AC : ON TEACHING THE OPERATING PRINCIPLES OF PIEZORESISTIVE SENSORS

Transcription

1 AC : ON TEACHING THE OPEATING PINCIPES OF PIEZOESISTIVE SENSOS ichard ayton, os-hulman Institut of Tchnology ichard A. ayton is th Dirctor of th Cntr for th Practic and Scholarship of Education (CPSE) and an Associat Profssor of Mchanical Enginring at os-hulman Institut of Tchnology. H arnd a B.S. in Enginring from California Stat Univrsity, Northridg, and rcivd his M.S. and Ph.D., both in Mchanical Enginring, from th Univrsity of Washington, Sattl. His aras of scholarship includ studnt tam managmnt, assssmnt, ducation, and rmdiation, undrgraduat nginring laboratory rform focusd on studnt larning, data analysis and visualization, and nginring systm dynamics. His work has bn rcognizd with multipl bst-papr awards. H conducts workshops in studnt tam-building, tam-formation and pr valuation, in laboratory assssmnt, and in ffctiv taching. Prior to his acadmic carr, Dr. ayton workd for twlv yars in consulting nginring, culminating as a group had and a projct managr. H is a guitarist and songwritr and a mmbr of th rock band Whispr Down. Thomas Adams, os-hulman Institut of Tchnology Thomas M. Adams is an Associat Profssor of Mchanical Enginring at os-hulman Institut of Tchnology. H arnd a B.S. in Mchanical Enginring from os-hulman Institut of Tchnology, and rcivd his M.S. and Ph.D., both in Mchanical Enginring, from th Gorgia Institut of Tchnology. His aras of xprtis includ hat transfr and nrgy systms, MEMS, and microfluidics. H has workd xtnsivly to bring th fild of MEMS and microscal tchnology to an undrgraduat audinc. H is th rcipint of bst papr awards for both ducational and tchnical paprs and has bn awardd th Dan s Outstanding Taching Award at os-hulman Institut of Tchnology. H is an avid fingrstyl/jazz guitarist, an amatur body-buildr, and a yoga instructor. Amrican Socity for Enginring Education, 200 Pag 5.92.

2 On Taching th Oprating Principls of Pizorsistiv Snsors Abstract W prsnt an approach to taching th oprating principls of pizorsistiv snsors that addrsss many of th limitations of th tratmnts ncountrd in most instrumntation and MEMS txtbooks. Namly, w dirct th prsntation to an undrgraduat audinc rathr than a rsarch-lvl audinc and at th sam tim w avoid ovrsimplifying th dvlopmnt of th principls of opration. To this nd, w mak a discussion of bridg analysis cntral to th dvlopmnt, us a strain-formulation for gag factor and pizorsistor placmnt rathr than th mor common strss-formulation, and kp th associatd physics and mathmatics at an appropriat lvl for sophomor nginring undrgraduats. In so doing, w maintain accssibility and cohrnc throughout. W prsnt svral sts of larning objctivs and stratgis for taching th matrial that can b tailord to suit th nds of a particular cours. Introduction Pizorsistiv snsors ar commonplac th dominant commrcial applications ar pizorsistiv acclromtrs for automotiv airbag dploymnt and pizorsistiv prssur snsors for both automotiv and mdical applications. Bcaus of this widsprad us, particularly in micro-lctro-mchanical systms (MEMS) applications, undrgraduat nginring programs whos larning outcoms includ instrumntation tchnologis gnrally includ an introduction to th basic oprating principls of pizorsistiv snsors. In our opinion, howvr, th xposition of ths principls in popular txtbooks for instrumntation systms and MEMS ar gnrally inadquat authors tnd to ithr ovrsimplify, laving a studnt unawar of oprational dtails, or writ for a rsarch-orintd audinc, making th matrial inaccssibl to undrgraduats. In this papr w prsnt an approach to taching th oprating principls of pizorsistiv snsors that addrsss ths issus. Th distinguishing faturs of our approach ar its accssibility and cohrnc. First, th tchnical contnt and mathmatics ar appropriat for sophomor-lvl nginring undrgraduats. Scond, th tchnical matrial is prsntd cohrntly and compltly, that is, ach stp of th xposition is motivatd by th rsults of th prvious stp. Third, sinc mchanical strain is th physical phnomnon rlating input to output, a strain-formulation is usd for gag factor and for th placmnt and orintation of th pizorsistors instad of th strss-formulation found in most txtbooks. In this papr w shar our approach with th instrumntation ducation community in th hop that its accssibility and cohrnc will hlp improv th taching of th oprating principls of this on important typ of snsor. imitation W hav no studnt larning data to spcifically support our assrtion that th approach w prsnt has gratr cohrnc and accssibility for undrgraduats than any othr. Howvr w do mak th cas in th following sction that our work maks a contribution via a synthsis of th strngths of widly-usd txts. Also, in rcnt yars w hav sn a stady incras in our Pag

