An Accurate Study on Capacitive Microphone with Circular Diaphragm Using a Higher Order Elasticity Theory

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1 59 An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory Abtract Thi tudy ha been undertaken to invetigate the mechanical behavior of the capacitive microphone with clamped circular diaphragm uing modified couple tre theory in comparion to the claical one. Preence of the length cale parameter in modified couple tre theory provide the mean to evaluate the ize effect on the microphone mechanical behavior. Invetigating Pull-in phenomenon and dynamic behavior of the microphone are the matter provided due to the application of a tep DC voltage. Alo the effect of different air damping coefficient on dynamic pull-in voltage and pull-in time have been tudied. The output level or enitivity of the microphone ha been tudied by invetigating the frequency repone in term of magnitude for different length cale parameter to figure out how the length cale parameter affect on the enitivity of the capacitive microphone. To achieve thee end, the nonlinear differential equation of the circular diaphragm ha been extracted uing Kirchhoff thin plate theory. Then, a Step-by-Step Linearization Method (SSLM) ha been ued to ecape from the nonlinearity of the differential equation. Afterward, Galerkin-baed reduced-order model ha been applied to olve the obtained equation. Shakiba Dowlati a Ghader Rezazadeh b* Saeid Afrang a Mehrdad Sheykhlou b Ayan Madan Paandi a a Electrical Engineering Department, Urmia Univerity, Iran b Mechanical Engineering Department, Urmia Univerity, Iran * Author g.rezazadeh@urmia.ac.ir Received.7.5 In revied form 8..5 Accepted..5 Available online 5..6 Keyword MEMS, capacitive microphone, couple tre theory, length cale parameter, enitivity. INTRODUCTION A microphone i an electro-acoutic tranducer that convert the acoutic energy into electrical ignal. The microphone are broadly utilized in voice communication device, hearing aid, vibration control and urveillance military aim (Miao et al., ; Ma et al., ). Traditional microphone uffer from the diadvantage of high cot and large ize. Therefore there i a further need to tudy on technology which can overcome thee problem.

2 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory 59 Micro-electro-mechanical technology defined a miniaturized mechanical and electro-mechanical element i going to be more and more convenient in almot all field of indutrie (Rezazadeh et al., ; Vahdat et al., ; Rezazadeh et al., ). However, it application in electronic draw more attention, pecifically. Microphone can be mainly categorized into dynamic, optical and capacitor (Suzuki et al., 6). MEMS-baed capacitive microphone are ued where low power, low noie level, large band width and high enitivity are the crucial requirement. In the cae of MEMS capacitive microphone a diaphragm movement toward a ground plate i caued by the applied acoutic preure on the diaphragm leading to a changing capacitance (Kwon et al., 7). The ening capacitance i calculated by integration of the diaphragm deflection in repect to the fixed ground-plate (Chen et al., 8). The change in conductance produce a change in voltage needing to be amplified to obtain a meaurable ignal. By applying voltage acro the two plate, attractive electrotatic and elatic retoring force have been induced between them. Both force are increaed by raiing voltage until the applied voltage i touched to a critical value, in which the diaphragm collape on the fixed electrode which mean pull-in happen (Batra et al., 8). The critical voltage aociated with thi intability i called pull-in voltage (Batra et al., 6). In ome cae, delaying the onet of pull-in intability would be extremely deirable a it i conidered a a limiting factor of the functional range of capacitive microphone. Accordingly in the deign of electrotatically actuated MEMS microphone, cornertone i to evaluate the pull-in voltage accurately and adjut the electric load away from the pull-in intability (Abdel-Rahman et al., ). Many tudie have delved into the pull-in phenomenon (Abdel-Rahman et al., ; Tilman and Legtenberg, 994). The acoutic enitivity of the microphone ha been decribed a the output voltage repone to an acoutical preure input (Raichel, ). To put it in another word, the mechanical enitivity i defined a how much the deflection change per ound preure applying on the diaphragm (Scheeper et al., 994). In repect of the ame bia voltage and ound preure, a microphone with a high enitivity value ha a higher output level and need le preamplifier gain before the analog-todigital converion compared to a microphone with a low enitivity value. A microphone with high enitivity would be uitable in far-field application where there i a large ditance from the ource of ound to the microphone for intance conference phone and ecurity camera. Li et al. () developed a MEMS capacitor microphone to raie the mechanical enitivity over releaing the initial tre and diminihing the mechanical tiffne of the diaphragm over a deeply corrugated diaphragm. Alo everal invetigation have looked into the mechanical behavior of MEMS microphone; Quaegebeur and Chaigne (8) tudied effect of mechanical and electrical nonlinearity on the electroacoutical behavior of electrodynamic tranducer. It i perceived from hybrid atomitic-continuum model and experimental reult, there i a ize effect in micron and ub-micron cale which ha a key role in mechanical behavior of microtructure (Tiata, 9). It maintain that the claical theory of elaticity doe not touch the accurate and definite characterization of deformation occurrence due to not including Size effect. To overcome thi inadequacy and predict the accurate behavior of microtructure, non-claical theorie uch a train gradient theorie (Lazopoulo, 4; Lazopoulo, 9) nonlocal elaticity theorie (Eringen, 983; Reddy, ) and couple tre theorie (Toupin, 96; Mindlin and Tierten, 96; Koiter, 969; Yang et al., ) which take the ize effect into conideration uing length cale pa- Latin American Journal of Solid and Structure 3 (6) 59-69

