Nonlinear Identification of a Capacitive Dual ... - Semantic Scholar

Proceedings of IDETC/CIE Mechanical of Vibration and2005 Noise 20th ASME Biennial Conference onProceedings ASME 2005 InternationalSeptember Design Engineering Technical Conferences 24-28, 2005, Long Beach, CA, USA & Computers and Information in Engineering Conference September 24-28, 2005, Long Beach, California USA

DETC2005-84591

DETC2005-84591 NONLINEAR IDENTIFICATION OF A CAPACITIVE DUAL-BACKPLATE MEMS MICROPHONE Jian Liu1, David T. Martin2, Karthik Kadirvel2, Toshikazu Nishida2, Mark Sheplak1, and Brian P. Mann1 1 Department of Mechanical & Aerospace Engineering 2 Department of Electrical & Computer Engineering University of Florida, Gainesville, Florida 32611-6250 352-392-4550, 352-392-1071(fax), [email protected]

ABSTRACT This paper presents the nonlinear system identification of model parameters for a capacitive dual-backplate MEMS microphone. System parameters of the microphone are developed by lumped element modeling (LEM) and a governing nonlinear equation is thereafter obtained with coupled mechanical and electrostatic nonlinearities. The approximate solution for a general damped second order system with both quadratic and cubic nonlinearities and a non-zero external step loading is explored by the multiple time scales method. Then nonlinear finite element analysis (FEA) is performed to verify the accuracy of the lumped stiffnesses of the diaphragm. The microphone is characterized and nonlinear least-squares technique is implemented to identify system parameters from experimental data. Finally uncertainty analysis is performed. The experimentally identified natural frequency and nonlinear stiffness parameter fall into their theoretical ranges for a 95% confidence level respectively. Keywords: Nonlinear identification, MEMS, microphone, FEA, nonlinear least-squares INTRODUCTION Silicon micromachining technology provides the opportunity for batch fabrication of micro-electro-mechanical devices such as microphones. Silicon micromachining provides excellent process control (such as dimensions and material properties) and offers the potential of improved reproducibility, and high-degree miniaturization [1]. Moreover, the batch fabrication of silicon devices can lead to lower costs and smaller sizes since hundreds or thousands of devices can be fabricated together on a single silicon wafer [2]. A microphone is a transducer that converts acoustic energy into electrical energy and is widely employed in a variety of applications such as sound field measurements [3, 4], hearing

aids [5, 6], and noise monitoring [7]. Many transduction schemes, such as piezoelectric, piezoresistive, capacitive, electrodynamic and optical [8] have been developed for the microphone. In comparison with other types of silicon microphones, capacitive silicon microphones have many advantages of high sensitivity, large bandwidth, and inherently low power consumption [1]. However, capacitive silicon microphones have some potential issues - such as electrostatic “pull-in” instability, output signal attenuation due to the parasitic capacitance, and decreased sensitivity at high frequencies due to viscous damping of the perforated backplate [8]. Several researchers have developed various single backplate capacitive MEMS microphones [4, 9-12], however, the bandwidths of all those microphones are limited up to 20kHz, which is not suitable for aero-acoustics applications. Some performance specifications of those microphones are summarized in Table 1. Over single backplate MEMS capacitive microphones, dual-backplate capacitive MEMS microphones have the advantage of smaller size, higher sensitivity, larger dynamic range and broader bandwidth [14]. Force feedback is made possible and feasible by the modern silicon micromachining technology and has successfully been applied to MEMS devices (such as the accelerometer [15]). The implementation of force feedback in the microphones could further increase the bandwidth, dynamic range, and linearity [14]. For the dual backplate MEMS microphone, Bay et. al. [5, 6] proposed dual-backplate capacitive microphone designs for hearing aid applications with desired sound pressure levels (SPL) up to 120 [dB] (ref. 20 [µPa]) and bandwidths up to 20 [kHz]; however, no experimental results were reported. Rombach et. al. [13] demonstrated a dual back-plate condenser microphone with a bandwidth of 20 [kHz], a sensitivity of 13 [mV/Pa] and a linearity of up to 118 [dB SPL].

1

Copyright © 2005 by ASME

Researcher

Diaphragm Area [ mm 2 ]

Gap [ µm ]

Bandwidth [ kHz ]

Sensitivity [ mV / Pa ]

Noise Level [ dB ]

Bourouina et al. [9]

1.0

7.5

10

2.4

38

Scheeper [10]

4.0

3.0

14

*

7.8

30

Zou et al. [11]

1.0

2.6

9.0

14.2

--

Torkkeli et al. [12]

1.0

1.3

12

4.0

33.5

4.0

0.9

20

13

22.5

20

*

Top Backplate

Diaphragm

Bottom Backplate

Figure 3: Schematic diagram of microphone electrical model.