3 accrditation program-outcom masurs supportd by our masurmnt systms cours, although this matrial on pizorsistiv snsors would contribut at most two hours of contnt to th cours. Basd on ths broad masurs, w ar satisfid that a prsntation of snsor oprating principls lik th on dvlopd hr contributs to mting our larning objctivs. W plan to dvlop an approach for masuring succss for th nxt offring of th cours. Background In doing our litratur survy for a chaptr on pizorsistiv snsors for our rcntly publishd book 2 on introductory micro-lctro-mchanical systms (MEMS), w sought a tratmnt suitabl for a sophomor-lvl audinc. Though svral authors giv xcllnt dvlopmnts on particular aspcts of th topic, non wr quit what w wantd. In our opinion, th gnral problms ar that authors of txtbooks on masurmnt and instrumntation systms tnd to giv good covrag to mtallic strain gags but only a passing or no rfrnc to smiconductors. (In som indics, th trm pizorsistanc dos not vn appar.) In th cas of MEMS txtbooks, howvr, th authors tnd to giv good covrag to th pizorsistiv ffct, but writ for an advancd audinc. For xampl, masurmnt and instrumntation systms txts by Bckwith t al., Holman 4, Northrop 5, and Whlr and Ganji 6 all giv good dvlopmnts of mtal strain gags and bridg analysis, but only passing rfrnc to pizorsistanc and smiconductors. Txts by Figliola and Basly 7 and Doblin 8 ar similar, and though thy includ brif discussions of pizorsistiv cofficints, thy do so without a cohrnt connction to thir strain-gag matrial. Among ths txts, only Doblin and Northrop us th Taylor sris th propr mathmatical tool, in our opinion for xploring small changs in variabls such as snsor output voltag, lctrical rsistanc, and ara chang. And whil th txts by Holman and by Whlr and Ganji pay som attntion to th dimnsional gomtry associatd with strain, non of ths txts dvlops th gomtry in dtail. Th txt on snsors by Busch-Vishniac 9 is an xcption. Th author provids a fairly complt dvlopmnt of th pizorsistiv ffct and of mtal and smiconductor pizorsistors, though writtn for an advancd audinc. Th bst-known MEMS txts hav th sam dficincy (for our purposs) of tnding to b writtn for an advancd audinc. All covr th pizorsistanc principls of adquatly, somtims going dpr into molcular bhavior than is ndd by our audinc. Standouts in this ara includ authors Maluf 0 and Snturia. Also bcaus of thir advancd audinc, lik Busch-Vishniac 9, ths books tnd to omit or suprficially trat bridg analysis an important topic in our approach. W also find that MEMS txts tnd to covr dformation mchanics in mor dtail than that ndd by our audinc,.g., txts by Snturia and by Madou 2. Nvrthlss, som of ths MEMS txts provid important matrial for our approach that is gnrally missing from convntional tratmnts on masurmnt and instrumntation systms 8. Such attributs includ driving gnral mathmatical modls of pizorsistanc to includ both mtal and smiconductor pizorsistors 9, dvloping a stratgy for placmnt of pizorsistors on th mchanical systm subjctd to strain,2, dvlopmnt of th gomtry of pizorsistors Pag 5.92.

4 undr strain 0,, som dtails of th configuration of pizorsistiv snsors (how thy ar put togthr) 9,0,, and numrical valus for cofficints of pizorsistanc and lastorsistanc 9,0,2. Ths dvlopmnts tnd to us a strss formulation (using th π pizorsistanc cofficints) rathr than th strain formulation w rcommnd hr (using th γ lastorsistanc cofficints). On last small but notabl shortcoming of ths txts, in common with th masurmnt and instrumntation systms prviously dscribd, is thir tndncy to nglct th application of th Taylor sris for xploring small changs in variabls. W draw on th individual strngths of ths rfrncs to synthsiz a complt, cohrnt, and balancd approach to tach th oprating principls of pizorsistiv snsors. W us a systmlvl prspctiv and attmpt to dvlop ach topic to roughly th sam dpth of dtail using mathmatics, physical principls, and nginring analyss suitabl for our sophomor-lvl nginring audinc. Basic principls of opration W trat our snsor as an input-output systm, illustratd in Figur. A mchanical input (prssur, forc, or acclration for xampl) is applid to a mchanical structur of som kind (a bam, a plat, or a diaphragm) causing th structur to xprinc mchanical strain. Small pizorsistors scurd to th structur undrgo th sam mchanical strain, changing thir lctrical rsistancs. Ths rsistors ar wird togthr in a Whatston bridg an lctrical circuit dsignd for dtcting small changs in rsistanc. Th bridg rquirs a constant DC voltag input and producs a masurabl DC voltag output whos magnitud is proportional to th magnitud of th mchanical masurand. constant DC voltag input to bridg mchanical input pizorsistiv transducr mv output from bridg Figur : Systm inputs and outputs for a pizorsistiv transducr. 2 Th primary physical phnomnon that maks this possibl is pizorsistanc: th matrial proprty that th lctrical rsistanc of a matrial changs whn th matrial is subjctd to mchanical dformation, illustratd in Figur 2. An lctrical rsistor is fabricatd from a pizorsistiv matrial and wird into on lg of a bridg circuit. A forc f dforms th matrial, oftn in bnding, causing strain in th matrial, changing its lctrical rsistanc. calling Ohm s law, i, whr is voltag, i is currnt, and is rsistanc, th chang in rsistanc Δ du to mchanical dformation is dtctd via a chang in bridg output voltag Δ. Pag

