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An energy harvester driven by colored noise

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2011 Smart Mater. Struct. 20 025011

(http://iopscience.iop.org/0964-1726/20/2/025011)

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IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 20 (2011) 025011 (6pp) doi:10.1088/0964-1726/20/2/025011

An energy harvester driven by colorednoiseLars-Cyril Julin Blystad and Einar Halvorsen

Department of Micro and Nano Systems Technology, Faculty of Technology and MaritimeSciences, Vestfold University College, PO Box 2243, N-3103 Tønsberg, Norway

E-mail: Einar.Halvorsen@hive.no

Received 30 September 2010, in final form 9 December 2010Published 14 January 2011Online at stacks.iop.org/SMS/20/025011

AbstractThis paper presents experiments on a piezoelectric energy harvester driven by broadbandvibrations. The energy harvester is investigated both with and without a mechanical end stopthat limits proof mass motion. The end stop increases the effective bandwidth at largeacceleration amplitudes. For sinusoidal vibrations, the mechanical end stop causes the outputpower to reach a plateau at a critical acceleration amplitude and does not increase significantlywith increasing acceleration amplitude. In contrast, the output power increases with increasingspectral density of the broadband vibration, though at a smaller rate in the presence of an endstop. The optimal load for broadband vibrations is different from either of the two optimal loadsfor sinusoidal vibrations.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Piezoelectric transduction is one of the main technologiesfor vibration energy harvesting [1–3]. Most vibrationenergy harvesters reported in the literature are based onresonant behavior. This applies especially for piezoelectricones. A few electrostatic energy harvesters are genuinelynonresonant [4, 5]. Resonant energy harvesters are expectedto work in environments where the main frequency componentof the vibration matches the harvester’s resonance frequency.On the other hand it is expected in many applications that theambient vibrations fluctuate in amplitude and phase. Theycan even contain a band of frequencies with a relatively flatpower spectral density over a large bandwidth, comparedto the bandwidth of the energy harvester. It is thereforeof interest to design energy harvesters that can respondeffectively to more broadband vibrations. Several meanshave been proposed to extend the bandwidth of harvesters,such as multiple resonators [6, 7], beam prestress [8, 9],mechanical nonlinearities [10, 11] and magnetically inducednonlinearities [12].

The use of impacts on mechanical end stops to achievea bandwidth extension appears particularly interesting forseveral reasons. Mechanical end stops that limit proof massmotion must be present, either to avoid beam fracture orbecause space is limited. The method requires no control

overhead. It is completely passive. Piezoelectric beamstypically are multilayer structures with uniform thickness andwith a cross section that cannot easily be modified to designa nonlinear stiffness. Nonlinear springs pose at least afabrication challenge for other structures as well. An exceptionis high aspect ratio micromachined silicon devices with in-plane motion.

The promising use of impacts on mechanical end stopshas been demonstrated for an electromagnetic device [11, 13].It was subject to frequency sweeps and steps. A largeincrease of bandwidth on frequency up-sweeps was obtained.Simulation showed improved performance with end stopswhen the vibration frequency was randomly changed at regularintervals. A cantilever type vibro-impacting energy harvesterwith symmetrical piezoelectric end stops was driven by asinusoidal force and presented in [14]. Frequency sweepsshowed an increased bandwidth during vibro-impacts with thepiezoelectric end stops. The same group presented a similarharvester driven by a band-pass-filtered random waveformin [15]. They show that it is possible to harvest energy fromsuch an excitation, but do not present the correspondencebetween input spectral density and output power. Neither dothey compare the output power with a case without end stops.

It is useful to analyze an harvester system when it isdriven by broadband random vibrations in the form of whiteor colored noise. This gives a realistic impression of the

0964-1726/11/025011+06$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA1

Smart Mater. Struct. 20 (2011) 025011 L-C J Blystad and E Halvorsen

(a) (b)

Figure 1. The measurement set-up. (a) Schematic overview. (b) The lab set-up.

