BSTJ 40: 5. September 1961: The Effect of Driving Electrode Shape on the Electrical Properties of Piezoelectric Crystals. (Lewis, J.A.)

8.2 MB·English

The Effect of Driving Eleetrode Shape
on the Electrical Properties of
Piczoclectric Crystals

By JAC TEWIS.

os the povart separ genoa! cela

snc evil i Torte of 3 reso

Imatonelcapacianee, ane dori lpi rte ineatiptin of the

"fit of electri shape on te seca af resnauere and Be eparonce

Fula ath ergata. Pactoaar etention te aie i fo anne af acta
ty amatl pcm rng an hen sya ate

she vee edit of
2 frupuncice and cae an

Pivalelae expat so
ecilatons, Fg. {shoes type itt rupve fae sch a esa
ful Fig, 2 shows tar eornspording ec low
Ctwencins the expel tchaves lke a eapaeitor, mth a eapacitanes sp
Dsinetly wil to the state eapactanee bel the deving elie

or tsst wri lems fs.

Tews. Drs the panrestie ete, a upp aleraing tere
Fe causes the evatl to vibrate ane, st cova naar rece of
fee vibration, it riven inn weston deal nace iy the apple

vlog

Ts the wsighberbood of wae stn Foes, Me a an 0
the syst ie heey approximated the imple euler
Fig 3 This he quivalent svt emamonly need in he aplietions
Tis a gua apprestmition ove «freuen nage popertinsal tothe
staring bwtween the resonant fequery, ah whieh Uae ae tane fe
invite Cn abectes of dissipation jy ed the atisenant Beene,
ewe thease eda

Tr soll cll suns rl uig hs sping ert to
the vapaeanve matin rs fora trae he rao a a
josie Besta al Parse bav shown hove this mage be dome
I280 er MrT SvEMTAL THe. ZOLA, AHURA ek

A hgh frequents, where many neonsnoes may ofeus within 8 er
ow frrpieney range, she simple eqnvaient eve of Fig 3 may teas
tobe apabiala, However, i always poet al eat in pincpl,
fin! ar okrtnale eonfigaral aie es a ease eee tans of
those reser, A wipe exarpl ie exude eee ei
sion, estiking only svimmetie modes of free vibvarion. Vormet™ tas
sSioon theseetinay yd experientally what tae appnprite eleateode
Supe fs forthe longitdindvradions ow pemeletre hae

Tn the fellowing, ve consider both the question of capacitance mio
nannies und thal af een suppres in an deta fom che
theortral point view. ‘The “nverigation Fe aividad into cbree pets

Tx Sarton IL clugugh Ve eonsider the gxeral pmblem of sti
‘throne ofa pisorostie hed, The eign tbe i the =
tious are the pigorloctie analogs of surousjntegra Hheorans of che
Seal chat, Fosa, for insane, sn Love? We prs, fe expe
tha the material particle dsplvemens eormsqnining (a tve diferent
lwcrntex) muORETERS oF ruegoLuenW eRseRALS 1201

rredos of foe eibmition of peaeleetsie ody are ural [ash as
Mey ase for an ordinary cli Ia

“Te prinipal results of tess scion ee yw cxpressons for cho
‘eqivaten eloiien adnitanen of 47 piosleteie body end for che
notion repalianmes, he ater expres iadiates how the eles
‘mfguzaton mst be chogen to suppress n given ronan,

‘Section VII is devoted to the case of ll pvorlestie coupling, in
whish, 9 4 fins eppronimatinn, we mag separa Ue aehanfeal su
esta! problems complete, has making i powble to obtain more
Caplets han in the gonéral es A sample reset sates that the
Snir rahe 470 36 egal br de equate of that pion of The
init electieal energy shih ys Faro eRe he my divide
iy the prod of the chloe easy the desing Tell and the
slvcineseegy of this mse of res stim. cinta by prodt of
fn saealetons we tenn pper an foc Tao the chess
Snnant feqetes. prose hy a eatnae in rere conigceation
ste upper bol on te razasitanro mtn

"rally i Srolavos VII ee 1X, star ase of che simplifications
plo whee pace bly rn thin ple, i particular obtein-
Sha mn eapheitselston forthe slectonde configunsion ximisag the
Capscitane ratio. Ac sip althoush somewhat rll example,
tre consider the low fournty haiti! wiration of a pceoleet ss
‘ar the example tated fy Wouter in se decal using thi example
to point out sont ot Ne advantages, wl as Ue felt, in the
Iseical pplirasion of Uh atrging tecuiqus,

‘Bevorecommeneing on dionesion of (hos problems, few general
remoris may be upp. Tn ender 0 apy the tebniguespreeently
Drones, soresdhntle detailed infraction sont the modes of ins

