Ž .Sensors and Actuators 84 2000 213–219www.elsevier.nlrlocatersna

Effects of electrostatic forces generated by the driving signal oncapacitive sensing devices

Minhang Bao), Heng Yang, Hao Yin, Shaoqun ShenDepartment of Electronic Engineering, Fudan UniÕersity, Shanghai 200433, People’s Republic of China

Received 9 September 1999; received in revised form 9 December 1999; accepted 21 December 1999

Abstract

In measuring the capacitance of a variable mechanical capacitor used in a capacitive mechanical sensor, an electrical driving signal isusually needed. The electrostatic forces caused by the driving signal on the mechanical capacitor may interfere with the measurement andthe normal operation of the devices significantly. In this paper, quantitative analyses on the effects of driving signal are made for

Ž .single-sided driving, double-sided driving and double-sided driving with voltage feedback i.e., force-balanced measurement schemes .Ž . Ž .The effects caused by the driving signal are found to be: 1 the zero offset of the sensors for single-sided driving signal, 2 the change of

Ž .the measurement sensitivity, and 3 the reduction of the critical measurand signal level causing the pull-in effect that hampers the normaloperation of the device. The levels of critical measurand signal for specific driving signal levels are found quantitatively.

Ž .Based on the analyses, the conclusions are: 1 the level of driving signal can be selected by the compromise among the requirementsŽ .on the sensitivity, the accuracy and the reliability of the sensors devices for a specific configuration, 2 the side effects of the driving

signal can be minimized by using the testing scheme of double driving with voltage feedback. q 2000 Elsevier Science S.A. All rightsreserved.

Keywords: Electrostatic forces; Capacitive sensors; Driving signal

1. Introduction

Capacitive sensors have been getting more and morepopular in recent years for microsensor applications or forelectromechanical systems due to its process compatibilitywith most mechanical structures, high sensitivity and lowtemperature drift.

As it is well-known, for capacitive sensing, a certainform of voltage driving signal is usually necessary for themeasurement of capacitance. The voltage driving signalcauses electrostatic forces on the movable electrode andinterferes with the movement of the movable electrode.Therefore, the accuracy of the measurement or even thenormal operation of the devices can be affected by thelevel of driving voltage used for the measurement.

In operation, the movable electrode in a capacitivesensor is subjected to four forces: the force caused by the

) Corresponding author. Tel.: q86-21-6564-2763; fax: q86-21-6564-8783.

Ž .E-mail address: mhbao@fudan.edu.cn M. Bao .

Ž .measurand acceleration, for example , the elastic recoveryforce of the flexure structure, the damping force and theelectrostatic force generated by the driving signal. In thispaper, the damping effect is not considered.

Generally, the driving voltage consists of a DC compo-w xnent and an AC component 1,2 . A commonly used form

Ž .of the driving signal is "V "V sinv t , where the fre-0 1Ž 6 .quency, v, in the order of 10 rs of the AC component is

usually much larger than the frequency of the measurandsignal and the natural vibration frequency of the mechani-

Ž 4 .cal structure both are in the order of 10 rs . Therefore,the force applied on the movable electrode is the averageof the electrostatic force of the driving voltage. The aver-age force is:

A´´ 10 2 2F s V q V ,e 0 12 ž /22 d yxŽ .0

where A is the area of the electrodes, d is the original0

distances between two electrodes, ´ and ´ are permitiv-0

ity of the vacuum and the relative permitivity of the

0924-4247r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved.Ž .PII: S0924-4247 00 00312-5

( )M. Bao et al.rSensors and Actuators 84 2000 213–219214

medium in between the electrodes of the capacitor,respectively. If we define the effective voltage, Veff

12 2s V q V , we have the general form of the( 0 1ž /2

electrostatic force on the electrode caused by the drivingvoltage:

A´´0 2F s V . 1Ž .e eff22 d yxŽ .0

Sometimes, the alternative component of the drivingsignal is a square pulse with a duty cycle of 50% and anamplitude of "V . In these cases, the effective voltage is1

2 2 Ž .V s V qV and Eq. 1 is still effective.(Ž .eff 0 1w xPuers and Lapadatu 3 noticed that the electrostatic

force by the driving signal may cause offset, sensitivityvariation and pull-in effect to hamper the normal operationof the sensor devices. The voltage for pull-in to occur wasfound for single-sided driving scheme and a double-sidedQ-mode driving was suggested. However, the offset andthe variation of sensitivity for a specific structure with aspecific driving signal were not analyzed quantitativelyand no further suggestion was made for the implementa-tion of the Q-mode driving scheme.

