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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS
1
An Analytical Model for Capacitive Pressure
Transducers With Circular Geometry
Ira O. Wygant , Member, IEEE, Mario Kupnik, Senior Member, IEEE, and Butrus T. Khuri-Yakub, Fellow, IEEE
Abstract— This paper derives an analytical model of a capacitive pressure transducer based on equations for deflection and
fundamental mode shape of a clamped circular plate. The derived
model enables efficient large-signal electromechanical simulations
and calculation of an equivalent circuit. The model shows
good agreement with finite element analysis and experiments.
The spring-mass-damper system used in the model is calculated for uniform, anisotropic, and layered plate materials.
A cubic spring constant captures large plate deflections. Use
of the clamped-plate shape function leads to a simple expression for capacitance as a function of deflection and to closedform equations for pull-in voltage. Comparison of the model
with finite element analysis for several air-coupled capacitive
micromachined ultrasonic transducer designs shows better than
10% agreement for deflection, resonant frequency, and pullin voltage. The model also compares well with characterized
devices. The analytical model could be further improved with
additional damping sources and methods to model a compliant
plate boundary.
[2017-0161]
Fig. 1. The capacitive pressure transducer modeled in this paper (threequarter section view). Force on the plate causes it to deflect, which changes
the capacitance between the plate and substrate. Analytical equations that are
a function of the dimensions and material properties of the plate, cavity, and
insulator accurately describe the transducer’s behavior.
Index Terms— Capacitive micromachined ultrasonic transducer (CMUT), capacitive pressure sensor, equivalent circuits,
microelectromechanical devices, sensor device modeling.
I. I NTRODUCTION
L
EVERAGING expressions for the deflection of a clamped
circular plate, this paper derives an analytical model of a
capacitive pressure transducer. The derivation results in simple
closed-form modeling equations that are a function of transducer geometry and material properties. These equations show
good agreement with simulated and measured data. Furthermore, they yield convenient design expressions, straightforward calculations of an equivalent electromechanical circuit,
and efficient frequency- and time-domain simulations.
While the model could be extended to other plate shapes,
this paper considers capacitive transducers with circular
uniform-thickness plates. This type of transducer, shown
in Fig. 1 and Fig. 2, is widely used in applications such as
pressure and ultrasonic sensing.
Manuscript received July 20, 2017; revised March 30, 2018; accepted
March 31, 2018. Subject Editor C. Mastrangelo. (Corresponding author:
Ira O. Wygant.)
I. O. Wygant was with Texas Instruments Inc., Santa Clara, CA 95051
USA. He is now with Swift Sensing Inc., Palo Alto, CA 94301 USA (e-mail:
ira.wygant@swiftsensing.com).
M. Kupnik is with the Department of Electrical Engineering and Information
Technology, Technische Universität Darmstadt, 64289 Darmstadt, Germany.
B. T. Khuri-Yakub is with the Department of Electrical Engineering,
Stanford University, Stanford, CA 94305 USA.
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JMEMS.2018.2823200
Fig. 2. Axisymmetric cross section of the transducer in polar coordinates r
and θ . Key dimensions for the transducer model are the plate radius a, post
width w post , plate thickness h, cavity height h cav , and insulator thickness
h ins . Force applied to the plate causes it to deflect. Peak plate deflection,
w pk , occurs at the plate’s center (r = 0). Deflection averaged over the plate
area, wavg , always equals one-third peak deflection.
When designed as a pressure sensor, the transducer can
measure static or slowly varying pressure and has applications
in automobiles, home appliances, industrial equipment, and
consumer products. Many categories of pressure sensors exist.
A pressure sensor with a vacuum-sealed cavity, such as a
barometric pressure sensor, measures absolute pressure; a
sensor with a second pressure port for the plate’s back side
measures differential pressure. MEMS (microelectromechanical systems) pressure sensors are small and often tightly
integrated with electronics. Whereas pressure sensors manufactured with conventional methods, like those constructed
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2
JOURNAL OF MICROELECTROMECHANICAL SYSTEMS
as a function of plate displacement; (4) implement the model
as a set of differential equations or as an equivalent circuit.
