HAL Id: hal-02042560 https://hal.archives-ouvertes.fr/hal-02042560 Submitted on 3 Jan 2020 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Nonlinear Conditioning Circuits for Piezoelectric Energy Harvesters Adrien Badel, Elie Lefeuvre To cite this version: Adrien Badel, Elie Lefeuvre. Nonlinear Conditioning Circuits for Piezoelectric Energy Harvesters. Nonlinearity in Energy Harvesting Systems, 25 (2), Springer International Publishing, pp.321-359, 2016, 10.1007/978-3-319-20355-3. hal-02042560
Transcript
HAL Id: hal-02042560https://hal.archives-ouvertes.fr/hal-02042560
Submitted on 3 Jan 2020
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Nonlinear Conditioning Circuits for Piezoelectric EnergyHarvesters
Adrien Badel, Elie Lefeuvre
To cite this version:Adrien Badel, Elie Lefeuvre. Nonlinear Conditioning Circuits for Piezoelectric Energy Harvesters.Nonlinearity in Energy Harvesting Systems, 25 (2), Springer International Publishing, pp.321-359,2016, �10.1007/978-3-319-20355-3�. �hal-02042560�
INTRODUCTIONDesign and analysis of piezoelectric vibration energy harvesters is a complex multi-physicsproblem related to mechanics, materials science and electronics. The analysis of workspublished in the field of piezoelectric energy harvesting over the last decade shows thatnumerous papers focused onmechanical optimizationwithout taking into account the actualconstraintsandrequirementson theelectrical sideof thesystems(i.e. theelectriccircuitwasoftenmodeledasasimpleresistor).Conversely,otherworksaimedatoptimizingsystemsfromtheelectricalpointof viewwithout taking intoaccount themechanical effects inducedby theenergyconversionprocess.Consequently, inbothcases thesolutionsproposedwerenot trulyoptimalorremainedveryfarfrompracticalapplications.To highlight themain aspects of thismulti-physics problem, this chapter beginswith generalconsiderationsaboutharvestedpowerbasedonthesimpleandwell-knownmodelproposedbyWilliam and Yates [1]. Starting from this model, the maximal power and the frequencybandwidthofthesystemisanalyzed,andafigureofmerittakingintoaccountboththepowerandthebandwidthoftheenergyharvesterisproposed.Thismodelisthenrefinedtoincludethedescriptionofpiezoelectricelectromechanicalcoupling,leading toaccurateand reliablebehavioral representationofmost linear inertialpiezoelectricvibrationenergyharvesters.Usingtheclassicalanalogiesbetweenelectricalandmechanicalfigures,anequivalentelectricalcircuitrepresentingthewholeelectromechanicalsystemisdescribed.Suchcircuitmaybeveryconvenient to study the system associated to its electronic interface using SPICE-typesimulations. In order to get the model parameters representing an actual vibration energyharvester,anidentificationprocedurebasedonthemeasurementofthecomplexadmittanceispresented.Theperformancesofthesystemusingoptimalimpedancematchingapproachisthenanalyzed,followedbyabriefoutlineofapossibleimplementationusingaPWMinvertercircuit.Theassociationofthepiezoelectricvibrationenergyharvesterwiththeclassicalrectifiercircuitisthenanalyzed,showingtheneedforanadditionalmaximumpowerpointtrackingsystem.Afterashortdescriptionofthemainnonlinearinterfacecircuitsdevelopedoverthelastdecade,adetailedanalysisoftheso-calledOSECEinterfacecircuitisexposed.Finally, this chapter discusses the possibility of tuning the resonant frequency of thepiezoelectric energy harvester through its interface circuit. The theoretical analysis of a newinterfacedevelopedforthispurposeandtermedFTSECEispresented.
1 AFIRSTVERYSIMPLEMODELFORKINETICENERGYHARVESTERSThepowergeneratedbyavibrationenergyharvesterdependsonthe transducerused for theelectromechanical energy conversion and on the way the transducer is implemented in amechanicalstructurethatissuitabletocaptureambientmechanicalenergy.
Inmostvibrationenergyharvesters(VEH)aninertialmassdrivenbytheambientaccelerationisusedtotransmitmechanicalenergytotheelectromechanicaltransducer.Asimple{mass-spring-damper}system, initiallyproposed in [1] can be used to model this behavior. A schematic of such an inertial vibration energy harvester is shown in Figure 1, where DL is a damper that embodies mechanical and electrical losses and DH is a damper that corresponds to the electromechanical transducer. The assumption of modeling the transducer by a simple damper is valid in the case of sinusoidal motion of the inertial mass, choosing the value of DH such as the energy dissipated into the damper during one mechanical period corresponds to the harvested energy.
Figure1.Inertialvibrationenergyharvester
1.1 ConsiderationontheharvestedpowerThe equation governing themotion of the inertial mass is given by (1.1), where𝛾 = �̈�is theambientacceleration.
𝑀�̈� + (𝐷* + 𝐷+)�̇� + 𝐾𝑥 = −𝑀𝛾 (1.1)
The relativemotion x of themass with respect to the housing can then be expressed in thefrequencydomainas:
(1.2)
Wherew is the operation angular frequency,w0 is the mechanical oscillator natural angularfrequency,WisthenormalizedoperationfrequencyandxLandxHarethedampingratiosduetointrinsiclossesandenergyharvestingrespectively.
Thispowerlimitisindependentfromthetransductionprincipleitself.Thismeansthatwhateverthe transduction mechanism, the maximal power can be harvested provided that theelectromechanicalcouplingishighenoughtoreachtheintrinsicdampingratio[2].
The harvested power can be normalized as shown in (1.6). This allows to compare theperformances of different energy harvesters independently from their resonant frequenciesinertialmasses and from the ambient accelerationmagnitude. The normalized power cannotexceed whichisthequalityfactorofthemechanicaloscillator.
(1.6)
It isworthyofnote that thisresult isonlyvalid for inertial linearvibrationenergyharvestersdrivenaroundtheirresonancefrequenciesbysinusoidalvibrations.