3 59 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory rameter have been applied to develop the non-claical model. The claical couple tre theory wa originated by the Coerat brother (99), Toupin (96), Mindlin and Tierten (96), and Koiter (969) to delve into the ize-dependent behavior of microtructure. They utilized two length cale parameter to capture the ize effect. Yang et al. () developed an additional equilibrium relation auming the couple tre tenor to be ymmetric. Thi work led to the number reduction of material length cale parameter from the two in claical couple tre theory to only one, etablihed in the modified couple tre theory (MCST). The two main advantageou which make the MCST more preferable over the claical couple tre theory are uing only one length cale parameter beide Lame contant and ymmetry of couple tre tenor (Park and Gao, 6; Ma et al., 8). The vat majority of the tudie around the microphone are retricted to the claical theory (CT) which have not been aociated with material length cale parameter therefore they may lead to the inaccurate prediction of the mechanical behavior. Accordingly, uing non-claical theorie i inevitable for approaching the more accurate mechanical behavior of thee device. Thi paper aim to carry out the intability behavior and the enitivity of the capacitive microphone expoed to an electrotatic force baed on the MCST for variou material length cale parameter and compared with the CT. In addition the effect of variou damping coefficient on dynamic behavior i invetigated. The differential equation of microphone are formulated by a Kirchhoff thin plate theory applying MCST. A tep by tep linearization method (SSLM), and the Galerkin baed reduced order model are ued for evaluation of the mechanical behavior of microphone. MECHANICAL MODEL AND GOVERNING EQUATIONS. The Modeling of Sytem Figure : Schematic illutration of the capacitive microphone with electrotatic actuation. A MEMS capacitive microphone generally can be viewed a two conductive circular micro plate with a voltage acro them a illutrated in figure The top plate i a thin deformable elatic plate with thickne h (- h / z h / ), radiu R (- R r R), Poion ratio n and young' modulu E which i held fixed along it boundary and play the role of diaphragm. According to low weight of the diaphragm, it can vibrate back and forth and produce energy. The lower plate of the Latin American Journal of Solid and Structure 3 (6) 59-69

