Analytical modelling of a MEMS transducer composed of a rigid micro-beamattached at one end to a flat spring moving against a reduced-size backplatePetr HONZÍK(1) , Antonín NOVÁK(2) , Stéphane DURAND(2) , Nicolas JOLY(2) , Michel BRUNEAU(2)(1) Czech Technical University in Prague, Faculty of Transportation Sciences, Konviktská 20, 110 00 Praha 1, Czech Republic,honzikp@fd.cvut.cz(2) Laboratoire d’Acoustique de l’Université du Mans (LAUM, UMR CNRS 6613), avenue Olivier Messsiaen 72085, Le Mans,France, antonin.novak@univ-lemans.fr, nicolas.joly@univ-lemans.fr, stephane.durand@univ-lemans.fr,michel.bruneau@univ-lemans.fr

Abstract
The use of planar micro-beams as moving parts of acoustic and electroacoustic devices has increased recently
because of their geometrical simplicity, hence lowering fabrication costs. The precise modelling of such devices
is then of interest. The miniaturized transducer proposed herein is composed of a planar rigid micro-beamattached at one end to a flat spring (the other end remaining free) surrounded by thin slits and loaded by a thin
fluid layer (situated in the gap between the micro-beam and a reduced-size backplate) and a small cavity, bothbeing placed behind the beam. Such a configuration reduces the overall size of the device (no need of an externalcavity) and enables to adjust more parameters comparing to the case of the backplate of the same size as theone of the micro-beam. The thermoviscous damping effects originating in the fluid-filled parts of the device(slits, air-gap, and cavity) are taken into account. As a result of the model, the displacement of the micro-beam
is calculated and compared with the reference finite element solution, the acoustic pressure sensitivity of thetransducer is finally presented and discussed.
Keywords: MEMS transducer, micro-beam, analytical modelling

1INTRODUCTION

Precise models of the transducers, particularly of the miniaturized ones, are of interest when an optimization isneeded. Such models of the devices containing the micro-beams are of rising importance while some realizationsof such transducers has appeared recently [1].
Herein the reduced-size backplate has been employed because of its advantages in terms of sensitivity andbandwidth of the transducer [2]. Since the model of the behaviour of the rigid beam attached at one end to aflat spring loaded by a thin fluid film of uniform thickness [3] and the model of elastically supported rigid beam
loaded by fluid film with discontinuity in thickness [4] have been published recently, it is logical to combine
herein both approaches together.
Note that in such models the damping originating in the thermal and viscous boundary layers present in thethin airgap and in the slits has to be taken into account [5, 6, 7, 8, 9] along with the coupling between theincident acoustic pressure and the acoustic pressure field inside the transducer through the slits [3, 4].

2ANALYTICAL SOLUTIONThe system considered herein is shown in figure 1. The planar rigid micro-beam of length L, width b andthickness hb is attached at the left side to a deformable support acting as a spring (the right side remainingfree). The acoustic system surrounding the micro-beam consists of a thin airgap of thickness hg opened into asmall cavity of volume Vc (both being placed behind the micro-beam) and two thin lateral slits of thickness hs
and one thin slit of thickness hL at the right end of the micro-beam (at x = L) coupling the acoustic pressure7375

Figure 1. Geometry of the system: a) side view, b) cross-sectional view of the device cut at x > lc
field in all above described acoustic elements with the incident acoustic pressure pinc .
2.1 The displacement of the beam
The displacement η(x) of a planar rigid micro-beam attached at one end to a flat spring is given by the angle
θ through the relation η(x) = θ x, the angle being governed by the equation [3]ZLZ LdFs,y (x)−ω 2 J + iωDb +Cb θ = b [p(x) − pinc ] x dx − 2x dx − LFL,y ,dx0(1)0Jb = mb L2 /3

whereis the mass moment of inertia, mb and L being the mass and the length of the beam
respectively, Cb is the force moment and Db is the damping coefficient of the spring.
2.2 The acoustic pressure inside the device
The acoustic pressure p(x) inside the device consists of the pressure field in the airgap pgap (x) and the pressurein the cavity pc which is supposed to be uniform
pcx ∈ (0, lc )p(x) =(2)pgap (x) x ∈ (lc , L).
Wave equation governing the acoustic pressure field in the thin airgap (between x = lc and x = L) loading themicro-beam is given as follows [3, 4]
 2
∂
2+
χ
pgap (x) = − [U1 pinc +U2 η(x)] ,(3)∂ 2xwhere η(x) is the displacement of the beam, pinc being the incident pressure, the other parameters such as thecomplex wavenumber χ and the coefficients on the right-hand side of the equation (3) U1 and U2 being takenfrom the literature [3].
The solution of the equation (3) takes the classical form
pgap (x) = A cos(χx) + B sin(χx) −1(U1 pinc +U2 θ x).
χ2(4)The acoustic pressure in the cavity pc = wtot Zc is given by the impedance of the cavity Zc = ρ0 c20 /(iωVc )(i being the imaginary unit, ω representing the angular velocity, ρ0 and c0 denoting the air density and the7376

adiabatic speed of sound respectively) and the total volume velocity entering to the cavity
wtot = −iωb

