MEMS electrostatic energy harvesters with end-stop effects

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2012 J. Micromech. Microeng. 22 074013

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IOP PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING

J. Micromech. Microeng. 22 (2012) 074013 (12pp) doi:10.1088/0960-1317/22/7/074013

MEMS electrostatic energy harvesterswith end-stop effectsCuong Phu Le and Einar Halvorsen

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

E-mail: Einar.Halvorsen@hive.no

Received 3 April 2012, in final form 2 June 2012Published 27 June 2012Online at stacks.iop.org/JMM/22/074013

AbstractIn micro scale energy harvesting devices, end-stops that limit the proof mass motion areinevitable from reliability concerns and can even be exploited as a functional element toachieve a broadband response. To investigate how these can be modelled, both characterizationand modelling of vibration energy harvesters with end-stop effects are presented in this paper.A Hertz contact model of the impact force between the proof mass and the end-stops isanalysed and compared to a linear stiffness model. The resulting impact force model is thenincluded into a SPICE model of an electrostatic harvester. The performance prediction of themodel is validated by comparing simulations and measurements on two different prototypes,one with mechanical quality factor Qm = 5.7 and one with Qm = 203.5. The electromechanicalcoupling factors of the two devices are respectively k2 = 1.44% and 2.52%. Both devicesdisplay the well-known jump phenomenon and output voltage saturation during, respectively,frequency and amplitude sweeps. Under low-level broadband excitations, the high-Qm deviceperforms in agreement with linear theory at an efficiency of 71.8%. For sufficiently highacceleration power spectral density (PSD), it displays a soft limit on the output power and abandwidth increase, e.g. a factor 3.7 increase of 3 dB bandwidth when increasing theacceleration PSD from 0.34 × 10−3 to 0.55 × 10−3 g2 Hz−1. The end-stop effects reduce thedevice efficiency down to 35.4% at 1.69 × 10−3 g2 Hz−1. A comparison between model andexperiment shows that a model with end-stop stiffness extracted from the contact analysis canadequately model the nonlinear end-stop effects both for narrow- and broadbandaccelerations.

(Some figures may appear in colour only in the online journal)

1. Introduction

Energy harvesting from vibrations is a potential means toobtain electrical energy and avoid use of batteries for wirelesssensors. By this, one hopes to extend lifetime or maintenancecycles of the system or, in some cases, to make sensor nodespossible in environments where batteries cannot survive. Theenergy conversion typically uses either of the three commonmechanisms: piezoelectric, electromagnetic and electrostatictransduction. A spring–dashpot–mass configuration is widelyused for harvesting power from ambient motion [1–9]. Onechallenge in the design of the energy harvester is to obtainsubstantial output power over a range of accelerations. Forincreasing acceleration, the proof mass displacement increases

resulting in a corresponding output power increase. In practice,the proof mass displacement can be limited under highacceleration due to the restricted device dimensions. Also,proof mass motion must be limited by design to avoid beamfracture at a large amplitude displacement.

This is accomplished by the design of mechanical end-stops. The proof mass hits the end-stops for a sufficientacceleration level. The end-stop impacts cause the device tobehave nonlinearly, resulting in the jump phenomenon andother nonlinear phenomena. The impacts must be modelledto predict the performance at high accelerations. A linearimpact force model with very large, in some cases somewhatarbitrarily chosen, spring stiffness has been considered byseveral authors [10–15]. A microscale piezoelectric energy

0960-1317/12/074013+12$33.00 1 © 2012 IOP Publishing Ltd Printed in the UK & the USA

J. Micromech. Microeng. 22 (2012) 074013 C P Le and E Halvorsen

Table 1. Device structure parameters of prototypes 1 and 2.

Parameters Prototype 1 Prototype 2

Active area 8 × 4 mm2 5 × 4 mm2

Device thickness, t 60 μm 25 μmLength of capacitor fingers, lf 25 μm 25 μmWidth of capacitor fingers, wf 4 μm 2 μmGap between capacitor, g0 3.2 μm 2 μmNumber of capacitor fingers on each electrode, Nf 438 530Initial capacitor finger overlap, x0 10 μm 10 μmMaximum proof mass displacement, xmax 7 μm 10 μmBump radius, R 30 μm 30 μm

harvester prototype for low-frequency-vibration applicationsdemonstrated the utilization of mechanical end-stops forwideband operation under sinusoidal excitations [16].Recently, there has also been focus on exploiting internalimpacts on transducing structures both in piezoelectric [17, 18]and microelectromechanical systems (MEMS) electrostaticdevices [19]. In order to model such devices, models of impactsare needed.

