Sensors and Actuators A 123–124 (2005) 240–248
A non-invasive capacitive sensor strip for aerodynamic
pressure measurement
M. Zagnoni
a∗
, A. Golfarelli
a
, S. Callegari
a
, A. Talamelli
b
,
V. Bonora
c
, E. Sangiorgi
a
, M. Tartagni
a
a
ARCES, University of Bologna, Via Fontanelle 40, 47100 Forl
´
i, Italy
b
DIEM, University of Bologna, Via Fontanelle 40, 47100 Forl
´
i, Italy
c
DICASM, University of Bologna, Via Fontanelle 40, 47100 Forl
´
i, Italy
Received 13 September 2004; received in revised form 15 March 2005; accepted 18 March 2005
Available online 23 May 2005
Abstract
This paper presents a capacitive pressure sensor strip implemented in general purpose printed circuit board (PCB) technology based on a
thin 3D structure composed of polyimide, woven glass reinforced epoxy resin (FR4) and metal layers. Multiphysics finite elements method
(FEM) simulations have been performed over the proposed structure in order to develop a time-dependent electrical and mechanical model
that can be easily used to tailor the characteristics to the application. The device targets a wide class of fluid dynamics applications, being
non-invasive, comformable and smart for placement. The device simulations are herein validated by experimental wind tunnel measurements
and compared with figures obtained on a wing profile by conventional electromechanical pressure transducers. This approach is one of the
first example of fully embedding and electronically controlled fluid flow monitoring apparatus that could be used in replacement of state of
the art mechanical systems.
© 2005 Elsevier B.V. All rights reserved.
PACS: 81.05.Lg; 83.60.Bc; 84.37.+q; 89.20.-a
Keywords: FEM; Polyimide; Capacitive sensors; Pressure sensors; Fluid dynamics
1. Introduction
Pressure measurements are of great importance in almost
all field of engineering and industrial application. The recent
development of numerical codes to calculate fluid flow has
not diminished the need of detailed space and time resolved
measurements, both to provide boundary condition and to
validate the results. Despite the large amount of literature on
the subject [1], whenever sensors are used in situ to monitor
pressuresonlargedomainsinhighlyunsteadyflows, there are
still problems in using classical techniques due, for instance,
to the cost of each single transducer, their intrusiveness or
their time response.
In transport related industry,sensors are essentialformon-
itoring the fluid dynamic environment, required for instance
∗
Corresponding author. Tel.: +39 0543 786924; fax: +39 0543 786926.
E-mail address: mzagnoni@deis.unibo.it (M. Zagnoni).
in aerospace, ground vehicle and nautical applications, and
for the aerodynamic body optimization during the design
phase. Although similar, the above mentioned applications
are characterized by very different environments, requiring
sensor features and specifications that are very different from
each others. For instance, sensors for aircraft design need
high accuracy and precision, working in ranges up to 2 kPa.
Conversely, sensors for internal airflow for automotive ap-
plications require a reduced sensitivity with respect to the
previous ones, operating in a range of pressure from 10 to
about 30kPa and requiring fast dynamic response with re-
gard to the fluid to be monitored. Finally, sensors for nautical
applications must be able to resist to wet environments and
they must detect pressure ranges up to 300Pa.
A common specification in most of the above applications
is related to the large size of the surfaces that has to be moni-
tored, leading to the use of a large number of sensors in order
to achieve the required spatial resolution. In this scenario, a
0924-4247/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.sna.2005.03.049
M. Zagnoni et al. / Sensors and Actuators A 123–124 (2005) 240–248 241
real-time pressure distribution represents an important mean
for the analysis of the aerodynamic behavior of the body and
for its correct trim.
Modern pressure sensors that are used in this environment
are silicon based capacitive sensor [2–4]. Silicon sensors can
reach high sensitivity and accuracy, however they scarcely
address all the requirements of the above industrial applica-
tions, such as high robustness and low manufacturing costs.
On the other hand, standard fluid dynamics techniques, as
Pitot tube, Prandtl tube or other optic techniques [5,6], are
not able to satisfy, in many cases, either the low invasiveness
(due to the presence of the pressure tubes) or the required
accuracy, reducing their range of applications.
2. Aims and sensor structure
In the last few years printed circuit board (PCB) technol-
ogy has greatly improved, achieving the photolithographic
resolutions of silicon planar technology in the earlier 1970s,
thus allowing the design of low cost precision transducers
[7,8]. Materials such as polyimides and polyesters are now
available in thin films of tens of m, allowing PCB de-
vices to be used other than connecting electronic compo-
nents but also as means for mechanical transduction. By
using PCB technology it is possible to build up devices
achieving most of the specifications required by fluid dy-
namic applications at low cost. Herein the sensors structure
is presented together with the simulation methodology by
which these devices can be optimally designed for the tar-
geted application. Furthermore, the use of PCB technology
has advantages over other approaches: it allows to naturally
host electronic sensing and signal processing components by
means of smart packaging such as the chip on board (COB)
technology.
