piezoelectric energy harvesting and dissipation on structural...

Piezoelectric Energy Harvesting and Dissipationon Structural Damping

J. R. LIANG AND W. H. LIAO*

Smart Materials and Structures Laboratory, Department of Mechanical and Automation EngineeringThe Chinese University of Hong Kong, Shatin, N. T., Hong Kong, China

ABSTRACT: This article aims to provide a comparative study on the functionsof piezoelectric energy harvesting, dissipation, and their effects on the structural dampingof vibrating structures. Energy flow in piezoelectric devices is discussed. Detailed modeling ofpiezoelectric materials and devices are provided to serve as a common base for both analysesof energy harvesting and dissipation. Based on these foundations, two applicationsof standard energy harvesting (SEH) and resistive shunt damping (RSD) are investigatedand compared. Furthermore, in the application of synchronized switch harvesting on inductor(SSHI), it is shown that the two functions of energy harvesting and dissipation are coexistent.Both of them bring out structural damping. Further analyses and optimization for the SSHItechnique are performed.

Key Words: energy harvesting, energy dissipation, vibration damping, piezoelectric material,SSHI.

INTRODUCTION

WIRELESS sensor networks (WSNs) can be dep-loyed to monitor the health of military and civil

infrastructures (Liao et al., 2001). Different from thetraditional sensor systems, these widely distributed net-works put more emphasis on the energy issue because oftheir nature of wireless connections (Raghunathan et al.,2002). Yet, nowadays, most of the devices used in WSNsare still battery-driven. The use of batteries in thesedevices restricts both their lifespan and installation inplaces where batteries are hard to replace. This situationwould be completely changed if such devices couldscavenge energy themselves from their ambient environ-ment. Starting with this motivation, over the past fewyears the techniques of energy harvesting have been putunder the spotlight (Paradiso and Starner, 2005; Beebyet al., 2006; Anton and Sodano, 2007).One of the promising sources from which people could

collect energy is mechanical vibration. Three transduc-tion mechanisms (i.e., piezoelectric, electromagnetic, andelectrostatic) were studied in order to reclaim the ambientvibration energy and turn it into useful electrical power.Among generators based on these three mechanisms, thepiezoelectric ones are the simplest to fabricate (Beebyet al., 2006); therefore, they are particularly suitable forimplementation in microsystems.

Up to now,most of the research on piezoelectric energyharvesting has been mainly concerned with the absoluteamount of energy that can be harvested from vibratingstructures (Ottman et al., 2002; Badel et al., 2005; Antonand Sodano, 2007). The effect, which results from energyharvesting and reactions to the vibrating structure, wasseldom discussed in these studies. Lesieutre et al. (2004)discussed such an issue and claimed that the harvesting ofelectrical energy from the piezoelectric system brings outstructural damping. On the other hand, it has beenknown for a long time that the effect of structuraldamping can be caused by energy dissipation. In most ofthe shunt damping treatments, energy dissipation wasregarded as the only function that contributes tostructural damping (Moheimani, 2003). The two applica-tions of standard energy harvesting (SEH) and resistiveshunt damping (RSD), even though their main functionsare energy harvesting and energy dissipation, respec-tively, can be compared in terms of damping capabilities(Lesieutre et al., 2004).

Referring to the comparison between the two applica-tions, we note that it is possible that the two functions cancoexist in a certain condition and both effect structuraldamping. This phenomenon happens in the applicationof synchronized switch harvesting on inductor (SSHI).In this article, the relationship among the functions ofenergy harvesting, dissipation, and their effects onstructural damping will be investigated. This under-standing is crucial towards developing an adaptive energyharvesting technique. In the section ‘Energy Flow in

*Author to whom correspondence should be addressed.E-mail: [email protected] 9, 10 and 15 appear in color online: http://jim.sagepub.com

JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, Vol. 20—March 2009 515

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Piezoelectric Devices’, we give an explanation of the twofunctions of energy harvesting and dissipation in terms ofenergy flow, in order to clarify some terms for thefollowing discussions. In the section ‘Piezoelectric DeviceModeling, in terms of Impedance’, we propose animpedance-based piezoelectric device model, which canserve as a common base for energy harvesting anddissipation analyses. In the section ‘Energy Harvestingand Dissipation’, the key concepts and results of SEHand RSD optimizations are reviewed and discussed.Then a comparison between these two applications interms of damping capabilities is made, concerningcoupling coefficients. In the section ‘Energy Harvestingand Dissipation of SSHI’, after proposing a model thatgeneralizes SEH and SSHI, we discuss the effect ondamping contributed by both energy harvesting anddissipation, and further analyze the energy harvestingand damping capabilities for SSHI. Finally, a conclusionis given in the section ‘Conclusion’.

ENERGY FLOW IN PIEZOELECTRIC DEVICES

In most of the literature, damping is the dissipationof the energy of a mechanical system (Harris, 1996;de Silva, 2005), and dissipation usually means the lostenergy is converted into heat (Wikipedia). However,de Silva (2005) also pointed out that damping is theprocess that converts and dissipates mechanical energyinto other forms of energy. Considering piezoelectricenergy harvesting, which also results in structuraldamping, the previous definitions need to be clarified.In this section, energy flow in piezoelectric devices isstudied; afterwards, terms are specified in order toinvestigate various effects in the following sections.

