit spontaneouspolarization. In the next decades, it was discovered that the spontaneous po-larization can be reversed under the action of a strong applied electrical field insome of the pyroelectric crystals. By analogy with ferromagnetism, this effecthas been designated as ferroelectricity.

The work of Voigt became a standard reference for piezoelectricity andthe tensor algebra and notations introduced are today still widely in use. Thecomplexity of the mathematics required to design piezoelectric systems led to adelay in the development of practical applications. The first engineering appli-cation of the piezoelectric effect was made by Langevin in 1917. In the WorldWar I context, his goal was to develop a system able to detect submarinesand his work on piezoelectric transducers led to the development of SONAR(SOund Navigation And Ranging) and to further advances in the field of ul-trasonics. During the 1920-40’s, most of the classic piezoelectric applications(microphones, accelerometers, ultrasonic transducers, bender element actua-tors, phonograph pick-ups, signal filters, etc.) were conceived and brought intopractice. Some important names associated with this period are Cady, whodeveloped the quartz oscillator and published reference books in the field [100–103], Butterworth and Van Dyke, who proposed the first electrical networkmodel of a piezoelectric resonator thus simplifying the design of piezoelectrictransducers [104–106], Mason who extended the electromechanical represen-tation of piezoelectric systems and developed alternative formulations of the


piezoelectricity

piezoelectric theory [107–111].

Until the 40’s, piezoelectricity had only been observed in crystals. A tech-nologic revolution occurred when it was discovered during World War II (inthe U.S., Japan and the Soviet Union), that certain ceramic materials exhib-ited dielectric and piezoelectric constants having the same order of magnitudethan those of common cut crystals. Piezoelectric ceramics are prepared byheat treatment of metallic oxide powders and have a polycrystalline structure.They require an electrical poling process in order to exhibit a piezoelectric ef-fect. Indeed, as described in Figure 2.3(a), the material consists before poling ofmisaligned domains. When the material is deformed, electrical charges appearslocally at the boundaries between the domains, but due to charge cancellation,no net piezoelectric effect is observed at the surface of the sample. The polingprocess is illustrated in Figure 2.3(b): metallic electrodes are attached at twoopposite faces of the sample and a strong electrical field is applied on the elec-trodes so that the different domains partially align their polar axis along theelectrical field. After removing the electrical field, the different domains keepa certain alignment and a noticeable piezoelectric effect is observed. Note thatthe poling process is only relevant for polycrystalline ferroelectric materials.A piezoelectric but not ferroelectric polycrystalline material (as for examplequartz) with randomly oriented grains can not be poled; it can only exhibitmacroscopic piezoelectric properties if the growth orientation of the differentgrains is well controlled. Independently of the required poling step, manufac-turing technologies much cheaper than those existing for growing crystals weredeveloped for piezoelectric ceramics in the 40’s and a renewed interest was ob-served at the level of scientific and engineering research. In the following years,the barium titanate and lead zirconate titanate (PZT) class of materials weredeveloped. During that period, the relations between the electromechanicalcoupling and the perovskyte crystalline structure corresponding to PZT mate-rial were also understood. Doping of these materials with metallic impuritieswas also investigated and successfully implemented. Some of the most impor-tant 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 successeswere achieved in Japan. The field of applied piezoelectricity shifted from purelymilitary and academic interests to a wide variety of everyday life applications asfor example smoke and intrusion alarms or TV remote controls. The 80’s havethen seen the opening of new commercial markets for piezoelectric applicationsin each part of the world. Applications were found in the field of automotive,actuation, aeronautics and more recently of energy harvesting. A new field ofapplication for piezoelectric materials has been induced recently by the industryof MEMS. In terms of energy density, the piezoelectric effect is not affectedby miniaturization, so that it constitutes a principle of choice for designingsmall scales actuators, sensors or vibration energy harvesters. Strong effortshave been done in order to integrate piezoelectric materials in the form ofthick or thin films into silicon wafer based batch processes. Non ferroelectric


Monocrystal with single polar axis

Polycrystal with random polar axis

(a)

Random dipoles

Polarization

Surviving polarity

Electrodes

(b)

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

and piezoelectric materials such as for example aluminum nitride (AlN) or zincoxide (ZnO) are fairly easy to integrate into a conventional IC process flow. Onthe opposite, strong difficulties are encountered for PZT type materials whichrequire high temperature processing. Furthermore, contamination problemsare encountered with PZT.

In modern applications, piezoelectric transducers exist in a wide variety ofshapes and are excited in a wide variety of vibrations modes. Some of theseshapes, modes of vibration, corresponding range of frequency and applicationsare presented in Figure 2.4. Flexural vibrations piezoelectric transducers (can-tilever, membranes) allow designing systems performing in a range of a fewHz till tens of kHz. Typical applications related to this mode of vibration arebuzzer, cooler, motion actuators and vibration energy harvesters as it will bediscussed in this thesis. Length, thickness and area vibrations of piezoelectricplates or discs are used for filters and resonators in the range of tens of kHztill a few MHz. Filters and resonators performing from a MHz till tens of MHzare obtained by designing thickness shear or thickness trapped vibration. Fi-nally, in order to operate at higher frequencies (above GHz), surface acousticwaves are implemented. The two types of vibration energy harvesters that areanalyzed in this thesis are based on flexural piezoelectric elements and moreprecisely on beam structures. In the remainder of this chapter, the basis of themodel of the piezoelectric benders which will be implemented later to analyzeand optimize the performances of the energy harvesters is developed.


piezoelectricity

Vibration mode Frequency (Hz) 1K 10K 100K 1M 10M 100M 1G

Applications

Flexural vibrations

Piezoelectric buzzers

Lengthwise vibrations

kHz filters

Area vibrations

kHz resonators

Radius vibrations

kHz resonators

Thickness shear

vibrations

MHz filters

Thickness trapped vibrations

MHz resonators

Surface acoustic wave

SAW filters

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 classi-cal mechanics and thermodynamics. The approach proposed in this subsectionis based on the classical continuum approach. Relatively large transducers aredealt with (≥5 mm2) and there are no particular needs of describing the mi-croscopic theory of piezoelectricity (such a theory was proposed by Born andHuang [112]). The different steps required to derive the classical constitutiveequations of piezoelectricity are given. Details have been extensively given inthe literature as for example by Mason [111] or Cady [103].

By definition, piezoelectric materials exhibit elastic, dielectric and coupledelastic-dielectric phenomena, so that it is necessary to discuss first the elasticand dielectric continuum. The classical elastic continuum is described by thewell-known relations between stresses and strains in a solid. For a Cartesianelement of volume, the force acting on each surface of the cube can be decom-posed into three components directed along the different axis x, y and z of theCartesian coordinates system. Weak equilibrium considerations (i.e. the bodyforces are neglected) impose that the forces applied on two opposite surfacescancel out so that from the 18 stress components obtained from the decompo-sition 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 torespectively the x, y and z axis for the convention used here) tend to change


the elementary volume without distorting it (tensile stresses). The three othercomponents of the stress tensor tend to distort the elementary cube of materialwithout changing its volume (shear stresses). The stresses can be consideredas intensive thermodynamic variables. To each stress component correspondsan extensive variable usually defined as the elongation or distortion per unitlength along a relevant axis, also known as strain Si. The relation betweenthe stresses and strains in the absence of piezoelectric effect, also known asconstitutive equation of elasticity, depends on the properties of the materialconsidered. The constitutive relation is formally established using thermody-namics principles considering adiabatic or isothermal (differences between thematerial constants in adiabatic or isothermal situations are negligible for solids[110]) reversible transformations. For linear elasticity, the material propertiestensor relating the stresses to the strains is denominated as the compliancetensor (rank 2, symmetric). Its components are labeled as sij with i=1..6 andj=1..6.

The fundamental laws governing the physics of dielectrics are the quasi-static form of Maxwell’s equations. These equations involve 4 thermodynamicvariables which are the electrical field, the electrical displacement, the mag-netic field and the magnetic strength. In the classical theory of dielectrics, themagnetic field and its conjugated variable are neglected (because of the dimen-sions of the considered system and of the relatively low involved frequencies),so that the dielectric continuum is totally described by the electrical field Eand displacement D. In case of linear dielectrics, the electrical field is relatedto 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 thestate of the system in an adiabatic or isothermal situation. From thermody-namics, each variable can be expressed as an explicit function of two othersin the same way as proposed for the dielectric and elastic continuum. Be-cause of the number of variables, it is possible to derive different forms of theconstitutive equations. Throughout this thesis the convention and notationsproposed in the IEEE standards on piezoelectricity [68] are used. The com-mon strain-charge form is given in (2.1) and (2.2) using the compressed matrixnotation, in which sEij represents a compliance term of the material under con-stant electrical field (this condition is indicated by the superscript E), dij isa charge constant relating the amount of dielectric displacement created bya given stress and εTij is the electrical permittivity of the medium under con-stant 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 thefollowing parts of this chapter.

Si = sEijTj + dijEj (2.1)

Di = dijTj + εTijEj (2.2)


piezoelectricity

The order 3 tensor containing the electromechanical coupling terms can berepresented by a 6*3 matrix so that one can obtain the matrix expanded formof the converse and direct linear constitutive equations of piezoelectricity.

S1

S2

S3

S4

S5

S6

=

sE11 sE12 sE13 sE14 sE15 sE16






T1

T2

T3

T4

T5

T6

+

d11 d21 d31

d12 d22 d32

d13 d23 d33

d14 d24 d34

d15 d25 d35

d16 d26 d36

E1

E2

E3

(2.3)

D1

D2

D3

=

d11 d12 d13 d14 d15 d16

d21 d22 d23 d24 d25 d26

d31 d32 d33 d34 d35 d36

T1

T2

T3

T4

T5

T6

+

εT11 εT12 εT13

εT12 εT22 εT23

εT13 εT23 εT33

E1

E2

E3

(2.4)

Some common alternative forms of the constitutive equations of piezoelec-tricity are given below. The superscripts S, T, D and E represent respectively aconstant strain, stress, electric displacement and electrical field condition. Thedifferences between the values of the material properties under these differentconditions are not negligible as it was the case under adiabatic and isothermalsituations. It is then important to use the superscripts describing these condi-tions. Explicit relations between some of the material constants are given laterwhen dealing with the particular case of laminated piezoelectric beams.

Ti = cEijSj − eijEjDi = eijSj + εSijEj

(2.5)

Ti = cDijSj − hijDj

Ei = −hijTj + βSijDj(2.6)

Si = sDijTj + gijDj

Ei = −gijTj + βTijDj(2.7)


2.1.3 Dissipative and non linear effects in piezoelectricmaterials

Different kinds of dissipative and non linear effects occur in piezoelectric mate-rials. Some are common to both ferroelectric and non ferroelectric piezoelectricmaterials, others are peculiar to ferroelectric materials. The losses have beenneglected until here by considering a reversible, adiabatic or isothermal situa-tion.

Linear behaving dissipation mechanisms induce a phase shift between theinput and the response of the system. This situation is easily described froma formal point of view. The dissipations and the resulting phase shift between”input and output” are introduced in the system by setting complex valuedcomponents in the elastic, dielectric and electromechanical tensors. The useof complex components to represent losses at a phenomenological level canbe justified from relaxation and domain wall motion phenomena described bymicroscopic theories [113–115]. A detailed analysis of the implications of thismethodology on the modeling of piezoelectric materials is proposed by Mezher-itsky [116].

A first type of non linear effect observed in both ferroelectric and non fer-roelectric piezoelectric materials comes from the high order terms neglected inthe derivation of the linear equations of piezoelectricity. Some of them can nolonger be neglected when relatively large signal inputs are considered. In theparticular case of ferroelectric materials, a fundamental reorganization of themicrostructure occurs under high signals and it gives rise to hysteretic effects.As the poling process necessary to obtain piezoelectric ceramics is based on thiseffect, it is important to give a concise and simple explanation of the observedhysteretic behavior. A detailed analysis of hysteresis effects in piezoelectric ma-terials is proposed by Damjanovic [117]. As introduced previously, ferroelectricmaterials are a special case of pyroelectric materials possessing a spontaneouspolarization. In ferroelectrics, the spontaneous polarization possesses two ther-modynamic states of equilibrium and its direction can be reversed through theapplication of a strong electrical field. Each grain in a ferroelectric polycrys-talline material is made of several so-called Weiss domains consisting of a col-lection of crystalline cells with aligned spontaneous polarization (Figure 2.5).Before a poling process is carried on, the misalignment of the different Weissdomain in a crystal results in a low value of the macroscopic polarization, ifany. This effect has even more influence on ceramics, as the influence of themisalignment of Weiss domains is reinforced by the different orientations of thegrains.

The influence of the poling process on the microstructure of a ferroelectricceramic is illustrated by Figure 2.6. In a ”virgin” state, i.e. before any elec-trical field has been applied to the prepared material, the polarization in thedifferent Weiss domains of each grain are misaligned, so that no net electricalcharges are observed at the surface of the ceramics. This situation corresponds


piezoelectricity

Figure 2.5 - Weiss domains in a grain.

to the point a in Figure 2.6. As the applied electrical field increases, the ma-terial behaves as a dielectric (linear or non linear) in the path (ab). More andmore different grains and domains tend to align their polarization during thisphase until saturation occurs at point b. The maximum number of aligned do-mains is obtained at this point and a further increase of the electrical field doesnot affect the surface charges. The influence of the irreversibility of the rear-rangement process is observed when one starts decreasing the applied electricalfield: the electrical displacement does no longer follow the path (ab). Becauseof the strong coupling that now exists between the microscopic polarizationsand because of the thermodynamic stability of this state, the different domainsoppose a strong resistance to any change in their orientations. Therefore, thematerial conserves a permanent (also known as remanent) polarization whenthe electrical field is set to zero (point c): the material is now polarized. Be-tween the point b and d, the material again behaves as a normal dielectric. Inorder to reverse the orientation of the electrical dipoles, the magnitude of theapplied electrical field should be decreased further till reaching the so calledcoercive field Ec corresponding to point d in Figure 2.6: the system reachesat this moment an unstable state in which the dipoles already reoriented im-poses a cascade reversal of polarization on the remaining of the unmodifieddomains and the system reaches the state described by the point e. As in thepreviously discussed saturation state, a further decrease of the electrical fielddoes not influence the net charges developed. A behavior equivalent to the onepreviously described occurs when the amplitude of the applied electrical fieldagain increases and a new reversal of polarization is observed when the positivevalue of the coercive field is reached. Practically, several cycles are applied onprepared ceramics in order to increase the values of the remanent polarizationand piezoelectric properties.

Some of the characteristic dissipative and non linear effects occurring inpiezoelectric and ferroelectric materials have been briefly discussed in this para-graph for sake of completeness. However, for the applications presented in thisthesis, the applied and resulting fields are small with respect to the coercive


D

Ea

b

cd

e

Ec ‐Ec

Dr

‐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 lineareffects due to hysteresis and to high order terms in the constitutive equationscan be neglected. In other words, it is considered in the following of this workthat a relevant model of our devices can be developed under the assumption ofthe linear constitutive equations given in (2.3) and (2.4).

2.1.4 Relevant piezoelectric materials and correspondingsimplifications of the constitutive equations

The general characteristics of the piezoelectric materials implemented in thedevices developed for this thesis and the corresponding simplifications of theconstitutive equations are presented in this section. Depending on the consid-ered type of energy harvester, different families of piezoelectric materials areimplemented for the manufacturing of prototypes. For the human environmentenergy harvester, solely commercial PZT obtained by conventional ceramicstechnologies are used. For the machine environment energy harvester, twodifferent piezoelectric materials consisting of commercial thin film PZT andthin film AlN grown at IMEC, Belgium (http://www2.imec.be) and the HolstCentre, The Netherlands (http://www.holstcentre.com/) are used. The goalof this work is to develop prototypes of energy harvesters based on relevantphenomenological models, so that little time is devoted to the study and op-timization of the manufacturing of the different materials. However, as thepresented thesis is done in parallel with the development of a deposition pro-cess for thin film AlN dedicated to energy harvesting situation, some words andfurther analysis of the produced AlN are proposed in Chapter 3. On the otherhand, all PZT based materials used for this work are obtained from commercialcompanies and no details about their processing are given.

The materials belonging to the PZT family exhibit a perovskyte type crys-tal structure. They are solid solution of general formula PbZr1−xTixO3 and


piezoelectricity

the ratio of Ti4+ to Zr4+ ions determines the phase of the solution. The ratiois generally chosen close to 50% so that the so called morphotropic bound-ary is reached. It has been shown that the piezoelectric properties are max-imal for this composition [119]. The underlying phenomena occurring dur-ing this phase transition are yet not well understood. It has even be sug-gested that a new phase exists at the morphotropic boundary [120]. Two PZTmaterials are implemented: thick ceramic PZT obtained from Piezo Inc, US(http://www.piezo.com) and thin film PZT deposited on silicon wafers fromInostek, Korea (http://inostek.com/). The former material is a ferroelectricceramic. Before poling, it is isotropic because of the randomly aligned grains.The process of electrical poling influences the texture of the film and induceselements of symmetry similar to those of hexagonal polar crystal of the symme-try group 6mm [121]. The latter material is grown by the sol gel method andconsists in a columnar polycrystalline arrangement. The growth of the mate-rial is controlled so that the polar axes of the grains are aligned in the samedirection and the averaged anisotropy of this type of PZT is also equivalent tothe one of crystals belonging to the 6mm group [122].

PZT thin films exhibits high piezoelectric constants but are not easily in-tegrated into conventional CMOS process, which have to be considered if it isdesired to integrate the conditioning electronics with the manufacturing of thedeveloped MEMS harvesters. These difficulties result from the high requiredprocessing temperatures and from contamination problems. For this reason,other thin films materials, such as zinc oxide and aluminum nitride, were in-vestigated in the recent years. AlN is a good dielectric which has been reportedto grow for temperatures between 100 and 900C [123]). The suitability of itspiezoelectric properties depends of the application considered: AlN is a poorcandidate for actuators, but relevant for sensing or energy harvesting devices,as demonstrated along this thesis. Research on AlN deposition processes witha focus on energy harvesting is ongoing at the Holst Centre and the AlN basedharvesters studied during this work are manufactured by this institute. A SEMphotography of an example of deposited AlN is given in Figure 2.7. As for PZTthin films, the growth direction of the grains is perpendicular to the seed plane.Again, symmetry elements similar to the ones found in hexagonal crystals areobserved [122].

The different piezoelectric materials used belong to the same symmetrygroup. Symmetry considerations allow reducing the number of independentcomponents in the material properties tensor [124] and the constitutive equa-tions 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 thepiezoelectric layers. It is considered that they are either isotropic (brass alloy,stainless steel) or transverse isotropic (silicon) so that their compliance matricescan be written in the same form as the one given in (2.8) (with s33=s11 ands12=s13 for the isotropic case).


AlN

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

S1

S2

S3

S4

S5

S6

=

sE11 sE12 sE13 0 0 0sE12 sE11 sE13 0 0 0sE13 sE13 sE33 0 0 00 0 0 sE44 0 00 0 0 0 sE55 00 0 0 0 0 sE66

T1

T2

T3

T4

T5

T6

+

0 0 d31

0 0 d31

0 0 d33

0 d15 0d15 0 00 0 0

E1

E2

E3

(2.8)

D1

D2

D3

=

0 0 0 0 d15 00 0 0 d15 0 0d31 d31 d33 0 0 0

T1

T2

T3

T4

T5

T6

+

εT11 0 00 εT11 00 0 εT33

E1

E2

E3

(2.9)

32 2.2 The constitutive equations of piezoelectric laminated beams

2.2 The constitutive equations of piezoelectriclaminated beams

The equations describing the problem at the mesoscopic scale have been de-rived. A macroscopic model is required for analyzing the performances of thepiezoelectric transducers implemented in our energy harvesters. Representa-tions of the macroscopic behavior of simple piezoelectric vibrators was initiatedby the work of Mason [109–111], Butterworth [104] and van Dyke [105, 106].However, few relevant models were proposed for piezoelectric laminated beamuntil the work of Smits [125, 126], who initiated a renewal of interest for thetheoretical analysis of such structures. In the following year, a large amount ofpublications adapting and slightly refining the results of Smits were proposed[127–130].

Most of the derivations proposed in this section are based on the workof Smits, but new theoretical elements are also included. In the following,the equations describing the macroscopic behavior of elastic laminated beamare first derived. Then, the obtained equations are adapted for the case oflaminated 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 conditionsof the beam allowing a reduction of the number of relevant stress and straincomponents are first defined. Second, the assumed distribution of the stressand strains is presented. Then, the mesoscopic variable stress and strains arerelated to corresponding macroscopic variables (efforts and displacements) us-ing the stress resultants method and kinematic considerations in the frameworkof a small displacement approximation. The equation governing the dynamicalbehavior of an elastic beam is derived by introducing the constitutive equationsof elasticity in the obtained relations. Finally, the results are extended to in-homogeneous 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 tobe 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 thestructure belonging to the z axis of the chosen coordinates system. As hs ismuch smaller than the two other dimensions, the thickness shear and strainscan be neglected: S3=T3=0. Second, a beam is characterized by the boundaryconditions applied on the lateral faces parallel to the yz plane: both faces areconstrained by a couple of compatible boundary conditions which can not beboth free. At the opposite side, the two faces belonging to the zx plane arenot constrained and free to deform, except in the close neighborhood of theyz constrained surfaces. The most common example of a compatible couple ofboundary conditions corresponds to a clamped-free (cantilever) configuration.


Free face

Constrained face

Loading P(x)z

xy

W

(a)

P(x)

hs

x

z

z0

(b)

Figure 2.8 - (a) Schematic representing the definition of a mechanical beam, (b) simplified2D 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 direc-tion perpendicular to the constrained faces and the width W as illustrated inFigure 2.8(a). It is considered in beam theory that the distributed mechanicalload P is applied on the faces belonging to the xy plane. In the classical anal-ysis of mechanical beams, it is also assumed that the cross sections along yzplanes retains the same shape and are symmetric with respect to the xz planeat W/2. These cross sections are taken as rectangular. The applied load isuniform along the width of the structure and is directed solely along the z axisin 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 inthe 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, sothat the faces in the xz plane are free to expand (except close to the boundaryconditions) and the stress developed along y is negligible: T2=0. This conditionis referred to as plain stress. In a following section, a simple refinement isproposed to adapt the problem to plain strain condition. From all previoussimplifications, the problem is independent of the position on the y axis. Thebeam can be described by the 2D model of Figure 2.8(b). Furthermore, theconstitutive equations of elasticity can be reduced to (2.10). The reductionof the coordinate dependency of the different stress and strains is also given.Note that the lateral strain S2=s12T1 is not a variable of primary interest inthe analysis of beams.[

S1 (x, z)S5 (x, z)

]=[s11 00 s55

] [T1 (x, z)T5 (x, z)

](2.10)

The modeling work is developed assuming a small displacement approxima-tion: the spatial frame of reference (Eulerian formulation) is supposed to re-main equivalent to the material frame of reference (Lagrangian formulation) inthe deformed state. An important consideration resulting from this assump-tion consists in the fact that an effort applied in a direction perpendicular totop surface of the beam in the non deformed state is supposed to remain per-


M(x) T1(x) T1(x+dx)

P(x)dx

T5(x+dx)J(x+dx)

J(x) P(x)dx

z x

T5(x) hs z0

dx dx

P(x)

hs

x

z z0

z0

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

pendicular to this surface in the deformed state. The 2D beam described byFigure 2.8(b) can conceptually be represented as a superposition of straightlongitudinal fibers deformed into continuous curves under the application ofa load. The assumption of continuous displacements and of a curved shapesuggests 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 isdiminished, i.e. the sign of the strain depends of the position along z. By ana-lyzing the curvature of the collection of fibers, it is possible to show first thatthe maximum and minimum longitudinal tensile strains S1 occur in the mostouter fibers of the beam, and second that a specific fiber referred to as neutralaxis undergoes zero longitudinal strain. The neutral axis location is denomi-nated as z0. In homogeneous beams, the neutral axis is the central fiber of thestructure. The theory of Timoshenko [132–134] is implemented for the nextderivations, i.e. it is assumed that the variation of the longitudinal and shearstrains S1 and S5 show respectively a linear and constant behavior along thethickness of the beam, with S1 being zero on the neutral axis. As illustrated byFigure 2.9, the stresses T1 and T5 can be represented in the same way becauseof the considered simplified linear constitutive equations.