4 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory 593 capacitor i fixed, rigid and thick with no movement introduce a the reference. Thee two plate are eparated by a dielectric ubtance like air. Applying a DC voltage to microphone caue an electrotatic pull down force that can be repreented a follow: ev FwV (, ) = ( g - w( r, q) ) () Where e i the dielectric contant of the air, V i the applied bia DC voltage, g i the initial gap between the diaphragm and the back plate, t i the time and wrq (, ) i the deflection of the diaphragm. A change in the ditance of upper and lower plate owing to electrotatic pull-down force lead to the capacitance value changing. Thu the tructure act a a variable capacitor. The air preure change contantly with time, but at a certain point of time it ha an exact value. A ound preure wave can be repreented by a random time dependent mathematical function with a wide range of frequencie approximated by Fourier erie a a combination of imple inuoidal function. In thi paper, we conider a pure inuoidal ound wave a the actuation ource of the microphone a follow: In which P and P ( w, t ) = P i n( w t ) () w are the amplitude and frequency of the ound preure wave, repectively. During the diaphragm ocillation normally to the ubtrate, the air gap i queezed leading to a coniderable change in the air volume. Owing to the vicou flow of air and o doe the preure changing, type of force are come into exitence. One i the retoring force known a air pring caued by the air compreion. The total tiffne of the ytem i given by the rigidity of the diaphragm and the pring effect of the air gap. In the audio frequency range, the air gap pring coefficient i very low comparing to the diaphragm pring coefficient. Therefore, the air gap tiffne can be neglected (Eteve et al., ). The other one i damping force caued by the vicou flow of the air which i effective in the microphone operation, particularly in the high frequencie (Eteve et al., ). A circular plate damping coefficient i decribed a follow by analytically olving the Reynold equation (Bao and Yeng, 7). C 3 = S 3 pg h (3) Where S i the plate area and g i the air gap between the two plate and h i the air vicoity in the ambient condition. A imple model (Whelan and Hodgon, 978) a h = r a v l /3 can determine the air vicoity in which r a i the ma denity of air, l i the mean free path of the molecule and v i the average velocity of the molecule. Baed on experimental reult, different amount of the air vicoity are derived from ambient condition including preure and temperature. Alo during the deflection, the diaphragm carrie a certain amount of the fluid volume along itelf. Thi entrained fluid i called added or virtual ma. On the ubject of microphone, the virtual Latin American Journal of Solid and Structure 3 (6) 59-69

5 594 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory ma and the inertia effect are often neglected ince the diplaced fluid ma i negligible (De Silva, 5).. Modified Couple Stre Formulation With reference to the modified couple tre theory introduced by Yang et al. (), in an iotropic elatic material occupying a volume of W bounded by the urface A, the train energy denity i a function of train and gradient of the rotation vector a following: = ( ij : eij + mji : cji ) dw ò (4) W Where ij i the Cauchy (claical) tre tenor, e ij i the train tenor, m ij i the deviatoric part of the ymmetric couple tre tenor, and χ ij i the ymmetric curvature tenor a given: = le d + me e = ( u + u ) ij kk ij ij ij i, j j, i ij = m c ij m c = ( qij, + qji, ) ij (5) In which l and m are lame' contant, d ij i the kronecker delta, u i i the diplacement vector, i a material inertial length cale parameter and the parameter q i i the rotation vector defined a follow (Yang et al. ) E E q = eijkuk, j m = l = i ( + n) ( + n) (- n) (6) Where e ijk i the permutation ymbol..3 Dynamic Model of a Micro-circular Diaphragm The relationhip between the diplacement component u ( r, q,z), v ( r, q,z) and wrq (,, z) along the r, q and z direction, repectively, baed on Kirchhoff thin plate theory can be written a (Sun and Tohmyoh, 9) wr (, q) wr (, q) u ( r, q, z) =- z v ( r, q, z) =- z w( r, q, z) = w( r, q) r r q (7) Auming a two-dimenional formulation ytem by dropping the z dependency the train and curvature tenor are, repectively, decribed by: Latin American Journal of Solid and Structure 3 (6) 59-69