In this paper, the effects of the electrostatic force on themeasurement and the operation of the sensor devices areanalyzed quantitatively. For the sake of simplicity, we takecapacitive accelerometers as examples in the discussion.According to the structures used for practical sensor de-vices, analyses will be made for three commonly usedconfigurations for capacitive sensors: single-sided driving,double-sided driving and a double-sided driving with anelectromechanical feedback in the following sections.Based on the analyses, the effects of driving signal on theoffset of output, on sensitivity and on the pull-in effect ofthe movable electrode will be discussed. It is found thatthe side effects can basically be reduced by using theconfiguration of double-sided driving with voltage feed-back.

2. Single-sided driving

For a single-sided driving configuration as schemati-cally shown in Fig. 1, the equation to decide the displace-ment of the movable plate is:

A´´ V 20

qmaykxs0, 2Ž .22 d yxŽ .0

where ma is the inertial force in x-direction and d the0

original gap between the two electrodes. For simplicity,

Fig. 1. Schematic structure for a single-sided driving accelerometer.

the effective voltage of the driving signal is designated byV instead of V . By using the notation of xsxrd and˜eff 0

2 2 Ž .F sA´´ V r2 d , Eq. 2 can be written as:e0 0 0

F mae0q yxs0. 3Ž .˜2 kdkd 1yxŽ .˜ 00

Ž .If psF rkd and qsmarkd are defined, Eq. 3e0 0 0

can be written as:

pxy sq. 4Ž .˜ 21yxŽ .˜

Ž .Based on Eq. 4 , discussions are made for the followingthree conditions.

2.1. Zero Õ, i.e., ps0

In this case, xsq; the relation between displacement˜and acceleration is linear.

2.2. Non-zero Õ, i.e., p/0

Ž .For small x, after expanding Eq. 4 in series and˜holding just the first terms, the following expression isobtained:

3 px 2 y 1y2 p xq pqq s0. 5Ž . Ž . Ž .˜ ˜

Ž .For small p and q, the stable solution of Eq. 5 can beapproximated as:

p 1y4 pq7p2 1y4 pq10 p2Ž .xs q˜ 3 31y2 p 1y2 pŽ . Ž .

=3 p

q 1q q . 6Ž .2ž /1y4 pq10 p

Therefore, the following conclusions are derived.Ž .a The driving voltage causes an offset displacement at

zero acceleration as indicated by the first term on the rightŽ .side of Eq. 6 . Obviously, the larger the p-value, the

larger the offset displacement.Ž .b The sensitivity of the accelerometer is proportionalŽ 2 . Ž .3to 1y4 pq10 p r 1y2 p , which is dependent on the

driving voltage. The larger the p-value, the larger thesensitivity of the accelerometer.

( )M. Bao et al.rSensors and Actuators 84 2000 213–219 215

Fig. 2. Graphic solution for single-sided driving.

Ž .c The driving voltage causes additional nonlinearitybetween the displacement and the acceleration. According

Ž .to Eq. 6 , The nonlinearity is:

3 pqmaxNLsy , 7Ž .24 1y4 pq10 pŽ .where q is the q-value corresponding to the maximummax

w xacceleration. The readers are referred to Ref. 4 for thedefinition of nonlinearity.

As an example, if ps0.05, we have xs0.056q˜Ž .1.13q 1q0.18q If the q is 0.1, the nonlinearity causedm

by the driving signal is y0.45%.

2.3. General solution

For general situation, the equation can be solved numer-ically or by a graphic method given as follows. By defin-

Ž .ing a function of f x, p :˜p

f x , p sxyŽ .˜ ˜ 21yxŽ .˜( )and drawing the curves for f x relation with p as a˜

parameter, we obtained the curves in Fig. 2.Ž .The solutions to Eq. 4 for specific p and q values can

be found by the cross points between a horizontal lineparallel with the x-axis for the specific q-value and thecurve for the specific p-value.

