Design and validation of piezoelectric energy harvesting systems

FACULTEIT INGENIEURSWETENSCHAPPEN
Vakgroep Toegepaste Mechanica
Design and validation of piezoelectric energy harvesting systems
Ontwerp en validatie van piëzo- elektrische energy harvesting systemen
Academiejaar 2010-2011
Promotor: Prof. dr. ir. Patrick Guillaume
Prof. dr. Steve Vanlanduit
Eindwerk voorgelegd voor het behalen van de academische graad
van Master in de Ingenieurswetenschappen door:
Ilyas Caluwé

iii
Dankwoord
Mijn dank gaat uit naar prof. dr. ir. Patrick Guillaume en naar prof. dr. Steve Vanlanduit voor het aanvaarden van dit eindwerk. Van beiden kreeg ik voldoende ruimte om zelfstandig aan deze proef te werken, terwijl ik steeds op hun begeleiding, hulp en advies kon rekenen. Tevens dank ik Jean-Paul Schepens voor zijn kundige hulp bij de experimenten. Ook ir. Mohammad Ahmadi Bidakhvidi wens ik te danken voor zijn advies.
Verder dank ik mijn vriendin Mona, mijn ouders en mijn broer voor hun steun gedurende heel mijn studietijd. Zonder hen had ik dit alles nooit tot een goed einde kunnen brengen.
Ilyas Caluwé
juni 2011

iv
Samenvatting
De doelstelling van dit werk is te onderzoeken hoe kleine hoeveelheden mechanische energie kunnen worden omgezet in bruikbare elektrische energie. Mogelijke energiebronnen zijn trillingen en windstroming.
Hiertoe wordt eerst een literatuurstudie uitgevoerd. Hieruit blijkt dat het piëzo-elektrisch effect interessant is om zowel energie uit trillingen als uit stromingen om te zetten. Een opstelling met een plaatje en een piëzo-elektrische laag of MFC piëzopatch werd gekozen. Een dergelijke setup heeft goede energie omzettingskarakteristieken, is robuust en veelzijdig.
Een opstelling waarmee trillingsenergie kan worden omgezet werd gebouwd. Hiervoor werden zowel een commercieel beschikbare harvester met twee lagen piëzokeramisch materiaal als een met composietmateriaal versterkte piëzopatch op een structureel plaatje gebruikt. De parameters die belangrijk zijn om de setup te optimaliseren worden besproken.
Vervolgens worden opstellingen voorgesteld waarmee energie uit een windstroming kan worden omgezet. Hiertoe werden verschillende aëroelastische fenomenen bestudeerd. Een aantal van de ontwerpen werd getest. De mogelijkheid om energie om te zetten met behulp van aëroelastische effecten wordt aangetoond. Verder worden er suggesties tot mogelijke verbeteringen gedaan.
Summary
The aim of this study is to examine how small amounts of ambient energy, such as in vibrations or wind flow, can be converted to electrical energy and to build a working design.
The different energy harvesting principles found in literature are studied first. Piezoelectric energy harvesting was found suitable for both energy harvesting out of ambient vibrations and wind flow. A cantilevered beam setup with a piezopatch (MFC patch) is chosen because it has good power conversion characteristics, it is robust and versatile. Both vibration and wind flow harvesting devices can be constructed with this setup.
Vibration harvesting setups were constructed with both a commercially available bimorph piezoceramic harvester and with an unimorph harvester consisting of a aluminum plate and a composite-reinforced piezoceramic patch attached to it. The power output is reported. The parameters that are of importance to optimize the setup are discussed.
The possibilities to use the beam for wind flow harvesting were explored. Different aeroelastic phenomena were studied to give insight into possible working principles. A number of designs are proposed and some are tested using the aluminum plate with the MFC patch. The possibility of using aeroelastic stability to harvest energy is shown, and suggestions for further improvements are given.

v
Résumé
L'objectif de ce travail est d'étudier comment des petites quantités d'énergie mécanique peuvent être converties en énergie électrique utilisable.
D'abord, une étude de la littérature est conduite. Cette étude montre que l’effet piézoélectrique est intéressant pour convertir l’énergie des vibrations mécaniques et de l’écoulement du vent. Une configuration avec une plaque piézoélectrique disponible commercialement et avec une plaque en métal avec un patch piézoélectrique <MFC> a été choisie. Une telle configuration présente de bonnes caractéristiques de conversion d'énergie et est robuste et polyvalente.
Une configuration pour convertir l’énergie présente dans des vibrations mécaniques est construite. Les paramètres qui sont importants pour l'optimisation de la configuration sont ensuite discutés.
Puis, les possibilités de conversion de l’énergie présente dans un flux de vent sont étudiées. Différents phénomènes aéroélastiques sont étudiés pour donner un aperçu des principes de travail possibles. Quelques modèles sont testés. La possibilité de convertir l'énergie en utilisant des effets aéroélastiques est démontrée. En outre, des suggestions d'améliorations pour des travaux futurs sont données.

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Contents
Dankwoord ........................................................................................................................................ iii
Samenvatting .................................................................................................................................... iv
Summary ........................................................................................................................................... iv
Résumé ............................................................................................................................................... v
Contents ............................................................................................................................................ vi
List of symbols ................................................................................................................................. viii
Abbreviations .................................................................................................................................... ix
1 Introduction ............................................................................................................................... 1
1.1 The use of small amounts of harvested energy ................................................................. 1
1.2 Structure of this work......................................................................................................... 1
2 Overview of energy harvesting principles and devices .............................................................. 2
2.1 Piezoelectric ....................................................................................................................... 4
2.1.1 Beam bending ............................................................................................................ 5
2.1.2 Energy harvesting eel ................................................................................................. 7
2.1.3 Piezoelectric windmill ................................................................................................ 8
2.1.4 Diaphragm piezoelectric transducer .......................................................................... 8
2.1.5 Microfiber nano wire .................................................................................................. 9
2.2 Electrostatic ...................................................................................................................... 10
2.2.1 Coulomb-damped resonant generator (CDRG) ........................................................ 10
2.2.2 Coulomb-force parametric generator (CFPG) .......................................................... 12
2.3 Electromagnetic ............................................................................................................... 13
2.3.1 Electromagnetic, vibration-powered generator ...................................................... 13
2.3.2 Velocity-damped resonant generators (VDRGs) ...................................................... 15
2.3.3 Rotary electromagnetic generator ........................................................................... 15
2.3.4 Flutter mill ................................................................................................................ 16
2.4 Other principles and concepts ......................................................................................... 17
2.4.1 Transverse Galloping ................................................................................................ 17
2.4.2 Flapping foil generator ............................................................................................. 18
2.4.3 Leading edge flutter wind power generator ............................................................ 18
2.4.4 VIVACE (vortex induced vibration aquatic clean energy) ......................................... 19
2.5 Advantages and disadvantages of the different harvesting principles ............................ 19
2.6 Harvesting powers found in literature ............................................................................. 20
3 Choice of the harvesting principle and setup .......................................................................... 21
3.1 The piezoelectric effect .................................................................................................... 21
3.2 Some theory describing piezoelectricity .......................................................................... 21
3.2.1 The use of subscripts ................................................................................................ 21
3.2.2 Piezoelectric charge constant .................................................................................. 21
3.2.3 Piezoelectric voltage constant ................................................................................. 22
3.2.4 Electromechanical coupling factor ........................................................................... 22
3.2.5 Linear equations describing the piezoelectric effect ............................................... 22
3.2.6 Equivalent electrical circuit, electrical load and voltage output .............................. 23
3.2.7 Polarization of a bimorph generator ........................................................................ 24
3.3 Piezoelectric materials ..................................................................................................... 24

vii
3.3.1 Lead Zirconate Titanate Ceramic (PZT) .................................................................... 25
3.3.2 Polyvinylidene Fluoride (PVDF) ................................................................................ 26
3.3.3 Other materials ........................................................................................................ 27
3.4 The harvesting devices used in this work......................................................................... 27
4 Piezoelectric energy harvesting out of vibrations .................................................................... 29
4.1 Experimental setup .......................................................................................................... 29
4.2 Bimorph harvester ........................................................................................................... 32
4.2.1 Resonant frequency ................................................................................................. 32
4.2.2 Damping ratio ........................................................................................................... 34
4.2.3 Power output ........................................................................................................... 35
4.3 MFC patch on a plate ....................................................................................................... 42
4.3.1 Resonant frequency ................................................................................................. 42
4.3.2 Power output ........................................................................................................... 42
4.4 Comparison ...................................................................................................................... 44
4.4.1 Power density of the bimorph and the MFC ............................................................ 44
4.4.2 Comparison with literature ...................................................................................... 44
4.5 Conclusions ...................................................................................................................... 45
4.5.1 The use of self adaptive structures .......................................................................... 46
5 Flow induced vibration ............................................................................................................. 47
5.1 Overview .......................................................................................................................... 47
5.1.1 Vortex shedding ....................................................................................................... 49
5.1.2 Flutter ....................................................................................................................... 51
5.1.3 Other instabilities ..................................................................................................... 54
5.2 Power available in wind flow ........................................................................................... 55
5.3 Proposed designs to extract energy from wind flow ....................................................... 55
5.3.1 Plate placed perpendicular to the flow .................................................................... 55
5.3.2 Piezoplate with a transverse tip plate ...................................................................... 56
5.3.3 Plate with a stationary upstream cylinder ............................................................... 58
5.3.4 Oscillating tip cylinder .............................................................................................. 61
5.3.5 Wing profile .............................................................................................................. 67
5.4 Conclusions ...................................................................................................................... 67
6 General conclusions ................................................................................................................. 70
7 Future work .............................................................................................................................. 71
Appendix A: Properties of the Piezo harvesters.............................................................................. 72
Appendix B: Shaker measurements ................................................................................................. 74
Appendix C: MatLab lsqnonlin code ................................................................................................. 75
Appendix D: Impedance head properties ........................................................................................ 76
Appendix E: St in function of Re for cylinders .................................................................................. 76
Appendix F: MFC harvester dimensions ........................................................................................... 77
Appendix G: PZT material properties at room temperature ............................................................ 78
Appendix H: Power output flow induced vibration ......................................................................... 80
Appendix I: Mean wind speed in Belgium at 10 m height ............................................................... 85
References ........................................................................................................................................ 86

