Download - Theoretical analysis of the sensor effect of cantilever piezoelectric benders

JOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 3 1 FEBRUARY 1999

Theoretical analysis of the sensor effect of cantilever piezoelectric bendersQing-Ming Wang,a) Xiao-hong Du, Baomin Xu, and L. Eric CrossIntercollege Materials Research Laboratory, The Pennsylvania State University,University Park, Pennsylvania 16802-4801

~Received 26 May 1998; accepted for publication 13 October 1998!

Piezoelectric bending mode elements such as bimorph and unimorph benders can be used as bothactuation and sensing elements for a wide range of applications. As actuation elements, thesedevices convert electric input energy into output mechanical energy. As sensing elements, theyconvert external mechanical stimuli into electrical charge or voltage. In this article, the sensingeffect of cantilever mounted piezoelectric bimorph unimorph and triple layer benders subjected toexternal mechanical excitations are discussed. General analytical expressions relating generatedelectric voltage~or charge! to the applied mechanical input excitations~momentM, tip forceF, andbody forcep! are derived based on the constitutive equations of these bending devices. It is foundthat the clamping effect of each component in the bender devices decreases the dielectric constant.The bimorph bender has a higher voltage sensitivity than the unimorph or triple layer bender withthe same geometrical dimensions. The dependence of voltage and charge sensitivities on thethickness ratio and the Young’s modulus ratio of the elastic layer and piezoelectric layer underdifferent conditions are discussed and compared for the unimorph and triple layer benders. ©1999American Institute of Physics.@S0021-8979~99!05302-5#

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I. INTRODUCTION

Many novel solid state devices based on piezoelecmaterials have been developed in recent years for a wrange of electromechanical applications.1–3 These devicescan perform either actuator or sensor functions by utilizthe direct or converse piezoelectric effect. Two classicalamples of piezoelectric devices are bimorph and unimoconfigurations, which have been widely used for acousensing, loudspeaker, relay, micropumps, micropositionand many other applications. With simple structures, the umorph and bimorph are capable of producing large strounder low electric voltage as actuation elements or providhigh mechanical force/load sensitivity as sensory device

As schematically shown in Figs. 1~a! and 1~b!, a bi-morph bender consists of two piezoelectric thin plates wpolarization normal to the interface. The application of eletric field forces one plate to expand and the other to contrSince there is constraint at the interface of these two plabending deformation occurs in the whole structure. Simlarly, bending can be produced in the unimorph wheretransverse deformation of the piezoelectric plate is cstrained by the nonpiezoelectric elastic plate@Fig. 1~c!#.Therefore, bimorphs and unimorphs can be used as actuelements. Two types of connections are commonly usebimorph design and fabrication: one is a series connecand the other is a parallel connection. In the series conntion, the two piezoelectric layers have opposite polarizatdirections, and an electric field is applied across the tothickness of the bimorph. While connected in parallel,two piezoelectric layers have the same polarization dir

a!Author to whom correspondence should be addressed; [email protected]

1700021-8979/99/85(3)/1702/11/$15.00

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tions and the electric field is applied across each individplate with opposite polarity. In both cases, the applied eltric field is parallel to the polarization direction in one plabut antiparallel in the other plate so that the transverse stcan be converted into bending deformation. The driving voage for a bimorph actuator in the parallel connection canreduced to half the value for a bimorph in the series conntion while remaining the same electric strength and keepthe same actuation capability. However, since the dieleccapacitance in the parallel case is four times that in the secase, power consumption,P51/2CV2 is the same in bothcases. With the same driving voltage, the actuation capabof a parallel bimorph is twice that of a series bimorph. Factuator applications, low driving voltage is usually desable, therefore a bimorph in parallel connection can be us

On the other hand, the unimorph and bimorph bendcan be used as mechanical sensing elements since elecharge or voltage can be generated on the electrodes obimorph or unimorph devices when an external mechanmoment, force or load is applied. As an example, if a forceFis acting perpendicularly on the tip of a cantilever mountseries bimorph, bending deformation occurs, thus tenstresses are induced in the upper plate while compresstresses are induced in the lower plate in the direction oflength of the beam. The distribution of these stresses is nuniform with a maximum value at the top surface~positive!and the bottom surface~negative! and zero at the interface~neutral plane! @Fig. 1~a!#. Consequently, negative chargeare produced at the top electrode while positive chargesat the bottom electrode, since at the top surface,

P35d31T1 , ~1!

whereP3 is polarization in the thickness direction in unitsil:

2 © 1999 American Institute of Physics

IP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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1703J. Appl. Phys., Vol. 85, No. 3, 1 February 1999 Wang et al.

FIG. 1. Structure of bimorph, unimorph and triple layebenders:~a! bimorph bender in series connection,~b!bimorph bender in parallel connection,~c! triple layerbender,~d! unimorph bender, and~e! the RAINBOWdevice.

