Research Article Experimental Analysis of a Piezoelectric Energy...

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2013, Article ID 241025, 12 pageshttp://dx.doi.org/10.1155/2013/241025

Research ArticleExperimental Analysis of a Piezoelectric Energy HarvestingSystem for Harmonic, Random, and Sine on Random Vibration

Jackson W. Cryns, Brian K. Hatchell, Emiliano Santiago-Rojas, and Kurt L. Silvers

Health Monitoring and Radio Frequency Sensing Team, Pacific Northwest National Laboratory, P.O. Box 999,Richland, WA 99352, USA

Correspondence should be addressed to Jackson W. Cryns; [email protected]

Received 17 April 2013; Revised 28 June 2013; Accepted 1 July 2013

Academic Editor: K. M. Liew

Copyright © 2013 Jackson W. Cryns et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Harvesting power with a piezoelectric vibration powered generator using a full-wave rectifier conditioning circuit is experimentallycompared for varying sinusoidal, random, and sine on random (SOR) input vibration scenarios; the implications of source vibrationcharacteristics on harvester design are discussed. The rise in popularity of harvesting energy from ambient vibrations has madecompact, energy dense piezoelectric generators commercially available. Much of the available literature focuses on maximizingharvested power through nonlinear processing circuits that require accurate knowledge of generator internal mechanical andelectrical characteristics and idealization of the input vibration source, which cannot be assumed in general application. Variationsin source vibration and load resistance are explored for a commercially available piezoelectric generator. The results agree withnumerical and theoretical predictions in the previous literature for optimal power harvesting in sinusoidal and flat broadbandvibration scenarios. Going beyond idealized steady-state sinusoidal and flat random vibration input, experimental SOR testingallows for more accurate representation of real world ambient vibration. It is shown that characteristic interactions from morecomplex vibration sources significantly alter power generation and processing requirements by varying harvested power, shiftingoptimal conditioning impedance, inducing voltage fluctuations, and ultimately rendering idealized sinusoidal and random analysesincorrect.

1. Introduction

Modular devices requiring no external power supply havebecome commonplace inmany industries. Every day, wirelessmonitors gather information on hazardous processes andremote equipment, and consumer electronics take advantageof self-contained designs. These devices are often limitedin their capabilities, size, and weight by the power sup-ply, typically electrochemical batteries. Batteries are heavy,environmentally hazardous and require regular charging orreplacement. To alleviate the constraints of batteries, thepower demand of the device must be reduced or externalenergy sources need to be implemented. Modern radiofrequency sensor nodes and wireless monitors only requiremilliwatts (mW) of power; many drop into microwatts (𝜇W).A growing alternative to batteries is to harness energyfrom the surrounding environment, a concept known asenergy harvesting. Ambient energy exists in many forms

including thermal energy, kinetic energy, electromagneticradiation, and vibration. Transducers convert this environ-mental energy into usable electrical energy for an electricaldevice, effectively reducing or removing the power demandof a system [1–3].The amount of energy that can be extractedfrom the environment varies greatly by application, trans-duction method, and processing circuitry, but many studieshave shown that properly developed harvesting applicationscan produce ones to tens of mW of power, depending ontransducer size [1, 4–8].

Vibration-powered generators are types of transducersthat convert vibrational energy into electric energy. Numer-ous studies have been conducted on harvesting vibrationenergy, from powering electronics by walking [1] to charginga battery with the vibrations of a car [7]. A vibrationenergy harvester (VEH) incorporates both the transducerand electronics necessary to deliver power to the targetelectronics. Vibration transducers convert energy through

2 Advances in Acoustics and Vibration

three transduction mechanisms: electrostatic transducersharness vibration energy against the electric field of a variableplate capacitor, electromagnetic transducers induct powerfrom kinetic energy as the vibrations move a magnet, andpiezoelectric transducers convert mechanical strain into acurrent or voltage through the piezoelectric effect [4, 7, 9].The piezoelectric effect is the collection of charge in responsetomechanical stress in certain solidmaterials [10].This paperconsiders piezoelectric generators as they are the most com-pact and have the highest energy density [7, 11]. Moreover,electrostatic transducers require a separate voltage sourcein order to harvest energy and electromagnetic transducerstypically produce voltages below 1 volt (V), making themnonideal for most applications [3, 12].

Power conditioning circuits control how the power isdelivered to the electric load and come in many forms.Designs can range from just a few analog components tocomplex architectures controlled by firmware loaded ontomicrocontrollers [1, 6, 13–15]. Passive conditioning circuitscontain only the components necessary to provide a DCsignal at specified voltages and intervals. Other circuits utiliz-ing nonlinear techniques like synchronous charge extraction(SCE) and synchronized switch harvesting with inductors(SSHI) make use of active circuit components such as com-parators and switch controllers to maximize the extractedpower. High-efficiency harvesting circuits that take advan-tage of extreme low power microprocessors to maximize theconverted power have become commercially available.

VEHs are application-specific and require careful opti-mization for effective power conversion. Proper characteri-zation of the source vibration and the transducer allows fortuning in order to achieve resonance. Much of the avail-able literature has focused on idealized sinusoidal vibrationsources, which may be due to the cross-discipline nature ofthe application [4–6, 8, 11, 12, 15, 16]. Unfortunately, mostreal-world applications incorporate broadband signals andnonlinear vibration influences that stray significantly fromsinusoidal signals [17–19]. Tang et al. [8] found that SCEharvested 3.6 times the maximum power of the passiveconditioning circuit and Lefeuvre et al. [4] found that theSSHI could theoretically reduce the amount of piezoelectricmaterial needed to harvest sufficient energy by a factor of16. Both of these studies, however, did not account for theadditional power input required to implement the activecontrol circuits. Another study [6] found that the powernecessary to implement active control circuits for SCE andSSHI would be on the order of hundreds of 𝜇W, which isnonnegligible when compared to the energy harvested innonidealized conditions [1, 6, 19]. Due to their reliance onvibration frequency, SSHI is not considered in nonsinusoidalconditions in these studies and SCE harvesting effectivenessdrops significantly, in some cases was less than that of thepassive conditioner [8, 17, 19].