5 Taching this matrial Figur 2: Concptual schmatic of pizorsistanc: lctrical rsistanc varis with mchanical strain. 2 Our prsntation is adaptabl to a numbr of taching stratgis dpnding on th larning objctivs of a particular cours. Instructors can slct th lvl of dtail appropriat to thir cours and studnts. At any lvl of dtail howvr, w suggst that th lsson covr th major aspcts of th transducr opration shown in Figur, illustrating our concpt of a cohrnt dvlopmnt, that is, ach stp of th xposition is motivatd by th rsults of th prvious stp. Input-output systm, Fig. Pizorsistanc proprty of matrials, Fig. 2 Bridg to dtct d, Fig. 4 d rlatd to mchanical strain, Eq. 4 Signal conditioning Placmnt of pizorsistors,.g., Fig. 7 Modl rlating bridg output to strain, Eq. 29 Gag factor F as a masur of snsitivity of d to strain, Eq. 22 Suggstd larning objctivs Figur : Topic flow chart for prsnting th oprating principls of pizorsistiv snsors. Objctivs st : To tach this matrial in a brif xposition, with th last amount of dtail, on can prsnt just th outlin of th oprating principls shown in Figur. Using th suggstd figurs and quations, an instructor can covr th basic principls of opration in about 5 20 minuts, providing studnts an ffctiv ovrviw without dtaild modling or analysis. Appropriat larning objctivs for this sort of lsson might includ: ist th inputs and outputs of th snsor. Dfin pizorsistanc. Explain th purpos of th bridg. Explain how a pizorsistiv snsor works. Dscrib th purpos of signal conditioning. This 5 20 minut xposition is th approach w us in taching th mini-labs of our masurmnt systms cours. In a mini-lab, a convntional lctur is rplacd with a short lctur commingld with a guidd hands-on xprinc in a 2-hour studio format 7. Whil w hav not masurd th fficacy of th mini-lab spcifically compard to a traditional lcturhomwork format, our masur of program outcoms supportd by this cours has stadily Pag

6 improvd ovr th yars. W assrt, thrfor, that vn a brif xposition of a snsor s oprating principls can mt cours larning objctivs if th prsntation is cohrnt and accssibl. Objctivs st 2: To tach this matrial in gratr dtail, on might tach th bridg analysis and parts of th cas study in dtail in class with othr aspcts assignd as homwork. Appropriat larning objctivs for this sort of lsson might includ all of thos listd abov plus: Givn a manufacturr s spcification sht, dtrmin th snsitivity of th snsor. Driv a modl for a half-bridg configuration. Givn rprsntativ valus of gag factor and rprsntativ strss and strain valus in th rgions of maximum strss for a prssur diaphragm, dtrmin th snsitivity of th prssur transducr in mv/v/mpa. Explain th possibl purposs of signal conditioning in a pizorsistiv snsor. Objctivs st : In a lsson at th highst lvl of dtail, on might tach most of th matrial prsntd in this papr, laving som contnt as xrciss for th studnt. For xampl, on could tach th chang-in-gomtry matrial for a rctangular cross sction, but lav th circular-cross sction matrial as an xrcis for studnts. Appropriat larning objctivs for this sort of lsson might includ all of thos listd abov plus: Driv th chang-in-gomtry ΔA/A xprssion for a conductor of circular cross-sction. Givn a circuit of a Whatston bridg with a null-offst or tmpratur compnsation, driv th bridg modl. Givn a diaphragm gomtry and locations and magnituds of maximum strss and strain, dtrmin th position and orintation th pizorsistors and thir wiring configuration in th bridg to produc maximum snsitivity. Givn appropriat matrial proprtis, stimat th gag factor of a pizorsistor. Givn a configuration of p-typ Si rsistors on a squar diaphragm whr two rsistors ar locatd sid by sid to sns th maximum strss σ C, and givn dsign valus and strss-strain valus (lik in th cas study that concluds th papr), dtrmin th nw bridg quation, th valu of Δ/ for ach rsistor, th valu of Δ o / i, and th snsor snsitivity. Th mini-lab In our junior-lvl masurmnts cours, w prsnt this matrial in a 2-hour mini-lab on prssur snsors. W spnd th first 5-20 minuts dvloping th oprating principls to mt th first st of larning objctivs dscribd abov. Th rmaindr of th lab priod is a handson, collaborativ activity in which th studnts xplor th opration of th transducr by informal xprimntation, dtrmining th snsor s snsitivity and rsolution followd by an applications qustion and an lmntary uncrtainty analysis. Th mini-lab concpt is simpl on: instad of lcturing about a transducr or class of transducrs, rplac th lctur with a hands-on activity with on particular transducr from th class,.g., a load cll, a prssur transducr, a potntiomtr, tc. 8 W ask th studnts, working in pairs, to connct th transducr to appropriat input and output dvics, but w do not provid dtaild procdurs. Our goal is to hav studnts intract with th snsor in a mod w might call guidd discovry studnts ar givn gnral guidlins about what thy ar to find Pag

7 and th profssor circulats and answrs qustions as thy work. Studnts dtrmin charactristics such as snsitivity, rang, and rsolution, compar thir informal xprimntal findings to th manufacturr s spcifications (giving thm practic at rading and intrprting spcifications), and answr qustions rgarding an application, an uncrtainty analysis, and at last on qustion rquiring critical thinking. Th prssur mini-lab apparatus is illustratd in Figur 4. Studnts assmbl th apparatus, connct th transducr to a powr supply and a digital multimtr, and answr ths qustions:. Obtain radings for an informal calibration curv and stimat snsitivity. Quantitativly compar th xprimntal snsitivity to th xpctd snsitivity you found in th prlab. 2. mov th plastic tub from port 2 and connct it to port, laving port 2 opn to th atmosphr. Discovr if th transducr can b usd this way. Explain.. Idntify a way to incras th snsitivity of th transducr (not th manomtr). B spcific and quantitativ. 4. Dtrmin th rsolution of th masurmnt systm. Explain your approach. 5. Suppos this prssur transducr wr connctd to a Pitot-static tub to masur airspd (s th txt, p. 4-2, assum C ). Excitation voltag is 0 V and air dnsity of slug/ft. What is th maximum airspd this transducr could masur? 6. For an airspd of 88 ft/sc (60 mph), dtrmin th uncrtainty in th airspd. Nglct any uncrtainty in dnsity. Show all your work. This concluds our discussion of th contxt in which w tach transducr oprating principls gnrally. W turn now to th dtaild modling of th pizorsistiv snsor in particular. port, opn to atmosphr port 2 t tubing diffrntial prssur transducr clip opn to atmosphr bllows for applying prssur manomtr: for th rfrnc prssur masurmnt Figur 4: Apparatus for th prssur mini-lab. Th diffrntial prssur transducr is a pizorsistiv typ. Dtaild modling Considr th pizorsistiv snsor as th input-output systm shown in Figur. W bgin by modling th bridg to obtain Δ as a function of Δ. This rsult motivats th nxt stp rlating Δ to mchanical strain, producing a modl of rsistanc as a function of gomtry and rsistivity. Th gomtry trm motivats a dvlopmnt of th gag factor for mtal rsistors Pag