An Overview of Energy Involved

First, the energy involved in piezoelectric devices willbe clarified. Figure 1 shows the single degree-of-freedom (SDOF) schematic representation of piezo-electric devices. Regardless of the purposes of structuraldamping or energy harvesting, their mechanical parts in

the structures are similar. The main differences liein their shunt circuits. Figure 2 provides an overviewof the three forms of energy involved in these devices.These three forms are: mechanical, electrical, andthermal. The first two are linked by the bi-directionalpiezoelectric transducer. At the same time, eithermechanical or electrical energy can be converted intothermal energy by dissipation elements such as mechan-ical dampers or electrical resistors. Once the energy isdissipated, i.e., transformed into heat, it will not berecovered in the devices, therefore dissipative transfor-mation is uni-directional.

From the energy flow shown in Figure 2, we can sortout three paths. With path A, mechanical energy isdirectly dissipated, thus it represents the function ofmechanical energy dissipation. Path B and path C bothpass through the ‘piezoelectric bridge’, converting aportion of the mechanical energy into electrical energy.They differ in that some of the electrical energy isconverted into heat via path B, while some of it isconverted via path C into energy storage. Therefore,path B is related to energy dissipation in electrical way,or briefly electrical energy dissipation; path C is relatedto electrical energy harvesting. Mechanical dissipationseems unrelated to piezoelectric transducers, but in factthey have some relation. Since piezoelectric transducersare bi-directional, in some cases, such as active con-strained layer damping treatment (Stanway et al., 2003)and active-passive hybrid piezoelectric networks (Tangand Wang, 2001), piezoelectric elements are used tosuppress the mechanical vibration so as to enhancemechanical energy dissipation.

Term Specification

The three paths in Figure 2 represent three functionsthat can remove energy from the vibrating structure with

Mechanicalenergy

Thermal energy

Electricalenergy

Energystorage

Mechanicaldissipation

element

Piezoelectrictransducer

Harvestingcircuit

Ambientvibration

1 Electricaldissipation

element

2

3

Figure 2. Energy flow chart of piezoelectric devices.

Mass

Damper SpringPiezoelectricelement

F

x

Shuntcircuit

Figure 1. Schematic representation of piezoelectric devices.

516 J. R. LIANG AND W. H. LIAO

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the piezoelectric transducer. One or more of thesefunctions can take place in certain applications andcause the effect of structural damping. Table 1 gives theterm specification for four functions that can removemechanical energy from vibrating structures. To have acomplete classification, this table includes the item D,mechanical energy harvesting. For example, in automaticwatches, with an elaborately designed mechanism,mechanical energy can be stored in the mainsprings todrive the watches. This technique has been successfullyapplied for more than 80 years.With the above specification, when considering

the damping applications, e.g., RSD, we emphasize thetotal effect of the involved functions on the structure.On the other hand, when considering the applications ofenergy harvesting, e.g., SEH or SSHI, we focus on theutilization of the electrical energy harvesting function.However, this does not mean that there is no otherfunction in the system. On the contrary, in the appli-cations of energy harvesting, parasitic mechanical andelectrical energy dissipations usually exist. In general,these were not considered in the previous research. Butsince they partake and dissipate some of the energy thatcould be harvested, these functions would becomeimportant for the purpose of harvesting energy fromvibration sources where exploitable energy is limited.The three factors within the parentheses in Table 1

are indices to evaluate the corresponding functions oreffect. For traditional damping, the term loss factor wasusually used to evaluate the total damping capability.It was defined as the ratio between the energydissipated per cycle and the energy associated withvibration (Warkentin and Hagood, 1997). Here, inorder to continue using this term for dampingevaluation, wemake a subtle change in this old definition.The new defined loss factor is the ratio between theenergy removed per cycle and the energy associated withvibration. Moreover, considering energy harvesting,it is not enough to show the detailed energy relationswith the use of loss factor only. Therefore, two additionalfactors are defined with respect to energy harvestingand energy dissipation, as follows.

For energy harvesting, the new term harvesting factoris defined to evaluate the harvesting capability as:

�h¼

Eh

2�Emaxð1Þ

where Eh denotes the harvested energy in one cycle,Emax is the energy associated with vibration.

For energy dissipation, the term dissipation factor isused to evaluate the dissipation capability as:

�d¼

Ed

2�Emaxð2Þ

where Ed is the dissipated energy in one cycle.With Equations (1) and (2), we can define the loss

factor as:

��¼

�E

2�Emax¼ �

hþ �

dð3Þ

where �E is the summation of Eh and Ed, whichrepresents the total removed energy from the vibratingstructure in one cycle.

PIEZOELECTRIC DEVICE MODELING, IN

TERMS OF IMPEDANCE

To compare the damping effect between energyharvesting and energy dissipation, we should first havea common representation of piezoelectric elements.As given in IEEE Standard on Piezoelectricity (1987),the linear piezoelectricity can be represented by four setsof constitutive equations, as follows:

Tp

Di

" #¼

cEpq �ekp

eiq "Sik

" #Sq

Ek

" #ð4Þ

Sp

Di

" #¼

sEpq dkp

diq "Tik

" #Tq

Ek

" #ð5Þ

Sp

Ei

" #¼

sDpq gkp

�giq �Tik

" #Tq

Dk

" #ð6Þ

Tp

Ei

" #¼

cDpq �hkp

�hiq �Sik

" #Sq

Dk

" #ð7Þ

where T, S, D, and E denote the stress, strain, electricdisplacement, and electric field, respectively; d, e, g, and hare piezoelectric constants; c is elastic stiffness constant;s is elastic compliance constant; " is permittivity constant;� is impermittivity constant; the subscripts are tensornotations; the superscripts T, S, D, and E denote thecorresponding parameters at constant stress, strain,

Table 1. Term specification.