6 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory 595 é u æ u v vö æ u wöù r çr q r r ç z r è ø è ø æ u v vö u v æ v w ö e = = çèr q r r ø r r q çè z r q ø æ u wö æ v w ö w + + ê ç z r ç z r q ë è ø è ø z úû é w æ w ö w ù -z - z r r ç r q r è ø æ w ö æ w w ö w z z rç r q r q r q r q è ø çè ø w w r r q êë úû é æ w ö w w w ù rç r q r r r q r è ø w w w æ wö χ = = r r r q r r q ç r è ø êë úû é w w w w w ù r r r r r q q r q r w w w w r r r q r r r q ê ë ú û (8) (9) By ubtituting of the Lamé contant with the modulu of elaticity E and the Poion ratio n the tre tenor and the couple tre tenor take the following form, repectively: é E æ u æu v öö æ u vö ù + n + G + n ç r çr r q ç q r - è è øø è ø æ u vö E æu v uö = G n = èç q r ø n èçr r q r ø - êë úû é - E æ w æ w w z nz öö æ w ö ù + + Gz - n è r ç è r r r q - ø rç r q ç ø è ø w Ez æ æ w wö wö æ ö Gz - z nz rç r q n r r è ø - çè çè r q ø r ø êë úû () Latin American Journal of Solid and Structure 3 (6) 59-69

7 596 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory é æ ö ù æ wö w w w G G r r q r r r q r èç ø ç è ø æ ö w w w æ wö m =- G G r r r q r r q çè r () èç ø ø ê ú ë û In which G = E /( + n) i the hear modulu. é æ ö ù Ez w w w æ w ö nz Gz n r r r r q rç r q - çè ø è ø + h / æ wö Ez æ w wö w M = Gz nz ò zdz = rç r q n r r r q r -h / è ø - çè ø êë úû é é w æ w wöù æ wö ù - D + n D( n) + r r r r r - - rç r q êë çè øúû è ø w éæ w wö wù æ ö D( n) D n rç r q r r r r r è ø êë çè ø úû êë úû é æ wö æ w w wö ù G G ê ç - ç - + ú rç r q r r r q r è ø èç ø + h / m æ w w wö æ wö M = G G ò dz r r - r q r r q r = -h / èç ø ç è ø ê ë ú û é æ w ö D ç æ w w wö ù D rçèr q ø r r ç è r q r ø æ w w w ö w æ ö -D D r r r q r r q ç r èç ø è ø ê ë ú û () (3) 3 Where D = Eh /(- n ) i the bending rigidity of the diaphragm l and D = El h /( + n) = Gl h i the contribution of rotation gradient to the bending rigidity. According to Hamilton principle, the actual motion diminihe the difference between the kinetic energy and total potential energy for a ytem with precribed configuration at t = to T a follow (Reddy, ): Latin American Journal of Solid and Structure 3 (6) 59-69

8 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory 597 T ò éδk ( δ δw) ù ë - - û dt = (4) Where k, P and W are the kinetic energy, the train energy and the work of external load of the micro plate, repectively. The train energy denity can be obtained from eq. (4), () and (3) a follow: = ò ì M w rr M æ w wö ü - - qq + r r r ç r q è ø w w ï æ ö æ ö ç ç ý ïda ï q ç r r ïî è ø çèr r r q r øþï m í- Mrq + Mrr rçèr q ø rçèr q ø A m æ wö æ m w w wö ï- ï M qq M ï ç - rq ç - + (5) Moreover, the firt variation of the train energy in the plate on the time interval [, T] can be expreed a: d ìï ü ï ï ýdwdadt ï r r r r ïî r q r ïþ ï Mrr Mrr Mqq Mqq Mrq r r r r q r r r r q T T m m m Mr Mrr Mrr M dt ï q qq ò = ò ò í r r r q q r q r q r A m m m m Mrq Mrq Mrq Mrq ï (6) The work done by the external force can be obtained a: W ext = ò Q(r) w(r, t)da (7) The firt variation of the work on the time interval [, T] i: T A T δòwdt = ò ò Q(r ) d wdadt (8) Total kinetic energy of the circular diaphragm i given by: A K é ù æ uö æ vö æ wö = r d ò + + W êçè t ø çè t ø çè t ø ú W ë û (9) Where r i the ma denity of material. Applying u / t = v / t = the firt variation of total kinetic energy of the plate on the time interval [, T] take the form: d T T æ w ö Kdt =- rh ç δwdadt çè t ø A ò ò ò () Latin American Journal of Solid and Structure 3 (6) 59-69