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C/m2, T1 is stress in the length direction in units of N/m2,and d31 is the transverse piezoelectric coefficient~C/N!; atthe bottom surface,

P35d31~2T1!. ~2!

d31 is a negative value for piezoelectric ceramics suchbarium titanate BaTiO3 and lead zirconate titanatPb~ZrxTi12x)O3 ~PZT!. Therefore, a voltage differencebuilt up across the top and bottom electrodes. For a parconnected bimorph, electric charges with the same signgenerated on the top and bottom electrodes. Electric volis built up between the surface electrodes and the interelectrode of the bimorph. With the same external force,electric charges generated by a parallel bimorph is twicevalue generated by a series bimorph. However the generelectric voltage in the parallel bimorph is half the value pduced by a series bimorph, since the capacitance of theallel bimorph is four times that of the series bimorph,

Vpara5Qpara

Cpara5

2Qseries

4Cseries5

1

2Vseries. ~3!

For sensory applications, high generated voltage isquently desired, therefore the bimorph in series connectiopreferred.

However, since bimorphs are composed of two thinezoelectric ceramic plates, and ceramic is fragile in genereliability is of concern in practical applications. Triple laybenders, which consist of two piezoelectric layers andsandwiched central elastic layer@Fig. 1~d!#, can be used toimprove the mechanical reliability. However, in any case,interfaces in the bimorph, unimorph or triple layer bendare mechanically weak locations. Delamination may ocafter being driven for a period of time under periodic extations due to the relaxation, debonding of the epoxy boing materials.


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Recently, a new piezoelectric bending mode device,so-called reduced and internally biased oxide wa~RAINBOW! was developed by Haertling at ClemsoUniversity.4,5 This device consists of a piezoelectric actilayer and a chemically reduced nonpiezoelectric laformed by a special high temperature processing. Basicthe RAINBOW device is a modified unimorph bender. Whthe piezoelectric layer is driven under electric field, benddeflection will be generated due to the constraint ofchemically reduced piezoelectric inactive layer. On the othand, when an external force or load is applied to the RABOWs, electric charges~or voltage! will be generated in thepiezoelectric layer due to direct piezoelectric effect. Onethe advantages of the RAINBOW device over other bendmode piezoelectric devices is that RAINBOW has a monlithic structure. Moreover, compared with bimorph and umorph devices, large thermal stresses exist in RAINBOdevices. These internal stresses are formed whenRAINBOW is cooled down to room temperature after thigh temperature treatment, since the chemically redulayer and the remaining piezoelectric layer have differthermal expansion coefficients. These residual therstresses are believed to be beneficial to the electromechaperformance of RAINBOW devices.

Although bimorph, unimorph and triple layer bendehave been widely used in many applications and theresignificant amount of literature available detailing their oerational principles and applications, it is found, howevthere is a lack of analysis of the sensing performance of thdevices; for example, no systematic comparison has bmade on these devices. Most previous research has focon actuator applications. In this article, the sensing effectthe above mentioned devices subjected to external mechcal excitations are discussed based on their constitutive etions. Comparison is made between these devices by ta


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1704 J. Appl. Phys., Vol. 85, No. 3, 1 February 1999 Wang et al.

into account their geometrical dimensions and the elasticelectromechanical properties of their components.

II. SENSOR PERFORMANCE ANALYSIS OF ACANTILEVER BIMORPH

The constitutive relations for the cantilever mountedmorph have been derived by Smitset al.,6 and they describethe behavior of the bimorph under static conditions. Foseries connected bimorph bender subjected to the followexcitations: an electric voltageV across its thickness, a unformly distributed external body loadp, an external tip forceF perpendicular to the beam, and an external momentM atthe free end, the generated electrical charge then can bepressed by the following equation:

Q53d31L

t2 M13d31L

2

2t2 F1d31wL3

2t2 p

1«33

X Lw~12k312 /4!

tV, ~4!

whereL, w, and t are the bimorph length, width and thickness, respectively;«33

X is the dielectric constant of the piezoelectric material under a free condition;d31 is the transversepiezoelectric coefficient;k31

2 is the transverse piezoelectrcoupling coefficient, andk31

2 5d312 /s11

E «33X . The total capaci-

tance of the bimorph bender can be obtained by

C5]Q

]V, ~5!

i.e.,

C5«33

X Lw~12k312 /4!

t. ~6!

Therefore, the dielectric constant of the bimorph bender awhole is

«b5«33X ~12k31

2 /4!, ~7!

which is smaller than the free dielectric constant of pieelectric material since in a bimorph bender two piecespiezoelectric material are mechanically bonded together.clamping effect exists in the bimorph actuator because ofconstraint of the two piezoelectric layers at the interfaThis clamping effect reduces the dielectric constant ofbimorph bender.

When only an external moment is applied on the frend of the bimorph, the generated electric charge becom

Q53d31L

t2 M . ~8!

The open circuit electric voltage generated between theand bottom electrodes of the bimorph can be calculated

V5Q

C5

3d31

«33X wt~12k31

2 /4!M . ~9!