Path Function Effect

A Mechanical energydissipation

Energydissipation

B Electrical energydissipation

(dissipationfactor, �

d)

Vibration damping(loss factor, �

�Þ

C Electrical energyharvesting

Energyharvesting

D Mechanical energyharvesting

(harvestingfactor, �

h)

Piezoelectric Energy Harvesting and Dissipation on Structural Damping 517



electric displacement, and electric field, respectively.Either one of these four can be used to describethe same coupling characteristics of piezoelectric materi-als. In the studies of traditional damping, Equation (5)was usually used (Hagood and von Flotow, 1991;Clark, 2000; Moheimani, 2003); while in the studies ofenergy harvesting, Equation (4) was more popular (Badelet al., 2005; Shu et al., 2007); but still, there wereexceptions (Lesieutre and Davis, 1997; Ng and Liao,2005; Roundy, 2005).This section begins with the dynamic representations

of these four sets of constitutive equations, in terms ofimpedance. Based on these, we select one as the commonbasis of our analysis and then obtain the device modeland equivalent circuit.

Dynamic Representations

Suppose a piezoelectric element, whose dimensionsare shown in Figure 3, is bonded on a vibrating beam,and is working under 3–1 mode. Assuming the motionwavelength is much larger than l; and l, w are muchlarger than the thickness t, four dimensional relationscan be obtained:

F ¼ twT1, x ¼ l S1, U ¼ �tE3, Q ¼ wlD3 ð8Þ

where F, x denote the force and displacement in the ‘1’direction; U, Q denote the voltage across and chargestored in the ‘3’ direction. Substituting Equation (8) intoEquation (4) yields the macroscopic piezoelectricequations:

F

Q

" #¼

tw

lcE11 we31

we31 �wl

t"S33

2664

3775 x

U

" #: ð9Þ

Equation (9) does not explicitly show the dynamicbehavior of the piezoelectric patch. To study thedynamic behavior, two derivative relations betweenelectrical current and charge, mechanical velocity, anddisplacement are needed, i.e.:

I ¼ �Qs, v ¼ xs ð10Þ

where I denotes current, v denotes velocity, and s is theLaplace operator. Substituting Equation (10) intoEquation (9) yields:

F

I

" #¼

KE s= �e

��e sCS

" #v

U

" #ð11Þ

where:

KE ¼tw

lcE11, CS ¼

wl

t"S33, �e ¼ we31 ð12Þ

are the short circuit stiffness, clamped capacitance, andforce–voltage coupling factor of the piezoelectric patch,respectively.

For illustration, Figure 4(a) shows the schematicdiagram corresponding to Equation (11). Note thatKE s= represents the mechanical impedance of theshort circuit stiffness in ‘1’ direction, and 1=ðsCSÞ

represents the electrical impedance of the clampedcapacitance in ‘3’ direction. This model regardsthe mechanical part as the patch’s stiffness in parallelwith a force source, and the electrical part asthe capacitance in parallel with a current source.We call this P–P model in brief, where ‘P’ stands forparallel relation. Similarly, for the other three constitu-tive equations, i.e., Equations (5)–(7), we can derive theP–S, S–S, S–P models corresponding to Figure 4(b)–(d),where ‘S’ stands for series relation, and �g isvelocity–current coupling factor.

Since those four dynamic models illustrated inFigure 4 are compatible with the analyses of mecha-nical impedance networks and electrical ones, theycan provide us with a guideline for selection ofconstitutive equations in the analyses of piezoelectricdevices. For instance, from the electrical point of view,it would be more convenient to use P–P or S–P modelsto analyze devices whose shunt circuit network isbuilt up by parallel connecting impedances; P–S orS–S models are preferred for shunt networks built upin series.

Device Model

Based on Equation (5), Hagood and von Flotow(1991) also drew a macroscopic representation for theelectrical part of the piezoelectric materials, andincluded shunt impedance in the governing equations.This process altered the equations to ‘half device level’,thus making it possible to consider the piezo-electric patch and its shunt circuit as a whole. Also,consider the dynamic definition of device couplingcoefficient given by:

k2d ¼!D� �2

� !E� �2

!Dð Þ2

ð13Þ

where !D is the open circuit natural frequency, !E isthe short circuit natural frequency. Compared to the

l

t

w

Direction of F, x

3

1

2

U

Figure 3. Schematic diagram of piezoelectric patch.




electromechanical coupling coefficient of the piezo-electric element (Badel et al., 2005):

k2e ¼�2e

KECS þ �2eð14Þ

this dynamic definition is again in ‘half device level’, sincethe measurements of !D and !E regard the mechanicalpart of the device as a whole, but exclude the shunt circuitfrom the device.Referring to these two analyses at either electrical or

mechanical device level, and based on our represe-ntations for piezoelectric materials, we form an inte-grative device model by extending the P–P modelof piezoelectric materials. The mounting piezoelectricpatch on the beam structure can be modeled withmechanical impedances in parallel (Badel et al., 2005),and electrical shunt circuit appears in parallel tothe inherent piezoelectric capacitance (Hagood andvon Flotow, 1991). As illustrated in Figure 5, thedevice model is given as:

F

I

" #¼

PZE

mech �e

��eP

YSelec

" #v

U

" #ð15Þ

where:

XZE

mech ¼KE

sþ Zmech ð16Þ

XYS

elec ¼ sCS þ Yelec ð17Þ

and Zmech is the total external mechanical impedance;Yelec ¼ Z�1elec, is the total external electrical admittance.

