high sensitivity electret accelerometer with integrated fet

1770 IEEE SENSORS JOURNAL, VOL. 14, NO. 6, JUNE 2014

High Sensitivity Electret AccelerometerWith Integrated FET

Joachim Hillenbrand, Tilman Motz, and Gerhard M. Sessler, Fellow, IEEE

Abstract— Electret accelerometers with high sensitivity andlow equivalent noise level are presented. An integrated fieldeffect transistor is used as a high impedance voltage amplifier.The voltage is generated between a metallic seismic mass anda backplate, covered by a charged electret film. These twoelectrodes are separated by a soft cellular polymer ring. Uponapplication of an external acceleration, the inertial forces ofthe seismic mass change the thickness of the ring and thusthe distance of the electrodes. Sensitivities of up to 600 mV/gand equivalent noise voltages as low as 80 µg were measuredwith the electret accelerometers. High quality accelerometers withflat frequency responses can either be obtained with completelyclosed or opened air gap volumes. For each of these twoimplementations, an analytical model is presented, assuming theabsence of damped air streaming effects. Both model variantsare discussed and calculated data from the models are comparedwith measured results obtained with an accelerometer operatedwith various internal static pressures on the cellular spacer ring.

Index Terms— Accelerometer, electret, electret transducer,cellular polymer, ferroelectret, piezoelectret, fluoropolymer.

I. INTRODUCTION

ACCELEROMETERS [1] are often used as acousticalsensors [2], [3] for structure-borne sound. In this appli-

cation, an accelerometer is placed on a solid object and theacceleration of the surface, which is due to waves propagatingin the solid, is measured. From the measured acceleration,velocity and deflection of the surface can be obtained byintegration. The majority of the accelerometers produced todayare, however, used in numerous non-acoustic applications suchas airbag deployment systems of cars, smartphones, tabletPCs, radio-controlled quadro- and helicopters and controllersof video game consoles. In these applications often very lowfrequency or even static accelerations, originating from thegravity of earth, have to be measured. The most common typesused today are capacitive, piezoresistive, and piezoelectricaccelerometers [4].

Some years ago, a special type of piezoelectric accelerom-eter, based on piezoelectrets [5]–[7] or ferroelectrets [8], was

Manuscript received September 25, 2013; revised January 14, 2014;accepted January 15, 2014. Date of publication January 23, 2014; dateof current version April 14, 2014. This work was supported in part bythe Hessische Ministerium für Wissenschaft und Kunst and in part bythe Deutsche Forschungsgemeinschaft. The associate editor coordinating thereview of this paper and approving it for publication was Prof. Sang-Seok Lee.

The authors are with the Institute for Telecommunications Technology,Darmstadt University of Technology, Darmstadt D-64283, Germany (e-mail:[email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSEN.2014.2302300

introduced [9]. These piezoelectric materials are made ofmechanically very soft open or closed cell polymer electretsand show, after corona charging, comparable d33 and muchhigher g33 constants than lead zirconate titanate (PZT).

Various such piezoelectret accelerometers, types with chargeor voltage output, with an open housing or with a closed metalhousing and integrated field effect transistor (FET), were builtand investigated [9], [10]. It was shown that piezoelectretaccelerometers have high charge and even higher voltagesensitivities and low equivalent noise levels, can be usedat very low frequencies, and can be built lightweight andinexpensively.

However, the thermal stability of piezoelectret accelerom-eters based on common polypropylene (PP) ferroelectrets isunsatisfactory for temperatures above 60 °C and thus insuf-ficient for many potential applications. Layered ferroelectretsconsisting of fluoropolymer films are available since severalyears and show increased thermal stability of up to about90 °C [11]–[13]. With various such film types piezoelectretaccelerometers were built and tested. For comparable sensitiv-ities of the PP and fluoropolymer accelerometers significantlylower resonance frequencies and additional mechanical stabil-ity problems of the latter types were found. This is due to thelower Young’s modulus and the mechanical structure of thelayered ferroelectret films.