To determine the equations describing the behavior of the beam, equilibriumconsiderations applied on a differential cross sectional element of volume Whsdxof the beam have first to be elaborated. An applied mechanical load inducesstresses in the beam. A common method of establishing relations between thestresses and the load consists in introducing the so called stress resultants: thedistribution of the stresses in the beam is statically equivalent to forces andmoments applied on the different cross sections. Because of the restrictionsimposed on the geometry and loading of the beam, the only non zero stress


resultants are a force directed along the z axis and a moment inducing a rota-tion around y. These elements are referred to as shear force Jz and bendingmoment My respectively. The bending moment is the static equivalent of thedistribution of the longitudinal stress T1, while the shear force is the one of thedistribution of the shear strain T5. The bending moment and shear force canbe written as (2.11) and (2.12). The origin of the z axis is taken at the lowestfiber 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 byRoark [131] in the small displacement approximation. In the remainder of thisthesis, the classical sign convention defined by Timoshenko [133] for the shearforce and bending moment in a beam is used: if the fibers located above theneutral axis undergo compression, My is defined as positive; if the shear forcesinduce a negative vertical motion of the cross section, Jz is defined as positive.

My (x) = ±Whs∫0

zT1 (x, z) dz (2.11)

Jz (x) = ±Whs∫0

T5 (x, z) dz (2.12)

A relation between the stresses and the relevant intensive macroscopic vari-ables is obtained. The macroscopic variables corresponding to the tensile andshear strain S1 and S5 are classically defined as the displacement and distortionfields of a segment belonging to the elastic continuum. The kinematics of a por-tion of a beam according to Timoshenko theory is illustrated by Figure 2.10. Inthis theory, it is assumed that a segment (belonging to the yz plane) of materialremains a segment after deformation. Practically, the initial segment is rathertransformed into a curve by the deformation process, but Timoshenko’s theoryhas been proven to give a good approximation of the behavior of beams. Therelations between displacements and strains are obtained by a geometrical anal-ysis of the transformation undergone by the previously discussed segment. Letsdefine 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 thepoint A belongs to the neutral axis. After deformation, the points A and B arerespectively 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 negligi-ble. 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 smalldeformations, θ0 can be approximated by dw/dx-θy. Positive angles correspondto counter clockwise rotations. A geometrical analysis of Figure 2.10 combinedwith the previous assumptions leads to the so called kinematic equation of thebeam.

u (x, z) ≈ u (x, z0)− (z − z0)dw (x)dx

(2.13)


A

B

A’

B’

u(x,z)

w(x,z)

u(x,z0)

w(x,z0)

θy θ0

Neutral axis before and after deformation

Portion of a beam before and after deformation

x

z

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 thex direction and can be written as S1=du/dx+(z-z0)dθy/dx. Combining thisrelation with (2.13) leads to

S1 (x, z) = − (z − z0)(d2w (x)dx2

− dθy (x)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 isthe main displacement of interest in the analysis of beams and not attention isgiven to the longitudinal one u.

Both relations between the stresses and the bending moment/shear forceand between the strains and displacement fields are obtained. It is now pos-sible to link the displacement to the mechanical efforts by introducing theconstitutive equations of elasticity relating stress and strain in the previousresults. Combining (2.10), (2.11), (2.12) and (2.14):

My (x) =(d2w (x)dx2

− dθy (x)dx

)W

s11

hs∫0

z (z − z0)dz (2.15)

Jz (x) = −Whss55

θy (x) (2.16)

In (2.15), the termhs∫0

z (z − z0)dz (2.17)


is referred to as the area moment of inertia Ii of a beam cross section. Forhomogeneous rectangular beams, Ii=Whs3/12. In order to obtain a theoreticalmodel predicting accurately the behavior of a beam, Timoshenko has shownthat the compliance term s55 involved in the definition of the shear force givenin (2.16) has to be weighted by a factor κ depending on the cross section ofthe beam. This correction is required to compensate the error induced on thecalculations 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 ob-tained by combining (2.15)and (2.16) with dynamical equilibrium considera-tions on the elements of Figure 2.9 for the vertical and rotary motions. Theequations can be rearranged in (2.18) and (2.19), in which ρ represents thedensity (assumed uniform) of the material and A is the cross section area. Anexternal viscous type damping is present through an additional term Dadw/dt,while the internal damping is represented through complex values of the com-pliances involved in the equation. The right-hand side of (2.18) contains allelements related to an eventual distributed load applied on the surface of thebeam. In the case of concentrated load, all vanish. The terms related to theshear correction are easily identifiable through the presence of the complianceterm s55. The larger the value of this compliance, the smaller the influenceof the shear effects. The coupled time-space derivative term and the one indx4w/dt4 are related to the rotary inertia of the cross sections of the beam. If aquasi static situation without distributed load is considered, one easily obtainsthe well known static beam equation Ii/s11dx4w/dt4=0. According to (2.19),if shear is considered in the static and load free situation, it becomes constantover the length of the beam. In the considered case, it can be shown for ex-ample by FEM simulations that the value of the shear is constant over most ofthe length of the beam, but varies abruptly near the constrained boundaries.However, the differences in the prediction of the general behavior due to thiseffect are negligible.

P (x, t)− Iis11

κs55

A

∂2P (x, t)∂x2

+ ρIiκs55

A

∂2P (x, t)∂t2

=Iis11

∂4w (x, t)∂x4

− ρIi(

1 +κs55

s11

)∂4w (x, t)∂t2∂x2

+ρA∂2w (x, t)

∂t2+Da

∂w (x, t)∂t

(2.18)

1κs55

∂θy (x, t)∂x

+ P (x, t) = ρ∂2w (x, t)

∂t2(2.19)

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


h1

h2 z0 T5

S5 S1

T1

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 islimited to a simple theory in which the phenomena occurring in the interfaceare neglected and it is assumed that the different layers do not slip towards eachother so that the shear and longitudinal strains are continuous along the thick-ness of the beam. The thicknesses of the interfaces are considered negligible.In this simple theory, the shear and longitudinal stresses are discontinuous atthe interfaces, because of the different compliances of the materials. Further-more, the stress gradient of T1 and the value of T5 (constant in Timoshenkotheory) are not equal in the different layers. This situation is illustrated byFigure 2.11. The relevant compliances terms are labeled s1,11 and s1,55 for thefirst 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 de-termined by imposing the conditions that the longitudinal force resulting fromthe distribution of the stress T1 over a cross section is zero and that T1(z0)=0from the definition of the neutral axis. Simple closed form expressions of z0 areproposed by Weinberg [129]. An explicit form of this parameter is given laterfor the particular structures studied.

In the framework of the proposed theory, the single layer beam problem isadapted to multilayer by reconsidering the expression of the bending momentand shear force given in (2.15)and (2.16). These two equations can be rewrittenfor the structure depicted in Figure 2.11 as

My (x) = W

(d2w (x)dx2

− dθy (x)dx

) 1s1,11

h1∫0

z (z − z0)dz +1

s2,11

h2∫h1

z (z − z0)dz

(2.20)

Jz (x) = −Wθy (x)(

h1

κ1s1,55+

h2

κ2s2,55

)(2.21)

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


longitudinal compliance and the quotient of the cross section area over theeffective shear compliance.

EIeq = W

n∑i=1

1si,11

hi∫hi−1

z (z − z0)dz (2.22)

GAeq = W

n∑i=1

hiκisi,55

(2.23)

Following these definitions, the equations describing the dynamical equilibriumof a homogeneous elastic beam given in (2.18) and (2.19) can finally be rewrittenfor multilayered structures as

P (x, t)− EIeqGAeq

∂2P (x, t)∂x2

+ ρIi

GAeq

∂2P (x, t)∂t2

=EIeq∂4w (x, t)∂x4

− ρ(Ii +A

EIeqGAeq

)∂4w (x, t)∂t2∂x2

+ρ2IiA

GAeq

∂4w (x, t)∂t4

+ ρA∂2w (x, t)

∂t2+Da

∂w (x, t)∂t

(2.24)

GAeqA

∂θy (x, t)∂x

+ P (x, t) = 0 (2.25)

in which ρ represents the average density along the thickness of the laminate.Note that the expressions A and Ii representing the cross section area and thearea moment of inertia now have to be written considering the total thicknessof the structure.

The equations describing the behavior of an elastic beam in terms of relevantmacroscopic variables are obtained by combining the constitutive equations ofelasticity with equilibrium and kinematics consideration. In the next subsec-tion, an equivalent approach for deriving the representative equations of thedynamics of a piezoelectric laminated beam is followed.

2.2.2 Piezoelectric laminated beams

In the previous subsection, the constitutive equations of elasticity are sim-plified and an equation describing the macroscopic behavior of a multilayerelastic beam is derived. Equivalent manipulations are performed in order toderive the constitutive equations of beams made of a superposition of elas-tic and piezoelectric rectangular layers. In this section, the simplifications ofthe constitutive equations of piezoelectricity related to the particular typesof piezoelectric laminates studied are presented. Then, the transverse elec-tromechanical coupling factor is discussed before presenting the polarization


Piezoelectric layer

Metallic electrodes

x y

z

Metallic electrodes

Elastic layer

x y

z

Elastic layer

Piezoelectric layer

(a)

Piezoelectric layer

Metallic electrodes

x y

z

Metallic electrodes

Elastic layer

x y

z

Elastic layer

Piezoelectric layer

(b)

Figure 2.12 - (a) Common configuration of a piezoelectric laminated beam, (b) interdigitatedelectrodes configuration.

and electrodes arrangement scheme related to piezoelectric unimorphs and bi-morphs. Finally, the equations describing the macroscopic electrodynamics ofpiezoelectric laminates are derived.

In this work, it is considered that the piezoelectric layer(s) is configuredas a classical capacitor, i.e. sandwiched between a pair of electrodes. Thisallows imposing electrical boundaries conditions by fixing the voltage or thecharge on the electrodes. The electrodes are assumed to be located on the xyplane (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 grainsis oriented along the z axis (non ferroelectrics). If fringing effects occurring onthe lateral sides of the element are neglected, the components of the electricalfield and displacement vanished along y and x, so that the constitutive equationof dielectrics is reduced to the simple form D3=εT33E3 or D3=εS33E3. Although itis not discussed further, it is interesting to present an alternative and relativelyunconsidered electrode configuration, shown in Figure 2.12(b). This designrequires depositing electrodes on a single surface of the device, which can beadvantageous in the case of micro fabrication. It also involves piezoelectriccoefficients different than those appearing in our applications. However, theperformances in terms of energy harvesting obtained from it do not competefor the moment with the ones resulting from the classical type of electrodesconfiguration [97].

The constitutive equations of piezoelectricity for the piezoelectric laminatedbeam of Figure 2.12(a) can be obtained by combining the simplified equationsof dielectrics and of elasticity. Assuming all the elements of symmetry thatwere introduced in the analysis of the elastic beam, the electrical field can bewritten as E3(x,z) and the dielectric displacement as D3(x). The constitutiveequations of piezoelectricity given in (2.8) and (2.9) can be simplified to theform given in (2.26) and (2.27). It can be seen that a single piezoelectric chargeconstant is involved in the problem; the piezoelectric material is excited in the


so called d31 mode.[S1 (x, z)S5 (x, z)

]=[sE11 00 sE55

] [T1 (x, z)T5 (x, z)

]+[d31

0

]E3 (x, z) (2.26)

D3 (x) = d31T1 (x, z) + εT33E3 (x, z) (2.27)

Before continuing the derivation of the equations of laminated piezoelectricbeams, the concept of the electromechanical coupling factor is now introduced.As described above, several forms of the constitutive equations of piezoelec-tricity do exist. The relations between the materials constants involved in thedifferent forms are too complex to be of practical use in the non simplified ex-pressions, but they can now be established in the framework of the simplifiedequations (2.26) and (2.27). Not all the relations between the constants areexpressed but only those which are of primary interest for the understanding ofthe problem. Combining the different forms of the constitutive equations, onecan obtain the relation between compliance and permittivity under constantstress (also referred to as ”free” material properties) and constant strain (alsoreferred to as ”clamped” material properties).

sD11 = sE11

(1− d2

31

sE11εT33

)(2.28)

εS33 = εT33

(1− d2

31

sE11εT33

)(2.29)

The term k312=d31

2/(εT33sE11) in (2.28) and (2.29) is referred to as transverseelectromechanical coupling factor. Its physical meaning can be understood byconsidering a simple and unsupported axial piezoelectric transducer configuredfor d31 mode operation undergoing a tension or compression directed along itsaxis 1. An ideal quasi-static thermodynamic cycle, as defined in the IEEEstandards 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 acompressive 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 thetransducer (slope equal to sE11). In a second phase, the piezoelectric transduceris open-circuited and free to return to a zero stress configuration (slope equal tosD11). During this phase, a quantity Em of mechanical energy is ”consumed’ anda quantity Eel of electrical energy is developed in the structure. Finally, thecycle is completed by shunting the electrodes of the piezoelectric element to aperfect load in which Eel is integrally dissipated. Obviously EM=Em+Eel. Onecan easily compute the fraction Eel/EM found to be equal to the piezoelectricmaterial transverse coupling factor k31

2.

The analysis is focused on specific cases of piezoelectric laminates. Bothfamilies of structures considered are illustrated in Figure 2.14. In both situa-tions, the thickness of the electrodes is assumed negligible. The first structureis referred to as a unimorph [60] and consists in an elastic layer supporting a


Em

Eel

S1

T1

Shorted

Shunted

Opened

231

2311

el

m

E kE k

=−

231

el

el m

Ek

E E=

+

Figure 2.13 - Quasi-static thermodynamic cycle illustrating the definition of k312 and

k312/(1-k31

2).

single piezoelectric capacitor. One of the electrodes is grounded. Due to man-ufacturing convenience it is generally the one located at the interface betweenthe elastic and piezoelectric layers. The other(s) are connected to a prescribedcharge or voltage. The piezoelectric laminate presented in Figure 2.14(b) isknown as bimorph and is manufactured by attaching a piezoelectric capaci-tor on each opposite sides of the elastic layer. It is common to connect thecapacitors in order to limit inputs and outputs of the system and to improvethe performances. The bimorph is symmetric with respect to the x axis andthe same piezoelectric materials are used for the two capacitors. Dependingon their position in the beam and on the direction of their poling, the piezo-electric layers in the bimorph can develop either positive or negative electricalcharges. In this case, appropriate schemes of polarization orientation and ca-pacitors connection have to be implemented so that the charges developed inthe different piezoelectric layers do not cancel. The two basic possible schemesare presented for the bimorph in Figure 2.14(b). Series operation imposes op-posite directions of poling Π (also known as X poled) in the two piezoelectriclayers so that the developed electrical fields have the same directions. Paralleloperation requires the same direction of poling Π (also known as Y poled).

The sign of the longitudinal strain in a given piezoelectric layer can varydepending on the boundary conditions and on the type of load. In this casepoling and connection scheme must be reconsidered. For example, consideringa clamped-clamped unimorph loaded by a concentrated force applied at thecenter of the structure, some of the upper fibers of the beam undergo tensilestrain while others undergo compressive strain. It can be shown that the totalstrain along the x axis cancels, so that no net voltage or charges are developedif the piezoelectric layer consists of a single capacitor and has constant polar-ization. The solution consists in either dividing the layer in several capacitors


E

x

z

V

Π hs

hp

(a)

E

x

z

E

E

E

V V

Π

Π

Π

Π

hs

hp

hp

(b)

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

or implementing a non constant polarization scheme. In this thesis, cantileverbeams are dealt with and they require no special adjustment of the polarizationor electrodes configuration.

The constitutive equations of piezoelectricity are now rewritten in terms ofthe voltage developed across the arrangement of piezoelectric capacitors. Thefollowing 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 arediscussed.

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 T 1 represents the average longitudinal stress along thethickness of the piezoelectric layer. Because T1 is considered to vary linearly,T 1(x)=T1(x,hs+hp/2).

V = −hpD3 (x)εT33

+hpd31

εT33

T1 (x) (2.30)

The simplified constitutive equations of piezoelectricity can be rewritten interms of the potential difference as

S1 (x, z) = sE11T1 (x, z)− d31V

hp− d2

31

εT33

(T1 (x, z)− T1 (x)

)(2.31)

D3 (x) = d31T1 (x)− εT33

V

hp(2.32)

In the following derivation, the term in d312/(εT33) in (2.31) is neglected. This

simplification constitutes the so-called small piezoelectricity approximation andis valid for material with low electromechanical coupling coefficient [135]. Sim-ple refinements based on literature results are proposed later.

The relevant expressions of the mesoscopic constitutive equations in thecase of piezoelectric laminated beams are obtained. The next step consists inderiving relations describing the macroscopic behavior of the unimorph. In the


311

11= E

p

d VThs

z0 hs

hp

V

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

previous subsection, the macroscopic variables corresponding to the stress andstrain in the case of laminate elastic beams are introduced. For the consideredpiezoelectric 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 beredefined by considering the piezoelectric effect. The expression of the bendingmoment due to the piezoelectric effect can be found by analyzing the stressesoccurring in a cross section of a unimorph undergoing solely an applied voltage(Figure 2.15). According to (2.31), the longitudinal stresses in the piezoelectriclayer resulting from an applied voltage is d31/(sE11hp)V. These stresses existsolely in the piezoelectric layer and result in a bending moment applied on thecross section of the beam. This moment exists because of the opposition thatthe elastic layer exhibits against longitudinal deformations. For the materialsconsidered, an applied voltage does not induce shear stress: the beam undergoespure bending.

The bending moment Mv developed under the action of an applied voltageis found by integrating the product of the distance from the neutral axis of afiber belonging to the piezoelectric layer and the stress corresponding to thisfiber. Its expression is

Mv = −αV (2.33)

with

α = Wd31

sE11

(hs +

hp2− z0

)(2.34)

According to the conventions used in this thesis, a positive voltage corre-sponds to a downward curvature.

The dynamic equation of the beam including the piezoelectric bending mo-ment represents somewhat a macroscopic equivalent of the mesoscopic converseequation and an equivalent for the direct equation should be elaborated. Themacroscopic variable corresponding to the electrical field has already been de-fined. It is the potential difference V. The dielectric displacement can be linked


to the surface electrical charges Q accumulated on the electrodes. Q is foundby integrating the dielectric displacement given by the direct equation (2.32)over the surface of the electrodes. To perform the integration, the expression ofT 1(x) derived from the SPE form of (2.31) is first introduced in (2.32). The ob-tained equation depends on the strain S1 replaced by its expression in terms ofthe vertical displacement field w as given in the kinematic equation (2.13). In-tegration of the latter leads to the macroscopic direct equation of piezoelectricbeams:

Q = −α((

dw (l)dx

− dw (0)dx

)− (θy (l)− θy (0))

)− CpfV (2.35)

In this equation, Cpf=εS33Wl/hp is the clamped (motion restrained) capac-itance of an unsupported slab of the piezoelectric layer. It is shown later thatthe theoretical expression of the clamped capacitance of the same piezoelectriclayer coupled to an elastic one is different from Cpf .

The fundamental equations describing the macroscopic behavior of a piezo-electric laminated beam are derived. These equations allow a correct represen-tation of the behavior in a large range of frequencies and for a large variety ofgeometries respecting the beam definition given previously. However, analyti-cal solutions for these equations are not easily found. The purpose is to developrelatively simple closed form formulas allowing the understanding of the influ-ence of the geometry and of the material properties on the performances ofthe energy harvesters. Shifting right now to numerical analysis would not ful-fill this requirement. Therefore, the analysis is focused on a simplified formof the fundamental equations of the piezoelectric beam. The devices are ei-ther excited by a harmonic signal in the neighborhood of their fundamentalresonance frequency (industrial environment harvesters) or undergo an impactwhich, as shown in Chapter 4, excites mainly the fundamental vibration modeof the beam (human environment harvesters). Shear and rotary inertia effectsgenerally occur at high frequencies, well above the fundamental mode [136],and they can be reasonably neglected in the analysis. Also, it is assumedthat the parasitic damping could be represented by the sole use of complexmaterial properties (complex EIeq for internal dissipations and complex ρ forexternal ones). The simplified fundamental equations that are investigated inthe following considering time dependant variables are

P (x, t) = EIeq∂4w (x, t)∂x4

+ ρA∂2w (x, t)

∂t2(2.36)

Q (t) = −α(dw (l, t)dx

− dw (0, t)dx

)− CpfV (t) (2.37)

θy (x, t) = 0 (2.38)


Before solving the equations derived for representative cases, simple meth-ods for relaxing the SPE and for shifting the problem from a plain stress to aplain strain approximation are proposed. Also, expressions of EIeq and z0 aregiven for unimorph and bimorph.

The approximations used in the derivation of the constitutive equations tobe relaxed in this subsection are plain stress and small piezoelectricity assump-tions. Plain stress simplification is used when deriving (2.10). This assumptionis valid for long and slender beam. However, in some cases, short and widebeam are implemented. The problem can be represented in this case by a moremeaningful 2D assumption consisting in neglecting the lateral strain S2 ratherthan the lateral stress T2. This assumption is referred to as plain strain. Tad-mor [137] has shown that the results reached for plain stress are easily adaptedto plain strain by introducing effective values of the material properties. Theeffective longitudinal compliances are indicated for the elastic and piezoelectricmaterial respectively as ss and sEp . The effective permittivity under constantstress and the effective charge constant of the piezoelectric layer are labeledrespectively as εTp and dp. Their expressions in plane stress and plane strainare

ss = ss,11 sEp = sEp,11

εTp = εT33 dp = d31

Plane stress

ss = ss,11 −

s2s,12ss,22

sEp = sEp,11 −sEp,12

2

sEp,22

εTp = εT33 −d2

32

sEp,22

dp = d31 − d32

sEp,12

sEp,22

Plane strain

(2.39)

The effective transverse electromechanical coupling factor has to be com-puted with the effective values of the material properties and is labeled askp=dp/(spεTp )1/2.

When deriving the macroscopic form of the constitutive equations, the smallpiezoelectricity approximation is used by omitting the electrical field inducedby mechanical deformation in the converse equation (2.31). This simplificationcan have repercussions in modeling modern piezoelectric materials which ex-hibit large electromechanical coupling effect. As shown by Tadmor [137], theSPE can be relaxed by defining an effective moment of inertia for the piezo-electric layer: the value of Ii has simply to be replaced by Ii(1+ξ) in whichξ=(kp)2/(1-kp2). This consideration has an incidence on the expression of EIeqand z0 involved in the constitutive parameters of the model. Combining theresults of Tadmor with those of Weinberg [129], one can obtain the closed formexpressions of EIeq, GAeq and z0 for the piezoelectric structures depicted inFigure 2.14. They are given in (2.40) for the unimorph and in (2.41) for thesymmetric bimorph. The reference of the z axis is taken at the lowest fiber of


the beam.

z0 =sEp h

2s + ssh

2p + 2sshphs

2(sEp hs + sshp

)

EIeq = W

(sEp)2h4s + (1 + ξ) (ss)

2h4p+

sEp sshphs((4 + ξ)h2

p + 4h2s + 6hphs

)

12sEp ss(sEp hs + sshp

)(2.40)

z0 = hp + hs2

EIeq = WsEp h

3s + 2sshp

((4 + ξ)h2

p + 3h2s + 6hphs

)12sEp ss

(2.41)

A comparison of the values of EIeq obtained from the basic and refined theoryis proposed in Figure 2.16. For the material properties, values representative ofstrong coupling structures that will be investigated in later chapters are used(PZT-5A and generic brass alloy). The total thickness and the width of thebeam are set to a fixed value so that a comparison can easily be achieved. Itis clearly seen that for both unimorph and bimorph the plain stress and plainstrain approximations results in important differences. The former simplifi-cation results in lower values of the predicted stiffness and of the fundamen-tal resonance frequency. In accordance with common sense, the SPE has anoticeable influence for structures in which the bulk of the beam consists ofpiezoelectric material. Relaxing the SPE induces higher predicted values of thestiffness because of the additional stress term in (2.31). According to the con-clusions presented in this paragraph, one should carefully analyze the geometryand boundary conditions of a given problem and determine according to it thecorrect expressions of the effective material properties.