In particular, if the electrical part is purely composedof impedances, i.e., the current I in Equation (15) andFigure 5 equals to zero, simplifying Equation (15) withthis condition yields the expression of the equivalentmechanical impedance:

Zmech�eq ¼ F=v ¼X

ZEmech þ �

2e ZS

�elec ð18Þ

where ZS�elec ¼

PYS

elec

� ��1is the total electrical

impedance.

Equivalent Circuit

With the equivalent mechanical impedance given inEquation (18), the piezoelectric device can be regarded

Piezoelectric patch

+

_

U

IF v

F~1

sCSKE

sαeU αev

(a)

Piezoelectric patch

+

_

U

IF v

+

F~

1

s

(b)

KE

Piezoelectric patch

F v

v~

+

_

U

I

+

1

s

(c)

KD

sCT

Piezoelectric patch

+

_

U

IF v

v~

1s

(d)

KD

sCSαgI αgF

αev

αgI

αeU

αgF

sCT

Figure 4. Schematic diagrams corresponding to four sets of constitutive equations. (a) P–P model. (b) P–S model. (c) S–S model. (d) S–P model.

Piezoelectric patch

+

_

U

I

F

v

+F~ ZelecZmech

SelecY∑E

mechZ∑

1

sCSKE

sαeU αev

Figure 5. Piezoelectric device model.




as a pure mechanical device. On the other hand, in orderto study the electrical behavior of the device, we can alsomake the device equivalent to a pure electrical circuit.To keep voltages across and currents through allelectrical elements unchanged, the equivalent circuitcan be characterized in form of the Ohm’s Law as:

ueqieq¼

PZE

mech

�2eþ ZS

Selec ð19Þ

where ueq and ieq are equivalent voltage and current asso-ciated with mechanical force and velocity. Their relation-ships were given by Warkentin and Hagood (1997) as:

ueq =F=�eieq =v�e

�: ð20Þ

In most of the previous analyses and also our studypresented in this article, the external mechanicalstructure is regarded as a part of the vibration source,which excites the piezoelectric patch. Therefore, themechanical part only includes the patch’s short circuitstiffness. With this, Equation (19) can be specified as:

ueqieq¼

1

sCeqþ

1

sCS þ Yelecð21Þ

where Ceq is the equivalent capacitance corresponding tothe short circuit stiffness of the piezoelectric patch, withthe following relation:

Ceq ¼�2eKE

: ð22Þ

According to Equation (21), the equivalent circuitdiagram is shown in Figure 6(a). In the figure, Seq isthe equivalent voltage source representing the sinusoidalforce excitation applied to the patch structure. Thisequivalent circuit can serve as a common base fordamping analyses with both energy harvesting andenergy dissipation. It depends on whether a shuntcircuit designed for energy harvesting or energy dissipa-tion is connected.Moreover, the circuit can be further simplified with an

additional approximation that the coupling coefficientk2e ! 0. From Equation (14), such approximationresults in Ceq � CS. Therefore, 1=ðsCeqÞ, as electricalimpedance, is dominant. In this case, Seq and Ceq

together form a current source S0eq. The simplifiedequivalent circuit is shown in Figure 6(b).In traditional passive damping, the simplified

equivalent circuit shown in Figure 6(b) was seldomused, since the approximation k2e ! 0 contradicts thepurpose of extracting as much mechanical energyas possible. Hagood and von Flotow (1991) proposedan inspired analysis for shunt damping optimization,

which can also be derived from the equivalent circuitin Figure 6(a). Later work in this area mainly focusedon multiple mode vibration damping methods(Moheimani, 2003).

In the research of energy harvesting, up to now, mostof the analyses were based on the simplified equivalentcircuit as shown in Figure 6(b) (Ottman et al., 2002;Lesieutre et al., 2004; Guyomar et al., 2005; Badel et al.,2005; Makihara et al., 2006; Anton and Sodano, 2007).As a result, the counteraction of the shunt circuit to themechanical structure was usually neglected. In fact,more universal analysis based on SEH circuit and thepiezoelectric equivalent circuit as shown in Figure 6(a)was once proposed with the title ‘nonlinear shuntdamping’ (Warkentin and Hagood, 1997), before therecent research on energy harvesting.

The concept of loss factor is important for evaluatingdamping performance. But in energy harvesting, sincemost of the studies were based on the simplified equi-valent circuit, only the absolute harvested energy can beknow. Up to now, different piezoelectric generators wereusually compared with their absolute harvesting power(Beeby et al., 2006). In order to provide a betterunderstanding, we suggest that the relative energyharvesting capability should also be considered. Basedon the equivalent circuit shown in Figure 6(a), in thefollowing analysis we discuss the relative energy harvest-ing performance with the new term ‘harvesting factor’,which was defined in the previous section.