1.2 ConsiderationonthefrequencybandwidthWhenconsideringaninertialvibrationenergyharvester,notonlythemaximalpowerbutalsothe frequency bandwidth has to be studied. Equation (1.5) shows that intrinsic lossesrepresented by xL should beminimized to increase themaximal power. The goal of this sub-sectionistohighlighttheeffectofxLonthefrequencyresponseoftheVEH.
WedefinetheangularfrequencybandwidthDwastheangularfrequencyrangewherethepoweris at least 50% of its maximal value ( ). It is obtained seeking W1 and W2 so that
. The normalized bandwidth is then given by (1.8), and thebandwidthby .
ItispossibletoincreasethebandwidthifxHcanbeadjustedasafunctionofthefrequency.Anidealcaseisnowconsidered,wherexHcanbetunedwithoutanylimitation.TheoptimalvalueofxH thatmaximizes theharvestedpower canbeobtainedas a functionofW by looking for therootsofthederivativeofEq.(1.6)withrespecttoxH.Itisgivenby:
(1.9)
Andtheexpressionofthenormalizedpoweristhen:
(1.10)
The new expressions of the half-power normalized angular frequencies are given by (1.11),togetherwith thenormalizedbandwidth,which is shown to be times larger than the casewherexHiskeptequaltoxL.
(1.11)
An interesting result is shown here:whatever the strategy used to tunexH, the bandwidth isproportionaltoxL,whereasthemaximalpowerisinverselyproportionaltoit.
Figure 3 exhibits the normalized power and the optimal value of xH as a function of thefrequency,fordifferentvaluesofQM=1/(2xL).It isshownthatthebandwidthcanbeenlargedprovidedthatxHcanbeadequatelyincreasedwhileWgetsawayfrom1.
Figure4(upperplot)showsthenormalizedmaximalpowerandbandwidththatcanbederivedfromWilliamsandYatesmodelasafunctionofthedampingratioxHmax/xL,wherexHmax isthemaximal energy harvesting induced damping. It is shown that for xHmax < xL, the normalizedpower limit(QM)cannotbeobtained. IfxHmax≥xL, thepower limit isreachedat theresonancefrequencyandforxH=xL.Thenormalizedbandwidthlimit( )isreachedforxHmax≥3xL.Inthiscase,xHhastobesetequaltoxLforW=1,andhastobeincreasedforW���������rlowerthan1.xHwillthusbeequalto3xLforW= W1andW= W2(Eq.(1.11)).
In conclusion of this sub-section, Williams and Yates model is very simple but allowsunderstanding the relationship between themaximal power, the bandwidth and the dampinginducedbyboth the energyharvestingprocess and the intrinsic losses. It however exhibits amajorlimitation:Inthismodel,theelectricalloadhasnoeffectonthenaturalfrequencyoftheVEH;itonlyaffectsthedamping.Yet,inpracticalVEH,especiallyinthecaseofpiezoelectricVEH,
the electrical side of energy harvesting (i.e. the interface circuit) also impacts the resonancefrequency.Thiseffectwillbefurtherevidencedinthenextsectionsofthischapter.
2.1 ModelforpiezoelectricvibrationenergyharvesterBasedonelectromechanicaltransductionprinciplesandtheelectricalandmechanicalequationsofequilibrium,lumpedelectromechanicalmodellingofelectromechanicaltransducersisknownasaveryeffectivemethod[3], [4]. In theory,suchmodelsareexactwithnorestrictionsotherthan linearity, within the limits of the assumptions on boundary conditions and within thefrequencyrangecoveredbythemodeledresonantfrequencies.
Most inertial piezoelectric vibration energy harvesters are based on a linear mechanicaloscillator(acantileverbeamwithatipmassforinstance)includingoneorseveralpiezoelectricpatches.
The very simple PVEH lumped model presented in Figure 5 then provides an accurate andreliable behavioral representation of such PVEH excited around one of their resonantfrequencies,asdescribedin[2].
A piezoelectric element represents the piezoelectric transducer(s) (electrically connectedtogether in series or parallel if several are used). It is characterized by its electromechanicalforcefactora,itsclampedcapacitanceCPanditsparasiticresistanceRP;AP=1/RPistheparasiticconductance.M is theequivalentdynamicmass,D is thedampingcoefficientcorrespondingtothe mechanical losses and K represents the stiffness of the system when the piezoelectricelementisshort-circuited.
Three dimensionless parameters are used for the characterization of PVEH: theelectromechanicalcouplingcoefficientsquaredk2, themechanical lossesdampingratioxMandtheresistivelossescoefficientxE,whoseexpressionsare:
(2.3)
k2 describes the effectiveness of quasi-static energy conversion between electrical andmechanicalforms.ForaPVEHinopen-circuitsubjectedtoaquasi-staticstress,itisequaltotheelectrostatic energydividedby the total energy in the system.Amodified coupling coefficient𝑘04 ,definedas(2.4),isintroducedtosimplifytheoreticalexpressions.Itis,too,anindicatorfortheelectromechanical coupling, insofaras it represents thequotientbetween theelectrostaticenergyandtheelasticenergyinthegenerator(alsoinquasi-staticoperation).Notethatdespitek2 remains always lower than 1 (k2 = 1 meaning that all the input mechanical energy isconverted into electrostatic energy),𝑘04 is not limited. For weakly coupled generators, thevaluesofk2and𝑘04 arecloseonetotheother.
(2.4)
2.2 ElectricalmodelforSPICE-typesimulationsUsing the classical analogies between electrical and mechanical figures (force and voltage;velocityandcurrent),theequivalentelectricalnetworkshowninFigure6canbederivedforthemodeling of PVEH. The αV voltage source and the𝛼�̇�current sources correspond to theelectromechanical transduction.Themechanical power absorbed in the voltage source equalstheelectricalpowerprovidedbythecurrentsource,reflectingalosslessenergyconversion.Themechanicalbranch includesanelectricaloscillator {Lm,Cm,Rm} thatrepresents themechanicalresonance.Theelectricalbranch is thesameas theonedepicted inFigure5b).ThegoverningequationsfromthiselectricalnetworkareidenticaltoEqs.(2.1)providedthatLm=M,Cm=K-1andCm=D.
Figure6.EquivalentelectricalnetworkforaPVEH
This model can be easily implemented in SPICE-based software (SPICE: Simulation Programwith IntegratedCircuitEmphasis) and can thenbeused for efficient simulationofPVEHwithvariouselectricalinterfaceandambientaccelerationprofiles.
