3rd International Conference on Sensing Technology, Nov. 30 - Dec. 3,2008, Tainan, Taiwan

Modeling of Capacitive Sensor filled with ElasticDielectrics and its Advantages

o. P. ThakurSchool of applied Sciences,

Netaji Subhas Institute of TechnologyNew Delhi, India

opthakur@yahoo.com

Anjani Kumar SinghSchool of applied Sciences

Netaji Subhas Institute of TechnologyNew Delhi, India

anjaninsit@gmail.com

have high mechanical flexibility, low acoustic impedance, lowmanufacturing cost, more robust and can be easily molded intodesirable shape. The electrostriction properties of the materialimprove the sensorN response. The linear elastic responsegoverned by HookeN law, the Maxwell electrostatic stressgoverned by coulomb law and the electrostriction stress(dielectric response) are illustrated in figure 1.

AbstractU Electrostriction phenomena and mathematicalmodeling in respect of capacitive sensor filled with elasticdielectric material have been described. A comparativetheoretical analysis of the traditional air gap capacitive sensorwith the electrostriction based solid-state capacitive sensor hasbeen carried out thoroughly. Difficulties and limitations with airgap capacitive sensor are highlighted. Solid-state capacitivesensor with elastic dielectric material sandwiched between thecompliant electrodes exhibits higher mechanical and electricalstability, sensitivity, higher tolerance with environment andwithstands large load in comparison to the air gap capacitivesensor. An experimental setup with proper solution has beendescribed for the measurement of relative capacitance variation.An alternative method of current-mode bridge has also beensuggested for the purpose of sensing pressure and strain withhigh accuracy.

E=O

----------_ ..

+++++++++

E~OMaxwell Stress

lb.~

------------

Ie eeel+++++++++

E'tOElectrostriction

~d

Keywords- Capacitance sensor; electrostriction; elasticdielectrics; strain; stress.

I. INTRODUCTION

The variation in capacitance due to variation in geometry ofthe capacitor has been studied by the various researchers in thepast. However the change in caJBcitance due to variation indielectric constant (due to electrostriction) has been poorlyexplored in the literature. In this paper the relative advantageand disadvantage of electrostriction based solid statecapacitance sensing and simple geometry (area and separation)based capacitance sensing have been analysed for variouspressure, stress/strain, and tactile sensing. In most of thetraditional capacitor designs, the dielectric constant does notchange with the applied field and the change in capacitance dueto variation in geometry can be measured by various methods[I, 2]. However, the variation in geometry is restricted in micromachining of the capacitive sensor arrays with a large area andsmall gap for increasing capacitance. The electrostriction basedcapacitive sensor with enhanced capacitance overcomes thislimitation and exhibits additional advantage particularly at highload. The real challenge is to identify the key parameters andthe environmental tolerance of the elastic dielectric materialbeing used for specific application. Electromechanical couplingeffect such as electrostriction and piezoelectricity can beapplied in various type of transducers and sensors [3,4].Theelastic dielectric materials incorporated between the electrodes

Figure 1. Schematic diagrams illustrating (a) Pure linear elastic responsegoverned by HookeN law, (b) Maxwell stress produced by surface charge& governed by CoulombN law, and (c) electrostriction stress due to dipolesallignment.

The choice and fabrication of the polymer-electrodestructure is also the area of interest for various appl ications.When the polymeric material (elastic dielectrics) is sandwichedbetween the compliant electrodes (made up of colloidal carbon)in a polymer binder and subjected to the external electric field,the Maxwell stress effect (figure Ib) and the electrostrictioneffect (figure Ic) would be resulted. The overall performance[5, 6] of elastic dielectric material can be estimated on the basisof evaluation ofbasic parameters like response time, theoreticalefficiency, mechanical & electrical stability and stress-strainresponse etc. Acrylic elastomers have demonstrated actuatedstrains of over 200 %, stresses up to 7 MPa, and estimatedenergy density of over 3 Jig [5]. Dielectric elastomers areclosely related to the electrostrictive polymers likepolyurethanes. It is observed that all electrostrictive polymersexhibit a component of Maxwell stress ranging from 10% to500/0 of the total response [7].