When only an external tip force is acting on the tip of tbimorph bender, the generated electric charge in the bimobecomes


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es

ps

ph

Q53d31L

2

2t2 F. ~10!

Therefore, the open circuit electric voltage generated inbimorph bender by the applied external tip force is

V53d31L

2«33X wt~12k31

2 /4!F. ~11!

Therefore, the bimorph bender can be used as a force seIf the dielectric constant«33

X , piezoelectric constantd31 andcoupling coefficientk31

2 of the piezoelectric material and thdimensions of the bimorph are known, by measuringelectric voltage generated across the bimorph bender, theing external tip force can be calculated by Eq.~11!. Obvi-ously, from Eq.~11!, to increase the sensitivity~V/F!, a bi-morph with a largeL/t ratio is preferred.

The external tip force can be related to the tip deflectd by7,8

F53EpId

L3 , ~12!

whereEpI is the flexural rigidity of the bimorph bender. Ishould be noted that hereEp is the Young’s modulus of thepiezoelectric material under the open circuit condition (Ep

51/s11D ), and I is the moment of inertia of the bimorp

bender. For beam structure, the moment of inertia is7

I 5wt3

12. ~13!

Substituting Eqs.~12! and ~13! into Eq. ~11!, we have

V53t2

8L2

d31

«33X s11

D

1

12k312 /4

d. ~14!

Therefore, the bimorph bender can also be used as aplacement sensor.

Similarly, when only an external loadp is applied to thebimorph bender, the generated electric charge in the bimois

Q5d31wL3

2t2 p, ~15!

and the open circuit electric voltage generated in the bimobender in this case becomes

V5d31L

2

2«33X t~12k31

2 /4!p. ~16!

Therefore, a bimorph bender can be used for acoustic seapplications. A bimorph with a largeL2/t ratio, high k31,and a larged31/«33

X ratio has a high sensitivity to the externload p.

By combining Eqs.~9!, ~11! and ~16!, the voltage gen-erated in the bimorph bender under various external mchanical excitations can be expressed as


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V53d31

«33X wt~12k31

2 /4!M1

3d31L

2«33X wt~12k31

2 /4!F

1d31L

2

2«33X t~12k31

2 /4!p. ~17!

Obviously, to have high sensitivities~which can be de-fined asV/M, V/F, or V/p! for the bimorph bender to externamechanical excitations, piezoelectric materials with a hpiezoelectric coefficientd31, high electromechanical coupling coefficientk31, and low dielectric constant«33

X shouldbe used for fabricating bimorph displacement, force or prsure sensors. A figure of merit,

S5d31

«33X ~12k31

2 /4!~18!

can be defined for materials selection for bimorph mechacal sensor design and fabrication. The properties and theure of merit of several commercially available piezoelectmaterials are listed and compared in Table I. For ferroelecceramics such as PZTs, an increase of piezoelectric proties is accompanied by a dielectric constant increase,improvement of the electromechanical coefficientk31 thenbecomes more critical in increasing the figure of merit.

III. SENSOR PERFORMANCE ANALYSIS OFCANTILEVER TRIPLE LAYER BENDERS

The constitutive equations for a symmetrical triple laybender have been derived by Wang and Cross9 recently. Fora series connected symmetrical triple layer bender, whensubjected to the above mentioned excitations, the eleccharge generated can be expressed as

Q56s11

md31~ tm1tp!L

DM1

3s11md31~ tm1tp!L2

DF

1s11

md31~ tm1tp!L3w

Dp1

Lw

2tp«33

X

3S 12D26s11

mtp~ tm1tp!2

Dk31

2 DV, ~19!

where D52s11m (3tm

2 tp16tmtp214tp

3)1s11E tm

3 . The totalthickness of the triple layer bender ist52tp1tm . As a spe-cial case, iftm50, Eq. ~19! becomes Eq.~4!.

The capacitance of the triple layer bender is dependon the thickness of the piezoelectric and elastic layers,

TABLE I. Comparison of piezoelectric materials properties for bendmode sensor applications.

Materials K3X a d31 (310212 C/N) k31

Figure of meritS (31023)

BaTiO3 1900 279 0.21 24.746Soft PZT 3203HD 3800 2320 0.44 210Soft PZT 5A 1700 2171 0.34 211.74Undoped PZT 730 293 0.31 214.75Hard PZT 4 1300 2122 0.33 210.90Hard PZT 8 1000 297 0.30 211.21

a«33X 5«0K3

X and«058.85310212 F/m.


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nt

Ctriple5Lw

2tp«33

X S 12D26s11

mtp~ tm1tp!2

Dk31

2 D . ~20!

For analyzing the dielectric property of the triple laybender, we define

A5s 11

p

s11m 5

Em

Ep, B5

tm

2tp, ~21!

where Em and Ep are the Young’s modulus of the elast~metal! layer and the piezoelectric ceramic layer, i.e.,A is theYoung’s modulus ratio of these two layers andB is the thick-ness ratio of the elastic layer and the piezoelectric layer.capacitance and dielectric constant of the triple layer bencan be written as

Ctriple5Lw

2tp«33

X F12S 123

4

~2B11!2

AB313B213B11D k312 G ,

~22!