8 (strain-gag typ snsors) and th rsistivity trm motivats a discussion of gag factor for smiconductor rsistors (MEMS snsors). W us gag factor to obtain a modl rlating output voltag to input strain. Th strain-formulation of th snsor modl is usd to motivat a discussion about th physical placmnt and orintation of pizorsistors on th mchanical structur. Th strain formulation is, to our knowldg, uniqu. Bridg analysis A Whatston bridg is an lctrical circuit that nabls th dtction of small changs in rsistanc. Bckwith and Holman 4 dvlop modls of svral diffrnt typs of Whatston bridgs. Figur 5 illustrats th bridg w xamin: a voltag snsitiv, dflction typ circuit with a constant voltag DC input, and idal rsistancs (i.., no impdanc lmnts) in th arms of th bridg. pizorsistiv transducr constant DC voltag input to bridg + i A 2 i m mchanical input causs a chang in on or mor of th rsistancs B i 2 i i i 4 D + o mv output from bridg 4 C Figur 5: Whatston bridg circuit insid a pizorsistiv snsor. On or mor of th rsistors through 4 may b pizorsistors affctd by th mchanical input. 2 Th input voltag i is supplid by a constant DC sourc. Th four arms of th bridg ach contain a rsistor, through 4 (at last on of which is mad of a pizorsistiv matrial). Th output voltag o is th diffrnc btwn th voltags at nods A and C, that is, o. () A To obtain an xprssion for th output, thrfor, w nd xprssions for th voltags at A and C. Using nodal analysis (applying Kirchhoff s currnt rul) at thos two nods, C at A : i + im i2, at C : i + im i4. (2) Th output voltag is usually masurd by a voltmtr with a high rsistanc, making th currnt i m small nough to b ngligibl. Thus, i, i i4. () i 2 Pag

9 Th bhavior of rsistors is dscribd using Ohm s law, Δ i, whr Δ is th diffrnc in voltag across th two nds of th rsistor. Substituting for ach currnt in () yilds, A D B 2 A, B C C 4 D. (4) Th voltag at B is th input voltag and th voltag at D is th rfrnc zro voltag. Making ths substitutions yilds A i A, 2 i C C 4. (5) arranging ths two quations to solv for th voltags at A and at C and substituting into () yilds th input-output rlationship. (6) 4 o i Th first conclusion w can draw from this analysis so far is th condition for bridg balanc. If thr is no mchanical input to th snsor, w would lik to hav zro voltag output. Zro output occurs if th parnthtical trm in (6) is zro, that is, + + W can rarrang this rlationship to obtain (7). (8) This condition can b mt by th dsignr in mor than on way. For instanc, in what is calld a full-bridg, all four rsistors ar idntical pizorsistors with idntical valus of. Or in a halfbridg, and 2 could b idntical pizorsistors with and 4 bing idntical fixd rsistors. In ithr cas, slcting, 2,, and 4 such that (8) is tru, w hav o 0 whn thr is no mchanical input to th transducr. Continuing our bridg analysis, w know that th snsor is dsignd such that th rsistors undrgo a small chang in rsistanc Δ that producs a small chang in output voltag Δ o. Small changs lik ths ar radily (and rigorously) xprssd mathmatically using a Taylorsris xpansion about th balancd condition. If w assum that all th rsistancs ar subjct to chang du to applid strain, thn th Taylor sris has th form, o o o o Δo Δ + Δ2 + Δ + Δ4 + highr ordr trms. (9) 2 W assum that th highr-ordr trms ar ngligibl bcaus thy involv products and intgr powrs of Δ. Th magnitud of Δ is small and so th products and powrs of Δ ar smallr still hnc ngligibl. 4 Pag

10 W obtain th partial drivativ trms from (6), substitut thm into (9), nglct th highr-ordr trms, and divid by i to obtain Δ i o 2 4 Δ Δ + Δ Δ ( + ) ( + ) ( + ) ( + ). (0) This gnral form of th bridg modl accommodats any combination of valus of th rsistancs through 4. To dvlop insight into th dsign of th snsor, howvr, it hlps at this point to study a particular dsign th full-bridg with four idntical pizorsistors, that is, 2 4. With ths substitutions, (0) bcoms Δ i Δ 4 Δ Δ + Δ o 2 4. () As a point of pdagogy: th bridg analysis rlats th lctrical input i, th lctrical output Δ o, and th rlativ chang in rsistanc Δ/ of th pizorsistiv matrial. Our nxt logical stp, thrfor, is to xamin how th rsistanc trm Δ/ rlats to th mchanical proprtis of th pizorsistiv matrial. lating lctrical rsistanc to mchanical strain. Givn a physical matrial of lngth and constant cross-sctional ara A, its lctrical rsistanc is givn by ρ, (2) A whr ρ is th matrial s rsistivity. Th gomtry of th rsistor is illustratd in Figur 6 for both a rctangular cross-sction (as in a thin plat) and a circular cross-sction (as in a thin wir). uniform cross-sctional ara A rctangular conductor circular conductor Figur 6: Gnralizd rsistor gomtry. 2 A chang in rsistanc Δ is producd by changs in any of th thr quantitis rsistivity, lngth, or ara. Small changs ar modld using a Taylor-sris xpansion. From (2) w obtain Δ Δρ + Δ + ΔA + highr ordr trms. () ρ A W obtain th partial drivativ trms from (2), nglct th highr-ordr trms of th sris, and divid by to obtain Δ Δρ Δ ΔA +. (4) ρ A Pag