The following sections describe these steps. FEA and experiment verify the model for typical air-coupled CMUT designs
with three different radii. These designs range from a CMUT
with moderate linear deflection to a CMUT with significantly
nonlinear deflection and large deflection relative to the gap.
II. P LATE S PRING AND M ASS C ONSTANTS
Fig. 3. Schematic representation of the transducer model. A spring and
mass model the plate’s compliance and modal mass. Damping is due to
radiated sound and other sources of energy loss. The transducer capacitance,
C, changes with the plate’s average deflection, wavg . The model has two
ports. The electrical port relates voltage, V , and current, I . The mechanical
port relates force, F, and plate velocity, U .
from a metal foil, are rugged and suitable for harsh environments. The model developed in this paper applies to pressure
sensors from all categories.
When designed as a CMUT (capacitive micromachined
ultrasonic transducer), the same transducer structure radiates
and senses high-frequency sound pressure. CMUT applications
are similar to those of traditional bulk piezoelectric ultrasound
transducers. These applications include medical imaging, level
and fluid flow sensing, and proximity sensing.
Most of the model also applies to capacitive microphones.
However, this work neglects squeeze film damping, which is
critical to microphone performance. Extending the model to
microphones would require combining it with a squeeze-filmdamping model like that described in [5].
For all types of capacitive transducers, analytical models
help transducer development. Without analytical models,
engineers typically use FEA (finite element analysis) for
transducer design and analysis. However, analytical models
are faster and more efficient than FEA; and furthermore, they
reveal the parameter relationships and physics that govern
the transducer’s operation. Even when FEA is required, for
example to validate a final design, analytical models help
verify the FEA’s correctness.
Despite the utility of analytical models, existing capacitive
transducer models can be improved. This paper extends on the
model first described by Wygant et al. [6]. Models published
prior to [6] depended in part on either FEA [7] or numerical
iteration [8] to derive an equivalent circuit. Since [6], others
have followed a similar analytical modeling approach, and
incorporated anisotropic material properties [9], rectangularshaped plates [9], two-layered plates [9], and implementations
in hardware descriptive language [10], [11].
Unlike other models, this work accounts for the nonlinearity
of large plate deflection. Large plate deflection is important to
CMUT air transducers where wider bandwidth requires a thin
plate and for pressure sensors where linear deflection with
pressure is desired.
Fig. 3 gives a schematic representation of the transducer
model. The steps to derive this model are: (1) calculate
lumped spring and mass constants for the plate; (2) estimate
damping due to radiation impedance; (3) derive capacitance
The transducer’s plate is a continuous structure that can take
on any shape; thus, it has infinite degrees of freedom. One
approach for simplifying analysis of the plate is to express
the plate’s deflection as the sum of its orthogonal mode
shape functions [12]. Each mode and its shape function corresponds to a simple independent spring-mass-damper system
governed by
Fm (t) = m m ϕ̈m − bm ϕ̇m − km ϕm ,
(1)
where Fm (t) is the modal force acting on the plate for a
mode m. The modal mass, damping, and spring constants
are m m , bm , and km , respectively. Numerical techniques exist
for deriving and solving (1) for a given force and initial
conditions. Summing just a few modal solutions often gives
an answer close to the exact solution.
This paper uses just the plate’s fundamental mode.
Neglecting all higher-order modes simplifies the analysis yet
yields accurate results for frequencies up to and exceeding the
plate’s fundamental resonant frequency. This frequency range
satisfies the requirements for many transducer analyses.
A. Mode Frequencies and Shapes
The plate’s mode frequencies and mode shapes are known
assuming the plate is thin, perfectly clamped, and subject to
a uniform pressure [12], [13]. These assumptions introduce
relatively small errors when compared to a more realistic
finite element analysis (Section VII), even though an actual
transducer plate is not perfectly clamped and the electrostatic
force acting on it is not uniform.
For the plate geometry in Fig. 2, the free undamped modal
frequencies, i.e. the plate’s resonant frequencies, equal
λ2ns D
(2)
ωns = 2
a
ρh
for plate radius a and plate thickness h. The plate’s density
and flexural rigidity are ρ and D, respectively.