†Rombach et al. [13]

≈ 12

Scheeper et al. [4]

20

22

23

Table 1: Performance specifications for capacitive microphones from various researchers (*open circuit sensitivity, † dual backplate microphone). The dual-backplate capacitive MEMS microphone presented here is developed for noise source localization and characterization in aero-acoustic applications. Figure 1 shows a microscope photograph of the microphone, its die size is approximately 1mm × 1mm , bond pads are fabricated to realize the electrical connections to its backplates and diaphragm.

1 mm

1 mm

Bond pad

Figure 1: Top-view photograph of the microphone. The cross-section of the microphone structure is shown in Fig. 2. To realize the energy transduction, the top and bottom backplates are biased oppositely with respect to middle diaphragm by an AC voltage. When the acoustic wave impinges the microphone, the applied pressure deflects the middle diaphragm and therefore alters the capacitances of two nonlinear capacitors, which are formed by two backplates and middle diaphragm as shown in Fig. 3. The resulting differential electrical output signal can be picked up from the diaphragm, and it can be further demodulated by the supporting circuitry to determine the incoming pressure. Backplate Holes

The microphone was fabricated using the SUMMiT V process at Sandia National Laboratories. It has a 2.25 [µm] thick circular diaphragm with a 230 [µm] radius and 2 [µm] gap between each circular perforated backplate as shown in Fig. 2. This microphone demonstrated a linear response up to 160 [dB SPL] and over 180 [kHz] bandwidth, but it possessed a low sensitivity of 282 [µV/Pa] as a tradeoff result of high bandwidth [16]. The overall performance of this microphone is expected to be further improved via the implementation of the force feedback loop. This paper presents a method of nonlinear parameter identification for this capacitive dual-backplate MEMS microphone. First, system parameters of the microphone are developed by lumped element modeling (LEM) and a nonlinear dynamic governing equation with coupled mechanical and electrostatic nonlinearities is thereafter obtained. Nonlinear finite element analysis (FEA) is performed to verify the accuracy of the theoretical lumped stiffnesses of the diaphragm. In order to identify system parameters, an approximate solution for the nonlinear oscillations of the diaphragm is obtained using the method of multiple time scales. The microphone is characterized and nonlinear least-squares technique is implemented to identify system parameters from experimental data. Finally, an uncertainty analysis is performed and the theoretical and experimental results are compared and some conclusions are drawn. THEORETICAL MODELING Considering the middle circular diaphragm under uniform transverse pressure loading, it is assumed that the diaphragm is linearly elastic, and that the boundary condition is clamped with zero residual stress inside the diaphragm. From the theory of elasticity, it is assumed that the deflected surface is symmetrical with respect to the origin of the circular diaphragm. The general small displacement solution in the polar coordinate system is given as follows [17] 2

⎡ ⎛ r ⎞2 ⎤ w ( r ) = w0 ⎢1 − ⎜ ⎟ ⎥ , ⎣⎢ ⎝ a ⎠ ⎦⎥

Top Backplate Diaphragm Bottom Backplate Cavity Anchors

Gaps

Figure 2: 3D cross-sectional view of microphone structure.

(1)

where r is the distance of any radial point along the radius, a is the radius of the diaphragm, w0 is the center displacement of the diaphragm

2


w0 =

pa 4 , 64 D

(2)

where p is the applied acoustic pressure, and D is the flexural rigidity of the diaphragm and is defined as D=

Eh

3

12 (1 −ν 2 )

(3)

,

of the diaphragm to that of an equivalent lumped system [18, 19], 1 Ame = π a 2 . (6) 3 Equation (4) indicates that the diaphragm behaves like a Duffing spring under the large displacement case, the Duffing spring model and lumped element modeling are used to obtain the following expression for the static case

where E and ν are Young's modulus and Poisson's ratio of the diaphragm respectively, and h is the thickness of the diaphragm. For the large displacement solution, by assuming the same general solution form defined in Eq. (1), an energy approach [17] is utilized to obtain the following approximate large displacement relation for the center displacement (specific Poisson's ratio ν = 0.22 is used). w0 =

pa 4 ⋅ 64 D

1 w2 1 + 0.4708 02 h

.

