Distributed Sensing in Capacitive Conductive
Composites
Edward L. White, Michelle C. Yuen, and Rebecca K. Kramer
School of Mechanical Engineering
Purdue University
West Lafayette, IN, USArebeccakramer@purdue.edu
Abstract—In this paper, we describe a capacitance-based
strain field sensor. The device is interrogated at a single location,and uses multiple excitation frequencies to resolve the spatial
distribution of strain. The system we model is representative of acapacitor composed of a graphite-based conductive elastomer
composite in a silicone binder with a silicone dielectric layer. We
utilize an artificial neural network to reconstruct the strain basedon the magnitude of the measured impedance. Our analysis uses
finite element simulation to demonstrate the concept in a onedimensional application. We demonstrate that multiple elements
can be interrogated using a single electrical interface, reducing thecomplexity of high deformation sensor arrays.
Keywords—distributed sensing, dielectric elastomer sensors,high-deformation sensing, capacitive sensor impedance

I.

INTRODUCTION

Measuring large strains and deformations is required inmany applications involving soft bodies [1], including both soft
robot and wearable applications [2]. In traditional robots andrigid mechanisms, deformations are concentrated at joints. In
soft-bodied systems, however, deformations are distributed
throughout the structure, requiring that sensors likewise bedistributed. However, distributing sensors poses two challenges.
First, it increases the total number of sensors which must befabricated and integrated. Second, it increases the total numberof interfaces between the sensors and the associated signal
conditioning electronics. In many cases, these two requirementswork against the goal of the soft system, which is to maintainhigh deformations and low elastic modulus. Furthermore,increasing the number of devices and interfaces can reduce theoverall reliability of the soft system.
Instead of a traditional paradigm where a single sensor isused to measure strain at a single point, we demonstrate amethod to measure strain at multiple locations with a single
physical device. This method allows us to reduce the total
number of sensors which must be fabricated, and consequently
drives down the total number of interface electrodes. As notedby previous authors, a one-dimensional capacitive sensor withfinite electrode resistance can be modeled as an RC network[3,4]. By adjusting the excitation frequency of the sensor, we areable to control the distribution of the electric field within theequivalent RC network, which in turn allows us to control the
spatial sensitivity of the device. Therefore, by exciting thesensor at multiple frequencies and measuring the impedance, wecan uniquely reconstruct the strain at multiple locations along978-1-5090-1012-7/17/$31.00 ©2017 IEEE

Fig. 1. Simulation representation. a) A 1-D electromechanical element. The
length of the element is l, the width is w, and the electrode and dielectric
thicknesses are te and td, respectively. Note the directions of the Cartesian
coordinate system e. b) The corresponding electrical element of the mechanicalelement shown in a). Note that the red and green dots shown in a) and b) are
shown in the same relative locations. c) The complete 5-element sensor. The
deformable regions are shown in blue, while the reinforced mounting regions
are shown in white.

the length of the sensor. Previously, researchers haveconstructed a keyboard by using multiple frequencies tomeasure deformation due to pressure at different locations on a
two dimensional monolithic substrate [5]. Additionally, a single
interface has been used to interrogate multiple capacitive strain
sensor elements connected in series [6]. Our work extends theseprevious examples by demonstrating how continuous values ofstrain can be measured in multiple elements along the length ofa single sensor body.
II.

FINITE ELEMENT SIMULATION

A. Model Development
The conductive elastomer composite sensors we discuss inthis work are designed for applications with large deformations(>50% strain). To accurately capture these finite deformations,
we used a Lagrangian mesh throughout our analysis, since a
linear approximation was insufficient to capture the non-linear
geometry of the sensor. Instead of solving for the complete
electrical field around and within the sensor, we used a direct

analysis technique to construct an element which represented
both the resistance of the electrode and capacitance of thedielectric layer within a finite thickness slice of the sensor. The
physical and electrical representations of a single element are
shown in Fig. 1(a-b). The resistance, / , of the electrodeelement is defined by:
= ( )

/2

where ( ) is the resistivity as a function of the strain, is thecurrent length of the element, is the current width, and is
the current thickness of the electrode. The subscript 1/2 indicatesthat the resistance only covers one half of the length of theelement. The capacitance of the element is:=whereis the permittivity of free space,is the relative
permittivity of the dielectric layer, and is the thickness of thedielectric. Assuming that the material is incompressible andisotropic, these equations become:(1 + )

=and
= ( )

/

(1 + )
2

where the 0 subscript denotes initial conditions. The governing
equation of a single element is:=(1)

where:=

1 3 +24( + )14(1+14(+2+ )=−=−)where= / and= 1/() represent the complex
impedances of the resistor and capacitor, respectively.
B. Physical Concept
To demonstrate the concept of measuring strain at multiple
points within a single sensor, we propose a five-element sensor
as shown in Fig. 1(c). The sensor is composed of inextensibleregions, shown as white segments, and stretchable regions,shown as blue segments. We intend to build this device fromelastomer-based conductive composites, where expanded
intercalated graphite is used as a conductive phase in theelectrode material [7]. The notional dimensions used in themodel are a width of 1cm, an inextensible segment length ofFig. 2. Change in resistance (top) and capacitance (bottom) relative to baseline
values when individual sensor elements are strained by 10%. Each strain isapplied individually, i.e. the element 1 test applied a 10% strain to element 1,leaving all other elements unstrained.

