Journal of Sensor Technology, 2013, 3, 63-69 Published Online September 2013 (
A Wireless Inductive-Capacitive Resonant Circuit
Sensor Array for Force Monitoring
Andrew J. DeRouin, Brandon D. Pereles, Thadeus M. Sansom, Peng Zang, Keat Ghee Ong
Department of Biomedical Engineering, Michigan Technological University, Houghton, USA
Received July 11, 2013; revised August 11, 2013; accepted August 19, 2013
Copyright © 2013 Andrew J. DeRouin et al. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A wireless passive sensor array based on inductive-cap acitive (LC) resonant circuits capable of simultan eously tracking
two points of force loading is described. The sensor consisted of a planar spiral inductor connected to two capacitors
forming a resonant circuit with two resonant frequencies. When a load was applied to one or both of the parallel plate
capacitors, the distance between the plates of the capacitor was altered, thus shifting the observed resonant peaks. Test-
ing illustrated that applied loading to a particular capacitor caused a significant shift in one of the resonant peaks and
also a smaller shift in another resonant peak. This interdependence resulted from each capacitive element being con-
nected to the same inductive spiral and was accounted for with a developed analysis algorithm. To validate the experi-
mental observation, a circuit simulation was also generated to model the sensor behavior with changing force/displace-
ment. The novelty of this syste m lies not only in its wireless passive nature, bu t also in the fact that a single LC sensor
was fashioned to detect more than one point simul taneously.
Keywords: Force Sensors; RFID Tags; Sensor Array; Wireless Sensors
1. Introduction
Generally, wireless sensors consist of sensor components
for taking local measurements placed at the areas of in-
terest and a remotely located system to receive informa-
tion wirelessly for processing and presentation [1-3].
Since these systems lack physical connections between
the sensing components and the processing apparatus,
they are highly versatile in terms of sensor deployment,
and are thus ideal for wide area monitoring. Additionally,
wireless sensors have the capacity to operate, depending
on the particular method of data acquisition and trans-
mission, in an active or passive mode. Active sensors are
powered with an internal source whereas passive devices
receive power remotely. While active wireless sensors
can monitor a wide range of parameters encompassing a
large region, they can also suffer from significant instal-
lation and maintenance costs and battery lifetime limita-
tions [4]. Passive wireless sensors, on the other hand,
have limited functionality compared to their active coun-
terparts but are generally cheaper, easier to implement
and last longer. There are many types of wireless passive
sensors, and most of them are powered magnetically or
electromagnetically. Examples of these sensors include
magnetoelastic, inductive-capacitive, and surface acous-
tic sensors [5-9].
A wireless passive sensor, known as the inductive-
capacitive (LC) sensor, was developed to monitor chang-
ing environmental conditions such as temperature, hu-
midity and pressure [10]. This type of sensor altered its
capacitance depending on the experienced environmental
parameter, resulting in a shift in its resonant frequency.
The change in resonant frequenc y was remotely captured
with a coil antenna by monitoring its impedance change
[10]. Unfortunately, the limitation of the existing LC
sensors is their inability to monitor multiple parameters.
For applications that require simultaneous detection of
multiple targets, more than one LC sensor is necessary,
significantly increasing the overall sensor size with no
improvement to sensor detection range. Therefore, a two
element sensor was developed comprised of a planar
square spiral inductor and two parallel plate capacitors.
While the operating principle of the presented sensor
is the same as current LC sensor technologies [10], it
features multiple sensing elements (capacitors) sharing
the same inductor. As a result, there are interferences
among the capacitive elements and further investigations
were performed to realize the new, two-element sensor
opyright © 2013 SciRes. JST
The electrical representation of the two element LC
sensor is shown in Figure 1, where the inductor is re-
sponsible for coupling to a nearby antenna, receiving
power and transmitting information about changes in
resonant frequencies, and the capacitors are responsible
for sensing parameters of interest in terms of changing
capacitance values. The inductor is separated into differ-
ent segments, for convenience, named L1, L2, etc., that
are connected to capacitors C1, C2, etc., respectively. To
remotely detect the sensor’s response, a network analyzer
generates a frequency-varying electromagnetic field
through the antenna to the sensor’s inductor and then
monitors the change in the antenna’s impedance. This
work focused on a sensor with two inductor-capacitor
pairs for two-parameter monitoring. However, more in-
ductor-capacitor pairs can be added to the circu it to form
a more complex sensor capable of simultaneously meas-
uring more than two parameters.