For example, for ps0.05 and qs0.2, there are twoŽ .cross points, A and B, between the curve for f x, ps0.05˜

and the straight line qs0.2. The displacement corre-Ž .sponding to the left cross point, A, at x(0.3d is the0

stable solution, while the displacement corresponding toŽ .the right cross point, B, at x(0.67d is an unstable0

solution.For a specific p, there is a maximum f-value, f ,max

which corresponds to a critical acceleration, a :c

f kdmax 0a s .c m

ŽFor a q-value larger than f i.e., for an accelerationmax.a larger than a , there would be no solution for thec

equation. This means that the pull-in effect occurs due tothe combined effect of the electrostatic force and the

Žinertial force. As the larger the p-value i.e., the larger the

.driving voltage, V , the smaller the f , p should be asmax

small as possible for a reliable operation. However, thesmaller the p-value, the smaller the sensitivity of thesensing circuit. Therefore, there should be a compromisefor a practical application. This poses restrictions on thedesign of capacitive sensors.

For example, for ps0.05, f is found to be aboutmax

0.3. Therefore, the critical acceleration that causes pull-ineffect is a s0.3kd rm. For ps0.1, the critical accelera-c 0

tion is reduced to about 0.12 kd rm. There would be no0Žstable displacement at all if p is larger than 0.15 the exact

. w xvalue is 0.148 3 .

3. Double-sided driving

For a double-sided driving structure as shown in Fig. 3,the equation to decide the displacement of the movableplate is:

2A´´ V 1 10y qmaykxs0, 8Ž .2 2 22 d 1yx 1qxŽ . Ž .˜ ˜0

where ma is the inertial force in x-direction and d the0

original gaps of the two capacitors. By using the notationof xsxrd , F sA´´ V 2r2 d2 and p and q as defined˜ 0 e0 0 0

Ž .before, Eq. 8 can be written as:

4 px 1y sq. 9Ž .˜ 221yxŽ .˜

Ž .Based on Eq. 9 , the following discussions are made.

3.1. Zero Õ, i.e., ps0

In this case, xsq and the relation between displace-˜ment and acceleration is linear.

3.2. Non-zero Õ, i.e., p(0

Ž .As Eq. 9 can be written in the form of:q

xs , 10Ž .˜ 4 p1y 221yxŽ .˜

we can come to the following conclusions.

Fig. 3. Schematic structure for double-sided driving accelerometer.

( )M. Bao et al.rSensors and Actuators 84 2000 213–219216

Ž .a As xs0 for qs0, no offset displacement is˜caused by the driving voltage due to the symmetric electro-static forces on the mass.

Ž .b For very small q and x, the approximate relation˜Ž .between x and q is x(qr 1y4 p . Therefore, the larger˜ ˜

the driving voltage, the larger the sensitivity of the ac-celerometer. And the maximum p-value for stable opera-tion is 0.25.

3.3. General situation

For general situation, the equation can be solved by aŽgraphic method as follows. By defining a function of f x,˜

.p :

4 pf x , p sx 1y , 11Ž . Ž .˜ ˜ 221yxŽ .˜

Ž .and drawing the curve for f x relation with p as a˜parameter, the plots are shown in Fig. 4.

Ž .The solutions of Eq. 9 can be found by the crosspoints between a horizontal line parallel with the x-axis fora specific q-value and the curve for a specific p-value.

For example, for ps0.05 and qs0.3, there are twocross points between the horizontal line for qs0.3 and

Ž .the curve of f x for ps0.05. The displacement corre-˜Ž .sponding to the left cross point at x(0.43d is the0

stable solution and the displacement corresponding to theŽ .right cross point x(0.61d is the unstable solution.0

It can also be found that, for a specific p, there is aŽ .maximum value for the curve of f x , f . For a q-value˜ max

larger than f , there is no stable solution. This meansmax

that the mass is pulled-in by the combined effect of theelectrostatic and the inertial forces: the larger the p-valueŽ . Ži.e., the larger the V , the smaller the f i.e., themax

.smaller the ma . For example, for the curve for psmax

0.05, f is about 0.32. Therefore, the critical accelera-max

tion, a , that causes pull-in effect is 0.32 kd rm. For0 0

ps0.1, the critical acceleration is reduced to about

Fig. 4. Graphic solution for double-sided driving.