‘imation i reaied Cn voll, each dnfermatien see salar
Fly for few spel ruse, Hoe, of let for crystal plates, i ape
Sears shat sith iors say be wba y Phe approssmete thee
Tole teninces eve Ey Malin wl Mis cooorkess* over the

couplet pre infarc, Hae 2y smi ota ee Ce
lust dese, Beperimen:al meted sine an clots probe whieh mey
Ihe more aver Ube efaes othe ryt te ashe metho dovlace
ne Fs Dyk acd ayia, Play sant Tees te abo yield te
ested in‘ormton, Conversely, newsorement of the cleteehadimt-
Ses of give enya’ for worn olssrede eorfiguations and driving
Frequencies ney sie owisderabe ium abo: mode shapes.

‘The esations governing the slay vilnatone uf a pkomeletrin
ceystal at angle Srcpeney aay bs wit inthe fw

Tost ous =O. w
Da =O, 2
‘whore Tse thertoss emo,» the ma dense, the materiel

Aislasement soetor, aa! Dy the elestedplarement vector.
in Te following, we nae Ca-asian tri note lion (xe, [or exemple,
Ieee! in shiek. cominne dance lifer lain wih espe to He
rts coomnctte (ay et} and & repeated eubeenipt adcatez
sramatio, ower al peetble slurs of tbat sutserns. Thus for example,
12) ie ues tho ural quasiteie eet id oyion, stating that he
Aivergise of the sectie ioplacement ete wisiehes, The =pmboin
need for stows, sic, nt ex thos ned by Meson.”

In the ear of ¢ piozoclovsie moditan the ste 7, end lets die
placement Dare given iv tore ot the strain Sy ad elesie Held

fay the liner aati nett ave elaine
Ty = eB eas (i
De wu'Fet eoeSony uw

‘and ace given i rms of th partile displacement wand
letra potential Vy the reicions

By = Bae tech 6)
om

‘The elastic aonteane oven, piezelztre sonst ri, end dcteone
constants «sat sho semmetey rations
pence, Pontes 6 tnanmuxeme eHnerie 1208

We sti lan have aoension to ye the allemate constitutive relations

Tein Bia — ee @
B= BAP. hw ish
Sy — Toe m
0)
ssbene
rink

The net of the parler cel especie hy the phescelon tie
ronplingeoeieiene fy here A” ey Le glen by any ono af the hn
Miserstolnes ratio, ee 18 ae, WML, «4, ee Being mpm

fevvative valies ofthe verenponding materia” sonstants Por all ral
Dismeleteir materi ss compan wich unity

ye pitiele diegcement asl corte poentiel setting the
reading eiatiasin a pleruclririe nly 1, ae compistely eterna
Ty te npeitvaion of ert conditions on ie unbaxe oe Pi
Wie shall sive seeunce tht (ve bly supported i eich Fashion
Ua is sia eee of tions ey

Mou = 0 ay
1264 vin Yet. ste TRERITENE. JOHN PTE 108

fon S, where ay is the outwssd anrmal vector. This assumption is nt
essential ae nl onside asta well the eas of iid or ni)
appr, It coe, hnwever, rake the eiheeennt aigebra tomcat
super

"The body in nana tab dive bi
stant emplitude 1, applied betwaen clectendea S
its mirfoee, Ts

i voltage of eon
fan, plat on

re a2
‘the remaining sztee i cesamed to be feof plating, On Us por
ion of the surf, then, one hs two conctionenequiing tht the po-
tata and nureal elect digpacement bo equal o the potenti aad
ormel clvtrie diplacement sn the externa eld, We may
ster lug fel rts asta formally sop requis that
Reanim Ue preredg sorton hold shaihout sll spe Cesc
tay Shand 8), with the sey lain, aud paaceleote constants at
‘shing dentally ouside 1! and the deere constant caside being
‘hat of free space. nthe following we shal inate tha a inte ie
fo be taken overall epoce by using Me in plane af B. Ty complete the
et af bourdaryeosdtions, we seme het 1 vaiahes a lar isn
fom B.he esteraal bel i ookzded hete oly few frmal hal vm
plete; in amos all pencil problems i sf mele inpertanee

Supper thol se and 1 ste any: Calah'y cantina) veer wa
‘raat functions, dined on # 8. Note shat, bees previo cane
enti, V is actually defined threngant all space. We Stal enste
that Vi eoatinnons actors $e ena infty. Ts geval vous,
‘uss wll oe sti the equstion of the poveeding sot. Hiner
bb introducing susable volume and surface fore, P,, Ts, and vole
sd nace chase, go go 8 anty conatractabrandary val pulon
‘ise by nes fanton