11 Th first trm on th right-hand sid of (4) rprsnts th pizorsistiv proprty a chang in rsistanc du to a chang in rsistivity Δρ du to th application of mchanical strss. Th scond two trms rprsnt changs in rsistanc du to changs in th gomtry (lngth and ara) of th rsistor. In smiconductors, th first trm dominats. In mtals, th gomtric trms dominat 8,9. In th nxt two sctions w dvlop th modl for both mtals and smiconductors. Gag factor for mtal rsistors. In mtals, th rlativ chang in rsistanc Δ/ is du primarily to th changing gomtry. In this cas w can nglct th Δρ/ρ trm in (4) and modl th rlativ chang in rsistanc using Δ Δ ΔA. (5) A Pizorsistors mad of mtals ar most commonly ncountrd in th form of strain gags. Strain gags ar fabricatd to b most rsponsiv to uniaxial strain. Thus our analysis of th gomtric ffct in (5) is basd on uniaxial strain applid to an isotropic matrial. Whn a matrial is subjctd to strss in on dirction (w ll us a rctangular cross-sction to illustrat), th lngth of th matrial incrass by a small amount Δ and th hight h and width w dcras by th amounts Δh and Δw, as illustratd in Figur. Strain ε in th dirction of th applid strss is dfind as th ratio of th chang in lngth to th original lngth, that is, ε Δ/. original ara A, hight h and width w nw ara, hight hδh and width wδw + Δ applid strss σ Figur 7: Dimnsional changs du to applid strss in on dirction. 2 Th cross-sctional ara of th unstrssd spcimn is A hw. A small chang in ara ΔA is modld using (yt again) a Taylor-sris xpansion, A A ΔA Δh + Δw + highr ordr trms. (6) h w Th highr-ordr trms involv products and intgr powrs of Δh and Δw that ar ngligibl in magnitud compard to th first-ordr trms. Nglcting thm and dividing by A yilds ΔA Δh Δw +. (7) A h w Pag 5.92.

12 Th transvrs strain trms Δh/h and Δw/w for mtals and cubic crystals can b xprssd in trms of Poisson s ratio ν and th axial strain ε by Δh νε and h Δw νε, (8) w whr th ngativ signs indicat that both h and w dcras with positiv axial strain ε. Substituting (8) in (7) and simplifying yilds ΔA 2νε. (9) A To show that this rlationship is not uniqu to th rctangular conductor, w outlin th analysis for a circular conductor. Cross-sctional ara A for a circular conductor is givn by A πr 2, whr r is th conductor radius. Applying a Taylor sris as bfor yilds ΔA Δr 2. (20) A r adial strain is a function of axial strain, Δr/r νε. By substitution in (20) w obtain ΔA 2νε. (2) A This rsult (2) for a circular conductor is idntical to rsult (9) for a rctangular conductor. turning to our chang-of-rsistanc rlationship (5), substituting (2) and ε Δ/ yilds Δ ( + 2ν ) ε. (2) This rlationship modls th chang in rsistanc in mtals usd in strain gags as a function of a matrial proprty, Poisson s ratio, and th applid mchanical input, uniaxial strain. Snsitivity, in a snsor, is th ratio of chang in output to chang in input. For th rsistiv snsing lmnt th output is th rlativ chang in rsistanc Δ/ du to th input strain ε. Thus th snsitivity of th lmnt dpnds on th ratio of Δ/ to ε. In pizorsistiv applications, this ratio is calld th gag factor, F, a dimnsionlss numbr dfind as Δ F. (22) ε Applying this dfinition to (2), w obtain th xprssion for gag factor for mtal strain gags F + 2ν. (2) Gag factor is usd to compar th prdictd prformanc of candidat matrials as pizorsistors (highr F mans gratr snsitivity) and to guid us in positioning th pizorsistors on th mchanical structur that is subjctd to th input strain a topic to which w rturn aftr dvloping an xprssion gag factor for smiconductors. Pag