Fig. 5. f vs. p relation for single-sided and double-sided driving.max

0.17kd rm. There would be no stable displacement if p is0

larger than 0.25. The mass is not stable even without anacceleration signal. It will always be pulled into contactwith one of the fixed electrodes.

Curves in Fig. 5 shows the dependence of f on pmax

for single-sided and double-sided driving schemes. Ac-cording to the curves in Fig. 5, f drops very fast withmax

increasing p-value. Therefore, for a reliable operation, pshould be very small. The cost would be a reduced sensi-tivity for the sensing circuit. Therefore, there should be acompromise in a practical design. For most practical appli-cations, p should be smaller than about 0.05. This is therestriction on the driving signal for the double-sided sens-ing schemes.

4. Double-sided driving with feedback voltage

For a forced balanced accelerometer with a feedbackvoltage, V , the simplified model is shown in Fig. 6. For ar

small displacement, the feedback voltage is proportional tothe displacement of the mass: V sgV x, where g is a˜r 1

constant designating the strength of the feedback opera-

Fig. 6. Double-sided driving with electromechanical feedback.

( )M. Bao et al.rSensors and Actuators 84 2000 213–219 217

tion. Obviously, the maximum value of V is restricted byr

the supply voltage of the electronic system.With the feedback voltage, the electrostatic force on the

mass is:

2A´´ V qV sinv tyVŽ .0 0 1 rF se 2 22 d 1yxŽ .˜0

2V qV sinv tqVŽ .0 1 ry , 12Ž .21qxŽ .˜

which can be rearranged as:

A´´0 2F s y4V V qV sinv t 1qxŽ . Ž .˜e r 0 122 22 d 1yxŽ .˜0

2 2 2 2q4 x V q2V V sinv tqV sin v tqV . 13Ž .˜ Ž .0 0 1 1 r

As the driving frequency, v, is much larger than thesignal frequency and the natural vibration frequency of themechanical structure, the average force applied on themass is:

A´´0F se 22 22 d 1yxŽ .˜0

=1

2 2 2 24 V q V xq4V xy4V V y4V V x .˜ ˜ ˜0 1 r r 0 r 0ž /2

Using the notation of V saV , one gets:1 0

4F 1e0 2F s 1q a x̃e 2 ž /2 21yxŽ .˜

2 2y a gx q a gx xy a gx x , 14Ž . Ž . Ž . Ž .˜ ˜ ˜ ˜ ˜

Ž 2 . 2where F is defined as A´´ r2 d V instead ofe0 0 0 0Ž 2 . 2Ž 2 .A´´ r2 d V 1qa . Therefore, the force balance0 0 0

equation for the mass with an acceleration a becomes:

4F 1e0 2maykxq 1q a x̃2 ž /2 21yxŽ .˜

2 2y a gx q a gx xy a gx x s0.Ž . Ž . Ž .˜ ˜ ˜ ˜ ˜

By using the same notations for p and q as before, thefollowing equation is obtained:

4 p 12qsxy 1q a x˜ ˜2 ž /2 21yxŽ .˜

2 2y a gx q a gx xy a gx x . 15Ž . Ž . Ž . Ž .˜ ˜ ˜ ˜ ˜

Ž .According to Eq. 15 , the q;x relation can be discussed˜as follows.

4.1. Zero p

In this case, xsq for small signals and the relation˜between displacement and acceleration is linear.

4.2. Non-zero p and small x̃

In this case, we have:

qx( .˜ 1

21y4 p 1q a q4 pa gž /2

The q;x relation is related to the value of g as˜follows.

qŽ .a For gs0, as x( , the larger˜ 1

21y4 p 1q až /2the p the larger the displacement for the same accelerationdue to the effect of electrostatic force.

Ž .b For g/0, the larger the g, the smaller the displace-ment due to the electromechanical feedback that is nega-tive in nature.