The sensor presented in this paper is a capacitive differ-
ential pressure transducer built in PCB technology as shown
in Fig. 1. The sensitive unit consists of a three layer struc-
ture in a stack (Fig. 2): a rigid copper-clad glass-fiber base, a
rigidglass-fiber spacer and a25mthick deformable copper-
clad Kapton
®
polyimide layer. Layers are attached to each
other by means of a 50m thick biadhesive tape, patterned
in the same shape as the spacing layer. The device length and
width can be set according to the application: the measure-
Fig. 1. PCB pressure sensor strip.
Fig.2. PCB pressure sensor strip structure: (a) deformable copper-clad poly-
imidelayer, (b) rigid glass-fiber spacer, (c) rigid copper-clad glass-fiber base.
Exploded top view (up left side). Exploded bottom view (up right side). As-
sembled view (bottom).
ments and simulations of this paper are related to devices
that are from 13 to 16 cm long and from 1.5 to 3 cm large.
The total thickness is below 1 mm. Each unit can be elec-
tronically readout in a multiplexed fashion in order to collect
a set of surface pressure points, depending upon the appli-
cation (Fig. 3). All chambers are connected by miniaturized
pipes, patterned in the spacing layers, in order to share the
same internal pressure. Small holes, drilled on the sides of
Fig. 3. Application example: monitoring pressure distribution over a wing
profile. The pressure distribution over the profile depends by free stream
velocity V and angle of attack α: a changing in the (α, V) field leads to a
different pressure pattern.
242 M. Zagnoni et al. / Sensors and Actuators A 123–124 (2005) 240–248
Fig. 4. PCB differential pressure sensor strip working principle. Membranes
deflect upward or downwardwithrespect to the gradient of pressure between
outside and inside the chamber.
the first unit membrane or obtained in the spacing layer, act
as a reference. As illustrated in Fig. 4, the membrane at each
point of sensing deforms itself downward or upward with re-
spect to the static pressure reference taken by means of the
holes. Since the membrane area is usually much smaller than
the aerodynamic surface to be monitored, the corresponding
pressure distribution over the deformable film can be con-
sidered constant with a good approximation. The differen-
tial pressure sensors approach is very useful in aerospace
environment: it overcomes altitude problems whereas ab-
solute pressure sensors are affected by barometric pressure
gradients. Membrane displacements due to the pressure in-
duce variations of electrical capacitance between upper and
lower conductive plates that can be easily readout by elec-
tronic circuitry. Capacitive sensors offer advantages as low
power consumption, high sensitivity and low temperature
dependence.
3. Multiphysics simulations
The sensor design is a fundamental task which involves
understanding the range and sensitivity of the PCB units and
tomakecomparisons tocommonlyusedsiliconMEMSstruc-
tures. With the aid of FEM simulation, it is possible to de-
scribe most of the physical and structural sensor behavior
in order to find optimum geometry, and to choose best ma-
terials to satisfy the specific application. Since the output
of the sensor is a capacitive information, coupled electrical
and mechanical simulation have to be taken into account.
Fortunately, simple hand calculations show that, in contrast
with typical MEMS structures, electrostatic force is negligi-
blewithrespecttoexternalforces inducing mechanicaldefor-
mation: for a parallel circular plate capacitor with a radius of
1cm, the distance among armatures should be tens of m for
obtaining an electrostatic force, in air, equivalent to 5 Pa for
an applied difference of potential of 5 V. As a consequence,
electrostatic simulations may follow mechanical ones with-
out coupling. On the other hand, an important difficult issue
is due to viscoelastic behavior of polymers. This problem is
usually referred to as creep and its modeling is described in
Section 3.3.
3.1. Approximated reference model
Modeling of plate deformations with respect to pressure
should be carefully taken into account for design purposes.
Whenever small mechanical perturbations are applied, clas-
sical mechanical theory of deformation can be used, where
a linear stress–strain relationship for matters (Hooke law) is
used. In this case, linear deformation of plates with respect
to pressure is expected. However, whenever large deforma-
tions are applied, a specific theory has to be considered [9].