This paper analyzes a VEH with a commercially availablepiezoelectric generator and a passive conditioning circuit,here referred to as the “standard circuit,” and builds arelationship between vibration source characteristics anddesign considerations; it is organized as follows. Section 2provides background on the theory and architecture of

Transducer

Proof mass

Rigid clamp

Vibration source

Figure 1: Transducer, proof mass, and clamping mechanism underbase excitation.

energy harvesting devices and information on the harvesterimplemented in this study. Section 3 presents the powerharvesting results and design implications for sinusoidal,random, and SOR vibration sources.

2. Energy Harvesting Architecture

2.1. The Piezoelectric Generator. Materials exhibiting thepiezoelectric effect have been widely developed for applica-tion as actuators. Two prominent materials have emerged,lead-zirconate-titanate (PZT) and the macrofiber composite(MFC) developed by NASA [20, 21]. Sodano et al. [7] foundMFC transducers to have lower energy density and powergeneration than PZT. PZT conversion effectiveness is dueto the high electromechanical coupling factor and moduluselasticity [5]. RawPZTmaterial is very brittle and resonates atfrequencies significantly higher than that found in industrialapplication, making it difficult to implement for customharvesters. To lower the natural frequency and increaserobustness, the study [7] attached raw PZT to a much largeraluminum cantilever, which severely limits applicability.Thispaper assesses a commercially available Quick Pack PZTactuator as a transducer, the V25Wmodel produced byMideTechnology Corporation [22]. The V25W forms a bimorphcantilever plate; two piezoceramic layers are bonded arounda flexible dielectric center shim and between two laminateoutside layers for a compact and robust transducer [23]. Thetransducer is mounted with a clamping mechanism rigidlyattached to the vibration source. The clamp overlaps thepiezoelectric material to maximize active bending stressesand thus power generation in the transducer, as seen inFigure 1.

Seismic masses are attached to the transducer to tuneit to a desired natural frequency. Piezoelectric transducershave tunable frequency range due to physical limitations.Firstly, an appropriate transducermust be selected for a rangeof interests because the transducer cannot be tuned to afrequency higher than its bare natural frequency by addingmass. Secondly, the transducers typically have vibration dis-placement limitations to prevent damaging the piezoelectricmaterial, which imposes a lower limit on the tunable frequen-cies. The V25W has an approximate tunable range 50Hz to125Hz with a maximum peak to peak tip displacement of0.15 in. [23]. Modal analysis techniques are used to determine

Advances in Acoustics and Vibration 3

Piezo

I

VCr

RL VDC

Figure 2: Standard power conditioning circuit adopted from Lefeu-vre et al. [4].

the natural frequencies of the transducer for a given proofmass configuration. The analysis is conducted by providinga very short impulse from the shaker to the clamped baseof the cantilever and measuring the response as the system“rings out.” Without any mass, the bare transducer has anatural frequency of 124.5Hz; in this paper, specifying abare transducer implies a 124.5Hz natural frequency and viceversa. Proof masses used in this study are of comparable sizeand weight to the generator itself, as seen in Figure 1. Sincethe piezoelectric material spans the entire width and lengthof the generator, the attachment of mass may alter the modeshape of the beam for larger displacements, thus, affecting thebeam dynamics and harvested power.

2.2. Power Conditioning Circuit. Piezoelectric generatorsprovide an alternating current (AC) signal as they oscillatein response to vibration, while microelectronics typicallyrequire direct current (DC).The standard conditioner imple-mented in this paper incorporates two components in parallelwith the load, a full wave rectifier for current conversionand a temporary storage capacitor to smooth out the signalof the time-dependent input as shown in Figure 2. Theload represents the target electronics that require powerand is here considered part of the conditioning circuit.The electronic load characteristics and power demand ofapplication define the input power needed to supply thesystem. Low power target circuits are often represented aspurely resistive loads, although they are likely significantlymore complex.Thenet transfer of energy is null through idealtransient circuit components such as capacitors and inductorsas they do not dissipate energy. Real transient componentstend to leak energy, but this loss can often be represented as avoltage drop and thus as an equivalent resistive load; diodesand transistors can similarly be modeled with equivalentresistances [24]. This study holds the load capacitance 𝐶

𝑟

constant at 110 𝜇F and alters the resistance of a variableresistive load 𝑅

𝐿to determine the optimum impedance and

the maximum available power. Measuring the voltage acrossthe resistor and using Ohm’s law allow for calculation ofthe power delivered to the load, otherwise referred to asharvested power.

Devices harvesting from intermittent vibration sourceslike helicopters or other vehicles might not be able tosupply power to the load for extended periods of time. Inthis case one might look to charging secondary batteries.

Rechargeable batteries have varying voltage demands andcan absorb varying amounts of power [25]. Rechargeablebatteries follow drastically different circuit characteristicsthan resistors and can adapt intake current with availablepower. NiMH batteries have high energy density and chargearound 1.2 V; however, low voltage applications are limited[26]. Other cathode compositions such as lithium and thinfilm solid-state batteries can continuously provide upwardsof 4V but require strict charge control circuitry [25]. Purelyresistive loads allow one to find the maximum availablepower, while implementation of control components such ascharge pumps, boost converters, and comparators to meetspecific load demands always dissipates extra-power.