4.3. General conditions

Generally, the relation between x and q can be found˜by a graphical method. By defining the function ofŽ .f x, g, p,a :˜

f x , g , p ,aŽ .˜

4 p 12sxy 1q a x˜ ˜2 ž /2 21yxŽ .˜

2 2y a gx q a gx xy a gx x 16Ž . Ž . Ž . Ž .˜ ˜ ˜ ˜ ˜

Ž .and drawing the curve for f x, g, p,a with different g-˜Ž .values for specific p and a values, the curves of f x can˜

be obtained. Two curves are shown in Fig. 7 for two setsof p and a . In the calculation for the curves, a ceiling ofV is set for Vr.0

It can be seen from Fig. 7 that for small g the maximaŽ .of the curves of f x , f , are much smaller than unity˜ max

due to the effect of the electrostatic force of the drivingsignal. This means that the critical acceleration that causespull-in effect, i.e., a s f kd rm, is small. With in-c max 0

creased g, f ’s increase with g until g is about 5 whenmaxŽthe curves have the highest linearity for the two sets of p

.and a considered . With further increased g, the displace-ment of the mass decreases for the same acceleration dueto the effect of electromechanical feedback. All the curveswith g-values larger than 5 merge for large x. This is˜

( )M. Bao et al.rSensors and Actuators 84 2000 213–219218

Fig. 7. Graphic solutions for double-sided driving with electromechanicalŽ . Ž .feedback. a For ps0.05, a s0.2 and V FV b for ps0.1, a s0.2r o

and V FV .r o

because the feedback voltages are restricted to a maximumvalue of V .0

Ž . Ž .It can be seen by comparing Fig. 7 a and b that forsmall g, the larger the p-value the smaller the maxima ofthe curves due to the larger electrostatic force of thedriving signal. This means that the larger the p-value, thesmaller the critical acceleration, a , that causes the pull-inc

Žproblem. However, if the feedback is large enough say,.g)5 for the two conditions considered , f can be quitemax

close to 1. This means that the pull-in effect caused by thedriving signal can be mostly eliminated by the feedbackconfiguration.

Now let us consider the accelerometer described in Ref.w x2 as an example. As the mass of the seismic mass isms1.6=10-10 kg and the resonant frequency of thestructure is 17.7 kHz, the elastic constant of the flexure isfound to be about 1.98 Nrm. As the original distancesbetween the electrodes are d s1.3 mm, the original ca-0

pacitances are C s0.1 pF and the driving signals have a0Ž .DC voltage of V s1.6 V and a high frequency 1 MHz0

Ž .square wave with an amplitude of V s0.3 V V s2V ,1 p-p 1

F is found to be 9.85=10-8 N. According to the defini-e0

tion for p, q and g, the parameters are found to beŽps0.038, qs0.265 and g(8.5. Therefore, f formax

. Ž .g(8.5 is found by Eq. 15 to be about 0.87. However, ifthe feedback is taken away, according to Fig. 5, the f ismax

only about 0.37.Normally, the level of the acceleration signal for the

devices is usually not larger than its nominal operationrange of 50 g. However, once a working device is sub-jected to a large acceleration caused by an inadvertentcrash of the device to a hard surface, the seismic mass mayenter into a permanent pulled-in state and stop its normaloperation. As f is about 0.87, the maximum accelera-max

tion, a , that causes a permanent pull-in effect is aboutc

1400 g. This is considered to be adequate for most applica-tions. However, if the feedback is taken away, the corre-sponding a is reduced to about 600 g. This would be tooc

small for many applications. A measure of some kind,such as oxide knotsrbumpers on the mass, has to be takento restrict the movement of the mass to a certain distanceso that no permanent pull-in can occur.

5. Conclusion

According to the analyses given above, the drivingsignal for capacitive sensing causes side effects on thecapacitive sensing.

5.1. The zero offset of the sensors

This effect appears only for single-sided capacitivestructures. The larger the driving signal the larger the zerooffset as shown by the curves in Fig. 2. There would be nozero offset if the capacitance is double-sided and symmet-ric. Therefore, the double-sided driving is preferred formost applications.