Since it was, and still is, questionable whether the mechan-ical drawbacks of layered fluoropolymer ferroelectrets can besolved in the future, an alternative path for obtaining thermallystable accelerometers, for temperatures even higher than the90 °C, was taken and electret accelerometers were built andinvestigated. In these accelerometers, uncharged cellular PPand charged fluoropolymer are employed. The mechanicallysuperior cellular PP film is used as a spring and supplies therestoring force while the electrically superior fluorinated ethyl-ene propylene (FEP) film stores the electret charges. Thus, thebasic design of the electret accelerometers was inspired by thecommon electret microphones [14]: Spacer ring and membraneof the microphones were replaced in the accelerometers by asoft cellular PP ring and a metallic seismic mass, respectively.Due to these similarities, the thermal stabilities of electretmicrophones and electret accelerometers are expected to becomparable. Since electret microphones can be used withoutsignificant loss of sensitivity in locations with elevated tem-peratures and high humidity [15], a broad field of potentialapplications is also expected for electret accelerometers.

In the past, some work on electret accelerometers withouta cellular ring has been done: Early investigations on electret

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HILLENBRAND et al.: HIGH SENSITIVITY ELECTRET ACCELEROMETER WITH INTEGRATED FET 1771

microphones showed that their vibration sensitivity is muchsmaller than that of other microphone types [16]. However,by simply adding additional mass to the electret micro-phone membrane the vibration sensitivity can be drasticallyincreased, as required in accelerometers [17]–[19].

For the first electret accelerometers with a cellular spacerring a relatively large and heavy experimental design without ashielding housing was used [20]. With this design, the seismicmass, the static force onto the mass, and geometric quantitiesof the accelerometer, such as the cellular ring outer and innerdiameters and the air gap distance, can be easily changedand their influence on the sensitivity and basic properties ofthe accelerometer can be rapidly tested. However, the designdoes not facilitate the determination of typical accelerometerquantities like cross-axis sensitivity, microphone sensitivity,and noise level.

For the present paper, therefore, a smaller and lighteraccelerometer design with an integrated FET and a metal-lic housing was developed and investigated. The design,an analytical model, and measurements made with theseaccelerometers are presented in the following.

II. ACCELEROMETER DESIGN

In contrast to the older design, which provides a voltage ora charge output, the new accelerometer design is limited to avoltage output. For voltage amplifiers, best results concerningsensitivity and noise level are obtained when the amplifier,or at least its first high impedance transistor stage, is bothplaced as close as possible to the electret transducer and iscarefully shielded from electromagnetic noise. Close place-ment ensures low parasitic capacitances and highest possibleinput voltage at the amplifier. Since the first transistor stageof a high impedance electret transducer is primarily used asan impedance converter, the shielding of the high impedanceinput side is by far more important than that of the lowimpedance output side. In view of these requirements, a smallPCB board, containing a junction gate field-effect transistor(JFET) and two resistors, was integrated in the metal housingof the electret accelerometer.

The design and a photograph of an electret accelerometerare presented in Fig. 1. The housing consists of two partswhich are connected by a thread. Turning the two housingparts relative to each other changes the total length of thehousing, and thus allows to continuously adjust the length ofthe spring element which consists of a soft polymer foam. Theadjustable compression of the spring element generates a staticforce onto the seismic mass which in turn is pressed onto thecellular ring. This serves as a second spring and adjusts also,in conjunction with a recess in the backplate, the height ofthe air gap. The seismic mass is thus anchored between twosprings which allow movement and vibration in measuring(vertical) direction but restrict movements in other directions.Electrically, the seismic mass is connected to the groundedhousing by the conducting spring element. The cylindricalbackplate electrode, with a conical cavity on its rear side,covers the component side of the PCB and is electricallyconnected to the gate of the JFET. A metalized FEP-film is

Fig. 1. Cross-section (left) and photograph (right) of an electret accelerom-eter. The housing consists of two parts which are connected by a thread.

cemented onto the front side of the backplate electrode and iscorona-charged to about 400 to 500 V before assembling.

The magnitude of the adjustable static force generated bythe above mentioned spring element is essential for the oper-ation of the accelerometer since it influences both sensitivityand resonance frequency due to several effects. These effectsare the nonlinear and pressure dependent Young’s modulus ofthe cellular ring, the magnitude of the spring constant of theair volume enclosed by the cellular ring, the variation of theeffective contact area of the two cellular ring surfaces andthe leakage of air through these interfaces as a function ofthe applied static pressure. These effects will be discussed indetail in the Sects. V and VI of the paper.