In the next section, the simplified fundamental equations of the piezoelectricbeam are solved for the particular case of a mass loaded cantilever which isrepresentative of the devices manufactured during this thesis. The obtainedsolutions are arranged in the form of an impedance matrix, which is particularlyuseful for phenomenological analysis. Also, an alternative representation of theimpedance matrix based on electrical equivalent networks is presented.

2.3 Constitutive matrix and electrical networkrepresentation of piezoelectric beams

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

482.3 Constitutive matrix and electrical network representation of piezoelectric

beams E

I eq

1.00.80.60.40.2

hp/(hp+hs)

(a)

EI e

q

1.00.80.60.40.2 2hp/(2hp+hs)

(b)

Figure 2.16 - Values of EIeq computed for (a) unimorph, (b) symmetric bimorph. Solidand dotted lines correspond respectively to the plain stress and plain strain cases consideringthe SPE, dashed and dotted dashed lines correspond to the plain stress and plain straincases when the SPE is relaxed. The scale of the ordinates axis is linear and arbitrary inboth graphics. The material properties are ss,11= ss,22=10 pPa−1, ss,12=3 pPa−1, sE

p,11=

sEp,22=16.4 pPa−1, sE

p,12=5.7 pPa−1, d31=d32=175 pC/N, εT33/ε0=1700.

The general method of resolution of the equations developed in the previoussections and the representation of the solutions in the form of an impedancematrix or equivalent electrical network are then presented. Finally, the conceptof generalized electromechanical coupling factor is introduced.

2.3.1 Geometry of the harvesters and applied mechanicalefforts

The energy harvester consists, in the most general way, of a piezoelectric can-tilever beam with a mass mt attached at its tip (Figure 2.17). A unimorphis considered in the derivation of the model, but results of the analysis for bi-morphs are also presented later. The attached mass mt has a large thickness Hcompared to the one of the beam, so that it is assumed that no strains occuralong the length L of the mass. Therefore, the piezoelectric layer(s) covers onlythe length l of the laminated beam. The deflection along the top surface of thebeam and of the mass are denoted respectively as w and δ. The deflectionw of the beam is defined by a curve, while the deflection along the attachedmass follows a straight line. According to the kinematic relations introducedpreviously and considering a small beam tip angle, the different deflections arerelated by δ(x)=w(l)+(x-l)dw(l)/dx. Finally, mb represents in Figure 2.17 thetotal mass of the piezoelectric beam (mb=ρWl(hp+hs)).

The mechanical efforts that are applied on the piezoelectric structures arenow defined. They are different in the case of the machine and human envi-ronment harvester.

In the former situation, the clamped end of the cantilever undergoes a har-


L

hs

H

x

z

hp l

mt

w(x)

Piezoelectric layer

Elastic layer

Clamped boundary

δ(x)mb

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

monic motion Z(t)=Z0sin(ωt)which results in apparent z directed forces perunit volume in the beam and in the tip mass when the computations are real-ized in the frame of reference attached to the bender. Steady state is consideredin the analysis of this harvester. The inertial forces acting on the beam canbe represented by a uniform pressure P applied all along the length of thebeam (Figure 2.18(a)). In the following, it is supposed that the mass mb of thebeam is small compared to the one of the tip mass mt, so that the discusseddistributed pressure is ignored in the analysis of the machine environment har-vester and P=0. In this case, the left hand side of (2.36) vanishes. The forcedistributed over the volume of the mass can be represented by a resultant Facting at the centre of gravity G.

In the case of the human environment harvester, the mechanical effortsconsist in the result of the impact of a moving object on the beam. It is shownlater that the collision can be represented within some approximations as aconcentrated pulse type force F=U(t). It is assumed that this force is appliedat middle of the mass (or at the tip of the beam if no additional mass isattached). For both inertial and impact situations, the mass-cantilever systemcan be represented by a simple cantilever of the same length, while the tip ofthis equivalent beam undergoes a vertical force F and a torque FL/2, assumingsymmetry of the mass and small displacements (Figure 2.18(b)).

It is judicious to define now characteristic dimensions of the beams ana-lyzed. The particular types of piezoelectric cantilever configurations that areinvestigated can be described by three different sets of characteristic dimen-sions and values of the beam and attached masses: in the case of the machineenvironment harvester, MEMS fabricated short and wide cantilevers support-ing a long attached mass are investigated. PZT and AlN materials are used forthe piezoelectric layer. For this type of harvester, long and slender PZT basedcommercial beams supporting a short attached mass are also considered. In thehuman environment harvesters, the last type of described cantilever withoutattached mass is used. PZT is the sole piezoelectric material implemented in


beams

z

G

x

z

x

F

F L/2

L/2

F=U(t)

2

2bm d ZPl dt

= −

2

2

δ= − GdF M

dt

z

x (a)

z

G

x

z

x

F

F L/2

L/2

F=U(t)

2

2bm d ZPl dt

= −

2

2GdP M

dtδ

= −

z

x

(b)

Figure 2.18 - (a) Mechanical efforts applied on the energy harvesters (blue symbols refer tothe 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 thevibration energy harvesters.

Case label A B C

Illustration

Application field Machine environment Machine environment Human environment

Human environment

Manufacturing MEMS Conventional Conventional Material AlN, PZT PZT PZT Representative dimensions

W=7 mm, l=2 mm, hp=1 μm, hs=100 μm L=7 mm, T=650 μm

W=5 mm, l=4 cm, hp=600 μm, hs=200 μm,

L=5mm, T=1 cm

W=5 mm, l=4 cm, hp=600 μm, hs=200 μm

L=5 mm, T=1 cm Representative beam and tip mass

mt =100 mg mb =5 mg

mt =3 g mb =500 mg

mt =0 g mb =500 mg

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

2.3.2 Concepts of the constitutive matrix and electricalequivalent network

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


electromechanical coupling factor of the piezoelectric beams is introduced.

It is important to define what is meant by constitutive matrix of the piezo-electric beam. The idea consists in relating local values of the intensive macro-scopic variables describing the system to the local values of the correspondingextensive variables by simple linear relationships, which can be expressed inthe form of a symmetric matrix. It can be considered that a so called lumpedmodel 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 Fig-ure 2.18. In this situation, the extensive variables are limited to the force Fand the voltage V, while the intensive variables are the displacements w or δand the charges Q. Q and V have already been ”lumped” when deriving themacroscopic fundamental equations of piezoelectric beams. The location of theapplied force has already been defined (Figure 2.18(b)) and it is only requiredto define a point of observation for the deflection. It is chosen in the followingas the position δg of the center of mass of the attached body (or tip of thebeam when L=0). Rather than using a true impedance matrix representationas Smits, the model is developed in the form of a heterogeneous admittancematrix described in (2.42). This particular representation allow developing aconvenient form of the equivalent electrical network approach of the problem.In (2.42), kc represents the lumped mechanical stiffness of the beam, Cpc theclamped capacitance of the piezoelectric layer (this time coupled to an elasticlayer, Cpc 6=Cpf ) and Γc a macroscopic piezoelectric conversion factor, macro-scopic ”equivalent” of e31.[

FQ

]=[kc ΓcΓc −Cpc

] [δgV

](2.42)

The impedance matrix of (2.42) can be represented in the form of an electricalequivalent circuit. This type of representation has some general advantages: itfirst allows representing the problem in terms of lumped parameters belongingto a single engineering domain. Network representation provides a single do-main tool which requires only a basic understanding of the physical phenomenaoccurring in the other domain. These models are furthermore very general, asthey can be applied to many kinds of electromechanical transducers, as shownby Tilmans in [138][139]. Second and more important, the network represen-tation can be easily integrated into modern simulation software which allowbuilding models where the transducer is coupled to complex mechanical orelectrical structures. This type of model is particularly useful in the field ofenergy harvesting, as the transducer has to be connected to power managementelectronics.

Electrical network representations of piezoelectric transducers have beeninitiated by the work of Butterworth and van Dyke [104][105], where a modelrepresenting the behavior of the structure near to resonance was proposed. Ma-son was the first to introduce a representation taking into account wave propa-gation phenomena [108]. Redwood [140] and Krimholtz [141] proposed refinedrepresentations of Mason’s model for thickness mode piezoelectric transducers


beams

dδg/dt 1/kc

Cpc

Γc : 1 mt

F

dQ/dt

V F

Figure 2.19 - Electrical network representation of a piezoelectric beam.

using transmission lines analogies. The equivalent network representation ofpiezoelectric 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 andadding the dynamic component due to the attached mass mt, the dynamicalequilibrium of the piezoelectric beam can be represented in the form of theequivalent two ports electrical network shown in Figure 2.19. The left side ofthe circuit corresponds to mechanical parameters, the right to electrical. Themechanical stiffness kc is represented in the electrical domain by a capacitor1/kc and the attached mass by an inductance of value mt. The electromechan-ical conversion related to the piezoelectric effect is represented by a perfecttransformer of ratio Γc. In the particular situation of vibration and motionenergy harvesters, the mechanical ports of the network are connected to asource of mechanical energy, steady state vibrations or impulse, while the elec-trical ports are connected to an electrical load in which energy is stored ordissipated. Complex values of the different components allow representing theparasitic losses occurring in the system.

Explicit expressions of the different parameters involved in the constitutivematrix and in the equivalent electrical network should now be determined. Thisprocess 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 net-work 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 beamequation is available. The dynamical behavior corresponding to the piezoelec-tric benders implemented in the two types of investigated energy harvestersare different in essence. In the case of the machine environment harvester, thedevice operates according to a steady state principle, while for the human en-vironment harvester, the repeated impacts of the moving body on the benderresults in free-vibrations and a transient type behavior. In both cases, it can be


assumed that the displacement is separable in space and time. For the steadystate situation, the displacement can be written as w(x,t)=wx(ω,x)wt(ω,t), inwhich ω is the angular frequency of the input motion. The time dependantpart 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 tothe amplitude and frequency of the input. The solution for multi harmonicvibrations is found through Fourier decomposition of the input and superpo-sition principle. For the transient case, results of modal analysis [136] suggestthat w(x,t) can be expressed as w(x,t)= Σ wxi(ωi,x)wti(ωi,t) in which ωi isthe ith eigenfrequency, wxi(ωi,x) is the related modal shape of the beam andwti(ωi,t) is the related time component. It is possible to write wti(ωi,t)=Cexp((ξi+jωi)t) in which ξi is a dissipative factor. It is shown in Chapter 4 thatone can assume in the discussed situation that the impact on the beam excitessolely the fundamental mode at ω0 (this frequency depends on the electricalboundary applied on the electrodes in the case of a piezoelectric laminatedbeam). The solutions of the steady state and transient problems have then anequivalent form and can be elaborated by a single type of analysis describedin the following. The derivation of the impedance matrix does not require animmediate investigation of the time dependant part of the displacement. Thisanalysis is performed later in the chapters dealing with the optimization of thedifferent harvesters.

For both transient and steady-state cases, the time components in the beamequation can be eliminated yielding (2.43) with ω=ω0 for the transient situ-ation. λ represents the characteristic wavelength of the bending wave in thebeam.

d4wx (ω, x)dx4

− 1λ4wx (ω, x) = 0 (2.43)

λ =1√ω

4

√EIeqρA

(2.44)

General solutions of (2.43) can be written as

wx (ω, x) = C1 sinh(xλ

)+ C2 sin

(xλ

)+ C3 cos

(xλ

)+ C4 cosh

(xλ

)(2.45)

in which the Ci are real constants. These constants depend however on theelectrical and mechanical boundary conditions of the system.

Four boundary conditions are required for obtaining expressions of the Cicoefficients. Two different sets of boundary conditions are required for deter-mining the expressions of the constitutive matrix parameters. When consider-ing that the electrodes of the piezoelectric bender are short circuited, i.e. V=0,the matrix constitutive equation is reduced to[

FQ

]=[kcΓc

]δg (2.46)


beams

In this situation, the relevant mechanical boundary conditions are defined asfollow (in the frame of reference attached to the bender): at the clamped end, nodisplacement is possible so that wx(ω,0)=0. There are also no deflection angleat the clamped end so that dwx(ω,0)/dx should also be null. At the free end ofthe equivalent cantilever of Figure 2.18b, a shear force F and a moment FL/2result from the force applied on the attached mass. The boundary conditionsfor this situation are

w (0) = 0dw (0)dx

= 0

d2w (l)dx2

= − F

EIeq

L

2(2.47)

d3w (l)dx3

=F

EIeq

In order to simplify the expressions of the equations developed in the remainderof this chapter, the following definitions are used.

Ac = cos(lλ

)Ach = cosh

(lλ

)As−ch = sin

(lλ

)cosh

(lλ

)As = sin

(lλ

)Ash = sinh

(lλ

)Ac−ch = cos

(lλ

)cosh

(lλ

)Ac−sh = cos

(lλ

)sinh

(lλ

)As−sh = sin

(lλ

)sinh

(lλ

)(2.48)

It is now possible to determine the expressions of the coefficients Ci of (2.45)by solving the system of equations defined by the boundary conditions (2.48)combined with (2.45):

C1 = −C4 =λ2(λ (Ac +Ach) + L

2 (As −Ash))

2EIeq (1 +Ac−ch)F (2.49)

C2 = −C3 =λ2(−λ (As +Ash) + L

2 (Ac +Ach))

2EIeq (1 +Ac−ch)F (2.50)

As expressed by (2.46), F=kcδg in the considered situation. As δg=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=Γcδg for a short circuited bender. By inserting the expressionof wx obtained above in the charge equation (2.37) and by performing someadditional 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)


Two components of the constitutive matrix have been obtained by consideringthe behavior of a short circuited piezoelectric bender. The constitutive matrixis symmetric and a single parameter is missing, being the clamped capacitanceof the piezoelectric layer. It is obtained by analyzing the behavior of a beamundergoing an applied voltage without applied mechanical force (F=0). In thiscase, the matrix constitutive equation is reduced to the form given in 2.53.[

δgQ

]=[

−Γc/kc−Cpc − Γ2

c/kc

]V (2.53)

The relevant mechanical boundary conditions are defined as follow: no dis-placement neither rotation are allowed at the clamped end, i.e. wx(ω,0)=0 anddwx(ω,0)/dx=0. As described by (2.33), a voltage applied across the electrodesof 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) = 0dw (0)dx

= 0

d2w (l)dx2

= − αV

EIeq(2.54)

d3w (l)dx3

= 0

By the same method used for the short circuited bender, the coefficients Ci arefirst derived:

C1 = −C4 =αλ2 (Ash −As)

2EIeq (1 +Ac−ch)V (2.55)

C2 = −C3 = − αλ2 (Ac +Ach)2EIeq (1 +Ac−ch)

V (2.56)

The value of the dynamic clamped capacitance Cpc is obtained by solving thecharge equation (2.37):

Cpc = Cpf −4α2λ3 (1 +Ac−ch)

EIeq ((L2 + 4λ2)As−ch + (L2 − 4λ2)Ac−sh + 4λLAs−sh)(2.57)

It was verified that the expression of Γc found with the boundary conditionscorresponding to the clamped bender was the same than the one obtained withshort circuit condition.

The three different components of the constitutive matrix are now available.The computations are done assuming mechanical wave propagation phenomenaand the so called dynamic constitutive matrix of the piezoelectric bender hasbeen obtained. When the length of the attached mass is null, the componentsof the derived matrix correspond to those found by Smith [126]. The obtainedformulas are tedious and developed for the purpose of studying the responseof the harvester excited around high order modes of oscillation. However, in


beams

the present work, the devices are excited around their resonance frequencyand a quasi-static approximation (mechanical wave propagations neglected,i.e. λ→∞) is justified. It was verified by FEM analysis that the quasi staticformulas give reasonable estimates of the constitutive parameters for the threetypes of bender described in Table 2.1. The quasi static constitutive parame-ters are denoted as k0, Γ0 and Cp0. In order to introduce the effects of parasiticdamping in the quasi static form of the model of Figure 2.19, complex valuesof the parameters are used in the analysis proposed in the next chapters. Theapproach is limited to complex values of k0 and Cp0 which can be rewrittenas k0=k(1+j/Qm) and Cp0=Cp(1+j/Qe) in which Qm and Qe are the qual-ity factors related to respectively the mechanical and dielectric dissipations(Qm>0, Qe<0). The real part k of the quasi static stiffness, the quasi statictransformation factor Γ and the real part Cp of the quasi static capacitance are

k =3EIeq

l3(1 + 3

2Ll + 3

4L2

l2

) (2.58)

Γ =3α(1 + L

l

)2l(1 + 3

2Ll + 3

4L2

l2

) (2.59)

Cp = Cpf −α2l

4EIeq(1 + 3

2Ll + 3

4L2

l2

) (2.60)

The equivalent network describing the dynamic equilibrium of the piezoelectricbender has to be slightly rearranged when the quasi static approximation isconsidered. Indeed, in the dynamical form of the circuit, the mass of the beamis taken into account by the component kc. The quasi static version of thisparameter is not related to the mass of the beam mb. This problem can besolved by replacing the mass mt of the attached body by an effective massme in the circuit of Figure 2.19. An expression of me can be obtained by firstelaborating a formula for the mechanical angular resonance frequency ω0 of thebeam. It can be understood from the equivalent network that a piezoelectricbeam disposes of two characteristics frequency: one corresponding to a shortand the other to an open circuit arrangement. In order to avoid confusionbetween the two values, the short and open circuit resonance are labeled as ωs0and ωo0 respectively. It will be shown in later chapters that the relation betweenthe two values can be approximately written as ωo0= ωs0(1+Γ0

2/(k0Cp0))1/2.From the circuit of Figure 2.19 (with mt replaced by me), it is possible to writeωs0=((k0/me)1/2, so that the determination of ωs0 allow finding an expressionfor the effective mass of the beam.

In this case, ωs0 is not easily found from the eigenvalues problem becauseof the attached mass resulting in a highly non linear characteristic equation.Rather, the Rayleigh quotient method is used [136]. It gives an upper boundto the fundamental frequency. The Rayleigh quotient R can be defined asR=Wp/Wk=(ωs0)2, in which Wp and Wk represent respectively the maximumpotential and kinetic energy present in the system. For the situation described


by Figure 2.18(b) and in short circuit configuration, (ωs0)2 can be written as

(ωs0)2 =

l∫0

EIeq

(d2w (x)dx2

)2

mbl

l∫0

w (x)2dx+

mt

L

l+L∫l

δ (x)2dx

(2.61)

The quasi static form of the expression of wx(ω,x) found when computing theparameters kc and Γc of the impedance matrix have to be used in (2.61) inorder to determine the closed form expression of (ωs0)2 given below.

(ωs0)2 =3EIeq

(1 +

32L

l+

34L2

l2

)

33140mbl

3

(1 +

9166L

l+

2144L2

l2

)+mtl

3

(1 + 3L

l+ 63

16L2

l2+ 21

8L3

l3+ 3

4L4

l4

)

(2.62)

The correctness of (2.62) was investigated by performing 3D finite elementssimulation (FEM) on a simple singled layer elastic beam. We observed thatthe proposed closed form expression yields very small error for the geometricalsituations studied in this work. A noticeable case where (2.62) does not predictcorrectly the resonance frequency is related to structures for which a short andextremely thick mass is attached to the cantilever (”L shaped”).

By computing k0/(ωs0)2, the expression of the effective mass given below isobtained.

me =33140

mb

(1 + 91

66Ll + 21

44L2

l2

)(1 + 3

2Ll + 3

4L2

l2

)2+mt

(1 + 3Ll + 63

16L2

l2 + 218L3

l3 + 34L4

l4

)(1 + 3

2Ll + 3

4L2

l2

)2 (2.63)

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 theattached 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 piezoelectricunimorphs (Figure 2.14(a)). The proposed results can easily be adapted forthe symmetric bimorphs of Figure 2.14(b) by reconsidering the expression ofEIeq, 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


beams

bimorph parallel configuration. Cp0 has to be replaced by Cp0/2 for the bi-morph series configuration. The bimorph parallel or series arrangement has astrong influence on the performances of the bimorph in terms of actuation orsensing applications. On the opposite, it is shown later that using of a series orparallel bimorph configuration does not have important consequences in energyharvesting situations.

2.3.4 Generalized electromechanical coupling factor

The longitudinal electromechanical coupling factor k312 and the related effec-

tive coefficient k312/(1-k31

2) have been introduced previously. They define theamount of mechanical energy transformed into electrical energy for a piezo-electric axial transducer during a quasi static thermodynamic cycle. Equiv-alent variables exist for multilayered piezoelectric beams. The generalizedelectromechanical coupling factor K which is the equivalent of k31

2/(1-k312)

[143]. Because of physical considerations, the generalized electromechanicalcoupling factor of a bimorph or unimorph can only be smaller than the coeffi-cient k31

2/(1-k312) characteristic of its piezoelectric layer. In these structures,

the piezoelectric layer(s) is supported by a purely elastic one, so that only afraction of the developed mechanical strain produces electrical charges. Also,the structure undergoes bending deformations and not purely tensile or com-pressive ones. K can be expressed in terms of the equivalent network parametersas

K2 =Γ2

kCp(2.64)

It is shown in the chapters dealing with the optimization of the harvesters thatthe generalized electromechanical coupling factor is an important parameterrelated to the output power of the devices. It has been demonstrated by theauthor of this thesis in [56] that the generalized electromechanical couplingfactor depends solely on the thicknesses ratio, on the compliances ratio ss/sEpand on the transverse electromechanical coupling factor k31

2. Furthermore,K2 is almost directly proportional to k31

2. The generalized electromechanicalcoupling factor allows representing the performances of the beam in terms ofenergy harvesting for any piezoelectric material (transverse isotropic) and inde-pendently 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 thecase of the bimorph, a series or parallel arrangement of the piezoelectric layersdoes not have any influence on the value of the generalized electromechanicalcoupling factor and then on the energy harvesting capabilities of the device.The transverse electromechanical coupling factor k31 is in the range of 0.4 formodern PZT ceramics and the maximum values of K that can theoretically beobtained are approximately 0.2 for unimorphs and 0.35 for bimorphs.


2.4 Conclusion

In this chapter, the history and the basic concepts of the piezoelectric effectare presented. Then, the linear constitutive equations of piezoelectricity arederived, based on the classical continuum description. Non linear phenomenaand simplifications of the constitutive equations are also described. Piezoelec-tric cantilevers constitute the electromechanical transduction element in thevibration energy harvesters studied during this thesis. Therefore, the constitu-tive equations of piezoelectricity are combined with Timoshenko’s beam theoryin order to elaborate the fundamental equations of piezoelectric multilayeredbenders. This complete derivation had not been proposed in the literature.Both dynamic and quasi static solutions of the latter equations are derived andarranged in the form of a constitutive matrix and of an equivalent electricalnetwork. A complete representation of a cantilever loaded by a distributedmass which is the basis of many harvesters currently investigated is developed.Finally, the generalized electromechanical coupling factor, a crucial parameterfor the performances of the piezoelectric beams in terms of energy harvesting,is defined and a clear interpretation of the difference between the material andstructure coupling factor is proposed.

The derived equivalent electrical circuit constitutes the base of the modelused to analyze the characteristics and performances of the energy harvesters.Experimental methods for measuring the values of parameters involved in theelectrical circuit model are described in Chapter 3. The optimization of thehuman environment energy harvester is presented in Chapter 4 and the one ofthe machine environment harvester in Chapter 5.

60 2.4 Conclusion

Chapter3Manufacturing and primarycharacterization of MEMSpiezoelectric harvesters

The fabrication by MEMS technologies and the primary characterization ofpiezoelectric harvesters are presented. As described in Table 2.1, differenttechnologies were involved in the manufacturing of the piezoelectric bendersimplemented in the harvesters. However, the devices obtained by conventionalprocessing (cases B and C) are bought from commercial companies and theirfabrication is not presented. The manufacturing process for the MEMS devicesis described in a first part of this chapter. In a second part, static, steadystate and transient methods of characterization for piezoelectric benders arepresented. The characterization methods presented in this chapter aim at de-termining the values of the parameters involved in the impedance matrix andthe corresponding electrical network model presented in Chapter 2. The out-put power characteristics of the harvesters are not discussed in this part, butin the subsequent ones.