(7)

where k1 and k3 are linear and cubic stiffnesses of the diaphragm respectively. By comparing Eq. (7) with Eq. (4), k1 and k3 are obtained as follows 64 Dπ k1 = , (8) 3a 2

(4) k3 =

The factor 0.4708 w0 2 h 2 in Eq. (4) represents the nonlinear effect of in-plane stretching, which is significant and can not be neglected under large displacement. If the displacement is much smaller than the thickness of the diaphragm, it follows that Eq. (4) can be reduced to Eq. (2). Since the system dynamics of the microphone is highly coupled and it is difficult to obtain the closed-form solution. A simple way to approximate the coupled system dynamics is through the lumped element modeling (LEM). Lumped element modeling is based on the assumption that the device length scale of interest is much smaller than the characteristic length scale (for example, wave length) of the physical phenomena [18, 19]. By using lumped element modeling, the spatial variations of the quantities of interest can be decoupled from the temporal variations and the coupled system can be then divided into many lumped impedances (mass, stiffness and dissipation). After choosing the center of the diaphragm as a reference lumping point, and assuming the static mode shape defined in Eq. (1), the lumped mass is determined by computing the total kinetic energy stored in a diaphragm and then equating it to the work-equivalent lumped system. In the mechanical domain, the work-equivalent lumped mass M me [18, 19] is given as

1 M me = π a 2 h ρ d , 5

p ⋅ Ame = k1 ⋅ w0 + k3 ⋅ w03 ,

(5)

(9)

The cavity of the microphone impedes the movement of diaphragm by storing potential energy and acts as a spring, by assuming only plane wave exists in circular cavity cylinder, its work-equivalent mechanical stiffness kc can be calculated as follows [18-20] kc =

ρa c2 πρ a c 2 ac 2 2 2 = , π a ( ) c π ac 2 d c dc

(10)

where c is the speed of sound in air, ρ a is the density of air, ac and d c are the radius and depth of the cavity cylinder respectively, and

(π a )

2 2

c

is used to convert the acoustical

stiffness into the mechanical stiffness. Therefore, the first linear resonance frequency ω0 is given by

ω0 =

k1 + kc . M me

(11)

The acoustical holes in the top and bottom backplates dissipate acoustic energy via friction when the diaphragm vibrates, and the equivalent lumped damping coefficient b [21] for the squeeze-film damping in the gaps is given by

where ρ d is the density of the diaphragm. Similarly, the lumped area of the diaphragm Ame is determined by equating the calculated total volumetric velocity

10.044 Dπ . a 2 h2

b=

4γπ a 4 4γπ a 4 B ( Abp ) + B ( Atp ) , 3 3nbp d 0 3ntp d 03

(12)

where γ is the dynamic viscosity of air, ntp and nbp are total number of holes in the top and bottom backplates respectively,

3


Atp and Abp are ratios of total hole areas to the top and bottom

backplate areas respectively, d 0 is the nominal gap between backplates and diaphragm, and B is a known function with the following form 1 ⎛1⎞ 3 1 1 B ( y ) = ln ⎜ ⎟ − + y − y 2 . 4 ⎝ y⎠ 8 2 8

(13)

Table 2 summarizes some properties of the diaphragm.

a h E ρ

230e-6 [m] 2.25e-6 [m] 1.6e11 [N/m2] 2.23e3 [kg/m3] 0.22

ν

The top and bottom backplates are assumed to be rigid and have equal areas with the diaphragm. Each backplate is separated by a nominal gap d 0 from the diaphragm. The diaphragm is modeled by a Duffing spring (two spring constants k1 and k3 ), a lumped mass M me with a lumped area Ame , which results in the removal of the clamped boundary. The stiffness of the cavity and resistance of the acoustical holes in the backplates are represented by kc and b respectively. In general, if the top and bottom backplates are biased with respect to middle diaphragm by two electrical signals ( ±V ( t ) ) with equal magnitude and opposite sign respectively, and the microphone is impinged by a normal incident acoustic wave with amplitude p , the open-loop equation of motion (E.O.M.) can be written as

Table 2: Properties of the diaphragm (material: polysilicon). Table 3 shows the calculated theoretical parameters for the microphone under study. k1

202.2 [N/m]

k3

1.88e13 [N/m3]

kc

24.27 [N/m]

M me

1.67e-10 [kg]

Ame

5.54e-8 [m2] 3.05e-5 [N⋅s/m]

b

lumped

2 ε 0 Ame ⎡ V ( t )

V (t )

⎤ − ⎢ ⎥ 2 ⎢⎣ ( d 0 + x )2 ( d 0 − x )2 ⎥⎦ − p ⋅ Ame , (14)

M me x + bx + ( k1 + kc ) x + k3 x = − 3

2

where x represents the center displacement of the diaphragm (positive indicates an upward direction) and the parallel plate assumption is used when calculating each electrostatic force. Obviously, Eq. (14) represents a nonlinear system with coupled mechanical and electrostatic nonlinearities. Equation (14) can be rewritten as

Table 3: Theoretical lumped parameters of the microphone.