III. SPACER RING OF CELLULAR POLYPROPYLENE

An important element of the electret accelerometers is thespacer ring made of soft cellular polymers [21]. In contrastto spacer rings in conventional electret microphones, thecellular spacer rings fulfill several functions: Firstly, the height(or thickness) of the air gap is adjusted by the ring. Sincesmaller air gap thicknesses than the cellular ring thicknessesare desirable, a ring-shaped recess in the backplate wasimplemented. Secondly, the cellular ring is compressed andexpanded by the accelerating forces of the seismic mass, i.e. itacts as a spring. Thirdly, surfaces of cellular polymers are quiterough. When such materials are in contact with other materials,such as FEP or brass in the case of the present accelerometer,and a relatively low contact pressure is applied, the interfacesare not airproof for low frequencies. The contact pressure,in the case of the present accelerometer the static force ofthe spring element, thus determines whether the enclosed airvolume inside the cellular ring acts as a spring or as a damper.

In the present accelerometer, a cellular polymer made ofPP, with an air volume content of about 50% and a closedcell structure with lens-like air voids was used. These foamsare about three orders of magnitude softer than the basematerial PP [21]. Because of the lens-like air voids and thelow stiffness, cellular PP can be rendered highly piezoelec-tric by corona charging [6]–[8]. Pressure expansion meth-ods were developed [22], [23] to increase the piezoelectric


Fig. 2. Frequency response of Young’s modulus of an expanded cellularPP film. The sample was electrically excited with AC- and DC-voltagesand the frequency response of Young’s modulus was calculated from theinterferometrically measured vibration amplitude.

d33-coefficients mainly by decreasing Young’s modulus Y ofthe cellular films. These methods not only affect the size ofYoung’s modulus but also its frequency response. Since coronacharging has only little or no influence on the mechanicalproperties of the cellular films, the results of the formerpressure expansion studies with PP piezoelectrets can bedirectly used to obtain optimized cellular ring materials forelectret accelerometers.

As a typical example, measurements of the frequencyresponse of Young’s modulus for an expanded and unchargedPP film are shown in Fig. 2.

The measurements were made by using an interferometer.An AC-voltage (method 1) or a DC-voltage plus an AC-voltage (method 2) were applied to the metallized films andthe second harmonic or the fundamental, respectively, of theperiodic thickness variations were measured interferometri-cally [24]. Since the forces generated by the applied voltagesare known, Young’s modulus of the film can be calculatedfrom the interferometric measurements. For both measuringmethods essentially the same response of Young’s modulus,namely a slight increase from 0.8 to about 1 MPa withincreasing frequency, as shown in Fig. 2, was obtained. Theslight increase of Young’s modulus with increasing frequencyis essentially due to the viscoelastic properties (cf. Sect. VI)of the polymer and will eventually lead to slightly decreasingsensitivities of the electret accelerometers.

Besides their small Young’s modulus, the relatively flatfrequency response is nevertheless a further advantage of theexpanded cellular PP materials.

IV. MEASUREMENTS

Based on the design presented in Sect. III, variousaccelerometers with different air gap thicknesses, electretvoltages and spring elements were built and characterized.For the characterization, acceleration measurements, both inthe nominal working direction and perpendicular to this direc-tion, acoustical measurements and noise measurements wereperformed.

TABLE I

PROPERTIES OF THE ELECTRET ACCELEROMETERS

For the acceleration measurements the electret accelerom-eters are mounted on an electro-dynamic vibration exciter(B&K 4809), which is driven by a power amplifier(B&K 2713). An audio analyzer (R&S UPD) is used togenerate the signal for the power amplifier and to record andanalyze the output signals of the electret accelerometers.

A typical measuring program of the audio analyzer was usedfor the acceleration measurements: Sinusoidal excitation sig-nals, gradually increasing frequency in the range from 10 Hz to5 kHz, digital band-pass filtering of the accelerometer outputsignal, and calculation of the amplitude. In some experiments,however, white noise was used as the excitation signal and theoutput of the electret accelerometer was analyzed with a FastFourier Transform (FFT). As expected, very similar frequencyresponses were obtained with both measuring methods.