Asanexample,Figure7isascreenshotofaschematicimplementedintheLTSPICE™Software(LinearTechnologies Corporation),where a classical full bridge rectifier is used as anAC/DCelectricalinterfacebetweenthePVEHandtheresistorRL,whichmodelstheinputresistanceofthecircuittobepowered.
2.3 ModelidentificationprocedureTheparametersofthemodelpresentedintheprevioussubsectioncanbederivedfromagivenPVEHstructureusinganalyticalorfiniteelementmodelingapproaches[5][6],providedthatthecharacteristics and dimensions of the used materials are known with sufficient accuracy. Inpractice,themechanicallossesrelatedtomechanicalassemblyofthedifferentpartsofaPVEHarenoteasytopredict(i.e.bonding,clamping,etc.).Thepiezoelectriccharacteristicsprovidedby manufacturers may also exhibit important uncertainty and variability. This explains whysignificant discrepancies are usually observed between theoretical PVEH characteristics andexperimentalones.
This subsection details a very convenient procedure to determine experimentally the actualelectromechanical parameters of the model based on a simple measurement of the PVEHcomplex admittance. The measurement of the complex admittance of the PVEH has to beperformedarounditsnaturalfrequency.Thiscanbedoneusinganimpedanceanalyzer.Duringthismeasurement,thePVEHshouldnotbeexcitedbyambientacceleration.
ItisshownthattheadmittanceistheoneofaCPcapacitormultipliedbyadimensionlessfactoronly function ofW,𝑘04 , xE and xM. Figure 8 shows the comparison of the experimental andmodeledadmittanceof a realPVEH.The identificationof theparametershasbeenperformedusingoptimizationmethodstogetthebestmatchbetweenthemeasurementsandthemodel.
Fiveparameterscanbeobtainedfromthemeasurementoftheadmittance:CPw0,𝑘04 ,xEandxM.Fromthem,itisnotpossibletogetallthesixparametersoftheconstitutiveequations(2.1).TogetM,K,Danda,anadditionalmeasurementis indeedrequired.Forinstance,theratiointhefrequencydomainoftheopen-circuitvoltagev0tothedisplacementofthedynamicmassxcanbe used to determinea�� as shown by Eq. (2.6). From Eqs (2.2) and (2.3),M,K andD caneventuallybecalculated.
Amongotheridentificationprocedurewhichcanbefoundinliterature,aclassicaloneconsistsindetermining the coupling coefficient k2 from the open-circuit and short-circuit resonancefrequencies of the PVEH, and the mechanical damping ratio xM from the -3dB frequencybandwidth of the displacement x of the PVEHwith piezoelement short-circuited as electricalboundarycondition[7].
��� electrical losses coefficient xE is usually much lower than 0.01 and its effect on theadmittanceandmoregenerallyonthePVEHperformancesisweakinmostcases.Consequently,xEisneglectedinmostoftheliterature.
3 OPTIMALIMPEDANCEMATCHINGSeveral works used the well-known optimal impedance matching strategy to maximize thepoweroutputofPVEH[5],[8]. Inthissectiontheoptimal linearelectrical loadthatmaximizesthe PVEH power generationwill be analytically determined. As a first approach, no technicallimitationwillbetakenintoaccount.Resultsthatarenotachievableinpracticemaybeobtained,butthisstudywilltheoreticallydefinethePVEHperformanceupperboundary.
which can be rewritten as Eq. (2.8), where n and u are two dimensionless coefficientrespectivelycorrespondingtothein-phaseandthequadrature-phasecomponentsofthevoltagewithrespecttothedisplacement.
(2.8)
Substituting Eq. (2.8) in the mechanical constitutive equation (2.1) written in the frequencydomainleadstotheexpressionofthedisplacement:
(2.9)
Normalizing the displacement with respect to𝛾0/𝜔34(the ambient displacement magnitude)gives:
The harvested energy is considered to be the energy dissipated in the real component of theelectrical load(AL).Ofcourse, thisdoesnotcorrespondtoarealisticenergyharvestingcircuitsince the voltagewouldneed to be rectified.AL canhowever be considered as the equivalent
As previously mentioned if xE=0, the optimal power reaches Plim whatever the operationfrequency,providedthatALandBareadequatelytuned.IfxEisnotneglected,itisshownthattheoptimal power reached a maximum forW=1. Equation (2.16) gives the normalized maximalpower.
ThenormalizedpoweranddisplacementareplottedinFigure10asafunctionoftheoperatingfrequency fordifferent valuesofxE. The corresponding real and complexpartsof theoptimalelectricalloadadmittanceareshowninFigure11.It isclearlyshownthatxEdrasticallyaffectsthebandwidthofthePVEH.FromFigure11,itcanbeseenthattheoptimalcomplexpartoftheloadadmittanceisindependentonxEandthatitssignismainlynegativeinthefrequencyrangebutcanalsobepositive.Thisconcretelymeansthattheoptimalloadisaresistorinparallelwithaninductorinmostofthefrequencyrangebutthatitissometimesaresistorinparallelwithacapacitor(foroperationfrequencyslightlyabovethenaturalfrequencyofthePVEH).
Figure12exhibitsthevoltageonthepiezoelectricelementwhentheoptimalcomplexelectricalloadisselectedasafunctionoftheoperatingfrequency.Itisclearlyshownthatincreasingthebandwidthleadtoveryhighpiezoelectricvoltagewhentheoperatingfrequencyisshiftedawayfrom the resonance frequency (up to six times the maximal open-circuit voltage in the
consideredcases).This isdue to the increaseof reactivepoweralternatively flowingbetweenthecapacitanceofthepiezoelectricelementandthecomplexpartoftheoptimaladmittance.
Wherecis dimensionlessparameterthatcanbeseenasanindicatoroftheperformanceofthePVEH (it increases with the coupling coefficient and decreases with the electrical andmechanicallosses).