The nondimensional modal frequency λns for a wide range
of mode numbers n and s can be calculated or found in standard references [12]. For the fundamental (s = 0, n = 0) and
first axisymmetric (s = 1, n = 0) mode, λns is 3.197 and
6.306, respectively. The plate’s density and flexural rigidity
are ρ and D, respectively.
The plate’s mode shape functions equal
r Jn (λns ) r In λns
Wns (r, θ ) = Jn λns −
cos(nθ ), (3)
a
In (λns )
a
where Jn and In are Bessel and modified Bessel functions
of the first kind [12]. Note that axisymmetric modes (n = 0)
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WYGANT et al.: ANALYTICAL MODEL FOR CAPACITIVE PRESSURE TRANSDUCERS WITH CIRCULAR GEOMETRY
are more likely to be excited in a transducer where the
impinging pressure is uniform and the electrostatic actuation
is axisymmetric.
The shape function
2
r2
W0 (r ) ≈ 1 − 2
(4)
a
is a good approximation of (3) for the plate’s fundamental
mode. Compared to (3), the benefits of using (4) are that it
equals the exact solution for the plate’s static deflection [13]
and it yields a closed-form equation for the transducer’s
capacitance as a function of deflection. For design calculations,
(3) and (4) are close enough to be interchangeable. However,
(3) is slightly more appropriate for resonant frequency and
modal mass calculations whereas (4) is better for static calculations.
For a given peak plate deflection, w pk , the plate’s displacement as a function of radial position equals
2
2
r2
r2
(5)
w(r ) = w pk W0 (r ) = w pk 1− 2 = 3wavg 1− 2 .
a
a
Averaging (5) over the plate area shows that the plate’s average
deflection, wavg , is always one-third its peak deflection.
Because the plate’s shape function is independent of amplitude, the plate’s continuous deflection is representable with
a single amplitude parameter—this representation is what
enables modeling the plate with the single-degree-of-freedom
system described by (1). The choice of amplitude parameter or reference point, e.g. wavg or w pk , does not affect the
solution to the plate’s vibration but it does affect scaling of
the spring, mass, and damper constants. Acoustic calculations
favor the use of average deflection because acoustic impedance
depends on volume displacement, which is the product of wavg
and the plate’s area. As a result, we use wavg for the modal
amplitude parameter.
B. Modal Mass
The kinetic energy of the continuous plate must equal the
kinetic energy of its modal representation according to
1
KE=
2
1
2πρh ω0 w pk W0 (r ) r dr = m 0 (ω0 wmod )2 ,
2
2
(6)
0
where wmod is the modal amplitude parameter.
Solving (6) for the fundamental mode’s modal mass yields
a
m 0 = 2πρh
0
w pk
W0 (r )
wmod
2
r dr = 1.88πa 2ρh.
plate’s center. The mass ratio is greater than one because the
plate’s center has more kinetic energy than the reference point
wavg . Using w pk for the amplitude parameter results in a ratio
of 0.2 If the plate moved as a perfect piston, then the mass
ratio would be one.
As described in [14], the medium in which the CMUT
operates, e.g. water, contributes additional mass. With this
addition, the modal mass equals
ρf a
),
(8)
m 0,liquid = m 0 (1 + ρ h
where ρ f is the medium’s density.
The theoretical value of is between 0.47 and
0.67 depending on the acoustic boundary conditions. One
study reports an empirical value of 0.55 [15]. This added mass
from the medium is approximate as these studies assume the
plate diameter is large relative to the wavelength, which is not
true for many transducer applications.
C. Modal Spring Constant
Calculating the modal spring constant from the modal mass
m 0 and frequency ω0 gives
192.2π D
.
(9)
a2
Alternatively, we can derive the spring constant from the
expression for the plate’s static deflection [13]. A clamped
circular plate subject to a uniform pressure, q, has a shape
function equal to (4). The peak deflection equals,
k0,mod = m 0 ω02 =
qa 4
,
64D
which results in a spring constant equal to
w pk =
k0 =
F
3
192π D
= πa 2 q
=
,
wavg
w pk
a2
(7)
This solution uses (3) for the shape function and wavg for the
modal amplitude parameter.