Mainly due to the electro-dynamic principle of the vibrationexciter, the measuring setup does not have a flat frequencyresponse. For a constant input voltage, the resulting accelera-tion of the shaker significantly decreases below 100 Hz and forfrequencies above 2 kHz. Therefore, calibration measurementsfor the setup were performed with commercial accelerome-ters (B&K 4332 and 4344) by using a substitution method.Different loads on the shaker were compensated by mount-ing additional masses on the shaker during the calibrationmeasurements. This exactly ensures the same accelerationsduring the measurements with the commercial accelerome-ter and during the measurements with the various electretaccelerometers.

In the following, various measurements with different elec-tret accelerometers are presented. In Table I values or valueranges for a number of quantities of these electret accelerom-eters are summarized.

In Fig. 3, six measured frequency responses of the voltageoutput sensitivity of an electret accelerometer with a 25 μmthick FEP electret film, charged to 400 V, an air gap thicknessof 26 μm, and a seismic mass of 2.13 g are shown.

The six different frequency responses were obtained byvarying the static pressure generated by the spring elementonto the seismic mass and the cellular ring. For all staticpressures, the expected (cf. Sect. III) slightly decreasingresponse for lower frequencies and a resonance peak at higherfrequencies was observed. Furthermore, increasing the static


Fig. 3. Six frequency responses of the sensitivity of an electret accelerometerfor different static pressures on the cellular PP ring. The FEP film was chargedto 400 V. The six short dashes on the left-hand side are calculated sensitivitiesand the numbers are calculated Young’s moduli in MPa.

Fig. 4. Three frequency responses of the sensitivity of an electret accelerom-eter for different static pressures on the cellular PP ring. The FEP film wascharged to 400 V and the air gap thickness is nominally 5 μm.

pressure lowers the sensitivity of the microphone and increasesits resonance frequency. The sensitivities measured for thisaccelerometer range from about 60 to 300 mV/g while res-onance frequencies from 1800 to 3800 Hz were observed.It should be noted, that after turning the two housing partsfor changing the static pressure in the accelerometer, creepingeffects both in the cellular ring and the spring element lead toslow variations of the sensitivity and the resonance frequencyeven after the turning process is finished. After about 0.5 h theaccelerometers are stable over time and measurements weretypically performed after such a period of time. The six shortdashes shown on the left-hand side of the figure are calculatedsensitivities and the numbers are calculated Young’s moduli.These calculated values will be discussed in Sect. VI.

Another electret accelerometer with a higher sensitivity wasused for the frequency response measurements presented inFig. 4. The electret film was also charged to 400 V, theair gap thickness is nominally only 5 μm and the seismicmass is again 2.13 g. The higher sensitivity, compared to theaccelerometer presented first, is thus due to the small air gapand the resulting high electric field in the gap. Again, thestatic pressure inside the accelerometer was varied for the

Fig. 5. Cross axis sensitivity in x- and y- direction of an electret accelerom-eter with an air gap thickness of 31 μm and an electret voltage of 500 V.For comparison, the sensitivity of the accelerometer in normal (z-) directionis also shown in the figure.

three measurements presented. For the lowest static pressure asensitivity of nearly 700 mV/g and a resonance frequency ofabout 2 kHz was measured. For the highest static pressure, thesensitivity is reduced to about 150 mV/g while the resonancefrequency is increased to 4.5 kHz. Calculated sensitivities andYoung’s moduli for this accelerometer are not shown in Fig. 4nor discussed in Sect. VI since the air gap thickness of 5 μm,which is required for these calculations, is only a nominalvalue and is subject to a significant uncertainty.

The presented frequency responses in the last two figuresshow a special feature of electret accelerometers: A higherresonance frequency and a larger operating frequency rangecan be obtained at the expense of a lower sensitivity bysimply increasing the static pressure of the spring element.Thus, tailored accelerometers for specific applications can beobtained by this approach.

Besides the sensitivity (in working direction) and theresonance frequency of an accelerometer the – unwanted –sensitivities to some other disturbance variables is of impor-tance for its quality. Such variables are accelerations perpen-dicular to the working direction and the sound pressure level.Frequency responses of the sensitivity for these variables arepresented in the next two figures.

Fig. 5 shows three frequency responses of an electretaccelerometer measured in z-, x-, and y-direction. Theaccelerometer has an air gap thickness of 31 μm and anelectret voltage of about 500 V. The sensitivity in normal (z-)direction of about 200 mV/g and the resonance frequency ofalmost 3 kHz are comparable to data of the accelerometerspresented before. The sensitivities of the two measurementsperpendicular to the working direction (x- and y-) are smallerthan 5 % of the normal sensitivity for medium frequencies. Forlower and higher frequencies the apparent cross axis sensitivityis higher. However, a significant portion of these cross axissensitivity values is caused by cross axis excitation due to theimperfect shaker, especially near some of its resonances.