Prior to actual data collection, the background imped-
ance (measurement with no sensor present) was collected
so that all measurements could be subtracted from the
background coil impedance to obtain the pure sensor
response. Figure 2 shows the measured resonance spec-
trum of the fabricated sensor. Two resonant frequencies
Figure 1. Circuit representation of the two-element LC sen-
sor and the detection antenna.
5 1015202530354045
Impedanc e (O hms)
Frequency (MHz)
Resonance 1Resonance 2
Figure 2. Resonant spectrum of the two-element LC sensor
measured with the network analyzer showing two resonant
peaks corresponding to C1 (Capacitor 1) and Resonance 1,
and C2 (Capacitance 2) and Resonance 2. The values of C1
and C2 were both measured as 12.5 pF, and L1 and L2 were
measured as 3.3 µH and 4.4 µH, respectively.
are shown on the plot, where Resonance 1 and Reso-
nance 2 resulted primarily from C1 and C2, respectively.
During testing, loading C1 caused a decrease in Reso-
nance 1 due to an increase in the capacitance value of C1.
However, there was also a slight decrease in Resonance 2
even though loading on C2 remained constant. This in-
terference is expected and explained as a result of the
interdependence between capacitors physically connected
to one inductive spiral (see discussion for additional de-
tails). To account for this, an algorithm was developed to
determine the actual force based on the changes in both
resonant frequencies.
2. Experiments
2.1. Analytical Model
An analytical model was developed to simulate the sen-
sor response. The model was based on the equivalent
circuit in Figure 1, assuming for simplicity that the cou-
pling magnitudes of each inductor loop were equal at all
frequencies (see Figure 3). Note the addition of R1 and
R2 to more accurately predict the losses in the inductors
and capacitors. The inductance, capacitance, and resis-
tance values of the sensor were measured using an im-
pedance analyzer (Agilent 4192) and were used for all
simulations. The analytical circuital equations were plot-
ted with MATLAB.
To determine the change in capacitance due to force
loading, the compressive strain of the sensor’s parallel-
plate capacitors during force loading was determined ex-
perimentally using a TestResources Inc. 100 Series me-
chanical testing device. The device exerted a compres-
sive load (0 - 100 N) onto the sensor’s capacitors while
measuring the compression of the capacitors. This al-
lowed calculation of the change in the distance between
the two plates of the capacitor for determination of the
capacitance by the equation [11]:
where ε is the permittivity of the medium between the
capacitive plates, A is the overlapping surface area of the
conducting plates, and d is the distance between them.
Figure 3. Circuital model used in the MATLAB simulation.
Copyright © 2013 SciRes. JST
The permittivity of the sensor medium was determined
using the capacitance value, plate areas, and distance
initially measured prior to loading. The capacitance cal-
culated from change in displacement was then used as an
input for the circuit model of the sensor impedance to
find its resonant frequencies.
The simulation model was developed by determining
the impedance of the resonant sensor, Z, when looking
into the spiral inductor terminals (top of L1, bottom of L2)
as illustrated in Figure 4. The branches containing R2
and L2 in parallel with C2 can be lumped as:
2222 22
 (2)
where j is the imaginary number and ω is the radian fre-
quency. Then, the series combination of Z2 and L1 can be
Figure 4. (a) Isometric illustration of the two-element force
sensor and (b) the top view of the sensor, as well as (c) a
photograph of the fabricated sensor. The wire connections
between the top capacitor plates and the spir al inductor are
combined with th
also illustrated.
e parallel C, resulting in:
211 12
jL R Z
 (3)
Equation (3) was then evaluated in MATLAB for fre-
2.2. Sensor Fabrication
or design. Sensors were fab-
rformance of the two-element sensor
2.3. Experimental Setup and Procedure
to evalu-
encies from 10 MHz to 50 MHz at intervals of 1 kHz
to determine its resonant frequencies at capacitance val-
ues corresponding to those acquired from the force-dis-
placement curve. The values of the electrical components
used in the simulation were measured as R1 = R2 = 5 ,
C1 = C2 = 12.5 pF, L1 = 3.3 µH, and L2 = 4.4 µH. It
should be noted from Equation (3) that because the poles
of the resonant circuit are dependent on both capacitors,
C1 and C2 each affect both resonant frequencies. The
same analysis can be extended for additional capacitive-
inductive pairs, allowing design of sensors with higher
numbers of sensing parameters.