In this work, MEMS electrostatic energy harvester devicesare designed and characterized to investigate the influenceof a practical impact model represented by a parallel-connected spring and damper as in previous work, but withthe stiffness value obtained analytically from Hertz’s contactproblem [20, 21]. The damping coefficient is fit to the jumpphenomenon observed in the measurements. The impact forcemodel is then incorporated into the mechanical domain of theelectrostatic energy harvester model so that it can handle alarge amplitude displacement of the proof mass. The modelpredictions are compared to measurements. This approach toparameter estimation can be valuable for modelling of rigidend-stop effects in energy harvesters operating both in narrow-and wideband regimes.

2. Analysis and modelling

We present two device prototypes made in multi-project wafer(MPW) foundries. One is designed for a process that giveshermetically packaged devices and is henceforth referred to asprototype 1. The other is not hermetically encapsulated and isreferred to as prototype 2.

2.1. Device structures

The prototype 1 MEMS electrostatic energy harvester withoverlap varying capacitors and end-stops was designed asshown in figure 1. The proof mass is suspended by foursprings with folded shape to achieve mechanically linearperformance. The end-stops are designed as bumps on theanchors of the springs that contact a flat surface of the proofmass upon impact. As the bumps are drawn as semicirculardiscs in the layout, and the etching has a high aspect ratio, theresulting geometry in the fabricated device is approximatelycylindrical with a semicircular cross section. We choose thiscross-sectional shape because it is conveniently made in anin-plane device and limits the contact area, which is good foravoiding stiction and possibly also for reducing squeezed film

Table 2. Model parameters for device prototypes 1 and 2.

Parameters Prototype 1 Prototype 2

Proof mass, m 2.8 mg 1.2 mgSpring stiffness, km 166.5 N m−1 21.1 N m−1

Damping constant, b 3.75e-3 Ns m−1 2.5e-5 Ns m−1

Nominal capacitance, C0 1.6 pF 1.3 pFParasitic capacitance, Cp 17.3 pF 7.5 pFLoad resistance, RL 4.9 M� 15.2 M�Load capacitance, CL 4.0 pF 5.0 pF

damping. The bumps have a radius R = 30 μm and a lengthequal to the device layer thickness t = 60 μm. The proofmass motion is limited to a maximum displacement amplitudexmax = 7 μm as the mechanical end-stops become engagedunder sufficiently high vibration amplitudes.

Figure 2 shows a picture of the device prototype 1 whichis fabricated in Tronics MPW services using 60 μm thicknessSOI high aspect ratio micromachining. The die dimension is4 × 8 mm2 and the device is encapsulated by an additionalcap wafer. All device structure parameters of the electrostatictransducer are listed in table 1. In order to achieve accuratebeam width, protection beams are designed in parallel withthe springs to reduce the risk of extensive spring over-etchingduring fabrication. The point is to decrease the discrepancybetween spring dimensions in the layout and the fabricateddevice. The harvester is operated by an external bias voltageconnected to pads deposited on the anchors and the outputpower is simply obtained by connection of the fixed electrodesto external loads RL.

Figure 3 shows a MEMS device prototype 2 fabricatedin the SOIMUMPS process with a device layer thickness of25 μm. In this case, the end-stops are implemented as bumps onthe proof mass impacting on a flat surface. The overall designis otherwise similar to prototype 1. The device is measuredin a non-hermetic package, but is mounted on a chip carrierwith a plastic lid for dust protection. The proof mass motionis limited to a maximum displacement of xmax = 10 μm. Alldesign parameters are listed in table 1.

2.2. Impact force analysis

The end-stops confine the proof mass motion to therange −xmax �x �xmax. For displacement amplitudes beyondthese limits, the end-stop force Fim can be modelled as

2

J. Micromech. Microeng. 22 (2012) 074013 C P Le and E Halvorsen

Figure 1. Prototype 1 energy harvester design with capacitive transduction and use of anchors as end-stops.

