Modelling of a MEMS transducer using approximate eigenfunctions of asquare clamped plate
Karina Abramova(1) , Petr Honzík (1) , Nicolas Joly(2) , Stéphane Durand(2) Michel Bruneau(2)(1) Czech Technical University in Prague, Czech Republic, abramkar@fd.cvut.cz, honzikp@fd.cvut.cz(2) Laboratoire d’Acoustique de l’Université du Mans UMR CNRS 6613, France, nicolas.joly@univ-lemans.fr,stephane.durand@univ-lemans.fr, michel.bruneau@univ-lemans.fr

Abstract
The model of electroacoustic MEMS transducer with a moving square shaped clamped plate loaded by a thin
ﬂuid gap and a peripheral cavity is presented herein. The behaviour of the transducer, namely the thermal andviscous boundary layers effects originating in the ﬂuid gap between the moving electrode and the ﬁxed one andthe strong coupling between the moving electrode displacement and the acoustic pressure ﬁeld in the ﬂuid gap,
have to be described correctly by the model. The modelling approach proposed herein involving the integral
method for describing the acoustic pressure in the ﬂuid gap requires an analytical expression of eigenfunctionsof the square shaped clamped plate. Such an approximate expression in form of two-dimensional cosine series
proposed recently suffers from slow convergence and inaccurate boundary conditions. The solution proposed
herein is based on the series expansion over the system of functions satisfying exactly the boundary conditions, which leads to faster convergence, hence lowering computational costs. The proposed eigenfunctions
are described and the difference from the previous approximations is discussed. Finally, the acoustic pressure
sensitivity of the transducer is presented and compared to the results of a reference ﬁnite element model.

Keywords: MEMS transducer, analytical model, plate eigenfunctions, thermoviscous losses

1INTRODUCTION

The importance of precise models of miniaturized electroacoustic devices has rised recently while the use ofsuch devices is growing in many domains such as the consumer audio, the noise monitoring using wirelesssensor networks [1], but also in many other scientiﬁc applications which request high precision of such devices
used as sensors of acoustic ﬁelds [2].
Such models usually employ lumped-element approach [3, 4], full analytical approach [5, 6, 7, 8, 9, 10, 11]or numerical (FEM) simulation [12] in order to take into account thermoviscous losses originating from theboundary layers effect inside the narrow ﬂuid-ﬁlled elements (such as the ﬂuid gap between the moving andﬁxed electrode in the electrostatic transducer) and the strong coupling between the ﬂuid and moving parts of thetransducer. In the present work these effects are expressed using a recently published approach [10] employing
an integral formulation for the acoustic pressure in the thin ﬂuid layer. Eigenfunctions of the moving part of
the transducer (whose approximate analytical expression is searched for herein) are needed by such formulationin order to experess the ﬂuid-structure coupling.
Even though the moving electrodes of miniaturized transducers can have very complex shapes, a simple squareplate clamped at all edges has been chosen herein in order to show the ability of the method. The present
approach is based on the previous work [11] where the eigenfunctions of the plate have been expressed in theapproximate form of two-dimensional cosine series approximating the result of simple numerical (FEM) modelof the plate (without ﬂuid). Since such approach suffers from the slow convergence and inaccurate boundary
conditions, the modal functions of 1D beam satisfying exactly the boundary conditions have been used hereinin the series expansion instead of the cosine functions.

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2THEORETICAL MODEL

The transducer considered herein consists of a square elastic clamped plate with a halfside a. A thin ﬂuid layer(an air gap herein) of thickness hg is trapped between the plate and a back rigid electrode having the same
shape as the elastic plate. This ﬂuid layer is loaded at its periphery by a small cavity of volume Vc (see ﬁgure1).

Figure 1. Geometry of the system: a) the dimensions of the square clamped plate and b) geometry of thetransducer in the 1st quadrant.

2.1 Eigenfunctions of the plate
The governing equation of motion for a plate loaded by acoustic ﬁelds on both sides pinc and p(x, y) can be
expressed as [13][ΔΔ − k4p ]ξ (x, y) =1(−pinc + p (x, y)) ,D

(1)

Eh3p

where D = 12 1−ν 2 is the ﬂexural rigidity, ν being the Poisson’s ratio, E the Young’s modulus, h p is the()thickness of the plate, k4p = MDs ω 2 , ω is the angular velocity, Ms = h p ρ p is the mass per unit area, ρ p being thedensity of the plate.
The solution of the equation (1) can be written as an expansion over the orthonormal eigenfunctions
ξ (x, y) = ∑ ξmn ψmn (x, y) ,(2)mnwhere the functions ψmn (x, y) are the solutions of the homogeneous equation associated with eq. (1)4ψmn (x, y) = 0,ΔΔ − kmn24 = k2 + k2
and the modal coefﬁcients of the serie are given byand kmn
xmyn
ξmn =D1

2 + k2kxmyn

2− k4p

· a a−a −aψmn (x, y)[p(x, y) − pinc ] dxdy.