Acoustical measurements with the electret accelerometerswere carried out in a normal laboratory room to obtainthe microphone sensitivity of the accelerometers. A dynamic


Fig. 6. Microphone sensitivity of an electret accelerometer measured for twodifferent orientations of the accelerometer.

loudspeaker at a relatively short distance of 0.5 m fromthe accelerometer as sound source and the audio analyzer(R&S UPD) were used. The reference sound pressure levelwas determined by substituting the electret accelerometer witha 1/2”-condenser microphone (B&K 4191), a preamplifier(B&K 2669) and a measuring amplifier (B&K 2636). Typicalresults for the acoustic sensitivity of an electret accelerometerare depicted in Fig. 6. The small difference between the twomeasurements is caused by a 90°-rotation of the accelerometerprior to the second measurement.

With increasing frequency, the acoustic sensitivity firstdecreases up to about 1 kHz, then increases and reachesa maximum at approximately 3 kHz. The response at lowfrequencies is caused by tiny leaks of the accelerometerhousing. As is well known, this effect decreases with increas-ing frequency and thus the acoustic sensitivity shown in Fig. 6also decreases. This lower part of the frequency response couldbe further minimized by the use of a thicker and heavierhousing with diminished leakage. At higher frequencies, theaccelerometer resonance at 3 kHz determines the frequencyresponse. This resonance is excited by the vibrations of thetransducer element, caused either directly by an excitation ofthe housing or indirectly by excitation of its mounting. Whilethe first part is an intrinsic problem of any accelerometer,which is difficult to solve, the second part is independent ofthe accelerometer and could be further minimized by bettervibration isolation of the mounting.

The high sensitivity of electret accelerometers was shownat the beginning of this section. However, the sensitivity ofa sensor is related to the noise produced by this sensor.Thus, noise measurements were performed and they provedthat the output noise of the investigated electret accelerom-eters is dominated by the noise of the integrated FET. TheFFT spectrum of the output voltage of a typical electretaccelerometer in the absence of any acceleration and in a quietsurrounding is shown in Fig. 7. The electronic circuit used forthe measurements, with all electronic components inside andoutside the accelerometer housing, is presented as an inset inthe figure.

The FFT spectrum corresponds to a noise voltage of 50 μVin the frequency range from 2 to 20 kHz. For the most sensitive

Fig. 7. FFT noise spectra of a typical electret accelerometer. The spectrumcorresponds to a noise voltage of 50 μV in the frequency range from 2 to20 kHz. The electronic circuit used for the measurements is presented asan inset. FET and two resistors are on a small PCB encapsulated inside themetallic housing of the accelerometer.

Fig. 8. Magnified cross section of the accelerometer components around theair gap. The formula symbols used for the model calculations are given inbrackets.

accelerometer presented in Fig. 4, with a sensitivity of about600 mV/g, the equivalent noise voltages is approximately80 μg. This is a very low value and adequate for mostapplications.

V. MODEL CALCULATIONS

The part of the electret accelerometer, relevant for thediscussion of a model, is depicted again in Fig. 8. The formulasymbols used in the following are given in brackets. First, theworking principle of the electret accelerometer will be brieflyexplained.

Seismic mass, air gap, FEP film, and the backplate forma capacitor. Since the FEP-electret film is charged, also thiscapacitor, with its two electrodes seismic mass and backplate,is charged. Its capacitance is inversely proportional to the airgap thickness. The air gap thickness is controlled by severalstatic and dynamic forces.

These forces will be discussed below in more detail. Whenthe accelerometer housing is accelerated, the inertia of the seis-mic mass works against these forces and eventually changesthe height of the air gap. This changes the capacitance andgenerates a voltage across the grounded seismic mass and thebackplate, which is connected to the gate of the FET on theinternal PCB. The FET, electrically shielded by the groundedmetallic housing, is used as first stage of an impedanceconverter or a high-impedance amplifier.