13 Gag factor for smiconductor rsistors. In smiconductors, th rlativ chang in rsistanc Δ/ is du primarily to changs in rsistivity, not gomtry. Nglcting th gomtric trms Δ/ and ΔA/A in (4) yilds Δ Δρ. (24) ρ For a pizorsistor subjctd to longitudinal and transvrs strsss, th rsistivity chang is Δρ π σ + π Tσ T. (25) ρ whr π and π T ar th longitudinal and transvrs pizorsistanc cofficints of th matrial. In practic, longitudinal mans in th dirction of currnt and transvrs mans prpndicular to th dirction of currnt. Altrnativly and slightly mor usful in our discussion of gag factor rsistivity can b xprssd in trms of strain, Δρ γ ε + γ TεT. (26) ρ whr γ and γ T ar th longitudinal and transvrs lastorsistanc cofficints of th matrial. Th two modls, (25) and (26), ar rlatd by th rlationships btwn strss and strain in th longitudinal and transvrs dirctions. Howvr, bcaus pizorsistiv smiconductor matrials ar gnrally anisotropic, th linar Young s modulus rlationship, σ Eε, dos not apply hr. Substituting (26) in (24) w obtain Δ γ ε + γ TεT. (26) This rlationship modls th chang in rsistanc in smiconductors usd in pizorsistiv applications as a function of matrial proprty, th lastorsistanc cofficints, and th applid mchanical inputs, longitudinal and transvrs strains. Dividing by ε, w obtain an xprssion for th gag factor, F γ + γ ε T T. (28) ε Both th lastorsistanc cofficints and th strains dpnd on th orintation of th rsistor. Gag factors for diffrnt matrials li on th approximat rangs givn in Tabl (s, for xampl 9, ). Th tabl shows that smiconductors ar mor snsitiv (highr F) that mtal strain gags. Th trad-off is that smiconductors ar mor brittl and hav lowr valus of fractur strss than mtals. Pag 5.92.

14 Tabl : Gag factors for mtals and smiconductors 2 F 5 for mtals, 5 F 50 for crmts (cramic-mtal mixturs) 70 F 5 for silicon and grmanium Physical placmnt and orintation of pizorsistors. Using th dfinition of bridg factor (22), w rarrang trms to obtain an xprssion for th rlativ rsistanc chang: Δ/ Fε. This xprssion holds for both mtal and smiconductor pizorsistors. Substituting Fε for ach Δ/ trm in th bridg modl () yilds an xprssion for bridg output in trms of strain only, assuming F is known and constant, Δ F ( ε ε + ε ε ) o 2 4. (29) i 4 By introducing th concpt of gag factor, this modl unambiguously rlats th snsor s basic mchanical input (strain) to th snsor s basic lctrical output (Δ o ) for both mtal and smiconductor pizorsistors. Th modl shows too that for a givn lvl of strain, a highr gag factor (or snsitivity) producs highr output voltag from th bridg. Th modl also lads us to insights rgarding placmnt of pizorsistor on th mchanical structur of th snsor lmnt. Equation (29) indicats that if th strains ar qual in magnitud and hav th sam sign, thn th bridg will produc zro output not a usful outcom. To obtain a usful voltag output, w thrfor position rsistors 2 and 4 to undrgo strain in th opposit sns to th strain of rsistors and. For xampl, considr th pizorsistiv acclromtr shown in Figur. Th acclromtr housing is scurd to a body undrgoing th acclration w want to masur th vibration of a machin, for xampl. In rspons to th acclration of th body, th sismic mass acclrats causing th cantilvr bam to bnd. Th bnding applis strain to th pizorsistors. Physical placmnt and orintation of th rsistors is influncd by thr considrations: th bridg configuration (which rsistor is wird into which lg of th bridg), th rgion of maximum strss, and th dirction of th strain to which w want th rsistor to rspond. rsistors orintd to rspond to longitudinal strain ε cantilvr bam sismic mass acclration snsitiv axis 2 4 (undr) snsor housing rgion of maximum strss Figur 8: Placmnt of rsistors on a bam-typ pizorsistiv acclromtr snsitiv to acclration in on dirction i 4 bridg configuration + o Pag

15 Th bridg configuration in this xampl is dtrmind by th sign of th strain in ach rsistor location. If th bam bnds downwards, th two rsistors on top of th bam ar in tnsion (a positiv ε for p-typ smiconductors) at th sam tim th rsistors undrnath th bam ar in comprssion (a ngativ ε for p-typ smiconductors). If th acclration changs dirction, th rgions of tnsion and comprssion swap. For th strains to add up and not cancl ach othr out in (29) w plac rsistors and sid by sid on on sid of th bam (hr both ar shown on top of th bam) and rsistors 2 and 4 on th othr sid of th bam. Thn ε and ε hav th sam sign, opposit that of ε 2 and ε 4, and th bridg producs a total positiv or a total ngativ output voltag Δ o. Th four rsistors ar placd at th bas of th bam bcaus this is th rgion of maximum strss. Any othr placmnt rducs th snsitivity of th snsor. Th rsistors ar orintd to b rsponsiv to longitudinal strain, that is, th dirction along th longitudinal axis of th cantilvr bam. Transvrs ffcts ar ngligibl in this configuration. Ths thr factors bridg configuration, location and orintation of maximum strss and strain, and th orintation of th rsistors that rspond to th strain ar usd to dtrmin th optimum placmnt and orintation of pizorsistors in a transducr. Dvic cas study: a pizorsistiv prssur snsor In this sction w study a pizorsistiv prssur snsor with oprational paramtrs similar to thos sn in automotiv applications. W us paramtrs rprsntativ of a class of prssur snsors similar in siz and us to th Omga PX409 prssur transducr. To giv our numrical xampl som smblanc to rality, w us physical paramtrs from th litratur 4,5. stainlss stl housing lctrical connction prssur inlt 80 mm Extrior viw silicon oil rsrvoir stainlss stl diaphragm structur to support th Si-wafr (00) Si-wafr prssur inlt i o Cross-sctional schmatic signal conditioning compartmnt Figur 9: A commrcially availabl pizorsistiv prssur snsor. 2 (Basd on th Omga PX409 prssur transducr 6.) Pag