Figure 2. Picture of device prototype 1 fabricated in Tronics MPW services using 60 μm SOI high aspect ratio micromachining with diedimension of 4 × 8 mm2: movable and fixed electrodes for the capacitive transduction, anchor and folded flexure spring with a protectionbeam. The anchor with bumps functions as end-stops for the inertial proof mass motion (Photograph: Tronics Microsystems SA).

a parallel-connected spring–damper system as indicated infigure 4 and given by

Fim = kimδ + bimδ for |x| > xmax, (1)

where kim is the end-stop stiffness, bim is the dampingcoefficient and δ is the relative displacement between the proofmass and the end-stops during impact

δ = |x| − xmax. (2)

To use this model, the stiffness and damping of the end-stopneed to be determined.

The impact between the proof mass and the end-stops canbe considered as the contact of a half-cylinder on a flat surface.The contact area is then initially a line that grows to a rectanglewith a length equal to that of the device layer thickness t and awidth 2y0 that is dependent on the impact force. To avoid theneed for parameter fitting of at least the stiffness, we considera Hertzian line contact between a surface of the proof mass anda half-cylinder with a radius of R and a length of t as shown infigure 4.

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J. Micromech. Microeng. 22 (2012) 074013 C P Le and E Halvorsen

Figure 3. Device prototype 2 without encapsulation fabricated using SOIMUMPS process with a layer thickness of 25 μm: close-up view ofspring and end-stop designs in the device.

Figure 4. Model of the impact force between the inertial mass m and the end-stops. Geometry of two deformed bodies of the proof mass andthe stopper with the pressure force s(y) and the relative displacement between two deformed bodies δ.

We assume that the body surfaces are smooth and thatcontact does not exceed the elastic limit. The contacting bodiesare made from the same elastic material with Young’s modulusE and Poison’s ration υ . In order to determine kim, we use staticanalysis.

The impact between the proof mass and the end-stopscauses deformation of the two bodies. This is a result from anormal stress s(y) created by the impact force Fim. This stressis expressed by [20, 21]

s(y) = 2Fim

πy0t

√1 − y2

y20

, (3)

where

y0 =√

8λRFim

tand λ = 1 − υ2

πE. (4)

We assume that the relative displacement δ of two deformedbodies is uniform on the contact area. Therefore, the value ofthe relative displacement δ can be determined at the centre ofthe contact area y = z = 0, given by

δ = 2λ

∫ y0

−y0

∫ t/2

−t/2

s(y′)√y′2 + z′2 dz′dy′

∣∣∣∣y=z=0

. (5)

The relative displacement δ between the proof mass and theend-stops is simplified as being uniform in the entire contact

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J. Micromech. Microeng. 22 (2012) 074013 C P Le and E Halvorsen

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0δ δ

0imF

F

Figure 5. Relationship between the normalized impact force Fim and the normalized relative displacement δ with F0 = t3

2λR and δ0 = t2

R .

Figure 6. Energy harvester model in the mechanical and electrical domains. The force Fim between the proof mass and the end-stops ismodelled as a behavioural source.

area and the small contacting surfaces with t � y0. Thus, therelative displacement δ can be calculated as [22, 23]

δ =[

1 + lnt3

2λRFim

]2λ

tFim. (6)

Figure 5 shows the nonlinear relationship between the impactforce Fim and the relative displacement δ. For our two devices,δ0 is respectively 120 and 20.8 μm, so it is only the far left-hand side of the curve that is relevant for these two cases.In the modelling that follows, a nonlinear impact stiffnesskim is calculated from equation (6) by kim(δ) = Fim/δ, whilethe damping constant bim in (1) is determined by a fit tomatch the simulations to the observed frequency of the jumpphenomenon in the measured device frequency response.