(3)

(4)

Herein, the eigenfunctions ψmn (x, y) are approximated employing the known symmetric eigenfunctions of the1D beam [7, 14] as the basis of the serie instead of cosine functions [11] in the following manner:NNψmn (x, y) = ∑ ∑ c(qr),(mn) φq (x) φr (y) ,q=1 r=17362(5)  cos αqs x
cosh αqs x1− , with tan αqs a = − tanh αqs a .
where φq (x) = √sscosh αq a2a cos αq a(6)The coefﬁcients c(qr),(mn) have been calculated from the simple numerical (FEM) solution n ψmn of the equation(3) asc(qr),(mn) = a an−a −aψmn (x, y)φq (x) φr (y)dxdy.

(7)

The eigenvalues kxm , kyn are expressed from the numerically calculated eigenfrequencies n fmn as follows
kxm =2π n fmm,2c p(8)where c p = D/Ms is the wave speed on the plate, kyn having the same values as kxm . Note that the numericallycalculated eigefunctions n fmm changes with the side of the plate a and have to be ﬁtted from several numericalsolutions for different plate sides using a polynom of order 3.
2.2 Coupling between the plate displacement and the acoustic pressure in the thin ﬂuid layer [10]The propagation of the acoustic pressure in the air gap is governed by the wave equation
 22∂xx + ∂yy+ χ 2 p (x, y) = −U (x, y) ,(9)where U (x, y) = ρ0 ω 2 ξ (x, y)/ (hg Fv ), with a boundary condition ∂n p|a = − jωρ0 pc /4Fv Zc hg a (∂n denoting thenormal derivative), ρ0 is the air density, Zc is the impedance of the peripheral cavity, the mean proﬁle of theparticle velocity over the thickness of the air gap Fv and the complex wavenumber χ accounting for the effectof thermoviscous losses in the air gap being given for example in [6, 7, 8, 9, 10]. The acoustic pressure ﬁeld
in the small cavity pc is assumed to be uniform.
The solution of the equation (9) can be expressed as follows:p (x, y) = a a00

G (x, x0 ; y, y0 )U (x0 , y0 ) dx0 dy0 + pc IG (x, y) ,where
IG (x, y) =jωρ08Fv Zc hg a+++

 a0

 a
0 a0

 a0

(10)

G(x, x0 ; y, a)dx0

G(x, a; y, y0 )dy0
(11)∂y0 G(x, x0 ; y, a)dx0∂x0 G(x, a; y, y0 )dy0 ,and where the Green’s function can be chosen as
G(x, x0 ; y, y0 ) = g(x, x0 ; y, y0 ) + g(x, −x0 ; y, y0 ) + g(x, x0 ; y, −y0 ) + g(x, −x0 ; y, −y0 ),withg(x, x0 ; y, y0 ) = −iH0− χ(x − x0 )2 + (y − y0 )2 ,(12)

(13)

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H0− denoting the Hankel function of the second kind of order "0". The pressure in the peripheral cavity pc =p (x, a)x is supposed to be equal to the mean value of the pressure at the air gap periphery (y = a for example)
over the x coordinate.
Using the equation (4) the modal coefﬁcients ξmn can be obtained from the matrix equation[−A + B] (Ξ) = (C),(14)aawhere (Ξ) and (C) are the column vectorsof elements ξmn and cmn = −pinc −a−a ψmn (x, y)dxdy respectively,
2 + k2 2 − k2 and the elements of the matrix A are found by substiB is a diagonal matrix of elements D kxm
ynptuting expression for pressure (10) into the integral

3a a−a −a ψmn (x, y) p (x, y) dxdy of equation (4).

RESULTS AND DISCUSSIONFirst, the approximated eigenfunctions are compared with the simple numerical (FEM) result n ψmn (withoutconsideration for the coupling effect between the plate and air ﬁlled acoustic elements) for the plate with theproperties given in table 1. As an example, the comparison between the approximated and the numericallycalculated eigenfunctions for the ﬁrst mode (m, n = 1) is shown in ﬁgure 2 and the differences between these
two results for N=1,2,3,7 (the corresponding number of members of the serie is N 2 ) are shown in ﬁgure 3.



