fabrication, characterization and modelling of electrostatic micro-generators
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Fabrication, characterization and modelling of electrostatic micro-generators
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2009 J. Micromech. Microeng. 19 094001
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J. Micromech. Microeng. 19 (2009) 094001 (11pp) doi:10.1088/0960-1317/19/9/094001
Fabrication, characterizationand modelling of electrostaticmicro-generators
Daniel Hoffmann1, Bernd Folkmer1 and Yiannos Manoli1,2
1 HSG-IMIT, Institute of Micromachining and Information Technology, Wilhelm-Schickard-Straße 10,78052 Villingen-Schwenningen, Germany2 Department of Microsystems Engineering (IMTEK), University of Freiburg, Georges-Koehler-Allee102, 79110 Freiburg, Germany
E-mail: [email protected]
Received 13 January 2009, in final form 30 March 2009Published 26 August 2009Online at stacks.iop.org/JMM/19/094001
Abstract
This paper presents an electrostatic energy-harvesting device for electrical energy extractionfrom vibrations. We successfully fabricated prototypes of completely packagedmicro-generators with a chip size of 5 mm by 6 mm. This was achieved using a modified SOItechnology developed for inertial sensors at HSG-IMIT. Micro-generators produce amaximum rms power of 3.5 μW when they are excited at their resonance frequency with aninput excitation of 13 g. During a long-term experiment over a period of 2 h, the electrostaticenergy harvester generated a total net energy of 13.38 mJ corresponding to an average powerof 1.58 μW. The effect of mechanical stoppers and the bias voltage on the generated power isalso evaluated. In order to get a more profound understanding of the dynamic behaviour of themicro-generator, we have developed a signal-flow model for numerical simulation of theelectrostatic transducer on system level. This model includes a mechanical and an electricaldomain which are coupled by electrostatic forces. The limited displacement of the proof massis also considered using an elastic stopper model. We show that the numerical model iscapable of providing good predictions of the device behaviour.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
The choice of a specific transducer mechanism for energyharvesting from vibrations is heavily dependent on boththe operating conditions (amplitude and frequency spectrumof the excitation source) and available space given by theapplication environment. If miniaturization of the generatoris an important issue and generator devices are required ata large production scale, electrostatic MEMS transducersare the preferred choice. On the one hand, this is dueto the fact that electrostatic forces (s2) scale at a differentrate than electromagnetic forces (s3) [1]. As a result,utilizing the electrostatic transduction mechanism seems tobe advantageous when miniaturization is required. However,in the field of vibration energy harvesting, it is morepractical to use the electromechanical coupling coefficient
(also transduction coefficient), describing the conversionefficiency of a generator, as an evaluation criterion. Itcan be shown that the electromagnetic coupling coefficientkem = N · l · B [2] for a moving conductor coil in a constantmagnetic field B scales with s2 if the internal resistance of thecoil and the remanence BR of the permanent magnet is keptconstant. In contrast, the electrostatic coupling coefficientkes = ε · l · E [2] of an area-overlap varying capacitor scaleswith s1 if the electric field E in the gap is constant. Therefore,when the system decreases by a factor of 100 in size, theelectrostatic coupling coefficient decreases by a factor of 100,whereas the electromagnetic coupling coefficient decreasesby a factor of 1000. From this it can be concluded thatthe electrostatic conversion mechanism is more efficient forvibration generators of the size of a typical MEMS device.
0960-1317/09/094001+11$30.00 1 © 2009 IOP Publishing Ltd Printed in the UK
J. Micromech. Microeng. 19 (2009) 094001 D Hoffmann et al
On the other hand, the technology for manufacturingelectrostatic transducers such as capacitive-based sensors andactuators is well established. It is therefore beneficial toutilize standard MEMS technology for manufacturing micro-generators based on electrostatic transduction. This will allowthe production of generator devices in large numbers at lowcost. Moreover, easy integration with electronics is alsoprovided since standard packages as used for inertial sensorsare employed for packaging. Therefore, electrostatic micro-generator devices can be handled and manipulated as any otherelectronic component providing a high level of integration.
The energy conversion mechanism of an electrostatictransducer is based on electrostatic forces coupling theelectrical and the mechanical domains. As reviewed in [3],there are two main schemes for operating an electrostaticmicro-generator: switched systems and continuous systems.In this paper we focus on continuous systems since the use ofcontrolled switches required for switch-type operated systemscomplicates the implementation of the generator and thecircuitry. In a continuous system, the variable capacitor iscontinuously connected to the circuitry, which includes theload and the polarization voltage for biasing the variablecapacitor. A change in capacitance will always result in acharge transfer through the load resistor, which causes powerto be dissipated. The amount of energy that can be extracted isheavily dependent on the balance between the strength of theelectrical damping force (specified by the transducer design)and the excitation force (given by the operating conditions).Therefore, an optimal value of the electrostatic force must bedesigned with respect to the excitation conditions in order toachieve a maximum of power generation.
