Solution to Theoretical Question 2. A Piezoelectric Crystal Resonator under an Alternating Voltage Part A

Transcription

1 Solion o eoreical Qesion A Piezoelecric Crysal Resonaor ner an Alernaing olage Par A a Refer o Figre A e lef face of e ro oves a isance v wile e ressre wave ravels a isance wi / ρ e srain a e lef face is S v v Fro Hooe s law e ressre a e lef face is Aa v S ρ v A Figre A / v e velociy v is relae o e islaceen ξ as in a sile aronic oion or a nifor circlar oion as sown in Figre A of anglar freqency erefore if ξ ξ sin en v ξ A e srain an ressre are relae o velociy as in Prole a Hence S v / ξ A3 ρ v ρ ξ S ξ A Alernaively e answers ay e oaine y iffereniaions: ξ v ξ ξ S ξ Figre A ξ ξ ξ v ξ

2 Par B c Since e anglar freqency an see of roagaion are given e waveleng is given y λ π / wi / e saial variaion of e islaceen ξ is erefore escrie y g B sin B B Since e ceners of e elecroes are asse o e saionary g/ is leas o B Given a e ai of g is we ave B ± an g ± sin s e islaceen is B ξ ± ξ sin B3 Since e ressre or sress s vanis a e en faces of e qarz sla ie an e answer o is role can e oaine y analogy fro e resonan freqencies of son waves in an oen ie of leng However given a e ceners of e elecroes are saionary all even aronics of e fnaenal one s e ecle ecase ey ave aninoes raer an noes of islaceen a e isecion lane of e sla Since e fnaenal one as a waveleng λ e fnaenal freqency is given y f / e see of roagaion is given y /s B4 ρ 3 65 an given a - e wo lowes saning wave freqencies are 3 f 73 Hz f 3 88 Hz 3 f B [Alernaive solion o Proles c an ]: A longiinal saning wave in e qarz sla as a islaceen noe a / I ay e regare as consising of wo waves raveling in oosie irecions s is islaceen an velociy s ave e following for ξ ξ sin sin B6 ξ sin v ξ ξ sin sin were an e firs an secon facors in e sqare races reresen waves B7

3 raveling along e an irecions resecively Noe a Eq B6 is ienical o Eq B3 if we se ξ ±ξ For a wave raveling along e irecion e velociy v s e relace y v in Eqs Aa an A so a we ave v S an ρ v waves raveling along B8 v S an ρ v waves raveling along B9 As in Prole e srain an ressre are erefore given y S ξ ξ B ρ ξ ρ ξ B Noe a v S an ay also e oaine y iffereniaing ξ as in Prole e sress or ressre s e zero a o ens an of e sla a all ies ecase ey are free Fro Eq B is is ossile only if / or πf nπ λf n 3 5 B In ers of waveleng λ Eq B ay e wrien as λ n n 3 5 B3 e freqency is given y is is ienical wi e resls given in Eqs B4 an B5 n n f n 3 5 B4 λ ρ e Fro Eqs 5a an 5 in e Qesion e iezoelecric effec leas o e eqaions S E B5 S E B6 Becase / s e a noe of islaceen for any longiinal saning wave in e sla e islaceen ξ an srain S s ave e for given in Eqs B6 an B ie wi ξ ξ sin φ B7 3

4 4 φ ξ S B8 were a ase consan φ is now incle in e ie-eenen facors By assion e elecric fiel E eween e elecroes is nifor an eens only on ie: E B9 Ssiing Eqs B8 an B9 ino Eq B5 we ave φ ξ B e sress s e zero a o ens an of e sla a all ies ecase ey are free is is ossile only if φ an ξ B Since φ Eqs B6 B8 an B9 ily a e srface carge ensiy s ave e sae eenence on ie an ay e eresse as B wi e eenence on given y ξ B3 f A ie e oal srface carge Q on e lower elecroe is oaine y inegraing in Eq B over e srface of e elecroe e resl is an an ] [ α α C w w w w Q B4 were w C α B5 e consan α is calle e elecroecanical coling coefficien

5 Noe: e resl C w/ can reaily e seen y consiering e saic lii of Eq 5 in e Qesion Since an wen << we ave li Q / C [ α α ] C B6 Evienly e consan C is e caaciance of e arallel-lae caacior fore y e elecroes of area w wi e qarz sla of icness an eriiviy serving as e ielecric ei I is erefore given y w/ 5

6 Maring Scee oal Scores Par A 4 s Par B 6 s eoreical Qesion A Piezoelecric Crysal Resonaor ner an Alernaing olage S Scores a 6 4 c e f 4 Maring Scee for Answers o e Prole e srain S an ressre on e lef face 4 for l v an l 4 for S -v/ for sign 4 for relaing o S as S for sign 4 for ρv for sign e velociy v srain S an ressre 3 3 sinsoial variaion wi correc ase consan for ase consan 3 3 for alie 3 for eenence on an as - e fncion g for a saning wave of anglar freqency 4 for g/ 3 for B ± for o signs 4 for B e wo lowes saning wave freqencies for waveleng of fnaenal one λ for ecling even aronics 3 for f / 73 Hz for vale 3 for f 3 3/ 88 Hz for vale e srface carge ensiy as a fncion of an for ξ an S eac a searale fncion of an for ξ an S eac eens on ie as wi φ 3 for saial ar ξ ξ sin / 3 for saial ar S ξ / 3 for 3 for [ ξ / / ] / / ξ 6 for D 3 an D 3 in e consans C an α for relaion eween an Q as Q w 3 for noing Q/ C as 4 for C w/ 3 4 for α / 98 for vale 6