Gate capacitance and other parasitic capacitances reducethe open circuit output voltage at the backplate. However,this reduction and thus the real voltage at the gate can becalculated by means of a capacitive voltage divider if allcapacitances are known. Gate voltage and output voltage ofthe accelerometer are related by the amplification factor ofthe circuit, consisting of the internal FET and two internaland some external resistors.

Therefore, the output voltage of the total circuit, whichactually is the output voltage of the accelerometer, can becalculated, if the open circuit voltage at the backplate or,in other words, the open circuit sensitivity of the electretaccelerometer is known. An analytical model for the lattercalculation will now be outlined.

First, the electrical properties of the accelerometer areconsidered.

After corona charging all electret charges with area den-sity σ are located nearby or directly at the surface of theFEP film. The surface potential VE generated by the electretcharges and σ are then related by σ = ε0εVE /tE . If backplateand seismic mass are temporarily short-circuited, the electricfield inside the air gap E A can be written as a function of σor VE [15]

E A = σ tE

ε0(ε tA + tE )= ε VE

(ε tA + tE ), (1)

where tE and tA are the thicknesses of electret film and airgap, respectively, ε0 is the vacuum permittivity, and εis therelative permittivity of the electret film. Under the assumedopen-circuit conditions, an infinitesimally small voltage dV isgenerated when the air gap thickness tA is changed by dtA.Since the electric field given by (1) remains constant one canwrite for the voltage generated across backplate and seismicmass:

dV = E AdtA = εVE

(εtA + tE )· dtA. (2)

Second, the mechanical properties of the accelerometer areconsidered.

The seismic mass ms and a certain number of springsand dampers form a damped mass-spring-system. It will bediscussed now, which springs and dampers should be takeninto account and which can be neglected without a significantloss of accuracy. Three springs are obvious. These are thecellular polymer ring with thickness tP , area AP , and Young’smodulus Y (spring 1), the enclosed air in the gap betweenback plate and electret film with thickness tA and area AA

(spring 2), and the spring element (spring 3). All three springsare mechanically in parallel. In contrast to springs 1 and 2,which define or depend on the air gap thickness tA , spring 3is independent of the electrically important tA. Therefore, itsspring constant can be chosen significantly smaller than thetwo others, in order to obtain maximum output signal.

In practice, this was achieved by using a soft material witha comparatively large thickness. The spring element (spring 3)can therefore be neglected in the model.

The modeling of spring 2 depends on the static pressure onthe cellular ring since its two non-glued contact interfaces withthe FEP film on one side and the seismic mass on the other

side can be either air-tight (high static pressure) or permeable(low static pressure) or something in between (medium staticpressure). Therefore, for high static pressure, the enclosed airin the gap acts as a spring, for low static pressure this springdoes not exist and for medium pressure the air gap has tobe modeled as a spring in series with a damper. Certainly, thediscussed air permeability is frequency dependent. This meansthat in a certain frequency band damping and the damperare always present for any given static pressure. However,experimentally such a frequency band, where the sensitivityof the accelerometer should theoretically decrease inverselyproportional to the frequency [25], always lead to unwantednon-flat frequency responses. Thus, for simplification in thefollowing, the enclosed air gap volume is either modeled as aclosed volume, which results in a spring, or as a completelyopened volume, then no such spring exists. “Completelyopened” means that air streaming between the air gap volumeand the housing exists without any damping effects.

While experimentally damping was not yet clearly observedbelow the resonance frequencies of the accelerometers, therelatively small Q-factors of the resonances are due to damp-ing. This damping is, however, dominated by the viscoelasticproperties of the cellular polymer ring and will not be treatedin the model.

Finally, spring 1 (the cellular polymer ring) is modeledin the following as a simple spring element. However, twofurther springs related to the cellular ring could be addedto the model, when very low static pressures are present inthe accelerometers. This is due to the fact, that the cellularring is just clamped between FEP film and seismic mass. Forvery small static pressure on the ring its two actual contactareas with FEP film and seismic mass are smaller than thegeometric areas. This is due to the relatively rough surfaceof the cellular material. Thus, two irregular air gaps exist atboth interfaces of the ring which are, however, compactedand finally disappear in the case of increasing pressure. Note,that these two interfaces were already mentioned above in thecontext of the air-tightness of the air gap volume. Formally,the two irregular air gaps are non-linear springs mechanicallyin series with the cellular ring (spring 1). Since these twosprings are only relevant for very low pressures and since theiranalytical description is difficult or even impossible, they willbe neglected in the following formulas.