In [4], we presented results from a theoretical design studyof electrode configurations (e.g. in-plane area overlap, in-planegap closing, etc) for electrostatic micro-generators with anexternal bias voltage source. In a static analysis (no feedbackconsidered), the capacitance as a function of displacement,the total capacitance change and the electrostatic force werecalculated analytically for different electrode geometries. Wefound, that a modified electrode geometry (trapezoidal shape),which combines sensitivity of both gap and area, provides thegreatest change in capacitance. However, dynamic stabilitybecomes an issue due to the pull-in effect, and hence the biasvoltage must be restricted significantly. Based on these results,we considered an in-plane area-overlap micro-generator withstraight finger electrodes for a first design implementation.
The idea of electrostatic energy conversion goes backto 1976 when O P Breaux filed a patent about a rotarynon-resonant conversion system [5]. About 13 years ago,Williams et al first published a concept about resonantelectrostatic micro-generators [6]. Since then only afew groups have published experimental data on resonantelectrostatic generator prototypes in contrast to piezoelectricand electromagnetic transducer types [7–13]. Most of thedevices presented are open systems which are not hermeticallyencapsulated and therefore vulnerable to environmentalinfluences. Moreover, in most of the cases, they are eitherlarge in size (�5 cm3) or require very high bias voltages(�800 V) in order to generate sufficient power in the range
Seismic Mass
C1(x) C2(x)
R1 R2
C
Figure 1. Schematic view of an electrostatic micro-generator model.
of μW [9, 12]. In the present work, we present a completelypackaged prototype device of an electrostatic micro-generatorwith a package volume of about 0.2 cm3. By integration ona PCB test board, we investigated the device behaviour withrespect to the excitation level and the bias voltage as well asthe frequency response.
We have also developed a numerical signal-flow model inMatlab/Simulink to gain a more profound understanding of thedevice behaviour. This model considers the nonlinear regime,where electrostatic forces couple back from the electricaldomain to the mechanical domain. In addition, the effect ofmechanical stoppers limiting the displacement of the proofmass is also taken into account. The simulation resultspredicting the device behaviour are in good agreement withexperimental measurements.
2. Design and analysis
2.1. Principle of operation
The design of the electrostatic transducer structure is basedon a mechanical resonator and two symmetrical variablecapacitors as shown in figure 1. The two variable capacitorsare connected to the proof mass in such a way that theircapacity changes in a complementary manner. This workingprinciple offers some advantages in contrast to a single variablecapacitor, and was proposed by Sterken et al [14]. One of theadvantages is that the transduction is quite insensitive to straycapacitances.
The variable capacitors are electrically connected by twoload resistors. The voltage source, which is required to biasthe device, is realized by a charged capacitor (figure 1). Whenmotion of the proof mass occurs, charge transport between thetwo variable capacitors is imposed, inducing a current throughthe load resistors. If the load resistors are infinitely high, thegenerator operates in constant charge mode. In contrast, aconstant voltage mode predominates if the generator is shortcircuited (R1 = R2 = 0). In both of these cases, no work can bedone since either the generated current or the generated voltageis zero. As a consequence, a continuous system can neitherbe purely charge constraint nor purely voltage constraint. Foran optimized load, the generator operates in between those
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J. Micromech. Microeng. 19 (2009) 094001 D Hoffmann et al
Table 1. Design parameters of the electrostatic transducer structure.
Parameter Symbol Value
Mechanical parametersMass of proof mass m 642 μgTotal spring stiffness k 72.6 N m−1
Displacement limit xmax 20 μmSuspension beam width WB 4 μmSuspension beam length LB 310 μmNumber of folded beams Nf 8
Capacitor parameters
Number of capacitor finger pairs NF 936Initial overlap L0 20 μmGap between fingers gF 2.5 μmFinger length LF 30 μmFinger height (device layer thickness) HF 50 μmTotal change of capacitance �C 13.3 pF
two modes. This type of generator is referred to as a velocitydamped generator because the damping force is approximatelyproportional to the relative velocity between the proof massand the device package [3].
2.2. Design of the transducer structure
There are a number of design parameters as well as application-specific design constraints to be considered, when designingcustomized electrostatic micro-generators. Our first goalwas to be compatible with microelectronic packaging andassembling tools; therefore, we chose a standard ceramicleadless chip carrier (CLCC) as a device package. Thispackage is commonly used for packaging inertial sensor units[15]. This particular choice restricts the chip size and thus thedesign space to a specific area due to the defined cavity size ofthe CLCC package. There may be other design constraints tobe considered when developing customized micro-generatordevices depending on the specific application. Our secondgoal was to keep the chip size as small as possible in order tokeep the cost moderate. The more chips that can be placed ona wafer layout, the lower the cost per chip. Due to the aboveconsiderations, a chip size of 5 mm by 6 mm with a design area(including proof mass, capacitors and suspensions) of 2.3 mm× 4 mm was chosen.