« triple5«33X F12S 12

3

4

~2B11!2

AB313B213B11D k312 G . ~23!

To visualize the variation of the dielectric constant of ttriple layer bender with the properties of its components,define a nondimensional parameter,

j512S 123

4

~2B11!2

AB313B213B11D k312 . ~24!

In Fig. 2~a!, j is plotted as a function of thickness ratioB for

FIG. 2. ~a! Normalized dielectric constantz plotted as a function of elasticceramic thickness ratio for different triple layer benders;~b! normalizeddielectric constantj plotted as a function of elastic/ceramic thickness rafor different unimorph benders.


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a series of triple layer benders made from soft PZT andferent elastic layers: silicon, brass, aluminum and acryThe Young’s modulus ratios of these materials to a soft Pceramic are listed in Table II.k3150.44 of soft PZT 3203HD~Motorola, Albuquerque, NM! is used in the calculation.j isactually the normalized dielectric constant of the triple laybender compared with the free dielectric constant of theezoelectric component. The dielectric constant of the trilayer bender is always lower than that of the piezoelecmaterial in the free condition due to the partial clampieffect of each component. A maximum dielectric constanobserved at a certain thickness ratio for the triple labender. The thickness ratio for maximum dielectric constcan be found by

dj

dB50. ~25!

If 1<A,`, the thickness ratio for maximum dielectric costant is given by

Bmax5cosF1

3arccosS 2

A21D G2

1

2, ~26!

and if A<1,

B51

23A2

A211AS 2

A21D 2

21

11

23A2

A212AS 2

A21D 2

2121

2. ~27!

The use of a stiffer elastic layer leads to a lower dielecconstant for the triple layer bender. For comparison pposes, the normalized dielectric constantz of a unimorphbender, which will be discussed in Sec. IV, is shown in F2~b!. It is found that at the same thickness ratioB, the triplelayer bender has a lower dielectric constant than themorph bender.

Substituting Eq.~21! into Eq. ~19!, we have

Q53d31L

4tp2

~2B11!

AB313B213B11M

13d31L

2

8tp2

~2B11!

AB313B213B11F

1d31L

3w

8tp2

~2B11!

AB313B213B11p1

Lw

2tp«33

X

3F12S 123

4

~2B11!2

AB313B213B11D k312 GV. ~28!

If only an external momentM is applied on the free endof the triple layer bender, the generated electric charge i

Q53d31L

4tp2

~2B11!

AB313B213B11M , ~29!

and the electric voltage in the open circuit condition is


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i-

V5Q

C

56d31

«33X wtp

3~2B11!

4~12k312 !~AB313B213B11!13k31

2 ~2B11!2 M .

~30!

Similarly, if only an external tip forceF is applied per-pendicularly at the triple layer bender, the electric chagenerated and open circuit voltage are

Q53d31L

2

8tp2

~2B11!

AB313B213B11F, ~31!

and

V5Q

C

53d31L

«33X wtp

3~2B11!

4~12k312 !~AB313B213B11!13k31

2 ~2B11!2 F.

~32!

As in the case of the bimorph, the open circuit electvoltage of the triple layer bender generated can be relatethe tip displacement. The flexural rigidity of a triple layebender can be obtained by the transformed cross secmethod of the composite beam,7,8,10

EpI c52wtp

3Ep

3~AB313B213B11!. ~33!

Substitution of Eqs.~12! and ~33! into Eq. ~32! leads to

V56d31tp

2

«33X s11

D L2

3~2B11! ~AB313B213B11!

4~12k312 ! ~AB313B213B11!13k31

2 ~2B11!2 d.

~34!

Therefore, the voltage generated in the cantilever mountriple layer bender is proportional to the tip displacement;measuring the open circuit voltage, the displacement canobtained.

If only uniform distributed external loadp is applied, theelectric charge and voltage generated in the triple labender become

Q5d31L

3w

8tp2

~2B11!

AB313B213B11p, ~35!


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V5d31L

2

«33X tp

3~2B11!

4~12k312 !~AB313B213B11!13k31

2 ~2B11!2 p.

~36!

Therefore, to achieve high sensitivity, materials with hid31/«33

X and highk31 should be used to fabricate triple laybenders, i.e., the previously defined figure of meritS shouldbe used for materials selection. To further clarify the senseffect of the triple layer bender to external moment, forceload, the dependence of the generated electric chargevoltage on the geometrical dimension and elastic properof each component will be discussed for the following thrcases.

~1! We can compare the generated electric chargeelectric voltage of a triple layer bender with a bimorpwith the same geometrical dimensions. Since 2tp1tm5tand tm /2tp5B, we have tp5t/2(B11). Substituting tp

5/2(B11) into Eqs. ~31! and ~32!, and comparing withEqs. ~10! and ~11!, we define the following normalizedcharge and voltage parameters:

qtriple5~2B11!~B11!2

AB313B213B11, ~37!

v triple54~2B11!~B11!~12k31

2 /4!

