THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 35, NUMBER 9 SEPTEMBER 1963 Electrostatic Microphones with Electret Foil G. M. SESSLER Bell TelephoneLaboratories, Inc., Murray Hill, New Jersey In this paperthe operationof electrostaticmicrophones with foil electretsis studiedtheoretically. It is assumedthat a foilelectret has equal (free or bound)constant charges of oppositesign on its twosurfaces. A formulais obtainedwhich shows the generatedoutput voltage of the microphoneto be a functionof the sur[acecharges of theelectret, the geometrical dimensions of the microphone, and the externallyconnected resistance.According to this formula,typical sensitivities of a few mV/ubarcan be expected. The electretmicrophone is alsocompared theoretically to a similarsystem with nonelectretfoil. The calcu- lationsshow that a surfacecharge of t0-s C/cm2 on the surfaceof «-milMylar foil correspondsto a biasof about 45 V in the nonelectretreference system. It is pointedout in this connectionthat the measurement of surfacecharges of foil electretsby the inductionmethod requires a correctionformula. INTRODUCTION in thedielectric, called tictire or boundcharges, or the "heterocharge."At the sametime, electronsare in- HEmicrophonesconstruction with and foilperformance electrets (electretofelectrostatic micro- jectedinto or extractedfrom the foil by the metal phones)have beendescribed previously2 These micro- plates,thus creating real or free charges,the "homo- phones utilize a permanently polarized thin foil charge." The term "electret" is thereforeused for a (typicallyk- to «-mil Mylar foil) whichhas a metal dielectricin whichthe heterocharge,the homocharge, layer on one side. The free charges(due to positive or both chargesare permanent. and negativeions or electrons)and the boundcharges For simplicity,we assume2.a that throughoutthe (due to a polarization)of sucha foil electretinduce volumethe polarizationP and the real charges chargesin the metal layer of the foil and in the back satisfydivP= 0 andp= 0. We allow,however, for finite plate. The numberof thesecharges is dependentupon fictive and real surfacecharge densities tr s and trr and the distancebetween the foil and the back plate. Since assumea effectivevalues for •rs and ,r which also take a soundwave which impingeson the foil will change into accountthe volumecharges. Since real and ticrive this distanceperiodically, a voltage is generatedbe- surfacecharges in electretsusually have opposite sign, tween the two electrodes. Thus, the electret micro- we can write for the total surfacecharge densities of phone convertsmechanical energy directly into elec- the two surfaces•= •,--• and tr2= trical energywithout usingan external bias. A foil electret with one of its two surfaces covered While all electret charges(whether free or bound) by a metallayer obviously is not capableof carrying are subject to decay due to finite relaxation times, a permanentsurface charge on this surface.Therefore, thesechanges are relatively slow,for goodelectrets and if wewant to ignorespace charges as postulated above, relaxation times are of the order of years. Therefore, wehave to assumean internal"surface" separated by the operationof the microphonewill be studiedin this a small distancefrom the metal layer. We thus con- paper under the assumptionthat electretcharges are sider the foil electretas consistingof two dielectric constant in time. layers. A crosssection of an electretmicrophone with such I. THEORY OF THE ELECTRET MICROPHONE a two-layerelectret is shownin Fig. 1. Verticaldimen- Electrets are usually formed in an electric field be- sionsare enlargedin the figure.In real cases,vertical dimensionsare sosmall that we canneglect effects due tweentwo metal plates.The field createsa polarization • G. M. Sesslerand J. E. West, J. Acoust. Soc. Am. 34, 1787 • B. Gross,J. Chem. Phys. 17, 886 (1949). (1062). aA. N. Gubkin,Soviet Phys.-Tech. Phys. 2, 1813(1958). 1354 ELECTROSTATIC MICROPHONES WITH ELECTRET FOIL 1355 a• is determinedby Ea: 4rai2=eaEa. (8) LAYER FOIL Eliminatinga• fromEqs. (7) and (8) yields ELECTRET TERMINATING RESISTOR V= RA (ea/4•r)dE•/dt. (9) AIR GAP If Eq. (5) is differentiatedwith respectto time, Eq. BACK PLATE (9) permitsthe eliminationof dEa/dt.This yieldsa lineardifferential equation for V: Fro. 1. Enlarged small sectionof electret microphone.Hori- zontal dimensionsof completemicrophone are much larger than --v+ v -&= o, (lO) vertical dimensions.Microphone is terminatedby resistorR. dtd (;C•D+ •t•d• •d3d ) •D+4•rad•d•d• dt where the capacitance to the finite horizontal size of the electret. The metal layer and the backplate are connectedby a resistorR. C= e•e•A/ 4•r(eaDq- e•da) (11) The surfacecharges a• and a2 of the electret,which are consideredto be constant,induce