One of the requirements for maximum power extraction ismaximizing the capacitance change per unit displacement [16].Therefore, we designed the proof mass as a fishbone structureto allow accommodation of a large number of electrodes(figure 2). A summary of relevant parameters is given intable 1. The capacitive electrodes comprise interdigitatedcomb structures with a constant gap varying area-overlapcharacteristic. Fixed and movable comb fingers are designedwith a gap g0 of 2.5 μm and an initial overlap L0 of20 μm. Our current fabrication technology provides an activedevice layer of 50 μm thickness to be used. Therefore, amaximum variation of capacitance �C of 13.3 pF (analyticallycalculated) for each capacitor is achieved utilizing 936 fingerpairs. The displacement amplitude xmax of the proof mass islimited to 20 μm by mechanical stoppers. The suspensionbeams have a width of 4 μm and a length of 310 μm. Each
Proof Mass WithMovable Comb
Electrodes
Fixed CombElectrode Units
MechanicalSuspension Units
Figure 2. Schematic view of the transducer layout. Bright dottedareas represent comb electrode units and cross-hatched areasrepresent mechanical suspension units.
suspension unit (figure 2) is designed with two folded beams;thus, the total spring constant k of the resonator is 72.6 kgs−2 (analytically calculated). With a total effective mass m of642 μg, the resonance frequency yields 1692 Hz. The effectivemass includes the proof mass as well as the mass of the beamsand the trusses.
2.3. Signal-flow model
In order to understand and predict the dynamic behaviour of theelectrostatic micro-generator, we have developed a signal-flowmodel in Matlab Simulink. The model is based on two energydomains, a mechanical domain and an electrical domain,which are electromechanically coupled by an electrostaticfield.
According to figure 1, a set of two nonlinear differentialequations can be established by means of Kirchhoff’s secondlaw applied to the meshes of the circuit. Thus, the state of thecharge qi on the two variable capacitors can be written as
R1dq1
dt+
q1
C1(x)+
q1 + q2
CBV− VBV = 0
R2dq2
dt+
q2
C2(x)+
q1 + q2
CBV− VBV = 0,
(1)
where Ri is the load resistance, qi is the charge on the variablecapacitors, CBV and VBV are the capacitance and the voltage ofthe bias voltage source, respectively, and Ci is the capacitanceof the two variable capacitors.
The capacitances Ci of the variable capacitors with anarea-overlap characteristic are given by
C1(x) = 2 · NF · ε · HF · xmax + x
gF
C2(x) = 2 · NF · ε · HF · xmax − x
gF
,
(2)
where NF is the number of fingers of the comb electrodes, ε isthe permittivity of the medium between the fixed and movableelectrodes, HF is the height of the finger equal to the thicknessof the device layer, gF is the gap between the fingers andxmax is the initial overlap of the fingers which is equal to themaximum displacement amplitude.
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J. Micromech. Microeng. 19 (2009) 094001 D Hoffmann et al
The electrostatic forces that provide the feedback in theelectromechanical system are calculated as follows:
FES1 = 1
2· V 2
C1 · dC1(x)
dx
FES2 = 1
2· V 2
C2 · dC2(x)
dx,
(3)
where VCiis the voltage over the variable capacitors. It
is evident from equation (3) that the electrostatic force isproportional to the bias voltage squared and to the rate ofchange of the capacitance. As previously described, thepresent electromechanical transducer system is a continuoussystem, where the voltage over the variable capacitors is notconstrained and thus does change with time. Consequently,electrostatic forces are not constant over time unlike the casefor constant voltage area-overlapping capacitors. The voltagefor each capacitor is described as
VC1 = q1
C1(x)
VC2 = q2
C2(x),
(4)
where qi is the charge on the variable capacitors Ci . Both thecharge stored in the variable capacitors and the capacitancevalue change with time.
We also included elastic stoppers in our model, becausethey have a significant impact on the dynamic behaviour andthe performance of the device. Since mechanical stoppers arenecessary to be implemented in a real device, they must alsobe considered in the model. The elastic stoppers are modelledas described by Tvedt [17], where the stoppers are representedby springs that come into effect when the displacement of theproof mass is larger than the predefined displacement limitxmax:
FS =⎧⎨⎩
0,
ks(x + xmax),
ks(x − xmax),
−xmax � x � xmax
x < −xmax
x > xmax
. (5)
In this expression, ks is the spring stiffness of the stoppers andshould be chosen much larger than k to account for the rigidcharacteristic of the stoppers. In equation (5), it is assumedthat there is no damping involved when the stoppers comeinto effect; hence, the stoppers act purely elastic. This idealbehaviour of the stoppers is justified in this case since the effectof the stoppers themselves (limitation of the displacementamplitude) on the device behaviour is the main focus of thisinvestigation.
Mechanical damping, e.g. due to internal friction withinthe material (suspension beams) and viscous gas flow inthe cavity, must be considered when modelling resonantelectromechanical systems. The mechanical dampingcoefficient b can be described as a function of the qualityfactor Q of the resonator:
b = 1
Q· ω0 · m, (6)
where ω0 is the mechanical eigen angular frequency and m isthe mass of the proof mass. The Q factor used in the simulation
was calculated from frequency response measurements of thefabricated prototype in accordance with equation (7):
Q = fR
�f, (7)
where fR is the resonance frequency and �f is the bandwidthof the frequency response curve. Finally, the motion of theproof mass is described by Newton’s second law:
mx = −bx − kx − FS + FES1 + FES2 − ma, (8)
where k is the spring stiffness and ma is the excitation forcedue to the acceleration of the device. The motion of theproof mass as described by equation (8) is inherently nonlinearsince electrostatic forces (equation (3)) depend on the voltagesquared (equation (4)) which varies over time. Also, thediscontinuity of the stopper force (equation (5)) contributesto the nonlinear behaviour of equation (8).