3.1 Manufacturing

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

61

62 3.1 Manufacturing

(a)

(b)

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

thesis is only marginally involved in the device fabrication. His contribution islimited to the mask design and to the development of some process step. Afterthis initial work, the investigations are concentrated on the design and char-acterization of the harvesters. The main part of the fabrication work on PZTdevices has been done by our colleague Andreas Schmitz in the framework of apost doctoral research. His results have been published in literature [144]. TheAlN based harvesters are manufactured by the WATS division of IMEC at theHolst Centre, The Netherlands (http://www.holstcentre.com). A descriptionof the work performed by the Holst Centre can be found in [78] and [24]. Be-cause the manufacturing process of the devices investigated during this thesisis already described in the literature, this section is limited to the presentationof the most important elements in the fabrication of the harvesters. A generaldescription of the devices is proposed in a first subsection which is followed bya 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 de-sign of bulk micromachined accelerometers and consists of a mass connected toa thin beam and a vibrating package. The process includes a zero level packagewhich is obtained by protecting the wafer holding the piezoelectric generator bya top and bottom wafer attached through benzocyclobutene (BCB) bonding.When the package moves under the action of an external vibration, the piezo-electric beam is deflected and electrical charges are produced. These chargesare allowed to flow in a load circuit connected to the contact pads. In thisway, electrical energy is harvested from vibrations. Pictures of the fabricatedharvesters are shown in Figure 3.1.

Several designs of piezoelectric devices differing in geometry and electricalconnections are manufactured. The devices are equipped with masses of dif-

3. MANUFACTURING AND PRIMARY CHARACTERIZATION OFMEMS PIEZOELECTRIC HARVESTERS 63

ferent lateral dimensions. Here, the thickness of the mass is limited to the oneof the device wafer. Alternative flows requiring the manual assembly of anexternal heavier mass were proposed in the literature [95]. The introductionof such a step has to be evaluated in terms of fabrication cost and generatedpower. Harvesters resonating at various frequencies are realized by modifyingthe length or the thickness of the bending beam. The targeted values of theresonance frequencies are in the range of 100 Hz to 2 kHz. The devices were de-signed either as a single piezoelectric capacitor or as a series connection of fourpiezoelectric capacitors. The latter devices allow a higher output voltage at theprice of a lower output current and of higher electrical output impedance. Anoverview of the geometrical parameters of the different fabricated harvesters isgiven in Table 3.1.

Table 3.1 - Dimensions of the fabricated devices.

Device label Number of Lateral dimensions Width of the Length of thecapacitors of the mass (µm*µm) beam (µm) beam (µm)

Dev1 1 3000*3000 3000 2670Dev2 4 3000*3000 3000 2670Dev3 1 3000*3000 3000 3365Dev4 4 3000*3000 3000 3365Dev5 1 5000*5000 5000 2235Dev6 4 5000*5000 5000 2235Dev7 1 5000*5000 5000 2832Dev8 4 5000*5000 5000 2832Dev9 1 7000*7000 7000 2526Dev10 4 7000*7000 7000 2526Dev11 1 7000*7000 7000 1996

3.1.2 Process flow

The first steps of the proposed process differ for PZT and AlN based harvestersbecause of the different chemical nature, reactivity and compatibilities of thetwo materials. The manufacturing methods for AlN and PZT based harvestersmerge into a single one after the patterning of the piezoelectric capacitors. Inthe following, the process of the AlN and PZT devices is described separatelytill this process step. Then, the last and common steps of the fabrication arepresented.

The silicon wafers used for the AlN based harvesters have a (100) surfaceorientation and are coated with 500 nm thermal silicon oxide and 100 nmLPCVD silicon nitride. In the first step of the process, a 100 nm platinumbottom electrode with a titanium adhesion layer is deposited by sputtering andpatterned by a lift-off technique [145] in order to define the bottom electrodesof the piezoelectric capacitors. In the second step, a layer of AlN (200-1000nm) is sputtered on the wafer according to the parameters defined in [146].


The AlN coating is then shaped by a wet etching process based on an OPD262solution. Because the AlN layer is thicker than the electrodes, it is etched insuch a way that the contact pad of the top electrode is held on top of thepiezoelectric material. In this way, the probability of step coverage relatedfailure is reduced. In the next step of the process, a 100 nm conductive layerof aluminum is sputtered on the wafer and etched according to classical wettechniques respecting the integrity of Pt and AlN, so that the top electrodeof the device is patterned without damaging the bottom electrodes and thepiezoelectric material. Finally, the top nitride, oxide and silicon are patternedand dry etched to predefine the shape of the cantilever beam. The depth of thetrench defines the thickness of the beam and it is between 20 and 100 µm forour devices. The steps described in this paragraph are illustrated in Figure 3.2.

For the PZT based harvesters, silicon wafers coated with 500 nm thermaloxide, 100nm nitride and a 1 µm PZT layer (sandwiched between metallic elec-trodes) are bought from Inostek, Korea (http://inostek.com/). Details of thedeposition process are given on the company’s website. Both electrodes onInostek’s wafers are made of platinum. For these devices, the first step of theprocess consists in patterning and etching the top Pt layer by reactive ion etch-ing in order to define the top electrode and first bond contact of the device. ThePZT layer is then etched by the successive action of a buffered HF/HCl mixtureand concentrated HF [147]. The thickness of the resist used in the lithographyprocess and the etch time must be appropriate for this step, as hydrochloricacid etch organic compounds. In the next step, the bottom platinum layer ispatterned and etched by the same technique as the top electrode to form thebottom one. Finally, as in the AlN process flow, the shape of the cantileverbeam is predefined by patterning and dry etching the top nitride, oxide andsilicon. 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 samemethod. The back silicon nitride and oxide are first patterned and etched todefine the limit of the KOH wet etching used in the next step for shaping thebeam and the attached mass. Si3N4 is KOH resistant and can be used as ahard mask for such a chemical etching. The top side of the wafer supportingthe piezoelectric capacitor is protected from KOH attack by being maintainedin a special vacuum holder. The hard mask of nitride should be carefullydesigned 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 aconsequence convex corners of the substrate are completely undercut (etchingperpendicular to the (110) planes). As the etching velocities in the <100>and <110> directions are similar, large parts of convex corners are etchedaway and the volume of the mass can be reduced drastically. In order to avoidthis problem, simple corner compensation structures have to be implemented[148][149]. The KOH etch is stopped a few micrometers before reaching the toptrenches 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


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

Deposition and patterning of the top electrode

Predefinition of the cantilever beam

Si

Si3N4

SiO2

Pt

AlN

Al

Figure 3.2 - Description of the first steps in the manufacturing of the AlN based harvesters.

and by wet etching based on tetramethylammonium hydroxide (TMAH) for theAlN harvesters. The beam and the mass are also slightly etched during thislast step. The process steps relative to beam shaping and release are picturedin Figure 3.4.

The last part of the manufacturing process consists in the packaging of thereleased devices. This is done by adhesive bonding of KOH preshaped wafersto the device wafer, as illustrated in Figure 3.5. BCB (BenzoCycloButene), aphotopatternable polymer, is used as adhesive layer. The harvesters are readyto be connected to an electrical load via the contact pads accessible throughthe 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 vibra-tion energy harvesters is proposed in this section. The process is based on bulkmicromachining of the silicon wafer. Sputtered AlN and PZT grown by sol-gel methods are used as piezoelectric materials. The devices are packaged by


Process step Side view Top view Silicon wafer coated with oxide, nitride and a Pt/PZT/Pt layer

Patterning of the bottom electrode

Patterning of the piezoelectric layer

Patterning of the bottom electrode

Predefinition of the cantilever beam

SiSi3N4

SiO2

PtPZT

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

KOH etching of back cavities

Release of the cantilevers

Si

SiO2

Si3N4

Figure 3.4 - Release of the cantilevers.


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

Top view

Patterning and etching of top oxyde and nitride

KOH etching of top cavities

Packaging of the device wafer

Si

SiO2

Si3N4

BCB

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

68 3.2 Characterization

being encapsulated between capping wafers. The developed process has beenproven to be reliable and to result in functional devices. Some flaws are how-ever present and are being eliminated: the precision of the KOH attack usedto shape the mass and the bending beam is limited by the fact that this stepis 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 layerand the thickness of the beam will be more precisely controlled. Also, it isshown in the next chapters that the proposed packaging approach introducesa large amount of mechanical dissipations, due to squeeze film damping in theenclosure. To eliminate this problem, future devices will be packaged undervacuum.

In the next section, experimental methods for measuring values of the equiv-alent network parameters of the fabricated devices are described and imple-mented on different samples.

3.2 Characterization

Abundant literature on the characterization of length or thickness piezoelectricvibrators is available [68, 121, 150]. This is however not the case for piezoelec-tric bending structures. Recently, some analyses related to this problem wereproposed in [151–153]. However, the methods described by these authors weredeveloped for the purpose of characterizing actuators and they do not allowobtaining all the information required. Therefore, specific methods of charac-terization fulfilling the requirements of this analysis are developed. Thanks tothese methods, it is possible to determine experimental values of the parametersof the equivalent network model of the harvesters (Chapter 2).

In this section, the general concept of the proposed characterization is firstdescribed. Then, static, steady-state and transient methods are presented inthree separate subsections. Finally, a summary of the characteristics values ofthe measured parameters are presented for the three types of studied piezo-electric benders (MEMS PZT and AlN devices and commercial ceramic PZTbeams).

3.2.1 General concept for the determination of the net-work parameters

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


Table 3.2 - The different possible combinations of measurements.

Constrained

V=0 (Short circuit)

Q=0 (Open circuit)

F=0 (Free bender)

δg=0 (Clamped bender)

F δg F δg F δg F δg V Q V Q V Q V Q Measured δg F Q Q δg F V V Q V δg δg Q V F F Case label 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 toconstrain two of the four variables. Due to the electromechanical nature of thesystem, it is in fact necessary to constrain one mechanical and one electricalvariable. For characterization purposes, one of the constrained variables is fixed(usually to 0) while the other is varied and the free variables are measured. Forexample, a force can be applied on a short circuited bender (F varied, V=0) andthe force-deflection and the force-charge relations can be measured. In total, 16different relations can be obtained. Some of them are however reciprocal and 12are found to be independent. The 12 possible characterization experiments arepresented 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 thesesituations. It is shown in the following that some of the experiments of Table 3.2are difficult to implement in practice. Furthermore, it is not necessary toperform all of them in order to obtain the full set of parameters (however,multiplying the number of performed experiments might increase the accuracyof the values obtained).

The different methods of characterization are discussed in the next sub-sections. Results of the realized measurements are presented in details for acommercial bender and MEMS PZT and AlN devices. The tested commercialbender 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 gis glued to its tip and the assumption of concentrated attached mass can rea-sonably be applied for this structure. The tested MEMS benders are Dev1sample using PZT and AlN. While the lateral dimensions of the characterizedMEMS 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 inTable 3.2 are given for a static situation in Table 3.3. All experiments involv-ing the measurement of the voltage or charges are complex to perform becauseof charges leaking phenomena: the charges developed in the piezoelectric layertend to be dissipated through the electrodes of the structure or through theread out electronics. The decay of the charges follows an exponential behavior


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

Case label X1 X2 X3 X4 X5 X6 Theoretical relation

=gFk

δ

Γ=

FQk

Q δ= Γ

( )21=

+

Fk K

δ ( )2

21=Γ +

K FVK

Γ= g

pV

Cδ

Case label X7 X8 X9 X10 X11 X12 Theoretical relation

( )21cQ C K V= +

Γ=

Vk

δ

( )2

21K Q

Kδ =

Γ +

= pQ C V

F V= Γ

Γ=

p

QFC

governed by a time constant estimated to be 10 ms for the commercial piezo-electric bender. For the MEMS devices, the low amplitude of the developedcharge or current in quasi static mode makes these measurements even moredifficult. In these conditions, extremely rapid and precise force and displace-ment actuators and dedicated read out electronics (such as the one used byDubois [154]) are necessary to perform the experiments X2, X3, X5 and X6.However, rapid actuators would induce transient phenomena and the expres-sions of Table 3.3 might no longer be valid. Similar limitations are encounteredfor experiment X4: indeed, the stiffening of the structure due to the generalizedelectromechanical coupling represented by K2 is annihilated by the leakage ofcharges. For the experiments X7 and X10, dedicated read out electronics arerequired. The goal is to provide easy to implement techniques of characteriza-tion and no efforts are devoted to the development of such elements. Finally,a charge generator was not available, so that the practical static methods ofcharacterization for the piezoelectric benders are limited to experiments X1and X8 (X11 does not bring additional information). It can be understoodthat the easily achieved static measurements only allow obtaining the values ofthe stiffness k and of the transformation factor Γ.

The experiments X1 and X8 are realized with a setup consisting of a capac-itive contact force sensor (1 mN resolution) attached to a micromotor (16 nmresolution)(Figure 3.6). Contact with the sample is done with a metallic needleattached to the force sensor. The controller of the micromotor monitors thedisplacement applied to the structure while the sensor measures the reactionforce of the sample. The chosen observation point is located near the tip of thebeam for the commercial element. For the MEMS bender, the displacementis applied at the tip of the beam (the limits of the beam correspond to thelimits of the piezoelectric capacitor easily found by observation through a mi-croscope). The deflection at the center of the mass δg is then estimated fromthe theoretical relation established in Chapter 2.

The results of X1 and X8 are proposed respectively in Figure 3.7 and Fig-ure 3.8. k=749N/m and Γ=4.6*10−4N/V is obtained for the ceramic PZT


Micromotor Force sensor

Sample

Needle

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

500

400

300

200

100

0

Dis

plac

emen

t (μm

)

4003002001000 Force (mN)

(a)

150

100

50

0

Dis

plac

emen

t (μm

)

30252015105Force(mN)

(b)

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

15

10

5

0

Dis

plac

emen

t (μm

)

2520151050 Voltage (V)

(a)

6

5

4

3

2

1

Dis

plac

emen

t (μm

)

5040302010 Voltage (V)

(b)

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


bender, while k=190N/m and Γ=2.2*10−5N/V is found for the thin film AlNbender. Γ is directly related to the performances of the device in actuationmode; because of the low thickness of the piezoelectric material compared tothe one of the substrate in the MEMS devices, this type of piezoelectric ben-der is much less adapted to actuation than the commercial ones. However,the situation is different for energy harvesting as demonstrated in the nextchapters.

It has been shown in this section that static methods of characterization arein practice limited for piezoelectric benders, principally due to the phenomenaof charge leakage. The only parameters of the network model easily measuredare the stiffness k and the conversion factor Γ. In spite of the limitations ofthe static methods of characterization, the static experiment X1 is extremelyimportant as it allows determining the stiffness k, which can not be directlyobtained 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 simi-lar to the static methods: one electrical and one mechanical variable are con-strained and the others measured. The measurements are realized by usingsinusoidal signals (voltage or mechanical vibrations) with a frequency near tothe fundamental resonance frequency. The obtained relations are then fittedwith the theoretical expressions obtained by analyzing the network model. Fit-ting is realized here by a characteristic point approach, i.e. the theoretical andexperimental relations are only fitted at frequencies corresponding to some re-markable points (maxima and minima) and not over a continuous range offrequency (except for the measurements which are not frequency dependant).Due to the steady state nature of the problem, one should determine for eachspecific experiment the frequency dependence of the amplitude and phase of themeasured variable. Amongst others, it is also possible to use a real and imagi-nary part representation rather than an amplitude-phase one. The choice of themost relevant representation depends on the considered experiment. Indeed,depending on the situation, the simplicity of the expressions corresponding tothe remarkable points of the measured curves depends on the chosen repre-sentation. Also, for some of the experiments of Table 3.2, it is advantageousto normalize or multiply the measurements to the frequency of oscillation inorder to compute the parameters of the network model. This information ispresented 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 frequencydependence is observed and theoretical expressions very similar to the onesobtained for the static case are found. They are given in Table 3.4 taking intoaccount the dielectric and mechanical parasitic dissipations by the intermediaryof the electrical and mechanical quality factors Qe and Qm. These experiments


are interesting as they allow almost direct measurements of Γ, Qe and Qm.However, several practical limitations are again encountered: the experimentsX11 and X12 require a dynamic force sensor with a high resolution (<100 µN forprecise measurements) and which does not perturb the dynamics of the testedsample. For the MEMS piezoelectric benders, which have a small effectivemass (<100 mg), such high end sensors might exist but are absolutely notstandard measurement equipment and these experiments are skipped. Theyare furthermore not crucial, as Γ and Qe can easily be determined by othermeans. 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 whilethe other end of the test sample is attached to a mechanical ground. X6 can bedone using the same experimental arrangement. However, the devices are smalland relatively fragile and the handling required for performing experiments X3and 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 typesof 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 topand bottom clamps. X10 allows obtaining the value of the clamped capacitanceand the electrical quality factor.

The results of X10 are given for the MEMS AlN and commercial benderin Figure 3.9. It can be seen that the value of the measured parameter isapproximately constant over frequency, even if a slight shift is observed for thereal 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 ofthe dielectric losses can be attributed to the electrical connections between thetested devices and the impedance analyzer but not all. Indeed, it can be seenthat the dielectric quality factor of the MEMS harvesters is higher than the oneof the commercial samples while the same electrical connections were used. It isthen clear that dielectric dissipations intrinsic to the harvesters exist. It can besupposed that these losses are larger for the commercial structures because theywere made of a larger volume of piezoelectric material, so that the probabilityof defect in the material is increased.

Until now, only the experiments leading to non frequency dependant rela-tions have been described. In the case of the experiments X1, X2, X4, X5,X7, X8 and X9, the theoretical relations between the applied and measuredvariables follow a different behavior, typical of resonant systems, described bymathematical functions disposing of maxima and minima. For X4, X5 andX9, the theoretical relations are extremely tedious when both dielectric andmechanical dissipations are considered and we disregard them as the preciseextraction of the parameters is made difficult by the complexity of the rela-tions. Therefore, the analysis is limited to X1, X7 and X8. For these threeexperiments, the theoretical relations corresponding to the maximum of theamplitude and to the maximum and minimum of the real part are derived. Forthese three remarkable points, the theoretical expressions of the elements listedbelow are summarized in Table 3.5 and Table 3.6.


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

X3 Measured value: I

X6 Measured value: V*ω

X10 Measured value: I/ω

Amplitude

vΓ 21

e

p e

Q vC Q

Γ

+ 21p e

e

C QV

Q+

Phase

0

( )1/ eArcTan Q

( )1/ eArcTan Q

Real part

vΓ

( )2

21e

p e

Q vC Q

Γ

+

p

e

CV

Q−

Imaginary part

0 ( )21

e

p e

Qv

C QΓ

+

pC V

X11

Measured value: F

X12

Measured value: F*ω

Amplitude

VΓ 21

e

p e

Q IC Q

Γ

+

Phase

0

( )eArcTan Q

Real part

VΓ

( )21e

p e

QI

C QΓ

+

Imaginary part

0 ( )

2

21e

p e

QI

C QΓ

+

46

10-11

2

46

10-10

2

46

10-9

I / ω

(A.s

)

500480460 ω(rad/s)

(a)

10-132

4

10-122

4

10-112

4

10-10

I /ω

(A.s

)

440042004000 ω (rad/s)

(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.


• 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 thebeam and of the attached mass. However, as explained in Chapter 2, theinertial force on the beam itself can be neglected for the tested samples andit is relevant to only consider the resulting force on the attached mass. Theapplied force can be expressed as F=meA0 in which me and A0 are the values ofrespectively the effective mass and the amplitude of the acceleration of the inputvibration. For experiment X1, the deflection of the beam was monitored usingan 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 resonancefrequencies ωs0 are found to be equal to 4153rad.−1 for the MEMS bender andto 469rad.s−1 for the commercial one. It is now possible to determine theeffective mass of the structure reminding that ωs0=(k/me)1/2. For the AlNMEMS device, k was previously found equal to 190 N/m, so that we can deriveme= 11 mg. The effective mass is in this case equivalent to the large massattached at the tip of the cantilever. For the PZT commercial element, weobtained 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 latterwas 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 andmaterial properties of the piezoelectric beam. It is also possible to determinethe mechanical quality factor from the maximum of the amplitude: Qm=640 isfound for the MEMS harvester and Qm=48 for the commercial structure. Thefrequencies of maximum and minimum of the imaginary part corroborated thesevalues.


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

X1 Measured value: δg


Maximum of the amplitude Frequency 0

sω 0sω

Value ( )20

m

se

Q Fm ω

( )20

m

se

Q Fm ω

Γ

Corresponding phase

0

0

Maximum real part Frequency

0sω 0

sω Value ( )20

m

se

Q Fm ω

( )20

m

se

Q Fm ω

Γ

Corresponding value of the imaginary part

0

0

Imaginary part Maximum Minimum Maximum Minimum

Frequency

011−s

mQω

011+s

mQω

011−s

mQω

011+s

mQω

Value

( )202

m

se

QF

m ω

( )202

m

se

QF

m ω−

( )202

m

se

QF

m ω

Γ

( )202

m

se

QF

m ω

Γ−

Value of the real part ( )202

m

se

Q Fm ω

( )202

m

se

Q Fm ω

( )202

m

se

Q Fm ω

Γ

( )202

m

se

Q Fm ω

Γ

2.0x10-4

1.6

1.2

0.8

0.4

Am

plitu

de o

f δg

(m)

500480460440 ω (rad/s)

-80

-40

0

40

80

Phase (°)

(a)

3.5x10-4

3.0

2.5

2.0

1.5

1.0

0.5

Am

plitu

de o

f δg

(m)

4180416041404120 ω (rad/s)

-80

-40

0

40

80

Phase (°)

(b)

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


Table 3.6 - Theoretical relations for experiments X7 and X8 in steady state mode.


X8 Measured value: δg

Maximum of the amplitude Frequency

0sω

Value

( )20

m

se

QV

m ω

Γ

Corresponding phase

0

Maximum real part Frequency

0sω 0

sω Value

21p m

eC K Q V

Q⎛ ⎞

− −⎜ ⎟⎝ ⎠

( )20

m

se

QV

m ω

Γ

Corresponding value of the imaginary part

pC V

0

Imaginary part Imaginary part Maximum Minimum Minimum Minimum

Frequency

011−s

mQω

011s

mQω +

011−s

mQω

011+s

mQω

Value

21

2⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠

mp

K QC V 2

12m

pK QC V

⎛ ⎞−⎜ ⎟⎜ ⎟

⎝ ⎠ ( )202

m

se

QV

m ω

Γ -( )202

m

se

QV

m ω

Γ

Value of the real part

212m

pe

K QC VQ

⎛ ⎞− −⎜ ⎟⎜ ⎟

⎝ ⎠

212m

pe

K QC VQ

⎛ ⎞− −⎜ ⎟⎜ ⎟

⎝ ⎠

( )202

m

se

QV

m ω

Γ

( )202

m

se

QV

m ω

Γ

X7 is easily performed with a classical impedance analyzer as for the clampedelectrical impedance measurement corresponding to experiment X10. The ob-tained measurements are illustrated in Figure 3.11. The real-imaginary partsrepresentation allows determining Qm=670 and Qm=58 for respectively theMEMS and commercial bender. Also, from the values of the clamped capaci-tance and of the electrical quality factor derived previously, one can determineK2=2.9*10−3 for the MEMS structure and K2=1.2*10−1 for the commercialone. 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 com-mercial element. These values are in agreement with those measured with staticexperiments.