The last paragraphs can be summarized as follows:A reasonable analytical model for an electret accelerometercan be simplified to either one or two springs. The modelvariant with one spring is appropriate when the air gap volumeis quasi opened and air can stream from the small air gapvolume to the large housing volume or vice versa withoutsignificant damping. The model variant with two springs inparallel is suitable, when the air gap volume is sealed dueto a high enough static pressure on the cellular ring, whichprohibits air exchange through the two interface layers of thecellular ring.

The stiffnesses of the two springs can be calculated bytaking into account the definition of Young’s modulus ofthe cellular ring and by considering an adiabatic change ofthe enclosed air volume, respectively. With Hooke’s law and


the stiffnesses of one or two springs in parallel the force can becalculated when one or both springs are deflected by the smalldistance dtA. The counterforce msda is generated by a smallacceleration da of the seismic mass ms. Thus, for small staticpressures and opened air gap volume in the context of the1-spring model variant one can write

msda = Y · AP

tPdtA, (3a)

while for higher static pressures, sealed air gap volume andthe 2-spring model variant the balance of forces is given by

msda =(

Y · AP

tP+ γ p0 AA

tA

)dtA, (3b)

where p0 is the atmospheric pressure and γ the ratio of specificheats (or the adiabatic index).

Now, by inserting (3a) or (3b) into (2) the voltage sensitivitySV of the electret accelerometer, well below resonance, can beobtained for low static pressures (1-spring model variant) as

SV = dV

da= ε VE

(ε tA + tE )·(

Y · AP

tP

)−1

· ms (4a)

and for medium and higher static pressures (2-spring modelvariant) as

SV = dV

da= ε VE

(ε tA + tE )·(

Y · AP

tP+ γ p0 AA

tA

)−1

· ms. (4b)

The resonance frequency fres = ωres/2π of the electretaccelerometer corresponds to the resonance frequency of amass-spring-system with one or two springs and can thus bewritten for low static pressures (1-spring model variant) as

fres = 1

2π

√1

ms

(Y · AP

tP

)(5a)

and for medium and higher static pressures (2-spring modelvariant) as

fres = 1

2π

√1

ms

(Y · AP

tP+ γ p0 AA

tA

). (5b)

Insertion of (5a) or (5b) into (4a) or (4b), respectively, yieldsthe voltage sensitivity

SV = ε VE

(ε tA + tE )· 1

ω2res

(6)

as a function of the resonance frequency of the accelerometer,independent of the applied static pressure and thus valid forboth model variants.

VI. DISCUSSION

The measured frequency responses of the two accelerome-ters presented in Figs. 3 and 4 are quite different. While inFig. 4 the responses of the more sensitive accelerometer arenearly flat below the resonance peaks, the sensitivities in Fig. 3significantly decrease with increasing frequency below theresonance frequencies. Certainly, the flat responses and highsensitivities of the second accelerometer are advantageous foran application of the electret sensors. The different behavior

is probably due to different cellular ring materials used in thetwo presented accelerometers. In general, viscoelastic creepingof polymers causes a decreasing frequency response. Thus, foroptimal accelerometers the creeping of the ring material shouldbe minimized by optimizing the expansion procedure of thecellular PP (cf. Sect. III).

Creeping is one possible source for decreasing frequencyresponses while damped air streams can be a second source.The flat response of the second accelerometer (Fig. 4) provethe absence of damped streaming, thus the air gap volumeis either closed or opened, and the 1-spring model variantor the 2-spring model variant is valid for each of the threefrequency responses with the related static pressure used inthe accelerometer.

In Fig. 3, the situation for the six responses of the firstaccelerometer is more complicated. For a better understand-ing of the measured sensitivities, (6) was used to calculatesensitivities from the related resonance frequency. Note thatthis equation is valid both for the 1-spring and the 2-springmodel variant. The calculated values are given as short dashesin the color of the related frequency response on the left-handside of the figure. The calculated values reasonably agree withthe measured sensitivities when, as (6) implies, extrapolatedsensitivity values at the resonance frequencies are taken.