charges trit and hasbeen substituted. In Eq. (10), C can be considered in the metal layersand generateelectric fields Eh E2, constantsince the variablepart da of da is smallcom- Ea in the three dielectriclayers with thicknessesdr, d•, paredto da0.Equation (10) hasthe solution doand dielectricpermittivities •1, e2, If a soundwave impinges on the metallizedfoil, the thicknessda of the air gapis changedperiodically, thus changingthe electricfields and the inducedcharges, V:EK+f,•d•daddtd'(exp/Fdt)l and generatinga voltageV acrossthe resistorR. This voltage will be determinedin the followinganalysis. At the two surfacesof the electret we have, because Xexpf (--FdZ), (12) where of Gauss'stheorem, 1 et d •E•-- •E•= 4•r•, (1) F= -- --d• (13) RC taD+ e•dadt •2E2-- •aEo=4/r•'2. (2) and K is a constant.Integration of the exponential In addition,feds= 0 gives functionscan be performedeasily. Further integration is simpleif da is substitutedfrom Eq. (6) and if the 3 smallvariable part of dais neglectedin e•D+ e•da.The V+Z d,E,= O. (3) result is EliminatingE• and E2 from Eqs. (1) to (3) yields V = K ({aDd-e•da0)exp - Gubkin's • formula 4rad•d• sim0t+(lfioRC) corot (4) -[ (14) •aD+ elda0 1+ (1/coRC)• Assuminga•=--a•=--a and •= t2 and setting Neglectingthe transient term and rewriting the steady- = D, this is state solution,we have finally 4qrad•dat V- sin(cot+•), (15) In Eq. (5) the quantitiesD and a are constant.Now (½aDq-t •da0) (1-3- tan a supposeds is changedby a soundwave accordingto da= dao+da•sinwt, (6) where tanq= 1/•oRC. (16) where da<<dao.Then V is determinedby the change in Equation(15) is the generalrelation for the voltage V generatedacross a resistorR by an electretmicro- V = RA dai•dt, (7) phoneif the air gap dois changedperiodically. The whereA i's the back-platearea. The inducedcharge open-circuitvoltage V• can be foundif R= m (or 1356 G. M. SESSLER tango=0)is introducedinto Eq. (15): V•= da•sinco/. (17) •aDq- elda0 In this case,the generatedvoltage is in phasewith the vibration of the air layer if •r is positive. We now want to determine the sensitivity of the electret microphone at frequenciesbelow resonance. The resonancefrequency is determinedby the restoring force and the mass of the foil. In electrostatic trans- ducers with solid dielectric, the restoring force is largely determined4 by the elasticityof the air layer. This elasticity dependsupon the effectivethickness s of the air cushionbehind the foil whichis mostlyde- terminedby an additionalcavity connectedto the air gap by small holes. The resonancefrequency cot = (po/sM)l (po is the atmosphericpressure, M the massper cm2 of the foil) is usually5 at the upperend of the audio range or higher. Therefore, confinement to frequenciesbelow resonanceis no seriouslimitation. The amplitudeof the foil displacementis then , da•=sp/po, (18) Fro. 2. Sensiti•ty of an electretmicrophone (with a=10 -• wherep is the soundpressure. C/cm•, s=0.1 cm, and q=3) as functionof thcknessof air Substitutingda• from Eq. (18) into Eq. (17) and gap/t•c•ess of dielectric(dao/D) with d•/D asparameter. The lowerabscissa (capacitance X thicknessof dielectric)pertains settingV•= Vn sincotgives the sensitivityp,• = Vn/p of to a foilarea of • cm•. At present,the best experimental systems the electretmicrophone below resonance: have a d,o/D of about 1. o,•---4•rrrd•s/ ( e3D+ e•da0)p0. (19) structure.Electric charges and fieldshave practically Usingthe capacitanceof the systemas definedin Eq. no influenceon the mechanicalproperties since they (11) for da--da0,the sensitivitycan be expressedas influenceonly the air gapda, not the totalair layers. p• = (4• )2,sCa•/• •4 po. (20) Electrically,nonelectret and electretmicrophones differand the questionis whatbias corresponds to a In Fig. 2, the sensitivityis plottedfor a= 10-s C/cm2 certainsurface charge density. A nonelectretmicro- =03 esu, s=0.1 cm, •1=3, {a=l, p0=106 dyn/cm•, phonewhich is mechanically equal to thesystem shown and with d2D as parameter.Abscissas are dao/Dand in Fig. 1 andwhich is terminatedby a largeresistor (for A=4r cm=) CD, referringto Eqs. (19) and (20), R>>l/coCgives, because of Q=CV=const(Q=total respectively.The sensitivity,which does not depend chargeon the electrodes, V= totalvoltage between the upon the area of the unit, is largestfor small values electrodes) of dao/D.Typically dao/D is about 1 and d•/D is close to 1 (i.e., the chargesare closeto the surfacesof the dV•'/dt=-- ( Vo/C)dC/dt, (21) dielectric)in experimentalmicrophones. Therefore sen- whereV0 is the applieddc voltageand V•= V-- V0 is sitivitiesof a few mV/ubarare expected.This agrees thevoltage change caused by thechange in capacitance. with measuredvalues. Figure 2 showsthat the sensi- If C fromEq. (11) is substitutedinto Eq. (21) andif tivity couldbe