2.3. Device model

A lumped model of the energy harvester is shown infigure 6. The mechanical and electrical domains are capturedby the following equations:

mx + bx + kmx + Fe + Fim = ma (7)

Vb = − q1/2

C1/2(x) + Cp+ VL1/L2, (8)

where q1 and q2 are the charges on transducers 1 and 2,respectively. The electrostatic force is

Fe = 1

2q2

1d

dx

(1

C1(x) + Cp

)+ 1

2q2

2d

dx

(1

C2(x) + Cp

), (9)

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J. Micromech. Microeng. 22 (2012) 074013 C P Le and E Halvorsen

Figure 7. Experimental setup including the devices glued onto theprinted circuit board, shaker, accelerometer and LabView control.

where

C1/2(x) = C0

(1 ± x

x0

)= 2Nf ε0

x0t

g0

(1 ± x

x0

), (10)

g0 is a gap between the capacitor, x0 is an initial capacitorfinger overlap and Nf is the number of capacitor fingers oneach electrode. A parasitic capacitance Cp in parallel with thevariable capacitance is also included in the model. The modelparameters of both devices are listed in table 2. The proofmass m, the nominal capacitance C0 and the spring stiffness km

are found from the designs. Loading effects in etching duringfabrication cause the resonant frequency to deviate about 1.6%and 0.4% from the designed values for devices prototypes 1and 2 respectively. Values of the damping coefficient and theparasitic capacitances were obtained based on fitting to resultsin a linear regime for sinusoidal excitations. The chosen loadsRL are the optimal values for sinusoidal vibrations, determinedby simulation for device prototype 1 and from impedanceanalysis for device prototype 2. The location of the end-stopsin the model is given by the design values in table 1.

2.4. Electromechanical coupling

In order to characterize the devices as electromechanicaltransducers, it is useful to consider their electromechanicalcoupling factors k2 in the linear regime. This can be donebecause the common mode and differential mode behaviourdecouples in linearization for this type of device havingtwo anti-phase electrostatic transducers [24]. The linearelectromechanical two-port for the differential mode can thenbe written in standard notation, see [25], as

FT = Kx + �

Cq, (11)

VT = �

Cx + 1

Cq, (12)

where

VT = V2 − V1, (13)

q = q2 − q1

2. (14)

With the load capacitances CL included in the transducermodel, the parameters read

C = C0 + Cp + CL

2, (15)

� = C0Vb

x0, (16)

K = km + 2C20V 2

b

x20(C0 + Cp + CL)

. (17)

The resulting electromechanical coupling factor k2 is then

k2 = �2

KC= 2C2

0V 2b

x20(C0 + Cp + CL)km + 2C2

0V 2b

. (18)

Values for k2 and the figure of merit k2Qm for the devices aregiven in table 3. In the linear regime of a two-port energyharvester, a clear distinction between the weak- and strong-coupling regimes can be identified based on whether k2Qm islarger than the critical value (kQm)c = 1 + √

1 − k2 ≈ 2or not, see [26] (be aware of notational differences). Clearly,prototype 2 is in the strong-coupling regime, while prototype1 is deep into the weak-coupling regime.

3. Measurement and simulation results

3.1. Experimental setup

The device is glued onto a printed circuit board (PCB)mounted onto a shaker system consisting of a TIRA vibrationexciter and its power amplifier as shown in figure 7. Thetransducer outputs are connected to the load resistances RL

and the voltage across the resistor is measured throughbuffer amplifiers on the PCB to avoid electrical influences ofexternal parts. The acceleration is measured by a PiezotronicsInc. model 352A56A accelerometer. National Instruments’LabView program with an NI-USB-6211 DAQ collects theacceleration and output voltage data.

3.2. Device prototype 1

Figure 8 shows the frequency response of the device prototype1 in simulations and measurements at a bias voltage Vb = 30 V.For an acceleration arms = 4.1 g, the proof mass displacementis excited to a small amplitude. Therefore, there is no impactbetween the proof mass and the end-stops, and no clearsignatures of nonlinear behaviour are observed.

With an increase of the acceleration to arms = 5.5 g, theexcitation level is sufficient to drive the proof mass into the end-stops. In the up-sweep frequency response, the output voltageincreases with increasing frequency up to f = 1135 Hz. Atthis point, the proof mass hits the end-stops, causing a kink inthe output voltage and a moderate further increase. The outputvoltage jumps down at a frequency f = 1425 Hz. This jumpphenomenon is similar to that seen for a hardening spring andis due to the abrupt increase in stiffness at the impacts.