Note that the modal mass equals 1.88-times the plate’s
physical mass (using (4) in place of (3) for the shape function
yields a ratio of 1.8). The modal mass does not equal the
plate’s physical mass because not all parts of the vibrating
plate contribute equally to its kinetic energy; for example,
regions close to the clamped edge contribute less than the
(10)
(11)
which is very close to (9).
The spring constant depends on the plate’s flexural
rigidity D. For uniform isotropic materials, flexural rigidity
equals
Diso =
a
3
Eh 3
,
12(1 − ν 2 )
(12)
where E is the material’s isotropic Young’s modulus and ν is
its Poisson ratio.
For anisotropic materials like silicon, [16] provides the
following equations for calculating the flexural rigidity:
s22
−s12
c11 =
, c12 =
,
2
2
s11 s22 − s12
s11 s22 − s12
s11
1
c22 =
, c66 =
2
s66
s11 s22 − s12
α3 = 3c11 + 3c22 + 2c12 + 4c66
α3 3
h ,
Daniso =
(13)
96
where sx x are the components of the inverse of the material’s
stiffness matrix.
A typical MEMS device is made from anisotropic (100)
silicon, where the X, Y, and Z directions correspond to
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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS
the [110], [1̄10], and [001] crystal directions, respectively [2].
For this type of silicon, it is sometimes convenient to derive
isotropic values of Young’s modulus and Poisson ratio that
yield the same flexural rigidity as (12). Applying (12) and (13)
yields an equivalent isotropic Young’s modulus of 150.5 GPa
for a Poisson ratio of 0.177. This value for Poisson ratio is
the radial average value and is the correct value for the largedeflection calculations made in the following section.
The plate may also comprise multiple layers of different
materials. For example, a silicon plate could be covered with
conducting or passivation layers. For multilayer plates, [17]
and [18] give a procedure for calculating the equivalent
flexural rigidity for an arbitrary number of layers.
D. Large Deflections
For most transducer applications, the plate’s deflection is
small relative to its thickness. For larger deflections, tensile
forces in the plate’s midplane become more significant relative
to the plate’s bending forces [19]. For these cases, the deflection no longer varies linearly with the applied force. Following
the procedure in [13] yields a modified spring constant for
these large-deflection forces.
Applying this procedure, the plate’s radial displacement, u r ,
due to midplane tension is assumed to take the form
∞
u r = r (a − r )
Cn r n ,
(14)
n=0
where Cn are constants.
Neglecting n greater than three and calculating Cn that
gives the minimum energy for the deflected plate results
in the following expression for the energy associated with the
midplane tension:
Ut = 6π D
4
wavg
Cν ,
a 2t 2
where Cν is a constant that depends on Poisson ratio:
896585 + 529610ν − 342831ν 2
.
29645
The energy associated with plate bending equals [13]
Cν =
Ub = 96π D
(15)
(16)
2
wavg
.
(17)
a2
Differentiating the total energy with respect to wavg gives
the following expressions for mechanical force in terms of k0
and a nonlinear spring constant k0,t :
Fspr =
d
(Ut + Ub )
wavg
= wavg
2
24π DCν wavg
k0 +
a2
h2
3
= k0 wavg + k0,t wavg
(18)
where,
k0,t =
24π DCν
.
a2h2
(19)
Taking the ratio of the cubic tensile force term to the linear
bending force term in (18) yields (20)
3
k0,t wavg
Cν w pk 2
Ft ension
=
=
.
(20)
Fbend
k0 wavg
72
h
This expression shows that tensile forces are negligible when
the plate’s deflection is small relative to its thickness. If the
plate’s peak deflection is less than about one-seventh its
thickness, the tensile force is less than 1% of the bending
force. Since Cν does not vary much between materials, this
effect is true for any plate material.
E. Damping
Damping equates to energy loss and equivalently mechanical noise [20]. An important and sometimes dominant
damping mechanism is the radiation impedance, which is a
result of energy dissipated to the surrounding medium.
If we approximate the radiation impedance with the planewave radiation impedance, then the damping constant equals
the product of the plate area and the specific acoustic
impedance of the medium, Z 0 , according to
b0 |plane wave = πa 2 Z 0 .