Moreover, on the left side of the dashes the values ofYoung’s modulus of the cellular ring are written; these valuesyield, according to (4a), the sensitivity of the dash for the1-spring model. Young’s modulus values according to (4b)for the 2-spring model are given on the right-hand side ofthe dashes. The values for Y range from 1.4 to 5.9 MPa forthe 1-spring model and from 0.6 to 5.2 MPa for the 2-springmodel.

One interpretation of the data is as follows: Above 50 Hz,and in particular in the resonances, the air gap volumes forall accelerometers and for all static pressures are completelyclosed, the air gap volume acts as a spring and the 2-springmodel variant is appropriate. The Young’s modulus range from0.6 to 5.2 MPa, which are all reasonable values, follows.Apart from the responses for lowest and highest static pres-sure, the slope of the sensitivity is significantly increased forfrequencies below 20 Hz. This may be interpreted as dampedair streaming from the air gap volume to the housing andtransition from the 2-spring model to the 1-spring model. Theabsence of streaming effects for the highest static pressure(magenta curve) is understandable since high forces flatten thecellular ring and seal the air gap volume. However, the absenceof air streaming also for the lowest static pressure (black curve)then has to be explained by the fact, that the lowest staticpressure was applied last, following the measurement with thehighest static pressures. Those high pressures were appliedfor several hours and could have flattened the cellular ringand sealed the air gap volume permanently.

VII. CONCLUSION

Electret accelerometers with integrated FET are introducedin this paper. The integrated FET is used as the shieldedfirst stage of a high impedance voltage amplifier. The volt-age is generated between the metallic seismic mass and the


backplate, both separated by a soft cellular polymer ring,which is expanded and compressed by inertial forces ofthe seismic mass. The backplate is coated with a chargedFEP electret film. Thus working temperatures up to 120 °C,comparable to those of common electret microphones, andhigh enough for many potential applications, are possible.Measurements with the electret accelerometers showed thathigh sensitivities of up to 600 mV/g, low equivalent noiselevels as low as 80 μg, and flat frequency responses below theresonance frequencies are achievable. A necessary conditionfor flat responses is the absence of damped air streamingfrom the air gap volume to the housing. Therefore, a highquality electret accelerometer must be constructed with eithera completely opened or closed air gap volume. For bothaccelerometer implementations an analytical model variantwas developed, based either on one spring, the cellular ring,or on two springs, the cellular ring and the closed air gapvolume.

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Joachim Hillenbrand received the Diploma degreein physics from the University of Ulm, Ulm, Ger-many, and the Ph.D. degree from the TechnicalUniversity of Darmstadt, Darmstadt, Germany, in1992 and 1997, respectively. He was with the Ger-man Heavy Ion Research Center, Darmstadt, in thematerial research group, from 1993 to 1997. In 1998,he developed video and audio software at AnimatedImage Systems Technology, Ampfing, Germany. In1999, he joined the Electroacoustics Group, Uni-versity of Darmstadt. His current research interests

include electrets, piezoelectric polymers, and the application of these materialsin the acoustic field.

Tilman Motz was born in Frankfurt am Main,Germany, in 1939. He received the Diploma degreein communication technology in 1966 and the Dr.-Ing. degree in 1971 from the University of Tech-nology in Darmstadt, Darmstadt, Germany. He wasthe Administration Manager of the Institute of Com-munication Technology and Electroacoustics at theDarmstadt University. He was involved in researchof network synthesis, mainly impulse forming allpass networks. His current research interests includeantenna design, ionospheric wave propagation, and

electret transducers.

Gerhard M. Sessler (F’77) was born in Rosenfeld,Germany, in 1931. He received the Diploma degreein physics from the Universities of Freiburg andMunich and the Dr.rer.nat. degree from the Univer-sity of Goettingen in 1959. He was a member of theAcoustics Research Department, Bell Laboratories,from 1959 to 1975, where he had responsibilities forwork on electret transducers, concert hall acoustics,and charge storage phenomena in solids. In 1975, hejoined with the University of Technology in Darm-stadt, Darmstadt, as a Professor of electroacoustics.

He has been involved in various fields of acoustics and solid-state physics,in particular, acoustic silicon transducers, electret and piezoelectric materials,and acoustic signal processing. He is an Inductee of the National InventorsHall of Fame, Corresponding Member of the Heidelberg Academy of Science,and a fellow or member of various other professional societies in the U.S.and Europe.

high sensitivity electret accelerometer with integrated fet

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