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List of symbols
a Acceleration [m/s²]
C Capacitance [F]
c Chord length [m]
c Stiffness coefficients matrix [N/m²]
d Diameter [m]
D Electric displacement components vector [C/m²]
d Piezoelectric coupling coefficients (strain-charge form) [C/N]
E Electric field components vector [N/C]
e Piezoelectric coupling coefficients (stress-charge form) [C/m²]
F Force [N] [kg.m/s²]
f Frequency [Hz]
fn Natural frequency [Hz]
fres Resonant frequency [Hz]
fs Vortex shedding frequency [Hz]
g Piezoelectric coupling coefficients (strain-voltage form) [m²/C]
I Current [A]
k Stiffness constant [N/m] [kg/s²]
K Stiffness constant [N/m] [kg/s²]
l Length [m]
m Mass [kg]
P Power [W]
Q Charge [C]
q Piezoelectric coupling coefficients (stress-voltage form) [N/C]
R Resistance [Ω]
Re Reynolds number [-]
s Compliance matrix [m²/N]
s Semispan [m]
S Strain components vector (εi’s) [-] [m/m]
St Strouhal number [-]
T Stress components vector (σi’s) [N/m²]
t Time [s]
U Speed [m/s]
V Velocity [m/s]
V Voltage root mean square [V]
V Voltage [V]
V Volume [m³]
Vcc Closed circuit voltage [V]
Voc Open circuit voltage [V]
y Displacement [m]
Greek
μ Dynamic viscosity [Pa.s] [kg/(s.m)]
ε Electric permittivity matrix [F/m]

ix
ν Kinematic viscosity [m²/s]
ρ Air density [kg/m³]
σ Material density [kg/m³]
φ Magnetic flux [Wb]
ω Angular frequency [rad/s]
ωN Natural angular frequency [rad/s]
ζ Damping ratio [-]
Abbreviations
AC Aerodynamic Center
CDRG Coulomb-damped resonant generator
CFPG Coulomb-force parametric generator
CG Center Of Gravity
CMOS Complementary Metal Oxide Semiconductor
EC Elastic Center
FRF Frequency response function
IEEE Institute of Electrical and Electronics Engineers
MDOF Multiple degrees of freedom
MEMS Micro Electro Mechanical Systems
MFC Micro Fiber Composite
SDOF Single degree of freedom
VDRG Velocity-damped resonant generator

1
1 Introduction
1.1 The use of small amounts of harvested energy
Many systems that require an electrical power supply operate in remote environments or in spaces that are difficult to reach. Some method is needed to supply these systems with electrical energy. The storage capacity of current technology ‘fixed energy devices’, such as batteries, is however insufficient to power most systems during their whole lifespan. So periodic replacement of batteries is necessary, which is very inconvenient and costly. To avoid this, energy harvesting from the ambient can be a solution.
A wireless sensor network is an example of such systems. Such a network consists of autonomous sensors that monitors physical conditions and pass them to a main location. Powering the individual nodes is the main concern in this application. These wireless sensor nodes have a power requirement in the order of magnitude of tens to hundreds of μW [1]. This makes them very suitable to be powered with ambient energy sources. Examples of such sources include solar energy, fluid flow, thermal gradients, vibrations (including acoustic energy)… The main challenges involving the powering of wireless sensor nodes are discussed in detail in [2].
Sensor nodes are devices that consist of a sensor, a transceiver and supporting electronics (processor). Each component uses a certain percentage of the total required power. Often, the energy produced or harvested is insufficient to power the device directly. If so, energy needs to be accumulated until a sufficient amount is present, which allows to power the device (for example, transmitting a signal) [3]. Thus, a lot of research is also done on the converter and accumulation circuits needed when using harvesters.
Sensor networks are used in many applications. These include environmental monitoring, building climate control, supply chain control, identification and personalization (RFID tags), child education, structural health monitoring (aerospace, automotive), warfare and surveillance, fleet monitoring, micro surgery, agricultural automation… Distances that can be covered by individual nodes range from approximately 1-10 meters. Higher distances can be covered by using multihop capabilities.
1.2 Structure of this work
This work consists out of three parts. In the first part, a literature study is performed. The possibilities of harvesting small amounts of energy out of ambient vibrations and vibrations caused by wind flow are explored. It is explained why harvesting out of vibrations and wind flow is chosen in this work.
The harvesting of energy out of vibration can be based on multiple principles. An overview of these principles is given in chapter 2. It is then examined which of these principles is the most promising. A motivation is given why the piezoelectric effect and a cantilevered setup was chosen. The piezoelectric effect is studied in more detail in chapter 4 which allows a good choice of materials and to make the construction of an optimal experimental setup possible.
In the second part, the conversion of mechanical vibration energy to electrical energy by using the chosen principle is studied in more detail in chapter 4. The parameters that are important to optimize power harvesting efficiency are extracted and studied in detail. Theoretical laws are

2
used to explain and validate the results that are found. Two designs are compared and the comparison with values found in literature is made.
In the third part, the possibility of extracting energy from air flow by using self-induced aeroelastic phenomena and external excitations (for example a vortex street) is examined in chapter 0. Some setups are proposed and the most promising are tested.
2 Overview of energy harvesting principles and devices
As mentioned in the introduction, there are multiple ambient power sources that can be used to power devices that need small amounts of electrical energy, such as wireless network sensors. Examples of energy sources were already given in chapter1, but they can all be classified as either solar, thermal or mechanical energy. An overview is given in [4].
In this text, the possibility to harvest mechanical vibration energy and to convert it to electrical energy is discussed. The choice of mechanical vibration energy harvesting is made by comparing the advantages and disadvantages of the different energy sources, in Table 2. Implementation of solar energy on small scales has the inherent disadvantage that the area that is lit remains small, and so does the harvested energy. Thermal energy needs a thermal gradient to function. In most conditions, this gradient is very small, which puts a limit of electrical energy that can be generated.
In some environments, vibrations are a reliable source of energy. Examples includes engines or pipe ducts. Vibrations can also be caused by wind flow, and structures can be designed specifically to make the wind flow induce vibrations. This makes vibration energy harvesting usable in a large number of environments. As the power that is expected to be obtained, seen in Figure 1, is also sufficient to power wireless sensors, vibration energy harvesting is interesting to be studied in more detail.
Figure 1: Mean supplied and used power of the different energy sources and applications [5][5]
There are three basic mechanisms [6-8] to convert mechanical vibrations to electrical energy: piezoelectrical, electrostatical or electromagnetical energy harvesting. Below, an overview with practical examples is given:

3
Table 1: Overview mechanical energy harvesting devices
Piezoelectric [3, 5, 9-13]
Electrostatic [14]
Electromagnetic
Others/Concept
Cantilever / beam
 Beam bending [15-24]
 Piezoelectric windmill [10, 25, 26]
 Coulomb-damped resonant generator (CDRG) [27, 28]
 Electromagnetic vibration-powered generator [29-34]
Vortex shedding / Von Kármán street
 Energy Harvesting Eel [10, 35]
 Vortex induced vibration aquatic clean energy (VIVACE) [36]
Diaphragm
 Diaphragm piezoelectric transducer [37, 38]
 Coulomb-force parametric generator (CFPG) [27, 39, 40]
 Velocity-damped resonant generators (VDRGs) [27, 41]
Rotating
 Rotating piezoelectric windmill [26]
 Rotary electro-magnetic generator [42]
Transverse galloping


 Transverse Galloping [43]
Flutter
 T-shaped piezoelectric cantilever
 Wind tree [44, 45]
 Flutter mill [46]
 Leading-edge-flutter wind power generator [47-49]
Flapping foil
 Flapping wing generator (2 DOF)
 Oscillating-wing windmill (1 DOF)
Other
 Nanowire [50]
 Planar Electret Based Electrostatic Micro- Generator [51]

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The power [8] that is expected to be harvest from ambient energy differs a lot for the different principles. In Table 2, three possible conversion methods of mechanical vibrations are shown, with the power they are expected to generate. These are compared with the values for solar and thermal energy harvesting.
Table 2: Overview of expected power
Energy source
Problems
Expected power
(for each cm³ or cm²)
Solar
Small area that is lit
10 μW - 15 mW
Outside: 0,15 - 15 mW
Inside: < 1 μW
Vibrations
Variable vibrations
1-200 μW
Piezoelectric ± 200 μW
Electrostatic 50-100 μW
Electromagnetic < 1 μW
Thermal
Small temperature gradients
15 μW
10°C gradient
These expected powers are merely an indication. Examples are available [29] where a much higher power is obtained with an electromagnetic generator. The table indicates that vibration energy is promising when used with wireless sensors. These sensors have a power usage of the same order of magnitude as the expected harvesting power of vibration energy, shown in Figure 1.
2.1 Piezoelectric
Piezoelectric materials have a crystalline structure that allows them to convert mechanical strain into electrical charge (direct effect). Inversely, they convert an applied electrical potential into mechanical strain (inverse or secondary effect). Much research has been done on how to convert mechanical vibration energy into electrical energy, the necessary circuits to do so, the conversion efficiency and such more. An overview is given in [9]. Specific research into charging a capacitor or a battery, with the needed circuits, is done in [11].
Various piezoelectrical materials are available. Unimorph, bimorph, composite (with or without active piezoelectric fibers or used with interdigitated electrodes [12]), round diaphragms and high strength reinforced materials are available. All have different possible applications. [10] Details on piezoelectric materials are given in §3.3.
The harvested power is generally too low (or too irregular) to be used directly to power wireless sensors or other devices continuously. So, a battery or capacity is charged. In [3], piezoelectric materials are tested for their ability to charge a battery or capacity.
If the system is not excited at its resonant frequency, one can assume the electrical plates act as a capacitor [10]. The optimal material parameters will be different then for a resonating system. Electrostatic systems are discussed separately in this text.