The last performed steady state measurement, X8, is done with the samePSD setup used for X1. The obtained results are depicted in Figure 3.12. Theextracted values of the different parameters were Qm=660, Γ=2.5*10−5N/Vfor the MEMS sample and Qm=64, Γ=5.5*10−4N/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


1.5x10-9

1.0

0.5

0.0

-0.5

I /ω

(A.s

)

490480470460450440 ω (rad/s)

(a)

100x10-12

80

60

40

20

0

I /ω

(A.s

)

4180416041404120 ω (rad/s)

(b)

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

2.4x10-4

2.0

1.6

1.2

0.8

0.4 Am

plitu

de o

f δg

(m)

490480470460450440

ω (rad/s)

-80

-40

0

40

80

Phase (°)

(a)

4.0x10-4

3.0

2.0

1.0

0.0

Am

plitu

de o

f δg

(m)

4180416041404120

ω (rad/s)

-80

-40

0

40

80

Phase (°)

(b)

Figure 3.12 - Results of X8 in steady state mode for (a) the PZT commercial bender (V=5V), (b) the AlN MEMS bender(V=5 V). The solid line corresponds to the amplitude of thedeflection, 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 charac-terization allow determining most of the parameters of the equivalent networkrepresentation of the piezoelectric benders. The different experiments lead tosome variation of the measured parameters, because of the deviations from themodel and because of experimental errors. However, the spread of the measure-ments was reasonable and their averages can be considered as a representativevalue. The steady state characterization approach proposed in this section isvery simple and can be refined by, for example, considering more characteristicpoints and also their derivatives. The obtained values might be processed byan algorithm performing statistical analysis. At the moment, such an algo-rithm has been written solely for the clamped and free impedance analysis. Inthe future, it will be combined with other routines analyzing the results of theother measurements.


3.2.4 Transient characterization methods

For piezoelectric benders, transient methods of characterization are limited dueto the complexity of the analytical treatment of the transient response of thedevices. As shown in Chapter 4, it is not possible to obtain a simple closedform of the transient behaviour in a general case. It is not relevant to moni-tor the open circuit voltage or short circuit current resulting from a shock, asthe possible presence of dielectric losses makes the behavior of these two vari-ables relatively complex. For the same reason, it is difficult to realize a preciseanalysis of the waveform of the deflection resulting from an initial voltage orcurrent. Therefore, the sole parameters that can be easily obtained from tran-sient techniques are the fundamental resonance frequency and the mechanicalquality factor. These parameters can be obtained by applying an impulse forceor by setting initial velocity or displacement conditions to the short-circuitedstructure and by monitoring the time dependence of the deflection after theimpulse: 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 condi-tions is equivalent to the one of a purely elastic beam. In this case, the freeoscillations follow a pseudo sinusoidal behavior described by

δg (t) = e−ωs0/2Qmt[

A cos

(ωs0

√1− 1

4Q2m

t

)+B sin

(ωs0

√1− 1

4Q2m

t

)](3.1)

in which A and B are real constants. Classical methods such as the logarith-mic decrement [151] or a simple fit with an exponential function are used todetermine the ratio ωs0/2Qm. From this ratio and from the free oscillationfrequency ωs0(1-1/4Qm)1/2 the values of ωs0 and Qm are obtained. ωs0=4150rad.s−1 and Qm=710 for the MEMS device, while ωs0=468 rad.s−1 and Qm=62for the ceramic PZT bender.

In the next section, typical values of the network parameters for the dif-ferent fabricated MEMS devices and commercial samples are summarized andcompared. Also, the material properties of the elastic and piezoelectric layersimplemented in the piezoelectric benders are estimated from the performedmeasurements.

3.2.5 Typical values of the network parameters and esti-mation of the material properties

The different characterization experiments done on the AlN based Dev1 samplehave been repeated for a large number of samples (PZT and AlN based MEMSharvesters and ceramic PZT commercial devices). The obtained results wererelatively homogeneous and corresponded to the expected values for most of


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 theAlN Dev1 sample and the commercial bimorph characterized in the previoussections. Also, the parameters for a PZT Dev1 sample are given in the sametable.

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 easilybe 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. Thedisplacement actuation performances are controlled by the ratio Γ/k similarfor 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 MEMSdevice performs in this case better than the PZT MEMS structure, but not aswell as the commercial bender.

The mechanical dissipations are mainly due to air damping and increasewith the size of the devices. It is then observed that the mechanical qualityfactor of the commercial device is much smaller than for the MEMS. TheMEMS devices tested in this chapter are unpackaged. The packaged deviceshave a lower mechanical quality factor, similar to the one of the commercialstructures. It is planned to package the MEMS devices under vacuum in orderto avoid this problem. The values of the electrical quality factor are in thesame range, but slightly smaller for the PZT devices. Also, it can be seen thatthe 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 materialcompared to the one of the substrate in the MEMS devices. Indeed, as shownin Chapter 2, the generalized electromechanical coupling factor depends solelyon the thicknesses and compliances ratio and on the piezoelectric constant k31.K2 is an important parameter for power generation and one could argue that theMEMS devices should have been designed for optimum K2. However, technicalproblems limit the thickness of the deposited piezoelectric layer and the beamshould not be too thin in order to avoid fracture phenomena. Furthermore,it is shown in Chapter 5 that, for the machine environment harvester, it isnot necessary to optimize the generalized electromechanical coupling factor ifparasitic dissipations can be maintained below a certain threshold.

Combining the measured values of the different parameters with their the-oretical expressions in terms of dimensions and material properties developedin Chapter 2 ((2.58), (2.59), (2.60), (2.40)and (2.41)), the effective complianceof the substrate ss, the effective permittivity εTp and the effective piezoelectricconstant ep=dp/sEp of the piezoelectric material can be estimated. In order tochoose the correct equations, it is important to remind that the MEMS devicesare unimorphs while the ceramic PZT beam are symmetric bimorphs. Also,the thickness of the piezoelectric material was neglected for the MEMS devicesand 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


found in the literature [78]. For the PZT MEMS harvesters, the complianceof the silicon differs slightly from the one found from the analysis of the AlNdevice, probably because the etching of the beam is imperfect and results inthickness unhomogeneities. Inostek (http://inostek.com/), the supplier of thePZT covered wafers, do not provide the material properties of their PZT sothat it was not possible to compare the obtained values with existing data. Forthe PZT commercial structure, the thickness of the piezoelectric material isnot negligible as it is the case for the MEMS ones. Therefore, the values of thepiezoelectric material compliance given on the manufacturer website (sp=17pPa−1) are used to estimate ss. The measured value of the permittivity andof ep corresponds approximately to the manufacturer’s data. Unfortunately, itis not possible to estimate the values of the electromechanical coupling factork31 in the case of the MEMS harvesters as the compliance of the piezoelectricmaterial is unknown. For the PZT commercial structure, k31=0.33.

Table 3.7 - Values of the network parameters and estimated material properties for thedifferent types of tested samples. The standard deviation of the measurements is indicatedwhen possible.

AlN MEMS PZT MEMS Ceramic PZTk0 (N.m−1) 190 830 749me (mg) 11.2±0.4 11.4±0.2 3390±100Γ (µN.V−1) 22±3 170±24 496±47

Measured Cp0 (nF) 0.64±0.01 71±1.6 2.58±0.03parameters Qm (-) 656±15 1200±23 57±8

Qe (-) -110±13 -63±9 -30±6K2 (-) 0.0036±0.0006 0.00057±0.00003 0.114±0.008ωs0 (rad.s−1) 4120±74 8691±89 470±7W (mm) 3 3 3.2l (mm) 2.37 2.37 31.8

Dimensions L (mm) 3 3 7hs (µm) 45 80 340hp (µm) 0.8 1 270

Estimated ss (pPa−1) 6.6 8.5 14.7material ep (C.m−2) 0.9±0.1 4.1±0.5 11.9±1.1

properties εTp /ε0 (-) 8.5±0.2 1130±25 1850±21

3.3 Conclusion

In this chapter, the manufacturing by MEMS technologies of piezoelectric har-vesters and the experimental determination of their corresponding equivalentnetwork parameters is presented. A robust process flow is developed. It re-sults in functional devices but still needs some improvements at the level ofetch homogeneity and packaging. Particularly, the proposed ambient atmo-sphere packaging approach results in low mechanical quality factors and will

82 3.3 Conclusion

be improved in the future by implementing a vacuum packaging method.

Static, transient and steady state experimental methods for determining theequivalent network parameters are then described and implemented on MEMSPZT and AlN based harvesters and on a commercial ceramic PZT structures.No complete procedure for the characterization of the network parameters ofpiezoelectric beams existed before this work and methods are developed to thisaim. From a theoretical point of view, a large amount of experiments can berealized. However, from a practical point of view, some of them are difficult tobe done and only experiments that can be realized with commonly availablelaboratory equipments are presented. It is shown that the static methods allowmeasuring the stiffness k and the transformation factor Γ. All the parametersbut k can be determined from steady state measurement, which are for mostof them easily implemented. Finally, simple transient methods allow only ex-tracting the resonance frequency ωs0 and the quality factor Qm. Characteristicvalues of the network parameters are presented for the three types of structuresstudied. The material properties of the piezoelectric materials are estimatedfrom these measurements. It is shown that PZT is more adapted to actuationsituation than AlN, but, at the opposite, that the latter material has betterperformances in terms of sensing applications. It is now important to analyzethe capabilities of the piezoelectric benders when energy harvesting is consid-ered. This is done in the next two chapters, first for the human environmentdevice and then for the machine environment.

Chapter4Design and analysis of the humanenvironment vibration energyharvester

In this chapter, the design and the characterization of the impact type harvesterdescribed in the introductory chapter of this thesis is presented. For sake ofclarity, the general principles of the presented device are repeated. Classicalresonant systems are not easily adapted to the human environment because ofthe low frequency of the ”vibrations” occurring in this case. Therefore, theenergy harvester presented is based on a different principle which allows ob-taining a reasonable amount of generated electrical power for low frequencyexcitation. A conceptual representation of the human environment generatoris depicted in Figure 4.1. It consists basically of a frame containing a channelwhich 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. Thebenders are cantilevers in our case. When the frame is shaken, the mass oc-casionally impinges on the piezoelectric structures and a part of the kineticenergy accumulated by the moving object is transformed into electrical energythrough the impact on the piezoelectric structures. The output voltages ofthe benders are processed by an electronic block which can also contain thepowered application. This type of harvester has been already discussed by afew authors: Umeda [53–55] pioneered the analysis of the energy generated bythe impact of a steel ball on a piezoelectric membrane. Keawboonchuay stud-ied high power impact piezoelectric generator that can be incorporated intoammunitions [26]. The author of this thesis presented some analysis relatedto the impact harvester for human applications in [56, 57, 155]. Cavallier [58]and Takeuchi [59] described an experimental analysis of an equivalent device.

83

84 4.1 Modeling of the impact based harvester

Processing electronics

& powered application

Free moving object Guiding channel

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 thedevice including the description of the impact mechanism and of the resultingbehavior of the piezoelectric bender. Therefore, a simple but complete modelof such a harvester is developed in a first part of this chapter. The modelis based on the equivalent network representation of the piezoelectric bendersdescribed in Chapter 2. The experimental validation of the model and thecharacterization 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 objectcan be represented by fundamental free oscillations. The details of these ap-proximations and relevant derived information are described in a first part ofthis subsection. The equations describing the dynamics of the full system arealso derived. In a second part, particular types of load circuitry are introducedin the problem and the output power of the harvester is theoretically optimizedfor 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 cannot be easily treated without considering specific applications. Low velocityimpacts are relevant in this thesis. They do not result in extreme reconfig-urations of the considered system and they can be classified along the fourcategories described by Stronge [156]. The simplest representation of impactphenomena is achieved by the so called stereo-mechanical model: in this case,

4. DESIGN AND ANALYSIS OF THE HUMAN ENVIRONMENTVIBRATION ENERGY HARVESTER 85

the dynamics of two perfectly rigid and smooth spherical particles is predictedby applying the momentum conservation principle established by Newton. Forbodies with a finite stiffness, one should consider the local deformations thatoccur in the neighborhood of the impact surface, assumed to remain small.The Hertz theory is used to describe this situation. A third category consistsin the transverse impact of a rigid body on a flexible element. This case is themost relevant in the analysis of the harvester. A fourth category of impactsdescribed by Stronge consists in axial impact on flexible bodies and is not ofinterest here.

The proposed analysis is limited to one dimensional collision on a flexiblebeam. It is assumed that the velocities of the objects and the efforts resultingfrom the collision are directed perpendicularly to the large faces the beam.The contact surfaces are supposed to be perfectly smooth so that the frictionmechanisms 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 directedalong the normal to surface of the bender. In this application, this velocitydepends on the particular excitation applied to the frame of the device ofFigure 4.1. The impact process itself is described by Figure 4.2(b) and consistsof two separate phases: in the first, the two objects tend to interpenetrate eachother and a local compression referred to as indentation δi is observed in thearea surrounding the contact surfaces. The contact area becomes larger as themagnitude of the indentation increases. The compression phase ends when theamplitude of the restoring elastic force F(t) is large enough to induce a localexpansion of the two bodies tending to repulse each other at this moment.During the contact time, a radial wave due to the indentation starts propagatingaway from the impact location. It is assumed in this work that the contact areaand the magnitude of the indentation are small so that the corresponding radialwave does not have an important influence on the macroscopic behavior of thetwo bodies and it is ignored. Another wave consisting in a vertical displacementis initiated in the beam during the impact process. It is referred to as bendingwave and results from the exchange of momentum between the two objects. Theimpact of the missile on the beam might excite a large number of oscillationmodes. However, for the configuration studied in this chapter, it is shownlater that most of the energy transferred to the beam during the impact isconfined to the fundamental mode of vibration, so that only the fundamentalbending wave is considered. After the expansion phase (Figure 4.2(c)), the twoobjects separate with velocities directed along opposite direction. The missilealso oscillates along its own eigenmodes after the separation. As the movingobject is assumed to be very stiff, these oscillations do not have influence on thegeneral dynamics. In the beam, the bending wave propagates further away fromthe impacted vertical segment. The behavior of the bender prior to a secondimpact consists in free oscillations. It is assumed that the time necessary fora ”standing” wave to be established is negligible compared to the fundamentalperiod of the beam so that the displacement induced at the impact location is


Bending wave

Indentation wave

Compression

F(t)

‐F(t) ‐F(t)

F(t)

Expansion

(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 ofthe beam. This last simplification might not be correct in case of an extremelyflexible or long bender. Also, the contact time tc is considered to be so shortthat 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 Hertzmodel [157]. In this approximated theory, it is assumed that the dynamics oftwo curved contact surfaces during an elastic impact can be represented by asingle degree of freedom model consisting of two lumped masses connected by anon linear spring ki creating a contact force equal to kiδi3/2. In a first approx-imation, the lumped mass associated with the impacted segment of the beamcorresponds to its effective mass me. For the impacting missile, it correspondsto its full mass M. The expression of the indentation stiffness ki obtained bythis method is given by (4.1), in which Rb and RM represent the curvatureof respectively the beam and missile impact surfaces at the initial contact (inthe 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 ten-sors of the two objects (the subscript b refers to the bender, M to the missile),according to the direction convention given in Chapter 2.

ki =43

√(R−1b +R−1

M

)−1((1−

(sb,13

sb,33

)2)sb,33 +

(1−

(sM,13

sM,33

)2)sM,33

) (4.1)

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


Fc and Ωc found by Lee’s method are

Fc = 1.437ki

0.5MU20

ki

(1 + M

me

)3/5

(4.2)

Ωc =

(k2iU0

1.306M2

(1 +

M

me

)2)1/5

(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 tothe description of the impact, numerical values of the different characteristicparameters are computed by using the material properties and geometries ofthe problem. Representative dimensions of the piezoelectric cantilevers imple-mented in the human environment harvester are given in Table 2.1. For thenumerical application, the radius of curvature of the impacting missile at thecontact surface is RM=5 mm while Rb tends to infinity (flat surface). M=4g and me=0.4 g corresponding to the case C of Table 2.1. U0=1 m.s−1 andthe fundamental resonance frequency of the beam ωs0=1700 rad.s−1. For theindentation related compliances, the relevant values of steel for the movingmass 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−1and Fc=60 N. The contact time tc and the maxi-mum indentation are equal to 20 µs and 5.5 µm respectively. The assumptionsof negligible duration of the contact and of negligible indentation magnitudeare then justified. The numerical application is carried out considering thegeometries of a C type bender, but the same conclusions can be reached withtype B.

The first assumption used in Chapter 2 when developing the equivalent net-work of the piezoelectric bender is proven; i.e. the duration of the contact ismuch smaller than the fundamental period of the beam. The impact force ofa moving object on the beam can then be represented by a pulse type load. Itis also assumed in the derivations of Chapter 2 that the impact process excitessolely the fundamental mode of vibration of the beam. The validity of thisassumption is verified by performing 2D plane stress FEM simulations of thebeam C of Table 2.1. A triangular mesh and quadratic elements are used. Theanalysis is purely mechanical and piezoelectric effects are not considered. Atransient solver is used. Elastic behavior and small deformations are assumed.The first boundary condition of the model corresponds to the clamped end ofthe 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 approxima-tion, it is implicitly assumed that this force is applied on the full width of thecantilever, rather than being concentrated on a single point. This considera-tion does not change the conclusions related to the single mode approximation.


Pos

ition

of t

he ti

p of

the

beam

(m)

5x10-343210Time(s)

Figure 4.3 - Position of the tip of the beam C obtained from FEM (solid line) and by thelumped 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 comparedto the one found by a single degree of freedom representation (”mechanical”part of the equivalent circuit developed in Chapter 2). It is observed that thehigh order frequency components of the response have a small effect on thefirst periods of oscillation of the beam, but they do not have any influenceafterwards. The single mode approximation can then be implemented withoutyielding noticeable errors in the general dynamics of the system. The resultsof the analysis are only presented for a C type beam, but they are also true fora B type one.

As the contact time is so short, it is also assumed that the impact processand the corresponding pulse results in a quasi instantaneous redefinition of thevelocity of the impacted segment in the beam and of the bulk of the movingobject. In this model, the dynamic behavior of the system is then discontin-uous at the moment of the collision. The velocity of the objects before (vgand vM ) and after (vg

′and vM

′) impact can be obtained by considering that

the variation of momentum occurring during the compression-expansion phaseshould equate the time integral of the pulse F(t):

me

(vg − v

′

g

)= M

(vM − v

′

M

)=

ti+tc∫ti

F (t) dt (4.4)

The momentum conservation principle is sufficient to determine an expressionof 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 noenergy dissipations occur during the compression-expansion phase. In practice,complex dissipation mechanisms result from the collision. The elastic energycan for example be transformed into heat by viscoelastic or internal frictionphenomena or induce local plastic deformations. In this case, the value of the


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

e2 =e2bWc,b + e2

MWc,M

Wc,b +Wc,M(4.5)

This representation is particularly useful here as the duration of the impactprocess is very small. e depends on the effective masses, on the curvatures andareas of the contact surfaces, on the material properties and on the relativecollision velocity (e decreases if U0 increases [159, 160]). No efforts are devotedfor finding an explicit expression of the coefficient of restitution. Instead, thevalues of e are determined experimentally.

Considering the fact that Wc,b+Wc,M is equal to the initial quantity ofenergy before impact, the energy Wd dissipated during the impact is

Wd =12

meM

me +M

(1− e2

)U2

0 (4.6)

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

v′

g =mevg +MvM − eM (vg − vM )

M +me(4.7)

v′

M =mevg +MvM + eme (vg − vM )

M +me(4.8)

The results of the developed model are very similar to those coming from thesimple approach of Newton for two impacting spheres [161].

All the elements necessary to establish the simplified system of equationsdescribing the dynamics of the human environment harvester are now avail-able. If a simple representation of the losses which occur during the slidingof the free mass is assumed, the dynamics of the missile are described in thereferential of the device’s frame by the simple ordinary differential equationMd2δM/dt2+DvdδM/dt=Fext in which Dv is a viscous damping factor, δM rep-resents the position of the mass respectively to one of the piezoelectric benders


1

•

gδ 1/k0

Cp

me

V2

Fext

M

Dv e

e

2

•

gδ

δM=δg2 ?

δM=δg1 ?

V1

1

•Q

2

•Q

Mδ•

Γ : 1

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

(δM is assumed one dimensional in our model) and Fext is the apparent forceresulting from the motion of the frame (the model is developed in the referentialattached to the device’s frame). The dynamics of the bender are representedby the quasi static solutions of the beam equation developed in Chapter 2 (itis assumed that the low frequency motion undergone by the frame do not af-fect noticeably the behavior of the beams). The behavior of the full systemis obtained by coupling the two previous equations with the developed impactrepresentation which consists in redefining the bodies velocities at the momentof collision. By similarities with the considerations expressed in Chapter 2, itis possible to develop the electrical network model of the impact energy har-vester depicted in Figure 4.4 in which the indices 1 and 2 correspond to the twobenders. The electrical equivalent of the piezoelectric beams has already beendiscussed previously. Dielectric and piezoelectric dissipations are neglected inthis 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 caseof the missile, its dynamics are represented by the series association of aninductance M and a resistor Dv. The impact coupling between missile andbenders is represented through sensing/actuating type elements (indicated bythe encircled symbol e) which determine if the moving mass enters into contactwith one of the benders. In this case, the velocities of the missile and of theimpacted bender are reinitialized according to (4.7) and (4.8).

The base of the model that used for the analysis of the human motion energyharvester has been developed in this section. In the next, specific types of loadcircuitry and of input forces are introduced so that the power delivered to theload can be analyzed and optimized from a theoretical point of view. Theremainder of the modeling part is divided in two sections: in a first one, effortsare devoted in order to establish analytical elements which allow obtaining


1/k0

Cp0

Γ: 1 me

F

'(0)δ•

=g gv

RL

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

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

4.1.2 Theoretical optimization of the generated power:analytical perspectives

It is impossible to develop an analytical model representing the complete be-havior of the human environment harvesters for a simple reason: the equationsof motion are transcendental. Indeed, sinusoidal components constitute themotion of the beam while the motion of the missile contains (at least) onelinear component vM

′after an impact. No closed form solutions are known

for transcendental relations, so that it is impossible to determine an analyticalexpression of the collision times (except for the first one).

The simplified situation considered in this part is illustrated in electricalnetwork form by Figure 4.5. It consists of a single impact approximation inwhich the missile hits one of the benders and does not interact with it after-wards. In this case, the piezoelectric beam undergoes after impact unperturbedfree oscillations resulting from the induced velocity vg

′(the bender is assumed

at rest before collision). Also, a purely resistive load RL is assumed in thisanalytical approach. It can be understood that the bender behaves after aninput 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 im-pulse can then be defined as Pd=E0/ta, in which ta represents the time intervalbetween two consecutive impacts or shocks. The bender should be designedin such a way that its output power is the closest possible to this theoreticallimit. The optimization process goes through the solution of the free oscillationproblem described above.

The detailed computations of the proposed analysis are given in [56] andonly the most important steps of the derivation are presented. Applying Kirch-hoff’s laws to the circuit of Figure 4.5 and a Laplace transformation to the


obtained equations leads to the expressions of the deflection δg and voltage Vgiven below.

L (δg) =mev

′g (1 + CpRLs) k + ( k

Qmωs0

+ kCpRL + Γ2RL)s

+(me + CpRL

k

Qmωs0

)s2 +meCpRLs

3

(4.9)

L (V ) =mev

′gΓRLs k + ( k

Qmωs0

+ kCpRL + Γ2RL)s+(me + CpRL

k

Qmωs0

)s2 +meCpRLs

3

(4.10)

s represents the Laplace variable and the L function indicates the Laplacetransform 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 theload. In order to reduce the number of variables involved in the denominatorof (4.10), the dimensionless parameters defined in Table 4.1 can be used andthe Laplace transform of the voltage can be rewritten as

L (V ) =v′gωs0

RLΓs

ωs0 1 +[Ψ(1 +K2) + 1

Qm

]sωs0

+

(1 + ΨQm

)(sωs0

)2

+ Ψ(sωs0

)3

(4.11)

Ψ represents the ratio of the fundamental resonance frequency ωs0 to the cut-offangular frequency ωe of the electrical RC circuit, Qm is the mechanical qualityfactor of the system and K is the GEMC of the piezoelectric unimorph (definedin Chapter 2). In order to approach the optimum output power per impulsedefined in the previous section, it should be insured that most of the initialamount of energy present in the bender after an impact is transferred to theload and not dissipated by parasitic mechanisms. Because the parasitic dis-sipations and the energy harvesting process compete in time for transformingthe initial amount of mechanical energy, it can be assumed from the definitionof the mechanical quality factor and of the GEMC that optimum harvestingperformance is achieved when K and Qm are maximized. Secondly, an electri-cal load appropriately matched to the system should be defined, so that theelectrical damping is optimum. It is possible to obtain an analytical expressionof the power dissipated into the load by applying an inverse Laplace transformto (4.11) in order to obtain the expression in the time-domain of the voltageV. The average power Pd dissipated into the load resistor can then be writtenas

Pd =1

RLta

∫ ta

0

V (t)2dt (4.12)


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

Short circuit mechanical angular frequency 0ω =s

e

km

Cut-off angular frequency of the RC circuit e

1=

L pR Cω

Ratio of the mechanical and electrical angular frequency 0

eΨ =

sωω

Generalized electromechanical coupling factor (GEMC)

22

p

Γ=KkC

in which ta represents the time interval between two consecutive impacts orshocks.