(21)
This approximation is valid when the diameter of the transducer, which may consist of multiple cells, exceeds about
two wavelengths. Assuming damping equals (21) leads to an
expression for the plate’s quality factor:
9 ρhω0
m 0 ω0
=
.
(22)
Q=
b0
5 Z0
This expression shows that for a given resonant frequency,
quality factor is proportional to plate thickness. For this reason,
a thinner plate results in a transducer with wider bandwidth.
Transducers are often less than one wavelength
in width or height. In this case, the radiation impedance is
a complex value that varies with frequency. The radiation
impedance for a circular piston radiator equals
J1 (2kar )
H1 (2kar )
,
(23)
+ j
Z = Z0 1 −
kar
kar
where H1 is the Struve function, ar is the piston radius, and k
is the wavenumber [21]. A similar expression exists for a
radiator with velocity profile like (4) [21].
A CMUT element usually consists of multiple cells operating in parallel. Tightly packed cells act roughly like a single
radiator with volume velocity equal to the sum of the cells’
volume velocities. For less tightly packed cells, accounting for
the cells’ mutual impedance gives a better estimate for the net
radiation impedance [22].
To model the complex radiation impedance as an equivalent circuit, we can use a circuit that matches (23) over
a wide frequency range [23]. Or we can add a mass equal
to the reactive part of the damping constant around a single
frequency [7].
Beyond radiation impedance, other forms of damping
include energy loss to the substrate and thermoeleastic
damping. These sources of damping are harder to predict
analytically and, without more analysis, are best derived
empirically or with finite element analysis.
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WYGANT et al.: ANALYTICAL MODEL FOR CAPACITIVE PRESSURE TRANSDUCERS WITH CIRCULAR GEOMETRY
III. C APACITANCE
Writing (30) in terms of the model parameters yields
Capacitance depends on the transducer’s gap height and
plate deflection. To account for an insulating layer in the
cavity, we use an effective gap height
g0 = h cav +
h ins
,
εr,ins
(24)
where, h cav is the cavity height, h ins is the insulator thickness
(Fig. 2), and εr,ins is the insulator’s relative permittivity.
Using shape function (4) and integrating over the plate area
gives
3wwavg
2 ε a tanh
πa
0
a
g0
2πr ε0
dr
=
C=
2
3g0wavg
r2
0 g −3w
0
avg 1− a 2
(25)
for the capacitance as a function of average deflection.
The first and second derivatives of capacitance equal
ε0 πa 2
dC
= C =
dwavg
2g0 wavg 1 −
3wavg
g0
−
C
,
2wavg
(26)
2
dwavg
= C =
−
πa 2
3ε0
2
3w
2g02 wavg 1− gavg
0
C
ε0 πa 2
1
+ 2 −
C , (27)
3wavg
2w
2w
2
avg
avg
2g0 wavg 1− g0
respectively. Calculation of electrostatic force and pull-in
require these derivatives.
IV. E LECTROSTATIC F ORCE AND P ULL -I N
Applying the principal of virtual work gives the electrostatic
force, Fe , on the plate for a voltage V :
1 2 V C.
(28)
2
The net mechanical force, Fm , equals the sum of the spring
restoring force and force due to impinging pressure:
Fe =
3
− πa 2 P.
Fm = k0 wavg + k0,t wavg
2
)C = 0,
−Fm C + (k0 + 3k0,t wavg
which we can solve for the average deflection at pull-in. We
denote this deflection as wavg, pi .
The pull-in voltage in terms of wavg, pi [24] equals
dUm (wavg, pi ) 2Fm (wavg, pi )
VP I = 2
/C =
. (33)
dwavg
C
Under certain conditions, closed-form solutions exist for
wavg, pi and Vpi . For some transducer designs we can neglect
large-deflection forces (k0,t = 0) and atmospheric pressure
(P = 0). With these assumptions, the transducer always pulls
in when the plate’s average deflection is 15% of the gap:
wavg,P I (34)
k0,t =0,P=0 = 0.15
g0
and the pull-in voltage equals
3k
g
g03 Qω0 Rb
0
0
V P I k0,t =0,P0 =0 = 0.39
=
0.39
.
