IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGY, VOL. 33, NO. 1, MARCH 2010 79

Capacitive Pressure Sensor With Very LargeDynamic Range

Ezzat G. Bakhoum, Senior Member, IEEE, and Marvin H. M. Cheng

Abstract— A new capacitive pressure sensor with very largedynamic range is introduced. The sensor is based on a newtechnique for substantially changing the surface area of theelectrodes, rather than the inter-electrode spacing as commonlydone at the present. The prototype device has demonstrated achange in capacitance of approximately 2500 pF over a pressurerange of 10 kPa.

Index Terms— Capacitive pressure sensor, pressure sensor,sensor dynamic range.

I. INTRODUCTION

PRESSURE sensors are widely used in automotive,aerospace, industrial, and medical applications. Capac-

itive pressure sensors [1]–[8] have attracted much attentionrecently because of their versatility, low temperature sensitiv-ity, and low power consumption. The fundamental problemin those sensors, however, is the poor dynamic range. Thedynamic range (maximum change in capacitance) that a ca-pacitive pressure sensor offers is typically less than 1 pF, whichusually necessitates the use of a sophisticated interface circuitto counteract the inherently poor resolution of the sensor.1

This paper introduces a new concept of a capacitive pressuresensor with a very large dynamic range. The prototype devicethat was built and tested by the authors have demonstrateda change in capacitance of approximately 2500 pF over apressure range of 0 to 10 kPa. Its resolution is thereforesubstantially higher than any of the known types of capacitivepressure sensors.

The idea of the new device is to mechanically deformtwo drops of mercury that are separated by a thin slab ofa dielectric material, so as to form a parallel-plate capacitorwhere the area of the electrodes is variable to a high degree.When the drops are deformed under pressure, the surface areaof the electrodes forming the capacitor is at its maximum.When no pressure is acting on the drops, the surface area of theelectrodes is nearly zero. This principle is illustrated in Fig. 1.

Manuscript received October 7, 2008; revised March 8, 2009. First versionpublished October 13, 2009; current version published March 10, 2010. Rec-ommended for publication by Associate Editor K.-N. Chiang upon evaluationof reviewers’ comments.

E. G. Bakhoum is with the Department of Electrical and ComputerEngineering, University of West Florida, Pensacola, FL 32514 USA (e-mail:ebakhoum@uwf.edu).

M. H. M. Cheng is with the College of Science and Technol-ogy, Georgia Southern University, Statesboro, GA 30460 USA (e-mail:hmcheng@georgiasouthern.edu).

Digital Object Identifier 10.1109/TCAPT.2009.20229491The resolution is usually defined as the ratio of the dynamic range to the

pressure range.

The principle of the new device, therefore, is to create acapacitor with a variable electrode area, rather than a variableinter-electrode spacing as commonly done in the devicesshown in the literature [1]–[8]. It should be pointed out that asimilar idea was recently published by Liu et al. [9]. In thatapplication, a mass of mercury is used to sense acceleration,and the contact area of the mercury with the surface of a fixedelectrode constitutes a variable capacitor.

The detailed structure of the new sensor, together with thetest data, are given in the following sections.

II. STRUCTURE OF THE NEW SENSOR

The basic structure of the new sensor is shown in Fig. 2.Two drops of mercury of a diameter of 5 mm each are placedon both sides of a slab of a ceramic material that has a veryhigh dielectric constant (such as barium titanate or one of itscompounds). The drops are held in place by means of twoaluminum disks that serve as the compression mechanism.The compression disks, in turn, are acted upon by meansof two metallic diaphragms with bellows, as shown (thesemetallic diaphragms are available from a number of industrialsuppliers). The compression disks are given a slight curvature,as shown in the figure, such that the spacing between each diskand the ceramic slab is exactly 5 mm at the center, but lessthan 5 mm everywhere else. In this manner, the mercury dropswill be forced to the center each time the diaphragms retract.

The two metal diaphragms are held in place by meansof two large aluminum rings, as shown. The rim of eachdiaphragm is inserted into a slot inside the ring which isfilled with a conductive paste so that an air-tight seal isformed. The two aluminum rings are finally separated fromeach other by a plastic washer, to which the ceramic slabis firmly attached. Pressure is applied to the sensor in thevertical direction, where it must act on both diaphragms.Since the air that surrounds the mercury droplets must beallowed to exit from the sensor and re-enter as the sensoris pressurized/depressurized, two atmospheric pressure reliefconduits are drilled in the plastic washer, as shown on theright-hand side of Fig. 2. In most applications, those conduitswill be connected to an atmospheric pressure environment via,for example, an extra tube to be connected to the sensor. Aphotograph of the components of the sensor is shown in Fig. 3.