9 598 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory Subtituting Eq. (6), (8) and () into Eq. (4), the governing equilibrium differential equation of motion of a circular micro plate become a: m Mrr Mrr Mqq Mqq Mrq Mrq Mrr r r r r q r r r r q r q r r q m m m m m m M rr Mqq Mrq Mrq Mrq Mrq w Q rh = r q r r q r r r r r q r t () Subtituting Eq. () and (3) into Eq. () the governing equation of the micro plate in term of the deflection i given by: 4 w ( D + D ) w + r h = Q t () Becaue it i aumed that the plate deflection i ymmetrical with repect to circumferential coordinate w / q =, the deflection only depend on the radial poition r. Therefore the operation and 4 in polar coordinate ytem for the axiymmetric circular micro plate are ex- preed a: = + r r r 4 = + + ç r r r ç r r r æ öæ ö è øè ø (3) Therefore for a circular micro plate actuated by electrotatic force and ound preure wave conidering an equivalent damping ratio, equation of motion take the following form: 4 w w ev r t t ( g -w) ( D + D ) w + h + C = + P in( w t ) (4) For convenience the below non-dimenional parameter are preented in order to rearrange the equation into dimenionle form: w r ˆ,, ˆ t w rh w = rˆ = t =, ˆ =, t =, = (5) g R t t * * w * R w * D * w æ D D ö + 4 wˆ wˆ av wˆ + + a 3 = + a in( w ) ˆ ˆ t ç è D ø t t ( -wˆ) (6) 4 4 er PR 4 CR 3 Dg Dg 3 * Dt a = a = a = (7) Latin American Journal of Solid and Structure 3 (6) 59-69

10 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory NUMERICAL SOLUTION 3. Nonlinear Equation of Static Deflection Due to the non-linearity of tatic deflection of the diaphragm ubjected to an applied DC voltage, an analytical olution i complicated and time conuming. Therefore, SSLM i ued (Rezazadeh et al., 9) to linearize the tatic motion equation. Auming that i w i the diaphragm deflection due to applied DC voltage i V : æ i D D ö + 4 i ( V ) ç wˆ = a çè D ø - i ( wˆ ) (8) By increaing the applied DC voltage to: The diplacement can be written a: V i + i + (9) V dv i + i + y() (3) w ˆ w ˆ r ˆ Rewriting Eq. (8) at the tep of (i+ ), uing the calculu of variation theory and Taylor expanion, neglecting the higher order term of Taylor erie, and ubtracting the tep (i) from tep (i+ ) lead to the following linearized equation to calculate y () ˆr : æ i i D D ö + 4 ( V + ) V + dv ç y() rˆ - a y() rˆ = a çè D ø - - i 3 i ( wˆ ) ( wˆ ) (3) y () ˆr can be approximated by the function pace in term of bai function a follow: m N y ( rˆ) = å a f ( rˆ) (3) m m m= Where f m(ˆ r ) i the hape function atifying the boundary condition and a m are the unknown Coefficient evaluated uing Galerkin Weighed Reidual Method in each tep. Subtituting Eq. (3) into Eq. (3), multiplying by the weight function f k(ˆ r ) in Galerkin method and then integrating the reult with repect to r over [, ], lead to a et of differential equation a follow: N å m= ( K ) a = N k =, ¼, N km m k (33) In which Latin American Journal of Solid and Structure 3 (6) 59-69