Harvesting a significant amount of themechanical energycauses a damping effect on the electromechanical system thatalters vibration amplitude [27–29]. Through the backwarddynamic effects of the electromechanical system, there areoptimal circuit characteristics for maximum power extrac-tion; here we vary resistance. For instance, transducers withhigh electromechanical coupling factors can saturate powergeneration at certain impedance values, causing a dip inpower generation, which can lead tomore than one optimumimpedance or frequency value [4, 11, 16]. Additionally, Liaoet al. [11] showed that other parameters such as oscillation fre-quency and circuit characteristics within the transducer alsoaffect the harvesting efficiency. Multiple equivalent circuitmodels for piezoelectric transducers have been developed[1, 8, 27]. The most common representation is a currentsource modeled in parallel with an internal capacitanceand resistance. The transfer of power between interactingcircuits is maximized when their equations decouple, thatis, when their impedances match. For systems with vary-ing input vibration or significant broadband characteristics,conditioning circuits that update their impedance with timecould increase power conversion for deviations from optimaloperating conditions [28]. However, self-updating powerconditioning is out of the scope of this study.

2.3. Vibration Source. The input power for the followingexperimentation is provided by the LDS V721/2–PA 1000Lshaker table. A closed-loop algorithm in software developedby LDS-Dactron controls the shaker table vibration. Withthe generator clamped and mounted to the shaker table, asseen in Figure 1, distributed dynamic inertial loads developon the cantilever as it vibrates in response to the inputforcing. A PCB Piezotronics impedance head model 288D01measures the acceleration and force at the base of thetransducer. Knowledge of these variables at the base allowsfor calculation of the input force when needed [30].TheVEHis tested under three different vibration signal conditions:sinusoidal, broadband, and sine tones superimposed on abroadband profile (SOR). Additionally, shock impulses willbe used in modal analysis to allow for transducer tuning andparameter identification. Although purely sinusoidal vibra-tions are rarely seen in application, studying the responseto sinusoidal input provides a starting point for harvesterdevelopment and allows for model validation. Similarly, flatrandom vibrations are rare in application, but investigatingthe ability of a harvester to capture broadband vibration


similarly aides in development. Most real-world ambientvibration sources can be faithfully reproduced with a properSOR profile, significantly aiding in harvester development forunique application.

2.4. Benchmark Theoretical Calculations. Exploitation of thepiezoelectric effect as a source of electric power generationmeans that more mechanical strain produces more useableelectrical energy. Sheets of piezoelectric material producethe largest strain when vibrating at the cantilever natu-ral frequencies, where displacements, and thus strains, arelargest.This paper considers only designing for vibration nearthe first natural frequency of the cantilever, which can bewell modeled in one dimension [4, 29, 31, 32]. Mechanicalvibration theory and the piezoelectricity constitutive equa-tion create the following electromechanical system for thepiezoelectric generator [4, 8, 17, 29, 32]:

𝐹ext (𝑡) = 𝑀�� (𝑡) + 𝐶�� (𝑡) + 𝐾𝑥 (𝑡) − 𝑘em𝑉 (𝑡) , (1a)

𝑄 (𝑡) = 𝜅𝑝𝑥 (𝑡) + 𝐶

𝑝𝑉 (𝑡) , (1b)

where 𝑀, 𝐶, and 𝐾 are the modal mechanical mass, modalmechanical damping coefficient, and the modal mechanicalstiffness, respectively. 𝜅

𝑝is the electromechanical coupling,

𝐶𝑝is the piezoelectric material capacitance, and 𝐹ext(𝑡) is the

external forcing function.As shown in (1a) and (1b), available energy is depen-

dent on both the induced strain and the electromechanicalcoupling factor. The electromechanical coupling factor is anintrinsic piezoelectric material property and a measure ofconversion between electrical and mechanical energy. Ignor-ing the electromechanical coupling factor, 𝜅

𝑝, linear vibration

theory allows for simple natural frequency and responsedisplacement calculation from (1a); however, ignoring elec-tromechanical coupling neglects backward dynamics of theVEH and would render inaccurate theoretical predictions[8]. Furthermore, deriving and simulating the responses ofa unique full coupled system requires accurate knowledge ofgeometries and circuit characteristics within the transducerand/or the use of numerical simulation packages or numeri-cal differential equation solvers [5, 11, 12, 16, 17, 27, 29, 32, 33].

Other studies employing various fundamental assump-tions about the piezoelectric electromechanical system andinput vibration have devised simpler analytical models forpredicting piezoelectric power generation [4, 31, 34]. Com-mon among almost every analytical and computational studyare limiting assumptions of the external vibration, such assteady-state sinusoidal vibration and broadband vibrations ofconstant spectral density, which allow for simplified vibrationand circuit analysis. Unfortunately, vibrations experienced inapplication are very rare in such convenient forms, whichmay compromise the validity of such calculations whenapplied to more complex and realistic scenarios. This studyuses two approximate analytical models for benchmark com-parison with experimental results, one for steady-state sinu-soidal vibration and the other for flat broadband vibration.

Rigid mass MF

u

I

V

Damper C

Structure stiffness KsPiezoelectric

element

Figure 3: Simple one dimensional dynamic model for the elec-tromenchanical system utilized by Lefeuvre et al. [4]. A similarmodel was used by Halvorsen [31].

Corresponding simple analytical models are not available formore complex vibration inputs such as SOR.