ε 0Γ ⎡ V (t )

V (t )

⎤ − ⎢ ⎥ 2 2 2 ⎢⎣ ( d 0 + x ) ( d 0 − x ) ⎥⎦ − p ⋅ Γ, (15)

x + 2ξω0 x + ω0 2 x + β x3 = −

Once all the lumped parameters are extracted, an openloop dynamic model for the microphone is shown schematically in Fig. 4.

2

2

where ξ = b ( 2 M meω0 ) is the damping ratio, β = k3 M me is

Fixed Top Backplate

the nonlinear stiffness parameter and Γ = Ame M me is the ratio of lumped area over lumped mass. k ,k 1 3

k

c

b

d0

V (t )

behavior of the microphone, therefore identifying system parameters becomes critical.

Diaphragm

x=0 M

me

As seen from Eq. (15), system parameters such as ξ , ω0 , β , and Γ play a very important role in determining the overall

,A me

d0

−V ( t )

Pressure p

Fixed Bottom Backplate Figure 4: Schematic of a dual backplate microphone with a cubic mechanical nonlinearity and an electrostatic nonlinearity.

Although system parameters can be calculated theoretically from the previous modeling, it is well-known that some of the modeling assumptions, such as a perfectly fixed boundary condition, equal-area parallel plates, and no fabrication variations associated with geometry properties of the microphone are probably not accurate. Therefore, some errors exist between the modeled system parameters and real system parameters; this paper examines the experimental system parameters identified through a series of calibration experiments.

4


Formulating Problem for Experimental Study

α4 = β −

When the microphone was calibrated to obtain system parameters experimentally, no acoustic pressure actually existed in the diaphragm and a uni-polar square wave V ( t ) was applied directly to either top or bottom backplate with diaphragm and the other backplate electrically grounded. The expression for the applied uni-polar square wave is given by ⎧ ⎪⎪V0 V (t ) = ⎨ ⎪0 ⎪⎩

nT ≤ t < nT + nT +

T 2

T ≤ t < ( n + 1) T 2

(16)

,

where V0 and T are the voltage amplitude and period of the square wave respectively, and n = 0, 1, 2, ... T ≤ t < ( n + 1) T , no electrostatic 2 force acts on the diaphragm, therefore the governing equation for the free vibration of diaphragm is

During the time nT +

x + 2ξω0 x + ω0 2 x + β x3 = 0.

(17)

T , an electrostatic force 2 acts on the diaphragm and forces it to vibrate, and if the voltage is applied to the top backplate, the governing equation for the forced vibration can be reduced to

α5 =

ε 0Γ 2

V0 2

( d0 − x )

2

APPROXIMATE SCALES

A Taylor’s series expansion around x = 0 for the nonlinear electrostatic force up to 3rd order results in a convenient polynomial expression

ε 0Γ 2

V0

2

( d0 − x )

2

≈

ε 0 ΓV0 ⎡ 2

2d 0 2

⎛ x ⎞ ⎛ x ⎞ ⎤ x ⎢1 + 2 + 3 ⎜ ⎟ + 4 ⎜ ⎟ ⎥ . (19) d0 ⎢⎣ ⎝ d0 ⎠ ⎝ d 0 ⎠ ⎥⎦ 2

x + α1 x + α 2 x + α 3 x + α 4 x = α 5 , 2

where

3

α1 = 2ξω0 ,

α 2 = ω0 2 − α3 = −

d0

3

3ε 0 ΓV0 2 , 2d 0 4

,

SOLUTION

BY

MULTIPLE

TIME

x (τ ,τ 1 ,τ 2 , ε ) = x0 (τ ,τ 1 ,τ 2 ) + ε x1 (τ ,τ 1 ,τ 2 ) + ε 2 x2 (τ ,τ 1 ,τ 2 ) , (26)

where x0 , x1 , and x2 are three unknown functions, and the multiple independent time scales are defined as

τ = t ,τ 1 = ετ , and τ 2 = ε 2τ .

(27)

Therefore, the time derivatives with respect to t become the following expansion terms of the partial derivatives with respect to the corresponding time scales d = D0 + ε D1 + ε 2 D2 , dt

(28)

d2 = D0 2 + 2ε D0 D1 + 2ε 2 D0 D2 + ε 2 D12 , dt 2

(29)

where D0 =

d d d , D1 = and D2 = . dτ dτ 1 dτ 2

(30)

The damping and nonlinear terms in Eq. (20) are further ordered to show up in the O ( ε 2 ) with the following change of parameters

(20)

α1 = ε 2 µ1 , α 2 = ω 2 ,α 3 = ε 2 µ3 and α 4 = ε 2 µ4 .

x + ε 2 µ1 x + ω 2 x + ε 2 µ3 x 2 + ε 2 µ4 x3 = α 5 .

(22) (23)

(31)

Equation (20) is changed into

(21)

ε 0 ΓV0 2

(25)

.