5
2.1.1 Beam bending
Consider a clamped beam built out of two layers of piezoceramic material and an intermediate structural layer. The tip of the beam can be made to move, so that the beam is prone to bending. The top piezolayer will be subject to compression and the lower piezolayers will be subject to tension, or vice versa. Due to the piezoelectric effect, a sinusoidal voltage is induced in each of the piezoelectric layers if the tip of the beam is made to vibrate sinusoidal [16]. As the piezo elements are compressed or strained, the polarization of the voltage changes. This means that the way an electrical load is connected to the different layers is important. An electrical load can be connected [18] via a series- or parallel connection with the piezolayers, as shown in Figure 2.
The piezolayers are inversely polarised (X-poled) when connected in series, and are polarised in the same direction (Y-poled) when connected in parallel. Advantages and disadvantages of both connections are studied in §3.2.7. A beam generator with two piezolayers as described above is called a bimorph generator. A beam with one piezolayer can be used as well, this is called a unimorph generator. For example, the influence of inderdigitated combs on unimorph generators is investigated in [23].
Figure 2: Working principle beam bending; above: series connection of piezoceramic layers; below: parallel connection
In research [15] the following optimal electrical loads and powers were found for the X-poled piezoceramic material T226-A4-503X from Piezo Systems1. When excited at 45.6 Hz the optimal load is 35 kΩ, with a maximum power output of 23.9 mW/g2. When excited at 48.4 Hz, the optimal load is 186 kΩ, which yields approximately the same power output.
The total volume of the overhanging part of the piezofan and the attached mass is 3,52 cm3. The Total mass is 20,6 gram. The power density (6.8 mW/g2 cm-3) specific power ( 1,15 W/g2 kg-1) can be calculated at excitation at 45,6 Hz and a load of 35 kΩ.
In another work, [17] with a design of a volume of 1 cm³ of PZT-5H material a specific power of 375 μW/cm³ is found for a resistive load when vibrating at 120 Hz and an acceleration of 2,5 m/s². For the same vibration, a specific power of 190 μW/cm³ is found for a capacitive load.
1 More info on the numbering and used materials by Piezo Systems and other manufacturers is given in appendix A

6
An example where the piezobeam principle is used is the implantation of a bimorph piezolayer in the heel of a shoe. [19] Another example is the energy supply of a wireless sensor, studied in [5]. In another work [13], the piezobeam was placed on the wall of a pipe. The fluid flow through the pipe made energy harvesting possible.
Microgenerators using the bending beam principle can be built as well. The dimensions of such generators are in the order of magnitude of several μm. Dimensions and obtained powers are given in [10, 20-22].
2.1.1.1 T-shaped piezoelectric cantilever
An endplate is attached to a piezoelectric bimorph cantilever as described above. A T-shape cantilever results. It is used for harvesting electric power from fluid flow. The T-shape fastens the occurrence of the flutter phenomenon, which is explained in §5.1.2, at low fluid speeds. A prototype was found to produce a continuous peak electrical power of 4,0 mW at a wind speed of 4 m/s. The dimensions of the prototype where 100*60*30 mm3. The simple T-shape of the cantilever makes the device cost-effective.
2.1.1.2 Wind tree
In research [44], it was tried to combine two aeroelastic effects. The vortex shedding and the self induced effect (flutter) were combined by using a flapping leaf generator. A bluff body at the clamping of the flexible piezoelectric PVDF stalk induces vortex shedding, which was based on the research done in [52]. The self induced effect takes place due to the vortex formation at the trailing edge of a plastic ‘leaf’.
Figure 3: Concept of the (vertical) leaf generator, top view [44]
Some different shapes and positions of the leaf were tried. It was found that the maximum power output was obtained for a vertical position of the leaf, shown on the right picture in Figure 4. This means that the best setup in this research had no bluff body at the base. So there was also no vortex shedding effect in front of the leaf, as was intended originally. The wind speed was in between 2 and 8 m/s. A maximum power output of 0,61 mW was found for a bimorph PVDF harvester at a winds speed of 8 m/s. The concept is to build a tree-like construction consisting of multiple of such leaf-like elements. More information is available in [45]. The principles on which this energy harvesting setup is based will be explained more thoroughly in Chapter 5.

7
Figure 4: Setup of a horizontal-stalk and vertical-stalk leaf [44]
2.1.1.3 Solar Ivy
The SMIT (Sustainably Minded Interactive Technology) has developed patches that combine piezoelectric vibration harvesting with solar energy conversion. The leaf-shaped patches are mounted to a roster that can be attached to a wall to resemble ivy. Early versions are commercially available at this moment.
Figure 5: Two views of solar ivy leaves
2.1.2 Energy harvesting eel
A piezoelectric membrane, or ‘eel’ is placed into the wake of a bluff body. [35] Due to vortex shedding, which is explained in depth in §5.1.1, the eel starts vibrating. The membrane has to be sufficiently flexible to be able to vibrate at the same frequency as the undisturbed wake would do. The oscillating of a downstream cylinder caused by the wake flow of an upstream cylinder is also called wake galloping. [53] A similar phenomenon is discussed in §2.4.1.
Figure 6: Oscillation of the membrane, induced by vortex shedding

8
The material that was chosen is PVDF, a semi-crystalline polymer that has undergone a surface treatment to obtain a piezoelectric effect. [10] The piezoelectric properties of this material remain good until 70°C, so the eel remains usable at this temperature. Furthermore, PVDF is mechanically strong, resistant to many chemicals and it can be produced as a continuous wire. Research is now done in improving the obtained electrical power, which is currently still low. More information on the PVDF material is given in the section on piezoelectric materials in §3.3.2.
2.1.3 Piezoelectric windmill
2.1.3.1 Non-rotating
A non-rotating windmill with piezoelectric blades is shown in . [25] The blades continuously oscillate when an airflow goes through the mill. The blades act as bending beams and induce an electric voltage, as explained in §2.1.1. Bimorph piezoceramic material is used here because of its higher mechanical strength, higher tension coefficient and higher maximal displacement (which results from its higher mechanical strength), and a lower production cost of the bimorph material. [10]
Figure 7: Working principle piezoelectric windmill
Tests [25] were done on a prototype with 10 blades of 60*20*0.6 mm3 with a resonant frequency of the blades of 65 Hz. At a rated wind speed of 10 mph (16 km/hr), a power of 7,5 mW was found for an electrical load of 6,7 kΩ.
2.1.3.2 Rotating
Experiments are also done with rotating designs. For example, a cup anemometer [26] was used to excitate piezoblades with three turning bulges. This turned out not to be a favourable setup. Not enough piezoblades where excited, and the excitation does not happen in phase. This means the signals of each blade has to be processed separately. An improvement was suggested [26], using parallel bimorph plates and a crank-shaft mechanism, powered by a traditional windmill.
2.1.4 Diaphragm piezoelectric transducer
A piezoceramic layer (PZT) is placed on top of a structural, non active brass layer. [37] These two layers form a membrane that is attached to an aluminum ring. This is shown in Figure 8. Mechanical vibrations of the aluminum base ring induce oscillations in the membrane, and energy is harvested by the piezoelectric effect of the PZT layer.

9
Figure 8: Top and side view of the diaphragm piezoelectric transducer
With a membrane of a diameter of 25 mm, a maximal power output of 1.8 mW was obtained at
an acceleration of 2 g and an electric load of 56 kΩ. This is obtained at a resonant frequency of
2,58 KHz.
The implementation of this principle on micro scale is also examined. (for a radius of
approximately 750 μm). A PZT layer is therefore placed on a silicone layer. Two pairs of electrodes
are used as can be seen on the top picture in Figure 8. An alternating voltage is put on the
primary electrodes, and the inverse piezoelectric effect induces a vibration. In the secondary
electrode, this vibration is reconverted into an electric tension. The ratio of energy conversion is
function of the polarization of the material and the ratio of the primary and secondary electrodes.
This system is only favorable when used at resonance.
Figure 9: Micro-diaphragm transducer
2.1.5 Microfiber nano wire
Vibration- or friction energy is converted to electrical energy by using piezoelectric nanowires
[50] that are placed radially around textile fibers. By winding two fibers around each other, and
by making them move with respect to each other, electrical energy is harvested by a coupled
piezoelectric semiconductor process.

10
2.2 Electrostatic
Mechanical energy can be converted to electrical energy by using a variable capacitor. [54] The plates of the capacitor are made to move away from each other and back as the result of an external vibration. While doing this, either the charge or voltage of the capacitor are kept constant. The capacitor needs to be precharged at its highest capacity point: This is the point where the distance between the plates is the smallest.
In a constant charge system, the plates of the capacitor are kept in an open circuit as they move apart. This makes the capacitance drop, which makes the voltage rises: 푄푐푡푒=퐶.푉
The relationship between energy and voltage in a capacitor is given by: 퐸=12⁄.퐶.푉2
Thus, the energy stored in the capacitor will rise due to the quadratic relationship between voltage and energy. It follows that it is very important to precharge the capacitor at its maximum capacitance point, and to unload it at its minimum capacitance point, in order to obtain a maximum energy transfer.
In a constant voltage system, the voltage is kept constant while the distance between the capacitor plates increases. By moving the plates away from each other, the capacitance drops. Charge is thus driven out of the capacitor: 푄=퐶.푉푐푡푒
The escaping energy is conducted to another capacitor, or battery. So the variable capacitor acts as a current source: 푖= 푑푄 푑푡 =퐶 휕푉 휕푡 +푉 휕퐶 휕푡 =푉 휕퐶 휕푡
If the capacitor plates move apart with constant electric tension or charge, a constant force is induced with displacement.
The main advantage of electric energy harvesting is its scalability. This can be seen in the examples below; most setups have microscale applications. The greatest disadvantage is the smaller power output when compared with other energy harvesting principles.
Below, two practical implementations are given. Both can work with either the constant voltage or constant charge principle. [27] A more thorough description of the concept and the different methods of electrostatic energy harvesting is given in [14].
2.2.1 Coulomb-damped resonant generator (CDRG)
The CDRG [27, 28] extracts energy by using a damper that exercices a constant force in the direction opposing the movement, as described before. In fact, Coulomb dampers are realized by doing this. Coulomb damping is most often used to model friction on a (dry) surface, but it can be used to describe electrostatic effects too. [27]

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Figure 10: Model of CDRG: y(t) absolute displacement of the reference; z(t) relative displacement with respect to the
reference
A possible realization is given in Figure 11. The system is built out of three main parts: a moving
mass, a folded spring at each site and two stationary combs. Both springs consist out of 4 bending
beams each, 1 rigid anchor and a rigid beam connecting the bending parts. By using this setup,
the system will theoretically vibrate in one degree of freedom: up and down.
The fingers of the interdigitated mass and the stationary part form variable capacities. As the
mass oscillates, the fingers move up and down, changing the surface of the capacitor, and thus
the capacity. Energy can be harvested by connecting electrical circuits to both the upper and
lower anchors of the moving and stationary part.
Figure 11: Possible implementation of CDRG, vertical movement
The setup described above was designed for a vibration frequency of 2250 Hz. The vibration
source that is aimed at are the (well known) harmonics of a motor.
It is possible to build a similar system [14] with a horizontal moving mass instead of using a mass
that moves up and down. This is shown in Figure 12. The distance between the capacitor plates
changes, and thus the capacity and/or voltage changes.