A schematic view of the signal-flow diagram of theelectrostatic micro-generator model is shown in figure 3. Themechanical domain, having one degree of freedom, is suppliedwith an acceleration signal a which can be harmonic orrandom. The displacement x of the proof mass is then used tocalculate the instantaneous capacitances Ci of the two variablecapacitors. In the electrical domain, having two degrees offreedom, the charge qi on the variable capacitors is determinedon the basis of the instantaneous capacitance values Ci bysolving equation (1). The voltages VCi
over the variablecapacitors can now be calculated using equation (4). Theinstantaneous voltage VRi
over the load resistor follows fromKirchhoff’s voltage law (all variables are time dependent):
VRi= VBV − VCi
− VCBV , (9)
where VCBV is the voltage over the bias capacitor. Thegenerated peak and rms power is calculated using the followingequation:
Ppeak, rms = V 2peak, rms
Ri
. (10)
By using the space derivatives of Ci together with the voltagesVCi
, the electrostatic forces are determined which couple backinto the mechanical domain. The total simulation time waschosen in such a manner that the system was well in thesteady state region when the simulation finished. The outputparameters of the system model are voltage, current and powerwith respect to the load resistor.
Three types of investigations were carried out: theinfluence of the bias voltage, the effect of the stoppers andthe dynamic frequency response. In the first type, the biasvoltage VBV was varied between 0 V and 60 V in steps of 1 V.The influence of this bias voltage was investigated for fivedifferent excitation levels. In the second type (effect of thestoppers), the excitation level a was varied between 0 g and25 g in steps of 1 g. This was done for five different biasvoltages. The dynamic frequency response was investigatedby performing individual frequency sweeps from 1000 to2000 Hz and vice versa. The model parameters used inthis simulation are based on both the design parameters ofthe transducer structure (as provided in table 1) as well asthe component parameters given by the circuitry (e.g. bias
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J. Micromech. Microeng. 19 (2009) 094001 D Hoffmann et al
x-x
C1(t)
R1 R2C
Calculation Capacitance
Values
C1(t)C2(t)x(t)
Calculation Electrostatic
Forces
Ci(x)
Fi(t)
dC1(x)/dx dC2(x)/dx
VC1(t)
VR(t)
C2(t)VC2(t)
VC1(t)VC2(t)
IR(t)
PR(t)
Feedback
Excitation
Electrical DomainTransducer DomainMechanical Domain
a
Figure 3. Schematic view of the signal flow model including the mechanical domain, the electrical domain and the transducer domain.
Table 2. Model parameters.
Parameter Symbol Value
Spring stiffness of stoppers ks 4000 N m−1
Spring stiffness of suspension k 54 N m−1
Damping coefficient b 5 × 10−5
Bias capacitance CBV 1 μFDisplacement limit of zmax 21 μmvariable capacitorsLoad resistor R1 = R2 = R 560 k�
capacitor, load resistor). The corresponding values of thecomponent parameters are given in table 2. The spring stiffnessof the suspension was re-calculated from frequency responsemeasurements of the prototype device in order to match theresonance frequency of the model to that of the device (table 2).Other parameters include the mechanical damping coefficient(calculated by equations (6) and (7)) and the stiffness ofthe mechanical stoppers. Results on the simulation will bepresented together with experimental data in section 4.
3. Fabrication and experimental approach
3.1. Fabrication process flow
The device is manufactured in silicon utilizing a process flow,which is adapted from a manufacturing process previouslyused at our institute to fabricate gyroscope sensors. Thisprocess uses a single active layer of 50 μm thickness. First, asubstrate wafer is dry-etched to create a cavity of 50 μm depthfor free movement of the proof mass (figure 4(b)). Then a2000 nm thermal oxide layer is produced to provide isolationbetween the substrate and the device layer. Subsequently,a highly p-doped device wafer is bonded onto the substratewafer and thinned to the required thickness by chemicalmechanical polishing (figure 4(c)). In order to realize acapacitive structure, fixed and movable electrodes requireelectrical isolation from each other. This was achieved usinga trench refill technology as described in [18] and is shownin figure 4(d). The trench refill technology also allows thefabrication of track crossings, which are necessary in thisgenerator design. Conductor tracks for contacting the fixedand movable electrodes are formed by wet etching of a 500 nm
thick aluminium layer (figure 4(e)). Prior to the deposition ofthe aluminium layer, a thermal oxide of 200 nm thickness isgenerated and structured to prevent short circuits between themetal tracks. As a last step, the device layer is dry-etchedto create the proof mass, suspensions and comb electrodes(figure 4(f )). For encapsulation of the device wafer, a capwafer is wet-etched to form cavities at the front side (foraccess to the bond pads) and the backside (to allow freemovement of the proof mass) of the wafer (figure 4(h)). Thedevice wafer and the cap wafer are then bonded together usinga glass frit bonding technology. This generates a hermeticseal, which protects the micro-structures from environmentalinfluences and allows operating the generator in a definedvacuum (figure 4(i)). Finally, the wafer is diced to separatethe micro-generator chips, which are then ready for packaging.