II. THEORETICAL ApPROACH

The simplest possible configuration of a parallel platecapacitor with a vacuum between the electrodes is typical formost micro machined capacitive sensors and for microscopicpressure & tactile capacitive sensors [8]. Considering the

978-1-4244-2177-0/08/$25.00 © 2008 IEEE 467

3rd International Conference on Sensing Technology, Nov. 30 - Dec. 3, 2008, Tainan, Taiwancontribution from a fringe effect (figure 2), the capacitance, C, separation between the plates (_ ~), neglecting both theof a parallel plate capacitor of an area, A, and separation, h, hbetween the plates is given [9] as variation in area by considering rigid electrode (8A = 0) as in

(figure 3c, d).

c = EEoA [I + Lh log .fA]h 21tA h'

(1)If the electrodes are rigid & incompressible (figure 3: c &

d), then 8A = 0 and equation (3) becomes

For air gap capacitor (figure 3a, b) with non variable

dielectric constant (i.e. 8E = 0), we have

For a large number of applications, the above equation (5)is used, but at high stress or at higher loading the air filledsensor may be damaged due to bending and twisting of theelectrodes. In tactile sensing applications [10], multiple layersare used for the compliance of electrodes to avoid bending andtwisting, and to increase effective area and decrease separationbetween plates to enhance both stability & capacitance.However the contribution due to relative change in dielectricconstant towards total capacitance change is maximum incomparison with the geometry based terms. It is very difficultto model the capacitance accurately as the sufficient test dataexhibiting the change in capacitive behaviour with the changein applied voltage for the elastic dielectric material is notavailable. The large change in the capacitance is mainlyattributed to the electrostriction effect as the capacitancevariation is restricted due to smaller separation and higher areaof the plates.

(4)

(5)

8E 8h---E h

8C 8h

C h

where E is the dielectric constant of the material between theplates and L is the perimeter of the plates. As the distribution ofcharge over the electrodes/plates (figure 2) is not uniformparticularly at a separation comparable to its lateral dimension,the correction due to edge effect is very much required for allpractical purposes. The contribution due to fringing effect cannot be neglected in case of micro capacitive sensor withseparation comparable to its lateral dimension. However thiscorrection term might be ignored in case of a thin filmcapacitor where the separation is very less than the root of itsarea.

For a parallel plate capacitor ignoring edge effect, thecapacity is given as

Figure 2. Schematic diagram exhibits fringing effect between the electrodesof a parallel plate capacitor

c = EEoA .h

A relative variation in capacitance is given by

8C 8h 8A 8E-=--+-+-.C h A E

(2)

(3)

III. ELELECTROSTRICTION PARAMETERS AND ITS

MEASUREMENTS

For very small deformation in elastic dielectric material,

only the first order terms in strain tensor Su have been

considered and the dielectric tensor E U in terms of the strain

tensor is given [9] by

Three terms on the right hand side of equation (3) represent

8hthe relative variation in separation, i.e. - -, between the

h_ 8A

plates, the relative variation in electrode~area, i.e. - , andA

8Ethe relative variation in dielectric constant, i. e. -, under

Edeformation.

Actually the sensor response is represented by the RHSterms of equation (3) depending on the type of sensor and itsapplication in a particular situation. Most of the traditional airgap capacitance sensor is based on the relative variation in

Eij =E 8ij +alSij +a2Skk8jj' (6)

where E is the permittivity of the undeformed body and

at & a2 are two parameters describing the variation in

dielectric properties of the material in shear and bulkdeformation respectively.

For a linear elastic dielectric material tmdergoingmechanical deformations (described by HookeN law), the

general expression [11] for elastic strain tensor Sij in terms of

elastic stress tensor ~j' Young modulus Y, and Poisson ratio

cr is given by

468

(8)

The relative change in thickness of an elastic dielectricmaterial as the coupling expression (SacerdoteN formula) forelectromechanical phenomena [12] and is given by

3rd International Conference on Sensing Technology, Nov. 30 - Dec. 3, 2008, Tainan, Taiwan1 strains exist [15] in the capacitor electrode even if the dielectric

8" =- [(1 + (j )T.. , - (j Tkk8 .. ] . (7) medium is empty space. So the contribution from lateral strains1/ y r II U d' . f I. should not be neglecte for the correct estllnahon 0 resu ts.