11 6 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory K N k k æ + ö = () rˆ () rˆ () rˆ () rˆ çç ç - drˆ ç ø = æ D D ö 4 av ò ç fm fk f 3 m fk D ( wˆ ) èèç ø - i av dv ò f () ˆ ˆ k r dr ( - wˆ ) (34) Repeating thee tep lead to the deflection of the diaphragm to an applied electrotatic force. 3. Nonlinear Equation of Dynamic Deflection Toward olving Eq. (6) numerically, a Galerkin baed model can be ued (Nayfeh and Mook, 979). The approximated olution for olving the dynamic equation of diaphragm deflection i propoed a: N w (, rˆtˆ) = () ˆ å b t f () rˆ (35) d n n n= In which a linear combination of time b ˆ n () t and hape function f n(ˆ r) are ued. The conidered hape function atified all boundary condition of the clamped circular diaphragm. Subtituting Eq. (35) in Eq. (6) lead to following error function: N N N 4 f () ˆ 3 () ˆ () ˆ (, ˆ ˆ n rbn a fn r bn fn r bn ftw,v, wˆ ) Er n= n= n= æ D + D ö ç = çè D å å å ø (36) Where αv f (, twv ˆ ˆ,,ˆ) w in(ˆ ˆ = + a wt) ( - wˆ ) (37) Baed on Galerkin method: ˆ ˆ ˆ ò f j( re ) r( rtdr, ) ˆ = j =, ¼, N (38) to Uing the weight function E r lead to the following ordinary differential equation: f (r) ˆ imilar to the hape function and applying Galerkin procedure j N æ ö M n n ( ˆ, ˆ jn b Cjn b Kjnb å + + n = Fj w t) j =, ¼, N Mb + Cb + kb = F ç è ø n= ( ) (39) In which Latin American Journal of Solid and Structure 3 (6) 59-69

12 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory 6 M = f () rˆf () rˆdrˆ jn j n C = a f () rˆf () rˆdrˆ jn 3 j n æ D D 4 av jn fj ( ò ç fn f 3 n ç D èç ( - wˆ ) ö ç æ + ö K = rˆ) çç ( rˆ) ( rˆ) çç ç - drˆ ç è ø ø F j = a ò ò ò f () rˆdrˆ j (4) 4 NUMERICAL RESULTS AND DISCUSSION Owing to tudy the mechanical behavior of microphone including a clamped circular diaphragm with electrotatic actuation, we conider a cae with the material and aigned geometrical propertie a illutrated in table. In the cae of circular diaphragm, the following hape function can be utilized atifying all boundary condition. (n - ) p f() rˆ = co ( rˆ) (4) A noted before to evaluate the ize effect, an intrinic material length cale parameter mut be incorporate into the contitutive equation. Thi paper i concerned with three different material length cale parameter a. m m,.47 mm and.73 m m. The of. (Bin and Wanji, ) refer to a microphone with aluminum diaphragm (Ganji and Majli, 8) and the two other length cale parameter belong to a gold diaphragm microphone (Kim et al., 7) with thickne of.5 and μm repectively (Cao et al., 7). Deign variable Value R (µm) 5 g (µm) 3 h (µm) E (GPa) 69 r (Kg/m 3 ) 33 e (F/m) * - n.3 Table : Material propertie and geometrical of the diaphragm. 4. Stable Region of the Microphone to a DC Voltage Applying the bia DC voltage to a capacitive microphone reduce the tiffne of the diaphragm, cauing the diaphragm to be deflected. By approaching the bia voltage to the tatic pull-in voltage, Latin American Journal of Solid and Structure 3 (6) 59-69