According to the above, it should be therefore clear thatthis new sensor is essentially a differential pressure sensor(an equation for calculating the absolute pressure will bederived, however). In applications where it is desired to detect

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80 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGY, VOL. 33, NO. 1, MARCH 2010

Dielectric

Mercury

(a)

Dielectric

Mercury

(b)

Fig. 1. (a) Two drops of mercury are flattened against a slab of a dielectricmaterial. A parallel-plate capacitor with liquid electrodes is formed. (b) Underzero pressure, the two drops of mercury return to their nearly sphericalshape. The change in capacitance between the two configurations (which isproportional to the change in the area of the electrodes) can be more than600-fold.

pressures that are lower than the atmospheric pressure at sealevel (like in aircraft altitude applications for example), asuitable vacuum can be initially applied to the pressure reliefconduits (in which case the mercury droplets will be initiallyflattened at sea level).

III. BASIC THEORY OF OPERATION

A. Pressure–Geometry Relationship in the Deformation of aDrop of Mercury

Fig. 4 shows the geometry of a drop of mercury that isdeformed between two solid surfaces. The vertical pressurethat is acting on the drop is P , and the lateral pressure is theatmospheric pressure Patm. Pint is the internal pressure, andR is the radius of curvature of the part of the surface of theliquid that is not flattened, as shown.

The internal pressure Pint in the liquid must be balancedby the atmospheric pressure plus the Laplace pressure, or thepressure due to surface tension [10]; that is

Pint = Patm + 2γ

R(1)

where 2γ /R is the Laplace pressure with γ the surface tensionof mercury. As the drop of mercury is flattened, the differencein the internal pressure will be equal to the applied pressureP , i.e.,

P = Pint − (Pint)0 (2)

where (Pint)0 is the internal pressure at zero applied pressure.By using (1), (2) can be written as follows:

P = Patm + 2γ

R−

(Patm + 2γ

R0

)(3)

where R0 is the original unflattened radius of the drop. Hence

P = 2γ

(1

R− 1

R0

). (4)

In the sensor described here, the total pressure that is appliedto the device must be equal to the pressure P plus an additional

Plastic washer

Aluminum ring Mercury drop

Patm

Patm

P

P

Air-tight sealDiaphragm with bellows

Compression disk Ceramic dielectic slab

Fig. 2. Mechanical structure of the sensor (scale: 2:1).

Fig. 3. Components of the sensor. The overall diameter of the package is42 mm, and the overall height is 18 mm. The figure shows the two diaphragms,a compression disk, one mercury drop, and the remaining components as oneintegrated unit.

pressure that is needed to deform the metal diaphragm. Thatadditional pressure was determined experimentally and isdiscussed further below.

B. Capacitance–Geometry Relationship Between TwoDeformed Drops of Mercury

The capacitance of a parallel plate capacitor is given by thewell-known equation [11]

C = ε A

d(5)

where ε is the permittivity of the dielectric medium, A isthe surface area of the electrodes, and d is the thicknessof the dielectric. It is to be noted from Fig. 2 that thecontact area, or “wetting” area, A of the undeformed dropof mercury is very small, and that an additional capacitance

BAKHOUM AND CHENG: CAPACITIVE PRESSURE SENSOR WITH VERY LARGE DYNAMIC RANGE 81

P

RPint Patm

Fig. 4. Pressures and geometry in the deformation of a drop of mercury.

[beside the one defined by (5)] exists between the twometal spheres, with air being the dielectric. That parasiticcapacitance can be calculated from theoretical models givenin references such as [12]. A more accurate calculation ofthat capacitance, however, was obtained with finite-elementanalysis (see the Appendix). As shown in the Appendix, theparasitic capacitance is quite negligible in comparison withthe main capacitance given by (5) (this is due to the veryhigh dielectric constant of the ceramic slab used between themercury electrodes. That is, the air capacitance is negligible incomparison with the capacitance of the ceramic). Equation (5)is therefore the capacitance equation that is relied upon in thepresent work.