Researcher [7] has not considered the contributions from lateralstresses in the estimation of strain. Neglecting the shear

components ( 8 ij ) of strain and their corresponding

components (E u) of permittivity is a matter of great concern

for correct mathematical mooeling.

On comparing equation (8) with the general equation,S =v" E E , for electrostriction [13], we get the relative

IJ I IJkl k I

change in separation between plates as

(d)

STRESS(e)

STRESS(c)

(a) STRESS

Mlh

(9)

(10)dE 33 dSkk )-- =a] +a2-- =at +a')(1-2cr ,dS

33ciS

33~

On differentiating equation (6) with respect to S33 ' we get

8h ')S33 =h = -y 33 E - ,

where y33 is the electrostrictive coefficient.

if the normal force is acting only along the direction ofelectric field E, then we have Sll = S22 = 0 and if the sides ofdielectric material are also fixed by the rigid wall then we

have 8 32 =8 3] =O. We have, for the constrained situations

as in figure 3, Figure 3. Parallel plate capacitive sensors subjected to one and twodirectional stresses, indicating (a) an elastic electrode undergoes deformationin air gap capacitive sensor, (b) both electrodes undergo bending & canwithstand smaller load to avoid breaking, (c) & (d) rigid and compliantelectrodes undergo uniform deformation, and (e) differential variation in area& thickness due to non compliant electrodes with the material.

IV. SELECTION OF MATERIALS

The selection of materials is very significant for gettinggood response with higher mechanical and electrical stability,sensitivity and better compliant with the electrodes. Materialsmust exhibit smaller modulus of elasticity and a largerdielectric constant for greater strain & mechanical flexibility asindicated in equation (8) and high dielectric strength to preventdisruptive electrical discharges at a large potential difference.

Among the widely used elastic dielectric materials i.e.VHB-4910, TC-5005 (B1B Enterprises) and CF19-2186 (NuSil Technology), VHB-491a exhibits high viscoelasticbehaviour with low modulus of elasticity (i.e. higher strain &flexibility) and low dielectric breakdown strength with creepeffects than TC-5005. Due to sticky nature of VHB 4910, it ispreferred for the compliant electrodes. Generally conductivecarbon grease and a very finely ground carbon powder areused as compliant electrodes.

Most adverse property of VHB 4910 is its lower dielectricstrength, which can be improved by prestraining the material[5]. The VHB-4910 shows superior properties than the TC-

(12)

(11 )dE_3_3=a+a.ciS 1 2

33

On substituting the value of second term under bracket ofSacerdate formula (equation 8) from equation (10), we have

dz 1 2 [( ') 1 dE 33 ]S33 = - = - - E oE E 1+ 2cr . - - -- .z 2 E dS33

The relative change in capacitance, thickness and thedielectric parameters, a1 & a2, are experimentally obtained [14]

by determining dE 33 / dS33 in constrained and unconstrainedsituation, with and without dielectric material, in a capacitorusing LC oscillator circuit. The equation (8) is formulated onthe assumptions which are valid for linear elastic dielectricmaterial and it is certainly not justified to use this equation forthe material like VHB 4910 exhibiting viscoelastic behaviorparticularly at high stresses. Secondly the neglect of shearcomponents of stresses and corresponding components ofpermittivity even in constrained situations [14] in equations (8­12) does not formulate the correct mathematical model forelectrostriction. However if the deformations are not quadraticin the applied electric field particularly at higher field strength,the linearity of theory for electrostriction fails. But very fewreliable measurements of electrostrictive coefficients for elasticdielectrics have been reported in the literature. The lateral

469

3rd International Conference on Sensing Technology, Nov. 30 - Dec. 3, 2008, Tainan, Taiwan5005 if the applied electric field is less than the breakdownstrength of VHB-49IO. However TC-5005 exhibits higherdielectric breakdown strength and hence it is suitable at highelectric field but it has lower mechanical flexibility & tolerancethan VHB-49IO. The performance of an elastic dielectricmaterial can be estimated on the basis of evaluation of basicparameters like response time, theoretical efficiency,mechanical & electrical stability and stress-strain response etc.Some times we have to make compromise between themechanical and the electrical properties of the materialsdepending on requirements.