Fis wn rulelale stains nl electra eld cumponenis, Si, By
foun (6) and (9) an atieces env elesiie disgasetnent somponents
‘tam the eonsticntive latins, (3) and (4). The mainline
af volume fren and charge ams then gy Ue rllione

oh = Tay t pslag
whose isan aabitars pve eval Silas, the distributions of

unre fore and charge Sa
Te = Pay
n= Wales

‘vhero nis Hh enswatd noma to Sat (8h jumpin the lets
Gisplacement resto anos S, given by
[D1 = (ia = (Dade

Now suppose we Base any iv pins of meh fneions (a! 7,
fu, V") anal eaealate the sorreponding eee, eletie diane.
rents te. Then she divergence Ubewrer yields

[wnastan = futon as — [ssn an, ay

[vpsone—[v'ntnas tf wiocan, an

front 4 otoran—— [otreas +f

[ovata fracas [mene an,

or wssniple, fe fad slaatione previously considered,
AOE) NE = La, Phya eB EP maa = 8,
‘io ond rsarogerent of 151 and (46) yi

[ona edhe eeu tB TQ, 1D
‘ure Vises a het ehage Ss
oe focus ~ fides as)

[Note that, by our previo convention, oven? aud wan outside
Faunsion (17) eseiily stairs that the niga of the poten een,
mide up of the strain enong nd etvtatie energy, and the lsnotc
ceomgy i equal to tie ener paeing inte 8 theough 8, du to tho ap
ped voltage

Por fae vibrations, wis fm, V3) — CP %atoy (17) pies the
1266 on a. srRAC RARE, OMAK, ARPT 18
Rayleigh cuotient fr the sural freqeney ose

Tminran

‘his relation euggets thet th fect ef pimaelectcty i incre the

vubural frequencies over tbe values they mould ave in an ordinary

‘Shot body wich the elise constants ey", In Sersion VIE ae sal

prove thse this isthe caee fr emall paneer coupling elie
Ti ww mines (18) from (18) sal sesanage toms, 8 btain

[eer sare) rt tan + [oat — vas

=f esere — Biota

“This oquavign ail add sc wes in eta pean Goble prize
fuga ek Furr

ROT RSD! ST — Bue,
sex hs

~ * 420)

= [meat = stot) = vradyan = fovcey veces

‘This inthe anata frm of she ss nsiprralUheosem (20 Hef
43, Chapter VIE, ps 174) lie «peeled bly. Ln words it may be
‘ate fll

‘The dierence tenceon the wveanint sl rcinl vam done by the
force (ancllingKsviereccions} aed aoe th fe oy ating wr
‘He diplomat it putd pro My the Boon st opal te
aiferouenUetven Br ovchenral na cet! sens dane ly the Sores
‘ana charge of the sod dst, ating over the Aaplacements alert
roca by te fr

We nore mtimn to the problem of fered vibeatons, dessin Sor
Vin Tao TL. Tho equstons nd bors enn ene of Mos eto
uncnueh pxSHRTINA om Prnzmaneniue enyenacs 1207

ornpletely deteraine w atique partie digplocement yard pientiad
orp a contain ala rerio 2,2 = 1, 2, 0009 where 01
[elcid futon tc of the fr vibration peobiem, eh P= 0)
ye demote The ahstion o the slic pron with w = 0) by

wee nny satiny all of the peering equations and boundary

outa eseept {13g hy setting
Law, en
versdars, a

vere the asa to be determined to slify (1. Sie

1

= Pog! eau = 0,

ve tis ease (1) ead i

Ele oe )

‘Nowe euppose.we st (42,7) = ORAM (HPA) = Gn, wich
Ga rere it (20). Wirebiain

so thot displucenotssomrsponing to two diesen natural Feguences
tre nethoginal, jos as they ave in aed chsiiy. Thon, wa piying
22) hy and eteting over H, we hin

pied =a [what att = ot [a
se oti 20, hi wh C0) = GAT Co

r=

ow? [vinta =f viptias = vy fits

Thos

25)

pata? ~ eT

ae tan efron 86a note the identities
1268 mm ENT AYEHDON MFCMSTOWE. JOURRAT, AROUND 16h

pot [uracan - [ete on

sPH)a, Cry

Lf vaneingas =f rootSisst
blaine by di potion th vere theo

Yio mm 2qWaLONT REMIND ance

‘singh ayyganai develop i he proves tions, vw angle
‘atter to obtains general expression for the adrittence of the eral
body. The admittance Yi) is the ratio of the total input curren /,
int th ryt ha wll Fase erm Th erent
rae of nope of he lsal hana Qo Ube eartnale 8, wha

a [wives

sar arf thous

wows 1 Seno

v0 =

hove the stnia samen C, is given ay

wan

3)

toate f tases »
[toss
o,_{L) ”

of

we set