Z lc0

η(x)dx − wgap (lc ) − 2hs

Z lc
0vs,y (x)dx(5)

which consists of the volume velocity of the beam in the interval x ∈ (0, lc ), the volume velocity entering fromthe airgap wgap (x) = −∂x pgap (x)Fv,g hg b/(iωρ0 ), and the volume velocity entering through the lateral slits [3]vs,y (x) = −1 pinc − p(x)1Fv,s + iωη(x)Kv,s ,iωρ0hb2(6)where p(x) = pc in the interval x ∈ (0, lc ). Reporting the solution for the pressure in the airgap (4) to theexpression for the volume velocity at the boundary of the airgap wgap (lc ) gives the pressure in the cavity
Zcpc =C1 lc pinc − (C2 lc2 +C3U2 /χ 2 )θ +C3 (Bχ cos(χlc ) − Aχ sin(χlc )) ,(7)1 +C1 lc Zc
where

2hs Fv,s,iωρ0 hb
C2 =iω(b + Kv,s hs )/2,Fv,g hg b.
C3 =iωρ0C1 =(8)The boundary conditions at both sides of the airgap, namely the continuity of between the acoustic pressure in
the airgap and in the cavity pgap (lc ) = pc and the continuity between the volume velocity in the airgap and in
the slit at the end of the beam wgap (L) = vL,y hL b (the latter being expressed by the similar way as eq. (6))
enables to express the integration constants of the solution (4)A =θ H + pinc I,
B =θ E/D + pinc F/D,

(9)

with
H =(C4 + EC6 /D)/CA ,I =(C5 + FC6 /D)/CA ,C4 =U2 lc /χ 2 − Zc (C2 lc2 +C3U2 /χ 2 )/(1 + ZcC1 lc ),C5 =U1 /χ 2 + ZcC1 lc /(1 + ZcC1 lc ),C6 = − sin(χlc )) + cos(χlc )ZcC3 χ/(1 + ZcC1 lc ),CA = cos(χlc ) + sin(χlc )ZcC3 χ/(1 + ZcC1 lc ),C7 =iωKv,L hL bL/2,(10)

C8 =hL Fv,L b/(iωρ0 hb ),C9 =C8 cos(χL) −C3 χ sin(χL),
D = −C3 χ cos(χL) −C8 sin(χL) −C6C9 /CA ,E =C7 −U2 (C3 +C8 L)/χ 2 +C4C9 /CA ,F = −C8 (1 +U1 /χ 2 ) +C5C9 /CA .
The acoustic pressure in the cavity can be finally expressed as
pc =θ O + pinc P,with
O =Zc EC3 χ cos(χlc )/D − HC3 χ sin(χlc ) − (C2 lc2 +C3U2 /χ 2 ) / (1 +C1 lc Zc ) ,P =Zc [FC3 χ cos(χlc )/D − IC3 χ sin(χlc ) +C1 lc ] / (1 +C1 lc Zc ) .

(11)

(12)

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2.3 Coupling between the acoustic pressure and the beam displacement
When reporting the solution for the acoustic pressure inside the transducer into the equation for the displacementof the beam (1), the integral of the pressure in the gap over the airgap length has to be expressed asZ Llcwithpgap (x) =θ J + pinc N(13)J =HIC + EIS /D −U2 (L3 − lc3 )/(3χ 2 ),N =IIC + FIS /D −U1 (L2 − lc2 )/(2χ 2 ),
Z L

IC =
IS =lcZ L
lccos(χx)x dx = [χL sin(χL) + cos(χL) − χlc sin(χlc ) − cos(χlc )] /χ 2 ,(14)sin(χx)x dx = [sin(χL) − χL cos(χL) − sin(χlc ) + χlc cos(χlc )] /χ 2 .