Figure 4 illustrates the sens
ricated from a printed circuit board using a milling ma-
chine. The line width of the pattern was 0.3 mm and the
spacing between lines was 1.8 mm. The inductor part of
the sensor was a square spiral measuring 90 mm × 90
mm encompassing 10 turns. Addition ally, at the cen ter of
the sensor was a rectangular condu ctor measuring 48 mm
× 48 mm. A parallel plate capacitor was then formed by
adhering silicone foam between the conductor pad and
two pieces of copper clad FR-4 fiberglass PCB, measur-
ing 56 mm × 27 mm each. A multiple pole LC resonant
circuit was then fabricated from this setup by connecting
the top PCBs to the spiral inductor. The developed sensor
was monitored using a 105-mm-diameter single loop
detection coil co nnected to an Agilent Network/Sp ectrum
Analyzer (4396B).
To evaluate the pe
terms of signal strength, two single-element LC sen-
sors with resonant frequencies at 10 MHz and 38 MHz,
were fabricated. The single element sensor was similar in
design with the previous work [10]. The lengths of the
inner and outer loops of the spiral inductor of both sin-
gle-element sensors were 34.4 mm and 64 mm, respec-
tively. Additionally, the center capacitors of the sin-
gle-element sensors were 32 mm × 32 mm and the com-
bined footprint of these two single-element sensors was
equivalent to the footprint of a single two-element LC
sensor. Moreover, to ensure the desired resonance was
achieved (10 MHz and 38 MHz) the inducto rs of the sen-
sors were adjusted to have 10 and 7 turns, respectively.
Figure 5 illustrates the experimental setup used
ate the force response of the fabricated sensor. The sensor
Copyright © 2013 SciRes. JST
Figure 5. Experimental setup fo r testing the force sensity
as placed onto a Teflon loading platform of a custom-
valuate the two-element sensor against the single-
2.4. Analysis Algorithm
second LC pair (L2 and C2)
of the two-element LC sensor. A computer controlled pneu-
matic force applicator was used to apply independent force
loadings on the capacitive elements of the sensor.
ized automated pneumatic mechanical loader system.
The loader was controlled with a computer through stan-
dard RS232 communication using a customized Visual
Basic program, and the detection antenna was situated
around the loading platform. During experiments, the
loader applied force to C1 from 0 N to 89 N at 22.25 N
intervals while a constant force was held on C2. This
constant load at C2 was then changed, from 0 N to 89 N
at 22.25 N intervals, and full loading was reapplied to C1.
The resulting sensor output was collected via an antenna
with the network/spectrum analyzer. This process was
then repeated with C1 acting as the constant loading ele-
To e
ement sensors, the antenna was kept stationary and
connected to the network/spectrum analyzer during dis-
tance testing. The two-element sensor was adhered to a
beam with double sided adhesive tape and suspended
above the antenna, with the opposing end of the beam
adhered to an adjustable scaffold. During characteriza-
tion, the distance between the antenna and the sensor was
increased at 5.0 mm intervals until the resonant peak
could no longer be measured. The sensor was then re-
turned back to its original position at 5.0 mm intervals
with data being collected at each interval. This procedure
was then repeated with an antenna measuring 161 mm in
diameter to further characterize and compare the sensors
in relation to changes in sensor response as a result of
altering the antenna size. The same experiment was re-
peated for both single-element sensors to determine their
responses against separation distance from the antenna.
As indicated in Figure 1, the
was in series with inductor L1 and capacitor C1 (instead
of a parallel LC pair of L1 and C1). As a result, the im-
pedance of the sensor was not a simple superposition of
the two LC pairs but rather a complex relationship (as
indicated in Equation (3)) that allowed the value of C2 to
interfere with the resonance of the first LC pair, and vice
versa. Experimentally, the loading effect on each capaci-
tor was also found to be interdependent on the load ex-
perienced by the other capacitor. As a result, an iterative
algorithm was developed to determine the force loadings
on C1 (F1) and C2 (F2) based on the resonant frequency
shift of the associated resonant peaks (f1 and f2 respec-
tively). Since the sensor is a forth-order circuit, the reso-
nant frequency is expected to change with the capaci-
tance following an inverse of quartic root curve. How-
ever, for simplicity and also due to the narrow force
range, a 2nd order polynomial equation was applied to
represent the resonant frequency change with loading.
The 2nd order equation was found to provide a good fit
with the experimental results (see Figure 6). Therefore,
f1 could be expressed in terms of F1 as:
111 21
 (4)
However, to accommodate the depe1 2
th ndency of f on F,
e coefficients in Equatio n (4) were expressed as:
12 223 1,2,3iii ii (5)
0 20406080100
0 N
22.25 N
44.50 N
66.75 N
89.00 N
Resonant Frequency (MHz)
Force (N)
0 N
22.25 N
44.50 N
66.75 N
89.00 N
Resonant Frequency (MHz)
Force (N)
Figure 6. (a) The measuredonant frequency 1 (f1) as a
function of loading at C1 corresponding to five different C2
loading conditions; (b) The measured resonant frequency 2
(f2) as a function of loading at C2 corresponding to five dif-
ferent C loading conditions.