The value of the optimum load resistor is found by determining the max-imum of (4.12) in terms of RL. The exposed procedure is relatively complexfrom the mathematical point of view and it is preferred to follow a differentapproach, based on reasonable approximations and giving better insights in thephysics of the problem. It has been stated previously that the damping effects(parasitic and energy harvesting process) compete in time, so that the rateat which energy is extracted from the harvesting process (or in other words,the instantaneous harvested power) has to be maximized, independently of theamplitude of the parasitic dissipations. Furthermore, it can be seen from thedenominator of (4.11) that the parasitic damping and the energy harvestingprocess act independently, as the coefficients of s/ωs0 and of s3ωs0

3 do not con-tains cross terms of Qm and Ψ. Then, the optimization of the system can beperformed by analyzing the parasitic damping free behavior of the system andby finding the parameters which lead to the shortest possible time requiredfor dissipating integrally the initial quantity of energy E0. From (4.12), theshortest possible time to transform the initial amount of energy corresponds tothe shortest possible settling time for the voltage. The parasitic free expressionof the voltage is obtained by assuming Qm→∞ in (4.11):

L (V ) =v′

g

ωs0

RLΓs

ωs0

1 + [Ψ(1 +K2)] sωs0

+ s2

ωs02 + Ψ s3

ωs03

(4.13)

In order to proceed, it is necessary to determine the time domain form ofthe 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 caseof second-order systems, O(s) would be a quadratic polynomial in s and theinfluence of the nature of its roots on the transient characteristics of a system iswell-known: a couple of complex conjugates roots means that the system has a


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

<0

1 real r1

2 complex r2, r2*

( ) ( )1 2( )1 2 2 3 2( ) ( )⎡ ⎤+ +⎣ ⎦r t Re r tC e e C cos Im r t C sin Im r t

>0

3 single real r1, r2, r3

31 2

1 2 3r tr t r tC e C e C e+ +

=0

1 double real r1 1 single real r2

1 2

1 2 3( ) r t r tC t C e C e+ +

=0

1 triple real r1

121 2 3( )+ + r tC t C t C e

pseudo-oscillatory behavior, a double real root that it is critically damped andtwo real and different roots that the system is overdamped. In free oscillationscases, the critically damped state is the one for which the system comes themost rapidly at rest, without oscillating. In the following it is shown thatit is also theoretically possible to reach the equivalent of a critically dampedbehavior in the investigated system, which is a third order one.

The different forms of the time domain expression of (4.13) are given inTable 4.2 considering the possible values of the discriminant of O(s). In thistable, the symbols Re and Im indicate respectively the real and imaginary partof the corresponding variable. The coefficients Ci are real numbers.

It has been shown in the literature that the shortest settling-time of third-order systems is obtained when the system parameters are such that the rootsof O(s) are near the triple real root [162]. This state corresponds to the lastline of Table 4.2. The values of K and Ψ corresponding to a triple real rootr1 for O(s) can be obtained by comparing the coefficients of O(s) with thoseof the polynomial (Cs-r1)3, in which C is a real constant. They are K2=8 andΨ=(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 factorcan not be obtained, so that this behavior, although theoretically possible, cannot 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 criticallydamped one and it is then also practically impossible to reach it. Therefore, thetime domain expression V(t) of (4.13) has necessarily in practical applicationsthe form given in the first line of Table 4.2, equivalent of the pseudo-oscillatorybehaviour of second order systems. It consists in the sum of an exponentiallydecaying term and of a pseudo-oscillatory one. The roots of the characteristicequation of the system consist in a real one r1, and in two complexes conjugatedones r2 and r2

∗. The real parts of the different roots can only be negative, as


they represent the inverse of the time constants of a system which has nophysical possibilities to diverge. V(t) can then be written as

V (t) = C1er1t + eRe(r2)t C2Cos [Im(r2)t] + C3Sin [Im(r2)t] (4.14)

Now that the form of the time domain expression of the voltage across the loadis known, efforts should be done to find an explicit expression of the differentparameters involved in it, in order to define their influence on the settling timeof the structure. A method proposed in [161] is adapted to the problem. Thismethod allows obtaining analytical expressions of the value of the load resistorleading to the minimum settling time of (4.14) and of the different parametersinvolved for this electrical loading condition. This method is based on theassumption that the minimum settling time is obtained when one or both ofthe 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 thismethod, it is found that, in the given conditions, the expression of the optimumload Ropt is

Ropt =1

ωs0Cp

(1 +

K2

2

) (4.15)

The corresponding angular frequency ωs of the shunted piezoelectric bender is

ωs = ωs0

√1 +

K2

2(1− K2

8) (4.16)

It is finally possible to derive the explicit expression of V(t):

V (t) = −Γv′g

2Cpωs0

exp (−ωs0t)

+ exp(−K

2ωs04 t

)(

cos(ωst) +(

1− K2

2

)sin(ωst)

) (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 Math-ematica. V(t) is first computed by performing a numerical inverse Laplacetransform on (4.13) considering various values of the parameters Ψ, RL , K2,ωs0 and Γ and it is verified that (4.13) leads to a correct approximation of thevoltage when parasitic dissipations are ignored. It is observed that the ex-pression of the shunted resonance frequency is correct within a few %. Theenergy dissipated into the resistive load (Pd*ta) is then computed using (4.12)for ta→∞ (the expression of the voltage taking into account parasitic dissi-pations was used in this case). It is first observed that, even in the presenceof parasitic dissipations, the value of the optimum load resistor Ropt obtainedfrom 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


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 thetime constant of the exponential term is just equal to the inverse of the shortcircuit and parasitic free natural frequency of the structure ωs0. Second, thetime constant of the pseudo-oscillatory term depends on both the generalizedelectromechanical coupling factor and natural frequency. Finally, the three co-efficients related to the exponential term, to the exponentially decaying sinusterm and to the exponentially decaying cosines one are in practice almost equal.In this case, minimizing the settling time of the structure means minimizingboth the time constants of the exponential term and of the pseudo oscillatoryones. This can be done by maximizing the generalized electromechanical cou-pling factor and the natural frequency ωs0. However, this conclusion is notabsolute for ωs0, because the model is valid when the duration of the inputimpulse is much shorter than the oscillation period of the bender. With highnatural pulsation of the structure, it might be practically difficult to find ap-plications in which our model is relevant. Also, the addition of an extra massat the tip of the bender (see Table 2.1) leads to a reduction of the naturalfrequency, 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 bendersin terms of impact energy harvesting are optimized by maximizing the general-ized electromechanical coupling factor, by limiting the parasitic losses and byappropriately tuning the value of the load. This analysis is carried out in theframework of an over simplified situation. However, the conclusions that arereached gives useful insights for the more representative situations investigatedby 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 generalrules for optimizing the output power of the harvester. The behavior of thedevice is very complex (it might even become chaotic [163]) and involves animportant number of independent variables. It is not relevant to analyze in-dividually the influence of each of the constituent on the behavior. In theprevious section, it is shown that the performances of the system are optimizedwhen the generalized electromechanical coupling factor and the quality factorof the piezoelectric bender are maximum. This conclusion has been obtained inthe framework of a simplified situation but it appears intuitively that it shouldstill yield for multiple impact case. Therefore, the parameters defining the gen-eralized electromechanical coupling factor and the quality factor are set in thefollowing to fixed values representative of the benders that are implementedin the experiments. k, Γ, Cp and Qm are set to respectively 1000 N/m, 0.5


mN/V, 2.6 nF and 50 (and then K2=0.12). The dissipations which oppose themotion of the missile (Dv) are not taken into account in the present computa-tions as it is clear that they have a negative influence on the output power ofthe system and they have to be minimized. The assumed input motion of theframe in the numerical analysis consists of quasi instantaneous 180 rotationsoccurring every second. Between these rotations, the frame is supposed to beoriented in such a way that the sliding channel is aligned with the gravity fieldg, so that the missile undergoes a gravitational force Fext=-M*g which makesit moving from one of the benders to the other (the action of gravity on thebeams is neglected). The considered input motion is representative of a move-ment 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 thisparameter in the numerical simulations, that the performances of the harvesterin terms of conversion efficiency decrease slightly as Λ increases. This effect canbe 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 whenthe second rotation occurs. In this case, a part of the potential energy of themass has not been transferred to the piezoelectric bender and the efficiency ofthe energy conversion is reduced. It does not necessarily mean a lower outputpower but in any case a lower output power per unit volume. However, forvalues of Λ ranging from 10 mm till 50 mm, the differences in the predictionof the model in terms of electromechanical conversion efficiency are noticeablebut no exceeding a few %, so that the analysis is limited to the constant valueΛ=15 mm.

From all the previously expressed considerations, the variables that are leftfor a parametric analysis of the harvester consists of the masses M and me,the coefficient of restitution e and the characteristics of the load circuitry. It isdetermined numerically that when the natural period of oscillation of the beamis much smaller than the period of the rotations of the frame, the influence ofthe 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 thesame performances for constant M/me independently of the particular valueschosen for the different masses. Concerning the electrical load, a simple type ofcircuit consisting in a pure resistor is considered. The situation is complicatedby the ”two sided” nature of the system. If the piezoelectric benders are shuntedby the same load circuit, they have to be connected in a parallel or seriesarrangement. In this case, when one of the benders is impacted, a part ofthe electrical energy it develops flows towards the second bender rather thantowards the load. This effect reduces inevitably the efficiency of the conversion.This problem might easily be eliminated by designing for example electricalswitches which close and open in a relevant way. For matter of simplicity, itis considered that each bender is connected to its own load. Furthermore, themissile is assumed at rest on one of the bender before each rotation. It ispossible to estimate the power generated by the harvester by studying solely


6

5

4

3

2

1

0 Bea

m a

nd m

issi

le d

ispl

acem

ent (

mm

)

0.250.200.150.100.05Time (s)

(a)

-4x101

-2

0

2

Vol

tage

(V)

0.250.200.150.100.05Time (s)

(b)

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

the behavior of the system between two successive rotations. As the period ofthe considered rotations is 1 s, the average output power corresponds to theenergy dissipated or stored into the load during that time.

Before presenting the results related to the output power, some observationson the general behavior of the system are proposed. The motion of the missileand bender after a rotation of the frame is illustrated in Figure 4.6(a). It canbe seen in the inset of the figure that each ”genuine” impact is followed bya multitude of short time related collisions until the missile is ejected fromthe neighborhood of the bender. At this moment, the beam oscillates freelytill a second genuine impact occurs. Depending on the values of the differentparameters, the missile might apparently ”stick” to the bender during a shortduration. In a perfectly inelastic case (e=0), the missile and the beam mergeinto a single oscillator just after impact. They separate when the accelerationof 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 anddepend on all different characteristic parameters. However, when the effectivemass of the bender is negligible compared to the missile, the behavior is verysimilar to the inelastic one independently of the value of the coefficient ofrestitution. The voltage developed on the electrical ports (open circuited) ofthe piezoelectric beam is reported in Figure 4.6(b). Each genuine impact resultsin a sharp peak of the voltage followed by decaying sinusoidal oscillations. Thesecondary collisions tend to distort the initial peak and they can have a stronginfluence on this part of the waveform of the voltage.

For sake of clarity, the simplified circuit representation of the harvester cor-responding to the situation that is investigated in this part is given in Figure4.7. Multiple impacts are here taken in consideration (not being the case inthe previous section). The parametric analysis that is performed consists indetermining the influence of M/me and RL on the efficiency of the energy con-version. The initial energy E0 present in the system at the beginning of a cycle


•

gδ

1/k0

Cp

me

‐Mg

M e

δM=δg ?

V

Mδ•

Γ : 1

RL

( )( )0 0

0

=

= Λg

M

δ

δ

( )

( )

0 0

0 0

δ

δ

•

•

=

=

g

M

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

can be approximated by the potential energy of the missile Mgλ with g=9.8m.s−2 and λ=15 mm. The energy EL dissipated in the load resistor during aperiod of the motion is obtained by integrating V(t)2/RL. The efficiency η isdefined 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 beenfound for a single impact approximation in (4.15) and it is first checked if thisformula is valid in multiple impact situations. To this aim, the efficiency ηis computed for a large variety of values of the coefficient of restitution andof the masses ratio, while considering a constant range of values for the loadresistor. The chosen range was centered on the value resulting from the ex-pression of Ropt given in (4.15). A few of the obtained results are presentedin Figure 4.8. The abscissa of the presented graphics corresponds to the loadresistor normalized to the optimum value (4.15). It can be seen in Figure 4.8that the coefficient of restitution does not have a strong influence on the valueof the optimum load, but on the other hand, that M/me plays an importantrole in this matter. When this parameter increases, the optimum load resistorin multiple impact situations Rm

opt shifts towards higher values. This effect canbe explained in the following way: it is assumed in the analytical derivationsof the previous section that the missile is impacting the beam a single timeand that the objects do not have any interactions afterwards. In this case,the beam oscillates solely along its fundamental frequency and the transfer ofenergy to the load resistor is optimized for the value of Ropt (4.15). In the mul-tiple impacts situation, the mechanical behavior of the bender can be dividedinto three characteristic periods. In the first, the missile is not in the vicinityof the beam and the situation corresponds to the one just described. In thesecond, the beam is in the neighborhood of the beam and several short timerelated collisions occur (inset of Figure 4.6(a)). For the third type ones, themissile apparently sticks to the bender and the system oscillates at a frequency


30

20

10

0

Effi

cien

cy (

%)

60.1

2 4 61

2 4 610

2 4

RL/Ropt

30

20

10

0

Effi

cien

cy (

%)

60.1

2 4 61

2 4 610

2 4

RL/Ropt

40

30

20

10

0

Effi

cien

cy (

%)

60.1

2 4 61

2 4 610

2 4

RL/Ropt

40

30

20

10

0

Effi

cien

cy (

%)

60.1

2 4 61

2 4 610

2 4

RL/Ropt

(a) (b)

(c) (d)

Figure 4.8 - Efficiency of the energy conversion versus RL/Ropt, (a) e=0.7, (b) e=0.5, (c)e=0.2, (d) e=0. In each graph, the black solid line corresponds to M/m=0.2, the black dottedline 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.

ωs(1+M/me)1/2. The occurrence of these periods depends on most of the differ-ent parameters of the model. However, periods of the second type are observedin situations involving high coefficient of restitution and M/me≈1, while thethird type occur for high M/me and low coefficient of restitution. The thirdtype characteristic periods of the motion introduce low frequency componentsinto the frequency spectrum of the bender displacement. The expression ofthe optimum load resistor given in (4.15) is found considering only the periodsof the first type. In order to adapt it to the multiple impacts situation andto the corresponding additional low frequency components, one can assume arepresentative or ”average” frequency of the motion. This average frequencyωm is inevitably smaller than the shunted resonance frequency ωs of the freebender when M/me1. As Ropt is inversely proportional to the frequency ofthe oscillation undergone by the bender, the optimum load in multiple impactscases Rm

opt=Roptωs/ωm shifts towards higher values. This effect can clearlybe observed in Figure 4.8 for large M/me independently of the value of thecoefficient of restitution.


40

30

20

10

Effi

cien

cy a

t opt

imum

load

(%)

0.1 1 10M/me

A

B

(a)

Pos

ition

(m)

Time (s)

Pos

ition

(m)

Time (s)

A

B

(b)

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

From the obtained results, it is possible to estimate the maximum efficiencyof the system to be approximately 40-50%. It is however not straightforwardto establish a clear influence of the coefficient of restitution and of the massesratio from the curves of Figure 4.8. It is observed that for a fixed value ofe, the efficiency does not vary monotonically with M/me and does not followan equivalent behavior for different values of e. Also, it can be seen thathigher values of the efficiency can be obtained with low values of the coefficientof restitution, which is somewhat surprising. In order to clarify this point,the efficiency at the optimum load resistor versus the masses ratio and fordifferent 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 coefficientsof restitution are considered. The presence of these peaks is explained bythe complex dynamics of the system and by the repeated exchanges of energybetween beam and missile. The dynamic situations corresponding to the pointsA and B in Figure 4.9(a) are illustrated in Figure 4.9(b). It can be seen that forthe maxima of point A, the missile is ejected after the first impact in such a waythat his position is just slightly above the one of the beam when it has reachedits maximum amplitude. In this way, the quantity of energy transferred to thebeam during the first impact remains in the bender and can be dissipated duringthe free oscillations. At the opposite, for the point B, the configuration is suchthat, after the first collision, the beam hits the missile when it has reached itsmaximum velocity and it transfer back a large part of its initial energy to themoving object. The quantity of energy left in the beam for harvesting is thenmuch smaller for the point B than for the point A. Equivalent reasoning can bedeveloped 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 behaviordoes not exist and is reduced to a ”sticking” situation. It is also observedfrom Figure 4.9(a) that when M/me becomes very large, the efficiency for thedifferent values of the coefficient of restitution merges. Finally, for middle rangevalues of the masses ratio, it is advantageous (counter intuitively) to design asystem 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 en-ergy harvester is optimized by maximizing the generalized electromechanicalcoupling factor, limiting the parasitic losses, choosing an appropriate load andimplementing an appropriate masses ratio. In the best conditions and neglect-ing 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 ex-perimentally and a prototype of the impact harvester is tested on the humanbody.

4.2 Experimental measurements

Most of the parameters of the network model of the impact harvester can befound 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 e is proposed for the materials andstructures implemented in this thesis. Then, the predictions of the model arecompared to experimental measurements in the simple rotary motion describedpreviously. Finally, a prototype of the impact harvester is mounted on thehuman body and its performances are measured.

4.2.1 Coefficient of restitution

The value of the coefficient of restitution e is obtained using a simple methodin which the velocity of a steel missile was measured before and after impactby means of laser detection. The impacted piezoelectric beam is clamped sothat there is no need to measure the velocity of the bender. The principle ofthe method is illustrated by Figure 4.10(a). The missile is dropped from anarbitrary height (the chosen values is 15 mm for most of the experiments) onthe surface of the clamped piezoelectric bender. A laser beam coupled to a lightdetector allows detecting the successive times t1 and t2 at which the movingobject crosses and exits the beam, so that from the basic law of motion, itsvelocity 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 wavecreated at the moment of the collision and which results in a voltage developedacross the open circuit piezoelectric laminate. The distance between the laserbeam and the piezoelectric bender LL is chosen so that it is just a bit largerthan LM . In this way, the dissipations due to air damping can be neglected in


LM

LL

t1

t3

t4

t2

Laser PSD

Clamped piezoelectric beam

Impact

(a)

Vol

tage

(V)

30x10-320100-10 Time(s)

t1 t2 t3 t4

(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 re-spectively to the voltage across the light detector and across the electrodes of the piezoelectricbeam.

this portion of the motion of the missile and its velocity just before impact v3

can reasonably be approximated by (2g(LL-Lm)+v22)1/2. In the same way, the

velocity just after impact v′

3 can be estimated from the time interval betweenthe impact and t4 so that v

′

3=(LL-LM )/(t4-t3)+g(t4-t3)/2. If one considersnow the expression involving the coefficient of restitution given in (4.8), it isseen that in case of the impact of a missile on a clamped beam (vg=v

′

g=0and meM), it is possible to write e= -v

′

3/v3, i.e. the energetic coefficient ofrestitution is equivalent to the kinematic one [156]. A typical measurement plotobtained from this method is given in Figure 4.10(b). The voltage across thelight detector becomes null when the missile passes in front of the beam. Asindicated on the graphic, the different times related to the instants illustratedin Figure 4.10(a) are easily obtained from the described measurement. Inthis experiment, the angle of drop of the missile is not perfectly controlled sothat small variations are observed on the measured values of e. The standarddeviation of the measurements was approximately equal to 5%, so that it isreasonable to consider the average of the distribution as a representative valuefor the next comparisons. The average value of e was found to be equal to 0.55.

4.2.2 Comparison of the model predictions with experi-mental measurements

The prediction of the model concerning the voltage developed across the elec-trodes of the piezoelectric beam and the power dissipated in the load resistor arecompared with experimental measurements using the arrangement described in


Figure 4.11(a). The equivalent network parameters of the piezoelectric beamused 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 charac-terized in Chapter 3, without attached mass) and e=0.55 is assumed. In a firstseries of experiments, the efficiency of the energy conversion versus the valueof the load resistor is measured for a drop distances of 15 mm. The resultsare reported in Figure 4.11(b). It is observed that the theoretical and ex-perimental optimum values of the load coincide well. Rm

opt=400kΩ≈2/(ω0Cp),corresponding approximately to the value found in Figure 4.8(b) for e=0.5 andM/me=10), but the predicted efficiency is approximately 50% higher than themeasured one. This result is explained because of the losses occurring duringthe motion of the missile that are neglected in the numerical computations.They play an important role on the general behavior of the device. These par-asitic 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 andof bounce. Indeed, it can be easily understood from Figure 4.11(a) that, inthe measurement setup used, the missile after a bounce might not move in adirection perfectly perpendicular to the beam and it can hit the sides of theguiding channel. Friction phenomena resulting from this effect decrease dra-matically the efficiency of the system. This assumption is verified by performinga second series of experiments in which the theoretical and experimental timedependence of the voltage across the piezoelectric beam and of the energy dis-sipated in the optimum load are compared (Figure 4.12). The model gives avery good estimation of the voltage and efficiency for the first impact. How-ever, the second impact occurs after a shorter time than the predicted one andthe corresponding efficiency step is smaller than expected. This effect becomesmore pronounced for the following collisions. It appears that at each bounce,an important part of the kinetic energy of the missile is dissipated because ofthe parasitic phenomena described above. In spite of the presence of this par-asitic damping mechanism, the conclusions related to the optimum bimorph’sparameters remains valid.

The experiments performed in this section are meant to establish a compar-ison between the predictions of the model and the measurements in a simpleexperimental situation. In the next part, a similar characterization is performedon a prototype of the impact harvester.

4.2.3 Characterization of a prototype of the human envi-ronment harvester

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


Load and measurement electronics

PSDLaser

Piezoelectric bender

Release pad

Guiding channel

Vertical stage

Missile

(a)

30

20

10

0

Con

vers

ion

effic

ienc

y (%

)

1042 4 6 8

1052 4 6 8

106

Load resistor (Ω)

(b)

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

60

40

20

0

-20

-40

-60

Voltage(V)

0.50.40.30.20.10.0Time(s)

(a)

30

20

10

0 Con

vers

ion

effic

ienc

y (%

)

0.80.60.40.20.0Time(s)

(b)

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

mm. A part of the housing of the prototype might be eliminated or replaced bythe powered application and conditioning electronics, so that the output powerper unit volume or mass can be improved. The piezoelectric cantilevers imple-mented in the prototype are not optimized for energy harvesting because of alack of materials. The characteristics corresponding to the network model werek=6000 N/m, me=600 mg, Qm=40, Γ=1 mN/V and Cp=9.6 nF. The corre-sponding 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 efficiencyof the energy conversion. For the coefficient of restitution, the same value of0.55 that was considered in the previous experiments (the same piezoelectricmaterials are implemented). Reliability problems are observed in the testingof the prototype because of the large deflection and dynamical stress resultingfrom the impact. In order to limit this problem, small magnets are attachedon each cantilever and on the closing cap in such a way that they repulse


(a)

(b)

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

each other. This simple method allows implementing a non linear spring whichsmoothes the motion of the cantilever when it approaches the closing cap. Innormal mode of operation, the magnets remain relatively far from each otherand they do not have a strong influence on the general dynamics and efficiencyof the system.