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J. Micromech. Microeng. 22 (2012) 074013 C P Le and E Halvorsen

900 1000 1100 1200 1300 1400 1500 16000

0.1

0.2

0.3

0.4

0.5

0.6

Frequency [Hz]

RM

S O

utpu

t V

olat

ge [

V]

measurement

simulation: nonlinear without end-stop damping

simulation: nonlinear, end-stop damping of 1.5Ns/m

simluation: linear, end-stop damping of 2Ns/m

simulation: nonllinear, end-stop damping of 10Ns/m

RMS acceleration =5.5g

RMS acceleration=4.1g

Figure 8. Prototype 1 up-sweep frequency response of RMS output voltage for different acceleration levels at a bias voltage Vb = 30 V.

Table 3. Evaluation of electromechanical coupling for the device prototypes 1 and 2.

Device Electromechanical k2 (%) Mechanical quality factor Qm k2Qm

Prototype 1 at Vb = 30 V 1.19 5.7 0.068Prototype 2 at Vb = 15 V 2.52 203.5 5.125

A higher end-stop damping causes a smaller up-sweepfrequency bandwidth. Without the end-stop damping (bim = 0),the simulation gives a higher jumping frequency, i.e. 1475 Hz,than the measurement. A value of bim = 1.5 Ns m−1 leads toa fit of the nonlinear impact force model to the measuredresponse. A higher end-stop damping causes a smaller up-sweep frequency bandwidth. For example, a further increase inbim = 10 Ns m−1 results in the reduction of the jump frequencyto 1342 Hz.

The relative displacement between the proof mass andthe end-stops during impact is small, meaning that δ � δ0. Itis possible to approximate the nonlinear relation between theimpact force and the displacement by a linear relation over therelevant range of displacements. For a range of δ from 0 to0.5 μm, a constant end-stop stiffness kim can be modelled bythe approximate formula:

kim ≈ 0.058πEt

1 − υ2, (19)

obtained from a least-squares fit.As observed in figure 8, use of the linear stiffness model

gives a fitting damping coefficient bim = 2 Ns m−1. Both linearand nonlinear end-stop stiffness properly capture the behaviourunder high acceleration levels. The results of simulations andmeasurements are approximately the same regardless of thestiffness model. Hence, we have shown that a linear model isadequate and that the stiffness can be calculated.

Figure 9 shows output powers of the device prototype1 with an increase of the rms acceleration for different biasvoltages Vb. For a higher bias voltage Vb, the greater output

power is obtained. Both the simulation and measurementresults exhibit impact at the acceleration of arms = 4.5 g.At this excitation level, the end-stops confine the proof massmotion at the maximum displacement xmax, leading to saturatedoutput power or voltage. Furthermore, the proof mass impactsthe end-stops at the same acceleration level for various biasvoltages Vb = 20, 30 and 40 V. This means that the mechanicaldamping is still dominant in comparison with the electricaldamping for this range of the bias voltage Vb. Similar tothe linear performance without impact, the saturated outputpower is approximately proportional to the square of the biasvoltage Vb. For example, at the acceleration of arms = 5.5 g,both simulation and experiment results show that the saturatedoutput powers increased from 31 to 125 nW for increasing thebias voltages Vb from 20 to 40 V respectively.

The end-stop effects under random wideband excitationare also characterized and compared to the predictions of themodel used for the narrowband vibrations. Both the inputacceleration a(t) and output voltage V(t) are functions of time tand are non-periodic. Therefore, we characterize them by theirtwo-sided power spectral densities (PSDs). Mathematically,the PSD of a random time series X = X(t) which is a powersignal, i.e. has infinite ‘energy’ and finite average ‘power’, canbe defined by the Wiener–Khinchin relation [27–29] (otherequivalent definitions are possible). For a stationary stochasticprocess, it reads

SXX(ω) =∫ +∞

−∞E[X (t)X (t + τ )]e−iωτ dτ , (20)

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J. Micromech. Microeng. 22 (2012) 074013 C P Le and E Halvorsen

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

20

40

60

80

100

120

140

RMS acceleration [g]

Out

put p

ower

[nW

]

measurement

simulationVb=40V

Vb=30V

Vb=20V

Figure 9. Prototype 1 output power with the end-stop effects for increasing acceleration amplitude for various bias voltages at a resonantfrequency f 0 = 1220 Hz.