ENERGY HARVESTING AND DISSIPATION

Besides the majority of literature studying absoluteenergy or power that can be harvested from an idealcurrent source, Lesieutre et al. (2004) have discussed thestructural damping effect due to energy harvesting in the

Ceq

CS

CS

Seq

S ′eq

Shuntcircuit

Shuntcircuit

Piezoelectricequivalence


ieq

ieq

ueq

+

–

(a)

(b)

Figure 6. Equivalent circuit diagrams of a piezoelectric device.(a) Equivalent circuit. (b) Simplified equivalent circuit.




application of SEH. In terms of damping capability,which is evaluated by the loss factor, this effect can becompared to RSD. The comparison made by Lesieutre etal. (2004) was only valid under the condition thatk2e ! 0. In this section, a brief review of the optimiza-tions of SEH and RSD will be given; afterwards, generalcomparisons covering the whole range of k2e will be made.

Standard Energy Harvesting

In the SEH application, only the energy harvestingfunction contributes to the effect of structural damping.The shunt circuit is a nonlinear circuit, which iscomposed of a bridge rectifier, a filter capacitor, andother loads in parallel. Its equivalent circuit is shown inFigure 7. In analyzing this circuit, the DC filtercapacitor Crect is assumed to be large enough so thatthe voltage across this capacitor Vrect is nearly constant(Ottman et al., 2002). For the ‘nonlinear damping’circuit proposed by Warkentin and Hagood (1997), itdiffers from the circuit in Figure 7 in that the Crect isreplaced by a constant DC voltage supply. Yet, theiranalyses are compatible.The ratio between the harvested energy and energy

associated with vibration in one cycle, which is calledharvesting factor in this article, is a function of k2e and~Vrect (Warkentin and Hagood, 1997):

�h¼

4

�~Vrect

1� ~Vrect

1� k2ek2eþ ~Vrect

ð23Þ

where ~Vrect is obtained by non-dimensionalizing the recti-fied voltage to the amplitude of open circuit voltage, i.e.:

~Vrect ¼Vrect

Vocð24Þ

The coupling coefficient of the piezoelectric elementdepends on material and geometry properties, whichcannot be changed after the device is made. The maxi-mum harvesting factor can be obtained as follows:

�h, max¼

4

�

1�ffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2e

p� �2k2e

ð25Þ

at a non-dimensional rectified voltage:

~Vrect, opt ¼1� k2ek2e

ffiffiffiffiffiffiffiffiffiffiffiffiffi1

1� k2e

s� 1

!: ð26Þ

The optimum rectified voltage can be achieved byproperly choosing the load. This load can be an adaptive

DC–DC converter (Ottman et al., 2002) or optimalresistor (Guyomar et al., 2005). The extracted power istransferred or consumed so as to keep the optimum rectifiedvoltage constant. In addition, it can be proven that whenk2e ! 0, ~Vrect, opt ¼ 0:5, which is the optimal result givenby Ottman et al. (2002). In their study on the dampingeffect of SEH, Lesieutre et al. (2004) took ~Vrect, opt asconstant, i.e., equal to 0.5, regardless of k2e , therefore,their optimum non-dimensional rectified voltage andmaximum harvesting factor1 are only valid when k2e ! 0.

Resistive Shunt Damping

In the application of RSD (Hagood and von Flotow,1991), only the energy dissipation function contributesto the effect of structural damping. The equivalentcircuit is shown in Figure 8. It only connects a resistor asits shunt circuit to dissipate the extracted energy, thusresulting in damping. The dissipation factor is given by:

�d¼

� k2e1� k2e þ �

2ð27Þ

where � is the non-dimensional frequency:

� ¼ RCS! ð28Þ

and ! is the excitation angular frequency, R is theresistance of the shunt resistor. The maximum dissipa-tion factor can be obtained as follows:

�d, max¼

k2e

2ffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2e

p ð29Þ

at a non-dimensional frequency of:

�opt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2e

q: ð30Þ

1Loss factor was used in their study. In this paper, harvesting factor is used instead, while loss factor was defined in the section ‘Energy Flow in Piezoelectric Devices’.

Ceq

CS

Crect

SeqLoad


ieq

ueq

+

–

Figure 7. Equivalent circuit for SEH.

Ceq

CS R Seq


ieq

ueq

+

–

Figure 8. Equivalent circuit for RSD.




Comparison between SEH and RSD

To compare the characteristics between SEH andRSD, we employ the non-dimensional voltage–chargediagrams to illustrate their energy conversion cycles.Since the equivalent current in Figure 6(a) equals v�e,

the equivalent charge input is the integral of this current,which is x �e, where x is the displacement of themechanical source, i.e., the integral of the velocity v.Assuming X as the maximum displacement, the max-imum equivalent input charge should be X�e. Non-dimensionalizing x�e with X�e, we have:

~qeq ¼x�eX�e¼

x

X¼ ~x: ð31Þ

It is not only the non-dimensional equivalent inputcharge ~qeq, but also the non-dimensional displacement ~xinduced by the mechanical source. Similarly, we can alsonon-dimensionalize the equivalent input voltage F=�ewith respect to the maximum voltage across Ceq:

~ueq ¼F=�e

X�e=Ceq¼

F

KEX¼ ~F: ð32Þ

The non-dimensional equivalent input voltage ~ueq isalso the non-dimensional force ~F applied to the piezo-electric patch.Given the situation that k2e ¼ 0:3, for instance, the energy

conversion cycles for SEH with optimum ~Vrect and RSDwith optimum � are shown inFigures 9 and 10, respectively.The black solid curve in each diagram shows the relationbetween non-dimensional charge and non-dimensionalvoltage in one cycle. The areas of blue and green ellipsesrepresent 2� multiplying the maximum stored energy inthe devices: blue for electrical and green for mechanical.2

The areas enclosed by the ~q� ~u loci represent the energyremoved from the structures in one cycle. But in order todistinguish whether the extracted energy is harvested ordissipated, different patterns are used in the diagrams.The main differences between Figures 9 and 10 are the

shapes of the ~q� ~u loci and the patterns that fill the areasenclosed by the loci. But even though the flows of theextracted energy in these two applications are different,they can be compared in terms of damping capability,which is evaluated by the loss factor. Without energybeing dissipated, the harvesting factor in SEH is also theloss factor; similarly, without energy being harvested,the dissipation factor in RSD is also the loss factor.According to the relations given in Equations (25) and

(29), the two maximum loss factors, i.e., ð��, maxÞSEH and

ð��, maxÞRSD, in the two applications, as functions of k2e ,are compared in Figure 11. As k2e increases, theattainable maximum loss factors for both applications

increase; also, the ratio between the two factors of SEHand RSD decreases. Moreover, it should be noted that,when coupling coefficient of the piezoelectric elementapproaches zero, we can obtain:

limk2e! 0

ð��, maxÞSEH

ð��, maxÞRSD

¼2

�¼ 63:66% ð33Þ

which can also be observed from the dot curve inFigure 11. Lesieutre et al. (2004) found the same result

2Diagrams in color appear in online version. For grayscale printing, the electrical and mechanical energy is represented by darker and lighter gray patches, respectively.

Mechanical energy associated with vibration in one cycleElectrical energy associated with vibration in one cycle

Dissipated energy per cycleThe locus

q~

u~

q –~ u~

Figure 10. Energy cycle for RSD.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.840

50

60

70

0

0.5

1

1.5

0.9

(h∑,max)

SEH (h∑,max

)RSD

(h∑,max)

RSD

(h∑,max)

SEH

Rat

io o

f tw

o h ∑

,max

(%)

h ∑,m

ax

ke 2

Figure 11. Maximum loss factors in the two applications and theirratio.

Mechanical energy associated with vibration in one cycleElectrical energy associated with vibration in one cycle

Harvested energy per cycle The locus

q~

u~

q –~ u~

Figure 9. Energy cycle for SEH.




under this special condition. Indeed most of the previousanalyses on SEH took k2e ! 0 as their premise, explicitlyor implicitly. This premise constrains the endeavorto increase the material coupling in order to harvestmore energy. It simplifies the analysis; however,it confines the optimization to specific, rather thangeneral, conditions.

ENERGY HARVESTING AND DISSIPATION

OF SSHI

In the previous two applications, each has only onedominant function that contributes to the effectof structural damping. In the application of SSHI, thesituation is more complicated. Both of the two functions,energy harvesting and energy dissipation, coexist in thisapplication, and bring out damping effect. Previousresearch on SSHI was conducted for the only purpose ofharvesting energy; nevertheless, unlike the application ofSEH, considering its contribution to structural damping,the function of energy harvesting may not be dominantin all situations. Detailed study of the relationshipbetween energy harvesting, dissipation and their effectson SSHI can help us better understand the energy flowand conversion mechanism within the piezoelectricdevices.

The SSHI Technique

The technique of SEH provides a passive solution toharvest ambient vibration energy. It is simple andreliable; however, its harvesting capability is difficultto further enhance. As the electrical part of the device iscomposed of CS in parallel with the shunt circuit,Figure 12 shows the typical waveforms of thevoltage across it, the current (�ev) flowing into it,and its power input (product of voltage and current).In most of the cycle the power is positive, whichmeans that energy is converted from mechanicalinto electrical; but in some intervals it has negativevalue, which indicates that the energy returns fromelectrical part to mechanical part. We call this energyreturn phenomenon.In order to enhance the energy conversion efficiency,

Guyomar et al. (2005) proposed a solution called SSHI(Badel et al., 2005; Shu et al., 2007). The equivalentcircuit and typical voltage, current, and power wave-forms of this technique are shown in Figures 13 and 14,respectively. This technique was further specified as‘parallel-SSHI’ by Lefeuvre et al. (2006). By involvingthe shunt path composed of an active switch sw and asmall inductor L, with appropriate control of the switch,this circuit can overcome the energy return phenomenonso as to make sure the power always flows into theelectrical part.

The switch is off in most of the cycle; it takes action atthe time when the current equals to zero. Also at thisinstant, the charge stored in CS is at its extreme value.During the operation, the switch is first turned on tocreate a ‘shortcut’ for the charge stored in CS, and thenturned off to disconnect the shortcut again when thevoltage across CS alters to another extreme value. Theelectrical cycle, which is decided by the time constantLCS, is much shorter than the mechanical cycle. Theresponse time can be neglected, so the voltage waveformcan be regarded as changing from V1 to V2 steeply at theinstant when the current equals to zero. The zoom-inview of the action instant is also shown in Figure 14.

t

t

V1

V2

close

open

CurrentVoltage

Vrect

–Vrect

t

Pow

er

0

0

Power

Figure 14. Typical voltage, current and power waveforms in energyharvesting with SSHI technique.