 Figure 2. a) Numerical solution n ψ11 (x, y) and b) analytical approximation of ψ11 (x, y) for N = 3
While difference and comparison graphs provide visual impressions of the error between the test and comparative result, the estimates of these errors accountning varying number of members of the serie can be calculated
as follows:
Err =n2∑Mi=1 (ψmn i − ψmn i )· 100%M n2∑i=1 ψmn i(15)where ψmn i is the value of the approximated ψmn at the i-th node of the mesh (similarly for the reference
numericaly calculated eigenfunction n ψmn ) and M is the number of mesh nodes. The dependence of this error
on the summation limit N in equation (5) is shown in ﬁgure 4.
Figures 3 and 4 show clearly that the present method provides better approximation of the eigenfunctions comparing to the previous approach [11] and leads to faster convergence of the serie.

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Figure 3. Difference between numerical solution and a) present analytical approximation and b) approximationfrom [11]10 2
approximation from cosine function
presented method

Mean error, [%]

10 1

10 0

10 -1

10 -2

10 -3

1234

567NFigure 4. Mean error of the approximation for m,n=1,1

The acoustic pressure sensitivity of an electrostatic transducer is given by σ = −U0 ξ̄ /(hg pinc ), where ξ̄ is themean displacement of the moving electrode and U0 is the polarization voltage (herein U0 = 30V ). Figure 5
shows the frequency dependence of the pressure sensitivity obtained by using the present approach comparedwith the numerical result obtained from full 3D FEM model of the whole structure using Comsol Multiphysics
software [12] (all properties of the transducer and the ﬂuid are given in tables 1 and 2). A good agreement7365

between those results can be observed, the damping in the analytical approximate solution being slightly underestimated.
b)

-50

-80

|σ| [dB re 1V/Pa]|σ| [dB re 1V/Pa]

a)

-100

-150
102

104106-81-82-83
10 1

10 2

10 3

frequency [Hz]phase(σ) [rad]phase(σ) [rad]

-2-4
10 1

Reference numerical solution
Integral method with present eigenfunctions

10 2

10 3

10 4

10 5

10 6

10 2

10 3

10 5

frequency [Hz]c)d)|σ| [dB re 1V/Pa]   

10 4-pi/8

10 1

frequency [Hz]
 





 -120-1403









45

678

910×10 5

frequency [Hz]

-80-100phase(σ) [rad] !" 

10 5

0

0-1

-3

10 4

frequency [Hz]0-pi/8-pi/2-pi



3



45

frequency [Hz]678910×10 5Figure 5. The comparison between the acoustic pressure sensitivity obtained using the present approach (fullline) and the one obtained from FEM simulations (black points); a) module and phase of the sensitivity inthe whole frequency range, b) detailed view on the ﬂat part of the sensitivity, c) detailed view on the ﬁrstresonance, d) detailed view on the higher frequency range.
Table 1. Dimensions of the transducer
Parameter
Plate side
Plate thickness
Air-gap thickness
Cavity volume
Plate density
Young’s modulus
Poisson’s ratio7366Symbolahphg
VcρpE
νValue1 · 10−310 · 10−6
10 · 10−6
1 · 10−102330
160 · 109
0.27Unit
[m]
[m]
[m]
[m3 ]
[kg/m3 ][Pa][−]

Table 2. Air properties
Parameter
Adiabatic sound speed
Air density
Shear dynamic viscosity
Permittivity
Thermal conductivity
Speciﬁc heat coefﬁcient atconstant pressure per unitof mass
Ratio of speciﬁc heats

4

Symbolc0
ρ0με0λh
CpValue343
1.20
1.81 · 10−58.8542 · 10−1225.7 · 10−31005

Unit[m · s−1 ][kg · m−3 ][Pa · s][F · m−1 ][W /(m · K)][J/(kg · K)]

γ

1.4[−]

CONCLUSIONS

An approximation of the eigenfunctions of the square clamped plate (used as a moving electrode of a MEMStransducer) expressed in the form of two-dimensional series expansion over the modal functions of 1D beam,
the coefﬁcients of the serie being calculated from the simple numerical model of the plate, has been proposed
herein. Comparing to the previously explored method using the series of cosine functions this approach leads
to faster convergence of the series and enables to satisfy exactly the boundary conditions. The approximatedeigenfunctions has been used in a model of the MEMS transducer. In this model the integral formulation takenfrom the literature was employed to express the acoustic pressure ﬁeld in the thin ﬂuid layer coupled withthe displacement of the plate. The pressure sensitivity of the transducer shows very good agreement with thecomplete 3D numerical (FEM) simulation taken as reference, only slight underestimation of the damping in theanalytical approximate solution can be observed near the resonance frequencies.
The method presented herein shows the ability to provide correct models of miniaturized transducers containing
the moving electrode in form of square clamped plate. Future works will include the testing of this method on
more complex shapes of the moving electrodes.

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

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