# Effects of electrostatic forces generated by the driving signal on capacitive sensing devices

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.Sensors and Actuators 84 2000 213219www.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 Uniersity, Shanghai 200433, Peoples 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

.single-sided driving, double-sided driving and double-sided driving with voltage feedback i.e., force-balanced measurement schemes . . .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

.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.

.Based on the analyses, the conclusions are: 1 the level of driving signal can be selected by the compromise among the requirements .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.

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

.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

.of the driving signal is "V "V sinv t , where the fre-0 1 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-

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 /22 d yx .0where A is the area of the electrodes, d is the original0distances between two electrodes, and are permitiv-0ity of the vacuum and the relative permitivity of the

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

( )M. Bao et al.rSensors and Actuators 84 2000 213219214

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 1 /2

electrostatic force on the electrode caused by the drivingvoltage:

A0 2F s V . 1 .e eff22 d yx .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 .V s V qV and Eq. 1 is still effective.( .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, 2 .22 d yx .0

where ma is the inertial force in x-direction and d the0original 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 andeff 0

2 2 .F sA V r2 d , Eq. 2 can be written as:e0 0 0F mae0 q yxs0. 3 .2 kdkd 1yx . 00

.If psF rkd and qsmarkd are defined, Eq. 3e0 0 0can be written as:

pxy sq. 4 . 21yx .

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

2.1. Zero , i.e., ps0

In this case, xsq; the relation between displacementand acceleration is linear.

2.2. Non-zero , i.e., p/0

.For small x, after expanding Eq. 4 in series andholding just the first terms, the following expression isobtained:

3 px 2 y 1y2 p xq pqq s0. 5 . . . .For small p and q, the stable solution of Eq. 5 can be

approximated as:

p 1y4 pq7p2 1y4 pq10 p2 .xs q 3 31y2 p 1y2 p . .

=3 p

q 1q q . 6 .2 /1y4 pq10 pTherefore, the following conclusions are derived.

.a The driving voltage causes an offset displacement atzero acceleration as indicated by the first term on the right

.side of Eq. 6 . Obviously, the larger the p-value, thelarger the offset displacement.

.b The sensitivity of the accelerometer is proportional 2 . .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 213219 215

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

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

.to Eq. 6 , The nonlinearity is:3 pqmaxNLsy , 7 .24 1y4 pq10 p .

where q is the q-value corresponding to the maximummaxw xacceleration. The readers are referred to Ref. 4 for the

definition of nonlinearity.As an example, if ps0.05, we have xs0.056q

.1.13q 1q0.18q If the q is 0.1, the nonlinearity causedmby 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-

.ing a function of f x, p :pf x , p sxy . 21yx .

( )and drawing the curves for f x relation with p as aparameter, we obtained the curves in Fig. 2.

.The solutions to Eq. 4 for specific p and q values canbe 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 two .cross points, A and B, between the curve for f x, ps0.05

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

stable solution, while the displacement corresponding to .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 :cf kdmax 0

a s .cm

For 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

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

.driving voltage, V , the smaller the f , p should be asmaxsmall 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 aboutmax0.3. Therefore, the critical acceleration that causes pull-ineffect is a s0.3kd rm. For ps0.1, the critical accelera-c 0tion is reduced to about 0.12 kd rm. There would be no0

stable 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 10 y qmaykxs0, 8 .2 2 22 d 1yx 1qx . . 0where ma is the inertial force in x-direction and d the0original 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

.before, Eq. 8 can be written as:4 p

x 1y sq. 9 . 221yx . .Based on Eq. 9 , the following discussions are made.

3.1. Zero , i.e., ps0

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

3.2. Non-zero , i.e., p(0

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

xs , 10 . 4 p1y 221yx .

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 213219216

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

.b For very small q and x, the approximate relation .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 agraphic method as follows. By defining a function of f x,

.p :

4 pf x , p sx 1y , 11 . . 221yx .

.and drawing the curve for f x relation with p as aparameter, the plots are shown in Fig. 4.

.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

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

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

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

larger than f , there is no stable solution. This meansmaxthat the mass is pulled-in by the combined effect of theelectrostatic and the inertial forces: the larger the p-value . i.e., the larger the V , the smaller the f i.e., themax

.smaller the ma . For example, for the curve for psmax0.05, f is about 0.32. Therefore, the critical accelera-maxtion, a , that causes pull-in effect is 0.32 kd rm. For0 0ps0.1, 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 is0larger 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 pmaxfor single-sided and double-sided driving schemes. Ac-cording to the curves in Fig. 5, f drops very fast withmaxincreasing 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 arsmall displacement, the feedback voltage is proportional tothe displacement of the mass: V sgV x, where g is ar 1constant designating the strength of the feedback opera-

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

( )M. Bao et al.rSensors and Actuators 84 2000 213219 217

tion. Obviously, the maximum value of V is restricted byrthe supply voltage of the electronic system.

With the feedback voltage, the electrostatic force on themass is:

2A V qV sinv tyV .0 0 1 rF se 2 22 d 1yx .02V qV sinv tqV .0 1 ry , 12 .21qx .

which can be rearranged as:

A0 2F s y4V V qV sinv t 1qx . .e r 0 122 22 d 1yx .02 2 2 2q4 x V q2V V sinv tqV sin v tqV . 13 . .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:

A0F se 22 22 d 1yx .0

=1

2 2 2 24 V q V xq4V xy4V V y4V V x . 0 1 r r 0 r 0 /2Using the notation of V saV , one gets:1 0

4F 1e0 2F s 1q a xe 2 /2 21yx .2 2y a gx q a gx xy a gx x , 14 . . . .

2 . 2where F is defined as A r2 d V instead ofe0 0 0 0 2 . 2 2 .A r2 d V 1qa . Therefore, the force balance0 0 0equation for the mass with an acceleration a becomes:

4F 1e0 2maykxq 1q a x2 /2 21yx .2 2y a gx q a gx xy a gx x s0. . . .

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

4 p 12qsxy 1q a x 2 /2 21yx .

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

.According to Eq. 15 , the q;x relation can be discussedas follows.

4.1. Zero p

In this case, xsq for small signals and the relationbetween displacement and acceleration is linear.

4.2. Non-zero p and small x

In this case,...

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