Lefeuvre et al. employed a one-dimensional lumpedmass, spring, damper, and piezoelectric element model ofthe piezoelectric generators, as seen in Figure 3, to developsimple analytical formulae for harvesting capabilities understeady state harmonic vibration. Equation (2) gives the powerdelivered to the load (harvested power) for this model whenconsidering the standard circuit in Figure 2. This simplemodel approximates the responsewell when considering onlyone mode of vibration and small (linear) displacements [4]:

𝑃sin =𝑅𝐿𝛼2

𝜔2

𝑛

(𝑅𝐿𝐶0𝜔𝑛+ 𝜋/2)

𝑈2

𝑀. (2)

𝐶0and 𝛼 are the internal capacitance of the piezoelectric

generator and the force factor, calculated from the ratioof forcing amplitude to voltage output as described in [4],respectively. 𝑅

𝐿, 𝜔, and 𝑈

𝑀are the load resistance, driving

angular frequency and generator tip displacement, respec-tively. Even with the simplified electromechanical model, itis not unreasonable to expect the model to fit the simplesinusoidal data well at low amplitudes. In this regime, theinconsistencies between the models are minimized, such asmode shape alterations, imperfect clamp rigidity, and strayelectrical losses. In general, the theoretical data is expectedto be higher than measured data since the model doesnot account for electric losses of the bridge diodes or loadcapacitor.

Halvorsen derived the output power from a similar one-dimensional lumped mass, spring, damper, and piezoelectricelement model when subjected to white noise [31]. Thisapproximation is valid only in flat spectrum (constant spec-tral density) broadband scenarios with small displacements:

𝑃ran =1

2𝑀𝑆𝑎

𝑟𝜅2

𝑝𝑄

1 + (1/𝑄 + 𝜅2𝑝𝑄) 𝑟 + 𝑟2

. (3)

In (3), 𝑟 = 𝑅𝐿𝜔𝐶0,𝑀0is the inertial mass, and𝑄 is the quality

factor of the cantilever beam system. As in [17, 31], the modal


Table 1: System parameters.

Symbol Representation Value Unit𝑓𝑛 Natural frequency 58.3l; 124.5h Hz

𝑓𝑎 Antiresonant frequency 60.1l; 127.5h Hz

𝑀 Modal mass 9.5l; 1.1h g𝑄 Quality factor 60l; 120h —𝛼 Force factor 4.8 ∗ 10

−3 NV−1

𝜅𝑝

Electromechanicalcoupling factor 0.92 %

𝐶𝑝 Clamped capacitance 130 nf

𝐶𝑟 Load capacitance 600 𝜇f

The superscripts l, standing for low, and h, standing for high, represent thecorresponding values for the frequencies near the lower and upper boundsof tunable range, 58.3Hz and 124.5Hz, respectively.

mass is determined as a unimorph cantilever beam with zerotaper.One important difference from themodel derived in [4]is that (3) is derived for a standard AC conditioning circuit,with no rectifier or filter capacitor. The calculated harvestedpower results in (3) are still compatible with the experimentalsystem described in this section since the net power throughthe capacitor is null, save for stray electrical losses. In contrastto the harmonic model, random vibration has some inherentinconsistencies between models and they are not expected toagree as well. Parameters like modal mass and damping willalter with the dominant input frequencies, which fluctuatesignificantly in random vibration. Additionally, statisticalvariations in the input vibration from one moment to thenext can lead to drastic changes in displacement amplitudeand harvested power, and thus averaging is common practice[17, 19].

Due to the limited knowledge of exact internal geometriesand circuit characteristics of an externally manufacturedgenerator, all parameters necessary for experimental analysisand theoretical calculation are determined experimentallyor read directly from data sheets provided by the manufac-turer. With the theoretical considerations discussed herein,the electromechanical system presented in this section canbe fully described at specified natural frequencies by theparameters in Table 1.

3. Vibration Source Testing and Results

This section discusses the experimental and approximatetheoretical analyses of the energy harvesting architecturedescribed in Section 2. Power is harvested by the standardcircuit from a V25W piezoelectric generator subjected tovarying harmonic, random, and SOR vibration scenarios.

As in most experimental applications, this paper char-acterizes source vibrations by acceleration in the time andfrequency domains. In the following tests, input vibrationamplitude refers to the input acceleration amplitude at thebase. Controlling the vibration by acceleration allows forsubsequent experimental validation by omitting the correc-tion needed for input voltage variations for different shakers.However, this testing method assumes that the attachment

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2

Har

veste

d po

wer

(mW

)

Amplitude (g)

58.3 Hz124.5 Hz

58.3 Hz-theory124.5 Hz-theory

Figure 4: Theoretical and experimental harvested power as afunction of sinusoidal input amplitude for two natural frequenciesand constant 20 kΩ load.

of the harvester does not alter the source vibration char-acteristics or vice versa, which is a good assumption forshaker armatures masses and vibration sources much largerthan the harvester [35]. One should also note that inputpower is not constant for identical acceleration amplitudes atdifferent sinusoidal frequencies or bandwidths, but the inputmechanical power is not designed against because it is of nocost to the user in application.

3.1. Harmonic Excitation. Power harvesting under steady-state harmonic excitation is tested at multiple resonant fre-quencies, resistance (impedance) values, and vibration ampli-tudes. Unless otherwise stated, the harvester is driven at thetransducer natural frequency. Integration of the accelerationresponse, to consider displacement, reveals two importantresults for consideration in harvester design.

Firstly, the resulting displacement amplitude scalesapproximately linearly with input acceleration amplitudefor a given driving frequency. This approximation applieswhen operating within the small transducer displacementlimits. Applying Ohm’s law for power through a resistorresults in the harvested power scaling with the square of theinput acceleration amplitude [4, 5, 27, 29]. Figure 4 showsthe expected quadratic trend when tuned to 58.3Hz and124.5Hz, while operating at constant load resistance, 𝑅

𝐿=

20 kΩ. The trend at 58.3Hz is limited to 0.6 g amplitudeto avoid possibly exceeding the displacement limits anddamaging the transducer. Secondly, resulting transducer tipdisplacement amplitude scales inversely with the square ofthe tuned natural frequency. Clearly, the harvested poweris significantly larger at the tuned frequency than at thebare natural frequency for similar vibration amplitudes. Thisimportant result indicates that it is advantageous to tunethe harvester to lower frequencies for vibration sources withmultiple sine tones of similar acceleration amplitude.