12
Figure 12: Possible implementation CDRG, horizontal movement
Various other designs are possible. An example is the design where parallel interdigitated plates
move horizontally with respect to each other, thereby varying the effective capacitor surface. The
is described in [51]. A problem with all energy harvesting setups described above are their
stability. It is difficult to make the above described systems vibrate with only one degree of
freedom. If the systems vibrates in multiple degrees of freedom, there is a risk that the capacitor
plates touch.
2.2.2 Coulomb-force parametric generator (CFPG)
This generator differs from most other systems described in this work, as it does not necessarily
vibrate at its resonant frequency. [27, 39, 40] It is therefore usable for applications at lower
frequencies, below the resonant frequency of the system. An example are movements of the
human body.
Consider a system consisting of a spring and a damper vibrating at its resonant frequency. With
resonating systems, the work done on the damper and spring can be absorbed by the damper, or
stored in the spring. The energy stored in the spring can be released to the damper in another
stage of the vibration cycle. With a non-resonating system, this is not possible and work is done
to the damper only. Therefore, the work done has to be maximized during the whole cycle, which
requires a different design as for resonating systems.
Figure 13: Model CFPG: y(t) absolute displacement of the reference; z(t) relative displacement with respect to the
reference

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In Figure 14, a possible setup [39] is shown (width 28 mm, height 2 mm). The precharging of the capacitor takes place when it makes contact with the lowest contact points (‘input’) and is connected to a charging circuit by doing so. The stored energy in the capacitor is released when contact is made with the top contact points (‘output’). Research to optimize of the charging circuit with constant charge is done in [55].
Figure 14: Possible CFPG setup
The obtained power is smaller than with, for example, the CDRG, if vibration takes place at the resonant frequency. If the structures vibrate at a frequency below their resonant frequency, more energy is harvested by using the CFPG.
A comparable system can be build for a resonating system. The mass then oscillates up and down, changing the distance between the plates of the capacitor, and so does the capacity. Such a system would resemble the setup described in §2.2.1.
2.3 Electromagnetic
With electromagnetic energy conversion, a magnetic field is used to convert mechanical energy to electrical energy. This is done as stated by Faraday’s Law, which poses that an electrical voltage is created as the flux of a magnetic field changes. Due to the small displacements caused by the vibrations, the powers obtained with this method are inherently small, as shown in Table 2.
2.3.1 Electromagnetic, vibration-powered generator
This generator [30] consists of a mass-spring system that forces a coil (or magnet) to move up- and down with respect to a magnetic core. By doing so, the coil moves through the magnetic field of the core. The variation in magnetic flux induces an electric field, as stated by Faraday’s law of induction:
푉푒푚=−푁 푑휑 푑푡
With φ being the magnetic flux and N the number of windings of the coil. Either the coil or the magnet is fixed at the body, or vice versa. The fixed coil is often preferred, as it is easier to connect the wires if the coil does not move.

14
Figure 15: Model of a vibration-powered generator: y(t) absolute displacement of the reference, z(t) relative
displacement with respect to the reference, k spring constant, d damping, f=f(N,A,B) with N the number of windings, A
the surface and B the flux density.
A prototype Figure 16, was built in [29]. The core/magnet was attached to the far end of a
clamped beam (1,1*0,9*0,85 cm3). The beam resonates with respect to a stationary coil. At an
optimal load of 0,6 Ω, a maximal electrical power output of 180 μW was obtained with a
displacement at the end of the beam of 0,85 mm.
Figure 16: Working principle of a electromagnetic harvester, prototype A
In a second prototype, pictured in Figure 17, a magnetic field is created over a greater portion of
the coil, which increases the harvested power. The beam now measures 2,1*1,5*1 cm3.
Figure 17: Working principle of a electromagnetic harvester, prototype B
As a test, the generator was placed on the motor compartment of a car. While driving with an
average speed of 24 km/h, an average power of 157 μW was obtained. The produced power was
function of the rotating speed of the motor, with a resonance peak at 3000 rev/min. This was
attributed to a resonance of the engine mounting. The obtained peak power was 3,9 mW.

15
Research [32] is also done to use this principle for powering a micro generator. A spiral spring (Ø
4 mm) is used because of its compactness. Tests were done to obtain the best way to excite the
spring (vertically or horizontally) and to obtain the optimal force to obtain a maximum power
output for the lowest mechanical load to the spring. In other research [56], the working principle
of Figure 17 was used to build a micro generator, but with more coils. In other work, multiple
moving magnets and stationary coils were used to harvest energy in a micro generator. [34]
2.3.2 Velocity-damped resonant generators (VDRGs)
A mass-spring system is used to harvest energy with a damper, as seen in Figure 18. [27] Notice
the resemblance with Figure 15, with the function f replaced by damping.
Figure 18: Model mass-spring system: y(t) absolute displacement of the reference, z(t) relative displacement with
respect to the reference, k spring constant, dpar parasite damping, dgen damping by generator
This is realized by attaching a magnetic mass to a membrane, as in Figure 19. The membrane will
act as a spring as the structure is made to vibrate. This induces an electric voltage in the coil
below the silicon wafer.
Figure 19: Possible practical setup VDRG
This setup is mostly used in micro generators, as in [41].
2.3.3 Rotary electromagnetic generator
A rotating electromagnetic (micro) generator consists of a stator and a rotor. The rotor is a disc
(outer diameter 2,5 mm, inner diameter 1 mm) that is built from sectors of differently polarized
permanent magnets. The stator consists of sectors of copper windings, as drawn in Figure 20.

16
Stator and rotor are placed parallel to each other. When turning the rotor, electric tension is induced in the coils. In the research, 8 sectors are chosen for both the rotor and stator. Multiple stator disks with windings can be put on top of each other. In this research, 4 layers were used.
Figure 20: Rotary electromagnetic generator
The Total volume of this setup is approximately 5 * 5 * 2 mm³. In this research, with 4 layers, and a distance between the magnet and coil from 1 mm, a tension of 112,2 mV and a power of 0,412 mW were found at 149,3 Hz.
2.3.4 Flutter mill
The flutter mill consists of a clamped flexible plate with embedded conductors. This is shown in Figure 21 and in Figure 22. The plate is placed between two magnetic panels, in an axial flow. As the fluid velocity exceeds a certain critical value, flutter takes place. On average, energy is transferred [46] from the fluid to the plate in order to have a sustained flutter movement. This energy is harvested and converted to electric energy. At certain segments of the beam, the fluid will exercise work on the beam, but the opposite may occur as well. On some segments, the beam exerts energy to the flow. It is important to know this, and to know the best segments of the beam to extract energy from the fluid.
It was discussed in §2.1.1 how energy conversion can happen in a beam with a piezoceramic material. As the displacement from a clamped beam only results in small strains, the associated induced current will remain small as well, although a high potential difference can be obtained. The potential power production is limited by this fact. The flutter mill therefore tries to convert the energy in another way.
As flutter takes place, the flexible plate deforms in its bending modes, which is shown in Figure 21 for the second bending mode. Conductors are placed onto or in the plate, as can be clearly seen in Figure 22. As a result of the magnetic field of both magnets, a potential difference between the different conductors exists. By connecting the conductors, an electrical current circuit is obtained. It becomes clear that the position of the conductors is important (best at the highest deflection points), as mentioned before.

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Figure 21: Side view Flutter mill
Figure 22: Top view Flutter mill
It is expected [46] that approximately 10% of the energy captured by the plate can be converted
to electrical energy. For a device that measures (length*thickness*height) 0,58*0,2*0,58 m3 in
total, and has a mass ratio fluid/plate of 0,5 and a fluid speed of 12 m/s, it is expected that the
fluid exercises 10 Watt of power on the plate. This means that such plate would be able to supply
1 Watt of energy. At a lower mass ratio, an higher power could be obtained with a higher fluid
speed of 40 m/s. A system that has comparable gainings as a horizontal axis turbine could be
built.
2.4 Other principles and concepts
2.4.1 Transverse Galloping
Transverse galloping or Across-wind galloping takes place as the incident flow on a slender body
(such as a cable) exceeds a certain critical speed. [43, 53] A large-amplitude oscillation in a plane
normal to the oncoming flow results. The stabilizing effect of mechanical damping is hereby
overcome by the destabilizing effect of the fluid forces. A small transverse displacement of the
object leads to an interacting fluid force that leads to an higher vibration amplitude.
Once the stability threshold is passed, an oscillating movement develops. The amplitude increases
until the dissipated energy by damping equals the supplied energy of the fluid flow. This is a low
frequent movement, with an amplitude that rises with speed. By carefully choosing the elastic
properties of the body, the energy harvesting can be maximized. A mechanism needs to be
implemented to transfer the mechanical energy to electrical energy. This can be a crankshaft
mechanism, or a piezoelectric conversion.