The approximate solution of Eq. (20) is assumed as a second order expansion in terms of a small positive parameter ε , which is a measure of the amplitude of the motion,

3

It follows the general form of Eq. (18) can be written

2d 0 2

Since there is no closed form solution to Eq. (20), a multiple time scales approach [22] is used to obtain the approximate solution.

(18)

.

ε 0 ΓV0 2

(24)

Equation (20) represents a general damped second order system with both quadratic and cubic nonlinearities and a nonzero external step loading, its solution can be easily applied to Eq. (17) and other important applications, the following effort has been done to obtain its approximate solution.

During the time nT ≤ t < nT +

x + 2ξω0 x + ω0 2 x + β x3 =

2ε 0 ΓV0 2 , d 05

(32)

Substitution of Eqs. (26), (27), (28), (29) and (30) into Eq. (32) results in the following expressions after separating into O (ε )

5


O ( ε 0 ) : D0 2 x0 + ω 2 x0 = α 5 ,

A (τ 2 ) =

(33)

O ( ε 1 ) : D0 2 x1 + ω 2 x1 = −2 D0 D1 x0 ,

(34)

O ( ε 2 ) : D0 2 x2 + ω 2 x2 = −2 D0 D1 x1 − 2 D0 D2 x0 − D12 x0

− µ1 D0 x0 − µ3 x0 − µ4 x0 . (35) 2

α x0 = 5 + A (τ 1 ,τ 2 ) eiωτ + A (τ 1 ,τ 2 ) e− iωτ , α2

where R is the amplitude of x0 and φ is the phase angle. Substituting Eq. (42) into (41) and separating into real and imaginary components results in dR 1 + µ1 R = 0, dτ 2 2

(36)

(43)

dφ 2µ3α 5α 2 + 3µ 4α 5 2 3µ4 2 − − R = 0. 2ωα 2 2 8ω dτ 2

where A (τ 1 ,τ 2 ) and A (τ 1 ,τ 2 ) are complex conjugates. The following expression is obtained by substituting Eq. (36) into Eq. (34)

R (τ 2 ) = R0 e

The elimination of the secular terms from Eq.(37) requires that D1 A (τ 1 ,τ 2 ) and D1 A (τ 1 ,τ 2 ) are zero, which means that

φ (τ 2 ) =

A and A are only functions of τ 2 .

1 − µ1τ 2 2

(45)

,

2 µ3α 5α 2 + 3µ4α 5 2 3µ R 2 τ 2 − 4 0 e − µ1τ 2 + φ0 , (46) 2 8ωµ1 2ωα 2

where R0 and φ0 are constants.

Therefore, the solution for Eq. (37) can be written as (38)

Combining Eqs. (26), (36), (42), (45) and (46), the approximate solution is

where B (τ 1 ,τ 2 ) and B (τ 1 ,τ 2 ) are complex conjugates.

x (τ ,τ 1 ,τ 2 ) ≈

1 − µ1τ 2 ⎛ α5 2 µ α α + 3µ α 2 + R0 e 2 ⋅ cos ⎜ ωτ + 3 5 2 2 4 5 τ 2 α2 2ωα 2 ⎝

Substituting Eqs. (36) and (38) into Eq. (35) yields −

α D0 x2 + ω x2 = ⎡⎣ −2iω ( D1 B ) − 2iω ( D2 A ) − 2µ3 5 A α2 2

⎛α 2 2 ⎞ ⎤ −i µ1ω A −3µ 4 ⎜ 5 2 + A ⎟ A⎥ ⋅ eiωτ + O.H .T ., (39) ⎝ α2 ⎠ ⎥⎦

x (t ) =

⎛α 2 2 ⎞ −3µ4 ⎜ 5 2 + A ⎟ A = 0. ⎝ α2 ⎠

−

⎤ 3α 4 R0 2 −α1t e + φ0 ⎥ . 8ωα1 ⎦

(48)

If an initial displacement χ 0 is imposed and the system starts from rest, the resulting transient displacement becomes (40)

x (t ) =

It follows that B is only a function of τ 2 [22], therefore

⎡⎛ α 5 m − 12 α t 2α α α + 3α α 2 ⎞ + χ0 ⋅ e ⋅ cos ⎢⎜ ω + 2 3 5 2 4 5 ⎟ t α2 2ωα 2 ⎢⎣⎝ ⎠ 1

−

⎛ α α2 −2iω D2 A − i µ1ω A − ⎜ 2µ3 5 + 3µ4 5 2 α2 α2 ⎝

(47)

1 ⎡⎛ − α1t α5 2α α α + 3α α 2 ⎞ + R0 e 2 ⋅ cos ⎢⎜ ω + 2 3 5 2 4 5 ⎟ t α2 2ωα 2 ⎢⎣⎝ ⎠

The elimination of the secular terms in Eq. (39) requires

α5 A α2

⎞ 3µ 4 R0 2 − µ1τ 2 + φ0 ⎟ . e 8ωµ1 ⎠

Using Eqs. (27) and (31), the above equation can be further simplified as

where O.H .T . are other harmonic terms that are neglected in the following analysis.