The harvester prototype was first tested according to the simple rotary mo-tion described previously. In this case and for the proposed configuration, thekinetic energy of the mass when hitting one of the cantilevers is Mgλ=593 µW.The theoretical and experimental values of the conversion efficiency are plottedin Figure 4.14. Because of the low value of the generalized electromechanicalcoupling factored, the maximum theoretical value of the efficiency is reducedto 13% instead of 40% in Figure 4.14(a). The discrepancy between the predic-tion of the model and the measurements is slightly lower than for the situationstudied in the previous subsection and the maximum experimental value ofthe efficiency is found to be around 8%. As before, this discrepancy is dueto the friction of the missile when sliding in the guiding channel and to theangle of bounce of the moving object after impact. A single rotation of thedevice is performed in these measurements but one can estimate that duringthe repeated rotary motion at 1 Hz, the total power dissipated in the resistiveloads would be approximately 95 µW.

The device is also tested when it is attached to the hand of a person andforcibly shaken. The frequency of the applied motion is estimated to 7 Hzwhile the amplitude is approximately 10 cm. This extreme situation is ob-tained by simulating a strong ”scratching” motion. Such vibrations might alsobe observed in some sportive situations (off road motor biking) and profes-


12

10

8

6

4

2

Con

vers

ion

effic

ienc

y (%

)

1032 4 6 8

1042 4 6 8

105

Load resistor (Ω)

(a)

600

500

400

300

Out

put p

ower

(μW

)

1032 4 6 8

1042 4 6 8

105

Load resistor (Ω)

(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 forparticular industrial applications such as weaving machines. The output powerof the device under these conditions and versus the load resistor is presentedin Figure 4.14(b). A maximum of 600 µW was measured. In such erraticconditions of excitation, the behavior of the power does not follow a smoothcurve. 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 supplyenergy to low consumption applications.

4.3 Conclusion

In this chapter, the design and characterization of a prototype of harvester ableto produce energy from the motion of human limbs is presented. The proposedharvester is based on the impact of a moving object on piezoelectric bendingstructures. An output power of 600 µW is obtained for a device of dimensions3.5*2*2 cm3 weighting 60 g placed on the hand of a person and shaken ata frequency of approximately 7 Hz for a 10cm amplitude. Also, a power of47 µW is measured when the harvester is rotated over 180 each second. Alarge amount of the volume occupied by the prototype of the harvester canbe 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 rotarymotion. 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 torealize an efficient impact energy harvester, one should apply the followingdesign guidelines:

• In order to maximize the efficiency of the energy conversion, the general-ized electromechanical coupling factor and the mechanical quality factorof the piezoelectric transducer have to be made as high as possible by

108 4.3 Conclusion

choosing appropriate materials and dimensions. Also, the parasitic dissi-pations in the motion of the missile have intuitively to be minimized.

• The resonance frequency of the bender should be high enough so that theamount of energy transferred by the moving object during an impact canbe dissipated prior to a second impact.

• The missile should be made as stiff as possible in order to avoid stor-ing energy into vibrations of the missile rather than in vibrations of thepiezoelectric transducer.

• In order to limit reliability problems due to high stresses and deflectionsresulting from high amplitude excitations, a damping system has to beimplemented in the neighborhood of the piezoelectric elements (repulsingmagnets are used here).

Chapter5Design and analysis of themachine environment vibrationenergy harvester

The output power characteristics of the harvesters designed for generating elec-trical power from high frequency and low amplitude vibrations (machine envi-ronment) are presented. As explained in Chapter 1, the developed harvestersare resonant devices which have to be excited in the neighborhood of their fun-damental resonance frequency for optimizing the production of electrical power.In this thesis, the investigations are limited to single harmonic mechanical ex-citations. In real life applications, broadband spectrum of input vibrationsshould be considered. In the framework of a first tentative of optimization,it is however assumed that the spectrum of most ubiquitous vibrations con-tains a dominant frequency which can be roughly approximated by a singlefrequency. The basic principles of energy extraction from harmonic vibrationsby piezoelectric elements have been developed in the seventies for purposesdifferent than producing useful electrical power: piezoelectric elements wereprimarily used for attenuating parasitic and undesired vibrations in mechani-cal machinery. The goals of the vibration damping and energy harvesting fieldsare different by essence but most of the principles developed for the former do-main are applicable in the latter, particularly when the design of an effectiveload circuitry is considered.

The energy harvesting field received little attention till the beginning ofthe second millennium. The amount of publications related to the subjecthowever increased exponentially in the following years. Roundy [28, 164] wasthe first to propose an analytical approach to the optimization of the generatedpower. The expressions that he developed are often taken as a reference for

109

110 5.1 Theoretical analysis of the harvester’s output power

elaborating simple estimation but it is shown in this chapter that they are onlyvalid in particular and restrictive conditions. Approaches equivalent to the oneof Roundy and refinements of the obtained results were proposed by numerousauthors such as Richards [75], DuToit [97], Mitcheson [165, 166] or the authorof this thesis [24]. Advanced but purely numerical methods were also recentlyproposed by Liao in [167].

The approach proposed here is based on the analysis of the equivalent net-work circuit developed in Chapter 1. It is organized as follow: in a first part, atheoretical analysis of the expected performances of the harvesters is presented.The analysis is concentrated on purely resistive load circuits. Alternative lin-ear and non linear signal conditioning methods are only briefly discussed. In asecond part, the output power of the devices is measured experimentally. Boththe MEMS fabricated and commercial benders based harvesters (Chapter 3)are characterized.

5.1 Theoretical analysis of the harvester’s out-put power

The elements proposed here are based on the steady state behavior of the piezo-electric benders and classical methods of harmonic analysis are implemented.Rather than utilizing the Laplace transform which is particularly adapted fortransient situations (see Chapter 4), the complex transform is used. The quasi-static version of the equivalent circuit proposed in Chapter 2 is taken as a basisfor performing the proposed analysis and is repeated in Figure 5.1 for sake ofclarity. The force F acting on the mechanical side of the device has been re-placed by the inertial force mesin(ωt) due to the input vibration applied to theclamped end of the piezoelectric cantilevers. ZL represents the electrical loadcircuitry connected to the harvester and in which energy is stored or dissipated.When dealing with loads made of linear components, ZL also correspond to theimpedance of the load circuit. Dielectric and parasitic dissipations are takeninto account and k0=k(1+jQm) and Cp0=Cp(1+jQe). In the following, theanalysis of the power dissipated by the harvesters through a perfect resistoris first presented in details. In a second part, alternative power conditioningmethods 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, thecharges developed in the piezoelectric layer are allowed to flow through RL. Thecorresponding energy dissipated per cycle is equivalent to the output power ofthe harvester. Roundy [41] was the first to analyze this problem for the scopeof energy harvesting, even if the same principle was understood since long in

5. DESIGN AND ANALYSIS OF THE MACHINE ENVIRONMENTVIBRATION ENERGY HARVESTER 111

1/k0

Cp0

Γ: 1 me

F

( )0 sin ω− em A t

RL

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

the field of vibration absorption [143]. Assuming steady state behavior, theaverage power Pd dissipated in the resistor can be written as

Pd =ω

2π

2π/ω∫0

V (t)2

RL=|V (ω)|

2

2RL(5.1)

in which the upper score indicated the complex transform of the correspondingvariable.

It is then necessary to determine first an expression of the voltage dropacross the load resistor before obtaining the one of the output power. ApplyingKirschoff’s laws in the frequency domain to the circuit of Figure 5.1 leads to

meA0 =(meω

2 + k0

)δt + ΓV (5.2)

V

RL= jωΓδt − jωCp0V (5.3)

This system can be solved for the complex voltage and deflection.

V =meA0RLΓs k

(1 + j

Qm

)+(Cp

(1 + j

Qe

)k(

1 + jQm

)RL + Γ2RL

)s

+mes2 +meCp

(1 + j

Qe

)RLs

3

(5.4)

δg =meA0

(1 + Cp

(1 + j

Qe

)RLs

) k

(1 + j

Qm

)+(Cp

(1 + j

Qe

)k(

1 + jQm

)RL + Γ2RL

)s

+mes2 +meCp

(1 + j

Qe

)RLs

3

(5.5)

In order to simplify the expressions, the non dimensional parameters introducedin Table 4.1 are also used in this chapter. According to the definitions ofTable 4.1, Ψ represents the ratio of the mechanical resonance frequency ωs0 tothe cut-off angular frequency ωe of the electrical RC circuit corresponding to theload resistor coupled with the piezoelectric capacitor and K is the generalizedelectromechanical coupling factor of the piezoelectric unimorph. The symbol


Ω =ω/ωs0 is also used in the following. The amplitude of the voltage and ofthe deflection can now be written as

|V | = A0ΓCp (ωs0)2

∗

√Ω2Ψ2

(Ω2 − 1)2 + Ω2Ψ2 (Ω2 − 1−K2)2 +Dm +De +Dem

(5.6)

|δg| =A0

(ωs0)2

∗

√√√√√√ 1− 2ΩΨQe

+Ω2Ψ2

(1 +Q2

e

)Q2e

(Ω2 − 1)2 + Ω2Ψ2 (Ω2 − 1−K2)2 +Dm +De +Dem

(5.7)

with

Dm =1Qm

(1Qm

+ 2ΩΨ(K2 +

ΩΨ2Qm

))(5.8)

De = −2ΩΨ

(Ω2 − 1

)2Qe

(1− ΩΨ

2Qe

)(5.9)

Dem = − 2ΩΨQeQm

(1Qm− ΩΨ

(1

2QeQm−K2

))(5.10)

The terms defined in (5.8), (5.9) and (5.10) are related to the parasiticdissipations. Dm is solely linked to the parasitic mechanical dissipations, whileDe depends only on the dielectric losses. Dem is a coupled term and is differentfrom zero only if both mechanical and dielectric dissipations are considered. Itcan also be seen that at the short circuit resonance (Ω=1), the pure dielectricterm De vanishes while the coupled term Dem does not.

All the elements necessary for computing the output power Pd of the har-vester have been established. By combining (5.1) and (5.6), one can elaboratethe expression of Pd given below.

Pd =meA

20

2ωs0

Ω2K2Ψ(Ω2 − 1)2 + Ω2Ψ2 (Ω2 − 1−K2)2 +Dm +De +Dem

(5.11)

In the absence of parasitic losses, two ”resonance” frequencies expressed bythe terms ω2-1 and ω2-1-K2 are clearly identifiable. In short-circuit configura-tion, RL=0 and Ψ is then also null in (5.11), so that the term Ψ2Ω2(ω2-1-K2)vanishes from the denominator. (ω2-1) corresponds to the normalized short-circuit fundamental normalized frequency Ωs0=1. In a situation close to anopen-circuit one, Ψ is very large and Ψ2Ω2(ω2-1-K2) dominates the denom-inator of (5.11). It corresponds to the normalized open-circuit fundamental


frequency Ωo0=(1+K2)(1/2)=1 of the bender. Ωs0 and Ωo0 can also be approxi-mately defined as the frequencies at which the impedance seen from the elec-trical ports of Figure 5.1 are respectively minimum and maximum. The shiftbetween the short and open circuit characteristic frequencies is pronounced ifthe generalized electromechanical coupling factor is large. Without parasiticdissipations, a non physical result is obtained from (5.11) when the bender isshort circuited and excited at Ωs0 or when the bender is open circuited andexcited at Ωo0, i.e. Pd→∞. This non-physical result is explained by the factthat the deflection of the bender is also mathematically infinite in these cases,so that in a practical situation, even if the parasitic dissipations are extremelysmall, the output power is limited by physical constraints such as for examplethe 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 inthe devices and have to be incorporated in the model. It is interesting to getsome 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 normalizedfrequency Ω considering the values of K2, Qm and Qe measured in Chapter3 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 ofresonance, the dielectric term De dominates the losses for both devices, evenif the difference is more pronounced for the MEMS devices. However, nearthe short circuit resonance, the values of De decrease sharply till becomingnull at Ω=Ωs0=1. The effect of the dielectric dissipations is reduced as the theimpedance of the piezoelectric vibrator reaches a minimum at the resonancefrequency. In this case, the mechanical losses term Dm dominates. In anysituation, the electromechanical variable Dem is at least 10 times smaller thanDm and can be neglected. It would also be tempting to neglect De as the devicesshould intuitively be excited around their resonance to generate a large output.However, it is shown later that the frequency corresponding to the maximumoutput 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 beobserved so that neglecting De might lead to noticeable errors.

Assuming the simplification proposed in the previous paragraph, it is nowpossible to determine the frequency(ies) and the characteristics of the loadleading to the optimum output power. The mathematical manipulations areeasier by determining first the optimum value of the load resistor (or of Ψ). It isobtained by computing the derivative of (5.11) with respect to Ψ, equating thecorresponding expression to 0 and solving the obtained equation with respectto Ψ. The determined expression of the optimum of Ψ is given in (5.12). IfΩ=1 and if a small generalized electromechanical coupling factor is assumed,the familiar expression Ropt=(ωs0Cp)−1 is obtained.

Ψopt =1Ω

√√√√ 1Q2

m+ (Ω2 − 1)2

1Q2

m+ (Ω2 − 1−K2)2 + (Ω2−1)2

Q2e

(5.12)


10-8

10-6

10-4

10-2

Am

plitu

de o

f the

dam

ping

term

s

1.41.21.00.80.6Ω

(a)

10-7

10-5

10-3

10-1

Am

plitu

de o

f the

dam

ping

term

s

1.41.21.00.80.6Ω

(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 blacklines 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 behaviorsdepending on the values of the quality factor and of the generalized electrome-chanical coupling factor. They are illustrated in Figure 5.3 and Figure 5.4: forstructures with high Qm and high K2 (K2>0.01 for Qm=10000, K2>0.05 forQm=1000, K2>0.1 for Qm=100, K2>0.5 for Qm=10), two maximums of theoutput power are observed. As noted by DuToit [97], the presence of two powerpeaks (which also correspond approximately to displacement peaks) is relatedto the fundamental resonant and anti resonant characteristics of piezoelectricmaterials and transducers. The first maximum corresponds approximately tothe minimum impedance (seen from the electrical side) frequency (Ω≈Ωs0) andthe optimum load related to this situation has also a minimum value because ofpower transfer matching considerations. At the opposite, the optimum load cor-responding to the second maximum (Ω≈Ωo0) has a relatively large impedance.Therefore, the dielectric losses have a stronger influence on the anti resonancepower peak than on the resonance one. It can be seen that for |Qe|>100, thedielectric dissipations do not have an influence on the characteristics, but thatfor |Qe|<100, they tend to diminish the amplitude of the anti resonance powerpeak. At extremely low values of the electrical quality factor, the latter peakcan even vanish. A second type of behavior is observed for low Qm or lowK2 structures: a single maxima of power is observed and the anti resonancepeak does not exist (Figure 5.4(a)). This can be explained either because theamplitude of the resonance power peak is strongly reduced by the mechan-ical dissipations (low Qm) or because the frequency shift between resonanceand 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 hyperboleΨopt=Ω−1(1+K2)−1/2, except close to Ω=1 where a down scaled version of thebehavior observed in Figure 5.3(b) can be observed. From the results obtained


1

2

3

4

5

6

7

8

9

10

2

3

4

5

6

Opt

imum

out

put p

ower

1.41.31.21.11.00.90.8 0Ωs 0Ω

o

(a)

0.1

1

10

Opt

imum

load

par

amet

er Ψ

1.41.31.21.11.00.90.8 0Ωs 0Ω

o

(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 toQe=-100, the dotted lines to Qe=-10.

0.001

0.01

0.1

1

10

Opt

imum

out

put p

ower

1.41.31.21.11.00.90.8 0Ωs

(a)

8

9

1

Opt

imum

load

par

amet

er Ψ

1.41.31.21.11.00.90.8 0Ωs

(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 thiscase, it does not have any influence on the behavior.

in Chapter 3, it is possible to determine the type of behavior that should cor-respond to the different types of piezoelectric bender described in Table 2.1.From the values of the generalized electromechanical coupling factor and qual-ity factor given in Table 3.7, it can be concluded that the MEMS devices (AlNor PZT based) should follow the second type of behavior described above. Thesituation 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 andit is shown that the available samples do not allow a clear visualization of twodistinct peaks.

The expression obtained for the output power at optimum load is relativelytedious and some simplifications are introduced. The remainder of the analysisis focused on the resonance and anti resonance power peaks and it is assumedthat the former is obtained at Ωs0 and the latter at Ωo0. As the impedance of


the piezoelectric vibrator is minimum at Ωs0, the dielectric dissipations havea 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 optimumpower PRopt at Ωs0. The dielectric dissipations can however not be neglectedwhen considering the anti resonance power peak PAopt which corresponds to amaximum of the impedance of the piezoelectric vibrator (only relevant for highQm-high K2 structures). The expressions of PRopt and PAopt considering thesesimplifications and those of the corresponding optimum load parameter aregiven below.

PRopt =meA

20Qm

4ωs0

1

1 +√

1 + 1K4Q2

m

(5.13)

ΨRopt =

1√1 +K4Q2

m

(5.14)

PAopt =meA

20Qm

4ωs0

−K2QmQe√

1 +K2(−K2QmQe +K4Q2

m

+√

(1 +K4Q2m) (Q2

e +K4Q2m)

) (5.15)

ΨAopt =

√√√√√ 1 +K4Q2m(

1 +K4Q2

m

Q2e

)(1 +K2

) (5.16)

The expression found for PRopt corresponds to the one obtained by Roundy[28] except for the fact that, in the given reference, the material coupling factork31

2 is erroneously used instead of the generalized electromechanical couplingfactor K2. This confusion has a strong impact on the estimations of the deviceperformances. Indeed, it is shown in Chapter 2 that the generalized electrome-chanical coupling factor is only a fraction of k31

2, because the piezoelectricmaterial is coupled with elastic material in a piezoelectric composite structureand because it undergoes bending and not purely tensile deformations. Bothexpressions of the optimum power depend primarily on the effective mass ofthe structure and on the frequency and amplitude of the input vibrations. Thisdependence is expressed by the term meA0

2/(4ωs0). This term does not repre-sent an upper limit for the power as the additional multiplicative terms presentin the expressions of PRopt and PAopt are not necessarily lower than 1 (for theresonance, the limit is approximately Qm). However, me, A0 and ωs0 are thefirst parameters to be taken into consideration when designing the harvesters.

For analyzing the influence of the other parameters on the performancesof the device, numerical evaluations of (5.13) till (5.16) are performed. InFigure 5.5, the graphics describing the values of the optimum load parame-ter Ψopt are presented. The curves corresponding to the resonance behaviorare represented by red solid lines. The curves corresponding to the anti reso-nance behavior are represented by solid (Qe→∞), dashed (Qe=-100) and dot-ted (Qe=-10) black lines. According to (5.14), for a given Qm, the optimum


load parameter decreases as the generalized electromechanical coupling factorincreases in the resonance case. This is due to the fact that the minimum ofthe impedance of the piezoelectric vibrator observed at the resonance frequencydecreases when the generalized electromechanical coupling factor is increased.ΨRopt is very close to 1 for small values of K2 while the optimum resonance power

is obtained for very low load impedances when K2 is large. At the opposite ofthe characteristics observed at resonance, the optimum anti resonance load pa-rameter ΨA

opt increases continously with K2 when the dielectric dissipations areneglected. An increase of K2 results in larger absolute values of the impedanceof the system at the anti resonance, so that the optimum load resistor has alsolarger values. However, when dielectric dissipations are included, the optimumload parameter increases till reaching a maximum and decreases afterwards.This effect can be explained as follow: the presence of dielectric dissipationsis 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 capacitorCp and of a real valued resistor Rel=-Qe/(ωo0Cp). As K2 becomes larger, theimpedance of the piezoelectric vibrator at anti resonance increases while Rel

decreases (because ωo0=ωs0(1+K2)1/2). When the impedance of the piezoelec-tric element becomes too large, the load resistor RL only ”‘see”’ Rel as inputimpedance and the optimum load is at this moment equal to Rel.

In Figure 5.6, the values of PRopt and PAopt normalized to meA02Qm/(4ωs0)

are presented. In terms of output power at resonance, it is always worth todesign a low mechanical loss structure. Independently of the values of thegeneralized electromechanical coupling factor, the output power of the deviceincreases monotonically with Qm. When mechanical dissipations are large, it isimportant to make the generalized electromechanical coupling factor as high aspossible so that the process of energy extraction can compete with the parasiticdissipations. On the other hand, in low parasitic losses situation, the outputpower is constant over a large range of the generalized electromechanical cou-pling factor and it is only necessary in this case to design of the harvester soas to exceed the lower boundary limit of K2. It was shown in Chapter 2 thatthe value of K2 depends as well on the relative dimensions and on the materialproperties, particularly on the piezoelectric constant d31. As the output powerdoes not depend on the generalized electromechanical coupling factor above acertain range, it is then possible to adjust the other parameters (particularlyto match the frequency of the input vibrations) while not perturbing the per-formances. This observation is extremely important, as it suggest that low K2

structure (MEMS devices) can compete with high coupling structure (ceramicPZT bender) in terms of power generation when similar mass and resonancefrequency are considered.

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


10-1

100

Ψop

t

10-32 4 6

10-22 4 6

10-12 4 6

100

K2

10-2

10-1

100

101

Ψop

t

10-32 4 6

10-22 4 6

10-12 4 6

100

K2

10-2

10-1

100

101

102

Ψop

t

10-32 4 6

10-22 4 6

10-12 4 6

100

K2

10-310-210-1100101102

Ψop

t

10-32 4 6

10-22 4 6

10-12 4 6

100

K2

(a) (b)

(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 theanti resonance characteristics for respectively Qe→∞, Qe=-100 and Qe=-10. The red linesrepresent the resonance characteristics.

is small. For high values of K2, the behaviors are however totally different.Independently of the amplitude of Qm, PAopt increases monotonically with thegeneralized electromechanical coupling factor when the dielectric dissipationsare neglected and there is no saturation of the normalized power. When dielec-tric dissipations are included, a maximum of PAopt is found. For the same reasonsthat were expressed when discussing the optimum load parameter, above a cer-tain value of K2, the largest amount of the electrical energy produced by thepiezoelectric element is shared between the dielectric loss resistor Rel and theload resistor RL rather than being shared between the capacitor Cp and RL.

It is also interesting to investigate the output power per unit displacement ofthe system. Indeed, the power density of the harvester has to be maximized forminiature electronic applications. The graphics of the power density normal-ized to meA0ω

s0 and assuming the expressions of the optimum load that have

been proposed previously are given in Figure 5.7. At resonance, the powerdensity increases with K2 till saturating for high values of K2 as it was thecase for the power. The situation is different for the anti resonance: the powerdensity possesses an inflexion point but increases monotonically with the gener-


10-2

10-1

100

Nor

mal

ized

opt

imum

pow

er

10-32 4 6

10-22 4 6

10-12 4 6

100

K2

10-2

10-1

100

Nor

mal

ized

opt

imum

pow

er

10-32 4 6

10-22 4 6

10-12 4 6

100

K2

10-2

10-1

100

Nor

mal

ized

opt

imum

pow

er

10-32 4 6

10-22 4 6

10-12 4 6

100

K2

10-2

10-1

100

Nor

mal

ized

opt

imum

pow

er

10-32 4 6

10-22 4 6

10-12 4 6

100

K2

(a) (b)

(c) (d)

Figure 5.6 - Output power at optimum load normalized to meA02Qm/(4ωs

0). (a) corre-sponds 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 dielectricdissipations have a negligible influence on the behavior of this parameter.