13 6 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory the diaphragm become untable for any initial condition and collape on the ubtrate. Conidering that thi voltage limit the range of the applied DC voltage to the microphone the evaluation of tatic pull-in voltage i neceary. Figure illutrate the diaphragm non-dimenional center deflection veru voltage uing MCST for different length cale parameter. The calculated tatic pull-in voltage baed on MCST i more than the one calculated by CT and increaing the length cale parameter lead to increment of the tatic pull-in voltage. Reult imply that the tatic pull-in voltage of micron cale diaphragm i ize dependent. Alo the achieved pull-in voltage under the claical theory ( = ) i cloe to that calculated by Oterboerg (995). Figure : The tatic pull-in variation baed on modified couple tre theory for different length cale parameter in comparion to the claical theory. 4. Dynamic Repone of the Microphone to a Step DC Voltage (Vdc ; P=) The threhold of the tep DC voltage applied to the capacitive microphone i limited owing to the dynamic pull-in intability phenomenon. Thi critical voltage which i a low a 9-9% of tatic pull-in voltage introduced a dynamic pull-in voltage (Nayfeh et al., 7). The dynamic motion equation i integrated uing Runge-kutta fourth-order method by neglecting the ound preure wave. Figure 3 and 4 depict the dynamic behavior of the diaphragm uing MCST in comparion to CT, for a cae without damping. The predicted reult by the claical theory i almot equal to thoe predicted by applying MCST with a nanoized length cale parameter and increaing the length cale parameter make the more difference between MCST and CT. Latin American Journal of Solid and Structure 3 (6) 59-69

14 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory 63 Figure 3: Phae portrait of the diaphragm uing modified couple tre theory for different length cale parameter in comparion to the claical theory. Figure 4: Time hitory of the diaphragm uing modified couple tre theory for different length cale parameter in comparion to the claical theory. Figure 5 and 6 illutrate the phae portrait and dynamic pull-in variation for different damping coefficient uing MCST for =.47mm in comparion to CT. The mall damping ratio refer to the cae of larger gap and the lower air vicoity which can be even neglected. Under both theorie, increment of the damping coefficient to a critical value caue the dynamic pull-in voltage converge to the tatic pull-in voltage. A depicted in figure 6, the non-dimenional pull-in time increae by increment of damping ratio. Diminihing the air gap and increaing the air vicoity and plate radi- Latin American Journal of Solid and Structure 3 (6) 59-69

15 64 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory u may reult in higher damping ratio which caue inappropriate and impractical condition of uing microphone. Figure 7 how pull-in time over the applied voltage for variou length-cale parameter. Alo figure 8 depict the pull-in time over a range of length-cale parameter. A the length-cale parameter increae, the pull-in time decreae. Figure 5: Phae portrait of the diaphragm uing modified couple tre theory (l=.47 m) for different damping coefficient in comparion to the claical theory. Figure 6: Dynamic pull-in variation of the diaphragm uing modified couple tre theory (l=.47 m) for different damping coefficient in comparion to the claical theory. Latin American Journal of Solid and Structure 3 (6) 59-69

16 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory 65 Figure 7: Pull-in time of the diaphragm ubjected to the actuating tep DC voltage for different length cale parameter in comparion to the claical theory. Figure 8: Pull-in time of the diaphragm ubjected to the actuating tep DC voltage for different length cale parameter. 4.3 Senitivity (Frequency Repone) Figure 9 repreent frequency repone in term of magnitude in abence of electrotatic force applying MCST for different length-cale parameter. Increaing the length-cale parameter lead to the Latin American Journal of Solid and Structure 3 (6) 59-69

17 66 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory le deflection and the more fundamental frequency. A a reult of that the enitivity in output voltage term decreae. A can be een in figure by increaing the bia DC voltage under contant ound preure, the deflection increae, So doe the enitivity. In addition, it i evident that the higher bia DC voltage create the lower fundamental frequency which lead to the limitation in the upper band of the frequency repone. Figure 9: Frequency repone of the diaphragm baed on modified couple tre theory for different length cale parameter in comparion to the claical theory. Figure : Frequency repone of the diaphragm for variou bia voltage under modified couple tre theory (l=.47 m). Latin American Journal of Solid and Structure 3 (6) 59-69