4~12k312 !~AB313B213B11!13k31

2 ~2B11!2 ,

~38!

whereqtriple and v triple are the normalized electrical chargand voltage generated on the triple layer bender when exnal mechanical excitations are applied as compared withbimorph bender. In Figs. 3~a! and 3~b!, qtriple and v triple areplotted against the variation of the thickness ratioB for aseries of triple layer benders. It is found that if a stiff elaslayer~such as silicon! is used, the electrical charge generatincreases with the thickness ratioB initially, reaches a maxi-mum value and then decreases with theB value. However, ifa less stiff elastic layer~such as aluminum! is used, thecharge generated monotonically increases with theB value.WhenB becomes very large, the normalized charge paraeter approaches a limiting value of2/A. The electrical volt-age generated always decreases with theB values. Thereforethe voltage sensitivity of a triple layer bender is alwalower than that of a bimorph bender if the dimensions aresame.

~2! If the piezoelectric layer thicknesstp is a constantwhereas the center elastic layer is variable, from Eqs.~31!and~32! we can define two nondimensional charge and vage parameters

qtriple8 5~2B11!

AB313B213B11, ~39!

v triple8 5~2B11!

4~12k312 !~AB313B213B11!13k31

2 ~2B11!2 .

~40!


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e

d

r-e

d

-

e

-

The variations ofqtriple8 andv triple8 with the thickness ratioBfor differentA values are plotted in Figs. 4~a! and 4~b!. Bothqtriple8 andv triple8 decrease as the thickness ratioB increases.

~3! In some cases, the center elastic layer~substrate!thicknesstm is fixed, while the piezoelectric layer thicknesis variable. The dependence of the generative chargevoltage of the triple layer bender on the thickness ratio athe Young’s modulus ratio can be obtained by substituttp5tm /2B into Eqs.~31! and ~32!,

Q53d31L

2

8tm2

4B2~2B11!

AB313B213B11F, ~41!

V53d31L

«33X wtm

32B~2B11!

4~12k312 !~AB313B213B11!13k31

2 ~2B11!2 F.

~42!

Similarly, we can define the following two nondimensionparameters:

qtriple9 54B2~2B11!

AB313B213B11, ~43!

and

v triple9 52B~2B11!

4~12k312 !~AB313B213B11!13k31

2 ~2B11!2 .

~44!

FIG. 3. ~a! Normalized electrical charge as a function of the thickness rafor different triple layer benders;~b! normalized electrical voltage as a function of the thickness ratio for different triple layer benders.~Here the totalthickness of the triple layer bender is fixed.!


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qtriple9 andv triple9 are plotted as functions of the thickness rafor a series of triple layer benders, as is shown in Figs. 5~a!and 5~b!. It is found that the charge parameterqtriple9 increaseswith B very fast initially and then gradually becomes sarated whenB becomes very large. In addition, a maximuvoltage sensitivity can be obtained by adjusting the pieelectric layer thickness. This is clear if we consider two limiting cases.~i! When theB value approaches zero, i.e., vethick piezoelectric layers are used in the triple layer bendit becomes difficult to bend the device, and the deformatof piezoelectric layer will be very small. Therefore, the eletric voltage generated will approach zero.~ii ! When theBvalue becomes infinitely large, which means the piezoetric layer thickness is infinitely thin, the capacitance of tpiezoelectric layers will be infinitely large. Since the charggenerated on the piezoelectric layers approach a certain v~8/A! whenB→` @Eq. ~43!#, the voltage generated will theapproach zero. Therefore, a maximum generated voltagebe observed between the two limiting cases. By a compson of the different triple layer benders, it is observed tuse of a less stiff elastic layer always leads to a higher chaor voltage sensitivity, since more bending deformationcurs in the piezoelectric layers for a given mechanical extation.

IV. SENSOR PERFORMANCE ANALYSIS OFCANTILEVER UNIMORPH BENDERS

The actuation performance of a unimorph benderbeen analyzed by Steelet al.11 Later, the constitutive equa

FIG. 4. ~a! Nondimensional electrical charge parameterqtriple8 as a functionof the thickness ratio for different triple layer benders;~b! nondimensionalelectrical voltage parameterv triple8 as a function of the thickness ratio fodifferent triple layer benders.~Here the piezoelectric layer thicknesstp isfixed.!


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c-

slue

anri-t

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tions for a cantilever mounted unimorph have been deriby Smits and Choi,12 and they describe the behavior of thunimorph under static conditions. If an external momentM,an external forceF, a uniformly distributed loadp, and anelectric voltageV are applied to the unimorph, then the geerated electrical charge can be expressed by the followequation:

Q56d31s11

ms11p tm~ tm1tp!L

KM

13d31s11

ms11p tm~ tm1tp!L2

KF

1d31s11

ms11p tm~ tm1tp!L3w

Kp

1Lw

tpS «33

X 2d31

2 tm~s11mtp

31s11p tm

3 !