Copyright © 2013 SciRes. JST
where ai1, ai2, and ai3 were determined empirically.
Similarly, f2 was primarily dependent on F2 with a slight
dependence on F1 and could thus be described by:
212 223
BFB FB (6)
11213 1,2,3
ii i ii
BbF bFb
where bi1, bi2, and bi3 were determined e
te i and
ults of loading C1 from 0 N to
The iratve process began by setting F2 to 0 N
determining A using Equation (5), followed by so
for F1 using Equation (4). The calculated F1 was then
substituted into Equation (7) to solve for Bi, which was
then used to solve for F2 with Equation (6). The value of
F2 was then substituted back into Equation (5) and the
process was repeated until both F1 and F2 converged
within an acceptab le error.
3. Results and Discussi
3.1. Force Monitoring
Figure 6(a) depicts the res
89 N at 22.25 N intervals while C2 was heldnstant at 0
N, 22.25 N, 44.5 N, 66 .75 N , and 8 9 N . Th e senso r was 1
cm from the center of the coil. Similarly, Figure 6(b)
depicts the resu lts of loading C2 from 0 to 89 N while C1
was held constant at different loading conditions. As can
be seen, changing the constant load on the other capaci-
tor decreased the amplitude of the force loading curve.
Analyzing the changes in the curves revealed that their
behavior followed a 2nd order polynomial equation. By
curve fitting this data, a set of 2nd order polynomial co-
efficients was obtained. These coefficients were used in
Equation (4) and Eq uation (6) to solve for F1 and F2.
Figure 7 plots the calculated F1 and F2 when the val-
ues of f and f corresponding to F1 = F = 66.75 N w
1 22ere
used as the input parameters for the iteration algorithm.
The algorithm went through multiple iterations finding
0510 15 20 25 30 35 40
Force 1
Force 2
Force (N)
Iteration Step
Figure 7. Calculated F1 (force loading on Capacitor 1)nd
F (force loading on Capacitor 2) based on the values of f
expected values for F1 and F2 were 66.75 N.
Figure 9 shows the signal amplitudes for the two-ele-
nt single-element sen-
The result of the MATLAB simulation for the sensor
hown in Figure 10. As
2 1
(resonant frequency 1) and f2 (resonant frequency 2). The
values for F1 and F2 until converging to the correct solu-
tions. For this particular input condition, th e convergence
occurred at about 25 iteration steps, which was typical
for all loading conditions.
Figure 8 plots the percentage errors of the calculated
forces when F1 = F2. These errors were largely due to
errors in the collected f1 and f2 data when compared to
the 2nd order polynomial curve fit. Another observation
in Figure 8 is that the errors in F2 are larger than those of
F1. This is due to a poorer curve fitting of the depend-
ence of f2 on the force loading at C1 (as indicated in Fig-
ure 6(b) where the curves are not as well spread as the
curves in Figure 6(a)).
3.2. Sensor Characterization
ment sensor and the two equivale
sors with increasing sepa ration distance from the antenn a.
As expected, the resonant amplitudes of the sensors de-
creased with increasing distance between the antenna and
sensor. Additionally, it can be seen that the two-element
sensor had larger signal amplitude and, as a result, a lar-
ger detection range than the two single-element sensors
of equivalent footprint. Moreover, the differences be-
tween the two-element and single-element sensors were
also larger with the larger antenna since the coupling
between the antenna and the smaller single-element sen-
sors changed more with increasing antenna size than the
coupling between the antenna and the two-element sen-
3.3. Theoretical Validation
impedance at zero loading is s
depicted, the resonant peaks are at relatively similar fre-
quencies when compared to the experimentally measured
curve from Figure 2. The difference of 0.2 MHz on the
first resonant frequency and 0.3 MHz on the second
% Error of Calculated F
-20 020 40 60 80100
% Error of Calculated F
Percentage Error (%)
Applied Force on C
and C
Figure 8. The percentage errors of the calculated forces
compared to the actual force loadings when F1 = F2.