πa 2 ε0
πa 2 ε0
(29)
w pi, pp 1
k0,t =0,P=0 =
g0
3
and
where Um is the total mechanical energy.
=
0,t =0,P=0
(35)
8
27
and
=
3 =0
(30)
2(k0 wavg, pi − πa 2 P)
.
C (wavg, pi )
(37)
(39)
As described in [25] and [26], a nonlinear spring extends
the travel range of the transducer. Assuming k0,t dominates
(k0 = 0) and neglecting atmospheric pressure yields
(40)
wavg, pi k0 =0,P=0 = 0.24
and
V pi k
0 =0,P=0
(31)
g03 k0
g03 k0
= 0.54
.
2
πa ε0
πa 2 ε0
(36)
Assuming the expressions for a pressure transducer have the
same form as those for a parallel-plate transducer enables us to
derive approximate expressions that account for atmospheric
pressure:
wavg, pi k0,t =0 ≈ 0.15(1 + 1.18wavg,P )
g0
πa 2 P/k0
= 0.15 1 + 1.18
(38)
g0
V pi k
for wavg gives the plate’s static deflection for a given dc
voltage and applied pressure.
When the applied voltage exceeds the pull-in voltage, V pi ,
no wavg satisfies (29). For these voltages, the plate is unstable
and snaps to the bottom of the cavity. Following the procedure in [24], we can calculate V pi . At the brink of pull-in,
the deflection, wavg, pi , satisfies
d 2 Um dC
dUm d 2 C
−
= 0,
2
2 dw
dwavg dwavg
dwavg
avg
V pi, pp k
Solving the sum of these forces,
Fe + Fm = 0,
(32)
Interestingly (33) and (34) look like the corresponding
expressions for a parallel-plate transducer, which are
and
d 2C
5
= 0.08
g05 k0,t
.
πa 2 ε0
(41)
Comparing V with (34) we see that a nonlinear spring extends
the travel range from 15% to 24% of the gap.
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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS
V. G OVERNING D IFFERENTIAL E QUATION
Having derived the electrical and mechanical model parameters of the transducer, we can write the transducer’s governing
differential equations.
The net force, F, on the transducer plate equals the sum of
the mechanical and electrical forces:
1
3
− V 2C .
F = m 0 ẅavg + k0 wavg + k0,t wavg
2
(42)
The current, I , flowing into the transducer’s terminals equals
the sum of the currents from the changing voltage and the
moving plate:
I = C V̇ + V C ẇavg .
(43)
Together, (42) and (43) form the governing differential
equations for the transducer’s operation. Solving these equations numerically gives the transducer’s response to electrical and mechanical inputs. Implementing them in a hardware descriptive language enables simulation with electrical
circuits [10], [11]. Writing them as state equations provides
other options for simulation and analysis [26].
Fig. 4. Electromechanical equivalent circuits of the transducer. a) Basic twoport equivalent circuit. For a given transducer design and dc operating condition, expressions (48)-(53) give component values. The through and across
variables at the electrical port are current, I , and voltage, V , respectively.
At the mechanical port, they are velocity, U , and force, F. b) Equivalent
circuit with additional model components: input voltage source, Vs ; source
resistance, Rs ; parasitic capacitance, C par ; mass loading of the medium,
L med ; added damping, Rdamp ; and applied force, Fs . Based on (51), this
circuit divides the plate’s compliance into a mechanical part, Cm0 , and a
negative spring-softening part, Cs .
TABLE I
V ERIFICATION W ITH FEA FOR I SOTROPIC , O RTHOTROPIC ,
AND L AYERED P LATES
VI. L INEAR E QUIVALENT C IRCUIT
Expressions (42) and (43) support a two-port model of the
transducer. They relate the transducer’s electrical port (voltage
and current) to its mechanical port (force and velocity). For
this two-port model, following the procedure in [26] yields
the linear equivalent circuit in Fig. 4. This equivalent circuit
is valid for small variations in plate deflection at a specific
operating point. Solving (29) gives the plate deflection at the
specified operating point.
To calculate the equivalent circuit, we first write the Jacobian,
⎡ δV ⎤
g δV q dV
⎥ δq
⎢
δg
(44)
= AB = ⎣ δδq
F δ F ⎦ δg ,
dF
g
q
δq
δg
that gives the linearized relationship between the two ports of
the transducer.