K DV, ~45!

where

K5~s11m !2~ tp!414s11

ms11p tm~ tp!316s11

ms11p ~ tm!2~ tp!2

14s11ms11

p tp~ tm!31~s11p !2~ tm!4, ~46!

s11m ands11

p are elastic compliances of the elastic layer andpiezoelectric layer;tm and tp are the thickness of the elastlayer and the piezoelectric layer; andL andw are the lengthand width of the unimorph bender. The bonding layer thicness is usually very thin and its effect is ignored.

FIG. 5. ~a! Nondimensional electrical charge parameterqtriple9 as a functionof the thickness ratio for different triple layer benders;~b! nondimensionalelectrical voltage parameterv triple9 as a function of the thickness ratio fodifferent triple layer benders.~Here the center elastic thicknesstm of thetriple layer bender is fixed.!


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Similarly, for analyzing the effect of the elastic layer othe sensing performance of the unimorph bender, we de

A5s11

p

s11m 5

Em

Ep, B5

tm

tp, ~47!

and Eq.~45! can be rewritten as

Q56d31L

tp2

AB~B11!

114AB16AB214AB31A2B4 M

13d31L

2

tp2

AB~B11!

114AB16AB214AB31A2B4 F

1d31L

3w

tp2

AB~B11!

114AB16AB214AB31A2B4 p

1Lw«33

X

tpS 1

2k312 AB~11AB3!

114AB16AB214AB31A2B4DV. ~48!

The capacitance of the unimorph bender is

C5Lw«33

X

tpS 12k31

2 AB ~11AB3!

114AB16AB214AB31A2B4D .

~49!

Therefore, the dielectric constant of piezoelectric layer inunimorph bender becomes

«u5«33E S 12k31

2 AB~11AB3!

114AB16AB214AB31A2B4D , ~50!

which is a function of the thickness ratioB and the Young’smodulus ratioA of the elastic layer and the piezoelectrlayers. Obviously, the dielectric constant of a unimorbender is lower than the free dielectric constant of the pieelectric materials because of the constraint of the elalayer on the vibration of the piezoelectric layer.

To visualize the effect of the elastic layer on the dieletric constant of the unimorph bender, we can define

z512k312 AB~11AB3!

114AB16AB214AB31A2B4 , ~51!

and plotz against the thickness ratioB for a series of uni-morphs made up of silicon, brass, aluminum, and acrlayers with a soft PZT ceramic layer, as shown in Fig. 2~b!.The calculated result for the cantilever RAINBOW benderalso included using the Young’s modulus ratio of 0.8513

Obviously, the unimorph bender has a lower dielectric c

TABLE II. Materials properties used in the calculations.

d31 Young’s modulus AMaterials ~m/V! (N/m2) (5Em /Ep)

Soft PZT ceramic 320310212 6.231010¯

Silicon ¯ 1931010 3.05Brass ¯ 1131010 1.77Aluminum ¯ 6.531010 1.05Acrylic ¯ 0.3131010 0.05


e

e

-ic

-

c

-

stant than its free ceramic counterpart. The use of a stielastic layer leads to larger decreases in the dielectric cstant because the stiffer elastic layer can more effectivclamp the piezoelectric ceramic layer.

The clamping effect of the elastic layer on the dielectproperty of piezoelectric materials in either a unimor~monomorph! or a triple layer bender configuration is impotant in device design and performance evaluation. In recyears, PZT thin/thick films have been studied for microatuator and microsensor applications.14–19 The above resultsare useful in considering the dielectric properties of ththick film PZT materials. Thin film PZTs usually havelower dielectric constant than bulk ceramics with the sacomposition, which is often attributed to the grain sieffect.20 The size effect significantly affects the dielectrproperty of the PZT film. However, the clamping effectthe substrate on the pioezoelectric film should also be tainto account. Thin/thick film PZTs are usually depositedplatinized silicon substrates. After deposition, the substratetched~by dry or wet etching techniques21,22! to a certainthickness. If the thickness ratio of the silicon layer and tPZT film is about 1:1 in a cantilever unimorph bender, a 5reduction in dielectric constant can be expected. If the thiness is 5:1, the dielectric constant of the PZT layer willdecreased to about 80% of its free value due to the clampeffect of the silicon substrate, as shown in Fig. 2~b!.

When only an external momentM is applied on the uni-morph bender, the electric charge generated is

Q56d31L

tp2

AB~B11!

114AB16AB214AB31A2B4 M , ~52!

and the electric voltage under the open circuit condition

V5Q

C

56d31

«33X wtp

3AB~B11!

114AB16AB214AB31A2B42k312 AB~11AB3!

M .

~53!

Similarly, when only an external tip force is acting othe tip of the unimorph bender, the electric charge generaon the unimorph becomes

Q53d31L

2

tp2

AB~B11!

114AB16AB214AB31A2B4 F, ~54!

and the open circuit electric voltage produced due toapplied external tip force is

V53d31L

«33X wtp

3AB~B11!