The wetting area A of the deformed drop (see Fig. 4) can becalculated from the fact that the volume of the drop remainsconstant. The volume of the undeformed drop is equal to4π R3

0/3, where R0 is the radius of the undeformed drop.Calculating the wetting area of the deformed drop is a simple,but rather lengthy and uninformative exercise in geometry. Wejust give the result as

A = π

⎡⎣

√π2 R2

16+ 2R3

0

3R− π

4R

⎤⎦

2

(6)

where R is the radius of curvature of the part of the surfaceof the liquid that is not flattened (see Fig. 4). The capacitanceC between the two liquid electrodes can then be determinedfrom (5). In the device considered here, the capacitance isactually measured, and then other parameters such as radiusof curvature and, finally, pressure can be calculated from theequations. Equations (5) and (6) can be solved to give theradius of curvature R as a function of the capacitance C . Thisrelationship is found to be

R =√

Cd

επ3+ 4

3R3

0

( ε

πCd

)1/2 −√

Cd

επ3. (7)

C. Calculation of the Pressure Acting on the Sensor

From (4) and (7), the pressure P acting on the mercurydrops can be formulated as a function of the capacitance

P = 2γ /

(√Cd

επ3+ 4

3R3

0

( ε

πCd

)1/2 −√

Cd

επ3

)− 2γ

R0. (8)

The physical pressure acting on the sensor, however, isequal to P plus the pressure that is needed to deform themetal diaphragms. That additional pressure was determined

0

2.0

4.0

6.0

8.0

10.0

1 2 3 4 5 6 7 80

Pressure (kPa)

Displacement (mm)

Fig. 5. Pressure–displacement relationship for the metal diaphragm used inthe prototype sensor.

experimentally. Fig. 5 shows the relationship between thepressure (in kPa) and the displacement in millimeters forthe diaphragm that is used in the prototype described here(the dots in the graph represent the experimental values, andthe dashed curve is a least-squares best fit).

As can be seen from Fig. 5, those diaphragms have excellentlinearity in the region of small pressure (essentially, thistype of diaphragm acts like a spring. When a small pressureis applied, the spring action is linear. For large pressures,however, the spring action is nonlinear). The operation of theprototype described here is actually entirely within the linearregion. The slope of the linear relationship can be determinedfrom the data shown. Furthermore, since the capacitance ofthe device is directly proportional to the displacement of thediaphragm (specifically, a displacement of 5 mm correspondsto a capacitance change from 4 pF to 2500 pF), the relationshipbetween the applied pressure and the capacitance can be easilydetermined as well. That relationship can be represented as

Pdia = αC (9)

where Pdia is the pressure acting on the diaphragm and αis the constant of proportionality.2 For the prototype devicedescribed here, α was found to be equal to 8 × 10−4 kPa/pF.

The physical, or total, pressure acting on the sensor is equalto the sum of the two pressures in (8) and (9). That totalpressure is now finally given by

Ptotal = αC − 2γ

R0

+ 2γ

/ (√Cd

επ3+ 4

3R3

0

( ε

πCd

)1/2 −√

Cd

επ3

)(10)

By knowing the physical parameters of the device and by mea-suring the capacitance between the two external electrodes,the total pressure can therefore be directly calculated. For theprototype sensor, the diaphragm pressure Pdia was found toreach a maximum of 2 kPa (see Fig. 5), while the pressure Prequired to fully deform the mercury droplet was found to be

2Note from (5) that C is proportional to the contact area A only, since bothε and the dielectric thickness d are constants. A, in turn, is linearly relatedto the displacement of the diaphragm, since the volume of the mercury dropremains constant.

82 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGY, VOL. 33, NO. 1, MARCH 2010

315 630 945 1260 1575 1890 2205 252040

2.0

4.0

6.0

8.0

10.0Theoretical

Experimental

C(pF)

P(kPa)

Fig. 6. Total pressure acting on the sensor as a function of the measuredcapacitance.

about 84 kPa. The sensor can therefore handle a total pressureof about 10 kPa. Of course, for applications where a largerpressure range is needed, a stiffer diaphragm must be used.

IV. EXPERIMENTAL RESULTS

For the prototype device described here, the following arethe dimensions and the physical constants:

1) dielectric constant (εr ) of the barium titanate ceramicslab: 1200;

2) thickness of the ceramic slab: 2 mm;3) radius R0 of the undeformed mercury droplet: 2.5 mm;4) wetting area of the undeformed mercury droplet:

0.785 mm2;5) wetting area of the fully deformed mercury droplet:

490 mm2.

Notice that the ratio between the wetting areas in thedeformed and the undeformed configurations is more than 600!In fact, straightforward substitution into (5) gives capacitancesof 4.16 and 2500 pF in the two cases, and direct experimentalmeasurements have confirmed these numbers.3

Fig. 6 shows a plot of the total pressure calculated from (10)as a function of the capacitance C , along with the actualpressure that was measured in a pressure chamber.