3.2. Prototype device
The encapsulated micro-generator chips are packaged intoceramic chip carriers using a conductive epoxy adhesive(figure 5). This allows connecting the substrate to the groundin order to avoid any unwanted charging of the substrate. Thechip includes five bond pads for connecting the generator tothe circuitry. Conventional wire-bonding technology is usedto connect the bond pads to the CLCC package. For protectionof the wire bonds and the generator chip, a ceramic lid may beattached to the package. At this stage, the micro-generatorsare ready for integration on the PCB level together with theelectronic circuitry.
3.3. Experimental approach
The packaged devices were integrated onto a PCB test boardfor characterization (figure 6). The circuitry realized on thetest board complies with figure 2. Therefore, the test boardcontains two load resistors R1 and R2 (560 k� each) and a pairof multilayer ceramic capacitors with a total capacity of 1 μFfor biasing the device.
First, the frequency response was measured with acustomized circuit board in order to determine the resonancepeak fR and the quality factor Q of the micromechanicalresonator. This measurement method does not require ashaker or other instruments. The circuit board provides two
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J. Micromech. Microeng. 19 (2009) 094001 D Hoffmann et al
(a) Substrate wafer
(b) Creation of a cavity
(c) Bonding of a silicon device wafer and thinning
(e) Creation of metal tracks
(f ) Creation of proof mass, electrodes and suspension
(g) Cap wafer
(h) Creation of cavities at top and bottom
(i ) Bonding of cap and device wafer
(d ) Trench refill technology
Figure 4. Fabrication process flow.
Figure 5. Micro-generator chips packaged in CLCC packages.
complementary (positive and negative) drive signals which areapplied to the fixed electrodes of the two variable capacitors,respectively. The drive signal also contains an overlaid carriersignal. The response of the device is then measured usingthe signal taken from the movable electrodes together with thecarrier signal.
Second, the micro-generator test board was physicallyattached to a shaker (Tira GmbH) for experimentalcharacterization. The excitation profile was harmonicwith varying amplitudes and frequencies. Prior to eachmeasurement, the bias capacitors were charged to the requiredbias voltage. The induced peak voltage Vpeak across the
Figure 6. Prototype integrated on a PCB test board forcharacterization.
load resistor R1 was measured with a 10 M� voltage probeconnected to an oscilloscope. The peak power Ppeak wascalculated according to equation (10).
The experimental characterization included four types ofinvestigations (according to the simulation procedure): theinfluence of the bias voltage, the effect of the stoppers, thedynamic frequency response and a long-term measurement. Toinvestigate the effect of both the bias voltage and the stoppers,the voltage VBV was varied between 10 V and 50 V in stepsof 10 V, and the excitation a was varied between 1 g and 22 g.The micro-generator was excited at its resonance frequency.The dynamic frequency response was investigated byperforming individual frequency sweeps from 1400 to 2000 Hzand vice versa at a fixed excitation level but different biasvoltages. The long-term test was carried out in order toinvestigate the robustness of the device. The micro-generatorwas therefore continuously excited at resonance for 2 h. We
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J. Micromech. Microeng. 19 (2009) 094001 D Hoffmann et al
Mechanical Guidance
Interdigidated Comb Electrodes
Proof MassIsolation Trench
Metal tracks
Figure 7. Microscopic close-up view of the electrostatic transducerstructure. The inset in the right upper corner shows a more detailedview of the interdigitated comb electrode.
chose an excitation level high enough (a = 6.5 g, VBV = 30 V)so that the stoppers could come into effect. As a consequence,2920 impacts per second occurred between the proof mass andthe mechanical stoppers.
4. Results and discussion
4.1. Prototype fabrication
In figure 7, a microscopic close-up view of the fabricatedmicro-generator is shown including one of the four mechanicalsuspension units and a comb electrode unit. As can be seenfrom figure 2, there are ten comb electrode units at the longsides of the chip. Each unit comprises two comb electrodesthat change its capacitance in an opposite manner. Thecorresponding comb electrodes are interconnected on the chipvia conductor paths to form two variable capacitors. The trackscrossing the isolation trenches for contacting the movablecomb electrodes attached to the proof mass and the fixed combelectrodes are also visible. The inset in figure 7 providesa more detailed view of the interdigitated finger electrodes.On the basis of linear measurements, we observed a trench-widening effect that occurred due to technology imperfections.This effect produced a gap sizing between 200 nm and 400 nm,which means that the gap between the fingers is larger thandesigned. In the analysed structure (figure 7), the gap wasmeasured to be 2.9 μm on average resulting in a gap sizingof 400 nm. Consequently, the total change of capacitancewas reduced. Moreover, the trench-widening effect leadsto a beam sizing of the suspension beams between 300 nmand 500 nm, i.e. the beam width is smaller than the designvalue. As a result, the resonance frequency of the device shiftstowards lower frequencies. For instance, a beam sizing of400 nm will cause a frequency shift of 240 Hz. In order todevelop electrostatic micro-generators with the best possibletransduction performance, the natural frequency of the devicehas to match the specific vibration frequency of the respectiveapplication. This can be achieved primarily by reducing the
Q 100
Q 133
Q 76
Q 52
0
20000
40000
60000
80000
100000
120000
140000
160000
1100 1200 1300 1400 1500 1600 1700
Frequency (Hz)
Am
plit
ud
e (
arb
itra
ry)
Chip 4448
Chip 4749
Chip 4850
Chip 5246
Chip 5146
Design
Figure 8. Frequency response of five micro-generator devices.
trench-widening effect but also by considering this effect inthe design phase. However, there are MEMS foundries thatalready provide process technologies with very little trenchwidening.