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

1

2

3

4

5

6

7

8

0 20 40 60 80 100 120

58.3 Hz58.3 Hz-theory33 kOhm

124.5 Hz124.5 Hz-theory15.5 kOhm

Har

veste

d po

wer

@fn=58.3

Hz (

mW

)

Har

veste

d po

wer

@fn=124.5

Hz (

mW

)

Load resistance (kΩ)

Figure 5: Harvested power as a function of load resistance for twonatural frequencies at 0.6 g input amplitude.

The theoretical model and experimental data agree wellat low driving amplitudes at both frequencies. At higheramplitudes, the expected differences between the modeland experimental harvested power are more evident. Thetheoretical power trend at 58.3Hz has a slightly differentshape than that of measured power; most consequentially,theory predicted less harvested power than was measured.Considering that the theoretical predictions should be higherthan measured in all cases due to the omission of electriclosses, this disagreement suggests that the approximatemodelderived in [4] may not be a good fit for the architecture usedhere.

Another important design consideration is the optimaloperational resistance. Figure 5 displays harvested powerversus load resistance at 0.6 g acceleration amplitude at58.3Hz and 124.5Hz. Additionally, the optimal load resis-tances (maximum power generation) are marked verticallines. Theoretical maximum harvested power is seen at15.5 kΩ for the bare natural frequency and at about 33.0 kΩ for58.3Hz natural frequency.The optimal resistance predictionsfit the experimental data well as the data reported maximumvalues at 15 kΩ and 40 kΩ, respectively, although the exper-imental resolution is quite coarse. Analytically, an inverserelation between optimal resistance and natural frequency isexpected, and both findings agree [1, 11, 12, 15].The theoreticaltrends provide a similar over all shape to the experimentaldata, but over predict harvested power by as much as 17%(approximately 1.5mW) at 58.3Hz and 28% (approximately0.5mW) at 124.5Hz. Investingmore testing time for increasedresolution would be beneficial for more accurate optimalresistance determination in application.

Additionally, one will notice a dip in harvested powerwith optimal load resistance for the bare transducer.The dropin harvested power was seen at all vibration amplitudes islikely the saturation effect seen by Liao et al. [11] and Renno etal. [16] due to an electromechanical coupling and electrically

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4Deviation from natural frequency (Hz)

58.3 Hz85 Hz124.5 Hz

Frac

tion

of m

axim

um p

ower

(—)

−4 −3 −2 −1

Figure 6: Harvested power FRF for three natural frequencies atoptimal resistance and 0.6 g amplitude.

induced mechanical damping. The absence of saturation atlower frequencies may be due to the increase in availablemechanical power at lower frequencies with similar vibrationamplitudes.

Figure 6 depicts the frequency response function (FRF)relative to the maximum harvested power for 58.3, 85, and124.5Hz natural frequencies. Deviations in driving frequencyfrom the tuned natural frequency have a significant effecton harvested power. Generated power drops by 50% withinapproximately 1Hz and 90%within approximately 3Hz for allfrequencies. This frequency dependency establishes a usablebandwidth for the transducer of just over one cycle persecond, bidirectionally, outside of which power generationis severely limited. The FRF at 124.5Hz is slightly wideron the upper limits, which suggests that the FRF becomesmore generous with higher resonant frequencies but is nearlynegligible in the available frequency range.ThenarrowFRFofthe transducer reinforces the importance of accurate tuningin the design process.Theoretical data trends are not availablefor the frequency response function since (2) assumes that theharvester is driven at its natural frequency.

Designing for harvesting maximum power from a sinu-soidal source requires weighing the gains from sine compo-nent amplitude with those from tuning to lower frequenciesand finding the optimal conditioner impedance.

3.2. Random Excitation. The titles, broadband and randomexcitation, are often interchanged; however, random vibra-tions need not be broad. In application, random excitationrefers to vibration inputs incorporating a finite number(dependent on resolution) of oscillation frequencies overa specified bandwidth. Naturally, random vibrations arecharacterized differently than sinusoidal. Most commonly,random vibrations are represented by an acceleration powerspectral density (PSD) profile with units of g2/Hz. Integratingthe PSD over a frequency range and taking the square root


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Aver

age h

arve

sted

pow

er (m

W)

58.3 Hz58.3 Hz-Mod. theory58.3 Hz-Halv. theory

124.5 Hz124.5 Hz-Mod. theory124.5 Hz-Halv. theory

Spectral density (g2/Hz)

Figure 7: Average harvested power versus spectral density for500Hz bandwidth and constant 40 kΩ resistance.

result in the root mean square (RMS) level of vibration forthat filtered frequency range. In this study, the harvesteris subjected to PSD profiles of varying bandwidth. A newrandom signal, centered at the transducer natural frequency,is generated for each data point to avoid random statisticalcorrelations. Due to statistical and time variations, harvestedpower is averaged over a 100-second sample for each datapoint to increase repeatability [17, 19].

Halvorsen [31] derived (3) as a direct formula for har-vested power under white noise vibration input from proper-ties of the generator. The term with 𝑟, 𝜅

𝑝, and 𝑄 in this equa-

tion is always less than or equal to one; thus, an even simplerformula for maximum power can be extracted: 𝑃 ≤ 1/2𝑚𝑆

𝑎

[31]. In order to predict at least as much harvested power aswas measured at 𝑆

𝑎= 1𝑒 − 3 (seen in Figure 7), a modal mass

of at least 0.2 kg would be required at 124.5Hz. However, sucha mass is on the order one hundred times larger than themass of the generator and proof mass, thus a scaling factorwas considered for this study. The scaling factor required topredict reasonable values for harvested power increased withnatural frequency. It was found that multiplying (3) by thetransducer resonant frequency produced results more closelyaligned to experiment. Figure 7 shows both theoretical resultsfrom (3), directly from Halvorsen, and the modified model,scaled as mentioned previously, alongside the experimentalresults measured when subjected to white noise at variousspectral densities, set to a load resistance of 40 kΩ, and tunedto two natural frequencies. Each data point was subjected toflat spectrum random vibration with 500Hz bandwidth.