−2iω ( D1 B ) − 2iω ( D2 A ) − i µ1ω A − 2 µ3

(44)

The solutions for R and φ are

D0 2 x1 + ω 2 x1 = −2iω ⎡⎣eiωτ D1 A (τ 1 ,τ 2 ) + e −iωτ D1 A (τ 1 ,τ 2 ) ⎤⎦ . (37)

2

(42)

3

The general solution to Eq. (33) is

x1 = B (τ 1 ,τ 2 ) eiωτ + B (τ 1 ,τ 2 ) e− iωτ ,

1 R (τ 2 ) eiφ (τ 2 ) , 2

⎞ 2 ⎟ A − 3µ 4 A A = 0. (41) ⎠

⎤ m2 3α 4 χ 0 e −α1t − 1 ⎥ , 8ωα1 ⎥ ⎦

(

)

(49)

where

χm0 = χ 0 −

The polar form for the A (τ 2 ) can be written as

6

α5 . α2

(50)


If there is no external step loading and quadratic nonlinearity in the system, which occurs for the experimental case of transients about a zero equilibrium position, the following parameters can be changed to (51)

α 5 = 0.

(52)

Equation (49) is then reduced into x (t ) = χ0 ⋅ e

1 − α1t 2

⎡ ⎤ 3α χ 2 ⋅ cos ⎢ωt − 4 0 e −α1t − 1 ⎥ , 8ωα1 ⎣ ⎦

(

)

(53)

a5 , a3

(55)

0.6

0.4

0

500

1000

1500 2000 2500 3000 Applied Pressure (Pa)

3500

4000

4500

Figure 5: Transverse deflection of the diaphragm under the uniform applied pressure. The final results from nonlinear FEA are compared to ideal linear results and also the nonlinear analytical results as shown in Fig. 5. From Fig. 5, it can be concluded that the theoretical lumped parameters k1 and k3 are in a good agreement with the calculated values from nonlinear FEA. EXPERIMENT

2d 2α Γ = 0 25 , ε 0V0

ω0 = α 2 +

Ideal Linear Result Nonlinear FEA Result Analytical Result

0.8

0

(54)

Once α1 , α 2 , α 3 , α 4 and α 5 are estimated by a nonlinear least-squares curve-fitting technique based on the time history of experimental transient displacement. System parameters ξ , ω0 , β , Γ and d 0 can then be identified by solving Eqs. (21), (22), (23), (24) and (25) simultaneously as follows, d 0 = −3

-6

0.2

which can be easily applied to Eq. (17) to yield the following solution ⎡ ⎤ 3βχ 0 2 −2ξωt − 1) ⎥ . x ( t ) = χ 0 ⋅ e −ξωt ⋅ cos ⎢ωt − e 2 ( 16ξω ⎣ ⎦

x 10

1 Center Displacement (m)

α 3 = 0,

1.2

(56)

ε 0V0 2 Γ d 03

(57)

,

30 MHz Synthesized Function Generator DS345, Stanford Research Systems

α ξ= 1 , 2ω0

(58)

SYNC

OUTPUT

Polytec OFV 074 Microscope Adapter

2ε V 2 Γ β = α 4 + 0 50 . d0 NONLINEAR FEA VERIFICATION STIFFNESSES OF THE DIAPHRAGM

The calibration experiment of the microphone was conducted using the Polytec Scanning Laser Doppler Vibrometer (SLDV) in the Interdisciplinary Microsystems Laboratory at the University of Florida. The block diagram of the experiment setup is shown in Fig. 6.

Polytec MSV-Z-040 Scanner Controller

(59) OF

TRIG IN

REF

GATE IN

VEL

LUMPED

To determine the accuracies of theoretical lumped stiffnesses ( k1 , k3 ) of the diaphragm, nonlinear finite element analyses (FEA) are carried out in Coventor Ware 2004 [23] for different applied pressure cases.

Polytec OFV 3001S Vibrometer Controller

Polytec VibraScan DAQ PC

OLYMPUS BX60 Microscope

The properties of the diaphragm used in the FEA are shown in Table 2 and the uniform pressure applied to the bottom surface of the diaphragm is varying from 10 to 4000 Pa. The center displacement for each applied pressure is obtained.

Polytec OFV 511

With Test Microphone

Fiber Interferometer

Figure 6: Block diagram of experiment setup.