We would like to point out that the yield of workingdevices was rather low. Two major issues were identified.One is related to the dc resistance between the bond pads ofthe micro-generator chip. The dc resistance was measuredbetween the fixed and movable electrodes, the fixed electrodesand the substrate as well as between the movable electrodesand the substrate using a wafer prober and a dedicatedmeasurement unit. We observed that the dc resistance droppedwell below 100 M� for a large number of chips after the capwafer was bonded onto the device wafer. The other issueis related to the bond between cap wafer and device wafer.We found that the tightness of the bond was not uniformacross the wafer so water and sawdust were able to intrudeinto some of the chips destroying the transducer structureand its functionality. Micro-generator chips that showedacceptable dc resistance values (�1 G�) after dicing werefurther characterized.
4.2. Natural mechanical frequency
The resonance frequency was measured for five devices andwas between 1330 Hz and 1480 Hz, which is well belowthe designed value (figure 8). This frequency shift towardslower values is caused primarily by the beam sizing which alsovaries across the wafer. In order to increase the device yieldfor application-specific operation conditions, the beam sizingmust be reduced by further improvement of the fabricationprocess. From figure 8, it can be seen that the amplitude ofthe response curves varies significantly. Consequently, thequality factor Q (calculated by equation (7)) of the mechanicalresonators varies from device to device. We assume that thebond is not uniformly tight and therefore different air pressuresexist inside the chip cavities. This will cause different dampingconditions (e.g. viscous damping) and will thus affect thequality factor. One particular device having a quality factorof Q = 100 was further characterized since its resonancefrequency (1460 Hz) was closest to the design frequency. Thefollowing experimental data originate from this device only.
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J. Micromech. Microeng. 19 (2009) 094001 D Hoffmann et al
Table 3. Comparison of experimental measurements and simulation results (maximum values at resonance frequency).
gF VBV Vpeak Dev. Vrms Dev. Ppeak Dev. Prms Dev.(μm) (V) (V) (%) (V) (%) (μW) (%) (μW) (%)
Experiment 2.7–2.9 30 0.84 – 0.6 – 1.3 – 0.6 –50 1.37 – 1 – 3.4 – 1.75 –
Simulation 1—design according to table 1 2.5 30 0.98 14.3 0.7 14.3 2 35 0.9 33.350 1.6 14.4 1.2 16.7 5.47 37.8 2.56 31.6
Simulation 2—larger gap gF 2.7 30 0.9 6.7 0.65 7.7 1.7 23.5 0.8 2550 1.5 8.7 1.1 9.1 4.67 27.2 2.2 20.5
The spring stiffness k used in the simulation was adjusted to54 N m−1 in order to match the resonance frequency of theprototype device being characterized.
4.3. Maximum voltage and power levels
In a first study we investigated the maximum voltage and powerlevels of the prototype generator as a function of bias voltage.The measurements were compared to the simulation results toevaluate the accuracy of the numerical model (table 3).
In fact, the output power will always increase withincreasing bias voltage if the excitation level is raisedaccordingly. However, the maximum possible bias voltage thatmay be used in the experiments is restricted to the maximumallowable voltage of the ceramic capacitors. In order to avoidany structural damage of the device, we did not use voltagesbeyond 50 V. The maximum output power (applying a biasvoltage of 50 V) was achieved for an excitation level ofapproximately 13 g. Harmonic excitation at the resonancefrequency generated a voltage of 1 Vrms over the load resistor.Since the induced current of 1.8 μArms was driven through twoload resistors R1 and R2, a total maximum power of 3.5 μWrms
was generated.In table 3 the results of the experimental measurements
and the numerical simulations are summarized for biasvoltages of 30 and 50 V. The data are shown for one loadresistor and include the voltage (peak and rms measured) andpower (peak and rms calculated by equation (10)). Table 3 alsocontains the deviation of the measured data from the predictedvalues of the numerical model. Comparing the voltages (peakand rms) of the simulation (simulation 1) and the experiment,results differ by 14.3–16.7%. The deviation is much largerwhen comparing the power levels, where the results differ by31.6–37.8%. In this respect, however, it must be noted thatthe power is calculated using the voltage according to equation(10), which means that the power is proportional to the voltagesquared. It can be shown that the deviation of the calculatedpower is approximately twice the deviation of the voltage.Therefore, the accuracy of the numerical model appears to beinsufficiently low when comparing power levels.
Since the voltage is the measured variable, we suggestevaluating the numerical model on the basis of the voltage.Yet, a deviation as large as 17% is still comparatively high.The main reason for this discrepancy is the uncertainty ofthe parameter gF , the gap between movable and fixed fingerelements. As stated in section 4.1, the gap is affected bythe trench-widening effect, which results in a gap sizing of200–400 nm. This induces a reduction in the total change of
capacitance and hence a decrease in the generated voltage andpower levels. Table 3 also contains the simulation data for atransducer structure with a gap of 2.7 μm. Now the deviationbetween the simulation and experiment is only 6.7–9.1% whencomparing voltages and 20.5–27.2% when comparing powerlevels.
