plate electrodes where one electrode is fixed and the other is able to move toward or

Transcription

1 .1 lettati Paallel Plate Atuat The imlet elettati aallel late atuat (PPA) nit f tw aallel late eletde whee ne eletde i fixed and the the i able t mve twad away fm the fixed eletde, a illutated in Figue.8. The mtin f the mvable eletde i etained by ne me ing. When the vltage a the eletde i ze, the elettati fe between the eletde i al ze, eulting in a et eletde eaatin ditane, x, al alled the et ga ditane. A the vltage between the tw eletde i ineaed fm ze, the eulting elettati fe between the tw eletde ull the mvable eletde twad the fixed eletde until the elettati fe equal the ing fe. Ba [11] gave an exellent diuin n the deivatin f the equatin f the elettati fe dued in a PPA and n the limit f table eatin, whih ae ummaized belw. The enegy ted in the PPA aaitane,, at any itin, x x, i CV =. (.) I (t) + V _ (t). x, x Sing Mvable eletde F, ing fe x F e, elettati fe Fixed eletde

2 Figue.8. An illutatin f an elettati aallel late atuat. Subtituting (.1) int (.) yield AV =. (.3) d Hweve, ine d = x + x (.4) then AV = ( x +. (.5) If the t eletde i allwed t mve an infiniteimally mall ditane, Δx, twad the bttm eletde while the vltage a the eletde, V (t), i held ntant, the atuat will gain enegy and the we uly will give u intenal enegy. Theefe the enegy balane equatin beme d d Fe Δ x = Δx Δx, (.6)

3 whee F e Δx equal the enegy gained by the PPA aaitane minu the intenal enegy lt by the we uly. By dividing bth ide f (.6) by Δx, the equatin f F e an be imlified t F e d d =, (.7) whee d V C( V A = = x ( x + (.8) and d = V q(. (.9) x But ine q ( = C(, (.10) V theefe d = V C( x V A =. (.11) ( x +

4 Subtituting the eult f (.9) and (.11) int (.7) yield the equatin f the elettati fe F e V A = V A V A + = ( x + ( x + ( x + x ). (.1) Obeve that the elettati fe i tinal t the dive vltage quaed, but it i al inveely tinal t the eletde eaatin ditane quaed. Refeing bak t the illutatin f a imle PPA in Figue.8, whee the mvable eletde ha a ma, m, and the mehanial ytem ha daming whih an be mdeled by, the equatin deibing the ytem dynami i m& x&+ x& + kx =. (.13) F e The negative ign in fnt f F e eult fm the definitin that a itive elettati fe eult in a mvable eletde mtin in the negative x dietin, a defined in Figue.8. Afte a mall ntant nnze vltage, V, ha been alied t the atuat and the mvable eletde velity ha deayed t ze, the eulting fe balane equatin i AV kx = ( x +, (.14)

5 whee k i the ytem ing ntant. If the alied vltage i lwly ineaed fm ze, tw lutin exit f x, whih yield tw equilibium int, ne table and ne untable. F an alied vltage jut geate than ze, the table equilibium int eide almt at x = 0, while the untable equilibium int eide almt at the x = x. A the vltage i futhe ineaed, the tw equilibium int mve twad eah the and finally nvege int ne untable equilibium int. Thi equilibium int nvegene u when the eletde eaatin ditane equal ne thid f the et ga ditane, x. The vltage at whih thi u i alled the ull-in vltage, V i, whee V i 8kx 3 =. (.15) 7 A A vltage equal t geate than V i eult in an untable nditin whee the mvable eletde na int ntat with the fixed eletde. Thi u beaue the ing fe i nw alway le than the elettati fe. Theefe f en l eatin, the PPA ange f mtin i limited t le than ne thid f x. Seveal tehnique have been develed t ineae the table eating ange f mtin, but they uually eult in a enalty uh a highe equied eating vltage [1, 13].