18 S. Dowlati et al. / An Accurate Study on Capacitive Microphone with Circular Diaphragm Uing a Higher Order Elaticity Theory 67 5 CONCLUSIONS Thi paper tudied the ize dependent behavior of the capacitive microphone with circular diaphragm uing MCST. The numerical reult how that the pull-in voltage calculated by modified couple tre theory i more than that calculated by the claical one. It wa hown that conidering the air damping, the dynamic pull-in voltage ecape from that of obtained by ignoring the damping force and riing damping coefficient ha caued the dynamic pull-in voltage to converge to the tatic pull-in voltage. Alo a the length-cale parameter i increaed the pull-in time i decreaed. Moreover, it i interpreted that under the ame condition (the ame ound preure and the ame bia voltage) the deflection etimated by the propoed theory i maller than that by the claical theory. Conequently, the MCST predict the le output voltage or the le enitivity in comparion to CT. According to the reult, by increaing electrotatic force, the diaphragm deflection i increaed and the fundamental frequency i decreaed. Therefore, the enitivity i increaed. The main concluion to be drawn from thi paper i that uing the claical theory for diaphragm with coniderable material length cale parameter may return inaccurate reult and mechanical behavior of the capacitive microphone hould be tudy under modified couple tre theory. Reference Abdel-Rahman, E.M., Youni, M.I., Nayfeh, A.H. (). Characterization of the mechanical behavior of an electrically actuated microbeam. Journal of Micromechanic and Microengineering : Bao, M., Yeng, H., (7). Squeeze film air damping in MEMS. Senor and Actuator A: Phyical 36: 3-7. Batra, R.C., Porfiri, M., Spinello, D. (6). Analyi of electrotatic MEMS uing mehle local Petrov Galerkin (MLPG) method. Engineering Analyi with Boundary Element 3: Batra, R.C., Porfiri, M., Spinello, D. (8). Vibration and pull-in intabilitie of microelectromechanical von Kármán elliptic plate incorporating the Caimir force. Journal of Sound and Vibration 35: Bin, j., Wanji, C., (). A new analytical olution of pure bending beam in couple tre elato-platicity Theory and application. International Journal of Solid and Structure 47; Cao, Y., Nankivil, D.D., Allameh, S., Soboyejo, W.O. (7). Mechanical Propertie of Au Film on Silicon Subtrate, Material and Manufacturing Procee : Chen, J., Hu, Y.C., Lee, S.S., Mukherjee, T., Fedder, G.K. (8). Modeling and imulation of a condener microphone. Senor and Actuator A 45-46: 4-3. Coerat, E. and Coerat, F., (99). Theorie de Corp Deformable, Hermann et Fil (Pari). De Silva, C.W. (5). Vibration and hock handbook, CRC Pre (New York). Eringen, A.C. (983). On differential equation of nonlocal elaticity and olution of crew dilocation and urface wave. Journal of Applied Phyic 54: Ganji, B.A., Majli, B.Y., (9). Deign and fabrication of a new MEMS capacitive microphone uing a perforated alumunum diaphragm. Senor and Actuator A 49: Joue, E., Rufer, L., Rehder, G. (). Capacitive microphone fabricated with CMOS-MEMS urfacemicromachining technology. Deign, Tet, Integration and Packaging of MEMS/MOEMS (DTIP), Sympoium on. IEEE. Kim, H.J., Lee, J.W., Lee, S.K., Park, K.H. (7). A Miniature Condener Microphone for Portable Terminal Application. IEEE Senor Conference. Koiter, W.T. (969). Couple-tree in the theory of elaticity: I & II. Proc K Ned Akad Wet B 67: Latin American Journal of Solid and Structure 3 (6) 59-69

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