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Across-wind galloping can sometimes be observed with individual electricity cables that have become asymmetric, for example due to ice loading. With bundles of power transmission lines, wake galloping, already discussed in §2.1.1.3 with the energy harvesting eel using vortex shedding, can sometimes be observed. This treated in more detail chapter 0.
2.4.2 Flapping foil generator
A flapping foil generator, also named wingmill, is discussed in [57]. The working principle is as follows. A wing is mounted so that it is able to move in two degrees of freedom, as in Figure 23. A motor allows the wing to make a pitching oscillation. A positive angle of attack results, and the lift induces a heaving movement. Energy is extracted from the work resulting from the heaving movement. The pitching movement of the foil caused by the motor required 1% of the overall power.
Figure 23: Principle of the Wingmill [57]
2.4.3 Leading edge flutter wind power generator
The Leading Edge Flutter Wind Power Generator (LEFWPG) [47] uses the dynamic aeroelastic instability of a wing profile that rotates around its leading edge (leading-edge torsional flutter). This occurs once the fluid flow obtains critical incident fluid speed. [48] As the wing only rotates, this is a one dimensional movement. In this article, a conceptual crankshaft mechanism is used, as shown in Figure 24. The LEFWPG is a particular case of flutter: stall flutter. A general explanation of the (stall) flutter phenomenon is given in 5.1.2.
Figure 24: Working principle leading edge flutter wind power generator
The LEFWPG is derived from similar setups that use two degrees of freedom (rotation and translation) as, for example, the flapping wing power generator or oscillating-wing windmill do.

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2.4.4 VIVACE (vortex induced vibration aquatic clean energy)
The VIVACE [36] setup shows similarities with the above mentioned energy harvesting eel. It is however intended for use on larger scale and no piezoelectric materials are used. Different cylinders are placed horizontally after each other in a fluid flow (a river, for example). Due to the Von Kárman street that is formed after each cylinder, the cylinder in the wake of the preceding cylinder moves up and down. This movement is converted into electric energy, for example with the help of a crankshaft mechanism or with magnets and coils as in an electrical generator. The VIVACE is interesting as it can work with a flow of 2 to 4 knots (1 to 2 m/s), where classic turbines need a speed of 4 knots or more.
Figure 25: Working principle of the VIVACE [36]
2.5 Advantages and disadvantages of the different harvesting principles
The (dis-)advantages of the different energy transfer mechanisms that are found in the literature are summarized in Table 3:
Table 3: Advantages and disadvantages of the different vibration energy harvesting principles.
Advantages
Disadvantages
Challenges for micro- implementation
Piezoelectric
 No need for a voltage source
 High voltage output
 Difficult integration
 Moving parts
 Small coupling between small films.
Electrostatic
 Very scalable
 Compatible with current technology
 Voltage source needed for precharging
 Stability
Electromagnetic
 No need for a voltage source
 Low voltage output
 Moving parts
 Difficult to integrate the magnet

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2.6 Harvesting powers found in literature
Results that are found in literature for different energy harvesting mechanisms are shown in Table 4: Harvesting powers found in literature for various harvesters
. It is important to note that the power density and the specific power shown in the table are not dimensionless representations, the figures given below are hard to compare. For the same volume or mass, the aspect ratio or other properties can differ, yielding a totally different power output. In most works, the volume or mass are not well documented. Ideally, the input vibration amplitude and frequency should be known in order to calculate an efficiency, as the output power depends on the input power
Table 4: Harvesting powers found in literature for various harvesters
Name of the principle and year
Specs/dimensions
Power
PE
Beam bending [17]
2004
Material: PZT-5H
V = 1 cm³, a=2,5 m/s², fres= 120 Hz
0,375 mW
PE
Beam bending [15]
2009
Material: PZT-5A (Piezo Systems T226-A4-503X)
V = 3,52 cm3 fres= 45.6 Hz
Power output given per unit of base acceleration (g=9,81 m/s²)
23,9 mW/g² (peak)
PE
Beam bending [24]
Material: MFC (MFC 2814P2) (28*14 mm²)
5 m/s², fres = 12.5 Hz, with tip mass (5,32 gram)
3,5 mW
PE
T-shaped piezoelectric cantilever
2010
6 patches of MFC 2814P2 mounted as 3 bimorphs on a plate
Wind speed: 4 m/s
4 mW
continuous peak
PE
Flapping Leaf generator [44]
2011
Material: PVDF
wind speed: 8 m/s
300μW/cm³
80μW/gram
PE
Non rotating piezo windmill [25]
2005
10 bimorphs of 60*20*0.6 mm3
fres=65 Hz
Wind speed = 4,4 m/s
7,5 mW
PE
Piezo generators excited by wind power [26]
2007
18 bimorphs of 60*20*0.6 mm³
Rated wind speed = 4,4 m/s
15 mW continuous
PE
Piezo membrane
2005
Ø=25 mm, 2g acceleration, fres=2,58 KHz
1,8 mW
max output
PE
Flapping leaf
2009
Bimorph cross flow PVDF stalk
72 * 16 * 0.41mm³
Wind speed = 6,5 m/s
0,119 mW
max output
PE
Flapping leaf
2011
Bimorph cross flow PVDF stalk
72 * 16 * 0.41mm³
Wind speed = 8 m/s
0,61 mW
max output
EM
Moving magnet, stationary coil
2003
Engine mounted, V=3,15 cm³, cantilever length 2.1 cm, width 5,1 cm
3,9 mW
EM
Rotary generator
V = 5 * 5 * 2 mm³, frequency=149,3 Hz
0,412 mW

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3 Choice of the harvesting principle and setup
For a number of reasons, the piezoelectric effect is chosen to be studied in more detail. This effect is easily studied on larger prototypes. This makes it possible to build a rather large experimental prototype, that can be easily reduced in size for application in smaller devices. Another very important factor motivating this choice are the good energy conversion properties compared with electromechanical and electrostatic harvesting, which are seen in Table 2. Piezoelectric conversion leads to high voltages and low currents, but the energy conversion efficiency is good. It is also a solid-state technology, which make robust, reliable and compact structures possible.
The setup with a piezomaterial can range from very simple to very complex. A bending beam with a piezoelectric patch is seen as very robust and cost-effective. More complex setups are possible too, as we already saw with the piezoelectric windmill, for example. Also, the principle allows a lot of different setups regarding wind energy harvesting, which will be discussed in chapter 0.
3.1 The piezoelectric effect
A piezoelectric material has some distinctive characteristics. When it is subject to a mechanical force, the material becomes polarized. The voltage that is generated is in proportion with the applied force. The polarity of the voltage depends on whether the material is subject to tension or compression. This is called the piezoelectric effect. Conversely, if the material is exposed to an electric field, it lengthens or shortens in proportion with the strength of the field. Whether the material shortens or lengthens depends on the polarity of the field. This is called the inverse piezoelectric effect.
Piezoelectricity is used in many applications. It is used to build sensors, to measure force or displacement for example. The inverse piezoelectric effect is used to build actuators or to create generating acoustic sonic and ultrasonic signals. For each of the applications, different properties of the material are of importance. Therefore, the most important properties and materials are studied first to allow a good choice of materials and setup.
3.2 Some theory describing piezoelectricity
3.2.1 The use of subscripts
In the coefficients that will be used below, subscript ranging from 1-3 are used. These numbers correspond with the three axis: Direction 3 is parallel to the direction of polarization. For some coefficients, two subscripts are used. The first subscript is related to the direction of the applied voltage or the produced charge. The second subscript is related to the direction of mechanical strain or stress.
3.2.2 Piezoelectric charge constant
The piezoelectric charge constant, d, describes the relation between the polarization that results and the mechanical stress that is applied to the piezoelectric material. Alternatively, it describes the mechanical strain experienced by the material when a unit of electrical field is applied. Two effects or modes are generally used for piezoelectric energy harvesting: the d31 and d33 effect. Most manufacturers have both d31 and d33 types of generators are available. Both are piezoelectric properties of a material, and each one of the two is better suited for some particular applications.

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Figure 26: d33 and d31 effect on piezoelectric materials [58]
For d33, the induced polarization is in the direction parallel to the direction in which the ceramic element is polarized. Stress is applied in this same direction. The advantage to this type is its high energy conversion rate. However, the electrical output voltage is very high and the electrical current low. This makes this type less suitable for use with low energy consuming electronics, as the losses for converting the voltage and current to a usable values are high. This type is more suitable as sensor or actuator.
With d31, the induced polarization is in the direction parallel to the direction in which the ceramic element is polarized. Mechanical stress is applied perpendicular to the direction in which the ceramic element is polarized. The voltage output of this type is one order of magnitude lower then for the d33-type, while the electrical current is larger. This makes d31 better suited for energy harvesting, although the overall energy conversion is lower than for the d33 effect. In our search for suitable harvesters, we will look for d33-type harvesters.
3.2.3 Piezoelectric voltage constant
The piezoelectric voltage constant g describes the electric field that is generated by a unit of applied mechanical stress, or the mechanical strain per unit of electric displacement. Again, subscripts indicate the direction of electrical and mechanical properties. This property is very important for piezoelectric materials that are used as sensors.
3.2.4 Electromechanical coupling factor
The electromechanical coupling describes the conversion of mechanical energy to electrical energy in case of the piezoelectric effect, and vice-versa for the inverse piezoelectric effect. For the conversion of mechanical energy to electrical energy one writes: 푘=√ 푠푡표푟푒푑 푒푙푒푐푡푟푖푐푎푙 푒푛푒푟푔푦 푎푝푝푙푖푒푑 푚푒푐ℎ푎푛푖푐푎푙 푒푛푒푟푔푦
In appendix F, the effective electromechanical coupling factor keff is given for PZT piezoceramic materials. These materials are studied more deeply in §3.3.1. keff was obtained using the formula below. Here, fn is the resonant frequency (at minimal impedance) and fm the antiresonant frequency. The antiresonant frequency is the frequency at maximal impedance. 푘푒푓푓= √ 푓푛 2−푓푚 2 푓푛 2
3.2.5 Linear equations describing the piezoelectric effect
A constitutive relation relates two physical quantities specific to a material and approximates the response of that material to external forces. An example is Hooke’s law in mechanics. Here,