16 Th basic configuration of this pizorsistiv snsor is shown in Figur 9 (adaptd from 6 ). Th snsor housing is stainlss stl, about 80 mm long, with a prssur fitting at on nd and an lctrical connction for input and output voltag at th othr nd. Snsors of this typ can typically masur prssur rangs from kpa ( psi) to MPa (5000 psi). Th DC xcitation voltag is usually btwn 5 0 V. Whn slcting a particular snsor mak and modl, on usually has a choic of lctrical output signals: 0 00 mv, 0 5 V, or 4 20 ma. Th cross-sction shows that th working fluid whos prssur w want to masur ntrs th snsor through th prssur inlt fitting and imposs prssur on th stainlss stl diaphragm. A small volum of silicon oil transfrs th prssur from th stainlss-stl diaphragm to th (00) Si-diaphragm. Th inducd strss and strain of th wafr is dtctd by pizorsistors on th wafr in a bridg configuration, producing an lctrical output proportional to th inlt prssur. In this xampl, w xamin a prssur snsor with a 0 MPa (45 psi) full scal input, 0 00 mv full scal output, a 0 VDC xcitation, and p-si pizorsistors. W bgin by discussing th mchanical proprtis of th diaphragm. In this xampl, th (00) Si-diaphragm is a squar,.2 mm on ach sid, 80µm thick, orintd with th <0> dirctions biscting th squar as shown in Figur. Whn prssur is applid from blow th largst upwards dflction of th diaphragm is at th cntr of th squar. Consquntly, th largst strss σ C, locatd midway along ach dg, is dirctd towards this cntr. Strss σ B, at th sam location, is dirctd paralll to th dg, along th boundary of th diaphragm. Basd on publishd xprimntal rsults for a squar diaphragm of this typ and siz, w stimat σ C 45.0 MPa and σ B 22.5 MPa 4. Th strains in th sam two dirctions ar ε C 52 µε and ε B 7 µε, whr th symbol µε mans microstrain ( microstrain 0 6 strain). Th maximum strss σ C is lowr than th fractur strss of th Si-diaphragm (60 MPa) by safty factor of 8. <0> location of maximum strss and strain (all four dgs) location of maximum dflction (00) Si-diaphragm σ C σb σ C.2 mm ach sid σ B <0> 80 µm thick Figur 0: Orintation of th Si-diaphragm and points of maximum strss and strain. 2 Th four p-typ pizorsistors ar placd at th four locations of maximum strss and strain. From th bridg modl (29), w know w want rsistors and to undrgo a positiv strain at th sam tim rsistors 2 and 4 undrgo a ngativ strain. W accomplish this by orinting th rsistors as shown in Figur (adaptd from,2 ). sistors and ar orintd with thir Pag

17 longitudinal axs in th dirction of maximum strss (σ C ); ε and ε will b positiv. sistors 2 and 4 ar orintd with thir transvrs axs in th dirction of maximum strss (σ C ); ε 2 and ε 4 will b ngativ. pizorsistor conductors to bridg <0> 2 σ σ T i currnt runs longitudinally σ i 4 σ T 2 + i 4 bridg configuration + o (00) Si-wafr <0> Figur : ocation and orintation of four pizorsistors on th squar diaphragm. 2 call, from (26), that th chang in rsistanc Δ/ is a function of th longitudinal and transvrs lastorsistanc cofficints and strain. W group th analysis blow into two columns: th lft column for rsistors and and th right column for rsistors 2 and 4. W us (24) and (26) for ach cas: Δ, Δ γ ε + γ Tε, 4 T, γ ε + γ TεT 2. (0) Comparing Figur to Figur, w s that th longitudinal strain of rsistors and is ε C (towards th cntr of th diaphragm) whil th longitudinal strain of rsistors 2 and 4 is ε B (along th boundary of th diaphragm). Substituting for ε yilds Δ Δ 4 2. (), γ ( ε C ) + γ Tε, T, γ ( ε B ) + γ TεT whr parnthss hav bn usd to highlight th substitutions. Th convrs is tru for th transvrs strains. Th transvrs strain of rsistors and is ε B (along th boundary) whil th transvrs strain of rsistors 2 and 4 is ε C (towards th cntr). Substituting for ε T yilds Δ, γ ( ε ) + γ ( ε ),, γ ( ε ) + γ ( ε ) C T B Δ 4 2 B T C. (2) Th lastorsistanc cofficints of th rsistors in th <0> dirction ar found in publishd tabls 9 to b γ 20 <0> and γ T 54 <0>. Pag

18 W r rady to obtain a numrical valu for th two Δ/ trms: Δ Δ, 2,4 6 6 ( 20)( 52 0 ) + ( 54)( 7 0 ) ( 20)( 7 0 ) + ( 54)( 52 0 ) () Thus, as w d plannd, rsistors and s a positiv chang in rsistanc (of about 2%) and rsistors 2 and 4 s a ngativ chang in rsistanc (of about %). As a rality chck on th analysis to this point, w comput th gag factor F for rsistors and, orintd longitudinally in th dirction of maximum strss (σ C ). Th gag factor is computd blow, with a rsult that lis in th xpctd rang (70 F 5) for silicon. Δ F ε C (4) Nxt, w dtrmin th ratio of lctrical output to input at th full prssur load of.0 MPa using th full-bridg modl and th valus of Δ/, Δ i o Δ Δ mv/v. 2 Δ + [ 9. ( 0.2) + 9. ( 0.2) ] Δ 4 0 With a 0 VDC xcitation voltag, th bridg full load output is 47 mv. Sinc th full load is.0 MPa, th snsitivity η is givn by. (5) 47 mv η MPa. (6) 47 mv/mpa (.0 mv/psi) To mt our goal of a 0 00 mv full scal output, w add an amplifir circuit to attnuat (rduc) th 47 mv bridg output by a factor of 00/ This factor is calld a gain K and is oftn rportd in th units of dcibls (db) as follows, 00 K 20log0 47. (7).2 db Pag