(a) (b)

Figure 2. (a) Piezoelectric beam with proof mass. (b) Implementation with a mechanical end stop on one side.

energy harvester response in random vibration environments.We previously noted spectral broadening in simulations of anelectrostatic energy harvester with an end stop on one side [16].Theoretical arguments showed that the influence of end stopson output power is dependent on the electrical loading [17].In [18] we simulated in detail an MEMS aluminum-nitride-based resonant energy harvester with both sinusoidal andbroadband excitation. The system response was investigatedboth with and without mechanical end stops present in thesystem.

In this paper we experimentally investigate some ofthe main phenomena predicted in our previous theoreticaland computational work. We present measurements on alead–zirconate–titanate-based resonant energy harvester withsinusoidal and broadband excitation. The effect of amechanical end stop is investigated.

2. Experimental set-up

National Instruments’ LabView was used in all measurementsas the core in our data acquisition system. An accelerometerfrom PCB Piezotronics Inc. (model 352A56A) together witha power amplifier (type 2706) and a vibration exciter (type4809) from Bruel and Kjær constituted the main components inthe set-up. The data acquisition system simultaneously loggedboth the base acceleration of the device and its output voltage.A schematic and a picture of the measurement set-up are shownin figures 1(a) and (b), respectively. The transducer element is

Table 1. Dimensions of the piezoelectric beam with steel mass.

Symbol Value Units Description

lp 31.8 mm Length of piezoelectric elementlb 10.6 mm Length of piezoelectric beamwb 12.7 mm Width of piezoelectric beamtb 0.51 mm Total thickness of bimorphtp 0.19 mm Thickness of piezoelectric layerslm 6.25 mm Length of proof masswm 14.95 mm Width of proof masstm 9.45 mm Thickness of proof massms 6.75 g Steel proof massm 7.07 g Total proof mass including part of the beam

a three-layer lead–zirconate–titanate piezoelectric generator oftype PSI-5A-S4-ENH [19]. It has two identical piezoelectriclayers on each side of a metal shim, i.e. it is a symmetricbimorph. The piezoelectric generator is a cantilever beampoled for parallel operation. The beam was glued to a frameand a steel proof mass was attached at its end (figure 2). Thetransducer dimensions are presented in figure 2(a) and table 1.

A mechanical end stop is implemented on one side by theuse of a metal plate attached to the frame. It was positionedjust above the piezoelectric beam (figure 2(b)). The verticalposition of the mechanical end stop can easily be adjusted.It was set such that, by applying a practically convenientexcitation force to the system, the end stop was hit. In ourcase, that was within the range of 0.1g. We kept the settingof the vertical position fixed through all our experiments with

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Smart Mater. Struct. 20 (2011) 025011 L-C J Blystad and E Halvorsen

the end stop. The edge of the proof mass then has the samedisplacement in all cases of impact on the end stop. Thedistance was estimated to be of the order of 50–100 μm.

A variable resistor was connected between the electrodesof the piezoelectric transducer. We measured the value RL ofthe resistor and the voltage VL across the load. It gives usthe instantaneous power P = V 2

L /RL delivered by the energyharvester.

3. Results and discussion

The short-circuit and open-circuit resonance frequencies weredetermined by driving the system with a 1 Hz low amplitudesquare pulse and determining the frequency of the peak inthe output voltage spectrum. The measurements were madeat very low excitation levels to ensure linear behavior. Witha low load resistance (60 �) and a high (729 k�) loadresistance, we found respectively the short-circuit resonancefrequency at fr = ωr/2π = 173.0 Hz and the open-circuitresonance frequency at fa = ωa/2π = 181.0 Hz. The deviceelectromechanical coupling factor k2 is then given by

k2 = 1 − ω2r

ω2a

= 8.6%. (1)

Both the open and the short-circuit quality factors wereestimated from the measurements to be around 100.