The increase of bending of a circular plate creates a strain
in the middle plane that can not be neglected in cases where
the deflections are no longer negligible with respect to the
plate thickness. At the same time, large deflections in a plate
causesupplementarystresseswith respect to the conventional
elastic theory and they must be taken into consideration in
deriving the differential equations. This is a geometrical ef-
fect that causes non linearity between stress and deflection:
the more a plate is stressed, the less it deflects. One should
distinguish between two kindsof plates: thinplateswith large
deflections and thick plates. Sensors membrane, because of
its thickness, is supposed to be described by the large de-
flections theory for thin plates with a good approximation.
An useful relationship for an appropriate calculation of the
deflections can be obtained by applying the energy method
[9]. Considering a circular plate of radius a be clamped at the
edge and be subject to a uniformly distributed pressure p and
assuming the shape of the deflected surface represented by:
w = w
0
1 −
r
2
a
2
2
(1)
where r is the radius coordinate whose origin is set in the
center of the membrane and w
0
the maximum deflection, the
relationship of the displacement with respect to pressure is
given by:
w
0
=
pa
2
64D
1
1 + 0.488(w
2
0
/h
2
)
(2)
where D =
Eh
3
12(1−ν
2
)
, E is the modulus of elasticity, ν is the
Poisson ratio and h is the plate thickness. Eq. (2) is typically
used for w
0
greater than 0.5 h and it shows how the rigidity
of the plate increases with respect the deflection, so that w
0
is no longer proportional with respect to the pressure, as in
the elastic theory where w
0
=
pa
4
64D
. In the case of very thin
plates, where w
0
h a useful relationship is given by the
following [10]:
w
0
= 0.662a
3
pa
Eh
. (3)
This relationship shows that the deflection changes as the
cube root of the intensity p. The above equations are a first
useful tool for a rough estimation of the sensor dimensions.
Assuming to build the sensor membrane with homogeneous
materials and considering pressure values up to 2 kPa, one
M. Zagnoni et al. / Sensors and Actuators A 123–124 (2005) 240–248 243
can describe the membrane deflections with Eqs. (2) and (3)
for a plate thickness of tens of m (i.e., as provided by PCB
technology). Using the above expressions, it turns out that
circular shaped membranes having diameters of about 1cm
can show deflections in the order of hundreds of microns in
the given pressure ranges. This is precisely the amount of de-
flection needed for having the optimal electrical capacitance
dynamic range. Unfortunately, this theory is very satisfactory
in case of an homogeneous material, but not for composite
laminates, where FEM non-linear algorithms need to be used
to refine sensors design.
3.2. FEM models
Among the many possible FEM packages now available,
Femlab [11] has been used as finite elements software. The
simulation has been organized in the following mode: first,
the sensor geometryis solved for themechanical large deflec-
tion problems, static and time dependent, that gives as a re-
sult the membrane deflection, then the respective capacitance
value is calculated. With this approach one can establish the
sensor characteristic in absence of creep and add afterward
the time dependent creep contribution as an error, calculated
as a worst case. This procedure will be described in Section
3.3. The solution for the mechanical static large deflections
problem is obtained solving the equilibrium equations for
an axial symmetric problem, revealing that the creep behav-
ior can not be neglected because of the stresses present in the
membrane. Femlab solvesfor large deflections using a strain-
displacement relation, known as Green or Green-Lagrange
strains [12] and defined as:
GL
=
1
2
l
2
− l
2
0
l
2
0
, (4)
where is the deformation, l
0
the initial length and l the de-
formed length. This formula has been used in place of classi-
cal engineering strains, defined as
E
=
l−l
0
l
0
. Furthermore
Cauchy stresses, defined as σ
C
=
F
A
, are replaced by second
Piola Kirchoff stress:
σ
PK
=
l
0
l
F
A
, (5)
whereσ is the stress, F the force and A the surface.Thismodel
has been used for the whole structure, even for the parts that
are subjected to small deformation: this because the large
deflections model is valid with a good approximation also
for small displacement. By means of static simulations the
stresses on the membranes have been calculated. As it will
be better explained in the following paragraph, the values
obtained by simulation indicate the necessity to enrich the
model with viscoelastic behavior. To this aim, a time depen-
dent large deflections FEM model has been made, using an
exponential in time modulus of elasticity. As second step,
for any deformed geometry obtained by mechanical FEM
analysis, the Poisson equation
∇
2
V =−
δ
in the internal
Fig. 5. FEM simulations graphical results: (a) sensitive unit geometry, (b)
mechanical simulations: stress pattern, (c) electrostatic simulation: electric
potential pattern.
chamber of the sensor is solved for the space charge density
δ variable, where V is the electrostatic potential and the per-
mittivity. Integrating δ over the plate area of the sensor, the
total capacitance C is obtained as:
C =
Q
V
=
2π
V
R
0
δ(r)r dr (6)
where R is the radius of the electrode. To summarize, from
this analysis the sensor membrane deflection are calculated
asthe response to therespectiveapplied pressuredistribution,
finding out the capacitance variation, due to the geometrical
changing (Fig. 5).