As expected, harvested power scales linearly with spectraldensity, in contrast with quadratic scaling for sinusoidalvibration. The theoretical predictions from (3), representedby the dotted lines in Figure 7, are on the order of 50 and110 times smaller than the experimental data at 58.3Hz and

0

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0.015

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0.025

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0.01

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0.08

0 20 40 60 80 100 120

58. Hz-theory58.3 Hz63 kOhm

124.5 Hz-theory124.5 Hz28.8 kOhm

Aver

aged

har

veste

d po

wer

@fn=58.3

Hz

Aver

aged

har

veste

d po

wer

@fn=124.5

Hz


Figure 8: Harvested power versus load resistance for two frequen-cies, 𝑆

𝑎= 1𝑒 − 3.

124.5Hz, respectively, while the scaled theoretical model,represented by the dashed lines, are on the same order ofmagnitude. The modified model over predicts the harvestedpower by approximately 30% and 20% for 58.3Hz, and124.5Hz respectively. This over prediction is reasonable dueto the narrow resonant bandwidth of PZT piezoelectrics andthe unpredictability of random vibration that lead to lowharvested power.

Additionally, optimal load resistance is altered for ran-dom vibration from that of sinusoidal vibration. This vari-ation in optimal resistance is predicted in similar theoret-ical derivations and seen in numerical simulations [5, 17,19]. Figure 8 shows harvested power as a function of loadresistance random excitation at the two natural frequencies.Theory predicted an optimal resistance of about 63.0 kΩ and28.8 kΩ for 58.3Hz and 124.5Hz, respectively.This predictionagreed with the measured optimal resistance well at 58.3Hzbut did not agree with an approximate optimal resistance of45 kΩ at 124.5Hz.

Figure 9 compares harvested power at varying band-widths for three natural frequencies. In this case, each inputsignal has a spectral density of 10

−3 g2/Hz. With constantspectral density, input power increases proportional to thesquare-root of the bandwidth, but this is not seen in themeasured harvested power data. Except for random statisticaldeviations from one test point to the next [17], the harvestedpower remained invariant with bandwidth. We also seethat the harvested power is again inversely proportional toresonant frequency. Moreover, spectral densities outside ofthe transducer bandwidth negligibly influence the harvestedpower. To illustrate this, the harvester is subjected to varyingrandom vibration profiles that are 1𝑒 − 4 g2/Hz near thetransducer useable bandwidth of the bare transducer, asdepicted in Figure 10. The harvester yielded averaged power


0.01

0.1

1

1 10 100 1000

Aver

age h

arve

sted

pow

er (m

W)

Bandwidth (ΔHz)

58.3 Hz85 Hz124.5 Hz

Figure 9: Harvested power versus bandwidth for three naturalfrequencies at 1𝑒 − 3 spectral density.

60 80 100 120 140 160 180Frequency (Hz)

FlatGradualSharp

10−3

10−4

Acce

lera

tion

spec

tral

den

sity

(g2/H

/z)

Figure 10: Random profiles, all centered at 124.5Hz with 1𝑒 −

4 spectral density, with varying densities outside the transducerbandwidth.

of 0.65, 0.67, 0.71 𝜇W, respectively, which is accounted for bystatistical variations between tests [17].

In addition to low average energy conversion efficiencywhen compared to sinusoidal vibration, power harvestingfrom random or noise vibration is severely limited in appli-cation due to corresponding increases in voltage fluctua-tions with spectral density. Such voltage functions are notaccounted for in any theoretical predictions and must bedesigned against in application as they can damage targetelectronics. Figure 11 depicts 50 second samples of voltagesignal supplied to the load for varying spectral densities

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

Time (s)

Load

supp

ly v

olta

ge (V

)

0.0005 (g2/Hz)0.00025 (g2/Hz)

0.0025 (g2/Hz)0.005 (g2/Hz)

0.001 (g2/Hz)

Figure 11: 50 second sample time responses for 500Hz bandwidthrandom signals supplied to a transducer tuned to 58.3Hz at varyingspectral densities.

supplied to the transducer tuned to 58.3Hz. Maximum tominimum voltage fluctuations increases from 0.6V at 2.5𝑒 −

4 g2/Hz to 3.5 V at 5𝑒 − 3 g2/Hz; above and below which,scale larger and smaller respectively. Lefeuvre et al. reportedvoltage fluctuations of over 10V in order to complete afull implementation of the standard conditioner [19]. Thefilter capacitor in the standard circuit is able to smoothout oscillations on a small time scale; however, larger-scalechanges in vibration require more attention if they were to beattenuated.

These results exhibit that the average power harvestedfrom random vibrations is dependent only on three factors:the transducer natural frequency, the load resistance, andthe spectral density near the natural frequency. In order toharvest significant power from random vibrations, ratherlarge input vibrations are required.Moreover, the power gainsare overshadowed by the implications of large load supplyvoltage fluctuations induced at large spectral densities. Whentuned to the low endof the transducer frequency range,whichproduced more power, the harvester produced less than halfof amilliwatt of power at a spectral density 5𝑒−3 g2/Hz,whichis 10 to 100 time higher than ambient spectral densities seenin many industrial and vehicular applications [7, 18, 36].