23
Hooke’s law is expressed in its inverse formulation: S=s.T. In this equation, T is the stress vector, S the strain vector and s the compliance matrix. In electromagnetism, a similar law exists: D=ε.T. In this equation, D is the electric charge displacement vector, T the stress vector and ε the electric permittivity matrix.
The constitutive equations for piezoelectric materials combine the two preceding equations. The relations are linear within a certain range. The approximated linear relations between the stress (T), strain (S), electric field (E) and electric charge displacement (D) are given below: {푆}=[푠퐸].{푇}+[푑]푡.{퐸} {퐷}=[푑].{푇}+[휀푇].{퐸}
Symbols in [ ] represent matrices and symbols in { } vectors. The piezoelectric coupling terms are in the matrix d, and were already described in §3.2.2. The subscripts in sE and εT indicate that the data in the matrix was obtained for constant electrical field and stress respectively. The relations above are written in the Strain-Charge form. These relations can be rewritten to Stress-Charge, Strain-Voltage and Stress-Voltage formulations. Usually, the coefficients d and g are given by manufacturers of piezoelectric materials. d is seen in the equation above, g is used in the Strain- Voltage form of the equations.
3.2.6 Equivalent electrical circuit, electrical load and voltage output
The piezo beam can be modeled as an equivalent electrical circuit [59]. This is shown in its easiest form for an unloaded beam in Figure 27. Values for the capacitance of the piezo situate around 10 nF, while resistance values of the piezo exceed 40 MΩ. This parallel resistance can be neglected. One could say that power is drawn by the resistor Rl from a capacitor that is constantly recharged by the environment.
Figure 27: Simplified equivalent electrical circuit
With a resistive electrical load attached to the piezo, the equivalent impedance of this piezo (at a given amplitude and frequency) and the load form a resistor divider. In order to maximize transfer efficiency, the load must be matched to the equivalent impedance of the piezo. If so, the loaded voltage of the piezo corresponds with half its open circuit voltage (Voc). This way, optimization of the electrical power output can be obtained without knowing the equivalent impedance of the piezo.
This optimization can be achieved practically by using a circuit that uses a storage capacitor (C1) that charges until the optimum working voltage (Voc/2) is obtained. If this voltage is exceeded, a converter depletes the storage capacitor and provides power at a suitable voltage to the load. This can for example be a battery or a wireless sensor.

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3.2.7 Polarization of a bimorph generator
A piezo generator with two layers, named a bimorph generator, can have its layers poled in the
opposite (X-poled) or in the same direction (Y-poled). Depending on the polarization and wiring,
the layers will operate in either serial or parallel mode.
Figure 28: Left: X-poled for serial mode operation; right: Y-poled for parallel mode operation
When used as a generator, it is obvious that serial wiring provides an higher voltage output. This
leads to a lower current output. This is explained as follows: A piezoelectric generator is specified
by its closed-circuit current (ICC) and its open-circuit voltage (VOC). ICC is the total current obtained
at maximum recommended strain level and operating frequency, for a closed circuit with no
voltage buildup. VOC is the voltage measured at the electrodes if no current flows from one
electrode to another. The current will be at maximum if there is no voltage buildup, and the
voltage will be at maximum if there is no charge flow. The other values of current- voltage
combinations can be obtained by drawing a straight line at the voltage- current diagram. This is
shown in Figure 29 for two Y-poled piezo generators with different dimensions with values
obtained in the Piezo Systems catalog [60].
Figure 29: Voltage- current diagram for a piezo generator; circle: design point
Generally, a piezo generator has its operating point defined by a specified voltage and current it
has to deliver. The design will require the least power input if the generator delivers the required
voltage at half its closed circuit current, or half its open circuit voltage. This was already seen
using the equivalent circuit in §3.2.6. This design point is marked on the figure with a dot. The
intersection of the curves with the vertical axis corresponds with the open circuit voltage, the
intersection with the horizontal axis with the closed circuit current.
3.3 Piezoelectric materials
The piezoelectric effect was discovered by Jacques and Pierre Curie on crystalline minerals, such
as quartz, in 1880. Other piezoelectric crystals are Berlinite, Tourmaline and Gallium
orthophosphate. In the present, man-made materials can be used to benefit from the
0
5
10
15
20
25
0 10 20 30 40 50 60
Voltage [V]
Current [μA]
303YB
503YB

25
piezoelectric effect. Piezoceramic materials have been made: Barium titanate and Lead Zirconate Titanate (PZT). Other piezoelectric materials include Polyvinylidene fluoride (PVDF), zinc oxide and aluminum nitride.
However, PZT is the most used piezoelectric material in most applications. An overview of the most used materials for energy harvesting purposes is given below. Some techniques to increase the strength of brittle piezoceramics are also explained.
3.3.1 Lead Zirconate Titanate Ceramic (PZT)
Ceramics manufactured out of lead zirconate and lead titanate are the most widely used piezoelectric ceramics. The very strong piezoelectric effect of lead zirconate titanate was discovered in 1954. As the ceramic is macroscopically isotropic, the spontaneous polarization of the material must be oriented by a poling process, using an external electrical field. As the PZT ceramics are very brittle, they are usually adhered to an elastic material, such as stainless steel. Electrode patterns are necessary to connect the ceramic layers with an electrical circuit. The PZT layer(s) can be put in between multiple elastic layers. Some examples are given in the paragraphs below.
Different types of PZT have been developed. PZT-4 and PZT-5 are the most common. Two types of PZT-5 exist: type A and type H. The properties of both materials differ. This can be seen in appendix F. The most important difference is that PZT-5A has a higher Curie temperature. If a piezoelectric material is heated above this temperature, it loses its polarization and thus its piezoelectric properties. This means that PZT-5A can be used over a broader temperature range than PZT-5H.
PZT is not only piezoelectric, but also pyroelectric, which means that it generates a voltage difference over its surface if there is a change in temperature.
3.3.1.1 PZT layers with a structural inner layer
Two piezoceramic layers are attached to an inner structural layer most of the time. One of the fabricants selling this setup is Piezo Systems. Structural brass (economical), steel (high strength), composite (high performance) inner layers are available. Non magnetic versions with both non magnetic structural layers and electrodes are available too.
3.3.1.2 PZT fibers in a flexible composite material
Composites made from combinations of plastics and piezoceramics are also available. As an example with regard to energy harvesting, Smart Materials sells the MFC, or Macro Fiber Composite. It consists of a patch of parallel piezoceramic/epoxy fibers covered with epoxy layers on the top and bottom, as can be seen in Figure 30. The epoxy layers prevent crack propagation in the brittle piezoceramic layer. This setup leads to an improved flexibility and damage tolerance when compared to a single ceramic. The three inner layers are covered with an interdigitated electrode pattern. This allows fabrication of both longitudinal and transversal polarization of the piezoelectric fibers, as explained in 3.2.2.

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Figure 30: Product layers of the Macro Fiber Composite (MFC) [58]
As these patches are highly flexible, they are intended to be glued onto another, stiffer, surface. This can be, for example, a steel plate subject to bending.
Figure 31: Picture of the Smart Materials MFC [58]
3.3.1.3 Bimorph PZT in between structural layers
An example of a bimorph PZT material that is reinforced by placing it in between layers of other materials is the energy harvester manufactured by Midé. The Midé energy harvester consists of 5 layers that are bonded together with an epoxy adhesive. The inner layer is made of ‘Espanex’, which itself consists out of an isolating polyimide layer covered with electrical conducting copper layers. The piezoceramic layers (PZT material) are glued onto the Espanex, and an isolating FR4 layer protects these. FR4 is made of woven fiberglass, held together with an epoxy resin binding. This layer acts as electrical insulator and has a good mechanical strength.
Figure 32: Product layers of the Midé harvester [61]
The two piezo layers each have two separate connections going to a universal connecter. This allows to connect the layers in series or parallel simply by adjusting the wiring.
Figure 33: Picture of a Midé harvester [61]
3.3.2 Polyvinylidene Fluoride (PVDF)
Polyvinylidene Fluoride (PVDF) is a very flexible piezoelectric thermoplastic material. The material is given its piezoelectrical properties by mechanically stretching it and then poling it by placing the material under an electric field. This material is used in the T-shaped setup explained in §2.1.1.2 and in the energy harvesting eel, §2.1.1.3, because of its flexibility. The flexibility of this

27
material makes it very suitable for energy harvesting out of wind flow. The mechanical coupling factor k31 is about four times smaller than for PZT5 piezoceramic material (k31=0,10-0,15) [62]
3.3.3 Other materials
Some other materials and material combinations are also used. Examples include reinforced ceramics. Such are available under the following brand names: PFC (Piezo Fiber Composite) is comparable with the MFC patches treated earlier. LIPCA (Lightweight piezo-composite actuator) also uses PZT material and THUNDER (Thin layer composite unimorph Ferroelectric driver) has multiple PZT layers in between aluminum layers. PMN-PT is a piezoelectric crystal with strong dielectric and piezoelectric properties. It is used mainly in acoustic transduction devices, such as in the thermo-acoustic power conversion device discussed in [63].
3.4 The harvesting devices used in this work
Types of harvesting devices from two manufacturers were chosen. The first type is a ready to use bimorph harvester, sold by Midé, described in §3.3.1.3 and pictured in Figure 33. It is ready to use and easy to connect with a universal connector.
The second harvester is based on the MFC patch described in §3.3.1.2 and pictured in Figure 31. The Smart Materials patches are very thin and flexible and should be glued onto another surface. This gives more possibilities in choosing the shape and material to which the patch is mounted.
Figure 34: Top view of the MFC patches and the bimorph generators used in this work
The size of different types of harvesters that were chosen are compared in Figure 34. Two types of MFC patches will be used: The M8507 patch is very slender, its active length to width ratio is approximately 12. This type of patch can be useful as the beam that will be used for vibration and wind flow harvester will be very slender too. The M2814 patch has a length to width ratio of 2. The two layers of the V21B have a length to width ratio of approximately 2,5. The mounting of the patches on other materials for rigidity will be explained in §4.3.