114AB16AB214AB31A2B42k312 AB~11AB3!

F.

~55!


nit

tn

t

ghhthoiproin

anithth

nd.-hec

df

tio.

erap-andthe

di-ur-toptric

ll,ctric-mit-

ses.tedn

nd

ontio

-d-


The generated open circuit electric voltage of a umorph bender can also be related to the tip displacemenEqs. ~12! and ~55!. The flexural rigidity of a unimorphbender is given by10

EpI c5Epwtp

3

12 S 114AB16AB214AB31A2B4

AB11 D . ~56!

Substitution of Eqs.~12! and ~56! into Eq. ~55! leads to

V53d31tp

2

4«33X s11

D L2

AB~B11!

R2k312 AB~11AB3!

R

AB11d, ~57!

where R5114AB16AB214AB31A2B4. By measuringthe electric voltage generated on the unimorph bender,tip displacement can be obtained from the above equatio

When only an external loadp is applied, the electriccharge generated and the open circuit electric voltage inunimorph are

Q5d31L

3w

tp2

AB~B11!

114AB16AB214AB31A2B4 p, ~58!

V5d31L

2

«33X tp

3AB~B11!

114AB16AB214AB31A2B42k312 AB~11AB3!

p.

~59!

Therefore, to achieve high sensitivity, materials with hid31/«33

X and highk31 should be used to fabricate unimorpbenders. Following a similar procedure in discussingtriple layer bender to further clarify the sensing effectunimorph benders to external excitations, the relationshbetween the electric signals generated and the physical perties of the unimorph bender are discussed in the followthree cases.

~1! We can compare the electric charge generatedthe electric voltage of unimorphs with a bimorph bender wthe same geometrical dimensions. If the total thickness ofunimorph bender ist, i.e., tm1tp5t Since tm /tp5B, thentp5t/(B11). Substitutingtp5t/(B11) into Eqs.~54! and~55!, we can define

quni52AB~B11!3

114AB16AB214AB31A2B4 , ~60!

vuni52AB~B11!2~12k31

2 /4!

114AB16AB214AB31A2B42k312 AB~11AB3!

,

~61!

wherequni andvuni are the ratios of the electric charge avoltage generated by the unimorph and bimorph bendersFigs. 6~a! and 6~b!, quni andvuni are plotted against the thickness ratioB for a series of unimorph benders, including tcantilever RAINBOW device. It is found that if a stiff elastilayer ~such as silicon! is used, thequni value increases withthe thickness ratioB initially, reaches a maximum value anthen decreases with theB value. However, if a less stifelastic layer ~such as aluminum! is used, thequni value


-by

he.

he

efsp-g

d

e

In

monotonically increases with theB value. WhenB becomesvery large, thequni value approaches a limiting value of2/A.

From Fig. 6~b!, a maximum charge parametervuni can beobtained for a unimorph bender at a suitable thickness raThis result is clear if we consider two limiting cases.~i!When theB→0, the structure becomes a single cantilevpiezoelectric beam. If external mechanical excitations areplied, one half of the bender is subjected to tensile stressthe other half is subjected to compressive stress. Sincepolarization of the piezoelectric plate is in the thicknessrection from the bottom surface pointing toward the top sface, the same electric charge will be produced on thesurface and on the bottom surface. Therefore, the eleccharge generated and the voltage become zero.~ii ! WhenB→`, the thickness of the piezoelectric layer is very smahence its capacitance becomes very large. Since the elecharge generated approaches2/A, the electric voltage generated (V5Q/C) approaches zero. Therefore, a maximuvoltage sensitivity can be expected between these two liming cases. The thickness ratioB for the maximum voltagesensitivity decreases as the Young’s modulus ratio increa

It has been demonstrated that for a cantilever moununimorph actuator, the maximum generative tip deflectioand blocking force are half the value of that for abimorphactuator if they have the same geometrical dimensions aare driven under the same magnitude of electric field.10 Thethickness ratio for the maximum generative tip deflectiand blocking force is related to the Young’s modulus raby a simple equation,

FIG. 6. ~a! Nondimensional charge parameterquni plotted against the thick-ness ratio for different unimorph benders;~b! nondimensional voltage parametervuni plotted against the thickness ratio for different unimorph beners.~Here the total thickness of the unimorph bender is fixed.!


thueuti-

ethag

obmhng

ph

isthe

umtingyer

, ifstic

ials

. In

tricc

-nd

ph


Bmax5A1

A. ~62!

Here it is interesting to find that as a sensory elementmaximum voltage sensitivity of the unimorph is close to bless than half the value of the bimorph bender. The thicknratio B for maximum voltage sensitivity is very close to bhigher thanA1/A. The difference is very small and is neglgible.

~2! If the piezoelectric layer thickness is fixed, while thelastic layer thickness is variable, in order to visualizevariation of the electric charge generated and the voltwith the thickness ratioB and the Young’s modulus ratioAin the unimorph benders, we may define

quni8 5AB~B11!

