Lecture 6-3 Transductin Based n Changes in the Energy Stred in an Electrical ield Department f Mechanical Engineering
Example:Capacitive Pressure Sensr Pressure sensitive capacitive device With separatin diaphragm and variable capacitance single differential Direct pressurecnverting device single differential Department f Mechanical Engineering
Example:Capacitive Pressure Sensr Pressure sensitive capacitive device Expansin in series: r small diaphragm deflectin (h/a<<1) Defrmatin f flexible pressure sensitive diaphragm Department f Mechanical Engineering
Example:Capacitive Pressure Sensr Pressure sensitive capacitive device The value f capacitance C fr the incremental ring f width dr, that is lcated n the spherical surface when the ring is deflected a distance y, Department f Mechanical Engineering
Example:Capacitive Pressure Sensr Substitutin f y gives: Therefre: (πε/δ)a is capacitance befre deflectin The relative change f the capacitance The sensitivity is Department f Mechanical Engineering
Example:Capacitive Pressure Sensr Pressure sensitive capacitive device Defrmatin f stiff diaphragm Department f Mechanical Engineering
Example:Capacitive Pressure Sensr The capacitance crrespnding t the deflectin f the diaphragm under pressure P Substituting y and integrating, the capacitance change is The relative change f the capacitance and sensitivity are: Department f Mechanical Engineering
Example:Capacitive Pressure Sensr The resnance frequency fr the flexible diaphragm 0. 38 f k a σ ρ ρdensity f the diaphragm s materials kcefficient fr specific mde f vibratin, k1 fr fundamental mde σradial tensin, which shuld nt exceed a yield stress aradius f the diaphragm r rigid plane diaphragm, the fundamental resnance frequency is t E f 0.47 a ρ 1 µ ( ) E mdulus f elasticity tdiaphragm thickness µpissn s rati Department f Mechanical Engineering
Example:Capacitive Pressure Sensr Micrphne: amng the widely used cndenser micrphnes are the pressure-acustic transducers cntaining the diaphragm-capacitive element An electrical equivalent circuit f a micrphne cntains a dc vltage surce, variable gap acustic pressure sensitive capacitr C, and resistance f lad R L, and an ac utput vltage 1 prprtinal t the current change i 1 is develped acrss R L. 1 irl + idt C C CiRL + idt dc dt C R di + ir dt dc dt L L + i Department f Mechanical Engineering
Example:Capacitive Pressure Sensr C 1 s i1c RLs + irlc1s + i1 s jω C and i are capacitance and current crrespnding t steady state Current i 1 extracted frm the abve equatin is r the steady cnditins: >> i R L i 1 sc1 R ( i R ) L C s + 1 L The utput ac vltage is Therefre r the capacitr 1 dc 1 i1r L C1RL s R C s + 1 L εa δ 0 where δ is δ fr steady cnditin C dδ δ dδ Department f Mechanical Engineering
Example:Capacitive Pressure Sensr Replacing the differentials by increments and drpping the - sign C A time cnstant C 1 δ 1 δ τ C RL inally the transfer functin 1 δ1 δ jωτ i.e., 1 δ1 δ jωτ + 1 τs τs + 1 If C 1 /C<<1, which means δ 1 /δ<<1, and 1 δ 1 δ τ C R >>1/ ω L Insensitive t frequency Department f Mechanical Engineering
Example:Capacitive Pressure Sensr r τ <<1/ω The utput directly prprtinal t the frequency 1 1 δ ( jw) jδ ωτ τ changes can be prvided by cnnecting additinal resistances and/r capacitances parallel t r in series with C and R L The phase between 1 and δ 1 is: Many cntemprary micrphnes and vibratry sensrs cntain an electrlyte as a surce f plarizatin 1 ϕ arctan ωc R L An electrlyte is a plastic r ceramic slid dielectric material pssessing persistent electric plarizatin by virtue f a lng time cnstant fr decay f a charge instability : electrlyte Department f Mechanical Engineering
Relating rce and ltage A secnd Cmmn means f using a variable gap parallel plate capacitr in transductin is t cuple the frce and the vltage relatins Typically explited in Actuatr: An input vltage is used generate a frce which subsequently prduce mtin ( + x) Q d ε Q ε A A ( d x) + εa The relatinship between the frce and the vltage is nt linear, but quadratic Department f Mechanical Engineering
Department f Mechanical Engineering If we assume the vltage may be divided int a large bias vltage and a small signal vltage And the frce similarly has a dc cmpnent and a time varying prtin We may have + + + + + + + + + d x d x d x d x A d ε A d ε Zerth-rder respnse + d x A d ε irst-rder respnse Relating rce and ltage
Relating rce and ltage The DC term describe the static frce f attractin between the plate. There are tw linear AC terms, ne varying with displacement and ne with frce first rder. In rder fr the transducer t respnse t frce rather than t displacement, it is necessary that x << d d εa Since is assumed t be small cmpared with, We meet the inequality restrictin by insisting that x be small cmpared t d, i.e., we make the system very stiff mechanically s that there is little relative mtin f the plates. The quadratic term and third rder term can be ignred, there terms can be cnsidered as perturbatins t the system described by the zerth- and first-rder terms Department f Mechanical Engineering
Relating rce and ltage: r actuatr The quadratic frm f the frce/vltage transductin mechanism is ften used in actuatr. One example is that a cnductive film is used as ne plate between tw metal electrdes as the secnd plate fr either the tp r the bttm electrde plate t frm a bi-stable micrvalve actuatr. Department f Mechanical Engineering
Relating rce and ltage: r actuatr Anther example f an actuatr expliting nnlinear frm f the frce/vltage relatinship is the drive mechanism fr surface prfiling micrscpe Kenny et al Department f Mechanical Engineering
Relating rce and ltage: r actuatr Example: A linear actuatr by Minami et. al The actuatr uses hneycmb structure f electrdes rather than parallel plates. Applicatin f a vltage cmpresses the hneycmb, prducing a linear mtin Anther example: Electrstatic ludspeaker takes advantages f the linearized frce t vltage transductin mechanism in a parallel plate capacitr structure. Department f Mechanical Engineering
Relating rce and Charge A third means f ptentially using a variable gap capacitr as a transducer is t relate the frce and charge Q Recall ε A Nte that the frce is related t the square f the charge, s the relatinship is nnlinear. If we assume that there is a bias charge Q 0 and a small time varying charge Q Q QQ Q Q + + + ε A ε A εa ε A Static term QQ Prvide the linear relatinship between the time-varying charge εa and the time-varying frce. Other terms can be regarded as a perturbatin that leads t secnd harmnic distrtin. This transductin is hardly ever explited Department f Mechanical Engineering