We stepped the load resistance and measured thecorresponding output power for three different driving signals,see figure 3. The vibrations were not strong enough todrive the proof mass into the mechanical end stop in theseexperiments. The optimum load resistances were found byidentifying the value at maximum output power. They wereRLR = 2.60 k� for sinusoidal vibrations at the short-circuitresonance, RLA = 162.5 k� for sinusoidal vibrations at theopen-circuit resonance and RLW = 18.5 k� for colored noise.The colored noise was obtained by low pass filtering a whitenoise process as described in [16]. The signal has a −3 dBcutoff at 3 kHz and was sampled at 24 kHz. The cutoff is highenough to allow the vibration signal to be treated as white noisein the analysis.

For a linear energy harvester with one electrical port andone mechanical degree of freedom, and which is driven bywhite noise, the optimal load is [17]

RLW = 1/ωaCt (2)

where Ct is the clamped transducer capacitance. Using thisrelation, the measured open-circuit resonance frequency andthe measured optimal load resistance, we can estimate thetransducer capacitance to be Ct = 47.5 nF. When k2 Q � 1,we can estimate the optimal loads for sinusoidal vibrations as

RLR ∼ 1/ωaCtk2 Q = RLW/k2 Q (3)

andRLA ∼ k2 Q/ωaCt = k2 Q RLW (4)

where Q is the open-circuit quality factor and we have adaptedthe results of [20] to our notation. Using the values for RLW

103

104

105

0

0.2

0.4

0.6

0.8

1

R L [Ohm]

Nor

mal

ized

Out

put P

ower

reswbanti-res

Figure 3. Power versus load resistance. Excitation signals aresinusoidal at short-circuit resonance (res), colored noise (wb) andsinusoidal at open-circuit resonance (anti-res).

and k2 from the measurements, we get RLR ≈ 2.15 k� andRLA ≈ 159.1 k� in approximate agreement with the directlymeasured values given above.

We note that a rule of the thumb, RLW ∼ √RLA RLR,

follows directly from (2)–(4).To check that the measurements are reasonable with

respect to the geometry and materials of the device, wecompare the above results to a simple model. Low-ordermodels derived from eigenmode expansions of the loadedbeam can serve this purpose [21]. Most such models in theenergy harvesting literature treat the proof mass as a point massat the tip and do not take into account its rotary inertia. Anexception can be found in [22, 23] and could have been usedhere. Due to its simplicity, we preferred instead a model thattreats the beam quasi-statically, is therefore low order alreadyfrom the formulation and projects onto the lowest mode ofthe total beam/proof mass system [18, 24]. This is adequatehere due to the large ratio of proof mass to beam mass of13. The model is parameterized in terms of a modal massm1, a stiffness, the capacitance Ct and a coupling constant.From this the device electromechanical coupling factor k2

also follows. These parameters can be calculated using theprocedure in [24], the equations for a bimorph in [25], thepiezoelectric material parameters from the vendor [19] andtable values, E = 110 GPa, ρ = 8600 kg m−3 and ν = 0.34,for the brass shim [26].

Our beam is wider than it is long and is about 25 times aswide as it is thick. Therefore the 1D form of the constitutiveequations for the piezoelectric layers should be that pertainingto a wide beam and referred to as plane strain constitutiveequations in [27]. We found that the plane stress 1D equationsapplicable to a narrow beam give a device coupling factorless than half the measured value. The main reason for thelarge discrepancy is the increase of the effective piezoelectriccoupling constant for the wide beam due to the normal stressesthat prevents the Poisson expansion in the width direction. Anincrease of the (effective) Young modulus and a decrease of

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Smart Mater. Struct. 20 (2011) 025011 L-C J Blystad and E Halvorsen

155 160 165 170 175 180 185 1900

0.05

0.1

0.15

0.2

0.25

0.3

0.35

f [Hz]

V [V

]

Up chirp

Down chirp

Figure 4. Up and down frequency sweeps for accelerationamplitudes of 0.006g, 0.029g,0.053g, 0.057g, 0.068g and 0.087g.

the permittivity for the same reason contribute further to theincreased coupling in the wide beam. We refer to [27] for adetailed explanation.