Current

VoltageVrect

–Vrect

t

t

Pow

er

0

0

Power

Figure 12. Typical voltage, current and power waveforms in SEH.

Ceq

CS

CrectSeq

L

sw

rLoad


ieq

ueq

+

–

Figure 13. Equivalent circuit for energy harvesting with SSHItechnique.




General Representation for Shunt Energy Harvesting

Most references on SSHI gave the relation between V2

and V1 in terms of electrical quality factor Q. In order tomake a more general representation, we take:

V2

V1¼ �, � 15�40 or � ¼ 1: ð34Þ

When �¼ 1, it represents the SEH. When � is negative, itcan represent energy harvesting with SSHI techniqueunder any electrical Q value, since in this situation:

� ¼ �e��2Q: ð35Þ

For a practical inductor, there is always parasitic resi-stance, which is denoted as r in Figure 13. If no additionalenergy is provided to the CS and ‘shortcut’ loop, � cannever reach �1.

Coexistent Energy Harvesting and Dissipation in SSHI

Most of the research on harvesting with SSHItechnique focused on the optimization in order toharvest more energy from the mechanical source(Guyomar et al., 2005; Badel et al., 2005; Lefeuvreet al., 2006; Shu and Lien, 2006; Shu et al., 2007), whilethe electrical energy dissipation was considered in thesimulation by Badel et al. (2005). The energy dissipationcorresponds to the voltage change from V1 to V2 acrossCS. Till now, no further analytical result is given,especially of the relationship between energy harvestingand energy dissipation in this application.As mentioned above, the functions of energy harvest-

ing and energy dissipation coexist in this application. Theamount of energy harvested in one cycle is:

Eh ¼ 2CSVrect 2Voc � Vrect 1þ �ð Þ½ � ð36Þ

while the amount of energy dissipated in one cycle is:

Ed ¼ CSV2rect 1� �

2� �

: ð37Þ

Besides, when �40, no energy returns from the electricalpart to the source. The energy associated with vibrationonly includes the strain energy, which is:

Emax ¼aCSV2

oc

2ð38Þ

where a stands for the ratio of:

a ¼1� k2ek2e

: ð39Þ

The relations among Eh, Ed and 2�Emax can beillustrated in the non-dimensional voltage–charge diagram.

For example, given the situation where k2e ¼ 0:3,� ¼ � 0:2, i.e., Q � 1:0, the energy conversion cycle forharvesting energy with SSHI technique under optimumsituation is shown in Figure 15. The steep voltagechanges at maximum charge enable the ~q� ~u locus toenclose more area. Compared with the SEH and RSD,whose energy conversion cycles are shown in Figures 9and 10, energy harvesting with SSHI technique iscapable to extract more energy in one cycle. Within acycle, some of the extracted energy is converted intoheat, i.e., dissipated, while the rest is harvested and keptin some suitable electrical energy storage devices(Sodano et al., 2005a,b; Guan and Liao, 2007, 2008).

Substituting Equations (36)–(38) for Ed, Eh, and Emax

into Equations (1) and (2), the harvesting factor anddissipation factor of this device can be obtained as:

�h¼

4 ~Vrect � 2 ~V2rect 1þ �ð Þ

a�ð40Þ

�d¼

~V2rect 1� �

2� �a�

ð41Þ

where ~Vrect is given by Equation (24). According toEquation (3), the loss factor �

�is the sum of �

hand �

d:

��¼

4 ~Vrect � ~V2rect 1þ �ð Þ

2

a�: ð42Þ

In addition, since the harvesting factor �hgiven in

Equation (40) and the loss factor ��given in Equation

(42) should not be negative, there is a practical range of~Vrect, which is:

04 ~Vrect42

1þ �: ð43Þ

In this interval, when k2e and � are fixed, �dand

��

monotonously increase with ~Vrect, yet �h

is a

Mechanical energy associated with vibration in one cycle

Harvested energy per cycleDissipated energy per cycle

The locus

q~

u~

q –~ u~

Figure 15. Energy cycle for energy harvesting with SSHI technique.




non-monotonic function. The maximum value of �h

can be obtained:

�h,max¼

2

a� 1þ �ð Þð44Þ

at an optimum non-dimensional rectified voltage:

~Vrect, opt ¼1

1þ �: ð45Þ

It should be noted that the optimum rectified voltagesfor maximum harvesting factor are one half of itsapplicable range, as given in Equation (43).According to Equations (40)–(43), the harvesting

factor, dissipation factor and loss factor, as functions of~Vrect and k2e , are evaluated with four different values of �.The results of these three evaluating factors are shown inFigure 16. Figure 16(a) stands for the situation of SEH.3

Ideally, no energy is dissipated in this situation, thus thedissipation factor equals zero; the curves of harvestingfactor overlap with those of loss factor. As � decreases,under the same k2e and

~Vrect, all the three factors increase.In addition, it can also be seen from Figure 16 thatwhen ~Vrect reaches its upper limit, there is no energy

harvesting effect, i.e.,�h¼ 0, thus all structural damping

effect is due to energy dissipation. In this special case, thisdevice becomes a synchronized switch damping on inductor(SSDI) device (Guyomar and Badel, 2006). This dampingtreatment can achieve a large loss factor, which iscomparable to those of some high polymers, e.g., forhard rubber and polystyrene, whose loss factors are 1.0and 2.0, respectively (Cremer et al., 2005).