5.2. The Õariation of sensitiÕity

As the driving signal plays a role of positive feedback,the sensitivity of the devices is increased because of theeffect of electric driving signal. This effect is shown by the

Ž . Žslopes of the curves near the zero points in Figs. 2, 4.and 7 . Generally, the larger the driving signal the larger

the sensitivity. In case of a force-balanced device, theeffect of the driving signal is compensated by the negativeelectromechanical feedback.

5.3. Pull-in effect

There are two conditions of pull-in effect.Ž .a The electrostatic force generated by the electric

driving signal causes pull-in of the electrode if the drivingvoltage exceeds a certain limit. This limitation is desig-nated by p for the system. For a single-sided driving,max

( )M. Bao et al.rSensors and Actuators 84 2000 213–219 219

p is 0.148; while for a double-sided driving, p ismax max

0.25. The mass will be pulled into contact with a fixedelectrode when the corresponding p-value of the drivingsignal exceeds p . Once the mass is pulled into contactmax

with a fixed electrode, the devices will no longer befunctional if it is not prevented by an appropriate electron-ics or by using blockers to limit the mass movement.

Ž .b Even with a p-value smaller than p , the superpo-max

sition of the electrostatic force and the mechanical forcecaused by the measurand may still cause the pull-in of thestructures. There would be a critical mechanical force bythe measurand for a specific p-value during the operationof the devices. Once the force exceeds the critical value,the mechanical structure will enter into a pull-in state. Thedevices will stop its normal function in this state unless theelectric power of the system is cut off and the system isrestarted. The critical mechanical force can be found by

Ž .the maximum of the curves in Figs. 2, 4 and 7 , corre-sponding to a specific working condition. As far as thepull-in effect is concerned, p-value should be as small aspossible for the two open-loop configurations.

The overall conclusion is: of the three configurationsconsidered, the configuration of double-sided driving withan electromechanical feedback has the best performancesand is least affected by the side effects of the drivingsignal.

Acknowledgements

The research is supported by the National ScienceŽ .foundation of China Project No. 69876009 and by The

Key National Basic Research and Development ProgramŽ .of China No. G1999033199 .

References

w x1 H. Leuthold, F. Rudolf, An ASIC for high-resolution capacitiveŽ .microaccelerometers, Sens. Actuators, A 21–23 1990 278–281.

w x2 W. Kuehnel, S. Sherman, A surface micromachined silicon ac-celerometer with on chip detection circuitry, Sens. Actuators, A 45Ž .1994 7–16.

w x3 R. Puers, D. Lapadatu, Electrostatic forces and their effects onŽ .capacitive mechanical sensors, Sens. Actuators, A 56 1996 203–210.

w x4 K. Matsuda, Y. Kanda, K. Yamamura, K. Suzuki, Nonlinearity ofpiezoresistive effect in n-and p-type silicon, Sens. Actuators, A

Ž .21–23 1990 45–48.

Biographies

Minhang Bao graduated from the Fudan University, Shanghai, China, in1961 and joined the faculty of the Physics Department in the same year.He finished his graduate studies on Solid-State Physics in 1966. He was avisiting assistant professor to the EEAP Department of Case WesternReserve University, USA, from 1979 to 1981 and a visiting researchfellow to the EECS Department of UC Berkeley, USA, during 1985–1986.Since 1983, he has been with the Electronic Engineering Department ofthe Fudan University, where he is currently a professor and the director ofthe Microsensor Labs. He is on the board of directors of several Chineseacademic societies on sensors and actuators. He is the author of Inte-

Ž .grated Transducers book in Chinese , several chapters in books and anumber of research papers. His current research interest is in microme-chanical sensors and integrated circuits.

Heng Yang

Hao Yin

Shaoqun Shen graduated from the Physics Department, Fudan Universityof China, in 1964 and joined the faculty of the Physics Department thatyear, teaching courses in Physics. Since 1983, he has been with theDepartment of Electronic Engineering, Fudan University. He has beendoing research on semiconductor devices and integrated circuits since the1970s. He was a visiting research fellow on sensors to CNM , Spain, in1994-1995. He is currently the director of the Integrated Circuit Technol-ogy Lab. His current interest is in solid-state sensors and ASICs.