Copyright © 2013 SciRes. JST
0 20406080100120
Equivalent Res 1
Equivalent Res 2
Multi- L C Res 1
Multi- L C Res 2
Impedance at Resonance (Ohms)
Separation Distance (mm)
020 40 60 80100120
Equivalent Res 1
Equivalent Res 2
Multi-LC Res 1
Multi-LC Res 2
Impedance at Resonance (Ohm
Separation Distance ( mm)
Figure 9. The resonant ampdes of the two-element sen-
sor (indicated as Multi-LC Res 1 and 2) and two single-
element sensors (indicated as Equivalent Res 1 and 2) as a
function of separation distance between the sensor and the
antennas with diameters of (a) 105 mm and (b) 161 mm.
5 1015202530354045
Impedance (kOhm)
C sensor showing two resonant peaks corresponding to C1
(Capacitor 1) and Resonance 1 and C2 (Capacitance 2) and
Resonance 2. The measured sensor values were used for all
resonant frequency is due to stray inductances and para-
sitic capacitances not accounted for in the circuital model.
Another notable point is the larger second resonance
peak in the measurement (but not in the simulation). This
Frequency (MHz)
Figure 10. Simulated resonant spectrum of the two-elem
0 20406080
Peak 1 Simulated
Peak 1 Measured
Peak 2 Simulated
Peak 2 Measured
Resona n t Fre quency 1 (MHz )
Resonant Frequency 2 (MHz)
Force (N)
Figure 11. Comparison between simulated and measured
force-frequency characteristics holding C2 constant and
measuring Peak 1 as a function of C1 (lower two curves) and
holding C1 constant and measuring Peak 2 as a function of
C2 (top two curves).
is largely due to the self-resonan ce of the coil antenna (at
about 60 MHz), causing signals closer to 60 MHz to be
amplified. As a result, the amplification near the self-
resonance of the coil antenna reflects on the measured
sensor imnce. peda
Figure 11 depicts the simulated circuit results in cor-
respondence to the experimentally collected data. The
error between the simulated and experimental curves is
likely an effect created by parasitic capacitances and in-
ductances within the sensor circuit. In addition, the
change in permittivity of the silicone foam between the
capacitor plates due to compression was assumed to be
negligible, but may have been significant enough to
cause the observed error.
4. Conclusions
f loading these capacitors was tested
-to-reach civil engineering
A wireless passive LC multi-element sensing array was
presented. The fabricated system functioned by incorpo-
rating two parallel plate capacitors into a signal LC cir-
cuit, thus producing two resonant peaks from a single
ircuit. The effect oc
with a changing load of 0 N to 89 N at 22.25 N intervals
applied to one capacitor while a constant load was ap-
plied to the other capacitor. The response of each ca-
pacitor was found to be interdependent on the constant
load applied to the other capacitor. This interdependence
was handled through the use of an iterative algorithm. A
circuit model was also developed to validate the experi-
mental results. Additionally, when compared to an
equivalent setup of two single-element LC sensors, the
two-element LC sensor not only had a stronger signal,
and thus a larger detection range, but also exhibited less
change when coupled with different sized antennas. Due
to the limited detection distance (about 8 cm), the current
version of sensor would be useful in places such as moni-
toring force loading in hard
Copyright © 2013 SciRes. JST
Copyright © 2013 SciRes. JST
Figure 12. Illustration of the difference between single and
multi element LC sensors when applying for wide area mo-
nitoring. The shaded squares represent the sensing element
and the white squares the inductor. The detec tion points are
represented by arrows.
structures, for example, concrete and wooden beams in
This device, with further development, represents an
innovative method of multi-parameter/target sen sing with
a single LC circuit. One potential application that can
highlight the advantage of this sensor compared t
previous LC sensor is as an embedded sensor for pas
monitoring of stresses in a large area, such as under a
roadbed. If the single-element LC sensor is employed,
the user need s to monitor the response at each individual
sensor location, which could be time consuming for road
condition monitoring. However, for the multi-element
sensor design, a number of stress-sensitive capacito
pacitors can be placed at different locations and con-
nected to the inductor via wires as illustrated in Figure
12. As a result, a number of sensors can be md
tem. Additionally, by utilizing in-
o the
r ca-
simultaneously, which can significantly shorten the mo-
nitoring time.
With further development, the number of elements
could be expanded thus providing for an even more ro-
bust force mapping sys
terdigital capacitors functionalized toward certain che-
micals and environmental parameters, a sensor could
theoretically be developed and deployed for monitoring
multiple parameters such as humidity and volatile com-
pounds. The future works of this system include increas-
ing the number of elements, investigating methods to de-
crease overall sensor size, increasing the detection range,
and adapting the system to monitor other parameters
such as heat, chemical concentrations, moisture, etc.
5. Acknowledgements
The authors would like to acknowledge the National De-
fense Science and Engineering Grant for support of a
graduate studen t fellow working on this project.
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