With the intention to optimize SSHI, considering theproportional relations between harvesting factor anddissipation factor shown in Figure 16, two guidelines aresuggested. First, for SSHI with the same k2e and �, thereare two ~Vrect corresponding to the same harvestingfactor �

h; yet the smaller one should be the preference.

Its corresponding dissipation factor is smaller, therefore,less energy is dissipated within one cycle with the sameharvesting performance. The saved energy can beharvested in the future cycles. Second, for SSHI under~Vrect, opt, even though �

h, maxincreases against �, the

smaller � is not necessary the better. Since �dat ~Vrect, opt

also increases against �, its increasing rate is even largerthan that of �

h,max. In other words, the ratio between the

dissipated energy and harvested energy increases with �.Therefore, rather than merely enhancing the harvestingcapability, a more optimized SSHI should also take

3Equations (23)–(26) are used for this sub-figure.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

Eva

luat

ing

fact

ors

Eva

luat

ing

fact

ors

Eva

luat

ing

fact

ors

Eva

luat

ing

fact

ors

0.4

0.3

0.2

0.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

0.4

0.3

0.2

0.1

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.4

0.3

0.2

0.1

Dissipation factor hd ;Harvesting factor h

h ; Loss factor .

= 0.5ke 2

Vrect Vrect

VrectVrect

0.4

0.3

0.2

0.1

h∑

(a) (b)

(c) (d)

= 0.5ke 2

= 0.5ke 2 = 0.5k

e 2

Figure 16. Evaluating factors, as functions of ~Vrect and k2e , under different values of �. (a) �¼ 1 (SEH); (b) �¼ 0 ðQ ¼ 0Þ; (c) � ¼ �0:3 ðQ ¼ 1:3Þ;

(d) � ¼ �0:6 ðQ ¼ 3:1Þ.




these into account in order to make a good balancebetween the coexistent energy harvesting and energydissipation.

Energy Harvesting and Damping Performances of SSHI

Besides clarifying the relation of energy harvesting,dissipation, and damping in SSHI, we can alsotheoretically prove the advantages of the SSHItechnique. In terms of energy harvesting capability, wecan non-dimensionalize the maximum harvesting factorsof SSHI to those of SEH under the same couplingcoefficients, as:

~�h,max¼ð�

h, maxÞSSHI

ð�h, maxÞSEH

: ð46Þ

Figure 17(a) shows the non-dimensional maximumharvesting factor ~�

h,maxas function of k2e under different

values of �. It shows that, for the purpose of harvestingenergy, the harvesting capability of SSHI technique isbetter than that of SEH, regardless of k2e and �, since all~�h, max

in Figure 17(a) is larger than 1, i.e., 100. Moreover,more significant improvements can be achieved byintroducing the SSHI treatment to the harvestingdevices with higher coupling coefficient, i.e., larger k2e .On the other hand, with the same device using SSHI, thesmaller �, the larger improvement can be obtained.Similarly, in terms of damping capability, we can non-

dimensionalize the maximum loss factor of SSHI, whichis specified as SSDI, to that of RSD under the samecoupling coefficients, as:

~��, max¼ð�

�, maxÞSSDI

ð��, maxÞRSD

: ð47Þ

Figure 17(b) shows the non-dimensional maximum lossfactor ~�

�,maxas function of k2e with respect to different

values of �. The damping capability of SSDI is betterthan that of RSD. The curves of ~�

�, maxshow a similar

trend to ~�h, max

in Figure 17(a).

CONCLUSION

In this article, we proposed analyses to clarify therelationship among the functions of energy harvesting,energy dissipation, and their effects on structuraldamping in piezoelectric devices. Related conceptswere discussed and detailed models of piezoelectricdevices were provided. Two applications of SEH andRSD utilizing piezoelectric materials were investigated,and the similarities and differences between energyharvesting and energy dissipation were discussed.These two functions could be selected to achievedifferent objectives. Furthermore, coexistent energyharvesting and dissipation in the implementationof energy harvesting with SSHI technique wereinvestigated. These coexisting functions can bothcontribute to the effect of structural damping.The performances of the SSHI technique were alsoinvestigated. It has been shown that the SSHIwould outperform the SEH in terms of harvestingcapability and outperform the RSD in terms of dampingcapability. The results and discussion given in thisarticle will help us develop a more efficient andadaptive piezoelectric energy harvesting system as apromising power source.

ACKNOWLEDGMENTS

The work described in this article was supported bygrants from Innovation and Technology Commission(Project No. ITP/011/07AP) and Research GrantsCouncil (Project No. CUHK4152/08E) of Hong KongSpecial Administrative Region, China.

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0 0.2 0.4 0.6 0.8 1

103

102

101

100

g = 0 g = –0.3 g = –0.6 g = –0.9

g = 0 g = –0.3g = –0.6 g = –0.9

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Wikipedia, URL: http://en.wikipedia.org/wiki/Dissipation




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