28
The active areas of the piezoceramic layers are of the same order of magnitude for all chosen generators, which makes comparison easier and meaningful. This can be seen in Figure 35. Full details on the bimorph harvester and the patches is given in appendix A.
Figure 35: Active areas of the different harvesters
0
100
200
300
400
500
600
700
M8507
M2814
V21B & V21BL
active area [mm²]

29
4 Piezoelectric energy harvesting out of vibrations
4.1 Experimental setup
An experimental setup that enables us to excite the piezoelectric generator at different vibration frequencies is constructed. This is schematically shown in Figure 36. An electrical signal, coming from a function generator is amplified and fed to a shaker. The piezoelectric generator is mounted on the shaker via an impedance head (s/n 1231794, properties in appendix D) that is capable of measuring force and acceleration. The acceleration, force and voltage outputs are visualized on the digital oscilloscope.
A variable resistor, or different discrete resistors are attached to the harvester output. This allows us to have different electrical loading conditions. This also makes it possible to determine the output electrical energy with ease. By measuring the voltage over this resistor, the mean electrical power output is obtained (Prms=Vrms²/R). The excitation signal is chosen sinusoidal, so it is possible to excite the beam at a specific frequency. The excitation signal frequency will be matched to the resonant frequency of the beam in order to maximize the power output.
Figure 36: Experimental setup of the piezoelectric beam on a shaker
The charge sensitivity for acceleration and force of the impedance head need to be set in the charge amplifiers, as this sensitivity is different for every single impedance head. With the correct settings, the charge amplifiers give a voltage output related to the acceleration or force. (For example, 1 V/m/s² or 1 V/N)
Generally, the resonant frequency of a system is obtained by exciting the system with a constant force and looking for the frequency at which the maximum acceleration amplitude occurs. The resonant frequency of the harvester is determined here practically by varying the excitation frequency and by measuring the power output while keeping the mechanical power input constant. The frequency corresponding to the highest electrical power output will be very close to the resonant frequency, if not equal. This will be done in §4.2.1. The conditions to have an optimal power output can also determined by the experimental setup. The mutual interaction between the mechanical input and the piezoelectric effect can also be studied. This is done in §4.2.3.

30
Figure 37: Overview of the setup
Figure 38: Close up of the clamped harvester on the shaker

31
Care was taken to obtain a good clamping, as this is of great influence on the resonant frequency of the beam and on the reliability of the different measurements. The harvester plate was clamped in between two metal beams. Two bolts and nuts were used to hold the plates firmly together, as shown on the left in Figure 39. The clamping beams are long enough, so that they can hold two small parallel harvesters. This allows us to experiment with different setups of two fans next to each other. The bimorph harvester, the clamping and the magnet that is attached to the impedance head have a total mass of 31,5 grams.
This clamping also allows us to test another design. One larger sheet of metal can be clamped as well, this is shown on the right in Figure 39. A flexible MFC patch is glued onto the plate. The clamping, plate and MFC patch have a mass of 33,4 grams. A bimorph setup could be easily obtained by attaching a second patch on the lower side of the metal plate.
Figure 39: Clamping of the harvester; left: bimorph; right: MFC patch on a aluminum plate
A picture of the clamped plate with the MFC patch is shown in Figure 40. The electrical wires that are soldered to the MFC patch can also be seen. The electrical wires were chosen as thin as possible, to minimize their effect on the vibration. A picture of the clamped bimorph generator was already shown in Figure 38.
Figure 40: Clamped aluminum plate with the MFC patch
Care needs to be take in order to avoid exceeding the maximum strain level of the piezogenerator. If this strain level is nevertheless exceeded, the device may suffer deterioration due to fatigue, or crack and fail immediately. In the datasheet of the bimorph piezobeam, the strain is related to a maximum deflection limit, given below.
MFC patch
Steel plate
Bimorph harvester

32
Table 5: Max safe tip displacement for the given bimorph generator
Type
Max. tip displacement (in)
Max. tip displacement (mm)
V21B
0,06
1,52
V21BL
0,18
4,57
The maximum tip displacement is less of a problem for the MFC patch, as it is glued onto a large metal plate, which allows for large tip deflections without risk of damaging the piezolayers. Also, the piezopatch itself is very flexible, further reducing the risk of damage.
4.2 Bimorph harvester
4.2.1 Resonant frequency
The resonant frequencies of the bimorph harvester (V21B) are shown in Figure 41 for different masses attached to the tip of the beam. A tip mass of 4 grams can be seen in Figure 38. These resonant frequencies can be further tuned by moving a given tip mass from the tip towards the base of the beam or vice versa. The maximum tip mass is limited to prevent damage to the piezoceramic layer, as the input forces become higher if the mass is increased and the acceleration is kept constant. At some point, the stress on the piezoceramic will become too high and it will crack. This will deteriorate the harvester quickly, or even make it permanently ineffective immediately.
Figure 41: Resonant frequency in function of the tip loading
The resonant frequencies that are shown in Figure 41 are obtained for a piezobeam that is connected to a resistive electrical load. The electrical load is chosen so that it corresponds with the maximum obtainable electrical power output for a given mechanical input power. This electrical load changes for different tip loads. Each tip mass corresponds with a different optimal electrical load that is connected to the harvester and that makes the power output optimal.
The estimated resonant frequency is plotted as well in Figure 41. It is calculated by using the general formula for a mass-spring system without damping, shown below.

33
휔=√ 푘 푚
The values of the two unknowns k and m are obtained by expressing that the following relation should be minimal:
Σ(푓푛 푖− 12휋 √ 푘 푚푐푎푙푐+푖훥푚 ) 2 4푖 =0
In this formula, fn are the measured resonant frequencies and Δm is the increase in tip mass for each step (1 gram). mcalc and k are obtained by using the lsqnonlin function in MatLab, given in Appendix C. The following values were found for the unknowns: mcalc = 0,000985 [kg] and k = 785,89 [kg/s²] It can be seen that the measurements are in accordance with the theoretical formula without damping.
For a constant tip load, the resonant frequency changes slightly with different electrical loads as can be seen in Figure 42. This is related to the electromechanical coupling coefficient, which relates converted electrical energy to the input mechanical energy, as explained in §3.2.4. The shift in resonant frequency is not very large in this case, as the piezolayer is only a part of the beam structure, which consists for a large part out of FR4, polyimide and some copper and epoxy. The contribution of the piezolayer to the overall structure stiffness is not therefore very large. The shift would be much larger for a purely piezoceramic beam.
In Figure 42, discretization errors can be seen. These errors result from the method that is used to obtain the resonant frequency. The power output was measured, and it was assumed (as was explained in §4.1) that the resonant frequency is the frequency at which the power output is the highest for a given power input. However, by changing the frequency, the mechanical power input changes. So the input power needs to be changed by manipulating the amplifier for each frequency. This rather complex process makes the accuracy of the frequency measurement limited. Accurate measurements of the frequency could be done by performing laser vibrometer measurements.
Figure 42: Resonant frequency in function of electrical load for a beam without tip mass, connected in series
139,2
139,3
139,4
139,5
139,6
139,7
139,8
139,9
140
140,1
1000
10000
100000
1000000
10000000
Resonant frequency [Hz]
Electrical load [Ω]

34
The harvester has a small frequency range in which it resonates. It is clear that, in order to have a maximum power output, this resonant frequency should be matched to the (dominant) frequency at which the beam is excited. This can be very difficult in practical applications. Self adapting structures, that alter their resonant frequency to the excitation frequency could be designed. This is briefly discussed in §4.5.1.
The higher resonant modes of the beam are also of interest. Higher frequency components in an actual (environmental) excitation can induce strain cancellation. This would result in a power output reduction. The electrode pattern should be chosen with care if it is expected that multiple resonance modes will be excited.
4.2.2 Damping ratio
The damping ratio describes how an oscillation in a system damps away. It is a very important factor in maximizing the harvester efficiency, as will be seen in §4.2.3.1. The damping ratio of the harvester will be derived by using the time domain response to an impulse. The response is shown in Figure 43 with a timescale of 20 ms/div. The measurements were done several times to increase accuracy.
Figure 43: Response of the bimorph harvester to an impulse
The logarithmic decrement was measured several times, and an average value was found. 훿=푙푛( 푥푛 푥푛+1)
Another way to obtain the logarithmic decrement is to measure the decrement over multiple peaks. The same value for δ was found for calculation with several values of n. 훿= 1 푛 푙푛( 푥1 푥푛+1)=0,16
The damping is obtained from the logarithmic decrement: 휁= 1√1+( 2휋 훿 ) 2=0,025

35
4.2.3 Power output
The power output of the harvester for a given mechanical power input will be measured. This power output will be measured for different excitation amplitudes and frequencies, tip masses, positions of the tip masses and electrical loads. All these parameters are interrelated, as will be explained below.
The natural frequency of the cantilever decreases as the tip mass increases, because the stiffness stays the same. The open circuit voltage then increases [60]. The energy harvester and the electrical load attached to it need to be as such that the highest possible efficiency in converting the input mechanical energy is obtained. With a resistive electrical load, the voltage increases with increasing load, while the current decreases [15]. Thus, some electrical load will correspond with maximum power output. This electrical load will be predicted by using the equivalent electrical scheme of a piezobeam.
4.2.3.1 General model of a vibration harvester
In [64], the optimal power output for a general, simple vibration harvesting system is derived. The model used is shown in Figure 44. K is the spring stiffness, CM the mechanical damping and CE the electrical damping. The harvested electric energy is seen as damping by this system.
Figure 44: 1D model of a vibration harvester
The optimal obtainable power output is given by the equation shown below. Y is the amplitude of the base displacement (y(t)=Yeiωt). The amplitude of the base acceleration is 푌̈, which is related to the displacement amplitude by 푌̈=휔2푌. The mass is indicated by m and ζm is the mechanical damping ratio. The damping ratio ζ is related to the damping coefficient by C = 2mωNζ. It was also shown in [64] that the optimal operating frequency is close to the natural frequency ωN if the total damping ratio is small, which was already shown in §4.2.1 and measured in §4.2.2. This can be expressed alternatively by stating that the optimal frequency ratio 훺= 휔 휔푁 =1. It was also shown that the optimal electrical damping ratio ζe,opt equals the mechanical damping ratio ζm if Ω = 1 [8, 64]. |푃표푢푡|표푝푡= 푚푌̈216휁푚휔푁 = √푚3푌̈216휁푚√푘 푢푠푖푛푔 휔푁=√ 푘 푚