V. EXPERIMENTAL SET UP

In the past the change in capacitance with geometry of theair gap capacitor has been measured by various methods [1, 2].The relative change in capacitance due to electrostriction ismeasured by using capacitor bridge circuit [16] with a DSPlock-in amplifier as indicated in figure (4a & b). Usingexperimental set up of two independent LC oscillators withequal inductances, an electrostriction enhancement of 1.8 & 3.5has been determined [17] for solid state sensors with polyesterand urethane respectively up to a load of 140 Pa.

In the experiment, the current, I, after the bridge circuit(Figure 4.) is given by

circuit (AZKA cell) designed recently [18], which is thecurrent-mode alterative of the traditional voltage-modeWheatstone bridge, can also be used as experimental setup withhigher accuracy and sensitivity. The exact electrical advantageof an electrostriction based caJllcitive sensor over the air gapcapacitive sensor is not measured is not measured with highaccuracy and it is certainly an area of future work.

i = jVW(C2 -C1) = jwv8c, (13)

Mechanical!ElectricalSuess

Stress free identicalcapacitor sensor (Cv

/

SensingBridge Circuit

\\

Capacitor Sensor (Cl)Experiencing stresses

(a)

DSP Lock in OutputAmplifier

FunctionGenerator

and

Also, in the integrated circuit used for suppressing unwantedsignals, the current is

i = (0 - vout

)[_1_ + jwc f ]Rf

i(s) =-vout (s)[_I_ +scf] (14)Rf

=-vout (s)scf' assmning the condition

wcfRf »1, (15)

where s is complex frequency variable So, we have

dvi(t) =--!!!!!.... Cf ' and

dt

vo(t) = __1 fi(t)dt. (16)cf

The equation (16) is valid for integrator, if the condition (15) issatisfied. The above experimental set up can be used for thedetermination of capacitance variation due to deformation orelectrostriction. But it is very difficult to evaluate thecontribution from electrostriction and the contribution fromgeometrical change separately.

Alternately, the Wheatstone bridge is employed extensivelyfor sensing force, pressure, strain, displacement,concentrations, material content compositions etc. A new

v..,+

(b)

Figure 4. (a) Schematic block diagram indicating the lock-in amplifier,function generator and a sensing circuit with two identical capacitive sensorsin which one is experiencing additional stresses, and (b) circuit diagramindicating two identical capacitors of capacitor bridge and an integrated circuitfor suppressing unwanted signals.

VI. DISCUSSION AND CONCLUSION

In case of air gap capacitive sensor (governed by therelative change in separation and area), the electrodes can bedamaged at high stresses due to lack of mechanical supportfrom inside the gap. On the other hand, the capacitive sensorwith elastic dielectric material is strong, stable and is caJllbleof withstanding high mechanical/electrical stresses as requiredin various sensing applications. The relative change incapacitance in case of elastic dielectric caJllcitive sensor isextremely high due high change in dielectric constant, it exhibithigher sensitivity and enhances the sensorN response up to alarge extent. In case of requirement of an in-built sensor tosense unidirectional stress, one of the electrodes must be fixedon the base. The electrodes must be in compliance with thematerial to transmit the stress to the interior of material

470

3rd International Conference on Sensing Technology, Nov. 30 - Dec. 3,2008, Tainan, Taiwanuniformly. Keeping in mind the various advantages ofelectrostriction based capacitive sensor, they can be a bettersubstitute to the traditional air gap capacitive sensor forpressure, stress/strain, and tactile sensing particularly at highstress/strain. The main disadvantage of using elastic dielectricmaterial like YHB 4910 between the plates is the formulationof proper mathematical model primarily due to its viscoelasticbehavior and coupling equation (8), which is valid for linearmaterial, canN be applied in this situation. However, muchmore work has to be done for the correct estimation ofcontribution from electrostriction in elastic dielectrics. The realchallenge is to identify a suitable material and its keyparameters at molecular level for exhibiting excellentelectrical/mechanical performance in terms of sensitivity.stability, environmental tolerance and better compliant with theelectrodes. It is also important to note that while selecting thematerial for particular sensing application, we have to make acompromise between the dielectric properties and themechanical properties.

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[18] S. J. Azhari and H. Kaabi, FAZKA Cell, the Current-Mode Alternativeof Wheatstone Bridge,S IEEE Transaction on circuits and systems, Vol.47, No.9, pp. 1277-1284,2000.