Based on this we can estimate the device coupling factorto be k2 = 10.4% which is an overestimate and the transducercapacitance to be Ct = 42.4 nF which is an underestimate. Ittherefore seems likely that the material has a somewhat lowercoupling than the value d31 = −190 × 10−12 m V−1 given bythe vendor. Possible reasons can be uncertainties in the vendormaterial data due to processing variations and maybe someageing. If we reduce the value to d31 = −176 × 10−12 m V−1

to get agreement with the measured value of k2 = 8.6%, wealso get a capacitance value Ct = 46.0 nF which comparesmore favorably to the previous estimate of 47.5 nF determinedfrom the optimal load.

The modal mass is dependent only on geometry and massdensities. We found the value m1 = 1.40m, which means thatthe kinetic energy due to the rotary motion is 40% of that dueto center-of-mass motion at short-circuit resonance.

3.1. Sinusoidal vibrations

The response to narrow band vibrations was investigatedby sweeping the frequency up and down at a slow rate of0.8 Hz s−1. It was done in a frequency interval of 35 Hzaround the short-circuit resonance frequency (figure 4). Theload resistor was 2650 �. We estimated the total Q at lowamplitude to be approximately 35 based on the full width athalf-maximum.

A clear resonance is seen for small vibration amplitudeswhen the end stop is not hit. At first, an increasing excitationamplitude results in a slightly decreasing resonance frequency.This effect is similar to what was reported in [28] for other PZTcantilevers and could be attributed to material nonlinearities.

A strongly nonlinear behavior of the system appears forRMS acceleration amplitudes above 0.06g. At these levels thedisplacement amplitude is large enough for the proof mass toreach the end stop during part of the sweep. In part of thefrequency response, two possible stable orbits of the systemare seen in figure 4.

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1x 10

–4

Acceleration RMS [g]

P [W

]

without end stop

with end stop

Figure 5. Power versus acceleration amplitude at short-circuitresonance, both with and without end stop.

On up-sweep, a plateau is reached when the end stop ishit. The response continues within the plateau region into thehigher stable orbit of the region of two stable orbits. Theresponse stays on the high amplitude orbit until a certainfrequency where a jump phenomenon [29] occurs. The slowincrease in output power within the plateau region is due tothe increasing vibration frequency of the system. On down-sweeps, the system follows the lower stable orbit.

The bandwidth is effectively increased with furtherincrease of the vibration amplitude beyond 0.06g. A wideningoccurs on both sides of the resonance frequency. At thelower band edge, it is entirely due to clipping of the resonantresponse.

A slight difference between the up- and down-sweeps canbe seen in the frequency ranges outside the region with twostable orbits. It is especially clear on the left side of theresonance. This effect can be explained as a result of thefrequency sweep rate [13]. The system does not have time tofully develop a steady state response. Thus the up- and down-sweep will respectively under- and overestimate the amplitudeof motion.

Measured traces of the output power versus accelerationamplitude with and without the mechanical end stop inplace are shown in figure 5. The output power increaseswith increasing acceleration amplitude when there is no endstop. Linear behavior is expected at low accelerations andsmall proof mass displacements. For the linear case theoutput power has a quadratic relationship to the drivingacceleration. Nonlinear effects become increasingly significantas the proof mass displacement increases due to the resonanceshift previously seen in figure 4. Therefore the output powerversus acceleration in figure 5 have a less than quadraticincrease.

With a mechanical end stop present, the output powerreaches a plateau when the end stop becomes effective. Itdoes not increase significantly with further increase of theacceleration amplitude (figure 5).

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Smart Mater. Struct. 20 (2011) 025011 L-C J Blystad and E Halvorsen

0 2 4 6 8x 10

–4

0

1

2

3

4

5

6

7

8x 10 –5

Sa [g2/Hz]

P [W

]

without end stop

with end stop

linear theory

Figure 6. Power versus acceleration spectral density Sa with filteredwhite noise excitation, both with and without end stop, and comparedto linear theory.