Another reason for the inaccurate predictions from thesimulation is the fact that there are no losses considered in themodel, which apparently exist in the real device. There are,for instance, losses due to leakage currents through parasiticresistors parallel to the variable capacitors. Furthermore, thereis a deterministic measurement error due to the fact that theinput resistance of the measuring instrument is not infinite(10 M�).
4.4. Effect of mechanical stoppers
In figure 9, the effect of the mechanical stoppers on thegenerated power is shown. In general, both simulation andexperimental data suggest that the generated power increaseswith the square of the acceleration a at low excitation levels.Depending on the bias voltage, the generated power starts tolevel off at a critical excitation level. This is due to the fact thatthe proof mass starts to impact at the mechanical stoppers, andtherefore oscillates at its maximum possible displacement. Afurther increase in excitation does not cause a further increasein displacement or capacitance change, respectively. Instead,the stoppers cause the power to decline continuously at a verylow rate. From figure 9, it becomes evident that the maximumamount of energy that can be generated is strongly dependenton the strength of the electrical damping force, which againis dependent on the bias voltage (equation (3)). Therefore,the amount of mechanical energy that is required to operatethe device at its maximum inner displacement amplitude mustincrease with higher bias voltages. Consequently, the biasvoltage must be recognized as a significant design parameter,when designing customized micro-generators with a givenoutput impedance.
According to figure 9, there is a good qualitativeconformance between simulation results and experimentaldata. The device behaviour, in particular the effect of thestoppers, is well described by the numerical model. Theoccurrence of a plateau above a critical excitation level iswell predicted. The level of the plateau increases with thebias voltage in a nonlinear manner. This can be seen inboth simulation and experiment. The mismatch betweenthe simulation and experimental results with respect to theabsolute values has been discussed in the previous section.
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J. Micromech. Microeng. 19 (2009) 094001 D Hoffmann et al
0
1
2
3
4
5
6
0 5 10 15 20 25
Excitation (g)
R1 P
ea
k P
ow
er
(μW
)
50V
40V
30V
20V
10V
(a) (b)
0
0.5
1
1.5
2
2.5
3
3.5
4
0 5 10 15 20 25
Excitation (g)
R1 P
ea
k P
ow
er
(μW
)
50 V
40 V
30 V
20 V
10 V
Figure 9. Effect of mechanical stoppers: (a) simulation, (b) measurement.
0
1
2
3
4
5
6
0 10 20 30 40 50 60
Bias Voltage (V)
R1 P
ea
k P
ow
er
(μW
)
12.87g
8.586g
4.29g
2.12g
(a)
0
0.5
1
1.5
2
2.5
3
3.5
0 10 20 30 40 50 60
Bias Voltage (V)
R1 P
ea
k P
ow
er
(μW
)
12.87g
8.58g
4.29g
2.14g
(b)
Figure 10. Effect of bias voltage: (a) simulation, (b) measurement.
4.5. Influence of the bias voltage
The influence of the bias voltage is shown in figure 10. Fromboth simulation results and experimental data, an optimal biasvoltage can be found for a specific excitation level at which thegenerated power is at its maximum. The power increases ina nonlinear manner until the optimal point is reached. At thispoint of maximum power, the proof mass oscillates with themaximum possible displacement, without the occurrence ofan impact at the mechanical stoppers. Below the optimal biasvoltage, the generator operates in impact mode, i.e. the motionof the proof mass is impeded by the mechanical stoppers andthus impacts occur. For bias voltages higher than the optimalvalue, the displacement amplitude of the proof mass starts todecline since the influence of the electrostatic damping forcebecomes increasingly stronger. Therefore, when designingelectrostatic micro-generators, the bias voltage (e.g. providedby an electret) requires to be designed with respect to theapplication. The optimal bias voltage will also depend onthe value of the load resistor. However, in practice, theload is usually represented by a rather complex circuit andits impedance cannot be changed easily once the circuit ismanufactured. Therefore, it is more useful to adjust thepolarization voltage for different application scenarios while
using the same harvester circuit. This means that the circuitcan be fabricated in large numbers at low cost. The bias voltagecan be adjusted by the charging conditions of the capacitor orthe electret.
Again, according to figure 10, there is good conformancebetween simulation and experiment. The model is able topredict the device behaviour including the presence of anoptimum bias voltage. As in the previous case, the measuredpeak power is lower than the one predicted by the model. Also,the values for the optimal bias voltages are slightly higher thanpredicted. We assume the same reasons as described abovefor this discrepancy.
4.6. Long-term excitation
We also performed a long-term experiment, in which thegenerator was continuously excited at its resonance frequency(1460 Hz) over a period of 2 h. We chose an excitation levelof 6.5 g above the critical value in order to induce impacts ofthe proof mass at the mechanical stoppers. The bias capacitorwas initially charged to 30 V. The generated peak voltage wasmeasured every 15 min. In figure 11, the rms power is shownover time for the load resistor R1. After 2 h, the generatedrms power dropped by 50% from 1.26 to 0.6 μW. During this
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J. Micromech. Microeng. 19 (2009) 094001 D Hoffmann et al
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 20 40 60 80 100 120 140
Time (min)
RM
S P
ow
er
(μW
)
Figure 11. Long-term excitation: rms power over a time span of120 min.