3.3. Viscoelastic model
Viscoelastic behavior of polymers has to be considered
[13]. These properties depend upon their chemical structure
and morphology, the size of the applied load and, crucially,
temperature. An important implication ofviscoelastic behav-
244 M. Zagnoni et al. / Sensors and Actuators A 123–124 (2005) 240–248
iors is that the stress–strain characteristics can not be rig-
orously considered a static (i.e. memory-less, though non-
linear) relationship as depicted in Section 3.2. Conversely,
the stress–strain characteristic exhibits behaviors that appear
highly non linear, evenfor smalldeformations, and that, most
important, depends on the derivatives of the stress and strain
functions. This phenomenon is well evidenced by the analy-
sis of a sample case: if a stress function step is applied to a
sample of material subject to creep, a sudden elastic strain is
followed by a viscous and time dependent strain with an in-
creasingtrend. Conversely,if astrainstepis applied,thestress
decreases as a monotonic function. This type of behavior is
usually present in polyimides at ambient temperature and for
stress bigger than 1 MPa [14] and is conventionally known
as creep. Surprisingly, creep affects the proposed structure
and creep-caused membrane deflections of some m should
be expected, manifesting themselves in time-scales of tens of
minutes. First of all, observethat the mechanical deformation
of a body subject to creep phenomena is a function of the en-
tire loading history ofthebody itself. Inother terms,thanks to
viscoelasticity the system gains memory: each loading step,
appended in the past, contributes to the final response. The
Boltzmann superposition principle is a useful means of ana-
lyzing the creep deformation resulting from several distinct
loading or unloading steps of strain or stress [15]. The above
observation implies that in order to know the exact response
of a structuresubject to creep,a model ofits excitation should
be available, describing the evolution in time of input stress
(or strain). This is normally not possible in fluid dynamic
applications where the input loading and its dynamic is un-
known.Inthiscase,thebestthat can be achievedisaboundon
the maximal deviation that creep may introduce with regard
to static models such as those in Section 3.1. Such bound can
be roughly interpreted as an uncertainty that should be taken
into account when using the sensor as a measurement device
in a dynamic environment. A convenient way to obtain such
bound consists in realizing that creep can be approximately
classified as a low-pass phenomenon, so that a typical ex-
periment to estimate its extent consists in applying, at t = 0,
a step-like excitation in stress spanning the whole allowable
stress range and in evaluating the difference between the re-
sponse at t = 0
+
and the response at t →∞. Intuitively, any
structure subject to a slowly varying load will deform devi-
ating from a non-viscoelastic response by no more than such
quantity (the slower the load dynamics, the lower the devia-
tion). The major reason to practice this kind of analysis is to
understand how actions on the geometry and materials em-
ployed in the sensor fabrication can reduce the extent of the
viscoelastic response and thus tightening the error bounds.
In the modeling of creep [16], one should consider that by
taking into accountthe viscoelastic phenomena,the deforma-
tion model of a membrane changes from a static, non-linear,
time-invariantmodel to a dynamic, non-linear, time-invariant
model. In other terms, one could in principle model the vis-
coelastic behavior by introducing time derivatives into the
system of partial equations that rule the membrane deforma-
tion. In practice, it is generally convenient not to do so. In
many conditions it is handier to model creep by using equa-
tions where time-varying parameters take care of describing
the dynamical effects. A particularly effective way of doing
so is by the introduction of a time dependent modulus of
elasticity, obtained starting from Kapton
®
data sheets [14].
As shown in Fig. 6a, from strain versus time curves, given by
different applied stresses, the corresponding time dependent
modulus of elasticity have been calculated, interpolating the
strain curves, as:
E
i
(t) =
σ
i
i
(t)
, (7)
where
i
(t) is the time dependent strain, σ
i
is the corre-
sponding stress and the index i represents different values
of stresses and temperature conditions. The approach is
convenient because it leads to equation sets which fit more
easily into conventional FEM simulation framework than
models with explicit time derivatives. In other terms it allows
creep to be simulated by a sequence of static simulations
referring to different time instants. These curves (Fig. 6b)
have been fitted minimizing the root mean square difference
Fig. 6. Kapton creep: (a) strain versus time Kapton
®
creep behavior from
datasheetforparticulartemperatureandstress conditions, (b)timedependent
modulus of elasticity E(t), as described in Section 3.3 Eq. (7) for particular
temperature and stress conditions.