3.3. Sine on Random Excitation. Purely sinusoidal or randomvibrations are idealized simplifications of ambient vibration.Superposition of sinusoidal signals on broadband profilesallows for more accurate representation of environmentalvibrations experienced in application like the Apache heli-copter response seen in [18]. However, SOR vibration testsrequire control of more parameters; the driving frequency,amplitude, and number of sine tones need to be specified inaddition to the random profile shape and density. Using theresults of Section 2, experimental testing can be simplified.Power and voltage magnitudes scale directly with amplitudeand inversely with natural frequency, and spectral densities


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Aver

age p

ower

(mW

)

58.3 Hz-random58.3 Hz-SOR58.3 Hz-theory

124.5 Hz-random124.5 Hz-SOR124.5 Hz-theory

Spectral density (g2/Hz)

Figure 12: Average harvested power versus spectral density forrandom and SOR with sinusoidal power subtracted.

far from the transducer bandwidth negligibly influence har-vesting results. A random profile is implemented flat with150Hz bandwidth and, as with the random vibration tests,SOR harvested power is calculated over the time averageof a 100-second signal sample to increase experimentalrepeatability.

Driving the system with both sinusoidal and randomcomponents increases the input power within the bandwidthof the transducer, which in turn should increase the out-put power of the harvester. Figure 12 shows the increasein average harvested power for an SOR profile with 0.3 gsinusoidal component input and varying spectral densitiesagainst the harvested power of random vibration alone. Thespectral density profiles are plotted in Figure 13. Evidently, theharvested power increaseswith spectral density and thus boththe sinusoidal and random vibration components contributeto harvested power. Linear superposition of both componentsover infinite time would yield random vibration and spectraldensity contributions in SOR (sinusoidal power subtractedfrom the total power) having the same results in Figure 12:thus, theoretical calculations of harvested power from ran-dom vibration serve as a benchmark in this case as well.Time-dependent fluctuations and imperfect superpositionin the control software render varying results, especially atlarge spectral densities. Likewise, input voltage fluctuationsincrease with spectral density as well. In a similar manner toFigure 11, voltage fluctuations at 58.3Hz increase from about0.6V to 3.8 V as spectral density component increases from2.5𝑒 − 4 g2/Hz to 5𝑒 − 3 g2/Hz, which must be consideredin harvester design. Altered power generation seen in thesecases would be overlooked when simply ignoring the spectralcontent in application.

Tests conducted with varying sinusoidal componentamplitudes yielded nearly identical results for increases in

40 45 50 55 60 65 70 75Frequency (Hz)

10−1

10−2

10−3

10−4Acce

lera

tion

spec

tral

den

sity

(g2/H

z)

2.5e − 4 (g2/Hz)5e − 4 (g2/Hz)1e − 3 (g2/Hz)

2.5e − 3 (g2/Hz)5e − 3 (g2/Hz)

Figure 13: PSD of SOR input signals in Figure 17 for 58.3Hz.

average power and induced voltage fluctuations, as seenpreviously. The significance of power gains due to spectraldensity, however, remains relative to the sinusoidal amplitude(i.e., a 100 𝜇W increase in power from random content ismore significant at 0.1 g than at 0.6 g sinusoid amplitude),while voltage fluctuations remain equally significant in bothcases. As with harvesting from purely random vibrations,power gains from spectral content are dwarfed by thedemands of induced voltage fluctuation and correspondingcontrol circuitry power.

Sinusoidal and random vibration results revealed a gapin the dependence of optimal load resistance due to sourcevibration type. This variation is bridged when incorporatingboth sinusoidal and random components. This could furtherbe expanded upon by incorporating nonlinear and transientinputs found in application; however, such is beyond thefocus of this paper. Figure 14 shows how the optimal loadresistance varies with spectral density over a constant sinecomponent. The optimal load resistance increases with spec-tral density and the response shifts away from sinusoidalcomponent dominance, until it reaches the purely randomoptimal load resistance. This trend suggests that the optimalresistance shift is directly dependent on the ratio of spectraldensity to sinusoidal component. Interactions between thetwo components render harmonic and randompredictions ofoptimal resistance inaccurate for significant levels of spectraldensity.

Complex environmental vibration sources often com-pound numerous components with unique vibration signa-tures that result in sinusoidal components in close proximity.Flight tests for an Apache helicopter conducted in [9] showthat vibrations from the main and tail rotors give dominantsine tones of similar magnitude within approximately 2Hz ofeach other. The FRF for the transducer shows that interac-tions between sine tones sufficiently close is nonnegligible.Interactions between two nearby sine tones in the sourcevibration causes a significant “beating” in source vibrationamplitude as the two tones go in and out of phase and also


0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 10 20 30 40 50 60

0.3 g sine only

15.5 kOhm

25 kOhm

30 kOhm

40 kOhm

45 kOhm

Frac

tion

of m

axim

um p

ower

(—)


0.3 g, 1e − 5g2/Hz

0.3 g, 1e − 4g2/Hz

0.3 g, 1e − 3g2/Hz

0.3 g, 1e − 2g2/Hz

1e − 3g2/Hz only

Figure 14: Percent of maximum harvested power within each seriesas a function of load resistance under sinusoidal and spectral densityinputs to the bare transducer.

0 1 2 3 4 5 6 7 80

0.10.20.30.40.50.60.70.80.9

1

Time (s)

Arb

itrar

y in

put a

mpl

itude

(—)

Figure 15: Input vibration of arbitrary acceleration units.

increase the input vibration RMS value in the bandwidth ofthe transducer.