There is a consistently lower output power at smallamplitudes when the end stop is present. Increased air dampingdue to the narrow gap between the beam and the end stop canbe a reason for the lower output power.

A jump in the output power occurs when the end stopcomes into effect. It can be related to the resonance shift.During the sinusoidal excitation in figure 5 the harvester isdriven at its low amplitude short-circuit resonance (173 Hz).Upon increasing the acceleration amplitude, the resonanceshifts down, resulting in 173 Hz being within the region oftwo stable orbits. It is then possible for the system to jumpto the high amplitude orbit upon increase of the accelerationamplitude.

3.2. Broadband vibrations

Now we excite the system with a broadband signal consistingof colored noise. The output power versus the spectral densityof the input signal both with and without an end stop is shownin figure 6.

The average output power was calculated from the timeaverage of the instantaneous power. The same sequence of

sampled random acceleration values of duration 30s was scaledand used for each data point. Due to the random nature andfinite duration of the acceleration waveform, we would haveobtained somewhat different output power values with anotherrandom sequence of acceleration values. To quantify thisspread, we repeated the experiment for three selected pointsusing six independently generated sequences. The resultsof these measurements are shown in the form of standarddeviation error bars in figure 6.

The output power Pwb predicted by linear theory in theabsence of stray losses and with optimal load [17] is

Pwb = m2Sa

4m1

k2 Q

2 + 1/Q + k2 Q(5)

where m is the total proof mass including part of the beamand Sa is the one-sided acceleration spectral density. Forcomparison, we have plotted (5) together with the measuredpower in figure 6.

The measured output power closely follows the lineartheory at low input spectral densities. Deviations are foundaround Sa = 5–6g2 Hz−1 and indicate that the device does notbehave entirely linearly. We have already seen a resonant shiftin the sweeps in figure 4.

Comparing figures 5 and 6 it can be seen that the outputpower for the two cases behaves quite differently when theend stop comes into effect. While with sinusoidal excitationthe output power reaches a flat plateau even with increasingacceleration amplitude, this is not the case for broadbandexcitation. Here the output power continues to increase whenthe end stop is hit, but at a slower rate than with no end stoppresent.

Estimates of the power spectral densities of the outputsignal with and without the end stop are shown in figure 7.It can be seen that without the end stop the maximum powerspectral density increases with increasing spectral density ofthe input signal. Nonlinearities cause the peak frequency inthe output spectrum to slightly decrease with increasing inputspectral density. With the end stop present we see that thespectral density seems to reach a limit when the end stop comesinto effect. It is clear that the peak of the spectrum shiftsto higher frequencies and that the bandwidth broadens. Thisresults in a contribution of a larger range of higher frequencies

Figure 7. Power spectral density of the energy harvester output for increasing spectral density of the input signal. Without end stop (solidline) and with end stop (dashed line). Sa = 0.087, 0.82, 2.3, 8.0 × 10−4g2 Hz−1 starting from the lowest curve.

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Smart Mater. Struct. 20 (2011) 025011 L-C J Blystad and E Halvorsen

to the output power when the end stop is hit than without theend stop present. Looking at the output spectrum we find thatit has lower intensity with the end stop compared to the casewithout. Even if the response is shifted upwards in frequencyby including the end stop, it is still within the spectrum of thedriving force. Consequently, the output power can continue toincrease with increasing acceleration spectral density as shownpreviously in figure 6.

4. Conclusion

This paper presented an experimental study of a PZT-basedpiezoelectric energy harvester. The harvester was driven eitherby sinusoidal or broadband vibrations. We compared thebehavior of the harvester driven by these two forms of vibrationwhen it incorporates a mechanical end stop on one side.

The experiments display some of the key phenomenathat have previously only been predicted theoretically ornumerically. In the linear regime, we have demonstrated thatthe optimal load for broadband vibrations does not coincidewith either of the two possible optimal loads for sinusoidalexcitations. There is a difference in output power between thenarrow and wide band cases with increasing vibration strengthand limited proof mass motion. While the former saturates, thelatter does not. This effect can potentially be exploited.

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