time span, the voltage VBV of the bias capacitor pair declinedto 21 V. This voltage loss corresponds to an energy loss of460 μJ according to the energy function of a capacitor.The total energy generated by the micro-generator was twice5920 μJ. As a result, a total net energy of 11.38 mJ wasgenerated. This is equivalent to an average power of 1.58 μWover a period of 2 h. The energy loss of the bias capacitorsis due to leakage currents between the variable capacitorsand between the proof mass and the substrate wafer. Inconclusion, this experiment demonstrates effective conversionof mechanical to electrical energy by means of the electrostatictransduction mechanism. The use of a pre-charged capacitoras a bias voltage source is not practical for developing energyautonomous systems due to immanent losses (e.g. leakagecurrents because of finite resistances of capacitors) in thesystem. The use of permanent electrical charges buried ina dielectric layer (i.e. an electret) is essential for reliableoperation of the electrostatic micro-generator.
4.7. Dynamic frequency response
In order to investigate the dynamic behaviour of the micro-generator, a frequency sweep was carried out with a fixedexcitation amplitude y of 1.5 μm. The bias capacitor was
0
0.5
1
1.5
2
2.5
1000 1200 1400 1600 1800 2000
Frequency (Hz)
R1 P
ea
k V
olta
ge
(V
)
50V 16g up-sweep
50V 14g up-sweep
50V 16g down-sweep
30V 7g up-sweep
30V 6g up-sweep
30V 7g down-sweep
(a)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
1300 1400 1500 1600 1700 1800 1900 2000
Frequency (Hz)
R1P
eak V
olta
ge (
V) Frequency sweep up - 50V
Frequency sweep down - 50V
Resonance frequency
Frequency sweep up - 30V
Frequency sweep down - 30V
(b)
Figure 12. Dynamic frequency response: (a) simulation, (b) experiment.
pre-charged at different voltages to vary the bias of thedevice. Figure 12(b) shows measurement data (peak voltagegenerated over load) for a bias voltage of 50 V. At theresonance frequency (1460 Hz), a peak voltage of 1.45 V wasgenerated. When performing a frequency sweep upward, themicro-generator followed the excitation frequency continually,i.e. the amplitude of the voltage generated at the resonancefrequency increased slightly at a constant rate. This behaviouris also predicted by our model according to figure 12(a), andwas also observed for lower bias voltages (20 V and 30 V).Using a fixed excitation amplitude y, the acceleration levela increases with increasing frequency in accordance witha = ω2y. Consequently, if the amplitude of the proof massreaches the displacement limit at the resonance frequency, themicro-generator starts to operate in impact mode with furtherincreasing frequency, i.e. persistent collisions occur betweenthe proof mass and the mechanical stoppers. Assuming apure elastic characteristic of the mechanical stoppers, thedynamic behaviour of the generator is comparable with that ofa piecewise-linear oscillator [19]. Piecewise-linear oscillatorsexhibit a broader bandwidth characteristic for up-sweepexcitations. This increase in the bandwidth may enhancethe performance of the micro-generators since application-specific vibration profiles are usually random and broadband.From experimental data (figure 12(b), down-sweep), it is alsoevident that the micro-generator shows nonlinear Duffing-likebehaviour (softening). This however was not clearly predictedby the numerical model (figure 12(a)).
5. Conclusions
We successfully fabricated electrostatic micro-generators witha chip size of 5 mm by 6 mm in silicon using standard MEMSfabrication processes. The micro-generator devices showedresonance frequencies between 1300 Hz and 1500 Hz. Amaximum voltage of 1 Vrms and a current of 1.8 μArms weregenerated using a bias voltage of 50 V (harmonic excitationat resonance, 13 g). With respect to the load (2× 560 k�), acorresponding total output power of 3.5 μWrms was generated.The experimental data show that there is an optimal bias
10
J. Micromech. Microeng. 19 (2009) 094001 D Hoffmann et al
voltage where the output power is maximal. Consequently,the bias voltage has to be considered as a significant designparameter with respect to the excitation conditions of thecorresponding application. At larger excitations above acritical level, the mechanical stoppers come into effect causingthe output power to flatten and weakly decrease. In a long-term experiment, we demonstrated effective conversion ofmechanical to electrical energy. There was no evidence ofdegradation despite operating the device in impact mode. Wealso found that the bandwidth during a frequency up-sweepcan be significantly increased when the generator is operatedin impact mode.
In order to investigate the dynamic system behaviour ofour electrostatic micro-generators, a numerical model wasdeveloped. This nonlinear model considers electrostaticcoupling between the mechanical and electrical domainsas well as the effect of the mechanical stoppers at thedisplacement extremes. The predicted device behaviourprovided by the simulation results is in good agreement withthe experimental data. We will further enhance our numericalmodel to account for losses that exist in a real device. With aspecial focus on device optimization, parameter studies will becarried out with respect to relevant design parameters. We willalso investigate the effect of non-harmonic excitation profiles(random and wideband) on the power generation capabilitiesof our micro-generators. For further device development, it isplanned to investigate strategies for implementing a permanentbias voltage source (e.g. an electret).
Acknowledgment
The authors would like to acknowledge the funding of theZOFF III project ‘Energieeffiziente Autonome Mikrosysteme’by the government of Baden-Wurttemberg.
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