【正文】 mer equivalent circuit to model the energy conversion from the electrical domain to mechanical domain 2121 010 eeff????? ? ? ????? ? ? ???? ? ? ?????Electromechanical Equivalent Circuit ?E2=Fd1, e1=v1, just need η1: ?From the matrix: e2=ηe1 111dpCF V vx?? ? ??11 pCVx???? Output Current Into Ground ?When the mass moves with timedependent displacement x(t), the electrodetomass capacitors C1(x,t) and C2(x,t) vary with time ?This generates an output current: ? ?? ? ? ?? ?? ?? ?? ?? ? ? ?? ?222 2 222222222,I n ph a sor f or m : I Ip p pppdq V Cq C V i C Vdt t tdV t dC x ti t C x t V tdt dtdC C xV t V i V Vdt x tCj V j xxCj j V xx??????? ? ? ? ?????? ??? ? ? ? ? ? ? ?????????????? ?? ? ? ? ? ? ? ?22o290 ph a se l og t t w he n x = 1ppC CI j j V x V vxxI?? ? ?? ? ? ???? ? ?? ?2 1 1 2 221 2 2 2 221, pf f f ff I f v I vCVx????? ? ? ? ?? ? ? ? ?????Output Current Into Ground ?Again, model with a transformer: ? ?? ? ? ?? ?? ?? ?? ?? ? ? ? ? ? ? ?? ?11 1 11 1 1 1111 1 1 1 1111 1 1 1111 1 111 1 1 1Ge t I : ppppppjdV t dC x ti t C x t V tdt dtdV C xV t v V i C v vdt x tCCI j C j V V j x V j xxxCCj C V j V x j V xxxCv V I j j C V j V xx?? ? ? ?? ? ?? ? ???? ?? ? ? ?? ? ? ? ? ?? ? ? ?????? ? ? ?????? ? ?????? ? ? ????1 11 F e e dt hr ou gh M ot i on a l C urr e nt C urr e ntC1DC : xre sdpFx V Vkk???? ? ??????d1 11QFon a nc e : x=jkpCQVvj k x????Output Current Into Ground ? ?? ?211 0 1 1 0 120 1 1 0 1o11T hus: r e sona nc e : 90 pha se shi f t e d I n pha se / V f r om V T hi s i s a cpC QI j j C V j V Vx j kQj C V Vk? ? ?? ? ??????????????e0111 2 2 21 0 1 1a pa c i t y T hi s i s a n e f f e c t i v e i n shunt / k i nput r e si st a nc e se e n l ooki ng i nt o El e c t r ode 1Mot i ona l R e si st a nc e :T he e qui v a l e nt c kf . be t t e r ge t txxmV kbRRI ??? ? ? ?? ? ? ? ?hi s right !??????Input Current Expression (Cont) Input Current Expression (Cont) Static electrodetomass overlap capacitance 1111oe p pCCVVxd?????222 2oe p p CCVVxd? ????? ?21211 2 2 22 2 21 2 2 1122112e 0 2 1 1F r om our t r a ns f orm e r m ode l : 0 1/ 12 1 1 21 1 1 111iixix eexi x xxeeee e eefff f f fve e e Fzzf f f i xlz j l r jjC?? ?????? ? ? ??????? ? ?? ? ? ?????? ? ? ?????? ? ? ?? ? ? ? ???? ? ? ? ? ? ???????????? ? ? ? ?????? ?? ? ?22111xeexrjC ?????Output Current Into Ground ?What is the impedance seen looking into port 1 with port 2 shorted to ground? ? ?2 2 222 2 221 1 1i x xi x xix e e eexv l rz j l r ji j C jC?? ?? ? ???????? ? ? ? ? ? ?????????Input Impedance Into Port 2 ?What is the impedance seen looking into port 2 with port 1 shorted to ground? Note: there are not the same as Lx1,Cx1,1/2Rx1! ? ?2220111121 120 1 212 12 12 1 21212121111121 1e i eiee i exxxxxeeeex x x e e xixxxxx xeexveivzj l rxjCellij j l r C Cv j Cj l r rjC R?????????????? ? ? ????????????????? ? ???????????????? ???? ?? ? ? ? ? ?????? ???? ???Port 1 to 2 TransG Across the Circuit ?What is the transconductance from port 1 to port 2 with port 2 shorted to ground? ? ?? ?? ?? ?? ?? ?0220000//0 : 0 0:1:0sQss s Qss j js?????????? ? ?? ? ?? ? ? ? ?Port 1 to 2 vitoi0 Transfer Function ? ?121 120 1 212 12 12 1 212121211xxeeeex x x e e xixxx xx xeelLij j L r C Cv j Cj l r rjC R????? ? ? ????????? ???? ?? ? ? ? ?????? ???? ???? ?0 22111 1 1xxi x x x x xxxx x x xsCi sCsv s L C sC R RsL RsssCL C L??????? ? ??? ?????? ????Separate freq. response f magnitude: ? ?? ? ? ?02 0 0 00 22 00/1 1 1,/xxx x x x i x xsQL R iQsL C R L Q v R Rs s Q????????? ? ? ? ? ? ? ???????Condensed Equiv. Circuit (Symmetrical) Holds for the symmetrical case, where port 1 and p