It is now important to determine what the best option for power generationis between resonance and anti resonance operating frequency. The anti reso-nance peak does not exist for low values of the generalized electromechanicalcoupling 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 advan-tageous 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 in-fluence on PAopt but did not have noticeable effects on PRopt, so that the previousconclusion does not hold when Qe is not taken as infinite. Indeed, in high di-electric dissipation and low mechanical losses situations, PRopt has larger valuesthan PAopt. The optimum load parameter Ψopt for the anti resonance is alwaysmuch larger than for the resonance. In the former case, extremely large loadimpedance is required for optimum power generation so that the current deliv-ered to the load element is very small. At the opposite, low shunt impedanceis needed at resonance so that the generated voltage is relatively small in this


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Figure 5.7 - Power par unit displacement at optimum load normalized to meA0ωs0. (a)

correspond to Qm=10, (b) to Qm=100, (c) to Qm=500 and (d) to Qm=1000. The blacksolid, dashed and dotted lines represent the anti resonance characteristics for respectivelyQe→∞, Qe=-100 and Qe=-10. The red lines represent the resonance characteristics.

case. The relative amplitude of the deflection in the two discussed situationsdepends 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 thatcan be imagined and is useful for demonstrating experimentally the relevanceof the developed circuit model. However, the impedance of the piezoelectricharvester can not match fully the impedance of the load circuit as the formercontains imaginary components so that the transfer of energy is not optimized.One can then imagine that a ”complex” load consisting in a parallel or se-ries association of linear components such as inductors, capacitors or resistorsshould lead to better performances. The imaginary part of the impedance ofthe piezoelectric harvester seen from the electrical side is positive so that theimaginary part of the impedance of the load circuit should be negative in order


to reach the matching condition. Then, it is only possible to use inductors ina series or parallel association with a dissipative load resistor. Hagood [143]has shown in the field of vibration absorption that it was indeed possible tolargely increase the performances of the harvester with the use of an impedancematching circuit. However, the values required for the inductors, for both se-ries and parallel scheme, are so large (>1000H for our MEMS devices) that itis practically impossible to implement it. For real applications, the situationis then limited to the purely resistive load case described above when linearconditioning electronics is assumed.

In the situation described above, the delivered power is pure AC and theportable electronics applications that can be powered with it are limited. Forthis reason, efforts have been done in the literature for proposing more effectiveinterfaces based on non linear conditioning. Lefeuvre [92] proposed an inter-esting analysis of 4 different conditioning circuitries. The first one consists ina standard rectifying approach and includes a diode rectifier bridge and a filtercapacitor. The piezoelectric element is open circuited when the rectifier bridgeis blocking, i.e. when the absolute value of the voltage across the electrodes ofthe piezoelectric element is lower than the rectified DC voltage VDC . In steadystate operation, the average current through the filter capacitor Cr is null, sothat the absolute value of the electric charge outgoing from the piezoelectricelement during a period T is equal to the average current flowing through theload RL. The maximum power that can be obtained from this method is equalto meA0

2K2/(2πωs0). The delivered power is then in any cases smaller thanthe one obtained with a pure resistive load, but consists in a DC voltage whichis more adapted to practical applications. The second approach has been de-nominated as synchronous charge extraction principle and consists in removingperiodically the electric charge accumulated on the piezoelectric capacitor Cp,and then to transfer the corresponding amount of electrical energy to the loador to the energy storage element. The piezoelectric element is let most of thetime in open circuit configuration and the charge extraction phases occur whenthe electronic switch S is closed: the electrical energy stored in the piezoelectriccapacitor is then transferred into the inductor L. When the electric charges inthe piezoelectric bender vanish, the switch is re-opened and the energy storedinto the inductor L is transferred to the smoothing capacitor Cr through thediode D. The inductor L is chosen to get duration of the charge extraction phasemuch shorter than the vibration period. The output power corresponding tothis interface is equal to more than 5 times the one obtained by the basic recti-fication circuit. Based on the work of Guyomar [94], Lefeuvre also studied theso called SSHI (Synchronized Switch Harvesting on Inductor) interface whichcan be implemented in a series or parallel arrangement. The Series-SSHI inter-face leads to low matching load impedances, while the Parallel-SSHI interfaceleads to higher matching load impedances. The SSHI interface is composed ofa non-linear processing circuit connected with the piezoelectric electrodes andthe input of the rectifier bridge. The non-linear processing circuit is composedof an inductor L in series with an electronic switch S. The electronic switch

122 5.2 Experimental characterization of the harvesters

is briefly turned on when the mechanical displacement reaches a maximum ora minimum. At these triggering times, an oscillating electrical circuit L-Cp isestablished. The period of these oscillations is chosen much smaller than themechanical vibration period. The switch is turned off after half an electricalperiod, resulting in a quasi-instantaneous inversion of the voltage V. Theoret-ical and experimental investigations have shown that this method allows insome cases to multiply the delivered power by a factor 15 and is then a verypromising conditioning circuit. Shu [168] proposed a refined analysis of thismethod and demonstrated that the SSHI interface is particularly relevant forlow coupling structures. The MEMS harvesters that are studied during thisthesis belong to this category of devices and the SSHI interface appear partic-ularly adapted for these harvesters, even if its practical implementation is notstraightforward.

Some efforts have also been done in order to realize conditioning electron-ics based on IC manufacturing technologies. These approaches are particularlyuseful in the case of the MEMS harvesters, as the processing circuit and the har-vesters could be fabricated in the same clean room environment (in the future,monolithic integration may also become a relevant option). A few examples ofthe results published in the literature are the work of Dallago, who proposed ahigh efficiency integrated AC-DC converter [169], based on CMOS technologyand described in [170]. Han [88] proposed an interface based on charge pumpingrealized through switches, MOSFETS and capacitors. Ottman [171] describedan integrated step down converter which allows increasing the rate of chargingof a battery by three times compared to a standard interface. Finally, D’Hulstproposed in [172] a buck-boost topology, working in discontinuous conductionmode. The circuit performs an AC-DC conversion as well as presenting thecorrect electrical impedance to maximize the power output of the harvester.This design has been implemented in a 80 V CMOS technology and proved tohave a conversion efficiency of more than 60%.

Alternative load to a purely resistive one have been described in this subsec-tion. It was shown that is possible in this way to increase or adapt in a betterway the power produced by the harvester to a real life application. The experi-mental validation of the theoretical predictions is however limited in this thesisto the case of a purely resistive load and of the model described in the previoussubsection. This experimental analysis is presented in the next section.

5.2 Experimental characterization of the har-vesters

5.2.1 Output power

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


input vibration matches the short circuit resonance frequency in the case ofthe presented harvesters, so that the experimental analysis is limited to thisparticular frequency of excitation. In the following, the theoretical predictionsconcerning the voltage and the output power of the devices are first comparedwith experimental measurements. In a second subsection, some non linear ef-fects that are not included into the model, particularly the effect of the packageon the MEMS devices, are discussed.

The output voltage and corresponding power is measured at resonance forthe three tested devices while varying the value of the resistive load (and thenof Ψ ). The results are reported for the AlN MEMS, for the PZT MEMS and forthe commercial PZT harvester respectively in Figure 5.8, Figure 5.9 and Fig-ure 5.10. As the power is computed according to (5.1), the deviation betweentheory 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 theAlN devices, both for the amplitude of the expected signals and for the value ofthe optimum load parameter Ψopt. The amplitude of the input acceleration isequal to 1 m.s−2 and a maximum power of approximately 200 nW is measuredfor a load resistor of 110 kΩ (Ψopt=0.28). The fit is less convincing for thetwo PZT based devices and a relatively important deviation from the modelis observed in terms of amplitude. It can be seen from Figure 5.9 and Fig-ure 5.10 that the values of the optimum load resistor coincides approximatelywith the expected ones, but that the amplitudes of the output voltage and ofthe power are lower than those predicted by the model. This deviation hasalso been reported by most of the authors who have investigated PZT basedharvesters such as for example Gao [173] and Kasyap [66]. There might bemultiple possible reasons for the observed deviation but the fact that it occursonly for both thick and thin PZT suggests that it is linked to the materialitself. At the opposite of AlN, PZT is ferroelectric and one can imagine thatimportant phenomena occurring at the microstructural level are not taken intoaccount in our analysis. Also, linear constitutive equations of piezoelectricityare assumed. It is shown in the next subsection that non linear effects existfor large amplitudes of the input acceleration. Finally, it is also possible thatsome dissipation mechanisms neglected in our approach (such as for examplethe piezoelectric dissipations described by Mezheritsky [116]) do exist and dohave a non negligible influence on the output power and voltages.

Despite the deviation between the theoretical predictions and the experi-mental measurements observed for the PZT based devices, the developed modelstill results in a reasonable estimation of the performances of the harvesters,assuming small amplitude of the input vibrations. The level of the input accel-erations used in the presented experiments is quite low and resulted in outputpower below the µW for the AlN MEMS harvesters. For higher amplitude ofthe exciting vibrations, output power in the range of 50 µW are obtained forboth PZT and AlN MEMS devices, which is enough to power low consumptionsensor nodes [24].

124 5.2 Experimental characterization of the harvesters

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5.2.2 Non linear effects

As stated previously, large amplitude of the input vibration might results in anon linear behavior. This effect was investigated by measuring the amplitudeof the open circuit voltage of the harvesters at resonance for different valuesof the input acceleration. The results of this experiment are reported in Fig-ure 5.11(a) for the ceramic PZT device and in Figure 5.11(b) for the AlN MEMSharvester. In the proposed graphics, the solid line represents the theoreticallinear variation of the voltage versus the input acceleration. Experiments andtheory fit well for low levels of the input but a discrepancy is observed at largevalues of A0: the open circuit voltage is lower than expected from the linearmodel. The deviation from the model is higher for the PZT commercial devicethan for the AlN MEMS one. Furthermore, the non linear deviation occursfor lower amplitudes of the input. This phenomena results from the dimen-sions of the commercial bender. This piezoelectric bender was made of a longbeam whose behavior deviates quite rapidly from the small deformations EulerBernoulli beam theory on which the developed model is based.

It was introduced in Chapter 3 that the MEMS fabricated piezoelectricharvesters are meant to be packaged in order to limit the potential threats fromthe environment and to include a constraint to limit the maximum deflectionof the beam. The package consists in a top and bottom wafer containinga cavity. The air contained in the cavity is not allowed to flow smoothlyaround the piezoelectric beam in the case of a packaged device and additionalsqueeze damping exists in the system. In order to estimate the influence ofthe latter damping, the output power at optimum load of the same harvester ismeasured in the three different following conditions (the characterized device isan AlN MEMS device having dimensions different from the sample investigatedpreviously):

• Unpackaged device under atmospheric pressure

126 5.3 Conclusion

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Figure 5.12 - Output power of the same AlN MEMS harvester for the optimum resonanceresistive load versus the normalized frequency of the input vibration and for different environ-mental 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 apackaged 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 themeasurements corresponding to the packaged and unpackaged device underatmospheric pressure, it can be seen that the additional damping due to thepackage reduces the output power of more than 50% and in the same way, thequality factor of the device is reduced to half its original value. The vacuumexperiment was realized by placing a packaged device into a vacuum chamber,but it is planned in the future to realize vacuum packaged devices. However, themanufacturing process of the harvesters is not advanced to this level at this mo-ment, but it is believed that the experiments done in the vacuum chamber arerepresentative of the performances that will be obtained with vacuum packageddevices. The output power of the devices should be multiplied by a factor 4 inthese conditions. The parasitic dissipations are reduced in vacuum conditionand the output power for a same input acceleration is intuitively larger than forthe two others situations. Also, it can be seen that the frequency of maximumpower shifts towards lower values when the amplitude of the parasitic dissipa-tions increases (the reference Ω=1 was taken for the vacuum experiment). Thisresult is in accordance with the fact that parasitic mechanical dissipations tendto decrease the value of the resonance frequency.

5.3 Conclusion

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


in the previous chapters. When assuming a single harmonic input vibration,it is shown that the best performances are obtained at the resonance or antiresonance frequency of the devices. However, the anti resonance performancesdepend strongly on the generalized electromechanical coupling factor and onthe dielectric dissipations. It is not possible to study experimentally the antiresonance behavior for the available samples, either because of too low gen-eralized electromechanical coupling factor in the case of the MEMS devicesor because of too large dielectric dissipations in the case of the ceramic PZTpiezoelectric bender. Therefore, the analysis is focused on the resonance char-acteristics. The output power delivered to a purely resistive load at resonancedepends primarily on the effective mass of the bender, the frequency and theamplitude of the input vibrations, but also on the generalized electromechan-ical coupling and the mechanical quality factor, which have not necessarily tobe maximized, but only to be set above a certain value. This conclusion is im-portant as it means that low coupling structures (such as AlN or PZT MEMSharvesters) 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 SSHIinterface, allows strongly improving the performances of the harvesters. MEMSdevices based on thin film PZT and AlN materials and commercial piezoelectricbender made of ceramic PZT are experimentally characterized. It is demon-strated that the developed model gives a good estimation of the measured data,but that some discrepancies are observed, particularly at high level of inputacceleration for which non linear effects are observed. Finally, it is shownthat the packaging of MEMS AlN harvesters creates additional parasitic dissi-pation 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 afactor 4 in this case. At the present moment, output power in the range of 50µW are obtained with AlN MEMS harvesters, which is enough to power lowconsumption sensor nodes.

128 5.3 Conclusion

Chapter6General conclusions and futurework

In Chapter 1, it is demonstrated that energy harvesters are a promising re-newable source of electrical energy for portable electronics applications suchas wireless sensor networks. They act by converting the energy available inthe environment into useful electrical power. Common environmental energysources are light, heat, vibrations, human motion or wind. The work realizedduring this thesis is focused on harvesting energy from vibrations and humanbody motion. The energy produced by these sources can be transformed follow-ing a ”directly applied force” or an inertial scheme. The latter type of devicesis investigated, as they offer more freedom in terms of possible applications.Piezoelectric transduction is the principle chosen for the electromechanical en-ergy conversion, because of its relative simplicity of implementation comparedto electromagnetic or electrostatic transduction. It is also determined that dif-ferent designs of the harvesters are required for extracting efficiently energyfrom vibrations and from the motion of the human body. The former typeof 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 basedon a classical mass/spring/dashpot system which must be excited in the neigh-borhood of its resonance frequency to deliver maximum power. Such devicesare manufactured by MEMS technologies and characterized. In the case ofbody motion energy harvesting, a non resonant device based on the impact ofa rigid body on piezoelectric beams is investigated. A prototype is fabricatedand characterized.

Chapter 2 is focused on the modeling of piezoelectric beams. The constitu-tive equations of piezoelectricity are combined with Timoshenko’s beam theory

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in order to elaborate the fundamental equations of piezoelectric multilayeredbenders. From our knowledge, this complete derivation had not been proposedin the literature. Both dynamic and quasi static solutions of the latter equationsare derived and arranged in the form of a constitutive matrix and of an equiv-alent electrical network. The electrical network representation is particularlyuseful in the field of energy harvesting, as the devices has to be coupled withcomplex conditioning electronics. A complete representation of a piezoelectriccantilever loaded by a distributed mass, which is the basis of many harvesterscurrently investigated, is developed. Finally, the generalized electromechanicalcoupling factor defining the amount of mechanical energy transformed into elec-trical energy (or vice-versa) by a piezoelectric bender during a cycle is definedand a clear interpretation of the difference between the material and structureelectromechanical coupling factor is proposed. Confusion between these twofactors is often found in the literature. It can have great repercussions on theestimations of the model. It was also shown that the generalized electrome-chanical coupling factor depends only on the material properties of the supportand piezoelectric materials and on the thickness ratio of the two materials, butnot on the lateral dimensions of the structure.

In Chapter 3, the manufacturing by MEMS technologies of piezoelectricharvesters and the experimental determination of their corresponding equiva-lent network parameters is presented. A robust process flow is developed. Itresults in functional devices but they still need some improvements at the levelof etch homogeneity and packaging. Particularly, the proposed ambient atmo-sphere packaging approach results in low mechanical quality factors and will beimproved in the future by implementing a vacuum packaging method. Static,transient and steady state experimental methods for determining the equivalentnetwork parameters are then described and implemented on MEMS PZT andAlN based harvesters and on commercial ceramic PZT structures. No completeprocedure for the characterization of the network parameters of piezoelectricbeams 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 carriedout and experiments that can be realized with commonly available laboratoryequipments are presented. It is shown that all the parameters but the stiffnessof the beam can be determined from steady state measurements, which are formost of them easily implemented. The determination of the stiffness of thebeam requires a quasi-static experiment. Characteristic values of the networkparameters are then presented for the three types of structures (MEMS thinfilm AlN unimorph, MEMS thin film PZT unimorph and commercial thick filmPZT bimorph) studied during this thesis. The material properties of the piezo-electric materials are estimated from these measurements. The e31 piezoelectricconstant is found to be 0.9 C.m−2, 4.1 C.m−2 and 11.9 C.m−2 for respectivelythe thin film AlN, the thin film PZT and the thick film PZT. It is shown thatPZT, 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

6. GENERAL CONCLUSIONS AND FUTURE WORK 131

of sensing applications. The generalized electromechanical coupling factor isequal to 4.10−3 for the MEMS AlN harvester, 6.10−5 for the PZT MEMSdevice and 1.10−1 for the commercial bender. These values do however notcontain enough information to perform a direct comparison in terms of energyharvesting. This is done in the next chapters.

The human environment harvester, based on the impact of a rigid body onpiezoelectric beams, is investigated in Chapter 4. A complete model of theimpact of a mass on a piezoelectric bender based on the equivalent networkrepresentation previously obtained is established for the first time. Analyticaland numerical analysis of the developed model was realized. The followingconclusions are obtained:

• In order to maximize the efficiency of the energy conversion, the general-ized electromechanical coupling factor and the mechanical quality factorof the piezoelectric transducer have to be made as high as possible bychoosing appropriate materials and dimensions. Also, the parasitic dissi-pations in the motion of the missile have intuitively to be minimized.

• The influence of the masses ratio and of the impact coefficient of restitu-tion is very complex. However, when the mass of the impacting body issmall compared to the one of the beam, better performances are obtainedby implementing an elastic impact. At the opposite, the performancesare counter intuitively optimized for inelastic impact when the mass ofthe impacting body is large compared to the one of the beam.

• The resonance frequency of the bender should be high enough so that theamount of energy transferred by the moving object during an impact canbe dissipated prior to a second impact.

• In order to limit reliability problems due to high stresses and deflectionsresulting from high amplitude excitations, a damping system has to beimplemented in the neighborhood of the piezoelectric elements (repulsingmagnets are used in our case).

As the conditions of large generalized electromechanical coupling factor are re-quired for optimizing the performances, commercial ceramic piezoelectric ben-ders are implemented in the prototype of the human motion energy harvester.Because of their low generalized electromechanical coupling factor, the fabri-cated MEMS devices can clearly not give good results in this situation. Anoutput 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 ofapproximately 7 Hz for a 10 cm amplitude. Also, a power of 47 µW is mea-sured when the harvester is rotated of 180 each second. A large amount ofthe volume occupied by the prototype of the harvester can be eliminated andone can estimate an output power density of 10 µW/cm3 or 4 µW/g for anoptimized device undergoing the previously described rotary motion. These

132

figures can be multiplied by a factor 12 when a 7 Hz frequency, 10 cm ampli-tude linear motion is considered. These results are the highest published forinertial human motion energy harvesters.

In Chapter 5, the behavior of the output power of the piezoelectric vibra-tion energy harvesters (machine environment) is first theoretically investigated.For the first time, dielectric dissipations are included in the model. No publica-tions taking into account this mode of parasitic mechanism has been proposed.When assuming a single harmonic input vibration, it is shown that the bestperformances are obtained at the resonance or anti-resonance frequency of thedevices. In terms of power per unit displacement, it is demonstrated that itwas always advantageous to work at anti-resonance, at the price of a high imp-edance optimum load (low current). However, the anti-resonance performancesdepend strongly on the generalized electromechanical coupling factor and onthe dielectric dissipations. It is not possible to study experimentally the antiresonance behavior for the available samples, either because of too low gen-eralized electromechanical coupling factor in the case of the MEMS devicesor because of too large dielectric dissipations in the case of the commercialPZT piezoelectric bender. Therefore, the analysis is focused on the resonancecharacteristics. The output power delivered to a purely resistive load at res-onance is not affected by dielectric dissipations and depends primarily on theeffective mass of the bender and on the frequency and amplitude of the inputvibrations, but also on the generalized electromechanical coupling factor andon the mechanical quality factor. At the opposite of what is observed in thecase of the human environment harvester, it is shown that the generalized elec-tromechanical coupling factor and the mechanical quality factor do not havenecessarily to be maximized, but only to be set above a certain value. Thisconclusion is important as it means that thin film AlN or PZT MEMS devicescan achieve performances similar to those of ceramic PZT structures if theparasitic mechanical dissipations can be reduced below a certain threshold. Inthe remainder of Chapter 5, the behavior of the output power at resonanceof MEMS devices based on thin film PZT or AlN materials and commercialpiezoelectric bender made of ceramic PZT are experimentally characterized.It is demonstrated that the developed model gives a good estimation of themeasured data. Some discrepancies are however found, particularly at highlevel of input acceleration for which non linear effects were experimentally ob-served: the generated voltage predicted by the linear model is clearly largerthan the measured one in this situation. Finally, it is shown that the packagingof MEMS AlN harvesters creates additional parasitic dissipation mechanismswhich reduce drastically the output power of the devices. At the present mo-ment, output power in the range of 50 µW are obtained for an approximatedvolume of 0.3 cm3 with AlN and PZT MEMS harvesters, which is enough topower low consumption sensor nodes. These results are amongst the high-est published data related to the output power of MEMS piezoelectric energyharvesters.

The research on the developed MEMS piezoelectric harvesters continues at

6. GENERAL CONCLUSIONS AND FUTURE WORK 133

Imec and at the Holst Centre. Particularly, vacuum packaged devices havebeen produced. As mentioned in the last chapter, vacuum packaging allowsincreasing the produced power by a factor 4 by limiting the parasitic mechan-ical dissipations. New problems are however encountered: due to the largedisplacement of the mass, it impacts in most of the situations on the inner sideof the package. The effect of this phenomenon on the reliability of the devices iscurrently investigated. A general study of the fracture and fatigue behavior isalso 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 toremain functional for periods exceeding several years. From a literature review,it is determined that fatigue in micromachined piezoelectric laminated beams isthe most susceptible to occur in the interfaces between the different materials.

Geometries different than a rectangular beam such as for example taperedcantilevers or membranes are also investigated for the purpose of increasingthe power generated by the devices. To the same aim, structures based on33 piezoelectric coefficients rather than on the 31 coefficients are investigated.Also, efforts are done for developing viable conditioning electronics for rectify-ing the AC power delivered by the piezoelectric vibration harvesters. Finally,new designs of devices for harvesting efficiently the energy from broadbandspectrum mechanical vibrations are studied.

List of Publications

Journal Publications

1. M. Renaud, P. Fiorini and C. van Hoof, “Optimization of a piezoelec-tric 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 andR. Puers, “Fabrication, modeling and characterization of MEMS piezo-electric 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, “Harvestingenergy from the motion of human limbs: the design and analysis of animpact-based piezoelectric generator”, Smart Mater. Struct., vol. 18, pp.943–961, 2009.

4. K. Karakaya, M. Renaud, M. Goedbloed and R. van Schaijk, “The effectof the built-in stress level of AlN layers on the properties of piezoelectricvibration 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 trans-ducer”, 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, Modelingand Measurements”, in Proc. of the Conf. on Solid State Sens., Act. andMicrosys., Transducers’07, 2007, pp. 891–894.

3. M. Renaud, P. Fiorini, R. van Schaijk and C. Van Hoof, “An ImpactBased Piezoelectric Harvester Adapted to Low Frequency Environmen-

135

136 List of Publications

tal vibration”, in Proc. of the Conf. on Solid State Sens., Act. andMicrosys., 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 Simu-lation”, in Proc. of Power MEMS, pp. 61–64, 2005.

5. N. El Ghouti, R. Vullers, A. Schmitz and M. Renaud, “Piezoelectricenergy 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 andvalidation 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, “Vacuumpackaged MEMS piezoelectric vibration energy harvesters”, accepted forpresentation 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. Pendersand R. van Schaijk, “First autonomous wireless sensor node powered bya vacuum-packaged piezoelectric MEMS energy harvester”, accepted forpresentation at IEDM 2009.

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