Then, the viscous forces originating in the lateral slits and in the slit at the end of the beam (subscript s and Lrespectively) are given by [3]1(15)dFs,y (x) = − Kv,s hs (pinc − p(x)) + iωη(x)Πs dx,2

1FL,y = − Kv,L hL (pinc − p(L)) + iωη(L)ΠL b,(16)2where Πa = −kv µhb cot(kv ha ), the subscript a designates either s or L, µ being the shear dynamic viscosity of
the air, and where the pressure at the end of the airgap isp(L) =θ Q + pinc S,with

Q =H cos(χL) + E sin(χL)/D −U2 L/χ 2 ,S =I cos(χL) + F sin(χL)/D −U1 /χ 2 .

(17)

(18)

Reporting of the equations (11), (13), (15) and (16) to the equation (1) provides, after some calculation, thefinal expression for the angle of the beam
θ = pinc3(b − Kv,s hs )(lc2 P/2 + N − L2 /2) − Kv,L hL b2 L(S − 1)/2.
−ω 2 Jb + iωDb +Cb + (lc2 O/2 + J)(Kv,s hs − b) + Kv,L hL b2 LQ/2 + iωL2 (2Πs L/3 + ΠL b)(19)

RESULTS AND DISCUSSION

In this section the results of the method presented herein are compared with the results of 3D numerical (FEM)
simulation obtained using Comsol Multiphysics software [10]. The dimensions of the transducer considered andthe properties of the air are given in table 1. The force moment Cb = 1.0916 × 10−4 Nm has been estimatedfrom the numerical simulation of the beam motion without the surrounding acoustic elements, the damping
coefficient is neglected Db = 0.
The mean displacement of the beam over the surface of the fixed electrode (x ∈ (lc , L)), which is the variableof interest when searching for the sensitivity of the electrostatic transducer, given as η̄ = θ (L + lc )/2 is shownin figure 2 for the incident acoustic pressure pinc = 1Pa. The comparison between the result of the present
analytical method (solid line) and the result of the full 3D FEM simulation (black points) shows very goodagreement. Given the behaviour of the model near the resonance frequency, the damping in the system seems
to be slightly underestimated.

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Parameter
Beam length L
Beam width b
Beam thickness hb
Cavity length lc
Cavity thickness hc
Airgap thickness hg
Thickness of the slits hs

Value
3 × 10−30.4 × 10−3
50 × 10−6
L/4
150 × 10−6
50 × 10−6
10 × 10−6

Unitmmm
m
m
m
m

Parameter
Static pressure P0
Static temperature T0
Density ρ0
Adiabatic speed of sound c0
Shear dynamic viscosity µBulk dynamic viscosity µB
Thermal conductivity λh
Ratio of specific heats γ
Specific heat coefficient at constantpressure per unit of mass CP

Value101330
293.151.204343.2
1.814 × 10−51.088 × 10−525.77 × 10−3
1.4001005UnitPaK
kg/m3m/s
Pa sPa s
W/(mK)
J/(kgK)

Table 1. Dimensions of the system and parameters of the thermoviscous fluid (air).

Figure 2. Mean displacement of the beam over the fixed electrode (x ∈ (lc , L)) calculated analytically (solid line)
and using 3D FEM simulation (black points): a) magnitude, b) phase.
In figure 3 the acoustic pressure sensitivity σ = U0 η̄/(pinc hg ) of the transducer used as an electrostatic receiverwith polarization voltage U0 = 30V is shown. At the left side of the figure the dependence of the sensitivity on
the airgap thickness hg presents very small impact of the variation of this parameter on the frequency responseof the transducer. Variation of the thickness of the slits hs and hL (right side of the figure) brings, by contrast,
an important change in the damping, thus making the frequency response more flat, which is important for
MEMS microphones.

4CONCLUSIONS

A simple analytical model of the miniaturized transducer composed of the planar rigid micro-beam attached atone end to a flat spring moving against a reduced-size backplate has been developed. The comparison of theresults of the present model and the full 3D numerical (FEM) simulation in terms of the mean displacement of
the beam over the reduced-size backplate shows very good agreement, the damping in the system being slightlyunderestimated. Variation of the thickness of the slits has an important impact on the frequency dependence of
the acoustic pressure sensitivity of the transducer used as a receiver, while the impact of the variation of theairgap thickness remains negligible.

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Figure 3. Magnitude of the sensitivity of the transducer for different values a) of the thickness of the airgap hg ,b) of the thickness of the slits hs , hL .

ACKNOWLEDGEMENTS
This work was supported by the Grant Agency of the Czech Technical University in Prague, grant No.
SGS18/200/OHK2/3T/16.

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