Figures 15 and 16 show base input vibration and resultingload voltage waveforms experienced in the harvester, respec-tively, over two beating periods for two superposed sine tonesof 0.3 g amplitude: one on the bare natural frequency andone 0.25Hz above. In Figure 17, 𝑉

1and 𝑉

2correspond to

the voltage at the transducer electrode terminals while 𝑉𝑐

is the voltage delivered to the load. The electromechanicalcoupling factor alters the transducer vibration dynamics andvoltage processing, resulting in nonlinear voltage waveformsdelivered to the load. The charge accumulated on the capac-itor prevents the voltage delivered to the load from droppingto zero as vibration drops to zero and in turn alters the

0 1 2 3 4 5 6 7 80

0.20.40.60.8

11.21.41.61.8

2

Time (s)

Frac

tion

of si

nuso

idal

vol

tage

(—)

V1V2Vc

Figure 16: Load voltage response at 0.3 g amplitude for twosinusoidal components 0.25Hz apart.

0

0.2

0.4

0.6

0.8

1

1.2

1

1.05

1.1

1.15

1.2

1.25

1.3

0 1 2 3 4 5 6 7 8 9

Beat

ing

ampl

itude

as a

frac

tion

of si

nuso

idal

vol

tage

(—)

Aver

age p

ower

as fr

actio

n of

sinu

soid

al p

ower

( —)

Harvested powerBeating amplitude

ΔF (Hz)

Figure 17: Average harvested power as a fraction of sinusoidalharvested power and beating amplitude as a fraction of sinusoidalvoltage.

piezoelectric supply voltages,𝑉1and𝑉2.The voltage delivered

to the load fluctuates by 2.61 V from crest to trough, whichis just about 100% of the voltage supplied by a 0.3 g singlesine tone at this natural frequency. Additionally, the averageharvested power is 0.47mW, or about 28% higher than withjust a single sine tone. Assumptions of the input vibrationcommonly afforded to theoretical calculations and numericalsimulations would not account for these results as theyconsider the harvester to be driven at only one frequency ora spectral content of uniform density.

The FRF of the transducer (Figure 6) indicates thatbeating amplitude should approach zero and harvested powershould approach that of a sinusoid as the difference betweenthe two tones increases; Figure 17 exhibits these trends. Qual-itatively, as the secondary frequency recedes, its individual


0 10 20 30 40 50 60 70 80 90 1000

0.51

1.52

2.53

3.54

4.55

Time (s)

Volta

ge (V

)

Three sinusoidal components supplied to bare transducer

Figure 18: Resulting waveform for three unrelated sinusoidalcomponents near the transducer natural frequency.

impact on the harvester drops with the transducer FRF; thus,the phase and spectral contributions decay. By approximatelytwo Hz separation, the average power delivered to the loadis within a few percent of that of a single sinusoid and thebeating amplitude is less than 20% of the sinusoidal voltage.For this case, the peak-to-peak beating amplitude above2Hz separation is less than 0.35V, which is not damagingto most electronics. However, interactions from sine tonesseparated by more than two Hz could become significantbecause both the average power delivered to the load and thebeating amplitude scale directly with sinusoidal componentamplitude and inversely with natural frequency.

Incorporating more sinusoidal components within thebandwidth of the transducer increases the harvested powerand voltage fluctuations. Figure 18 exhibits the resulting loadsupply voltage waveforms for three unrelated 0.3 g sine com-ponents within the bandwidth of the transducer. In this case,the bare transducer saw voltage fluctuations of more than 3V,which only increase in amplitude as the tuning frequencylowers. These results exemplify the results of fluctuatingvoltages on target electronics due to interactions of a finitenumber sine tones, as might be seen in complex vibrationsources like the Apache helicopter [18]. As the number ofsine components within a specified bandwidth increases, thevibration begins to characterize random vibrations and lessorderly voltage fluctuations arise.

Harvesting power from complex vibration sourcesrequires incorporation of more characteristic factors thanidealized sinusoidal or random vibrations. As noted inSection 2, a strongly time-varying DC signal is disadvan-tageous and potentially damaging in application. Voltagecontrol circuitry can be implemented to stabilize the powerprovided to the load at the cost of power loss. Implementinga much larger filter capacitor or additional super capacitor asstorage will reduce the beating amplitude and random varia-tions due to spectral content but at the cost of lower voltages.Power or voltage loss from implementation of additionalcontrol circuitry to maintain specified voltage requirementsmust be weighed against power gains of harnessing vibrationfrom complex source vibrations. Otherwise, one can attemptto minimize such consequences by harvesting from isolatedsinusoidal components with relatively low spectral content.

4. Conclusion

This paper experimentally investigated the ambient powerharvesting ability of a commercially available piezoelectricvibration powered generator with a standard conditioningcircuit for sinusoidal, random, and SOR excitations. Testingwas conducted by characterizing the source vibration by itsacceleration response, experimentally determining pertinentelectrical and mechanical properties and measuring powerdelivered to the electric load resistor. The presented resultsfor the idealized sinusoidal and random excitations are com-pared against previous theoretical predictions and numericalexperiments for validation.

As was predicted analytically, the results show that lowertransducer natural frequencies result in more harvestedpower for similar input acceleration amplitudes. Optimalload resistance varies with both natural frequency and vibra-tion source characteristics. Vibrations with large spectralcontent near the transducer natural frequency inducedhigheraverage power and significant fluctuations in voltage suppliedto the load. Significant voltage oscillations were also seenwhen multiple sinusoidal components are in close proximity.

The results presented in this paper indicate that, whileuseful in select cases, considering only purely sinusoidal orrandom vibratory environments can severely limit harvesterperformance in application and can bring upon unexpectedand undesired consequences when applied too generally.Current theoretical and numerical simulation results areshown to predict incorrect results due to assumptions madeabout the input vibration and electromechanical model.The commercialization of vibration energy harvesting andapplication of such to complex vibration environmentsmakesSOR testing an invaluable tool in harvester developmentas one can implement configurations to accurately recreatealmost any ambient vibratory scenario.

Acknowledgment

This work was supported by the US Army Joint AttackMunition Systems (JAMS) Project Management Office underInteragency Agreement AGRD19D1021D1.

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