101

102

103

104

10-5

10-4

10-3

10-2

10-1

Frequency [Hz]

Spe

ctra

l den

sity

of

inpu

t ac

cele

ratio

n [g

*g/H

z]

BW=10kHz

Figure 10. Spectral density Sa = 10−2 g2 Hz−1 of a wideband excitation signal generated in 10 s with a bandwidth of 10 kHz and a samplingfrequency of 100 kHz.

where E[ · ] means the statistical expectation. The expectationin (20) is the autocorrelation function of X. In practice, for ameasured signal, the time-average autocorrelation function isused instead. SXX(ω)dω/2π gives the amount of the signal’spower in the frequency interval dω/2π . Sometimes the one-sided PSD, i.e. 2SXX restricted to positive frequencies, is usedinstead of SXX. Other normalizations are also in common use.To estimate the PSD from measured data, we use Welch’s

averaged modified periodogram method described in [29]and available in Matlab [30]. It splits the signal into severaltime windows with 50% overlap between successive datasegments, does properly normalized spectral estimation oneach time window and then averages the estimates over all timewindows. The input accelerations are generated as randomsignals with different spectral densities. In this work, low-pass-filtered white noise acceleration signals with bandwidth

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J. Micromech. Microeng. 22 (2012) 074013 C P Le and E Halvorsen

600 800 1000 1200 1400 1600 1800 200010

-10

10-9

10-8

10-7

10-6

Frequency [Hz]

Out

put P

ower

PS

D [u

W/H

z]

measurement

simulation3.49e-4g*g/Hz

1.12e-4g*g/Hz

4.42e-6g*g/Hz

Out

put P

SD

[µW

/Hz]

]

Figure 11. Output PSDs of the device prototype 1 for various acceleration PSDs at Vb = 30 V.

500 1000 1500 2000 2500 3000

10-6

10-5

Frequency [Hz]

Out

put p

ower

PS

D [u

W/H

z]

2.85e-2g*g/Hz

1.31e-2g*g/Hz

7.4e-3g*g/Hz

Out

put P

SD

[µW

/Hz]

]

Figure 12. Simulation results of bandwidth increase for the device prototype 1 due to end-stops for sufficiently high acceleration PSDs atVb = 30 V.

10 kHz, duration 10 s and sampled at 100 kHz are used. Theexcitation level is characterized by its average spectral densityin the flat region of its spectrum. Figure 10 shows an exampleof the acceleration spectral density with an average valueSa = 10−2 g2 Hz−1.

The lumped model gives good agreement between themeasurements and the simulations. Higher output power isachieved for larger PSDs of the acceleration as shown infigure 11 for a bias voltage Vb = 30 V. Here, the PSDs aresmall enough that the device responds linearly. The bandwidthsof the responses are consequently equal for the differentacceleration PSDs in this range.

Due to limits of the vibration level in the characterizationsetup, the output spectrum of the device prototype 1 wasonly evaluated from the simulation model at high accelerationPSD. A further increase of the acceleration PSD results infrequent impacts between the proof mass and end-stops,and the harvester behaves nonlinearly. The bandwidth nowincreases with increasing acceleration PSD. Therefore, thedevice bandwidth can be improved with the stopper effectsunder wideband excitation as shown in figure 12. For example,the 3 dB bandwidth is enlarged from 380 to 600 Hz forincreasing the acceleration PSD from 7.40 × 10−3 to2.85 × 10−2 g2 Hz−1.

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J. Micromech. Microeng. 22 (2012) 074013 C P Le and E Halvorsen

600 650 700 750 800 8500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Frequency [Hz]

RM

S O

utpu

t V

olta

ge [

V]

ExperimentSimulation

RMS acceleration=0.31g

RMS acceleration=0.12g

Figure 13. Up-sweep frequency response of RMS output voltage for the device prototype 2 at different acceleration levels and a bias voltageVb = 10 V, with the use of the end-stop model with the linear stiffness from equation (19), a fitted damping coefficient bim = 0.4 Ns m−1.

10-4

10-3

100

101

102

Acceleration PSD [g*g/Hz]

Out

put

Pow

er [

nW]

Linear theory calculation

Simulation

Experiment2

0.718a

P

mS=

Figure 14. Output power of the device prototype 2 under increase of the input acceleration PSDs at a bias voltage Vb = 15 V with the use ofthe end-stop model with the linear stiffness from equation (19), a fitted damping coefficient bim = 0.4 Ns m−1.