2cm and an extensible segment length of 3cm. We have foundthat a linear resistivity model of the form ρ(ϵ)=ρ0+ρ1ϵ adequately
captures the behavior of the conductive electrode material.
We conducted a preliminary FEM study with the model tounderstand the sensitivity to strain of each element when
deformed individually prior to applying simultaneousdeformations. We applied a 10% strain to each element in turnand simulated the overall impedance of the resulting system,with the results shown in Fig. 2. The plot of impedance
demonstrates several interesting effects. First, at the low
frequency limit, strain in every element results in a similarchange in impedance. Since at low excitation frequencies thefield is approximately uniform along the length of the sensor,changes in impedance at any point will have similar impact as
any other point. The differences between each element are dueto the changes in resistivity. If the resistivity was unchanging,the changes in impedance would be identical. Second, at thehigh frequency limit, no change in impedance is observed forstrain in any element. At high frequencies, any applied potential
is shunted through the sensor in the immediate vicinity of theelectrodes, and there is no potential between the upper and lower
electrodes along the body of the sensor. Finally, we observe thatwithin the range ∈ 10 , 10, applying the same strain toeach element of the sensor produces different responses, withthe element closest to the electrodes producing the largest
response and the element furthest away producing the smallest
response. Therefore, by probing the sensor across this frequency
range, we can differentiate the strains at different elements
within the sensor.

this, we selected 10 nodes and 10 frequencies as our nominal
ANN configuration.
Once trained, we used the ANN, with 10 frequencies and 10
hidden nodes, to reconstruct the strain in our simulated loading
conditions. The reconstruction of the 100 loading conditions
used to train the ANN are shown in Fig. 3. Testing with anadditional set of conditions showed slightly higher errors. In Fig.
3, we show the reconstructed strain by element, although thestrains were applied across all elements simultaneously in thetest. The plots show that Element 1, which is closest to theelectrical interface, has the lowest error and best fit across allvalues of applied strain. Moving away from the interface,Elements 2 through 5 have progressively larger error values. The
increasing error is due to the decreasing change in magnitude of
the impedance at elements further away from the interface, asshown in Fig. 2, coupled with the fact that strains are being
applied to all elements at once. The strain approximations atdistal elements show increasing error because the signal isoverwhelmed by the signal from proximal elements. Throughthis evaluation, we determined that it is possible to use an ANN
to uniquely reconstruct the spatial distribution of strain state ina capacitive sensor through a single interface.
III.

Fig. 3. Regression results of the trained artificial neural network. The plotsshow the reconstructed strain as points, with an ideal curve shown as ablack line. Error bars represent 95% confidence intervals. Data shownrepresent 100 loading conditions generated using a Latin hypercube
design. Individual tests include simultaneous loading of all elements withrandomly assigned strain values.

As shown in Fig. 2, the response of the sensor to strain is nota simple function. We found that an artificial neural network(ANN) was required to obtain accurate reconstruction of thestrain field. In order to train the ANN, we generated 100 test
conditions using a Latin hypercube experimental design. Insteadof considering the entire impedance spectrum, we limited oursamples to 10 logarithmically spaced frequencies between
15.9kHz and 1.59MHz, simulating the capabilities of a physical,
on-board measurement system. The impedance results from thefinite element simulation for all 100 conditions at the 10 sample
frequencies were used to train the ANN. 70 conditions were used
for training, 15 for validation, and 15 to compute theperformance. We used MATLAB's Neural Network Toolbox(MATLAB 2016a, NNT 9.0).
As a sensitivity study for the parameters of the ANN, weevaluated changes to the number of nodes in the hidden layer(3,5,7,10,15,20)
andthenumberoffrequencies(3,5,10,20,50,100). On average, we observed an order-ofmagnitude improvement between 5 and 10 frequencies, but only
marginal improvement beyond 10 frequencies. At 10frequencies, we found significant improvement up to 10 nodes,
but no significant improvement beyond that point. Because ofCONCLUSION

In this paper we have demonstrated a frequency-based
method for measuring distributed strains in a deformablecapacitive sensor. Our approach utilizes an electrode with finite
resistance, which can be realized in practice using conductive
elastomer composites. The method we have demonstrated canbe used to reduce the number of individual sensors required tomeasure the distributed strains found in deformable systems,
such as soft robots and wearable devices. This in turn improves
the manufacturability of these devices, decreases the number ofinterfaces required, and thereby increases the overall reliabilityof the deformable system.
IV.

ACKNOWEDGMENT

This work is partially supported under the NASA Early
Career Faculty Program, Space Technology Research Grants
Program (NNX14AO52G). ELW and MCY are supported by
the NSF Graduate Research Fellowship (DGE-1333468).
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