Thenormalizedpowerandbandwidth,aswellastheFoMareplottedinFigure13asafunctionofc������anbeseenthattheFoMcanlargelyexceed2√2,whichwastheupperlimitobtainedfromtheWilliamsandYatesmodel.Thisisbecauseofthefrequencytuningmechanisminducedbythecomplexpartoftheloadadmittance.FortypicalPVEH,theorderofmagnitudefor𝑘04 isaround10-2,whereas it is between10-2 and10-3��� xE �����xM. Thismeans thatc�typicalvalues ranges from10 to 100 about. In this case, the figure ofmerit canbe approximatedby𝐹𝑜𝑀 ≈ √𝜒.
Inpractices, this impedancematchingstrategycanhardlybe implementedbecause itrequirestherealizationofcomplexelectricalloadwithtuningmechanisms.Onecouldthinkaboutusingactive gyrator circuits to emulate large tunable inductors. Such synthetic inductors based onoperational amplifiers, capacitancesand resistanceshowever consumea lotofpowerandarenotaviableoptionforenergyharvestingapplications.
AnotherlimitationofthisapproachcanbeinferredfromFigure12:thevoltageincreaseduetothe reactive part of the complex optimal loadmay lead to very high piezoelectric voltage forwhichdepolingeffectoratleastnonlinearitiesinthepiezoelectriccoefficientsmaybeobserved.
3.2 PracticalimplementationModernpowerelectronicsoffersmanysolutionsbasedonswitching-modeconvertersenablinghigh-efficiency electrical power conversion. Input and output voltages can exhibit variousshapes and polarities, and the power transfer can be unidirectional or bi-directionnal,dependingontheconsideredcircuits[9].
In practice, electric loads powered by energy harvesters –such as wireless sensor nodes forinstance– don’t behave at all like the “optimal matching impedance” defined in the previoussubsection. Indeed, such electric loads require DC voltage-regulated power supply, and theirconsumptionmaybeextremelyvariableintime.Thus,AC-DCpowerconversion,energystorageandvoltageregulationaretheminimumrequirementsfortheelectronicinterface,whichwillbeusedtotransfertheelectricalenergyproducedbythepiezoelectricmaterialtotheelectricload.
PassiveAC-DCconverters,suchasdioderectifiers,areverysimpletoimplementbuttheshapeof theirAC inputvoltageandcurrent isnotsimilar to thatof linear impedances.Emulationoflinear impedance is however possible using PWM rectifiers. The proposed implementation oftheoptimalimpedancematchingstrategyisbasedontheinterfacecircuitrepresentedinFigure14,whichispresentedin[10].Inthiscircuit,theACinputofthePWMrectifierisconnectedtothepiezoelectricelement.ThroughthePWMcontroloftheswitchesoftheactiverectifier, it istheoreticallypossibletoemulateanylinearornonlinearload,includingtheoptimalimpedancedefined previously. The DC output of the active rectifier is connected to and energy storageelement (a capacitor or a supercapacitor)whose voltagemay vary, depending on the energystored.TheoutputDCvoltagedeliveredtotheelectricloadcanbethenregulatedusingabuck-boostconverter.Thissub-circuitenablestoensureoutputvoltageregulationwhethertheoutputDCvoltageVLislowerorhigherthantheinputDCvoltageVDC.
Thecontrolstrategyofthisinterfacecircuitisnotverycomplicated,butitisoutofthescopeofthischapteranditwillnotbedetailedhere.Amongimportantideastohaveinmind,practicalimplementation of the optimal impedancematching strategy is possible. However, in case ofvery lowharvestedpower level, typically intherangeofa fewtenthsofmicrowattsorbelow,theavailablepowermaybetoolowtoimplementsophisticatedcontrolcircuits.Therefore,“lessoptimal”approachesbasedonsimplercontrolprinciplesandsimplercircuitsmayturnouttobemuchmoreefficientinpractice.Thisistheobjectiveofthevarioustechniquesdevelopedinthenextsections.
For simplicity, it will be assumed that the electrical losses can be neglected (xE �= 0). ThisassumptionisvalidinmostofpracticalcaseswhereRLismuchlowerthanRP.
4.1 PowerandBandwidthThiscircuithasbeenstudiedindetailsbyShuandLienin[11].Sinusoidalambientaccelerationsaround the resonance frequency of the PVEH are considered, and it is assumed that thesmoothing capacitor is large enough so that VDC is ripple free. The value of VDC can then beobtainedasafunctionofthemagnitudexMofthedynamicmassrelativedisplacement:
(3.1)
Because of the nonlinear behavior of the full-wave rectifier, the piezoelectric voltage v isperiodic but not sinusoidal.However, since only its fundamental harmonic frequency is closefrom the resonance frequency of the VEH, it can be assumed that only this fundamentalcomponent v1, given in the frequency domain by (3.2), impacts the displacement of the VEHdynamicmass.
(3.2)
Theexpressioninthefrequencydomainofthedisplacementandthenormalizeddisplacementare then the same as in section 3.1, equations (2.9) and (2.10) respectively, except that theexpressionsofnanduaredifferent.
Itcanbeseenfrom(3.4)thatnisbetween0(whenr’=0)and1(whenr’tendstoinfinite).From(2.10), this means that the resonance frequency of the PVEH is between w0 (short-circuitresonancefrequency)and (open-circuitresonancefrequency).
Itcanalsobeseenthatu isbetween0(whenr’=0orwhenr’tendstoinfinite)and1/p(whenr’W = p/2�. From equation (2.12), thismeans that the optimal damping cannot be obtained if
.
Theharvestedpowercanbeexpressedas:
(3.5)
Andthenormalizedharvestedpowerisfinallygivenby:
(3.6)
ThemaximalnormalizedpowerandthecorrespondingoptimalnormalizedloadareplottedasafunctionofthefrequencyinFigure16, fordifferentvaluesof𝑘04 .When𝑘04 increases, itcanbeseen that whereas the bandwidth keeps on increasing, the maximal normalized power firstincreasesandthensaturatesatQM.Itisworthyofnotethatacontinuoustuningoftheelectricalload as a function of the operation frequency is required to get the best performance. Thecolored areas represent the load domains for which at least 50% of the maximal power isharvested.