19 Th simplst op-amp txtbook circuit that provids positiv attnuation compriss two invrting amplifirs in sris, as shown in Figur. W hav to slct th rsistors f and o to obtain th dsird gain of (or.2 db). f f voltag signal from bridg o + o + transducr output Figur 2: Simpl non-invrting attnuation circuit. 2 Th gain of an invrting amplifir is th ratio of f to o, that is, K f. (8) and thrfor th gain of th two invrting op-amps in sris is givn by o 2 f f f K +. (9) o o o W writ a computr program to sort through th combinations of radily obtainabl rsistors looking for valus of K clos to W find that slcting f 6.2 Ω and o 7.5 Ω producs a gain K 0.68, which is within 0.2% of th dsird valu, clos nough to b usful. This amplifir circuit is placd in th signal-conditioning compartmnt of th snsor. Our lctrical output-to-input ratio now includs th amplifir gain, Δ i o K Δ Δ mv/v. 2 Δ + Δ [ 9. ( 0.2) + 9. ( 0.2) ] 4 0 With a 0 VDC xcitation voltag, th bridg full load output is 00 mv and th snsitivity η is now, as dsird,. (40) η 00 mv/mpa (0.69 mv/psi). (4) As a point of pdagogy: w hav purposfully slctd an ovrsimplifid attnuation circuit to kp th analysis accssibl to our audinc. In taching this matrial to our studnts w not that ral signal conditioning is gnrally mor complx and accomplishs mor than just stting a gain. Snturia, for xampl, shows a schmatic of a signal conditioning circuit for a Motorola prssur snsor, though without mathmatical analysis. In summary, this xampl illustrats how ach of th important factors bridg configuration, location and orintation of maximum strss and strain, and th orintation of th rsistors that rspond to th strain affct th dsign and snsitivity of a pizorsistiv transducr. Pag

20 Conclusion Th goal of this papr is to prsnt an approach to taching th oprating principls of pizorsistiv snsors that addrsss th issus w v ncountrd in most instrumntation txts of ithr ovrsimplifying th prsntation or writing for a rsarch-lvl audinc. In our opinion, th lvl of dtail w provid and basing ach stp of th dvlopmnt on principls familiar to th sophomor-lvl undrgraduat nginring studnt mts this goal. First, th tchnical contnt and mathmatics ar appropriat for sophomor-lvl nginring undrgraduats. Scond, th tchnical matrial is prsntd cohrntly, that is, ach stp of th xposition is motivatd by th rsults of th prvious stp. Third, sinc mchanical strain is th physical phnomnon rlating input to output, a strain-formulation is usd for gag factor and for th placmnt and orintation of th pizorsistors instad of th strss-formulation found in most txtbooks. W shar this approach with th instrumntation ducation community in th hop that its accssibility and cohrnc will hlp improv th taching of th oprating principls of this on important typ of snsor. Bibliography. Bryzk J, oundy S, Bircumshaw B, Chung C, Castllino K, Stttr J, and Vstl M (Mar/Apr 2006) Marvlous MEMS, IEEE Circuits & Dvics Magazin, p Adams T and ayton (200) Introductory MEMS: Fabrication and Applications, Springr, Nw York, NY.. Bckwith TG, Marangoni D, and inhard JH (2007) Mchanical Masurmnts 6/, Parson Education, Parson Prntic Hall, Uppr Saddl ivr, NJ. 4. Holman JP (200) Exprimntal Mthods for Enginrs 7/, McGraw-Hill, Nw York, NY. 5. Northrop B (2005) Introduction to Instrumntation and Masurmnts 2/, CC Prss, Taylor & Francis Group, Boca aton, F. 6. Whlr AJ and Ganji A (200) Introduction to Enginring Exprimntation /, Parson Highr Education, Prntic Hall, Uppr Saddl ivr, NJ 7. Figliola S and Basly DE (2000) Thory and Dsign for Mchanical Masurmnts /, John Wily & Sons, Nw York, NY. 8. Doblin EO (2004) Masurmnt Systms : Application and Dsign 5/, McGraw-Hill, Nw York, NY. 9. Busch-Vishniac IJ (999) Elctromchanical Snsors and Actuators, Springr-Vrlag, Nw York, NY. 0. Maluf N and Williams K (2004) An Introduction to Microlctromchanical Systms Enginring 2/, Artch Hous, Norwood, MA.. Snturia SD (200) Microsystm Dsign, Kluwr Acadmic Publishrs, Norwll, MA. 2. Madou MJ (2002) Fundamntals of Microfabrication 2/, CC Prss, Boca aton, F.. Bby S, Ensll G, Kraft M, and Whit N (2004) MEMS Mchanical Snsors, Artch Hous, Norwood, MA. 4. Clark SK and Knsall DW (979) Prssur snsitivity in anisotropically tchd thin-diaphragm prssur snsors, IEEE Trans on Elctron Dvics, ED-26:2, Dc in, Chu H-C, and u Y-W (Dc 999) Pizorsistiv prssur snsors, J of Microlctromchanical Systms, 8:4. 6. Omga Enginring (2009) PX409 Gag and Absolut Prssur Spcifications, rtrivd from 7. ayton, (2006) Mini-abs: A Hands-On Substitut for cturs, workshop givn at th Frontirs in Educ. Conf., San Digo. 8. ayton A and Mayhw JE (2006), Mchanical masurmnts: writing th script, in proc. ASEE Annual Conf., Chicago Pag