7


The test microphone is put on the stage of the microscope (OLYMPUS BX60). A fiber interferometer (Polytec OFV 511) generates the input laser beam and receives the resulting interference optical signal through a microscope adapter (Polytec OFV 074). The resulting optical signal is converted into an electrical signal by the photo detector inside the fiber interferometer and subsequently decoded by a vibrometer controller (Polytec OFV 3001S) to give the velocity. The velocity output from the Vibrometer controller is then connected to the “VEL” channel of a scanner controller (Polytec MSV-Z-040). A data acquisition PC (Polytec VibraScan) acquires data from both “REF” and “VEL” channels in the scanner controller [24]. The output of the function generator (DS345, Stanford Research Systems) is used to provide the electrical signal to excite the microphone under the microscope, and also connected to the “REF” channel of the scanner controller. The “SYNC” signal coming out of the function generator is connected to the “TRIG IN” and “GATE IN” channels of the scanner controller to trigger the data acquiring process. Figure 7 shows that the laser beam spot was positioned inside the center hole of the top backplate so that the center velocity response of the diaphragm can be measured.

When the calibration experiment for the top single backplate microphone is conducted, the middle diaphragm and the bottom backplate are electrically grounded. A uni-polar 1 [kHz] square wave with 50% duty cycle is applied directly to the top backplate of the microphone, and the dynamic response of the center velocity of the diaphragm is recorded using the laser vibrometer system. The above process is repeated several times for the square waves with different voltage amplitudes. Similarly, the calibration experiment for the bottom single backplate microphone is conducted with the middle diaphragm and the top backplate electrically grounded. During the high part (up-stroke) of the square wave cycle, the vibration of the diaphragm is forced by the external electrostatic force, and free response happens during the low part (down-stroke) of the square wave cycle. ANALYSIS OF EXPERIMENTAL RESULTS

Once all the experimental velocity responses are obtained, simple trapezoidal rule is applied to numerically integrate the velocity to yield the displacement since the time step is very small (0.39 [µs]), the numerical integration error is very small and can be neglected. Results for Top Single Backplate Microphone

Shown in Fig. 8 is the experimental velocity response within one excitation period for a uni-polar 1 [kHz] square wave with the amplitude of 18 [V] and the corresponding calculated displacement response is shown in Fig. 9.

Center hole

0.025

10 µm

Laser spot 0.02 0.015

Figure 7: Laser beam spot (white dot) hits the diaphragm through the center hole of the top backplate. The measurement settings used in the laser vibrometer system are shown in Table 4. It can be calculated that the minimum detectable displacement is approximate 0.59 [pm]. Time Average Number Channels Filters Sampling Frequency Sample Time Sample Points Velocity Sensitivity Velocity Resolution Trigger

100 DC coupling No 2.56 [MHz] 3.2 [ms] 8192 HF 25 [mm/s/V] 1.5 [µm/s] External (TTL)

Table 4: Time measurement settings for the laser vibrometer.

Center Velocity (m/s)

Up-Strok e Response 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025

0

1

2 Time (s)

3 x 10

-4

Figure 8: Experimental velocity response for a uni-polar 1 [kHz] square wave with the amplitude of 18 [V]. The drift in the steady state of calculated displacement response is due to the numerical integration of measured noise because the laser vibrometer does not measure the static displacement.

8


By using Eq. (49), a nonlinear least-squares curve-fitting procedure is carried out in MATLAB to obtain the parameters ξ , d 0 , ω0 , β and Γ from the calculated displacement response. The results are listed in Table 5. -8

10

x 10

Results for Bottom Single Backplate Microphone

Similarly, the above analysis can be carried out for the bottom single backplate microphone, the final results are summarized in Table 6.

8

Center Displacement (m)

Shown in Fig. 10 is the comparison plot of the experimentally calculated displacement result and curve-fitting displacement result. The nonlinear least-squares curve-fitting result agrees very well with the experimentally calculated result.

6

4

Parameters ξ

Results 0.069

d 0 [m]

2.09e-6

ω0 2π [Hz] β [N/m3/kg]

188.0e3

Up-Strok e Response

2

*

0

-2

Γ [m /kg]

Table 6: Nonlinear least-squares curve-fitting results for bottom single backplate microphone (*value for 5V only). 0

1

2

3 -4

Time (s)

x 10

Figure 9: Calculated displacement response for a uni-polar 1 [kHz] square wave with the amplitude of 18 [V]. Parameters

*

UNCERTAINTY ANALYSIS

In the previous sections, no uncertainties are assumed with all the physical dimensions and material properties of the microphone. However, there always exist some uncertainty sources in the experiment, such as the variations of physical dimensions and material properties in the real microphone device, which are typically caused by the fabrication process.