Thenormalizedpower,bandwidthandthefigureofmeritdefinedin(1.12)areplottedinFigure17asafunctionof𝑘04 fordifferentvaluesofQM.TheblackcurveinFigure17acorrespondsto𝑘04 𝑄E = 𝜋. It isshownthat thenormalizedpowerequalsQM for𝑘04 𝑄E > 𝜋.Figure17bshowsthat the bandwidth is quasi linearly increasing with𝑘04 . For very large values of𝑘04 , thebandwidth however decreases because the power in the well between the two maxima (cf.Figure16with𝑘04 >6%)becomeslessthanhalfofthemaximalpower.FortypicalPVEH,𝑘04 isintheorderofafewpercentand𝑄Eisbetween10andafewhundreds.Thefigureofmeritisthenusuallylowerthan5.Figure17cclearlyshowsthatPVEHwithhigh𝑘04 andhigh𝑄Eexhibithigherperformance.
It is worthy of note that results presented in Figure 17 imply a continuous tuning of theelectrical load as a function of the operation frequency, which requires specific powerconversioninterfaceanddedicatedMPPTcontrolcircuitforpracticalimplementation.
4.2 PracticalimplementationSeveralinterfacecircuitbasedontheclassicalrectifiercircuitprinciplehavebeenproposedformaximizing the power transfer from the piezoelectric device to the load. In this optimizationapproach, themain function of the interface circuit is to emulate the optimal load resistancedefinedintheprevioussection.Ottmanetalstudiedoneofthefirstinterfacecircuits[12].ThiscircuitwasbasedonthebuckDC-DCconverterrepresentedinFigure18.UsingtheappropriatecontrollawoftheDC-DCconverter,itwasshownthattheperformancesofthesystemcouldbegreatlyimproved.Inordertoreducethepowerconsumptionofthecontrolcircuitsothatitcanbe self-powered, the authors proposed a new control principle of the buck converter thatexhibitedsimilarperformances,withmuchsimplerimplementation[13].
Followingthesameapproach,Lefeuvreetal.[14]proposedaninterfacecircuitbasedonabuck-boost DC-DC converter depicted in Figure 19. In discontinuous current mode, this circuitexhibits a constant input resistance for a given duty-cycle control of the electronic switch,makingpossibletheimplementationoftheinterfaceusingonlyanoscillatorwithconstantduty-cycleascontrolcircuit(IC1inFigure19).Foroutputpowersrangingfrom200µWto1.5mW,experimental results showed more than 70% overall efficiency of this circuit, including thecontrol circuit consumption. In this practical example, the input resistance of the interfacecircuit was predetermined to be as close as possible to the optimal value at the resonant
frequency. Consequently, the robustness was relatively weak with respect to variations ofelectromechanicalcharacteristics.
To overcome this drawback, Maximum Power Point Tracking (MPPT) circuits with ultra-lowpowerconsumptionwereimplemented.Yietal.proposedalow-powerinterfacecircuit,basedonaswitchedcapacitorDC-DCconverter, integrated ina0.35µmCMOSprocessASIC[15]. InthisASIC,anenergy-adaptiveMPPTallowedtoactivatedifferentoperationmodesaccordingtothe available power. Kong et al. presented an interface circuit based on a flyback DC-DCconverter[16].TheMPPTalgorithmwasimplementedusingalow-powermicrocontrollerunitMSP430 from Texas Instruments. Experimental results indicated that the proposed interfacecircuit achieved 72% efficiency around the resonant frequency and around 8.4 mW outputpower.
In this domain, the current trends are clearly to improve the efficiency of low-power DC-DCconverters and to design fast and effective MPPT control circuits with ultra-low powerconsumption[17],[18].
5.1 PrincipleNonlinearenergyextractionapproacheshavebeendevelopedtooptimizetheenergyextractedfrom PVEH. From the simple model presented in section 2.1, the energy equation (4.1) isobtained multiplying both terms of the dynamical mechanical equilibrium equation by thevelocityandintegratingoverthetimevariable.Itshowsthattheenergyprovidedbytheambientacceleration isdivided intokinetic energy, potential elastic energy,mechanical losses and theenergy extracted from the piezoelectric element. Nonlinear energy extraction circuits aim atincreasingthislastterm.
(4.1)
Thebasicelectronicelement fornonlinearenergyextractioncircuit is shown inFigure20a. Itconsists inconnectinga coil in serieswithanelectronic switch inparallel to thepiezoelectricelement. The switch is almost always open, except when a minimum or maximum of thedynamicmassdisplacementoccurs.Atthismoment,theswitchisclosedandCPinparallelwithLforms an electrical oscillating circuit. The switch is kept closed during half of the oscillatingperiodTI,sothatthevoltageisreversed.Becausetherearesomeelectricallossesinthe{L,CP}network,characterizedbythequalityfactorQI,theabsolutevalueofthevoltageafterinversionisslightlyreducedcomparedtotheonebeforetheswitchisclosed.Correspondingwaveformsforthevoltage,displacementandvelocityofthedynamicmassareshowninFigure20b.
Thissimplecircuithastwoeffects:First,forconstantdisplacementamplitude,theamplitudeofthe voltage is largely increased; second voltage and velocity are of the same sign. These twoeffectsclearlyinducetheincreaseoftheenergyextractiontermofequation(4.1).
AddingAC/DC and energy storage stages to this elementary circuit has been the basis of thefurther developments of nonlinear energy harvesting approaches. For instance, the SSHI(Synchronized Switch Harvesting on Inductor) simply consists in combining this switch-coilcircuit to the classical full-wave rectifier approach, connecting them in parallel to thepiezoelectricelement[7].
5.2 ComparisonofvariousnonlinearcircuitsBasedonthegeneralprincipleexposedpreviously,severalnonlinearenergyharvestingcircuitswithdifferent featureshavebeenproposedover the lastdecade.This subsectionoutlines the
mainpropertiesofeachcircuit.TheyhaveincommontoincreasethelasttermofEq.(4.1),thatis to say they increase themechanical damping induced by energy conversion. Low-couplingPVEH particularly benefit from these circuits. Indeed, such increase of the electromechanicalenergyconversioneffectivenessallowsgettingclosertotheoptimaldamping(seesection1.1).This beneficial effect also increases the PVEH performances in case of out-of resonance andpulsedexcitation[20]. In thecaseofstronglycoupledPVEHexcitedatresonant frequency,nogain can be expected if optimal damping is already attained. Specific features of the mainnonlinearcircuitsaredetailedhereafter.