114AB16AB214AB31A2B4 , ~63!

vuni8 5AB~B11!

114AB16AB214AB31A2B42k312 AB~11AB3!

.

~64!

In Fig. 7, quni8 and vuni8 are plotted against theB valuesfor a series of unimorph benders. Similarly, by choosingsuitable thickness ratio, a maximum sensitivity can betained. This result becomes clear by considering two extrecases:~i! whenB→0, the electric charge generated and tvoltage will approach to zero, as discussed above for a sipiezoelectric plate;~ii ! whenB→0, i.e., a very thick elasticlayer is used, it becomes difficult to bend the unimor

FIG. 7. ~a! Nondimensional charge parameterquni8 plotted against the thick-ness ratio for different unimorph benders;~b! nondimensional voltage parametervuni8 plotted against the thickness ratio for different unimorph beers.~Here the thickness of the piezoelectric layertp is fixed.!


etss

ee

a-e

ele

bender, and the deformation of the piezoelectric layertherefore very small. The electric charge generated andvoltage will again approach zero. Therefore, a maximgenerated voltage can be observed between the limicases. Also, it is seen that for a given piezoelectric lathickness,tp , the use of a stiffer elastic layer~such as sili-con! can increase the maximum sensitivity~V/M, V/F, orV/p!, and the thickness ratioB for the maximum sensitivityshifts to a lower value asA increases. TheB values for maxi-mum sensitivity are 0.29, 0.375, 0.5, 1.75, respectivelysilicon, brass, aluminum and acrylic are used as the elalayers.

~3! In some cases such as in piezoelectric materbased on microelectromechanical system~MEMS! devicefabrication, the elastic layer~substrate! thickness is fixed,while the piezoelectric layer is deposited~or bonded! to thesubstrate. The thickness of piezoelectric layer can varythese cases, substitutingtp5tm /B into Eqs. ~21! and ~22!,we may define

quni9 5AB3~B11!

114AB16AB214AB31A2B4 , ~65!

vuni9 5AB2~B11!

114AB16AB214AB31A2B42k312 AB~11AB3!

.

~66!

quni9 andvuni9 are plotted in Figs. 8~a! and 8~b! as functions ofthe thickness ratio for a series of unimorphs. The eleccharge increases with theB values, and a maximum electri

-

FIG. 8. ~a! Nondimensional charge parameterquni9 is plotted against thethickness ratio for different unimorph benders;~b! nondimensional voltageparametervuni9 is plotted against the thickness ratio for different unimorbenders.~Here, the thickness of the elastic layertm is fixed.!


f

reth

veiotiv

ee

esdthctihe

ernd

o-at

rpi

eog. Fityt

gethan

gebleh

sserle

is-

eq.

on-

s.,

–11

en,

-


voltage is found at a suitable thickness ratio. The use oless stiff elastic layer leads to high sensitivity. TheB valuesfor maximum voltage sensitivity are 1.32, 1.68, and 2.1,spectively, if silicon, brass, and aluminum are used aselastic layers.

V. SUMMARY

The sensing effects of three piezoelectric cantilebenders subjected to various external mechanical excitathave been analyzed in this article based on their constituequations. The following conclusions are obtained.

~1! Bimorph, unimorph, and triple layer benders have lowdielectric constants than the piezoelectric material itsdue to the clamping effect of each component in thdevices. For unimorph and triple layer benders, theirelectric constants also vary with the thickness ratio ofelastic and the piezoelectric layers. A maximum dieletric constant can be observed at certain thickness rafor the triple layer bender, while for the unimorpbender, dielectric constant decreases with the thicknratio monotonically. The use of a stiffer elastic layleads to lower dielectric constant in both unimorph atriple layer benders.

~2! A figure of merit is defined for the selection of piezelectric materials for bending mode sensor design; mrials with high d31/«33

X and highk31 are preferred forsensor design and fabrication.

~3! Theoretical calculations demonstrate that the bimobender has the highest voltage sensitivity compared wunimorph and triple layer benders with the same gmetrical dimensions. For the triple layer bender, voltasensitivity decreases as the thickness ratio increasesthe unimorph bender, a maximum voltage sensitivclose to half the value of a bimorph can be obtained athickness ratio of aboutA1/A @Figs. 3~b! and 6~b!#.

~4! With the piezoelectric layer thickness fixed, the voltasensitivity of the triple layer bender decreases withthickness ratio, while a maximum voltage sensitivity cbe obtained at a certain thickness ratio@Figs. 4~b! and7~b!#.


a

-e

rnse

rlfei-e-os

ss

e-

hth-

eor

,a

e

~5! With the substrate thickness fixed, maximum voltasensitivities can be achieved by choosing a suitathickness ratio for both the triple layer and unimorpbenders@Figs. 5~b! and 8~b!#.

~6! The variation of charge sensitivities with the thickneratio and the Young’s modulus ratio of the elastic layand the piezoelectric layer for both unimorph and triplayer benders at different conditions~constantt, constanttm and constanttp) have also been compared and dcussed.

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