The measurements were performed in a commercial-qualitypressure chamber that is pressurized with compressed air. Thechamber is equipped with two different types of commerciallyavailable precalibrated pressure sensors to eliminate any pos-sibility for errors in the measurements. The capacitance ofthe sensor was measured by directly connecting the sensorto a capacitance meter. As the graph in Fig. 6 shows, theagreement between the experimental and the theoreticallycalculated values of the pressure was excellent in the region oflow pressures. However, a discrepancy of approximately 7%was observed in the region of high pressure (7–10 kPa). Sincethe discrepancy is not the result of measurement errors, thebasic theoretical model of (10) will have to be refined. More

3It is to be noted that the capacitance between the two external electrodes(the aluminum rings) in the sensor is very nearly equal to C , or the capacitancebetween the two mercury electrodes; as other parasitic capacitances arenegligible due to the presence of air and low dielectric-constant materialsbetween the two external electrodes (see Appendix).

Fig. 7. Finite-element model for calculating the parasitic capacitance.

specifically, it should be noted that the relationship betweenthe displacement and the pressure acting on the diaphragm inFig. 5 is in fact not perfectly linear in the operating region of0–2 kPa; and hence a polynomial representation instead of thesimple constant α in (9) must be used. This refinement will notbe given here, however, as such a polynomial relationship isstrongly a function of the diaphragm used in the sensor (such apolynomial will vary depending on the type of metal, size, andstructure of the diaphragm used). Alternatively, a calibrationcurve for the sensor can simply be used.

A note concerning the response time of the sensor is nowin order. The response time of the sensor can essentially becharacterized by the time that the mercury drop takes to fullyretract after being compressed. That time is approximately50 ms, and is therefore negligible in the present application.

V. CONCLUSION

The new capacitive pressure sensor described in this reporthas a dynamic range that is substantially higher than any ofthe other capacitive pressure sensors known at the presenttime. The prototype device that was fabricated has shown achange in capacitance (dynamic range) of almost 2500 pF overa pressure range of 0 to 10 kPa (the sensitivity of the sensor istherefore 250 pF/kPa, which is substantially higher than any ofthe known types of pressure sensors). As described above, thepressure range that the sensor can handle can be increased bysimply using a stiffer diaphragm. The equations that relate theapplied pressure to the measured capacitance of the device arefairly simple, and testing has shown a very good agreementbetween theory and experiment.

APPENDIX: FINITE-ELEMENT CALCULATION OF THE

PARASITIC CAPACITANCE

The parasitic capacitance in the sensor was calculated with afinite-element analysis program. The model is shown in Fig. 7,where the metal components of the sensor are represented andthe ceramic dielectric slab is assumed to be nonexistent.

The capacitance calculated for the model shown in the figureis approximately 0.1 pF. Given that the capacitance providedby the presence of the ceramic slab is between 4.16 and2500 pF, as indicated in Section IV, it is clear that the parasiticcapacitance is negligible throughout the range of operation ofthis sensor.

BAKHOUM AND CHENG: CAPACITIVE PRESSURE SENSOR WITH VERY LARGE DYNAMIC RANGE 83

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Ezzat G. Bakhoum (SM’08) received the Bach-elor’s degree from Ain Shams University, Egypt,in 1986, and the Master’s and Ph.D. degrees fromDuke University, Durham, NC, in 1989 and 1994,respectively, all in electrical engineering.

From 1994 to 1996, he was a Senior Engineer andManaging Partner at ESD Research Inc., ResearchTriangle Park, NC. From 1996 to 2000, he was aSenior Engineer at Lockheed Martin/L3 Communi-cations Inc., Camden, NJ. From 2000 to 2005, heserved as a Lecturer in the Electrical Engineering

Department at the New Jersey Institute of Technology, Newark. He iscurrently an Assistant Professor at the Department of Electrical and ComputerEngineering, University of West Florida, Pensacola, FL.

Marvin H. M. Cheng received the B.S. and M.S.degrees in mechanical engineering from NationalSun Yat-sen University, Taiwan, in 1994 and 1996.In December 2005, he received the Ph.D. degreein mechanical engineering from Purdue University,West Lafayette, IN.

From 1997 to 1999, he was a Research and Devel-opment Engineer at National Synchrotron RadiationResearch Center in Taiwan, developing real-timemonitoring system for high-energy emission sys-tems. Currently, he is an Assistant Professor at the

College of Science and Technology, Georgia Southern University, Statesboro.His current research interests include mechatronics, controller synthesis withthe consideration of finite wordlength, precision, motion control, fast imagingof atomic force microscope, and embedded controllers.