Matrix A in (44) is expressible in terms of the transducer
model parameters as follows:
A11
A12
A22
1
d V d q =
=
=
g
dq
dQ C
C
2
Vdc
d V d F q
C = 2 C
= A21 =
=
=
q
g
dg
dq
q
C
d F d
F
e
2
=
q = k 0 + 3k 0,t wavg −
dg
dwavg
2
1 2 2C
C 2
= k0 + 3k0,t wavg + q
− 2 .
2
C3
C
parameters in terms of the number of parallel cells, N.
The circuit parameters are as follows:
C0 = NC,
A12
= N V C ,
n=N
A11
A211
1
1
1
ke2 = 3
=
,
N A12 A22
N nC A22
1
= N A22 (1 − ke2 ),
Cm
L m = Nm 0 ,
(45)
(46)
(47)
Next, we write the equivalent circuit parameters in terms
of the coefficients of matrix A. Since a transducer element
can consist of multiple cells in parallel, we write the circuit
(48)
(49)
(50)
(51)
(52)
and
Rb = Nb0 .
(53)
These are the basic equivalent circuit elements. With additional elements, we can also model other mechanical and
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WYGANT et al.: ANALYTICAL MODEL FOR CAPACITIVE PRESSURE TRANSDUCERS WITH CIRCULAR GEOMETRY
7
Fig. 5. Axisymmetric geometry for verifying pull-in voltage and analyzing
effects of post width, w post with finite element analysis. For the clamped-edge
condition, the plate displacement is fixed for r = a plate .
TABLE II
FEM V ERIFICATION OF P ULL -I N V OLTAGE
Fig. 6. The effect of post width on the plate’s deflection and resonant
frequency for the a = 560 μm design in Table II. Results are from finite
element analysis for the geometry in Fig. 5. Compared to the clamped-edge
assumption made for the analytical model, a plate with a finite post width is at
least 8% more compliant. This result suggests using a slightly more compliant
spring constant to model a more realistic plate boundary condition.
TABLE III
E XPERIMENTAL VALIDATION OF R ESONANT F REQUENCY
AND P ULL -I N V OLTAGE
electrical effects, e.g., mass loading given by [7], complex
radiation impedance described in [23], or parasitic capacitance.
Fig. 4(b) gives an example of an equivalent circuit with
additional model elements.
VII. C OMPARISON W ITH F INITE E LEMENT A NALYSIS
Two finite-element models implemented with ANSYS help
verify the analytical model. A three-dimensional quartersymmetry model constructed from SHELL281 elements verifies clamped plate deflection and resonant frequency for
isotropic, anisotropic, and layered plate materials. For the
transducer designs in Table I, the analytical model agrees with
the finite element model with less than 0.5% error.
An axisymmetric model (Fig. 5), constructed from
PLANE183 elements, verifies pull-in voltage and evaluates the
effect of a compliant post. Table II shows that for the clamped
plate, the analytical pull-in voltage matches the finite element
results to better than 10%. The biggest mismatch occurs for the
design with the largest deflection (a = 585 μm). This design’s
Fig. 7. Comparison of measured and modeled transducer admittance for the
a = 560 μm design in Table III and a 24-V dc bias. The model accurately
predicts the resonant frequency and magnitude of the impedance but underpredicts the amount of damping. a) Admittance curves. b) Equivalent circuit
and component values corresponding to the admittance curves. The measured
equivalent circuit is the result of a fit to the measured admittance curve.
The circuits match well except that the measured parallel resistance, R x ,
is about 3.8-times higher than the modeled R x . This mismatch indicates the
model does not capture all sources of damping.
plate-shape differs the most from the analytical plate shape (4)
because it has the most midplane tension.
The axisymmetric model also helps evaluate the effect of a
compliant post (Fig. 5). For the simulation results in Fig. 6,
post widths greater than 50-μm (roughly 3.5-times the platethickness) give deflections 8% larger than the perfectlyclamped plate. This result suggests the use of a slightly
reduced flexural rigidity in design calculations. Table II shows
that using a reduced flexural rigidity better predicts the pull-in
voltage of a device with a compliant post.