Theparallel SSHI circuit depicted on Figure 21 is one of the first nonlinear PVEH interfacesbased on the principle of “synchronized switching” [21], [7], [20]. This circuit is astraightforwardassociationoftheclassicalrectifiercircuitofFigure15andthecircuitofFigure20a. Compared to the classical rectifier circuit, the parallel SSHI circuit tends to increase thepiezoelectricvoltage.Theoptimalloadresistancetendsalsotobehigherthanwiththeclassicalrectifier circuit. This voltage magnification property may be used to get high voltages, or toreducetheenergylossesrelatedtothevoltagedropofthediodesintherectifierbridge.Thisisparticularly interesting in the case of low-voltage PVEH microsystems, whose open-circuitvoltageistypicallylowerthan1V.Shuetal.studiedindetailtheeffectofthiscircuitonPVEHasafunctionoftheexcitationfrequency[22].
IncaseoftheseriesSSHIcircuit(Figure22),thecoil-switchdipoleisconnectedinserieswiththePVEHinsteadofbeingconnectedinparallel.Thisinducesslightchangesonthepiezoelectricvoltagewaveform,andsignificantdifferencesonthecircuitoutputvoltageandtheoptimalloadresistance. Indeed, in this case theoutputvoltageand theoptimal load resistanceare smallerthanthoseoftheclassicalrectifiercircuit[23].Therefore,theseriesSSHIcircuitisparticularlyinteresting togetanoutputvoltage lower than thepiezoelectricvoltage.This circuitwas firstproposedforthisvoltagereductionpropertybyTayloretal.[24].
TheprincipleoftheSECEcircuitconsistsinextractingpromptlyandentirelytheelectricenergyconverted by the piezoelectric element on each extremumof the piezoelectric voltage. In the
Several works focused on efficiency improvement of the previous circuits for low-voltageapplications.Awayforreducingthevoltagedropsconsistedinreducingthenumberofdiodesusedforvoltagerectification.Inthisdomain,Makiharaetal.[27]proposedahalf-bridgecircuitfor theParallel SSHI circuit. Lallart et al. proposed another half-bridge circuit for low-voltageimplementationoftheseriesSSHIinterface[28]
TheSSHI-MRcircuitproposedbyGarbuioetal.[29](Figure24)broughtanultimatereductionofthelossesduetothethresholdvoltageofthediodes.Thetypicalwaveformsofthiscircuitarevery similar to thoseof the series SSHI circuit.Themaindifference comes from themagnetictransformer, which replaces the coil of the series SSHI circuit. The voltage gain of thetransformerinassociationwiththesingle-dioderectifierenablesoperationtowardsultra-low-voltage PVEH: experimental results showed effective energy harvesting from piezoelectricvoltagesaslowas30mV.
The single-supply pre-biasing circuit presented by Elliott et al. [30] has piezoelectricwaveformsidenticaltothatoftheparallelSSHIcircuit.TheuseofMOSFETelectronicswitchesinsteadofdiodesenablesefficientoperationevenwithultra-lowPVEHvoltages.
Figure24.SSHI-MRcircuitschematic(from[29]).
TheDSSHcircuitproposedbyLallartetal. isanassociationoftheSeriesSSHIcircuitandthebuck-boost DC-DC converter of the SECE circuit [31] (Figure 25). The intermediate energystoragecapacitorCintbringsanadditionaldegreeof freedomtocontroltheenergyconversion.This intermediate energy tank is used here for optimizing the trade-off between energyharvesting andmechanical damping. In addition, the buck-boost DC-DC converter makes theharvested power optimal whatever the load characteristics (i.e. no influence of the load
equivalent resistance). TheDSSH circuit implementation is a little bitmore complicated thanthatoftheSECEortheSSHIcircuits.Despitesthecumulatedlossesofthetwoconversionstages,experimental results exhibitedmuchbetterperformances than thatof the standardandSECEtechniquesinthecaseofPVEHwithsmallk2Qm.Theso-calledESSHcircuitproposedbyShenetal.[32]canbeconsideredasanimprovementoftheDSSHcircuit,whichallowsafinercontrolofthemechanicaldamping,inducedbyenergyconversion.
Basedonthisgeneralnonlinearapproach,severalotherinterfacecircuitshavebeenproposed.Wuetal.proposedtheso-calledSSDCIcircuit,basedonacircuitsimilartotheseriesSSHI,butwith amodified switch control [33]. The principle of themethod consists of transferring theelectrostatic energy available on the piezoelectric element to a storage capacitor through aninductance.
Dicken et al. proposed the so-called Pre-Biasing circuit [34], which enables to pre-bias thepiezoelectric element with the appropriate voltage to finely optimize the harvested power.However, the complexity of the circuit, which includes numerous switches and two differentpowersupplies,maybeanobstacletolow-powerstandaloneimplementation.
EnergyconversioncyclescanbeactivelycontrolledusingPWMinverters [10].This techniquewas named “active energy harvesting” by Liu et al. [35], and further analyzed for blood-pressureenergyharvesting[36].Suchactiveprincipletheoreticallyenablestogiveanyshapetothepiezoelectric voltagewaveform, including for instance the “optimal impedanceemulation”described in section 3.1, yielding outstanding power level in theory. However, powerconsumption of the PWM control and energy losses of the circuit may limit the actualperformances.
In summary, this subsection presented an overview of the main nonlinear circuits proposeduntil now.The readerwill findmoredetailed analysis of these circuits in theoriginal articlesgiven in the References list. The next subsection presents a detailed analysis of the so-calledOSECE circuit [37], which is one of the last proposed nonlinear circuits. It can be seen as avariantof theSSHI-MRcircuit,butwith simplifiedcontrolprinciple,makingmucheasier low-powerstandaloneimplementation.
5.3 TheOSECEapproachThe OSECE (Optimized Synchronous Electrical Charge Extraction) has been developed as animprovement of the SECE approach. It allows to keep its main feature, namely the lowdependencyoftheperformanceontheelectricalload,whilesimplifyingtheelectronicswitchesdrivingsignalsandenhancingtheenergyconversion[37].