36
Although the model is only valid for the situation in Figure 44, some general conclusions can be drawn from it. The optimal driving vibration frequency is equal to the natural frequency if the damping is small. In this case, the natural frequency approximately equals the resonant frequency. The power output will increase with rising harvester mass. So one should try to maximize the mass, while keeping the other parameters (resonant frequency, maximal strain) equal. The power output is proportional to the square of acceleration. The mechanical damping should be minimized. An important factor that is included in the mechanical damping of this cantilevered beam setup is the damping caused by the air surrounding the plate. So the size of the plate will also influence the amount of damping.
The power output is inversely proportional to the resonant frequency. Lower peaks in the vibration spectrum are thus preferred when designing the piezo system, if the acceleration amplitude is at least as large as the peaks on higher frequencies. Most ambient vibrations are low frequent (between 20 en 200 Hz).
4.2.3.2 Optimal power output of a bimorph harvester
In [17], a more specific and accurate model is derived from the constitutive relations to determine the power output of a bimorph harvester that is connected to a resistive load. Both the mechanical and electrical parts of the system were therefore modeled as electric circuit elements. For example, the electromechanical coupling is modeled as a transformer.
An equation that allows us to optimize the electrical power output was obtained. In the formula below, Ain is the Laplace transform of the vibration acceleration and tc the thickness of one piezoceramic layer. The parameter a equals 1 if the layers are connected in series and 2 if connected in parallel. Cb is the capacitance of the bender and Cp the elastic constant for the piezoelectric material. It was assumed that the driving vibration frequency was equal to the natural frequency of the system. 푃= 12휔2 푅퐶푏 2( 2퐶푝푑31푡푐 푘2푎휀 ) 2 퐴푖푛 2(4휁2+푘314)(푅퐶푏휔)2+4휁푘312(푅퐶푏휔)+4휁2
The factor k2 is given by: 푘2= 푙푏 23푏 (2푙푏+ 32 푙푚) (2푙푏+푙푚−푙푒)
lb is the length of the beam from the clamping to the tip mass. lm is the length of the tip mass and le is the length of the electrodes on the piezolayers. w is the width of the piezobeam.

37
Figure 45: Geometric parameters used in the model
By setting the derivative of the formula above with respect to R equal to zero, the optimal load
was obtained as well. It can be seen that the optimal electric load is inversely proportional to the
vibration frequency.
푅표푝푡 =
1
휔퐶푏
2휁
√4휁2 + 푘31
4
The maximal field strength, the surface charge, the maximal mechanical strain and stress are also
factors limiting the maximal (lossless) attainable power. Following calculations made in [65], the
mechanical stress is the factor limiting the (lossless) maximum power. The material PZT-5H would
be able to handle 330 W/cm³ at 100 kHz. Furthermore, the electrical control- and conversion
circuit that will be present in practical applications has a large influence tot the final obtained
output power.
4.2.3.3 Series and parallel connection of piezolayers
When using a bimorph harvester, both piezolayers can be connected in series or in parallel. With
the Midé harvester, the upper and lower side of both piezolayers are separately connected. This
results in four pins that can be connected with a universal connector, as in Figure 46. So, two
connectors were soldered to allow a quickly transition from series- to parallel connection.
Figure 46: Midé Series and parallel connection [61]
It is clear that the series connection of two piezolayers has double the open circuit voltage output
and the same current of a single layer wafer. The parallel connection will have the same voltage
of a single layer and should theoretically have double the current output of a single wafer. The
capacitance of a series connection is half that of a single wafer, while the capacitance of a parallel
connection is twice that of a single wafer.
A potentiometer with a maximum resistance of 20 kΩ is wired as a variable resistor. The optimal
power output is however obtained for resistances higher than 20 kΩ. So, discrete measurements
with fixed value resistors with a higher resistance are done. The voltage and current output for
different resistor values can be seen in Figure 47 and Figure 48. These results were obtained for a
beam without tip mass and for a constant acceleration rms of 2,5 g for both the series and
parallel connection.

38
Figure 47: Output voltage in function of the electrical load for a beam without tip mass and for an acceleration rms of 2,5 g; vertical lines: optimal power output for respectively series and parallel connection
Figure 48: Output current in function of the electrical load for a beam without tip mass and for an acceleration rms of 2,5 g; vertical lines: optimal power output for respectively series and parallel connection
As was already expected, the voltage rises with increasing electrical load, and current decreases with increasing load. It is clear that for some value of the resistance, power output will be optimal. To determine the load for which the electrical power output is optimal, the power is plotted in function of the electrical load.
The optimal electrical load to obtain a maximum power output differs for series and parallel connection. The power transfer to the resistor is theoretically optimal if the electrical load resistance equals the equivalent resistance of the piezo [60]. In this case, the voltage over the resistance will be half of the open circuit voltage of the piezo.
0,00
5,00
10,00
15,00
20,00
25,00
30,00
35,00
40,00
1000
10000
100000
1000000
10000000
Output voltage [V]
Resistive electrical load [Ω]
Parallel
Series
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1000
10000
100000
1000000
10000000
Stroom [mA]
Weerstand [Ω]
Parallel
Series

39
As can be seen in Figure 49, the optimal resistance for series connection (46,2 kΩ) is higher than for the parallel connection (12,5 kΩ). Indeed, if two resistors with half the optimal series connection resistance are put in parallel, the optimal load resistance for parallel connection is approximately obtained: ( 123,1 푘훺 + 123,1 푘훺 )−1=11,6 푘훺.
It can be seen very well in Figure 47 and Figure 48 that the optimal electrical voltage of the series connection is double that of the parallel connection. It can also be seen that the optimal electrical current of the series connection is half that of the parallel connection.
Figure 49: Electrical power output in function of load resistance; beam without tip mass; acceleration rms of 2,5 g
The maximum power output that is obtained is seen to be approximately the same for series and parallel connection of the piezo layers. The measured power output points are in good agreement with the values that are calculated using the model in §4.2.3.2.
The choice between parallel or series connection depends mainly on the application. For a series connection, the higher voltage and lower current at the optimal load resistance can be a disadvantage for energy harvesting. On a side note, he input impedance of the oscilloscope is 1 MΩ. This will somewhat affect the measurements. The effect becomes more important if the electrical load that is put in parallel with the oscilloscope rises. As the optimal load is much smaller, the input impedance is neglected.
4.2.3.4 Different tip masses
4.2.3.4.1 Change in optimal electrical load with tip mass
It was already explained that both the optimal electrical load and the corresponding voltage change when the tip mass is altered. This is validated as follows: For a given tip mass, the electrical load is changed until a maximum power output is obtained for a given acceleration input of 1 g rms. This is repeated with other tip masses, and the optimal electrical loads are derived in each case.
0
1
2
3
4
5
6
7
8
9
0
20000
40000
60000
80000
100000
Power output [mW]
Resistance [Ω]
Parallel theoretic
Series theoretic
Parallel measured
Series measured

40
The optimal electrical load is seen to increase with tip mass, which corresponds with a higher voltage over the resistive load. The increase in input and output power is reasonable, as by raising the tip mass, the input power increases.
Figure 50: Optimal electrical load (left scale) and corresponding voltage (right scale) in function of the tip mass for a constant acceleration rms of 1 g
4.2.3.4.2 Influence of the tip mass on the power output
The electrical power input rises with increasing tip mass. This is limited structurally however. The input acceleration was kept constant with an rms of 1 g. As the acceleration is kept constant, the input force and power will rise with increasing tip mass.
The force that is exercised on the harvester is now chosen much smaller than the force applied in the previous experiment with no tip mass, §4.2.3.3. This is done in order to prevent damaging the piezoceramic layer, as the load on the piezo layer will be much higher when using the tip loads. The power input ranges from 3,8 mW without tip mass to 14,2 mW with a tip mass of 4 grams. It is seen that both the output and input power increases linearly with the increasing tip load over the given range of tip loads. The input power rises faster than the output power, which means the energy conversion efficiency becomes lower. Note that the resonant frequency is different for every tip load.
Figure 51: Output electrical power in function of the tip load for a constant acceleration rms of 1 g
4
6
8
10
12
14
0
5000
10000
15000
20000
25000
30000
0
1
2
3
4
Voltage [V]
Electrical load [Ω]
Tip mass [gram]
Electrical load
Voltage
0
2
4
6
8
10
12
14
16
0
1
2
3
4
Power [mW]
Tip mass [gram]
Output power
Input power

41
4.2.3.4.3 Overview of the different values for different tip masses
An overview of measurements for different tip masses is given in Table 6. For a constant acceleration rms of 1 g at resonance, the optimal load resistance is seen to increase with increasing tip mass. As a result, the optimal output voltage increases. As expected, the input force increases and the resonant frequency decreases with increasing tip mass. The output power increases with increasing tip mass and input force as well, which is desirable.
Table 6: Measured values at resonant frequency for different tip masses with an acceleration rms of 1 g
Tip mass [grams]
Input power [mW ]
Optimal resistance [kΩ]
Voltage rms [V]
Output power rms [mW]
Resonant frequency [Hz]
Power out / Power in
0
3,84
12,5
4,21
1,42
142,4
0,37
1
5,90
13,75
5,41
2,13
99,3
0,36
2
8,57
15
6,19
2,55
80,9
0,30
3
11,07
24,85
9,30
3,48
70,5
0,31
4
14,17
24,85
10,18
4,17
65
0,29

Design and validation of piezoelectric energy harvesting systems

Design and validation of piezoelectric energy harvesting systems

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