Results 0.078

ξ d 0 [m] ω0 2π [Hz] β [N/m3/kg]

1.95e-6 196.0e3 1.09e23 232

Γ [m /kg] 2

The errors associated with the measured velocity and calculated displacement are very small and thereafter neglected in the analysis in this section. Also the error due to the approximate solution and nonlinear least-squares algorithm is proven to be very small and neglected in the following analysis.

Table 5: Nonlinear least-squares curve-fitting results for top single backplate microphone (*value for 18V only).

7.5

x 10

From the reproducibility data of the fabrication process, for the 95% confidence interval, the thickness of the diaphragm is 2.27 ± 0.02 [ µ m ]. The gap between diaphragm and bottom backplate is 2.2 ± 0.5 [ µ m ] and the gap between diaphragm and top backplate is 2.0 ± 0.5 [ µ m ]. For the Young’s modulus of the polysilicon, 173 ± 20 [GPa] is used [25].

-8

Experimentally Calculated Result Curve-Fitting Result 7

Center Displacement (m)

1.17e23 140

2

6.5

6

5.5

Parameters ξ

Mean 0.025

ω0 2π [Hz] β [N/m3/kg]

193.5e3

10.2e3

1.22e23 329.2

0.14e23 2.9

Γ [m2/kg]

Uncertainty 0.013

Table 7: Mean values and uncertainties given for a given 95% confidence level.

5

4.5 1.5

2

2.5

3

3.5 Time (s)

4

4.5

Figure 10: Comparison of displacement responses.

5 x 10

-5

Since the above uncertainties are uncorrelated, these uncertainties can be further propagated into the uncertainties of the system parameters by using the analytical method [26]. The final uncertainty analysis results are summarized in Table 7.

9


DISCUSSION

CONCLUSIONS

From Table 5, Table 6, and Table 7, the experimental results for resonance frequency and nonlinear stiffness parameter fall into their respective theoretical ranges for a 95% confidence level.

Nonlinear system identification of model parameters for a capacitive dual-backplate MEMS microphone is presented in this paper. Lumped element modeling is used to extract system parameters of the microphone and a governing nonlinear equation is obtained with coupled mechanical and electrostatic nonlinearities. The nonlinear finite element analysis (FEA) shows the lumped stiffnesses of the diaphragm are in good agreements with their theoretical values. The approximate solution for a general damped second order system with both quadratic and cubic nonlinearities and a non-zero external step loading is obtained by the multiple time scales method. Based on the approximate solution, a nonlinear least-squares algorithm is implemented to identify system parameters from the experimental data. Uncertainty analysis shows that experimentally identified resonance frequency and nonlinear stiffness parameter fall into their theoretical ranges for a 95% confidence level respectively. The experimentally estimated damping ratio is higher than its theoretical value because only squeeze-film damping is considered in the current model and other damping mechanisms will be included in the model. Experimental results of Γ show that Γ is a function of the applied voltage, and the discrepancy between its theoretical and experimental values is due to the inaccurate assumption of equal-area parallel plates when the electrostatic force is calculated.

The experimental damping ratio result is above its theoretical range since the current damping model only considers the squeeze-film damping in the gaps. The total viscous damping should include not only the squeeze-film damping but also the effective resistance coming from acoustical holes in the top and bottom backplates [27]. Also, the thermo-elastic damping inside the diaphragm and damping due to the energy loss around the boundary of the diaphragm could be included in the total damping [28]. It is shown that Γ is a constant for a set of given parameters from previous lumped element modeling based on a simple equal-area parallel plate assumption. However, experimental results shown in Fig. 11 indicate that it is a function of the applied voltage (center displacements of the diaphragm), which means that the equivalent area for calculating electrostatic force is actually changing with different applied voltages if the lumped mass is assumed to be fixed. 280 Experimentally Estimated Result Curve-Fitting Result

260

ACKNOWLEDGMENTS

Support from U.S National Science Foundation CAREER Award (CMS-0348288) is gratefully acknowledged.

2 Γ (m /kg)

240

220

REFERENCES 200

1. 180

160

2. 140 11

12

13

14 15 16 Applied Voltage (V)

17

18

Figure 11: Experimentally estimated Γ value for top single backplate microphone. The difference between theoretical and experimental results of Γ are due to the following reasons; 1) in physical device, the top backplate area is larger than the diaphragm area, and the diaphragm area is larger than the bottom backplate area, therefore the equal-area assumption is not accurate in practice [29]; 2) the backplates are perforated with holes, which decreases the overlapping area and generates extra fringing field effect when calculating the electrostatic forces; 3) the overlapping area and fringing field effect change due to the different bending shape of the plates when different voltage is applied between the backplate and diaphragm.

19

3. 4.

5.

6.

7.

10

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11