The switch S1 and S2 are complementary driven: S1 is closed and S2 opened after thedisplacement (voltage) reaches a maximum and S1 is opened and S2 closed after thedisplacement (voltage) reaches a minimum. This switching strategy allows the voltage to bepartlyinvertedtwotimesaperiodofvibration.
Thevoltageisonlypartlyinvertedbecausetheinversionphaseisstoppedassoonasthevoltageon the secondary reaches VDC (at this moment, the diode in series with the closed switchbecomesreversedbiased).Adetaileddescriptionandthemodelingofthisinterfacecircuitcanbefoundin[37].
Figure27 shows the typicalOSECEwaveforms for thedisplacement, thepiezoelectric voltageand theswitchcontrol signal for severalperiods.Detailsofvoltagesandcurrents close to theenergyextractionmomentsarealsoshown.VMandVmarethepiezoelectricvoltagevaluesjustbeforeandjustaftertheenergyextractionphasewhosedurationistm.
whereQIisthequalityfactoroftheprimary{L1,CP}oscillatingcircuit,mistheturnsratioofthetransformer, and q is the phase angle corresponding to the duration of the inversion phase,givenby:
As for theclassical rectifierapproach,only the fundamental componentv1 of thepiezoelectricvoltage,givenby(4.4),isconsideredtoimpactthemotionofthedynamicmass.
Since n =1, the natural angular frequency of the PVEH is the open-circuit angular resonancefrequency .Itcanalsobeseenthatu�isalwayslargerthat4/p��sinceq isbetweenp/2 and p) and depends on q���� QI. Compared to the classical rectifier case whereu�����������������p��itisclearthattheOSECEapproachinducesmoredamping.
As for the classical approach, the harvested power can be expressed as (3.5), and thenormalizedharvestedpowerisfinallygivenby:
(4.5)
ThemaximalnormalizedpowerandthecorrespondingoptimalnormalizedloadareplottedasafunctionofthefrequencyinFigure28, fordifferentvaluesof𝑘04 .When𝑘04 increases, itcanbeseen that whereas the bandwidth keeps on increasing, the maximal normalized power first
increasesandthendecreases.Thisdecrease isdue to the too largedampingeffect inducedbytheOSECE approach in the case of highly coupled PVEH. Consequently comparing the powerusing theOSECEapproachand theclassical rectifierapproach, it is found thatOSECE leads tobetterperformanceforstructureswithlowcouplingcoefficientsorstructuresdrivenoutoftheirresonancefrequency.Thecoloredareas in the lowerplot represent the loaddomains forwhichat least50%of themaximalpower isharvested. It is shownthat thedependencyon the load ismuch lower thanwhenusingtheclassicalrectifierapproach.
It is worthy of note that the performances plotted in Figure 29 are given for the optimalelectrical load. Practical implementations of the classical approach may lead to significantlypoorerresultsthantheoreticalonesbecauseofthestrongdependencyoftheharvestedpoweron the load, which imply the mandatory use of a complex MPPT (Maximum Power PointTracking)strategy.On theotherhand, theOSECEapproach ismuchmore tolerant to the loadvariations,whichmakesitspracticalperformancecloserfromtheory[38].
6.1 TheneedforwidebandPEHMostofPEHsreportedintheliteratureexhibitsquaredcouplingcoefficient𝑘04 lowerthan5%.As shown in the previous sections, their bandwidth can be increased using nonlinear energyextractioncircuits,butarestilllimitedtoafewpercentoftheirresonancefrequency.SuchPEHcanthusefficientlyoperateinanarrowfrequencybandtunedtomatchtheexcitationfrequency.
However, environmental excitations have broadband or time-dependent characteristics inwhichtheenergyisdistributedoveraspreadspectrumoffrequencies.Twostrategieshavebeeninvestigated: Developing nonlinearwideband oscillators and developing linear oscillatorwithresonance frequency tuning mechanisms. Nonlinear oscillators (hardening, softening orbistable)arewellsuitedforbroadbandvibrationspectrum[39],whereasresonancefrequencytuning is more appropriated for narrow band but time-dependent vibrations (which can befoundonamotorwhoserotationspeedisvarying).Nonlinearoscillatorsapproachesarebeyondthe scopeof this chapter since theyarenot related to theenergyextractioncircuitbut to thearchitecture of the PEH. Resonance frequency tuning is usually done through an additionalmechanical component that passively or actively changes the stiffness or inertia of a linearmechanicaloscillator[40].Thesemechanicalapproacheswillalsonotbedetailedhere.
This section reports a theoretical nonlinear energy extraction approach to tune the resonantfrequency of linear inertial PEH through the control strategy of the associated electronicinterfacecircuit.Whenassociatedwithpiezoelectricdevicesexhibitinghighelectromechanicalcoupling,itenablestovarytheresonantfrequencyinlargeproportionswithoutanyadditionalcomponent.
Seddiketalpreviouslyproposedaprincipleofcontrol through theelectronic interfacecircuit[41]. It consisted in connectingshunt capacitors to thepiezoelectricdevice.Consequently, theelectromechanical structure stiffness was varied, and the resonant frequency was changedaccordingly. They showed that itwas possible to significantly vary the resonant frequency ofhighly coupled piezoelectric energy harvesters. However, because the shunt capacitance wasvariedstepbystep,theresonantfrequencycouldnotbecontinuouslytuned.Withtheproposedtechnique,itisexpectedthatacontinuoustuningisachievableonanoticeablylargerfrequencyrange.
6.2 FrequencyTuningSECE(FTSECE)The FTSECE approach is also derived from the previously developed SECE (SynchronizedElectricalChargeExtraction)approach. It iscalledFTSECE forFrequencyTuningSECE[42]. Itconsists in letting the PEH in open circuit condition most of the time, and to extract thegeneratedelectricalchargestwotimesaperiodofvibration.IncontrasttotheSECEtechnique,the FTSECE circuit does not extract energy at the piezoelectric voltage extremum, butwith aphase shiftf.Moreover,whereas thepiezoelectricvoltage isnull aftereachenergyextractionphase in theSECEapproach(all theelectricalchargesareextracted), it canbe tunedwith theFTSECEtechnique.