3.3. Device prototype 2

Figure 13 shows the frequency response of the deviceprototype 2 with a narrowband excitation at small andhigh excitation levels. The impact stiffness model (19) isagain validated by agreement between the simulation andexperimental results in capturing the impact nonlinearity andjump phenomenon. Observation of the jumping frequencygives a fitted damping coefficient of bim = 0.4 Ns m−1. This

indicates that the approximate stiffness appropriately modelsthe impact behaviour. Using instead equation (6), we obtain afitted damping coefficient bim = 0.34 Ns m−1 and a frequencyresponse curve that fits equally well as those in figure 13.Therefore, the predicted performance of the harvester modelis somewhat insensitive to the detailed form of the impactmodel.

Both simulation and measurement results exhibit the end-stop effects of the device prototype 2 under an increase of

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J. Micromech. Microeng. 22 (2012) 074013 C P Le and E Halvorsen

500 600 700 800 900 100010

-8

10-7

10-6

10-5

10-4

10-3

Frequency [Hz]

Out

put

PS

D [

μW/H

z]

measurementsimulation

0.55e-3 g*g/Hz

1.69-3 g*g/Hz

0.34e-3 g*g/Hz

Figure 15. Output power PSDs of the device prototype 2 for different levels of the input acceleration PSDs at a bias voltage Vb = 15 V.

the input acceleration PSDs at a bias voltage Vb = 15 V, seefigure 14. For small vibration levels where the proof massdoes not reach the end-stops, the output power is linearlyproportional to the input acceleration PSD Sa. The theoreticaloutput power in the linear regime is [31]

P = mSa

2

rLk2Qm

1 + (1/Qm + k2Qm)rL + r2L

, (21)

where

rL = 2RLC

√K

m(22)

and is plotted together with the measured results.Since the input power is given by Pinput = mSa/2 [32],

2P/mSa is the efficiency. It is about 71.8% for prototype 2,while it is only about 3% for prototype 1. The main reasonbehind this is that the huge difference is damping between thetwo devices.

A further increase of the acceleration causes the proofmass to hit the end-stops more frequently. As a result, theoutput power grows only moderately and with less thanlinear growth. At the acceleration levels of the end-stopactivation, the simulation results give a slightly smaller outputpower than those of the measurement, but the simulationmodel qualitatively captures the performance of the deviceprototype 2. The measurement and simulation results exhibitalmost the same reduction in device efficiency when the proofmass frequently impacts on the end-stops. For example, theefficiency decreases from 71.8% to 35.4% when the inputacceleration PSD increases to Sa = 1.69 × 10−3 g2 Hz−1.

The enhancement of the response bandwidth seen infigure 12 is also present in the simulation and measurementresults of the device prototype 2 in figure 15. At a smallacceleration PSD of 0.34 × 10−3 g2 Hz−1, the frequencyresponse of the device gives a peak at the resonant frequencyof 675 Hz and a corresponding 3 dB bandwidth of 14.9 Hz.

The 3 dB bandwith is enlarged by up to a factor of 3.7 whenincreasing the acceleration PSD to 0.55 × 10−3 g2 Hz−1.For further increasing the acceleration PSD levels, thenonlinear behaviours of the end-stops effects become moreevident with an enlargement of the response bandwidthtowards increasing frequencies, e.g. at an acceleration PSDof 1.69 × 10−3 g2 Hz−1.

4. Conclusion

Two microelectromechanical systems electrostatic energyharvester prototypes with low and high figures of merit k2Qm

were characterized and modelled taking into account the end-stop effects under large amplitude displacement excitation.The impact force between the proof mass and the end-stops wasanalysed using a Hertz’s line contact model. The device modelwith impacts effectively captured the device behaviour givingagreement between the simulation and measurement results.Under both narrowband and wideband vibrations, the end-stopmodel with a linear stiffness fit to the nonlinear stiffness modelover a limited displacement range performed equally well. Inthe wideband regime, both simulation and measurement resultsshow that the bandwidth improvement is a positive effect ofthe end-stops for large acceleration power spectral densities(PSDs) and result in a moderated increase in power instead ofsaturation. The end-stop effects decrease the device efficiencyin comparison with the linear theory calculation under increaseof the acceleration PSDs.

Acknowledgment

This work was supported by the Research Council of Norwayunder grant no. 191282.

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J. Micromech. Microeng. 22 (2012) 074013 C P Le and E Halvorsen

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