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8
JOURNAL OF MICROELECTROMECHANICAL SYSTEMS
VIII. C OMPARISON W ITH E XPERIMENT
Comparison with characterized examples of airborne CMUT
devices further helps to validate the model. Table III compares
modeled and measured resonant frequency and pull-in voltage
for three different radii. The results show good agreement;
however, the model slightly overpredicts the pull-in voltage,
which suggests using a further reduced flexural rigidity to
model the compliant post.
Fig. 7 compares modeled and measured electrical
impedance. The model accurately predicts the transducer’s
motional impedance except that it underpredicts the amount of
damping. Using a better model for radiation impedance than
(23) and including additional damping sources would improve
this mismatch.
IX. C ONCLUSION
This work shows that starting from the geometry and
material properties of a capacitive transducer, we can derive
an accurate model that agrees with finite element analysis and
experiment. Critical modeling equations are (7), (11), (19),
and (21) for the spring-mass-damper model of the plate and,
(4) and (25) for the plate’s deflection and capacitance. With
these equations, we can calculate an equivalent circuit or solve
(42) and (43) for arbitrary time-varying electrical and mechanical inputs.
While these equations meet basic analysis needs, the model
could be improved in several ways. For example, including
higher-order modes would give a better prediction of a
CMUT’s wideband frequency response in liquid. Film stress,
temperature, and additional sources of damping are important
to typical sensing applications but are not included in the
model. These and other effects are topics for continued
research.
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Ira O. Wygant received the B.S. degree in electrical engineering and computer science from the
University of Wyoming, Laramie, in 1999, and
the M.S. and Ph.D. degrees in electrical engineering from Stanford University, in 2002 and
2008, respectively. His Ph.D. research at Stanford
was focused on capacitive micromachined ultrasonic
transducer technology and its integration with frontend electronics. From 2008 to 2017, he was with
Texas Instruments Inc., Santa Clara, CA, USA,
in research and engineering roles to develop microelectromechanical transducer technologies.
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WYGANT et al.: ANALYTICAL MODEL FOR CAPACITIVE PRESSURE TRANSDUCERS WITH CIRCULAR GEOMETRY
Mario Kupnik received the Dipl.Ing. degree in electrical engineering from the Graz University of Technology, Austria, in 2000, and the Ph.D. degree
in electrical engineering from the University of
Leoben, Austria, in 2004. From 2005 to 2011, he
was a Post-Doctoral Researcher, a Research Associate, and a Senior Research Scientist with the
Edward L. Ginzton Laboratory, Stanford University,
USA. From 2011 to 2014, he was a Full Professor of
electrical engineering with the Brandenburg University of Technology, Cottbus, Germany. Since 2015,
he has been a Full Professor with the Technische Universität Darmstadt,
Germany, heading the Measurement and Sensor Technology Group.
9
Butrus (Pierre) T. Khuri-Yakub (F’95) received
the B.S. degree from the American University of
Beirut, the M.S. degree from Dartmouth College,
and the Ph.D. degree from Stanford University,
all in electrical engineering. He is currently a
Professor of electrical engineering with Stanford
University. He has authored over 600 publications and has been a principal inventor or coinventor of 102 U.S. and international issued
patents. His current research interests include
medical ultrasound imaging and therapy, ultrasound
neuro-stimulation, chemical/biological sensors, gas flow and energy flow
sensing, micromachined ultrasonic transducers, and ultrasonic fluid ejectors.
He received the Medal of the City of Bordeaux in 1983 for his contributions to
nondestructive evaluation, the Distinguished Advisor Award from the School
of Engineering, Stanford University, in 1987, the Distinguished Lecturer
Award from the IEEE UFFC Society in 1999, the Stanford University
Outstanding Inventor Award in 2004, the Distinguished Alumnus Award
from the School of Engineering, American University of Beirut, in 2005,
the Stanford Biodesign Certificate of Appreciation for commitment to educate
mentor and inspire Biodesign Fellows in 2011, and the IEEE 2011 Rayleigh
Award. He was elected as a Fellow of the AIMBE in 2015.