The FTSECE approach has not been practically demonstrated yet, but a possible electronicinterfacecircuitforitsrealizationisschematicallydepictedinFigure30.Thiscircuitincludesanelectronic switch S, whose control is synchronized with the piezoelectric voltage. TwoparametersofSarevariedthroughthecontrolcircuit:theonstatetimeduration(tON)andthelagtimeduration(tlag)betweenthevoltageextremaandtheinstantwhereS isturnedon.Thephase shiftf =wtlag canbepositive (S is closedafter thevoltageextremum)ornegative (S isclosedbeforethevoltageextremum).IfVMisthepiezoelectricvoltageatthemomentwhereSisturnedon,varyingtONwillallowtosetthevoltagetobVMaftertheenergyextractionphase.IfQIis the quality factor of the {L,CP} circuit,b�����be tuned between 1 (tON=0) and−𝑒NO/(4PQ)(tON≃π(LCP)1/2)
The loadresistancevaluehasno influenceon theharvestedpowerprovided that theDC loadvoltageislargerthanthepiezoelectricvoltageamplitude.Inthisway,energytransfersfromtheinductor to the rectifier only occur right after S is turned off. Varying tON enables to tune theamountofelectricalenergyextractedfromthepiezoelectricelements,and,ifneeded,toreversethe polarity of the piezoelectric voltage. Varying tlag enables tomodify the electromechanicalstructurestiffnessbytuningthephaseshiftbetweenthepiezoelectricvoltageandthestraininthepiezoelectricelement.
Equations (5.1) confirm thatf andb (tlag and tON) affect both the resonance frequencyof thesystemanditsdamping.FrequencytuningaswellasoptimizationoftheenergytransferisthenachievableusingtheFTSECEapproach.
The expression of VM is given by (5.3) where xM is the magnitude of the dynamic massdisplacement.
(5.3)
Taking intoaccount the losses in the inductorL thenormalizedharvestedpower can thenbeapproximatedby:
(5.4)
ThenormalizedpoweraswellastheoptimalfandbareplottedinFigure32asafunctionoftheoperatingfrequency,fordifferentvaluesof𝑘04 .HighcouplingcoefficientshavebeenconsideredbecausetheFTSECEapproachisespeciallypromisinginthiscase.Suchcouplingcoefficientscanbe practically obtained using single crystals piezoelectric materials. For instance, a PEH forwhich𝑘4equals53%(𝑘04 =112%)waspresentedin[42].
Fromtheevolutionoffandb,itcanbeseenthattheextractiontimescoincidewiththevoltageextrema (f=0) only at the open-circuit resonance frequency of the PEH. In this case, if thecoupling coefficient is high (typically larger than 5%), only a small amount of the generatedelectricalchargesareextracted(btendsto1)topreventatoolargedampingofthePEH.Whenthevibrationfrequencyisslightlyshiftedawayfromtheopen-circuitresonancefrequency,theswitchingphaseshiftfisadjustedtotunetheresonancefrequencyofthePEH.Thisphaseshiftinduces a decrease of the generated electrical charges,which is compensated by extracting alarger percentage of them. If the vibration frequency is shifted further, β becomes negative,whichmeansthataportionofthegeneratedelectricalchargesisinjectedbacktothePEHwithareversed polarity. This effect increases the piezoelectric voltage amplitude and enhances theenergyextraction.
Figure33ashowsthatcomparingtotheclassicalrectifierapproachthepowerusingtheFTSECEapproach is larger when𝑘04 𝑄E < 0.9and slightly lower when𝑘04 𝑄E > 0.9(the black curvecorrespondsto𝑘04 𝑄E = 0.9).Figure33bconfirmsthebandwidthenhancementinducedbytheFTSECEcircuit.Finally,Figure33cshowsthattheFTSECEapproachgiveshigherfigureofmeritwhateverthevalueof𝑘04 .
The FTSECE circuit has not been practically implemented yet. The real-time tuning of twoparameters(tonandtlag)requiresthedevelopmentofadedicatedswitchcontrolcircuit,whosecomplexity may hinder the overall performances. Moreover, the effect of the piezoelectricvoltage switching on highly coupled PEH may generate non-sinusoidal displacement of thedynamic mass, in which case the theoretical calculation detailed above become inaccurate.Becauseofthelackofexperimentaldemonstration,thislastsectionshouldthenbeconsideredwith caution. It however suggests that resonant piezoelectric structures with high
electromechanical coupling coefficient combined with dedicated nonlinear energy extractioncircuit could lead to large bandwidth PEH (up to 20% of the resonance frequency couldreasonablybeobtained).
TocomparetheFTSECEapproachandtheidealimpedancematchingstrategy(cf.section2),itisassumedthatxE=0.005.Taking𝑘04 =100%andxM=0.005(QM=100)leadstoc������.TheFoMusing the impedancematching strategywould then be√𝜒 = 200, whereas it is only 20 usingFTSECE (see red curve at𝑘04 =100% in Figure 33c). This result confirms that the idealimpedancematchinggives anupper limit for aPVEHperformance. It also suggests that someadvancedenergyharvestingcircuitswithenhancedbandwidthcapabilitymightbedevelopedbyfurtherresearch.
CONCLUSIONThis chapter presents a broad analysis of the existing power conditioning techniques forpiezoelectric energy harvesting devices and the related circuits enabling practicalimplementation.
The classical impedance matching technique would enable in theory to get the bestperformancesintermsofpowerandfrequencybandwidth.However,practicalimplementationofthistechniquewouldrequirecomplicated,powerconsumingcontrolalgorithms.Todate,veryfew experimental results can be found about this technique [8], confirming difficultimplementation.
Overall, the so-called nonlinear interfaces bring several advantages. One of their mostremarkable properties is the drastic power improvement of PVEH exhibiting low k2Qm.. ForPVEH with higher k2Qm, they give enhanced performance for pulsed excitation or out ofresonanceexcitation.
Current trends in thedomainofpowerconditioningcircuits forpiezoelectricvibrationenergyharvesting tend to push the limits towards “high” voltages (above 50 V) and “low” voltages(below1V)with ultra-lowharvestedpower, typically in the range of 10nW to 10µW. Suchdevelopments raise new challenges in the domain of ultra-low power AC-DC and DC-DCconverters,andinthedesignofultra-lowpowerASICs.
